Report No. 77-2 COMPLETE STRESS-STRAIN CURVE OF CONCRETE

COMPLETE STRESS-STRAIN CURVE OF CONCRETE AND ITS EFFECT ON DUCTILITY OF REINFORCED CONCRETE MEMBERS

by

Pao-Tsan Wang Research Assi$tant

Surendra P. Shah Professor of Civil Engineering

Antoine E. Naaman Associate Professor of Structural Design

October 1977

Report No. 77-2

COMPLETE STRESS-STRAIN CURVE OF CONCRETE AND ITS EFFECT ON DUCTILITY OF RF: I ;~IORCED CONCRETE MEMBERS

by

P. T. Wang, S. P. Shah and A. E. Naaman Department of Materials Engineering

University of Illinois at Chicago Circle Chicago, Illinois 60680

ABSTRACT

A relatively simple experimental technique is developed to obtain the

entire stress-strain curve of concrete including the descending portion. The

experimental stress-strain curves of normal weight concretes of strengths up

to 11,000 psi and lightweight concretes of strengths up to 8,000 psi are

reported.

A general analytical expression for the stress-strain curve of concrete

is proposed to reflect accurately experimental results. The expression has

four constants which depend on the properties of both the ascending and the

descending portions of the stress-strain curve and can be evaluated from the

knowledge of four key points of the curve, namely; on the ascending portion,

the peak point and the point corresponding to a stress of 45% of the peak,

and, on the descending portion, the inflection point, and another point

symmetric to the peak with respect to the inflection point. Furthermore, the

coordinates of the four key points were expressed in function of the compres-

sive strength of concrete so as to allow prediction of the entire curve solely "

from the knowledge of the compressive strength.

Similarly, an expression was developed to represent the entire stress-

strain curve of Grade 60 reinforcing bars with yield strengths varying from

about 60 to 75 ksi. This expression was based on the analysis of results

from 50 random reinforcing bar specimens tested according to ASTM standards.

I~

Based on the above results, a computerized nonlinear analysis progr~~

was developed to predict the structural response of reinforced concrete

beams, and columns under combined axial load and uniaxial and biaxial

bendings. The method takes into consideration actual material properties

for both concrete and steel, and uses the general assumptions of equilibrium

and compatibility. Using the above program, the comparison of theoretically

generated information with some experimental data reported in technical

literature is found acceptable.

The effects of high strength concrete, tensile and compressive re-

inforcement ratios, axial load and eccentricity on the structural behavior

of reinforced concrete sections, such as ultimate load capacity, curvature

and ductility are thoroughly explored. Load-moment-curvature-strain

interaction diagrams are automatically generated and plotted.

An extensive parametric study of the concrete compression zone is

undertaken and leads to recommendations on the appropriate values of the

rectangular stress block parameters (s , al ) for high strength normal cu

weight and lightweight concretes.

"

It

iii

ACKNOWLEDGEMENTS

This research was at first initiated by the need to characterize the

stress-strain properties of portland cement mortar as used inferrocement.

The study on ferro cement was supported by a National Science Foundation

Grant No. ENG 74-20829 to the University of Illinois at Chicago Circle.

However, the completion of the first phase of this continuing investigation

would not have been possible without the support of the Department of

Materials Engineering and an internal grant from the University of Illinois

Research Board. The help of the above three sponsors is greatly appreciated.

iv

TABLE OF CONTENTS

Page

ACKNOWLEDGEMENTS. • • • • • • • • • • • • • . • • • • • • • • • • • • • • • • • • • • • • . • • • • • • • . • • . • iii

TABLE OF CONTENTS. . . . . • . . • • • . • • • • • . • • • • • • • • • • • • • • • • • • • • • . • • • . • . . . • i v

LIST OF FIGURES................................................... vii

LIST OF TABLES.................................................... xiii

NOTATIONS ...•••.•••••••••••••.•.•••...•••.•••••••.••••••• -. . • • • • • • . xiv

CHAPTER

I

II

III

IV

INTRODUCTION ••.•.••..••...•.•••••••.•.•••...•••.•..••.. 1

COMPLETE STRESS-STRAIN CURVE OF CONCRETE ••.•.•...•.••.. 6

2.1 REVIEW OF EXISTING STRESS-STRAIN CURVE OF CONCRETE. • • . • . • • . . . • . . . • . • . • . . . . • • . . • . • . • • • . . • • . . • 6

2.2 PROPOSED ANALYTIC STRESS-STRAIN CURVE OF CONCRETE. • • . • . . . . . • • . . . • • . . • . • • • • . • • • . . • • . • . • . • . . . 25

STRESS-STRAIN CURVES OF REINFORCING STEEL •.•.•••.•..•.. 43

3.1 INTRODUCTION. • . . • • . • . • • • • • • . • • • . . . • • • . • • • . • • • . • • . . 43

3.2 STRESS-STRAIN CURVE OF MILD STEEL WITH DEFINITE YIE LD PLATEAU •••••••••••••••.•.••••••••. 44

3.3 STRESS-STRAIN CURVE OF STEEL WITHOUT DEFINITE YIELD POINT SUCH AS HIGH STRENGTH REINFORCING BARS. . • • • • • . • • • • • • • • • . . • • • . . • • . . • • • • • . . . • • . • • • . • . • 50

3.4 &~ALYSIS OF AVAILABLE EXPERIMENTAL DATA........... 60

3.5 CONCLUSIONS.......... • • • • • • • • • • • • • • • • • • • • • • • • • • • . • 66

TEST PROGRAM AND VALIDATION OF RESULTS .....•........•.• 74

4.1 _ EXPERIMENTAL PROGRAM.··· ••. ··· ... · •.... ·....•..•.• 74

4.2 TESTING METHOD ••••• · .•••• • •.• ·••••••. . • • • . • . • . . • . • 74

4.3 FABRICATIONS AND MATERIALS·....................... 77

4.4 TEST RESULTS...................................... 78

4.5 GENERATING THE CONSTANTS OF THE ANALYTIC EXPRESSION FOR THE STRESS-STRAIN CURVE............ 83

v

TABLE OF CONTENTS (Cont'd)

CHAPTER Page

IV 4.6 VALIDATION OF THE ANALYTICAL EXPRESSION.. •• ••• ••• • 103

4. 7 CONCLUSIONS •.•.•.•••••••.••••••....•••••.••.••••. 105

V A COMPUTER PROGRA~ FOR THE NONLINEAR ANALYSIS OF REINFORCED CONCRETE SECTIONS .• ·· .• ·· •. ·•· •. · ••. ·· ••• 109

5.1 INTRODUCTION. • • . . . . . . • • • • . • • . . • . . . • . . • . . . • . • • • . • . • 109

5.2 GENERAL PROCEDURE FOR THE NONLINEAR ANALYSIS OF BEAM AND COLUMN SECTIONS .•..••.••.•.••..•..•••. 109

5.3 BEAM UNDER PURE BENDING.· •. ·• •. · •.• ··•· ••. · •.• ·••. 111

5.4 COLUMN UNDER AXIAL LOAD AND BENDING.· ............• 123

VI DUCTILITY OF REINFORCED CONCRETE MEMBERS. . • . . • • . . . • • • . • 143

6.1 INTRODUCTION··.··· •• ·••••·•·•• •• ·• •• •··.··· .• ··.·. 143

6.2 DUCTILITY OF PLAIN CONCRETE ••• ·• ••. ·•·•·•· .•.••.. · 143

6.3 DUCTILITY OF REINFORCED-CONCRETE BEAM SECTIONS.... 147

6.4 DUCTILITY OF REINFORCED-CONCRETE COLUMN SECTIONS .• 152

6.5 ROTATIONAL CAPACITY OF HINGING REGION AND MID-SPAN DEFLECTION OF REINFORCED CONCRETE BEAMS ..... · 169

VII STUDY OF THE ULTIMATE STRENGTH PARAMETERS FOR THE DESIGN OF REINFORCED-CONCRETE SECTIONS................. 179

7.1 INTRODUCTION................ • . • • • • . . . . . . . • . • • . . . . • 179

7.2 PARAMETERS ~ AND 6 OF THE CONCRETE COMPRESSION ZONE. • . • . . . • • . • . . • • • • • • • . . . . . • • . • • . . . • . . . • • • • • . . . • 179

7.3 CONCRETE STRAIN E AT MAXIMUM MOMENT •.•••••••••.• 189 ell

7.4 STRESS BLOCK DEPTH PARAMETER 61

FROM THE ACI CODE. • • • . • • • . • • • . . • • • . • . . • . . • . • • . . • • . • . • • . • . • . • • . • 189

7.5 THE ULTIMATE MOMENT, ULTIMATE CURVATURE AND DUCTILITY FACTOR...... . . . . . . . . . . . . • • . • • • . . . . • . . • • • 196

VIn SillvlMARY AND CONCLUSIONS •.......••.•.••..•••.•.•..••.••• ·210

vi

TABLE OF CONTENTS (Cont'd)

Page

REFERENCES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

APPENDIX I COMPUTER PROGRAM FOR ANALYZING RECTANGULAR AND T- BEAM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

APPENDIX II COMPUTER PROGRAM FOR ANALYZING UNIAXIALLY LOADED COLUMN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

APPENDIX III COMPUTER PROGRAM FOR ANALYZING BIAXIALLY LOADED COLUMN ................................... , . . . . . .. .. 246

VITA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

Figure

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

2.10

2.11

2.12

2.13

2.14

2.1Sa

2.lSb

LIST OF FIGURES

Typical stress-strain curve.

Smith and Young's exponential function.

Polynomial functions.

Sturman, Shah and Winter's curve.

Desay and Krishnan's curve.

Fractional functions by Popovics and Saenz.

Fractional function with six parameters by Saenz

Gamma distribution function.

Sixth degree polynomial.

Fractional function with a pole.

Typical ascending portion.

Descending portion through two points.

Descending portion through inflection point.

Descending portion through inflection and arbitrary points.

Comparison between a1ternatesl, 2 and 3 for descending portion.

Comparison between alternates 1, 2 and 3 for descending portion.

3.1 Bresler and Heimdahl's stress-strain curve for grade 60 steel.

3.2

3.3

3.4

3.5

Sargin's stress-strain curve for grade 60 steel.

Comparison between Sargin's and revised formulas for representing the stress and strain curve of grade 60 steel up to a strain of 0.200.

Typical stress-strain curve of steel without definite yield point.

Sargin curve for high strength reinforcing bar.

vii

Page

8

11

13

16

18

20

23

27

28

29

31

34

36

39

41

42

45

46

49

51

53

Figure

3.6

3.7

3.8

3.9

3.10

3.11

3.12

3.13

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

4.10

4.11

4.12

Goldberg and Richard's curve for high strength reinforcing bars.

Comparison between proposed formula and actual data of grade 60 steel.

Comparison between proposed formula and actual data of steel having no definite yield point.

fsu' fsf vs. fy

Esh' E ,E f vs. f su s Y

E , E h vs. f s s y

Generalized stress-strain curves of grade 60 steel with yield strengths of 60, 65, 70 and 75 ksi.

Variation of data for f = 68 ksi. y

Test set-up.

Typical stress-strain curves for normal weight concrete.

Typical stress-strain curves for lightweight concrete.

Failure patterns for normal weight concrete.

Failure patterns for lightweight concrete.

Comparison between normal weight and lightweight concrete stress-strain curves.

The four key points for the proposed analytic stressstrain curve.

Linear relation between E and f for normal weight concrete. 0 0

Linear relation between El and f for normal weight concrete. 0

Linear relation between f. and f for normal weight concrete. l. 0

Linear relation between f2i and f for normal weight concrete. 0

Linear relation between E and f for normal weight c 0 concrete.

viii

Page

58

64

65

68

69

70

71

72

76

80

81

82

82

84

90

91

92

93

94

95

Figure

4.13

4.14

4.15

4.16

4.17

4.18

4.19

5.1

5.2

5.3

5.4

Linear relation between e: 0'

e:. and f for lightweight concrete. 1 0

Linear relation between f. and f for lightweight concrete. 1 0

Linear relation between E c' f 2. and f for lightweight 1 0 concrete.

Comparison of experimental data with the generalized analytic curve for normal weight concrete.

Comparison of experimental data with the generalized analytic curve for lightweight concrete.

Comparison of stress-strain curves for concrete from different sources.

Validation of the generalized curve up to a strain of 0.020.

Stress-strain curve of concrete.

Stress-strain curve of grade 60 steel.

Upwards shifting of neutral axial under loading for underreinforced beams.

Downwards shifting of neutral axial under loading for overreinforced beams.

5.5 Parameters a and S for stress block of concrete compression zone.

5.6

5.7

5.8a

5.8b

5.9

5.l0a

5.l0b

5.ll

5.12

Forces in rectangular beam.

Relationship of M-e:c '

M-e: relationship for rectangular beam. c

M-¢ relationship for rectangular beam.

Forces in T-beam.

M-e: relationship for T-beam. c

M-¢ relationship for T-beam.

Column under axial and flexural loads.

Column under uniaxially eccentric load.

ix

Page

96

97

98

102

104

106

107

ll2

ll2

113

113

116

ll7

ll7

120

121

122

124

125

126

126

Figure

5.13

5.14

5.15

5.16

5.17

5.18

5.19

5.20

5.21

5.22

5.23

5.24

5.25

5.26

6.1

6.2

6.3

6.4

6.5

6.6

6.7

6.8

6.9

Interaction P-M diagram.

Forces in uniaxially eccentrically loaded column.

Interaction P-M diagram of uniaxially loaded column.

Column cross section.

Neutral axial position.

Finite element mesh.

Determining the compression zone.

Numbering sequence in compression zone.

Strain distribution.

Concrete stress distribution

Concrete force acting on each element.

Iteration on position of neutral axis.

Iteration on inclination angle of neutral axis

Typical interaction diagram of biaxially loaded column.

Calculation of curvatures.

Typical cross sections.

Concrete stress-strain curve.

Stress-strain curve of grade 60 steel with f = 60 ksi. y

Comparison of ductility between normal and lightweight concrete.

Comparison of ductility between theoretical and ACI values for normal weight concrete.

Comparison of ductility between theoretical and ACI values for lightweight concrete.

Effect of compression reinforcement on ductility for normal weight concrete.

Effect of compression reinforcement on ductility for lightweight concrete.

x

Page

126

129

134

135

135

135

135

135

138

138

138

141

141

142

144

144

145

150

151

153

154

155

156

Figure

6.10

6.11

6.12

6.13

6.14

6.15

6.16

6.17

6.18

Effect of axial loads on ductility: M-¢ diagram for normal weight concrete. -

Effect of axial loads on ductility: M-¢ diagram for lightweight concrete.

Interaction diagram for normal weight concrete.

Interaction diagram for lightweight concrete.

Effect of axial loads on ductility: M-¢ diagram for e = 00 •

Effect of axial load on ductility: M-¢ diagram for e = 22.5 0 •

Effect of axial load on ductility: M-¢ diagram for e = 450 •

Effect of eccentricity angles on ductility: M-¢ diagram for f~ = 5000 psi.

Effect of eccentricity angles on ductility: M-¢ diagram f I = 9000 psi.

c

6.19 Effect of concrete strengths on column interaction P-M

6.20

6.21

6.22

diagram.

Comparison of theoretical and the ACI column interaction P-M diagram for f' = 5000 psi. c

Comparison of theoretical and the ACI column interaction P-M diagram for f' = 9000 psi.

c

Approximate straight lines for M-¢ diagram.

6.23 Calculation of mid-span deflection for beam loaded at mid-span.

6.24

7.1

7.2

7.3

Calculation of mid-span deflection for beam loaded at one-third span.

Stress-strain Curves of normal weight concrete with f' = 3,000 to 13,000 psi. c

Stress-strain curves of lightweight concrete with f' = 3,000 to 7,000 psi. c

Relationship of a, S vs. E for stress-strain curves of c normal weight concrete.

xi

Page

157

158

159

160

162

164

165

166

167

168

170

171

173

175

177

180

181

182

Figure

7.4 Relationship of a, S vs. s for stress-strain curves of c lightweight concrete.

xii

Page

183

7.Sa Moment-strain relationship for normal weight concrete with f' = 5,000 psi to 13,000 psi. 185

c

7.Sb Moment-curvature relationship for normal weight concrete with f' = 5,000 psi to 13,000 psi. 186 c

7.6a Moment-strain relationship for lightweight concrete with f' = 3,000 psi to 7,000 psi. 187 c

7.6b Moment-curvature relationship for lightweight concrete with f' = 3,000 psi to 7,000 psi. 188

c

7.7 Relationship of s vs. f' for normal weight concrete. 190 ~ c

7.8

7.9

7.10

7.lla

7.llb

7.12

7.13

7.14

7.15

7.16

7.17

Relationship of s vs. eu

Relationship of Sl vs.

f' for lightweight concrete. e

f' for normal weight concrete. c

Relationship of Sl vs. f~ for lightweight concrete.

Effect of compressive reinforcement on Mu vs. p relationship for normal weight concrete.

Comparison of the theoretical and ACI values of M vs. P u relationship for normal weight concrete.

Effect of compression reinforcement on M vs. p relationu ship for lightweight concrete.

Effect of compression reinforcement on ¢u vs. p relationship for normal weight concrete.

Effect of compressive reinforcement on ¢ vs. p relationu ship for lightweight concrete.

Comparison of ultimate curvatures from the theoretical ACI and proposed values.

Comparison of ductility factors between the theoretical and ACI values for normal weight concrete.

Comparison of ductility factors between the theoretical and ACI values for lightweight concrete.

191

193

194

197

198

199

200

201

206

207

208

Table

3.1

3.2

3.3

3.4

4.1

4.2

4.3

4.4

4.5

5.1

5.2

5.3

7.1

7.2

7.3

LIST OF TABLES

Characteristic Parameters of Grade 60 Steel

PCA Steel Curve for Steel A6l5 Grade 60

Regression Equations for the Characteristic Variables of the Stress-Strain Curve of A6l5 Grade 60 Steel from WJE

Characteristics and Constants of Equation for Generalized Stress-Strain Curves of Grade 60 Steel with Yield Strength of 60, 65, 70 and 75 ksi

Mix Proportions (lb/cu yd)

Characteristic Points of Normal Weight Concrete

Characteristic Points of Lightweight Concrete

Regression Equations for the Key Parameters of the StressStrain Curve

Constants of Equation for Generalized Stress-Strain Curves

Flowchart - Rectangular and T-Beam Under Pure Bending (Generate Beam M-¢ Curve

Flowchart - Uniaxially Loaded Column (Generate Column Interaction Diagram)

Flowchart - Biaxial1y Loaded Column (Generate Column Interaction Diagram for Given Angle of Eccentricity)

Moments, Curvatures and Ductility Factors at Yield and Ultimate for p = 0.005

Moments, Curvatures and Ductility Factors at Yield and Ultimate for p = 0.5 Pb

Moments, Curvatures and Ductility Factors at Yield and Ultimate for P = 0.75 Pb

xiii

Page

61

63

63

67

79

87

89

100

101

119

132

133

203

204

205

A,B,C,D

a

A c

A g

A s , A s

h

b w

c

C

d

d"

d'

e

E c

E 0

E s

f' c

f. 1

f2i

or b'

or f 0

xiv

NOTATION

parameters used in Y = CAX+BX2)jCl+CX+DX2)

depth of equivalent rectangular stress block, (= f\ c in ACI code)

area of concrete at the cross-section considered (depending on the particular case it may be the net area, the gross area or the transformed area)

gross area of concrete at the cross section considered

area of non-prestressed tension reinforcement

area of compression reinforcement

width of compression face of member

web width of a flanged member

distance from extreme compression fiber to neutral axis

resulting compressive force on the concrete section due to the prestressing force and applied external forces

distance from extreme compression fiber to centroid of tension reinforcement

distance from extreme compression fiber to centroid of compressive reinforcement

thickness of concrete cover measured from the extreme tension fiber to the centroid of tension reinforcement

eccentricity of applied external load or of the C force on the concrete section parallel to axis measured from the centroid of the section

modulus of elasticity of concrete

f jE ) secant modulus of elasticity at the peak stress 00·

modulus of elasticity of non-prestressed steel

compressive strength of concrete

stress at the inflection point of stress-strain curve of concrete

stress at E2i

f y

f' y

h

M

M u

p

p' u

t

T

u

X

y

Ct.

or P 0

xv

stress at arbitrary point after the inflection point

stress in the non-prestressed tensile reinforcement

stress in the compressive reinforcement

yield strength of non-prestressed tensile reinforcement

yield strength of compressive reinforcement

overall thickness or depth of member

flange thickness of a flanged member (t is also used)

ratio of depth of neutral axis to d

moment in general

bending moment capacity at balanced conditions, i.e., at simultaneous assumed ultimate strain of concrete and yielding of tension steel

applied ultimate moment at section cor.sidered

concentrated external load in general

axial load capacity at balanced conditions, i.e., at simultaneous assumed ultimate strain of concrete and yielding of tension steel

axial load capacity of compression member subject to pure compression (no moment)

thickness of the upper flange of flanged section (see also hf )

tensile load

used as subscript for "ultimate"

EIEo' strain in unit system

fifo' stress in unit system

parameter used related to the resultant force of stress block of compression zone of concrete

stress block of compression zone of concrete

E C

E cu

E. 1.

EO

e

11

p

p'

Pb

Pmin

Pmax

(J s

(J or f

¢

xvi

factor used to define the depth of the equivalent rectangular· stress block at ultimate as a function of the neutral axis. According to ACI:

, , 61 = 0.85 f for f < 4000 psi c c -

61 = 0.85 - 5xlO- 5 (f~ - 4000) for 4000 ~ f~ $ 8000 psi

61 = 0.65 for f~ ? 8000 psi

deflection in general

strain in general

concrete strain in general

concrete strain at ultimate on extreme compression fiber

strain at arbitrary point after th.e inflection point

strain at the inflection point of the stress-strain curve of concrete

2Ei-Eo' strain at twice the distance of the inflection pomt with respect to the peak point

strain at peak stress of the stress-strain curve of concrete

eccentricity angle for biaxial loading on column

¢u/¢y' ductility factor

As/bd, ratio of non-prestressed tension reinforcement

I As/bd, ratio of compression reinforcement

reinforcement ratio producing balanced condition

minimum specified reinforcement ratio

maximum specified reinforcement ratio

ratio of volume of spiral reinforcement to total volume of core

stress in general

curvature of section

1.1 GENERAL

CHAPTER I

INTRODUCTION

In analyzing the behavior of any structural member, it is essential to

have the accurate knowledge of the constitutive equations of the materials

concerned. In reinforced-concrete structures, the material properties of

the two major components, concrete and steel, can be best described by their

stress-strain relationships, in compression for concrete and in tension for

steel.

The stress-strain curve of concrete shows generally an ascending por

tion up to a maximum stress called peak stress, and a S-shaped descending

portion after the peak stress. The ascending portion can be considered

linear for all practical purposes up to about 45% of the peak stress, after

which it becomes nonlinear with a decreasing slope. The test data regarding

this portion can be obtained in any compression test machine. However,

little data on the complete stress-strain curve of concrete including the

descending portion, especially for high strengths, are available in existing

technical literature. This is because it is difficult to actually record

the descending portion of the curve. Many testing machines used for standard

compression test apply increasing loads rather than increasing deformations.

This results in an uncontrolled, sudden failure of the specimen immediately

after the peak load because of the release of energy from the testing

systems when the specimen is unloading. Even with the deformation controlled

loading, the release of the energy stored in the testing machine, when the

specimen is unloading, will influence the shape of the descending portion.

In this research, a simple technique was developed to obtain the stress-strain

curve of concrete up to a strain of 0.006. Higher strains can also be

achieved by slightly modifying the testing procedure.

2

The stress-strain curve of reinforcing steel of Grade 60 exhibits a

strain-hardening portion up to the peak load. For a reinforced-concrete

section with high percentage of compressive steel, the strain in the tensile

steel always reaches the strain-hardening portion at ultimate; therefore,

the accurate knowledge of the stress-strain relationship of reinforcing

steels including the strain-hardening portion is needed. Furthermore, once

the experimental stress-strain curves have been determined, an analytical

expression must be developed in order to be used in computerized models of

structural analysis.

In designing reinforced-concrete structures, there is a tendency to

design the structural members with sufficient ductility, especially in an

earthquake zone. Ductility implies the ability to sustain significant in

elastic deformations after the peak load without a significant variation

in the resisting capacity prior to collapse. The ductility of reinforced

concrete members can be improved by adding compressive steel in the concrete

section, or by confining the concrete compressive zone which leads to an

improvement in the ductility of the material. Note that improving the

ductility of the materials generally leads to an improvement in the ductility

of the reinforced-concrete section and/or member.

It is generally believed that higher strength concrete as a material

has less ductility than lower strength; however, in analyzing the load

moment-curvature-strain interaction diagrams for typical reinforced-concrete

sections, there is evidence that the same ductility factors, for the section

and/or the member, can be obtained for all strengths of concrete.

3

Although high strength concrete does not significantly improve the

moment capacity of beam sections under pure bending, it significantly im

proves the load capacity of column sections under combined flexural and

compressive loadings. A thorough and careful investigation of the effects

of using high strength concrete on the structural response of reinforced

concrete members should be undertaken. There is also a definite need to

carry out a detailed study of the current design approaches, such as the

ACI recommended stress block parameters at ultimate behavior, as to their

applicability to structures made with high strength reinforced concrete.

1.2 OBJECTIVE AND SCOPE

The main objectives of this investigation are:

(1) To develop, if necessary, and assess methods for obtaining the complete

stress-strain curve of concrete including the descending portion,

(2) To generate experimental information on the complete actual stress

strain curves of high strength normal weight and lightweight concretes,

(3) To evaluate existing analytical expressions for the stress-strain

curves of concrete, to check if they represent actual experimental

observations, to propose a new expression if others are found inade

quate,

(4) To propose an analytical expression for the complete stress-strain

curves of Grade 60 reinforcing bars which may show yield strengths

between 60 to 7S ksi,

4

(5) To formulate a computerized nonlinear analysis model for the analysis

of reinforced-concrete beams and columns under uniaxial and biaxial

bending,

(6) To evaluate the effects of using high strength concrete on the struc

tural response of reinforced-concrete members such as ultimate moment,

curvature, ductility and deflections, and

(7) To conduct a parametric evaluation of high strength reinforced-concrete

members in order to define appropriate values for the stress block

parameters of the concrete compression zone to be recommended for design

purposes.

The information from this investigation will provide a better under

standing of the characteristics and behavior of high strength concrete, and

its effects on the structural response of reinforced-concrete members. It

will allow the designer to use high strength concrete with more confidence

and take advantage of its properties when they are desired.

1.3 STRUCTURE OF THIS THESIS

Chapter II is devoted to study the existing analytic expressions for

the stress-strain curve of concrete, to analyze the characteristic points

of these curves, and to formulate an analytic relationship that is most

representative of the observed behavior of various concretes. Chapter III

is a similar study on reinforcing steels. Chapter IV describes a test pro

cedure for obtaining the complete stress-strain curve of concrete in

compression. The experimental program undertaken in this investigation is

also described in Chapter IV and test results are presented. Furthermore,

5

the observed characteristic points of the curve are linearly related to the

peak stress using regression analysis. Chapter V explains the nonlinear

computerized analysis technique developed for reinforced-concrete sections

under pure bending and bending combined with uniaxial and biaxial compres

sive loadings. Chapter VI explores the structural behavior of reinforced

concrete sections, namely, the ultimate moment, curvature, ductility and

deflection for various concrete strengths and amounts of reinforcement.

Chapter VIr is a quantitative study of the ultimate strength parameters of

the concrete compression zone to be used for the design of reinforced high

strength concrete strengths. Chapter VIII presents the summary of conclu

sions.

6

CHAPTER II

COMPLETE STRESS-STRAIN CURVE OF CONCRETE

2.1 REVIEW OF EXISTING STRESS-STRAIN CURVES OF CONCRETE

2.1.1 Introduction

A complete stress-strain curve consists of two portions: an

ascending portion before the peak, and a descending portion after the peak.

Several expressions have been proposed to describe the cr-s curve of con-

crete; unfortunately they have one or more shortcomings, which will be

pointed out in each reviewed expression. In order to review the existing

stress-strain expressions systematically, these expressions are grouped

into several categories. They include: polynomial functions, exponential

functions, fractional functions, and statistical distribution functions.

For each of these analytic expressions, the following items were investi-

gated:

(a) the boundary conditions, namely the peak stress, f, the o

strain at peak point, so' the secant modulus of elas-

ticity, Ec' at 0.45 f , the stresses and strains at the o

inflection point and another far point on the descending

portion.

(b) the shape of the curve over its entire range.

(c) the comparation of the analytic curve with experimental

data and its shortcomings if any.

2.1.2 Description of Mathematical Procedure

In order to review the various equations and their representative

curve, a system for describing each curve is set up as follows:

7

A. Unit stress-strain system.

Unit stress and unit strain are used to express the stress-strain

relationship. By this manner, the expression ensures itself a correct

dimension, and all the parameters involved are dimensionless. A typical

cr-E curve is shown in Fig. 2.1 and the following notations will be used

throughout this report:

f X E Y 0 = = -

Eo Eo

f E E

0 A c = - = 0 Eo E 0

where

f and S = stress and strain in general

f and S -' peak stress and corresponding strain 0 0

f t and Et = stress and strain at 0.45 f 0

f. and s. = stress and strain at inflection point J. J.

ff and Sf = stress and strain at far point on descending

portion which was arbitrarily selected

that Sf-E. = E. -s for this study J. J. 0

E = secant modulus of elasticity at 0.45 f c 0

B. Boundary conditions for ascending portion.

BCI - Curve passes through the peak point:

(e: = e: f = f ) 0' 0

or (X = = Y = 1) o

BC2 - At the peak point, the slope equal zero:

df\ = 0 de: (e: f) ,

0' 0

or dYI - 0 dX (1,1) -

such

(2.1)

8.0~,----------------------------------------------------------~

6.0

(/) ~ .. 4

.0

(/)

(/)

W

0:: ......

(f)

2.0

o

45

f-,-

• 0

I I I I I '

X=

-I

~

. E

I

,--f

j E

o I

1 I

I I

I I

I I

I I

I

I l

I '

I •

1 14

/I

~14

¥ )1

IT

r

I I

I I

I I

I . I

.,

1

I I

I Eo

.

lE

i. .

.:t.f

0.0

02

0

.00

3

0.0

04

0

.00

5

STR

AIN

Fig

. 2

.1

Ty

pic

al st

ress

-str

ain

cu

rve.

t y=

...L

fo

0.0

06

0.

007

00

BC3 - At the origin, the slope is equal to the initial modulus of

elasticity:

dfl _ E dE (0,0) -

or dY\ = A dX (0,0)

BC4 - Curve passes through the point at 0.45f : 0

(E 0.45f

= 0.45fo) Et

0 f f = = = or E t

(x Xt

0.45 Y Y

t 0.45) = = = = A

Note: E could be taken to be equal to Ec as per ACIo

C. Boundary conditions for descending portion.

BCl ( Same as for the_ascending portion to ensure continuity at the

BC2 ) peak point.

BCS - Curve passes through the inflection point:

(E = E., f = f.) ]. ].

or (X = X., Y = Y.) ]. ].

BC6 - Curve passes through a far point on the tail portion:

or

BC7 - Zero curvature at the inflection point:

d2

f\ = 0 dE2

(E.,f.) ]. ].

or = 0

2.1.3 Categories of Stress-Strain Curves

As mentioned earlier, the expressions to be reviewed have been

9

divided into four categories: exponential functions, polynomial functions,

10

fractional functions, and statistical distribution functions. For each

expression, the original equation was transferred to the unit stress-

strain system and the representative curve was plotted by using that

system; then the boundary conditions were checked to see if they are

satisfied.

A. Exponential functions.

1. Smith and Young [35] has proposed the following expression with two

parameters: fo and Eo

(1 - EIE ) f = f (E/E )e 0

o 0 (2.2)

which leads to the following dimensionless form (plotted in Fig. 2.2):

Y = Xe (I-X)

and for which

~~ = (I-X) e (I-X)

(X-2) e (I-X)

Checking boundary conditions:

BCI X = 1, Y = I -+ 1 - I identity

BC2 X = 1, dY = 0 -+ a - 0 identity dX

BC3 X = 0, dY = A -+ A = e = 2.71828 dX

Location of inflection point:

+ X. = 2 , J.

2 Y. = - = 0.73576

J. e

(2.3)

(2.4)

(2.5)

O.K.

O.K.

Constrained

lOI:------r------r--~~::==::~~~--~~-=~~::======~--~------,-------r---------

1_

_

CJ

__

__

_ I

·8

.(,

y t'4 ·2

Infl

ecti

on

po

inf .,

/

X=

2 ~

Y=

·73

6

Dr

, ,

, «

, ,

« ,

, ,

' ..

o ·2

.J

! .b

·8

1.

0 '·

2

I,~

l,tO

/.

8

2.0

2.

2->X

Fig

. 2

.2

Sm

ith

and

You

ng1s

ex

po

nen

tial

fu

nct

ion

. f-

' f-

'

12

It can be seen that this expression has the following characteristics:

It satisfies B.Cl and B.C2.

It leads to an initial modulus fixed by A = 2.71828 for all values

of f . o

It leads to an inflection point fixed by X = 2, Y = 0.73576 for all

values of f . o

B. Polynomial functions.

1. The European Concrete Committee has proposed the following expression

(second-degree polynomial) with two parameters: E and So

f = Es (1 - ~) (2. 6) 2so .


Y = A (X - ; X2) (2.7)

and for which

dY dX = A(l-X) (2.8)


BCl X = 1, Y = I -+ A = 2 Constrained

BC2 X = 1, dY = 0 -+ 0 - 0 identity dX O.K.

BC3 X = 0, dY = A dX -+ A - A identity O.K.


It satisfies BCl and BC2

It shows no inflection point

It leads to an initial modulus fixed by A = 2 for all values of f o

/.0,

"E

O

I --

........

... =

"'

--.....

, " ',,~European C

oncr

ete

Com

miH

ee

" Y

=2X

_X2

" ·8

, '\ '\

'\

., , \

Sa.e

n} T

hird

Def

Jree

fol

ynom

ia I

------

"\

y= A

X+

(3-2

A)

X2+

(A-2

) X

~ '"

y t ·4

A=

/.5

\ \ \ \ ,

\

·If

-b

-8

J.CJ

J-~

/.'1

2,2

>

)(

Fig

. 2

.3

Pol

ynom

ial

fun

ctio

ns.

~

CN

14

It is symmetric with respect to the peak point (it may eventually

lead to a negative stress for large strains).

2. Saenz [29] has proposed the following expression (third-degree poly-

nomial) with three parameters: E, Eo' and So

(2.9)

which lead to the following dimensionless form (plotted in Fig. 2.3):

Y = AX + (3-2A)X2 + (A-2)X3

and for which

dY 2 dX = A + (6-4A)X + (3A-6)X

-- = (6-4A) + (6A-12)X


BCl X = 1, Y = 1 + 1 - 1 identity O.K.

BC2 X = 1, dY = 0 + 0 - 0 dX

BC3 X = 0, dY = A + A :: A dX

Location of inflection point

o + x. 1

= 3-2A 6-3A '

identity O.K.

identity O.K.

