Frequency Domain Assessment of Direct Shear in Normal and Ultra-High-Performance Concretes

MISSING IMAGE

Material Information

Title:
Frequency Domain Assessment of Direct Shear in Normal and Ultra-High-Performance Concretes
Physical Description:
1 online resource (112 p.)
Language:
english
Creator:
Kim, Jae Yoon
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
Publication Date:

Thesis/Dissertation Information

Degree:
Master's ( M.S.)
Degree Grantor:
University of Florida
Degree Disciplines:
Civil Engineering, Civil and Coastal Engineering
Committee Chair:
KRAUTHAMMER,THEODOR
Committee Co-Chair:
ASTARLIOGLU,SERDAR

Subjects

Subjects / Keywords:
frequency -- impact -- shear -- uhpc
Civil and Coastal Engineering -- Dissertations, Academic -- UF
Genre:
Civil Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract:
Direct shear is recognized as a possible response mechanism in structural concrete systems subjected to severe dynamic loads, and it could lead to catastrophic failure. Behavioral models for direct shear in normal strength concrete (NSC) were introduced in the 1970s, and their adaptation for the analysis of structural response under blast and ground shock effects was presented in the 1980s. The introduction of ultra high performance concrete (UHPC) for protected facilities has created the need to reevaluate direct shear in both NSC and UHPC, and to characterize this response mechanism more accurately. Therefore, direct shear impact tests were conducted on NSC and UHPC specimens with three reinforcement ratios, and the results were analyzed in both the time and frequency domains. This paper is focused on the assessment of direct shear in the frequency domain to identify the relationships between parameters in the direct shear resistance functions for NSC and UHPC from impact tests. The data for each type of specimen were analyzed by using the experimental time histories for impact force, displacement, velocity and acceleration. Finally, the results were compared also with those obtained from both the previous and modified direct shear models to identify the behavioral parameters that could explain the direct shear behavior under impact loading.
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Jae Yoon Kim.
Thesis:
Thesis (M.S.)--University of Florida, 2014.
Local:
Adviser: KRAUTHAMMER,THEODOR.
Local:
Co-adviser: ASTARLIOGLU,SERDAR.

Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
lcc - LD1780 2014
System ID:
UFE0046851:00001


This item is only available as the following downloads:


Full Text

PAGE 1

1 FREQUENCY DOMAIN ASSESSMENT OF DIRECT SHEAR IN NORMAL AND ULTRA HIGH PERFORMANCE CONCRETES By JAEYOON KIM A THESIS PRESENTED TO THE GRAUDATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULF ILLMENT OF THE REQUIREMENTS FOR THE DEGR EE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 201 4

PAGE 2

2 201 4 J aeyoon K im

PAGE 3

3 To my wife and son

PAGE 4

4 ACKNOWLEDGMENTS I would like to express my special appreciation and thanks to my advisor Professor Dr. Theodor Krauthammer who ha s been a tremendous mentor for me. I would also like to thank my thesis committee member Dr. Serdar Astarlioglu for his assistance. Moreover, I would like to acknowledge the Defense Threat Reduction Agency (DTRA) for funding this research; Michael Stone for teaching me with t he physical operation of the machinery in the structures lab and helping me with data processing by MATLAB ; Robin French for academic advice on direct shear model ; and all the other staff and students at Center for Infrastructure Protection and Physical Security (CIPPS). I would also like to thank my wife for supporting me throughout this entire endeavor.

PAGE 5

5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF FIGURES ................................ ................................ ................................ .......... 8 ABSTRACT ................................ ................................ ................................ ................... 11 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 13 Problem Statement ................................ ................................ ................................ 13 Objective and Scope ................................ ................................ ............................... 14 Research Significance ................................ ................................ ............................ 14 2 BACKGROUND ................................ ................................ ................................ ...... 15 Frequency Domain Analysis ................................ ................................ ................... 15 Fourier A nalysis ................................ ................................ ................................ 15 Fourier Series ................................ ................................ ................................ ... 17 Discrete Fourier Transform ................................ ................................ ............... 21 Fast Fourier Transform ................................ ................................ ..................... 23 Fourier Transform in Structural Dynamics ................................ ........................ 24 Transfer F unction ................................ ................................ ............................. 27 Mechanical Imp edance ................................ ................................ .................... 28 Normal Strength Concrete ................................ ................................ ...................... 31 Compressive Strength and Behavior ................................ ................................ 31 Tensile Strength and Behavior ................................ ................................ ......... 32 Ultra High Performance Concrete (UHPC) ................................ ............................. 33 Definition of UHPC ................................ ................................ ........................... 33 Ultra High Performance Concrete Composition and Mixing Methodology ........ 34 Cement ................................ ................................ ................................ ...... 34 Sand ................................ ................................ ................................ .......... 35 Silica F ume ................................ ................................ ................................ 35 Superplasticizer ................................ ................................ ......................... 35 Fibers ................................ ................................ ................................ ......... 35 Mixing Methodology ................................ ................................ ......................... 37 Curing ................................ ................................ ................................ ............... 37 UHPC Mechanical Properties ................................ ................................ ........... 38 Stress S train R elationship ................................ ................................ .......... 38 UHPC Rate and Size E ffects ................................ ................................ ..... 40 Direct Shear Behavior ................................ ................................ ............................. 40 Original and Modified Hawkins Shear Models ................................ .................. 42 Hofbeck et al. (1969) and Mattock and Hawkins (1972) ................................ ... 46

PAGE 6

6 3 RESEA R CH APPROACH AND METHODOLOGY ................................ ................. 50 Signal Processing ................................ ................................ ................................ ... 50 Data Acquisition ................................ ................................ ............................... 50 Aliasing ................................ ................................ ................................ ............. 51 Resolution and Confidence ................................ ................................ .............. 52 Filtering ................................ ................................ ................................ ............. 54 Load vs Time ................................ ................................ ................................ .... 55 Slip vs Time ................................ ................................ ................................ ...... 56 Natural Frequency Ra nge ................................ ................................ ....................... 57 Limitation ................................ ................................ ................................ ................ 62 4 RESULTS AND DISCUSSION ................................ ................................ ............... 65 NC Specimens ................................ ................................ ................................ ........ 65 NSC 1A 0 D ................................ ................................ ................................ ..... 65 NSC 1 1 D ................................ ................................ ................................ ........ 66 NSC 1 2 D ................................ ................................ ................................ ........ 67 COR TUF1 (CT1) Specime ns ................................ ................................ ................. 69 CT1S 1A 0 D ................................ ................................ ................................ .... 69 CT1S 1 1 D ................................ ................................ ................................ ...... 70 CT1S 1 2 D ................................ ................................ ................................ ...... 71 COR TUF2 (CT2) Specimens ................................ ................................ ................. 72 CT2S 1A 0 D ................................ ................................ ................................ .... 72 CT2S 1 1 D ................................ ................................ ................................ ...... 73 CT2S 1 2 D ................................ ................................ ................................ ...... 75 Comparison of 0% Speci mens ................................ ................................ ................ 76 Comparison of 0.8% Specimens ................................ ................................ ............. 76 Comparison of 1.6% Specimens ................................ ................................ ............. 76 Load vs Time ................................ ................................ ................................ .......... 93 5 CONCLUSIONS AND RECOMMENDATIONS ................................ ....................... 96 Conclusions ................................ ................................ ................................ ............ 96 Recommendations for Future Research ................................ ................................ 97 APPENDIX A: MATLAB CODES ................................ ................................ ................................ ...... 98 Filtering Code ................................ ................................ ................................ ......... 98 FFT Code ................................ ................................ ................................ .............. 103 B: TEST SPECIMENS ................................ ................................ ................................ 106 C: EXPERIMENTAL TEST SUMMARY ................................ ................................ ....... 108 LIST OF REFERENCES ................................ ................................ ............................. 111 BIOGRAPHICAL SKETCH ................................ ................................ .......................... 112

PAGE 7

7 LIST OF TABLES Table page 2 1 Basic transfer functions ................................ ................................ ...................... 28 2 2 Quasi static uniaxial compressive performance of UHPCC at 60 d .................... 36 3 1 NSC, CT2 and CT1 Dynamic Modified Hawkins direct shear models. ............... 60 3 2 NSC, CT2 and CT1 Modified Hawkins direct shear models. .............................. 61 4 1 Summary of spike distribution ................................ ................................ ............. 92 B 1 Push off specimen geometry ................................ ................................ ............ 106 B 2 Push off specimen reinforcement and compressive strengths. ........................ 106 C 1 Summary of static test results. ................................ ................................ .......... 108 C 2 Summary of impact test results. ................................ ................................ ....... 109 C 3 Approximate ratio of dynamic to static shear strength for UHPC. ..................... 110

PAGE 8

8 LIST OF FIGURES Figure page 2 1 Principle of Fourier Series ................................ ................................ .................. 17 2 2 Concept of orthogonality ................................ ................................ ..................... 19 2 3 Sampled periodic function ................................ ................................ .................. 22 2 4 Relationship between time and frequency domain of load data ......................... 25 2 5 Fourier decomposition ................................ ................................ ........................ 26 2 6 Magnitude and phase of impedance for some values of damping ...................... 29 2 7 Mechanical impedance for mass, spring and damper ................................ ........ 30 2 8 Typical stress strain curves for concrete in compression. ................................ .. 32 2 9 Typical stress strain curve for concrete ................................ .............................. 33 2 10 Crack patterns in reinforced concrete elements subjected to tension. ................ 34 2 11 Bekaert Dramix ZP305 fibers. ................................ ................................ .......... 36 2 12 UHPC example mix proportion. ................................ ................................ .......... 37 2 13 Mechanical properties of NSC and fiber reinforced UHPC. ................................ 38 2 14 UHPFRC stress crack opening curve ................................ ................................ 39 2 15 Effect of fibers on UHPFRC on compressive stress strain curve ........................ 40 2 16 Direct shear failure of a reinforced concrete slab. ................................ .............. 42 2 17 Original and modified shear stress slip relationship for direct shear.. ................. 43 2 18 Push off specimen ................................ ................................ .............................. 46 2 19 Typic al load slip curves Mattock ................................ ................................ ......... 47 2 20 Effect of concrete strength on shear strength in initially cracked specimens. ..... 48 2 21 Shear transfer in initially uncracked concrete. ................................ .................... 49 3 1 Slip vs Time of CorTuf 2 with 0.8% shear reinforcement ................................ .... 51 3 2 Example of Aliasing ................................ ................................ ............................ 52

