Biax

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BIAX 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"

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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.

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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

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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

Function

About

Gives general information about the program

Input

Invokes the text editor

Run

Invokes analysis program and prompts the user to the data file name. During execution, the program displays messages indicating the analysis progress

Plot

Gives different plotting options

Help

Provides online help

Exit

Terminates the program

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Several output files are created by the program. they are explained in the following table

(1)

Filename.out

Output file of program EXECUTION including Calculated results.

(2)

Filename.dat

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)

Filename.dbg

Debug file

(4)

Conc.dat

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)

Mesh.dat

Concrete element coordinates

(7)

Sects.dat

Steel bar coordinates for plotting check of input file

(8)

Sectc.dat

Concrete section corner coordinates for plotting check of input file

(9)

Checkx.dat

Non-linear strain distribution along x-axis

(10)

Checky.dat

Non-linear strain distribution along y-axis

(11)

Strx.dat

Strain distribution factor for each bar along x-axis

(12)

Stry.dat

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.

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The program calculates section properties. Inertias (Ixx, Iyy and Ixy) are calculated for both the neutral axis location and the global midpoint of all the sections defined. The moments are given about the midpoint because it is easy for the user to locate. For circular cross section, the midpoint is alway at the origin (i.e. point (0,0)). A batch file is provided to delete output files created by the BIAX program. Use the file EAT.BAT and the input filename to delete the files created by the program (execute by typing EAT followed by the filename, e.g. EAT EX1)

Program Input The following several pages describe the input data file requirements for the BIAX program. The input file is separated into eight blocks.Separators are used within the input file to specify the location of the eight data blocks. The separators may appear in any order. The separators and the information obtained within each data block are described in the following table.

SEPERATOR

DESCRIPTION OF DATA BLOCK

SYSTEM

Data block to define global analysis parameters.

SECTION

Data block to define rectangular subsections that make up the R.C. section. Data block to define location, area, and type of steel reinforcing bars. Data block to define any holes (voids) that may exist in R.C. section. Data block to define portions of the R.C. section that are confined. Data block to define the non-linear strain distribution

STEEL HOLES CONFINED STRAIN PROPERTIES DATA

Data block to define material properties for the reinforceing steel and concrete. Data block to define the parameters for the solution or solutions desired.

NOTES: (1) For the analysis of a simple unconfined R.C. section, only data blocks (2), (3),(6) and (7) are required. All the parameters for the SYSTEM data block have default values, and therefore, this block is not required in this case. (2) The only required data block is SECTION. If only this data block is specified, the program will compute the section properties (Area, Inertia, etc). (3) In the CAL/SAP subroutines a colon is used to denote the end of data input for each line. Comment statements are allowed following the use of a colon. (4) The following pages describe the block.

input

file requirements for each data

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System Input Data block used to specify global analysis parameters. Line 1

D=

i,j,k

i,j,k=0 no debug messages (default) i,j,k =1 debug messages given for different options within the program

U=

1

Input specified in data file (default)

0

Interactive input

0

No generation of P-M surface (default)

1

Generation of P-M surface

2

Generation of p-m surface (dimensionless)

0

Area of bars specified

R

Where R= steel ratio for section

0

Automatic itereation

1

Interactive iteration

0

(Default)

1

Plot material stress--strain relations

0

(Default)

1

Limit maximum concrete strain so rebar does not fracture

2

Do not allow rebar to fracture

TOL=

#

Tolerence of convergence to specified axial load [0.001]

C=

0

If rebar is located within both confined and unconfined

T=

R= S= P= E=

concrete then the program will search to determine type of concrete closest to each rebar (to subtract off concrete area taken up by steel).

M=

1

If all concrete at rebar is unconfined

2

If all concrete at rebar is confined

0

Default

1

Rotate entire section by 180 degrees For non-symmetric section to change location of compression zone. See Fig. A-5 for the definition of the compression reference point.

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Section Input Data block used to specify RC section geometry and mesh. Line 1 to END

One line for each subsection defining the complete R.C. section. NS is the total number of sections I= The section # in sequential order (e.g. 1,2,3, 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, Xr,Yr Angle = (0-360 degrees) to rotate about point Xr, Yr. A positve angle represents counterclockwise rotation. Xr,Yr = Point to rotate section about (default = 0,0). See note (1). 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, Y coordinate system, and rotate them to an arbitrary position. Slight overlapping of sections will occur at points where sections that are rotated are ``attached'' to sections that are not rotated. Xr and Yr define global coordinates about which to rotate the section. The 'A=' option defaults to the previous specified value ('A=0,0,0' to stop section rotation).

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Steel Input Data block used to specify reinforcing bar information. 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 #, j bar #, bar increment # For linear generation of reinforcing bars between bars i and j, at the specified bar increment. For example, G=1,5,1 generates bars 2, 3, and 4 at equal increments ( ) between bars 1 and 5. R= Angle, Xr, Yr Where: angle is the angle of rotation in degrees (0-360) and Xr, Yr is the center of rotation. A positive angle bars will be rotated until 'R=0,0,0' is specified. END SECTION WITH BLANK LINE

NOTES: (1) nbar must begin each line (2) scalar multiples can be used on any term except nbar, e.g. X=10*12 or X=25/12 (3) the default value for 'A' is the previously specified value. (4) area of bar can be omitted if 'R=' is specified in SYSTEM data block. (5) bar increment # defaults to [1] (6) values for nbar need not be sequential or continous. This provides an easy means to add or delete bars without renumbering. (7) When the 'R=' option is used, all rebar specified on subsequent lines will also be rotated. To deactivate this option use 'R=0,0,0'.

