Blast Resistant Design for Roof System

BLAST RESISTANT DESIGN FOR ROOF SYSTEMSA Thesis presented to the Faculty of the Graduate School of the University of Missouri – Columbia In Partial Fulfillment of the Requirements for the Degree Master of Science by MARK ANDREW MCCLENDON Dr. Hani Salim, Thesis Supervisor DECEMBER 2007 The undersigned, as appointed by the Dean of the Graduate School, have entitled the thesis entitled BLAST RESISTANT DESIGN FOR ROOF SYSTEMS Presented by Mark A McClendon a candidate for the degree of Master of Science in Civil Engineering and hereby certify that in their opinion it is worthy of acceptance. ________________________________________________ Dr. Hani Salim ________________________________________________ Dr. Sam Kiger ________________________________________________ Dr. Craig Kluever ACKNOWLEDGEMENTS I would like to express my sincere gratitude to my advisor, Dr. Hani Salim, Assistant Professor of Civil Engineering, University of Missouri-Columbia. His guidance has been invaluable during the course of this study. His good nature, patience, frankness, and technical expertise had a profound impact on my academic experience and personal goals. I would also like to thank Dr. Sam Kiger, C.W. La Pierre Distinguished Professor and Director of the Center for Explosion Resistant Design, University of MissouriColumbia. His enthusiasm and technical knowledge helped guide this investigation. A special thanks is given to Dr. Perry Green, Technical Director, Steel Joist Institute and especially Tim Holtermann from Canam Steel for donating testing materials and continuous design input. Gratitude is extended to my fellow students, Aaron Saucier, Rhett Johnson, John Hoemann, Tyler Oesch, and others whose efforts were in my favor. Without their hard work and dedication, testing would have been overwhelming. Furthermore, I would like to thank all the members at the University of Missouri who have provided assistance. Finally, I extend my sincerest thanks to my friends who have kept me sane throughout my academic experience, and my family, especially my mother Verna and my aunt Wilhelmina. ii .....................1 2.................................................................................................1 1..................................3 Thesis Overview...........................................1 Problem Statement...................................................................................................5 Roof Systems......................................3 Effects on Structures................................16 iii ....4 2..ii LIST OF ILLUSTRATIONS..................................3 2 Review of Literature 2..........................................................2 Single-Degree of Freedom Models....................................................2 Equivalent Blast Load....................................................5.........4 Blast Resistant Design..9 2........................................................................................5.................xii ABSTRACT...................................................4.................14 2....14 Resistance Functions.................................................4 2...........................2 Types of Blast Loads....................................................................................10 CONWEP.2 Thesis Objective.....................................................TABLE OF CONTENTS ACKNOWLEDGEMENTS.........1 2...xiii 1 Introduction 1....................................................................................................2 1......8 2...........................4........1 Introduction................vi LIST OF TABLES....................................................................................14 2............................................... ...........................................4....32 Loading and Solution..........2 3.........3........4 Dynamic Load Simulations..............................................4 Application of Blast Loads.....1 3........................51 Equivalent Blast Loading.......73 4......4....................2 3.......72 4 Static Resistance Function of Open Web Steel Joists 4.4 Preprocessing........................................74 iv .3 3...........21 3........................................31 3..................................................43 Results (Positive Phase Only) ....20 3.........................3 3......3..........3.....4.........6 Field Test......1 3.......22 Loading and Solution...45 Results (Positive and Negative Phase) .............5...................3...........1 3................35 3..............................2 3..................55 3..................................................24 Results........................................................................................2 Review of ANSYS LS-DYNA..................7 Summary....60 3..................................1 Introduction....................................................2 Analytical Resistance Function.........1 Introduction........3 Verification of Equivalent Blast Load Procedure 3......................................................34 Post Processing..........................3 3..............................................................................................................................43 3...........35 Results...5 Blast Load Simulations.......5..................19 3..........3 Static Load Simulations..................................................................................................................5......................................................4 Preprocessing...24 Post Processing...............5.......................................4................................................................24 3................................................................ .....................3...........1 Conclusions....................................2 4.108 v .....93 4................................4.................3..3.....................................80 Testing Apparatus..........................................................................85 Results................................3 Experimental Verification..........4...................4 Summary.............................................................................4......................................78 4......1 16K5 Joist Test...................................................86 4................................................................2 26K5 Joist Test..................................107 REFERENCES.......................3...1 4....3..................................99 4................................3.........................................104 5 Conclusion and Recommendations 5......................................................3 32LH06 Joist Test.....2 Recommendations................................105 5..3.....3 4..86 4........................................4.....78 Test Set-Up.........................4 Testing Samples. ............................................................................................................................16 Figure 2-8: DAHS Equivalent Load..............................33 Figure 3-10: Dynamic Pulse Load #1......8 Figure 2-3: Damped.......................................................................................28 Figure 3-6: Static Load #5..........................................................................................................26 Figure 3-4: Static Load #3........13 Figure 2-6: Equivalent Load Factor and Blast Wave Location Ratio......27 Figure 3-5: Static Load #4....................................................................................16 Figure 3-1: BEAM 4 – ANSYS element..........25 Figure 3-3: Static Load #2.........................................................................31 Figure 3-9: BEAM 161 – ANSYS LS-DYNA element.........................................................................................38 Figure 3-12: Dynamic Load #3.............................................. Maximum loading = 375 lb/ft.... Maximum Loading = 375 lbs/ft................................30 Figure 3-8: Percentage of Error of Max Deflection Location............ Maximum Load = 375 lb/ft.................................... Single-Degree-of-Freedom System...10 Figure 2-4: Idealized Dynamic Response Curves for Triangular Loading.............15 Figure 2-7: DAHS Equivalent Loading Technique....................................................23 Figure 3-2: Static Loading #1....................................12 Figure 2-5: Equivalent SDOF System....................................29 Figure 3-7: Percentage Error of Maximum Deflection........37 Figure 3-11: Dynamic Load #2......................6 Figure 2-2: Blast Loading on Structure...................................................................................................LIST OF ILLUSTRATIONS Figure 2-1: Generalized Blast Pressure History..............................................................................................................................................................................39 vi ............ ....................................................53 Figure 3-29: Negative Impulse Comparison....57 vii ...............42 Figure 3-16: Locations of CONWEP measurements................................................41 Figure 3-15: Percentage Error in Response Period..................................................................................54 Figure 3-31: Response Comparison of Positive Only and Pos............................................................52 Figure 3-28: Loading Scenario #2 – Positive and Negative Phase...........................................................................49 Figure 3-24b: Response Comparison of Loading Scenarios using LS-DYNA (without Scen........................................................................... #1)........53 Figure 3-30: Response of Scenario #2 – Positive and Negative Phase......................................................................................................49 Figure 3-24a: Response Comparison of Loading Scenarios using LS-DYNA.............................................................................50 Figure 3-25: Dynamic Response for Loading Scenario #2................................................44 Figure 3-17: Blast Pressure Distribution....................................................................51 Figure 3-26: Actual Blast Load – Positive and Negative Phase.47 Figure 3-21: Loading Scenario #2...............................................................................Figure 3-13: Percentage Error in Maximum Response........................................................................40 Figure 3-14: Percentage Error in Time of Maximum Response........ & Negative Data for Loading Scenario #2......................................................................................................................................................................................55 Figure 3-32: TM-855 Equivalent Blast Load..............................46 Figure 3-20: Impulse Comparison..........................44 Figure 3-18: Actual Blast Loading........................46 Figure 3-19: Loading Scenario #1...............................47 Figure 3-22: Loading Scenario #3.................................52 Figure 3-27: Loading Scenario #1 – Positive and Negative Phase..............................48 Figure 3-23: Impulse Comparison......................... ..63 Figure 3-38: Pressure-time history at 35 ft...75 Figure 4-3: Resistance Function for 32LH06 Joist....77 Figure 4-5: Bearing seat plates for 16K5 and 26K5 Joists......................................................................................................................................................Figure 3-33: Verification of Equivalent Load Response.........................72 Figure 4-1: Resistance Function for 16K5 Joist...................................................................................................................71 Figure 3-48: Deflection of R2 Panel at far quarter-point.........................................................................................................................................................................................................................................................67 Figure 3-43: Impulse-time history at 55 ft.................. one-degree-of-freedom system for triangular pulse load..............66 Figure 3-42: Pressure-time history at 55 ft.......................................81 viii ...68 Figure 3-45: Deflection of R1 panel at far quarter-point...................................................................................................................................................60 Figure 3-36: Static Resistance Function for FRP Panels..................................................70 Figure 3-47: Deflection of R2 Panel at near quarter-point..............................................................................71 Figure 3-49: Deflection of R2 Panel at midpoint............................................................66 Figure 3-41: Impulse-time history at 45 ft................................................................................58 Figure 3-34: Comparison of Dynamic Response.....................................................................................................................75 Figure 4-2: Resistance Function for 26K5 Joist......67 Figure 3-44: Deflection of R1 panel at near quarter-point....61 Figure 3-37: Roof Panel Schematic............64 Figure 3-39: Impulse-time history at 35 ft........................59 Figure 3-35: FRP Panel Test.............65 Figure 3-40: Pressure-time history at 45 ft......................69 Figure 3-46: Deflection of R1 panel at midpoint...................76 Figure 4-4: Maximum deflection of elasto-plastic............................................................................... ..................................89 Figure 4-17: 16K5 Joist – Failure Sequence 4 of 5 – Continued out-of-plane bending.........86 Figure 4-12: 16K5 trusses prior to testing.........................................................82 Figure 4-7: Steel Joist Institute Specifications for horizontal bridging..................................................................................................84 Figure 4-11: 16-Point Loading Tree.....90 Figure 4-18: 16K5 Joist – Failure Sequence 5 of 5 – Failure of bearing seat weld..................83 Figure 4-9: Lateral bracing – placement.......................................................................94 ix ...............94 Figure 4-22: 26K5 Joist – Failure Sequence 1 of 5 – Deformation of Tension Chord............................................Figure 4-6: Bearing Seat Plates for 32LH06 Joist.....................................................................................................................................................89 Figure 4-16: 16K5 Joist – Failure Sequence 3 of 5 – Failure of horizontal bridging......................87 Figure 4-13: 16K5 Joist – Failure Sequence 1 of 5 – Initial bending....................91 Figure 4-20: Midpoint Static Response for an individual 16K5 joist compared to existing methods......................................................................................82 Figure 4-8: Lateral bracing – welded to strong floor............................................................................................................................................88 Figure 4-14: 16K5 Joist – Failure Sequence 2a of 5 – Failure of lateral bracing.........................................88 