report no. 77-2 complete stress-strain curve of concrete
TRANSCRIPT
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 .
which leads to the following dimensionless form (plotted in Fig. 2.3):
Y = A (X - ; X2) (2.7)
and for which
dY dX = A(l-X) (2.8)
Checking boundary conditions:
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 can be seen that this expression has the following characteristics:
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
Checking boundary conditions:
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 can be seen that this expression has the following characteristics:
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.
Checking boundary conditions:
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 can be seen that this expression has the following characteristics:
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
Checking boundary conditions:
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 )
which leads to the following dimensionless form (plotted in Fig. 2.6):
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)
Checking boundary conditions:
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 can be shown that this expression has the following characteristics:
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)
which leads to the following dimensionless form (plotted in Fig. 2.6):
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
Checking boundary conditions:
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 can be shown that this expression has the following characteristics:
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)
which leads to the following dimensionless form (plotted in Fig. 2.7):
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
Checking boundary conditions:
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 can be shown that this expression has the following characteristics:
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'
, ,
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·1,'
.1
Fig
. 2
.11
: ! \
!
~'Xi
Typ
ical
as
cen
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ort
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.
. 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
Checking boundary conditions:
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.
Checking boundary conditions:
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' ..
'.
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I I
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. 2
.13
D
esce
ndin
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th
rou
gh
in
flecti
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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
Checking boundary conditions:
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.
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: Ii
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D
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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.
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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
, .... "
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1~t1,
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i ", '~P'
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. 3
.1
Bre
sler
and
H
eim
dah
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stress
-str
ain
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rve
for
grad
e 60
st
eel.
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DO
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• I'
ii, I
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'"
,I
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I I
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, '.
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i I'
i
:! !
I'
• !
:
1-
J , ,I I' ~
j .
' "I 'I
I,
I
I /
fro.~t
ur-e
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-fS
f
, ;~sr
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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)
It can be seen that this expression has the following characteristics:
(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
Characteristic Parameters of Grade 60 Steel
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)
Characteristic Parameters of Grade 60 Steel
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
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Fig
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Com
pari
son
betw
een
pro
po
sed
for
mul
a an
d actu
al
dat
a o
f g
rad
e 60
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ill~'------------------------------~C~H~R~R~R~C~T~E~R~l~S~T~I~C~'~S----'
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Fig
. 3
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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); •
_.1 1-
---- ---~------__ 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: , ~ ,- "
-.••. ci~~_~m~-~t[ff-~~~~I •• ·~iI:-f~ .. ·.-~t.~ •.• ~ .• ~_~~ •.• ;:. -~r_~f~·~~=t~~~_;~~X:Fr~~~--!~~~lF~ ~~~~l~~~-L:~-~t~~i~J:fiL-J-:~~~F-~1;-Lf~~:~:~::~:lI:;:i~LL3If.SL:i
•. J._ ; __ _ -_ .... _--_.--~-::::-:=-~-.. --~-----.---: --_._---.::---, ----~-~--~--~~-.---:...-=-::::-~.-. -'-~--
---_. ____ --~-~ ~-.---~--~ ,---.;...~-C" ~. ___ .C. ___ _ -- .. -:--
_. -: __ :~=~~;:~:-.. :-. :---=-: r-~ - _,"~' :.:::".-= .c __ :-::.:. __ :_:~-=~~~=. :.:~ _______ .. . .. . __ .: __ ==--~=---l. __ '_---:-:-:-:-____ .~ _____ .~~ ____ ~_.~. __ .
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
U)-rl--------~-------------------------------------------------------------------.
~
o :::!'
-,-i
L)
I---
t (\J
-
Ul
~-<
Yo
en ,
-i
(f)
LL
C~
CCCO
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L.J
-en
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IU
(:;.; (RX+BX~~2) / (1+CX+UX~~2)
o
CE
NE
RR
LIZ
[[J
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RL
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UR
V[-
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01
I :
; :
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0
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.12
0
.16
U
.2U
STR
RIN
Fig
. 3
.12
G
ener
aliz
ed s
tress
-str
ain
cu
rves
o
f gr
ade
60
steel
wit
h
yie
ld s
tren
gth
s o
f 60
, 6
5,
70 a
nd
75
ksi
.
