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Title: Design Considerations for Integral Abutment/ Jointless Bridges in the USA
Authors: Ralph G. Oesterle
Principal
Building and Environmental Consultants (BEC)
1019 Airpark Drive
Sugar Grove, IL 60554, USA
Phone: (630) 556-4950
Fax: (630) 556-9710
E-Mail: roesterle@building-investigations.com
Habib Tabatabai (Presenter and contact person)
Associate Professor
Department of Civil Engineering
University of Wisconsin-Milwaukee
3200 N Cramer Street
Milwaukee, WI 53211, USA
Phone: (414) 229-5166
Fax: (414) 229-6958
E-Mail: ht@uwm.edu
Proceedings of the 1st International Workshop on
Integral Abutment/Jointless Bridges, Fuzhou University, China,
March 2014, pp. 71-101
1
Design Considerations for Integral Abutment/ Jointless
Bridges in the USA
Ralph G. Oesterle and Habib Tabatabai
ABSTRACT
This paper summarizes results of a major study on jointless bridges sponsored by the U.S.
Federal Highway Administration. This study included extensive laboratory and field experiments as
well as detailed analytical studies. A set of design recommendations were provided. Jointless
bridges have reduced maintenance, improved riding quality, lower impact loads, reduced snowplow
damage, and structural continuity for live load and seismic resistance. However, the thermal
movements of the bridge and restraint forces from the abutments and piers must be considered and
accommodated. The general design philosophy is to build flexibility into the support structures to
the extent feasible while providing sufficient strength for restraint forces that cannot be completely
eliminated. The experimental phase of the research addressed thermal movements and stresses;
creep and shrinkage movements, including the effects of exposure to the outside environment; and
pile behavior. The overall analytical program consisted of studies on abutment soil-structure
interaction, pier behavior, longitudinal bridge movement, secondary continuity forces, skewed
bridge behavior, and construction sequences. In some cases, the research demonstrates that existing
design procedures and engineering data can be used to adequately quantify the structural response
and design forces for the structure. In other cases, the results of more complex analyses were used
to develop simplified design relationships and procedures.
_____________
Ralph G. Oesterle, Principal, Building & Environmental Consultants, 1019 Airpark Drive, Sugar Grove, IL 60554, USA
Habib Tabatabai, Associate Professor, University of Wisconsin-Milwaukee, 3200 N Cramer St., Milwaukee, WI 53211,
USA
2
BACKGROUND
Over the years, many types of expansion joints, expansion bearings, and other structural release
mechanisms have been used on long, multi-span highway bridges to accommodate thermal
movements. The desirable characteristics of an expansion joint are watertightness, smooth
rideability, low noise level, wear resistance, and resistance to damage caused by snowplow blades.
The actual performances of many joint systems, however, are disappointing. When subjected to
traffic and bridge movement, they typically fail in one or more important aspects, notably
watertightness. (3)
Jointless bridges have the advantages of improved riding quality, lower impact loads, reduced
snowplow damage, and structural continuity for live load and seismic resistance. On a jointless
bridge, however, special considerations are required for movement and/or restraint stresses resulting
from creep, shrinkage, and thermal strains in the design and detailing of the piers, abutments, and
approach slabs. The general design philosophy is to build flexibility into the support structures to
the extent feasible while providing sufficient strength for the restraint forces that cannot be
completely eliminated.
The uncertainty and complexity in behavior and design have made some States and designers
reluctant to use the jointless concept and may have led to some inefficiency in design by those
States and designers that do use it. The design and construction of jointless bridges started with
relatively short structures having only a few spans. The success of these structures in eliminating
joints has resulted in the use of jointless bridges for increasingly longer structures. While these
structures have generally performed well, their design and construction have been based principally
on experience obtained over the years. Where problems have occurred, the causes have been
identified and the procedures improved for the next structure. The design of jointless bridges has,
for the most part, been based on empirical rules rather than rigorous engineering principles.
Major questions identified in this study included the effect of annual temperature variations,
including the influence of internal restraint and thermal mass on temperature movement; effect of
diurnal temperature variations; effective coefficients of thermal expansion; effects of creep and
shrinkage on thermal expansion; foundation stiffness, particularly the relationship between passive
earth pressure and abutment movement and the capacity of abutment piles to accommodate the
movement; pier stiffness and load transfer to the piers; and response of skewed jointless bridges to
annual and diurnal temperature variations.
OBJECTIVES AND SCOPE
The overall objectives of this research effort were to develop a greatly expanded knowledge of the
behavior of jointless bridges, provide a scientific basis for design, and make design
recommendations. This research work consisted of an experimental program and an analytical
program. The scope of work included jointless bridges with composite steel, prestressed concrete
and reinforced concrete beams subjected to gravity loads combined with thermal loads, and the
time-dependent effects of concrete creep and shrinkage.
The experimental work included material and girder component tests, including constructing and
monitoring two full-scale, two-span continuous girders in an outdoor environment for a period of 20
3
months. The experimental work also included field surveys of 15 jointless bridges and field
monitoring of a long, heavily skewed jointless bridge.(6) The analytical work included assessment of
the interaction of jointless bridge superstructures with integral abutments and piers, and the effects
of secondary continuity forces in combination with end-restraint forces.(7) Skew effects were
included and retrofitting of existing jointed bridges to eliminate expansion joints was addressed.
The field survey of jointless bridges also included curved bridges.
SUMMARY OF FINDINGS AND RECOMMENDATIONS
In this section, findings and recommendations related to various issues that were studied are
summarized and discussed.
Abutment-Soil Interaction
The soil pressure and abutment movement data shown in figure 1, as developed by G.W.
Clough and J.M. Duncan(5) and presented in the NCHRP Report 343,(8) represent a reasonable
upper-bound to determine the portion of full passive Rankine pressure to be used as a design passive
pressure at the expected maximum abutment movement.
Figure 1. Relationship between abutment wall movement and earth pressure.(5,8)
The starting point for calculating maximum design passive pressure should conservatively be at
the point of maximum contraction (from creep, shrinkage, and thermal strain) of the bridge
superstructure. As contraction decreases the soil pressure to the minimum active pressure, the
granular soil will follow the contacting abutment wall and re-compact. The re-compaction is
4
sufficient to shift the soil pressure and abutment movement curve so that passive pressure starts to
build immediately upon thermal re-expansion.
