numerical and experimental investigations of anti …
TRANSCRIPT
The Pennsylvania State University
The Graduate School
College of Engineering
NUMERICAL AND EXPERIMENTAL INVESTIGATIONS
OF ANTI-RAM BARRIERS
UNDER VEHICULAR IMPACT
A Dissertation in
Acoustics
by
Tae Kwang Yoo
2018 Tae Kwang Yoo
Submitted in Partial Fulfillment
of the Requirements
for the Degree of
Doctor of Philosophy
May 2018
ii
The dissertation of Tae Kwang Yoo was reviewed and approved* by the following:
Tong Qiu
Associate Professor in Civil Engineering
Dissertation Advisor
Co-Chair of Committee
Victor W. Sparrow
United Technologies Corporation Professor of Acoustics
Director, Graduate Program in Acoustics
Co-Chair of Committee
Sean Brennan
Associate Professor in Mechanical and Nuclear Engineering
Daniel A. Russell
Teaching Professor of Acoustics
Robert L. Campbell
Senior Research Associate, Applied Research Laboratory
Head, Structural Acoustics Department
*Signatures are on file in the Graduate School
iii
ABSTRACT
As terrorist attacks have frequently occurred in recent years, security measures to stop
or mitigate the damage they cause have been increasingly discussed and researched.
Vehicle anti-ram systems have been widely used for protecting sensitive buildings and
facilities against vehicular impacts. Numerous researchers have investigated vehicle anti-
ram systems under vehicular impact using the LS-DYNA research/commercial code and
field-scale crash tests. However, comparisons between different formulations in LS-
DYNA for the interaction between soil and embedded anti-ram systems involving large
soil deformation are remarkably sparse in the literature, particularly when the comparison
is validated using instrumented, field-scale crash tests. The aim of this study is to
numerically investigate several vehicle anti-ram systems with comparisons to experimental
results to improve the accuracy and efficiency of the simulation models and the impact
performance of the systems.
In the first part of the dissertation, two field-scale crash tests of Streetscape Vehicle
Anti-Ram (SVAR) barrier systems and LS-DYNA simulations to predict the global
response of each system under vehicular impact were conducted. Tests 1 and 2 consisted
of a five-post welded bus stop and a welded bollard, respectively. Test 1 resulted in a P1
rating, where minimal foundation uplift and rotation were observed; however, test 2 failed
to result in a P1 rating, where significant foundation uplift, rotation, concrete cracking, and
large deformation of surrounding soil were observed. For each test, two LS-DYNA
models, namely an FEM-only model and a hybrid FEM-SPH model, were created to predict
iv
the global response of the system. The hybrid FEM-SPH model did a much better job in
matching the crash test than the FEM model did. This research suggests that the hybrid
FEM-SPH approach is more appropriate in simulating the field performance of embedded
structures under impact loading when large deformation of the surrounding soil is expected.
In the second part of the dissertation, a series of experimental testing and numerical
modeling studies to optimize the parameters of a constitutive material model were
conducted to accurately simulate the behavior of polystyrene crushable concrete during
impact loading using LS-DYNA. Quasi-static compression tests and confined drop impact
tests were performed. To model the quasi-static compression tests, the response surface
methodology was used to optimize the Poisson’s ratio and friction angle in the pseudo-
tensor model in LS-DYNA. Using the optimized model parameters, the simulated
compression stress vs. strain relationship showed an excellent agreement with those from
the compression tests. To model the confined drop impact tests, the strain rate sensitivity
parameter in LS-DYNA was optimized by comparing the drop impact simulations at
different strain-rate sensitivity values with the drop impact tests. This study suggests that
the pseudo-tensor material model is suitable for modeling crushable concrete. Although
the optimized constitutive model parameters are specific for the polystyrene concrete mix
used in this study, a similar approach can be used to optimize model parameters for other
polystyrene concrete mixes.
In the third part of the dissertation, a series of tests were designed and conducted to
determine the angle of a boulder face at which the impact of a vehicle changes from
preventing override to allowing override. Medium-scale pendulum tests were performed
v
for the vehicular override research of a boulder with the impact face angled at 55°, 60°,
and 65° from the horizontal plane. A dimensionless analysis was conducted to properly
relate the pendulum test results to full-scale field situation. LS-DYNA simulations were
conducted to yield the input parameters needed for the dimensionless analysis. Vehicle
override is predicted to occur for the 55° override angle under both M30 and M50
scenarios, but not to occur for the 65° override angle under both scenarios. This prediction
is consistent with the results of high-fidelity LS-DYNA simulations of the full-scale crash
tests.
vi
Table of Contents
List of Figures………………………………………………………....……...……..……ix
List of Tables…………………………………………………...……………......……...xiii
Acknowledgments……………………………………………………………….…...…xiv
Chapter 1. Introduction…………………………………………………………………….1
1.1. Background…………………….......…...…….………….…………………….1
1.2. Problem Statement………………………..……...……………………………..2
1.3. Objectives………………..……………………………………………………..5
1.4. Dissertation Scope and Layout …………..……………………………………..5
Chapter 2. Field Testing and Numerical Investigation of Streetscape Vehicular Anti-Ram
Barrier under Vehicular Impact using FEM-Only and Coupled FEM-SPH
Simulations ……………………..…..…………………………………..……..7
2.1. Introduction………………………………...…………..………………………8
2.2. Field-Scale Testing…………………………………..………………………..11
2.2.1. SVAR Barriers………………………………………………...…………..13
2.2.2. Test Results………………………………………………………………..19
2.3. LS-DYNA Model……………………………………………………………..21
2.3.1. Truck Model……………………………………………………………….22
2.3.2. Material Models and Properties………………………..…………………..23
2.3.3. Soil Domain Size…………………………………………………………..26
2.3.4. SPH and FEM-SPH Coupling………………………….………………….28
vii
2.3.5. FEM and Hybrid FEM-SPH Models……………………………………….31
2.4. Results and Discussion………………………………………………………..33
2.5. Additional Verification of Improved Simulation Method……………………..39
2.6. Conclusions………………………………………………………...………….40
Chapter 3. Optimization of Constitutive Model Parameters for Simulation of Polystyrene
Concrete Subjected to Impact…...………….………..……………………….43
3.1. Introduction……………………………………………………………………44
3.2. Confined Compression Tests for Quasi-Static Analysis……………………….48
3.2.1. Crushable Concrete Mix Design……………..…………………………….48
3.2.2. Mixing Procedure………………………………………………….………50
3.2.3. Confined Compression Tests……………………..………………….…….51
3.3. Simulation and Optimization of Parameters for Quasi-Static Tests…………..52
3.3.1. Simulation of Confined Compression Tests………………………….……52
3.3.2. Parametric Study for Optimizing Material Parameters………..…...………58
3.3.3. Simulation of Unconfined Compression Test…………………….………..62
3.4. Confined Drop Impact Tests……………………………………….…….……64
3.5. Drop Impact Simulation and Optimization of Strain Rate Parameter…………68
3.6. Conclusions ……………………………………………………….…………..73
Chapter 4. Pendulum Override Tests and Simulations for Dimensionless Analyses
…………………………………………………………………………..….75
4.1. Pendulum Tests ……………………………..…………………………………...76
4.1.1. Facility or Equipment Description……………………………..…………..76
viii
4.1.2. Test Procedure…………………………………………………….……….77
4.1.3. Test Results…………………………………………………………..……80
4.2. Dimensionless Analysis…………………………………………...………...…..82
4.3. Pendulum and Full-Scale Crash Simulations……………………………………83
4.4. Dimensionless Analysis Results…………………………………………………91
4.5. Conclusions……………………………………………………..…………...…..93
Chapter 5. Conclusions and Recommendations………………………..…………………94
5.1. Conclusions…………………………………………………..………………….94
5.2. Recommendations for Future Research……………………...…………………..99
References………………………………………………………………….…………...100
Appendix A. Raw Data from Compression Test 1 and 13 Simulations…………………111
Appendix B. LS-DYNA Input Code of Simulation Case 1……………………….…….115
Appendix C. Analysis Results of Response Surface Methodology……………………..126
ix
List of Figures
Figure 2.1 Description of P1 rating according to ASTM F2656-07…………..……..……12
Figure 2.2 Location of high-speed video cameras during field-scale test (not to scale)…..13
Figure 2.3 Dimensions of five-post welded bus stop: (a) plan view; and (b) side view….15
Figure 2.4 Photographs showing installation of five-post welded bus stop……………….16
Figure 2.5 Dimensions of welded bollard: (a) plan view; and (b) side view……..………18
Figure 2.6 Photographs showing installation of welded bollard……………………...…..19
Figure 2.7 Final positions of barrier and truck after impact: (a) side and top views of Test
1; and (b) side and top views of Test 2………………….…....……………...21
Figure 2.8 Numerical models: (a) Test 1; and (b) Test 2…………………………..……..22
Figure 2.9 Horizontal and transverse tubes………………………………………...……..26
Figure 2.10 Soil domain in the field and in an initial LS-DYNA model………………….27
Figure 2.11 LS-DYNA model with different soil domain sizes……………………….….28
Figure 2.12 Particle approximation based on kernel function W in influence domain Ω with
radius kh………………………………………………………….…………29
Figure 2.13 LS-DYNA models for five-post welded bus stop: (a) FEM model; (b) hybrid
FEM-SPH model…………………………....………………………..……..32
Figure 2.14 LS-DYNA models for welded bollard: (a) FEM model; (b) hybrid FEM-SPH
model………………………………………………………………………..32
Figure 2.15 Modeling concrete slab using a single layer of fully-integrated solid
elements………………………………….………………………………….33
x
Figure 2.16 Comparison of crash test and simulations from FEM and hybrid FEM-SPH
models at various time for five-post welded bus stop………………………..34
Figure 2.17 Simulated mode of deformation of five-post welded bus stop: (a) FEM model;
(b) hybrid FEM-SPH model………………………………………….……..35
Figure 2.18 Comparison of crash test and simulations from FEM and hybrid FEM-SPH
models at various time for welded bollard…………………………..………36
Figure 2.19 Simulated deformation of welded bollard at t=0.5 second: (a) FEM model; (b)
hybrid FEM-SPH model…………………………………………………….36
Figure 2.20 Point tracking in Photron FASTCAM software……………………………..38
Figure 2.21 Comparison of displacements of front corner of cargo bed…………….……38
Figure 2.22 One-Post Lamppost SVAR system with foundation W110″×D18″×L48″
……………………………………………………...……….………………39
Figure 2.23 Comparison of test and simulation (One-Post Lamppost, 2011)………..…..40
Figure 3.1 Polystyrene concrete sample showing that a polystyrene volume fraction of 40%
is sufficient to produce a closed cell foam…………………………………..49
Figure 3.2 Expanded polystyrene spheres………………………………………………..50
Figure 3.3 Testing setup for confined compression tests………………………..………..52
Figure 3.4 Modeling confined compression tests in LS-DYNA…………………………54
Figure 3.5 Three mesh sizes for mesh-size convergence test: (a) mesh size = 13 mm; (b)
mesh size = 6.5 mm; and (c) mesh size = 3 mm……………………………..54
Figure 3.6 Compressive stress vs. strain of three mesh size models………………………55
xi
Figure 3.7 Simulated compressive stress vs. strain for four tensile cutoff values…….….58
Figure 3.8 Applying RSM for parametric study to optimize material parameters…….….59
Figure 3.9 Values of variables and sequence of randomly designed simulations…….…..60
Figure 3.10 Results from compression Test 1 and 13 simulations…………………….…61
Figure 3.11 Optimizing results of 13 simulation cases…………………………….……..61
Figure 3.12 Comparison between optimized simulation and confined compression tests..62
Figure 3.13 Unconfined compression test setup and LS-DYNA model……………….….63
Figure 3.14 Comparison between optimized simulation and unconfined compression
test…………………………………………………………………………..63
Figure 3.15 Drop impact test setup for crushable concrete specimens……………………65
Figure 3.16 An image recorded by high-speed camera just before impact………………..65
Figure 3.17 Point tracking in motion analysis using Photron FASTCAM software……...66
Figure 3.18 Loading piston displacement vs. time for drop impact Tests 1 and 2……..…67
Figure 3.19 Impact force measured by load cell vs. time for drop Tests 1 and 2………...67
Figure 3.20 Modeling drop impact test in LS-DYNA…………………………………….69
Figure 3.21 Impact force vs. time from drop impact tests and simulations………..……..71
Figure 3.22 Displacement of loading piston vs. time from drop impact tests and
simulations…………………………….…………………………………….71
Figure 3.23 Displacement of loading piston vs. time from an independent drop impact test
and simulation……………………………….…………………………...….72
Figure 4.1 Representation of full-scale vehicle and embedded boulder ………...………..75
Figure 4.2 Description of cable mounted pendulum …………………...………..…...…..76
xii
Figure 4.3 Image of impact sled …………………………..……………………………..77
Figure 4.4 An American Black granite boulder placed on compacted and amended
AASHTO soil ………………………………...…………………...………..79
Figure 4.5 Angled boulder (65°) and impact sled pulled to maximum height ………..…..79
Figure 4.6 Override for 55° incline face - 0.05 seconds between images …………….…..80
Figure 4.7 Override for 60° incline face - 0.05 seconds between images …………….…..81
Figure 4.8 Override for 65° incline face - 0.05 seconds between images ……………..….81
Figure 4.9 Two consecutive sequential images for investigating effect of cables ………..84
Figure 4.10 Simulation models: (a) With cables; (b) Without cables ………………...…84
Figure 4.11 Measurement point for comparing test results ……..…………………….….87
Figure 4.12 X displacement from pendulum test and simulations of different conditions..88
Figure 4.13 Y displacement from pendulum test and simulations of different conditions..88
Figure 4.14 Full-scale crash simulation model ……...…………….……………………..90
Figure C.1 Analysis results for response surface regression…………………....……….125
Figure C.2 Contour plot of response vs. friction angle, Poisson’s ratio………..……….126
Figure C.3 Contour plot of response vs. friction angle, Poisson’s ratio……….…….…..127
Figure C.4 Optimization plot……………………………………………………………128
xiii
List of Tables
Table 2.1 Model parameters for soil……………………………………………...………24
Table 2.2 Model parameters for concrete………………………………………….……..24
Table 2.3 Model parameters for rebar, stiffeners, studs, and tubes……….………………25
Table 3.1 Polystyrene concrete mix design details (per m3 of concrete)……………..…..50
Table 3.2 Model parameters for crushable concrete…………………………..………….58
Table 4.1 Model parameters for boulder and soils …...…………………………………..86
Table 4.2 Model parameters for impact sled ……………………...……………..……….87
Table 4.3 Results of dimensionless calculations based on LS-DYNA simulations ………91
xiv
Acknowledgments
I would like to thank my family for their love and support. This study was a big challenge
in my life because I had to quit my job and leave my family in my country. A lot of people
asked me why you are going to quit your job. However, my family believed that I made a
right decision for my life and it will be a great chance to get better in my future life. Also,
they were always willing to support me. So, I am truly grateful to my wife, children, father,
mother, and brother for their unstinting help.
I have participated in a large and multidisciplinary project for my research. So, I have
worked with researchers and faculty members in various fields such as Civil Engineering,
Mechanical Engineering, and Acoustics, which has been very helpful and efficient to
conduct my research and a pretty good opportunity to learn different fields’ knowledge. In
terms of that, Dr. Tong Qiu has given me a lot of great advice about his field and the project
and I am very grateful for his effort and patience. Also, Dr. Victor W. Sparrow has tried to
specifically check my academic and research progress and I am pretty grateful for his
consideration and support. I am very thankful to Dr. Sean Brennan, Dr. Daniel A. Russell,
and Dr. Robert L. Campbell, who offered their expertise and advice during my research.
Finally, I would like to thank my fellow graduate students for their support.
Chapter 1. Introduction
1.1. Background
As terrorist attacks have frequently occurred in recent years, security measures to stop
or mitigate the damage they cause have been widely discussed and researched. With raised
national security awareness due to these terroristic threats, safety systems that can prevent
the loss of life and structural damage as a result of impact and blast loading have been
considered. One of the critical research topics has been perimeter security systems (Griebel
and Phillips, 2001; Grosskopf, 2006; Benton-Short, 2007). Sensitive buildings and
facilities have been frequently protected by vehicle anti-ram systems against terrorists’
vehicular impacts. These systems are typically categorized as Streetscape Vehicle Anti-
Ram (SVAR) systems and Rock Vehicle Anti-Ram (RVAR) systems. The SVAR system
is usually applied for urban regions such as bollards typically consisting of artificial
materials involving steel and concrete. On the other hand, the RVAR system is mainly
applied for suburban regions, generally consisting of natural materials usually including
rock and soil such as boulders.
The crashworthiness of anti-ram systems should be thoroughly investigated, which is
commonly conducted by field-scale testing and numerical modelling. In order to
effectively utilize the two methods, crash simulations are first conducted by using explicit
dynamic solvers to obtain an initial design of the system, and the initial design is then
validated by full-scale crash tests which serve to further calibrate simulation parameters.
Typical SVAR systems under vehicular crash have been examined by many researchers
using the LS-DYNA research/commercial code (Hallquist, 2006) and field-scale crash tests
2
(Krishna-Prasad, 2006; Omar et al., 2007; Liu et al., 2008; Ferdous et al., 2011; Hu et al.,
2011; O’Hare et al., 2012; Uzzolino et al., 2012; Reese et al., 2014, 2016; Keske et al.,
2015; Yoo et al., 2016). A variety of SVAR systems with shallow foundations (e.g.,
lampposts, bus stops, and street signs) were designed by O’Hare et al. (2012), which can
be selected based on site restrictions, availability, cost of materials, and various
architectural aesthetics. The designs of these SVAR systems were optimized through
numerous LS-DYNA simulations and validated through field-scale crash tests (O’Hare et
al., 2012; Yoo et al., 2016). On the other hand, there has been little research about RVAR
systems (Reese et al., 2014 and 2016). Research on various RVAR systems with a boulder
embedded in soil was conducted at the Larson Transportation Institute affiliated with the
Pennsylvania State University. The overall global response of the systems under vehicular
impact was predicted through LS-DYNA simulations and compared with collected field
test data.
1.2. Problem Statement
In order to simulate vehicular crashes, LS-DYNA which is based on the finite element
method (FEM) is frequently used. The FEM can efficiently simulate large-scale dynamic
problems. However, it is difficult to handle large deformations due to severely distorted
meshes, imprecise results, and failure of convergence. Adaptive remeshing (Khoei and
Lewis, 1999) and Arbitrary Lagrangian–Eulerian method (Hughes et al., 1981) have been
applied to solve this issue. However, these developed techniques can be problematic when
complex constitutive models are employed (Bui et al., 2008). On the whole, continuum-
3
scale numerical techniques without a mesh (i.e., mesh free) are considered more desirable
for the simulation of problems involving both large scale and large deformations. Many
researchers recently developed various mesh-free methods tracking materials through a set
of particles rather than grids. A detailed discussion of various mesh-free methods is
presented by Liu and Liu (2003). The Smoothed Particle Hydrodynamics (SPH) method
which is one of these mesh-free continuum-scale methods is a relatively mature one. The
SPH method was originally developed for astrophysical applications by Lucy (1977) and
Gingold and Monaghan (1977). It has been broadly applied to simulate free surface flows
and multiphase flows (Monaghan, 1994; Monaghan and Kocharyan, 1995; Monaghan et
al., 2003) and flow through porous media (Zhu et al., 1999). The SPH method has been
recently used to simulate the elastic response of solids (Libersky et al., 1993; Gray et al.,
2001) and the elasto-plastic behavior of geomaterials (Bui et al., 2008; Chen and Qiu, 2012,
2014). In addition, coupled SPH and FEM formulations have been developed to utilize the
benefits of both methods (Beal et al., 2013; Bojanowski, 2014). However, published
comparisons between traditional FEM and coupled FEM-SPH simulations for soil-
structure interaction involving large soil deformation are remarkably sparse in literature,
particularly when the comparison is validated using instrumented, field-scale tests (Reese
et al., 2012; Reese et al., 2014; Zhou et al., 2016).
Although a reasonable simulation model can be obtained through applying the
appropriate simulation method, which can efficiently deal with the large deformation, the
major aim of these barriers is to prevent an attack vehicle ramming into buildings. In order
to achieve this aim, the barrier should absorb a large amount of kinetic energy from the
attack vehicle. This kinetic energy is generally absorbed by the deformation of the barrier
4
and the surrounding soil. In addition, due to increased requirements of minimizing the
barrier size, it has become much more difficult to design a barrier capable of stopping an
attack vehicle. Crushable materials for the barrier can be incorporated to dissipate some
of an attack vehicle’s kinetic energy, potentially reducing the size and cost of the barrier
without any loss of ability to stop the vehicle. Crushable concrete has been applied for
aircraft arrestor beds (White and Agrawal, 1993; Heymsfield and Halsey, 2008; Zhang et
al., 2013) and absorption of blast energy from explosions outside of structures (Li and
Muthyala, 2008). Many researchers have conducted numerical investigations on the
behavior of concrete and reinforced concrete under impact loading through various reliable
material models in LS-DYNA (Tai and Tang, 2006; Teng et al., 2004; Tu, 2009). However,
few studies about the numerical simulation of crushable concrete have been conducted.
