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THEORETICAL AND NUMERICAL STUDY OF A CONCRETE CYLINDER SUBJECTED TO AN IMPACT LOAD

By

AVSHALOM GANZ

A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE

UNIVERSITY OF FLORIDA

2011

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© 2011 Avshalom Ganz

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To my great family

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ACKNOWLEDGMENTS

I would like to express my heartfelt gratitude to my thesis advisor, Professor

Theodor Krauthammer for his time, guidance and advice throughout the whole period of

time. I would also like to thank my thesis committee member, Dr. Serdar Astarlioglu for

his time and assistance. Next, I would like to thank Dr. Long Hoang Bui for his advice

and teaching on the finite element software. I would also like to thank my friend and

research partner, Mr. Liran Hadad. Furthermore I am honored and grateful for Ministry

of Defense, Israel, on the research founding and for University of Florida on the

scholarship granted to me. I am grateful to Professor Oren Vilnay from Ben-Gurion

University, Israel, for his unlimited time and knowledge. In addition, I thank all my

colleagues at the Center for Infrastructure Protection and Physical Security (CIPPS), the

Chabad family, and my friends in Gainesville for their love and support which made my

stay in USA a wonderful experience. I owe my loving thanks to my family back in Israel

for their endless love and support.

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TABLE OF CONTENTS page

ACKNOWLEDGMENTS .................................................................................................. 4

LIST OF TABLES ............................................................................................................ 7

LIST OF FIGURES .......................................................................................................... 8

ABSTRACT ................................................................................................................... 11

CHAPTER

1 INTRODUCTION .................................................................................................... 12

1.1 Problem Statement ........................................................................................... 12

1.2 Objectives and Scopes ..................................................................................... 13 1.3 Research Significance ...................................................................................... 13

2 LITERATURE REVIEW .......................................................................................... 15

2.1 Failure Based on Material Performance ............................................................ 15 2.2 Fracture Mechanics .......................................................................................... 15

2.2.1 Background ............................................................................................. 15 2.2.2 Linear Fracture Mechanics. ..................................................................... 16

2.3 Size Effect ......................................................................................................... 17

2.3.1 Background ............................................................................................. 17

2.3.2 Energy Theory and Size Effect. ............................................................... 18 2.4 Strain Rate Effect .............................................................................................. 18

2.4.1 Background ............................................................................................. 18

2.4.2 Experimental Techniques ........................................................................ 19 2.4.3 Inertia Effect ............................................................................................ 22

2.5 Size and Rate Effect as a Coupled ................................................................... 22 2.6 Propagation of Waves in Elastic Solid Media. ................................................... 25

2.6.1 Wave‟s Equation ..................................................................................... 25

2.6.2 Superposition of Waves ........................................................................... 26 2.6.3 Reflection of Waves................................................................................. 26

3 METHODOLOGY ................................................................................................... 37

3.1 Failure Due to Dynamic Buckling. ..................................................................... 37

3.2 Strain Rate Effect Approach ............................................................................. 38 3.2.1 Mass-Spring Model Approach ................................................................. 38 3.2.2 Queries Regarding the Inertial Effect Explanation. .................................. 38 3.2.3 The Kinetic Energy of a Specimen in Split Hopkinson Pressure Bar ....... 39 3.2.4 Finite Elements Model of SHPB .............................................................. 40

3.3 The Suggested Approach for Buckling .............................................................. 41 3.3.1 Dynamic Buckling of a Single Column. .................................................... 42

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3.3.2 Fracture Mechanics and Energy Methods ............................................... 44

4 RESULTS AND DISCCUSIONS ............................................................................. 51

4.1 Energies in Strain Rate Effect ........................................................................... 51

4.1.1 Applied Load ........................................................................................... 51 4.1.2 Energies and Strain in the Specimen ...................................................... 52 4.1.3 A Theoretical Explanation for the Results Above .................................... 53 4.1.4 Strain Energy and Strain Distribution....................................................... 55 4.1.5 Summary of the Results by Time Sequence ........................................... 56

4.2 Buckling Load of a Group of Columns .............................................................. 59 4.2.1 The Number of Rods in a Group ............................................................. 59 4.2.2 Buckling as a Failure Criteria ................................................................... 60

4.2.3 Examination of Bending as a Possible Post Failure Effect ...................... 60 4.2.4 A Possible Explanation the Post Failure Behavior ................................... 63

5 CONCLUSIONS AND RECOMMENDATION ......................................................... 75

APPENDIX

A MATHCAD CALCULATION SHEET FOR THE SHPB PROPERTIES .................... 77

B MATHCAD CALCULATION SHEET FOR THE BUCKLING ................................... 80

C MATHCAD CALCULATION SHEET FOR POST-FAILURE BENDING .................. 82

LIST OF REFERENCES ............................................................................................... 88

BIOGRAPHICAL SKETCH ............................................................................................ 90

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LIST OF TABLES

Table page 3-1 Properties of the parts ........................................................................................ 46

‎4-1 The shape of the applied load ............................................................................ 64

4-2 Properties of the hammer and the specimen ...................................................... 64

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LIST OF FIGURES

Figure page ‎2-1 Strain-stress curves ............................................................................................ 27

‎2-2 Stress distribution along the hole with respect to the material brittleness ........... 28

‎2-3 Semi-infinite plate with a hole made in it ............................................................ 29

‎2-4 Size effect law .................................................................................................... 29

‎2-5 Concrete DIF vs. strain rate ................................................................................ 30

2-6 Charpy impact test device .................................................................................. 30

‎2-7 Dropped weight impact test device ..................................................................... 31

2-8 Split-Hopkinson pressure test device ................................................................. 31

2-9 Typical direct-compression test data from the SHPB ......................................... 32

2-10 Scheme of the setup and principle functioning of the spalling technique ............ 32

2-11 Equivalent mass-spring system .......................................................................... 32

2-12 A 600x1200 mm specimen ready for soft impact test ......................................... 33

2-13 Compression failure modes observed ................................................................ 34

2-14 Illustration of wave‟s function shifting ................................................................. 35

2-15 Two waves traveling in an opposite direction, after time t .................................. 35

2-16 A bar subjected to a sudden compressive load .................................................. 35

2-17 Superposition of waves ...................................................................................... 35

2-18 Reflection of wave from free end ........................................................................ 36

2-19 Reflection of wave from fixed end ....................................................................... 36

3-1 Failure modes due to Poisson‟s effect ................................................................ 46

3-2 Failure due to dynamic buckling ......................................................................... 46

3-3 Spring-mass model for strain rate effect ............................................................. 47

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3-4 A bar subjected to a stress wave ........................................................................ 48

‎3-5 An illustration of the finite elements model of a SHPB ........................................ 48

‎3-6 An illustration of the finite elements model of SHPB, close view of the specimen area .................................................................................................... 49

3-7 An illustration of the specimen and the incident bar ........................................... 49

3-8 Set of elements along the specimen .................................................................. 50

‎3-9 Illustration of a set of two nodes of the specimen ............................................... 50

4-1 The applied load shape ...................................................................................... 64

‎4-3 Strain energy and kinetic energy at the specimen .............................................. 65

4-4 Average strain of the specimen vs. time ............................................................. 65

4-5 Strain and kinetic energies at the specimen, with time marks(close view from the wave‟s arrival time) ....................................................................................... 66

4-6 Average strain of the specimen (compression is positive) vs. time, with time marks .................................................................................................................. 66

4-7 A bar subjected a rectangular stress wave at time t=l/c...................................... 67

4-8 A bar subjected a rectangular stress wave at time t=3/2⋅l/c................................ 67

4-9 The incident wave and the wave which was reflected due to a medium change ................................................................................................................ 67

4-10 SE/(G⋅ε^2) vs. time ............................................................................................. 68

4-11 Standard deviation of the strains in the specimen vs. time ................................. 68

‎4-12 Standard deviation/average of the strains in the specimen vs. time ................... 69

4-13 Strain and kinetic energies at the specimen ....................................................... 69

4-14 Average strain of the specimen (compression is positive) .................................. 70

4-15 SE/(G⋅ε^2) vs. time ............................................................................................. 70

‎4-16 Standard deviation of the strains in the specimen .............................................. 71

4-17 Standard deviation divided by average of the strains in the specimen vs. time .. 71

4-20 Model for bending of the cylinder ........................................................................ 72

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4-21 The components of the velocity .......................................................................... 72

4-22 The radial velocity in a single columns and the bending direction. ..................... 72

4-23 An approximated model to obtain the average velocity in the bending direction. ............................................................................................................. 73

‎4-24 The strain energy required for failure (uMcrack) of a single column *10^3 and the kinetic energy that can use for bending vs. the number of column in a group. ................................................................................................................. 73

4-25 A model for the post failure behavior consist of masses and springs. ................ 74

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Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science

THEORETICAL AND NUMERICAL STUDY OF A CONCRETE CYLINDER

SUBJECTED TO AN IMPACT LOAD

By

Avshalom Ganz

August 2011

Chair: Theodor Krauthammer Major: Civil Engineering

In the past few decades, many studies have shown an increase in the nominal

strength at failure of vary type of materials when subjected to high-strain rates. In this

study, a concrete cylinder subjected to an impact load was examined concentrating on

the kinetic characterization of the process and examination of the strain rate effect. In

addition, the observations of several modes of failure were investigated as well by

considering the subsequences of the kinetic analysis. The methodology included the

theory of static and dynamic buckling, equation of motion for a mass-spring system, the

theory of elastic waves and a finite elements model (FEM) of a Split Hopkinson bar to

examine the strains and the energies of a specimen subjected to an impact. By using

waves‟ analysis and FEM model, It was shown that at the dynamic domain, part of the

kinetic energy converts to strain energy, but some remains in the system until the

failure. the kinetic energy converts to strain energy due to superposition of waves, which

can cause an non-uniform distribution of strains. The subsequences of this strains‟

distribution is not completely clear and require further investigation. By considering the

conclusions above and by using numeric approximated model, the modes of failures

were investigated and are believed to be a post-failure phenomenon.

