CAMBER GROWTH PREDICTION IN PRECAST PRESTRESSED CONCRETE … · precast prestressed concrete bridge...
Transcript of CAMBER GROWTH PREDICTION IN PRECAST PRESTRESSED CONCRETE … · precast prestressed concrete bridge...
CAMBER GROWTH PREDICTION IN PRECAST
PRESTRESSED CONCRETE BRIDGE GIRDERS
A Dissertation
Presented in Partial Fulfillment of the Requirements for the
Degree of Doctor of Philosophy
with a Major in Civil Engineering
in the
College of Graduate Studies
University of Idaho
by
Krista M. Brown
December 1998
Major Professor: Edwin R. Schmeckpeper, Ph.D.
'OMX Humber: 9914955
Copyright 1999 by Brown, Krista Maretta
All rights reserved.
UMI Microform 9914955 Copyright 1999, by UMI Company. All rights reserved.
This micro!orm edition is protected agaimt unauthorized copying under Title 17, United States Code.
UMI 300 North Zeeb Road Ann Arbor, MI 48103
ii
AUTHORIZATION TO SUBMIT DISSERTATION
This dissertation of Krista M. Brown, submitted for the degree of Doctor of
Philosophy with a major in Civil Engineering and titled "Camber Growth Prediction in
Precast Prestressed Concrete Bridge Girders," has been reviewed in final form, as
indicated by the signatures and dates given below. Permission is now granted to
submit final copies to the College of Graduate Studies for approval.
Major Professor
Committee Members
Department Administrator
Edwin R. Schmeckpeper
f:.f7~ Fouad Bayomy
;.&µ m-MJLIL Donald M. Blackketter
~ R~ J~ Nielsen
Dean ~f Col_lege ~' w0 ~ , ) of Engmeenng ~~ / ~
RichardJ~
Final Approval and Acceptance by the College of Graduate Studies
Date /2. /+/I'M?
rz/1/11 Date ___ _
Date iz/z/98
Date 7 Pc?t:: ft3
Date I ?/If (rf;-
iii
ABSTRACT
In this study the author examines existing models for predicting camber growth of
precast prestressed concrete bridge girders including the present method utilized by
the Idaho Transportation Department (ITD). The camber growth of these girders is
affected by time-dependent concrete properties which include modulus of elasticity,
creep and shrinkage. The types of girders and manufacturers' practices are also
factors.
To allow a detailed analysis, this study focuses on the types of bridge girders
designed by ITD and manufactured by local prestressed concrete plants. As a
result, the author proposes a time-dependent model for predicting the camber of
precast prestressed concrete bridge girders at any age. The author also develops a
simple formula for estimating the camber at erection. The camber that is predicted
by both methods is compared to data that were provided by girder manufacturers.
The coefficients presented in the simple formula are appropriate for the types of
bridge girders designed by ITD and manufactured locally. However, the general
procedures described in the time-dependent model provide for the derivation of
coefficients suitable elsewhere.
ACKNOWLEDGMENTS
I thank my major professor, Dr. Edwin Schmeckpeper, and committee members
Dr. Richard Nielsen, Dr. Donald Blackketter and Dr. Fouad Bayomy, both for their
professional and personal support. Also, I am thankful to Dr. Howard Peavy who
provided the encouragement to pursue this degree and the opportunity to teach at
University of Idaho while I completed my research.
I wish to acknowledge my fellowships funded by EpsCOR and Idaho Space Grant
Consortium and the funding provided by Idaho Transportation Department for this
research.
I am appreciative to Chuck Prussack and his staff at Central Pre-Mix Prestress
Company in Spokane, WA, and Tracy McGillick at Monroe, Inc. in Boise, ID, who
continually answered my questions regarding the precast prestressed concrete
industry.
To my friends Marti Ford, Dr. Michael Kyte, Dr. and Ruth Finnie, Carol Ruyf and
Mona Klinger, I give my gratitude for bolstering my spirits throughout this arduous
task.
My heartfelt thanks to my husband, Mal, and children, Brian, Eric and Lizzie, who
persevered throughout this process, including my year-long absence.
iv
v
TABLE OF CONTENTS
AUTHORIZATION TO SUBMIT DISSERTATION .......................... ii
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
TABLE OF CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....... v
UST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
LIST OF ILLUSTRATIONS ........................................... x
NOTATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
CHAPTER 1 INTRODUCTION AND OBJECTIVE . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Objective and Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Current Models, Theories and Practices . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Time-step methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.2 Multiplier methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.3 Practices of other state bridge design departments . . . . . . . . 8
1.2.4 Current practice of the Idaho Transportation Department . . . 8
1.3 Objective and Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
CHAPTER 2 FACTORS AFFECTING INITIAL CAMBER AND CAMBER GROWTH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1 Modulus of Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.1 Stress-strain behavior of concrete . . . . . . . . . . . . . . . . . . . . 10
2.1.2 Existing models for the modulus of elasticity of concrete ..... 11
2.1.3 Recent experimental data . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Prestress Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.1 Loss of prestress force due to elastic shortening . . . . . . . . . 18
2.2.2 Creep of concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
vi
2.2.3 Shrinkage of concrete . . • . • . . . . • . . . . . . . . . . . . . . . . . . . 24
2.2.4 Relaxation of steel strands . . . . . . . . . . . • . . . . . . . . . . . . . . 27
2.2.5 Regain of prestress due to application of permanent loads . 27
2.2.6 Case studies . . . • . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2. 7 Summary of prestress losses . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3 Precast Manufacturers' Practices and PCI Guidelines . . . . . . . . . . 30
2.3. 1 Girder tolerances • . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3.2 Jacking . . . . . . . . . . . . . . • . . . • . . . . . . . . . . . . . . . . . . . . . . 31
2.3.3 Harping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
CHAPTER 3 DEVELOPMENT OF THE CAMBER PREDICTION MODEL . . . . . 36
3.1 Prediction of Camber at Release . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 Prediction of Camber at Erection . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3 Prediction of Long-term Camber . . . . . . . . . . • . . . . . . . . . . . . . . . . 52
3.3.1 Non-composite sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.3.2 Composite sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.4 Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
CHAPTER 4 COMPARISON OF PREDICTION MODEL AND CAMBER DATA 59
4.1 Validity of Existing Camber Measurements . . . . . . . . . . . . . . . . . . . 59
4.2 Comparison of Camber at Release . . . . . . . . . . . . . . . . . . . . . . . . 60
4.3 Comparison of Camber at Erection . . . . . . . . . . . . . . . . . . . . . . . . 63
4.4 Comparison of Long-term Camber . . . . . . . . . . . . . . . . . . . . . . . . . 68
CHAPTER 5 CONTROL OF CAMBER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.1 Methods of Camber Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.2 Effects of Camber Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.3 Necessity of Camber Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
CHAPTER 6 CONCLUSIONS AND FUTURE STUDY . . . . . . . . . . . . . . . . . . . . . 73
6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6.2 Future Study . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
vii
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
APPENDIX I SURVEY OF STATE BRIDGE DESIGN SECTIONS . . . . . . . . . . . 86
APPENDIX II PREDICTING CAMBER AT RELEASE BY NUMERICAL INTEGRATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
APPENDIX Ill PREDICTING CAMBER AT RELEASE AND ERECTION WITH A MATHCAD WORKSHEET. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
APPENDIX IV PREDICTING CAMBER AT RELEASE FOR A BOX SECTION BY NUMERICAL INTEGRATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
APPENDIX V CAMBER MEASUREMENT DATA ....................... 108
Table
Table 1
Table 2
Table 3
Table4
Table 5
Table6
Table 7
Table 8
Table 9
Table 10
Table 11
Table 12
Table 13
Table 14
Table 15
viii
LIST OF TABLES Page
Multipliers for predicting camber in non-composite girders . . . . . . . . 7
Formulae for calculating modulus of elasticity for normal weight, mature concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Comparison of ACI recommendations and results from FHWA bridge girder projects with HPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Case studies of precast prestressed bridge girders . . . . . . . . . . . . 29
Adjustments to jacking force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Effects of manufacturers' tolerances on midspan camber at release 34
Comparison of moments of inertia for a girder with harped strands 40
Relative humidity of selected cities . . . . . . . . . . . . . . . . . . . . . . . . . 48
Predicting camber at erection by time-step method versus proposed multiplier method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Comparison of predicted and measured camber at release . . . . . . 61
Predicted prestress losses and camber at erection . . . . . . . . . . . . 67
Survey of bridge design departments of various states . . . . . . . . . 87
Measured camber of 14 girders from the Kamiah project . . . . . . . 109
Average measured camber of the girders from the Cole-Overland project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Camber measurements of eight girders from the Cole-Overland project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
Figure
Figure 1
Figure 2
Figure 2
Figure4
Figure 5
Figure 6
Figure 7
Figure 8
Figure 9
Figure 10
Figure 11
Figure 12
Figure 13
Figure 14
Figure 15
Figure 16
Figure 17
Figure 18
Figure 19
Figure20
Figure 21
Figure 22
Figure23
ix
UST OF FIGURES Page
Anatomy of a precast prestressed girder . . . . . . . . . . . . . . . . . . . . . 1
Typical non-composite girder cross-section . . . . . . . . . . . . . . . . . . . 3
Typical composite girder cross-section . . . . . . . . . . . . . . . . . . . . . . . 3
Nomenclature for calculation of camber . . . . . . . . . . . . . . . . . . . . . . 6
Stress-strain behavior of concrete . . . . . . . . . . . . . . . . . . . . . . . . . 11
Comparison of various methods of calculating Ee . . . . . . . . . . . . . . 13
ACI and CEB normalized creep functions . . . . . . . . . . . . . . . . . . . . 22
ACI and CEB normalized shrinkage functions . . . . . . . . . . . . . . . . 26
Creep coefficient for a girder with a volume-to-surface area ratio of 80 mm (3.2 in) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Shrinkage strain for a girder with a volume-to-surface area ratio of 80 mm (3.2 in} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Creep coefficient for a girder with RH=60% . . . . . . . . . . . . . . . . . . 47
Shrinkage strain for a girder with RH=60% . . . . . . . . . . . . . . . . . . . 47
Predicted prestress losses for a girder of the Cole-Overland project 50
Predicted camber for a girder of the Cole-Overland project . . . . . . 51
Ratio of predicted camber to predicted camber at release for a girder of the Cole-Overland project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Factor for adjusting the creep coefficient for loadings added after release . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Effect of girder length-to-depth ratio (UD) . . . . . . . . . . . . . . . . . . . 62
Effect of girder length-to-depth ratio (UD) with box girder omitted . 63
Measured and predicted camber of 14 girders of the Kamiah project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Average measured and predicted camber for girders of the ColeOverland project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Measured and predicted camber for eight girders of the Cole-Overland project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Predicted camber for a temporarily loaded girder of the Cole-Overland project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Flovvchart for calculating camber at release and erection . . . . . . . 76
Illustrations
Illustration 1
Illustration 2
Illustration 3
Illustration 4
x
UST OF ILLUSTRATIONS Page
Individual jacking of strands . . . . . . . . . . . . . . . . . . . . . . . . . 31
Gang harping of strands . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Harped strands after gang harping . . . . . . . . . . . . . . . . . . . . 33
Girders temporarily loaded to control camber growth . . . . . 69
NOTATIONS
a - distance from centerline of girder to harping point
a1, a2, • •• - coefficient
A - cross-sectional area of the girder
Ac - gross concrete cross-sectional area of the girder
~ - cross-sectional area of the strands
Av- - transformed cross-sectional area of the girder
b - a constant; b=1.0 for straight strands, b= 0.85 for harped strands
c. g. - center of gravity
Cdeckplacement - creep coefficient at the time of deck placement
Cerection - creep coefficient at the time of erection
Ci - creep coefficient in the ith time step
Ci - creep coefficient at the time added dead load is placed
Ck - creep coefficient at the time a temporary load is removed
en - creep coefficient in the nth or last time step
Cu - ultimate creep coefficient
C0 - notational creep coefficient
C(t) - creep coefficient as a function of time
C{t,RH,VS) - creep coefficient as a function of time, relative humidity and volume-to surface area ratio
e - distance from e.g. of the strands to e.g. of girder
eenc:1 - distance from e.g. of the strands to e.g. of girder at the centerline of bearing
emid - distance from e.g. of the strands to e.g. of girder at midspan
E - modulus of elasticity
Ee - modulus of elasticity of concrete at 28 days
Ec(t) - modulus of elasticity of concrete at time, t
Eci - modulus of elasticity of concrete at transfer
Em - modulus of elasticity of matrix
EP - modulus of elasticity of particles
Eps - modulus of elasticity of low-relaxation strands; 197 GPa (28600 ksi)
xi
fcir - stress in the girder at the location of the c. g. of the strands
f es - loss of prestress due to the elastic shortening of the girder
f1oss - prestress loss
f 1oss - prestress loss at the end of the nth time step
fregain - prestress loss due to dead loads applied after release
fre1ax - prestress loss due to relaxation of strands
f sH - loss of prestress due to shrinkage
fst - stress in strands at jacking; 0.75·fu for low-relaxation strands
f u - ultimate strength of 13 mm (0.5 in} dia steel strands; 1860 MP a (270 ksi) for low-relaxation strands
f CR(t) - loss of prestress due to creep; a function of time
f sH(t) - loss of prestress due to shrinkage; a function of time
r c - 28 day compressive strength of concrete
f c(t) - compressive strength of concrete at time t days
f ci - compressive strength of the concrete at the time of transfer
F adi - a constant; refer to b
I - moment of inertia
le - gross concrete moment of inertia of the girder
l1rans - transformed moment of inertia of the girder
k - a variable
l - length of the girder between centerlines of bearing
Maddec1 dead load - moment due to an added dead load
Md1 - moment due to girder weight
-P - prestress force in strands
pelf - effective prestress force at any time
Pi - effective prestress force in the ith time step
Pi.Joss - cumulative loss of prestress force in the ith time step
P 1oss - loss of prestress force at time of erection
P 0 - prestress force at jacking
P 1 - prestress force after immediate tosses
~ - factor applied to creep coefficient for loads that are applied after release
xii
RH - relative humidity, %
s - a constant
t - age of girder, days
~ - age of girder Y.lhen a load is added after release, days
ta - time of initial loading, days
V m - volume fraction of matrix
VP - volume fraction of particles
VS - volume-to-surface area of a girder, mm (in)
w - unit YJeight of dry concrete, lbs/ft3
wdl - YJeight per unit length of girder
~ - a coefficient based on relative humidity
r3sc - a coefficient based on type of cement
xiii
~deadloadtolhecomposil&sec:tion - camber at midspan due to dead load that is applied to the composite section, positive upward
~deadload - camber at midspan due to added dead load, positive upward
~ - camber at midspan due to dead load of deck, positive upward
~ - predicted camber at midspan at release, positive upward
~ - predicted final camber at midspan, positive upward
~iw11emp - predicted final camber at midspan when a temporary load has been applied, positive upward
~1 etrective prestre:ss - camber at midspan due to final or long-term effective prestress force, positive upward
~ 1oss - camber at midspan due to loss of prestress force in the ith time step, positive upward
~idspan - camber at midspan at any time after release, positive upward
~ - camber at midspan due to prestressing force, positive upward
~ - predicted camber at midspan at release, positive upward
~ - camber at midspan due to initial prestress force, positive upward
~1 - camber at midspan due to initial prestress force less immediate losses, positive upward
~ - camber at midspan due to temporary load, positive upward
f1wt - camber at midspan due to girder weight, positive upward
& - strain
&e1astic - elastic strain
BcR(t) - strain due to creep as a function of time t
estt(t) - strain due to shrinkage as a function of time t
xiv
estt(t,RH,VS) - shrinkage strain as a function of time, relative humidity and volume-to surface area ratio
Estti - cumulative strain due to shrinkage in the ith time step
esttu1t - ultimate strain due to shrinkage
<f>x - curvature of the girder at distance x from the end
p - unit mass of dry concrete, kg/m3
a - stress
1
CHAPTER 1
INTRODUCTION AND OBJECTIVE
In prestressed precast concrete bridge girders, camber or upward deflection is
induced by the eccentric location of the pretensioned steel strands. This camber is
reduced by the self-weight of the girder, but not completely negated. Since
concrete is subject to creep, the camber increases with time. This increase in
upward deflection is termed camber growth. Figure 1 shows the layout and includes
notes on the production of a typical precast prestressed concrete girder.
Steel strands ere stressed
~ ~. j ~ed~a:U~s ~---~ placedandWienithasreached
Figure 1
fci (appr0>arrate1y 18-24 hou"s) Bevatioo Vlem of Grder the sfJ aids ere aJ.
VVhen the stnn1s are cut the eccentricity of the presfress fcrce inci.ces an UJMBrd deflection mch usualy ex:eeds the deflectiai due to grder wei~
Anatomy of a precast prestressed girder
The prediction of initial camber and its growth has proven troublesome to bridge
engineers, manufacturers and contractors. Engineers set the vertical alignment or
grade of the bridge deck and roadway approach based on the anticipated camber of
the girders. Contractors prepare their bids by estimating the labor and material
quantities necessary to construct the bridge per the engineers' plans.
Manufacturers are responsible for providing the prestressed precast concrete
bridge girders as specified by the engineers' plans.
When the camber of the girders is different than predicted it results in a design
change or modification of the girders. Any change to the original design is time
consuming and expensive.
Due to the aforementioned reasons, all parties desire a reasonable estimate of a
girder's initial camber, camber at erection and long term camber. The purpose of
this study is to develop an accurate and rational camber prediction method for use
by bridge designers of the Idaho Transportation Department (ITD).
1.1 Background
Precast prestressed concrete bridge girders have been routinely produced in the
United States since the early 1950's (Dunker and Rabbat 1992). By 1989, almost
40% of the new bridges were constructed of prestressed concrete.
The early prestressed concrete bridge girders had solid rectangular cross-sections
and only a few pretensioned steel strands. The spans were often less than 15
meters (50 feet). The design compressive strength of the concrete was
approximately 28 MPa (4000 psi) and the concrete was cured several days before
the stresses from girder weight and prestress force were transferred to the girder.
With so few prestressed strands and such short spans, camber and its growth were
not concerns.
Presently, girders have a variety of cross-sections such as bulb-tees, I-shapes,
hollow rectangular boxes, and voided and solid slabs. Figure 2 shows a
representative cross-section of a non-composite precast prestressed bridge girder.
Figure 3 shows a bulb-tee girder with a cast-in-place deck, which makes the girder
2
a composite section.
Precast, prestressed concrete girder
Figure 2 Typical non-composite girder cross-section
Cast-in-place concrete deck
Haunch or carrber strip (depth varies)
Precast prestressed girder
Figure 3 Typical composite girder cross-section
Most modern girders have the maximum number of prestressed steel strands that
can physically fit in the girder and may have concrete compressive strengths in
excess of 70 MPa (10,000 psi). The girders are typically steam cured less than 24
hours and may span over 40 meters (130 feet}. The use of slenderer sections
coupled with longer spans and a shorter curing period have resulted in girders
having significant initial camber and camber growth.