Even though the improved simulation method suggested above could save much time
in predicting the impact performance, the simulations usually require more than 10 hours
for just one case and it would take much more time if parametric studies are needed to
improve the system design. Dimensional analysis can be applied to low-order models as a
means of developing dimensionless equations of motion and scaling laws. The scaling
laws can then be used to scale up the results of small-scale crash tests such as pendulum
tests to the full-scale field conditions. The major appeal in small-scale testing and
simulating is that an expensive full-scale crash can be duplicated in a much smaller and
cheaper manner. Industry has already verified this concept with the simple advent of wind
tunnels (Howell et al., 2010; Burdett and Van Treuren, 2012; Van Treuren, 2015).
5
1.3. Objectives
This Ph.D. research has three main objectives. The first objective is to quantitatively
compare the performances of traditional FEM simulations and coupled FEM-SPH
simulations in modeling full-scale vehicular crash tests involving large vehicle and soil
deformations. The second objective is to optimize the constitutive model parameters for a
crushable concrete material that can be incorporated into the design of a barrier system.
The third objective is to properly relate small-scale pendulum test results to full-scale field
situation through a dimensionless analysis.
1.4. Dissertation Scope and Layout
In order to accomplish the aforementioned objectives, this dissertation consists of the
following chapters.
Chapter 1 presents the background, motivation of this study, a brief review of available
literature summarizing relevant work, and dissertation objectives, scope, and layout.
Chapter 2 is related to the first objective. In this chapter, two field-scale crash tests of
SVAR barrier systems and LS-DYNA simulations to predict the global response of each
system under vehicular impact are presented. Tests 1 and 2 consisted of a five-post welded
bus stop and a welded bollard, respectively. Test 1 resulted in a P1 rating, where minimal
foundation uplift and rotation were observed. Test 2 failed to result in a P1 rating, where
significant foundation uplift, rotation, concrete cracking, and large deformation of
surrounding soil were observed. For each test, two LS-DYNA models, namely an FEM-
only model and a hybrid FEM-SPH model, were created to predict the global response of
6
the system. The performances of these models in capturing the global responses of the
crash tests are discussed and compared.
Chapter 3 is related to the second objective. This chapter presents the results of a series
of experimental testing and numerical modeling studies to optimize the parameters of a
constitutive material model to accurately simulate the behavior of polystyrene crushable
concrete during impact loading using LS-DYNA. Quasi-static compression tests and
confined drop impact tests were conducted. To model the quasi-static compression tests,
the response surface methodology was used to optimize the Poisson’s ratio and friction
angle in the pseudo-tensor model in LS-DYNA. Using the optimized model parameters,
the simulated compression stress vs. strain relationship showed an excellent agreement
with those from the compression tests. To model the confined drop impact tests, the strain
rate sensitivity parameter in LS-DYNA was optimized by comparing the drop impact
simulations at different strain-rate sensitivity values with the drop impact tests.
Chapter 4 is related to the third objective. A series of tests were designed and
conducted to determine the angle of a boulder face at which the impact of a vehicle changes
from preventing override to allowing override. This chapter presents the setup, testing and
preliminary results of the pendulum tests performed for the vehicular override research of
a boulder with the impact face angled at 55°, 60°, and 65° from the horizontal plane. A
dimensionless analysis was conducted to properly relate the pendulum test results to full-
scale field situation.
Chapter 5 draws final conclusions and provides recommendations for future research
on designing similar anti-ram barriers.
7
Chapter 2. Field Testing and Numerical Investigation of Streetscape
Vehicular Anti-Ram Barrier under Vehicular Impact using FEM-Only
and Coupled FEM-SPH Simulations
This chapter is published as: Yoo TK, Qiu T, Reese L, et al. (2016) Field Testing and
Numerical Investigation of Streetscape Vehicular Anti-Ram Barriers under Vehicular
Impact using FEM and Coupled FEM-SPH Simulations. International Journal of
Protective Structures 7(2): 213–231.
Keywords: anti-ram system, barrier, bollard, crash test, large deformation, numerical
model, smoothed particle hydrodynamics method, soil-structure interaction
ABSTRACT
This paper presents two field-scale crash tests of SVAR barrier systems and LS-DYNA
simulations to predict the global response of each system under vehicular impact. Tests 1
and 2 consisted of a five-post welded bus stop and a welded bollard, respectively; both
were in a steel and concrete composite foundation embedded in compacted AASHTO
aggregate. Test 1 resulted in a P1 rating, where minimal foundation uplift and rotation
were observed. Test 2 failed to result in a P1 rating, where significant foundation uplift,
rotation, concrete cracking, and large deformation of surrounding soil were observed. For
each test, two LS-DYNA models, namely an FEM-only model and a hybrid FEM-SPH
8
model, were created to predict the global response of the system. In the FEM-only model,
traditional FEM approach was used for the entire soil region; in the hybrid FEM-SPH
model, the near-field soil region was modeled using the SPH approach, whereas the far-
field soil region was modeled using the FEM approach. For Test 1 both the FEM-only
model and the hybrid FEM-SPH model were able to match the recorded global response
of the system. For Test 2, however, the FEM-only approach was not able to accurately
predict the global response of the system; on the other hand, the hybrid FEM-SPH approach
was able to capture the global response including the bollard pullout, soil upheaval, and
vehicle override. This research suggests that the hybrid FEM-SPH approach is more
appropriate in simulating the field performance of embedded structures under impact
loading when large deformation of the surrounding soil occurs.
2.1. Introduction
Vehicle anti-ram systems have been widely used for protecting sensitive buildings and
facilities against vehicular impacts. These systems generally include Streetscape Vehicle
Anti-Ram (SVAR) systems and Landscape Vehicle Anti-Ram (LVAR) systems. The
SVAR system (e.g., anti-ram bollards) is generally used in urban areas, which are typically
comprised of man-made materials including steel and concrete. The LVAR system is more
often used in suburban areas and typically made of natural materials such as boulders. It
is important to examine the crashworthiness of anti-ram systems, and field-scale testing
and numerical modelling are two common methods. An integrated computational and
experimental approach is particularly attractive. In this approach, implicit/explicit
9
dynamic solvers are first utilized to run crash simulations and arrive at an initial design of
the system, full-scale crash tests are subsequently utilized to validate the initial design and
provide the needed data for calibrating simulation parameters.
Numerous researchers have investigated typical SVAR systems under vehicular impact
using the LS-DYNA research/commercial code (Hallquist, 2006) and field-scale crash tests
(Ferdous et al., 2011; Hu et al., 2011; Krishna-Prasad, 2006; Liu et al., 2008; O’Hare et al.,
2012; Omar et al., 2007; Uzzolino et al., 2012). O’Hare et al. (2012) developed various
SVAR systems with shallow foundations, including street benches, bus stops and street
signs, which can be selected based on site restrictions, availability and cost of materials,
and varying architectural aesthetics. Designs of these SVAR systems were optimized using
LS-DYNA and then validated through field-scale crash testing (O’Hare et al., 2012).
LS-DYNA is known to be a reliable program for modeling vehicular impact and is
based on the finite element method (FEM). The FEM is efficient for large-scale explicit
dynamic problems. One shortcoming associated with the method, however, is the difficulty
in dealing with large deformations, which may lead to severe distortion of meshes,
inaccurate results, and failure of convergence. Advanced techniques such as adaptive
remeshing (Khoei and Lewis, 1999) and Arbitrary Lagrangian–Eulerian method (Hughes
et al., 1981) have been used to remediate this problem. However, these remeshing
techniques become problematic when complex constitutive models are employed (Bui et
al., 2008). Overall, continuum-scale numerical methods that do not require a mesh (i.e.,
mesh free) are considered more desirable for the simulation of problems involving both
large scale and large deformations. In recent years, several mesh-free methods tracking
materials by a set of particles instead of grids have been developed. A detailed discussion
10
of various mesh-free methods is presented by Liu and Liu (2003). Among these mesh-free
continuum-scale methods, the Smoothed Particle Hydrodynamics (SPH) method is a
relatively mature one. Originally developed for astrophysical applications by Lucy (1977)
and Gingold and Monaghan (1977), SPH method has been widely used to simulate free
surface flows and multiphase flows (Monaghan, 1994; Monaghan and Kocharyan, 1995;
Monaghan et al., 2003) and flow through porous media (Zhu et al., 1999). More recently,
the SPH method has been used to simulate the elastic response of solids (Gray et al., 2001;
Libersky et al., 1993) and elasto-plastic behavior of geomaterials (Bui et al., 2008; Chen
and Qiu, 2012, 2014). Additionally, coupled SPH and FEM formulations have been
developed to utilize the benefits of both methods (Beal et al., 2013; Bojanowski, 2014) so
that coupling of the two domains, one consisting of particles and the other consisting of
meshes, is improved and the computation time is optimized.
Published comparisons between traditional FEM and coupled FEM-SPH simulations
for soil-structure interaction involving large soil deformation are remarkably sparse in
literature, particularly when the comparison is validated using instrumented, field-scale
tests (Reese et al., 2012; Reese et al., 2014; Zhou et al., 2016). This paper presents the
numerical simulations and field-scale crash tests of two SVAR systems: a five-post welded
bus stop and a welded bollard, both embedded in soil. For the former, little deformation
of soil was observed from the crash test; whereas large soil deformation was observed for
the latter. For each crash test, two LS-DYNA models, namely an FEM-only model and a
hybrid FEM-SPH model, were created to predict the global response of the system. In the
following sections, the field-scale crash tests are first discussed, followed by descriptions
of the FEM-only and coupled FEM-SPH models. The simulations and crash test results
11
are compared. Finally, conclusions are reached regarding the predictive capabilities and
limitations of the FEM-only and coupled FEM-SPH formulations in LS-DYNA
simulations of SVAR barriers.
2.2. Field-Scale Testing
Vehicular impact tests were completed according to ASTM F2656-07 - Standard Test
Method for Vehicle Crash Testing of Perimeter Barriers (ASTM 2007), which establishes
penetration ratings for perimeter barriers subjected to vehicular impact, at the Larson
Transportation Institute affiliated with The Pennsylvania State University. For M50 impact
(i.e., vehicular speed of 80.5 km/hr or 50 miles/hr), a penetration distance of equal or less
than 1 m is required to achieve a P1 rating. Figure 2.1 shows a description of the P1 rating
according to ASTM F2656-07. The penetration distance is measured from the inside face
or non-impact surface of the test article (blue dotted line in Figure 2.1) to the front of the
cargo bed (red circle in Figure 2.1) when the test vehicle has reached its final positon. In
Figure 2.1, the red circle should not exceed the red dotted line for P1 rating. The test
facility uses a rigid rail to provide vehicle guidance, a reverse towing system to accelerate
the test vehicle to the required speed, and a release mechanism that disconnects the tow
cable and steering guidance prior to impact. For a detailed description of the system, please
refer to Reese et al. (2012, 2014). The test vehicles used in this study were flatbed medium-
duty diesel trucks. Barrels filled with ballast were secured on the bed of the truck making
the test mass of approximately 6750 kg. The height of the lower and upper edges of the
front bumper was 0.48 m and 0.75 m, respectively.
12
High-speed cameras (1,000 frames-per-second, 1016 by 1016 pixels resolution) were
implemented during testing to record pertinent information such as barrier translational and
rotational displacements and global response of the system which includes truck
deformation. Figure 2.2 shows where the high-speed cameras were placed. Camera 1 was
positioned at a 90° angle to the center of the test article to measure dynamic penetration.
Camera 2 was positioned at a 90° angle above the center of the test article to capture enough
surface area prior to and after impact to determine impact speed, impact angle, exit angle
and debris field. Camera 3 was positioned behind the test article centered along the guide
rail to record the approach of the test vehicle to track its alignment with the center of the
test article during impact.
Figure 2.1 Description of P1 rating according to ASTM F2656-07
Test Article
Test Vehicle
1 m
13
Figure 2.2 Location of high-speed video cameras during field-scale test (not to scale)
2.2.1. SVAR Barriers
Two full-scale crash tests of SVAR barriers for M50 impact were conducted for this
research. Test 1 consisted of a five-post welded bus stop in a steel and concrete composite
foundation. Figure 2.3 shows the dimensions of the device. The foundation has a
dimension of 3,556 mm × 1,219.2 mm × 457.2 mm (140 in × 48 in × 18 in). The device
utilized a fully welded design and E70XX (½ inch) welds were used at all connections.
The five vertical posts consisted of 152.4 mm × 101.6 mm × 12.7 mm (6 in × 4 in × ½ in)
A500 Gr. B steel tubes protruding above grade to a height of 2,641.6 mm (104 in), with a
clear spacing of 711.2 mm (28 in). To enable above-grade load sharing between the
vertical tubes, a 254 mm × 101.6 mm × 12.7 mm (10 in × 4 in × ½ in) A500 Gr. B steel
horizontal girt tube spanned between the vertical tube members. Vertical members were
also connected below grade to horizontal 304.8 mm × 304.8 mm × 15.9 mm (12 in × 12 in
× ⅝ in) A500 Gr. B steel tubes through member penetration. The vertical members passed
Camera 1
Test Vehicle
Test Article
Camera 2
Camera 3
14
completely through the horizontal members and protruded 25.4 mm (1 in) below these
members, forming an “L” shape in elevation (see Figure 2.3). To strengthen and
interconnect each vertical and horizontal tube system, a pair of stacked transverse 76.2 mm
× 76.2 mm × 9.5 mm (3 in × 3 in × ⅜ in) A500 Gr. B steel tubes penetrated both the
horizontal and vertical members orthogonally to ensure the transfer of bending, shear, and
torsion, and prevent vertical member pullout. The transverse tubes had a vertical center-
to-center spacing of 15.24 cm (6 in). Concrete was then poured into the foundation and fill
the horizontal and vertical steel tubes. Concrete on the day of testing was recorded having
a compressive strength of 29.6 MPa (4,296 psi). There were no aesthetic features attached
to this device. Figure 2.4 shows photographs of the test article installation.
15
(a) Plan view
(b) Side view
Figure 2.3 Dimensions of five-post welded bus stop: (a) plan view; and (b) side view
16
Figure 2.4 Photographs showing installation of five-post welded bus stop
Test 2 consisted of a steel tube in a steel and concrete composite foundation. Figure
2.5 shows the dimensions of the device. The foundation has a dimension of 1,400 mm ×
1,220 mm × 480 mm (55 in × 48 in × 18 in). The device utilized a fully-welded design and
the impacted member consisted of a vertical 254 mm × 254 mm × 15.9 mm (10 in × 10 in
× ⅝ in) A500 Grade B steel tube protruding above grade to a height of 1000 mm (39 in).
This vertical tube was internally stiffened with a W8×48 (203.2 mm × 1,219.2 mm) A992
stiffener member. Vertical members were connected below grade to a horizontal 304.8
17
mm × 304.8 mm × 15.9 mm (12 in × 12 in × ⅝ in) A500 Grade B steel tube through member
penetration. The vertical members passed completely through the horizontal members and
protruded 25.4 mm (1 in) below, forming an “L” shape in elevation (see Figure 2.6). To
strengthen and interconnect the vertical and horizontal tube system, a pair of stacked
transverse 50.8 mm × 50.8 mm × 6.4 mm (2 in × 2 in × ¼ in) A500 Grade B steel tubes
penetrated both the horizontal and vertical members orthogonally to ensure the transfer of
bending, shear, and torsion, and prevent vertical member pullout. The transverse tubes
have a vertical center-to-center spacing of 152.4 mm (6 in). There were no aesthetic
features added to this device. Figure 2.6 shows photographs of the test article installation.
The vehicle used in Test 1 was a 1997 NAVSTAR 4700 and used in Test 2 was a 1995
Chevrolet Kodiak. These vehicles conform to the ASTM F2656 requirements for the
medium-duty diesel truck. The test vehicles were structurally sound, having no major rust
or weaknesses noted. No structural modifications or additions were observed that might
enhance or otherwise affect test performance. Five 55-gallon drums with removable lids
were filled with ballast consisting of quarry waste/gravel and placed in two rows at each
truck’s front end of the bed. The drums were secured with ratchet straps to each truck’s
front bed bulkhead. The weight of the vehicle in Test 1 was 6,812 kg (15,020 lbs) and in
Test 2 was 6,831 kg (15,060 lbs), which is within the test weight range of 6,660 – 6,940 kg
(14,691 – 15,309 lbs) as specified by ASTM F2656.
18
(a) Plan view
(b) Side view
Figure 2.5 Dimensions of welded bollard: (a) plan view; and (b) side view
19
Figure 2.6 Photographs showing installation of welded bollard
2.2.2. Test Results
Figure 2.7 shows the final positions of the truck and device for both tests. For Test 1,
the impact was centered on the device’s centerline. Large deformations of the vertical
posts were observed. The vertical posts showed similar deflections averaging at
approximately 1,120 mm (44.1 in) and good above-grade load sharing capability. The
impacted vertical post showed double-curvature deflection, whereas the remaining four
20
posts showed single-curvature deflection. Minimal foundation uplift and rotation was
observed (less than 12.7 mm of vertical foundation uplift on the attack side). As seen in
Figure 2.7 (a), the front of the truck bed stopped at 1,320 mm (52 in) before the back face
of the impacted post. Therefore, Test 1 resulted in a P1 penetration rating.
Test 2 however did not result in a P1 penetration rating. As shown in Figure 2.7 (b),
final penetration of the front corner of the cargo bed was 9,100 mm (358 in) beyond the
pre-test inside edge of the barrier. The vertical post, stiffener, horizontal tube, rebar and
concrete primarily remained intact as one unit but was pulled out of the ground. Significant
foundation uplift, rotation, and concrete cracking were observed; the steel and concrete
barrier translated 8,100 mm (319 in). Concrete spalling was observed over a majority of
the top surface.
21
(a) Side and top views of Test 1 (five-post welded bus stop)
(b) Side and top views of Test 2 (welded bollard)
Figure 2.7 Final positions of barrier and truck after impact: (a) side and top views
of Test 1; and (b) side and top views of Test 2
2.3. LS-DYNA Model
The LS-DYNA research/commercial code (Hallquist, 2006) was utilized to simulate
the two crash tests. In this section, the numerical model is first described, including
material properties and descriptions of the FEM-SPH contact algorithms used within LS-
DYNA. The general numerical model consists of three major parts which are the medium-
1,320 mm
9,100 mm
22
duty truck, the SVAR barrier device, and the surrounding soil and concrete slab. Figure
2.8 shows the numerical models for the two tests.
Figure 2.8 Numerical models: (a) Test 1; and (b) Test 2
2.3.1. Truck Model
The truck model used for the simulations was modified from a model readily-available
in the National Crash Analysis Center (NCAC) database (2008). The NCAC truck model
was developed to ensure that the load transfer between the truck and hardware, the
deformation of the truck, and the overall behavior of the truck during impact simulations
could be as accurate as feasible given the model computational requirements. Based on
requirements from ASTM F2656-07 (2007), the modified truck model consisted of 1,606
eight-node constant stress solid elements, 20,333 four-node Belytschko-Tsay shell
elements, and 377 Hughes-Liu beam elements with cross section integration. The truck
model was validated in Reese et al. (2014) using checks of equilibrium, conservation of
energy principles, and the amount of energy absorption that occurred through plastic
Medium-duty truck
Concrete slab
Soil
Post
Foundation
Medium-duty truck
Concrete slab
Soil
Bollard
Foundation
Concrete
Rail
Foundation
Concrete
Rail
Foundation
23
deformation of truck components. The numerical simulation in Reese et al. (2014)
indicated that the truck absorbed approximately 69% of the impact energy and the barrier
and soil absorbed the rest of the impact energy when hitting the barrier, which is consistent
with the field-scale test conducted by Omar et al. (2007).
2.3.2. Material Models and Properties
The LS-DYNA Material Type 173, “Mohr-Coulomb (M-C)” (Hallquist, 2013) was
utilized to model the soil beneath the concrete slab (see Figure 2.8). The soil is AASHTO
(American Association of State Highway and Transportation Officials) uniformly-graded
coarse aggregate and was compacted to 90% relative compaction as evaluated by ASTM
D698-12 (2012). The M-C model was used due to its simplicity and found adequate in
modeling the behavior of AASHTO aggregate when an embedded boulder was subject to
vehicular impact by Reese et al. (2014). The M-C model characterizes failure of a material
based on its cohesion, normal stress on an element and friction angle as follows (Hallquist,
2006, 2013):
)tan(max nc (2.1)
where max is the shear strength on any plane, n is the normal stress on that plane, c is
cohesion, and is friction angle. Based on the gradation and angularity of the AASHTO
coarse aggregate, the friction angle is estimated to be 45° and the dilation angle is estimated
to be 15°, based on an empirical relation between friction angle and dilation angle from
Bolton (1986). Model parameters for the soil are summarized in Table 2.1, which were
calibrated and validated by Reese et al. (2014).