= (2A-3) (A2 + 6A-18) Y. 2 1 27 (A-2)

(2.10)

(2.11)

(2.12)

(2.13)


It satisfies BCl, BC2, and BC3

15

It shows an inflection point fixed by X. = X. (A), Y. = Y. (A) which 1 1 1 1

may not exist depending upon the value of A.

It is asymmetric with respect to the inflection point, therefore

it may lead to either negative stresses or very large stresses for

large strains.

3. Sturman, Shah, and Winter [37] have proposed the following expressions

with. three parameters: AI' A2, and n

(2.16 )

which cannot be written in a dimensionless form before some conditions

are satisfied.


BCl e: = e: f = f 0 0

df + e: :;: e: -= 0

0 de: BC2

BC3 e: = 0 df E + , de: =

From Eqs. (2.17), it yields

E n = E-E '

o

f Al = (n~l) gO

0 (2.17) fo

A2 = (l-n)e:~

Al = E

(2.18)

After substituting Eqs. (2.17) into Eq. (2.14), it can be rewritten in

the following dimensionless form (plotted in Fig. 2.4).

Y = (--E-) X _ (_1 ) Xn n-1 n-1

(2.19)

("'"

/.0.

--

@ ~

·8 ."

y i .lj

·2

Fig

. 2

.4

St~rman,

Sha

h an

d W

inte

r's

curv

e.

I-'

C]\

17


It satisfies BCl, BC2, and BC3

It shows an inflection point at the origin; there is no inflection

point on the descending portion, its representative curve is con-

cave downwards for all positive strains, therefore it leads to

negative stresses for large strains.

The value of n can be determined by a least square analysis which

leads to: n = 1.9 for axial loading and n = 1.6 for eccentric

loading.

C. Fractional functions.

1. Desay and Krishnan [12] have proposed the fOllowing expression with two

parameters: E and Eo

EE f = ----

1 + (E/Eo) 2

(2.20)

which leads to the following dimensionless forms (plotted in Fig. 2.5):

AX Y = 1+X2

and for which

dY ACl-X2) dX = (1+X2)2

A( -2X) (3_X2)

Cl+X2) 3


BCl x = 1, Y = 1 -+ A = 2

BC2 dY X = 1, dX = 0 0_0

(2.21)

C2.22)

(2.23)

Constrained

identity O.K.

I. 0

I :s

:;::::

:::="

Ej

) :::

t:::::

:

·8

-b

Y 1-4

·2

·2

-4

.~

·8 Y=

2

X

'-f X

2

1.0

-~>X

Infl

ecti

on

pa

in!

X=I.

73~

Y=.

87

/.2

'·

4

'·6

Fig

. 2

.5

Des

ay a

nd

Kri

shn

an's

cu

rve.

1.8

2.0

2.2

......

00

BC3 dY X = 0, dX = A -+ A = A

Location of inflection point

o -+ X. = 1.73205 , 1

identity O.K.

Y. = 0.86603 1

19

It can be shown that this expression has the following characteristics:

Its initial modulus is fixed by A = 2 for all values of f . o

Its inflection point is fixed by X. = 1. 73205, Y. = 0.86603 for 1 1

all value of f . o

Furthermore, it satisfies BCl and BC2

2. Saenz [29] has proposed the following expression with three parameters:

E, . E , and e: o 0

(2.24 )


AX Y = ------~----1 + (A-2)X +X2

(2.25)

and for which

dY A(1-X2) dX = h + (A-2)X + X2}2

(2.26)

d2y 2A[X 3 - 3X - (A-2)] --- =

{l + (A-2) X + X2 } 3 dX2 (2.27)


BCl X = 1, Y = 1 -+ 1 - 1 identity O.K

BC2 X = 1, dY = ° dX -+ ° - 0 identity O.K

BC3 X = 0, dY = A dX

-+ A :: A identity O.K

1.0,

~., c::~

--"

~:;:;;

o-':O'

·8

'b

t.4

·2

//

;1 ';1

,f'

//

//

//

~

,1/

,/

.,

" "

".

,/'

·8

-"""~.-C I

nfle

ctio

n p

oin

f /

-.....,

, Sa

en} ~

'~"""

Y_

AX

'"

-.....

IT

(A

-2)X

t X

2 ......

. , '-

A=

I.5

/-_

'-P

opov

icS

"

y=

nx

(n-I

) -t

Xn

fl.:

:;

/. 0

/.2

1.

4 I. ~

1·8

2.0

2

·2

~

Fig

. 2

.6

Fra

cti

on

al

fun

ctio

ns

by P

op

ov

ics

and

Sae

nz.

N

o

21

Location of inflection point:

d2 y 0 -+- X3

- 3X - (A-2) 0 (2.28) --= = dX 2

for A = 1. 5 , X. = 1.64 , Y. = 0.86 1 1

for A = 2.0 , X. = 1.73 , Y. = 0.866 1 1


It satisfies BCl, BC2, and BC3.

Its inflection point is fixed by a value of A determined from

Eq. (2.28). This value of A simultaneously determines the

initial modulus.

3. Popovics [26] has proposed the following expression with three parameters:

n (2.29)


nX Y = -----

(n-l) + Xn

and for which

dY _ n(n-l) (l_Xn) dX - (n-l + Xn) 2

n-1 n (nX ) (-n-l + X )

(n-l + Xn) 3


BC 1 X = 1, Y = 1 -+- 1 == 1

(2.30)

(2.31)

(2.32)

identity O.K.

BC2 dY X = 1, dX = 0

BC3 dY X = 0, dX :: A

Location of inflection

d2y --= 0 -+ X. dx2 1

For example: A = 1.5,

0-0

A n :: n-l

point:

identity O.K.

A -+ n=--

A-I

log (n+ 1) = e n

n = 3, X. = 1.587, Y. 1 1

22

(2.33)

(2.34)

= 0.794


It satisfies BCl, BC2, and BC3.

Its inflection point is fixed by the value of n from Eq. (2.34)

and the value of n is directly related to the initial modulus.

4. Saenz [29] has proposed the following expression with six parameters:

fo' Eo' n, A, B, and C

f A(E/Eo)

fa = 1 + B(E/Eo) + C(E/Eo)n (2.35)


AX Y = ----

1+ BX + CXn

and for which

dY A + (l-n)ACXn

dX = (1+BX+CXn)2

d2y -2AB - 2ACXn- l { n (n+ 1) + (n+ 1) (n- 2) BX + n (l-n) CXn } --=

(2.36)

(2.37)

(2.38)

/.0 I

I :;J:

Iijj:>"

'" A~

::::

:!::

·8

"7n=Q

A

., fJ /

AX

A

Y =

I T

e

x 1-

C xn

A

A-

E

-Eo

11

-B

= A

-n

-I

·2 f-

A

_ J

C -

n-J

A A

A A

D

a.:fa

. po

in15

1

/1

0

'2

·4

.~

·8

,.0

/·2

/·

4

I. fa

/·8

2

.0

2.2

~

N

0-l

Fig

. 2

.7

Fra

ctio

nal

fu

nct

ion

wit

h s

ix p

aram

eter

s b

y S

aenz

.

24


BCl X = 1, Y = 1 I -+ B = A - (n/n-1) , C = l/n-l (2.39)

BC2 X = 1, dY = 0 dX

BC3 X = 0, dY = A -+ A - A identity O.K. dX

BCS X = Xd, Y = Yd (curve through arbitrary point on decending portion)

AXd

n ) -- X + n-l d

(2.40)

The solution of Eq. (2.40) can be obtained by trial and error. For

example A = 1.5, Xd = 1.4, Yd = 0.65, n = 7.021.


It satisfies BCl, BC2, BC3, and BCS.

The value of n can only be obtained by solving Eq. (2.40) by trial

and error.

Its inflection point is fixed by Eq. (2.38) for all values of f . o

Sa1se [30] proposed an expression which can be derived from Saenz's

equation after satisfying boundary condition.

2.1.4 Sununary

(l)" Most of the reviewed expressions have one or more of the following

shortcomings: a) they are based only on the properties of the

ascending portion [12,26,29,34,35]; b) they contain constants which

can be changed to fit a given curve, but these constants are not

based on any particular physical characteristic of the stress-

strain curve [30,37]; and c) the descending portion does not

always have an inflection point or it does not show a long tail

which is characteristic of concrete behavior [29].

25

(2) Generally, polynomial equations of order three or above exhibit

an upward tail which cannot characterize actual concrete behavior.

(3) Saenz's expression has one parameter, n, which must be determined

to fit the descending portion; however this parameter also affects

the ascending portion. Since the descending portion is very

important, one prescribed boundary condition does not seem

sufficient. This expression by Saenz seems to be the best so far,

however n has to be determined arbitrarily. More important, it

was not possible to fit his expression to our experimental data

(Fig. 2.7 shows one of our curves). This is because changing n

changes both ascending and descending portions in such a way that

the stiffer the ascending portion, the flatter the descending

portion, which appears contrary to the observation.

(4) It seems desirable that an equation be developed which will accommo

date independently the ascending and the descending portions of the

curve, i.e., independent parameters will be used to describe each

portion. Furthermore, for the descending portion, there should be

at least two boundary conditions prescribed.

2.2 PROPOSED ANALYTIC STRESS-STRAIN CURVE OF CONCRETE

In a preliminary investigation, several forms of equations were

tried to describe the stress-strain curve of concrete. For example, a

gamma distribution function was used; this distribution function leads

26

to a family of exponential equations, one of which is the same as the

equation proposed by Smith and Young [34]. It has. the advantage however

to allow for a variety of choices in the shape of the representative

curve as can be seen in Fig 2.8; a six-degree polynomial function satis-

fying six boundary conditions was also used, but the intrinsic property

of oscillation of polynomials between prescribed points leads to an un-

realistic description of the a-s curve of concrete in the descending

portion as shown in Fig. 2.9. Fractional functions were also tried, but

generally the poles of these .functions,if any, reduce the denominator to

zero which creates a discontinuity at that pole (Fig. 2.10); therefore,

such functions cannot be used to describe the entire stress-strain curve

of concrete without special precautions; for example, they can be made to

represent one portion of the curve only as shown in the following section.

After an extensive search for a suitable expression, the following

fractional stress-strain relations were found to be most acceptable:

and

Y = AX + BX2

1 + ex + DX2

AX + BX. 2

Y = -------1 + ex + DX2 + HX 3

where A, B, e, D, and H = constants to be determined.

(2.41)

(2.42)

The accuracy of these expressions in representing the stress-strain

curve was much improved when two separate sets of values of constants were

used for the ascending and the descending portions. For the ascending

portion, Eq. (2.41) was used and the values of the four constants were

evaluated from the four boundary conditions described in Section 1.1.2.

For the descending portion, Eqs. (2.41) and (2.42) were used and the

27

~ IQJ r'\

~ · ~ I -...J

~:::J 0

t.... .r-! +..l u

r< ~

tB " ~

"...... 0 ~

.r-!

-....J +..l ;:j

~ ..0 'r-!

::J 1-; .f-I

i til

'r-! "0

cd

~ cd '-'

CO · N

· CO 'r-! (.I..,

'" X ~

+ 1.0

>< u.. + ~ X 0 + co ><. U

~; cD + X « II

>-

........

.t; -~ '-' -ID--!!l N" ---..---'"

• '-J . cO

O'i> s: '-:r')

4-0 en ~ d

CIl

N N

..,g ,

~ ..:.

N ..:.

0 ..:.

00

'-.9

o~----~----~~----~----~----~o '" 00 '? :::r

>-~(-

28

~

cd .r-!

~ ~ ~

a p..

(!) (!) $-I OJ) (!)

X "0

..c:

i +J ><

.r-! Cf)

O'l . N

. OJ)

'r-! ~

I. 0

I _

__

EH

:=

::::::

::

'8

'b

y i .lj

'2

·2

./....

(

y= A

><+

8X

2+

CX3

''''

DX

t

H X

2

A=2

.. 8

= -

2,/

73

6{;

,1

C::

0

.57

84

31

D

= -

o.57

8l13

/ /-1

= -

0.0

16

807

Spec

.' r~ed

po

iht -

?o

'b

·8

/.0

/.

2

1·4

~X

J'b

Fig

. 2

.10

F

ract

ion

al

fun

ctio

n w

ith

a p

ole

.

Pole

Spet

ifl'ed

Po

,Ont

/

l.O

2

,2

N ~

30

values of the constants were evaluated from the boundary conditions

described in Section 2.1.2. After comparing the two expression Eq~ (2.41)

was finally selected because it was simpler and led to about the same

accuracy as explained in Section

2.2.1 Determination of the Four Constants for the Ascending Portion

The equation and its derivative are expressed as follows:

Y = AX+BX2 1+CX+DX2

dY (A+2BX)(1+CX+DX2) - (AX+BX2)(C+2DX) dX = (1+CX+DX2) 2

Checking boundary condition:

BCI X = 1, Y = 1 -7 A+B = l+C+D

BC2 X = 1, dY = 0 dX

-7 (A+2B) (l+C+D) = (A+B) (C+2D)

BC3 X = 0, dY =A -7 A - A identity dX 2 AXt + BXt

BC4 X = Xt , Y = Y -7 Yt = t 2 l+CXt + DXt

From Eqs. (2.44) and (2.45)

B = 0 1

C = A 2

From Eqs. (2.47) and (2.46)

where

o = (-A). + [Yt - 2Xt Yt +X~] Xt X2(1_Y )

t t

X = 0.45 t A Yt = 0.45

(2.41)

(2.43)

(2.44)

(2.45)

(2.46)

(2.47)

(2.48)

Some typical ascending portions, represented by this expression, are shown

in Fig. 2.11.

I': I: ii'

',.

ii"

."'i~

10

,.,

·"·i'·'· Ii

, ,

[ L--~~

','I'

, ,

I

·1,'

.1

Fig

. 2

.11

: ! \

!

~'Xi

Typ

ical

as

cen

din

g p

ort

ion

.

. I··

, I

' I;

,

.. !

VI .....

32

2.2.2 Determination of the Constants for the Descending Portion

Three alternates for representing the descending portion are dis-

cussed in the following sections:

1. Alternate 1 - The curve is constrained to pass through two points,

namely the inflection point and a far point on the tail portion of the

curve, but the zero curvature condition is not prescribed at the

inflection point. This approach is the one finally selected in this

investigation for the analysis of experimental results. The equation

and its derivatives are expressed as follows:

or

Y =

dY dX -

AX+BX2

1+CX+DX2

(A+2BX)(1+CX+DX2) - CAX+BX2)(C+2DX)

(l+CX+DX2) 2

dY A+2BX + (BC-AD) X2 dX = (1+CX+DX2) 2

d2y -2{(CA-B) + (BC+3AD)X + 4BCX2 + D(BC-AD)X 3 }

dX2 = (1 +CX+DX2) 3


BCl X = 1, Y = 1

\ B = D 1 +

Be2 dY = a C = A 2 X = 1, dX

BCS X X. , Y Y. A c l b2 - c2b1

= = = l. l. al b2 - a2bl

+

BC6 X Xf ' Y Yf D a l c2 - a2c1

= = = al b2 - a2bl

(2.41)

(2.43)

(2.49)

(2.47)

(2.50)

where

a l = X. - X. Y. , 111

bl =X7-X7Y., 111

(X.,Y.) = inflection point 1 1

c l = Y. - 2X. Y. + X7 111 1

(Xf , Yf ) = arbitrary far point on tail portion (chosen

in this study such that Xf - X. = X. - X == X. - 1) 110 1

33

(2.51)

Two typical descending portions represented by this expression are

shown in Fig. 2.12 for an arbitrary far point for which Xf = 2.

2. Alternate 2 - The curve is constrained to pass through the inflection

point with zero curvature prescribed at that point. The equation and

its derivatives are the same as that for Alternate 1.


BCl X = 1, Y = 1 I B = D 1 +

X 1, dY = 0 C = A 2 = dX

(2.47)

BC2

X = X., Y = Y. + A = qD+P 1 1

RC5

Y.-2X.Y.+X~ (2.52)

where X. , P 1 1 1 1 q = =

1 X. - X. Y. 1 1 1

X X. d2 y

0 = --= + 1 dx2

BC7 (2.53)

Substituting Eq. (2.47) into Eq. (2.53) leads to a quadratic equation

for D as follows:

Fig

. 2

.12

D

esce

ndin

g p

ort

ion

th

rou

gh

tw

o p

oin

ts.

'l

v:.

.j::,.

35

mD2 + nD + Q, = 0 (2.54)

and the two solutions are

D -n + Ini: - 4mQ,

first solution = 2m

or (2.55)

D -n - In2 - 4mQ, .second solution = 2m

where

n = -2X~+3X7+l+2q+p(X~-3X.)-2pq 1. 1. 1. 1.

Q, = -1 + 2p _ P 2 (2.56)

q = -X. l.

Y. - 2X. Y. + X7 1. 1. l. l.

P = X. - X. Y. l. l. l.

The first solution of the quadratic equation for D yields a lower

bound solution, and is selected in this expression. The curves for

both solutions are shown in Fig. 2.13. It can be seen that these two

solutions are very close.

3. Alternate 3 The curve is constrained to pass through two points,

namely the inflection point and a far point on the tail portion the

curve, and the zero curvature condition is prescribed at the inflec-

tion pOint. Its equation has one more constant as compared to

Alternate 1 or 2. The equation and its derivatives are expressed

as follows:

Y = C~. 42)

i' ..

'.

I i

I I

'

,',

.. ,

....

i.

I ..

! [!

. I 'J .iAXtBX~!

Y i

.' I:

'1"III~XtDX~'

! : .

~ :

" \ '

'\

! .

!' \

. ~, .

i

II I:'

;.

'I I·

I .

~I' .

: ' ,

. '.

i'::

.:

i· i"

,

i '.vvnfle~tJ'ol1, p

oi~-

t I, !.

·1.

I ,

/I/i

th S

Pfc

ifie

d,

'I i

. !

~ ize

roc~

rv:a

t:ur

e I!

'<I~

" I

• .

! I

! !:

[ i

,

Fig

. 2

.13

D

esce

ndin

g p

ort

ion

th

rou

gh

in

flecti

on

po

int.

(.N

0

\

or

dY (A+2BX) (1+CX+DX2+HX 3) - CAX+BX2) CC+2DX+3HX2)

dX = (I+CX+DX2+HX 3)

dY dX =

A+2BX+CBC-AD)X2 - 2AHX 3 - BHX4

(I+CX+DX2+HX3) 2

the equation for satisfying zero curvature, i. e.

{B+(BC-AD)X - 3AHX2 - 2BHX 3} (1+CX+DX2+HX 3)

- (C+2DX+3HX2){A+2BX+(BC-AD)X2_2AHX3_BHX4} = 0


BCI X = 1, Y = 1

BC2 X 1, dY

0 = dX -

BC5 X = X. , Y = Y. 1 1

BC6 X = Xf , Y = Yf

BC7 X X. , d 2y

0 = --= 1 dX2

37

(2.57)

0, is

(2.58)

By using boundary conditions, the results obtained are summarized as

follows:

A = t + vH

B=m+nH

C = f + gH

D = P + qH

and

aH3 + bH2 + cH + d = a

where

t :: - X.p + r. , 1 1

m=p-l,

v=s.-X.q 1 1

n = q + 2

(2.59)

(2.60)

f = t-2 g = v+l ,

r i - r f si - sf P = , q =

Xi - Xf Xi - Xf

x: - 2X. Y. + Y. 2 1 1 1 1

Xf - 2Xf Yf + Yf r. = r f = 1 X· - X· y. Xf - XfYf 111

Y. +X~Y. - 2X. Yf + X~Yf - 2Xf 1 1 1 1 s. = Sf = 1 1 - Y. 1 - Yf 1

a = X~(-ngq+vq2+2vg)+X~(2ng)+X~(-3V-3ng+3\)q)+X~C-2n+6V)+X~(3n)

b = xi (-mgq-fnq-png+2pqv+tq2+2tg+2vf) +2Xlf (mg+nf)

+3X~(-t-mg-nf+tq+vp)+Xr(-2m+6t)+3x?m-(\)g) 1 . 1

-3xvq-3X~(\)+nq)-Xr(n+3\))-6nX~

c = X:(-mfg-fpn-mpg+2pqt+Vp2+2tf)+2mfX~+3X~(-mf+tp) 1 . 1 1

-(tg+vf)-3X. (tq+vp)-3X~(t+mq+np)-X:(m+3t)-6mX~+n . 1 1 1 1

d = X~(-mfp+tp2)_tf-3tpX.-3mpX~+m 111

38

The solutions of the cubic Eq. (2.60) yields three roots for H; in most

cases, one real root and two imaginary roots are obtained; if however

three real roots are encountered one of them should be selected to

better fit the data. Typical curves are shown in Fig. 2.14. It can be

seen that when three real roots are obtained, the representative curves

are very close to each other.

!:'

i

: Ii

i "i

1'1

I

,

11

-'!

1

f \

(Cci.

~e o

f one

I re

a./

root

if

or

, ru

bie'

~9l/

atio

n I

of(

H:

I :

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/,2

1.1

1

j !

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yd !A

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: i

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'

i;/ fC

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: ' ,,

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i '!

: iii

I':

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C~, se,

q,f ,t

hree,)'

iii'

real

roo

ts' ,

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i' i

: ,

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: j

,

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Fig

. 2

.14

D

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th

rou

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on

and

arb

itra

ry p

oin

ts.

tN

\D

40

2.2.3 Comparison of the Three Alternatives Used to Describe the Descending

Portion of the Curve

Three different approaches were described in the preceding section

to determine the parameters of the descending portion. The first two

required four parameters while the third one required five parameters. In

order to select the most appropriate alternative, the three generated curves

were compared to actual stress-strain relationships. In Figs. 2.lSa and

2.lSb,Alternates 1 and 2 were compared, and it can be seen that Alternate 1

leads to a better fit. Also Alternates 1 and 3 are very comparable and

lead to about the same fit. Therefore, Alternate 1 was finally selected

for the following reasons:

(1) It is superior to Alternate 2.

(2) The result in fitting the data is as good as Alternate 3.

(3) The number of parameters is less than that of Alternate 3.

(4) It has the same form as the equation representing the ascending

portion.

I.·.; 8

. ·1

'-:1

I

'\

·4··

,.1 ':'i'

,: If> I

'.'! 15

.... V

;I

..lIi:: . ~4

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, i

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. 2

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a C

ompa

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n be

twee

n alt

ern

ate

s 1

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nd

3 fo

r d

esce

nd

ing

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n.

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2.

y::

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2

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: ;

,-.cr" ""

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: ....

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Fig

. 2.

1Sb

C

ompa

riso

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etw

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3 fo

r d

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nd

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.f::.

N

43

CHAPTER III

STRESS-STRAIN CURVES OF REINFORCING STEEL

3.1 INTRODUCTION

Reinforcing steels, used in reinforced concrete structures, will be

classified here into two categories as regard to their 0-€ curves: the

first with a definite yield plateau, and the other with no definite yield

point. For the first category, the appearance of the 0-€ curve over its

entire range up to failure will be described as follows:

(a) an initial elastic portion up to the yield point,

(b) a yield plateau up to the strain-hardening portion,

(c) a strain-hardening portion.

For the second category, the 0-E curve consists essentially of two por

tions:

(a) an approximately linear portion up to the proportional limit

which cannot generally be identified except as specified in

codes of practice such as 0.2% offset,

(b) a strain-hardening portion.

A third category of steel, which will not be described here, consists of

prestressing steels for which the 0-€ curve shows three portions:

(a) an initial elastic portion up to the proportional limit,

(b) a curvilinear portion up to about 95% of ultimate st.rength,

(c) an asymptotic straight-line portion up to ultimate.

The main purpose of this chapter is to find a suitable analytic

expression for the 0-E curves of the first two groups of reinforcing

steels described above. A discussion of the available expression is

also included.

44

3.2 STRESS-STRAIN CURVE OF MILD STEEL WITH DEFINITE YIELD PLATEAU

Various expressions to represent the O-E curve of this category of

steel have been proposed. The ACI (318-71) proposed a design curve con-

sisting of only two portions: an initial elastic portion and a yield

plateau, assuming that in the design the steel does not reach the strain-

hardening stage. Heimdahl and Bianchini [18] and Bresler [4] both proposed

a curve consisting of three portions: an initial elastic portion, a yield

plateau, and a strain-hardening portion represented by a straight-line up

to ultimate. Their curves are shown in Fig. 3.1 for A6l5 Grade 60 steel.

Sargin [32] proposed a curve consisting of three portions: an

initial elastic portion, a yield plateau, and a strain-hardening portion

up to failure which was represented by a parabola. This expression is

more representative of the real behavior of this category of steels, and

will be discussed in detail next.

3.2.1 Sargin's O-E Curve for Mild Steel with Definite Yield Plateau

The proposed expression has five parameters: Es' Esh ' fy' f su ' and

Esh ' which are identified in Fig. 3.2 where a typical O-E curve is plotted.

The three portions of the curve are given by the following equations:

f = E E for 0 < E ~ E (3.1) s s s s Y

f = f for· E < E < Esh (3.2) s y y s

fs = f + E h (E - E h) [ _ E sh ( S s - E sh) ]

for E >E (3.3) y s s s 1 4 (f - f ) s sh su Y

The first derivative of Eq. (3.3) leads to:

(3.4)

"i'+

. :

60

, .... "

~I:t

1~t1,

... rr:

I:'

i ", '~P'

~':'

I~:::

t I.

,

1~40

I.E i::

, V

'\.!

";;.

11-.,

l' ,I

.

I 'I'~;

: \.

, ,;

1

1::

:

1

Fig

. 3

.1

Bre

sler

and

H

eim

dah

1's

stress

-str

ain

cu

rve

for

grad

e 60

st

eel.

·1)~iJ

t·

;

oj::.

ti

l

[~T:

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:1:1

: :l

l:iii

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:

,"I":~'

1 .

.•

• I'

ii, I

. /

' ..

'"

,I

, , !,

I I

r, ..

,

, '.

I

! I

I .1

,

i I'

i

:! !

I'

• !

:

1-

J , ,I I' ~

j .

' "I 'I

I,

I

I /

fro.~t

ur-e

V,

!"--

-fS

f

, ;~sr

:i j

I !

Fig

. 3

.2

Sa

rgin

's s

tress

-str

ain

cu

rve

for

grad

e 60

st

eel.

i I.·." i I !

I

I ~, 1

'I : '1 "

1ztii

l'ij;

! : ~ !

: <

~:. '

I,

.j::o

. 0

\

47

and the strain at the peak point is given by equating Eq. (3.4) to zero,

which leads to:

E = E + su sh

2 (f - f ) su Y

(3.5)


(a) the strain-hardening portion is represented by a parabola

which is symmetric to the peak point,

(b) the value of strain at the peak point is determined by

Eq. (3.5).

3.2.2 Proposed Expression to Represent the Strain-Hardening Portion for Mild Steel with Definite Yield Plateau

The parabolic curve for representing the strain-hardening portion

proposed by Sargin [32] does not seem, as will be shown latter, to properly

fit the observed experimental data after the peak load. Although we are

seldom interested in reinforcing bars in the strain beyond the peak, and

although the ultimate strength of concrete sections always occur at a

tensile strain of steel less than the strain at the peak, it may be impor-

tant to know more exactly their behavior up to the exact failure point;

the failure point is a characteristic point of the complete stress-strain

curve of steel and can be obtained in a tensile test of reinforcing bars;

therefore, the following expression is proposed when such need arises

Y = AX + BX2

1 + ex + DX2 (3.6)

Its derivative is given by:

dY (A + 2BX) (1 + ex + DX2) - (AX + BX2) (C + 2DX)

dX = (1 + CX + DX2) 2 (3.7)

where

x =

Y =

E - E h s S

E o

f - f s Y

f o

f and E = stress and strain in general (Fig. 3.3) s s

f and E = peak stress su su

fsf and Esf = stress and

Esh and E = strains at su peak point

E = Young's modulus s

and corresponding strain

strain at failure point

onset of strain-hardening

Esh = strain-hardening modulus

A,B,C,D = constants

E = E - E 0 su sh

f = f - f 0 su Y

E = f IE 0 o 0

A = E hiE 5 0

and

48

(3.8)

The curve is constrained to satisfy the following four boundary conditions.

BC1 Curve passes through the peak point

or ex = x - 1, Y = Y - 1) o 0

BC2 At the peak point, the slope equals zero

(:E = E dfs = 0)

s su' dE S

or (x = X 0 - 1, ~~ = 0)

BC3 At the onset of strain-hardening, the slope is equal to the

strain-hardening modulus

(Es = df

ESh) (x dY A)

s Xsh ' Esh ' --= or = dX = dE

S

'fl:::

I:::L

:: I',;

ii,

",i

"'"

I

"~, 120

i'1

1; ,I

'l:\

!r~+

i,

ri","

'1 ',',

\:

': :'1 ; <

i4 ,::

'\i,;"

i' ':

lloo

, '

tiill'"

iili!~:'i

',I

'"

.1 1

'-'

Fig

. 3

.3

f:'

Com

pari

son

bet

wee

n S

arg

in's

and

re

vis

ed f

orm

ula

s fo

r re

pre

sen

tin

g t

he

stress

an

d st

rain

cu

rve

of

grad

e 60

st

eel

up

to

a st

rain

of

0.2

00

.

, , !

.p..

\0

50

BC4 Curve passes through the failure point

(€ = E f' f = f f) s s s s or

By using the boundary conditions, the four constants are determined as

follows:

E A .sh =E

o

B = D - 1

c = A - 2

D = Ysf + (A - 2)Xsf Ysf - A\f + X~f

X~f - X~f Ysf

(3.9)

where

Y = sf

f - f sf y f

o

The curve is plotted in Fig. 3.3 as well as that proposed by Sargin. It can

be seen that this new expression can represent more accurately the observed

data.

3.3 STRESS-STRAIN CURVE OF STEEL WITHOUT DEFINITE YIELD POINT SUCH AS HIGH STRENGTH REINFORCING BARS

A typical stress-strain curve (Re~ 39) for a reinforcing bar in this

category is shown in Fig. 3.4. It can be seen that in the strain-hardening

portion there are an ascending part and a descending part, and that a high

value of strain (0.12) can be reached. Several stress-strain equations

have been proposed to represent the experimental behavior of these steels.

They are discussed next.

i,

i .-

j

Fig

. 3

.4

I 1£5

i

I: ,

I

"i I I I . ~

..

[ . 'i I

'.··.··1

£.,· ..

-;--

'Jsf

i

!

Typ

ical

st

ress

-str

ain

cu

rve

of

steel

wit

hou

t d

efi

nit

e y

ield

po

int.

'I ;\

U1

I-'

52

3.3.1 Sargin's Equation

Sargin [32] has proposed the following two-part stress-strain relation-

ship in which the nonlinear part is the inverse of the Ramberg-Osgood

polynomial [27]; the expression is as follows:

where

and

f = E e: for O<e: <e: s s s

f = f + (f - f ) Z s e su e (1 + Zv) l/v

Z = (e: - e: )E s e s

f - f su e

f = ultimate strength su

for

f ,e: = stress and strain at elastic limit e e

E = Young's modulus . s

v = constant

- s - e (3.10)

(3.11)

(3.12)

By translating the coordinate system as shown in Fig. 3.5, Eqs. (3.11) and

(3.12) can be rewritten as follows:

Y = Y Z o (1 + Zv) l/v

(3.13)

E X Z = ; (3.14)

o

The first-order derivatives are as follows:

dY E s (3.15) -= dX (1 + Zv) l/v

dZ E s (3.16) dX = y-o

where

X = e: - e: s e

Y = f - f s e

Y = f - f 0 su e

/1 ;{

Fig

. 3

.5

i I

·1

:1

: ,I

i.

i I

i, i

.00

6

'r

t I

' i, i

. .!

~. , 1

I,,:

,I·,'"

!,

i, '

I ,.

1,:

\

('!

" 'i

sl!

.. ~

.4

X i i i

i I I

.~ '.

{ X.

)( i "

1: I

I I • pe

tta..

i ,j

. i

, l'

,j

! I

!.008

:' i

~OIO

----+

) ~ fr

ai n

\

, t'

l i I i I

i

i

Sar

gin

cu

rve

for

hig

h s

tren

gth

rei

nfo

rcin

g b

ar.

, i'

, '

I

01

0.

l

54

The curve is constrained to satisfy the following two boundary conditions:

BCl At the point of the proportional limit (s ,f ), the slope e e is equal to Young's modulus, E

s

(

dfS ds

s or

BC2 Curve passes through any specified point (Ssl,fsl )

(S = S l' f = f 1) s s s s or (X = Xl == Ssl - Se'

Y=Y ==f -f) 1 sl e

Checking boundary conditions to determine v

BCI E = E s s identity

BC2 Y 1 Zl

Yo = (1 + ZI) l/v

and

From Eq. (3.18)

~ log (1 + 2~) = log 21 + log C~)

(3.17)

(3.18)

(3.19)

(3.20)

The constant v is determined by trial and error from Eq. (3.20); for

example, given

E = 27000 ksi, f = °115 ksi, f = 52 ksi, s su e

Se = 0.002, fsl = 105 ksi, ssl = 0.012

then

Yo = 63 ksi, Yl - 53 ksi, Xl = 0.010, Zl = 4.28571

and

v = 1. 085

55

The curve for the above example is plotted in Fig. 3.5. Its charac-

teristic points are taken from the data of Heimdahl and Bianchini [18].

The following results are drawn:

For the range of strain considered (up to 0.012), the fit is

quite good.

The curve is monotonously increasing, i.e., E = 0.2, s

f = 114.5 ksi and E = 00, f = f = 115 ksi; so, there s s s su

~s no peak point.

For the entire range of the strain (up to 0.200), as shown

in Fig. 3.4, this expression cannot predict the real stress-

strain behavior of steel, since there is no descending

portion.

3.3.2 Goldberg and Richard's Equation

Goldberg and Richard [16] have proposed the following expression:

where N

and

E E f s s =

[1 + e:::n l/N s

can be determined from

N = log 2

log r:;) f ,E = stress and strain in the steel s s

f = ultimate strength su

fy = yield strength

E = Young's modulus s

(3.21 )

(3.22)

Equation (3.21) can be rewritten as follows:

where

f = f s su

E E S S

Z = -f-su

Z

The first derivatives of Eqs. (3.23) and (3.24) are given by:

df E s s --= (1 + ZV) (1 + 1/v) dE s

dZ E S

dE = f s su

56

(3.23)

(3.24 )

(3.25)

(3.26)

It can be seen that Eqs. (3.23), (3.24), (3.25), and (3.26) are similar to

Eqs. (3.13), (3.14), (3.15), and (3.16) given by Sargin in the preceding

section.

The value of N in Eq. (3.21) is determined from Eq. (3.22); N can

also be determined independently by satisfying the following two boundary

conditions:

BC1 At the origin, the slope is equal to

( ::s = Es) s (0,0)

E , s

the Young's modulus

BC2 Curve passes through a specified point such as (Es1 ,fs1 )

(E = E l' f = f 1) s s s 5

Checking the boundary conditions to determine N:

BCl E = E identity s s

BC2 fsl f Zl

(3.27) = [1 + Z~] liN su

57

where E E

2 = s sl 1 fs1

(3.28)

Equation (3.27) can be rewritten as:

1 N su (

f ) N log (1 + 21) = log 21 + log fs1 (3.29)

The constant N is then determined by trial and error from Eq. (3.29); for

example, given

then

Es = 27000 ksi, f = l15 ksi, su

N = 1. 7165

N = 1. 7169

from Eq. (3.29)

from Eq. (3.22)

fSl = 105 ksi, Es1 = 0.012

The curve for the above example is plotted in Fig. 3.6. Its charac-

teristic points are taken from the data of Heimdah1 and Bianchini [18].

