dynamic response of plates and buried structures

Graduate Theses, Dissertations, and Problem Reports

2005

Dynamic response of plates and buried structures Dynamic response of plates and buried structures

Chee Heong Tee West Virginia University

Follow this and additional works at: https://researchrepository.wvu.edu/etd

Recommended Citation Recommended Citation Tee, Chee Heong, "Dynamic response of plates and buried structures" (2005). Graduate Theses, Dissertations, and Problem Reports. 1883. https://researchrepository.wvu.edu/etd/1883

This Thesis is protected by copyright and/or related rights. It has been brought to you by the The Research Repository @ WVU with permission from the rights-holder(s). You are free to use this Thesis in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you must obtain permission from the rights-holder(s) directly, unless additional rights are indicated by a Creative Commons license in the record and/ or on the work itself. This Thesis has been accepted for inclusion in WVU Graduate Theses, Dissertations, and Problem Reports collection by an authorized administrator of The Research Repository @ WVU. For more information, please contact [email protected].

https://researchrepository.wvu.edu/

https://researchrepository.wvu.edu/

https://researchrepository.wvu.edu/etd

https://researchrepository.wvu.edu/etd?utm_source=researchrepository.wvu.edu%2Fetd%2F1883&utm_medium=PDF&utm_campaign=PDFCoverPages

https://researchrepository.wvu.edu/etd/1883?utm_source=researchrepository.wvu.edu%2Fetd%2F1883&utm_medium=PDF&utm_campaign=PDFCoverPages

mailto:[email protected]

Dynamic Response of Plates and Buried Structures

By

Chee Heong Tee

Thesis submitted to the College of Engineering and Mineral Resources

at West Virginia University in partial fulfillment of the requirements

for the degree of

Master of Science in

Civil Engineering

Roger H.L. Chen, Ph.D., Chair H. Ilkin Bilgesu, Ph.D. Lian-Shin Lin, Ph.D.

Department of Civil and Environmental Engineering

Morgantown, West Virginia 2005

Keywords: Dynamic Response, Plates, Buried Structures, Soil-Structure Interaction, Finite Element Analysis

ABSTRACT

Dynamic Response of Plates and Buried Structures

CHEE HEONG TEE

Soil is a particulate material, which provides a unique behavior when interacting with its adjacent structure. Emphases of this study are placed on the simulation of the behavior of a target plate and a buried structure. An analytical modeling technique, explicit Finite Element Method (FEM) by ABAQUS was used to simulate the dynamic responses of the plate and buried structure: 1) a plate resting on elastic half space under impact loading; 2) buried structure under a shock impulse system.

In the first analysis, a free-drop impact system was considered to generate the dynamic loading on the plate free surface. Two FEM models were built, one with a slide line underneath the target plate and another one without the slide line. The numerical results (FEM) of the radial strain at the bottom of the target plate were compared with the experimental measurement. The numerical results show good agreement with the experimental results.

In the second analysis, a cylindrical buried structure was considered, and low-velocity impact loadings were generated into the soil. A FEM model with slide line underneath the target plate and each sand layer was built. Under dynamic loading, the buried structure and the soil medium will be traveling at different speeds and will separate from one another at certain moment in time. By modeling soil and structure as separate entities and allowing soil/structure separation, the numerical models are shown to have good correlation with experimental observation of the peak displacement of the buried roof. The results show that smaller amounts of sand-layers beneath the aluminum plate experienced larger displacements but shorter durations than with large amounts of sand-layers.

DEDICATION…

This thesis is dedicated to my parents, sisters, brother,

sister-in-law, niece, and friends for their never-ending love,

support and encouragement.

iii

ACKNOWLEDGEMENT

The author wishes to express his sincere gratitude to his advisor,

Professor Roger H. L. Chen, who provided valuable support and guidance

throughout the duration of his study. He is particularly appreciative for his

enthusiasm and encouragement in the preparation of this thesis. Appreciations

are extended to Professor H. Ilkin Bilgesu and Professor Lian-Shin Lin, members

of committee, for their constructive comments, suggestions and helps in this

research.

The author would like to thank his friends, Ziheng Yao and Jeong-Hoon

Choi, for their helps in FEM modeling. The author also offers grateful thanks to

his friends, Joseph Sweet, Yong Zhang, Mei Zhao and Ying Wooi Wan, for their

friendship, supports and helps in preparation of this thesis. Appreciation is also

extended to the host family that the author had stayed with, Mr. Vincent LaFata

and his lovely family for their kindness and encouragement.

The final note of appreciation is reserved for his parents, Mr. Ah Keong

Tee, Mrs. Siew Hong Lee Tee, for their endless love, understanding and support,

and to his sister, Hui Lian Tee, his brother, Chee Seng Tee and his little sister

Angel Tee for their understanding and encouragement.

iv

TABLE OF CONTENTS

ABSTRACT........................................................................................................... ii

ACKNOWLEDGEMENT ...................................................................................... iv

TABLE OF CONTENTS........................................................................................ v

LIST OF TABLE..................................................................................................viii

LIST OF FIGURES .............................................................................................. ix

CHAPTER I...........................................................................................................1

INTRODUCTION ..................................................................................................1

1.1. GENERAL REMARKS ...............................................................................1

1.2. OBJECTIVES AND SCOPE.......................................................................3

1.3. LAYOUT.....................................................................................................4

CHAPTER II..........................................................................................................5

LITERATURE REVIEW ........................................................................................5

2.1. GENERAL REMARKS ...............................................................................5

2.2. PLATE ON ELASTIC HALF-SPACE ..........................................................5

2.3. BURIED STRUCTURES ............................................................................8

2.4. WAVE PROPAGATION IN SOIL MEDIUM ................................................9

2.5. SOIL-STRUCTURE INTERACTION (SSI)................................................10

2.6. FINITE ELEMENT METHOD....................................................................12

2.6.1. History of Nonlinear Finite Elements..................................................12

CHAPTER III.......................................................................................................15

ANALYSIS ..........................................................................................................15

3.1. GENERAL REMARK................................................................................15

3.2. INTRODUCTION OF ABAQUSTM SOFTWARE .......................................15

3.3. ABAQUS/STANDARD AND ABAQUS/EXPLICIT ....................................16

3.4. EXPLICIT DYNAMIC ANALYSIS .............................................................16

3.4.1. Integration Time Increment ................................................................17

3.5. AXISYMMETRIC ELEMENTS..................................................................18

3.6. DAMPING.................................................................................................19

3.6.1. Material Damping...............................................................................19

v

3.6.1.1. Alpha Damping Factor ....................................................................19

3.6.1.2. Beta Damping Factor ......................................................................19

3.6.1.3. Composite Damping Factor ............................................................20

3.6.2. Contact Damping ...............................................................................20

3.7. CONTACT MODELING............................................................................21

3.7.1. Slide line and Non-slide line...............................................................21

3.7.2. Small-Sliding Interaction between Bodies..........................................21

3.7.3. Finite-Sliding Interaction between Deformable Bodies ......................22

3.7.4. Contact Interaction Property ..............................................................22

CHAPTER IV ......................................................................................................24

DYNAMIC RESPONSE OF PLATE RESTING ON SAND ..................................24

4.1. GENERAL REMARKS .............................................................................24

4.2. PLATE RESTING ON SAND....................................................................24

4.3. DISCUSSION ON RESULTS ...................................................................28

4.4. COMPARISON OF FEM RESULTS AND EXPERIMENTS......................29

4.4.1. ABAQUS FEM (with non-slide line) Strains, SAP4 FEM and

Experiments.................................................................................................29

4.4.2. ABAQUS FEM (with slide line) Strains and Experiments...................30

4.5. CONCLUSION .........................................................................................31

CHAPTER V .......................................................................................................42

DYNAMIC RESPONSE OF BURIED STRUCTURES.........................................42

5.1. FINITE ELEMENT MODELING OF BURIED STRUCTURES ..................42

5.2. DISCUSSION OF RESULTS....................................................................43

5.3. COMPARISON OF EXPERIMENTAL AND NUMERICAL RESULT.........46

5.4. CONCLUSION .........................................................................................47

CHAPTER VI ......................................................................................................73

CONCLUSIONS..................................................................................................73

CHAPTER VII .....................................................................................................75

RECOMMENDATIONS.......................................................................................75

REFERENCES ...................................................................................................76

APPENDIX A ......................................................................................................79

vi

ABAQUS BASICS...............................................................................................79

APPENDIX B ......................................................................................................80

ABAQUS/CAE MODULES..................................................................................80

APPENDIX C ......................................................................................................82

INPUT MATERIALS FOR PLATE.......................................................................82

APPENDIX D ......................................................................................................84

INPUT MATERIALS FOR BURIED STRUCTURE..............................................84

VITA....................................................................................................................87

vii

LIST OF TABLE

Table 4.1. Material Properties (Chen, 1988) .......................................................27

viii

LIST OF FIGURES

Figure 2.1. Attenuation of Stress Wave with Depth (Newmark, 1964) ................14

Figure 4.1. Idealization of the Impact Loading (Chen et al., 1996)......................32

Figure 4.2. Finite Element Mesh for Circular Plate on Elastic Half-Space ..........33

Figure 4.3. Loading Function for Impact Duration = 0.238 ms ............................34

