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PAGE 1 1 THEORETICAL AND NUMERICAL STUDY OF A CONCRETE CYLINDER SUBJECTED TO A N IMPACT LOAD By AVSHALOM GANZ A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR TH E DEGREE OF MASTER OF SCI E NCE UNIVERSITY OF FLORIDA 201 1 PAGE 2 2 201 1 Avshalom Ganz PAGE 3 3 To my great family PAGE 4 4 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 sof tware. 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 f riends in Gaine sville 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 PAGE 5 5 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 P erformance ................................ ............................ 15 2.2 Fracture Mechanics ................................ ................................ .......................... 15 2.2.1 Background ................................ ................................ ............................. 15 2.2.2 L inear 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 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 PAGE 6 6 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 Dis tribution ................................ ....................... 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 Examina tion of Bending as a Possible Post Failure Effect ...................... 60 4.2.4 A Possible Explanation the Post Failure Behavior ................................ ... 63 5 CONCLUSIONS AND R ECOMMENDATION ................................ ......................... 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 .................. 8 2 LIST OF REFERENCES ................................ ................................ ............................... 88 BIOGRAPHICAL SKETCH ................................ ................................ ............................ 90 PAGE 7 7 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 PAGE 8 8 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 Spl it Hopkinson pressure test device ................................ ................................ 31 2 9 Typical direct compression test data from the SHPB ................................ ......... 32 2 10 Scheme of the setup and princi ple 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 ................................ ................................ 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 fix ed end ................................ ................................ ....... 36 3 1 ................................ ................................ 46 3 2 Failure due to dynamic buckling ................................ ................................ ......... 46 3 3 Spring mass model for strain rate effect ................................ ............................. 47 PAGE 9 9 3 4 A bar subjected to a stress wave ................................ ................................ ........ 48 3 5 An illustr ation 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 Illustrati on 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 t ................................ ................................ ....................... 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 ) 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 ................................ ................................ ............................. 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 cyli nder ................................ ................................ ........ 72 PAGE 10 10 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 (uMcra ck) 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 PAGE 11 11 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 CONCRET E 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 subjec ted 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 mo del (FEM) of a Split Hopkinson bar to examine the strains and the energies of a specimen subjected to an impact. By using 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 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. PAGE 12 12 CHAPTER 1 INTRODUCTION According to the strength theory the nominal st rength of a material is a unique property of the material. However, t he size effect law ( & Planas 1998) indicates a direct dependency of the strength of a material on the geometrical dimensions. This law is explained by using nonlinear fracture mec hanics, 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 stren gth 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 S ize and rate effects the two phenomena above, were widely dis cussed 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 str ength. 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 dependen cy of the strain rate on the displaced mass is believed to be the first step PAGE 13 13 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. Fo r this purpose, the observation s of Krauthammer and Elfahal ( 200 2) experiments are analyzed In addition to the strain rate effect, Krauthammer and Elfahal ( 200 2) observed some failure modes. These modes were classified by Krauthammer and Elfahal ( 200 2) in to 7 modes of failure Since the phenomenon of several modes of failure is not clear, it is suggested to ex plain each mode separately In t his study the buckling fa ilure modes are examined, and the subsequences of the kinetic energy presence during the loa ding 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 Signif i cance Man y studies in the recent century