24
Table 2.1 Model parameters for soil
Density
(kg/m3)
Elastic shear
modulus
(MPa)
Poisson’s
ratio
Cohesion
(kPa)
Friction
angle
(degrees)
Dilation
angle
(degrees)
Soil 2,100 20 0.25 4.8 45 15
The LS-DYNA Material Type 159, “Smooth or Continuous Surface Cap Model” was
utilized to model the concrete, in which the essential material properties include density,
elastic modulus, Poisson’s ratio, uniaxial compressive strength, tensile strength, and tensile
fracture energy. Instead of fitting a set of material model parameters to experimental data,
typical material properties were used in this study. Specifically, with a given uniaxial
compressive strength of 21 MPa and a maximum aggregate size of 38 mm, the other
material properties such as elastic modulus and tensile fracture energy were calculated
based on empirical equations available in the literature (Murray, 2007). The material
parameters for the concrete are summarized in Table 2.2.
Table 2.2 Model parameters for concrete
Density
(kg/m3)
Elastic
modulus
(GPa)
Poisson’s
ratio
Uniaxial
compressive
strength
(MPa)
Tensile
strength
(MPa)
Tensile
fracture
energy
(N/m)
Concrete 2,320 23 0.15 21 2 54
25
The LS-DYNA Material Type 24, “Elasto-Plastic Material” (Hallquist, 2013) was
utilized to model the rebar, stiffeners, studs, and tubes, and the material parameters for each
part are summarized in Table 2.3. In addition, the horizontal and transverse tubes, as shown
in Figure 2.9, were modeled by using reduced yield strength and plastic strain to failure (as
compared to the properties of vertical tube) to account for the welding between them.
Table 2.3 Model parameters for rebar, stiffeners, studs, and tubes
Density
(kg/m3)
Elastic
modulus
(GPa)
Poisson’s
ratio
Yield
strength
(MPa)
Tangent
modulus
(GPa)
Plastic
strain
to
failure
Rebar 7,830 200 0.3 476 2.1 0.12
Stiffeners 7,830 200 0.3 344 2.1 0.25
Studs 7,830 200 0.3 400 2.1 0.12
Vertical tube 7,830 200 0.3 315 2.1 0.25
Horizontal
and
transverse
tubes
7,830 200 0.3 283 2.1 0.20
26
Figure 2.9 Horizontal and transverse tubes
2.3.3. Soil Domain Size
The size of soil domain surrounding the SVAR device plays an important role in the
LS-DYNA simulations and was determined through a parametric study by gradually
increasing the soil domain size until convergence in simulation results was obtained.
Figure 2.10 presents the soil domain in the field and in an initial LS-DYNA model. A
small domain size was utilized in the initial simulation for computational efficiency,
although the domain is much larger in the field.
Transverse tube
Horizontal tube
27
Figure 2.10 Soil domain in the field and in an initial LS-DYNA model
For the parametric study, the size of the soil domain varied from 2-times to 8-times the
initial model in horizontal directions, as shown in Figure 2.11, to examine the effects of
reflected compression waves that may interfere with the impact response. The fixed
boundary conditions for all of directions were utilized for the four side boundaries and
bottom boundary. In addition, the non-reflecting boundary conditions, which are important
for limiting the spatial extent of the finite element mesh and thus the number of solid
elements for geo-mechanical problems, were used to reduce the effect of reflected
compression waves from the four side boundaries and bottom boundary. Based on the
parametric study, a soil domain that is 5-times the initial domain size was used to simulate
the crash tests.
Y=0.68m X=0.68m
Impact
directionFoundation
Impact
direction
28
Figure 2.11 LS-DYNA model with different soil domain sizes
2.3.4. SPH and FEM-SPH Coupling
Traditional FEM formulations have been discussed extensively in the literature and
hence are not presented herein. The SPH formulation used in this study is briefly discussed
in this section. In SPH, the computational domain is discretized into a finite number of
particles, each representing a certain volume and mass of the material (fluid or solid) and
carrying simulation parameters such as velocity, acceleration, density, and pressure/stress.
The particles interact according to a set of rules. Figure 2.12 shows a description of particle
approximation based on a kernel function 𝑊 for particles within an influence domain Ω
defined by a radius 𝑘ℎ, where ℎ is the initial particle spacing and 𝑘 is a constant (1.2 was
1.37m 1.37m
2.06m 2.06m
2.74m 2.74m3.43m 3.43m
x2 Size x3 Size
x4 Size x5 Size
29
used in this study). The widely-used cubic spline kernel function (Monaghan and
Lattanzio, 1985) was used in this study. There has been an issue of FEM mesh truncating
the influence domain of SPH particles in the vicinity of a FEM-SPH boundary. To address
this issue, Sakakibara et al. (2008) conducted analyses of the effects of changing
formulation type while keeping the smoothing length constant. In this study, to account
for the issue, a renormalization technique readily available in LS-DYNA was used for all
SPH analyses.
Figure 2.12 Particle approximation based on kernel function 𝑊 in influence domain
Ω with radius 𝑘ℎ
Both SPH and FEM formulations in LS-DYNA are based on the Lagrangian approach.
Therefore, it is possible to link both methods at an interface. The interface should ensure
continuous bonding of the two methods. At the interface, the SPH particles are constrained
and move with the FEM elements. The influence domain of the particles at/near the
Κh
ΩW
i
Sphere of influence for particle i
30
interface zone covers both FEM elements and SPH particles and, hence, certain
considerations are required in the computation. For strain and strain rate calculations of
each particle, only those from the SPH particles within the influence domain are considered,
whereas the contributions from both SPH particles and FEM elements inside the influence
domain are included to calculate forces (Johnson, 1994).
LS-DYNA allows FEM and SPH to exist and interact in one simulation allowing users
to take advantage of both procedures. The interaction or coupling between FEM and SPH
can be defined using traditional tied- or penalty-based contact definitions (Beal et al., 2013).
Since there is no mesh connectivity for the SPH particles, it is imperative that only
“nodes_to_surface” contact definitions are utilized in which SPH particles are always
defined to be the slave nodes and finite elements are defined as the master surface.
Tied-based contact consists of “tying” SPH slave nodes to FEM surfaces to connect the
two domains. LS-DYNA ties translational degrees of freedom of nodes to a specified
surface. The constraints are only imposed on the slave nodes, so the more coarsely meshed
side of the interface should be the master surfaces (i.e. FEM) (Hallquist, 2006). Ideally,
each master node should coincide with a slave node to ensure complete displacement
compatibility along the interface, but this is difficult, if not impossible, to achieve.
Consequently, the standard penalty-based contact formulation was utilized for this
study. In this formulation, a contact consists of placing normal interface springs with
stiffness factor of ik between all penetrating nodes and the contact surface. The interface
stiffness is chosen to be approximately the same order of magnitude as the stiffness of the
interface element normal to the interface particle. In applying the penalty method, each
slave node is checked for penetration through the master surface. If the slave node does
31
not penetrate, nothing is done. If it does penetrate, an interface force is applied between
the slave node and its contact point. The magnitude of this force is proportional to the
amount of penetration (Hallquist, 2006).
The stiffness factor, ik , is determined in LS-DYNA several ways including: minimum
of the master segment and slave node stiffness, the master segment stiffness, the slave node
stiffness or the area-/mass-weighted slave node value. Since the same material is across
the boundary between the SPH particles and the solid FEM segments, the stiffness will be
identical and therefore the default of using the minimum of the master segment and slave
node stiffness was used.
2.3.5. FEM and Hybrid FEM-SPH Models
Two numerical models were created for each test to compare their performance at
capturing global response of the SVAR system when varying magnitudes of soil
deformation occurred. The first model consisted solely of finite elements for the SVAR
barrier and soil domain; whereas the second model used a hybrid FEM-SPH approach for
modeling soil. In the hybrid approach, SPH formulations were used in the near-field soil
region to take advantage of SPH’s capabilities in modeling large deformations; whereas
finite elements were used in the far-field soil region to take advantage of FEM’s
computational efficiency. The same constitutive material models and parameters were
used in both models. Figures 2.13 and 2.14 show the two LS-DYNA models for the five-
post welded bus stop and welded bollard, respectively.
32
Figure 2.13 LS-DYNA models for five-post welded bus stop: (a) FEM model; (b)
hybrid FEM-SPH model
Figure 2.14 LS-DYNA models for welded bollard: (a) FEM model; (b) hybrid FEM-
SPH model
In the FEM only model, eight-node constant stress solid elements were used for the
surrounding soil. The soil element size was approximately 100 mm based on a parametric
Foundation
Soil using FEM Posts
Soil using SPH
Impact
direction
Soil using FEM
(a)
Concrete rail
foundation
(b)
Soil using SPH
BollardSoil using FEM
Concrete rail
foundation
Foundation
Soil using FEM
(a)
Impact
direction
(b)
33
study conducted by Reese et al. (2014). A single layer of fully-integrated solid elements
was used to model the thin concrete slab as shown in Figure 2.15. The fully-integrated
solid elements are capable of modeling bending and torsional mode of deformation
(Hallquist, 2006, 2013).
Figure 2.15 Modeling concrete slab using a single layer of fully-integrated solid
elements
2.4. Results and Discussion
Figure 2.16 shows comparisons of recordings from Camera 1 (see Figure 2.1) against
simulation results from the FEM and hybrid FEM-SPH models at various time for the five-
post welded bus stop. Figure 2.16 illustrates that both models were able to satisfactorily
capture the global response of the device. Both models predicted P1 rating of the crash
test. Figure 2.17 shows a more detailed view of the simulated mode of deformation of the
Slab
Soil
34
bus stop from the two models. Figure 2.17 illustrates a double-curvature deformation mode
for the impacted center post. The maximum deflection of the vertical posts in both of the
FEM and hybrid FEM-SPH models was approximately 830 mm, whereas the vertical posts
in the crash test showed similar deflections averaging at approximately 1,120 mm.
Although there was a minor difference, the simulation results were generally consistent
with post-test field observations.
Figure 2.16 Comparison of crash test and simulations from FEM and hybrid FEM-
SPH models at various time for five-post welded bus stop
Crash test FEM only Hybrid FEM-SPH
t=0s
t=0.09s
t=0.15s
t=0.28s
35
Figure 2.17 Simulated mode of deformation of five-post welded bus stop: (a) FEM
model; (b) hybrid FEM-SPH model
Figure 2.18 shows comparisons of recordings from Camera 1 (see Figure 2.1) against
simulation results from the FEM and hybrid FEM-SPH models at various time for the
welded bollard. Figure 2.18 shows that the FEM model drastically underpredicted the
global deformation of the welded bollard and truck, whereas the hybrid FEM-SPH model
was able to capture the pullout of the welded bollard and subsequent truck overrun. The
FEM model predicted a P1 rating of the crash test; however, the crash test failed to meet
the P1 rating, which was predicted by the FEM-SPH model. Figure 2.19 shows a more
detailed view of the simulated deformation of the welded bollard from the two models.
Figure 2.19 illustrates that the FEM model was able to capture the initial uplift of the
welded bollard; however, the bollard didn’t flip out of the ground, likely due to the lack of
large deformation in the surrounding soil. The hybrid FEM-SPH model was able to capture
the large deformation of the surrounding soil and, hence, pullout of the welded bollard.
Impact direction
(a)
Impact direction
(b)
36
Figure 2.18 Comparison of crash test and simulations from FEM and hybrid FEM-
SPH models at various time for welded bollard
Figure 2.19 Simulated deformation of welded bollard at t=0.5 second: (a) FEM
model; (b) hybrid FEM-SPH model
Crash test FEM only Hybrid FEM-SPH
t=0s
t=0.17s
t=0.34s
t=0.52s
Impact
direction
Impact
direction
(a) (b)
37
A motion analysis software Photron was used to calculate displacement of the front
corner of the cargo bed in Test 2 in order to compare with simulation results of FEM and
hybrid FEM-SPH models. Figure 2.20 shows a screen shot of the point tracking in Photron
analysis. Figure 2.21 shows a quantitative comparison of truck displacement versus time
from crash test and model simulations. Figure 2.21 shows that the crash test resulted in a
gradual increase of the front corner’s displacement to over 4,500 mm at t = 0.4 second.
The penetration of the front corner of the cargo bed exceeded the limitation of P1
penetration rating at about t = 0.33 second. The FEM model simulation yielded a maximum
displacement of approximately 2,600 mm that was reached at about 0.35 second and
remained constant afterwards. The hybrid FEM-SPH model was able to capture the
gradual increase of truck displacement, although the magnitude of displacement was about
4,100 mm at t = 0.4 second and was smaller than that observed in the crash test. From the
hybrid FEM-SPH simulation, the penetration of the front corner of the cargo bed exceeded
the limitation of P1 penetration rating at about t = 0.37 second, which was late by 0.04
second as compared to the crash test. Nevertheless, the hybrid FEM-SPH model did a
much better job in matching the crash test than the FEM model did.
38
Figure 2.20 Point tracking in Photron FASTCAM software
Figure 2.21 Comparison of displacements of front corner of cargo bed
39
2.5. Additional Verification of Improved Simulation Method
The coupled FEM-SPH simulation method presented above was further verified by
using a SVAR system that was tested at the Larson Transportation Institute in 2011. Figure
2.22 shows that the system consists of a single post with a reinforced concrete foundation
and is referred to as One-Post Lamppost. Fig. 2.23 shows the comparison of the test and
simulation results and an excellent agreement on the global response is obtained.
Figure 2.22 One-Post Lamppost SVAR system with foundation W110″×D18″×L48″
Width
Depth
Length
Concrete
Foundation
Steel
Tubes
40
Figure 2.23 Comparison of test and simulation (One-Post Lamppost, 2011)
2.6. Conclusions
This paper presents two field-scale crash tests of SVAR barrier systems and LS-DYNA
simulations to predict the global response of each system under vehicular impact. Tests 1
and 2 consisted of a five-post welded bus stop and a welded bollard, respectively; both
were in a steel and concrete composite foundation embedded in compacted AASHTO
aggregate. For each test, two LS-DYNA models, namely an FEM-only model and a hybrid
FEM-SPH model, were created to predict the global response of the system under vehicular
impact. In the FEM-only model, a traditional FEM approach was used for the entire soil
region. In the hybrid FEM-SPH model, the near-field soil region was modeled using the
Passed
Passed
Soil using SPH
41
SPH approach, whereas the far-field soil region was modeled using the FEM approach.
Based on the results of this study, the following conclusions can be made:
Test 1 resulted in a P1 rating of the device, where minimal foundation uplift and
rotation were observed. The vertical posts showed similar deflections averaging at
approximately 1,120 mm (44.1 in) and good above-grade load sharing capability.
The front of the truck bed stopped at 1,320 mm (52 in) before the back face of the
impacted post.
For Test 1 both the FEM-only model and the hybrid FEM-SPH model were able to
match the recorded global response of the system. The crash test and both models
showed a double-curvature deformation mode for the impacted center post. Although
there was a minor difference, the simulation results were generally consistent with
post-test field observations in terms of maximum deflection of the vertical posts and
displacement of the front of the truck bed.
Test 2 failed to result in a P1 rating for the device, where significant foundation uplift,
rotation, concrete cracking, and large deformation of surrounding soil were observed.
Final penetration of the front corner of the cargo bed was 9,100 mm (358 in) beyond
the pre-test inside edge of the barrier. The steel and concrete barrier translated 8,100
mm (319 in).
For Test 2, however, the FEM-only approach was not able to accurately predict the
global response of the system. The FEM-only model simulation resulted in a
maximum displacement of approximately 2,600 mm that was reached at about 0.35
second and remained relatively constant afterwards, whereas the crash test resulted
in a gradual increase of the front corner’s displacement to over 4,500 mm at 0.4
42
second. On the other hand, the hybrid FEM-SPH model was able to capture the
gradual increase of the truck displacement, although the magnitude of displacement
was about 4,100 mm at 0.4 second and was smaller than that observed in the crash
test. The hybrid FEM-SPH model did a much better job in matching the crash test
than the FEM model did. This research suggests that the hybrid FEM-SPH approach
is more appropriate in simulating the field performance of embedded structures under
impact loading when large deformation of the surrounding soil is expected.
43
Chapter 3. Optimization of Constitutive Model Parameters for
Simulation of Polystyrene Concrete Subjected to Impact
This chapter is published as: Yoo TK and Qiu T (2017) Optimization of Constitutive
Model Parameters for Simulation of Polystyrene Concrete Subjected to Impact.
International Journal of Protective Structures Published online before print June 20, 2017,
DOI: 10.1177/2041419617716496.
Keywords: Crushable concrete; Drop impact; Quasi-static compression; Response
surface methodology; Strain rate effect
ABSTRACT
This paper presents the results of a series of experimental testing and numerical
modeling studies to optimize the parameters of a constitutive material model to accurately
simulate the behavior of polystyrene crushable concrete during impact loading using LS-
DYNA. Quasi-static compression tests and confined drop impact tests were conducted.
To model the quasi-static compression tests, the response surface methodology was used
to optimize the Poisson’s ratio and friction angle in the pseudo-tensor model in LS-DYNA.
Using the optimized model parameters, the simulated compression stress vs. strain
relationship showed an excellent agreement with those from the compression tests. To
model the confined drop impact tests, the strain rate sensitivity parameter in LS-DYNA
was optimized by comparing the drop impact simulations at different strain-rate sensitivity
44
values with the drop impact tests. This study suggests that the pseudo-tensor material
model is potentially suitable for modeling crushable concrete. Although the optimized
constitutive model parameters are specific for the polystyrene concrete mix used in this
study, similar approach can be used to optimize model parameters for other polystyrene
concrete mixes.
3.1. Introduction
As terrorist attacks have frequently occurred in recent years, security measures to stop
or mitigate the damage they cause have been widely discussed and researched. With raised
national security awareness due to these terroristic threats, safety systems that can prevent
the loss of life and structural damage as a result of impact and blast loading have been
considered. One of the critical research topics has been perimeter security systems (Griebel
and Phillips, 2001; Grosskopf, 2006; Benton-Short, 2007). These systems consist of
protective barriers to secure buildings and structures from potential vehicular attacks. The
primary goal of these blockades is to prevent an attack vehicle from ramming through the
barrier and into the building.
To accomplish this goal, a large amount of kinetic energy from the vehicle must be
absorbed by the barrier. This kinetic energy is typically absorbed through deformation
and/or displacement of the barrier and the surrounding soil. In addition, due to increased
requirements of minimizing the barrier size, it has become much more difficult to design a
barrier capable of stopping an attack vehicle. A method for dissipating some of a vehicle’s
high kinetic energy is to incorporate crushable materials into the barrier design. Crushable
45
materials capable of absorbing large amounts of energy could potentially reduce the size
and cost of a barrier without any loss of performance in stopping moving objects.
Typically, crushable materials consist of a stiff matrix material and a crushable cell
(foam) material, and behave differently under compressive and tensile loadings. For the
purpose of this research, the focus will be the material’s behavior under compressive
loading. Generally, if a crushable material is loaded to sufficiently high strains, it will
show three phases of behavior: a linear-elastic phase, a crushing phase, and a densification
phase (Gibson and Ashby, 1997). At low strains, the material will first deform in a linear-
elastic manner (if the matrix material itself is linear-elastic) (Patel and Finnie, 1970; Abd
El-Sayed et al., 1979; Warren and Kraynik, 1987), which is known as the linear-elastic
phase. The material will then reach a stress high enough to cause the cell walls to collapse.
At this point, the material enters the crushing phase. During this phase, the material is
loaded to higher strains but the stress remains relatively constant. This stress value is
known as the plateau stress. The behavior of the crushing phase is primarily a function of
the type of cell wall failure that occurs, including non-linear elastic buckling, plastic
yielding, and brittle fracture. These failure modes are typically associated with the specific
type of material. For example, elastic buckling is the general failure mode of elastomer
foams (Barma et al., 1978; Christensen, 1986); plastic yielding typically occurs when the
matrix material is composed of a metal, such as aluminum or zinc (Thornton and Magee,
1975a, 1975b); and ceramic foams, such as glass or cement-based foams, normally exhibit
a brittle fracture failure mode (Rusch, 1970; Morgan et al., 1981; Maiti et al., 1984). As
the volume fraction of the cell material increases, the plateau stress will decrease. Finally,
after most of the cells have been crushed, the material will begin to behave more like a
46
solid and less like a foam. When the material reaches its densification strain, the cellular
volume approaches zero and the material enters the densification phase. At this point, the
stress will begin to increase rapidly with a small increase in strain. As the volume fraction
of the cell material increases, the densification strain will increase.