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CHAPTER 1 INTRODUCTION

According to the strength theory the nominal strength of a material is a unique

property of the material. However, the size effect law (Bažant & Planas, 1998) indicates

a direct dependency of the strength of a material on the geometrical dimensions. This

law is explained by using nonlinear fracture mechanics, and was well based by

experiments.

In addition, while the nominal strength of a material is traditionally determined to

be independents on the loading conditions, many researches (Lindhom, 1964) show an

increase of strength with respect to load or strain rate. Although some explanations

were offered, the cause of this phenomenon, which is known as strain rate effect, is still

not clear.

1.1 Problem Statement

Size and rate effects, the two phenomena above, were widely discussed

separately. However, recent studies suggest a dependency (Park & Krauthammer,

2006).

Although the strain rate effect was not theoretically well based, the inertia effect

seems to be the main cause for the increase in strength. Those two facts in addition to

the fact that volume and mass are proportional for a certain material imply a

dependency for the size and rate effect. Since a theoretical approach to the strain rate

effect was not established yet, basing the dependency of the strain rate on the

displaced mass is believed to be the first step.

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1.2 Objectives and Scopes

This study aims to achieve a better understanding of the strain rate effect, by

analyzing the failure of concrete cylinders subjected to an impact. For this purpose, the

observations of Krauthammer and Elfahal (2002) experiments are analyzed. In addition

to the strain rate effect, Krauthammer and Elfahal (2002) observed some failure modes.

These modes were classified by Krauthammer and Elfahal (2002) into 7 modes of

failure. Since the phenomenon of several modes of failure is not clear, it is suggested to

explain each mode separately.

In this study the buckling failure modes are examined, and the subsequences of

the kinetic energy presence during the loading phase is explored. This mode is

assumed to consists two phases that are analyzed separately. During the first phase the

cylinder is compressed and split to a group of columns, and the second phase

describes the buckling of the columns. The presence of kinetic energy during the

loading time is investigated and bases the initial conditions for the buckling phase.

1.3 Research Significance

Many studies in the recent century based the strain rate effect, mainly

empirically. Subsequently, the concept of increase of strength due to high strain rate

was adopted for a design purposes and materials assumed to have higher nominal

strength when they are subjected to high strain rate. However, the explanation for this

phenomenon is not clear and therefore, misinterpretation of tests‟ results may occur. It

seems that some concepts in the static domain that were adopted in the dynamic

domain and may need to be reconsidered.

The strain - stress curve, which is being widely used in the static domain, was

adopted in the dynamic domain as well. While in the static domain, this curve

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represents the material‟s conditions at each specific point, at the dynamic domain the

velocity of the particles is not accounted for in chart. This absence is significant since

the kinetic energy is potentially strain energy, as mentioned above, indicating that in the

dynamic strain-stress curve there might be an additional strain that is not taken into

account.

Another concept that was adopted from the static domain is the determination of

the nominal strength according to the maximum load that was measured. By

considering the neglected kinetic energy that was mentioned above together with the

fact that the specimen is constantly moving toward the failure point, it can be concluded

that in the dynamic domain the maximum load is measured when the specimen already

has enough energy to reach the maximum nominal strain.

The proposal above is supported by the observations of the dynamic failure

process. In contrast to the quasi-static tests, where the specimen‟s failure is

characterized by several crack planes, in the dynamic failure many crack planes are

observed and particles of the specimen are moving out with a certain velocity. Thus, the

specimen at the failure time has more energy than needed for failure.

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CHAPTER 2 LITERATURE REVIEW

2.1 Failure Based on Material Performance

The type of failure undergone by a material is largely depends on its brittleness.

Materials can be roughly categorized as either: brittle, ductile or quasi-brittle (Figure

‎2-1). While in brittle materials (Figure ‎2-1A), stresses suddenly drops down to zero after

fracture, ductile materials (Figure ‎2-1B) maintains constant stresses when yielding.

Quasi-brittle materials (Figure ‎2-1C) demonstrate “softening,” characterized by

gradually decreasing of stresses after the peak stress is attained.

The failure characteristics greatly depend on the brittleness of the material. These

can be described (Shah, Swarz, & Ouyang, 1995) by considering an infinitely wide plate

with an elliptical hole subjected to a far-field tensile stress, as in (Figure ‎2-2). For a

perfectly brittle material (Figure ‎2-2A), a sudden failure occurs when the maximum

normal stress on the holes‟ edge reaches the material capacity. However, for a ductile

material, after the maximum stress, ft, is reached, stress will be redistributed, and failure

occurs when the entire cross section A-A is yielding. The failure of a Quasi-brittle

material is depicted in (Figure ‎2-2C), where after the stress reaches the material‟s

strength, the stresses will be redistributed, whereas material that reaches capacity is

damaged, and hence, has diminished strength.

2.2 Fracture Mechanics

2.2.1 Background

It is common knowledge that all materials contain flaws. The theory of elasticity

introduces the linear stress distribution for the case of a semi-infinite plate with a hole

made in it (Timoshenko, 1951):

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Assuming the flaw‟s shape to be elliptical for elastic analysis leads to the case of

an elliptical hole, where 2a is the axis of the ellipse, perpendicular to the tension S.

where 2b represents the other axis, the maximum tension on the hole‟s edge is:

( ⋅

) (2-1)

It can be seen that for a large ratio a/b, the tension on the edge becomes infinite

for any external load. Thus, another method must be considered to describe the

crack/flaw.

2.2.2 Linear Fracture Mechanics.

An energy based failure criterion was developed (Griffith, 1920), which stipulates a

model for the failure by crack propagation. The crack will propagate if the energy

released rate at the crack tip equals the energy rate required for the crack to be

extended unit length. Assuming the only energy required for crack extension is the

surface energy, The crack propagation criterion can be expressed as:

( ) (2-2)

Where:

is the material constant defining specific surface energy required to break atomic

bonds,

a is the crack length,

Ue is the external work,

Us is the crack„s strain energy.

For more general cases, this criterion can be expressed as:

Where G is the energy release rate and Gc is the critical energy release rate.

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2.3 Size Effect

2.3.1 Background

Until the last few decades, it has been believed that the nominal strength of the

concrete is independent the specimen‟s dimension. However, experimental

investigations conducted by Bažant and Planas (1998) and Krauthammer and Elfahal

(2002), indicate clearly that the concrete behavior is high dependent on the size of the

specimen. These results led to attempts to explain this phenomenon. A few approaches

have been conducted: statistical, numerical, and theoretical.

Weibull (1939) offered a statistical explanation for the size effect, which has been

widely accepted. This theory considers the randomness of the material‟s strength as the

cause of the size effect, based on a chain model. Since the failure strength of a chain

determines by the weakest link, the longer the chain is, the greater the reduction in

strength. The probability of failure of a concrete structure can be expressed by the term

(Krauthammer & Elfahal, 2002)

( ) * ∫ * ( )

+

( )

(2-3)

∫[ ( )] (2-4)

Where V is the volume of the structure, Vr is a constant, m and σ0 are material

constants representing the Weibull modulus and the threshold stress. And σ(p,x) is the

stress caused by load p at location x.

while this explanation is appropriate for most of the metals, which fail at initiation of

the crack propagation, it is found to be inapplicable for reinforced concrete (Bažant &

Planas, 1998).In addition, experiments conducted on diagonal shear failure of

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reinforced concrete, provided results which contradicted the statistical theory (Bažant &

Planas, 1998).

2.3.2 Energy Theory and Size Effect.

Bažant (1986) Expressed the failure stress of geometrically similar structures of

different sizes as the following infinite series: (Krauthammer & Elfahal, 2002)

[ ]

(2-5)

Where B, , , , etc. are constants,

is the relative structural ratio, and

is the size-independent tensile strength of the material under consideration. For small

specimens,

, Equation 2-5 can be simplified to the following form, named the

Sized Effect Law(Bažant & Planas, 1998):

√

(2-6)

It can be seen in Figure ‎2-4, that for small specimens, the size effect law

approaches the strength theory‟s results, whereas for large specimens, it approaches

the results of Liner Elastic Fracture Mechanics.

2.4 Strain Rate Effect

2.4.1 Background

During the last few decades, a strengthening was observed, for the most the

materials, as the strain rate or the load rate was increased. Whereas the common

European scientific community‟s term is “load rate” or “stress rate”, using units of

, the common term used in United States is “strain rate”, using units of . In

contrast to Weerhijm (1992), (Tedesco, Powell, Ross, & Hughes, 1997) reported the

Young‟s modulus to not be strain rate sensitive and therefore, the static Young‟s

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modulus can be used to convert stress rate into strain rate, and vice versa (Tedesco et

al., 1997). The term DIF, which stands for Dynamic Increase Factor, describes the

strengthening due to the strain rate property of a material. The DIF is defined as the

dynamic strength, as a function of strain rate, over the quasi-static strength.