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Holes Input Data block used to specify holes or voids in RC section.

Line 1 to END

OR

i,j,inc

i, F=

i

element # for first element

j

element # for last element

inc

element increment for generation (default=1)

i,inum,jnum,iinc,jinc i

first element for frontal generation

inum

number of elements in i-direction

jnum

number of elements in j-direction

iinc

element increment for i-direction

jinc

element increment for j-direction

END SECTION WITH BLANK LINE

NOTES: (1) A line to specify values for confined concrete is required; however, it may be a left blank when confined concrete is not required. (2) The only values required for the reinforcing steel are the yield stress, and the initial modulus. If only these values are specified, an elasto-plastic steel stress-strain is assumed. (3) The relation suggested by Vecchio and Collins [ACI Journal, Vol. 83 No. 2, March-April 1986.] is used to describe the stress-strain for concrete in tension. (4) Unconfined and confined concrete masonry based on work done at the University of Colorado. (5) A maximum of 150 points can be used to define the user defined stress-strain relations (fewer than 150 points may be used). Terminate the user defined stressstrain relations with a blank line, followed by a line (or lines) defining the reinforcing steel properties.

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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=1.0) 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. (2) If generating P-M surface only 'EC=' and 'A=' need to be specified once, 'P=' is not required (and will be ignored if specified). (3) If P-M surface is not being generated, use as many lines as necessary to define the desired solutions. For moment-curvature analyses specify as many values of extreme fiber compressive strain (monotonic) as desired. (4) 'EC=' (compression reference point) is the point on the section furthest from the neutral axis (at angle specified). See Fig. A.5. (5) For P < 0.10, the Phi factor is determined by using linear interpolation between the specified phi factor and 0.9 (at P=0). [Note: Pb is not considered].

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Confined Concrete Input Data block used to specify location (elements) of confined concrete.

Line 1 to END

OR

i,j,inc

i, F=

i

element # for first element

j

element # for last element

inc

element increment for generation (default=1)

i,inum,jnum,iinc,jinc i

first element for frontal generation

inum

number of elements in i-direction

jnum

number of elements in j-direction

iinc

element increment for i-direction

jinc

element increment for j-direction

END SECTION WITH BLANK LINE

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Properties Input Data block used to specify material properties.

UNCONFINED CONCRETE AND MASONARY REALTIONS Line 1

Unconfined concrete properties I=

1, for modified Kent-Park or Sheikh-Uzumeri (default)

FC=

Peak compressive stress

E1=

Strain at peak compressive stress

E2=

E50u strain (Modified Kent--Park) Note: E2 does not need to be if Fc is specified in units KSI (kips per square inch.)

FR=

FR,I; where FR is the rupture stress and I=1 includes the effects of tension stiffening, See Note (1). Default: I=0, no tension stiffening

OR Unconfined concrete properties

Line 1 I=

2, High-Strength Concrete (Yong's Model)

FC= E=

Modulus of elasticity

FR=

FR,I; where FR is the rupture stress and I=1 includes the effects of tension stiffening, See Note (1). Default: I=0, no tension stiffening

OR Line 1

Unconfined masonary properties I=

3, Masonary

FM=

Peak compressive stress

E1=

Strain at peak compressive stress

CONTINUED ON THE NEXT PAGE

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Line 1

Unconfined concrete properties I=

5, Saatciglu and Razvi (Normal Strength Concrete)

FC=

Peak compressive stress

E1=

Strain at peak compressive stress(Saatcioglu and Razvi) Note: E2 does not need to be specified if Fc is specified in KSI units KSI

E2=

E85u strain (Saatcioglu and Razvi) Note: E2 does not need to be if Fc is specified in units KSI (kips per square inch.)

FR=

FR,I; where FR is the rupture stress and I=1 includes the effects of tension stiffening, See Note (1). Default: I=0, no tension stiffening

OR Line 1

Unconfined concrete properties I=

6, Saatciglu and Razvi (High Strength Concrete)

FC=

Peak compressive stress

E1=

Strain at peak compressive stress(Saatcioglu and Razvi) Note: E2 does not need to be specified if Fc is specified in KSI units KSI

E2=

E85u strain (Saatcioglu and Razvi) Note: E2 does not need to be if Fc is specified in units KSI (kips per square inch.)