based the strain rate effect, mainly empirically. Subsequently, the concept of in crease 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, misinte rpretation 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 t he dynamic domain as well. While in the static domain, this curve PAGE 14 14 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 t he 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, wher 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 neede d for failure. PAGE 15 15 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 1 A ) stresses suddenly drops down to zero after fracture, ductile materials ( Figure 2 1 B ) maintains constant stresses when yielding. Quasi brittle material s ( Figure 2 1 C ) gr adually decreasing of stresses after the peak stress is attained The failure character istics greatly depend on the brittleness of the material. These can be described (Shah, S warz, & 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 2 A ) a sudden failure occurs when the maximum normal stress on the hole edge reaches the material capacity However, for a ductile material, after the maximum stress, ft, is reached, stress will be redis tributed, and failure occurs when the entire cross section A A is yielding. The f ailure of a Quasi brittle material is depicted in ( Figure 2 2 C ) 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 sem i infinite plate with a hole made in it (Timoshenko, 1951) : PAGE 16 16 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 t (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 A n energy based failure criterion was developed (Griffith, 1920) which stipulate s a model for the failure by crack propagation. The crack will propagate if the energy release d rate at the crack tip equals the energy rate required for the crack to be extended unit length. Assuming the only en ergy required for crack extension is the surface energy, T he 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 For more general case s this criterion can be expressed as: Where G is the energy release rate and Gc is the critical energy release rate. PAGE 17 17 2.3 Size Effect 2.3.1 Background Until the last few decades, it has been believe d that the nominal strength of the concrete is independent the specim investigation s conducted by and Planas (1998) and Krauthammer and Elfahal ( 200 2) indicate clearly that the co ncrete behavior is high dependent on the size of the specimen. These results led to attempts to expl ain 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. T his theory considers the randomness of the material 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) constan 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 inappl icable for reinforced concrete Planas, 1998) .I n addition, experiments conducted on diagonal shear failure of PAGE 18 18 reinforced concrete, provided results which contradicted the statistical theory Planas, 1998) 2.3.2 Energy Theory and Size Eff ect ( 19 86) 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 : (2 6) It can be seen in Figure 2 4 that for small specimens, the size effect law approaches the strength theor res ults, whereas for large specimen s it approaches the results of Liner Elastic Fracture Mechanics. 2.4 Strain Rate Effect 2.4.1 Background During the last few decades, a strengthen ing was observed, for the most the materials, as the strain rate or the load rate was increase d Whereas the common European scientific community term is load rate or stress rate using units of the common term used in United State s is strain rate using units of In contrast to Weerhijm ( 1992) ( Tedesco, Powell, Ross, & Hughes, 1997) reported the not be strain rate sensitive and therefore, the static Y PAGE 19 19 modulus can be used to convert stress rate in to strain rate, and vice versa (Tedesco et al. 1997) The term DIF, which stand s for Dynamic Increase Factor, describes the strengthen ing 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 strengthen ing (DIF becomes about 8) observed, whereas the compression DIF is about 1.7 W hile 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 pa ste, 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 i mpact and split Hopkinson pressure bar test. C harpy 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 rotate s and strikes the specimen which is positioned horizontally, with a constant parallel velocity ( Figure 2 6 ). A drop weight impact test is usually pe rformed by dropping a weight from a tower down to ward the specimen, striking it with a constant velocity, and continually applying a gravitational force ( Figure 2 7 ). PAGE 20 20 Split Hopkinson pressure bar testing is u sed to perform a compression impact test and measure the compression wave. When the striker bar strikes the in cident bar (bar No. 1 in Figure 2 8 ), a compression stress wave is generated and start s traveling toward the specimen, measured at the strain ga u ge 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 reflecte d 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 b y using superposition of the waves and the formals that are used to calculate the strains and stresses are given (Lindhom, 1964) : (2 7) wave strain. The displacement of the incident bar, u 1 is the results of both the incident pulse tra veling in the positive x direction and the reflected pulse traveling in the negative x direction. Thus: (2 8) i r is the reflecte