Página 1 de 51BIAX for MS WINDOWS Strength Analysis of Reinforced Concrete Sections By John W. Wallace Yasser A. Ibrahim March 1996 INTRODUCTION PROGRAM CAPABILITIES REFERENCES SYSTEM REQUIREMENT PROGRAM INSTALLATION PROGRAM EXECUTION PROGRAM INPUT System Section Steel Holes Confined Properties Data APPEDIX A "SECTION AND MATERIAL MODELING" APPENDIX B "EXAMPLES AND APPLICATIONS" APPENDIX C "CONCRETE STRESS-STRAIN RELATIONS" Página 2 de 51 Introduction. A general purpose computer program to evaluate uniaxial and biaxial strength and deformation characteristics of reinforced concrete (RC) sections is described. The program was developed for use on MS-DOS personal computers and has been used extensively at the University of California for both research and teaching purposes. Although other programs are available for this purpose NISEE 1987, many have limitations (such as uniaxial loading), are not available for personal computers, or are cumbersome to use. The intent of the current upgrade is to make the program more user-friendly by introducing a graphic interface that saves both time and effort, improve the program capacity by improving the memory configuration, update the confined concrete models available with the program and incorporate circular sections. Program Capabilities. The program computes strength and deformation characteristics based on the assumption that plane sections remain plane after the application of loading. Based on this assumption, the program can be used to compute strength or moment-curvature relations for uniaxial or biaxial monotonic loading of reinforced concrete sections. The strength can be computed for a single loading case, or interaction diagrams can be generated (e.g. P-Mxx, P-Myy, or Mxx-Myy). New features includes a windows version of the original FORTRAN code that is compatible with MS Widows 3.1 and later, new memory configuration that allows the program to utilize all the available extended memory and a feature to plot all the output files such as P-M diagrams and moment-curvature diagrams. Nonlinear material models are used for both the reinforcing steel and the concrete. The model for the stress-strain behavior of the reinforcing steel is versatile, allowing relations that closely approximate experimentally observed behavior. Models for the concrete stress-strain behavior include. Sheikh and Uzumeri (1982), and Yong et al. (1988) relations. A stress-strain relationship for unconfined and confined masonry and user defined stress-strain relationships are also incorporated. The relationship suggested by Vecchio and Collins (1986) is used to describe the stress-strain relation for concrete in tension. The tensile strength of the concrete beyond the rupture stress may be included or neglected. The current upgrade includes Saatcioglu and Razvi Model for both normal and high strength concrete (1995), Tanaka, Park and Li (1994) and Mander, Priestly and Park (1988). Presently, the program allows seven stress-strain diagrams for concrete (unconfined and confined), and four relations for reinforcing steel (for any given problem). The current upgrade has the capability of analyzing circular cross-sections and non-linear strain distribution for both steel and concrete in tension. To make the program even easier to use, new default options for steel were added. The RC section is described as a combination of subsections; therefore, the program allows easy generation of T, L, circular or barbell shaped sections. The program user specifies a mesh for each subsection. An iterative procedure (simple bisection algorithm) is used to obtain a solution for the prescribed problem. Página 3 de 51 The CAL/SAP library of subroutines (developed by E. L. Wilson at the University of California at Berkeley) is used to read the data from the input file. The subroutines allow the use of free-form input which facilitates the task of creating an input data file. Referances Hognestad, E. (1951) " A Study of Combined Bending and Axial Load in Reinforced Concrete Members" University of Illinois Engineering Experimental Station, Bulletin Series No. 399. November 1951, 128 pp. NISEE: National Information Service for Earthquake Engineering (1987) "Computer Software for Earthquake Engineeing," Earthquake Engineering Research Center, University of California at Berkeley, June 1987 Park, R., Priestley M. J. N.,Gill, W. D. (1982) "Ductility of Square Confined Concrete Columns" Journal of the Structural Division" ASCE, Vol. 108, No. ST4, April 1982, pp. 929-950 Saenz, L. P. (1964), "Equations for the Stress-Strain Curve of Concrete," ACI Journal Proceedings, Vol. 61, No. 22, September 1964, pp.1229-1235. Sheikh, S. A., Uzumeri, S. M. (1982), "Analytical Model Concrete Confinement in Tied Columns," Journal of the Structural Division, ASCE, Vol. 108, No. ST12, pp. 2703-2722. Vecchio, F. J., Collins, M. P. (1986), "The Modified Compression-Field Theory for Reinforced Concrete Elements Subjected to Shear, ACI Journal, Vol. 83 No. 2, March-April 1986, pp.219-231. Yong, Y. K., Nour, M. G., Nawy, E. G. (1988), "Behavior of Laterally Confined HighStrength Under Axial Loads," ASCE Structural Journal, Vol. 114, No. 2, Feb. 1988, pp. 332-351. Mander, J. B., Priestly, J. N., Park, R. (1988), "Theoritical Stress-Strain Model for Confined Concrete," ASCE Journal of Structural Engineering, Vol. 114, No. 8, August, 1988, pp. 1804-1826 Tanaka, H., Park, R., Li, B. (1994), "Confining Effects on The Stress-Strain Behavior of High- Strength Concrete," Second US-Japan-New Zealand-Canada Multilateral Meeting on Structural Performance of High-Strength Concrete in Seismic Regions, Honolulu, Hawaii, 29 November-1 December 1994 System Requirement - IBM or 100% compatible - Intel 486 CPU or higher - MS Windows 3.1 or later - 7 MB of free disk space Página 4 de 51 Program Installation To install the program, the user will need PKZIP 2.04 or later (preferably WINZIP 6.0). 1- Unzip the contents of the disks titled DLL.ZIP and VBX.ZIP to your system directory. ( that is x: where x is your hard drive ). 