'-I ~
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-_
_. !"~:_=-,-. -i- . -:)- [" -f·_ -~·:-=-:::i.::-=:·.·
----;---,--.,:-. -.. .>\~:;~.~:.'.~b,~~:\;=;: ;~-_.,
.·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-, -.,
.---,.
..--,-
--r----r--.-
----r--.-
-r---r-.-
i
10.0
8.0
NO
RM
AL
W
EIG
HT
INF
LE
CT
ION
P
OIN
T
80
60
(\J
6.0
4
0
E
CJ)
~
CJ)
C
J)
w
a::
4.0
I- (f
)
2.0
20
DA
TA
P
OIN
TS
•••••
L-_
_ ~ _
_ ~ _
_ ~ _
_ ~ _
_ ~ _
_ ~ _
_ ~ _
_ ~ _
__
_ L_
__
~ _
_ ~ _
_ ~IO
a 0.
001
0.0
02
0
.00
3
0.0
04
0
.00
5
0.0
06
ST
RA
IN
Fig
. 4
.2
Ty
pic
al st
ress
-str
ain
cu
rves
fo
r no
rmal
w
eigh
t co
ncr
ete.
" z ~
(Xl
o
8.0
Cfl 60
0r ~
.
.. (f
) (f
) ~ 4.o~ I
2.0
I
o
LIG
HT
WE
IGH
T
60
/ ~
\/IN
FL
EC
TIO
N
PO
INT
-1
40
/b
\ I \
11 :?!
20
//~
" "''''
'-.'"
I ////
0.00
1
DA
TA
P
OIN
TS
····
·
0.0
02
0
.00
3
ST
RA
IN
0.0
04
0
.00
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
Batch 2 (Cast on February 21, 1976)
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)
Batch 2 (Cast on February 21, 1976)
...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
The tensile strain in the steel at ultimate moment capacity of the
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
The tensile strain in the steel at ultimate moment capacity of the
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
Compatibility Equations
d-c E = --E 5 C c
Es = c-d" --E c c
c-t Efb = --E
c c
Constitutive Equations
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
Equilibrium Equations
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.
Compatibility Equations
e: s c-d" = -- e: c c
Constitutive Equations
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
Equilibrium Equations
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.
Compatibility Equations
where
e: s
c-d = -- e:
c c
c-d" e:' =--e: s c c
is the strain at bottom fiber of section.
Constitutive Equations
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
com
pre
ssio
n r
ein
forc
emen
t on
du
cti
lity
fo
r no
rmal
w
eigh
t co
ncr
ete.
!t
.....
01
0
1
Fig
. 6
.9
Eff
ect
of
com
pres
sion
rei
nfo
rcem
ent
on d
ucti
lity
fo
r li
gh
twei
gh
t co
ncr
ete.
t-
' V
I 0
\
I : ~ ,
t:·iL
~I
le:J
r""
I ,
! ,
r Q"i ,
.,~.
r ....
5 ~ ...
. tQ
'l::
o'i 'y i'
",
I,; i
i ,. ; I
'I,
Fig
. 6
.10
E
ffec
t o
f ax
ial
load
s on
du
cti
lity
: M-~
diag
ram
fo
r no
rmal
wei
gh
t co
ncr
ete.
!:
j' '-1
J .....
U1
-....J
Fig
. 6
.11
E
ffec
t o
f ax
ial
load
s on
du
cti
lity
: M-~
diag
ram
fo
r li
gh
twei
gh
t co
ncr
ete.
, ,
.-.. ;
·2.2
.... tJ1
00
a o ",'I~---------------------------------------------------------------------------------------------------------,
,..., o <
II) " - co
<CI
o ::J' '" C\I
:x:
](0
ro~
:.:
o lL.
"- 0...
0 "" C.J
o '" CJ ~
CJ ~ " <>
o
NORM
AL
WEI
GH
T R
NR
L,T
ICR
L--
--+
++
+
RC
I C
OO
E--
----
****
CO
LUM
N NO
. B
( I N
) o
(I N
) H
(I
N)
RS
(SO
.IN
) R
SP
(S
O.
I N
l FL
J (K
SIl
F
,(K
SIl
10"
•••
::: 1
• =
10
.00
=
18
.00
=
20
.00
::
:1.8
0 =
1.8
0
=4
.00
=
60
.00
ben
din
.J-_
--+
20'
(A
)( ;S
I · · .-r
--r 21
1
c·
I i
I ,..