Figure ‎2-5 describes the DIF of concrete vs. strain rate, for tension and

compression. It can be seen that for characterized conventional weapon strain rate, a

very significant tensile strengthening (DIF becomes about 8) observed, whereas the

compression DIF is about 1.7.

While the strain rate effect has been well supported by mainly experimental

evidences, a theoretical explanation has not been supplied. However, several theories

offered the proposed the root cause to be the viscosity of the hardened cement paste, a

thermally activated crack growth, the limit of crack propagation velocity, and/or the

inertia effect.

2.4.2 Experimental Techniques

There are three experimental techniques apply an impact which are commonly in

use: Charpy impact test, drop weight impact, and split Hopkinson pressure bar test.

Charpy impact test is characterized by a horizontal impact, which causes the acting

force to be applied only during the impact. This is done by a pendulum with a weight at

the free end. Upon being released, it rotates and strikes the specimen, which is

positioned horizontally, with a constant parallel velocity (Figure ‎2-6).

A drop weight impact test is usually performed by dropping a weight from a tower,

down toward the specimen, striking it with a constant velocity, and continually applying

a gravitational force (Figure ‎2-7).

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Split-Hopkinson pressure bar testing is used to perform a compression impact test

and measure the compression wave. When the striker bar strikes the incident bar (bar

No. 1 in Figure ‎2-8), a compression stress wave is generated and starts traveling

toward the specimen, measured at the strain gauge on bar No.1. When the

compression wave reaches the specimen, part of it will be reflected back to strain gage

No.1, and the rest will continue through the specimen until the far end of the specimen.

There, the wave splits again. A part of it is reflected back, and the rest transmits through

the contact to the transmitter bar (bar No. 2 in Figure ‎2-8), and will be measured by the

strain gauge. A typical result can be seen in Figure ‎2-9

The stress wave in the specimen can be estimated by using superposition of the

waves, and the formals that are used to calculate the strains and stresses are given

(Lindhom, 1964):

∫ ⋅

(2-7)

Where u is the displacement at time t, c is the elastic wave velocity, and ε is the

wave strain. The displacement of the incident bar, u1, is the results of both the incident

pulse traveling in the positive x direction and the reflected pulse traveling in the negative

x direction. Thus:

∫ ⋅

∫ ⋅ ∫ ( ) ⋅

(2-8)

Where εi is the incident strain, and εr is the reflected strain. Similarly, the

displacement u2 of the face of the transmitter bar is obtained from the incident pulse εt:

∫ ⋅

(2-9)

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The nominal strain in the specimen is then:

∫ ( ) ⋅

(2-10)

Where l0 is the initial length of the specimen. By assuming the stress distribution along

the specimen to be constant,

(2-11)

And therefore, the expression above can be simplified to:

∫ ( ) ⋅

(2-12)

The applied loads p1 and p2 on each face of the specimen are:

⋅ ( ) (2-13)

⋅ (2-14)

Thus, the average stress in the specimen is:

Where E is the modulus of elasticity of the bars, A is the area of the pressure bars, and

As is the area of the specimen. The simplified expression will be therefore:

⋅ (2-16)

Brara & Klepaczko (2006) conducted a direct tensile impact test using on Split-

Hopkinson bar, as shown in Figure ‎2-10. When the projectile strikes the bar, a

compression wave start propagating toward the reaction, is reflected from the reaction

as a compression wave traveling to the free end. The compression wave is reflected

from the free end as a tensile wave, moving toward the specimen.

⋅

⋅ ( ) (2-15)

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2.4.3 Inertia Effect

One main explanation for the strain rate effect is the inertial forces acting on the

body during impact. The inertial effect can be classified into two types: the structural

inertial effect, and the inertial effect on failure criteria. The inertial forces acting on a

body under impact resist the applying force, an effect which yields a reduced strain. For

a maximum strain failure criterion for concrete, this leads directly to strength increase.

Chandra and Krauthammer (1995) explained strain effect in a flawless material, using

the approach of the structural inertial effect, modeled by spring-mass SDOF and 2-DOF.

This approach is also supported by the observation of an increase in the modulus of

elasticity as the strain rate increases, summarized by Weerhijm (1992).

Attempts to examine the way inertial effect influences the crack expansion criteria,

have been done as well. This strain rate‟s dependent strengthening can be expressed in

terms of fracture mechanics, as a decrease in the stress intensity factor as a function of

strain rate. To obtain the dynamic intensity factor, Freund (1998) developed a strain rate

dependent reduction factor, to be multiplied by the static intensity factor. This factor has

been defined to be zero when the crack expansion velocity reaches the theoretical limit

which is the Rayleigh Wave velocity.

2.5 Size and Rate Effect as a Coupled

Park and Krauthammer (2006) developed a dynamic crack propagation criterion,

by adding a kinetic energy component. This dynamic criterion, which is based on LEFM,

is expressed as:

(2-17)

Where Ue, Us, and Uk are the external work, strain energy rate , and kinetic

energy rate, B is the specimen width, and is the surface energy required to propagate

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a crack,per unit area. By adopting Marur‟s (1996) equivalent mass-spring (Figure ‎2-11)

principle Park and Krauthammer (2006) simplified the criterion to be the following:

( ) (2-18)

(

( ) ( ) )

(2-19)

(

( ) ( ) )

(2-20)

From the equation of motion:

( )

(2-21)

Substituting Equation 2-20 yields:

[

] (2-22)

the criterion for LEFM dynamic crack propagation can be expressed as:

[

] (2-23)

It can be seen that on the right hand side of the eq., a mass depended component

was added, and it should reflect the strain rate strengthening.

Krauthammer and Elfahal (2002) Conducted experimental research and testing,

which included four different sizes of geometrically-similar cylinders, of normal and high

strength concrete. These specimens were subjected to an impact load, produced by a

weight dropped with hitting speed of either 0 m/s(static), 5 m/s, or 7 m/s. each case

was performed 3 times to obtain a variety of results. To eliminate the effect of drying

rate, the specimens were tested after two years of completely dry storage. The tests

were also divided into soft and hard impact: in the soft impact (Figure ‎2-12) tests, rubber

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pads were placed on top of the cylinder. In contrast, in the hard impact tests, the

specimens were impacted directly by the hammer. In total, 124 tests were conducted for

HSC and NSC, both static and dynamic.

The main achievements yielded by this research are:

Existence of the size effect was observed for cylindrical NSC specimens under

both static and dynamic axial compressive loads.

The size effect law predictions were found to be very accurate for the static NSC

and HSC tests.

The strain rate effect was clearly observed for both NSC and HSC testing.

In addition, by using a high frame rate camera, the failures were recorded by

video, and the failure shapes were classified by the authors into seven failure modes:

Vertical splitting: the cylinder splits through several vertical planes (Figure ‎2-13A. and Figure ‎2-13B). The split can be started from the top (Figure ‎2-13C) or from the bottom (Figure ‎2-1D).Cone-shaped shear failure: the sides are pushed out, leaving two cone-shaped remnants, at the top and at the bottom, forming an axis-symmetric hourglass shape (Figure ‎2-13E).

Diagonal shear failure: failure occurs under a diagonal failure plane (Figure ‎2-13F). In some cases, the failure combined vertical splitting or shell brusting (Figure ‎2-13G).

Buckling failure: inflation of the cylinder, from the bottom (Figure ‎2-13H) or from the half of the length (Figure ‎2-13I).

Compressive belly failure: inflation of the center of the cylinder, forming a belly shape (Figure ‎2-13J).

Shell-core failure: happens by the bursting of the shell of the cylinder, leaving a core at the center (Figure ‎2-13K)

Progressive collapse: gradual transfer of the failure from the top to the bottom, as the hammer goes down inside the cylinder. It‟s distinguished from the others failure by the fact that the specimen stays intact and in place while the hammer is moving down, and progressively eroding it from the top to the bottom(Figure ‎2-13I and Figure ‎2-13M).

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2.6 Propagation of Waves in Elastic Solid Media.

2.6.1 Wave’s Equation

By considering a rod under impact, Timoshenko (1951) shows the wave‟s equation

derivation. The wave‟s equation can be derived from the equation of motion of a random

section of the rod. By neglecting the lateral displacement of the particles, the provided

equation is:

(2-24)

where: √

.

It can be seen that the wave equation is a second order linear differential equation,

and the solution for this equation is given by:

( ) ( ) ( ) (2-25)

For both functions, f and f1, it can be seen that for each for a certain time T they both

will be only functions of x. the shape of the functions stays unchanged and only depend

on the functions f and f1. When t is equal to t+Δt the shape will be offset by Δx=cΔt as

described in Figure ‎2-14.Thus, the function ( ) describes two waves traveling in

opposite directions at a constant speed c (Figure ‎2-15).

The relationship between the particle‟s velocity, ν to the stress, σ, can be

developed by the impulse law

∫ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ (2-26)

which yields:

⋅ ⋅ (2-27)

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2.6.2 Superposition of Waves

Because of the linear characteristic of the wave equation, the superposition

principle is valid. For two waves traveling in opposite directions (Figure ‎2-17A), the

resultant of the stress and particle„s velocity are obtained by superposition with sign

sensitivity (Figure ‎2-17B). After passing, the waves return to their original shape (Figure

‎2-17C).