2- Unzip the contents of the disk titled BIAX.ZIP to the directory of your choice. Warning: Do NOT overwrite any pre-existing files in your system directory Program Execution Double click the program called BIAXW2.EXE, the main window appears with the main menu. Following is the menu items and the corresponding functions. Menu Item About Input Run Function Gives general information about the program Invokes the text editor Invokes analysis program and prompts the user to the data file name. During execution, the program displays messages indicating the analysis progress Plot Help Exit Gives different plotting options Provides online help Terminates the program Página 5 de 51 Several output files are created by the program. they are explained in the following table (1) (2) Filename.out Filename.dat Output file of program EXECUTION including Calculated results. File for plotting the P-M diagrams (P-M analysis) Column (1) X-moment, Mxx Column (2) Axial load, P Column (3) Y-moment, Myy Column (4) Curvature, ec/d Column (5) Maximum steel tensile strain (3) (4) Filename.dbg Conc.dat Debug file Concrete material stress-strain Column (1) Concrete Strain Column (2) Unconfined Stress Column (3) Confined Stress Column (4) Tensile Stress (5) Steel.dat Steel material stress-strain Column (1) Steel Strain Column (2) Type 1 Steel Stress Column (3) Type 2 Steel Stress (6) (7) (8) (9) (10) (11) (12) Mesh.dat Sects.dat Sectc.dat Checkx.dat Checky.dat Strx.dat Stry.dat Concrete element coordinates Steel bar coordinates for plotting check of input file Concrete section corner coordinates for plotting check of input file Non-linear strain distribution along x-axis Non-linear strain distribution along y-axis Strain distribution factor for each bar along x-axis Strain distribution factor for each bar along y-axis By using the plot menu,the input file can be checked for errors by plotting the stressstrain diagrams for both concrete and steel, the cross section and the generated mesh. section. only data blocks (2).Separators are used within the input file to specify the location of the eight data blocks. (3). section. All the parameters for the SYSTEM data block have default values. (2) The only required data block is SECTION.C.C. input file requirements for each data . etc). area. Data block to define any holes (voids) that may exist in R. For circular cross section. Inertias (Ixx. NOTES: (1) For the analysis of a simple unconfined R. point (0. A batch file is provided to delete output files created by the BIAX program. Data block to define rectangular subsections that make up the R.e. Data block to define portions of the R. and therefore. The separators and the information obtained within each data block are described in the following table. Iyy and Ixy) are calculated for both the neutral axis location and the global midpoint of all the sections defined.Página 6 de 51 The program calculates section properties. If only this data block is specified.0)). e. Data block to define the parameters for the solution or solutions desired.BAT and the input filename to delete the files created by the program (execute by typing EAT followed by the filename. Data block to define location. the midpoint is alway at the origin (i. The separators may appear in any order.g.C. The moments are given about the midpoint because it is easy for the user to locate. SEPERATOR SYSTEM SECTION STEEL HOLES CONFINED STRAIN PROPERTIES DATA DESCRIPTION OF DATA BLOCK Data block to define global analysis parameters.C. section that are confined. the program will compute the section properties (Area. Comment statements are allowed following the use of a colon. Use the file EAT. this block is not required in this case. Data block to define the non-linear strain distribution Data block to define material properties for the reinforceing steel and concrete. and type of steel reinforcing bars. The input file is separated into eight blocks. Inertia. section.(6) and (7) are required. (4) The following pages describe the block. EAT EX1) Program Input The following several pages describe the input data file requirements for the BIAX program. (3) In the CAL/SAP subroutines a colon is used to denote the end of data input for each line. j.k i. Line 1 D= i. 1 2 M= 0 1 If all concrete at rebar is unconfined If all concrete at rebar is confined Default Rotate entire section by 180 degrees For non-symmetric section to change location of compression zone.j.001] If rebar is located within both confined and unconfined concrete then the program will search to determine type of concrete closest to each rebar (to subtract off concrete area taken up by steel). . A-5 for the definition of the compression reference point.Página 7 de 51 System Input Data block used to specify global analysis parameters.j. See Fig.k =1 debug messages given for different options within the program U= T= 1 0 0 1 2 R= S= P= E= 0 R 0 1 0 1 0 1 2 TOL= C= # 0 Input specified in data file (default) Interactive input No generation of P-M surface (default) Generation of P-M surface Generation of p-m surface (dimensionless) Area of bars specified Where R= steel ratio for section Automatic itereation Interactive iteration (Default) Plot material stress--strain relations (Default) Limit maximum concrete strain so rebar does not fracture Do not allow rebar to fracture Tolerence of convergence to specified axial load [0.k=0 no debug messages (default) i. Xr. Line 1 to END One line for each subsection defining the complete R. See note (1). section. etc) S= Section type S=1 for rectangular section HI = Length of section in the I-direction HJ = Length of section in the J-direction N1 = # of elements in the I-direction (Default = 1) N2 = # of elements in the J-direction (Default = 1) X= X Starting location for bottom left corner of rectangle Y= Y Starting location for bottom left corner of rectangle A= angle.0).3. 