Herath Thilakarathna Thesis

Vulnerability assessment of reinforced concrete columns subjected to vehicular impactsi Vulnerability Assessment of Reinforced Concrete Columns Subjected to Vehicular Impacts By HMI Thilakarathna. MSc, BSc (Hons.) A THESIS SUBMITTED TO THE SCHOOL OF URBAN DEVELOPMENT QUEENSLAND UNIVERSITY OF TECHNOLOGY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY December 2010 Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts iii Dedication Dedication Dedication Dedication To my parents To my parents To my parents To my parents with love with love with love with love Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts v Abstract Columns are one of the key load bearing elements that are highly susceptible to vehicle impacts. The resulting severe damages to columns may leads to failures of the supporting structure that are catastrophic in nature. However, the columns in existing structures are seldom designed for impact due to inadequacies of design guidelines. The impact behaviour of columns designed for gravity loads and actions other than impact is, therefore, of an interest. A comprehensive investigation is conducted on reinforced concrete column with a particular focus on investigating the vulnerability of the exposed columns and to implement mitigation techniques under low to medium velocity car and truck impacts. The investigation is based on non-linear explicit computer simulations of impacted columns followed by a comprehensive validation process. The impact is simulated using force pulses generated from full scale vehicle impact tests. A material model capable of simulating triaxial loading conditions is used in the analyses. Circular columns adequate in capacity for five to twenty story buildings, designed according to Australian standards are considered in the investigation. The crucial parameters associated with the routine column designs and the different load combinations applied at the serviceability stage on the typical columns are considered in detail. Axially loaded columns are examined at the initial stage and the investigation is extended to analyse the impact behaviour under single axis bending and biaxial bending. The impact capacity reduction under varying axial loads is also investigated. Effects of the various load combinations are quantified and residual capacity of the impacted columns based on the status of the damage and mitigation techniques are also presented. In addition, the contribution of the individual parameter to the failure load is scrutinized and analytical equations are developed to identify the critical impulses in terms of the geometrical and material properties of the impacted column. In particular, an innovative technique was developed and introduced to improve the accuracy of the equations where the other techniques are failed due to the shape of the error distribution. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts vi Above all, the equations can be used to quantify the critical impulse for three consecutive points (load combinations) located on the interaction diagram for one particular column. Consequently, linear interpolation can be used to quantify the critical impulse for the loading points that are located in-between on the interaction diagram. Having provided a known force and impulse pair for an average impact duration, this method can be extended to assess the vulnerability of columns for a general vehicle population based on an analytical method that can be used to quantify the critical peak forces under different impact durations. Therefore the contribution of this research is not only limited to produce simplified yet rational design guidelines and equations, but also provides a comprehensive solution to quantify the impact capacity while delivering new insight to the scientific community for dealing with impacts. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts vii Keywords Dynamic analysis; Numerical simulation; Lateral impact; Circular column; Eccentric loading; Bi-axial bending; Residual capacity; Analytical equations; Vulnerability assessment. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts ix TABLE OF CONTENTS Abstract.........................................................................................................................v Keywords....................................................................................................................vii Table of contents..........................................................................................................ix List of abbreviations....................................................................................................xv List of symbols...........................................................................................................xvi List of tables..............................................................................................................xxii List of figures............................................................................................................xxii Publications..............................................................................................................xxix Statement of original authorship..............................................................................xxxi Acknowledgements...............................................................................................xxxiii 1. INTRODUCTION ................................................................................................ 1-1 1.1 Background ................................................................................................... 1-1 1.2 Aims and objectives ...................................................................................... 1-5 1.3 Hypotheses and research problems ............................................................... 1-6 1.4 Methodology ................................................................................................. 1-8 1.5 Thesis outline .............................................................................................. 1-12 2. LITERATURE REVIEW ................................................................................... 2-15 2.1 Characteristics of impact pulses .................................................................. 2-15 2.2 Behaviour of structural elements under impact loading .............................. 2-16 2.3 Dynamic impact tests on reinforced concrete columns ............................... 2-17 2.3.1 Columns subjected to soft impact ......................................................... 2-17 2.3.2 Columns subjected to hard impact ........................................................ 2-18 2.3.3 Columns subjected to axial impact ....................................................... 2-19 2.3.4 Shortcomings of the individual column tests ........................................ 2-20 2.4 Dynamic tests on reinforced concrete beams .............................................. 2-20 2.5 Behaviour of concrete under impact loads .................................................. 2-24 2.6 Dynamic properties of concrete and steel ................................................... 2-27 2.6.1 CEB-FIP specifications for concrete ..................................................... 2-27 2.6.2 Dynamic properties of steel .................................................................. 2-34 Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts x 2.7 Interaction between reinforcement and concrete ......................................... 2-36 2.7.1 Static bond slip analysis ........................................................................ 2-36 2.7.2 Dynamic bond slip analysis................................................................... 2-38 2.8 Factors affecting ductility of concrete columns .......................................... 2-39 2.8.1 Effects of confinement on enhancement of the ductility and strength .. 2-39 2.8.2 Effects of concrete cover ....................................................................... 2-42 2.8.3 Compressive axial load level................................................................. 2-43 2.8.4 Combined effects of axial load and flexure........................................... 2-43 2.9 Shear capacity calculations and effects of axial load .................................. 2-45 2.10 Energy absorption characteristics under impact loads .......................... 2-47 2.11 Design practices and provisions of RF in critical sections .................... 2-49 2.11.1 Influence of the various parameters on confinement ............................ 2-52 2.11.2 Theoretical stress strain curves for confined concrete by transverse RF2-55 2.12 Effects of impact induced torsion in eccentrically loaded columns ...... 2-57 2.13 Impact reconstruction ............................................................................ 2-58 2.13.1 Application to accident reconstructions ................................................ 2-60 2.14 Design guidelines .................................................................................. 2-61 2.14.1 Dynamic design for impact ................................................................... 2-63 2.15 Knowledge gaps and literature review findings .................................... 2-64 3. FE MODELLING OF SHORT RC COLUMNS UNDER LATERAL IMPACT3-67 3.1 Introduction ................................................................................................. 3-67 3.2 Finite element modelling for impact problems ........................................... 3-68 3.3 Finite element modelling and selection of material models ........................ 3-69 3.3.1 Evaluation of Constitutive material models in LS-DYNA ................... 3-69 3.3.2 Theory on Mat_Concrete_Damage model ............................................ 3-70 3.3.3 Definition of compressive and tensile meridians at p < f c /3 ................. 3-72 3.3.4 Pressure cut-off and softening ............................................................... 3-73 3.3.5 Strain rate effect .................................................................................... 3-74 3.3.6 Equation of state .................................................................................... 3-75 3.3.7 Evaluation of LS-DYNA material model Mat_Brittle_Damage ........... 3-75 3.4 Development and validation of a numerical model of a RC column .......... 3-77 Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts xi 3.4.1 Introduction ........................................................................................... 3-77 3.4.2 Experimental test set up ........................................................................ 3-78 3.4.3 Numerical simulation of the physical testing ........................................ 3-79 3.4.4 Convergence study and mesh discritization .......................................... 3-80 3.4.5 Contact algorithm and prevention of initial penetration ....................... 3-82 3.4.6 Validation of the finite element model using Mat_Concrete_Damage . 3-84 3.4.7 Material properties of steel.................................................................... 3-85 3.4.8 Load simulation for a dynamic system ................................................. 3-86 3.4.9 Hourglass energy and damping effects ................................................. 3-87 3.4.10 Procedure for axial load application ..................................................... 3-88 3.4.11 Confinement effects under strain gradient ............................................ 3-90 3.4.12 Numerical and experimental results for Mat_Brittle_Damage ............. 3-91 3.4.13 Comparison of numerical and experimental results for Mat_Concrete 3-92 _Damage .......................................................................................................... 3-92 3.5 Conclusions ................................................................................................. 3-95 4. IMPACT RECONSTRUCTION AND PARAMETRIC STUDIES ................... 4-97 4.1 Introduction ................................................................................................. 4-97 4.2 Impact reconstruction by using crash test data ............................................ 4-98 4.2.1 Vehicle-Column Interaction .................................................................. 4-99 4.2.2 Effects of Impact Pulse Parameters ..................................................... 4-100 4.2.3 Impact pulse modelling and effects of the impact pulse parameters .. 4-103 4.2.4 Simulation of impact of axially loaded columns in medium rise buildings .............................................................................................. 4-104 4.3 Impact behaviour of the columns and possible damage modes ................ 4-106 4.4 Vulnerability prediction ............................................................................. 4-107 4.5 Bending moment and resultant shear due to impact ................................. 4-110 4.6 Effects of the diameter of the column, concrete grade and steel ratio ...... 4-111 4.7 Effects on the slenderness ratio ................................................................. 4-112 4.8 Energy absorption due to the impact ......................................................... 4-113 4.9 Effects of impact duration ......................................................................... 4-113 4.10 Conclusions ......................................................................................... 4-115 Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts xii 5. CAPACITY OF THE AXIALLY LOADED COLUMNS UNDER LATERAL IMPACTS .................................................................................................. 5-117 5.1 Introduction ............................................................................................... 5-117 5.2 Design against accidental loads ................................................................. 5-118 5.3 Parametric studies of impacted columns ................................................... 5-119 5.3.1 Finite element analysis of confined circular columns ......................... 5-119 5.3.2 Effects of the confinement .................................................................. 5-119 5.4 Effectiveness of confinement under impact loads ..................................... 5-120 5.5 Effects of the unconfined cover and use of external wrapping ................. 5-124 5.6 Effects of the slenderness ratio on capacity enhancement ........................ 5-125 5.7 Comparison of the dynamic and static shear capacities ............................ 5-126 5.8 Impact capacity of partially loaded circular columns ................................ 5-127 5.8.1 Introduction ......................................................................................... 5-127 5.8.2 Damage criterion ................................................................................. 5-128 5.8.3 Effect of axial load on the duration of the impact ............................... 5-129 5.8.4 Simplified method to investigate the residual capacity of columns .... 5-129 5.8.5 Effects of transverse reinforcement on capacity enhancement ........... 5-133 5.8.6 Effects of longitudinal reinforcement ratio ......................................... 5-138 5.8.7 Effects of the slenderness ratio ............................................................ 5-139 5.8.8 Anomalous behaviour of columns under post impact loading ............ 5-140 5.8.9 Buckling of reinforcement under impact ............................................ 5-144 5.9 Derivation of empirical relationships to predict critical impulse .............. 5-145 5.10 Derivation of simple linear regression equations ................................ 5-145 5.10.1 Descriptions of the outputs .................................................................. 5-145 5.10.2 Pearson Correlation ............................................................................. 5-146 5.10.3 Coefficient of Determination and Analysis of Variance ...................... 5-147 5.10.4 Interpretation of partial (regression) plots ........................................... 5-148 5.10.5 Regression coefficients and derivation of the linear equations ........... 5-150 5.11 Derivation of Polynomial equations .................................................... 5-152 5.12 Conclusions ......................................................................................... 5-155 Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts xiii 6. IMPACT ON COLUMNS UNDER UNIAXIAL BENDING .......................... 6-159 6.1 Introduction ............................................................................................... 6-159 6.2 Behaviour of the impacted columns under single axis bending ................ 6-160 6.2.1 The load application procedure ........................................................... 6-161 6.2.2 Material models and mesh generation................................................. 6-162 6.2.3 Axial load and eccentric load applications in an explicit environment6-163 6.3 Deflection profiles and resultant bending moment ................................... 6-165 6.4 Deformation characteristics of the impacted column ................................ 6-167 6.5 Impact behaviour of the eccentrically loaded column .............................. 6-167 6.6 Behaviour of eccentrically loaded confined columns under impact ......... 6-169 6.7 Selection of the load combinations ........................................................... 6-170 6.8 Parametric studies and discussion of the finite element results ................ 6-171 6.8.1 Impact behaviour of eccentrically loaded columns under maximum allowable capacity ............................................................................... 6-172 6.8.2 Impact behaviour under reduced load eccentricities ........................... 6-173 6.9 Impact behaviour of columns under positive eccentric loading ................ 6-174 6.9.1 Impact response under positive eccentric moments ............................ 6-176 6.10 Confinement effects on eccentrically loaded columns under impact .. 6-177 6.11 Effects of the longitudinal steel ratio on the impact behaviour of columns .............................................................................................................6-178 6.12 Confinement effects on the impacted columns with high steel ratio .. 6-179 6.13 A comparison of the confined columns with different steel ratios ...... 6-180 6.14 Effects of the slenderness ratio and intensity of loading on impact capacity .............................................................................................................6-181 6.15 Strain rate sensitivity of eccentrically loaded columns ....................... 6-182 6.16 Linear equations for 20% and 50% loaded columns ........................... 6-183 6.16.1 Linear equations for 20% loaded columns .......................................... 6-184 6.16.2 Linear equations for 50% loaded columns .......................................... 6-184 6.17 Conclusions ......................................................................................... 6-185 7. IMPACT ON COLUMNS UNDER BIAXIAL BENDING ............................. 7-187 7.1 Introduction ............................................................................................... 7-187 Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts xiv 7.2 Numerical simulation of biaxial loaded columns ...................................... 7-189 7.3 Characteristics of the simulated columns .................................................. 7-191 7.4 Selection of load combinations ................................................................. 7-192 7.4.1 Impact capacity of columns under biaxial bending ............................. 7-192 7.5 Effects of the direction of the impact ........................................................ 7-195 7.6 Effect of reduced axial load on biaxial bending ........................................ 7-196 7.7 Effects of longitudinal steel ratio on biaxial bending ................................ 7-198 7.8 Effects of biaxial bending on 20% loaded impacted columns with a 4% steel ratio. ................................................................................................................... 7-199 7.9 Damage mitigation of the impacted columns under single axis bending .. 7-201 7.10 Effects of the confinement on biaxial bending .................................... 7-201 7.10.1 Impact behaviour of 50% loaded columns with 4% steel under biaxial bending ................................................................................................ 7-201 7.11 Behaviour of 20% loaded confined columns with 4% steel under biaxial bending.. ............................................................................................................ 7-203 7.12 Impact behaviour of 50% loaded columns with 1% steel under biaxial bending.. ............................................................................................................ 7-203 7.13 Impact behaviour of 20% loaded columns with 1% steel under biaxial bending.. ............................................................................................................ 7-204 7.14 Effects of the steel grade and diameter of the hoops ........................... 7-204 7.15 Effects of slenderness ratio on impact capacity of columns under biaxial bending.. ............................................................................................................ 7-208 7.16 Effects of the concrete grade on impact behaviour of columns .......... 7-210 7.17 Development of equations for biaxially loaded columns under lateral impact.... ............................................................................................................ 7-212 7.17.1 Linear equations for 50% loaded columns (Impact angle 0 o to 90 o ) ... 7-213 7.17.2 Linear equations for 20% loaded columns (Impact direction 0 to 90 o )7-215 7.18 Conclusions ......................................................................................... 7-218 8. CONCLUSIONS AND FURTHER DEVELOPMENTS ................................. 8-221 8.1 Introduction ............................................................................................... 8-221 8.2 Main contributions of the thesis ................................................................ 8-221 Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts xv 8.3 Practical significance ................................................................................. 8-222 8.4 Recommendations for future work ............................................................ 8-225 LIST OF ABBREVIATIONS ALS - Accidental Limit State ACI - American Concrete Institute AJI - Architectural Institute of Japan BS - British Standards BM - Bending moment CPU - Central Processing Unit CS - Cross Section CEB-FIP -European Committee for Concrete-International Federation for Prestressing DIF - Dynamic Increasing Factor DTEI - Department for Transport, Energy and Infrastructure EAF - Energy of Approach Factor EAF o - On set Energy of the vehicle EoS - Equation of State FE - Finite Element EFA - Finite Element Analysis EFM - Finite Element Model HSC - High strength concrete HSRC - High Strength Reinforced Concrete LSC -Low strength concrete NHTSA - National Highway Traffic Safety Administration N.A. - Neutral axis RC - Reinforced Concrete REL - Release (version) SDoF - Single degrees of freedom SLS - Serviceability Limit State S/R - Selective Reduced solid NZS - New Zealand Standards Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts xvi ULS - Ultimate Limit State VC - Viscous Friction VDC - Viscous Damping Coefficient LIST OF SYMBOLS LOWER CASE LATIN CHARACTORS b - web width of a section c/c - centre to centre b 1 ,b 2 ,b 3 -user defined scale multipliers b c - breadth of the core concrete for square/circular column b” - width of the confined core c - concrete cover thickness c s - sound speed d - effective depth of a section / diameter of a hoop d s - distance between bar centres e x - load eccentricity along the X axis e y - load eccentricity along the Y axis f ’ cc - compressive strength of confined concrete f c,imp - dynamic compressive strength (mean) f cm - mean compressive strength of concrete f ’ co - compressive strength of unconfined concrete f ct,imp- dynamic tensile strength f ctm - tensile strength of concrete f c - compressive stress f ’ c - concrete compressive cylinder strength f l - confining stress k i f α - hourglass resisting force vector ' l f - effective lateral confining stress f t -tensile stress f y - yield stress Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts xvii ' sy f - yield strength of confined reinforcement f y - yield strength of the transverse reinforcement f yh - maximum strength of transverse steel h - smaller dimension of the rectangular section α i h - nodal coordinates k - spring stiffness k d -internal scale multiplier k e -confinement effectiveness coefficient m - gross mass of a vehicle n - modular ratio p - mean stress in the tri-axial compression failure test/pressure q - strain rate sensitivity of the material r - pearson correlation r f -modification factor to account for dynamic strength of concrete s - spacing of transverse steel s l - spacing between laterally confined longitudinal bars t - duration w - crushed width x - crushed length / variable v - velocity of the vehicle v& - rate of deformation v c - reduced shear stress e v - element volume v r - object velocity at impact UPPER CASE LATIN CHARACTORS A - shape function A b - area of the transverse reinforcements A c - net area of a concrete core A g - gross concrete area of a section Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts xviii A i - cross sectional area of an equivalent impacting object A sh - transverse steel area across a section A sh,c - ACI code provision of transverse reinforcement content A sw - transverse reinforcement area within a spacing A ws - area of the longitudinal steel in the section B - shape function C - a material constant C o, C 1- bulk viscosity coefficients D - diameter D c - characteristic strain rate D i - Damage index D o - diameter of circular section L - length of the element L i - length of the impacting object L e - element length E - modulus of elasticity E c,imp - mean impact modulus of elasticity (compression) E ci - modulus of elasticity at 28 days E c - reduced modulus of elasticity E t - tangent moduli E o - absorbed energy by a crushed vehicle E m - mesh density F - impact or breaking force F o - plastic strength of a structure G - shear modulus G f - fracture energy of concrete H - effective height H s -softening modules I - impulse I o - critical impulse I c - critical impulse (corrected) I e - confinement index Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts xix I 1 - first stress invariant of a stress tensor J 2 - second invariant of deviatoric stress tensor J 3 - third stress invariant L i - length of the impacting object M - moment M x - moment on a column corresponding to d xP axial load ) 1 0 ( < < x M xs - moment about X axis M ys - moment about Y axis P - axial load / amplitude of an impact pulse P a - the axial capacity P c - critical impact force (corrected) P d - design axial load capacity P e - confining pressure P o - critical impact force P r - residual axial load carrying capacity R 2 - coefficient of determination R d - design resistance R k - characteristic resistance S d - design load effects S k - characteristic load effect S y.x - standard error of estimate β S -standard error of the slope T limit -tensile limit V c - shear capacity of concrete V d - dynamic shear capacity V s - static shear capacity V n - shear strength corresponding to the maximum moment capacity V r - volumetric ratio of confined reinforcement V s - shear capacity of steel X i - independent variable Y i - dependent/response variable Y o - deformation capacity of a crushed vehicle Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts xx X,Y - two orthogonal (principle) axes GREEK CHARACTORS β - shear retention factor h β - kinematic hardening parameter σ - stress c σ& - stress rate (compression) ct σ& - stress rate (tension) ' d σ - dynamic flow stress s σ - static flow stress c ε& - strain rate (compression) ε& - strain rate c ε - compresive strain s ε - strain in steel ε v - volumetric strain ε v,yield - volumetric strain at yield p dε -effective plastic strain p ij dε -plastic strain increment tensor λ -damage function σ d - stress under dynamic conditions σ s - stress under static conditions η - yield scale factor/viscosity f γ - partial factor for loads m γ - material factor φ - diameter φ’ - strength reduction factor α - a factor dependent on the tie configuration Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts xxi α’ - tangent of a crack angle α s - shear-bending capacity ratio µ φ - curvature ductility factor ρ - mass density / poisons ratio ρ cc - ratio of area of longitudinal reinforcement to area of core of section s ρ - ratio of volume of transverse reinforcement to volume of concrete core ρ total - steel ratio of the section w ρ - transverse reinforcement ratio v ρ - longitudinal steel ratio k α Γ - nodal velocity ∆ - Impact angle in degrees l ∆ - deceleration length ∆t e - critical time step size ∆t - duration of the plus ∆ t - small time interval λ ∆ -volumetric plastic strain σ ∆ - stress difference c y σ ∆ - compressive meridian of the initial yield surface c m σ ∆ - maximum failure surface c m σ ∆ - residual surface (compression) t m σ ∆ - tensile meridian L σ ∆ -loading surface after yielding pf σ ∆ - post failure surface ABBRIVIATIONS OF UNITS kg kilogram km/h kilometres per hour kN kilo Newton Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts xxii m Metres mm Millimetres MPa mega Pascal ms Milliseconds mph Miles per hour s Seconds t tonne cm centimetre LIST OF TABLES Table 2.1: Tangent moduli and reduced moduli of elasticity .................................. 2-30 Table 2.2: Mass and stiffness coefficient for impact reconstructions ..................... 2-61 Table 3.1: Material properties used for the concrete ............................................... 3-77 Table 3.2: Characteristics of the Feyerabend’s test specimens (Feyerabend 1988) 3-79 Table 3.3: Material properties used for the main reinforcement ............................. 3-86 Table 3.4: Material properties used for the Rigid Body .......................................... 3-86 Table 5.1: Descriptive Statistics ............................................................................ 5-146 Table 5.2: Pearson correlations ............................................................................. 5-147 Table 5.3: Coefficient of determination of the equation ....................................... 5-147 Table 5.4: Analysis of Variance ............................................................................. 5-148 Table 5.5: Regression Coefficients for linear equations ........................................ 5-151 Table 7.1: Biaxial load combinations on the 300mm column under 50% axial load ............................................................................................................................... 7-192 LIST OF FIGURES Figure 1.1: Severely damaged columns due to a vehicle impact .............................. 1-1 Figure 2.1: Sequence of an impact (El-Tawil et al. 2005) ....................................... 2-15 Figure 2.2:Impact force vs. time histories for Chevy truck at various speeds ........ 2-16 Figure 2.3: Comparison between different modes of vibration with equal potential energies (Hughes and Speirs 1982) ......................................................................... 2-21 Figure 2.4: Strain-rate sensitivity for concrete in compression, tension and flexure Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts xxiii ................................................................................................................................. 2-26 Figure 2.5: Model for the strain rate dependency of concrete elastic modulus ...... 2-29 Figure 2.6: Strain-rate influence on the compressive strength of concrete ............. 2-31 Figure 2.7: Strain-rate influence on the tensile strength of concrete ...................... 2-32 Figure 2.8: Model for the strain rate dependency of concrete in compression and tension according to the CEB-FIP model code (CEB 1993) and with the modified model according to Malvar and Crawford (1998-a) ............................................... 2-33 Figure 2.9: Distribution of confining pressure produced by various shapes of ...... 2-41 Figure 2.10: Stress-strain model for confined concrete proposed by Mander et al. (1984) ...................................................................................................................... 2-55 Figure 2.11: Confining stress provided by the transverse reinforcements .............. 2-56 Figure 3.1: Failure surfaces in Mat_Comcrete_Damage_REL3 (Malvar et al. 1997) ................................................................................................................................. 3-71 Figure 3.2: The test set-up by Feyerabend (1988) .................................................. 3-78 Figure 3.3: The simplified test set-up and the cross section of specimen ............... 3-79 Figure 3.4: Convergence of the numerical model ................................................... 3-80 Figure 3.5: Mesh generation for the impacted column & rigid body ...................... 3-81 Figure 3.6: Single element under uni-axial tensile test ........................................... 3-84 Figure 3.7: Lateral pressure distribution and the resultant strain gradient .............. 3-90 Figure 3.8: Comparison of the resultant deflections ................................................3-91 Figure 3.9: Interface forces during the impact ........................................................ 3-92 Figure 3.10: Comparison of the resultant deflections ............................................. 3-93 Figure 3.11: Crack Propagation of the impacted column and numerical simulation ................................................................................................................................. 3-93 Figure 3.12: Comparison of the resultant impact force...........................................3-94 Figure 3.13: Comparison of the resultant reactions ................................................ 3-94 Figure 4.1: Front and side views of an impacted column (NHTSA) ...................... 4-99 Figure 4.2: Lateral pressure distribution across the diameter of the 300mm column ............................................................................................................................... 4-100 Figure 4.3: Force Time histories of full scale crash tests (NHTSA 1997) ............ 4-101 Figure 4.4: Iso-damage pulses............................................................................... 4-103 Figure 4.5: Effects of the strain rate or frontal stiffness ........................................ 4-103 Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts xxiv Figure 4.6: Cross sectional areas of the circular concrete columns........................4-105 Figure 4.7: Support conditions and external load applications ............................. 4-105 Figure 4.8: Plan view of the half models .............................................................. 4-106 Figure 4.9: Comparison of impact capacities of columns with full scale crash tests (NHTSA) ............................................................................................................... 4-108 Figure 4.10: Honda Accord in a frontal collision at a speed of 48.3km/h ........... 4-108 Figure 4.11: Ultimate capacities of columns..........................................................4-109 Figure 4.12: Support reaction and Impact pulse ................................................... 4-109 Figure 4.13: Resultant bending moments...............................................................4-110 Figure 4.14: Damaged column under vehicle impact ........................................... 4-110 Figure 4.15: Effects of the diameter of the column.................................................4-111 Figure 4.16: Effects of the concrete grade ............................................................ 4-111 Figure 4.17: Effect of the slenderness ratio............................................................4-112 Figure 4.18: 500mm column with 4% steel .......................................................... 4-112 Figure 4.19: Equivalent impulse diagrams.............................................................4-113 Figure 4.20: Iso-damage pulses for 600mm column .............................................4-113 Figure 4.21: Force vs reaction histories for pulses with different durations ......... 4-114 Figure 4.22: A typical Pressure impulse curve......................................................4-114 Figure 4.23: Iso-damage contours for impact ....................................................... 4-114 Figure 5.1: Effects of confinement under lateral impacts ..................................... 5-119 Figure 5.2: Stress difference at the cover-core interface ....................................... 5-120 Figure 5.3: Capacity enhancement for 30MPa concrete ....................................... 5-122 Figure 5.4: Capacity enhancement for 50MPa Concrete ...................................... 5-123 Figure 5.5: Confined strength for different concrete grades ................................. 5-124 Figure 5.6: Columns confined with 12mm links at 100mm spacing .................... 5-125 Figure 5.7: Comparison of the dynamic and static shear capacities ..................... 5-126 Figure 5.8: Rehabilitation of a bridge after catastrophic failure of a column ....... 5-127 Figure 5.9: Axial pressure application on 300mm diameter column .................... 5-130 Figure 5.10: Deflection characteristics of 450mm column ................................... 5-131 Figure 5.11: Axial load sensitivity of impacted columns ...................................... 5-132 Figure 5.12: Deformation characteristics of 300mm column ............................... 5-134 Figure 5.13: Impact capacity of 300mm column under varying axial loads ......... 5-135 Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts xxv Figure 5.14: Enhanced capacities for confined columns with different axial loading ............................................................................................................................... 5-136 Figure 5.15: Capacity reduction due to axial load ................................................ 5-137 Figure 5.16: Capacity enhancement due to confinement ...................................... 5-137 Figure 5.17: Impact capacity under varying axial load..........................................5-139 Figure 5.18: Capacity increment under varying axial load ................................... 5-139 Figure 5.19: Impact capacities of short columns .................................................. 5-139 Figure 5.20: Different failure characteristics of structural columns ..................... 5-140 Figure 5.21: Axial load sensitivity of the counterintuitive effect .......................... 5-141 Figure 5.22: Typical failure pattern of rectangular columns under eccentric loading ............................................................................................................................... 5-142 Figure 5.23: Failure of columns by concrete crushing .......................................... 5-144 Figure 5.24(a-f): Partial regression plots of each parameter against Log P .......... 5-149 Figure 5.25: Accuracy of the prediction by linear equations ................................ 5-151 Figure 5.26: Accuracy of the polynomial equations ............................................. 5-153 Figure 5.27 (a-g): Steps of the derivation of polynomial equations ..................... 5-153 Figure 6.1: Plan view of the column head (under uni-axial bending) ................... 6-160 Figure 6.2: The Bulk head of columns used for the application of moment ......... 6-162 Figure 6.3: Numerical simulation of eccentrically loaded columns ..................... 6-163 Figure 6.4: Interaction diagram for the 300mm......................................................6-164 Figure 6.5: Extreme strain in the 300mm ............................................................. 6-164 Figure 6.6: A typical Interaction diagram of a column ......................................... 6-165 Figure 6.7: Contours of effective plastic strain ..................................................... 6-166 Figure 6.8: Time histories of BM of 300mm eccentrically loaded half column with 1% steel ....................................................................................................................... 6-166 Figure 6.9: Deflection of 300mm diameter column with 4% steel at the near collapse stage....................................................................................................................... 6-167 Figure 6.10: Resultant bending moment at different locations on the 450mm column ............................................................................................................................... 6-167 Figure 6.11: Resultant shear forces at different locations on the 450mm column 6-168 Figure 6.12: Lateral pressure distribution and the corresponding stress-strain relationships .......................................................................................................... 6-169 Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts xxvi Figure 6.13: Interaction diagrams for 450mm column according to AS3600 and ACI: 318 ......................................................................................................................... 6-170 Figure 6.14: locations of the selected loading points on the interaction diagrams6-171 Figure 6.15: Eccentrically loaded columns with 1% steel ratio ............................ 6-172 Figure 6.16: Peak force under different load combinations .................................. 6-173 Figure 6.17: Comparison of the Impact capacities under positive and negative moments ................................................................................................................ 6-175 Figure 6.18: Cracks on 20% loaded 300mm column with 1% steel.......................6-176 Figure 6.19: Cracks on 20% loaded 450mm and 600mm columns with 1% steel 6-176 Figure 6.20: Resultant bending moments of the 20% loaded 300mm column ..... 6-177 Figure 6.21: Capacity of eccentrically loaded confined columns under impact ... 6-177 Figure 6.22: Effect of longitudinal steel ratio on impact capacity ........................ 6-179 Figure 6.23: Impact capacity enhancement due to confinement ........................... 6-180 Figure 6.24: Effects of the longitudinal steel ratio on impact capacity enhancement ............................................................................................................................... 6-181 Figure 6.25: Effects of the slenderness ratio on capacity enhancement for columns ............................................................................................................................... 6-181 Figure 6.26: Strain rate sensitivity of a ductile column ........................................ 6-182 Figure 6.27: Residuals of the predicted values (20%)............................................6-183 Figure 6.28: Residuals of the predicted values (50%) .......................................... 6-183 Figure 7.1: Impact capacity prediction for intermediate load combinations ......... 7-189 Figure 7.2: Typical interaction diagram for circular columns under biaxial bending ............................................................................................................................... 7-190 Figure 7.3: Numerical model of the column under biaxial bending ..................... 7-191 Figure 7.4: Impact capacities of the columns under biaxial bending .................... 7-193 Figure 7.5: Simulation of the effects of the direction of the impact ..................... 7-195 Figure 7.6: Failure characteristics of 50% loaded 300mm columns under biaxial bending .................................................................................................................. 7-198 Figure 7.7: Peak force vs slenderness ratio for 4m high columns made of 50MPa concrete ................................................................................................................. 7-201 Figure 7.8: Impact capacity of 20% loaded columns under varying hoop spacing ............................................................................................................................... 7-205 Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts xxvii Figure 7.9: Impact capacities of 20% and 50% loaded columns under varying hoop spacing................................................................................................................... 7-206 Figure 7.10: Impact capacities of 20% and 50% loaded columns under varying hoop diameter ................................................................................................................. 7-206 Figure 7.11: Impact capacities of 20 and 50% loaded columns under varying yield strength .................................................................................................................. 7-207 Figure 7.12: Peak force vs. Slenderness ratio for columns of 50MPa concrete with 1% steel ....................................................................................................................... 7-209 Figure 7.13: Peak force vs. Slenderness ratio for columns of 50MPa concrete with 4% steel ....................................................................................................................... 7-209 Figure 7.14: Ultimate capacity of 3m columns made of 50MPa concrete ............ 7-210 Figure 7.15: Ultimate capacity of 2m columns made of 50MPa concrete ............ 7-210 Figure 7.16: Comparison of peak force of 4m high columns made of Grade 30 and 50 concrete ................................................................................................................. 7-210 Figure 7.17: Peak force vs. Slenderness ratio for columns of 30MPa concrete with 1% steel ....................................................................................................................... 7-212 Figure 7.18: Peak force vs. Slenderness ratio for columns of 30MPa concrete with 4% steel ....................................................................................................................... 7-212 Figure 7.19(a-h): Partial regression plots of each parameter against Log P ......... 7-214 Figure 7.20: Residual of the predicted values ....................................................... 7-215 Figure 7.21 (a-h): Partial regression plots of each parameter against Log P ........ 7-216 Figure 7.22: Accuracy of the predicted values ...................................................... 7-217 Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts xxix Publications Refereed International Journal Papers HMI Thilakarathna, DP Thambiratnam, M Dhanasekar, N Perera. Numerical simulation of axially loaded concrete columns under transverse impact and vulnerability assessment. International Journal of Impact Engineering 2010; 37(11): p. 1100-1112. HMI Thilakarathna, DP Thambiratnam, M Dhanasekar, N Perera. Impact response and vulnerability assessment of concrete columns under vehicle impacts. Advances in Structural Engineering 2010; (under review). HMI Thilakarathna, DP Thambiratnam, M Dhanasekar, N Perera. Capacity of biaxially loaded circular reinforced concrete columns under transverse impacts. Engineering structures 2010; (under review). International Conference Papers HMI Thilakarathna, DP Thambiratnam, M Dhanasekar, N Perera. Behaviour of Axially Loaded Concrete Columns Subjected to Transverse Impact Loads. 34 th Conference on Our World in Concrete & Structures ‘Green Concrete’; 2009. V28. p 359-366. Park Royal Hotel on Kitchener Road, Singapore HMI Thilakarathna, DP Thambiratnam, M Dhanasekar, N Perera. Impact Response and Parametric Studies of Reinforced Concrete Circular Columns. 4 th international conference on ‘Protection of structures against hazards’. October 2009, p 347-354. Tsinghua Unisplendour International Centre, Beijing, China Book Chapter Yigitcanlar, T., Ed. (2010). Sustainable Urban and Infrastructure Development: Management, Engineering and Design. Chapter 15. Infrastructure sustainability: vulnerability of axially loaded columns subjected to transverse impact loads. Brisbane. Local Conference Paper HMI Thilakarathna, DP Thambiratnam, M Dhanasekar, N Perera. Vulnerability of Axially Loaded Columns Subjected to Transverse Impact Loads. March 2009. The second infrastructure theme Postgraduate Conference. Gardens Point campus, QUT. p.22-34. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts xxxi Statement of original authorship The work contained in this thesis has not been previously submitted for a degree or diploma at any other higher education institution. To the best of my knowledge and belief, the thesis contains no material previously published or written by another person except where due reference is made. Indika Thilakarathna 6 December 2009 Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts xxxiii ACKNOWLEDGEMENTS I wish to express my heartfelt gratitude to the principle supervisor Professor David Thambiratnam for his motivation, patience and endless support throughout the period of my PhD studies. I also want to express an especial thankyou to the associate supervisors Professor Sekar Dhanasekar and Professor Nimal Perera for their encouragement and professional guidance during the entire period. My sincere thanks go to Mr. Mark Barry and the other staff members at the High Performance Computer Unit for their assistance and cooperation during the research and for enthusiastic responses to my numerous requests for assistance. I extend my thanks to the LEAP support, the QUT software provider for their great effort to assist me with the problems during the Finite Element Modelling. It is a true pleasure to express my gratitude to staff of the Document Delivery Unit at QUT for their continuous assistance and cooperation whenever needed. Many thanks also go to Dr. Greg Nagel and Department of International Student Services at QUT for revising and proof reading of the thesis. I gratefully acknowledge the financial support granted to me from the IPRS scholarship and the President’s Fund of Sri Lanka to conduct the research. I am also grateful to the personnel and my fellow doctoral students at the Faculty of Built Environment and Engineering and also to my colleagues at QUT for sharing knowledge and for contributing to a friendly and fruitful atmosphere. Finally, I also wish to express my deepest gratitude to my wife, parents and family members for their patience and understanding during the completion of this research. Queensland University of Technology, Brisbane, December – 2010. Indika Thilakarathna. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 1-1 1. INTRODUCTION 1.1 Background Increased industrialisation has brought to the forefront the susceptibility of concrete columns in both buildings and bridges to vehicle impacts. Catastrophic failure of a bridge or a building, as a result of a vehicle collision, worsens the consequences and requires special attention to design and detailing. For example, the Department for Transport Energy and Infrastructure (DTEI, 2004) in their annual report state that vehicle impact with road-side objects alone costs the Australian society $4.6 billion per year. As a whole, these crashes represent 30% of the total fatal motor vehicle crashes. Figure 1.1: Severely damaged columns due to a vehicle impact In fact, there are several techniques that can be used to mitigate the damage due to impact. Crash barriers, Fencing and Bollards are the most common methods that can be used to prevent the direct collusion with structural columns or to reduce the approaching speed of the impacting vehicle. Additionally, key elements can be avoided by providing alternative load paths. However, there may be restrictions applied on such methods due to aesthetic reasons and limitations on space. For instance, these options are hardly adopted or inadequate to prevent the vehicle impact on bridge underpass (see Fig 1.1). In the absence of suitable passive resistance, structural strengthening will be the most appropriate solution. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 1-2 In fact, structural columns are seldom designed for vehicle impacts due to inadequacies of design guidelines. The behaviour and vulnerability assessment of columns subjected to lateral impact is, therefore of an interest. This is requires a knowledge on the behaviour of concrete under extreme loading conditions, which is very complex. In fact, there are few experimental investigations conducted on laterally impacted columns that demonstrate the effects of strain rate and confinement, particularly under mid span hard impact conditions (Louw et al. 1992). However, the hard impacts usually represent the possible upper bound of the typical vehicle impacts (Tsang et al. 2005) and hence generate over conservative results due to the exaggerated strain rate effects. In contrast, the energy based vulnerability assessment techniques presented in the literature (Tsang et al. 2005) limit their application to columns that are fail in flexure under mid span low velocity impacts where strain rate and inertia effects are not predominant. The typical low elevation vehicle impacts initiate flexural-shear failure, a condition which is substantially different to flexural failure events under mid span impacts. Thus, research on shear critical RC columns under low elevation impact remains unexplored. Furthermore, since the impact response of columns is associated with higher modes of vibrations, strain rate effects, confinement effects, as well as various other vehicle specific parameters, the analysis process becomes quite complex (Hughes and Speirs 1982; Varat and Husher 2000). Consequently, the impact reconstruction techniques based on the deformable body assumption (Campbell 1974; Prasad 1990) are largely simplified while the rest of the methods are limited to simulate an impact between a specific vehicle and a column (El Tawil et al. 2005). These methods may not be reliable to assess the vulnerability of columns against a general vehicle population or new generation vehicles. As a result, the use of numerical methods for the vulnerability predictions of RC columns is exceptionally limited. This is also reflected by the generic and limited design guidelines provided in the current codes of practices (AASHTO-LRFD 1998; EN 1991-1-7:2006) which do not provide adequate design information on impacted columns. Additionally, different design codes specify significantly different magnitudes for the expected quasi-static impact loading, which indicates a lack of understanding of the dynamic behaviour of both the column and colliding vehicle (AASHTO-LRFD 1998; EN 1991-1-7:2006; Vrouwenvelder 2000). In fact, the equivalent static force given in AASHTO-LRFD Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 1-3 (1998) is based on information resulting from early full-scale crash tests of barriers used for redirecting tractor trailers and truck collisions. This indicates that the design forces were not derived directly from head-on collision tests. At present there are no comprehensive guidelines on how to detail a vulnerable column to ensure that it will survive in a catastrophic impact situation (El Tawil 2005). Consequently, there is a pressing need for the development of some simplified yet rational method to quantify the effects of impact. Another aspect of the present work was to determine the vulnerability of the structural column during the construction process when the applied load is only a portion of the design load and hence the shear capacity and the stiffness have not reached their full potential (Abrams 1987; Zeinoddini 2008). Proper impact damage assessment is vital to determine whether the column has to be replaced or can be repaired for further use. Design guidelines developed on partially loaded columns subjected to earthquake (Esmaeily and Xiao 2004) and blast (Shi et al. 2008) loading may not be adequate in this circumstance where mode of failure, strain rate effects and inertia effects are substantially deferent. Moreover, a decision on the portion of total load that can be allowed during the rehabilitation process has to be made. Proper damage assessment will be essential to minimise the risk to rescue workers who enter into the building following an impact, or when the affected bridge structure has to be used as a vital supply line. This thesis addresses these questions by investigating the impact capacity of columns particularly in low to medium raised buildings designed according to the Australian standards. Numerical simulation techniques were used in the process and validation was performed using experimental results published in the literature. The analysis extended to investigate the influence of critical parameters that govern the vulnerability of columns under lateral impact loads. Numerical simulations were conducted using the Finite Element program LS-DYNA (2006), incorporating steel reinforcement, confinement effects and strain rate effects. A simplified method based on impact pulses generated from full scale impact tests was used for impact reconstruction and the effects of the various pulse loading parameters were investigated under low to medium (30-80km/h) velocity impacts. A constitutive Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 1-4 material model which can simulate failure under a tri-axial state of stress was used for concrete. The sensitivity of the material model parameters used for the validation was also scrutinised and numerical tests were performed to examine their suitability to simulate the shear failure conditions. Columns made of 30 to 50MPa concrete with a longitudinal steel ratio ranging from 1% to 4% under pure axial loading were analysed in the first phase. The study was then extended to analyse columns subjected to single axis and biaxial bending. The impact capacity reduction under varying axial loads was also quantified. Suggestions on the residual capacity of the columns based on the status of the damage and mitigation techniques are presented. Empirical equations have been developed to quantify the critical impulses in terms of the geometrical and material properties of the impacted columns. Equations are provided for explicit and reliable vulnerability assessment. Moreover, a universal technique which can be applicable to determine the vulnerability of the impacted columns against collision with new generation vehicles under most common impact modes is proposed. The investigation has confirmed that columns in medium storey buildings located in urban areas are vulnerable to medium velocity car and truck impacts and hence these columns need to be re-designed for retrofitting. The proposed equations can be used to quantify the critical impact pulses for 300mm to 750mm diameter circular columns under axial loading. This allows the impact capacity of the columns to be improved by optimum use of the key design parameters without relying on external energy absorbers or wrapping. According to the overall results, the vulnerability of the axially loaded columns can be mitigated by reducing the slenderness ratio and concrete grade, and by choosing the design option with a minimal amount of longitudinal steel. Results also indicated that the ductility as well as the mode of failure under impact can be changed with the volumetric ratio of lateral steel. Moreover, to increase the impact capacity of the vulnerable columns, a higher confining stress is required. The general provisions of current design codes do not sufficiently cover this aspect and hence this research provides additional guidelines to overcome the inadequacies of code provisions. In addition, an extensive numerical simulation was conducted on uniaxially and biaxially loaded columns. In particular, the analysis procedure became quite complex Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 1-5 under biaxial loading due to large number of load combinations involved in the analysis process which includes the direction of the impact. Consequently, non-critical cased excluded from the analyses. Analytical equations were derived for 300mm to 600mm eccentrically loaded columns that are valid in the range of 0 o to 90 0 impact angles. The remaining impact angles can be treated separately based on the observation that columns under the Positive Eccentric Loading are non-critical. This procedure allows defining critical impulses for three consecutive points on the interaction diagram for a one particular column so that linear interpolation can be used to quantify the critical impulses for the points in-between. Having provided a known force and impulse pair for an average impact duration, this method can be extended to assess the vulnerability of columns for a general vehicle population. Consequently, this investigation delivered new insight and comprehensive technique to the scientific community for dealing with impacts. 1.2 Aims and objectives The overall aim of this research is to generate design guidelines and equations on the impact capacity of axially and eccentrically loaded columns while allowing optimum use of key design parameters without relying on energy absorbers or wrappings. Additionally, the impact capacity of the columns during construction will be assessed comprehensively where the applied axial load is only a portion of the total allowable load on the column and hence the shear capacity may be crucial under low elevation impacts. In particular, this thesis is concerned the global behaviour of the impacted columns and the influence of the critical parameters on their performance in order to mitigate damage. The research also has developed a numerical modelling technique to lessen the need for full scale vehicle impact testing. The aims of this research are achieved by addressing the following enabling objectives: a) Development of a finite element model for an impacted column, and validating the model using existing experimental results reported in the literature. