Integral construction is used to avoid problems associated with bridge deck
joints and reduce the construction and maintenance costs. One of the major types
of construction of integral abutment bridges is a continuous jointless deck
connected integrally to the abutment. The end diaphragm or the abutment is cast
monolithically with the superstructure and may be directly supported on strip
footing or on a single row of piles. The structural components of a typical
integral bridge consist of superstructure, abutment wall, abutment foundation,
abutment backfill and wing wall if necessary (England and
Tsang, 2001). Due to design guidelines, that limit the maximum thermal movement
of abutment within the range of ±20 mm, the importance of study of bridge
abutment movement in such these bridges could be felt significantly (Anonymous,
2003). Integral Bridge that considered in this study was assumed to have
full height frame abutments supported on strip footings (Anonymous,
2003). As shown in Fig. 1, the frame abutment supports
abutment supports the vertical loads from the bridge superstructure and acts
as a retaining wall for embankment earth pressures.
In addition, the frame abutment is connected structurally to the deck to transfer
the bending moments, shear forces and axial loads to the foundation system.
Moreover, the frame abutment walls rotate about their foundations and have no
translation at the bottom (Anonymous, 2003).
Daily and seasonal temperature fluctuations cause longitudinal displacements
in integral abutment bridges. Resistance to expansion and contraction of a bridge
is provided by abutment backfill and the interactive substructure restraint
(Civjan et al., 2007).
||Abutment wall active and passive states (Horvest, 2005)
As the length of integral bridge increases, the temperature-induced displacement
in bridge components and surrounding soil may become larger and consequently
the backfill soil would be densified in greater amount as compared to initial
conditions (Arockiasamy and Sivakumar, 2005). When a
bridge contracts due to decrease in temperature, the abutment wall moves away
from the backfill soil. This may cause the soil loose its lateral support, subsequently
slide over the wall and apply an active earth pressure behind the abutment wall
(Horvath, 2000). On the other hand, when the bridge elongates
due to increase in temperature, the abutment wall moves toward the backfill
soil and hence, a passive earth pressure would be developed behind the abutment
wall (Horvath, 2000). Depending on the amount of temperature-induced
displacement of abutment, as shown in Fig. 2, earth pressure
can be as low as minimum active or as high as maximum passive pressures (Arsoy
et al., 1999). In this study, the interaction of soil-abutment due
to positive temperature changes is under investigation. Therefore, only the
passive modes of abutment wall movements were considered.
MATERIALS AND METHODS
This research was conducted since 2006 and the latest available correlations and theories were deployed. The research methodology consisted of three phases:
||Former formula citation
For the first phase, the previous formulas and correlations assessing the soil
behavior due to lateral pressure were represented. In this phase, the strengths,
advantages, weaknesses and deficiencies of these correlations were explained.
For the second phase, the new method of calculation for the abutment wall displacement
profile according to the bridge deck elongation and the soil lateral resistance
In this phase, those correlations cited in phase one were applied. For the
final phase, numerical modeling, a Finite Element (FE) model was deployed and
subsequently the corresponding structural and geotechnical bridge components
were built in SAP 2000. The aim of running FE model was to verify the integrity
of the obtained results from the previous phases. Also to investigate whether,
the deployed method is in a close agreement with the numerical data or not.
In continue, each phase is explained in details consecutively.
Formula citation: The ratio between the lateral and vertical principal
effective stresses when an earth retaining structure moves away or toward the
retained soil is defined as the soil lateral earth pressure coefficient. If
the wall has no movement, then it would be called the at rest position and the
earth pressure coefficient symbol for this state is Ko (Budhu,
2007). There are some theories and correlations for calculation of soil
lateral pressure that were proposed in the past researches. Some coefficients
were defined just as functions of soil properties like in Coulombs and
Rankins theories while in others such as British Standard, Massachusetts
manual, Canadian manual and Husain-Bagnaroil, they were proposed either as functions
of soil properties or abutment wall displacement (Anonymous,
2003). Figure 3 shows the distributions of the soil lateral
coefficient and the earth pressure along the abutment height.
According to Fig. 3, Eq. 1 expresses the resultant force applied on the back of the wall.
In Eq. 1, Fs is the soil resultant force, γ is the soil bulk unit weight, We is the effective girders width, H is the abutment height, Ko is the at rest soil lateral coefficient and K* is the passive soil lateral coefficient that is explained in continue.
As mentioned earlier, there are some correlations for calculation of soil lateral pressure coefficient. Some of them are presented below, respectively. In these equations, d is the bridge deck final displacement and H is the abutment wall height.
