Dynamic response of plates and buried structures
Transcript of 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
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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.
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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
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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
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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
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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
2.1. GENERAL REMARKS
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
4.1. GENERAL REMARKS
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
Figure 5.3. Buried Structure with 6 layers below the plate
51
Figure 5.4. Buried Structure with 12 layers below the plate
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
Contact for each layers=25
Contact for each layers=30
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
63
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
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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