2.6.3 Reflection of Waves

By considering a compressive wave traveling in the positive x direction and a

tension wave with the same length and stress magnitude moving in the opposite

direction (Figure ‎2-18A), it can be seen by using superposition that for the middle cross

section m-n, the stress magnitude will be always zero. Thus, it can be determined to be

equivalent to a free end (Figure ‎2-18C). It can be concluded that the compression wave

will be reflected from a free end as a tension wave with the same length and stress

magnitude as the compression wave, and vice versa (Timoshenko, 1951)

By Considering a case where two identical compression waves which are traveling

in opposite directions(Figure ‎2-19A), it can be seen that the velocity of the particles in

the cross section m-n, is always zero, which yields zero displacement of this cross

section. Hence, the cross section can be replaced by fixed end, as demonstrated in

(Figure ‎2-19C).From this case, it can be concluded that a wave is reflected from a fixed

end completely unchanged.

27

Figure ‎2-1. Strain-stress curves. (Shah, Swarz, & Ouyang, 1995)

28

Figure ‎2-2. Stress distribution along the hole with respect to the material brittleness

(Shah, Swarz, & Ouyang, 1995)

29

Figure ‎2-3. Semi-infinite plate with a hole made in it (S.Timoshenko, 1951)

Figure ‎2-4. Size effect law (Bažant & Planas, 1998)

30

Figure ‎2-5. Concrete DIF vs. strain rate. (Tedesco, Powell, Ross, & Hughes, 1997)

Figure ‎2-6. Charpy impact test device (Gopalarm, Shah, & John, 1984)

31

Figure ‎2-7. Dropped weight impact test device (Banthia, Mindess, Bentur, & Pigeon, 1989)

Figure ‎2-8. Split-Hopkinson pressure test device (Ross, Tedesco, & Kuennen, 1995)

32

Figure ‎2-9. Typical direct-compression test data from the SHPB (Tedesco, Powell, Ross, & Hughes, 1997)

Figure ‎2-10. Scheme of the setup and principle functioning of the spalling technique(Brara & Klepaczko, 2006)

Figure ‎2-11. Equivalent mass-spring system

33

Figure ‎2-12. A 600x1200 mm specimen ready for soft impact test (Krauthammer & Elfahal, 2002)

34

Figure ‎2-13. Compression failure modes observed by (Krauthammer & Elfahal, 2002)

35

Figure ‎2-14. Illustration of wave‟s function shifting (Timoshenko, 1951).

Figure ‎2-15. Two waves traveling in an opposite direction, after time t (Timoshenko, 1951).

Figure ‎2-16. A bar subjected to a sudden compressive load (Timoshenko, 1951).

Figure ‎2-17. Superposition of waves (Timoshenko, 1951).

36

Figure ‎2-18. Reflection of wave from free end (S.Timoshenko, 1951)

Figure ‎2-19. Reflection of wave from fixed end (Timoshenko, 1951).

37

CHAPTER 3 METHODOLOGY

Krauthammer and Elfahal, (2002) classified the failure of concrete cylinders

subjected to a dynamic load into 7 modes of failures, as mentioned in Section 2.5. The

main causes of the failures are believed to be the tensile strain due to Poisson‟s effect

(Figure ‎3-1), and the buckling phenomena. The inertia effect, which is assumed to be

the governing cause of the strain rate effect, must therefore be taken into account. A

fundamental assumption in this study is defining the failure of concrete under a tensile

strain of 0.0002, for a static or dynamic case. In addition, by adopting the conclusion

that the Young‟s modulus changes with respect to the load rate (Weerhijm, 1992), the

structural inertial forces are believed to be the cause of the reduced strain, due to the

dynamic load. According to those assumptions, the buckling failures, which are

characterized by large lateral displacements, must be influenced by the inertia of the

mass that accelerates laterally before the failure occurs. A fundamental concept in this

study states that the way to understand the failure of concrete subjected to a dynamic

load is by analyzing each mode separately. Therefore, this study will concentrate on the

failure due to buckling (Figure ‎3-2A and Figure ‎3-2B).

3.1 Failure Due to Dynamic Buckling.

Buckling failure is believed to consist of two phases of failure. At the first phase,

the cylinder is impacted by the hammer and as a result compressed in the axial axis.

Due to Posssion‟s effect, the cylinder is expanded in the radial direction and split to a

collection of concrete columns with varying sizes and shapes. The cross section of each

column varies according to the position on the longitudinal axis of the column. The

38

splitting phase is characterized by high strain rate due to the mass acceleration, and

therefore the strain rate effect should be investigated deeply.

The second phase is the buckling phase: the columns buckle under the dynamic

load, as a collection of columns. The boundary conditions of each column depend on

the friction between the cylinder and the testing machine, and are assumed to be simply

supported for the experiments of Krauthammer and Elfahal (2002).

3.2 Strain Rate Effect Approach

3.2.1 Mass-Spring Model Approach

Chandra & Krauthammer (1995) describes the inertial forces explanation of the

strain rate effect by using a single degree of freedom model. In this study the

explanation states that the increase in strength is a result of the inertial forces which are

resisting the external force. Thus, the internal force is equal to the external force minus

the inertia force of the mass, and therefore smaller than the force which is measured

(Figure ‎3-3). The observation of low strain caused by high stress (in comparison to the

static domain) is explained by this concept. In addition, by adopting the failure criterion

for concrete to be the maximum strain, the increase in the nominal strength is explained

as well.

3.2.2 Queries Regarding the Inertial Effect Explanation.

The inertial effect explanation raises some questions that require a further

investigation. The inertial effect explanation indicates that the internal force is lower than

the external force due to the stationary mass that creates resistance against the

displacement. However, it seems that this fact is ignored later. In the dynamic case, due

to the fact that the mass creates a resistance against displacement, it is accelerated

and this fact must be taken into account. This can be seen from an energetic point of

39

view as well. Since part of the impact energy is transferred to strain energy and the rest

is transferred to the kinetic energy of the mass, the kinetic energy will subsequently be

converted to strain energy (in perfectly elastic case).

The strain - stress curve, which is being widely used in the static domain, was

adopted in the dynamic domain as well. While in the static domain, this curve

represents the material‟s conditions at each specific point, at the dynamic domain the

velocity of the particles is not accounted for in chart. This absence is significant since

the kinetic energy is potentially strain energy, as mentioned above, indicating that in the

dynamic strain-stress curve there might be an additional strain that is not taken into

account.

Another concept that was adopted from the static domain is the determination of

the nominal strength according to the maximum load that was measured. By

considering the neglected kinetic energy that was mentioned above together with the

fact that the specimen is constantly moving toward the failure point, it can be concluded

that in the dynamic domain the maximum load is measured when the specimen already

has enough energy to reach the maximum nominal strain.

The proposal above is supported by the observations of the dynamic failure

process. In contrast to the quasi-static tests, where the specimen‟s failure is

characterized by several crack planes, in the dynamic failure many crack planes are

observed and particles of the specimen are moving out with a certain velocity. Thus, the

specimen at the failure time has more energy than needed for failure.

3.2.3 The Kinetic Energy of a Specimen in Split Hopkinson Pressure Bar

By considering the model of Chandra & Krauthammer (1995), the kinetic energy of

the specimen causes the mass to keep moving as a rigid body and by that converting

40

the kinetic energy to strain energy. At the end of motion, the internal force in the spring

will include all of the inertial forces of the mass. However, modeling this physical case

as single degree of freedom might not be a good approximation. Therefore, the effect of

kinetic energy of the particles will be investigated by using the theory of elastic waves.

By considering an elastic wave in a bar as illustrated in Figure ‎3-4, it can be seen that

the strain energy (SE) of the bar is:

⋅ ⋅ ⋅

(3-1)

Where A is the cross-sectional area of the bar, c is the wave‟s speed in the bar, σ is the

magnitude of the stress wave, and E is the Young‟s modulus of the bar‟s material. The

work (W) that has been done on the bar is given by:

⋅ ⋅ ⋅ ⋅

⋅ ⋅ ⋅

(3-2)

It can be seen that the strain energy of the wave is only half of the total energy of the

wave and the kinetic energy (KE) of the wave is:

⋅

⋅ ⋅ ⋅ ⋅

⋅ ⋅ ⋅ (

⋅ )

⋅ ⋅ ⋅

(3-3)

This is the other half of the total energy and equal to the strain energy. Thus, when

the specimen is subjected to a stress wave, it also contains kinetic energy. The effect of

this energy will be investigated by a finite elements model (FEM) that is described in the

next chapter.

3.2.4 Finite Elements Model of SHPB

While many studies ( (Brara & Klepaczko, 2006), (Lindhom, 1964), (Ross,

Tedesco, & Kuennen, 1995) and etc.) investigated the increase of strength due to high

strain rate by using SHPB. However, the purpose of this model is to clarify the

41

subsequences of the strain energy that resulted from the presence of kinetic energy at

the dynamic loading. Thus, the shape of the incident wave is considered, but the

magnitudes of stresses are not analyzed.

An elastic FEM model for a SHPB was developed using Abaqus 6.10, and SI units

were used. The model consists of an incident bar, a specimen, and a transmitter bar,

where all of the elements are isoperimetric, elastic, and solid (Figure 3-5, Figure 3-6,

and Figure 3-7) with 8 integration points. The properties of the parts are detailed in

Table 3-1.