The following results are drawn:

For the range of strain considered (up to 0.012), the fit is

good, since there is only one expression used in comparison

to the two-portion curve proposed by Sargin in the preceding

section.

The curve is monotonously increasing, i.e., E = 0.2, s

f = 114.9 ksi and E = 00, f = f = 115 ksi; so, there is s s S su

no peak point.

For the entire range of the strain (up to 0.200), as shown

in Fig. 3.4, this expression cannot predict the real stress-

strain behavior of the steel since there is no descending

portion.

.+

.O(J

b·

r'

! '"!

Fig

. 3

.6

Gol

dber

g an

d R

ich

ard

's

curv

e fo

r h

igh

str

en

gth

rein

forc

ing

bars

, U

1 0

0

59

3.3.3 Proposed Expression to Represent the Entire Stress-Strain Relation Up to a Strain of 0.200 for Steel without a Definite Yield Point Such as High Strength Reinforcing Bars

The conclusion from the preceding two sections shows that both Sargin' s

expression and Goldberg's expression cannot predict well the typical curve

over its entire range, as shown in Fig. 3.4, for this category of steel.

The proposed expression consists of two parts as follows:

(a) Initial elastic portion

f = E E S S S

for 0 < E < E S Y (3.30)

(b) Strain-hardening portion

AX + BX2 Y =

1 + CX + DX2 (3.31)

The expression for the strain-hardening portion has the same equation

as that in Section 2.2.2 for mild steel with a definite yield point. The

constants A, B, C, D and the boundary conditions are determined by the

same manner described in that section, and will not be repeated here.

The curve represented by Eqs. (3.30) and (3.31) has the following

characteristics:

The representative curve is an idealized curve, so there is a

gap existing between the analytical curve and the actual data

points near the yield point, as shown in Fig. 3.4. Consequently,

the discontinuity in the slope at the yield point is a result

of that idealization.

To improve the fit on the entire strain-hardening portion, the

values of fy and Esh can be adjusted.

60

3.4 ANALYSIS OF AVAILABLE EXPERIMENTAL DATA

The data used here for analysis were obtained from Weiss, Janney,

Elstner and Associates, Inc. and was submitted in a report to the Technical

Committee of the Associated Reinforcing Bar Producers [41]. The testing

variables are as follows:

(a) Type of steel: A6l5 Grade 60

(b) Bar size: #3, #4, #5, #8, #11 (from various manufacturers)

(c) Strain measurement: ASTM Standard

3.4.1 Synthesis of Results

About two hundred specimens of reinforcing bars from different manu-

facturers were tested and recorded for various bar sizes. In order to see

the trend in the variability of the data, 46 specimens of bar sizes from

#3 to #11 were studied. Two categories of stress-strain curves were

observed from these data: one with a yield plateau, one without a definite

yield point. For the first category, a typical stress-strain curve for

Specimen No.9 from Table 3.1 isshown in Fig. 3.7; for the second category,

a typical stress-strain curve for Specimen No. 44 is shown in Fig. 3.8.

The characteristic variables of the data are summarized in Tables 3.1 and3.2

for these specimens, and are explained below:

f = yield strength y

f s = stress and strain at the peak point su' su

fsf' ssf = stress and strain at the fai lure point

Esh = strain at the onset of strain-hardening

Es = Young's modulus

Esh = strain-hardening modulus

61

TABLE 3.1


Wiss, Janney, E1stner & Associates, Inc.

Specimen Code f f fsf Esh E Esf Es Esh y su su No. No. ksi ksi ksi 0.0001 0.0001 0.0001 10 3 ksi 10 3 ksi

1 110503 70 120.0 105 62 600 1400 29.4 1.07 2 10301 74 114.0 106 46 600 787 27.9 1.50 3 10303 65 91. 0 76 125 687 1470 28.6 0.76 4 20501 64 110.0 100 75 700 1330 29.4 1. 36 5 20502 67 114.0 106 87 870 1340 29.1 1. 36 6 20503 67 117.0 106 40 725 1180 29.1 1.50 7 20504 69 116.0 104 57 500 1120 29.9 1. 25 8 20505 68 114.0 105 120 912 1490 29.2 1.12 9 30501 65 109.0 96 85 750 1190 28.9 1.45

10 30502 73 122.0 96 28 687 1150 28.6 1.50 11 40401 68 97.0 78 100 625 1400 29.0 0.64 12 40503 72 100.0 81 100 662 1710 29.8 0.72 13 40504 71 98.0 85 120 725 1710 29.0 0.72 14 50503 72 97.0 88 125 750 1520 29.2 0.64 15 50504 74 105.0 90 90 750 1210 29.9 0.80 16 60401 69 114.0 92 60 780 1400 30.1 1.32 17 60504 64 106.0 84 80 687 1590 29.6 1. 28 18 60505 72 117.0 99 65 725 1150 29.5 1. 28 19 60506 68 117.0 102 80 687 1090 30.2 1.44 20 70302 74 120.0 108 65 820 1190 28.3 1. 28 21 80501 62 108.0 89 85 635 1610 28.6 1. 28 22 80503 62 108.0 90 75 650 1410 29.4 1. 28 23 80505 62 109.0 92 60 660 1460 28.6 1. 28 24 110501 64 114.0 101 75 625 1490 29.6 1. 28 25 110502 64 115.0 105 65 620 1370 30.1 1.44 26 130401 68 107.0 98 103 580 1130 28.1 1.44 27 130402 68 108.0 98 75 662 1320 28.9 1. 04 28 180401 68 118.0 109 70 650 1070 29.8 1.48 29 200501 68 102.0 86 75 625 1770 30.0 0.88

30 170803 64 113.1 105 62 . 780 1410 29.98 1. 28 31 150812 64 122.3 116 50 720 1350 29.84 1. 36 32 140810 66 100.2 92 80 660 1540 30.34 1.04 33 90808 60 104.1 93 75 750 1590 30.46 1. 20 34 80808 64 '100.0 88 100 820 1710 29.81 0.96 35 80806 63 94.3 88 130 780 1420 29.86 0.96

36 81111 64 . 107.8 93 100 840 1330 29.65 1. 04 37 81112 63 101. 2 90 110 850 1570 29.69 0.96 38 81115 64 96.9 80 130 810 1650 30.03 0.88 39 91101 66 110.0 102 100 700 1420 29.42 1.44 40 131115 60 103.2 98 80 710 1480 29.76 1. 28 41 141111 68 98.4 90 100 750 1710 28.64 0.96 42 141112 66 100.1 92 80 750 1400 29.28 0.88

62

TABLE 3.1 (Cont'd)


Wiss, Janney, E1stner & Associates, Inc.

Specimen Code f f fsf Esh E Esf E Esh y su su s No. No. ksi ksi ksi 0.0001 0.0001 0.0001 10 3 ksi 10 3 ksi

43 210401 55 100.0 80 * 840 1500 28.8 * 44 210810 55 100.0 89 * 690 1770 31.2 * 45 210709 52 99.0 90 * 500 1580 28.0 * 46 210708 55 113.0 106 * 850 1510 28.7 *

*The stress-strain curve has no definite yield point

TABLE 3.2

PCA Steel Curve for Steel A615 Grade 60

Specimen f f fsf Esh £ Esf E y su su s

10 3 ksi No.

1

2

3

y*

a

b

*y =

ksi ksi ksi 0.0001 0.0001 0.0001

74.2 110.0 103 1250 27.8

64.9 102.6 79 1290 28.2

60.0 100.8 65 1440 30.2

TABLE 3.3

Regression Equations for the Characteristic Variables of the Stress-Strain Curve of A615 Grade 60 Steel from WJE

f su

ksi

+ 0.523

+ 73.2

+ 0.274

+ 77.0

a[f Cksi)] + b Y

E s

ksi

- 54.39

+ 33023

- 9.44 - 0.00009

+ 1788 + 0.0145

£ su

- 0.00023

+ 0.0867

63

Esh 10 3 ksi

1. 22

1. 25

1. 24

- 0.00221

+ 0.2868

o W-'~I -------------------------------~C~H~R~R~R~C~,T~E~.r~1~I~S~T~I~C~~)~--~

-.-1

o .:::tI

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o I-

---i

(\

J -

.-

(f)

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~ ..

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Uj

W

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CC

co

f-

Ul

c::.L

.. (£

1

0-'

::.T'

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(\J

Y:c: (RX+f3Xx~2) / (1+CX+UX~~2)

FY

(KS

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. Cl

OO

FS

U

(KS

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09

,,4

00

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RIC

IN

RT

STR

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=:;

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71

73

1

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FN I

NG F

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(K

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) :J

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40

0

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Fig

. 3

.7

Com

pari

son

betw

een

pro

po

sed

for

mul

a an

d actu

al

dat

a o

f g

rad

e 60

st

eel.

Q\

+0-'

,-..J

ill~'------------------------------~C~H~R~R~R~C~T~E~R~l~S~T~I~C~'~S----'

';--

i

(:::: (RX+BX~o~2).I (1+CX+[1X7o~2)

FY

(KS

I) :

:{JC

J. 0

00

F

S U

( K

S 1

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.. S D

0 ~ -1

-· . (

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R I T

[.5 T

R R

I N

H

R R

IJ E

N I

N G

F S

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50

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(,J -

,-

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~-<

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C)

l" 0

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(j)

~-I

c.n

W

0-'

cr-.

co

I--

T;·

.(j)

~-

L··

LO

0-

::T'

0-'

eLl

C= 1

, 3C

J340

6 EY

~. 0

0 1

9B

[J

::.;

0.7

52

24

8

[S

H::{). 0

01

98

E~)U

:c.ll

.Of:)

900

E~ ~)

F

=0.

1 7

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PO

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No.

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) G

ENER

RLI

ZEO

R

NR

LYTI

C

CU

RV

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o.

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4

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8

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2

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6

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STR

RIN

Fig

. 3

.8

Com

pari

son

bet

wee

n p

rop

osed

fo

rmu

la a

nd

act

ua

l d

ata

of

steel

hav

ing

no

defi

nit

e y

ield

po

int.

0\

U1

66

The proposed analytic expressions for these two categories of the o-€

curve described in Sections (3.2.2) and (3.3.3), are applied to fit the data

of Specimens No. 9 and No. 44; the fittings are considered excellent as can

be seen in Figs. 3.7 and 3.8.

For analyzing the first category of steel with a yield plateau, the

next step is to relate the characteristic variables of the curve to the

yield strength, so as to make it possible to generate a stress-strain curve

solely from the knowledge of the yield strength. For this purpose, the

characteristic variables were statistically related to the yield strength.

In the statistical analysis, a linear relation is used to see the trend of

the scattering of the data. The data points ,and the linear regression lines

for these characteristic variables are plotted in Figs. 3.9, 3.10 and 3.11.

The corresponding coefficients of the regression equation are summarized in

Table 3.3. The large scatter of the data points is due to the various bar

sizes and the different manufacturers as mentioned before (a more detailed

analysis would be to separate the effect of bar sizes).

Based on the above results, a computer program was written to generate

a stress-strain curve for any given yield strength. Such generated curves

are shown in Fig. 3.12 and Table 3.4 for yield strengths equal to 60 ksi,

65 ksi, 70 ksi, and 75 ksi, respectively. In Fig. 3.13, seven experimental

curves whose yield strengths were 68 ksi are compared with an analytical

curve generated for a yield strength equal to 68 ksi. From this comparison,

it appears that the analytic expression is quite satisfactory.

3.5 CONCLUSIONS

(1) Although available expressions (Sargin, Goldberg, et al.) for the

stress-strain curve of reinforcing bars made of steels with no yield

TABLE 3.4

Characteristics and Constants of Equation for Generalized Stress-Strain Curves of Grade 60 Steel with Yield

Strength of 60, 65, 70 and 75 ksi

Characteristics

f f fsf E:sh E: E:sf Es Esh

y su su 10 3 103

ksi ks.i ksi ksi ksi

60 104.6 93.4 0.0091 0.0729 0.1542 29.76 1.222

65 107.2 94.8 0.0086 0.0717 0.1431 29.49 1.174

70 109.8 96.2 0.0082 0.0706 0.1321 29.22 1.127

75 112.4 97.5 0.0077 0.0694 0.1210 28.94 1.080

Constants of Equation for Representing Strain-Hardening Portion*

f Y A B C 0

ksi

60 1.748272 0.173674 -0.251726 1.173672

65 1. 756240 -0.145637 -0.243758 0.854363

70 1. 766823 -0.416587 -0.233175 0.583412

75 1. 780521 -0.655189 -0.219478 0.344811

*y =

67

1, ., ,

68

·~·~~=rFf:~~TI~~·~T~=J

-- .. _~~.-=_n~-=~~:?~~.~L-.. c-·-"E ~~~~- •.. . ~:=-:}~.~~-~~;~l..~- .. :~·;~t; n~:~=~~~~ 0_:~~~~~~':l~jj~~~ -=~. --1su-ff(sc}":' 73.2 :f-e;:S23 f4·--(KSt)--··~·--~--~~: ~~~·-~·-~~l--·~==--~~-··

- . -- ... : _. - :J .. :,. ; :. -:: ... 1 .. _

70····· 75

Fig. 3.9

69 r-- --:- I -: r:~ :~:-J: ::~::: -F::-:

- __ ~()ILJ,~~~~~';;';';';';';;"";"';""'~~"";"';""'-~~~---';;';;';;';";"~~~~"";"';""'----r:

E~h--~:~O: [f! 5 ~-:~:o:o ~-~~'l_ f~ c K~ i J:~_~-

.08

·/2

::-.::x.~-::-:.., :-))(:~: 0~_ --- --=-~~----

.JSf:;·~8(;8 .•..• 'Of:Z21 fi}ksi); •

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---- ---~------__ x:-:--~ ---- -------: x. :-~:::-;::.:x-_=_~--=-~.::

___ -1_~ _______ ~ __ .. ___ . __ _

WJE--7-XXX'A ' _ -I ,eX:' --.-.. ~----.~---. -----~----.------- ----------_.-.----""'--.....--- -------.. _--L1?CA~~~-:.:i::-·Ap..AA- --___ ~:~~: --

___ . l_ ~_

S r- -~---.:: : :- . _ ••. -. __ ~ __ r •

Fig. 3.10

-- -

~~

\lJ~: •

-~ ~/'lf ~. ~.

, ,5:

70

-1 •..

:~L.~~_: . ---~-------- '---.~ -.,

. E s (}(si i~~-p3-''-2~3·~~~-~.q.~~9:f9XkS tX_~ ~-~~~~~_:~~_~:~_~:~L~ . ~~ X: , ~ ,- "

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ESh:(ks~()~~Z ~ 8 8 ~ ~~ LJ '1-: fi'(kS:~J-

~

XX~ . :~:-~~ .. -~.:::-::-~~ X ~::::-x --~- .------.-.

x'· ~:- ::-- ::- ~~_:_:-·-x~

:.._~~....: :.::.~J(:.:: X : . 'r"~--~

---;-:---T:-: :--:::-.--::-":--. -:-::-: ~:.;.::~~~~-.-::=r ~ __ ~~L __ ._::-...:~ . =~~-- ~~~~~--.-:~~~

W .J E ~ --:.:.---XXX'!. ._---- .. __ . ,'---' -.-~------.-------.....:.-.----- ----<---- - --- ".-:--' -~'----'-~-"-' -.~-. ~-.. -~-... --~ --- .. ~-.-

~_~.:...-_: : PDAL=:._:~::c~JAAIA~ . --~ -

. '65 ':: (~:-

Fig.

'-j~ (Ksl) _ ft. ~~~ __ ~ ~ ~_ .

3.11

··-~~~·~7{J:.··

f Y

80.

CJ

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ener

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steel

wit

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nd

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.

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72

-::1"'~----'---~T--~~'-T-~""""';"';;;';";'-""""';;';';';;;;""';~~""";;-~~;;';';;";;~";"';;"'';';;'''';~;.;;.;.r.'~.---

-------~-=--->----~---. ---:-- --,~- ___ ~=;~: l~-:::~-::=:--------=-t~:--~~===:=-._~ ___ ._~ _____ . ____ -:_. ~-.- . ~ .. • "" - ---- "-----\.-.-.-. ___ I •..•. __ . "_ .. --~---.- .•• - .-.:-=-_:~~.:.---- --~ . .t.._-:-- r __ -

----~~------------------.,------

''-; rJ')

~

00 \0

II

>. '-I-!

r.-0

'-I-!

cO .j-l cO

"0

'-I-! 0 ---- c-I::: 0

''-; .j-l cO

''-; r.-eO > t') .-1

t')

Ol) ''-; >;r., -i--·--!-" -·1

.. _- .4". __ ,L.o - "'J-.- t-_

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.·i~=t··.:=.:·~;~r-:~; E:~~~S~~~~l?ITf~T~~~ •. i~~=~~:_· t--,----,----'---,--..;....--;--,.,.,-"-'''''''-<'<-'--''+-...... --+----;-- ::~~~;j.:-:~-=;~u -:-r -.: ".: i : - - -I:: :.: ~l. _ -1" ----. -_-1---

_ >- ·c~¥t=~t~~ L--_._:_-~~:·\·: :-::i,-~ l- -~L, ~ ~=-=--:~E~=j ____ _ --=..:c' "---;.....,. _ .. , .

plateau adequately describe their actual behavior well after the

yield point (say up to a strain of 0.08), they do not seem to be

representative at a higher value of strains (say up to a strain

of 0.20).

(2) For steels whose stress-strain curves show a yield plateau, the

expression proposed by Sargin [32] seems to be descriptive of the

actual behavior, especially before the peak point. As the expres

sion needs five parameters, none of which describe the fracture

point, the curve is not necessarily representative after the peak.

73

(3) The expression proposed to represent the strain-hardening portion

of the stress-strain curve of steels with yield plateau, and the

nonlinear portion of steels without yield plateau, can be made to

exactly describe the actual behavior up to the fracture point. It

needs four constants which can be derived from the characteristic

parameters of the curve; namely, the strain-hardening modulus, the

stresses and strains at the peak point and the fracture point.

(4) The regression equation proposed in this chapter to predict the

characteristic parameters of the stress-strain curve from the

knowledge of the yield strength only, seems to lead to an adequate

prediction of the entire stress-strain curve of steels with a yield

plateau. (Note, however, that some of the regression equations

show relatively poor correlation as the variation in bar size was

not taken into consideration in this analysis.)

74

CHAPTER IV

TEXT PROGRAM AND VALIDATION OF RESULTS

This chapter is comprised of two major parts: the first describes

the experimental program and results obtained on the stress-strain curve

of concrete specimens in compression; the second compares actual curves

with those obtained from the analytical expression described in Chapter II,

and checks if the analytical expression is representative of the actual

behavior of concrete from this invest~gation and other investigations.

4.1 EXPERIMENTAL PROGRAM

The primary purpose of the experimental program was to experimentally

obtain the stress-strain curve in compression of normal weight and light

weight concrete specimens, and to define, from the observed data, the

regression equations to predict the characteristic parameters needed in

generating the analytical stress-strain curves for any concrete. The

stress-strain curves were obtained from testing, up to a strain of 0.006,

normal weight concrete cylinders with compressive strength varying from

3000 to 11000 psi (21 to 77 MN/m2), and lightweight concrete cylinders with

strength from 3000 to 8000 psi (21 to 56 MN/m2).

4.2 TESTING METHOD

One of the reasons why there is insufficient experimental or analytic

information on th~ shape of the stress-strain curve beyond the peak stress

is the difficulty in experimentally measuring the descending portion of the

curve. Many testing machines used for standard compression test apply in

creasing loads rather than increasing deformations. This results in an

uncontrolled, sudden failure immediately after the peak load. Even with

75

the deformation controlled loading, the release of the energy stored in the

testing machine, when the specimen is unloading, will influence the shape

of the descending portion. Several investigators have developed techniques

to obtain the complete stress-strain curves of concrete [2,6,31,38]. Some of

these techniques are costly, require testing machines which may not be

available in a normal quality control laboratory or require extensive modi

fications to a standard testing machine. In this resear~h, a simple

technique was developed to obtain th~ stress-strain curve of concrete up to

the strain of 0.006 (Fig. 4.1).

Concrete cylinders were loaded parallel with a steel tube. The steel

tube was made of 4142 alloy steel and was case-hardened to 43-45 RC

(Rockwell Hardness NO.) so that its stress-strain curve was linearly

elastic up to the strain of 0.006. The specimens were capped in the test

ing machine to assure that the load will be shared simultaneously by both

the tube and the cylinder immediately on loading. The thickness of the

tube wall was such that the sum of the load carried by the steel tube and

the concrete cylinder was always increasing up to the strain of 0.006.

Thus, there was no release of energy from the testing machine. During

loading, the strains in the steel tube were measured with two foil-type

resistance strain gages. These strains gave not only the amount of load

taken by the steel tube, but were also used to obtain nominal strains in

concrete. Thus,knowing the total load and the corresponding steel strain,

the stress-strain relationship for concrete can be obtained. In a pre

liminary study, the ascending portions of the stress-strain curves of two

sets of 15 specimens each, one tested with the steel tube and one without,

were compared and were found similar. With this procedure, reproducible

76

co ~

·~L....-t,~"""~-"'~.-4I~~~~_~~:~A~:--.::~IN~~~~~.~~~_~~A_<:,~c--D=--<""~~.~-...:.,,~"-~~~~~~-~~J--J

c .-CD C\J

CD

x W 0 Z ~ g g-CJ) o C O::r<) o >-0 ::c

C CD

x o ~

Co

..,. ' \ t:a I

,~

0" '~, ;0.. , <J,'

, , '/J " I

3in. lOlA. " I

I~ICONCRETE CYLINDER::.,

, I

4in. biA. STEEL TUBE (CASE - H ~ RDENED)

0:: Co ~ 0 ,I

::c c ' 1 )0,

-.H-oI1-- :}32 . 10.

v-STRAIN GAUGE

n. .- b. ," HARDENED .' , ~ ~ ',b STEEL PLATE...lo!./:}-+--. CJ) 0 ,. ,rj _ ... ' ' <J : '4 (

~ t> , I • I

~ /.-~~ -~ . --=- ---=:-'\

Fig. 4.1 Test set-up.

77

descending portions of the stress-strain curves were obtained for more than

150 concrete specimens. The rate of loading was 10 micro-strain per second.

Although the method developed is simple and gives satisfactory results,

it has certain limitations: (1) the testing machine must apply a load to

both the steel tube and concrete cylinder; thus the size of the cylinder is

limited by the capacity of the machine; (2) the limit of 0.006 for final

strain may be too small when the concrete is confined with lateral reinforce

ment; (3) the definition of strain is such that the deformations of the

thin capping materials and those of the end zones of concrete specimens

(where purely uniaxial state of stress does not exist) are inCluded; (4) the

presence of the steel tube precludes any observation of the failure modes of

the specimen during testing.

4.3 FABRICATIONS AND MATERIALS

The specimens used were 3 x 6 in. (7.5 x 15 cm) cylinders. For normal

weight concrete, three batches of thirty cylinders each were made. To obtain

different values of compressive strength, the age of testing was varied from

2 days to 125 days. The specimens were cured in a fog room (70°F - 21°C,

100% R.H.) until a day before testing. At each testing age, six cylinders

were tested. After each set of tests, the steel tube was also tested by

itself up to the strain of 0.006, in order to check its linear elastic re

sponse. The specimens from the first batch were used for developing the

testing method and for the preliminary testing mentioned earlier. For the

statistical analysis of the results, the first batch was not included. All

normal weight concrete specimens were made with Type I cement, silicious

sand and 3/8 in. (9.5 mm) maximum size of dolomitic limestone coarse

aggregates. The water-cement, sand-cement and aggregate-cement ratios by

weight were 0.45,2.0 and 3.0.

Thirty lightweight concrete specimens were supplied by the Material

Service Corporation of Chicago. These 30 cylinders were made with five

different mix proportions (Table 4.1), all containing expanded shale with

78

3/8 in. (9.5 mm) maximum size aggregates and normal weight sand. They were

cured for 28 days in a fog room and tested within three days after curing.

Each set of six cylinders were designed to give a 28-day compressive

strength varying from 3000 to 8000 psi (21 to S6 MN/m2).

4.4 TEST RESULTS

Typical stress-strain curves for normal weight and lightweight concrete

specimens are shown in Figs. 4.2 and 4.3, respectively. The following remarks

can be made about these stress-strain curves: (1) every stress-strain curve

showed an inflection point; at about the strain corresponding to this point,

a slight drop in the load was observed and a cracking noise was often heard;

(2) the failure of all the specimens was gradual, even the normal weight

specimens with compressive strength of 11000 psi (77 MN/m2) and the light

weight specimens with compressive strength of 8000 psi (56 MN/m2) showed

extensive vertical splitting cracks around the surface of the cylinders; the

failure patterns for normal weight and lightweight concrete specimens are

shown in Figs. 4.4 and 4.5; (3) for normal weight concrete, the strain at

the peak stress as well as that at the inflection point did not seem to vary

substantially with the compressive strength; (4) for lightweight concrete,

both of these strains seem to increase with the compressive strength.

79

TABLE 4.1

Mix Proportions (lb/cu yd)

Normal Weight

Source of Type I Fly Ash Sand Aggregate Water Admixture Specimen Cement (3/8" max.)

UICC 62S 1256 1884 282

Material Water Reduc. Service 850 100 1070 1700 320 + Corp. Air Entraining

Lightweight

Design Type I Fly Fine Medium Strength Sand Materia1ite Water Admixture

(psi) Cement Ash Materiali te ( 3 / 8" max.)

Water Reduc. 3000 470 - 1180 900 300 300 +

Air Entraining

Water Reduc. 4000 560 - 1100 900 300 315 +

Air Entraining

Water Reduc. 5000 635 100 940 900 300 330 +

Air Entraining

6000 752 100 1000 900 300 340 Water Reduc.

7000 785 100 1040 860 300 335 Water Redu.c.

12.0

r-, -.,

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10.0

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_ L_

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_ ~IO

a 0.

001

0.0

02

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.00

3

0.0

04

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06

ST

RA

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Fig

. 4

.2

Ty

pic

al st

ress

-str

ain

cu

rves

fo

r no

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ncr

ete.

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o

8.0

Cfl 60

0r ~

.

.. (f

) (f

) ~ 4.o~ I

2.0

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o

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WE

IGH

T

60

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EC

TIO

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//~

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0.00

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····

·

0.0

02

0

.00

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RA

IN

0.0

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5

Fig

. 4

.3

Ty

pic

al st

ress

-str

ain

cu

rves

fo

r li

gh

twei

gh

t co

ncr

ete.

0.0

06

00

I-

'

-#/0· ____ 2 I

Fig. 4.4 Failure patterns for normal weight concrete.

Fig. 4.5 Failure patterns for lightweight concrete.

82

83

In Fig. 4.6, a selected comparison is made between the stress-strain

curves of normal weight and lightweight concrete having essentially the

same compressive strengths of about 4300 and 7400 psi (30.1 and 51.8 MN/m2).

It can be seen that for a similar strength, lightweight concrete exhibits

a more steeper drop of the descending part of the stress-strain curve than

normal weight concrete. This means that if an analytic stress-strain curve

is expressed as a function of the compressive strength alone, a separate

expression would be requir~d for each type of concrete. The two stress-

strain curves for normal weight and lightweight concrete with the compressive

strength of about 4300 psi (30.1 MN/m2) have ascending portions which are

quite similar but have widely varying descending portions. This means that

the expression for the analytic stress-strain curves cannot be based just on

the parameters of the ascending portions alone, as have often been done in

the past.

4.5 GENERATING THE CONSTANTS OF THE ANALYTIC EXPRESSION FOR THE STRESSSTRAIN CURVE

The observed stress-strain relationship of concrete has to be expressed

analytically in order to be useful in the analysis and design of structural

members. The expression discussed in Chapter II, Section2.2 is used here to

check its validity in respresenting the actual behavior of concrete. As it

is also desirable that the constants of this expression be evaluated solely

from the key physical parameters of the stress-strain curve, these key param-

eters were expressed in terms of the readily available material properties

su~h as compressive strength, type of aggregate (normal or lightweight) and

amount of confinement. In this manner, the entire stress-strain curve can

be generated only from the knowledge of these few material properties.

8.0

.. 6.0

1

(J) ~

(J)

(J) w g:

4.0

(J

)

2.0

o

LIG

HT

WE

IGH

T

NO

RM

AL

W

EIG

HT

..

,.,

LIG

HT

WE

IGH

T

DA

TA

PO

INT

S

•••••

0.00

1 0

.00

2

0.0

03

0

.00

4

0.0

05

ST

RA

IN

Fig

. 4

.6

Com

pari

son

betw

een

norm

al w

eigh

t an

d li

gh

twei

gh

t co

ncr

ete

stre

ss-s

train

cu

rves

.

60

40

(\J

20

0.0

06

E ~

,.

z :E

00

~

where

The stress-strain relation used is given by:

Y = AX + BX2

1 + ex + DX2

Y = flf 0

X = E/E 0

f and E = stress and strain in general

f and E = peak stress and corresponding strain 0 0

A,B,e,D = constants

E = f IE = secant modulus of elasticity at the peak stress 000

85

(4.1)

Two different sets of constants A, B, e and D were used for the ascend-

ing and descending portions of the curve. For the ascending portion, the

values of the four constants were evaluated from the following four conditions,

for which E represents the secant modulus of elasticity at 0.45 f : c 0

Y = 0.45

Y = 1

dY = 0 dX

at the origin (Y = 0, X = 0)

for X 0.45 = E IE c 0

for X = 1

for the peak point (Y = 1,'X = 1)

(4.2)

For the descending portion, the conditions for determining the four constants

were:

Y = 1 for X = 1

dY 0 for the peak point (Y 1, X 1) dX = = =

f. E. (4.3)

Y .2:. for X 1 = =-

f E 0 0

Y f2i

for X E2i

= = f E 0 0

86

where f. and E. are the stress and the strain at the inflection point, 1 1

and f2i and E2i refer to a point which was arbitrarily selected

(Fig. 4.7) such that: E2·-E.=E.-E. 1 110

With this approach an analyt ic stress- strain curve can be generated from

the knowledge of four key points of the experimental curve, namely: stress

and strain at the peak; stress and strain at 0.45 f (or the secant modulus . 0

of elasticity at that point); stress and strain at the inflection point,

and stress and strain at a point symmetric to the peak with respect to the

inflection point. The characteristics of the four key points from the test

data have been tabulated for normal weight and lightweight concrete in

Tables 4.2 and 4.3. Due to the definition of strain used in this investi-

gation, the characteristics obtained may have to be adjusted for comparison

with data from other techniques.

It can be seen from Fig. 4.7 that Eq. (4.1) fits a given curve rather

well when the four constants are determined according to the above proce-

dure.

The next step was to relate the four key points of the curve to the

compressive strength, so as to make it possible to generate a stress-strain

curve solely from the knowledge of the compressive strength. For this

purpose, the above four points (Fig. 4.7) were statistically related to the

compressive strength (f ) for the experimental data of both the normal o

weight and lightweight concrete specimens. The corresponding least square

fitting lines relating the characteristics of the four key points to the

peak stress are shown in Figs. 4.8 to 4.15 for both lightweight and normal

weight concrete. In the statistical analysis, it was generally observed

that a linear relation is quite acceptable and a relation either in terms

87

TABLE 4.2

Characteristic Points of Nonna1 Weight Concrete

Batch 1 (Cast on February 5, 1976)

,..c: ~ 4-!.f.J (!)

fo E:o f. E:. f2i E:2i E W o 0.0 S !-l ~ ''''; (!) 1 1 C

til (!) U,.c 103 1b >'!-l Q.) S CI!.f.J 0..:::1 psi 0.001 psi 0.001 psi 0.001 psi cu ft QrJ) cnz

6 5425 3.20 3380 4.90 1720 6.60 2222 147.3 30 5990 2.85 4140 4.60 2405 6.35 2974 147.3

4 5240 2.95 3100 4.70 850 6.45 2447 147.9 23 5190 2.85 3820 4.40 2270 6.45 2583 147.3

7 13 4960 3.00 2855 4.65 820 6.20 2343 147.5 25 5515 2.80 4130 4.30 2945 5.80 3001 146.9

2 8065 3.05 5545 3.95 3000 4.85 3812 151.3 8 4980 3.20 2830 4.70 835 6.20 1887 148.3 1 5350 3.20 3535 4.95 1980 6.70 2372 148.9

12 6505 2.95 4695 4.05 3185 5.15 2585 149.7 14 5940 2.60 4175 3.60 2955 4.80 2808 148.0

14 10 5870 2.90 4245 3.95 2745 5.00 2460 150.0 15 6365 3.20 4525 4.10 2660 5.00 2320 150.6 11 5615 2.50 4030 3.60 2545 4.70 2829 148.9 9 5940 3.20 4525 4.10 3255 5.00 2146 150.5

17 8500 3.70 5660 4.70 2675 5.70 2829 148.2 16 7370 3.55 4880 4.85 2455 6.15 2770 148.2

28 27 7500 3.55 5095 4.90 2715 6.25 2705 148.2 20 7370 3.60 5405 4.75 3580 5.90 2716 148.2 28 6760 3.45 5375 4.90 3720 6.35 2476 148.0

26 7995 3.90 5090 4.85 2530 5.80 2587 150.7 22 7355 2.95 5090 3.75 2815 4.55 3339 149.1

56 24 7285 3.65 5305 4.80 3325 5.95 2711 150.4 18 6805 3.70 5090 4.65 3550 5.60 2274 150.4 31 6670 3.65 4810 4.60 3100 5.55 2360 148.7 19 5770 2.95 4245 4.45 3100 5.95 2493 148.7


76 2945 3.05 2545 4.50 2065 5.95 1700 149.1 61 3610 2.70 3110 3.95 2690 5.20 2000 150.9

2 74 3140 2.90 2830 3.80 2460 4.70 1785 150.0 58 3040 2.75 2545 4.10 2090 5.45 1555 149.6 64 3085 2.90 2660 4.05 2263 5.20 1905 150.0 65 2945 2.60 2545 3.95 2135 5.30 1650 150.0

TABLE 4.2 (Cont'd)


...c I=! f f. f2i ~-I-l (l) E E1 o bO S ~ 0 0 ~ I=! ''''; (l)

CIl (l) U,D >.~ (l) S ro -I-l 0..::3 psi 0.001 psi 0.001 psi Ot/) t/)Z

56 4175 2.55 3255 4.10 2390 54 3790 3.25 2945 4.75 2220

3 53 4215 2.75 3000 4.40 1950 52 4145 2.75 3680 4.00 3135 51 4355 2.70 3395 4.35 2405 55 4100 2.60 3110 4.10 2265

79 7510 2.70 5515 3.45 3820 70 7215 2.60 4810 3.70 2065

28 80 7640 2.70 4690 3.75 2290 69 7285 2.85 4950 3.85 2900 77 7355 2.65 5020 3.85 2690 83 7355 3.00 5660 3.85 3775

90 60 8405 3.33 5235 4.20 1415

72 9480 3.05 5660 4.25 1785

125 67 9870 2.90 6790 3.60 3395 62 10645 3.05 6860 4.00 2545 73 11035 3.20 7780 4.05 3465

Specimens from Materials Service Co.