Figure 4-15: 16K5 Joist – Failure Sequence 2b of 5 – Failure of lateral bracing.........................................................90 Figure 4-19: Static Response for 26K5 Joist System.......................................................................................................................84 Figure 4-10: String Potentiometer.................92 Figure 4-21: 26K5 Joist System prior to test..................................................................................................................................... .....100 Figure 4-33: 32LH06 Joist System prior to test....................................................................................................................................................97 Figure 4-28: 26K5 Joist – Failure Sequence 5a of 5 – Connection Plate Failure.............................................................................................................................................102 Figure 4-36: 32LH06 Joist – Failure Sequence 1c of 1 – Continued Buckling and Bending of Tension Chord....................................................96 Figure 4-27: 26K5 Joist – Failure Sequence 4b of 5 – Failure of End Tension Member.................99 Figure 4-32: Initial Deformation of 32LH06 Joist.................................................96 Figure 4-26: 26K5 Joist – Failure Sequence 4a of 5 – Failure of End Tension Member...................................................................................................................98 Figure 4-31: Midpoint Static Response for an individual 26K5 joist compared to existing methods.97 Figure 4-29: 26K5 Joist – Failure Sequence 5b of 5 – Connection Plate Failure.......................102 x .............................................................95 Figure 4-25: 26K5 Joist – Failure Sequence 3 of 5 – Failure of Secondary Web Member................................................................................95 Figure 4-24: 26K5 Joist – Failure Sequence 2b of 5 – Failure of End Tension Member...................................101 Figure 4-35: 32LH06 Joist – Failure Sequence 1b of 1 – Buckling of Secondary Web Member...................98 Figure 4-30: Static Response for 26K5 Joist System............................................................................................................................................101 Figure 4-34: 32LH06 Joist – Failure Sequence 1a of 1 – Compression Web Member Buckling..........................................................................................................................................................................................................Figure 4-23: 26K5 Joist – Failure Sequence 2a of 5 – Failure of End Tension Member............................. ..........................................103 Figure 4-38: Midpoint Static Response of individual 32LH06 Joist comparing existing methods.......Figure 4-37: Static Response of 32LH06 Joist System...................................................................103 xi .......................... .......................................................................................36 Table 3-2: Aspect Ratio...............................................LIST OF TABLES Table 3-1: Minimum Time Steps by Element Size... Range...............................................................................................................56 Table 4-1: Resistance Data from Engineering Calculations...................................................................................................36 Table 3-3: Peak Pressure vs.....................................44 Table 3-4: CONWEP Blast Data.................77 Table 4-2: Resistance Data from SBEDS Calculations.....78 xii . Different structural members. impulsive and non-simultaneous over the length of a roof. Blast loads are dynamic. xiii . The objective of this research is to test this procedure and compare its results to the deflections from blast loads. a procedure has been developed to devise a uniform dynamic load on a roof that matches the response from blast loads. Also. have been thoroughly evaluated under blast loads. the equivalent loading procedure did not adequately predict the initial peak deflection or the maximum deflection. the response of experimentally measured roof blast pressures is compared to the equivalent loading response. a common roof component. 1992) and the Single-degree-of-freedom Blast Effects Design Spreadsheet (SBEDS) from the Army Corps of Engineers Protection Design Center.ABSTRACT The design of structures to resist explosive loads has become more of a concern to the engineering community. While the responses from finite element modeling matched the experimental responses. The numerical pressures are calculated using the Conventional Weapons Effects Program (CONWEP) (Hyde. such as walls. This research focuses on the design techniques for the loading on roof structures and the resistance of open web steel joists. To design against explosions. This research uses finite element analysis to evaluate the responses from numerically calculated blast loads and compares them to the equivalent loading response. The current resistance methods calculate larger maximum loads than the experimental values and the assumption of a perfect plastic post-peak response ignores the buckling failure of web members. Three tests consisting of different steel joist pairs are performed. It is also recommended that an analytical resistance function for OWSJ be clearly defined. such as hot-rolled steel beams and reinforced concrete slabs. and current methods extrapolate techniques used in the design and analysis of hot-rolled steel beams and reinforced concrete. The resistance function is computed from these results and compared to current methodologies. Open web steel joists (OWSJ) are other types of common roof components. The resistance function currently used for these members are linear elastic and perfectly plastic after the elastic deflection limit. It is believed that the failure mechanisms of OWSJ significantly are not accurately being taken into account. Their responses under loading are not clearly defined.The response of several structural members used in roof construction. It is recommended that additional research is to be done on the prediction of blast pressures on roofs and on the development of an equivalent uniform dynamic load. which includes all failure limit states. xiv . are well documented and understood. && + R = F (t ) My where M = mass R = resistance F(t) = applied load (1.1 Problem Statement Many research has been performed on the design of structures under explosive threats. The design and response for a wall system subjected to blast loading and its components are very well known. there is a gap in the knowledge for the roof system component under explosive loads. the resistance. such as blast pressure due to a bomb. The analysis of any system with dynamic loads. 1967). can be generalized with an equation of motion as shown in Eqn. The parameters in this equation are the mass. This equation can be solved by various numerical procedures to calculate the dynamic response (Biggs.Chapter 1 Introduction 1. 1.1. and the load. However.1) 1 . open web steel joists are also common roof components. In addition to concrete and hot-rolled steel members. and they are in a high demand of use due to their low weight and relatively high resistance. In roof systems. 1. Explosive loads are highly impulsive. the resistance and load are more complex. this procedure has not been verified using experimental or numerical data. current design practices should be researched. For walls. Therefore. pure plastic load-deflection curve. and nonuniform. Current techniques used in design assume a linear elastic. the blast pressures are mostly uniform on the entire face of the wall. The resistance of many structural systems used in construction of roof slabs is known.2 Thesis Objective The overall objective of this research is to develop an analysis and design procedure for open web steel joist roof system under blast loading. there is a need to explore and validate the methods and results this technique uses. 2002). To achieve this goal. these loading pressures change with respect to time and with respect to distance. Due to their high use in military and commercial buildings. and they can be accurately estimated using available codes such as CONWEP (Hyde. 1992). as far as the author is aware. It is believed that such truss systems exhibit different failure modes than those currently used in design. However. non-simultaneous.While the mass of any system can be easily calculated. There is currently a loading procedure developed by the Army Corps of Engineers that equate the response from a blast load to the response from an equivalent uniform dynamic load that is more suited to design (UFC. the following two specific tasks are realized: 2 . 3 . It also contains numerical integration techniques for dynamic response calculation. The results from the static tests are compared to current resistance function techniques used by design engineers. Chapter 5 summarizes the analysis of the research and presents recommendations for future work. In Chapter 3 the ANSYS LS-DYNA program is verified for static and dynamic loads before using the program to develop responses for roof models subjected to blast loads. the equivalent blast loading procedure is compared to field test data of a roof subjected to explosive loads. as well as methods for determining the resistance function for open web steel joists. Finally. the static testing procedure for open web steel joist samples is presented. The failure mechanisms present in open web steel joists are also discussed. In Chapter 4.• Analyze the roof loading procedure specified by the United Facilities Criteria (UFC 3-340-01. 2002) and compare its validity using numerical simulation and field test data. Also. 1. The equivalent blast load procedure is discussed. These responses are compared to the response from the equivalent blast loading. • Analyze and compare current methods of developing the resistance function for open web steel joists with experimental data using static tests.3 Thesis Overview Chapter 2 covers a literature review of the methods and techniques used in the thesis. This chapter includes an explanation detailing the current knowledge of explosives and blast waves. 2 Blast Waves Explosions can be caused by physical.Chapter 2 Literature Review 2. Nuclear explosions generate kinetic energy. In this chapter current techniques used to describe blast effects will be discussed. 2.1 Introduction Understanding and solving the problem statement requires basic knowledge of blast effects and the responses for roof structures. Nuclear fission divides the nuclei of heavy atoms. internal energy. along with fragments of the vessel. and thermal energy. Vast amounts of energy can be released in short periods by breaking the bonds between protons and neutrons. Pressure is released. Nuclear fusion combines nuclei from light atoms. several design methods for roof structures will be introduced. Changing the structure of atomic nuclei produces nuclear explosions. 4 . nuclear. In addition. or chemical events. A common example of a physical explosion occurs when a pressure vessel fails. Important factors pertinent to burst pressures include the peak pressure. Examples of some of these equations are shown in Eqns. This causes the pressure in the shock wave to drop below the atmospheric pressure. 3. the velocity of the shock front.3 (Smith and Hetherington. ambient air pressure. High explosives (chemical and nuclear) in a surrounding medium.1. These gases naturally expand. a drag force. There are several derived equations that calculate the shock front velocity Us. the duration. 3. the state returns to the atmospheric pressure. which expands. peak dynamic pressure qs. cause shock waves in the medium.2. 3. The shock front travels in a radial direction. The compressed medium. The air behind the shock front also places a load. the amount of “overpressure” the shock front carries decreases. on objects encountered (Smith and Hetherington. more expansion is necessary to actually reach equilibrium. and the surrounding medium is consequently compressed (Smith and Hetherington. After sufficient “underpressure” is expended. As the explosive gases cool and slow their movement. or for the specific case of air. due to the high pressure and mass of the gases.The rapid oxidation of fuel elements develops chemical explosions. the air density behind the shock front. 1994). However. such as air or water. and the impulse of the blast pressure. and the speed of sound in air at the ambient pressure. 5 . and air density behind the shock front ρs based on the peak overpressure. 1994). The gases release energy to reach equilibrium towards the atmospheric pressure. 1994). The blast releases high-pressure gases at high temperatures. forms a shock front. This reaction releases heat and produces gas. The general shape of a pulse shape is shown in Figure 2-1. Low-end explosives create quasi-static loads. Figure 2-1: Generalized Blast Pressure History (TM 5-1300, 1990). Us = 6 p s + 7 p0 ⋅ a0 7 p0 (2.1) ρs = 6 p s + 7 p0 ⋅ ρ0 p s + 7 p0 2 (2.2) 5 ps qs = 2( p s + 7 p0 ) where ps = peak static overpressure (2.3) p0 = ambient air pressure in front of shock wave ρ0 = air density in front of shock wave a0 = speed of sound in air at air pressure 6 The peak overpressure is related to a factor called the scaled distance, Z (Eqn. 2.4). This is proportional to the distance from the charge and the cubed root of the charge mass. Typically the charge mass is measured in terms of TNT, and other types of explosives are converted to this mass type. As the distance increases, the maximum pressure of the shock wave decreases. The total duration of the shock burst actually increases. It should also be noted that at any particular range, the peak overpressure of the blast wave decays exponentially to the atmospheric pressure (Biggs, 1967). Z= R W 1/ 3 (2.4) where R = distance from blast source W = mass of charge in terms of TNT When blast waves strike a surface, the overpressure increases. The pressure from the expanding gases build up since there is no medium to compress and displace. Therefore, the burst pressures from a surface explosion are larger than an explosion occurring in the air. The overpressure of a surface burst is approximately twice that of a free-air burst (Smith and Hetherington, 1994). Next, the discussion of blast loads continues to how they affect buildings and objects. 7 2.3 Effects on Structures Three main loading conditions are available, as explained by Smith and Hetherington, 1994. In the first type a relatively large shock wave reaches a structure relatively small enough that the blast wave encloses the entire structure. The shock wave effectively acts on the entire structure simultaneously. Additionally, there is a drag force from the rapidly moving wind behind the blast wave. The structure is, however, massive enough to resist translation. The second condition also involves a relatively large shock wave and a target much smaller than the previous case. The same phenomena happen during this case, but the target is sufficiently small enough to be moved by the dynamic, drag pressure. In the final case, the shock burst is too small to surround the structure simultaneously and the structure is too large to be shifted. Instead of simultaneous loading, each component is affected in succession. For a typical building, the front face is loaded with a reflected overpressure. Figure 2-2: Blast Loading on Structure (Forbes, 1999). 