1. Y coordinate system.2.Yr Angle = (0-360 degrees) to rotate about point Xr. OR S= 2 for circular section R= Radius NR = Number of rings NP = Number of poles END SECTION WITH BLANK LINE NOTE: (1) By using the 'A=' option it is possible to create all sections in a global X. The 'A=' option defaults to the previous specified value ('A=0. and rotate them to an arbitrary position.0. Slight overlapping of sections will occur at points where sections that are rotated are ``attached'' to sections that are not rotated. .Página 8 de 51 Section Input Data block used to specify RC section geometry and mesh.0' to stop section rotation). A positve angle represents counterclockwise rotation.C. Xr. NS is the total number of sections I= The section # in sequential order (e.Yr = Point to rotate section about (default = 0. Yr. Xr and Yr define global coordinates about which to rotate the section.g. 0. R= Angle. (5) bar increment # defaults to [1] (6) values for nbar need not be sequential or continous. Xr.5. 3.1 generates bars 2. END SECTION WITH BLANK LINE NOTES: (1) nbar must begin each line (2) scalar multiples can be used on any term except nbar. X=10*12 or X=25/12 (3) the default value for 'A' is the previously specified value. all rebar specified on subsequent lines will also be rotated. Line 1 to END nbar X= Y= A= T= G= Bar number X-ccordinate Y-coordinate area of the reinforcing bar steel type (two types allowed) i bar #. To deactivate this option use 'R=0.0' is specified. G=1. Yr Where: angle is the angle of rotation in degrees (0-360) and Xr. and 4 at equal increments ( ) between bars 1 and 5. Yr is the center of rotation.0'.Página 9 de 51 Steel Input Data block used to specify reinforcing bar information. (4) area of bar can be omitted if 'R=' is specified in SYSTEM data block.g. . (7) When the 'R=' option is used. For example. e. A positive angle bars will be rotated until 'R=0. This provides an easy means to add or delete bars without renumbering.0. at the specified bar increment. j bar #. bar increment # For linear generation of reinforcing bars between bars i and j. jnum. (4) Unconfined and confined concrete masonry based on work done at the University of Colorado. If only these values are specified. Line 1 to END i. March-April 1986. 2.iinc. Terminate the user defined stressstrain relations with a blank line.inc i j inc element # for first element element # for last element element increment for generation (default=1) first element for frontal generation number of elements in i-direction number of elements in j-direction element increment for i-direction element increment for j-direction OR i. an elasto-plastic steel stress-strain is assumed. (2) The only values required for the reinforcing steel are the yield stress. Vol.Página 10 de 51 Holes Input Data block used to specify holes or voids in RC section. however. (3) The relation suggested by Vecchio and Collins [ACI Journal.inum.jinc i inum jnum iinc jinc END SECTION WITH BLANK LINE NOTES: (1) A line to specify values for confined concrete is required. it may be a left blank when confined concrete is not required. 83 No. and the initial modulus. .] is used to describe the stress-strain for concrete in tension.j. (5) A maximum of 150 points can be used to define the user defined stress-strain relations (fewer than 150 points may be used). followed by a line (or lines) defining the reinforcing steel properties. F= i. use as many lines as necessary to define the desired solutions.Página 11 de 51 Data Input Line 1 to end EC= A= P= PHI= Extreme fiber strain for concrete Neutral axis angle desired (defined from global X-axis) Axial load desired Phi factor to apply to produce design P-M interacion curve (default=1. For moment-curvature analyses specify as many values of extreme fiber compressive strain (monotonic) as desired. (2) If generating P-M surface only 'EC=' and 'A=' need to be specified once.10.9 (at P=0). (5) For P < 0. (4) 'EC=' (compression reference point) is the point on the section furthest from the neutral axis (at angle specified). the Phi factor is determined by using linear interpolation between the specified phi factor and 0.5.0) N= Number of points used to compute the P-M interaction diagram (default = 11) END SECTION WITH BLANK LINE NOTES: (1) This data block is not required when interactive input is specified. See Fig. 'P=' is not required (and will be ignored if specified). . A. (3) If P-M surface is not being generated. [Note: Pb is not considered]. jinc i inum jnum iinc jinc END SECTION WITH BLANK LINE .inc i j inc element # for first element element # for last element element increment for generation (default=1) first element for frontal generation number of elements in i-direction number of elements in j-direction element increment for i-direction element increment for j-direction OR i.jnum.j.iinc.inum. Line 1 to END i.Página 12 de 51 Confined Concrete Input Data block used to specify location (elements) of confined concrete. F= i. for modified Kent-Park or Sheikh-Uzumeri (default) Peak compressive stress Strain at peak compressive stress E50u strain (Modified Kent--Park) Note: E2 does not need to be if Fc is specified in units KSI (kips per square inch.) FR. Default: I=0. where FR is the rupture stress and I=1 includes the effects of tension stiffening. See Note (1).I. Default: I=0. See Note (1). where FR is the rupture stress and I=1 includes the effects of tension stiffening. no tension stiffening Unconfined concrete properties 2. Masonary Peak compressive stress Strain at peak compressive stress CONTINUED ON THE NEXT PAGE .I.Página 13 de 51 Properties Input Data block used to specify material properties. High-Strength Concrete (Yong's Model) OR Line 1 I= FM= E1= Unconfined masonary properties 3. no tension stiffening OR Line 1 I= FC= E= FR= Modulus of elasticity FR. UNCONFINED CONCRETE AND MASONARY REALTIONS Line 1 I= FC= E1= E2= FR= Unconfined concrete properties 1. Saatciglu and Razvi (High Strength Concrete) Peak compressive stress Strain at peak compressive stress(Saatcioglu and Razvi) Note: E2 does not need to be specified if Fc is specified in KSI units KSI E85u strain (Saatcioglu and Razvi) Note: E2 does not need to be if Fc is specified in units KSI (kips per square inch.) FR.Página 14 de 51 Line 1 I= FC= E1= E2= FR= Unconfined concrete properties 5. no tension stiffening OR Line 1 I= FC= E1= E2= FR= Unconfined concrete properties 6. See Note (1). where FR is the rupture stress and I=1 includes the effects of tension stiffening. no tension stiffening CONTINUED ON THE NEXT PAGE . where FR is the rupture stress and I=1 includes the effects of tension stiffening.I. See Note (1).) FR.I. Default: I=0. Default: I=0. Saatciglu and Razvi (Normal Strength Concrete) Peak compressive stress Strain at peak compressive stress(Saatcioglu and Razvi) Note: E2 does not need to be specified if Fc is specified in KSI units KSI E85u strain (Saatcioglu and Razvi) Note: E2 does not need to be if Fc is specified in units KSI (kips per square inch. for Modified Kent-Park Peak Unconfined compressive stress (ksi) Strain at peak unconfined compressive stress Spacing of transverse reinforcing (inches) Dimension of confined core (inches) Transverse steel volumetric reinforcing ratio Yield stress of transverse steel (KSI) Rupture stress CONTINUED ON THE NEXT PAGE . for modified Kent-Park or Sheikh-Uzumeri (default) Peak compressive stress Strain at peak compressive stress E50u +E50h strain (Modified Kent--Park) or stain for 1/2 of drop from fcmax to fcmin from E1 (or E3 if specified) Yield strain plateau (Default=E1.Página 15 de 51 CONFINED CONCRETE AND MASONARY REALTIONS Line 2 I= FC= E1= E2= E3= FR= FM= I= FC= E1= S= H= RHO= FY= FR= Confined concrete properties (must be specified .See note(1)) 1. no plateau) Rupture stress Minimum compressive stress at high strain OR 1. AY where NY= number of tensverse reinforcing bars in the Ydirection and AY=area of one transverse reinforcing bar in the Y-direction B1= BXG.BYN where BYG=length of transverse bar in the Y-direction BYN=clear confined dimension in the Y-direction FYT= Yield stress of longitudinal reinforcement FM= Factor multiplied by peak compressive stress to give the minimum compressive stress at high strain S= Spacing of transverse steel CONTINUED ON THE NEXT PAGE . for High Strength Concrete (Yong's Model) Peak compressive stress Yield stress of longitudinal reinforcement Factor multiplied by peak compressive stress to give the minimum compressive stress at high strain Longitudinal volumetric reinforcing ratio Transverse volumetric reinforcing ratio Rupture stress Number of longitudinal reinforcing bars Diameter of longitudinal reinforcing bars Diameter of lateral (transverse) bars Length of one side of the transverse tie Spacing of the transverse steel CONTINUED ON THE NEXT PAGE Line 2 CONFINED CONCRETE AND MASONARY REALTIONS Confined concrete properties (must be specified .BXN where LXG=length of transverse bar in the X-direction BXN=clear confined dimension in the X-direction B2= BYG. for Saatcioglu and Razvi (Normal Strength Concrete) X= NX. AX where NX= number of tensverse reinforcing bars in the Xdirection and AX=area of one transverse reinforcing bar in the x-direction Y= NY.See note(1)) I= 5.Página 16 de 51 CONFINED CONCRETE AND MASONARY REALTIONS Line 2 I= FC= FY= FM= RL= RP= FR= Line 3 N= D= DT= H= S= Confined concrete properties (must be specified .See note(1)) 2. =10%) Strain at stress FU (def.BYN where BYG=length of transverse bar in the Y-direction BYN=clear confined dimension in the Y-direction Yield stress of longitudinal reinforcement Factor multiplied by peak compressive stress to give the minimum compressive stress at high strain Spacing of transverse steel CONTINUED ON THE NEXT PAGE Line 2 I= FM= E1= R= S= H= Confined concrete masonary properties (must be specified .=0) OR 1 for design grade 60 steel 2 for probable steel stress-strain 3 for strain hardening steel stress-strain END SECTION WITH BLANK LINE . = FY) Failure stress (def. for Saatcioglu and Razvi (High Strength Concrete) NX.=10%) Strain at stress FF (def.See note(1)) 6. fcconfined.BXN where LXG=length of transverse bar in the X-direction BXN=clear confined dimension in the X-direction BYG.=10%) Modulus of elasticicty Initial modulus of elasticity for strain hardening region (def.Página 17 de 51 Line 2 I= X= Y= B1= B2= FYT= FM= S= CONFINED CONCRETE AND MASONARY REALTIONS Confined concrete properties (must be specified . for user defined relations See Note (5) strain. fcunconfined. AX where NX= number of tensverse reinforcing bars in the Xdirection and AX=area of one transverse reinforcing bar in the x-direction NY.See note(4)) 3. fcunconfined. for confined concrete masonary Factor multiplied by peak compressive stress to give the minimum compressive stress at high strain Peak concrete masonary compressive stress Transverse steel volumetric reinforcing ratio Vertical spacing of confinement combs Width of confined core CONTINUED ON THE NEXT PAGE Line 1 I= Line 2 : Line 150 line 3 to 7 FY= FU= FF= E1= E2= E3= E= ET= Line 1 SY= USER DEFINED CONCRETE STRESS-STRAIN User defined stress-strain relationship 4. = FY) Strain for onset of strain hardening (def. fcconfined. ft : strain. AY where NY= number of tensverse reinforcing bars in the Ydirection and AY=area of one transverse reinforcing bar in the Y-direction BXG. ft REINFORCING STEEL RELATIONS Reinforcement steel properties Yield stress Ultimate stress (def. C. . Section Modeling Material Modeling Solution Reference Angle Solution Reference Point Solution Scheme. followed by available material stress-strain relationships R.Página 18 de 51 Appendix A SECTION AND MATERIAL MODELING Modeling capabilites for the materials and section geometry are described. Section definition is described first. YS ) of each section (Fig. for each layer in the j-direction (Fig. A.2 T-Beam Cross Section Modeling . several rectangular sections can be used to describe the element geometry (Fig. The location of reinforcing is defined by specifying coordinates in the global coordinate system (X. Fig. and a possible mesh. A reference point on each section (I=0.2)). (A. For more complicated sections.Y). A. The elements are numbered in the i-direction from left to right. A.2) in the global (X.2).Página 19 de 51 Reinforced Concrete Section Modeling The modeling of rectangular reinforced concrete sections is accomplished by subdividing the section into elements.1) presents a beam cross-section.J=0) is used to define the starting location ( XS . Fig.Y) coordinate system. A. (A.1 Beam Cross Section Modeling Fig. The mesh of elements is defined by specifying the number of elements in each of the principal directions of the section. such as T-Beams or walls with boundary elements. or the Saatcioglu and Razvi Model (1995). Nour. A. Experimental studies of rectilinearly confined high-strength columns under combined axial load and uniaxial bending have been conducted by Thomsen and Wallace at Clarkson University to evalaute relations for more typical loadings.3 Example Concrete Stress-Strain Relations A relationship for rectilinearly confined high-strength concrete is also available. The columns were subjected to axial loads only. The concrete stress-strain behavior can be described by either the modified Kent and Park model (Park et al. The relation suggested by Hognestad (1951) is used in this program. Relationships for unconfined and confined concrete masonry have recently been incorporated. CONFINED W/PLATEAU COMPRESSIVE STRESS (KSI) UNCONFINED CONFINED W/O PLATEAU STRAIN Fig. (1982)). the Sheikh and Uzumeri model (1982) (Fig. a user defined stress-strain curve may be specified. In addition. .56 ksi. and are based on experimental studies of twenty-four columns with compressive strength ranging between 12. and Nawy (1988) are used. The initial portion of the stressrelation is described by a second-degree parabolic shape. (A.13 and 13. The relationships developed by Yong.3)).Página 20 de 51 Material Modeling The program allows stress-strain curves for unconfined and confined concrete and concrete masonry. The relationships are based on work done at the University of Colorado. Care must be exercised when using the strain hardening curve because the relationship used to produce the curve is sensitive. the ultimate stress and strain.4 Example Steel Stress-Strain Relations .4)).Página 21 de 51 The stress-strain model for the reinforcing steel allows a bilinear curve. or the consideration of strain hardening effects (Fig. The equations describing the relationship are based on the equation presented by Saenz (1964). and the fracture stress and strain. (A. It is suggested to always plot the material stress-strain curves to ensure they are reasonable. The strain hardening curve is defined by the initial slope of the stress-strain curve at the onset of strain hardening. NO YIELD PLATEAU YIELD PLATEAU W/STRAIN HARDENING STRESS (KSI) ELASTO-PLASTIC STRAIN Fig. A. 5).5 Solution Referance Parameters . Fig. Therefore. a user specified angle of zero (0.0) coincides with the global X-axis. Angles other than zero (0 to 90 degrees) are measured counterclockwise from the global X-axis (Fig. A. A.Página 22 de 51 Solution Reference Angle The solution is computed with respect to the global X-axis. Página 23 de 51 Solution Reference Point The analysis of the R.5 Solution Referance Parameters .5).C. A. A. section is based on a user specified extreme fiber concrete compressive strain. The location of the compression reference point is determined by the program to be the point furthest on the defined section perpendicular to the defined orientation of the neutral axis in the positive Y direction (Fig. Fig. Appendix B Examples and Applications Fifteen examples are presented to facilitate program use and detail program applications. The moments (X-direction and Y-direction) are computed as the product of the element axial force and the perpendicular distance to a line through the geometric centroid of the concrete section parallel to the specified orientation of the neutral axis.Página 24 de 51 Solution Scheme The strain at the centerline of each element is computed from the reference point strain assuming plane sections remain plane after loading. The axial load is computed as the sum for all elements (the elements are defined by the specified mesh) of the element stress times the element area.Unconfined Concrete" Example 5 "High-Strength Column With Confined Concrete Example 6 "Masonary Column With Confined Masonary" Example 7 "T-Beam w/ Unconfined Concrete" Example 8 "Column w/ unconfined Concrete" Example 9 . Example 1 "COLUMN w/ Unconfined Concrete P-M Interaction Diagram" Example 2 "COLUMN w/ Unconfined and Confined Concrete Moment-Curvature Analysis" Example 3 "Shear Wall w/ Confined Boundary Elements" Example 4 "L-Section Rotated 30 degrees . The accuracy of a given solution is determined in part by the number of elements used to model a cross section. No detailed calculations were made to determine precise values for the material property variables. The intent of the examples is to detail program options. The user should verify that the mesh refinement is adequate. 5 : 2 in. 7 : assume #4 hoops 8 X= 2.4 : Rupture strength = 10% f'c : Blank line for confined concrete FY=60 E=29000 : Design Steel Stress : : DATA: EC=0.70 : compressive strain of 0.1 Example 1: Column Cross Section Example 1: COLUMN w/ Unconfined Concrete P-M Interaction Diagram .Based on Design Steel stress-strain SYSTEM: T=1 E=1 C=1 P=1 TOL=0.003.5 A=0.5 Y= 2.5 G= 3.#8 Bars 5 X= 21. 3 : 8 .5 G= 5.85 E1=0. B.5 G= 1.5 Y= 21.0001 : T=1 for P-M Diagram : P=1 to check stress-strain : C=1 all concrete is unconfined : E=1 do not allow any rebar to fracture : SECTION: 1 HI= 24 HJ= 24 N1= 12 N2= 12 X= 0 Y= 0 : 24"x24" Column : 12x12 mesh : STEEL: 1 X= 2. Grade 60 elasto-plastic steel and fc' = 4 ksi are used. Fig.5 Y= 12 : : : : PROPERTY: FC=4*0. with and : without capacity reduction factors END .003 A=0 N=20 PHI=0.004 FR=0.Página 25 de 51 Example 1 Compute the P-M interaction diagram for the section shown below.5 Y= 2.003 E2= 0.003 A=0 N=20 : Moment about global X-axis for Ultimate EC=0. Steel Type 1 3 X= 21. clear cover to bars 7 X= 2.5 Y= 21.79 T= 1 : Long. B. B. The stress-strain relationship for confined concrete is based on Modified Kent-Park (Appendix C).1 Example 2: Column Cross Section .Página 26 de 51 COMPUTED P-M AXIAL LOAD (KIPS) DESIGN P-M MOMENT (IN-KIPS) Fig.2 Example 1: Computed P-M Interaction Diagram Example 2 Is the same as Example 1 except that the the moment-curvature diagram is computed and confined concrete is specified for the column ore. Fig. clear cover to bars 7 X= 2.5 Y= 12 : : PROPERTY: FC=4*0.0008 EC=0.4 : FC=4*0. Steel Type 1 3 X= 21.5 G = 5.0002 : Moment about global X-axis EC=0.002 E2= 0.131 : STEEL: 1 X= 2.0006 EC=0.5 Y= 2.5 Y= 21.79 T= 1 : Long.1.