1984 the deflections of reinforced and partially
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University of WollongongResearch Online
University of Wollongong Thesis Collection University of Wollongong Thesis Collections
1984
The deflections of reinforced and partiallyprestressed concrete box beams under repeatedloadingYen Wen WongUniversity of Wollongong
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Recommended CitationWong, Yen Wen, The deflections of reinforced and partially prestressed concrete box beams under repeated loading, Doctor ofPhilosophy thesis, Department of Civil and Mining Engineering, University of Wollongong, 1984. http://ro.uow.edu.au/theses/1275
THE DEFLECTIONS OF REINFORCED AND PARTIALLY PRESTRESSED CONCRETE BOX BEAMS UNDER REPEATED LOADING
A thesis submitted in fullfilment of the requirements for the award of the degree of
DOCTOR OF PHILOSOPHY
from
THE UNIVERSITY OF WOLLONGONG
by
Yen Wen WONG B.E.
DEPARTMENT OF CIVIL AND MINING ENGINEERING 1984
iii
ACKNOWLEDGEMENTS
This study was conducted in the Department of Civil and Mining
Engineering, University of Wollongong. The author is indebted to
Dr. Y. C. Loo, his major supervisor, for the close supervision,
fruitful discussions and invaluable suggestions he has given for
many years. The author also greatly appreciates the beneficial
training in research skills given by him during the course of this
study.
The author is deeply grateful to Associate Professor R. W.
Upfold, for his guidance at many stages of this study, especially
for the generous help he gave at the most difficult stage of the
study - the setting up of the experiments.
Special acknowledgement is due to Professor C. A. M. Gray for
his encouragements given during the early period of the study.
The author also wishes to express his sincere gratitude to
the Chairman of the Department, Professor L. C. Schmidt, for the
encouragements and convenience he provided at the final stages of
the preparation of this thesis.
Much appreciation is expressed to the entire technical staff
of the Structural Laboratories for their technical and physical
assistance in the experimental work.
Acknowledgement is also made to the staff of the Wollongong
University Computing Centre, for their frequent advice on programming
and the usage of the statistical package SPSS and the document
processing package TEX. The latter package was used to compose this
thesis.
iv
Thanks are extended to Mrs. J. Fullerton for her typing of
the tables and figures in this thesis.
The author's stay at Wollongong from January 1979 to December
1982 was made possible by a Wollongong University Scholarship. He
greatly appreciates this generous grant.
Finally, the author is indebted to his family for the under
standing and suffering during the rather long period of this study.
SUMMARY
v
Several major types of civil engineering structure including
off-shore platforms and road and railway bridges are subjected to
repeated loading. Earlier studies have indicated that such loading
causes higher deflections in concrete structures than static loading.
This thesis is devoted to the study of the deflection behaviour of
reinforced and partially prestressed concrete box beams. In addition
to presenting details and results of a comprehensive experimental
programme, a simple and reliable procedure is proposed for the
analysis of concrete beam deflections under repeated loading.
Thirty reinforced and five partially prestressed concrete box
beams (of i scale) were fabricated and each tested up to 105
repetitions of load. For each beam, the repeated loading range was
kept constant with the lower and upper limits set at 30 percent
and 50 percent of the yield load to simulate the dead load and
service load respectively. For some beams 70 percent or 90 percent
of the yield load was used as the upper limit, simulating overloading
conditions. The deflections at both loading limits were measured
immediately after 1, 10, 102, 103, 104 and 105 cycles of load.
It was found that after the initial loading cycle the bending
rigidity of the beams Qience the instantaneous deflection) was not
affected by further repeated loading. The dead load deflection on
the other hand increased with increasing number of loading cycles
although the rate of increase reduced rapidly as the number of
cycles became larger. This amplification of dead load deflection by
repeated load can be seen as a parallel phenomenon to the time creep
of concrete under sustained loads. Because large deflection can
accumulate in a short time under repeated loads, the accumulated
deflection is referred to herein as the 'intensive creep' deflection.
vi
For some lightly reinforced beams the intensive creep deflection was
as high as eight times the initial dead load deflection.
The steel ratio, degree of prestressing and the maximum loading
level are found to be the important factors influencing the deflection
behaviour of concrete beams under repeated loading.
A simple procedure is presented for the analysis of the total
deflections of reinforced and prestressed concrete box beams under
repeated loading. The total deflection is obtained by summing
the instantaneous live load deflection and the accumulated dead
load deflection. The latter may be computed as the product of
the initial dead load deflection and the intensive creep factor.
Following a comparative study, the most suitable method is chosen
from 9 well-known procedures for the computation of initial dead
load deflection. Based on the experimental data of the initial
and accumulated dead load deflections an empirical formula for the
intensive creep factor is derived using statistical means. Parallel
to the logarithmic time-creep equation, the formula is considered to
be more realistic than an alternative hyperbolic model. For computing
the instantaneous live load deflection of any loading cycle, a new
equation for the effective moment of inertia is proposed. This
equation is shown to be superior to the existing formulas.
All equations recommended for use are simple and explicit. In
light of the experimental data obtained herein and those by other
researchers, comparisons are made with two other recently published
methods. It is concluded that the proposed procedure is more
versatile in that it accounts for more variables and allows for
the predictions of instantaneous deflection and permanent set in
addition to the total deflection. In most cases it also gives more
accurate results.
TABLE OF CONTENTS
TITLE PAGE i
DECLARATION ii
ACKNOWLEDGEMENT iii
SUMMARY v
TABLE OF CONTENTS vii
LIST OF TABLES xii
LIST OF FIGURES xiv
NOTATION xvii
1 INTRODUCTION 1
1.1 Background 1
1.2 The Characteristic and Application of Box Beams 4
1.3 Objectives and Scope 5
1.3.1 Objectives 5
1.3.2 Scope 5
1.3.3 Outline of thesis 6
2 REVIEW ON REPEATED LOADING TESTS OF CONCRETE BEAMS 7
2.1 General Remarks 7
2.2 Strength Fatigue 9
2.2.1 Plain concrete 9
2.2.2 Reinforcing and prestressing steel 10
2.2.3 Structural members 12
2.2.4 Conservative fatigue design
recommendations 12
2.3 Effects of Repeated Loading on Serviceability 15
viii
2.3.1 High level repeated load tests 17
2.3.2 Tests at service and other loading levels 19
2.3.3 Comments on previous tests 29
3 THEORETICAL CONSIDERATION 31
3.1 Pilot Study 31
3.1.1 Dead load deflection and Intensive Creep 32
3.1.2 Instantaneous live load deflection 32
3.1.3 Major variables for deflections under
repeated load 33
3.2 Possible Intensive Creep Model 36
3,3 Existing Equations for !r 39
3.4 The Total Deflection Equation 40
4 EXPERIMENTAL PROGRAMME 42
4.1 General Remarks 42
4.2 Design of Test Specimens 43
4.2.1 Number of beams 43
4.2.2 Design of reinforced concrete box
beams 43
4.2.3 Design of prestressed concrete box
beams 44
4.3 Beam Details and Fabrication 49
4.4 Test Equipment and Instrumentation 55
4.5 Test Procedures 60
4.6 Experimental Results 63
4.7 Observations and Analysis of Beam Behaviour 64
IX
4.7.1 General behaviour 64
4.7.2 Mechanism of deflection accumulation 66
5 INITIAL DEAD LOAD DEFLECTION 71
5.1 General Remarks 71
5.2 Existing Methods for Predicting the
Initial Deflection 72
5.2.1 Moment of inertia methods 72
5.2.2 Bilinear moment-curvature or
moment-deflection methods 76
5.3 Comparative Study 80
6 INTENSIVE CREEP FACTOR 88
6.1 General Remarks 88
6.2 Statistical Analysis 89
6.2.1 The use of SPSS package 89
6.2.2 Selection of parameters 90
6.2.3 Multiple regression 91
6.3 The Proposed Formulae 97
6.4 Effects of Main Variables on Intensive Creep 100
6.4.1 Effect of load repetitions 100
6.4.2 Effect of load level 100
6.4.3 Effect of steel ratio 101
6.4.4 Effect of prestressing 101
6.4.5 Discussion 112
7 INSTANTANEOUS LIVE LOAD DEFLECTION AND THE EFFECTIVE
MOMENT OF INERTIA 118
7.1 Repeated Loading and Beam Rigidity 118
7.2 Existing Formulae for lrep 121
7.3 The Proposed Equations for lre_ 123
^ 7.4 Comparison of Results 126
8 TOTAL DEFLECTION OF CONCRETE BOX BEAMS
UNDER REPEATED LOADS 137
8.1 The Proposed Computational Procedure 137
8.2 Other Prediction Procedures 138
8.2.1 Balaguru and Shah's method 138
8.2.2 Lovegrove and El Din's formula 141
8.3 Comparison with Experimental Data 142
8.3.1 Reinforced concrete box beams
(150 points) 142
8.3.2 Partially prestressed concrete box beams
(30 points) 142
8.3.3 Sparks and Menzies' reinforced concrete beams
(10 points) 143
8.3.4 Bennett and Dave's prestressed beams
(9 points) 143
8.3.5 Overall accuracy 144
8.4 Comparisons with Other Prediction Methods 154
9 CONCLUSIONS 162
V 9.1 Deflection Behaviour 163
9.2 The Proposed Computational Procedures 165
9.3 Versatility and Accuracy of Proposed Procedure 167
XI
9.4 Recommendations for Further Study
REFERENCES
APPENDIX III Beam Surface Strains
APPENDIX IV Variations of Maximum Crack Width
APPENDIX V Measured Total Deflections and
Moment-deflection Curves
APPENDIX VI Examples for the Use of SPSS Package
and Equations of Statistical Coefficients
Used in SPSS
APPENDIX VII Numerical Examples and Computer Programs
for the Computation of 8T
Using the Proposed Method
PAPERS PUBLISHED BASED ON THIS THESIS
168
169
APPENDIX I Equations for Computing kd, Mcr. M and M^ 181
APPENDIX II Computer Program for Designing Partially
Prestressed Concrete Box Beams 187
197
216
223
256
274
300
AUTHOR'S CURRICULUM VITAE 301
LIST OF TABLES
TABLE PAGE
la Permissible compressive stress of Grade 40 concrete 3
lb Permissible tensile stress of mild-steel reinforce- 3 ment
2 A summary of published serviceability tests on 16 concrete beams under repeated loading
3 Details of reinforced box beams 47
4 Details of partially prestressed box beams 48
5 Upper and lower limits of repeated loading moment 62
6 The comparison of measured and computed Mcr, 65 My, and Mu
7 Existing methods for predicting initial deflections 74
8a Calculated and measured initial dead load def lee- 84 tions of reinforced box beams
8b Calculated and measured initial dead load deflec- 85 tions of partially prestressed box beams
9 Deviations of calculated 8^ 86
10 Frequency distributions of calculated dd 87
11 Multiple regression of equations for V 96
12 Correlations of measured kp and computed k_ by 99 Eq. 6.10 with various P values
13 Measured instantaneous live load deflection at 128 mid span of r.c box beams
14 Measured instantaneous live load deflection at 129 mid-span of partially prestressed box beams
15 Calculated and measured irep of reinforced box 130 beams
Calculated and measured lrep of partially prestressed 131 box beams
Comparisons of Eq. 7.3 with measured l.„ of 132 reinforced concrete box beams
Deviations of calculated lrep of reinforced con- 133 crete box beams
Deviations of calculated lrep for partially pre- 134 stressed concrete box beams
Frequency distributions of calculated I of 135 reinforced concrete box beams
Frequency distributions of calculated lrep for 136 partially prestressed concrete box beams
Properties of beams tested by Sparks and Menzies 148 [1973]
Properties of beams tested by Bennett and Dave 150 C19693
Frequency distribution of the errors of total 152 deflections computed using Eq. 8.1
Prediction accuracy of Eq. 8.1 153
Statistics of correlations presented in Figs. 159 35, 36 and 37
Statistics of correlations presented in Fig. 38 161
j values for reinforced concrete beams 185
\j values of reinforced concrete beams 186
Measured total deflection of reinforced box beams 224 at mid-span
Measured total deflection of prestressed box 225 beams at mid-span
LIST OF FIGURES
PAGE
Goodman diagram for plain concrete 14
Similar effects of sustained and repeated loads 28
on deflections
Typical load-deflection diagram of beams under 34
repeated loading CWong and Loo, 1980]
Typical load-deflection diagram obtained by Ben- 34
netts and Atkins [1977]
Typical load deflection diagram obtained by 34
Snowdon [1971]
Deflection of concrete beam under repeated loading 35
Test beams series 46
Details of test beams 51
Fabrication of concrete box beams 52
A typical stress-strain curve for concrete 53
Stress-strain curve of mild-steel deformed bars 54
Stress-strain curve of hard-drawn prestressing 54
wires
XV
10 Test set-up 57
11 Repeated loading control panel of the Dartec 58
System
12 Positions of dial gauges and Demac gauges 59
13 Typical increases of strain and crack width 69
under repeated loading
14 The bond mechanism of deformed bars 70
15 Bending rigidities of reinforced or prestressed 75
concrete beams
16 Idealized bilinear moment-curvature or moment- 78
deflection curve
17 Stress and strain distribution of a cracked 79
section suggested by CP110 [1974]
18 Regression line 95
19 k versus T for Mt/My = 0.53 (R200 series) 102
20a k versus T for Mt/My = 0.5 (R300 series) 103
20b k versus T for Mt/My = 0.7 (R300 series) 104
20c k versus T for Mt/My - 0.9 (R300 series) 105
21 k versus T for Mt/My = 0,53 (R450 series) 106
XVI
22 k versus moment ratio 107
23 k versus p for Mt/My = 0.53 (R200 series) 108
24a k versus p for Mt/My = 0.5 (R300 series) 109
24b k versus p for Mt/My = 0.7 (R300 series) 110
24c k versus p for Mt/My = 0.9 (R300 series) 111
25 k versus p for Mt/My = 0.53 (R450 series) 112
26 Effect of prestressing on intensive creep factor 113
27 Comparison between measured k and k calculated 116
by Eq. 6.8
28 Comparison between measured kp and kp calculated 117
by Eq. 6.10
29 Load-deflection curve showing change in beam 125
rigidities [see Burns and Seiss, 1966]
30 Moment-deflection curve of concrete beams 125
31 Correlations of measured and computed total 146
deflections for 30 reinforced box beams
32 Correlations of measured and computed total 147
deflections for 5 partially prestressed box beams
33 Correlations of measured and computed total 149
deflections for Sparks and Menzies' reinforced
concrete beams
XV11
Correlations of measured and computed total 151
deflections for Bennett and Dave's prestressed
concrete beams
Correlations of measured total deflections and 156
computed values using author's Eq. 8.1
Correlations of measured total deflections and 157
computed values using Balaguru and Shah's method
Correlations of measured total deflections and 158
computed values using Lovegrove and El Din's
method (T>105 cycles)
Comparisons between measured total deflections 160
of reinforced and prestressed box beams (105
cycles only) and computed total deflections by
three methods
Typical box section 183
Stress distribution of under-reinforced concrete 184
beams
Increases of surface strain under repeated 198
loading
Variation of maximum crack width at maximum load 217
with number of loading cycles
Moment vs deflection 226
XV111
NOTATION
intercept of regression line
area of tensile reinforcing steel
area of prestressing steel
overall width of beam section
web width of beam section
slope of regression line
creep strain in micro-strain
creep coefficient at any time t
ultimate (in time) creep coefficient
effective depth of beam section, distance from extreme compression fibre to the centriod of tension steel
distance from extreme compression fibre to the centriod of tensile reinforcing steel
distance from extreme compression fibre the centriod of prestressing steel
napierian base = 2.7183
eccentricity of prestressing steel
modulus of elasticity of concrete
modulus of elasticity of concrete under fatigue loading
modulus of elasticity of steel
concrete compressive stress
concrete tensile stress
yield strength of prestressing steel
tensile strength (modulus of rupture) of concrete under static loading
XIX
tensile strength (modulus of rupture) of concrete under fatigue loading
yield strength of reinforcing steel
concrete cylinder strength of reinforced beams
concrete cylinder strength of prestressed beams
tensile strength of concrete calculated according to AS1480-1982
moment of inertia of cracked transformed section
cracked transformed moment of inertia calculated using Ee.T
effective moment of inertia as defined in Eq. 5.6a
effective moment of inertia calculated using lcr-p and Mcr,T
moment of inertia of gross section
effective moment of inertia at T th loading cycle
moment of inertia at first yielding of the beam
internal lever arm by the elastic theory
internal lever arm by the ultimate strength theory
intensive creep factor of reinforced beams
depth of neutral axis by the elastic theory
depth of neutral axis by the ultimate strength theory
intensive creep factor of prestressed beams
span of the beam
cracking bending moment
cracking bending moment calculated using fr T
dead load moment or the bending moment caused by the lower limit of repeated load
decompression moment of prestressed beams
XX
bending moment increment caused by the passage of live load
bending moment caused by the total load, which includes dead and live loads or the bending moment caused by the upper limit of the repeated load
yield bending moment of the beam
ultimate bending moment of the beam
moment for calculating lrep, see Eq. 7.3
modular ratio = Es/Ec
ratio of steel reinforcement = Agt/bd
equivalent steel ratio = (Ast + Asp)/bd
modification coeffcient of intensive creep factor for partially prestressed beams
effective prestressing force
slope of straight line relating k and log^T, see Eq, 6.8
Standard error of estimate
time of loading in case of sustained load
number of repeated loading cycles
an independent variable in a regression
distance from the bottom fibre to the neutral axis of concrete beam
observed value of a dependent variable
estimated value of a dependent variable
multiplication factor for deflection governed by the loading conditions
angle of bilinear moment-deflection curve at the bifurcation point
Pearson's correlation coefficient
XXI
accumulated dead load deflection after the initial and the first cycle of repeated load
accumulated dead load deflection (permanent set) caused by the initial and the previous T load cycles
initial dead load deflection
the increment of deflection caused by live load passage at the T th loading cycle
total deflection after T cycles of repeated load
initial total deflection
concrete strain, static and cyclic, in micro-strain
steel strain, in micro-strain
degree of prestressing
stress range of concrete expressed as a fraction of the compressive strength
mean fibre stress of concrete expressed as a fraction of the compressive strength
maximum fibre stress of concrete
minimum fibre stress of concrete
curvature
coefficient representing moment ratio in Eqs. 3.8, 3.9 and 5.5a
1
Chapter 1
INTRODUCTION
1.1 Background
Concrete structural members should be designed not only
for strength but also for serviceability requirements. Better
understanding of the rigidities and deformations of concrete beams
under different kinds of loading will improve the design for
serviceability. Research on serviceability of concrete structures
and structural members is becoming more and more important these
days. This is because of the increasing use of higher design stress
in materials, coupled with many significant changes in the approaches
to design.
The specifications of higher design stresses for concrete members
in many codes of practices have been justified either by better
quality control of existing materials or by the use of new. higher
strength materials.
'A comparison of the current British Concrete Structures Code with
its earlier versions as given in Table 1 is a good example showing
the progressive increases in design stresses for both concrete
and reinforcing steel. Note that the table was compiled from data
given by Scott and Glanville [1934 and 1965]. Bate [1968]. and British
Standard Institution [19721
The use of higher design stresses or higher-strength materials
in concrete structures leads to a lower-cost structure but with
more slender sections or relatively higher deformation.'
The methods used in designing concrete members have also been
changed within the past 20 years. Ultimate strength design procedure
2
has been widely accepted in many countries as a replacement for the
conventional elastic design methods. This less conservative design
method is based on better understanding of the failure mechanics of
concrete structural members. The use of ultimate strength design
method results in more accurate sections as far as the strength
capacity is concerned. However, such advance is acceptable only
if the design for serviceability is carried out with reasonable
accuracy.
Extensive theoretical and experimental studies on the deflection
of both reinforced and prestressed concrete members over many
years have led to fairly well established methods of design for
serviceability under static loads. However, the effects of repeated
loading on the rigidities and deflections of concrete members are
still scarcely understood.
Loadings on offshore structures, highway and railway bridges
and some other structures are often repetitive in nature. Repeated
loading causes higher deflections than static loads in concrete
structural members. These deflections usually include some con
siderable permanent sets which increase with the number of load
repetitions. This phenomenon has been observed by many researchers.
However, relevant experimental dataarestill rather scarce especially
for beams under large number of loading cycles. Furthermore, at
the time when this study began there was no reliable procedure for
the prediction of the deflections of concrete beams under repeated
loading. Neither was there any work done on box beams under the
same loading. It appeared that the deflections of concrete mem
bers under repeated loading deserved detailed investigations. This
thesis project was initiated with the aim of developing simple but
accurate formulae for analysing both the instantaneous deflection
and permanent set of concrete box beams under repeated loading.
3
TABLE la PERMISSIBLE COMPRESSIVE STRESS OF GRADE 40 CONCRETE
Year of the Codes
CP114-1957
CP114-1965
CPllO-1969
CPllO-1972
Max. Permissible Stress
1500 psi (10.0 N/mm2)
1830 psi (12.2 N/mm2)
2050-2340 psi (13.7-15.6 N/mm2)
2400 psi (16.0 N/mm2)
TABLE lb PERMISSIBLE TENSILE STRESS OF MILD-STEEL REINFORCEMENT
Year of the Codes
DSIR -1934
CP114-1965
Max. Permissible Stress
18000 psi (124 N/mm2)
20000 psi (138 N/mm2)
4
U2 The Characteristics and Application of Box Beams
The box section is one of the most popular cross sectional shapes
for medium span bridges,As compared to the I-sectionit has better
lateral stability and a much higher torsional rigidity.
Standardized concrete box beams are used for bridge decking
in many countries. In the United States. AASHO-PCI standard box
beams can be used with prestressed strands to span up to 30 metres
[Prestressed Concrete Institute. 1975], As discussed by Somerville
and Tiller [1970], in Britain, standard box beams having a fixed width
of 970mm but with varying depth can be used for bridges of up to
36m span.
Because of their relatively low cracking strength, the behaviour
of box beams under repeated loading can be different from beams
with thick webs. In this thesis, this characteristic is incorporated
in the development of prediction equations for deflections.