When the actual camber is substantially different from the predicted camber, it
creates a problem for both the contractor and design engineer. In the case of non-
3
composite sections, the shear keys and dowels of adjacent girders must be aligned
or physically manipulated to be joined. In composite sections, if there is too much
camber, the girders extend into the bridge deck or the girder stirrups are above the
deck. If there is too little camber, the stirrups do not extend into the deck as
designed. When one girder has too much camber and the adjacent girder has too
little, the problem is compounded. These problems can be remedied by
• Controlling the camber of the girder prior to or at erection,
• Adjusting thickness of the haunch,
• Adjusting the grade of the deck, or
• A combination of the above.
Nevertheless, any alteration to the original design could impad the schedule or
budget of the entire project
1.2 Current Models, Theories and Practices
Existing camber prediction models can be classified into one of two categories:
• Time-step methods
• Multiplier methods
1.2.1 Time-step methods
Time-step methods allow for the time-dependent variability of creep, shrinkage,
modulus of elasticity of the concrete and the relaxation of the prestressing steel
(ACI 1992; Ghali et al. 1974). They offer more versatility but were previously
thought to be too arduous for ordinary bridge designs. The advent of personal
computers has made this method less tedious.
For the basic time-step method the life of the girder is divided into finite time
intervals. These intervals may be equal or unequal, but should consider the
following events: release of prestress, erection and placement of the deck or other
permanent loads. Since the eccentricity of the steel strands, dead load moment,
4
5
and moment of inertia vary along the length of the girder, it is preferable to calculate
the curvature of sections of the girder. The curvature, 4>x• of a portion of the girder
at the end of time step n is determined by summing the curvature due to initial
prestressing force, P 0 , girder weight and the increase or decrease in curvature due
to loss of prestress, creep or shrinkage during each interval. ACI 435 (ACI 1995)
gives the following format for computing the curvature, 4>x• of every segment of a
girder:
where, for any time step, i, Ci is the creep coefficient at time step i, Pi is the
prestress force at time step i, n is the final time step, e is the eccentricity of the
strand and Md, is the moment due to the girder weight.
The curvature is then numerically integrated twice to determine the camber at any
point along the girder.
In most cases, only a prediction for camber at midspan is desired. For a girder with
constant cross-sectional properties and a harped strand pattern as illustrated in
Figure 4, Equation 1 can be manipulated to determine the midspan camber as
follows (PCI 1992):
ti . = Po ·[emid ·L2 - emid .a2 eend .a2]- 5wd,L4 ·(1+C )-midspan E .{ 8 6 + 6 384E -I n
a a
(2)
where wd1 is the weight per length of the girder and emid• eend and a are as shown in
Figure4.
T e.g. of girder
Figure 4
Centerline of girder
I
0rnid
Elevation Viemof Girder
Nomenclature for calculation of camber
e.g. of strands
Others have modified the time-step method shown by Equation 1 or 2. Ghali and
Favre (1986) employ an age-adjusted effective modulus, E/(1+C0 ), to compute the
strains, then relate these to the stresses and curvature. Tadros et.al. (19n) also
utilize the age-adjusted effective modulus, but compute creep, shrinkage and
prestress losses at the middle of the interval rather than the beginning or end.
As shown in Equations 1 and 2, the time-dependent relationships of prestress loss,
creep, shrinkage and steel relaxation must be known as well as the times at which
erection and subsequent loadings occur. The accuracy of the time-step method
may be significantly undermined by the uncertainty of these components.
1.2.2 Multiplier Methods
Several researchers (ACI 1992; Branson 1977; Ghali 1985) have adapted or
abbreviated the time-step method. Branson uses large time increments such as
casting to erection and erection to long-term or ultimate. For example, his equation
6
7
for predicting the midspan camber at erection has the following format:
A erection = Aprestress + kC u · Awt + Aprestress ·[-Ploss + kC u (1- Ploss)] (3) P0 2P0
where Cu is the ultimate creep coefficient, P1oss is the loss of prestress force at
erection and k is a time dependent variable between zero and one.
Multipliers are merely coefficients that have been developed from a time-step
analysis (Martin 1977; Branson 1977; Tadros et al. 1985). Assumptions are made
regarding length of time intervals, creep and shrinkage functions and prestress
losses. The time-step computation then reduces to coefficients (a1, a2, a3) that are
multiplied by the individual curvatures or cambers. Equation 4 illustrates the use of
these coefficients.
(4)
Table 1 lists multipliers that have been derived for non-composite precast
prestressed girders without non-prestressed reinforcement. This list is merely for
comparison. It is assumed that the girders are stored under average conditions as
specified by the individual author (PCI 1992; Tadros et al. 1985; Branson 1977).
The multipliers recommended by Precast/Prestressed Concrete Institute (PCI) were
developed by Martin (1977).
Table 1 Multipliers for predicting camber of non-composite girders
Method Camber at Erection Final Camber
a, a2 a3 a1 a2 a3
Tadros 1.96 1.96 1.00 2.88 2.88 2.32
PCI 1.80 1.85 - 2.45 2.70 -Branson - - - 2.53 2.88 -
Aswad (1991) approached multipliers differently. He determines the coefficients of
Equation 4 by using camber measurements to set up a system of linear equations.
8
Although the multipliers are simple to use, they were developed to predict camber
only at specific times such as 60 days or five years. Often design engineers are not
cognizant of the assumptions inherent in the formulation of the multipliers. Site
conditions, curing period, concrete properties, loss of prestress, girder type and
plant practices may be different than those assumed. Other researchers (Huo and
Tadros 1997) have shown that multipliers do not accurately predict camber growth.
1.2.3 Practices of other state bridge design departments
There is no uniform camber prediction procedure among the states surveyed. A few
states use multipliers, while others predict neither the initial nor long-term camber.
Several states are currently reevaluating their methods or in-house computer
programs for predicting camber growth. Appendix I shows the results of a survey of
thirteen state bridge design departments concerning their current practices of
predicting prestress losses and camber growth of precast prestressed bridge
girders.
1.2.4 Current practice of the Idaho Department of Transportation
Currently, Idaho Transportation Department estimates initial camber of a bridge
girder by superimposing the deflection due to girder weight, ~, and the camber
due to final, long-term effective prestress force, f1rina1 effective prestress· The predicted
midspan camber at time of girder erection, fl. erection• is calculated by multiplying the
estimated initial camber by 1.5. No theoretical or empirical basis was found for
the1 .5 multiplier.
A erection = 1 .5 · { Afinal effective prestress + Awt) (5)
9
For many cases, the above procedure underestimates the camber at erection. Even
with ITD tolerances of 13 mm (± 0.5 in) for spans 24.5 m {80 ft) or less and 25 mm
(± 1 in) for spans over 24.5 m {80 ft), manufacturers frequently need to take
measures to control the camber growth of the girders.
1.3 Objective and Scope
The objective of this study is to develop a rational and accurate method for
predicting the camber growth of precast prestressed concrete bridge girders that
are designed by Idaho Transportation Department (ITD).
The prediction of camber growth is complicated by variability of concrete
constituents, girder cross-sections, design assumptions, manufacturers' practices
and weather conditions. In this study the author proposes a practical, time
dependent model for predicting the camber growth of precast prestressed concrete
bridge girders. In addition, a multiplier method is developed for estimating camber
at erection. Camber predictions based on both methods are then compared to
camber measurements that had been previously recorded by girder manufacturers.
Unlike previous models, the author gives consideration to the practices and
properties of the concrete mixes used by the local prestressed plants, including
Central Pre-Mix Prestress Company in Spokane, WA; Monroe, Inc. plants in Boise,
ID; Idaho Falls, ID and Salt Lake City, UT and Buehner Prestressed in Salt Lake
City, UT. This study is limited to the types of girders (e.g. bulb-tees, modified bulb
tees, AASHTO sections and boxes) most frequently designed by the Idaho
Transportation Department. However, the methods proposed may be applied to
other similar cross-sections.
10
CHAPTER2
FACTORS AFFECTING INITIAL CAMBER AND CAMBER GROWTH
Both initial camber and camber growth are influenced by material properties,
ambient conditions and manufacturers' procedures.
2.1 Modulus of Elasticity
The modulus of elasticity of concrete, unlike that of steel, is not a consistent, readily
assessed number. The quality and quantity of the concrete constituents including
admixtures varies among plants and mix designs for the required f ci and f c· There
are also day-to-day differences caused by lack of consistency among suppliers' lots
and curing routines.
2.1.1 Stress-strain behavior of concrete
The stress-strain curve of concrete is not linear. The modulus of elasticity, whether
it is the tangent or secant modulus, is not a constant. As illustrated in Figure 5 the
curve is approximately linear up to the vicinity of 0.5·f c or 0.8·f c (Neville 1997;
Ahmad and Shah 1985; Mindess and Young 1981 ). In some cases, initial plastic
deformation is reported (CEB 1993; Mindess and Young 1981 ). When concrete
cylinders are tested per the ASTM (1994) testing procedure, this plastic deformation
is not accounted for in the experimental determination of Ee.
For the design of precast prestressed bridge girders, AASHTO (1996; 1997) limits
stresses due to dead loads to 0.60·f ci at release and 0.40·f c to 0.60·f c after
prestress losses. Up to these limits the stress-strain curve is relatively linear and
therefore the modulus of elasticity of the concrete is considered a constant for a
given f c·
0
Figure 5
0.002 Strain
Stress-strain behavior of concrete
2.1.2 Existing models for the modulus of elasticity of concrete
11
Even though a linear relationship exists for the stress-strain curve at low stresses,
Ee: is not a constant for all concretes. For many years it was presumed that Ee: was a
function of unit vveight of the dry concrete and the square root of the 28 day
compressive strength of the concrete (ACI 1995). As the usage of higher strength
concretes has increased, other relationships have been presented. Most notable in
these relationships is the influence due to the type of coarse aggregate (Aitcin and
Mehta 1990; Alexander 1996; Arioglu 1997; Attard and Setunge 1996; Baalbaki et
al. 1991; Gutierrez and Canovas 1995; CEB 1993; lravani 1996) and amount of
coarse aggregate (Cook 1989; Myers and Carrasquillo 1997).
Table 2 lists several formulae (Alexander 1996; CEB 1993; Gutierrez and Canovas
1995; Ahmad and Shah 1985) for calculating the modulus of elasticity. This list is
representative of the various formats for calculating Ee. A graphical comparison is
shown in Figure 6.
Table2
Method
ACl-318
ACl-363
Gutierrez &Canovas (based on CEB)
Alexander
Ahmad& Shah
CEB 1990
12
Formulae for calculating modulus of elasticity for normal weight, mature concrete
Metric U.S. Customary Comments (MPa) (psi)
0.0043p1.S./f c: 33w15./f -c:
(3300./f c: + (40,000./f c: + fc~41MPa 6900)(p/2320)1.s · 1,000,000)(w/145)1.s (6000 psi)
a 8480(f c:l 113 - Adjustments for aggregates: quartzite a =1.15 sandstone a=0.60 limestone a=0.90 basalt (dense) a=1.2
Ko+ afcu Ko and a YJere developed for 23 aggregates and t'NO concrete ages
- v;.s f o.325 -c:
21 SOO((f c: +8)/10) 113 - LoYJer value allows or for initial plastic 18300((f c:+S)/10)113 strain. The 8 MPa
adjusts for the difference bet'Neen design f c: and cylinder f c:·
13
-e-
40 CEB
ca Q..
C> ACl-318 0 w 35 ~ -(j Gutierrez & canovas ;: fl) CIJ iii --0 30 Ahmad&Shah "' :J "'3 "O -0 :i ACl-363
25
20--~~--~---~--~---__..____._~--~--___.
20 30 40 50 60 70
re MPa
Figure 6 Comparison of various methods of calculating Ee
The previous empirical relationships vvere developed for mature concrete. There is
relatively little research on elastic modulus of immature concrete. Gardner (1989)
has found that, for water-cured cylinders, the association betvv'een f c and Ee
ct1anges after 28 days. Alexander (1996) reports that a higher modulus is found
after 180 days, even if the compressive strengths at 28 days and 180 days are the
same. Apparently, Ee is a function of concrete maturity in addition to strength.
Gutierrez (1995) uses Equation 6, which correlates the elastic modulus, Ec(t), for
concrete at any age, to its 28 day modulus and its compressive strength, f c(t), at
that age and at 28 days.
14
(6)
The CEB-FIP Model Code 1990 (1993) recommends
(7)
where s is a coefficient for type of cement and ~ is the number of days adjusted for
temperature.
Neville (1997) and Baalbaki et al. (1992) contend that there is no simple
relationship between the compressive strength of high-strength concrete and its
modulus of elasticity. Concrete is not homogeneous. In simple terms, it can be
described as a composite of coarse aggregate (particles) in mortar or hardened
cement paste (matrix). There are several existing models for determining the
modulus of elasticity of just such a composite (Mindess and Young 1981; Sellvold et
al. 1994). The law of inverse mixtures, which assumes no bonding between the
particles and matrix is:
(8)
where EP is the elastic modulus of the particles; Em the modulus of the matrix; VP is
the volume fraction of particles; and V m is the volume fraction of the matrix.
Due to the bonding that does take place, the following equation is considered a
more logical model (Mindess and Young 1981 ).
15
(9)
However, the above assumes a perfect bond is developed betvv'een the matrix and
spherical particles. It may be more appropriate to regard Equation 8 as a lower
bound and Equation 9 as an upper bound.
Although either of the above equations is a simple calculation, typically neither the
modulus of the aggregate nor that of the mortar are known. Furthermore, neither
equation allows for the microcracking which develops when the moduli of the
aggregate and hardened cement paste approach one another.
2.1.3 Recent experimental data
In a testing program initiated by the Indiana Department of Highways, Kaufman and
Ramirez (1989) found that the ACI 318 relationship adequately predicted Ee for
concretes with compressive strengths up to 62 MPa (9000 psi).
Recent studies {Bums et al. 1997; French et al. 1997; Myers et al. 1997; Roller et
al. 1995; Stanton et al. 1997; Huo and Tadros 1997; Tadros 1997) funded by the
Federal Highway Administration (FHWA} on the use of High Performance Concrete
(HPC) for bridge girders have included Ee measurements on 28 day or 56 day test
cylinders. The results were inconclusive. Table 3 summarizes these results.
16
Table 3 Comparison of ACI recommendations and results from FHWA b "d "rd • cts "th HPC n geg1 erpro1e WI
Research Institute Cu eSHultt Measured Prestress Coarse or Code µ Ee Loss% Aggregate
ACI 2.35 780 ACI 318 or - -Recommendations ACI 363for
fc>6 ksi
LA Trans. 1.11t 409 ACI 318 11t -Research Center
University of - - 25% below 20-26 Limestone Minnesota ACl-318 or gravel
University of =1.80 =425 Close to ACI 15* -25 Limestone Nebraska 318, but mix
dependent
University of Texas 1.95 510 Slightly above 13* -22 Dolomitic ACI 318 limestone
University of 1.84§ =625§ 3.5% below 20-25 Gravel Washington ACI 318 (basalt)
t At one year * Conventional girders § Calculated from researchers' data
Notes: 1. Due to higher prestress force, ES is usually larger in the HPC girders as
compared to conventional (non-HPC) girders. 2. All girders were less than 2 years old at testing. 3. All strands were low-relaxation.
Of special interest are the results by researchers (Stanton et al. 1997) at the
University of Washington. The concrete for their girders and cylinders was
produced by Central Pre-Mix Prestress Company in Spokane, WA. This plant is
also a supplier of girders for ITD. Although the HPC was not identical to that
typically used for ITD girders, both mixes contain basaltic gravel which constitutes
41 % of the volume. From the HPC mix, concrete having a 56 day f c of about 76
MPa (11,000 psi) had a measured Ee of 4560·fco.s MPa (55000·f co.s psi) (Stanton et
al. 1997). This is only 4 % below that calculated by the ACI 318 fonnula. It is
important to note that this experimental detennination of Ee is based on tests from
cylinders, not full-sized girders.
2.2 Prestress Loss
17
In the stress analysis of prestressed concrete girders it is considered conservative
to overestimate prestress losses. Unfortunately this same design practice will
underestimate camber and camber growth.
Prestress loss is the reduction of the original prestress force or stress due to the
following:
• Elastic shortening of girder (f Es)
• Creep of concrete (f cR)
• Shrinkage of concrete (fsH)
• Relaxation or creep of steel strands (frux)
• Regain of prestress due to the application of pennanent loads
excluding girder weight (f regain)
The components due to creep, shrinkage and steel relaxation are interrelated, time
dependent and best modeled with a time step method (PCJ 1975). The specific
mechanisms of creep and shrinkage will not be discussed, only the outcomes.
A representative time-dependent expression (Zia et a. 1979) for total prestress loss,
f to!.al ross. 0 , at the end of the nth time step is
n n n
ftotallossn = fes + L [fcRi -fcRi.,]+ L [tsHi -fsHi.,]+ L [tretaxi -fre1ax;.,]+frega1n (10) 1=1 1=1 i=1
The following sections will address each of these components of prestress loss.
18
2.2. 1 Loss of prestress force due to elastic shortening
Elastic shortening is the shortening of the girder's length upon application of the
initial prestress force. If elastic behavior and uniaxial stress are assumed, the strain
can be calculated by Hooke's Law.
CT &=-
E (11)
Since the steel strand and concrete are bonded, the strain must be the same in
both the concrete and the strand. The normal stress in the concrete at the strand
location is composed of
• Axial stress from the prestress force
• Bending stress from the eccentricity of the prestress force
• Bending stress from the moment created by girder weight.
The stress in the concrete at the location of the strands is the sum of the above.
P Pe2 Md1e u. =-+-----
c A I I (12)
The strain in both the concrete and the strand is
(13)
Therefore the accompanying loss of stress in the steel strands is
P Pe2 Md,e -+-----f - A I I stee1 1oss - E .
-9!...
(14)
Eps
19
The above loss of prestress results in a reduced effective prestress force. The
lower effective prestress force leads to a lesser strain, thereby reducing the loss.
Thus, a more accurate assessment of the loss of prestress due to elastic shortening
would be found by an iterative solution.
An alternative to iteration is Equation 15 which was developed by Tadros et al.
(1977). Its derivation is based upon the strain compatibility between the strands
and the concrete. This loss is merely referred to as elastic shortening {fES):
(15)
The author found that the prestress loss due to elastic shortening as predicted by
Tadros' equation closely matches that of Equation 14 after two iterations.