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 1-6 b) Development of an impact simulation (reconstruction) technique which allow assessing the vulnerability of the column for a general vehicle population over most common modes of collisions under low to medium velocity impacts. c) Quantification of the effects of column diameter, axial load and moment, impact height, impact duration, strain rate effects, concrete grade, impact direction, support conditions, confinement and lateral steel ratio. It also investigates the method of enhancing the performance of the impacted columns by optimum usage of the critical parameters. d) Evaluation of the impact behaviour of reinforced concrete columns under single axis bending by extending the validation process to simulate the impacted columns under eccentric loading. Assess the vulnerability of biaxially loaded columns under serviceability conditions by excluding the non-critical load combinations from the analyses to generate conservative outcomes. e) Developing equations and design information to assess the vulnerability of the columns both under pure axial and eccentric loadings. Additionally, generate design information to assess the vulnerability of structures under construction while providing mitigation techniques for partially loaded columns through the enhancement of confinement effects. f) Examination of the static and dynamic shear capacity of the columns and quantify its correlation with the static shear capacity of the columns so that it can be used for approximate vulnerability assessments for impacted columns. g) However effects of the surrounding temperature variations and columns under submerged conditions will not be investigated due to the complexity of the simulation process. 1.3 Hypotheses and research problems Damage due to lateral impact loads applied on columns can be minimised if there is a Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 1-7 satisfactory amount of research information on their impact performance. To strengthen this argument, the following hypotheses will be tested throughout this research project: I. The lateral impact capacity increases substantially with the axial load. Consequently, the vulnerability of the structural column during the construction process may be high where the applied load is only a portion of the design load and hence the shear capacity and the stiffness have not reached their full potential. On the other hand, shear strength enhancement, resulting from the increase of axial load during impact, can effectively increase the lateral load carrying capacity of short columns. II. The impact capacity substantially changes with the bending moments due to initial deformation present in the columns prior to the impact. In particular, columns under mild tension and reduced compression have the tendency to reduce the shear capacity under lateral impacts. III. The impact capacity of the reinforced concrete columns can be increased by improving their ductile characteristics. In general, the ductile capacity depends on the amount and distribution of transverse reinforcement within the plastic hinge region and this concept was particular effective under earthquake loading. IV. Concrete grade, steel area, diameter of the column, hoop spacing, area of the hoops and yield strength and effective height are the key design parameters which determine the vulnerability of columns under shear critical quasi-static loading. The relation between these parameters can be been expressed in terms of static and dynamic shear capacities of the columns. V. Impact induced torsional moments may significantly change the internal stress and deformation capacity of structural columns by changing the failure mode particularly when the impact force is applied perpendicular to the direction of eccentric loading. There are no analytical models developed combining the effects of shear-flexure-torsion interaction for vulnerability assessment of the impacted columns. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 1-8 VI. A strength enhancement of 30% could be expected in structural beams under mid span impact loads, particularly when the strain rate effects accompanied by higher flexural strains (Louw et al. 1992). This enhancement would be undesirable for columns susceptible to fail in shear under vehicle impacts. On the other hand, most designers tend to adopt oversimplified procedures in vulnerability assessment and impact reconstructions. These procedures may not be rational and hence, underestimate or overestimate the consequences. For instance, Tsang et al. (2005) quantified the capacity of reinforced concrete columns under mid-span head-on collisions by using a displacement based failure criteria. However this energy based method limits its application to columns that fail in flexure under low velocity impacts where strain rate and inertia effects are not predominant. Alternatively, El Tawil et al. (2005) conducted a dynamic impact simulation to demonstrate the inadequacies of the AASHTO-LRFD (1998) code provisions by using a validated numerical model of a Chevy truck. The drawback of such methods is that the outcomes cannot be applicable to a general vehicle population or new generation vehicles. In fact, the equivalent static force given in AASHTO-LRFD (1998) is based on information resulting from early full-scale crash tests of barriers used for redirecting tractor trailers and truck collisions. This indicates that the design forces were not derived directly from head-on collision tests. At present there are no comprehensive guidelines on how to detail a vulnerable column to ensure that it will survive in a catastrophic impact situation (Tsang et al. 2005). Consequently, there is a pressing need for the development of some simplified yet rational method to quantify the effects of impact. 1.4 Methodology The finite element method has been widely used to perform numerical analysis as physical testing can be expensive and time consuming. Once validated, the numerical models allow detailed investigation of the stresses, strain and failure modes by changing the parameters where physical testing may be difficult due to practical limitations. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 1-9 In general, structural columns can be conveniently subdivided into two categories; axially loaded columns and columns with eccentric loading. Validation of the numerical models for each category of columns is performed at the beginning by using existing experimental testing and existing theoretical and numerical results. All numerical modelling in this research project was conducted using LS-DYNA (2006); using the pre-processor MSc Patran. LS-PREPOST was used to visualise the compiled results. The finite element code LS-DYNA is capable of demonstrating and simulating the nonlinear deformation of the impacted columns with the added advantage of simulating contact between the impacting bodies where two materials with different stiffness interact under high velocities. The selection of proper material models is also important in the validation process as the reliability of the result mainly depends on the capacity of the material formulation to simulate the observable behaviour of the impact. Two material models were selected for the validation process by considering their capacity to simulate the failure modes and their reliability of predicting the properties of different concrete grades in the parametric studies. After the validation, the most reliable model was used for the rest of the analyses. In addition, simulating the impacting mass in a realistic manner in the validation process allows comparison of the interface forces generated during the impact so that boundary conditions can be suitably adjusted to simulate the actual constraint applied on the column. The impact reconstruction process also plays an important role. Producing realistic finite element models of the impacting vehicle is very time consuming process and the conclusions drawn for a specific vehicle cannot be extended to a general vehicle population. Thus it was decided to use impact pulses generated from full scale vehicle impact tests to simulate vehicle collision. Having observed that the strain rate effects and the pulse shape (with same peak and duration) have negligible effects on the vulnerability of the columns, triangular pulses were used in the impact reconstruction process. The average duration of the typical vehicle impact was observed to be 100ms. The advantage of this method is that it can be applied to determine the vulnerability of the columns to new generation vehicles. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 1-10 Circular reinforced columns of 2m to 4m in height made of 30MPa to 50MPa concretes were considered in the analysis. Columns that are sufficient in capacity for 5 to 20 storey buildings designed as per AS3600 (2007) were considered in the first phase of the analysis under pure axial loading. Having observed that the typical 20 storey building columns are less vulnerable to medium velocity (80km/h) vehicle impacts, the second phase of the analysis was limited to the 15 storey building columns where eccentric loads were taken into account. In the process, priority was given to optimise the column design using key design parameters such as the longitudinal and vertical steel ratios, concrete grade, effective height and support conditions without relying on external energy absorbers or wrappings. Confinement effects were introduced to the model by using equations by Mander (1988). A parametric analysis was conducted on axially loaded columns in the first phase by varying the key design parameters. The analysis was also extended to investigate partially loaded columns to assure their safety during construction. The Damage Index D i was used to identify the capacity degradation of the impacted columns. Dependency of the duration of the impact on the enhanced stiffness characteristics of the columns due to axial load variation was neglected in the analysis. The investigation was conducted by reducing the axial load and then restoring the load at post impact stage. A simplified method was introduced to apply the post impact loading by avoiding complex explicit-implicit transformation based methods presented in the literature. The collapse of the impacted columns is investigated by varying the parameters so that there are substantial warnings before collapse following the post impact loading. The hoop spacing was of main interest as it was successful under earthquake loading. Consequently it was concluded that it would be more appropriate to replace the impacted columns rather than repair them for further use. In addition, a set of equations for explicitly determining the critical impact force (P c ) and critical impulse (I c ) that has been developed using the results from the parametric study and further simulations. In particular, an innovative technique was developed and introduced to make sure the accuracy of the equations developed for predicting the critical impact force and impulse where the other techniques are failed due to the shape of the error distribution. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 1-11 The analysis was extended to investigate the impact under uniaxial bending by applying the impact force in the plane of bending under full service load. Numerical simulation of half of the column was used for axially loaded columns where loads are symmetrical about the direction of the impact. The columns that deflect against the direction of the impact force are always safe compared to their counterpart. Thus further analysis was conducted by excluding such conservative load combinations. Having observed that the capacity drop can be around 30% when the load eccentricity reaches the ‘balanced failure’ point under uniaxial loading (compared to the axially loaded columns), further analyses have been limited to the 50% to 20% loaded columns with corresponding moments. Consequently design guidelines are generated to quantify the impact capacities of 50% and 20% loaded columns. The impact angle was taken into account in the third phase of the analysis. A full column was used in third phase where the biaxial moment application lead to unsymmetrical loading. Load combinations were selected based on the interaction diagrams for columns of grade 30to 50MPa concrete with 1% to 4% longitudinal steel. It was observed that three load combinations are sufficient for the vulnerability analyses of circular columns under biaxial bending. Vehicle impacts in the direction of the resultant moment were excluded from the analyses. Conservative results were generated by considering the columns with maximum (allowable) load eccentricities. A software program based on the least square method was used to generate simplified linear equations for all the three phases of the investigation. Polynomial equations were developed to quantify the critical impact pulses where the percentage error of the linear equation exceeded the acceptable limit. In particular, polynomial relationships were developed in stages by varying one parameter at a time. These parameters were then combined to produce an equation based on the least square method, which could be used to quantify the peak force and the associated impulse at the near collapse stage for fully loaded columns. In fact, the main aim was to define three consecutive points on the interaction diagram so that linear interpolation can be used to quantify the critical impulses for points in-between. In the process, effects of longitudinal and lateral steel ratios, concrete grade, direction of the impact, strain rate sensitivity of columns and slenderness ratio were also quantified. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 1-12 1.5 Thesis outline This section briefly summarizes the format of this thesis. Chapter 2 provides a literature review on analytical and experimental research to-date related to laterally impacted columns. Chapter 3 describe the numerical modelling and simulations of reinforced concrete columns under axial loading subjected to lateral impact. As the validation process mainly focuses on the vulnerability assessment of axially loaded columns it is extended to eccentrically loaded columns by using some of the investigation reported in literature. This chapter also describe an experimental setup reported in the literature the data from which have formed the basis for the validating the Finite Element (FE) model. Chapter 4 presents the methodology used for impact reconstruction with the aim of developing a comprehensive method for vulnerability assessment that can be applied to a general vehicle population including new generation vehicles. Impact pulses generated from full scale impact tests are used for impact reconstruction. The dynamic response of the impacted column and method for vulnerability prediction are also discussed in this chapter. Chapter 5 investigate the effects of enclose lateral confining steel reinforcement in columns subjected to axial load and lateral impact loads. The confining model proposed by Mander (1988) is used as the basis for the study. This chapter also describe sensitivity analyse of key design parameters to lateral impact of axially loaded columns. Based on the sensitivity analyses, analytical equations are developed to quantify the critical impact pulse parameters for columns of specific diameters and concrete grades. Investigation extended to quantify the residual capacity of the partially loaded columns to make a decision whether or not the impacted column has to be replaced or repaired for further use. Damage index D is used to identify the capacity degradation of the impacted columns. The collapse of the impacted columns is investigated by varying the parameters so that there are substantial warnings before collapse following the post impact loading. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 1-13 Chapter 6 describes the severity of eccentrically loaded columns subjected to transverse impacts. The eccentrically loaded columns are divided into two groups based on the direction of the impact in the plane of bending. To facilitate application of the eccentric load the column is provided with a bulk head; the impact behaviour is then investigated in detail. Chapter 7 reports the impact behaviour of RC columns under biaxial bending and extend the parametric analysis to columns under biaxial bending. Analytical equations are derived for the bi-axially loaded columns; single axis bending is treated as special case of a biaxial bending with zero moment about one axis. This chapter deals exclusively with the orientation of the impact for 50% and 20% fully loaded columns with the aim of defining three consecutive points on the interaction diagram for one particular column so that linear interpolation can be used to predict the critical impulse for points in-between. These equations are exclusively valid between 0 o to 90 o impact angles. In fact, having provided that columns under positive eccentric loads can be treated as non-critical, the equations can be extended to account for the other impact angles. Chapter 8 Summarises the main conclusions that have emerged from this thesis along with their practical implications. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-15 2. LITERATURE REVIEW 2.1 Characteristics of impact pulses The duration of the typical vehicle frontal impact is around 100ms. Separation occurs soon after the impact if there is no post collision speed or further impact. However, in the real world, depending upon the circumstances of the impact and the type of vehicle involved, the duration of the impact can vary. If the crash is off-set, that means if the vehicle hit at an angle, then the duration of the impact will be longer. Depending upon the angle of crash, offset collisions can result in approximately a 200ms impulse. If the vehicle has a stiff front, the duration of the impact will be less because of smaller deformation. An example of an impact sequence of a vehicle (truck) is shown in Figure 2.1, where the total duration of the impact is around 0.18 seconds. The duration and the force generated from an impact can also be taken as a measure of the vulnerability caused by the impact. Figure 2.1: Sequence of an impact (El-Tawil et al. 2005) The force versus time response generated by the transverse impact of a Chevy truck for various approaching speeds for two different pier configurations namely pier I and II, is shown in Figures 2.2(a) & 2.2(b). The impact force versus time functions records contain several small amplitudes followed by a large spike irrespective of the approach velocities of the vehicle. The sharp peaks occur when the stiff and heavier components such as chassis or engine block reach the pier and interact with it. As the approach Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-16 speed increases the first peak occurs earlier in time. It is also important to note that pier I and II has two different pier configurations, and hence, different stiffness. Comparatively higher peak force is observed in Figure 2.2(a), as the pier ‘I’ has the higher level of stiffness. Figure 2.2:Impact force vs. time histories for Chevy truck at various speeds (El-Tawil et al. 2005) According to El-Tawil et al. (2005) the peak force generated at the impact is not representative of the design structural demand, as the structures do not have enough time to respond to a rapid change of loading. Chopra (2001) suggested that the equivalent static force, which is defined as the static force necessary to produce the same deflection at the point of impact as that produced by dynamic event, is a more appropriate measure of the design structural demand. However, this displacement based criteria for failure is distinguished from the traditional strength or strain based criteria at the threshold of damage, and applicability of these criteria to assess column response under high velocity impact may be questionable and requires needs careful further considerations (Tsang et al. 2005). 2.2 Behaviour of structural elements under impact loading Behaviour of structural elements under dynamic loading conditions is quite different from the behaviour under static loading conditions. Dynamic loads, such as impact loads, give rise to accelerations of the structural elements and kinetic energy and the inertia effects must be considered in the analysis. Structural element subjected to dynamic loading conditions must have higher energy absorption capacity, and therefore, should be designed to allow for plastic deformations. The plastic deformation capacity will improve the ductility of the element and hence prevent the Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-17 occurrence of a brittle failure mode under severe impact conditions. In addition, it is important to utilize more flexible elements, which can undergo larger deformations and thereby, obtain a larger energy absorption capacity. Reinforced concrete (RC) elements subjected to dynamic loading conditions will exhibit two forms of failure modes namely, flexure and shear (Johnny 2003). Flexural failure will often result after the formation of plastic hinges at the locations where the ultimate moment capacity is reached. This failure mode is characterised by initial cracking of the concrete, subsequent yielding of the tensile reinforcement and ultimately compression failure of the concrete. Also, this failure mode is rather ductile and absorbs energy during impacts. Contrary to the flexural failure mode, the shear failure mode is catastrophic and brittle in nature, which severely hampers the energy absorption capacity of the element. The ultimate moment capacity of an element cannot be obtained and diagonal tension cracks will form close to the supports, followed by initial tensional cracks that develop at the points where the maximum moment is reached. Premature failure will result. 2.3 Dynamic impact tests on reinforced concrete columns Previous research on columns has mainly focused on improving the axial load carrying capacity and stiffness, while improvement of impact resistance has been largely unexplored. The few investigations conducted on laterally impacted columns highly emphasised the importance of the stain rate effects. Some of the test results indicated that the increased structural resistance is somewhat greater than the commonly accepted maximum increase of 30% of the static resistance (Louw et al. 1992). Strain rate effects, as well as the behaviour of the vehicle during the impact, are of primary importance as far as the structural response is concerned (Prasad 1990). Thus the impact is classified as soft or hard, based on the way that impact energy absorbs during an impact. Generally, in a soft impact the striker absorbs most of the kinetic energy through plastic deformation, while the structure experiences minor deformations. 2.3.1 Columns subjected to soft impact Leodolft (1989) tested thirty-nine 350x150x1600 mm reinforced concrete columns Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-18 under soft impact conditions. The soft impact condition was achieved by inserting a pipe buffer system in between the pendulum and the column. The columns were axially preloaded with 100 kN and 200 kN forces and subjected to an impact velocity of about 7ms -1 . The applied loads were sufficient to permanently damage the impacted columns. In this experiment, the peak load occurred later and is more likely to influence the flexural shear resistance of the element, than its pure shear and inertia stiffness. During the first 10 ms, the buffer system was subjected to elastic-plastic deformations. By that time, substantial energy had been transferred to the column, which deflected significantly. The generated axial load from the impact was increased as the column increases in length and subsequently decreases and remained compressive. In addition, the strain rate of up to 10 -2 was generated at the rear surface of the column. It was observed that the partially damaged columns exhibited the same static lateral capacity as the undamaged columns. Moreover, the impacted columns were subjected to a series of peak shear and corresponding moments and peak moments and corresponding shear. According to the test results, it was concluded that the dynamically loaded slender columns are considerably stronger than the ultimate load predicted by the modified ACI equation for the slenderness ratio. 2.3.2 Columns subjected to hard impact However, during a hard impact, the kinetic energy of the striker is mostly absorbed by the structure and the striker itself suffers small deformations. Fererabend (1988), conducted an experimental investigation on 300x300x4000mm reinforced concrete columns subjected to lateral impact at mid span. The columns were tested in a horizontal position, where one end was restrained using a 20t mass to simulate the inertial restraint provided by a bridge deck. The axial load was applied by pulling the free sliding end using external prestressing bars towards the stationary end. The impact load was generated by dropping a 1.14t mass onto the column at mid span and the shear reinforcements were provided to ensure a flexural failure of column. An important feature of the impact behaviour of that column was the initial increase in axial force as the column lengthened along its centre line. The authors also observed that the initial peak of the applied impact load depended on the inertial characteristics Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-19 of the column and the boundary conditions. Under the effects of the impact load, the column experienced shear deformations while local deformations occurred at the point of impact. Even though these deformations were relatively small, the initial impact force had a high initial peak. The initial force was opposed primarily by the inertia forces of the element. The shear stiffness of the column was the main parameter that controlled its response. As the shock wave progressed through the cross sections of the elements, they were subjected to fluctuating moments, shear forces and axial loads. After observing these responses the author has emphasised the impotency of the all these forces in determining the critical section. By assuming a 10% increment of the material properties due to strain rate effects, the dynamic moment capacity of the tested column exhibited 20% increment compared to that of its static value. On the other hand, observed dynamic shear capacity of the column was substantially greater than the ultimate static shear capacity of the column. Therefore, it was concluded that the initial peak shear force generated during hard impact, is not an indication of the ultimate structural resistance of the column when adequately reinforced in shear. In addition, under the hard impact condition the moment-shear combination moves from a low moment high shear value to a higher moment much lower shear value. Therefore there is a possibility to generate initial shear cracks in a section which probably diminishes the flexural resistance that follows. 2.3.3 Columns subjected to axial impact The dynamic buckling response of columns under axial impact loads has been subjected to extensive investigation over the past decades. The interest was mainly focused on the behaviour of the short and slender columns under eccentric loads. In addition, parametric studies have been conducted on element aspect ratio, element formulation, boundary conditions and geometric imperfections. Most of the researchers selected low velocity hard impact conditions with fairly large masses. Kenneth et al. (1964) conducted research on a total number of 205 plain and RC columns under concentric and eccentric loading conditions. In addition, two foundation conditions were also simulated by using rigid pads and rubber supports. In this experiment, the rise time was defined as the time between the commencement of Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-20 the load and the ultimate load. The average rise time in all the dynamic tests was about 30ms. Longitudinal inertial forces were neglected because the longitudinal natural period of the columns was less than 4.0 ms for all the specimens. The results of this test showed that even though the ultimate load on statically loaded columns agreed well with the ACI 318-02 (2002) code specifications, dynamically loaded columns exhibited a 30 to 40% increment compared to the statically loaded columns. In addition, columns on elastic foundations, such as rubber or soil, were stronger when loaded dynamically than similar columns on rigid supports. The test results further indicated that, the ultimate strength of short dynamically loaded columns can be computed from the ACI equation, after being modified to account for the strength increment of the material due to strain rates. However this was valid only for the strain rates in which the inertial forces are negligible. 2.3.4 Shortcomings of the individual column tests There are several disadvantages associated with individual column tests (Gebbeken et al. 2007). The main disadvantage being the idealized boundary conditions. The flexibility of the realistic support conditions was not taken into account in these tests. This factor can shift the location of the plastic hinge and consequently, the failure mechanism would be different from the usual fixed assumption. In addition, the effects due the wave reflection at the boundaries cannot be neglected. At free boundaries, the compressive wave is reflected as a tensile wave, while at fixed boundaries, the reflected wave becomes a compressive wave. The small models are the ones that suffer most due to the boundary conditions (Gebbeken et al. 2007). Even though the shear cracks, spalling of the concrete cover, and confinement failure are the ideal failure modes for the individual columns, the effects of the global structural configuration cannot be neglected. 2.4 Dynamic tests on reinforced concrete beams Depending on the nature of the transient dynamic load, an element can be subjected to higher modes of vibrations. Even though the amplitudes of the higher modes are relatively small, they can give rise to larger shear forces in the element. Hughes and Speirs (1982) performed a theoretical and experimental investigation on the behaviour Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-21 of concrete beams subjected to mid span impact by a falling mass. They mainly focused on the static and the dynamic vibration modes of the beams under the pin-ended conditions. By comparing the static mode with first and third free vibration modes under equal potential energies, a relationship between the mode of vibration and the mode of failure was derived. The mode profiles considered are illustrated in the Figure 2.3. The figure shows reduction in displacement and bending moments for the first and third modes, compared to that of the static mode. However, the shear forces are greater in the dynamic modes and as a consequence, beam could fail in flexure under static loading, whereas an identical beam might fail in shear under dynamic loading. The experimental results of Takeda et al. (1977) support these arguments. They have demonstrated that the reinforced concrete beams with high shear could fail in a ductile manner under static loading, but in a brittle manner under dynamic loading with some reduction in shear strength of concrete. Figure 2.3: Comparison between different modes of vibration with equal potential energies (Hughes and Speirs 1982) Niklasson (1994) investigated the impact behaviour of simply supported reinforced concrete beams with different amounts of reinforcement and different concrete grades. The impact loads were applied as symmetrical point loads on either side of the mid-span. In this experiment, the rise time and the amplitude of the applied loads were controlled by placing rubber pads at the interface between the beam and the impacted mass. This diminished the magnitude of the applied load, while increasing the rise Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-22 time. The increased rise time led to low impact frequencies, compared to a short rise time. Therefore, the number of modes of vibration excited by the loads differed from each other, which led to a different failure mode. In addition to the rise time, the natural frequency of vibration may also influence the response of the structural elements (Niklasson 1994). Therefore, in order to determine the impact response of a beam, the correlation among the load duration, rise time of the load and the natural period of vibration must be taken in to account. When the loading pulse has a short duration and a high peak value, the probability of shear failure increases (Kishi et al. 2002). Other than the load intensity, Kishi et al. (2002) also showed that beam stiffness played an important role in the change of failure mode. By conducting research on concrete beams without stirrups, the authors concluded that beams with a low amount of reinforcement may fail in flexure, whereas beams fail in shear when the lateral reinforcement amount was doubled. However, when the velocity of the impact increased further, beams carrying a low amount of lateral reinforcement also failed in shear. Based on similar experimental results, Palm (1989) also made similar conclusions. Ansell (2005) points out that a concrete structure, designed to fail in flexure under static loads, may fail in shear when loaded dynamically. Moment and shear waves generated by the impact loads will travel from the impacted zone towards each support, as described by Hughes and Speirs (1982). Under these circumstances, large bending moments and shear forces will generate along the beam, which differs from the shear and moment distribution in a static load case. This will result in local failure with the mass punching a cone out of the beam, due to the concentration of large shear forces in that area, usually referred to as punching shear failure (Anzell 2005). Bentur et al. (1986) conducted a series of tests on conventionally reinforced concrete beams under impact loading conditions. The impact load was generated by dropping 345kg mass from a 3m height onto the 1.4x0.1x0.125m beam with a spanning length of 0.960m. According to their observations, concrete can withstand a higher bending load under impact than under static conditions and can absorb more energy under Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-23 impact loading conditions. Further, steel deformation under the impact load was confined to a region only few centimetres long, located beneath the point of impact. The deformation caused by the impact exceeded the maximum strain capability of the steel. This means that there was not enough time to develop extensive bond slip along the length of the bar under impact conditions. Most of the time, the steel failed in a ductile manner at the point of the impact. However, under quasi-static loading, beams which were deflected to the same degree showed no evidence of reinforcement failure. Instead, significant cracking and debonding of concrete along the reinforcing bar was observed. Kulkarni and Shah (1998) investigated the strain rate effects of concrete beams by applying the impact force using hydraulic system. The beams were loaded at a rate of 0.00071 cm/s and 38 cm/s. The tests on seven pairs of singly reinforced simply supported concrete beams (without shear reinforcements), showed some reduction of the total number of cracks at high strain rates. Also, there wasn’t any sharp ‘Yield Point’ or a ‘Yield Plateau’ in the load-deflection curves for beams failing in flexure at the high strain rate. Standard sectional analysis using a rate-dependent constitutive relationship did not adequately predict the shape of the high-rate load deflection curve, and the localised yielding of steel at higher strain rate was believed to be one of the reasons behind this observation. In addition, the final failure mode shifted from shear failure at the static rate, to flexural failure at the high rate, contrary to what was found by other researchers. Small-scaled cantilever beams, having different shear span to depth ratios and stirrup spacings, were tested by Chung and Shah (1989) to investigate the effect of loading rate on bond in beam-column joints. The generated strain rates were in the range of 0.004-0.08s -1 . Reduction of the cracks in the specimens was observed due to increased tensile strength of the concrete and bond strength between concrete and steel. The improved bond strength at high strain rates also led to a stress concentration in the reinforcements. As a result, the steel yield earlier and lead to a lower ductility ratio at failure. The stiffness of the beams also affected by the loading rate and different failure modes were observed at different loadings rates. It is evident that the confinement effects act as a governing factor of the failure modes. For stirrup spacing of s/2 the Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-24 specimens failed in brittle and ductile modes under dynamic and static loading conditions respectively. For specimens with stirrup spacing of s/4, the loading rate appeared to have no effect on the failure mode. 2.5 Behaviour of concrete under impact loads The behaviour of concrete under the effects of high strain rate has been extensively studied over the past few years. According to the Watstein (1953), the initial investigations carried out by Jones and Richard (1936) and Granville (1938) concluded that the compressive strength of concrete increased with the rate of loading. Watstein (1953) suggested that there was an increase of over 80% in compressive strength for concrete loaded at a strain rate of 10s -1 . Tests carried out on plain concrete specimens under uniaxial loading conditions revealed that compressive impact strength can be as much as 85-100 % higher than the static strength of concrete (Bischoff and Perry 1995). However, with respect to the critical axial strain values at peak stress, there is no general agreement about changes in the deformation. Material properties such as the compressive strength, the poison’s ratio, the volumetric strain and the ductility, may increase as the strain rate increases beyond the static value. Particularly, uniaxial compressive strength of plain concrete increases linearly with the logarithmic increase in strain rate (Mainstone 1975; Suaris and Shah 1982). The secant modulus of elasticity may also be changed under high strain rates. But there is no evidence of change of elastic modulus or initial tangent modulus due to the effects of strain rate (Bischoff and Perry 1995). Concerning the energy absorption, it appears to depend to a large extent on the strain capacity of concrete, which in turn is governed by the failure mode. This means that the higher grades of concrete will show less absorption of energy at failure contrary to what would be expected (Georgin 2003). In addition, Fu et al. (1991a) observed that concrete fails in an explosive manner under very high strain rates. Theoretically, energy absorbed per unit volume of a material during an impact, can be written as; E 2 2 σ Eq. 2.1 where σ is the stress of the material and E is the modulus of elasticity of the material. This means that small portion of a member, where the highly localised stress occurs, Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-25 absorbs an excessive amount of energy before the main portion of the member can be stressed appreciably. As a result, the small portion where the localised stress occurs is likely to be stressed above the yield stress of the material. Therefore, even if the material is relatively ductile, the energy absorbed by the local area of the member may be subjected to rupture. In fact, this may account for explosive manner under which concrete fails in very high strain conditions. Hughes et al. (1972) investigated the compressive strength and ultimate strain of concrete by using the drop hammer test. They investigated the effects on compressive strength, ultimate strain, energy absorption and deformation modes of concrete cubes under impact loading. Trapezoidal load function was assumed for both, the hammer impact pulse and cube impedance function and hypothetical load and strain records were derived. Concrete with varying mix proportions and two different course aggregates were examined and the resultant compressive strength and ultimate strain were reported. The compressive strength of cubes tested at a stress rate of less than 160 kPa/s and rates of strain of less than 8s -1 was almost the same as the static compressive strength. Increment of the strain rate beyond 8s -1 caused considerable effects on the compressive strength. For example, an average increment of the compressive strength of 28 days concrete cube was 11% greater when it was tested at a strain rate of 10s -1 . When the strain rate was increased to 14s -1 , the strength was increased to 25%. From these test results, it can be concluded that the concrete gains a substantial increase in compressive strength under the effects of high rate loading. On the other hand, this enhancement of strength may change the failure mode of concrete from ductile to brittle. Therefore, omission of the loading rate may cause substantial errors in both, prediction of failure mode and magnitude of impact response. According to Fu et al. (1991b), early studies on split cylinder test indicate that tensile strength of concrete increases with increasing stain rate. Comparing results between dynamic and static tests, Cowell (1966) observed an increasing tensile strength of 18-65%. Tekeda and Techicawa (1971) obtained a 70% increment in the tensile strength of concrete. These test data were obtained using different test set ups, Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-26 specimen dimensions and material properties, and hence direct comparison would be difficult. However, the general perception is that the strength increment due to strain rate effects is more pronounced in tension than in compression (Barpi 2004). According to Figure 2.4 the considered strain rates are less than 1s -1 . However the difference is become even large beyond this strain limit (CEB-FIP 1990). Figure 2.4: Strain-rate sensitivity for concrete in compression, tension and flexure ( Suaris and Surendra 1985) Analytical stress-strain curves derived by Suaris and Surendra (1985) also indicated the higher strain-rate sensitivity in tension compared to the compression. The graphical representation of the suggested curves is shown in the Figure 2.4. Test results on compressive, tensile and flexural responses of concrete also indicated that the amount of strength increase is the highest for concrete under tension and lowest for concrete under compression (Tekeda et al. 1977). The strength increment under flexure lies in-between the increments gained under tension and compression. This means that concrete elements under flexure will exhibit higher strain rate sensitivity than the elements under compression (Sukontasukkul and Mindess 2003). The water/cement ratio has a considerable influence on concrete behaviour at high strain rates. Kaplan (1980) tested a large number of concrete specimens to investigate the relationship between the concrete strength and loading rate for concrete with various moisture contents. He concluded that the moisture content in concrete is one of the principle variables affecting the relationship between strength and the loading rate. When concrete cubes with a low water/cement ratio were tested under high strain rate, Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-27 a considerable increment of the compressive strength was observed compared to that of the cubes tested under static loading conditions (Cowel 1966). Decreasing the water/cement ratio from 0.98 to 0.63 increased the static compressive strength only by 3.5 times the average standard deviation. However for the impact tests, it increased the dynamic compressive strength by 14 times the average standard deviation. More recently Reinhardt et al. (1990) extensively investigated the influence of the free water content on the behaviour of micro-concrete, maximum aggregate size of which was 2mm, under strain rate from 0.25 to 1.25s -1 . Dry specimens showed a small increase of strength of 1.45, while wet specimens showed a larger increase of 4.1. 2.6 Dynamic properties of concrete and steel 2.6.1 CEB-FIP specifications for concrete Concrete is very strain rate sensitive. In the CEB-FIB Model Code (1990), there is a relationship for DIF (Dynamic Increase Factor) for compression and tension at varying strain rates. The DIF in the code is a design value, which means that the given strength increments are lower than the values obtained from the experiment. For a given stress rate, the compresive strength under high rates of loading may be estimated from the following equations (CEB-FIP 1990). 2.6.1.1 Modified strain rate for concrete in compression ( ) s co c cm imp c f f α σ σ & & / , = for s MPa c / 10 6 ≤ σ& , Eq. 2.2 ( ) 3 / 1 , / co c s cm imp c f f σ σ β & & = for s MPa c / 10 6 > σ& , Eq. 2.3 cmo cm s f f 6 10 1 + = α , Eq. 2.4 and 2 6 log − = s s α β , Eq. 2.5 where, f c,imp is the mean impact compressive strength, c σ& is the stress rate (MPa/s) valid in the range 1 MPa/s < c σ& < 10 7 MPa/s, f cm is the mean concrete compressive strength, f cmo = 10 Mpa, s Mpa co 1 − = σ& . Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-28 Similarly, for a given strain rate the compressive strength may be estimated from, ( ) s co c cm imp c f f α ε ε 026 . 1 , / & & = for 1 30 − ≤ s c ε& , Eq. 2.6 ( ) 3 / 1 , / co c s cm imp c f f ε ε γ & & = for 1 30 − > s c ε& , Eq. 2.7 2 15 . 6 log − = s s α γ , Eq. 2.8 where, c ε& is the strain rate valid for 30x10 -6 < c ε& < 3x10 2 s -1 , co ε& = -30x10 -6 s -1 , Eq. 2.9 cmo cm s f f 6 10 1 + = α . Eq. 2.10 2.6.1.2 Modified strain rate for concrete in tension Compared to other materials, concrete exhibits higher strain rate sensitivity under impact loading, due to scale size of the heterogeneity (Weerheijm and Doormaal 2007). Especially, the tensile strength exhibits a strong increase beyond loading rates in the order of 10MPa/s. For a given stress rate, the tensile strength under high rates of loading may be estimated from the following equations. ( ) s cto ct ctm imp ct f f δ σ σ & & / , = for s MPa ct / 10 6 ≤ σ& , Eq. 2.11 ( ) 3 / 1 , / cto ct ctm imp ct f f σ σ & & = for s MPa ct / 10 6 ≥ σ& , Eq. 2.12 with cmo cm s f f 6 10 1 + = δ , Eq. 2.13 3 7 7 log − = s δ λ , Eq. 2.14 where, f ct,imp is the mean impact tensile strength, ct σ& is the stress rate (MPa/s) valid for 0.1 MPa/s < ct σ& < 10 7 MPa/s, f ctm is the mean tensile strength, = cto σ& 0.1MPa/s, f cmo = 10 MPa. Similarly, for a given strain rate the tensile strength under high rate of loading may be Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-29 estimated from, ( ) s cto ct ctm imp ct f f δ ε ε 016 . 1 , / & & = for 1 30 − ≤ s ct ε& , Eq. 2.15 ( ) 3 / 1 , / cto ct s ctm imp ct f f ε ε β & & = for 1 30 − > s ct ε& , Eq. 2.16 33 . 2 11 . 7 log − = s s δ β , Eq. 2.17 where ct ε& is the strain rate (s -1 ) valid for 1 2 1 6 10 3 10 3 − − − × < < × s s ct ε& , o ct ε& = 3×10 -6 s -1 . 2.6.1.3 Modulus of elasticity The strain rate dependence of the elastic modulus is also included in the CEB-FIP Model Code 1990, as presented in Figure 2.5 with the governing equations. The effect of stress and strain rate on modulus of elasticity may be estimated from, ( ) 025 . 0 , / co c ci imp c E E σ σ & & = , Eq. 2.18 ( ) 026 . 0 , / co c ci imp c E E ε ε & & = , Eq. 2.19 where, c σ& is the stress rate (MPa/s), ε& is the strain rate (s -1 ), Figure 2.5: Model for the strain rate dependency of concrete elastic modulus according to the CEB-FIP Model Code 1990 (CEB 1993) Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-30 co σ& = -1.0 MPa/s and ε& co = -30×10 -6 s -1 for compression, cto σ& = 0.1 MPa/s and ε& cto = 3×10 -6 s -1 for tension, E c,imp is the impact modulus of elasticity, E ci is the modulus of elasticity (MPa) at a concrete age of 28 days, obtained from the following equations: ( ) [ ] 3 / 1 / ' com c co ci f f f E E ∆ + = , Eq. 2.20 Where, f ’ c is the characteristic strength (MPa), ∆f = 8 MPa, f com = 10 MPa, E co = 2.15×10 4 MPa. Similarly, when the actual compressive strength of concrete at an age of 28 days f ’ c is known, E ci may be estimated from, [ ] 3 / 1 / cmo cm co ci f f E E = . Eq. 2.21 In the situations where only an elastic analysis is carried out for a structure, the reduced modulus of elasticity E c should be used in order to account for the initial plastic strain. The reduced modulus of elasticity can be calculated as, E c = 0.85E ci Eq. 2.22 Values of the tangent moduli E ci and the reduced moduli E c for different grades of concrete are given in the Table 2.1. Table 2.1: Tangent moduli and reduced moduli of elasticity According to the CEB-FIP Model Code 1990 (CEB 1993), the strain rate of 3×10 -5 s -1 is considered as the datum where the dynamic loading rates begin. As a comparison, a strain rate of around 0.01s -1 can be expected for concrete columns subjected to soft impact loading (Louw et al. 1992). Many researchers have comprehensively investigated the compressive and tensile strength of concrete at different strain rates (Magnusson 2007). In their works, it was observed that there is an increasingly large scatter in the test results for increasing strain rates (see Fig. 2.6). The reason for this Concrete grade C12 C20 C30 C40 C50 C60 C70 C80 E ci (GPa) 27 30 34 36 39 41 43 44 E c (GPa) 23 26 29 31 33 35 36 38 Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-31 relatively large scatter would have been the differences of the experimental techniques and the method of analysis that they employed in the testing procedure (Bischoff and Perry 1991). On the other hand, the results may have been influenced by factors such as specimen size, geometry, aspect ratio and the moisture content in the tested specimens (Magnusson 2007; Bischoff and Perry 1991). However, in contrast to this scattered behaviour, there is a relatively sharp transition zone with different strain rates. The sharp transition zone is common for both compressive and tensile strengths and each portion can be represented by straight lines, which consist of a moderate increment followed by a steep increment (see Fig. 2.6 & 2.7). Figure 2.6: Strain-rate influence on the compressive strength of concrete (Bischoff and Perry 1991) Many researchers commented on the first moderate increment in strength, as summarised by the Johansson (2000). Many of them observed that wet concrete specimens are more strain-rate sensitive than dry specimens during the first moderate increment. Therefore, most of the explanations were based on the viscose effects produced by the free water inside the micropores. For example, when the specimen is loaded in compression, the free water in the specimen is forced to move inside, which results in a build-up of internal pressure. This pressure improves the capacity of the materials to resist external loads, delays the crack initiation and improves the compressive strength. The water trapped inside the micropores not only helps to increase the compressive strength of the concrete, the thin films of water trapped between the particles aid the reduction of movement between aggregates and hence Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-32 increase the tensile strength. Figure 2.7: Strain-rate influence on the tensile strength of concrete (Malvar and Crawford 1998-a). As the strain rate exceeds the transition zone, both compressive and tensile strength show a sharp increment (see Fig. 2.6 & 2.7). This is mainly caused by the inertia effects and the lateral confinements of concrete. Weerheijm (1992) investigated the effects of changes in the strain rate on tensile strength of concrete. They suggested that changes in the stress and energy distributions due to inertia effects around the crack tips are the cause of the sharp strength increment. However, under compressive loading conditions the process is affected by the propagation of micro-cracks around the crack-tips, and hence, the explanation may not be valid under compressive loading conditions. A reasonable explanation of the behaviour observed under the high compressive strain rate conditions was given by Bischoff and Perry (1995). Their argument was that when it comes to the crack-propagation, the crack-propagation velocity must have an upper physical limit. When the stress rises faster than the time needed for propagation of cracks, an apparent delay in the crack propagation occurs and hence an increase of strength can be expected. They also considered the effects of lateral inertia confinements. This effect can be described by the following hypothesis. An elastic material subjected to axial compression should exhibit lateral expansion due to the effects of Poison’s ratio. However due to inertial confinements, a cylindrical specimen subjected to a rapid axial load increments is not able to expand in the radial direction at Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-33 an instant. This creates conditions similar to those in a specimen that is subjected to a uniaxial stress conditions under increased lateral confinements. Even though this condition prevails only until the material accelerates in the radial direction, a substantial increase in compressive strength can be generated (Bischoff and Perry 1991). Figure 2.8: Model for the strain rate dependency of concrete in compression and tension according to the CEB-FIP model code (CEB 1993) and with the modified model according to Malvar and Crawford (1998-a) A model for the strain rate dependency of concrete in compression and tension was presented in the CEB-FIP Model Code 1990 (CEB 1993) and is valid for strain rates of up to 300s -1 . According to the CEB-FIP code, the rapid increase in the material properties for both, compressive and tensile loading starts at strain rates greater than 30s -1 (see Fig. 2.8(a) & (b)). However, based on the extensive amount of reliable experimental results, Malvar and Crawford (1998-a) suggested that the rapid change of the tensile strength should start around 1s -1 , instead of 30s -1 . Therefore, they proposed a formulation similar to the CEB-FIP model, which was fit against the test data given in Figure 2.7 and their results with the CEB-FIP specifications are shown in Figure 2.8(b). Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-34 2.6.2 Dynamic properties of steel The stress-strain behaviour of steel is particularly sensitive to the loading rate and this phenomenon is known as strain rate sensitivity. As far as energy absorption is concerned, the strain rate sensitivity plays an equally important role to that of the inertia effect of the material. It clearly reflects from the load-displacement curve of the material, which was tested under various uniaxial compression strain rates (Marsh and Campbell 1963). Information from the existing literature on the effects of strain rate on the yield strength of reinforcing steel has been summarised by Lu et al. (1991b). According to Lu et al. (1991b), Cowell (1966) found that strength increment for steel with static strength 351MPa when tested under strain rates of 0.03s -1 , 0.1 s -1 , 0.3s -1 and 1.0s -1 respectively was 10%, 13%, 17% and 19%. However, for steel with yield strength of 264 MPa loaded at similar strain rates, the corresponding increment rose to 25%, 33%, 38% and 53% respectively. Similar observations were also made by Norris et al. (1959) concerning static yield strength of 330 MPa and 278 MPa, when tested under similar conditions. Wakabayashi et al. (1980) performed a tensile test on round and deformed steel bars with a 13mm diameter. The measured stress-strain curves showed that with increasing strain rate both, the upper and lower yield stress increased. Compared to the yield strength at quasi-static rate, the average increase in lower yield strength was 7-8% at a strain rate of 0.005s -1 and 16-18% at a strain rate of 0.1s -1 . A similar increment for the upper yield strength of the material was also observed. To account for the strain-rate effects in numerical applications, it is more desirable to have an explicit rate-dependent constitutive equation. Numerous attempts have been made to generate an effective constitutive model to describe the strain-rate sensitivity of the material. Cowper-Symond’s relation is the most widely employed rate-dependent constitutive equation, applicable particularly for solving impact problems. It was also found to be reliable for considering strain rate effects. This relationship basically represents a perfectly plastic material with a dynamic yield or flow stress that depends on strain rate. According to Reid and Reddy (1986), the Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-35 Cowper-Symond’s relationship can be widely used to account for strain rate effects in dynamic structural plasticity problems. In addition, this equation gives dynamic flow stresses, which agree well with the dynamic uniaxial tension and compression test results on several materials (Jones and Wierzbicki 1983). Other than the factors mentioned before, the major advantage of this constitutive equation is its portability with finite element programs. This means that the data required to generate strain rate effects can be directly fed in to the finite element program and hence be extensively used for research in this thesis. The Cowper-Symond’s constitutive equation is noted as follows: ( ) q c s d D 1 1 ε σ σ & + = , Eq. 2.23 where, D c represents a characteristic strain rate, q is a measure of the rate sensitivity of the material, ε& is the strain rate and σ d and σ s represent the dynamic and static stress of the material respectively. Malver (1998) proposed another equation for steel reinforcing bars produced under ASTM standards. The equation is particularly valid for the steel bars with yield stress ranging from 290 MPa to 710 MPa and for strain rates between 10 -4 and 10s -1 . The Dynamic Increasing Factor (DIF), which is defined as the ratio of the dynamic to static yield stress, was used to represent the influence of strain rate on strength enhancement under dynamic conditions. To derive these equations Malvar (1998) used several test results available in the literature. It is evident that under dynamic loading for a strain rate of up to 10s -1 , the strength properties of the reinforcing bars increased up to 60 %. For determining the yield strength and ultimate strength for reinforcing bars at different strain rates, he proposed the following formulation of the DIF: α ε | ¹ | \ | = −4 10 & DIF , Eq. 2.24 where for yield stress α = α fy | | ¹ | \ | − = 414 040 . 0 074 . 0 y fy f α , Eq. 2.25 and for the ultimate stress α = α fu | | ¹ | \ | − = 414 009 . 0 019 . 0 y fu f α , Eq. 2.26 where f y is in MPa. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-36 Based on the existing DIF data for both yield and ultimate strengths, it is concluded that the DIF is inversely proportional to the yield strength. This means that the same argument of higher strength gain for lower strength materials under dynamic loads appears to be applicable in both concrete and steel. Generally, the yield strain, the strain at which strain hardening begins, as well as the length of the yield plateau in the stress-strain diagram of steel, will increase at higher strain rates (Lu et al. 1991b). But there are no significant effects of the loading rate on the modulus of elasticity and ultimate strain for steel (Wakabayashi et al. 1980; Malvar 1998). 2.7 Interaction between reinforcement and concrete Chemical adhesion, frictional resistance and rib-bearing are the main components of interaction between reinforcement and concrete. According to Luts et al. (1967), chemical adhesion is the main resisting mechanism for very small values of bond stress in order of 200psi (1.38MPa). When bond stresses increase further, chemical adhesion is replaced by the wedging action of the ribs (Malvar 1992). Longitudinal and radial cracks can be generated under the influence of the wedging action of the ribs and if adequate confinement is not provided, bond failure occurs soon after the cracks propagate to outer layers of the concrete. If proper confinement is provided, bond stress reaches a maximum of 3 c f ′ and meanwhile the frictional type of forces are involved gradually to provide the required bond. 2.7.1 Static bond slip analysis Menzel (1939) conducted a series of tests to investigate the effects of surface conditions, cement ratio, embedment length and position of the bar relative to placement direction of the concrete. Menzel identified the marked effects of surface conditions on bond resistance and concluded that increased cement ratio or increased embedment resulted in increased bond resistance. Experiments conducted by Ferguson and Thompson (1962) revealed that bond between reinforcement and concrete was a function of development length, and not bar size. They emphasised the adequacy of bar cover to acquire the required development length and found that ultimate bond stress is proportional to squire root Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-37 of compressive strength of concrete. Later in 1965, they observed that increased embedment length resulted in decreased the bond stresses. Also, crack growth in concrete tended to be more severe for larger bars than for smaller bars. Finally, increased cover caused increase in bond resistance, however was not helpful in reducing crack width. McDermott (1969) investigated the influence of steel strength and reinforcement ratio on the mode of failure and strain energy capacity of reinforced concrete beams. It was observed that within the ductile range, the yield strength of the steel bars had no effect on the strain energy of beams of equal static bending strength that were subjected to moderate strain rates. Therefore, it was concluded that ductility of reinforced concrete beams is independent from yield strength of the reinforcement. Malvar (1992) conducted a series of tests to investigate the effects of confinements on the bond of reinforcement. In this experiment, 12 specimens were subjected to confining stresses in the range of 3.45-31MPa. Instead of selecting a conventional two dimensional surface to simulate the bond between steel and the concrete, a 73.5mm process zone was defined surrounding the reinforcement. The confinement effects and the bond slip condition were described with respect to this arbitrarily selected process zone. When the confinement stress increase from 3.45 to 31.0MPa, almost 200 percent increment of bond strength was observed. However, for higher confining stresses, the effects of confinement on bond behaviour appeared less pronounced. The same conclusion was made by Robins and Standish (1982), who conducted pulled out tests on 8 and 12mm bars in a 100mm concrete cube. The cubes were laterally loaded on two opposite sides and 100 percent increment was observed under the confining pressure of about 10 N/mm 2 . As in the earlier observation, additional application of lateral pressure of up to 28 N/mm 2 did not increase the failure load considerably. By conducting extensive research on effects of confinement in concrete, Hungspreug (1981) concluded that increasing cover and transverse reinforcement increase the confinement effects on the bar, and hence can be treated as factors which increase the bond strength. In addition, it was observed that an increase of bond strength is a direct result of increase in concrete with concrete tensile strength or in other words, the bond Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-38 strength rises proportionately with the increase of the square root of the compressive strength. Similar to the observations of Malvar (1992), the bond stress linearly increased up to a confinement stress of 2.8 MPa, however severe radial cracking prevented the linear increment for higher confinement stresses. 