FR=

FR,I; where FR is the rupture stress and I=1 includes the effects of tension stiffening, See Note (1). Default: I=0, no tension stiffening

CONTINUED ON THE NEXT PAGE

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CONFINED CONCRETE AND MASONARY REALTIONS Line 2

Confined concrete properties (must be specified - See note(1)) I=

1, for modified Kent-Park or Sheikh-Uzumeri (default)

FC=

Peak compressive stress

E1=

Strain at peak compressive stress

E2=

E50u +E50h strain (Modified Kent--Park) or stain for 1/2 of drop from fcmax to fcmin from E1 (or E3 if specified)

E3=

Yield strain plateau (Default=E1, no plateau)

FR=

Rupture stress

FM=

Minimum compressive stress at high strain

OR I=

1, for Modified Kent-Park

FC=

Peak Unconfined compressive stress (ksi)

E1=

Strain at peak unconfined compressive stress

S=

Spacing of transverse reinforcing (inches)

H=

Dimension of confined core (inches)

RHO=

Transverse steel volumetric reinforcing ratio

FY=

Yield stress of transverse steel (KSI)

FR=

Rupture stress

CONTINUED ON THE NEXT PAGE

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CONFINED CONCRETE AND MASONARY REALTIONS Line 2

Confined concrete properties (must be specified - See note(1)) I=

2, for High Strength Concrete (Yong's Model)

FC=

Peak compressive stress

FY=

Yield stress of longitudinal reinforcement

FM=

Factor multiplied by peak compressive stress to give the minimum compressive stress at high strain

Line 3

RL=

Longitudinal volumetric reinforcing ratio

RP= FR=

Transverse volumetric reinforcing ratio Rupture stress

N=

Number of longitudinal reinforcing bars

D=

Diameter of longitudinal reinforcing bars

DT=

Diameter of lateral (transverse) bars

H=

Length of one side of the transverse tie

S=

Spacing of the transverse steel

CONTINUED ON THE NEXT PAGE

Line 2

CONFINED CONCRETE AND MASONARY REALTIONS Confined concrete properties (must be specified - See note(1)) I= 5, 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, 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,BXN where LXG=length of transverse bar in the X-direction BXN=clear confined dimension in the X-direction B2= BYG,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

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Line 2 I= X= Y= B1= B2= FYT= FM= S=

Line 2 I= FM= E1= R= S= H=

CONFINED CONCRETE AND MASONARY REALTIONS Confined concrete properties (must be specified - See note(1)) 6, for Saatcioglu and Razvi (High Strength Concrete) NX, AX where NX= number of tensverse reinforcing bars in the Xdirection and AX=area of one transverse reinforcing bar in the x-direction NY, AY where NY= number of tensverse reinforcing bars in the Ydirection and AY=area of one transverse reinforcing bar in the Y-direction BXG,BXN where LXG=length of transverse bar in the X-direction BXN=clear confined dimension in the X-direction BYG,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

Confined concrete masonary properties (must be specified - See note(4)) 3, 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, for user defined relations See Note (5) strain, fcunconfined, fcconfined, ft : strain, fcunconfined, fcconfined, ft REINFORCING STEEL RELATIONS Reinforcement steel properties Yield stress Ultimate stress (def. = FY) Failure stress (def. = FY) Strain for onset of strain hardening (def.=10%) Strain at stress FU (def.=10%) Strain at stress FF (def.=10%) Modulus of elasticicty Initial modulus of elasticity for strain hardening region (def.=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

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Appendix A SECTION AND MATERIAL MODELING Modeling capabilites for the materials and section geometry are described. Section definition is described first, followed by available material stress-strain relationships R.C. Section Modeling Material Modeling Solution Reference Angle Solution Reference Point Solution Scheme.

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Reinforced Concrete Section Modeling The modeling of rectangular reinforced concrete sections is accomplished by subdividing the section into elements. Fig. (A.1) presents a beam cross-section, and a possible mesh. The mesh of elements is defined by specifying the number of elements in each of the principal directions of the section. For more complicated sections, such as T-Beams or walls with boundary elements, several rectangular sections can be used to describe the element geometry (Fig. (A.2)). A reference point on each section (I=0,J=0) is used to define the starting location ( XS , YS ) of each section (Fig. A.2) in the global (X,Y) coordinate system. The elements are numbered in the i-direction from left to right, for each layer in the j-direction (Fig. A.2). The location of reinforcing is defined by specifying coordinates in the global coordinate system (X,Y).

Fig. A.1

Beam Cross Section Modeling

Fig. A.2 T-Beam Cross Section Modeling

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Material Modeling The program allows stress-strain curves for unconfined and confined concrete and concrete masonry. In addition, a user defined stress-strain curve may be specified. The concrete stress-strain behavior can be described by either the modified Kent and Park model (Park et al. (1982)), the Sheikh and Uzumeri model (1982) (Fig. (A.3)), or the Saatcioglu and Razvi Model (1995). The initial portion of the stressrelation is described by a second-degree parabolic shape. The relation suggested by Hognestad (1951) is used in this program.

COMPRESSIVE STRESS (KSI)

CONFINED W/PLATEAU

UNCONFINED

CONFINED W/O PLATEAU

STRAIN Fig. A.3 Example Concrete Stress-Strain Relations A relationship for rectilinearly confined high-strength concrete is also available. The relationships developed by Yong, Nour, and Nawy (1988) are used, and are based on experimental studies of twenty-four columns with compressive strength ranging between 12.13 and 13.56 ksi. The columns were subjected to axial loads only. 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. Relationships for unconfined and confined concrete masonry have recently been incorporated. The relationships are based on work done at the University of Colorado.

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The stress-strain model for the reinforcing steel allows a bilinear curve, or the consideration of strain hardening effects (Fig. (A.4)). The strain hardening curve is defined by the initial slope of the stress-strain curve at the onset of strain hardening, the ultimate stress and strain, and the fracture stress and strain. The equations describing the relationship are based on the equation presented by Saenz (1964). Care must be exercised when using the strain hardening curve because the relationship used to produce the curve is sensitive. It is suggested to always plot the material stress-strain curves to ensure they are reasonable.