8 This can range from simplified hand calculations to utilizing computer programs programmed to deal with complex loading and resistances. This reflected pressure decays over time. Again. 1999). In addition to this blast wave diffraction. the incident pressure attenuates as it transverses the roof and sides. drag forces load the structure (UFC. This means that while the blast pressures change with respect to time. Any method used takes into account the knowledge of blast loads described in the previous section and the component resistances that make up the structure (Morison. Similar to how the peak pressure decays and load duration grows as the range from the charge increases. as shown in Fig 2-2. 2. 2006). 2002). these pressures are evenly distributed on every portion of the wall. due to the perpendicular orientation of the front wall to the shock wave. After the roof and sides are surpassed. The roof and sides of the building react instead to incident or side-on pressures. Blast pressures simultaneously affect walls with the same pressure-time history on the entire wall area. the pressure magnifies due to reflection. the pressures converge on the back of the building. 9 .As stated in the previous section. Also. a reflecting effect amplifies the overpressure. In the presence of a sloped roof.4 Blast Resistant Design Engineers employ several different methods for the structural design of structures resisting blast loads. the blast pressure is reflected and magnified (Forbes. Air molecules are stopped by this material and compressed further by the shock front behind them. reflected pressure accumulated when the blast wave meets material denser than the medium in which it is traveling. 1967). This model idealizes an entire structure or structural component as one point in the structure.2. 10 . The equation of motion for a linear elastic. && + ky + cy & = F1 [ f (t )] My where M = mass of structure k = structural stiffness c = damping coefficient F1 = constant force value f(t) = nondimensional time value (2. Single-Degree-of-Freedom System.5) y M F1 Figure 2-3: Damped. damped single-degree-offreedom system follows (Biggs.1 Single-Degree-of-Freedom Model The first explosive design methods are based on the single-degree-of-freedom (SDOF) model.4. The resistance at this point is also taken as the resistance for the entire structure. This method only analyzes simplified systems and was not ideal for more general loading histories and resistance functions.6). The first technique. 1967).6) where y0 = initial displacement & = initial velocity y ω = circular frequency ≡ k/M yst = static deflection ≡ F1/k f(τ) = non-dimensional load-time function The military has used two main SDOF methods to deal with explosive threats. Normalized curves were created to aid in calculation of maximum deflection for various dynamic loads and linear elastic. An example of this non-dimensional chart is shown. The most notable dynamic load used was a triangular load with zero rise time. the damping coefficient is assumed to be zero. two dots mean acceleration. The equation simulates the response of lumped mass-spring system (Biggs. 11 . Solution of this equation (without damping) leads to Duhammad’s.The dot superscripts represent partial derivatives with respect to time. the response period is taken as the natural period of the first mode under free vibration. In this method. y = y 0 cosωt + ω & y sin ωt + y st ∫ f (τ ) sin ω (t − τ )dτ 0 t (2. the Modal SDOF method. This gives the deflection-time history. It also undercalculates the deflection response and reaction response (Morison. 2006). On dot signifies velocity. pure plastic resistance curves. reissued in 1965 as TM 5-855-1 and not superseded until 1986. turns up in a 1946 manual “Fundamentals of Protective Design (Non-Nuclear)” EM 1110-345-405. 2. Integral (Eqn. For simplicity. or Convolution. Figure 2-4: Idealized Dynamic Response Curves for Triangular Loading (Morison. 1967). resistance. 12 . 2006). this alternative method determines equivalent values of mass. The equivalent SDOF method was widely published in 1957 in parts of the USACE manual “Design of Structures to Resist the Effects of Atomic Weapons”. and EM 1110-345-416 “Structural Elements Subjected to Dynamic Loads”. Using the conservation of kinetic energy. transformation factors can be found to change the system’s mass. EM 1110-345-415 “Principles of Dynamic Analysis and Design”. These transformation factors are also based on the deflected shape function of the system as a whole from a particular reference point. resistance and loading into their lumped mass counterparts. internal strain energy and external work. and loading for the lumped mass-spring system based on the distribution of the structure’s mass and the system’s loading. While the Modal SDOF method uses the actual mass and loading in the system. usually the middle point of a structure (Biggs. the solution becomes more approximate and more errors are entered. It should also be noted that the transformation factors for load and resistance are the same (Biggs. & = Fe (t ) M e && y + ke y or (2. This can be solved in the same manner as the Modal SDOF models. 1967).Accurate shape functions result in accurate solutions. When the shape function is not as well known.6b. The equation of motion for this equivalent SDOF system is given in Eqns. The static deflection shape under the same loading distribution as the dynamic load is relatively easy to calculate and results in very accurate answers.6b) where KM = mass transformation factor KL = load/resistance transformation factor M = total mass k = stiffness F = total load p k Δ Figure 2-5: Equivalent SDOF System.6a and 2. the ratio of the mass factor to the load factor. 13 y . Analysis of the equation allows for the use of only one transformation factor KLM.6a) && + K L ky & = K L F (t ) K M My (2. 2. positive phase duration. The load factor is a function of the Lwb/L. more importantly. The drag coefficient is a function of the dynamic pressure. 2. The equation is shown in Eqn.7.2.2 CONWEP The Conventional Weapons Effect Program (CONWEP) (Hyde.5 Roof Systems This section discusses design techniques for roof systems subjected to blast loads. The loading function and. incident and reflected impulse. the mass of a structure is a well-known value. time of arrival. The shape of this load is a triangular pulse with an unequal rise and decay times. This equivalent load is calculated using values from the CONWEP program.5. This program outputs parameters of the pressure-time history including side-on and reflected pressure. which is the ratio of the length of the blast wave at the back of the roof to the length of the roof. Looking at the general equation of motion. 1992) uses blast test data and empirical equations to calculate load data for different types of explosives with varying loading sequences. 2. Different methods for determining these quantities are presented. 14 . 2. and peak dynamic pressure.4. This program only computes data corresponding to the positive pulse of a blast load. shock front velocity. the resistance are the less realized components.1 Equivalent Blast Load The Unified Facilities Criteria 3-340-01 (which supersedes Army Manual TM 5855-1/Airforce AFPAM 32 1147/NAVYFAC P-1080/DSWA DAHSCWEMAN-97) formulates an equivalent uniform blast loading for the roof and sidewalls of a structure. 2-6. the load factor and the ration of D/L can be computed.7) CONWEP data is taken at a range equal to the radial distance from the blast source to the back of the roof. tb and tofb. The time to reach the peak pressure point is defined as td. 2-8. 2-8. 15 . In Fig.Por = C E Psob + Cd q ob where Por = peak pressure Psob = peak incident pressure at back of roof CE = equivalent load factor qob = dynamic pressure at back of roof Cd = side-on element dynamic drag coefficient (2. Using Fig. The loading ends after the blast wave reaches the end of the roof and the duration of the loading expires. tf is the time the shock wave hits the roof. Figure 2-6: Equivalent Load Factor and Blast Wave Location Ratio (DAHS. as shown in Fig. 2002). where D is the point on the roof where the peak pressure is said to occur. demonstrates that the maximum pressure on the roof takes time to build. The shape of the loading function. The length of the shock wave is calculated from the shock front velocity and the positive phase duration of the shock front. the changing pressure-time history accumulates to a peak value. respectively. As the shock front transverses the roof. 1990). Figure 2-8: DAHS Equivalent Load (TM 5-1300. 16 .2 Resistance Function There are several different types of roof structures. open web steel joists (OWSJ) and metal decking.Figure 2-7: DAHS Equivalent Loading Technique. purlins and decking. steel joists.5. 2. Some types include concrete slab and I-beam composites. For these systems, the I-beam and the steel truss are the significant components with respect to design. The behavior of I-beams is well-known and documented. However, the response of open web steel joists to failure is not well documented. Conventional engineering designs assume a simple bilinear linearly elastic, perfect plastic resistance function. The Steel Joist Institute (SJI) has developed charts to ascertain the maximum allowable load per length for every truss as a function of joist type and length (SJI, 2005). The allowable deflection is L/360, where L is the effective joist length. In addition, the Steel Joist Institute provides an equation for the approximate joist moment of inertia, based on the maximum live load and the effective length of the truss. This equation is described in Eqn. 2.8. The derivation for this moment of inertia originates from the stiffness of a simply supported beam under a uniform load and includes a 15% increase in deflection due to elongation in the truss web. Along with the equivalent SDOF method, current engineering practices define a resistance function shape for use in design. I j = 26.767( w LL )( Lspan ) 3 (10 − 6 ) (2.8) where wLL = maximum allowable live load of joist (lb/ft) Lspan = effective length (ft) A current engineering method of deriving the resistance formula takes into account the properties of the joist members and the Steel Joist Institute load specifications. The truss can be designed as a simply supported beam with a uniformly distributed load. Using the design tensile strength of the web members as a limit, the elastic section modulus of the joist cross-section can be computed. 17 Using this value and the joist material characteristics, an effective moment capacity can be computed. The maximum moment for a simply supported beam under uniform load can be determined from the load and the length of the beam. Included into this maximum moment calculation are strength increase factors for the joist materials. Using this relationship, the ultimate resistance can be computed. Similarly, using the deflection equation for the same system, the stiffness and elastic deflection limit can be calculated. The maximum response is calculated using numerical techniques and/or idealized charts based on the ratios of load duration to the joist period and resistance magnitude to the forced load magnitude. SBEDS (2004) uses a slightly different approach. For one, the moment of inertia is calculated from the actual cross-sections of the joist, including the top chord, the bottom chord, and some factor for the web. These calculations prove to result in higher moments of inertia for a given steel truss compared to the Steel Joist Institute’s approximate calculation. The program defines a value Lshear, which is the maximum joist length designed according to SJI with the maximum total allowable load. For K-Series joists this maximum total load is 550 lb/ft. For the deeper joists LH and DLH, this maximum shear load is based on individual truss and is detailed in the SJI Load Tables document. The effective moment capacity is calculated from the maximum shear load and the Lshear quantity, assuming a simply-supported beam with a distributed load. SBEDS program back-calculates a resistance value from the joist length and effective moment capacity. This resistance value is also factored using strength increase factors. The elastic deflection limit and stiffness are calculated using the simply supported beam assumption mentioned earlier. 18 Chapter 3 Verification of Equivalent Blast Load Procedure 3.1 Introduction Blast loads are dynamic, impulsive, and non-uniform. An experimental blast load has been formulated to emulate the response of a blast load acting on a roof system. This loading resembles a triangular pulse with a rise time and decay time. This loading shape has been in use for over fifty years. With the advent of more advanced computer technology and finite element analysis programs, current techniques for expressing blast loads on the roofs of structures can be evaluated. This chapter explains the programs and procedures used in blast and dynamic loading estimation. 19 20 . Inc.0 is used in this research.3. The program ANSYS ED Release 9. also called finite elements. and electromagnetic analyses. compatibility equations and constitutive relations. ANSYS is able to construct static and dynamic structural analysis (both linear and non-linear). ANSYS was created in 1970 by Dr. simpler discrete regions. acoustics. This is applicable for structures experiencing large deformations and short time durations. and shape of the elements control the program results. The product ANSYS LS-DYNA is a result of the partnership between ANSYS. are linked and controlled by several governing equations. including impact test simulations. and degrees of freedom. It is an explicit non-linear structural simulation used for dynamic analyses. and metal forming. including equations of equilibrium. material types. and Livermore Software Technology Corporation and was first introduced in 1996.). drop test problems.2 Review of ANSYS Program Finite element analysis is a numerical method of deconstructing a structure or system into smaller. fluid flow problems. The computer solves these equations simultaneously using different numerical schemes to express the behavior of the structure as a whole. Inc. In general the number. size. explosive simulations. heat transfer problems. This is a student level version meant for academic use. It supports many of the capabilities present in the full version of ANSYS Multiphysics with some limitations on types and size of elements. John Swanson (Swanson Analysis Systems. These regions. graphs. size and type of element is based on the analysis type and the user’s judgment. three main steps must be performed – preprocessing. The number. or contour plots. The system can now be discretized into elements. The user directs the program to solve the problem and output results. Symmetry can also be used to simplify the construction of the model. A linearly elastic.3 Static Load Simulations The ANSYS program was used to compare deflection curves derived from kinematic boundary conditions with the results obtained from ANSYS finite element modeling. 3.For ANSYS use. First. 