Normally, for simply-supported girders, elastic shortening is only computed at the
midspan, where it is maximum. For girders with a harped strand pattern as
illustrated in Figure 4, the author ascertained that the average fes along the length
of the girder is approximately 85% of that calculated at midspan. See Appendix II.
Regardless of which equation is used, the prestress loss due to elastic shortening is
only an estimate. The following variability must be considered:
• Ee is estimated, not measured.
• At jacking, the prestress force, P 0 , has a .± 5% tolerance.
• Both the moment due to prestress force and the moment due to
girder weight are a function of strand center of gravity (e.g.),
which has a tolerance and varies along the length of the girder.
The range of Ee has previously been presented. Manufacturers' tolerances and
their effects on the initial prestress force will be addressed in Chapter 3.
2.2.2 Creep of concrete
20
Creep is the time-dependent plastic deformation that occurs at a constant stress
which is below the material's yield strength. Although creep is frequently associated
with elevated temperatures, concrete exhibits creep even at normal temperatures.
In prestressed concrete, the creep due to the compressive stress will continually
shorten the girder. Since the steel strands are bonded to the concrete they will also
shorten, thereby decreasing the prestress force. Equation 16. which is based on
Hooke's Law, is the loss of prestress due to the creep of concrete.
The stress, fcir• in the concrete at the e.g. of the strands is
Md1e I
(16)
(17)
where Pelf is the effective prestress force, which is the jacking force minus elastic
shortening and any time-dependent losses that have occurred. For a girder with
harped strands, fcir varies along the length. As with elastic shortening, the author
recommends using 85% of the fcir at midspan as an estimate for average fcir·
The creep coefficient, C(t), is the time-dependent ratio of creep strain to the elastic
strain:
C(t) = &creep (t) Ee1ast1c
(18)
21
For a member or specimen experiencing constant stress it is a simple relationship.
However, for a prestressed member with variable load history and time-dependent
prestress losses, C(t) is a more elaborate formulation.
There has been considerable qualitative and quantitative research, primarily on lab
specimen in controlled conditions, regarding the rate of creep and ultimate creep
strain. As a result of an intensive study, ACI 209 (1992) recommends the following
empirical formula:
ct - 0 c [ (t-t )0·6 ]
()- 10+(t-to)o.s. u (19)
Tadros (1997) recommends the substituting 12 - 0.0725·f c (12 - 0.0005·f J for 10 in
the denominator.
Per ACI 209 (1992) an average value for the ultimate creep coefficient, Cu, is 2.35
but it can vary between 1.30 and 4.15. Coefficients for relative humidity, age at
loading, concrete composition and volume-to-surface ratio are used to modify the
ultimate creep value. PCI (1997) uses the same formula but suggests 1.88 for the
ultimate creep with adjustments applied for the various components of the concrete
mix.
The creep model presented by the CEB-F/P Model Code 1990 (1993) is
Ct -c t-to [ ]0.3
( ) - o . At + (t - to ) {20)
where C0 is the notational creep coefficient and ~H is dependent upon relative
humidity and volume-to-surface ratio. CEB specifies that additional adjustments,
not only coefficients, for concrete compressive strength, cement type, and
22
temperature history, be applied to both the notational creep coefficient and the
creep rate. The CEB notational creep coefficient, C0, is not necessarily equivalent
to the ACI 209 ultimate creep value, Cu.
The time-dependent portions of Equations 19 and 20 are calculated for a girder with
a volume-to-surface area ratio of 80 mm and a relative humidity of 60%. These
values are normalized and compared in Figure 7. There is essentially no difference
in the time-dependent shape of the ACI and CEB equations for creep.
1.0
0.8 a. Cl> !
0.6 0 ACI "O Cl> .!:! (ii 0.4 CEB E ... 0 z
0.2
0.0 0 60 120 180 240 300 360
days
Figure 7 ACI and CEB normalized creep functions
Neither CEB nor ACI considers the type of coarse aggregate. Since the coarse
aggregate comprises about 40% of the concrete, the aggregate's creep properties
are not insignificant. Typically aggregate with a high modulus of elasticity also has
low creep characteristics (Alexander 1996; Khan et al. 1997). This further
emphasizes the importance of performing creep tests on individual concrete mixes.
Creep is not as simple as the previous equations imply. One cause is particle
shifting due to loading. As the internal moisture decreases and the age of concrete
increases the number of bonds increases, thus slowing the creep process. Pore
humidity, not ambient. is the controlling factor. However, pore humidity and thus
creep are influenced by relative humidity {Bazant and Panula 1980; Smerda and
Kristek 1988; Rusch et al. 1983) except when relative humidity {RH) is less than
60% {Wittmann 1982).
23
Due to the relationship between creep and shrinkage, it is difficult to accurately
measure or model them separately. It has been shown that total deformation of a
nonprestressed concrete specimen is greater than the sum of the creep and
shrinkage components {Wittmann 1982). The combined creep and shrinkage are
affected by the compressive strength of the concrete, elastic modulus, and the type,
amount and size of coarse aggregate {Bazant 1982). Not only are creep and
shrinkage interrelated, but creep even influences deformations from short duration
loads, thereby affecting elastic shortening (Bazant 1982).
Numerous rheological and empirical models {Bazant 1982; Branson 1977; Rusch et
al. 1983; Smerda and Kristek 1988) for creep have been developed and compared.
No one method or equation is preferred. For example, Bazant and Kim {1991) claim
an uncertainty of ±8%. Their model is convoluted and requires Ee. shrinkage and f c
in addition to the formulation of the concrete mix. Estimating Ee and shrinkage
increases the bounds on the uncertainty to ±24%.
In an earlier model, Bazant and Panula (1980) did not explicitly include the concrete
mix. They compared their model to former versions of the ACI 209 and CEB-FIP
Model Code. Although their model compared favorably to the others, the outcome
was dependent upon knowing Ee. In their example of a prestressed segmental
bridge, they showed that error in estimating prestress losses would yield large
errors in the creep effect of camber. It is ineffectual to spend time calculating creep
with a complicated, meticulous method if the variables are mere estimates.
Several researchers {Bazant and Panula 1980; Goodyear and Smith 1988) have
24
noted a difference between the results of laboratory experiments and those of
structures. They observed that even with the same concrete mix, using lab data to
predict structural effects can result in up to an 100% error.
As shown earlier, research has been conducted on full-size bridge girders and their
concrete properties. Refer to Table 3 for creep and shrinkage results.
2.2.3 Shrinkage of concrete
There are two types of shrinkage (Mindess and Young 1981). Plastic shrinkage
occurs in fresh concrete and drying shrinkage in hardened concrete. In precast
prestressed concrete plastic shrinkage is limited by the controlled curing during the
initial 18 hours, therefore only the effects of drying shrinkage are considered.
When the concrete shrinks, the strands shorten accordingly. The equation for loss
of prestress due to shrinkage is very similar to that for creep.
fsH(t) = &sH(t)·Eps (21)
Many of the same factors that influence creep also affect shrinkage, but not to the
same degree. Volume-to-surface area, relative humidity, amount of aggregate and
water-to-cement ratio govern both the rate of shrinkage and total shrinkage.
Admixtures and creep itself influence shrinkage (Mindess and Young 1981).
Some drying shrinkage is recoverable. Rewetting reverses the shrinkage process.
Like creep, research is usually perfonned on small lab samples rather than full
sized members. Total or ultimate shrinkage is typically extrapolated rather than
measured.
For steam cured concrete, which is the method used by the manufacturers of ITD
25
girders, ACI 209 recommends the following formula for shrinkage strain:
(22)
Tadros (1997) recommends using 45 - 0.36·f c: (45 - 0.0025·f Jin lieu of 55 in the
denominator of the above equation. However, this change along with the one
suggested for Equation 19 may vary for each individual concrete mix and should be
verified prior to being incorporated into the equations.
Ultimate shrinkage (esHu1t) can range between 415µ and 1070µ. It is suggested that
780µ be substituted for esHu1t when a measured value is unavailable (ACI 1992). As
with creep, correction factors are used to accommodate the ambient relative
humidity (RH) and constituents of the concrete mix.
CEB-FIP (1993) presents the following equation for shrinkage at any time, t:
&sH(t)=[1.5+-(~~)')]· __ t __
350( 2VS) 2 + t 100
{ 160 + 10 P SC ( 9 - : ~)] X 10-6
0.5
(23)
where VS is the volume-to-surface area in mm and '3sc is a coefficient for cement
type. In the case of high early strength, normal strength cement, 13sc = 5.
To compare the ACI and CEB shrinkage functions of Equations 22 and 23, the
shrinkage strain of a hypothetical girder having a volume-to-surface ratio of 80 mm
26
{3.2 in), f c= 41 MPa (6.0 ksi), Type Ill cement and relative humidity of 60% is
calculated. As with the creep functions, the shrinkage values are then normalized.
Figure 8 presents the ACI and CEB normalized shrinkage strains.
1.0
c g 0.8 en ID Cl as 0.6 ACI .::ie. c ·c: s= en
CEB "O 0.4 ID ~ ti e 0.2 0 z
0.0 0 60 120 180 240 300 360
days
Figure 8 ACI and CEB normalized shrinkage strains
Although the ACI and CEB plots of normalized strain are not identical, the difference
is not substantial. At most, the author determined that the difference between the
two equations represents approximately a 14 MPa (2.0 ksi) discrepancy in the
prestress loss due to shrinkage. This is less than 5% of the anticipated total
prestress losses. Neither equation can be considered accurate. The CEB equation
was developed from data having a coefficient of variation of 0.35.
As discussed earlier, it is difficult to separate the shrinkage and creep strains. Two
researchers might develop different creep and shrinkage relationships, but the
combined effect could be the same. Without experimental corroboration, a designer
should not use a creep function from one source and the shrinkage function from
another.
27
2.2.4 Relaxation of steel strands
Currently in precast prestressed concrete girders, low-relaxation steel strands are
used exclusively. The relaxation of the stress in the steel strands is significantly
reduced over that of the older, stress-relieved strands. To calculate the loss of
prestress due to the relaxation of low-relax strands, AASHTO (1996; 1997) and PCl
(1975; 1997) recommend:
f = f [log 24t]. [ fst _ 0_55] relax st 10 0.85fu
(24)
where fst - 0.55 ~ 0.05, fst is the jacking stress and fu is the ultimate strength 0.85fu
of low-relaxation strands.
Typically, the strands are jacked a few hours prior to placing the concrete. In this
case the loss due to strand relaxation is only about 11 MPa (1600 psi). When the
strands are jacked the previous day or the girder is not released until after a
weekend, the losses are calculated at 16 MPa (2300 psi). When the effects of all
losses are considered, this formula would predict a steel relaxation loss of about
30 MPa (4000 psi) after 30 years.
The majority of prestressed concrete plants that produce girders for ITD use strands
manufactured by Sumiden Wire Products Corporation. Sumiden (1997) predicts the
loss to be approximately 0. 7% at 20 hours and 1.6% at 30 years. This corresponds
to 10 MPa (1400 psi) and 22 MPa (3200 psi), respectively.
With either of the methods, the prestress losses due to steel relaxation are a
fraction of the other losses.
2.2.5 Regain of prestress due to application of permanent loads
When a non-composite girder has a wearing surface or other permanent dead load
28
applied, or when the deck is placed on a composite girder, the stress in the
concrete at the level of the strands increases. Hence, the force in the strands is
also increased. This increase in prestress force is occasionally addressed as a
decrease in the prestress losses. Most codes (AASHTO 1997; AASHTO 1996; PCI
1992) and engineers have neglected this change in prestress force. Normally this
gain of prestress is small, but one researcher (Stanton 1997} estimated it as high as
60 MPa (9000 psi). The stress regained as a result of the moment from an added
dead load is counted as a decrease in the prestress losses and is calculated by:
f. . = -Madded dead road • e . Eps regain I E
c
As with many of the previously discussed prestress losses, freomn is typically only
calculated at midspan where it is a maximum.
(25)
By the time a cast-in-place deck is poured, most of the shrinkage has already taken
place in the girder. According to Branson (1977), the differential shrinkage between
the deck and girder may also produce a meaningful gain in prestress.
2.2.6 Case studies
In the last 10 years, several existing precast, prestressed bridge girders have been
removed from service and tested under laboratory conditions until failure. The
results of these experiments (Gustaferro et al. 1983; Halsey and Miller 1996; Labia
et al. 1997; Leon et al. 1990; Pessiki et al. 1996; Shenoy and Frantz 1991) are
summarized in Table 4. These bridges had been in service long enough for
essentially all losses to have taken place. Except for the box girder in Nevada the
measured prestress losses were significantly below the predicted losses. All of
these girders were in good condition and had stress-relieved strands, which have
higher losses than the low-relaxation strands currently used.
Table4 Case Studies of Precast Prestressed Bridge Girders
Location Type Span Age Design Meas. Meas. Strand Predicted Meas. Meas. to of (m) (yrs) f c f c Ee type prestress prestress predicted
girder (MPa) (MPa) (GPa) loss loss loss (MP a) (MPa) (%)
East Box 16.5 27 34.5 49.0 28.3 Stress- 255 124 49 Hartford, CT relieved
Mercer I-beam 27.0 28 35.2 58.4 34.1 ----* 280 187 67 County, PA
Athens Inverted 8.25 40 ---- 45 27.4 Stress- 280 223 80 County, OH tee relieved
Reno, NV Box 21.3 20 38 55.5 37.9 Stress- 241 427 177 relieved
Minneapolis, AASHTO 19.7 20 47.5 58 -- Stress- 275-410 310 76-113 MN Type Ill relieved
Illinois I-beam ----- 25 27.6 69.6 ---- Stress- 280 140 50 Tollway relieved
*Type of strands was not documented. However, only stress-relieved strands were in common use at the time of construction.
~
30
2.2. 7 Summary of prestress losses
Some engineers choose whichever method predicts the highest total prestress loss.
HoYJever, the case studies and an analysis by Lwin et al. (1997) indicate that
standard methods may overestimate losses. As stated earlier, overestimating
prestress losses will underestimate camber and camber growth.
AASHTO (1996; 1997) guidelines for the calculation of prestress losses are not
discussed in this study. Lump sum prestress losses cannot be used in a time step
analysis and the time-dependent equations presented in AASHTO overestimate the
losses (Lwin et al. 1997; Dorgan 1997).
2.3 Precast Manufacturers' Practices and PCI Guidelines
Precast prestressed bridge girders are manufactured for ITD by several companies
and plants. The author interviewed personnel and observed operations at Central
Pre-Mix Prestress Company in Spokane, WA-, Monroe, Inc. plants in Boise, ID;
Idaho Falls, ID and Salt Lake City, UT and Buehner Prestressed in Salt Lake City,
UT. Contact via mail and telephone was made with Morse Brothers in Harrisburg,
OR and Montana Prestressed Concrete in Helena, MT.
All the precast prestressed concrete plants that manufacture girders for ITD are
certified by PCI, which establishes standards for the industry. Some of these
standards have the potential to influence camber.
2.3.1 Girder tolerances
The Precast/Prestressed Concrete Institute (PCI 1992) establishes tolerances on
member dimensions and strand location. The member tolerances are dependent
upon the type of girder. The forms are commercially produced and assembled such
that there is minimal variation from one girder to the next. Each individual strand
has a location tolerance of .±6 mm (.±0.25 in) and the center of gravity of bundled
strands can vary up to 13 mm (±0.5 in). The tolerances of the member do not
significantly alter its area or moment of inertia. However, the tolerances of the
strand location can result in variations, albeit minor, on the eccentricity of the
prestress force. This change in eccentricity will consequently change the moment
in the girder and the accompanying camber.
2.3.2 Jacking
31
The strands are either jacked individually or in groups to 0. 75·fu (MSHTO 1996;
AASHTO 1997). The strands have an ultimate strength of 1860 MPa (270 ksi), the
nominal jacking force for each 13 mm (1/2 in) diameter strand is 139 kN (31 kips).
Illustration 1 shows an individual strand being stressed with a pneumatic jack.
Illustration 1 Individual jacking of strands
The jacking force is verified by both strand elongation and jacking force. Both of
these must be within ±5% of the specified values. Since camber due to prestress
force is directly proportional to the prestress force, a 5% increase in jacking force
results in a 5% increase in camber due to the prestress force.
32
After all strands are jacked, some plants employ a second pass, which reduces
losses due to steel relaxation. This second jacking is only performed on a case-by -
case basis and not on every girder.
Table 5 displays adjustments that are applied to the jacking force by suppliers of
girders for ITD. The strands are jacked an additional amount to allow for anchorage
slipping and sometimes for the rotation of the end supports of the casting bed. Both
of these are nominal adjustments. On long girders, an overestimation of strand
slippage or bed rotation is negligible. However, on girders less than 18 m (60 ft), a
.±3 mm (±1/8 in) differential in anchorage slippage could change the prestress by
35 MPa (5.0 ksi).
Table 5 Adjustments to jacking force
Purpose Adjustment
Temperature differential between ambient air and fresh concrete Varies
Anchorage seating loss 6.4mm (0.25 in)
Rotation of end supports of the casting bed 3.2mm
(0.125 in)
2.3.3 Harping
With the exception of voided and solid slabs, most girders designed by ITD have
harped strands. Gang harping is accomplished by gathering the strands with a
channel-like fixture and jacking this fixture along a single, vertical strand.
Illustrations 2 and 3 show gang harping in progress and at completion.
33
Illustration 2
Gang harping of strands
Illustration 3
ffilrJ>ecl Sfr.tn<ts after 9ang h"'J>ing
34
Another harping technique employs unique, commercially produced hardware called
"harps." These "harps" can either be pulled down or in some cases, elevated, to
produce the desired strand pattern.
In either harping method the strands are not fully stressed initially. A lower initial
stressing force is calculated such that, when the strands are harped, the geometry
of the harping pattern results in fully stressed strands. This calculation is based on
elementary mechanics of materials and the prestress is not independently
measured. Some plants neglect friction at the hold-down points while others further
increase the prestress force to compensate for friction at these locations.
Since the actual initial prestressing force in the harped strands is not measured, it
is possible to accumulate the tolerances on jacking force and the location of the e.g.
of the harped strands. The tolerances on the location of the e.g. of the harped
strands will also change the eccentricity of the prestress force, thus impacting the
moment and the camber. Table 6 displays the effect manufacturers' tolerances and
their accumulation have on midspan camber at release.
