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Albhaisi and Nassif 1

SIMPLE APPROACH TO CALCULATE DISPLACEMENTS AND ROTATIONS IN

INTEGRAL ABUTMENT BRIDGES

Suhail Albhaisi*, P.E., Ph.D.

Jacobs Engineering Group

2 Penn Plaza Suite 603, New York, NY 10121

Phone: (212) 946-2325, Fax: (212) 302-4645

[email protected]

Hani Nassif, P.E., Ph.D., Professor

Rutgers Infrastructure Monitoring and Evaluation (RIME) Laboratory

Department of Civil and Environmental Engineering

Rutgers, The State University of New Jersey

96 Frelinghuysen Road, Piscataway, NJ 08854

Phone: (848) 445-4414, Fax: (732) 445-8268

[email protected]

* Corresponding Author

Revision No. 1

Word count: 3197

Abstract: 249 < 250

Figures & Tables: 9 x 250 = 2250

Total: 5696

Submission Date: 11/15/2014

mailto:[email protected]

mailto:[email protected]


ABSTRACT

This paper presents a simple approach to calculate the displacements and the rotations induced 1

by thermal loading in Integral Abutment Bridges (IABs). The approach was derived from the 2

results of a parametric study that investigated the effect of substructure stiffness on the 3

performance of short and medium length steel IABs built on clay and sand under thermal load 4

effects. Various parameters such as pile size and orientation, pile material, and foundation soil 5

stiffness were considered in the study. Detailed three-dimensional (3D) Finite Element (FE) 6

models using the software LUSAS were developed to capture the overall behavior of IABs. The 7

developed 3D FE model was calibrated using field measurements obtained from a previous study. 8

Using the calibrated models, a parametric study was carried out to study the effects of the above 9

parameters on the performance of IABs under thermal loading using the American Association 10

of State Highway Transportation Officials (AASHTO) Load and Resistance Factor Design 11

(LRFD) temperature ranges. The study showed that most of the parameters have significant 12

effects on the displacement and rotation of the abutment and the supporting piles. Also, for 13

relatively wide IABs, there are significant variations in the displacement and rotations in the 14

substructure elements between interior and exterior locations. This approach, which consists of 15

using simple equations and charts and including parameters such as the length of the bridge, the 16

stiffness of the foundation soil, and the pile location provides results that are comparable to those 17

and in lieu of using a detailed FE analysis. 18

Key Words: 19

Integral Bridge 20

Finite Element 21

H-Piles 22

Simple Approach 23

Soil-Structure Interaction 24


INTRODUCTION

Expansion joints and end bearings in conventional (jointed) bridges are expensive and require 1

special handling during construction. They also require periodic inspection and maintenance and 2

may need to be replaced several times throughout the bridge life. This is especially true for areas 3

with considerable snow amounts where deicing chemicals are used throughout the cold season 4

and where snowplows could repeatedly hit and damage the joints. Furthermore, water and 5

deicing chemicals would penetrate through the expansion joints to cause extensive deterioration 6

to the bearings, superstructure, and substructure components. Leakage at joints accounts for 70% 7

of the deterioration at the end of the girders (1). Consequently, expansion joints and bearings in 8

bridges have provided considerable construction and maintenance challenges for most 9

transportation agencies. For the above reasons, integral abutment (Jointless) bridges are 10

becoming increasingly popular in the USA and in many parts of the world and are considered as 11

a more economical alternative to conventional bridges. A sketch for a typical single-span IAB is 12

shown in Figure 1. 13

14

FIGURE 1. Typical single-span integral abutment bridge.