......
. I
tL
I I
I j
I I
I 'b
.OO
O.
QI.~
0
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:;
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0
.20
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6
C.3
Z
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6
c.1W
O.
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Ff:h
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IIH)
Fig
. 6
.12
In
tera
cti
on
dia
gram
fo
r no
rmal
wei
gh
t co
ncr
ete.
......
til
lD
c o N"-
----
----
----
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D o N '. " <>
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HTW
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HT
RN
RL
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L--
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++
+
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CO
O[-
----
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t1N
NO.
B (
I N
) o
(I N
) H
(I
N)
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(~)Q
, IN
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SP
(S
Q.
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tKS
IJ
FY (
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I)
10"
-eo •
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r)
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.
=1
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8.0
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80
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.00
=G
O.O
O
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0
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MI
(Ftl*
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Fig
. 6
.13
In
tera
cti
on
dia
gram
fo
r li
gh
twei
gh
t co
ncr
ete.
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-
' (]
'I
.;:..
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
-ej>
dia
gra
m f
or
f'
= 5
000
psi
. c
'j
.•
;
.....
0\
0\
i i , ' i
.1./
Fig
. 6
.18
. , '" : .
6
'-
• 8
I I
! /. 0
'
I 1/
.2.
¢ (.
00
1/ In
)
I
Eff
ect
of
eccen
tric
ity
an
gle
s on
du
cti
lity
:
: "
. ,
M-~
dlag
ram
f
=
c 9
00
0 p
si.
, ..,..
(]
\ ,T'~
, i J I,
, ,
, !,
I
I'
,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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140
Fig
. 6
.20
C
ompa
riso
n o
f th
e<;l
reti
cal
and
the A
CI
colu
mn
inte
racti
on
P-
M d
iagr
am f
or
fc =
5000
p
si.
0.4
4
.,
I-'
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,--
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CD
[~.
o '" c;, a ::J!
o o N
()
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Cl 't. C
O
NORM
AL
WEI
GH
T AN
AL ,(
T IC
AL
----
, +
++
+
RCI
CO
O[-
----
-**
**
C.O
Il
0.0
8
1:'.1
2 0
.16
D
.20
0
.24
M
I (FO~B~H~H)
COLU
MN
NO.
B (
I N)
o
(I N
) H
(I
N)
RS
(5Q
, IN
) R
SP
(S
Q.
IN)
FO (
KS
I)
FY
(KS
IJ
10
"
--~)
-c:..
_ Q
=1
0.0
0
=::8
.00
==10
.CJO
=
1.2
0
=1
.20
=
9.0
0
=GO
.OO
lo'/l~
71 •
t I
ED
• I
•
0.2
6
C.:3
2 0
.36
D
·lfO
Fig
. 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
, ,
I' i
' '
'10 :
: :0
;::\
' !,"l
i !I,t
II
l'
Ie I
'
r!!~"
II'
,
'I::
: ,I
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:
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, I'::
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i .....
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:'1
. ,,'
,
i ',
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in
ll!'
, !~i!:
, .... ' ,
'
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IO
il.
. i!
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il I!,
I I\i!
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00
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n lJ
i
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Fig
. 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
5
1641
3
.33
0
.0
860
0.1
70
0.
0011
10
77
1. 2
85
3 0
.5
867
0.1
64
0
.00
10
12
28
1.9
40
1
.0
871
0.1
60
0
.00
09
13
18
2.2
7
0.0
.
869
0.1
60
0
.00
09
12
99
1. 9
5 5
0.5
87
4 0
.16
0
.00
09
13
71
2.3
9
1.0
87
6 0
.15
0
.00
08
14
09
2.5
7
0.0
87
6 0
.16
0
.00
08
14
63
2.5
7
7 0
.5
879
0.1
5
0.0
00
8
1476
2
.78
1
.0
880
0.1
5
0.00
07
1486
2
.85
No
te:
M,M
in
kip
-in
; ~.~
in
(0.0
01
ra
d/i
n)
y u
y u
AC
I
At
Ult
imat
e A
t U
ltim
ate
£ £
~u/~y
M
~u
~u/~y
cu
s u
0.0
06
0
0.0
27
2
11
.00
91
5 1
.27
5
7.6
0
0.0
06
0
0.0
34
5
13
.83
91
5 1
.36
6
8.3
0
.00
60
0
.03
84
1
5.5
91
5 1
.40
8
.8
0.0
06
0
0.0
37
1
15
.00
93
7 1
.88
9
11
.82
0
.00
60
0
.04
12
1
6.8
94
1 1
.71
0
10
.9
0.0
06
0
0.04
31
17
.8
942
1.6
5
10
.7
0.0
06
0
0.0
46
5
18
.92
94
7 2
.31
4
15
.00
0
.00
60
0.