The data from the computational analysis was obtained for four pre-defined sets:

A set of ten elements along the specimen‟s longitudinal axis (Figure ‎3-8) was defined to determine the strain distribution along the specimen for each time point.

A set of two nodes, one at each face of the specimen (Figure ‎3-9), was defined to determine the axial displacement and velocity at each face of the specimen.

A set of all the elements in the specimen is used to obtain the total kinetic energy and the total strain energy in the specimen.

A set of all the elements in the model is used to calculate the total work that has been done on the model.

3.3 The Suggested Approach for Buckling

This study models the failure process as a dynamic buckling of a group of

concrete columns. To simplify the solution, the problem must be approximated under

the following assumptions:

The cross-sectional area is constant along the longitudinal axis of each column.

All of the columns are assumed to have the same cross-sectional area.

The cross-sectional area of each column is assumed to be circular.

All of the columns in the group are failing simultaneously and not as progressive buckling.

42

The capacity of the group of the columns is equal to the summation of all the individual column‟s capacities.

3.3.1 Dynamic Buckling of a Single Column.

The static buckling of a column, also known as Euler‟s buckling, has been detailed

by Timoshenko (1930). The calculation of the buckling load is reached by equating the

external moments to internal moments, for the deformed rod. The second derivative of

this equation, with respect to x, yields the forces equation of equilibrium, which is:

( ) (3-4)

Where E is the modulus of elasticity, I is the moment of inertia, P is the applied

force, y is the lateral displacement of the rod, and y0 Is the initial displacement of the

rod.

This equation yields the well-known solution for the critical load (Pc) of a simply

supported rod:

⋅

(3-5)

For a dynamic case, the mass of the rod accelerates laterally, and structural

inertial forces must therefore be taken into account. The problem of dynamic buckling of

a rod has been deeply discussed (Lindberg & Florence, 1987) and is briefly summarized

in the section below.

By adding the Inertial force component ( ⋅

) to the static equation, the

dynamic forces equation is given by:

⋅

( )

(3-6)

43

Where E is the Young‟s modulus of the material, I is the area moment of inertia, P

is the buckling load, y is the deformation of the rod, y0 is the initial deformation of the

rod, is density of the material, A is the cross sectional area, and t is the time variable.

For convenience, the equation can be divided by EI and the following parameters will be

substituted:

,

,

This yields:

(3-7)

The solution for the differential equation can be expressed as a product of

functions in the form of a Fourier sine series of x:

( ) ∑ ( ) ⋅ (

)

(3-8)

The initial displacement is also expressed in series form as:

( ) ∑ ⋅ (

)

(3-9)

Where the coefficients can be found from:

∫ ( ) ⋅ (

)

(3-10)

Substituting equations 3-9 and equation 3-8 into 3-8 to find ( ) yields:

(

)

(3-11)

Rearranging the equation yields:

44

(

)

(3-12)

It can be seen that a trigonometric solution, which reflects a bounded solution, will

be given only for

. The solution for

is hyperbolic, and therefore unstable

with respect to time. The mode number

is the first trigonometric mode. This mode,

as well as all of the trigonometric modes, demonstrates a half buckling wavelength:

, which corresponds to a half wavelength of a rod subjected to a quasi-static load

P, with a length

and identical properties of E,I. This conclusion is significant for this

study because it emphasizes that the buckling wavelength is what is important and not

the length of the rod, properties which are not equivalent in the dynamic buckling case.

Moreover, this conclusion may explain mode number 4 of cylinder failure (Figure

‎2-13B). In this mode the cylinder splits to a group of columns that were buckled only at

a limited portion of their total length.

To apply this concept of dynamic buckling to a cylinder, the separation of the

cylinder into a group of columns, which represents the first phase of failure, must be

investigated as well. It seems that the two governing causes that initiate the separation

are Poisson‟s effect and the buckling phenomenon, but the contribution of each of them

is indeterminate and will be probably need to be approximated. By using these

concepts, an approximated analytical calculation and a finite element model can be

developed to describe the failure of a concrete cylinder under compression.

3.3.2 Fracture Mechanics and Energy Methods

Timoshenko (1930) proposes an approximated method to calculate the static

buckling load of a column by using an energy-based method. To attain the buckling load

45

through method, the added bending strain energy due to the buckling curvature should

be equated to the decrease in the potential energy of the axial load P. The bending

strain energy (U) is given by:

∫(

)

(3-13)

and the decrease in the potential energy (U1) due to the lowering of the load P is given

by:

⋅

∫(

)

(3-14)

Where l is length of the column and y is the buckling shape. While for a known buckling

shape this method yields an accurate solution, a good approximated solution will be

provided by assuming an approximate buckling shape, as long as this shape obeys the

boundary conditions.

Modifying this method to a dynamic buckling equation of a dropped weight striking

a single column requires adding two more components:

1. The kinetic energy of the striking weight:

2. The kinetic energy of the column during the buckling:

∫

Where: is the mass of the striking weight, is velocity of the striking weight, is

the density of the column, is the surface area of the column, and y is the buckling

shape as function of position and time.

By adding these two components, the modified energy equation becomes:

(3-15)

46

Table ‎3-1. Properties of the parts

part Length[m] Diameter[m] E[Pa] ν ρ[KG/m3]

Bars

specimen

1

0.02

0.02

0.02

2*1011

2*109

0.3

0.2

7800

2000

Figure ‎3-1. Failure modes due to Poisson‟s effect (Krauthammer & Elfahal, 2002).

Figure ‎3-2. Failure due to dynamic buckling (Krauthammer & Elfahal, 2002), buckling length was emphasized.

47

Figure ‎3-3. Spring-mass model for strain rate effect. (Chandra & Krauthammer, 1995).

48

Figure ‎3-4. A bar subjected to a stress wave.

Figure ‎3-5. An illustration of the finite elements model of a SHPB.

49

Figure ‎3-6. An illustration of the finite elements model of SHPB, close view of the specimen area.

Figure ‎3-7. An illustration of the specimen and the incident bar.

50

Figure ‎3-8. Set of elements along the specimen.

Figure ‎3-9. Illustration of a set of two nodes of the specimen.

51

CHAPTER 4 RESULTS AND DISCCUSIONS

This chapter consists of results and discussions for the strain rate effect and for

the buckling phase. Since some results demanded further investigations, those

investigations are described with discussions of the results.

The results for the buckling phase indicate that a failure due to buckling is not

possible, and the modes observed seem to be a post-failure behavior. Thus, the modes

were analyzed as post-failure effects and the research was concentrated on the

energies in the strain rate effect.

4.1 Energies in Strain Rate Effect

4.1.1 Applied Load

The load applied is an approximately rectangular stress wave with the magnitude of 1

MPa, and time duration of 2E-5 seconds. Since it is not possible to define a load‟s

amplitude with two values for one time point in the Abaqus software package, an exact

rectangular load cannot be defined and therefore it was well approximated using the

data shown in Table ‎4-1 and drawn in Figure ‎4-1.

When the load mentioned above is applied to this specific SHPB, a stress wave with

a wave length of λb is produced and is given by (Appendix A):

⋅ (4-1)

Where cb is the wave‟s speed in the bar, and tl is the duration of the loading. This

wave is propagating toward the specimen and when it reaches the face of the specimen

the reflection process begins. This process‟s duration (tr) depends on the time at which

the incident wave exists in the transition face, and will therefore be:

52

(4-2)

Subsequently, the wave length that is generated in the specimen (λs) is given by:

⋅ (4-3)

Where cs, is the speed of the wave in the specimen.

It is important to observe that the wavelength in the specimen is equal to the length

of the specimen and is different than the wave length measured in the incident bar. By

considering the linear character of elastic waves, findings yielded by this loading case is

valid for longer waves as well by using the superposition principle.

4.1.2 Energies and Strain in the Specimen

The strain energy and the kinetic energy of the specimen are drawn together in

Figure ‎4-2. In a brief view, the curves seem like the classic harmonic motion of a mass-

spring system. Where at the beginning of the process the kinetic energy is equal to the

strain energy, kinetic energy is then converted to strain energy, it reaches a peak and

finally the strain energy is converted back to kinetic energy. However, it can be seen

that the kinetic energy does not decrease to zero before it increases back. Moreover,

the specimen is always in compression, and the sign of the strains does not change

(Figure ‎4-3).

For a better understanding of the results above, two time points were marked on

the last two figures to produce Figure ‎4-4 and Figure ‎4-5. The first time point, which is

called “max SE”, indicates the time of maximum strain energy in the specimen, which is

2.288154647E-04 sec. The second time point, which is called “max NE”, indicates the

time when maximum strain of the specimen has been achieved, which is

2.250545076E-04sec.

53

By examining Figure ‎4-4, it can be seen that after the time point “max NE”, the

strain energy keeps increasing. In addition, Figure ‎4-5 shows that the strain of the

specimen decreases before the maximum strain energy has been achieved. In other

words, the surprising conclusion is that between those two points, the strain energy

increases but strain is decreasing.

4.1.3 A Theoretical Explanation for the Results Above

The explanation for the results above might be found in the theory of elastic

waves. It seems that this phenomenon is a result of two causes: the superposition of the

wave that is reflected from the face of the transmitted bar and the fact that the wave is

reflected from the transmitted bar with the absence of the transmitted wave.