Wing

Wall

Single Row of

Vertical Piles

Continuous Deck slab

Cycle Control

Joint Approach

Slab Girder

Stub

Abutment


More IABs are built every year in the United States and all over the world. According to the 1

Tennessee department of transportation (TDOT), 85% of the new bridges built in the State are 2

integral abutment bridges (2). In the United Kingdom, British Highways Agency Design Manual 3

for Roads and Bridges recommends that all new bridges less than 60 m (200 feet) in length and 4

skews not exceeding 30° shall be designed as integral bridges. A 2004 survey suggests that the 5

number of IABs has increased in the past decade with most transportation agencies planning to 6

replace jointed bridges with integral bridges when conditions permit (1). The survey also shows 7

that 70% of the States use bearing type steel H-Piles to support integral bridges without 8

consensus on the orientation of the piles with respect to the centerline of the bearings. To further 9

reduce the stiffness of the substructure, many States enclose the top part of the H-Piles by a 10

sleeve filled with loose sand or crushed stones. Some States (e.g. Iowa) consider, in addition to 11

steel H-Piles, prestressed concrete piles to support the abutment (3). Although drilled shaft 12

foundations are considered much stiffer than other deep foundation types and are not allowed to 13

be used by many States in the foundation of integral bridge, Hawaii used drilled shafts to support 14

integral abutments because of the severe corrosion conditions in the State that prohibits the use 15

steel H-Piles (4). 16

Given the variability in the state-of-practice among various states, there is a need to 17

provide simplified guidelines that would be applicable to the design of IABs. Currently, neither 18

AASHTO-LRFD bridge design specifications nor the Standard AASHTO bridge design 19

specifications have provided such design criteria. In the absence of unified design criteria for 20

integral bridges, most States developed their own design criteria and geometric limits. These 21

limits are mainly imposed on the maximum bridge length, skew angle, pile type, pile orientation 22

and the type and compaction level of the backfill material behind the abutment. In general, these 23

limits vary considerably between States and are based mainly on the State’s experience with 24

existing integral bridges and limited research. 25

Substructure stiffness has a major impact on the performance of integral bridges. When 26

designing IABs, bridge engineers try to reduce the stiffness of the substructure to accommodate 27

the movement of the superstructure while minimizing the stresses in the superstructure and 28

substructure during thermal expansion and contraction of the bridge. They usually use a stub 29

abutment supported by a single row of piles to support the bridge ends. To further reduce the 30

stiffness of the single row of piles, many States enclose the top part of the piles by a sleeve filled 31

with loose sand or crushed stones. Researchers have studied the effect of substructure stiffness 32

on the performance of concrete IABs using validated three-dimensional (3D) FE models (5, 6). 33

Researchers have also studied the effect of substructure stiffness on the performance of steel 34

IABs under thermal loads (7, 8). The majority of these studies were carried out using simplified 35

2D models without verification. Useful guidelines are available for the design of IAB’s (9, 10, 36

11). These guidelines provide useful design examples based on experience in the design of IABs, 37

but do not provide theoretical approach for the analysis. The authors conducted a detailed 38

parametric study to investigate effect of substructure stiffness on the performance of steel IAB’s 39

using 3D FE models (12, 13). Based on the results of this parametric study, a simple approach 40

was derived to calculate the displacement and the rotation induced by thermal loading in IABs. 41

THREE-DIMENSIONAL (3D) FINITE ELEMENT MODEL 42

Two IABs were considered in the study. The bridges depict two integral bridges recently 43

constructed in New Jersey. The two bridges were slightly modified to suit the parametric study. 44

The first bridge is a 38-meter (127 foot) single span steel plate girder bridge and the second 45


bridge is a two-equal span steel plate girder bridge with a total length of 90 meters (300 feet). 1

The cross sections of the single-span bridge and the two-span continuous bridge are shown in 2

Figures 2a and 2b, respectively. The lengths of the two bridges cover a substantial range of 3

common IABs’ lengths. 4

5

6

FIGURE 2. Cross section of (a) Single-span bridge (b) Two-span bridge.


To accurately capture the behavior of IABs, the entire parametric study was carried out using 3D 1

Finite Element (FE) Models. Using 3D models captures the behavior of integral bridges in the 2

transverse direction and gives better results than using 2D models. The software LUSAS was 3

used for the analysis throughout the research (14). A typical 3D FE model for the single-span 4

bridge is shown in Figure 3. AASHTO LRFD recommended temperature ranges for steel 5

bridges in cold climates were used in the study (15). 6

ment. 7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

FIGURE 3. A typical 3D FE model for the single-span bridge.