Figure 4.4. Deformed State of slide line model of Aluminum Plate Separated from

Sand at 0.000775 sec .........................................................................................35

Figure 4.5. Radial Strain of Circular Plate at r = 12.7 mm with material damping

factors α (mass proportional damping) and β (stiffness proportional damping)

(slide line model).................................................................................................36

Figure 4.6. Radial Strain of Circular Plate at r = 12.7mm with different contact

damping (slide line model) ..................................................................................37

Figure 4.7. Radial Strain of Circular Plate at r = 12.7 mm with different α (mass

proportional damping) damping factors (slide line model)...................................38

Figure 4.8. Radial Strain at r = 12.7 mm from Center of Target Plate (with non-

slide line).............................................................................................................39

Figure 4.9. Radial Strain at r = 12.7mm from Center of Target Plate with contact

damping of 20%, and no α and β damping factors (with slide line).....................40

Figure 4.10. Radial Strain of Circular Plate at r = 12.7 mm (Chen, 1990)...........41

Figure 5.1. Finite Element Mesh for Elastic Buried Structure..............................49

Figure 5.2. Buried Structure with 3 layers below the plate ..................................50

Figure 5.3. Buried Structure with 6 layers below the plate ..................................51

Figure 5.4. Buried Structure with 12 layers below the plate ................................52

Figure 5.5. Loading Function for Loading Duration = 0.77 ms ............................53

Figure 5.6. Deformed State of Buried Structure at 0.00069 sec .........................54

Figure 5.7. Close Up of Buried Structure Separated from Aluminum Plate and

Sand (12 Layered) at 0.00102 sec......................................................................55

Figure 5.8. Close Up of Aluminum Plate Separated from Sand at 0.0054 sec....56

Figure 5.9. Displacement at the center of the roof of buried structure (no layered)

with different kind of damping .............................................................................57

ix

Figure 5.10. Numerical Results in Comparison with Experimental Results with α

damping factor of 1650 s-1 and contact damping of 30% (Displacement at the

center of the roof of buried structure with different layers beneath the target plate)

............................................................................................................................58

Figure 5.11. Displacement at the center of the roof of buried structure (12

layered) with different α (mass proportional damping) damping factors and

contact damping of 20% .....................................................................................59

Figure 5.12. Displacement at the center of the roof of buried structure (12

layered) with α (mass proportional damping) damping factor of 1500 s-1 and

different contact damping....................................................................................60

Figure 5.13. Time History of the Stress Field with a Buried Structure (Nonlinear

Analysis): (a) Stress at Bottom of Target Plate; (b) Stress along z-Axis; (c) Stress

at Buried Roof .....................................................................................................62

Figure 5.14. Time History of the Stress Field with a Buried Structure (Linear

Analysis): (a) Stress at Plate’s Bottom; (b) Stress along z-Axis (Chen, 1990); (c)

Stress at Buried Roof (Chen, 1990) ....................................................................64

Figure 5.15. Normal Stress along Bottom of Target Plate (Nonlinear Analysis)..65

Figure 5.16. Schematic Plot of Normal Stress along Bottom of Target Plate

(Linear Analysis) (Chen, 1990) ...........................................................................66

Figure 5.17. Vertical Pressure along Buried Roof (Nonlinear Analysis) ..............67

Figure 5.18. Schematic Plot of Vertical Pressure along Buried Roof (Linear

Analysis) (Chen, 1990) .......................................................................................68

Figure 5.19. Displacement at the center of the roof of buried structure with

different Poisson’s ratios of sand, α (mass proportional damping) damping ratio

of 1650 s-1 and contact damping of 30%.............................................................69

Figure 5.20. Numerical Results in Comparison with Experimental Results

(Displacement Measurement) .............................................................................70

Figure 5.21. Displacement at the center of the buried roof and the sand on top of

buried roof (12 layered model) ............................................................................71

Figure 5.22. Displacement at the center of the aluminum plate and the sand

below the plate (12 layered model) .....................................................................72

x

Figure B.1. Selecting a module...........................................................................81

xi

CHAPTER I INTRODUCTION

1.1. GENERAL REMARKS

In the past three decades, dynamic soil-structure interaction problems for

underground protection structures under explosive threats have been

investigated extensively. It was found that the importance of the interaction

effects at burial depths as shallow as 20-50% of the clear span with dynamic

loading. In addition, a temporary load relief was experienced by the roof of the

buried structures, which actually increased the protective capacity of the buried

structures.

Understanding such interaction has extensive applications in improving

the design of buried structures that are likely subjected to transient loading. The

load relief is related to the interaction process between the soil medium and the

embedded structure, which may involve several coupled mechanism: the wave

propagation through the soil medium, the separation between the soil and the

structure, the dynamic arching within the soil medium, the tension waves

reflected from bottom of the buried roof, the rigid-body displacement of the buried

structures.

An impulsive loading can be produced by impact where the duration of the

impulse is on the order of microseconds. The particular problem of a circular

1

aluminum plate impacted at its center by a steel ball was considered in this

study. The contact duration as well as the radius of the contact area were

measured and compared with an estimation based upon a Hertzian contact

assumption (Chen and Chen, 1996). A review of the works related to

experimental studies of wave propagation due to low velocity impact was given

by Al-Mousawi (1986). A detailed study of low velocity impact of circular plates

was conducted by Greszczuk (1982).

Soil, which is particulate in nature, is composed of “a system of discrete

mineral particles more or less free to move relative to one another, subjected to

the forces of adhesion and friction between the particles and to the geometric

constraints imposed by the arrangement of the particles”. (Whitman, 1970) Sand,

a type of soil, is principally produced by the grinding action of waves.

The dynamic response of an elastic plate resting on sand is a practical

topic for investigation that covers a wide range of applications. First, the plate

can be viewed as a footing of a structure, which has significance for foundation-

vibration studies. Secondly, the finite plate response due to an impact force is a

fundamental problem in structural design. Moreover, investigating the impact

force itself is also important and interesting since an accurate knowledge of the

live load on the impacted structure enables a better design and prediction of the

damage to the structure. Finally, it is important to understand fully the impact

loading transmission through the soil.

Earlier, Chen et al. (1990) used SAP4 a linear-elastic finite element model

to solve the plate bonded with sand. Recently, more advanced finite element

2

modeling programs like ABAQUS, FEMAP, and ANSYS have been developed.

Using ABAQUS to model the separation of the plate and sand, and compare the

result with the experimental result can be more accurate than using SAP4 since

the plate actually separated from sand during the experiment.

1.2. OBJECTIVES AND SCOPE

The objective of this research is to use ABAQUS software to model the

separation between the plate and the sand, and between the buried structure

and the surrounding soil medium. Therefore, the objectives of this research are:

1. To establish finite element modeling by ABAQUS to account for the

separation of aluminum plate and sand, and compare the result

from ABAQUS with the previous results from SAP4 analysis and

experiment by Chen et al. (1990).

2. To establish finite element modeling by ABAQUS to solve the

separation of aluminum plate, sand and buried structure, and

compare the results with the previous findings from linear-elastic

analysis and experiments by Chen et al. (1990).

3

1.3. LAYOUT

This thesis is organized in a format such that different approaches and

modeling techniques are presented in separate chapters: A broad but brief

review of related literature is presented in Chapter 2; Chapter 3 describes the

analysis; Chapter 4 describes the dynamic response of plate resting on sand;

Chapter 5 describes the dynamic response of buried structures; Chapter 6 is the

conclusions; Chapter 7 is the recommendations.

4

CHAPTER II

LITERATURE REVIEW


This chapter will discuss: 1. Plate on Elastic Half-Space; 2. Buried

Structures; 3. Wave Propagation in Soil Medium; 4. Soil-Structure Interaction; 5.

Finite Element Method.

2.2. PLATE ON ELASTIC HALF-SPACE

The dynamic response of a plate on viscoelastic medium has drawn the

attention of many researchers. In Lin’s (1978) paper, “dynamic response of

circular thin plates resting on viscoelastic half space”, he studied the dynamic

response of circular thin plates resting on viscoelastic half space under the

conditions of harmonic vertical and rocking excitations. He assumed that the

contact between the plate and the surface of the half space is frictionless. The

dynamic mixed boundary- value problem leads to a set of dual integral equations.

By numerical procedures, those equations are reduced to Fredholm’s integral

equations of the second kind. The foundation is treated as an elastic thin plate to

account for the flexibility of the foundation. The viscoelastic half space medium is

5

assumed to be homogeneous and isotropic; the viscosity is taken into account by

means of the complex shear modulus.

He obtained the following results:

The flexibility of the plate (the δ value) strongly affected the stiffness

coefficients, or the real parts of the impedance function. The damping

coefficient, or the imaginary part, is approximately independent of the δ

value.

For rocking excitation, both stiffness and damping coefficients are almost

independent of the δ value.

The maximum deflection of the plate becomes larger when the excitation

frequency increases. In his case, Lin found the amplification factor for the

vertical excitation may be as large as 20, but for rocking excitation, less

than 3.

For rocking excitation, the phase angles are approximately independent of

the δ value.

