Lateral Torsional Buckling of Rectangular Reinforc

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Lateral Torsional Buckling of Rectangular Reinforced Concrete Beams
Article in ACI Structural Journal · January 2014

DOI: 10.14359/51686431 · Source: OAI
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LATERAL TORSIONAL BUCKLING OF RECTANGULAR
REINFORCED CONCRETE BEAMS
A Dissertation
Presented to
The Academic Faculty
by
Ilker Kalkan
In Partial Fulfillment
Of the Requirements for the Degree
Doctor of Philosophy in the
School of Civil and Environmental Engineering
Georgia Institute of Technology

December 2009
Copyright © 2009 by Ilker Kalkan

LATERAL TORSIONAL BUCKLING OF RECTANGULAR
REINFORCED CONCRETE BEAMS
Approved by:
Dr. Abdul-Hamid Zureick, Advisor Dr. Bruce R. Ellingwood

School of Civil and Environmental School of Civil and Environmental
Engineering Engineering
Georgia Institute of Technology Georgia Institute of Technology
Dr. Lawrence F. Kahn Dr. George Kardomateas

School of Civil and Environmental School of Aerospace Engineering
Engineering Georgia Institute of Technology
Dr. Kenneth M. Will

School of Civil and Environmental
Engineering
Date Approved: September 28th,2009

To my family…
ACKNOWLEDGEMENTS
The author is grateful to his advisor Dr. Abdul-Hamid Zureick for his help, guidance,
patience and encouragement throughout this project.
The author would like to express his sincere gratitude to Dr. Lawrence Kahn,
whose technical expertise and guidance contributed greatly to the success of the
experimental program. The author is also grateful to the other thesis committee members
Dr. Kenneth M. Will, Dr. Bruce R. Ellingwood and Dr. George Kardomateas for their
time and suggestions.
The experiments of the present study were carried out at the Structural
Engineering and Materials Laboratory of Georgia Institute of Technology. The author
feels grateful to the facility manager Jeremy Mitchell; former research engineer Marcus
Millard; mechanical specialist Michael R. Sorenson; senior facilities manager Andrew
Udell; fellow graduate students Yavuz Mentes, Jong Han Lee, Jonathan Hurff, Victor
Garas, Robert Moser, Brett Holland, Jennifer Dunbeck, Katherine Snedeker and Kennan
Crane; former fellow graduate students Murat Engindeniz and Felix Kim; former
undergraduate assistants Andrew Cao and Luis Fajardo and other people without whose
helping hand the author could not conduct the experiments. Special thanks are due to
fellow graduate students Mustafa Can Kara and Towhid Bhuiyan for their tremendous
help and support during the course of the experiments.
This dissertation is dedicated to the family of the author, whose support and love
made this dream possible. The author feels indebted to his parents Müzeyyen and İsmail
iv
Kalkan and his brother Murat Kalkan for their unconditional love and unlimited support
particularly at difficult times.
Financial support provided for the experiments by Georgia Department of
Transportation (GDOT) is gratefully acknowledged. The author would like to express his
appreciation to Kırıkkale Üniversitesi, Turkey for providing financial support to the
author during the course of his dissertation. Thanks are also due to Dr. Mustafa Y. Kılınç,
Dr. Osman Yıldız and Dr. Orhan Doğan for their support during this study.
v
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS iv
LIST OF TABLES x
LIST OF FIGURES xii
LIST OF SYMBOLS xx
LIST OF ABBREVIATIONS xxvii
SUMMARY xxviii
CHAPTERS
I INTRODUCTION 1
1.1 Introduction 1
1.2 Project Objectives 4
1.3 Organization of the Study 4
1.4 Previous Studies 6
1.4.1 Review of Previous Experimental Work 7
1.4.2 Analytical Methods for Predicting Lateral Torsional Buckling 28
1.4.3 Summary 56
II SPECIMENS AND MATERIAL PROPERTIES 57
2.1 Specimens 57
2.1.1 Specimen Descriptions 57
2.1.2 Experiment Design 58
2.2 Concrete Material Properties 62
III EXPERIMENTAL SET-UP, INSTRUMENTATION AND PROCEDURE 69
3.1 Experimental Setup 69
vi
3.1.1 Loading Mechanism 69
3.1.2 Supports 75
3.1.3 Load, Deflection and Strain Measurements 88
3.1.3.1 LVDT Strain Measurements 93
3.1.3.2 Strain Measurements through Electrical Resistance Strain

Gauges 95
3.2 Test Procedure 98
3.3 Summary of the Test Results 99
IV LATERAL BENDING RIGIDITY OF RECTANGULAR REINFORCED

CONCRETE BEAMS AND INFLUENCE OF SHRINKAGE CRACKING
ON THE RIGIDITY 101
4.1 Introduction 101
4.2 Available Lateral Bending Rigidity Expressions 102
4.3 Proposed Lateral Bending Rigidity Expression 105
4.3 Influence of Shrinkage Cracking on the Lateral Bending Rigidity 121
V TORSIONAL RIGIDITY OF RECTANGULAR REINFORCED

CONCRETE BEAMS 131
5.1 Torsional Behavior of Reinforced Concrete Beams 131
5.2 Torsional Rigidity of Rectangular Reinforced Concrete Beams 137
5.2.1 Uncracked Torsional Rigidity Expressions 137
5.2.2 Post-Cracking Torsional Rigidity 141
5.3 Experimental Torsional Rigidities of the Test Beams 145
5.4 Proposed Torsional Rigidity Expression 151
VI CRITICAL MOMENT CALCULATIONS AND INFLUENCES OF

THE INITIAL GEOMETRIC IMPERFECTIONS ON THE LATERAL
STABILITY OF REINFORCED CONCRETE BEAMS 153
6.1 Introduction 153
6.2 Critical Moment Calculations 153
vii
6.3 Influences of Sweep and Initial Twisting Angle on the Lateral
Stability of Reinforced Concrete Beams 156
VII EXPERIMENTAL RESULTS AND OBSERVATIONS AND

CORRELATION OF THE ANALYTICAL AND EXPERIMENTAL
RESULTS 173
7.1 Experimental Results and Observations 173
7.1.1 Crack Patterns of the Specimens 173
7.1.2 Experimental Results 181
7.2 Correlation of the Analytical and Experimental Results 183
VIII SUMMARY AND CONCLUSIONS 195
8.1 Summary 195
8.2 Conclusions 200
8.3 Futures Research 203
APPENDIX A: NOMINAL AND MEASURED DIMENSIONS OF THE

SPECIMENS 205
APPENDIX B: MEASURED INITIAL GEOMETRIC IMPERFECTIONS AND

PERMANENT DEFORMATIONS OF THE SPECIMENS 213
APPENDIX C: MIDSPAN STRAIN DISTRIBUTIONS OF THE BEAMS AT

DIFFERENT LOAD LEVELS 229
APPENDIX D: EXPERIMENTAL LOAD-DEFLECTION PLOTS OF THE

SPECIMENS 249
APPENDIX E: METHOD FOR THE EVALUATION OF THE CENTROIDAL

DEFLECTIONS AND ROTATION OF A BEAM 265
APPENDIX F: CRITICAL MOMENT CALCULATIONS OF THE SPECIMENS 273
APPENDIX G: CONSTRUCTION DETAILS OF THE SPECIMENS 283
G.1 Compression Reinforcement 283
G.2 Tilt-up and Lifting Mechanisms for the Specimens 284
APPENDIX H: DETERMINATION OF THE EXPERIMENTAL BUCKLING

LOADS OF THE SPECIMENS 289
viii
REFERENCES 297
VITA 304
ix
LIST OF TABLES
Page
Table 1.1: Nominal dimensions of the beams tested by Hansell and Winter (1959). 7
Table 1.2: Results of the tests by Hansell and Winter (1959). 8
Table 1.3: Beams tested by Sant and Bletzacker (1961). 15
Table 1.4: Results of the tests by Sant and Bletzacker (1961). 16
Table 1.5: Specimens tested by Massey and Walter (1969). 21
Table 1.6: Beams tested by Konig and Pauli (1990). 23
Table 1.7: Midspan initial imperfections of the beams tested

by Konig and Pauli (1990). 23
Table 1.8: Loads, midspan deformations and rotations at failure of the beams
tested by Konig and Pauli (1990). 23
Table 2.1: Specimens of the present experimental program. 58
Table 2.2: Mechanical properties of concrete. 64
Table 3.1: Experimental results of the specimens. 100
Table 4.1: Experimental and calculated cracking moments of the second set of
specimens. 113
Table 4.2: Descriptions of the shrinkage specimens from the concrete mixtures
used in B44 and B36L. 126
Table 5.1: Maximum torsional moments of the specimens at the initiation

of buckling and the cracking torques. 149
Table 5.2: Torsional rigidities of the specimens. 150
Table 7.1: Experimental and analytical critical load values of the specimens. 186
Table 7.2: Experimental–to-analytical load ratios of the specimens. 187
Table 7.3: Measured sweeps and angles of twist at limit load of the
specimens at midspan. 189
Table A.1: Nominal and measured heights of the first set of specimens. 206
x
Table A.2: Nominal and measured heights of the second set of beams. 207
Table A.3: Nominal and measured widths of the first set of specimens
along the span. 209
Table A.4: Widths of the first set of specimens along the depth
of midspan section. 209
Table A.5: Nominal and measured widths of the second set of beams. 211
Table A.6: Nominal and measured span lengths of first set of specimens. 212
Table A.7: Nominal and measured total lengths of the second set of specimens. 212
Table B.1: Measured sweeps of Specimens B18. 214
Table B.2: Measured sweeps of Specimens B30 and B36. 214
Table B.3: Measured sweeps of Specimens B44. 216
Table B.4: Measured sweeps of Specimens B36L. 217
Table H.1: Critical loads from the classical and Meck’s (1977) version of
the Southwell (1932) plots for the second set of beams. 295
xi
LIST OF FIGURES
Page
Figure 1.1: (a) Lateral torsional buckling of a beam subjected to a concentrated

load at midspan; (b) Lateral and vertical deflections and torsional
rotation of the midspan section 2
Figure 1.2: Load-lateral deflection curve of a reinforced concrete beam. 3
Figure 1.3: Comparison of the vertical deflections at yield point of the companion
beams tested by Hansell and Winter (1959). 9
Figure 1.4: Comparison of the lateral top deflections at yield point of the companion
beams tested by Hansell and Winter (1959). 9
Figure 1.5: Comparison of the lateral bottom deflections at yield point of the
companion beams tested by Hansell and Winter (1959). 10
Figure 1.6: Experimental-to-calculated ultimate moment ratios of the beams

Tested Hansell and Winter (1959). 10
Figure 1.7: Loading mechanism used by Hansell and Winter (1959). 12
Figure 1.8: Undeflected and deflected configurations of the loading

mechanism. 12
Figure 1.9: Direction of the torsional moments induced by the eccentric

application of the load. 13
Figure 1.10: Web sidesway buckling failure of the specimen due to the lateral
restraining forces in the loading mechanism. 15
Figure 1.11: Experimental-to-predicted buckling moment ratios of the beams in the

first three specimen groups, B36, B30 and B24. 17
Figure 1.12: Loading frame used by Sant and Bletzacker (1961). 18
Figure 1.13: Deflected configuration of the loading mechanism

used by Sant and Bletzacker (1961). 19
Figure 1.14: Cross-sectional details of the specimens tested by

Konig and Pauli (1990). 22
Figure 1.15: Ratio of the buckling load of each Specimen to the buckling load
of Specimen 2. 25
xii
Figure 1.16: Loading mechanism used by Konig and Pauli (1990). 27
Figure 1.17: Stress-strain curve of normal-strength concrete and the tangent

moduli of elasticity at different stress levels. 31
Figure 1.18: Secant modulus of elasticity corresponding to the extreme

compression fiber strain. 34
Figure 1.19: (a) Stress distribution in the compression zone of the beam section
in the plastic state; (b) Stress-strain curve of concrete;
assumed by Siev (1960). 40
Figure 1.20: Strain distribution at midspan section and the corresponding

reduced modulus of elasticity. 42
Figure 1.21: Definition of the variables in the expressions proposed

by Massey (1967). 45
Figure 1.22: Effect of the vertical location of the applied load with respect to
the shear center of beam section. 48
Figure 1.23: Definition of area and perimeters in Equations (1.30) - (1.32) 54
Figure 2.1: First set of specimens (B36, B30, B22, and B18) 59
Figure 2.2: Second set of specimens (B44, B36L) 60
Figure 2.3: Congested reinforcement in B36 63
Figure 2.4: Application of self-consolidating concrete 63
Figure 2.5: Stress-strain curves of concrete in the first set of beams 67
Figure 2.6: Stress-strain curves of concrete in B44 67
Figure 2.7: Stress-strain curves of concrete in B36L 68
Figure 3.1: Undeflected and deflected configurations of the loading frame

and loading cage. 70
Figure 3.2: Loading frame 71
Figure 3.3: Undeflected and deflected shapes of the Gravity Load Simulator 72
Figure 3.4: (a) Undeflected; and (b) Deflected configurations of the Gravity Load
Simulator. 72
Figure 3.5: Ball-and-socket joint 73
xiii
Figure 3.6: Vertical orientation of the loading cage (a) Before the test;
(b) After buckling. 74
Figure 3.7: Minor-axis rotations at the supports 75
Figure 3.8: Lateral deformations and torsional rotations restrained

at the supports. 76
Figure 3.9: In-plane support conditions: (a) In analysis models;

(b) Hinged-hinged; (c) Roller-hinged; (d) Roller-roller. 77
Figure 3.10: Roller supports at the beam ends in the second set of experiments. 81
Figure 3.11: Behavior of the beams with: (a) Roller supports;

(b) Hinged supports in lateral direction. 82
Figure 3.12: (a) Support frame in the first set of experiments; (b) A ball roller
in contact with the beam. 83
Figure 3.13: Bending of the ball roller 84
Figure 3.14: Lateral support frame in the second set of experiments. 84
Figure 3.15: Rigid caster in contact with the specimen. 85
Figure 3.16: Support frame in the second set of tests: (a) B44-1; (b) B44-2. 87
Figure 3.17: Distortion in the cross-section at the beam end. 87
Figure 3.18: Deviation of the Initial Orientations of the Potentiometers. 89
Figure 3.19: Coupling between the in-plane and out-of-plane deflection

measurements for a lateral string potentiometer with varying distances
from the specimen. 90
Figure 3.20: Lateral torsional buckling (a) with; (b) without distortions in the
cross-sectional shape of the beam. 91
Figure 3.21: Lateral deflection potentiometers in the first set of experiments. 92
Figure 3.22: Lateral deflection potentiometers in the second set of experiments. 92
Figure 3.23: Strain measurement using LVDT’s in the first set of tests. 93
Figure 3.24: Strain measurement through strain gages in the second set of tests. 94
Figure 3.25: 2-element cross strain gage on the side face of B44-1. 96
xiv
Figure 3.26: Longitudinal strain gages along the depth of north face of Specimen
B44-2 at midspan. 97
Figure 3.27: Strain gages on an aluminum strip to measure the longitudinal strain
in the tension zone. 98
Figure 4.1: Spring models defining (a) Branson’s (1963); (b) Bischoff’s (2005)
effective moment of inertia expression. 109
Figure 4.2: In-plane deflections of Beams B44-1 at midspan. 110
Figure 4.5: In-plane deflections of Beams B36L-1 at midspan. 111
Figure 4.6: In-plane deflections of Beams B36L-2 at midspan. 112
Figure 4.7: Moduli of elasticity corresponding to the fibers in the compression

zone of a beam section. 114
Figure 4.8: Proposed spring model for the lateral bending behavior
of reinforced concrete beams. 117
Figure 4.9: Shrinkage cracking in B30 prior to the test. 123
Figure 4.10: Length changes of specimens with and without SRA

from the concrete mixture of B44. 126
Figure 4.11: Delta rosette for principal strain measurement at a point. 128
Figure 4.12: Principal strains on the side face of B36L-2. 129
Figure 4.13: Principal strains on the side face of B36L-3. 129
Figure 5.1: Torque-twist curve of a reinforced concrete beam

with shear reinforcement. 132
Figure 5.2: Components of the axial torque on the failure surface of a concrete
beam according to the skew-bending theory. 134
Figure 5.3: Comparison of the coefficients βc calculated from different equations. 142
Figure 5.4: Thin-walled tube space truss model. 144
Figure 5.5: Experimental torque-twist curve of Specimen B44-1. 146
Figure 5.6: Experimental torque-twist curve of Specimen B36L-1. 146
xv
Figure 5.7: Approximation of the torque-twist curve of B44-2
into a series of line segments. 148
Figure 6.1: Lateral centroidal deflections of B44-1 and B44-2 at midspan. 158
Figure 6.2: Lateral centroidal deflections of B36L-1 and B36L-2 at midspan. 158
Figure 6.3: Longitudinal strain distributions in a cross-section from major-axis

and minor-axis bending moments. 161
Figure 6.4: Extreme compression fiber strains of B44-1 and B44-2 from
. major-axis bending 163
Figure 6.5: Extreme compression fiber strains of B36L-1 and B36L-2 from
. major-axis bending 163
Figure 6.6: Extreme top strains on the convex faces of B44-1 and B44-2 caused by
. minor-axis bending 165
Figure 6.7: Top strains on the convex faces of B36L-1 and B36L-2 caused by
. minor-axis bending 167
Figure 6.8: Rotation of the major and minor axes of a section due to twist. 169
Figure 6.9: (a) Southwell (1932) Plot; (b) Load-Deflection Plot for
Specimen B44-1. 169
Figure 7.1: Typical crack pattern on the convex faces of the specimens
after buckling 174
Figure 7.2: Typical crack pattern on the concave faces of the specimens
after buckling 174
Figure 7.3: Flexural cracks on the concave face of B44-3 at midspan

before buckling 175
Figure 7.4: Vertical cracks on the convex face of B44-2 at midspan

after buckling 176
Figure 7.5: Directions of the shear and principal stresses due to the shear forces. 178
Figure 7.6: Directions of the shear and principal stresses due to the torsional
moments. 179
Figure 7.7: Diagonal tension cracks on the convex face of B36L-1 after buckling. 180
Figure 7.8: Diagonal tension cracks on the concave face of B18-2 after buckling. 180
xvi
Figure 7.9: Diagonal tension cracks continuing on the top surface of B44-2
after buckling 181
Figure 7.10: Maximum compressive strains in the first set of beams,

illustrated on the stress-strain curve of concrete. 182
Figure 7.11: Maximum compressive strains in B44 at the instant of buckling. 182
Figure 7.12: Maximum compressive strains in B36L at the instant of buckling. 183
Figure 7.13: Experimental to analytical critical load ratios of the specimens

according to different formulae. 188
Figure A.1: Height measurement points along the lengths of the beams. 208
Figure A.2: Width measurement points along the lengths of the specimens. 210
Figure A.3: Length measurement depths of the specimens. 212
Figure B.1: Imperfection measurement points on Beams (a) B18; (b) B30 and B36;
(c) B44 and B36L. 215
Figure B.2: Sweep at midheight of B18-1. 219
Figure B.4: Sweep at midheight of B30. 220
Figure B.5: Sweep at midheight of B36. 220
Figure B.9: Sweep at midheight of B36L-1. 222
Figure B.10: Sweep at midheight of B36L-2. 223
Figure B.11: Permanent lateral deformation at midheight of B30. 223
Figure B.12: Permanent lateral deformation at midheight of B44-1. 224
Figure B.15: Permanent lateral deformation at midheight of B36L-2. 225
xvii
Figure B.16: Permanent torsional rotations of B30. 226
Figure B.17: Permanent torsional rotations of B44-1. 226
Figure B.20: Permanent torsional rotations of B36L-1. 228
Figure C.1: Loads and lateral deflections corresponding to the strain distributions
in Figures C.2 to C.4. 230
Figure C.2: Midspan strain distributions of B44-1 at the initial stages of loading. 231
Figure C.3: Midspan strain distributions of B44-1 close to buckling. 232
Figure C.4: Midspan strain distributions of B44-1 after buckling. 233
in Figures C.6 and C.7. 233
Figure C.7: Midspan strain distributions of B44-2 at the initiation of buckling. 235
Figure C.10: Midspan strain distributions of B44-3 at different load levels. 237
Figure C.11: Midspan strain distributions of B44-3 close to buckling. 238
in Figures C.13 and C.14. 239
Figure C.13: Midspan strain distributions of B36L-1 at the initial stages of loading. 240
Figure C.14: Midspan strain distributions of B36L-1 close to buckling. 241
Figure C.16: Midspan strain distributions of B36L-2 at the initial stages of loading. 243
Figure C.17: Midspan strain distributions of B36L-2 at different load levels. 244
xviii
Figure C.18: Midspan strain distributions of B36L-2 prior to and after buckling. 245
Figure C.19: Depthwise strains along the midspan section of B44-1 at the
initial stages of loading. 246
Figure C.20: Depthwise strains along the midspan section of B44-1 at the
final stages of loading. 247
Figure C.21: Depthwise strains along the midspan section of B44-1 during
unloading. 248
Figure D.1: Out-of-plane deflections of B18-2 at midspan. 250
Figure D.4: Out-of-plane deflections of B30 at midspan. 251
Figure D.5: Out-of-plane deflections of B36 at midspan. 252
Figure D.9: Out-of-plane deflections of B36L-1 at midspan. 254
Figure D.10: Out-of-plane deflections of B36L-2 at midspan. 254
Figure D.11: Lateral centroidal deflections of Beams B44 at midspan. 255
Figure D.12: Lateral centroidal deflections of Beams B36L at midspan. 255
Figure D.13: In-plane deflections of Beam B18-2 at midspan. 258
Figure D.14: In-plane deflections of Beams B22 at midspan. 258
Figure D.17: In-plane deflections of Beams B44-1 at midspan. 260
xix
Figure D.20: In-plane deflections of Beams B36L-1 at midspan. 261
Figure D.21: In-plane deflections of Beams B36L-2 at midspan. 262
Figure D.22: Experimental torque-twist curve of Specimen B44-1. 262
Figure D.25: Experimental torque-twist curve of Specimen B36L-1. 264
Figure D.26: Experimental torque-twist curve of Specimen B36L-2. 264
Figure E.1: Potentiometer configuration in the tests. 266
Figure E.2: Angle of twist calculations. 269
Figure E.3: Centroidal deflection calculations. 272
Figure G.1: Reinforcement in Specimen B36. 285
Figure G.2: Lifting mechanism in the first set of beams. 285
Figure G.3: Use of spreader beams to lift the beams. 287
Figure G.4: The lifting point in the second set of beams

connected to the spreader beam. 288
Figure H.1: Southwell (1932) Plots for Specimen B44-2. 292
Figure H.2: Meck’s (1977) Version of the Southwell (1932) Plots

for Specimen B44-2. 293
Figure H.3: Massey’s (1963) Version of the Southwell (1932) Plots

for Specimen B44-2. 294
xx
LIST OF SYMBOLS
Ac Gross area of the cross-section (Figure 1.24)
Ae Area bounded by the centerline of the effective wall
Ao Area bounded by the shear flow zone
As Total cross-sectional area of the longitudinal reinforcement
At Cross-sectional area of one leg of a stirrup
A1 Area bounded by the centerline of a closed stirrup (Figure 1.24)
A2 Area of the rectangle formed by the lines connecting the centroids of the
corner longitudinal bars (Figure 1.24)
B Lateral bending rigidity
E Modulus of elasticity
Ec Elastic modulus of concrete
Eit Initial tangent modulus of elasticity
Eo Overall modulus of elasticity [= (Esec+Ec)/2]
Reduced modulus of elasticity   4  Ec  Etan   

2
Er Ec  Etan

Es Modulus of elasticity of steel
Esec Secant modulus of elasticity
Etan Tangent modulus of elasticity
Etano Tangent modulus of elasticity corresponding to the extreme compression fiber

strain (Figure 1.17)
ECw Warping rigidity
EIx In-plane bending rigidity
EIy Out-of-plane bending rigidity
xxi
G Modulus of rigidity
Gc Modulus of rigidity of concrete
G’c Reduced modulus of rigidity of concrete   Esec 2  1    
GcrCcr Post-cracking torsional rigidity (Figure 5.1)
Go Overall modulus of rigidity of concrete
Gs Modulus of rigidity of steel
(GC) Torsional rigidity
Icr Cracked moment of inertia about the major axis
Ie Effective moment of inertia
Ig Moment of inertia of the gross cross-section about the major axis
Ix Moment of inertia about the major axis
Iucr Uncracked moment of inertia about the major axis considering the contribution
of the longitudinal reinforcement
Iy Moment of inertia about the minor axis
J Torsional constant
L Unbraced length
Mx In-plane bending moment
My Out-of-plane bending moment
Ma Maximum in-plane bending moment along the span of a beam
Mc Calculated ultimate moment
Mcr Critical moment
Mcra Cracking Moment
ML Limit moment
Mex Experimental ultimate moment
P Applied load
xxii
Pan Analytical ultimate or critical load
Pcr Critical load
PL Limit load
T Torsional moment
Ta Applied torque
Tb Maximum torsional moment in the beam at the instant of buckling
Tcr Cracking torque
Tmax Maximum torsional moment in a beam
Tn Torsional strength of a reinforced concrete member
Tnp Torsional strength of a plain concrete member
a Moment lever arm of a section
b Beam width
bs Width of the longitudinal reinforcement layer (Figure 1.18)
b1 Width of the area bounded by the centerline of a closed stirrup (Figure 1.18)
b2 Width of the rectangle formed by the lines connecting the centroids of the
longitudinal reinforcing bars (Figure 1.24)
c Neutral axis depth from the compression face
cu Neutral axis depth at the ultimate flexural moment level
cv Percent coefficient of variation
d Depth of the centroid of the tension reinforcement from the compression face
d1 Depth of the area bounded by the centerline of a closed stirrup (Figure 1.18)
e Vertical distance of the point of application of load from the centroid of the
beam section
f Stress
fc Concrete stress
f’c Compressive strength of concrete
xxiii
fr Modulus of rupture of concrete
ft Splitting tensile strength of concrete
h Beam height
kcr Lateral bending rigidity of the cracked portion of a concrete beam (Figure 4.1)
keq Lateral bending rigidity of a concrete beam (Figure 4.1)
kucr Lateral bending rigidity of the uncracked portion of a concrete beam (Figure 4.1)
n Modular ratio of steel to concrete [=Es/Ec]
pe Perimeter of the area bounded by the centerline of the effective wall
po Perimeter of the area bounded by the shear flow zone
p1 Perimeter of the area bounded by the centerline of a closed stirrup (Figure 1.24)
p2 Perimeter of the rectangle formed by the lines connecting the centroids

of the corner longitudinal bars (Figure 1.24)
s Spacing of the stirrups
ti Wall thickness of a tube
ts Thickness of the longitudinal reinforcement layer (Figure 1.18)
uc Out-of-plane deflection of the centroid of midspan section
uo Sweep at midheight of a beam at midspan
ut Out-of-plane deflection at the top of midspan section
uto Sweep at the top of a beam at midspan
u(z) Centroidal sweep at a distance z from the end of a beam
v Lateral deflection of the centroid of midspan section in the direction of the

major axis of the twisted configuration of a beam (Figure 6.8)
vc In-plane deflection of the centroid of midspan section
vo Initial imperfection at the center of a beam in the direction of the major axis
of the initial configuration of midspan section
w Self-weight per unit length of a beam
xxiv
wcr Critical self-weight per unit length of a beam causing buckling
y Depth of the center of gravity of the transformed beam section from

the compression face (Equation 4.6)
ΣIsy Moment of inertia of the longitudinal reinforcement about the minor axis of the
beam section
β Coefficient for St. Venant’s torsional constant
ε Strain
εc Strain of the compression fibers at an arbitrary depth from the compression

face created by the major-axis bending (Figure 1.17)
εco Extreme compression fiber strain from major-axis bending
εcl Compressive strain on the concave face of the beam originating from the
lateral bending moment only (Figure 1.17)
εcr Cracking strain of concrete
εo Strain at peak stress
εto Strain at the centroid of the tension reinforcement from major-axis bending
εtl Tensile strain on the convex face of the beam originating from the lateral
bending moment only (Figure 1.17)
εy Yielding strain of steel
θ Twist
θcri Twist at the initiation of diagonal cracking (Figure 5.1)
θcrp Twist at the end of diagonal cracking (Figure 5.1)
θu Twist at ultimate torque (Figure 5.1)
μ Mean value
ν Poisson’s ratio
νc Poisson’s ratio of concrete
ρl Volumetric ratio of the longitudinal reinforcement   As Ac 
ρt Volumetric ratio of the transverse reinforcement   At  p1  Ac  s 
xxv
ρto Total volumetric ratio of the transverse and longitudinal reinforcement [= ρl + ρt]
σ Standard deviation
σc Extreme compression fiber stress
φc Angle of twist of the beam at midspan
φo Initial twisting angle of a beam at midspan
φult Twisting angle of a beam at midspan at the instant when the limit load is
reached
φ(z) Twisting angle at a distance z from the end of a beam
xxvi
LIST OF ABBREVIATIONS
DEMEC Demountable mechanical strain gage
HRWR High range water reducing admixture
LVDT Linear variable differential transducer
NA Neutral axis
OC Conventionally vibrated ordinary concrete
SCC Self-compacting concrete
SRA Shrinkage reducing admixture
WWR Welded wire reinforcement
xxvii
SUMMARY
The study presents the results of an investigation aimed at examining the lateral stability
of rectangular reinforced concrete slender beams. A total of eleven reinforced concrete
beams having a depth to width ratio between 10.20 and 12.45 and a length to width ratio
between 96 and 156 were tested. Beam thickness, depth and unbraced length were 1.5 to
3.0 in., 18 to 44 in., and 12 to 39.75 ft, respectively. The initial geometric imperfections,
shrinkage cracking conditions and material properties of the beams were carefully
determined prior to the tests.
Each beam was subjected to a single concentrated load applied at mid-span by
means of a gravity load simulator that allowed the load to always remain vertical when
the section displaces out of plane. The loading mechanism minimized the lateral
translational and rotational restraints at the point of application of load to simulate the
nature of gravity load.
Each beam was simply-supported in and out of plane at the ends. The supports
allowed warping deformations, yet prevented twisting rotations at the beam ends.
In the experimental part of the study, reinforced concrete beams with initial
imperfections (sweep) failed under loads lower than the critical loads corresponding to
the geometrically perfect configuration of the respective beams. The maximum load
carried by an imperfect beam is known as the limit load (PL). In the present study, the
limit load (PL) and the critical load (Pcr) were distinguished.
In the first part of the analytical investigation, a formula was developed for
determining the critical loads corresponding to the lateral torsional buckling of
xxviii
rectangular reinforced concrete beams. The effects of shrinkage cracking and inelastic
stress-strain properties of concrete and the contribution of longitudinal reinforcement to
the lateral stability are accounted for in the critical load formula. The second part of the
investigation focused on developing a formula for the estimation of limit loads of
reinforced concrete beams with initial lateral imperfections. The proposed limit load
formula was obtained by introducing the destabilizing effect of sweep as a reduction term
to the critical load equation.
Finally, the experimental results were compared to the proposed analytical
solution and to various lateral torsional buckling solutions in the literature. The
formulation proposed in the present study was found to agree well with the experimental
results. The good correlation with the experimental results and the incorporation of the
geometric and material nonlinearities into the formula makes the proposed solution, given
below for a simply supported rectangular reinforced concrete beam loaded with a
concentrated load at midspan, practical for design purposes:
Pcru  

4  M cr uto  48  Ec  I y  (1)
L sin(ult )  L3
where PL is the limit load; L is the unbraced length of the beam; uto is the sweep at the top
of the beam at midspan; Ec is the elastic modulus of concrete; Iy is the second moment of
area of the beam section about the minor axis; φult is the angle of twist of the beam at
midspan corresponding to the limit load (PL). Mcr is the critical moment corresponding to
the geometrically perfect configuration of the beam, obtained from Equation (2):
xxix
4.23  e Bo 
M cr   1  1.74     B   GC o (2)
L 

L  GC o  o
where Bo is the lateral bending rigidity, obtained from Equation (3); (GC)o is the torsional
rigidity, calculated from Equation (4); e is the vertical distance of the load application
point from the centroid of the midspan cross section.
 