Diaphragm

Soil Springs Abutment

H-Pile Deck Underside

Isometric View

Steel Plate Girder

Stiffener


SOIL-PILE INTERACTION 1

Generally, the soil-pile interaction for a particular point along the pile is defined by a nonlinear 2

load (P) versus deformation (Y) curve or P-Y curve, where P is the lateral soil resistance per unit 3

length of pile and Y is the lateral deflection. The computation of the lateral force-displacement 4

response of a pile involves the construction of a full set of P-Y curves along the pile to model the 5

force deformation response of the soil. A typical P-Y curve for soft clay is shown with a solid 6

line in Figure 4 (16). Figure 4 shows that the P-Y relationship is nonlinear and can be expressed 7

as follows: 8

p

pu

0.5

y

y50

1

3

(1) 9

Where Pu = the ultimate lateral soil resistance per unit length of pile and y50 = 50% (i.e., one-half) 10

of the deflection of soil at ultimate resistance and it is expressed as: 11

y50 2.550b (2) 12

Where 50 is the strain corresponding to one-half the maximum principle stress 13

difference and b is the width of the pile. The value of p is assumed constant after 508 /y y . 14

0.5

1.0

1.0 8.0

Approximate

Elasto-Plastic Curve

FIGURE 4. P-Y curve for soft clay.


This nonlinear behavior is simplified using an elasto-plastic curve displayed on Figure 4 with the 1

dashed line. The elastic portion is defined with a slope equal to the secant soil modulus, Es, and 2

the soil modulus for clay can be calculated using the following expression: 3

Es

pu

2

2.550b

pu

5.050b (4) 4

For a foundation supported by clay, the soil was approximated with elastic spring elements 5

having a spring constant equal to the soil modulus, Es (12). A modified Reese procedure (16) 6

was followed to construct the p-y curves for sand (12). 7

ABUTMENT-BACKFILL INTERACTION 8

When an integral bridge contracts due to a decrease in temperature, active backfill pressure will 9

immediately develop behind the abutment at a very small displacement. The intensity of this 10

active backfill pressure can be directly calculated using Rankine’s theory (17). Thus, only active 11

earth pressure needs to be considered under negative thermal variation. However, when the 12

bridge elongates due to an increase in temperature, the intensity of the backfill pressure behind 13

the abutment depends on the magnitude of the bridge displacement towards the backfill soil. 14

The actual backfill pressure coefficient, K, may change between the at-rest backfill pressure 15

coefficient, K0 , and the passive backfill pressure coefficient, K p , depending on the abutment 16

displacement. Clough and Duncan (18) obtained the variation of backfill pressure coefficient, K , 17

as a function of abutment displacement from experimental data and finite-element analyses. 18

Uniform compacted backfill was assumed in the analysis. The spring constant for the springs in 19

the model was calculated as follows (12): 20

Kspring 300 z / H (5) 21

Where = the backfill soil unit weight, z = the spring depth measured from the top of 22

the abutment, and H = abutment height. Interaction between the backfill and the approach slab 23

was ignored in the models. 24

MODEL VALIDATION 25

The field measurements from the new Scotch Road Bridge (19) were used to calibrate and 26

validate the 3D FE model. Table 1 presents a comparison between the field measurements and 27

the calibrated 3D FE model results for the top of abutment movement and top of pile stresses. 28

TABLE 1 Comparison between field measurements and 3D FE model results (12) 29

Parameter Case* Field Data

(19) 3D FE Model

Bridge movement

range (inches)

Elongation T=25 oC (45

oF) 0.40 - 0.52 0.44 - 0.50

Contraction T=-25oC (-45

oF) 0.38 - 0.50 0.44 - 0.50

H-Pile stresses (ksi) Elongation T=25

oC (45

oF) 17-18 (Compression) 18 - 19

Contraction T=-25oC (-45

oF) 9-12 (Tension) 9 - 11

*Base Temperature = 10 o

C (50 oF)


SUBSTRUCTURE DISPLACEMENT

The total displacement of the bridge, B, can be divided into two main displacements. Those are 1

the displacement of the pile, P, and the relative displacement of the abutment, A, which is the 2

displacement of the top of the abutment relative to the top of the pile (bottom of the abutment). 3

The parametric study results showed that, the movement of the abutment can be approximated by 4

rigid body movement as shown in Figure 5. Thus, the total bridge displacement, B, of the 5

bridge deck is expressed as; 6

B P A (6) 7

8

FIGURE 5. Deformed shape of the abutment and the pile.