21 . the different material properties that make up the model have to be described and applied to the model’s geometry. The entire structure can be modeled or parts of the structure can be modeled. a two. The material of the beam is modeled as steel. The next sections depict the methods used for static and dynamic loads. completing a solution. isotropic BEAM-type element was used in the modeling. The ANSYS program includes a graphical user interface (GUI) to aid in executing these steps. Next. Its dimensions were 4” x 18” x 20’.or three-dimensional model of the structure or system has to be formed. These analyses are also beneficial in validating the use of ANSYS in analytical experimentation. These results can be illustrated using tables. and post processing. The beam was modeled as being simply supported. To gain confidence in ANSYS. Once the system is idealized into finite elements with their own properties. physical constraints and different loading types can be positioned on the model. static and dynamic analyses were performed using simple models. It is described as a “uniaxial element with tension.1 Preprocessing The element type used in this analysis is called BEAM4. y. 3-1) has six degrees of freedom from translations in the local x.3. Figure 3-1 shows the rotational aspects of the beam element. the cross-sectional dimensions. This technique is commonly used during the loading stage to determine on which face to apply the load. and the mass density per unit length. For this model. Different material models for several structural materials are available in the ANSYS program. This element is used in static analyses to simulate the reactions of an elastic. The modulus of elasticity (and also the mass density) is chosen for this particular material type. y-. compression. the ANSYS program supplies different ways of detailing each section. a linear elastic. untapered section in 3D. the weak and strong moments of inertia. The numbers depict the six different surfaces on the element. and z directions and rotations in the local x-. a set of real constants is inputted. torsion. For BEAM4 elements. isotropic material model was used. and bending capabilities. Each node (shown as I and J in Fig.” The element contains two nodes at either end. These constants include the cross-sectional area. and z-axes. 22 . For different elements. The GUI offers an automated section development that allows users to choose the shape of the element cross-section and it’s dimensions.3. A straight line is drawn between these keypoints. In the Figure 3-1 this point is denoted as node K. The user defines the number and size of the finite elements. For the creation of a simple beam. A third keypoint is also drawn to define the orientation of the element. The next step is to mesh the line. 2004). The user can apply the real constants and material attributes to the element mesh and use some node K to orient the elements. This will apply elements along the line’s geometry. keypoints are chosen as the locations of the beam’s ends. The line is then meshed into the number of prescribed elements.Figure 3-1: BEAM 4 – ANSYS element (ANSYS. lines. and elements. The ANSYS ED version limits the number of possible nodes. 23 . 3.3. Each beam was divided into a different number of segments. 3.3. the node at the other end of the beam is constrained in the x and y direction. loads are placed on one surface of the beam elements to simulate bending about the weak axis of the beam’s cross-section. A list of the nodes and positions in the global directions can be provided. 3. For each static loading. or a graphical representation of the original and deflected position of the model can be drafted. or 240 beam elements.3 Post-Processing ANSYS provides several post processing options. and z directions. Each configuration contained only distributed loads. y. Next. The tests were designed to determine how the finite element model deflection compared to expected results and how discretization of the model affects results. Each static load was tested on a beam model consisting of 1’ segments. 40 beam elements. 24 .2 Loading and Solution The structure under consideration is a simply supported beam. 6” segments. or conversely 20 beam elements. three test beams were modeled. A contour of the stresses in the beam model can be generated. To that effect. respectively. or 1” segments. the node at one end of the beam is constrained from movement in the x. This model is then solved.4 Results Five different static load configurations were tested.3. As can be seen in Figure 3-2. This loading also results in the greatest amount of deflection from all five curves. The shape of the results is consistent with known kinematics. The displacement of the beam reaches maximum in the center of the beam. Eqn.The first static load is a uniform static load. 6 5 4 3 2 1 0 0 5 10 Distance from Support (ft) 15 20 400 350 300 250 200 150 100 50 0 0 5 10 15 20 Distance from Support (ft) Defl. there appears to be little difference between the calculated values and each ANSYS test. From the graph. as expected. the expected deflection curve as well as the different curves from the three test beams are fairly accurate. The loading amount is designed to result in a discernable deflection curve. Load (plf) 25 . 1' Beam Elements 6" Beam Elements 1" Beam Elements Deflection (in) Figure 3-2: Static Loading #1. while keeping the stress of the beam in the elastic range. Maximum loading = 375 lb/ft. The next static load is a triangular loading that decays over the length of the beam. Little difference can be discerned from the different test beams in this graph. 26 . the centroid of the loading is at the center of the beam’s length and it is balanced about the center of the beam. The third static load is also triangular. The total load on the beam is half that of the uniform loading. however it rises to a peak halfway across the beam’s length and descends the rest of the length to zero.5 0 0 5 10 Distance from Support (ft) 15 20 400 Load (plf) 300 200 100 0 0 5 10 15 20 Distance from Support (ft) Defl. and the range of the displacement reflects this. The final two static loads are neither symmetrical nor are they continuous along the beam.5 1 0. The maximum loading is located in the center of the beam. The displacements for the three ANSYS tests also appear fairly close to expected answers. Maximum Load = 375 lb/ft. However. the maximum deflection is off the center of the beam’s length.Eqn 1' Beam Element 6" Beam Element 1" Beam Element Figure 3-3: Static Load #2. The total load of this curve is the same as that of Static Load #2. 3 2.5 Deflection (in) 2 1. Due to the centroid of the loading. the kinematic boundary conditions of the beam state that the slope and deflection must be continuous along the beam and at the changes in loading. 2001). Also.4 3.5 3 Deflection (in) 2. While the deflection curves of the previous three static loads can be easily found in a book or manual (AISC. Static Load #4 consists of three varying uniform loads acting on three different equal length sections of the beam. 1' Beam Element 6" Beam Element 1" Beam Element 2 1. The total load is little more than half of Static Load #1. the derivation of the deflections for Static Loads #4 and #5 involves calculating the internal moment in the beam and integrating twice. Eqn. 27 . and the deflection is in the right scale. The general shape of the displacement curves is customary to the expected results.5 0 0 400 300 200 100 0 0 5 10 15 Distance from Support (ft) 20 5 10 Distance from Support (ft) 15 20 Figure 3-4: Static Load #3. Maximum Loading = 375 lbs/ft.5 Load (plf) Defl. and the maximum beam dislocation is located about an inch from the center of the beam’s length.5 1 0. The load’s center of gravity is about 9’ from a support. 1' Beam Element 6" Beam Element 1" Beam Element Deflection (in) Figure 3-5: Static Load #4. The test with the smallest element length (conversely the largest number of elements) appears to coincide with that of the deflection equation. the 1’ beam segment overestimates and the 6” curve underestimates the expected result. It can also be seen that the difference in expected and calculated values decreases with increased discretization. the location of maximum deflection is offset from the center. The curves all underestimate the deflection. Again.5 1 0.The curves for the 1’ length beam element and for the 6” length beam element are discernable Figure 3-5 around the center of the beam. but they arrive closer to the expected value as the element size increases. 3 2. It is made up of three different uniform loads acting on equal length sections of the beam. along with one extra section that remains unloaded. 28 . As can be seen by the deflection curves of Static Load #4.5 0 0 5 10 Distance from Support (ft) 15 20 Load (plf) 300 200 100 0 0 5 10 15 20 Distance from Support (ft) Defl. and the curves reflect this.5 2 1. Static Load #5 has similar results. Eqn. due to the loading schematic. In addition. the symmetrical loads (Static Loads #1 and #3) have very small deviations from the expected values.5 0. the maximum deflection and the locations of maximum deflection are compared to the derived deflection equation. the continuous loads have less error than discontinuous loads. for the most part.25 Deflection (in) 1 0.1. And.5 1. 29 . 1' Beam Element 6" Beam Element 1" Beam Element 0.25 0 0 300 200 100 0 0 5 10 15 20 Distance from Support (ft) 5 10 Distance from Support (ft) 15 20 Figure 3-6: Static Load #5. To further examine this. as expected. The ANSYS results appear to resemble the expected values. Eqn. As can be seen from Figures 3-2 through 3-6.75 Load (plf) Defl. the percentage of error decreases as the element size decreases. the maximum deflection is located at the center of the beam. the more accurate the measurement is. The smaller the element. 30 .00% 1. The number of elements in the model determines the number of nodes in the beam.00% 3.00% Error 2.00% 0.00% 1' 6" Element Size 1" Static Load 1 Static Load 2 Static Load 3 Static Load 4 Static Load 5 Figure 3-7: Percentage Error of Maximum Deflection.50% 0.50% 3.50% 2. the location of the maximum deflection is no more than half a foot from the center of the beam.50% 1. so there is no error for any beam model. Each node is spaced the length of an element.50% 4. For each loading. 5.Finally. the location along the beam model of the maximum displacement is compared. and so accuracy of the displacement and location of displacement is limited by the length of an element. so there is a relatively larger error for every loading. The closest the 1’ beam element models can measure is 1’. For Static Loads #1 and #3.00% 4. 00% 3.00% 4. 3.50% 4.50% 1. it can be concluded that ANSYS provides good values for displacement. data points). ANSYS has more graphical support than LS-DYNA. LS-DYNA computes solutions faster than ANSYS and can supply more information (i. 31 . The following section details the process of evaluating the dynamic analysis capabilities of ANSYS. the smaller element results in close to exact answers.00% 0.5. However.50% 0. From these tests. a sub-program called LS-DYNA exists in ANSYS that is more suited towards explicit solutions and dynamic loads that are applied over short durations.50% 3.4 Dynamic Load Simulations The ANSYS program can be also be used to model structures subjected to dynamic loadings. but larger element sizes can give reasonable answers. with less than 5% error.00% 1. On the other hand.50% 2. If available.00% Error 2.e.00% 1' 6" Element Size 1" Static Load 1 Static Load 2 Static Load 3 Static Load 4 Static Load 5 Figure 3-8: Percentage of Error of Max Deflection Location. but are computed and stored as such. An equivalent single degree of freedom (SDOF) model was used to determine both the maximum deflection of the beam under a uniformly loaded dynamic load and the shape of the time-history curve of this deflection. A Hughes-Liu beam element assumes a constant moment along its length and detects stresses in the center.The LS-DYNA program is used to verify the maximum deflection of a simply supported beam under different pulse loadings. This maximum value and the shape of the graph as a whole were compared to the LSDYNA results. the material properties of the beam are defined. the magnitude and duration of the dynamic load were constant for each pulse loading. 32 . The modulus of elasticity and Poisson’s ratio can be inputted. Each node has six physical degrees of freedom – translation and rotation in three directions – and also three nodal velocity degrees of freedom and three nodal acceleration degrees of freedom. linear elastic materials. The BEAM 161 element has two main methods of calculation. The same beam dimensions from the static tests were used. Next. These latter are not technically degrees of freedom. This element is used for explicit dynamic analyses only. consists of a node at either end of the element (I and J) and an orientation node (node K) on a different line than that of the beam direction. 3. BEAM 161. It can model beams of different crosssection shapes.1 Preprocessing The beam element used in this analysis. The cross-sectional dimensions are defined as real constants. A Belytschko-Schwer element generates a linearly changing moment along the element and measures stresses at either end.4. and can be used to model isotropic. Also. the beam can be meshed into the desired number of elements with the set material and geometrical properties. So. with a Young’s modulus of 30 x 106 psi and a Poisson’s ratio of 0. This is necessary from a practical point to have a node in the center of the beam to measure the maximum deflections. at the very least the beam model has to be divided into 2 elements. the material was assumed to be steel. The beam can be modeled very simply with a line drawn between two keypoints. therefore. because this is an explicit solution. 33 .Figure 3-9: BEAM 161 – ANSYS LS-DYNA element. another material property is the mass density. Also. After construction of the line keypoints and line. The line that makes up the beam can be divided into a certain number of elements or a particular element size. the weight of the structure is considered a necessary value. A third keypoint that is not collinear with either endpoint is also necessary for element orientation.3. For this model. These pulse loads were inputted into the model as array parameters. uniform load. The pulse loads are uniformly distributed across the length of the beam. and are set to act in the center of the beam’s cross-section. A rectangular pulse loading with instantaneous acceleration was tested because it is a very simple. The two other loadings tested are triangular pulse loads. ANSYS solutions have a limit of 100 points and are usually used to examine the deflection of an entire structure at a specific time. The first triangular pulse load is an instantaneous loading with a linear decay to zero. The first triangular pulse loading allows for comparison to results with the rectangular loading and is similar to the pulse loading from a blast load.2 Loading and Solution Three different pulse loadings were used to test the dynamic load capabilities of LS-DYNA. 34 . and very few points need to be inputted. The second triangular load is also used for comparison and is similar to the equivalent loading of a roof structure subjected to a blast load. One array gives the time values and the other the load values. All three dynamic loads have equal magnitudes and durations. The program interpolates the points between those given. The loads were all linear. and so translational constraints are put on the endpoints.3. The elements that make up the beam can be defined as components where the loads will be acting. The second pulse loading has an equal time to rise from zero and time to decay to zero.4. The test beam is designed as a simply supported. Two arrays are defined for each load. The duration of the analysis can then be specified and the program can be directed to output ANSYS solutions or LS-DYNA solutions. and 1” respectively. 