 3 
  b c  1    Esec  Ec 
Bo     (3)
 12   
2
 M cra   c  
  2


 1      1 
  M cr   h  
Esec  Ec  b3  h  b 
(GC )o    1  0.63    (4)
4  1     3  h 
where b and h are the width and height of the beam, respectively; c is the depth of the
neutral axis from the compression face; Mcra is the cracking moment; ω is a constant,
which has a value of 1 in the absence of restrained shrinkage cracks in concrete and a
value of 2/3 in the presence of restrained shrinkage cracks and υ is Poisson’s ratio of
concrete. Esec is the secant modulus of elasticity of concrete corresponding to the extreme
compression fiber strain at midspan at the instant when Mcr is reached.
xxx
INTRODUCTION
1.1 Introduction
Due to the increasing use of slender structural concrete beams in long-span bridges and
other structures, lateral stability is becoming an important criterion in the design of
structural concrete girders. Lateral-torsional buckling of long-span precast concrete
girders is a matter of concern, particularly during bridge construction.
Bridge girders are laterally supported by diaphragms and the bridge deck after the
completion of a bridge. Nonetheless, lateral stability of the precast bridge girders should
be assured also during fabrication, lifting, transportation and erection stages.
Accordingly, precast concrete girders should be designed to remain stable even under the
most unfavorable loading and support conditions of the transitory phases of construction.
Lateral instability of a beam arises from the compressive stresses in the beam
resulting from flexure due to transverse loading. The compression zone of the beam tends
to buckle about the minor axis of the overall cross-section of the beam while the tension
zone tends to remain stable. When the load reaches a certain “critical” value, the beam
buckles out of plane and twists (Figure 1.1) as a result of the differential lateral
displacements of the compression and tension zones.
For assessing the stability, the critical moment of a concrete girder should be
evaluated for the loading and support conditions of different phases of construction. A
beam free from initial geometric imperfections does not undergo out-of-plane deflections
and rotations before reaching a critical moment value. When the maximum moment in
the beam reaches the critical moment value, the beam experiences sudden excessive out-
1
Figure 1.1 – (a) Lateral torsional buckling of a beam subjected to a concentrated load at
midspan; (b) Lateral and vertical deflections and rotation of the midspan section
of-plane deformations and torsional rotations. This type of buckling is known as
bifurcation instability and the moment at which the beam loses its stability and
experiences rapid and excessive deformations at a constant load level is known as the
critical moment (Mcr).
A beam having initial geometric imperfections, on the contrary, does not bifurcate
at the limit load. The beam undergoes deformations and rotations throughout the whole
2
course of loading, even prior to buckling. The moment carried by the beam reaches an
ultimate value, called the limit moment (ML), beyond which greater lateral deformations
and rotations take place while the moment-carrying capacity of the beam slowly
decreases (Figure 1.2). This type of instability is known as limit load instability.
Figure 1.2 – Load-lateral deflection curve of a slender reinforced concrete beam
In reinforced concrete beams, the difference between the critical moment (Mcr)
and the limit moment (ML) is more pronounced since cracking in an imperfect concrete
beam due to the lateral displacements prior to buckling decreases the moment-carrying
capacity of the beam significantly.
3
ACI 318-05 (2005) does not include an analytical method for the calculation of the
critical moment of a concrete beam. The only provision regarding the stability is given in
Section 10.4, which limits the ratio of beam span to beam width, L/b, to less than 50.
AASHTO LRFD (2005) specifies in Section 5.5.4.3 that: “Buckling of precast
members during handling, transportation, and erection shall be investigated.” However,
no analytical method is given for the calculation of the critical moment of a reinforced
concrete beam
1.2 Project Objectives
The research described herein investigates the lateral stability of rectangular reinforced
concrete beams experimentally and analytically. The analytical study was carried out to
develop an analytical method to estimate critical moments of rectangular reinforced
concrete beams. In the experimental part of the study, a total of eleven slender
rectangular reinforced concrete beams were tested to produce experimental data for
supporting the analytical methods proposed for examining the lateral-torsional buckling
of reinforced concrete beams.
Attention is given to the effects of the initial geometric imperfections and
shrinkage on the lateral stability of reinforced concrete beams.
1.3 Organization of the Study
Section 1.4 summarizes the previous studies in the literature on lateral stability of
reinforced concrete beams. Chapter II introduces the specimens of the experimental
program and mechanical properties of the concrete mixtures used in the specimens.
4
Chapter III presents the experimental setup used to test the beams and summarizes the
test procedure.
Chapter IV summarizes the previously developed formulae concerning the lateral
bending rigidity of rectangular reinforced concrete beams and introduces the new lateral
bending rigidity equation proposed in this study. The chapter also presents the spring
systems used to model a reinforced concrete beam when developing the flexural rigidity
expressions. Finally, the effect of restrained shrinkage cracking on the lateral bending
rigidity of a concrete beam is examined in the last section of Chapter IV, where a
modification to the proposed lateral bending rigidity expression is introduced to account
for the reduction in the rigidity due to the presence of possible shrinkage cracks in
concrete.
In Chapter V, the torsional rigidity expressions for rectangular reinforced concrete
beams available in the literature are presented. Later, the slopes of the experimental
torque-twist curves of the specimens are compared to the analytical values obtained from
the torsional rigidity expressions given in the chapter. The torsional rigidity expression
giving the closest agreement with the experimental results is modified to account for the
possible inelastic material behavior of concrete at the time of buckling.
Chapter VI presents the critical moment calculations of reinforced concrete
beams. In Section 6.2, effects of the initial geometric imperfections on the ultimate
moment and the out-of-plane deformations and twisting rotations of an imperfect
reinforced concrete beam are explained and modifications to the critical moment
expression are proposed to account for the effects of geometric nonlinearities.
5
In Chapter VII, the crack patterns of the specimens and some experimental
results are presented, and the analytical critical load values obtained from the formulae
given in Chapter VI are compared to the experimental buckling loads of the specimens to
determine the degree of correlation between the analytical and experimental results.
Finally, conclusions of the study are summarized in the last chapter.
In the present study, limit moments of the specimens were taken as the greatest
moments in the experimental load-deflection plots of the specimens. There are some
other methods given in the literature for obtaining the buckling moments of beams by
using the experimental data. The methods developed by Southwell (1932), Meck (1977)
and Massey (1963) and their applications to reinforced concrete beams are explained in
Appendix H.
1.4 Previous Studies

This section reviews the previous studies on lateral torsional buckling of rectangular
reinforced concrete beams. The experimental studies in the literature are presented in
Section 1.4.1. Next, the analytical methods in the literature for predicting the critical
loads of reinforced concrete beams are explained in Section 1.4.2. Finally, the
contributions of the previous studies to the field of lateral stability of reinforced concrete
beams are summarized in Section 1.4.3, where the factors that remained uninvestigated in
the literature are also emphasized to support the need for the present research.
6
1.4.1 Review of Previous Experimental Work
Hansell and Winter (1959) studied the lateral stability of reinforced concrete beams both
experimentally and analytically. The main goal of the experimental study was to
investigate any possible reductions in the flexural capacities of reinforced concrete beams
with increasing L/b ratios. Hansell and Winter (1959) tested five different groups of
beams, namely B6, B9, B12, B15 and B18. Two companion beams for each group of
specimens were made and tested to failure. Nominal dimensions of the beams are
presented in Table 1.1. All specimen groups except B6 violated the slenderness criterion,
given in the 1956 Edition of ACI Building Code, which limited the L/b ratio to less than
32 for reinforced concrete beams.
Table 1.1 – Nominal dimensions of the beams tested by Hansell and Winter (1959)
Height, h Width, b Length, L

Specimen d/b ratio L/b ratio
(in.) (in.) (ft)
B18 13 2.5 18 4.5 86.4
B15 13 2.5 15 4.5 72.0
B12 13 2.5 12 4.5 57.6
B9 13 2.5 9 4.5 43.2
B6 13 2.5 6 4.5 28.8
All specimens tested by Hansell and Winter (1959) failed in in-plane bending
compression failure after the yielding of tension reinforcement and developed their
ultimate flexural strength prior to lateral torsional buckling. Experimental ultimate
moments of the specimens are presented in Table 1.2 together with the calculated
ultimate flexural moments. The experimental-to-calculated ultimate moment ratio of each
test beam is also given in the table. The experimental ultimate moments are in good
7
Table 1.2 – Results of the tests by Hansell and Winter (1959)
Experimental Calculated Experimental-

Ultimate Ultimate to-Calculated
Specimen Failure Mode
Moment, Mex Moment, Mc Moment Ratio
(in-kips) (in-kips) Mex/Mc
B6-1 Flexure 216 196.7 1.10
B6-2 Flexure 199 196.7 1.01
B9-1 Flexure 201 196.7 1.02
B9-2 Flexure 205 196.7 1.04
B12-1 Flexure 193 197.0 0.98
B12-2 Flexure 199 197.0 1.01
B15-1 Flexure 192 195.9 0.98
B15-2 Flexure 198 195.9 1.01
B18-1 Flexure 190 196.2 0.97
B18-2 Flexure 196 196.2 1.00
agreement with the calculated moment values. The mean and the coefficient of variation
of Mex/Mc are 1.01 and 3.5%, respectively.
Hansell and Winter (1959) also reported the midspan vertical, lateral top and
lateral bottom deflections of the specimens at the onset of yielding of the flexural
reinforcement. These are shown in Figures 1.3, 1.4 and 1.5, respectively. The test results
of the identical (companion) beams are also shown in the figures. It is to be noted in these
figures that the vertical deflections corresponding to the onset of steel yielding and the
ultimate moments (Figure 1.6) are in close agreement among the companion beams while
the lateral top and bottom deflections show significant variations among the companion
beams. For instance, out-of-plane deflections of the companion beams in Specimen
Groups B12, B15 and B18 display considerable variation.
Hansell and Winter (1959) loaded the specimens at quarter points to have constant
in-plane flexural moment over the middle part of the span. Under the loading and support
conditions reported by Hansell and Winter (1959), the bottom portions of the beams at
8
Figure 1.3 – Comparison of the vertical deflections at yield point of the companion
beams tested by Hansell and Winter (1959)
Figure 1.4 –Comparison of the lateral top deflections at yield point of the companion
9
Figure 1.5 – Comparison of the lateral bottom deflections at yield point of the companion
Figure 1.6 – Experimental -to- calculated ultimate moment ratios of beams tested by
Hansell and Winter (1959)
10
midspan were subjected to tensile stresses from major-axis bending while the top portions
were subjected to compressive stresses. In lateral torsional buckling, the compression
zone of a beam is prone to undergo greater lateral deflections than the tension zone due to
the stabilizing effect of the tensile stresses from in-plane bending. Nevertheless, the
midspan lateral bottom deflections of some test specimens of Hansell and Winter (1959)
exceeded the lateral top deflections.
Figure 1.7 shows the mechanism used by Hansell and Winter (1959) to convey
the load from the head of a universal testing machine to the test beam. Hansell and
Winter (1959) used a loading ball for the rotational freedom and a roller assembly to
provide lateral translational freedom at the loading point. When the test beam deflected
out of plane, the parts of the loading mechanism below the roller assembly were
supposed to move with the rollers in the lateral direction (Figure 1.8), preventing any
lateral restraint to the test beam. Furthermore, the loading cage around the beam was
expected to rotate with the beam about the loading ball, preventing any torsional restraint
to the beam at the loading point. The specimens were loaded using a universal testing
machine. The load was transmitted to the loading points (quarter points of the span)
through a steel beam connected to the specimen at each loading point, through the
loading mechanism shown in Figure 1.7.
An examination of the loading fixture (Figure 1.8) used by Hansell and Winter
(1959) reveals that the steel beam transmitting the load to the specimen does not displace
in the lateral direction while the beam deforms out of plane. Therefore, the line of action
11
Figure 1.7 – Loading mechanism used by Hansell and Winter (1959)
Figure 1.8 –Undeflected and expected deflected configurations of the loading mechanism
12
of the vertical load, initially passing through the shear center of the beam section,
becomes eccentric with respect to the shear center as the specimen deforms out of plane.
The eccentricity of the applied load creates larger and larger torsional moments in the
beam as the applied load increases in the course of the test.
Figure 1.9 depicts the position of the line of the applied load relative to the beam
when the beam undergoes lateral deflections and torsional rotations. The roller
assemblies allow free out-of-plane deflections in the beam at the loading points while the
steel beam remains stationary in the lateral direction. Hence, the line of action of the
applied load stays in its original position, rendering the applied load eccentric relative to
the shear center, which results in torsional moments in opposite direction to the torsional
rotations from instability. The accidental torsions constitute a restraint to lateral torsional
buckling.
Figure 1.9 –Direction of the torsional moments induced by the eccentric application of
the load
13
In the loading fixture used by Hansell and Winter (1959), the loading ball right
above the specimen, the socket plate, the lower roller block and the roller assembly
(Figure 1.7) move with the specimen in the lateral direction when the specimen
undergoes out-of-plane deflections. The loading ball, moving with the specimen in lateral
direction, applies lateral forces to the socket plate (Figure 1.10). If there is rolling friction
in the roller assembly, the lateral translation of the roller assembly and the socket plate is
restrained and the socket plate applies reaction forces to the loading ball, which restrains
the lateral deflection of the top portion of the beam. Significant friction forces in the
roller assembly can cause the top portion of the beam to be more stable than the bottom
portion, which has no lateral translational restraint. In this case, the bottom portion
undergoes greater lateral deformations than the top portion (Figure 1.10) and the beam
experiences a different type of buckling called the web sidesway buckling. Hansell and
Winter (1959) stated that all rolling surfaces in their setup was cleaned and oiled prior to
each test to minimize the rolling friction in the loading fixture and the lateral translation
restraint to the top portion of the beam.
Considering the good agreement between the experimental ultimate moments and
the analytical values calculated according to Eq. (A.1) in 1956 Edition of the ACI
Building Code, Hansell and Winter (1959) concluded that there were no reductions in the
experimental ultimate moments of the beams due to the slenderness effects.
Sant and Bletzacker (1961) tested four different groups of beams, denoted B36,
B30, B24 and B12 whose nominal dimensions are specified in Table 1.3. Three identical
beams of each of the first three groups, B36, B30 and B24 and two identical beams of the
fourth group, B12 were tested to failure. Table 1.4 summarizes the test results. The mean
14
Figure 1.10 –Web sidesway buckling failure of the specimen due to the lateral restraining
forces in the loading mechanism
Table 1.3 – Beams tested by Sant and Bletzacker (1961)
Specimen Number of Height, h Width, b Length, L

d/b ratio L/b ratio
Group Samples (in.) (in.) (ft)
B36 3 36 2.5 20 12.45 96
B30 3 30 2.5 20 10.20 96
B24 3 24 2.5 20 8.13 96
B12 2 12 2.5 20 3.78 96
value of the test results of identical beams are included in the table.
15
Table 1.4 – Results of the tests by Sant and Bletzacker (1961)
Test Moment,
Group Test Specimen Failure Mode
Mtest (in-kips)
B36-1 Stability 1620
I
µ* 1605
II B30-2 Stability 2160
µ 1867
III B24-2 Stability 1350
µ 1350
B12-1 Flexure 300
IV B12-2 Flexure 210
µ 255
* - Mean value of the test moments of the beams in the same group
Test results show considerable variation. For instance, the experimental buckling
moment of Specimen B30-2 is 54% larger than the experimental moment value obtained
by testing its companion, B30-1. In Fig. 1.11, the experimental-to-predicted buckling
moment ratios of the beams in specimen groups B36, B30 and B24 are shown to reveal
the variation in the test results of companion beams. Since B12-1 and B12-2 did not
experience lateral torsional buckling, they are not included in the figure.
Sant and Bletzacker (1961) used a steel loading ball to provide rotational freedom
and a rolling mechanism to provide lateral-translational freedom at the point of
application of load (Figure 1.12). The specimens were loaded through a hydraulic load
cylinder, placed right above the beam and connected to the beam through threaded rods.
16
Figure 1.11 – Experimental-to-predicted buckling moment ratios of the beams in the
first three specimen groups, B36, B30 and B24
The loading ball was located on the head of the load cylinder in a ball-and-socket joint.
Finally, the roller assembly was placed above a load cell, which was located adjacent to
the top surface of the socket plate of the ball-and-socket assembly.
A ball-and-socket joint allows free angular motion of the connecting parts relative
to each other. When a beam experiences torsional rotations in the test setup used by Sant
and Bletzacker (1961), the beam and the load cylinder, connected to it, rotate relative to
the top portion of mechanism above the loading ball. The loading ball rotates in the
socket with the specimen and cylinder, preventing any rotational restraint to the beam at
loading point. As illustrated in Figure 1.13, the load cylinder is also free to rotate relative
17
Figure 1.12 – Loading frame used by Sant and Bletzacker (1961)
18
Figure 1.13 – Deflected configuration of the loading mechanism
used by Sant and Bletzacker (1961)
to the top portion of the mechanism, since the cylinder is placed between the specimen
and the ball-and-socket joint. Therefore, the cylinder ceases to be oriented vertically once
the specimen experiences torsional rotations. The deviation of the applied load from the
vertical axis induces lateral restraining force to the loading mechanism, which prevents
the rollers in the rolling mechanism to move freely in the lateral direction. The lateral-
translational restraint at the loading point increases the buckling load of a beam and
decreases the out-of-plane deformations and the torsional rotations.
The verticality of the applied load in a lateral-torsional buckling test is crucial,
particularly in the case of geometrically imperfect beams. Theoretically, a geometrically
19
perfect beam experiences little or no out-of-plane deformations and torsional rotations
prior to buckling. On the contrary, a beam with initial geometric imperfections undergoes
lateral deformations and torsional rotations throughout the entire course of loading.
Therefore, the vertical orientation of the load applied by the loading mechanism used by
Sant and Bletzacker (1961) is lost at the very early stages of the test of an imperfect
beam. The inclination of the applied load with respect to the vertical axis continuously
increases in the course of loading, introducing greater and greater lateral restraining
forces to the roller mechanism. This lateral-translational restraint affects the experimental
results.
Massey and Walter (1969) tested five small-scale beams with the details given in
Table 1.5. The table also includes the experimental buckling loads of the specimens. The
specimens were simply-supported in plane and out of plane and subjected to a
concentrated load at mid-span. Massey and Walter (1969) used a special method of
loading. The specimens were loaded through a water tank connected to the beam at the
centroid of the mid-span section. Using dead weights hung from the specimen is a proper
method of loading in lateral-torsional buckling experiments for two reasons. First, a dead
weight hanging from the beam travels with the beam and does not induce any lateral-
translational and rotational restraint to the beam at the load application point. Secondly,
the vertical orientation of the dead weight does not change regardless of the rotations in
the beam, since the gravitational forces are always vertical. Loading a beam with dead
weights is also a quite economical and convenient method of loading. Nevertheless, this
method of loading has a limitation preventing it to be applicable to large-scale beam tests,
particularly if water is used as the means of loading. Water has a low unit weight (0.0624
20
kip/ft3). Therefore, large volumes of water are needed to load large-scale beams up to the
failure.
Table 1.5 – Specimens tested by Massey and Walter (1969)
Effective Experimental
Width, b Length, Tension
Specimen Depth, d Buckling
(in.) L (ft) Reinforcement
(in.) Load, Pcr
1 12 1 10 ½ x ½ square bar 3.77
2 12 1 12 ½x½ 3.68
3 15 ¾ 12 1x¼ 2.20
4 15 ¾ 12 ¾ x¼ 1.42
5 12 ¾ 14 ¾ x¼ 0.60
Due to the limitations of the loading method, Massey and Walter (1969) tested
small-scale beams, which are also easier to fabricate and to test, compared to the large-
scale ones. Nevertheless, due to their relatively small lateral-flexural and torsional
rigidities, the experimental results of the beams with smaller scales are more sensitive to
the parameters associated with the test setup (tolerance errors). For instance, restraints
from the loading mechanism, accidental deviations from vertical and eccentricities of the
applied load have more pronounced influences on the behavior and test results of a small-
scale beam, as opposed to a beam with larger scale.
Konig and Pauli (1990) carried out an extensive experimental study in which they
tested six reinforced and prestressed concrete beams. The first five beams were T-shaped
and they were designed in a way that each specimen was distinct from the other four
specimens in an individual parameter, influencing the lateral stability of concrete beams.
The sixth beam was totally different from the other five specimens in cross-sectional
shape, dimensions and reinforcement (Figure 1.14). In the following discussion, the
21
Figure 1.14– Cross-sectional details of the specimens tested by Konig and Pauli (1990)
specimens are introduced by emphasizing the individual parameters whose effects were
examined. Then, the influence of each parameter will be discussed in the light of the
experimental results obtained by Konig and Pauli (1990). The nominal dimensions and
the flexural reinforcement of the test specimens are presented in Table 1.6 and the initial
geometric imperfections are tabulated in Table 1.7. Finally, the experimental buckling
loads and deformations of the specimens are given in Table 1.8.
22
Table 1.6 –Beams tested by Konig and Pauli (1990)
Top
Span Beam
Flange Tension Compression
Specimen Length, Height,
Width, Reinforcement Reinforcement
L (ft) h (in)
b (in)
1 59 10.4 51.2 6 M25 4 M12 & 4 M8
2 59 10.2 51.2 6 M25 4 M12 & 4 M8
3 59 14.2 51.2 6 M25 4 M12 & 4 M8
4 59 10.2 51.2 6 M25 4 M25 & 4 M8
5 59 10.2 51.2 14 M12.5 strands 4 M12 & 2 M12.5
6 84 14.2 53.1 24 M12.5 strands 4 M12 & 4 M8
Table 1.7 – Midspan initial imperfections of the beams tested by
Konig and Pauli (1990)
Angle of twist, φo
Specimen Initial Sweep at Midheight, uo (in.)
(radian %)
1 0.79 0
2 0.12 0.30
3 0.24 0.30
4 0.10 0.15
5 0.63 0.30
6 0.43 0.40
Table 1.8 – Loads, midspan deformations and rotations at failure of the beams
tested by Konig and Pauli (1990)
Mid-span Deformation (in.) Midspan Angle

Critical Load,
Specimen of Rotation
Pcr (kips) Lateral Top Vertical (%)
1 42.7 6.4 4.6 0.5
2 44.5 3.3 2.4 0
3 57.0 5.6 4.6 1.0
4 53.4 1.9 3.7 0
5 45.1 7.2 2.8 0
6 50.9 8.8 5.5 0
23
Specimens 1 and 2 had the base nominal dimensions, reinforcement and cross-
sectional details. They only differed in the initial geometric imperfections. Specimen 3
was identical to the first two specimens, except the width of the top flange. The top
flange of the third specimen was made stockier than the first two specimens by increasing
the breadth from 10.4 in. (25 cm) to 14.2 in. (35 cm). Specimen 4 had heavier
compression reinforcement than the first three beams. The M12 bars in the top flanges of
the first three beams were replaced with M25 bars in the fourth specimen while keeping
the nominal dimensions identical to the first two specimens. Specimen 5 was reinforced
with prestressing strands instead of rebars to examine the influence of prestressing of
reinforcement on the lateral stability of concrete beams. The M8 bars in the top flanges
and the M25 bars in the bottom portions of the first four specimens (Figure 1.14) were
replaced with M12.5 strands in the fifth specimen. Specimen 6 was tested to investigate
the lateral stability of prestressed concrete beams with I-section to observe the influence
of the cross-sectional shape on the stability.
Specimen 2 had a smaller initial lateral deformation, sweep, than the first
specimen. Accordingly, the test results of the second specimen are closer to reflect the
behavior of an initially-perfect beam with the base dimensions. Therefore, the results of
each of the five specimens are compared to the experimental values of Specimen 2
(Figure 1.15) when discussing the influence of an individual parameter on the lateral
stability of concrete beams.
Specimen 1 with the greater sweep (0.79 in. at mid-height of the mid-span
section) failed at an applied load, 4 % lower than the buckling load of Specimen 2, whose
sweep was measured as 0.12 in. Although the midspan sweep of one of the specimens
24
Figure 1.15 – Ratio of the buckling load of each specimen to the buckling load of
Specimen 2
was almost seven times greater than the sweep of the other beam, the reduction in the
buckling load was only 4 %, implying that the influence of the initial imperfections was
not quite significant.
The loading and support conditions reported by Konig and Pauli (1990) suggests
that the top flanges of the specimens were subjected to compressive stresses while the
bottom portions were under tension in the tests. Specimen 3, which had a wider top
flange than the first two beams, buckled under an applied load of 57 kips, which is 28 %
greater than the buckling load of the second specimen. The significant increase in the
failure load depicts the major stabilizing effect of a wider compression flange on the
beam.
25
The test results of Specimen 4 indicated that the stabilizing effect of the
compression reinforcement was two-fold. First, the buckling load increased to 53.4 kips,
corresponding to an increase of 20 % with regard to the second specimen. Secondly, the
lateral-top deflection of Specimen 4 at failure was smaller than the top deflections of the
first three beams. Both the increase in the failure load and the decrease in the lateral-top
deflection were bound up with the increase in the out-of-plane rigidity of the top flange.
The M12 bars in the top flanges of the first three beams were placed 3.5 in. away from
the weak axis of the section (Figure 1.14). Owing to the distance from the minor axis, the
reinforcing bars significantly contributed to the lateral bending rigidity. The use of M25
bars in Specimen 4 in replacement of the M12 bars increased the resistance of the beam
to lateral-torsional buckling by further constraining the top flange from deforming out of
plane.
The buckling load of Specimen 5 was only 1.3 % greater than the buckling load of
Specimen 2. Accordingly, Konig and Pauli (1990) concluded that the stabilizing effect of
prestressing was not as pronounced as the effects of the top flange width and the
compression reinforcement. However, the type of reinforcement was not the only
difference between Specimen 5 and Specimen 2. Specimen 5 had a significantly larger
sweep than Specimen 2. The buckling load of Specimen 5 might have been reduced by
the major sweep, causing the experimental results not to reflect the actual degree of
stabilization provided by prestressing. To evaluate the influence of prestressing on the
lateral stability of concrete beams, more experimental results are needed.
Konig and Pauli (1990) used the loading mechanism illustrated in Figure 1.16 in
their experiments. A water tank was connected to the beam at the one-third points of the
26
Figure 1.16 – Loading mechanism used by Konig and Pauli (1990)
span. Steel plates were attached to the tank. The use of steel plates in addition to the
water tank reduced the need for excessive volumes of water in the tests and enabled the
researchers to use a tank of smaller capacity to attain the buckling loads of the beams.
The water tank was connected to a loading cage through cables. The loading cage
transmitted the load to the top of the beam. A pivot bearing, joining the beam and the
loading cage, provided the beam with the rotational freedom at the point of application of
load.
The loading mechanism used by Konig and Pauli (1990) was clearly superior to
the mechanisms used by Hansell and Winter (1959) and Sant and Bletzacker (1961).
When introducing the loading mechanism used by Massey and Walter (1969), the
27
efficiency of dead weights, hung from the specimen, in minimizing the lateral-
translational and rotational restraining forces at the loading point was discussed. Konig
and Pauli (1990) also overcame the need for a spacious water tank in the setup by using
steel plates. By considering the nature of the dead loads, the test results obtained by
Konig and Pauli (1990) can be considered reliable to be used in the analytical studies.
The beams tested by Konig and Pauli (1990) were simply supported in and out of
plane at the ends. The boundary conditions of a simply-supported beam are explained in
the third chapter of the present text in detail. One of the conditions that need to be
fulfilled to achieve the simple support conditions is the absence of a major restraint from
the supports against the displacement in longitudinal direction. The lateral supports used
by Konig and Pauli (1990) allowed the longitudinal displacements at the beam ends. The
top flanges of the beams were supported laterally through ball-bearings. The bottommost
portions of the beam ends were also supported in lateral direction to preserve the integrity
of the support sections. Sliding pads were placed between the beam and the bottom
supports. The ball bearings at the top and the sliding pads at the bottom minimized the
longitudinal friction forces from the lateral supports and allow the ends to rotate in plane
with no major restraint.
1.4.2 Analytical Methods for Predicting Lateral Torsional Buckling

One of the first investigations on the lateral stability of reinforced concrete beams was
conducted by Marshall (1948). The analytical study aimed at developing critical load
expressions for a laterally-unsupported beam under three different loading conditions:
1. Concentrated load at mid-span;
2. Uniformly distributed load throughout the span;
28
3. Equal and opposite bending moments at the beam ends;
Marshall (1948) obtained the critical load equations (1.1) and (1.2) for the loading cases 1
and 2, respectively. Equation (1.3) gives the critical moment of a beam subjected to the
equal and opposite end moments (loading case 3):
16.93
Pcr  2
 B   GC  (1.1)
L
28.6
wcr  3
 B   GC  (1.2)
L
8.47
M cr   B   GC  (1.3)
L
where Pcr, wcr and Mcr are the critical concentrated load, the critical unit load and the
critical moment of a laterally unsupported beam, respectively; L is the unbraced length of
the beam; B and GC are the out-of-plane flexural and the torsional rigidities of the beam,
respectively. Marshall (1948) proposed the use of the following lateral flexural and
torsional rigidity expressions for rectangular reinforced concrete beams:
b3  d
B  2.5  10 6
 (1.4)
12
b3  d
GC  0.9  10 6
 (1.5)
3
where b and d are the width and the effective depth of the beam, respectively. The
multipliers 2.5x106 and 0.9x106 in Equations (1.4) and (1.5) are the modulus of elasticity
and the modulus of rigidity of concrete, respectively. Marshall (1948) assumed the
29
modulus of elasticity and the modulus of rigidity to be constant for the concrete fibers
throughout the length and depth of the beam at the time of buckling. This assumption
disregards the inelastic lateral-torsional buckling behavior of reinforced concrete beams.
Figure 1.17 illustrates the stress-strain curve of a normal-strength concrete. The moduli of
elasticity corresponding to different stress values are shown on the curve. The first
portion of the curve up to the proportional limit stress (0.4.fc’ for normal-strength
concrete) is linear. The slope of this line represents the initial tangent modulus of
elasticity (Eit), and it is calculated according to Equation (1.6), given in ACI 318-05
(2005) for normal-weight concrete.
Eit  57000  f c' (1.6)
where Eit and fc’ are the initial tangent modulus of elasticity and the compressive strength
of concrete in psi, respectively.
If all the compression fibers throughout the depth and length of a beam are
stressed below the proportional limit (elastic limit in many cases) of concrete at the
instant of buckling, the beam experiences elastic lateral-torsional buckling. In the case of
elastic buckling, the initial tangent modulus of elasticity of concrete provides a good
estimate for the rigidity of all the compression fibers in the beam. The constant modulus
of elasticity value, 2.5x106, proposed by Marshall (1948) corresponds to a concrete
compressive strength of 1920 psi according to Equation (1.6). 1920 psi is a low concrete
strength compared to the compressive strength values encountered in today’s practice.
Therefore, the constant modulus of elasticity value proposed by Marshall (1948) will
30
Figure 1.17 – Stress-strain curve of normal-strength concrete and the tangent moduli of
elasticity at different stress levels (from Nawy 2005)
result in low estimates when computing the load associated with the elastic lateral
torsional buckling of a reinforced concrete beam.
In the case of inelastic lateral-torsional buckling, however, some compression
fibers of the beam are stressed beyond the proportional limit of concrete at the time of
buckling. Figure 1.17 indicates that the slope of the stress-strain curve reduces as the
stress and strain increase beyond the proportional limit [tan(α1)>tan(α2)>tan(α3)].
Particularly, if the concrete is stressed to more than 0.7fc’, it loses its rigidity to a major
extent and the modulus drops drastically vanishing at ultimate stress. Depending on the
stresses reached at the initiation of buckling, the modulus of elasticity values for highly-
stressed fibers at the outermost parts of the compression zone can be significantly lower
than the initial tangent modulus of elasticity of concrete (Eit>Et2>Et3), reducing the
overall modulus of the beam used in the evaluation of the lateral bending rigidity.
31
To summarize, the modulus of elasticity value used in the calculation of the
flexural rigidity about the minor axis varies significantly along the length and depth of
the beam at the instant of buckling. Therefore, a constant value of the elastic modulus
should not be used when computing the lateral flexural rigidity.
The multiplier 0.9x106 in Equation (1.5) is the assumed modulus of rigidity value
of the concrete. By using a Poisson’s ratio of 0.3, this value can be calculated from
Equation (1.7):
E
G (1.7)
2  (1   )
where E and G are the modulus of elasticity and the modulus of rigidity, respectively: ν is
the Poisson’s ratio. Similar to the elastic modulus term in the lateral bending rigidity
expression, Marshall (1948) proposed the use of a constant modulus of rigidity value in
the torsional rigidity calculations, which disregards the inelastic material behavior of
concrete and the variation of the modulus of rigidity along the length and depth of the
beam at the time of buckling.
The use of the effective depth d in Equations (1.4) and (1.5) suggests that
Marshall (1948) assumed that all fibers of the beam from the compression face to the
centroid of the tension reinforcement contribute to the resistance of the beam against
lateral-torsional buckling. Only the portion of the beam below the centroid of the tension
reinforcement is neglected in the critical load calculations. Using d in the critical load
calculations is based on a very general assumption that the concrete above the centroid of
the tension reinforcement remains uncracked until buckling. Depending on the strain
distribution in the tension zone of the beam, the flexural cracks may propagate upward
32
close to the compression zone before buckling, rendering the cracked zone ineffective in
resisting buckling. Particularly, in the case of inelastic lateral-torsional buckling, many
tension fibers in the beam reach strains higher than the cracking strain of concrete in
tension. Therefore, extension of the flexural cracks in the tension zone should be well-
established to determine the portion of the beam providing rigidity against buckling. The
use of d in the critical load calculations may overestimate the portion of the beam
effective in resisting lateral buckling at the time of failure.
Marshall (1948) used uncracked, elastic and homogeneous material assumption in
the critical load calculations. Consequently, the rigidity expressions given in the study do
not reflect the true behavior of reinforced concrete beams, especially if the buckling takes
place close to the ultimate flexural load levels. Marshall (1948) also inferred that the
stability criteria based on L/b ratio only is not factual and the lateral stability of a beam
should be evaluated based on d/b ratio as well as the L/b ratio. The study included the
stability analysis of both singly- and doubly-reinforced concrete beams. Marshall (1948)
did not investigate the effects of initial geometric imperfections on the lateral stability of
reinforced concrete beams.
Hansell and Winter (1959) conducted an analytical study to investigate the lateral
stability of an initially perfect rectangular reinforced concrete beam. In their study,
Hansell and Winter (1959) found that the secant modulus of elasticity corresponding to
the extreme compression fiber strain at the instant of buckling reflects the material
behavior of concrete in the compression zone, and therefore, the secant modulus of
elasticity should be used as the material rigidity term when evaluating the critical
moments of rectangular reinforced concrete beams. Secant modulus of elasticity is the
33
slope of the line on the stress-strain curve connecting the origin to the point
corresponding to the extreme compression fiber strain (Figure 1.18). The modulus of
rigidity used in the assessment of the torsional rigidity of a beam is calculated from the
secant modulus of elasticity according to Equation 1.7.
Figure 1.18 – Secant modulus of elasticity corresponding to the extreme compression
fiber strain
Hansell and Winter (1959) conservatively assumed that the concrete below the
neutral axis is fully cracked at the time of buckling and its contribution to the resistance
to lateral-torsional buckling should be disregarded. By only taking the compression zone
of the section into account, Hansell and Winter (1959) obtained the following lateral
bending and torsional rigidity expressions for rectangular reinforced concrete beams:
34
b3  c
B  Esec  (1.8)
12
Esec  b3  c  b 
2
GC     1  0.35    (1.9)
2  1     3  d 
where c is the neutral axis depth, b is the beam width, d is the effective depth to the
centroid of reinforcement and Esec is the secant modulus of elasticity corresponding to
the extreme compression fiber strain at the instant of buckling.
The use of the neutral axis depth in the equations is appropriate in the case of
inelastic lateral-torsional buckling. When a beam fails in inelastic buckling, many
portions along the depth of the tension zone exceed the cracking strain of concrete.
Hence, the flexural cracks in the tension zone propagate towards the compression zone
rendering an important portion of tension zone ineffective at the time of buckling. On the
other hand, a considerable portion of the tension zone can be still effective in resisting the
instability failure in the case of elastic lateral-torsional buckling, particularly if the
buckling takes place at the early stages of loading when only some fibers in the outermost
portion of the tension zone reach the cracking strain of concrete. Thus, use of the neutral-
axis depth results in low buckling load estimates for slender concrete beams subject to
elastic lateral-torsional buckling.
Another noteworthy detail in the rigidity expressions proposed by Hansell and
Winter (1959) is the use of a constant c value for different sections along the span. When
the in-plane bending moment is constant throughout the span of a beam, the neutral axis
depth (c) of each section along the span is the same. Nevertheless, the neutral axis depths
of different cross-sections of a beam are different in the case of non-uniform in-plane
35
bending moment along the span. For design purposes, Hansell and Winter (1959)
proposed the use of a single c value for the entire beam independent of the in-plane
bending moment distribution in the span. Hansell and Winter (1959) recommended the
use of the rigidity values corresponding to the beam section with maximum bending
moment along the span (for example the midspan section of a beam subjected to a
concentrated load at midspan), since the rigidities in a concrete beam are minimum at the
maximum moment locations.
Siev (1960) identified three different states of a reinforced concrete beam along
the loading history; uncracked elastic, cracked elastic and cracked plastic states. A
different lateral-bending rigidity expression was developed for each state. Nonetheless,
Siev (1960) advocated the use of a single torsional rigidity expression for reinforced
concrete beams independent of the state of the beam at the time of buckling.
In his study, Siev (1960) analyzed a beam simply supported in and out of plane at
the ends and subjected to a concentrated load at mid-span. Under the specified loading
and support conditions, the largest in-plane bending moments occur at mid-span of the
beam, while the end portions of the beam are subjected to minor bending moments.
Therefore, few or no flexural cracks form in the tension zone of the beam near the end
supports. Since the largest torsional moments are resisted by the end portions of the
beam, Siev (1960) included the contribution of the tension zone to the torsional rigidity
and proposed the following rigidity expression:
Ec  b3  h  b 
GC     1  0.63    (1.10)
2  1     3  h 
where h is the overall depth of the beam and ν is the Poisson’s ratio.
36
In the uncracked state, Siev (1960) considered the reinforced concrete as a
homogeneous material and disregarded the contribution of the flexural and shear
reinforcement to the lateral-flexural rigidity. The uncracked flexural rigidity (Bu) is given
by Equation (1.11):
b3  h
Bu   Ec (1.11)
12
The second state of a reinforced concrete beam was identified as the cracked
elastic state. In the cracked elastic state, flexural cracks form and propagate in the tension
zone of the beam while the concrete in the compression zone is still linearly elastic. Siev
(1960) approximated the stress-strain curve of concrete into a linearly elastic and a plastic
portion. The lateral-bending rigidity of the beam in the cracked elastic state was obtained
by dividing the out-of-plane bending moment to the out-of-plane bending curvature
induced by the moment. The curvature of the beam was determined from the stresses and
strains in the cross-section. Siev (1960) considered the fact that the neutral axis of a
cross-section of the beam deviates from horizontal in the presence of biaxial moments,
namely the in-plane and out-of-plane bending moments. Based on a linear stress-strain
relationship in the compression zone of the section and a rotated neutral axis due to the
presence of lateral bending moments in addition to the major-axis bending moments, Siev
(1960) developed the following lateral-flexural rigidity expression for the cracked elastic
state of the beam:
M c  Ec  b 2 bo2 
Bc      (1.12)
c a  6  c 4 d  c 
37
where M is the in-plane bending moment; σc is the extreme compression fiber stress
corresponding to M; bo is the horizontal distance between the centroids of the reinforcing
bars and a is the internal moment arm of the section. As a result of assuming a triangular
stress distribution in the compression zone of the section, a is equal to d-c/3.
The lateral-flexural rigidity in the cracked elastic state (Bc) is a function of the in-
plane bending moment (M) the extreme compression stress (σc) and the neutral axis depth
(c) corresponding to M. Therefore, the rigidity value at the time of buckling can only be
calculated by knowing the critical moment and the stress and strain distributions in the
section corresponding to the critical moment. The evaluation of the critical moment based
on the rigidity expressions given by Siev (1960) requires an iterative approach. First, an
initial value of M is assumed and the lateral-flexural rigidity corresponding to the initial
value of M is calculated. Subsequently, the critical moment is computed from the
calculated lateral-flexural and torsional rigidity values, using Equation (1.13).
C1
M cr   B C (1.13)
C2  L
where C1 and C2 are the constants corresponding to the loading and support conditions of
the beam, respectively. The iterations are then continued until the moment value
converges.
Finally, Siev (1960) proposed a lateral-flexural rigidity expression for the cracked
plastic state of the beam. In the plastic state, some fibers in the compression zone of the
beam are strained beyond the elastic limit of concrete (Figure 1.19). Since an elastic-
perfectly plastic stress-strain behavior was assumed for concrete, a uniform stress
38
distribution is reached within the outermost portion of the compression zone. Siev (1960)
derived the following lateral-flexural rigidity expression for the plastic state of the beam:
b 2  M c p  ce
Bp   (1.14)
12   c  a c  ce
p
2
where cp and ce are the depths of the plastic and elastic portions of the compression zone,
respectively (Figure 1.19); εc is the strain at the extreme compression fibers.
Sant and Bletzacker (1961) also carried out an analytical study on the lateral
torsional buckling of rectangular reinforced concrete beams. The following lateral-
flexural and torsional rigidity expressions were proposed:
b3  d
B  Er  (1.15)
12
Er b3  d
GC   (1.16)
2  1    3
where Er is the reduced modulus of elasticity of concrete corresponding to the extreme
compression fiber strain.
Equations (1.15) and (1.16) are different from the rigidity expressions adopted by
Hansell and Winter (1959) mainly in two aspects. First, Sant and Bletzacker (1961)
assumed that only the concrete above the centroid of the tension reinforcement
contributes to the resistance of a beam against buckling. Therefore, the effective depth (d)
is used in the rigidity expressions instead of the neutral axis depth (c) presuming that the
concrete below the centroid of the tension reinforcement is ineffective at the time of
39
Figure 1.19 – (a) Stress distribution in the compression zone of the beam section in the
plastic state; (b) Stress-strain curve of concrete; assumed by Siev (1960)
(* σe = elastic limit stress)
buckling due to cracking. The portion of a beam effective in resisting lateral torsional
buckling is determined according to the strain distribution in the section and in the span
of the beam at the onset of buckling. The fibers in the tension zone strained beyond the
cracking strain of concrete are not taken into consideration in critical load calculations.
According to the moment levels reached prior to buckling and the cross- sectional and
material properties of the beam, the use of c or d or a value between them might be more
appropriate to account for the effective portion of the beam in buckling resistance.
40
However, the use of c in critical load calculations results in lower buckling load estimates
than the use of d, proposed by Sant and Bletzacker (1961).
In addition, Sant and Bletzacker (1961) argued that the reduced modulus of
elasticity reflects the material rigidity of concrete at the time of buckling. The reduced
modulus of elasticity expression used by Sant and Bletzacker was first derived by
Considère (1891) and Engesser (1895) and later supported by the experimental and
analytical studies on inelastic column buckling by Von Karman (1910). The application
of the reduced modulus theory to lateral torsional buckling is briefly explained in the
following discussion.
In the double modulus theory for lateral-torsional buckling, the strain distributions
along the depth as well as the width of the midspan section of a beam at the time of
buckling are established as in Figure 1.20. For simplification, the beam is assumed not to
experience out-of-plane bending deformations prior to buckling. Accordingly, the beam
is only strained as a result of the in-plane bending deformations at the onset of buckling.
When the beam loses its stability and bends out of plane, the fibers in the concave half of
the section are compressed further. On the other hand, the out-of-plane deformations after
buckling introduce tensile stresses and strains to the fibers in the convex half of the beam.
In other words, the compressive strains resulting from the in-plane and out-of-plane
bending moments add up in the concave side of the compression zone, while the tensile
strains caused by the lateral bending cancel the compressive strains resulting from the
vertical bending in the convex side of the compression zone of the section. The formation
of the tensile strains in the compression zone of the beam is named as strain reversals. As
shown in Figure 1.20, the further loading of the fibers in the concave side of the section
41
Figure 1.20 – Strain distribution at midspan section and the corresponding reduced
modulus of elasticity
takes place along the line tangent to the curve at point A. On the other hand, the
unloading of the fibers in the convex side takes place along a line parallel to the initial
straight portion of the stress-strain curve. Therefore, the modulus of elasticity of the
loading fibers is the tangent modulus of elasticity, Etan corresponding to the compressive
strain of the fibers at the onset of buckling, while the modulus of elasticity valid for the
unloading fibers is the initial tangent modulus of elasticity of concrete, Ec.
The reduced modulus (double modulus) theory is based on the presumption that
the load increase in the beam due to additional compressive strains in the concave side of
the beam is equal to the decrease in the load due to tensile strains developed in the
42
convex side of the beam after buckling. Thus, the load-carrying capacity of the beam is
constant during buckling. Based on the constant load assumption, the following
expression is developed for the reduced modulus of elasticity:
4  Ec  Etan
Er  (1.17)
 