There are opposing philosophies as to the degree of compaction in the backfill soil adjacent to
the abutment. One concept is to use loose granular backfill to minimize passive pressure forces.
The opposite concept is to use highly compacted backfill based on the observation of voids beneath
the approach slab and settlement of the approach slab as a common problem with integral
abutments. Tests on a large-scale abutment specimen(9) indicated that voids under the approach
slabs develop as a result of contraction beyond the initial starting point, even with 97 percent
relative compaction of the backfill adjacent to the abutment. Therefore, there does not appear to be
an advantage to using high compaction. Also, a high degree of compaction is difficult to attain in
the location adjacent to the abutment backwall.
Parametric computer analyses indicated that the passive pressure can be significantly decreased
by using relatively uncompacted backfill and/or by increasing the horizontal length of the backfill
zone behind the abutment. The calculated resultant passive soil reaction decreased by a factor of
approximately 2.5, with a decrease in compaction from 90 percent to 80 percent. Also, with backfill
at 80 percent compaction, the calculated resultant passive pressure decreased by a factor of
approximately 2 when the slope of the backfill/in situ soil interface was changed from 45° to 30°
(from horizontal). However, extending the backfill zone behind the abutment increases the length
needed for the approach slab to span the backfill region and/or increases the settlement of the
approach slab.
The shape of the pressure distribution is dependent on the rotation of the abutment wall, but
typically includes a large increase in pressure at the base of the abutment. The concentration of
pressure at the base increased with a decrease in rotation. The rotation of the abutment is dependent
on the relative bending stiffness of the superstructure and the rotational stiffness of the
soil/pile/abutment interaction. The pressure distribution shown in figure 2 is recommended for the
design of stub abutments on piles. The pressure distribution shown in figure 3 is recommended for
abutments on spread footings and the pressure distribution shown in figure 4 is recommended for
full-height abutments.
Abutment Pile Capacity
The analytical approach selected for this study was based on the simplified model developed by
Abendroth, et al.(1) Three criteria for determining abutment pile capacity were incorporated in this
model. The first criterion was based on the geometric stability of the pile (elastic or inelastic
buckling as a function of its vertical load and moment associated with lateral movement). The
second criterion was related to material strength (usually governs in case of short or stubby piles).
The third criterion was based on the rotational capacity of the pile (a ductility issue related to local
hinging of the pile). The calculations associated with these three pile capacity criteria were based
on an equivalent cantilever length of the pile. The equivalent cantilever length can be determined
for either fixed or pinned conditions between the top of the pile and the bottom of the abutment.
The results of the parametric study are typically in the form of ultimate vertical load capacity
versus lateral displacement of piles. A comparison was made with respect to the maximum
assumed design vertical load obtained from AASHTO specifications,(28) considering no lateral
5
displacement and no soil data available. Examples of the results of analyses are shown in figures 5
through 7 for piles without overdrilled holes. The analyses were terminated either when the
ductility limit was reached or when 100 mm (4 inches) of deflection were reached.
Figure 2. Proposed passive pressure distribution for design of integral stub abutments on
piles
* Resultant of triangular distribution.
Pp = ½ γ Kp H2
Kp = Passive pressure coefficient dependent on wall movement
according to Clough and Duncan(5,8)
Figure 3. Proposed passive pressure distribution for design of integral stub abutments
on spread footings
6
Figure 4. Proposed design envelope for full-height wall abutments based on passive
and at-rest pressures
The data presented in figures 5 through 7 indicate that displacement capacities increase as soil
stiffness decreases and as section size increases. The results of analyses for H-piles shown in figure
5 are based on weak-axis bending. Based on surveys of State DOT practices, it appears that many
States prefer to orient H-piles for weak-axis bending under longitudinal movement, while other
States prefer to use strong-axis bending. The rationale for weak-axis bending is that the flexibility
of the pile is greater and the pile can therefore accommodate larger movements. However, the
deformation capacity of the pile depends not only on the pile flexural stiffness, but on the combined
pile and soil system stiffness. As the pile stiffness increases, the soil is affected to increasing
depths. The effective cantilever length in bending therefore increases, tending to decrease the
overall stiffness of the pile-soil system.
Analyses presented in the analytical report demonstrate that bending about the strong axis
results in a larger displacement at first yield than bending about the weak axis.(7) For an HP10x42,
the ratio of yield displacement for strong axis bending to yield displacement for weak-axis bending
is approximately 1.7. It should be noted, however, that while strong-axis bending provides a larger
displacement capacity, design forces for the pile-to-pile-cap connection are increased.
A summary of the findings and conclusions based on the analyses, literature review, and
available test data for piles in integral abutments is as follows:
7
1. Vertical load capacity of an abutment pile is a function of the abutment’s horizontal
movement associated with bridge expansion or contraction. Vertical load capacities
decrease when displacements increase.
2. Longitudinal displacement capacity increases as pile flexural stiffness increases and/or soil
stiffness decreases. An increase in the ratio of pile stiffness to soil stiffness increases the
effective cantilevered length of the pile. Therefore, the curvature demand associated
with a particular lateral displacement is spread over a longer length and the ductility
demand at that displacement is decreased. Based on these analyses, situating an H-pile
for bending about the strong axis with abutment movement will allow larger
displacement.
3. The steel H-pile test specimen demonstrated that slight local buckling of the flanges
occurred during the tenth cycle of lateral displacement of the inflection point of ±61 mm
(±2.4 inches). This displacement is the calculated deformation limit corresponding to
the ductility criterion presented by Abendroth, et al.(1) Therefore, the test results
indicated that the ductility criterion proposed by Abendroth et al., was appropriate.
Also, the pile was able to sustain the load capacity through 50 cycles at this level of
deformation without further deterioration of the section. Therefore, this criterion is
recommended for the design of H-piles.
4. The analyses indicate that concrete-filled pipe piles have very large displacement capacities
and generally lose a lower percentage of vertical load capacity at large displacements.