d strain. Similarly, the displacement u 2 t : (2 9) PAGE 21 21 The nominal strain in the specimen is then: (2 10) Where l 0 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 an d p2 on each face of the specimen are: (2 13) (2 14) Thus, the average stress in the specimen is: Where E is the modu lus 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 H opkinson 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 th e free end as a tensile wave, moving toward the specimen. (2 15) PAGE 22 22 2.4.3 Inertia Effect One main explanation for the strain rate effect is the inertial force s acting on the body during impact. The inertia l effect can be classified into two types: the structural ine rtial 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 str ength increase. Chandra and Krauthammer ( 1995) e xplained 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 increase s summarized by Weerhijm ( 1992) Attempts to examine the way inertia l effect influences the crack expansion criteria, ing can be expresse d 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 multipl ied 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 cr iteri on by adding a kinetic energy component. This dynamic criterion, which is based on LEFM, is expressed as: (2 17) W here Ue, Us, and Uk are the external work, strain energy rate and kinetic energy rate, B is the spe cimen width, and is the surface energy required to propagate PAGE 23 23 a crack ,per unit area. By adopting Marur ( 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 eq uation 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 se en that on the right hand side of the eq., a mass depended component was added, and it should reflect the strain rate strengthen ing Krauthammer and Elfahal (2002) Conducted experimental research and testing which included four different sizes of geometr ically similar cylinders, of normal and high strength concrete. The se 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 in to soft and hard impact: in the soft impact ( Figure 2 12 ) tests, rubber PAGE 24 24 pads were place d 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 effe ct was observed for cylindrical NSC specimens under both static and dynamic axial compressive load s The size effect law predictions were found to be very accurate for the static NSC and HSC tests. The s train rate effect was clearly observed for both NSC a nd HSC testing In addition, by using a high frame rate camera, the failures were r ecorded by video and the failure shapes were classified by the authors into seven failure modes: Vertical splitting: the cylinder splits through several vertical plan e s ( Fi gure 2 13 A and Figure 2 13 B ) The split can be started from the top ( Figure 2 13C ) or from the bottom ( Figure 2 1 D ) .Cone shaped shear failure: the sides are pushed out, leaving two cone shaped remnants, at the top and at the bottom forming an axi s symmet ric hourglass shape ( Figure 2 13 E ) Diagonal shear failure: fail ure occurs u nder a diagonal failure plane ( Figure 2 13 F ) In some cases the failure combined vertical splitting or shell brusting ( Figure 2 13 G ) Buckling failure: inflation of the cylinde r, from the bottom ( Figure 2 13 H ) or from the half of the length ( Figure 2 13 I ) Compressive belly failure: inflation of the center of the cylinder, forming a belly shape ( Figure 2 13 J ) Shell core failure: hap pens by the bursting of the shell of the cy linder, leaving a core at the center ( Figure 2 13 K ) Progressive collapse: gradual transfer of the failure from the top to the bottom, as the hammer goes failure by the fact that the specimen sta ys intact and in place while the hammer is moving down, and progressively eroding it from the top to the bottom( Figure 2 13 I and Figure 2 13M ) PAGE 25 25 2.6 Propagation of Waves in Elastic Solid Media Equation By considering a rod under impact, Tim oshenko ( 1951) shows equation derivation 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) F or both functions f and f1, it ca n be seen that for each for a certain time T they both will be only function s of x. the shape of the functions stay s unchanged and only depend on the functions f and describe d in Figure 2 14 Thus the function describe s two waves traveling in opposite directions at a constant speed c ( Figure 2 15 ) The relation ship the stress, can be developed by the impulse law (2 26) which yields: (2 27) PAGE 26 26 2.6.2 Superposition of W aves Because of the linear characteristic of the wave equation the superposition principle is valid. For two waves traveling in opposite directions ( Figure 2 17 A ) the resultant of t sensitivity ( Figure 2 17 B ) After passing, the waves return to their original shape ( Figure 2 17 C ) 2.6.3 R eflection of Waves By considering a compressive wave traveling in the pos itive x direction and a tension wave with the same length and stress magnitude moving in the opposite direction ( Figure 2 18 A ) 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 18 C ) I t 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 ( Timoshen ko, 1951 ) By Considering a case where two identical compression waves which are traveling in opposite directions( Figure 2 19 A ) it can be seen that the velocity of the particles in the cross section m n, is always zero, which yields zero displacement of t his cross section. Hence, the cross section can be replace d by fixed end, as demonstrated