2
Ec  Etan
Since the tangent modulus of elasticity depends on the strain of the fibers at the
onset of buckling, the reduced modulus of elasticity is a function of the strain in the fibers
at the initiation of buckling. Equation (1.17) was developed for the buckling of a column,
subjected to equal concentrated loads at the ends. In column buckling, the axial strain is
assumed to be constant across the width and the length of the column. Therefore, all
fibers in the concave side of the column have the same tangent modulus of elasticity at
the onset of buckling. On the other hand, the strains resulting from the in-plane bending
vary throughout the depth and the length of a beam, subjected to a concentrated load at
mid-span. Consequently, the tangent and reduced moduli of elasticity change along the
depth and the length of the beam. Therefore, the use of a constant tangent modulus of
elasticity along the depth of the concave half of the compression zone does not actually
reflect the material rigidity of the beam at the time of buckling. However, the rigidity and
critical load calculations considering the variations in the reduced modulus of elasticity in
the section and in the span of the beam are not practical and quite time-consuming. Thus,
Sant and Bletzacker (1961) proposed the use of the smallest reduced modulus of
elasticity corresponding to the most-strained compression fibers in the beam, which are
the extreme compression fibers of the midspan section, in the case of midspan loading.
As indicated in Figure 1.20, the smallest reduced modulus of elasticity corresponds to the
43
outermost side of the compression zone of the midspan section because the slope of the
loading line (the tangent modulus of elasticity) reduces as the strain in the fibers
increases.
The double modulus theory makes use of the assumption that strain reversals take
place in the convex part of the compression zone of the beam. The strain measurements
taken by Sant and Bletzacker (1961) validated the presence of the strain reversals, i.e. the
formation of the additional tensile strains, in the convex side of the compression zone
after buckling.
In contrast to the previous researchers, Massey (1967) included the contribution
of the longitudinal reinforcement to the lateral-bending and torsional rigidities and the
contribution of the shear reinforcement to the torsional rigidity of a reinforced concrete
beam and proposed the Equations (1.18) and (1.19):
b3  c
B  Esec   Es  I sy (1.18)
12
1   b12  d1  At  Es
GC  Gc'    b3  h    Gs  Gc'   bs3  ts  (1.19)
3 2 2 s
where h is the height of the section; ΣIsy is the moment of inertia of the longitudinal steel
about the minor axis of the section; bs and ts are the width and the thickness of the
longitudinal reinforcement layer, respectively, as illustrated in Figure 1.21; γ is a constant
defined by Cowan (1953); b1 and d1 are the breadth and the depth of the cross-sectional
area enclosed by a closed stirrup, respectively (Figure 1.21); s is the spacing of the
stirrups; Ao is the cross-sectional area of one leg of the stirrup; β is the coefficient for St.
Venant’s torsional constant; Es and Gs are the modulus of elasticity and the modulus of
44
Figure 1.21 – Definition of the variables in the expressions proposed by Massey (1967)
rigidity of steel, respectively; G’c is the reduced modulus of rigidity of concrete,
calculated according to Equation (1.20):
Esec
Gc'  Gc  (1.20)
Ec
where Ec and Gc are the modulus of elasticity and the modulus of rigidity of concrete,
respectively.
Massey (1967) modified the lateral-bending rigidity expression developed by
Hansell and Winter (1959) by adding the second term, Es.ΣIsy, corresponding to the out-
45
of-plane bending resistance provided by the longitudinal reinforcement. According to
Equation (1.18), the contribution of the longitudinal reinforcement is only of concern if
the lateral-torsional buckling takes place prior to the yielding of the flexural
reinforcement. When the steel yields, its modulus of elasticity becomes zero and the
second term vanishes. After yielding of the flexural reinforcement, only the uncracked
concrete above the neutral axis provides the out-of-plane flexural resistance.
According to Massey (1967), the torsional rigidity of a reinforced concrete beam
is calculated by the summation of the three different rigidity terms given in Equation
(1.19). The first two terms correspond to the contributions of the concrete and the flexural
reinforcement, respectively. The rectangular concrete section and the thin-walled
rectangular reinforcement layer (the gray area in Figure 1.21), consisting of the
longitudinal reinforcing bars, are considered as the two main components of the non-
homogeneous concrete beam. The last term, on the other hand, is the contribution of the
shear reinforcement and is taken into account only if the stirrups are closed.
The inelastic behavior of the concrete is taken into consideration by the use of the
secant modulus of elasticity and the reduced modulus of rigidity in the lateral-flexural
and torsional rigidity expressions, respectively. Similar to Equation (1.18), yielding of the
longitudinal reinforcement nullifies its contribution to the torsional rigidity of the beam.
Stiglat (1991) investigated the agreement of the critical moment predictions
based on an approximate method proposed by Stiglat (1971) with the experimental results
obtained by Konig and Pauli (1990). The approximate method suggests that the critical
moment calculated for an elastic and uncracked concrete beam should be modified using
the stresses at the extreme compression fibers at the onset of buckling to account for the
46
inelastic material properties. To begin with, an initial critical moment value is calculated,
neglecting the inelastic material properties of concrete. The following initial critical
moment expressions were presented for a simply-supported beam with three different
loading conditions: uniformly-distributed load along the span (Equation 1.21), a single
concentrated load at mid-span (Equation 1.22) and equal concentrated loads at one-third
points of the span (Equation 1.23).
3.54  e 2.5  I y  Ix  I y
M cri    1  1.44     E c Gc  I y  J  (1.21)
L  L J 
 Ix
4.23  e 2.5  I y  Ix  I y
M cri   1  1.74     E c Gc  I y  J  (1.22)
L  L J 
 Ix
3.25  e 2.5  I y  Ix  I y
M cri   1  1.44     E c Gc  I y  J  (1.23)
L  L J 
 Ix
where Mcri is the initial (uncorrected) critical moment; Ix and Iy are the moments of
inertia about the major and minor axes, respectively; J is the torsional constant; L is the
unbraced length of the beam; e is the initial vertical distance of the load from the shear
center of the beam section.
The terms in the parenthesis in each equation correspond to the stabilizing or
destabilizing effect of the vertical location of the load with respect to the shear center.
Figure 1.22 illustrates the deflected and undeflected configurations of a beam with a
concentrated load, applied at the top, at the shear center and at the bottom of the cross-
section. In all three cases, the line of action of the applied load passes through the shear
center prior to torsional rotations. When the beam experiences torsional rotations, the line
47
Figure 1.22 – Effect of the vertical location of the applied load with respect to the shear
center of beam section
of action of the load continues to pass through the shear center in case (b). In cases (a)
and (c), on the other hand, torsional rotations in the beam render the applied load laterally
eccentric with respect to the shear center. A load acting above the shear center creates
torsional moments, in the same direction as the existing torsional rotations due to
instability. Therefore, the applied load increases the rotations in the beam, having a
destabilizing effect. On the contrary, the torsional moments induced by the load applied
below the shear center oppose the torsional rotations due to instability. Consequently, a
load acting below the shear center has a stabilizing effect on the beam.
In Equations (1.21) - (1.23), the term “e” is taken positive, when the load acts
above the shear center. For a positive value of e, the expression in the parenthesis is less
than unity, so the critical moment is reduced due to the destabilizing effect of the load.
48
When the load acts below the shear center, on the other hand, e is negative, increasing
the buckling moment in account for the stabilizing influence of the applied load in the
deflected configuration of the beam.
Stiglat (1991) recommended using a value of 60 % of the uncracked torsional
constant of the beam section in Equations (1.21) - (1.23). Using reduced values for the
torsional constant takes into consideration the zones in the beam which are already
cracked at the onset of buckling.
Next, the critical moment calculated based on the elastic material properties
should be corrected for the material nonlinearities of concrete. The correction is done
using the comparative slenderness parameters. First, the stress at the extreme
compression surface of the most-stressed beam section (for instance, the midspan section
in the case of a concentrated load at midspan) is calculated according to Equation (1.24):
eo
 cri  M cri  (1.24)
Ix
where σcri is the stress corresponding to the extreme compression fibers of the most-
stressed section along the span; eo is the vertical distance of the outermost compression
fibers from the centroid of the section. The comparative slenderness parameter is
obtained using σcri in Equation (1.25):
Ec
v    (1.25)
 cri
where λv is the comparative slenderness parameter. The slenderness parameter, defined by
Stiglat (1991), is a function of the modulus of elasticity of concrete and the maximum
49
compressive stress in the beam. The initial critical moment, calculated with the modulus
of elasticity and modulus of rigidity corresponding to the initial portion of the stress-
strain curve of concrete below the proportional limit stress, is corrected using λv to
account for the reduced modulus of elasticity, valid for the fibers stressed beyond the
elastic limit of concrete. Later, an equivalent stress value, σT, is obtained from the
comparative slenderness parameter using tables presented by Stiglat (1991). Finally, the
critical moment of the beam is calculated by correcting the initial critical moment
according to Equation (1.26):
T
M cr  M cri  (1.26)
 cri
Stiglat (1991) reported that the analytical critical moment values according to the
proposed approximate method were in close agreement with the experimental results
obtained by Konig and Pauli (1990). Moreover, Stiglat (1991) stated that the use of the
reduced torsional constant in Equations (1.21) - (1.23) resulted in conservative critical
moment predictions since all analytical critical moment values were smaller than the
experimental buckling moments.
Revathi and Mennon (2006) modified the effective moment of inertia expression
in ACI 318-05 (2005) Section 9.5.2.3 for the case of out-of-plane bending. The original
form of the expression is given in Equation (1.27):
 M cra 
3
  M 3 
Ie     I g  1      I cr  I g
cra
(1.27)
 Ma    M a  
50
where Ig, Icr and Ie are the uncracked, the cracked and the effective moments of inertia,
respectively; Ma is the maximum moment in the span at the particular applied load level;
Mcra is the cracking moment of the beam.
Equation (1.27) is the weighted average of the uncracked and cracked moments of
inertia of a concrete beam. The uncracked moment of inertia corresponds to the early
stages of loading when the cracking moment of the beam is not exceeded and the entire
beam section contributes to the in-plane bending resistance. When the cracking moment
is exceeded, flexural cracks form in the outermost layers of the tension zone of a beam.
Later, the flexural cracks propagate in the tension zone towards the compression zone and
the moment of inertia of the beam decreases as the applied load increases. When the
flexural cracks render the entire tension zone ineffective, the moment of inertia reaches a
minimum limit, called the cracked moment of inertia. Equation (1.27), which is the
moment of inertia of a concrete beam when the maximum moment in the span is Ma,
reflects the variation in the moment of inertia of a concrete beam from the uncracked
state to the fully cracked state as the flexural cracks propagate in the tension zone.
The out-of-plane bending rigidity expression proposed by Revathi and Mennon
(2006) is given in Equation (1.28):
3
 M cra   b3  h  
    
 0.8  M ult   12  
B  Ec    (1.28)
   M cra    b3  cu
3
 Es  
 1   0.8  M     12     E  I sy   
   ult     c  
where Mult is the ultimate flexural moment of the beam; cu is the neutral axis depth of the
beam at ultimate flexural load; ΣIsy is the moment of inertia of the longitudinal
51
reinforcement about the minor axis; ψ is a multiplier, which is taken 0 for under-
reinforced beams and 1 for over-reinforced beams. The first term in the
parenthesis, b3 h 12 , is the uncracked moment of inertia of a reinforced concrete beam
about the minor axis. The term b3cu 12      Es / Ec  I sy  is the moment of inertia of a
concrete section at the ultimate flexural load. Accordingly, Revathi and Mennon (2006)
proposed a lateral bending rigidity at the time of buckling, which is a weighted average of
the uncracked moment of inertia and the moment of inertia of the beam at ultimate load.
Equation 1.27 uses the maximum moment in the span at a particular load to
average the uncracked and cracked moments of inertia. In the case of lateral bending, the
maximum moment at the time of buckling is the buckling moment of the beam. Hence,
the buckling moment should be known to calculate the out-of-plane flexural rigidity of
the beam at the buckling moment. To avoid an iterative procedure, Revathi and Mennon
(2006) proposed the use of the lateral bending rigidity corresponding to 80% of the
ultimate flexural moment. Although the use of 0.8.Mult is a close approximation in the
case of inelastic lateral-torsional buckling, it can underestimate the lateral bending
rigidity in the case of elastic lateral torsional buckling. When the buckling moment is
much smaller than 0.8.Mult as in the case of elastic lateral torsional buckling, there are less
flexural cracks in the beam and the rigidity of the beam at the instant of buckling is
significantly greater than the value calculated from Equation (1.28).
Revathi and Mennon (2006) also proposed the use of initial tangent modulus of
elasticity, Ec, in the rigidity calculations. As shown in Figure 1.17, the initial tangent
modulus of elasticity is used when concrete is in the elastic range of stress-strain curve.
Since all compression fibers throughout the beam are stressed below the elastic limit, the
52
use of Ec is appropriate when the beam undergoes elastic lateral-torsional buckling. In the
case of inelastic lateral-torsional buckling, nonetheless, some compression fibers in
highly-stressed portions of the beam are strained beyond the elastic range, where the
modulus of elasticity of concrete is smaller than Ec. Consequently, Ec is not applicable for
all compression fibers in the beam, buckling inelastically, contrary to the assumption in
Equation (1.28).
Finally, Revathi and Mennon (2006) proposed the following torsional rigidity
expression:
4   '  Es  A22  Ac
C (1.29)
 1 1 
p22    
 l t 
where Ac is the area of the gross cross-section of the beam; A2 and p2 are the area and the
perimeter of the rectangle connecting the centers of the corner longitudinal bars (Figure
1.23); μ’ is a rigidity multiplier taken as 1.2 for under-reinforced and 0.8 for over-
reinforced sections; ρl and ρt are the volumetric ratios of the longitudinal and transverse
reinforcement, respectively, calculated from Equations (1.30) and (1.31):
As
l  (1.30)
Ac
At  p1
t  (1.31)
Ac  s
where As is the area of the longitudinal reinforcement in the cross-section; At is the cross-
sectional area of one leg of a stirrup; p1 is the perimeter of the centerline of a stirrup
(Figure 1.23); s is the spacing of the stirrups.
53
Figure 1.23 – Definition of area and perimeters in Equations (1.29) - (1.31)
The torsional rigidity of a reinforced concrete beam before torsional cracking is
given by Equation (1.32) according to St. Venant’s theory.
C  Gc    b3  h (1.32)
where β is the coefficient for St. Venant’s torsional constant, obtained from Equation
(1.33):
192 b  1 (2n  1) h

  1    tanh (1.33)
 5
h n 0 (2n  1) 5
2b
The torsional rigidity expressions, adopted by Hansell and Winter (1959) (Equation 1.9),
Siev (1960) (Equation 1.10) and Sant and Bletzaker (1961) (Equation 1.16), all
correspond to the uncracked stage of a concrete beam, and they are derived from
54
Equation 1.32. Equation 1.29, on the other hand, is the torsional rigidity of a reinforced
concrete beam in the early post-cracking stage, meaning right after the formation of
diagonal tension cracks due to torsion.
The post-cracking torsional rigidity of a concrete beam is provided by the outer
thin-walled layer of concrete surrounding the corner longitudinal bars and stirrups. A
three-dimensional model called the thin-walled tube, space truss model was used by
Lampert (1973) and Hsu (1973) to develop the post-cracking torsional rigidity of
rectangular reinforced concrete beams. Later, Tavio and Teng (2004) simplified the
rigidity expression developed by Hsu (1973) and proposed a new expression. Equation
(1.29) is a modified version of the torsional rigidity expression proposed by Tavio and
Teng (2004), which is presented in Chapter V (Equation 5.27).
Equation (1.29) is the torsional rigidity of a concrete beam right after cracking.
Using the equation in critical moment calculations suggests that the whole concrete beam
is cracked diagonally at the time of buckling. In the presence of lateral supports at the
beam ends, torsional moments resulting from the out-of-plane deformations increase
from zero at midspan to a maximum value at the ends. Therefore, the middle portion of
the beam remains diagonally uncracked throughout the loading while the end portions,
under significant torsional moments, are cracked to a major degree at the time of
buckling, particularly in the case of a geometrically-imperfect beam, experiencing major
lateral deformations prior to buckling. The uncracked torsional rigidity of the beam
reasonably reflects the torsional resistance of the middle portion of the beam, yet the ends
possess torsional rigidities close to or even smaller than the post-cracking torsional
rigidity. Adopting the post-cracking torsional rigidity for the whole beam in the
55
calculations is overly-conservative while using the uncracked torsional rigidity
overestimates the resistance of a beam, leading to unsafe results. If a single torsional
rigidity expression is desired to be valid for the whole span, it should be an average of the
maximum and minimum values of the torsional rigidity of the beam along the span.
1.4.3 Summary
The analytical methods for predicting lateral torsional buckling loads of reinforced
concrete beams, presented in Section 1.4.2, considered the elastic-inelastic stress-strain
behavior of concrete and steel, the contribution of the longitudinal and shear
reinforcement to the stability and the flexural cracking of concrete. Nevertheless, the
influences of the initial geometric imperfections and the restrained shrinkage cracking of
concrete on the lateral stability of reinforced concrete beams have not been studied yet.
The analytical part of this study aimed at incorporating all the factors, including the
initial geometric imperfections and restrained shrinkage cracking of concrete, into the
formula.
For an exact analysis of the lateral stability of a reinforced concrete beam, the initial
geometric imperfections, the initial cracking condition (presence or absence of shrinkage
cracks), the experimental stress-strain curves of concrete and steel and the cross-sectional
details of the beam should be fully known. In none of the experimental studies presented
in Section 1.4.1, all of the aforementioned properties of the test specimens were reported.
Therefore, reinforced concrete beams, whose geometric and material properties are fully
known, were tested in the present study for a better evaluation of the analytical methods
presented in Section 1.4.2 and the method proposed in the present study.
56
CHAPTER II
SPECIMENS AND MATERIAL PROPERTIES
2.1 Specimens
2.1.1 Specimen Descriptions
In the experimental program, two sets of specimens were tested. The first set of
specimens was composed of six beams of four types, B36, B30, B22 and B18. The
second set of beams consisted of five beams of two different types, B44, B36L. Table 2.1
presents the specimens of the entire experimental program. Figure 2.1 and Figure 2.2
illustrate the nominal cross-sectional details of the specimens.
Each beam is denoted with the letter “B”, followed by two numbers. The first
number corresponds to the depth of the specimen in inches, while the second number is
used for the identification of the specimen. For instance, B44-1 corresponds to the first of
the identical beams having a depth of 44 inches. Additionally, specimen group B36L has
the letter “L” (representing the longer span) to distinguish it from the specimen B36.
Companion beams were identical to each other in dimensions and amount of
flexural and shear reinforcement. Furthermore, concrete from the same batch was used in
companion beams to minimize the influence of the mechanical properties of concrete on
the experimental results. Similarly, reinforcing steel of the companion beams was from
the same batch with the exception of specimen groups B22 and B18. Flexural reinforcing
bars in specimens B22-1 and B18-1 were Grade 60 while the bars in specimens B22-2
and B18-2 were Grade 40 (ASTM A615/A, 2007). Since Beams B22 and B18 buckled
57
Table 2.1 – Specimens tested in the experimental program
Specimen Number of Height, h Width, b Span Length, d/b L/b

Group Samples (in) (in) L (ft) ratio ratio
B36 1 36 2.5 20 12.45 96
B30 1 30 2.5 20 10.20 96
B22 2 22 1.5 12 12.45 96
B18 2 18 1.5 12 10.20 96
B44 3 44 3.0 39 12.45 156
B36L 2 36 3.0 39 10.20 156
before yielding of flexural reinforcement, grade of the reinforcing bars had no influence
on the buckling behavior of the beams.
Shear reinforcement was needed in the specimens to prevent shear failure. Due to
the small widths of the specimens, welded wire reinforcement (WWR) sheets were used
instead of bent reinforcing bars. Two 2x6-W2.5xW3.5 sheets, one on each side of the
flexural reinforcement, constituted the shear reinforcement of each specimen (Figures 2.1
and 2.2).
2.1.2 Experiment Design

The first set of tests was carried out to evaluate the performance of the experimental
setup. Thus, any potential shortcomings in the loading and support systems could be
discovered and corrected before the second set of experiments. Another goal of the first
set of tests was to observe the lateral-torsional buckling behavior of reinforced concrete
beams and to detect the factors affecting the lateral stability. Therefore, the first set of
specimens was designed to be quite slender so that the beams would certainly fail by
lateral-torsional buckling.
58
Figure 2.1 – First set of specimens (B36, B30, B22, and B18)
Beams B30 and B36 were tested in the first stage of the experimental program.
They had similar dimensions and cross-sectional details to the beams tested by Sant and
Bletzacker (1961), whose experimental work formed the basis of the slenderness
limitation specified in Section 10.4 of ACI 318-05 (2005) together with the experimental
study carried out by Hansell and Winter (1959). To understand scale effects, four smaller
beams of two types, B22 and B18 were tested in the first stage.
59
Figure 2.2 – Second set of specimens (B44, B36L)
In the first stage of the experimental program, several observations were made
leading to the design of the second set of specimens. The second set of beams was
constructed at a larger scale than the first set for two reasons. First, tests on B22 and B18
demonstrated that small-scale beams were extremely sensitive to experimental errors,
such as eccentricities in the applied load and deviations from vertical in the orientation of
60
the load. The smaller lateral-flexural and torsional rigidities of small-scale beams cause
the experimental results of such beams to be excessively influenced by the accidental
torsions resulting from the slight eccentricities and deviations of the applied loads. As the
scales of the specimens were increased, the slight tolerance errors became less influential.
Secondly, all specimens were constructed in the same lab environment, using the same
type of materials. Therefore, the initial geometric imperfections of the specimens of
different sizes were of the same order of magnitude. Since large-scale beams were
expected to be less affected than the small-scale beams by imperfections of the same
order of magnitude, testing beams with greater scales was preferred in the second stage of
the experimental program.
All specimens were designed to undergo elastic lateral torsional buckling, so that
the influences of the factors other than inelasticity on the lateral stability of reinforced
concrete beams could be examined. Elastic lateral torsional buckling of reinforced
concrete beams takes place when both concrete and reinforcement in the beams are
strained in the elastic portions of their respective stress-strain curves. In a simply-
supported beam, subjected to a concentrated load at midspan, the extreme compression
fibers at midspan are the most-stressed portion of the concrete beam. If the extreme
compression fiber strain at initiation of buckling is within the initial elastic portion of the
stress-strain curve of concrete, the entire compression zone of the beam behaves
elastically at initiation of buckling. Similarly, the strain in the tension reinforcement at
initiation of buckling does not reach the yield strain of steel in elastic lateral torsional
buckling of reinforced concrete beams.
61
2.2 Concrete Material Properties
The small dimensions and congested reinforcement (Figure 2.3) in the first set of beams
rendered the mechanical vibration of concrete difficult. To overcome the consolidation
problems, Self-Consolidating Concrete (SCC) was used in the first set of specimens. SCC
is a flowable type of concrete which spreads into the form and consolidates under its own
weight (Figure 2.4). The high-range water-reducing (HRWR) admixtures in SCC
decrease the viscosity of concrete and eliminate the need for mechanical vibration. The
spread of SCC was measured as 25 in. according to the slump flow test, described in
ASTM C1611 (2005). The SCC used a 3/8-in maximum size aggregate.
Mechanical properties of concrete and reinforcing steel influence the lateral
buckling behavior of reinforced concrete beams significantly. Concrete from the same
batch and reinforcing bars from the same batch of steel were used in the companion
beams to reduce differences.
For the concrete used in the first set of beams, three 6 in. x 12 in. cylinder samples
were tested on the 7th day, on the 28th day and on each test day to obtain the compressive
strength of concrete (f’c) according to ASTM C39-05 (2005). Furthermore, three more
cylinder tests were conducted on each day to determine the modulus of elasticity (Ec) and
the Poisson’s Ratio (υc) of the concrete according to ASTM C469 (2002). Different from
the first set of beams, cylinder tests were only conducted on the test days in the second
set of beams. Table 2.3 tabulates the means and the standard deviations of the test results
of each test day.
62
Figure 2.3 – Congested reinforcement in B36
Figure 2.4 – Application of self-consolidating concrete
63
Table 2.2 – Mechanical properties of concrete
Age f’c (psi) Ec (ksi) υc

Test at Test
Day Sample μ1 Sample μ Sample μ
(days) Size Size Size
B22-1 119 3 11730 3 5200 3 0.16
B22-2 129 3 11000 3 4850 3 0.17
B18-1 145 3 11460 3 5000 2 0.13
B18-2 160 3 11320 3 5000 3 0.16
B30 220 3 12220 3 5950 3 0.20
B36 249 3 12780 3 5850 3 0.17
B44-1 179 3 8470 3 4450 3 0.16
B44-2 225 3 8540 3 4450 3 0.15
B44-3 234 3 8560 3 4550 3 0.14
B36L-1 192 3 7900 3 4300 3 0.15
B36L-2 201 3 7940 3 4500 3 0.15
1
Sample Mean
In critical and ultimate bending moment calculations, the stress-strain curves of
concrete used in the specimens were needed. Therefore, compression tests were
conducted on 6 in. x 12 in. concrete cylinders to determine the experimental stress-strain
curves of concrete. Figure 2.5 to Figure 2.7 illustrate the experimental stress-strain curves
of concrete, obtained from cylinder tests. Critical moment and ultimate flexural moment
calculations of the specimens are simplified if the stress-strain curve of concrete is
expressed in a mathematical form. For this purpose, several analytical models for the
stress-strain curve of high-strength concrete were examined.
Analytical stress-strain curves from the models proposed by Carreira and Chu
(1985), Tomaszewicz (1984) and Wee et al. (1996) were included in the plots to
determine the model giving the best agreement with the experimental stress-strain curves.
64
Carreira and Chu (1985) proposed Equation (2.1) for the stress-strain relationship
high-strength concrete.
   
    

f c  f c'    o  
(2.1)
 
  1     
   
 o 
where ε and fc are the concrete strain and stress, respectively; εo is the strain at peak stress
and f’c is the compressive strength of concrete according to the cylinder tests; β is a
material parameter, given by
1
 (2.2)
1  f  o  Ec
c
'
The model proposed by Tomaszewicz (1984) adopts Equation (2.1) for the
ascending branch of the stress-strain curve. For the descending branch of the curve, on
the other hand, Equation (2.3) was developed with the introduction of a new parameter, k
to Equation (2.1).
   
    

f c  f c'    o  
(2.3)
k  
  1     
   
 o 
where k = f’c/2.90 with f’c given in ksi.
Similar to the formulation given by Tomaszewicz (1984), Wee et al. (1996)
recommended the use of Equation (2.1) for the ascending branch and a modified form of
Equation (2.1), given below, for the descending branch.
65
   
 k1      

f c  f c'    o  
(2.4)
k2  
 k   1    
 1   
 o 
where k1 = (7.26/f’c)3.0 and k2 = (7.26/f’c)1.3 with f’c given in ksi.
All three models adopt the same equation (2.1) for the ascending branch of the
stress-strain curve. Figure 2.5 to Figure 2.7 indicate that Equation (2.1) closely estimates
the ascending portions of the experimental curves. Since the stresses in all specimens
were within the initial portions of the stress-strain curves of concrete, only the ascending
portions of the curves were determined in the cylinder tests.
66
Figure 2.5 –Stress-strain curves of concrete in the first set of beams
Figure 2.6 –Stress-strain curves of concrete in B44
67
Figure 2.7 –Stress-strain curves of concrete in B36L
68
CHAPTER III
EXPERIMENTAL SET-UP, INSTRUMENTATION AND

PROCEDURE
3.1 Experimental Set-up
3.1.1 Loading Mechanism
The loading frame used for testing the beams of the present study consisted of a loading
mechanism, called the gravity load simulator, a tension jack mounted to the center pin of
the simulator, a loading cage and a ball-and-socket joint conveying the load from the
cage to the beam (Figures 3.1 and 3.2).
Gravity load simulator was first developed by Yarimci et al. (1967) and used in
sway-permitted testing of large scale frames, later in lateral torsional buckling tests of
steel I-beams by Yura and Phillips (1992) and lateral stability of polymer composite I-
shaped members by Stoddard (1997). In the present experimental program, the gravity
load simulator designed and applied by Stoddard (1997) was used.
The gravity load simulator is composed of two inclined arms and a rigid
triangular frame connected to the arms through pins. The pin connections at both ends of
the arms cause the mechanism to be unstable. The instantaneous center of the mechanism
at any configuration of the simulator is the intersection point of the extensions of the
inclined arms (Figure 3.3). The center pin (bottom pin) of the triangular frame moves in
an approximately horizontal line for certain limits of mechanism motion. Since the center
pin is directly below the instantaneous center at any configuration of the simulator, the
line of action of the load applied by a loading device connected to the center pin has a
vertical orientation passing through the instantaneous center. The applied load has
69
Figure 3.1 – Undeflected and deflected configurations of the loading frame and loading
cage
insignificant deviations from the vertical orientation in a certain range of lateral
displacement of the center pin. In the present study, a hydraulic cylinder mounted to the
center pin of the triangular frame loaded the beams vertically throughout the entire test
and did not restrain the out-of-plane translation of the loading point owing to the lateral
motion of the center pin of the simulator with the beam (Figure 3.4).
The rotational freedom of the loading point was achieved with the help of the
ball-and-socket joint (Figure 3.5), which was composed of two steel plates and a steel
70
Figure 3.2 – Loading frame
ball between them, positioned in a socket. Bottom plate of the joint, which was epoxied
to the top surface of the beam, rotated with the beam allowing the loading cage (Figure
3.6) and the hydraulic cylinder to preserve their vertical orientation. Consequently, the
applied load, transferred from the loading cage to the beam by the steel ball, continued to
have a vertical orientation even after the rotations of the beam. The socket was lightly
oiled prior to each test to diminish the friction between ball and plates.
71
Figure 3.3 – Undeflected and deflected shapes of the gravity load simulator
(a) (b)
Figure 3.4 – (a) Undeflected and (b) Deflected configurations of the gravity load
simulator
72
Figure 3.5 – Ball-and-socket joint
73
(a) (b)
Figure 3.6 –The vertical orientation of the loading cage (a) before the test; (b) after buckling
74
3.1.2 Supports
In the design of the experimental setup, the in-plane and out-of-plane supports were
selected and designed to obtain simple support conditions about the major and minor axes
of the beam. The end supports allowed rotations about the major and minor axes (Figure
3.7) while restraining rotation about the longitudinal axis of the beam (Figure 3.8).
Furthermore, the end supports restrained the in-plane (vertical) and out-of-plane (lateral)
translations (Figure 3.8), yet allowed longitudinal translation and warping deformations.
Figure 3.7 – Minor-axis rotations at the supports
75
Figure 3.8 – Lateral deformations and torsional rotations restrained at the supports
Longitudinal deformations of the beams played an important role in the design of
in-plane and out-of-plane supports. Restraining the displacements of the support sections
in longitudinal direction changes the behavior of a beam completely. When the
longitudinal displacements of a beam are prevented at the support locations, the in-plane
support conditions deviate from simple support conditions.
When analyzing beams subjected to in-plane loading, through one-dimensional
models, the supports are located at the centroids of the support sections [Figure 3.9(a)]. In
76
* C – Centroid of the Mid-pan Section of the Undeflected Beam
C’ – Centroid of the Mid-pan Section of the Deflected Beam
Figure 3.9 – In-plane support conditions: (a) in analysis models;