Iguchi and Luco (1982) also studied the dynamic response of a massless

flexible circular plate with a rigid core supported on a layered viscoelastic half-

space under the conditions of harmonic vertical and rocking excitation. In their

study, they considered the out-of-plane deformation of the plate when harmonic

vertical forces and rocking moments are applied on the rigid core. They also

assumed that the relaxed contact condition between the plate and the underlying

half-space is frictionless. But slippage between them is allowed. In the mean

time, they ignored the mass of the plate, and attenuations in the half-space are

6

included by use of complex shear moduli. Similarly, solutions to the mixed

boundary-value problem are reduced to Fredholm’s integral equations of the

second kind by using numerical method. The emphasis is given to the description

of the effect of flexibility of the plate and relative size of the internal rigid core on

the force-displacement relationships, total motion of the plate, and contact

stresses.

By analysis of the dynamic response, they evaluate the effects of flexibility

of the plate on the impedance functions: the out-of-plane motion of the plate, and

the contact stress distribution at the interface between the plate and the

underlying half-space. In regards to the impedance functions, they analyze two

conditions: homogenous half-space and layered half-space. In both half-space

conditions, the same conclusion can be drawn: the flexibility of the plate leads to

a significant reduction of the equivalent radiation damping coefficients for both

vertical and rocking vibration. By analyzing the vertical displacement, they

concluded that at low frequencies the equivalent stiffness coefficients for a

flexible plate are lower than those for a Regis disk. At high frequencies this trend

is reversed. Finally, they found that the distribution of contact stresses is highly

sensitive to the flexibility of the plate. In the case of a flexible circular plate with

an internal rigid core, the concentration of contact stresses migrates from the

edge of the plate to the vicinity of the rigid core as the flexibility of the plate

increases.

7

2.3. BURIED STRUCTURES

Specific soil-structure interaction problems for underground protection

structures under explosive threats have been received considerable attention in

the past four decades. It was found that the importance of the interaction effects

at burial depths as shallow as 20-50% of the clear span with dynamic loading. In

addition, a temporary load relief was experienced by the roof of the buried

structures, which actually increased the protective capacity of the buried

structures.

Understanding such interaction has extensive applications in improving

the design of buried civil structures that are likely subjected to transient loading.

The load relief is related to the interaction process between the soil medium and

the embedded structure, which may involve several coupled mechanism: the

wave propagation through the soil medium, the separation between the soil and

the structure, the dynamic arching within the soil medium, and the tension waves

reflected from bottom of the buried roof and the rigid-body displacement of the

buried structures.

Baylot (2000) used dynamic finite-element analyses, which including the

detonation of high explosives in the soil, to determine the changes in soil flow

and structure loads. H.L. Chen and S.E. Chen (1996) investigated a shallow-

buried flexible plate under impact loading. They observed different degrees of

load-relief phenomenon for the buried plates of three different stiffness. They

8

used a decoupled SDOF model to simulate the dynamic response and got a

good agreement with the experimental results.

Chen et al. (1990) also studied the dynamic response of shallow-buried

cylindrical structures. They used a small-scale model test to simulate some of the

major aspects of the full-scale field observations. Through small-scale model

testing, the generation of an impulse loading on the free surface can be well

controlled; thus, the measured displacement at the center of the buried structural

roof and the corresponding radial strain can be predicted by a linear-elastic

analysis with approximately 75% accuracy of both the peak amplitude and the

peak arrival.

2.4. WAVE PROPAGATION IN SOIL MEDIUM

Explosions at ground zero cause surface cratering and air-blasting, which

result in stress waves propagating into the soil medium. The wave propagation

phenomenon through the soil medium has been addressed by Newmark (1964),

Selig (1964) and Ginsburg (1964). The stress wave behavior is dependent on the

wave velocity as well as the seismic velocity of the soil. The vertical stress wave

is subjected to attenuation through depth, due to the hysteretic nature of soil. The

depth effect on the pressure wave resulted in longer risetime, lower peak and

longer decay of the wave (Figure 2.1). Using a burster slab on the soil surface,

the explosion resulted in a distributed loading pressure. The distribution of the

loading pressure may influence the responses of the buried structure.

9

The original Lamb’s solution by assuming a point load propagating

through an elastic halfspace has been presented by Graff (1975) and Achenbach

(1980). Using a linear-elastic FEM model, Chen et. al (1988) has shown that the

wave generated underneath the target plate will not create an evenly distributed

pressure and may have significant energy dissipation due to the geometric

dissipation effect. He found that the peak amplitude remaining upon arrival at the

buried structure is about 70% of the peak load beneath the target plate. Material

damping due to internal frictions may also occur during propagation through sand

medium. It is therefore essential to determine the wave propagation effects which

may result in some of the load-relief on the roof structure.

2.5. SOIL-STRUCTURE INTERACTION (SSI)

The problem of soil-interaction has always been a subject of research with

particular practical interest. Today, the problem may be solved theoretically with

refined analytical techniques.

Stavridis (2002) studied the analysis for an arbitrary static loading by using

stiffness matrices of the soil surface and of the structure with respect to their

contact nodes. He proposed a practical procedure for the evaluation of the state-

of-stress of any structure under a static loading, resting on an elastic soil

simulated by an elastic layered half-space.

Trifunac and Todorovska (2001) described an approximate model in which

the soil-structure interaction phenomena are viewed in simplest possible form,

10

via a rigid foundation model. They measured the instantaneous apparent

frequency by two methods: windowed Fourier analysis and zero-crossings

analysis

Budek et al. (2000) studied an inelastic finite-element analysis on the

structure using as the pile constitutive model the section moment-curvature

relationship based on confined stress-strain relationships for the concrete. Rizos

and Loya (2002) developed a direct time domain formulation for the analysis of

unbounded media and foundations that treats dynamic excitation and ground

motion in a uniform manner. Maeso et al. (2002) modeled the foundation rock as

a uniform viscoelastic boundless domain where the incident traveling wave field

is defined by its analytical expression, which may include any spatial variation.

Chen et al. (1988 and 1990) developed a small laboratory test using low-

velocity impact technique. This technique has several advantages over actual

field testing for the study of shallow-buried structures:

1. relatively inexpensive

2. well controlled dynamic loading

3. accessible dynamic loading and responses

Using the experimental technique on buried cylindrical structures, the effects of

dropping heights and different buried depths were studied. By comparing the

measured dynamic responses of the buried structure and the loadings at the

center of the buried structure, Chen et al. (1990) concluded that dynamic soil

arching, as a form of load release at the roof center, occurred.

11

2.6. FINITE ELEMENT METHOD

Finite element method (FEM) is the most commonly used method for

complicated soil-structure interaction. Wong and Weidlinger (1983) simulated the

SSI problem by using FEM to model the buried structure and used a decoupled

scheme to simulate the separation process. Zaman et al. (1984) proposed an

interface element for FEM modeling of SSI problems.

2.6.1. History of Nonlinear Finite Elements

Belytschko et al. (2000) described the methods of nonlinear analysis for

solid mechanics. Nonlinear analyses have shortened design cycles and

dramatically reduced the need for prototype tests. Oden’s (1972) work is

particularly noteworthy since it pioneered the field of nonlinear finite element

analysis of solids and structures. Ed Wilson’s liberal distribution of his first

program fueled the excitement in the 1960s. The first generation of these

programs had no name. SAP (Structural Analysis Program) was the second

generation of linear programs developed at Berkeley. NONSAP was the first

nonlinear program which evolved from this work at Berkeley, which had

capabilities for equilibrium solutions and the solution of transient problems by

implicit integration.

Argyris (1965) and Marcal and King (1967) were among the first papers on

nonlinear finite element methods. Pedro Marcal set up a firm to market the first

12

nonlinear commercial finite element program, MARC, in 1969, it is still a major

player. In the same year, John Swanson developed a nonlinear finite element

program, ANSYS, which for many years dominated the commercial nonlinear

finite element scene. David Hibbitt worked with Pedro Marcal until 1972, and then

co-found HKS, which markets ABAQUS. This program was one of the first finite

element programs to introduce gateway for researchers to add elements and

materials models. Jurgen Bathe launched his program, ADINA, shortly after

obtaining his PhD at Berkeley. The commercial finite element programs marketed

until about 1990 focused on static solutions and dynamic solutions by implicit

methods.

Costanito (1967) developed what was probably the first explicit finite

element program. Sam Key completed HONDO, which featured an element-by-

element explicit method. The first release of the DYNA code was by John

Hallquist in 1976.

13

Figure 2.1. Attenuation of Stress Wave with Depth (Newmark, 1964)

14

CHAPTER III

ANALYSIS

3.1. GENERAL REMARK

With the advent of modern computers and advanced numerical methods,

Finite Element Method has become a powerful tool to simulate various

indentation procedures and quite significant conclusions have been withdrawn

from the simulation results.

3.2. INTRODUCTION OF ABAQUSTM SOFTWARE

Commercial FEM simulation software package ABAQUSTM version 6.3 is

used in the modeling. ABAQUSTM is developed by Hibbitt, Karlsson & Sorensen,

Inc. It is a suite of engineering programs, based on the finite element method,

which can solve problems ranging from relatively simple linear analysis to

challenging nonlinear simulations.

ABAQUS consists of two main analysis modules –ABAQUS/Standard and

ABAQUS/Explicit. The ABAQUS/Explicit is mainly used in this study.

ABAQUS/CAE is the complete ABAQUS environment that includes capabilities

for creating ABAQUS models, interactively submitting and monitoring ABAQUS

15

jobs and evaluating results. In this study, the ABAQUS/CAE is used as the

preprocessor (Part, Property, Assembly, Step, Interaction, Load, Mesh, Job

modules) and postprocessor (Visualization module).