1102 10185 3.30 6365 4.20 2265 56* 1103 9905 3.65 6365 4.50 2690

1104 10470 3.80 6790 4.75 2120

125* 1106 10750 3.50 7500 4.60 2545

210+ 1201 10270 3.85 6735 4.70 3165 1202 10100 4.10 6230 5.00 1765

*Materia1s Service Co. Concrete Laboratory Specimen

+Riverside Plaza Column Core

88

E2i Ec W

103 1b 0.001 psi cu ft

5.65 2315 150.9 6.25 1705 150.9 6.25 2475 150.0 5.25 1865 150.0 6.00 2330 150.4 5.65 2120 151.3

4.20 3615 150.0 4.80 3660 150.0 4.80 3590 150.0 4.80 3395 150.2 5.05 3445 150.0 4.70 3110 150.0

5.10 3605 150.9

5.45 3885 150.9 4.30 4570 150.4 4.95 4580 151.3 4.90 4500 150.4

5.10 3735 147.6 5.35 3715 149.6 5.70 3825 149.1

5.70 3850 152.0

5.55 3440 151.9 6.00 3495 151.4

89

TABLE 4.3

Characteristic Points of Lightweight Concrete

Specimens from Materials Service Co. (Tested on June 17, 1976)

~ ~ f f. E. f2i E2i EC W .jJ (!) E ~ Ol) S ~ 0 0 l. l. Ol)~ .!"'i (!)

10 3 .r-! (!) U,D 1b U') ~ (!) S (!) . .jJ o..;:j psi 0.001 psi 0.001 psi 0.001 psi cu ft ClU:> u:>z

301 3535 2.85 1910 3.90 635 4.95 1885 118.4 302 3610 2.85 2265 3.80 820 4.75 1915 119.1

3000 303 3090 3.00 1700 4.00 850 5.00 1570 118.7 304 3655 3.15 2265 4.10 850 5.05 1625 117.9 305 4105 2.95 2830 4.10 1910 5.25 1670 110.6 306 4355 3.00 2845 4.10 1700 5.20 1835 110.6

401 4385 2.90 2545 4.00 705 5.10 2160 119.2 402 4540 3.10 2690 4.05 1060 5.00 1980 118.7

4000 403 4555 3.05 2545 3.85 705 4.65 2065 120.0 404 4555 3.15 2405 4.10 850 5.05 2065 118.6 405 5095 3.25 3270 4.00 1625 4.75 1850 1l0.2 406 4400 2.90 2860 3.65 1685 4.40 1840 1l0.2

501 5235 3.25 2545 4.05 425 4.85 2155 118.9 502 5660 3.25 3110 4.10 1555 4.95 2315 118.2

5000 503 5870 3.45 3395 4.30 565 5.15 2425 118.2 504 5955 3.20 3535 3.95 1130 4.95 2445 116.9 505 5095 3.15 3185 . 3.80 1485 4.50 1885 109.5 506 5305 3.25 3535 3.90 1625 4.55 1860 109.5

6000 605 7780 3.75 5165 4.55 1910 5.35 2560 116.8 606 6510 3.55 4030 4.30 1840 5.05 2220 116.8

702 7780 3.55 4950 4.40 2150 5.25 2565 125.2 703 7355 4.00 4810 4.80 1585 5.60 2450 124.4

7000 704 7780 4.00 5095 4.80 1840 5.60 2375 125.4 705 8065 3.80 5235 4.55 2195 5.30 2580 118.6 706 7500 3.50 5095 4.35 2690 5.20 2425 118.6

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Fig

. 4

.12

L

inea

r re

lati

on

bet

wee

n Ec

an

d fo

fo

r no

rmal

w

eig

ht

con

cret

e.

to

til

Fig

. 4.

13

Lin

ear

rela

tio

n b

etw

een

E

, E

. an

d f

for

lig

htw

eig

ht

con

cret

e.

o 1

0

i;

, ;

«.0

0\

97

II' !! I

I,

:"

:. :

r'l

t! i

Fig

. 4.

15

Lin

ear

rela

tio

n b

etw

een

Ec~

:t:2i

an

d fo

fo

r li

gh

twei

gh

t co

ncr

ete,

\0

00

99

of f2 or ~ did not improve the fit. The linear regression equations o 0

relating the peak stress (independent v~riable) to the four key points of

the curve are shown in Table 4.4. These lines are of the form y = ax + b

where y is the dependent variable, x = f the peak stress, o a the slope

of the line and b the intercept. Also shown are some statistical charac-

teristics of the data including the coefficients of correlation which

indicate how well the data are approximated by a straight-line. It can be

seen that for normal weight concrete the coefficients of correlation are

poor for and E .• 1

This seems to confirm the experimental observation

that these parameters are almost constant for all values of f. Note also o

that the predicted values of the modulus of elasticity E c given in Table

4.4 are generally smaller than those resulting from a standard ACI-ASTM

test to determine the modulus using a compressometer attached to the middle

portion of the CYlinder; this is due to the different definitions of strain

. as used in this investigation.

Based on the above results, a computer program was written to generate

a stress-strain curve for any given compressive strength and for a given type

of aggregate (normal weight or lightweight). First the program calculates

the coordinates of the four key points of the curve for the given compressive

strength, from the relations shown in Table 4.4. Once these four points are

determined, the four constants A, B, C and D of Eq. (4.1) are calculated

separately as shown in Table 4.5 for the ascending and the descending

portion of the .curve by satisfying the conditions mentioned earlier. In

Fig. 4.16, eight experimental curves whose compressive strength was about

7400 psi (51.8 MN/m2) are compared with an analytic curve generated for a

compressive strength equal to their average experimental compressive

y*

E 0

f· 1 +-l (psi) ,!:; Oil

• .-4 (])

:== E. r-f 1 ro IS !-I f2i 0 z (psi)

Ec (psi)

So

fi +-l (psi) ,!:; Oil

• .-4 (])

E. ::: +-l 1 ,!:; Oil

• .-4 f2i >oJ

(psi)

Ec (psi)

TABLE 4.4

Regression Equations for the Key Parameters of the Stress-Strain Curve

a b Std. Dev.* Coefficient of Variation*

0.000125 0.00230 0.000326 0.105

580 774 326 0.073

0.00005 0.00401 0.000422 0.097

85 2024 656 0.256

271,000 978,000 331,528 0.123

0.00020 0.00217 0.000138 0.042

704 -471 243 0.073,

0.00016 0.00329 0.000176 0.043

262 -47 443 0.322

180,900 1,127,000 143,778 0.068

100

Correlation Coefficient

0.636

0.966

0.239

0.263

0.865

0.913

0.975

0.801

0.666

0.885

*y = a[fo(psi)/lOOO] + b; Std. Dev. = 1L:(Ydata - y)2/n-1 coefficient of

variation = (Std. Dev.) divided by the mean value of the dependent variable

over the range considered.

fo

e: 0

psi

3000

.0

0267

4000

.0

028

5000

.0

0292

6000

.0

0305

~

...c::

7000

.0

0317

b.O

''';

Q

) 80

00

.003

30

:s::

rl

9000

.0

0342

cd

~ 10

000

.003

55

0 z 11

000

.003

67

1200

0 .0

0380

1300

0 .0

0392

3000

.0

0277

~

...c::

4000

.0

0297

b.O

''';

Q

) 50

00

.003

17

::: ~

...c::

6000

.0

0337

bO

''';

....

.:I 70

00

.003

57

--

TABL

E 4

.5

Co

nst

ants

o

f E

qu

atio

n f

or

Gen

eral

ized

Str

ess

-Str

ain

Cur

ves

Asc

endi

ng

Des

cend

ing

A

B

C

D

A

B

C

1.59

6973

-0

.35

20

41

-0

.40

30

25

0.

6479

59

-1.2

75

76

9

1.4

79

52

0

-3.2

75

76

8

1.44

3399

-0

.64

25

42

-0

.55

66

01

0.

3574

58

0.1

15

28

6

0.30

6384

-1

.88

47

14

1.36

4803

-0

.75

80

33

-0

.63

51

95

0.

2419

66

0.32

6434

0.

0479

69

-1. 6

7356

5

1.32

3699

-0

.80

94

90

-0

.67

63

01

0.

1905

10

0.36

5932

-0

.05

07

30

-1

.63

40

68

1.30

4016

-0

.83

19

52

-0

.69

59

82

0.

1680

47

0.35

7600

-0

.09

55

50

-1

.64

23

98

1.29

7725

-0

.83

88

36

-0

.70

22

75

0.

1611

64

0.33

2631

-0

.11

62

74

-1

.66

73

69

1.30

0356

-0

.83

59

73

-0

.69

96

41

0.

1640

26

0.30

1863

-0

.12

44

28

-1

. 698

136

1.30

9239

-0

.82

61

30

-0

.69

07

61

0.

1738

70

0.26

9654

-0

.12

53

15

-1

.73

03

46

1.32

2664

-0

.81

07

04

-0

.67

73

34

0.

1892

95

0.23

8022

-0

.12

17

64

-1

.76

19

77

1. 3

3949

9 -0

.79

04

39

-0

.66

05

01

0.

2095

61

0.20

7840

-0

.11

53

54

-1

. 792

160

1.35

8955

-0

.76

57

31

-0

.64

10

44

0.

2342

68

0.17

9552

-0

.10

70

95

-1

. 820

446

1.54

1688

-0

.46

64

98

-0

.45

83

11

0.

5335

02

0.17

8520

-0

.04

62

08

-1

. 821

479

1.3

74

06

9-0

.74

55

87

-0

.62

59

31

0.

2544

13

0.22

3037

-0

.08

40

08

-1

. 776

963

1.28

7968

-0

.84

92

26

-0

.71

20

30

0.

1507

73

0.23

5274

-0

.10

32

81

-1

. 764

724

1. 2

4263

1 -0

.89

29

65

-0

.75

73

69

0.

1070

35

0.23

1553

-0

.11

20

01

-1

. 768

447

1.22

0581

-0

.91

15

35

-0

.77

94

18

0.

0884

64

0.21

9928

-0

.11

44

48

-1

. 780

071

I

D

2.47

9519

.

1.30

6384

1.04

7969

0.94

9270

0.90

4450

0.88

3726

0.87

5572

0.87

4685

0.87

8235

0.88

4646

0.89

2904

0.95

3791

0.91

5992

0.89

6719

0.88

7999

0.88

5551

I-'

o I-'

,

10

.0,

AS

CE

ND

ING

A=

I.3

00

50

1

8.0

r B

= -

0.8

35

81

8

C =

-0

.69

94

98

0

=0

.16

41

82

-6.0

fo

= 7

35

9P

SI

U) ~

0-=

76

0 P

SI

.. U

)

V=

0.1

67

U

) w

0::

I-4

.0

U)

2.0

r V

-

o 0.

001

Y;:

(A X

+ B

X 2

) /

( I +

ex

+ 0

X2)

I I

• I

• J

' -

• •

• •

•

I •

• •

!I

• •

• • •

• •

•

DA

TA

PO

INT

S

•••••

GE

NE

RA

LIZ

ED

A

NA

LY

TIC

AL

CU

RV

E

0.0

02

0

.00

3

ST

RA

IN

•

0.0

04

DE

SC

EN

DIN

G.,

60

A=

0.3

49

77

7

B =

-0

.10

49

63

C

=-

1.6

50

22

2

D=

0.8

95

03

6

•

fO}

• • I .. • •

• •

• • •

• I

• • •

• •

• •

• •

• •

• 1

20

•

• •

• •

• •

•

0.0

05

0

.00

6

Fig

. 4.

16

Com

pari

son

of

expe

rim

enta

l d

ata

wit

h t

he

gen

eral

ized

an

aly

tic

curv

e fo

r no

rmal

w

eigh

t co

ncr

ete.

I-'

o N

103

strength. From this comparison, it appears that the analytic expression is

quite satisfactory. The experimental curves in Fig. 4.16 also illustrate

the reproducibility of the descending portion of the stress-strain curve

and the type of scatter that might be expected. A similar comparison is

shown in Fig. 4.17 for lightweight concrete specimens whose compressive

strength was about 4500 psi (31.5 MN/m2).

4.6 VALIDATION OF THE ANALYTICAL EXPRESSION

Is the analytical procedure described here well suited to include the

effect of lateral reinforcement which influences primarily the descending

portion of the stress-strain curve? The answer is yes because the four

constants for each the ascending and the descending portions of the curve

are separately controlled as shown earlier.

Can the analytic curves generated from the experimental data of this

investigation be valid for concrete made with different mix proportions?

Or, how valid is the assumption that the stress-strain curve for a given

type of concrete (normal weight or lightweight) can be expressed solely as

a function of compressive strength, regardless of the mix proportions and

age at testing? Although more experiments are needed to completely examine

this assumption, some partial verification was obtained from the results of

tests on six additional high strength, normal weight concrete cylinders.

These cylinders were supplied by Material Service Corporation of Chicago

and their mix proportions are shown in Table 4.1. Four of the cylinders

were made in their laboratory, cured in the fog room for 55 days and tested

at 56 days. The other two were cylindrical cores cut from an experimental

column cas~ a~ che site of the Riverside Plaza structure in Chicago [8J.

10.Oi~---------------------------------------'

(f)

:::::e:::

.. (f

) (f

) w

8.0

6.0

n:: 4

.0

r (f)

2.0

a

Y=

(AX

+ B

X?

) i (

I +

CX

+ D

X2 )

AS

CE

ND

ING

A

= 1

.32

53

30

6

=-0

.80

75

64

C

=-0

.67

46

68

0

=0

.19

24

35

fo =

448

7 P

SI

(T=

2

66

PS

I

V=

0.1

I5

/ fI,

v D

AT

A

PO

INT

S

•• •

• •

G

EN

ER

ALI

ZE

D

•

/ A

NA

LYT

IC

CU

RV

E·-

-

• •

•

0.0

01

0

.00

2

0.0

03

0

.00

4

ST

RA

IN

DE

SC

EN

DIN

G

A=

0.2

31

74

0

B =

-0

.09

50

92

c=

-1

.76

82

58

0=

0.9

04

90

8

• •

• 0.0

05

60

C\I

40~

20

• • 0.C

06

z ~

Fig

. 4.

17

Com

pari

son

of

expe

rim

enta

l d

ata

wit

h t

he

gen

eral

ized

an

aly

tic

curv

e fo

r li

gh

twei

gh

t co

ncr

ete.

.....

o .j:::.

105

The cores were tested at 210 days after casting. The compar~son of some of

the data from these cylinders with a cylinder of comparable strength re

ported previously (UICC) can be seen in Fig. 4.18 along with the analytical

curve generated for a compressive strength equal to the average of the data

shown (10,366 psi or 72.5 MN/m2).

How well the analytic expression would represent concrete behavior

beyond a strain of 0.006 (which was the limiting strain of the experimental

results)? Sangha and Dhir [31] have reported some experimental stress

strain curves of concrete cylinders up to a strain of 0.020. These curves

were obtained by loading 2 in. (5 cm) diameter specimens at a rate of 2.5

. micro-strain per second, in a very stiff machine of 5.85 MN (1200 kips)

capacity. Equation (4.1) was fitted to the four key points of their curve

and a comparison is shown in Fig. 4.19. It can be seen that the analytic

expression is acceptable beyond the strain of 0.006.

The stress-strain curves reported here were obtained by testing con

crete cylinders with length to diameter ratio of 2. Testing longer

cylinders will reduce the effects of end restraints and may lead to

different curves especially in the descending portion, as reported in [31,

36] .

4.7 CONCLUSIONS

It has been shown in this investigation that:

1. High strength, normal weight and lightweight concretes do not fail in

a brittle manner (provided special precautions are taken during test

ing) and exhibit a reproducible descending portion of the stress-strain

curve.

12.0

'Oool

8.0

'

(f)

6.0

~ ..

(f)

(f)

W

0:: .... (f)

4.0

2.0

a 0.

001

, I'

,

GE

NE

RA

LIZ

ED

A

NA

LY

TIC

AL

-I 8

0

CU

RV

E

, ....

MA

TE

RIA

L

SE

RV

ICE

C

O.

LAB

OR

AT

OR

Y

I I \

\ , ~ C

OLU

MN

C

OR

E

60

40

(\J

20

E

........

-

z 2

DA

TA

PO

INT

S

••• •

•

0.0

02

0

.00

3

0.0

04

0.

005

0.0

06

STR

AIN

Fig

. 4

.18

C

ompa

riso

n o

f st

ress

-str

ain

cur

ves

for

con

cret

e fr

om d

iffe

ren

t so

urc

es.

I-" o 0'\

.. 6.

0~

~

14

0 C\

J E

.........

(/)

4.0

, I

\ D

ATA

P

OIN

TS

•

• •

• •

(FR

OM

S

AN

GH

A

AN

D

DH

IR) I

z ~

~

.. (/

) ~ 2J I

~-120

~ -

I- en

1 • -

. •

• 0

,.

0 0.

004

0.00

8 0.

012

0·:0

16

0.0

20

ST

RA

IN ~

Fig

. 4.

19

Val

idat

ion

of

the

gen

eral

ized

cur

ve u

p to

a s

train

0:1;

0

.02

0.

......

o -...J

2. The analytic expression for the stress-strain curve should contain

constants which reflect not only the properties of the ascending

portion but also those of the descending portion. The presence of

lateral reinforcement can then be accounted for more rationally.

108

3. An inflection point was generally observed along the descending por

tion of the stress-strain curves. For normal weight concretes

strains at the inflection point and at the peak did not change

substantially with the compressive strength; whereas for lightweight

concrete these strain values increased almost linearly with strength.

4. There is some indication that the coordinates of the inflection

point may be a property of the material and thus are useful in the

analytic characterization of the descending portion of the stress

strain curve.

s. The analytic expression developed here for the stress-strain curves

of concretes in compression correlates well with experimental

observations up to strains of 0.020.

CHAPTER V

A COMPUTER PROGRAM FOR THE NONLINEAR ANALYSIS OF REINFORCED CONCRETE SECTIONS

5.1 INTRODUCTION

A computer program was written to perform a n?n1inear analysis of

reinforced concrete sections, which include rectangular and T-beam, uni-

axia11y-eccentrica11y-1oaded and biaxially eccentrically-loaded columns.

A general procedure for analyzing typical beam and column sections is

explained, in which the actual stress-strain curves of concrete and steel

109

were used, and no limiting concrete strain criterion was applied. The com-

puter program accommodates the above variables. Theoretically predicted

values of moments and curvatures are compared with some experimental

results from the technical literature, and with the 1971 ACI code values

to check the safety margin provided by the code.

5.2 GENERAL PROCEDURE FOR THE NONLINEAR ANALYSIS OF BEAM AND COLUMN SECTIONS

5.2.1 Assumptions

The following assumptions are used:

(a) Linear distribution of strains across the section (i.e., plane

section remains plane after loading).

(b) Concrete has no strength in tension.

(c) Perfect bond exists between concrete and reinforcement (i.e.,

the same change in strain in the steel and concrete at the level

of the steel).

(d) No limiting concrete strain criterion was assumed.

110

(e) The stress-strain relationships of concrete and steel are

known.

(f) At all times, equilibrium of forces and moments must be

satisfied.

5.2.2 General Equations in Determining Ultimate Capacity

Equilibrium of internal forces and external loads leads to:

C - T = P (5.1)

where

C = compression on concrete and steel

T = tension on steel, tension on concrete being neglected

P = external load

Equilibrium of internal and external moments leads to:

M. = M lnt ext (5.2)

The compatibility of strains on concrete and steel, and the linear distri-

bution of strain along the depth of the section even after cracking leads

to:

d-c e: = --e: s c c

(5.3)

, c-d" e: = --e: s c c

where

c = depth of neutral axis

d = distance from extreme fiber to centroid of tension steel

d" = distance from extreme fiber to centroid of compression

steel

e: c = strain on extreme compressive fiber

ES = strain at centroid of tension steel

E' = strain at centroid of compression steel s

The constitutive equations of concrete in compression and steel in

tension are assumed to be of the general forms:

111

f = fCE ) C C

C5.4)

f = fCE ) S . S

(5.5)

The above general equations will be applied to the cases of beams and

columns in the following sections.

5.3 BEAM UNDER PURE BENDING

5.3.1 Behavior of Beam

The position of the neutral axis, the strain on the extreme compres-

sive fiber, and the tensile strain in the steel are the key characteristics

for describing the beam behavior during loading. The maximum compressive

strain on the concrete, E , C

increases with the applied moment and takes

on values such as Eel' Ec2 ' ... until failure as shown in Fig. 5.1; simul

taneously the neutral axis shifts upwards or downwards as shown in Figs. 5.3

and 5.4; the tensile strain in the steel at ultimate observed capacity,

determine the failure patterns of the beam, i.e. either a ductile or a

brittle failure. Due to the different failure patterns, beam sections are

classified as follows:

1. Under-reinforced Beams

The tensile strain in the steel at ultimate moment capacity of the

section is greater than the yield strain, i.e. E > E as shown in sy

Fig. 5.2, the failure pattern is ductile, and the neutral axis shifts

...

t:z Ei" to E ~ 5traJn, Ec

112

-ThiS post of the curve is importa,nf (n uitimttte strength ca../cula.Ji.on

Fig. 5.1 Stress-strain curve of concrete .

underreintorced bea:m

-ov~tr(inforc.ed bea.M

Fig. 5.2 Stress-strain curve of grade 60 steel.

-, ; -

C<A.sf~

fs< f~ M<Mu

Fig. 5.3 Upwards shifting of neutral axial under loading· for underreinforced beams. .

Fig. 5.4 Downwards shifting of neutral axial under loading for overreinforced beams.

113

114

upwards under loading toward the extreme compressive fiber as shown

in Fig. 5.3.

2. Overreinforced Beams


section is smaller than the yield strain, i.e. E < E, as shown in s y

Fig. 5.2 the failure pattern is brittle and the neutral axis shifts

downwards toward the extreme tensile fiber as shown in Fig. 5.4,

3. Balanced Beams


section is exactly equal to the yield strain, i. e. E = E as shown s y' .

in Fig. 5.2; the balanced beam in ultimate strength design is not a

practical beam, but the concept is fundamental to guide the design

procedure to avoid a brittle failure pattern, i.e. overreinforced

beam design.

Two types of beam cross sections, i. e. rectangular and T - beam, will be

analyzed next according to the theoretical procedure described in

Section 5.2. The ACI code formula will not be repeated here, although

they will be used to compare the results.

5.3.2 Rectangular Beam

Rectangular beams can be either singly reinforced or doubly reinforced

as shown in Fig. 5.6. The nonlinear analysis procedure as applied to rec-

tangular beams is explained below; the corresponding general equations are

given first as:

E.quilibrium equations

A f = 0 s s (S.6)

Compatibility Equations

d-c e: =--e: s c c

I c-d" e: =--e: s c c

Constitutive Equations

CL = CL(e: ) c

6=6Ce:) c

115

(5.7)

(5.8)

(5.9)

(5.10)

where CL and 6 are explained in Fig. 5.5

Curvature

<P = e: c c (5.11)

In the above equations, there are only two independent.parameters,

namely the depth of neutral axis, c, and strain at extreme compressive

fiber, e:c ' Generally, e:c is assumed first, then the depth of neutral axis

c is determined by trial an~ error in order to satisfy equilibrium equations

(5.6), through the use of compatibility and constitutive equations; then the

corresponding moment M, and curavture <p are calculated from Eqs. (5.7)

and (5.11). Typical variations between M and e: are shown in Fig. 5.7 c

from which the maximum moment capacity Mu can be determined. The logical

flowchart for generating the moment-curvature relationship of a given beam

d h

-'-

116

. f3c

__ C .J:...-_TI)- af~bc > '/// / / / / / / / / / / / / / / / /.

CJ)

en w 0: Ien

o

STRAIN 01 STRIBUTION

CONCRETE COMPRESSION BLOCK

STRESS DISTRIBUTION

PROPORTIONAL TO a

LOCATION OF CENTER OF GRAVITY PROPORTIONAL TO f3

PROPORTIONAL TO C .. I

Fig. 5.5. Parameters a and 8 for stress block of concrete compression zone.

117

Fig. 5.6 Forces in rectangular beam.

, . -~'--------:"'-----=-----' -----_ .......

J:tf .~: ...·ITI:.'~c~F~Icfcf,' 4 . ~ ..• ~--~"'::7=:::=::_::':::::

-~-.-- ------. - -+-~ -__ . __ . __ --I

--:.1 ~-~--=-==-=-:-.r.

Fig. 5.7 Relationship of M-EC'

118

cross section is given in Table 5.1, the corresponding computer program

(Appendix I), and for typical relationships of M-E and M-¢ as generated c

by the program for a given beam cross section are shown in Figs. 5.8a and

5.8b.

5.3.3 T-Beam

When a T-beam does not act as a rectangular beam, its compression

zone may be divided, for the purpose of this analysis, into two parts as

shown in Fig. 5.9; part (1) with dimension (bxc), and part (2) with dimen:"

sion (b-b') x (c-t). The corresponding general equations for the T-sections

are given below:

Equilibrium Equations

a. l f'bc(d-S1c) +A'f' (d-d")

c s s


d-c E = --E 5 C c

Es = c-d" --E c c

c-t Efb = --E

c c


f = f (E )

I 5 5 s

f' = fs (E5 ) 5

A f = a s 5

(5.12)

(5.13)

(5.14)

(5.15)

Input

E = E u C = C

TABLE 5.1

Flowchart - Rectangular and T-Beam Under Pure Bending (Generate Beam M-¢ Curve)

section properties, stress-strain curves of concrete and steel

I Given Eu t

• I Assume N.A. C = 0.4 D I t ....

Compute internal force, XF I

if XF > 0, C =

XF < ETS = 0.001 No

Yes if XF < 0, C =

Compute internal moment, XM curvature ¢

~ + 0.0001 u

remain the same position of N.A. as trial value for next ultimate strain

t I Eu = 0.006 I

t Compute maximum moment Mu and yield

moment My

t Compute ACI moment .1

+ I Plot M-¢ and M-E curves I u

+ I END I

119

C - DELTA

C + DELTA

r

CJ

~ ~~

i---

----

----

----

----

----

----

----

----

----

----

----

----

----

----

----

----

----

----

----

-'

zm

l-

-I

I ~CJ

CJ

l--I

CJ

~(D

'---

-'

r- 7 C

J ~

CJ

wo ::

j1

~

0 2:=

CJ

CJ

0 (\J

CJ

B C

I Nl

o (l

N)

BP

(I N

) T

(I

N)

RS

(SQ

. IN

) A

SP

(S

Q.

IN)

=1

0.0

00

=

18

.00

0

=0

.00

0

=0

.00

0

=1

.80

0

=0

.00

0

FO (

KS

I)

=l!

. 0

00

F

r (K

SI)

=

60

.00

0

RECT

ANGU

LAR

BERM

LI

GH

TWEI

GH

T -;lI

'.;

RN

RL

YT

ICA

L--

---+

+r+

RC

I C

OO

[---

----

**

**

BE

RM

NO

.=

2.

B·

T

D l

I.A1

-@

O~I----_r----_r-----+-----+-----+----_+----_+----_+----~----~----~----~

0.0

0

.00

1

0.0

02

0

,,0

03

O

uOO

L!

0.0

05

S T

R R

IN,

Ec

Fig

. 5

.8a

M-E

re

lati

on

ship

fo

r re

ctan

gu

lar

beam

. e

0.0

06

I-'

N o

· 0 0

r-'.

0

2CD

I---

-l

I B

(I

N)

=1

0.0

00

B

o

(I N

) =

18

.00

0

[La

BP (

1 N

) =

0.0

00

T

0

T

( IN

) =

0.0

00

I-

---l

0

AS

(50.

IN

) =

1.8

00

~(D

RSP

(50.

IN

) =

0.0

00

D

L!A

'-

---.r

FL

J(K

Sl)

=L!.O

OO

FY

(KSI

) =

6C

J.0

00

~ 0

1 RE

CTRN

GU

LRR

BERM

LI

GH

TWEI

GH

T 0

WO

:::

t'

>

0 ~

0 0 0 (\j

I ;-

RN

RL

YT

ICR

L--

---+

++

+

ReI

C

OO

E--

----

-**

**

o

· -

BEAM

N

LJ.=

2

.

a l~--___+----1----4----+---

I +

0

.0

0.0

00

4

0.0

00

8

0.0

01

2

0.C

J01

6

0.0

02

0

CU

RV

RT

UR

E (

1/

IN)

Fig

. S

.8b

M-~ re

lati

on

ship

fo

r re

ctan

gu

lar

beam

.

0.0

02

4

~

N ~

122

e <:0 (l,)

Ell' ,!:l I

E-t

~ or-!

tn (l,) U ~ 0 ~

0)

Lf)

0

b.C or-! ~

___ • _______ ~ .. ' ___ r. .• --------- .. -

123

a1 = aCe ) c

Sl = S(ec)

(5.16) a2 = aCefb )

S2 = S(efb)

where

Curvature

<P = e: c c

efb is the strain at the joint of flange and web

(5.17)

The logical flowchart is the same as that for the rectangular section as

shown in Table 5.1, and the corresponding computer program is given in

Appendix 5.1. Typical computer outputs are shown in Figs. 5.l0a and 5.l0b.

5.4 COLUMN UNDER AXIAL LOAD AND BENDING

5.4.1 Behavior of Columns

A reinforced-concrete column may reach its strength under enumerable

combinations of bending moments and axial loads. These may vary from a

maximum axial load, Po' which is associated with zero bending moment, to

a maximum moment, M , o associated with a zero axial load. It can be assumed

that axial load and bending moment act independently of each other on a

given cross section of a column. Figure 5.11 shows a schematic representa-

. tion of a column under a bending moment, M, and an axial load, P, and

Fig. 5.12, a statically equivalent system in which M = Pee The locus of

the combinations of axial load, P, and moment, M, at which a column

attains its strength is usually represented by an interaction diagram or

P-M diagram. A typical diagram is shown schematically in Fig. 5.13 for a

rectangular cross section. Because the eccentricity e = M/P may be

u C.J

,r--

---.

. C

)

:z ee

,

I---

---i

CL-

e'J

C.J

I-

----

-i C

J

~

IL'

'---

--'

f-

Z

CJ

CJ

lLJ

CJ

-:.1"

'

~

0 ~

I'

, . .J

CJ

,:.:;

('J

C:J

Cj

--

r-~---

' , , ,,,,, , ,

,-

+-+

-.

--/.

-

/ 1:[

1 (K

~ Il A

I. C

1oq,

T

[ 0

=\ 8

" ,-

I ~

'1 ~~ ~ 1

) :.c

GC

l. U

L)d

+

I-bL

RM

I

LIC

HTW

EIG

HT

.-/

'1N

fiL iT

I r;R

L --

---+

++

+

. R

CI

[ODt----~--****

I BE

"AM

NO.~

2"

: ,

, ,

, ,

, ,

,

Cl.

[\

[I

'L-IO

1

It;:}

1

[I

L-' 1-,

'") r;

J

l) L

. D

.DD

3 O.ClO~

ST

R R

I N

~ Cc.

Fig

ure

5.l

0a

M-E

re

lati

on

ship

fo

r T

-bea

m.

c

~

8=

30

" J

T=

4" .2

-

~ .1 j[

A5=

b.'l

lf-tn

B~ /

2"

, ,

, ,

[I

'-"IL

)' r;

t_

J L

....

'I

I-II

I f::

I.

" _J

L

! 0

I-'

N

.j::

.

,.---

-...

z f---

i

(L

f---

i

Y

'---

-"

~

:z

lLJ

2:=:

0 ~

C)

C)

C)

CC> " -.-!

" .. .-!

CJ

L0

CJ

C)

0 ~-ji

c·'

-.-!

C

)

0 ('J

CJ

t-+-+~\

F (1

::K

5 IJ

r:'

1' (K

SI)

T

-LiF

F,M

l_

I CHT

WE

I CHT

II C

q,.',

:..

:. -J:

-

ll' l'

:..:.

G l:l

" CJ

0 C!

qNRLYTICRL-----+++~

R ['

I .,

-:, '-1

r.-'

. ll~c-------KKKK

1:3 E-

C M

N

(1 ..

'-

j L

_.1:

•

--c

.,

6=

30

"

TI-

---,

~ T

=4"

D

= IS

"

i __

I. As

= ro. 2

Lf ;n

2.

B'='

~"

t::J

! ::

----

+--

---1

:-. ---f-----+--~-+---___l

U .

C! 0

" C!

0 0 ~

(J ,

CJ 0

0 3

0

. [!

0 1

2

CI ,

[! 0

1 6

U

, CJ

[) 2

U

0 u

Cl 0

;2 l!

C·U

RV

RT

UR

E (1

/ I

NJ

Fig

. S.

IOb

M-~

rela

tio

nsh

ip f

or

T-b

eam

.

.....

N

VI

126

.-" . . . _<._. __ -4- +~_ •••• _____ ~

. r .

i-:-·':.:: t:-:-~~.'::~-:=:-:-:-:~:-:-- • __ ~ __ -J'~~ __ ~. - ~ •• -- ._.. , +_ ••

----. ---.--.~-- .

-+ • _ •• ,--

-, - .---~.-+-- ~---...

Fig. 5.11 Column under axial and flexural loads.

Fig. 5.12 Column under uniaxially eccentric load.

- +--_ ..... --- -- .. -.~.-----.-- •. _- --, ---.- -- . ... ... _-- \_.-

.. - .....---..... _. -- _. --------.......,....---.--<-~--------- -.u .. __ . __ ._ .. _. _ ~ _ _ .... ·'1++---·----·-

. - --- '" - -.- - - ~ . -

_ .•.... , _.-.. -.. -.-.-"'-'-"' _ .. ;-- ---::::-:- .-- -t·"· --~ -:.::-: .:-" . T

-- _ .. -- --,- .. ---.-.-"-~ .. ----- ..... _--_.- -_ .. -- . . --; - -'--

6<l.t Glnt~lJ1:J.iJure:.=~:~., . B(M~~Pg)':" ....... --.-:: .......

. .;..-... ---.-.-.~-..... --....... - .. --~- .. __ .......... ' .

e.e!,dl"$:·.~T ... . ::- I ..... : .. : ..'" ..:;:mom~ht' ; tvl :::: .

-.~~.:~~~~~:- .. ~:.::.~ :-:~~~'~T·~::.::-.::.::t=---::~· ... -, .. :.: : .. ~ i:-:-~"-'::'::- :.:.-.

. ~AxiiCt t~:·· :--:..-.: .. - ... -... -.~ -.~:~

:"h>t\5 i () n ' ..... ~:. • . ..'_ :~~: ... ~ .. __ . ______ ._._

Fig. 5.13 Interaction P-M diagram.

127

obtained from any two values of M and P, an interaction diagram can

also be defined as the locus of the maximum values of axial load which can

be applied to a short column, when the eccentricity of the load varies from

zero (concentric compression) to infinity (pure bending). Line OA in

Fig. 5.13 represents a case in which the external load varies in such a way

that the values of axial load and moment increase in the same proportion.

The slope of this line is the eccentricity e = MjP. The strength of the

column for this case would be MA, PA. Line OC represents an arbitrary

loading history where axial load and bending moment do not increase in the

same proportion. Point B represents the balanced condition; i.e., under

simultaneous action of the axial load, Pb , and moment, r~, the column will

reach its ultimate strength, simultaneously with the tension steel reaching

its yield stress.

The interaction diagram shown in Fig. 5.13 corresponds to a column

with a given cross section under single bending. For the cross section

shown in Fig. 5.12, the plane of bending coincides with a plane of symmetry.