#8 Bars 5 X= 21.23 : ELEMENTS 26 .5 Y= 21.0025 EC=0.01707 : FY=68 FU=100 FF=95 E1=0.0010 EC=0.0060 EC=0.12: CONFINED CONCRETE CORE: ELEMENTS 14 .12 E=29000 ET=1500: : DATA: EC=0.5 G = 1.5 Y= 2.85 E1=0.0035 EC=0.004 E2=0.004 FR=0.0030 EC=0.002 H=20 S=4 FY=60 RHO=0.0004 EC=0. 3 : 8 .0050 EC=0. 5 : 2 in.10.5 G = 3.35 : : ELEMENTS 122 .85 E1=0.08 E3=0.0040 EC=0. 7 : assume #4 hoops 8 X= 2.0001 : T = 1 for P-M Diagram : P = 1 to check stress-strain : C = 1 all concrete is unconfined : E = 1 do not allow any rebar to fracture : SECTION: 1 HI= 24 HJ= 24 N1= 12 N2= 12 X= 0 Y= 0 : 24"x24" Column : 12x12 mesh : CONFINED: 14 F=14.10.0020 EC=0.0070 : END .Página 27 de 51 Example 2: COLUMN w/ Unconfined and Confined Concrete Moment-Curvature Analysis: Based on Probable Steel stress-strain SYSTEM: T=0 E=1 C=1 P=1 TOL=0.5 A=0.0015 EC=0. Página 28 de 51 Fig. B. B. In addition confined boundary elements are specified. and the reinforcing steel behavior is based on typical observed experimental relations for grade 60 steel. The concrete is 4000 psi.3 Example 2: Computed Moment-Curvature Relations Example 3 Compute the Moment-Curvature diagram for the section shown below. Fig.4 Example 3: Wall Cross Section . 01 FR= .5 G= 13.32 : 33 X= 33 Y= 8 A= 0. 8: BOUNDARY ELEMENT 362 F = 362. 6.0050 A=-90 P=1000 EC=0.004 FR= . 5 : 16 #11 Bars 9 X= 21 Y = 21 G= 5.#5 @ 18 in.0025 A=-90 P=1000 EC=0.0002 A=-90 P=1000 : Moment about global X-axis EC=0. 9 : 13 X= 3 Y = 21 G= 9.0004 A=-90 P=1000 EC=0.12 E=29000 ET=1500: : DATA: EC=0.0008 A=-90 P=1000 EC=0.0001 : SECTION: 1 HI = 24 HJ = 24 N1 = 8 N2 = 8 X= 0 Y = 0 : Boundary Element 2 HI = 216 HJ = 12 N1 = 72 N2 = 4 X= 24 Y = 6 : Web 3 HI = 24 HJ = 24 N1 = 8 N2 = 8 X=240 Y = 0 : Boundary Element : CONFINED: 10 F = 10.5 : : PROPERTY: FC=4 E1= 0.0020 A=-90 P=1000 EC=0.1.16 : 17 X= 243 Y= 3 : Boundary element steel 21 X= 243 Y = 21 G= 17.0015 A=-90 P=1000 EC=0. 8: BOUNDARY ELEMENT : STEEL: 1 X= 3 Y= 3 A= 1.0006 A=-90 P=1000 EC=0.0070 A=-90 P=1000 END . 6.0035 E2= 0. 44 X= 231 Y= 8 G= 33.31 : Web steel .4 FM= .003 E2= 0. 6.0040 A=-90 P=1000 EC=0.21 : 25 X= 261 Y = 21 G= 21.1.0035 A=-90 P=1000 EC=0.08 E3=0.Curvature Calculations for Probable Steel stress-strain SYSTEM: T=0 E=0 P=1 TOL=0.0030 A=-90 P=1000 EC=0.4 : Unconfined Concrete FC=5 E1= 0.56 T= 1 : Boundary element steel 5 X= 21 Y= 3 G= 1.25 : 29 X= 261 Y= 3 G= 25.0060 A=-90 P=1000 EC=0.80 : Confined Concrete FY=65 FU=90 FF=85 E1=0.13 : 16 X= 3 Y = 7.29 : 32 X= 246 Y= 3 G= 29.004 E2=0.0010 A=-90 P=1000 EC=0.44 : 45 X= 33 Y = 16 : 56 X= 231 Y = 16 G= 45.Página 29 de 51 Example 3: Shear Wall w/ Confined Boundary Elements Moment . 6. Special options include multiple sections.0. 9 : 10 X = 98 Y = 2 : 11 X = 98 Y = 18 : 19 X = 98 Y = 98 G=11.0.Página 30 de 51 Example 4 Compute the P-M interaction diagram for the section shown below.6 Example 4: Wall Cross Section Example 4: L . and steel rotation. B.0001 : : : SECTION: 1 HI = 100 HJ = 20 N1 = 50 N2= 10 X = 0 Y = 0 A=30.31 R=30.Unconfined Concrete P-M Interaction Surface : : SYSTEM: T=1 E=0 C=1 P=1 M=0 TOL=0.Section Rotated 30 degree .37 : : .0.19 : 20 X = 2 Y = 18 : 29 X = 82 Y = 18 G=20.10: 2 HI = 20 HJ = 80 N1 = 10 N2= 40 X =80 Y = 20 A=30. section rotation. and grade 60 reinforcing is used.29 : 37 X = 82 Y = 98 G=29. The concrete is 4000 psi. Fig.10: Rotate all steel 9 X = 82 Y = 2 G = 1.10: : STEEL: 1 X = 2 Y = 2 A = 0. 005 E2=0.10 E=29000 ET=1500: : DATA: EC = 0.7 Example 4: Computed P-M Interaction Diagram . B.003 A = 90 : Moment about y-axis EC = 0.070 E3=0.0030 E2= 0.004 FR= .003 A = -90 : Moment about X-axis : END AXIAL LOAD (KIPS) Fig.Página 31 de 51 PROPERTY: FC=4 E1= 0.4 : : Blank Line for Confined Concrete FY=70 FU=110 FF=105 E1=0. 79 T= 1 : Long.0. and ``PROPERTIES'' for unconfined and confined highstrength concrete.10.5 G= 3.000 psi and grade 60 reinforcing is used.0 : : .0 S=0 T=1 E=1 C=2 P=1 TOL=0. Special options used include ``CONFINED elements''.5 Y= 21. The concrete is 12.5 G= 5.5 A= 0.5 Y= 12.5 Y= 2.5 Y= 2. Fig.10. B. clear cover to bars 7 X= 2.1. 5 : 2 in. 7 : assume #4 hoops 8 X= 2.8 Example 5: Column Cross Section Example 5 : HIGH-STRENGTH COLUMN WITH CONFINED CONCRETE P-M Interaction Diagram . Steel Type 3 X= 21. 3 : 8 #8 Bars 5 X= 21.5 Y= 21.0001 : T=1 for P-M Diagram : P=1 to check stress-strain : C=2 all concrete is confined at rebar : E=1 do not allow any rebar to fracture : SECTION: 1 HI=24 HJ=24 N1=12 N2=12 X=0 Y=0 : 24"x24" Column : 12x12 mesh : CONFINED: 14 F=14.5 G= 1.Based on Design Steel stress-strain SYSTEM: D=0.Página 32 de 51 Example 5 Compute the P-M interaction diagram for the section shown below.12 : : STEEL: 1 X= 2. Página 33 de 51 PROPERTY: I=2 FC=10.003 A=0 N=11 : Moment about global X-axis for Ultimate EC=0.5 H=19 S=4 : FY=60 E3=0.9 Example 5: Computed P-M Interaction Diagram . B.70 : compressive strain of 0.1 FY=60 FM=0.3 RL=0.120 E=29000 : Fracture of steel at 12% strain : DATA: EC=0.2 E=5400 FR=1.2 E=5400 FR=1.003.003 A=0 N=11 PHI=0.011 RP=0.02 : I=2 FC=11.01 N=8 D=1 DT=0. with and : without capacity reduction factors : END Fig. 5 Y= 21.120 E=29000 : Fracture of steel at 12% strain : DATA: EC=0.0001 : T= 1 for P-M Diagram : P= 1 to check stress-strain : C= 1 all concrete is unconfined : E= 1 do not allow any rebar to fracture : SECTION: 1 HI=24 HJ=24 N1=12 N2=12 X=0 Y=0 : 24"x24" Column : 12x12 mesh : STEEL: 1 X= 2.5 G= 3. 5 : 2 in.5 G= 1. clear cover to bars 7 X= 2.001 S=12 H=20 : CONFINED MASONRY FY=60 E3=0. with and : without capacity reduction factors : END . Example 6: MASONRY COLUMN w/ Confined Masonry P-M Interaction Diagram Based on Design Steel stress-strain SYSTEM: S=0 T=1 E=1 C=1 P=1 TOL=0. 7 : 8 X= 2.5 Y= 2.003. 5 Y= 2.003 A=0 N=12 : Moment about global X-axis for Ultimate EC=0. Steel Type 3 X= 21.5 Y= 12 : : : PROPERTY: I=3 FM=4 E1=0.014 R=0.5 Y= 21.79 T= 1 : Long.5 A= 0. 