Traditionally, bending tests on concrete beams are usually
conducted on I-sections or rectangular beams. Tests on box beams
are often done under torsional or combined loading. Very few reports
can be found dealing only with flexural tests on box beams except the two
programmes reported by Drew and Leyh [1965] and Arthur and Mahgoub
[1975] in which full scale box beams were tested under statical loads
only.
1.3 Objectives and Scope
5
1.3.1 Objectives
The objectives of the present work may be summarised as follows:
a. To investigate the effects of repetitive loading on
the deflections of reinforced and partially prestressed
concrete box beams', to identify the deflection behaviour
of such beams from results of a comprehensive and
carefully planned test programme.
b. To establish computational procedures for the deflections
of concrete box beams under repeated loading which will
include the effects of all major variables and cover
concrete beams from reinforced to fully prestressed
beams.
1.3.2 Scope
The present study is limited to singly, under-reinforced beams
only. It is believed that the theory of singly reinforced beams is
fundamental to the development of a more general theory. Thus it
should be attempted first.
Similar to beams under static and sustained loading, the
derivations of prediction equations for deflections under repeated
loading are done with the aid of experimental data. A review of
published or related works has been conducted which includes about
150 papers, reports and books. Although the qualitative behaviour
of beams under repeated loading can be understood, to quantify
such behaviour a well planned test programme has to be carried
6
out. Thirty five box beams were fabricated and tested under a
large number of repeated loading cycles. The deflection results of
these tests are analysed by statistical means through which the
prediction equations are derived. The reliability of the equations
are checked. In light of the test results obtained herein and those
by other researchers, the proposed procedure is compared with two
other recently published approaches. The proposed equations are
shown to be superior.
L3.3 Outline of thesis
In Chapter 2, a historical review on fatigue strength tests
of concrete beams is presented together with a detailed survey
conducted of previous tests on the effects of repeated loading on
serviceability of concrete beams. The analytical models for the
'intensive creep' deflection (or permanent set), the instantaneous
live load deflection and the total deflection of concrete box beams
under repeated loading are expounded in Chapter 3. To bring
the proposed models to a usable stage, deflection data on both
reinforced and prestressed beams are required. In chapter 4 the
experimental programme is described in detail: the test results are
also presented. In Chapter 5 to Chapter 8 the derivation of the
proposed computational procedure is given in detail, and the accuracy
of this procedure is checked using the deflection data obtained
by the author and by other researchers. Finally, conclusions and
recommendations for further work are given in Chapter 9.
In view of the large number of tables and figures, they are
located at the end of sections (or sub-sections as convenient) in
which they are first mentioned. Tables and figures are numbered
separately but consecutively: the two separate lists are given in
page xii to xiii and xiv to xvii respectively.
7
Chapter 2
REVIEW ON REPEATED LOADING TESTS OF CONCRETE BEAMS
2.1 General Remarks
Research on concrete beams under repeated loads started as
early as in 1906 as noted by Nordby [1958] and the American Concrete
Institute [I960], About 150 technical papers and reports have been
published since then. Because repeated loading tests take so much
time and effort, the early reports are limited to studies on two
or three beams [see Berry. 19081
The effects of repeated loads on concrete beams may be divided
into two major types. Firstly, it is the effect on the strength
capacity. This is usually called strength fatigue. The second type
is the effect on the serviceability conditions of the beam, such
as deflection, cracking and effective rigidity. Strength fatigue
was most important in the early days as in earlier design methods
strength was always a governing criterion. Serviceability problems
under repeated loading were frequently neglected in most of the
early investigations. The negligence might not be a risk then as
the beams were usually stocky. However, the problems are much
more important today and can no longer be overlooked. A review
of available literature indicates that, among the previous research
work only a small portion has dealt extensively with serviceability.
In this chapter previous works on strength fatigue are briefly
described. Emphasis is given to summarizing the reports on
serviceability studies. A total of about 20 papers and reports.
which are fully or partly concerned with rigidity and deflections
under repeated loads, have been reviewed. From these publications
the properties of test specimens, characteristics of the repeated
8
test loads and the test results are summarised in this chapter
together with brief comments as necessary.
9
22 Strength Fatigue
*
Interests shown in non-metallic fatigue especially in concrete
and concrete members became evident nearly 40 years later than in
metal fatigue. Unlike metal fatigue, fatigue in concrete is much
more complex. Concrete structural members can fail in a number
of ways such as failure in the compressive zone, diagonal tensile
failure in the shear zone, fracture of reinforcing or prestressing
steel, bond failure between steel and concrete etc. The interaction
of these different failure modes is also very complicated even for
static loading conditions. Needless to say it is more difficult to
fully understand the failure mechanics under fatigue loads.
Strength fatigue tests can be divided into two groups: on
the component materials (concrete and reinforcing and prestressing
steel) and on the structural members. Fatigue tests on structural
members included models of different scales and even on real concrete
structures as reported by Rosli [1968],
Considerable variability always exists in fatigue test results
because too many variables can and do influence the behaviour of the
test specimen. Thus, theories based on these results are usually
very restrictive in applications. However, by analysing large amount
of test results some common conclusions can still be drawn in spite
of the individuality of different test cases as discussed by Nordby
[19581 Based on these conclusions some useful recommendations can
also be given to improve the design for fatigue problems [see ACI
Committee 215. 1974],
2^.1 Plain concrete
Hilsdorf and Kesler [1966] and Ople and Hulsbos [1966] found that
fatigue life of plain concrete is much dependent on the stress level
10
and stress gradient, while McCall [1958], Murdock and Kesler [1958]
and Struman. Shah and Winter [1965] suggested that the effect of
applied stress range is also significant. A higher maximum stress
level usually gives a shorter fatigue live. The S-N curve has become
a standard expression of the effect of maximum stress level on the
fatigue life of concrete. Note that the maximum stress level. S.
is often given as a percentage of the ultimate stress while the
number of loading cycles to failure. N. is in logarithmic scale. The
influence of the stress range can be represented by the so-called
Goodman diagram [see ACI Committee 215. 19741
Several other variables also affect the fatigue properties of
plain concrete including the rate of loading [see Murdock. 1965:
Awad and Hildorf. 1974: Raithby and Galloway. 1974 and Kesler. 1953],
loading history [see Hilsdorf and Kesler. 1966: Shah and Chandra.
1970], material properties and environment conditions [see Awad and
Hildorf, 19741 An individual fatigue test usually incorporates one
or two variables and neglects or limits the effects of all others.
2.2.2 Reinforcing and prestressing steel
Fatigue tests on reinforcing bars, prestressing wires and
prestressing strands are carried out on steel specimens either in
exposed condition [see Warner and Hulsbos. 1966: Jhamb and MacGregor.
1974] or embedded in concrete [see Hanson. Some and Helagson. 19741
Some test results show that steel embedded in concrete has a longer
fatigue life than the ones exposed. The other results lead to
the opposite conclusion! However, studies by MacGregor. Jhamb and
Nuttall [1971] indicate that for reinforcing bars there was little
difference for the two conditions if good bond is provided between
steel and concrete. ACI Committee 215 [1974] suggested on the other
hand that prestressing steel embedded in concrete with a poor bond
11
would have a lower fatigue strength than the strength of some wires
exposed in the air.
Most fatigue tests on reinforcing steel used specimens cut from
either hot rolled or cold- worked deformed bars. Different types
of bars have different fatigue properties as discovered by Kokubu
and Okamura [1965.1968] and Snowdon [19711 Jhamb and MacGregor [1974]
suggested that the ribs on the deformed bar can cause high stress
concentration under fatigue loading. The stress range is certainly
a predominant factor on the fatigue life of reinforcing bars. But
Hanson. Some and Helagson [1974] concluded that the effect of stress
range on the strength of the bar becomes insignificant after one
million cycles of load. This means that reinforcing bars have a
practical fatigue limit [see ACI Committee 215. 19741
Other factors affecting the fatigue life of reinforcing steel
are bar size [see MacGregor, Jhamb and Nuttall. 1971 and Kokubu
and Okamura, 1965, 1968]. minimum stress level [see Hanson. Some and
Helagson. 1974 and MacGregor, Jhamb and Nuttall. 1971]. prebending of
bars [see Pfister and Hognested. 1964] and welding on the bars [see
Burton and Hognested. 1967].
Two types of prestressing steel are commonly used in prestressed
concrete. They are the wires and the strands. Even for the
same type of prestressing steel, different countries have different
specifications. The results of fatigue tests on prestressing steel
are highly dependent on the types of steel and the manufacturers.
Furthermore, anchorage type, steel treatment and the degree of
bond also influence the fatigue life of prestressing steel [see ACI
Committee 215, 1974 and Warner and Hulsbos. 1966].
2J2.3 Structural members
12
Nearly all fatigue tests on structural members were performed
on beams under flexural fatigue loading. The specimens for fatigue
tests were mainly rectangular and T-beams of different sizes. The
common fatigue failure exhibited by the beams which had the fracture
of reinforcing or prestressing steel coupled with severe cracking
in concrete at the tension zone [see Soretz, 1974 and Hankins. 1964],
Fatigue failure caused by the failure of concrete in compression is
very rare. But some examples of shear failure or bond breakdown
have been reported in the past by Hanson. Hulsbos and Van Horn
[1970].
Analyses of available data indicate that stress range is
still the major factor influencing the fatigue behavior of concrete
structural members. Because of the complexity of the behaviour of
concrete structural members and the variety of test specimens and
conditions, more fatigue test data is still needed.
2^.4 Conservative fatigue design recommendations
In view of the lack of generality in published fatigue test
results. ACI Committee 215 [1974] has given some preliminary and rather
conservative recommendations on the design of concrete structures
subjected to fatigue loadings.
For non-prestressed members the Committee recommends that:
a. The maximum stress range in concrete shall be 40% of F'c
when the minimum stress is zero, or a linearly reduced
stress range if the minimum stress is increased. However.
the stress range shall be zero (i.e. static load) if the
minimum stress reaches 75% of F'c. This is shown as the
shaded area in Fig 1.
13
b. The stress range for straight deformed bars is limited
to 140 MPa for a minimum stress level up to 40 % of the
yield strength of the bars.
c. The stress ranges for prestressing strands (also bars)
and wires respectively are limited to 10 % and 12 % of
their tensile strength. These limitations may be imposed
only on the minimum stress level up to 60 % of the tensile
strength of the prestressing steel.
The Committee also suggests that if the recommended limitations
are not exceeded, the possibility of fatigue failure can be ignored.
14
* a
4-1 0
X 6
100
80
60
40
20
20
y
IflTjijJJ::::'
\- AC" : 215
20 40 60 80 100
lower stress limit, S . m m
of F') c
Figure 1 Goodman diagram for plain concrete [See ACI Committee 215, 1974]
15
2.3 Effects of Repeated Loading on Serviceability
Among the rather extensive publications on strength fatigue of
concrete structural members, few reports can be found dealing with
serviceability behaviour. Some researchers treated serviceability
behaviour as a by-product in their fatigue tests while others were
only interested in strength. A small amount of serviceability tests
under repeated loading can be found as the additional features in
some static load tests e.g. tests conducted by Stevens [1969]. and
Bennett and Chandrasekhar [19721
The repeated loading used in these tests usually has a constant
minimum loading level either at zero or the equivalence of dead load:
the upper limit of the loading varied at different levels. These
tests can be divided into two categories according to the maximum
loading level used. The first category has a very high maximum
level, frequently close to or even exceed the static yield load. In
the second category the maximum loading level varied but mainly
at the service load level. Within the same category of tests the
rate of repeated load (i.e. frequency) may vary in different tests.
Generally, most tests in category 1 tended to have slower rate and
less number of repetitions while in category 2 some tests involved
several million cycles of load at a very fast rate. In order to give
a general picture of these two test categories, the serviceability
tests under repeated loading on reinforced and prestressed concrete
beams are summarised in Table 2.
16
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2.3.1 High-level repeated loading tests
The main aim of this category of tests is to study the post-
elastic or post-yield behaviour of the structural members. This is
needed for ultimate strength design and the design for earthquake
and cyclone conditions. Most of the beams in this group were tested
under high intensity "one way" repeated loads. But some of them
also have a reversed loading component.
Sinha, Gerstle and Tulin [1964a] tested 9 singly-reinforced
concrete beams with deformed bars. Time creep of concrete in their
tests was eliminated by using a high rate of repeated load (0.5Hz).
Beams were tested up to 30 cycles of load, having the maximum load
at a post-yield level. The writers suggested that a shakedown limit
exists and the intersection between an unloading curve and the
corresponding reloading curve gives a point on the shakedown limit.
They also believed that, if the maximum repeated load level is below
this limit, the load repetition will not cause additional curvature
of the beam. A prediction procedure was established which was
based on the envelopes obtained from the repeated loading tests
of plain concrete conducted by the same writers [1964b]. However.
the shakedown limits are considerably lower than the experimental
results.
Later, Agranal. Tulin and Gerstle [1965] tested 3 doubly reinforced
concrete beams using the same method but included some reversed
loading components. Similar conclusion was drawn as in the tests
of singly-reinforced beams. In addition, they also found that the
beams exhibited highly nonlinear behaviour when subjected to load
reversals which were at the post yield level.
Burns and Siess [1966] studied the behaviour of under-reinforced
concrete beams subjected to both one-way and reversed cyclic loadings.
18
They tested 18 beams under one-way repeated load: three additional
beams were under reversed loads.
In the one-way repeated load tests the beams were unloaded:
a. shortly after first cracking.
b. at about 50% of the yield load.
c. immediately following first yield and
d. at several points within the plastic range.
They observed that the rigidity of the beams was gradually reduced
as the repeated load level increased: it was also related to the
amount of plastic deformation.
The results of the three beams tested under repeated reversed
loads were compared with those of a control beam loaded repeatedly in
one direction only. The ultimate strength of the beams after repeated
reversed loading was not reduced but a significant reduction in
ductility was evident. It was also found that the concrete confinement
due to closed stirrups substantially increased the ductility of the
beams.
Ruiz and Winter [1969] carried out high-level repeated loading
tests on 18 reinforced concrete beams. The beams were grouped
into five series namely A, B, C. D and E. Beams in series A and
B fabricated with compression reinforcements and closed stirrups
were called bonded beams.The remaining three series having no closed
stirrups and compression steel at mid-span region were referred
to as unbonded beams. Within each series all beams were similar.
The first beam of each series was tested statically to failure in
one cycle. The remaining beams were tested at repeated load levels
ranging from 85% to 100% of the static ultimate load (obtained from
19
the first beam). One complete cycle of load took approximately 15
minutes. Small numbers of load repetitions, from five to ten cycles.
was applied at each maximum load level. The maximum number of load
cycles on one single beam was 26.
It was observed that repeated loading always caused some in
creases in deformation in consecutive cycles. However the increments
of deflection with repeated loads were always found to decrease and
becoming almost negligible after a few loading cycles. The total
increase in deflection after as many as ten cycles of load ranged
from 5% to 25% of the initial deflection in series A.B.C and D and
from 20% to 45% in series E.
Bennetts and Atkins [1977] investigated 23 reinforced concrete
beams under four types of repeated loading. The minimum and maximum
loading levels respectively were about 20% and 90% of the beams'
static yield load. The four types of repeated loading included three
square functions and one ramp function. The maximum number of
cycles was 100 and the frequency of loading was on average 0.5 cycle
per minute. The beams were reinforced by two 16 mm deformed bars.
No significant effects were noted on the ultimate strength of the
beams under repeated loads. On the other hand substantial increases
in deformation were observed. Increases in residual deflection were
recorded and they varied between 20% to 200% of the corresponding
static deflections. A slight increase in rigidity was also noted as
a typical effect of repeated loading.
2.3.2 Tests at service and other loading levels
While post-yield level repeated loading tests provide basic
information for ultimate strength design, tests at service and
other levels facilitate the study of deformation behaviour which
20
are important for serviceability design. The tests in this category
often have a maximum load at service load level, but some of the
tests also deal with over-loading. Although tests of beams at
different maximum load levels (i.e. upper limits) were reported, no
systematic investigation on the effects of cyclic loading range on
deformation was evident. Comparing all the available test data from
different tests it is obvious that they are far from being mutually
supportive. Further, the prediction procedures proposed thus far
for the effects of repeated loading on deformation of beams are
usually unsatisfactory. A summary of the works reviewed is given
in the following pages.
Bate [1963] made a comparison between reinforced and prestressed
concrete beams tested under repeated loading. He conducted tests
on two series of reinforced and six series of prestressed concrete
beams. The latter included one series cast with high- alumina cement.
The total number of beams tested was 42 for which three types of
loading were used namely:
a. Static load
b. Repeated load with an increasing loading range, failure
occurred.
c. Repeated loading with a constant loading range.
The following conclusions were drawn by the writer:
a. Repeated loading within the working load range did not
lead to cracking of fully prestressed beams.
b. Repeated loading within the working load range caused
only slight deflection both in prestressed and reinforced
21
concrete members. However, reinforced concrete beams
cracked under repeated loading at this loading range.
c. Under repeated over-loading the increase in the propaga
tion of cracks was more rapid in prestressed beams than
in similarly over-loaded reinforced beams.
d. Repeated loading did not have a more adverse effect
on the behaviour of prestressed beams of high-alumina
cement concrete than on that of Portlant cement concrete.
Magura and Hognested [1966] tested four full-sized partially
prestressed concrete beams with concrete slab decking. Two beams
were post-tensioned and the other two were pretensioned. The beams
were designed using partial prestress so that the maximum concrete
tensile stress under design load would be approximately one half of
the modulus of rupture. The maximum repeated load varied from 1.0
to 2.5 times of the design live load. The beams were tested up to
about 5 million cycles. Deflection and crack results indicate that
the pretensioned beams with a tensile stress of about 700 psi (4.8
MPa) in concrete suffered no significant detrimental effects from
flexural cracks under repeated loading. On the other hand, similarly
stressed and cracked post-tensioned beams showed serviceability
deterioration and reduced flexural capacity by load repetitions.
Stevens [1969] tested three partially prestressed concrete beams
under repeated loading as one of the features of his research
programme which included five other partially prestressed, two
fully prestressed and three reinforced concrete T-beams. One beam
sustained one million cycles, while the other two beams failed after
750,000 and 470,000 cycles of load respectively. Failure was caused
by snapping of the prestressing wires or the steel reinforcement.
He observed that the central deflection increased as much as 44%
over the initial value after one million cvcles of load. He also
22
found that the change of stress in the non-prestressing steel in
partially prestressed beams when subjected to repeated load can be
much larger than that in a reinforced beam.
Hanson. Hulsbos and Van Horn [1970] looked at the fatigue life of
six fully prestressed concrete I-beams which had been overloaded (to
80% of the flexural strength) to cause flexural and shear cracking
prior to repeated loading. These six beams were identical in all
details except for the vertical shear reinforcement.
Repeated loading test on each beam was carried out in two
stages. First, the upper limit was set within 19% and 45% of the
ultimate strength. This was referred to as the design repeated
load. At this stage all the beams sustained 2 million cycles of
load and the total deflection tended to increase only slightly.
Repeated loading was then applied again on each beam, this time
with the maximum load increased to between 48% and 54% of the
strength. This was referred to as the above-design repeated load
and it invariably caused fatigue damage; in flexure and shear. The
deflection measurements were sensitive to such fatigue damage. Hence
they may not be reliable for use as the basis for the establishment
of empirical formulae.
Kripanarayanan and Branson [1972] conducted a test programme
consisting of 15 fully prestressed concrete beams, six of which were
tested under repeated loading. Each of the six was tested under
only three cycles of load with constant or increasing upper limit
at up to 70% of the ultimate load. Note that the upper limit was
always greater than the cracking load. Some of the beams were with
a cast in-situ slab.
For the instantaneous deflection under repeated loading the
writers proposed a formula for the equivalent moment of inertia.
lreo. After comparing with their own test data and those of Burns
23
and Siess [1966] (i.e. from three reinforced beams, see Section 2.3.1).
they concluded that their prediction procedure would be accurate.
provided that the maximum load is around the working load level.
Discussion on the shortcomings of this prediction procedure is given
in Section 7.2.
Sparks and Menzies [1973] carried out an extensive test programme
aimed at comparing the results of repeated loading tests on reinforced
concrete beams to those of sustained loading tests on similar
beams. They reckoned that repeated loads may produce both time-
and cyclic-dependent creep whilst a sustained load normally only
produces the former effect. They observed that at service load
level a beam under repeated loading increases its deflection by
about the same amount as a similar beam subjected to a sustained
load at the maximum level of the repeated load. At greater repeated
loading level the increase of deflection is more rapid in the early
stage. However, after about one million cycles of load, a beam under
sustained load at the maximum level of the repeated loading and
for the same length of time has developed a similar increase in
deflection. A linear relationship was found to exist between the
logarithm of the increase in maximum deflection and the logarithm
of the number of cycles of load or of the length of time under
loadings.
More details about Sparks and Menzies' tests will be given in
Section 8.3 in which some of their results are used to check the
accuracy of the prediction procedure developed in this thesis,
Abeles, Brown and Hu [1974] reported on a huge fatigue test pro
gramme which covered 40 fully prestressed and partially prestressed
concrete beams. The minimum load used constantly in the tests
was 30% of the static failure load, which was considered to be the
dead load of a bridge. The maximum load ranged from 50% to 90% of
24
the failure load which corresponded to the design service load and
design over-load of a bridge respectively. The plotted deflections
under repeated loading were much higher than the values of static
deflections.
Kulkarin and Ng [1979] tested 8 partially prestressed concrete
beams with the degree of prestressing n=0.3 under various levels of
repeated load. The main aim of their investigation was to study
the effects of repeated loading on deflections (including permanent
set), and on the ultimate load.
Enormous increase in deflections caused by repeated load was
found. However, the effects of repeated loading were apparent mainly
in the first one million cycles, after which the rate of increase in
deformation was reduced considerably.
The effects of repeated loads on the cracking and ultimate
strength of the beams were also investigated. It was found that
once the crack spacing is "stabilized" further addition of load
repetitions had little influence on the widening and deepening of
the cracks. It was also observed that up to two million cycles of
load, the loading repetitions had very little or no effect on the
ultimate load carrying capacity of the beams.