PAGE 9

9 3 3 Digital filter code in MATLAB ................................ ................................ .............. 56 3 4 Load vs Time after filter processing ................................ ................................ .... 5 6 3 5 Load, Slip, Velocity and Acceleration in time and frequency domain .................. 57 3 6 Actual weight of Cor Tuf1 with 0% shear reinforcement ................................ ..... 58 3 7 Static and Dynamic modified Hawkins direct shear model ................................ 59 3 8 Sequence frequency of the modified Hawkins model ................................ ......... 59 3 9 Example of voltage change in U(t) ................................ ................................ ...... 62 3 10 Typical response domains ................................ ................................ .................. 63 3 11 Example of frequency distribution on direct shear model ................................ ... 64 4 1 NCS specimens with 0% reinforcemen t after dynamic testing ........................... 66 4 2 NCS specimens with 0 .8 % reinforcement after dynamic testing. ........................ 67 4 3 NCS specimens with 1.6 % reinforcement after dynamic testing. ........................ 68 4 4 CT1 specimens with 0 % reinforcement after dynamic testing. ........................... 70 4 5 CT1 specimens with 0 .8 % reinforcement after dynamic testing ......................... 71 4 6 CT1 specimens with 1.6 % reinforcement after dynamic testing ......................... 72 4 7 CT2 specimens with 0% reinforcement after dynamic testing ............................ 73 4 8 CT2 specimens with 0 .8 % reinforcement after dynamic testing ......................... 74 4 9 CT2 specimens with 1.6 % reinforcement after dynamic testing ......................... 75 4 10 Time and frequency responses for NSC 1A 0 D 1 ................................ ............. 77 4 11 Time and frequency responses for NSC 1A 0 D 2 ................................ ............. 78 4 12 Time and frequency responses for NSC 1 1 D 1 ................................ ............... 79 4 13 Time and frequency responses for NSC 1 2 D 1 ................................ ............... 80 4 14 Time and frequency responses for NSC 1 2 D 2 ................................ ............... 81 4 15 Time and frequency responses for CT1S 1A 0 D 1 ................................ ........... 82 4 16 Time and frequency responses for CT1S 1A 0 D 2 ................................ ........... 83

PAGE 10

10 4 17 Time and frequency responses for CT1S 1 1 D 1 ................................ .............. 84 4 18 Time and frequency responses for CT1S 1 1 D 2 ................................ .............. 85 4 19 Time and frequency responses for CT1S 1 2 D 3 ................................ .............. 86 4 20 Time and frequency responses for CT2S 1A 0 D 2 ................................ ........... 87 4 21 Time and frequency responses for CT2S 1A 0 D 3 ................................ ........... 88 4 22 Time and frequency responses for CT2S 1 1 D 2 ................................ .............. 89 4 23 Time and frequency responses for CT1S 1 1 D 3 ................................ .............. 90 4 24 Time and frequency responses for CT2S 1 2 D 1 ................................ .............. 91 5 1 FFT of Load vs Time in Nyquist frequency domain for CT1S 1 1 D 1 ................ 93 5 2 FFT of Load vs Time in Nyquist frequency domain for CT2S 1 1 D 1 ................ 94 5 3 FFT of Load vs Time in Nyquist frequency domain for NSC 1 1 D 1 ................. 94 B 1 Push off specimen geometry ................................ ................................ ............ 107

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 FREQUENCY DOMAIN ASSESSMENT OF DIRECT SHEAR IN NORMAL AND ULTRA HIGH PERFORMANCE CONCRETES By Jaeyoon Kim May 2014 Chair: T h e o d or Krauthammer Major: Civil Engineering Direct shear is recognized as a possible response mechanism in structural concrete systems subjected to severe dynamic loads, and it could lead to catastrophic failure. Behavioral models for direct shear in normal strength concrete (NSC) were introduced in the 1970s, and their adaptation for the analysis of structural response under blast and ground shock effects was presented in the 1980s. The introduction of ultra high performance concrete (UHPC) for protected facilities has created the need to reevaluate direct shear in both NSC and UHPC, and to characterize this response mechanism more accurately. Therefore, direct shear impact tests were conducted on NSC and UHPC specimens with three reinforcement ratios, and the results were analyzed in bo th the time and frequency domains. This paper is focused on the assessment of direct shear in the frequency domain to identify the relationships between parameters in the direct shear resistance functions for NSC and UHPC from impact tests. The data for ea ch type of specimen were analyzed by using the experimental time histories for impact force, displacement, velocity and acceleration. Finally, the results were compared also with those obtained from both the previous and

PAGE 12

12 modified direct shear models to id entify the behavioral parameters that could explain the direct shear behavior under impact loading.

PAGE 13

13 CHAPTER 1 INTRODUCTION Problem Statement High strength concrete provides new structural opportunities, in particular the use of slender structural members. It allows for engineers to build structures that we never have imagined before. However, i t has led to increasingly lighter structures which are more prone to vibration problems. These large structures can also be attractive targets of terrorism, and critical infrastructure facilities such as nuclear plants could be also be targeted. It is essential to assess the contribution of these new stron ger materials to their survivability. D irect s hear is a sudden and catastrophic failure type commonly observed in reinforced concrete structures under highly impulsive dynamic loads. This failure type can severely compromise the integrity of a structure a nd can result in progressive collapse. Performing blast tests on a large number of specimens to test many variables is expensive and is not without risk of injury. Impact testing is a viable alternative to full blast design as it allows experimental testin g on specimens with similar peak loads and durations to blast loading. Normal strength concrete (NSC) and Ultra high performance concrete (UHPC) have been tested and analyzed in time domain under static and frequency r esponse to direct shear conditions is also critical to the proper design of structures under impact loading as well as blast or seismic loadings.

PAGE 14

14 Objective and Scope This paper is focused on the assessment of direct shear in the frequency domain to identif y the relationships between parameters in the direct shear resistance functions for NSC and UHPC and the power spectral density of the captured data from impact tests. The data for each type of specimen were analyzed by using the experimental time historie s for impact force, displacement, velocity and acceleration. Finally, the results were compared also with those obtained from both the previous and modified direct shear models to identify the behavioral parameters that could explain the direct shear beha vior under impact loading. Research Significance This analytical study will provide a measurement and signal processing technique to characterize the direct shear behavior of COR TUF UHPCs in the frequency domain. T his knowledge is particularly essential for the effective design of protective structures under any dynamic loading conditions including impact, blast, and seismic Previous time domain analysis on NSC and UHPC ha s been performed This research will allow for investigation into the fre quency response of direct shear that could explain the modified direct shear model either on static or dynamic loadings.

PAGE 15

15 CHAPTER 2 BACKGROUND This literature provides two parts of background pertinent to this research. First a brief summary of the principle of Fourier analysis will be presented. Then the behavior of NSC and UHPC will be discussed. Lastly, previous research conducted relating to direct shear model will be reviewed. Frequency Domain Analysis Fourier A nalysis The methods of time doma in analysis can be effectively used to obtain th e response of any single degree of freedom system. It is, however, possible in certain situations to greatly improve the efficiency of the analytical procedure by employing the transform method of mathematics Transform methods have proved to be very useful in many engineering applications. In the analysis of dynamic response of linear structures, Fourier transform can play an important role. Brief l y, this transform converts the task of evaluating the convolution integral of into that of finding the product of the Fourier transforms of the two functions involved in the convolution. The inverse Fourier tran s form of the product then gives the desired response function of time. The method of analysis is commonl y referred to as dynamic analysis in the fr equency domain It would appear that like other transforms, the success of the method should depend on the ease with which the Fourier transform and the inverse Fourier transform of the function s involved in the analysis can be evaluated. Unfortunately, except for simple cases, where the direct method of solving the convolution integral is as effective, the

PAGE 16

16 application of Fourier transform involves evaluation of integrals that are not always easy to solve. In most practical cases, therefore, Fourier transform is used in its discrete form, which permits the replacement of the integrals by equivalent numerical computations of areas. In fact, the efficiency of Fourier methods is most apparent in case s where the excitation is specified in terms of n umerical values at regular interval s of time rather than as a mathematical function. In such a situation, even the time domain analysis, which eventually depends on the evaluation of a convolution integral, must rely on numerical methods, and the alte rn ative method of analysis through discrete Fourier transform proves to be most effective. Ordinarily, th e computations involved i n obtaini ng the discrete Fourier transforms of the functions bei ng convolved, taking the product of these transforms and then evaluating the discrete inverse Fourier transform are no less than those in a direct eval uation of the discrete convolution. However, the development of a special algorithm called fast Fourier transform (FFT) has completely altered this position. The FFT algorithm, which derives its efficiency from explo i ting the harmonic property of a discrete transform, cuts down the computations by several orders of magnitude and makes frequency domain analysis highly effic ient. The FFT has found widespread application in several areas of engineering analysis. Its use in dynamic analysis of structures is, however, more recent. Besides computational efficiency, analysis in the frequency domain possesses several other advanta ges. By providing a clearer representation of the frequency content of the forcing function, it enables one to evaluate the potential for it to excite a given structure. In addition, for problems involving infinite domains such as those of wave motion in