3.3. ABAQUS/STANDARD AND ABAQUS/EXPLICIT

ABAQUS/Standard is a general-purpose analysis module that can solve a

wide range of linear and nonlinear problems involving the static, dynamic,

thermal, and electrical response of components.

ABAQUS/Explicit is a special-purpose analysis module that uses an

explicit dynamic finite element formulation. It is suitable for modeling transient

dynamic events, such as impact and blast problems, and is also very efficient for

highly nonlinear problems involving contact conditions, such as creating

simulations.

3.4. EXPLICIT DYNAMIC ANALYSIS

The explicit dynamics analysis procedure in ABAQUS/Explicit is based

upon the implementation of an explicit integration rule together with the use of

diagonal or “lumped” element mass matrices.

16

3.4.1. Integration Time Increment

The explicit method integrates through time by using many small time

increments. The stability limit with no damping is given in terms of the highest

eigenvalue in the system as

max

2ω

≤∆t .

In ABAQUS/Explicit a small amount of damping is introduced to control high

frequency oscillations. With damping, the stable time increment is given by

)1(2 2

max

ξξω

−+≤∆t ,

where ξ is the fraction of critical damping in the highest mode.

In ABAQUS/Explicit, the time incrementation scheme is fully automatic

and requires no user interference. In general, constraints such as boundary

conditions and contact have the effect of compressing the eigenvalue range,

which the element by element estimates do not take into account.

ABAQUS/Explicit determines the maximum frequency of the entire model, and

initially uses the element by element estimates.

A conventional estimate of the stable time increment is given by the

minimum taken over all the elements. The above stability limit can be rewritten as

17

)min(d

e

cL

t =∆ ,

where is the characteristic element dimension and is the current effective,

dilatational wave speed of the material. The time increment has to be shorter

than the time required for wave to pass by one element. Time increment is also

relative to the resolution of the FEM output.

eL dc

3.5. AXISYMMETRIC ELEMENTS

Two kinds of solid element are included in ABAQUS, CAX and CGAX,

which are axisymmetric (bodies of revolution) and which can be subjected to

axially symmetric loading conditions. Also, CGAX elements support torsion

loading. Therefore, CGAX elements will be referred to as generalized

axisymmetric elements, and CAX elements as torsionless axisymmetric elements.

In both cases, the body of revolution is generated by revolving a plane cross-

section about an axis (the symmetry axis).

18

3.6. DAMPING

3.6.1. Material Damping

3.6.1.1. Alpha Damping Factor

The α factor introduces damping forces caused by the absolute velocities

of the model and so simulates the idea of the model moving through viscous

"ether" (a permeating, still fluid, so that any motion of any point in the model

causes damping). This damping factor defines mass proportional damping, in the

sense that it gives a damping contribution proportional to the mass matrix for an

element. If the element contains more than one material, the mass average value

of α is used to multiply the element's lumped mass matrix to define the damping

contribution from this term. α has units of (1/time).

3.6.1.2. Beta Damping Factor

The β factor introduces damping proportional to the strain rate, which can

be thought of as damping associated with the material itself. The factor β defines

damping proportional to the elastic material stiffness. Damping can be introduced

for any nonlinear case and provides standard Rayleigh damping for linear cases;

for a linear case stiffness proportional damping is exactly the same as defining a

damping matrix equal to β times the stiffness matrix. β has units of (time).

19

3.6.1.3. Composite Damping Factor

This parameter applies only to ABAQUS/Standard analyses. Set this

parameter equal to the fraction of critical damping to be used with this material in

calculating composite damping factors for the modes (for use in modal

dynamics). This value is ignored in direct-integration dynamics in

ABAQUS/Standard.

3.6.2. Contact Damping

ABAQUS/Explicit damping is applied only when the surfaces are in

contact. This option is used to define viscous damping between two interacting

surfaces. It must be used in conjunction with either the surface interaction, the

gap, or the interface. In ABAQUS/Explicit this option is used to damp oscillations

when using penalty or softened contact.

20

3.7. CONTACT MODELING

3.7.1. Slide line and Non-slide line

A slide line model is a model with sliding interaction. In other words, it is a

nonlinear analysis. When forming a FEM simulation, two parts were created. One

is aluminum plate, and the other one is sand. Also, two different material

properties were applied, aluminum and sand, respectively. Contact interactions

were included to model sliding contact of two bodies with respect to each other.

A non-slide model is a model without sliding interaction. In other words, it

is a linear analysis. Only one part was created in the FEM simulation, but two

different material properties were applied which were aluminum and sand. No

contact interaction was applied to the model.

3.7.2. Small-Sliding Interaction between Bodies

In ABAQUS/Standard a capability is included to model small-sliding

contact of two bodies with respect to each other. With this formulation the

contacting surfaces can go through only relatively small sliding relative to each

other, but arbitrary rotation of the bodies is permitted. The small-sliding capability

can be used to model the interaction between two deformable bodies or between

a deformable body and a rigid body in two and three dimensions. ABAQUS will

21

automatically cover a slave surface with the appropriate element type, based on

the nature of the corresponding master surface.

3.7.3. Finite-Sliding Interaction between Deformable Bodies

A finite-sliding formulation where separation and sliding of finite amplitude

and arbitrary rotation of the surfaces may occur. Depending on the type of

contact problem, two approaches are available to the user for specifying the

finite-sliding capability: (1) defining possible contact conditions by identifying and

pairing potential contact surfaces and (2) using contact elements. With the first

approach ABAQUS automatically generates the appropriate contact elements.

3.7.4. Contact Interaction Property

A contact interaction property can define tangential behavior (friction and

elastic slip) and normal behavior (hard, soft, or damped contact and separation).

Also, a contact property can include information about damping, thermal

conductance, thermal radiation, and heat generation due to friction. A contact

interaction property can be referred to by a general contact, surface-to-surface

contact, or self-contact interaction.

22

Surface-to-surface contact interactions describe contact between two

deformable surfaces or between a deformable surface and a rigid surface. Self-

contact interactions describe contact between different areas on a single surface.

For the contact modeling of this study, a contact interaction property were

used. Critical damping factor of 30 % was applied. Surface-to-surface contact

was chosen. Finite-sliding interaction was used, and penalty contact method for

mechanical constraint formulation was chosen.

23

CHAPTER IV

DYNAMIC RESPONSE OF PLATE RESTING

ON SAND


FEM method, ABAQUS, which is capable of handling separation, is used

for the current research. Nonlinear effect is included in the study, allowing

separation between the soil and structure. Two-dimensional axisymmetric

elements and a direct time integration method were used. This analysis can

properly simulate the physical phenomena of the soil-structure system, and

provide an understanding of the loading wave propagating and the corresponding

structural response.

4.2. PLATE RESTING ON SAND

The problem of the dynamic response of a finite circular plate resting on

sand was studied by using ABAQUS in order to understand more about the

measured response of a target plate. The loading mechanism used in the

experiment to generate a dynamic loading into the soil medium is a low-velocity

impact system (Greszczuk, 1982). Since the measured contact area was much

24

smaller than the size of the target plate, the low velocity impact can be modeled

as a point load (Figure 4.1) applied at the center of the target plate. The

magnitude of the impact loading can be obtained from the momentum balance.

The rebound velocity of the ball is assumed to be negligible. Therefore, the

impact loading, P(t), can be defined as a product of the loading magnitude and a

time-varying function (Chen et al, 1988):

)()( 0 tfPtP = (4.1)

where the time function is assumed as a Hanning’s function for a monopeak,

smooth-shaped curve:

)2cos(5.05.0)(0Tttf π

−= (4.2)

where T0 is the impact duration. Assuming momentum balance, the peak

amplitude, P0, of the loading can be calculated as:

∫

=0

0

2/1

0)(

)2(Tb

dttf

gHMP (4.3)

where Mb is the mass of the ball, g is the gravitational acceleration; and H is the

dropping height.

25

As shown in Figure 4.2. a total of 6528 elements and 6746 nodes were

used for one half of the system. Four-node axisymmetric elements were used for

both plate and sand. The material properties of the aluminum plate and sand are

shown in Table 4.1. The aluminum plate had a 3 in. (7.62 cm) radius and was

0.5-in. (1.27 cm) thick. The outer boundaries along r = 6 in. (15.24 cm) and z = 4

in. (10.16 cm) were fixed. For a ball mass of 443.3 g and a 44.5 cm dropping

height, the impact force with peak amplitude is calculated as 2,450 lb (10.9 kN).

The measured impact duration was 0.238 ms. The loading function is shown in

Figure 4.3. The wave length of this dynamic loading is in the same order of

magnitude as the burial depth, hence can provide the same kind of wave

propagation and the interaction phenomenon as in the actual field test. Two FEM

models were built for this test, one with a slide line underneath the target plate

and another one with non-slide line. The FEM model with the slide line also

included two kinds of damping effects: material damping and contact damping.

For the material damping, mass proportional damping, α, and stiffness

proportional damping, β, were zero. For contact damping between the aluminum

plate and sand, the critical damping coefficient was 20%.