in ( Figure 2 19 C ) From this case, it can be concluded that a wave is reflected from a fixed end completely unchanged PAGE 27 27 Figure 2 1 Strain stress curves. ( Shah Swarz & Ouyang, 1995) PAGE 28 28 Figure 2 2 Stress distribution along the hole with respect to the material brittleness (Shah, Swarz, & Ouyang, 1995) PAGE 29 29 Figure 2 3 S emi infinite plate with a hole made in it (S.Timoshenko, 1951) Figure 2 4 Size effect law ( ) PAGE 30 30 Figur e 2 5 Concrete DIF vs. strain rate. (Tedesco, Powell, Ross, & Hughes, 1997) Figure 2 6 Charpy impact test device (Gopalarm, Shah, & John, 1984) PAGE 31 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) PAGE 32 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 PAGE 33 33 Figure 2 12 A 600x1200 mm specimen ready for soft impact test (Krauthammer & Elfahal, 2002) PAGE 34 34 Figure 2 13 Compression failure modes observed by (Krauthammer & Elfahal, 2002) PAGE 35 35 Figure 2 14 Illustration hifting ( 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 ). PAGE 36 36 Figure 2 18 Reflection of wave from free end (S.Timoshenko, 1951) Figure 2 19 Reflection of wave from fixed end ( Timoshenko, 1951 ). PAGE 37 37 CHAPTER 3 MET H ODOLOGY Krauthammer and Elfahal, ( 2002) classified the failure of concrete cylinders su bjected to a dynamic load into 7 modes of failures, as mention ed in Section 2.5. The ( Figure 3 1 ), and the buckling phenomena. The inertia effect, which is assumed t o 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 th e conclusion 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 subje cted to a dynamic load is by analyzing each mode separately. Therefore, this study will concentrate on the failure due to buckling ( Figure 3 2 A 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. ct, 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 PAGE 38 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 bound ary 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 Approa ch 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 r esisting 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 comparis on 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 d isplacement. 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 poi nt of PAGE 39 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 velocity of the particl es 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 speci men 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 observat ions of the dynamic failure process. In contrast to the quasi characterized by several crack planes, in the dynamic failure many crack planes are observed and particles of the specimen are moving out with a cer tain velocity. Thus, the specimen at the failure time has more energy than needed for failure. 3. 2 3 T he Kinetic Energy of a Specimen in Split Hopkinson Pressure Bar By considering the model of Chandra & Krauthammer (1995) the kinetic energy of the speci men causes the mass to keep moving as a rigid body and by that converting PAGE 40 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 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 en ergy. The effect of this energy will be investigated by a finite elements model (FEM) that is described in the next chapter 3. 2 4 Fin ite 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 S HPB. However, the purpose of this model is to clarify the PAGE 41 41 subsequences of the strain energy that resulted from the presence of kinetic energy at the dynamic loading. Thus, the shape of the incident wav e 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 un its 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: 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 th e 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 prog ressive buckling. PAGE 42 42 The capacity of the group of the columns is equal to the summation of all the 3. 3 .1 D ynamic Buckling of a Single Column The static bucklin g of a column, also known as Eu buckling, has been detailed by Timoshenko ( 1930) The calculation of the buckling load is reache d 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 y 0 Is the initial displacement of the rod. This equation yie lds the well known solution for the critical load (P c ) 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 d ynamic forces equation is given by : (3 6) PAGE 43 43 is the buckling load, y is the deformation of the r od, y 0 is the initial deformation of the rod, is density of the material, A is the cross section al area, and t is the time variable. For convenience, the equation can be divided by EI and the following parameters will be substituted: , This y ields: (3 7) The solution for the differential equation can be expressed as a product of functions i n 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 equation s 3 9 and equation 3 8 into 3 8 to find yields: (3 11) Rearrang ing the equation yields : PAGE 44 44 (3 12) It can be seen that a trigonometric solution, which reflects a bounded solution, will be given only f or 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 wave length: which correspond s 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 wave length 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 13 B ) In this mode the cylinder split s to a grou p 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 in to a group of columns which represents the first phase of failure must be investigat ed as well. It seems that the two govern ing causes that initiate the separation ffect 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 a n approximated analytical calculation and a finite element model can be developed to describe the failure of a concrete cylinder under compression. 