PAGE 17

17 r eservoirs of large extent or in unbounded soil media, and in other situations where the physical characteristics of the system are dependent on the vibration frequency, frequency domain analysis may prove to be more effective than time domain analysis. It should, however, be noted that frequency domain analysis applies only to time invariant, linear systems for which the principle of superposition holds. Fourier Series A periodic function can be represented as a sum of infinite number of sinusoidal compon ents at equally spaced frequencies with the interval of 1/T, where T is the period of the function, the so called Fourier Series. Figure 2 1 Principle of Fourier Series ( M Sek 2009)

PAGE 18

18 A function g(t) of time t is said to be a periodic function of t with a period equal to T 0 if it satisfies the following relationship: (2 1) W here n is an integer with negative or p ositive values. A periodic function with a finite number of discontinuities and a finite number of maximas or minimas within a range of time equal to its period T 0 can be represented by an infinite trignometric series as follows: (2 2) where a n and b n are constants to be determined and f 0 = I/T 0 is the frequency in cycles per second. The trignometric series of Eq. 2 2 is known as the Fourier series. Setting where w 0 is the frequency in radians per second, the series can be expressed in the alternative form (2 3) To obtain the coefficients a n we multiply the left a nd the right hand sides of Eq. 2 3 by and integrate over a period. Then noting that (2 4a) (2 4b)

PAGE 19

19 Integrate with = non zero Integrate with = zero Figure 2 2. Concept of orthogonality (Liangcai H 2003) and for all m and n (2 4c) we obtain the following expression for coefficients a: (2 5a) (2 5b) Coefficients b n are obtained in a similar manner by multiplying the left a nd the right hand sides of Eq. 2 3 by and integrating over a period. Then, because of the relationships (2 6) and that given by Eq. 2 4c, the coefficient b n is obtained as (2 7) It can be shown that at a discontinuity such as the one shown in, the Fourier series of Eq. 2 2 or Eq. 2 3 converges to a value that is the average of the

PAGE 20

20 values of the funct ion g(t) immediately to the left and the right of the discontinuity. In developing the frequency domain analysis procedure, it is convenient first to obtain an exponential form for the Fourier series of Eq. 2 the sine and cosine functions in Eq. 2 3 can be expressed in te rm s of complex exponentials as follows: (2 8) (2 9) Substitution of Eq. 2 9 in Eq. 2 5b yields (2 10a ) I n in Eq. 2 10a we get (2 10b) In a similar manner, b n is obtained by substituting Eq. 2 8 in Eq. 2 7: (2 11a) (2 11b) The Fourier series of Eq. 2 3 can be expressed as

PAGE 21

21 (2 12) Using relationships Eq. 2 10 b and Eq. 2 11 b, Eq. 2 12 becomes (2 13) in which coefficient c n is given by (2 14a) A nd (2 14b) Equations 2 13 and 2 14 together represent the exponential form of the Fourier series, which is seen to be much more compact than the alte rn ative form given by Eq. 2 3 2 5 and 2 7. Discrete Fourier Transform Except for very simple cases, Fourier transfo rm s are calculated numerical l y. This

PAGE 22

22 requires that the continu ou s values of t be expressed discretely and that the discretization be over a finite time interval T 0 This implies that (2 15) in which is the uniform time increment, such that (2 16) where N is the number of points in the time series approximation to g (t). To this point, the discussion of Fourier series has used the radian frequency n = 0 It is now somewhat more convenient, and also common pract i ce, to express frequency in h ertz. Thus, the Fourier frequencies are written (2 17) Also (2 18) Figure 2 3. Sampled periodic function (HumarJ.R 1990) and t he discrete Fourier transform is derived from Eq. 2 13 m=0,1 1 (2 19) and the frequency coefficients are derived from Eq. 2.14a

PAGE 23

23 C 0 =Cn 1 (2 20) The total number of discrete time values and discrete frequency values are the same. Fast Fourier Transform Modem frequency domain solutions algorithms are based on the fast Fourier Transform (FF T ). The FFT is predicated on the observation that if the number of points is a power of 2 (that is, N = 2 ^ L, where L is an integer), then the computations can be made using a recursive algorithm The development of the FF F has had a substantial influence on the solution of engineering problems because it significantly decreases the computatio n time for evaluating Fourier transfo rm s. The ratio of computation time using an FFT relative to the standard discrete Fourier transfo rm (DFT) is approximately (2 21) For example for a time series with N = 210 = 1024 points, the computation factor is 1/1024. For a larger ti me series with N = 220 = 1,048,576 points, the c omputation factor is 1/52,428.8. If this calculation took 10 m inutes to execute using an FFT, it would

PAGE 24

24 require 1 year to compute using a DFT. A million points is a very long time series, but it demonstrates that the FFT provides the means to routinely use Fourier analysis by rendering the problem numerically tra ctable. In practice, several variations of the FFT are routinely employed. The algorithm developed by Cooley and Tukey is one of the more popular. However, an alternative given by Press et al. is presented since its development is more intuitive Fourier Transform in Structural Dynamics In order to prevent resonance of a structure, first we need to find two parameters. One of them is natural period of the structure and the other one is the fundamental frequency of an excitation. Any time varying signal can be constructed by adding together sine waves of appropriate frequency, amplitude, and phase. Fourier tran s form is a technique that is used to determine which sine waves a given signal is m ade of, i.e. to deconstruct the signal into its constituent sine waves. The result is expressed as sine wave amplitude as a function of frequency. If a frequency has a very small or zero amplitude associated with it, then it does not contribute to the sign al. Thus, a signal consisting of a single sine wave will have a peak matching the frequency and amplitude of the sine wave, and zero contributions from all other frequencies.

PAGE 25

25 Figure 2 4 Relationship between time and frequency domain of load data (D Kim 2009) A fund a mental frequency can be determine d in the figure 2 1 If the natural period of the s tructure is equal to T = 1/f1, a resonance will occur. meaning of y value in P( w)? It is a sum of amplitudes at each corresponding frequency over the entire time duration. To be specific, an amplitude in the time domain indicates the amount of a force. So it can be said that the y value indicates how many times excitation oscillate w ith a certain amount of force at corresponding frequency until the excitation is over. This means that it is the sum of force of the same frequency which is distributed over the entire time length. This is the reason why the unit of y axis is "force*time" for example, kip*sec. So the y value consists of an amplitude and a number of oscillations. For example, if P(t) above has more times of oscillation with frequency f2. y value in right plot might be equal or beyond that of f1. f y value in U(w) ? Using the same principle of P(w), it is a sum of amplitudes of the same frequency over the entire time duration. In other words, it is a superposition of amplitudes which are from each time period. That's why unit of y value is distance* time for example in*sec. Basically, displacement is from a correlation between force and stiffness. If we find how two frequency functions of excitation and displacement are, we can track a dominant frequency of the structure.

PAGE 26

26 For example, Let's assume tha t f1 is the fundamental frequency of an excitation P(w) and it has another very small y value at frequency f3. Then let us consider displacement function U(w) If the highest y value occurs at f3 even though y value at f3 in P(w) is very small, f3 might be natural frequency of the structure because the resonance or the maximum response is subject to natural frequency of the structure. Figure 2 5. Fourier decomposition (D Kim, 2009)

PAGE 27

27 Transfer F unction the steady state response of the SDOF system to complex e xcitation For an SDOF system with viscous damping, the transfer function is given by (2 22) Where is system natural frequency, and is dam ping ratio. If an SDOF system is subjected to a complex excitation where P is the amplitude of requency, then the steady state responses of the SDOF system to this complex excitation are: In frequency domain : Displacement (2 23) Velocity (2 2 4) Acceleration (2 25) In time domain : Displacement (2 26) Velocity (2 27) Acceleration (2 28) 2 22 depending on the type of damping used. Similar transfer functions can be developed for acceleration responses.

PAGE 28

28 Table 2 1 Basic transfer functions Dimension U(w) / P(w) V(w) / P(w) A(w) / P(w) Name Admittance, Mobility Acceleration, Compliance, Inertance Receptance Dimension P(w) / U(w) P(w) / V(w) P(w) / A(w) Name Dynamic Stiffness Mechanical Impedance Apparent Mass, Dynamic Mass Mechanical Impedance This quantity is generally defined as the pressure or force divided by some velocity. In this manner one can distinguish between mechanical and acoustic impedance, the latter of which comes in many forms. For the mechanical impedance it is commonly the ratio between the force and the velocity that is understood and this means that the input impedance of the classical blocked (grounded) single degree of freedom, mass spring damper system is given by, (2 29 ) readily derived from the governing differential equation. Herein of course, M is the mass and K the spring stiffness whilst R is the damping In Figure 2 6 is shown the typical signature of the single degree of freedom system to magnitude and phase for some different values of the damping R

PAGE 29

29 Figure 2 6. Magnitude and phase of impedance for some values of damping ( ) R =1, ( ------) R =5 and ( ) R =50 ( Skudrzyk E., 1971) Observed in Figure 2 6 are three distinct regions, At low frequencies, the system is governed by the spring stiffness, corresponding to the third term in (1) i.e., when the is mass excited at a frequency to a given velocity, the reaction force is tha t produced by the spring. This force decreases linearly with frequency. For high frequencies, on the other hand, the mass governs the system corresponding to the first term and the