Warner and Pulmano [1980] described two serviceability testing
programmes conducted at the University of New South Wales by Tansi.
Heaney and Warner [1979] and Pulmano and Warner [1980]. One of
these programmes has included 8 limited prestressed concrete beams
(with only prestressing steel) while the other programme involved 8
partially prestressed beams (with non-prestressing steel as well).
Thirty cycles of repeated load up to the service load level were
applied on each beam. Increases in deflection due to repeated loading
were measured. These increases were expressed as a percentage of
25
the corresponding initial deflection. The highest increase was about
30% of the initial value.
Warner and Pulmano also found that the unloading-reloading
curves were near-linear with near-constant slopes which can be
approximately predicted by modifying the lreD equation given by
Kripanarayanan and Branson [19721 The two writers claimed that the
increase in the permanent set caused by the early load repetitions
is a significant phenomenon. However, they have not attempted to
predict this permanent set. In their report they had suggested a
method for predicting the static initial deflection. This is done
by finding the moment-curvature relationship. The method is but a
bilinear method offering certain advantage in that besides giving
more accurate results, it integrates well into the overall design
procedure for partially prestressed members. More details on this
method will be given in Chapter 5.
At about the same time as Warner and Pulmano published their
report, the author in collaboration with Loo [1980] presented the pilot
test results of ten reinforced concrete box beams under repeated
loading. The minimum and maximum loading levels were set at 30%
and about 55% of the static yield load respectively. Each beam was
tested up to 105 cycles. The effects on the bending rigidity was
found to be similar to that found in Warner's tests [see Warner and
Pulmano, 19801 In addition, it was also found that the permanent set
or accumulated deflection caused by repeated loads could be related
to the steel content of the beams and the number of load repetitions.
A hypothesis called 'intensive creep' was established with an aim
to predict the accumulated deflections. The intensive creep concept
forms the backbone of the prediction procedure developed in this
thesis for the total deflection of concrete beams under repeated
loading.
26
Bennett and Dave [1969] presented the test results of 40 partially
and limited prestressed concrete beams. Nine of the beams were
tested under repeated loading with the objectives of discovering
whether fatigue failure of wires would occur and of examining the
effects of this form of loading on the deflection and crack width.
Each beam was tested under at least three million cycles of load in
which the upper limit was the design load (i.e. 50% of the ultimate
load) and the lower limit was one half of the design load. The
increase of central deflections were recorded as being from 60%
to 100% of their initial values. The effects of both repeated and
sustained loads on deflections were tabulated and plotted. Very
similar effects have been found on the deflections under these two
types of loading. Fig. 2 shows an example of the similarity of the
two effects.
A similar experimental program, but with a smaller number of
beams, was conducted few years later by Bennett and Chandrasekhar
[19721 The programme involved 12 partially prestressed concrete
beams. The repeated loading had 55% of the ultimate load as the
maximum load and half of this value as the minimum load. Each beam
was tested up to 4 million cycles at the rate of 190 repetitions per
minute. It was found that in beams subjected to repeated loading
the final total deflection including the effects of time creep and
permanent set was up to 60% greater than the initial value. It was
also observed that the deflections increased rapidly at first but
after a million cycles of load, there was hardly any further change.
Snowdon [1971] published the test results of a massive program
of fatigue tests on reinforced concrete beams. His main interest was
to investigate the influence of the type of reinforcing steel .such as
reinforcing bars with different surface characters, on the fatigue
behaviour of the beams. Sixteen different types of reinforcement
were used. For each type of reinforcement at least two beams were
27
cast, one each was tested under static and repeated loads. Both
constant range and step-increased repeated loadings were applied
in his tests. Fatigue life, deflections and other behaviour of the
beams were recorded. Significant increase in deflections caused by
repeated loads were found. He concluded that for a given type of
loading the crack width and deflection of reinforced concrete beams
are controlled entirely by the upper level of the repeated loads:
they vary only very slightly even for bars of radically differing
degrees of surface deformation.
More recently, Lovegrove and El Din [1982] tested 12 reinforced
concrete rectangular beams under repeated loading. The loading
range was varied for different beams but was kept constant for
each beam. The long-term cyclic deflection at maximum loading level
was found to increase with the number of loading cycles. These
increases were as high as 35% , 57.5% and 80% at 106. 107 and 108 load
cycles respectively comparing to the initial deflections. The writers
have also given a simple logarithmic equation for predicting the
cyclic deflections. The empirical equation has shown good agreements
with their own test results and the results obtained by Sparks and
Menzies [1973] and Snowdon [1971]. The reliability of their equation
is discussed in Section 8.4.
Bishara [1982] presented the results of thirteen rectangular
reinforced concrete beams subjected to cyclic loading within service
load level. His study was aimed at finding the effects of concrete
area, amount of tensile steel and the ratio of compression to tension
reinforcements on crack width under repeated loading. However.
during the investigation of crack width, the increases in deflections
caused by 50,000 cycles of repeated load were also recorded. He
found that the maximum increase of mid-span deflection was about
10% and the variation of the compression reinforcement had vary
little or no effect on the deflection.
100 Days
(a) sustained loading
1000
50
25
•
_ — — • " " "
1
5R -
8R
I AR ^X^.P3R
/ ^ 3 R
9R
i<r ioH
Number of repetitions of load
(b.) repeated loading
10
Figure 2 Similar effects of sustained and repeated loads on deflections [See, Bennett and Dave, 1969]
29
2.3.3 Comments on previous tests
From the above review and the summary in Table 2. one can note
that the trend of research on serviceability under repeated loading
has changed from the use of high loading level and a small number
of cycles to the use of service load level and a large number of
repetitions.
The selections of the type of test specimens, loading procedure.
loading range, frequency etc are dependent on the researcher's
interests and the time, financial support and laboratory facilities
available to him. Thus, it is understandably difficult for any single
investigation to include all the important variables. However.
amongst the 21 investigations scrutinized in this review only those
of Bate [1963]. Abeles. Brown and Hu [1974] and Wong and Loo [1980]
can be considered to have included the major variables. They are
the maximum loading levels, loading range, number of cycles and
steel ratio. All the other investigations were merely noting or
discovering the existence of the effects of the repeated load on
serviceability. In addition, only selected variables were included
in their studies.
Thin-walled sections and flanged beams, although very popular in
practice, have not been investigated as extensively as rectangular
sections. The sizes of test beams in nearly all the above studies
were of one sixth to one quarter scale.
When repeated loading is continuously applied for a long period
of time, the time-creep effect would creep in. It is very difficult
to separate this time dependent effect from the cyclic effects in
the experiment. This left no other alternative but to select high
loading frequencies. In previous tests except those for a few cycles.
the frequency used were as high as 0.5 to 4 Hz.
30
All the investigations invariably concluded that repeated loading
caused increases in deflections of concrete beams. Several of the
researchers noted that the rate of increase was very high in the
early load cycles but it slowed down as the number of repetitions
increased [see Bennetts and Atkins. 1977: Kulkarin and Ng. 1979: Warner
and Pulmano. 1980: and Wong and Loo. 1980]. Bate [1963]. Kripanarayanan
and Branson [19721 Warner and Pulmano [1980], and Wong and Loo [1980]
also discovered that the permanent set or the residual deflection
was increased by load repetitions.
The rigidity of cracked concrete beams was found to have
increased by the first cycle of repeated load [see Bennetts and
Atkins, 1977: Kripanarayanan and Branson. 1972: Warner and Pulmano.
1980: and Wong and Loo. 19801 However further repetitions of load
would have very little effect on the rigidity provided that the
loading range and the maximum load level remain constant. As
reported by Bennetts and Atkins [1977] and Wong and Loo [1980] the
increase of maximum load level appeared to reduce the rigidity.
The observations made in this section have influenced the work
of the author which forms the main part of this thesis.
31
Chapter 3
THEORETICAL CONSIDERATIONS
3.1 Pilot Study
It is evident that repeated loading causes higher deflections
than static load in concrete beams. However, it can not be found in
the literature as to how to quantify such increase in deflection.
In view of this, a comprehensive experimental programme possibly
including all the major variables should be conducted. With the
results from these comprehensive experiments, one may be able to
formulate a computational approach for predicting the effects of
repeated loading. Before attempting the comprehensive programme
preliminary studies are advisable so as to identifying the major
variables.
A pilot test programme was carried out by the author in
collaboration with Loo [1980]. It involved a total of 10 singly-
reinforced box beams of gross section 305x305 mm (see Table 2). The
beams were tested up to 10 cycles of service load.
A typical plot for load versus deflection is given in Fig.
3a which is similar to those presented by Bennetts and Atkins
[1977] and Snowdon [1971] (see Fig. 3b and 3c). These together with
earlier findings of Sparks and Menzies [19731 Bennett and Dave [1969].
Kripanarayanan and Branson [1972] and Warner and Pulmano [1980] (see
Section 2.3.2) led the author to believe that:
a. there is an analogy between the residual beam deflection
under repeated loading and the time creep of concrete:
b. repeated loading does not affect the instantaneous (live
load) deflection: and more importantly,
* short-term 3tatic load
32
c. • the major variables are identifiable thus the effects
of repeated loading may be quantifiable.
3.1.1 Dead load deflection and Intensive Creep
For clarity. Fig. 3a may be idealized as Fig. 4, It is evident
that the repeated loading has an accumulative effect on the minimum
or dead load deflection. That is. the dead load deflection increases
as the number of load repetitions increases. It can also be seen
that the rate of increase is higher in early load cycles but rapidly
reduced as the number of cycles becomes greater. Thus, there
appears to be a parallel between the accumulation of dead load
deflection and time creep of concrete with T. the number of loading
cycles being analogous to the time in time creep. Depending on
the frequency of the repeated loading, the residual deflection may
accumulate in a very short time. Thus it is referred to herein as
the 'intensive creep' deflection.
Creep in concrete as a phenomenon has been widely studied for
over five decades. Many well tried analytical models have been
developed. In view of the above-mentioned analogy, such models may
be suitably modified for the analysis of the accumulated dead load
deflection under repeated loading.
3.1.2 Instantaneous live load deflection
The parallel (repeated live load versus deflection) lines in Figs.
3a, 3b and 3c clearly indicate that the instantaneous response of
the beam remains unchanged after the initial cycle. This means
that the live load deflection is not affected by repeated loading.
It should be noted that in these and the idealized Fig. 4. single
lines are used to approximate the hysteresis loop of the unloading
33
and reloading sequence. This is based on the assumption that the
fast rate of live load passage on the structure does not cause any
significant time creep effect and that the average of the unloading
and reloading loops can be reasonably represented by a straight
line. This assumption is close enough in real structures such as
bridge decks through which live loads passes within a very short
period.
From Fig. 4 it can be concluded that the instantaneous live
load deflection can readily be computed if an equivalent moment of
inertia. lrep can be determined.
3.1.3 Major variables for deflections under repeated loading
Very obviously, the number of loading cycles is the first
major variable. In addition, the following which affect the cracking
strength, bending rigidity and the extension of cracking, thereby
influencing the deflection, are considered to be important variables:
a. steel ratio.
b. repeated loading range, and
a cracking and yield moments of the beam.
o
34
Figure 3a Typical load-deflection diagram of beams under repeated loading [Wong and Loo,
1980]
Mid-span deflection
25
a < o
B E A M 24
20
15
monotonic loading before cyclic loading
duration of • mechanical strain
gauge readings
' load to failure after cyclic loading (zero shifted to remove residual)
response during cyclic loading.
Figure 3b Typical load-deflection diagram obtained by Bennetts and Atkins [1977]
10 20
CENTRAL DEFLECTION mm
Figure 3c Typical deflection diagram obtained by Snowdon
[1971]
T ~
8 10 MuKp.-in rl.-fl.-rt.nn m m
"T —
12 14 "™l If
35
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36
3.2 Possible Intensive Creep Models
As discussed earlier, the amplification of dead load deflection
caused by repeated loads may be seen as a parallel phenomenon to
the time creep of concrete under sustained loading. Similar to time
creep, an amplification factor may be defined as
Sri* k = - ^ (3.1)
where k is referred to as the "intensive creep" factor and dda and 5dj
are the initial and accumulated dead load deflections respectively.
Thus the time creep equations may be suitably modified for the
analysis of intensive creep, with the number of loading cycles. T
corresponding to the time in time creep.
In Eq. 3.1, 5dj may be computed by many published methods. The
choice of a suitable approach is discussed in Chapter 5 in light
of the box beam test results presented in Chapter 4.
There are many equations developed for calculating the creep
factor in time creep. Most of them were based on extensive
experimental studies. Generally, time creep equations fall into two
categories: equations with and without an ultimate creep. The first
category assumes that after a certain period of time the effect
of the sustained load will no longer increase with time. In this
category, ihe exponential equation given by Ulitskii in 1962 [see
Branson. 1977] has the form
Ct = (1-e-Bt)Cu (3.2)
37
where. Cu is the ultimate creep factor, t is the loading time in
days, and B is an experimental constant. The hyperbolic equation
originally takes the form [see Neville.1977]:
C = — — (3.3) a + bt
Later it has been modified by Branson and Cristiason [1971] and
Branson and Kripanarayanan [1971] as
tc
Ct = —- Cu (3.4) d + tc
where, Cu is the ultimate creep factor, and c and d are experimental
constants.
In the second category no ultimate value is assumed to exist.
although creep diminishes with time. In this category, the logarithmic
equation as formulated by the US Bureau of Reclamation [1953] can
be expressed as:
C = FOO logp(t + 1) (3.5)
where. K is the age at which the load is applied. FflO is a function
representing the rate of creep deformation with time, and t is the
time under load in days. A power equation [see Shank. 1936].
38
C = Kt1/r (3.6)
can also be expressed in a logarithmic way as
logC = log K + (1/r) log t (3.7)
where. C is the time creep strain, t is the time under load in days.
and r and K are experimental constants.
In view of the analogy between the repeated loading effects
on dead load beam deflection and time creep and with adequate
repeated loading test data the above analytical models may be
suitably calibrated. This would yield the equation for the intensive
creep factor, k.
39
3.3 Existing Equations for I
For evaluating the equivalent moment of inertia of rectangular
reinforced and prestressed beams under a limited number of load
cycles, Kripanarayanan and Branson [1972] presented the following.
Irep=^e+(1-tf')lg
_ M u~M t
Iv -M (3.8)
cr
where, le is the effective moment of inertia. lg is the moment of
inertia of gross section, M c r is the cracking moment. M u is the
ultimate moment, and Mt is the moment caused by the upper limit of
the repeated loading.
For partially prestressed solid beams. Warner and Pulmano [1980]
recommended an equation similar to Eq. 3.8. except that.
Mt " Mcr = _J 21 (3.9) M y - M c r
The suitability and accuracy of the above equations are discussed
in Chapter 7 in light of the box-beam deflection data presented in
Chapter 4. An improved formula is also proposed.
40
3.4 The Total Deflection Equation
From Fig. 4. for a beam under Tth cycle of load, the total
deflection can be expressed as
5 T = d d a + d l f3-l°>
From Ea. 3.1
Sda=kSdi (3.11)
In Eq. 3.10, d can be expressed in terms of the beam rigidity
and the external moment due to live load. As discussed in Section
3.1.2, the instantaneous live load deflection is inversely proportional
to the equivalent beam rigidity (Fig. 4), i.e..
OCM.L2
5t = — J — o.i2) Ec'reD
where a is a multiplication factor governed by the loading and
support conditions, L is the span. Ec is the initial modulus of
concrete and lrep is the equivalent moment of inertia at Tth loading
cycles. Thus- Eq. 3.10 can be rewritten as.
* Time creep and shrinkage deflections may be additive but are beyond the scope of this study.
41
OfM.L2
cc'reD
In Eq. 3,13 the following are the unknown quantities:
a. the initial dead load deflection. 5di.
b. the intensive creep factor, k. and
c. the equivalent moment of inertia under repeated loading,
'rep*
To help quantify these unknowns, a total of 30 reinforced and
5 partially prestressed box beams have been tested. Details are
given in Chapter 4. The selection of an accurate and reliable method
for computing 8da from amongst 9 published works is discussed in
Chapter 5. With the experimental data, calibrations of the intensive
creep methods (Section 3.2) are carried out in Chapter 6. A formula
for the intensive creep factor, k is recommended after an in-depth
and wide-ranging comparison of results. Chapter 7 compares the two
formulae for lrep published by Kripanarayanan and Branson [1972] and
Warner and Pulmano [1980] (see Section 3.3) and that proposed by the
author. Finally in Chapter 8 the accuracy of Eq. 3.10 for computing
the total deflection is checked.
42
Chapter 4
EXPERIMENTAL PROGRAMME
4.1 General Remarks
The purpose of the experimental programme was to observe the
behaviour of the reinforced and partially prestressed box beams
and to obtain deflection data for use in the establishment of the
computational procedure.
The programme involved three major variables, namely the number
of loading cycles, steel ratio and repeated loading range (or maximum
loading level) for the reinforced concrete beams. In addition, the
degree of prestressing is added for the tests of prestressed
concrete beams.
Details of all the box beams, fabrication, instrumentation and
test procedures are presented in this Chapter. Observations and
analysis of some beam behaviour are also given. For reference
purposes, the measured cracking, yield and ultimate moments are
compared with the corresponding values computed using procedures
given in the Australian Concrete Structures Code, AS1480-1982.
42 Design of Test Specimens
43
4.2.1 Number of beams
Three series of reinforced and one series of prestressed box
beams were designed for the programme: these included 30 reinforced
and 5 partially prestressed beams.
Each series designated by a code started with letter R (for
reinforced) or PP (for partially prestressed) followed by 200. 300. or
450 denoting the gross beam width . A beam in series R200. R450
and PP300 is identified by a number attached to the series code, for
example R200-1. However.series R300 contains five sub-series from 1
to 5. Thus the first number following R300 denotes the sub-series
and the second number denotes the beam, for example R300-1-4.
All the beams tested and the associated variables are shown
in Fig. 5.
4^2 Design of reinforced concrete box beams
The design includes the strength and rigidity of the beams.
The strength characteristics of a beam can be represented by the
cracking, yield and ultimate moments. The rigidity of- a beam is
reflected by of the (initial) modulus of elasticity and the effective
moments of inertia at different loading stages.
For the test beams, the design procedure follows that adopted
by the Australian Code [Standard Association of Australia. 1982]. The
equations used are given in Appendix I. Load factors and capacity
factors, were not included in the design calculations. Note that
* All beams were 305 mm high.
44
yield moment is not a significant value in practical design. However.
it is an important parameter for under-reinforced concrete beams
as far as research is concerned since yielding always occurs before
collapse.
The properties and the calculated values of beam strengths of
all reinforced concrete box beam are given in Table 3. Note that for
the heavily reinforced (but still under-reinforced) beams the value
of ultimate moment could be lower than the value of yield moment.
The calculation of the ultimate moment in the Australia Code is
based on the Whitney stress block which is rectangular, replacing
the actual parabolic stress block. This approximation causes the
anomaly (also see Appendix I).
42.3 Design of prestressed concrete box beams
In the design of prestressed concrete box beams, a trial and
error method based on strain compatibility [see Warner and Faulkes.
1979] is used instead of the simplified method adopted by Australian
Code [Standard Association of Australia. 1978], The trial and error
method is more accurate but the procedure is rather lengthy.
The prestressed concrete box beams in Series PP300 were designed
to have the degree of prestressing varied from 25% to 100%. The
term of "Degree of Prestressing" (DOP) was first used by Bachmann
[1979,1980]: it is used in order to be consistent with CEB/FIP Model
Code [CEB-FIP 1978. Hill 1980]. The DOP is defined as the ratio of
moments at decompression and at full service load i.e.
_ Decompression Moment Service Load Moment
45
In Series PP300 all the beams have the same yield strength as
the beams in sub-series R300-3. Therefore the value of n for Series
R300-3 can be considered as zero. Thus combining PP300 and R300-3
we have n varying from 0 to 100%.
The amount of non-prestressing steel and the effective prestress
ing force were determined by the trial and error method. A computer
program was written to implement the rather tedious calculations.
Table 4 gives the properties of the prestressed concrete box beams.
More details and some design examples are presented in Appendix
II.
R200
STEEL RATIO VARIES
R200-1
R200-2
R200-3
R200-4
R200-5
WIDTH OF BEAM VARIES
R300 R450
V STEEL RATIO
VARIES
R300-1
R300-2
R300-3
R300-4
R300-5
DEGREE OF PRESTRESSING VARIES
J2L
PP300-1
PP300-2
PP300-3
PP300-4
PP300-5
STEEL RATIO VARIES
R450-1
R450-2
R450-3
R450-4
R450-5
I LOAD LEVEL VARIES *
R300-1-1
R300-1-2
R300-1-3
R300-1-4
I LOAD LEVEL VARIES
R300-2-1
R300-2-2
R300-2-3
R300-2-4
I LOAD LEVEL VARIES
R300-3-1
R300-3-2
R300-3-3
R300-3-4
I LOAD LEVEL VARIES
R300-4-1
R300-4-2
R300-4-3
R300-4-4
1 LOAD LEVEL VARIES
R300-5-1
R300-5-2
R300-S-3
R300-5-4
* Note: last digit, 1 = static load, 2 = 30%-*50% yield load,
3 = 30%~70% yield load, 4 = 30%-"90% yield load.
Figure 5 Test beam series
47
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48
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fa s 4J w CO
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CO CO cu u 4J CO
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4H
n
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00
a •H a u o M-t C •H CU rH
MH 0
CO CO CU u 4J CO
-a H cu •H >H
+-
49
4.3 Beam Details and Fabrication
The sectional dimensions of the reinforced and prestressed box
beams are given in Fig. 6. They may be considered as a quarter
scale to those of standard box sections described by Somerville and
Tiller [19701 Stirrups were placed between the loading points and
the supports to prevent possible shear failure. Two 6mm bars were
also used in the top flange area to hang the stirrups. Between
the loading points, that is pure bending region, only tensile steel
was provided.
All beams were cast in steel moulds with ready-mixed concrete.