The failure patterns of column under loading can be characterized as

compression failure, balanced failure, and tension failure as shown in

Fig. 5.13. Their failure patterns correspond to those for overreinforced

beams, balanced beams, and underreinforced beams, respectively, as discussed

in the previous section.

Two types of loadings on columns will be considered in the following

sections: uniaxiallyeccentrical load, i.e. axial load plus single bending,

and biaxially eccentrical load, i.e. axial load plus biaxial bending.

128

5.4.2 Column with Uniaxially Eccentrical Load

For a rectangular column reinforced by A and A' along the two s s

faces parallel to the axis of bending as shown in Fig. 5.14, the analysis

procedure can be divided into two parts: one corresponds to the case where

the neutral axis falls inside the column section, and one where the neutral

axis falls outside the section. The corresponding general equations are

given below:

(a) The neutral axis falls inside the column section


af'bc + A'f' c 5 5

A f = P s 5

(S .18)

(5.19)

where the plastic center and the centroid of the cross-section are

assumed to coincide, and e is the eccentricity of the equivalent

axial load with respect to the centroid.


e: s c-d" = -- e: c c


f = f (e: ) s s s

f' = f (e: I ) s s . s

a = a(e: ) c I

(S.20)

(5.21)

(5.22)

. -- ~ -.. --.~ '-'. --,.

_.-.. !

----. ----.---_ ...

.. _----- ------.-~ .. ----.. ,

... ---_ -.. - -e __ .e_ ..

~ --.---~------~-- --

_____ . __ c __________ , _ ~_. ____ ._

Fig. 5.14 Forces in uniaxially eccentrically loaded column.

129

130

Curvature

(5.23)

(b) The neutral axis falls outside the column section


alf'bc - a2f'b(c-h) + A' f' + A f = P c c ss ss (5.24)

(5.25)

+ A'f' (h/2 - d") - A f (h/2 - d') = s s 5 S Pe = M

where e is the eccentricity with respect to the centroid of the cross

section.


where

e: s

c-d = -- e:

c c

c-d" e:' =--e: s c c

is the strain at bottom fiber of section.


f = f (e: ) s s s

f' = f (e:') s s s

a l = a(e: ) c

<X2 = a(~)

Sl = see: ) c

S2 = S(~)

(5.26)

(5.27)

(5.28)

131

Curvature

¢ = e: c' c (5.29)

A typical computer graphical output for a typical column load-moment inter-

action diagram is shown in Fig. 5.15 where the values corresponding to the

ACI-197l code procedure are also plotted for comparison. The locigal flow-

chart for generating the interaction diagram is given in Table 5.2 and the

corresponding computer program is listed in Appendix II. For the column

with reinforcing bars placed along four faces will be treated as a special

case of biaxially eccentrically-loaded columns with the angle of

eccentricity equal to zero. The biaxially-loaded column will be discussed

next.

5.4.3 Column with Biaxially Eccentrical Load

A. Introduction

For a given column of square cross-section, the most complicated

calculations result when the load is applied at an angle of eccentricity

between the symmetric case of 0° and 45° measured from an axis of symmetry

of the cross section. The major difficulty inVolves the location of the

neutral axis which is not necessarily perpendicular to the moment arm of

the applied load in the general. case; furthermore, its location translates

across the cross section with variations in the magnitude of the applied

eccentricity.

A typical biaxially eccentrically loaded column is shown-in Fig. 5.16,

and an arbitrary neutral axis is defined in Fig. 5.17. The analysis pro-

cedure is described in the following sections; the corresponding logical

flowchart is shown in Table 5.3 and the computer program is listed in

Appendix III. A typical computer output is shown in Fig. 5.26.

TABLE 5.2

Flowchart - Uniaxially Loaded Column (Generate Column Interaction Diagram)

Input section properties (steel located along two faces), stress-strain curves of concrete and steel

Given Pfix = 0.05 Pmax

132

Assume N.A. C = 0.4Dt--------E--------..... ~

Compute internal force XP r-------------~~----------------~

ABS (XP - Pfo ) < EST = 00 Oll--~.c. lX

Yes

Compute internal moment, XM curvature cp

o

Yes

c = C - DELTA

if XP < Pfo , C = C + DELTA lX

Compute maximum moment M ,E from M vs. E curve u u c

PfO = Pf . + 0.05 P lX lX max

No

Yes

Plot interaction curve

TABLE 5.3

Flowchart - Biaxially Loaded Column (Generate Column Interaction Diagram for Given Angle of Eccentricity)

Input section properties, material properties, angle of eccentricity, 8

. 0

't I Compute Pmax I '" I Given P fix = 0.1 P I max

" I Given Ec = 0.002 : ... .. , Assume m = -1/tan8 , b = o l

0 I ,

+ ~-

I N.A. Y = mx + b I

t Compute total internal force XP, and internal

moments Mx' My' 8t = tan- 1 (M/My )

~ 11 xp > P fix' 0 = 0 + t>>-lABS (XP - P r ) < ETA = 0.1 No lX

if XP < P fix' b = b - 6b Yes

lABS (8 t - 80

) < ETB = 0.10 No ICI. = tan- 1 m, 6C1. = 8 ~ 8 , , t 0

Yes ,

f Compute M = iN +M ¢ [m = CI. - 0.2 6C1. , x y ,

I +

rEc = Ec+O.OOOll

+ I Ec>0.0061 No .. , • Yes

J Compute maximum moment from M-E curve J c -+

I Pr =Pr + 0.1 P lX lX max ." I P. ~ P I No flX max I p-

Yes

IPlot interaction diagram

+ I END I

133

~ I'

(">

Co r.

c, '" o U

\

C)

.:I'

C) '"

::r:e:

. X

c>

!D

. -;

( -<

EJ

u... >u

'-

'-(t\

o c>

<D

<:>

o '.'" o D

(.

o C>

o D

O.O

C

NORM

RL

WE1

GH

T R

NR

LlT

JCR

L--

--+

++

+

RC

I C

OO

E--

----

~**~

O.O~

0.0

8

0.1

6

O.2

u

0.2

4

MI

(Ff:1

*BxH

*Hl

COLU

MN

NB.

S

(J

N)

o ( 1

N)

H (1

N)

RS

(5

0.

IN)

RS

P (

50

. IN

) FO

(K

SI)

FY

(K

S J

) 10"

L#7

-

jt I

~.

-I

::: 1

. =

=10

.00

=8

.00

=

10

.00

=

1.2

0

=1

.20

=

4.0

0

=5

0.0

0

1/

10

O.2

:l

C.~2

o. jr

;.

c.~o

Fig

. 5

.15

In

tera

cti

on

P-

M d

iag

ram

of

un

iax

iall

y l

oad

ed c

olum

n.

0.4

.4

f-'

<.N

.j::.,

.~.-- -I -------~--- _ .• --- .• , _. --

-- ~ -- --. - .. ._--- '. - ---

--.-- .. --~~

.. ~--~@ ,AS2 .• - : ~: ~ ~S I ~~.~~-;~-~~--.-.. - --- - .

. -- -- - . . --- -.----- -------- ---_. -----.-. . - .. ...

_ pi": ~~~ ~ ~~~-:: . ..;-::--. • -: ~ -~~-+ ~.-

- ~ . - - -- - ~ - - -. H .

- - ------- - -- -~ - -- .-~~ .... --~

Fig. 5.16 Column cross section .

. -~+---

.. ~

o -. ~.· •. -:~X

~ I· I~.·. __ ::~:'I ::-~~~~~~+I_"-l~ ~~-+._._

6~ 63 62 hI 60 5958 57

Fig. 5.18 Finite element mesh.

.o~ \:..:,r. ,

135

y

. ~. X ' .. - K f'tA. · . ~:::-~ ~ ~ .·:g=~m )C-~'b';o~

t\1~ ... ~-'-':.'-~-. -..... ---~.-- ..... _._,_ .. . . - .,

-::-'-:O~ .:.-~~..:..::~.. .. .-.. _. -_._. :

Fig. 5.17 Neutral axial position.

.y. -.;-.

Fig. 5.19 Determining the compression zone .

.... -:.~.y- . .. r--:-:

I . 2 I 8 7

Fig. 5.20 Numbering sequence in compression zone.

B. Analysis procedure for generating the interaction diagram for a given angle of eccentricity e (Fig. 5.16)

The following steps are used:

1. Divide the column cross section into finite elements as shown in

136

Fig. 5.18 for each element an identity number is given; the coordinates

of the element are computed with respect to the coordinate system

fixed at the centroid of the cross section.

2. Compute the cross section area and the coordinates for each rein-

forcing bar.

3. Assume a starting value for the concrete strain, sc' acting at

the extreme compression corner.

4. Assume a slope, m, for the neutral axis as shown in Fig. 5.17. A

good initial approximation is .obtained by positioning the neutral

axis perpendicular to the moment arm of the applied load. For the

~quare column in which the angle of eccentricity is 0° or 45°, the

neutral axis will be always perpendicular to the moment arm.

5. Assume a value, b, for the intercept of the neutral axis \vith the

y-axis as shown in Fig. 5.17.

6. Determine the concrete elements wh~ch are in cDmpression, i.e. those

elements located on one side of the neutral axis by the method

explained in Fig'. 5.19, and renumber the elements under compression

in sequence as shown in Fig. 5.20. Note that the tensile strength

of concrete is neglected.

137

7. Determine the strains at the center of each reinforcing bar and each

concrete element from the assumed planar strain distribution as shown

in Fig. 5.21. A corresponding concrete stress distribution is shown

in Fig. 5.22.

8. Determine the forces on each concrete element and each reinforcing

bar, and their corresponding moments with respect to the two coordinates

axes, Le.:

Force on each concrete element (see Fig. 5.23).

c. = A.f. 1 1 1

(5.30)

where C. = force on concrete incremental area

1

A. = concrete incremental area 1

f. = stress on the centroid of element 1

Moment about coordinate axes due to concrete element.

= c.y. 1 1 (5.31)

= C.x. 1 1

Total force and moments from concrete elements.

Cc = L C. i=l,n 1

M = I (Mxc) i xc i=l;n (5.32)

M = . L (Myc)i yc l=l,n

Force and moment on each reinforcing bar.

(5.33)

Fig. 5.21 Strain distribution.

---- . \ -, --. ...... \ .......... \ ..... \ \ \ N.A.

Fig. 5.22 Concrete stress distribution.

Fig. 5.23 Concrete force acting on each element.

138

(Mxs\ ~ (CsJ i * (Ysli I (MyS) i (C s ) i * (xs ) i

Total force and moments from reinforcing bars.

Cs = I (C ). i=l,k s 1

M = . I (Mxs)i xs l=l,k

M = . I (Mys) i ys l=l,k

Total force and moments from concrete and steel.

p = C + C c s

M = M + M x xc xs

M = M + M Y yc ys

Check if P is sufficiently close to (P ) given o

Resultant of internal moments.

The position of eccentricity from centroid of cross section.

The inclination angle of eccentricity.

Check if 6t is sufficiently close to (60) given

139

(5.33) Cont'd

(5.34)

(5.35).

(5.36)

(5.37)

(5.38)

140

9. Iteration procedures - the triple iteration procedure will be described

as follows:

(a) Iteration on interc~pt, b, of the neutral axis with the y-axis as

shown in Fig. 5.24:

i) b = 0 for first cycle analysis

ii) b= lb+Ll.b b - 6b

if

if P > (P) given ~

P < (P) given ~ for 'second cycle analysis

(b) Iteration on slope, m, of the neutral axis as shown in Fig. 5.25:

(c)

i) Assume that the neutral axis perpendicular to the

ii)

applied moment arm as shown in Fig. 5.25 for the first

cycle analysis, i.e. m = -1/tan6 and compute the ino -1 clination angle of the neutral axis, a = tan (m), as

shown in Fig. 5.25.

-1 m = ---;--:---:-tan (a-6a) for second cycle analysis

where Ila = (0.2) (8t - 80

) from Eq. (5.38) .

Iteration on extreme compressive strain, s , for'determining the c

maximum-moment capacity at the given loading.

i)

ii)

s = 0.002 (arbitrary starting value but not greater c

than s ). o

s = s + 0.001 for the next aSSigned strain up to 0.006. c c

.... _.1 __ _ j. -

---J -

---I. . - {.~-.--.----.. ~~~~ f··· - ~.- .

.~ p

--+----+--~~--~--;----x

Fig. 5.24 Iteration on position of neutral axis.

, , . . ------- .. ~----....----~---.--.-- -.~ .-.-_.-_ ..

~~=;~¥:~~y=f~:. ,~.{-;= ------+

'::=F:

.-_ .-1-----~ .. ----

:-:-~-;=:~::::l::-~·.::~~:~=-r·.-::~ -- -

. f:~~~=~~f=~:~~~~~~~~-~NA~f.tOOY···

Fig. 5.25 Iteration on inclination angle of neutral axis.

141

. i I . . .

I : i I

:)

) ,

.1,

I _

i

Fig

. 5.

26

Typ

ical

in

tera

cti

on

dia

gram

of

bia

xia

lly

loa

ded

colu

mn.

...... ~

N

143

CHAPTER VI

DUCTILITY OF REINFORCED-CONCRETE MEMBERS

6.1 INTRODUCTION

A ductile material is one that can undergo large strain while resist

ing loads. When applied to reinforced-concrete members and structures, the

term ductility implies the ability to sustain significant inelastic defor

mations without a significant variation in the resisting capacity prior to

collapse. This ability is, generally, quantitatively described by a

parameter called "ductility factor" or "ductility ratio."

The objectives of this chapter are: to investigate the ductility

behavior at the level of the materials, sections, and structural concrete

members; to formulate a precise method for computing the corresponding

ductility factors; and to compare these factors with the ACI code require

ments.

6.2 DUCTILITY OF PLAIN CONCRETE

6.2.1 Unconfined (Uniaxial Loading)

The typical stress-strain curves of plain concrete under uniaxial

loading for normal weight and lightweight concretes are shown in Fig. 6.3.

It can be seen that for the same type of concrete the higher the strength

the lower the ductility, and,for the same level of strength, normal weight

concrete is more ductile than lightweight concrete. For a given material

the ductility can be measured, for example, by the toughness, i.e., the

area under the complete stress-strain curve, or by the strain ESO at 50%

of the peak load on the descending portion of the stress-strain curve [28].

Since the ductility of concrete depends on the steepness of the descending

As

section

section

, I ~- ty----.l

strains

strains

144

Fig. 6.1 Calculation of curvatures.

_. _____ :::.':"' __ i.~--"-_._-__ ,

1-: '-----. -~-==~:-~-.-- -- ~.~~.~~- ~----:.-:.~~-:--. ; ~:...:~.~---+-~-::-~~- .. -- ~-:. r:.-~:_ t -_~~-t .. - --~--- i

Fig. 6.2 Typical cross sections.

I !

'

i " I

i :--

Fig

. 6

.3

Con

cret

e st

ress

-str

ain

cu

rve.

i. o~l

i !

I :

i I

.....

.j::.

V

I

146

portion of the curve, it may also be defined as the slope at a certain

characteristic point on the descending portion; such a point may be taken

as the inflection point as described in Chapter II.

6.2.2 Effect of Confinement

Roy and Sozen [28] showed that although the load-carrying capacity

of axially-loaded concrete specimens is not significantly improved by the

use of rectangular ties, there is a considerable increase in ductility as

expressed in the function of s50 (the strain at 50% of peak stress on

the descending portion) as follows:

or

where

= 0.015 h s

~ s

h = dimension of square column

s = tie spacing

p" = confinement reinforcement ratio

The above investigation was based on compressive specimens with the dimen-

sions of 5" x 5" X 25" and the concrete strength on prisms of about 3500 psi.

In a related investigation Shah and Rangan [33] showed that stirrups

are most effective in increasing the ductility of overreinforced beams as

compared to fibers and compression reinforcement. The investigation was

based on the flexural test pf a beam specimen with the dimension of 2"x

3" x 34" and the concrete strength of 4000 psi.

147

6.3 DUCTILITY OF REINFORCED-CONCRETE BEfu~ SECTIONS

Using the moment-curvature diagram of reinforced concrete sections,

Cohn and Ghosh [10] defined the ductility factor as the ratio of ultimate

curvature to yield curvature, i.e., 11 = <P 1</>. The yield curvature <p u y Y

is defined as the curvature at which the tension steel reaches its yield

stress, and the ultimate curvature <Pu is the one associated with the

strain €, load P or moment M at the ultimate stage. The ultimate u u u

stage, depending on each particular case, is defined as the point of maxi-

mum moment or load, or the point at which a specified strain is reached.

6.3.1 Calculation of Yield and Ultimate Curvatures According to ACI Code

A typical reinforced-concrete section with tension reinforcement

only and its corresponding strain distribution at the yield and ultimate

stages are shown in Figs. 6.la and 6.lb. The curvature calculations are

given by the following equations:

At Yield

€ Y

d (l-k) (6.1)

where k can be determined by assuming the linear stress distribution on

the concrete compression zone and satisfying the equilibrium condition of

forces. This leads to:

A s p = bd (6.2)

E s n = E c

148

At Ultimate

£

<fiu u (6.3) = c

where

£ = concrete compression strain at crushing of concrete u or at ultimate moment

c = depth of compression zone at ultimate, can be deter-

mined by satisfying equilibrium equation of forces

pf d = -.X. (6.4) f' 0.85 Sl c

where

{0.85 -0.05 C:~O -4) for f' < 8000 PSi} c Sl = (6.5)

0.65 for fl > 8000 psi c

The addition of compression reinforcement to a beam will shift the

neutral axis upwards, and increase the ultimate curvature substantially,

although it has little effect on its yield strength or yield curvature.

The term c in Eq. (6.4) then becomes

where

(p -p::~) f d c = l

0.85 Slf~

A' p' s

= bd

fl = stress in the compression reinforcement s

From Eqs. (6.1) and (6.3), the ductility factor, 11, is defined as:

(6.6)

(6.7)

149

For research purposes, it is possible to determine ~ and ~u ~ from the ~y

moment-curvature diagram which was derived in Chapter V using an iterative

calculation procedure based on the actual'stress-strain curves of concrete

and steel, and for which a computer program was written.

6.3.2 Effects of Concrete Stress-Strain Curve on the Ductility of Reinforced Concrete Beam Sections

It has been shown in previous investigations that the confining

reinforcement increases the ductility of concrete at the material level, and

in turn, increases the flexural ductility of reinforced-concrete sections.

Furthermore, most investigators agree that lateral confinement does not

significantly affect the peak stress of the 0-E curve and its ascending

portion, but it influences the descending portion drastically. A case

study is presented here to explore possible inferences between material

and sectional ductilities for two different concrete stress-strain curves

with identical ascending portions but different descending portions. The

section properties for the above case study and the actual stress-strain

curves of concretes and steel are shown in Figs. 6.2, 6.3 and 6.4. The

computer program described in Chapter V was used to generate information

on the curvature and ductility factors.

The effect of different stress-strain curves on the ductility of

beam sections are shown in Figs. 6.S for various tension and

compression steel ratios. For underreinforced beams, the ultimate moment

always occurs at a concrete compressive strain far beyond the peak strain;

thus the descending portion of the stress-strain curve of concrete becomes

important for ductility purposes. The effect can be seen in Fig. 6.S for

I---

-i

•

U) ~r

~~--

----

----

--~-

----

----

----

----

---

~~

0-

• o

(n

0 .-

--i

cn

lLJ

u

cc~

~

cn Q~

o CD

Q

o ::fI

-Q

o (\j

8

CHRR

ACT

ERIS

TIC

POIN

TS

FY

(KSI

J=

60.0

00

FSU

(KSI

;=

103.

753

FSF

(KS

l)==

89.0

30

EM I

(KS I

) ==

29

35

2 ~

EMSH

(KSI

l==

1208

. EY

=

0.

0020

4 ___ .

_.ES

H.

. ==

O

._Q

0860

IS

O --

.---.-

-=

O.

068

15

[SF

= 0

.151

93

GENE

RRLI

ZED

RNRL

YTIC

CURVE--~--

01

I I

I I

I I

I I

I

•

o. OGO~

0.0

8

0.1

2

0.1

6

0.2

0

Fig

. 6

.4

STRR

IN

-..... -

Str

ess

-str

ain

cu

rve

of

grad

e 60

st

eel

wit

h f

=

60

ksi

. y

.....

U1 o

~~~------~------------------~.

--- ----12.

. - '-'---'~-'-'--'----'" ....• -

fO"

•• As-

As •••

--- ____________ ~~ _ _O_ Liqhtwei9ht

2.0"

151

Fig. 6.5 Comparison of ductility between normal and lightweight concrete.

~, I' ..

152

the beam with tension steel only (note that the two concretes have the

same ascending portion). The effect of compression steel on the improve

ment of ductility can be observed from Fig. 6.5. The comparison

of ductility between theoretical and ACI code values for normal and light

weight concretes are shown in Figs. 6.6 and 6.7. Note that the higher the

reinforcement ratio, the higher the influence of compression steel on

ductility. The safety margin on ductility as provided by the ACI code for

lightweight concrete is very small as can be seen from Fig. 6.7a for p' = o.

The effect of compression reinforcement on the ductility of the

section as analytically derived is shown in Figs. 6.8 and 6.9 for normal

and lightweight concretes where the moment-curvature diagrams are plotted.

It can be seen that the addition of compression reinforcement to a beam

has relatively little effect on its yield moment or yield curvature. It

does, however, greatly increase the ultimate curvature and lead to a

substantial increase in the ultimate moment.

The effect of axial loads on the ductility of the section is shown

in Figs. 6.10 and 6.11 for normal and lightweight concretes, respectively.

These figures show that the ductility decreases drastically when the axial

load approaches about 30% of the axial load-capacity of the section. The

corresponding load-moment interaction diagrams for the section with

p'/p = 1 are shown in Figs. 6.12 and 6.13.

6.4 DUCTILITY OF REINFORCED-CONCRETE COLUMN SECTIONS

Reinforced-concrete column sections under uniaxial and biaxial bend

ing are examined here. The effect of axial loads on the moment-curvature

diagrams of the reinforced-concrete sections (uniaxial bending) are shown

1'1...-----r--..;----r----r------,-------r

ICf' , --,------.,

->-. - -" _ ••• _-_ •• -.<.~ --- .• _" .. ~- ~-. -

• • As'

As -... 2

II a

-- -----'--~_I__----'----...... ~-----'-----'-----_t

---..:-~~..:.-~-4~-_ ___I~_-~~--~-~~::_"_::=__~~"'E!o.. _ ____ .5_ _,' l.(? __ ,_-__ '.5 ----- ______________ _____ Re in forcemen t

Fig. 6.6 Comparison of ductility between theoretical and ACI values for normal weight concrete.

153

/4~----~----~~----~~---T------'

_.- ::... _.-t-_ '--::::: ---4 ....... . ---0- --c--:---

10"

• • A' - 5-

As ••••

2D" -

~ 2 -c:r-....:/B=+---~~~----'------'-----'----...., ... -.. - -~ -- .~

-----___ --::.-~.6· -~ --~-~--.. _:~~'4_

l....S"='. ·~~/~.1-:0';"'.=::;;';"~:_;';';" __ ~ __ ~l+.5= __ -:-_~:-'-;:';"'.:~-:_2:-+-#O'::"'" __ -"';" _ _ -. __ -___ ~ ___ :=I-__ -;::-==. __ ~3~_ ,0_ -- -------.:--.-------:geinforcement-rafcQ"r --S------:---:--:----~---:::-

Fig. 6.7 Comparison of ductility between theoretical and ACI values for lightweight concrete.

154

1,

I I,

Fig

. 6

.8

Eff

ect

of

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pre

ssio

n r

ein

forc

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t on

du

cti

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r no

rmal

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eigh

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Fig

. 6

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ect

of

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sion

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Fig

. 6

.10

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ial

load

s on

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diag

ram

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r no

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Fig

. 6

.11

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s on

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ram

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r li

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=

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In

tera

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gram

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r li

gh

twei

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t co

ncr

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I-' '" o

161

in Figs. 6.10 and 6.11. It can be seen that for axial loads greater than

30% of the axial load-capacity, very little ductility remains.

The effects of biaxial bending on the ductility of column sections

are investigated in the following example. The materials properties are:

for normal weight concrete, f' = 5 ksi and 9 ksi; for steel, f = 60 ksi. c y

The sectional properties are shown in Fig. 6.2b for a square column having

the dimensions of 10" x 10" with four #7 bars. The eccentricity angles due

to biaxial bending are 0°, 22.5° and 45° measured from the centroidal axis

of the cross section.

The effects of axial loads on the ductility can be observed from the

moment-curvature diagrams, for e = 0°, 22.5° and 45°, respectively, plotted

in Figs. 6.14, 6.15 and 6.16 for normal weight concrete with f' = 5 ksi. c

It can be seen that when the axial load approaches about.30% of the axial

load-capacity, the resulting moment capacity reaches a maximum value which

corresponds to the balanced moment Mb , of the column load-moment inter

action surface, while the ductility factor reduces to a value of one. As

the axial load continues to increase, the resulting moment decreases; the

moment-curvature diagram becomes steeper on the descending portion, and

the section is considered less ductile.

The effect of the eccentricity angles on the ductility can be

observed from the moment-curvature diagram for f' = 5 ksi c and 9 ksi,

respectively, in Figs. 6.18 and 6.19. Therefore, everything else being

equal, the higher the eccentricity angle the smaller the ductility of

the section.

The effect of concrete strength on ductility can be observed by

comparing Figs. 6.17 and 6.18, where the moment-curvature diagrams in

biaxial bending are plotted. It can be seen that the higher strength

'. I": i .1

·1

; 1

Fig

. 6

.14

Eff

ect

of

axia

l lo

ads

on d

ucti

lity

: M-

~ d

iagr

am f

or

e =

0°.

! ,i;

.....

0-

N

163

•

,,-

I

i .i

Fig

. 6

.15

. E

ffec

t o

f ax

ial

load

on

du

cti

lity

: M

-<P

diag

ram

fo

r e

o =

22

.5

. I-

' (]

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.;:..

i

Fig

. 6

.16

E

ffect

of

ax

ial

load

o

n d

ucti

lity

: M

-¢

dia

gra

m fo

r e

= 4

50

• .....

C

l\ tJ

l

Ii! : !!

. t

L::

. i

Fig

. 6

.17

E

ffect

of

eccen

tric

ity

an

gle

s on

d

ucti

lity

: M

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dia

gra

m f

or

f'

= 5

000

psi

. c

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i i , ' i

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. 6

.18

. , '" : .

6

'-

• 8

I I

! /. 0

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I 1/

.2.

¢ (.

00

1/ In

)

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ect

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tric

ity

an

gle

s on

du

cti

lity

:

: "

. ,

M-~

dlag

ram

f

=

c 9

00

0 p

si.

, ..,..

(]

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, i J I,

, ,

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I

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,1

Fig

. 6

.19

E

ffec

t o

f co

ncr

ete

stre

ng

ths

on

colu

mn

inte

ract

ion

P-M

dia

gram

.

;J il; "

I i

'I' J i"jl·,

'i::

1 I !

I-'

0\

00

169

concrete section (9 ksi) shows about the same ductility as the· lower

strength section (S ksi). This result is different from that observed

at the material level where the higher the strength the lower the

ductility.

The effects of concrete strength on the column load-moment inter-

action diagram are shown in Fig. 6.19. For pure bending, the higher

strength concrete does not significantly improve the moment capacity of

the section. However, as the axial load increases, the higher strength

concrete shows a much more beneficial effect by improving the column load

and moment especially at axial loads greater than about 30% of the axial

load-capacity for pure compression.

The comparison of column load-moment interaction diagrams derived

theoretically and according to the ACI code (where € =0.003) are shown u

in Figs. 6.20 and 6.21 for the uniaxial bending case. It can be observed

that there is little difference in the diagrams especially at lower axial

loads. This result strongly suggests that the ACI predictions for moment

and load capacities are accurate; however, it was noted earlier for beams

that although the moment capacity predicted by the code is satisfactory,

there may be substantial error in estimating the curvature of the section

at ultimate capacity.

6.5 ROTATIONAL CAPACITY OF HINGING REGION AND MID-SPAN DEFLECTION OF REINFORCED-CONCRETE BEAMS

6.5.1 Introduction

Both the limit design of structural concrete and the plastic design

of structural steel stem from a recognition of the inelastic behavior of

structures at high loads. The plastic design methods developed for steel

o g N~I--------------------------------------------------------------------------------------'

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. 6

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.16

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B (

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(I N

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(I

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(5Q

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(KS

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t I

ED

• I

•

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. 6

.21

C

ompa

riso

n o

f th

eo

reti

cal

and

the

AC

I co

lum

n in

tera

cti

on

,

P-M

dia

gram

fo

r fc

=

9000

p

si.

O.l

lll

I-'

--..J

I-'

172

concentrate on the formation of sufficient hinging mechanisms in a struc-

ture. Little attention is paid to the magnitude of the strains at the

individual hinging sections as the redistribution of the moment proceeds.

The strains that can be developed in the concrete and reinforcing steel

in a reinforced-concrete member are considerably less than those which

can be developed in a mild steel member. It is therefore necessary to

consider the 'deformation of the hinging region in any theory of limit

design for structural concrete.

6.5.2 Ultimate Strain at Extreme Fiber of Concrete and the Approximation of Moment-Curvature Diagram of a Section

The 1971 ACI code assumes that the ultimate moment capacity of a

reinforced-concrete section always occurs when the compressive strain at

the extreme fiber of concrete reaches a value of 0.003. This assumption

is considered to be a good approximation for calculating the ultimate

moment capacity, but it is quite different in calculating the ultimate

curvature especially for underreinforced beams. The strain at ultimate

moment may range from 0.0040 to 0.0048 as indicated, for example, in

reference [39].

It is apparent that the knowledge of the moment curvature relation-

ships for reinforced-concrete sections is necessary if calculations are

to be made of the total inelastic or plastic rotations for a structural

concrete member. In Fig. 6.22, two moment-curvature relationships are

shown for the column case under study (pure bending) with concrete

strength equal to 5 ksi and 9 ksi, respectively. A straight-line approxi-

mation for the portion between yield moment and ultimate moment is

considered to be representative, and will be used in the following analysis

, ,

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i !I,t

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: ,I

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00

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. 6

. 22

Approx~mate

stra

igh

t li

nes

for

M-'<j

> d

iagr

am.

I :, .-.

-...

j

VI

174

of the rotation capacity and the mid-span deflection for reinforced-concrete

beams [23].

6.5.3 Mid-Span Deflection of a Reinforced-Concrete Beam at Ultimate Capacity

A. Mid-span loading (Fig. 6.23a).

The steps suggested to calculate the mid-span deflection (see also

[23 n are as follows:

1. Draw the moment diagram for the simply supported beam as shown in

2.

Fig. 6.23b.

Determine the shear span z, and the plastic hinge length x , p

indicated in Fig. 6.23b where

z .Q, d' b . d ; 2; lstance etween maXlmum moment an zero moment

M ; ultimate moment u

M : yield moment y

as

(6.8)

3. Determine the moment-curvature relationship. The actual M-¢ diagram

for a given sectional and material property has been simplified by two

straight-lines meeting at the yield point as shown in Fig. 6.22.

4. Draw the curvature distribution diagram along the beam as shown in

Fig. 6.23c.

5. Calculate the elastic deflection at the mid-span (Fig. 6.23d)

(6.9)

,._-.. --------- --.------~-- -.- -- ._"-.... -- -- -;--- - .

---j-

----~ - , -

.-t~uu ,-~c,-~-'-. '"'rh-' C·:.-,.---··· ."-. -. -.-: _-i_Cr' .-.. ---

---tiT':' --:~=-~f - -----_.. - _. -

-. . ..... -

-=-- ~'~--:¢f;j--~-~~-. . - . - .. - -

"r:._ ---- -------.---- ---------. - .. -..:..--=-:----....:.:..::----.-~-----~-. ---,-- -

.. -. -.---_.- ..:..--:-...-~=-=-:---~--~~ .--- .---:------_ .. _----- --.----------~------~- -----_________ .---C-. __ _

ir·--~,:-~~~¢~···· ----~c-~_-- •. -_;~.~-; . __ --=~rtd-}~Ela-s-+t'Z---rifcif.,~n-c---~~--~

---------~------~-~.---~.-=-~-~~- --..:...-.-- ._-- ---:-:...:: . ..:.:.---"---- -----.-::.... .. ---~----.--.-.. -

.iC----'-------"'-'-_________________ .

---t- --~ -

Fig. 6.23

.~----'---- -~. ---- ...

Calculation of mid-span deflection fOT beam loaded at mid-span.

,175

where

~y = yield curvature

6. Calculate the inelastic deflection at the mid-span (Fig. 6.23e)

~. = [$U -(:~) $yJ Xp 1

1::.. = ~2i (;) = G) (Xp)ru -(:~)$y] 1

where

~u = ultimate curvature

7. Calculate the total deflection at the ultimate stage

I::. = I::. + 1::.. u e 1

8. Calculate ,the total deflection at the yield stage by substituting

M = Minto Eq. (6.9) which leads to: u y

9. Calculate the ductility factor in terms of deflections where

Ductility Factor I::.

u = Il

y

B. Third-point loading (Fig. 6.24a)

176

(6.10)

(6.11)

(6.12)

(6.13)

(6.14)

The steps to calculate the mid-span deflection are similar to those

for the beam loaded at the mid-span. The calculation steps are as follows:

1. The total mid-span deflection at the ultimate stage (Fig. 6.24b)

177

_. --" - -- ---- ---------_ -_-__ :-:-, _----___ -:--------------7-------__ -___ -_ -----~-___ -__ -----------

do--------'---- -- ---------------- -:~)tp~r-- ---

~-------~--......;....-....;;:;;",. f--.. --.--.---~--- .. - .. -...

- ------- --- -----------~--. "-_._-----_._--_ .... _-- -'-- ------~------.--- -_._----_ .. _-- ~- -_._-------- "---- ---_._ .. _----- .. . _.- -

----------- - -------- -'----'--

Fig. 6.24 Calculation of mid-span deflection for beam loaded at one-third span.

178

Deflection due to elastic rotation

I::, = [(:~) ty m + (:~) ty m] e2a) e

- (:~) ty m [(I + ;) + (:)]

or

I::, = (:~) tya2 (~!) = D. 9583 (:~) t ya2 (6.15) e

Deflection due to inelastic rotation

1::,. = [tu - (:~) t y] (a +2 Xp) [~ _ (a +/p)] (6.16) 1

where

(Mu -M ) x = y z P M u

z = a = shear span

Total deflection

I::, = I::, + 1::,. u e 1

(6.17)

2. The total mid-span deflection at the yield stage obtained by substitut-

3.

ing M = M u Y into Eq. (6.15) which leads to:

I::, = 0.9583<1> a 2 y y .

Ductility factor = I::, II::, u y

(6.18)

CHAPTER VII

STUDY OF THE ULTIMATE STRENGTH PARAMETERS FOR THE DESIGN OF REINFORCED-CONCRETE SECTIONS

7.1 INTRODUCTION

179

In analyzing the flexural behavior of reinforced-concrete sections,

several key parameters are considered, namely: the strain at the extreme

compressive fiber of concrete, E ; the properties of the compression zone cu

of concrete described by a and 8 (see Fig. 5.5); the strain in the

tensile steel, E ; the load-moment-curvature interaction diagram and the s

ductility factor. The above parameters will be investigated for normal

weight concrete strength from 3 ksi to 13 ksi, and lightweight concrete

strength from 3 ksi to 7 ksi. The stress-strain curves of concrete for

the various strengths considered are shown in Figs. 7.1 and 7.2. The

stress-strain curve for Grade 60 steel with a yield strength of 60 ksi

is shown in Fig. 6.4.

7.2 PARAMETERS a AND 8 OF THE CONCRETE COMPRESSION ZONE

For a given concrete stress-strain curve, two important parameters a

and 8 can describe the properties of that curve at any specified strain.

These parameters are explained in Fig. 5.5. The parameter a relates to

the area under the curve for a given strain, while 8 relates to the

moment arm of that area wi·th respect to the extreme compressive strain, E • c

Since the compression zone .of the reinforced-concrete section can be repre-

sented by these two' parameters, a detailed study on their relationship with

the entire range of strain is needed.

In Figs. 7.3 and 7.4, the variations of a, 8 (or 81) and E C

are

plotted for various stress-strain curves of normal and lightweight concretes.

180

g~--------------------------------------------------------~~----~ -

"

C) c-o

'"

Cl

'" '"

tl C.

.,; en

" " .,; co

C1

'" C r-

a .... " n

'-'

:i;g 0-. <J)

(f)n Wu CC{...; ;nil>

a c-O ::

n

'" .,; tTl

n CO>

0 ('II

a c;

~

o

. 13000 PSt

~NRL IT I C CURVE--

o~ ____ ~ __ ~ ____ ~ ____ ~ ____ ~ ____ r-__ ~ ____ ~~ __ ~~ __ ~ ____ ~~~

"'b. 00 0.05 0.10 0.15 0.20 0'.25 0.30 0.35 0.110 O.US O.SO 0.55 0.60

Fig. 7.1

5TRAIH.I~/lH _10-2

Stress-strain curves of normal weight concrete with f' = 3,000 c to 13,000 psi.

Cl o Q

181

N,~ __________________________________________________________________ --,

C) o <:>

n <>

C) a ,g

" o C> II)

Cl <:>

<,-, Ij)C> WU c;:: •

~~~

c)

'"

'=b.OG G.IO 0.15

7°oopsi.

