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THE P-DELTA AND SOIL–STRUCTURE INTERACTION EFFECTS ON BRIDGE PIERS.
by
Toun L Wu
B.S., Southern Illinois University, 2008
A Thesis Submitted in Partial Fulfillment of the Requirements for the
Master of Science Degree
Department of Civil Engineering
in the Graduate School Southern Illinois University Carbondale
August 2010
UMI Number: 1482679
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THESIS APPROVAL
THE P-DELTA AND SOIL–STRUCTURE INTERACTION EFFECTS ON BRIDGE PIERS.
By
Toun Loin Wu
A Thesis Submitted in Partial
Fulfillment of the Requirements
For the Degree of
Master of Science
in the field of Civil Engineering
Approved by:
Dr. J. Kent Hsiao, Chair
Dr. Aslam Kassimali
Dr. Sanjeev Kumar
Graduate School Southern Illinois University Carbondale
May 10, 2010
i
AN ABSTRACT OF THE THESIS OF
TOUN LOIN WU, for the Master of Science degree in Civil Engineering,
presented on the 10th of May, 2010, at Southern Illinois University Carbondale.
TITLE: THE P-DELTA AND SOIL-STRUCTURE INTERACTION EFFECTS ON
BRIDGE PIERS.
MAJOR PROFESSOR: DR. J. Kent Hsiao
The purpose of this study is to investigate the P-Delta effect on bridge
piers with consideration of soil-structure interaction effect. The traditional P-Delta
effect was calculated assuming the base of the structure is fully fixed. However,
all structures in real life are supported by soil, and soil is deformable. Therefore,
under wind or earthquake loads, the additional lateral deflection of a bridge pier
caused by the deformation of the soil, which supports the pier, should be
considered. The traditional method regarding the computation of the deflection
of a pier caused by wind or earthquake loads does not take soil-structure
interaction effects into account. In this study finite element analysis method will
be used to investigate the additional overturning moment introduced by the soil-
structure interaction effect.
ii
ACKNOWLEDGMENTS
First of all, I would like to thank my advisor, Dr. Kent Hsiao, for his
guidance and support throughout my undergraduate and graduate career. Also, I
would like to thank Dr. Kumar and Dr. Kassimali for their support. I am grateful
to all the faculty and staff in the Departments of Civil and Environmental
Engineering who taught me everything I need to know and helped me to become
a good engineer. I wish to thank my colleagues and friends who have helped me
in my undergraduate and graduate career here at Southern Illinois University
Carbondale. Finally, I would like to thank my family for the love and support they
have given me at every step of the way!
iii
TABLE OF CONTENTS
CHAPTER PAGE
ABSTRACT ...................................................................................................... i
ACKNOWLEDGMENTS .................................................................................. ii
LIST OF TABLES ............................................................................................. v
LIST OF FIGURES .......................................................................................... vi
CHAPTERS
CHAPTER 1 – Introduction .................................................................... 1
CHAPTER 2 – Finite Element Methods ................................................. 4
Section 2.1 Soil Pressure Bulb Verification ................................ 4
Section 2.2 Immediate settlements Verification ......................... 5
CHAPTER 3 – P-Delta Effect Verification ............................................ 11
CHAPTER 4 – Finite Element Model Development ............................. 17
Section 4.1 Introduction ............................................................. 17
Section 4.2 Dimensions ............................................................. 17
Section 4.3 Material Properties .................................................. 19
Section 4.4 Loads ...................................................................... 20
Section 4.5 Boundary Conditions ............................................... 21
Section 4.6 Finite Element Models for the Pier .......................... 21
Section 4.7 Results .................................................................... 26
CHAPTER 5 – Discussions .................................................................. 35
CHAPTER 6 – Conclusion ................................................................... 38
iv
REFERENCES ............................................................................................... 39
VITA ............................................................................................................. 41