3 : 8 #8 Bars 5 X= 21.003 : UNCONFINED MASONRY I=3 FM=4 E1=0.70 : compressive strain of 0.relation for the concrete masonry section shown below. Special options used include the relations for unconfined and confined masonry and the ``HOLES'' option for voids in a section.Página 34 de 51 Example 6 Compute the M.5 G= 5.003 A=0 N=12 PHI=0. B12 Example 7: Beam Cross Section .relation for a T-beam. Special options used include user specified stress-strain relations and multiple steel types. Fig.Página 35 de 51 Example 7 Compute the M. 12 E3=.5 A=0.5 : 8-#4 Bars in Flange 9 X=26 Y=24.5 Y=2.5 A=0.0060 A=0 P=0 : END . Clear Cover to Bars 6 X=19.79 T=1 : Long.0040 A=0 P=0 EC=0.4 : Unconfined Concrete : Blank Line for Confined FY=68 FU=100 FF=95 E1=.004 FR=.12 E=29000 ET=1200 FY=45 FU=70 FF=65 E1=.004 E2=.08 E3=.0002 A=0 P=0 : Moment about Global X-Axis EC=0.5 : 10 X=30 Y=24.0015 A=0 P=0 EC=0.0020 A=0 P=0 EC=0.0025 A=0 P=0 EC=0.5 G=4.002 E2=.5 : 12 X=8 Y=33.Página 36 de 51 Example 7 : T-Beam w/ Unconfined Concrete Moment-Curvature Calculations for Probable Steel Stress-Strain : SYSTEM: T=0 P=1 E=1 C=1 TOL=0.5 : 2 in.5 : 11 X=4 Y=33.5 : : PROPERTY: I=1 FC=4 E1=.0004 A=0 P=0 : No Axial Load EC=0.20 E=29000 ET=1100 : DATA: EC=0.5 Y=33.0010 A=0 P=0 EC=0.20 T=2 : Additional Steel 8 X=8 Y=24.5 : 13 X=26 Y=33.0001 M=0 : M=1 for Negative : Moment-Curvature : SECTION: 1 HI=10 HJ=22 N1=5 N2=11 X=12 Y=0 : 10"X22" Web 2 HI=34 HJ=8 N1=17 N2=4 X=0 Y=22 : 34"X8" Flange : STEEL: 1 X=14.0006 A=0 P=0 EC=0.0030 A=0 P=0 EC=0.6 : 7 X=4 Y=24.5 Y=2.3 : 3-#8 Bars in Top and Bottom 4 X=14.030 E2=.5 : 14 X=30 Y=33.5 Y=33.0035 A=0 P=0 EC=0.5 G=1.0008 A=0 P=0 EC=0. Steel Type 3 X=19.0050 A=0 P=0 EC=0. ) Fig. B.Página 37 de 51 MOMENT (IN-KIPS) CURVATURE (/IN.13 Example 7: Computed Moment-Curvature Diagram . with and : without capacity reduction factors : END .003 A=0 N=12 PHI=0. : SYSTEM: S=0 T=1 E=1 C=1 P=1 TOL=0. STRESS STRESS: USER DEFINED .330236E+00 .119500E-03 .REQUIRED TO SPECIFIY END OF USER DEFINED STRESSSTRAIN FY=60 E=29000 : DESIGN STEEL STRESS-STRAIN : : DATA: EC=0. 12x12 Mesh : STEEL: 1 X= 2.Página 38 de 51 Example 8 Same as Example 1 except that user specified concrete stress-strain relations are specified (as an example the file format is given below.000000E+00 : BLANK LINE .5 G= 5.5 Y= 21.763358E+00 .User specified concrete stress-strain relations. 5 : 2 in.5 Y= 12 : : PROPERTY: UNCONFINED CONFINED TENSILE I=4: STRAIN COMP.5 G= 3.646763E+00 .391772E+00 .003 A=0 N=12 : Moment about global X-axis for Ultimate EC=0.239000E-03 .5 G= 1.000000E+00: MAX 150 POINTS .5 Y= 21. however.003.70 : compressive strain of 0.79 T= 1 : Long.0001 : T=1 for P-M Diagram : SECTION: 1 HI=24 HJ=24 N1=12 N2=12 X=0 Y=0 : 24"x24" Column. 3 : 8 #8 Bars 5 X= 21.5 Y= 2. Steel Type 3 X= 21. the user specified stress-strain relation was truncated so that it would fit on this page Example 8: COLUMN w/ Unconfined Concrete : P-M Interaction Diagram .5 Y= 2. clear cover to bars 7 X= 2. STRESS COMP. 7 : assume #4 hoops 8 X= 2.5 A=0. 003 A=0 N=20 : Moment about global X-axis EC=0.56 T= 1 : Long. of rings : S=section type 1 for rectanglar .4 : : SY=1 : : DATA: EC=0.5 : 3 X= -12.000001 : : SECTION: 1 S=2 D=30 NP=10 NR=10 A=0: D=diameter : NP= no.003 A=0 N=20 PHI=0. 2 for circular : STEEL: 1 X= 12.003 E2= 0.85 E1=0. Steel Type 1 2 X= 0 Y= 12.5 : #4 hoops : PROPERTY: FC=5*0.004 FR=0.5 Y= 0 A= 1.Página 39 de 51 EXAM 13 : -----------------------------------------------------------: :Circular column without confinement : SYSTEM: T=1 E=2 C=1 P=1 TOL=0.5 Y= 0 : 4 X= 0 Y= -12.7: : END . of poles : NR= no. Página 40 de 51 . Página 41 de 51 Appendix C Concrete Stress-Strain Relations Modified Kent-Park (Park et al. (1982)) Sheikh and Uzumeri (1982) Yong's High-Strength Concrete Masonary Concrete Saatcioglu and Razvi . h is the width of the concrete core measured to the outside of the hoops.a) k=1+ ρ h f c' ' fc (C.1).5 +ε − kε 50u 50h 0 (C.1 Modified Kent-Park Stress-Strain Relations fc = k fc [ 2ε c ε .3) ε50u = 3 + ε 0 f c' f c' − 1000 (C. Fig. s is the center-to-center spacing of the hoops.kε0)2 ] ≥ 0. C.75ρ h / s (C.2 k fc′ (C.a) fc = k fc [ 1 . fyh is the yield stress for the hoop reinforcement.2) Zm = ε 0.Página 42 de 51 Modified Kent-Park C.5) to the outside of the hoops.Zm(εc . Units of psi are used for stress.( c )2 kε 0 kε 0 ] εc≤ kε0 εc > k εc (C. and ρ is the ratio of the volume of hoop reinforcement to volume of concrete core measured . and ε0 is typically assumed to be 0.002.1. Where fc is the longitudinal concrete stress.4) ε50h = 0. C.1 Modified Kent-Park [Park et al. ε is the longitudinal concrete strain. (1982) The following equations are used to describe the stress-strain relations of confined and unconfined concrete (See Fig.1. 0 + 2.Página 43 de 51 Sheikh and Uzumeri The following equations are used to describe the stress-strain relation for confined concrete (See Fig. C.6) .2) for a square column with uniformly distributed longitudinal reinforcement Fig C.73B 2 Pocc (C. B ⎠ ⎦ ⎣⎝ 55 KS = 1.2 Sheikh and Uzumeri Stress-Strain Relation ⎡⎛ nC 2 ⎞ s ⎞2⎤ ⎟⎛ ⎥ p s f s' ⎢⎜ 1 − 1− ⎜ ⎟ ⎝ 2B⎟ 2 ⎠ ⎥ ⎢⎜ . s is the spacing of the hoops. s is the ratio of the volume of hoop reinforcement to volume of concrete core.10.3Ks . C.Página 44 de 51 In equations C. and fs is the yield stress of the hoops. Pocc is ' f c 0.6 to C.002 is typically assumed for oo The minimum stress at high strains ' f c can be taken as 0. B is the center-to-center distance of the perimeter hoops (to define the confined core).6. C is the center-to-center spacing of the longitudinal reinforcing bars.85 Acore. A value of 0. n is the number of longitudinal reinforcing bars (in Eq. the quantity nC2 assumes equal spacing of longitudinal reinforcement). 