Since the FEM program is based on explicit scheme, very small time steps

were used (typically ≤ 0.1 µsec) to ensure stability in calculation. The element

sizes are no larger than 1.27 mm in the structures. The soil element sizes are

allowed to be larger in the slide models. The calculation duration is about 1 msec,

such that sufficient time is allowed for the loading waves to reach the bottom

plate.

26

Material v ρ

(kg/m3)

CL

(m/s)

CT

(m/s)

E

(MPa)

Aluminum 0.33 2,821 6,054 3,027 68,928

Ottawa 20-

30 sand

0.25

(assumed)

1,776 241

263

(measured)

139

152

86

103

Plastic 0.35 1,015 2,217 1,065 3,111

Table 4.1. Material Properties (Chen, 1988)

27

4.3. DISCUSSION OF RESULTS

Figure 4.4. shows a deformed state of aluminum plate separated from

sand at 0.000775 sec., and a partial separation is observed. Overlapping of

elements is observed. It should be noted that the actual deformations are very

small, and in these figures they are exaggerated by 150 times.

Figure 4.5. shows radial strain of the circular plate at r = 12.7 mm with

material damping factors α and β in the sand and no contact damping. It shows

that there were lots of noises appearing when the stiffness proportional damping,

β, was entered. It also took a very long time to complete the analysis of the FEM

model with the stiffness proportional damping, approximately a week, indicating

that it is not efficient to include the stiffness proportional damping in the current

FEM models.

Figure 4.6. presents radial strain of circular plate at r = 12.7 mm with

different contact damping, showing that results from contact damping with 20%

had better agreement with the experimental result, and neither α damping factor

nor β damping factor are used in the sand. Figure 4.7. shows radial strain of

circular plate at r = 12.7 mm with different α damping factors with no β damping

factor and no contact damping. It can be seen from Figure 4.6. and Figure 4.7.

that contact damping has more effect than material damping in these ranges,

because there was no differences when α damping factor was applied. Reason

being, there was no contact between the target plate and sand when separation

occurred, and α damping factor was only applied to sand, not the target plate.

28

4.4. COMPARISON OF FEM RESULTS AND EXPERIMENTS

4.4.1. ABAQUS FEM (with non-slide line) Strains, SAP4 FEM and

Experiments

In Figure 4.8., the measured radial strain (dashed line) at r = 0.5 in. (12.7

mm) on the bottom of the plate is plotted and compared with the SAP4 FEM

results, marked by a dotted line (Chen et al. 1990), and the ABAQUS FEM

results, marked by a solid line. The radial strain calculated by FEM was

computed by the difference of the radial displacement of two adjacent nodal

points. The comparisons show good agreement between experiments, ABAQUS,

and SAP4 FEM calculations. All techniques show the same arrival time. The

durations of the dominant signals are reasonably close and the difference of the

strain magnitude between FEM calculation and the experiments is within 10%.

This is understandable because the interface between the plate and the sand

was assumed to be continuous (perfect bond); no interfacial elements were

added in the FEM programs. In addition, the linear-elastic assumption of sand

was made. Hence, a direct inversion for getting the impact forcing function is not

reliable.

29

4.4.2. ABAQUS FEM (with slide line) Strains and Experiments

In Figure 4.9, the measured radial strain (dashed line) at r = 0.5 in. (12.7

mm) on the bottom of the plate is plotted and compared with ABAQUS FEM

results with contact damping of 20%, and no α or β damping factors, marked by a

solid line. The radial strain calculated by FEM was computed by the difference of

the radial displacement of two adjacent nodal points. The comparisons show

good agreement between experiments, and ABAQUS FEM calculations. The

durations of the dominant signals are reasonably close and the strain magnitude

between FEM calculation and the experiments is almost the same. This is

understandable because the interface between the plate and the sand was

assumed to be discontinuous (separation occurred); contact elements were

added in the interface between the plate and the sand. In addition, the linear-

elastic assumption for sand properties was made, but nonlinear behavior

(separation) due to slide line is allowed for the interface between the sand and

the adjacent structure, and also between the sand layers.

30

31

4.5. CONCLUSION

Results shown in Figure 4.8, Figure 4.9 and Figure 4.10 indicate that the

FEM and analytical (Chen et al., 1988) results for the radial strain underneath the

target plate at r = 0.5 in. are close to the experimental results. However, the

analytical solution, which assumed infinite plate, has a much lower peak

magnitude than the experimental measurement. The magnitudes between the

slide line and the non-slide line FEM models are comparable, and the slide line

model with contact damping of 20% has a lower peak than the non-slide line

model with no damping, and has a better agreement with the experimental

results.

Figure 4.1. Idealization of the Impact Loading (Chen et al., 1996)

32

Figure 4.2. Finite Element Mesh for Circular Plate on Elastic Half-Space

33

f(t)=0.5-0.5cos(2πt/T0)

0.0000

0.1000

0.2000

0.3000

0.4000

0.5000

0.6000

0.7000

0.8000

0.9000

1.0000

0 0.05 0.1 0.15 0.2 0.25

Time (msec)

Uni

t For

ce f(

t)

Figure 4.3. Loading Function for Impact Duration = 0.238 ms

34

Figure 4.4. Deformed State of slide line model of Aluminum Plate Separated from Sand at 0.000775 sec

35

Radial Strain

-0.0002

-0.0001

0

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 0.001

Time (s)

Stra

in (m

m/m

m)

FEM (a=3, b=0.003)FEM (a=2, b=0.002)FEM (a=4, b=0.004)Experimental Result

Figure 4.5. Radial Strain of Circular Plate at r = 12.7 mm with material damping factors α (mass proportional

damping) and β (stiffness proportional damping) (slide line model)

36

Radial Strain

-0.0003

-0.0002

-0.0001

0

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 0.001

Time (s)

Stra

in (m

m/m

m)

Contact=60

Contact=10

Contact=3

Experimental Result

Contact=0.2

Contact=20

Figure 4.6. Radial Strain of Circular Plate at r = 12.7mm with different contact damping (slide line model)

37

Radial Strain

-0.0003

-0.0002

-0.0001

0

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 0.001

Time (s)

Stra

in (m

m/m

m)

Alpha=300

Alpha=30

Alpha=3

Experimental Result

Figure 4.7. Radial Strain of Circular Plate at r = 12.7 mm with different α (mass proportional damping) damping

factors (slide line model)

38

Radial Strain

-0.0003

-0.0002

-0.0001

0

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 0.001

Time (s)

Stra

in (m

m/m

m)

ABAQUS (Linear)

Experimental Result

SAP4 (Chen, 1990)

Figure 4.8. Radial Strain at r = 12.7 mm from Center of Target Plate (with non-slide line)

39

Radial Strain

-0.0003

-0.0002

-0.0001

0.0000

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0.0000 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 0.0010

Time (s)

Stra

in (m

m/m

m)

Experimental Result

ABAQUS (Nonlinear)

Figure 4.9. Radial Strain at r = 12.7mm from Center of Target Plate with contact damping of 20%, and no α and β

damping factors (with slide line)

40

41

Figure 4.10. Radial Strain of Circular Plate at r = 12.7 mm (Chen, 1990)

CHAPTER V

DYNAMIC RESPONSE OF BURIED

STRUCTURES

5.1. FINITE ELEMENT MODELING OF BURIED STRUCTURES

The finite element mesh of a buried structure system is shown in Figure

5.1, a total of 7552 elements and 8900 nodes were used for one half of the

system. Four-node axisymmetric elements were used to model the plate, buried

structure and sand. Quadrilateral element (type=CAX4R) is used. ‘CAX4R’

means the 4-node, axisymmetric, reduced-integration with hourglass control,

solid element. Three different materials were used: aluminum, sand, and plastic.

The material properties used were the same as those shown in Table 4.1. Linear

elastic, axisymmetric elements with separable, frictionless interfaces are used to

simulate the interfaces between the buried structure and soil, and between the

target plate and the soil. The aluminum plate had a 6-in. radius and was 0.5-in.

(1.27 cm) thick. The outer boundaries along r = 9 in. (22.86 cm) and z = 13.5 in.

(34.29 cm) were fixed. Slide lines were added between the target plate and soil,

and between the soil and the buried roof-structure to allow for separation.

Damping was also included; α damping factor was 1650 s-1, β damping factor

was zero, and the critical contact damping was 30 % between each layer. Also,

42

finite element models of three different configurations were constructed: 3 layers,

6 layers, and 12 layers below the aluminum plate which are shown in Figure 5.2,

Figure 5.3, and Figure 5.4, respectively.

A concentrated load, vertically applied at the center of the target plate,

was used to simulate the impact loading. The shape of the loading in the time

domain was assumed to be a Hanning function similar to the one given in

Chapter 4. The only difference was that the loading duration was 0.77 ms, and

the loading magnitude was 3,170 lb (16.5 kN, dropping height H = 62.5 cm). The

loading function is shown in Figure 5.5.

5.2. DISCUSSION OF RESULTS

Figure 5.6 is a loaded figure, which shows a deformed state of the buried

structure at 0.00069 sec. at 23 frames. It should be noted that the actual

deformations are very small, and in these figures they are exaggerated by 100

times. Figure 5.7 is a rebounded figure, shows a close up of the buried structure

deformation at 0.00102 sec. at 34 frames, and a partial separation is observed.

Overlapping of elements is observed. Total duration of the response calculated is

200 frames = 0.006 sec.