v
LIST OF TABLES
TABLE PAGE
Table 2.1 Finite Element Model 2B Dimensions ............................................... 6
Table 4.3.1 Materials Properties for Finite Element Models .......................... 20
Table 4.4.1 Applied Loads for Finite Element Models .................................... 21
Table 4.7.1 Maximum Lateral Displacements at Each step for Model D ....... 27
Table 5.1 Comparison of Maximum Lateral Displacements .......................... 35
vi
LIST OF FIGURES
FIGURE PAGE
Figure 2.1 SYY Stresses of Model 2A ............................................................. 8
Figure 2.2 Three Dimension View of Finite Element Model 2B......................... 9
Figure 2.3 Maximum Y-Displacement of Model 2B ......................................... 10
Figure 3.1 Three Dimensional View of Model 3A & B3B ................................ 14
Figure 3.2 Maximum Lateral Displacement of Model 3A ................................ 15
Figure 3.3 Maximum Lateral Displacement of Model 3B ................................ 16
Figure 4.1 Section View of Design Example No.6 Bridge Pier Details ........... 18
Figure 4.2 Footing Dimensions used for Finite Element Models..................... 19
Figure 4.6.1 Finite Element Model for Piers ................................................... 22
Figure 4.6.2 Three Dimensional View of Finite Element Model 4A ................ 23
Figure 4.6.3 Three Dimensional View of Finite Element Model 4B ................. 24
Figure 4.6.4 Side View of Finite Element Model 4C ....................................... 25
Figure 4.6.5 Three Dimensional View of Finite Element Model of 4C ............. 26
Figure 4.7.1 Maximum Lateral Displacement of Model 4A ............................. 28
Figure 4.7.2 Maximum Lateral Displacement of Model 4B ............................. 29
Figure 4.7.3 Maximum Lateral Displacement of Model 4C ............................. 30
Figure 4.7.4 Maximum Lateral Displacement of Model 4C at the Footing ...... 31
Figure 4.7.5 Model C: Time vs. Maximum Lateral Displacement .................... 32
Figure 4.7.6 Model C: Maximum Lateral Displacements vs. Loads ................ 33
Figure 4.7.7 Deformed Geometry of Model C ................................................ 34
1
CHAPTER 1
INTRODUCTION
The P-Delta effect occurs when a structure, acted upon by a lateral load,
becomes laterally displaced, and the applied vertical loads become eccentric,
with respect to the bases (Lindeburg & Baradar, (2000)). This results in
additional forces, moments and increased lateral displacements. The P-Delta
effect tends to reduce the overall stiffness and strength of a building and could
result in the collapse of the structure. Therefore it is important to study the
effects. This effect can occur on any structure, especially tall structures or
structures subject to large lateral loads. Highway bridges are one type of
structure in which the P-Delta effect may have significant impact, due to heavy
traffic loads, wind or earthquakes.
There are a number of methods can be used to calculate P-Delta effect.
Four different analytical methods were commonly used to calculate the P-Delta
effects (Dobson 2002). The pseudo load method, pseudo displacement method,
the two cycle iterative method, and non-linear static analysis. The first three
methods do not take “stress stiffening” into account for the P-delta effect
calculations. The non-linear static analysis method is a method carried out in an
incremental step by step analysis with total applied loads divided into a number
of load steps. The non-linear static analysis allows for all sorts of non-linear
2
conditions to be accounted for simultaneously, including “stress stiffening”.
Therefore non-linear static analysis was used in this study.
In practice, P-delta effect can be calculated by using hand calculations
with the assumption that the column is fully fixed at the base. Earlier studies
(Poston 1986, Main 2004, and Ger & Yen 2004) also used the same approach by
assuming the columns to be fully fixed at the base when studying the lateral
displacements of bridges and bridge piers. In bridge design, engineers also use
the same procedure for the calculation the P-Delta effect. In reality, bridge piers
are connected to a footing while the footing is supported by soil. In fact, concrete
footing and soil are both deformable materials. When the materials start to
deform, the overall displacement of the structure will also increase. Therefore
the traditional equation used to calculate this displacement could be insufficient.