2.7.2 Dynamic bond slip analysis Yan (1992) studied the bond slip under impact loading conditions. Tests were conducted in the dynamic as well as in the static range. A drop weight impact hammer was used to apply the impact load and the effects of different surface roughness, compressive strength, amount of fibre content and two different fibres (polypropylene and steel) were investigated. It was found that for smooth rebars, the bond slip relationship was linear for both, static and dynamic loading conditions. For deformed bars, the effects of the surface roughness were found to play an important role in bond slip resistance. The bond-slip relationship under dynamic loading changes with time and location along the reinforcing bar and is greatly influenced by the above mentioned parameters. Specifically, the surface roughness affects the stress distribution in the concrete, the slip at the interface between rebar and concrete and the crack development. Series of experimental tests were carried by Weathersby (2003) to investigate the (a) chemical adhesion between smooth steel bars and concrete (b) bond resistance of smooth bars and deformed bars and (c) influence of concrete confinements and bar diameter. Three modes of failure were identified from the experiment. The failure of smooth bars mainly occurred due to pullout and the failure mode was independent of the loading rate. The resistance to pullout mainly provided by static and dynamic frictions with chemical adhesion. Strength of the chemical adhesion and the static friction increased with increasing loading rates. For example, the combined static friction and the chemical adhesion were 6.62 MPa for the quasi-static loading and 22.1 MPa for the impact loading. In deformed bars, failure occurred due to radial cracking regardless of the loading rates. Compared to the smooth bars (10φ and 8φ), 70%-77% increment was observed due to Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-39 the deformations and this increment was independent of the loading rate. The ultimate load at failure increased as the loading rate increased. It was observed that the ultimate load at failure was 70% to 100% higher than that under quasi-static loading conditions. Confinement of concrete increases the bond stress between the reinforcement and concrete over all loading rates (Malvar 1992). As a result, under impact loading conditions the failure mode between concrete and steel shifted from concrete cracking to steel yielding. However, with increasing loading rate the increment of bond resistance dropped significantly. It was observed that, as long as the failure mode remained constant, the bond stress of impacted specimens was nearly twice the value of quasi-statically loaded specimens. 2.8 Factors affecting ductility of concrete columns 2.8.1 The effects of confinement on enhancement of the ductility and strength In general, both ductility and strength of columns increase with the confinement provided by transverse steel. In the absence of sufficient transverse steel, the behaviour of the column is governed by the strength of unconfined concrete, which is caused by inactivation of transverse reinforcement in providing required confinement. Nevertheless, through experimental results, it has been proved that columns with a larger volumetric ratio of confinement steel have larger deformability and flexural strength capacity (Saatciglu and Razvi 1992; Mandar et al. 1984). Therefore, under the impact loads, the effects of confinement play an important role in terms of ductility and energy absorption. One function of confinement steel is to provide passive confining pressure to the concrete core by lateral expansion. Passive confining pressure depends on many factors such as diameter, dimension of the core, and spacing of the confinement reinforcement. The diameter of confinement steel affects the total lateral force acting on the concrete core and the spacing affects the distribution of pressure in the concrete core. The confinement pressure acting on the concrete core is highest at the contra-flexure point and gradually reduces in a parabolic manner (Mandar et al. 1984). Therefore, close distribution of the transverse reinforcement will increase uniformity of the lateral pressure. Previous experimental tests showed that adverse effects of the Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-40 large transverse spacing can be overcome by using smaller diameter steel bars placed at closer distance. The distance should be no longer than the distance at which the premature buckling starts in the longitudinal reinforcements (Saatcioglu and Razvi 1998). However, the relationship between spacing and the diameter is not linear (Johnny 2003). Because of the parabolic distribution of the confinement pressure in horizontal and vertical directions, the effectiveness of the confinement greatly reduces as the transverse steel spacing increases in the vertical direction. Therefore, columns with larger link spacing fail due to the lack of confinement effects, irrespective of the transverse steel content. Hence, the British Code, BS 8110 (1985) limits the maximum spacing of the transverse steel to 12 times the smallest longitudinal compression bar diameter. Confinement effect depends on the yield strength of transverse reinforcement. It is expected that steel of higher strength provides greater confining pressure in the concrete core. However, lateral expansion of the concrete core will not be greater and the tensile capacity of the steel may not be fully developed under working conditions (Johnny 2003). It is shown that for ordinary high yield deformed bars of yield strength, f y = 460MPa, strain hardening of the transverse steel is unlikely to occur under serviceability conditions. ACI 318(2002) and NZS 3101 (1995) also specify an upper limit for the yield strength of transverse reinforcements. ACI limits the allowable maximum yield stress to 420MPa and NZS to 800MPa. However, highly localised stresses generated under impact loading conditions may yield the transverse reinforcement (Memari et al. 2005). Hence, the above restrictions underestimate the possible maximum strain that can be developed in transverse reinforcement under impact loading conditions. In other words, hoops having higher yield strength should be used for columns susceptible to impact loads. Saatcioglu and Razvi (1998) introduced an important relationship between volumetric ratio V r and yield strength ' sy f of confined reinforcement so that a compromise between each parameter is possible. ie. ' sy s r f V ρ = Eq. 2.27 This equation implies that a reduction in yield strength can be compensated by increasing the amount of transverse reinforcement. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-41 The confinement effect of concrete also depends on the configuration of transverse steel and distribution of the longitudinal steel. Confinement provided by longitudinal reinforcement is particularly effective if the reinforcement consist of larger diameter bars, which are less prone to inelastic buckling. Nevertheless, it was found that irrespective of the distribution of longitudinal reinforcement, columns with closer links have better confinement effects and consequently an increased flexural strength and ductility (Cusson and Paultre 1994). Therefore, in providing the required confining pressure, the distribution of both transverse and longitudinal reinforcement is equally important. Figure 2.9: Distribution of confining pressure produced by various shapes of transverse steel (Razvi and Saatcioglu 1999) The confining pressure provided by various kinds of links is shown in Figure 2.9. When the lateral force produced by the links is well distributed across the perimeter, as in the case of circular columns, the efficiency of the confinement effects is improved. However, for square columns the confinement pressure reduces from the corner to the mid point of each side of the links and efficiency of the confinement decreases. As far Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-42 as the rectangular columns are concerned, the pressure distribution is not uniform in each direction and hence the efficiency greatly decreases. The following equation quantifies the relationship between confinement effects due to transverse steel spacing and distance between laterally tied longitudinal steel (Razvi and Saatcioglu 1999); 1 15 . 0 2 ≤ | | ¹ | \ | | ¹ | \ | = l c c s b s b k , Eq. 2.28 where b c is the breadth of the core concrete for square column, s is the spacing of the transverse steel along the height of the column and s l is the spacing between laterally confined longitudinal bars. According to Razvi and Saacioglu (1999), parameter k 2 is proportional to the effectiveness of the confinement provided by both, transverse and longitudinal steel. It was observed that the effectiveness of the confinement can be improved by reducing spacing ‘s’ between the transverse steel and the distance between the longitudinal steel ‘s l ’. However, the addition of cross ties at a fixed volumetric ratio may or may not improve the confinement effects. Because, while improving the confinement effects in the lateral direction, the cross ties will on the other hand increase the transverse steel spacing. 2.8.2 Effects of concrete cover The amount of transverse steel required for confinement increases when the concrete cover to thickness ratio c/D o for a circular section or c/h for a rectilinear section, increases (where c is the concrete cover thickness, h is the smaller dimension of the rectangular section and D o is the diameter of circular section). This is because when the concrete cover is high, the column loses significant flexural strength as the cover concrete spalls off quickly under comparatively low strain conditions. The resultant loss of the flexural strength can be recovered by providing adequate transverse reinforcements to confine the core concrete (Johnny 2003). For convenience, New Zealand Standard NZS 3101 (1995) expresses the concrete cove thickness ratio c/D o or c/h by A g /A c , where A g and A c are gross concrete area of the section and area of the concrete core respectively. According to NZS 3101 (1995) specifications the ratio A g /A c lower than 1.2 should be avoided. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-43 2.8.3 Compressive axial load level Flexural ductility reduces significantly with the increase in compressive axial load on the column (Paultre et al. 2001). As the axial load level increases, the concrete becomes subjected to higher compressive stress levels and the moment capacity of the column is then dependent mainly on the compressive strength of the concrete. Since concrete is brittle in nature, flexural strength reduces rapidly after reaching the maximum moment capacity of the column. Moreover, depth to the neutral axis increases as the axial load level increases and the extreme concrete fibre is subjected to higher compressive strains. Under these conditions the concrete will reach its ultimate strain sooner and as a result, the concrete cover will spall off rather quickly, causing a decline in the flexural capacity of the section. If sufficient transverse reinforcement is provided, the reduction in flexural capacity can be compensated by increasing the capacity of the concrete core such that the concrete core could dilate properly under large compressive axial loads (Johnny 2003). On the contrary, in the presence of adequate amount of transverse steel in RC columns, the concrete core dilates and may induce transverse radial confining pressure in the concrete core. Under this circumstance, the ultimate strain of the concrete core is considerably larger than that of the unconfined concrete cover. The actual stress-strain behaviour of the concrete core, therefore, could vary from that of the outer cover of the column (Anselm 2005). This may cause considerable deviation of the results if a common stress-strain diagram is assumed for a non-linear impact analysis of the column. 2.8.4 Combined effects of axial load and flexure Extensive experimental research has been conducted to investigate the behaviour of columns under the combined action of axial load and flexure. The investigation mostly covered flexural strength and flexural ductility performance under the combined action of axial load and flexure. However, the flexural behaviour of reinforced concrete columns also depends upon many other factors such as column slenderness (Kim and Yong 1995), load eccentricity (LIoyd and Rangan 1996), boundary conditions at the ends (Majewski 2007), area and the shape of the cross section of Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-44 concrete, area and spacing of the vertical and horizontal reinforcement (Razvi and Saatcioglu 1999) and the reinforcement ratio. On the other hand, improved flexural behaviour of concrete columns will help minimise the adverse effects of lateral impact loads, by converting the failure mode from brittle to ductile. Axial load level could be a critical factor that governs the above phenomenon. Research, which covers the flexural ductile behaviour of concrete columns and the factors affecting the ductile behaviour is summarised in the following paragraphs. By conducting an analytical study of flexural strength and ductility of RC columns, having various arrangements and quantities of transverse reinforcements, Watson et al. (1994) concluded that more transverse steel is required for confining the core of square and rectangular columns than that of circular columns. This is due to variation of the lateral pressure distribution inside the concrete core. On the other hand, the quantity of transverse reinforcement required for confinement to meet any particular curvature ductility factor demand, increased with increasing axial load level, increasing concrete strength, decreasing longitudinal reinforcement ratio and increasing concrete cover thickness. Additionally, large flexural strength enhancement was detected for columns subjected to medium or high axial loads. Sheikh and Khoury (1997) proposed a performance based approach for designing confining steel in tied columns. By comparing the provisions of the ACI code (1995) with their findings, it was concluded that the behaviour of column design according to the ACI code may vary from unacceptably brittle to very ductile. To overcome the problem, a modification was introduced to the ACI (1995) equation and when compared with the experimental results, the proposed equation showed excellent agreement. The final equation is as follows; c h a h A P P A , 5 29 13 1 φ µ α ( ( ¸ ( ¸ | | ¹ | \ | + = , Eq. 2.29 where A h,c is the ACI code provision of transverse reinforcement content, α is a factor dependent on the tie configuration, µ φ is the curvature ductility factor, P is the axial load applied and P a is the axial capacity of the section. It is interesting to note that this equation accounts for the axial load effects on the column. The curvature ductility factor is included in the equation, because the amount of transverse reinforcement Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-45 according to the current provision in ACI code, is too low for high axial load levels and too high for a low axial load level ( ) 4 . 0 ' < c g f A P . 2.9 Shear capacity calculations and effects of axial load Columns subjected to impact loading may fail in either shear or flexure. Shear capacity of structural columns under dynamic impact loads are therefore important to determine the threshold of a shear failure. Axial load in particular, is one of the deterministic factors that can affect the shear capacity of impacted columns. On the other hand energy dissipation characteristics (Li et al. 1991), sequence of the load application (Saadeghveziri 1997) and confinement effects (AIJ 1994) are also play an important role in the shear capacity determination. Following are the information that is found from the literature which can be used to determine the behaviour of shear in columns under impact loading conditions. According to the Louw et al. (1992), when a column is subjected to a hard impact, the maximum dynamic shear force generated just after the impact, and the ratio of maximum dynamic shear to ultimate static shear strength of the column, vary from 0.87 to 1.67. The variation of the axial load in the column due to the impact load is quite similar to that of the dynamic shear force. As a result, the shear capacity of the column can change and hence the dynamic shear capacity can differ from the static shear capacity of the column. In contrast with a hard impact, the peak load occurs much later under soft impact conditions. In addition, it is more likely to affect the flexural shear resistance of the column rather than the pure shear and inertial stiffness. However, the actual contribution of the axial load in resisting the dynamic shear force is uncertain. Kreger and Linbeck (1986) reported a test result of three double curvature specimens under various lateral and axial load variations. Two specimens were subjected to axial loads proportional to the lateral loads. The other specimen was tested using uncoupled axial and lateral loads. The test results reveal that strength of specimens increased with increasing axial load. In addition, the energy dissipation characteristics of the columns depended largely on the axial load history. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-46 Based on an analytical investigation of columns with proportionally and non-proportionally varying axial loads, Saadeghveziri (1997) suggested that the energy dissipation capacity of the columns could be reduced significantly under uncoupled variation in axial and lateral loads. Li et al. (1991) came to similar conclusions by conducting extensive experiments on seventeen cantilever columns, under constant axial loads and proportionally and non-proportionally varying axial loads. Moreover, in non-ductile columns, the proportionally varying axial load pattern resulted in significant shear strength degradation. The result also showed that the variation in magnitude of the axial load had significant effects on the stiffness, strength and deformation capacity of the column. Different formulations and parameters have been proposed over the past few decades to calculate the shear strength of the concrete columns. The shear strength is calculated as a summation of the strength contributions from the concrete and transverse steel. However, representation of the effects of various parameters such as axial load, displacement ductility and aspect ratio are vary depending on the formulation or some times are not included. The equation given in ACI 318-02 considered the effects of axial load on the shear strength enhancement. The shear strength V n , is calculated as the summation of contribution of the concrete V c and the transverse steel V s . The contribution of the concrete to the shear strength of the members subjected to shear and axial compression is given as, bd f A P V c g c ' 2000 1 2 | | ¹ | \ | + = , Eq. 2.30 where P is the axial load, A g is the gross cross sectional area, f ’ c is the compressive strength of concrete and b and d are the web width and effective depth of the section, respectively. Here all the units are in lb, in and psi. The contribution of the transverse reinforcement was calculated as, s d f A V y sw s = , Eq. 2.31 where A sw is the transverse reinforcement area within a spacing, s and f y is the yield strength of the transverse reinforcement. Special provisions of the ACI 318-02 further Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-47 state that at column ends or in the possible plastic-hinge regions, the concrete contribution, V c should be taken as equal to zero, if the factored axial compressive force including earthquake effects is less than 20 ' g c A f and if the earth quake induced shear force is large. The ASCE-ACI committee report (1977), provides a broad insight into the shear transfer mechanism. The report describes the important mechanisms as: (a) shear transferred by the uncracked concrete; (b) interface shear transfer in the cracked concrete, ie. aggregate interlock; (c) dowel shear carried by the longitudinal reinforcement; (d) arch action in deep members and (e) shear transferred by the transverse reinforcements. The most critical mechanisms identified were the shear transfer by the transverse reinforcement and concrete. For members with short span-to-depth ratio, the committee recommends the use of reduced shear stress v c . Aschheim and Moehle (1992), used laboratory test data from cantilever column tests to identify the shear strength characteristics. The data indicated that the column shear strength is a function of displacement ductility demand, the quantity of transverse reinforcement (confinement), and axial load. The proposed approach is similar to the ACI 318-02 code equation, except when the equation is used to calculate the concrete contribution V c . The concrete contribution is defined as, bd f bd f A P V c c g c ' ' 5 . 3 ) 2000 1 ( ' ≤ + = α . Eq. 2.32 For design and evaluation of rectangular hoop reinforced concrete columns, δ µ ρ α yw w f 006 . 0 ' = , Eq. 2.33 where w ρ is the transverse reinforcement ratio, bs A sw w = ρ . 2.10 Energy absorption characteristics under impact loads Structural columns subjected to dynamic loading conditions must have higher energy absorption capacity, and hence, should be allowed certain plastic deformation to avoid premature shear failures. In addition, rate of energy absorption is among the factors that determine the mode of failure. In brief, the energy absorption capacity of a column, Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-48 under impact conditions may depends on the factors such as material characteristics, relative masses, stiffness, area of contact, and frequency of loading. Bischoff and Perry (1995) investigated the energy absorption capacity of concrete by conducting a series of tests on air-dried plain concrete cylinders. The cylinders were subjected to slow static loading rates of 10 micro strains per second, as well as much higher rates of about 5 to 10 strains per second. Two grades of concrete with design strength of 30 and 50 MPa were tested. This investigation showed that absorbed energy is greater for specimens subjected to strain rates higher than 250 micro strains, compared to specimens subjected to quasi-static rates. Therefore, the energy absorption capacity of plain concrete seems to depend on the strain capacity of concrete, which in turn is governed by the failure mode. In addition, according to the experimental results, the weaker mix absorbed more energy than its counter part. It was, therefore, concluded that both strength and strain rate capacity play an important role in determining energy absorption capacity. According to Mindess et al. (1986), energy absorption capacity can be improved by adding extra fibre volume to concrete. Also, maximum flexural load and fracture energy increased significantly with the increase in fibre volume. The enhancement of the moment rotation characteristics of the flexural compression zone and the increment of local deformation characteristics due to the fibre reinforcement could be the possible reasons for those observations. In general, fibre reinforcement will increase impact resistance or dynamic toughness of plain concrete, owing to the significantly increased maximum beam deflection. This effect was predominant only after the fibre content was increased up to 0.75% (Hughes and Al-Dafiry 1995). The most significant effect of the additional fibre content is the delay in failure which means, for higher fibre content, the duration of the impact of 5ms compared to the duration of the impact event with lower fibre content of only 1ms (Wang 1996). The average deflection also increases under these circumstances and results in a lower rate of energy absorption, which reduces the possibility of a shear failure of an element. However, improvements in the peak load and fracture energy under impact loading may be considerably lower compared to the improvements Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-49 obtained under static loading conditions (Banthia et al. 1989). Also, the addition of nominal shear reinforcement with fibre did not show any significant impact on energy absorption (Wang 1996). Fracturing effects can also contribute to the energy absorption characteristics under the impact loads. Concrete can absorb more energy under impact loading conditions (Remennikov and Kaewunruen 2006). The energy consumption for plasticity and other non-liner deformations under the unstable conditions can be one possible reason. In addition, boundary conditions may also affect the energy absorption process (Xu, 2001). Due to the influence of the boundary conditions, the resultant energy consumption closer to the boundary will vary from that in the region far away from the boundary. On the other hand, tensile strength of concrete gained the highest consideration among the parameters that influence energy dissipation under the impact loading conditions (Weerheijm and Doormaal, 2007). Conditions at the supports and contact interface may also have considerable effect on energy absorption under laboratory conditions. Since the displacement of the supporting devices in experimental set-ups was found to be very small, energy absorption at the supports can be neglected compared to that of the contact interface, especially under hard impact conditions, where significant deformations take place (Hughes and Al-Dafiry 1995). 2.11 Design practices and provisions of RF in critical sections In general, existing reinforced concrete columns contain additional amount of longitudinal reinforcement, which provides excessive strength against simultaneous compressive axial load and flexure. However, provisions of shear reinforcement in structural columns were just sufficient to withstand the ultimate shear force in the columns (Ho and Pam 2003). Additionally, codes such as BS 8110 contain few provisions for the detailing of transverse reinforcement. As a result, using current design methods, post elastic behaviour and failure mechanisms cannot be anticipated. Consequently, the resulting failure modes under the impact could be brittle and sudden. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-50 Most of the current design provisions include the inelastic behaviour of the structure under seismic attacks and overloading. Existing design philosophy is based mainly on the energy dissipation through extensive inelastic deformation within the potential plastic region, especially under earthquake conditions. However, the nature of the damage caused by impact loads is quite different from the damage caused by earthquake loadings. In either case, proper detailing of longitudinal and transverse reinforcement within the potential plastic hinge region is needed to avert the following undesirable modes of failures in structural columns (Ho and Pam 2003): (a) shear failure (b) bond and anchorage failure at the member joints (c) buckling of longitudinal steel and (d) premature failure of concrete core due to inadequate transverse steel. All of the above are classified as brittle and sudden failures, which must be avoided. In the absence of more specific information on plastic hinge formation in columns under lateral impact loads, especially for low velocity hard impact conditions, the experimental results in the following studies can be used to identify the factors affecting the formation of plastic hinges. Mendis (2001) conducted a series of tests on simply supported beams with a point load at mid span, which he subjected to various levels of axial load to determine the plastic hinge length. It was concluded that plastic hinge length remained constant at 0.40d or 0.33h, where d and h are the effective and overall depth of the beam respectively. According to the observations, the plastic hinge length is independent from the compressive axial load level on columns. Based on the tests of four full size confined RC columns with a 550mm square cross section, Park et al. (1982) also showed that the equivalent plastic hinge length was insensitive to the compressive axial load level and had an average value of 0.5d, where d is the effective depth of the column. The tested columns contained transverse steel ratio of up to 4%. However, the effect of the transverse steel ratio on the length of plastic hinge was not demonstrated. Contrary to the above observations, Clause 8.5.4.1 of NZS 3101 (1995), specifies that the potential plastic hinge length is influenced considerably by the applied compressive axial load level. For an RC member subjected to an axial load level (Refer Eq. 2.34) of 0.25 or smaller, the potential plastic hinge length is equal to 1.0h or Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-51 a region where the moment exceed 0.8 of the maximum moment, whichever is larger: ' ' c g f A P φ Eq. 2.34 where P is the compressive axial load, φ’ is the strength reduction factor, A g is the gross cross sectional area and ' c f is the concrete compressive cylinder strength. The plastic hinge length is increased to 2.0h or over a region, where the moment exceed 0.7 of the maximum moment, whichever is larger, for an axial load level larger than 0.25 but smaller than or equal to 0.5. It is further increased to 3.0h or over a region, where the moment exceed 0.6 of the maximum moment, whichever is larger, for axial load level larger than 0.5 but smaller than or equal to 0.7. Based on the test results of eight large scale high strength reinforced concrete (HSRC) columns with a square cross section of 305x305mm, Paultre et al. (2001) showed that the potential plastic hinge zone could range from 1.0h to 2.0h, depending on the volumetric ratio of transverse steel and the compressive axial load level. The plastic hinge length was found to increase with an increase in compressive axial load level and concrete compressive strength, but with an increase in the transverse steel volumetric ratio, it was reduced. Ho and Pam (2003) also recommend that the potential plastic hinge region of HSRC columns can be taken as h, which is the overall depth of the column cross section. However, Ho and Pam (2003) observed that the plastic hinge length of columns was insensitive to the spacing of the transverse steel, provided it was less than 0.75d. Bayrak and Sheikh (2001) presented an analytical procedure to predict the behaviour of plastic hinges in RC columns. The analysis incorporated complex behaviours, such as softening of longitudinal bars due to inelastic buckling and reinforcement cage–concrete core interaction. According to their observations, the concrete core-reinforcing cage interaction, which caused outward deflection in longitudinal bars, did not only reduce the ductility of longitudinal bars under compression, but it also reduced the maximum stress the bars were able to achieve. They also proposed that for high curvature ductility demand, ratio of tie spacing to longitudinal bar diameter should be kept under 6, which is the same as that proposed by Mander et al. (1988). However, for moderate curvature ductility demand, the ratio should be kept below 8. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-52 While there is no general agreement about the effects of the volumetric ratio, axial load and spacing of transverse steel, it is quite clear that the height, where the plastic hinge is formed is increased with the compressive strength of concrete. That is, under impact loads, where strain rate effects are predominant, and hence the compressive strength is increased, significant changes of the height, where the plastic hinge is formed, can be expected. In addition, based on the fact that inelastic buckling of the longitudinal reinforcement and yielding of confinement steel is excessive within the plastic hinge region, it is proposed that the end hooks of transverse reinforcement within that region are 45 o , while those outside the plastic hinge region be 90 o (Ho and Pam 2003). Moreover, the average length of the plastic hinge region affected by the ‘Stub Effect’ was observed to be around 50mm (Paultre et al. 2001; Ho and Pam 2003). 2.11.1 Influence of the various parameters on confinement 2.11.1.1 Effect of the cover concrete and volumetric ratio of steel on confinement It was found that when concrete strength ' c f of the column increased, the column strength achieved a proportional increase in capacity. The ultimate strength was gained at larger axial deformations than the yield point. Concrete cover spall off just before the peak strength was achieved without achieving its full compressive strength and in general, it was approximately at 0.75 ' c f (Cusson et al. 1994). Some of the testing procedures have taken in to account the effect of the longitudinal steel while many of them did not consider that effect. The limited experiments with longitudinal steel revealed that when concrete strength and the yield strength of longitudinal steel were constant, the strength of the columns increased with an increase of spiral reinforcement. The effects of spacing were more pronounced for steel ratios from 1.1% to 1.7% and s/D c appeared to be around 1.2% and 0.24% respectively for stable unloading behaviour (Toklucu et al. 1992). In addition, columns that contained volumetric ratio of steel spiral around 3.2% were capacitated to replace the loss of load carrying capacity due to cover spalling. Moreover, typical columns with more than 3.2% volumetric ratio of spiral developed a second maximum load after the first peak but never exceeded the first maximum load at spalling (Richart 1946). Thus, it can be Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-53 concluded that confinement steel increases both the strength and deformation capacity of concrete in compression while increasing the stresses in spirals at peak confined concrete strength (Tomosawa et al. 1990). Even though lateral reinforcement has very significant effects on the confined core, increasing lateral steel results in less proportional increase in strength and ductility (Sheikh and Uzumeri, 1982). It was also revealed that the volumetric ratio of lateral steel ranging from 0.55% to 1.64% is not sufficient to substitute for the loss of load-carrying capacity resulting from the failure of the protective cover. On the other hand experimental tests on 150x150mm specimens made out of Grade 59 to 68MPa concrete revealed that the stress-strain curves were not significantly influenced by the area of the longitudinal steel or the spacing of the lateral ties including the one with 50mm spacing (Hwee et al. 1990). However, strength due to the confinement effects was enhanced with further reduction of the lateral spacing from 50 to 15mm in 150x300mm specimens made out of Grade 50 to 120 concrete (Tomosawa et al. 1990). Therefore the above strength category can be treated under HSC category where the behaviour is significantly different from the LSC. Thus, the limit of 76mm for the spiral spacing in large diameter column was considered to be too restrictive (Toklucu et al. 1992). 2.11.1.2 Changes in the stress-strain curve due to confinement According to the Yong et al. (1988), the stress-strain curves exhibited a relatively linear ascending branch below the maximum axial load for the tested specimens of plane concrete, and confined concrete with or without cover. Also the relative enhancement of the confined strength and the corresponding strain were decreased as the grade of concrete is increased. Furthermore, the slope of the descending branch of the stress-strain curve was deteriorated more rapidly with higher concrete cylinder strengths. Therefore the effectiveness of the confinement will decrease as the concrete grade increase (Ahmad and shah, 1982). Apart from that, decreasing the spiral spacing by keeping the volumetric ratio as a constant will not improve the slope of the falling branch of the stress-strain curve. It is also concluded that the number of longitudinal bars had little influence on stress-strain behaviour (Mander et al. 1988). In general, as the volumetric ratio of confining reinforcement increased, the peak stress increased, Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-54 the slope of the falling branch decreased, and the longitudinal strain at which spiral fracture occurred increased (Mander et al. 1988). But not proportionally (Yong et al. 1988). 2.11.1.3 Effects of the axial load and grade of concrete on confinement In particular, lower grade of concrete exhibited large plastic deformation capacity without losing its load carrying capacity significantly while the slope of the descending branch is reduced with increased confinement. In contrast, the steepness of the ascending branch of the stress-strain curve was increased with higher grades of concrete. It is also observed that the confining effects of spirals were small for loads up to about 30% of the unconfined column strength. Above this load, rapid increment of the spiral stress and confinement pressure can be seen. Once the load level had reached unconfined compressive strength there was rapid disintegration of the concrete microstructure and hence greatly increases the confinement stress. Consequently, spirals yield before the maximum capacity of the column was reached. With the enhancement of the concrete strength the disintegration of the concrete at the maximum compressive strength was diminished and hence the rapid increment of spiral stresses was diminished. As a result, none of the spirals in HSC (Grade 55 to 83 MPa) column reached their yield strength at the maximum load (Martinez et al. 1984). Moreover, the failure of the protective cover of HSC columns was sudden and brittle compared with LSC columns. However, for HSC the second maximum load found to be greater than first contrary to what would be expected (Martinez et al. 1984). 2.11.1.4 Effects of yield strength of hoops on confinement As far as the yield strength of the spiral is concerned, it did not influence the compressive strength enhancement of concrete. Similar steel stresses were observed at peak load regardless the yield strength of the spirals for the specimens made out of same grade of concrete. Thus the ductility will increase with the enhancement of the yield strength. In fact, confining pressure exerted by spiral depends on the potential lateral expansion of the corresponding plane concrete. Thus, reluctant to the lateral expansion of the higher grade of concrete will excrete less lateral strain on spirals for a given axial strain. However despite the grade of the concrete the stress on spirals Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-55 rapidly increases once the peak stress is achieved (Ahmad and shah, 1982). It was also found that lateral steel reaches its yield strength at the maximum load even when very high strength steel (f y = 915MPa) was used (Polat 1992). 2.11.2 Theoretical stress strain curves for confined concrete by transverse RF 2.11.2.1 Stress-strain model for confined concrete Numerous empirical models have been proposed by various researchers to predict the non-linear behaviour of confined concrete columns under concentric loading. However numerical modelling of the nonlinear response of the confined concrete has not been addressed substantially in the literature. In the present study a material model compatible with nonlinear finite element analysis of three dimensional concrete models was used to investigate the confinement effects. The stress-strain model developed by Mander et al. (1988) simulates the confinement effects in this process. It was assumed that the passive lateral confining pressure exerted by the transverse reinforcement leads to a tri-axial state of stress in the core concrete and thus enhances the compressive strength compared to the unconfined concrete. Eventually the equal and opposite forces acting on the lateral reinforcement may rupture the hoops at one stage by bringing the useful ultimate longitudinal compression strain to a residual level (see Fig. 2.10). Longitudinal compression strain of confined concrete within the range of 0.02 to 0.08 should be maintained to satisfy this requirement (Watson et al. 1994). Figure 2.10: Stress-strain model for confined concrete proposed by Mander et al. (1984) The proposed model is applicable for both circular and rectangular shaped transverse reinforcement. The stress strain model illustrated in Figure 2.10 is based on an Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-56 equation suggested by Popovics (1973). The ' cc f is defined as compressive strength of confined concrete and ' cc f will depend on links spacing, area of effectively confined concrete core, diameter of the hoops and yield strength. The lateral confining pressure can be found by considering the stability of the half body confined by the transverse hoops as shown in Figure 2.11. In fact, the model assumes an arching action to occur in the form of a second degree parabola with an initial tangent slope of 45 o . Hence, the effectively confined area is calculated based on the confined concrete core midway between the levels of transverse reinforcement. In the model, the resultant confinement stress due to the various components such as yield strength, diameter and spacing of the hoops are expressed in terms of equivalent uniform confinement over the core concrete. Additionally, two sets of equations were developed for rectangular and circular columns separately by taking into account the variation of lateral confining stress across the sections. However the cross ties can excrete either equal or unequal confining stress along each of the transverse axis depending on the topology of the section. As far as the circular columns are concerned the stress distribution is uniform across the section and hence a single equation can be used for the stress-strain equation despite the axis of bending. c yh b l sb f A f 4 = Eq. 2.35 c yh b l sb f A f 2 = Eq. 2.36 Figure 2.11: Confining stress provided by the transverse reinforcements The effective lateral confining stress in each direction exerted on the core at yield strength is given by: l e l f k f = ' Eq. 2.37 Where f l is the confining stress calculated as in the Eq. 2.36 and k e is the confinement Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-57 effectiveness coefficient that takes into account the arching action. s is the links spacing and f yh is yield strength of transverse reinforcement. The coefficient k e for circular hoops is defined as the ratio of the area of effectively confined core concrete to the area of concrete within the centrelines of the peripheral hoop or spiral and mathematically expressed as, cc s e d s k ρ − | | ¹ | \ | − = 1 2 1 2 Eq. 2.38 where cc ρ ratio of area of longitudinal reinforcement to area of core of section and d s is the distance of spiral between bar centres. s is the clear vertical spacing between spiral or hoops. In addition, for a section with equal effective lateral confining stress in each direction, the ratio of the compressive strength of the confined concrete ' cc f to the comprehensive strength of unconfined concrete ' co f is given by, 254 . 1 ' ' 0 . 2 ' ' 94 . 7 1 254 . 2 ' ' − − + = co l co l co cc f f f f f f Eq. 2.39 Over the last few decades many experiment have been carried out on confined concrete columns under both concentric and eccentric loading conditions. In addition, numerous models exist to predict stress strain behaviour of normal and high strength confined concrete. However various limitations still persist when it comes to the column geometry and axial loading conditions. 2.12 Effects of impact induced torsion in eccentrically loaded columns The torsion is more likely in skewed bridges and bridges with outrigger bents though the structural columns with irregular configurations may also susceptible to in-built torsional moments. When the impact occurs in a direction perpendicular to the plane of bending, impact induced torsional moments are generated. Most of the investigations were focused only on the effects of bending moments and shear forces by neglecting the effects of torsion. In the absence of any detail on the effects of impact induced torsion, some insight may be provided by the tests conducted under seismic loading conditions. For instance, Tirasit and Kawashima (2008) reported that the torsion Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-58 induced damage tends to occur above the flexural plastic hinge region and consequently change the formation of plastic hinge zones. According to Otsuka et al. (2005) pitch of the lateral ties can be used to control the torsion induced damage. Thus, spiral reinforcement which might adequate for flexural design may not be adequate in the presence of torsional moments (Prakash et al, 2009). Further, they observed that the deformation characteristics and failure modes may also affected by the presence of torsion. The reduction of confinement effects of core concrete followed by the torsion induced spalling further support these observations. Consequently, columns failed without achieving the ultimate shear capacity (Belarbi et al. 2008). In general, three modes of failures can be identified under the combined effects of shear flexure and torsion depending on the initiation of failure. In under reinforced members the reinforcement will fail before concrete crushes while in partially over reinforced columns the failure will occur either by yielding transverse or longitudinal reinforcement. If concrete crushed before any steel yield they can be categorised as completely over reinforced members (Prakash et al. 2009). In addition, open hoops may unlock and become non-functional under torsional loads by causing significant spalling of cover concrete while reducing the confinement effects particularly under cyclic loading (Belarbi et al. 2008). Therefore close hoops or spirals may be more suitable for columns subjected to torsional moments. In addition, the torsional strength mainly depends on the amount of longitudinal and transverse reinforcement, the sectional dimensions and the concrete strength. For instance, in circular columns the torsional strength was reached first and then the flexural strength while in square columns the two strengths were achieved simultaneously. 2.13 Impact reconstruction Impact of a rigid object with a column is obviously a hypothetical case, which represents more extreme situations of a vehicle impact. For example, the rigid body induced deflection is highly concentrated to the point of impact and the bending moment varies even after the formation of the plastic hinges (Tsang et al. 2005). Therefore, associated shear force can be much higher than the actual impact and hence intends to model more extreme condition, usually the upper bound of possible impact Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-59 damage. On the other hand, simulation of an actual impact using a finite element model of a vehicle would be a tedious task and is beyond the scope of this project. Alternatively, impact induced vehicle deformation can be used to quantify the energy absorbed during a collision. Many impact phase reconstructions assume a linear relationship between an absorbed energy and the residual deformation (Campbell 1974; Varat et al. 1994; Neptune 1999). Based on this assumption, the stiffness of the vehicle frontal impact can be represented by constant spring stiffness. This Campbell Model (1974) was further improved by Parsad (1990) with the aid of repeated barrier impact testings. One assumption continuing through that reformulation is that a constant liner spring rate over the entire depth is applicable. However, available crash data indicates that vehicle frontal stiffness cannot be precisely modelled through the use of single linear springs for all vehicles (Varat et al. 1994). Hence it is prudent to be aware of the potential problems with this method. Varat et al. (1994) proposed the following method to categorise the existing multiple barrier crash test data, covering vehicle velocity of up to 50mph. When analysing the test data, energy of the impact should be used to account for the weight differences between different test vehicles. That is, speed of the vehicles alone may not be sufficient to differentiate the impact of vehicles belonging to different weight categories. On the other hand, rebound velocities were not available for all the test data and consequently, the mathematical formulation was developed based on the energy absorbed by the crushed vehicle. The following equation 2.40 was derived by assuming linear dissipative spring. Factor called Energy of Approach Factor ( EAF) was implemented to maintain a linear relationship. x B w E o = 2 , Eq. 2.40 where x is the crushed length and w k B = . k is the spring stiffness and w is the crushed width. Factor w E o 2 is called EAF and E o represents the absorbed energy by the crushed vehicle. The above equation describes a relationship with a zero y- intercept. However, vehicles do require some initial onset energy before permanent Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-60 crush takes place. Adding a onset energy factor to Equation 2.40 results in the following, x B EAF EAF o + = . Eq. 2.41 Existing records of the crashed test were then compared with the above model, using least squares method to determine how well the set of data fits the particular model. The liner and quadratic curve fits were considered at the initial stage and R 2 values were compared and analysed along with the percentage errors between the fits and the actual test data points. The vehicles were classified as non-linear, if noticeable improvement was achieved through the use of the second order fit. Some of the vehicles exhibited consistently linear relationship between EAF and residual crush of up to 35 mph tested speed. It was observed that the overall average percent error for all vehicles was approximately 5.0% for speeds from 15 to 35 mph (Varat et al. 1994). On the other hand, some of the vehicles exhibited non-linear relationship within this velocity region. This is an indication of the structural softening even for the low velocity levels. The second order fits yielded good correlation for this category of vehicles and overall average percent error was limited to 1.0%. Similarly, some of the vehicles exhibited linear correlation of up to 50 mph while others exhibited non-liner response within the same velocity region. 2.13.1 Application to accident reconstructions When performing an automotive accident reconstruction, the re-constructionist may not have access to impact test data to the required extent. In addition, he does not know whether a vehicle under analysis has a linear or non liner trend considering the EAF versus crush relationship. Therefore, the study of Varat et al. (1994) needs further generalisation by assuming a bi-linear relationship between all the crash test data. Varat et al. (1994) suggest that the bi-linear approximation takes the form of a straight line for the test data below 30 mph and another strait line to approximate the data from 30 to 50 mph. These two lines share the common 30 mph data point. Then the percentage error between the assigned linear relationships and the actual data points was calculated. The resultant overall average error was approximately 5.5% at the speeds between 15 and 35 mph. The average error was found to be less than 5% for the Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-61 velocity range from 30 to 50mph. However, the vehicles exhibiting liner relationship up to 50mph cannot be satisfactorily modelled with the presented bi-linear approximation used between 30 to 50 mph. A similar procedure was adopted by Wagstrom et al. (2004) to simulate a frontal collision of vehicles of different velocity categories. The vehicles were divided in to three mass class and stiffness coefficients suggested by Summmers et al. (2001) were assigned to find the deformation characteristics (Table 2.2). Wagstrom et al. (2004) was further suggested that the cars can be further simplified to two main categories based on the front stiffness. Table 2.2: Mass and stiffness coefficient for impact reconstructions Category Mass (kg) Front Stiffness (kN/m) Light 1200 1000 Medium 1600 1000 Heavy 2000 2000 Front stiffness values only represent the initial slope of the force deformation curve and by assuming liner springs, the impact forces are highly overestimated for larger measurement of deflections. In addition, it is important to note in the model, that there is no differentiation between the resultant elastic or plastic deformations, even though the term stiffness is usually associated with linear elastic deformation. In fact, after reaching their maximum deformations, linear spring elements would eventually act as a tension elements. This has to be prevented by setting the stiffness constant to zero, as the velocity of the mass changes the sign. This means that structural restitution effects are considered as negligible (Wagstrom et al. 2004). On the other hand, vehicles with bi-linear characteristics can be modelled as a two linear springs in series, where the second spring does not compress until the first spring bottoms out. In this case, the first spring represents the engine compartment and the second spring represents the occupant compartment (Neptune 1999). 2.14 Design guidelines Irish Standards, I.S. EN 1991-1-7-2006 suggests some useful guidelines for the assessment of impact and accidental loads on buildings. The guidelines considered accidental impact forces applied from rail or road traffic, ship impact and impact due Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-62 to helicopter emergency landing on a roof of a building. The code is recommending to use equivalent static force when the consequences are low to medium. On the other hand, more advanced analysis is required for more serious consequences. Special attention has been given to buildings used for parking, where vehicles are permitted to enter the inside of the buildings and the buildings that are located adjacent to the road or railway traffic. These buildings are highly susceptible to impact loads and the code recommends the use of dynamic analysis or equivalent static force method to calculate the effects on a structure. The equivalent static force is used for the verification of static equilibrium of the structure and for the determination of the deformations of the impacted structure. The code also considers the effects of variables, such as impact velocity, angle of impact, mass distribution, deformation behaviour and the damping characteristics of both, the impacting object and the structure. In addition, the code recommends the use of upper and lower characteristic values for the material properties of the impacting body and for the structure, respectively. Horizontal equivalent forces applied on column or wall in a structure are tabulated in the code for the type of road and vehicle. The maximum force 1000 kN is used to account for a truck impact and 500 kN for a car impact in a parking garage. No horizontal force needs to be considered on overhead elements, unless the clearance is less than 6m. If the clearance is less than 6m, prescribed horizontal force can be used in the design. Such forces can then be applied on the underside of the bridge over a traffic lane. According to the specifications given in AS 1170.1 (1989), columns in car parks should be designed to withstand the additional horizontal impact load arising from the movement of vehicles. The additional live load F can be calculated as follows, l mv F ∆ = 2 2 Eq. 2.42 where, F = impact or breaking force in newtons, m = gross mass of the vehicle in kilograms, v = velocity of the vehicles, in meters per seconds, Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-63 ∆ l = deceleration length in meters. ∆ l can be taken as the sum of the deflection of the vehicle and the barrier. In the absence of reliable data, ∆ can be considered to be 0.1m. The recommended value of m is, a) for domestic car parking - 1500 kg , b) for general car parking - 2000 kg , c) maximum expected gross vehicle mass exceeding 2500 kg is rare. Barriers facing ramps longer than 20.0m must be capable of withstanding an impact force applied by a moving vehicle having a velocity of 10.0ms -1 . In the absence of reliable data, the deceleration length ∆ l , can be taken as 0.15m. The code only considered the collision with one vehicle at a time. The impact load was considered at 0.5m above the floor level for cars and 1.0m for trucks. The impact force was considered to be distributed over a distance of 1.5m, along the barrier or full width of a column. 2.14.1 Dynamic design for impact Complex interaction between two objects takes place under impact process. Due to this complexity, simplified approximations are used to quantify the impact response. Dynamic effects as well as the non-linear material behaviour must be taken in to account in the impact analysis. EN 1991-1-7(2006) covers the dynamic aspects of the design briefly based on the simplified or imperial models. Based on the initial kinetic energy dissipation process, the impact is characterised as either soft impact or hard impact. For soft impact, approximate dynamic analysis may be performed by using the Equations from 2.43 to 2.45. By assuming that the impacting body deforms linearly during the impact phase, the maximum dynamic interaction can be calculated as, km v F r = , Eq. 2.43 where v r is the object velocity at impact, k is the equivalent elastic stiffness of the object and m is the mass of the colliding object. If required, force due to impact may be considered as a rectangular pulse with a non-zero rise time. The duration of the plus is be given by, Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-64 k m t or mv t F / = ∆ = ∆ , Eq. 2.44 k = EA i /L i and m = ρAL i , Eq. 2.45 where L i is the length, A i is the cross sectional area, E is the Modulus of Elasticity and ρ is the mass density of the equivalent impacting object. Hard impact Under hard impact conditions, the structure is assumed to be elastic and the colliding object as rigid. The condition is similar to the assumption made in the validation process presented in this thesis. The maximum dynamic force applied in this case can also be calculated using the same equation, by substituting structural stiffness value for the equivalent elastic stiffness k. The rest of the calculations can be carried out based on the assumption that structure has sufficient ductility to absorb the total kinetic energy noted as 2 2 1 r mv by plastic deformation. This requirement can be expressed by using the following expression, o o r Y F mv ≤ 2 2 1 Eq. 2.46 where F o is the plastic strength of the structure and Y o is its deformation capacity. Indicative probabilistic information for the basic variables used in the Equation 2.46 is given in the Table C.1 in EN 1991-1-7(2006). 2.15 Knowledge gaps and literature review findings In this chapter, the literature on impact loads applied on columns in the lateral and longitudinal direction was reviewed. • The investigations under dynamic loading conditions highly emphasised the effects of strain rate, and 30% strength enhancement was expected compared to the quasi-static conditions. However, this enhancement would be undesirable for columns susceptible to shear failure conditions. • A significant reduction of flexural ductility was noticed on structural columns caused by the increase in compressive axial load level. However, no particular effort was made in the past to identify the variation of the shear capacity of columns under instantaneous increment of axial load followed by lateral impact. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 2-65 • There is no evidence of research performed on laterally impacted columns under varying axial loading conditions. Strain rate effects and confinement effects are highly sensitive to the axial load level, and the resultant impact behaviour of columns has not been investigated to a satisfactory level. For instance, concrete elements under flexure exhibit higher strain rate sensitivity than the elements under compression. This means that the strain rate sensitivity of a column can be changed with the axial load. • Most of the tested beams were simply supported and hence, exhibited flexural type failures. As a result, the beams survived under higher bending moments and absorbed more energy compared to the static loading conditions. However, higher modes of vibration due to dynamic impact loads in impacted columns were inevitable. Consequently, the shear failures were unavoidable under impact loading and needs further investigation. • Confinement effects are a governing factor of the ductility of the columns, and bond stress can increase up to 200% due to confinement effects. Higher strain rate sensitivity of tensile strength of concrete reduced the amount of cracks under impact loads, while increasing the bond between reinforcement and concrete. Therefore prefect bond may be assumed between concrete and steel under impact conditions. • Material erosion criteria associated with the Lagrangian mesh discritization may not represent the realistic behaviour of concrete during an impact. • Methods proposed by Campbell (1974) and Summers et al. (2001) facilitate comparison of the hard impact data with more realistic vehicular impact events. These methods are highly depending on the stiffness of the impacting vehicle. Based on the literature review findings, a comprehensive research is proposed to bridge the gaps in the knowledge. In the process, axial load is chosen as one of the major parameters. Strain rate effects, confinement effects, elevation of the impact, slenderness ratio, steel ratio, confinement, the amplitude and duration of the impact are among the other parameters that are needed to be examined. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 3-67 3. FE MODELLING OF SHORT RC COLUMNS UNDER LATERAL IMPACT 3.1 Introduction In general, finite element analysis can be performed based on implicit or explicit algorithms. The predictor-corrector method is employed by the implicit algorithm to solve non-linear problems, in which the stiffness matrix is updated throughout the entire analysis. On the other hand, in explicit codes, the corrector method is omitted and no equilibrium check is preformed. The advantage of using the explicit method is that there is no need to calculate stiffness and mass matrices for the entire system, thus the solution can be carried out at the element level and relatively little storage is required. In addition to its computational efficiency, it can handle contact problems effectively. The explicit Finite Element program LS-DYNA was therefore used in the numerical simulation process (Hallquist 2007). The procedure used to solve the discrete equation of motion is called explicit, if the solution at some time t t ∆ + in the computational cycle is based on the knowledge of the equilibrium condition at time t (Homuda and Hashmi 1996). According to Newton’s axioms, the forces that act on a structure which is not in equilibrium, cause acceleration, that can be expressed in terms of velocity and displacement. The summation of all the forces drives the system to acquire a position of equilibrium. However, forces acting on the impacted structure change regularly during each time step and the system does not reach an equilibrium position. Therefore, the stability of the solution method strongly depends on the time step size. For a typical finite element mesh it may be necessary to use time steps in the order of microseconds (Mathisen et al. 1999) which indeed vary with the natural frequency of the structure and the characteristic length of the elements in discretisation. The explicit time integration method is conditionally stable, when the time step is sufficiently smaller than the travel time of the material stress wave across the smallest described element or zone width within a given FE spatial discretisation (Hallquist 2007). To avoid unpredictable errors, critical time step size based on the so-called Courant’s criterion must not be exceeded (Hallquist 2007). This time step, ∆t e is determined automatically in a Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 3-68 conservative way within the relationship of the sound speed, c s and the element lengths, L e . The sound speed is a function of Young’s modules, E mesh density, ρ and Poisson’s ratio, ν. Q is a function of the bulk viscosity coefficients C o and C 1 . The Equations 3.1 & 3.2 can be used for calculation of the time step for solid elements. ( ) [ ] 2 1 2 2 s e e c Q Q L t + + = ∆ Eq. 3.1 ( ) ( )( )ρ ν ν ν 2 1 1 1 − + − = E c s Eq. 3.2 Therefore, the time step sizes usually become rather small (10 -8 s) for reasonable FE discretisations (15mm). The solution for system of non-linear equations using this time integration scheme requires the inversion of the mass matrix within each time step (LS-DYNA 2006). The LS-DYNA explicit code has the ability to solve advanced, robust and computationally efficient contact algorithms. That includes crashes, drop tests and further contact related applications. Although contact zones can be defined to any level of precision, general contact algorithms are also available where the specified contact algorithm for the entire model or part of the model can be selected (Rust and Schweizerhof 2003). In LS-DYNA code, generally displacement control is preferable, as then the structure can be well controlled globally. In large deformation analysis, reduced integration should be used to reduce computational expenses. In particular, reasonable mesh size, boundary and contact conditions need to be chosen by considering the c s , L e , bulk viscosity and duration of the analysis in order to give comprehensive results. 3.2 Finite element modelling for impact problems When it comes to the impact problems, the Lagrange discritization is the state of the art method. The Lagrangian method, uses a mesh which is not fixed in space but fixed in body. Thus the cells can be distorted and change their volume (Gebbeken et al. 2007). To overcome this shortcoming, many numerical codes use erosion criteria and elements get deleted after achieving the predetermined distortion criteria (Hallquist Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 3-69 1997). However this phenomenon will not represent the actual behaviour of a material during an impact. For example, concrete may have 11% compressive strength at the residual state compared to its undamaged state. The advantages of the Lagrange method are that the mass of the each Lagrange cell is constant and hence, fulfils the principle of conservation of mass. In addition, modelling as well as the contact definitions are easy to implement in this method. The main disadvantage of the Lagrange method, however are very small time steps resulting from the element distortion within a short period of time. 3.3 Finite element modelling and selection of material models One of the biggest challenges associated with modelling the behaviour of reinforced concrete is the difficulty of incorporating realistic material models that can represent the observable behaviour of the physical system. This section discusses the material model formulation used for different materials such as concrete, reinforced steel, stirrups and rigid body used in the finite element simulation. 3.3.1 Evaluation of Constitutive material models in LS-DYNA Concrete is known to be ductile in nature under hydrostatic pressure conditions and may be subjected to brittle failure in tension under impact loading conditions. LS-DYNA contain several material models that can be used for concrete, however the actual behaviour of concrete under a three dimensional state of stress is extremely complicated. Concrete is highly strain rate dependent and any material model should incorporate these non-linear characteristics. However, due to changes of state within microseconds, dynamic experiments that determine these parameters are much more complicated and expensive than the static load test, and therefore data relevant to the dynamic experiments are limited in the literature. ‘Concrete Brittle Damage Model 96’ presented in the LS-DYNA library is extremely useful in this circumstance as it employs a fully anisotropic damage rule which is free from adjustable parameters for a given failure surface (Govindjee et al. 1995). At the same time the constitutive material model Mat_Concrete_Damage_REL3 offers an added advantage of only one user input parameter; ie. Unconfined compressive strength is sufficient in the Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 3-70 calibration process (Malvar et al. 1997; Schwer and Malvar 2005). Since the unconfined compressive strength of concrete can be easily determined from experimental testing, these two models are very useful in impact simulation process. Consequently, these two material models were implemented in the validation and the superior material model was implemented in the parametric study based on the accuracy of the outcomes. In fact, for simplicity some other constitutive models in LS-DYNA adopt highly restrictive assumptions and hence their applicability is limited to a certain class of problems. For example, material model Mat_Soil_and_Form uses a perfectly plastic flow to approximate the post-yield behaviour and is unable to capture the various softening behaviours of concrete under different loading conditions. Similarly Mat_Geologic_Cap_Model generates a circular deviatoric cross section where experimental data indicate that for brittle materials, the shape of the deviatoric section is circular only at the high pressure regime. In addition, this model cannot predict the softening behaviour of concrete and confinement effects due to the use of an associated flow rule (Yonten et al. 2005). The Mat_Soil_Concrete has similar weaknesses as this model uses only two stress invariants, I 1 and J 2 while neglecting third stress invariant J 3 . It has been also shown that the model does not give a smooth transition at the softening region due to tri-linear variation of f c which is used define material softening behaviour from plastic to residual (Yonten et al. 2005). Therefore, their capability of describing the actual nonlinear behaviour of concrete under dynamic loading can be different and hence this chapter intends to evaluate the similarities and distinctive features of the selected models by using the results generated from a numerical simulation of columns impacted by a rigid mass. 3.3.2 Theory on Mat_Concrete_Damage model This model uses three shear failure surfaces namely an initial yield surface, a maximum failure surface and a residual surface with consideration of all three stress invariants (I 1 , J 2 and J 3 ). The attractiveness of this model is it allows the general material properties to be generated based solely on the unconfined compressive strength of concrete and the density. The compressive meridian of the initial yield Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 3-71 surface c y σ ∆ , the maximum failure surface c m σ ∆ and residual surface c r σ ∆ are defined independently as; Figure 3.1: Failure surfaces in Mat_Comcrete_Damage_REL3 (Malvar et al. 1997) p a a p a y y oy c y 2 1 + + = ∆σ Eq. 3.3 p a a p a o c m 2 1 + + = ∆σ Eq. 3.4 p a a p f f c r 2 1 + = ∆σ Eq. 3.5 where σ ∆ is the stress difference and p is the mean stress in the triaxial compression failure test. The parameters f y y oy a a a a a a a 1 2 1 0 2 1 , , , , , , and f a 2 can be determined by a regression fit of the above equations to the available laboratory test data. Having provided the three separate strength surfaces, the corresponding loading surfaces representing strain hardening after yield are defined as follows; y m L σ η σ η σ ∆ − + ∆ = ∆ ) 1 ( Eq. 3.6 The post failure surface pf σ ∆ is defined by interpolating between the maximum failure surface and the residual surface; r m pf σ η σ η σ ∆ − + ∆ = ∆ ) 1 ( Eq. 3.7 The variable η is called the yield scale factor which is determined by a damage function λ which has two distinctive definitions for compression ( ) 0 ≥ p and tension ( ) 0 < p to account for the different damage characteristics of concrete in Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 3-72 compression and tension: η varies from “0” to “1” when the stress state advances from the initial yield surface to the maximum failure surface and visa versa when softening begins. [ ] [ ] ¦ ¦ ¹ ¦ ¦ ´ ¦ < + ≥ + = ∫ ∫ p p p f p d p f p d b t p b t p ε ε ε ε λ 0 0 0 1 0 1 2 1 Eq. 3.8 where f t is the static tensile strength of concrete, p dε is the effective plastic strain increment, and ( ) p ij p ij p d d d ε ε ε 3 2 = , where p ij dε is the plastic strain increment tensor. The yield stress factor ( ) λ η governs the nonlinear behaviour of the material and follows a piece-wise linear relationship whose control points must be pre-defined in the data input file. 3.3.3 Definition of compressive and tensile meridians at p < f c /3 As the compressive meridian of the failure surface is determined based on the experimental data with pressures equal or above f c /3, the pressure values below f c /3 predicted by extrapolating the existing values generally over estimate the compressive strength in that region. The tensile meridian (image) obtained by using the compressive meridian will also become inappropriate. To eliminate this problem, the tensile meridian below the pressure range f c /3 is defined as a linear curve derived using experiments such as uniaxial and triaxial hydrostatic extension tests. The compressive meridian (image) will then be derived by dividing the tensile meridian by a factor ψ. The tensile meridian ( ) t m σ ∆ can be obtained from the following equation for the range of pressures less than f c /3; ( ) t t m f p + = ∆ 3 2 σ Eq. 3.9 The corresponding effective stress ( ) σ ∆ and the tensile strength (f t ) pairs will be determined from the following testing procedures; (i) Biaxial tension (on the compressive meridian) (ii) Plane stress pure shear (θ = 30 o ) Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 3-73 (iii) Uniaxial compression, The tensile to compressive meridian ψ has the following values. ¹ ´ ¦ = + ≤ − = 3 2 3 2 1 0 5 . 0 c c t t f p f f f ψ Eq. 3.10 For higher compression pressures, the model chooses three additional points as follows: ( ) ( ) ( ) ( ) [ ] ¦ ¹ ¦ ´ ¦ ≥ = = + + = c c c c c o c f p f p f p f a a f a f p 453 . 8 0 . 1 3 753 . 0 3 / 2 3 / 2 3 / 2 / 2 1 α α α α ψ Eq. 3.11 For computational purposes the above function is made continuous by a piece-wise linear interpolation of the discrete points above. 3.3.4 Pressure cut-off and softening In the model, the maximum principle stress of concrete under tension is limited by the uniaxial tensile strength f t considering the quasi-static loading rate. In the stress softening procedure in compression based on pf σ ∆ , the yield scale factor η is controlled by the accumulation of a scaled effective plastic strain p dε . This may cause a problem when the stress path is closed to the triaxial extension path because there are no stress deviators and hence no plastic strain accumulation in the model. Therefore both the damage function λ and scale factor η remain zero. The pressure return from EoS will decrease from 0 to –f t and will remain at that level thereafter, contrary to what would happen in concrete after the failure surface is reached. To overcome this problem the model implements a volumetric damage term by considering volumetric plastic strain λ ∆ in conjunction with the damage function λ; ( ) yield v v d d k f b , 3 ε ε λ − = ∆ Eq. 3.12 where b 3 is a user defined scalar multiplier, k d is an internal scalar multiplier, v ε represents the volumetric strain and yield v, ε is the volumetric strain at yield. The scalar f d is used to restrict the effect of this additional term λ ∆ , only to the triaxial tensile Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 3-74 stress path region by limiting the f d so that 0 = d f when 1 . 0 3 2 ≥ p J . With the implementation of λ ∆ , the damage is incorporated for the tri axial tension stress state. However, post-peak softening behaviour will still not be achieved since there is no interconnection between the post failure surfaces and the triaxial tension stress path for pressure above –f t . To solve this problem, the model shifts the pressure cutoff when softening occurs in the negative pressure range by scaling the original pressure cutoff by the factor η. 3.3.5 Strain rate effect In the concrete damage model, the failure surface and the damage function λ are modified to account for the strain rate effects. Based on the fact that rate enhancement is experimentally obtained from the unconfined compression and tension tests, and that they follow the radial paths from the origin in the principal stress difference verses pressure plane, the model implements a radial rate enhancement on the concrete failure surface as given in the following equation; ( ) f c m f c me r p r × ∆ × = ∆ σ σ Eq. 3.13 Referring to the maximum failure surface meridian, the modified equation becomes p a r a pr r a f f f o c m 2 1 + + = ∆σ Eq. 3.14 After the modification to account for the strain rate effects the variation of λ becomes: [ ] [ ] ¦ ¦ ¹ ¦ ¦ ´ ¦ < + ≥ + = ∫ ∫ p p p f r p r d p f r p r d b t f f p b t f f p ε ε ε ε λ 0 0 0 1 0 1 2 1 Eq. 3.15 where r f is the dynamic increasing factor for concrete. Thus, the current model (REL3) allows input of the different rate enhancement for tension and compression solely based on one strain rate curve. Equations given in the CPF-FIP (1990) model code were implemented to account for the strain rate effects in concrete. For a given strain rate and concrete grade, the compressive and tensile strength enhancement can be estimated from these equations. The Dynamic Increasing Factor (DIF) given in the Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 3-75 code is a design value, which means that the given strength increments are lower than the values obtained from experiments. 3.3.6 Equation of state An Equation of State (EoS) is a formula describing the interconnection between various measurable properties of a system. For the physical state of matter, this equation usually relates the thermodynamic variables of pressure, temperature and the volume of a system to one another. An EoS is particularly important to define isotropic concrete material model Mat_Concrete_Damage (Hallquist 2006). A failure criterion for isotropic material should be an invariant function of its state of stress. This means that it must be independent of the choice of the defined coordinate system. Therefore, a failure criterion is defined using stress invariant. The state of stress is subdivided into two components: hydrostatic and deviatoric. An EoS is used to describe hydrostatic material behaviour and relate quantities such as pressure, density and energy. In contrast, the deviatoric stress state corresponds to the shear stress and represents the change of shape. Generally, the constitutive model relates the stress vs. strain behaviour, while the EoS relates the hydrostatic pressure with density and energy. In addition, the shear stress and stiffness is increased with pressure. The deviatoric behaviour also depends on the temperature, strain rate and degradation of material (Gebbeken et al. 2007). This means that these parameters cannot be considered as constant for a wide range of impact scenarios and hence are not unique to the particular system. 3.3.7 Evaluation of LS-DYNA material model Mat_Brittle_Damage 3.3.7.1 Material characteristics This model (second selection) is primarily formulated for evaluating brittle damage in concrete and hence is useful in the impact simulations. It is well known that numerical results are very sensitive to the nonlinear material properties. In the absence of more specific data for the dynamic properties of concrete, the following equation presents an approximate value for its tensile limit (CEB-FIP, 1990); Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 3-76 ( ) | | ¹ | \ | = o c it a f T 2 ' lim 58 . 1 Eq. 3.16 where the compressive strength ' c f should be in MPa and a o is 145 for SI units. Once the principal tensile stress has been reached at a point, a smeared crack is initiated at that point with a normal that is co-linear with the first principal direction. Once initiated, the crack is fixed at that location though it will connect with the motion of the body. Allowed tensile traction normal to the crack plane is progressively degraded as the loading progresses. This behaviour is implemented by reducing the material modules normal to the smeared crack plane by using the program based internal parameter. In addition, the shear strength basically governs the initial shear traction that may be transmitted across a smeared crack plane. According to Halliquist (1998), in the absence of more specific data, the shear traction can be calculated by using the following equation. [ ] ) exp 1 )( 1 ( α β s s H f − − − Eq. 3.17 According to Equation 3.17 it is interesting to note that the shear degradation is coupled to the tensile degradation through the internal variable alpha (α) which measures the intensity of the crack field. In general, the shear degradation factor accounts for the reduction in the shear stiffness of the material parallel to the smeared crack plane. The evolution of alpha is governed by a maximum dissipation argument. Here β is the shear retention factor and βf s represents the shear traction that is allowed across the smeared crack plane as the damage progresses. The parameter H s represents the softening modulus. Govindjee et al. (1995) present a full description of the tensile and shear damage part of this material model. As far as the viscosity of the material is concerned, the Perzyna regularization method was used to implement the viscose behaviour of the material. In order to avoid error termination, values of viscosity (η ) between 0.71 and 0.73MPa are recommended (Hallquist 2007). Apart from serving as a regularizing parameter which stabilises the calculations, the viscosity of the material also allows the inclusion of first order rate effects during the simulation. Other than the viscosity of the material, the stability of the calculation process will also depend on the fracture toughness of the material. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 3-77 Once the facture toughness is entered, the softening modulus (H s ) is automatically calculated based on element and crack geometries. Instead of using stress-strain relationship, the impact behaviour of concrete was defined by using various parameters which simulate its non-linear behaviour. Table 3.1 shows the reasonable values selected for Grade 47 concrete. In the absence of definite data, the modulus elasticity of concrete (E) was calculated by using the following equation (CEB-FIP 1990); ( ) [ ] 3 / 1 ' / cmo c co ci f f f E E ∆ + = . Eq. 3.18 where ' c f is the characteristic cylindrical strength of concrete, ∆f is 8 MPa, f cmo is 10 MPa and E co is 2.15x10 4 MPa. The shear retention factor was selected by using a trial and error method. The cylinder compressive yield strength (σ y ) should be used here; and can be directly calculated from the uni-axial cubic strength by using Table 3.1 of the CEB-FIP code (CEB-FIP, 1990). Table 3.1: Material properties used for the concrete To achieve damage degradation the model employs three damage surfaces which evolve with material damage. In order to cater for compressive failure, the model adopts a very simple J 2 flow correction method which is not capable of representing the enhancement of material shear strength due to the presence of high pressure. The use of a simplified resistance function for concrete elements may ignore the nonlinear behaviour of concrete specifically near the ultimate conditions. 3.4 Development and validation of a numerical model of a RC column 3.4.1 Introduction With a view to assessing the vulnerability of columns to low elevation vehicular impacts, a non-linear explicit numerical model has been developed and validated using existing experimental results. The numerical model accounts for the effects of strain rate and confinement of the reinforced concrete, which are fundamental to the successful prediction of the impact response. The sensitivity of the material model parameters used for the validation is also scrutinised and numerical tests are performed Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 3-78 to examine their suitability to simulate the shear failure conditions. Conflicting views on the strain gradient effects are discussed and the validation process is extended to investigate the ability of the equations developed under concentric loading conditions to simulate eccentrically loaded columns. Two of the most sophisticated material models presented in the LS-DYNA library were compared and one of them was selected for parametric studies based on its capacity to generate most accurate results. Experimental data on impact force time histories, mid span and residual deflections and support reactions have been verified against corresponding numerical results. 3.4.2 Experimental test set up Feyarabend (1988) tested a square RC column in a horizontal position as shown in Figure 3.2. Fixed support conditions were achieved at one end by stationary steel sections fixed at the ground and the other end was attached to a 20t mass that can slides over horizontal low friction rails simulating roller support. The axial load was applied through a system of prestressing wires located on either sides of the column and the impact was generated by dropping a 1.14t mass on to the column at mid span. In order to account for the shear critical conditions, a separate numerical test was conducted on the material model and parameters such as fracture toughness and shear retention was proved to be adequate in representing the shear critical condition by using CPF (1990) model code and the existing literature on material properties. In the absence of experimental results from testing of a concrete member with axial load impacted nearer the support, the above validation was carried out (Thilakarathna et al. 2010). Figure 3.2: The test set-up by Feyerabend (1988) Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 3-79 The testing procedure involved impact test on 0.3×0.3m square column specimens made of 47MPa concrete. The properties of the test specimen are given in Table 3.2 and detailed description of the experimental program can be found in Feyerabend (1988). Table 3.2: Characteristics of the Feyerabend’s test specimens (Feyerabend 1988) Details of test No.SB2 Cross-section (m×m) Span (m) Concrete cube strength, f cu (MPa) Steel Yield stress, f y (MPa) Main bars, d s (mm) Shear stirrups, A vs (mm/mm) Restraining mass (kN) Initial axial load (kN) Striker mass (kN) Impact velocity (m/s) Velocity at which f y was reached (m/s) 0.3×0.3 4.0 47 548 4φ25 12φ@150 196.2 -197 11.18 3.0 ±1.8 3.4.3 Numerical simulation of the physical testing According to El-Tawil et al. (2005) the peak force generated at the impact is not representative of the design structural demand, as the structures do not have enough time to respond to a rapid change of loading. Consequently, it is assumed that the lateral movements of the restraining mass and the elongation of the prestressing cable system during the impact do not affect the impact behaviour of the column. Hence the prestressed cable was excluded and the axial load applied as a ramped up surface pressure over the cross section. The small fluctuation of the axial load due to the impact is also neglected as the fluctuation is unlikely to alter the flexural or shear capacity of the column. Figure 3.3: The simplified test set-up and the cross section of specimen } Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 3-80 The experimental set-up of the impacted column is simplified as shown in Figure 3.3. Half of the column was modelled and appropriate boundary conditions were introduced to maintain the symmetric conditions. One support was restrained against rotation while allowing translation along the longitudinal direction at the other end to simulate partial restrained conditions at that end as shown in Figure 3.3. These boundary conditions allow representation of the much complex fixed and free sliding supports with constraints as close as possible. 3.4.4 Convergence study and mesh discritization Discretization is the process that transfers continuous models into discrete counterparts. The discrete elements generated in the simulation process is solved by assuming linear equations where displacement at each node is calculated for the given load and its constraints, and is then used to approximate the stress contours of each element. As the stress approximation is based on the relative displacement of the individual nodal locations, the stress contour will not necessarily be continuous from one element to the next thus causing an error. This discritization error can be minimised by decreasing the size of elements. However, small mesh discritization will significantly increase the duration of the analysis and demand a compromise between the duration and accuracy. Figure 3.4: Convergence of the numerical model A convergence study is used to solve this problem and a typical rule in convergence study is to double the element density for each iteration in the area of interest. According to Burnett (1987), drastic local refinement should be avoided in mesh Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 3-81 convergence studies. Therefore, in this investigation, the element size homogeneously varies from 50mm to 5mm at each stage. The total energy accumulation in the impacted column is shown in Figure 3.4 under the 3.0ms -1 impact. Figure 3.4 illustrates that the total energy of the model converges towards 2565J as the size of the elements decrease. With the convergence of the total energy, peak displacement is also converging by limiting the hourglass energy accumulation to an acceptable level. According to the analysis the 25mm mesh discritization provides the optimum solution. It is also evident that a further decrease of the mesh size had very little effect on the accuracy of the results, while at the same time, increased the duration of the analysis dramatically. The error due to the selected mesh size is only 2% compared to the 5mm mesh, and hence negligible. Figure 3.5: Mesh generation for the impacted column & rigid body Consequently, the concrete column was modelled using 25mm eight node hexagonal ‘constant stress’ solid elements with one point integration (see Fig. 3.5). Similarly 25mm long beam elements were used for both vertical and the lateral reinforcements with 2×2 Gauss integration. The vertical reinforcements are defined as truss elements and the links are defined as a Hughes-Liu beam element with cross sectional integration (Flanagan and Belytschko 1981). 35mm cover is assumed for the main reinforcement. Eight noded hexagonal solid elements with one point integration were employed for the drop mass. No attempt was made to simulate the guide rail or restraint when the drop mass bounces back after the first impact. A nominal radius was maintained at the bottom of the mass to avoid stress concentration along the perimeter and initial analysis was carried out by assigning rigid body conditions (Hallquist 2007). Rigid body Impacted column with 25x25mm hexagonal elements Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 3-82 Movement caused by the gravitational acceleration were not considered in the analysis as the main impact pulse lasted for a maximum of 10ms. As far as the contact between concrete and steel is concerned, chemical adhesion, frictional resistance and rib-bearing are the main components of the bond between concrete and the steel. Under the influence of the small stress difference, the chemical adhesion predominates over the other two components. Wedging action becomes predominant when chemical adhesion alone is not sufficient to resist the stress difference. Longitudinal and radial cracks are generated under the influence of wedging action. Restraining effects due to wedge action are replaced by the frictional forces only after propagation of the longitudinal and radial cracks. In addition, the entire process may alter due to the confinement effects, strain rate, phase transformations and the pressure variations. However, due to their complexity, all these factors cannot be taken into account in a simulation. As far as deformed reinforcement bars are concerned, the ultimate dynamic bond at failure was 70-100% higher than that under quasi-static loading conditions (Weathersby 2003). Also the steel deformation under the impact load was limited to a region only few centimetres long beneath the point of the impact (Bentur et al. 1986). This means that there was not enough time to develop extensive bond slip along the length of the bar under impact condition. Therefore perfect bond was assumed between the reinforcement and concrete. Under these circumstances, strain hardening characteristics of the longitudinal steel may not alter the numerical results significantly. 3.4.5 Contact algorithm and prevention of initial penetration Segments with high velocity conditions often come into contact under impact conditions. This results in large deformation of the segments. In the finite element modelling, instead of being deformed after the collision, one segment penetrates the other segment, especially if the stiffness of the segments is different. This is known as initial penetration. The initial penetration occurred at the contact interface between the column and the rigid body, leading localised stresses and strains that were quite unusual. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 3-83 Suitable contact interfaces themselves can be effectively used to eliminate the overlap or penetration between the interacting surfaces. The detection of a penetration is also an important part in this elimination process. In brief, to detect penetration due to contact, the LS-DYNA code firstly performs a global search followed by a local search using incremental search techniques. After detecting any penetration condition, the amount of penetration is calculated and a force is applied to remove the penetration. This force can in some instances be very large and may create negative energy conditions. To avoid negative energy conditions, a suitable contact definition can be used as a default treatment. Detailed description of the contact definition including penetration detection and elimination process can be found in LS-DYNA theoretical manual (LS-DYNA 2007). In order to prevent contact break downs similar mesh discretisation was used in this simulation for the drop weight and the column including the contact treatment AUTOMATIC_SURFACE_TO_SURFACE (Hallquist 2007). This standard penalty based formulation consists of interface springs between all the contact surfaces which apply interface forces proportionate to the amount of penetration. In a typical discrete environment the nodes of the softer body (slave) will be penetrated by the harder body (master). The selection of the slave and master surface can however be arbitrary with this formulation, since there are two checks for possible penetration between slave and master surfaces and vice versa. Contact stiffness can also be effectively used to handle the initial penetration conditions. Since the default stiffness option depends on the material properties and the size of the segment, it is inadequate to handle the initial penetration between dissimilar materials. In addition, the default treatment may distort the original geometry at location, where the penetrations are detected. The soft constrained-based approach (SOFT=1) was effective in this circumstance where contact stiffness calculations are based on stability considerations by taking into account the time step. This option was particularly useful to address the dissimilarities of the mesh orientation and stiffness of the two bodies. The contact nonlinearity was further stabilised by assigning a value of 30 for the Viscous Damping Coefficient. In addition, friction coefficients were introduced to simulate the frictional forces that were Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 3-84 transmitted across the contact interface due to the nominal radius that was maintained at the bottom of the impacting mass. The friction coefficients were assumed to be dependent on the relative velocity of the column (concrete) and rigid body (steel) that were in contact and consequently the selected values for the static and dynamic friction coefficients were 0.6 and 0.5 respectively (McCormick 2009). 3.4.6 Validation of the finite element model using Mat_Concrete_Damage 3.4.6.1 Material characteristics Impact loads generate tri-axial state of stress in concrete columns. For instance, from a material point of view, spalling occurs at the contact interface as a result of tri-axial extension stress conditions. Subsequently impacts on the column generate tri-axial compression stress conditions in the core concrete. In the meantime, uni-axial tensile stresses will be generated at the opposite face by scabbing the concrete as a result of wave reflection at the boundaries. Therefore, a material model that can replicate the results of tri-axial tension tests, tri-axial compression tests, and uni-axial tensile tests may be much suited in the impact simulation process. Mat_Concrete_Damage fulfils this requirement. Each of these tests will represent different damage modes. However, the possibility of combined mode of failures may not be negligible. 3.4.6.2 Elimination of mesh dependency of the fracture toughness Figure 3.6: Single element under uni-axial tensile test As the softening part of the unconfined uni-axial tension stress-strain curve is governed by the values of two parameters b 2 and b 3 (Refer to Equations 3.12 & 3.15), Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 3-85 these parameters have to be adjusted to minimise the differences between the numerical and the experiment results (Schwer and Malvar 2005). For example, the softening behaviour becomes mesh-dependent if it is not governed by a localisation limiter or characteristic length. Thus, the mesh dependency of the fracture energy has to be corrected by changing the parameters assigned for the tensile softening of the material. The mesh dependency can be eliminated by selecting ‘h’, ie. the size of the finite element so that the ratio h G f is equal to the area under the stress-strain curve for the uniaxial unconfined tensile test, where f G is the fracture energy of the concrete. This procedure eliminates the mesh size dependency on the fracture toughness. Figure 3.6 shows the stress-strain response of a single element (1×1×1mm) subjected to uni-axial tension test. According to the CPF-FIP model code, the fracture energy of Grade 47 concrete should be in the range of 100 Nm/m 2 to 125 Nm/m 2 (CEB-FIP 1990). Hence, the default value of b 2 (b 2 =1.35) will overestimate the fracture toughness of 47MPa concrete. Therefore, the value of the b 2 is adjusted until the area under the stress-strain curve becomes 120 Nm/m 2 . However, no specific guidelines are found in the CEB-FIP code related to the strain rate effects on fracture toughness. Therefore strain rate effects are exempted from the fracture toughness and the selected value is based on the static material characteristics similar to the one used by Unosson (2001). Similarly, parameter b 3 is determined from the hydrostatic tri-axial tensile test. The default value of b 3 (b 3 =1.15) leads to desired results and hence accepted without alteration. The effects of these parameters are crucial where tensile and shear failure characteristics are more predominant. 3.4.7 Material properties of steel Longitudinal reinforcement and hoops were modelled as elastic perfectly-plastic materials by using Mat_Plastic_Kinematic model which has the reputation of minimising the duration of the analysis and is available for Hughes-Liu beam elements and truss elements (Hallquist 2007). Kinematic hardening is implemented for reinforcement with strain rate effects. Table 3.3 represents the material properties adopted for main reinforcement and Strain rate was incorporated using the Cowper-Symonds model given in Equation 3.19. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 3-86 P s d C 1 1 | ¹ | \ | + = ′ ε σ σ & , Eq. 3.19 where, d σ′ is the dynamic flow stress at a uni-axial plastic strain rate ε& , and σ s is the associated static flow stress. C and P represent the material constants. The relevant values for steel can be found in Table 8.1 of Stouffer and Dame (1996). Failure strain is not defined here as there is no evidence of a steel rupture. The hardening parameter is β h =0 to represents kinematic hardening characteristics. Table 3.3: Material properties used for the main reinforcement Density (kg/m 3 ) E s (GPa) Poisson’s ratio ρ σ (MPa) E t (GPa) Hardening Parameter (β h ) C p 7800 210 0.30 548 2.0 0 40 5 As the falling weight of 1.14t did not experience excessive deformations, rigid material model was implemented for the impacted mass. Rigid elements are bypassed in the element processing and no storage is allocated for storing history variables. However the inertial properties of the materials are calculated from the geometry of the elements and hence realistic values for the density and the modules of elasticity must be provided. In addition when the rigid body interacts with the column, the contact interface parameters are determined by using the given Young’s modules ‘E’ and Poisson’s ratio ‘ρ’ values of the material (Hallquist 2007). The material characteristics used for the drop mass are given in Table 3.4. The displacement and rotational constraints for the rigid body were restrained about global X and Y directions. Table 3.4: Material properties used for the impacting body Density (kg/m 3 ) Young’s Modulus (GPa) Poisson’s ratio 7800 210 0.3 3.4.8 Load simulation for a dynamic system When imposing static loads in explicit environment the ramp up loading would be the better solution to avoid stress fluctuation. The ramp up load must be increased from zero to its final value and the load curve should extend beyond the termination time for Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 3-87 stability. In general, the ramping duration should be greater than the fundamental vibration period (0.004s) of the column. Furthermore, it is observed that the stabilisation depends on the wave speed and the superimposing effects of the reflected waves at the boundaries. The minimum fluctuation was observed when the ramping duration was around 0.02s, above which there was instability again. The lateral impact force was applied after achieving the vertical load stabilisation. The rigid body is accelerated from zero velocity to 3ms -1 at impact over a 10mm gap. Realistic value for the density is essential for the rigid body since conventional equations of motion are involved in the velocity generation. 3.4.9 Hourglass energy and damping effects To determine the stiffness, damping and mass matrices and the external force vector, it is necessary to integrate the different terms that are present in the equation of motion, with respect to the volume of the object. The method typically used for the integration is the Gauss quadrature rule. In particular, in LS-DYNA code the integration is conducted with a low order Gauss method. For example, reduced integrated solid elements use only one integration point, located at the centre of the element. This low order integration is beneficial in terms of CPU time but may come across some deformation modes, which create zero strain at the Gauss point, determining a zero internal energy. This is usually known as hourglass problem. Since these zero-energy-modes have no stiffness, they may cause numerical errors, decrease the time step size and interrupt the calculation. If this formulation is implemented in a simulation, an hourglass control algorithm is mandatory (Schwer et al. 2005). The other types of formulations, such as, fully integrated selective reduced (S/R) solid and quadratic eight node element with nodal rotation, generally do not exhibit any hourglass problems. However, these types of element formulation often increase computational cost. Additionally, they are unstable in large deformation where some tendency to 'shear-lock' and thus behave too stiffly in applications particularly when the element shape is poor. Elements with reduced integration are more robust in impact simulations because the strain terms evaluated at the integration point remain well conditioned at larger Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 3-88 deformations. The hourglass resisting force vector k i f α is given in Equation 3.20 and descriptive details of each parameter can be found in (LS-DYNA 2007). In brief, the parameter ακ Γ depends on nodal velocity and h iα depends on nodal coordinates respectively. k i h k i h a f α α α Γ = Eq. 3.20 Where 4 3 2 s e hg h c v Q a ρ = . Eq. 3.21 In which e ν is the element volume, ρ is the density, c s is material sound speed, and Q hg is a user defined constant. When Q hg is equal to 0.05, improved results have been observed. As this equation contains a component of the volume v e , theoretically the hourglass error should reach zero with the mesh refinements. Moreover, the default setting of LS-DYNA which is given by the Equation 3.20 is not orthogonal. Therefore the orthogonal approach, as described by Flanagan and Belytschko (1981) is used in this simulation. As far as the damping effects are concerned, damping forces do not impose significant effects during impacts where duration of the event is shorter than the fundamental natural period of the specimen (Zeinoddini 2008; Jones 1997; Strong and Yu 1993). However, when the damping effects are introduced to the system based on ‘mass weighted damping’ method which is ideal for damped low frequencies and rigid body motion (LS-DYNA 2007), the deflection of the column and impact force reduced slightly with negligible increment of the contact duration. Since the procedure does not significantly improve the post peak behaviour of the column the damping effects were excluded. Moreover, the application of a specific damping coefficient cannot be justified in a realistic situation where the supporting structure has many elements that are interconnected. 3.4.10 Procedure for axial load application Axial load on a column can be applied as a uniform surface pressure over the gross cross-sectional area of the column or as combination of loads on the steel and concrete areas separately by assuming uniform strain distribution. In the absence exact knowledge on load sharing between the concrete and steel in a column, researchers Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 3-89 used the first option based on the assumption that perfect bond between the concrete and steel can generate a uniform strain condition (Shi et al. 2008). However, there is an uncertainty in the axial force transferred to the longitudinal steel when the total axial load is applied as a surface pressure over the gross surface cross-sectional area of the column. Therefore it is prudent to investigate the above options in detail and select the one that minimises the error due to the contact enforcement. In order to achieve the uniform strain condition a column subjected to pure axial compression must fulfil the following requirement. s c ε ε = Eq. 3.22 where ε c and the ε s represent the longitudinal strain in concrete and steel respectively. For linear elastic behaviour, stresses in concrete and steel are in proportion to the modular ratio, c s E E n / = . Then by considering axial load compatibility, it is possible to derive the following equation. total c c n A P ρ σ + = 1 1 Eq. 3.23 where P is the axial load on the column and A c is the cross sectional area of the concrete column. ρ total steel ratio of the section. This equation is based on the assumption that strain of concrete and steel vary in an identical manner under the axial loads, which is not particularly true at the ultimate stage. However the longitudinal steel yields before concrete. In addition, with the implementation of the axial load reduction factor ' φ =0.6 according to AS 3600 (2004) the error induced due to the assumption of strain compatibility can be neglected. Deflection of the column under the separate application of axial load on steel and concrete areas is different from the one with uniformly distributed load. This may cause a substantial difference of the failure load at near ultimate stage under lateral impacts and hence the full bond between concrete and steel will not ensure the strain compatibility particularly close to the load application region. In fact, this would affect the shear capacity of the column under the lateral impact conditions. On the other hand wave propagation effects through the material may alter due to the artificial stress concentration. Because of the same reason separate applications of axial load perform well as far as hourglass is concerned. Consequently, the axial loads must be applied on Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 3-90 steel and concrete areas separately and the numerical simulation process must contain a contact enforcement verification phase other than the mesh discretisation. By this way, the numerical simulation can represent the actual behaviour of structural columns particularly when it forms a part of a structure. 3.4.11 Confinement effects under strain gradient The confinement effects can improve the strength and the deformability of the concrete columns under impact loading conditions while mitigating the damage. The confinement effects were therefore taken into account in the numerical simulation by assigning enhanced concrete characteristics to the core concrete and unconfined characteristics to the cover concrete based on the equations proposed by Mander et al. (1988). a) Section under concentric loading b) Section under eccentric loading Figure 3.7: Lateral pressure distribution and the resultant strain gradient It is worth investigating the capacity of this method to simulate concentrically as well as eccentrically loaded columns under impact. In fact, numerous models have been proposed to simulate the confined characteristics of concrete both under concentric (Mander et al. 1988; Sheikh and Uzumeri 1982) and eccentric loading conditions (Sargin 1971; Lokuge et al. 2003). Enhanced stress-strain characteristics that develop under concentric loading conditions may not be applicable to eccentrically loaded Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 3-91 conditions where the stress-strain distribution across the section substantially differs from that under concentric loading conditions (see Fig. 3.7). Therefore the resultant strain gradient in eccentrically loaded columns was numerically simulated by dividing the cross section into a number of strips and then assigning various stress-strain characteristics based on the depth to the neutral axis (Lokuge et al. 2003). The validity of such a method may be uncertain as impacted columns may generate higher vibration modes where the compression and tension sides interchange over the height. However, in similar investigations, the confinement characteristics developed under concentric loading conditions were applied to eccentrically loaded columns (Lokuge et al. 2003; Saatcioglu et al. 1995). The results confirmed that the capacity predicted from the confinement characteristics developed under concentric loading conditions was adequate for evaluating the behaviour under eccentric loading conditions, particularly for low strength (56MPa or less) concrete columns. One of the assumptions of such investigation was that strength decay is essentially a function of confinement stress and does not vary with the strain gradient (Sargin et al. 1971). In addition, the investigation considered the fact that the stress-strain curves in the strips closer to the neutral axis may not be substantially different to the one that under fully confined conditions particularly in the preloading strain range (Saatcioglu et al. 1995). Moreover, it is important to note that the flexural cracks appearing on the column at the ultimate stage under the impact will minimise the stress differences in various layers across the section. All these observations lead to the conclusion that vulnerability assessments of the eccentrically loaded columns are independent of confinement characteristics resulting from strain gradient. Therefore the present vulnerability assessment techniques could be extended to the eccentrically loaded columns by assigning uniform confined compressive characteristics to the core section. 3.4.12 Numerical and experimental results for Mat_Brittle_Damage The aim of the validation process is to investigate the ability of the material model Mat_Brittle_Damage to simulate the dynamic response of the impacted columns. As an axial load is present on the column, the effects of axial stress propagation, axial shortening and shear deformation characteristics must also be reflected in the Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 3-92 simulation process. The results of the numerical simulation follows the displacement characteristics of the experimental results accurately up to t=60ms as shown in Figure 3.8. However, the residual deflection characteristics of the impacted column were not simulated by the numerical model where non-linear characteristics of the concrete play a vital role. This deviation reflects the error introduced by the use of a simplified resistance function for concrete elements which ignores the nonlinear behaviour of concrete especially near the ultimate stage. In fact, this material model implements only first order strain rate effects by using a viscosity parameter for concrete. However, the major deflection characteristics are well reflected by the model. This indicates that the second order strain rate effects can be exempted from the numerical simulations of the impacted columns. This is a reasonable assumption as it was observed that the average strain rate across the impacted column is less than 0.1s -1 . The reaction force generated at the interface also agrees reasonably well with the experimental results (see Fig. 3.9). As a whole, these results are an indication of the accuracy of the stiffness and axial inertia characteristics of the model as there is strong interaction between the axial and lateral inertia effects for the dynamic elastic-plastic deformation of a column with axially moving and stationary ends (Karagiozova and Jones, 1996). Figure 3.8: Comparison of the resultant deflections Figure 3.9: Interface forces during the impact 3.4.13 Comparison of numerical and experimental results for Mat_Concrete _Damage Time histories of the mid span deflection of the impacted column as predicted by the FE model is compared with the published experimental data as shown in Figure 3.10. The resultant maximum deflection and duration of the impact are well reflected in the Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 3-93 results. Hence the inertia and stiffness of the impacting bodies, boundary conditions and effects of confinement are accurately simulated in the numerical model. Also, it is important to note that the residual displacement of the column was reasonably approximated by the numerical results even after cracking and slight crushing of the concrete occurred simultaneously during the impact as shown in Figure 3.11. Local crushing is important as this can reduce available energy and consequently modify the post impact behaviour of the impacted column. In addition the result confirmed the fact that the stress-strain characteristic developed under concentric loading conditions can be assumed over the entire core section to simulate the flexural failure conditions of impacted columns made of low grade concrete. Figure 3.10: Comparison of the resultant deflections Figure 3.11: Crack Propagation of the impacted column and numerical simulation In Figure 3.12, the continuous line represents the contact force history obtained through the numerical simulation and the dotted line represents the one that was obtained from the experimental test. It can be seen that the differences between those two graphs are insignificant and the impact-time histories are almost identical. On the other hand, exact simulation of the peak contact force is evidence of the accurate Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 3-94 representation of the stiffness and boundary conditions as it would mainly depend on the inertia characteristics of the column for given boundary conditions. The comparison is quite good for both of these global parameters (Impact force and deflection). Therefore the model as well as the input data used for the simulation can be regarded as satisfactory. Figure 3.12: Comparison of the resultant Figure 3.13: Comparison of the resultant reactions impact force Some differences are observed in reaction forces generated by the numerical simulation compared to the experimental results (see Fig. 3.13). The sources of the error may be explained as below: In the experiment, the load cells were placed in between the bearing plates and the supports to measure the reaction. In the absence of descriptive details of the bearing system, the load cells were not modelled in the numerical simulation. However, the reaction forces are found to be very sensitive to the recording position (Zeinoddini 2008). Other possible reasons would be the inevitable error induced by the partial fixity of the supports due to the stationary steel sections and the filtering procedure used to extract the data from the data logger system (Feyerabend 1988). In spite of the differences in local peaks, the FE model prediction appears to be tracing the trend quite well. The FE model predicted crack pattern in the impacted columns and those observed in the experiment are presented in Figure 3.11. Tension cracks initiated at the bottom and top of the beam followed by the crushing of the material beneath the impacted zone. The simulation reproduced the tension cracks at the bottom while displaying dense tensile crack propagation at the top surface of the beam. The dense crack concentration occurred directly beneath the impacted zone with the partial separation of the material represented as a region with higher stress accumulation. Based on these factors it can Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 3-95 be concluded that the FE numerical simulation reasonably predicts the impact behaviour of the axially loaded column. On the whole, LS_DYNA material model Mat_Concrete_Damage_REL3 generated better results. It has been also proven that the default parameters generated by the software based on the unconfined compressive strength of concrete satisfy the majority of the well characterised tests results for 45.6MPa concrete (Schwer and Malvar 2005). In addition, many researchers have proven that this material model generates better results in dynamic simulations (Zhenguo and Yong 2009; Yonten et al. 2005; Bao and Li 2009). However, the model is still subject to the possible deviations that can occur with other concrete grades. To minimise the associated errors the parametric study will be limited to 30MPa to 50MPa concrete by avoiding High Strength Concrete (HSC). 3.5 Conclusions In fact, validated computational models can dramatically simplify the analysis process and greatly reduce the cost and time involving in physical testing. The underlying challenge in this method is the capability of the constitutive models of concrete to represent the realistic response of the columns under the impact loading conditions. Based on its distinctive features, reliability and capacity to simulate contact impact problems, the explicit finite element software LS-DYNA was selected for the numerical simulation process. Two of the most sophisticated material models present in the LS-DYNA library were compared and one of the material models was selected for parametric studies based on its capacity to generate the most accurate results. The main conclusions of this chapter are summarised below: 1. It is found that the numerical simulations of the column tests can be simplified greater by isolating the impacted column from the connecting structure and by assuming perfect bond between steel and concrete. However axial load must be applied separately on steel and concrete in order to maintain uniform strain distribution. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 3-96 2. A material model that can replicate tri-axial state of stress must be used in the impact simulation process. LS_DYNA material model Mat_Concrete_Damage_REL3 fulfils the requirement. However the mesh dependency of the fracture toughness has to be corrected by a single element analysis to enable simulation of tensile and shear failure conditions. In addition, contact algorithms, hourglass problems and initial penetration condition must be treated carefully in the simulation process. Conversely, strain hardening and the strain rate effects of the longitudinal steel may not alter the result significantly. Damping effects may also be exempted from the numerical simulation. 3. For impact simulation, it is suggested that the application of the stress-strain models developed under concentric loading conditions is valid under eccentric loading conditions particularly for low grade of concrete. 4. Comparison of the results shows that the both concrete material models are satisfactorily describes the behaviour of the impacted columns with some limitations under specific conditions. As the Mat_Concrete_Damage generated the most accurate results it will be used for the further analyses. 5. A better understanding of the impact behaviour of the column is reached with the supplementary information from the numerical simulation. An extended model generated from the validation process can be used for a parametric study of typical columns in multi-storey buildings which are highly susceptible to vehicle impacts. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 4-97 4. IMPACT RECONSTRUCTION AND PARAMETRIC STUDIES 4.1 Introduction Analyses of concrete columns under vehicle impact loads are limited in the literature. Therefore noticeable differences can be seen in the magnitude of the specified loads prescribed by different authorities for columns subjected to lateral impact loads (Vrouwenvelder 2000). Another reason for this difference is the complex behaviour of the vehicle column interaction (Louw et al. 1992). For instance, Reinschmidt et al. (1964) and Feyerabend (1988) conducted a series of tests by applying hard impact loads in their experiments whereas experimental studies on the structural behaviour of the rail and the road vehicles demonstrated that the colliding vehicle would exert soft impacts (Dodd and Scott 1984; Sutton and Lewis 1984; Varat et al. 2000). As far as soft impacts on concrete columns are concerned, there are several questions that remain unanswered, namely; (a). What is the overall dynamic strength enhancement factor of the impacted column compared to its static capacity? (b). What are the effects of the boundary conditions on the strength enhancement? (c). What are the influences of the parameters such as steel ratio, concrete grade, effective height on ultimate capacity (Louw et al. 1992)? According to Popp (1965) the ratio of dynamic to static loads at failure could be as high as 2.7 for a 3.8m high hinged column that is struck at 1.2m elevation by an 18t truck. However, this enhancement may not be representative of the entire impact loading particularly when the flexural shear is critical. Nevertheless, the DIF of the shear capacity always exceeds that of the moment. Also, it must be kept in mind that during a hard impact the high strain rates occur when the flexural failures are small, whereas during a soft impact the greater flexural strains are accompanied by high strain rates (Louw et al. 1992). Therefore selection of a realistic impact duration is mandatory in vehicle impact simulations, in addition to the elevation of the impact. Even though non-linear finite element analysis is a speedy process compared to Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 4-98 experimental testing, impact reconstruction is still a challenging task. Therefore, finite element methods have been directed towards the development of a proper simulation technique, which, in turn, enables researchers and designers to have better insight of vehicle impacts. To this end, an extensive parametric study based upon the finite element code LS-DYNA proves to be a useful tool and the findings can be easily summarised to develop a rational basis for the design of impacted columns. Consequently, a comprehensive study has been conducted to determine the appropriate shape of the impact pulses, the effect of the pulse characteristics, mass of the vehicle, strain rate, contact area and duration of the impact (or the stiffness). As a result, a universal technique which can be applied to determine the vulnerability of the impacted columns against collisions with new generation vehicles under the most common impact modes is proposed. Additionally, the observed failure characteristics of the impacted columns are explained using extended outcomes. Columns having circular sections have been comprehensively studied under low to medium velocity impacts. 4.2 Impact reconstruction by using crash test data Vehicle-column interaction plays a vital role in the impact simulation process. However, generation of realistic numerical models of vehicles for impact simulation is quite complex and difficult to obtain. On the other hand, specified methods (or models) used in the past are limited in their application to assess the vulnerability of columns to impacts from a general vehicle population (El Tawil et al. 2005; Tsang et al. 2005). The crash response depends on the mode of impact, rate sensitivity of the vehicle, dynamic crush characteristics, restitution, collision partner and vehicle specific parameters (Varat and Husher 2000). Therefore the force history of an impact at one particular velocity is different from vehicle to vehicle even though the mass remains the same. Consequently impact simulation with a rigid body or even with the simplified deformable body assumptions may not be adequate. For instance, a typical contact force history of a rigid body impacting a concrete column has two force pulses (see Fig. 3.12). The main pulse lasts only for about 6ms and this will highly exaggerate the strain rate effects. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 4-99 In fact, a linear relationship between the absorbed energy and residual deformation of the vehicle can be assumed under the deformable body assumption (Campbell 1974; Varat et al. 1994; Neptune 1999). Based on this assumption, the stiffness of the vehicle in frontal impact can be represented by a constant spring stiffness. This Campbell Model (Campbell 1974) was further improved by Prasad (1990) with the aid of repeated barrier impact testing. One assumption maintained during the reformulation is that a constant liner spring rate over the entire depth is applicable. However, available crash data indicates that vehicle frontal stiffness cannot be precisely modelled through the use of single linear springs for all vehicles, in particular when the crash propagates to the passenger compartment (Varat et al. 1994). On the other hand, the maximum resulting dynamic interaction force calculation based on the ‘Hard Impact’ assumption (EN 1991-1-7: 2006) is exceptionally sensitive to the equivalent elastic stiffness of the impacting object (i.e. the ratio between force F and total deformation) (EN 1991-1-7: 2006). Hence it is inappropriate to represent the impact response of a vehicle population having a unique mass but different stiffness, due to the inherent relationship between the force (amplitude) and the total deformation. Having observed the limitations of the existing methods, this research implements an impact pulse generated from a typical car to rigid barrier impact to reconstruct the vehicle collisions so that the generated results can be applicable to a general vehicle population over common modes of collisions. 4.2.1 Vehicle-Column Interaction Figure 4.1: Front and side views of an impacted column (NHTSA) Pressure transducers Location of the impact Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 4-100 The area of contact has some correlation with the mode of failure and the dissipation of kinetic energy through structural deformation. Hence, the vulnerability will depend on the distribution of the pressure over the contact interface. Euro code EN 1991-1-7 (2006) suggested that the contact area should be 0.25m high for car impacts. However, there is no guidance on the lateral distribution of the pressure across the section, particularly for circular columns. Figure 4.1 shows a rigid pole used to conduct several full scale impact tests on cars (NHTSA). It is evident that the effective contact area is around 25% of the perimeter if the frictional effects are neglected. Therefore, uniform normal pressure distribution is assumed across the 25% of the perimeter and the resultant lateral pressure distribution used in this study is shown in Figure 4.2. Figure 4.2: Lateral pressure distribution across the diameter of the 300mm column When the column diameter increases, the contact area also increases and contact pressure will reduce any particular impact. The resultant pressure distribution affects only the local damage. On the other hand, to account for the possible changes to the resultant impact duration, a method is proposed to calculate the peak forces under 50ms to 150ms impact durations. 4.2.2 Effects of Impact Pulse Parameters As impact pulse parameters play a vital role in the generation of a universal method to determine the vulnerability of columns under all modes of collision, it is intended to investigate the effects of pulses generated from full scale vehicles to rigid barrier impact in the impact reconstruction process. To investigate the shape of the various curves to simulate the collision pulses, force histories generated from several full scale frontal collision scenarios are studied. The ‘MATHLAB’ program can be used to Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 4-101 generate the ideal curve by using accelerometer data alone and hence derivation of mathematical model for different curves by using parameters such as velocity profile, frontal stiffness, residual deformation and restitution is not considered at this stage. Based on this method several force time histories derived by using accelerometer data presented in National Highway Traffic Safety Administration (NHTSA) are compared in the following paragraph. Figure 4.3: Force Time histories of full scale crash tests (NHTSA 1997) Comparisons of the Force -Time history of the Honda Accord, Ford Taurus & Renault Fuego are shown in Figure 4.3. Durations of the impacts were 100ms and triangular pulse shape is best fitted with the force history diagrams. This pulse shape has already been identified as a useful collision pulse model to simulate the frontal impact conditions (Breed et al. 1991). Sine, Haversine (Tsang et al. 2005; Campbell 1974), and square formats (Brach 1991; EN 1991-1-7: 2006; Vrouwenvelder 2000) are the other standard formats used widely to represent the frontal vehicle impacts. Comparative analysis revealed that the peak forces generated by the side impacts are less significant as the peak force is small and the impact duration is high and consequently the strain rate effects are also minimized. Numerical approaches to determine impact pulse parameters are rare in the literature as collision pulse characteristics depend on the mode of impact, collision partner and vehicle specific parameters. In fact, equations derived based on velocity profile, frontal stiffness, amount of deformation, and restitution may not be applicable to a general vehicle population (Varat and Husher 2000). Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 4-102 Table 4.1: Comparison of the force time history data with properties of the Impulse Vehicle Velocity v(ms -1 ) Mass m (kg) Impact Duration (s) (mv) Area of the Equivalent Triangle Percentage variation Ford Explorer 16.9 2242 0.150 37890 39573 1.04 Ford Taurus 15.6 1619 0.100 25274 27794 1.09 Renault Fuego 13.3 1329 0.100 17793 16687 0.94 Honda Accord 13.4 1329 0.100 17830 19500 1.09 A comparison of the area of the triangular impulse generated from the full scale impact tests (NHTSA 1997) with the product of mass and the velocity of the respective vehicles is given in Table 4.1. The force time history was obtained using the data generated by the accelerometer placed at the centre of gravity of the impacted vehicle. It is evident that the force pulse generated from the realistic impact events agree well with the product of mass and the velocity for the range of impact scenarios. This will ensure the accuracy of the test data and eliminate the uncertainty of the amplitude from the Force-Time history diagram for that particular velocity range which is specified differently by various authorities (Vrouwenvelder 2000). This will also enable accurate prediction of the vulnerability of the column for the respective vehicle impact. In fact, force time history data generated from vehicle impact with rigid barriers are always conservative for the vulnerability assessment of deformable bodies such as concrete columns. Therefore in the absence of realistic numerical models of vehicles, the vehicle column interaction can be conservatively regenerated by using the force time history diagram resulting from the rigid barrier-vehicle impact events. This observation is also effective in determining the basic pulse characteristic in the absence of precise numerical methods to quantify the collision pulses. Based on these results, duration of the typical impact pulses can be assumed to be 100ms on average as they belong to a group of vehicles with various masses, stiffness and velocities. On the other hand it is worth to note that frontal stiffness of a vehicle must be satisfied both passenger safety and better driving performance. Therefore the practical range of the frontal stiffness is cannot be changed dramatically. Therefore average impact duration of around 100ms may also be applicable to a new generation vehicles. These force pulses will also be used for comparison purposes once the collapse load of the column is determined. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 4-103 4.2.3 Impact pulse modelling and effects of the impact pulse parameters Effects of the pulse shapes are investigated in this subsection. Figure 4.4 shows the various pulses that lead to equivalent damage (or Iso-damage) conditions. Triangular (1.0875MN) and Haversine (1.080MN) failure amplitudes are almost equal and the square pulse with equivalent ramp up duration is the one that causes the same damage with minimum peak. The observed impact behaviour is more sensitive to the peak impact force than the associated impulse similar to quasi-static loading. It seems that the impact force is located close to the quasi static region. The strain rate sensitivity of the impact can be examined by reducing the slope of the iso-damage functions. Figure 4.4: Iso-damage pulses Figure 4.5: Effects of the strain rate or frontal stiffness When the slope of the ramp function is increased, the column fails at a lower amplitude (see Fig. 4.5) even though the triangular force pulses implied the opposite. In fact, Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 4-104 inertia effects and the strain rate effects are the main parameters that govern the failure as the damping and deformation characteristics have negligible effects on energy dissipation during the impact. The inertia effects are more predominant than the strain rate effects for rectangular pulses in the quasi-static region (see Fig. 4.5) and the opposite is true for triangular pulses. As the resultant variations are negligible, any of these characteristic curves for the vehicle impact generated force history space can be used to define a force-impulse diagram for the impacted columns when the structural damage is controlled by the shear capacity of the section. Therefore, in this research only the triangular pulse is used to reconstruct the vehicle impact due to its simplicity. The peak force has been varied until the columns reach near ultimate stage while keeping the duration constant in order to account for the various masses and velocities. A comprehensive discussion of the duration will be provided at a later stage. 4.2.4 Simulation of impact of axially loaded columns in medium rise buildings Impact capacity of typical columns of five to twenty storied buildings made of 50MPa concrete was investigated by using comprehensive numerical simulations. The columns support 6m spanning slabs in each direction, subject to 3kN/m 2 imposed (live) load identical to the design load capacity of an office building, classrooms or lecture theatres at each floor level (AS3600 2004). Vulnerability of a ground floor column was assessed for a typical frontal collision of light weight vehicles such as cars or vans. The structural design is based on the Australian Standard AS3600 (2004). The optimum column diameters are rounded to the nearest 50mm and consequently, to facilitate ease of comparison of various columns made of one particular concrete grade, the axial stresses on the columns are maintained constant, rather than the axial load. This has lead to smoother graphs at the later stage. In addition, two alternative design options with two different steel ratios were considered and Figure 4.6 represents the cross sections of the selected ground level columns supporting different number of stories. The design axial load capacity P d was calculated as, ( ) 6 . 0 85 . 0 ' × + = y s c c d f A A f P Eq. 4.1 Where ' c f is the compressive strength of concrete, A c is the net area of concrete, A s is the area of steel and f y is the yield strength of steel. Even though this analysis was Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 4-105 conducted by assuming 500MPa for longitudinal steel, the results can be extended to other steel grades by using the equivalent steel area method proposed by Shi et al. (2009). The spacing between the longitudinal steel bars was kept close to 100mm as much as possible even though it inevitably varied from 70mm to 110mm depending on the diameter and steel ratio of the columns. No further effort has been taken to minimise the distance between the bars as it violates the general procedures used in practice. Figure 4.6: Cross sectional areas of the circular Figure 4.7: Support conditions and external concrete columns loads applications As the entire structure did not have enough time to react under impulsive loading (El Tawil et al. 2005), fixed support conditions were assumed in this analysis as shown in Figure 4.7. The translations as well as rotations of all the nodes in all directions were restrained at the bottom. However, only the outer vertical faces of the column head were constrained against horizontal movements (ie. X and Y directions) in order to permit axial shortening during the impact. The restrained height was maintained equal to the diameter of the column. If this condition is satisfied, the support condition can be treated as fixed (BS8110 1985). Based on a convergence study, 25mm×25mm quadratic solid elements with one point integration were used for concrete while 25mm long beam elements with 2×2 Gauss integration were used for reinforcement. Hourglass control is mandatory with reduced integrated elements, and the method described by Flanagan and Belytschko (1981) was implemented in the hourglass Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 4-106 control. A detailed description about the selection can be found in (Schwer et al. 2005). Moreover, a 35mm cover was provided for the reinforcements and at least two cubic elements were used across the thickness. Figure 4.8: Plan view of the half models Quadratic solid elements were used at the core concrete zone (see Fig. 4.8) as the wedge element has caused stability problems under severe deformations including element interlocking. Obviously, the number of elements generated inside the rectangular segment is quite large and hence, stability of this model is high when it undergoes large deformations. These quadratic solid elements are the smallest elements in the entire model and the time step size has therefore been governed by the size of these elements. 4.3 Impact behaviour of the columns and possible damage modes Once a triangular pulse was applied, the displacement of the column increased simultaneously with the impulsive load until its peak is reached. Then the displacement was decreased with several minor peaks in the post peak region until the residual displacement was achieved. This behaviour accounted for the axial load acting on the column, which has developed second order bending effects. In an actual impact event there could be some contact losses due to the relative movements of the bodies in this region as the speed of the deformation of the column exceeds the velocity of the vehicle even though they are moving in the same direction. According to further investigations a rapid change of residual deflection occurred with a small increment of the impact force at a later stage. Consequently the permanent damage to the column can be identified by the continuous increment of the residual deflection without recovering. However, failure due to vehicle impact varies from the usual flexural type of failures during a mid span impact and hence, a conventional hypothesis based on the energy absorption capacity of the column may not be applicable as the energy absorption Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 4-107 characteristics mainly depend on the flexural deformation of the column. Since the column has not been subjected to flexural deformations, a small portion close to the impacted region has undergone highly localised stress and has absorbed an excessive amount of energy. This localised stress has exceeded the crushing stress of the concrete and hence may fail abruptly during the impact. This will considerably reduce the effective area of the column and the resultant eccentric axial load has finally diminished the axial load carrying capacity of the column. Under these circumstances, the column has failed due to shear failure initially and subsequently by flexural failure leading to collapse. The observed failure modes can be categorised as shear or shear flexural type of failures depending on the test variables as observed during the many simulations. Accordingly, the strain rate effects are more predominant in the vicinity of the impact and hence localised effects on the dynamic material characteristics could be significant. However the resultant overall capacity enhancement seems to be insignificant as the average strain rate was only marginal. For instance, the standard definition of the strain rate, t δ ε δ ε = & can also be expressed as L t l L ν δ δ ε & & = = 1 , where L is the original length of the element (column) and v& is the rate of deformation. This definition generates a much broader view of the strain rate effects while allowing the average strain rate to be calculated. According to the Feyerabend (1988) column test, the maximum axial deformation of the column is around 1.3mm and the duration of the impact is approximately 6ms. This leads to an average strain rate of approximately 5.4×10 -5 s -1 . As far as medium velocity vehicle impacts are concerned, the duration of the impact is around 100 ms and hence the average strain rate is well below 0.1s -1 . 4.4 Vulnerability prediction The analyses include 300mm to 500mm diameter axially loaded columns which are adequate in capacity for five to ten story buildings with two different steel ratios as shown in Figure 4.6. The critical impact pulses for each column are given in Figure 4.9 along with the collision force-time histories of some cars - Honda Accord, Ford Taurus & Renault Fuego which are in frontal impacts similar to the one that shown in Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 4-108 Figure 4.10. Calculations based on the areas under the curves show that the axially loaded five-story building column design for gravity loads cannot withstand impact velocities more than 15ms -1 or 40 km/h (see Fig. 4.9). This velocity range is common in urban areas and hence axially loaded column having diameters equal to or less than 340mm are likely to collapse under medium velocity car impacts. Though statement cannot be generalised, this method will allow prediction of the vulnerability of the impacted columns against new generation of vehicles for most common modes of impacts and hence provide a common base for comparison purposes. Figure 4.9: Comparison of impact capacities of columns with full scale crash tests (NHTSA) Figure 4.10: Honda Accord in a frontal collision at a speed of 48.3km/h (NHTSA) If the impact duration remains constant, the peak force of the tri-angular pulse will determine the severity of the impact due to the negligible effects of the slope of its legs (see Fig. 4.5). This observation implies that the peak forces can be interpolated to quantify severity of an impact. Based on this argument, it can be concluded that Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 4-109 collision severity can be predicted by interpolating a known collision pulse as the influence of the slope of triangular pulse (or the strain rate effect) is negligible. Comparison of the maximum impact capacities of impacted columns up to twenty-story buildings in terms of Force-Time history is given in Figure 4.11. According to the Figure 4.9, the maximum impact force applied by medium velocity car impact is around 675kN. Therefore it is evident that axially loaded columns made of 50MPa concrete in ten-story and above will not susceptible to collapse under medium velocity car impact. Figure 4.11: Ultimate capacities of columns Figure 4.12: Support reaction and Impact pulse Figure 4.12 represents the support reaction and the applied lateral impact pulse on the column during an impact simulation. The reaction forces are nearly equal to the applied impact pulse as inertia effects and damping effects are less pronounced under shear failure conditions. By definition, for ‘hard impact’ it is assumed that the structure is rigid and immovable and that the colliding object deformed linearly during the impact phase (EN 1991-1-7: 2006). Based on that assumption an equation was derived in EN 1991-1-7 (2006) to calculate the maximum resulting dynamic interaction force applied on the impacted column. However, the aforementioned difference actually represents the probable error that might occur in interaction force determination based on the undeformable impacted body assumption. Therefore this assumption may only be true for the shear failure predominant columns. Consequently the analysis based on this assumption may need amplification factor which can account for the inertia and damping effects under flexural failure predominant column. Therefore undeformable body assumption used in EN 1991-1-7 (2006) is conservative for flexural failures predominant columns. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 4-110 4.5 Bending moment and resultant shear due to impact Variation of the resultant bending moment due to impact was measured at three different locations along the height of the column in order to identify the impact behaviour of the column. Variation of the resultant bending moment due to impact can be measured using ‘SECFORC’ in LS-DYNA code (LS-DYNA 2007). The ‘SET’ option under the ‘SECFORC’ uses a set of nodes to define a cross section. Forces from the elements belonging to the node set are summed up to form the moment in this method and hence interconnected elements belonging to only one side of the nodes should be defined at one particular section. The measured moments at each cross section (CS) are given in Figure 4.13 with the relevant distances measured from the bottom. The highest moments are generated in the vicinity of the impact (0.975m) and close to the support (0.100m), but with opposite signs. The moment is gradually reduced away from the bottom support beyond the point of the impact. However another directional change can be seen close to the top support (3.9m) which signifies the generation of third order vibration mode in the impacted column. Consequently, excessive shear forces are generated at the contraflexure points located close to the supports. This observation may be cited as a potential reason for the failure of the column shown in Figure 4.14 which indicates typical shear critical situation. It is also important to note that laps forming in this region worsen the consequences. Thus, conventional design and detailing practices, which lead to impact damage, need modification. Laps should be avoided and transverse reinforcement should be provided close to the support where shear strength is vital for survival. Figure 4.13: Resultant bending moments Figure 4.14: Damaged column under vehicle impact Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 4-111 4.6 Effects of the diameter of the column, concrete grade and steel ratio The impact capacities of the columns supporting different number of storeys and the capacity variation with the grade of concrete are shown in Figure 4.15 along with two steel ratios. In fact, this plot is based on the assumption that one particular structure with certain number of stories constructed by using different concrete grades. In other words, variation of concrete grade and steel ratio will automatically change the diameter of the column to resist the constant axial loading. The concrete grade and steel ratio have profound effects on the impact capacity of the columns. For instance, the impact capacity of a column made of lower concrete grade is higher compacted to an equivalent (capacity) column made of higher concrete grade. Conversely, equivalent column with a high longitudinal steel ratio becomes more vulnerable to the impact loads. The diameter of the column has to be increased as the steel ratio decreases to maintain the same amount of axial load and the capacity enhancement is therefore partially due to the increase in the diameter of the column. Figure 4.15: Effects of the diameter of the column Figure 4.16: Effects of the concrete grade Comparative average increments of the impact capacities, with respect to the Grade 50MPa concrete columns with 4% steel, are given in Figure 4.16. The area in between the two curves shows the range of capacity increment that can be achieved by changing the longitudinal steel ratio and the concrete grade. The diagram clearly exhibits the influence of the steel ratio and concrete grade while emphasising the possibility of improving the impact capacity up to 2.4 times by selecting various combinations of concrete grades and longitudinal steel. The rate of capacity enhancement of columns with 1% steel towards the 30MPa concrete reflects the greater influence of the diameter of columns made of lower concrete grades and the Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 4-112 resultant shear capacity enhancement. Diameter of the column reduced with the increment of the steel ratio and hence the columns having higher steel ratios become increasingly vulnerable. In other words the dynamic impact capacity of columns mainly depends on the area of the concrete, and conversely the influence of longitudinal steel is not so significant compared to the behaviour under static loading conditions. This observation confirms that there is not enough time to develop extensive bond failure along the length to activate the steel under the impact conditions, particularly for circular columns. 4.7 Effects on the slenderness ratio Figure 4.17: Effect of the slenderness ratio Figure 4.18: 500mm column with 4% steel The effectiveness of slenderness ratio is investigated in Figure 4.17. As the slenderness ratio decreases, the failure plane will change its inclination. Consequently, the diagonal cracks are formed between the support and the point where maximum moment is reached followed by initial tension cracks (see Fig. 4.18). This change will increase the fracture energy dissipation through the cracked surface while increasing the number of effective links in preventing the crack propagation. Thus, noticeable 50% increase in the critical (collapsed) peak force can be achieved on average. Apart from that, improvements are increased with the columns having larger diameters (see Fig. 4.17). As far as the effect of boundary conditions are concerned, there is no significant contribution from the supporting conditions to resist the impact force. However, slight increments can be seen when the slenderness ratio is higher and the difference will be decreased with the enhancement of the diameter. 2 m Crack formation Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 4-113 4.8 Energy absorption due to the impact Energy absorption will primarily depend on the failure mode which in turn governed by the specific parameters such as slenderness ratio, steel ratio, concrete grade, contact interface parameters etc. However, energy consumption for plasticity and nonlinear deformation are comparatively low under impact conditions. Due to the localised damages and the micro-cracks, most of the energy dissipation will takes place closer to the contact interface. Therefore, fracture toughness of the concrete gained the highest consideration among the parameters that influence the energy dissipation of concrete rather that the tensile strength. Consequently, the energy absorbed by the concrete is comparatively low and the most of the internal energy is stored as strain energy in the longitudinal steel. 4.9 Effects of impact duration Figure 4.19: Equivalent impulse diagrams Figure 4.20: Iso-damage pulses for 600mm column Figure 4.19 exhibits the impact characteristics of a 600mm diameter column with 4% steel. The column was subjected to equivalent impulses (F∆t = mv = constant) with different durations (∆t) and the 100ms impact represents the critical pulse in which the column reached near failure conditions. In fact, these impulses replicate the cars with constant mass (m) and velocity (v) but with different frontal stiffness. Numerical results revealed that the column failed under 50ms impact yet remained unaffected under 150ms impact. This observation confirmed the importance of the frontal stiffness characteristics of the vehicle that govern the impact behaviour of columns. At the next stage, the peak forces of the triangular pulses are adjusted until the iso-damage (or equivalent damage) at near failure conditions are achieved as shown in Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 114 Figure 4.20. This proves that the column is highly sensitive to the amplitude of the impact and not to the associated impulse (Shi et al. 2008). Thus, the vehicle impact generated forces are theoretically closer to the quasi-static loading category where the response becomes somewhat insensitive to impulse (Shi et al. 2008). This comment is applicable to all the other column diameters varying from 300mm to 750mm. Another observation is that the impact pulses and resultant support reactions are nearly equal in terms of shape, duration and amplitude as shown in Figure 4.21. According to Baker et al. (1983), if the load and response functions terminate simultaneously, such a scenario can be included in the dynamic region which is between the impulsive and quasi-static regions. However, as the average strain rate was shown to be well below 0.1s -1 , it can be concluded that the strain rate effects are less pronounced in the vehicle impact analysis. Figure 4.21: Force vs reaction histories for pulses with different durations Figure 4.22: A typical Pressure impulse curve Figure 4.23: Iso-damage contours for impact By considering all these factors, it is suggested that the vehicle impacts belong to a specific domain in the pressure impulse diagram (see Fig. 4.22). To investigate this Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 4-115 hypothesis, force and the relative impulses that lead to the iso-damage conditions are plotted in a log scale for different column sizes (see Fig. 4.23). In fact, the curve reasonably agrees with the typical shape of the Pressure-Impulse diagram which is commonly adopted to predict the structural damage under blast loading conditions. Hence, the impact capacity (in terms of amplitude P, impulse I) can be expressed by the following mathematical expression which is valid between the dynamic and the quasi static regions. ( )( ) B c c c c I P A I I P P | ¹ | \ | + = − − 2 2 Eq. 4.2 In the above equation, P c and I c represent a known force and impulse pair located on the iso-damage curve, A and B are constants to be determined based on the shape of the iso-damage contours. P c and I c can be expressed in terms of the column diameter, concrete grade, steel ratio, slenderness ratio and hoop spacing. A parametric study and the derivation of equations for the impacted column based on this observation will be presented in the following chapters. Polynomial equations will be provided for more accurate estimations along with linear equations for approximate predictions. 4.10 Conclusions Numerical model of an axially loaded column subjected to transverse impact loads has been validated and extended to simulate the behaviour of vehicle impacted columns. This analysis has confirmed the feasibility of using numerical simulation techniques in vulnerability assessment of impacted columns while enhancing our understanding of the impact behaviour of columns under vehicular impacts. The main outcomes and findings of the investigations are summarised below. 1. Impact pulse generated from a typical car to rigid barrier impact is successfully used to reconstruct the vehicle collision. This method can be used as the foundation to generate a data base which can be used to determine the vulnerability of column against most common modes of collision. Additionally, the rigid barrier-vehicle impact data are conservative in the impact reconstruction due to the low deformation of the column under shear predominant conditions. 2. If the duration of impact and peak force remain identical, the effects of the shape of Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 4-116 the pulses are insignificant in vulnerability assessment. Strain rate also has a minor effect if the pulse characteristics belonging to the vehicle impact generated force history space. Inertia effects are more predominant than the strain rate effects for the rectangular pulses and the opposite is true for triangular pulses even though the comparative advantages are insignificant. Hence triangular pulses can be effectively used for impact simulations and the average duration can be taken as 100ms. In addition, collision severity can be predicted by interpolating a known collision pulse as the strain rate effect is negligible. 3. The generated results will be more reliable if the peak force occurs at a time of 25ms or more, which is derived by assuming 50ms triangular impact pulses with equal legs. If this condition can be satisfied for particular impact duration, the arrangement of the mechanical components of the impacting vehicle is immaterial as the effects of the shape of the impact pulses are insignificant. 4. The analysis also revealed typical columns adequate in capacity for five-storey buildings made of 30MPa to 50MPa concrete with 1% to 4% steel are vulnerable to medium velocity car impacts. However at the design stage, impact capacity of the columns can be increased by 20% by selecting the alternative design method with the low amount of steel. 5. Resulting bending moment revealed the generation of third mode of vibration in the impacted column and generation of maximum shear forces at contra flexure points close to the supports. Consequently, the impacted column may tend to fail close to the supports under shear critical conditions. Thus, laps should be avoided close to the supports and maximum transverse reinforcement should be provided in the vulnerable regions to avoid shear failures. 6. The impacts of columns treated herein are theoretically close to the quasi-static loading region where the response becomes less sensitive to impulse but more sensitive to the peak force. Iso-damage curves of the impacted columns will follow the shape of the conventional pressure impulse diagrams under blast loading conditions. Hence analytical equations can be derived to quantify the impact effects. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 5-117 5. CAPACITY OF THE AXIALLY LOADED COLUMNS UNDER LATERAL IMPACTS 5.1 Introduction Structural columns are seldom designed for vehicle impacts due to inadequacies of design guidelines. This chapter presents an extensive numerical study on the response of exposed concrete columns in multi-storey building to vehicle impacts with a focus on evaluating their impact capacity. Due to the low elevation impact conditions, most of the impacted columns failed in flexural shear mode. Localised stress concentration exceeded the compressive and tensile capacities of the concrete at the contact interface and diminished the strength of cover concrete by reducing the effective area of the column. Under this circumstance, the columns failed due to shear initially and flexure subsequently, causing collapse. Based on the above observations, it is expected that the parameters that govern flexural shear type failures under quasi-static loading conditions can be effectively used to mitigate the damage under impact conditions. A comparative analysis revealed that the flexural-shear failure conditions of the columns are well predicted by the AASHTO equations for bridge piers, ATC/MCTTER equations, equations by Ascheim and Moehle, and equations by Priestly et al. (Lee et al. 2003). In these equations, the total shear capacity of a column was calculated by adding together the contribution of the steel and concrete. It can be seen that concrete grade, steel area, diameter of the column, hoop spacing, area and yield strength of hoop are the key design parameters and therefore the effect of these parameters are further investigated in detail. As analyses have been conducted in the previous chapters by assuming nominal confined conditions, the effects of the confinement are yet to be determined. Another aspect of was to determine the vulnerability of the structural column during the construction process when the applied load is only a portion of the design load and hence the shear capacity and the stiffness have not reached their full potential (Abrams 1987; Zeinoddini et al. 2008). Proper impact damage assessment is vital to determine whether the column has to be replaced or can be repaired for further use. Design guidelines developed on partially loaded columns subjected to earthquake (Esmaeily Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 5-118 and Xiao 2004) and blast (Shi et al. 2008) loading may not be adequate in this circumstance where mode of failure, strain rate effects and inertia effects are substantially different. Moreover, a decision on the portion of total load that can be allowed during the rehabilitation process has to be made. Proper damage assessment will also minimise the risk to rescue workers and those who enter into the building following an impact, or when the affected bridge structure has to be used as a vital supply line. 5.2 Design against accidental loads As the low elevation impacts initiate the shear failure conditions, the limit states formulated based on the kinetic energy or deformation capacity (EN 1991-1-7: 2006) may not be adequate. In the absence of specific guidelines for accidental design, Accidental Limit State (ALS) design was considered as an alternate solution. In particular, the ALS mainly used for floating and fixed offshore structures made of steel which are subjected to Ship Collisions and Explosions (DNV-RP-C204). Possible extension of the same philosophy for the concrete columns is discussed here in brief. In the process, the inherent uncertainties associated with the frequency and magnitude of the accidental loads should be determined in advance. To demonstrate the concept of Accidental Limit State (ALS) a column with fixed geometric properties can be taken in to consideration. Then the Design resistance (R d ) is a constant and allowable design load (S d ) can be varied depending on the particular loading. The requirement to be fulfilled in the Accidental Limit States may be written as; d d R S ≤ , Eq : 5.1 where, f k d S S γ = = Design load effect, m k d R R γ = = Design resistance, S k is the characteristic load effect, γ f is the partial factor for loads, R k is the Characteristic resistance and γ m is the Material factor. For ALS design, the load and material factor should be taken as 1.0 (DNV-RP-C204). Thus it seems to be closely related to the serviceability limit state (SLS). However, it should be in conjunction with the factors involved in the safety evaluations (Moan 2009). For instance, substantial residual capacity should remain in the impacted columns to avoid damage to the supported structure or vital component to be used in rescue missions. Consequently, the structure Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 5-119 should be checked for all relevant limit states and in particular accidental limit state can be declared in-between the serviceability and ultimate limit states depending on the requirement. 5.3 Parametric studies of impacted columns 5.3.1 Finite element analysis of confined circular columns The effects of the confinement as a damage mitigation technique were investigated at the initial stage of the analysis. The simulation was conducted by assigning unconfined material characteristics to the cover concrete and confined material characteristics to the core concrete to account for the various confinement stresses due to the hoop spacing, hoop diameter and steel grades. The stress-strain model developed by Mander et al. (1988) was used in the simulation to account for the confinement effects. In particular, this equation expresses the ratio of the compressive strength of the confined concrete ' cc f to the compressive strength of unconfined concrete ' co f , for a section with equal effective lateral confining stress in each direction. 5.3.2 Effects of the confinement (a) Hoop distribution (b) Resultant strain variation in the column Figure 5.1: Effects of confinement under lateral impacts Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 5-120 According to the Liu and Foster (1998) early cover spalling in confined high strength concrete is caused by the tri-axial state of stress at the cover-core interface and high tensile strength is observed between cover-core interfaces under low axial loading conditioned. To provide an explanation for similar observations under impact loading conditions where cover concrete spall off quickly, the 12mm spirals are placed at 100mm spacing in 300mm column as shown in Fig. 5.1(a). This arrangement generates higher confinement stresses in the core concrete and hence the resultant stresses in core and cover concrete vary in all directions. The extensive lateral stresses generated at the core-cover interface has measured and it has exceeded the tensile strength of concrete during impact (see Fig. 5.1(b)). This is one of the reasons why concrete cover fails abruptly during impact. However there is no excessive stress difference under the serviceability conditions which represented by the stresses up to 20ms where the impact begins (see Fig. 5.2). The cover concrete in cracked sections did not contribute to the axial capacity of the concrete column at the residual stage. In fact, these sections are the weakest sections which govern the residual capacity of the concrete columns. However the stress difference between cover and core concrete gradually reduced towards the top support as the stress in the vertical direction did not excessively increase in those sections due to impact as shown in (see Fig. 5.1(b)). This observation also supports the argument that sufficient stress must apply on the core concrete in order to activate the confinement effects. Stress diffrence in lateral direction -6 -5 -4 -3 -2 -1 0 1 0 0.05 0.1 0.15 0.2 Time (s) S t r e s s ( N / m m 2 ) Stress diffrence Figure 5.2: Stress difference at the cover-core interface 5.4 Effectiveness of confinement under impact loads In general, transverse reinforcement confines the compressed concrete and prevents premature buckling of compressed longitudinal bars while acting as shear Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 5-121 reinforcement. One of the objectives of this investigation is to quantify the enhancement of the impact capacity due to the transverse reinforcement present in the vulnerable region of the impacted column. In particular, the effects of the hoop spacing s, diameter d and yield strength ' sy f are investigated in this section for columns made of 30MPa and 50MPa concrete. 6mm hoops with 350MPa yield strength at 250mm spacing are used as the datum of the analysis as this arrangement will not be effective for improving the confinement effects. The diameter of the hoops is varied from 6mm to 12mm while spacing is varied from 50mm to 250mm. In fact, Hwee and Rangan (1990) as well as many others are varied the lateral steel distribution within this range in their experiments. The maximum and minimum yield strength of the hoops are limited to 500MPa and 250MPa respectively. Confinement effects are particularly effective when the hoops are below 100mm spacing. It was observed that for the 350mm diameter columns made of 30MPa concrete, nearly 32% improvement can be achieved by reducing the hoop spacing to 100mm and it can be further improved up to 44% by reducing the spacing to 50mm with 6mm diameter bars. However this enhancement is limited to 10% for 900mm diameter columns due to the lack of confinement stress. This indicates that confinement has to be increased with the diameter of the column by a suitable means to achieve the same level of capacity enhancement. As far as the effectiveness of the individual parameters is concerned, the vulnerability of the column reduces as the hoop spacing decreases and as the hoop diameter and yield strength increase. In other words, individual or collective use of these parameters is possible to enhance the impact capacity of the columns. In particular, provision of hoops with a larger diameter is more effective for damage mitigation compared to reducing the hoop spacing for the range of values under consideration. This observation is effective when the hoop spacing cannot be reduced due to practical issues. However, the capacity enhancement due to the increment of yield strength of the hoops is negligible. In fact, yielding of the lateral reinforcement depends on the axial load, and hence provision of hoops having higher yield strength may not be effective under serviceability loading conditions (Janke et al. 2009). As a whole, the highest confinement stress is gained by the small diameter columns and hence their Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 5-122 effectiveness is more pronounced. ACI-ASCE Committee 441 (1997) recommended the ratio c sy s f f ' ' ρ , referred to as the confinement index, to evaluate the effectiveness of the confinement. However, it was revealed that the confinement index cannot capture the effects of yield strength of the confinement steel, particularly under P-∆ effect (Paultre et al. 2001). Consequently another confinement index, which can also account for the distribution of confinement steel and yield strength is introduced as, ' ' co l e f f I = Eq: 5.2 where ' l f is the effective lateral confining stress and ' co f is the unconfined strength of the concrete. A detailed description can be found in Mander et al. (1988). Paultre et al. (2001) also demonstrated that this method was very efficient in predicting the effectiveness of yield strength of lateral steel. Impact capacity increment vs f' l /f' co 1.0 1.1 1.2 1.3 1.4 1.5 1.6 0.00 0.10 0.20 0.30 f' l /f' co C a p a c i t y i n c r e m e n t 350mm; G30 500mm; G30 600mm; G30 700mm; G30 900mm; G30 Figure 5.3: Capacity enhancement for 30MPa concrete The capacity enhancements with 30MPa and 50MPa concrete are shown in Figure 5.3 and 5.4 respectively for the range of parameters under consideration. It is evident that columns made of a lower grade of concretes gain the highest capacity enhancement due to confinement compared to those with a higher concrete grade. Flexural shear failure characteristics of columns made with 30MPa concrete leads to an additional capacity enhancement as the column diameter reduces. This is because flexural characteristics become more predominant as the column diameter decreases. However, almost all the columns made of 50MPa concrete show shear failure characteristics where the impact capacity is based entirely on shear strength of the columns. Hence, Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 5-123 the capacity increases proportionately with the diameter of the column and hence there is no proportional change of the capacity increment with the diameter of the column as shown in Figure 5.4. Consequently the shear capacity increment is responsible for observed behaviour of the columns with larger diameters while tensile strength increment is responsible for the capacity improvement in the 300mm column. Impact capacity increment vs f' l /f' co 1.00 1.05 1.10 1.15 1.20 1.25 1.30 0.00 0.05 0.10 0.15 0.20 f' l /f' co C a p a c i t y i n c r e m e n t 300mm; G50 600mm; G50 750mm; G50 800mm; G50 Figure 5.4: Capacity enhancement for 50MPa Concrete As the effectiveness of the confining stress is further reduced with the increase in concrete grade, the columns made of higher grade concrete need an even higher confining stress compared to those with lower grade concrete. For instance, improvements due to the reduction of hoops spacing down to 100mm is not effective for 50MPa concrete and the increment is nearly 4%. Under these circumstances the hoop spacing needed to achieve the required level of confinement under impact loading conditions may be highly underestimated by the general provisions of the codes (BS8110 1985; AS3600 2004), which are based on the maximum diameter of the longitudinal steel. Moreover, diameter of the hoops also has a profound effect on the impact capacity of the columns. Impact capacity can be improved by 12% and 22% on average by increasing the diameter of hoops from 6mm to 8mm and 12mm respectively. Consequently overall 35% improvement can be made if the 12mm hoops with 50mm spacing are selected for 300mm columns. On the other hand yield strength of the lateral reinforcement has a similar effect. The investigation is conducted by assuming hoops having 350N/mm 2 yield strength and it is observes that almost 10% increment of the impact capacity can be achieved with 500MPa steel. Hence the final capacity Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 5-124 can be increased up to 49% if 12mm bars with 500MPa placed at 50mm spacing. Thus the diameters of the hoops and the hoop spacing have the most profound effects on enhancement of the impact capacity of the columns made of higher grade concrete. These conclusions were made based on the assumption that hoop spacing is constant throughout the column. 5.5 Effects of the unconfined cover and use of external wrapping (a) 30MPa concrete (b) 50MPa concrete Figure 5.5: Confined strength for different concrete grades In general, compressive strength increment due to the confinement effects can be expressed in term of concrete grade as the hoop spacing, diameter of the hoops and steel grade will finally contribute to increase the compressive strength of the core concrete. A point on graphs in Figures 5.5(a) & 5.5(b) represents the improved grade of core concrete that can be achieved by changing the confinement effect alone. It can be observed that the rate of the change of enhancement is more pronounced for columns with larger diameter even though the range of variation of the grade of concrete not as much as widespread as for smaller diameter columns. These graphs can also be used to estimate the impact capacities of the columns at an intermediate combination which is not covered by the selected spacing and bar diameters in the present analyses. The 600mm diameter column is common for both the concrete grades and comparison of the impact capacity at a particular concrete grade would indicates the strength reduction caused by the unconfined concrete properties assigned to the cover concrete. For instance, comparison of the impact capacities for confined compressive strength of 50MPa concrete indicates that the cover concrete can enhance the impact capacity around 13%. The contribution of the cover concrete may increase Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 5-125 around 20% for confined compressive strength of 60MPa concrete. This means that there is a potential for external wrapping can be used to enhance the impact capacity of the confined concrete columns significantly where full cross section experiences a tri-axial state of stresses. 5.6 Effects of the slenderness ratio on capacity enhancement Figure 4.17 of the Chapter 4 shows the impact capacity (in terms of peak force) for the columns made of 50MPa concrete with 4% steel and nominal hoop spacing. As the slenderness ratio decreases, the shear failure plane increases its inclination. Consequently, this change will increase the fracture energy dissipation through the cracked surface while increasing the number of effective hoops in preventing crack propagation. Hence, the investigation continued to examine the impact behaviour of columns with 50mm hoops spacing and 6mm bars. Peak force vs Slenderness ratio 0 1 2 3 4 5 6 7 3 6 9 12 15 Slenderness ratio (L/D) F o r c e ( M N ) 300mm Col 450mm Col 500mm Col. 600mm Col. 300mm Col. Uncon. 450mm Col. Uncon. 500mm Col. Uncon. 600mm Col. Uncon. Figure 5.6: Columns confined with 12mm links at 100mm spacing Capacity enhancement due to the confinement effect is shown in Figure 5.6 along with the capacities of the columns with nominal confinement. The enhancement is more predominant in larger diameter columns and the average increment of 38% is observed except for the 300mm column which shows a lesser increment of about 8 to 20% depending on the effective height. Therefore, the confinement effects may not provide substantial extra capacity particularly for the 300mm diameter short columns. On the other hand, 450mm and 500mm diameter columns changed the mode of failure from flexural-shear to flexure, even for low slenderness ratios, by enhancing the capacity due to the confinement. However, the failure mode of 600mm column remains Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 5-126 unchanged and hence 500mm is the limiting diameter which changes the response due to the confinement effects. 5.7 Comparison of the dynamic and static shear capacities Dynamic shear capacity enhancement 1.5 2.0 2.5 3.0 3.5 4.0 4.5 0.30 0.40 0.50 0.60 0.70 0.80 Diameter (mm) V d / φ V s G30 100c/c G30 50c/c G50 100c/c G50 50c/c Figure 5.7: Comparison of the dynamic and static shear capacities The shear capacity enhancement under dynamic loading condition is compared in Figure 5.7. The design static shear capacity (φ'V s ) calculated by using AASHTO specification for circular bridge piers where the strength reduction factor φ' can be taken as 0.85 (AASHTO-LRFD 2002). Fully loaded columns of 30MPa and 50MPa concrete with 50mm and 100mm hoop spacing were selected in the analyses. The dynamic shear capacity (V d ) was equal to the peak force (collapse) of a 100ms triangular impulse. It can be observed that the dynamic to static shear capacity ratio varies in between 1.6 and 4.2 depending on the diameter and concrete grade of the columns. This clearly indicates that the static shear capacity is not an indication of the maximum allowable (peak) force during an impact, even though the peak force has some correlation with the static shear capacity of the column. If the correlation factor is known, the static shear capacity can be used for approximate vulnerability assessment. According to these results the dynamic amplification factor ϕ dyn (varies from 1.0 to 2.0) given in EN 1991-1-7:2006 highly under estimate the dynamic impact capacity of the larger diameter columns. In general, the columns made of lower grade concrete have the highest dynamic capacity enhancement due to their flexural-shear failure characteristics. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 5-127 5.8 Impact capacity of partially loaded circular columns 5.8.1 Introduction Axial load is one of the key design parameter in the analysis and substantial discussion is therefore provided on impact behaviour of partially loaded columns. Comprehensive understanding of the status of the damage to an impacted column is vital for prevention of progressive collapse of the supporting structure as well as to determine whether the column has to be replaced or can be repaired for further use. Moreover, a decision on the portion of total load that can be allowed during the rehabilitation process has to be made. Proper damage assessment will also minimise the risk to rescue workers and those who enter into the building following an impact, or when the affected bridge structure has to be used as a vital supply line. Each of these decisions has to be made based on the residual capacity of the column after an impact. Current knowledge on damage assessment is incomplete and major decisions are made based on personal experience. Figure 5.8: Rehabilitation of a bridge after catastrophic failure of a column It was evident that the behaviour of reinforced columns under simultaneous axial loading and lateral impact has not been given significant consideration in the literature. In fact, there is very limited data on the dynamic failure of pre-loaded concrete columns subjected to lateral impacts. Bao and Li (2009) and Shi et al. (2008) have studied the residual strength of concrete columns under blast loading. The studies emphasised the importance of pre-loading when the effects of the damage is determined. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 5-128 To bridge the gap, a numerical analysis was conducted on partially loaded concrete columns. This analysis included 300mm to 800mm columns which are adequate in capacity for five to twenty storied buildings. Two steel ratios were considered in the analysis and the effects of the hoop spacing, slenderness ratio, support fixity and load eccentricity were extensively investigated in the parametric studies. The finite element analysis continued to find out the residual capacity of the columns under arbitrary impact velocities and was extended to investigate the residual capacity of the columns under specific velocity conditions. The damage criteria used by Shi et al. (2008) was implemented for damage assessment. 5.8.2 Damage criterion According to the requirements it is clear that the selected damage criteria for the impacted columns should express the residual capacity of the impacted columns in terms of its design load capacity. Therefore the damage index D is defined as; d r i P P D − =1 , Eq: 5.3 where P r is the residual axial load carrying capacity of the damaged RC column and P d is the design capacity of the circular concrete columns according to the Australian standards. As will be seen later, P r is the ultimate on-factored residual axial load carrying capacity. The degree of capacity degradation was defined as follows (Shi et al. 2008): ( ) . ) 0 . 1 8 . 0 ( , ) 8 . 0 5 . 0 ( , ) 5 . 0 2 . 0 ( , 2 . 0 0 collapse D damage high D damage medium D damage low D i i i i − = − = − = − = There is no comprehensive definition given for each term. However the physical meaning is that the supporting building is increasingly at high risk at each stage and the column has to be replaced when the damage index is in the 0.8 - 1.0 range. The advantage of this index is none of the commonly used damage criteria, such as residual deflection, maximum stress and strain conditions, satisfy the above requirements. On the other hand, the axial load capacity degradation will reflect the shear damage, flexural damage or local damage conditions due to impact while expressing the global Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 5-129 behaviour of the impact, and these factors are easily obtain from numerical simulation techniques. 5.8.3 Effect of axial load on the duration of the impact It is worth identifying the effects of the varying axial loading on the duration of the impact, in a realistic vehicle collision, even though the impact simulation was conducted in this thesis by applying an equivalent impact pulse of constant duration at a specific height. The duration of the impact may depend on the percentage of the axial load on the column as axial compression can influence the strength, stiffness, and deformation capacity of reinforced concrete columns. In addition, it has long been known that axial force has significant effects on stiffness (Zeinoddini, et al. 2002), flexural strength and behaviour of reinforced concrete columns (Abrams 1987). For instance, the rebound velocity of the striker decreases as the axial load on the column increases, particularly for mid span impacts (Zeinoddini et al. 2002). It was also observed that the first natural frequency decreases with an increase in the level of axial pre-loading due to a decrease of the flexural stiffness. On the other hand, the occurrence of plastic deformation under mid span impacts causes the impact duration to be further extended. Moreover, reduction of the time lag between successive impacts caused by the impacting object as it rebounds can be observed with the increase of the axial load level. However, the effect of the percentage of axial loading on the duration could be negligible under low elevation vehicle impacts where the damage mode is predominantly shear. On the other hand, the consequences due to the varying impact duration with the percentage of axial loading are not very significant as the peak force is the governing factor of the vulnerability under the vehicle impact. Therefore the duration of the impact is kept constant in further analyses. 