6”.4. The purposes of these tests do not necessitate a very large number of output points. and because of their large size are usually limited to observe the time history of important points or nodes on the beam.4. 40 elements.The LS-DYNA solutions have a limit of 1000 points. The LS-DYNA program calculates the time steps used in the numerical integration based on the dimensions and properties of the beam and the element length (ANSYS. or graph the displacement time history of a particular node on the beam.1) where L = length of element c = wave propagation velocity 35 . 20 elements. 6 elements.9 L c (3. 1’. Δt = 0.4 Results For each dynamic load the model divides the beam into sets of 2 elements. 3. or element lengths of 10’. For these tests the time history of the center node of the beam is graphed. 2004). 3.3 Post-Processing ANSYS has the capabilities to display the deflection of the entire beam model at a particular time. The length of time under investigation should be the length of at least half a response period. and 240 elements. 3’ 4”. 10 elements. This was not encountered in this investigation.c= E ρ (3.1) used in the program is the ratio of the smallest element length in the model and the propagation velocity. For all models the element length is uniform.0001068 20 12 5. the smaller the time steps are.671E-05 240 1 4. Table 3-1: Minimum Time Steps. Six models were used with different lengths of beam element.451E-06 Table 3-2: Aspect Ratio.2) is calculated from the Young’s Modulus and density of the beam. 36 . the more elements in a beam (i. and the smallest time step (Eqn.0005342 6 40 0.e.342E-05 40 6 2. It would be assumed that the smaller the time steps used in numerical integration. Number Element Smallest Length Time Step of (sec) (ft) Elements 2 120 0.2) where E = modulus of elasticity ρ = mass density The wave propagation velocity (Eqn. the more accurate the solution would be. The following table shows the number of beam elements. It must be noted that too small of a time step could lead to a numerical instability in which the solution is not bounded and the resulting deflection approaches infinity. the shorter the element length). respective lengths.0001781 10 24 0. 3. Number Length: of Depth Elements Ratio 2 30:1 6 10:1 10 6:1 20 3:1 40 3:2 240 1:4 As can be seen. 3. and used and corresponding time steps. Subsequent decreases in element length arrive to a more accurate response. 37 .6 0.4 0. 2. Coincidentally.2 Time (msec) 10 20 30 40 2 elem (LS-DYNA) 6 elem (LS-DYNA) 20 elem (LS-DYNA) 240 elem (LS-DYNA) Duhammad's Integral Load (plf) 400 200 0 0 2 4 Time (msec) 6 50 60 70 80 90 Figure 3-10: Dynamic Pulse Load #1. the more warped the graphs become. this is the result without including the load and mass factors.The ANSYS test data was compared to a time history calculated using a single degree of freedom (SDOF) model and load and mass factors to correct for the shape of the load and the shape of the beam (refer to Sec. The beginning curvature of the graphs demonstrates zero initial displacement and zero initial velocity.2 0 0 -0. However.1 for details).4. The period response also becomes more and more exact.8 Deflection (in) 0. Dynamic Load #1 is a rectangular pulse load with an instantaneous rise and decay. the more elements modeled. Several deflection time history curves are shown in Figure 3-10. Using six elements results in a displacement within 20% of the theoretical response. The beam model with two elements and only one node between the endpoints drastically underestimates the response.2 1 0. 1. Dynamic Load #2 is a triangular pulse load with an instantaneous rise and a linear decay to zero.4 Deflection (in) 0. 0. In addition to the shape.5 0. the curves also become more and more wavy with the addition of elements. the graphs show a zero initial deflection and a zero initial velocity (Fig. which is the expected result. The accuracy of the answer increases with the decrease in element length. 38 . The maximum displacement is less than that of the first dynamic loading.1 0 0 10 20 400 300 200 100 0 0 2 4 Time (msec) 6 30 40 50 60 70 80 90 -0. Again. the waviness affects the time of maximum deflection. This displays the precision of the ANSYS program. Unfortunately.3 Load (plf) 2 elem (LS-DYNA) 6 elem (LS-DYNA) 20 elem (LS-DYNA) 240 elem (LS-DYNA) Duhammel's Integral 0. This causes a difference between the calculated and expected time of maximum response.2 0. 3-11).6 0. The time history curves resemble those for Dynamic Load #1. The response period is the same. which is consistent with the period of Dynamic Load #1 and with the natural period of the beam model.1 Time (msec) Figure 3-11: Dynamic Load #2. 2 0. and the aspect ratio of the smallest beam element is 1:4. 39 . The ANSYS code chosen was designed only for regular beams.5 0. and have different kinematic and equilibrium conditions than those of regular beams. The elements with aspect ratios less than 5:1 are specified as deep beams. The time history curves are very similar to that of Dynamic Load #2 in terms of shape and magnitude. The similar shapes are due to the accuracy of the analysis.6 0. Due to the beam model’s properties. and thus the depth of each element. The close responses result from the equal impulse of the pulse loads. measures 4 inches. 0. The shape of the curves relates to the aspect ratio of the beam elements.3 0. The aspect ratio of the largest beam element (as shown in Table 3-2) is 30:1. The resistance of the beam is less related to the peak load than to the pulse’s impulse.4 Deflection (in) 0. it is an impulsive sensitive system.1 0 0 -0. The depth of the beam.Dynamic Load #3 is also a triangular pulse load with an equal time to rise and time to decay.1 10 20 30 40 50 60 70 80 90 400 Load (plf) 200 0 0 2 4 Time (msec) 6 2 elem (LS-DYNA) 6 elem (LS-DYNA) 20 elem (LS-DYNA) 240 elem (LS-DYNA) Duhammel's Integral Time (msec) Figure 3-12: Dynamic Load #3. The error in the magnitude of maximum displacement for the 20-element beam is less than 2% for all loads. 40 . the accumulation of elements past that level also increases the amount of error. and period are compared to the expected values. time of maximum response. the maximum deflection. Figure 3-13 shows a clear decrease in maximum deflection error with an increase in element number. Figure 3-14 presents the error in calculating the time of maximum deflection. This trend is common for every pulse load. Unfortunately. 60% 50% Dynamic Load #1 40% Error 30% 20% 10% 0% 2 6 Dynamic Load #2 Dynamic Load #3 10 20 Number of Elements 40 240 Figure 3-13: Percentage Error in Maximum Response.To further explore the ANSYS results. They are compared according to the number of finite elements and the type of loading. The beam model results in a value within 3% of the SDOF model’s prediction. 41 .35% 30% 25% 20% 15% 10% 5% 0% 2 6 10 20 40 240 Number of Elements Dynamic Load #1 Dynamic Load #2 Dynamic Load #3 Error Figure 3-14: Percentage Error in Time of Maximum Response. while the error for the 240-element beam is close to zero. Figure 3-15 shows a marked decrease in error with the increase in element number. The error for the 20element beam is about 3%. the period of response is examined for the test beams. Finally. the response of beams under blast loads is explored. the ANSYS/LS-DYNA models result in answers close to the expected values. and response period. 42 . In conclusion. In the next section. an aspect ratio between 3:1 and 3:2 appears to result in the best solutions in terms of maximum deflection. An element length of 1’ is used for the next section. For the current purposes. time of maximum deflection. and additional programming with the LS-DYNA and other finite element analysis software needs to be performed to determine if these are currently the most accurate outcomes.20% 16% Dynamic Load #1 Dynamic Load #2 Dynamic Load #3 12% Error 8% 4% 0% 2 6 10 20 Number of Elements 40 240 Figure 3-15: Percentage Error in Response Period. None are the answers are perfect. the error is small enough to be applicable. For these analyses. 1 Application of Blast Loads The pressures on a structure from a blast are non-uniform and highly impulsive. The response of the equivalent loading is also derived using Duhammad’s integral. Finally data from an actual blast test is used in analysis.5. For each subsequent section of the beam. there are five sets of time and load arrays. The test beam is divided into five separate components. To reflect the variable loads. peak load. the same process is repeated using both positive and negative phase data. 43 . The loads on each section have a different initial load time. The load pulses on a roof contain a positive downward phase and a negative section phase. and final time. Using ANSYS. 3. The peak pressures decay and the load duration increases as the blast wave traverses the roof length. The loading response is analyzed in stages.3. the beam model is divided into five differing loading sections. corresponding to an element length of 1’ each. different sections of the beam require different loading characteristics at different times. Twenty beam elements make up the beam. First. 3.3. where each set of time and load data is applied to each component. Instead of one set of time and load arrays.5 Analysis of Blast Load Response Using LS-DYNA The same steps as before are used to generate a model of a simply-supported beam. the positive pressure phase is idealized using a triangular pulse with an instantaneous rise time and a time to decay. The responses of the blast loads are compared to the response of the equivalent dynamic blast load using ANSYS. the peak load decreases and the time history of the loading changes to represent the accentuation of the blast pulse. The loads are inputted as time and load array parameters similar to the pulse loadings in Sec. Next. 23 41. 1992) generates time of arrival.92 69. In CONWEP. 35 feet. and 50 feet (corresponding to the back end of the beam).30’ 35’ 40’ 45’ 50’ Figure 3-16: Locations of CONWEP measurements.75 Range of Data Distribution 30-32' 32-37' 37-43' 43-48' 48-50' 44 . Table 3-3: Peak Pressure vs.0 Peak Pressure (psi) 134. Range of Data Origin (ft) 30.0 45. positive phase duration. Figure 3-17: Blast Pressure Distribution. Range.2 94.0 50. 45 feet. incident peak pressure. These ranges are shown in Figure 3-16.0 40. and incident impulse output used in the arrays. The application CONWEP (Hyde.83 53. Pressure information is recorded at ranges of 30 feet (the front end of the beam).0 35. a hemispherical surface explosive of 1000 lbs of TNT is analyzed at a range of 30 feet from the simply-supported beam. 40 feet. One shape follows the pattern of decreasing primary and secondary peaks. The 40. From the simple dynamic models performed earlier. 3-18.5. it can be ascertained that the beam model is impulse sensitive. The beam model supports line loads instead of surface loads. 45. The pressure from the 30 feet range is distributed (as shown in Fig.2 Results (Positive Phase Only) The actual blast loads can be seen in Fig. To correct this. 3-21 and 3-22). and 2 feet. respectively. The first trial positive phase pressures decay linearly over the same length of time (see Fig 3-19). the pulses were modified to a bilinear decay. 5. This decay model is performed with two separate examples (Figs. Table 3-3 displays the peak pressure and the range to which it relates. The pressures are multiplied by the width of the cross-section to calculate the associated distributed load. 45 . 3-20). In the actual blast pressures.These five loading patterns are distributed along the beam to replicate actual blast characteristics. 3-17) over the first two feet of the beam. The pressure from the 35 feet range is allocated from 32 to 37’. The third scenario keeps the bilinear decay form without following a discernable configuration. the peak decays in a nonlinear manner to zero. 3. the actual shape should not matter. An important check in this analysis compares the impulses from the actual data to the approximate data. and the arrangement of the bilinear decay should be irrelevant. It can be shown that the first trial impulses are from 2 to 7 times that of the actual pulses (see Fig. and 50 feet range data is applied over the next 6. from 30 to 32’. along with increasing durations. Therefore. 02 0.140 120 Incident Pressure (psi) 100 80 60 40 20 0 0 5 10 15 Time (msec) 20 25 30 Beam Segment 1 Beam Segment 2 Beam Segment 3 Beam Segment 4 Beam Segment 5 Figure 3-18: Actual Blast Loading. 2500 2000 Load (lbs/in) 1500 Beam Segment 1 Beam Segment 2 Beam Segment 3 Beam Segment 4 Beam Segment 5 1000 500 0 0 0.03 Figure 3-19: Loading Scenario #1.01 0.025 0. 46 .005 0.015 Time (sec) 0. 1600 1400 Incident Impulse (psi-msec) 1200 1000 800 600 400 200 0 30.02 0.015 Time (sec) 0.0 Distance (ft) 45. 47 .0 50.0 40.01 0.0 Real Data Trial #1 Data Figure 3-20: Impulse Comparison.005 0.0 Figure 3-21: Loading Scenario #2. 2500 2000 Load (lb/in) 1500 Beam Segment 1 Beam Segment 2 Beam Segment 3 Beam Segment 4 Beam Segment 5 1000 500 0 0 0.025 0.0 35. 2500 2000 Beam Segment 1 Beam Segment 2 Beam Segment 3 Load (lbs/in) 1500 Beam Segment 4 Beam Segment 5 1000 500 0 0 0. 40. Also. 35.005 0.025 0. and 50 feet range. This occurs because the pulses closer to the blast source are larger than the pulses farther away. Four different pulses load these sections. From these results. The deflection-history for Scenario #1 greatly exceeds the deflection of the other tests. To further continue validation of the input data. 40.015 Time (sec) 0.03 Figure 3-22: Loading Scenario #3. the beam is divided into four equal-sized components instead of five different sized components.01 0. respectively (Figs. Scenario #4 and Scenario #5 represent the upper and lower bounds. 3-23) and the impulses from the three main load sets are compared to the actual positive impulses. This is repeated in Scenario #5 using loads from the 35. and 45 feet range act on the four segments. 3-24a and b). 45. 48 . loads from 30. For Scenario #4. Scenarios #2 and 3 barely differ discernibly. it can be seen that the impulse for Scenario #2 matches the actual impulses.02 0. Scenarios #2-5 agree with other more closely. the best loading data to use is Scenario #2. The dynamic responses from these five loading combinations are plotted together (Fig. 0 35.0 50.240 230 Incident Impulse (psi-msec) 220 210 200 190 180 170 160 150 140 Exact Data Trial #2 Trial #3 30.0 Distance (ft) 45.0 Figure 3-23: Impulse Comparison. 49 . 8 6 4 Deflection (in) 2 0 0 -2 -4 -6 -8 Time (msec) 50 100 150 200 250 Scenario #1 Scenario #2 Scenario #3 Scenario #4 Scenario #5 Figure 3-24a: Response Comparison of Loading Scenarios using LS-DYNA.0 40. 5 0 -0.2. The beam takes longer than the time of the load to fully react.5 1 Deflection (in) 0. although the left has a higher deflection than the right.5 Time (msec) 0 50 100 150 200 250 Scenario #2 Scenario #3 Scenario #4 Scenario #5 Figure 3-24b: Response Comparison of Loading Scenarios using LS-DYNA (without Scen.5 -2 -2. #1).5 2 1.5 -1 -1. This outcome is due to the high natural period of the beam relative to the short duration of the blast. As expected. The maximum deflection happens after all the load pulses have come and gone. Figure 3-25 displays the deflection-history of the beam model at various points. The rightmost quarter-point encounters a maximum before the left quarter-point. the midpoint deflection measures the highest of the beam. 50 . 