(b) Hinged-hinged; (c) Roller-hinged; (d) Roller-roller.
the following discussion, the centerline of the beam is assumed to be coincident with the
neutral axis. According to this assumption, the centerline continues to be unstrained as
the beam deforms in plane. In a simply-supported beam, the support sections rotate in
plane about the line of contact of the support with the beam. In a beam with support
77
conditions as in Figure 3.9(a), the end sections rotate about the horizontal centroidal axis
(major axis) of the section. Therefore, the portion above the centroidal axis displaces
longitudinally towards the mid-span of the beam while the portion below the centroidal
axis displaces outwards. In other words, the top portion of the beam shortens while the
bottom portion elongates due to the in-plane bending rotations. The centroids of the end
sections remain at their initial positions maintaining the longitudinal distance between
them. Furthermore, the centerline and the initial midspan of the beam do not undergo
translations in longitudinal direction. The in-plane flexural deformations of the beam take
place symmetrically about the mid-span of the undeflected configuration of the beam.
The in-plane support conditions are somewhat different in an experiment. The
supports are located underneath the beam. Using rollers or hinges or a combination of
them at the support locations influences the in-plane behavior and deformations of a
beam when the supports are beneath the beam. Figures 3.9(b)-(d) illustrate the in-plane
deformations of a beam with hinges at both ends, a hinge at one end and a roller at the
other end and rollers at both ends, respectively.
A beam exhibits completely different in-plane flexural behaviors under the three
different boundary conditions shown in Figures 3.9(b)-(d). The three cases are
investigated to find the boundary conditions under which a beam has an in-plane flexural
behavior in closest agreement with the case considered in the one-dimensional analysis
[Figure 3.9(a)].
In Figure 3.9(b), both ends of the beam are supported with hinges. Since the
hinges at the ends restrain the longitudinal displacement, the bottommost portions of the
end sections remain at their original positions as the beam bends in plane. The end
78
sections undergo major-axis rotations about the supports and the centroids of the end
sections displace towards mid-span. The longitudinal displacements of the centroids of
the end sections constitute a clear distinction from the in-plane flexural deformations of
the beam supported as in Figure 3.9(a). Hence, providing fixtures simulating hinges at
both ends was not adopted in the experiments.
In Figure 3.9(c), the beam is hinge-supported at one end and roller-supported at
the other end. The roller support translates in longitudinal direction while the hinge
remains in its original position when the beam bends in plane. The roller-supported end
of the beam experiences longitudinal translations as well as flexural rotations; yet the
hinged end of the beam only rotates in-plane about the support. As the beam ends
undergo bending rotations, the roller translates in longitudinal direction to assure that the
initial distance between the centroids of the end sections is preserved. The in-plane
flexural behavior of a beam with a hinge at one end and a roller at the other end matches
with the behavior in Figure 3.9 (a) in the preservation of the initial longitudinal distance
between the centroids of the end sections. Nevertheless, there are differences between the
in-plane deformations of a beam supported as in Figure 3.9(c) and a beam supported as in
Figure 3.9(a). The midspan section of the beam in Figure 3.9(c) translates in the
longitudinal direction towards the roller-supported end as the beam flexes. Hence, the
location of the midspan load also shifts longitudinally with the beam, constituting a
significant difference from the behavior of the beam supported at the centroids of the end
sections.
In Figure 3.9(d), both ends of the beam are roller-supported. The beam is
statically unstable since there is no restraining force at the support locations preventing
79
the beam from undergoing rigid-body translation in the longitudinal direction. However,
the lack of the longitudinal restraining force at the beam ends was not significant in the
present study since the longitudinal displacements and in-plane bending deformations of
the specimens were estimated to be small due to the large major-axis bending rigidities
possessed by the slender beams. Furthermore, the loading frame prevented significant
longitudinal translation of the beams. The bending rigidity of the loading frame would
provide a longitudinal force at the load point, restraining the tilting of the loading frame
due to the longitudinal translation of the beam.
Figure 3.9(d) depicts that the roller supports at both ends of the beam translate in
longitudinal direction, allowing the stretched bottom portions of the end sections to
displace outwards as the beam bends in plane. Since the centroids of the end sections do
not displace in longitudinal direction, the initial longitudinal distance between the
centroids of the end sections is maintained as the beam ends rotate in plane. Unlike a
beam with a hinge at one end and a roller support at the other end, the centerline of a
beam with roller supports at both ends does not move in longitudinal direction.
Based on the above discussion, the in-plane flexural behavior of a beam with
roller in-plane supports at both ends is closest to the behavior of a beam with the support
conditions as in a one-dimensional analysis. Therefore, each specimen was roller-
supported at both ends to have similar support conditions to a one-dimensional analysis
(Figure 3.10).
Lateral stability of a beam is also influenced by the longitudinal restraint at the
lateral supports. Figure 3.11 illustrates the two lateral support conditions in the aspect of
longitudinal restraint. Lateral support (shown with crosses in the figure) was provided at
80
Figure 3.10 – Roller supports at the beam ends in the second set of experiments
five points along the depth of the beam ends. In Figure 3.11 (a), lateral supports allowed
free translation in longitudinal direction while preventing the beam from deflecting in
lateral direction. On the other hand, lateral supports in Figure 3.11 (b) restrain the
longitudinal displacements as well as the lateral displacements at the beam ends. Hence,
lateral supports are shown as rollers in Figure 3.11 (a) and as hinges in Figure 3.11 (b).
If the lateral supports prevent the ends from rotating in plane by restraining the
longitudinal displacements, the beam ends become fixed rather than simply-supported.
Since the lateral stability of a beam is closely related to its in-plane flexural behavior, the
longitudinal restraining forces at the lateral supports should be minimized to achieve
simple support conditions in and out of plane.
In the present experimental program, out-of-plane supports were designed in a
way that the points in contact with lateral supports were allowed to translate in
longitudinal direction with insignificant levels of restraint. In that way, the beam ends
were provided with rotational freedom about the major axis.
81
Figure 3.11 – Behavior of the beams with: (a) roller supports; (b) hinged supports
in lateral direction
Two different support frames were built in the two stages of experimental
program to achieve the aforementioned lateral support conditions. In the first stage, ball
rollers were employed to support the beams laterally (Figure 3.12). A ball roller [Figure
3.12(b)] is a special type of caster, whose wheel is a steel ball capable of swiveling freely
in a socket. The rotational freedom of the ball allows free motion in any direction. The
82
use of ball rollers in the first set of experiments assured that the points on the beam in
contact with the lateral supports were not restrained from translating in longitudinal
direction. So, the lateral supports provided the support sections of the beams with in-
plane rotational freedom to achieve the simple support conditions. The ball rollers were
mounted to the support frames through threaded studs [Figure 3.12(b)].
Although the ball rollers were observed to prevent the beam ends from rotating
about the longitudinal axis of the beam and from deflecting laterally, the support forces
transferred from the beam to the ball rollers were noticed to cause the threaded rods of
the ball rollers to bend during the tests (Figure 3.13). Therefore, a new lateral support
frame (Figure 3.14) was designed prior to the second set of experiments. Rigid casters
(Figure 3.15) were used instead of the ball casters.
Figure 3.12 – (a) Support frame in the first set of experiments; (b) A ball roller in contact
with the beam
83
Figure 3.13 – Bending of the ball roller
Figure 3.14 – Lateral support frame in the second set of experiments
84
Figure 3.15 – Rigid caster in contact with the specimen
The rigid casters had a wheel rotating about an axle passing through the center of
the wheel. The casters were mounted to the lateral support frames in a way that the wheel
rotations allowed longitudinal displacements of the points of contact of the beams with
the casters (Figure 3.15). Therefore, the in-plane flexural rotations of the end sections
were not restrained to satisfy the simple support conditions about the major axis.
The casters had a mounting plate with four corner holes to bolt the caster to a
frame. Instead of bolting the casters directly to the support frame, the mounting plate of
each caster was connected edge to edge to a steel plate adjacent to the other side of the
frame (Figure 3.15) to allow the casters to move to the desired level along the height of
the frame to accommodate different beam depths. The four ½-in diameter bolts
connecting the casters to the support system provided adequate rigidity to the casters
85
against the bending moments induced by the vertical friction forces between the beams
and the caster wheels.
The support frames in the second set of experiments were mainly composed of
two HSS 3x3x1/4 structural tubes, one on each side of the beam (Figure 3.14). Each of
these tubes was supported by two diagonal knee braces. One of these braces was
extended to the top of the support member (HSS 3x3x1/4) while the other brace was
connected to the tube at one-third of the height of the tube.
In the first test of the second stage (Specimen B44-1), two casters were used on
each side of the beam to support the beam ends laterally [Figure 3.16(a)]. One of the
casters supported the topmost portion of the beam while the other caster was touching the
beam at the two-third of the height. Although two casters had sufficient capacity to
withstand the lateral forces in the tests, problems associated with deformations and
distortions at the beam ends were encountered. Since lateral support was provided at the
top halves of the beam ends only, the bottom portions of the ends displaced in the
opposite direction to the lateral deformations, after buckling (Figure 3.17). The top
portions, on the contrary, remained in their initial positions owing to the adequate lateral
support at the top. Displacement of the bottom parts of the end portions relative to the top
resulted in distortions in the cross-sectional shape of the beam. Figure 3.17 illustrates the
distortion. The distortions in the support regions did not affect the buckling moment and
the deformations in the beam prior to buckling since the bottom parts of the end sections
started moving laterally as a result of the excessive out-of-plane deformations in the post-
buckling stage. Two additional casters on each side, supporting the bottom halves of the
beam ends were used in the following experiments [Figure 3.16(b)]. The beam ends were
86
supported by four casters on each side of the beam to provide lateral translational and
rotational restraint along the depth of the beam.
Figure 3.16 – Support frame in the second set of tests: (a) B44-1; (b) B44-2
Figure 3.17 – Distortion in the cross-section at the beam end
87
3.1.3 Load, Deflection and Strain Measurements
A 50-kip load cell in the first set of experiments and a 100-kip load cell in the
second set of experiments were used. The load cells were placed in line with the jack and
loading cage in order to measure the applied load. String potentiometers were utilized to
determine the in-plane and out-of-plane deflections, the torsional rotations and distortions
at midspan.
In a lateral-torsional buckling test, the in-plane deflections of a beam are
accompanied by out-of-plane deflections. Each point along the span of a slender beam
undergoes lateral displacements as well as vertical displacements. Therefore, the cable of
a potentiometer, having horizontal or vertical orientation at the beginning of the test,
deviates from its initial orientation once the beam starts deforming out of plane (Figure
3.18). Since a geometrically imperfect beam deflects out of plane and experiences
rotations even at the initial stages of loading, uncoupled lateral and vertical deflections
cannot be measured directly by using horizontally- and vertically-oriented potentiometers
even prior to the buckling of the beam.
In the first set of experiments, the coupled deflection measurements from the
potentiometers were converted into in-plane and out-of-plane deflections and rotations at
the shear center (centroid in rectangular sections), through a modified approach presented
by Zhao (1994) and extended by Stoddard (1997). This approach, presented in Appendix
E, is based on geometric relations linking the deflection measurements from three
potentiometers to deflections and rotation of the centroid.
88
Figure 3.18 – Deviation of the Initial Orientations of the Potentiometers
In the second set of experiments, the distances of the lateral and vertical
potentiometers to the beam were increased to minimize the coupling between the lateral
and vertical deflection readings. This is illustrated in Figure 3.19, which shows that the
angle of the measuring cable from horizontal (vertical in the case of a vertical string
potentiometer) in the twisted configuration of the beam decreases as the distance between
the potentiometer and the beam increases. Accordingly, the difference between the
horizontal component of the measuring cable (L.cosα in Figure 3.19) and the length of
the measuring cable (L) decreases with an increase in the distance of the potentiometer to
the beam. This means that the change in length of the measuring cable, measured by the
89
Figure 3.19 – Coupling between the in-plane and out-of-plane deflection measurements
for a lateral string potentiometer with varying distances from the specimen
potentiometer, becomes closer to the lateral deflection of the cable-attachment point on
the beam, as the distance of the potentiometer increases.
The cross-section of a concrete beam might distort when the beam buckles in a
lateral torsional mode. As shown in Figure 3.20(a), two lateral potentiometers are
adequate to determine the rotated configuration of the midspan section after buckling
when the cross-section of a beam does not distort. Nonetheless, distortions in the cross-
section of a beam cannot be detected by only measuring the out-of-plane deflections at
two different depths along the midspan section. Therefore, lateral deflections were
measured at three or more different points along the depth of each specimen at midspan
[Figure 3.20(b)] to assess the shape of the midspan section throughout the test and to
detect any possible distortion in the cross-section. Three lateral potentiometers in the first
90
Figure 3.20 – Lateral-torsional buckling (a) with; (b) without distortions in the cross-
sectional shape of the beam
stage of the experimental program (Figure 3.21) and five lateral potentiometers in the
second stage of the experimental program (Figure 3.22) were used.
During the first stage of the experimental program, Linear Variable Differential
Transducers (LVDT’s) were used for obtaining the strain distributions through the depth
of the convex and concave faces of each specimen at midspan (Figure 3.23). These
LVDT’s were replaced with electrical resistance strain gauges (Figure 3.24) during the
second stage of the experimental program.
91
Figure 3.21 – Lateral deflection potentiometers in the first set of experiments
Figure 3.22 – Lateral deflection potentiometers in the second set of experiments
92
Figure 3.23 – Strain measurement using LVDT’s in the first set of tests
3.1.3.1 LVDT Strain Measurements
Each LVDT was placed in an aluminum box glued to the side face of the specimen
(Figure 3.23). The extension rod of the LVDT core (armature) was attached to an
aluminum plate bonded to the side face of the specimen. The initial longitudinal distance
between the box and plate was the gage length, over which the strain was measured. As
93
Figure 3.24 – Strain measurement through electrical resistance strain gauges in the
second set of tests
the beam bent in and out of plane, the longitudinal distance between the box and plate
changed, causing the armature to slide inside the LVDT tube. The strain was calculated
from the slide of the armature. Nevertheless, it was found out that the slide of the
armature was not equal to the change in the longitudinal distance between the box and
plate. The out-of-plane bending deformations in the beam caused the extension rod,
connecting the armature to the plate, to bend and lose its initial straightness, which
94
caused the measurement taken by the LVDT to be different from the axial elongation or
shortening of beam at the LVDT location. Therefore, LVDT’s were not used for
measuring the strains in the second set of experiments.
3.1.3.2 Strain Measurements through Electrical Resistance Strain Gauges
In the first test of the second stage of the experimental program (Specimen B44-1),
the longitudinal strains from in-plane and out-of-plane bending and the depthwise strains
from the possible distortions in the cross-section of the beam were measured through
two-element cross strain gauges, attached to the side faces of the beam at mid-span
(Figure 3.25). The strain gage oriented in the depthwise direction was used for detecting
the possible distortions in the cross-section of the beam. Three-element rosettes were not
needed since the longitudinal and depthwise strains were estimated to be the principal
strains due to negligible shear stresses from shear forces and torsional moments at
midspan.
Strain was measured at five points along the depth of the beam (Figure 3.26) to
determine the strain distributions on the convex and concave faces of the beam. Appendix
C presents the longitudinal strain distributions along the convex and concave faces of the
second set of specimens at midspan. The depthwise strains measured at midspan of B44-1
are also given in the appendix.
The strain measurements in the first test (B44-1) indicated that the depthwise
strains did not reach significant levels prior to buckling. Therefore, in the remaining tests
individual gauges, measuring the longitudinal strains only, were used instead of cross
gauges (Figure 3.26).
95
Figure 3.25 - 2-element cross strain gauge on the side face of B44-1
In the test of B44-1, strain measurements in the tension zone were greatly
influenced by flexural cracking. Cracks which formed directly under the gauges caused
the measured strain values to be extremely high. To measure the tensile strains in the
remaining tests, the strain gauges on the tension side of the beam were installed on
aluminum strips, which were attached to the face of the beam by means of concrete drop-
in anchors and bolts (Figure 3.27) to prevent the slip of the strips during the tests. The
strain gauges installed on the aluminum strips measured the average tensile strain
between the two points, where the strip was attached to the side face of the beam.
96
Figure 3.26 – Longitudinal strain gauges along the depth of north face of specimen B44-2
at midspan
Consequently, the tensile strain measurements were not affected from the flexural
cracking in the tension zone. Full-bridge strain gauge circuits (Figure 3.27) composed of
two transverse and two longitudinal gages were installed on some of the aluminum strips
to cancel the accidental bending strains in the strips and to measure the axial strains only.
97
Figure 3.27 –Electrical resistance strain gauges on an aluminum strip for measuring the
longitudinal strain in the tension zone
3.2 Test Procedure
The beams were positioned on their sides during the construction stage. After the
concrete was set, each specimen was tilted into the vertical position and moved to the test
setup through a special lifting method which is explained in Appendix G. The sweep and
initial twisting angles of the specimens were measured prior to the tests (Appendix B).
The beams were loaded to failure. To detect the experimental cracking load of
each specimen and to explore the extension of in-plane flexural cracks, loading was
stopped at every 1-to-2 kip load increment at the initial stages of the test. Once the rate of
increase in the out-of-plane deflections and torsional rotations became large, the beams
were loaded to failure without interruption.
98
3.3 Summary of the Test Results
All specimens of the present experimental program failed in lateral torsional
buckling. Table 3.1 presents the experimental buckling loads of the specimens and the
centroidal lateral and vertical deflections and the torsional rotations at midspan at the
instant of buckling.
In all specimens, the typical crack pattern of lateral torsional buckling, which is
explained in Chapter VII, was observed. The experimental load-lateral (out-of-plane)
deflection, load-vertical (in-plane) deflection and torque-twist curves of the specimens
are presented in Appendix D. The midspan strain distributions of the beams throughout
the test are presented in Appendix C.
The tests indicated that the initial geometric imperfections, sweep (initial lateral
deflection) in particular, significantly influence the lateral stability of a reinforced
concrete beam. Section 6.3 explains the effects of the initial geometric imperfections on
the lateral stability and load-lateral deflection behavior of reinforced concrete beams in
the light of the results of the present experimental program.
99
Table 3.1 – Experimental Results of the Specimens
Buckling Lateral Vertical Angle of

Specimen Load Deflection at Deflection at Twist
(kips) Centroid (in.) Centroid (in.) (degrees)
B18-1 12.4 1.12 0.37 1.17
B18-2 12.0 1.18 0.35 0.45
B22-1 8.7 2.06 0.24 1.66
B22-2 11.0 1.44 0.22 0.90
B30 22.0 1.82 0.48 0.86
B36 39.2 0.39 0.40 0.52
B44-1 15.2 2.81 0.55 0.77
B44-2 12.0 2.12 0.48 0.55
B44-3 21.0 2.58 0.78 0.66
B36L-1 13.5 2.82 0.84 0.70
B36L-2 21.7 1.48 1.37 0.65
100
CHAPTER IV
LATERAL BENDING RIGIDITY OF RECTANGULAR

REINFORCED CONCRETE BEAMS AND INFLUENCE OF
SHRINKAGE CRACKING ON THE RIGIDITY
4.1 Introduction
Rigidity of a beam against bending moments about the minor axis is termed as the lateral
(or out-of-plane) bending rigidity. Lateral bending rigidity is the product of two
variables: (1) the second moment of area about the minor axis of the section (Iy); and (2)
the modulus of elasticity (E), reflecting the overall material resistance of the beam at the
initiation of buckling. Determination of the lateral bending rigidity of a reinforced
concrete beam is not straightforward due to the variation in Iy and E as loading
progresses. The flexural cracks in a beam render some portions of the beam ineffective in
resisting flexural moments. Therefore, sectional response of a concrete beam to lateral
bending (Iy) is not constant throughout the test. Secondly, the stress-strain behavior of
concrete is linear and elastic only up to the elastic limit, assuming that the proportionality
limit of concrete is equal to the elastic limit. If a reinforced concrete beam or some
portions of it is strained beyond the elastic limit of concrete, the material response of the
beam cannot be reflected through E and another modulus of elasticity should be used to
account for the inelastic material behavior of concrete. Accordingly, the lateral bending
rigidity expression proposed for reinforced concrete beams should take into account the
elastic-inelastic material behavior of concrete, the non-homogeneous nature of a
reinforced concrete beam, and the reduction due to cracking in the cross-sectional area of
the beam providing the bending rigidity.
101
Different lateral bending expressions for reinforced concrete beams in the
literature are summarized in Section 1.3. In the following section, only the lateral bending
rigidity expressions that are used in the analysis of the experimental results are explained
in more detail. In Section 4.3, the lateral bending rigidity expression proposed in the
present study is presented. The proposed rigidity expression is developed based on
modeling a reinforced concrete beam with a system of springs. Section 4.3 also discusses
the spring system models used by Bischoff (2007) and Bischoff and Scanlon (2007) in
explaining the differences between the effective moment of inertia expressions proposed
by Branson (1963) and Bischoff (2005). Finally, influence of restrained shrinkage
cracking on the lateral bending rigidity of reinforced concrete beams is explained in
Section 4.4, where the lateral bending rigidity expression, proposed in the present study,
is modified to account for the effect of shrinkage cracks. Furthermore, factors that
promoted the formation of the shrinkage cracks in the first set of beams and the measures
taken to prevent shrinkage cracking in the second set of beams are also discussed in
Section 4.4.
4.2 Available Lateral Bending Rigidity Expressions
In the analytical study, four different lateral flexural rigidity expressions were used in
addition to the rigidity expression proposed in the present study. The first expression is
the lateral flexural rigidity of a homogeneous and elastic beam:
b3  h
Beh  Ec  (4.1)
12
102
where Ec is the elastic modulus of concrete; b and h are the width and height of the beam,
respectively.
Equation (4.1) takes into account the contribution of the entire cross-section of a
beam to lateral bending rigidity and therefore neglects the reduction in the bending
rigidity due to flexural cracking. Furthermore, the use of Ec in the equation reveals that
the entire beam is assumed to be stressed in the elastic range of the stress-strain curve of
concrete, up to buckling, which is true in elastic lateral torsional buckling of reinforced
concrete beams only. In the case of inelastic buckling, Ec does not represent the overall
material rigidity of concrete in a beam at the time of buckling.
Although Equation (4.1) neglects the contribution of the reinforcement to the
bending rigidity, ignores the reduction in the rigidity due to the presence of flexural
cracks and considers the elastic buckling only, the equation was included in the analytical
study to determine the influence of flexural cracking, reinforcement and inelastic material
behavior of concrete on the lateral stability of concrete beams.
Equations (4.2), (4.3) and (4.4) are the lateral bending rigidity expressions
proposed by Hansell and Winter (1959), Sant and Bletzacker (1961) and Massey (1967),
respectively. The expressions were previously presented in Section 1.3 of Chapter I.
b3  c
Bhw  Esec  (4.2)
12
b3  d
Bsb  Er  (4.3)
12
b3  c
Bm  Esec   Es  I sy (4.4)
12
103
where Bhw, Bsb, Bm are the lateral flexural rigidities according to Hansell and Winter
(1959), Sant and Bletzacker (1961) and Massey (1967), respectively; c is the neutral axis
depth of the midspan section of a beam at the initiation of buckling; ; d is the effective
depth of the beam; Es is the elastic modulus of the reinforcing steel; ΣIsy is the moment of
inertia of the longitudinal reinforcing bars about the minor axis of the section; Esec and Er
are the secant and reduced modulus of elasticity of concrete corresponding to the extreme
compression fiber strain at the instant of bifurcation, respectively. Er is calculated from
Equation (4.5):
4  Ec  Etan
Er  (4.5)
 
2
Ec  Etan
where Etan is the tangent modulus of elasticity of concrete corresponding to the extreme
compression fiber strain at the instant of buckling.
Equations (4.2)-(4.4) offer different approaches to account for the possible
inelastic material behavior of concrete at the instant of buckling, by proposing the use of
different types of modulus of elasticity (Esec and Er). Equations (4.2) and (4.4) account
for the destabilizing effect of flexural cracks, by considering the minor axis moment of
inertia of the compression zone only. Finally, Equation (4.4) accounts for the contribution
of the longitudinal reinforcement to the lateral bending rigidity through the use of the
second term on the right hand side of the equation (Es.ΣIsy). Equations (4.1)-(4.4) are
included in the study to compare the results obtained from these equations to the lateral
bending rigidity values obtained from the rigidity equation proposed in the present study.
104
4.3 Proposed Lateral Bending Rigidity Expression
The proposed lateral bending rigidity expression was developed through spring models.
The idea of using springs in modeling the bending behavior of beams originated from the
works of Bischoff and Scanlon (2007) and Bischoff (2007), who used spring models to
justify the effective moment of inertia (Ie) expression developed by Bischoff (2005). For
a better understanding of the spring model used in the present study, the effective
moment of inertia expression proposed by Bischoff (2005) and the spring model
corresponding to this expression is explained in the following discussion.
Prior to the formation of flexural cracks in the tension zone of a beam, the entire
cross-section of the beam contributes to the moment of inertia, which is obtained from
Equation (4.6) by also considering the contribution of the flexural reinforcement:
2
1  h
  b  h3  b  h   y     n  1  As   d  y 
2
I ucr (4.6)
12  2
where As is the cross-sectional area of the flexural reinforcement; n is the modular ratio of
steel to concrete; y is the depth of the center of gravity of the transformed section from
the top surface of the beam. When calculating the uncracked moment of inertia, Iucr, the
flexural reinforcement is transformed into an equivalent concrete area in accordance with
the modular ratio of steel to concrete, n. The gross moment of inertia of a concrete beam
(Ig), on the other hand, is calculated from Equation (4.7):
1
Ig   b  h3 (4.7)
12
105
The contribution of the flexural reinforcement to the moment of inertia can be neglected
and Iucr can be simplified to Ig in reinforced concrete beams with low reinforcement
ratios.
When the bending moment at a cross-section reaches the cracking moment (Mcra),
flexural cracks form in the outermost layers of the tension zone. As the bending moment
increases, the flexural cracks propagate upwards, rendering a greater area in the tension
zone ineffective in resisting bending. Therefore, moment of inertia of the section
decreases as loading progresses and the moment of inertia reaches a minimum limit,
called the cracked moment of inertia (Icr) in serviceability limits. Icr is calculated from
Equation (4.8):
1
 b  c 3  n  As  d  c 
2
I cr  (4.8)
12
where c is the neutral axis depth when all fibers in the compression zone are stressed
below the elastic limit of concrete.
Bending moments exceeding Mcra result in discrete cracks along the length of a
concrete beam. The difference in the moments of inertia of the cracked parts and the
uncracked parts of a beam causes variation in the flexural rigidity along the span.
Concrete between the discrete cracks contributes to resist the tensile stresses in the beam
and increases the overall flexural rigidity. The tensile contribution of the concrete
between the cracks is called tension stiffening. Formation of discrete flexural cracks
along the span and tension stiffening raise a gradual transition of the moment of inertia of
a beam from the uncracked moment of inertia (Iucr) to the cracked moment of inertia (Icr),
as the applied moment (Ma) increases beyond Mcra. The gradual transition in the post-
106
cracking stage was taken into account by Branson (1963), who proposed an effective
moment of inertia expression, which is a weighted average of the moment of inertia of
the gross cross-section (Ig) and the moment of inertia of the fully cracked transformed
cross-section (Icr):
 M cra 
3
  M 3 
I eb     I g  1      I cr
cra
(4.9)
 M   M  
a   a
where Ieb is the effective moment of inertia according to Branson (1963); Ma is the
maximum bending moment along the span; and Mcra is the cracking moment of the beam.
Equation 4.9 is the effective moment of inertia expression recommended in ACI 318-05
(2005) Section 9.5.2 to compute the immediate vertical deflections of reinforced concrete
beams.
Branson’s (1963) expression concerning the effective moment of inertia is an
empirical equation, which is based on test results of simply-supported rectangular
reinforced concrete beams with a reinforcement ratio of 1.65 %. Later, Bischoff (2005)
found that Equation (4.9) overestimates the effective moment of inertia of concrete
beams with low steel reinforcement ratios (ρl<1%) and concrete beams reinforced with
fiber-reinforced polymer bars. Using the tension stiffening strain approach, Bischoff
(2005) was able to develop the following alternative effective moment of inertia
expression:
 M cra  1   M cra   1
2 2
1
    1     (4.10)
I ebi  M a  I g   M a   I cr
107
Equation (4.10) is different from the expression of Branson (1963), which is an
average of the rigidities of the uncracked and cracked portions of a beam. Bischoff’s
(2005) effective moment of inertia was developed through averaging the flexibilities of
the uncracked and cracked parts.
According to Bischoff and Scanlon (2007), the difference between Branson’s
(1963) effective moment of inertia (Ieb) and Bischoff’s (2005) effective moment of inertia
(Iebi) can be explained through spring models. Ieb, which is the weighted average of the
rigidities, can be obtained by modeling the uncracked and cracked parts of a beam
through springs in parallel [Figure 4.1(a)]. Iebi, nevertheless, is obtained by modeling the
uncracked and cracked portions of a beam with springs in series [Figure 4.1(b)].
Figure 4.1 (b) illustrates that springs in series carry the same load (applied
moment, Ma in this case), whereas the parallel connection of springs in Figure 4.1(a)
implies that the load is distributed to the springs in accordance with their rigidities. A
discrete crack in the span and an uncracked portion right adjacent to it are subjected to
approximately the same bending moment and therefore modeling the uncracked and
cracked parts of a beam with springs in series is more appropriate.
Later, the experiments carried out by Gilbert (2006) on simply-supported
rectangular one-way slabs revealed that the sectional resistance of reinforced concrete
flexural members with low reinforcement ratios (ρl<1%) was overestimated by Branson’s
(1963) effective moment of inertia expression, while Bischoff’s (2005) effective moment
of inertia expression produced immediate vertical deflections in close agreement with the
experimental deflections of the specimens.
108
(a)
(b)
Figure 4.1- Spring models defining (a) Branson’s (1963); (b) Bischoff’s (2005) effective
moment of inertia expression
For the present study, Figures (4.2) to (4.6) compare the experimental vertical
deflections of the second set of specimens (B44 and B36L) with the analytical values
calculated using Branson’s (1963) and Bischoff’s (2005) effective moment of inertia
expressions.
According to Bischoff and Scanlon (2007), the analytical deflection estimates
based on both Equations (4.8) and (4.9) are in close agreement with each other when the
steel reinforcement ratio of a concrete beam is above 1%. Specimens B44 and B36L had
109
Figure 4.2 – In-plane deflections of B44-1 at midspan
110
Figure 4.5 – In-plane deflections of B36L-1 at midspan
111
Figure 4.6 – In-plane deflections of B36L-2 at midspan
reinforcement ratios of 2.5% and 2.8%, respectively. Although in Figures 4.5 and 4.6, the
analytical deflection curves corresponding to Ieb and Iebi are only slightly different from
each other, Figures 4.2 to 4.4 reveals that Bischoff’s (2005) effective moment of inertia
expression (Iebi) produces closer estimates to the experimental values.
The experimental curves of B36L-1 and B36L-2 in Figures 4.5 and 4.6 are in
close agreement with the analytical curves corresponding to Ieb and Iebi. Nevertheless, the
experimental curves of B44-1, B44-2 and B44-3 in Figures 4.2 to 4.4 do not show a good
agreement with the analytical curves due to the significant differences between the
experimental cracking moment values of the specimens (Table 4.1) and the cracking
moment values obtained from Equation (4.11), which was used for obtaining the
analytical curves corresponding to Ieb and Iebi:
112
M cra 

I ucr  7.5  f c'  (4.11)
h y
where 7.5  f c' is the modulus of rupture of normal-weight concrete, given in ACI 318-
05 (2005) Section 9.5.2.3.
Table 4.1 – Experimental and calculated cracking moments of the second set of
specimens
Specimen Experimental Mcra Calculated Mcra

(in-kips) (in-kips)
B44-1 470 870
B44-2 530 870
B44-3 480 870
B36L-1 420 610
B36L-2 540 600
Previously, Bischoff and Scanlon (2007) and Bischoff (2007) showed that spring
models well represent the in-plane bending behavior of reinforced concrete beams.
Accordingly, the lateral bending behavior of reinforced concrete beams can also be
represented by a spring system, if the contributions of different portions of a concrete
beam can be reasonably evaluated.
The proposed spring model for the lateral bending behavior of reinforced concrete
beams makes use of the reduced modulus theory [Considère (1891) and Engesser
(1895)]. Here, the reduced modulus theory and its use in the proposed model are
explained in detail with the help of Figure 4.7.
The proposed model is based on a geometrically perfect beam, which does not
experience lateral deformations and torsional rotations prior to bifurcation. Figure 4.7 (c)
113
Figure 4.7 – Moduli of elasticity corresponding to the fibers in the compression zone of a
beam section
is the longitudinal strain distribution along the depth of a cross-section of the beam before
buckling. The longitudinal strains in the pre-buckling stage of the beam solely originate
from in-plane bending moments. The compressive strain varies linearly from zero at the
neutral axis to maximum (εco) at the extreme fibers and the strain at an arbitrary depth y
from the compression face of the beam is denoted as εc. Figure 4.7(d) is the stress-strain
curve of concrete in compression. Since the longitudinal strain is not constant along the
depth of the compression zone, compression fibers at different depths are at different
points on the stress-strain curve. For instance, Point A on the curve corresponds to the
fibers at a depth y while Point B corresponds to the outermost fibers. When bifurcation
takes place, the concave part of the section is subjected to additional compressive strains
from lateral bending while the convex part is subjected to tensile strains, as shown in
114
Figure 4.7(a). The longitudinal strain from out-of-plane bending increases from zero at
the minor axis, which is the vertical centroidal axis in symmetric sections, to maximum
(εtl and εcl) at sides. Tensile strains from lateral bending cause the fibers on the convex
side of the compression zone to be unloaded, while the additional compressive strains
result in further loading of the compression fibers on the concave side of the section.
Figure 4.7(d) illustrates that unloading of the compression fibers takes place along a line
parallel to the initial linear portion of the stress-strain curve of concrete. In other words,
the elastic modulus, Ec is valid for all unloading fibers in the compression zone,
independent of the longitudinal strain (εc) of a fiber prior to buckling. The further loading
of a compression fiber at an arbitrary depth y, on the other hand, takes place along a line
tangent to the stress-strain curve of concrete at Point A. Since the slope of the line
tangent to the curve changes along the stress-strain curve, the tangent modulus of
elasticity corresponding to the loading fibers changes along the depth of the compression
zone of the section.
Hansell and Winter (1959) analytically showed that the secant modulus of
elasticity corresponding to the extreme compression fiber strain (εco) should be used as
the material rigidity term if the entire compression zone of a section continues to be
loaded after buckling. Secant modulus of elasticity (Esec) corresponding to the extreme
compression fiber is the slope of the line connecting Point B on the stress-strain curve to
the origin O as shown in Figure 4.7 (d). The origin of the stress-strain curve corresponds
to the fibers at the neutral axis depth, which have zero longitudinal strain at the initiation
of buckling. Point B, on the other hand, corresponds to the most-stressed fibers of the
compression zone. Therefore, the line connecting Point B to the origin represents the
115
entire compression zone if all compression fibers of the section are further loaded in the
post-buckling stage.
In the present study, the compression zone of a section is divided into a loading
and an unloading portion after buckling, according to the reduced modulus theory. The
secant modulus of elasticity, Esec corresponding to the extreme compression fiber strain at
the instant of bifurcation is used as the modulus of the loading part of the compression
zone.
The spring model proposed in the present study is shown in Figure 4.8. A
reinforced concrete beam is composed of uncracked and cracked parts along the length.
Each of the uncracked and cracked portions of the beam is partitioned into a loading and
an unloading segment when buckling takes place. Applied moment is distributed to the
loading and unloading segments of a portion. Since a cracked portion along the span and
an uncracked portion adjacent to it bear approximately the same lateral bending moment,
the cracked and uncracked parts of the beam are modeled with springs in series. The
loading and unloading segments of a cracked or an uncracked portion of the beam
contribute to the resistance of the lateral bending moments in accordance with their
flexural rigidities about the minor axis of the beam section. Hence, loading and unloading
segments of a portion are modeled with springs in parallel.
In an uncracked section, the entire section contributes to resistance to the minor-
axis bending moments. Therefore, k1 and k2 contain the term h. In a cracked section, on
the other hand, concrete below the neutral axis is assumed not to contribute to the flexural
rigidity of the beam due to the flexural cracks in the tension zone of the section.
116
Figure 4.8 – Proposed spring model for the lateral bending behavior of reinforced
concrete beams
Therefore, the neutral axis depth of the section at the initiation of buckling (c) is used in
k3 and k4.
In the rigidity expressions k1, k2, k3 and k4, b/2 was adopted as the width of each
of the loading and unloading segments of a section. The widths of the loading and
unloading segments are equal if the secant modulus of concrete (Esec) corresponding to
the extreme compression fiber strain at the initiation of buckling is equal to the elastic
modulus of concrete (Ec). Having Esec equal to Ec is possible only if the entire beam is
stressed in the linear elastic range of concrete, which is known as the elastic lateral-
torsional buckling. In the case of inelastic lateral-torsional buckling, nevertheless, Esec
can be much lower than Ec, causing the widths of the loading part and the unloading part
of a section to be different. However, b/2 was used in the equations to simplify the
lateral-flexural rigidity expression.
117
The equivalent rigidity (keq) of the spring system in Figure 4.8 is obtained from
Equation (4.12) by also using the weight factors for the cracked and uncracked parts of a
beam, previously employed by Branson (1963) and Bischoff (2005):
1  M cra 
m
1   M m  1
    1   cra    (4.12)
keq  M cr  k1  k2   M cr   k3  k 4
 
where Mcra and Mcr are the cracking moment and the critical moment of a beam,
respectively. Using the expressions for k1, k2, k3 and k4, given in Figure 4.8
1  M cra 
m
1   M m  1
   3 3
 1   cra    3 3
(4.13)
keq  M cr  h b hb   M cr   E  c  b  E  c  b
Esec   Ec  sec c
24 24 24 24
After simplifications, the lateral flexural rigidity of a reinforced concrete beam, keq is
obtained from Equation (4.14):
b3  c Esec  Ec
keq   m
(4.14)
24 M  c 
1   cra     1 
 M cr   h 
ACI 318-05 (2005) suggests the use of a value of 3 for the power m in Equation
4.9 to obtain an average rigidity for the entire span of a reinforced concrete beam with
discrete cracks along the span. Bischoff (2005), on the other hand, stated that a value of
m=2 in his effective moment of inertia expression (Equation 4.10) correlates well with
Branson’s original equation. In the present study, the spring system (Figure 4.8) models
the cracked and uncracked portions of a concrete beam with springs in series, similar to
118
the spring model used by Bischoff (2005). Therefore, using a value of m=2 was assumed
to be more appropriate.
The lateral bending rigidity in Equation (4.14) can be formulated as the product of
the modulus of elasticity of concrete and the effective moment of inertia of the beam
about the minor axis, leading to Equation (4.15):
 
 3 
  b c  1    Esec  Ec 
keq    (4.15)
 12  M   c   
m 
2


 1   cra    h  1
  M cr    
The expression in the square brackets in Equation (4.15) is the effective moment of
inertia of the beam about the minor axis. The remaining part of the equation, on the other
hand, is the overall modulus of elasticity of concrete (Eo) of the beam, calculated from
Equation (4.16):
 E  Ec 
Eo   sec  (4.16)
 2 
Considering the contribution of the longitudinal reinforcement to the lateral
bending rigidity is meaningful if two criteria are satisfied. First, the longitudinal rebars in
a beam should remain unyielded till the buckling moment to contribute to the lateral
bending resistance. Secondly, longitudinal rebars should be located close to the sides of
the beam to increase the lateral distance from the minor axis, which constitutes the
moment arm of the bars in lateral bending. Previously, the contribution of the
longitudinal rebars to the lateral bending rigidity was taken into account by Massey
(1967) and by Revathi and Mennon (2006). The second term in Equation (4.4), proposed
119
by Massey (1967), and the term ψ.((Es/Ec).ΣIsy in Equation (1.29), proposed by Revathi
and Mennon (2006) correspond to the longitudinal reinforcement. In the spring model
employed in the present study, rigidity contributions of the longitudinal rebars to the
lateral bending rigidities of the uncracked and cracked sections of a beam can be
represented by a spring connected in parallel to the other two springs of each of the
cracked and uncracked parts of the beam. In other words, the number of parallel springs
in each of the cracked and uncracked portions should be increased to three if the
contribution of the longitudinal reinforcement is desired to be included in the rigidity
expression. Accordingly, the lateral bending rigidity expression is modified, giving
Equation (4.17):
keq  
   
 Esec  Ec   h  b3 24  Es  I sy    Esec  Ec   c  b3 24  Es  I sy 
   (4.17)
   