0471
1
9.5
95
9 1
.19
7
12
.6
0.0
06
0
0.0
47
3
19
.8
962
1. 7
9 12
. 0
.00
56
0
.05
53
2
2.5
3
953
2.7
63
1
8.3
8
0.0
06
0
0.0
52
7
22
972
2.1
16
1
4.2
0
.00
60
0
.05
13
2
1.7
97
9 1
.93
1
3.2
0
.00
51
0
.06
37
2
5.9
7
956
3.3
76
2
2.9
3
0.0
05
5
0.0
55
6
23
.3
986
2.3
6
16
.2
0.0
05
8
0.0
54
0
23
997
2.1
0
14
.6
0.0
04
7
0.07
11
29
.12
95
8 3
.99
0
27
.57
I
0.0
05
2
0.0
59
1
24
.9
999
2.5
9
18
.06
0
.00
53
0
.05
48
2
3.4
10
13
2.2
6
15
.9
0.0
04
2
0.0
18
9
7.5
5
914
1.2
75
7
.49
I
0.0
05
9
0.0
29
0
11
.78
91
5 1
. 36
8.3
I
0.0
06
0
0.0
34
8
14
.1
915
1.4
1
8.8
0
.00
49
0
.03
02
11
. 9
938
1.8
89
1
1.5
I
0.0
06
0
0.0
37
1
IS
941
1. 7
1 1
0.7
!
0.0
06
0
0.0
40
2
16
.4
942
1.6
5
10
.5
0.0
05
3
0.0
41
0
16
.1
947
2.3
14
1
4.5
0
.00
60
0
.04
42
1
7.9
95
8 1
. 91
12
.3
0.0
06
0
0.0
45
3
18
.6
962
1. 7
9 11
. 7
N o VI
.f-l ..c:
bO
• ..-i
(l
) ~
r-I CIS
~ 0 :z:
.f-l ..c:
bO
• ..-i
(l
) ~
.f-l ..c:
bO
• ..-i
....
:l
TABL
E 7
.2
Mom
ents
, C
urv
atu
res
and
Du
cti
lity
Fac
tors
at
Yie
ld a
nd
Ult
imat
e fo
r p
= 0
.5 P
b
Rec
tan
gu
lar
beam
, f..
=
60
ksi
(b
=
10
",
d =
18
")
No
te:
M ,
Moo
in
kip
-in
; <1>
..
,<1>
..
in
(0.0
01
ra
d/i
n)
. .
Th
eo
reti
cal
AC
I
ft ~
At
Yie
ld
At
Ult
imat
e A
t U
ltim
ate
c P
(ksi
) M
<l>
y E
M
<l>
u E
E
<l>
u/<I>
y M
<l>
u <l>
u/<I>
y y
cy
u cu
s
u
0.0
18
48
0.2
08
0
.00
17
20
20
1.0
4
0.0
06
0
0.0
12
7
5.0
19
07
0.5
6
2.7
3
0.5
19
08
0.1
90
0
.00
14
24
55
1. 5
0 0
.00
60
0
.02
09
7
.8
1988
0
.84
4
.4
1.0
19
37
0.1
78
0
.00
12
28
74
2.0
8
0.0
06
0
0.0
31
5
11
.6
2002
1
. 04
5.8
0
.0
2733
0
.21
4
0.0
01
8
2965
0
.92
0.