The strain rate of loading could influence the mechanical behavior of crushable
materials. In general, the faster the load is applied, the higher the initial peak strength will
be. This is due to three main mechanisms. First, micro-inertial effects on the cell wall will
suppress its tendency to buckle (Klintworth, 1989). In other words, there will not be
enough time for the cell wall to deflect horizontally, which is necessary for buckling to
occur. Second, local densification will occur in the regions where the load is being
transmitted. This denser region will be capable of carrying a much higher load than the
less dense, uncompacted regions in the material (Reid and Bell, 1984). Finally, since
crushing tends to occur closer to the region of loading, the local strain rate in the region
will be much higher than the global, nominal strain rate. This last mechanism will further
amplify the effect of the first two mechanisms.
Crushable concrete has been used for various purposes, including aircraft arrestor beds
(White and Agrawal, 1993; Heymsfield and Halsey, 2008; Zhang et al., 2013) and for
absorbing blast energy from explosions outside of structures (Li and Muthyala, 2008).
Numerical investigations on the behavior of concrete and reinforced concrete during
impact loading have been widely conducted using various robust material models readily-
available in LS-DYNA (Teng et al., 2004; Tai and Tang, 2006; Tu, 2009). Few studies
have focused on the numerical modeling of crushable concrete. For modeling crushable
materials under impact loading, many studies on the behavior of polymeric foam material
47
have been conducted (Zhang et al., 1998; Du Bois et al., 2006; Mamalis et al., 2009). A
number of tests (compressive strength tests, split tensile tests, absorption tests) were
performed with different polystyrene concrete mix designs to investigate the mechanical
properties by changing the polystyrene volume fraction or the polystyrene sphere size
(Babu and Babu, 2003). Impact tests by dropping a hammer at different velocities were
performed on three mixes with different polystyrene contents (Bischoff et al., 1990). In
these tests, as the polystyrene volume fraction in the mix design increased, the hammer
penetration depth and load time increased. When the crushable concrete is loaded in an
unconfined setting, very few cells are crushed and the sample fails primarily in shear as
frequently seen in compressive strength tests of regular concrete (Doyle, 2015). This
suggests that unconfined condition is not an efficient way to incorporate a crushable
concrete element into a barrier. Adequate confinement (e.g., encapsulated element) has
been shown to eliminate shear failure and maximize cell crushing (Doyle, 2015).
It is crucial to investigate the crashworthiness of barrier systems, and field-scale testing
and numerical modelling are two common methods. Implicit/explicit dynamic solvers are
widely utilized to run crash simulations first and arrive at an initial design of the system,
full-scale crash tests are then subsequently utilized to validate the initial design and/or
provide the needed data for calibrating simulation parameters. LS-DYNA is known to be
a reliable program for modeling vehicular impact and is based on the finite element method
(FEM). Numerous researchers have investigated typical barrier systems under vehicular
impact using the LS-DYNA research/commercial code (Hallquist, 2006) and field-scale
vehicular crash tests (Krishna-Prasad, 2006; Omar et al., 2007; Liu et al., 2008; Ferdous et
48
al., 2011; Hu et al., 2011; O’Hare et al., 2012; Uzzolino et al., 2012; Keske et al., 2015;
Reese et al., 2014, 2016; Yoo et al., 2016).
The primary objective of this research is to optimize the constitutive model parameters
for a crushable concrete material that can be incorporated in the design of a barrier system.
In this paper, a series of experimental tests and parametric studies using a statistical
analysis tool have been conducted to optimize the parameters of a constitutive material
model to accurately simulate the behavior of crushable concrete during quasi-static
compression and impact loading using LS-DYNA. In the following sections, confined
compression tests and optimization of model parameters for quasi-static analyses are first
presented, followed by drop impact tests and optimization of model parameters for high-
strain-rate analyses.
3.2. Confined Compression Tests for Quasi-Static Analysis
3.2.1. Crushable Concrete Mix Design
The polystyrene concrete’s properties will be much more consistent and predictable if
the cement paste fully covers each polystyrene sphere creating a closed-cell foam. This
can only be achieved if the polystyrene’s volume fraction in the concrete mix design is less
than the polystyrene’s bulk density percentage when it is in a loose state. For the confined
compression tests, a polystyrene volume fraction of 40% was used. This value is well
below the minimum bulk density of 52% for uniform spheres that are loosely packed and
ensured that the polystyrene concrete behaved as a closed-cell foam. A photograph taken
49
of a sample that was cut open is shown in Figure 3.1 and demonstrates that the sample has
closed-cell foams at a polystyrene volume fraction of 40%.
Figure 3.1 Polystyrene concrete sample showing that a polystyrene volume fraction
of 40% is sufficient to produce a closed cell foam
Confined compression tests were performed on a concrete mixture with a water-to-
cement ratio of 0.4. Table 3.1 shows the mixing proportions. The polystyrene spheres
were 5 mm in diameter on average and had a unit weight of approximately 41.6 kg/m3 (2.6
lb/ft3). A photograph of the polystyrene spheres is shown in Figure 3.2. For detailed
discussions on the mix design, please refer to Doyle (2015) and Doyle et al. (2017). This
mix design resulted in a small amount of strain hardening during the crushing phase, which
is beneficial to maximizing energy dissipation and reducing the peak load transferred to
the vehicle barrier.
50
Table 3.1 Polystyrene concrete mix design details (per m3 of concrete)
Component (per m3) Polystyrene mixture (kg)
Polystyrene 16.7
Water 334.3
Cement 835.8
Air Entrainer 1.63
Water Reducer 2.34
Figure 3.2 Expanded polystyrene spheres
3.2.2. Mixing Procedure
For this study, all concrete mixing was done in general accordance with the ASTM
C192 standard for making and curing concrete test specimens in the laboratory (ASTM,
2013) with a few noted exceptions to accommodate mixing with expanded polystyrene.
First, the materials were placed in the mixing bowl and hand-mixed for approximately one
minute. This was done to cover the polystyrene spheres in cement paste so that they were
not ejected from the bowl when the automatic mixer was turned on. Then, as prescribed
51
by ASTM, a paddle style mixer was used to mix the concrete for three minutes, followed
by a two-minute break, and three additional minutes of mixing. Plastic, watertight molds
with a 100 mm (4 in) diameter and 200 mm (8 in) height were used to form the test samples.
Form oil was applied to the inside of each mold, so that the samples could be demolded
without being damaged. Casting was done in two layers and the concrete was consolidated
by tamping each layer 25 times with a 10 mm (3/8 in) tamping rod. The sides of the mold
were then lightly tapped 10 times to remove any additional large air voids. Vibration
methods were not used because of their tendency to cause the polystyrene to rise to the top
of the cylinder. The cast samples were covered with plastic sheets and then placed in a
humidity-controlled curing chamber with a constant temperature of 23°C and a relative
humidity greater than 98%. After 24 to 48 hours, the samples were demolded by drilling
a small hole in the bottom of the mold and pushing the concrete out with a compressed air
gun. The samples were then placed back in the curing chamber until their test day.
3.2.3. Confined Compression Tests
Two confined compression tests were performed to determine the stress-strain behavior
of polystyrene concrete in a confined setting. The confined tests were performed by
placing the concrete cylinders in a steel tube with a 103 mm (4 1/16 in) inner diameter and
12.7 mm (½ in) wall thickness. This tube provided the rigid confinement needed to prevent
shear failure and maximize polystyrene cell crushing during the test. An oil lubricant was
first applied to the interior walls of the steel tube to reduce side friction. The sample was
then loaded into the steel tube. A piston was subsequently placed on top of the sample to
52
apply a quasi-static axial load at a strain rate of 3.48×10-4 mm/mm/s. The force and
displacement values were automatically recorded at a rate of 5 Hz. A cross section of the
testing setup is shown in Figure 3.3.
Figure 3.3 Testing setup for confined compression tests
3.3. Simulation and Optimization of Parameters for Quasi-Static Tests
3.3.1. Simulation of Confined Compression Tests
The quasi-static compression tests were simulated using LS-DYNA to optimize
parameters of a constitutive material model to accurately simulate the stress-strain behavior
of the polystyrene concrete. Figure 3.4 shows that the LS-DYNA model consists of a
polystyrene concrete specimen, a confining steel cylinder, a steel ram at the top, and a steel
plate at the bottom. Four layers of constant-stress eight-node solid elements were used to
53
model the steel cylinder, steel ram, and bottom steel plate. The polystyrene concrete
specimen was modeled using 32 layers of the same elements. The steel cylinder and bottom
plate were fixed, whereas the steel ram had a prescribed vertical displacement boundary
condition consistent with the loading rate of the actual physical tests. Contact algorithms
readily available in LS-DYNA were used to mimic frictional interaction between the
concrete specimen and steel components (i.e., cylinder, ram, and base plate). Two-way
contact sliding interfaces were used, which prevented nodes of selected elements from
penetrating the surfaces of others. The static and dynamic coefficients of friction at the
interfaces were taken as 0.5 and 0.4, respectively. The contact between the steel ram and
cylinder was not activated and, hence, the concrete specimen provided the only resistance
to the ram’s vertical movement.
A mesh-size convergence test was conducted to identify an optimal mesh size for the
simulations in this study. The convergence test consisted of three mesh sizes which were
13, 6.5, and 3 mm, respectively, as shown in Figure 3.5. The total number of elements
were 2,612, 20,888, and 167,104, respectively. The numerically simulated compression
stress vs. strain relationships of the crushable concrete specimen from the three mesh sizes
were compared. There was a good agreement and convergence between the 3 mm and the
6.5 mm mesh sizes, whereas the 13 mm mesh size did not converge with the other two as
shown in Figure 3.6. Hence, the 6.5 mm mesh size was used for simulating the confined
compression tests in this study.
54
Figure 3.4 Modeling confined compression tests in LS-DYNA
Figure 3.5 Three mesh sizes for mesh-size convergence test: (a) mesh size = 13 mm;
(b) mesh size = 6.5 mm; and (c) mesh size = 3 mm
Computer
Simulation
Polystyrene
Concrete
Steel
Ram
Steel
Cylinder
Bottom
Steel Plate
200 mm
100 mm
(a) (b) (c)
55
Figure 3.6 Compressive stress vs. strain of three mesh size models
In the LS-DYNA model, the readily available pseudo-tensor model (i.e., MAT_016)
was used to model the polystyrene concrete. This material model was selected because it
is the next progression from LS-DYNA Material Type 5, “Soil and Foam” and only
constitutive model for foam to incorporate strain rate effects. From previous quasi-static
simulations completed by Du Bois et al. (2006), Material Type 5 was successfully used to
predict the force-displacement curves of crushable materials from laboratory unconfined
and confined compression tests. Two different response modes exist for the pseudo-tensor
model: (1) Mode I, which is well suited for implementing standard geological models; and
(2) Mode II, which uses two shear strength curves with damage and is best used for
0
2
4
6
8
10
12
14
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
Str
ess
[MP
a]
Strain
Mesh size 3 mm
Mesh szie 6.5 mm
Mesh size 13 mm
56
modeling concrete. Response Mode I was selected herein because it implements the
widely-used Mohr-Coulomb yield surface with a Tresca limit (Hallquist, 2013). Details of
the constitutive model and its parameters are discussed below.
In the pseudo-tensor model, the Mohr-Coulomb yield criterion is rewritten as:
sin
2cos
2
3131
c (3.1)
where c is the cohesion, is the friction angle, and 1 and 3 are the major and minor
principal stresses, respectively. The first and second stress invariants 1J and 2J are:
pJ 33322111 (3.2)
2
1 1
2 2ij ij ij ijJ S S p p (3.3)
where p is the mean stress. The Mohr-Coulomb yield criterion is rewritten in terms of
stress invariants:
2 121 sin cos 1 sin cos cos sin
3 3 3
J Jc
(3.4)
where is the similarity angle, convenient for expressing the principal stresses in terms
of invariants. The failure strength depends on the mean stress as:
2
2 0 1 2J A A p A p (3.5)
where 0A , 1A , and 2A are constants for the yield function and can be written as functions
of the friction angle and cohesion:
222
0sin3
cos12
cA ,
21sin3
cossin24
cA , and
22
2sin33
sin4
A (3.6)
57
The perfectly plastic yield function, , is described as:
2
2 0 1 2J A A p A p (3.7)
On the yield surface 2
2
1
3yJ where y is the uniaxial yield stress, i.e.,
21
2
2103 pApAAy (8)
The volumetric strain vs. mean pressure relationship from the confined compression
tests was calculated and tabulated as input information for the pseudo-tensor model. Table
3.2 shows parameters of the material model which were used for simulating the crushable
concrete. In the pseudo-tensor model, there is a tensile cutoff parameter. The original
formulation of the Mohr-Coulomb material model is extended allowing for the reduction
of tensile strength of polystyrene concrete, which in the case of the standard Mohr-
Coulomb model is given by cotc . This value can be reduced by specifying the value of
tensile cutoff. A simple parametric study was conducted to investigate the effect of tensile
cutoff on the simulation results of the polystyrene concrete under confined compression.
For the standard Mohr-Coulomb model, the tensile cutoff value was 0.056 MPa. Figure
3.7 shows the simulated compressive stress vs. strain curves for four tensile cutoff values:
0, 0.028, 0.056, and 0.112 MPa. Figure 3.7 shows that the simulation results are insensitive
to the tensile cutoff values for confined compression tests. A tensile cutoff value of 0.028
MPa, which was calibrated against the results of unconfined compression tests, was used
in this study (more details are provided later in this chapter). A parametric study to
optimize the Poisson’s ratio and friction angle was conducted and discussed below.
58
Table 3.2 Model parameters for crushable concrete
Density
(kg/m3)
Elastic
modulus
(MPa)
Cohesion
(MPa)
Tensile
cutoff
(MPa)
Poisson’s
ratio
Friction
angle
(degree)
1,122 177 0.06 0.028 Unknown Unknown
Figure 3.7 Simulated compressive stress vs. strain for four tensile cutoff values
3.3.2. Parametric Study for Optimizing Material Parameters
The Response Surface Methodology (RSM) was used to conduct a parametric study to
optimize the Poisson’s ratio and friction angle. In statistics, the RSM explores the
relationships between several explanatory variables and one or more response variables.
In this study, the explanatory variables are the Poisson’s ratio and friction angle, and the
0
2
4
6
8
10
12
14
0 0.1 0.2 0.3 0.4 0.5 0.6
0
0.028
0.056
0.112
Str
ess
(MP
a)
Strain
Tensile Cutoff
(MPa)
59
response variable is the difference in stress vs. strain relationship between the confined
compression test and LS-DYNA simulations. As shown in Figure 3.8, using the second
order regression, the optimum values of the Poisson’s ratio and friction angle can be
identified by minimizing the difference in stress vs. strain relationship.
Figure 3.8 Applying RSM for parametric study to optimize material parameters
The main idea of the RSM is to use a sequence of designed simulations to obtain an
optimal response. The values used for each variable are shown in Figure 3.9. A total of
13 simulation cases were conducted as a sequence of randomly designed simulations. For
efficient estimation of the quadratic terms in the second-order model, additional exterior
values were used in the designed simulations (Montgomery, 2010); these exterior values
Difference in Compression Stresses
between Test 1 and LS-DYNA Simulations
Minimum Point
60
were automatically selected by the software (Minitab 17) and included 0.038 and 0.462 for
the Poisson’s ratio, and 4° and 46° for the friction angle.
Figure 3.9 Values of variables and sequence of randomly designed simulations
Figure 3.10 shows the compression stress vs. strain of the specimen from the 13
simulations, as well as the recorded results from confined compression Test 1. Appendix
A shows the raw data from the simulations and test used for calculating the difference in
compression stresses. Appendix B presents the LS-DYNA input code of the simulation
case 1 in Figure 3.9. Using the second order regression obtained from the RSM as shown
in Appendix C, the regions of minimum difference in compression stress vs. strain between
the compression test and LS-DYNA simulations can be found by optimizing the contour
plot and surface plot in Figure 3.11. The optimized values of the Poisson’s ratio and
friction angle are found to be 0.038 and 46°, respectively. Figure 3.12 shows a comparison
Factors Low Middle High
Poisson's Ratio 0.1 0.25 0.4
Friction Angle (degrees) 10° 25° 40°
Simulation Cases Poisson's Ratio Friction Angle
1 0.4 40°
2 0.25 25°
3 0.1 40°
4 0.1 10°
5 0.25 25°
6 0.462 25°
7 0.25 25°
8 0.25 25°
9 0.25 46°
10 0.038 25°
11 0.4 10°
12 0.25 25°
13 0.25 4°
Input :
Ranges of Variables
Output :
Randomly
Designed
Simulations
Required
13 Simulation Cases
61
of the compression stress vs. strain from the simulations using the optimized parameters
and compression Tests 1 and 2, and an excellent agreement is observed.
Figure 3.10 Results from compression Test 1 and 13 simulations
Figure 3.11 Optimizing results of 13 simulation cases
0
5
10
15
20
25
30
0 0.1 0.2 0.3 0.4 0.5
RSM01RSM02RSM03RSM04RSM05RSM06RSM07RSM08RSM09RSM10RSM11RSM12RSM13Test 1
Str
ess
(MP
a)
Strain
Response (Difference in Compression Stresses between Test 1 and LS-DYNA Simulations)
Region of
Minimal
DifferencesRegion of
Minimal
Differences
62
Figure 3.12 Comparison between optimized simulation and confined compression
tests
3.3.3. Simulation of Unconfined Compression Test
The calibrated LS-DYNA model with optimized material parameters was used to
simulate an unconfined compression test to further validate the optimized material
parameters. The uniaxial unconfined compression test was conducted in accordance with
the ASTM C39 standard for compressive strength of cylindrical concrete specimens
(ASTM, 2012) with some modifications. To simulate the unconfined test, the same model
used for the confined compression simulation was used after removing the steel cylinder.
Figure 3.13 shows the unconfined compression test setup and the LS-DYNA model. Figure
0
2
4
6
8
10
12
14
16
0 0.1 0.2 0.3 0.4 0.5
Test 1
Test 2
Optimized Model
Str
ess
(MP
a)
Strain
63
3.14 shows a comparison of the compression stress vs. strain from the simulation and the
unconfined compression test, and an excellent agreement is observed.
Figure 3.13 Unconfined compression test setup and LS-DYNA model
Figure 3.14 Comparison between optimized simulation and unconfined compression
test
0
0.2
0.4
0.6
0.8
1
0 0.01 0.02 0.03 0.04 0.05 0.06
Unconfined Compression Test
LS-DYNA Simulation
(Tensile Cutoff = 0.028 MPa)
Str
ess
(MP
a)
Strain
64
3.4. Confined Drop Impact Tests
Confined drop impact tests were performed on 28-day, 100-mm diameter and 190-mm
high specimens. Each specimen was placed in a confining steel tube with a base plate on
the bottom and a 160-mm high loading piston on top, as seen in Figure 3.15. A 305-mm
long by 89-mm diameter steel cylinder, weighing 14.8 kg, was dropped on each specimen
through a clear cylinder PVC pipe guide, hitting the top of the loading piston as shown in
Figure 3.15. The loading piston was used to distribute the load evenly on the concrete
specimen and to confine the top of the specimen. The drop weight was manually elevated
by a rope connected to the top of the steel weight and released from a height of 1,092 mm,
achieving an initial impact velocity of approximately 4.5 m/s. An IDT MotionXtra NR-
4S2 high-speed camera (2,500 frames-per-second, 1016 by 1016 pixels resolution) was
utilized to record pertinent information such as translational displacement and global
response of the specimens. Figure 3.16 shows an image recorded by the high-speed camera
just before the impact. A ruler was used to convert image scaling from pixels to real units
(e.g., mm).
65
Figure 3.15 Drop impact test setup for crushable concrete specimens
Figure 3.16 An image recorded by high-speed camera just before impact
A motion analysis software Photron was used to track the displacement of the top of
the loading piston to monitor the compression of the specimen during impact as shown in
14.8 kg Steel Drop
Weight
Loading Piston
with Load Cell
Concrete Specimen
Confinement Form
Clear PVC Pipe Guide
Base Plate
Load Cell
Ruler
Steel
Drop
Weight
Steel
Confinement
Clear
PVC
Pipe
66
Figure 3.17. Figure 3.18 shows the recorded loading piston displacement vs. time for two
drop impact tests using different specimens. Figure 3.18 shows that the two tests resulted
in a similar maximum compression of 12 mm and a similar permanent compression of 9
mm for the specimens; however, the rebound displacements of the piston were different,
which is likely due to the piston not being placed at the exact same location and hitting the
steel confinement after impact in the two tests. Figure 3.19 shows the force vs. time
measured by the load cell (204,800 samples/second) for the two drop impact tests, which
indicates that the two tests resulted in a similar maximum force of approximately 13.7 kN.