The concrete mix proportions were 1 : 1.5 : 3.2 (cement : sand :
aggregate) by weight and the maximum size of aggregates was 10
mm. Type A cement (ordinary Portland) was used, and the water-
cement ratio was 0.5. For each batch of concrete delivered to the
laboratory a slump test was carried out before the casting of the
beams. Average slump was between 80 mm to 100 mm. For the identical
beams in each group the same batch of concrete was used in order to
limit the variation on strength properties. The void in each beam
was created by embedding polystyrene prisms. The bottom flange of
the beams was first cast and then polystyrene prisms were placed
in position. Small concrete spacers and wires were used to fix the
polystyrene prisms to the mould to avoid possible movements and
floating of the prisms. After fixing the prisms, more concrete was
placed for the webs and the top flange of the beam. An electrical
internal vibrator was used to ensure proper compaction.
All the prestressed beams were pretensioned in a self-contained
steel prestressing rig. An IBISONET 4-ton monowire jack with a 41-mm
piston and a 200-mm stroke was used for the pretensioning. After
tensioning, the prestressing force in each wire was anchored by
a grip having three split-cone wedges. Strains in the wires were
50
first checked by a 50mm demountable mechanical (Bemec) strain gauge
to ensure there is no slip between the wires and the grips. Then
concrete was poured. After concrete had the required strength.
transfer was made by cutting the prestressed wires (using an oxygen
cutting torch). The tensioning of wires and placing of concrete
into moulds may be seen in Fig. 7. To reduce possible variation in
the loss of prestressing forces in different beams, each was tested
immediately after transfer.
All beams were cured under moist hessian for at least seven
days after casting. At least twelve cylinders were cast from
each batch of concrete for the strength and elastic modulus tests.
Four of the cylinders were kept in a standard fog room and were
tested at 7 days and 28 days. The rest of the cylinders were
kept beside the beams and then tested during the repeated loading
tests. The cylinder tests were according to SAA Concrete Test Codes
[19781 A typical stress-strain curve of concrete obtained from the
cylinder tests is shown in Fig. 8. Samples of reinforcing bars and
prestressing wires were also tested for their strengths. Fig. 9
shows typical stress-strain curves of steel plotted automatically
by the plotter on an AVERY testing machine. All reinforcing bars
used were hot-rolled mild-steel deformed bars. The yield plateau of
these bars can be clearly seen from the stress-strain curves. The
yield stresses of these bars averaged about 300 MPa. The actual
values of the yield stress are given in Table 3. The prestressing
wires were 5 mm hard-drawn wires having the 0.2% proof stress as
the yield stress. All prestressing wires used in Series PP300 were
from the same batch and the yield stress was 1610 MPa.
51
250
1450 14. (1900)
E (19Q0)
1450 (1900)
f
4350 (5700) 250
4850 (6200)
Note: Figures in brackets are for series R200 and R450
200 305 450
R300
305
) • o o o •
m ~o en
• • . ) • o • o • o •
PP300-1 PP300-2 PP300-3
o reinforcing bars
• prestressing wires
• o • • o • V f
• • • o • * •
PP300-4 PP300-5
Figure 6 Details of test beams (See Tables 3 and 4 for other information)
^tV^^;Sfe^:W5;9l ^^SiNCjN«^"-'-""~' ran
m ^^^^« -Jill
(a)
MMHm • m
I mum
(b)
Figure 7 Fabrication of concrete box beams
53
0.001 0.002 c
strain
Figure 8 A typical stress-strain curve for concrete
500.00
431.32
400.00
300.00
v—'
CO
cu 4J CO
272.05
200.00
54
100.00
16mm deformed bar cross 2
section area: 201.1mm
0 0.05 0.10 0.15 0.20 0.25 Strain
Figure 9a Stress-strain curve of mild steel deformed bars
2000 r-
co
co CO
cu u 4-1 CO
Diameter 5.00mm 2
Cross section area 19.63mm
0 0.200 1.000 2.000 Strain (%)
Figure 9b Stress-strain curve of hard-drawn prestressing wires
55
4.4 Test Equipment and Instrumentation
Beams were tested under third point loading provided by a
steel I-beam distributor. The tests were carried out in a massive
self-contained space frame. The set-up can be seen in Fig. 10.
The repeated loading was applied through a Dartec testing
machine. The machine is a servo-hydraulic feed back system including
a 600 kN actuator, a 75 Hp pump unit and an electronic fatigue
control panel M1000/R. (see Fig. 11). Provided that the input data
such as the mean load, frequency and the command input are correctly
set on the control panel, the Dartec system would give accurate
control for the repeated loading tests. The accumulated numbers
of loading cycles were also recorded by the fatigue control panel.
The fatigue rated load cell of the machine was calibrated to 0.01 kN
accuracy. During the repeated loading tests the upper and lower
limits of the load were also checked by a dual-beam oscilloscope on
the control panel.
Deflections of the beams were measured by dial gauges with
0.01mm graduation. An independent supporting system not connected
to the loading frame, was used to mount the dial gauges. Deflections
were measured at seven positions along each beam as shown in Fig.
12a,
Surface strains of the beams were also measured. A Demac
strain gauge having gauge length of 200 mm was used to measure the
strains. Stainless steel reference discs were glued on both sides
of the beam at the mid-span region. The positions of the strains
measured are also shown in Fig. 12b.
All box beams were painted with white wash for easier discovering
of cracks. Crack widths were also measured in the pure bending
56
region. A microscope with 40-time magnification and an accuracy of
0.02 mm was used.
57
Figure 10 Test set-up
58
6 cu 4-1 CO CO CJ
cu 4-1 U CO p CU rC 4J
CU
a CO CX
o u 4J C!
o CJ 60
C •H T3 CO O
T3 CU 4J CO CU P-. CU Pi
cu JH
00 •H fa
59
L/3
^
L/: L/3
L/ 6 J L / 6 I L/6 I L/6 I L/6 l L/6
(a)
Dial gauges
m o cn
o
o vO
to
m ID
200
^-L±
200
i
Stainless steel Demec points
(b)
Figure 12 Positions of dial gauges and Def&ec gauges
60
A3 Test Procedures
For the beams in series R300. different levels of repeated
loading were applied. The first beam of each group in this series
was tested statically to failure to check the accuracy of the design
equations for Mcr. My and Mu. The remaining three beams were tested
under repeated loads having a lower limit at 30% of the yield load
(to simulate the dead load) and with the upper limit set at 505s. 70%.
and 90% of the yield load respectively. The five beams in series
PP300. being the extension of series R300-3. were only tested with
the upper limit at 50% of the yield load. In the tests in series R200
and R450. the lower and upper limits respectively were set at 30%
and 53% of the corresponding yield loads. The minimum and maximum
loads for beams tested under repeated loading are summarised in
Table 5.
All beams under repeated loading were each tested up to 100.000
repetitions of load. In view of the fact that the frequency of
a structure member such as a main bridge girder being subjected
to full service load is low. a test up to 105 cycles is believed
to be sufficient in serviceability studies. Before the start of
the repeated loading test, the beam was tested up to the upper
load limit statically. Referred to as the initial loading test
(see Fig. 4) it was to facilitate the determination of the initial
response of the beam which would form the basis for comparisons
with post-repeated loading behaviour. It should be noted that the
repeated load variation was of a sinusoidal nature and the frequency
was between 100 and 150 cycles per minute depending on the upper
limit of the load. In order to detect changes in beam behaviour.
unloading then reloading was carried out immediately after 1. 10. 10".
103, 104 and 10 cycles of load. Deflections, strains and crack widths
at various specified points were measured during these static tests.
The increments of static load depended on the maximum load level.
61
After the static load test further load repetitions were applied.
To minimize possible effects of time creep, all tests for a beam were
carried out continuously and completed within twenty four hours.
TABLE 5 UPPER AND LOWER LIMITS OF REPEATED LOADING MOMENT (kNm)
Beam
R200-1
R200-2
R200-3
R200-4
R200-5
R300-1-2
R300-1-3
R300-1-4
R300-2-2
R300-2-3
R300-2-4
R300-3-2
R300-3-3
R300-3-4
R300-4-2
Minimum Moment
(Md)
14.15
23.51
28.31
34.33
40.27
9.30
9.30
9.30
16.07
16.07
16.07
25.61
25.61
25.61
32.80
Maximum Moment
(Mt)
25.01
41.54
50.01
60.65
71.15
15.51
21.71
27.91
26.79
37.50
48.21
42.69
59.76
76.83
54.67
Beam
R300-4-3
R300-4-4
R300-5-2
R300-5-3
R300-5-4
R450-1
R450-2
R450-3
R450-4
R450-5
PP300-1
PP300-2
PP300-3
PP300-4
PP300-5
Minimum Moment
(Md)
32.80
32.80
48.93
48.93
48.93
33.28
47.06
57.96
78.77
92.95
25.61
25.61
25.61
25.61
25.61
Maximum Moment
(Mt)
76.54
98.41
81.55
114.16
146.78
58.79
83.13
102.37
139.17
164.20
42.69
42.69
42.69
42.69
42.69
63
4.6 Experimental Results
The raw data from the tests conducted in this study are the
surface strain, crack width and deflections. Strains were recorded
over the depth of beam at initial loading and after the repeated
loading cycles. Crack widths were also recorded in the same way.
Results of strain and crack width are given in Appendices III and IV
respectively. The initial dead load deflections ddj were measured on
each beam before the repeated loading tests started. After 1. 10.
102, 103, 104 and 10 cycles of loading the dead load deflections were
also recorded. Plots of mid-span deflections against the bending
moments are presented in Appendix V. The values of total deflection.
8T, at the initial loading and after the repeated loading cycles
were also recorded. They are tabulated in Tables AV-1 and AV-2 for
reinforced and prestressed box beams
Two types of derived results were obtained from the raw results
of deflection data. The first is the intensive creep factor, k = ^
(see Eq. 3.1). The difference between total deflection and dead load
deflection at the same loading cycle gives the instantaneous live
load passage deflections, i.e. d = 8T - 5da. Using d the second
quantity can be derived. This is the effective moment of inertia
of the beam under repeated load i.e. lrep = g£g- .
64
4.7 Observations and Analysis of Beam Behaviour
4.7.1 General behaviour
Under repeated loading the beam deformations accumulate with
increasing number of loading cycles. The increases in strain and
crack width resulted in the increase in the deflection. Typical
examples of strain and crack-width increases are illustrated in Fig.
13 (for more records see Appendices III and IV respectively).
The static behaviour of all box beams tested was typical of any
under-reinforced concrete beam. Bilinear M - S relationship with
cracking point as a bifurcation point was observed. As sufficient
shear reinforcements were provided, beams tested statically were
all failed in tension mode. This means that yielding occurs in the
steel prior to the crushing of concrete in the compressive zone.
Comparisons of observed Mcr, My and Mu and their computed values
are given in Table 6. In general the correlations are close.
Both dead-load and total-load deflections accumulated under
repeated loading. After 105 cycles of load the total increase in
dead-load deflections ranged from 40% to 300% of the initial value.
For total deflections the increases were lower, from 1% to 60%. The
increases were significant in earlier loading cycles but tended to
slow down in later cycles. For example, for beam R300-3-2. which
was tested under constant repeated load with maximum load at 50%
of the yield load, the increase in dead load deflections after 105
cycles was 76%. Of the 76% increase. 66% occurred in the first 10
cycles of load. All beams tested followed a similar pattern.
65
XI
C CO
u CJ
p w H P
1 u o p w PC!
P CO
<3 w S O is o CO H U
w p H vO
w p « <! H
1 a3
1 r* v—'
! >—•
r4
a°
/— V
in \ v—' N
XI
cu cu 4-» 3
/-v 3 rH vO p* CO
w g > O XI
cu cu U 3
/-N 3 iH
m co co ^ co >
CU
a
XI cu CU 4-1 3
•-s 3 rH si- p. CO
w g > o CJ XI
cu cu U 3
•-v 3 rH
cn co co *-* CO >
CU
S / — V
\ ^
i-H \
XI
cu cu /-v 4J 3 CN 3 rH w p. cfl
e > o CJ
XI
cu cu '-s U 3 rH 3 rH ^ CO CO
CO > CU
a
1 CU PQ
o cn r-H
cn •
CN
cn
o •
CN
O
i—i
i—i
o .
i-H
en
00 CN
• cn cn
CO •
o
o cn cn . — i
m CN .
rH i—1
i—1
1 i—1
1
o o cn oi
i—i
r—1
m CN
•
m vO
r-1
cn m vO
cn o r-l
vO
m •
cn m
vO
m <oo m
cn 00 •
o
o cn •
cn i—i
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r—i r H
F H
1 CN 1
o o cn oi
cn CN
rH
vO 00 •
00
O O •
cn r-l
O .
i-H
cn •
m 00
00 CN
i-H
cn
00 00 •
o
o cn •
cn i—i
m CN
CM rH
rH
1 cn 1
o o cn Pi
r-H r-H
. r-H
m cn *
o rH 1—1
00
o »
cn CN i-H
m o .
rH
<* cn •
cn o i-H
00 S3-
• st i—4 i-H
cn 00 •
o
o cn *
cn I—I
o m .
rH i—1
r-H
1 S3-
1 O O cn Pd
cn o .
r-H
o o m m r-H
00
m •
cn vO
00
cn «
o
o rH
. cn vO rH
VO rH .
o vO i-H
cn 00 •
o
o cn •
cn i-H
o vO .
i-H i-H
r-H
1
m l
o o cn P4
CO
u 3 4-1 CJ
3 U 4-J CO CU 4-1 CJ
u o a o CJ
c CO •H l-i CO
u 4J
co
3
00
c •H XI
u o a a co x cu 4J cfl rH 3 U rH CO CJ CU
U cfl CO •
cu /-v 3 CN rH 00
co cn > rH
I XI o CU 00 W S3-3 rH
& <
o CJ at x o CJ
w H o
66
Significant differences have been found between the results of
the tests under different ranges of repeated loads. Provided that
the dead load (minimum load) is constant, a higher maximum load gave
a higher increase in dead load deflections. For example, in the tests
of the three identical beams in Series R300-5. with maximum load at
50%, 70% and 90% of the yield load the increases in deflections were
found to be 29%, 47% and 79% respectively after 1000 cycles of load.In
other series of beams, which included tests under different ranges
of repeated load, a similar trend was found.
If other conditions remain unchanged, it was observed that
heavily reinforced or highly prestressed beams tended to have a
smaller increase in deflections under repeated loading. For instance.
Beam R200-1 had a steel ratio of 1%. After 105 cycles of repeated
load a 200% increase in dead load deflections was recorded, whereas
only a 62% increase was found in Beam R200-5. which had a 3% steel
ratio. For prestressed concrete beams. Beam PP300-1 with n=0.25. a
138% increase of dead load deflection was recorded after 10 cycles
of repeated load. On the other hand, only 61% increase was found
in Beam PP300-4 having n=0.80.
4.7.2 Mechanism of deflection accumulation
The deterioration of bond between reinforcing bars and the
surrounding concrete is the main reason for the increase in
deflections under repeated loads. The behaviour described in the
previous section can be explained by scrutinizing the nature of bond
resistance in concrete beams especially under repeated loading.
The bond mechanisms of deformed bars may be illustrated in Fig.
14. Bond between steel and concrete is initially provided by chemical
adhesion between mortar paste and bar surface. Even low stresses
will cause sufficient slip to break the adhesion between the concrete
67
and the steel. Further bond will be provided by friction and the
wedging action of small dislodged sand particles between concrete
and steel, and also by the interlocking of ribs with surrounding
concrete in the case of deformed bars.
Under repeated loading, bond deterioration occurs between steel
and the surrounding concrete. The mechanisms of bond deterioration
have been observed by Bresler and Bertero [1968] in repeated loading
tests on steel bars embedded in cylindrical concrete specimens which
may be used to explain qualitatively the deflection behaviour of
the beams tested in this thesis:
a. During loading the deterioration is mainly caused by:
i. failure in concrete 'boundary layer' adjacent to the
steel-concrete interface (breaking of adhesion):
ii. slippage of steel relative to concrete:
iii. inelastic deformation and local crushing (or con
solidation of mortar paste) at the steel-concrete
interface: and
iv. inelastic extensional deformation in concrete result
ing from microcracking and release of shrinkage.
b. During unloading:
i. Reverse motion between steel and concrete is
resisted primarily by the wedging action of the
rugged surfaces in the boundary layer. This
results in the resistance to slip being greater
than that during the preceding loading stage.
68
ii. Further unloading will overcome the wedging action.
Resistance to slip (due to friction) is about the
same as that during loading.
iii. After loading is removed, full recovery of steel
elongation is prevented by the shear resistance
at the interface between concrete and steel.
Residual state of tension in reinforcing steel
and of overall net compression in concrete remain.
Local cracks in the boundary layer, as well as
those extending fully through the section do
not close completely after unloading, this causes
irrecoverable deformation.
c. During the reloading procedure and the further loading
cycles, further disruption of bond occurs. But (for a
constant maximum loading level) the rate of disruption
diminishes and the process appears to stabilize after
a certain large number of loading cycles.
The deterioration of the adhesive bond under repeated loading
contributes to the increase in dead load deflection (or permanent
set) in concrete beams. The stabilization or consolidation of the
mechanical bond resistance, on the other hand, is responsible for
the diminishing rate of increase in this irrecoverable deformation.
(IJ > CJ
CJ
cu 4J CO •-v 60 6
51 o 6 rl UH I
c o •H rH
cu X
CO rl 4J
co
1000.
900"
800
700
69
10 10 10
Number of loading cycles
(a)
10 10-
c •H CJ rl O
>4H
c •H CU -U 4J < CO rS CJ 4J > X CJ
Jd CJ CJ CJ CO 4J rl cn
a e 3 6 •H X CO
2
0.20-
0.10 10 10 10
Number of loading cycles
10 10"
Figure 13. Typical increase of strain and crack width under repeated loading, Beam R300-2-3
70
Nominal diameter -
Figure 14 The bond mechanism of deformed
bars [see Park and Paulay, 1975]
71
Chapter 5
INITIAL DEAD-LOAD DEFLECTION
5.1 General Remarks
With the initial deflection data obtained in Chapter 4 it is
possible to determine the most suitable computational procedure for
concrete box beams from amongst published methods. In this Chapter.
a review is first made of the existing methods of calculation. This
is followed by a comparative study involving nine more popular ones.
In the comparisons, statistical approaches are adopted.
72
52 Existing Methods for Predicting the Initial Deflection
Many methods have been established to predict the initial
deflection for reinforced and prestressed concrete beams. The key
step of calculating initial deflections is to find, with reasonable
accuracy, the bending rigidity which is the product of the initial
concrete modulus of elasticity. Ec and the effective or nominal
moment of inertia of the cracked composite section. In existing
methods there are two major approaches for finding the moment of
inertia:
a. to introduce a factor to modify either the moment of
inertia of the gross section or that of the "fully"
cracked section (i.e. cracking up to the neutral axis):
b. to calculate curvature from the idealized moment-curvature
curve, the bending rigidity being the moment divided by its
corresponding curvature.
A chronological listing of published methods together with the
originators (or the Codes which adopted them) is given in Table 7. In
addition to these two basic approaches many simplified or graphical
methods have also been established [see Lutz 19731 Moreover, some
methods predict the plastic deflections after the beam has yielded
[see for example Baker. 1965: Institute of Civil Engineers. 1962: Corley.
1966: and Hsu and Mirza. 19741 As neither of these two latter groups
of methods is relevant to the present work, they are not discussed
herein.
5.2.1 Moment of inertia methods
73
The deflection calculations for the two extreme cases. i.e. the
moments of inertia of uncracked and "fully" cracked cross sections.
are very straight-forward.
The moment of inertia of an uncracked cross section is usually
denoted as I . It is calculated using the entire cross sectional area y
ignoring or including the steel area. In most cases the difference
between these two values is very small. Therefore, for consistency.
the steel area is ignored in this thesis.
The moment of inertia of "fully" cracked section is denoted as
LP. In reality most cracked beams are not "fully" cracked. The cr
cracks usually do not reach the neutral axis. Also, along the length
of the beam only a certain number of cracks will occur. Concrete
between the cracks are still taking some tensile stress. Therefore
the moment of inertia of these sections should be a value between
I and Icr. This moment of inertia is referred to as the effective
moment of inertia and denoted by le.
With the bending rigidity of the beam given as Ecle. the initial
dead load deflection may be given in general as.
6d = ^ (5„ cc'g
The formulae or procedures for computing le given by the various
authors may be found elsewhere (see Table 7 for sources). Some of
the more renowned methods are used in Section 5.3 for a comparative
study. Moments of inertia lg. lcr and le may be represented by the
polar lines in the moment-deflection curve as shown in Fig. 15.
74
TABLE 7 EXISTING METHODS FOR PREDICTING INITIAL DEFLECTIONS
Authors
Meney
Swain
ACI Committee 307
Murasher
PCA
Dunham
Yu & Winter
CEB
Branson
ACI 318-63
Burus
Corlay & Sozen
BPR
Beeby
CEB
ACI 318-71
CP110 or Unified Code
ACI Committee 443 AASHO
AS1480-1974
Filing
ACI 318-77
Warner
Direct Bilinear
Year
1914
1924
1931
1940
1947
1953
1960
1961
1963
1963
1964
1966
1966
1968
1968
1971
1972
1974
1974
1974
1977
1980
Moment of Inertia Method
Icr
/
/
/
Method A
pfy>500
/
P50.005
M>2M cr
X8
/
Pfy*500
°SfsSfy/3
M<M cr
^Icr
/
•
Method B
*Ig
M <M<2M cr cr
^lIcr+^2Ig
•
/
fy/3*Vfy
/
/
/
/
Bilinear m-cf)
Method
/
/
/
p<0.005
/
/
/
75
Deflection, 6
Figure 15 Bending Rigidities of Reinforced or Prestressed Concrete Beams
76
5.2.2 Bilinear moment-curvature or moment-deflection curve methods
These methods adopt an idealized bilinear moment-curvature or
moment-deflection curve, such as the one shown in Fig. 16. The
deflection calculations in this category take a general form:
8 = c51 + S2 < d3 (5.2)
where.