5. Comparison of the actual impulses and the approximate triangle impulse (Fig.5 Left Quarter-point 1 0. a contour using three lines in implemented. To keep the design and input simple. There exist a decay time (rise in the absolute sense) to the maximum negative load and a rise time (decay in the absolute sense) back to zero (Fig.5 -1 -1. 3-27). 3.5 -2 50 100 Midpoint Right Quarter-point Deflection (in) 150 200 250 Time (msec) Figure 3-25: Dynamic Response for Loading Scenario #2. the negative phase of the blast loading starts at zero. 3-29) concludes that the approximate negative impulse is less than the actual by about 11%. Initially. This phase is designed to have the same impulse as the actual load.2 1.5 0 0 -0. 51 . the estimation of the negative phase begins with the shape of a triangle.3 Results (Positive and Negative Phase) Similarly. 2500 2000 1500 1000 500 0 -500 Beam Segment 1 Beam Segment 2 Beam Segment 3 Beam Segment 4 Beam Segment 5 Load (lbs/in) 0 0.02 0.1 0. 52 .12 0.06 0.08 Time (sec) 0.04 0.140 120 Incident Pressure (psi) 100 80 60 40 20 0 -20 0 20 40 60 80 Time (msec) 100 120 140 Beam Segment 1 Beam Segment 2 Beam Segment 3 Beam Segment 4 Beam Segment 5 Figure 3-26: Actual Blast Load – Positive and Negative Phase.14 Figure 3-27: Loading Scenario #1 – Positive and Negative Phase. 14 Time (sec) Figure 3-28: Loading Scenario #2 – Positive and Negative Phase.04 0. 53 .1 0.0 35.12 0.0 Figure 3-29: Negative Impulse Comparison.0 40.2500 2000 Beam Segment 1 Beam Segment 2 1500 Load (lb/in) Beam Segment 3 Beam Segment 4 Beam Segment 5 1000 500 0 -500 0 0.08 0. -350 -300 Negative Impulse (psi-msec) Actual Trial 1 -250 Trial 2 -200 -150 -100 -50 0 30.0 50.02 0.0 Distance (ft) 45.06 0. The dynamic response of Figure 3-30 is the result of the positive phase and the aforementioned negative phase from Scenario #2. The deflection bounces back to a higher value that the positive phase loading. 54 . The same event happens at the left and right quarter-points. the right point experiences higher loading before the left point (Figs. 3-28 and 3-30). 3 2 Left Quarter-point Midpoint Right Quarter-point Deflection (in) 1 0 0 -1 50 100 150 200 250 -2 -3 Time (msec) Figure 3-30: Response of Loading Scenario #2 – Positive and Negative Phase. From Figure 3-31. The suction pressure from the negative phase decreases this deflection and enhances the deflection in the rebound. Again. as can be seen from the midpoint results. it can be seen that the initial maximum deflection is less than the deflection from only the positive phase. 5. & Negative Phase Figure 3-31: Response Comparison of Positive Only and Pos.3 2 1 0 -1 -2 -3 0 50 100 150 Time (msec) 200 250 Deflection (in) Positive Phase Only Pos. This data is shown in Table 3-4. 3. In this example. The method is discussed and the response is compared to both the positive phase response and the deflection from the positive and negative blast pressure. namely 30 feet. Blast data from this range of 50 feet is used in the analysis. there is no structure height. The range to the back of the structure equals the range to the target plus the length of the beam model.4 Equivalent Blast Loading The procedure to define the blast loading on a roof structure is used for the beam model. The next section covers the equivalent loading defined by TM-855-1. The CONWEP application can calculate the necessary information from known bomb size and distance. & Negative Data for Loading Scenario #2. 55 . The range to the target structure is the same range used in the blast data. 83 150 2068 30. The length of time the blast affects a structure greatly increases when including the negative phase. while blast loads have both positive and negative phase. The equivalent loading contains only a positive phase. the design of the equivalent loading applies the pulse to the entire beam. The first noticeable characteristics of the equivalent blast loading and the load from blast data are the differences in time of arrival and loading duration. The total impulse from the positive and negative phase of the blast loading differs 15% from the equivalent impulse. Peak Pressure (psi) Positive Phase Duration (msec) Time of Arrival (msec) Incident Impulse (psi-msec) Shock Front Velocity (ft/s) Peak Dynamic Pressure (psi) 41. While the equivalent load begins at a relatively much later time than the actual arrival time of the blast pressure.41 The results of CONWEP are used to describe the pulse loading in Figure 3-32. the durations vary by about eight percent for the positive phase only of the blast loading. The positive and negative blast pressure has a much larger duration than the equivalent load pressure. while the actual blast loads have different effects on different portions of the beam model. To keep these details in perspective.95 15. the sum of the positive impulses from the blast loads equals three times that of the equivalent loading. 56 . The peak loading of the blast pressures are all greater than the peak load for the DAHS equivalent load.73 12.Table 3-4: CONWEP Blast Data. Additionally. 02 0. First.005 0. There is a 3% difference between the maximum deflections and also between the response periods. the deflection time history can be derived using Duhammad’s Integral. 57 .04 0. we will evaluate how the ANSYS program reproduces the deflection time history by comparing it to the hand numerical integration methods (Fig.01 0.045 Time (sec) Figure 3-32: TM-855 Equivalent Blast Load. due to the shape of the load and how it is applied to the entire beam.500 450 400 350 Loading (lb/in) 300 250 200 150 100 50 0 0 0.03 0. As expected. Also. 3-33).025 0. The response of this load on the beam model can be calculated using ANSYS.015 0. there is a very good correlation between the hand calculation and the ANSYS results.035 0. Since it has been verified that ANSYS calculates the equivalent response correctly and again verifies that ANSYS accurately calculates dynamic responses. Figure 3-34 contains the deflection from the positive phase blast data. 58 . the positive and negative phase blast data.4 3 2 1 0 0 -1 Duhammel's Integral Deflection (in) 20 40 60 80 100 120 140 160 -2 -3 -4 ANSYS Time (msec) Figure 3-33: Verification of Equivalent Load Response. and the equivalent pulse load. the deflection for the equivalent load can be compared to the approximate blast loading data. Analyzing the curves. several points can be observed between the three simulations. As noted earlier. The next step in the research is to apply the equivalent load procedure to field test data. The period response for the equivalent loading almost matches the positive phase response data. the first peak is the most valued by design engineers. There is a large variation between the first peak of the equivalent response and the positive phase only response. & Negative Phase Equiv. Load. the time of the maximum response differs between curves.4 3 2 Deflection (in) 1 0 -1 -2 -3 -4 0 50 100 150 Time (msec) 200 250 Positive Phase Only Pos. 59 . Figure 3-34: Comparison of Dynamic Response. and the equivalent response first peak is almost twice that of the positive and negative response. the starting time of the equivalent loading is about ten milliseconds after the blast wave is calculated to affect the beam model. Finally. It is believed that the data including the positive and negative blast phase is more accurate than the positive phase only. While the subsequent peaks match for the positive and negative phase and the equivalent load. and the positive and negative phase data varies by only six percent. 3.000-lbs ANFO 30-ft Standoff to block face Figure 3-35: FRP Panel Test. The test setup is shown in Figures 3-35 and 3-37. 33-ft Standoff to panel face 1.000 lbs of ammonium nitrate and fuel oil (ANFO). The roof components are structural panels made up of honeycomb fiber reinforced laminations. The charge weight is 1. The structure consists of six feet high concrete support blocks with earth berms along the sides to prevent blast pressure from forming underneath the roof. The cross-section of an FRP panel contains core layers consisting of an FRP material formed into a sinusoidal wave bonded to a flat piece of similar makeup. The results of these tests can be used to further evaluate the DAHS load and procedure.6 Field Test A research project by Hoemann and Salim (2007) was performed to analyze the resistance and dynamic response of roof members consisting of fiber reinforced polymers (FRP). 60 . and the standoff has a range of 30 feet to the face of the support block and 33 feet to the roof component. 61 . One FRP sample has a crosssection depth of 12 inches. The experimental static resistance functions of these structural members are shown in Figure 3-36. The second sample. The sample roof was designed against blast loads considering only the elastic range of the FRP material’s resistance. R2. Static Tree Testing 12″ and 15″ Deep Panel Sections 20 16 Equivalent Section of Panel R2 Pressure (psi) 12 8 Equivalent Section of Panel R1 4 0 0 2 4 6 8 10 12 Δ . It is denoted as R1. Both roof panels were 4 feet wide with a simply supported span of twenty feet.Displacement (in.Two FRP components were used in the explosive field test. is 15 inches deep.) Figure 3-36: Static Resistance Function for FRP Panels. Most likely the blast wave loses more energy from striking the roof. This schematic is shown in Figure 3-37. Additionally. It should also be noted that the test explosive is placed so that the resulting blast wave travel along roof direction. or that blast pressures are not equal from radial distances at varying incident angles. which proves that this program correctly calculates pressures at the ground level but not at higher elevations.The test setup includes pressure gages on the roof and on the surface at equal radial distances. 62 . Comparing the blast pressures measured on the roof and on the surface. For this research project. There are also deflection gages at the midpoint and two quarter-points for both FRP panels. The response is measured against field deflections. the measured pressures are inputted into an ANSYS LSDYNA model. the dynamic response from the DAHS equivalent load is compared to the experimental values. The surface pressures do match the pressures and loading characteristics computed from CONWEP. it is noted that the loads on the ground surface are larger than the loads on the roof. Panel R1 FF8 FF7 Panel R2 FF6 D5 D 5-ft FF5 FF4 D3 D4 5-ft FF3 D1 10-ft FF1 5-ft D2 5-ft FF2 10-ft 35-ft 30-ft Displacement Gauges Free Field Pressure Gauges Figure 3-37: Roof Panel Schematic (Hoemann and Salim. 63 . 2007). For Figures 3-38 through 3-43.pressure FF2 . Also. and calculated pressures from CONWEP and SBEDS. 105 90 75 FF1 . the measured negative phases appear to have the same durations. 64 .Figures 3-38 through 3-43 show blast pressure-time histories from the ground. although they are not well defined. It is also of interest that the blast pressures measured on the ground and at the rooftop have the same positive phase durations. Impulses are also calculated from the pressure-time histories. The CONWEP data was used to determine the time of arrival of the shock front and SBEDS was used to generate simulated pressure-time history.pressure CONWEP/SBEDS Data Pressure (psi) 60 45 30 15 0 -15 -30 0 8 16 24 32 40 48 56 64 72 80 Time (msec) Figure 3-38: Pressure-time history at 35 ft. rooftop. the CONWEP/SBEDS simulation has the same peak pressure measured at the ground level. The peak incident pressures for the 35 ft range computed by CONWEP is almost twice the impulse measured at the roof (Fig. This same trend occurs in the impulse diagram (Fig. While this does occur. as shown in Figures 3-40 and 3-42. there is still appreciable error between the measured and calculated blast pressures. The CONWEP/SBEDS simulation should closer predict the measured pressure levels. Comparable to the pressure history. the impulse-time history shows lower values on the roof than on the ground. 3-38). The cause of the shock wave reduction when it reaches the roof is expected to affect the pressures less as the blast wave transverses the roof. 65 .200 160 Impulse (psi-msec) 120 80 40 0 -40 -80 -120 0 8 16 24 32 40 48 56 64 72 80 Time (msec) FF1 FF2 CONWEP/SBEDS Data Figure 3-39: Impulse-time history at 35 ft. The actual impulses for the simulation are greater than those measured on the rooftop or on the ground. 337). 180 150 Impulse (psi-msec) 120 90 60 30 0 -30 -60 -90 0 8 16 24 32 40 48 56 64 72 80 Time (msec) Figure 3-41: Impulse-time history at 45 ft. 66 FF4 FF5 CONWEP/SBEDS Data .60 50 Pressure (psi) 40 30 20 10 0 -10 0 8 FF4 FF5 CONWEP/SBEDS Data 16 24 32 40 48 56 64 72 80 Time (msec) Figure 3-40: Pressure-time history at 45 ft. 40 32 FF7 FF8 CONWEP/SBEDS Data Pressure (psi) 24 16 8 0 -8 -16 0 8 16 24 32 40 48 56 64 72 80 Time (msec) Figure 3-42: Pressure-time history at 55 ft. 180 150 Impulse (psi-msec) 120 90 60 30 0 -30 -60 -90 0 8 FF7 FF8 CONWEP/SBEDS Data 16 24 32 40 48 56 64 72 80 Time (msec) Figure 3-43: Impulse-time history at 55 ft. 67 Figures 3-44 and 3-45 display the response of the FRP roof panels from the field data and the simulated deflections from the measured roof pressures. The deflections at the quarter-points predicted by ANSYS LS-DYNA match closely to those measured in the field. In the model, damping is not considered, whereas the effect occurs in the experiment. These models use the measured roof blast pressures. Normalized Deflection (delta/EI) 2.4 1.8 1.2 0.6 0 -0.6 -1.2 -1.8 -2.4 0 50 100 150 Time (msec) 200 250 Field Test Response ANSYS LS-DYNA Response Figure 3-44: Deflection Response of R1 panel at near quarter-point. 68 Normalized Deflection (delta/EI) 2.4 1.8 1.2 0.6 0 -0.6 -1.2 -1.8 -2.4 0 50 100 150 Time (msec) 200 250 Field Test Response ANSYS LS-DYNA Response Figure 3-45: Deflection Response of R1 panel at far quarter-point. From Figure 3-46, it can be seen that the ANSYS LS-DYNA model continues to closely match the field response, not including damping effects. The DAHS procedure is calculated using CONWEP data. Due to the differences between measured pressures, a corrected DAHS procedure is also used. The only factor changed in the corrected procedure is the peak pressure. The equivalent load response, however, measures almost three times the initial peak deflection and twice that of the rebound deflection. In design, it has been customary to focus on the first maximum deflection point. The rebound effects are similar to the model in Sec. 3.5, where the subsequent peaks are larger than the first. The corrected DAHS response predicts values closer to measured deflections, but the first peak deflection is still twice of that measured. In that regard, the DAHS procedure is still very conservative. 69 70 .5 3 1. The ANSYS LS-DYNA simulations approximately measure the primary peak deflections. but again. The deflections during rebound are larger because of a tie-down anchor failure during the negative pressure phase duration. Panel R2 displays similar results. The corrected DAHS procedure does more closely approach the measured deflections for this test. but this may be inconclusive due to the anchor failure. It does come close to the peak responses due to rebound. but appear to significantly underestimate the rebound peaks (Fig. The DAHS procedure predicts higher deflections than is measured in the first peak.Normalized Deflection (delta/EI) 6 4.5 0 -1. that may be inconclusive due to the anchor failure.5 -6 0 Field Test Response LS-DYNA Response DAHS Response Corrected DAHS 50 100 150 Time (msec) 200 250 Figure 3-46: Deflection Response of R1 panel at midpoint. 3-49).5 -3 -4. 