Figures 3.18 and 3.19 suggest that the instrumentation and test setup are capable of
measuring consistent test results in terms of maximum and permanent compressions of the
specimen, and maximum impact force.
Figure 3.17 Point tracking in motion analysis using Photron FASTCAM software
67
Figure 3.18 Loading piston displacement vs. time for drop impact Tests 1 and 2
Figure 3.19 Impact force measured by load cell vs. time for drop Tests 1 and 2
-15
-10
-5
0
5
10
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
Drop Impact Test 1
Drop Impact Test 2
Time (s)
Ver
tica
l D
ispla
cem
ent
of
Ram
(m
m)
Rebound
-15
-10
-5
0
5
0.00 0.05 0.10 0.15 0.20
Drop Impact Test 1
Drop Impact Test 2
Time (s)
Impac
t F
orc
e (k
N)
68
3.5. Drop Impact Simulation and Optimization of Strain Rate Parameter
The drop impact tests were modeled in LS-DYNA to optimize a strain rate parameter
based on the optimized constitutive material model for the quasi-static condition as
discussed earlier. Figure 3.20 shows that the LS-DYNA model consists of a polystyrene
concrete specimen, a confining steel cylinder, a steel plate at the bottom, a loading piston,
and a steel drop weight. The same simulation models from the quasi-static compression
simulations were used for the polystyrene concrete specimen, the confining steel cylinder,
and the steel plate at the bottom. A total of 32 and 24 layers of constant-stress eight-node
solid elements were used to model the steel drop weight and the loading piston,
respectively. The steel cylinder and bottom plate were fixed, whereas the loading piston
was free to move vertically to transfer the impact load to the polystyrene concrete
specimen. The steel drop weight had an initial velocity of 4.5 m/s which is consistent with
the impact velocity of the actual tests. The same contact algorithms used in the quasi-static
simulations were used in the drop impact simulation.
69
Figure 3.20 Modeling drop impact test in LS-DYNA
High strain rate behavior of polystyrene concrete materials is crucial in the design and
modeling of polystyrene concrete structures subjected to impact loading conditions. The
strain rate sensitivity index is an important variable in characterizing structural superplastic
deformation (Kumaresan, 2014). There are various methods of assessing the strain rate
sensitivity but broadly all lead to a variation of the sensitivity with strain rate; the strain
rate sensitivity also varies with temperature and grain size (Hosford, 2010). Given the
dependency of geomaterial’s resistance to penetration on the rate of loading, various
material models in LS-DYNA were considered by Wright (2012). He suggested that a
simple yield surface scaling approach, as used by the pseudo-tensor model, would be
appropriate for the addition of rate sensitivity to an FE model. The strain rate sensitivity
Steel
Confinement
Initial vel.:
4,522 mm/s
Weight:
14.8 kg
Loading Piston
with Load Cell
Steel Drop
Weight
70
in the pseudo-tensor model is accounted for by the use of a strain rate multiplier curve,
defining the increase in yield strength as a function of strain rate. Given that the yield
surface as a whole is scaled while pressure dependency remains constant, this approach
allows control over the polystyrene concrete’s cohesive response as a function of strain
rate. In LS-DYNA, the strain rate sensitivity is entered as tabular data to be used as a strain
rate multiplier for increasing the yield strength (Hallquist, 2013). The strain rate sensitivity
was optimized for the drop impact simulation utilizing the optimized constitutive material
model for quasi-static condition. In this study, the strain rate sensitivity is assumed to be
a constant for simplicity.
The strain rate sensitivity was optimized by comparing the drop impact simulations at
different sensitivity values with the drop impact tests. Figures 3.21 and 3.22 show
comparisons of impact force vs. time and displacement of loading piston vs. time,
respectively, from the drop impact tests and simulations with different sensitivity values.
As the sensitivity value increased from 0.7 to 2.7, the simulated maximum impact force
increased from 12.3 kN to 16.4 kN and the simulated maximum compression decreased
from 14 mm to 10 mm. Both figures indicate that a sensitivity value of 1.7 resulted in the
best overall match between impact test recordings and simulations.
71
Figure 3.21 Impact force vs. time from drop impact tests and simulations
Figure 3.22 Displacement of loading piston vs. time from drop impact tests and
simulations
-20
-15
10
-5
0
5
0.00 0.01 0.02 0.03 0.04 0.05 0.06
Drop Impact Test 1
Drop Impact Test 2
Sensitivity Value: 0.7
Sensitivity Value: 1.7
Sensitivity Value: 2.7
Time (s)
Impac
t F
orc
e (k
N)
-15
-10
-5
0
5
10
15
0.00 0.01 0.02 0.03 0.04 0.05
Drop Impact Test 1
Drop Impact Test 2
Sensitivity Value: 0.7
Sensitivity Value: 1.7
Sensitivity Value: 2.7
Ver
tica
l D
isp
lace
men
t o
f R
am (
mm
)
Time (s)
72
An independent drop impact test was conducted, where the steel drop weight was
released from a height of 787 mm after being manually elevated by the rope. The
developed LS-DYNA model with the optimized strain rate sensitivity value was used to
simulate this drop impact test to further validate the optimized numerical model. Figure
3.23 shows the displacement of loading piston vs. time from the drop impact test and LS-
DYNA simulation. In the simulation model, the steel drop weight had an initial velocity
of 3.8 m/s which is consistent with the impact velocity of the actual test. Figure 3.23 shows
an excellent agreement between the test and simulation results. The LS-DYNA model was
able to capture the maximum compression of 10 mm and the rebound of the specimen.
Figure 3.23 Displacement of loading piston vs. time from an independent drop
impact test and simulation
-15
-10
-5
0
5
10
0.00 0.01 0.02 0.03 0.04 0.05
Drop Impact Test
LS-DYNA Simulation
Time (s)
Ver
tica
l D
ispla
cem
ent
of
Ram
(m
m)
73
3.6. Conclusions
This chapter presents the results of a series of experimental tests and numerical
modeling studies to optimize the parameters of a constitutive material model to accurately
simulate the behavior of polystyrene crushable concrete during impact loading using LS-
DYNA. Based on the results of this study, the following conclusions are reached:
For the quasi-static compression tests, the Response Surface Methodology can be
used to optimize the Poisson’s ratio and friction angle in the pseudo-tensor model in
LS-DYNA. The optimized values for the Poisson’s ratio and friction angle are 0.038
and 46°, respectively. The simulated compression stress vs. strain using the
optimized parameters showed an excellent agreement with those from the
compression tests conducted under both confined and unconfined conditions.
Two confined drop impact tests were conducted to examine the behavior of the
polystyrene concrete during impact loading. The two tests resulted in a similar
maximum compression of 12 mm and a similar permanent compression of 9 mm of
the specimens, and a similar maximum impact force of 13.7 kN. These results show
that the instrumentation and test setup are capable of producing repeatable results.
For the confined drop impact tests, the strain rate sensitivity parameter in LS-DYNA
was optimized by comparing the drop impact simulations at different sensitivity
values with the drop impact tests. As the sensitivity value increased from 0.7 to 2.7,
the simulated maximum impact force increased from 12.3 kN to 16.4 kN and the
simulated maximum compression decreased from 14 mm to 10 mm. By comparing
74
with the rest results, the optimized strain rate sensitivity value was determined to be
1.7.
The developed LS-DYNA model with the optimized strain rate sensitivity value was
used to simulate an independent drop impact test, where the steel drop weight was
released from a different height, and an excellent agreement was observed. The
developed LS-DYNA model was able to capture the maximum compression and
rebound of the specimen.
This study suggests that the pseudo-tensor material model is potentially suitable for
modeling crushable concrete. Although the optimized constitutive model parameters are
specific for the polystyrene concrete mix used in this study, a similar approach can be
used to optimize model parameters for other polystyrene concrete mixes.
75
Chapter 4. Pendulum Override Tests and Simulations for Dimensionless
Analyses
A series of tests were designed and conducted to determine the angle of a boulder face
at which the impact of a vehicle changes from preventing override to allowing override.
This chapter presents the setup, testing and preliminary results of the pendulum tests
performed for the vehicular override research of a boulder with the impact face angled at
55°, 60°, and 65° from the horizontal plane. A dimensionless analysis was also conducted
to properly relate the pendulum test results to a full-scale field situation. For the purpose
of this study, override is defined to occur when the height of the lowest point at the front
of a vehicle elevates above the highest point of the barrier. Figure 4.1 shows a
representation of a full-scale vehicle and an embedded boulder. The override height (OH)
is measured from the bottom of the front axle to the top of the barrier. The impact face
angle θ is measured from the horizontal plane.
Figure 4.1 Representation of full-scale vehicle and embedded boulder
Override
Height
RockSoil
Truck OH
Ɵ
Override
Angle
76
4.1. Pendulum Tests
4.1.1. Facility or Equipment Description
The Crash Safety Research (CSR) impact pendulum at Penn State was utilized for this
research. The pendulum has a height of 15.2 m (50 ft) which allows for a maximum arc
radius of 13.7 m (45 ft) for the impact sled. The design of the impact pendulum allows for
a maximum overall vertical elevation change of 9.8 m (32 ft) for the impact sled. When
released from maximum height, the resulting maximum horizontal speed is approximately
13.4 m/s (30 mph) upon impact. A description of the pendulum can be seen in Figure 4.2.
Figure 4.2 Description of cable mounted pendulum
15.2m
(50ft)
Dumpster
Boulder
Dumpster
544kg
(1,200lbs)
Impact
Sled
77
The impact sled consists of a steel bin, 1.01 m (3.5 ft) wide, 1.2 m (4 ft) long, and 0.55
m (1.8 ft) tall that is suspended by four cables from the horizontal beams of the pendulum.
On the front face of the sled is a bumper-like assembly used both to model a vehicle bumper
and to protect the main part of the impact sled from damage. The bumper assembly consists
of a 1.2 m (4 ft) long, 0.25 m (10 in) steel C-channel with 0.006 m (1/4 in) thickness bolted
to a 1.2 m (4 ft) long 0.1 m by 0.15 m (4-in by 6-in) steel tube with 0.006 m (1/4 in)
thickness. An image of the impact sled can be seen in Figure 4.3.
Figure 4.3 Image of impact sled
4.1.2. Test Procedure
The tests used an American Black granite boulder as shown in Figure 4.4. The height
of the right side of the boulder is 1.6 m (63 in) (see left picture in Figure 4.4), the horizontal
dimension is 1.2 m (47 in), and the length of the angled face is 1.9 m (75 in). The left side
78
of the boulder has a smaller cross section, thus the boulder is not symmetric. The width of
the impact face is approximately 0.8 m (34 in) and the boulder weighs about 1,750 kg
(3,858 lbs). The impact face is considered to be natural face rather than smooth or cut face.
As with the impact sled, checkered tape was placed on the side of the rock and sled for
camera-tracking purposes. The boulder was embedded in compacted and amended
AASHTO soils. The AASHTO is an acronym for American Association of State Highway
and Transportation Officials uniformly graded coarse aggregate and was compacted to 90%
relative compaction as evaluated by ASTM D698-12:2012 (2012). The amended
AASHTO soil had a cement content of 5% (by weight). The boulder was placed on
compacted AASHTO soil and then amended AASHTO soil was placed around the boulder
as seen in Figure 4.4. The extent of the amended soil was 0.3 m (11.8 in) from all sides.
As the boulder is asymmetric it was placed in angle to present respectively the 55°, 60°,
and 65° tilted faces perfectly parallel to the impacting pendulum (see Figure 4.5). The
boulder was embedded 0.45 m (17.7 in) in the engineered AASHTO soil. As shown in
Figure 4.4, a large steel container (dimensions: 2.2 m (7.2 ft) wide, 6.7 m (22 ft) long, and
1.5 m (4.9 ft) high) was used for confining the soils and embedding the boulder. The
container was secured by a couple of steel piles to the ground restricted from lateral
movement from the pendulum impact.
79
Figure 4.4 An American Black granite boulder placed on compacted and amended
AASHTO soil
An impact sled weighing 544 kg (1,200 lbs) was used as an impact ram equipped with
a steel fabricated bumper. For each test, the impact sled was lifted to a height of 2.9 m (9.5
ft) to achieve a target impact speed of 7.5 m/s (16.77 mph). During the installation the
boulder was adjusted to ensure that the impact faces of the granite were 55°, 60°, and 65°
measured from the horizontal plane respectively (see Figure 4.5 below). The impact was
recorded with a high-speed camera and motion analysis software was used to analyze the
images and evaluate the potential for override.
Figure 4.5 Angled boulder (65°) and impact sled pulled to maximum height
65°
80
4.1.3. Test Results
Figures 4.6, 4.7, 4.8 show the test results through sequential images with an interval of
0.05 second for the 55°, 60°, and 65° override angle, respectively. Based on analyses
conducted using the motion analysis software, the maximum vertical lift of the impact ram
was 0.945 m (37.2 in) at 0.3 second after impact for the 55° override angle, 0.296 m (11.7
in) at 0.56 second after impact for the 60° override angle, and 0.327 m (12.87 in) at 0.42
second after impact for the 65° override angle.
Figure 4.6 Override for 55° incline face - 0.05 seconds between images
The maximum vertical lift for the 60° override angle did not follow the trend
established by those for the 55° and 65° override angles. A careful examination of the
high-speed video revealed that, during the impact for the 60° override angle, one side of
the steel bumper was suddenly disconnected from the sled due to a failure of two bolts,
which is depicted as a red-colored tracking line shown in Figure 4.7. Given this abnormal
81
test condition, the test results for the 60° override angle were discarded from further
analysis.
Figure 4.7 Override for 60° incline face - 0.05 seconds between images
Figure 4.8 Override for 65° incline face - 0.05 seconds between images
82
4.2. Dimensionless Analysis
The pendulum and boulder configuration provided a smaller scale version of the full-
scale situation depicted in Figure 4.1, allowing researchers to perform multiple tests more
efficiently and cost effectively. However, a method to relate the parameters between the
two different scales: the pendulum scale and the field scale, was needed to ensure proper
interpretation of the results. Earlier research conducted by Keske (2012) developed and
validated the scaling laws that were used in these experiments. In this research, a
dimensionless parameter is used to relate the two different scales to each other. The
research conducted herein is concerned with the override height ( OH ). When an object
rises after collision, some amount of initial kinetic energy is converted to its gravitational
potential energy. Therefore, the relationship between both energies can be expressed as:
)(2
1 2
0 OHmgmV
(4.1)
where OH represents the override distance, m is the mass of the object, g represents the
acceleration of gravity, 0V is initial velocity, represents an energy conversion ratio that
corresponds to the percentage of initial kinetic energy converted to override energy. Based
on Eq. (4.1), all of variables are moved to the right side of Eq. (4.1) and m can be canceled
and the constant number in the left side of Eq. (4.1) can be removed for the dimensionless
equation. As a result, the dimensionless parameter OH is defined as:
2
0
2
0 )()(
V
gOH
V
gOHOH
(4.2)
83
where subscripts and indicate large- and small-scale, respectively. In Eq. (4.2),
OH = 0.44 m will lead to override due to the proposed boulder size and height of truck
chassis; )( 0V = 13.4 m/s for M30 scenario (i.e., 30 miles/hour) and 22.4 m/s for M50
scenario (i.e., 50 miles/hour); and )( 0V = 7.5 m/s. Hence, the critical override height
(maximum vertical displacement) in the pendulum test that would correspond to the
override situation in the field can be obtained as:
44.04.13
5.72
2
OH for M30 scenario (4.3a)
44.04.22
5.72
2
OH for M50 scenario (4.3b)
Parameters and are not readily available, but can be obtained from LS-DYNA
simulations. In the LS-DYNA simulations, the system kinetic energy and override energy
were monitored and used to calculate and .
4.3. Pendulum and Full-Scale Crash Simulations
LS-DYNA simulations of the pendulum tests were conducted to monitor the system
kinetic energy and override energy for calculating the parameter . Initially, the
pendulum was modeled without cables for holding and guiding the sled to reduce the
simulation time. However, as shown in Figure 4.9, during the pendulum tests, it has been
suspected that the cables could affect the behavior of the sled due to the inertia effect of
the cable mass. Using the 55° override angled boulder, two pendulum override tests with
84
and without cables were modeled in LS-DYNA to investigate the effect of the cables.
Figure 4.10 shows that the LS-DYNA model consists of an American Black granite
boulder, compacted and amended AASHTO soils, the impact sled, and four steel cables.
Figure 4.9 Two consecutive sequential images for investigating effect of cables
Figure 4.10 Simulation models: (a) With cables; (b) Without cables
Inertia effect
Y
X
(a) With cables
<Specifications of cables>
- Diameter : 0.0254 m
- Total length : 53.6 m
- Total mass : 108 kg
(b) Without cables
Sled
Boulder
Amended AASHTO
Compacted AASHTO
85
A single boulder was embedded in compacted and amended AASHTO soils as shown
in Figure 4.10. Eight-node constant-stress cubic solid elements were used for the boulder
and soils in both of the models. The solid element size for the boulder and soils was 50
mm. The LS-DYNA Material Type 173, “Mohr-Coulomb (M-C)” (Hallquist, 2013) was
utilized to model the boulder and soils behavior in all simulations. The M-C model was
used to represent the boulder and soils due to its ability to effectively and simplistically
capture the behavior of these geomaterials in the field testing (Reese et al., 2014) as
validated in Chapter 2. The Mohr-Coulomb model characterizes failure of a material based
on its cohesion, friction angle, and normal and shear stresses at a point as follows
(Hallquist, 2006 and 2013)
tanmax nc (4.3)
where max is the shear strength on any plane, n is the normal stress on that plane, c is
cohesion, and is the friction angle. Material properties for the boulder and soils are
summarized in Table 4.1. The LS-DYNA Material Type 24, “Elasto-Plastic Material”
(Hallquist, 2013), was utilized to model the impact sled. Eight-node constant-stress cubic
solid elements with a size of 100 mm were used for the impact sled in both of the models.
The material parameters are summarized in Table 4.2. Please note that detailed structure
of the sled and its components were not modelled; instead, the sled was modelled as an
equivalent homogeneous solid for simplicity. The LS-DYNA Material Type 71, “Elastic
Cables” (Hallquist, 2013), was utilized to realistically model steel cables for holding the
impact sled. Two-node discrete-beam/cable elements were used for the steel cables. The
86
element size was 224 mm. The mass density and elastic modulus were 4,710 kg/m3 and
200 GPa, respectively.
Contact algorithms readily available in LS-DYNA were used to mimic frictional
interaction between the sled and boulder, and between the boulder and soils. Two-way
contact sliding interfaces, which prevented nodes of selected elements from penetrating the
surfaces of others, were used for contacts between the boulder and amended soil, and
between the amended soil and compacted soil. The static and dynamic coefficients of
friction at the interfaces were taken as 0.4. One-way contact, in which only the specified
slave nodes were checked for penetration of the master segments, was used for the contact
between the impact sled and boulder because the one-way contact is appropriate when the
master side is a rigid body. The static and dynamic coefficients of friction at the interfaces
were taken as 0.16, which was optimized by comparing the simulation results with the
pendulum override tests.
Table 4.1 Model parameters for boulder and soils
Density
(kg/m3)
Elastic shear
modulus (MPa)
Poisson’s
ratio
Friction
angle (degree)
Cohesion
(MPa)
Dilation
angle (degrees)
Boulder 2,356 9,480 0.25 37.8 18.3 0
Compacted soil 2,100 38.4 0.25 40 0.0144 10
Amended soil 2,100 38.4 0.25 40 1 10
87
Table 4.2 Model parameters for impact sled
Density
(kg/m3)
Elastic modulus
(GPa)
Poisson’s
ratio
Yield
strength (MPa)
Plastic strain
to failure
Impact sled
824.2 210 0.3 350 0.35
In order to investigate the effect of the cables, a parametric study was conducted. The
mass density of the cables was varied to have 75%, 50%, and 25% of the original mass
density. For these simulations, the X and Y displacements of the left bottom corner of the
sled (see Figure 4.11) were monitored. Figure 4.12 shows the X displacement time
histories from the pendulum test and simulation results for different conditions, and Figure
4.13 shows the corresponding plot for the Y displacement time histories. Both Figures
4.12 and 4.13 show that: 1) the developed model can adequately reproduce the pendulum
test results if the cables were properly accounted for; and 2) the inertia of the cables plays
a significant role in the pendulum tests. As the mass density of the cables decreases, the X
displacement of the sled decreases (see Figure 4.12) but the Y displacement increases (see
Figure 4.13).