Table 6 Effect of manufacturers' tolerances on midspan camber at release (camber due to prestress force +camber due to girder weight)
Project Gottt Cole-Overland*
Predicted camber at release 49mm 52mm
Jacking force +5% 45-53 mm 47-56 mm
Strand e.g. at harp point ±13 mm 46- 51 mm 51 -53 mm
Anchorage loss ± 3 mm 47-50 mm 51 -53 mm
Strand e.g. at end and e.g. at harp point 46-51 mm 50-53 mm
±13mm
All the above tolerances 42-57 mm 46-57mm
t AASHTO Type Ill girder with 36 strands and 24.4 m span * 1830 mm bulb-tee with reduced top flange, 40 strands and 32.4 m span
35
In the case of the shorter Goff bridge girders, accumulated tolerances can result in
a midspan camber at release of 86% to 116% of the predicted value. In the longer
Cole-Overland bridge girders, a range of to 88% to 110% is possible. Since the Goff
and Cole-Overland girders are typical of those designed by ITD, the author
maintains that manufacturers' tolerances may significantly effect midspan camber at
release and result in variations in camber among girders of the same design.
CHAPTER3
DEVELOPMENT OF THE CAMBER PREDICTION MODEL
In this chapter, the rationale for the camber prediction model is presented.
Chapter 4 compares the model to measured camber data.
36
In developing the camber growth prediction model for precast prestressed concrete
bridge girders, the author makes the following assumptions:
• Only camber at midspan is predicted.
• The girder is supported at centerlines of bearing during storage.
• The strands are bonded to the concrete between the centerlines of
bearing.
• The amount of non-prestressed reinforcement in the girders typically
designed by ITD is insufficient to affect creep and shrinkage.
• The girders are AASHTO, bulb tee or modified bulb tee girders, but
except as stated later, can be applied to any precast prestressed
concrete girder type.
Only the camber at midspan is required. The bridge design engineers at ITD use a
camber strip to accommodate the differences in elevation between the bearing
points and midspan of a girder. Refer to Figure 3.
Although the location of the supports while the girder is stored does not influence
the camber due to the prestress force, it does affect the camber due to the girder's
weight. If the supports are 2 m (6 ft) inside of the bearing points, the camber due to
girder weight may be reduced as much as 13 mm (0.5 in). In general, girders are
supported close to their centerlines of bearing, but on one plant visit the author
observed a set of girders that were supported up to 1. 7 m (5.5 ft} inside of the
centerlines of bearing.
37
Per ACI 318 (1995) and PCI (1992), 13 mm (0.5 in) diameter steel strand requires
630 mm (25 in) to transfer the stress from the strand to the concrete. ITD girders
usually have 150 - 250 mm (6 -10 in) between the end of the girder and the bearing
points. This indicates that the strands have not fully bonded at the centerlines of
bearing. However, most ITD girders are longer than 25 m (80 ft) and therefore the
effect that incomplete bonding at the bearing points has on midspan camber is
negligible.
Researchers {Ghali 1989; Ghali and Trevino 1985; Tadros et al. 1985; Tadros et al.
19n; Rao and Jayaraman 1989; Moustafa 1986; Naaman and Hamza 1993) have
stated that creep, shrinkage, prestress losses and camber are lessened when non
prestressed reinforcement is present. To make a noticeable difference, the amount
of non-prestressed steel should be greater than 50% of the prestressed steel or
more than 1 % of the concrete area. For most ITD girders, there is minimal non
prestressed reinforcement. Typically, the non-prestressed steel accounts for only
0.2% of the concrete area and less than 20% of the prestressed steel area.
The majority of the girders designed by ITD are AASHTO sections, bulb tees or
modified bulb tees. Voided slabs, solid slabs, double tees and box sections are not
as common.
3.1 Prediction of Camber at Release
After the concrete is placed, the girder has a 3 to 4 hour initial concrete set period;
then a 12 hour steam cure with an intended maximum temperature of 70°C (160°F);
followed by a cool-down period. On occasion, the girder is left in the forms over a
weekend with or without steam.
When the girder is cast, concrete cylinders of either 152 mm x 304 mm (6inx12 in)
or 102 mm x 203 mm (4 in x 8 in) are made. These cylinders are either match-cured
38
alongside the girder or cured in an on site laboratory by the Surecure™ system. In
the Surecure™ method the ambient temperature of the cylinders is the same as that
of a thermocouple installed inside the curing girder. Approximately 18 hours after
casting the cylinders are tested for compressive strength. When the specified initial
concrete compressive strength (f c:J is achieved, form removal and the transfer of
stresses from the jacked strands to the concrete can proceed.
To meet production schedules this transfer, destress or release occurs
approximately 18 to 24 hours after concrete placement. The release is
accomplished by alternately flame cutting the strands on the "deadn end and saw
cutting them on the jack end.
After release, the prestressed girder is lifted by its "pick pointsn and transported by
gantry crane to a storage area where it is set on dunnage or timbers. The girders
remain there until they are shipped for erection.
At release, the deflection of the girder due to its weight is
~wt= (26)
where wd1 is the weight per unit length of the girder, Ec:i is the modulus of elasticity of
the concrete at release and I is the moment of inertia.
The average density of fresh concrete at local plants producing prestressed girders
for ITD is approximately 2400 kg/m3 (150 pcf). During curing this density decreases
a few percent to about 2320 kg/m3 (145 pcf). The author found that in several
bridge projects designed by or for ITD, the strands, stirrups and non-prestressed
reinforcement increased the overall density by 3% to 5%. Based on this, 2400 to
2500 kg/m3 (150 to 156 pct) is a reasonable estimate for the density of a
prestressed girder when calculating wdi· This overall girder density should not be
39
confused with the density of dry concrete which appears in the ACI formula for Ee.
At release, the camber due to the prestressing force is
e id • a2 e d . a2 ] m +-en __
6 6 (27)
where P1 is the prestress force at release (i.e. P1 = P0 - P1ossesatre1ease ) and emid• eenc1
and a are as shown in Figure 4.
As shown in Equations 26 and 27 the camber and deflection of a girder are both
indirectly proportional to the product of the modulus of elasticity, which is a material
property, and moment of inertia, which is a cross-sectional property.
Neither ITD nor any local prestress manufacturers have any data relevant to the
modulus of elasticity of concrete at release or at 28 days or their coarse aggregate.
However, as discussed in Chapter 2, Ee was measured by researchers at the
University of Washington and reported at 4560·f c 0·5 MPa (55000·f c 0·5 psi) (Stanton
et al. 1997). Their value of Ee was based on 56 day test cylinders that had a
concrete mix similar to that used for girders produced for ITD and that were cast at
Central Pre-Mix Prestress Company in Spokane, WA. As stated earlier, the ASTM
testing procedure for determining Ee does not address the existence of initial plastic
deformation.
Girder manufacturers begin form removal as soon as their test cylinders attain f ci·
Thus destress or release takes place within one to two hours after the cylinder tests
are completed. If the test cylinders are truly indicative of the compressive strength
of the girder, the actual fa of the girder at the time of release is slightly greater than
the design fa·
Although the author feels the ACI 318 method may overestimate Eg, the formula
provides a reasonable estimate. Therefore, assuming a dry concrete density of
2320 kg/m3 (145 pct)
40
Ea = 4730.Jf'; MPa
(Ea = 57000.Jf'; psi)
(28)
Theoretically, the moment of inertia in a deflection formula should consider the
prestressed strands and non-prestressed reinforcement. This is the transformed
moment of inertia, I1ranS. However, for harped strands I1rans varies along the length of
the girder and is highest at midspan. In a such a girder which is representative of
those designed by ITD, the author calculated and compared the transformed
moment of inertia at midspan, with the average (along the length of the girder)
transformed moment of inertia and the gross concrete moment of inertia, Ic·
Table 7 summarizes the computations of Appendix II.
Table 7 Comparison of moments of inertia for a girder with harped strands
Moment of inertia 106 mm4
% oflc (in4)
Gross concrete moment of inertia, le 187,400 100%
(450,300)
Transformed moment of inertia at midspan (without 199,900 107%
non-prestressed reinforcement) (480,200)
Transformed moment of inertia at midspan (with non- 204,800 109% prestressed reinforcement) (492,000)
Transformed moment of inertia averaged along length 195,000 104%
of girder (without non-prestressed reinforcement) (468,500)
Transformed moment of inertia averaged along length 199,800 107%
of girder (with non-prestressed reinforcement) (480,100)
Since using the ACI 318 formula (Equation 28) most likely overestimates Ea. it is not
41
unreasonable to use the gross concrete moment of inertia in lieu of the transformed
moment of inertia when calculating the deflection and camber at release per
Equations 26 and 25. The author does not ordinarily condone the practice of
underestimating one variable to allow for overestimating another. However, given
the uncertainties associated with Ea. P0 , e.g. of the strands and the prestress loss
due to elastic shortening, the calculation of camber at erection can be nothing more
than an estimate. Considering the above-stated reasons and due to the ease of
calculation, the author recommends using the gross concrete moment of inertia, fe, in calculating deflection and camber.
The strand stress at release is the jacking stress less the losses due to elastic
shortening and steel relaxation. For low-relaxation strands with an ultimate strength
of 1860 MPa {270 ksi), AASHTO (1996; 1997) limits the jacking force to 0.75fu or
1400 MPa {203 ksi).
As discussed in Chapter 2, the loss due to elastic shortening can be calculated by
Equation 15, as recommended by Tadros et al. (1977). As shown by the author in
Appendix II, for a girder with harped strands the average elastic shortening is
approximately equal to 85% of the elastic shortening at midspan.
Midspan camber at the time of release is the sum of the deflection due to the
girder's weight and the camber due to the initial prestress force, P 0 , less the losses
due to elastic shortening and steel relaxation. Due to the uncertainties of Ea. the
jacking force and the relatively small magnitude of the prestress loss due to steel
relaxation, the author contends that the loss attributed to steel relaxation at the time
of release may be neglected. The effect of shrinkage is also considered negligible
at release.
For the majority of girder types the cross-section does not vary along the length of
the girder. The notable exception is the box girder with internal diaphragms. For
42
prismatic girders, the midspan camber at release is determined by
(29)
where flwt and flp1 are calculated per Equations 26 and 27.
Appendix Ill shows a sample calculation of predicting camber at release by using
Mathcad (MathSoft 1995) .
3.2 Prediction of Camber at Erection
From release to erection the compressive stress is not constant across the cross
section of the girder. Therefore creep will cause the girder's curvature and ensuing
camber to increase.
The age of the girder at erection is unknown at the time of design. Erection of the
girders usually occurs between 30 to 120 days after casting but sometimes as late
as one year. The design engineer can only estimate the age of the girder at
erection and cannot anticipate a manufacturer's schedule or construction delays.
The camber at erection can be calculated by either the time-step metliod or the
multiplier method. The author has chosen the time-step method that may also be
used to derive a single or range of multipliers.
For a prismatic section, Equation 2 can be modified to calculate midspan camber at
erection as follows:
n
+ L .::ii.loss ·(1+ Ci -Cj..1) i=1
(30)
43
where Ceraction is the creep coefficient at erection, C. is the creep coefficient at the
end of each time step, and Ai.1oss is the midspan deflection in each time step due to
the prestress losses after release. The ratio of EJEc is inserted because the creep
occurs on the mature girder, whereas Ap.. and A,,,. are calculated at release. An
alternative is to determine the modulus of concrete, l;(t), as a function of time.
However, with Type Ill cement the concrete strength and stiffness develop quickly.
Since Eci and Ee are estimates, introducing Ec(t) does not add accuracy to the
camber prediction.
As the girder cures, the concrete compressive strength and the modulus of elasticity
increase. The 28 day compressive strengt,1, r c• specified by the design engineer is
the result of stress analyses from several different load combinations as required
by AASHTO (1996, 1997) rather than the re that will be reached as a result of
required fa· For example, the design requirements for a girder may be fa = 34 MPa
(5.0 ksi) for release and f c = 38 MPa (5.5 ksi) at 28 days f e· A concrete mix with
Type Ill cement that has a compressive strength of 34 MPa (5.0 ksi) at 18-24
hours is very likely to reach 45 - 55 MPa (6.5 - 8.0 ksi) by 28 days. The author
recommends that Ee be based on the anticipated re rather that than the design f c·
In reviewing compressive strength data from local prestressed manufacturers, the
author found that the ratio of compressive strength at release to that at 28 days
ranges from 0.65 to 0.80. The author suggests that a ratio of 0.75 be utilized in
calculations. In most cases the error introduced by an incorrect estimation off Jf e
will be smaller than that incurred from calculating Ee based on the design f c· The
design engineer should periodically verify the ratio off a to f c by examining
manufacturers' concrete compressive strength records.
Using Equation 6 and substituting 0. 75 for f Jf c , Ee is estimated as follows:
Solving the above equation for Ee
E = Ea c 0.85
For prismatic girders with harped strands Equations 29, 30 and 32 can then be
combined to calculate the midspan camber at erection as follows:
~erection= ~release ·(1+0.85Cerection}-
44
(31)
(32)
where P i.1oss is the cumulative time-dependent prestress loss that exists in the ith
time step and is defined in Equation 38.
Noticeably missing from Equations 30 and 33 are the computations of the time
dependent creep coefficient, Ci, and prestress loss, Pi.1oss· Without any original
experimental data on full-sized precast prestressed bridge girders, the author
suggests using the ACI 209 creep and shrinkage functions but with different
ultimate creep and shrinkage values. The author recommends an ultimate creep
coefficient, Cu, and ultimate shrinkage, eSHutt> of 1.85 and 620µ, respectively. These
values are based on the research summarized in Table 3 and the camber data
acquired from local prestressed concrete manufacturers that is presented in
Chapter4.
45
An acceptable alternative to the above is to use the creep and shrinkage functions
from the CEB-FIP Model Code 1990 (1993). A comparison of the CEB and ACI
functions is discussed in Chapter 2.
Rather than use separate correction factors, creep and shrinkage are expressed not
only as a function of time, but of relative humidity (RH) and volume-to-surface area
ratio (VS). The other correction factors presented in ACI 209 are associated with
concrete composition. For the concrete mixes used locally, these characteristics do
not vary significantly and their effects are included in Cu and Ss.iu1t· Since the girders
are usually released at age one day or less, the ta term appearing in the creep
function is insignificant and is therefore omitted. Thus the author recommends the
following creep and shrinkage functions:
and
&sH(t,RH, VS)= 620x1 o..a( SSt+ J[ 1.2e-0.0047.vs](1.4-0.01 ·RH) (35)
Figures 9 and 10 illustrate the effects that time and relative humidity have on the
creep coefficient and shrinkage strain. Both creep and shrinkage start leveling--off
at about 90 days and do not appreciably increase after 180 days. According to this
model, at one year 85% of the creep and 90% of the shrinkage have occurred.
1.2
1.0 - RH=40% - --- --- ----~ ,,,,- - - -(.) --..: 0.8 ./
/ ,,,,- - RH=50% c / , ---Q) , , -u ,
----E 0.6 ·.·/ RH=60% Q)
0 0 Q. 0.4 Q) RH=70% Q) ....
(.)
0.2 RH=80%
0.0 0 60 120 180 240 300 360
days
Figure 9 Creep coefficient for a girder with a volume-to-surface area ratio of 80 mm (3.2 in)
Figure 10 Shrinkage strain for a girder with a volume-to-surface area ratio of 80 mm (3.2 in)
46
Figures 11 and 12 show the influence that the volume-to-surface ratio has on the
creep coefficient and shrinkage strain. Both of these figures assume a relative
humidity of 60%. The majority of girders designed by ITD have volume-to-surface
ratios that range from 60 to110 mm (2.4 to 4.3 in).
1.2
1.0
~ - I 0 O.B I c
CD ·5 E o.s I Q)
8 0.4 I 0.
CD CD ... 0 I
0.2,
0.0 0
Figure 11
500
<.O 400
I w 0 ..--c 300 -~ -(f) a> 200 O> as .:.:. c ·c: .c
100 en
0 0
- - -- - ---- -- VS=60 mm
VS=80 mm
VS=100 mm
60 120 180 240 300 360
days
Creep coefficient for a girder with RH=60%
60 120 180
days
-- - VS=60mm .~----1
VS=80mm
VS=100 mm
240 300 360
Figure 12 Shrinkage strain for a girder with RH=60%
47
48
Usually the design engineer does not know what manufacturer will produce a girder
or the time of year it will be produced. Prior to erection the relative humidity should
reflect where a girder is produced, not the location of the bridge project. Table 8
shows relative humidity data for cities with or near precast prestressed plants that
supply girders for ITD (Gale Research 1996). When the location of the
manufacturer is unknown it is reasonable to estimate a relative humidity of 60%.
Table 8 Relative humidity of selected cities
City Monthly high Monthly low Average yearly relative humidity relative humidity relative humidity
(%) (%) (%)
Billings, MT 61 40 55
Boise, ID 76 37 56
Pocatello, ID 76 41 58
Salt Lake City, UT 75 36 55
Spokane, WA 86 45 65
Time-dependent prestress losses begin immediately after release. The time
dependent losses occurring at any time, t, are expressed as:
fross(t,RH, VS)= b· C(t,RH, VS)·fcir +esH(t,RH, VS)-~ +freiax(t) (36}
where b = 0.85 for harped strands
b = 1.0 for straight strands
Since fc:in the stress in the concrete at the e.g. of the strands, is typically calculated
at the midspan where it is maximum, the coefficient b is added by the author to
adjust this maximum stress with one that is averaged along the length of the girder.
The creep and shrinkage functions were previously defined in Equations 34 and 35.
The loss due to elastic shortening, fes, does not appear in the above equation
because it occurs at release and is not time-dependent
49
The prestress losses affect fcirl therefore it is time-dependent as are the losses. A
preferred and more accurate method of calculating time-dependent losses is the
time-step method. The creep coefficient, shrinkage coefficient and far are calculated
at the end of each time-step per Equations 34, 35 and 17, respectively. Ten day
time-steps are suggested between release and erection but the size of each time
step need not be equal, especially since creep and shrinkage effects diminish with
time. Utilizing the time-step method with n steps, the total time-dependent prestress
losses, f1oss , can be expressed as: n
(37)
Using the same nomenclature as above, Pi.loss• the cumulative time-dependent loss
of prestress force that exists in the ith time-step and appears in Equation 33, can be
calculated by
(38)
The author recommends using 90 days for the girder age at erection. Only
occasionally are girders erected prior to 60 days. After 90 days the rate of camber
growth decreases. The design engineer should be able to expect and
accommodate a reasonable range of camber values at erection.
50
Although Equations 33 through 38 appear complicated, they are easily implemented
with Mathcad (1995) or a spreadsheet Appendix Ill contains a detailed example for
predicting the camber at erection for a 1830 mm (72j bulb tee with a reduced
flange, 40 harped 13 mm {0.5 in) diameter strands and a span of 32.55 m (106.75
ft). From this example, Figures 13 and 14 illustrate predicted total prestress losses
and predicted camber growth, respectively, as a function of time and relative
humidity. Figure 13 shows that predicted prestress losses of 210 to 275 MPa (30 to
40 ksi) occur between 60 and 120 days.
(Q c.