Although no pipe piles were tested in the experimental program for this project, testing
of concrete-filled steel tubular columns has shown a high degree of ductility with stable
loops.(11) A conservative design approach would include limiting the maximum strain to
0.01 and the maximum aspect ratio of pile diameter to wall thickness to 39.
5. Analytical studies indicated that prestressed concrete piles would have significant
displacement capacity if ductile behavior based on a maximum compressive strain limit
is assumed. However, testing indicated that a prestressed concrete pile would suffer
severe cracking and spalling if cycled at a deformation limit based on the ductility
criteria used in the analytical study. Therefore, it is recommended that displacement
limits for prestressed concrete piles be based on an allowable stress limitation as initially
presented by Abendroth, et al.(2)
8
Figure 5(a). Ultimate load capacity for HP10x42 (without overdrilled holes)
Figure 5(b). Ultimate load capacity for HP12x74 (without overdrilled holes)
9
Figure 6(a). Ultimate load capacity for 406- by 406-mm (16- by 16-inch)
prestressed concrete piles (without overdrilled holes).
Figure 6(b). Ultimate load capacity for 610- by 610-mm (24- by 24-inch)
prestressed concrete piles (without overdrilled holes).
10
Figure 7(a). Ultimate load capacity for PP 12-3/4 (without overdrilled holes).
Figure 7(b). Ultimate load capacity for PP 16 (without overdrilled holes).
Effective Stiffness of Piers
There are two important factors in the effective response of piers to superstructure movement.
The foundation response and nonlinear pier response (cracking, creep, and shrinkage) are
considered and discussed below.
Foundation Response
The rotational stiffness of the foundation is an important factor in the response of piers to
longitudinal deck movement. Foundation type and size, as well as soil stiffness, influence the
rotational stiffness. The study indicated that the moment-rotation behavior was essentially linear
and the rotational stiffnesses were consistent with a simplified analysis presented by J.
Zederbaum.(12) Rotational stiffness, Kθ, using Zederbaum's approach, is based on the elastic
11
behavior of an ideal earth prism of a depth equal to one-third of the footing width, b, and can be
determined as:
Kθ =
b
IE3
fs (1)
where, Es = modulus of elasticity of the soil
If = moment of inertia of the foundation base
b = footing width
The rotational stiffness of the pile foundation was based on an elastic approach presented by
A.A. Witecki and V.K. Raina.(13)
Pier Response
The structural stiffness of the pier wall or columns is another important factor in estimating the
restraining forces associated with longitudinal thermal movement of a bridge superstructure. Pier
stiffness is a function of the slenderness ratio (i.e., height to thickness), concrete properties, applied
vertical loads, inelastic behavior (i.e., creep and cracking of concrete), and the type of connection at
the top of the pier (i.e., integral versus semi-integral or pinned piers). Integral piers will transfer
both moment and shear forces at the top, whereas semi-integral piers transfer shear forces only.
Based on nonlinear parametric studies, the following relationship for the effective moment of
inertia, Ie, for the pier is recommended:
cr
25.1
a
cr
g
25.1
a
cr
eI
M
M
1 I
M
M
I
−+
=
g
I≤
(2)
in which,
Ma = calculated elastic moment based on assumed bridge movement
Ig = moment of inertia of gross concrete section about the centroidal axis, neglecting
reinforcement
Icr = moment of inertia of cracked section transformed to concrete
Mcr = cracking moment, can be calculated as follows:
Mcr =
y
I
f
A
Pg
r
+
(3)
where,
P = applied vertical load
A = cross-sectional area of concrete, neglecting reinforcement
y = half-depth of pier section
fr = modulus of rupture of concrete
12
= 0.62
'c
f
(MPa)
= 7.5
'c
f
(lbf/inch2)
'
c
f
= compressive strength of concrete
Results of the analyses for pier behavior indicate that foundation rotation contributed to a
significant portion of the longitudinal movement at the top of the pier. For the range of
parameters used in the study, the foundation rotation accounted for 20 percent to 79 percent of
this movement.
The primary factors affecting the contribution of bending in the pier to the longitudinal
movement at the top of the pier are the slenderness of the pier (height-to-thickness ratio) and
cracking. Axial stress in the pier affects the cracking moment and, therefore, the effective
stiffness. Concrete creep and the age of the concrete at loading had a relatively insignificant
effect on the results.
Use of an integral pier with a moment connection between the pier cap and the superstructure
will induce significantly higher restraint forces in the pier as compared to the case of semi-
integral pier with a pinned connection. Since pier stiffness has a negligible effect on the total
bridge movement, the stiffer connection does not decrease the expected movement at the top of
the pier. Use of an integral pier eliminates the need for the elastomeric bearing pads and details
to accommodate rotation between the pier cap and the superstructure. Also, integral piers
eliminate the mass of the pier cap for seismic design and provide increased clearance. However,
the details for a semi-integral pier are simple compared to detailing a moment connection at the
top of the pier.
Expected Bridge Movement
The factors involved in predicting the expected bridge movements include an effective temperature
range, including seasonal and diurnal components; coefficient of thermal expansion; creep, and
shrinkage; and the restraint from piers and abutments. The overall variability of these factors causes
uncertainty in the determination of bridge movements.
Effective Temperature Range
Emerson developed a relationship between instantaneous shade temperature and effective bridge
temperature by first developing a relationship between the mean shade temperature and the effective
bridge temperature.(3) Emerson then related the mean shade temperatures to the instantaneous shade
temperatures based on meteorological data recorded throughout the United Kingdom. Imbsen
applied and extended Emerson's approach to effective bridge temperatures in NCHRP Report 276.(4)
To cover the larger range of minimum and maximum shade temperatures experienced within the
United States, Imbsen extrapolated Emerson's mean and effective bridge temperature relationships.
However, isotherms in NCHRP Report 276 only represent normal daily minimum and maximum
values instead of the extreme values of shade temperatures.
13
The AASHTO LRFD Bridge Design Specifications (15) include two separate procedures for
estimating bridge temperature ranges (Procedure A and B). This study considers Procedure A to be
inadequate. Although Procedure B is closer to the actual bridge response than Procedure A, it is
recommended that the temperature ranges be estimated using localized temperature data as
proposed here. The following procedures are recommended to determine the range of the effective
bridge temperatures:
1. Obtain the minimum and maximum shade temperatures from the ASHRAE weather
data(14) for the State and weather stations nearest the bridge site, and interpolate as
needed. An example of such data for major US cities is shown in table 1. The shade air
temperatures given in table 1 should be adjusted at a rate of 1°C/100 m (1°F/200 ft) of
elevation change.