NO YIELD PLATEAU

STRESS (KSI)

YIELD PLATEAU W/STRAIN HARDENING ELASTO-PLASTIC

STRAIN Fig. A.4 Example Steel Stress-Strain Relations

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Solution Reference Angle The solution is computed with respect to the global X-axis. 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.5).

Fig. A.5 Solution Referance Parameters

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Solution Reference Point The analysis of the R.C. 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. A.5).

Fig. A.5 Solution Referance Parameters

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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. 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. The accuracy of a given solution is determined in part by the number of elements used to model a cross section. The user should verify that the mesh refinement is adequate.

Appendix B Examples and Applications

Fifteen examples are presented to facilitate program use and detail program applications. No detailed calculations were made to determine precise values for the material property variables. The intent of the examples is to detail program options. 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 - 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

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Example 1 Compute the P-M interaction diagram for the section shown below. Grade 60 elasto-plastic steel and fc' = 4 ksi are used.

Fig. B.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=0.0001 : 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 Y= 2.5 A=0.79 T= 1 : Long. Steel Type 1 3 X= 21.5 Y= 2.5 G= 1, 3 : 8 - #8 Bars 5 X= 21.5 Y= 21.5 G= 3, 5 : 2 in. clear cover to bars 7 X= 2.5 Y= 21.5 G= 5, 7 : assume #4 hoops 8 X= 2.5 Y= 12 : : : : PROPERTY: FC=4*0.85 E1=0.003 E2= 0.004 FR=0.4 : Rupture strength = 10% f'c : Blank line for confined concrete FY=60 E=29000 : Design Steel Stress : : DATA: EC=0.003 A=0 N=20 : Moment about global X-axis for Ultimate EC=0.003 A=0 N=20 PHI=0.70 : compressive strain of 0.003, with and : without capacity reduction factors END

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AXIAL LOAD (KIPS)

COMPUTED P-M

DESIGN P-M

MOMENT (IN-KIPS)

Fig. B.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. The stress-strain relationship for confined concrete is based on Modified Kent-Park (Appendix C).

Fig. B.1

Example 2: Column Cross Section

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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=0.0001 : 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=14,10,10,1,12: CONFINED CONCRETE CORE: ELEMENTS 14 - 23 : ELEMENTS 26 - 35 : : ELEMENTS 122 - 131 : STEEL: 1 X= 2.5 Y= 2.5 A=0.79 T= 1 : Long. Steel Type 1 3 X= 21.5 Y= 2.5 G = 1, 3 : 8 - #8 Bars 5 X= 21.5 Y= 21.5 G = 3, 5 : 2 in. clear cover to bars 7 X= 2.5 Y= 21.5 G = 5, 7 : assume #4 hoops 8 X= 2.5 Y= 12 : : PROPERTY: FC=4*0.85 E1=0.002 E2= 0.004 FR=0.4 : FC=4*0.85 E1=0.002 H=20 S=4 FY=60 RHO=0.01707 : FY=68 FU=100 FF=95 E1=0.004 E2=0.08 E3=0.12 E=29000 ET=1500: : DATA: EC=0.0002 : Moment about global X-axis EC=0.0004 EC=0.0006 EC=0.0008 EC=0.0010 EC=0.0015 EC=0.0020 EC=0.0025 EC=0.0030 EC=0.0035 EC=0.0040 EC=0.0050 EC=0.0060 EC=0.0070 : END

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Fig. B.3

Example 2: Computed Moment-Curvature Relations

Example 3 Compute the Moment-Curvature diagram for the section shown below. The concrete is 4000 psi, and the reinforcing steel behavior is based on typical observed experimental relations for grade 60 steel. In addition confined boundary elements are specified.

Fig. B.4 Example 3: Wall Cross Section

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Example 3: Shear Wall w/ Confined Boundary Elements Moment - Curvature Calculations for Probable Steel stress-strain SYSTEM: T=0 E=0 P=1 TOL=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, 6, 6,1, 8: BOUNDARY ELEMENT 362 F = 362, 6, 6,1, 8: BOUNDARY ELEMENT : STEEL: 1 X= 3 Y= 3 A= 1.56 T= 1 : Boundary element steel 5 X= 21 Y= 3 G= 1, 5 : 16 #11 Bars 9 X= 21 Y = 21 G= 5, 9 : 13 X= 3 Y = 21 G= 9,13 : 16 X= 3 Y = 7.5 G= 13,16 : 17 X= 243 Y= 3 : Boundary element steel 21 X= 243 Y = 21 G= 17,21 : 25 X= 261 Y = 21 G= 21,25 : 29 X= 261 Y= 3 G= 25,29 : 32 X= 246 Y= 3 G= 29,32 : 33 X= 33 Y= 8 A= 0.31 : Web steel - #5 @ 18 in. 44 X= 231 Y= 8 G= 33,44 : 45 X= 33 Y = 16 : 56 X= 231 Y = 16 G= 45,5 : : PROPERTY: FC=4 E1= 0.003 E2= 0.004 FR= .4 : Unconfined Concrete FC=5 E1= 0.0035 E2= 0.01 FR= .4 FM= .80 : Confined Concrete FY=65 FU=90 FF=85 E1=0.004 E2=0.08 E3=0.12 E=29000 ET=1500: : DATA: EC=0.0002 A=-90 P=1000 : Moment about global X-axis EC=0.0004 A=-90 P=1000 EC=0.0006 A=-90 P=1000 EC=0.0008 A=-90 P=1000 EC=0.0010 A=-90 P=1000 EC=0.0015 A=-90 P=1000 EC=0.0020 A=-90 P=1000 EC=0.0025 A=-90 P=1000 EC=0.0030 A=-90 P=1000 EC=0.0035 A=-90 P=1000 EC=0.0040 A=-90 P=1000 EC=0.0050 A=-90 P=1000 EC=0.0060 A=-90 P=1000 EC=0.0070 A=-90 P=1000 END