PAGE 30

30 reaction force is the inertia of the mass. At some intermediate frequency t he imaginary part of the impedance cancel and a resonant behavior occurs. The system is governed by the damper i.e. the real part of the impedance. This means that only in a narrow band around the resonance frequency the damper plays a role. With respect to the phase, the stiffness controlled region is featured by a negative phase equal to / 2 i.e., the force lags the velocity (but not the displacement) in time. The mass controlled range on the other hand has a positive phase to / 2 and hence th e force leads the velocity (but not the acceleration). In the resonant range, close to the resonance frequency, the phase approaches zero and precisely at resonance, the force and the velocity are in phase and hence maximum mechanical power is fed to the s ystem. As is further seen from the graphs, the impedance has its minimum at resonance and this minimum gets shallower the more damping is applied. Also the range in which the system behaves actively i.e., the impedance has a non negligible real part is wid ened. The opposite behavior is called reactive for which the impedance is essentially imaginary. Figure 2 7. Mechanical impedance for mass, spring and damper ( H. P. Olesen and R. B. Randall 1982)

PAGE 31

31 Normal Strength Concrete Concrete is a composite material composed of aggregate, most commonly sand and gravel, portland cement and water. The aggregate is usually graded in size from sand to gravel and in structural concrete the maximum aggregate size is most commonly in. but can be larger or smaller depending on the application. Compressive Strength and B ehavior Concrete is mainly a compression material. The use of reinforcing steel in Louis Lambot, a farmer in southern France, used iron bars and wire mesh to reinforce small rowboats he constructed. While concrete is made from elastic, brittle materials, its stress strain curve under compression is somewhat ductile. This behavior can be described by microcrack ing in the material that redistributes the stresses from element to element. As concrete strength increases, the slope of the descending branch of the compressive stress strain curve increases. This behavior is illustrated in Figure 2 8. For concrete st rengths up to 6,000 psi, the slope of the descending branch tends to be less than the ascending branch. Once the concrete strength reaches 10,000 psi, the slope of the descending branch is nearly vertical. This behavior can be explained by major longitud inal cracking of the concrete structure. It can also be seen for the stress strain relationship that the maximum concrete strain decreases with increasing strength.

PAGE 32

32 Figure 2 8. Typical stress strain curves for concrete in compression (Wight and MacGreg or 2012). Tensile Strength and B ehavior Tensile strength is very difficult to capture experimentally. In general, tensile strength is approximated as 10% of its compressive strength though this value is dependent on the type of tensile test, the compressive strength of the concrete and the pres ence of compressive stress transverse to the tensile stress (Wight and MacGregor 2005) There are two commonly used tests to determine tensile strength; the split cylinder test (ASTM C496) and the modulus of rupture or flexural test (ASTM C78 or C293).

PAGE 33

33 Figure 2 9. Typical stress strain curve for concrete (Hsu et al. 1987) Ultra High Performance Concrete (UHPC) Definition of UHPC Ultra high performance concrete (UHPC) has been the topic of much research in the last few decades. Researchers continue to study the material and generate the necessary information to maximize its usefulness in structural applications. UHPC is commonly characterized by increased strength, durability and ductility (Astarlioglu, Krauthammer, & Felice, 2010) In this section of the literature review, you will find a comprehensive background of the development and material characterization of UHPC followed by a brief overview of the current products developed as UHPC. Further, you will find a brief discussion on its applications in the field of protective structures.

PAGE 34

34 Ultra High Performance Concrete Composition and Mixing Methodology UHPC can be characterized as a reactive powder concrete (RPC). RPCs do not include coarse aggregates typically found in conventional concrete but use fine aggregates and pozzolanic powders to achieve a densely packed mixture. The main constituents of UHP C are cement, sand, silica fume, superplasticizer, water and fibers. Below you will find a short description of these constituents most commonly used in UHPC mix design. Further, the mixing and curing methodology currently employed is reviewed. Figure 2 10. Crack patterns in reinforced concrete elements subjected to tension (Brandt 2008). Cement The high strength compressive strength as well as the ductility and durability are attributed to the very low water to cement (w/c) ratios present in UHPC. T he w/c ratio of UHPC typically ranges from 0.17 0.21. This is very low when compared to conventional concrete w/c ratios of 0.30 0.45. There is general agreement in the

PAGE 35

35 literature that the cement should have a low alkali and tricalcium aluminate (C3A ) content (Richard and Cheyrezy 1995) Sand Quartz sand is typically used as the aggregate in UHPC with the mean particle size often being less than 1 mm although mixes with much higher particle sizes, between 8 and 16 mm, have been produced (Habel 2004). Silica F ume The use of silica fume in UHPCs has three functions. It increases the strength through dense particle packing and its reaction with calcium hydroxide (Habel 2004, Richard and Cheyrezy 1995). The silica fume with its small particle size fills v oids in the concrete limiting porosity. Due to the particles perfect sphericity the fluidity of the mix is improved (Richard and Cheyrezy 1995). Superplasticizer Superplasticizers are used for two main reasons in UHPC; to increase the workability of the fresh concrete for placing without disturbing the mix composition and to reduce the w/c ratio in order to increase the strength and improve durability at a given workability (Collep ardi 1998). The most common type of superplasticizers used are based on polycaroxyletes which can reduce the water requirement by up to 40% while maintaining flowability (Hirschi and Wombacher 2008, Lallemant Gamboa, et al. 2005). Fibers Metallic or synthetic fibers are used in UHPC mixes to increase the ductility of the otherwise brittle cementitious matrix. The fibers are randomly distributed and orientated in the cementitious matrix during mixing. The effect of percent volume of fiber s in the mix has been investigated by (Rong, Sun and Zhang 2010). Three volume percentages

PAGE 36

36 of 0%, 3% and 4% fiber content were selected to study the mechanical properties of UHPC. Figure 2 1 1 Bekaert Dramix ZP305 fibers (Williams et al. 2009). Tab le 2 2 shows the results of quasi static uniaxial compression tests performed on the three fiber contents. The compressive strength, peak value of strain and the toughness index of the specimens all increase with an increase of fiber content. Although th ese attributes increase with an increase in fiber content, a limit of roughly 5% by volume of fibers is the upper limit due to workability issues. Table 2 2 Quasi static uniaxial compressive performance of UHPCC at 60 d (Rong et al. 2010) Number Compressi ve Strength, MPa Peak Value of Strain, 10 3 Elastic Modulus, GPa Toughness Index c5 c10 c30 UHPCC(V 0 ) 143 2.817 54.7 2.43 2.43 2.43 UHPCC(V 3 ) 186 3.857 57.3 3.59 5.08 5.57 UHPCC(V 4 ) 204 4.165 57.9 4.57 6.32 7.39

PAGE 37

37 Mixing Methodology The procedure for preparing UHPC specimens are outlined below and are typical of most UHPC. First, the dry constituents, cement, sand, and silica fume and/or silica flour, are weighed and dry blended together in a mixer for 2 5 minutes. Water and superpl asticizer are mixed together and gradually added to the dry mix allowing it to become a wetted, flowable paste, 5 15 minutes. Steel fibers are then added to the mixer and allowed to mix for 3 10 minutes. Casting is recommended to be done within 20 minute s of mix completion and most commonly is done on a vibrating table or accomplished with vibrating rods (Graybeal 2005, Williams, et al. 2009) Figure 2 1 2 UHPC example mix proportion (Park et al. 2008). Curing The curing procedure can be quite drastic and intensive and varies considerably depending on the manufacturer. Williams et al. (2009) proposed the curing procedure for Cor demolded after 24 hrs. Then the specimens ar 13 days. Lafarge recommends its commercially available mix marketed under the name

PAGE 38

38 Ductal % relative humidity for 48 hours directly after demolding (Graybeal 2005) Others have investigated a less intensive curing regime with results of lower compressive strength. UHPC Mechanical P roperties Stress S train R elationship As previously stated, the most significant benefits of UHPC are its high compressive and tensile strengths when compared to normal strength concrete. Another important feature of UHPC is the added ductility of the material post peak stress is reached. Th is ductility plateau is made possible by the matrix of fibers slowly yielding. The compressive and tensile stress strain curves along with the demonstration of the added ductility are shown in Figure 2 13 which compares UHPFRC to NSC. Another observation is the gradual softening of the tensile stress strain curve for UHPFRC compared to the rather abrupt decrease in stress post peak of the NSC. Figure 2 13. Mechanical properties of NSC and fiber reinforced UHPC (Wu et al. 2009). The descending branch of the stress strain curve for UHPFRC depends on the variables related to the fibers present such as the fiber content, geometry, length in regards to the maximum aggregate size and orientation. The effect of fiber orientation

PAGE 39

39 has been explored by Fehling e t al. (2004). The influence of the fibers has a more dramatic effect on the tensile behavior of the concrete as illustrated in Figure 2 14 where tensile stress is plotted against crack opening. Of interest is the dramatic difference in the descending slo pe depending on fiber content, type and orientation. Figure 2 14. UHPFRC stress crack opening curve (Fehling et al. 2004). Fiber orientation and volume determine the descending branch for compression also but with less pronounced differences. Typical ly, a 10 15% gain in compressive strength can be achieved by the addition of fibers. Like the tensile stress strain relationship, the descending branch can vary depending on fiber content and orientation as depicted in Figure 2 15

PAGE 40

40 Figure 2 15. Effect of fibers on UHPFRC on compressive stress strain curve (Fehling et al. 2004). UHPC Rate and Size E ffect s As with normal strength concrete, UHPC has a definite increase in strength form the application of a dynamic load such as blast or impact. Smaller specimens tend to have increased strength as well when compared to similar specimens of larger dimensions. Millard et al. (2010) has concluded through testing that the enhancement of she ar strength, known as the dynamic increase factor, DIF, in push off specimens is fairly insignificant for strain rates less than 10 s 1 Dynamic shear testing was done using a drop hammer and push off specimens with geometry The push off specimens used a re very small when compared to other direct shear tests in the literature. Hopkinson bar tests are required to determine the dynamic increase factor for strain rates higher than those tested. Direct Shear Behavior Direct shear is classified as a sliding type failure along a well defined plane where slip occurs perpendicular to the axis of the member. Shear failure in structural