The results of this study would also determine whether the soil-structure
interaction effect needs to be considered in P-Delta effect calculations.
The traditional method for calculate P-Delta effect can only be used when
the moment of inertia of the structural element is a constant. In cases where the
moment of inertia of a structural element is not a constant, the traditional
equation would require excessive time to calculate. Due to this excessive time
requirement, a finite element program was used in this study to construct an
analysis. Finite element modeling has been proven to be an accurate modeling
of bridge structures according to Huang & Zhu 2008 recent study on “Finite
element model updating of Bridge structures based on sensitivity Analysis and
optimization Algorithm”. In their study the authors used field data to compare the
3
results obtained from finite element model. The finite element model has been
proven to model bridge structure accurately. Within Huang & Zhu’s study there
were some assumptions made in establishing the models, such as neglecting the
dynamic interaction of soil-structure, not considering P-Delta effects and non-
linear material behavior.
In this study, the main focus was to take into account the different key
factors that other studies (Poston 1986, Main 2004, Ger & Yen 2004, and Huang
& Zhu 2008) had neglected, such as soil-structure interaction, P-Delta effect and
non-linear material behavior. Therefore, three models were constructed to
investigate the actual displacement due to P-delta and soil-structure interaction
effects. The first model was subjected to lateral load only while the second
model takes into account the P-Delta effect. The last model takes into account
both P-Delta and soil-structure interaction effects. Non-linear analysis was used
to analyze all three models. In addition, two simple models were created
throughout this study to ensure the finite element program could be used to
calculate the P-Delta and soil structure interaction effect accurately. Two
mathematical equations were used to verify the results obtained from the simple
models.
4
CHAPTER 2
FINITE ELEMENT METHODS
2.1 Soil Pressure Verification
All structures are supported by some sort of soil or rock, therefore
soil has a significant influence on the performance of a structure. This study
focuses on how soil affects the lateral displacement of bridge piers due to lateral
and vertical loads. In order to study the soil-structure interaction effect soil
behavior must be understood. In this study, finite element program NISA (1999)
has been used to study the soil-structure interaction.
The stress distributions for soil under dead and live loads have
been well studied for years, so charts have been developed describing how
stress is distributed at different depths of soil. Therefore, a finite element model
(referred as Model 2A as shown in Figure 2.1) was created to study the stress
distributions in different depth of the soil. The model uses the same
configurations (width and depth of soil profile) as the soil pressure bulb (Day
1999); each element on Figure 2.1 represents one half B (where B is the width of
the footing). The soil layers were taken three times of width of footing from
each side and six time of footing width toward bottom of soil. A unit load of 1 kip
was applied to the top of the soil at the middle of the soil model. Since the main
purpose for this model was to verify whether the finite element program is
5
capable of analysis the soil correctly for the finial models, therefore in this model
same material properties for the soil will be used. A very stiff clay soil was
selected for this model. The typical values of Young’s Modulus were ranged
from 1000 to 2000 kip per square foot. The modulus of elasticity of 1500 kip per
square foot (average of highest and lowest values) was selected for the soil. The
1500 kips per square foot was converted to 10.4 ksi. The modulus of elasticity of
10.4 ksi and the possion ratio of 0.4 was used for this model. After the model
was complete, linear analysis was performed to determine the stress on the
model. The stresses from the finite element model shown in Figure 2.1, was very
close to the stresses shown in soil pressure bulb presented by Day (1999).
2.2 Immediate Settlement Verification
This verification consists of taking the vertical displacement from the finite
element model and comparing it to the hand calculation. In order to do so, a
simple computer model (referred to as Model 2B as shown figure 2.2) was
created. The model contains a simple footing supported by a very stiff clay soil
which is strong enough to support the vertical loads (this same type of soil will be
use for the finial model). The model was constructed according to the dimension
shown on Table 2.1.