0091 ⎜ 1 − ρ⎟ ⎟⎜ρ + ⎝ 8sd ⎠ f ' h '' ⎠ ⎝ c '' (C.Página 45 de 51 Yong's High-Strength Concrete The following equations were developed by Yong .00265 + f c' ( ) (C. f c by the factor K given in Eq. C. d '' is the nominal diameter of the longitudinal steel bars in inches.0035⎜ 1 − ρ fy '' ⎟ ⎝ ⎠ h ε0 = 0. '' 0.734 s ⎞ '' '' 2 / 3 ⎛ 0. (C. d '' is the nominal diameter of the lateral ties in inches. The strain at peak stress (confined) is computed as: 0. h is the length of one side of the rectangular ties in inches. n is the number of longitudinal steel bars.245s ⎞ ⎛ '' nd '' ⎞ f y ⎛ K = 1.3).13 to 13. and Nawy to describe the axial stress-strain relation for rectilinearly confined high-strength concrete (See Fig.11) Where s is the center-to-center spacing of the lateral ties in inches.11). ρ is the volumetric ratio of the lateral reinforcement. and f y'' is the yield stress of the lateral steel in psi. The relations are based on experimental studies of twenty-four square columns with compressive strength ranging from 12. ρ is the volumetric ratio of the longitudinal reinforcement.0 + 0.12) . Nour. The peak stress of the confined concrete is obtained by multiplying the peak ' compressive stress of the unconfined concrete.56 ksi . (C.45 . The relations are linear (Fig.3) until a ' f c compressive stress of 0.Página 46 de 51 The remaining values used to produce the stress-strain curve are determined using the following equations: by ⎡ ⎛ f i = f 0 ⎢0.13) ⎡ ⎛ ε0 ⎞ ⎤ ε i = K ⎢14 .14) (C.3 f 0 ⎝ 1000 ⎠ ⎣ ⎦ (C.12) through (C. The two polynomial equations intersect at the point of peak stress at zero slope (Fig.0003⎥ ⎣ ⎝ k ⎠ ⎦ ⎡ ⎤ ⎛ f ⎞ f 2i = f 0 ⎢0. C.4 ⎥ f0 ⎟ ⎥ ⎠ ⎦ (C.15) (C.16) ε 2i = 2 ε i − ε 0 Two polynomial equations are used to produce a smooth continuous curve for the stress-strain behavior of rectilinearly confined high-strength concrete given Eq. ⎜ ⎟ + 0. (1988)].16) [Yong et al. .3).025⎜ 0 ⎟ − 0. C.25⎜ ⎜ ⎢ ⎝ ⎣ ⎤ f c' ⎞ ⎟ + 0.065⎥ ≥ 0. f m is the maximum compressive stress of the concrete masonry.046 f m + 0. and S h is the vertical spacing of confinement combs.4).26) Where. B. Unconfined Concrete Masonry ' ε u = 0.4( hc Sh ) − 01 . concrete masonry strain.014 A= B= 480000 ' fm ' fm 0. The parameters A. ρ s is the volumetric ratio of confinement reinforcement to the concrete masonry core (measured to the outside of the combs).8 (C.18) Where.055 [ ] ( f m in ksi ) ' (C.01 0. stress.01 0.055 ε u is the strain corresponding to maximum compressive ε m is the [ ] ( ' in ksi ) fm (C.131 (C.Página 47 de 51 Masonary Conctere The following stress-strain relations were developed based on work done at the University of Colorado for unconfined and confined concrete masonry (See Fig. and C depend on the degree of confinement. B = 574.003 A = 176700.6. hc is the width of the confined core measured to the outside of the confinement comb.25) 4.23) (C. C.19) (C.3ρ s ( hc S h ) C = 0. (C. ⎛ 1 ⎞ fm = − Aε 2 0 ≤ εm ≤ εu (C.21) Confined Concrete Masonary ' ε u = 0.046 f m + 0. C = 0.20) ε mc = 0.17) + Aε m ⎟ ε m m +⎜ ' ⎝εu ⎠ fm fm ' fm = C + (1 − C )e ' − B( ε m − ε u ) ε u ≤ ε m ≤ ε mc (C.22) ε mc = 0.24) (C. . ε mc is the maximum usable strain. tie spacing for buckling specified in the ACI Building Code ( ACI-318-89) should be satisfied ).Página 48 de 51 Saatcioglu and Razvi Model The following equations are used to describe the stress-strain relations of confined and unconfined concrete (See Fig.5.27) (C. The column should conform to the building code limitation for maximum spacing of laterally supported longitudinal reinforcement.1 Square Column ' ' f cc = f co + k1 f le (C.17 (C. ⎛ bc ⎞ ⎛ bc ⎞ ⎛ 1 ⎞ k 2 = 010036 .29) f le = k 2 + f l f l = ∑ AS f yt Sinα Sbc k1 = 4.30) ≤ 10 . .84( f le ) −0. max.1 Normal-Strength Concrete C.31) Where: S l = Spacing of laterally supported longitudinal reinforcement S = Spacing of ties f l is in ksi units • Premature buckling of longitudinal reinforcement is prevented ( i.1.5. this may be taken as the upper limit for Sl. Therefore. C.28) (C.5) C.e. ⎜ ⎟⎜ ⎟⎜ ⎟ ⎝ S ⎠ ⎝ Sl ⎠ ⎝ f l ⎠ (C. 2 Rectangular Column ( ) f ly = ( ∑ AS f yt sin α bcy ) f lx = ∑ AS f yt sin α bcx ⎛ bcx ⎞ ⎛ bcx ⎞ ⎛ 1 ⎞ k 2 x = 010036 .3 Ductility of Confined Concrete ε 1 = ε 01 (1 + 5k ) ' k = k1 f le f co (C.38) Where : f lex and f ley are the effective lateral pressures acting perpendicular to core dimensions bcx and bcy respectively and f le is the overall equivalent lateral pressure C.34) ≤ 10 .33) (C. A value of 0.42) (C.40) Where: ε 01 = Strain corresponding to peak stress of unconfined concrete.36) (C.37) f le = f lex bcx + f ley bcy ( ) (bcx + bcy ) (C. (C.41) ε 85 = 260ρ c ε 1 + ε 085 ρ c = ∑ AS (C.5.002 is appropriate for ε 01 ε 1 = Strain corresponding to peak stress of confined concrete ' f co = 0.43) [S (bcx + bcy )] .39) (C.35) f lex = k 2 x f lx f ley = k 2 x f ly (C.85 f c' (C.32) (C.Página 49 de 51 C.1.5. ⎟⎜ ⎟ ⎟⎜ ⎜ ⎝ S ⎠ ⎝ S l ⎠ ⎝ f lx ⎠ ⎛ bcy ⎞ ⎛ bcy ⎞ ⎛ 1 ⎞ ⎟ k 2 y = 010036 ⎜ ⎟⎜ ⎟⎜ ⎟ ⎝ S ⎠ ⎝ Sl ⎠ ⎜ ⎝ f ly ⎠ ≤ 10 .1. (C. A value of 0. (C.5. ∑ AS = The summation of transverse reinforcement area in two directions crossing bcx and bcy . (C.4 Stress-Strain Relation for Confined Concrete ⎡ ' ⎢ ⎛εc ⎞ ⎛εc ⎞ ⎥ f c = f cc 2⎜ ⎟ − ⎜ ⎟ ⎢ ⎝ ε1 ⎠ ⎝ ε1 ⎠ ⎥ ⎣ ⎦ 1 2 ⎤ 1+ 2 k ' ≤ f cc (C. ≤ 10 (C. ⎜ ⎟⎜ ⎟ ⎝ S ⎠ ⎝ Sl ⎠ ≤ 10 . For square column :⎛ bc ⎞ ⎛ bc ⎞ k 2 = 015 .45) • For a rectangular column :⎛ bcx ⎞ ⎛ bcx ⎞ k 2 x = 015 . ≤ 10 .Página 50 de 51 Where: ε 085 = Strain at 85% strength level of unconfined concrete. ρ c reduces to transverse reinforcement in one direction only C. ⎟ ⎟⎜ ⎜ ⎝ S ⎠ ⎝ Sl ⎠ ⎛ bcy ⎞ ⎛ bcy ⎞ k 2 y = 015 .46) (C. ⎜ ⎟⎜ ⎟ ⎝ S ⎠ ⎝ Sl ⎠ ≤ 10 .0038 is appropriate for ε 085 . • If the column is spirally reinforced or has identical reinforcement arrangement in each direction.5.96 ' f co .47) k3 = 5.44) C.2 Modifications Introduced for High-Strength Concrete • • The yield strength of lateral steel is limited to 1000 MPa (149 ksi).48) .1. (C.5 ≤ 10 . ' ⎛εc ⎞ f cc ⎜ ⎟r ⎝ ε1 ⎠ (C. f c' + 10281 . = cc ksi ε1 Ec r= E c − E sec.53) (C.0018k 3 ε 85 = 260k 3 ρ c ε 1[1 + 0.55) (C.0028 − 0.0008k 3 ε 1 = ε 01 (1 + 5k 3 k ) 2 ε 085 = ε 01 + 0. and present applications and examples.50) (C.5k 2 ( k 4 − 1)] + ε 085 E c = 1281537 . f' E sec.Página 51 de 51 k4 = f yt 74. .51) (C.57) Report Outline The program manual is presented in the following section.54) ε 01 = 0.49) (C. Appendices are included to detail the material and section modeling capabilities of the program.56) fc = ⎛ε ⎞ r −1+ ⎜ c ⎟ ⎝ ε1 ⎠ r (C.52) (C. Pertinent notes are included to facilitate user understanding.