3. 3 .2 F racture Mechanics and Energy Methods Timoshenko ( 1930) proposes an approximated method to calculate t he static buckling load of a column by using an energy based method. To attain the buckling load PAGE 45 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 T he 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) W here l is length of the co lumn and y is the buckling shape. While f or 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 Modify ing 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) PAGE 46 46 Table 3 1 Properties of the parts part Length[m] Diameter[m] E[Pa] 3 ] Bars specimen 1 0.02 0.02 0.02 2*10 11 2*10 9 0.3 0.2 7800 2000 Figure 3 1 Failure (Krauthammer & Elfahal, 2002) Figure 3 2 Failure due to dynamic buckling (Krauthammer & Elfahal, 2002) buckling length was emphasized PAGE 47 47 Figure 3 3 Spring mass model for strain rate effect. (Chandra & Krauthammer, 1995) PAGE 48 48 Figure 3 4 A bar subjected to a stress wave Fi gure 3 5 An illustration of the finite elements model of a SHPB PAGE 49 49 Figure 3 6 An illustration of the f inite eleme n ts model of SHPB close view of the spec imen area Figure 3 7 An illustration of the specimen and the incident bar PAGE 50 50 Figure 3 8 Set of elements along the specimen Figure 3 9 Illustration of a set of two nodes of the specimen PAGE 51 51 CHAPTER 4 RESULTS AND DISCCUSI ONS T his chapter consists of results and discussions for the strain rate effect and for the buckling phas e. Since some results demanded fur ther 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 m odes 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 M Pa and time duration of 2E 5 seconds Since it is not possible to define a load 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 dat a 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 b is produced and is given by (Ap p endix A): (4 1) Where c b is the wave speed in the bar, and t l 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 duration (t r ) depends on the time at which the incident wave exists in the transition face, and will therefore be: PAGE 52 52 (4 2) Subsequently, the s ) is given by: (4 3) Where c s is the speed of the wave in the specimen. It is important to obse rve that the wave length 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, finding s yielded by this loading case is valid for longer wave s as well by using the s uperposition principle. 4. 1 .2 E nergies 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 energ y is then converted to strain energy, it reaches a peak an d 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 chang e ( Figure 4 3 ) For a better understanding of the results above, two time point s were marked on the last two figures to produce Figure 4 4 and Figure 4 5 The first time point which is called indicates the time of maximum strain energy in the specimen which is 2.288154647E 04 sec time when maximum strain of the specimen has been achieved which is 2.250545076E 04 sec PAGE 53 53 By examining Figure 4 4 strain energy keep s 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 ener gy 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 mention ed above can be understood by considering a stress wave propa gating 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 as illustrated in Figure 4 6 1 ) and the strain energy in the bar (SE 1 ) are given by: (4 4) (4 5) At time t= ( 3/2 ) ( l/c ) the reflected wave reaches the the incident wave as illustrated in Figure 4 7 Hence, it is trivial that the stress and the of bar are both equal to zero. However, t he stress and at the r ight half of the bar are achieved by superimposing those PAGE 54 54 respectively. Subsequently the shortening of 2 ) and the strain energy in bar (SE 2 ) 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 the se two time points are representing the two extreme cases, a general conclusion can be made: for a case of a cantilever subjected to a rect angular stress wave with a wave l ength equal to the length of the ba r, the shortening of the bar at each time point between l/c to 2l/c, 1 and the strain energy of the bar varies between SE 1 and 2SE 1 By taking into account the second that part of the wave was transmitted to the transmitted bar, the case disc ussed is slightly different than the case of the cantilever above. A t time t= ( 3/2 ) ( l /c ) the 2 ) is lower than the incident ( Figure 4 8 ) Hence the shortening of the bar is now lower than in the case above, a nd the strain energy (at this time point) of the bar can vary between SE 1 /2 and 2 SE 1 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 is indeed never dropped to zero. PAGE 55 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. A fter 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 V 2 3. V 2 4. V Where V is the magnitude of the velocity due to the stress and V 2 is the magnitude of 2 4. 