6 0.4 -3. Normalized Deflection (delta/EI) 3 1.5 -3 -4.6 -2.2 -4 0 50 100 150 Time (msec) 200 250 Field Test Response ANSYS LS-DYNA Response Figure 3-47: Deflection Response of R2 Panel at near quarter-point.8 -1.4 1.8 0 -0.5 -6 0 50 100 150 Time (msec) 200 250 Field Test Response ANSYS LS-DYNA Response Figure 3-48: Deflection Response of R2 Panel at far quarter-point. 71 .5 0 -1.Normalized Deflection (delta/EI) 2. 3.4 1.Normalized Deflection (delta/EI) 3.6 -2. More research is needed to verify this and change the program to account for the load reduction.8 0 -0.4 -3.6 0.7 Summary It appears from the ANSYS LS-DYNA curves that the equivalent loading procedure does not predict the maximum deflection well.2 2. The first deflection curve from the “exact” blast loading is more than 50% less than the deflection response from the equivalent loading. but is still conservative. The DAHS procedure more closely approaches the peak deflections from rebound. The field test data highlights an interesting phenomenon in blast pressures on the tops of structures. 72 .8 -1.2 0 50 100 150 Time (msec) 200 250 Field Test Response LS-DYNA Response DAHS Response Corrected DAHS Figure 3-49: Deflection Response of R2 Panel at midpoint. but encounters greater error on roofs. The curves define linear elastic materials and do not include any dampening effects. CONWEP accurately predicts the ground surface pressures. 1 Introduction Current resistance functions for open web steel joists include a linear elastic curve close to the elastic limit and a fully plastic curve after yield. deflection. 73 . and amount of energy able to be absorbed of a typical joist system. It is believed that the failure mechanisms in a steel joist contradict this behavior.Chapter 4 Static Resistance Function of Open Web Steel Joists 4. and static tests are performed to qualify this theory. The steel trusses in this research are tested to failure to evaluate the maximum load. The engineering resistance ranges from 94 to 99% of the SBEDS resistance. for the largest joist.1 (Biggs.2 Analytical Resistance Function Three types of open web steel joists are considered in this research. the elastic deflection limit also has to change with a similar trend. ye = Ru ke (4. the ratio is about 75%.1) where ye = elastic deflection limit Ru = ultimate resistance ke = effective stiffness 74 . and 32LH06 (SJI. The ultimate resistances for the trusses almost coincide for the SBEDS calculation and the engineering practice. The truss types are 16K5. Before actual experimentation of the joists.4. it is more than 27%. Their sizes are chosen to represent three differing aspect ratios. Figures 4-1 through 4-3 display the response curves for all three sets of trusses according to SBEDS (2004) calculation and an example of current engineering practice. Since the ultimate resistances are almost equal with respect to the procedure and the stiffness ratio changes. However. 4. For the smallest truss. 1967). Stiffness and resistance are related according to Eqn. 26K6. 2005). other resistance methods can be employed. the stiffness and elastic deflection limit did not match as well. The stiffness ratio of engineering calculation to SBEDS decreases as the aspect ratio decreases. 8 10 Deflection (in) 12 14 16 Figure 4-2: Resistance Function for 26K5 Joist. Figure 4-1: Resistance Function for 16K5 Joist. Ultimate Load. Defl. 75 .900 800 700 600 Load (lb/ft) 500 400 300 200 100 0 0 2 4 6 8 10 Deflection (in) 12 14 16 Engineering Calc SBEDS Max Allowable Load. Ultimate Load. Defl. 1200 1000 Engineering Calc 800 Load (lb/ft) 600 400 200 0 0 2 4 6 SBEDS Max Allowable Load. Defl. Defl. The SBEDS calculation can generate pressure-impulse diagrams that calculate the maximum deflections each truss can resist and the associated pressures and impulses.1600 1400 1200 Load (lb/ft) 1000 800 600 400 200 0 0 2 4 6 8 10 Deflection (in) 12 14 16 Engineering Calc SBEDS Max. Allowable Load. This relationship is graphed for single-degree-of-freedom models with an elastic-plastic resistance shape subjected to triangular pulse loads in Fig. The calculation for stiffness utilizes an approximate formula from the Steel Joist Institute (2004) to calculate joist moment of inertia. The SBEDS calculations use data from actual joist cross-sectional properties to calculate the stiffness. and the ratio of pulse duration to the joist natural period. The maximum deflection in the engineering procedure is based on the ratio between the ultimate resistance. Defl. The calculation for ultimate resistance in the engineering procedure uses wellknown values. Defl. 4-4. Figure 4-3: Resistance Function for 32LH06 Joist. Ultimate Load. the loading. 76 . This maximum deflection depends on the burst load. 34 499. Tables 4-1 and 4-2 summarize the results of the engineering calculation and SBEDS.1902 2.349 T (msec) 53.03 903. one-degree-of-freedom system for triangular pulse load (Biggs. 1967).93 77 . respectively.057 622. Table 4-1: Resistance Data from Engineering Calculations.23 Ye (in) 3.07 39.The maximum load and ultimate load based on allowable stress design from the Steel Joist Institute are also noted in Figures 4-1 through 4-3.72 1128. 16K5 26K5 32LH06 Stiffness (lb/ft/in) 231. Figure 4-4: Maximum deflection of elasto-plastic.333 1462. For the LH-Series truss the maximum load is about 77% for both the engineering calculation and the SBEDS calculation.25 38.7 Ru (lb/ft) Ru (SJI) 774. For the K-Series joists the maximum resistance is 83% of the engineering calculation and about 78% of the SBEDS calculations.411 640 1093.3475 2. This is probably due the smaller stiffnesses derived from the engineering method.64924 44. both procedures allow for the calculation of joist natural periods.Table 4-2: Resistance Data from SBEDS Calculations. 4. The joists were loaded until ultimate failure.40354 30. the natural period is about 77% more than the SBEDS calculations. 78 . All trusses are 24 feet long.3.532127 Ru (lb/ft) Ye (in) T (msec) 812.5155286 2302. 16K5 26K5 32LH06 Stiffness (lb/ft/in) 308. The natural periods from the engineering calculations are larger than the SBEDS calculations. For the LH-Series. The results from the testing are compared to the analytical resistance functions.1 Testing Samples Three pairs of joists are examined in this research. The natural periods for the engineering calculations are 20-30% more than the SBEDS calculations for the K-Series trusses.03 1168.64483 22.28 1. 4.689 2.3155217 838.3 Experimental Verification This section will discuss the testing of the joists systems to establish the resistance function. Attention was kept to the Steel Joist Institute specifications for placing and loading OWSJ. Two joist sets are from the KSeries type joists and one is a LH-Series truss.05 0.01 Additionally.52 1478. The compression (top) chord consists of two equal leg angles 1.5” wide with thickness of 0.02 inches.5 inches wide with a thickness of 0. The web members connect to make four panel points in the top chord and three panel points in the bottom chord. The diameters for the round bars in the truss include 11/16” and 19/32”.130 in thick.30 inches is required. The secondary web members are also solid round bars 11/16” in diameter and are designed to react in compression.30 inches. in construction of the joist a camber of 0.090 inches. The web members delineate 9 panel points in the top chord and 8 panel points in the bottom chord. The crimped-end members are slightly thicker than the other web members.166 inches.155 in. The end tension diagonal is a round steel bar 15/16” in diameter. They are designed to be in compression These U-shaped members are 1” wide and 1” tall with a thickness of 0. The truss had a camber of 0. Also.5 inches wide and 0. The 26K5 truss has an effective depth of 25. The tension (bottom) chord is also made up of two equal leg angles 1. 79 . The end tension members are made up of solid round bars ¾” in diameter. The web consists of crimped-end U-shaped members near the outside of the web and regular U-shaped members in the web set. The regular web members have the same dimensions as the secondary web members.The 16K5 steel joist has an effective depth of 15. It should also be noted that the diameters of the web members decrease closer to the center of the joist length.109 inches. The compression chord contains two equal leg angles 2” wide with thickness of 0. The tension chord is made up of two equal leg angles 1. The secondary web members consist of U-shaped members in diagonal near the ends and vertical in the web. The rest of the web is made up of tension and compression solid round bar members.14 in from the top chord centroid to the bottom chord centroid. 4. and a minimum of 2 . There are no crimped members.75” wide with thickness of 0.3. the open web steel joists (OWSJ) are designed for resting on steel supports. The bottom (tension) chord is made up of two equal leg angles of 1. The end tension diagonal is a solid round bar with a 1” diameter. The secondary (compression) members have 1” width and 1” height and thickness of 0. According to the Steel Joist Institute 42nd Edition Standard Specifications (2005). no less than 4” of the joist end may be placed on the support.30 inches. The rest of the web members consist of U-shaped members. The necessary camber is 0.109 inches thickness.143 inches.09 inches.118 inches. The compression chord is made up of two equal leg angles of 1.The 32LH06 steel truss has an effective length of 31. Note the top chord has a larger cross-sectional area compared to the bottom chord for all truss types. no less than 2 ½” of the end of a K-Series OWSJ may lay on the steel support.090 inches. For LH and DLH-Series OWSJ. 80 .2 Test Set-Up For our testing purposes.5” wide with 0. This trend continues for the other two joists.¼” fillet welds 2 inches long each must be used. The slenderness ratio of the web members grows larger closer to the center of the joist length. The compression web members have width of 13/8” and height of 1-3/8” and thickness of 0. The tension web members have the same dimensions. For the 16K5 steel truss the cross-sectional areas became smaller closer to the center of the joist length. The end of the OWSJ is welded to the steel supports using a minimum of two 1/8” fillet welds one inch long each. This joist contains three panel points in the top chord and 2 panel points in the bottom chord. the width of the fillet welds for the bearing seats will be ¼” and 3 inches long.5 inches. the width of the plate is limited to 13 inches. The support system itself is bolted to a strong floor.Figure 4-5: Bearing seat plates for 16K5 and 26K5 Joists. The bearing seats are laid out on the plate to allow room for the welds on each side and for a joist center to joist center distance of 6. The bearing seats of the open web steel joists are welded to a 3/8” thick plate that is bolted to the support system. The length of the plate is also 13 inches. Due to the plate thickness and LRFD standards. Due to the dimensions of the support. Six 1” diameter and two ¾” diameter regular-strength (A307) bolts are used to connect the plate to the strong floor support. 81 . Each attachment will be made by welding and is to have a slenderness ratio (between attachments) of less than 300.Figure 4-6: Bearing Seat Plates for 32LH06 Joist. Figure 4-7: Steel Joist Institute Specifications for horizontal bridging. horizontal steel members (equal leg angles) attached to both the top and bottom chord. horizontal bridging consists of continuous. Each weld should resist at least 700 lbs of horizontal force. and specifications approved by the Steel Joist Institute. length. 82 . Diagonal bracing is not necessary for any truss. Due to the joist type. 5” equal leg angles. smaller effect on resistance of truss system. These connections can be fixed. hinged. The latter connection type was chosen for several reasons: stability of the joists. The equal angle horizontal bracing can be welded to the third points on the truss. The angles were welded to the strong floor and grease was applied to the sides of the channels to simulate a roller-type connection. The lateral bracings were placed in the center and spaced at 4 feet intervals along the length of the steel joist system. i. and ease of construction. Five sets of lateral bracing were constructed from 12 inches long angles with channel members welded to either end. The horizontal bridging consists of 1. The equal angle horizontal bracing for this pair of trusses can be welded at the middle of the trusses. The spacing has to be uniform throughout the joist. Figure 4-8: Lateral bracing – welded to strong floor. They are welded according to the figure with a 1/8” wide weld one half an inch long. at 8’ from either end. The LH-Series joist only requires one row of horizontal bracing.Both K-Series joists need a minimum number of two rows of horizontal bracing. the ends of joist systems require connections at wall or any other structural component capable of resisting lateral loads. 83 . Figures show these bracings.e. Including the horizontal bridging between trusses. or rollers allowing movement in the vertical direction. These deflection gages are connected to the plywood boards that run the length of the joists.Figure 4-9: Lateral bracing – placement. String potentiometers are placed in the center of the joist system length and at the quarter. 84 . Figures 4-10: String Potentiometer.points of the joist system length. Figures show the connection of the strain pots. They will be connected at the supports. by the horizontal bridging.3 Testing Apparatus Each test will consist of a pair of OWSJ. Two plates are connected to each of these smaller beams. Each of the final plates in the loading tree has connections at the ends to apply load to a testing sample. A general schematic of the loading tree is shown below. The loading tree has a maximum load of 110 kips.3.4. 4-11) with 16 loading points acting along the joists. The equipment pulls up to simulate a downward distributed force. and two smaller plates are connected to the previous plates. Straps were used to connect the truss system to the loading tree due to their relatively high depths. The variation in lengths between the joist structure and the approximate distributed loading arises due to equipment restraints and available trusses. The joist pair will be tested using an 18’ loading tree (Fig. 85 . and finally the loads from the four points are distributed to sixteen points. The net effect is that the load from the pneumatic pump is distributed to two points. and so the test sample must be placed upside-down. The loading tree consists of a pneumatic pump that pulls a beam made up of two channel sections back-to-back. the loads from those two points are distributed to four points. Two smaller beams each made up of two channel sections back-to-back are connected to the larger beam. and by ½” boards of plywood the entire length and width of the joist pair. This was determined by observing the failure mechanisms of the joist during the test and the decay of load resistance with an increase in deflection. 86 .5 inches per second. As stated in Section 4. 4.3.3.4. Each joist system was tested until it could not sustain any more load.1 16K5 Joist System Figure 4-12 show transverse and side views of the 16K5 truss system. due to the camber in the truss and their own inherent flexibility out of the plane of loading.4 Results The following section presents the results of the steel joist testing.3. 