81 = <XL2<*~1
52 = cci_2c*~1
in which £, and 4>2 are the curvatures of the beam at different
stages of loading.
By calculating the curvatures of the beam. <£=M(Ecl)~ .- the
deflections can be computed without any difficulties. Different
values of angle B at the bifurcation point (see Fig. 16) may be
adopted in the curvature calculations [see CEB. 1968 and Bate. 19681
Alternatively, curvatures and deflections can be calculated by
determining the co-ordinates of the bifurcation points on the curve
(i.e. A, C and D in Fig. 16). Beeby [1968] and Warner [1980b] methods
adopt this procedure. Their equations are given in the following
Section.
77
Note that in the CP110 method [1972], the tensile stress of
concrete between the cracks (known as tensile stiffening effect) is
taken into account. The magnitude of this tensile stress may be
obtained by assuming a triangular tensile stress block with 1 MPa
at the steel level and zero at neutral axis of every section along
the beam, this is illustrated in Fig. 17.
78
4-1 ti CJ
e o
4>c (fic)
B =0.75 to 0r85 E I
Curvature (Deflection)
(9 <f>u
(5J
Figure 16 Idealized bilinear moment-curvature or moment-deflection curve
79
Section Strain Stress
Figure 17 Stress and strain distribution of a cracked section suggested by CP110 [1974]
5.3 Comparative Study
80
The more popular amongst the methods reviewed in 5.2 are used
in the present comparative study with an aim to select the most
suitable method of computing $dj as required in Eq. 3.13. The methods
and their respective equations are listed below.
a. I Method [PCA, 1947].
«M d L2
Sri, = -=-2— (5.3) 'di E c'g
b. Lr Method [Meney. 1914: Swain. 1924: ACI Committee 307. 1931].
sdi = y i - (5.4)
c. Yu & Winter's Method [I960].
2
8dj = -^- (5.5) cc'e
where, Jcr
'e *1
in which
and
Md
M1 = 0.1(F'c)3bw(h-kd)
d. Branson's Method [1963],
tfMHL 2 2
c'e
(5.5a)
SHi = — (5.6) di g i
where,
'e^N^-'^'crS
CEB Method [1968],
5 .. = «l-_[_cr + 4 _d cr 3 (5J)
"PI 3 I t c 'g J cr
The direct bilinear method [see Branson. 1977].
d" E c 'g 'or
The CP110 Method [1972],
*di = " i f " = a<t,L (5'9)
where 4>. or L r is calculated assuming a concrete tensile cr
stress of 1 MPa between cracks (see discussion in Section
5.2.2). The solution is obtained by a trail and error
method.
Beeby's Method [1968].
dl " 5Wi = — - - L - (5.10)
-c'e
where.
,.=.,-[1-1 JilL^SL (5.10a) 'e 'g u,g 'crjM _ M
mu cr
82
Warner's Method [1980b].
where.
caM-.L2
SHi = — — (5.11) dl ££ |
MH - Mor
"e = 'a ~ C'q " •v3— ~ (5-11a) e ^ 9 v M y - M c r
Note that Warner's method closely resembles that of Beeby's
except that, for reinforced concrete beams, lcr and Mu in Beeby's
equation are replaced by ly, the moment of inertia at yield, and My
respectively. The calculation of ly is based on the assumption of
elastic behaviour of the beam at the first yield stage. For the
cases tested, the two methods also give almost identical results
but the latter is more convenient to use for partially pretsressed
beams.
Dead load, which is represented by the lower limit of the
repeated loading, caused cracking in most of the reinforced beams
tested except Series R300-1. All the prestressed beams tested did
not crack under dead load. For beams, which were not cracked under
dead load, the initial dead load deflections can only be calculated
using the lg method.
The comparisons of measured and calculated dead load deflections
are presented in Tables 8a and 8b respectively for reinforced and
prestressed beams. Tables 9 and 10 give the deviations and frequency
distribution of the calculated 5dj for the reinforced beams. It can be
seen in Table 8a that both the direct bilinear method and Branson's
method give the best average accuracy. On average, the lcr method
is the most conservative followed by the CEB 1968 method. Beeby's
and Warner's methods give values only slightly higher than the lg
method. By examining in Tables 9 and 10. it is clear that all except
the l„ and Lr methods have low deviations and/or reasonably sharp u cr
83
distribution curves. The lcr method tends to be over-conservative
and as expected, the lg method does not perform well for beams
cracked under dead load. However, it is quite apparent that from an
engineering point of view Branson's method is the most satisfactory.
being consistent and slightly on the safe side. For these reasons.
the author recommends the use of this method in calculating the
initial dead load deflection. §dj. as required in Eq. 3.13,
For all the partially prestressed beams the cracking moments
were greater than corresponding dead load moment. As a result
Branson's equation for initial dead load deflection reverts to the I
method. However, the validity of Branson's procedure for computing
the initial deflection of cracked prestressed beams has been well
established [see Branson. 19631
TABLE 8a CALCULATED AND MEASURED INITIAL DEAD LOAD DEFLECTIONS (IN MM) OF REINFORCED BOX BEAMS
Bean
IUOO-1
R20O-2
rUOO-3
R200-4
R200-5
B300-1*
R300-1-2*
R300-1-3*
WOO-1-4*
R300-2-1
R300-2-2
1300-2-3
R30O-2-4
R30O-3-1
RJOO-3-2
B300-3-3
R30O-3-4
R30O-4-1
R30O-4-2
R300-4-3
R300-4-4
uoo-s-i
R30O-S-2
R30O-5-3
R300-5-4
8450-1
R450-2
R450-3
R450-4
M50-3
Daad Load
Banding Hoaanc
14.13
23. SI
28.31
34.33
40.27
9.30
9.30
9.30
9.30
16.07
16.07
16.07
16.07
23.61
25.61
23.61
25.61
32.SO
32.90
32. SO
32.80
48.93
48.93
48.93
48.93
33.28
47.06
57.96
78.77
92.95
-
Heaaured Initial Dead Uad
Deflections Sdl
4.42
6.85
9.03
9.74
12.57
0.70
0.84
0.90
0.78
2.10
2.21
1.83
1.B0
4.74
5.24
4.25
5.15
4.83
4.57
4.81
4.66
5.79
6.44
5.94
6.24
7.20
8.97
9.35
12.36
13.43
Calculated Initial Dead Load Deflections
Effective Moment of Inertia Method!
\ Sdl
3.69
6.12
7.37
8.94
10.49
1.09
1.09
1.09
1.09
1.88
1.88
1.88
1.S8
2.99
2.99
2.99
2.99
3.83
3.83
3.83
3.83
5.71
5.71
5.71
5.71
4.35
6.16
7.58
10.30
12.16
Meal Cal
1.20
1.12
1.23
1.09
1.20
0.64
0.77
0.83
0.72
1.12
1.18
0.97
0.96
1.58
1.75
1.42
1.72
1.26
1.19
1.26
1.22
1.01
1.13
1.04
1.09
1.66
1.46
1.23
1.19
1.10
Xcr
Sdl
8.84
10.07
10.66
11.38
12.08
4.35
4.35
4.35
4.35
5.06
5.06
5.06
J.06
5.08
5.08
5.08
5.08
6.28
6.28
6.28
6.28
9.07
10.01
10.73
12.08
12.99
Mea Cal
0.50
0.68
0.85
0.86
1.04
0.48
0.51
0.42
0.41
0.94
1.04
0.84
1.02
0.95
0.90
0.95
0.92
0.92
1.05
0.95
0.99
0.79
0.90
0.87
1.02
1.03
Yu 4 Winter.
5di
3.03
6.49
7.66
8.87
9.91
2.27
2.27
2.27
2.27
3.67
3.67
3.67
3.67
4.10
4.10
4.10
4.10
5.54
5.54
5.54
5.54
6.59
8.22
9.27
10.98
L2.04
Mea Cal
1.46
1.06
1.18
1.10
1.27
0.93
0.97
0.81
0.79
1.29
1.43
1.16
1.40
1.17
1.11
1.17
1.14
1.05
1.16
1.07
1.13
1.09
1.09
1.01
1.13
1.12
_
Branson
6dl
5.52
9.49
10.41
11.29
12.04
2.35
2.35
2.35
2.35
4.56
4.56
4.56
4.56
4.96
4.96
4.96
4.96
6.27
6.27
6.27
6.27
7.07
9.46
10.52
12.04
12.90
Mea Cal
0.80
0.72
0.87
0.86
1.04
0.89
0.94
0.78
0.77
1.04
1.15
0.93
1.13
0.97
0.92
0.97
0.94
0.92
1.03
0.95
1.00
1.02
0.95
0.89
1.03
1.03
Bilinear Methods
CEB
Jdi
5.67
10.11
11.63
13.24
14.61
2.40
2.40
2.40
2.40
4.71
4.71
4.71
4.71
5.53
5.53
5.53
5.53
7.62
7.62
7.62
7.62
7.15
10.10
11.84
14.54
16.14
Mea Cal
0.78
0.68
0.78
0.74
0.86
0,87
0.92
0.76
0.75
1.01
1.11
0.90
1.09
0.87
0.83
0.87
0.84
0.76
0.83
0.78
0.82
1.01
0.89
0.79
0.85
0.83
Direct Bill
5di
4.95
8.27
9.42
10.62
11.65
2.21
2.21
2.21
2.21
3.94
3.94
3.94
3.94
4.55
4.55
4.55
4.55
6.12
6.12
6.12
6.12
6.06
8.27
9.58
11.60
12.80
Mea Cal
0.89
0.83
0.96
0.92
1.08
0,95
1.00
0.83
0.81
1.20
1.33
1.08
1.31
1.06
1.00
1.06
1.02
0.95
1.05
0.97
1.02
1.19
1.08
0.98
1.07
1.05
CP110
*di
6.96
8.97
9.76
10.66
11.47
3.19
3.19
3.19
3.19
4.31
4.31
4.31
4.31
4.57
4.57
4.57
4.57
5.91
5.91
5.91
5.91
7.60
8.98
9.91
11.47
12.47
Mea Cal
0.64
0.76
0.93
0.91
1.10
0.66
0.69
0.57
0.56
1.10
1.22
0.99
1.19
1.06
1.00
1.05
1.02
0.98
1.09
1.01
1.06
0.95
1.00
0.94
1.08
1.08
Warner
Sdl
3.90
6.61
7.89
9.40
10.83
1.94
1.94
1.94
1.94
3.20
3.20
3.20
3.20
4.02
4.02
4.02
4.02
5.84
5.84
5.84
5.84
4.68
6.64
8.09
10.68
12.36
Mea Cal
1.13
1.04
1.14
1.04
1.16
1,08
1.14
0.94
0.93
1.48
1.64
1.33
1.61
1.20
1.14
1.20
1.16
0.99
1.10
1.02
1.07
1.54
1.35
1.16
1.16
1.09
Beeby
4dl
3.89
6.60
7.87
9.40
10.84
1.93
1.93
1.93
1.93
3.20
3.20
3.20
3.20
4.02
4.02
4.02
4.02
5.84
5.84
5.84
5.84
4.66
6.63
8.08
10.70
12.38
Mea Cal
1.14
1.04
1.13
1.04
1.16
1.09
1.13
0.95
0.93
1.48
1.64
1.33
1.61
1.20
1.14
1.20
1.16
0.99
1.10
1.01
1.07
1.54
1.35
1.16
1.16
1.08
Average 1.16 0.84 1.13 0.94 0.85 1.03 0.92 1.19 1.19
Beaa aerlaa I 300-1 ia uncracked under H^
^ Mae m Heaaured value ' Cal Calculated value
TABLE 8b CALCULATED AND MEASURED INITIAL DEAD LOAD DEFLECTIONS OF PARTIALLY PRESTRESSED BOX BEAMS
(M = 25.61 kNm)
Beam
PP300-1
PP300-2
PP300-3
PP300-4
PP300-5
Measured Initial Dead Load Deflection,
<5,. (mm) di
2.47
2.60
2.23
2.12
2.37
Calculated Initial Dead Load Deflection,
6 1 ^mm)
2.99
2.99
2.99
2.99
2.99
NOTE: The depth of neutral axis 'kd' are calculated using trial and error method.
TABLE 9 DEVIATIONS OF CALCULATED 6 di
Methods
I Method g
I Method cr
Yu & Winter's Method
Branson's Method
CEB 1968 Method
Direct Bilinear Method
CP 110 Method
Beeby's Method 2
Warner1s Method
Maximum Deviation
- 0.66
+ 0.52
- 0.46
+ 0.28
+ 0.25
- 0.33
+ 0.44
- 0.64
- 0.64
Average Deviation
- 0.16
+ 0.16
- 0.13
+ 0.06
+ 0.15
- 0,03
+ 0.08
- 0.19
- 0.19
Standard Deviation
0.32
0.26
0.20
0.12
0.17
0.13
0.19
0.27
0.27
Note: 1. Maximum Deviation = the maximum value of (Calculated 5,. - Measured 6,.)/Measured 6,.
di ai di
Average Deviation = the average value of (Calculated <5,. - Measured 6 .)/Measured 6 di
Standard Deviation = Cal.6,. - Mea.5,. .
[EC fe-s ~) ] Mea. 6 di
|/N
87
TABLE 10 FREQUENCY DISTRIBUTIONS OF CALCULATED 6
Method
I Method g
I Method cr
Y & Winter's Method u
Branson's Method
CEB 1968 Method
Direct Bilinear Method
CP110 Method"
Beeby's Method 2
Warner's Method
Over-estimating
40-60%
0
19%
0
0
0
0
8%
0
0
20-40%
10%
8%
4%
11.5%
34.5%
0
15%
0
0
0-20%
10%
50%
11.5%
54%
50%
38%
23%
11%
11%
Under-estimating
0-20%
37%
23%
65%
34.5%
15.5%
54%
50%
65%
65%
20-40%
23%
0
11.5%
0
0
8%
4%
8%
8%
40-60%
10%
0
8%
0
0
0
0
8%
8%
60-80%
10%
0
0
0
0
0
0
8%
8%
88
Chapter 6
INTENSIVE CREEP FACTOR
6.1 General Remarks
As defined in Eq. 3.11. the dead load deflection accumulated
under repeated loading may be computed as the product of the
intensive creep factor k. and the initial dead load deflection. §dj.
In this chapter the most suitable equation for calculating k will be
derived by calibrating the creep models (see Section 3.2) against the
test results. The variables which have major influence on k. such as
the loading cycles, steel ratio, loading range and maximum loading
level are included in the study. Because of the variability of the
test results of concrete beams, statistical methods are necessary
for the calibration and establishment of the suitable formulae for
k.
6.2 Statistical Analysis
89
6.2.1 The use of SPSS package
A statistical package called SPSS (Statistical Package for the
Social Sciences) developed by Nie. Hull and Bent [1975] was used in
the analysis of the test results. The package was first developed
in 1965 at Stanford University in the form of several individual
statistics programs. It was to serve the research and teaching needs
of the political scientists. By 1970 it became an integrated system
which can handle normal statistical analyses as well as various
other data processing tasks. The package used in this study is
the second edition published in 1975.
The used of SPSS has been successful for the social scientists.
In the engineering field it was also found to be an effective tool in
some areas such as in hydraulic engineering. The author believes
that this is the first time the package is used to analyse the
test results of concrete engineering.
The statistical procedures in SPSS selected for this study
included the correlation analysis, scattergram plotting, multiple
regression and frequency distribution. These procedures were used
in an attempt to find the relationship between the deflections of
the beams and the major variables which influence such deflections.
The correlation analysis provides the researcher with a means
for measuring the linear relationship between two variables and
produces a single summary statistic describing the "strength of
the association; this statistic is known as Pearson's correlation
coefficient 'r' or simply the correlation coefficient.
90
While the correlation analysis provides a single summary
statistic describing the relationship between two variables, the
'scatterplot' shows graphically such relationship in detail.
As mentioned previously, there are more than one variable
affecting deflections under repeated loading. A multiple regression
procedure is therefore needed for deriving the relationship between
the deflection and the various variables. The multiple correlation
and regression programs in the SPSS package allow the study of
the linear relationship between a set of independent variables and
a dependent variable, at the same time taking into account the
interrelationship amongst the independent variables. The basic
goal of multiple regression is to produce a linear combination of
independent variables which will correlate as closely as possible
with the dependent variable. This linear combination can then be
used to "predict" the value of the dependent variable.
An example of the operation of the SPSS package is presented
in Appendix VI.
d22 Selection of parameters
A regression analysis only gives the closest linear relationship
between two variables. From the test results and the pilot study
it has been noted that the relationship between the intensive creep
factor, k and other influential variables are not in linear forms.
To be able to use the regression procedure, it is necessary to
transform the original variables. For each variable, parameters in
different forms have been selected for the trials in the preliminary.
single-variable regression process. Correlations were compared in
order to find the most accurate regression line.
91
In searching for the relationship between the intensive creep
factor, k and the number of cycles, T, the latter has been transformed
into logarithmic and hyperbolic forms. Single-variable regression
analyses had shown that both forms gave reasonable accuracy where
other variables were held constant.
For the steel ratio, p, the plots of k versus D [see Wong and
Loo, 1980] showed an inverse relationship. Therefore parameters such
as - \ 4r -T, \i —• -• etc have been tried in the regression P p ' P p! p3 P P
with k. It was shown that both -1 and \ aave reasonable results. P P "
However for simplicity but without sacrificing accuracy, only - was retained eventually.
Similarly, several parameters representing the repeated loading
range or the maximum loading level were tried in different forms.
For example J , ^ ^ and ^*±L. The last form, which not onlv My My Mv-Mp.
represents the loading range and level but also reflects the strength
properties of the beams, gave the best results.
The trial of each parameter using SPSS gave a scattergram
allowing the closeness to the linear line to be shown clearly.
Regression of k with single parameters can not give the inter
relationship between the independent variables. To find a usable
prediction equation incorporating all the variables, the multiple
regression procedure has to be used.
6.2.3 Multiple regression
From single regression analyses three parameters have been
chosen, they are:
a. for steel ratio, 1, P
92
b. for number of cycles. T. loqinT or -L=B. and ",u a+T
c. for loading range or maximum loading level. Mt"M". My-Me.
Due to the interdependence between the different parameters.
good correlation in single regression does not necessarily give good
correlation in multiple regression. In simple bivariate-regression
analysis, values of the dependent variable (k in this case) are
predicted from a linear function in the form of
Y' = A + BX (6.1)
where Y' is the estimated value of the dependent variable. Y. B is
a constant by which all values of the independent variable X are
multiplied and A is a constant which is added to each case. The
difference between the actual and the estimated value of Y for each
case is called the residual i.e. the error in prediction, and my
be represented by the expression: Residue= Y - Y\ The regression
strategy involves the selection of A and B in such a way that the
sum of the squared residues is smaller than any possible alternative
values.
S(Y - Y'r = minimum (6.2)
The regression line is shown as in Fig. 18. Since the sum of squared
residues is minimized, the regression line is called the least-square
line of best fit.
93
The basic principles of regression analysis used in bivariate
regression may be extended to situations involving two or more
independent variables. This procedure is called multiple regression.
The general form of multiple regression is.
Y' = A +B1X1 + B2X2 + ... + BjXj (6.3)
Again, A and Bj are selected in such a way that the sum of squared
residues 2(Y' - Yr is again the minimum.
The goodness of fit of the regression equation can be evaluated
by examining the correlation coefficient, r or its squared form r .
The SPSS package used here on an UNIVAC 1102 main-frame computer
carries out the regression calculation in considerable speed. More
equations of regression coefficients, correlation coefficient and
worked examples of multiple regression are presented in Appendix
VI.
Numerous regression calculations were carried out to find the
best equations for k. The calculation and comparison tasks were
proceeded in two parallel ways. One aimed at producing a hyperbolic
equation for T, the other, a logarithmic equation.
The hyperbolic equation takes a general form as.
1 M t - M d Tb 1 P 2 M y - M c r -
3a + T b
or
94
k = A + B., (-!•)(•—* 2-1—! (6.5) P " M v - M c r - a + T b
The logarithm equation has the form.
1 M t - M H
k = A + B 1 ( - ) + B 9 ( — 3-) + B3(log10T) (6.6) P M y - M c r '
or
1 Mi — M-i 1 M t — M_i k = A + B H C ^ - X — ^ 2_1 + B,(I)f — I -Vlog10T) (6.7)
1 0 My-Mcr' 2 O My-Mcr-
10
For Eqs. 6.4 and 6.5 different values of a and b had also been
tried. The comparisons of the correlation coefficients for the four
multiple regression equations are shown in Table 11. It is concluded
that, Eqs. 6.5 and 6.7 give the best predictions as they gave a
value of r closest to 1.
95
• • • . • Regression line
• • 6 • • • •
iX.Y) . . " * • • • •
• •
X CD
' . » • • •
• • tx.r)
• • • •
• •
Xaxis
Figure 18 Regression line
TABLE 11 MULTIPLE REGRESSION OF EQUATIONS FOR 'k'
• - — • • • -
Equations
6.4
6.5
6.6
6.7
Constants
Correlation Coefficient, y
Constants
Correlation Coefficient, y
Constants
Correlation Coefficient, • y
Constants
Correlation Coefficient, y
A
-1.79060
-
1.184150
-
-0.393309
-
1.189640
-
Bl
0.024218
0.90764
0.014608
0.95148
0.024218
0.90872
0.001490
0.95310
B2
1.824430
0.93603
-
-
1.824430
0.93709
0.028837
0.94830
B3
0.732744
0.09174
-
-
0.092446
0.10193
-
-
97
6.3 The Proposed Formulae
Substituting the relevant coefficients in Table 11 to Eq. 6.7.
the logarithmic equation becomes.
k = k1 + R log T (6.8)
where, T is the number of loading cycles.
1 Mi — M_i k1 = 1.18 + 0.029X- —- — (6.8a)
P My - MCf
and
1 Mi - Md R =0.0015x- — (6.8b)
DM -M • ,v,y m c r
In Eq. 6.8, k., represents the ratio of dead load deflection to initial
dead load deflection after the 1st cycle of repeated loading. As
in time creep calculations, the logarithmic equation does not have
an ultimate creep value.