300
~ 250 gf .2 200 Cl) Cl) CD .;; 150 ! a. - 100
~ 50
- - -- -- -_ - --- - - -· -. .--- --- - - - - - -_::._:_ - - -
~:-: ~ -:::-= :_.:_: --- -... ::::.--.,,,..,
O'--~~ ...... ~~-'-~~-'-~~--~~--~~ ....... 0 60 120 180
days 240 300 360
RH=40%
RH=50%
RH=60%
RH=70%
RH=80%
Figure 13 Predicted prestress losses for a girder of the Cole-Overland project
For a relative humidity of 60%, Figure 14 indicates that the predicted camber at
erection might range from 70 to 85 mm (2. 75 to 3.375 in). Regardless of the relative
humidity there is no significant camber growth predicted after 180 days.
E E ..: Cl> .0 E as (,)
"Cl Cl> -(,)
:0 CD ..... a.
100
~
80 ~ - -~ - -
- - - - - - - -~~-~-- - - -
~ -r 60
-40 -
-20 -
~
o.__ ____ ...__ _____ ,__. ___ ~·-----------------~ 0 60 120 180
days 240 300 360
RH=40%
RH=60%
RH=80%
Figure 14 Predicted camber for a girder of the Cole-Overland project
51
Another method of viewing predicted camber is to express the predicted camber as
a non-dimensional ratio of the predicted camber at time, t, to the predicted camber
at release. Figure 15 presents the same predicted camber as Figure 14 using this
ratio. However, for girders with a camber less than 25 mm (1.0 in) this ratio is not
the preferred format for presenting predicted or measured camber. In this case, the
measurement tolerance could result in an unusually small or large ratio.
-.....
1.8 -----------------------
1.6
--1.4
.8 E 1.2 B
1.0 ...._ ______ __,_ __ ___..._ ___ ....__ __ _._ __ __.
0 60 120 180 days
240 300 360
RH=40%
RH=60%
RH=80%
Figure 15 Ratio of predicted camber to predicted camber at release for a girder of the Cole-Overland project
52
3.3 Prediction of Long-term camber
The camber of bridge girders is rarely measured after erection or deck placement
The following predidion models for the long-term camber growth of non-composite
and composite sections are presented merely as a hypothesis.
3.3.1 Non-composite sections
The long-term prediction of camber for non-composite sections is a continuation of
the method developed for predicting camber at erection. If necessary, a term
should be added for the deflection due to a 'Nearing surface, sidewalk or guard rail.
Equation 39 predicts the long-term camber of non-composite girders.
A final = Arelease • ( 1+0.85Cu )-
f [(~Joss -Pi-1,loss l . [emid · L2 _ emid · a2 + eend · a2 J]-i=1 ECIC ) 8 6 6
f [(ci-Ci-1)· ~Joss ·[emid·L2 - emid.a2 + eend·a2]]+ i=1 Eclc 8 6 6
(39)
[ 1+Radj(Cu - Cj)]· A Added dead load
where Cu is the ultimate creep coefficient, C1 is the creep coefficient at the beginning
of the ith time-step as determined by Equation 34, Ci is the creep coefficient at the
time of the added dead load as determined by Equation 34, 6.Mded dead load is the
elastic deflection caused by the added dead load and ~ is an adjustment factor
recommended by ACI 209 for delayed loadings, such that
Rad; =1.13-t~<;f (40)
where ~ is the age of the steam-cured girder at the time the load is applied.
Figure 16 indicates the effect ~ has on the creep coefficient when loads are
added after release.
0.9
'6' 8!. 0.8
0.7
0.6 ~__.._____.. _ __.. _ __._ _ __._ _ _.__.....__....___...__ _ _.____,_____,
0 60 120 180 240 300 360 girder age at time of added load
Figure 16 Factor for adjusting the creep coefficient for loads added after release
53
Although Figure 13 presents predicted prestress losses for a particular girder, its
shape is indicative of the losses for any girder. By erection, 70% to 85% of the total
prestress losses have occurred. The deflection resulting from the continued
prestress loss will be negligible for most girders.
After erection the time intervals can be larger since the rate of change has slowed
significantly. Time intervals of six months or one year would not be excessive for
predicting long-term camber. Recall that the predicted camber is already based on
estimated values for Ee, shrinkage and creep.
Neglecting the deflection due to losses after erection and since the wearing surface,
sidewalk or guard rail are installed soon after the erection of the girders, the long
term camber for a non-composite girder can be calculated from the camber at
erection as follows:
54
&final = &erection • [ 1 + {Cu - C erection)] +
[ 1 +R.,i( Cu - Cerection)] ·&Added dead load
(41)
where C-=tion is the creep coefficient at the time of erection as calculated per
Equation 34.
The above equation is very similar to one developed by Branson (19n). Since the
added dead load occurs at an older girder age, the creep of the added dead load is
often negligible.
3.3.2 Composite sections
After erection, some girders have a cast-in-place concrete deck or slab, thus
becoming composite girders. Since the concrete of the girders and the deck are of
different ages and strengths, the girder and deck form a composite section.
This study does not address the placement of a continuous deck over tvvo or more
spans of simply supported girders to form a single, continuous girder.
The concrete deck is placed from one to six months after erection of the girders.
Therefore the deck placement occurs somewhere between tvvo to ten months after
the girder is cast. As with the age at erection, the design engineer can merely
estimate the age of the girder when the concrete deck is poured. Prior to the time of
deck placement, the camber is calculated in the manner of Equation 30 or 33. The
deck placement will cause an immediate elastic deflection to the girder.
For structural analyses, the composite girder is typically modeled as a single
section by using transformed section properties based on the relative moduli of
elasticity of the two components, the original girder and the deck. HOVJever,
analyzing the camber growth of a composite girder is complicated by:
• T \VO different sets of creep and shrinkage functions due to two
completely different concrete mixes;
• Two different sets of creep and shrinkage rates due to different
concrete maturities;
• Stresses induced by the differential shrinkage between the
girder and the deck;
• Regain of prestress due to deck placement
55
The concrete for the cast-in-place deck is usually from a ready-mix concrete plant
that is close to the bridge project. This deck concrete is of an entirely different mix
design than the girder. In general, it has a lower compressive strength, lower
elastic modulus and a different coarse aggregate than the precast prestressed
girder. These properties result in higher creep for the deck which lessens the
tensile stresses from shrinkage of the deck (Krauss and Rogalla 1996).
Due to differential shrinkage of the deck and girder, an additional compressive
stress will be induced in the top of the precast girder (Silfwerbrand 1997; Joo and
Tadros 1991). This compressive stress reduces the camber, but the deck will crack
before the stresses become too high.
The gain of prestress due to deck placement varies in each situation, but averages
less than 25 MPa (3.5 ksi) for the girders commonly designed by ITD. This increase
in prestress has a negligible effect on both the camber and total prestress losses.
After deck placement the creep occurs on the entire, composite section. The
moment of inertia of the composite section is usually several times greater than the
moment of inertia of the girder without the deck. Also, the stress distribution from
the top to the bottom of the precast girder is relatively uniform once the deck is cast
Both of these conditions temper the camber growth.
As previously discussed with the non-composite sections, creep is reduced when
56
the load is added at a later girder age.
Most long-term camber prediction models ignore the aforementioned concepts and
merely apply a creep coefficient to the elastic deck deflection as calculated with
either gross or transformed section properties and add it to the camber at erection.
The author regards this as incorrect
Assuming that the creep has stabilized by the time of deck placement, the long-term
camber for a composite girder can be predicted by:
~final = derection • [ 1 + ( C deck placement - C erection)] + 6.deck +
d Added dead load to the composite section
where the elastic deflection due to the added dead load must be based on the
moment of inertia of the composite section.
If the deck is to be placed shortly after erection, Equation 42 reduces to:
Annal = 6.erection + Adeck +A Added dead load to the composite section
(42)
(43)
The author cautions that this prediction is approximate and is based upon estimates
for Ee, Eci, creep, shrinkage and time at erection or deck placement.
3.4 Multipliers
Multipliers should only be used for preliminary estimates of camber. Creep,
shrinkage and prestress losses other than elastic shortening losses are functions of
relative humidity, volume-to-surface area and time. The author derived multipliers
for use by ITD with the following assumptions:
• Erection occurs at 90 days.
• Relative humidity is 60%.
• The volume-to-surface area ratio is 80 mm (3.1 in).
• The prestress loss due to elastic shortening is 100 MPa (15 ksi)
and additional time-dependent losses of 100 MPa (15 ksi) occur
prior to erection.
E· • ~ = 0 .85 per Equation 32 Ee
• The strands are low-relaxation and 13 mm (0.5 in) diameter.
• The ultimate creep coefficient, Cu, is 1.85.
• One time step of 90 days is used, but the creep coefficient for
the losses is calculated at 45 days.
When the above assumptions are incorporated in Equations 33 through 35 the
author derived the following equation to estimate midspan camber at erection:
Aerection = 1.65 · Arelease + 1.49 · Aross
where a1oss is the elastic camber due to the assumed 100 MPa (15 ksi) loss of
prestress between release and erection and is negative.
Since a,oss occurs on the mature concrete section and is proportional to the
assumed 1300 MPa (188 ksi) prestress force at release, P1, it becomes:
[ Eci] (-100) ~loss = - . . ~P1 = -0.065~?1 Ee 1300
57
(44)
(45)
Equation 44 may also be expressed in terms of the camber due to girder weight and
prestress force, P1•
58
Aerection = 1.65 · Awt + 1 .55 • Ap1 (46)
The author emphasizes that Equation 46 includes many assumptions and is
intended only as a preliminary estimate of midspan camber at erection.
Table 9 compares the results of predicting camber at erection by the time-step
method from Equation 33 and the proposed multiplier method of Equation 46. For
the.c;e girders, the camber predictions are commensurate.
Table 9 Predicted camber at erection by time-step method versus propose d lti r th d mu p 1erme 0
TI me-step Multiplier
~ Ap1 Are1ease method method
ITD Project (Equation 33) (Equation 46) (mm) (mm) (mm)
A erection AerectiOn (mm) (mm)
Cole-Overland -30 82 52 76 78
Glendale -32 81 49 71 73
Goff -25 73 48 71 72
Isaac's Canyon -68 120 52 71 73
Kamiah -34 88 54 78 80
S.H. 55 at Dry Creek -9 31 22 32 33
59
CHAPTER4
COMPARISON OF PREDICTION MODEL AND CAMBER DATA
A mathematical or empirical model is useful only if it can be confirmed by legitimate
data.
4.1 Validity of Existing Camber Measurements
The author gathered camber measurement data for precast prestressed bridge
girders from four different manufacturers. However, much of this data was not
useful due to the following:
• Measurements were made from either the top or the bottom of
the girder. The top surface can be very rough and uneven.
Hence, it is not an ideal datum for measurement.
• The girders were not always supported at their centerlines of
bearing and actual support locations were not documented.
• Not all girders were measured at release.
• Camber measurements were missing or incomplete for some
girders.
• Many bridge project have five or fewer girders. This makes the validity
of statistical analysis or comparison questionable.
• A set of voided slabs had been rejected due to improper
location of the voids.
• Some data was questionable. A one inch increase or decrease
in a 24 hour period is highly unlikely. This may have been due
to either human error or thermal effects (Gunnison 1997).
• Some girders were moved during storage or had weights placed
on them to control camber growth.
All the camber measurement data must be viewed with some reservations.
60
4.2 Comparison of Camber at Release
The midspan camber at release is predicted per the proposed time-step method
(Equation 33) and compared to camber measurements of the girders manufactured
for eight bridge projects and the current ITD prediction method. Table 10 shows the
results. With the exception of the box girders, there is good correlation between the
predicted and measured camber at release. The current method used by ITD
underestimates camber at release.
Box girders have internal diaphragms that add stiffness and girder weight In
Appendix IV the camber is computed by numerical integration to account for the
irregularity of the stiffness and weight This computed camber of 39 mm (1.6 in) is
closer to the measured camber, but still not an adequate prediction. If the actual f 0
of the concrete is considered, the predicted camber is 33 mm. Since only four
girders were measured, it is unknown whether low camber is typical of all box
sections.
Table 10 Comparison of predicted camber at release and measured camber at release
Project Type #of #of Span LID Design Meas. Pred. ITD Model Average Meas. Name of girders strands (m) f cl f cl ES pred. pred. meas. camber
girder (MP a) (MPa) (MP a) camber camber camber at at at at release
release release release (mm) (mm) (mm) (mm)
Cole- 1830 mm Overland bulb-tee 49 40 32.4 17.7 35.2 116 39 52 55 41-75 w/reduced -
flanges
Glendale AASHTO 5 34 25.7 22.5 38 46.2 110 36 49 47 43-52 Type Ill
Goff AASHTO 3 36 24.4 21.3 41.4 45.5 116 38 48 50 49-52
Type Ill
Isaac's 1830 mm 24 42 40.2 22.0 34.5 39.5 99 32 52 47 32-59 Canyon bulb-tee
Kamiah 1880 mm 3 42 35.2 18.8 41.4 47.8 112 43 54 56 40-76
bulb-tee
S.H. 55 at AASHTO 8 24 17.9 16.2 27.6 33.7 84 17 22 24 19-29 Dry Creek Type HI
South 36" box 49 Idaho 4 50 24.0 26.2 34.7 43.4 110 38 (39)* 26 19-29 Falls l.C.
South 36" box Idaho 2 14 12.0 13.1 34.7 41.7 38 3 5 12 10- 13 Fallsl.C.
* Camber predicted by numerical Integration O> ~
62
To investigate the effect of span length-to-girder depth (UD) on camber the ratio of
measured to predided camber at release is plotted as a fundion of UD. Figure 17
includes all girders from Table 10 except the short box girder, while Figure 18 omits
both box girders. The best fit line is shown in both figures. Although the figures
seem to indicate lower camber for higher span-to-depth ratios, there is insufficient
data from a variety of girder types to form a conclusion. However, Figure 18 does
show that the predided and measured camber at release are within 10%.
UD
Figure 17 Effect of girder length-to-depth ratio (UD)
UD
Figure 18 Effect of girder length-to-depth ratio (UD) with box girder omitted
4.3 Comparison of Camber at Erection
63
The camber data available after release is sparse and irregular. Figure 19 shows
the camber measurements recorded for 14 girders from the Kamiah project. These
girders have the same design. The camber of each girder was only measured once
and each girder had a different age when the measurement was taken. For
comparison, the predicted camber is also shown. However, as established in the
previous chapter, it is not the intent of the camber prediction model to predict
camber at girder ages less than 30 days.
Unfortunately, the data presented in Figure 19 is representative of the camber
measurements recorded by manufacturers of girders for ITD. Due to the lack of
data, it was impractical to track camber growth on the girders of most ITD projects.
100
80 -- -- - -E
~-•
E 60 - measured ...: ID - -- -.Q E 40 predicted m 0
20
o.__~.._~_._~......_~__._~_._~_._~__.~__....._~..___,
0 10 20 30 40 50 t, days
Figure 19 Measured and predicted camber of 14 girders of the Kamiah project
Unlike other projects, the Cole-Overland project presented an abundance of well
documented camber data. On this project there were 120 girders, 98 having the
same design. The camber of each girder was recorded at several times during its
storage.
64
Figure 20 plots the camber as predicted by the author's time-step model and the
average measured camber at release, 30 days, 60 days, etc. When linear
regression is performed to investigate the relationship between average measured
camber and predicted camber, it is found that the predicted camber lies within the
95% confidence interval of the average measured camber. The slope of this
regression line is 1.02. As good as this correlation appears, the author cautions
that it is not known if any of the girders were loaded to control camber and that not
all girders were measured at each 30 day interval.
65
100
-80 -- -E -E 60 measured (average) ..: (I) .0 E 40 predicted ca ()
20
0 0 30 60 90 120 150 180
t, days
Figure 20 Average measured and predicted camber for girders of the Cole-Overland project
Figure 21 shows the measured camber for eight girders. These girders are
randomly selected from the group presented in Figure 20. Figure 21 illustrates the
wide scatter of camber that is not evident in Figure 20, which only presents the
average measured camber at 30 day intervals.
80
E E 60 ..: (I) ..c
~ 40 ()
20
-
0'--~~'--~~'--~--''--~--'~~--'~~---'~~---~~--'
0 50 100 t, days
150 200
Figure 21 Measured and predicted camber for eight girders of the Cole-Overland project
66
Based on linear regression of the measured camber of these eight girders, at an
age of 30 days or more the author's predicted camber is within the 95% confidence
interval and the slope of the regression line is 1.01. Thus, the model developed by
the author accurately predicts the camber growth of the Cole-Overland girders.
The girders for the Cole-Overland project were cast in Boise, Idaho, over a period of
seven months. Although the average monthly relative humidity varied from 40% to
70% during this period, there is no discernible difference in the observed camber
growth among the girders during different months and seasons. The author offers
the following hypothesis:
The ACI 209 creep and shrinkage equations are based on laboratory
conditions where the relative humidity and temperature were held constant
throughout the testing program. When girders are stored outside they are
subjected to varying relative humidity throughout the day and rewetting when
it rains. Also, in Boise the available water in the air is greater when relative
humidity is lower during summer months than when relative humidity is
higher in winter months (Moran and Shapiro 1992). This is due to the
difference in the average monthly temperatures. These ambient conditions
may effect the creep and shrinkage differently than as treated by the constant
relative humidity and temperature models of either ACI 209 or CEB.
Table 11 summarizes predicted prestress losses and measured and predicted
camber for several girder types and spans. Since there is little camber data
available after release, comparisons with measured camber are made at various
times. The relative humidity is based on the yearly average at the plant location
where the beams were cast In the majority of projects, the predicted camber is
close to the measured camber. As with the camber at release, the current ITD
method underestimates camber at erection. In all but one case the PCI (1992)
multiplier method significantly overestimates the measured camber. The predicted
camber as determined by the multipliers of Tadros et al. (1985) are not shown, but
equal or exceed those predicted by the PCI method.
Table 11 Predicted prestress losses and camber at erection
Project Type uo Design Design Total Pred.* Average Pred.* Average Pred.* Pred. Pred. name of fcl f c predicted camber measured camber measured camber camber at camber
girder (MPa) (MPa) prestress at 30 camber at at 60 camber at at 90 erection at losses at days 30 days days 60 days days by current erection 90 days (mm) (mm) (mm) (mm) (mm) ITD byPCI (MPa) method method
(mm) (mm)
Cole. 1830mm Overland bulb-tee 17.7 35.2 41.4 266 70 66 74 70 76 59 92 w/reduced
flanges
Goff AASHTO 21.3 41.4 41.4 256 65 58 69 71 56 85 Type Ill -Isaac's 1830mm 22.0 34.5 41.4 238 67 50 69 60 71 48 90 Canyon bulb-tee
Kamiah 1680mm 18.8 41.4 44.8 253 73 65 76 79 65 96 bulb-tee -S.H.55 AASHTO at Dry Type Ill 16.2 27.6 34.5 209 30 35 31 42 32 26 39 Creek
• Predicted by proposed time-step method
~
4.4 Comparison of Long-term Camber
No data was available to compare with the long-term prediction model. Per the
discussion presented in Chapter 3, the author contends there is no appreciable
camber growth after the deck is placed or the bridge is placed in service. Figures
14, 15, 20 and 21 support this position.