PAGE 41

41 members can occur suddenly and have catastrophic effects. This is especially true for critical sections, near supports for exa mple, in systems where reinforcement placement may be extremely difficult to place in reality. Failure in direct shear can happen under static loading conditions or during dynamic events such as blast loading. Direct shear or dynamic shear as it is also referred, can occur in structural systems where very high, short duration impulsive loads are generated (Slawson 1984) The failure behavior is essentially a vertical shear of the section that results in a ruptu re of the reinforcing bars crossing the shear plane. The direct shear failure response occurs when the member subjected to the load has no time to react in a flexural mode. It is usually occurs very early in the loading, roughly one millisecond, before a ny significant flexural response can occur. If the member does not fail in direct shear over the initial loading phase, a flexural mode of failure dominates (Slawson 1984) In contrast, diagonal or flexural shear is characterized by cracks that form at an angle to the horizontal plane and occurs where both shear and flexural stresses exist in the member. Diagonal shear will occur when the primary response of the system is flexure. Krauthammer et al. (1986) proposed the uncoupling of the direct shear resp onse from the flexural response and concluded that the shear mechanism in direct shear is similar in nature to shear transfer across an uncracked concrete interface as described by Park and Pauley (1975). This behavior has been reported in slabs subjected to severe and rapid loading by Kiger and Getchell (1980 1982) and Slawson (1984). While some slabs exhibited flexural failure, others failed in direct shear. Figure 2 16 shows a photograph of the roof slab of a RC buried box structure that failed due to d irect shear. The shear failure

PAGE 42

42 created a vertical failure plane at the supports. While the side walls appear practically undamaged, the roof slab sheared off completely at the side walls. The top and bottom reinforcement experienced necking before being severed nearly flush with the failure plane. Figure 2 16. Direct shear failure of a reinforced concrete slab (Slawson 1984). Original and Modified Hawkins Shear M odels Hawkins (1972) proposed a model for direct shear based on the shear stress slip rel ationship. The model describes the shear transfer of reinforced concrete members with well anchored main reinforcement without compressive forces in the static domain.

PAGE 43

43 Krauthammer et al. (1986) modified Hawkins model to account for compression and rate e ffects produced by severe dynamic loads. Hawkins original model can be found described in great detail by Murtha and Holland (1982). The modification proposed by Krauthammer et al. (1986) was employed by applying an enhancement factor of 1.4 to account f or the effects of compression and rate effects. The original Hawkins model and the modified model are represented in Figure 2 17 Figure 2 1 7. Original and modified shear stress slip relationship for direct shear (Krauthammer et al. 1986).. A direct shear failure occurs at the last point on the model, point E, where the slip max The difference between the Hawkins model and the enhanced or modified model by Krauthammer is a factor, K, designated to account fo r compressive effects and strain rate effects. The factor was determined to be equal to 1.4. This includes the dynamic effects stated above and modifies the original static model to be applicable in the dynamic domain. Below, the various line segments o f the model are described.

PAGE 44

44 Segment OA: The response in this region is elastic and the slope K e is defined e for a slip of 0.004 in. (0.1 mm). The resistance is given by the expression, (2 30 ) w e and f` c are in psi. The initial elastic limit should be no greater than m /2. Segment AB: The slope of the curve decreases continuously with increasing m is reached at a slip of 0.012 in. (0.3mm). Th m is given by the expression, (2 31 ) m c y vt is the ratio of total reinforcement area to the y is the yield strength of the reinforcement crossing the plane. Segment BC : The shear capacity remains constant with increasing slip. Point C corresponds to a slip of 0.024 in. (0.6 mm). Segment CD : The slope of the curve is negative, constant and indepen dent of the amount of reinforcement crossing the shear plane. The slope, in units of psi/in., is given by the expression, (2 3 2 ) Where A sb s is the tensile strength of the reinforcement, and A c is the cross sectional area. Segment DE : The capacity remains essentially constant until failure occurs at a max For a well anchored bar the slip at failure is given by the expression, :

PAGE 45

45 (2 33 ) Where, (2 34 ) and d b is the bar diameter in inches. L is given by the expression, (2 35 )

PAGE 46

46 Hofbeck et al. (1969) and Mattock and Hawkins (1972) Thirty eight push three were initially cracked along the shear plane. The push off specimens had a shear plane equal to 50 in. 2 (32,258 mm 2 ). The geometry is shown in Figure 2 1 8 Figure 2 1 8 Push off specimen (Mattock, Hofbeck and Ibrahim 1969). The main objectives of the study were to determine the influence of: a pre existing crack along the shear plane on the shear transfer strength, the reinforcement crossing the shear plane (strength, size and arrangement), and the concrete strength on the s hear transfer strength. Other objectives included examining dowel action of the theory presented by Birkeland and Birkeland (1966) and Mast (1968) on specimens that were initially cracked along the shear plane. Typical load slip curves are shown in Figure 2 19. The specimens that were initially uncracked generally have less slip than specimens with initially cracked shear

PAGE 47

47 planes. The ultimate shear stress generall y reduces with initially cracked specimens as y section of the stirrups crossing the shear plane, A s divided by the area of the shear plane, bd. It was concluded that strength. Figure 2 19. Typical load slip curves Mattock, ( Hofbeck and Ibrahim 1969) Two series, 2 and 5, were used to compare the effects of the concrete strength on the shear transfer strength. They were both initially cracked and were identical in every way except concrete strengths, Series 2 having 4,000 psi concrete and Series 5 havi ng 2,500 psi concrete. Figure 2 20 shows the results. They concluded that the concrete y below which the relationship u y is the same for concretes of strength equal to or greater t han that of the concrete being considered, and above which the shear transfer strength increases at a much reduced rate.

PAGE 48

48 Figure 2 20. Effect of concrete strength on shear strength in initially cracked specimens (Hofbeck et al. 1969). The different crack ing patterns of the uncracked and initially cracked specimens were determined to be the cause of the differing effects of dowel action. The uncracked specimens, where cracking happens via short diagonal cracks crossing the shear plane, as illustrated in Figure 2 21. Shear transfer in initially uncracked concrete (Mottock and Hawkins 1792). 21 create concrete struts that rotate causing the measured sli p. This puts the reinforcement into tension as a truss like action develops, instead of in a shearing action at the shear plane. Mattock believes this is why no dowel action is observed in the uncracked specimens. In the initially cracked specimens, the a shearing action on the reinforcement from the concrete on either side of the crack. This allows dowel action to be developed in the bars crossing the shear plane.

PAGE 49

49 Figure 2 21. Shear transfer in initially uncracked concrete (Mottock and Hawkins 1792).

PAGE 50

50 CHAPTER 3 RESEA R CH APPROACH AND METHODOLOGY Analytical investigation for test data was required to determine the frequency response of direct shear of NSC, C OR TUF1 and COR TUF2. Various software was used to make the process fast and more accurate. First, certain data ranges after impact were removed and meaningful data in time domain were extracted using DPlot and Excel. Then signal processing and fast Fourie r transforms were performed using MATLAB. Finally, the sequence frequency domain of the resistance functions along the entire slip was computed using Mathcad. The procedures were developed using previous research and analytica l investigations. Signal Processing This section will outline the technique that was used during signal processing. Data Acquisition 1. Sampling rate: 1 / 1e 6 sec = 1000 kHz 2. Number of data: 1.048561 x 10^6 3. Total time length: 1.05 sec 4. Laser resolution: 300kHz 5. Load cell frequency resolution: 1MHz

PAGE 51

51 Figure 3 1 Slip vs Time of CorTuf 2 with 0.8% shear reinforcement Aliasing The highest frequency that can be sampled in the data is half the Nyquist frequency. If the time series has high frequencies, the errors can be r educed by sampling at a faster rate (i.e., smaller t) or by pre filtering the data to remove high frequencies. If the data is sampled at a rate of at least twice the highest frequency in the signal, no high frequency information will be lost due to sampling. Pre filtering the analog data with an anti aliasing filter is a good practice because this will ensure there is no high frequency information in the sampled time series. Oversampling the data at a high rate while digitizing and then numerica ll y f iltering the data does not ensure that aliasing will be eliminated. Aliasing is the phenomenon by which frequencies greater than the Nyquist frequency are shifted erroneously to lower frequencies. The Nyquist frequency is c alculated with the following form ula: f Nyquist = sampling rate / 2

PAGE 52

52 Figure 3 2 Example of Aliasing Aliasing occurs when the data is generated or sampled. A liased components cannot be remove from the data without detailed knowledge of the original signal. In general, true frequency components cannot be distinguish ed in aliased frequency components. Therefore, accurate frequency measurements require adequate alias protection. Fortunately, DAQ at CIPPS has automatic aliasing protection in any acquisition. Additionally, sampling rate of impact tests was 1MHz which is long enough to eliminate aliasing. Resolution and Confidence There is a trade off in spectral estimates between resolution and co the spacing of the Fourier frequencies is very c l ose, then the resolution in frequency will be high. Since this can be accomplished by taking a longer length of data. However, because the frequencies are c lo sely spaced, there may be energy a t one frequency, none at the next, etc. This leads to an estimate of the spe ctru m that appears

PAGE 53

53 very rough or jumpy. For this case, the resolution is high but the confidence is low. Confidence is an indicator of how well the spec tr al estimate at a given fre quency agrees with the actual value. Usually the actual value is in a band or confidence interval somewhere around the estimated value. The confidence can be increased by having more widely spaced frequencies so that each Fourier component represents an av erage over a range. This requires T 0 to be smaller. Thus, resolution and confidence are at odds. One way to improve results is to pre filter the data with a low pass filter. The filter is chosen to optimize results. This generally requires some type of insight into the physical system to ascertain what ranges of frequencies are meaningful. One alternative is to take a long record length, but then analyze the data as a number of shorter segments. In this way, several independent estimates of the spectral coefficients are provided at each Fourier frequency. The mean, variance, and confidence may be deter m ined at each frequency. The deter m ination of the record length and number of segments again depend on the physical system. For a SDOF system with light dam ping, the length of the data record for estimating a spectrum is approximately (3 1) w here T 0 record length f = undamped natural fr equency = damping factor, and The required number of segments is approximately (3 2)