9

It was noticed throughout the study that the total displacement of the bridge, B, is not sensitive 10

to the stiffness of the substructure for both the contraction and expansion cases. It was noticed 11

that for both the expansion and contraction cases, the bridge exterior displacement, Bex, is 12

almost equal to the thermal displacement demand of the bridge and it can be expressed as: 13

(7) 14

Where is steel thermal expansion coefficient, L is half the bridge length, T is the change in 15

temperature. 16

P

B

Abutment

Pile

H

Le

Original Shape

Actual Deformed Shape

A

(Can be approximated by Rigid Body movement)

Approximate Deformed Shape


It was also noticed that for both the expansion and contraction cases, the bridge interior 1

displacement, Bin, is less than the thermal displacement demand of the bridge and it can be 2

expressed as: 3

(8) 4

Where m is a factor ≤ 1.0 5

Throughout the study, the interior displacement was observed to be about 10% to 15% less than 6

the thermal displacement demand of the bridge and therefore the value of m is between 0.85 and 7

0.9. Assuming an average value of m=0.87, Equation 8 can be rewritten as: 8

(9) 9

10

On the other hand, the displacement of the pile, P, and relative displacement of the abutment, A, 11

are sensitive to the stiffness of the substructure. The pile displacement is almost equal to the 12

total displacement of the bridge when the bridge is under contraction and supported by soft clay 13

or loose sand (12). 14

The summation of the ratio of the relative displacement of the abutment to the total displacement 15

of the bridge and the ratio of displacement of the pile to the total displacement of the bridge 16

equals unity, or: 17

1A P

B B

(10) 18

Figure 6a shows the ratio of the relative displacement of the abutment and displacement of the 19

pile to the total displacement of the bridge in clay during bridge contraction. Figure 6a can be 20

utilized to estimate the relative displacement of the abutment and the displacement of the pile 21

during bridge contraction for various bridge lengths. Figure 6a shows that during bridge 22

contraction, the stiffness of the clay is the dominant factor in determining the displacement ratios 23

for the abutment and the pile. In the graphs the letters A and P refer to the locations at the top of 24

Abutment and at the top of the pile respectively. 25

Figure 6b shows the ratios of the relative displacement of the abutment and displacement of the 26

pile to the total displacement of the bridge in clay during expansion. Figure 6b can be utilized to 27

estimate the relative displacement of the abutment and the displacement of the pile during bridge 28

expansion for various bridge lengths. The figure shows that during bridge expansion, although 29

the stiffness of the clay is the dominant factor in determining the displacement ratios for the 30

abutment and the pile, other factors including pile stiffness and location also contribute in 31

determining these ratios. 32

33

Figures 6c and 6d show the ratios of the relative displacement of the abutment and displacement 34

of the pile to the total displacement of the bridge in sand during bridge contraction and expansion 35

respectively. Similar observations were noticed in the sand case for both contraction and 36

expansion. 37


FIGURE 6a. Displacement ratios for the abutment and the pile (Contraction, Clay)

FIGURE 6b. Displacement ratios for the abutment and the pile (Expansion, Clay)

P/B

A/B

P/B

A/B


FIGURE 6c. Displacement ratios for the abutment and the pile (Contraction, Sand)

FIGURE 6d. Displacement ratios for the abutment and the pile (Expansion, Sand)

P/B

A/B

P/B

A/B


SUBSTRUCTURE ROTATION 1 A simple presentation of the rotation along the abutment is shown in Figure 7. The 3D FE 2

models study showed that the abutment movement can be approximated by a rigid body 3

movement. Based on that observation, the average rotation along the abutment, A, can be 4

utilized to calculate the rotation at the top of the abutment and at the top of the pile. 5

FIGURE 7. Simplified model for the abutment displacement.