4.Figure 4-11: 16-Point Loading Tree. the loading tree does not cover the entire length of the joist. The test samples were loaded using a deflectioncontrolled rate of 0. it can be seen that the trusses are slightly warped. Also.3. the welding connecting the compression chord to the bearing seats fails (Fig. Figures show this sequence of events. the tension chord begins to twist. Then. As the top chord (in figures the bottom is the top) continues to bend. 4-18). Initially the truss system displayed very little noticeable displacement. 4-16). 4-13). 87 . Finally. The truss system is now characterized as being unable to support any further load. 4-14 and 4-15). This bending is mainly due to failure in the compression flange. The lateral bracing at the center of the joist length fails due to this deflection (Figs. The welds on the opposite side of the transverse bending fail. Eventually the horizontal bracing in the tension chord fails (Fig. the middle of the truss began to bow out-of-plane (Fig.Figure 4-12: 16K5 trusses prior to testing. Figure 4-13: 16K5 Joist – Failure Sequence 1 of 5 – Initial bending. 88 . Figure 4-14: 16K5 –Failure Sequence 2a of 5 – Failure of lateral bracing. Figure 4-16: 16K5 Joist – Failure Sequence 3 of 5 horizontal bridging. 89 .Figure 4-15: 16K5 Joist – Failure Sequence 2b of 5 – Failure of lateral bracing. 90 . Figure 4-18: 16K5 Joist – Failure Sequence 5 of 5 – Failure of bearing seat weld.Figure 4-17: 16K5 Joist – Failure Sequence 4 of 5 – Continued out-of-plane bending. 000 30.000 25. while D2 and D3 are the quarter-points (Fig. The more noticeable bending of the compression chord out of the loading plane releases some of the energy absorbed.000 Load (lbs) 20. The load resistance also increases in a fairly linear manner. D1 measures the midpoint. The failure of the horizontal bracing releases more built-up energy.000 15. Following this. 91 . As can be seen. and finally the bearing seat weld failure finishes the usability of the truss system. the system is able to receive more energy.000 0 0 5 10 Displacement (in) D1 D2 D3 15 20 Figure 4-19: Static response for 16K5 truss system. 35.000 5.000 10. 4-19). Failures are denoted in the chart as areas where there is a large gap between data points and a downward slope.The load-deflection responses of the 16K5 truss system are shown in Figure 4-19. the system experiences linear displacement initially. It can be seen that the experiment measures a lower yield than existing analytical methods. the maximum experimental load coincides with the ultimate load defined by SJI (2004) and steel joist manufacturers. The original stiffness of the 16K5 truss corresponds well to the engineering calculation stiffness. The analytical methods assume failure mainly by bending. Ultimate Load. 92 . Figure 4-20: Midpoint Static Response for an individual 16K5 joist compared with existing methods.900 800 700 600 Load (lb/ft) 500 400 300 200 100 0 0 2 4 6 8 10 Deflection (in) 12 14 16 Experiment Engineering Calc SBEDS Max Allowable Load. Additionally. The midpoint static response for an individual 16K5 truss was determined by dividing the system response by two times the effective joist length (Fig 4-20). Defl. Defl. 4-23 and 4-24). buckling of a compression web member.4. and the secondary web member and another web member from the other joist buckles (Figs. The results of the end tension member break were a large release of energy and redistribution of forces. The truss system continues to absorb energy and the behavior has a linear shape (4-30). The source is evident from the next failure mode. it can be seen that the weld for the tension member failed for one angle in the tension chord. When the end of the tension chord begins to bend. the truss system was also slightly askew (Fig. 4-22). 93 . 4-21). The first noticeable deflection was the bending of the ends of the tension flange in both trusses (Fig.2 26K5 Joist System In this test. the load-deflection behavior was fairly linear (Fig 4-30). or a combination of the two. This phenomenon would be caused by high tension in the end tension member. From Figure 424. Figure 4-24 shows the weld from this new tension member fails. Since the end tension member in one truss failed. A part of the other angle in the tension chord was ripped. the response curves. The failure mechanisms were also different. This is the most likely cause of the web compression member buckling. It should be noted that this member was originally designed as a compression member. This was the same portion that was bending at the start of the test.3.4. The end tension member of one joist fails. the next web member in that truss takes its place as a tension member. Up to this failure. It was noticed afterwards that the welds bonding the bearing seats to the connection plates failed (Fig. The load-deflection behavior after this point continued to drop without the anticipation of rebound. 94 . 4-29). this failure seems to be caused by the twisting of the tension chord more than the bending of the member. Figures 4-21: 26K5 Joist System prior to test.The end tension member of the other truss fails. From Figure 4-26. Figure 4-22: 26K5 Joist – Failure Sequence 1 of 5 – Deformation of Tension Chord. Figure 4-24: 26K5 Joist – Failure Sequence 2b of 5 – Failure of End Tension Member.Figure 4-23: 26K5 Joist – Sequence 2a of 5 – Failure of End Tension Member. 95 . Figure 4-25: 26K5 Joist – Failure Sequence 3 of 5 – Failure of Secondary Web Member. Figure 4-26: 26K5 Joist – Failure Sequence 4a of 5 – Failure of End Tension Member. 96 . Figure 4-28: 26K5 Joist – Failure Sequence 5a of 5 – Connection Plate Failure. 97 .Figure 4-27: 26K5 Joist – Failure Sequence 4b of 5 – Failure of End Tension Member. 000 20.000 10.Figure 4-29: 26K5 Joist – Failure Sequence 5b of 5 – Connection Plate Failure.000 40. 98 .000 0 0 5 10 Displacement (in) 15 20 Figure 4-30: Static Response for 26K5 Joist System. 45.000 35.000 5.000 Load (lbs) D1 D2 D3 25.000 30.000 15. This was not expected to affect the strength of the joist system. Again. compares well with the engineering calculation value.4. Ultimate Load. The initial slope of the elastic response curve of the joist.1200 1000 800 Load (lb/ft) 600 400 200 0 0 2 4 6 8 10 Deflection (in) 12 14 16 Experiment Engineering Calc SBEDS Max Allowable Load. Again the ultimate load of the test closely matches the ultimate resistance of the joist defined by the SJI and the manufacturer. Defl. 99 . 4. The tension chord in one of the trusses is slightly warped and twisted (Figure 4-32). Figure 4-31 demonstrates the multiple failure modes a steel joist has to resist before ultimate failure. it can be seen from Figure 4-31 that the current methods calculate more resistance than is experimentally observed.3 32LH06 Joist System The test set-up for this truss system is displayed in Figure 4-33.3. Figure 4-31: Midpoint Static Response for individual 26K5 Joist compared with existing methods. the stiffness. Defl. After the multiple buckles. Again the initial load-deflection behavior changes linearly. 4-34 through 4-36). the response showed no sign of rebounding. This resulted in a loss of energy from the system. The test was stopped. The buckling of these members redistributed the forces and caused the secondary web (compression) members to buckle as well. 100 . Like the previous joist types. The primary failure mechanism is the buckling of web compression members near the end of the truss (Figs. there was little noticeable deformation at the beginning of the test. Figure 4-32: Initial Deformation of 32LH06 Joist.This failure system is similar to that of the 26K5 joists in that it is localized in an area other than the midpoint. 101 . Figure 4-34: 32LH06 Joist – Failure Sequence 1a of 1 – Compression Web Member Buckling.Figures 4-33: 32LH06 Joist System prior to test. 102 .Figure 4-35: 32LH06 – Failure Sequence 1b of 1 – Buckling of Secondary Web Member. Figure 4-36: 32LH06 – Failure Sequence 1c of 1 – Continued Buckling and Bending of Tension Chord. 000 20.000 10.000 0 0 2 4 6 8 10 Displacement (in) Figure 4-37: Static Response of 32LH06 Joist System. Defl. 103 . 1600 1400 1200 Load (lb/ft) 1000 800 600 400 200 0 0 2 4 6 8 10 Deflection (in) 12 14 16 Engineering Calc Experiment SBEDS Max. Figure 4-38: Midpoint Static Response of individual 32LH06 Joist comparing to existing methods.000 25.30.000 5.000 Load (lbs) D1 D2 D3 15. Allowable Load. Defl. Ultimate Load. The current techniques used to define the resistance function of open web steel joists do not take into account the buckling of web members and the resultant strain softening from the loss of resistance and energy. This might have resulted in a sheardominated behavior instead of bending. the joist is designed to span longer lengths of about 30 ft or more (SJI. The joist came close to reaching the maximum allowable load (SJI.The steel joist in this test did not reach the expected resistance value as shown in Figure 3-38. 4. 2005). it can be concluded that more research is necessary to develop the static resistance functions of open web steel joists. 104 . It is possible that the experimental resistance was much lower because truss span to depth ratio is too small. Current blast design methodologies assume that the resistance function of open web steel joists behave similar to hot-rolled beams and reinforced concrete slabs. From the third test it is recommended for LH-Series Joists that longer spans be tested. Since this truss is a longspan steel joist. The experiments for two out the three tests showed ultimate resistances close to the Steel Joist Institute’s value of ultimate load. The full resistance of the joist and its members was not utilized.4 Summary From these static tests. 2005). The results of the equivalent blast did not adequately predict the response of the simulated blast load. The experimental resistance function did not match these analytical resistance functions currently used in SBEDS.Chapter 5 Conclusions & Recommendations 5. However. The equivalent load procedure was also compared to experimental blast data. The finite element model closely predicted the response from the field test. Three different open web steel joists were tested under static uniform loads. The resistance function for each steel joist was calculated from these results and compared to current methods for developing truss resistance function.1 Conclusion The equivalent uniform blast procedure defined by DAHS was compared to numerical blast data simulated using finite element analysis. the equivalent load did not adequately predict the deflection measured in the field test. The following are the conclusions that can be drawn from this research. 105 . Therefore. In summation. but do not reasonably evaluate blast pressures on roofs. • The equivalent load procedure does not adequately predict response. • Current methods for calculating blast pressures such as CONWEP and SBEDS correctly computes pressures at the ground level. • The current methodology to develop resistance function for open web steel joists measures a higher peak load than experimentally tested. current methods to calculate the blast loading on a roof and to design for the resistance of open web steel joist are not accurate. the following recommendations are provided to develop an engineering method for blast design of open web steel joist roof system. • The SBEDS assumption of perfect plastic behavior post-peak does not take into account the various failure mechanisms observed in open web steel joists. 106 . even after adjusting the pressure input calculated from CONWEP or SBEDS.• The DAHS procedure is very conservative and does not compute accurate responses for roof systems. • The equivalent uniform blast load procedure requires more research to better predict response.2 Recommendations The following recommendations are made based on the observations of this study. 107 . The calculation of blast pressures on rooftops needs to be further researched. Higher deflections can occur after this. This can be accomplished by conducting an array of numerical solutions using various roof and blast parameters.5. • The resistance function for open web steel joists should be developed based on more analytical and experimental research. • Current methods to measure blast pressures achieve good results at the ground level. Engineering analysis and design should consider the first peak deflections and subsequent maximum deflections. • Existing and additional field test in roof systems should be evaluated to verify developed engineering analysis and design. It has been observed that roof systems affect the blast load and effectively reduce the loading. • Currently engineers only focus the first peak deflection in dynamic analysis. H. AISC Manual of Steel Construction – Load and Resistance Factor Design. “Design and Analysis of Hardened Structures to Conventional Weapon Effects (DAHS). C. Illinois.. Texas. McGraw-Hill Book Co. 1. Salim. M.. (2001). (1992).REFERENCES American Institute of Steel Construction (AISC). “Dynamic Response of Walls and Slabs by Single-Degree-ofFreedom Analysis – A Critical Review and Revision. K.” Structural Mechanics Division. USAE Engineer Research and Development Center (ERDC). 108 .. Chock. Vol. D. p. the Navy. Q. Morison. C. Florida. pp... (2002). Meng. 82. Kapania. (1991). 2. “Experimental Evaluation of Structural Composites for Blast and Fragmentation Resistance Design.” The Shock and Vibration Digest. R. Structures Laboratory. J. pp. (1999). H. No. USA.A. Nebuda. Li. Department of the Air Force Manual AFM 88-22.Application of TM5-855-1.. “Blast Loading on Petrochemical Buildings.” International Journal of Impact Engineering. Oswald. Vol. (2001). Departments of the Army. 3. M.. Galambos. 32. San Antonio. J.” Proceedings from the 31st DoD Explosives Safety Seminar. J. (2004). and the Air Force.“Review of Two Methods for Calculating Explosive Air Blast. 94-102. Hyde. SAS IP. pp. Tyndall AFB. Forbes. “SBEDS (Single-Degree-of-Freedom Blast Effects Design Spreadsheets). 1214-1247. “ConWep . C.. No. “The Effect of Lateral Bracing on the Stability of Steel Trusses.” Air Force Research Laboratory. D. Biggs.” United Facilities Criteria (UFC) 3-340-01. pp. K.J. D. (2002). 125..” Journal of Energy Engineering. Vol. Vicksburg. J. (2004). (1990). M. Department of Defense. Xykis. Hoemann. Structures to Resist the Effects of Accidental Explosions. Department of the Navy Publication NAVFAC P-397.. Vol. Vol. 91-102. 251-258.” Journal of Construction Steel Research.0.. 33. (2007). “Pressure-Impulse Diagram for Blast Loads Based on Dimensional Analysis and Single-Degree-of-Freedom Model. AFRLML-TR-2006-4578.” Journal of Engineering Mechanics. 20. T. (1967). (2006). No. ANSYS ED Code Version 9. Department of the Army Technical Manual TM 51300. Introduction to Structural Dynamics. Mississippi. Chicago. 128.. 5. Geotechnical and Structures Laboratory (ERDC-GSL) Technical Report. Hetherington. 42nd Edition Standard Specifications – Load Tables and Weight Tables for Steel Joists and Joist Girders. Sunshine. “Overview of Simplified Methods and Research for Blast Analysis.S.. Steel Joist Institute (SJI).. P.S. (2004). Yost. (2004).G.. B. D. Pote. Steel Joist Institute. J. (2005). A. M. (2002). D. Army Corps of Engineers. J.S. Butterworth-Heinemann.” ASCE. Army Engineering Research and Development Center. U. J. P. TR-02-14. Gross. 715. Army Corps of Engineers. Dinehart. Vol.. TR-05-02. U. D. 130. “Lightweight Steel-Deck Roof System – BlastResponse Evaluation. Blast and Ballistic Loading of Structures. No. R.Smith.S. 109 ..” U. (1994).. “Component Explosive Damage Assessment Workbook. Protective Design Center (PDC) Technical Report. W... Swanson. S. Army Corps of Engineers. “Strength and Design of Open Web Steel Joist with Crimped-End Members. (2005).. p. Gargan.. J.” Journal of Structural Engineering.” U. Amini.