The first cause mentioned above can be understood by considering a stress wave

propagating in a cantilever toward the fixed end. At time t=l/c the wave reaches the fixed

end, and the loading is stopped. Now, the stress is uniformly distributed along the bar

with the magnitude σ(l is the length of the bar and c is wave‟s speed in the bar), as

illustrated in Figure ‎4-6.Therefore, the shortening of the bar (Δl1) and the strain energy

in the bar (SE1) are given by:

⋅

(4-4)

⋅ ⋅ ( )

⋅ ⋅

(4-5)

At time t=(3/2)⋅(l/c), the reflected wave reaches the bar‟s midpoint as well as the end of

the incident wave, as illustrated in Figure ‎4-7. Hence, it is trivial that the stress and the

particles‟ velocity at the left half of bar are both equal to zero. However, the stress and

the particles‟ velocity at the right half of the bar are achieved by superimposing those

54

two waves and are equal to 2σ and zero, respectively. Subsequently, the shortening of

the bar (Δl2) and the strain energy in bar (SE2) are given by:

⋅

(4-6)

⋅ ( )

⋅ ⋅

(4-7)

By comparing the results at the two time points above, it can be seen that while

the shortening of the bar is equal for both of the time points, the strain energy for the

second time point is double the strain energy in the first time point. Thus, by observing

that these two time points are representing the two extreme cases, a general conclusion

can be made: for a case of a cantilever subjected to a rectangular stress wave with a

wavelength equal to the length of the bar, the shortening of the bar at each time point

between l/c to 2l/c, is equal to Δl1, and the strain energy of the bar varies between

SE1and 2SE1.

By taking into account the second that part of the wave was transmitted to the

transmitted bar, the case discussed is slightly different than the case of the cantilever

above. At time t=(3/2)⋅(l/c), the reflected wave magnitude (σ2) is lower than the incident

wave magnitude (σ) (Figure ‎4-8). Hence, the shortening of the bar is now lower than in

the case above, and the strain energy (at this time point) of the bar can vary between

SE1/2 and 2⋅SE1.

This conclusion can provide a good explanation for the non-constant relationship

between the average strain and the strain energy that were evaluated by Abaqus.

Moreover, it can be seen that particles‟ velocity is indeed never dropped to zero.

55

This case yields one more important principle. While in the case of the cantilever

above the velocities of all of the particles are in the same direction, this case is different.

After the time t=(3/2)⋅(l/c), the velocity of a certain particle can be one of the of the

values below:

1. 0

2. V-V2

3. -V2

4. –V

Where V is the magnitude of the velocity due to the stress σ, and V2 is the magnitude of

velocity due to the stress σ2.

4.1.4 Strain Energy and Strain Distribution

In section 4.2.3 it is shown that results from section 4.2.2 can be explained by the

changing relationship between strain energy and average strain with respect to strain

distribution. Thus, this relationship changing is investigated in the SHPB.

Axial strain energy (SE) in a bar subjected to uniformly distributed strain (ε) is

calculated by:

⋅ ⋅ ⋅

⋅ ⋅ ⋅

(4-8)

Hence, the relationship between the strain energy and the strain is quadratic and can be

simplified to:

⋅ (4-9)

Where G is a constant and is equal to AEl/2.

The ratio SE/(G⋅ε2), which is equal to one for a uniform distribution of strains, is

shown in Figure ‎4-9 together with the two time points “max SE” and “max NE”. It can be

56

seen that this ratio increases between the maximum strain recorded (“max NE”) and the

maximum strain energy(“max SE”). Moreover, this ratio approaches 1, indicating a

uniform distribution of strains, at the point corresponding to time t=l/c, as described

before (Figure ‎4-6).

Another indication for the amount of disorder in the strain distribution might be the

standard deviation of strains. The changing of this function with respect to time, and the

two time steps mentioned before are shown in Figure ‎4-10. Again, it can be seen that

between these two points the strain distribution increases.

A difference can be seen at the left end of the two figures. While Figure

‎4-9provides high values close to the left end, the corresponding values in Figure ‎4-10

are very small. This difference is a result of the strains‟ magnitude, which is accounted

for in Figure ‎4-10, but is not taken into consideration in Figure ‎4-9. In other words, in

Figure ‎4-9 small noises can provide high ratio values, such as in the case for the strains

at the left end which are approaching zero. This effect is vanished when the standard

deviation values are divided by the average strain values, as illustrated in Figure ‎4-11.

4.1.5 Summary of the Results by Time Sequence

For a better understanding of the discussions above, the results are provided with

three time step marks and a summery for each time step. The wave‟s propagation time

steps were calculated analytically (Appendix A), and therefore some minor discrepancy

is observed.

At time t= 1.975E-04 sec, which is noted “wave 1st face” in Figures 4.12-4.16, the

front of the incident wave reaches the first face of the specimen. Subsequently, the

wave starts propagating in the specimen, and as result the kinetic and the strain

energies in the specimen increase (Figure ‎4-12), along with the average strain in the

57

specimen (Figure ‎4-13). During the wave propagation in the specimen, the strains‟

disorder decreases (Figure ‎4-16), and therefore, the efficiency (an average strain per

strain energy unit) of the strain energy increases (Figure ‎4-14). However, the total

amounts of strains at the beginning of this phase are low, as indicated in Figure ‎4-15.

At time t= 2.175E-04 sec, which is noted “wave 2nd face”, the front of the incident

wave reaches the second face of the specimen, and the specimen is uniformly loaded

along all of its length. At this point, the efficiency of the strain energy reaches the

maximum, which is indicated by a ratio of 1 in Figure ‎4-14. Subsequently, the reflection

process begins and the disorder increases again. Due to the superposition of the

reflected wave and the incident wave, kinetic energy is converted to strain energy

(Figure ‎4-12), and the strains are high close to the reflection point and are zero close to

the other face. This disorder (Figure ‎4-15 and Figure ‎4-16) decreases the efficiency of

the strain energy (Figure ‎4-14). This, combined with the fact that the total energy in the

specimen decreases due to the transmitted wave, the average strain decreases.

When the reflected wave‟s front and the incident wave‟s rear reach the half point

of the specimen‟s length as illustrated in Figure ‎4-8, the two waves are fully

superimposed. Thus, the average velocity of the particles is minimal and strain energy

reaches a maximum (Figure ‎4-12).This point is characterized by high strain disorder

(Figure ‎4-15 and Figure ‎4-16), leading to inefficiency of the strain energy (indicated as a

peak in Figure ‎4-14).

From this point, the reflected wave keeps propagating toward the first face of the

specimen, and the overlapped length of the two waves decreases. Subsequently, strain

58

energy is converted back to kinetic energy, and for the first time, the velocity of all of the

particles is not in the same direction.

At time t= 2.375E-04 sec, which is noted “reflected wave 1st face”, the front of the

reflected wave reaches the first face of the specimen. As result, the specimen‟s strains

are uniformly distributed, with the magnitude of the reflected wave.

In this section it was shown that the kinetic energy converts to strains energy

where waves are super imposed. For long specimens and short waves, the result is a

non-uniform distribution of strains along the specimen. This conclusion agrees with the

study of Wu and Gorham (1997) yielded that in a long specimen the difference of

stresses at the two faces becomes high, and therefore average of these two stress is

not a good approximation. However, for a short specimen subjected to a long wave,

many waves are superimposed and the non-uniformity of strains becomes negligible.

This is a result of the fact that the superposition of two waves with the length of the

specimen produces a uniform distribution of strains. Thus, for several reflections in the

specimen, the non-uniformity is a product of the only last reflected wave.

Another important conclusion from this section is concerning the kinetic energy. It

was shown that some amount of kinetic energy remains in the system, due to the fact

that the wave is not fully reflected from the bars. The amount of remaining kinetic

energy depends on the properties ratio, E and ρ, of the specimen the bars. The

presence of this energy is important for the analysis of the buckling phase. The

curvature of the columns due to an axial force was assumed to be a buckling behavior

since no bending force was applied. However, the inertia of the mass due to this

velocity has a radial component, due to Possion‟s effect, that can bend the columns.

59

4.2 Buckling Load of a Group of Columns

4.2.1 The Number of Rods in a Group

The number of columns in the group was determined according to the static

buckling load, which was calculated using the software Mathcad (Appendix B). The

results of this analysis are shown below with brief key principles.

The total base area of the cylinder is divided by the number of columns in the

group, n, to determine the cross sectional area of each column.

( ) (4-10)

Where a is the cross sectional area of a column from a group which consists of n

columns, and A is the base area of the cylinder. Therefore, the radius of each column is

given by:

( ) √ ( )

(4-11)

Despite the fact that the cylinder is subjected to a dynamic load, the capacity of

each column in the group is determined as the static buckling load and this for two

reasons described below.

An experiment done by (Gladen, Handzy, Belmonte, & Villermaux, 2005), where

brittle rods were struck by an accelerated mass, shows that buckling length changes as

a function of the stress wave magnitude. However, despite the fact that all of the rods

were buckled corresponding to different loads, they all eventually failed. This fact

demonstrates that the buckling capacity is different from the load corresponding to the

buckling shape which is observed, and this capacity is supposed to be taken as Euler‟s

first mode.