PAGE 54

54 The segments can also overlap with each other. Overlapping the segment means they are no longer independent. However, it does increase the number of estimates at each frequency. Optimum results occur at about a 50% overlap. Figure illustrates a 50% overlap with the preceding segments and a 50% overlap with the following seg m ents. For this case the length of each segment T s is det er m ined by (3 3) I n this research, the resolution in frequency could be determined by how many zero padding data exists. Since the number of sample points at every 200us was different based on duration of slip, the same power of 2 was used so that the frequency response can be easily compared with others. Additionally, exact damping ratio of the material could not be computed so, the only option to enhance resolution and confidence was to take a long record length. For load time history, 2^16 of record length was used and 2 ^8 for slip time history was used and t his was found adequate to express frequency response in detail. Filtering In signal processing a filter is a device or process that removes from a signal some unwanted component or feature. Filtering is a class of signal processing, the defining feature of filters being the complete or partial sup pression of some aspect of the signal. Most often, this means removing some frequencies and not others in order to suppress interfering signals and reduce background noise Correlations can be removed for certain frequency components and not for others without having to act in the fr equency domain .The drawback of filtering is the loss of information associated with

PAGE 55

55 it In this research, impact peak load is an important value. If the filtered data has a reduced peak load in comparison with original one, the filtered frequency might not be noise, but has important information. Since all of impact tests data was digital, di gital filters were developed by using MATLAB. Digital filters operate on signals represented in digital form. The essence of a digital filter is that it directly implements a mathematical algorithm, corresponding to the desired filter transfer function, in its programming or microcode. Load vs Time First, the load and time vectors were imported to MATLAB and the noise section at the end or beginning of the time axis was extracted. Next, a real valued time domain FFT of the data supplied was performed by pad ding the record with zeros to the closest next power of 2 of the length used for generating the FFT of zero mean data to determine the noise in data acquired from actual tests. Third, the frequency that contributes to the noise was determined and eliminate d. Then the filtered data is converted to the time domain. Finally, the process is repeated until the peak value is quite different when c ompared to the original one

PAGE 56

56 Figure 3 3 Digital filter code in MATLAB Figure 3 4 Load vs T ime after filter processing Slip vs Time First, a file that is filtered with the above filter code is imported to DPlot and the time axis between starting and ending slip is truncated. By zooming in on the U(t) plot, stair shapes became visible. A time point at which the voltage change starts is

PAGE 57

57 determined. Next, one point on the middle of the step in U(t) is extracted at every 200 s. Then, connected points were differentiated to get velocity V(t) and accelerat ion A(t) time history and supply zero padding to the same power of 2. Finally, U(t) V(t) A(t) were transformed into the frequency domain by FFT. Figure 3 5 Load, Slip, Velocity and Acceleration in time and frequency domain Natural Frequency Range First, the mass of shear block was determined. The actual weight of the block was measured with an electronic scale. Only half of the block was considered as mass M, because the support is composed of a steel plate on the top of concrete during impact test s which means it is rigid. So, the bottom of the block can also be treated as a part of the support.

PAGE 58

58 Figure 3 6 Actual weight of Cor Tuf1 with 0% shear reinforcement Table 5 1 Actual Weight of Specimen Specimen Reinforcement Weight (lbf) CT1S 0% 162.8 CT1S 0.8% 156 CT1S 1.6% 160 CT2S 0% 149.3 CT2S 0.8% 143.8 CT2S 1.6% 147.8 NSC 0% 142 NSC 0.8% 137.4 NSC 1.6% 138.2 Next, the sequence frequency of the resistance function was computed. During the impact tests, the natural frequency can be varied along the slip.

PAGE 59

59 Figure 3 7 Static and Dynamic modified Hawkins direct shear model The e lastic limit of displacement in dynamic model is significantly smaller than s tatic one. This indicates that the natural frequency during impact testing can be varied significantly faster than the static case. Figure 3 8 Sequence frequency of the modified Hawkins model

PAGE 60

60 The higher slope in the dynamic model caused higher sequence frequency and maximum in dynamic is approximately 20000Hz which exists during only elastic range. On the other hand, the maximum in the static model was about 4500Hz. Table 3 1. NSC, CT2 and CT1 Dynamic Modified Hawkins direct shear models. NSC CT2 CT1 K c K u 2 3 4 max

PAGE 61

61 Table 3 2 NSC, CT2 and CT1 Modified Hawkins direct shear models. Original Hawkins NSC CT2 CT1 1 K e NA NA K c K u 1 0.004 2 0.012 3 0.024 4 max Units are psi, in and psi/in 1 The value for should not be taken larger then

PAGE 62

62 Limitation A DAC (digital to analog converter) was used to output a reference voltage and that is how the slip distances were measured by the lasers. The issue was that the DAC 200uS. Between these points, an unchanging voltage could be seen. So, an accurate time history of the rapidly increasing slip was not recorded, as seen in Figure 3 9. The recorded slip increased in steps due to the voltage change. Since the laser resolution was 300kHz, the noise frequenc y of the laser could not be captured. Figure 3 9 Example of voltage change in U(t) This could lead to lower sampling frequency to the inverse of 200uS which is 5000Hz. Also, based on the Nyquist principle, the absolute of FFT is symmetric, so only the half of the frequency 2500Hz could be used. quasi static domains. The load time history and the slip time history can be used to

PAGE 63

63 determine which response regime can be applied to the test, as shown in Figure 3 10. This al lows us to determine which direct shear model should be used between the dynamic and static cases. To be specific, if the response is in dynamic regime, both of the static and dynamic direct shear models would be needed to accurately predict the response. shear behavior given a certain loading condition. Figure 3 10 Typical response domains (a) impulsive, (b) quasi static and (c) dynamics (T. Krauthammer et al. 2008) As previously discussed, sequence frequency range in the dynamic Hawkins model is approximately 20000Hz to 900Hz and the static case is about from 4500Hz to 600Hz. Since the sampling frequency is limited to 2500Hz, the spike frequencies in a response funct ion could not provide any information on the dynamic direct shear model. For example, Figure 3 11 shows that spike frequencies of V(w) are distributed on the static and dynamic direct shear models. They are concentrated in the similar regime due to the lim itation of frequency range in V(w). In this case, any information between two plots could not be captured.

PAGE 64

64 Figure 3 11 Example of frequency distribution on direct shear model

PAGE 65

65 CHAPTER 4 RESULTS AND DISCUSSION Generally a t low frequencies, the system is governed by the spring stiffness the reaction force is that produced by the spring. For high frequencies, on the other hand, the mass governs the system and the reaction force is the inertia of the mass The analysis results of U(w) and A( w) followed the above principle, so mechanical impedance V(w) / P(w) could be an index to track the fundamental frequency domain. To illustrate, if the mechanical impedance is relatively low at a certain frequency, resonance would occur which means that th e frequency could be treated as the natural frequency in SDOF system In this research P(w) is relatively low at high frequency during impact testing. If the V(w) is high at the corresponding frequency domain, that should be the fundamental frequency do main that is needed to be found. Additionally, A(w) is quite similar with V(w) so it was difficult to capture valuable information from the plot. Therefore, the discussion below is mainly focused on graphical characteristics of V(w). During the analysis, s pikes in V(w) were only considered beyond 500Hz because all of P(w) have high spikes ranging from 0Hz to approximately 500Hz NC Specimens NS C 1A 0 D First, the Fourier transform of velocity time history V(w) exhibits several spikes in the high frequency domain. This indicates that concrete alone does not contribute to any shear strength once it starts to crack. The concrete absorbed the impact energy through cracking Concrete cracking is visible in U(t) as several sawtooth shapes. U(t)

PAGE 66

66 also shows that the total slip duration is relatively short in comparison with 0.6% NSC which indicates that it has short periods which lead to high frequencies in V(w) Deep sawtooth shapes indicate the rate of slip change and it means an increase in amp litude in velocity time history V(t) which are displayed as sharp spikes in the time history. A B Figure 4 1 NCS specimens after dynamic testing. A) NSC 1A 0 D 1 B) NSC 1A 0 D 2 (Robin French. Gainesville. FL: Powell laboratory, 2013.) N S C 1 1 D The largest difference in NSC 1% is the presence of many spikes in the velocity time history V(t) This indicates rapid changes of slip occur frequently. Resistance of the shear reinforcement after concrete cracking during failure results in large sawtooth shapes in the U(t) When compared to V(w) of 0% NSC, more spikes are distributed in

PAGE 67

67 the low frequency domain. This is attributed to ductility of the reinforcement. Figure 4 2 l resisted the impact in the plastic range until the gap is fully closed. Moreover, the load time history P(t) shows that it has a relatively long load duration in comparison with that of 0% NSC and some spikes exists. This could lead to relatively long duration of the load indicating more ductile behavior. A B Figure 4 2 NCS specimens after dynamic testing. A) NSC 1 1 D 1 B) NSC 1 1 D 2 (Robin French. Gainesville. FL: Powell laboratory, 2013.) N S C 1 2 D Three NSC 1.6% specimens were tested, none of which failed on the shear plane, even though the blocks were placed a half inch off of center. Figure 4 19 shows a

PAGE 68

68 failure in the top cantilever of NC specimen with 1.6% shear reinforcement and shows flexural cracks tha t formed at the bottom corner. Moreover, P(t) shows graphically that once impact load reaches peak and decrease during 0.02s which is relative long duration than others because the over reinforced specimen resists impact until the hammer brake release. U(t ) has also jagged shape elements during 0.02s and this shape indicates flexural cracks during the tests. Due to the long time duration of slip, V(t) is also relatively smoothed shape which means low spikes values in V(w) Since there are no fibers, spallin g occurred during impact loading. As seen in Figure 4 3 there are no cracks on the shear plane. A B Figure 4 3 NCS specimens after dynamic testing. A) NSC 1 2 D 1 B) NSC 1 2 D 2 (Robin French. Gainesville. FL: Powell laboratory, 2013.)