0051
0
.01
15
4
.3
2874
0
.56
2
.6
5 0
.5
2823
0
.19
4
0.0
01
5
3605
1
.48
0
.00
60
0
.02
06
7
,6
2966
0
.84
4
.3
1.0
28
66
0.1
81
0
.00
12
42
46
2.0
7
0.0
06
0
. 0.0
31
3
11
.4
2980
1
. 04
5.7
0
.0
3357
0
.21
3
0.0
01
8
3761
0
.96
0
.00
49
0
.01
24
4
.5
3580
0
.56
2
.6
7 0
.5
3461
0
.19
4
0.0
01
5
4499
1
.51
0
.00
58
0
.02
14
7
.8
3661
0
.84
4
.3
1.0
35
12
0.18
1 0
.00
12
52
30
2.0
9
0.0
06
0
0.0
31
7
11
.5
3668
1
. 04
5.7
0
.0
4008
0
.21
3
0.0
01
8
4591
1
. 00
0.0
04
9
0.01
31
4.7
43
08
0.5
6
2.6
9
0.5
41
32
0.1
94
0
.00
15
54
45
1.5
2
0.0
05
5
0.0
21
9
7.8
43
86
0.8
3
4.3
1
.0
4193
0
.18
1
0.0
01
2
6264
2
.11
0
.00
60
0
.03
20
1
1.6
43
90
1.0
4
5.7
0
.0
4882
0
.21
7
0.0
01
9
5609
0
.98
0
.00
48
0
.01
29
4
.5
5266
0
.56
2
.6
11
0.5
50
42
0.1
96
0
.00
15
66
36
1.4
8
0.0
05
3
0.0
21
5
7.5
53
61
0.8
4
4.2
1
.0
5121
0
.18
3
0.0
01
3
7640
2
.10
0
.00
60
0
.03
18
1
1.5
53
66
1.0
4
5.7
0
.0
5755
0
.21
9
0.0
01
9
6639
0
.98
0
.00
48
0
.01
28
4
.5
6224
0
.56
2
.6
13
0.5
59
53
0.1
98
0
.00
15
78
39
1.4
7
0.0
05
2
0.0
21
3
7.4
63
36
0.8
4
4.2
1
.0
6049
0
.18
4
0.0
01
3
9016
2
.09
0
.00
6
0.0
31
7
11
.4
6341
1
. 04
5.6
0
.0
1841
0
.21
2
0.0
01
8
1925
0
.52
0.
0031
0
.00
62
2
.4
1907
0
.56
2
.7
3 0
.5
1905
0
.19
2
0.0
01
5
2232
1
. 08
0.0
04
2
0.0
15
3
5.6
19
88
0.8
4
4.3
1
.0
1935
0
.18
0
0.00
12
2755
1
. 93
0.0
05
9
0.2
89
1
0.7
20
02
1. 0
4 5
.7
0.0
27
11
0.2
2
0.0
02
0
2887
0
.63
0
.00
36
0
.00
78
2
.8
2874
0
.56
2
.5
5 0
.5
2814
0
.20
0
.00
16
34
27
1. 2
1 0
.00
47
0.
0171
6
.3
2966
0
.84
4
.2
1.0
28
61
0.1
8
0.0
01
3
4160
2
.00
0
.00
60
0
.03
01
1
0.8
29
80
1.0
4
5.6
0
.0
3319
0
.22
0
.00
20
36
91
0.8
7
0,0
04
6
0.0
11
1
3.8
35
80
0.5
6
2.5
7
0.5
34
45
0.2
0
0.0
01
6
4383
1
. 35
0.0
05
0
0.0
19
3
6.7
36
61
0.8
4
4.2
1
.0
3503
0
.18
0
.00
13
51
68
2,0
5
0,0
06
0
0.0
31
1
1.0
36
68
1. 0
4 5
,6
tv
o ~
.j...l
..c:
b.O
'M
<i) :s:
...;
cd
~ 0 z .j...l
..c:
b.O
'M
<i) ~
.j...l
.L:: b.O
'M
~
TABL
E 7
.3
Mom
ents
, C
urv
atu
res
and
Du
cti
lity
Fac
tors
at
Yie
ld a
nd U
ltim
ate
for
p =
0.7
5 P
b
Rec
tan
gu
lar
beam
, f
= 6
0 k
si
(b =
10
",
d =
18"
) y
Th
eo
reti
cal
f'
E..