5.8.4 Simplified method to investigate the residual capacity of columns The investigation conducted by reducing the axial load and then restoring the load at the post impact stage in consecutive steps. In fact, this will simulate the axial loads applied during construction and at the serviceability stage on respective columns. The initial dead weight on the column is considered as 20% of the design load in this study. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 5-130 However it has been shown that columns behave in a ductile manner if the axial load is less than 20% of the axial load capacity, particularly for HSC (Li et al. 1994; Patrick et al. 2009). Here axial load capacity was measured as ' c g f A , where A g is the gross concrete area and ' c f is the compressive strength of concrete. Two other initial loading cases, namely 0.6P d and 0.4P d are considered in this analysis. The 60% axial load will ensure that failure will occur by crushing the concrete above the balanced point while 40% load will tend to initiate flexural-shear failure conditions. Staged ramp up loading 0 5 10 15 20 25 30 35 0 0.1 0.2 0.3 0.4 0.5 Time (s) A x i a l p r e s s u r e ( M P a ) Figure 5.9: Axial pressure application on 300mm diameter column The initial axial load is applied gradually as a ramped up pressure on cross section and then the impact load is applied over the front surface of the column, which takes approximately 120ms as shown in Figure 5.9. Due to the close proximity of the impact pulse to a quasi-static load, the column gained its residual state just after the impact. From this time onwards a slight fluctuation of the residual deformation of the column was seen even though its consequences on the post impact response were negligible. Thus the axial load is applied gradually on the column in steps as separate ramped up functions following the impact. In particular, this method avoided the explicit-implicit transformation required in similar analyses (Shi et al. 2008). At each stage, the capacity of the column to resist the respective axial load increments is investigated. If the column can withstand several consecutive axial loading steps at the post impact stage, it is an indication of considerable axial load capacity remaining in the column. In fact, this indicates a low damage state. Thus, the analysis was conducted by increasing the impact load gradually in steps while increasing the axial load at the post Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 5-131 impact stage. For example, the 20% loaded 450mm column completely collapsed under a 1.75 MN impact (D i =0.8), and was partly damaged under a 1.7 MN impact (D i =0.2). This is illustrated in Figure 5.10. As the difference has not much practical significance the damage index D i , would not be sensitive as far as impacted columns are concerned. D i =0.2 represents low damage conditions and D i =0.8 represents a near collapse stage (Shi et al. 2008). The column failed under 1.7MN in flexure and 1.75 MN in shear critical conditions. Therefore the local failure due to impact caused the collapse and the difference between the two critical impulses represents the energy absorbed that causes local damage to the concrete. This damage reduces the effective area of the column at the contact interface. Deformation due to the post axial load -0.45 -0.30 -0.15 0.00 0.15 0.30 0.05 0.15 0.25 0.35 0.45 Time (s) D e f l e c t i o n ( m ) D=0.2 ; 1687kN D=0.8 ; 1750kN D=0.4 ; 1725kN D=1.0 ; 1812kN Figure 5.10: Deflection characteristics of 450mm column Even though the 400mm column with 20% axial load clearly demonstrated the various stages of failure under the post impact loading, the 600mm and 750mm columns immediately collapsed and hence the critical impulses for each damage index were hardly noticeable. Therefore, if the columns are damaged due to impacts with small deflections reflecting shear failure, the only option is to replace the damaged column as it may catastrophically fail under further loading. Consequently only the mean failure impact force is used for comparison purposes under 20% to 60% axial loading. In general, the impact capacity of the columns reduced by 10%, 20% and 30% under the 60%, 40% and 20% axial loads respectively. This is evidence for the improved capacity of the impacted column with the axial load, due to its improved stiffness as shown in Figure 5.11. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 5-132 Axial load sensitivity of impacted columns 0 1 2 3 4 5 6 0.3 0.45 0.6 0.75 Diameter (m) P e a k f o r c e ( M N ) P=100% P=60% P=40% P=20% Figure 5.11: Axial load sensitivity of impacted columns However, it was evident that flexural ductility reduces significantly in flexure dominated columns with the increase in compressive axial load (Paultre et al. 2001; Zeinoddini et al. 2002; Gopalaratnam et al. 1984). As the axial load increases, the concrete is subject to higher compressive stress levels, such that the moment capacity of the column depends mainly on the compressive strength of the concrete. Thus the impact capacity has decreased with the axial load enhancement in flexure dominated columns. Conversely the shear capacity of the columns is increased with the enhanced axial load (see Fig. 5.11). In other words, the compressive stress enhancement will reduce the lateral impact capacity of flexure dominated columns while in shear dominated columns the opposite occurs. Either of these two factors can govern the impact capacity of the column depending on the elevation of the impact. For instance, shear failure will be initiated by a low elevation impact and hence the impact capacity will be improved with the axial load increment as a result of shear capacity enhancement. Since concrete is brittle in nature, flexural strength reduces rapidly after reaching the maximum moment capacity of the column. Moreover, the depth to the neutral axis increases as the axial load level increases and the extreme concrete fibre is subjected to higher compressive strains conditions under the lateral impact. Under these circumstances the concrete will reach its ultimate strain sooner and as a result, the concrete cover will spall off rather quickly, causing a decline in the flexural capacity of the section. Thus the moderately damaged column fails in flexure under the axial load at the post impact stage. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 5-133 When concrete strength increases, the amount of transverse reinforcement has to be increased to acquire the same level of ductility if the axial load level remains the same (Paultre et al. 2001). If sufficient transverse reinforcement is provided, the reduction in flexural capacity can be compensated by increasing the capacity of the concrete core, such that the concrete core could dilate properly under large compressive axial loads (Johnny 2003). However if the axial load is further increased, the same ductility level has to be achieved by increasing the amount of transverse reinforcement due to premature dilation of the core concrete. This will result in congested transverse reinforcement and using lateral steel with a high yield strength is suggested to overcome the problem. This may not necessarily increase the ductility when the lateral steel ratio remains constant (Azizinamini 1994) since the lateral expansion of the concrete core is not greater and the tensile capacity of the steel may not be fully developed under working conditions (Johnny 2003). Therefore the parametric study was extended to investigate the effect of the yield strength of transverse reinforcement, hoop diameter and spacing under varying axial loading conditions. 5.8.5 Effects of transverse reinforcement on capacity enhancement To investigate the influence of the transverse steel characteristics the hoops spacing, diameter and yield strength were varied while the configuration was kept constant. The hoop spacing varied from 50 to 250mm and an initial investigation was performed using 6mm diameter hoops with 350MPa yield strength. The 250mm spacing generated the nominal confinement conditions while 50mm allowed the behaviour of the impacted column to be investigated under enhanced ductile conditions. 5.8.5.1 Effects of hoops spacing on partially loaded columns According to the observations there is a substantial improvement of the impact capacity of the 450mm column under 20% loading. The formation of plastic hinges near the top and bottom supports of the column is the reason behind this observation and the numerical simulation was extended to check whether the desired ductile behaviour could be generated in the rest of the columns by increasing the transverse reinforcement. In general, the ductile capacity depends on the amount and distribution of transverse reinforcement within the plastic hinge region and this concept is particularly effective under earthquake loading conditions. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 5-134 Displacement characteristics of the column 15 20 25 30 35 0.2 0.4 0.6 0.8 1 Axial load ratio D i s p l a c e m e n t ( m m ) 300mm Col.; L250 300mm Col.; L50 Figure 5.12: Deformation characteristics of 300mm column As observed in the present study, the ductile behaviour can not be improved to a greater extent in the impacted columns by providing hoops at closer intervals. Behaviour of the 450mm column can be considered as an exceptional case where the ductile behaviour of the column was further improved by the closer hoop spacing. As far as the overall behaviour of the impacted columns is concerned, there is no plastic hinge formation of the columns under low elevation impacts. In this circumstance, the effects of transverse reinforcement are limited to enhance the shear capacity of the columns and to provide more confinement to the core concrete. Such enhancement is still desired for the impacted columns as they need to bear a certain amount of axial load despite the damage caused by the impact. In contrast, the ductility is slightly improved when the axial load is enhanced. This is the desired behaviour for shear critical columns and concurs with the hypothesis that dilation of concrete will be further improved with the enhancement of the axial load. This argument is confirmed by the fact that the displacement characteristics of the fully loaded column are improved by the closer transverse steel spacing (see Fig. 5.12). For instance, 65% improvement of the ultimate deflection is observed under confined conditions compared to only 15% improvement under the nominal conditions with varying axial load. A 300mm column is considered for comparison purposes as it magnified the deflection characteristics. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 5-135 Impact capacity of the 300mm column 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.2 0.4 0.6 0.8 1 Axial load ratio P e a k f o r c e ( k N ) 300mm Col.; 4% steel; L250 300mm Col.; 4% Steel; L50 Figure 5.13: Impact capacity of 300mm column under varying axial loads In fact, both the ductility and shear capacity improvement contribute to the impact capacity enhancement of the 300mm column (see Fig. 5.12) and consequently the capacity improves by 10% with the confinement (see Fig. 5.13). The contribution of the displacement (via tensile strength activation) may be predominant with the axial load enhancement as the deflection of the column increases. This will reflect the secondary moments generated by the eccentric axial loads and in practice the deflection characteristics may be even higher as concrete can dilate properly under high axial loading conditions and resist higher compressive stresses generated by flexure before failure. However, the capacity improvement due to the enhanced displacement is insignificant and consequently the two lines in Figure 5.13 are almost parallel. Thus the confinement characteristics have improved the ductility of the column, while the shear capacity is the main factor that contributes to the impact capacity enhancement. These comments are applicable to all the other columns except the 450mm diameter column. 5.8.5.2 Role of hoops spacing as a early warning system prior to collapse Figure 5.14 compares the impact capacity increment of the columns with the hoops spacing. Columns are axially loaded 20% to 100% of their design capacities. In general the percentage reduction of the impact capacity is 10%, 20% and 30% when the axial load decreases by 40%, 60% and 80% respectively. There is a series of critical impulses where the 300mm and 450mm confined columns behave similar to the slightly damaged and highly damaged conditions according to the intensity of the Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 5-136 pulses. However there is no significance improvement of the critical range so that each column can exhibit some warning signs at the post loading stage before collapse. In other words, collapse of the impacted columns will be brittle and sudden upon post impact loading even though the residual deflections of the columns are insignificant. The condition remains unchanged even after introducing hoops at closer intervals even though it contributes to the overall capacity enhancement. As far as all the confined columns are concerned there is a trend to enhance the capacity with the enhancement of axial loading although the percentage increment randomly varies from 10% to 18%. Figure 5.14: Enhanced capacities for confined columns with different axial loading In fact, typical structural columns may not be subjected to substantial shear forces during their serviceability state and hence their full shear capacity is available to supply the demand. It is also worth to note the probability of catastrophic shear failures is increase with the intensity of the vertical loading. However compromise between axial load intensity and the allowable impact load may not be worth as there is no residual capacity remains in the impacted columns. 5.8.5.3 Effects of hoop diameter and yield strength on capacity enhancement The effects of the transverse reinforcement on columns with reduced axial load were comprehensively investigated. The aim is to check whether the capacity reduction can be compensated by shear strength enhancement, by increasing the diameter or yield stress of the transverse reinforcement as opposed to reducing the spacing. If the Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 5-137 transverse reinforcements are effective, the impact capacity will suddenly change when the shear capacity exceeds the flexural capacity of the columns or vice versa. Impact capacity vs f' l /f' co 0.0 1.5 3.0 4.5 6.0 7.5 0.00 0.05 0.10 0.15 0.20 f' l /f' co I m p a c t c a p a c i t y ( M N ) P=1.0; 300mm P=1.0; 450mm P=1.0; 600mm P=1.0; 750mm P=0.6; 300mm P=0.6; 450mm P=0.6; 600mm P=0.6; 750mm P=0.4; 300mm P=0.4; 450mm P=0.4; 600mm P=0.4; 750mm P=0.2; 300mm P=0.2; 450mm P=0.2; 600mm Figure 5.15: Capacity reduction due to axial load A range of possible reinforcement ratios are achieved by changing the diameter from 6 to 12mm and the yield strength from 250MPa to 500MPa, with the hoop spacing changing from 50 to 250mm similar to the earlier analyses. However, further improvement of the ductility is not possible and all the results are summarised in Figure 5.15. As mentioned in the previous paragraph, the small improvement of the ductility is not effective for enhancing the capacity of the columns. Thus the shear capacity governed the failure throughout the analyses. Therefore, it can be concluded that the transverse reinforcement does not have a significant influence on the failure mode of the columns under impact loading, particularly when the axial load ratio exceeds 20%. Impact capacity improvement vs f' l /f' co 0.65 0.75 0.85 0.95 1.05 1.15 1.25 1.35 0.00 0.05 0.10 0.15 0.20 f' l /f' co C a p a c i t y i n c r e m e n t P=1.0; 300mm P=1.0; 600mm P=1.0; 750mm P=0.6; 300mm P=0.6; 450mm P=0.6; 750mm P=0.4; 300mm P=0.4; 450mm P=0.4; 750mm P=0.2; 300mm P=0.2; 600mm P=0.2; 750mm Figure 5.16: Capacity enhancement due to confinement Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 5-138 The impact capacity increment due to the reduction of the axial load and confinement is given in Figure 5.16. The results are compared with fully loaded columns having 6mm hoops with 350MPa at 100mm spacing. As the axial load reduces, the impact capacity decreases and the maximum capacity drop is around 65% under 20% axial load, as designated by the bottom line of Figure 5.16. With confinement, the impact capacity increases and up to a minimum of 75% of the capacity drop can be recovered by providing 12mm bars with 500MPa at 100mm spacing. If the axial load on the column can be maintained around 40%, then 90% of the impact capacity drop can be recovered as shown in the same figure. 5.8.6 Effects of longitudinal reinforcement ratio Equivalent columns in capacity for 5 to 15 storey buildings made of 50MPa concrete with 1% and 4% steel ratios are simulated in this study to investigate the influence of the steel ratio on the impact behaviour of columns under varying axial loading conditions. Axial load varied from 20% (0.2P d ) to 100% (P d ) in the columns with nominal hoops spacing as shown in Figure 5.17. In general, the vulnerability reduces with reduction of longitudinal steel ratio and capacity enhancement is more pronounced in fully loaded 5 storey column. However there is no considerable change in the ductility due to steel ratio even for 5 storey building columns and hence there is no abrupt change in the capacity enhancement except in 450mm column in which the exceptional ductile behaviour occurs as discussed previously in detail. However 350mm column shows some brittle behaviour with the axial load increment and hence the entire shear capacity of the column is fully utilised. Therefore highest increment of around 84% is observed in 5 storey building columns under fully loaded conditions while 15 storey columns shows only 43% increment (see Fig. 5.18). Due to abrupt behaviour of the 450mm column, 10 storey building columns excluded from the comparison. As far as 15 storey columns are concerned, ductile behaviour of the columns do not affected by either steel ratio or axial load ratio on the columns. Therefore capacity increment is remained constant throughout the analyses. In addition, the range of impulses that can causes the damage to the columns designated by the parameter D i , varying from minor (D i =0.2) to severe (D i =0.8) conditions is further shrink with the reduction of the steel ratio. In fact this is a sign of flexural Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 5-139 capacity reduction with the reduction of longitudinal steel ratio even though the overall impact capacity is enhanced. Therefore columns with low steel ratio fail abruptly without giving any warnings prior to collapse. Figure 5.17: Impact capacity under varying Figure 5.18: Capacity increment under axial load varying axial load 5.8.7 Effects of the slenderness ratio Figure 5.19: Impact capacities of short columns In this analyses columns with diameters 350, 500 and 700mm with 1% steel are investigated. Transverse reinforcement is provided at 250mm spacing and 6mm bars with 350MPa yield strength was used in the process. The main objective is to investigate the impact behaviour of short columns under low axial loading conditions and effective height is varied from 2m to 4m range. The results are shown in Figure 5.19. Because of the various diameters are in use the slenderness ratio provides a common base for comparison purposes. Overall capacity of the columns increases with the reduction of the effective height as expected. According to the observations Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 5-140 higher capacity enhancement is noticed in small columns. On average it is 10% to 15% for 700mm column and 350mm column respectively when the effective height is decrease to 3m from 4m height. The enhancement is further increased 35% to 50% with reduction of the effective height to 2m. However there is no firm evidence for interrelationship between the reduced axial load, diameter and the percentage of capacity enhancement. Moreover, mode of failure sticks to the shear type of failures under 2 to 4m effective height despite the diameter of the column and the percentage of the axial loading. Thus the columns fail due to bulging at the impacted point as the volumetric strain exceed the elastic limit under the application of post axial loads and this behaviour is remained unchanged for a range of impulses ranging from D i =0.2 to 0.8. Again the failure will be catastrophic even though the capacity of the column enhanced with reduction of the effective height. 5.8.8 Anomalous behaviour of columns under post impact loading (a) Flexural failure (b) Shear failure with concrete crushing Figure 5.20: Different failure characteristics of structural columns The following observations were made on the impact behaviour of a 20% loaded 300mm diameter column under subsequent increase of the axial load at the post impact stage. The main observation was that the residual displacements change direction rapidly with the intensity of the impact loading even though the deflection characteristics during the impact are almost the same. Bulging at the impacted point Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 5-141 Figure 5.21: Axial load sensitivity of the counterintuitive effect From a concrete mechanics point of view, micro cracks are generated at the opposite side, due to the displacement caused by impact forces. Once these cracks are formed the concrete has reached its residual state as the elements do not carry any stress afterwards. At the same time, the compressive stress on the impacted surface does not exceed the compressive strength of the concrete and thus the surface remains unharmed. When the axial load is applied at the post impact stage, the column tends to deflect inwards (towards impacted face) as there is load eccentricity due to the element failed in tension in the opposite face of the impact. Figure 5.20(a) shows the points where cracks initiated in the opposite face. This behaviour becomes more predominant when the amplitude of the impact force is increased, where the column subjected to brittle failure under 20% axial load deflecting towards its impacted face. When the impact force is further increased, the compressive strength of the impacted surface is also reached in the residual stage due to crushing of the concrete. Therefore there is no substantial load eccentricity for buckling and the concrete core will tend to bear the axial load without any initialisation for buckling. Thus the column will fail by crushing the core concrete, and as the volumetric strain exceeds the elastic limit, compaction will occur and the concrete will tend to disintegrate (see Fig. 5.20(b)). When the initial axial load on the column is further increased, it will tend to increase the moment capacity of the columns and hence the residual deflection towards the opposite direction of the impact is minimised. Further axial load enhancement will minimise tensile stress development in the opposite side while crushing the concrete in Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 5-142 the vicinity of the impact. Resultant load eccentricity will deflect the column towards the opposite face of the impact (see Fig. 5.21). This behaviour gradually reduces as the column diameter increases. In addition, provision of lateral steel at closer intervals will also minimise this behaviour by enhancing the moment capacity of the columns. Figure 5.22: Typical failure pattern of rectangular columns under eccentric loading Similar observations were made when rectangular columns were subjected to eccentric loading as shown in Figure 5.22 (Nemecek et al. 2005). The collapse of the columns was initiated by concrete softening accompanied by symmetric buckling of reinforcement bars on the compression side around the mid height. The bars always buckled between the hoops and failure was localised. The damaged zone was larger for dense (50mm c/c) stirrups spacing and small for coarse (150mm c/c) stirrups spacing. From a comparison point of view, the similarity is the eccentric loading conditions where the load eccentricity is generated by the localized damage to the concrete due to the impact. The location of the affected region differs as the damage is initiated within the impacted region. There have been cases where the observed deflection was on the same side of the impact under impulse loading. This is also known as counterintuitive or anomalous column response and is supported by the experimental results published by Li et al. (1991) and Kolsky et al. (1991). It has also become evident that counterintuitive behaviour is extremely parameter sensitive and is constrained to a particular region of impulse loading. In other words a narrow zone of loading sensitivity can be identified for such a system within which the dynamic rebounding can occur and the final deflection of the column (or beam) rests in the opposite direction of the loading. It has also been shown that this transition is abrupt with respect to the change of loading Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 5-143 parameters. Similar phenomena were also reported for several other dynamic systems under radial pulse pressure loading (Forrestal et al. 1994), blast loading (Galiev 1996) and underwater explosion (Galiev 1997). According to Li et al. (2006), the time increment, element type or a change of simulation software may influence the existence of the counterintuitive phenomenon. These factors can also change the width of the sensitivity region. Apart from that, the anomalous behaviour is predominant closer to the region of transition from elastic to relatively plastic. The deviation observed here can be categorised as a probabilistic response of the system due to variation of the parameters. That is, the uncertainty of the response occurs mainly due to the randomness of the system parameters. If the system parameters can be determined accurately to a certain extent and the system itself is not parameter sensitive, the results can be predicted with sufficient accuracy. Although the parametric sensitivity of this system is comparatively less than that of a chaotic system, the uncertainties encountered in the practice may still be able to cause some response uncertainty through the parametric sensitivity. However, research conducted up to date has provided the solution only for very simple beams and plates. Therefore the data accumulated up to now is more restricted to the mathematical and theoretical aspects of the problem rather than engineering and practical applications (Galiev 1997). Formulations developed for a generalised event may not be valid in the region within which the parametric sensitivity is predominant. The sensitivity normally occurs closer to the nonlinear region and hence the equilibrium equations based on dynamic formulation under impact may need further modifications to calculate the column’s residual strength particularly under the effect of axial load. Therefore the structural response cannot be predicted by solving the equation of motion. On the other hand, parameters dealing with this problem are also complicated. Hence the finite difference method may be more reliable in the sensitivity analysis process. However either truncation errors or condition errors may occur under the finite difference method depending on the selected perturbation step length. For example, the truncation error may be excessive if the selected load increment step size is too large. Alternatively if the step size is too small, condition errors may occur due to inaccuracies in the Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 5-144 calculation of the displacement and round off errors in the finite difference calculations. Therefore selection of a proper step size to capture the dramatic variation itself would be a difficult task (Adelman and Haftka 1986; Li et al. 2006). 5.8.9 Buckling of reinforcement under impact (a) Concrete crushing (b) Internal forces generated in longitudinal steel Figure 5.23: Failure of columns by concrete crushing When the column fails as a result of concrete crushing under post impact loading conditions, the reinforcement in the vicinity of the impacted area tends to buckle outward by opening the longitudinal steel. A typical example is shown in Figure 5.23(a). All the longitudinal reinforcement tend to buckle simultaneously and the resultant tensile forces on the hoops tend to detach the hoops from their anchorage by releasing all the confinement effects on the surrounding concrete. Consequently the column catastrophically fails from the point of impact with concrete crushing. This behaviour has been reflected in numerical simulations with magnificent accuracy. The axial forces developed in the longitudinal and lateral steel at the point of impact are shown in Figure 5.23(b). Once the longitudinal steel buckles, the hoops tend to fail in tension. Compressive forces that develop in the longitudinal steel are shown with a negative sign and tensile forces in the hoops are shown with a positive sign. It is clear that all the reinforcements in the non-linear region activate simultaneously within a short period of time. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 5-145 5.9 Derivation of empirical relationships to predict critical impulse Whilst the individual parametric studies reveal some interesting features of the impacted columns, they do not reveal their collective contribution as far as routine column designs are concerned. To comprehensively understand the cumulative effects of key design parameters, a thorough statistical multivariate regression analysis was carried out. This section reports part of the regression analysis. Empirical relationships are developed in stages by varying each parameter at a time and then combined to produce an equation based on the least square method, which can be used to quantify the peak force and the associated impulse at the near collapse stage for fully loaded columns. Some of the terms in the empirical relationships have theoretical explanations and therefore the relationships can be considered as semi-empirical. The final results are approximate values for the characteristics of the critical impulse in terms of logarithm of Peak Force Log P c , and logarithm of Impulse Log I c , particularly for a 100ms impact. The relationship is valid under specific conditions as discussed later in this paper. 5.10 Derivation of simple linear regression equations A simple linear correlation between the parameters namely the diameter of the column D, steel ratio v ρ , concrete grade ' c f , effective height H, and yield strength ' sy f , area h A and spacing s of the hoops, is determined by using a statistic program ‘StatistiXL’. The objective of the statistic analysis is to check whether the equation can take into account cumulatively the key variables to predict the peak force and the associated impulse which leads to a near collapse of a column during a 100ms impact. A multiple regression analysis is performed on the data set to obtain the correlation coefficients of a possible linear relationship. The outcomes of the analysis are described in the following section. 5.10.1 Descriptions of the outputs Table 5.1 gives the descriptive statistics for the dependent variable Log P, followed by all independent variables in the order of the column entry. A total of 141 data records are used, and the standard deviation of each term is also given. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 5-146 Table 5.1: Descriptive Statistics Variable Mean Std. Dev. Log P 6.468 0.349 Steel ratio; ρ v 0.032 0.012 Com. Strength; f ’ c 39.299 9.208 Height; H 3.674 0.660 Diameter; D 0.589 0.176 Yield strength of hoops f sy 0.240 0.129 Hoop spacing s 353.546 48.050 Area of hoops; A h 35.846 22.152 5.10.2 Pearson Correlation The Pearson correlation is a number between -1 and +1 which measures the degree of association between two variables such as X and Y. A positive valve indicates a positive association of large values of X and Y pairs or small values of X and Y pairs. Conversely, negative values indicate that a negative or inverse association and large values of X tend to be associated with small values of Y and vice versa. The Pearson Correlation is computed as; ( )( ) ( ) y x i i x i S S n Y Y X X r 1 1 − − − ∑ = = Eq : 5.4 where X i and Y i are two arbitrary variables, and X and Y are the associated mean values. x S and y S are the standard deviations respectively. The term ( )( ) Y Y X X i i − − governs the sign of the correlation depending on the respective values of i X and i Y . The correlation coefficients measure the strength of a linear relationship between two variables. A value of ±1 indicates a perfect linear relationship and the relationship tends to decrease when the coefficient decreases. In general, a valve between ±1.0 and ±0.7 indicates a strong association and a value between ±0.7 to ±0.3 represents a negative association. This definition is however somewhat arbitrary and can be deviated. In addition, the correlation also depends on the sample size and will not reflect the practical significance. As there are a number of independent variables the correlations between every pair can be arranged into a matrix as shown in Table 5.2. It can be seen that there is an inverse correlation between the logarithm of peak force Log P, and Height H. The relationship of Log P, to the parameters such as s, ' sy f and A h also agree with the general perception. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 5-147 On the other hand, considerable correlations that exist among the individual parameters indicate that alternate options may be adopted. However, for a general case of axial loading on a column and in the absence of design guidance for impact, some of the other variables such as spacing of hoop steel and its diameter can be considered as arbitrary. Therefore the correlation factors do not reflect the actual effect on the capacity enhancement and hence the correlation does not affect the regression. Table 5.2: Pearson correlations 5.10.3 Coefficient of Determination and Analysis of Variance According to Table 5.3 the Coefficient of Determination (R 2 ) indicates that 94% of the variation in Log P, is explained by variation in the independent X variables, and the R value 0.97, which is the square roof of R 2 , indicates a strong correlation between Y and X variables. The Standard Error of Estimate, 0.084 is only 1% of the mean of Log P, 6.47 and thus indicates that the Multiple Regression model has accurately calculated a large amount of the Log P values. The adjusted coefficient of determination (R 2 adj ) provides an unbiased estimate of the coefficient of determination by allowing for the degrees of freedom of R 2 particularly with the numerous independent variables. Table 5.3: Coefficient of determination of the equation R 2 R Adj. R 2 S.E. of Estimate 0.944 0.972 0.942 0.085 Analysis of Variance is used to determine whether there is a difference between three or more categorical sets of values while t-test is used to compare two groups. First row of the Table 5.4 indicates the significance of the multiple regression model. The much larger mean square for the regressing 2.29 than the residual error 0.007 indicates that Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 5-148 the model is highly significant with zero probability of error. F statistic for linear regression indicates the statistical probability of the partial regression coefficients for a multiple linear regression is equal to zero. Table 5.4: Analysis of Variance Source Sum Sq. D.F. Mean Sq. F Prob. Regression 16.078 7 2.297 321.157 0.000 Residual 0.951 133 0.007 Total 17.029 140 5.10.4 Interpretation of partial (regression) plots A partial plot is a graphical representation of the relationship between a given independent variable (X i ) and the response variable (Y i ), with adjustment to the independent variable to reflect the effect of other independent variables in the model. Thus it plots the residuals of regressing response variable (Y i ) predicted from all other independent variables except X i verses the residuals of that independent X i variable regressed against all the remaining independent variables. For instance, in Figure 5.24(a) the Y axis represents the residuals from a multiple regression of Log P against all the other independent variables such as height, concrete grade etc. except D. The X axis represents the residuals of a multiple regression of diameter D against the other independent variables such as concrete grade, steel ratio etc. When performing a linear regression analysis with a single independent variable, each scattered plot provides a good indication of the nature of the relationship between the response variable and each independent variable. As there is more than one independent variable in this analysis, the partial regression plots specifically indicate the proper relationship when the plotted independent variable has a strong correlation with other independent variables in the model. A linear trend indicates a significant relationship between Y and the X i . Also the regression of the sets of residuals should pass through the origin and have the same slope as the regression coefficient for that particular independent variable X i . The influence of individual data values on the estimation of regression coefficient is clearly seen on these plots and the partial plots also enables examination of non-linearity and outlying points or points that contribute heavily to the regression. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 5-149 Partial Plot of D -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 -0.4 -0.2 0.0 0.2 0.4 Column diameter D L o g ( P ) Partial Plot of H -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 -2 -1 0 1 Height H L o g ( P ) (a): Partial regression plot of diameter (b): Partial regression plot of height Partial Plot of ρ v -0.30 -0.25 -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 -0.02 -0.01 0.00 0.01 0.02 Steel ratio ρv L o g ( P ) Partial Plot of s -0.3 -0.2 -0.1 0.0 0.1 0.2 -0.2 -0.1 0.0 0.1 0.2 Hoop spacing s L o g ( P ) (c): Partial regression plot of steel ratio (d): Partial regression plot of hoop spacing Partial Plot of Ah -0.30 -0.25 -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 -40 -20 0 20 40 60 80 Area of hoop Ah L o g ( P ) Partial Plot of f' sy -0.30 -0.25 -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 -200 -100 0 100 200 Yield strength of hoops f' sy L o g ( P ) (e): Partial regression plot of A h (f): Partial regression plot of ' sy f Figure 5.24(a-f): Partial regression plots of each parameter against Log P The outlying points can be identified by examination of the X-Y plots. It is not unusual that one or more data points in a sample do not comply with the chosen model. However, a formal statistical test is needed to identify these outlying points to avoid classifying too many points as outlying. As the test results are based on finite element analyses, the deviations occurred as a result of exceptional outcomes resulting from Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 5-150 the higher order vibrations and other inherent properties of the impacted columns. Therefore no outlying points have been declared in this analysis as they do not result from numerical errors. 5.10.5 Regression coefficients and derivation of the linear equations Table 5.5 represents the regression coefficients of the linear regression equation. S y.x is the Standard Error of Estimate which is defined as the square root of the error mean square; it is the variance in Y after accounting for the dependency of Y on X. S β is the Standard Error of the Slope (β) given in dotted lines in Figures 5.24(a) to (f) and is defined as ∑ 2 . X S x y . 95% confidence intervals (C.I.) are given in the next column and t represents the t Statistic. Here the t is based on the assumption that each partial regression coefficient (β i ) follows a normal distribution. The t value tests whether a partial regression coefficient differs from a particular value b and is calculated as; ( ) ii X Y i c S b t . − = β , Eq : 5.5 where c ii is an element in the inverse matrix of the corrected sum of cross products. As the probability of the regression coefficient of steel ratio ρ has a substantial deviation, the steel ratio does not have a strong co-relationship with the impact capacity of the concrete columns. One of the main reasons would be the non-identical configurations of the longitudinal steel across the circumference of the various columns. This is unavoidable in construction practice, and hence to account for this in the finite element model, the peripheral spacing between the longitudinal steel was kept close to 100mm as much as possible. However, the spacing is inevitably varied from 70mm to 110mm depending on the configuration of the longitudinal steel. No further effort has been taken to minimise the distance between the bars as it violates the general procedures used in practice. Consequently, the depth to the neutral axis deviated from one column to the other, so that no comparison can be made between columns having various diameters. Even though this database does not reflect the influence of steel ratio, it is included as a variable because the selected steel configurations are much closer to the practical applications. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 5-151 Table 5.5: Regression Coefficients for linear equations Parameter Coefficient S y.x S β -95% C.I. +95% C.I. t Prob. Intercept 5.458 0.097 0.0 5.266 5.649 56.416 0.000 ρ v -0.181 0.727 -0.006 -1.620 1.257 -0.249 0.803 f ’ c 0.004 0.001 0.100 0.002 0.005 4.754 0.000 H -0.071 0.012 -0.134 -0.095 -0.047 -5.848 0.000 D 1.897 0.041 0.958 1.816 1.978 46.349 0.000 s -0.209 0.075 -0.077 -0.357 -0.061 -2.790 0.006 f ’ sy 0.0004 0.000 0.014 0.000 0.000 0.674 0.502 A h 0.001 0.000 0.042 0.000 0.001 1.911 0.058 46 . 5 001 . 0 21 . 0 9 . 1 07 . 0 004 . 0 18 . 0 ' + + − + − + − = h c v A s D H f P Log ρ Eq : 5.6 Equation 5.6 can be used to calculate the predicted Peak Force P of the critical impulse for a typical 100ms vehicle impact. The standard errors of the regression coefficients are also given and the significance of the regression coefficient is determined by a t-test. For instance, for diameter, D the multiple regression coefficient is +1.897±0.041 and ranges from its -95% confidence limit of +1.816 to its +95% confidence limit of 1.978. The t-value for this coefficient of 46.35 is significant with zero probability of error. In contrast, the coefficient for the longitudinal steel ratio is not significant since t = -0.25 and hence the effect on the final value is not considerable even though the probability is 0.803. The standard partial regression coefficient, S β signifies the relative importance of each independent variable. A variable with a high S β is relatively more important than a variable with a lower S β. The S β essentially takes into account the possible variation in scale of the different X variables. If ' c f is measured in Pascal, Pa rather than in MPa, then very different partial regression coefficients would be expected due to the difference in scale, whereas the standardised partial regression coefficients take this difference in scale into account in the comparison. Residuals vs Predicted Log (P) y = -0.5595x 2 + 7.2421x - 23.373 R 2 = 0.6323 -0.30 -0.20 -0.10 0.00 0.10 0.20 5.5 6.0 6.5 7.0 7.5 Predicted Log (P) R e s i d u a l s Figure 5.25: Accuracy of the prediction by linear equations Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 5-152 Figure 5.25 represents the accuracy of the predicted data points compared to the observed values. The residuals are calculated as (Observed (Log P) - Predicted (Log P)) and hence the positive and negative residuals indicate an under and over-prediction of the data points respectively. By considering the distribution of the residuals the accuracy of the Log P can be further improved. The corrected Log P c and Log I c are given by; Eq : 5.7 Eq : 5.8 The final over and under prediction of the Peak Force, P is within the range of ±15 %. Hence polynomial equations are generated for more accurate estimation. 5.11 Derivation of Polynomial equations Polynomial equations are derived by assuming that all the parameters are independent and there is no correlation among them. The mathematical relationships between individual variables and the peak of the impulse force are determined by changing one variable while keeping the others fixed. As this procedure involves an enormous amount of modelling, the relationship between the column diameter and the peak impact force is determined first by keeping all the other parameters fixed. Once this relationship is derived as a polynomial equation of (D/0.5), the constant term of the equation should represent the effects of all other parameters if it is assumed that there is no correlation between the diameter and the other independent variables. This assumption will considerably reduce the total number of simulations needed to form the database for derivation of equations. Once the correlation between the diameter and the peak force is known, the known terms can be transferred to the left hand side of the equation except the constant term. The correlation between peak impact force and steel ratio is then determined by varying the steel ratio of individual columns. For the remaining calculations the same procedure is repeated (see Fig. 5.27(a) to (g)). In these Figures the X axes values are normalised with respect to typical values of the respective parameters. As there is no correlation between steel ratio and effective height, concrete grade, and hoop characteristics, the correlation between the individual parameters and peak collapse force can be calculated as a second order polynomial 30 . 1 37 . 23 242 . 8 ) ( 56 . 0 2 − = − + − = c c c P Log I Log P Log P Log P Log Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 5-153 equation which generates much more accurate results compared to the linear equations. Log P vs (D/0.5) y = -0.3715x 2 + 1.8787x + 4.7598 R 2 = 0.9936 5.5 5.8 6.0 6.3 6.5 6.8 7.0 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 (D/0.5) L o g P Log P vs (D/0.5) Poly. (Log P vs (D/0.5)) Log P-f(D/0.5) vs (ρ v /0.04) y = -0.0893x 2 + 0.0568x + 4.7903 R 2 = 1 4.75 4.76 4.77 4.78 4.79 4.80 4.81 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (ρ v /0.04) L o g P - f ( D / 0 . 5 ) Log P-f(D/0.5) vs (r/0.04) Poly. (Log P-f(D/0.5) vs (r/0.04)) (a): Log P vs. diameter (b): Log P-f(D) vs. steel ratio Log P-f(D/0.5)-f(ρ v /0.04) vs (f' c/45.6) y = 0.2374x 2 - 0.2803x + 4.8334 R 2 = 1 4.74 4.75 4.76 4.77 4.78 4.79 4.80 0.50 0.60 0.70 0.80 0.90 1.00 (f' c / 45.6) L o g P - f ( D / 0 . 5 ) - f ( ρ v / 0 . 0 4 ) Log p-f(D/0.5)-f(r/0.04) vs (f/45.6) Poly. (Log p-f(D/0.5)-f(r/0.04) vs (f/45.6)) Log P-f(D/0.5)-f(ρ v /0.04)-f(f' c/45.6) vs (H/4) y = 0.3138x 2 - 0.7969x + 5.3111 R 2 = 1 4.80 4.84 4.88 4.92 4.96 5.00 0.50 0.60 0.70 0.80 0.90 1.00 (H/4) L o g P - f ( D , ρ v , f ' c ) Log(P)-f(D/0.5)-f(r/0.04)-f(f'c/45.6) vs (H/4) Poly. (Log(P)-f(D/0.5)-f(r/0.04)-f(f'c/45.6) vs (H/4)) (c): Log P-f(D,ρ) vs. compressive strength (d): Log P-f(D,ρ,f ’ c ) vs. height Log P-f(D/0.5)-f(ρ v /0.04)-f(f' c /45.6)-f(H/4) vs (s /0.35) y = -0.0565Ln(x) + 5.3098 R 2 = 0.9999 5.30 5.32 5.34 5.36 5.38 5.40 5.42 5.44 0 0.2 0.4 0.6 0.8 1 (s /0.35) L o g P - f ( D , ρ v , f ' c , H ) Log P-f(D/0.5,f'c/45.6,H/4,r/0.04) Log. (Log P-f(D/0.5,f'c/45.6,H/4,r/0.04)) Log P-f(D/0.4)-f(ρ v /0.04)-f(f' c /45.6)-f(H/4)- f(s /0.35) vs (f' sy /350) y = 0.225x 2 - 0.3433x + 5.429 R 2 = 1 5.28 5.32 5.36 5.40 0.7 0.9 1.1 1.3 1.5 (f' sy /350) L o g P - f ( D , ρ v , f ' c , H , s ) Log p-f(D/0.5,f'c/45.6,r/0.04,H/4,s/0.35) Poly. (Log p-f(D/0.5,f'c/45.6,r/0.04,H/4,s/0.35)) (e): Log P-f(D,ρ,f ’ c ,H,s) vs. hoop spacing (f): Log P-f(D,ρ,f ’ c ,H,s) vs. yield strength of hoops Log P-f(D/0.5)-f(ρ v /0.04)-f(f' c /45.6)-f(H/4)- f(s /0.35)-f(f' sy /350) vs (Ah /28.27) y = 0.0497Ln(x) + 5.424 R 2 = 0.9987 5.42 5.44 5.46 5.48 5.50 1.0 1.5 2.0 2.5 3.0 3.5 4.0 (Ah /28.27) L o g P - f ( D , ρ v , f ' c , H , s , f ' s y ) Log P - f(D/0.5,r/0.04,f'c/45.6,H/4,s/0.35,f'sy/350) Log. (Log P - f(D/0.5,r/0.04,f'c/45.6,H/4,s/0.35,f'sy/350)) Residuals vs predicted Log(P) y = -0.2232x 2 + 2.831x - 8.924 R 2 = 0.3406 -0.20 -0.10 0.00 0.10 0.20 5.5 6.0 6.5 7.0 7.5 Predicted Log(P) R e s i d u a l s (g): Log P-f(D,ρ,f ’ c ,H,s,f sy ) vs. area of hoops Figure 5.26: Accuracy of the polynomial equations Figure 5.27 (a-g): Steps of the derivation of polynomial equations Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 5-154 Eq : 5.9 Eq : 5.10 Eq : 5.11 Where Log P is the logarithm of the Peak Force P (uncorrected), Log P c is the (corrected) logarithm of the Peak Force P, Log I c is the (corrected) logarithm of Impulse I, D is the diameter of the column in m, ρ is the longitudinal steel ratio, ' c f is the compressive strength of concrete in N/mm 2 , H is the height in m, ' sy f is the yield strength of hoops in N/mm 2 , A h is the area of hoop in mm 2 and s is the hoop spacing in m. With the introduction of the corrected equation, the over and under prediction of the Peak Force, P c is reduced up to ±12%. This is a significant improvement as far as the uncertainties associated with the impact behaviour of the concrete columns are concerned. Once the Corrected Impulse I c is known, the critical velocity, v can be calculated for a known impacted mass m (kg) of a vehicle, in meters per second (ms -1 ) by using the relationship given in Equation 5.12. For instance, Eurocode EN 1991-1-7 (2006) suggested that mean mass of 1500kg for cars and 20,000 kg for trucks. v m I c = Eq : 5.12 In general, polynomial models are among the most frequently used empirical models for curve fitting functions. They are popular because of their simple form and moderate flexibility of shapes. However, polynomial models do not extrapolate reliably. The valid range provides good fits, but deteriorates rapidly outside that range. Higher degree polynomials are notorious for oscillations between exact-fit values. Therefore, complex relationships which lead to higher degree polynomials are avoided to allow interpolation of values within the valid range. 301 . 1 92 . 8 83 . 3 ) ( 223 . 0 35 . 5 25 . 0 ln 056 . 0 27 . 28 ln 05 . 0 350 ' 34 . 0 350 ' 225 . 0 4 8 . 0 4 31 . 0 6 . 45 ' 28 . 0 6 . 45 ' 24 . 0 04 . 0 06 . 0 04 . 0 09 . 0 5 . 0 88 . 1 5 . 0 37 . 0 2 2 2 2 2 2 − = − + − = + | ¹ | \ | − | ¹ | \ | + | | ¹ | \ | − | | ¹ | \ | + | ¹ | \ | − | ¹ | \ | + | ¹ | \ | − | ¹ | \ | + | ¹ | \ | + | ¹ | \ | − | ¹ | \ | + | ¹ | \ | − = c c c h sy sy c c v v LogP I Log p Log p Log P Log s A f f H H f f D D P Log ρ ρ Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 5-155 The valid range of the equations is given as follows; 5.12 Conclusions This chapter has confirmed the feasibility of using numerical simulation techniques in vulnerability assessment of impacted columns while enhancing our understanding of the optimum usage of critical parameters. The main findings are summarised below. 1. It has been shown that numerical simulation techniques can be used for quantification of the critical impulse of axially loaded circular columns. Empirical equations are developed in the process to predict the critical (collapse) load and the associated impulse. Polynomial equations are provided for more accurate estimation along with linear equations for approximate assessment. 2. Shear failure characteristics are initiated by the low elevation impacts and hence the concrete grade, diameter of column, steel ratio, slenderness ratio and confinement effects become the key design parameters that govern the vulnerability of the columns. In fact, there is a correlation between the dynamic and static shear capacities that can be used for approximate vulnerability assessment. According to the limited investigation, dynamic amplification factor suggested in EN 1991-1-7:2006 generates over conservative results. 3. The columns susceptible to impacts should be checked for all relevant limit states. In particular the accidental limit state can be declared in-between the serviceability and ultimate limit states depending on the expected level of safety. Low shear demand under serviceability conditions strengthen this argument. However, the limit states formulation based on the kinetic energy or deformation m s m mm A mm MPa f MPa m H m MPa f MPa m D m h sy c v 25 . 0 050 . 0 1 . 113 27 . 28 500 ' 250 4 2 50 ' 30 04 . 0 01 . 0 75 . 0 3 . 0 2 2 < < < < < < < < < < < < < < ρ Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 5-156 capacity (EN 1991-1-7:2006) may not be appropriate under low elevation impact conditions. 4. Confinement effects are particularly effective when the hoop spacing is less than 100mm. The confinement has to increase particularly with the diameter of the columns and with concrete grade to achieve the same level of capacity enhancement. Therefore, a method based on the maximum diameter of the longitudinal steel is not effective for determining the lateral steel spacing of the columns susceptible to vehicle impacts. 5. The confinement induced capacity enhancement is more predominant in short columns as the number of effective hoops increase as the inclination of the failure plane increases with the confinement effects. Increasing the diameter of the hoops is recommended rather than yield strength, where restrictions may apply on the minimum allowable spacing of the hoops. Additionally, the impact capacity enhancement due to reduction of effective height is more pronounced in columns exceeding 500mm in diameter. 6. The collapse of the impacted columns will be brittle and sudden upon post impact loading even though the residual deflections of the impacted columns are insignificant. Therefore the damage index D, is not a sensitive index for impacted columns. In particular, there are no substantial warnings before collapse and the condition remains unchanged even after introducing hoops at closer intervals even though this contributes to the overall capacity enhancement. Under these circumstances, it may be more appropriate to replace the impacted columns rather than repair them for further use. However, the decrease in impact capacity resulting from partial axial loading can be recovered by providing transverse steel at closer intervals. 7. The impact capacities reduced by 10%, 20% and 30% under the 60%, 40% and 20% axial loading. This behaviour is typical for shear critical columns where shear capacity is decreased with the axial load reduction. Even though the possibility of catastrophic shear failures are increase with the axial load increase, compromise Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 5-157 between axial load intensity and the allowable impact load may not be worth as there is no residual capacity remaining in the impacted columns. 8. It was concluded that the transverse reinforcement does not have a significant influence on the failure mode of the columns under impact loading, particularly when the axial load ratio exceeds 20% where capacity drop is around 65%. However, the capacity drop can be recovered by increasing the confinement effects and almost 90% of the impact capacity drop can be recovered, if the axial load on the column can be maintained around 40%. 9. Reduction of the steel ratio minimised the band width the region defined from D=0.2 to D=0.8. Therefore columns with low steel ratio fail abruptly without giving any warnings prior to collapse. Due to low slenderness ratios most of the columns failed due to concrete crushing at the impact location where volumetric strain exceeds the elastic limit under the application of post axial loads. 10. It is observed that there is an extremely parametric sensitive region where the residual deflection is in the same side of the impact which is known as counterintuitive behaviour of columns. A narrow zone of loading sensitivity can be identified for such a system within which the dynamic rebounding can occur and the final deflection of the column rest in the same side of the loading. Such response cannot be predicted by solving the equation of motion. 11. Spalling of the cover concrete under impact can be explained using the stress variation in the core-cover interface resulting from the confinement effects. In addition, possible alteration to the duration of the impact due to variation of stiffness resulting from partial loading conditions can be neglected under shear critical quasi-static loading conditions where peak force determines the vulnerability of the column. 12. An innovative technique was developed and introduced to ensure the accuracy of the equations developed for predicting the critical impact force and impulse where the other techniques are failed due to the shape of the error distribution under logarithmic scale. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 6-159 6. IMPACT ON COLUMNS UNDER UNIAXIAL BENDING 6.1 Introduction Columns on the front of buildings are a typical example of columns under uniaxial bending, and they are always adjacent to main roads. Eccentric loads may also result from misalignment, initial imperfections, movements of joints or support settlements, while the possibility of eccentric loading induced by vehicular acceleration or deceleration cannot be neglected in bridge type structures. Eccentrically loaded columns under static conditions have gained sufficient attention from the scientific community. However their dynamic capacity under lateral impact loading is yet to be determined. Initial deformation present in the eccentrically loaded columns may enhance or reduce the impact capacity of the columns depending on the direction of the impact. Some insight may also be provided by the column tests under mid-span impacts where flexural failures are predominant (Remennikov & Kaewunruen 2006). However there is little or no knowledge on the exact amount of the capacity reduction due to the presence of eccentric loading. Axially loaded steel columns under transverse mid span impact have already been discussed in the literature (Zeinoddini et al. 2008). As the flexure is critical in those columns there may be some similarity to the eccentrically loaded impacted columns at the ultimate stage. It was also observed that strain rate, geometry, axial load, the shape of the impacting body, its velocity and impact position have considerable effects on impact capacity. Moreover, structural stiffness drops suddenly with transverse impacts (Sastranegara et al. 2005). It has also been observed that the duration of the transverse impact has a significant influence on ultimate capacity while buckling of columns can be controlled by applying transverse impact (Adachi et al. 2004). However, the behaviour of eccentrically loaded concrete columns impacted at a shear critical height has not been considered in the past and there is a significant room for improvement using both experimental and numerical analyses. In addition, the dynamic buckling of columns under axial impact has attracted much attention in recent decades (Cui et al. 2002). Lateral impacts are likely to have a secondary component along the longitudinal direction particularly when the eccentrically loaded column deforms Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 6-160 against the impact. This means transverse impact can be considered as a reverse scenario of axial impact. In fact, dynamic elastic buckling analysis of simply supported steel columns under intermediate velocity axial impact has also confirmed the importance of both amplitude and the duration to determine the severity of the buckling effects (Cui et al. 2002). Obviously, information presented in the literature has only little contribution to understanding the lateral impact on concrete columns. To address this perceived need, an investigation has been conducted on reinforced circular columns made of different concrete grades, steel ratio, slenderness ratio and effective height. The duration of the impact was kept constant at 100ms by assuming shear critical columns do not have much effect on stiffness change. The load combinations were selected according to the Australian standard AS3600 (2004). Comparison based on the number of stories will not be considered in the selection of load combinations which depend on the column spacing rather than the storey height. The main aim was to investigate the sensitivity of the moment present on the columns to transverse impact and to derive a numerical equation that can be used to quantify the impact capacity of the eccentrically loaded columns. In the process different load combinations were used and initial analysis was limited to impact loads applied in the plane of bending. Out of plane bending will be investigated in detail in the next chapter due to its complexity. The numerical equations proposed at the end of this chapter to quantify the uni-axial bending are the first of their kind, known to the author, on this matter. 6.2 Behaviour of the impacted columns under single axis bending Figure 6.1: Plan view of the column head (under uni-axial bending) The main objective of this chapter is to investigate the impact behaviour and capacity change in the impacted columns under eccentric loading conditions. The columns Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 6-161 under single axis bending were considered in the first phase. Moment was applied about Y axis so that the impact induced deflection and the deflection due to the moment are acting in the same direction (see Fig. 6.1). This is the most vulnerable situation for columns under single axis bending. At a later stage, the impact behaviour of the column will be investigated by applying the moment in the opposite direction. 