Figure 4.11 Measurement point for comparing test results
Measurement point
YX
88
Figure 4.12 X displacement from pendulum test and simulations of different
conditions
Figure 4.13 Y displacement from pendulum test and simulations of different
conditions
0
50
100
150
200
250
300
350
400
450
500
0.00 0.05 0.10 0.15 0.20 0.25 0.30
Dis
pla
cem
ent
[mm
]
Time [second]
With cables
With cables (75% density)
With cables (50% density)
With cables (25% density)
Without cables
Test
Time after impact [second]
0
200
400
600
800
1000
1200
0 0.05 0.1 0.15 0.2 0.25 0.3
Dis
pla
cem
ent
[mm
]
Time [second]
With cables
With cables (75% density)
With cables (50% density)
With cables (25% density)
Without cables
Test
Time after impact [second]
89
In addition, full-scale crash simulations were conducted to monitor the system kinetic
energy and override energy for calculating the parameter at the field scale. A wedge-
like boulder was embedded in engineered AASHTO soil (see Figure 4.14). The wedge-
like boulder is similar to the original DS-309D RVAR system, which was tested on June
1, 2016 and passed P1 criteria under M50 scenario. The DS-309D boulder has an
inclination angle of 90° (i.e., vertical) and approximate dimensions of 0.79 m W × 1.14 m
L × 2.44 m H (2.6 ft × 3.75 ft × 8.00 ft) with a weight of approximately 7,257 kg (16,000
lbs) and an embedment depth of 1.3 m (4.28 ft) in engineered AASHTO soil. The
engineered AASHTO soil extends 0.6 m (23.6 in) from all sides of the boulder while the
boulder sits on compacted soil. The engineered AASHTO soil contains a minimum of 5
% cement by weight and is surrounded by compacted soil.
In the full-scale crash simulations, the truck was modeled and modified to meet the
criteria of the ASTM F2656-15 standard (Reese et al., 2014). The modified truck model
consisted of 1,606 eight-node constant stress solid elements, 20,318 four-node Belytschko-
Tsay shell elements and 201 Hughes-Liu with cross section integration beam elements.
More details of the LS-DYNA model for the truck can be found in Chapter 2. Eight-node
constant-stress cubic solid elements were used for the boulder and soils in the full-scale
crash simulations. The LS-DYNA Material Type 173, “Mohr-Coulomb (M-C)” (Hallquist,
2013) was utilized to model the boulder and soils behavior in all the simulations because
of its ability to effectively and simplistically capture the behavior of these geomaterials in
the field testing (Reese et al., 2014). Contact algorithms were used to simulate frictional
90
interaction between the boulder and soil, truck and RVAR system, and self-contact of truck
components. One- or two-way contact sliding interfaces, which prevented nodes of
selected elements from penetrating the surfaces of others, were used to represent boulder-
to-soil and vehicle-to-RVAR barrier contact. The static contact friction coefficient
between the boulder and soil was 0.414 and the dynamic coefficient of friction is assumed
to be the same as the static coefficient. The static coefficient of friction between the truck
and RVAR system was 0.4 (Reese et al., 2014). In order to identify the minimum
inclination angle of the boulder to prevent override, several LS-DYNA simulations with
different boulder inclination angles (while keeping other dimensions the same) were
conducted.
Figure 4.14 Full-scale crash simulation model
Compacted soil
Granite boulder
Truck
Amended AASHTO soil
91
4.4. Dimensionless Analysis Results
Based on Equations (4.3a) and (4.3b) using and obtained from the validated
simulations of the pendulum tests and full-scale override crash tests, the critical override
heights in the pendulum tests respectively for the M30 and M50 scenarios were calculated.
Table 4.3 lists the results of dimensionless calculations based on the LS-DYNA
simulations.
Table 4.3 Results of dimensionless calculations based on LS-DYNA simulations
55° override
angle
65° override
angle
Dimensionless
calculation for
M30 scenario
of 30 mph full-crash simulation 4.5% 3.5%
of pendulum simulation 27% 19%
OH for 30 mph full-crash test 0.83 m 0.75 m
Dimensionless
calculation for
M50 scenario
of 50 mph full-crash simulation 2.2% 1.9%
of pendulum simulation 27% 19%
OH for 50 mph full-crash test 0.61 m 0.49 m
Maximum vertical displacement of pendulum test 0.945 m 0.327 m
92
The results suggest that as the override angle increases, a smaller percentage of system
kinetic energy is converted to the override energy and override is less likely to occur. The
pendulum simulation has a higher percentage of kinetic energy converted to override
energy, which is due to the lack of plastic deformation of the impact sled (lack of energy
loss). As the vehicle speed increases from M30 scenario to M50 scenario, a smaller
percentage of kinetic energy is converted to override energy because more kinetic energy
is dissipated through plastic deformation of the vehicle under higher impact velocity. Table
4.3 suggests that M50 scenario is more likely to result in override than M30 scenario since
the OH values for M50 scenario are more stringent (i.e., smaller). Although a smaller
percentage of kinetic energy is converted to override energy, the absolute amount of
converted override energy is still higher under M50 scenario since the initial kinetic energy
is 2.78 times higher under M50 scenario than under M30 scenario.
Given the recorded maximum vertical displacement in the pendulum test was 0.945 m
for the 55° override angle, which is larger than the critical values of 0.83 m and 0.61 m,
override is predicted to occur for the 55° override angle under both M30 and M50
scenarios. On the other hand, the recorded maximum vertical displacement in the
pendulum test was 0.327 m for the 65° override angle, which is smaller than the critical
values of 0.75 m and 0.49 m, override is not predicted to occur for the 65° override angle
under both M30 and M50 scenarios.
93
4.5. Conclusions
This chapter presents the setup, testing and preliminary results of a series of pendulum
tests performed to determine the angle of a boulder face at which the impact of a vehicle
changes from preventing override to allowing override. A dimensionless analysis was also
conducted to properly relate the pendulum test results to full-scale field situation. LS-
DYNA simulations of the pendulum tests and full-scale crash tests were also conducted to
derive parameters needed for the dimensionless analysis. Based on the results of this study,
the following conclusions are reached:
The developed LS-DYNA model can adequately reproduce the pendulum test results
if the cables are properly modeled. The inertial effect of the cable plays a significant
role in the pendulum tests.
As the override angle increases, a smaller percentage of system kinetic energy is
converted to the override energy and override is less likely to occur. As the vehicle
speed increases from M30 scenario to M50 scenario, a smaller percentage of kinetic
energy is converted to override energy because more kinetic energy is dissipated
through plastic deformation of the vehicle under higher impact velocity, however,
override is more likely to occur.
Vehicle override is predicted to occur for the 55° override angle under both M30 and
M50 scenarios, but not to occur for the 65° override angle under both M30 and M50
scenarios. This conclusion is consistent with the results of high-fidelity LS-DYNA
simulations of the full-scale crash.
94
Chapter 5. Conclusions and Recommendations
5.1. Conclusions
Physical security systems to prevent truck attacks from terrorists have been
increasingly investigated as frequent vehicular impacts have been recently threatening
sensitive facilities and structures. One of the physical security systems is the vehicle anti-
ram system which has been broadly applied to prevent vehicular impacts into sensitive
structures and buildings. In order to research the vehicle anti-ram systems against the
vehicular crash, the LS-DYNA explicit code and full-scale crash tests have been widely
used. The LS-DYNA simulations generally using the FEM formulation have been
compared and validated by the field-scale crash tests. However, comparisons between
different formulations in LS-DYNA for the interaction between soil and embedded anti-
ram systems involving large soil deformation are remarkably sparse in literature,
particularly when the comparison is validated using instrumented, field-scale crash tests.
The goal of this research is to numerically and experimentally investigate several vehicle
anti-ram systems to improve the accuracy and efficiency of the simulation models and the
impact performance of the systems.
In Chapter 2, two SVAR barrier systems were numerically investigated to compare the
performance of FEM-only and coupled FEM-SPH formulations in LS-DYNA in predicting
the global response of each barrier system under vehicular impact. Two field-scale crash
tests were conducted. Tests 1 and 2 consisted of a five-post welded bus stop and a welded
bollard, respectively; both were in a steel and concrete composite foundation embedded in
95
compacted AASHTO aggregate. For each test, two LS-DYNA models, namely an FEM-
only model and a hybrid FEM-SPH model, were created to predict the global response of
the system under vehicular impact. In the FEM-only model, traditional FEM approach was
used for the entire soil region. In the hybrid FEM-SPH model, the near-field soil region
was modeled using the SPH approach, whereas the far-field soil region was modeled using
the FEM approach. Based on the results of this investigation, the following conclusions
are reached.
1. Test 1 resulted in a P1 rating of the barrier system, where the foundation was
slightly raised and rotated. Both the FEM-only model and the hybrid FEM-SPH
model of Test 1 showed a good agreement with the recorded overall response of
the barrier system. The impacted middle post was transformed into a double-
curvature shape in the crash test and both models. The simulation results were
general consistent with the test results with respect to the maximum displacements
of the perpendicular posts and the front of the truck bed.
2. Test 2 did not result in a P1 rating for the barrier system, where the foundation was
significantly raised and rotated in addition to concrete fracture and massive
deformation of surrounding soil. The front corner of the cargo bed penetrated the
non-impact surface of the barrier by 9,100 mm (358 in) from the final position of
the truck. However, the FEM-only model of Test 2 failed to precisely predict the
overall response of the barrier system. The simulation result of the FEM-only
model showed a maximum displacement of nearly 2,600 mm at about 0.35 second,
which remained constant afterwards, whereas the test results showed a continuous
increase of the front corner’s displacement to over 4,500 mm at 0.4 second. On the
96
contrary, the simulation results of the hybrid FEM-SPH model showed the steady
increase of the truck’s displacement, even though the maximum displacement of
approximately 4,100 mm at 0.4 second was smaller than that observed in the crash
test. The hybrid FEM-SPH model simulation showed a much better agreement with
the crash test than the FEM model simulation.
3. This research suggests that the hybrid FEM-SPH approach is more appropriate in
simulating the field performance of embedded structures under impact loading
when large deformation of the surrounding soil is expected.
In Chapter 3, a simulation model for characterizing polystyrene composite material
which can effectively absorb a lot of impact energy was developed in LS-DYNA. A series
of experimental tests and numerical modeling studies to optimize the parameters of a
constitutive material model were conducted to accurately simulate the behavior of
polystyrene crushable concrete during impact loading using LS-DYNA. Based on the
results of this investigation, the following conclusions are reached.
1. The Poisson’s ratio and friction angle in the pseudo-tensor model of LS-DYNA
were optimized by using the Response Surface Methodology for the quasi-static
compression tests. The optimized Poisson’s ratio is 0.038 and the optimized
friction angle is 46°. The compression stress vs. strain curve in the simulation
model using the optimized parameters demonstrated an excellent match with that
in the confined and unconfined compression tests.
2. The physical response of the polystyrene concrete under impact condition was
investigated in two confined drop impact tests. The maximum and permanent
97
compressions were 12 mm and 9 mm, respectively, and the maximum impact force
was 13.7 kN, which was pretty similar in the two drop impact tests. As a result, it
is determined that the instrumentation and experimental setup are able to create
reproducible results.
3. The confined drop impact simulations with various sensitivity values were
compared with the confined drop impact tests so that the strain rate sensitivity
parameter in LS-DYNA could be optimized. As the strain rate sensitivity value
increased from 0.7 to 2.7, the maximum impact force in the simulation increased
from 12.3 kN to 16.4 kN and the simulated maximum compression decreased from
14 mm to 10 mm. As a result, the optimized strain rate sensitivity parameter was
determined to be 1.7.
4. The optimized strain rate sensitivity value was validated by simulating a separate
drop impact test at a different height of the steel drop weight. The simulation model
developed in LS-DYNA showed an excellent agreement with the separate drop
impact test and was able to capture the maximum compression and rebound of the
specimen.
5. This research suggests that the pseudo-tensor material model is potentially suitable
for modeling crushable concrete. Although the optimized constitutive model
parameters are specific for the polystyrene concrete mix used in this study, similar
approach can be used to optimize model parameters for other polystyrene concrete
mixes.
98
In Chapter 4, a series of pendulum tests were performed to determine the angle of a
boulder face at which the impact of a vehicle changes from preventing override to allowing
override. A dimensionless analysis was also conducted to properly relate the pendulum
test results to full-scale field situation. LS-DYNA simulations of the pendulum tests and
full-scale crash tests were also conducted to derive parameters needed for the
dimensionless analysis. Based on the results of this investigation, the following
conclusions are reached.
1. The inertia of the cable significantly affects the pendulum test results. Therefore,
if the cables are properly modeled, the simulation model developed in LS-DYNA
would be able to reproduce the pendulum test results.
2. A series of pendulum tests showed that as the override angle increases, a smaller
percentage of system kinetic energy is converted to the override energy and the
override is less likely to occur. As the vehicle speed increases from 30 mph to 50
mph, a smaller percentage of total kinetic energy is converted to override energy
because more kinetic energy is dissipated through plastic deformation of the vehicle
under higher impact velocity; however, the override is more likely to occur.
3. Vehicle override is anticipated to occur at the 55° override angle, but not to occur
for the 65° override angle under both M30 and M50 scenarios. This conclusion is
in a good agreement with the results of high-fidelity LS-DYNA simulations of the
full-scale crash tests.
4. Through dimensionless analyses to link pendulum test/simulation and full-scale
vehicular crash test/simulation, the override of full-scale RVAR system can be
99
predicted and prevented, which can significantly save time and cost of full-scale
crash tests and simulations.
5.2. Recommendations for Future Research
Based on the results of the research conducted in this dissertation, the following is a
list of suggested areas where additional research is needed.
In Chapter 3, the constitutive parameters needed to model the polystyrene concrete
material in LS-DYNA were obtained. However, the material model was not
applied in anti-ram barrier systems. In a complex anti-ram barrier system, the
effectiveness of such energy-absorbing material depends on how it is used. It is
recommended that the impact performance of a SVAR system with steel tubes filled
with the crushable polystyrene concrete instead of the normal concrete be simulated
and investigated. Ideally, such simulations should be validated by full-scale crash
tests.
In Chapter 4, vehicle override is predicted to occur for the 55° override angle under
both M30 and M50 scenarios, but not to occur for the 65° override angle under both
M30 and M50 scenarios. Although this conclusion is consistent with the results of
high-fidelity LS-DYNA simulations of the full-scale crash, such prediction is not
validated by field-scale crash tests. It is recommended that a full-scale crash test
with a 65° boulder impact angle be conducted to verify the dimensionless analysis.
100
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Appendix A. Raw Data from Compression Test 1 and 13 Simulations
This appendix presents the stress vs. strain data of the compression Test 1 and 13
simulations used to calculate the difference in compression stresses.