68
CHAPTER 5
CONTROL OF CAMBER
5.1 Methods of Camber Control
There are three common methods for controlling camber:
• Loading the girders with concrete blocks while they are in
storage, as illustrated in Illustration 4.
Supporting the girders as far inward as possible while they are
in storage. This technique increases camber and is termed
"hanging.n
• Jacking one girder against an adjacent girder at erection. This
procedure is reserved for non-composite sections which have
shear keys.
Illustration 4 Girders temporarily loaded to control camber growth
69
5.2 Effects of Camber Control
There is no literature regarding the effects of camber control. Some of the states
surveyed allow the manufacturer to control camber as needed prior to erection.
Currently, ITD approval is required before a manufacturer can implement any
camber control method. This approval process is merely a verification of stresses
and not a review of the implications to either short-term or long-term camber.
70
The long-term effect of a temporary load depends greatly upon when the load is
applied and removed. Because the creep factor is constantly changing, the
temporary load is modeled as a permanent load with a reverse load being added to
represent the removal of the temporary load. Using the principle of superposition
and modifying Equation 34, the predicted camber, .6ooa,w"8mp• for a girder due to
temporary loading is
Afinalw/temp = Arelease ·(1+0.85C0 )-
i [(Ci -Ci-1). ~Joss • [ emid • L2 - emid • a2 + eend • a2 ]] + i=1 Eclc 8 6 6
(41)
[1+RadjiC0 -Ci))·Atemp -[ 1 +Radj.k(Cn - Ck)] ·Atemp
'Nhere ~ is the elastic deflection due to the temporary load; ~j and ~i.k are the
adjustments at the times of applying and removing, respectively, the temporary load
and similarly for C1 and Ck.
Figure 22 illustrates the effect of a temporary load on the predicted midspan camber
71
of a girder from the Cole-Overland project. Calculations are included in Appendix
Ill. The applied temporary load is a uniform 11 kN/m (0. 75 kips/ft) over the middle
8.0 m (26 ft) of the 32.55 m (106. 75 ft} girder. In the figure, the load is applied at 30
days and removed at 180 days. The overall effect of the temporary load is minimal,
even when applied as early as 30 days.
100
E 80 E
--------------- ~------1 ---.....--ID
60 .a E
No temporary load
af 0 'O
40 ID - With temporary load
0 :a ~
Cl. 20
o~~---~.._~....._~_,_~--~_._~_._~_._~--
o 30 60 90 120 150 180 210 240 270 days
Figure 22 Predicted camber for a temporarily loaded girder of the ColeOverland project
As the figure shows, temporary loading at a late girder age has little effect on
controlling or countering camber growth. However, loading at a very early age
could produce an under-cambered girder if the concrete had not reached its 28 day
modulus, Ee·
Since the typical temporary loading is not applied uniformly along the girder, the
final shape of the girder may be altered.
The "hanging" of girders is difficult to model. Changing the location of the supports
creates an intricate problem. In temporary loading situations, how much of the
72
creep deformation is recoverable and how long its effect remains is unclear
(Mindess and Young 1981; Wittmann 1982). Every time the girder is handled, such
as shipping prior to erection, the camber changes. This occurs even when the
support locations remain the same.
Due to the expense and inconvenience, jacking non-composite girders to align
them during erection is not used frequently. Up to about a 6 mm (0.5 in) mismatch
of camber between adjacent girders can be accommodated without jacking. Jacking
should only be considered on a case-by-case basis and with a structural analysis.
5.3 Necessity of Camber Control
A girder that is greatly under-cambered or over-cambered might indicate an
underlying problem and should be investigated.
The model presented in this report should adequately predict camber, thereby
minimizing the need to adjust the camber of a girder. Occasionally camber control
techniques may be needed. This is especially true when either or both of the
following occur:
• Construction is delayed and the girders are to be stored for an
extremely long time or
• A girder is cured over a weekend and, as a result, did not
camber as much as the others.
However, personnel at two plants claim that when girders cure for extended periods
prior to release, such as over a weekend, they often will eventually camber within
tolerances. No recorded data is available to support or dispute this observation.
Girders requiring camber control should be the exception, not a frequent
occurrence.
CHAPTER 6
CONCLUSIONS AND FUTURE STUDY
6.1 Conclusions
For precast prestressed concrete bridge girders manufactured for Idaho
Transportation Department, the author recommends the following:
73
1. At release the modulus of elasticity of concrete should be calculated
by the ACI 318 formula:
( Eci = 57000,Jf"; psi) (28)
This is based on a dry concrete density of 2320 kglm3 (145 pcf) which
was found at local prestress concrete plants. Ea should not be revised
upward due to higher wet concrete density or increased girder density
due to the inclusion of strand or non-prestressed reinforcement. If
f ci > 48 MPa (7000 psi) or f c > 70 MPa (10000 psi) the use of the ACI
318 formula should be reevaluated.
2. The modulus of elasticity of mature concrete, f:c, should not be based
on the design f c• but rather the f c the concrete mix is capable of
attaining. Based on records from manufacturers that produce girders
for ITD, fa If c-0.75, which yields:
E = Ea c 0.85
The ratio off a If c should periodically be verified by the design
engineer.
(32)
74
3. At release the prestress loss due to elastic shortening is much greater
than the loss due to steel relaxation of low-relaxation strands.
Therefore prestress losses at release may be estimated as the loss
due to elastic shortening, fes· The prestress loss due to elastic
shortening at midspan can be calculated using the gross concrete
area and moment of inertia.
In girders with harped strands the average f es along the girder is
calculated by multiplying the fes at midspan by 85%.
(15)
4. The current ITD practice of assuming 100 MPa (15 ksi) losses at
release and a total 310 MPa (45 ksi) final prestress losses is
empirically justified. However, in girders with relatively few strands,
this lump-sum practice may overestimate the losses. Prestress losses
after release should be calculated by the time-step method presented
in Equation 37 and 38. Refer to the flowchart in Figure 23.
5. The midspan camber at release should be calculated by
where 6.p1 = 1}_. [emid . L2 - emid . a2 + eend . a2 ] (27) E~c 8 6 6
6.wt = 5wd,L4 (26) 384Edlc
The exception to the above is a non-prismatic section such as a box
75
section. In this case, midspan camber should be calculated by
numerically integration along the length of the girder to account for the
added stiffness and weight of the internal diaphragms and forms.
6. At the time of erection or deck placement. camber can be predicted by
the time-step method with an ultimate creep coefficient, Cu =1.85, and
ultimate shrinkage, €sHuit=620µ. Refer to the flowchart in Figure 23.
7. A preliminary estimate for midspan camber at erection is:
derection = 1.65 · Awt + 1 .55 · Ap1 (46)
This formula incorporates several assumptions such as, but not limited
to, girder age of 90 days at erection and prestress losses at release of
100 MPa (15 ksi).
8. No correlation was observed between camber growth and relative
humidity. This is most likely due to the outdoor storage of girders.
Relative humidity is only an indicator of the relative amount of water
vapor when the temperature is constant. However, using a relative
humidity of 60% in the ACI creep and shrinkage formulae yields
reasonable results.
9. No appreciable camber growth is anticipated after the bridge is placed
in service for non-composite girders or after deck placement for
composite girders.
The current range off ci and f e specified by design engineers at ITD is approaching
the upper limit of normal strength concrete as produced by the precast prestressed
girder manufacturers surveyed for this study. As new concrete mixes are
developed, the assumptions made in this report should be reevaluated.
Start
I Calculate Ea
Eci = 4730.Jf; MPa (Eci = 57000.Jf; psi)
I Determine prestress loss, fes. due to elastic shortening.
P0 P0e 2 Md1e -+-----f, - A I I ES -
Eci Aps Apse2 -+--+
or estimate fe5=100 MP a (15 ksi)
~s Ac le
*Multiple by calculated f es by 85% if strands are harped.
I Calculate A.,. and .6p1•
2 2] emid ·a + eend ·a 6 6
Note: If section is not prismatic use numerical integration.
I Predict camber at release.
l cont.
Figure 23 Flowchart for calculating camber at release and erection
76
Estimate camber at erection.
If girder age =90 days at erection, f c1/fc =0.75, VS =80 mm (3.1 in}
fes = 100 MPa (15 ksi) and final total losses = 31 O MPa ( 45 ksi}
Use Multiplier Method.
Multiplier Method
Aerection = 1 .65 • Awt + 1.55 • Ap1
( Stop J
Figure 23 (cont.)
or Use Time-step Method
--
cont.
n
78
Time-step Method
Estimate age of girder at erection.
Determine step size (e.g. 1 O days) and number of steps to erection.
Calculate creep coefficients with Cu=1.85 and shrinkage strain with £sHu.=620µ for each time step.
C(t RH VS)= 1.as( to.s J[2<1+1·1 3e-0.021-VS) 1-1 .27 -0.0067 ·RH)
' ' 1 0 + t0·6 3 J
Calculate f c1r and time-dependent prestress losses in each time step.
f . = P1 - Pi. ross + (P1 - ~. 1oss) • e2 Md1
cir; A A I
F\.,.. = [ C; · f.,., · b {~) + e5"' • E,. +f.-. ]-A,. where O.~fu - 0.55 ~ 0.05
where b=1.0 for straight strands and 0.85 for harped strands
f _ f [log 24t]·[ fst relax; - st 10 0.85fu
Compute midspan camber at erection:
.!\erection = L\ release • ( 1 + 0 · 85C erection )-
Stop
Figure 23 (cont.)
79
6.2 Future Study
The camber prediction model and the formula for the preliminary determination of
camber at erection is based on estimates for the moduli of elasticity, ultimate creep
coefficient and ultimate shrinkage strain of the concrete of the precast prestressed
bridge girders manufactured for ITD. A testing program should be initiated to
investigate these concrete properties.
There is limited camber measurement data after release to verify the model
presented in this study. Consistent and timely camber measurements must be
made on a variety of girder cross-sections. If camber measurement is to be
meaningful, the location of supports must be recorded for all girders. To perform a
proper statistical analysis these measurements must be made on bridge projects
having a sufficient number of girders. Measurements need to continue after
erection and, on composite girders, after deck placement.
80
REFERENCES
Ahmad, Shuaib and Shah, S. P. (1985). •structural Properties of High Strength Concrete and Its Implications for Precast Prestressed Concrete." PC/ Journal, PCI, 30(6), 92-119.
Aitcin, P. C. and Mehta, P. K (1990). Effect of Coarse-Aggregate Characteristics on Mechanical Properties of High-Strength Concrete." AC/ Materials Journal, ACI, 87(2), 103-107.
Alexander, Mark G. (1996). "Aggregates and the Deformation Properties of Concrete.· AC/ Materials Journal, ACI, 94(6), 569-577.
American Association of State Highway and Transportation Officials (AASHTO). (1997). AASHTO LRFD Bridge Design Specifications, AASHTO, Washington, D.C.
American Association of State Highway and Transportation Officials (AASHTO). (1996). Standard Specifications for Highway Bridges, AASHTO, Washington, D.C.
American Concrete Institute (ACI) Committee 209. (1992). Prediction of Creep, Shrinkage, and Temperature Effects in Concrete Structures, Detroit, ML
American Concrete Institute (ACI) Committee 318. (1995). Building Code Requirements for Structural Concrete and Commentary, Detroit, Ml.
American Concrete Institute (ACI) Committee 435. (1995). Control of Deflection in Concrete Structures, Detroit, Ml.
Arioglu, Ergin. (1997). "Discussion of 'Stress-Strain Relationship of Confined and Unconfined Concrete,' by M. M. Attard and S. Setunge." AC/ Materials Journal, ACI, 95(5), 445-447.
ASTM. (1994). "Standard Test Method for Static Modulus of Elasticity and Poisson's Ratio of Concrete in Compression." C 469-94, Philadelphia, PA.
Aswad, A. (1991). "Time-Dependent deflections of Prestressed Members: Rational and Approximate Methods.· Computer Analysis of the Effects of Creep, Shrinkage and Temperature Changes on Concrete Structures SP-129, ACI, 101-114.
Attard, M. M., and Setunge, S. (1996). "Stress-Strain Relationship of Confined and Unconfined Concrete." AC/ Materials Journal, ACI, 94(5), 445-447.
Baalbaki, Walid, Benmokrane, Brahim, Chaallal, Omar and Aitcin, Pierre-Claude. (1991 ). '"Influence of Coarse Aggregate on Elastic Properties of HighPerformance Concrete: AC/ Materials Journal, ACI, 88(5), 499-503.
Baalbaki, Walid, Aitcin, Pierre-Claude and Ballivy,Gerard (1992) ... On Predicting Modulus of Elasticity in High-Strength Concrete." AC/ Materials Journal, ACI, 89(5), 517-520.
81
Bazant, Z. P. (1982). "Mathematical Models for Creep and Shrinkage of Concrete." Creep and Shrinkage in Concrete Structures, Z. P. Bazant and F.H. Wittmann, eds., John Wiley & Sons, New York, 163-256.
Bazant, Z. P. and Kim, Jeong-Koo. (1991). "Improved Prediction Model forTimedependent Deformations of Concrete: Part 2 - Basic Creep." Materials and Structures, RILEM, 24(144), 409-421.
Bazant, Z. P. and Panula, Liisa. (1980). "Creep and Shrinkage Characterization for Analyzing Prestressed Concrete Structures." PC/ Journal, PCI, 25(3), 86-118.
Branson, Dan E. (1977). Deformation of Concrete Structures, New York. ••I J
Burns, Ned H., Gross, Shawn P. and Byle, Kenneth A. (1997). "Instrumentation and Measurements - Behavior of Long-Span Prestressed High Performance Concrete Brioges." Proc., Int. Symposium on High Performance Concrete, PCl/FHWA, New Orleans, LA. 566-577.
Comite Euro-International du Seton (CEB). (1993). CEB-FIP Model Code 1990. Lausanne, Switzerland.
Cook, James E. (1989). "10,000 psi Concrete." Concrete International, ACI, 11(10), 67-70.
Dorgan, Daniel. (1997). "Implementation of High Strength Concrete Research for Prestressed Girders-A DOTs Perspective." Proc., Int. Symposium on High Performance Concrete, PCl/FHWA, New Orleans, LA, 467-47 4.
Dunker, Kenneth F. and Rabbat, Basile G. (1992). "Performance of Prestressed Concrete Highway Bridges in the United States - The First 40 Years." PCI Journal, PCI, 37(3), 48-64.
French, Catherine, Shield, Carol and Ahlborn, Theresa. (1997). "Tests of T'MJ High Performance Concrete Prestressed Bridge Girders." Proc., Int. Symposium on High Performance Concrete, PCl/FHWA, New Orleans, LA, 394-405.
Gale Research. (1996). Weather of U.S. Cities, R. A Wood, ed., International Thomson Publishing Co., Detroit, Ml.
Gardner, N. J. (1989). "Effect of Temperature on the Early-Age Properties of
82
Type I, Type Ill, and Type I/Fly Ash Concretes.D PC/ Journal, PCI, 87(1 ), 68-75.
Ghali, Amin. (1989). "Stress and Strain Analysis in Prestressed Concrete: A Critical Review.• PC/ Journal, PCI, 34(6), 80-93.
Ghali, Amin and Favre, R. (1986). Concrete Structures: Stresses and Deformations, New York.
Ghali, Amin and Trevino, Jose. (1985). "Relaxation of Steel in Prestressed Concrete." PC/ Journal, PCI, 34(5), 82-95.
Ghali, Amin, Sisodiya, Ronny G. and Tadros, Gamil S. (1974). 'Displacements and Losses in Multistage Prestressed Members." J. Struct. Engrg., ASCE, 100(11 ), 2307-2322.
Gilbert, R I. (1988). Time Effects in Concrete Structures, Amsterdam.
Goodyear, David and Smith, Monte J. (1988). "A Practical Look at Creep and Shrinkage in Bridge Design.· PC/ Journal, PCI, 33(3), 108-121.
Gunnison, John. (1997). Pine Bluff Bridge Report. Unpublished Report for Spokane County Public Works, Spokane, WA
Gustaferro, Armand, Hillier, Marc A and Janney, Jack R. (1983). "Performance of Prestressed Concrete on the Illinois Tollway Aft.er 25 Years of Service." PC/ Journal, PCI, 28(1 ), 78-89.
Gustaferro, Armand, Hillier, Marc A and Janney, Jack R. (1983). "25 Year-Old Prestressed Concrete Bridge Girders Tested." PC/ Journal, PCI, 29(1), 177-179.
Gutierrez, Pilar Alaejos and Canovas, Manuel Fernandez. (1995). "The Modulus of Elasticity of High Performance Concrete." Materials and Structures, RILEM, 28(184), 559-568.
Halsey, J. Todd and Miller, Richard. (1996)." Destructive Testing of Two FortyYear-Old Prestressed Concrete Bridge Beams: PC/ Journal, PCI, 41(5), 84-93.
Huo, Xiaoming and Tadros, Maher K (1997). "Time-Dependent Effects in High Performance Concrete Bridge Members." Proc., Int. Symposium on High Performance Concrete, PCl/FHWA,NewOrleans, LA, 419-431.
83
lravani, Said. (1996). ·Mechanical Properties of High-Performance Concrete." AC/ Materials Journal, ACI, 93(5), 416-426.
Joo, Y. S. and Tadros, M. K (1991). "Serviceability of Prestressed Composite Members.· Computer Analysis of the Effects of Creep, Shrinkage and Temperature Changes on Concrete Structures SP-129, ACI, 171-188.
Khan, Arshad A, Cook, William D. and Mitchell, Denis. (1997). "Creep, Shrinkage, and Thermal Strains in Normal, Medium, and High-Strength Concretes during Hydration." PCI Journal, PCI, 94(2), 156-163.
Kaufman, M. Keith and Raimirez, Julio. (1989). "Use of Higher Strength Concrete for Prestressed Beams in the State of Indiana." PCI Journal, PCI, 34(3), 78-93.
Krauss, Paul D. And Rogalla, Ernest A. (1996). "Transverse Cracking in Newly Constructed Bridge Decks." National Cooperative Highway Research Program, TRB, Report 380, Washinton, D.C.
Labia, Yolanda, Saiidi, M. Saiid and Douglas, Bruce. (1997). "Full-Scale Testing and Analysis of 20-Year-Old Pretensioned Concrete Box." AC/ Structural Journal, ACI, 94(5), 471-482.