~NRLYTIC CURVE----~

c.zo 0.25 ~.~O O.jS STRR!N. !N/IN .. W-

0.115 <:.50 G.S5 o.!O

Fig. 7.2 Stress-strain curves of lightwe~ght concrete with f~ = 3,000 to 7,000 psi.

-..,; -8 ~ tn co c:5 II' -~

~.

·2

o.

P.f-- die' be (actuAl· stress blod<) .

f~= 3000 ps': --- ---~

5000 pSL·

/-00 -'>i 0(, ~I+" If I ~ __ g(Ip,fc'hc (ACISJre" bl,?<-k; 0(,::e5) ,- £

n.G\.. fe' = 130oops~ _._ .. - -- - . 'Db, · 90 . _....//,

.. S2.. t\l II

tr:t .. _- .. 80

-.. ,strOlI n.-

182

Fig. 7.3 Relationship of a, 8 vs. Ec for stress-strain curves of normal weight concrete.

-2

"---0 •. /,,/0

183

rf-:-~ p,c 0( for f~= 30oo·ps~ (AC!) r ,I ~f:Q.c- - .-::5~o~00-:t.P2.!SL~.I"' ;.,.----c ~~~--

/.00 --

.qO

-c:Q..

........ go

0- _ _.

- .. f3rfor·~ fc.'= 8{)oo psi and over--.-.--.-. . .. - ----~-.:.·,·.:.-:-·~::.:··:-...::;;f:=.:::-c:::-~i ::.~::-~:--,-c-"~ .. --.-c~::c{Ac..r}·: -.-~.--... --- .. ~- _.

Fig. 7.4 Relationship of a, S vs. Ec for stress-strain curves of light- . weight concrete.

.. , .

184

The corresponding values using the ACI assumptions are also plotted for

comparison. In order to determine the maximum resisting moment of-the

compression zone with respect to the tensile steel, one attempts to maxi-

mize the value of a and minimize the value of S. From the a, E c

relationship, the maximum a occurs at about E = 0.0045 c for most normal

weight concret~ strengths from 3000 psi to 13,000 psi, and at about

E = 0.0035, 0.004, 0.0045 , respectively, for lightweight concrete strengths cu

of 3000, 5000 and 7000 psi. Therefore, the location of the maximum value

of a can be written as follows:

E = 0.0045 ca for normal weight concrete of

f' = 3000 psi to 13,000 psi c

Eca = 0.0035 + 0.00025[(f~/1000) - 3] (7.1)

for lightweight concrete of

f' = 3000 psi to 7000 psi c

In most cases, the resisting moment of the compression zone of con-

crete, using E as the strain at the extreme compressive fiber, is not ca

much different from that obtained using the value of E cu This can be

verified from the following example with p=0.015, p' =0 and f' = 3000 psi c

to 13,000 psi for normal weight concrete and f' = 3000 psi to 7000 psic

for lightweight concrete. Figs. 7.5a and 7.6a compare the resisting

moments obtained according to Eqs. (7.1) for the approximate value of E cu

and the actual value of E It can be seen that Eqs. (7.1) is a good cu

approximation for the value of E as far as the ultimate moment is con-cu

cerned. Figures 7.5b and 7.6b show the corresponding moment-curvature

relationships. It can be seen that using E leads to a curvature closer ca

to the exact curvature at ultimate than the ACI approximation.

I

a a ,-

-...

. C-

~

zro

/-

---l

I CL~

a /-

---l

0

~m

'----/

~

za

0

wo :::

:tt

2:::=

0 L

0 0 0 (\J

o

B lI

N)

=1

0.0

00

o

(I N

) =

1f3

.00

0

SP

(IN

) =

0.0

00

T

( I

N)

=0

.00

0

RC'

(~jO.

IN)

=2

.70

0

J RS

P (5

0. I

N)

=0

.00

0

FY (

K5I

) =

6U

.00

0

RECT

RNG

ULR

R BE

RM

NCJR

MRL

W

EIG

HT

II

20

lO'l

2 7·

1-I

.. ; '+

A~':

:I

n

L.!...

.!J f

~=bOKii

j:= 13

000P

5~

c;9

~--~

T~----_aT~···

RN

RL

YT

ICR

L--

--t+

++

+

RCI

CC

JOE

----

--**

**

50

00

p~t

tc.=O.OOq5

~1

Eeu

o I

I I

0.0

0

.00

1

0.0

02

0

.00

3

0.0

04

0

.00

5

0.0

06

STR

RIN

Fig

. 7

.5a

Mom

ent-

stra

in r

ela

tio

nsh

ip f

or

norm

al w

eigh

t co

ncr

ete

wit

h f'

=

5,0

00 p

si t

o

13,0

00 p

si.

c

I-'

00

(J

1

0 0 ~

0 zm

I-

--l ~ gl

t-1

0

.~

CD

I

'----/

~

ZO

0

wo ::::

tt ~

0 ~ ~I

0 0 (\j

o

0 (.

.urv

at"'t~

£tt

ul

h\'t

(.\t

~

II

GU"'

vatu

\~ a

t tc

::: .

oolt

5

* A

C I

eu

rV'a

. tu

re

Q;t

ul t

i m tl

te.

~~IIIIIIIIIIJ\\:II""""