2. Determine the solar zone for the bridge location using AASHTO LRFD specifications,(15)
and the corresponding solar increment,
solar
Τ∆
from table 2.
3. Calculate minimum and maximum effective bridge temperatures from the following
relationships:
For concrete bridges:
Tmin = 1.00 Tmin + 5°C (9°F) (4)
eff shade
Tmax = 0.97 Tmax – 2°C (3°F) +
solar
Τ∆
(5)
eff shade
For composite steel bridges:
Tmin = 1.04 Tmin + 2°C (3°F) (6)
eff shade
Tmax = 1.09 Tmax – 0°C (3°F) +
solar
Τ∆
(7)
eff shade
where,
Tmin = minimum effective bridge temperature
eff
Tmax = maximum effective bridge temperature
eff
Tmin = minimum shade temperature from the weather data based
shade on bridge location (table 1)
Tmax = maximum shade temperature from the weather data based
shade on bridge location (table 1)
solar
Τ∆
= uniform temperature change from direct solar radiation
based on girder type and bridge location (see table 2)
14
Table 1. Climatic conditions for major cities in the US
Location
Shade Temperature
Mean*
Latitude
Longitude
Elev.
Min.
Max.
Constr.
State & Station
°
lN
°
lW
m
°C
°C
°C
Birmingham, Alabama
33
34
86
45
189
-8
36
17
Anchorage, Alaska
61
10
150
1
35
-31
22
11
Phoenix, Arizona
33
26
112
1
339
-1
43
21
Los Angeles, California
33
56
118
24
30
5
28
17
Denver, Colorado
39
45
104
52
1610
-21
34
17
San Francisco, California
37
37
122
23
2
2
28
13
Miami, Florida
25
48
80
16
2
7
33
24
Atlanta, Georgia
33
39
84
26
308
-8
34
17
Washington, DC
38
51
77
2
4
-10
34
17
Seattle-Tacoma,
Washington
47
27
122
18
122
-6
29
11
NYC, New York
40
39
73
47
4
-11
32
15
Boston, Massachusetts
42
22
71
2
5
-14
33
14
Chicago, O'Hare AP
41
59
87
54
201
-22
33
16
Minneapolis/St. Paul, MN
44
53
93
13
254
-27
33
17
St. Louis, Missouri
38
45
90
23
163
-17
36
18
Dallas, Texas
32
51
96
51
147
-8
39
19
Kansas City, Missouri
39
7
94
35
241
-17
37
18
Las Vegas, Nevada
36
5
115
10
664
-4
42
19
Charlotte, North Carolina
35
13
80
56
224
-8
35
16
Cincinnati, Ohio
39
9
84
31
231
-17
33
17
Baltimore, Maryland
39
11
76
40
45
-12
34
17
*Mean construction season shade air temperature (16).
Table 2. Solar increment values based on girder type and bridge location.
Zone
T
1
Concrete ∆
T
Composite Steel ∆
T
1 30°C 8°C 6°C
2 26°C 7°C 5°C
3 23°C 6°C 4°C
4 21°C 5°C 4°C
1°C = 1.8°F
Determination of Maximum Movements
In jointless bridges, it is important to estimate the maximum expansion and contraction at each end
of a bridge to determine the longitudinal displacement expected for the abutment piles. It is also
important to predict the movement at each pier and the joint width needed between the approach
slab and the pavement. Another important movement is the maximum total thermal movement at
each end resulting from the total effective temperature range. As discussed earlier, the starting point
15
to determine the maximum passive pressure should conservatively be at the maximum contraction.
The maximum passive pressure is related to the end movement, with re-expansion for the full
effective temperature range.
Calculation of the length change for a prestressed concrete bridge can be accomplished through use
of typical design values for the coefficient of thermal expansion combined with creep and shrinkage
strains from ACI 209.(17) However, the overall variability of these factors adds uncertainty to the
calculated end movements. Although a coefficient of thermal expansion for concrete is typically
assumed to be 9.9 to 10.9 millionths/°C (5.5 to 6.0 millionths/°F), it is known that this value can
range from approximately 5.4 to 12.6 millionths/°C (3.0 to 7.0 millionths/°F).(18) Also, the
variability of creep, shrinkage, and modulus of elasticity of concrete is known to be significant.(18)
In addition, resistance to length change from abutments and piers, combined with the variability of
the restraint (primarily caused by the variability of the soil), leads to unequal movement at each end
of a bridge and uncertainty as to the magnitude of the movement at each end. Finally, the effective
setting temperature of the bridge and the age of the prestressed concrete girders at completion of the
superstructure are typically unknown, making the relative magnitude of expansion and contraction
and the starting point for creep and shrinkage calculations uncertain.
To investigate the effects of the variability of the parameters and to provide guidance in formulating
recommendations for design calculations, Monte Carlo studies were carried out. Two standard
four-span bridge models were selected for these Monte Carlo studies. The first model was a
prestressed concrete bridge that was modified to simulate various conditions. The second model
included steel wide-flange stringers.
Table 3 presents the values of the magnification factors, referred to as Γ factors, for the various
conditions based on the results of the Monte Carlo studies. These Γ factors are intended to modify
the calculated values to account for uncertainty in the calculations. Case 1 includes magnification
factors for the maximum expected movement from the assumed "as constructed" condition. A
primary factor affecting the magnitude of these magnification factors is the uncertainty of the
construction temperature. The Γ values for total movement account for uncertainty for the
Table 3. Values of Γ magnification factor.
Case No.
Design Condition For Bridge Expansion For Bridge Contraction
Total
End
Total
End
1(a) Conventional Design of
Prestressed Bridge 1.50 1.60 1.30 1.35
1(b) Cast-in-Place Concrete
Bridge 1.50 1.60 1.30 1.40
1(c) Composite Steel Bridge 1.50 1.70 1.45 1.50
2 Re-expansion After Full
Contraction 1.10 1.20 ─ ─
16
calculation of the overall change in the length of the bridge. However, because of uncertainty for
the stiffness of the abutments and piers, the Γ’s for the calculation of the movement at each end are
somewhat larger. Case 2 addresses re-expansion from full contraction.