British standard formula (Anonymous, 2003):
where, φ is the soil internal friction angel that was assumed as 30° for the loose granule backfill.
Massachusetts manual formula (Abendroth and Greimann,
All the parameters are as same as Eq. 2.
Canadian manual proposed formula:
All the parameters above are like Eq. 2.
Husain-Bagnaroil formula: (Abendroth and Greimann,
All the parameters above are similar to Eq. 2.
Theoretical approach: When a bridge elongates due to increase in temperature,
the backfill soil will resist by applying earth pressure on abutment wall. The
intensity of earth pressure behind of the abutment is a function of magnitude
of the bridge deck displacement toward the backfill soil as demonstrated in
equations above and is equal to the products of soil lateral coefficient and
the soil normal effective stress. As appeared in the mentioned-correlations,
the magnitude of actual earth pressure coefficient, K*, is not constant and
would vary according to the amount of bridge deck movement. The soil structure
interaction model due to positive temperature changes could be best modeled
as Fig. 4 (Dicleli, 2000).
Figure 4 shows the structural model used to formulate the
effect of positive temperature variation on magnitude of earth pressure coefficient.
The structural model is obtained by conservatively neglecting the resistance
of piers, abutments stiffness against the structure longitudinal movement. If
there was no resistance against the bridge deck elongation, the bridge deck
could elongate freely under positive temperature changes. The structural model
for the bridge free longitudinal displacement, do, due to positive
temperature change is shown in Fig. 5 (Dicleli,
In Fig. 5, do is the free bridge elongation assuming no constrain against bridge expansion. The bridge free elongation is expressed by Eq. 8:
The fact is that the soil at the back of bridge abutment would resist against
deck elongation. Therefore, the actual bridge deck elongation should be less
than do. The structural model for the deck final displacement is
shown in Fig. 6 (Dicleli, 2000).
In Fig. 6, do is the bridge deck free elongation, dc is the amount of deck contraction due to backfill resistance and dfinal is the final position of bridge deck due to temperature-induced expansion force. The bridge deck contraction, dc, is defined by Eq. 9:
In addition, according to the bridge final displacement, the bridge deck axial force, Fd, applied on the abutment wall could be obtained by equation below:
|| Bridge superstructure structural model
||Bridge deck free elongation due to temperature change, no
soil resistance considered
||Deck final elongation due to temperature change and the soil-abutment
In Eq. 10, Kd is the bridge axial stiffness which was defined as:
All the parameters above were defined in Table 1. By substituting the deck axial stiffness, Kd, from Eq. 11 into Eq. 10, the deck axial force could be expressed as Eq. 12:
If do was replaced from Eq. 8, the bridge axial force could be expressed as below:
Assuming nearly identical abutment configurations at both sides of a bridge, the earth pressure force acting on abutment is completely transferred to the bridge deck. Therefore, to satisfy the equilibrium of forces in the longitudinal direction, the axial bridge deck force, Fd, should be equal to the earth pressure force, Fs.
By substituting the K* in Eq. 1 with the mentioned formulas presented in Eq. 2, 5, 6 and 7, the bridge deck final displacement could be calculated. These procedures are presented below:
Deck final displacement using British Standard:
Deck final displacement using Massachusetts:
Deck final displacement using Canadians:
Deck final displacement using Husain and Bagnaroil:
Final deck displacement, dfinal, could be obtained by solving each
equation from Eq. 15 to 18. As stated
before, in integral bridges, the connection of deck and abutment is fixed. Therefore,
the deck and abutment would move in the same direction and the same magnitudes.
It means, the abutment wall displacement at the top elevation is equal the bridge
deck final displacement.
Further more, in full height frame abutments, the walls are rigid, which rotate about their foundations. This would lead to linear deformations of walls with zero displacement at bottom level.
Figure 7 shows the abutment displacement profile along its height. The abutment wall displacement at each elevation can be obtained by Eq. 21.
||Linear deformation of full height frame abutment wall
Numerical modeling: In order to study the bridge behavior under temperature-induced
elongation, a model according to critical structural and geotechnical conditions
was selected. With this regard, an integral bridge with a three-span-continuous,
318 ft long, PC girder was modeled in SAP 2000 computer software. This bridge
had a U-shaped frame abutment supporting on a spread Reinforced-Concrete (RC)
backwall with strip footing. A summary of the geometric characteristics of the
modeled bridge is shown in Table 1.