Test 1 RSM01 RSM02 RSM03 RSM04 RSM05 RSM06 RSM07 RSM08 RSM09 RSM10 RSM11 RSM12 RSM13
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0.000502 0.029456 0.024519 0.025106 0.025503 0.025503 0.025106 0.024297 0.025106 0.025106 0.025106 0.025528 0.024518 0.025106 0.025106
0.001005 0.26387 0.077331 0.078321 0.081333 0.076273 0.078321 0.076286 0.078321 0.078321 0.078443 0.080955 0.073981 0.07832 0.072493
0.001507 0.414867 0.122089 0.123057 0.127872 0.115055 0.123056 0.119891 0.123056 0.123056 0.122563 0.12641 0.112456 0.123057 0.107909
0.00201 0.584567 0.144916 0.144745 0.150795 0.131106 0.144745 0.141688 0.144745 0.144745 0.14431 0.147327 0.129188 0.144746 0.121973
0.002512 0.814524 0.149234 0.147224 0.154407 0.131075 0.147224 0.145251 0.147224 0.147223 0.148137 0.149017 0.130346 0.147223 0.121722
0.003015 1.045677 0.15032 0.148434 0.156821 0.131872 0.148434 0.146178 0.148433 0.148433 0.14917 0.152027 0.130726 0.148434 0.122077
0.003517 1.190609 0.149982 0.149057 0.156916 0.132199 0.149055 0.146159 0.149057 0.149055 0.149347 0.152281 0.131144 0.149055 0.121994
0.00402 1.28532 0.149999 0.148722 0.156567 0.132638 0.148722 0.146079 0.148722 0.148722 0.14916 0.152434 0.131212 0.148722 0.122266
0.004522 1.392162 0.150343 0.148662 0.156488 0.132823 0.148662 0.146243 0.148662 0.148661 0.149417 0.15208 0.131338 0.148662 0.12244
0.004818 1.489469 0.15089 0.148877 0.15695 0.132896 0.148877 0.146664 0.148877 0.148875 0.150064 0.151805 0.131489 0.148877 0.122587
0.004315 1.489382 0.151403 0.149112 0.157727 0.133042 0.149113 0.147163 0.149112 0.149113 0.15065 0.152137 0.131586 0.149113 0.122816
0.003813 1.526016 0.15154 0.149629 0.158099 0.133113 0.149629 0.147467 0.149629 0.149628 0.150738 0.152695 0.131809 0.149628 0.122897
0.00331 1.528594 0.151557 0.149814 0.158435 0.133141 0.149814 0.147644 0.149814 0.149814 0.150854 0.153099 0.131926 0.149814 0.123012
0.002808 1.551857 0.151748 0.149956 0.158351 0.133303 0.149957 0.147789 0.149957 0.149957 0.150908 0.153412 0.132093 0.149956 0.123127
0.002305 1.604698 0.152031 0.150016 0.158378 0.13346 0.150018 0.14798 0.150018 0.150016 0.151195 0.153509 0.132227 0.150016 0.123203
0.001803 1.645303 0.152319 0.15008 0.15861 0.133601 0.150082 0.148235 0.15008 0.15008 0.151561 0.153511 0.13235 0.15008 0.123289
0.0013 1.683923 0.152547 0.150226 0.158961 0.133747 0.150227 0.148523 0.150226 0.150226 0.151801 0.153534 0.132486 0.150226 0.123413
0.000798 1.67128 0.152769 0.150429 0.159354 0.133883 0.150431 0.148789 0.150429 0.150429 0.152005 0.153533 0.132605 0.150429 0.123501
0.000295 1.657379 0.156512 0.150845 0.15958 0.133984 0.150853 0.149725 0.15085 0.150849 0.154237 0.153907 0.132778 0.150849 0.123622
0.000207 1.633771 0.164211 0.156052 0.164675 0.136137 0.156082 0.155431 0.156065 0.156223 0.161991 0.156639 0.136707 0.15609 0.125701
0.000709 1.620918 0.173114 0.161704 0.172148 0.139306 0.161676 0.162252 0.161695 0.162176 0.170287 0.161358 0.140875 0.161691 0.128272
0.001212 1.639543 0.184062 0.167542 0.180918 0.143139 0.16744 0.170187 0.167471 0.168359 0.178699 0.167647 0.145325 0.167275 0.131167
0.001714 1.607695 0.200125 0.174819 0.191554 0.147949 0.174711 0.181521 0.17452 0.176293 0.188716 0.175406 0.151985 0.174096 0.13505
0.002217 1.605154 0.225782 0.184611 0.206494 0.154821 0.184551 0.200087 0.183638 0.187887 0.201908 0.185267 0.16371 0.183424 0.141126
0.002719 1.65522 0.270907 0.199597 0.230495 0.168718 0.199274 0.236375 0.19853 0.20599 0.218544 0.202993 0.189106 0.198243 0.153461
0.003222 1.697996 0.302627 0.219729 0.252097 0.194743 0.218981 0.309053 0.217124 0.22711 0.234803 0.230742 0.244235 0.217614 0.176685
0.003724 1.712502 0.328055 0.241127 0.26529 0.240321 0.239597 0.386562 0.23782 0.247944 0.250994 0.262 0.338592 0.239641 0.214358
0.004227 1.728512 0.371913 0.264521 0.280513 0.304311 0.262104 0.418818 0.261952 0.26992 0.26987 0.283873 0.450404 0.263429 0.263452
0.004729 1.768291 0.444005 0.291385 0.311758 0.37853 0.288406 0.428565 0.290833 0.29613 0.293805 0.294672 0.550805 0.289245 0.316079
0.005232 1.784351 0.520266 0.321807 0.366729 0.44987 0.319152 0.555364 0.323941 0.328943 0.322986 0.314601 0.622722 0.31874 0.361909
0.005734 1.819122 0.595593 0.356806 0.397031 0.51176 0.358406 0.692763 0.353905 0.361922 0.358062 0.382398 0.711916 0.353721 0.412852
0.006237 1.837031 0.710264 0.401774 0.438177 0.566303 0.39952 0.857451 0.39441 0.404877 0.402472 0.444054 0.787657 0.400159 0.457953
0.006739 1.853412 0.836381 0.44839 0.507022 0.613912 0.44803 1.020504 0.443271 0.453395 0.451318 0.485477 0.862207 0.448003 0.502207
0.007242 1.88336 0.968753 0.506515 0.575063 0.651431 0.505752 1.185659 0.502952 0.506643 0.505499 0.530418 0.927526 0.504109 0.547074
0.007744 1.866461 1.109448 0.575449 0.625624 0.678953 0.570228 1.349546 0.56955 0.567206 0.566238 0.587226 0.979986 0.57192 0.592885
0.008247 1.888429 1.257407 0.632831 0.684206 0.702835 0.635878 1.481612 0.629939 0.640609 0.63223 0.624061 1.032056 0.642082 0.639568
0.008749 1.911298 1.35934 0.720597 0.727175 0.724887 0.719711 1.566214 0.715255 0.728286 0.701943 0.680586 1.053298 0.727307 0.676151
0.009251 1.870446 1.478429 0.804588 0.765533 0.744563 0.802046 1.63229 0.795058 0.803182 0.762004 0.722447 1.089039 0.803844 0.713173
0.009754 1.90201 1.564796 0.876649 0.807929 0.762867 0.874522 1.688696 0.86813 0.870772 0.824459 0.75295 1.142873 0.873609 0.747759
0.010256 1.905796 1.62695 0.938441 0.85503 0.780081 0.93472 1.740415 0.931288 0.931061 0.889691 0.77731 1.196573 0.9355 0.783387
0.010759 1.806836 1.681407 0.987904 0.900575 0.796739 0.982589 1.789778 0.979975 0.979663 0.953987 0.799146 1.239571 0.986476 0.820533
0.011261 1.768649 1.727057 1.014383 0.933147 0.813771 1.018614 1.836563 1.010394 1.010291 1.014121 0.818283 1.282816 1.022623 0.852796
0.011764 1.779071 1.768612 1.037643 0.957463 0.827229 1.042182 1.880005 1.046616 1.047546 1.084521 0.835704 1.312641 1.045981 0.881561
0.012266 1.751565 1.807231 1.066602 0.979479 0.835605 1.067706 1.919488 1.067662 1.071257 1.135343 0.852144 1.345031 1.072004 0.904639
0.012769 1.76517 1.844087 1.09206 0.99855 0.842456 1.092269 1.95786 1.091321 1.092875 1.173827 0.868005 1.376916 1.094806 0.923616
0.013271 1.781341 1.879339 1.113874 1.016232 0.846894 1.113942 1.995271 1.115417 1.114251 1.204043 0.883249 1.406605 1.116522 0.941194
0.013774 1.772435 1.913123 1.133878 1.032921 0.84908 1.132979 2.031855 1.135584 1.134189 1.228549 0.897905 1.434691 1.137817 0.958228
StrainStresses [MPa]
112
…
0.014276 1.778751 1.945785 1.152742 1.046906 0.852252 1.152076 2.068735 1.152985 1.151393 1.248785 0.912225 1.463912 1.154633 0.972643
0.014779 1.816605 1.977312 1.167071 1.065574 0.860255 1.16818 2.103408 1.171065 1.168986 1.268027 0.926199 1.490752 1.172057 0.985671
0.015281 1.869015 2.007963 1.183471 1.079344 0.867928 1.186709 2.136773 1.190759 1.180998 1.289797 0.939621 1.519281 1.189997 0.996701
0.015784 1.848034 2.037813 1.202606 1.092214 0.876742 1.199347 2.170101 1.208603 1.199929 1.308435 0.952808 1.545085 1.223218 1.020673
0.016286 1.826276 2.067009 1.225153 1.103714 0.886097 1.212887 2.202615 1.227213 1.217962 1.33916 0.965793 1.569002 1.261058 1.040422
0.016789 1.840966 2.095637 1.252535 1.112429 0.89583 1.23108 2.234055 1.244135 1.229895 1.379766 0.97856 1.589847 1.293214 1.050487
0.017291 1.842644 2.123661 1.285443 1.118091 0.906935 1.247872 2.265459 1.251005 1.242026 1.399871 0.991109 1.600924 1.320116 1.071232
0.017794 1.883545 2.151032 1.301935 1.131251 0.916987 1.257308 2.295432 1.278388 1.257185 1.419446 1.003329 1.583791 1.344538 1.082924
0.018296 1.929158 2.177921 1.300861 1.141135 0.92133 1.285234 2.325035 1.315589 1.273861 1.431632 1.015332 1.616761 1.365901 1.097217
0.018798 1.963534 2.20444 1.318956 1.15056 0.93214 1.303267 2.354391 1.346499 1.301663 1.453353 1.027179 1.648177 1.383774 1.106968
0.019301 1.95622 2.230713 1.338001 1.161672 0.945246 1.320017 2.383599 1.367061 1.329182 1.474149 1.038928 1.658193 1.399784 1.116719
0.019803 1.940247 2.256738 1.346142 1.175671 0.955969 1.3399 2.41261 1.378927 1.349016 1.487532 1.050559 1.659624 1.416843 1.129208
0.020306 1.937126 2.28232 1.357822 1.191898 0.964653 1.348855 2.441214 1.387771 1.365494 1.499916 1.061979 1.689695 1.437614 1.141074
0.020808 1.974586 2.307433 1.371292 1.200843 0.970894 1.327319 2.469114 1.408271 1.369084 1.494292 1.073185 1.70261 1.455043 1.15475
0.021311 2.024627 2.332423 1.384699 1.20297 0.985745 1.331451 2.49646 1.418385 1.369108 1.505417 1.084288 1.717658 1.46311 1.171712
0.021813 2.057733 2.357142 1.391323 1.21189 0.983521 1.33525 2.52346 1.433433 1.382072 1.529038 1.09525 1.723813 1.470091 1.182381
0.022316 2.0984 2.381613 1.397737 1.223139 0.971222 1.347794 2.550177 1.448962 1.401511 1.561725 1.10613 1.733064 1.480403 1.1926
0.022818 2.098967 2.405764 1.419335 1.236044 0.975657 1.384946 2.576585 1.471682 1.428536 1.605845 1.116938 1.758263 1.502617 1.211209
0.023321 2.112733 2.429484 1.471596 1.258233 1.026238 1.453279 2.602611 1.521872 1.47616 1.667074 1.1276 1.805282 1.547416 1.243222
0.023823 2.141892 2.452993 1.514582 1.277932 1.054744 1.500064 2.628353 1.558024 1.515593 1.704361 1.138159 1.879314 1.588355 1.268237
0.024326 2.161294 2.476256 1.5384 1.292708 1.073088 1.51901 2.653873 1.582064 1.542298 1.727895 1.14854 1.919697 1.61295 1.288009
0.024828 2.189491 2.49942 1.552881 1.300516 1.089319 1.537043 2.679085 1.602786 1.562107 1.745632 1.158834 1.94449 1.634276 1.304599
0.025331 2.191143 2.5224 1.56466 1.308336 1.10067 1.557136 2.703989 1.62299 1.584469 1.760779 1.169021 1.965964 1.656096 1.320128
0.025833 2.209658 2.545132 1.578031 1.319647 1.109256 1.578586 2.728621 1.642565 1.608929 1.777542 1.179133 1.981567 1.674536 1.334498
0.026336 2.247118 2.567704 1.594337 1.330218 1.123318 1.601417 2.75308 1.659315 1.624581 1.801002 1.189271 1.978064 1.682529 1.345624
0.026838 2.25231 2.589894 1.611778 1.326036 1.131124 1.6148 2.777391 1.67239 1.645969 1.827065 1.199282 2.003103 1.703387 1.357933
0.02734 2.222276 2.611887 1.632167 1.322879 1.143681 1.632414 2.801444 1.687734 1.673463 1.849921 1.209151 2.026033 1.720248 1.369404
0.027843 2.229862 2.633805 1.637989 1.320029 1.153469 1.643626 2.8252 1.706865 1.693198 1.866079 1.218992 2.049518 1.733705 1.382097
0.028345 2.237965 2.655428 1.647955 1.328701 1.164888 1.653407 2.848734 1.724417 1.707506 1.882089 1.228758 2.068476 1.74472 1.395134
0.028848 2.251447 2.670254 1.667802 1.349608 1.178546 1.668295 2.872096 1.737936 1.720914 1.901837 1.238412 2.086485 1.75608 1.407975
0.02935 2.302709 2.671277 1.680777 1.368899 1.188584 1.691471 2.895223 1.750159 1.73516 1.923089 1.247465 2.115101 1.771313 1.421012
0.029853 2.325911 2.70759 1.687302 1.372636 1.195337 1.717275 2.917832 1.761519 1.743153 1.942405 1.250845 2.135774 1.784782 1.431756
0.030355 2.350111 2.734689 1.706261 1.371205 1.205581 1.741439 2.918104 1.768858 1.749271 1.961314 1.254484 2.156853 1.78778 1.439933
0.030858 2.008901 2.755214 1.725502 1.367258 1.215999 1.768266 2.945264 1.780897 1.765244 1.979211 1.262032 2.170989 1.80374 1.450183
0.03136 1.584605 2.776479 1.739502 1.371267 1.228841 1.792837 2.979678 1.794477 1.787101 1.996591 1.268311 2.185877 1.823007 1.461198
0.031863 1.460914 2.800013 1.749098 1.385291 1.243543 1.811289 3.005704 1.808946 1.807219 2.013612 1.272504 2.206611 1.838104 1.470819
0.032365 1.432545 2.820513 1.760829 1.398909 1.255014 1.824351 3.028498 1.828631 1.820096 2.028883 1.279905 2.231107 1.85203 1.478651
0.032868 1.429535 2.841408 1.780095 1.387327 1.254965 1.840929 3.05128 1.853979 1.833627 2.032879 1.289933 2.254247 1.867411 1.49333
0.03337 1.435049 2.862352 1.793811 1.398551 1.266707 1.856754 3.073642 1.867202 1.848182 2.050098 1.301626 2.259958 1.882114 1.50855
0.033873 1.453094 2.882679 1.810463 1.435283 1.281274 1.870076 3.095807 1.881448 1.865092 2.06839 1.308262 2.282505 1.896348 1.515754
0.034375 1.477122 2.902883 1.817604 1.470091 1.292943 1.884556 3.117948 1.898248 1.882879 2.086744 1.312937 2.306915 1.908621 1.523204
0.034878 1.500373 2.923124 1.816988 1.488581 1.301861 1.900616 3.13999 1.915282 1.897545 2.105172 1.317846 2.324097 1.920931 1.535107
0.03538 1.526731 2.943057 1.827139 1.498831 1.312801 1.915565 3.161723 1.929935 1.905537 2.122847 1.322632 2.339861 1.937804 1.547688
0.035882 1.553645 2.962607 1.85145 1.512584 1.321978 1.936904 3.183284 1.950669 1.920055 2.139252 1.331895 2.348803 1.953642 1.558851
0.036385 1.589662 2.981306 1.874023 1.531764 1.330736 1.949744 3.20482 1.970368 1.936189 2.155127 1.339752 2.358757 1.964459 1.563488
0.036887 1.627283 2.997538 1.886135 1.537561 1.339247 1.965187 3.226257 1.985823 1.953839 2.171137 1.34946 2.378258 1.978262 1.57322
0.03739 1.664607 3.017039 1.900381 1.544839 1.347992 1.98005 3.247522 2.000279 1.969085 2.186863 1.356564 2.398623 1.986341 1.585172
0.037892 1.687179 3.040166 1.915195 1.559874 1.356675 1.992557 3.268664 2.015512 1.982332 2.202133 1.359512 2.415336 1.98935 1.596582
0.038395 1.696356 3.060013 1.916441 1.574244 1.364791 2.004867 3.289854 2.029832 1.998009 2.217256 1.361868 2.431272 1.998971 1.607338
0.038897 1.709788 3.065637 1.924779 1.581583 1.376299 2.018953 3.310737 2.036629 2.013304 2.232982 1.372624 2.445185 2.02031 1.616872
0.0394 1.721716 3.08964 1.94428 1.589958 1.383848 2.034927 3.330965 2.057129 2.026021 2.248277 1.395048 2.461997 2.025367 1.625728
0.039902 1.742314 3.11553 1.969677 1.605598 1.390188 2.046558 3.349788 2.071141 2.041452 2.262832 1.410614 2.475121 2.040366 1.635238
0.040405 1.750998 3.13719 1.989486 1.619759 1.399513 2.058942 3.368524 2.082575 2.058177 2.276819 1.423245 2.491773 2.058226 1.645636
0.040907 1.769857 3.154433 2.002104 1.63102 1.411108 2.073484 3.389246 2.092307 2.074779 2.291115 1.435148 2.512076 2.074767 1.656602
0.04141 1.79465 3.171628 2.015117 1.645069 1.420297 2.087435 3.413187 2.098104 2.09 2.306607 1.44425 2.528678 2.091986 1.667789
0.041912 1.800607 3.193509 2.023714 1.653259 1.427883 2.106504 3.434563 2.120109 2.095871 2.321643 1.455685 2.543159 2.102051 1.680864
0.042415 1.828471 3.213491 2.020002 1.663238 1.435777 2.115668 3.455088 2.13776 2.09618 2.336444 1.464627 2.551953 2.115348 1.689227
0.042917 1.845628 3.233152 2.027279 1.669713 1.444263 2.127534 3.475723 2.15002 2.108058 2.350999 1.470782 2.559477 2.129125 1.698181
0.04342 1.880856 3.25195 2.037023 1.671576 1.453847 2.136489 3.495866 2.161306 2.111068 2.365443 1.474963 2.577794 2.140917 1.706495
0.043922 1.920425 3.26965 2.040588 1.67503 1.462111 2.142829 3.51565 2.172654 2.102569 2.379788 1.478392 2.603462 2.151019 1.714882
113
…
0.431327 9.16441 18.61639 11.81027 9.985939 7.138154 11.81598 20.32016 11.83348 11.81055 13.55158 8.166472 13.20732 11.82857 8.009602
0.43183 8.968574 18.69175 11.8401 10.01848 7.156891 11.84769 20.35543 11.86721 11.84189 13.58106 8.190376 13.24223 11.8626 8.027289
0.432332 9.199526 18.74664 11.87241 10.04243 7.176638 11.886 20.36172 11.90437 11.87614 13.61054 8.210753 13.27171 11.90055 8.0442
0.432835 9.10777 18.794 11.90891 10.06416 7.196867 11.92684 20.41415 11.94656 11.91666 13.64162 8.235348 13.27307 11.93966 8.059038
0.433337 9.159945 18.83841 11.94886 10.08746 7.219612 11.96598 20.48408 11.98469 11.95373 13.67505 8.246943 13.31056 11.97834 8.072212
0.43384 9.292331 18.88368 11.96134 10.11005 7.239976 11.98307 20.5402 11.99676 11.96905 13.70971 8.242872 13.34769 11.9986 8.084781
0.434342 9.359258 18.9287 11.97042 10.13118 7.260871 11.99854 20.58646 12.00671 11.97948 13.74116 8.256527 13.3826 12.01255 8.097547
0.434844 9.290173 18.97285 11.99453 10.15252 7.283751 12.02876 20.63099 12.03452 12.00366 13.77003 8.291039 13.41664 12.03757 8.109955
0.435347 9.399716 19.01738 12.03179 10.17683 7.308383 12.07288 20.67786 12.08074 12.04832 13.80197 8.315868 13.4497 12.07226 8.12023
0.435849 9.359567 19.06253 12.0636 10.20036 7.346411 12.10379 20.72646 12.11437 12.08221 13.83898 8.340401 13.49127 11.98861 8.12752
0.436352 9.309982 19.10779 12.09423 10.22064 7.403976 12.12517 20.77345 12.13984 11.73067 13.87277 8.358953 13.54023 11.84151 8.133971
0.436854 9.408325 19.15331 12.00334 10.24059 7.493142 12.14084 20.82007 12.16504 11.62911 13.90398 8.375653 13.65593 11.86287 8.139126
0.437357 9.480248 19.1971 11.80821 10.26011 7.599269 12.1582 20.86756 12.18364 11.55515 13.93333 8.387063 13.79901 11.82894 8.136376
0.43786 9.537394 19.24286 11.67467 10.27882 7.715065 12.17179 20.91616 12.18488 11.44955 13.96158 8.385829 13.95837 11.75099 8.126755
0.438362 9.501451 19.29676 11.80843 10.29737 7.832835 12.16799 20.96513 12.16519 11.52123 13.98995 8.368623 14.09072 11.8765 8.124387
0.438865 9.377563 19.34659 11.94588 10.32202 7.851127 12.16681 21.01175 12.17617 11.71793 14.01783 8.371361 13.61288 11.96115 8.17486
0.439367 9.598141 19.39346 12.02758 10.34585 7.546304 12.19355 21.06208 12.20408 11.84023 14.04681 8.396079 13.52889 12.02881 8.116369
0.43987 9.620812 19.43688 12.08365 10.36756 7.353626 12.22303 21.11166 12.23269 11.92355 14.0774 8.421464 13.5242 12.0804 8.110942
0.440372 9.647775 19.48437 12.13075 10.39081 7.30493 12.25195 21.1589 12.26075 11.9842 14.11009 8.444529 13.42774 12.12361 8.145849
0.440875 9.643286 19.5363 12.17491 10.41542 7.348532 12.28035 21.20652 12.28844 12.03179 14.14315 8.465819 13.33881 12.162 8.190191
0.441377 9.711582 19.58502 12.21375 10.43778 7.41541 12.3083 21.25807 12.31524 12.07073 14.17435 8.486566 13.50335 12.19472 8.226776
0.441879 9.652796 19.62942 12.25119 10.4606 7.447221 12.3349 21.30741 12.34033 12.10373 14.20975 8.506745 13.63287 12.22649 8.261818
0.442382 9.583969 19.67358 12.28612 10.48597 7.47094 12.36118 21.35774 12.36636 12.13559 14.24787 8.52595 13.69935 12.25543 8.297354
0.442884 9.74116 19.71922 12.3184 10.5128 7.491724 12.38831 21.40819 12.39263 12.16601 14.28968 8.545364 13.75929 12.28643 8.328573
0.443387 9.652611 19.76535 12.34921 10.53956 7.510349 12.41458 21.45777 12.41828 12.19722 14.3331 8.565346 13.80937 12.31587 8.358953
0.443889 9.669694 19.81308 12.37955 10.56595 7.526766 12.44024 21.50612 12.44258 12.22884 14.37676 8.585538 13.84934 12.34341 8.390566
0.444392 9.821236 19.86477 12.40953 10.5925 7.541309 12.46688 21.55373 12.46577 12.25864 14.41956 8.605446 13.88807 12.37203 8.422303
0.444894 9.904494 19.91373 12.43444 10.61738 7.558417 12.49328 21.6048 12.49291 12.29033 14.45484 8.627044 13.93407 12.4004 8.449747
0.445397 9.951772 19.95888 12.46368 10.64216 7.576524 12.52152 21.66302 12.51893 12.31942 14.49234 8.644892 13.97379 12.4284 8.478807
0.445899 9.902508 20.00908 12.49217 10.66736 7.593656 12.54878 21.71926 12.54743 12.34761 14.52811 8.665441 14.01561 12.45664 8.509705
0.446402 10.03148 20.06076 12.52029 10.69223 7.611282 12.57518 21.77378 12.57395 12.37536 14.56252 8.686249 14.05508 12.48415 8.539123