Once the relative displacement of the abutment, A, is estimated, the average rotation 6

along the abutment, A, can be calculated as follows: 7

A

A ATANH

(11) 8

The rotations at the top of the abutment, t, and at the bottom of the abutment (top of the pile), 9

b, can be estimated as follows: 10

1 *t Am (12) 11

2 *b Am (13) 12

Where m1 and m2 are constants and can be estimated from Figure 8. 13

H

A

θt

θb

θA


FIGURE 8a. Rotation ratios for the abutment and the pile. (Contraction, Clay)

FIGURE 8b. Rotation ratios for the abutment and the pile. (Expansion, Clay)

mt/A

mb/A

mt/A

mb/A


FIGURE 8d. Rotation ratios for the abutment and the pile. (Expansion, Sand)

FIGURE 8c. Rotation ratios for the abutment and the pile. (Contraction, Sand)

mt/A

mb/A

mt/A

mb/A


EXAMPLE 1

Given 2

Steel bridge with total length = 90 meters (L=0.5*90=45 m), Thermal rise T= 40o

C, Piles are 3

supported on clay with Cu =100 KPa, Steel thermal coefficient is equal to 11.0 x10-6

4

Needed 5

Displacement at the top of the external pile 6

Using Eq. 7, the total displacement of the bridge, B=0.000011*45*40=0.0198 m=19.8 mm 7

Using Figure 6a (Use total bridge length), the ratio of the relative displacement of the abutment 8

to the total displacement of the bridge =0.29. The relative displacement of the abutment A = 9

0.29*19.8=5.7 mm. The displacement at the top of the pile P = 19.8-5.7=14.1 mm. 10

Alternatively, the displacement at the top of the pile P can be calculated directly by estimating 11

the ratio of the relative displacement of the pile to the total displacement of the bridge from 12

figure 6a which equals 0.71. The displacement at the top of the pile P = 0.71*19.8=14.1 mm. 13

Using Figure 7, similar steps can be followed to estimate the rotation at the top of the pile. Given 14

the displacement and the rotation at the top of the pile, the designer is now capable of checking 15

the stresses in the pile using hand calculations or commercial software like LPILE. 16

LIMITATIONS 17

The simplified approach is recommended for steel IABs with total bridge length between 10 and 18

120 meters and supported on H-Piles with weak axis orientation. Interpolation can be used to 19

obtain results for bridges with lengths between 38 and 90 meters. Extrapolation can be used for 20

bridges longer than 90 meters or shorter than 38 meters. The approach is also recommended for 21

typical bridges where the bridge length (longitudinal direction) is larger than the width 22

(transverse direction). 23

CONCLUSIONS 24

This paper presents an approximate simple approach to calculate the displacement and the 25

rotation induced by thermal loading in IABs. The simple approach, which consists of equations 26

and charts, takes into consideration the length of the bridge, the stiffness of the foundation soil, 27

and pile location for both the bridge expansion and contraction cases. Based on the analysis 28

results, the following conclusions could be made: 29

- The displacement at the top of abutment is equal to the thermal demand of the superstructure at 30

exterior locations and approximately 87% of the demand at interior locations. 31

- The displacement of the abutment in IAB’s can be approximated by Rigid Body Movement. 32

- The soil stiffness has a minor effect on the displacement at the top of the abutment but a 33

significant effect on the displacement at the top of the pile especially during bridge contraction. 34

- The rotation of the abutment and the supporting piles is not sensitive to soil stiffness. 35

- There are significant variations in abutment displacement and rotation between interior 36

locations and exterior locations in wide IABs when wingwalls are not attached to the abutment. 37

- The effect of substructure stiffness on the displacement and rotation of the abutment as well as 38

the supporting pile is greater during bridge contraction. 39


ACKNOWLEDGEMENTS 1

The 3D model in this research was validated using the data from a study sponsored by the New 2

Jersey Department of Transportation (NJDOT) (19) that is gratefully acknowledged 3

4

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