Leon, RT., Olson, S. A, French, C. A. (1990). "TYJenty-Year-Old Prestressed Bridge Girders: Assessment of Current Condition and Evaluation of Impact Damage Repair." Proc. Bridge Engrg. Research in Progress, NSF and Civil Engrg. Dept. Univ. of Nevada, Reno, NV, 151-154.
Lwin, M. Myint, Khaleghi, Bijan and Hsieh, Jen-Chi. (1997). "Prestressed I-Girder Design Using High Performance Concrete and the New AASHTO LRFD Specifications.n Proc., Int. Symposium on High Performance Concrete, PCl/FHWA, New Orleans, LA, 406-418.
Martin, Leslie D. (1977). "A Rational Method for Estimating Camber and Deflection of Precast Prestressed Members.n PC/ Journal, PCI, 22(1), 100-108.
MathSoft, Inc. (1995). Mathcad User's Guide. Cambridge, MA.
Mindess, Sidney and Young, J. Francis. (1981 ). Concrete. Engle'NOod Cliffs, NJ.
Moran, Michael J. and Shapiro, Howard N. (1992). Fundamentals of Engineering Thermodynamics. New York.
Moustafa, Saad E. (1986). "Nonlinear Analysis of Reinforced and Prestressed Concrete Members." PC/ Journal, PCI, 31{5}, 126-147.
84
Myers, John J. and Carrasquillo, Ramon, L. (1997). •Quality Control & Quality Assurance Program for Precast Plant Produced High Performance Concrete UBeams: Proc., Int. Symposium on High Performance Concrete, PCl/FHWA. New Orleans, LA. 368-382.
Naaman, Antoine E. and Hamza, Ali M. (1993). •prestress Losses in Partially Prestressed High Strength Concrete Beams.• PC/ Journal, PCI, 38(3), 98-114.
Neville, Adam M. {1997). "Aggregate Bond and Modulus of Elasticity of Concrete.• PC/ Journal, PCI, 95(1), 71-74.
PCI Committee on Prestress Losses. {1975). "Recommendations for Estimating Prestress Losses: PC/ Journal, PCI, 20(4), 43-75.
Pessiki, Stephen, Kaczinski, Mark and Wescott, Herbert. (1996). "Evaluation of Effedive Prestress Force in 28-Year-Old Prestressed Concrete Bridge Beams.· PC/ Journal, PCI, 41(6), 78-89.
Precast/Prestressed Concrete Institute (PCI). (1997). Precast Prestressed Concrete Bridge Design Manual. Chicago, IL.
Precast/Prestressed Concrete Institute {PCI). (1992). PC/ Design Handbook. Chicago, IL.
Rao, Prasada and Jayaraman, R. {1989). "Creep and Shrinkage Analysis of Partially Prestressed Concrete Members.· J. Struct. Engrg., ASCE, 115(5), 1169-1189.
Roller, John, Russell, Henry G., Bruce, Robert N. And Martin, Barney T. (1995). "Long-term Performance of Prestressed, Pretensioned High Strength Concrete Bridge Girders.• PC/ Journal, PCI, 42(6), 48-58.
Rusch, Hubert, Jungwirth, Dieter and Hilsdorf, Jubert. (1983). Creep and Shrinkage, New York.
Sellevold, Erik J., Justnes, Harald, Smeplass, Sverre and Hansen, Einar Aasved. (1994). "Seleded Properties of High Performance Concrete.· Advances in Cement and Concrete, M. W. Grutzeck and S. L. Sarkar, eds., Amer. Soc. of Civil Engineers, NewYork, 562-609.
Shenoy, Chandu V. and Frantz, Gregory C. (1991). "Strudural Tests of 27 YearOld Prestressed Concrete Bridge Beams.• PC/ Journal, PCI, 36(5), 80-90.
85
Silfwerbrand, Johan. (1997). "Stresses and Strains in Composite Concrete Beams Subjected to Differential Shrinkage: ACI Structural Journal, ACI, 94(4), 347-353.
Smerda, Zdenek and Kristek, Vladimir. (1988). Creep and Shrinkage of Concrete Bements and Structures, New York.
Stanton, John F., Eberhard, Marc 0., Barr, Paul and Fekete, Elizabeth A {1997). "Behavior of High Performance Concrete in the SR18/SR516 Covington Way Undercrossing: Preliminary Report.• Dept Of Civil Engr, Univ. of Washington, Seattle, WA
Stanton, John F., Eberhard, Marc 0., Barr, Paul and Fekete, Elizabeth A (1997). "Evaluation of long-Term Behavior of High Performance Prestressed Concrete Girders.· Proc., Int. Symposium on High Performance Concrete, PCl/FHWA, New Orleans, LA, 612-622.
Sumiden Wire Products Corporation. (1997). TEC-8504. Stockton, CA
Tadros, Maher K (1997). "Impact of HPC on Design Criteria." Dept. of Civil Engr., Univ. of Nebraska, Lincoln, NE.
Tadros, Maher K, Ghali, Amin and Dilger, Walter H. (1977). "Effect of NonPrestressed Steel on Prestress Loss and Deflection.• PC/ Journal, PCI, 22(2), 50-64.
Tadros, Maher K, Ghali, Amin and Dilger, Walter H. (1977). "Time-Dependent Analysis of Composite Frames.· J. Struct. Engrg., ASCE, 108(ST4), 871-884.
Tadros, Maher K, Ghali, Amin and Dilger, Walter H. (19n). "Time-dependent Prestress loss and Deflection in Prestressed Concrete Members." PC/ Journal, PCI, 20(3), 86-95.
Tadros, Maher K, Ghali, Amin and Meyer, Arthur W. (1985). "Prestressed Loss and Deflection of Precast Concrete Members." PC/ Journal, PCI, 30(1 ), 114-141.
Wittmann, F. H. (1982). "Creep and Shrinkage Mechanisms." Creep and Shrinkage in Concrete Structures, Z. P. Bazant and F.H. Wittmann, eds., John Wiley & Sons, New York, 129-162.
Zia, Paul, Preston, H. Kent, Scott, Norman L. and Workman, Edwin B. (1979). "Estimating Prestress Losses." Concrete International, ACI, 1 (6), 32-38.
APPENDIX I
SURVEY OF STATE BRIDGE DESIGN SECTIONS
Table 12 summarizes the results of the survey of the bridge design sections of
seleded state departments of transportation.
86
Table 12 Survey of Bridge Design Departments of Various States (U.S. Customary Units)
State Camber and deflection Determination of Calculation of Comments calculation practices prestress losses Ee
Alaska Use multipliers from Low-relax: For camber Typical design Is r ar6.5 ksl and Concrete Technology, a @transfer=23 ksl deflection: f'0=7.5 ksl. precast prestressed concrete @final=40 ksi 64000Jf; vendor In Alaska. (i.e. 189 ksl effective b. 3 man1111=1.7 b.p1+1.8b.wt prestress) For LL
nnaf 2. Ob.p1 +2.2b.wt+ .t\ieec1 Stress-relieved: deflection: @transfer=26 ksl 57000.jf'; @final=45 ksi
Arizona Multipliers: AASHTO refined 57000.Jf; Most girders are AASHTO Type 111-VI. A.r.ct1on=1.B·(b.p0+bwt+A1oae1) method. Assume Typical design Is f'01=4-5 ksl and f'0=5-Ann.1=3.0'(APo +bwt+Aro-)+1.2 tosses at erection (30- 6ksl. Aliltad) 60 days) are
ES+0.5'(time-dependent tosses)
California Multiplier: Elastic shortening is not 57000.Jf; There Is not much camber If calculated. Assume designers use a depth/span ratio of
A.rec11on=1.5·(0.89APo +bwt) 11 % loss at erection 0.05. Caltrans uses a California Type (60 days). I section which Is similar to AASHTO Total losses: Type Ill. A 2" camber strip ls provided low-relax = 35 ksl to accommodate camber differences. stress-relieved=45 ksl On exception, girder can be 1" Into
deck. Predicted camber Is not placed on plans. It's contractor's responslblllty to control camber.
~
State Camber and deflection Determination of calculation practices prestress losses
Colorado An in-house computer AASHTO method, but program calculates max and steel creep Is min camber and creep. calculated from
information (circa 1980) from Florida Wire & Cable Co.
Idaho Multiplier: @transfer-15 ksi l1.rect1on = 1. 5. (~pefl+ L\....t) @final=45 ksi where ~Pell is the camber Sometimes AASHTO due to prestress force after refined method is used. final losses.
Iowa Use PCI multipliers for AASHTO method. erection. After composite deck Is poured, camber growth Is Insignificant.
Minnesota Currently under discussion. AASHTO method, but Use LEAP™ software for moving to LEAP™ camber at erection. Feel which uses PCI (1975) long-term camber prediction method. is futile.
Calculation of Ee
62000Jf; Lower this by 10% due to poor local aggregate (weathered granite) or useACI 363.
ACI 318
ACI 318
LEAP™wlth w= 155 pcf.
Comments
Typically box girders are used. They try to design for <1" camber by keeping girder spacing small. Therefore camber growth Is Insignificant.
-
Camber strip ls used for camber differences. Occasionally the grade Is adjusted. Iowa Investigated the possibility of changing multipliers, but camber data was very scattered and thought to be unreliable due to unscientific measuring.
MNDOT uses camber strip to compensate for camber. MNDOT always uses a CIP deck.
<» 0)
State Camber and deflection Determination of calculation practices prestress losses
Montana Camber at erection is not AASHTO method. calculated.
Nevada For camber at erection, AASHTO method. Due camber at release is to low RH, lump-sum increased by 50-150%. No probably under-set guideline. estimates losses
Oregon Use PSBEAM™ program for AASHTO or lump-sum deflection calculation. Use as follows: recommendations from 14 days=20 ksi Modem Prestressed 90 days=30 ksi Concrete by Libby (1984) 9999 days=40 ksi for creep and steel relaxation.
Utah Calculate camber at release AASHTO except use only, but It is not included in 35 ksi for final losses the plans. but allow zero tension
in girder per former edition of AASHTO.
Calculation of Ee
ACI 318
ACI 318
ACI 318 with w= 155 pcf
ACI 318
Comments
Camber is only measured at release. Contractor loads girders, as needed, to match camber.
Nevada uses a 1 "-2" camber strip. Typically the camber of girders is within 0.5". There Is no prestressed concrete vender within the state, therefore only about one precast prestressed bridge Is designed every 2 years.
Sometimes they shave the deck if cambers don't match. Usually there is not a problem.
Utah uses a camber strip. If necessary, the grade Is adjusted. Typically there Is not a problem with adjacent girders matching.
CX> co
State Camber and deflection calculation practices
Washington Multipliers: Composite sections: f1nna1=2. 06191ea18
Non-composite sections: ~.,=3.0~-This method is being discussed.
Wyoming Use LEAP™. Are not concerned with camber growth after deck placement.
Determination of prestress losses
Lump-sum: @release=20 ksi @erection=35 ksi flnal(>1 yr)=48 ksi WashDOT is moving to calculating individual losses.
AASHTO method with PCI (1975) for steel relaxation.
Calculation of Ee
ACI 318
ACI 318
Comments
Specifications state that erection takes place prior to 120 days. Adjustment to the camber strip or grade is responsibility of contractor.
Wyoming uses a 2" camber strip or diaphragm placement to compensate for different cambers.
co 0
APPENDIX II
PREDICTING CAMBER AT RELEASE BY NUMERICAL
INTEGRATION
91
In the following spreadsheet a typical girder from the Cole-Overland project is
divided into tenth points along its length for the computation of prestress loss due to
elastic shortening and camber. Since the girder and loading are symmetric about
the centerline, the calculation is made from the right end to the midspan.
The spreadsheet also includes a comparison of the prestress loss due to elastic
shortening and camber as a function of transformed and gross concrete section
properties. The camber predicted by using gross concrete section properties
closely agrees with the actual camber measurements from the Cole-Overland
project. Furthermore, this predicted camber at release is essentially the same as
that predicted by a less rigorous method in Appendix Ill.
In Appendix IV a similar computation is made with a box section.
Using Numerical Integration to Calculate the Prestress Loss Due to Elastic Shortening and Camber at Release
Girder from Cole-Overland Project L (ft)= 106.375 rc1 (ksi)= 5.1 Yb (In)= 33.28 lc(ln114)= 450272 Ect (ksl)= 4115 UD= 17.7 Ac {in112)= 723 Eps (ksi)= 28600 Apa (ln114)= 6.12 n= 6.95 Po (kips)= 1239 A trans (ln112)= 759 #of strandt 40 A trans (ln112)= 765 w/rebar wdl (kif)= 0.763
Calcultion of Elastic Shortening and camber w/transformed section properties considering the strands and no rebar.
Position strand Ytrans (In) ltnn (ln114) Md1 Eqn 15 Eqn 15 fes Slope camber
(ft) cg (In) e (In) (In kip) Num Den om (ksl) (In) OL 0 17.25 32.51 459181 15.26 0 2.26 0.16 14.58 0.0055 0.00 0.1L 10.64 14.28 32.37 462787 18.09 4662 2.33 0.16 14.88 0.0044 0.63 0.2L 21.28 11.32 32.23 466991 20.91 8288 2.42 0.16 15.35 0.0033 1.12 0.3L 31.91 8.35 32.08 471819 23.73 10879 2.56 0.16 16.10 0.0023 1.48 0.4L 42.55 5.39 31.94 477239 26.55 12433 2.77 0.16 17.21 0.0012 1.70 0.5L 53.19 3.88 31.87 480249 27.99 12951 2.90 0.16 17.90 0.0000 1.77
Weighted 468482 Weighted 15.76 average average
Calculating fes and camber across the span w/gross concrete properties only
Position strand Ytrans (In) le (in114) Mdl Eqn 15 Eqn 15 fes Slope camber
(ft) cg (In) e (In) (In kip) Num Denom (ksl) (In) OL 0 17.25 33.28 450272 16.03 0 2.42 0.16 15.57 0.0061 0.00 0.1L 10.64 14.28 33.28 450272 19.00 4662 2.51 0.16 16.00 0.0049 0.71 0.2L 21.28 11.32 33.28 450272 21.96 8288 2.64 0.16 16.63 0.0038 1.26 0.3L 31.91 8.35 33.28 450272 24.93 10879 2.82 0.16 17.59 0.0026 1.66 0.4L 42.55 5.39 33.28 450272 27.89 12433 3.08 0.16 18.98 0.0013 1.91 0.5L 53.19 3.88 33.28 450272 29.41 12951 3.25 0.16 19.84 0.0000 2.00
Weighted 17.13 average
co N
Calculating fEs and camber across the span by Including rebar and strands into section properties
OL 0.1L 0.2L 0.3L 0.4L 0.5L
Summary:
Position strand Ytrsns (in) ltrans (ln"4) Md1 Eqn 15 Eqn 15 fEs Slope camber
(ft) cg (In) e (in) (In kip) Num Den om (ksl) (In) 0 17.25 33.06 470576 15.81 0 2.28 0.16 14.68 0.0056 0.00
10.64 14.28 32.91 474274 18.63 4662 2.34 0.16 14.99 0.0044 0.64 21.28 11.32 32.77 478566 21.45 8288 2.44 0.16 15.46 0.0034 1.13 31.91 8.35 32.63 463484 24.26 10679 2.58 0.16 16.22 0.0023 1.50 42.55 5.39 32.49 488994 27.10 12433 2.79 0.16 17.33 0.0016 1.72 53.19 3.88 32.42 492049 28.54 12951 2.92 0.16 18.02 0.0000 1.79
Weighted 480075 Weighted 15.87 average average
1. Ec1=4115 ksl Is based upon ACl-316. 2. Average fEs Is approximately 85% of midspan fes. 3. Dead load of 152 pcf Is assumed. 4. Integration by trapezoid rule. 5. Using the conventional (not integrating across the span) deflection formulae and section properties at midspan
yields an upward camber of 2.03" (52 mm). This closely matches the actual average field measurement of 2.12" • Numerical integration using gross concrete section properties Is the closest value.
co w
APPENDIX Ill
PREDICTING CAMBER AT RELEASE AND ERECTION WITH A
MATHCAD WORKSHEET
94
The author created a Mathcad (MathSoft 1995} YJOrksheet to calculate the camber
at release and erection. This YJOrksheet also shows the effect of a temporary load
on camber.
This "VVOrksheet uses a girder from the Cole-Overland project, but can be adapted
for any girder. However, as discussed in Chapter 4, this method may not accurately
predict camber for a box section.
Prediction of Camber at Raleaae for girder on the Cole-Overland Prvject 1. Using bare concrete section properties 2. Assumed concrete dead load of 152 pct 3. Bastic shortening loss calculated per Equation 15 4. Ed per ACI 318
5. 72" Bulb tee w/shortened flanges 6. U.S. customary units
Girder and Strand Properties:
# of 0.5" dia low-relax strands
fu =270 ksi
Aps =tun-0.153
f st =o.7Sfu stress at jacking
Eps =286QO ksi
Po =f st·Aps jacking force
L =106.375 ft Center line bearing distance
a :.!:- _ 5.25 ft harping point from end of girder 2
f c1 =s.1 ksi
E cl =33-1451.5. {f c:;t1000)0.5 1000
E cl= 4.115°103
Bare Concrete Section Properties
I c :450272
vs =3.17
Yb =33.28
2 wc1·L.:
Met =--·12 8-1000
Ac =m.3 in2
in volume-to-surface area ratio
in centroid of section (from bottom of girder)
girder deadload, plf
midspan moment due to girder weight, in kip
95
ksi
96
Strand locations
cg mid =3.875 in cg end =1725 in
e end =Yb- cg end e mid =Yb - cg mid
eend •16.03 in amid =29.41 in
Calculate prestress loss due to elastic shortening per Equation 15:
fes= 19.79 ksi
For girder with harped strands, average fes is approximately 85% of midspan fes
0.85-f ES "' 16.82 ksi
Let f es = 1s.a ksi
Prestress force at release:
p 1 = 1.135·1a3 kips
For two-point harping, midspan camber due to prestress force at release:
Midspan camber due to girder weight
in
Predicted midspan camber at release:
t.. release = 2.03 in
Prediction of Camber at Erection for girder from the Cole-Overland Project
Shrinkage
The ACI 209 shrinkage function includes coefficients that account for various components of the concrete mix. For the concrete mixes used by Central Prestressed Co. in Spokane, WA, and
97
Monroe, Inc. in Boise, ID, in the manufacture of girders for rro, most of these coefficients are close to unity and others are potentially misleading due to the use of admixtures. In this model, the following coefficients are set to 1.0:
Ks5 a coefficient for the slump of the concrete
~ a coefficient for the number of bags of cement
Ks AC a coefficient for the % of air
~ a coefficient for the % of fines in the concrete
The coefficients for the shrinkage function are defined as follows:
Ks s =1.0
Ks b =1.0
KsAc =1.0
Ks F =1.0
Ks vs(VS) =1.2-2.n·0.12-VS
KsH(RH) "1.4-0.01-RH
All the coefficients are incorporated into one function, YSH·
T sh{ RH, VS) =Ks H(RH).l<s vs(\IS)-Ks F·Ks b'Ks AC
The ultimate shrinkage strain is written as:
Note: 780µ is suggested by ACI 209, but 620µ is used as discussed in Chapter 3.