In Figure 5.8, the target plate is seen to “fly away” from the soil at 0.0054

sec. at 180 frames. Figure 5.9 shows the displacement at the center of the roof of

the buried structure (with just one big sand layer between the target plate and the

buried roof) with different kinds of damping. It shows that duration is small

43

compared with the ones with sand divided into many layers underneath the target

plate. Figure 5.10 shows the FEM results in comparison with experimental results

with α damping factor of 1650 s-1 and contact damping of 30%. It can be seen

that duration of 3 layered sand beneath the target plate is smaller than the

duration of 12 layered sand beneath the target plate with the same values of α

damping factor and contact damping. It also shows that using 12 layered sand

beneath the plate has a lower peak value of displacement than with the 3 layered

sand beneath the plate. Figure 5.11 shows displacement at the center of the roof

of buried structure (12 layered) with different α damping factors and a contact

damping of 20%, demonstrating that using an α damping factor of 500 s-1 allows

larger displacements than using an α damping factor of 1650 s-1 but a shorter

duration. Figure 5.12 shows displacement at the center of the roof of buried

structure (12 layered) with different contact damping and α damping factor of

1500 s-1, from which it can be seen that a contact damping of 30% has a shorter

duration than a contact damping of 20%. According to Fazio et. al. (2003), the

damping coefficient of sand is in the range of 33 % to 47 %.

Figure. 5.13(a) shows the time history of the normal stress (σz) of the sand

element, located at different radii, along the bottom of the target plate from the

results of the 12 layer model. The negative stress represents compression, and

the peak magnitude is gradually decreasing away from the center. Figure 5.13(b)

shows time history of the normal stress (σz) along the z-axis from the nonlinear

analysis. The propagation of compressive waves in sand can be recognized by

inspecting the arrival time of the stress wave propagates through the depth

44

(sand). The waves speed in sand was about 263 m/s (10,345 in./sec). Figure

5.13(c) shows the time histories of the normal vertical stress (σz) along the buried

roof for nonlinear analysis. The magnitude and the duration of the compressive

normal stress around the center of the roof (r = 0.13 in.) are shown to be smaller

than those of the stress at the roof’s edge (r = 2.63 in.). Figure 5.14(a), (b) and

(c) show linear analysis of stress at the bottom of the target plate, stress along

the vertical axis, and stress at the buried roof, respectively. Figures 5.13 and

5.14 show that nonlinear analysis has a much lower peak value than the linear

analysis, mainly caused by the material damping in the sand. Figure 5.13 shows

lots of oscillations compared to Figure 5.14 due to the nonlinear analysis where

separation occurred.

The spatial distribution of the normal stress can be calculated at any

different reference time. Figure 5.15 shows normal stress along the bottom of the

target plate from the nonlinear analysis at different times (t = 0.12 ms, 0.24 ms,

and 0.39 ms). Figure 5.16 shows the schematic plot of normal stress along the

bottom of target plate from linear analysis. This stress was then propagated to

the structural roof at a later time. Figure 5.17 shows vertical pressure along

buried roof from the nonlinear analysis at different times (t = 0.39 ms, 0.51 ms,

0.6 ms and 0.66 ms). The pressure at the center of the roof increased

continuously for times from 0.39 ms to 0.6 ms, and then decreased continuously

for time up to 0.66 ms; the pressure at edge of the roof increased continuously

for times from 0.39 ms to 0.66 ms. Figure 5.18 shows the schematic plot of

45

vertical pressure along buried roof from the linear analysis. The pressure on the

buried roof is lower as shown in the nonlinear case.

In this research, only one stiffness of buried structure was considered.

According to Chen (1996), the experimental results observed that the thinner

buried roof structures experienced smaller interaction loadings but larger

displacements, and the thicker roof structures experienced larger interaction

loading and smaller displacements. Study of the separation process shows that

the thinner roof enjoyed larger separation times than the thicker roof, which

indicates that separation as a result of soil-structure interaction can significantly

contribute to the structural dynamic responses. The dynamic effects of the impact

loading on the buried roofs of different stiffnesses showed that thickest roof had

the highest dynamic magnification factors and the longest impulse length ratio.

The more flexible roof, on the other hand, would have lower dynamic effects.

5.3. COMPARISON OF EXPERIMENTAL AND NUMERICAL RESULT

The experimental results in comparison with the FEM results are

discussed. In Figure 5.10, the displacements at the center of the roof of buried

structure is plotted. The calculated response has good agreement in some of its

quantities with the measured response of those three different sand-layers model.

All the models have α damping factor of 1650 s-1, β damping factor of zero, and

contact damping of 30%. There was good agreement between the calculated and

46

the measured arrival times. The peak displacement at the roof center (Figure

5.10) can be predicted by the current nonlinearly elastic analysis for both peak

amplitude and peak arrival. Moreover, the response subsequent to the peak also

shows good agreement.

Figure 5.19. also shows the displacement at the center of the roof of the

buried structure with two different Poisson’s ratios of sand. Both of the models

have α damping factor of 1650 s-1, β damping factor of zero, and contact

damping of 30%. It can be seen that Poisson’s ratio of 0.4 has lower peak value

of displacement than Poisson’s ratio of 0.25 and smaller duration. There was

good agreement between the calculated and the measured peak arrival times.

5.4. CONCLUSION

The responses of the buried structure are a function of the number of

separated sand-layers beneath the aluminum plate, which can be seen in Figure

5.10 where using 12 layered sand beneath the plate had a lower peak value of

displacement than with the 3 layered sand beneath the plate. The FEM results

have higher magnitudes, but shorter peak duration than the experimental result.

Separation between the center of the buried roof and the sand is observed at

around 0.84 ms which can be in Figure 5.21, and it is about at the time of the

peak buried roof displacements. Separation between the target plate and the

sand is observed at 0.93 ms which can be seen in Figure 5.22.

47

48

The comparison of FEM results with slide lines and with non-slide lines

(Chen, 1990) can be seen in Figure 5.19 and Figure 5.20, respectively. The

comparison of current results (with slide lines) show better agreement between

experiments and FEM calculations than previous study with non-slide line (Chen,

1990). In Figure 5.19 and Figure 5.20, it can also be seen that the Poisson’s

ratios has different effect to the nonlinear elastic analysis with 12 layered sand

beneath the target plate and linear elastic analysis with just one big sand layer

between the target plate and the buried roof. In the nonlinear model, using

Poisson’s ratio of 0.4 has lower peak and smaller duration than using Poisson’s

ratio of 0.25. Conversely, using Poisson’s ratio of 0.4 has larger peak and larger

duration than using Poisson’s ratio of 0.25 in linear elastic analysis.

0.25 in.

4.5 in.

5.5 in.

0.25 in.

3.5 in.

6 in. 3 in.

13.5 in.

2.75 in.

0.25 in.

0.5 in.

z

r

Figure 5.1. Finite Element Mesh for Elastic Buried Structure

49

Figure 5.2. Buried Structure with 3 layers below the plate

50


51


52

f(t)=0.5-0.5cos(2πt/T0)

0.0000

0.1000

0.2000

0.3000

0.4000

0.5000

0.6000

0.7000

0.8000

0.9000

1.0000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Time (msec)

Uni

t For

ce f(

t)

Figure 5.5. Loading Function for Loading Duration = 0.77 ms

53

Figure 5.6. Deformed State of Buried Structure at 0.00069 sec

54

Figure 5.7. Close Up of Buried Structure Separated from Aluminum Plate and Sand (12 Layered) at 0.00102 sec.

55

Figure 5.8. Close Up of Aluminum Plate Separated from Sand at 0.0054 sec.

56

Buried Structure with no layers

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0 0.001 0.002 0.003 0.004 0.005 0.006

Time (s)

Dis

plac

emen

t (in

)

no damping

Alpha=5

Contact=30

Alpha=50

Alpha=500, Contact=30

Experimental Result

Figure 5.9. Displacement at the center of the roof of buried structure (no layered) with different kind of damping

57

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0 0.001 0.002 0.003 0.004 0.005 0.006

Time (s)

Dis

plac

emen

t at C

ente

r (in

)

3 layers

6 layers

12 layers

Experimental Result

Figure 5.10. Numerical Results in Comparison with Experimental Results with α damping factor of 1650 s-1 and

contact damping of 30% (Displacement at the center of the roof of buried structure with different layers beneath the target plate)

58

Buried Structure with 12 layers and Contact Damping for each layers=20

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0 0.001 0.002 0.003 0.004 0.005 0.006

Time (s)

Dis

plac

emen

t (in

)

Alpha=500

Alpha=1500

Alpha=1650

Experimental Result

Figure 5.11. Displacement at the center of the roof of buried structure (12 layered) with different α (mass

proportional damping) damping factors and contact damping of 20%

59

60

Buried Structure with 12 layers and Alpha=1500

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0 0.001 0.002 0.003 0.004 0.005 0.006

Time (s)

Dis

plac

emen

t (in

)

Contact for each layers=20



Experimental Result

Figure 5.12. Displacement at the center of the roof of buried structure (12 layered) with α (mass proportional

damping) damping factor of 1500 s-1 and different contact damping

(a) Stress at plate bottom

-120

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014

Time (s)

Nor

mal

Str

ess

(psi

)

r = 0.125 inr = 0.625 inr = 1.125 inr = 2.125 inr = 3.125 inr = 5.75

(b) Stress along z-axis

-150

-100

-50

0

50

100

150

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014

Time (s)