 Esec  Ec   b3 24  c   M cra M cr   1   M cra M cr   h   Es  I sy

m m

where Es is the elastic modulus of the reinforcing steel; ΣIsy is the total moment of inertia
of the longitudinal reinforcing bars about the minor axis of the section. When the
longitudinal reinforcement yields prior to buckling, Es becomes zero and Equation (4.17)
reduces to Equation (4.15). Similarly, the contribution of the longitudinal reinforcement
(EsΣIsy) vanishes and the equation simplifies to Equation (4.15) if the longitudinal
reinforcing bars are located along the minor axis of the beam section (ΣIsy=0). In the
specimens of the present experimental program, for example, the longitudinal rebars were
located along the vertical centroidal axis of the beam, which coincides with the minor
axis in the case of elastic lateral-torsional buckling. Therefore, Equation (4.15) was used
in the critical moment calculations of the beams, buckling elastically.
120
4.4 Influence of Shrinkage Cracking on the Lateral Bending Rigidity
Shrinkage is defined as the volume change in a concrete member due to the loss
of water arising from the difference in relative humidity between concrete and the
surrounding environment. If a concrete member is allowed to shrink freely, it will
experience longitudinal deformations. If the shrinkage deformations of a beam are
restrained, on the other hand, tensile stresses develop in the beam, resulting in cracking of
concrete. Restrained shrinkage cracks may reduce the lateral bending resistance of a
concrete beam considerably.
Influence of shrinkage restraint stresses on the flexural rigidity of concrete beams
was studied by Scanlon and Bischoff (2008), who stated that shrinkage restraint stresses
in a beam reduce the cracking moment. Scanlon and Bischoff (2008) proposed the use of
a reduced effective cracking moment in Equation (4.10) in the presence of restrained
shrinkage cracks in a beam. The reduced cracking moment value proposed by Scanlon
and Bischoff (2008) is equal to 2/3 of Mcr. Scanlon and Bischoff (2008) also stated that
the influence of shrinkage cracks on the flexural rigidity becomes more pronounced when
a beam has a low longitudinal reinforcement ratio (ρl<0.8%).
Considering the influence of the shrinkage cracks on the cracking moment,
Equation (4.14) can be modified, leading to Equation (4.18):
 
 3 
  b c  1    Esec  Ec 
keq     (4.18)
 12   
2
 M cra   c  
  2


 1      1 
  M cr   h  
121
where ω is equal to 1 in the absence of restrained shrinkage cracks in a beam and ω is
equal to 2/3 in the presence of restrained shrinkage cracks.
Shrinkage restraint comes from several sources. For instance, free shrinkage of a
beam can be prevented by the structure (slab, beams) surrounding the beam. Longitudinal
reinforcement in a beam and the formwork in the construction stage also have restraining
effects on the shrinkage deformations of a beam.
In the first set of specimens of the present experimental program, shrinkage
cracking of concrete was observed (Figure 4.9). To overcome the cracking problem in the
second set of beams, the reasons for the formation of the shrinkage cracks were
investigated. According to the investigation, the use of self-compacting concrete (SCC)
instead of the conventionally vibrated ordinary concrete (OC) in the first set of beams
might have enhanced the degree of shrinkage cracking of the beams.
Previously, various researchers investigated the vulnerability of SCC to shrinkage
cracking. Loser and Leemann (2008) stated that shrinkage of a concrete mixture is
primarily related to the volume of the paste in the mixture. Owing to the higher paste
volume and lower aggregate content, SCC has greater total shrinkage and a higher
shrinkage rate, and therefore, an earlier age of cracking than OC with comparable
compressive strength if rapid drying of concrete takes place. Loser and Leemann (2008)
also recommended the use of shrinkage reducing admixtures (SRA) in SCC to reduce
shrinkage and increase the age of cracking of concrete. Turcry et al. (2006) conducted an
experimental study, through which they concluded that an SCC mixture cracks earlier
than the OC mixture with the same compressive strength due to the higher shrinkage rate.
Turcry and Loukili (2006) explained the higher shrinkage rate of SCC with its lower
122
Figure 4.9 – Shrinkage cracking in B30 prior to the test
bleeding capacity than OC as a result of the higher binder content in SCC. Similarly,
Leemann and Hoffmann (2005) found out that SCC has a shrinkage rate 30% higher than
OC with the same compressive strength.
The high shrinkage rate of SCC might have reduced the age of shrinkage cracking
of concrete and caused shrinkage cracks to form before the removal of the first set of
beams from the forms. Surfaces of the beams exposed to air were kept moist using wet
burlaps. However, the lower bleeding capacity and the higher shrinkage rate of SCC
might have resulted in rapid drying of the surface and induced tensile stresses to concrete
before the subsequent rewetting of the burlaps on the beams.
123
Another potential stimulus for the shrinkage cracking of the first set of beams was
the late removal of the beams from their forms. The first set of beams was kept in the
forms for approximately two weeks. Formwork of a concrete beam constitutes a restraint
for the free shrinkage deformations. Although the open surfaces of the beams were
maintained wet till the dismantling of the forms, the forms might have caused restrained
shrinkage cracks to form due to higher shrinkage rate of SCC.
The early cracking age of the first set of beams in the forms was also related to
the specimen geometry. The beams were cast on their sides to facilitate the concrete pour.
Position of the beams in the formwork caused one of the lateral faces of each beam to be
uncovered, providing a large surface for the evaporation of the bleeding water.
Furthermore, the small widths of the specimens facilitated the drying to reach the internal
regions and affect the entire beam rapidly. Weiss and Shah (2001) carried out an
experimental study in which they observed that thinner concrete sections are less resistant
to shrinkage cracking and the age of cracking decreases as the specimen thickness
decreases.
In the second set of specimens, some measures were taken to prevent shrinkage
cracking of concrete. First, conventionally vibrated ordinary concrete (OC) was used
rather than SCC to increase the age of shrinkage cracking of concrete through the lower
shrinkage rate. Secondly, the beams were removed from the forms in less than a week to
eliminate the shrinkage restraint for concrete as early as possible.
Addition of the Eclipse Shrinkage Reducing Admixture (SRA), produced by
Grace Construction Products, was another protective measure against restrained
shrinkage cracking of concrete. Studies done by several researchers indicated the
124
favorable influence of SRA on the reduction of the total shrinkage and the shrinkage rate
of concrete. The experimental study conducted by Shah et al. (1992) indicated that
addition of SRA’s to concrete greatly reduced the free shrinkage deformations and the
widths of the shrinkage cracks in the case of restrained shrinkage. Lura et al. (2007)
experimentally showed that the addition of SRA’s to mortar reduces the evaporation of
water from the surface of the mortar and causes smaller tensile stresses to develop at the
surface. Therefore, mortar mixtures with SRA have fewer and narrower shrinkage cracks
than the mixtures without SRA under the same environmental conditions.
Efficiency of the measures taken to avoid shrinkage cracking of concrete was
examined through some methods. First, shrinkage cracks could not be detected in any of
the beams constructed in the second phase of experimental program. Nevertheless, the
presence of micro-cracks in concrete cannot be observed through visual inspection.
Hence, two more methods were used to measure the shrinkage strains in the beams to
investigate shrinkage cracking of concrete at the micro level. First, prismatic specimens
with and without SRA were prepared from the concrete mixtures used in the beams.
Sampling of concrete was done according to ASTM C192 (2007). Length changes of
specimens were measured according to the test method described in ASTM C157 (2006).
Six specimens were prepared from each of the concrete mixtures used in Beams B44 and
B36L. The SRA contents and curing conditions of the specimens are tabulated in Table
4.2.
In Figure 4.10, the percent length changes of Specimens 3, 4 and 5 are compared
to illustrate the influence of SRA on the volume change of concrete. According to the
plot, the length change of the specimen without SRA (Specimen 5) was measured to be
125
Table 4.2 – Descriptions of the shrinkage specimens from the concrete mixtures used in
B44 and B36L
Addition Curing
Specimen
of SRA Conditions
Same Conditions
1 Yes
as the Beams
Same Conditions
2 Yes
as the Beams
In the moist room
3 Yes
for 28 days
In the moist room
4 Yes
for 28 days
In the moist room
5 No
for 28 days
In the moist room
6 No
for 28 days
Figure 4.10 – Length changes of specimens with and without SRA from the concrete
mixture of B44
126
close to the length changes of the specimens with SRA (Specimens 3, 4) in the first 120
days after the concrete pour. Later, Specimen 5 experienced greater length changes than
the other two specimens.
Secondly, strains on the lateral faces of the specimens were continuously
measured through DEMEC (Demountable Mechanical) gages to determine the restrained
shrinkage stresses in the beams for monitoring for the formation of shrinkage cracks in
concrete. Directions of the principal stresses originating from restrained shrinkage are not
known. Hence, three independent strain measurements in different directions are needed
to determine the principal strains and stresses at a certain point. To measure the stresses
at the surfaces of the specimens, delta strain rosettes were formed at two different
locations on the side face of each specimen in specimen group B36L.
A DEMEC gage is a mechanical device which measures the distance between two
points. The gage has two conical points, one at each end of an invar bar. One of the
conical points is fixed and the other conical point can move in a certain range. To
measure the distance between two fixed points on a surface, the conical points of the gage
are inserted into the holes drilled at the fixed point. The initial distance between the two
fixed points on the beam is the gage length over which the strain is measured.
In the present study, four screw anchors were embedded into the fresh concrete at
each strain measurement location according to the pattern shown in Figure 4.11. Four
screws positioned in this pattern form a delta strain rosette. Strain in each direction is
obtained by dividing the change in the distance between two points to the initial distance
between the points, measured on the concrete pour day.
127
Figure 4.11 – Delta rosette for principal strain measurement at a point
Variation of the two principal strains in time is illustrated in Figures 4.12 and 4.13
for Specimens B36L-2 and B36L-3, respectively. Cracking strain of concrete (εcr) in
uniaxial tension is also shown in each plot. εcr is calculated from Equation (4.19):
ft
 cr  (4.19)
Ec
where ft is the splitting tensile strength of concrete, which is obtained from Equation
(4.20):
ft  6.4  f c' (4.20)
128
Figure 4.12 – Principal strains on the side face of B36L-2
Figure 4.13 – Principal strains on the side face of B36L-3
129
Equation (4.20) is the tensile strength of concrete in uniaxial tension according to
Mirza et al. (1979). Although the tensile strength of concrete reduces in the presence of a
compressive stress in the perpendicular direction the reduction is ignorable considering
the small values of the compressive principal strains in Figures 4.12 and 4.13 on the day
when the tensile principal strains reached the peak values.
The plots indicate that tensile strains developed at the surfaces of the beams till
the removal of the beams from the formwork, which constituted a restraint for the free
shrinkage deformations. After the removal of the beams from the forms, on the other
hand, the beams were subjected to compressive strains originating from the free
shrinkage deformations of concrete. The tensile principal strains in the beams prior to the
dismantling of the forms only slightly exceeded the cracking strain of concrete for a short
period of time. Therefore, the potential shrinkage cracks in the beams are expected to be
narrow and small in number.
Since the specimens of the present study did not experience significant restrained
shrinkage cracking according to the aforementioned measurements, the multiplier ω in
Equation (4.18) was taken 1 in the evaluation of the lateral bending rigidities of the
specimens. ω can be taken 2/3 as a conservative assumption when the restrained cracking
condition of concrete in a beam is not known.
130
CHAPTER V
TORSIONAL RIGIDITY OF RECTANGULAR REINFORCED

CONCRETE BEAMS
Resistance of a beam to lateral torsional buckling is determined by the lateral bending
rigidity and the torsional rigidity of the beam. The present chapter briefly introduces the
torsional behavior of reinforced concrete beams and explains the evaluation of the
torsional rigidity of a concrete beam in the light of the experimental torque-twist curves
of the test specimens.
5.1 Torsional Behavior of Reinforced Concrete Beams
The torsional behavior of reinforced concrete beams is explained with the help of Figure
5.1, which is the typical torque-twist curve of a reinforced concrete beam with shear
reinforcement. The torque-twist curve in the figure can be divided into three distinct
segments: OA, AB and BC. The initial linear segment (OA) ends at point A, which
corresponds to the initiation of the diagonal cracking in the beam. The slope of the line
OA is termed as the uncracked torsional rigidity of the beam, (GC)u. Prior to the
formation of the diagonal tension cracks, the torsional rigidity is related to the shear
strains around the perimeter of the cross-section of a beam. The entire beam behaves as a
solid and homogeneous body and the contribution of the flexural and shear reinforcement
to the torsional rigidity can be neglected. The uncracked torsional rigidity expressions
existing in the literature are presented in Section 5.2.1.
131
Figure 5.1 – Typical torque-twist curve of a reinforced concrete beam with shear
reinforcement
The second segment (AB) starts when the applied torque reaches the cracking
torque, Tcr. According to ACI 318R-05 Section 11.6.1, Tcr can be determined from
Equation (5.1):
 Acp
2 
Tcr  4  fc'   (5.1)
 pcp 
 
where f’c is the cylinder compressive strength of concrete in psi; Acp is the gross cross-
sectional area of the beam and pcp is the perimeter of the cross-section.
132
Cracking torque of a reinforced concrete beam is the torsional strength of the
plain concrete beam with the same dimensions. Equation (5.1) was developed based on
the assumption that a plain concrete beam fails in torsion when the principal tensile stress
in the beam becomes equal to the tensile strength of concrete (f’t), which can be obtained
from Equation (5.2):
ft '  4  f c' (5.2)
Equation (5.2) is the tensile strength of concrete under biaxial tension and compression. It
was used instead of Equation (4.20), which is the tensile strength of concrete under
uniaxial tension, to account for the compressive and tensile principal stresses in a beam
under pure torsion.
Hsu (1984) developed a criterion for the torsional failure of plain concrete
members, which is based on the skew-bending theory, developed by Hsu (1968) to
explain the torsional behavior of concrete beams. According to the skew-bending theory,
the failure plane of a concrete beam, loaded in pure torsion, makes a 45-degree angle
with the longitudinal axis of the beam. The applied torque can be decomposed into two
components: a component parallel to the failure surface (bending component, Tb in
Figure 5.2) and a component perpendicular to the surface (twisting component, Tt in
Figure 5.2). According to Hsu (1984), torsional failure of a plain concrete beam takes
place when the tensile stress on the lateral face of the beam (σt in Figure 5.2) induced by
the bending component of the applied torque reaches the modulus of rupture of concrete.
Accordingly, the torsional strength of a plain concrete member is obtained from Equation
(5.3).
133
Figure 5.2 – Components of the axial torque on the failure surface of a concrete beam
according to the skew-bending theory
b2  h
Tnp    0.85  f r  (5.3)
3
where fr is the modulus of rupture of concrete, expressed in terms of f’c according to
Equation (5.4):
 10 
f r  21  1  2  3 f c' (5.4)
 b 
The multiplier 0.85 in Equation (5.3) is the reduction in the modulus of rupture of
concrete resulting from the stresses induced by the twisting component of the applied
torque. The compressive stresses (σc in Figure 5.2) from the twisting component are in
perpendicular direction to the tensile stresses from the bending component on the lateral
face of the beam. Interaction of the tensile and compressive stresses induces a 15-percent
reduction in the modulus of rupture.
134
Based on previous experimental results [Hsu (1968)], Hsu (1984) established that
torsional reinforcement in a beam (including both longitudinal and shear reinforcement)
increases the cracking torque of the beam, although it does not influence the uncracked
torsional rigidity. The cracking torque of a concrete beam reinforced with longitudinal
reinforcement and closed stirrups is obtained from the torsional strength of the plain
concrete beam with the same dimensions, Tnp (Equation 5.3), using Equation (5.5):
Tcr  (1  4  t )  Tnp (5.5)
where ρt is the total volumetric reinforcement ratio of the beam, calculated from Equation
(5.6):
t  l   s (5.6)
Volumetric ratio of the longitudinal reinforcement, ρl, and volumetric ratio of the shear
reinforcement, ρs, are determined according to Equations (5.7) and (5.8), respectively.
Al
l  (5.7)
Acp
At  p1
s  (5.8)
Acp  s
where Al is the total cross-sectional area of the longitudinal reinforcement; At is the cross-
sectional area of one leg of a stirrup; p1 is the perimeter of the area bounded by the
centerline of a stirrup; s is the spacing of the stirrups.
135
In Section 5.3 of the present chapter, the maximum torsional moments in the
specimens at the initiation of buckling will be compared to the cracking torques, obtained
from Equation (5.1) and (5.5), to determine the cracking conditions of the beams at the
instant of buckling.
In the uncracked stage, the entire solid section is effective in resisting the
torsional moments. Upon diagonal cracking, the concrete core of the section becomes
ineffective. In the post-cracking stage, therefore, torsional rigidity of a reinforced
concrete beam is provided by the outer skin of the section enclosing the closed stirrups
and the longitudinal corner bars. The horizontal plateau (AB) in Figure 5.1 corresponds
to the redistribution of the shear forces in the beam as the transition from the uncracked
condition to the post-cracking condition takes place. Segment AB becomes less
pronounced as the torsional reinforcement ratio of a concrete beam increases.
Initial portion of the post-cracking segment (BC) of the torque-twist curve is
linear. The slope of the initial linear portion is denoted as the post-cracking torsional
rigidity of the beam (GcrCcr in Figure 5.1). The post-cracking torsional rigidity
expressions in the literature are introduced in Section 5.2.2.
Beyond the linear portion of BC, the torque-twist curve curves gradually to
horizontal until the applied torque reaches the torsional strength of the beam, Tn. In the
curved portion of BC, torsional rigidity decreases with an increase in the torque. At the
torsional strength level, the beam does not possess torsional rigidity, and thus, torsional
failure takes place after a short time.
136
5.2 Torsional Rigidity of Rectangular Reinforced Concrete Beams
Torsional cracking changes the behavior of a reinforced concrete beam completely.
Different distributions of the strains from torsion in the pre- and post-cracking stages of
loading create different equilibrium conditions. Owing to the differences in the torsional
behavior of a concrete beam before and after diagonal cracking, the following discussion
classifies the torsional rigidity expressions, existing in the literature, into two separate
groups: the uncracked torsional rigidity expressions and the post-cracking torsional
rigidity expressions for reinforced concrete beams.
5.2.1 Uncracked Torsional Rigidity Expressions
A reinforced concrete beam is considered as an elastic and homogeneous body prior to
the formation of the diagonal tension cracks. The torsion of elastic and homogeneous
beams was studied by St. Venant (1856), who developed a semi-inverse method to solve
the equations from the theory of elasticity, defining the torsion of noncircular sections.
Using Fourier series, St. Venant (1856) reached the torsional rigidity expression for the
rectangular sections:
(GC )u  c  b3  h  Gc (5.9)
where Gc is the modulus of rigidity of concrete, calculated from Equation (5.10) and βc is
the coefficient for St. Venant’s torsional constant, obtained from Equation (5.11):
Ec
Gc  (5.10)
2  (1  )
where υ is the Poisson’s ratio of concrete and Ec is the elastic modulus of concrete.
137
1  192 b  1 (2n  1) h 
 c   1  5
  5
 tanh  (5.11)
3   h n0 (2n  1) 2b 
Equation (5.11) indicates that St. Venant’s torsional constant depends on the height-to-
width (h/b) ratio of a cross-section.
In discussing the torsional rigidity of rectangular sections, Wang (1953) stated
that the first term of the infinite series in Equation (5.11) gives the value of the sum to
within 0.5 percent. Therefore, for practical purposes, βc can be approximated to a simpler
form, considering the first term of the series only:
1  192 b h 
c   1  5
  tanh  (5.12)
3   h 2b 
According to Timoshenko and Goodier (1970), for narrow rectangular cross-sections
h
tanh 1 (5.13)
2b
Accordingly, Equation (5.12) can be simplified to Equation (5.14), if the beam has a
narrow rectangular cross-section:
1  b
 c   1  0.63   (5.14)
3  h 
The torsional rigidity expression adopted by Siev (1960) uses the above form of βc.
Assuming that the modulus of rigidity of concrete (Gc) obtained from Equation (5.10) is
valid at the time of buckling, Siev (1960) proposed Equation (5.15):
138
 b3  h  b 
(GC ) s  Gc     1  0.63    (5.15)
 3  h 
Another approximate form of Equation (5.11) was presented by Yen (1975), based on the
studies of Kollbrunner and Bassler (1969):
1  b b5 
 c   1  0.630   0.052 5  (5.16)
3  h h 
Hansell and Winter (1959) simplified Equation (5.11) to the following form:
2
1  b
 c   1  0.35   (5.17)
3  d 
Using the above from of βc, Hansell and Winter (1959) proposed a torsional rigidity
expression for rectangular reinforced concrete beams:
 b3  c  b 
2
(GC ) hw  Gc'   1  0.35    (5.18)
 3  d  
where G’c is the reduced modulus of rigidity of concrete according to Hansell and Winter
(1959), calculated from Equation (5.19):
Esec
Gc'  (5.19)
2  1  
where Esec is the secant modulus of elasticity of concrete corresponding to the extreme
compression fiber strain at the initiation of buckling.
139
Equation (5.19) takes into account both elastic and inelastic lateral torsional
buckling of reinforced concrete beams. Elastic modulus of concrete (Ec) and the modulus
of rigidity calculated from Ec (Equation 5.10) do not reflect the true material rigidity of a
beam if some fibers of the beam are stressed beyond the elastic limit of the stress-strain
curve of concrete, as in the case of inelastic lateral torsional buckling. Hansell and Winter
(1959) suggested to use the reduced shear modulus (G’c) to account for the reduction in
the overall material rigidity of the beam when the beam buckles inelastically. In the case
of elastic lateral torsional buckling, on the other hand, G’c becomes equal to Gc since all
fibers throughout the beam are stressed within the elastic range of the stress-strain curve
of concrete and Esec is equal to Ec
Sant and Bletzacker (1961) approximated the parameter βc to 1/3, which is
commonly used in thin-walled sections and proposed the following torsional rigidity
expression for narrow rectangular reinforced concrete beams:
b3  d
(GC ) sb  Gr  (5.20)
3
where Gr is the reduced modulus of rigidity of concrete according to Sant and Bletzacker
(1961), calculated from Equation (5.21):
Er
Gr  (5.21)
2  1  
where Er is the double modulus of concrete corresponding to the extreme compression
fiber strain at the instant of buckling, calculated from Equation (5.22):
140
4  Ec  Etan
Er  (5.22)
 
2
Ec  Etan
where Etan is the tangent modulus of concrete corresponding to the extreme compression
fiber strain at the instant of buckling.
Equation (5.20) depicts that Sant and Bletzacker (1961) preferred to relate the
shear modulus of concrete to the double modulus of elasticity, Er to account for the
possible inelastic material behavior at the initiation of buckling.
In the above discussion, simplified versions of St. Venant’s torsional constant
were presented. Equations (5.12), (5.14), (5.16) and (5.17) are the simplified forms of
Equation (5.11). Figure 5.3 compares the values obtained from the simplified forms of
Equation (5.11) to the actual values of βc obtained from Equation (5.11).
Figure 5.3 shows that Equations (5.12), (5.14), (5.16) and (5.17) are in good
agreement with Equation (5.11) for b/h smaller than unity. However, the use of a constant
value of 1/3 for βc, proposed by Sant and Bletzacker (1961), is meaningful only when the
section is quite narrow. For b/h>0.1, assuming βc=1/3 will introduce significant errors to
the calculations. To conclude, the aforementioned simplified versions of Equation (5.11)
can be used instead of Equation (5.11) to facilitate the uncracked torsional rigidity
calculations.
5.2.2 Post-Cracking Torsional Rigidity
In the post-cracking stage of loading, the torsional rigidity of a reinforced concrete beam
is provided by the outer skin concrete, since the concrete core is rendered ineffective by
141
Figure 5.3 – Comparison of the coefficients βc calculated from different equations
the diagonal tension cracks. The outer skin concrete is assumed to form a thin-walled
tube including the closed stirrups and the longitudinal corner bars.
The post-cracking torsional rigidity expressions developed by previous
researchers are based on a 3-D model, denoted as the thin-walled tube space truss model,
which is based on Rausch’s (1929) space truss analogy. According to the model, a solid
beam turns into a thin-walled tube after formation of the diagonal cracks. The thin-walled
tube, providing the post-cracking torsional rigidity, is a space truss, which is composed of
three different types of members (Figure 5.4). Helical concrete strips between the
diagonal tension cracks form the compression struts which are assumed to be connected
to the closed stirrups and the longitudinal reinforcing bars at the joints through hinges.
142
Figure 5.4 –Thin-walled tube space truss model
Compressive stresses in the tube are carried by the compression struts while the tensile
stresses are carried by the shear and longitudinal reinforcement. Using the equilibrium of
forces and compatibility of strains in the space truss, Lampert (1973) was able to develop
a post-cracking torsional rigidity expression (Equation 5.23) for rectangular reinforced
concrete beams.
4  Es  A23
Gcr  Ccr  (5.23)
 4  n    A2 1 1 
p22     
 p2  ti l  s 
where A2 is the area bounded by the lines connecting the centers of the longitudinal
corner bars; p2 is the perimeter of the area bounded by the lines connecting the centers of
the corner bars; λ is a multiplier for the concrete strain [λ=3 according to Lampert
143
(1973)]; n is the modular ratio of steel to concrete; ti is the wall thickness of the tube. ti is
the smaller of b/6 and b2/5; b is the width of the beam and b2 is the smaller dimension of
the rectangle formed by the lines connecting the centers of the longitudinal corner bars.
The three terms in the denominator of Equation (5.23) correspond to the
contributions of the concrete struts, the longitudinal reinforcing bars and the closed
stirrups, respectively.
Hsu (1973) proposed a similar equation using the thin-walled tube space truss
model:
4  Es  Ae2  Acp
Gcr  Ccr  (5.24)
 4  n  Acp 1 1 
pe2     
 pe  te l  s 
where Ae is the area bounded by the centerline of the effective wall; pe is the perimeter of
the area bounded by the centerline of the effective wall; Acp is the gross area of the
section; te is the effective wall thickness. Based on the previous experimental results, Hsu
(1973) proposed an empirical equation to obtain te:
te  1.4   l   s   b (5.25)
Later, Hsu (1990) introduced the concept of shear flow zone. According to the
concept, thickness of the thin-walled tube, providing the post-cracking torsional rigidity,
is the thickness of the shear flow zone (td) in the post-cracking stage. Thickness of the
shear flow zone depends on the applied torque according to the following equation:
4  Ta
td  (5.26)
Acp  f c'
144
where Ta is the applied torque. Equation (5.26) was obtained by Hsu (1990) using the
softened truss model.
Finally, Tavio and Teng (2004) developed an equation for the torsional rigidity of
a reinforced concrete beam at cracking, using the shear flow zone concept:
4    Es  Ao2  Acp
 GC cr  (5.27)
 1 1 
p   
2
 l  s 
o
where Ao is the area bounded by the centerline of the shear flow zone; po is the perimeter
of the area bounded by the centerline of the shear flow zone and μ is a multiplier. Tavio
and Teng (2004) stated that a value of μ= 1.5 matches well with the experimental data in
the literature.
Equations (5.23) and (5.24) correspond to the post-cracking torsional rigidity
(GcrCcr in Figure 5.1), which is the slope of the initial linear portion of the post-cracking
segment of the torque-twist curve. Equation (5.27), on the other hand, corresponds to
torsional cracking at rigidity [(GC)cr in Figure 5.1], which is the slope of the secant line
connecting the end point of the horizontal plateau of the torque-twist curve (Point B in
Figure 5.1) to the origin. The lack of the first term in the denominator of Equation (5.27)
depicts that Tavio and Teng (2004) neglected the contribution of the concrete
compression struts to the torsional rigidity at cracking in order to simplify the expression.
5.3 Experimental Torsional Rigidities of the Test Beams
Figures (5.5) & (5.6) illustrate the experimental torque-twist curves of B44-1 and B36L-1
to explain the torsional behavior of a reinforced concrete beam in a lateral-torsional
145
buckling test. The experimental torque-twist curves of the remaining specimens are
presented in Appendix D.
Figure 5.5 –Experimental torque-twist curve of Specimen B44-1
Figure 5.6 – Experimental torque-twist curve of Specimen B36L-1
146
Application of a single concentrated load at mid-span and simple support
conditions in and out of plane at the beam ends resulted in the non-uniform distribution
of the torsional moment along the span of each specimen, as previously shown in Figure
3.34(b). Due to the non-uniform moment distribution, the torque-twist curve of a test
beam is somewhat different from the typical torque-twist curve of a reinforced concrete
beam under uniform torque throughout the span. In Figure 5.1, the horizontal plateau
(AB) corresponds to the diagonal cracking throughout the entire span of a beam when the
applied torque reaches the cracking torque. In other words, the entire beam is subject to
diagonal cracking at the same stage of loading and the redistribution of the strains
throughout the whole span creates a noticeable softening in the beam beyond the initial
linear portion of the torque-twist curve. According to the experimental results obtained
by Hsu (1968), the horizontal plateau is distinct in reinforced concrete beams with closed
stirrups up to a total volumetric reinforcement ratio (ρt) of 0.04-0.05.
In Figures (5.5) & (5.6), on the other hand, the torque-twist curve does not have a
pronounced horizontal plateau due to the progressive reduction in the torsional rigidity of
the beam. In Figure 5.7, the torque-twist curve of B44-2 is approximated with a series of
linear segments with decreasing slopes to illustrate that the overall torsional rigidity of
the beam reduces gradually as the diagonal tension cracks, existing in the support zones
earlier in the test, spread towards the inner portions of the span in the further stages of
loading.
The maximum torsional moment in the beam at the initiation of buckling (Tb) is
shown with a heavy solid line on Figures (5.5) & (5.6). In the torque-twist curve of each
specimen, Tb falls into the first linear segment of the curve, which has the greatest slope
147
Figure 5.7 – Approximation of the torque-twist curve of B44-2 into a series of line
segments
among all segments. The fact that progressive reduction in the slope of the curve starts
beyond Tb manifests the absence of the diagonal tension cracks in the entire beam at the
time of buckling. In other words, all of the specimens were diagonally uncracked at the
initiation of buckling. The same conclusion can be drawn from Table 5.1, which tabulates
cracking torques of the specimens, according to Equations (5.1) and (5.5), together with
the maximum torsional moments in the beams at the start of buckling (Tb). Tb of each of
the specimens, except B18-2 and B30, is smaller than the cracking torques obtained from
both equations. Tb values of B18-2 and B30, on the other hand, are slightly larger than
Tcr, obtained from Equation (5.1). However, B18-2 and B30 are accepted as completely
uncracked at the initiation of buckling, since Tb of each specimen is smaller than Tcr
148
according to Equation (5.5), which is developed based on the results of several tests,
carried out by Hsu (1968).
Table 5.1 – Maximum torsional moments at the initiation of buckling and the cracking
torques of the specimens
Tcr (in-kips)
Specimen Tb* (in-kips)
Equation (5.1) Equation (5.5)
B18-1 12.6 8.5 34.9
B30 48.5 41.1 78.3
B36 27.6 50.8 95.9
B44-1 45.9 70.4 114.1
B44-2 36.2 70.6 114.5
B44-3 30.7 71.1 114.9
B36L-1 53.6 60.0 96.7
B36L-2 49.8 60.0 97.0
*
Maximum Measured Torsional Moment in the Beam at the Initiation of
Buckling
Table 5.2 tabulates the torsional rigidities of the beams, calculated from Equations
(5.9), (5.18), (5.20), (5.24) and (5.27). The table also includes the slopes of the initial
linear segments of the experimental torque-twist curves of the specimens, for the sake of
comparison. The slope of the experimental torque-twist curve of a beam under the
loading and support conditions of the present study cannot be directly compared to the
torsional rigidity values calculated from the aforementioned equations, due to the non-
uniform torsional moment distribution along the beam span. The experimental torque-
twist curves were obtained by plotting twist per unit length of the beam (θ) against the
maximum torsional moment along the span (Tmax). The torsional moment in the beam
decreases from maximum at the ends to minimum at mid-span. Therefore, Tmax
corresponds to the laterally-supported beam ends only. The slope of a torque-twist curve
is the torsional rigidity of a beam if the torsional moment is constant through the entire
149
Table 5.2 – Torsional rigidities of the specimens
Torsional Rigidity (x106 in2-kips)

Specimen (GC)m (GC)u (CG)hw (GC)sb GcrCcr (GC)cr
Fig. 5.4 Eq. 5.9 Eq. 5.18 Eq. 5.20 Eq. 5.24 Eq. 5.27
B44-1 1.067 0.751 0.284 0.669 0.045 0.115
B44-2 0.890 0.768 0.289 0.685 0.046 0.115
B44-3 1.166 0.798 0.297 0.712 0.047 0.116
B36L-1 1.059 0.685 0.273 0.614 0.043 0.101
B36L-2 0.988 0.720 0.282 0.646 0.044 0.101
B36 0.859 0.500 0.192 0.453 0.032 0.069
B30 0.612 0.410 0.151 0.369 0.026 0.056
B22-1 0.178 0.059 0.023 0.051 0.005 0.010
B22-2 0.198 0.048 0.019 0.042 0.004 0.008
B18-2 0.181 0.041 - 0.034 0.003 0.007
span. In the case of the test specimens, the slope of the experimental torque-twist curve,
plotting Tmax vs. θ, is greater than the actual torsional rigidity of the beam, since only the
support zones resist large torsional moments in the order of Tmax. However, the slopes of
the experimental curves are shown in Figures (5.5) & (5.6) and tabulated in Table 5.2 to
compare the order of magnitude of the experimental torsional rigidities of the beams with
the analytical values calculated from the uncracked and post-cracking rigidity
expressions.
To conclude, the experimental torque-twist curves of the specimens reveal that the
torsional behavior of a reinforced concrete beam, buckling elastically, is closely predicted
by St. Venant’s theory, and thus, the torsional rigidity of a concrete beam at the buckling
instant can be obtained from Equation (5.9). The profound difference between the slopes
of the initial linear portions of the experimental torque-twist curves [(GC)m] and the post-
150
cracking torsional rigidities of the beams (last two columns in Table 5.2) clearly indicates
that concrete beams, buckling elastically, do not undergo diagonal cracking. In the next
section, some modifications to Equation (5.9) are proposed to account for the case of
inelastic lateral-torsional buckling in reinforced concrete beams.
5.4 Proposed Torsional Rigidity Expression
As explained in the previous section, the torsional constant (C) of a reinforced concrete
beam is closely estimated by St. Venant’s theory prior to the formation of diagonal
tension cracks. The material rigidity term in Equation (5.9) is the shear modulus of
rigidity of concrete (Gc), which is obtained from the elastic modulus (Ec) through
Equation (5.10).
The use of Ec in the critical moment calculations is appropriate only if the beam
buckles elastically. To account for both elastic and inelastic lateral torsional buckling of
concrete beams, another modulus of elasticity, termed as the overall modulus of elasticity
[Equation (4.16)], was proposed. Using a simplified form of St. Venant’s torsional
constant, presented in Section 5.2.1, and accounting for the possible inelastic material
behavior of concrete at the instant of buckling, the following torsional rigidity expression
is proposed for the rectangular reinforced concrete beams:
 b3  h  b 
(GC )o  Go    1  0.63    (5.28)
 3  h 
where Go is the overall modulus of rigidity of concrete, calculated from Equation (5.29):
151
Eo
Go  (5.29)
2  1  
where Eo is the overall modulus of elasticity of concrete, obtained from Equation (5.30):
 E  Ec 
Eo   sec  (5.30)
 2 
152
CHAPTER VI
CRITICAL MOMENT CALCULATIONS AND INFLUENCES OF