Similarly, the hyperbolic equation can be obtained as,
k=11 8 +0£343, M t - M d T0.6 ^
p My ~ Mcr 0.3 + T0"6
98
The equations for intensive creep factor k can be modified to
suit both reinforced and prestressed beams with a smooth transition
between these two. From the test results it is noted that an
increase in the degree of prestressing reduces the intensive creep
effect. If we denote the intensive creep factor of prestressed beams
as k_, using the abovementioned regression procedures a best-fit
curve can be found for kp. We have
k„ = - (6.10) o p
in which k is given by Eq. 6.8 or 6.9. and
P = 1 + 3.2T>3,0 (6.10a)
where the degree of prestressing.
Decompression Moment _ 2_P_ (6.10b) Full Service Load Moment Mi,
The regression analysis and values of r for the trials of different
n are presented in Table 12. The factor 3.2 and index 3.0 in Eq,
6.10a are chosen for the comparatively favourable values of r. A. B
and Se they given.
99 TABLE 12 CORRELATIONS OF MEASURED k AND COMPUTED k
BY EQ. 6.10 WITH VARIOUS P VALUES P
p
1+2.3n2,5
1+2.4n,2-5
1+2.5n2-5
1+2.6n2*0
1+2.6n2'5
1+2.6n3'°
1+2.6n3-5
1+2.8n2-0
1+2.8n2*5
1+2.8n3-0
1+2.8n3*5
1+3.On2-0
1+3.on2-5
1+3.on3*0
1+3.on3-5
1+3.2n2-0
1+3.2n2-5
1+3.2ri3*0
1+3.2n3-5
1+3.4n2-0
1+3.6n2-0
1+3.8n2-0
Correlation Coefficient
Y
0.88616
0.89181
0.89437
0.82201
0.89507
0.88438
0.82912
0.84083
0.89362
0.88060
0.83175
0.85077
0.89060
0.87697
0.83325
0.85588
0.88717
0.87377
0.83426
0.85827
0.85909
0.85901
Intercept A
-1.64682
-1.40533
-1.19009
-0.95224
-1.00070
-0.87776
-0.66388
-0.73798
-0.68912
-0.56.963
-0.40503
-0.54322
-0.44790
-0.33785
-0.20719
-0.37468
-0.25790
-0.15819
-0.05186
-0.23101
-0.10873
-0.00420
Slope B
1.97466
1.85491
1.74681
1.74600
1.65093
1.48423
1.29496
1.65030
1.49223
1.33014
1.17095
1.55844
1.36905
1.21444
1.07634
1.47710
1.27220
1.12513
1.00235
1.40721
1.34776
1.29729
Std.Err. of Estimate
se
0.16818
0.16420
0.16236
0.20669
0.16185
0.16941
0.20292
0.19647
0.16290
0.17199
0.20149
0.19074
0.16507
0.17442
0.20067
0.18771
0.16748
0.17652
0.20012
0.18627
0.18577
0.18582
100
6.4 Effects of Main Variables on Intensive Creep
The accumulation of dead load deflection and the underlying
reasons for such phenomenon have been discussed in Section 4.7. The
effects of the various variables on the intensive creep factor are
presented herein.
6.4.1 Effect of load repetitions
One of the main concern of this study is to investigate the
effect of repeated loading on the accumulated dead load deflection
of reinforced and prestressed concrete box beams. The results for
the intensive creep factor, k. are plotted against the number of
load cycles for the three beam series respectively in Figs. 19. 20
and 21. It may be noted that the relationship between k and log10T
is generally linear for various beams tested. From Fig. 20c it is
also obvious that the linear relationship holds for r-r ratio of uo My
to 90*.
6.4.2 Effect of load level
Fig. 22 shows the results for k for the beams in series R300
under the three levels of repeated loading. As described earlier.
the lower limit of each level was set at 30% of the yield load: the
upper limits were 50%, 10% and 90% respectively. For convenience, k
is plotted against the moment ratio. *~ d instead of the customary My-M^
J£. For clarity, onlv results at the 1st, 10th. I03th and I05th and My
those for R300-1, R300-2 and R300-4 are presented. Those for the
I02th cycle are bounded by those for the 10th and I03th. and those
for the I04th, by I03th and io5th; the points for beams R300-3 and
R300-5 are very close to but on either side of those for beams
R300-4. It may be seen in Fig. 22 that k largely varies linearly with
101
the moment ratio or the load level. The linearity is particularly
obvious for the early few cycles and for beams with higher steel
content. As expected the higher the load level the larger the
accumulated dead load deflection.
6.4.3 Effect of steel ratio
The results for k plotted against steel ratio, D in Figs. 23.
24 and 25 are respectively for the beams in series R200. R300 and
R450. It is apparent that the relationship between k and D follows
a hyperbolic trend. The higher the steel ratio, the lower is the
aggravation by repeated loading: this is true up to p=2* beyond
which the aggravation remains constant with k equals to about 2
or lower. On the other hand, for a beam with a low steel ratio of
0.5%, the accumulated dead load deflections can be as high as 7 or
8 times the corresponding initial values, if the repeated load is
about 90* of the yield load (see Fig. 24c). However, if it is at about
the service load level (50*). the value of k stays at 4.5 or lower. It
may thus be concluded that for box beams under repeated loading.
the steel ratio should not be lower than 2%. if serviceability is
an important consideration.
6.4.4 Effect of prestressing
The prestressed concrete beams in series PP300 can be seen as # k
an extension of beam Series R300-3. A plot of measured -£ versus the degree of prestressing is given in Fig 26 which clearly shows
that a higher degree of prestressing results in a smaller intensive
creep effect. However, even for fully prestressed beams (i.e, n = 1.0)
the intensive creep effect may not be ignored.
* k for reinforced beam R300-3-2 and k for prestressed beams.
102
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1.20
1.00
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0.60
0.40
0.20
0.20 0.40 0.60 0.80
Degree of prestressing n
1.20
Figure 26 Effect of prestressing on intensive creep factor
6.43 Discussion
114
Comparisons are made between the two analytical solutions and
the measured k in Figs. 19. 20 and 21 showing the effects of T.
It may be seen that the two solutions are close to each other
especially for beams with higher steel ratios (or lower k). However.
it may be seen in Figs. 19. 20 and 21 that the hyperbolic equation
gives very little change in k value after about 103 cycles. This does
not reflect the observed trend which shows a continuous increase.
For this reason, the logarithmic model is considered superior and
will be retained for further discussions below.
It may be seen in Fig. 19 that Eq. 6.8 underestimates all but
the most heavily reinforced beams 03200-5): in general the predicted
values are around 20% below the measured ones. For the R300 series,
the correlations between the computed values and the measured ones
are very satisfactory for the beams under repeated loading at 70%
and 90% of the yield loads (see Figs. 20b and c). For those under 50%
of the yield load, the formula underestimates the intensive creep
for the lighter beams but reverse for the heavier beams (Fig. 20a).
The correlations for beams of R450 series are generally good to
slightly conservative as may be seen in Fig. 21.
In Fig. 22 the computed k values (Eq. 6.8) are compared with
the measured ones for the R300 series beams emphasizing the effects
of moment ratio (load level) and load cycles, the comparison may be
considered satisfactory.
The computed k values are superimposed on Figs. 23. 24 and 25
which demonstrate the effect of steel ratio. Comments made in the
second paragraph of this section are similarly valid here as these
figures are another version of those given in Figs. 19. 20 and 21
respectively.
115
The overall performance of Eq. 6.8 may be represented by the
scattergram shown in Fig. 27.
Using a modified form of Eq, 6.8 i.e. Eq. 6.10 to calculate
k for prestressed beams, the effects of prestressing are taken
into account. The accuracy of calculated k in comparison with to
measured values obtained from 5 prestressed box beams is given by
a scattergram shown in Fig. 28. It can be seen in this scattergram
that most of the correlation points are well within ±20%.
On reviewing the comparisons made in this section between
the computed and measured values of k. it is concluded that Eqs.
6.8 and 6.10 while by no means perfect are able to reflect the
complicated nature and trend of the intensive creep of reinforced
and prestressed concrete box beams.
116
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118
Chapter 7
INSTANTANEOUS UVE LOAD DEFLECTION
AND THE EFFECTIVE MOMENT OF INERTIA
7.1 Repeated Loading and Beam Rigidity
In many major structures such as bridge decks live load is
a portion of the total load and only acts on the structures for
a very short period of time compared to dead or sustained load
which in many cases is permanent. The passage of live load causes
further deflection in addition to the dead load deflection. This
additional deflection is defined herein as the instantaneous live load
deflection, St. It may be seen from Fig. 4. that the instantaneous
live load deflection is but the deflection due to the increment of
load from the lower to upper limit of the repeated loading. Further.
8t is inversely proportional to the post-initial load equivalent
moment of inertia. irep. Therefore to be able to predict the value
of lreD is vital for computing 8t.
For beams under a few cycles of repeated load. Kripanarayanan
and Branson [1972] concluded that repeated loading with a constant
level of maximum load does not change the bending rigidity of the
beam. This conclusion was made on the basis of their observation on
the behaviour of prestressed concrete beams under repeated loading
in tests conducted by themselves and by Paranagama and Edwards
[19691 They also incorporated similar observations made by Burns
and Siess [1966] on reinforced concrete beams.
Kripanarayanan and Branson's conclusion has been corroborated
by Warner and Pulmano's [1980] from tests of 16 prestressed concrete
beams for up to 30 cycles of load. Wong and Loo [1980] from tests
of 10 reinforced concrete box beams have found that while the
119
beams underwent considerable permanent set. the rigidity remained
unchanged for up to 100.000 load cycles.
The effects of repeated loading range and maximum loading level
on beam rigidity varied for different observations. Most of the
test results however indicate that the increase of maximum load
level or the range of repeated load would reduce the rigidity. This
definite trend was observed in the test results obtained by Burns
and Siess [1966] and Soretz [1957] as well as those in the present
study. As an example, a typical load deflection curve obtained from
Burns and Siess is reproduced in Fig. 29 showing the change of
rigidity caused by the change of maximum repeated loading levels.
Similar curves have also been produced by Soretz. These curves show
very clearly that the rigidity reduces as a result of an increase
in maximum load level.
Kripanarayanan and Branson however reported in their 1972
paper that they observed an opposite trend. They suggested that
the increase of maximum loading level will also increase the rigidity
of the beam under repeated loading. They further suggested that
the severely cracked beams (under very high level of repeated load)
will have the same rigidity as the uncracked beams. This suggestion
was the basis of their proposed equations for computing ireD which
could not be corroborated by their test results. More discussion
is given in subsequent sections.
120
Q <
3
25
20
10
0.
/FIR
J
/,
ST C
/ '
RAO
/
EING
/
YElLDIIs G — —* *> . /
0.10 0.20 0.30 0.40 0.50 0.60 0.70
MIDSPAN DEFLECTION-inches
Figure 29 Load-deflection curve showing change in beam rigidities
[see Burns and Siess, 1966]
121
72 Existing Formulae for reo
The prediction of the bending rigidity of reinforced and
prestressed concrete beams under repeated loads was first given by
Kripanarayanan and Branson [19721 They defined the beam rigidity
as Eclrep, where Ec is the initial modulus of elasticity of concrete
and lrep, the moment of inertia of the cross section of the beam
under repeated loading at a given upper limit.
Kripanarayanan and Branson proposed that the value of ireD
after any number of loading cycles should be interpolated between
the effective moment of inertia of the cracked section under static
load. Ie and the moment of inertia of the uncracked gross section.
la. Their recommendation is as follows:
lreD = le+C1-tf)lg <7.1)
where.
* = M u ~ M * (7.1a) M u-M c r
in which, le may be calculated using Eq. 5.6a. Ig is based on the
gross section neglecting the steel area. Mt is the bending moment
caused by the upper limit of repeated loading, Mu is the ultimate
bending moment of the beam, and Mcr is the cracking bending moment.
The reliability of Eq. 7.1 was checked by the two writers with
the test results of six prestressed concrete beams under three
cycles of repeated load and of three reinforced concrete beams (of
122
Burns and Siess ) under two cycles of repeated load. They claimed
that the results predicted by Eq. 7.1 normally agree with the test
results to within ±20% percent for loads up to 60 to 70 percent of
the ultimate load for non-composite prestressed beams and up to
75 to 85% for composite prestressed beams. For reinforced concrete
beams the loading was up to only 30 to 40 percent. The accuracy
would be generally better than ±20% for normal working load levels.
Such claims were also corroborated by the author in his pilot study
[1980] for beams tested at service load level.
Despite the apparent reliability of Kripanarayanan and Branson's
approach, there are certain discrepancies. According to Eq. 7.1. for
a given value of Mu and Mcr. a higher value of Mt. i.e. the total
moment, gives a lower rp which in turn, yields a higher value of lreD.
This contradicts the observed trend (see Fig. 29). which indicates
that the higher the total moment, the lower the value of lreD.
An alternative equation has been presented by Warner and
Pulmano [1980] in which </> in Eq. 7.1a is replaced by
M y -M c r
for partially prestressed beams. However, no detailed comparison
with experimental results has been carried out. It is shown in
Section 7.4 that this alternative equation leads to an improvement
in accuracy at higher load levels, but the accuracy deteriorates
somewhat at the service load level.
123
7.3 The Proposed Equations for I
For better accuracy at all levels of load, the author proposes
a new equation for lrep. The basis of Kripanarayanan and Branson's
equation is to interpolate for lrep between l_ and le by proportion
of moment ratios. There are many different moment ratios for such
a purpose. The author suggests that lreD should be interpolated
between ig and le as follows:
'r.p-<SE>f,VC,-<SEyX <™>
where.
(Mt - M o r )2
M = —* £!— + M_. (7.3a) * M y - M c r
cr
and m=2. 3 or 4. Other quantities are defined in Fig. 30 and the
derivations are given in the following paragraphs. The choice of
an appropriate value for m is discussed in Section 7.4.
Referring to Fig. 30 line DE representing irep can be moved
to intercept point 0 and be represented by the parallel line. OC.
Consequently, line OC is an interpolation line between line OA (for
lg) and line OD (for le). Note that, line OD is the interpolation line
between line OA and line OB (for lcr) and according to Branson [1963].
»e = Tr^'a + Cl-<Tr5fin3lor (5.6a)
124
For L „ on the other hand we can easily obtain: rep
reD M X 'g L M X
J e (7,3)
By working out the moment segments as illustrated in Fig. 30.
finding Mx is not a difficult task. There are two possible moment
ratios to be used:
M x -M c r = M v - M t
Mt-Mc r M y -M c r
(7.4)
and
Mx " Mcr _ Mt"Mcr (7 5)
M t-Mc r M y - M c r
Eq. 7.4 is similar to the moment ratio used hy Kripanarayanan and
Branson which is shown to contradict the observed trend. Eq. 7.5
is more logical and is adopted for Eq. 7.3. Rearranging terms Eq.
7,5 gives
(Mt - M o r )2
= —i 2 L _ + M _ (7.3a) My- Mcr ""Cr
The index m in Eq. 7.3 can be calibrated using relevant experimental
data. This is done in the next section.
125
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2 11 5 1 •
rC _fd 4_J i
CD c M —
O 4-1
C CD
E 0 5 11 ex a
CD
E cfl 0) rO
01 4J 0) 14 u 3 O CJ
O
01
> u 3 U
3 0 •rl 4J U 0) rH 4-1 0) •3 1 4J
3 01 B 0 2
0 CO
01 1-1 3 00 •rl fa
3 >•>
2 2 2 2 2
W' ;uaujow
7.4 Comparisons of Results
126
The mid-span deflections of the 30 reinforced and 5 prestressed
box beams due to the passage of live load 5, are given in Tables
13 and 14 respectively for the 1st. 10th. I02th. I03th, I04th and I05th
repeated loading cycles. It may be seen that the changes in live
load deflection are not significantly affected by repeated loading
cycles even up to 105 cycles. As a result, the mean values of the
'measured' lrep can be obtained by substituting the average measured
8t into Eq. 3.12. The results are given in column 3 of Table 15 and
column 2 of Table 16 respectively for the reinforced and prestressed
concrete box beams.
The 'measured' values of lrep are also compared with the computed
values obtained by Eqs. 7.1. 7.2 and 7.3. These are included in
Tables 15 and 16 for the two types of beams respectively. In the
applications of Eq. 7.3 three values of m are found to give reasonable
results as evident in Tables 16 and 17. In order to find the best
prediction equations, statistical analyses have been carried out.
The deviations of the ratios of calculated ireD to measured irep
for all the three methods are tabulated in Tables 18 and 19. The
frequency distributions of errors which give an indication of the
accuracy of the predictions are presented in Tables 20 and 21 for
all the prediction equations.
In the various comparisons it was observed that Eq. 7.1 gives
the worst overall accuracy amongst the three methods. While giving
reasonable accurate predictions at service load level. Eq. 7.1 is
seen to have considerably over-estimated lrep at higher levels of
repeated load i.e. at rr=0.7 and 0.9. Eq 7.2 imoroves the accuracy My
at higher loading levels, but gives less accurate lrep at service
load level. Both Eq. 7.1 and 7.2 are generally overestimating.
127
Comparing to Eqs. 7.1 and 7.2. the equation proposed by the
author (Eq. 7.3) gives better predictions for all levels of load
regardless of the different values used for m. This can be clearly
seen in Tables 15. 16 and 17. Further evidence may also be found in
Tables 18. 19. 20 and 21 where various error statistics are tabulated.
With m=3 and m=4 Eq. 7.3 gives the smaller deviations for
reinforced beams (see Table 17) which appear to be more desirable
then with m=2. However, a different conclusion may be reached
with regard to reliability when carefully examining the frequency
distributions of errors. It is shown in Table 20 that with m=2 the
equation gives the highest probabilities in the accuracy ranges
of ±5% ±10% and ±20%. Similar trends may be found in Table 21 for
partially prestressed beams. For these reasons. m=2 is concluded
to be the most satisfactory.
In view of the above discussion Eq. 7.3 with m=2 is recommended
for the evaluations of lrep which is required in Eq. 3.13 for computing
the total deflections.
TABLE 13 MEASURED INSTANTANEOUS LIVE LOAD DEFLECTION AT MID SPAN OF R.C. BOX BEAMS, <$0 (in mm)
Beam
R200-1
R200-2
R200-3
R200-4
R200-5
R300-1-2
R300-1-3
R300-1-4
R300-2-2
R300-2-3
R300-2-4
R300-3-2
R300-3-3
R300-3-4
R300-4-2
R300-4-3
R300-4-4
R300-5-2
R300-5-3 '
R300-5-4
R450-1 *
R450-2
R450-3
R450-4
R450-5
No. of Loading Cycles
1
4.25
6.28
7.34
9.37
10.10
1.30
3.01
5.34
1.90
4.04
6.89
2,67
5,95
10.76
2.91
6.44
9.90
3.85
7.71
13.07
4.24
6.40
8.82
10.73
12,08
10
4.43
6.32
7.26
8.65
9.54
1.33
3.35
5.55
1.88
4.08
6.95
2.64
5.89
11.20
2.88
6.45
10.00
3.74
7.71
13.09
4.08
6.37
8.09
9.73
10.99
IO2
4.42
6.29
7.22
8.69
9.41
1.30
3.30
5.51
1.89
4.27
7.44
2.65
5.95
11.00
2.88
6.59
10.12
3.67
7.71
13.17
4.23.
6,25
8.16
9.81
11.05
103
4.50
6.30
7.20
8.72
9.42
1.29
3.43
5.70
1.86
4.38
7,32
2.65
6.07
11.21
2.91
6.60
10.18
3.69
7.70
13.61
4.29
6.43
8„34
9.81
11.23
• 10"
4.49
6.27
7.54
8,84
9.25
1.29
3.44
5.88
1.89
4.63
7.20
2.65
6,08
10.89
2,93
7,04
9.78
3.70
8.07
13.46
4.26
6.42
8.30
9.96
11.23
10s
3.88
6.10
7,38
9.01
8.64
1.30
3.54
5,89
1.88
4.40
7,11
2.60
6,02
11,07
2.89
6,91
10.89
3.68
7.77
13.49
3.96
6.39
8.38
9,95
11.11
Ave.