1 .4 S train 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 s train energy and average strain with respect to strain distribution. Thus, this relationship changing is investigat ed in the SHPB Axial strain energy (SE) in a bar is calculated by: (4 8) Hence, the relationship between the strain energy and the strain is quadratic a nd ca n be simplified to: (4 9) Where G is a constant and is equal to AEl/2. The ratio SE/(G 2 ), which is equal to one for a uniform distribution of strains, is shown in Figure 4 9 together with the two time poin max NE It can be PAGE 56 56 seen that this ratio increases between the maximum strain recorded max NE 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 th e se two points the strain distribution increase s A difference can be seen at the left end of th e two figures. While Figure 4 9 provides high values close to the left end, the corresponding values in Figu re 4 10 a re very small. This 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 i n the case for the strains at the left end which are approaching zero This effect is vanis hed when the standard deviation values ar e divided by the average strain values, as illustrated in Figure 4 11 4. 1 .5 S ummary of the Results by Time Sequence For a better understanding of the discussions above, the results are provided with three time step steps were calculate d analytically (Appendix A), and therefore some minor discrepancy is obser ved At time t= 1.975E 04 sec 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 start s propagating in the specimen, and as result the kinetic and the strai n energies in the specimen increase ( Figure 4 12 ), along with the average strain in the PAGE 57 57 specimen ( Figure 4 13 ) During the wave propagation in the specimen 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 wave 2nd face ncident 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, th e reflection process begins and the disorder increase s 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 c lose to the other face. This disorder ( Figure 4 15 and Figure 4 16 ) decrease s the efficiency of the strain energy ( Figure 4 14 ) This, combined with the fact that the total energy in the specimen decrease s due to the transmitted wave, the average strain decreases. reach the half point of the Figure 4 8 the two waves are fully superimposed. Thus, the average velocity of the particles is minimal and str ain energy reaches a maximum ( Figure 4 12 ). This point i s 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 p ropagating toward the first face of the specimen, and the overlapped length of the two waves decreases. Subsequently, strain PAGE 58 58 energy is converted back to kinetic energy, and f or the first time, the velocity of all of the particles is not in the same directi on. At time t= 2.375E 04 sec reflected wave 1st face reflected wave reaches the are uniformly distributed, with the magnitude of the reflected wave. In this s ection 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 an d Gorham (1997) yielded that in a long specimen the difference of stress es 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 superimp osed 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 u niformity 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 refle cted from the bars. The amount of remaining kinetic energy depends on the properties The presence of this energy is important for the analysis of the buckling phase. The curvature of the columns due to an axial for ce was assumed to be a buckling behavior since no bending force was applied. However, the inertia of the mass due to this PAGE 59 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 to tal 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 consist s 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 t his for two reasons described below. An e xperiment 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 wa ve 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 first mode. PAGE 60 60 The number of columns in the group was calculated using properties of the cylinders that had a buckling fail the cylinde r (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 I t can be seen that capacity of the group decreases as the number of columns in the group increase s 4. 2 .2 B uckling as a Failure Criteria T he 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 th is eq uation the minimum number of columns that are needed for buckling failure can be found. Subsequently, the minimum number of column s needed for a buckling failure to occur in the cylinder mentioned above is 86. This result was calculated using Mathcad (App endix 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 cylinder s 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 PAGE 61 61 might b e the Hence, an approximated model was develop in Mathcad ( Appendix C) to check this possibility using the experiment al proper ties [reference] ( Table 4 3 ) and it is described below. The cylinder is axially loaded by t he force P and radial velocity Vr, as described in Figure 4 19 A Thus, any single column in the group i s loaded by axial force P/n and lateral velocity v ( Figure 4 19 B ) 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 column s in the g roup is limited by the maximum curvature prior to failure of the column, and it is calculated in Appendix C. T herefore, bending is possible if the energy that causes bending i n a column is greater than the critical bending strain energy of column The def lection shape (g) of a column is assumed to be: (4 13) w here 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 (mxC rack(x,n)) is given by: (4 14) w here is the amplitude of the