THE INITIAL GEOMETRIC IMPERFECTIONS ON THE LATERAL
STABILITY OF REINFORCED CONCRETE BEAMS
6.1 Introduction
A geometrically perfect beam buckles when the applied moment reaches a critical value,
denoted as the critical moment (Mcr). In the presence of initial geometric imperfections,
on the other hand, the ultimate moment-carrying capacity, also termed as the limit
moment (ML), of a beam is smaller than the critical moment (Mcr) corresponding to the
perfect initial configuration of the beam.
In Section 6.2, the critical moment calculations of beams are presented.
Determination of the critical moment of a beam includes the evaluation of its torsional
and lateral bending rigidities. Therefore, Section 6.2 is linked to Chapters IV and V.
In Section 6.3, the influences of the initial geometric imperfections on the lateral
stability of reinforced concrete beams are explained. Section 6.3 also presents an equation
to calculate the limit moment (ML) of a concrete beam with initial lateral imperfections
(sweep) and initial twisting angle from the critical moment (Mcr) corresponding to the
initially perfect configuration of the beam.
6.2 Critical Moment Calculations
Timoshenko and Gere (1963) developed critical moment expressions for beams with
different cross-sectional shapes, loading and support conditions. Equation (6.1) is a very
general form of the critical moment expression, developed by Vacharajittiphan et al.
153
(1974) considering the influence of the in-plane (vertical) deformations of a beam prior to
buckling on the lateral stability:
  2 ECw 
EI y GJ  1  
C1  GJL2 
M cr   (6.1)
C2  L  EI   GJ   2 ECw  
y
1    1   1  
 EI x    EI x  GJL2  
where C1 is a constant corresponding to the loading conditions of a beam; C2 is a constant
corresponding to the support conditions; Mcr is the critical moment; L is the unbraced
length; EIx, EIy ,GJ, ECw are the in-plane, out-of-plane, torsional and warping rigidities of
a beam, respectively.
Smitses and Hodges (2006) stated that the effect of warping rigidity (ECw) is
considerable in thin-walled open cross-sections only. According to Timoshenko and Gere
(1963), Cw can be taken zero in a beam with narrow rectangular cross-section. Hence,
Equation (6.1) simplifies to Equation (6.2):
C1 EI y GJ
M cr   (6.2)
C2  L  EI y   GJ 
1    1  
 EI x   EI x 
The expression in the square root in the denominator of Equation (6.2)
corresponds to the in-plane (vertical) deformations of a beam prior to buckling. In deep
beams, the in-plane flexural rigidity (EIx) is significantly greater than the out-of-plane
flexural rigidity (EIy) and the torsional rigidity (GJ). Therefore, the square root term in
the denominator is very close to unity in deep beams. Ignoring this term does not change
154
the calculated values to a major extent. For instance, Beams B44 of the present study had
a EIy/ EIx ratio of 0.0048 and a GJ / EIx ratio of 0.0080. Using these values, the square
root term in the denominator becomes 0.994, which corresponds to a 0.6% change in the
critical moment. When the square root term in the denominator is ignored, Equation (6.2)
reduces to Equation (6.3):
C1
M cr   EI y GJ (6.3)
C2  L
According to Allen and Bulson (1980), the constant C1 has a value of 4.23 for a
single concentrated load at midspan. The effective length ratio C2 has a value of 1.00
when a beam is simply-supported in and out of plane.
The applied load has an additional destabilizing effect on the beam when it is
applied above the centroid of the section (Figure 1.22 of Chapter I). On the contrary, the
load has a stabilizing effect on the beam when it is applied below the centroid. Equations
(1.21) – (1.23), proposed by Stiglat (1991), account for the influence of the location of
the load application point with respect to the centroid of the section. Timoshenko and
Gere (1963) developed a critical load expression considering the influence of the location
of the point of application of load with respect to the centroid of the midspan cross-
section. Accordingly, Equation (6.3) can be modified to Equation (6.4) to account for this
effect:
4.23  e EI y 
M cr  1  1.74     EI y GJ (6.4)
L  L GJ 
155
where e is the initial vertical distance of the load from the shear center of the beam
section.
EIy and GJ are the rigidities of a homogeneous and elastic beam. The lateral
bending rigidity (Bo) and the torsional rigidity [(GC)o] of a reinforced concrete beam are
different from EIy and GJ due to the differences in behavior between a reinforced
concrete beam and a homogeneous and elastic beam, such as cracking of concrete,
elastic-inelastic material behaviors of concrete and reinforcing steel, etc. In the present
study, Equation (4.17) and Equation (5.28) are proposed for calculating the lateral
bending rigidity and the torsional rigidity of a reinforced concrete beam, respectively.
Considering all the aforementioned changes, Equation (6.5) is proposed:
4.23  e Bo 
M cr   1  1.74     B   GC o (6.5)
L 

L  GC o  o
6.3 Influences of Sweep and Initial Twisting Angle on the Lateral Stability of
Reinforced Concrete Beams
Initial geometric imperfections play a crucial role in the stability of beams. Concrete
girders possess three different types of geometric imperfections: camber (initial in-plane
deformation), sweep (initial out-of-plane deformation) and initial twisting angle.
Influence of sweep on the lateral stability of reinforced concrete beams is two-
fold. First, the out-of-plane deformations of a beam are affected by the sweep. A
geometrically perfect beam does not experience lateral deformations and twisting
rotations until bifurcation buckling takes place. When the buckling moment is reached, a
156
beam free from sweep undergoes very large lateral deformations and rotations at a
constant moment level. Nevertheless, the load-deflection behavior of a beam is different
in the presence of sweep, which causes the beam to undergo lateral deformations in the
pre-buckling stage of loading. Lateral deformations start with the initiation of loading and
grow at a relatively low rate in the pre-buckling stage. Once the beam buckles, the lateral
deformations and twisting rotations grow at much higher rates while the moment carried
by the beam is maintained at an approximately constant level.
Similarly, the initial twisting angle in a beam causes the beam to experience
twisting rotations even prior to buckling. The twisting rotations, growing slowly in the
pre-buckling stage, become very large after buckling takes place.
The second effect of sweep is the reduction in the ultimate load carried by a
concrete beam. A geometrically perfect beam buckles when the maximum moment
carried by the beam reaches the critical moment. Nonetheless, the moment carrying
capacity of a beam with sweep is smaller than the critical moment (Mcr). The maximum
moment on the load-deflection curve of an imperfect beam is termed as the limit moment
(ML) of the beam, which should be distinguished from the critical moment.
To clarify the above discussion, the experimental load-lateral deflection curves of
the beams in specimen groups B44 and B36L are illustrated in Figures 6.1 and 6.2,
respectively. The load-deflection curves do not start from the origin. The sweep of each
beam at the centroid of midspan section was taken as the initial lateral deflection
(deflection at zero load). The load-lateral deflection curves of the other specimens are
presented in Appendix D.
157
Figure 6.1 – Lateral top deflections of B44-1 and B44-2 at midspan
Figure 6.2 – Lateral top deflections of B36L-1 and B36L-2 at midspan
158
According to Figures 6.1 and 6.2, the load-lateral deflection curve of a reinforced
concrete beam has an initial linear portion and a curved portion. The curved portion turns
into an approximately horizontal line beyond the limit load, meaning that the lateral
deflections in the beam increase excessively at a constant load level when the beam
buckles. The two-fold influence of sweep on the lateral stability of reinforced concrete
beams can be observed in Figures 6.1 and 6.2. The beam with the greatest sweep
experiences larger out-of-plane deformations than its companion before reaching the
ultimate moment. Furthermore, the ultimate moment carried by the beam having the
largest sweep is smaller than the ultimate moment carried by its companion.
The influences of sweep on the limit moment and the load-deflection behavior of
a concrete beam can be understood by considering the differences between the load-
lateral deflection curves corresponding to the identical beams. First, the load-deflection
curves of the companion beams differ in the slope of the initial linear portion of the
curve, which increases as the girder sweep decreases. Secondly, sweep affects the
sharpness of the curved portion of the load-deflection curve. When the girder sweep
increases, the linear portion of the curve ends at lower load levels and the slope of the
curve decreases from a maximum to zero along a greater portion of the curve, creating a
smoother curved portion.
The differences between the load-lateral deflection curves of identical beams with
different sweeps can be clearly observed in Figure 6.2. B36L-1 and B36L-2 were
identical in nominal dimensions, cross-sectional details and nominal material strengths,
and they only differed in initial geometric imperfections (Table B.4 in Appendix B). The
initial lateral deformations of B36L-1 and B36L-2 were measured as 15/16 and 3/8
159
inches, respectively, at the top of the beams at midspan. The initial linear portion of the
load-deflection curve of Specimen B36L-2 is steeper than the linear portion of the curve
of Specimen B36L-1. Furthermore, the load-deflection behavior of the beam with greater
sweep (B36L-1) ceased to be linear at earlier stages of loading than the beam with
smaller sweep (B36L-2). B36L-1, which buckled at a load of 13.5 kips, has a linear load-
deflection behavior up to 5 kips. On the other hand, the load-deflection curve of B36L-2,
with a buckling load of 21.6 kips, remains linear up to 15 kips. Since the linear portion of
the curve of B36L-1 ends at smaller loads and the slope of the curve gradually decreases
up to the buckling load, the load-deflection curve of B36L-1 has a smoother curve
beyond the linear portion. B36L-2, on the contrary, has a sharp curve beyond the linear
portion due to the rapid decrease in the slope of the curve beyond the longer linear
portion.
The reduction in the limit moment (ML) of a reinforced concrete beam due to
sweep has not been studied extensively in the literature. Burgoyne and Stratford (2001)
stated that the additional stresses associated with the initial minor-axis curvature created
by sweep are responsible for the reduction in the ultimate moment of a concrete beam.
Longitudinal strains in a beam with an initial lateral curvature originate from in-plane and
out-of-plane bending moments. Figure 6.3 illustrates the longitudinal strain distributions
in the cross-section of a beam caused by the major-axis and minor-axis bending
moments. In a geometrically perfect beam, strains from minor-axis bending [Figure
6.3(a)] are not present up to buckling. In a beam with initial lateral deformations,
nevertheless, the minor-axis curvature created by the sweep produces longitudinal strains
even prior to the application of load. Since the beam undergoes lateral deformations as
160
Figure 6.3 – Longitudinal strain distributions in a cross-section from major-axis and
minor-axis bending moments
loading progresses, the additional stresses associated with the minor-axis curvature
increase. According to Burgoyne and Stratford (2001), cracking of concrete led by the
longitudinal stresses from the minor-axis bending moments results in the reduction of the
lateral bending rigidity of the beam, which decreases the limit moment (ML). As the
sweep of a concrete beam increases, the initial longitudinal strains related to the minor-
axis curvature increase and the reduction in the buckling resistance of a beam due to
cracking takes place earlier in the loading history. Hence, the limit moment of a concrete
beam decreases with the increasing sweep.
161
To inspect the correctness of the above statements, the strain data obtained in the
second set of experiments was examined. As previously explained in Section 3.1.3,
longitudinal strains were measured on the lateral faces of each beam at midspan.
Assuming that minor axis of the cross-section is coincident with the vertical centroidal
axis, the compressive strain on the concave face of a beam (εcl in Figure 6.3) resulting
solely from the lateral bending moment is equal to the tensile strain on the convex face
(εtl) originating from lateral bending. Therefore, the longitudinal strain from major-axis
bending at a particular depth can be obtained by averaging the two strains measured on
the convex and concave faces of the beam at that depth. The difference between the strain
measured on the concave face and the average of the two strains is the compressive strain
(εcl) created by the minor-axis bending only while the difference between the strain
measured on the convex face and the average of the two strains is the tensile strain (εtl)
from lateral bending.
Figures 6.4 and 6.5 illustrate the extreme compression fiber strains of Beams B44
and B36L at midspan resulting from the in-plane bending moments only. The extreme
compression fibers at midspan are the most stressed compression fibers of a beam. The
load-strain curves in Figures 6.4 and 6.5 are linear up to the limit load (PL). The linear
relationships in the figures imply that the in-plane bending moments created elastic
material response in the beams. The load-strain curves in Figures 6.4 and 6.5 show a
different character from the load-lateral deflection curves of the beams, shown in Figures
6.1 and 6.2. The load-deflection curves do not remain linear up to the limit load. Beyond
a certain limit, the slope of the load-deflection curve starts decreasing until vanishing at
162
Figure 6.4 – Extreme compression fiber strains of B44-1 and B44-2 from major-axis
bending
Figure 6.5 – Extreme compression fiber strains of B36L-1 and B36L-2 from major-axis
bending
163
the ultimate load. The continuous decrease in the slope of the load-deflection curve refers
to the reduction in the lateral bending rigidity of the beam. This reduction is not related to
the increase in major-axis bending strains since the linear load-strain relationship of each
beam is preserved up to buckling, unlike the load-lateral deflection relationship.
Figures 6.6 and 6.7 depict the minor-axis bending strains on the convex, tension,
faces of the specimens at the top. The load-strain curves in the figures do not start from
the origin. The strain at zero load corresponds to the initial strain created by the minor-
axis curvature associated with the sweep. Each curve ends at the point corresponding to
the limit load carried by the specimen prior to buckling. Both figures indicate that the
minor axis bending strains of the companion beams at the limit load were approximately
equal to each other. Furthermore, a comparison of Figures 6.6 and 6.7 with Figures 6.1
and 6.2 indicates that the load-minor axis bending strain curve of each beam has the same
character as the load-lateral deflection curve of the beam. The same character of the two
curves implies that the reduction in the lateral bending rigidity of a beam, which leads to
instability, is related to the increase in the strains from the minor-axis bending moments.
Moreover, the approximately equal values of the in-plane bending strains of the
companion beams at limit load imply that a concrete beam loses its stability when the
minor-axis bending strains in the beam reach certain levels. Therefore, the experimental
results of the present study agree with the statements of Burgoyne and Stratford (2001),
who associated the instability failure of an imperfect concrete beam with the reduction in
its lateral bending rigidity due to the cracking of concrete caused by the increase in the
strains from minor-axis curvature as the load increases.
164
Figure 6.6 –Extreme top strains on the convex faces of B44 caused by minor-axis
bending
Figure 6.7 –Extreme top strains on the convex faces of B36L caused by minor-axis
bending
165
In the present study, an equation was developed to calculate the ultimate load-
carrying capacity of a reinforced concrete beam with initial out-of-plane deformations.
The limit load of an imperfect beam is obtained by reducing the critical load of a perfect
beam in an amount equal to the influence of sweep to the load-carrying capacity of the
beam. A similar approach was previously used by Burgoyne and Stratford (2001) to
obtain the minor-axis curvature of a beam under a certain load.
To understand the following discussion, twisting angles in a beam accompanying
the lateral deformations should be taken into consideration. A geometrically imperfect
beam experiences twisting rotations and lateral deformations prior to buckling when
loaded. Owing to the twisting rotations, the major and minor axes of a beam rotate about
the longitudinal axis passing through the centroid of the cross-section (Figure 6.8). ν,
which will be used in the following equations, is the lateral deflection of the centroid of
the midspan section in the direction of the major axis of the twisted configuration of the
section (x”x” in Figure 6.8). ν is measured from a longitudinal axis passing though the
centroids of the end sections of a beam. In other words, ν measured with respect to the
perfect configuration of a beam.
Burgoyne and Stratford (2001) expressed the load-deflection behavior of a beam
under its self-weight as follows, when the initial lateral imperfection and the critical self-
weight of the beam are known:
o
 (6.6)
w
1
wcr
166
Figure 6.8 – Rotation of the major and minor axes of a section due to twist
where w is the self-weight per unit length of the beam; wcr is the critical self-weight per
unit length, which causes buckling of the beam; νo is the initial imperfection at the center
of the beam in the direction of the major axis of the initial configuration of midspan
section (Figure 6.8).
167
Equation (6.6) is modified to account for the concentrated midspan loading used in the
present study:
o
 (6.7)
P
1
Pcr
where P is the applied load and Pcr is the critical load corresponding to the beam free
from initial geometric imperfections.
Equations (6.6) and (6.7) are based on a method developed by Southwell (1932).
Southwell’s (1932) method is used for obtaining the buckling load and the initial
imperfections of a beam by examining the experimental load and deflection
measurements of the beam under loads much lower than the critical load. Figure 6.9(a) is
the Southwell plot for the lateral deflection data of Specimen B44-1. The inverse slope of
the plot gives the buckling load of the specimen, while the absolute value of the x-
intercept of the plot is the initial lateral centroidal deflection of the beam at midspan. In
Appendix H, Southwell’s (1932) method and the modified versions of the method
proposed by Meck (1977) and Massey (1963) are discussed in more detail and the
application of the three methods to the data obtained in the present experimental program
are described.
Figure 6.9(b) compares the experimental load-lateral deflection curve of
Specimen B44-1 to the analytical curve obtained by using Equation (6.7). Due to the
close agreement between the experimental and analytical curves, Equation (6.7) was used
for expressing the load-deflection behavior of a reinforced concrete beam with initial
geometric imperfections in closed form when developing the limit load (PL) formula.
168
Figure 6.9 - (a) Southwell (1932) Plot; (b) Load-Deflection Plot for Specimen B44-1
169
It is to be noted that P is applied at the top of the beam. The deflection of the
point of application of load (midwidth of the top face) in the direction of the major axis
of the twisted configuration (νt) is related to the deflection of the centroid (ν) according to
Equation (6.8):
h
 t     tan( ) (6.8)
2
where φ is the angle of twist of the beam at midspan corresponding to the load P.
Similarly, the initial deflection of the load application point in the major-axis direction
(νto) is related to the initial centroidal deflection according to Equation (6.9):
h
 to   o   tan(o ) (6.9)
2
where φo is the initial angle of twist of the beam.
Using Southwell’s (1932) method, a relation between φ and φo similar to Equation (6.7)
is obtained:
o
 (6.10)
P
1
Pcr
Secondly, the relation between the vertical applied load and the lateral deflection should
be assessed. The additional lateral deflection of the point of application of load (νt - νto)
is created by the component of the vertical applied load (P) in the direction of the major
axis of the twisted section (P.sinφ):
170
 h  h  P  sin( )  L3
  tan( )   
  o  tan(o )   (6.11)
 2  2 48  Ec  I y
where L is the unbraced length of the beam; Ec is the elastic modulus of concrete and Iy is
the second moment of area about the vertical centroidal axis of the beam section.
Although the minor-axis bending rigidity of a concrete beam can be much lower than EcIy
right before buckling, the use of EcIy in Equation (6.11) was found out to agree much
better with the experimental results of the present study.
Combining Equations (6.7), (6.10) and (6.11) and using the small angle
assumption, the limit load PL is related to the critical load of a beam according to
Equation (6.12):
 o  o  h 2    48  Ec  I y 
PL  Pcr  (6.12)
ult  L3
where φult is the angle of twist of the beam at midspan corresponding to PL. The sweep of
the centroid (uo in Figure 6.8) is related to νo according to Equation (6.13):
uo   o  cos(o ) (6.13)
Using Equation (6.13) and the small angle assumption, Equation (6.12) is modified to
Equation (6.14):
 uo  o  h 2    48  Ec  I y 
PL  Pcr  (6.14)
ult  L3
171
In Chapter VII, analytical load estimates from Equation (6.14) are compared to
the experimental results and the analytical estimates from other methods in the literature.
Furthermore, simplifications to Equation (6.14) for design codes are presented in Chapter
VIII.
172
CHAPTER VII
EXPERIMENTAL RESULTS AND OBSERVATIONS AND

CORRELATION OF THE ANALYTICAL AND EXPERIMENTAL
RESULTS
Section 7.1 presents the crack patterns of the specimens after buckling and some of the
experimental results. Section 7.2 compares the analytical estimates from different
formulations to the experimental limit loads of the specimens measured in the tests.
7.1 Experimental Results and Observations
7.1.1 Cracks Patterns of the Specimens
Restrained shrinkage cracks of the specimens were marked prior to the tests to
distinguish the initial cracks from the cracks formed after the application of load.
Vertical flexural cracks extending through the entire depth of the beam at midspan and
diagonal tension cracks outside the midspan region constituted the typical crack pattern
on the convex faces of the specimens (Figure 7.1) after buckling. Few diagonal cracks in
the vicinity of the end supports and vertical flexural cracks only in the bottom portion of
the beam at and around midspan were observed on the concave faces of the beams after
lateral torsional buckling (Figure 7.2).
All specimens of the experimental program were failed by lateral torsional
buckling. Therefore, similar crack patterns were observed in all specimens. At the initial
stages of loading, flexural cracks initiated and propagated in the tension zone of each
beam around midspan. These vertical cracks were visible both on convex and concave
faces of the beam (Figure 7.3). As the applied load was increased, approaching to the
173
Figure 7.1 –Typical crack pattern on the convex faces of the specimens after buckling
Figure 7.2 –Typical crack pattern on the concave faces of the specimens after buckling
174
Figure 7.3 – Flexural cracks on the concave face of B44-3 at midspan before buckling
critical load value, lateral deflections and rotations in the beam increased and lateral
bending and torsion became more dominant on the crack patterns of the beams. The
flexural tension cracks in the outermost fibers of the tension zone, initiated by the in-
plane flexural moments, extended upwards on the convex side of the beam (Figure 7.4)
due to the tensile strains introduced by lateral bending. On the contrary, the flexural
cracks stopped propagating and closed up to a certain extent on the concave side as a
result of the compressive strains originating from the lateral bending moments (Figure
7.3). Since the lateral bending moment reached its maximum value at midspan, the
vertical flexural cracks extending through the entire depth of the beam were encountered
around midspan on the convex faces of the beams.
175
Figure 7.4 –Vertical cracks on the convex face of B44-2 at midspan after buckling
Cracking outside the midspan region in the test beams was in the form of diagonal
tension cracks resulting from the torsional moments and shear forces due to the large
lateral displacements after buckling. Figures 7.5 and 7.6 illustrate the directions of the
shear and principal stresses in the beams due to the shear forces and torsional moments,
respectively. Direct shear stresses (shear stresses due to the shear forces) coincided with
the shear stresses from torsion on the convex sides of the beams while the direct shear
stresses opposed the shear stresses from torsion on the concave sides. Since the shear
stresses from both sources added up on the convex side, the diagonal tension cracks were
pronounced on the convex faces of the beams (Figure 7.7). Nevertheless, few or no
diagonal cracks could be spotted on the concaves face of the beams (Figure 7.8).
176
Figure 7.6(b) shows the typical torsional moment diagram of the beams tested in
the present study. Ignoring the location of the point of application of load with respect to
the shear center of the midspan section, the torque induced by the lateral deflections in
each beam increased from zero at midspan to a maximum value at the beam ends.
Accordingly, the torsional moments were greater in the vicinity of the laterally-supported
beam ends. The greater shear stresses from torsion around the supports overcame the
direct shear stresses and few diagonal tension cracks became visible on the concave faces
of the beam at and around the supports (Figure 7.8). The diagonal tension cracks on the
concave side in the vicinity of the end supports were in perpendicular direction (reversed)
to the diagonal tension cracks on the convex side due to torsional restraint at the beam
ends. The diagonal cracks on the convex and concave sides were connected to each other
at the top of the beam, where the extensions of the diagonal tension cracks were visible
(Figure 7.9).
177
Figure 7.5 –Directions of the shear and principal stresses due to the shear forces
178
Figure 7.6 –Directions of the shear and principal stresses due to the torsional moments
179
Figure 7.7 –Diagonal tension cracks on the convex face of B36L-1 after buckling
Figure 7.8 –Diagonal tension cracks on the concave face of B18-2 after buckling
180
Figure 7.9 –Diagonal tension cracks continuing on the top surface of B44-2 after
buckling
7.1.2 Experimental Results
The experimental load-lateral (out-of-plane) deflection, load-vertical (in-plane) deflection
and torque-twist curves of the specimens are presented in Appendix D. The midspan
strain distributions of the beams throughout the test are presented in Appendix C.
All specimens of the present study experienced elastic lateral-torsional buckling.
Figures 7.10 to 7.12 illustrate the measured greatest compressive strains of the specimens
at the initiation of buckling on the experimental stress-strain curve of concrete. The point
corresponding to each of the specimens is on the initial portion of the stress-strain curve
of concrete, implying that concrete in each specimen behaved elastically at initiation of
181
buckling. Similarly, the reinforcing bars in the specimens were measured to be strained
within the elastic range of the stress-strain curve of steel (Appendix C).
Figure 7.10 – Maximum compressive strains in the first set of beams, illustrated on the
stress-strain curve of concrete
Figure 7.11 – Maximum compressive strains in B44 at the time of buckling
182
Figure 7.12 – Maximum compressive strains in B36L at the time of buckling
7.2 Correlation of the Analytical and Experimental Results
In Table 7.1, analytical estimates according to four different formulae in the literature and
the formula proposed in the present study are tabulated together with the experimental
limit loads of the specimens. Table 7.2 presents the experimental to analytical load ratios
of the specimens according to the formulae used in Table 7.1. The experimental to
analytical load ratios corresponding to different analytical formulae are also compared in
Figure 7.13 for each specimen. Specimen B44-3 is not included in Tables 7.1 and 7.2 and
Figure 7.13, since the experimental data of this beam, which buckled in the opposite
direction to its sweep, was not considered reliable to be compared to the analytical
estimates from different solutions.
The equations used for obtaining the analytical estimates in Tables 7.1 and 7.2
and Figure 7.13 are presented here. The limit load estimates according to the method
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proposed in the present study are obtained from Equation (7.1), which was previously
given in Chapter VI [Equation (6.14)]:
16.92  Bo   GC o  e Bo  uto   48  Ec  I y 
PL   1  1.74    (7.1)
L2

 L  GC o 
 sin(ult )  L3
where L is the unbraced length of the beam; Bo is the lateral bending rigidity, obtained
from Equation (7.2); (GC)o is the torsional rigidity, calculated from Equation (7.3); e is
the vertical distance of the load application point from the centroid of the midspan cross
section; uto is the sweep at the top of the beam at midspan; Ec is the elastic modulus of
concrete; Iy is the second moment of area of the beam section about the minor axis; φult is
the angle of twist of the beam at midspan corresponding to the limit load.
 
 3 
  b c  1    Esec  Ec 
Bo     (7.2)
 12   
2
 M cra   c  
  2


 1      1 
  M cr   h  
neutral axis from the compression face; Mcra is the cracking moment; Mcr is the critical
moment; Esec is the secant modulus of elasticity of concrete corresponding to the extreme
compression fiber strain at midspan at the instant when Pcr is reached; ω is a constant,
which has a value of 1 in the absence of restrained shrinkage cracks in concrete and a
value of 2/3 in the presence of restrained shrinkage cracks.
Esec  Ec  b3  h  b 
(GC )o    1  0.63    (7.3)
4  1     3  h 
184
where υ is Poisson’s ratio of concrete.
First, the torsional and lateral bending rigidities of a beam should be calculated
from Equations (7.3) and (7.2), respectively. Next, the limit load (Pult) can be calculated
from Equation (7.1), using the calculated values of Bo and (GC)o. Equations (7.2) and
(7.3) correspond to the bending and torsional rigidities of a rectangular reinforced
concrete beam when the beam is subjected to an applied load of Pcr. For example, Esec in
the equations is the secant modulus of elasticity of concrete corresponding to extreme
compression fiber strain at midspan when the maximum in-plane bending moment in the
beam is equal to Mcr. Furthermore, c in Equation (7.2) is the neutral axis depth of the
beam section resisting a moment of Mcr. Considering the terms Esec, c and Mcr in the
rigidity expressions, Equations (7.2) and (7.3) also depend on Equation (7.1). Due to this
interdependence, an iterative procedure is needed to calculate the limit load of a beam. In
Appendix F, the critical moment calculations of one of the specimens are shown as an
example.
The limit load of a beam depends on the girder sweep and the angle of twist at the
instant when the limit load is reached. Table 7.3 tabulates uto and φult values, which were
used in Equation (7.1) to calculate the limit loads of the specimens. uto values in the table
are the sweep values measured at the top of each specimen at midspan. The initial lateral
imperfections of the specimens measured at different points along the length of each
beam are presented in Appendix B. φult values in Table 7.3 on the other hand, are the
twisting angles calculated according to the method described in Section 3.1.3 of Chapter
III from the lateral deflection measurements taken in the tests.
185
Table 7.1 – Experimental and analytical critical load values of the specimens
Analytical Load, Pan (kips)
Experimental Sant &
Specimen Load, Elastic & Homog. Hansell& Present
Material, Pel, Winter (1959), Bletzacker (1961), Massey (1967), Study,
Pex (kips) Psb, Eq. (7.10) Pm, Eq. (7.14)
Eq. (7.4) Phw, Eq. (7.7) Pult, Eq. (7.1)
B18-1 12.4 26.7 10.2 19.8 17.7 9.0
B18-2 12.0 23.7 9.4 17.6 15.6 8.9
B22-1 8.7 33.2 12.8 25.4 21.4 8.0
B22-2 11.0 27.5 10.7 21.0 18.7 *
B30 22.0 86.6 31.9 69.1 51.9 17.0
B36 39.2 101.5 39.1 80.6 67.3 40.6
B44-1 15.2 38.8 13.2 31.2 24.1 15.6
B44-2** 12.0 40.1 13.7 32.6 24.2 7.2
B36L-1 13.5 36.7 13.4 30.0 22.9 11.4
B36L-2 21.7 38.7 14.0 31.7 24.5 18.2
* Sweep of Specimen B22-2 is not known.
** Specimen B44-3 buckled in opposite direction to its sweep. Due to this unusual situation, the experimental data of B44-3
was not considered reliable to compare to the analytical estimates.
186
Table 7.2 – Experimental–to-analytical critical load ratios of the specimens
Elastic & Homog. Hansell& Winter Sant & Bletzacker Massey Present Study,
Specimen
Material, Pex/Pel (1959), Pex/Phw (1961), Pex/Psb (1967), Pex/Pm Pex/Pult
B18-1 0.46 1.22 0.63 0.70 1.38
B18-2 0.51 1.28 0.68 0.77 1.35
B22-1 0.26 0.68 0.34 0.41 1.09
B22-2 0.40 1.03 0.52 0.59 *
B30 0.25 0.69 0.32 0.42 1.29
B36 0.39 1.00 0.49 0.58 0.97
B44-1 0.39 1.15 0.49 0.64 0.97
B44-2** 0.30 0.88 0.37 0.50 1.67
B36L-1 0.37 1.01 0.45 0.59 1.18
B36L-2 0.56 1.54 0.68 0.88 1.19
Mean 0.39 1.05 0.50 0.61 1.23
S.D. 0.10 0.26 0.13 0.15 0.22
COV% 26 25 27 24 18
* Sweep of Specimen B22-2 is not known.
** Specimen B44-3 buckled in opposite direction to its sweep. Due to this unusual situation, the experimental data of
B44-3 was not considered reliable to compare to the analytical estimates.
187
Figure 7.13 – Experimental-to-analytical critical load ratios of the specimens according to different formulae
188
Table 7.3 – Measured sweeps and angles of twist at limit load of the specimens at
midspan
Angle of Twist
Specimen Sweep, uto (in) at Limit Load
φult (deg)
B18-1 7/16 1.17
B18-2 1/8 0.45
B22-1 11/16 1.66
B30 5/8 0.86
B36 7/32 0.52
B44-1 9/16 0.77
B44-2 25/32 0.55
B36L-1 3/4 0.70
B36L-2 11/32 0.65
The column denoted as the elastic and homogeneous material in Table 7.1 gives
the analytical values calculated by assuming that a reinforced concrete beam is a
homogeneous and elastic body, not subjected to cracking under loads. Regarding these
assumptions, the ultimate load (Pel) of a concrete beam can be calculated from Equation
(7.4):
16.92  Bel   GC u  e Bel 

Pel    1  1.74    (7.4)
L2 

L  GC u 
where Bel is the lateral flexural rigidity of an elastic and homogeneous beam with a solid
rectangular cross section; (GC)u is the uncracked torsional rigidity of a reinforced
concrete beam (Equation 5.9 in Chapter V). Bel and (GC)u are calculated from Equations
(7.5) and (7.6), respectively:
189
b3  h
Bel  Ec  (7.5)
12
Ec
(GC )u    c  b3  h (7.6)
2  1  
where βc is the coefficient for St. Venant’s torsional constant, obtained from Equation
(5.11).
Although Equations (7.4) - (7.6) do not reflect the true behavior of a reinforced
concrete beam, the results obtained from Equation (7.4) are included in Table 7.1 to
compare the limit load of each specimen to the limit load of a homogeneous and elastic
beam with the same dimensions as the specimen.
The analytical ultimate load estimates according to Hansell and Winter (1959)
were obtained from Equations (7.7):
16.92  Bhw   GC hw  e Bhw 

Phw   1  1.74    (7.7)
L2 

L  GC hw 
where Bhw and (GC)hw are the lateral flexural and torsional rigidities according to the
expressions proposed by Hansell and Winter (1959), given as
b3  c
Bhw  Esec  (7.8)
12
Esec  b3  c  b 
2
 GC hw    1  0.35    (7.9)
2  1    3  d  
where d is the effective depth of the beam from the compression face.
190
The ultimate load values according to Sant and Bletzacker (1961) are obtained
from the following equation:
16.92  Bsb   GC  sb  e Bsb 

Psb   1  3.48    (7.10)
L2 

L  GC sb 
where Bsb and (GC)sb are the lateral flexural and torsional rigidities according to the
expressions proposed by Sant and Bletzacker (1961), given as
b3  d
Bsb  Er  (7.11)
12
Er  b3  d 
 GC sb     (7.12)
2  1    3 
where Er is the reduced modulus of elasticity of concrete corresponding to the extreme
compression fiber strain at midspan at the instant when the applied load is equal to Psb. Er
is calculated from
4  Ec  Etan
Er  (7.13)
 
2
Ec  Etan
where Etan is the tangent modulus of elasticity of concrete corresponding to the extreme
compression fiber strain at midspan at the instant when the applied load is equal to Psb.
The expression in parenthesis in Equation (7.10), which corresponds to the
destabilizing or stabilizing effect of the load applied above or below the centroid of the
midspan section, is different from the respective expression in Equations (7.1) and (7.7).
In their study, Sant and Bletzacker (1961) considered the influence of the distance of the
191
load application point from the centroid of the section and included the expression in
parenthesis in Equation (7.10) to account for this influence. In the present study, the
original critical moment expression [Equation (7.10)] proposed by Sant and Bletzacker
(1961) was used together with the torsional and lateral bending rigidity expressions
developed by Sant and Bletzacker (1961).
Finally, the ultimate load values according to Massey’s (1967) formulation were
obtained from the following equation:
16.92  Bm   GC m  e Bm 
Pm    1  1.74    (7.14)
L2 

L  GC m 
where Bm and (GC)m are the lateral flexural and torsional rigidities according the
expressions proposed by Massey (1967), given as
b3  c
Bm  Esec   Es  I sy (7.15)
12
 Gc'  c  b3  h  1   Gs  Gc'   bs3  ts    b1  d1  Ao  Es

2
 GC m (7.16)
3 2 2  s
where ΣIsy is the moment of inertia of the longitudinal steel about the minor axis of the
section; bs and ts are the width and thickness of the longitudinal reinforcement layer,
respectively (illustrated in Figure 1.18 in Chapter I); γ is a constant defined by Cowan
(1953); b1 and d1 are the breadth and depth of the cross-sectional area enclosed by the
closed stirrup, respectively (Figure 1.18); s is the spacing of the stirrups; Ao is the cross-
sectional area of one leg of the stirrup; Es and Gs are the modulus of elasticity and
192
modulus of rigidity of steel, respectively; G’c is the reduced modulus of rigidity of
concrete, calculated according to the following equation:
Esec
Gc'  (7.17)
2  (1   )
Tables 7.1 and 7.2, and Figure 7.13 indicate that the analytical estimates produced
by the proposed method are in good agreement with the experimental results.
Furthermore, the analytical estimates from the proposed method never exceeded the
experimental ultimate loads, with the exception of Specimens B36 and B44-1, for which
the analytical estimates are only 3-4% greater than the experimental values. The
experimental to analytical load ratios corresponding to the proposed method were in the
range of 0.97-1.67 with a mean value of 1.23 and coefficient of variation of 0.18 (Table
7.2).
Among the formulae proposed by the previous researchers, the equation given by
Hansell and Winter (1959) closely estimated the limit loads of the specimens with the
experimental to analytical load ratios in the range of 0.68-1.54 (a mean value of 1.05 and
coefficient of variation of 0.25). The formulae proposed by Sant and Bletzacker (1961)
and Massey (1967) constantly overestimated the limit loads of the specimens. The
analytical estimates from Sant and Bletzacker’s (1961) formula sometimes reached 2.5-3
times the limit loads measured in the tests.
As mentioned before, uo and φult are needed to calculate the limit load of a beam
according to Equation (7.1). φult, in particular, is a quantity which is determined by
testing a beam to failure. Since testing a beam to failure is not always possible,
particularly in a real construction, a value for φult should be assumed in the limit load
193
calculations. In Chapter VIII, Equation (7.1) is modified by assuming constant values for
φult to simplify the equation.
194
CHAPTER VIII
SUMMARY AND CONCLUSIONS
8.1 Summary
The present study investigated the lateral stability of rectangular reinforced concrete
beams both analytically and experimentally. The experimental part of the study provided
the experimental results of reinforced concrete beams, whose initial geometric
imperfections, shrinkage cracking conditions and material properties are completely
known. In the analytical part of the study, the following formula was developed for
estimating the limit loads (PL) of simply-supported rectangular reinforced concrete beams
with initial geometric imperfections, subjected to a concentrated load at midspan:
4  M cr uto   48  Ec  I y 
PL   (8.1)
L sin(ult )  L3
where PL is the limit load; L is the unbraced length of the beam; uto is the sweep at the top
of the beam at midspan; Ec is the elastic modulus of concrete; Iy is the second moment of
area of the beam section about the minor axis; φult is the angle of twist of the beam at
midspan corresponding to the limit load (PL). Mcr is the critical moment corresponding to
the geometrically perfect configuration of the beam, obtained from Equation (8.2):
4.23  e Bo 
M cr   1  1.74     B   GC o (8.2)
L 

L  GC o  o
195
where Bo is the lateral bending rigidity, obtained from Equation (8.3); (GC)o is the
torsional rigidity, calculated from Equation (8.4); e is the vertical distance of the load
application point from the centroid of the midspan cross section.
 