4.33
6.26
7.32
8.88
9.39
1.30
3.34
5.65
1,88
4.30
7,15
2,64
5.99
11.02
2.90
6.67
10,15
3.72
7.78
13.32
4.18
6.38
8.35
10.00
11.28
129
TABLE 14 MEASURED INSTANTANEOUS LIVE LOAD DEFLECTION AT MID-SPAN OF PARTIALLY PRESTRESSED BOX BEAMS
$£ (mm)
Beam
PP300-1
PP300-2
PP300-3
PP300-4
PP300-5
Number of Loading Cycles
1
2.59
2.34
1.92
1.69
1.55
10
2.58
2.47
1.89
1.68
1.64
102
2.63
2.29
1.96
1.68
1.63
IO3
2.69
2.69
2.06
1.71
1.62
IO4
2.73
2.62
2.19
1.70
1.63
105
2.65
2.47
2.04
1.74
1.60
Average
2.65
2.48
2.01
1.70
1.61
130
TABLE 15 CALCULATED* AND MEASURED Ir OF REINFORCED CONCRETE BOX BEAMS
Beam
R200-1
R200-2
R20O-3
R2Q0-4
R200-5
R300-1-2
R30O-1-3
R300-1-4
R300-2-2
R300-2-3
R300-2-4
R30O-3-2
R30O-3-3
R300-3-4
R30O-4-2
R300-4-3
R300-4-4
R300-5-2
R30O-5-3
R300-5-4
R450-1
R450-2
R450-3
R450-4
R450-5
M y
<*)
53
53
53
53
53
50
70
90
50
70
90
50
70
90
50
70
90
50
70
90
53
53
53
53
53
Measured I rep
(xlO9 mro")
0.251
0.297
0.322
0.324
0.369
0.311
0,275
0.212
0.398
0.308
0.332
0.464
0.410
0.323
0.582
0.476
0.424
0.607
0.483
0.477
0.558
0.619
0.620
0.675
0.712
Calculated I r e p
by Branson's (Eq. 7.1)
I rep
(x 10s mm")
0.274
0.326
0.346
0.369
0.388
0.507
0.427
0.521
0.411
0.474
0.558
0.473
0.528
0.586
0.536
0.569
0.604
0.594
0.607
0.620
0,581
0.653
0,698
0.768
0.804
Calculated I •
A v e r a g e Measured 1 rep
,c*l- lrep
Measu. I rep
1.09
1.10
1.07
' 1.14
1.05
1.63
1.55
2.46
1.03
1.54
1,68
1.02
1.29
1.81
0.92
1.20
1.42 '
0.98
1.26
1.30
1.04
1.05
1.13
1.14
1.13
1.28
Calculated Ire p
by Warner's (Eq. 7.2)
I rep
(xio'inm")
0.328
0.343
0.356
0.373
0.389
0.612
0.470
0.297
0.524
0.423
0.324
0.524
0.462
0.400
0.559
0.523
0,487
0.598
0.585
0.573
0.656
0.684
0.713
0.768
0.801
Measu. I r e p
1.31
1.15
1.11
1.15
1,05
1,96
1.71
1.40
1.32
1.37
0.98
1.13
1.13
1.24
0.96
1.10
1.15
0.99
1.21
1.20
1.18 •
1,11
1.15
1.14
1.12
1.21
Calculated I_„ rep
by Authors' (Eq. 7.3) I rep
(xlO'mm'')
0.288
0.285
0.306
0.335
0.365
0.621
0.500
0.344
0.499
0.370
0.313
0.450
0.397
0.380
0.502
0.480
0.474
0.572
0.568
0.567
0.553
0.572
, 0.617
0.715
0.777
Cal- ^ep
Measu. I rep
1.15
0.96
0.95
1.03
0.99
2.00
1.82
1.62
1.25
1.20
' 0.94
0.97
0.97
1.18
0.86
1.01
1.12
0.94
1.18
1.19
0.99
0.92
1.00
•1.06
1.09
1.14
* E used were obtained from concrete stress-strain curves,
sr
II
E en
m
r-. e
cr w
cn
II
e cn r>.
e
cr w
CN
II
E cn e
i-
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CN •
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. cr w u CO c t-i 0) S
*—( •
i-*
* cr
o CO « CO iH CQ
T3 (IJ J-l 3 CO cd OJ a
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rH cd
a CU u H » cd cu S
cr sr o E •—I s X a CD u
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cr. sr o E •—' E X Ou cu U
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cn sr o E "—' B X a CU u
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cr o . r-H
r o cn • o
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cr cn in • o
r»» i—i
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sr cr • o
cr cn sr • o
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•
o
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m CN . rH
r vo m e O
r CN .
i-H
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o cn .
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rH
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PH PH
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r-~ 00 • o
m cn vO
d
vO 00
e
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CN cn vO • o
vO 00 • o
cr CN vO • o
CN 00 • o
r cr m • o
sr i-H
. i—l
m cn 00 » o
rH
m r>» » o
sr i o o en
PH PH
cr O .
z—i
m LO 00
d
CN
o .
i — 1
vO O 00
d
vO
cr . o
CN in r-~ • o
o sr • o
sr rH
cn • o
i-
cr .
rH
o m m .
i-H
r 00
r-. • o
m l o o cn
PH PH
in o rH
m o .
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cr I—I
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131
TABLE 17 COMPARISONS OF EQ. 7.3 WITH MEASURED I REINFORCED CONCRETE BOX BEAMS
Beam
R20O-1
R200-2
R200-3
R20O-4
R200-5
R300-1-2
R300-1-3
R300-1-4
R30O-2-2
R300-2-3
R300-2-4
R300-3-2
R300-3-3
R300-3-4
R300-4-2
R300-4-3
R300-4-4
R300-5-2
R300-5-3
R300-5-4
R450-1
R450-2
• R450-3
R450-4
R450-5
Measured
^ep (xlO9 mm")
0.251
0.297
0.322
0.324
0.369
0.311
0.275
0.212
0.398
0.308
0.332
0.464
0.410
0.323
0.582
0.476
0.424
0.607
0.485
0.477
0.558
0.619
0.620
0.675
0.712
Calculated I
Measured Irep
Eq. 7.3 (m»4)
(xlO9 mm")
0.233
0.261
0.291
0.328
0.361
0.618
0.422
0.260
0.425
0.308
0.281
0.399
0.374
0.370
0.478
0.471
0.470
0.567
0.567
0.567
0.462
0.525
0.590
0.706
0.775
Cal. Mea.
0.93
0.88
0.90
1.01
0.98
1.99
1.53
1.23
1.07
1.00
0.85
0.86
0.91
1.15
0.82
0.99
1.11
0.93
1.17
1.19
0.83
0.85
0.95
1.05
1.09
1.05
Eq. 7.3 (m=3)
(xlO9 mm")
0.255
0.268
0.295
0.329
0.362
0.619
0.457
0.290
0.457
0.328
0.289
0.417
0.380
0.372
0.485
0.473
0.470
0.569
0.567
0.567
0.495
0.539
0.597
0.708
0.775
Cal. Mea.
1.02
0.90
0.92
1.02
0.98
1.99
1.66
1.37
1.15
1.06
0.87
0.90
0.93
1.15
0.83
0.99
1.11
0.94
1.17
1.19
0.89
0.87
0.96
1.05
1.09
1,08
Eq. 7.3 (m=2) (xlO9 mm")
0.288
0.285
0.306
0.335
0.365
0.621
0.500
0.344
0.499
0.370
0.313
0.450
0.397
0.380
0.502
0.480
0.474
0.572
0.568
0.567
0.553
0.572
0.617
0.715
0.777
Cal. Mea.
1.15
0.96
0.95
1.03
0.99
2.00
1.82
1.62
1.25
1.20
0.94
0.97
0.97
1.18
0.86
1.01
1.12
0.94
1.18
1.19
0.99
0.92
1.00
1.06
1.09
1.14
133
"TABLE 18 DEVIATIONS OF CALCULATED I„a OF rep
REINFORCED CONCRETE BOX BEAMS
Method
Eq. 7.1
Eq. 7.2
Eq. 7.3 (m=2)
Eq. 7.3 (m=3)
Eq. 7.3 (m=4)
Maximum Deviation
+ 1.46
+ 0.97
+ 1.00
+ 0.99
+ 0.99
Average Deviation
+ 0.2813
+ 0.2124
+ 0.1355
+ 0.0802
+ 0.0802
Standard Deviation
0.4387
0.3035
0.3059
0.2686
0.2527
NOTE: 1. Maximum Deviation = the maximum value of (Calculated I - Measured I )/Measured I
rep rep rep
Average Deviation = the average value of (Calculated I - Measured I )/Measured I
rep rep rep
3. Standard Deviation = /[£(-
Cal. I - Mea.I rep rep_ Mea. I
rep )2]/N
TABLE 19 DEVIATIONS OF CALCULATED Ir e p FOR PARTIALLY PRESTRESSED CONCRETE BOX BEAMS
Methods
Eq. 7.1
Eq. 7.2
Eq. 7.3
m =
m =
m =
2
3
4
Maximum Deviation
+ 0.97
- 0.60
+ 0.30
4- 0.27
+ 0.25
Average Deviation
+ 0.19
- 0.07
+ 0.06
+ 0.05
+ 0.05
Standard Deviation
0.44
0.32
0.17
0.14
0.13
NOTE: 1. Maximum Deviation = the maximum value of (Calculated I - Measured I )/Measured I
rep rep rep
Average Deviation = the average value of (Calculated I - Measured I___)/Measured I
rep rep rep
/ C a l . I - Mea.I 2 ,
[Z( zS—i ^ ]/ rep '
N
a. cu u
Q Cd
52
CJ
Pn O CO
S3 O H H & PH
H Pi H CO H
o CJ
w c w PH PH
O CN
59
B-S o ^ i
+ 1 &-« o rH
+ 1 B-8 Ln, + 1
00
•H 4-1 cd
E •H
4-1 CQ CU 1
u cu T3 C P
00 c •H 4J
cd E •H 4J cn cu l u cu > o
s-e o sr (H
OJ > 0
B-S o sr i o cn
S-9 o cn I o CN
6-S O CN 1 O rH
B-9 O r-i
1 m
B»S in i O
B-9
m I o
B-« O r-< 1 m
B-S o CN 1 o rH
6>« O cn l o CN
B*S O sr i o cn
B*« O sr
u cu > 0
Tj o Xi 4-1
.2! S
r>S o VO
B-S o sr
B-S sr CN
O
o
o
o
B>8 sr
6>8 sr
B*8 O CN
B-8 CN rH
B>8 O CN
&•« vO •-H
B*8 CN i-i
B*8 CN rH
rH
• r> e
cr w
B«S vO r>.
6-S 00 CN
&•? vO •—I
O
o
o
o
o
B-S CN i-l
BS sr
B«S CN rH
6*9 CO sr
6-5 vO i-i
O
B-« 00
CN • r
•
cr w
B-S oo 00
B-S vO m
6-S vO cn
o
o
o
B-S sr
B-8 CN rH
B-S sr CN
B>8 CN rH
S-8 CO
B-S 00 CN
O
B-S sr
B-S 00
-\ CN II E <>-/
cn • r*.
•
cr Id
B-S 00 00
B-S 00 sr
B^ sr CN
o
o
o
B-S sr CN
B-S CN i-4
S-S CN r-l
B-S CN rH
B-S CN rH
B-8 vO •-H
S-S sr
B-S sr
B-S sr
/—\ en M S >»/
cn • r*« • cr w
B-S sr 00
B-S 00 sr
B-S sr CN
o
o
B-8 00
B-S o CN
B-S CN rH
B-S CN i—l
B-S CN rH
B-5 CN rH
B-8 vO rH
o
B-8 sr
B-8 sr
/—N
sr !L e >—>
cn • r-~ • cr Ed
135
136
TABLE 21 FREQUENCY DISTRIBUTIONS OF CALCULATED I FOR PARTIALLY PRESTRESSED CONCRETE BOX BEAMS
Method
Eq. 7.1
Eq. 7.2
Eq. 7.3
m=2
m=3
m=4
Over-estimating
>30%
20%
20-30%
20%
20%
20%
20%
10-20%
20%
20%
20%
0-10%
20%
20%
20%
60%
60%
Under-estimating
0-10%
40%
20%
10-20%
20%
20%
20%
20%
20-30% >30%
20%
137
Chapter 8
TOTAL DEFLECTION OF CONCRETE BOX BEAMS UNDER REPEATED LOADS
8.1 The Proposed Computational Procedure
The formula for computing the total deflection at the Tth cycle,
of load may be rewritten from Eq. 3.13 as.
UJVJiL
cc reD
where §dj and lrep can be computed using Eqs. 5.6 and 7.3 (with
m=2y. k is given by Eq. 6.8 for reinforced beams and by Eq. 6.10
for partially prestressed beams. Note that all the equations are
simple and explicit and manual calculations can readily be carried
out. Numerical examples for both reinforced and prestressed box
beams are given in Appendix VII. However, because of the large scale
investigation in this study, requiring the results of 5T for many
dozens of beams at various loading cycles, computer programs have
been written for carrying out the calculations. Details of the
programs are also given in Appendix VII.
The accuracy of Eq. 8.1 is checked in Section 8.3 in light of the
box beam data and those available in the literature. Section 8.2
reviews in detail two other prediction methods published by Balaguru
and Shah C1982] and Lovegrove and El Din [19821 Comparisons with
these two prediction procedures are made in Section 8.4.
138
8.2 Other Prediction Procedures
Two other methods for predicting the total deflections of
reinforced and prestressed concrete beams have been published very
recently. They are due to Balaguru and Shah [1981.1982] and to Lovegrove
and El Din [1982]. Balaguru and Shah developed their procedure
based on published methods for predicting fatigue properties of
the component materials including concrete, prestressing and non-
prestressing steel. They did not carry out any experimental study.
Lovegrove and El Din on the other hand derived their equation from
their own test results on 12 reinforced concrete beams,
8.2.1 Balaguru and Shah's method
Balaguru and Shah introduced a concept of cyclic creep. They
suggested that concrete beams under fatigue loading not only undergo
time creep but also 'cyclic creep'. Cyclic creep is cyclic or repeated
loading dependent. The two main steps of their procedure are the
calculations of the concrete strain considering both time creep and
cyclic creep, and the 'cracking moment under fatigue load'. The
cracking moment of concrete beams in Balaguru and Shah's proposal
is also cyclic dependent. The cyclic strain in concrete is.
£c = 129amt3 + 17.8c7mc7T
3 (8.2)
where, ec is the cyclic creep strain in micro-strain, a is the
stress range expressed as a fraction of the compressive strength
=(amax-aminVF'c, am is the mean fibre stress expressed as a fraction
of the compressive strength =(o\„a + o\„:_V2F' , T is the number of
cycles, and t is the duration of loading in hours.
139
According to Balaguru and Shah, the cracking moment under
fatigue load is dependent on the tensile strength of concrete under
fatigue load. They suggested that the tensile strength of concrete
under fatigue load.
loa1flT 'r.T"rt1-io55r) ^
where, fr is the static tensile strength of concrete.
Thus, the cracking moment.
M 0 , T = !ail (8.4)
where yt is the distance form the bottom fibre to the neutral axis.
After calculating the cracking moment the effective moment of
inertia is calculated by the ACI equation [1963]. as follows.
k T ^ . T + C ^ 1 ) V'cr.T'S « *
Balaguru and Shah indicated that the calculation of lcrT is the
same as calculating lcr only that E c is now replaced by E c T , where
p _ = g m a x (8.6) CC,T rt
Ec c
140
which is a function of the creep strain. ec (given by Eq. 8.2) and
the maximum stress level. o"max.
Finally, the total deflection under fatigue load is computed as.
GCM+L2
5 = I (8.7) T Ec.T'e.T
The accuracy of Eq. 8.7 is compared in Section 8.3. In the mean
time the following comments are relevant:
a. Cracking moment should not be cyclic dependent as a
concrete beam would crack in the initial load cycle if
the load exceeds the cracking moment capacity of the
beam. Thus, how can repeated loading influences the
cracking moment if the beam was already cracked?
b. If T>1011. Eq. 8.3 gives negative tensile strength of
concrete.
c The basis for calculating lcr#T is not clear. Balaguru and
Shah suggested that lcr#T shall be calculated as lcr but
using E c T instead of Ec. However, the calculation of lcr
needs to use the depth of the neutral axis. kd. They did
not say whether Ec or Ec T shall be used to calculate
kd. If E c T is to be used, because it is dependent on
the creep strain ec (Eq. 8.6) then to calculate this ec.
the depth of the neutral axis is needed. Needless to
say, we would have a chicken and egg situation! In the
subsequent comparisons in Section 8.4. Ec is used for
computing kd and E a T for lcrT.
141
8.Z2 Lovegrove and El Din's Formula
Lovegrove and El Din [1982] proposed a very simple method. They
ignore all variables except T and simply relate dT and the initial
total deflection 5Tj as follows:
8T = 0.225STi tog10T (8.8)
where STi is defined in Fig. 7 and can be calculated using any
equation listed in Section 5.2.
The validity of Eq. 8.8 is checked in Section 8.4 and some
comments are offered herein:
a. The equation is not valid for T< 27826 cycles as it would
yield ST < djr
b. The proposed procedure by Lovegrove and El Din is very
rough. In their paper [1982], test results of Snowdon.
[1971] were used to compare with Eq. 8.8. However,
in Snowdon 's tests, there were two types of repeated
loads i.e. with constant and step-increasing loading
ranges. In the comparisons they failed to distinguish
the difference of these two types of loading.
142
8.3 Comparisons with Experimental Data
In order to verify the proposed prediction method, the author
not only made comparisons with his own experimental data but also
with those obtained by other researchers. However, despite the
large amount of published work on repeated loading tests, only a
few results are acceptable for comparisons. This is due to the
fact that most published materials lack the beam details, material
properties and/or loading histories. Furthermore, tabulated results
of deflections can not always be found. As a result, apart from
the author's own data, only the total mid-span deflections of the
beams tested by Sparks and Menzies [1973] and Bennett and Dave
[1969] are included. The calculated total deflections are compared
with the corresponding measured values. A total of 199 data points
are compared and the correlations are analysed statistically.
8.3.1 Reinforced concrete box beams (150 points)
The measured total deflections at mid-span of all the reinforced
concrete box beams tested by the author are plotted in Fig. 31
against the values calculated using Eq. 8.1. Note that different
symbols are used for the different series of beams. In total. 150
points are included. It may be seen that a good correlation exists
between the calculated and the measured values. The scattering is
reasonably low. Most of the correlation points are well within the
±20 percent limits. In general, the prediction tends to be slightly
conservative.
8.3.2 Partially prestressed concrete box beams (30 points)
A total of 30 measured values of total deflection were obtained
from the author's repeated loading tests on 5 partially prestressed
143
box beams. The calculated deflections are compared with the measured
values in Fig. 32. The correlation, in general, is not as good as for
the reinforced box beams although the majority of the points are
still within ±20 percent limits and the scattering is acceptably low.
It may be seen from Fig. 32 that the proposed method gives better
predictions for the beams with a lower degree of prestressing.
8.3.3 Sparks and Menzies' reinforced concrete beams (10 points)
The test results obtained by Sparks and Menzies [1973] are used
here to check the accuracy of the proposed method of analysis.
Details of the 10 reinforced concrete beams are given in Table 22. All
the beams had the same rectangular sections and were tested under
repeated loading of constant range. Only the total deflections for
10° cycles of load were recorded (except for beam T. it was for 10
cycles). Plots of the calculated deflections versus the measured
values are presented in Fig. 33. As can be seen, the agreement is
favourable, with all except two correlation points fall within the
±20 percent limits. However, the prediction generally tends to be
on the underestimating side.
8.3.4 Bennett and Dave's prestressed beams (9 points)
The details of the 9 partially prestressed concrete beams
tested by Bennett and Dave [1969] are given in Table 23. All the
beams had the same rectangular section of 127 mm x 203 mm. The
combined amount of prestressing and non-prestressing steel was
different for different beams. Thus the degree of prestressing also
varied. Note that the measured total deflections were reported
only at 3x106 cycles of repeated load. They are compared in Fig. 34
with the calculated values using Eq. 8.1. It is clear in Fig. 34
that the method underestimates most the of beam deflections. It
144
is difficult to attribute the exact reasons for such discrepancy.
However, the time required to apply 3X10? cycles of load could be
quite long. Thus Bennett and Dave's deflection results contain a
component of time-creep. If such time dependent deflections are
included in the computed values, the correlation would be improved.
It should also be noted that the proposed method is derived from
beam tests up to 105 cycles only. From 105 to 3X106 cycles represents
a considerable and may be inaccurate extrapolation. Fortunately, as
discussed in Section 4.5. the number of full service loads experienced
by a main bridge girder, for example, is more likely to be in the
hundred-thousand level than in the million level.
8.33 Overall accuracy
An overall picture of the accuracy of the proposed prediction
method can be seen in Tables 24 and 25.. where the frequency
distributions of the errors and four statistical coefficients are
presented respectively. The four coefficients are the correlation
coefficient, r, the intercept, A the slope. B and the standard error
of estimation. The correlation coefficient, r indicates the spread
of the correlation points: the closer the value of r to 1 the smaller
the scattering. Intercept, A and slope. B denote the closeness of
the regression line of the correlation points to the perfect 45°
line. Therefore the smaller the value of A and the -closer B to
1 indicate a better accuracy of the prediction. The equation for
the standard error of estimation is given in the footnote of Table
25. All other being equal, the smaller the standard error the more
accurate is the prediction method.
It may be seen in Table 24 that if presented graphically the
frequency distributions would generally have a tall and narrow bell
shape. This indicates that the probability of the proposed method
in giving accurate prediction is high. The possible reasons for the
145
method to perform poorly for the beams of Bennett and Dave have
been discussed in the preceding section. Table 25 shows that, except
for Bennett and Dave's results, the regression lines are close to
the 45° line which further confirm the reliability of the proposed
method of analysis based on Eq. 8.1.
146
-
-
1/5
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152
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153
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154
8.4 Comparisons with Other Prediction Methods
In light of all the available test results (a total of 199 points)
the proposed method is also compared with the methods proposed by
Balaguru and Shah [1981.1982] and Lovegrove and El Din [1982], The
details of the two methods have been given in Section 8.2. In order
to have a fair comparison, Lovegrove and El Din's method was used
only for deflections at or over 105 cycles. Figs. 35. 36 and 37
respectively give the correlations of the measured total deflections
and those computed using Eq. 8.1. Balaguru and Shah's method and
Lovegrove and El Din's formula.
It may be seen in Fig. 36 that Balaguru and Shah's method
generally underestimates the reinforced bos beam deflections and it
does not correlate well at all with Bennett and Dave's test results.
For Lovegrove and El Din's formula, the predictions tend to be
slightly unsafe as evident in Fig. 37, Despite these observations,
it is not easy by comparing Figs. 35 36 and 37 to identify the best
amongst the three prediction procedures. However, the superiority
of the proposed Eq. 8.1 is revealed from a statistical scrutiny.
Table 26 gives the correlation coefficients and the intercepts and
slopes of the regression lines of the three correlations. It is
evident that Eq. 8.1 is statistically more reliable than Balaguru
and Shah's method. The probability of Eq. 8.1 in giving accurate
results for 199 points is as high as Lovegrove and El Din's method
for only 49 points. For the same number of predictions. Eq. 8.1
would be better.
A further comparison of the three methods is made in Fig.
38 in light of the author's own deflection data for T=105 only
(i.e. 30 points). The correlation statistics are given in Table 27.
Predictions by Eq. 8.1 appears to be the most satisfactory, closely
followed by Lovegrove and El Din's method. The method of Balaguru
155
and Shah, the most complicated amongst the three, again ranks third
in this comparison.