6
Table 2.1 Finite Element Model 2B Dimensions
Dimensions Footing Soil
Length ( L) 80" 560" Width (B) 80" 560" Depth (H) 32" 480"
Modulus of elasticity (Es) 3150 Ksi 10.4 Ksi Possion ratio (v) 0.2 0.4
Material properties, load and boundary conditions were also applied to the
model. The footing was subjected to 4 kips per square foot of loading on top of
the footing. The boundary conditions for this model were as following: no
movements were allowed in the Y-direction at the bottom of the soil base. For
each material used in this model, the modulus of elasticity and the possion ratio
are shown in Table 2.1. A complete graphic representation of the model is
shown in Figure 2.2. A static analysis was performed to obtain the displacement
in the Y-direction. The displacement was taken from each node between the
footing and the soil layer in the model as shown in Figure 2.3. All the
displacement obtained from the each node were added up and divided into the
number of node being added to get the average displacements. The average
displacement was calculated to be 0.153 inches. This displacement was
compared to the results obtained using the immediate settlement equation (Das
2007). The equation used to determine the immediate settlement follows.
7
Where:
B = width of the foundation
qo = contract pressure
Es = modulus of elasticity of soil
µ = poisson’s ratio of the soil
αav = a factor for rectangle footing base on values of L/B. (Das 2007)
The displacement obtained by using immediate settlement equation was 0.161
inches as shown above. The displacements obtain from the finite element
program and the immediate settlement equation were similar and within a
reasonable range.
8
Figure 2.1 SYY Stresses of Model 2A.
P = 1Kips
Fixed at the Bottom
Width of each element: 0.5 B
Length of each element: 0.5 B
B
9
Figure 2.2 Three Dimension View of Finite Element Model 2B.
Vertical Load:
4 Kips per Square Feet
80”x80” Footing
560 Inches
480 Inches
560 Inches
Fixed Fixed Fixed
10
Figure 2.3 Maximum Y-Displacement of Model 2B.
80” x 80” Footing
Fixed Throughout Bottom
Vertical Displacement
11
CHAPTER 3
P-DELTA EFFECT VERIFICATION
Two finite element models (3A and 3B) were created for the study of P-
Delta effect. Both models were constructed in such a way that both
configurations were the same as shown in Figure 3.1(excluding the type of
analysis performed on each model). The only different that between the two
models was the applied loads. The first model (referred to as Model 3A and
shown in Figure 3.2) didn’t take into account for the P-Delta effect. The second
model (referred to as Model 3B shown in Figure 3.3) took the P-Delta effect into
account for the calculation of overall lateral displacement. A linear static analysis
was performed on Model 3A. A Non-linear static analysis was performed on
Model 3B.
The overall dimensions of the models were shown in Figure 3.1. Material
properties such as Modulus of elasticity and possion ratio were assigned to each
model. The modulus of elasticity assigned to the model was 3640 ksi whereas
the possion ratio is 0.2 for concrete. Two different loading conditions were
applied to the models; vertical and lateral. For Model 3A a 2 kips lateral load was
applied at the left side to the top. The second model also subjected to the same
vertical force as the first. However, the lateral force applied on Model B was
different than the first model. For the second model a total 2 kips lateral force
12
was divided evenly into 10 steps and applied as incremental loads on the top of
the model. In addition, a vertical load of 20 kips was applied at the midpoint of
the top surface.
After the models were constructed, a static linear analysis was performed
on Model 3A and the results were shown in Figure 3.2. For Model 3B as shown
in Figure 3.3, a nonlinear analysis was performed by included a stress-stain
curve for material properties and divided the load into 10 steps applied as
incrementally. The maximum displacement of 1.04 inches was obtained from
Model 3A (Figure 3.2) and 1.12 inches for the Model 3B as shown in Figure 3.3.