At
Yie
ld
c (k
si)
P M
~y
M
~u
£
y cy
u
0.0
26
48
0.2
47
0
.00
24
26
93
0.3
8
3 0
.5
2824
0
.20
8
0.0
01
7
3278
1
.18
1
.0
2887
0
.19
0
.00
14
41
21
1. 9
0 0
.0
3953
0
.25
0
0.0
02
5
4073
0
.45
5
0.5
41
88
0.2
1
0.0
01
8
4813
1
.16
1
.0
4278
0
.19
0
.00
14
60
75
1.8
9
0.0
48
76
0.2
47
0
.00
24
51
02
0.5
2
7 0
.5
5137
0
.21
2
0.0
01
8
6016
1
.16
1
.0
5243
0
.19
0
.00
14
75
30
1. 9
3 0
.0
5829
0
.24
7
0.0
02
4
6160
0
.58
9
0.5
61
34
0.2
12
0
.00
18
72
88
1.1
7
1.0
62
59
0.1
9
0.0
01
4
9053
1
. 97
0.0
70
97
0.25
1 0
.00
25
75
25
0.5
9
11
0.5
74
90
0.2
1
0.0
01
8
8893
1
.15
1
.0
7648
0
.19
0
.00
15
11
021
1. 9
4 0
.0
8364
0
.25
0
.00
26
88
93
0.6
2
13
0.5
88
46
0.2
1
0.0
01
9
1051
1 1
.14
1
.0
9036
0
.19
0
.00
15
12
990
1. 9
3 0
.0
2642
0
.25
0.
0025
26
82
0.3
4
3 0
.5
2823
0
.21
0
.00
18
30
32
0.8
2
1.0
28
86
0.1
9
0.0
01
4
3857
1
.62
0
.0
3928
0
.26
0.
0027
40
50
0.4
2
5 0
.5
4181
0
.22
0
.00
19
46
31
0.9
3
1.0
42
75
0.1
9
0.0
01
5
5846
1
.71
0
.0
4826
0
.26
0
.00
27
50
77
0.5
2
7 0
.5
5123
0
.22
0
.00
19
58
97
1.0
5
1.0
52
37
0.2
0
0.0
01
5
7365
1
.82
No
te:
M,M
in
kip
-in
; ~,~
in
(0.0
01
ra
d/i
n)
y u
y u
--
AC
I I
At
Ult
imat
e A
t U
ltim
ate
I i
£ £
~u/~y
M
~u
~uNy
! eu
5
u i I
0.0
03
3
0.0
03
6
1.6
26
41
0.3
7
1.6
0
.00
60
0
.01
53
5
.6
2925
0
.65
3
.1
I
0.0
06
0
0.0
28
3
10
2972
0
.91
4
.8
. 0
.00
36
0
.00
45
1
.7
4021
0
.37
1
.5
0.0
06
0
0.1
50
5
.4
4370
0
.65
3
.1
0.0
06
0
0.0
28
0
9.8
44
22
0.9
1
4.7
0
.00
38
0
.00
56
2
.1
5059
0
.37
1
.5
0.0
05
4
0.0
15
4
5.4
53
98
0.6
5
3.1
0
.00
60
0
.02
89
10
54
40
0.9
1
4.7
0
.00
40
0
.00
65
2
.4
6120
0
.37
1
.5
0.0
05
2
0.0
15
9
5.5
64
71
0.6
5
3.1
0
.00
60
0
.02
94
1
0.2
65
08
0.9
1
4.7
0
.00
41
0
.00
65
2
.3
7480
0
.37
1
.5
0.0
05
1
0.0
15
7
5.3
79
09
0.6
5
3.0
0
.00
59
0
.02
90
10
79
55
0.9
1
4.7
0
.00
43
0
.00
69
2
.4
8839
0
.37
1
.5
0.0
05
0
0.0
15
5
5.3
93
47
0.6
5
3.0
0
.00
59
0
.02
88
9
.9
9401
0
.91
4
.7
0.0
03
1
0.0
03
1
1.3
26
41
0.3
7
1.5
0
.00
40
0
.00
23
3
.9
2925
0
.65
3
.1
0.0
05
3
0.0
24
0
8.5
29
72
0.9
1
4.7
-0
.00
36
0
.00
40
1
.6
4021
0
.37
1
.4
0.0
04
5
0.0
12
2
4.2
43
70
0.6
5
3.0
0
.00
55
0
.02
54
8
.7
4422
0
.91
4
.6
0,0
04
0
0,0
05
3
2.0
50
59
0.3
7
1.4
0
.00
49
0
.01
47
4
.7
5398
0
,65
2
.9
0.0
05
7
0.0
27
2
9.2
54
40
0.9
1
4.6
N o V1
-!
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
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