156
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44
s
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Computed 6T (mm)
Computed 6 (mm)
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Computed 6 (mm)
(c) Lovegrove and El Din's method
160
44
Figure 38 Comparisons between measured total deflections of reinforced and partially prestressed box beams (10^ cycles
161
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162
Chapter 9
CONCLUSIONS
Conclusions of this study are mainly drawn in the following
three areas:
a. the deflection behaviour of reinforced and prestressed
concrete box beams under repeated loading:
b. the derivation of the proposed analytical procedure for
the total deflections under repeated loading: and
c. the accuracy of the proposed procedure.
They are given in Sections 9.1. 9.2 and 9.3 respectively with
recommendations for further study listed in Section 9.4,
9.1 Deflection Behaviour
163
It is obvious from the test results that while the live load
deflection remains constant after the initial loading, the dead load
deflection accumulates under repeated loading. The increase of
dead load deflection is significant in the early loading cycles but
it slows down rapidly as the number of loading cycles increases.
The accumulation of dead load deflection can be seen as a parallel
phenomenon to the time-creep of concrete under sustained loads. The
similarity of the effects of repeated load and sustained load have
been observed by the author as well as by some other researchers.
However, the author's is the first attempt to make use of such
parallel to establish a prediction procedure. It is the recognition
of this analogy between the repeated loading and time-creep effects
that led to the development of the proposed analytical procedure.
Live load deflection is caused by the fluctuating part of the
applied load. It was observed that this portion of the deflection
remained constant throughout the repeated loading cycles provided
that the maximum level and the range of the repeated load are kept
constant. This is because the bending rigidity of the beam is not
affected by the number of loading cycles after the initial loading.
However, the increase in maximum loading level or repeated loading
range leads to a deduction in the beam rigidity. These were
consistently observed in the author's tests.
Steel content in both reinforced and prestressed concrete beams
affects the amplification of the dead load deflection. It was found
that the higher the steel ratio the smaller the effects of the
repeated loading. Depending on the steel content, the intensive
creep factor can be as high as about 8. This represents a seven-fold
increase in dead load deflection due to repeated loading. However.
164
with a steel ratio of 2%. the increase was found to be limited to
about 100% or less for most of the reinforced box beams tested.
In addition to the steel content, the higher the degree of
prestressing the lower the increase in dead load deflection under
repeated loads.
165
9.2 The Proposed Computational Procedures
A simple procedure is presented for the prediction of the
total deflection of concrete box beams under repeated loading.
The procedure involves the computations of the initial dead load
deflections. §dj. the intensive creep factor, k and the instantaneous
live load deflections. St.
For computing 8dj. many proposals have been published previously
by other reseachers. Nearly all proposals are ssmi-empiricai. In light
of the author's own test results on box beams, a comparative study
is carried out involving 9 well-known methods. It is concluded
that, Branson's effective moment of inertia method gives the most
satisfactory results. Thus, his method is recommended for computing
the initial dead load deflections, 8dj.
Multiple regression analyses have been carried out on the
results of dead load deflections at various loading cycles. This
led to the establishment of a logarithmic equation for
calculating the intensive creep factor, k. This logarithmic equation
is found to be more realistic than a similarly derived hyperbolic
formula. To account for the effects of prestressing. a modified
formula is recommended for k.
A reliable evaluation of the live load (instantaneous) deflection.
8( is dependent on the accurate assessment of the effective moment
of inertia of the beam under repeated loads, lreD. A new formula
for lreD is proposed, which gives better all-round accuracy than
the existing methods due to Kripanarayanan and Branson [1972] and
Warner and Pulmano [19801
The combined use of Branson's procedure for §dj. the logarithmic
equation for k and the new formula for lreD allows the total deflection
of reinforced and prestressed concrete beams at any number of
166
repeated loading cycles to be calculated. All equations are simple
and explicit which are suitable for adoption in design codes.
167
9.3 Versatility and Accuracy of the Proposed Procedure
The proposed procedure covers directly the effects of all major
variables which influence the deflections of concrete beams under
repeated loading. In comparison. Balaguru and Shah's method ignores
the steel ratio and the degree of prestressing while Lovegrove and
El Din's formula only accounts for the number of loading cycles
(which must be greater than 27826).
The proposed method is also more versatile in that it may be
used for computing the instantaneous deflection and/or the permanent
set as necessary. The other two methods only calculate the total
deflections.
Based on experimental data obtained herein and published by
other researchers, a detailed comparison is made between the proposed
procedure and the methods of Balaguru and Shah, and of Lovegrove and
El Din. Statistical analyses of the correlations between measured
and predicted total deflections indicate that the probability of the
proposed procedure in giving more accurate results is the highest
amongst the three procedures.
168
9.4 Recommendations for Further Study
In future attempts to study further the deflection behaviour
of concrete beams under repeated loading, the following should be
investigated:
a. the effects of compressive reinforcement:
b. the effects of negative reinforcement at supports of
continuous beams:
c, the effects of repeated loading range particularly the
influences of the lower load limit: and
d. the applicability of the proposed procedure for beams
reinforced with plain bars, especially the case of
overloading conditions.
169
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181
Appendix I
EQUATIONS FOR COMPUTING kd, Mcr, M AND Mu
182
EQUATIONS FOR COMPUTING kd. M . M AND M * cr ' y '»"*' t
Depth of Neutral Axis, kd
For the box section detailed in Fig. AI-lfit can be shown that
kd =/[np-r(8-l)r]2 + 2[np +(B-l)r2/2 ] - [np + (8-l)r]d (Al)
where
n = E /E , p = A /b d, 8 = b/b and r = t/d s c' *i st w ' w
After computing kd, jd can be determined by simple statics.
Cracking, Yield and Ultimate Moments
The formulas used in this paper are those given in the Australian Code, The cracking moment,
M = FT /y<. (A2) cr t g Jt
where
F is the tensile strength of concrete « 0.62 /IP MPa, t c
I is the moment of inertia, neglecting reinforcement, of gross concrete section about the centroidal axis, and
Y is the distance from centroidal axis of gross cross-section, neglecting reinforcement, to extreme fibre in tension.
The yield moment,
i y sy st M__ = f___ A _ j d ( A 3 )
where
f is the yield or proof stress of reinforcing steel,
A is the cross sectional area of tensile reinforcement, and st
jd is depicted in Fig. Al-land can be obtained via Eq. (Al) ,
183
The ultimate moment,
M
0.6 A , f = A f d[i-__2£.«Z]
st sy bd F c
(AA)
if A < 0.85 F' bt/f 11 st c sy
And if Ast > 0.85 F^ bt/fgy, then
V f Mu = 0.85 fj t(b-bw)(d-0.5t)-rfsyAht d [ l - 0 . 6 n - f ]
w c
(A4a)
where F' c A - A - 0.85 -£- (b-b )t
Tit st f w sy
and the variables b, b , d and t are illustrated in Fig. AI-1 w
N.A.
Figure AI-1 Typical box section
The Explanation of M < M for Heavily Reinforced Beams — u y ~ .
184
f.
kd
r f„
hkd 'C
jd
+-T
0-85F;
rkd -a\
fsy
<r^~C
jud
->r
(a) Stress distribution (b) Stress distribution at at first yield failure, using Whitney
rectangular stress block Figure AI-2 Stress distribution of under-reinforced concrete beams
Assumptions:
(i) My is taken as the moment when, steel first yields. Measured My is usually obtained from the Moment-deflection or Moment-curvature curves.
(ii) M^j is taken as the maximum bending moment the beam can take before it collapses.
(iii) In both stages tensile stress in reinforcement are taken as f . sy
(iv) For under-reinforced beams when it first yields the concrete stress fc should be less than maximum stress 0.85 F .. The compressive stress block is a triangle.
(v) When the beam collapses, compressive concrete stress block is taken as the so called Whitney stress block (rectangular stress
block with depth yk^d less than the depth of N.A.).
In both stages tension force:
T - f A sy st
A M = f A jd y sy st
M = f A j d u sy st u
Provided p ^ p , the beam is still under-reinforced If i > i , then M > M . " " y u
185
Tabulated j values and j u values for rectangular beams are given in TablesAI-1 and AI-2Jt can be seen from the Tables that j > j u for beams having p close to pma x. Note that comparisons between j and j u
shall be made for beams with same np values. Therefore, for these beams M > M .
y u
TABLE AI-1 j VALUES FOR REINFORCED CONCRETE BEAMS
pn
0010 0020 0030 0040 0050 0060 0070
0072 0074 0076 0078 0080
0082 0084 0086 0088 0090
0092 0094 0096 0098 0100
0102 0104 0106 0-108 0110
0-112 0114 0116 0-118 0120
0122 0-124 0126 0128 0130
k
0132 0181 0-217 0-246 0-270 0-291 0-311
0-314 0-318 0-321 0-325 0-328
0-332 0-336 0-338 0-341 0-344
0-347 0-350 0-353 0-356 0-358
0-360 . 0-363 0-366 0-369 0-372
0-375 0-378 0-380 0-382 0-384
0-387 0-389 0-392 0-394 0-396
./'
0-956 0-940 0-928 0-918 0-910 0-902 0-896
0-895 0-894 0-893 0-892 0-891
0-889 0-888 0-887 0-886 0-885
0-884 0-883 0-882 0-881 0-881
0-880 0-879 0-878 0-877 0-876
0-875 0-874 0-873 0-872 0-872
0-871 0-870 0-870 0-869 0-868
pn
0-132 0134 0136 0138 0140
0142 0144 0146 0-148 0150
0152 0154 0156 0158 0160
0162 0164 0166 0168 0-170
0172 0-174 0176 0178 0180
0182 0184 0186 0188 0-190
0192 0194 0196 0198 0-200
0-210
k
0-398 0-401 0-403 0-405 0-407
0-410 0412 0-414 0-416 0-418
0-420 0-422 0-424 0-426 0-428
0-429 0-431 0-433 0-435 0-437
0-439 0-440 0-442 0-444 0-446
0-448 0450 0-452 0-453 0-455
0-457 0-458 0-460 0-462 0-463
0-471
./
0-867 0-867 0-866 0-865 0-864
0-864 0-863 0-862 0-861 0-861
0-860 0-860 0-859 0-858 0-857
0-857 0-856 0-856 0-855 0-854
0-854 0-853 0-852 0-852 0-851
0-850 0-850 0-849 0-849 0-848
0-848 0-847 0-847 0-846 0-846
0-843
186
TABLE AI-1 (Cont'd)
pn
0-220 0-230 0-240 0-250
0-260 0-270 0-280 0-290
k
0-479 0-485 0-493 0-499
0-506 0-513 0-519 0-525
0-840 0-838 0-836 0-834
0-831 0-829 0-827 0-825
pn
0 300
0-350 0-400 0-450 0-500
A
0-531
0-557 0-580 0-600 0-618
/
0-823
0-815 0-807 0-800 0-794
TABLE AI-2 j VALUES OF REINFORCED CONCRETE BEAMS
*sy (MPa)
230
410
450
F ' rc (MPa) 20 25 30 40
20 25 30 40
20 25 30 40
p max
0.034 0.042 0.051 0.061
0.016 0.020 0.024 0.028
0.014 0.017 0.021 0.025
np
0.323 0.357 0.383 0.397
0.152 0.170 0.180 0.182
0.133 0.145 0.158 0.163
Ju (mm)
0.770 0.773 0.770 0.794
0.807 0.807 0.807 0.831
0.815 0.820 0.815 0.835
187
Appendix II
COMPUTER PROGRAM FOR DESIGNING
PARTIALLY PRESTRESSED CONCRETE BOX BEAMS
188
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Appendix III
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216
Appendix IV
VARIATIONS OF MAXIMUM CRACK WIDTH
217
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223
Appendix V
MEASURED TOTAL DEFLECTIONS AND MOMENT-DEFLECTION CURVES
224
TABLE AV-1 MEASURED TOTAL DEFLECTION OF REINFORCED BOX BEAMS AT MID SPAN, 6 (IN MM)
Beam
R200-1
R200-2
R200-3
R200-4
R2Q0-5
R300-1-2
R300-1-3
R300-1-4
R300-2-2
R300-2-3
R300-2-4
R300-3-2
R300-3-3
R300-3-4
R300-4-2
R300-4-3
R300-4-4
R300-5-2
R300-5-3
R300-5-4
R450-1
R450-2
R450-3
R450-4
R450-5
No. of Loading Cycles
1
12.03
20.39
23.68
23.47
25.06
3.90
7.65
11.13
6.64
9.03
13.40
9.56
12.76
19.42
9.02
13.67
17.14
11.50
15.82
23.12
15.83
19.14
20.69
26.06
28.46
10
15.39
23.03
26.31
25.50
26.38
5.18
8.10
11.40
7.81
9.09
13.74
9.69
13.00
20.62
9.16
13.88
17.53
11.74
16.05
23.69
16.54
20.09
21.18
26.42
28.85
IO2
15.56
23.42
26.90
25.73
26.63
5.14
8.39
11.44
8.19
9.41
14.53
9.84
13.26
21.06
9.31
14.29
17.83
11.81
16.24
24.21
17.05
20.69
21.44
26.99
29.18
IO3
15.77
23.49
27.07
25.86
26.89
5.15
8.54
11.73
9.09
9.87
14.84
9.96
13.71
21.58
9.44
14.50
18.22
12.00
16.43
24.80
17.52
21.32
21.80
27.61
29.62
IO4
15.98
23.67
27.32
26.56
27.10
5.16
8.72
11.95
9.38
10.56
15.01
10.11 •
13.88
21.73
9.64
15.04
18.72
12.25
17.13
25.97
18.16
22.03
22.04
28.01
30.30
IO5
16-. 96
23.70
27.97
27.22
29.00
5.23
9.52
12.10
8.53
10.61
15.37
10.41
13.90
22.63
10.14
15.60
20.16
12.80
17.76
26.20
18.36
22.39
23.06
28.49
30.72
225
TABLE AV-2 MEASURED TOTAL DEFLECTION PRESTRESSED BOX BEAMS AT MID-SPAN, 5 (IN MM)
Beam
PP300-1
PP300-2
PP300-3
PP300-4
PP.300-5
Number of Loading Cycles
1
7.04
6.10
4.94
4.19
4.32
10
7.18
6.14
5.09
4.28
4.41
102
7.37
6.60
5.26
4.34
4.48
103
7.59
7.38
5.49
4.45
4.52
io4
7.83
7.68
5.72
4.63
4.70
IO5
8.54
8.36
6.25
5.16
5.13
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Appendix VI
EXAMPLES FOR THE USE OF SPSS PACKAGE
AND EQUATIONS OF STATISTICAL COEFFICIENTS USED IN SPSS
REGRESSION EQUATIONS USED IN STATISTICAL ANALYSIS PACKAGE 257 SPSS [See, NIE, HULL AND BENT, 1975]
(1) For bivariate regression:
Y' = A + BX
where,
A =
N v T-
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B =
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in which, X£ is the <Lth observation of variable X, Y^ is the .tth observation of variable Y and N is the number of observations.
The equation for computing Pearson correlation coefficients is
$LIXX? - ( O W E . , * - ft., y)2M 1/2
(2) For multiple regression
where,
Y' = A + B]Xi + B'2X2
SPvl(SS2)-SP,,,(SP12) B ~ SS,(SS2)-SP22
SS1(SS2)-SP2:
A = Y- BlXl - B2X2
in which, SS and SP stand for sum of squares and sum of products, or variation and covariation respectively, e.g.
ssi= I (xu - V2 and spi2= £(xirV(x2rV
The equation for computing partial correlation coefficients is
ry\~ry2r\2 ry\.2
258
/^(/^I) where Y ,» Y o > Y,.-. and Y _ are the Pearson correlation
yl y2 '12 y2 coefficients between two variables.
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Appendix VII
NUMERICAL EXAMPLES AND COMPUTER PROGRAMS
FOR THE COMPUTATION OF <5T USING THE PROPOSED METHOD
283
NUMERICAL EXAMPLES
For beam R 450-1^
b = 450 mm, bw = 120 mm, t = 50 mm, D = 305 mm, d = 270 mm, L = 5,700 mm;
Ast = 1'400 ™2'P = Ast/bd = °-0115. fs = 323 MPa, Eg = 2 x 105 MPa;
F' = 40 MPa, F = 0.62 /F~= 3.92 MPa, E = 31975 MPa-c. u c c
n = 6.5
The values of kd, M , M and M can be obtained using Eq. (Al), (A2), (A3) and (A4) respectively. Thus, ^ u
kd = 92.18 mm
M = 21.26 kNm cr
M = 110.93 kNm y M = 115.28 kNm
u For T = 10s cycles, M. = 0.3 M = 33.28 kNm and M^ = 0.53 M - 58.79 kNm,
d y t y ' the total deflection 6-, can be computed following the steps given below. (1) Moments of inertia
The gross and cracked moments of inertia are,
I = bD3/12 - (b-b )(D-2t)3/12 ft W
and
= 8,27 x 10s mm1*
I = b(kd)3/3 - (b-b)(kd-t)3/3+nAe rd-kd) LI W o L.
= 3.97 x 108 mm'4
The effective moment of inertia at M = M , d
I . - (M /M,)3 I +I1-(M /M,)3]I e,d cr d g cr d J cr = 5.09 x IO8 mm"
284
and at M = M
Xe,t " (Mcr/Mt)3 Ig+U"(Mcr/V,]1cr
= 4.17 x IO8 mm"
For computing I , Eq. 7.3a gives
M = 36.97 kNm x
Then with I = 1 , Eq. 7.3(with m=2) yields
I = 5.53 x 10B mm" rep
(2) Initial dead load deflection
For the third point loading, the deflection at mid-span,
6,. = 0.10648 ML2/E I . di d c e,d
= 7.07 mm
(3) Intensive creep factor k
The values of k. , and R are determined using Eq. 6.8a and 6.8b respectively.
They are,
k = 1.897
and R = 0.0371
Thus for T = IO5 cycles, Eq. 6.8 yields
k = 2.083
(4) Total deflection
Finally, employing Eq. 8.1, the total deflection is computed as,
6 - 2.083x7.07 + P J 0648(58,79 - J3.28) xlO6 x 570Q2 T 31975 x 5.52x10s
= 14.7 + 5.00
= 19.7 mm
285
For Beam PP300-3:
b = 305 mm; bw = 120 mm; t - 60 mm; D = 305 mm; L = 4350
mm; Ast -- 603 mm2; Ap = 118mm2; Aeq = 721mm
2; P -
0.0088; F' -33.11*.; F.-3.56MP,; Ec = 28.1 x loVa;
H=0.6; Pe = l41 kN; M ^ = 39.51 kNm; My = 85.37 kNm;
and kd = 83.16 mm
For T = 105 cycles, Md = 0.3 My = 25.61 kNm and M -0.5
My = 42.69 kNm, the total deflection 6T can be com
puted by applying the following steps:
(A) Moments of Inertia:
(a) For gross section,
bD3 (b-b.w)(D-2t)3
g " 12 12 = ^*2^ x 1® mm k
(b) For cracked section,
_ _ b(kd)3 (b"dw:
3 o + n A (d-kd) 2 _ b(kd)3 ( b - y (kd-t 3
c r 3 3 eq
= 2.44 x 108 mm"
(c) For the effective moment of inertia at M=M
Eq. 5.6 gives,
I .. = 5.45 x IO8 mm" e,t
(d) For computing 1 ^ , Eqn. y>3a g i v e g
M = 39.73 kNm - "
Then from Eq. 7.3 (with m=2)
I = 6.23 x 10s mm" rep
(B) Initial dead load deflection:
Under dead load, the beam is uncracked, thus
5,. = 0.10648 M-L2/E I =2.99 mm. di d c g
Note that if the section was cracked, the effective moment of inertia at M = MJ should be used.
(C) Intensive creep factor, k:
With n = 0.60, Eq. 6.10a gives P = 1.69
Then for T = 10s cycles Eq. 6.10 yields k = 1.61
(D) Total deflection:
Finally, 6T is given by Eq. 8.1 as,
oMgL2
5T =k5di + E-T- = !-61x2-99
c rep 0.10648 x(42.69 - 25.61) x 43502 x106
+ 28.1 x 103x6.23 x IO8
= 4.81 + 1.98
= 6.79 mm
287
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PAPERS PUBLISHED BASED ON THIS THESIS
Wong,Y.W. and Loo.Y.C. "The Accumulative Effects of Repeated
Loading on the Deflection of Reinforced Concrete Box
Beams", proceedings, 7th Australiasian Conference on the
Mechanics of Structures and Materials, May 1980, Perth,
pp97-102.
Wong,Y.W. and Loo.Y.C. "A Study of Deflection Behaviour of
Reinforced Concrete Box Beams under Repeated Loading".
AIT Research Report Mo. 153. Asian Institute of Technology.
May 1983. Bangkok, 40pp.
Loo.Y.C. and Wong,Y.W. "Intensive Creep Deflection of Rein
forced Concrete Box Beams under Repeated Loads". Proceed
ings. 8th Conference on Concrete & Structures. August
1983. Singapore. ppYCL 1-12 .
Loo.Y.C. and Wong.Y.W. "Analysis of Total Deflection of
Reinforced Concrete Box Beams under Repeated Leading".
Journal of the American Concrete Institute. Proceedings
Mo. 1 Vol. 81 January-February 1984, Detroit. pp87-94.
Wong.Y.W. and Loo.Y.C. "Intensive Creep Deflection of Par
tially Prestressed Concrete Box Beams" Department of
Civil and Mining Engineering Research Report ST 84/2.
University of Wollongong. August 1984. Wollongong. 4pp.
301
AUTHOR'S CURRICULUM VITAE
The author was born in 1944 in Shanghai. China where he also
received his early education. In 1967 he obtained his Bachelor of
Engineering degree in Structural Engineering from Tianjin University.
Tianjin China. From 1967 to 1973 the author worked as a structural
and construction engineer for the Department of Industry in Tianjin
County. From 1973 to 1974 he worked in Hong Kong and Macao as a
structural and construction engineer for Chee Lee Investment Co
Ltd. The author came to Australia in 1977. where he joined the
University of Wollongong and started his Ph.D. study. Since March
1983 he has been a lecturer in the Department of Civil and Mining
Engineering, University of Wollongong teaching subjects mainly in
the areas of structural design and surveying.