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Example 4 Compute the P-M interaction diagram for the section shown below. The concrete is 4000 psi, and grade 60 reinforcing is used. Special options include multiple sections, section rotation, and steel rotation.

Fig. B.6

Example 4: Wall Cross Section

Example 4: L - Section Rotated 30 degree - Unconfined Concrete P-M Interaction Surface : : SYSTEM: T=1 E=0 C=1 P=1 M=0 TOL=0.0001 : : : SECTION: 1 HI = 100 HJ = 20 N1 = 50 N2= 10 X = 0 Y = 0 A=30,0,10: 2 HI = 20 HJ = 80 N1 = 10 N2= 40 X =80 Y = 20 A=30,0,10: : STEEL: 1 X = 2 Y = 2 A = 0.31 R=30,0,10: Rotate all steel 9 X = 82 Y = 2 G = 1, 9 : 10 X = 98 Y = 2 : 11 X = 98 Y = 18 : 19 X = 98 Y = 98 G=11,19 : 20 X = 2 Y = 18 : 29 X = 82 Y = 18 G=20,29 : 37 X = 82 Y = 98 G=29,37 : :

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AXIAL LOAD (KIPS)

PROPERTY: FC=4 E1= 0.0030 E2= 0.004 FR= .4 : : Blank Line for Confined Concrete FY=70 FU=110 FF=105 E1=0.005 E2=0.070 E3=0.10 E=29000 ET=1500: : DATA: EC = 0.003 A = 90 : Moment about y-axis EC = 0.003 A = -90 : Moment about X-axis : END

Fig. B.7

Example 4:

Computed P-M Interaction Diagram

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Example 5 Compute the P-M interaction diagram for the section shown below. The concrete is 12,000 psi and grade 60 reinforcing is used. Special options used include ``CONFINED elements'', and ``PROPERTIES'' for unconfined and confined highstrength concrete.

Fig. B.8

Example 5:

Column Cross Section

Example 5 : HIGH-STRENGTH COLUMN WITH CONFINED CONCRETE P-M Interaction Diagram - Based on Design Steel stress-strain SYSTEM: D=0,0,0 S=0 T=1 E=1 C=2 P=1 TOL=0.0001 : 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=14,10,10,1,12 : : STEEL: 1 X= 2.5 Y= 2.5 A= 0.79 T= 1 : Long. Steel Type 3 X= 21.5 Y= 2.5 G= 1, 3 : 8 #8 Bars 5 X= 21.5 Y= 21.5 G= 3, 5 : 2 in. clear cover to bars 7 X= 2.5 Y= 21.5 G= 5, 7 : assume #4 hoops 8 X= 2.5 Y= 12.0 : :

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PROPERTY: I=2 FC=10.2 E=5400 FR=1.02 : I=2 FC=11.2 E=5400 FR=1.1 FY=60 FM=0.3 RL=0.011 RP=0.01 N=8 D=1 DT=0.5 H=19 S=4 : FY=60 E3=0.120 E=29000 : Fracture of steel at 12% strain : DATA: EC=0.003 A=0 N=11 : Moment about global X-axis for Ultimate EC=0.003 A=0 N=11 PHI=0.70 : compressive strain of 0.003, with and : without capacity reduction factors : END

Fig. B.9 Example 5: Computed P-M Interaction Diagram

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Example 6 Compute the M- 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.

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=0.0001 : 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 Y= 2.5 A= 0.79 T= 1 : Long. Steel Type 3 X= 21. 5 Y= 2.5 G= 1, 3 : 8 #8 Bars 5 X= 21.5 Y= 21.5 G= 3, 5 : 2 in. clear cover to bars 7 X= 2.5 Y= 21.5 G= 5, 7 : 8 X= 2.5 Y= 12 : : : PROPERTY: I=3 FM=4 E1=0.003 : UNCONFINED MASONRY I=3 FM=4 E1=0.014 R=0.001 S=12 H=20 : CONFINED MASONRY FY=60 E3=0.120 E=29000 : Fracture of steel at 12% strain : DATA: EC=0.003 A=0 N=12 : Moment about global X-axis for Ultimate EC=0.003 A=0 N=12 PHI=0.70 : compressive strain of 0.003, with and : without capacity reduction factors : END

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Example 7 Compute the M- relation for a T-beam. Special options used include user specified stress-strain relations and multiple steel types.