PAGE 69

69 COR TUF1 (CT1) Specimens CT1S 1A 0 D For the 0% CT1, U(t) has stair shape elements which have no specific interval. This lack of consistent intervals could be attributed to steel fibers which are randomly distributed across the shear plain. This indicates that the steel fibers are effective from the moment of impact to full slip. Another characteristic of the 0% is that the slip takes only 0.01s to reach its maximum in U(t) This indicates that time slope of slip is very high, and it leads to high amplitude in V(t) which finally results in strong spikes in V(w) Furthermore, j agged shapes are randomly distributed along the entire duration in U(t) It would indicate pullout of the steel fibers because there are no other contributions of resistance during fu ll slip such as major concrete spalling or mild st eel as shown in Figure 4 4 .Several spikes in P(t) also indicate fiber pullout as well. Overall the ductility of steel fibers extends the effective frequency domain which means the main resistance would occurs beyond the concrete elastic limit

PAGE 70

70 A B Figure 4 4 CT1 specimens after dynamic testing. A) CT1S 1A 0 D 1 B) CT1S 1A 0 D 2 (Robin French. Gainesville. FL: Powell laboratory, 2013 ) CT1S 1 1 D First, U(t) has many and more deep sawtooth shapes than that of 0% CT1. This is caused by addition of resistance of shear reinforcement crossing the shear plane. Mild steel could increase the ductility of the specimen, which results in a relatively long time duration of slip. Ultimately, if the tim e rate of displacement is low, it could lead to low amplitudes in V(t) To be specific, the maximum velocity in V(t) is approximately 255in/sec, which is significantly lower than that of the 0%. This indicates relatively small spikes in V(w) when compared to 0%. The only physical difference between 0% and 0.8%

PAGE 71

71 CT1 specimens is the addition of shear reinforcement. This could explain the fact that the ductility of shear reinforcement could significantly reduce spikes values in V(w) A B Figure 4 5 CT1 specimens after dynamic testing. A) CT1S 1 1 D 1 B) CT1S 1 1 D 2 (Robin French. Gainesville. FL: Powell laboratory, 2013 ) CT1S 1 2 D First, U(t) has the similar sawtooth shapes with 0.8% and 0% of CT1, but has a shorter time duration to reach maximum. Tha t means the rate of displacement is relatively small, it could lead to lower amplitudes in V(t) than others. Finally, it can lower the spike value in V(w) This could be attributed to increased ductility caused by the addition of shear reinforcement. Also, D ifferent periods of sawtooth in U(t) would result in the extension of effective frequency domain. Figure 4 6

PAGE 72

72 effective until full slip. A B Figure 4 6 CT1 specimens after dynamic testing. A) CT1S 1 2 D 3 B) CT1S 1 2 D 4 (Robin French. Gainesville. FL: Powell laboratory, 2013 ) COR TUF 2 (CT 2 ) Specimens CT2S 1A 0 D T he slip increased rapidly with several sawtooth shaped elements in U(t) More deeply jagged shape in U(t) has shorter period but with larger amplitude. U(t) has many sawtooth, it could lead to high spikes in V(w) due to the superposition. To illustrate physically, the block has no shear reinforcement. Therefore, once conc rete crack occurs,

PAGE 73

73 most impact energy would be absorbed through spalling concrete until the gap is closed This can be verified with P(t) which shows that it has only one peak in comparison with other specimen s of CT2. If the specimen has ductility properti es, P(t) could display some spikes as seen in P(t) of CT2 0.8% or CT2 1.6%. A B Figure 4 7 CT2 specimens after dynamic testing. A) CT1S 1A 0 D 2 B) CT2S 1A 0 D 3 (Robin French. Gainesville. FL: Powell laboratory, 2013 ) CT2S 1 1 D First, there are randomly distributed sawtooth shapes in U(t) The presence of these shapes indicates which material is resisting the direct shear, and it could build many amplitudes with various periods in V(t) Ultimately these amplitudes are superposed with the same periods and shows a value on y axis along the entire

PAGE 74

74 frequency domain of V(w) That is why it has more strong spikes than 0% CT2. For CT2, the sawtooth indicates spalling concrete. Compared to CT1, CT2 cannot transfer impact energy to the shear plain totally. The addition of mild steel could not fully enhance the concrete strength because CT2 is so brittle. Therefore, shear reinforcement of CT2 would be one of the disadvantages with aspect of dynami c stabilization. Moreover, F igure 4 8 shows rotated shear reinforcement after failure. This indicates that the mild steel was ductile during full slip. As mentioned previously about CT1, ductility can extend fundamental frequency domain from high to low. T hat would correspond to the elastic and plastic range in the direct shear model. A B Figure 4 8 CT2 specimens after dynamic testing. A) CT2S 1 1 D 2 B) CT2S 1 1 D 3 (Robin French. Gainesville. FL: Powell laboratory, 2013 )

PAGE 75

75 CT2S 1 2 D The most characteristic is the amplitude in V(t) It has many which are higher than 1%. This is also because the increase in shear reinforcement making the concrete spalling smaller. Resistance of shear reinforcement can also appear in P(t). In contrast to 0% and 0.8%, the second pulse of P(t) is very large. This could indicate the moment of mild steel failures at almost simultaneously. The addition of shear reinforcement would increase brittle characteristic only for CT2. Furthermore, it can be seen as man y jagged shape elements in U(t) Considering the superposition properties of V(w) this physical results could directly reflect to the shape of spike distribution of V(w) A B Figure 4 9 CT2 specimens after dynamic testing. A) CT2S 1 2 D 1 B) CT2S 1 2 D 2 (Robin French. Gainesville. FL: Powell laboratory, 2013 )

PAGE 76

76 Comparison of 0% Specimens First, V(w) of CT 1 has significant higher values than the other couple V( w ) of CT2 and NSC. As previously discussed, it is attributed to the very fast time duration of full slip. The addition of steel fibers makes the rate of displacement change very high. So this could reflect to V(t) as high amplitudes. Also, steel fibers on th e shear plane resist the impact without any regular periods and extend the effective frequency domain of V(w) On the other hand, material without ductility such as CT2 and NSC has spikes in only high frequency range. Finally CT2 has more extended spikes r age than NSC This is caused by more spalling, not ductility in contrast with CT1 Comparison of 0.8% Specimens Steel fibers make distributed spikes in V(w) more dense than others. It allows the plot to show specific dense spectrum more clearly. On the ot her hand, CT2 has more striking individual spikes because of spalling concrete. Comparison of 1.6% Specimens The main difference between the three types of model is amplitude in V(t) CT2 has significantly higher amplitude of V(t) when compared to the other materials. This directly i ndicates spike values in V(w) CT2 is over reinforced so impact energy This concrete spalling could reflect as stro ng spikes in V(w) On the other hand, CT1 has clearly failed on shear plane as shown F igure 4 16 This means steel fibers could achieve reinforcement balance. Ultimately this pure direct shear failure could lead to lower spikes than CT2 in V(w)

PAGE 77

77 Figure 4 10 Time and frequency responses for NSC 1A 0 D 1

PAGE 78

78 Figure 4 1 1 Time and frequency responses for NSC 1A 0 D 2

PAGE 79

79 Figure 4 1 2 Time and frequency responses for NSC 1 1 D 1

PAGE 80

80 Figure 4 1 3 Time and frequency responses for NSC 1 2 D 1

PAGE 81

81 Figure 4 1 4 Time and frequency responses for NSC 1 2 D 2

PAGE 82

82 Figure 4 1 5 Time and frequency responses for CT1S 1A 0 D 1

PAGE 83

83 Figure 4 16 Time and frequency responses for CT1S 1A 0 D 2

PAGE 84

84 Figure 4 17 Time and frequency responses for CT1S 1 1 D 1

PAGE 85

85 Figure 4 18 Time and frequency responses for CT1S 1 1 D 2

PAGE 86

86 Figure 4 19 Time and frequency responses for CT1S 1 2 D 3

PAGE 87

87 Figure 4 20 Time and frequency responses for CT2S 1A 0 D 2

PAGE 88

88 Figure 4 21 Time and frequency responses for CT2S 1A 0 D 3

PAGE 89

89 Figure 4 22 Time and frequency responses for CT2S 1 1 D 2

PAGE 90

90 Figure 4 23 Time and frequency responses for CT1S 1 1 D 3

PAGE 91

91 Figure 4 24 Time and frequency responses for CT2S 1 2 D 1

PAGE 92

92 Table 4 1. Summary of spike distribution Specimen Number of Spikes Peak Spike Location of Peak Spike (in/sec*sec) (Hz) NSC 1A 0 D 1 2 1 1900 NSC 1A 0 D 2 3 1.5 1800 NCS 1 1 D 1 19 2.7 1600 NCS 1 2 D 1 12 1.2 1100 NCS 1 2 D 2 14 1.2 1900 CT2 1A 0 D 2 9 3.7 2200 CT2 1A 0 D 3 5 2.9 2200 CT2 1 1 D 2 14 4 1650 CT2 1 1 D 3 18 3.3 1600 CT2 1 2 D 1 10 4.5 2000 CT1 1A 0 D 1 8 5.5 1400 CT1 1A 0 D 2 10 6.8 1400 CT1 1 1 D 1 5 3.2 1300 CT1 1 1 D 2 7 3.9 1000 CT1 1 2 D 4 6 2 1200