The following procedures are recommended to determine end movements in the longitudinal
direction while accounting for the uncertainty of calculations. It is assumed that the date and
temperature when the deck was cast are unknown and that no specific data on the material
properties of the concrete are available other than 28-day compressive strength, 'c
f. To determine
the maximum end movements of a prestressed concrete bridge:
4. From table 1, determine the mean construction temperature for locations nearest the
bridge, and interpolate to determine a mean construction temperature for the bridge
location. Alternatively, determine a mean construction temperature from data for stations
in the same part of the country with similar minimum and maximum shade temperatures.
5. To calculate end movements:
a. Determine the minimum and maximum effective bridge temperatures from
equations 4 and 5, respectively.
b. Use typical design values of 10.8 millionths/°C (6.0 millionths/°F) for the coefficient
of thermal expansion of concrete, values for creep and shrinkage from ACI 209,(16)
and 4700 ' c
f, MPa (57,000 ' c
f, lbf/inch2) for the modulus of elasticity of concrete.
c. Use the procedure presented by Zederbaum(12) to determine the point of zero
movement or the point of fixity within the bridge based on the stiffness of the piers
and the abutments. The equivalent cantilever method can be used to estimate the
lateral stiffness of the abutment piles. Since the responses of the piers and
abutments are nonlinear, the stiffnesses should be based on a preliminary estimate of
the end movements. It should be noted, however, that for a symmetrical bridge, the
point of fixity would be at the longitudinal center of the bridge.
d. The following equations are used to determine the strain values for a prestressed
concrete bridge. Changes in length are determined by multiplying the strain by the
total initial length.
εth = α ∆T (8)
deck
girder
shsh
shsh
)EA(
)EA(
1
girder
deck
girder
+
ε
−
ε
+ε=ε
(9)
17
+
ε=ε
deck
girder
crcr
)EA(
)EA(
1
1
girder
(10)
∆ = Γεtotal (11)
where,
εth = thermal strain
εsh = shrinkage strain
εcr = creep strain
α = coefficient of thermal expansion
E = modulus of elasticity
A = cross-sectional area
= length from calculated point of fixity to end of bridge. Note that, for a
nonsymmetrical bridge, two different lengths are involved.
Γ = magnification factor to account for uncertainty
εtotal = εth – εsh – εcr for expansion (12)
εtotal = -εth – εsh – εcr for contraction (13)
∆ = maximum end movement
6. Maximum expansion typically occurs shortly after construction. For this maximum
expansion (Case 1(a) expansion in table 3), use the temperature differential between the
maximum effective bridge temperature and the mean construction temperature for
thermal expansion, and a time span equal to one-quarter of the construction season for
creep and shrinkage contraction with the beams assumed to be 90 days old when the deck
is cast. Based on the Monte Carlo studies, the calculated end movements should be
increased by a Γ factor of 1.60 (see table 3) to account for the uncertainties with 98
percent confidence that the movement will be less than that calculated.
7. For maximum contraction after several years of service (Case 1(a) contraction in table 3),
use the temperature differential between the minimum effective bridge temperature and
the mean construction temperature for thermal contraction, and ultimate creep and
shrinkage values with the beams assumed to be 10 days old at the time of casting of the
deck. Based on the Monte Carlo studies, the calculated end movement should be
increased by a Γ factor of 1.35 (see table 3) to account for uncertainties with 98 percent
confidence that the movement will be less than that calculated.
8. For maximum thermal re-expansion from a starting point of full contraction (Case 2 in
table 3), the full range of effective bridge temperatures should be used without any creep
18
or shrinkage movement. The resulting calculated end movements should be multiplied
by a Γ factor of 1.20 (see table 3) to account for uncertainties in the calculation.
For reinforced concrete bridges, similar procedures can be used to determine the maximum
expansion and contraction end movements. However, shortening caused by creep is not a factor.
Calculated end movements for reinforced concrete bridges should be increased by the Γ factors for
Case 1(b) of 1.60 for maximum expansion and 1.40 for maximum contraction. A Γ factor of 1.20
can be used for Case 2, maximum thermal re-expansion from a starting point of full contraction.
Note that the magnitudes of the multipliers, Γ’s, for Cases 1(a) and 1(b) are significantly larger than
the multiplier, Γ, for Case 2 because of the uncertainty and variability of the construction
temperature.
For composite steel bridges, the procedures used to estimate the maximum end movements are also
similar to the procedures outlined above for the prestressed concrete bridges, except that a modulus
of elasticity of 20x104 MPa (29x106 lbf/inch2) and a coefficient of thermal expansion of 11.7
millionths/°C (6.5 millionths/°F) should be used for the steel girders. These values are
recommended by AASHTO for structural steel. The results of the Monte Carlo study for composite
steel bridges indicated that calculated end movements should be increased by the Γ factors for Case
1(c) of 1.70 for maximum expansion and 1.50 for maximum contraction. A Γ factor of 1.2 can be
used for Case 2, maximum thermal re-expansion from a starting point of full contraction.
If sufficient information regarding the composition of the concrete is available to the designer, a
more accurate value for the coefficient of thermal expansion of concrete can be obtained by using
the Emanual and Hulsey model(20) to estimate the value of the coefficient of thermal expansion of
concrete. Calculated end movements using the more accurate Emanual and Hulsey coefficient of
thermal expansion of concrete should be increased by Γ factors of 2.05 for maximum expansion and
1.45 for maximum contraction. These Γ's are greater than the values calculated for the Case 1(a) or
1(b) condition. This is caused by the difference in calculating the design thermal movements rather
than the variability of the predicted movements. The coefficient of thermal expansion design value
of 10.8 millionths/°C (6.0 millionths/°F) recommended by AASHTO and used for Cases 1(a) and
1(b) is conservatively high when compared to the average of the data base.(7) The average
coefficient of thermal expansion for concrete in the data base used in this study(7) is 8.8
millionths/°C (4.9 millionths/°F). Therefore, the conventional design calculated value of movement
already includes some margin of safety.