6.2.1 The load application procedure The vulnerability of typical columns adequate in capacity for 5 to 15 storied buildings with load eccentricities are investigated in the analysis. Axial load and bending moments are applied with load reduction factors according to the AS3600 code. Several options are considered at the initial stage of the numerical simulation of the eccentric loading. There is no single standard way of applying a moment across a section so that the applied moment distribution along the column follows the rules of moment distribution. Therefore, indirect ways of applying the moment are investigated in the numerical simulation which does not influence the initial conditions of the impacted column. For instance, applying a displacement to a plane or a section of the column to generate the moment introduces an artificial shear distribution across the column. Similarly, generating a moment by assigning a moment directly to a set of selected nodes is also not possible as the hexagonal elements without rotational degrees of freedom used in this analysis do not allow proper moment distribution. Consequently, the moment simulation was performed by using a coupling action of axial loads applied on a bulk head of the column (see Fig. 6.1 & 6.2). The bulk head of the columns was designed to have flexural and shear strengths well exceeding the external loads applied on the columns. This was to ensure that failure occurred in the column as intended. Hence, the top face of the column was projected 0.5m at 45 0 to form the lower part of the head and then projected another 0.25m in the vertical direction. Bearing plates were placed on the top of the head so that the load application area and eccentricity of the load were certain. The axial load required to produce the moment was applied as an eccentric load while the remaining load was applied directly on the plate at the centre. This bulk head reduced the stress concentration in the concrete due to the applied eccentric load. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 6-162 Figure 6.2: The Bulk head of columns used for the application of moment 6.2.2 Material models and mesh generation Due to symmetry of the structure and the loading, only half of the column was modelled and out of plane displacements were eliminated along the symmetric edge to maintain the symmetric conditions. The column consists of three separate parts; the column, column head and bearing plates, according to their intended purposes. The circular column was simulated using quadratic solid elements with a rectangular core area to minimise errors due to element distortion, as described in Chapter 5. Uniform element distribution across the height was maintained as this mesh generation derived better results. According to Xie et al. (1996), the behaviour of the concrete cover is significantly different from the core concrete. Therefore, at least two elements were used to simulate the thickness of the cover concrete. The column head was simply a projection of the top surface and hence its mesh distribution was very similar to that of the circular section. The mesh generation of the vertical portion of the head was conducted in a similar manner. It is important to note that the sizes of the elements increase towards the top of the head as the diameter increases. In fact, deformation of this portion, due to the load application, is negligible as the inertia is high compared to the circular portion. Hence ‘Rigid material’ characteristics were assigned to the bulk head, which is known as a very cost effective material as far as the duration of the analysis is concerned. The rigid material behaviour is mesh independent and this will also allow global or local constraints to be applied to the mass centre. This is an ideal way to transfer the bending action from the head to the column without applying constraints directly on the column, as this may cause unusual stress concentrations as well as constraint column rotations. However, the effective height of the column was slightly changed due to the bulk head as it translates and rotates about its centre of Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 6-163 gravity. Thus the centre of gravity was maintained at a certain height for all column heads for comparison purposes. It is also observed that the initial deflection at the head-column joint is negligible at the time of impact, as will be discussed later in detail. Both the longitudinal and transverse reinforcement were assumed to be elastic-perfectly plastic. The same material parameters used for steel in the Chapter 5 were used in this simulation and complete strain compatibility was assumed between the embedded bars and the concrete. 6.2.3 Axial load and eccentric load applications in an explicit environment Figure 6.3: Numerical simulation of eccentrically loaded columns Monotonically increasing axial compression was applied on the columns with constant end eccentricity during the simulation. The eccentric load simulating moment was gradually increased followed by the direct axial load as a ramped up surface pressure over the bearing plates to avoid stress fluctuation. This procedure also avoids the premature flexural failures due to the eccentric load alone acting on the column, while minimising the potential effects of cyclic loading as the eccentric load is applied. In particular, a separate ramp function has to be defined for applying the moment. One important observation is that the pitch of the fluctuation is minimised substantially when the duration of the ramp function (eg. 8ms) exceeds twice the duration of the fundamental period of vibration of the column (4ms). However, the optimum ramp duration reduces as the column diameter increases. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 6-164 Three different hoop spacings (50mm, 100mm, 250mm) and three different hoop diameters (6mm, 8mm, 12mm) were considered in the analysis without altering the yield strength ( ' y f =500MPa). However, lateral ties were not included as they may complicate the problem, and the addition of cross ties at a fixed volumetric ratio may or may not improve the confinement effects. That is, cross ties will increase the transverse steel spacing, while improving the confinement effects in the lateral direction. Figure 6.4: Interaction diagram for the 300mm Figure 6.5: Extreme strain in the 300mm column column Figure 6.4 shows the nominal interaction diagram of a 300mm column made of 30MPa concrete. Interaction diagrams indicate the nominal column capacities under most and least favourable confinement characteristics according to AS3600, including two other options in between. However, the design capacity remains well below the nominal capacity due to the usage of capacity reduction factors at the design stage. The load eccentricity e, may vary in the range of about 0.02 to 0.25 which corresponds to the axial load ratio of 20% to 80% compared to the concentric loading conditions which includes the upper level of load eccentricity used in practice. According to Figure 6.5 the maximum strain in steel is increased with low amount of steel content. Therefore contribution of the longitudinal steel could be limited in damage mitigation of columns under transverse impact loads. For instance, Figure 6.6 shows a typical interaction diagram of a typical circular column. It can be subdivided into compression control and tension control zones as shown. Characteristic points such as pure compression, balanced failure and pure bending can be identified using the relation to the strain in extreme tension steel. The strain in extreme tension steel Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 6-165 reaches a maximum value under the zero axial loading conditions where pure bending takes place. Columns with low steel ratios show higher tensile strain development and fail more quickly by crushing the concrete at the compressive side. Yielding of longitudinal reinforcement can be expected at the collapse stage depending on the test variables. Figure 6.6: A typical Interaction diagram of a column 6.3 Deflection profiles and resultant bending moment The axial loading system consists of two vertical loads applied on the bearing plates that simulate axial load and the bending moment. Half the design load is applied on the bearing pads as one half of the column is used in the numerical simulation. To investigate the accuracy of the selected method, 64kNm moment and 407kN axial force were applied on the 300mm column, which is equal to 20% of the design load under pure compression. According to the theory of moment distribution, half the applied bending moment (BM) must be transferred to the fixed end with an opposite sign (see Fig. 6.7 (a)). Under the impact loading the column generate cracks distributed through the column while shear is critical at the bottom. Consequently the column failed under flexural shear critical conditions. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 6-166 (a) Moments at serviceability stage (b) Ultimate stage after impact Figure 6.7: Contours of effective plastic strain Figure 6.8 shows the moment generated some of the selected cross sections (CS) at the top, bottom and at the point of impact. It can be seen that the numerical simulation accurately follows the conventional theories of moment distribution and the fluctuation of the applied moment can be minimised by choosing a desirable duration for the ramped up loading function. The reason for the slight difference between theoretical moment transferred and the actual moment present at the top is the elevated position of the centre of gravity of the bulk head where rotation takes place and hence external moment is fully applied. That is, the moment at the top of the column slightly deviates from the applied moment. However, this minor deviation can be negligible under low elevation impacts where the moment at the bottom is the governing factor which influences the overall behaviour of the column under impact. Figure 6.8: Time histories of BM of 300mm eccentrically loaded half column with 1% steel Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 6-167 6.4 Deformation characteristics of the impacted column Figure 6.9: Deflection of 300mm diameter column with 4% steel at the near collapse stage Figure 6.9 compares the deflection characteristics of the 300mm column under load eccentricities and pure compression. It is observed that the deflection characteristic at the ultimate stage increases with application of the moment. Residual deflection also increases with the associated fluctuation under eccentric loading conditions. The maximum deflection was observed in the 300mm column, and the 600mm column has the least residual deflection as the mode of failure changes from flexure to flexural shear as the column diameter increases. 6.5 Impact behaviour of the eccentrically loaded column Figure 6.10: Resultant bending moment at different locations on the 450mm column As far as the impact generated bending moments are concerned, the resultant bending moment diagrams follow the same triangular loading pattern which simulates vehicle Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 6-168 impact. Under pure axial loading the residual moment of the impacted column reaches zero. However, it was observed that the moment at the top of the column remains steady without changing while the moment generated at the bottom and at the point of impact increases to a greater extent during the impact and reaches its initial valve at the residual stage. These findings are summarised in Figure 6.10. This observation implies that the post impact behaviour of the column remains unchanged. If adequate shear reinforcement is provided, an impacted column can display the same static lateral capacity of an ordinary column (Loedolff’s 1989). Nevertheless, under eccentric loads, the impact capacity reduces due to the secondary moments applied by the enhanced residual deflection generated by the impact at the ultimate stage. However it can be seen that the dynamic moment (325kNm) generated under the impact does not have a substantial influence even though it exceeds two times the ultimate (nominal) static moment capacity of the 450mm column under the given axial load. Thus, implementation of SDoF system to quantify the impact behaviour is doubtful as localised strain rate effects and the corresponding strength enhancement may not be detected using a SDoF system. Also, considerable shear force variation generated between the point of impact and bottom support cannot be captured by such a simplified method. Figure 6.11: Resultant shear forces at different locations on the 450mm column The shear force generated at the bottom support under pure axial loading is larger than the shear force at the point of impact as shown in Figure 6.11. Consequently there is a gradual variation of shear force between the point of impact and the bottom support, which is unlikely to occur under static loading conditions. Thus the critical stress Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 6-169 distribution changes linearly in between those two sections and combination of moment and shear force are involved in determining the failure mode of the columns. Additionally, the initial deflection of the eccentrically loaded columns greatly reduces the ultimate shear force generated at the bottom. The stiffness characteristics of the column change with the eccentric load and thus eccentrically loaded columns are increasingly vulnerable and collapse under low velocity impacts. In other words, shear capacity under pure axial loads greatly reduced with the application of the moment. However, the residual value of the shear force remains unchanged from its original value after the impact, confirming the observation of Loedolff (1989). 6.6 Behaviour of eccentrically loaded confined columns under impact a) Concentric loading b) Eccentric loading c) Near Failure conditions Figure 6.12: Lateral pressure distribution and the corresponding stress-strain relationships Even though the validation process mainly focuses on the vulnerability assessment of axially loaded columns, it can be extended to assess the vulnerability of eccentrically loaded columns. The main difference between these two loading conditions is the formation of a stress gradient across the sections of the eccentrically loaded column as shown in Figure 6.12(b) & (c), which influences the lateral strain distribution based on the Poisons ratio of concrete. The lateral strain distribution will determine the intensity of the confinement in the lateral direction and consequently it decides the failure point due to the hoop fracture (Mander 1988). Thus it may not be always possible to apply the theories generated under concentrically loaded (uniform strain) conditions to Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 6-170 eccentric loading conditions in impact simulations. However, based on the observations of (Mander 1988, Lokuge 2003, saatcioglu 1995), it can be concluded that the results of this numerical simulation can be extended to assess the vulnerability of the eccentrically loaded columns made of the lower grade concrete. In fact, it was considered that the stress-strain curves in the strips closer to the neutral axis may not be substantially different to the one that under fully confined conditions particularly in the preloading strain range (Figure 6.12(b)) (Saatcioglu 1995). On the other hand, the strength decay is essentially a function of confinement stress and does not vary with the strain gradient (Sargin 1971). Moreover, it is important to note that the flexural cracks appearing on the column at the ultimate stage under the impact will minimise the stress differences in various layers across the section. Consequently, the vulnerability assessment techniques are extended to the eccentrically loaded columns by assigning uniform confined compressive characteristics to the core section based on the equations proposed by Mander et al. (1988). 6.7 Selection of the load combinations Figure 6.13: Interaction diagrams for 450mm column according to AS3600 and ACI: 318 Figure 6.13 shows the P-M interaction diagrams generated for circular columns that are subjected to bending moment and axial compression for two extreme lateral reinforcement ratios. Figure 6.14 represents specific points, including pure axial compression, balanced point and pure bending. Axial tension is neglected as it has limited applications. The interaction diagram depends on the configuration of the longitudinal reinforcement particularly for a circular column. However, for Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 6-171 convenience, the amount of reinforcement is expressed as the percentage of gross concrete area which generally varies from 1% to 4% according to AS 3600. The number of reinforcement bars in a cross section is 8, 12 and 16 for 300, 450 and 600mm column respectively. Nominal interaction diagrams for 1% and 4% steel ratios are represented in Figure 6.13 along with their design values and strength reduction factors which vary from 0.6 to 0.8 according to the AS3600 standard. It is interesting to note that allowable load at the serviceability stage is significantly high in ACI: 318 (1999) compared to AS 3600 (2004). As the Accidental Limit State was defined in between the ULS and SLS, the structural design according to the ACI: 318 (1999) (and for other codes) must be investigated separately under the impact loading. Figure 6.14: locations of the selected loading points on the interaction diagrams 6.8 Parametric studies and discussion of the finite element results During their service life, concrete columns experience numerous loading conditions due to load eccentricities, differential settlements or external loads such as wind. The impact behaviour of such columns could be complex due to various possible load combinations and hence difficult to predict. To simplify the numerical simulation, columns with single axis bending are considered in the first phase and the effects are further subdivided into positive and negative moments depending on the direction of the moment application. The negative moment will deflect the column in the direction of the impact and reduce the impact capacity, while positive moments do the opposite. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 6-172 6.8.1 Impact behaviour of eccentrically loaded columns under maximum allowable capacity The load combinations applied on the impacted columns are selected based on the corresponding interaction diagrams. Columns under pure axial load are represented as P=P d and no moments are acting on those columns. The other load combinations are represented by the ratio of the axial load to the pure design axial load capacity of the column and the corresponding moment. A point on an interaction diagram is represented using a notation such as P=0.2P d ; M=M 20 which means the column carries 20% of its pure (design) axial load and the total moment corresponding to that axial load according to the AS3600 (2004) standards (see Fig. 6.14). The selected loading points are marked on the interaction diagram and sensitivity of the moments is further investigated by reducing the moment by half. The selected load combinations represent to the load carrying capacities of columns with 1% and 4% steel. Figure 6.15: Eccentrically loaded columns with 1% steel ratio The vulnerability of impacted columns with 1% steel is illustrated in Figure 6.15. The peak force represents the critical (maximum) impact force that can be withstood by the impacted columns. It is clear that the bending moment present in the columns substantially reduces the impact capacity compared to its pure axial load capacity, despite the columns carrying low axial loads. The 20% loaded (P=0.2P d ) column with the corresponding full moment will reduce the impact capacity by about 50%, while the impact capacity reduction of columns with 50% axial load (P=0.5P d ) with the corresponding full moment is approximately 33%. Therefore load combinations Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 6-173 beyond this point may not be recommended. It is also evident that the 450mm columns with 20% loading are now under threat of the low velocity (<40km/h) vehicle impacts. 6.8.2 Impact behaviour under reduced load eccentricities Figure 6.16: Peak force under different load combinations One of the reasons for the impact capacity reduction under the flexural loading conditions could be the full moment (eg. M 50 ) applied on the columns under the corresponding 0.5P d axial loading. However, the effects of reduced eccentric loading were yet to be determined. Substantial impact capacity improvement may be expected due to the reduction of the depth to the neutral axis under the reduced moments where extreme concrete fibre is subjected to lower compressive strains. Consequently, the analysis extended to investigate the impact capacity of eccentrically loaded columns with partial moments. In the process, the maximum allowable moment on the column was reduced by 50% (eg. 0.5M 50 ) while maintaining the same axial load (0.5P d ) as shown in Figure 6.14. However, it was evident that reducing the moment does not have a considerable effect on the impact behaviour of the columns. This was partly due to the small moment present (transferred) in the column close to the bottom support where lateral impact force was applied. Based on this observation, columns with reduced eccentric loading were omitted in further analysis. Thus the empirical equations were generated by applying axially load with the maximum allowable moment. This reduces the large number of load combinations that are involved in the analysis process. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 6-174 As the eccentricity of the axial load decreases, the impact capacity of the columns increases as shown in Figure 6.16 except for the 50% loaded 300mm column. The moment reduction is more effective for columns with low axial loads rather than columns with high axial loads, as far as the resultant impact capacity improvement is concerned. On the other hand it is important to note that the 20% and 50% loaded columns carry almost equal bending moments even though there are significant differences in their impact capacities (see Fig 6.14). That is, when the applied moment remains constant, the impact capacity of the columns significantly increases with axial loading. A simple calculation based on load eccentricity revealed that the compressive stress generated on the column is significantly high with the axial load increase compared to the moment reduction. The enhanced axial load increases the stiffness of the column and hence increases the impact capacity to a greater extent. In particular, the column’s sensitivity to the axial load depends on the duration of the impact and this relationship is effective only if the duration of the transverse impact is equal or greater than the time taken by the buckling process particularly under flexural failure conditions. 6.9 Impact behaviour of columns under positive eccentric loading Positive eccentric loading is a hypothetical term used to describe a moment on a column which deflects the column towards the impacted side. It was generated by applying pressure on the same bearing plates in the upward (opposite) direction. It is important to note that most of the columns in building edges, supporting eccentrically loaded beams located perpendicular to the edges, are subjected to similar (positive) load eccentricities and most of the columns therefore generate counter moments against the impact. The aim of this study was to investigate the effects of the counter moment in the vulnerability analysis. Figure 6.17 presents the effect of the positive moment on capacity enhancement. The positive moments always enhance the impact capacity of the columns compared to the negative moments irrespective of the diameter of the column. The initial deformation was in the impacted face of the column and as a result the column resists the impact Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 6-175 induced deflection. Consequently, compressive stresses dominate the impact behaviour of the columns by increasing the stiffness. Thus, deformation during the transverse impact can be controlled by applying a positive moment on the column and the resultant changes to the stiffness can increase the capacity against lateral impact. Figure 6.17: Comparison of the Impact capacities under positive and negative moments In the comparison point of view, flexure is predominant throughout the column under negative bending moments compared with under positive moments, in which limited area is subjected to flexural conditions. The resultant maximum deflection of the column is also high under the negative moments. These factors imply that the strength enhancement due to the strain rate effects is high under negative moments compared with under positive moments. However the strength enhancement due to the strain rate effects is not reflected in the results. Therefore the main factor behind this strength enhancement under the positive moment should be the change in stiffness due to formation of an arch. Therefore the results will still be conservative if the positive load combinations are excluded from future analyses. However, the capacity reduction due to the bending stresses cannot be compensated by the strength enhancement due to the formation of the arch. Consequently the maximum impact capacity is obtained under the pure axial loading conditions (P d , M o ). Typical cracks that appeared at three consecutive time steps in the impacted columns with positive moment are shown in Figures 6.18(a) and (b) along with 6.19(a). This behaviour is different from double curvature bending and the continuous oscillation of the impact generated moments along the column leads to this crack pattern. The Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 6-176 motivation of this oscillation is the axial load fluctuation of the impacted 300mm column from 180% to 20% compared to the applied axial load. The intensity of the load fluctuation decreases with the diameter of the columns. According to Papazoglou and Elanashai (1996) the shear failure is likely when accounting for axial load increment followed by vertical ground motion particularly under earthquake loading. However, the axial load fluctuation and considerable preliminary deflection initiate the flexural failure conditions in the 300mm column. Consequently, impact generated shear force at the bottom of the 450mm column is 3.5 times higher than that of the 300mm column even though the ratio of the cross sectional areas and the ratio of critical amplitudes of the impact are 2.25 and 1.67 respectively. This observation implies that the impacted columns tend to fail in shear as their diameter increases, despite the eccentric loading conditions. This hypothesis is confirmed by the impact characteristics of the 600mm diameter column which failed in shear as shown in Figure 6.19(b). (a) During impact (b) Post impact (a) 450mm column (b) 600mm column Figure 6.18: Cracks on 20% loaded 300mm Figure 6.19: Cracks on 20% loaded 450mm and column with 1% steel 600mm columns with 1% steel 6.9.1 Impact response under positive eccentric moments As far as the impact induced shear forces are concerned, there are no specific changes due to the directional change of the moment except the impact generated shear forces act in the opposite direction compared with the shear forces applied at the service stage. The resultant shear force at the service stage is comparatively small and shear Primary cracks Secondary cracks Tertiary cracks Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 6-177 forces generated by the impact are concentrated close to the bottom support. These forces vary linearly from 20kN at the point of impact to 105kN at the bottom support, at the near collapse stage for the 300mm column with 20% loading. It is important to note that the shear force as well as the bending moment at the bottom of the support simultaneously increases with the impact (see Fig. 6.20). Therefore a combination of these two should be taken into account when determining the failure mode. Figure 6.20: Resultant bending moments of the 20% loaded 300mm column 6.10 Confinement effects on eccentrically loaded columns under impact Figure 6.21: Capacity of eccentrically loaded confined columns under impact The main objective of this section is to investigate the capacity (peak force) of eccentrically loaded confined columns under the impact loading conditions. As the peak forces generated by the positive eccentric moments are conservative, positive load combinations are excluded from here onwards. Mainly two eccentric loading Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 6-178 conditions namely (0.5P d , M 50 ) and (0.2P d , M 20 ) are investigated and compared with axially loaded conditions (P d , M o ) under nominal confined conditions (see Fig. 6.21). The nominal confined condition is achieved by proving 6mm hoops at 250mm spacing as the strength enhancement of the concrete due to this arrangement can be negligible. At the next stage, the hoop spacing ‘s’, is decreased down to 100mm and 50mm respectively and the resultant confined conditions are simulated by assigning enhanced strength characteristics to the core concrete according to the equations of Mander et al. (1988). It is observed that the 20% loaded (0.2P d , M 20 ) concrete columns are always highly vulnerable under these circumstances compared to the 50% loaded columns (0.5P d , M 50 ), irrespective of the hoop spacing. Consequently, it is not recommended use larger diameter columns than required to mitigate the impact damage (Note: M 20 ≈ M 50 ). In fact, impact capacity can be significantly enhanced by providing hoops at 100mm spacing. The impact capacities of the 50% loaded column and axially loaded (P d , M o ) column are equal when hoops are provided at 50mm spacing for the 50% loaded column. Consequently, when the axial load is greater than 50%, the impact capacity of eccentrically loaded columns can be increased beyond the impact capacity of fully axially loaded (nominally confined) columns by reducing the hoop spacing s. This is not the particularly true for 300mm column due to flexure initiated failure conditions. Thus, if the hoops can be provided at 50mm spacing, further analysis may not be required for columns carrying over 50% of their full axial load and the corresponding moment under single axis bending. 6.11 Effects of the longitudinal steel ratio on the impact behaviour of columns In this investigation, the longitudinal steel ratio is increased by increasing the diameter of the longitudinal reinforcement while keeping the diameter of the column and steel configuration constant. Two load combinations namely (0.8P d , M 80 ) and (0.2P d , M 20 ) were selected in the analyses along with the axially loaded column. The effective load carrying capacity of the column increases with the longitudinal steel ratio even though the axial stresses on the columns are maintained constant. It is observed that the impact capacity of concrete columns increases as the longitudinal steel ratio ‘R’ increases Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 6-179 from 1% to 4%. This behaviour is contrast to that under (pure) axial loading conditions. Figure 6.22: Effect of longitudinal steel ratio on impact capacity In detail, the columns with 1% steel fail prior during the impact compared to the columns with 4% steel. The effects of steel ratio are considerable for load combination below the balanced point (0.2P d , M 20 ) where flexure is predominant and hence steel can yield under lateral impacts. On the other hand the longitudinal steel ratio has very little effect closed to the pure axial loading conditions (0.8P d , M 80 ) where columns fail primarily in shear. The lateral deflection of the concrete column increases with applied moment and hence the moment has considerable effects on the failure mode of the impacted columns. However the axial load on the column is the governing factor of the impact capacity of these columns. Therefore it is concluded that the impact capacity enhancement of the eccentrically loaded columns relates not only to tensile strength but also to shear characteristics and fracture toughness. Consequently DIF of tension, compression, shear and fracture energy must be taken into account for predicting the impact capacity of eccentrically loaded columns. 6.12 Confinement effects on the impacted columns with high steel ratio To investigate the effects of confinement, columns having 4% steel ratio with three different lateral steel spacing (s = 250, 100, 50) were considered. As the 80% loaded column (0.8P d , M 80 ) fail in shear critical conditions and the 20% loaded (0.2P d , M 20 ) columns fail in flexural failure conditions, the effects of the confinement on each Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 6-180 column could be different. Therefore 20% and 80% loaded columns were selected in the investigation. It is evident that the impact capacity increment due to the confinement effects is 17% and 19% on average for 20% and 80% loaded columns with hoops at s=50mm intervals respectively as shown in Figure 6.24. When the hoop spacing increases up to 100mm the corresponding improvements are 8% and 11% respectively. This means that the hoop spacing can improve the impact capacity of the columns equally irrespective of the failure mode. In other words strain rate effects accompanied with flexural failure modes have equivalent effects compared to the shear capacity enhancement of the columns. Consequently DIF for strain rate and shear may be equal under eccentric loading conditions where critical sections of the column section are under low compression. Having provided that the higher mode of vibration does not take place due to a vehicle impact, the enhanced capacity of the columns may be directly calculated by multiplying the impact capacity under nominal confined conditions by a factor within 1.08 to 1.19 within this axial loading range. Figure 6.23: Impact capacity enhancement due to confinement 6.13 A comparison of the confined columns with different steel ratios According to the above analyses the impact capacity of columns are minimum for the axial load combinations below the balanced point. However the following analyses revealed that the impact capacity could be even low when the steel ratio drops to 1% from 4%. For instance, considerable capacity drop encountered for 250mm hoop spacing under 20% axial loading (0.2P d , M 20 ). The columns with 4% steel ratio have the highest capacity particularly for 20% loaded columns. The main reason for this Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 6-181 observation is the flexure initiated failure conditions of the 20% loaded column despite the steel ratio where steel can yield and contributes to the capacity enhancement. Figure 6.24: Effects of the longitudinal steel ratio on impact capacity enhancement 6.14 Effects of the slenderness ratio and intensity of loading on impact capacity Peak force vs Slenderness ratio 0.25 0.75 1.25 1.75 2.25 2.75 3.25 3.75 3.0 5.0 7.0 9.0 11.0 13.0 Slenderness ratio P e a k f o r c e ( M N ) 300mm; P=20% 300mm; P=50% 450mm; P=20% 450mm; P=50% 600mm; P=20% 600mm; P=50% Peak force vs Slenderness ratio 0.00 0.50 1.00 1.50 2.00 2.50 3.0 5.0 7.0 9.0 11.0 13.0 Slenderness ratio P e a k f o r c e ( M N ) 300mm; P=20% 300mm; P=50% 450mm; P=20% 450mm; P=50% 600mm; P=20% 600mm; P=50% (a) 50MPa concrete with 1% steel (b) 30MPa concrete with 1% steel Figure 6.25: Effects of the slenderness ratio on capacity enhancement for columns The impact capacity improvement with the concrete grade (50MPa and 30MPa) and axial load intensity are shown in Figure 6.25. The 20% and 50% loaded columns {(0.2P d , M 20 ) and (0.5P d , M 50 )} were selected in the analyses. It is evident that there is no appreciable improvement of the impact capacity with reduction of the slenderness ratio irrespective of the concrete grade. A similar conclusion has been made for circular columns with low steel ratio (2.1%) subjected to the combined action of bending and torsion under earthquake loading conditions (Prakash et al. 2010). Despite the torsion the failure was predominantly flexural. According to the present analyses the capacity enhancement remains almost the same even when the steel ratio increases to 4%, although the failure mode changes from flexure to shear with the Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 6-182 improved longitudinal steel content. Therefore substantial capacity improvement cannot be expected by reducing the slenderness ratio. However, substantial improvement can be obtained with the enhancement of the axial load intensity on the columns. For instance, the impact capacity substantially increases when the axial load increases from 20% to 50% even though the enhancement slightly reduces with the effective height. 6.15 Strain rate sensitivity of eccentrically loaded columns Figure 6.26: Strain rate sensitivity of a ductile column Strain rate sensitivity of the eccentrically loaded columns could be different from the axially loaded columns due to the change of failure mode from shear to flexural shear with the application of the moment. To investigate this hypothesis, impact pulses ranging from 50ms to 150ms were applied on axially and eccentrically loaded (0.2P d ) 300mm columns (Figure 6.26). Amplitudes of the impact at near failure conditions were recorded over the 50ms and 150ms duration impacts. The improvement of the impact capacity is 18% for the column under eccentric loading conditions, compared with only 7% for the axially loaded column. This is a significant improvement of the impact capacity as there is initially a 70% reduction of the capacity due to the eccentric loading conditions. Even though the overall capacity is reduced, eccentrically loaded columns are less vulnerable to hard impact conditions and thus the impact capacity will be proportionately enhanced with the stiffness of the vehicle. Additionally the kinetic energy stored in the column decreases from 1200J to 700J due to the initial deflection. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 6-183 6.16 Linear equations for 20% and 50% loaded columns Having provided the impact capacities of columns for a range of different parameters, empirical relationships, based on the last square method, are developed in this section. These relationships can be used to quantify the critical impact force and the associated impulse for eccentrically loaded columns. Residual analysis is used to demonstrate the accuracy of the predicted equations and for further improvement of the accuracy of the outcomes. Residuals vs Pred Log P y = -0.4064x 2 + 4.8565x - 14.455 R 2 = 0.4651 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 5.0 5.5 6.0 6.5 7.0 Predicted Log P R e s i d u a l s Residuals vs Pred Log P y = -0.5038x 2 + 6.1004x - 18.401 R 2 = 0.4542 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 5.0 5.5 6.0 6.5 7.0 Predicted Log P R e s i d u a l s Figure 6.27: Residuals of the predicted Figure 6.28: Residuals of the predicted values (20%) values (50%) In fact, residuals were used in many procedures designed to detect various types of disagreement between data and an assumed model. In early days, the residual analysis was used only to produce stronger compelling conclusions. However, interest in residual analysis was renewed later by developing methods for assessing the influence of individual parameters. For example, the scatter plot of residuals versus fitted values that accompanies a linear least square fit is a standard tool used to diagnose nonconstant variance, curvature, and outliers. In order to identify such deficiencies associated with the data points, different residual based expressions were used. In this analysis, the residuals are calculated as (Observed (Log P) - Predicted (Log P)) and hence the positive and negative residuals indicate an under and over-prediction of the data points respectively (see Fig. 2.27 & 6.28). The important point is, a polynomial plot to the residual valves can be used to improve the accuracy of the results further and consequently, the corrected Log P c and Log I c are given by the Equations 6.2 and 6.3. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 6-184 6.16.1 Linear equations for 20% loaded columns The linear equations were derived based on least square method to assess the vulnerability of the 20% loaded columns subjected to impacts under negative eccentric moment in the plane of bending. Eq. 6.1 Eq. 6.2 Eq. 6.3 Where Log P is the logarithm of the Peak Force P (uncorrected), Log P c is the (corrected) logarithm of the Peak Force P, Log I c is the (corrected) logarithm of Impulse I, D is the diameter of the column in m, ρ v is the longitudinal steel ratio, H is the height in m and A h is the area of hoop in mm 2 . Even though the yield strength of hoops, ' sy f and hoop spacing, s included in the parametric analysis they are not appear in the final equation. This means that effect of the above two terms are negligible compared to the other parameters. With the introduction of the corrected equation, the over and under prediction of the Peak Force, P c is reduced up to ±10%. 6.16.2 Linear equations for 50% loaded columns Similar set of equations can be derived for 50% loaded columns with the corresponding moment as given in Equations 6.4 to 6.6. Once the critical impulse for 20% and 50% loaded columns are calculated, linear interpolation can be used to calculate the critical impulse for columns loaded in-between. Impact capacity of the column under pure axial loading already discussed in Chapter 5 so that comprehensive vulnerability assessment for all the loading points located on the interaction diagram are possible. Eq. 6.4 Eq. 6.5 Eq. 6.6 The valid range of the equations is given as follows; 6 . 4 001 . 0 856 . 2 051 . 0 ' 008 . 0 721 . 0 + + + − + = h A D H c f v P Log ρ 30 . 1 401 . 18 10 . 7 ) ( 504 . 0 2 − = − + − = c c c P Log I Log P Log P Log P Log 53 . 4 001 . 0 885 . 2 051 . 0 ' 007 . 0 645 . 1 + + + − + = h A D H c f v P Log ρ 30 . 1 455 . 14 856 . 5 ) ( 406 . 0 2 − = − + − = c c c P Log I Log P Log P Log P Log Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 6-185 6.17 Conclusions The main aim of this chapter was to analyse the impacted columns under single axis bending in order to identify the severity of the moment present in the columns. The following conclusions can be drawn from the findings. A sequence of axial load combinations were used in the analyses and it was observed that the impact capacity reduction (or enhancement) is gradual for the considered loading range. This allows linear interpolation to be implemented in the vulnerability assessment process. 1. Consequently, linear equations are generated to predict the critical impulse for two consecutive points on the interaction diagram namely; (0.5P d , M 50 ) and (0.2P d , M 20 ). Having provided the impact capacity of axially loaded columns, a comprehensive vulnerability assessment is possible for all the intermediate load combinations (maximum allowable design) on the interaction diagram. 2. Reducing the moment down to 50% (eg. 0.5P d , 0.5M 50 ) without changing the axial load (0.5P d ) has only minor effects on the impact capacity of the columns. On the other hand, columns subjected to positive eccentric loading generate conservative results compared to columns subjected to negative eccentric loading. Therefore reduced eccentric loading conditions and columns under positive eccentric loading can be excluded from the analysis process. 3. It was observed that the impact capacities of 50% loaded (0.5P d , M 50 ) confined columns with a corresponding moment, and 100% axially loaded (P d ) columns are equal if hoops are provided at 50mm spacing. Thus the hoop spacing alone can recover the capacity drop due to eccentric loading by up to 50% of the design axial m s m mm A mm MPa f MPa m H m MPa f MPa m D m b sy c v 25 . 0 050 . 0 1 . 113 27 . 28 500 250 4 2 50 30 04 . 0 01 . 0 60 . 0 3 . 0 2 2 ' ' < < < < < < < < < < < < < < ρ Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 6-186 load. However, it is not recommended to use larger diameter columns than that required to mitigate the impact damage. 4. The impact capacity of the columns increases with the longitudinal steel ratio of the columns. The effects of the longitudinal steel ratio are predominant in lightly loaded columns where considerable deflection takes place. Overall the results have shown that the yielding of the steel as well as the shear characteristics and fracture toughness of the concrete have considerable influence on the impact behaviour of the eccentrically loaded columns. Additionally, the DIF for strain rate and shear may be equal under eccentric loading conditions where the critical sections of the columns are under low axial compression. 5. The impact induced bending moment is predominant close to the bottom support and reaches its initial values at the residual stage. However, the peak moment generated in the columns during the impact does not have a significant effect even though it exceeds two times the ultimate static moment capacity of the columns. The resultant shear force also follows a similar behaviour and therefore it is concluded that the impacted column can display the same static lateral capacity as an undamaged column. However, the initial deformation due to load eccentricity greatly reduces the ultimate dynamic shear capacity of the impacted columns. 6. The cracks generated under a positive moment and the second mode of vibration are identical for smaller diameter columns even though the mechanisms behind the two processes are totally different from each other. Additionally, the combination of moment and shear force must be taken into account when determining the failure mode. This cannot be identified by analysing a simplified SDoF system under equivalent static loads. 7. Higher axial load intensity on columns would be a better solution for impact capacity enhancement compared to height (slenderness) reduction. Even though the overall capacity is reduced under eccentrically loaded conditions compared to axially loaded columns, eccentrically loaded columns are less vulnerable to hard impact conditions where the duration of the impact is less than 100ms. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 7-187 7. IMPACT ON COLUMNS UNDER BIAXIAL BENDING 7.1 Introduction Generally, structural columns are loaded with biaxial bending, rather than pure axial compression. Biaxially loaded columns in underground car parks, overpass bridges and medium to low rise buildings located close to major roads are extremely vulnerable to vehicle impacts due to the initial deformation present in the columns. The bi-axially loaded columns behave in a complex manner in response to the impact loading at a shear critical height due to the complexity of the load combinations. However, there exits minimal data on dynamically tested columns under biaxial bending (Zahn et al. 1989) while the existing experiments on bi-axially loaded columns are limited to quasi-static or pseudo-dynamic loading such as earthquake (Wong et al. 1993). On the other hand, most of the experiments focused on nonlinear flexural deformation which is usually decoupled from shear (Xiao and Zhang 2007) and torsion (Prakash et al. 2010) even though columns in mild tension or reduced compression have the tendency to reduce the shear capacity (Papazoglou and Elnashai 1996). In fact, presence of torsion along with shear and bending initiates shear failure (Prakash et al. 2009). These observations implied that the impact capacity of bi-axially loaded columns may be substantially different from the columns under pure axial loading. On the other hand, impact induced torsional moments can affects the flow of internal forces and deformation capacity of impacted columns particularly when the impact force is applied perpendicular to the direction of bending by changing the failure mode. Therefore equation developed to quantify the impact capacity under pure axial loading or uni-axial bending may not be adequate in routine column design. Moreover, design codes (AASHTO-LRFD 1998; EN 1991-1-7:2006) and existing guidelines (Tsang et al. 2005) do not account for the bi-axially loaded columns under lateral impact loading. Consequently, there is a pressing need for the development of some simplified yet rational guidelines to quantify the impact capacity under biaxial bending. To address this issue, extensive numerical simulation have been conducted to study the Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 7-188 response of biaxially loaded impacted columns and the results used to develop empirical equations to quantify the critical impulse. Having provided that the vulnerability assessments of the eccentrically loaded columns are independent of confinement characteristics resulting from strain gradient, the validation process was extended to assess the vulnerability of biaxially loaded columns. The parametric study is limited to 30MPa to 50MPa concrete by avoiding High Strength Concrete (HSC). The 300mm to 600mm diameter columns adequate in capacity for 5 to 15 storey buildings with the longitudinal steel ratio ranging from 1% to 4% are investigated in detail. The structural design was based on the Australian Standard AS3600 (2004). Additionally, dependency of the duration of the impact on the enhanced stiffness characteristics of the columns due to axial load variation was neglected in the analysis. In particular, impact angle was taken into account in the analyses and full column was used in the numerical simulations where the moment application lead to unsymmetrical loading. One of the main objectives was to identify the severity of the moments present in those columns. However, the analyses process for biaxially loaded concrete columns under lateral impact is difficult due to large number of load combinations involved in the analyses process. If the influence of the configuration of the longitudinal steel is negligible, the bi-axially loaded columns can be numerically simulated by using two eccentric axial loads acting in two orthogonal directions. Thus, the effects of the direction of the impact within 0 o -90 o can be accounted by varying the eccentric load on the two perpendicular planes. The remaining impact angles were treated separately and the non-critical cases excluded from the analysis to simplify the process. Consequently, the entire analyses were focused on two load combinations on the interaction diagram namely; (0.5P d , M 50 ) and (0.2P d , M 20 ) where P d is the design axial load capacity of the column and M 50 and M 20 are the corresponding moments as shown in Figure 7.1. The aim was to define three consecutive points on the interaction diagram along with respective critical impulses so that linear interpolation can be used to quantify the critical impulses for points in-between (see Fig. 7.1). In the process, effects of longitudinal and lateral steel ratios, concrete grade, direction of the impact, strain rate sensitivity of columns and slenderness ratio were also quantified. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 7-189 Figure 7.1: Impact capacity prediction for intermediate load combinations 7.2 Numerical simulation of biaxial loaded columns A typical interaction diagram derived for circular columns under biaxial bending is shown in Figure 7.2. Cases (a) and (b) are the uni-axial bending about the two principle directions, X and Y. The interaction curve represents the failure envelope for various combinations of axial loading and bending moments. As far as the biaxial bending of the column is concerned, the methods of equivalent uni-axial eccentricity will give a better understanding of the depth and inclination to the neutral axis. Once the depth and inclination are determined, the corresponding interaction curve can be easily established. By establishing such curves for various radial distances L, the failure or allowable stress surface for biaxial bending can be constructed. The interaction diagram may be constructed by interpolation of the uni-axial bending cases if the differences introduced by the configuration of the lateral steel are neglected. Based on this assumption, the neutral axis can be taken as perpendicular to the direction of eccentricity of the resultant force. Thus different values for the bending moments are selected along the maximum allowable service stress contour for one particular axial load. Two of the selected values (Cases (a) and (b)) are located directly on the uni-axial bending planes X and Y, while the other one is located in-between (Case (c)). The Case (c) represents the resultant allowable moment which requires one particular steel ratio about the X or Y axis. According to these observations, the influence of the longitudinal steel configuration on the orientation of the neutral axial is higher for small diameter columns, and reduces as the diameter increases. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 7-190 Figure 7.2: Typical interaction diagram for circular columns under biaxial bending The biaxial loading was introduced to the columns by using two eccentric loads acting in two perpendicular directions X and Y (see Fig. 7.3(a) and (b)). The plate placed at the centre of the bulk head was used to apply the axial load, while the eccentrically placed plates were used to apply the moments. Since the end block was modelled to have excessive shear and flexural strengths, rigid material characteristics were assigned to the bulk head. The plates had known surface areas and eccentricities, thus allowing known eccentric loads to be applied about each axis. Fully fixed conditions between the head and plates were assumed, to avoid loss of contact during the lateral deformation of the column during impact. The moments were applied after the axial load by using two separate ramp functions to avoid flexural failure conditions at the load application stage. The impact force on the column was applied parallel to the X direction after the vertical loads stabilised. The impact can be considered as oblique as far as the resultant eccentric load on the column is concerned. The uni-axial load combinations were also taken into account along with the biaxial moments for comparison purposes. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 7-191 (a) Front elevation of the biaxially loaded column (b) Plan view of the top Figure 7.3: Numerical model of the column under biaxial bending Due to non symmetry of the axial loading, the entire column was modelled in the numerical simulation instead of using a half model (see Fig. 7.3(a)). A 50mm cover was provided for the reinforcement and closed hoops having yield strength 350MPa were varied from 250mm to 50mm intervals. Complete strain compatibility was assumed between the embedded steel bars and concrete, while an elastic perfectly plastic material model was used for both the longitudinal and transverse reinforcements. Axial loads were applied as ramp up surface pressures while translations and rotations were constrained at the bottom nodes of the columns to simulate fixed end conditions. Movement of the bulk head was constrained only in the lateral directions X and Y, while allowing rotations about both these directions so that it could move in the vertical direction and deflect as the column deformed. 7.3 Characteristics of the simulated columns Columns having diameters from 300mm to 600mm with 1% steel were considered in the initial stage of the analysis while 6, 8 and 12 longitudinal steel bars were equally distributed along the perimeter of the 300mm, 450mm and 600mm columns respectively. The effective height of the columns was 4m and 50MPa concrete with nominal transverse reinforcement was used. Approximately 25mm long hexagonal solid elements with one point integration used for concrete while 25mm long beam Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 7-192 elements with 2×2 Gauss integration were used for longitudinal steel. The bulk head was discretized so that the mesh generation of the head was compatible with that of the column along the lateral direction. The direction of the displacement of the column is not exactly in the X direction as in pure axial loading. Thus an additional set of nodes in the perpendicular direction is needed to obtain the torsion based deformation pattern. Table 7.1: Biaxial load combinations on the 300mm column under 50% axial load Load combination 0.5P d (kN) M xs (kNm) M ys (kNm) Steel ratio (X-X) Steel ratio (Y-Y) 1 1000 0 47 0.40 1.12 2 1000 18 40 0.40 1.14 3 1000 37 37 0.40 1.25 4 1000 40 18 1.14 0.40 5 1000 47 0 1.16 0.40 The structural design is based on the Australian Standard AS3600 (2004). Moments M xs and M ys were applied about the X and Y directions of the columns so that the longitudinal steel requirement along the major axes remained close to 1%. During the initial analysis, 50% of the design axial loading capacity (0.5P d ) was maintained with the corresponding maximum moment (M 50 ) so that eccentricities in loading comprehensively agree with the interaction diagram. The design axial load capacity P d was calculated from Eq. 4.1. Even though this analysis was conducted by assuming 500MPa for longitudinal steel, the results can be extended to other steel grades by using the equivalent steel area method proposed by Shi et al. (2008). This allows the impact behaviour of the columns to be investigated under conditions in which flexural failure predominates. Typical load combinations and steel ratios for the 300mm column are given in Table 7.1. The load combinations may occur even in a three storey building column and hence a comparison based on the number of storeys is not considered. 7.4 Selection of load combinations 7.4.1 Impact capacity of columns under biaxial bending Figures 7.4(a) to (c) compare the impact capacities of the columns for the five different combinations of moments about the X and Y axes, which bring the columns to the ultimate limiting stress under lateral impact conditions. As far as the 300mm column is Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 7-193 concerned, the impact capacity is highest when the moment acts about the Y axis. However, for the 450mm and 600mm columns the maximum capacity is obtained when the moment acts about their X axes even though the difference is small. In addition, the minimum capacity is obtained when the moment is applied perpendicular to the direction in which the maximum capacity is obtained for each column, particularly with 0.5P d loading. Under the remaining allowable biaxial moment combinations, the impact capacities vary almost linearly between the maximum and the minimum. This is evidence of the accuracy of the theoretical approach described in Figure 7.2 for determining the capacity of the columns under biaxial loading. Once the impact capacity of the column is determined in two perpendicular directions, the impact capacity under biaxial bending can be interpolated from the known capacities under single axis bending. Here the impact force has to be applied along either the X or Y axes. 1 2 3 4 5 0 5 10 15 20 25 30 35 40 45 50 Impact capacity of 300mm column under biaxial bending Peak force (x25kN) Moment about X axis Moment about Y axis 1 2 3 4 5 0 50 100 150 200 250 300 Impact capacity of 450mm column under biaxial bending Peak force (x25kN) Moment about X axis Moment about Y axis (a) Impact capacity of the 300mm column (b) Impact capacity of the 450mm column 1 2 3 4 5 0 100 200 300 400 500 600 700 Impact capacity of 600mm column under biaxial bending Peak force (x25kN) Moment about X axis Moment about Y axis (c) Impact capacity of the 600mm column Figure 7.4: Impact capacities of the columns under biaxial bending In fact, the impact capacities of the columns under each load combination do not Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 7-194 deviate considerably, contrary to what would be expected. The shear failure characteristics do not allow the impacted columns to deform substantially, particularly for larger diameter columns. Hence the secondary moments generated by the load eccentricities due to the buckling effects are minimised. Therefore, the 450mm and 600mm columns behave identically under the impact loading. Consequently, their maximum and minimum impact capacities under single axis bending in the two perpendicular directions differ by only 6% for both columns. Thus, only the impact capacity under single axis bending would be sufficient for preliminary design calculations of the circular columns susceptible to shear failure conditions. In fact, this is an indication of the dilute coupling action between the biaxial moments, in addition to the coupling effects between axial force and bending moments, particularly for shear critical impacted columns. However, the response is substantially different for the 300mm column where significant bending takes place with all the moment combinations. The impact force enhances the lateral deformation of the column which had already initiated under the single axis bending. With the substantial flexural deformation, the longitudinal steel yields and enhances the impact capacity while activating the tensile strength of the concrete. Consequently, the impact capacity of the column is greatest when the moment is applied about Y axis. The minimum deflection takes place when the impact force is perpendicular to the direction of bending (X) and the ratio between the maximum and the minimum impact capacities is around 16% for the 300mm column. The torsional moment generated during the impact was responsible for this deviation other than the coupling action between the biaxial moment and axial load. Thus, comprehensive investigation is recommended for the columns with flexural characteristics. It is also worth noting that the technique used to simulate the eccentric load on the 300mm column also contributes to the capacity reduction when the excessive lateral deformation takes place. For instance, even though the lever arm used to generate the moment is nearly the same for all columns, it produces secondary moments that would increase the deflection of small diameter columns. Therefore this can be considered as error induced by the modelling techniques and the error can be minimised by Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 7-195 increasing the lever arm. However despite all the deviations, three load combinations including single axis bending about two perpendicular axes along with one intermediate load combination are sufficient to predict the impact capacities of the columns for the entire range of load combinations. Thus the number of load cases reduces to three, which includes the one with equal moments about both perpendicular axes. As far as critical velocities are concerned, the 300mm column with the biaxial bending will be vulnerable for velocities ranging from 12.5ms -1 to 14.5ms -1 (45 to 52 km/h) generated by typical car impacts, while 450mm and 600mm columns in urban areas will not be vulnerable to car impacts under these conditions. 