Figure 8 shows the bridge overview. It was assumed that the two North and South abutment walls had identical conditions. The two intermediate piers were supported on strip walls, which inherently produced excessive resistance against bridge deck elongation.
Figure 9 shows the bridge girders arrangement. Full-composite action was assigned between the slab and girders. Constraint equations were used to create rigid links to connect the vertically-aligned nodes of finite elements for the slab and girders.
|| Full 3-D bridge model overview built in SAP
|| Bridge slab-girders connection
|| Bridge properties
These constraint equations coupled the translational and rotational, degrees-of-freedom
between nodes of slab and girders.
Table 2 shows the values obtained for equations multipliers.
These multipliers are obtained from British standard manual expressed in Eq.
15, Massachussetts manual expressed in Eq. 16, Canadian
manual in Eq. 17 and Husain-bagnaroil method in Eq.
18. For all the methods, the A* multipliers were 7.77E+08, the B* multiplier
for British Standard was 1.09E+06, for massachusetts was -4.52E+05, for Canadaians
was 4.82E+05 and for Husain-Bagnaroil was 1.17E+06. The C* multiplier for Massachusetts
was -7.42E+01. The other mutilpiers were shown in Table 2.
To obtain the bridge deck final displacement, the multipliers values were substituted
from Table 2 into Eq. 15 to 18
consecutively again. Afterward, the derived dfinal values were classified
due to low, mid and high temperature change ranges in Table 3-5,
Table 3 shows the bridge deck longitudinal displacements for the low-range temperature changes varies from 0 to 30°F.
Figure 10 shows the data of Table 3 in the columns pattern.
|| Deck final displacement under low temperature increase
|| Deck final displacement under mid temperature increase
|| Deck longitudinal displacement (dfinal) vs. temperature
||Deck longitudinal displacement (dfinal) vs. temperature
changes (mid- range)
Table 4 shows the bridge deck longitudinal displacements
for the mid-range temperature changes varies from 40 to 70°F.
Figure 11 shows the data of Table 4 in the columns pattern.
Table 5 shows the bridge deck longitudinal displacements for the high-range temperature changes varies from 80 to 110°F.
Figure 12 shows the data of Table 5 in the columns pattern.
By taking a close look to the obtained results, it can be seen that the SAP has the lowest rangewhile the free bridge displacement the largest sets of dfinal. This is because, the effects of existing piers were considered in SAP and hence it had the lowest ranges of results while for the other methods the effects of piers were ignored. In addition, in all the methods, the soil ressitance was taken into account while in the free bridge displacement method it was neglected and therefore, the free bridge displacement had the largest ranges of results.
For the low-range temperature changes as shown in Fig. 10, Massachusetts method underestimated the bridge deck elongation. In this temperature range, British Standard led to the closest results to SAP model as compared with the others. For the mid-range temperature changes as shown in Fig. 11, Massachusetts again underestimated the bridge deck elongation until approximately 55°F.
|| Deck longitudinal displacement (dfinal) vs. temperature
After this temperature, Massachusetts was the closest method to SAP. For the
high-range temperature changes, as shown in Fig. 12, all
the results were in the range between SAP and free bridge deck displacement
which this shows the integrity of the utilized methods. In this temperature
range, Massachusetts method was the closest method to SAP.
With regard to the presented materials in this study, these items were concluded:
||In study of abutment wall displacement, the interaction of
bridge deck elongation and backfill soil should be explicitly considered
||As the bridge deck and abutment wall are constructed integrally,
the abutment wall movement at its top elevation is equal to the amount of
||Abutment wall in Integral Bridges are mostly constructed in
reinforced concrete, it is assumed as a rigid mass, which has a linear deformation
||In full-height frame abutments, the walls rotate about their
foundations. Thus, the abutment wall movement at the bottom elevation could
||Rankin and Coulomb theories may not consider the effects of
deck elongation and soil resistance in their proposed formulas. Hence, they
may not be proper to be used in the corresponding calculations
||In low-range temperature changes, Massachusetts method underestimates
the deck elongation as compared to the other methods
||Approximately all the results obtained from British Standard,
Massachusetts, Canadian and Husain-Bagnaroil, except for the Massachusetts
low-range and mid-range temperature changes, were in the range between SAP
and the deck free displacement. This can assure the integrity of these methods
||It was seen that in loose granule backfill, for the low-range
and the mid-range temperature changes, British Standard was the closets
method to SAP, while in the high-range temperature changes, Massachusetts
was the closets one
||It is recommended to use British Standard method for calculation
of bridge deck elongation and abutment wall movement in loose granule backfill
under temperature changes, while the other methods are not denied