 3 
  b c  1    Esec  Ec 
Bo    (8.3)
 12   
2
 M cra   c  
  2


 1      1 
  M cr   h  
Esec  Ec  b3  h  b 
(GC )o    1  0.63    (8.4)
4  1     3  h 
neutral axis from the compression face; Mcra is the cracking moment, obtained from
Equation (8.5); ω is a constant, which has a value of 1 in the absence of restrained
shrinkage cracks in concrete and a value of 2/3 in the presence of restrained shrinkage
cracks and υ is Poisson’s ratio of concrete. Esec is the secant modulus of elasticity of
concrete corresponding to the extreme compression fiber strain at midspan at the instant
when Mcr is reached. Esec is calculated from Equation (8.6).
M cra 

I ucr  7.5  f c'  (8.5)
h y
where 7.5  f c' is the modulus of rupture of normal-weight concrete, given in ACI 318-
05 (2005) Section 9.5.2.3; y is the depth of the center of gravity of the transformed
196
section from the top surface of the beam; Iucr is the moment of inertia of the transformed
section about the major axis, obtained from Equation (8.7).
fc
Esec  (8.6)
c
where fc and εc are the extreme compression fiber stress and extreme compression fiber
strain at midspan corresponding to the critical moment Mcr.
2
1  h
  b  h3  b  h   y     n  1  As   d  y 
2
I ucr (8.7)
12  2
where As is the cross-sectional area of the flexural reinforcement; n is the modular ratio of
steel to concrete; d is the effective depth of the centroid of tension reinforcement from
compression face. When calculating the uncracked moment of inertia, Iucr, the flexural
reinforcement is transformed into an equivalent concrete area in accordance with the
modular ratio of steel to concrete, n.
Due to the interdependence of Mcr, Bo and (GC)o, an iterative process is needed
when calculating the limit load of a beam. This interdependence is also present in the
methods proposed by Hansell and Winter (1959), Sant and Bletzacker (1961) and Massey
(1967). The primary reason for the interdependence of the critical moment and the lateral
bending and torsional rigidities is that the modulus of elasticity and modulus of rigidity
terms in the rigidity expressions depend on the maximum compressive strain in the beam
corresponding to the critical moment, as explained in Appendix F in more detail.
The experimental stage of the study consisted of testing eleven reinforced
concrete beams with d/b ratios between 10.20 and 12.45 and L/b ratios between 96 and
197
156. Beam thickness, depth and unbraced length were 1.5 to 3.0 in., 18 to 44 in., and 12
to 39.75 ft, respectively. The test beams were simply-supported in and out of plane and
subjected to a single concentrated load at midspan. The end supports allowed warping
deformations in the beams while restraining the torsional rotations at the beam ends. The
loading mechanism used in the tests provided lateral translational and rotational freedom
at the load application point and ensured that the vertical orientation of the applied load
was maintained throughout the entire test. The in-plane (vertical) and out-of-plane
(lateral) deformations, the torsional rotations and the strain distributions in the beams
were measured at midspan, continuously along the tests.
There are several factors influencing the lateral stability of reinforced concrete
beams. Initial geometric imperfections, shrinkage cracking, contribution of the
longitudinal and shear reinforcement to the torsional and lateral bending rigidities, creep,
inelastic stress-strain properties of concrete and reinforcing steel, loading and support
conditions are some of the major factors affecting the lateral stability. Investigating the
influences of several factors in the same experiments renders the analysis and
interpretation of the experimental data cumbersome. In the present experimental program,
the main factor whose effects were investigated is the initial lateral imperfections (sweep)
of a beam. To detect the effects of sweep, the influences of some of the other factors were
minimized or eliminated through the following ways:
 The specimens of the present experimental program were designed in such a way that
both concrete and reinforcing steel in the beams remained in the elastic ranges of their
stress-strain curves throughout the loading process. By eliminating the inelastic
stress-strain properties of concrete and steel, pure elastic material behavior was
198
attained and the effects of inelasticity on the lateral stability of the specimens were
eliminated. Nevertheless, the lateral and torsional rigidity expressions proposed in the
present study account for both elastic and inelastic material behaviors of concrete and
reinforcing steel to be applicable for all reinforced concrete beams with different
dimensions, reinforcement details and material properties.
 Restrained shrinkage cracking of concrete was tried to be prevented through the
measures described in Section 4.4. Addition of Shrinkage Reducing Admixtures
(SRA) to concrete and early removal of the beams from the forms minimized the
amount and extent of shrinkage cracking in the second set of specimens.
 The specimens were designed in a way that the contributions of the longitudinal and
shear reinforcement to the lateral stability of the beams were negligible. The
longitudinal reinforcing bars were located along the vertical centroidal axis of the
beam section. Presuming that the minor axis of the beam is coincident with the
vertical centroidal axis throughout the test, the longitudinal reinforcement did not
contribute to the lateral bending rigidities of the beams since the second moment of
area of the longitudinal reinforcement (ΣIsy) is equal to zero in this design.
Furthermore, the specimens did not experience diagonal tension cracking up to
buckling. Prior to the formation of diagonal tension cracks, a reinforced concrete
beam behaves as a solid and homogeneous body, whose torsional rigidity is provided
by the entire cross-section. In the pre-cracking stage, contributions of the flexural and
shear reinforcement to the torsional rigidity are negligible. Due to the absence of
torsional cracks in the beams up to buckling, contribution of the reinforcement to the
resistance of torsional moments was disregarded.
199
 All specimens of the experimental program were tested under identical loading and
support conditions. Consequently, the effects of the loading and support conditions on
the test results of companion beams (beams with identical nominal dimensions,
reinforcement details and material properties) were eliminated.
 Concrete from the same batch and reinforcing steel from the same batch were used in
the companion beams to keep the material properties constant so that the
experimental results of the companion beams were not affected from the differences
between the mechanical properties of the beams.
 To eliminate the influence of creep on the lateral stability, the test beams were loaded
continuously up to the ultimate load without interruption.
Although the experimental program aimed at eliminating the influences of the
aforementioned factors on the lateral stability of the specimens, all factors affecting the
lateral stability of a reinforced concrete beam were taken into consideration in the
development of the proposed analytical method. For instance, Equation (4.18) in Chapter
IV gives the lateral bending rigidity of a reinforced concrete beam, considering the
influence restrained shrinkage cracking. For the possible contribution of the longitudinal
reinforcing bars, on the other hand, Equation (4.16) was developed. Furthermore, both
torsional and lateral bending rigidity expressions developed in the present study are
applicable to the cases of elastic and inelastic lateral-torsional buckling.
8.2 Conclusions
Some of the conclusions drawn from the present study for the lateral stability of
rectangular reinforced concrete beams follow:
200
 The load predictions from the proposed analytical method showed good correlation
with the experimental results. The analytical to experimental limit load ratios of the
specimens ranged from 0.97 to 1.67 for the proposed analytical method. The
analytical method developed in the present study is superior to the analytical methods
proposed by Hansell and Winter (1959), Sant and Bletzacker (1961) and Massey
(1967) in incorporating the effects of sweep, shrinkage cracking and inelastic stress-
strain properties of concrete into the buckling formula.
 Among the former methods considered in the present study, the analytical method
proposed by Hansell and Winter (1959) produced better correlation with the
experimental results than the methods proposed by Sant and Bletzacker (1961) and
Massey (1967), which constantly overestimated the ultimate loads of the beams.
 In contrast to the methods in the literature, the load estimates produced by the
proposed method were smaller or slightly larger than the experimental buckling loads
of the specimens (Table 7.2), which makes the proposed method more conservative
than the methods proposed by Hansell and Winter (1959), Sant and Bletzacker (1961)
and Massey (1967). Even some of the load estimates from the method proposed by
Hansell and Winter (1959), which was in closer agreement with the experimental
results than the methods proposed by Sant and Bletzacker (1961) and Massey (1967),
were greater than the experimental buckling loads of the specimens.
 In the case of elastic lateral torsional buckling, reinforced concrete beams do not
undergo diagonal tension cracking before buckling. Therefore, the uncracked
torsional rigidity closely reflects the torsional resistance of a reinforced concrete
beam at the instant of buckling, if the beam buckles elastically. Moreover,
201
contributions of the longitudinal and shear reinforcement to the torsional rigidity of a
reinforced concrete beam is negligible in elastic buckling since the reinforcement of a
concrete beam has a major influence on the torsional behavior of a concrete beam
only after diagonal cracking.
 The influence of the longitudinal reinforcement on the lateral stability of a concrete
beam originates from its contribution to the lateral bending rigidity. Similarly,
restrained shrinkage cracking of concrete affects the lateral stability of a beam by
reducing its lateral bending rigidity. Lateral bending rigidity expression given in
Equation (4.17) considers the increase in the lateral bending rigidity of a reinforced
concrete beam due to the contribution of the flexural reinforcement while the rigidity
expression given in Equation (4.16) was developed regarding the negative influence
of shrinkage cracking of concrete on the lateral bending rigidity.
 A geometrically imperfect concrete beam does not reach the critical moment
corresponding to the geometrically perfect configuration of the beam. The additional
stresses originating from the initial minor-axis curvature from sweep cause the
imperfect beam to crack earlier in the loading process (at smaller load levels).
Cracking on the convex side of the beam greatly reduces the lateral bending rigidity
and the load-carrying capacity of the beam starts decreasing before reaching the
critical load. The maximum load on the load-lateral deflection curve of an imperfect
beam is denoted as the limit load, which should be distinguished from the critical
load. The limit load formula proposed in the present study is based on reducing the
critical load by an amount equal to the destabilizing effect of the sweep on the beam.
202
 Results of the present experimental program indicated that an increase in the sweep
can greatly reduce the ultimate load resisted by a reinforced concrete beam before
losing its stability. The significant differences between the ultimate loads of the
companion beams mainly originated from the different initial lateral imperfections of
the beams. For instance, the experimental ultimate load of Specimen B36L-2 (11/32
in sweep) was 38% greater than the experimental ultimate load of Specimen B36L-1
(3/4 in sweep).
 To calculate the limit load of an imperfect reinforced concrete beam from Equation
(7.1), the angle of twist (φult) of the beam needs to be known. φult is a parameter
indicating the torsional rotations in a beam till buckling and it is determined by
testing a beam to failure. In the design of a concrete beam, a value for φult should be
assumed. The φult values of the test specimens ranged from 0.45 degrees to 1.66
degrees. However, most of the specimens had a midspan angle of twist between 0.55
degrees to 0.75 degrees at the limit load level. As φult decreases, the reduction in the
limit load of a beam increases. Therefore, assuming a value between 0.55 and 0.60
degrees for φult seems reasonable and safe according to the available experimental
data. However, more experimental data is needed to make more accurate
assumptions.
8.3 Future Research
The analytical method, proposed in the present study, incorporates several factors
influencing the lateral stability of reinforced concrete beams. The factors with
considerable influence were restated in Section 8.1. The experimental stage of the study,
nonetheless, aimed at investigating the effects of the initial lateral imperfections on the
203
buckling behavior of reinforced concrete beams. To achieve this goal, the effects of the
other influential factors were tried to be minimized, if not eliminated. Hence, further
experiments are needed to investigate the effects of the factors, which were disregarded
in the present experimental program. For example, all specimens of the experimental
program were failed by elastic lateral torsional buckling. Further experiments on
reinforced concrete beams subject to inelastic lateral torsional buckling are needed to
investigate the degree of agreement between the experimental ultimate loads of the beams
and the estimates produced by the proposed analytical method in the case of inelastic
lateral torsional buckling. Similarly, the accuracy of the analytical estimates needs to be
explored experimentally when the longitudinal reinforcement of a concrete beam
contributes to the lateral stability to a major extent. As stated before, the contribution of a
longitudinal rebar to the lateral bending rigidity of a beam increases as the lateral distance
of the bar from the minor axis increases. Further experiments on reinforced concrete
beams with longitudinal reinforcing bars distributed along the sides of the beams can be
useful to examine the accuracy of the estimates from the proposed analytical method
when the reinforcement plays an important role in resisting the lateral bending moments.
Finally, more experimental data is needed to determine the common values of the
twisting angle at limit load (φult) of reinforced concrete beams, which is used in Equation
(8.1) for calculating the limit load of a reinforced concrete beam with initial geometric
imperfections. Based on a statistical analysis on the results of a large number of lateral
torsional buckling tests of reinforced concrete beams, the appropriate values of φult can be
determined and recommended in the structural concrete codes for using in Equation (8.1).
204
APPENDIX A
NOMINAL AND MEASURED DIMENSIONS OF THE SPECIMENS
The actual dimensions of a concrete beam can be significantly different than the nominal
dimensions. For a more precise analytical study, the actual dimensions of a beam,
determined from several measurements throughout the beam, should be used when
calculating the analytical critical loads. In this concern, the dimensions of the specimens
were measured at several locations throughout each beam. The tables of the present
appendix tabulate the nominal and measured dimensions of the specimens together with
the means (μ), standard deviations (σ) and the percent coefficients of variation (cv %) of
the measurements. Standard deviations and coefficients of variation of the measurements
are included in the tables to reflect the degree of variation in the measurements of a
dimension.
Table A.1 and A.2 present the heights of the specimens, measured at several
locations along the lengths of the beams. The locations of the measurements are shown in
Figure A.1.
Tables A.3 and A.5, on the other hand, present the widths of the specimens,
measured at several locations along the depth and length of each beam. The locations of
the measurement points are shown in Figure A.2.
Each of the first set of beams was cut at mid-length after the test to determine the
actual locations of the longitudinal reinforcing bars in the beams. The widths of the
beams measured along the cuts are tabulated in Table A.4.
205
Table A.6 presents the unbraced lengths of the first set of beams. Table A.7, on
the other hand, presents the total lengths of the second set of beams, measured at five
different depths, as shown in Figure A.3. The unbraced span lengths of B44 and B36L
were measured as 39.33 ft.
Table A.1 – Nominal and measured heights of the first set of specimens
Specimen B 36 B 30 B 22-1 B 22-2 B 18-1 B 18-2

Nominal Height (in) 36.000 30.000 22.000 22.000 18.000 18.000
Measurement
Point
(Fig.A.1a&b)
1 35.906 30.000 22.000 22.063 18.000 18.000
2 35.969 30.063 22.031 22.125 18.094 18.094
Measured Height (in)
3 36.063 30.063 22.031 22.094 18.156 18.125

4 36.094 30.031 22.000 22.063 18.125 18.094
5 36.063 29.969 21.969 22.063 18.125 18.063
6 36.031 29.875 22.000 22.094 18.063 18.094
7 36.031 29.938 22.000 22.094 18.125 18.094
8 36.000 29.875 22.031 22.000 18.063 18.000
9 36.031 29.938 22.000 22.000 18.125 18.000
10 36.094 30.000 22.000 22.000 18.156 18.000
11 35.875 29.969 21.938 22.188 18.000 18.188
μ 36.014 29.975 22.000 22.071 18.094 18.068
σ 0.068 0.063 0.026 0.055 0.053 0.059
cv % 0.19 0.21 0.12 0.25 0.29 0.33
206
Table A.2 – Nominal and measured heights of the second set of beams
Specimen B44-1 B44-2 B44-3 B36L-1 B36L-2

Nominal Height (in) 44 44 44 36 36
Measurement Point
(Figure A.1c)
1 44 44 44 1/16 36 5/32 36 1/16
2 44 43 31/32 44 1/16 36 36
3 44 1/8 44 43 15/16 36 3/32 36 1/8
4 44 1/16 44 1/32 44 3/32 36 1/32 36 1/32
5 44 3/32 44 44 36 36
6 44 1/8 44 1/32 44 1/16 36 1/32 36
7 44 3/16 44 44 1/8 36 36 1/16
8 44 1/16 44 44 5/32 35 7/8 36

9 44 1/8 43 29/32 44 3/32 36 35 7/8
10 44 1/8 43 29/32 44 3/32 36 36 1/16
11 44 43 31/32 44 3/32 36 1/16 36
12 43 15/16 44 1/32 44 1/8 36 36
13 43 7/8 44 1/32 44 3/32 36 1/8 36 1/16
14 43 27/32 44 1/32 44 3/16 36 36 1/32
15 43 13/16 44 1/16 43 7/8 36 1/8 36 1/16
16 43 13/16 44 44 36 1/8 36 1/16
17 43 13/16 43 29/32 44 3/32 36 3/32 36
18 43 25/32 44 1/64 44 1/16 36 1/8 36 1/32
19 43 13/16 44 3/32 44 36 3/32 36
20 43 27/32 44 1/8 44 1/32 36 1/16 36 1/16
21 43 15/16 44 1/8 44 1/32 36 1/32 36
μ 43 31/32 44 1/64 44 1/16 36 3/64 36 1/32
σ 1/8 1/16 5/64 1/16 3/64
% cv 0.30 0.17 0.16 0.18 0.13
207
Figure A.1–Height measurement points along the lengths of the beams
(All dimensions are in feet)
208
Table A.3 – Nominal and measured widths of the first set of specimens along the span
Specimen B 36 B 30 B 22-1 B 22-2 B 18-1 B 18-2

Nominal Width (in) 2.500 2.500 1.500 1.500 1.500 1.500
Meas. Point
(Fig. A.2a&b)
1 2.438 2.563 1.578 1.563 1.563 1.484
2 2.500 2.500 1.578 1.563 1.578 1.500
3 2.438 2.438 1.563 1.563 1.578 1.453
Measured Width (in)
4 2.438 2.500 1.594 1.547 1.516 1.500

5 2.469 2.531 1.625 1.563 1.547 1.547
6 2.469 2.438 1.531 1.469 1.547 1.609
7 2.531 2.469 1.578 1.500 1.594 1.594
8 2.469 2.500 1.547 1.500 1.516 1.516
9 2.438 2.500 1.609 1.531 1.516 1.516
10 2.438 2.531 1.500 1.531 1.516 1.516
11 2.469 2.500 1.453 1.516 1.516 1.594
12 2.406 2.500 1.547 1.484 1.516 1.578
μ 2.458 2.498 1.559 1.527 1.542 1.534
σ 0.032 0.035 0.046 0.032 0.029 0.045
cv % 1.301 1.401 2.951 2.096 1.881 2.934
Table A.4 –Widths of the first set of specimens along the depth of midspan section
Specimen B 36 B 30 B 22-1 B 22-2 B 18-1 B 18-2

Nominal Width 2.500 2.500 1.500 1.500 1.500 1.500
1 2.575 2.586 1.576 1.447 1.567 1.632
2 2.595 2.540 1.630 1.469 1.557 1.647
3 2.617 2.537 1.590 1.474 1.550 1.643
4 2.655 2.557 1.583 1.495 1.529 1.592
5 2.652 2.583 1.588 1.497 1.539 1.611
6 2.617 2.567 1.565 1.512 1.535 1.605
7 2.643 2.614 1.549 1.524 1.547 1.612

8 2.616 2.622 1.540 1.494 1.587
9 2.641 2.639 1.517
10 2.643 2.660 1.527
11 2.613 2.637
12 2.605 2.619
13 2.602
14 2.552
15 2.550
16 2.496
17 2.496
18 2.498
19 2.490
μ 2.587 2.597 1.578 1.496 1.546 1.616
σ 0.056 0.039 0.026 0.025 0.012 0.016
cv % 2.147 1.508 1.652 1.646 0.788 0.990
209
Figure A.2–Width measurement points along the lengths of the specimens
(All dimensions are in feet)
210
Table A.5 – Nominal and measured widths of the second set of beams

Nominal Width (in) 3.000 3.000 3.000 3.000 3.000
Measurement Point
(Figure A.2c)
1 3.015 3.000 3.028 3.111 3.030
2 3.038 3.000 3.049 3.080 3.037
3 3.070 3.063 3.100 3.116 3.068
4 3.028 3.031 3.077 3.102 3.162
5 3.094 3.000 3.070 3.070 3.060
6 3.043 3.012 3.074 3.287 3.150
7 3.000 3.003 3.042 3.283 3.280
8 3.051 2.999 3.105 3.159 3.189
9 3.063 3.043 3.017 3.125 3.346
10 3.038 3.002 3.011 3.206 3.336
11 3.067 2.996 3.084 3.325 3.165
12 3.032 3.031 3.144 3.265 3.268
13 2.997 3.054 3.005 3.150 3.303
14 2.963 3.039 3.009 3.172 3.297
15 3.033 2.960 2.909 3.159 3.238
16 3.012 3.055 3.054 3.170 3.244
17 3.031 3.028 3.086 3.130 3.333
18 3.103 3.037 3.122 3.144 3.179
19 3.149 3.009 3.042 3.143 3.218
Measured Width (in)
20 3.019 3.015 3.016 3.270 3.164

21 3.042 2.982 3.027 3.163 3.214
22 3.063 2.992 3.127 3.093 3.298
23 3.055 3.072 3.034 3.245 3.273
24 3.067 3.036 3.040 3.219 3.337
25 3.107 3.122 3.061 3.160 3.147
26 3.109 3.122 2.973 3.283 3.266
27 3.064 3.042 2.859 3.284 3.296
28 3.051 3.050 2.892 3.205 3.186
29 3.036 3.120 2.982 3.281 3.103
30 3.106 3.094 3.031 3.258 3.365
31 3.070 3.074 3.096 3.142 3.125
32 3.022 3.121 3.115 3.121 3.111
33 3.033 3.022 3.090 3.245 3.066
34 3.053 3.056 3.100 3.216 3.195
35 3.095 3.026 3.101 3.239 3.105
36 3.076 3.100 3.137 3.171 3.065
37 3.065 3.006 3.111 3.280 3.070
38 3.085 3.012 3.020 3.123 3.107
39 3.025 3.084 3.089 3.207 3.062
40 3.073 2.982 2.993 3.161 3.162
41 3.082 3.096 3.048 3.061 3.166
42 2.998 3.083 3.115 3.068 3.197
43 3.054 3.037 2.988 3.192 3.109
44 3.128 3.031 3.001 3.310 3.243
45 2.944 3.063 3.072 3.190 3.145
46 3.047 3.156 3.128 3.149 3.208
47 3.046 3.156 3.180 3.160 3.226
48 3.017 3.188 3.147 3.143 3.185
μ 3.051 3.048 3.054 3.184 3.190
σ 0.039 0.050 0.068 0.070 0.091
% cv 1.3 1.6 2.2 2.2 2.8
211
Table A.6 – Nominal and measured span lengths of first set of specimens
Specimen B 36 B 30 B 22-1 B 22-2 B 18-1 B 18-2

Beam Length (in) 252.000 252.000 156.000 156.000 156.000 156.000
Span (in) 240.000 240.000 144.000 144.000 144.000 144.000
Measurement
Depth
(Figure A.3)
1 239.500 239.875 143.750 143.750 143.750 143.719
239.375 239.813 143.750 143.781 143.781 143.688
Measured
2
Span (in)
3 239.375 239.750 143.750 143.750 143.750 143.688

4 239.375 239.688 143.750 143.813 143.781 143.688
5 239.500 239.688 143.750 143.813 143.750 143.688
μ 239.425 239.763 143.750 143.781 143.763 143.694
σ 0.061 0.073 0.000 0.028 0.015 0.012
cv % 0.025 0.030 0.000 0.019 0.010 0.008
Table A.7 – Nominal and measured total lengths of the second set of specimens

Nominal Length (ft) 39.75 39.75 39.75 39.75 39.75
Measurement
Depth
(Figure A.3)
1 39.76 39.75 39.75 39.78 39.78
Length (ft)
Measured
2 39.77 39.74 39.76 39.79 39.77

3 39.78 39.74 39.76 39.79 39.77
4 39.77 39.75 39.77 39.80 39.78
5 39.77 39.75 39.77 39.79 39.78
μ 39.77 39.75 39.76 39.79 39.78
σ 0.01 0.01 0.01 0.01 0.01
% cv 0.02 0.01 0.02 0.02 0.01
Figure A.3–Length measurement depths of the specimens
212
APPENDIX B
MEASURED INITIAL GEOMETRIC IMPERFECTIONS AND

PERMANENT DEFORMATIONS OF THE SPECIMENS
The initial lateral deformations, sweeps, of the test beams are tabulated in Tables B.1-
B.4. The tables present the lateral deformations at the extreme top, mid-height and
extreme bottom of each beam. Sweep was measured at three different levels along the
depth of each beam to obtain the initial angles of twist of the specimens. Furthermore, the
initial lateral deflections of the beams were measured at several points along the length of
each beam to determine the exact curved shapes of the specimens out of plane.
Knowledge of the lateral bows of the test beams prior to loading was essential in the
analytical stage of the present study. The number and locations of the measurement
points along the lengths of the beams are illustrated in Figure B.1. Southward deflections
in Tables B.1-B.4 are positive while the northward ones are negative according to the
sign convention shown in Figure B.1.
213
Table B.1 – Measured sweeps of Specimens B18
Measurement Sweep (in)

Point along
B18-1 B18-2
the Span
(Figure B.1a) Top Bottom Top Midheight Bottom
1 -1/8 -3/16 -1/32 -1/32 -1/32
2 -5/16 -5/16 -1/16 -1/16 -1/16
3 -7/16 -7/16 -1/8 -1/8 -1/8
4 -7/16 -7/16 -1/8 -1/8 -1/8
5 -7/16 -7/16 -1/8 -1/8 -1/8
6 -7/16 -7/16 -3/32 -3/32 -3/32
7 -1/4 -5/16 -1/16 -1/16 -1/16
Table B.2 – Measured sweeps of Specimens B30 and B36

Point along
B30 B36
the Span
(Figure B.1b) Top Midheight Bottom Top Midheight Bottom
1 -1/4 -3/16 -3/16 1/32 1/64 1/32
2 -7/16 -3/8 -1/4 1/8 5/32 1/8
3 -5/8 -9/16 -1/2 1/8 5/32 1/8
4 -5/8 -9/16 -1/2 7/32 3/16 7/32
5 -5/8 -9/16 -1/2 7/32 3/16 7/32
6 -1/2 -7/16 -3/8 7/32 3/16 7/32
7 -1/4 -3/16 -1/8 1/8 3/32 1/8
214
(a)
(b)
(c)
Figure B.1 – Imperfection measurement points on beams (a) B18; (b) B30 and B36;
(c) B44 and B36L (All dimensions are in feet.)
215
Table B.3 – Measured sweeps of Specimens B44

Point along
B44-1 B44-2 B44-3
the Span
(Figure B.1c) Top Midheight Bottom Top Midheight Bottom Top Midheight Bottom
1 1/16 1/8 1/8 1/32 1/8 5/32 -11/32 -3/16 -1/4
2 3/16 1/8 1/8 1/16 5/32 1/8 -9/16 -3/8 -5/16
3 3/16 3/16 3/16 3/16 3/16 3/16 -5/8 -9/16 -1/2
4 1/8 1/8 1/8 9/32 7/32 7/32 -29/32 -13/16 -3/4
5 3/16 3/16 3/16 13/32 11/32 3/8 -1 1/8 -7/8 -1
6 1/4 1/4 3/16 17/32 1/2 9/16 -1 7/32 -1 5/32 -1 3/16
7 3/8 3/16 3/16 21/32 11/16 5/8 -1 3/8 -1 5/32 -1 1/4
8 1/4 3/16 1/4 23/32 5/8 19/32 -1 1/2 -1 9/32 -1 7/16
9 3/8 3/16 1/8 3/4 11/16 5/8 -1 9/16 -1 3/8 -1 1/2
10 9/16 3/8 1/4 25/32 23/32 13/16 -1 3/8 -1 3/8 -1 1/2
11 5/16 5/16 5/16 3/4 13/16 27/32 -1 5/16 -1 5/16 -1 1/2
12 5/16 1/4 5/16 25/32 27/32 7/8 -1 3/8 -1 5/16 -1 1/2
13 1/4 1/4 5/16 27/32 27/32 29/32 -1 9/32 -1 1/4 -1 7/16
14 1/4 3/16 1/4 7/8 13/16 27/32 -1 1/4 -1 5/32 -1 3/8
15 1/4 3/16 1/4 3/4 11/16 3/4 -1 1/8 -1 1/8 -1 1/4
16 3/16 3/16 3/16 19/32 9/16 5/8 -7/8 -13/16 -1 1/16
17 3/16 1/8 3/16 17/32 17/32 17/32 -5/8 -9/16 -3/4
18 1/4 1/8 1/4 3/8 7/16 3/8 -7/16 -3/8 -3/8
19 1/4 1/8 1/4 1/4 1/4 1/4 -1/4 -3/16 -1/4
216
Table B.4 – Measured sweeps of Specimens B36L

Point along
B36L-1 B36L-2
the Span
(Figure B.1c) Top Midheight Bottom Top Midheight Bottom
1 -3/16 -3/16 -3/16 -1/8 -1/4 -1/16
2 -3/8 -7/16 -3/16 -3/16 -1/8 -1/4
3 -9/16 -5/8 -5/16 -1/4 -1/8 -9/32
4 -5/8 -5/8 -1/4 -1/8 -3/16 -5/16
5 -11/16 -7/8 -5/16 -5/32 -3/16 -11/32
6 -7/8 -3/4 -5/16 -3/8 -3/8 -7/16
7 -7/8 -5/8 -7/16 -9/32 -1/2 -1/2
8 -3/4 -7/8 -9/16 -11/32 -1/2 -3/8
9 -15/16 -3/4 -19/32 -11/32 -9/16 -3/8
10 -15/16 -7/8 -11/16 -3/8 -9/16 -5/16
11 -1 -13/16 -5/8 -1/4 -3/8 -3/8
12 -3/4 -1 -11/16 -13/32 -1/2 -3/8
13 -3/4 -3/4 -1/2 -15/32 -1/2 -11/32
14 -13/16 -5/8 -9/16 -1/2 -1/2 -11/32
15 -11/16 -5/8 -1/2 -1/2 -1/2 -3/8
16 -3/4 -9/16 -3/8 -5/16 -1/4 -3/16
17 -5/8 -9/16 -5/16 -1/4 -5/16 -3/16
18 -1/4 -1/2 -1/4 -3/16 -5/16 -1/8
19 -1/8 -1/4 -1/8 -5/32 -1/4 -1/16
217
Figures B.2 to B.10 illustrate the sweeps of the specimens at mid-height. Figures
B.11-B.15 depict the permanent lateral deformations of the specimens at midheight,
measured after the complete removal of the applied load while Figures B16-B.20 show
the permanent angles of twist of the specimens along the beam length. The sinusoidal
curves, obtained from Equations B.1 and B.2, are included in Figures B.2-B.20 to
compare the initial lateral bow and the permanent deformed shape of the centerline of
each beam to the sinusoidal curve.
 z 
u  z   uo  sin   (B.1)
 L 
 z 
  z   o  sin   (B.2)
 L 
where z = the longitudinal distance from the end of a beam, L = the total length of the
beam; u(z) and uo = sweep of the beam at mid-height, at a distance z from the end and at
mid-span respectively; φ(z) and φo = the angles of twist at a distance z and at mid-span,
respectively. Figures B.11-B.20 reveal that the permanent deformations of a buckled
beam are in close agreement with the sinusoidal curve which implies that the buckled
shape of a beam can be best approximated by the sinus function.
218
Figure B.2 - Sweep at midheight of B18-1
219
Figure B.4 – Sweep at midheight of B30
Figure B.5 - Sweep at midheight of B36
220
221
Figure B.9 - Sweep at midheight of B36L-1
222
Figure B.10 - Sweep at midheight of B36L-2
Figure B.11 – Permanent lateral deformation at midheight of B30
223
Figure B.12 – Permanent lateral deformation at midheight of B44-1
Figure B.13 – Permanent lateral deformation at midheight of B44-2
224
Figure B.14 - Permanent lateral deformation at midheight of B44-3
Figure B.15 - Permanent lateral deformation at midheight of B36L-1
225
Figure B.16 – Permanent torsional rotations of B30
Figure B.17 – Permanent torsional rotations of B44-1
226
227
Figure B.20 – Permanent torsional rotations of B36L-1
228
APPENDIX C
MIDSPAN STRAIN DISTRIBUTIONS OF THE BEAMS AT

DIFFERENT LOAD LEVELS
The present appendix presents the longitudinal strain distributions along the depth of the
midspan section of each of the second set of specimens. As explained in Chapter III, in
the second set of tests longitudinal strains were measured continuously throughout the
tests, through strain gages attached on the convex and concave faces of the beams at mid-
span. Convex and concave faces of a beam are at equal distances from the mid-width of
the beam. The minor axis of the cross section of a beam coincides with the vertical
centroidal axis, if the beam section is symmetric about the midwidth. Longitudinal strains
originating from out-of-plane bending increase from zero at the minor axis to a maximum
at the outermost fibers of the section. In other words, the minor axis of a section is only
strained by major-axis bending because the lateral bending stresses vanish at minor axis.
The specimens of the present study were designed symmetrically about the
midwidth. Therefore, the minor axis of each specimen was coincident with the vertical
centroidal axis, and thus, the concave and convex faces of the beam were at equal
distances from the minor axis, assuming that the beams were perfectly symmetric about
the vertical centroidal axes, as designed. Accordingly, the compressive strains from
lateral bending on the concave faces were equal to the tensile strains from lateral bending
on the convex faces of the beams at mid-span. The longitudinal strain distributions along
the minor axes of the specimens, which originate solely from in-plane bending, were
obtained by averaging the longitudinal strains measured on the convex and concave faces
229
Figure C.1 – Loads and lateral deflections corresponding to the strain distributions in
Figures C.2 to C.4
of the beams. The longitudinal strains distributions on the convex and concave faces and
along the minor axes of the beams were illustrated in the following figures for different
load levels along the tests. The applied loads and lateral centroidal deflections
corresponding to the strain distributions in the figures are shown on the load-lateral
deflection curves of the specimens (Figures C.1, C.5, C.8, C.12 and C.15).
230
Figure C.2 – Midspan strain distributions of B44-1 at the initial stages of loading
231
Figure C.3 – Midspan strain distributions of B44-1 close to buckling
232
Figure C.4 – Midspan strain distributions of B44-1 after buckling
Figure C.5 – The Loads and lateral deflections corresponding to the strain distributions in
Figures C.6 and C.7
233
234
Figure C.7 – Midspan strain distributions of B44-2 at the initiation of buckling
Figure C.8 – The Loads and lateral deflections corresponding to the strain distributions in
Figures C.9 to C.11
235
236
Figure C.10 – Midspan strain distributions of B44-3 at different load levels
237
Figure C.11 – Midspan strain distributions of B44-3 close to buckling
238
Figure C.12 – The Loads and lateral deflections corresponding to the strain distributions
in Figures C.13 and C.14
239
Figure C.13 – Midspan strain distributions of B36L-1 at the initial stages of loading
240
Figure C.14 – Midspan strain distributions of B36L-1 close to buckling
241
Figure C.15 – The Loads and lateral deflections corresponding to the strain distributions
in Figures C.16 to C.18
242
Figure C.16 – Midspan strain distributions of B36L-2 at the initial stages of loading
243
Figure C.17 – Midspan strain distributions of B36L-2 at different load levels
Figure C.19-C.21 illustrate the depthwise strains on the convex and concave faces
of Specimen B44-1 at different load levels. The depthwise strains at mid-span of B44-1
were measured to determine the exact state of stress at mid-span and to detect any
distortions in the beam during the test. The measured strain values indicate that the
depthwise strains did not reach significant levels prior to buckling. The relatively higher
values of depthwise strains in the post-buckling stage (Figure C.21) are related to the
excessive out-of-plane deformations after buckling.
244
Figure C.18 – Midspan strain distributions of B36L-2 prior to and after buckling
245
Figure C.19 – Depthwise strains along the midspan section of B44-1 at the initial stages of loading
246
Figure C.20 – Depthwise strains along the midspan section of B44-1 at the final stages of loading
247
Figure C.21 – Depthwise strains along the midspan section of B44-1 during unloading
248
APPENDIX D
EXPERIMENTAL LOAD-DEFLECTION PLOTS OF THE

SPECIMENS
This appendix is a collection of the experimental load-deflection plots of the specimens.
The load-lateral (out-of-plane) deflection, load-vertical (in-plane) deflection and the
torque-twist plots of the specimens are presented in the appendix.
Each of Figures D.1 to D.10 shows the lateral deflections of a specimen,
measured at different points along the depth of the beam at mid-span. The depths of the
measurement points are also shown on the figures for the sake of comparison. As
previously explained in Section 3.1.3 of this dissertation, the goal of measuring the lateral
deflections of the beams at different depths was to evaluate the torsional rotations and
distortions in the beams.
In Figures D.11 and D.12, the load-lateral centroidal deflection curves of the
companion beams are compared for the Specimen Groups B44 and B36L, respectively.
The load-lateral deflection behavior of a reinforced concrete beam is greatly influenced
by its initial lateral imperfections. Due to the significant influence of sweep on the
stability, the centroidal sweep of each beam at midspan is illustrated on the respective
load-lateral deflection curve.
Each of the load-deflection curves in Figures D.1 to D.12 does not start from the
origin. The initial deflection value of each curve (the deflection at zero load) corresponds
to the initial lateral deformation of the beam at the particular depth. Including the initial
lateral imperfections in the plots was important particularly in Figures D.11 and D.12 to
illustrate the influence of sweep on the buckling behaviors of the companion beams. In
249
Section 6.3 of this dissertation, the effects of sweep on the load-lateral deflection
behaviors of the companion beams in Specimen Groups B44 and B36L were explained.
Figure D.1 – Out-of-plane deflections of B18-2 at midspan
250
Figure D.4 – Out-of-plane deflections of B30 at midspan
251
Figure D.5 – Out-of-plane deflections of B36 at midspan
252
253
Figure D.9 – Out-of-plane deflections of B36L-1 at midspan
Figure D.10 – Out-of-plane deflections of B36L-2 at midspan
254
Figure D.11 – Lateral centroidal deflections of Beams B44 at midspan
Figure D.12 – Lateral centroidal deflections of Beams B36L at midspan
255
Figures D.13 to D.21 illustrate the load-vertical deflection curves of the
specimens. The plots also include analytical load-deflection curves, obtained by using the
cracked moment of inertia and two different effective moment of inertia expressions in
Equation (D.1), which gives the in-plane deflections at midspan of a simply-supported
beam, subjected to a concentrated load at midspan.
P  L3
vc  (D.1)
48  EI x
where P is the applied load; L is the span length; EIx is the in-plane flexural rigidity.
Two of the analytical curves in each figure correspond to the effective moment of
inertia expressions proposed by Branson (1963) and Bischoff (2005). The third analytical
curve, on the other hand, corresponds to the cracked moment of inertia, which is the
moment of inertia of a beam section when the entire tension zone of the section is
rendered ineffective in resisting bending moments due to flexural cracking. The effective
moments of inertia according to Branson (1963) and Bischoff (2005) and the cracked
moment of inertia are calculated from Equations (D.2), (D.3) and (D.4), respectively.
 M cra 
3
  M 3 
I eb     I g  1      I cr
cra
(D.2)
M  M
 a    a  
 M cra  1   M cra   1
2 2
1
    1     (D.3)
I ebi  M a  I g   M a   I cr
1
 b  c 3  n  As  d  c 
2
I cr  (D.4)
12
where Ig is the gross moment of inertia (Equation D.5); Ma is the maximum in-plane
256
bending moment in the beam; Mcra is the cracking moment of the beam; c is the neutral
axis depth from the compression face when all fibers in the compression zone are stressed
below the elastic limit of concrete; b is the width of the beam; As is the total cross-
sectional area of the longitudinal reinforcement; n is modular ratio of steel to concrete.
1
Ig   b  h3 (D.5)
12
where h is the height of the beam.
In Specimens B22 and B30, restrained shrinkage cracking was detected. Based on
the studies of Scanlon and Bischoff (2008), the term Mcra in Equations (D.2) and (D.3)
was replaced with 2Mcra/3 to account for the reduction in the effective moments of inertia
of B22 and B30 due to the presence of shrinkage cracks in concrete.
The experimental load-deflection curves of Beams B18-2, B36, B44 and B36L
are in close agreement with the analytical curves corresponding to the effective moments
of inertia proposed by Branson (1963) and Bischoff (2005). The load-deflection
behaviors of Beams B22 and B30, nonetheless, are not closely estimated by Equation
(D.1) when effective moment of inertia is used in the equation. Figures D.14 and D.15
indicate that the initial linear portions of the experimental load-vertical deflection curves
of Specimens B22 and B30 are coincident with the analytical line corresponding to the
cracked moment of inertia (Icr), most probably due to shrinkage cracking of concrete.
Figures D.22 to D.26 illustrate the torque-twist curves of the test specimens. The
ordinate axes represent the maximum torsional moment in a beam (Tmax), meaning the
torque at the beam ends, while the x-axes represent the twist at mid-span (θ), calculated
by dividing the angle of twist of the beam at midspan to the longitudinal distance from
257
support to mid-span. The red line in each figure indicates the maximum torsional moment
(Tb) in the beam at the instant of buckling.
Figure D.13 - In-plane deflections of Beam B18-2 at midspan
Figure D.14 - In-plane deflections of Beams B22 at midspan
258
Figure D.15 - In-plane deflections of Beam B30 at midspan
Figure D.16 - In-plane deflections of Beam B36 at midspan
259
Figure D.17 – In-plane deflections of B44-1 at midspan
260
Figure D.20 – In-plane deflections of B36L-1 at midspan
261
Figure D.21 – In-plane deflections of B36L-2 at midspan
Figure D.22 –Experimental torque-twist curve of Specimen B44-1
262
263
Figure D.25– Experimental torque-twist Curve of Specimen B36L-1
Figure D.26– Experimental torque-twist curve of Specimen B36L-2
264
APPENDIX E
METHOD FOR THE EVALUATION OF THE CENTROIDAL