T

I 5000

psi

RN

RL

YT

ICR

L--

--j+

+++

ReI

C

GD

E--

----

~~~~

BER

M

NO

.:::

I 2"

-

II

20

10"

• .~

As=2

.7 ira

2.

t~=

{,o

\<Sl

..

C. I

, _

As

<f -

Jc. :

: 13 0

00

rs l.

01

I I

I I'

II I

I I

I I

I I

I 0

.0.

0.000~

0.0

00

8

0.0

01

2

0.0

01

6

0.0

02

0

0.0

02

4

Fig

. 7

.5b

CU

RV

RT

UR

E (

1/

I N

)

Mom

ent-

curv

atur

e re

lati

on

ship

fo

r no

rmal

w

eig

ht

con

cret

e w

ith

f'

= 5,

00

0 p

si t

o 1

3,00

0 p

si.

c

I---"

0

0

()'\

.

C)

~ g+I------------------------------------------------------~--------~

zoo

I-

---i

I 0

..-0

I-

---i

~

'---/

o o ill

~

zg

w~

2:

o 2::

o o o (\

.1

o ~I/-

o M

O\'\.\

.eVlt

oX

U\t

l'MO

.:t e

A

ty\O

\'\le

llt

Clt'

Ec =

feo<

.

"* M

Ot}

H'tt

t ce

t ~G

= o.

oc>3

R~RLYTICRL----t++++

RC

I C

OO

E--

----

KK

K*

B

ER

M

NO

. ==

1 .

II I

" 20

10/1

I. .:r A

5~'l·'

1 i:!

t!j=

bO

K~l

f;=

7000

psi

5DO

() p

si

30

00

Psi

I I

\

0.0

0

.00

1

0"0

02

0

.00

3

0"0

04

0

"00

5

0.0

06

Fig

. 7

.6a

S-TR

R I

N

Mo

men

t-st

rain

rela

tio

nsh

ip f

or

lig

htw

eig

ht

con

cret

e w

ith

fl

= 3

,00

0 p

si t

o

7,0

00

psi

. c

.....

00

....

...

j

~ ~JI __

____

____

____

____

____

____

____

____

____

____

____

____

____

____

~:: __

____

____

____

____

__ __

zro

10

" I-

--i I CL

a

I---

i

~

'----/

a a co

~

za

a LL

J a =

t'

L o ~

a a o (\1

a

20"

o c..u

r"Q.

TuV-

-e ct

X lL

Itit

n<L

t'€

A

c...urvClht~-e

Qt

tc =

tee(

.

* A

C.I

CMYV

~+u,

{>

oJ:

u..\-

\-ln'

\0-1

e

5.!.=

7000

psi

. 50

0 0ps

ICR

L- 1-

--+

++

+

<0

[---

---~~~~

o. ~

1 "

• A

5=2:

1 "n

~ f~

=bO

KSl

0'

'I'

'I'

I I

I I

I I

0.0

O

.OO

Oij

0.0

00

8

0.0

01

2

0.0

01

6

0.0

02

0

0.002~

CU

RV

RT

UR

E (

1/

IN)

Fig

. 7

.6b

M

omen

t-cu

rvat

ure

rela

tio

nsh

ip

for

lig

htw

eig

ht

con

cret

e w

ith

f~

=

3,0

00 p

si t

o 7

,00

0 p

si.

.....

00

0

0

189

7.3 CONCRETE STRAIN Ecu AT MAXIMUM MOMENT

The strain at the extreme compressive fiber of concrete at ultimate

are plotted for various concrete strengths and reinforcement ratios in

Figs. 7.7 and 7.8 for normal and lightweight concretes, respectively,

assuming a rectangular section. It can be seen that except for the case

of singularly reinforced sections with a high reinforcement ratio, the

ACI constant value E = 0.003 is over-conservative. The use of cu

E = E = 0.0045 for normal weight concrete and Eqs. (7.1) for lightweight cu ca

concrete seem more appropriate.

7.4 STRESS BLOCK DEPTH PARAMETER Bl FROM THE ACI CODE

The actual compression zone of concrete at ultimate (maximum moment)

is represented by an equivalent rectangular stress block by the ACI code

as shown in Fig. 7.3. Two conditions should be satisfied if this trans-

formation is equivalent in terms of the statics of the section, namely,

the magnitude of the resultant force and the location of that force should

remain unchanged. These conditions are expressed as follows:

For the equilibrium of forces:

alBl = a

or

Bl a =

eLl (7.2)

and

For the equilibrium of moments:

B = 1 2B (7.3a)

The value of as stipulated by the ACI code can be substituted into

Eq. (7.2) and leads to:

190

...... -.- -. ..,....-------------------------------r.

• 005

. . '-.00'1

- -.- .()() 3

.()O I

\\1--005 -4..J .- - ..

~ .- F;.{)Olf ~"------:s .003

.....,

~::O.005.~

.~.~~.~~~ ........ -.-•. -se_-... -_-.-.. --i.~~ .•..•• ~_=~~~~ . ---tE.].I---------·---=-·~- .. '

~ = o.lsfh \H q -._~. __ A ...& -.. ' -.~ . . ~--~ -.. . - - Acr- ---_________________ L ____ _

Rect'Clt13IA.l<lr beQ.t'Yl ( b::=\ 0" 'J d = , 8 II )

~'=o ~~60 l<s~-

~ ~--~ ~. ---_.. -~-~~--.

--~~-... ' 7 1 - - --~--____ £-.51_" ________________ _

.. . . ACI . -- --- .. .... _. _. __ .. _.:-.. __ ._ .. _ .. _-. --------_£_-----~--- ......

c:t .(JOt, .- ... -.---- ..... _-... - .. - -- .. --... -.. ----.... - .. -

.. -- .005

-. -. . , -- .. • f) o~ --- --. __ .. _ ... _ .. _. __ ~ .. _,,_~.'C';'--.. _ _c:'--.. ~~----.. --.. - .. -.. ---- ...... -.. -. -.

... ____ :.9!) I ~O~~---.·Z-· .. -··~-·.-... ~+~.~~-b~. ~---B±-~~-/~o~~--~--~-/~~·

--'--." -_.,._.-_. ---.--~--~------. ---.. -----.------.-.~ - ---,... .. ------.~.-.---- ... -.-.---- .. ---_ .. ~--.-.-.-.-

-.-. __ . _____ :~. __ :.~)~~---coticr .. ere-strenJth,. fc/~ ksi--- -. -:..:1 J._ . ~.. - - • ---

Fig. 7.7 Relationship of s vs. f' for normal weight concrete. cu c

'.000

.. • D05 ~ ...

.. • 004

.. 002

.. ~ .006 C,.\,)

r:; e oo5 .. '+-

d . ~·.OD4

. 9 . .:£~... .. ~ . • 003

0' " ,.~ 05" 1"-...... .

...... ~.~ .-- . .......... _ ... _.-_._--t. A C '[

"d~~'OD2 s: .

';::; ,O:;..o;:;..:/:...J-_""--_.L----1_--J.._-l-_-l-_.l..-----li----i._-t ~ .

. i- . if.:) .. ~.ODb

"·-.005

." ~"'." · .. ~._ ... ~~.1.~). _::-~--.'---. . :--.- . ~ Ac.r __ .. ~. _ .. ~~.. . ... ~ _____ L __ _ ·~···-- ... o03

... ~ ..... ~. !)O~

Fig. 7.8 Relationship of E vs. fe' for lightweight concrete. cu

191

192

(7.3b)

Furthermore, the ACI code stipulates the value of 81

as follows:

81 = 0.85 for f' < 4000 psi c

81 = ( f'

0.85 - 0.05 10~0 - 4) for 4000 psi :::: f~ :::: 8000 psi (7.4)

81 = 0.65 for f' > 8000 psi c

The comparison of ~\ values as given by Eqs. (7. 3a) , (7.3b) and

(7.4) are shown in Figs. 7.9 and 7.10 for various concrete strengths and

reinforcement ratios. For normal weight concrete, it can be seen that

the ACI formula (Eqs. (7.-4)) gives a good prediction when the value of 81

is derived from the force equilibrium (Eq. (7.3b)) as. indicated by dashed

lines, but is much less accurate when 61

is derived from the location

of the resultant force (Eq. (7.3a)) as indicated by solid lines. Since

there is no way to narrow the differences of 61 as obtained from

Eqs (7.3a) and (7.3b) for a rectangular stress block representation, it

may be possible to examine some average represention.

In order to explore the effect of /\ on the ultimate moment and

curvature, one should consider the equilibrium equations of forces and

moments of the section. From the ACI code approach and for underreinforced

beam, these equations are as follows

For force equilibrium:

A f = 0.85 f' ba s y c (7.5)

which leads to:

A f a =

s y 0.85 ff b

C

(7.6)

193

. /;0

·8

- 0 . f=o.oo5··· - .. j(. ~:'O.S~b

A- ~=o.15fb

1.0 f' -=0.5 f

.. ----

_ .•. 'f.t..."......;..,..j...-.;...~.-.;...~-4-.-.;.....I.----J---'--=----t.--:-I=--'-----:-'::'--.;:---:-!. _ 0 2. B 10 I~

-- .-. -.-.-.-------.,----. - --_ .. -----:"-- -l---=:--~--··~---:-----··~·:-·-·-~· -- -.---~-- .. _-" ----_ .. - .-~.---.--- -,--.-'--- -.

_._ .... ___ ...... __ cc_ ConcreteS1re"9ch J. fc . ,ksL.-.

Fig. 7.9 Relationship of $1 vs. f~ for normal weight concrete.

194

---·8

-- , -

. !L: o.5 ... --~ -- ---

/~2 .,?' ..... . -.~=I.O

·:-··-.~~8

Fig. 7.10 Relationship of B1 vs. f~ for lightweight concrete.

195

where

or (7.7)

For moment equilibrium:

M = A f (d _ a) u s y 2 (7.8)

The corresponding curvature at ultimate is given by:

(a)

(b)

¢ = u

E CU -- = c

SlECU

a (7.9)

From the above expressions the following remarks are derived

The ultimate moment M from Eq. (7.8) is determined by the value of u

.a and the sectional properties, and the value of a is determined by

A , f , f' and b only (Eq. (7.6) ); therefore, changing the value s y c

does not affect the calculation of M for the same properties. u

The ultimate curvature is determined by E and cu from Eqs. (7.9)

and (7.7), so changing the value of 81 does change the calculation

of ¢. A detailed study on the ultimate curvature will be discussed u

in. the following sections.

(c) From Figs. 7.9 and 7.10, a better fitting value for 81 representing

an average can be expressed as follows:

Sl = 0.85 for f~ ~ 4000 psi

81 = 0.85 - O. 05(i~io - 4) for 4000 psi ~ f~ ~ 6000 psi (7.10)

for f~ ? 6000 psi

Eqs. (7.10) are also shown in Figs. 7.9 and 7.10.

196

7.5 THE ULTIMATE MOMENT, ULTIMATE CURVATURE AND DUCTILITY FACTOR

The effect of concrete strengths and reinforcement ratios on the

ultimate moment of a given section is shown in Figs. 7.11a, 7.llb and

7.12 for normal and lightweight concrete, respectively. In the case of

low tension reinforcement ratio, i. e. , p = O. 005, the presence of com-

pressive reinforcement does not significantly increase the moment capacity

of the section; the reason is that the section is in tension failure.

As the tension reinforcement increases (see Tables 7.1, 7.2 and 7.3), the

effect of compressive reinforcement on the moment capacity becomes apparent

in the theoretical analysis, but not in the ACI code approach (Fig. 7.1lb).

This is due to the fact that the ACI code does not take into consideration

the strain-hardening portion of the stress-strain relationship of the steel.

As indicated in Table 7.3, the strain in the tensile steel at ultimate for

the case of f~ = 13,000 psi, P = 0.7SPb and p'/p = 1 reaches the value

of 0.029, the corresponding stress in the steel is 84 ksi, while the ACI

code still uses the yield stress f = 60 ksi in calculating the ultimate y

moment. Similar results are observable when lightweight concrete is used

(Fig. 7.12).

The effects on the ultimate curvature are shown in Figs. 7.13 and

7.14 for normal and lightweight concrete, respectively. Note that the

more the tension reinforcement, the lower the ultimate curvature. The

presence of compression reinforcement increases the ultimate curvature

especially for high tensile reinforcement ratios. It can also be seen

in these figures that the ACI code always gives the lower bound values

of ultimate curvatures. A comparison, of formulas for calculating the

ultimate curvature (Eq. (7.9)) according to the ACI code and to the equa-

tions proposed herein for s cu and is described next.

114

---12 .~ .... ~-

~ 10 ~

- -- - f~ =1'3 kS(

. _ =-~~~_: JL~_l kSL. ---5{=-:3 KS~

197

\if -

·0-. --

" .• 20

--_Q~--~.I.----~~--~---.L...:-~--~--~ o · 0 I .02. . 03 .04 . 05 .06 ... __ .. _-... _--._--- --- ---_ .... -.. -.... ..... .. "'. -- -

-Reih{orc.ement m,tio; .f

Fig. 7.lla Effect of compressive reinforcement on Mu vs. p relationship for normal weight concrete.

.. '. ... ..... . fie1nforcement

TIl eore tica./ . ACI··

Fig. 7.11bComparison of the theoretical and ACI values of Mu vs. p relationship for normal weight concrete.

198

6

4

2 .-.1

<r.)

~-

o 00 ~8

-2

----- rheoretic~1

---ACT

--f~=lKSl: - - --.: fc :-3 ~s;-

ratc'o, f--

Fig. 7.12 Effect of compression reinforcement on ~ vs. p relationship for lightweight concrete.

199

5~----~----~------~-----+-------------

.. -. 3

----2

.S: .".-.1l .. _

<::J ~--I -:0

----- Theoretica;.l

--ACI"

.... fc~.L~ KSL._. __ ....... ___ · .. · -"? -.-- ---_._ .. _._-_._--_ ... _ ... _-... _-_ ........... ---- .. - . ~

.. 0 ';"5 --~ " .. ---.-- .. - ... ---- .. -.-.... ---. ---. _ ...... -~~ f(=\'3 \<'Sl

..-=-.:~!~: 1l<S{ ... .~

"'~'4 ~ .. :-+- -~

.. ~- .... --~ ~ ...

u "'-3

..~~.~~.:.:. h' :: j kS(

. ·A· '~if= I -"" ... X' . ~'f -:;:. 5' ,'"

...__._ CD o/~ ::: 0

0,

___ ._. ____ Q._ .. .:::~_'O'_L~~.~~.:.f!~_~~_~_~ ._. ____ .. _____ . __ ..... -05

Fig. 7.13

.,_ rte,inforcemen t ra..1io~ f

Effect of compression reinforcement on ¢ vs. p relationship for normal weight concrete. u

200

----,--4-~---,.....--.;.,--..----.,....----,.....-----r

Fig. 7.14 Effect of compressive reinforcement on <p vs. p u relationship for lightweight concrete.

201

202

For normal weight concrete of f' = 7 ksi, p' /p = 0.5, the analytical c

value of curvature at maximum moment is

¢ = 0.00151 rad/in u

(Table 7.2)

The corresponding ACI code value is

(E:) - 0.003 cu ACI -

CEq. (7.4))

(Table 7.2)

The proposed value is

CE:cu)prop. = 0.45 (Eq. (7.1) )

(Sl)prop. = 0.75 (Eq. (7.10))

(¢u)prop. = (SlE:cu)EroE·

(¢u) ACI (SlE:cu)ACI or

(0.75)(0.0045) = (¢u)prop. = (0.7)(0.003) (0.00084) 0.00135 rad/in

(7.11)

Figure 7.15 shows the comparison of ultimate curvatures calculated from the

above two methods for normal and lightweight concretes. It can be seen that

the proposed method is generally superior to the ACI code in predicting the

ultimate curvature.

The effects of concrete strengths on the ductility factor are shown

in Figs. 7.16 and 7.17 for normal and lightweight concretes. Note that for

the case of p proportional to Pb' the ductility factors are not affected

by the variation of concrete strengths if the ACI code approach is used.

This seems to be also the trend in the theoretical results. The ACI code

~ t1

• ..-1 Q)

:=;: ~

ell ~ 0 z ~

...c:

OJ)

• ..

-1 Q) ::: 4-l ...c:

OJ)

• ..

-1 ....

:l

-----

TABL

E 7

.1

Mom

ents

, C

urv

atu

res

and

Du

cti

lity

Fac

tors

at

Yie

ld a

nd U

ltim

ate

for

p =

0.0

05

Rec

tan

gu

lar

beam

, f

= 6

0 k

si

(b

=

10

",

d =

18"

) y

Th

eo

reti

cal

f' ~

At

Yie

ld

c (k

si)

p

M

~y

£ M

~u

y

cy

u

0.0

86

3 0

.16

8

0.0

01

0

1223

1

.84

6

3 0

.5

869

0.1

63

0

.00

09

13

40

2.2

5

1.0

87

3 0

.15

0

.00

08

13

90

2.4

7

0.0

87

5 0

.16

0

0.0

00

9

1392

2

.39

2

5 0

.5

878

0.1

56

0

.00

08

14

40

2.6

2

1.0

88

0 0

.15

0

.00

7

1460

2

.73

0

.0

883

0.1

54

0

.00

08

15

09

2.9

17

7

0.5

88

5 0

.15

2

0.0

00

7

1514

2

.95

1

.0

886

0.1

49

0

.00

07

15

15

2.9

7

0.0

89

0 0

.15

0

0.0

00

7

1587

3

.38

6

9 0

.5

890

0.1

48

0

.00

06

15

70

3.2

6

1.0

89

0 0

.14

6

0.0

00

6

1562

3

.18

0

.0

895

0.1

47

0

.00

06

16

33

3.8

22

11

0

.5

895

0.1

46

0

.00

06

16

14

3.4

1

.0

895

0.1

44

0

.00

06

16

02

3.3

0

.0

899

0.1

45

0

.00

06

16

54

4.2

12

13

0

.5

899

0.1

43

0

.00

06

16

53

3.5

7

1.0

89

8 0

.14

2

0.0

00

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N o V1

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206

l<S~_~_~~ ____ ~::::= ____ ~.JbeoreJ£ca, L _______ _ -': -: - ~-.:- .' -- - - _ .. -.. : . _. - ~

,..---_ •.. '--- A C[-~~::.~----=--- ---

-~-- -.~ :.~-=: ::: ~·-~.-==~-~~.-.:::::::-l-=:·~-~~~~:~~~:·~~- - -_ .. -.-. . .. 7

------ _._----- -.-=-~...:--, --.--:....--~---:-=-::~~;~-~~ .. -=.---,.---. -~-:-=-:--:-:-.-- ---:--_.: -- --------~--.----.---- ._-o ... .-.-, -- L - -~~_Cr::~C~:_:

' __ .O~

Fig. 7.15 Comparison of ultimate curvatures from the theoretical ACI and proposed values.

207

---- :c~=L~J~-~~----TheofeticaJ -'.- -------------~~--c_~_--7_-~=~ ~~~~~~.::±-:- _cc--:

IiJ.ACI --- .. ~-.--. ----.- .--~- _. __ ._--_._----_.,

'-~--IO

._---- --. -----

.-- 2· --- ---------

o

f'l.

----~ /0 -~-~, 8

l .

0&-::0. _'" . __ .. __ . ---~ --- - -----::~ ~ ~:.:::-=-~-.:: :.:--:... :..-~...;;. - ~ ..;...~-::.::.....::..::-....: -* ...:.-=:..-.=-:;----

--=}t~-~= • s::- -: -- --.:.---=:--=------- --'----- -, _ I - - -, . . _ ----- -

A-~; \-0. ;"'<.:: ... _ ~~-=:.c~c-='i::.:::i =~~-_ <D~----~e~------~0~------~0~------~0~------0

4 ~_L 6~ - - B-c .. -c.!.t!~ __ ~ __ _ -- ~-~C6n~rete-·Stre;;9i:h~-fc'} ~-'<Sl '.

/LI

Fig. 7.16 Comparison of ductility factors between the theoretical and ACI values for normal weight concrete.

. ~ .' . ---. --- .. - - .. . --- - - . _. --":-·-~'l----- -=-:-::.-~ ... --.---.-.-.-------~---.---~-.- -_.-. _.,,- - .----~ .. -.- ---- -----. ---, -".- - .. --- -

-- ---~-~-~--. - --'-.-... ;-.--.. ~.----:::----. ---'.~ --.-------------

--. ~: .-. __ ~:-~ __ .~;; .:~:~------;.t~~~~~_·_ c. ___ -~ .:.---.~.::: OtS8b --.-. -:..·-~~~:.~-c~ -----.... --.---~ ..

- 0 - -

(.) ~-. 0 ~ =. 0 15·· . --- -.-. --- _. -- .

. ~~+_----~r_~~~------~------+_----~

~_J~-~~=~;;~~+r;_rL~;-IE;;.f:~

/0

ksi

Fig. 7.17 Comparison of ductility factors between the theoretical and ACI values for lightweight concrete.

208

209

values give a lower bound of ductility factors as expected due to lower

values of ultimate curvatures. The additions of compression reinforcement

increases the ductility factor. For a given reinforcement ratio, the

higher the concrete strength the larger the ductility factor. The lower

ductility of higher strength concrete at the material level as compared

to regular concrete does not imply a smaller ductility factor at the

sectional level.

CHAPTER VIII

SUMMARY AND CONCLUSIONS

Based on the results of this study, the following conclusions can

be drawn:

1. The complete stress-strain curve of ccncrete in compression including

the descending portion can be obtained using the simple experimental

technique developed in this investigation; this technique eliminates

the effect of the testing system during unloading of the specimen

and is adaptable to any testing machine.

210

2. High strength, normal weight and lightweight concretes do not fail in

a brittle manner (provided special precautions are taken during

testing) and exhibit a reproducible descending portion of the stress

strain curve.

3. Most analytical relationships representing the stress-strain curve of

concrete in compression are based on the characteristics of the

ascending portion only and do not consider the characteristics of

the descending portion. The analytic expression for the stress-strain

curve should contain constants which reflect not only the properties

of the ascending portion but also those of the descending portion.

The presence of lateral reinforcement can then be accounted for more

rationally.

4. An inflection point was generally observed along the descending portion

of the stress-strain curves. For normal weight concretes strains at

the inflection point and at the peak did not change substantially with

the compressive strength; whereas for lightweight concrete these strain

values increased almost linearly with strength.

211

5. There is some indication that the coordinates of the inflection point

may bea property of the material and thus are useful in the analytical

characterization of the descending portion of the stress-strain curve.

6. The analytical expression developed here for the stress-strain curves

of concretes in compression correlates well with experimental observa

tions up to strains of 0.020.

7. Based on data of 50 random specimens of Grade 60 reinforcing bars

tested according to ASTM standards, the observed yield strength of

reinforcing bars with a specified yield strength of 60 ksi can vary

from about 60 to 75 ksi.

8. The analytical expression·proposed to describe the entire stress-strain

curve of reinforcing bars with yield strength of 60 to 75 ksi shows

close correlation with experimental observed curves.

9. The computerized nonlinear analysis model (developed in this study)

which uses the actual stress-strain properties of steel and concrete

with general assumptions of equilibrium and compatibility, leads to

acceptable predictions of the structural behavior of reinforced

concrete sections; this includes ultimate loads, moments, curvatures

and ductility factors as reported in the technical literature for

beams and columns.

Based on the analytical results derived from the use of the computerized

model to predict the structural response of reinforced concrete section,

the following conclusions are drawn:

J

212

10. Although the shape of the descending portion of the stress-strain

curve of concrete has little influence on the ultimate moment of

reinforced concrete sections, it does significantly affect curva

tures and ductility factors for both normal weight and lightweight

concretes.

11. For the same specified concrete strengths and everything else being

equal, members reinforced with normal weight concrete exhibit a higher

ductility factor than those with lightweight concrete. Similar

ductilities can be achieved by adding compressive reinforcement.

12. The presence of compressive reinforcement in beams increases the

ductility factor but has little effect on the yield moment and curva

ture for both normal weight and lightweight concrete sections.

13. Given a reinforcement ratio of the section, the use of high strength

concrete does not significantly improve the moment capacity of

flexural members. It has however a substantial influence on the

load and load-moment capacity of column.

14. As the tensile reinforcement increases in reinforced concrete sections,

the effect of compressive reinforcement on the moment capacity becomes

apparent in the theoretical analysis, but is not reflected in the ACI

code approach. This is due to the fact that the ACI code does not

take into consideration the strain-hardening portion of the stress

strain relationship of the tensile steel. This strain always exceeds

the onset of strain-hardening for high compressive reinforcement

ratio; thus leading to a stress higher than the yield stress.

213

15. The increase of axial load leads to a decrease in the ductility factor

for columns. When the axial load approaches about 30% of the axial

load capacity of the column, the ductility factor in bending approaches

one, and the resisting moment capacity reaches its maximum value which

corresponds to the balanced moment, ~, in the column load-moment

interaction diagram.

16. For a symmetrically reinforced column under biaxially eccentric loading,

the effect of the eccentricity angle becomes significant only around

the balanced conditions as shown in the load-moment interaction diagram;

the eccentricity angle tends to decrease the load capacity and the

ductility of the column section.

17. Although high strength concrete as a material shows lesser ductility

than low strength concrete, the ductility at ultimate of reinforced

concrete sections made with both materials are not significantly

different.

18. There is evidence that the strain on the extreme compressive fiber of

concrete at ultimate behavior of flexural members can be approximated

by the value of 0.0045 for normal weight concrete with strength from

3 to 13 ksi and by the values of 0.0035, 0.0040 and 0.0045, respectively,

for lightweight concrete of 3, 5 and 7 ksi strengths. This conclusion

is substantially different from the ACI recommended value of 0.0030

for all concretes.

19. The parameter Bl for the depth of the rectangular stress block defined

in the ACI code, does not affect the calculation of the ultimate

moment of under-reinforced beams, but affects the calculation of the

ultimate curvature. Since two values of Bl can be derived from the

214

actual stress block by satisfying two static conditions, an average

value of Sl = 0.75 for both normal and lightweight concretes with

strengths greater than 6 ksi, was found adequate in predicting

ultimate curvature more accurately than the ACI recommended value.

215

REFERENCES

[1] ACI Committee 318, Building Code Requirements for Reinforced Concrete (ACI 318-71), American Concrete Institute, Detroit, 1971.

[2] Barnard, P. R., "Researches into the Complete Stress-Strain Curve for Concrete," Magazine of Concrete Research, Vol. 16, No. 49, Dec. 1964, pp. 203-210.

13] Behr, R., Goodspeed, C., and Romualdi, J., "Analytical and Experimental Investigation of Ferrocement Sandwich Panel,"

[4] Bresler, B., "Reinforced Concrete Engineering," Vol. 1, John Wiley & Sons, Inc., New York, 1974.

[5] Bresler, B., "Lightweight Aggregate Reinforced Concrete Columns," ACI Symposium on Lightweight Aggregate Concrete, New York ACI SP 2 9-7, pp. 81-130, April 1971.

[6] Brock, G., "Concrete: Complete Stress-Strain Curves," Engineering, London, Vol. 193, No. 5011, 1962, p. 606.

[7] Burns, N. H. and Siess, C. P., "Plastic Hinge in Reinforced Concrete," Proceeding of ASCE, J. STD., Oct. 1966, pp. 45-64.

[8] "High Strength Concrete in Chicago High-Rise Buildings," Preliminary report published by Chicago Committee on High-Rise Buildings, 10 South Wabash Avenue, Chicago, 1976, 63 pp.

[9] Cismigiu, A. and Dougariu, L., "Automatic Design of Strength and Ductili ty of Reinforced Concrete Members," Rev. Roum. Sci. Tech.Mech. App1., Tome 21, No.2, Bucarest, 1976, pp. 165-181.

[10] Cohn, M. Z. and Ghosh, S. K., "The Flexural Ductility of Reinforced Concrete Sections," S. M. Report No. 100, Solid Mech. Div., University of Waterloo and Publications International Association of Bridge and Structural Engineers, Vol. 32-2, 1972.

[11] Corley, W. G., "Rotation Capacity of Reinforced Concrete Beam," Proceeding of ASCE, J. STD., Oct. 1966, pp. 121-146.

[12] Desayi, P. and Krishnan, S., "Equation for Stress-Strain Curve of Concrete," Journal of the American Concrete Institute, Proc. V. 61, March 1964, pp. 345-350.

[13] Ferguson, P. M., "Reinforced Concrete Fundamentals," John Wiley & Sons, Inc., New York, 1973.

[14] Fintel, M., "Handbook of Concrete Engineering," Van Nostrand Reinhold Company, 1974, p. 229 - p. 245.

[15] Ghosh, S. K. and Cohn, M. Z., "Computer Analysis of Reinforced Concrete Sections under Combined S,ending and Compression," International Association of s'ridge and Structural Engineering, Proceedings, Vol. 34- I, Zurich, 1974.

216

[16] Goldberg, J. E. and Richard R. M., "Analysis of Nonlinear Structures," ASCE, SDJ, VoL 89, No. ST4, Aug. 1963.

[17] He imd ah 1 , P. D. and s'ianchini, A. C., "Ultimate Strength of Beams Reinforced with Steel Having No Definite Yield Point," ACI, J. Dec. 1974.

[18] Heimdahl, P. D. and Bianchini, A. C., "Ultimate Strength of s'iaxially Eccentrically Loaded Concrete Columns Reinforced with High Strength Steel," ACI SP50-4, pp. 93-117.

[19] Hognestad, E., "A Study of Combined Bending and Axial Load in Reinforced Concrete Members," University of Illinois Engineering Experiment Station Bulletin No. 399, 1951.

[20] Hognestad, E., Hanson, N. W., and McHenry, D., "Concrete Stress Distribution in Ultimate Strength Design," ACI Journal, Proceeding Vol. 52, No.4, Dec. 1955, pp. 455-480.

[21] Kulicki, J. M. and Kostem, C. N., "Inelastic Response of Prestressed Concrete Beams," International Association of Bridge and Structural Engineering, Vol. 35-11, Zurich, 1975.

[22] Mattock, A. H., Kriz, L. s'., and Hognestad, E., "Rectangular Concrete Stress Distribution in Ultimate Strength DeSign," ACI Journal Proceedings, Vol. 32, No.8, Feb. 1961, pp. 875-920.

[23] Mattock, A. H., "Rotational Capacity of Hinging Regions in Reinforced Concrete Beams," Proceedings of the International Symposium, Flexural Mechanics of Reinforced Concrete, Miami, Fla., Nov. 1964, pp. 143-181.

[24] Naaman, A. E., "Ultimate Analysis of Prestressed and Partially Prestressed Sections by Strain Compatibility," to be published in the Journal of the Prestressed Concrete Institute.

[25] Popovics, S., "A Review of Stress-Strain Relationships for Concrete," Journal of the American Concrete Institute, Proc. 67, March 1970, pp. 243-248.

[26] Popovics, S., "A Numerical Approach to the Complete Stress-Strain Curve of Concrete," Cement and Concrete Research, Vol. 3, 1973, pp. 583-599.

[27] Ramberg, W. and Osgood, W. R., "Description of Stress-Strain Curves by Three Parameters, NACA TN 902, July 1943.

[28] Roy, H.E.H. and Sozen, Mete A., "Ductility of Concrete," Flexural Mechanics of Reinforced Concrete, Proceedings of the International Symposium, Miami, Fla, Nov. 1964.

217

[29] Saenz, L. P., Discussion of Reference [38], Journal of the American Concrete Institute, Proc. Vol. 61, No.9, Sept. 1964, pp. 1229, 1236.

[30] Salse, E.A.B. and Fintel, M., "Strength, Stiffness and Ductility of Slender Shear Walls,"Fifth World Conference on Earthquake Engineering, Rome 1973, Session 3A.

[31] Sangha, C. M. and Dhir, R. K., "Strength and Complete Stress-Strain Relationships for Concrete Tested in Uniaxial Compression under Different Test Conditions," Materiaux et Construction, RILEM, Vol. 5, No. 30, 1972.

[32] Sargin, M., "Stress-Strain Relationship for Concrete and the Analysis of Structural Concrete Sections," Study No.4, Solid Mechanics Division, University of Waterloo, Waterloo, Ontario, Canada, 1971, 167 pp.

[33] Shah, S. P. and Rang an , B. V., "Effects of Reinforcements on Ductility of Concrete," ACI Journal, June 1970.

[34] Smith, G. M. and Young, L. E., "Ultimate Flexural Analysis Based on Stress-Strain Curve of Cylinders," Journal of the American Concrete Institute, Proc. Vol. 63, Dec. 1956, pp. 597-609.

[35] Smith, G. M. and Young, L. E., "Ultimate Theory in Flexure by Exponential Function," ACI Journal, Proc. Vol. 52, No.3, Nov. 1955, pp. 349-36Q.

[36] Spooner, D. C., "Progressive Damage and Energy Dissipation in Concrete in Uniaxial Compression," Ph.D. Thesis, University of London, Kings College, 1974, 262 pp.

[37] Sturman, G. M., Shah, S. P. and Winter, G .. , "Effect of Flexural Strain Gradients on Microcracking and Stress-Strain Behavior of Concrete," ACI Journal, Proc. Vol. 62, No.7, July 1965, pp. 805-822.

[38] Turner, P. W. and Barnard, P. R., "Stiff Constant Strain Rate Testing Machine," The Engineer (London), Vol. 214, No. 5557, July 1962, pp. 146-148.

[39] Wang, P., Shah, S. P. and Naaman, A. E., Dicussion of "Flexural Behavior of High Strength Concrete Beam," by Leslie, K. E., Rajgopalan, K. S. and Everard, N. J., ACI Journal, Proc. Vol. 74, No.9, March 1977, pp. 143-145.

[40] Winter, G. and Nilson, A. H., "Design of Concrete Structures," McGraw-Hill Book Company, New York, 1972.

[41] Wiss, Janney, Elstner & Associates, Inc., "Tensile Test Load-Elongation Autographic Plots for Associated Reinforcing Bar Producers Technical Committee," WJE No. 74458, Oct. 31, 1975.

218

[42] Berwanger, C., "Effect of Axial Load on the Moment-Curvature Relationship of Reinforced Concrete Members," ACI SP-50, pp. 263-288.

[43] Leslie, K. E., Rajagopalan, K. S. and Everard, N. J., "Flexural Behavior of High-Strength Concrete Beams," ACI Journal, Sept. 1976, pp. 517-521.

APPENDIX I

COMPUTER PROGRAM FOR THE NONLINEAR ANALYSIS OF RECTANGULAR AND T-SECTIONS

219

RELEASE 2.0 MAIN DATE = 77161

CCC CCC PURE BENDI NG qF RECTANGULAR AND T-BEAM (Append \'X \ ) ccc

105 240

250 CCC CCC CCC

III

222 CCC CCC

DIMENSION FF(900),EE(900),XALPHA{900),XBETA(900) DIMENSION 8XM(100),8CURV(100),BEU{100),8FS(100) READ(5,240) JOB FCRMAT(I5) IF{JOB.EO.O) STOP WRITE{6,250) Joe FORMAT( ////. BEAM NO.· ,15//, FO=CONCRETE PEAK STRESS(KSI), FY=STEEL YIELD STRE{KSI) ICONC=2 •••• NORMAL WEIGHT ICONC=l •••• LIGHTWEIGHT READ(S.111) FO,FYolCONC FCRMAT(2FIO.4,I5} READ(S,222) B.D,H, AS. ASP,DPP,8P,T FORMAT(SFIO.4) END OF DATA

IF( ICCNC.EO. 1) GO TO 11 IF(ICONC.EO. 2) GO TO 22

11 wRITE(6, 15)

CCC

CCC

GO TO 50 22 WRITE(6,25) 50 CCNTINUE 15 FORMAT(/t LIGHTWEIGHT-/) 25 FORMAT(/' NORMAL WEIGHTt/)

PF I X=Q.

BTl=0.S5-0.05*{FO-4.) IF(FO.GT. 8.) BTl=0.65

CCC PBAL= BALANCED REINFORCEMENT RATIO FROM ACI CODE PBAL={0.85*8Tl*FO/FY)*(S7./(S7.+FY»

CCC

CCC

IF{AS.NE. 0.) GO TO 10 AS=O. 50*~BAL*8* 0 AS=0.75*PBAL*8*D AS=0.005*B*D AS=0.015*B*D

ASP=O .5*AS ASP=O. ASP=AS

10 CONTINUE CCC·

PAS=AS/(8*D) PAS P= ASP / ( 8* D )

CCC BP=T=O.O FOR RECTANGULAR BEAM ETS=O.OOI

CCC COMPUTE CONCRETE PROPERTIES CCC

CCC

CCC

CALL ALPBET(FO;EO,FF,EE,.XALPHA,XBETA,DE,ICONC,EMC)

IF(BP.NE.O.) GO TO 1000

CCC RECTANGULAR BEAM CCC

w RITE ( 6 , S S 8)

220

20/37/15

888 FORMAT(/' ~ECTANGULAR·BEAM'//) WRI TE (6.666)

666 FORMAT(/' SECTION PROPERTIES'// 1 4X.IHB.IOX,lHD.IOX,lHH.IOX,3HDPP, 9X.2HBP,9X.IHT,9X

221

1 3HPAS. 8X.4HPASP.7X.2HAS, 9X,3HASP, 8X,2HFO, 9X.2HFY/) WRITE(6.777) B,D,H,DPP,8P.T.PAS,PASP,AS,ASP,FO,FY

777 FORMAT{12(FIO.4,lX» WRITE(6,444) ,

333 FORMAT(4(FIO.4,lX),2{FIO.5.1X}.2(FIO.3,lX),FI0.4.1X,2(FIO.3,lX), 1 5X,F6.5/)

444 FORMAT(//' ANALYSIS RESULTS'//

CCC

1 4X,' EU',7X,' ALPHA',5X,' BETA', 6X,' C{INS)',4X. 2· f E S' , 8 X,, ESP', 6 X, , F 5 ( K S I ) • , 3 X,, F S P (KS I ) • ,IX,' X P ( KIP) '. 3 2X,' XM(K-IN)',lX,' CURV(.OOl/IN}',' XM/FOBOO'//)