Note that when using the Emanual and Hulsey model(20) to estimate α, the concrete for the deck slab
is commonly different than the concrete for the beam. Therefore, an effective coefficient of thermal
expansion, αe, for the composite section can be calculated as:
deckgirder
deckgirder
e)EA( )EA(
)EA( )EA(
+
α+α
=α (14)
The end movements, determined from Case 1 for maximum expansion and maximum contraction,
are recommended for use in determining a design range of abutment movements. The larger of
these two movements should be used for comparison with the limiting pile displacement, ∆, when
designing abutment piles.
19
The end movements, ∆, determined from Case 2 for maximum re-expansion from a starting point
of full contraction, should be used to determine passive soil pressure on the abutment. In addition,
the end movements determined for Case 1 and Case 2 should be used to obtain the forces on the
abutment piles to be used in designing the connection between the piles and footing. Note that the
pile may not yield under the initial expansion from Case 1, which is typically a smaller movement,
but may do so with Case 1 contraction or the re-expansion from Case 2. It should also be noted that
the above procedures can be used to estimate design joint movements in bridges with expansion
joints.
Transverse Movement in Skewed Bridges
A skewed bridge is a bridge with the longitudinal axis at an angle other than 90° with the piers and
abutments. With skewed bridges, the soil passive pressure developed in response to thermal
elongation has a component in the transverse direction as illustrated in figure 8. Within certain
limits of skew, soil friction on the abutment will resist the transverse component of passive pressure.
However, if the soil friction is insufficient, either significant transverse forces or significant
transverse movements, depending on the transverse stiffness of the abutment, could be generated.
Figure 8. Components of abutment soil passive pressure response to thermal elongation in skewed
bridges with integral abutments.
20
Because of potential problems and uncertainty related to the response of skewed integral abutments,
many US State DOT’s limit the skew in this type of bridge. A typical limit for the maximum skew
angle for the jointless bridges used by many States is 30°. However, maximum skew angle limits in
various States range from 0° to no limit.(21) There is a need to define a rational basis for limiting
and/or accommodating skew angles for bridges with integral abutments. Therefore, an analytical
study was carried out to investigate transverse movement in skewed bridges. This work was
accomplished by developing equilibrium and compatibility equations for end abutment forces and,
in the case of a typical stub abutment, solving these equations for varying skew angles and bridge
length-to-width ratios.
Skew Angle Limit for Limiting Transverse Effects
Figure 9 shows the relationship between passive soil pressure response to thermal expansion and
soil/abutment interface friction assuming no rotation of the superstructure.
Figure 9. Soil pressure load, Pp, and soil/abutment interface friction, Faf.
For rotational equilibrium:
Faf (L cos
θ
) = Pp (L sin
θ
) (15)
Faf = Pp tan
δ
(16)
21
where,
Pp = Resultant normal soil pressure force
FaF = Soil/abutment interface friction force
tan
δ
= friction factor for interface of formed concrete and soil
Substituting equation 16 into 15:
tan
δ
=
θcos
θsin
= tan
θ
δ
=
θ
(17)
Therefore, the bridge superstructure will be held in rotational equilibrium until the skew angle,
θ
,
exceeds the angle of backwall friction,
δ
. Integral abutments are typically backfilled with granular
material. NCHRP Report 343, Manual for the Design of Bridge Foundations, lists a friction angle
of 22° to 26° for formed concrete against clean gravel, gravel sand mixtures, and well-graded rock
fill with spalls.(8) Based on the data, 20° represents a reasonably conservative skew angle below
which special consideration for transverse forces or transverse movement is not needed.
Forces Required to Resist Transverse Movement
With larger skew angles, the integral abutment can either be designed to resist the transverse force
generated by the soil passive pressure in an attempt to keep the abutment movement predominantly
longitudinal, or the abutment can be detailed to accommodate the transverse movement. Figure 10
shows the relationship for rotational equilibrium, including lateral resistance of the abutment, Fa, in
addition to the wall/soil interface friction, Faf. For rotational equilibrium:
(Fa + Faf) (L cos
θ
) = Pp (L sin
θ
) (18)
Faf = Pp tan
δ
Fa = Pp (tan
θ
– tan
δ
) (19)
22
Figure 10. Abutment lateral force, Fa, required in combination with interface
friction, Faf, for rotational equilibrium of heavily skewed bridge.
Figure 11 shows the ratio of Fa and Pp as a function of skew angle, assuming that the interface
friction angle,
δ
, is 20°. As shown in figure11, the force required to resist transverse movement is a
significant portion of the passive soil pressure response, Pp. It should be noted that Pp is not
necessarily the full passive pressure response, but can be determined for the end movement using
the Clough and Duncan relationships(5). The end movement to be considered for ∆/H in figure 1 is
the end movement normal to the abutment, ∆
n. This end movement is:
∆
n = ∆
cos
θ
(20)
where ∆
is the maximum expected end movement for thermal re-expansion from the starting point
of full contraction for the full range of effective bridge temperatures.
23
Figure 11. Relationship between force required for abutment lateral resistance,
Fa, and passive pressure response, Pp, to restrain lateral movement.
For relatively short bridges in locations with small effective temperature ranges, it may be feasible
to design the abutment substructure to resist Fa. It should be understood though that, for whatever
means are used to develop Fa (batter pile and/or lateral passive soil resistance), lateral movements
are required to develop the resistance, Fa. Therefore, details anticipating some transverse movement
should be used. The expected movements are a function of the relative stiffnesses of resistance to
Pp and Fa. It should also be noted that adding battered piles to an integral abutment for lateral load
will also increase the stiffness in the longitudinal direction, which induces more demand on the
superstructure and connections between the deck and abutments.
Expected Transverse Movement With Typical Stub Abutment
To investigate the relationship between skew angle and expected transverse movement for a typical
integral stub abutment, a set of relationships were derived based on the equilibrium and
compatibility of end abutment forces in the plane of the bridge superstructure. For this analysis, the
superstructure is assumed to act as a rigid body with rotation, β, about the center of the deck (for a
longitudinally symmetrical bridge) as illustrated in figure 12. The rotation occurs to accommodate
the thermal end movement, ∆. Forces considered in response to this movement include soil
pressure on the abutment and wingwalls, wall/soil interface friction on the abutment, and pile forces
normal to and in line with the abutment and wingwalls.