Fig. B12 Example 7: Beam Cross Section

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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=0.0001 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=14.5 Y=2.5 A=0.79 T=1 : Long. Steel Type 3 X=19.5 Y=2.5 G=1,3 : 3-#8 Bars in Top and Bottom 4 X=14.5 Y=33.5 : 2 in. Clear Cover to Bars 6 X=19.5 Y=33.5 G=4,6 : 7 X=4 Y=24.5 A=0.20 T=2 : Additional Steel 8 X=8 Y=24.5 : 8-#4 Bars in Flange 9 X=26 Y=24.5 : 10 X=30 Y=24.5 : 11 X=4 Y=33.5 : 12 X=8 Y=33.5 : 13 X=26 Y=33.5 : 14 X=30 Y=33.5 : : PROPERTY: I=1 FC=4 E1=.002 E2=.004 FR=.4 : Unconfined Concrete : Blank Line for Confined FY=68 FU=100 FF=95 E1=.004 E2=.08 E3=.12 E=29000 ET=1200 FY=45 FU=70 FF=65 E1=.030 E2=.12 E3=.20 E=29000 ET=1100 : DATA: EC=0.0002 A=0 P=0 : Moment about Global X-Axis EC=0.0004 A=0 P=0 : No Axial Load EC=0.0006 A=0 P=0 EC=0.0008 A=0 P=0 EC=0.0010 A=0 P=0 EC=0.0015 A=0 P=0 EC=0.0020 A=0 P=0 EC=0.0025 A=0 P=0 EC=0.0030 A=0 P=0 EC=0.0035 A=0 P=0 EC=0.0040 A=0 P=0 EC=0.0050 A=0 P=0 EC=0.0060 A=0 P=0 : END

MOMENT (IN-KIPS)

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CURVATURE (/IN.) Fig. B.13

Example 7:

Computed Moment-Curvature Diagram

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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; however, 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 - User specified concrete stress-strain relations. : SYSTEM: S=0 T=1 E=1 C=1 P=1 TOL=0.0001 : T=1 for P-M Diagram : SECTION: 1 HI=24 HJ=24 N1=12 N2=12 X=0 Y=0 : 24"x24" Column, 12x12 Mesh : STEEL: 1 X= 2.5 Y= 2.5 A=0.79 T= 1 : Long. Steel Type 3 X= 21.5 Y= 2.5 G= 1, 3 : 8 #8 Bars 5 X= 21.5 Y= 21.5 G= 3, 5 : 2 in. clear cover to bars 7 X= 2.5 Y= 21.5 G= 5, 7 : assume #4 hoops 8 X= 2.5 Y= 12 : : PROPERTY: UNCONFINED CONFINED TENSILE I=4: STRAIN COMP. STRESS COMP. STRESS STRESS: USER DEFINED .119500E-03 .391772E+00 .330236E+00 .000000E+00: MAX 150 POINTS .239000E-03 .763358E+00 .646763E+00 .000000E+00 : BLANK LINE - REQUIRED TO SPECIFIY END OF USER DEFINED STRESSSTRAIN FY=60 E=29000 : DESIGN STEEL STRESS-STRAIN : : DATA: EC=0.003 A=0 N=12 : Moment about global X-axis for Ultimate EC=0.003 A=0 N=12 PHI=0.70 : compressive strain of 0.003, with and : without capacity reduction factors : END

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EXAM 13 : -----------------------------------------------------------: :Circular column without confinement : SYSTEM: T=1 E=2 C=1 P=1 TOL=0.000001 : : SECTION: 1 S=2 D=30 NP=10 NR=10 A=0: D=diameter : NP= no. of poles : NR= no. of rings : S=section type 1 for rectanglar ; 2 for circular : STEEL: 1 X= 12.5 Y= 0 A= 1.56 T= 1 : Long. Steel Type 1 2 X= 0 Y= 12.5 : 3 X= -12.5 Y= 0 : 4 X= 0 Y= -12.5 : #4 hoops : PROPERTY: FC=5*0.85 E1=0.003 E2= 0.004 FR=0.4 : : SY=1 : : DATA: EC=0.003 A=0 N=20 : Moment about global X-axis EC=0.003 A=0 N=20 PHI=0.7: : END

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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

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Modified Kent-Park C.1 Modified Kent-Park [Park et al. (1982) The following equations are used to describe the stress-strain relations of confined and unconfined concrete (See Fig. C.1).

Fig. C.1 Modified Kent-Park Stress-Strain Relations fc = k fc

[

2ε c ε - ( c )2 kε 0 kε 0

]

fc = k fc [ 1 - Zm(εc - kε0)2 ] ≥ 0.2 k fc′

k=1+

Zm =

ε

ε50u =

ρ h f c' f c'

0.5 +ε − kε 50u 50h 0

3 + ε 0 f c' f c' − 1000

ε50h = 0.75ρ h / s

εc≤ kε0 εc > k εc

(C.1.a)

(C.1.a)

(C.2)

(C.3)

(C.4)

(C.5)

Where fc is the longitudinal concrete stress, ε is the longitudinal concrete strain, fyh is the yield stress for the hoop reinforcement, h is the width of the concrete core measured to the outside of the hoops, s is the center-to-center spacing of the hoops, and ρ is the ratio of the volume of hoop reinforcement to volume of concrete core measured

to the outside of the hoops. Units of psi are used for stress, and ε0 is typically assumed to be 0.002.