60

The number of columns in the group was calculated using properties of the

cylinders that had a buckling failure. This cylinder‟s diameter (D) was 0.3m, the height of

the cylinder (h) was 0.6m, and the modulus of elasticity (E) of 25.06 GPa, and static

nominal strength of 44.81 MPa.

Based on the data above, the capacity of the group was drawn as function of the

number of columns in the group, and is illustrated in Figure ‎4-17. It can be seen that

capacity of the group decreases as the number of columns in the group increases.

4.2.2 Buckling as a Failure Criteria

The specimen in the experiment fails as a group of buckling columns, if the

buckling capacity function is smaller than the static capacity of the cylinder:

( ) ⋅ ⋅ (4-12)

Where p(n) is the buckling capacity of each column in a group of n columns, is the

ultimate compressive stress, and A is the area of the base of the cylinder. By equating

the two sides of this equation, the minimum number of columns that are needed for

buckling failure can be found. Subsequently, the minimum number of columns needed

for a buckling failure to occur in the cylinder mentioned above is 86. This result was

calculated using Mathcad (Appendix B), and is graphically illustrated in Figure 4-18.

By comparing the result above to the number of columns in Figure 2-13(i), which is

estimated to be about 30 columns, it yields that those cylinders did not fail due to

buckling. However, this mode is characterized by curved lateral motion of the mass due

an axial load and therefore another explanation is required.

4.2.3 Examination of Bending as a Possible Post Failure Effect

In the previous section it was concluded that the group of columns are not

buckling. By combining this conclusion with the existence of kinetic energy, bending

61

might be the cause for the columns‟ curved shape. Hence, an approximated model was

develop in Mathcad (Appendix C) to check this possibility using the experimental

properties [reference] (Table ‎4-3), and it is described below.

The cylinder is axially loaded by the force P and radial velocity Vr, as described in

Figure ‎4-19A. Thus, any single column in the group is loaded by axial force P/n and

lateral velocity v (Figure ‎4-19B). Since each column is subjected to an axial force and to

a lateral velocity, the solution becomes complicated and therefore an energy based

method is offered.

The bending strain energy for each column as a function of the number of columns

in the group is limited by the maximum curvature prior to failure of the column, and it is

calculated in Appendix C. Therefore, bending is possible if the energy that causes

bending in a column is greater than the critical bending strain energy of column.

The deflection shape (g) of a column is assumed to be:

( ) (

) (4-13)

where x is the position along the longitudinal axis, h is height of the cylinder, and d is

the amplitude of the deflection. Therefore, the moment distribution along the column

(mxCrack(x,n)) is given by:

( ) ( )(

) (

) ( ) (4-14)

where ( ) is the amplitude of the curved shape at failure for each n, and is given by:

( ) ( )

( ) (4-15)

where is the ultimate tensile strength of the concrete. Finally, the critical energy

(Um(n)) as function of the number of columns in the group is:

62

( )

( )∫ ( )

(4-16)

This function is compared to the energy of the bending strain applied and was

approximated as described below.

The total kinetic energy in the system (Uk) is equal to the initial energy of the system

minus the critical axial energy of the specimen (Ecr):

(4-17)

By averaging the velocities, the average velocity (Vavg) is given by:

√

(4-18)

The average velocity consists of the axial (Va) and the radial (Vr) (Figure ‎4-20)

velocities, where:

⋅ (4-19)

Therefore, the radial velocity is given by:

⋅ ( ( )) (4-20)

Finally, the column is bending around one axis but the velocities‟ directions are

determined according to position of the column in the cylinder as illustrated in Figure

‎4-21.Hence, another approximation has been done to calculate the component of the

inertial forces only in the bending direction. A slice with the same area as the column

(a(n)) with an angle θ, is assumed to represent the velocities‟ distribution. Averaging the

component of the velocities in the bending direction velocities is given by:

∫ ⋅ (

) (4-21)

Hence, the kinetic energy that causes bending in a column is:

63

⋅ ⋅ ( ) ⋅ (4-22)

The kinetic energy that causes bending in a single column (uKn) is plotted with the

critical bending energy of a column (uMcrack), multiplied by 103 in order to fit in the

chart (Figure ‎4-23). It can be seen that when the number of columns approaches 1, the

energy required for a failure of the column (which is the cylinder in that case) in bending

is high. In the other hand, uKn is zero, because if the cylinder was not split the radial

inertia cancels off and bending is not possible. In addition, it can be seen that the extra

energy in the system is about 103 times than required to introduced post-failure

buckling, which indicates that this explanation is possible.

4.2.4 A Possible Explanation the Post Failure Behavior

Basing on the extra energy that observed in the system, an explanation of post

failure modes, is suggested. Figure ‎4-24 represents the cylinder as finite number of

masses and springs, and a line which represents a plan of failure in the specimen. At

time of failure, the model splits into two parts where the masses are accelerated and the

springs are still loaded. Subsequently, particles keep moving and more plans of failure

are produced. According to this model, the conditions of the springs and the masses

after the first plan of failure is produced seem to be dependent on the extra energy and

the loading energy rate.

64

Table ‎4-1. The shape of the applied load

Time Amplitude

0.00000E+00

1.00000E-10 2.00000E-05 2.00001E-05

0

1

1

0

Table ‎4-2. Properties of the hammer and the specimen

property value

Hammer‟s mass (KG) 578

Impact velocity (m/s) 5

Cylinder height (m) 0.6

Cylinder diameter (m) 0.3

Static capacity (Mpa) 44.8

E (Gpa) 30

ν 0.17

ρ (kg/m3 2400

εcr 0.002

εcr+ 8.00E-05

Figure ‎4-1. The applied load shape

65

Figure ‎4-2. Strain energy and kinetic energy at the specimen

Figure ‎4-3. Average strain of the specimen (compression is positive) vs. time

time (sec)

En

erg

y (

J)

0x100 30x10-6 60x10-6 90x10-6 120x10-6 150x10-6 180x10-6 210x10-6 240x10-60x100

2.5x10-6

5x10-6

7.5x10-6

10x10-6

12.5x10-6

15x10-6

17.5x10-6

20x10-6

22.5x10-6

SEKE

time (sec)

avera

ge s

train

0x100 30x10-6 60x10-6 90x10-6 120x10-6 150x10-6 180x10-6 210x10-6 240x10-6-10x10-6

-5x10-6

0x100

5x10-6

10x10-6

15x10-6

20x10-6

25x10-6

30x10-6

35x10-6

40x10-6

45x10-6

50x10-6

66

Figure ‎4-4. Strain and kinetic energies at the specimen, with time marks(close view from

the wave‟s arrival time)

Figure ‎4-5. Average strain of the specimen (compression is positive) vs. time, with time

marks (close view from the wave‟s arrival time)

time (sec)

En

erg

y (

J)

195x10-6 200x10-6 205x10-6 210x10-6 215x10-6 220x10-6 225x10-6 230x10-6 235x10-6 240x10-60x100

2.5x10-6

5x10-6

7.5x10-6

10x10-6

12.5x10-6

15x10-6

17.5x10-6

20x10-6

22.5x10-6

ma

x N

E

ma

x S

E

SEKE

time (sec)

avera

ge s

train

195x10-6 200x10-6 205x10-6 210x10-6 215x10-6 220x10-6 225x10-6 230x10-6 235x10-6 240x10-6-10x10-6

-5x10-6

0x100

5x10-6

10x10-6

15x10-6

20x10-6

25x10-6

30x10-6

35x10-6

40x10-6

45x10-6

50x10-6

ma

x N

E

ma

x S

E

67

Figure ‎4-6. A bar subjected a rectangular stress wave at time t=l/c.

Figure ‎4-7. A bar subjected a rectangular stress wave at time t=3/2⋅l/c.

Figure ‎4-8. The incident wave and the wave which was reflected due to a medium change

68

Figure ‎4-9. SE/(G⋅ε^2) vs. time (close view from the wave‟s arrival time)

Figure ‎4-10. Standard deviation of the strains in the specimen vs. time (close view from the wave‟s arrival time)

time (sec)

SE

/SE

(avg)

195x10-6 202.5x10-6 210x10-6 217.5x10-6 225x10-6 232.5x10-6 240x10-60

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

ma

x N

E

ma

x S

E

time (sec)

sta

nd

ard

de

via

tio

n o

f s

rta

ins

195x10-6 202.5x10-6 210x10-6 217.5x10-6 225x10-6 232.5x10-6 240x10-60x100

4x10-6

8x10-6

12x10-6

16x10-6

20x10-6

24x10-6

28x10-6

32x10-6

36x10-6

40x10-6

ma

x N

E

ma

x S

E

69

Figure ‎4-11. Standard deviation/average of the strains in the specimen vs. time (close view from the wave‟s arrival time)

Figure ‎4-12. Strain and kinetic energies at the specimen

time (sec)

sta

nd

ard

devia

tio

n/ avera

ge s

tria

n

195x10-6 202.5x10-6 210x10-6 217.5x10-6 225x10-6 232.5x10-6 240x10-60

0.25

0.5

0.75

1

1.25

1.5

1.75

ma

x S

E

ma

x N

E

time (sec)

En

erg

y (

J)