24
Figure 12. Elongation, , and rigid body rotation, β.
The results in figure 13 demonstrate the increase in the transverse movement with increasing skew
angle. The data in figure 13 also demonstrate the increase in transverse movement with decreasing
L/W ratio. The data in figure 13 show that increasing the wingwall length relative to the abutment
wall length (which includes increasing the number of wingwall piles relative to the number of
abutment wall piles) can significantly decrease the transverse movement. However, the wingwalls
and abutment have to be designed to transmit the restraint forces on the wingwalls into the
superstructure.
25
Figure 13. Relationship between transverse movement at the acute corner, t1,
and thermal expansion, , for an expansion of 25.4 mm (1 inch)
with constant-length bridge, L = 126.77 m (415.92 ft), and varying
L/W.
Figure 14 shows the resulting total longitudinal restraint force for these analyses and demonstrates
the decrease in longitudinal restraint with increasing skew angle. For the full-width bridge with a
L/W ratio = 3.15, the longitudinal restraint at a skew angle of 60° is approximately 60 percent of the
longitudinal restraint at θ = 25°. For the larger L/W ratio = 9.45, the ratio of the longitudinal
restraint at θ = 60° is approximately 70 percent of the restraint at θ = 25°. This demonstrates the
increase in restraint resulting from the increase in the resistance to lateral moment caused by the
larger ratio of wingwall length to abutment length.
26
Figure 14. Relationship between resultant longitudinal restraint force and skew angle for thermal
expansion, , of 25.4 mm (1 inch) with constant-length bridge, L = 126.77 m (415.92 ft), and
varying L/W.
It should be noted that the transverse movement, ∆t1, discussed above is the movement of the acute
corner of the bridge deck related to rigid body rotation of the superstructure caused by abutment
passive restraint of longitudinal thermal expansion, ∆. There will also be transverse thermal
expansion of the abutment. This transverse thermal expansion will be limited as compared to the
longitudinal expansion, because the temperature change is moderated by the fact that abutment
breastwalls are exposed to ambient air temperature on one side only and the abutment is not
appreciably exposed to solar radiation. Therefore, depending on the width of the bridge and the
skew angle, the transverse thermal expansion may or may not add significant additional transverse
movement to the abutment wingwalls. This additional transverse movement would add to ∆t1 and
subtract somewhat from ∆t2.
To detail the abutments for the transverse movement of the corners, all interfaces of the integral
abutment with other components, such as approach pavement, barrier walls, pavement for slope
protection, and drainage components, should be detailed to accommodate this movement. In
addition, relatively flexible connections to piers should be considered in the direction parallel to the
pier caps. The analyses indicated that, for right bridges, the restraint forces from the piers into the
27
superstructure, resulting from longitudinal expansion, have relatively minor effects. This is for
movement perpendicular to the pier cap. Foundation rotation and cracking in the pier (cantilevered
from the foundation) contribute to a relatively flexible pier response for this direction of movement.
The foundation and pier structure stiffness will probably be more significant for movement parallel
to the pier cap. Therefore, it is recommended that the connection between the bottom of the
superstructure and the pier caps be flexible in this direction. (This approach, however, may not be
appropriate for seismic design.) In this case, design of the diaphragms should consider the piers'
restraint of the rigid body rotations that result from passive abutment restraint of longitudinal
thermal expansion.
Temperature Gradient Effects
Strains within a continuous bridge cross section subjected to a nonlinear temperature gradient are a
function of two components: internal restraint strains and continuity strains. Internal restraint
strains result from the difference between the unrestrained free temperature strains and the actual
final restrained strain profile. The final restrained strain profile is typically assumed to be linear
based on one-dimensional beam theory and the Navier-Bernoulli hypothesis that initially plane
sections remain plane after bending. The internal restraint strains, which occur because of restraint
within the cross section, are shown graphically in figure 15. If the nonlinear temperature gradient of
figure 15(b) is applied to the cross section of figure 15(a), and if each individual fiber of the cross
section is allowed to move freely, the strain profile of figure 15(c) will result. However, the final
strain profile is assumed to be linear, as shown in figure 15(d). The difference between the free
strain profile and the final strain profile, shown by the shaded portion of figure 15(e), is the internal
restraint strains.
Figure 15. Diagram of restraint strains in a cross section.
28
These restraint strains can be calculated for a section without external axial and flexural restraints
from equations developed by M.J.N. Priestley.(22) Priestley's equations for average strain and
curvature, as modified for an irregular cross section, consist of the following:
εtavg = dy E b
dy E b t
yy
yyyy
∫
∆α∫ (21)
φ =
dy y E b
dy y E b t
2
yy
yyyy
∫
∆α∫
(22)
where, εtavg = strain at the neutral axis of the cross section caused by the imposed nonlinear
temperature variation along the depth and referred to as average strain in the
cross section
φ = Curvature of the cross section due to temperature gradient
αy = coefficient of thermal expansion as a function of depth, αy = F(y)
Δty = change in temperature of the cross section as a function of depth (the applied
thermal gradient),
∆
ty = F(y)
by = width of cross section as a function of depth, by = F(y)
Ey = modulus of elasticity as a function of depth, Ey = F(y)
y = distance from neutral axis of cross section
If the cross section is divided into n discrete subsections, and a transformed cross section is used
through application of a modular ratio, the equations simplify to:
εtavg =
c
ii
i
A
AΔtα
1i
n
=
∑
(23)
φ =
c
iii
i
I
yAΔtα
1i
n
=
∑
(24)
where,
αi = coefficient of thermal expansion in subsection i
∆
ti = change in temperature in subsection i
Ai = cross-sectional area of transformed subsection i
yi = distance from neutral axis of cross section to subsection i
29
Ac = cross-sectional area of the entire composite transformed section
Ic = moment of inertia of the entire composite transformed section
The average strain and curvature are shown graphically in figure 16. The restraint strains are then
calculated as the difference between the final strain profile and the free strain profile, also shown in
figure 16, through application of the following equation:
εry = εtavg + φy – αy
∆
ty (25)
where, εry = restraint strain as a function of depth
εtavg = average strain
φ = curvature
y = distance from neutral axis
αy = coefficient of thermal expansion as a function of depth
∆
ty = change in temperature as a function of depth (the applied thermal gradient)
However, the above calculations assume a cross section without external axial and flexural
restraints. For a simple-span bridge, the curvature resulting from the imposed thermal gradient,
calculated from either equation 22 or 24, will result in a bowing of the section along the span. Only
the internal restraint strains as discussed previously are present in the cross section because of the
imposed nonlinear temperature variation. For a two-span continuous bridge, on the other hand, the
same bowing is restrained by the center support. This external restraint gives rise to additional
forces acting on the cross section. These additional forces are often referred to as secondary
continuity forces since they result from the continuous nature of the structure.