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Sheikh and Uzumeri The following equations are used to describe the stress-strain relation for confined concrete (See Fig. C.2) for a square column with uniformly distributed longitudinal reinforcement

Fig C.2 Sheikh and Uzumeri Stress-Strain Relation

KS = 1.0 +

2.73B 2 Pocc

⎡⎛ nC 2 ⎞⎟ ⎛ s ⎞2⎤ ⎥ p f' ⎢⎜ 1 − 1− ⎜ ⎟ ⎝ 2 B ⎟⎠ ⎥ s s 2 ⎢⎜⎝ 55 . B ⎠ ⎦ ⎣

(C.6)

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In equations C.6 to C.10, B is the center-to-center distance of the perimeter hoops (to define the confined core), C is the center-to-center spacing of the longitudinal reinforcing bars, n is the number of longitudinal reinforcing bars (in Eq. C.6, the quantity nC2 assumes equal spacing of longitudinal reinforcement), Pocc is ' f c 0.85 Acore, s is the spacing of the hoops, s is the ratio of the volume of hoop

reinforcement to volume of concrete core, and fs is the yield stress of the hoops. A value of 0.002 is typically assumed for oo The minimum stress at high strains ' f c can be taken as 0.3Ks

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Yong's High-Strength Concrete The following equations were developed by Yong , Nour, and Nawy to describe the axial stress-strain relation for rectilinearly confined high-strength concrete (See Fig. C.3). The relations are based on experimental studies of twenty-four square columns with compressive strength ranging from 12.13 to 13.56 ksi . The

peak stress of the confined concrete is obtained by multiplying the peak '

compressive stress of the unconfined concrete, f c by the factor K given in Eq. (C.11), '' 0.245s ⎞ ⎛ '' nd '' ⎞ f y ⎛ K = 1.0 + 0.0091 ⎜ 1 − ρ⎟ ⎟⎜ρ + ⎝ 8sd ⎠ f ' h '' ⎠ ⎝ c

(C.11)

''

Where s is the center-to-center spacing of the lateral ties in inches; h is the length of one side of the rectangular ties in inches; n is the number of longitudinal steel bars; d

''

is the nominal diameter of the lateral ties in inches; d

is the nominal

diameter of the longitudinal steel bars in inches; ρ is the volumetric ratio of the lateral reinforcement; ρ is the volumetric ratio of the longitudinal reinforcement; and ''

f y''

is the yield stress of the lateral steel in psi.

The strain at peak stress (confined) is computed as:

(

)

0.734 s ⎞ '' '' 2 / 3 ⎛ 0.0035⎜ 1 − ρ fy '' ⎟⎠ ⎝ h ε0 = 0.00265 + f c'

(C.12)

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The remaining values used to produce the stress-strain curve are determined using the following equations: by ⎡ ⎛ f i = f 0 ⎢0.25⎜⎜ ⎢⎣ ⎝

⎤ f c' ⎞ ⎟ + 0.4 ⎥ f 0 ⎟⎠ ⎥⎦

(C.13)

⎡ ⎛ ε0 ⎞ ⎤ ε i = K ⎢14 . ⎜ ⎟ + 0.0003⎥ ⎣ ⎝ k ⎠ ⎦

(C.14)

⎡ ⎤ ⎛ f ⎞ f 2i = f 0 ⎢0.025⎜ 0 ⎟ − 0.065⎥ ≥ 0.3 f 0 ⎝ 1000 ⎠ ⎣ ⎦

(C.15)

ε 2i = 2 ε i − ε 0

(C.16)

Two polynomial equations are used to produce a smooth continuous curve for the stress-strain behavior of rectilinearly confined high-strength concrete given Eq. (C.12) through (C.16) [Yong et al. (1988)]. The relations are linear (Fig. C.3) until a ' f c compressive stress of 0.45 . The two polynomial equations intersect at the point

of peak stress at zero slope (Fig. C.3).

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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. C.4). ⎛ 1 ⎞ fm = − Aε 2m + ⎜ 0 ≤ εm ≤ εu (C.17) + Aε m ⎟ ε m ' ⎝εu ⎠ fm

fm f m'

= C + (1 − C )e

− B( ε m − ε u )

ε u ≤ ε m ≤ ε mc

'

(C.18)

Where, f m is the maximum compressive stress of the concrete masonry, concrete masonry strain,

ε m is the

ε u is the strain corresponding to maximum compressive

stress, ε mc is the maximum usable strain. The parameters A, B, and C depend on the degree of confinement. Unconfined Concrete Masonry

[

]

ε u = 0.01 0.046 f m' + 0.055

(

f m' in ksi )

(C.19)

ε mc = 0.003

(C.20)

A = 176700;

B = 574.6;

C = 0.131

(C.21)

Confined Concrete Masonary

[

]

ε u = 0.01 0.046 f m' + 0.055 ε mc = 0.014 A= B=

(C.22)

(C.23)

480000

(C.24)

f m' f m' 4.3ρ s ( hc S h )

'

( f m in ksi )

0.8

C = 0.4( hc Sh ) − 01 .

(C.25)

(C.26)

Where, ρ s is the volumetric ratio of confinement reinforcement to the concrete masonry core (measured to the outside of the combs), hc is the width of the confined core measured to the outside of the confinement comb, and S h is the vertical spacing of confinement combs.