0x100 30x10-6 60x10-6 90x10-6 120x10-6 150x10-6 180x10-6 210x10-6 240x10-60x100

2.5x10-6

5x10-6

7.5x10-6

10x10-6

12.5x10-6

15x10-6

17.5x10-6

20x10-6

22.5x10-6re

fle

cte

d w

ave

1st

face

wave 2

nd f

ace

wave 1

st

face

SEKE

70

Figure ‎4-13. Average strain of the specimen (compression is positive)

Figure ‎4-14. SE/(G⋅ε^2) vs. time (close view from the wave‟s arrival time)

time (sec)

avera

ge s

train

0x100 30x10-6 60x10-6 90x10-6 120x10-6 150x10-6 180x10-6 210x10-6 240x10-6-10x10-6

-5x10-6

0x100

5x10-6

10x10-6

15x10-6

20x10-6

25x10-6

30x10-6

35x10-6

40x10-6

45x10-6

50x10-6

wave 1

st

face

wave 2

nd f

ace

refle

cte

d w

ave

1st

face

time (sec)

SE

/SE

(avg)

195x10-6 202.5x10-6 210x10-6 217.5x10-6 225x10-6 232.5x10-6 240x10-60

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

wave 1

st

face

wave 2

nd f

ace

refle

cte

d w

ave

1st

face

71

Figure ‎4-15. Standard deviation of the strains in the specimen (close view from the

wave‟s arrival time)

Figure ‎4-16. Standard deviation divided by average of the strains in the specimen vs. time (close view from the wave‟s arrival time)

time (sec)

sta

nd

ard

de

via

tio

n o

d s

rtain

s

192x10-6 198x10-6 204x10-6 210x10-6 216x10-6 222x10-6 228x10-6 234x10-6 240x10-60x100

4x10-6

8x10-6

12x10-6

16x10-6

20x10-6

24x10-6

28x10-6

32x10-6

36x10-6

40x10-6

wa

ve 1

st

face

wa

ve 2

nd

fa

ce

refle

cte

d w

ave 1

st

face

time (sec)

sta

nd

ard

devia

tio

n/a

vera

ge s

train

195x10-6 202.5x10-6 210x10-6 217.5x10-6 225x10-6 232.5x10-6 240x10-60

0.25

0.5

0.75

1

1.25

1.5

1.75re

fle

cte

d w

ave

1st

face

wave 2

nd f

ace

wave 1

st

face

72

Figure ‎4-17. Model for bending of the cylinder (a) and a single column from the cylinder

(b) due to radial velocity.

Figure ‎4-18. The components of the velocity

Figure ‎4-19. The radial velocity in a single columns and the bending direction.

73

Figure ‎4-20. An approximated model to obtain the average velocity in the bending

direction.

Figure ‎4-21. The strain energy required for failure(uMcrack) of a single column *10^3

and the kinetic energy that can use for bending vs. the number of column in a group.

74

Figure ‎4-22. A model for the post failure behavior consist of masses and springs.

75

CHAPTER 5 CONCLUSIONS AND RECOMMENDATION

In this study, the buckling mode of failure that was observed by (Krauthammer &

Elfahal, 2002) was investigated by using the strain rate effect and buckling principles. It

was found that in contrast to the initial assumption, the modes of failure observed are a

post-failure behavior due to kinetic energy in the system. This kinetic energy is

converted to strain energy by producing no additional shortening in the specimen.

The following conclusions were determined based on this work:

The dynamic domain is characterized by particles‟ velocity at certain time. This property is not represented in the traditional strain-stress curve, but must be taken into account.

In contrast to the static domain, determining the nominal strength in the dynamic domain as the maximum load that was measured seems to be incorrect. It seems that an energy-based criterion should develop for the dynamic domain.

Part of the kinetic energy due to the particles‟ velocity is converted to strain energy. Thus, any inertial-based explanation for the strain rate effect must provide an explanation for the phase of where the kinetic energy converted to strain energy.

A single degree of freedom model, which predicts an additional increase in the average strain due to kinetic energy that converts to strain energy, is not a good model to describe the behavior of a specimen under an axial impact.

The part of the kinetic energy in SHPB‟s specimen is converted to strain energy by producing no additional increase in the average strain. The amount of remaining kinetic energy depends on the properties ratio, E and ρ, of the specimen the bars.

Since the strain distribution along the specimen during most of the time is not uniform, using of average strain of the specimen should be considered carefully.

The ratio of the strain energy and the squared strain‟s average provides a good indication for the strains‟ distribution along the specimen. In addition, the indication obtained by the standard variation function seems to be a good indication as well, by considering the fact that it‟s weight sensitive (the magnitudes of the values influence the result. This ratio and its effect should be investigated more deeply, and advanced statistics tools might be investigated.

76

The modes of failure that were observed by (Krauthammer & Elfahal, 2002), are believed to be a post failure behavior. Those post-failure modes might be influenced by the extra energy and the loading energy rate.

Based on these conclusions, the next steps recommended for this work are:

The results of a non-uniform distribution of strains should be investigated more deeply. Specifically, determining the maximum strain as the failure criterion of a specimen might require a modification. This necessity is raised due to the possible case where part of the specimen reaches the maximum tensile strain while the other part does not.

The strain distribution on the specimen should be analyzed by using more advanced statistical tools.

An energy-based failure criterion might be developed to take into account the kinetic energy in the system.

Since the contribution of the kinetic energy to failure is not clear, it is suggested to dynamically load a specimen with energy that corresponds to the strain energy needed for static failure. By doing that, the kinetic energy which will be converted to strain energy can be explored better.

77

APPENDIX A MATHCAD CALCULATION SHEET FOR THE SHPB PROPERTIES

78

79

80

APPENDIX B MATHCAD CALCULATION SHEET FOR THE BUCKLING

81

82

APPENDIX C MATHCAD CALCULATION SHEET FOR POST-FAILURE BENDING

83

84

85

86

87

88

LIST OF REFERENCES

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Bažant, Z. P. (1986). Fracture mechanics and Starin Softening in Concrete. Proceedings of US-Japan Seminar on Finite Element Analysis of Reinforced Concrete Structures, ASCE, 121-150.

Bažant, Z. P., & Planas, J. (1998). Fracture and Size Effect in Concrete and Other Quasibrittle Materials. Boca Raton: CRC Press LLC.

Brara, A., & Klepaczko, J. R. (2006). Experimental Characterization of Concrete in Dynamic Tension. Mechanics of Materials, 38(3), 253-267.

Chandra, D., & Krauthammer, T. (1995). Strength Enhancement in Particular Solids under High Loading Rates. Earthquake Engineering and Structural Dynamics, 24, 1609-1622.

Freund, L. B. (1998). Dynamic Fracture Mechanics. New York: Cambridge University Press.

Gladen, J. R., Handzy, N. Z., Belmonte, A., & Villermaux, E. (2005). Dynamic Buckling and Fragmantation in Britlle Rods.

Gopalarm, V. S., Shah, S. P., & John, R. (1984). A Modified Instrumented Charpy Test for Cement-Based Composites,”. Experimental Mechanics, SEM, 102-111.

Griffith, A. A. (1920). The Phenomena of Rupture and Flow in Solids. Transactions of the Royal Society of London, 163-198.

Krauthammer, T., & Elfahal, M. M. (2002). Size effect in normal and high-strength concrete cylinders subjected to static and dynamic axial compressive loads. The Pennsylvania state University.

Lindberg, H. E., & Florence, A. L. (1987). Dynamic Pulse Buckling, theory and experiments. Martinus Nijhoff publishers.

Lindhom, U. S. (1964). Some experiments with the split hopkinson pressure bar. J. Mech. Phys. Solids, 12, 317-335.

Marur, P. R. (1996). On the Effects of Higher Vibration Modes in the Analysis of Three Point Bend Testing. International Journal of Fracture, 77, 367-379.

89

Park, J. Y., & Krauthammer, T. (2006). Fracture-based size and rate effects of concrete members.

Ross, C. A., Tedesco, J. W., & Kuennen, S. T. (1995). Effects of Strain Rate on Concrete Strength. ACI Material Journal, 92(1), 37-47.

Shah, S. P., Swarz, S. E., & Ouyang, C. (1995). Fracture mechanics of concrete.

Tedesco, J. W., Powell, J. C., Ross, A. C., & Hughes, M. L. (1997). A strain-rate-dependent concrete material model for Adina. Computers & Structures, 64, NO. 5/6, 1053-1057.

Timoshenko S., J. N. (1951). Theory of Elasticity. McGraw-Hill Book Company.

Timoshenko, S. (1930). Strength Of Materials, Part II - Advanced Theory and problems. D. Van Nostrad Company, Inc.

Weerhijm, J. (1992). Concrete under impact tensile loading and lateral compression.

Weibull, W. (1939). A Statistical Theory of the Strength of Materials. Royal Swedish Academy of Engineering, 1-45.

Wu, X. J., & Gorham, D. A. (1997). stress Equlibrium in the Split Hopkinson Pressure Bar Test. Jornal De Physique, C391-C396.

90

BIOGRAPHICAL SKETCH

Avshalom Ganz was born in Israel in 1981. He was drafted into the army for his

national service from 2000 to 2003. He began his undergraduate studies at Ben-Gurion

University, Israel, in October 2004 and obtained his Bachelor of Science degree in

structural engineering on July 2008. In 2006, he joined Ortam-Malibu Engineering LTD,

Israel, in a student position and later as a structural engineer. In 2009, he began his

master‟s degree in civil engineering at the University of Florida, specializing in protective

structures.