In order to verify the results obtained from the finite element analysis, the
deflection equation (Lindeburg & Baradar 2001) for was used. The equations
used to calculate the lateral displacement without the P-delta effect were the
following:
Where:
H = lateral force
L = height of the structure
13
E = modulus of elasticity of material
I = moment of Inertia of structural element
The equation was used to calculate the lateral displacement with P-Delta effect
was shown as following (ACI, 2008):
Where:
Q = modification factor
P = vertical force
∆1 = lateral displacement without P-Delta effect
The displacement calculated by using the deflection equations matched the
results obtained from the finite element analysis.
14
Figure 3.1 Three Dimensional Views of Model 3A & 3B.
2 Kips Incremental Loads
Fixed
20 Kips Vertical Load
Width of Model: 10 Inches
Length of Model: 10 inches
14 Feet
15
Figure 3.2 Maximum Lateral Displacement of Model 3A.
Fixed
Maximum Lateral Displacement
14 Feet
16
Figure 3.3 Maximum Lateral Displacement of Model 3B.
Maximum Lateral Displacement
With P-delta Effect
Fixed
17
CHAPTER 4
FINTE ELEMENT MODEL DEVELOPMENT
4.1 Introduction
In order to investigate the P-delta and soil-structure interaction
effects on the bridge pier a total of four models were constructed. In these
models, all the key factors were taken into account i.e., dimensions, material
properties, loads, and boundary conditions. All the keys factors were taken base
on real life applications. At the end, varies models would allow compare and
check the final results.
4.2 Dimensions
The dimensions of the bridge superstructure were obtained from the
design example (U.S. Dept. of Transportation, Federal Highway Administration,
1996). The detailed dimensions for the pier were shown in Figure 4.1 (note the
thickness of the pier did not show here). The thickness of the pier is taken as 3
feet. In addition, the height of the pier for all three models was modified to 50
feet.
The focus of this study is to show how soil-structure interaction affected
respect to the bridge pier. In addition, for this study the footing was redesigned
18
in according to the needs of the study. The footing was redesigned in a way
according to the load that used in this study. Therefore no uplift pressure would
occur in this case. The final dimensions of the footing selected are as follows:
length as 22 feet, width as 20 feet, and the thickness as 32 inches. The footing
dimensions are shown in Figure 4.2.
The finial model in this study involved a soil profile. The dimensions of the
soil profile were based on the recommendation from the soil pressure bulb study
in chapter 2 and the pressure bulb beneath the strip footing and square footing
from Day (1999). Therefore, the width and depth of the soil profile were decided
as follows: 3 times the width of the footing on each side of the footing and 6
times the width of the footing in the vertical direction.
Figure 4.1 Section View of Design Example No 6, Bridge Pier Details. (ASSHTO
1997)
19
Figure 4.2 Footing Dimensions used for Finite Element Models.
4.3 Material Properties
Apart from the dimensions, other factors (such as material properties) are
also essential when analyzing the models. Material properties were assigned to
each model accordingly. In this study, an f’c equal to 4000 psi concrete was
used for the bridge pier and 3000 psi for the footing. One of the objectives of this
research is to study soil-structure interaction. In order to meet this objective, a
soil that is strong enough to support the load without failure was chosen. A very
stiff soil was determined to be appropriate for this study. The shear strength of
the soil was choosing to be 4 kip per square foot. The allowable bearing capacity
20
was calculated to be 9108 kips based on the soil properties and the applied loads
in this study. Before using this soil in the study, the assumption was made that
the soil condition was over consolidated and undrained. The material properties
(Modulus of elasticity and Poisson’s ratio) are shown in Table 4.1.