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Saatcioglu and Razvi Model The following equations are used to describe the stress-strain relations of confined and unconfined concrete (See Fig. C.5)

C.5.1 Normal-Strength Concrete C.5.1.1 Square Column ' ' f cc = f co + k1 f le

(C.27)

f le = k 2 + f l

(C.28)

f l = ∑ AS f yt Sinα Sbc

(C.29)

k1 = 4.84( f le )

−0.17

⎛ bc ⎞ ⎛ bc ⎞ ⎛ 1 ⎞ k 2 = 010036 . ⎜ ⎟⎜ ⎟⎜ ⎟ ⎝ S ⎠ ⎝ Sl ⎠ ⎝ f l ⎠

(C.30) ≤ 10 .

(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.e. max. tie spacing for buckling specified in the ACI Building Code ( ACI-318-89) should be satisfied ). The column should conform to the building code limitation for maximum spacing of laterally supported longitudinal reinforcement. Therefore, this may be taken as the upper limit for Sl.

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C.5.1.2 Rectangular Column

( ) f ly = ( ∑ AS f yt sin α bcy ) f lx = ∑ AS f yt sin α bcx

(C.32) (C.33)

⎛ bcx ⎞ ⎛ bcx ⎞ ⎛ 1 ⎞ k 2 x = 010036 . ⎟⎜ ⎟ ⎟⎜ ⎜ ⎝ S ⎠ ⎝ S l ⎠ ⎝ f lx ⎠

≤ 10 .

(C.34)

⎛ bcy ⎞ ⎛ bcy ⎞ ⎛ 1 ⎞ ⎟ k 2 y = 010036 ⎜ ⎟⎜ ⎟⎜ ⎝ S ⎠ ⎝ S l ⎠ ⎜⎝ f ly ⎟⎠

≤ 10 .

(C.35)

f lex = k 2 x f lx

(C.36)

f ley = k 2 x f ly

(C.37)

(

f le = f lex bcx + f ley bcy

) (bcx + bcy )

(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.5.1.3 Ductility of Confined Concrete

ε 1 = ε 01 (1 + 5k )

(C.39)

' k = k1 f le f co

(C.40)

Where: ε 01 = Strain corresponding to peak stress of unconfined concrete. A value of 0.002 is appropriate for ε 01 ε 1 = Strain corresponding to peak stress of confined concrete ' f co = 0.85 f c'

(C.41)

ε 85 = 260ρ c ε 1 + ε 085

(C.42)

ρ c = ∑ AS

[S (bcx + bcy )]

(C.43)

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Where: ε 085 = Strain at 85% strength level of unconfined concrete. A value of 0.0038 is appropriate for ε 085 . ∑ AS = The summation of transverse reinforcement area in two directions crossing bcx and bcy . •

If the column is spirally reinforced or has identical reinforcement arrangement in each direction, ρ c reduces to transverse reinforcement in one direction only

C.5.1.4 Stress-Strain Relation for Confined Concrete 1 2 ⎤ 1+ 2 k

⎡ ' ⎢ ⎛εc ⎞ ⎛εc ⎞ ⎥ f c = f cc 2⎜ ⎟ − ⎜ ⎟ ⎢ ⎝ ε1 ⎠ ⎝ ε1 ⎠ ⎥ ⎣ ⎦

' ≤ f cc

(C.44)

C.5.2 Modifications Introduced for High-Strength Concrete •

The yield strength of lateral steel is limited to 1000 MPa (149 ksi).

•

For square column :⎛ bc ⎞ ⎛ bc ⎞ k 2 = 015 . ⎜ ⎟⎜ ⎟ ⎝ S ⎠ ⎝ Sl ⎠

•

≤ 10 .

(C.45)

For a rectangular column :⎛ bcx ⎞ ⎛ bcx ⎞ k 2 x = 015 . ⎟ ⎟⎜ ⎜ ⎝ S ⎠ ⎝ Sl ⎠

≤ 10 .

(C.46)

⎛ bcy ⎞ ⎛ bcy ⎞ k 2 y = 015 . ⎜ ⎟⎜ ⎟ ⎝ S ⎠ ⎝ Sl ⎠

≤ 10 .

(C.47)

k3 =

5.96 ' f co

. ≤ 10

(C.48)

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k4 =

f yt

≤ 10 .

(C.49)

ε 01 = 0.0028 − 0.0008k 3

(C.50)

ε 1 = ε 01 (1 + 5k 3 k )

(C.51)

ε 085 = ε 01 + 0.0018k 32

(C.52)

ε 85 = 260k 3 ρ c ε 1[1 + 0.5k 2 ( k 4 − 1)] + ε 085

(C.53)

E c = 1281537 . f c' + 10281 .

(C.54)

74.5

f' E sec. = cc

(C.55)

ε1 Ec r= E c − E sec.

fc =

(C.56)

' ⎛εc ⎞ f cc ⎜ ⎟r ⎝ ε1 ⎠

⎛ε ⎞ r −1+ ⎜ c ⎟ ⎝ ε1 ⎠

ksi

r

(C.57)

Report Outline The program manual is presented in the following section. Pertinent notes are included to facilitate user understanding. Appendices are included to detail the material and section modeling capabilities of the program, and present applications and examples.