Figure 16. Restraint strain components.
30
The magnitude and distribution of the secondary continuity forces, and the corresponding continuity
strains, are a function of the bridge cross section and the number and relative length of each span
comprising the bridge. For example, the two-span bridge shown in figure 17(a) is subjected to a
positive thermal gradient. The interior redundant support is removed, and the girder is allowed to
bow upward as shown in figure 17(b). The curvature, φ, is then calculated from either equation 22
or 24, and the corresponding deflection, δ, is calculated from the second moment area theorem as φ
L2/8, where L is equal to the total length of the bridge, or a+b. To restore the interior support, a
force P must be applied at the location of the previously removed redundant support, which will
result in a deflection equal and opposite to that caused by the initial bowing, as shown in
figure 17(c). The magnitude of this force can be calculated by setting the deflection for a
concentrated load at midspan of a simple-span beam equal to the curvature deflection δ, or
PL3/48Ec Ic = φ L2/8. Solving for P yields a value of 6Ec Ic φ/L, where Ec and Ic are the modulus of
elasticity and moment of inertia, respectively, of the composite transformed section. The
corresponding midspan moment (moment at the center support) is then equal to PL/4 or (6Ec Ic φ/L)
L/4, which simplifies to 1.5 Ec Ic φ, and the complete secondary continuity moment diagram is as
shown in figure 17(d).
The same approach can be used to derive secondary continuity moments for any span and support
condition combination. The term Ec Ic φ is often referred to as the restraint moment, Mr, since it is
the result of the internal cross-section restraint of the section when subjected to a nonlinear thermal
gradient. Various span combinations will result in the secondary continuity moments ranging from
a minimum of 1.0 to a maximum of 1.5 times this restraint moment. By the principle of
superposition, the total thermal strains are then the combination of the internal restraint strains, as
calculated from equations 21 through 25, and the continuity strains, as calculated from the one-
dimensional beam theory for the continuity moments resulting from the specific support conditions
of the bridge in question. Thermal stresses are then calculated directly from the thermal strains.
Positive Secondary Moments in Prestressed Concrete Beams
Positive secondary moments are affected by beam creep and differential shrinkage between the
beam and the deck, positive diurnal temperature gradients, variation of the coefficient of thermal
expansion between the beam and the deck, and deck heat of hydration (i.e. locked-in temperature
differential between the deck and the beam).
The results of the analyses and the comparison with data from the test girder in this study indicate
that accurate creep and shrinkage parameters, along with the heat of hydration effects, are necessary
to reliably predict the actual continuity moments and the associated reactions. The analytical results
also demonstrated the effects of cracking in relieving continuity moments and reactions; however,
in doing so, the cracking induced high strains in the positive moment continuity connection (over
the pier).
31
(a) Two span bridge.
(b) Bowing after removal of redundant support.
(c) Restoration of redundant support.
(d) Resultant secondary continuity moment diagram.
Figure 17. Two span bridge secondary continuity moments.
Observations of the test bridge indicated that positive continuity moments are significantly affected
not only by creep and shrinkage in the girder and deck, but also by temperature gradients caused by
daily and seasonal temperature changes, and residual stresses resulting from the heat of hydration in
the deck. The level of shrinkage strain actually observed was low because of the outdoor
environmental effects. Data from the test girder demonstrated that all of these factors can be
additive and that the positive moment reinforcement provided in the diaphragm reached the yield
stress under the combination.
Connections are made in the diaphragm and deck concrete for this type of construction so that the
beams, which are simply supported for dead load, are made continuous for live load. Structural
capacity for live load can be gained, particularly with the development of negative moments over
the piers. However, by providing a positive moment connection, the secondary effects will
probably cause positive secondary moments and cracking at the bottom of the diaphragms. The
32
magnitude of the positive moment that can develop depends on the amount of positive
reinforcement provided and the adequacy of the anchorage of this reinforcement.
Considering the uncertainty of the factors affecting the potential for positive secondary moment, a
simplified procedure to account for these effects was a prime objective. To accomplish this
objective, two simplified design philosophies are proposed. The first involves eliminating the
positive moment connections in the bottom of the diaphragm over the piers. This essentially
eliminates positive secondary moments and the girders can be designed as simply supported for
service-level stresses from dead and live loads. For ultimate strength design, as long as sufficient
reinforcement is provided in the deck and a concrete diaphragm is cast between the ends of the
girders so that compression exists when the crack (or control joint) closes, there will be a negative
moment capacity for extreme loads. This approach eliminates the uncertainty of positive moment
service-level stresses in prestressed concrete girders, but provides a jointless deck and continuous
structure for strength design. The consideration of details that allow rotation to occur at the ends of
the girders and to connect the piers to the superstructure are discussed below.
The second philosophy for simplifying procedures to account for positive moment effects is to
design the positive moment connection to yield prior to developing stresses that could crack the
prestressed beams. With this philosophy, the positive moment connection acts as a fuse to eliminate
the uncertainty of the positive secondary moment effects by designing an upperbound for these
effects.
ACKNOWLEDGEMENTS
This study was sponsored by the U.S. Federal Highway Administration. The support of FHWA is
greatly appreciated. A large number of other researchers participated in this effort. These include
T.M. Refai, T.J. Lawson, J.S. Volz, and A. Scanlon.
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