Table 4.3.1 Materials Properties for Finite Element Models. Materials Properties
Pier Footing Soil Modulus of elasticity (ksi) 3640 3150 10.4 Poisson's ratio 0.2 0.2 0.4
4.4 Loads
Three different loading scenarios were applied to the models. Each
model was subjected to a lateral and vertical load. Wind loads were used as
lateral load in this study. Live and dead loads were use as vertical load. The
wind load serves as a lateral load applied to the side of the pier. The lateral load
was determined using the ASSHTO (Standard Specifications for Highway
Bridges, 17th Edition) as reference and was calculated to be 50 kips. In addition
to lateral load, live and dead loads were also calculated according to the
ASSHTO guidelines. The total vertical load was calculated to be 1137 kips. In
order to investigate the P-Delta effect, the 50 kips lateral load was divided into 10
load steps (in 5 kips increment) and applied to the models. Table 4.2 shows the
applied loads for the models.
21
Table 4.4.1 Applied Loads for Finite Element Models.
Applied Loads
Model’s 50 Kips Incremental 1137 Kips Vertical
Load Horizontal load Model 4A
Yes No (No P-Delta Effect) Model 4B
Yes Yes (with P-Delta Effect) Model 4C
Yes Yes (with Soil Profile)
(with P-Delta Effect)
4.5 Boundary Conditions
Boundary conditions were the last input parameters before the analysis
take place. All three different models were restrained at the bottom base on
model configurations. For Model 4A and Model 4B, since both models were
consider the pier only therefore restrains were applied the bottom of the pier. For
model 4C, restrain are applied throughout the bottom soil profile.
4.6 Finite Element Models for the Pier
Dimensions, material properties, loads and boundary conditions were
determined. A total of three models were constructed using the finite element
program “NISA/DISPLAY IV” (1999). NISA was used to analyze the models. In
the process of model analysis, the program took all the input data such as
22
dimensions, material properties, loads and boundary conditions into account.
The three different models referred to as Model 4A, Model 4B, Model and Model
4C are shown in Figures 4.6.2 through 4.6.5.
Figure 4.6.1 Finite Element Model for the Pier.
Length of the Pier: 5.5 Feet
Width of the Pier: 3 Feet
50 Feet
38 Feet
23
Figure 4.6.2 Three Dimensional View of Finite Element Model 4A.
50 Kip Incremental Loads
Fixed
24
Figure 4.6.3 Three Dimensional View of Finite element Model 4B.
1137 Kips Vertical Loads
50 Kips incremental Lateral Loads
Fixed
25
Figure 4.6.4 Side View of Finite Element Model 4C.
22’x 20’ Footing
Fixed at bottom
50 Kips incremental Lateral Load
1137 Kips Vertical Loads
120 Feet
53.5 Feet
144 Feet
26
Figure 4.6.5 Three Dimensional View of Finite Element Model 4C.
4.7 RESULTS
Three models with a variety of parameters were constructed for
analysis in this study. All the results were obtained and reviewed for further
discussion. The focus of this study was to investigate how P-Delta and soil-
structure interaction effects could affect lateral displacement. The maximum
1137 Kips Vertical Loads
50 kips incremental
Lateral Loads
140 Feet
27
displacements were taken from each model to determine whether the effects
were significant. The results were obtained from the finite element analyses for
all three different models and are shown from Figure 4.7.1 to Figure 4.7.3. The
displacement of Model 4A was due to a lateral load only. For Model 4B,
displacement was caused by P-Delta effect. Model 4C was due to both P-Delta
and soil-structure interaction effects with a soil profile and normal footing. Since
applied lateral loads for Model 4C was divided into 10 load steps, therefore the
maximum lateral displacement were also obtained in each steps. Table 4.7.1,
Figure 4.7.4 and Figure 4.7.5 were created base on the results obtained from
Model 4C.
Table 4.7.1 Maximum Lateral Displacement at Each Step for Model C.
Pseudo Time
(second)
Maximum Lateral Displacement
(inches)
Applied Lateral Loads
(Kips) 1 0.1835 5 2 0.3696 10 3 0.5586 15 4 0.7503 20 5 0.945 25 6 1.143 30 7 1.343 35 8 1.547 40 9 1.754 45 10 1.965 50
28
Figure 4.7.1 Maximum Lateral Displacement of Model 4A.