Nor

mal

Str

ess

(psi

)

z = 0.625 in

z = 1.125 in

z = 1.625 in

z = 2.125 in

z = 2.625 in

z = 2.875 in

61

(c) Stress at buried roof

-60

-50

-40

-30

-20

-10

0

10

20

30

40

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014

Time (s)

Nor

mal

Str

ess

(psi

)

r = 0.125 inr = 0.625 inr = 1.125 inr = 1.625 inr = 2.125 inr = 2.625 in

Figure 5.13. Time History of the Stress Field with a Buried Structure (Nonlinear Analysis): (a) Stress at Bottom of Target Plate; (b) Stress along

z-Axis; (c) Stress at Buried Roof

62

Figure 5.14. Time History of the Stress Field with a Buried Structure (Linear Analysis): (a) Stress at Plate’s Bottom; (b) Stress along z-Axis (Chen,

1990); (c) Stress at Buried Roof (Chen, 1990)

64

Normal Stress along Target Plate's Bottom

-120

-100

-80

-60

-40

-20

0

20

0.000 1.000 2.000 3.000 4.000 5.000 6.000

r(in)

(psi

)

t = 0.12 mst = 0.24 mst = 0.39 ms

Figure 5.15. Normal Stress along Bottom of Target Plate (Nonlinear Analysis)

65

Figure 5.16. Schematic Plot of Normal Stress along Bottom of Target Plate (Linear Analysis) (Chen, 1990)

66

Vertical Pressure along Buried Roof

0

5

10

15

20

25

30

35

0.000 0.500 1.000 1.500 2.000 2.500 3.000

r (in)

(psi

)

t = 0.39 ms

t = 0.51 ms

t = 0.6 ms

t = 0.66 ms

Figure 5.17. Vertical Pressure along Buried Roof (Nonlinear Analysis)

67

Figure 5.18. Schematic Plot of Vertical Pressure along Buried Roof (Linear Analysis) (Chen, 1990)

68

Buried Structure of 12 layers with Alpha 1650 Contact 30

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0 0.001 0.002 0.003 0.004 0.005 0.006

Time (s)

Dis

plac

emen

t at C

ente

r (in

)

poisson's ratio of 0.25

Experimental Result

poisson's ratio of 0.4

Figure 5.19. Displacement at the center of the roof of buried structure with different Poisson’s ratios of sand, α

(mass proportional damping) damping ratio of 1650 s-1 and contact damping of 30%

69

Figure 5.20. Numerical Results in Comparison with Experimental Results (Displacement Measurement) (Chen, 1990)

70

-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0 0.001 0.002 0.003 0.004 0.005 0.006

Time (s)

Dis

plac

emen

t at c

ente

r (in

)Plastic Buried Roof

Sand on top of buried roof

Separation occur

Figure 5.21. Displacement at the center of the buried roof and the sand on top of buried roof (12 layered model)

71

Figure 5.22. Displacement at the center of the aluminum plate and the sand below the plate (12 layered model)

-0.005

0

0.005

0.01

0.015

0.02

0.025

0 0.001 0.002 0.003 0.004 0.005 0.006

Time (s)

Dis

plac

emen

t at c

ente

r (in

)Aluminum plate

Sand below the plate

72

Separation occur

CHAPTER VI

CONCLUSIONS

The soil-structure interaction phenomenon involving buried structures has

been reproduced successfully. FEM modeling techniques are shown to be

capable of simulating the nonlinear problem and the following can be

summarized from the study:

1. The Finite Element Method model can be used to model wave

propagation through soil, and current assumption for material properties of

Ottawa 20-30 sand is a good approximation.

2. FEM models provide realistic modeling of the plate vibration resting on soil

and is able to show separation between the soil and the structure. The

FEM model can also include slide lines between the soil layers to simulate

soil separation.

3. For the plate resting on sand, the magnitudes between the slide line and

the non-slide line models are comparable, and the slide line model with

contact damping of 20% has a lower peak than the non-slide line model

with no damping, and has a better agreement with the experimental

results.

4. For the buried structure, the FEM results observed that smaller amounts

of sand-layers beneath the aluminum plate experienced larger

displacements but shorter duration than with large amounts of sand-layers.

73

5. Simulations are completed by a 900 MHz DuronTM AMD Processor with

192 KB On-Chip Cache Memory, 30 GB UltraDMA Hard Drive, 256

Megabytes RAM computer under the Windows XP environment. The

typical computation time for the computation is approximately 5 hours to

run a typical case of buried structure response of 6 msec.

74

CHAPTER VII

RECOMMENDATIONS

The following are recommendations for future studies:

1. It has been mentioned that Finite Element Method can be used to study

soil-soil separation. It is recommended that future studies include more

investigations on the Finite Element Method modeling of soil.

2. In this research, the Poisson’s ratios of sand were assumed to be 0.25

and 0.4. It is recommended that future studies compare the results from

different Poisson’s ratios of sand.

3. Moisture content and void ratio in the sand were not considered in this

study. It is recommended that future studies consider these parameters

with respect to any changes in soil properties such as damping that may

occur when these properties vary.

4. Material damping and contact damping were used in this research. It is

recommended that the damping coefficients used in this study need to be

experimentally verified in the future.

5. In order to prevent instability, it is recommended that future studies use

smaller ∆t.

75

REFERENCES ABAQUS Version 6.3 User’s Manual Achenbach, J.D., 1980, Wave Propagation in Elastic Solids, North-Holland Publishing Co., Amsterdam. Al-Mousawi, M. M., 1986, “On Experimental Studies of Longitudinal and Flexural Wave Propagations: An Annotated Bibliography,” Applied Mechanics Reviews, ASME, Vol. 39, No. 6, pp. 853-865. Argyris, J. H., 1965, “Elasto-plastic matrix displacement analysis of three-dimensional continua”, J. Royal Aeronautical Society, No. 69, pp. 633-635. Baylot, J. T., 2000, “Effect of Soil Flow Changes on Structure Loads,” Journal of Structural Engineering, Vol. 126, pp. 1434-1441. Belytschko, T., Liu, W. K., Moran, B., “Nonlinear Finite Elements for Continua and Structures,” John Wiley & Sons Ltd, England, 2000. Budek, A. M., Priestley, M. J. N., Benzoni, G., 2000, “Inelastic Seismic Response of Bridge Drilled-Shaft RC Pile/Columns,” Journal of Structural Engineering, Vol. 126, No. 4, pp. 510-517. Chen, H. L., Lin, W., Keer, L. M., and Shah, S. P., 1988, “Low Velocity Impact of an Elastic Plate Resting on Sand,” Journal of Applied Mechanics, Vol. 55, pp. 887-894. Chen, H. L., Shah, S. P., and Keer, L. M., 1990, “Dynamic Response of Shallow-Buried Cylindrical Structures,” Journal of Engineering Mechanics, Vol. 116, No. 1, pp. 152-171. Chen, H. L., and Chen, S. E., 1996, “Dynamic Response of Shallow-Buried Flexible Plates Subjected to Impact Loading,” Journal of Structural Engineering, Vol. 122, No. 1, pp. 55-60. Chen, S. E., 1996, The Dynamic SSI of A Shallow Buried Plate Under Impact Loading, Ph.D. Dissertation, West Virginia University, Morgantown, WV. Costanito, C. J., 1967, “Finite element approach to stress wave problems,” Journal of Engineering Mechanics Division, ASCE, No. 93, pp. 153-166. Fazio, C., Sperandeo-Mineo, R. M., Tarantino, G., 2003, “How did Roman Buildings survive earthquake?,” Physics Education, Vol. 38, No. 6, pp. 480-484. Ginsburg, T., 1964, “Propagation of Shock Waves in the Ground,” Journal of

76

Structural Division, ASCE, Vol. 90, No. ST1, pp.125-163. Graff, K.F., 1975, Wave Motion in Elastic Solids, Dover Pub., New York. Greszczuk, L. B., 1982, “Damage in Composite Materials Due to Low Velocity Impact,” Impact Dynamics, J. A. Zukas et al., eds., John Wiley, New York. Iguchi, M., and Luco, J. E., 1982, “Vibration of Flexible Plate on Viscoelastic Medium,” Journal of the Engineering Mechanics Division, Proceedings, ASCE, Vol. 108, No. EM6, pp. 1103-1120. Lin, Y. J., 1978, “Dynamic Response of Circular Plates Resting on Viscoelastic Half-Space,” ASME JOURNAL OF APPLIED MECHANICS, Vol. 45, pp. 379-384. Maeso, O., Aznarez, J. J., Dominguez, J, 2002, “Effects of Space Distribution of Excitation on Seismic Response of Arch Dams,” Journal of Engineering Mechanics, Vol. 128, No. 7, pp. 759-768. Marcal, P. V., King, I. P., 1967, “ Elastic-plastic analysis of two dimensional stress systems by the finite element method,” Int. J. Mechanical Sciences, No. 9, pp. 143-155. Newmark, N.M., 1964, “The Basic of Current Criteria for the Design of Underground Protective Constructions,” Symposium on Soil-Structure Interaction, University of Arizona, Engineering Research Lab, pp. 1-24. Oden, J.T., 1972, Finite Elements of Nonlinear Continua, McGraw-Hill, New York. Rizos, D. C., Loya, K. G., 2002, “Dynamic and Seismic Analysis of Foundations based on Free Field B-Spline Characteristic Response Histories,” Journal of Engineering Mechanics, Vol. 128, No. 4, pp. 438-448. Selig, E.T., 1964, “Characteristics of Stress Wave Propagation in Soil,” Symposium on Soil-Structure Interaction, University of Arizona, Engineering Research Lab, pp. 27-61. Stavridis, L. T., 2002, “Simplified Analysis of Layered Soil-Structure Interaction,” Journal of Structural Engineering, Vol. 128, No. 2, pp. 224-230. Trifunac, M. D., Ivanovic, S. S., Todorovska, M. I., 2001, “Apparent Periods of A Building. II: Time-Frequency Analysis,” Journal of Structural Engineering, Vol. 127, No. 5, pp. 527-537. Whitman, R. V., 1970, The Response of Soils to Dynamic Loadings. Final Report, No. 3-26, U.S. Army Engineering Waterway Experiment Station,