The time-dependent shrinkage function is:
•sH(t.RH.VS) ::55l ·ESt-fUt(RH.VS) +t
The loss of prestress due to shrinkage is
f SH(t.RH, VS) =t SH(t,RH, VS)·E ps
which is the same as Equations 21 and 35, but in a different form.
98
Creep
The ACI 209 creep function has coefficients similar to those of the shrinkage function. They are summarized as follows:
K 5 =1.0
K AC =1.0
Kb =1.0
K F =1.0
K H(RH) ::1.27-0.0067-RH
K vs(VS) -2·'1~1.13-2.72·0.5'4-VS 3
All the coefficien:.-S are incorporated into one function, yCR"
The time-dependent creep function is
(t)0.6 C(t,RH.VS) :;: oa·{1.85}-r CR<RH.VS}
10.,.- (t) .
where 1.85 is the ultimate creep coefficient ACI suggests 2.35 and PCI (1997) uses 1.88. The 1.85 value is used here per the discussion in Chapter 3. The above is a modified form of Equation 34.
The loss of prestress due to creep is
f cfi=C<t.RH,VS)·' ~:.f cir
where fr.1r is the stress in the concrete at the level of the strand e.g. For harped strands, 85% of the midspan fcir is an estimate of an average fcir along the length of the girder.
Computation of Presbess losses
The total prestress losses are a result of elastic shortening and the time-dependent losses due to shrinkage, creep and steel relaxation. The prestress force, P1, at release includes the elastic shortening. The time-dependent losses occur on the mature concrete where
E =Ee! c 0.85
From Equation 32
The time-dependent losses are interrelated and defined as follows:
Loss due to steel relaxation - . w.r2+t' - f st frelar;(t) -fst· ~, .. --· - .55 45 ~ 0.9-f l1'
This predicts higher relaxation losses than Sumiden {1997).
Losses due to creep are a function of fr:ir However, fr.1r is also a function of the effective prestress force, P,.,.. which is a function of all losses. Therefore, fc.R is calculated based on the effective prestress during the previous time interval. In this example, each time step is 10 days. The size of a time step is easily adjusted.
Let ~t be the duration of each time step in days.
The stress at the level of the e.g. of the strands in determined at midspan then multiplied by F Mi• 0.85, for an average fc1r·
F adj =o.85
Defining the effective prestress force for each time step:
Let - 1 emkJ2 z1 -----·
\Ac I c .
P eft(t,RH, VS) = P 1 if t<4t
P 1-Aps·ff~t)+f5H(l,RH,VS) ... _
·1· · ~I !+ fz1·Peft(t-4t,RH,VS) ... ,·C(t,RH,VS) E ·Fadj ': emkl I c I
' ! '.+·Mdl- , · I , I c 1 ~
if t241
99
Defining fcir and fSH as a function of Pelf:
2 . =t P eft(t.RH,VS) -. P eft(t,RH,VS)-emld _ Mc11-emld -F-f cfr{t,RH,VS) -..,
\ Ac le le
fCR(t.RH,VS) =C(t RH VS)·"~'·fQr{t RH VS) , . \Ee: , •
Total prestress losses at any time:
f1o5s(t,RH,VS) =r ES.,..(f reiax(thf CR(t,RH.VS) ~f SH(t.RH.VS})
At 90 days and 55% relative humidity: {60% may also be used)
f1oss(90,55,VS) =38.6 ksi or 266 MPa
t =&.2·~ .. 360 days RH ::40,45 .. 80
--,,,....,..,.._, ... .,,,,,,.----- -------·· .,. .,. --. ---~----1 ""--: .-· ..
/ .-/.•*"
1.· 1.·
1: .
100
Let \nc be the age of the girder at erection.
t erec =a.za .. 360 days
The constantz2 is introduced to abbreviate the camber erection formula.
The predicted camber at erection is:
Aerec:tionCt erec,RH,VS) =Are1ease· 1- ~ :·C<lerec,RH.VS) ...
tloor. ~ erec . !JI.I
+ 1 <-C{t erec,RH, VS) .•• ·· P eft(Al·l,RH,VS) ••. -22 -i= 1 + C(l!l.·i,RH,VS' • + P eft(llt·(i -1),RH,VS)
Predicted Camber for a Girder from Cole-Overland Project with A,.19ae=2.0" and Prior to Placement of Deck
3.5 ..-----.,..----~-----..----...-----------.
3.25
2.25
-·-··--------·· ·······-· -· ··-··· ... _. ...... --.... ........... ______ ........ ------... - .,,...-.--.. ~ .... -· __ _... ... _..,,,,. .·:.-,.,, .·,,,,,. ·~
.~ "I
1
2.._ ___ _._ ____ .._ ___ __._ ____ .._ ________ ___.
0 60 120 180 t (days)
240 300 360
101
102
Predicting camber from Temporary Loading
Let the temporary load be a 0.75 kip/ft applied to the mid 26 ft of the girder. For simplicity, this load will be modeled as a concentrated 19.5 kip load at the center. The midspan camber is less than 5% different between the concentrated load and the temporary load.
Calculation of concentrated load, P temp:
p temp =2fJ.0.75 Ptemp=19.5 kips
From beam deflection tables, a,A'M is the midspan camber {upward positive) for a concentrated load at the center
3 ·Ptemp·L.: A = ·1728
temp 48-E c·I c
A temp = -o.39 inches elastic camber due to temporary load
Let ~cad be the age of the girder when the temporary load is applied
t 1cad =J0.60 .• 90 days
Since the temporary load is applied after release, the adjustment factor, RAtii, is applied to the creep coefficient
Let RH =ss
Creep of temporary Load
Camber due to temporary load only
Camber for all prestress, girder weight and temporary load
Predicted Camber when a Temporary Load Is Applied
3.25
2.25
2'--~~~.._~~~..__~~~~~~~"""-~~~~~~~~
0 60
load @ 30 days load @ 60 days loed @ 90 days
120
Removal of Temporary Load
Let the ~ = girder age at unloading
t load :::30 days
Rebound due to unloading
180 days
240
Computation of camber due to loading and/or unloading
300 360
Atot<t erec• t loed• t tnoari> =1u lri:lacl<t erec.cmnber temp{t erec·t load) .•• ,ca'l'lber temp<t erec·t load>\ +A unload{t erec • t unload) ,
103
3.25
3
2.5
2.25
Effect of Temporary Loading on Predicted camber for a Girder of the Cole-Overland Project
... ... .. ·· ...
----······-····-······-····_··----1 ....
2..._~~--'---~~ ........ ~~~--'~~~ ....... ~~~--~~~~ 0 60
No temp load • • wJtemp load
120 180 240 300 360 t. days
104
105
APPENDIX IV
PREDICTING CAMBER AT RELEASE FOR A BOX SECTION BY
NUMERICAL INTEGRATION
Box sections have internal diaphragms at both ends and the mid-section. A non
removable plywood form shapes the hollow box in the remainder of the girder. This
form adds both weight and stiffness to the section. As a results of both the
diaphragms and form, the box section is not a prismatic shape.
The format of this spreadsheet is the same as that presented in Appendix II.
Calculation of Camber for South Idaho Falls 36" Box Girder by Numerical Integration Stiffness and weight of interior wood form is included
L (ft)= 76.58 rel (ksl)= 5.0 yb (in)= 16.43 le: (ln4)= 143574 Eel (ksl)= 4030 Per ACI 318 Ac (in2)= 657 Eps(ksl)= 28600 UD= 26 ~(ln4)= 7.65 n= 7.10 Po (kips)= 1549
cone 152 #of 50 wt{pct)= strands
strand wt 0.0181 (kif)= llGlld (1114)= 186624 Total 0.0319 steel wt .
stirrups 10.2 lbs/ft In excess of concrete wt long. 3.6 lbs/ft rebar
Calculation of Elastic Shortening and camber w/gross concrete properties but deadload of cone + strands Deadload moment ch3nges because the x-sectlon changes
Position Area Section DL shear DL e (In) I Eqn 15 Eqn 15 fes P1 Total Slope camber (ft) (ln2) wt (kips) (kip) moment Num Denom (ksl) (kips) moment (In)
(kip•ln) (In kip) OL 0 1728 0 39.56 0 8.35 186624 1.48 0.15 10.0 1473 6162 0.00628 0.00 0.05L 3.929 857 6.03 33.53 1723 6.38 143574 2.17 0.15 14.3 1440 7458 0.00582 0.29 0.1L 7.858 657 3.63 29.90 3218 7.18 143574 2.20 0.16 14.4 1439 7108 0.00523 0.55 0.15L 11.787 857 3.63 26.28 4543 7.98 143574 2.24 0.15 14.6 1437 6926 0.00465 0.78 0.2L 15.716 857 3.63 22.65 5697 8.78 143574 2.29 0.15 14.9 1435 6909 0.00409 0.96 0.25L 19.645 657 3.63 19.03 6679 9.59 143574 2.35 0.15 15.2 1433 7054 0.00352 1.16 0.3L 23.574 857 3.63 15.40 7491 10.39 143574 2.43 0.16 15.6 1430 7359 0.00293 1.32 0.35L 27.503 857 3.63 11.78 8132 11.19 143574 2.52 0.16 16.1 1426 7821 0.00232 1.44 0.4L 31.432 857 3.63 8.15 8602 13.20 143574 2.90 0.16 18.2 1410 10010 0.00159 1.53 0.45L 35.361 857 3.63 4.53 6901 13.20 143574 2.67 0.16 18.0 1411 9728 0.00079 1.59 0.475L 37.3255 857 1.81 2.72 8986 13.20 143574 2.86 0.16 18.0 1411 9648 0.00039 1.60 0.5L 39.29 1728 2.72 0.00 9018 14.77 186624 1.99 0.15 12.9 1450 12404 0 1.61
39.56 Weighted ave.= 15.17
.... 8
Calculation of Elastic Shortening and camber w/transformed concrete properties (except I) and deadload of cone + strands+rebar + stirrups Position Area Section DL shear DL Strand Y bot (in) e le Eqn 15 Eqn 15 fEs (ksi) P1 Total Slope camber (ft) (in2) wt (kips) (kips) moment cg (in) (in4) Num Denom (kips) moment (In)
(klp*ln) (In kip)
OL 0 1726 0 39.56 0 9.65 17.75 8.10 186624 1.44 0.15 9.7 1475 5972 0.0066 0.00 0.05L 3.929 857 6.09 33.46 1722 8.64 15.98 7.13 143574 2.27 0.15 14.9 1435 8517 0.0061 0.30 0.1L 7.658 857 3.68 29.80 3213 8.04 15.93 7.89 143574 2.30 0.15 15.0 1434 8099 0.0054 0.57 0.15L 11.787 857 3.66 26.12 4531 7.24 15.88 8.64 143574 2.34 0.15 15.2 1433 7851 0.0048 0.81 0.2L 15.716 857 3.68 22.44 5676 6.44 15.83 9.40 143574 2.39 0.15 15.5 1431 7770 0.0041 1.02 0.25L 19.845 857 3.68 18.76 6647 5.63 15.79 10.15 143574 2.45 0.16 15.8 1428 7854 0.0035 1.20 0.3L 23.574 657 3.66 15.08 7445 4.83 15.74 10.91 143574 2.53 0.16 16.2 1425 8101 0.0028 1.35 0.35L 27.503 857 3.68 11.40 8069 4.03 15.69 11.66 143574 2.62 0.16 16.7 1421 8507 0.0022 1.47 0.4L 31.432 857 3.68 7.72 8520 3.23 15.84 12.41 143574 2.73 0.16 17.3 1417 9067 0.0015 1.55 0.45L 35.361 857 3.68 4.04 8797 3.23 15.e4 12.41 143574 2.71 0.16 17.1 1418 8804 0.0007 1.60 0.475L 37.3255 857 1.84 2.20 8871 3.23 15.64 12.41 143574 2.70 0.16 17.1 1418 8734 0.0004 1.62 0.5L 39.29 1728 2.74 -0.54 8890 3.23 17.55 14.32 186624 1.92 0.15 12.5 1454 11926 0.0000 1.62
40.11 Weighted ave.= 15.29
Calculation of Elastic Shortening and camber w/transformed concrete properties and deadload of cone + strands+rebar + stirrups
Position Area Section DL shear DL Strand Y bot (in) e l1ra111 Eqn 15 Eqn 15 fEs (ksl) P1 Total Slope camber (ft) (ln2) wt (kips) (kips) moment cg (in) (in) Num Denom (kips) moment (in)
(klp*ln) (In kip)
OL 0 1728 0 39.56 0 9.65 17.75 8.10 190297 1.43 0.15 9.7 1475 5974 0.00635 0.00 0.05L 3.929 857 6.09 33.48 1722 8.84 15.98 7.13 147311 2.26 0.15 14.8 1436 8521 0.00566 0.29 0.1L 7.858 657 3.68 29.60 3213 6.04 15.93 7.89 146551 2.29 0.15 15.0 1434 8103 0.0052 0.55 0.15L 11.787 857 3.68 26.12 4531 7.24 15.88 8.64 147210 2.33 0.15 15.1 1433 7856 0.00457 0.78 0.2L 15.716 857 3.68 22.44 5676 6.44 15.83 9.40 147934 2.37 0.15 15.4 1431 7777 0.00395 0.98 0.25L 19.845 857 3.68 18.76 6647 5.63 15.79 10.15 148723 2.43 0.16 15.6 1429 7664 0.00333 1.15 0.3L 23.574 857 3.68 15.08 7445 4.83 15.74 10.91 149579 2.50 0.16 16.0 1427 8114 0.0027 1.29 0.35L 27.503 857 3.68 11.40 8069 4.03 15.69 11.66 150500 2.58 0.16 16.5 1423 8525 0.00206 1.41 0.4L 31.432 857 3.68 7.72 8520 3.23 15.64 12.41 151487 2.69 0.16 17.0 1419 9091 0.00137 1.49 0.45L 35.361 857 3.68 4.04 8797 3.23 15.84 12.41 152470 2.66 0.16 16.9 1420 8830 0.00068 1.54 0.475L 37.3255 857 1.64 2.20 8871 3.23 15.64 12.41 152470 2.65 0.16 16.8 1420 8761 0.00035 1.55 0.5L 39.29 1728 2.74 -0.54 6890 3.23 17.55 14.32 196107 1.88 0.15 12.1 1456 11964 0 1.55
40.11 Weighted ave.= 15.13
Summary: 1. Predlctlon of camber per method of Appendix Ill (non-segmented) = 1.93" 2. 1.0" <Actual camber < 1.25" ......
0 ......
108
APPENDIXV
CAMBER MEASUREMENT DATA
The following tables list camber measurement data as supplied by manufacturers.
The averages of the Cole-Overland measurements were calculated by the author.
Table 13 Measured camber of 14 girders from the Kamiah project
Girder age Measured camber Girder age Measured camber (days) (in) (days) (in)
1 3.00 16 2.6875
3 2.0625 17 2.0625
9 2.4375 19 3.125
10 2.0625 21 2.4375
11 2.00 24 3.00
12 2.25 37 2.6875
15 2.8125 41 2.75
Note: Data is shown in Figure 19.
Table 14 Average measured camber of girders from the Cole-Overland project
Girder age Measured camber Std deviation Sample size (days) (in) (in)
1 (at release) 2.16 0.30 49
30 2.60 0.28 40
60 2.76 0.24 36
90 2.75 0.31 25
120 2.84 0.23 28
150 3.11 0.25 19
180 3.34 0.19 4
Notes:
1. Data from a total sample size of 98 girders with the same design. 2. Data is shown in Figure 20.
109
Table 15 Camber measurements of eight girders from the Cole-Overland project
Girder A age Meas. camber Girders age Meas. camber GirderC age Meas. camber (days) (In) (days) (In) (days) (In)
1 1.625 1 2.00 1 2.00
7 2.4375 7 2.875 7 2.25
43 2.9375 36 2.8125 22 3.4375
79 3.125 42 3.125 27 2.5
100 3.25 56 2.875 37 2.3125
135 3.25 77 3.00 41 2.3125
148 3.125 98 3.375 48 2.50
164 3.375 135 3.125 56 2.6875
63 2.8125
77 2.75
Girder E age Meas. camber Girder F age Meas. camber GirderG age Meas. camber (days) (in) (days) (In) (days) (in)
1 2.25 1 2.25 3 2.375
6 2.375 7 2.6875 7 2.75
41 2.6875 41 2.75 31 3.00
77 2.625 85 2.6875
98 2.50 97 2.75
135 2.75
Note: Data is shown In Figure 21.
Girder Dage (days)
3
9
49
85
99
111
GlrderH age (days}
2
10
23
34
Meas. camber (In)
2.25
2.50
2.875
2.75
2.876
3.25
Meas. camber (in)
3.4375
3.00
2.75
3.1875
..... ..... 0
150mm _____ _J
,-----
APPLIED~ IMAGE I - .no ..=-_ 1653 East Main Street -===" - Rochester NY 14«>9 -=--= Phone: 71Ev4S2-0300 USA
- ..= Fax: 7161288-5989
mage. Inc.. All Rights RllSBM!d Cl 1993. Applied I
INFORMATION TO USERS
This manuscript has been reproduced from the microfilm master. UMI
films the text directly from the original or copy submitted. Thus, some
thesis and dissertation copies are in typewriter face, while others may be
from any type of computer printer.
The quality of this reproduction is dependent upon the quality of the
copy submitted. Broken or indistinct print, colored or poor quality
illustrations and photographs, print bleedthrough, substandard margins,
and improper alignment can adversely affect reproduction.
In the unlikely event that the author did not send U.MI a complete
manuscript and there are missing pages, these will be noted. Also, if
unauthorized copyright material had to be removed, a note will indicate
the deletion.
Oversize materials (e.g., maps, drawings, charts) are reproduced by
sectioning the original, beginning at the upper left-hand comer and
continuing from left to right in equal sections with small overlaps. Each
original is also photographed in one exposure and is included in reduced
form at the back of the book.
Photographs included in the original manuscript have been reproduced
xerographically in this copy. Higher quality 6' x 9" black and white
photographic prints are available for any photographs or illustrations
appearing in this copy for an additional charge. Contact UMI directly to
order.
UMI A Bell & Hawdl Information Company
300 Notth z.ccb Road, Ann Arbor MI 48106-1346 USA J1Jn61-4100 soo1s21.()6()()