7.5 Effects of the direction of the impact So far the analysis has been conducted by assuming that the direction of the impact is along either one of the major axes X or Y (see Fig. 7.5(a)). However a crucial situation is encountered when the direction of the impact is in the same direction as the resultant biaxial moment (see Fig. 7.5(b)). This is particularly the case when the equal (biaxial) moments are applied about the two perpendicular directions where the vector summations of the two moments may exceed the maximum allowable moment under single axis bending. On the other hand, no firm prediction can be made due to the variations in initial deflection and stress distribution under the biaxial bending. Hence the analysis was extended to investigate this isolated case and Figure 7.5 compares the plan view of the two column heads. (a) Impact is along the X axis (b) Impact is along the resultant moment Figure 7.5: Simulation of the effects of the direction of the impact Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 7-196 Columns made of 50MPa concrete having an effective height of 4m with 1% steel were considered in the analysis. The initial simulations were conducted on 300mm, 450mm and 600mm diameter columns by assigning equivalent axial loads on the eccentric plates to simulate the equal moments. The plates were placed at 45 o on either side of the impact so that the resultant force acted along the direction of the impact. Comparison was made with single axis bending about the Y axis (see Fig. 7.5(a)), as the two scenarios are almost identical. The results revealed that the impact capacity under the symmetric biaxial moments is always higher than the capacity under single axis bending about the Y axis. The resultant capacity increase varies from 3% to 1.2% for the 300mm and 600mm columns respectively. It is also evident that all the columns fail in flexure, and the resultant deflection characteristics are identical for each column under single axis bending (Y axis) and biaxial bending. Based on these results it is concluded that the oblique impacts do not significantly contribute to the vulnerability of the columns. Consequently the vulnerability under oblique impact loads can be predicted by applying the impact force along the principle (X or Y) axes. Thus this observation significantly reduces the number of cases that must be taken into account in the analysis process, particularly for circular columns under biaxial bending. 7.6 Effect of reduced axial load on biaxial bending In the next stage the axial load was reduced from 0.5P d to 0.2P d and the impact behaviour under biaxial loading was investigated with corresponding moments (0.2P d , M 20 ) (see Fig. 7.1). According to the results, except in the 300mm diameter column, there is considerable capacity reduction in the 450mm and 600mm columns. Moreover, the impact capacities under biaxial bending and single axis bending about the Y axis are equal in the 20% loaded 450mm column. In contrast, the capacity under biaxial bending exceeds the capacities under single axis bending for the 300mm and 600mm columns. Here the biaxial moment consists of equivalent maximum moments applied about both axes. On the other hand, the 600mm column is slightly more vulnerable under uni-axial bending when the moment is applied about the Y axis while the 300mm column is more vulnerable when the moment was applied about the X axis. Consequently it is quite difficult to explain the exact reasons for this behaviour particularly with the 20% axial loading, due its interaction with torsion, secondary Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 7-197 moments and axial load fluctuation during the impact. In fact, when the stress reaches the elastic limit under an axial force and biaxial bending, the columns tend to generate curvatures about each principle axis. The initiation for the translation and rotation is more significant for small diameter columns than for larger diameter columns, particularly after cracking occurs under the impact. Once the cracking begins, the axial force and the bending moments acting on that section will tend to shift towards the gross section with an additional secondary moment resulting from the eccentric loading. The secondary moment will depend on the amount of deformation and consequently the deformation about each principle axis of the gross section is affected by the respective moment about each axis. As the biaxial load combinations depend on the axial load level, the impact behaviour of the columns is changed with the axial load level. On the other hand, the hypothetical neutral axis of the deformed column due to the impact does not coincide with that of the axial force and biaxial moments. Hence there is no simple technique which can predict the impact behaviour of the column under biaxial bending by interpolating the capacities under single axis bending, particularly for the small diameter columns where the effects of secondary moments and torsion are severe due to the larger deformations and relatively small gross area remaining in the column after cracking. In fact, the 300mm column under biaxial bending with 1% steel has almost the same capacity under 20% and 50% loading even though there is a substantial difference in the ultimate deflection between each case The ultimate deflection increases from 20mm to 55mm as the axial load reduces from 50% to 20%, and this is the expected behaviour in flexure dominated columns. However, the enhanced strain rate due to the deflection does not always compensate the capacity reduction resulting from the diminished axial load. For instance, the impact capacity is reduced in the 20% loaded 300mm column with single axis bending (either about the X or Y axis) despite its enhanced deflection characteristics. However, despite all these differences, the impact capacity of the column under biaxial bending can be conservatively taken as the minimum of the capacities under single axis bending without further investigation. In particular, this comment applies when the direction of the impact is along one of the principle axes of bending, and it can be applicable to both 50% and 20% loading. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 7-198 7.7 Effects of longitudinal steel ratio on biaxial bending The effects of the longitudinal steel ratio were investigated by increasing the steel ratio from 1% to 4% without changing the configuration. The axial load on the column was 0.5P d and the corresponding biaxial moment was applied on columns having diameters of 300mm, 450mm and 600mm. Increasing the steel ratio to 4% enhances the axial load capacity by 20% to 30%, and the moment capacity by 40% to 50% compared to the columns with 1% steel. When the steel ratio increases to 4%, the impact capacity of the 300mm column increases around 17%, while the 450mm and 600mm columns show 3% and 5% reduction respectively. As far as the 300mm column with 1% steel is concerned, the impact capacity primarily results from the flexural-shear behaviour of the column where the longitudinal steel can yield under flexure and contribute to the capacity enhancement (see Fig. 7.6(a)). However, the 300mm column with 4% steel fails in shear due to the extra stiffness introduced by the additional steel content. Similarly, the 450mm and 600mm columns also stiffen with the 4% steel, and hence during the impact the longitudinal steel buckles due to shear deformation and the columns fail soon after this event (see Fig. 7.6 (d)). Figure 7.6: Failure characteristics of 50% loaded 300mm columns under biaxial bending The 450mm and 600mm columns always failed by shear despite the biaxial moment induced deflections or the effects of the longitudinal steel ratio. The longitudinal steel buckles prematurely due to shear failure induced lateral deformations. Consequently, there is no substantial contribution from the longitudinal steel to the impact capacity of Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 7-199 the columns. The results also show that the resultant deflection characteristics also reduce with the increase of the longitudinal reinforcement. Consequently, the extra axial load and moment applied on the column, which are supposed to be carried by the additional steel content, will transfer to the concrete once the longitudinal steel fails. Thus the columns with high longitudinal steel ratios are more susceptible to shear failure and hence their impact capacity will be further reduced. Therefore, increasing the longitudinal reinforcement ratio will not always improve the impact capacity of the columns under biaxial bending. In general, all the columns with 4% steel fail without developing their full flexural capacity due to premature buckling of the longitudinal steel at an early stage of the impact. Consequently, the 17% capacity enhancement observed in the 300mm column is due to the change of the failure mode from flexure to shear. However it may not always be true to expect that there is a capacity enhancement when the failure mode changes from flexure to shear as in the 300mm column. The motives are different in this particular case where there is extra axial load and moment applied on the column with 4% steel compared to the column with 1% steel, which may alter the internal stresses at the ultimate stage even though theoretically the axial stress on the concrete is the same. 7.8 Effects of biaxial bending on 20% loaded impacted columns with a 4% steel ratio The mode of failure also depends on the amount of the axial load present in the column at the time of impact. For instance, columns with 20% axial load may tend to fail in flexure while columns with 50% axial load may tend to fail in shear. Therefore it is worth identifying the impact behaviour of the column with low axial loading. However, enhancing the longitudinal steel will further stiffen the columns as discussed in the previous paragraph. Consequently, the impact behaviour of lightly loaded columns with a high steel ratio leads to a complex situation. In addition, higher concrete grades are also responsible for brittle failures and hence the effects of biaxial bending on 20% loaded impacted columns made of 50MPa concrete are investigated in the following section. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 7-200 The same columns having diameters from 300mm to 600mm with 4% steel are selected and their impact capacities are compared with 50% loaded columns. In general, the shear failures of the 50% loaded columns with 4% steel transform into flexural failures under 20% axial loading and corresponding moment. The flexural failure of the 300mm column occurs without any longitudinal steel buckling. Even though the longitudinal steel yields at the time of failure, the impact capacity reduces by 30 to 40% compared to the columns with 50% loading. The reason for this is the substantial reduction of the axial stress in the 20% loaded columns, and the conditions further worsen due to the corresponding moment which is nearly two times that under 50% loading. The initial deflection of the column is an indication of its vulnerability, and is more predominant under single axis bending about the Y axis where the deformation is in the same direction as the impact. Thus it represents the most critical case. As far as 300mm column is concerned, applying the moment about the X axis is also more critical than applying a biaxial moment. The secondary moments generated by the load eccentricity that corresponds to the moment about the X axis is the reason for this observation. The impact capacities of the 450mm and 600mm columns also decrease under the 20% loading. It is observed that there is longitudinal steel buckling followed by flexural deformation of the 450mm columns. This signifies the transition stage of the failure mode from flexure to shear depending on the diameter of the column. Thus the 600mm column always fails in shear as expected despite the axial load reduction. Thus far the analysis has confirmed that the impact capacity under bi-axial bending can be conservatively taken as the minimum of the two capacities under single axis bending if the impact force is applied along one of the principle axes of bending. The calculated value based on this assumption may be increasingly over-conservative when the percentage of the axial load decreases, the diameter of the column decreases or when the longitudinal steel ratio decreases. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 7-201 7.9 Damage mitigation of the impacted columns under single axis bending Peak force vs Diameter 0.25 0.75 1.25 1.75 2.25 2.75 3.25 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 Diameter (m) P e a k f o r c e ( M N ) X-X; 1%; P=20% X-X; 1%; P=50% X-X; 4%; P=20% X-X; 4%; P=50% Peak force vs Diameter 0.25 0.75 1.25 1.75 2.25 2.75 3.25 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 Diameter (m) P e a k f o r c e ( M N ) Y-Y; 1%; P=20% Y-Y; 1%; P=50% Y-Y; 4%; P=20% Y-Y; 4%; P=50% (a) Moment about X-X axis (b) Moment about Y-Y axis Figure 7.7: Peak force vs. slenderness ratio for 4m high columns made of 50MPa concrete The impact capacities of the columns bending about two orthogonal axes discussed so far are summarised in Figures 7.7(a) and (b). The figures reflect the effects of the diameter on individual columns under the varying steel ratio and axial stresses. It can be seen that the impact capacity reaches its maximum and minimum when the columns with 1% steel carry 50% and 20% loads respectively. The impact capacity of columns with 4% steel is in-between the capacities for 1% steel, particularly for larger diameter columns. It is important to note that both the axial load and moment carrying capacity of the columns increase with the longitudinal steel content. Thus, 20% loaded larger diameter columns with 4% steel are more effective compared to the columns with 1% steel. However the opposite is true for 50% loaded columns. At this stage it would be better to rely on axial loading rather than the steel content for damage mitigation as the axial stress increment generated the highest capacity enhancement. 7.10 Effects of the confinement on biaxial bending 7.10.1 Impact behaviour of 50% loaded columns with 4% steel under biaxial bending An investigation was conducted to improve the ductile characteristics of 50% loaded columns with 4% steel, by providing transverse steel at a closer spacing where the columns failed without developing their full flexural capacity due to premature buckling of the longitudinal steel. The hoop spacing was reduced from its nominal value of 250mm to 50mm and capacity enhancement due to the closer hoop spacing was simulated by assigning enhanced concrete characteristics to the core concrete by using the equation proposed by Mander et al. (1988). Single axis bending about the X Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 7-202 and Y axes was also considered for comparison purposes. The impact force was applied along the X axis as shown in Figure 7.5. The impact capacities are compared with 50% loaded columns with nominal confinement. With the enhancement of the confinement characteristics, the impact capacity of the 300mm column increased by 20% and 30% when the moment was applied about the X and Y axes respectively. Therefore the strength enhancement due to the confinement is not proportionate to the initial impact capacity of the column under nominal confined conditions. According to the numerical results, the ductile characteristics of the impacted column are improved with the confinement particularly when the moment is applied about the Y axis. However, the enhancement of the ductility may not be the only reason for the observed enhancement of the impact capacity of the column. In fact, the failure mode also changes from shear to flexural shear due to the confinement effects. Consequently the steel buckled at a later stage only at one point compared to the nominally confined columns where the longitudinal steel buckled at two separate sections simultaneously. In addition, the location of the shear failure plane moved further downwards while the remaining part of the column was subjected to flexural conditions. Thus, the 20% capacity enhancement occurred mainly due to the enhancement of ductility and a change of the failure mode. However, the 30% capacity enhancement mainly results from the enhancement of ductility alone as there is no significant change in the failure mode. Thus, in general, the reasons behind the capacity change of the 50% loaded 300mm column are two-fold. The first reason is the change of the failure mode from shear to flexure, and the second reason is the enhancement of ductility due to the confinement effect. As far as the 450mm and 600mm columns are concerned, there is no considerable variation of the capacity enhancement compared to the 300mm column. The average enhancement is around 15% and remains constant despite the axis of bending as there are no changes of the shear failure mode due to the confinement effects. Therefore the capacity enhancement solely depends on the shear capacity enhancement. Consequently is it concluded that the ultimate shear capacities of the columns do not depend on the orientation of the moment even though the resulting shear forces may increase with the confinement effects. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 7-203 7.11 Behaviour of 20% loaded confined columns with 4% steel under biaxial bending Impacted columns with 4% steel were investigated to identify the confinement effects on partial loading conditions which are associated with biaxial bending moments. The moments are also applied about the X and Y axes separately for comparison purposes while the impact force is applied along the X axis. The impact capacities are compared with 20% loaded columns with nominal confinement. The results showed that the impact capacities of the columns significantly reduce with the 20% loading. The capacity enhancement due to the confinement is around 16% on average for all the 20% loaded columns despite the diameter. The capacity enhancement of the 20% loaded 300mm columns is reduced compared to that of the 50% loaded 300mm columns. However, as far as the 450mm and 600mm columns are concerned there is a slight increase of the capacity compared to that under 50% loading. Therefore there is no general rule that can be used to determine the capacity enhancement due to the confinement effects as it depends on the diameter and loading conditions of the columns. All these individual cases were scrutinised to identify the source of the variation. In fact, the 300mm columns fail in flexure without significant shear deformation despite the level of confinement and the axis of bending. Consequently the strength enhancement is solely due to the longitudinal steel yielding under flexural deformation. On the other hand there is no change in the failure mode for the 450 and 600mm columns due to the confinement. Thus the capacity enhancement is constant for the 20% loaded columns with 4% steel. 7.12 Impact behaviour of 50% loaded columns with 1% steel under biaxial bending The effects of the confinement were investigated for columns with a low steel content. The strength enhancement was compared with nominally confined 50% loaded columns with 1% steel. It was observed that the results follow the conventional theories of confinement effects. For instance, there is a gradual decrease of the Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 7-204 capacity enhancement as the diameter increases. However there are no proper relationships as far as each axis of bending is concerned. The average capacity enhancement in the 300mm column is around 24% and gradually decreases to 18% and 16% for 450mm and 600mm columns respectively. This observation arises from the fact that the strength enhancement due to the confinement effects does not effectively change the mode of failure of the columns. The failure mode is predominantly flexure irrespective of the diameter of the columns. Moreover, there is a considerable ductility enhancement with the confinement effects particularly for 300mm columns. The considerable ductile behaviour with low steel content is a unique feature of the bi-axially loaded impacted columns. 7.13 Impact behaviour of 20% loaded columns with 1% steel under biaxial bending The investigation continued with the 20% loaded confined columns with 1% steel. The 6mm hoops made of 350MPa steel were placed at 50mm intervals. The capacity enhancement was compared with nominally confined columns with 20% loading. The behaviour of 20% loaded columns totally differs from 50% loaded columns. For instance, the average capacity enhancement is 6%, 17% and 29% for 300mm, 450mm and 600mm columns respectively. This indicates an increase of the capacity enhancement with the increase of column diameter. The 300mm column with 1% steel achieved a substantial ductile capacity with 20% loading which cannot be improved further by providing hoops at closer intervals. Therefore the capacity enhancement due to confinement is only marginal. However, the limited ductile characteristics of the 450mm and 600mm columns substantially improved with the closer hoop spacing. Thus the enhanced ductility increases the capacity of those columns to a considerable level. 7.14 Effects of the steel grade and diameter of the hoops So far the investigation has been conducted by assuming either nominally confined concrete columns with 250mm hoop spacing or confined concrete columns with 50mm hoop spacing. Instead of changing the hoop spacing, the impact capacity can Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 7-205 also be increased by increasing the yield strength or the diameter of the hoops. In fact, it was identified that the collapse load rapidly increases as the hoop spacing decreases, particularly less than 100mm. Therefore 100mm is considered as the optimum hoop spacing. Thus the impact capacity enhancement of the columns was estimated by changing the yield strength and diameter of the hoops while keeping the hoop spacing at 100mm. The diameter of the hoops was changed from 6mm to 12mm while the yield strength of the hoops was changed from 250MPa to 500MPa. Impact capacity vs Hoop spacing 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0 50 100 150 200 250 300 350 400 Hoop spacing (mm) P e a k f o r c e ( M N ) 300mm; X-X 450mm; X-X 600mm; X-X 300mm; Y-Y 450mm; Y-Y 600mm; Y-Y Figure 7.8: Impact capacity of 20% loaded columns under varying hoop spacing A comparison of collapse loads of 300mm, 450mm and 600mm columns under varying hoop spacing for 20% loading is shown in Figure 7.8. The capacity is highest when the moment is applied about the X axis, which is parallel to the direction of the impact. The difference between the collapse loads for bending about the X and Y axes increases with the diameter of the column. Peak force is substantially higher for hoops spacing less than 100mm and peak force is decrease as hoop spacing increase. Moreover, the impact capacity under biaxial loading with equal moments about both axes either falls in-between the peak forces under single axis bending (ei. X and Y), or else exceeds the peak forces under single axis bending. Other than the biaxial moment, the axial load plays a major role in determining the impact capacities of the columns. The impact capacity increases with axial load and the corresponding moment on the column. For instance, the impact capacity increases from 16% to 52% in 600mm and 300mm columns respectively, when the axial load increases from 20% to 50% along with the corresponding moment. The capacity Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 7-206 increase due to axial load is insensitive to the hoop spacing as shown in Figure 7.9. Thus the medium biaxial loadings do not effectively enhance the confinement effects. Impact capacity vs Hoop spacing 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0 50 100 150 200 250 300 350 400 Hoop spacing (mm) P e a k f o r c e ( M N ) 300mm; X-X; 20% 450mm; X-X; 20% 600mm; X-X; 20% 300mm; Y-Y; 20% 450mm; Y-Y; 20% 600mm; Y-Y; 20% 300mm; X-X; 50% 450mm; X-X; 50% 600mm; X-X; 50% 300mm; Y-Y; 50% 450mm; Y-Y; 50% 600mm; Y-Y; 50% Figure 7.9: Impact capacities of 20% and 50% loaded columns under varying hoop spacing The continuous lines in Figure 7.10 express the impact capacity enhancement of the 20% loaded columns when the diameter of the hoops is increased from 6mm to 12mm. The highest capacity enhancement is obtained by the 300mm diameter column even though its impact capacity does not depend on the axis of bending. On average, 12% to 35% capacity enhancement is possible by increasing the bar diameter alone. The impact capacity enhancements of the 50% loaded columns are shown by the dotted lines in Figure 7.10. Columns carrying larger axial load and corresponding moment have the highest capacity irrespective of the diameter of the columns. From a comparison point of view, the relative capacity enhancement is highest when the hoop diameter is increased compared with when the hoop spacing is reduced. Hence increasing the diameter of the hoops is more effective than reducing the hoop spacing. Impact capacity vs Hoop diameter 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 5 6 7 8 9 10 11 12 13 Hoop diameter (mm) P e a k f o r c e ( M N ) 300mm; X-X; 20% 450mm; X-X; 20% 600mm; X-X; 20% 300mm; Y-Y; 20% 450mm; Y-Y; 20% 600mm; Y-Y; 20% 300mm; X-X; 50% 450mm; X-X; 50% 600mm; X-X; 50% 300mm; Y-Y; 50% 450mm; Y-Y; 50% 600mm; Y-Y; 50% Figure 7.10: Impact capacities of 20% and 50% loaded columns under varying hoop diameter Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 7-207 Columns with 50% axial loading are always safe compared to those with 20% loading despite the axis of bending. This behaviour is common for various load combinations studied so far by changing the confinement effects. However, the capacity enhancement obtained by changing the yield strength is only marginal. The average capacity enhancement due to yield strength is lower than the enhancement obtained by varying either the bar diameter or hoop spacing. On average it varies from 5% to 7% (see Fig. 7.11). Therefore, increasing the yield strength is not recommended as a damage mitigation technique for impacted columns. On the other hand, the potential for yielding of the lateral reinforcement is reduced with the reduction of the axial load (Watson et al. 1994). Consequently, hoops with a yield strength exceeding 500MPa are not recommended for columns subjected to impact loads. Therefore, the maximum capacity enhancement is gained by changing the diameter of the hoops, or alternatively the hoop spacing can also be changed. Impact capacity vs Yield strength 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 200 250 300 350 400 450 500 550 Yield strength (Nmm -2 ) P e a k f o r c e ( M N ) 300mm; X-X; 20% 450mm; X-X; 20% 600mm; X-X; 20% 300mm; Y-Y; 20% 450mm; Y-Y; 20% 600mm; Y-Y; 20% 300mm; X-X; 50% 450mm; X-X; 50% 600mm; X-X; 50% 300mm; Y-Y; 50% 450mm; Y-Y; 50% 600mm; Y-Y; 50% Figure 7.11: Impact capacities of 20 and 50% loaded columns under varying yield strength On the whole, the direction of the impact still has considerable effect particularly for the confined columns. When the moment is applied in two orthogonal directions, the percentage variation of the capacity is around 15% on average for 600mm columns. In general, the difference is marginal for 300mm columns and therefore the direction of the impact has only minor effects on the impact capacity of the 300mm diameter columns. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 7-208 7.15 Effects of slenderness ratio on impact capacity of columns under biaxial bending The effectiveness of the reduction of slenderness ratio as a damage mitigation technique is also of interest, in particular for smaller diameter columns. The effective height of the columns was reduced over a range from 4m to 2m. Figures 7.12 and 7.13 represent the peak impact force that can be withstood by the columns with different slenderness ratios and steel ratios, and the capacity enhancements due to increasing the axial load intensity. The axial load is represented as a percentage of the design load capacity of the columns and the corresponding moment is applied on the column in two perpendicular directions. Figures 7.12(a) and (b) and Figures 7.13(a) and (b) represent the impact capacities when the moment is applied about the X and Y axes respectively. The load combinations consist of maximum moment about one axis and zero moment about the perpendicular axis simultaneously. The impact pulse is applied along the X direction as usual. The allowable moment on the 300mm column slightly increases due to the reduction of the effective height. However, the allowable moments on the 450mm and 600mm columns are not affected by the effective height. As a whole, the columns are relatively less vulnerable if the axial load and corresponding moment increase simultaneously. Conversely, the columns are more vulnerable if the moment is applied on an axis perpendicular to the direction of the impact (i.e. Y). In fact, the percentage increase of the impact capacity resulting from a decrease in the slenderness ratio significantly varies with diameter and direction of the applied moment. Therefore, a firm conclusion cannot be drawn on the capacity enhancement due to slenderness. However, reducing the effective height is more effective for small diameter columns. Thus, the most reliable way to enhance the impact capacity is to maintain a high axial load and corresponding moment on the columns. For instance, 50% loaded 450mm and 600mm columns will enhance the impact capacity by around 24% on average compared to 20% loaded columns. Consequently lightly loaded columns may not always be safe against impact loads. Moreover reducing the slenderness ratio will increase the shear failure characteristics of the impacted column. Therefore, columns with high longitudinal steel ratios are slightly less vulnerable compared to columns with low Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 7-209 steel ratios particularly for 2m columns. Peak force vs Slenderness ratio 0.25 0.75 1.25 1.75 2.25 2.75 3.25 3.75 3.0 5.0 7.0 9.0 11.0 13.0 Slenderness ratio P e a k f o r c e ( M N ) 300mm; P=20% 300mm; P=50% 450mm; P=20% 450mm; P=50% 600mm; P=20% 600mm; P=50% Peak force vs Slenderness ratio 0.25 0.75 1.25 1.75 2.25 2.75 3.25 3.75 3.0 5.0 7.0 9.0 11.0 13.0 Slenderness ratio P e a k f o r c e ( M N ) 300mm; P=20% 300mm; P=50% 450mm; P=20% 450mm; P=50% 600mm; P=20% 600mm; P=50% (a) Moment about X axis (b) Moment about Y axis Figure 7.12: Peak force vs. Slenderness ratio for columns of 50MPa concrete with 1% steel Peak force vs Slenderness ratio 0.25 0.75 1.25 1.75 2.25 2.75 3.25 3.75 3.0 5.0 7.0 9.0 11.0 13.0 Slenderness ratio P e a k f o r c e ( M N ) 300mm; P=20% 300mm; P=50% 450mm; P=20% 450mm; P=50% 600mm; P=20% 600mm; P=50% Peak force vs Slenderness ratio 0.25 0.75 1.25 1.75 2.25 2.75 3.25 3.0 5.0 7.0 9.0 11.0 13.0 Slenderness ratio P e a k f o r c e ( M N ) 300mm; P=20% 300mm; P=50% 450mm; P=20% 450mm; P=50% 600mm; P=20% 600mm; P=50% (a) Moment about X-X axis (b) Moment about Y-Y axis Figure 7.13: Peak force vs. Slenderness ratio for columns of 50MPa concrete with 4% steel The same data points were rearranged so that they reflect the dependency of the impact capacity on the effective height as shown in Figures 7.14 and 7.15. In general, the impact capacity can be increased by reducing the effective height. However, the capacity enhancement obtained by reducing the effective height is limited for eccentrically loaded columns compared to that of the axially loaded columns. For instance, the respective average capacity enhancement for 600mm and 300mm columns is around 15% to 25% under eccentric loading compared to 40 to 60% enhancement under (pure) axial loading. Therefore, it can be concluded that the reducing the height does not significantly reduce the vulnerability of the eccentrically loaded columns. However, the columns with 4% steel tend to increase the impact capacity at a higher rate as the height reduces compared to the columns with 1% steel. Moreover, columns are more vulnerable despite the height when the moment is applied about the Y axis, which is perpendicular to the direction of the impact. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 7-210 Peak force vs Diameter 0.25 0.75 1.25 1.75 2.25 2.75 3.25 3.75 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 Diameter (m) P e a k f o r c e ( M N ) X-X; 1%; P=20% X-X; 1%; P=50% X-X; 4%; P=20% X-X; 4%; P=50% Peak force vs Diameter 0.25 0.75 1.25 1.75 2.25 2.75 3.25 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 Diameter (m) P e a k f o r c e ( M N ) Y-Y; 1%; P=20% Y-Y; 1%; P=50% Y-Y; 4%; P=20% Y-Y; 4%; P=50% (a) Moment about X-X axis (b) Moment about Y-Y axis Figure 7.14: Ultimate capacity of 3m columns made of 50MPa concrete Peak force vs Diameter 0.25 0.75 1.25 1.75 2.25 2.75 3.25 3.75 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 Diameter (m) P e a k f o r c e ( M N ) X-X; 1%; P=20% X-X; 1%; P=50% X-X; 4%; P=20% X-X; 4%; P=50% Peak force vs Diameter 0.25 0.75 1.25 1.75 2.25 2.75 3.25 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 Diameter (m) P e a k f o r c e ( M N ) Y-Y; 1%; P=20% Y-Y; 1%; P=50% Y-Y; 4%; P=20% Y-Y; 4%; P=50% (a) Moment about X-X axis (b) Moment about Y-Y axis Figure 7.15: Ultimate capacity of 2m columns made of 50MPa concrete 7.16 Effects of the concrete grade on impact behaviour of columns The impact behaviour of the eccentrically loaded columns made of 30MPa concrete is investigated in this section by assigning 30MPa characteristics to the columns. The load carrying capacity of the columns with 1% steel reduces by 36% compared with the 50MPa concrete. In addition, the moment capacity of the column is also reduced by the same amount. Interestingly, it was identified that the overall impact capacity reduction is proportionate to the ratio of the concrete grades. Peak force vs Diameter 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.25 0.35 0.45 0.55 0.65 Diameter (m) P e a k f o r c e ( M N ) G30; 20%; X-X G30; 50%; X-X G50; 20%; X-X G50; 50%; X-X G30; 20%; Y-Y G30; 50%; Y-Y G50; 20%; Y-Y G50; 50%; Y-Y Peak force vs Diameter 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 Diameter (m) P e a k f o r c e ( M N ) G30; 20%; X-X G30; 50%; X-X G50; 20%; X-X G50; 50%; X-X G30; 20%; Y-Y G30; 50%; Y-Y G50; 20%; Y-Y G50; 50%; Y-Y (a) With 1% steel (b) With 4% steel Figure 7.16: Comparison of peak force of 4m high columns made of Grade 30 and 50 concrete Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 7-211 Figures 7.16(a) and (b) compare the peak impact force for columns made of 30 and 50MPa concrete with longitudinal steel content of 1% and 4%. It can be observed that the peak impact forces are slightly larger for columns with 4% steel. However, the comparative advantages are insignificant and the bearing capacity of the columns is also increased with the steel content. The failure load is low for columns made of the lower grade of concrete. The highest and lowest average capacity enhancement are observed in 50% loaded columns with 1% steel and 20% loaded columns with 4% steel respectively when the grade of concrete is changed, irrespective of the axis of bending. However, the difference between the maximum and the minimum rate of enhancement is insignificant. In addition, the 300mm and 600mm columns show the highest and the lowest capacity increase, due to the flexural and shear failure conditions of the respective columns. Thus, the overall capacity enhancement due to the grade 50 concrete is around 63% and this is proportionate to the ratio between the two concrete grades. However, it is not recommended to adhere to this hypothesis without further investigation, due to the capacity variations in individual columns. Figures 7.17 and 7.18 represent the peak impact force that can be withstood by the columns made of Grade 30 concrete with different slenderness ratios and steel ratios, and the capacity enhancements obtained by increasing the axial load. It is evident that the capacity enhancement due to the slenderness effects is significant for small diameter columns and rapidly deteriorates for larger diameter columns. For instance, the average capacity enhancement is around 56% for 300mm columns and 9% for 600mm diameter columns. The reason behind this enhancement is the flexural failure of 300mm diameter columns which yields the longitudinal steel. Strain rate effects may also have a significant contribution as the Dynamic Increasing Factor (DIF) increases when the concrete grade reduces. Therefore reducing the height is only effective for damage mitigation for eccentrically loaded small diameter columns made of lower grade concrete. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 7-212 Peak force vs Slenderness ratio 0.00 0.50 1.00 1.50 2.00 2.50 3.0 5.0 7.0 9.0 11.0 13.0 Slenderness ratio P e a k f o r c e ( M N ) 300mm; P=20% 300mm; P=50% 450mm; P=20% 450mm; P=50% 600mm; P=20% 600mm; P=50% Peak force vs Slenderness ratio 0.00 0.50 1.00 1.50 2.00 2.50 3.0 5.0 7.0 9.0 11.0 13.0 Slenderness ratio P e a k f o r c e ( M N ) 300mm; P=20% 300mm; P=50% 450mm; P=20% 450mm; P=50% 600mm; P=20% 600mm; P=50% (a) Moment about X-X axis (b) Moment about Y-Y axis Figure 7.17: Peak force vs. Slenderness ratio for columns of 30MPa concrete with 1% steel Peak force vs Slenderness ratio 0.00 0.50 1.00 1.50 2.00 2.50 3.0 5.0 7.0 9.0 11.0 13.0 Slenderness ratio P e a k f o r c e ( M N ) 300mm; P=20% 300mm; P=50% 450mm; P=20% 450mm; P=50% 600mm; P=20% 600mm; P=50% Peak force vs Slenderness ratio 0.00 0.50 1.00 1.50 2.00 2.50 3.0 5.0 7.0 9.0 11.0 13.0 Slenderness ratio P e a k f o r c e ( M N ) 300mm; P=20% 300mm; P=50% 450mm; P=20% 450mm; P=50% 600mm; P=20% 600mm; P=50% (a) Moment about X-X axis (b) Moment about Y-Y axis Figure 7.18: Peak force vs. Slenderness ratio for columns of 30MPa concrete with 4% steel 7.17 Development of equations for biaxially loaded columns under lateral impact Structural columns are seldom designed for vehicle impacts due to inadequacies of design guidelines. Empirical relationships are developed to bridge the gap based on the least square method, which can be used to quantify the critical force and the associated impulse for biaxially loaded columns. The empirical equations are particularly valid under serviceability design load combinations. Effect of the reduced load eccentricity can be neglected if it is within the 50% of the maximum allowable moment under one particular axial load. In fact, direction of the impact was varied from 0 to 90 0 by taking into account the most critical load combinations on two mutually perpendicular axes. Having provided that the positive eccentric loading conditions generated conservative results, the results can be extended to account for the other impact angles. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 7-213 7.17.1 Linear equations for 50% loaded columns (Impact angle 0 o to 90 o ) Partial Plot -0.2 -0.1 0.0 0.1 0.2 -0.03 -0.02 0.00 0.02 0.03 Steel ratio ρ L o g P Partial Plot -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 -20 -10 0 10 20 Compresive strength f' c L o g P (a): Partial regression plot of steel ratio (b): Partial regression plot of compressive strength Partial Plot -0.2 -0.1 0.0 0.1 0.2 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 Height H L o g P Partial Plot -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 -0.2 -0.1 0.0 0.1 0.2 Diameter D L o g P (c): Partial regression plot of height (d): Partial regression plot of diameter (e): Partial regression plot of A h (f): Partial regression plot of ' sy f Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 7-214 Partial Plot -0.2 -0.1 0.0 0.1 0.2 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 Impact angle (∆/45) L o g P (g): Partial regression plot of Impact angle (h): Partial regression plot of hoop spacing Figure 7.19(a-h): Partial regression plots of each parameter against Log P A simple linear correlation between the steel ratio v ρ , concrete grade ' c f (in N/mm 2 ), effective height H (in m), diameter of the column D (in m), area of the hoop h A (in mm 2 ), and impact angle ∆ (in degrees) is determined by using a statistic program ‘StatistiXL’. A total of 281 data records are used in the process. The Coefficient of Determination (R 2 ) of the data set indicates that 96% of the variation in Log P, is explained by variation in the independent X variables, and the R value 0.98, which is the square roof of R 2 , indicates a strong correlation between the Log P and independent X variables. The Standard Error of Estimate, 0.085 is only 1% of the mean of Log P, 6.06 and thus indicates that the Multiple Regression model has accurately calculated a large amount of the Log P values. The resultant linear regression expression is given in Equation 7.1. It can be used to calculate the predicted Peak Force P of the critical impulse for a typical 100ms vehicle impact. Eq. 7.1 ( ) 71 . 4 45 003 . 0 001 . 0 66 . 2 043 . 0 ' 007 . 0 062 . 0 + ∆ − + + − + − = h A D H c f v P Log ρ Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 7-215 Residuals vs Pred Log P y = -0.4077x 2 + 4.9385x - 14.909 R 2 = 0.3066 -0.20 -0.10 0.00 0.10 0.20 5.00 5.50 6.00 6.50 7.00 Predicted Log P R e s i d u a l s Figure 7.20: Residual of the predicted values The accuracy of the predicted data points compared to the observed values in the numerical simulation. Figure 7.20 represent the residuals that are calculated as (Observed (Log P) - Predicted (Log P)). The positive and negative residuals indicate an under and over-prediction of the data points respectively. By considering the distribution of the residuals, second order polynomial equation was used to improve the accuracy of the predicted Log P values. The final over and under prediction of the Peak Force, P is within the range of ±11 %. The corrected Log P c and Log I c are given by; Eq. 7.2 Eq. 7.3 7.17.2 Linear equations for 20% loaded columns (Impact direction 0 to 90 o ) Partial Plot -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 -0.04 -0.02 0.00 0.02 0.04 Steel ratio ρ L o g P Partial Plot -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 -20 -10 0 10 20 Compresive strength f' c L o g P (a): Partial regression plot of steel ratio (b): Partial regression plot of compressive strength 30 . 1 91 . 14 94 . 5 ) ( 408 . 0 2 − = − + − = c c c P Log I Log P Log P Log P Log Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 7-216 Partial Plot -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 -2.0 -1.0 0.0 1.0 2.0 Height H L o g P Partial Plot -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 -0.2 -0.1 0.0 0.1 0.2 Diameter D L o g P (c): Partial regression plot of height (d): Partial regression plot of diameter (e): Partial regression plot of ' sy f (f): Partial regression plot of A h Partial Plot -0.15 -0.05 0.05 0.15 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 Impact angle (∆/45) L o g P (g): Partial regression plot of Hoop spacing s (h): Partial regression plot of impact angle Figure 7.21 (a-h): Partial regression plots of each parameter against Log P Similar set of expressions can be derived for 20% loaded columns with the corresponding moment, as given in Equations 7.4 to 7.6. It was observed that 97% of the variation in Log P, is explained by variation in the independent X variables while Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 7-217 Standard Error of Estimate, 0.063 is only 1% of the mean of Log P, 5.98. The accuracy of the final over and under prediction of the Peak Force, P for the 20% loaded columns is also within the range of ±11% with the improvement of the accuracy (see Fig. 7.22). Residuals vs Pred Log P y = -0.3539x 2 + 4.2308x - 12.605 R 2 = 0.2794 -0.2 -0.1 0.0 0.1 0.2 5.0 5.5 6.0 6.5 7.0 Predicted Log P R e s i d u a l s Figure 7.22: Accuracy of the predicted values Eq. 7.4 Eq. 7.5 Eq. 7.6 Once the Corrected Impulse I c is known, the critical velocity, v can be calculated for a known impacted mass m (kg) of a vehicle, in meters per second (ms -1 ) by using the relationship given in Equation 7.7. For instance, Eurocode EN 1991-1-7 (2006) suggested that mean mass of 1500kg for cars and 20,000 kg for trucks. mv I c = Eq. 7.7 The valid range of the terms in the equations is given as follows; ( ) 635 . 4 45 006 . 0 001 . 0 698 . 2 045 . 0 ' 006 . 0 881 . 0 + ∆ − + + − + = h A D H c f v P Log ρ 30 . 1 605 . 12 231 . 5 ) ( 354 . 0 2 − = − + − = c c c P Log I Log P Log P Log P Log o h sy c v m s m mm A mm MPa f MPa m H m MPa f MPa m D m 90 0 25 . 0 050 . 0 1 . 113 27 . 28 500 ' 250 4 2 50 ' 30 04 . 0 01 . 0 60 . 0 3 . 0 2 2 < ∆ < < < < < < < < < < < < < < < ρ Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 7-218 The critical impulses for the 50% and 20% loaded columns can be calculated from the above equations while the impact capacity of axially loaded columns with nominal load eccentricities can be calculated by the empirical equations provided in Chapter 5. Hence, the critical impulses for three different locations (load combinations) on the interaction diagram can be identified for one particular column. Consequently, linear interpolation can be used to quantify the critical impulse for the loading points that are located in-between on the interaction diagram. Having provided a known force and impulse pair for an average impact duration of 100ms, this method can be extended to assess the vulnerability of columns for a general vehicle population based on the method suggested in Chapter 4. This involved an analytical method that can be used to quantify the critical peak forces under different impact durations. 7.18 Conclusions Design of RC columns for vehicle impact at shear critical height is a tedious task. Understanding of the impact behaviour of the columns and a knowledge on the order of importance of the key design parameters will have a significant contribution towards the routine column design. This research has provided comprehensive equations for this purpose in terms of the geometrical and material properties of the impacted column. The outcomes will greatly facilitate researchers and design engineers who desire to either cross check or validate their models. The main findings of this chapter are summarised below. 1. Consideration of three load combinations is sufficient for the vulnerability analyses of impacted circular columns under biaxial bending which include the single axis bending about two orthogonal directions and the one with equal moment about both principle axes. In fact, the impact capacity under single axis bending would be sufficient for preliminary design of larger diameter columns where shear failures are predominant while comprehensive investigation is recommended for the columns that initiate flexural failure characteristics. In particular, the application of the impact force should be along one of the major axes of bending. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 7-219 2. Vehicle impacts in the direction of the resultant moment can be considered as non-critical (see Fig. 7.5(b)). In addition, reasonably conservative results can be generated (from the equations) for load combinations that exceed 50% of the allowable moment (0.5M x ) for the corresponding axial loading( ) d xP . The columns that deflect against the direction of the impact force are always safe compared to their counterpart. Based on this conclusion the results generated from the equation can be extended to account for the other impact angles. 3. Effects of the longitudinal steel ratio are marginal for damage mitigation particularly for larger diameter columns. Thus the design (option) with low steel content may be the best alternative solution. Alternatively, the impact capacity rapidly increases with the intensity of the loading particularly under shear critical conditions. Thus, it would be better to rely on higher axial load intensities with low steel content when it comes to damage mitigation. When the effects of different concrete grades are concerned, the overall impact capacity variation is approximately proportionate to the ratio of the concrete grades. 4. Effectiveness of the confinement depends on axial load, axis of bending, diameter of column and longitudinal steel ratio. As the resultant changes will depend on the axis of bending, the resultant capacity enhancement due to the confinement for each axis of bending may not be proportional to the initial impact capacities of columns under unconfined conditions. This is particularly significant in small diameter columns where the failure mode changes with the confinement. 5. Confinement effects are mostly effective for columns with low longitudinal steel ratios as the deformation ductility will lessen with high longitudinal steel ratios under low elevation impacts, while the resultant axial load enhancement further increases the shear capacity of the column. However, the resultant impact capacity enhancement due to the confinement effects could be insignificant in lightly loaded small diameter columns with low longitudinal steel ratios as their ductility cannot be further improved by the confinement effects. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 7-220 6. From a comparison point of view, the relative capacity enhancement is highest when the hoop diameter is increased rather than the hoop spacing. However, increase of the yield strength may not be effective as a damage mitigation technique. 7. The impact capacity enhancement due to reduction of the effective height is substantially small in eccentrically loaded columns compared to that of the axially loaded columns. This option particularly effective only for small diameter columns made of lower grade concrete. In addition, even if the overall capacity is being reduced, the eccentrically loaded columns under negative moments are less vulnerable to the hard impact conditions. 8. Prediction of the impact capacities of the small diameter columns under biaxial bending is rather complex as it depends on the load intensity, initial deflection, secondary moment, torsional effects, coupling action, steel ratio and fluctuation of axial load during the impact. Conversely, larger diameter columns mostly fail in shear and hence their impact capacity is independent to a certain extent and thus stable. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 8-221 8. CONCLUSIONS AND FURTHER DEVELOPMENTS 8.1 Introduction The main objective of this thesis was to generate design information to quantify the vulnerability of columns subjected to vehicle impacts. In the process, a numerical model of an impacted column was developed and validated using experimental results. The validated numerical model was then used for parametric studies and development of equations, so as to minimise the cost and time of physical testing. 8.2 Main contributions of the thesis The most significant contributions arising from this research are listed below. 1. This research indicates that triangular impact pulses can be used in an impact simulation process based on the results of full scale impact tests. Implementing triangular impact pulses for frontal impact simulation is innovative as it generates outcomes in terms of critical velocity (v) of the impacting vehicle. The validated pulse characteristics also provide a cost-effective means of performing parametric studies by using numerical simulation techniques when conducting large amounts of full scale impact tests is difficult, if not impossible. 2. The parametric studies were conducted in three stages for axially loaded columns and for columns under uni-axial and biaxial bending. Partially loaded columns were also investigated in detail. Numerical equations were provided to quantify the peak force and the critical impulse for all load combinations exceeding 20% of the allowable axial load, with the corresponding moment. The equations are particularly valid for impact angles between 0 0 and 90 0 . The guidelines are also provided to extend the outcomes of these equations for other impact angles over the common mode of collisions. These equations are the first of their kind to predict the critical impact force and critical impulse in terms of the geometrical and material properties of the impacted columns. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 8-222 3. An innovative technique was developed and introduced to ensure the accuracy of the equations in predicting the critical impact force and impulse where other techniques failed due to the shape of the error distribution under a logarithmic scale. 4. New guidelines were introduced to determine the contact area between the colliding objects based on realistic vehicle impact tests on a circular column. Additionally, key design parameters were defined which can be used for damage mitigation while allowing for optimum usage in the design process. One other main contribution of this research is a better understanding of the dynamic response of reinforced concrete circular columns under both axial and eccentric loading conditions. 5. A new limit state (ie. ALS) was proposed to assure safety against impact loads. The columns susceptible to impacts should be checked for all conventional limit states. In particular the accidental limit state was declared in-between the serviceability and ultimate limit states depending on the expected level of safety. Low shear demand of structural columns under serviceability conditions strengthen this approach. 8.3 Practical significance The following exclusive research outcomes are recommended for routine column design. Numerical simulations The numerical simulations of the column tests can be simplified by isolating the impacted column from the connecting structure. This observation excludes the need to simulate the entire structure in the vulnerability analysis of structural columns. Additionally, it was shown that the material model Mat_Concrete_Damage_REL3 is more reliable for impact simulation compared to other material models available in the LS-DYNA library where columns are subjected to a tri-axial state of stress. Moreover, the application of stress-strain models developed under concentric loading conditions Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 8-223 was proved to be valid under eccentric loading conditions and thus the numerical simulation process itself can be largely simplified. These findings make the vehicle impact problems manageable at the industrial level where supercomputer facilities may not be available. Impact reconstructions The average duration of vehicle impact can be taken as 100ms. The duration of the impact depends on the stiffness of the vehicle which is the result of a compromise between passenger safety and better driving performance. Therefore, significant changes cannot be expected, and this method maybe extended to assess the vulnerability of columns against new generation vehicles where the impact duration varies from 50ms to 150ms. Consequently, the outcomes of the equations can be used as the foundation to generate a database to determine vulnerability assessment against a general vehicle population. In the process, the effects of the shape of the pulses and strain rate effects can be neglected and hence collision severity can be predicted by interpolating known collision pulses. Columns under concentric loading The impact capacity was investigated of reinforced circular columns up to 4m in height made of 30MPa to 50MPa concrete to suit 5 to 20 storey buildings. The longitudinal steel ratio varied from 1% to 4%. It is concluded that the vulnerability of axially loaded columns under vehicle impacts can be reduced by reducing the column slenderness ratio, and by choosing the design option with a low amount of steel and low concrete grade. Thus an equivalent (in terms of capacity) column of lower grade concrete with a low amount of steel and low slenderness ratio will offer the maximum protection against impact loads. Additionally, the confinement effects are particularly effective when the hoop spacing is closer than 100mm. The confinement has to increase particularly with the diameter of the columns and with concrete grade to achieve the same level of capacity enhancement. Therefore, a method based on the maximum diameter of the longitudinal steel is not effective for determining the lateral steel spacing of the Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 8-224 columns susceptible to vehicle impacts. Moreover, it is recommended to increase the diameter of the hoops rather than the yield strength when the minimum allowable spacing of the transverse reinforcement is restricted due to practical issues. In fact, the impact capacity reduced by 10%, 20% and 30% under the 0.6P d , 0.4P d and 0.2P d axial loading respectively. On the other hand, the capacity drop can be recovered by up to 90% by providing hoops at closer spacing if the axial load can be maintained around 0.4P d . Therefore, a limit should be imposed on the minimum axial load that has to be maintained during a rehabilitation process, so that risk of a progressive collapse can be minimised. In particular, there are no substantial warnings before collapse upon post impact loading and the condition remains unchanged even after introducing hoops at closer intervals even though this contributes to the overall capacity enhancement. Under these conditions, it may be more appropriate to replace the impacted column rather than repair it for further use. The catastrophic nature of the (shear) failure increases with the intensity of the axial loading. However, compromise between the axial load intensity and allowable impact force may not be of value as no residual capacity remains in the impacted columns. Static and dynamic shear capacities under impact have been compared. The investigations reveal that the static capacity is not an indication of the dynamic capacity, even though the peak force (dynamic shear capacity) has some correlation with the static shear capacity of the columns. If the correlation factor is known it can be used for approximate vulnerability assessments for impacted columns. However, existing guidelines highly underestimate the dynamic shear capacity of the larger diameter columns and hence may leads to non-aesthetic and expensive design options. Columns under eccentric loading Consideration of three load combinations is sufficient for the vulnerability analyses of impacted circular columns under biaxial bending which include single axis bending about either one or the other orthogonal direction and equal moment about both axes. In fact, the number of load combinations can be further reduced depending on the failure mode. Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 8-225 Since eccentric loading in a building cause bending in the columns about its principal (x – y) axes, the resultant moment about any other axis will not exceed the bending capacity of the column. Due to this vehicle impacts in the vectorial direction about which the resultant bi-axial moment is applied can be considered as non-critical. The columns that deflect against the direction of the impact force are always safe compared to their counterpart. In addition, reasonably conservative results can be generated (from the equations) for load combinations that exceed 50% of the allowable moment (0.5M x ) for the corresponding axial loading ( d xP ). The impact capacity variation is gradual for consecutive load combinations on the interaction diagram for one particular column. This allows linear interpolation to be used in a vulnerability assessment. The critical impact force for the intermediate load combinations can be easily calculated once the impact capacities of the columns are provided at three locations on the interaction diagram. This is the most significant practical outcome of this research as no other method in the literature allows quantifying the critical impulse under arbitrary load combinations. 8.4 Recommendations for future work The results of this research lead to several recommendations for future work. 1. The use of external confinement for damage mitigation is recommended where the strength of cover concrete can be fully utilised while minimising premature spalling which diminishes the confinement effects provided by transverse reinforcement. 2. The Damage Index D was used to identify the residual capacity of the partially loaded columns as it has a proven history of success against blast loads. However collapse of the impacted columns will be brittle and sudden upon post impact loading. Therefore the damage index is not a sensitive index for impacted columns and should not be used in future research. 3. This research is primarily based on the vulnerability assessment under low Vulnerability assessment of reinforced concrete columns subjected to vehicular impacts 8-226 elevation impacts. However impact behaviour of columns under train or ship collision could be significantly different, particularly the area of the impact while mass, duration and velocity may also affect the impact capacity and failure mode. Additionally a series of successive impacts can be expected under a train collision due to the significant momentum of passenger compartments. The number of the impacts, severity of post collisions and duration of the impacts are yet to be determined. 4. The typical dead weight of a bridge superstructure can be 5% to 10% of the capacity of the bridge piers (Prakash et al. 2009). Therefore a separate analysis is recommended for bridge piers where axial load is very limited and hence confinement effects may not be fully developed. In particular, loads and support conditions could be substantially different depending on the method of construction (ie. balanced cantilever, suspension, incremental launching, cable stayed). This is another area for further research. 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