77

Vicksburg, Mississippi. Whittaker, W. L., and Christiano, P., 1982, “Dynamic Response of Flexible Plates on Elastic Half-Space,” Journal of the Engineering Mechanics Division, ASCE, Vol. 108, pp. 133-154. Wong, F.S. and Weidlinger, P. 1983, “ Design of underground protective structures.” Journal of Structural Engineering, ASCE, vol. 109, no. 8, pp. 1972-1979. Zaman, M.M., Desai, C.S. and Drumm, E. C, 1984, “Interface Model for Dynamic Soil-Structure Interaction.” Journal of Geotechnical Engineering, ASCE, vol. 110, no. 9, pp.1257-1273.

78

APPENDIX A

ABAQUS BASICS

79

APPENDIX B

ABAQUS/CAE MODULES

ABAQUS/CAE is divided into modules, where each module defines an

aspect of the modeling process; for example, defining the geometry, defining

material properties, and generating a mesh. As you move from module to module,

you built the model from which ABAQUS/CAE generates an input file that you

submit to the ABAQUS/Standard or ABAQUS/Explicit solver for analysis.

An Example from ABAQUS manual: A cantilever beam

Entering the following ABAQUS/CAE modules and perform the following tasks:

Part Sketch a two-dimension profile and create a part representing the

cantilever beam.

Property Define the material properties and other section properties of the

beam.

Assembly Assemble the model and create sets.

Step Configure the analysis procedure and output requests.

Load Apply loads and boundary conditions to beam.

Mesh Mesh the beam.

Job Create a job and submit it for analysis.

Visualization View the results of the analysis.

80

Although the Module list under the toolbar lists the modules in a logical

sequence, you can move back and forth between modules at will.

Entering a module by selecting it from the Module list under the toolbar, as

shown in Figure B.

Figure B.1. Selecting a module

81

APPENDIX C

INPUT MATERIALS FOR PLATE

** MATERIALS ** *Material, name=Aluminum *Density 0.000264, *Elastic 9.99716e+06, 0.33 *Material, name=Sand *Damping, alpha=30. *Density 0.000166, *Elastic 12473.2, 0.25 ** ** INTERACTION PROPERTIES ** *Surface Interaction, name=Contact *Friction 0., *Contact Damping, definition=CRITICAL DAMPING, tangent fraction=1. 20., ** ---------------------------------------------------------------- ** ** STEP: Dynamic, Explicit ** *Step, name="Dynamic, Explicit" *Dynamic, Explicit, direct user control 2e-09, 0.001 *Bulk Viscosity 0.06, 1.2 ** ** BOUNDARY CONDITIONS ** ** Name: Axis Type: Symmetry/Antisymmetry/Encastre *Boundary _PickedSet8, XSYMM ** Name: Bottom Type: Symmetry/Antisymmetry/Encastre *Boundary _PickedSet9, PINNED ** Name: Edge Type: Symmetry/Antisymmetry/Encastre *Boundary _PickedSet10, PINNED ** ** LOADS **

82

** Name: Impact Loading Type: Concentrated force *Cload, amplitude="Loading Function" _PickedSet11, 2, -2450. ** ** INTERACTIONS ** ** Interaction: Int-1 *Contact Pair, interaction=Contact, mechanical constraint=PENALTY _PickedSurf7, _PickedSurf6 ** ** OUTPUT REQUESTS ** *Restart, write, number interval=1, time marks=NO ** ** FIELD OUTPUT: F-Output-1 ** *Output, field, variable=PRESELECT, number intervals=200 ** ** HISTORY OUTPUT: H-Output-1 ** *Output, history, variable=PRESELECT *End Step

83

APPENDIX D

INPUT MATERIALS FOR BURIED

STRUCTURE

** MATERIALS ** *Material, name=Aluminum *Density 0.000264, *Elastic 9.99716e+06, 0.33 *Material, name=Plastic *Density 9.49e-05, *Elastic 451212., 0.35 *Material, name=Sand *Damping, alpha=1650. *Density 0.000166, *Elastic 12473.2, 0.25 ** ** INTERACTION PROPERTIES ** *Surface Interaction, name=Contact *Friction 0., *Contact Damping, definition=CRITICAL DAMPING, tangent fraction=1. 30., ** ** BOUNDARY CONDITIONS ** ** Name: Axis Type: Symmetry/Antisymmetry/Encastre *Boundary _PickedSet126, XSYMM ** Name: Bottom Type: Symmetry/Antisymmetry/Encastre *Boundary _PickedSet127, PINNED ** Name: Edge Type: Symmetry/Antisymmetry/Encastre *Boundary _PickedSet128, PINNED ** ---------------------------------------------------------------- ** ** STEP: Dynamic, Explicit

84

** *Step, name="Dynamic, Explicit" *Dynamic, Explicit , 0.006 *Bulk Viscosity 0., 0. ** ** LOADS ** ** Name: Impact Load Type: Concentrated force *Cload, amplitude="Loading Function" _PickedSet45, 2, -3170. ** ** INTERACTIONS ** ** Interaction: Int-1 *Contact Pair, interaction=Contact, mechanical constraint=PENALTY "Top Sand 1", "Bottom Aluminum" ** Interaction: Int-10 *Contact Pair, interaction=Contact, mechanical constraint=PENALTY "Top Sand 10", "Bottom Sand 9" ** Interaction: Int-11 *Contact Pair, interaction=Contact, mechanical constraint=PENALTY "Top Sand 11", "Bottom Sand 10" ** Interaction: Int-12 *Contact Pair, interaction=Contact, mechanical constraint=PENALTY "Top Sand 12", "Bottom Sand 11" ** Interaction: Int-13 *Contact Pair, interaction=Contact, mechanical constraint=PENALTY "Top sand 13", "Bottom Sand 12" ** Interaction: Int-14 *Contact Pair, interaction=Contact, mechanical constraint=PENALTY "Bottom Sand 12", "Top Plastic" ** Interaction: Int-15 *Contact Pair, interaction=Contact, mechanical constraint=PENALTY "Inside Sand", "Outside Plastic" ** Interaction: Int-16 *Contact Pair, interaction=Contact, mechanical constraint=PENALTY "Top Sand 14", "Bottom Sand 13" ** Interaction: Int-17 *Contact Pair, interaction=Contact, mechanical constraint=PENALTY "Top Sand 14", "Bottom Plastic" ** Interaction: Int-2 *Contact Pair, interaction=Contact, mechanical constraint=PENALTY "Top Sand 2", "Bottom Sand 1" ** Interaction: Int-3 *Contact Pair, interaction=Contact, mechanical constraint=PENALTY "Top Sand 3", "Bottom Sand 2" ** Interaction: Int-4 *Contact Pair, interaction=Contact, mechanical constraint=PENALTY "Top Sand 4", "Bottom Sand 3" ** Interaction: Int-5 *Contact Pair, interaction=Contact, mechanical constraint=PENALTY "Top sand 5", "Bottom Sand 4" ** Interaction: Int-6 *Contact Pair, interaction=Contact, mechanical constraint=PENALTY

85

"Top Sand 6", "Bottom Sand 5" ** Interaction: Int-7 *Contact Pair, interaction=Contact, mechanical constraint=PENALTY "Top Sand 7", "Bottom Sand 6" ** Interaction: Int-8 *Contact Pair, interaction=Contact, mechanical constraint=PENALTY "Top Sand 8", "Bottom Sand 7" ** Interaction: Int-9 *Contact Pair, interaction=Contact, mechanical constraint=PENALTY "Top Sand 9", "Bottom Sand 8" ** ** OUTPUT REQUESTS ** *Restart, write, number interval=1, time marks=NO ** ** FIELD OUTPUT: F-Output-1 ** *Output, field, variable=PRESELECT, number intervals=200 ** ** HISTORY OUTPUT: H-Output-1 ** *Output, history, variable=PRESELECT *End Step

86

VITA

The author, the son of Ah Keong and Siew Hong Lee Tee, was born in

Klang, Malaysia on December 15, 1975. From 1998 to 2000, he attended West

Virginia University Institute of Technology, and received his Bachelor of Science

in Civil Engineering degree in May 2000.

In August 2001, he continued his education with Master of Science in Civil

Engineering degree specializing in structures at West Virginia University.

87

dynamic response of plates and buried structures

Documents