DEFLECTIONS AND ROTATION OF A BEAM
This appendix presents the approach presented by Zhao (1994) and extended by Stoddard
(1997), which was used in the present study to convert the coupled deflection
measurements from the potentiometers into in-plane and out-of-plane deflections and
rotations at the shear center (centroid in rectangular sections).
The direction at which the beam buckles changes the geometric relations.
Therefore, the equations given by Stoddard (1997) were modified to account for the
direction of buckling. In the present study, a beam experiencing out-of-plane
deformations towards the lateral potentiometers is assumed to buckle in positive direction
(Figure E.1). Conversely, buckling away from the lateral potentiometers is defined as
buckling in negative direction. Equations corresponding to the both directions of buckling
are presented here.
At the beginning of the test, two potentiometers, T and B (denoting the top and
bottom potentiometers, respectively) were positioned horizontally while a third
potentiometer, V (denoting the vertical potentiometer) was positioned vertically as shown
in Figure E.1.
uc, vc and φc are the out-of-plane and in-plane deflections and the angle of twist at
shear center, respectively. To, Bo and Vo are the initial string lengths while Tf, Bf and Vf
are the final string lengths of potentiometers T, B and V, respectively. The lateral and
vertical deflections of the point Bp are denoted as Bx and By, respectively. Using the
265
Figure E.1 – Potentiometer configuration in the test
Pythagorean theorem for triangles 1 and 2 (Figure E.1), Equations (E.1) and (E.2) are
obtained.
 Bo  Bx 2  By2  B 2f (E.1)
V 
2
O  By  Bx2  V f2 (E.2)
When the beam buckles in negative direction, Equation (E.3) should be used instead of
Equation (E.1) while Equation (E.2) remains unchanged.
 Bo  Bx 2  By2  B 2f (E.3)
266
The solution of Equations (E.1) and (E.2) yields two sets of solution, (Bx1, By1) and (Bx2,
By2).
Bo  A1  Vo  A2
Bx1  (E.4)
A3
Vo  A4  Bo  A2
B y1  (E.5)
A3
Bo  A1  Vo  A2
Bx 2  (E.6)
A3
Vo  A4  Bo  A2
By 2  (E.7)
A3
where,
A1  Bo2  Vo2  B 2f  V f2 (E.8)
A2 
  
 Bo4  B 4f  Vo4  V f4  2  Bo2  B 2f  Vo2  V f2  2  B 2f  Vo2  V f2 
(E.9)
2  Vo2  V f2

A3  2  Bo2  Vo2  (E.10)
A4  Bo2  Vo2  B 2f  V f2 (E.11)
Similarly, the solution of Equations (E.2) and (E.3) yields,
 Bo  A1  Vo  A2
Bx 3  (E.12)
A3
Vo  A4  Bo  A2
By3  (E.13)
A3
267
 Bo  A1  Vo  A2
Bx 4  (E.14)
A3
Vo  A4  Bo  A2
By 4  (E.15)
A3
After calculating Bx and By, the angle of twist in the beam can be obtained from
the unbuckled and buckled configurations of the beam. In Figure E.2, edges of triangle 1
are determined from geometry. Using the Pythagorean theorem for the triangle, Equation
(E.16) is developed.
2
x   
2 c  
 B  B  b  cos   h  sin   b  cos  
 o c 2 c 
   
2 (E.16)
     
  By  b 2  sin c  h  1  cos c   b 2  sin c 
 
 T f2
Equation (E.16) can be simplified to become
2
 Bo  Bx  h  sin c 2   B y  h  1  cos c    T f2 (E.17)
When the beam buckles in the negative direction, Equation (E.17) changes to Equation
(E.18).
2
 Bo  Bx  h  sin c 2   B y  h  1  cos c    T f2 (E.18)
The solution of Equation (E.17) yields
c1  a tan 2   a a a
, 

 1 C  D  E 1 Ga  h  B y  Da  Ea  

(E.19)
 2 Fa 2 Fa  Da 
268
Figure E.2 – Angle of twist calculations
c 2

 a tan 2  1 
Ca  Da  Ea 1 Ga
, 
  h  By  
Da  Ea 
 (E.20)
 2 Fa 2 Fa  Da 
where,
Ca  Bo2  B y  2  h  Bo  Bx  2  h3  4  By  h 2  h  Bo2  h  Bx2

(E.21)
 By3  3  h  By2  By  T f2  2  Bo  Bx  By  h  T f2  Bx2  By
2
Da    Bo  Bx  (E.22)
269
  B 4  4  B  B3  6  B 2  B 2  2  B 2  B 2  2  B 2  T 2 
 x o x o x x y x f 
 
 4  h  Bx  By  4  Bo  Bx  By  4  Bo  Bx  T f
2 2 2

 
Ea   8  h  Bo  Bx  By  4  Bo3  Bx  4  h  By  T f2  (E.23)
 
 4  h  Bo2  By  2  Bo2  By2  4  h  B y3  2  B y2  T f2 
 
 4  h 2  T 2  4  h 2  B 2  2  B 2  T 2  B 4  B 4  T 4 
 f y o f o y f 
 

Fa  h 2  By2  2  By  h  Bo2  Bx2  2  Bo  Bx  h  (E.24)
 4  h 2  B  B  2  h 2  B 2  2  h 2  B 2  B 4 
 o x o x o

  B 4  4  B3  B  6  B 2  B 2  B 2  B2 
 x o x o x o y 
 2 2 2 3 2 2
Ga   2  h  Bo  B y  Bo  T f  4  Bo  Bx  Bx  B y  (E.25)
 2 2 2 2

 2  h  Bx  B y  Bx  T f  2  Bo  Bx  B y 
 
 4  h  Bo  Bx  B y  2  Bo  Bx  T f2 
 
It is to be noted that the roots of Equation (E.18) can be obtained by changing the signs of
the terms containing Bx in Equations (E.21) to (E.25).
Finally, the lateral and vertical displacements of the shear center (uc and vc) are
determined from the following geometric relations in terms of the angle of twist at the
shear center (φc) and the lateral and vertical displacements of point Bp (Bx and By):
h b
uc  Bx   sin c  1  cos c  (E.26)
2 2
h b
vc  By   1  cos c   sin c (E.27)
2 2
270
h b
uc  Bx   sin c  1  cos c  (E.28)
2 2
h b
vc  By   1  cos c   sin c (E.29)
2 2
Equations (E.26) and (E.27) are valid when the beam buckles in the positive direction
(Figure E.3) while Equations (E.28) and (E.29) are used when the beam buckles in the
negative direction.
For each set of (Bx, By), two different twist angles are obtained according to
Equations (E.19) and (E.20). Similarly, two different sets of out-of-plane and in-plane
centroidal deflections are calculated using Equations (E.26) to (E.29) for each set of (Bx,
By). Since two different sets of roots are obtained by solving Equations (E.1) and (E.2),
there are four different sets of deflection and rotation values of the centroid. To choose
the correct solution set, each set was compared to the deflection measurements taken by
the string potentiometers. The solution set in closest agreement with the experimental
data was chosen.
271
Figure E.3 – Centroidal deflection calculations
272
APPENDIX F
CRITICAL MOMENT CALCULATIONS OF THE SPECIMENS
This appendix presents the procedures used in the critical moment calculations of the
specimens. Each section presents the critical load calculations according to one of the
methods described in Chapter VII. The equations used in the calculations are shown on
the left halves of the following pages. On the right halves of the pages, on the other hand,
the equations and the meaning of the terms used in the equations are explained.
The critical and ultimate load calculations presented in Sections F.2 to F.5 require
an iterative procedure because of the interdependence of the variables. The iterative
procedure was carried out through the programming tools of Mathcad 14.0 (2005). The
programs used in the iterative procedure are given in Section F.2.
In this appendix, εc denotes the extreme compression fiber strain of a beam at
midspan.
F.1 Critical Load Calculations Assuming that a Reinforced Concrete Beam is an

Elastic and Homogeneous Body, Free from Cracking
Lateral flexural rigidity of the

1 
Beh  Ec    b3  h  entire beam section with elastic
 12  material behavior
Elastic modulus of concrete
Ec
obtained from cylinder tests
Width, depth and span length of the

b, h, L
beam, respectively
1  192 b  1 (2n  1) h  Coefficient for St. Venant’s

 c   1     tanh 
3  5
 h n0 (2n  1) 5
2b  torsional constant
273
Ec
Gc  Modulus of rigidity of concrete
2  (1   )
Poisson’s ratio of concrete from

material tests
Uncracked torsional rigidity of the
3
(GC )u  c  b  h  Gc beam according to St. Venant’s
theory
C1  e Beh 
M el   1  1.74    B   GC u Critical moment of a beam
C2  L 

L  GC u  eh
The loading factor for a beam
C1  4.23 loaded with a single concentrated
load at midspan
The effective length factor
C2  1.00 accounting for the simple support
conditions in and out of plane
The expression accounting for the
 e Beh  destabilizing effect of the load,
1  1.74  


L  GC u  applied above the centroid of the
beam section
4 The critical load of a beam by also

Pel    M el  M s  taking into account the self-weight
L of the beam
Bending Moment at Midspan

Ms Originating from the Self-Weight
of the Beam
F.2 Critical Load Calculations according to Hansell and Winter’s (1959)

Formulation
The critical moment calculations according to Hansell and Winter (1959) require an
iterative procedure. Using an analogy with the tangent modulus theory in inelastic
buckling of columns, Hansell and Winter (1959) stated that the secant modulus of
elasticity of concrete corresponding to the strain at the extreme compression fibers is the
modulus of the compression zone of a beam section in bending. Since the secant modulus
274
(Esec) depends on the extreme compression fiber strain (εc), Esec was denoted as a function
of εc in the calculations, Esec (εc).
According to Hansell and Winter (1959), the lateral bending rigidity (Bhw) and the
torsional rigidity (Chw) of a reinforced concrete beam is provided by the compression
zone only. Ignoring the rotations in the neutral axis of a section due to the twisting
rotations in the beam, the compression zone is a rectangular area (bxc).
The neutral axis depth (c) of the beam section and the strain at the extreme
compression fibers (εc) depend on the critical moment (Mhw). To calculate Mhw, the lateral
bending rigidity (Bhw) and the torsional rigidity (Chw) of a beam are needed. Since Bhw and
Chw are functions of c and εc, there is interdependence between Mhw and c, εc. To calculate
Mhw, c and εc, programming tools of Mathcad 13.0 (2005) were used. c and εc were
obtained from the following programs:
c  for  c  0.0001  0.0002  0.0035
c c  0.002in

As Es  s c   c
while c 
 
0.5 fc  c b
 10
5
 
in  ( c  100in) if  s c   c   y
c  c  0.0001in
As fy
otherwise
 
0.5 fc  c b
 c
   
break if Mhw  c  c  As Es  s c   c  d    10lbin if  s c   c   y
 3
 
 c
 
break if Mhw  c  c  As fy d    10lbin if  s c   c   y
 3
 
return c
275
 c  for  c  0.0001  0.0002  0.0035
c c  0.002in
 
As Es  s c   c
while c 
 
0.5 fc  c b
 10
5
 
in  ( c  100in) if  s c   c   y
c  c  0.0001in
As fy
otherwise
 
0.5 fc  c b
 c
   
break if Mhw  c  c  As Es  s c   c  d    10lbin if  s c   c   y
 3
 
 c
 
break if Mhw  c  c  As fy d    10lbin if  s c   c   y
 3
 
return  c
The expressions used in the programs are explained as follows:
Strain in the centroid of the tension

 s  c,  c  reinforcement, which is a function
of c and εc
 
f c'     c 
fc  c    o  Stress at the extreme compression
k 
 c  fibers
 1   
 o 
1 0.00689  f c' The factor ‘0.00689’ is used to

 and k  convert the compressive strength in
f c' 20
1 psi to MPa, since the stress-strain
 o  Ec equation is given in terms of MPa.
The above stress-strain relationship is the stress-strain model proposed by Carreira and
Chu (1985) for high-strength concrete. In Section 2.2 of this dissertation, Carreira and
Chu’s (1985) model was shown to be in perfect agreement with the experimental stress-
strain curves of concrete used in the specimens. Therefore, the above equation was used
in the critical moment calculations to link the strains in concrete to the stresses.
276
Yield strain and yield stress of the
εy, fy reinforcing steel, determined from
material tests
Modulus of elasticity of the
Es reinforcing steel, determined from
material tests
 4.23 
 L  Bhw   c , c    GC hw   c , c  
M hw   c , c      Critical moment, which is a
e Bhw   c , c    function of c and εc
 1  0.74    


L  GC  hw
  c , c   
 
The lateral bending rigidity
b3  c
Bhw   c , c   Esec   c   expression proposed by Hansell
12 and Winter (1959)
Esec   c   b3  c  b   The torsional rigidity expression

2
 GC hw   c , c     1  0.35    according to Hansell and Winter
2  1    3  d  
(1959)
The secant modulus is the slope of
fc  c  the line connecting the point (εc, fc)
Esec   c  
c on the stress-strain curve to the
origin
4
Phw    M hw  M s  The critical load of the beam
L
The programs run until there is a negligible difference between the bending
moment obtained from the critical moment expression [Mhw (εc, c)] and the bending
moment obtained from the stress distribution in the cross-section. The strain
measurements in the experiments indicated that all compression fibers in the specimens
were stressed within the elastic limit of concrete (elastic lateral torsional buckling).
Therefore, bending moment resistance of the beam section was calculated based on a
triangular stress distribution in the compression zone of the section.
277
F.3 Critical Load Calculations according to Sant and Bletzacker’s (1961)
Formulation
When calculating the critical load of a beam from the formula proposed by Sant and
Bletzacker (1961), an iterative procedure is needed. Therefore, the programs for c and εc,
shown in Section F.2, are used in the critical moment calculations based on Sant and
Bletzacker’s (1961) formulation.
Sant and Bletzacker (1961) suggested that the reduced modulus of elasticity (Er)
corresponding to the extreme compression fiber strain (εc) is the modulus of a beam
section at the instant of buckling. The reduced modulus theory assumes that a portion of
the beam (the convex side) undergoes unloading while the remaining portion of the beam
(the concave side) is further loaded when the beam buckles. The lateral bending rigidity
(Bsb) and the torsional rigidity (Csb) expressions proposed by Sant and Bletzacker (1961)
include the reduced modulus of elasticity (Er) as the material term.
The following discussion presents the equations proposed by Sant and Bletzacker
(1961) and important details from the calculation procedure:
 4.23 
 L  Bsb   c , c    GC  sb   c , c   Critical moment expression
M sb   c , c     
proposed by Sant and Bletzacker
e Bsb   c , c   
 1  3.48     (1961)
 L  GC  sb   c , c   
 
The expression accounting for the
 e Bsb   c , c   location of the load application point
1  3.48    with respect to the centroid of the

 L  GC sb  c , c   beam section, given by Sant and
Bletzacker (1961)
b3  d
Bsb   c   Er   c   expression proposed by Sant and
12 Bletzacker (1961)
278
Er   c  b3  d The torsional rigidity expression
 GC sb   c    proposed by Sant and Bletzacker
2  1    3 (1961)
Reduced modulus of elasticity of
4  Ec  Etan   c  concrete which is a geometric
Er   c   average of the elastic modulus (Ec)
 
2
Ec  Etan   c  and the tangent modulus of elasticity
[Etan(εc)] corresponding to the
extreme compression fiber strain
Tangent modulus of elasticity is the
d slope of the line tangent to the stress-
Etan   c    f c   c   strain curve at the point
d c 
corresponding to the extreme
compression fibers.
F.4 Critical Load Calculations according to Massey’s (1967) Formulation
The iterative procedure explained in Section F.2 is used in the critical moment
calculations based on the formula proposed by Massey (1967). Similar to Hansell and
Winter (1959), Massey (1967) used the secant modulus theory. However, Massey (1967)
also included the contributions of the longitudinal and shear reinforcement of a beam to
the lateral bending and torsional rigidity expressions. The following discussion presents
the equations proposed by Massey (1967) and important details from the calculation
procedure:
 4.23 
 L  Bm   c , c    GC m   c , c  
M m   c , c      Critical moment, which is a function
m c    of c and εc
e B  , c 
 1  1.74   

 L  GC m   c , c   
 
b3  c
Bm   c , c   Esec   c    Es  I sy expression proposed by Massey
12 (1967)
279
The contribution of the longitudinal
Es  I sy reinforcement to the lateral bending
rigidity. When steel yields, Es = 0.
 
   b3  h  G '     
 c c c

1  The torsional rigidity expression
 GC m  c     Gs  Gc'   c    bs3  ts   proposed by Massey (1967)
3 
   b12  d1  Ao  Es 
 
 2 2 s 
Esec   c  The reduced modulus of rigidity of
Gc'   c  
2  1    concrete
1  Contribution of the Longitudinal

 Gs  Gc'   c    bs3  ts Reinforcement to the Torsional
3  Rigidity
  b12  d1  Ao  Es Contribution of the Shear
Reinforcement to the Torsional
2 2  s Rigidity
Width and thickness of the
bs , ts longitudinal reinforcement layer,
respectively, as illustrated in Figure
1.18
Width and depth of the cross-sectional
b1 , t1 area enclosed by a closed stirrup,
respectively (Figure 1.18)
Cross-sectional area of one leg of a
Ao, s stirrup and spacing of the stirrups,
respectively
γ a constant defined by Cowan (1953)
F.5 Ultimate Load Calculations according to the Proposed Method
Different from the other methods, the method proposed in the present study accounts for
the reduction in the ultimate load of a beam due to sweep. First, the critical moment of a
reinforced concrete beam is calculated using the lateral bending and torsional rigidity
expressions proposed in the present study. The critical load corresponds to the
280
geometrically perfect configuration of the beam. Next, the limit load of the imperfect
beam is calculated by reducing the critical load an amount equal to the influence of the
sweep on the load-carrying capacity. The equations used in the proposed method are as
follows:
 4.23 
 L  Bo   c , c    GC o   c  
  Critical moment, which
M p  c , c    
p c   
e B  , c  is a function of c and εc
 1  1.74   
 L  GC o   c   
  
 
 
 b3  c 1   Esec   c   Ec  The lateral bending
Bo   c , c     2   rigidity expression
 12    M cra   c    2  proposed in the present
 1      1 
 Mp  h  study
   
 E     Ec   b3  h  b  The torsional rigidity

(GC )o   c    sec c   1  0.63    expression proposed in
 4  1     3  h  the present study
M cra Cracking moment of

the beam
The factor accounting
2 / 3 in the presence of shrinkage cracks for the reduction in the
 lateral bending rigidity
1 in the absence of shrinkage cracks in the presence of
shrinkage cracks
Critical load
4 corresponding to the

Pp  M p  M s   L
geometrically perfect
configuration of the
beam
Limit load of the beam
Pult  Pp 

uto  48  Ec  I y  with a sweep of uto at
3
sin(ult )  L the top of the beam at
midspan
281

uto  48  Ec  I y  Reduction in the
buckling load due to
sin(ult )  L3 sweep
1 Second moment of area

I y   b3  h of the beam section
12 about the minor axis
The angle of twist of
the beam at midspan at
the instant when Pult is
ult reached. A value of
0.60 degrees was found
to be appropriate for
most of the beams.
The above equations indicate that the rigidity expressions [Bp and (GC)p] and the
critical moment (Mp) are interdependent. Therefore, the iterative approach, summarized
in Section F.2, is used to calculate the neutral axis depth (c) and the extreme compression
fiber strain (εc) at midspan at the instant when buckling initiates. Then, the critical
moment (Mp) and the limit load (Pult) are obtained.
282
APPENDIX G
CONSTRUCTION DETAILS OF THE SPECIMENS
This appendix presents some specific details about the specimens of the
experimental program.
G.1 Compression Reinforcement
Lateral torsional buckling arises from the differential behaviors of the tension and
compression sides of a beam. The compression side of a beam is subjected to
compressive stresses from in-plane bending. When the compressive stresses reach critical
levels, the compression side of the beam buckles out of plane. The tension side of the
beam, on the other hand, tends to remain stable. The out-of-plane deformations of the
compression side cause the tension side to deform out of plane due to the integrity of the
beam. However, the out-of-plane deformations of the tension side are much smaller than
the deformations of the compression side as a result of the stabilizing effect of the tensile
stresses from in-plane bending. The differential out-of-plane deformations along the
depth of the beam result in the rotation of the beam about its longitudinal axis. Hence,
lateral torsional buckling creates out-of-plane deformations and torsional rotations in a
beam.
The stresses in the compression side of a beam are the main cause for lateral
torsional buckling. Increasing the out-of-plane bending rigidity of the compression side
can restrain the excessive lateral deformations of the compression side, which can indeed
prevent lateral torsional buckling. Compression reinforcement contributes to the lateral
283
bending rigidity of the compression side. Konig and Pauli (1990) indicated
experimentally that the compression reinforcement significantly increases the buckling
load of a reinforced concrete beam and decreases the out-of-plane deflections of the
compression side at buckling. Considering the stabilizing effect of the compression
reinforcement, the specimens of the present experimental program did not contain
compression reinforcement to ensure that the beams failed in lateral torsional buckling.
The shear reinforcement of the specimens was composed of two layers of welded
wire reinforcement (WWR), separated by the longitudinal reinforcing bars (Figure G.1).
Due to the lack of compression reinforcement in the beams, spacers were needed to
maintain the distance between the WWR sheets in the compression side. For this purpose,
spacers cut from reinforcing bars were placed between the WWR sheets (Figure G.1).
The lengths of the spacers were smaller than the development lengths of the reinforcing
bars, from which the spacers were cut.
G.2 Tilt-up and Lifting Mechanisms of the Specimens
The concrete beams of the present experimental program were cast on their sides to
facilitate the mechanical vibration of concrete and to ensure the spread of concrete into
the entire form, flowing around the congested reinforcement. A mechanism was needed
to tilt up the beams, leaning on their sides, and move them to the test setup using the
crane.
Two different lifting systems were used in the two stages of the experimental
program. Each of the lifting points in the first set of beams consisted of a headed cast-in-
place anchor embedded 7 inches into concrete (Figure G.2). Two lifting points in Beams
284
Figure G.1 – Reinforcement in Specimen B36
Figure G.2 – The Lifting mechanism in the first set of beams
285
B18 and B22 and four lifting points in Beams B30 and B36 provided adequate shear
capacity to tilt up the beams and adequate tensile capacity to lift the beams.
The beams were lifted through cables attached to the beams at the lifting points.
An important consideration for a beam hanging from cables is the angle of inclination of
the cables lifting the beam. Stratford and Burgoyne (1999) found out that the buckling
load of a concrete beam increases as the angle of inclination of the cables increase and
the cables approach to the vertical orientation. Accordingly, the test beams were lifted
with vertically-oriented cables when moving to the test setup. As shown in Figure G.3,
the specimen was connected to a steel spreader beam with vertical ropes and the spreader
beam was connected to the hook of the crane with inclined ropes.
Beams in Specimen Groups B44 and B36L were heavier than the first set of
beams (B22, B18, B30 and B36). The lifting points used in the first set of beams were not
able to provide adequate shear and tension capacity to tilt up and lift the second set of
beams. Therefore, a new lifting point was designed and used in the second stage of the
experiments.
The lifting points in the second set of beams were composed of a steel channel
and two reinforcing bars welded to the channel. The channel section was included in the
mechanism to resist the shear forces at the lifting points during the tilt-up process. The
reinforcing bars, on the other hand, provided adequate tensile capacity for the lifting
mechanisms while lifting the beams in the vertical position. A nut was welded to the
inside of the channel, so that a bolt can be fixed to the lifting mechanism when
connecting the lifting points to the crane.
286
Figure G.3 – Use of spreader beams for lifting the beams
To tilt up and lift the specimens, swift lifting eyes (also known as hoist rings)
were attached to the steel channels of the lifting mechanisms by means of high-strength
steel bolts (Figure G.4). The bail of a swift lifting eye can pivot about the base of the eye
in order to compensate for the direction of lifting. The ability of the bail to pivot about
the base made it possible to tilt up and lift the beams continuously without the need for
rearranging the lifting system between the tilt-up and lifting processes. The lifting
mechanisms used in the specimens were designed according to ACI 318-05 (2005)
Appendix D. Different failure mechanisms in the appendix were considered in the design
of the lifting points to prevent any possible damage to the beams during the tilt-up and
lifting processes.
287
Figure G.4 – The Lifting point in the second set of beams connected to the spreader beam
288
APPENDIX H
DETERMINATION OF THE EXPERIMENTAL BUCKLING LOADS

OF THE SPECIMENS
This appendix introduces the techniques developed by Southwell (1932), Meck
(1977) and Massey (1963) for determining the critical loads (limit loads in the case of
geometrically imperfect beams) of beams by analyzing their experimental load-deflection
data.
In elastic flexural buckling, Southwell plot is a common technique used to obtain
the buckling load of a member from its experimental data. In an axially loaded column,
for example, there is a linear relationship between uc/P and uc, where P is the axial load
on the column and uc is the lateral deflection at midlength of the column. The slope of the
uc/P vs. uc plot is equal to 1/Pcr, where Pcr is the critical load of the column.
Lateral torsional buckling, nevertheless, is more complicated than flexural
buckling of columns. A beam subjected to lateral torsional buckling undergoes out-of-
plane deformations and torsional rotations at the same time. Cheng and Yura (1988) used
two different types of Southwell (1932) plots to analyze the data of their lateral buckling
experiments on coped steel beams. Accordingly, uc/P was plotted against uc and φc/P was
plotted against φc. uc and φc are the lateral centroidal deflection and the angle of twist at
midspan, respectively and P is the concentrated load applied at midspan of the beam.
Cheng and Yura (1988) found out that the critical loads obtained from both plots were
almost the same for each specimen. However, the critical loads obtained from the uc/P vs.
uc plots were used, since Cheng and Yura (1988) considered the lateral deflection data in
289
the tests more reliable than the twist data due to the localized distortions in the test
beams.
Meck (1977) proposed the use of a “skewed” version of Southwell plot for lateral
torsional buckling of beams. Accordingly, uc/P should be plotted against φc and φc/P
should be plotted against uc. The geometric mean of the inverse slopes of the two plots
gives the critical load (Pcr).
Massey (1963) proposed a modification to the original Southwell (1932) plot to
be applicable to lateral torsional buckling experiments. According to Massey (1963), the
term P in the ordinates of the original Southwell (1932) plots should be replaced with P2
for the case of lateral torsional buckling. Similarly, Stratford and Burgoyne (1999) stated
that a deflection/(load)2 vs. deflection plot is more appropriate for a beam subject to
lateral torsional buckling, based on the studies of Allen and Bulson (1980).
Mandal and Calladine (2002) investigated the use of classical Southwell (1932)
plot and the modified versions of Southwell (1932) plot proposed by Meck (1977) and
Massey (1963) in lateral torsional buckling experiments and reached several important
conclusions. In their study, Mandal and Calladine (2002) analytically showed that the
lateral deflection (u) and the twist (φ) of a beam are proportional to each other after the
initial stages of loading in a lateral torsional buckling experiment. The direct coupling
between u and φ becomes more pronounced as the load is increased. Consequently, the
critical loads obtained from original Southwell (1932) plot and Meck’s (1977) “skewed”
version of the Southwell (1932) plot should not be different to a major extent.
In a Southwell (1932) plot, the data points corresponding to the initial stages of
loading do not lie on the straight line, which is formed by the majority of the data points.
290
According to Cheng and Yura (1988), the deviation of the initial points from the ultimate
straight line is caused by the initial restraints in the test setup and other experimental
errors which are more influential at the initial stages of loading when the applied load is
small. Based on the analysis of the experimental data obtained by Cheng and Yura
(1988), Mandal and Calladine (2002) found out that the deviation of the initial data points
from the eventual straight line is greater in the Massey’s (1963) version of the Southwell
(1932) plot. This is most probably due to the use of P2 instead of P in Massey’s (1963)
plots.
For the sake of illustration, Figures H.1 to H.3 illustrate the standard Southwell
(1932) plots and Meck’s (1977) and Massey’s (1963) versions of the Southwell plots,
respectively, for Specimen B44-2.
Figures H.1 and H.2 agree with the observations of Cheng and Yura (1988), who
considered the lateral deflection data in their tests more reliable. Almost all the data
points in the first plot of Figure H.1 lie on a straight line. In the second plot of Figure
H.1, nonetheless, the data points are too scattered, causing the determination of a straight
line to be more complicated. In the Meck’s (1977) version of the plots, the data points in
both plots are scattered since the twist data is used in both of the plots. The experimental
data of the other specimens showed the same characteristic. The large dispersion of the
data points makes the determination of the experimental buckling load more difficult
when the twist data is used in any version of the Southwell (1932) plot. Therefore, the use
of lateral deflection data in the original Southwell (1932) plot is considered easier and
more reliable in the determination of the experimental buckling load of a reinforced
291
concrete beam. The dispersion in the twist data might have been caused by the distortions
in the beams, particularly in the midspan region close to the point of application of load.
Figure H.1 – Southwell (1932) plots for Specimen B44-2
292
Pcr  0.274  42.26  11.6kips
Figure H.2 – Meck’s (1977) version of the Southwell (1932) plots for Specimen B44-2
293
Figure H.3 – Massey’s (1963) version of the Southwell (1932) plots for Specimen B44-2
294
The plots in Figures H.1 to H.3 agree also with the conclusions drawn by Mandal
and Calladine (2002). The critical loads obtained from the classical Southwell (1932)
plots (Figure H.1) are close to the critical load value obtained from Meck’s (1977)
version of the plots (Figure H.2). In Table H.1, the critical loads of the second set of
specimens, obtained from the classical and Meck’s (1977) version of the Southwell
(1932) plots are tabulated together with the ultimate loads measured during the tests. The
last column in Table H.1 corresponds to the critical load values obtained from the
classical Southwell (1932) plot using the lateral deflection data. The tabulated values
show that the critical loads according to the classical and Meck’s version of the
Southwell (1932) plots are in close agreement for all specimens.
Table H.1 – Critical loads from the classical and Meck’s (1977) version of the Southwell
(1932) plots for the second set of beams
Experimental Critical Load (kips)

Specimen Limit Load Meck Southwell
(kips) (1977) (1932)
B44-1 15.2 18.7 17.3
B44-2 12.1 11.6 13.6
B44-3 21.0 21.5 20.8
B36L-1 13.5 18.0 17.3
B36L-2 21.6 26.9 26.8
Figure H.3 indicates that the data points in Massey’s (1963) version of the plots
are more scattered than the classical and Meck’s (1977) versions of the plots.
Furthermore, the data points corresponding to the initial stages of loading lie further from
the eventual straight line in Massey’s (1963) version of the plots, as previously
established by Mandal and Calladine (2002).
295
Finally, Table H.1 indicates that both the classical version and the Meck’s (1977)
version of Southwell (1932) plot overpredict the limit loads of the second set of
specimens. Therefore, the limit loads of the specimens measured in the tests were used in
the analytical study to be more conservative.
296
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VITA
İLKER KALKAN
İlker Kalkan was born on July 22nd, 1981 in Mardin, Turkey. He attended Middle
East Technical University in Ankara, Turkey and received a B.S. degree in Civil
Engineering with emphasis on Construction Management and Engineering in 2004. He
earned an M.S. degree in 2006 and a Ph.D. degree in 2009 in Structural Engineering,
Mechanics and Materials from Georgia Institute of Technology, Atlanta, Georgia. He
took classes from the Daniel Guggenheim School of Aerospace Engineering at Georgia
Institute of Technology as his minor field of study in Ph.D. In his doctoral studies, he
investigated the lateral stability of rectangular reinforced concrete beams analytically and
experimentally.
304
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Lateral Torsional Buckling of Rectangular Reinforced Concrete Beams

Article in ACI Structural Journal · January 2014

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Georgia Institute of Technology

Copyright © 2009 by Ilker Kalkan

Dr. Abdul-Hamid Zureick, Advisor Dr. Bruce R. Ellingwood

Dr. Lawrence F. Kahn Dr. George Kardomateas

Dr. Kenneth M. Will

Date Approved: September 28th,2009

patience and encouragement throughout this project.

time and suggestions.

Engineering and Materials Laboratory of Georgia Institute of Technology. The author

Millard; mechanical specialist Michael R. Sorenson; senior facilities manager Andrew

help and support during the course of the experiments.

particularly at difficult times.

Financial support provided for the experiments by Georgia Department of

appreciation to Kırıkkale Üniversitesi, Turkey for providing financial support to the

LIST OF FIGURES xii

LIST OF ABBREVIATIONS xxvii

1.2 Project Objectives 4

1.3 Organization of the Study 4

1.4 Previous Studies 6

1.4.1 Review of Previous Experimental Work 7

1.4.2 Analytical Methods for Predicting Lateral Torsional Buckling 28

II SPECIMENS AND MATERIAL PROPERTIES 57

2.1.1 Specimen Descriptions 57

2.1.2 Experiment Design 58

2.2 Concrete Material Properties 62

III EXPERIMENTAL SET-UP, INSTRUMENTATION AND PROCEDURE 69

3.1 Experimental Setup 69

3.1.3 Load, Deflection and Strain Measurements 88

3.1.3.1 LVDT Strain Measurements 93

3.1.3.2 Strain Measurements through Electrical Resistance Strain

3.2 Test Procedure 98

3.3 Summary of the Test Results 99

IV LATERAL BENDING RIGIDITY OF RECTANGULAR REINFORCED

4.1 Introduction 101

4.2 Available Lateral Bending Rigidity Expressions 102

4.3 Proposed Lateral Bending Rigidity Expression 105

4.3 Influence of Shrinkage Cracking on the Lateral Bending Rigidity 121

V TORSIONAL RIGIDITY OF RECTANGULAR REINFORCED

5.1 Torsional Behavior of Reinforced Concrete Beams 131

5.2 Torsional Rigidity of Rectangular Reinforced Concrete Beams 137

5.2.1 Uncracked Torsional Rigidity Expressions 137

5.2.2 Post-Cracking Torsional Rigidity 141

5.3 Experimental Torsional Rigidities of the Test Beams 145

5.4 Proposed Torsional Rigidity Expression 151

VI CRITICAL MOMENT CALCULATIONS AND INFLUENCES OF

6.1 Introduction 153

6.2 Critical Moment Calculations 153

VII EXPERIMENTAL RESULTS AND OBSERVATIONS AND

7.1 Experimental Results and Observations 173

7.1.1 Crack Patterns of the Specimens 173

7.1.2 Experimental Results 181

7.2 Correlation of the Analytical and Experimental Results 183

VIII SUMMARY AND CONCLUSIONS 195

8.1 Summary 195

8.2 Conclusions 200

8.3 Futures Research 203