curved shape at failure for each n, and is given by: (4 15) w here is the ultimate tensile strength of the concrete. Finally, the critical energy (Um(n)) as function of the number of column s in the group is: PAGE 62 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 (E cr ): (4 17) By averaging the velocities the average velocity (Vavg) i s given by: (4 18) The average velocity consists of the axial (Va) and the radial (Vr) ( Figure 4 20 ) velocities wh ere : (4 19) Therefore, the radial velocity is given by: (4 20) Finall y the column is bending around one axis but the vel ocities direction s are determined according to position of the column in the cylinder as illustrated in Figure 4 21 .H ence anot her 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 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 cause s bending in a column is: PAGE 63 63 (4 22) The kinetic energy that causes bending in a single column (uKn) is plotted with the critical bending ene rgy of a column (uMcrack), multiplied by 10 3 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 10 3 times than required to introduced post failure bu ckling, 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 cylind er 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 mo ving 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. PAGE 64 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 578 Impact velocity (m/s) 5 Cylinder height (m) 0.6 Cylinder diameter (m) 0.3 Static capacity (Mpa) 44.8 E (Gpa) 30 0.17 m 3 2400 cr 0.002 cr + 8.00E 05 Figure 4 1 The applied load shape PAGE 65 65 Figure 4 2 Strain energy and kinetic energy at the specimen Figure 4 3 Average strain of the specimen (compression is po sitive) vs. time PAGE 66 66 Figure 4 4 Strain and kinetic energies at the specimen, with time marks(close view from the Figure 4 5 Average strain of the specimen (compression is positive) vs. time, with time marks PAGE 67 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 PAGE 68 68 Figure 4 9 SE/(G Figure 4 10 Standard deviation of the strains in the specimen vs. time (close view from the PAGE 69 69 Figure 4 11 Standard deviation/average of the strains in the specimen vs. time (close view from Figure 4 12 Strain and kinetic energies at the specimen PAGE 70 70 Figure 4 13 Average strain of the specimen (compression is positive) Figure 4 14 SE/(G PAGE 71 71 Figure 4 15 Standard deviation of the strains in the specimen (close view from the time) Figure 4 16 Standard deviation divided by average of the strains in the specimen vs. time (close PAGE 72 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. PAGE 73 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. PAGE 74 74 Figure 4 22 A model for the post failure behavior consist of masses an d springs. PAGE 75 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 foun d that in contrast to the initial assumption, the modes of failu r e 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 follow ing conclusions were determined based on this work: The property is not represented in the traditional strain stress curve but must be taken into account. In contrast to the sta tic 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 t he 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 ad ditional 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 is converted to strain energy by producing no additional increase in the average strain. The amount of specimen the bars. Since the strain distribution along the specimen during most of the time is not uniform, u sing of average strain of the specimen should be considered carefully. In addition the indication obtained by the standard variation function seems to be a good magnitudes of the values influence the result. This ratio and its effect should be investigated more deeply, and advanced statistics tools might be investigated. PAGE 76 76 The modes of failure that were observed by (Krauthammer & Elfahal, 2002) are believed to be a post failure behavior. Th os e 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 spe cimen 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 adva nced 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 en ergy 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. PAGE 77 77 APPENDIX A MATHCAD CALCULATION SHEET FOR THE SHPB PROPERTIES PAGE 78 78 PAGE 79 79 PAGE 80 80 APPENDIX B M ATHCAD CALCULATION SHEET FOR THE BUCKLING PAGE 81 81 PAGE 82 82 APPENDIX C MATHCAD CALCULATION SHEET FOR POST FAILURE BENDING PAGE 83 83 PAGE 84 84 PAGE 85 85 PAGE 86 86 PAGE 87 87 PAGE 88 88 LIST OF REFERENCES Banthia, N., Mindess, S., Bentur, A., & Pigeon, M. (1989). Impact Testing of Concre te Using a Drop weight Impact Machine. Experimental Mechanics, 29(2) 63 69. Proceedings of US Japan Seminar on Finite Element Analysis of Reinforced Concrete Structures, ASCE 121 150. 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 M aterials, 38(3) 253 267. Chandra, D., & Krauthammer, T. (1995). Strength Enhancement in Particular Solids under High Loading Rates. 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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. PAGE 90 90 BIOGRAPHICAL SKETCH Avshalo m Ganz was born in Israel in 19 81 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 e ngin eering on July 200 8 In 200 6 he joined Ortam Malibu Engineering LTD Israel in a student position and later as a structural e ngineer. In 2009, he began his m stru ctures. 