Maximum Lateral
Displacement
Fixed
29
Figure 4.7.2 Maximum Lateral Displacement of Model 4B.
Fixed
Maximum Lateral Displacement
30
Figure 4.7.3 Maximum Lateral Displacement of Model 4C.
Maximum Lateral
Displacement Footing
Soils
Fixed at the Bottom Soil Profile
31
Figure 4.7.4 Maximum Lateral Displacement of Model 4C at the Footing.
Footing
Soil
The Maximum Lateral Displacement of Model
4C : at the Right Side of the Footing.
32
Figure 4.7.5 Model C: Time vs. Maximum Lateral Displacements.
0
0.5
1
1.5
2
2.5
0 2 4 6 8 10 12
Max
imiu
m L
ate
ral D
isp
lay
(in
che
s)
Time (Second)
Time vs. Maximum Lateral Displacements
33
Figure 4.7.6 Model C: Maximum Lateral Displacement vs. Loads.
0
10
20
30
40
50
60
0 0.5 1 1.5 2 2.5
Ap
plie
d L
oad
s(k
isp
)
Maximum Lateral Displacements (inches)
Maximum Lateral Displacement vs. Loads
34
Figure 4.7.7 Deformed Geometry of Model 4C.
Original Shape of the Pier
Deformed Shape of the Pier Notes: Deformations were modified to 5 times larger.
35
CHAPTER 5
DISCUSSION
All results obtained from the previous chapter are discussed in this
chapter. The results of this study show whether or not the P-Delta effect and the
soil-structure effect are significant. The maximum lateral displacements obtained
from finite element program for each models were shown in Table 5.1.
Table 5.1 Comparison of Maximum Lateral Displacements.
Model’s
Maximum Lateral-
Displacement Applied Forces Soil Under
Footing
Percent Different
Over Model 4A
Model 4A 1.27 Inches Lateral Force only
None
0% (Pier base
fixed)
Model 4B 1.35 Inches
Lateral and Vertical Force
None
6% (Pier base
fixed) Model
4C 1.97 Inches Lateral and
Vertical Force Yes 55%
Table 5.1 presents the maximum lateral displacement obtained from the
finite element program for all three different pier Models. The maximum Lateral
displacement (X-displacement) for the model 4A (due to lateral force) was 1.27
inches, without considering P-Delta effect and soil-structure interaction. Model
4B took the P-Delta effect into account and the maximum lateral displacement
was calculated as 1.35 inches which is increased 6% over Model 4A. The
36
method used for calculation of P-Delta effect on Model 4B was typical method
used by most engineers as well. The displacements obtain by only consider the
pier may be sufficient for the many designs. However, it is more important to
study exact displacement of the entire bridge pier. Therefore any factors that
could affect the displacement should be taken into consideration.
Model C was constructed while taking both soil-structure interaction and
the P-delta effects into account. The maximum lateral displacement of 1.97
inches was obtained at the top of the pier which is a 55% increase over Model
4A. It is evidence that the maximum lateral displacement of Model 4C was due
to three factors: the displacement at footing, rotation of footing, and P-Delta
effects. To identify the displacement due to each individual factor, two of the
displacements must distinguish. In this case the lateral displacement due to the
displacement at footing and P-delta effect were obtained from the finite element
program. Since the overall lateral displacement is 1.97 inches by subtracting the
1.35 inches which cause by the P-delta effect, then the displacement cause by
two other factors can be calculated. The 0.62 inches is displacement due to
displacement of footing and rotation of the pier. The lateral displacement of 0.01
inches as the displacement of footing due the 50 kips horizontal load inches at
the bottom right of footing as shown on Figure 4.7.4. Therefore the 0.61 inches
was determined that due to rotation of footing. The lateral displacement was
relatively small.