TXM=O. CPREV=0.4*D NPTS=60 DO 300 1= 1, NPTS EU=O.OOOI*I AN=EU/DE AAN=AN+O.5 N=AAN Nl=N+l ALPHA=X AL PHA (N ) BETA= XBE TA (N) FU=FF(Nl)

CCC TRY NEUTUAL AXlS.C CCC

C=CPREV DEL TA=.O.l YKK=lOOO. GO TO 370

310 C=C+DEL TA GO Te 370

320 C-=C-OEL TA 370 CONTINUE

IF(T.EQ. 0.) GO TO 200 IF(C.GT. T) GO TO 1340

200 CONTINUE E S=EU* ( (O-C) /C) CALL STEELA{FS.ES,FY,EMS) IF(ASP.EO. 0.) GO TO 325 ESP=EU* ((C-DPP) /C) CALL STEELA(FSP,ESP,Fy,EMS) GO TO 335

325 ESP=O. FSP=O.

335 CONTI NUE XP=ALPHA*Fo*e*C+ASP*FSP-AS*FS IF{YKK.EO. 1000.) GO TO 380 IF( (ABS( XP-PF I X) +ABS(XPP-PF I X» • NE. ABS (X P+XPP-2. *PFI X» GO TO 270

380 XPP=XP XCC=C YKK=2000. IF(ABS(XP-PFIX),LT. ETS ) GO TO 330 IF(XP-PFIX) 310.330.320

270DELTA=ABS(0.1*(C-XCC» C=(C+xCC)/2. GO TO 370

330 C ONT I NUE

400 XM=ALPHA*FO*S*C*(D-BETA*C)+ASP*FSP*(D-DPP) 390 CONTINUE

IF{T.EO. 0.) GO TO 392 WRITE{6,990)

990 FORMAT{) DESIGNED AS RECT.-SECTION) 392 CGNTII\UE

RATIO=XM/{FO*B*D*D) CURV=EU/C CURV~CURV* 1000.

222

WRITE(6,333) EU.ALPHA.BETA,C.ES.ESP,FS~FSP.XP.XM,CURV,RATI0 IF{T.NE. 0.) GO TO 1200 IF{ICODE.EO.2) GO TO 205 BXM( I )=XM 8CURV(I)=CURV BEU{ I )=EU . 8FS ( 1 )=FS IF(XM-TXM) 410.410,420'

420 TXM=XM TCURV=CURV TEU=EU

410 CONTINUE CPREV~C

290 C CNTI NUE 300 CONTINUE

CCC CCC COMPUTE yIELDING MOMENT BY INTERPOLATING ANALYTICAL RESULTS CCC

1=1 470 CONTINUE

IF{BFS{I)- FY) 450,460.460 450 I~1+1

GO TO 470 ~ 460 KK=I-I

KA=KK-l RR={ FY-BFS{KK»/(BFS(KK)-BFS(KA» REU=BEU(KK)+O.OOOl*RR RXM=BXM(KK1+RR*(BXM{KK)-BXM(KA}) RCURV=BCURV {KK} +RR* ( ECU RV {KK }-BCURV (KA ) ) WRITE(6.515)

515 FORMAT(//' AT YIELDING OF REINFORCEMENT'//) WRITE(6.535) REU.RXM,RCURV WRITE(6.525)

525 FORMAT(//' AT ULTIMATEI//) WRITE(6.535) TEU~TXM,TCURV

535 FORMAT(' EU =',FIO.5//' XM(K-IN) 1 • CURV(.OOI/IN.=l"FI0.6)

cec CCC bueTILITY FACTOR cee

DUCT=TCURV/RCURV WRITE(6,545) DUCT

545 FORMAT(/' DUCTILITY FACTOR=' .F10.4/) cce

:0 1 .FI 0.3//

cec ACI MOMENT FOR UNDERREINFOTCED BEAM(FOR THE CASE OF NO AXIAL LOAD) CCC

WRITE(6,2222) 2222 FORMAT(///' ACI CODE MOMENT FOP UNDERREINFORCED BEAMI/)

WRITE(6.444) EU=0.003 ALPHA=0.85*BTl BETA=BT1/2.

2000 CCC CCC CCC

2270 2250

2260 CCC CCC CCC

CCC

EY=FY/EMS XN=EMS/EMC X/'IoP=XN*PAS XNPP=XN*PASP CONTINUE

SECTION WITHOUT COMPRESSION STEEL

IF(A~F.NE. O.} GO TO 2200 PMA X= O. 75*PBAL IF{PAS-P~AX) 2260.2260.227d INRI TE (6 ,2250) . FORMAT(//' OVERREINFORCED BEAM----NO GO TO 205 CONTINUE

AT YIELD OF REINFORCEMENT

IVRITE(6.515) XK=-Xr...P+SQRT{XNP*XNP+2.*XNP) C=XK*D ES=EY FS=FY ESP=O. FSP=O. XP=O. XM=AS*FY*(O-C/3.) CUR V =E Y / ( D-C ) CURV=CURV*1000. EC=EY *( C/ (O-C) ) RATIO=XM/(FO*S*O*O) ALPHY=0.5 BETY=0.33333

223

SOL UTI ON '//}

WRITE(6,333) EC,ALPHY,BETY.C.ES.ESP,FS.FSP,XP.XM.CURV,RATIO XMYYY=XM CURYYY=CURV ECVYY=EC

CCC AT ULTIMATE CCC

WRITE(6.525) A=(AS*FY)/(0.85*FO*B) C=A/BTI XM=AS*FY*(O-A/2.) XP=O. ESP=O. FSP=O. ES=EU* ( (O-C) / C) CALL STEELA{FS ,ES .FY.EMS) CURV=EU/C CURV=CURV*1000. RATIO=XM/(Fo*e*o*o) WRITE(6.333) EU,ALPHA,BETA.C,ES,ESP,FS.FSP,XP.XM.CURV.RATIO XMCCO=XM EUCCO=EU CURCOD=CURV OUCTCD= CURCOO/CURYYY WRITE(6,545) OUCTCO GO TO-205

2200 CONTINUE CCC

CCC CCC

2240

2230 CCC CCC CCC

SECTION WITH COMPRESSION STEEL

PMA x'=0. 75* (P8AL+PASP) . PMIN=PASP+0.85*8Tl*(FO/FY)*(DPP/D)*(87./(87.-FY» IF(PAS-PMAX) 2230,2230,2240 IN RITE ( 6 , 22 50) . GO TO, 205 CONTINUE

AT YIELD OF REINFORCEMENT

WRITE(6,515) AA=XN~+XNPP -XK=-AA+SQRT(AA*AA+2.*XNP+2.*XNPP*(DPP/DJ) C=XK*D ' ES=EV FS=FY E C= EY * ( C/ ( 0-C) ) ESP=EC*(C-DPP)/C FSP=EMS*ESP XMS=ASP*FSP*(O-OPP) XMC=O. S*S*C*FY/XN*( C/( O-C»* (O-C/3.) XM=XMS+XMC . XP=O. CUR V=EY /( D-C) CURV=CURV*1000. RATIO=XM/(FO*S*O*O} ALPHY=O.5 SETY=0.33333 \

224

WRITE{6,333) EC,ALpHV,8ETY,C,ES,ESP,FS.FSP,XP,XM,CURV,RATIO XMYYY=XM CURYYY=CURV ECYYY=EC

CCC CCC AT ULTIMATE CCC

WRITE(6,525) IF(PAS-PMIN) 2210.2220,2220

2220 CONTINUE CCC CCC CCC

2210 CCC CCC CCC CC.C

COMPRESSION STEEL YIELD

A=(AS-ASP)*FY/(0.85*FO*S) C=A/STl XM=ASP*FY*<D-DPP)+{AS-ASP}*FY*(O-A/2.) XP=O. GO TO 2300 CaNT I NUE

COMPRESSION STEEL NOT YIELD

OETERMINE N.A. BY QUADRATIC EQUATION AA=0.8S*ST1*FO*8 SS=ASF*EU*EMS-AS*FY CC=ASP*EU*EMS*OPP ROOT2=BS*SS+4.*AA*CC RCOT=FCOOT2**0.5 C=(-SB+ROOT)/{2.*AA) A=C*STl FSP=EMS*EU*(C-OPPJ/C XM=0.8S*FO*A*S*(D-A/2.)+ASP*FSP*<O-OPP)

,

2300

CCC CCC CCC

1000

555

CCC CCC CCC

3100

1310

1320 1370

1340

CCC

XP=O. CONTINUE E5=EU*{(0-C)/C) ESP=EU*«C-OPP)/C) CALL STEELA(FS ,ES ,FY,EM5) CALL 5TEELA(F5P,ESP,FY.EMS) CURV=EU/C CURV=CURV*1000. RATIO=XM/(FO*B*O*O)

225

WRITE(6,333) EU,ALPHA.8ETA.C,ES,ESP9FS.FSP.XP,XM~CURV.RATIO CURCOO=CURV XMCOD=XM EUCOD=EU OUCTCD= CURCOD/CURYYY WRITE{6,545) DUCTCD GO TO 205

T-8EAM

CONTI NUE WRITE(6,555) " FORMAT(/' T-8EAM'//) WRITE(6.666) WRITE(6,777) 8,0,H.OPP,8P 9T,PAS,PASP.AS,ASP,FO,FY WRITE{6.444) TXM=O. C PREV= 0.4*0 NPTS=60 DO 1300 1= l,NPTS EU= 0.0001 * I AN=EU/OE AAN=AN+O.5 N=AAN Nl=N+l ALPHA=XALPHA(N) 8ETA=XBETA{N) FU=FF(Nl)

oJ

TRY NEUTUAL AXIS, C

CONTI NUE ITEST=1 C=CPREV OELTA=O.1 YKK=1000. GO TO 1370 C=C+DEL TA GO TO 1370 C=C-DEL TA CONTINUE ITE5T=ITE5T+1 IF{ I~EST.GT.30) GO TO 1375 IF{C.LT. T) GO TO 370

CONTINUE ES=EU*«O-C)/C) EFB=EU* ( (C-T )/C) EF8= STRAIN AT BOTTOM OF FLANGE IF(ICCOE.EQ.2) GO TO 3200 AN=EF8/0E AAN=AN+0.5 N=AAN

N1=N+ 1 ALPHAT=XALPHA(N) BETAT=XBETA{N) FF6=FF( Nl )

3200 CONTINUE CALL STEELA(FS.ES,FY,EMS) IF(ASP.EQ. 0.) GO TO 1325 ESP=EU*«C-DPP)/C) CALL STEELA(FSP.ESP,FY.EMS) GO TO 1335

1325 ESP=O. FSP=O.

1335 CONTINUE XP=ALPHA*FO*S*C+ASP*FSP-AS*FS-ALPHAT*FO*<B-BP)*(C-Tr IF(ABS(XP-PFIX} .LT. ETS ) GO TO 1330

226

IF{YKK.EC. 1000.) GO TO 1380 IF«ABS(XP-PFIX)+ABS(XPP-PFIX».NE.ABS{XP+XPP-2.*PFIX) GO TO 1270 IF(XP.GT. 0.) GO TO 1380

1380 XPP=XP XCC=C YKK=2000. IF(XP-PFIX) 1310,1310.1320

1270 DELTA=ASS{O.l*(C-XCC) C=(C+XCC)/2. GO TO 1-370

1330 CONTINUE 1400 XM=ALFHA*FO*S*C*(D-BETA*C)+ASP*FSP*(O-DPP)

1 -ALPHAT*FO*(B-BP}*{C-T)*(D-T-BETAT*(C-T» 1390 CONTINUE

WRITE(6.880) 880 FORMAT( I DE S I GNEO AS T-SECT ION' )

RATIO=XM/{FO*SP*O*O) CURV=EU/C . CURV=CURV*1000. WR1TE{6,333) EU,ALPHA.BETA,C.ES,ESP,FS,FSP,XP,XM,CURV.RATIC

1200 CONTINUE CPREV=C SXM(I)=XM SCURV ( 1 ) =CUR V BEU( I )=EU SF S{ I ) =F S IF(XM-TXM) 430,430.440

440 TXM=XM TCURV=CURV TEU=EU

430 CONT I NUE 1300 CONTINUE

GO TO 600 1375 CONTINUE

CCC CCC NO SOLUTION AT THAT PARTICULAR STRAIN CCC

NPTS=I-l 600 CONTINUE

CCC CCC COMPUTE YIELDING MOMENT BY INTERPOLATING ANALYTICAL RESULTS CCC

1=1 480 C 0 NT I NUE

IF(SFS{I}- FY) 485,495,495 485 1=1+1

495'

CCC CCC CCC

CCC

GO TO 480 KK= 1-1 KA=KK-l RR= ( FY-BFS(KK» /( BFS( KK)-BFS( KA» REU=BEU(KK}+0.0001*RR RXM=BXM{KK1+RR*(BXM{KK)-BXM(KA» RCURV=BCURV(KK)+RR*(BCURV(KK)-BCURV(KA» WRITE(6,515) / WRITE(6.535) REU,RXM,RCURV WRITE(6,525) WRITE(6.535) TEU.TXM.TCURV

DUCTILITY FACTOR

DUCT=(TCURV-RCURV)/RCURV ~RITE(6.~45) DUCT

227

CCC ACI MC~ENT FOR UNDERREINFOTCED BEAM(FOR THE CASE OF NO AXIAL LOAD) CCC

'fIRITE(6.2222) \II R I T E ( 6 • 44.4 ) EU=O.003 ALPHA=O .85*BTl BETA=8Tl/2. EY=FY/EMS XN=EMS/EMC X,,"P,:XN*PAS XNPP=XN*PASP

4000 CONTINUE PASW=AS/ (BP*D) PASPW=ASP/(8P*O) ASF=0.8S*FO*{B-BP>*T/FY P A SF = A SF / ( BP * D ) P8ALW=P8AL+PASF+PASPW PMAXW=0.75*P8ALw IF(PASW-PMAXW) 4230.4230,4240

4240 WRITE(6.2250) GO TO 205

4230 CONTINUE CCC CCC TESTING 8EAM DESIGNEe AS RECTANGULAR OR T-SECTION CCC ASSUME COMPRESSION STEEL YIELD CCC

A=(AS-ASP)*FY/{0.85*FO*B} IF(A-T) 4100,4200,4200

4100 CONTINUE CCC CCC BEAM DESIGNED AS REeT.-SECTION CCC

GO TO 2000 CCC CCC BEAM DESIGNED AS T-SECTION CCC

4200 CONTINUE CCC CCC AT YIELD OF REINFORCEMENT CCC

WRITE(6,S1S) BB=(X~F+XNPP)*(B/BP)-{T/D) CC=2.*XNP+2.*XNPP*(DPP/D)*(B/BP)-(T/D)**2 XK=-BB+SQRT(8B*88+CC)

CCC

C=XI<*O ES=EY FS=FY EC=EY*(C/(D-C) ) ESP=EC*<C-OPP)/C FSP=EMS*ESP EFB=EC*( C-T) /C FFB= E MC *EF B CA=O.S*B*C*FY/XN*C/(O-C) CB=0.S*(B-BP}*(C-T)*FF8

XM=C A* ( 0-C/3. ) + ASP*FSP* (D-OPP )-C8* (O-T- (C-") /3. ) ·XP=O. CURV=EY/{O-C) CURV=CURV*1000. RATIO=XM/(FO*BP*O*Ol ALPHY=O.S BETY=O.33333

228

WRITE{6~333) EC.ALPHY.8ETY.C.ES.ESP.FS.FSP.XP~XM.CURV.RATIO XMYYY=XM CURYYY=CURV ECYYY=EC

CCC AT UL TIMATE CCC

CCC

WRITE(6.525) ASF=O.85*FO*{8-BP)*T/FY CCF={AS-ASF-ASP)*FY A=CCF/(O.85*FO*SP) C=A/BTI XM=CCF *( O-A/2. ) +ASF *FV* (O"":T / 2. ) +ASP*FY* (D-OI?P) XP=O. ES=EU*( (O~C) IC) ESP=EU*{(C-OPP)/C} CURV=EU/C CURV=CURV*1000. RATIO=XM/(FO*SP*O*O) WRITE(6,333) EU,ALPHA,BETA.C,ES,ESP,FS.FSP,XP,XM.CURV,RATIO XMCOO=XM EUCOO=EU. CURCOO=CURV· OUCTCO= CU~COD/CURYYY WRITE{6.545) OUCTCO

205 CONTINUE CCC CCC

GO TO 105 END

229

PELEA5E 2.0 ALPBET DATE = 77161 20/37/15

CCC CCC

205 CCC CCC

3000 CCC

5000 CCC

7000 CCC

9000 CCC

SUBROUTINE ALPBET(FO.EO,FF.EE,XALPHA.XBETA.DE,ICONC,EMC) DIMENSION FF(900).EE(900).A(900).DX(900).ATOT(900).AMTOT(900) D I ME N 5 raN x A L PH A ( 9 0 0 ) , X 8 ET A ( 900 ) Yl(X)={Al*X+81*X*X)/(I.+Cl*X+Dl*X*X) Y2(X)={A2*X+82*X*X)/(I.+C2*X+D2*X*X) ICONC=2 •••• NORMAL WEIGHT ICONC=I •••• LIGHTWEIGHT IF(ICCNC.EC.2) GO TO 205 IF( ICCI\.C .EO.!) GO TO 105 CCNTII\.UE

NORMAL WEIGHT IF(FO.EQ. 3. IF(FO.EO. 5. IF(FO.EO. 7. IF(FO.EO. 9. IF(FO.EQ. 11. IF(FO.EO. 13. CONTI NUE

) GO ) GO ) GO ) GO ) GO ) GO

TO TO TO TO TO TO

FO=3. K5I •••• NORMAL EO=0.00267 Al=I.596974 B1=-0.352041 Cl=-0.403026 Dl=0.647959 A2=-1 .275770 B2=1.479521 C2=-3.275770 D~=2.479521 GO TO 9999 CONTINUE FO=5. K 51 •••• NORM PoL EO= 0.00292 Al=1.364804 Bl=-0.758034 Cl=-0.635196 Dl=0.241966 A2=0.326434 82= 0.047969 C2=-1.673566 D2=1 .047969 GO TO 9999 CONTI NUE FO=7. K5I •••• NORMAL EO=0.00317 A1=1.304017 81=-0.831953 Cl=-0.695983 01=0.168047 A2= 0 .357600 82=-0.095550 C2=-1.642400 D2=0.~04450 GO TO 9999 CONTINUE FO=9. K5I •••• NORMAL EO=0.00343 Al=1.300358 Bl=-0.835974

3000 5000 7000 9000 11000 13000

WEIGHT

wE IGHT

WEIGHT

WEIGHT

/'

<:1=-0.699642 o 1 = o. 1 64 026 A2=0.301863 82=-0.124428 C2=-1.698137 02=0.875572 GO TO 9999

11000 CONTINUE CCC FO:l1. KSI ••• • NORMAL WEIGHT

EO=0.00368 A 1=1.322665' 81 =-0.810704 Cl=-0.677335 01=0.189296 A2=0.238022 82=-0.121764 C2=-1 .761978 02=0.878236 GO TO 9999

13000 CONTII\UE· CCC FO=13. KSI •••• NORMAL WEIGHT

EO=O.00392 Al=1.358955 81=-0.765732 Cl=-0.641045 01=0.234268 A2=0.179553 82=-0.107095 C2=-1.820447 02=0.892905 GO TO 9999

CCC 105 CONT I NUE

CCC CCC LIGHTWEIGHT CCC

IF(FO.EQ .• 3. ) GO TO 30 .IF(FO.EQ. 5. ) GO TO 50 ·IF(FO.EQ. 7. ) GO TO 70

30 CONTINUE CCC FO=3000. PSI •••• LIGHTWEIGHT CONCRETE

E 0=0.00277 A 1 = 1 • 54 1689 81=-0.466498 Cl=-0.458311 01=0.533502 A2=O.178520 62=-0.046208 <:2';:-1.821480 02=0.953792 GO TO 9999

50 CONTINUE CCC FO=5000. PSI •••• LIGHTWEIGHTCONCRETE

EO=0.00317 Al=1.287970 81.=-0.849227 Cl=-0.712030 01=0.150773 A2=0.235274 62=-0.103281 C2=-1.764726

230

70 CCC

9999

CCC

CCC

500

200 300 100

CCO

400

700

800

D2=0.896719 GO TO 9999 CONTi NUE FO=7000. PSI •••• LIGHTWEIGHT CONCRETE EO=0.00357 A1=1.220582

. B1=-0.911536 C1=-0.779418 01=0.088464 A2=0.219928 82=-0.114448 C2=-1.780072 02=0.885552 GO TO 9999 CONTI NUE E MC=A 1 *FO/EO OE=INTERVAL IN STRAIN-AXIS EU=0.006 OE= o. 00001 AN=EU/OE AAN=AI'-I+0.5 N=AAN Nl=N+ 1 N=NCS. OF INTERVAL DO 500 I=l.Nl FF(I)=O. DO 100 I=1.N1 EE{ I )=DE*FLOAT (I-I) X=EE ( I) /EO IF(X.LT. 1.) GO TO 200 FF( 1 )=FO*V2{X) GO TO 300 FF { I )=F 0 *v 1 ( X) CONTINUE CONTINUE FU=FF(Nl) AREA AND MOMENT UNDER CURVE DO 400 I=l .. N A{ 1 )=O.5*OE* (FF (I )+FF{ 1+1) )

./

DOX={ DE/3.) * {{FF (I )+2. *FF( 1+1»/ (FF{ I )+F·F( 1+1)}) DX( I )=DDX+DE*FLOAT( I-l} CONTINUE DO 800 K=l.N EU=DE*K AMON= O. AREA=O. DO 700 I=l.K AR EA=AREA+A ( I) AMON=AMON+A(I)*DX(I) CONTI NUE ATOT(K)=AREA A MT OT ( K ) = A MaN XALPHA{K)=ATOT{K)/{FC*EU) BETA=AMTOT(K)/ATOT(K) BETA=8ETA/EU 8ETA=I.-8ETA X 8 ETA ( K ) = 8 ET A CONTI NUE RETURN END

231

RELEASE 2.0 STEELA \

SUBROUTINE STEELA(FS.ES.FY.EMI) IF(FY.EO. 60. ) GO TO 60000

60000 CONTINUE F SU=l 04.58 FSF=93.44 E MI =2976 O. EMSH= 1222. ESH=O .0091 ' ESU.=0.0729 ESF=0.1542 A=1.748272 B=0.173674 C=-0.251726 D=1.173672 GO TO 9999

9999 C aNT'I NUE EY=FY/EMI XO=ESU-ESH YO=FSU-FY

DATE = 77161

IF(ES.LT.EY ) GO TO 1300 IF(ES.GT.EY.AND.ES.LT.ESH) GO TO 1200 IF{ES.GT.ESH.AND.ES.LT.ESF) GO TO 1100 IF(ES.GT.ESF) GO TO 1400

1300 FS=EMI*ES GO TO 500

1200 FS=FV GO TO 500

1100 CONTINUE XX= (ES-ESH }/xa YY={A*XX+B*XX*XX)/(l.+C*XX+D*XX*XX) FS=YV*VO+FY GO TO SOD

14CO F 5=0. GO TO 500

500 CONTINUE RETURN END

232

20/37/15

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APPENDIX II

COMPUTER PROGRAM FOR NONLINEAR fu~ALYSIS OF UNIAXIALLY LOADED COLUMNS

236

237

qELEASE 2.0 MAIN DATE = 77ll! 16/24/23

CCC d "'\ CCC COLUMN •••• AXIAL LOAD PLUS UNIAXIAL BENDING (Appen \'X 2 J CCC REINFORCEMENT ALONG TWO FACES ONLY ~ CCC

DI~ENSION FF(900),EE(900),XALPHA(900},XBETA(900),BXP{lOO).BXM(lO0) 1 .BCURV{lCO).BC(lCC).HXP{lOO).HXM(100l.HCURV{100).HC(lOO). 2 8EU(100)

XPA(A)=O.85*FO*S*A+AS?*FSP-AS*FS XMA(A )=C.85*FO*S*A*{H/2.-A/2.)+ASP*FSP*(H/2.-DPP)+

1 AS*FS*(D-H/2.) XPB(FS)=O.85*FO*S*A+ASP*FY +AS*FS X~8{FS}=(ASP*FY-AS*FS)*{H/2.-DPP)

105 READ(5.240) JOB 240 FOR MA T( I 5 )

250 CCC CCC CCC CCC CCC

IF{JOQ.Ea.O) STO;:> WRITE(6.25C) JOB FORMAT(////' COLUMN NO.' d5//' UNIAXIAL BENDING'//) READ P~RAMETERS OF CONCRETE STRESS-STRAIN CURVE FO=PEAK STRESS IN CONCRETE. KSI FY=YIELD STRESS IN STEEL. KSI ICONC=2 •••• NORMAL WEIGHT ICONC=l •••• LIGrlTWEIGHT READ(5,111) FO,FY,ICONC

111 FORMAT(2F10.4.15) READ(5.222) B,D,H, AS. ASP.DPP

222 FORMAT(6FIO.4) DP=H-D PAS=AS/{S*D) PASP=ASP/ (8"~D) 'tJRITE (6,666)

666 FOR~AT(/4X,lH8.12X,lHD,12X,lHH,12X,3HDPP,lOX.

777 CCC CCC CCC

CCC

CCC

1 3HPAS,lOX.4HPASP.9X.2HAS,11X.3HASP,lOX,2HFO,11X,2HFY/) ~RITE(6.777) 8.D,H.DPP.PAS,PASP,AS.ASP,FO,FY FORMAT(lOCFIO.4.3x»

COMPUTE CONCRETE PROPERTIES

CALL ALPBET{FO,EO,FF,EE,XALPHA,XBETA.DE,ICONC.EMC)

ES=EO CALL STEELA{FS,ES.FY.EMI)

PMAX=FO*(S*H-AS-ASP)+FS*(AS+ASP) WRITE(6,999) PMAX

999 FORMAT(///' MAX. AXIAL LOAD(KIPS)='.F12.3/1. :::rS=O.OOl

CCC

CCC

€TS=C.Ol

DO 6')0 11=1.20

FACTOR=C.05*(Ir-l} ;:>FIX=P~~X*F~CTOR WRITE(6.888) PFIX

888 FORMAT(//' AXIAL LOAD(KIPS) ='.F12.3//) ~RITE(6,551) FACTOR

551 FORMAT(/I FACTOR OF LOAD ='.F5.2//) WRI TE (6,444)

444 FORMAT(///4X,' EU',7X.' ALPHA',5X,' BETA', 6X,' C(INS)'.4X, 1 'ES',8X,' ESP'.6X. I FS(KSIP.3X,' FSP(KSI)',lX,' XP{KIPS)I.

2 2X.4 XM{K-IN)·,lX_,-' CURV<.OOl/INl'//) , 333 FOR~AT{4{FI0.4,lX),<:{FI0.5tlX),5{F O.3,lX)/)

CCC

CCC

410 a.20

CCC CCC CCC

310

320 370

380

270

TXM=C. ' CPREV=O .4*D

DO 300 1=10.60

EU=O.OOOl*I AN=EU/DE AAN=AN+O .5 N=AAN Nl=N+l ALPHA=XALPHA(N} BETA=XBETA{N) FU=FF{Nl)

. ES=~U CALL STEELA(FS,ES,FY.EMI) DMAXl=FU*(B*H-AS-ASP)+FS*{AS+ASP) IF(EU-EO} 410,410.420 IF(PMAX1.LT. PFIX) GO TO 290 CaNT I NUE

TRY NEUTUAL AXIS. C

C=CPREV DELTA=O.l YKK=lOOO. C=C+DEL TA GO TO 370 C=C-DELTA CONTINUE IF(C.GT. D} GO TO 1000 ES=EU*( (D-C)/C) ESP=EU*«C-DPP)/C) CALL STEELA{FS,ES,FY,EMI) CALL STEELA(FSP~ESP.FY.EMI) XP=ALPHA*FO*B*C+ASP*FSP-AS*FS IF(YKK.EQ. 1000.) GO TO 380 A XP= XP-PF I X A XPP= XPP-PF I X IF( (A8S(AXP}+ABS(AXPP».NE. ABS{AXP+AXPP) GO TO 270 XPP=XP . XCC=C YKK=2000. IF(ABS{XP-PFIX).LT. ETS ) GO TO ,330 IF(XP-PFIX) 310,330,320 DELTA=ABS(O.l*(C-XCC» C=( C+XCC )/2. GO TO 310 CONTINUE XM= MOMENT ABOUT CENTER OF SECTION

238

330 CCC

400 XM=ALPHA*FO*i3*C* (H/2 .-8ETA*C) +ASP*FSP* (H/2 .-DPP) +AS*FS*( D-H/ 2.) CUR V=EU/C CURV=CURV*lOOO. WRITE(6.333) EU.ALPHA,BETA,C,ES,ESP,FS.FSP,XP,XM,CURV GO TO 290 I

CCC CCC C GREATER THAN D CCC

1000 CONTINUE ES=O.

~~k~to8o~5*EU GO TO 1340

1310 ES=ES+DELTA GO TO 1340

1320 ES=ES-DELTA 1340 CONTI NUE

IF(ES-EU) 1440.1450,290 1440 CONTINUE

IF(ES.LT. 0.) ~O TO 290 C=D*(EU/(EU-ES) ) ESP=EU*{(C-DPP)/C) EF3=EU*{(C-H)/C)

CCC EF3= STRAIN AT FACE OF LESS COMPRESSION GO TO.1490

1450 ESP=EU EFB=EU

1490 CONTINUE CALL STEELA(FS,ES,FY,EMI) CALL STEELA{FSP,ESP.FY,EMt) IF{C.GT. H) GO TO 1350 ALPHAT=O. BETAT=O. GO TO 1360

1350 CONTINUE AN=EF8/DE AAN=AN-f-O .5 N=AAN N1=N+l FF8=FF{Nl) ALPHA T=XALPH!~ (N) BETAT=X9ETA(N)

1360 CONT I NUE XP=ALPHA*FO*S*C+ASP*FSP+AS*FS-ALPHAT*FO*S*(C-H) A XP= XP-PF I X AXPP=XPP-PFIX IF(YKK.EQ. 1000.) GO TO 1380 IF({A8S(AXP)+ABS(AXPP».NE. ABS(AXP+AXPP}) GO TO 1270

1380XPP=XP YKK=2000. IF(A3S(XP-PFIX).LT. 'ETS ) GO TO 1330 IF{XP-PFIX) 1310.1330.1320

1270 ES=ES-DELTA DELTA=DELTA/I0. IF(DELTA.LT.O.OO<)COOOl) GO TO 1400 GO TO 1310

1330 CONTI NUE CCC ~~= MOMENT ABOUT CENTER OF SECTION

239

1400 XM=ALPHA*FO*S*C*CH/2.-BETA*C)+ASP*FSP*(H/2.-DPP)-AS*FS*(H/2.-DP) 1 +ALPHAT*FO*S*(C-H)*(BETAT*(C-H)+H/2.}

CURV=EU/C CURV=CURV*1000. ~RITF(6.333) EU,ALPHA.BETA.C.ES,ESP.FS.FSP,XP.XM,CURV

290 CONTINUE IF(XM-TXM) 2100.2100.2200

2200 TXM=XM TCURV=CURV TC=C TEU=EU

2 1 (\ 0 CON T I NU E 300 CONTINUE

8XM(IIl=TXM 8XP(l!)= PFIX BCURV( 1 I )=TCURV 9EU ( I I )=TEU 8C{ I I )=TC

!Seo CONTINUE NPTS= I 1+1 N1 =NPTS f3XM(Nl)=O. BXP(Nl)= PM AX \IIRITE(6~2222) DO 2300 11=1.;\11 WR I TE ( 6 .2111) S XP ( I I ) ~ 8X M ( I I ) .6 CU RV ( I I ) , BEU ( I I ) • BC ( I I}

2300 CONTINUE 2222 FORMAT(//' INTERACTION CURVE'//' AXIAL LOAD(KIPS)I.2X,

240

1 • MOMENT(KIP-INS)',2X,' CURV(.001/IN)'.5X,' EU',6X,' C(IN)'/j 2111 FORMAT(2{5X.FIC.3).6X,FIQ.4.4X.FI0.4~2X,FI0.4/)

CCC CCC ACI INTERACTION DIAGRAM CCC CCC MAXIMUM CONCENTRIC LOAD CCC

CCC

POP=O.85*FO*(S*H-AS-ASP)+FY*{AS+ASP) DP=H-D EU=C.003 BT1=O.85-0.C5*(FO/IOOO.-4.) EMIEU=EMI*EU P8AL=(C.85*8Tl*FO/FY)*{EMIEU /<EMIEU +FY}) EY=FY /EM I

CCC 8ALANCED CONTIUN CCC

C = D * { EU / ( E U + EY) ) A=C*S Tl FS=FY ESP=EU*(C-DPP)/C) IF(ESP.LT.EY) GO TO 3100 FSP=F Y GO TO 3200

31 CO F SP=E SP*E M I 32CO CONTINUE

XPBAL=XPA(A)

CCC

XM8AL=XMA(A) ECC8AL=XMBAL/XPBAL CUR8AL={EU/C>*1000. CBAL=C

CCC N.A. LOCATED AT TENSSIONSTEEL CCC

CCC

C=D A=C*8Tl FSP=FY H2=H/2. Cl=O.85*Fo*a*A C2=ASP*FY XP1=Cl+C2 XM1=Cl*(H2-A/2.)+C2*(H2-DPP) CNA1=C CURV1=(EU/C>*1000.

CCC N.A~ LOCATED AT FACE OF CONCRETE

/

C<;:C

CCC CCC CCC

5110

5120 5130

CCC CCC CCC

2240

2230

2220 CCC CCC CCC

2210 CCC CCC CCC CCC

C=H A=C*STl F,SP=FY ES=EU*<DP/H) FS-=ES*EMI C 1 -= 0 • 85 *F 0* B * A C2=ASP*FY C3=AS*FS XP2=C 1 +C2+C3 XM2=Cl*(H2-A/2.)+C2*(H2-DPP)-C3*(H2-DP) CNA2=C CURV2=(EU/C)*lCOO.

241

N.A. LOCATED OUTSIDE OF SECTION, DEPTH OF STRESS SLOCK EQUAL TO H

A=H C=H/BTI ES=EU*( (C-D'/C) IF{ES-EY) 5110.5110~5120 FS=ES *EMI GO TO 5130 FS=FY CONTINUE C1=C.85*FO*8*A \ C2=ASP*FY C3=AS*FS XP3=C 1'+C2+C3 XM3=Cl*{H2-A/2.)+C2*(H2-DPP)-C3*(H2-DP} CNA3=C CURV3=(EU/C)*10CO.

PURE BENDING FOR DOUeLE REINFORCED SECTION

PMAX=0.75*(PBAL+PASP) PMIN=PASP+0.8S*STl*(FO/FY)*{DPP/D)*(EMIEU /(EMIEU -FY» IF{PAS-PMAX) 2230,2230,2240 WRITE(6.2250) GO TO 5900 CONTINUE IF{PAS-PMIN) 2210,2220,2220 CONTINUE

COMPRESSION STEEL YIELD 1

A=(AS-AS~}*FY/{0.85*FO*B) C=A/B T1 XM=ASP*FY*(D-DPP)+(AS-ASP)*FY*{D-A/2.) XP=C. GO TO 2270 CONTINUE

COMPR~SSION STEEL NOt"YIELD DETERMINE N.A. BY QUADRATIC EQUATION

AA=O.85*8Tl*FO*8 88=ASP*EU*EMI-AS*FY CC=ASP*EU*EMI*DPP ROOT2=8S*S8+4.*AA*CC ROOT=ROOT2**0.5 C=(-88+ROOT)/(2.*AA)

( tFft~lt. DPP) GO TO 2260 Fso=EMI*EU*{C-DPP)/C FS=FY XM=XMA(A) XP=O. GO TO 227C

2260 CONTI NUE CCC CCC ASSUME COMPRESSION STEEL NOT EXIST CCC

A=(AS*FY)/{0.85*FO*B) C =A/B T1 XM=AS*FV*(D-A/2.) XP=O. '

2270 CONTI NUE XMBD= XM XDBD=XP CNABD=C CURVBD=(EU/C)*1000.

2250 FORMAT(//t OVERREINFORCED BEAM----NO SOLUTION'//} HXP( 1 )=0.

CCC

CCC

HXM( 1)= XM8D HCJRV( l)=CURVBD HC(l}=CNABO HXP(2l)=POP HXM(21)=0. -HCURV(21)=0. HC( 21 )=99999.

DO 6000 I 1=2 ,20

FA C T 0 R= 0 • 0 5 * ( I I - 1 ), PF I X=POP*F ACTOR HXP( I I )=PFIX IF(PFIX.LT. XPBAL) GO TO 5000 IF{PFIX.GT. XPBAL) GO TO 5100

5000 CONTINUE FS=FY FSP=FY A=(PFIX-ASP*FSP+AS*FS)/(O.85*FO*B) C=A/8Tl ' HXM( I I )=XMA(A} HCJRV(II)=(EU/C)*lCOO. HC ( I I ) =C GO TO 5900.

51CO CONTINUE CCC CCC FAILURE BY CO~PRESSION OF CONCRETE CCC

IF(PFIX.LT. XPl) GO TO 5200 IF(PFIX.LT. XP2) GO TO 5300 IF(PFIX.LT. XP3) GO TO 5400 IFCPFIX.GT. XP3) GO TO 5500

52<'0 CONTINUE CCC CCC LOCATE N.A. CCC

AA=0.85*BT1#FO*B 88=ASP*FY+AS*EJ*EMI-PFIX CC=AS*EU*EMI *0

242

~88t§RaBf~~t~:~AA*CC C=(-BB+ROOT)/(2.*AA) A=C*8Tl FSP=F Y E S = E U * ( ( D-:- C ) / C } FS=ES*EMI HXM( I I )=XMA{Aj HCURV(II)=(EU/C}*lOOO. HC(II)=C ' GO TO 5900

5300 CONTINUE RR=(PFIX-XPl)/(XP2-XPl) HXM(II)=XMl+(XM2-XMl)*RR C=CNA1+(CNA2-CNA1)*RR HCURV(II)=(EU/C)*1000. HC ( I I ) =C GO TO 5900

5400 CONTINUE RR=(PFIX-XP2)/(XP3-XP2} HXM( fI)=XM2+(XM3-XM2)*RR C=H+(CNA3-CNAZ)*RR HCURV(II)={EU/C)*lOOO. He ( I I )=C GO TO 5900

5500 CONT I NU'= ASFs=p~rX-O.85*FO*8*H-ASP*FY HXM(III=ASP*FV*(H2-DPP)-ASFS*(H2-DP) FS=ASFS/AS '=S=FS/EMI C=O*{EU/(EU-ES) ) HCURV(II)={EU/C)*lOOO. HC t I I )=C GO TO 5900

59CO CONT I NUE 6000 CONTI NVE

wRITE(6,6200) 6200 FORMAT(/I' ACI CODE'II)

WRITE(6,2222) DO 6300 11=1.21 WRITE(6,2111) HXP<Il),HXMCII),HCURV( lIlt EU

6300 CONTINUE . WRITE{6,6400)

6400 FORMAT{/I' BALANCED CONDITION'/) WRITE(6.2111) XPBAL.XM8AL,CUR8AL,EU,C8AL >'J cUTE(6,6500) -

6500 FORMAT(/' N.A. LOCATED AT TENSION STEEL'/) WRITE(6.2111) XPl,XMl,CURV1,EU,CNAl WRITE(6,5600)

66CO FORMAT{/' N.A. LOCATED AT FACE OF CONCRETE'/) WRITE(6,2111) XP2,XM2.CURV2,EU,CNA2 WRITE(6,6700)

6700 FORMAT(/' DEPTH OF STRESS BLOCK EQUAL TO H'/} WRIT~(6,2111) XP3,XM3,CURV3,EU,CNA3

CCC

CCC GO TO 105 END

243

• HC ( I 1)

COLUMN NO.

UNIAXIAL BENDING

1:3

10.000~

D

8.0000

H

10.0000

AS

1.2'000

ASP

1.2000

FG--·

9.0000

INr:::RA:TIO\t CU~V=

AXIAL :"'OAD(KIPS) MOMENT(KIP-INS)

0.0 642.095 -

51.12C 772.150

102.240 904.693

153.360 1049.393

2C4.480 1 U36. 809

255.600 1299.827

306.72.0_ 1374.959

357.840 1353.922

408.96 C 1315.765

460.080 1271.219

511.2CQ 1216.367

562.320 1147.709

513.439 1061.758

664.560 956.182

715.679 834.406

766.800 709.886

'817.919 582.224

869.039 453.228

920 -159 322.995

971.279 188.496

1022.399 C.O

OPP

2.0000

FY

60.0000

CJRV(.CO.1/IN)

2.9475

,2.3380

1.8655

1.3143

1.0943

0.9105

".7587

0.7241

0.6431

0.5874

0.5224.

0.4801

0.4430

C.4102

0.3671

0.3371

0.2929

0.26C8

0.2132

0.1678

0.0

PAS

0.0150

EU

0.0054

0.0049

O. C 046

0.0040

0.0 C 40

0.0040

0.0040

0.0042

0.0041

0.C041

0.0040

0.0040

0.0040

0.0040

0.0039

0.0039

0.0038

0.0038

0.0037

0.0037

0.0

244

PASP

0.0150

C( IN)

1.8320

2.0958

2.4658

3.0434

3.6554

4.3932

5.2719

5.8002

6.3755

6.9799

7.6573

8.3312

9.0300

9.7502

10.6243

11.5695

12.9755

14.5707

17.3561

22.0533

0.0

245 ACI CODE

INT:::RACTION CURVE

AXIAL LOAD(KIPS) MOMENT(KIP-INS} CURV(.OOI/IN) EU C( IN)

0.0 542.117 3.3454 0.0030 0.8967

44.532 641 .69,8 5.4090 0.0030 0.5546

89.064 825.473 2,.7045 0.0030 1.1093

133.596 983.326 1.8030 0.0030 1 .6639

178.128 1115.~56 1.3522 0.0030 2.2185 I

222.66C 1221.263 1.0818 C.0030 2.7732

267.192 1301.347 0.9015 C.r. 030 3.3278

311.724 1355.508 0.7727 0.0030 3.8824

356.25'6 1383.747 0.6761 0.0030 4.4371

400.788 1369.766 0~6099 0.0030 4.9192

445.320 1319.650 0.5645 0.0030 5.3146

489.852 1259.037 0.5239 0.0030 5.72,67

534.384 1185.975 0.4875 0.0030 6.1537

578.916 1098.707 0.4550 0.0030 6.5941

623.448 995.676 0.4258 0.C030· 7.0463

667.979 875.518 0.3995 0.0030 7.5089

712.511 737.057 0.3759 0.0030 7.9806

757.043 548.271 (1.3542 0.0030 8.470.2

801.575 357.773 0.3348 0.0030 8.9603

8~6.107 167.275 0.3174 0.0030 9.4505

890.639. 0.0 C.O 0.0030 99999.0000

RALANCED CONDITIJN

371.271 1355.999 0.6305 0.0030 4.7580

N.A. LOCATED AT TENSION STEEL

714.323 731 .01 7 C.3750 0.0030 8.0000

N.A. LJCATED AT FACE OF CONCRETE

8g6.038 -46.316 0.3000 0.C03Q 10.0000

DEPTH OF STRESS '3LCCK EQUAL TO H

'353.Q44 165.155 (1.3149 O.C03C 9.5279

APPEND IX I I I

COMPUTER PROGRAM FOR NONLINEAR ANALYSIS OF BIAXIALLY LOADED COLUMNS

246

247

D.ATE=7711~ 15/02/5'3

1} I '~P,I S 1 0 "I P)( a ~ 1 0') ) • R Y M ( 1 i)') ) .. '3 C tI q" ( 1 C ') ) , 8 E U ( 1 (.,,, ) • 8 e ( I,,·) ) ')! '~E";"J S r O!'I AS r c:; j } • X<::; ( ~~. , , Y 5 ( 5/) , • !") r 5T s ( t::q ) • ("''5 ( "; C) l • '\( Me; ( ") b , .. Y Me; ( '30 , •

1 ~S(~01.ES(~,).SIG~(5r,.,

2 '\(ELE{1~~O'~Y~L~{lC00' .. nISTE(100Ql,~ENA(lCOC) .. ~C(lQ0C),EC{lOOO). 3 CC(1000),,\(~C(100~1.Y~C(110C'.XCL~(1"CO'.Y~LA(1000\.DISTA{lDOC}

ecc

ecr !(~ ~FAD(c:;.112' JOg

11="( YJR.E0.(', 5T'")P WrlT~(A.2211 JCP

??1 cCl'<"~T( 1//' C8LIP,'f\I NO.='. i511' ~H AXI "L 9F:NrHNr; '1/' ccr: erC cO=CONC~ET~ DEA~ ST~=SS(~Sl). cY=STEEL vII="LD STRE(KSl) ere ~=L~~GTH IN X-1}I~ECTID~. P=WIDTH !~ Y-DIRECTION ecc

.-<- A ('I =1") .:'\ 1 ?'4";2?g nE(.;::r.:;7.2 c C:;77 c '51 ~EADfc:;.111) ~o.cv

111 cO::-~.~A~(8"'"1~.41

>~EAD(,3,111' 4,P qEAr)( ::::',1' 2) NRAq

112 COR~AAT(I'3'

ASTOL =:; • ecr ~~T~L::T07AL A~EA cc ~FIN~c~e~MEN79 P~S=PAT!Q OF qEIN~DPeF~~NT

,)iJ 1')1 !:::I.I'lRAP '., ~ t.. r f ::::. • 1 1 1 \ A, 5 ( ! ) • x S ( ! )', v <: r I ) ~STOL =A STnL + AS ( !'

! ,) n ceNT I 'JUE OAS=A,'3TCL/(R~!-I' ~EAnr5.111) T4ETA

e(r eee =:1':0 DC" I',IPl}T r)ATA eer ~CC=~((EN7~ICITY(!NCH1, TH=TA=AN~L~ CPOM X-AXlS(nEG~E~l ece THFTdR=THETA IN ~AnIAN cee ~vXV=RATI0 8= ~ceFNT~IC!7Y=ECeY/~ccx cee

TH~Taq=THCTA*O.0174512q Ic(THSTA.c.:0. D.) G:J ~c 1')2 I=(THETA.E0.00.1 GO TO 1~2 +VXY=Y~N(T'H;::,7"~' SL PR.A =- 1 • /r:: '.-XY

1 (2 eONT PJUE err eee CCNCQETE CCoo~oTIFS eee

E e A::,) • C " 1 CALL cnNCr::?(FCA.?CA.cC.E~l

eee eec STEEL PRac~RTIES ecr

EC,:A=:.:.rCl CALL STEFLA(FSA,FSA,CY,EV) wRIT~rF.2221 R.H.ASTOL.DaS.FQ.EO.~v.EY

??2 FO~i'~aTr/' SEeTTCN Ol=<r,PEPTIE~'I/ 1 5x.5H8rIN',Fx.~HHrIN1.4x.q~4s(sn.!Nl~5x.3HCAS.4X.7!-1F8(VS!l,

248

2 7Y.?~~O.~X.7~~Y(¥~!'.7X.2~EY" 3 5(=lJ.~tlYlt~lJ.5tlx.=lC.~tlY.Fln.~/' wprT=(~t?2~1 .

??3 ~O~MAT{/f 4R~A~G~~EN7 Cc ~~INF~RrIN~ AARS{ GP1~IN AT e~NTE~ ryF SE ITION"//lYt7~8Aq Nry •• 3Y.Q~~S(S0.IN't~X.5~X(IN)t~Yt5~Y(tN'//' ')0 1 .~ 1 ! = 1 • N9 .A R WF!T~(6t2?41 !.AS(I).Y5l!l.vS(I

1 ; 1 ceNT PH!E 224 FC~~AT(15.~x.3(F10.3.1X)/)

W~!T~(~,??51 THETAtT~ETAP ??5 =oRM~Tr/' LOCATION OF AODLIF') LOA~IN~'//

1 T~F"7'A {I')EC':;)F>::1='.=1,).5// 2 '-HFTAP fPADIAN1='.=10.S/)

ecc ect ~lE~~NT OC0o~QTr~s ecc ')~s=n!VFM5iO~ ~c ELFM~NT S0UA~E(INCH' err ~·~EY=NCS. Dr- F:LE'IENT Hi x-rI~::C""ION err- r-.1t::"V="r1S. or- C::L~N<=I·." !N y··'"'r r:t:;'c;'" rr;~; ccr AELE=A~EA [lc EAC~ €L~MENT C("C NTEL":=1'!GS. 0= i']"'- AL EU::M':?:f'l";-S ere X::=:L E='(= enG',:!);r I\!/\ TE OF Fl. F'~F'!T CCC vElE=Y--COO~1)1'\lAT=- n= EtE~·Ej<.Ji

eec nES =.) • ~ \lFY='-I/r.:::c. 'J::::Y=9/QES Ai7L E=OE S*I"\"~ S '.F':':LE =i'!~X *~{!,:V W~IT~(~,2?A' ~T~LE.NFx.~~Y.~FL=.nES

22 P ~O~M~~(/' ELEME~TS P~GPE?TIFS' //' ~~TAL NOS. OF FL~MENTS='915// 1 t\.!iJS. 'JI"" l'?L=·,,=,~,ITS I"I y,n!R=CTION='. 15// 2 N'JS. '1= c::U=,"~~"ITS P.I v-. ['1 Q2CT reN:::' • 15/ / .3 A~'':::A OF S0U1V~F e:L=~·~<:::N7fsn.I"q=, .Fl~.4/' 4 WI!")TH 0:= S0U.l\r:-F FLF~'~N""(!'4) -",'.Flr.4//l

ecr-eee Y-v CO~R!")INAT~S 0= ~AC4 ~L~VENT cee XELA( I' .Y:=LI\( 1 \ ,r:HSTA (!, •••• ·~IJ·,,,:~,::qI!'·!G· !N (JPIGINAL SYSTEM ccr XFLE(:\' ,YC:-LE':r \ .f")rSTt:::'(:r ~ ••• • '\!t.1'~8EPI!,lr, 1"J ",.IF'''' SYSTEM r-rr-~- - ..

,.., (1 1 1 0 J = 1 • N EV ~FX.Jl="'=')(*( J-'l) ')() llg l==l.NFX l<-=NFY J 1 +! VELA(K'=R/?+!")ES/2.-0ES*, XFl.Arvl=H/?+?2S/2.-0ES*I

11 A ceN'T'')U= 1 ! c. r: (1 1\, T : "H} ~

cr.r: O"'A X =r. i'"* r p* '-1.- t\ S """L ~ += v,;, A s-r:JL ~0IT~r~.qoq, PMay

~OQ =0~MA~(//lt MAX. AXIAL LOAD{l<ICS)=',F12.3/) ')I")T='") .1

err

crr

IST="! LST=4

:=A("TJ~~,)DT*(II-l' c=lX=DMAY*=ACT')q !1""(p=rY.~0.o~AX' GO TO 70C I~rD<=Ix.GT.oUAXJ G~ TO 7~ry

249

w~rT~rA~AAq~ p~tv qR9 eoru,Tf//' AXIQL L~AD(VIC~' =t.~12.3//'

'"~I ~, I 1"= ( ;<:, • C) c:; 1 ~ t= A C"" (' ':~ <:;<:;1 COCMA,TC/' l=ACT'10 (!~ LOA') =',"'"5.2//)

'IIF I r= ( ;<:, .22") , ~2~ C~RUAT(/t A~ALVSIS QE5UL T t//5v.2HEU,Rx.6HC(Kl o l.3x.lnHT MOV{K=INI.

cee:

cee

h 11 622

ccr: er:e: eee eer Ct""r ecr: cc:c: eee

eee: cr:e

1 ~'J(. 71-1VR (r~t),4.x.S)'-I XP (Il\n .4x.3H SLP .4X.I1 H ANGNA (D=G'. 2 3 Y .9H ~ t:~).~X.6HTHETAT,<:;X,13~CUDV(.001/IN'//l

XRO=::. V80='~ • ~ELTAD=0.~+f"'"AeTnF*6.1**~

':':U=')."C"Il*1JK It=(l=AeTT{.-:;T.'J.7;~; .AN!). ":::U.LT. r '.i),)25' r,n TO Lt.<::)t:>r;, 1~(!=!I,C"'Ij:'(.r,-r.").=.;:: .M,Jf'. -::l).LT.).(-=:C~~' GO TlJ49GIJ VI(' K = 1 'J ;'1 () • C "LL CO NC::,,£:: (l=U,;::U ,"'";]. EO' '=5 A =r.:u , CALL S7E~LhreSA,~SA.ev,~Ml' PMAX1=FU*(R*H-ASTCL l+I="SA*(AST~L il={F'J-~O' 611.;<:'11,622 It=(P~AXl.LT. PFIYl r,n T~ 4900 C,)P-JT '!' ',JI} ~

'nlY A l'Jf=."Tr- AL t\ X 7 S ~.A ••••••• ~~SLo*X-VP=~ SLC=~L00~ ~c ~~UTF\L AXIS V R = I \,;: ~ q.3 r:: C ... ! T'! I"i 1\, V- A v I S 'v iT H "l. A • -{',IA=C),::'"'p=:Nf'rr!JL~r: '1!STI\NCF I=:',O'.A LGAD!NG OGS!TICN TO N.A. $,A<;;THJr, V-\LU~ 1=0:: T~r~l~ OJ::' NEI.JT~AL ~XlS

SLO=SLPO ')(8=Xop VP=Yr-lD ')eLT!}.=f)~LThP

eec CCH.1'JT=C -'-H;::: :::L:-MFr-TS APO!)T Nr:1)T~"L A'l(IS er, f)1:STI\'Jr:;:: c~IJ'~ c=:.a.Ct-l ELEv=':~JT TO~,IEUTRAL AX!S eCr-

IJ::'(TH~TA.~~. ~.1 G0 T0 1~1

ll=rTH=:TA .;=I"\.~,').' r.1~ .... 0 1.32 ')"! 1 ~"::' ! =, • "1- ":L E x:xELA(!) V=vELA( I' ')ZSTA(t'=nI5T~N(X9V'

120 C:OI':TP~t'E r.;'J T:J 13C:;

1:71 ')" t~l !=l.",r=':L:=" X=XF:LAfl' ') 1ST A ( 1: 1-:: x- XP

1"'1 CG,,:TI-'lU=': '-;0 T(J 13"5

132 00 1?2 I=l,NT=:LE

12? 135

r:CC CCc ccc CC'-: CCC: CCC

1 ~,) 17C 1 ,(t r)

13('

1 c 1 CCC CCr: CCT CCC

?~-f'\

? 1 {~ Cr:C CCC ccc

V:VF:L" ( I' ,",!~T,,( 1 }~v-Y8 CO!,! T I ',j U E CONT r '·!'JE

'IEf'.!A{Il:!'V)S. 0<= e:U::v~NT'5 AAOVE DIST~{I'= ••••• A8CV~ N.A. nISTh{I\=- •••• 8ELOW N.A. ~rNAT=TCT~L NUU8~~ Oc= EL=MENTS

NFNAT=~ r)O 1 -,:') J= 1 9 N;::V 'lFNA ( J) =()

') D 1 4 J ! = 1 • "1? X ~'l~ X ,J 1 =NEX * ( ,J~-l ) l('=NEXJ 1 +1 l~(nISTArKl) lS~.1~)91~O :p'nEY=') GO Tn 17') I ~!i)E'J(=l NFNA(J)=N~NA(Jl+TNnEX CCNT!NIJ~ ?\Jl=NA-r=NEI>I"'"'"+"l::UA (Jl I~rNCNA{Jl.EQ. l' ~C T~ 1~0-C0N"":'-IUE GO TCl lev) I PGW=J~~ 1 ,,('1 1"') lql IKQ\'I=NFv CONTINUE

~J!JVR::q!Nr; ..,.~~ ~LFMFN;S A~O\/r:: N.A. !'" SF.:0IJEI'lCE KAA=NU M8 cc ING IN O~I~INAL9 KR9~NUMAE~ING IN NE- S~()UENCE

rCOL=NENA. {I l I)'J ?3-:) I:"l.IC'JL x: E LE ( r l = X =: L A ( ! ) vEL:-:{1.l=vELA(!) nIST={Il=DIST4fI' CONTINUE KK'K='j no ?1-:) J=?TQD'>J IC'JL=N~No.(,J) I(I(K=l('KK+~I':'lA(J'-l ) ~"F'J(Jl-:::"lE'J(*( J~-l) ')0 2?0 1=1. reOL Kj).A=KKK+'l K~P=N;:'<Jl +'1: XELE{KAA}=Y=LA(I(PP~ YELE{I(AA)=veL~(I(PB)

')ISTE(KAA'=0ISTA{I(PR) CDI'lT::"lIfE CDNTINUE

X=~/?

Y=R/~. IF(T~ET~.E0. C.) G~ TO 212 I~(TH~TA.Fn.qO.l GO TO ?13 ')ISTCN=DI5TAN(Y.V)

250

,,0 :'1 ?1lt. ?1? f1!STr:"l ~\(_YR

GO T"J ?la ?13 "JISTCN :Y-VR ':>14 CONT!'.JUE

cce cce s~~~ss A~n ST~AIN ~N ~A~4 CCNC~E~E EL~MENT

CC'-: 1)0 31;) l=l,N,-?N~'" FC(T}=~U*(f)TST=rt"DISTC~' ~ALL CONC~ErCCrI\.~crI'.~r.FQ'

"11 ij C:JNTT"JlJE ccr: cee P]i=< C>:: A t-F' '~OVE!\!T ('N E ACi-l EL EME"~T cc r

~? .. ~

cee crc cc r c~r

cee

I~;~ "1 l1DA

CCTGL =c) • '.( !I.~CT=~) • YMCT='~ • 00 32) I::l.NEN~T C C ( ! , '" A EL F,* 1= C ( I , XMC(Il=CC(I)*Y~L~(rl YMC(Tl=CCfI)*xEL=(!' CCTnL=CCTOL+ccrll YMC';'"=XMC-+YM('("! ) y~.ICT-=v',·CT+Y·AC{ I ) CONTI'IUE

STRF~S ANn STDA!N ~N ~F!N='lRC!~G PAR ~~~LACEM~N~ 01= CnNCGETC aeEA ON Cc~o~ESSICN gE 11\1(: C O"IS Y ,)E~E" ') r=(THETA.eo. ~.J GQ Tn 41? lcrT4E~A.F0.0~.' G~ TO 413

f"lO 4,')1 T=1.N8AR '\(=Y5(-;:) Y=YS{j) nISTS(I'",nIS~~N(Y.Y' CrlNT!"HJE St:l T.J 4.0 {~ ')(} l1J 2 !:: 1 • ~~R A '1 X::':lS( n 1)ISTS( I }=x-xq CCNTINUE SO Tn A~4 'J'] 4(l3 !=1,~J'1.\~ v=vS(!, ,)IST<:',( T }=Y,~"R en t,! .. 'I n tJ F C.::1NTI/,IUE ")0 l11j l=l.Nn'-'.,", ::S~=::IJ*{f)tS'S{ '{ l//"'II~TCNl r=(f"ss' '~?0 .t~~J .lL.l1,) 'S :I G 1',1( :r ) =~. 1 • FC"=). r;c T'] l15')

S:I G N ( :1 ,=:: • FCA=r~. t;C i") 4')"1 S! G~H '\ '=1 • C-LL CONC~E(eeA.ESS.=r.EO' C0.NT~ NI IE

251

:::Sf!)=A8S(-=SS) CALL s-rE'::LI\(CS(!'.r.:'S(T'.1:OV) '=':( I )=F~{ I '*S!(;N{ t )-Fed

It 1 0 r: Co N TIN \ .1 C r.cc ere I="(,FC~ At.-l) )"::J'~Et-!T C\I r.F-!"JI="01=:'CP'I(; 8A~ eee

CST0L=O. x·A5T=rj • YM5 T =J. ()C 4f,r; 1=l,NRA'::; C S ( T ) = AS ( ": '*'='H ! ) X~S(I)=csrl'*YS(Il YMsn )-:::CS( J: )*xSn) CSTGL=CS70L+CS(II xI\,1C::T=X'1S"'+':o~Sf I , v',~ST=YVST+v~~S (! )

ILt'C CONT!NUE ere CCC

~=CCTOL H- STC1L )( M=X~ACT +x M5 T Y ,~= Y VeT + Y •. ~ c:: T

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Documents

Report No. 77-2 COMPLETE STRESS-STRAIN CURVE OF CONCRETE