0.446904 10.05861 20.11084 12.54841 10.71634 7.629094 12.60145 21.82731 12.59825 12.40768 14.59607 8.705467 14.09245 12.51129 8.548325
0.447406 10.19909 20.16018 12.57604 10.74032 7.645844 12.62834 21.88097 12.62365 12.44949 14.62925 8.723561 14.13106 12.53867 8.514725
0.447909 10.1044 20.2179 12.6033 10.76717 7.665678 12.6572 21.93943 12.6519 12.49241 14.66218 8.743852 14.16818 12.56556 8.45362
0.448411 10.14095 20.26712 12.63291 10.79489 7.683945 12.68582 21.99568 12.67694 12.52781 14.69487 8.763414 14.20346 12.59578 8.37257
0.448914 10.07736 20.31966 12.66177 10.82411 7.704223 12.71283 22.05069 12.70136 12.56025 14.7283 8.782397 14.23837 12.62563 8.283095
0.449416 10.134 20.37356 12.69199 10.85354 7.725513 12.74145 22.10521 12.7264 12.59158 14.76148 8.801664 14.27253 12.65511 8.218289
0.449919 10.10826 20.42747 12.72356 10.88272 7.746321 12.7713 22.15381 12.75317 12.62156 14.79564 8.819931 14.30633 12.68434 8.189365
0.450422 10.24398 20.48174 12.75452 10.91184 7.765822 12.79991 22.19377 12.78203 12.65017 14.83215 8.836028 14.34037 12.71394 8.188452
0.450924 10.34241 20.5333 12.78536 10.93935 7.785508 12.82952 22.25655 12.80793 12.68249 14.86669 8.858094 14.37355 12.74416 8.223988
0.451427 10.2924 20.58621 12.81644 10.96628 7.805477 12.85764 22.32131 12.83137 12.71259 14.90554 8.87704 14.41056 12.76933 8.301856
0.451929 10.47014 20.63777 12.8458 10.99129 7.825015 12.88712 22.38298 12.63895 12.74478 14.95069 8.897022 14.44954 12.79732 8.349726
0.452432 10.34633 20.68945 12.87577 11.01494 7.843949 12.91734 22.4412 12.41717 12.77475 15.00385 8.917028 14.49086 12.75477 8.404393
0.452934 10.52351 20.74249 12.90661 11.03942 7.863339 12.94805 22.49745 12.37437 12.80115 15.05935 8.936394 14.53267 12.60824 8.475255
0.453437 10.59756 20.79602 12.90217 11.06626 7.884061 12.97963 22.55406 12.59306 12.82631 15.10869 8.954957 14.57177 12.43876 8.539974
0.453939 10.60926 20.84573 12.6667 11.08941 7.902587 13.01306 22.60932 12.73898 12.85431 15.15593 8.97114 14.60791 12.65437 8.577409
0.454441 10.4802 20.90296 12.44641 11.11725 7.92231 13.04698 22.66137 12.40398 12.88354 15.19503 8.993293 14.63517 12.80176 8.616202
0.454944 10.60622 20.9634 12.38843 11.15119 7.938012 12.87836 22.71836 12.39695 12.53349 15.23648 9.012794 14.6575 12.8886 8.654858
0.455446 10.56176 21.02372 12.33413 11.1841 7.950482 12.65782 22.77707 12.25953 12.39411 15.27607 9.032085 14.67242 12.69026 8.700841
0.455949 10.49865 21.08662 12.21569 11.21633 7.964469 12.29558 22.8348 12.03075 12.5716 15.31308 9.05261 14.69018 12.4601 8.756137
0.456451 10.60817 21.15533 12.12296 11.25355 7.983847 11.7274 22.88956 12.116 12.70704 15.34811 9.07422 14.72447 12.58332 8.823545
0.456954 10.77626 21.21971 12.39016 11.29673 7.99982 11.88078 22.93236 12.45788 12.41804 15.36944 9.097063 14.76456 12.45146 8.931719
0.457456 10.73674 21.28373 12.65979 11.33048 8.015275 11.94391 22.98824 12.69273 12.44801 15.38351 9.116996 14.79034 12.53793 8.986817
0.457959 10.85264 21.34565 12.84222 11.35875 8.034949 12.24665 23.05842 12.83766 12.63204 15.40571 9.138421 14.81661 12.73084 9.030259
0.458461 10.84321 21.40522 12.9573 11.39229 8.05577 12.57456 23.12318 12.92906 12.81953 15.44135 9.158391 14.84005 12.9013 9.063711
0.458964 10.7546 21.46344 13.03378 11.42968 8.075369 12.81262 23.1809 12.99986 12.94509 15.48514 9.177608 14.86422 13.00713 9.091809
0.459466 10.89771 21.52105 13.09496 11.46058 8.093427 12.96729 23.24233 13.06955 13.01824 15.52992 9.19923 14.89308 13.07596 9.121079
0.459969 10.93158 21.57581 13.1475 11.47667 8.112743 13.05771 23.30585 13.13134 13.06757 15.58481 9.223702 14.90628 13.13393 9.149695
0.460471 10.99678 21.63206 13.19388 11.51737 8.13048 13.12851 23.36666 13.18006 13.11531 15.63007 9.249 14.9196 13.18426 9.176695
0.460973 11.08807 21.68349 13.23594 11.55154 8.14834 13.18919 23.4292 13.22509 13.15848 15.66017 9.272399 14.93354 13.23212 9.203695
0.461476 11.11121 21.72321 13.27751 11.57703 8.164992 13.24383 23.48643 13.26887 13.20054 15.68138 9.296032 14.94205 13.2759 9.230017
114
0.461979 10.93577 21.76379 13.32167 11.60152 8.18083 13.29601 23.54107 13.31278 13.2442 15.71284 9.321231 14.95081 13.3171 9.255439
0.462481 11.10736 21.81115 13.3694 11.62896 8.19737 13.34744 23.60015 13.35793 13.29071 15.76847 9.348046 14.96981 13.35879 9.280083
0.462984 11.17584 21.79956 13.41676 11.63159 8.21491 13.39456 23.65615 13.40517 13.34128 15.81904 9.375133 14.99151 13.40048 9.303839
0.463486 11.19893 21.82509 13.45883 11.62271 8.228947 13.4423 23.71844 13.44575 13.38778 15.86973 9.403971 15.01322 13.44563 9.328274
0.463989 11.15031 21.9086 13.49867 11.65909 8.241824 13.48534 23.78813 13.48855 13.43379 15.90871 9.436115 15.04665 13.4851 9.351377
0.464491 11.33508 21.99691 13.54319 11.70614 8.256268 13.52852 23.80972 13.53111 13.47609 15.95102 9.468419 15.0131 13.52913 9.374109
0.464994 11.32982 22.06845 13.59019 11.7418 8.269798 13.5723 23.8318 13.5369 13.51692 15.99875 9.501365 14.8974 13.57292 9.396977
0.465496 11.3 22.12963 13.63361 11.76436 8.278026 13.61535 23.93047 13.50002 13.56059 16.04612 9.531782 14.66995 13.60783 9.419698
0.465999 11.39409 22.18822 13.67702 11.79184 8.281603 13.66087 24.03223 13.59105 13.60524 16.09373 9.535001 14.08271 13.6473 9.440814
0.466501 11.43267 22.2473 13.71403 11.82318 8.290323 13.70478 24.09958 13.6473 13.64939 16.13357 9.539762 13.80839 13.68874 9.464028
0.467003 11.37956 22.30774 13.76201 11.85066 8.2967 13.74647 24.1562 13.68652 13.6917 16.17193 9.575162 13.67949 13.73524 9.487094
0.467506 11.45931 22.36744 13.80567 11.87714 8.3032 13.78618 24.21133 13.72994 13.72846 16.21313 9.610093 13.64779 13.77891 9.50969
0.468008 11.68728 22.42813 13.84379 11.90131 8.314857 13.82208 24.27337 13.77718 13.76139 16.25753 9.634726 13.68899 13.81986 9.531905
0.468511 11.42644 22.49128 13.88165 11.92235 8.338971 13.85279 24.34467 13.81702 13.79235 16.30305 9.654991 13.77533 13.86179 9.55391
0.469013 11.61417 22.55012 13.91323 11.95068 8.387236 13.88338 24.41066 13.84786 13.81258 16.34868 9.672815 13.8259 13.90139 9.57515
0.469516 11.59656 22.61216 13.93778 11.983 8.418701 13.9078 24.48232 13.87043 13.82578 16.39457 9.680869 13.96158 13.91742 9.596131
0.470018 11.72 22.67371 13.95159 12.01464 8.443987 13.92606 24.54967 13.87968 13.82676 16.4444 9.669571 14.12871 13.84317 9.616939
0.470521 11.89098 22.73513 13.94148 12.04981 8.4281 13.92581 24.61541 13.88375 13.8317 16.49645 9.662343 14.45533 13.79075 9.636896
0.471023 11.71921 22.79779 13.93679 12.0856 8.372175 13.92334 24.68115 13.87142 13.85896 16.54813 9.672075 14.78306 13.8275 9.655608
0.471526 11.8756 22.86169 13.97046 12.11748 8.315732 13.9432 24.74665 13.85698 13.91101 16.59636 9.69741 14.90628 13.94135 9.673308
0.472028 11.85746 22.92287 14.01006 12.15008 8.338687 13.98255 24.80919 13.96627 13.94493 16.63842 9.719562 15.00422 13.99488 9.691514
0.472531 11.76839 22.98442 14.04743 12.18764 8.423043 14.01795 24.87912 14.02856 13.98008 16.67962 9.745218 15.1917 14.03522 9.709979
0.473033 12.03229 23.04769 14.08517 12.22526 8.478585 14.05249 24.94758 14.07617 14.01462 16.72057 9.76747 15.3581 14.07444 9.726791
0.473536 12.01094 23.11084 14.12242 12.26122 8.517673 14.08739 25.0153 14.11848 14.04755 16.75844 9.790128 15.49538 14.11243 9.742443
0.474038 11.97591 23.17449 14.15795 12.29646 8.551606 14.12353 25.08166 14.16004 14.07987 16.79655 9.813835 15.60121 14.15018 9.756134
0.474541 12.20849 23.23986 14.1915 12.33239 8.585624 14.16078 25.1432 14.20136 14.11317 16.8396 9.837394 15.68052 14.18878 9.766939
0.475043 12.16421 23.30326 14.22443 12.36747 8.617731 14.20013 25.19353 14.23639 14.14783 16.88388 9.860719 15.75379 14.22727 9.775981
0.475546 12.33067 23.36691 14.25835 12.39744 8.64615 14.23861 25.2742 14.27118 14.18484 16.92779 9.886424 15.82114 14.26427 9.783122
0.476048 12.34095 23.43093 14.29227 12.43247 8.673779 14.27685 25.34709 14.3141 14.22295 16.97589 9.909983 15.88737 14.23726 9.788056
0.476551 12.33848 23.49765 14.32742 12.46577 8.69666 14.31756 25.41691 14.35801 14.26168 17.02597 9.934233 15.95213 14.27118 9.79209
0.477053 12.37117 23.56401 14.36122 12.49809 8.71843 14.35838 25.48043 14.39921 14.30189 17.07778 9.958976 16.01281 14.35024 9.796493
0.477556 12.34625 23.62618 14.39255 12.53201 8.743223 14.3965 25.53865 14.43782 14.34407 17.1318 9.982954 16.07042 14.40501 9.801057
0.478058 12.53016 23.68859 14.42918 12.56408 8.765264 14.43375 25.61771 14.47544 14.38355 17.18817 10.00704 16.13776 14.44781 9.802229
0.478561 12.51388 23.75347 14.46939 12.59639 8.78627 14.46939 25.68691 14.51404 14.42474 17.24713 10.03344 16.25013 14.48617 9.81238
0.479063 12.40903 23.81971 14.51207 12.62994 8.804007 14.50775 25.75549 14.55216 14.46532 17.30375 10.05874 16.34067 14.52552 9.801525
0.479565 12.78696 23.88681 14.55537 12.66399 8.826037 14.54624 25.82407 14.5904 14.4779 17.35678 10.08328 16.41443 14.56314 9.794075
0.480068 12.514 23.95391 14.59767 12.69729 8.852556 14.58558 25.89401 14.61025 14.47519 17.40686 10.10928 16.47795 14.60051 9.796407
0.48057 12.71912 24.02089 14.63801 12.72985 8.879433 14.62728 25.96616 14.5883 14.49776 17.45534 10.13773 16.53629 14.63973 9.799244
0.481073 12.55779 24.08725 14.67809 12.76279 8.90256 14.66835 26.03931 14.51256 14.59989 17.50283 10.16442 16.59155 14.67846 9.782358
0.481575 12.7655 24.15509 14.60828 12.79412 8.912169 14.71325 26.11122 14.62506 14.65441 17.54809 10.18936 16.64336 14.66958 9.778115
0.482078 12.73467 24.22071 14.59989 12.83124 8.939539 14.75691 26.18165 14.72213 14.70042 17.59373 10.2184 16.6827 14.60199 9.801488
0.48258 12.87158 24.28583 14.67723 12.86381 8.969117 14.79947 26.25356 14.78997 14.74667 17.64159 10.24847 16.73599 14.65898 9.837937
0.483083 12.87614 24.35194 14.78454 12.8976 8.994563 14.84103 26.32584 14.83857 14.79256 17.68994 10.27733 16.79791 14.77911 9.874743
0.483585 12.9948 24.42077 14.85867 12.93991 9.017888 14.88198 26.39676 14.88285 14.83548 17.73644 10.30477 16.85502 14.84116 9.908059
0.484088 13.07399 24.4949 14.9106 12.9837 9.047404 14.92417 26.46929 14.9254 14.87791 17.7827 10.33385 16.91299 14.88716 9.950292
0.48459 13.0244 24.56829 14.95106 13.0212 9.065351 14.96586 26.54823 14.96759 14.92059 17.8292 10.3606 16.97219 14.93046 9.974579
0.485093 13.24902 24.6423 14.99904 13.05968 9.084618 15.0094 26.62606 15.00854 14.96327 17.88248 10.38864 17.03115 14.96981 9.985705
0.485595 13.22299 24.71285 15.04665 13.09594 9.115244 15.05245 26.69057 15.04998 15.00656 17.94675 10.41971 17.08876 15.01199 9.990861
0.486098 13.00775 24.7818 15.0913 13.13048 9.151508 15.09463 26.75693 15.0934 15.05072 18.01903 10.45201 17.14401 15.05676 9.991588
0.4866 13.3023 24.85236 15.13435 13.16625 9.183615 15.1362 26.84007 15.13817 15.09475 18.09316 10.48269 17.19619 15.10043 9.990305
0.487103 13.23261 24.92402 15.17863 13.20683 9.210183 15.17715 26.92369 15.18023 15.13731 18.1647 10.51039 17.24565 15.14015 9.99741
0.487605 13.38926 25.00062 15.22328 13.24445 9.209801 15.21983 26.99906 15.22118 15.17986 18.22662 10.53813 17.29079 15.18578 10.04809
0.488108 13.42182 25.07931 15.26707 13.2875 9.244646 15.26435 27.08096 15.26411 15.22303 18.28878 10.56532 17.33976 15.22636 10.09272
0.48861 13.40122 25.15714 15.31036 13.33166 9.278208 15.30827 27.16274 15.30765 15.26522 18.35021 10.59552 17.37603 15.26731 10.12756
0.489113 13.59278 25.24188 15.3539 13.3805 9.303013 15.35131 27.24415 15.35181 15.31012 18.41065 10.62656 17.37701 15.30962 10.15649
0.489615 13.4492 25.33575 15.39831 13.43662 9.327509 15.39448 27.32654 15.39646 15.35871 18.4685 10.6551 17.36184 15.35107 10.18081
0.490118 13.45932 25.42197 15.44259 13.49324 9.362761 15.43753 27.40832 15.44012 15.4025 18.5187 10.68127 17.43523 15.38819 10.18975
0.49062 13.51199 25.50892 15.48403 13.54973 9.380758 15.47996 27.48948 15.47959 15.44506 18.57852 10.70636 17.51886 15.42914 10.19024
0.491123 13.64495 25.5949 15.52609 13.59673 9.407425 15.52437 27.5699 15.5198 15.49069 18.63403 10.73336 17.5851 15.47083 10.20631
0.491625 13.69367 25.67988 15.5721 13.64101 9.437348 15.56815 27.6481 15.56544 15.53695 18.68854 10.76018 17.64467 15.51672 10.23917
0.492126 13.87092 25.76154 15.61971 13.68418 9.466446 15.61182 27.71927 15.6128 15.58123 18.74195 10.78713 17.70178 15.56223 10.26662
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Appendix B. LS-DYNA Input Code of Simulation Case 1
This appendix presents the LS-DYNA input code of the simulation case 1 in Figure 3.9
to obtain stress vs. strain data of the compression simulation.
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119
120
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121
…
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Appendix C. Analysis Results of Response Surface Methodology
Figure C.1 shows the analysis results of the response surface methodology to yield the
quadratic equation for optimizing the unknown parameters. In the model summary of
Figure C.1, the adjusted R-sq is used to decide the degree to which the input variables
explain the variation of the output. As shown in Figure C.1, 96.38 % of the variation in
the output variable is explained by the input variables. Based on the analysis, the regression
equation was obtained in order to optimize the unknown parameters.
Figure C.1 Analysis results for Response Surface Regression
127
A contour plot is used to show how fitted response values relate to two continuous
variables based on a model equation. A contour plot provides a two-dimensional view in
which all points that have the same response are connected to produce contour lines of
constant responses. Contour plots are useful for establishing desirable response values and
operating conditions. A contour plot consists of predictors on the x- and y-axes, contour
lines that connect points that have the same fitted response value, and colored contour
bands that represent ranges of the fitted response values. Figure C.2 shows a contour plot
of response vs. friction angle, Poisson’s ratio. This contour plot would be plotted by using
the quadratic regression equation obtained from the statistical analysis shown in Figure
C.1. As shown in Figure C.2, the lightest green-colored band represents the smallest
difference between the compression test and simulations. Therefore, the optimum regions
can be potentially expected as shown in Figure C.2.
Figure C.2 Contour plot of response vs. friction angle, Poisson’s ratio
128
3D surface and 3D wireframe plots are graphs to be able to use to explore the potential
relationship between three variables. The predictor variables are displayed on the x- and
y-scales, and the response variable (z) variable is represented by a smooth surface (3D
surface plot) or a grid (3D wireframe plot). A 3D surface plot displays a 3-dimensional
view of the surface. Like contour plots, 3D surface plots are useful for establishing
response values and operating conditions. 3D surface plots can provide a clearer concept
of the response surface than contour plots. Figure C.3 shows a surface plot of response vs.
friction angle, Poisson’s ratio. This surface plot provides a clearer trend of the response
surface than the previous contour plot. It shows that #1 region has a greater possibility of
the optimum point to minimize the difference than #2 region as shown in Figure C.3.
Figure C.3 Contour plot of response vs. friction angle, Poisson’s ratio
#1
#2
129
Response Optimizer can be used to identify the combination of input variable settings
that optimize a single response or a set of responses. Minitab calculates an optimal solution
and draws an optimization plot. This interactive plot allows to change the input variable
settings to perform sensitivity analyses and possibly improve upon the initial solution.
Figure C.4 shows an optimization plot to identify a set of input variables to optimize a
single response.
Figure C.4 Optimization plot
TAE KWANG YOO 101 Bernreuter, White Course Apartments University Park, PA 16802
[email protected], [email protected]
814-441-9360
SUMMAY OF QUALIFICATIONS Dynamic Soil-Structure Interaction in Shallow Foundation System
Finite Element Analysis for static and dynamic (vehicular crash and drop impact) simulations
Vehicular crash simulations and tests (Anti-ram barriers) based on ASTM Standard Acoustic tests and simulations (Boundary Element Method)
Applications of Six Sigma (Design of Experiments, Response Surface Methodology)
EDUCATION Ph.D. in Acoustics (Graduation: May 2018) The Pennsylvania State University, University Park, PA Cumulative GPA: 3.67/4.00
Research Co-Advisers: Dr. Victor W. Sparrow (Acoustics), Dr. Tong Qiu (Civil Engineering)
Master of Science in Department of Mechanical Design (Graduation: February 2003) SungKyunKwan University, Suwon, South Korea Cumulative GPA: 3.63/4.00
Bachelor of Science in Department of Mechanical Design (Graduation: February 2001)
SungKyunKwan University, Suwon, South Korea Cumulative GPA: 3.24/4.00
RELEVANT EXPERIENCES Research Assistant (PhD student), The Pennsylvania State University (May 2014–Present)
• Conducting development of anti-ram and counter-terrorism barriers funded by U.S.
Department of State : Simulating and testing full vehicular crash to develop Streetscape Vehicular Anti-Ram
(SVAR) and Rock Vehicular Anti-Ram (RVAR) systems
: Modeling and optimizing constitutive material model to simulate polystyrene crushable
concrete in LS-DYNA using statistical analyses (RSM of Six Sigma) : Conducting medium pendulum override tests and simulations using LS-DYNA for
dimensionless analysis to save cost and time of full scale crash tests and simulations
Mechanical and Acoustic Engineer, Samsung Electronics (February 2003–August 2010) • Implemented mechanical acoustic simulation for mobile phones with micro speakers using
lumped modeling and acoustic CAE (BEM using LMS Virtual.Lab Acoustics)
• Conducted drop impact and hinge stress simulations on mobile phones
Research Assistant (MS student), SungKyunKwan University (March 2001–February 2003) • Conducted fatigue life assessment about marine diesel engine of Doosan Engine and steel
manufacturing facilities of POSCO
PUBLICATIONS • Yoo, T.K., Qiu, T., Reese, L., and Rado, Z. “Field Testing and Numerical Investigation of
Streetscape Vehicular Anti-Ram Barriers under Vehicular Impact using FEM and Coupled
FEM-SPH Simulations.” International Journal of Protective Structures, vol. 7, no. 2, 2016, pp.
213-231. • Yoo, T.K., Qiu, T. “Optimization of Constitutive Model Parameters for Simulation of
Polystyrene Concrete Subjected to Impact.” International Journal of Protective Structures,
Article first published online: June 20, 2017, DOI: 10.1177/2041419617716496.