In order for the model to experience the P-delta effect, the applied 50 kips
lateral load was divided into 10 load steps and in a 5 kips incremental to the
37
structure. Therefore as the results, the maximum lateral displacement in each
steps were also obtained. Figure 4.7.5 and Figure 4.7.6 were generated to
investigate the change of the displacements at different time and different applied
loads. Both figures show the maximum lateral displacements increase in a way
that close to linear fashion as time and load increased.
38
CHAPTER 6
CONCLUSION
Three Finite element models were presented for the non-linear analysis of
bridge piers subjected to different loadings. Two of models were taken into
account for soil structure interaction effect. In additional, four simple finite
element models were created and two mathematical equations (maximum
deflection equation and immediate settlement equation) were used for
verifications to finite element program used in this study.
The results show a significant increase of lateral displacement when soil –
structure interaction was taken into account in this case. P-Delta effect
determined was not as significant as soil-structure interaction in this study.
However, soil-structure interaction effect was confirmed it has significant impact
on the overall lateral displacement of the bridge pier. The maximum lateral
displacement was concluded that due to lateral load on the bridge pier and the
influent between the footing and soils. This study suggested that the lateral
displacement due to soil-structure interaction effect should be taken into
consideration in the bridge design procedure. Overall, the maximum lateral
displacements can be predictable based on the results of this study.
39
REFERENCES
American Association of State Highway officials. (2002). Standard Specifications for Highway Bridges. (17th Edition). Washington, D.C: American Association of State Highway Officials.
American Concrete Institutes. (2008). Building Code Requirements for Structural Concrete (ACI 318-08) and Commentary Farmington Hills, MI: American Concrete Institutes. Das.B. (2007). Principles of Foundation Engineering. (6th edition). Toronto, Ontario: Thomson Canada Limited. Day, R. (1999). Geotechnical and foundation engineering: design and Construction. NY: McGraw-Hill Dobson, R. (2002). An overview of P-Delta Analysis. CSC Software and
Solutions for structural Engineers. Retrieved April 9, 2010, from http://www.cscworld.com
Dobson, R. & Kenny, A. (2002) A brief overview of 2nd order (or P-Delta) Analysis. CSC Software and Solutions for structural Engineers. Retrieved April 9, 2010, from http://www.cscworld.com Ger, J., & Yen. P. (2004). Nonlinear Static Analysis of Bridge Bents by Finite Segment Method. Structures Congress and Exposition. 351-359. Huang, M. S., & Zhu, H. P. (2008). Finite Element Model Updating of Bridge Structure Based on Sensitivity Analysis and Optimization Algorithm. Journal of Natural Sciences. 13(1). 87-92. Lindeburg, M., & Bardar, M., (2001). Seismic Design of Building Structures: a professional’s introduction to earthquake forces and design detail. (8th edition). Belmont, CA: Professional Publications, Inc. Main. J. (2004). Seismic Analysis of a Suspension Bridge Model. Retrieved
April 9, 2010, from http://www.ce.jhu.edu/jmain NISA/DISPLAY IV [Computer Software]. (1999). Troy, MI: Engineering Mechanics Research Corporation.
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Poston, R. W. (1986). Nonlinear Analysis of Concrete Bridge Piers. Journal of Structural engineering, 112 (9), 2041-2056 Retrieved from http://ascelibrary.aip.org
U.S Department of Transportation, Federal Highway Administration. (1996). Seismic design of bridge, design example no. 6: three-span continuous CIP concrete bridge.
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VITA
Graduate School
Southern Illinois University
Toun Loin. Wu Date of Birth: March 4, 1985 2627 S Princeton Ave, Chicago, IL 60616 [email protected] Southern Illinois University Carbondale Bachelor of Science, Civil Engineering, December 2008 Special Honors and Awards: Dean’s List Thesis Title:
THE P-DELTA AND SOIL-STRUCTURE INTERACTION EFFECTS ON BRIDGE PIERS.
Major Professor: Dr. J. Kent Hsiao