PAGE 93

93 Load vs Time Load time history was measured with a high sampling frequency. So, if there are spikes or something in high frequency domain beyond 2500Hz, it might be valuable information that needs to be researched. Since the dynamic modified shear model has sequence fr equency range from 20,000Hz to 500Hz. This means the material could exhibit the dynamic properties in the high frequency domain. The following figures are P(w) of full frequency domain which is Nyquist frequency of sampling Figure 5 1. FFT of Load vs T ime in Nyquist f requency d omain for CT1S 1 1 D 1

PAGE 94

94 Figure 5 2. FFT of Load vs Time in Nyquist f requency d omain for CT2S 1 1 D 1 Figure 5 3. FFT of Load vs Time in Nyquist f requency d omain for NSC 1 1 D 1

PAGE 95

95 CT1 has more spikes in the high frequency r ange than others. The fact that the peak load and duration of CT1 are higher than others and it has residual load might be the reason. Next, CT2 has also spikes in high frequency domain but its amount is less than CT1. These spikes might be able to represe nt brittle or any features of CT2. Finally, NSC has relatively small amount of spikes in high frequency than others. Even though P(w) has some spikes in the high frequency, any conclusion could not be made since V(w) has no information beyond 2500Hz.

PAGE 96

96 CHAPTER 5 CONCLUSIONS AND RECOMMENDATIONS Conclusions Due to the limitation of the sampling rate of slip time history, a limited frequency domain ranging from 0Hz to 2500Hz was analyzed. However, each individual plot has its own characteristics that could be visibly distinguished. Comparing the frequency response could partially explain material properties under impact loading. First, t he s hear the reinforcement could help to achieve dynamic stabilization only if sufficient ductility of concrete is guaranteed. Second, the s hear reinforcement could induce concrete spalling for C or T uf 2, and resistance through concrete spalling could exhibit increased spike s in V(w) Thus, s hear reinforcement in brittle concrete such as Cor Tuf2 has more possibilit y to cause resonance than other materials like Cor Tuf1 and NSC. Third, s teel fibers could move the location of peak spike in V(w) from high to low frequency. L ow frequency domain indicates residual capacity regime in re sistance function. Considering the addition of steel fibers could cause residual strength, increased ductility caused by addition of steel fibers could allow mild steels to reach full strength during slip and ultimately move the fundamental frequency domain from high to low. In other words, the addition of steel fibers causes resonance under an excitation with a long period Additionally most spikes in V(w) are distributed in the high frequency domain for all of 0% specimens. This means that the concrete without shear reinforcement is britt le, so this kind of concrete should be carefully designed under an excitation with high frequency

PAGE 97

97 Recommendations for Future Research F or future research, the voltage change increment should be shortened and laser resolution should be adjusted in order to obtain higher frequency domain. Furthermore, the trigger time of high speed camera should be adjusted to match other equipment so that a very short moment at certain displacement or time can be captured. It will be helpful to interpret the phe nomenon and identify relationship with other test results. These recommendations can build upon the base of knowledge for the direct shear behavior of NSC and UHPC

PAGE 98

98 APPENDIX A MATLAB CODES Filter ing Code function [ f,a, peak, fLoad ] = NFFTPvT( Y_vec, Time_vec) %NFFT Performs 2^n FFT of data supplied. % Performs a real valued time domain FFT of the data supplied by padding the record % with zeros to the closest next power of 2 of the length. Used for % generating FFT of zero mean data to figure out the noise in data % acquired from actual tests. % % Returns f = frequency, a = amplitude, peak = frequencies selected, % fLoad = filtered load data % % Dependencies: BandStop.m, tightfig.m %Set Frequency Fs=round(1/(Time_vec (end) Time_vec(end 1))); %round because of numerical issues interval = (Time_vec(end) Time_vec(end 1)); dataloc=questdlg( 'Get baseline data from beginning or end of data?' ... 'Baseline data Location' ... 'Beginning' 'End' 'End' ); if strcmp( dataloc, 'End' ) timebeforestart = input( ... 'Enter how many ms before the end of the test to start new data [default 250ms]? ); if isempty(timebeforestart) || isnan(timebeforestart) timebeforestart = 250; end %Set zero mean are a to FFT newTime = Time_vec(length(Time_vec) Fs*timebeforestart/1000:end); newY = Y_vec(length(Time_vec) Fs*timebeforestart/1000:end); else timeend = input( 'How many ms after the start of the file [default 40ms]?' ); if isempty(timeend) || isnan(timeend) timeend = 40; end newTime = Time_vec(1:Fs*timeend/1000); newY = Y_vec(1:Fs*timeend/1000); end maxY = max(abs(newY)); L = length(newTime); NFFT = 2^nextpow2(L); Y = fft(newY,NFFT)/L; f = Fs/2*linspace( 0,1,NFFT/2+1);

PAGE 99

99 a = 2*abs(Y(1:NFFT/2+1)); %plot the summary figure summary_figure = figure( 'Name' 'Summary Chart' ); %Data Supplied subplot(2,6,[1,2]); orig_data = plot(Time_vec, Y_vec); xlim([Time_vec(1) Time_vec(end)]); sub_data = rectangle( 'Position' [newTime(1)+interval 2*maxY newTime(end) newTime(1) interval 4*maxY]); set(sub_data, 'EdgeColor' [0.2 0.8 0.8]); set(sub_data, 'FaceColor' [0.2 0.8 0.8]); set(gca, 'Children' [orig_data sub_data]); title( 'Original Data' ); %Truncated Data subp lot(2,6,[7,8]); plot(newTime,newY); title( 'Truncated Data' ); xlim([newTime(1) newTime(end)]); %FFTs for 1 2000 Hz in 500 Hz Chunks subplot(2,6,[3 4 5 6]); plot(f,a); axis([ 50 500 0 30]); title( '0 500 Hz' ); %FFT from 0 Fs/2 Hz subplot( 2,6,[9,10,11,12]); plot(f,a); axis([ 100 500000 0 10]); str = sprintf( '0 %i Hz, Truncated Data' ,Fs/2); title(str); set(gcf, 'Position' ,get(0, 'Screensize' )); %maximize figure tightfig; %and minimize whitespace %now go into datacursor mode dcm_obj = datac ursormode(summary_figure); set(dcm_obj, 'DisplayStyle' 'datatip' 'Enable' 'on' ) uiwait(helpdlg( 'Select Peaks from frequency charts.' 'Peak Selection' )); freq_peaks = getCursorInfo(dcm_obj); peak = zeros(length(freq_peaks),1); for i = 1:length(freq_peaks ), peak(i)=round(freq_peaks(i).Position(1)); fprintf( 'Frequency selected at %i Hz. \ n' ,peak(i)); end set(dcm_obj, 'Enable' 'off' ); clean_data = questdlg( 'Do you want to filter the Original data at +/ 10 Hz at the points selected?' ); clear i if strcmp(clean_data, 'Yes' ) loadfilter = BandStop(peak(1) 20,peak(1) 10,peak(1)+10,peak(1)+20,Fs); fLoad=filter(loadfilter,Y_vec); for i = 2:length(peak), if (peak(i)>0) && (peak(i)
PAGE 100

100 loadfilter=BandStop(peak(i) 20,pe ak(i) 10,peak(i)+10,peak(i)+20,Fs); fLoad = filter(loadfilter,fLoad); end end else fLoad = Y_vec; end ; filtered_figure = figure( 'Name' 'Filtered and Raw Data' ); plot(Time_vec,Y_vec, ' ,Time_vec,fLoad, 'r ); legend( 'Unfiltered data' 'Filtered Data' ); end function hfig = tightfig(hfig) % tightfig: Alters a figure so that it has the minimum size necessary to % enclose all axes in the figure without excess space around them. % % Note that tightfig will expand the figure to completely encompass all % axes if necessary. If any 3D axes are present which have been zoomed, % tightfig will produce an error, as these cannot easily be dealt with. % % hfig handle to figure, if not supplied, the current figure will b e used % instead. if nargin == 0 hfig = gcf; end % There can be an issue with tightfig when the user has been modifying % the contnts manually, the code below is an attempt to resolve this, % but it has not yet been satisfactorily fixed % origwindowstyle = get(hfig, 'WindowStyle'); set(hfig, 'WindowStyle' 'normal' ); % 1 point is 0.3528 mm for future use % get all the axes handles note this will also fetch legends and % colorbars as well hax = findall(hfig, 'type' 'axes' ); % get the original axes units, so we can change and reset these again % later origaxunits = get(hax, 'Units' ); % change the axes units to cm set(hax, 'Units' 'centimeters' ); % get various position parameters of the axes if numel(hax) > 1 % fsize = cell2mat(get(hax, 'FontSize'));

PAGE 101

101 ti = cell2mat(get(hax, 'TightInset' )); pos = cell2mat(get(hax, 'Position' )); else % fsize = get(hax, 'FontSize' ); ti = get(hax, 'TightInset' ); pos = get(hax, 'Position' ); end % ensure very tiny border so outer box always appears ti(ti < 0.1) = 0.15; % we will check if any 3d axes are zoomed, to do this we will check if % they are not being viewed in any of the 2d directions views2d = [0,90; 0,0; 90,0]; for i = 1:numel(hax) set(hax(i), 'LooseInset' ti(i,:)); % set(hax(i), 'LooseInset', [0,0,0,0]); % get the current viewing angle of the axes [az,el] = view(hax(i)); % determine if the axes are zoomed iszoomed = strcmp(get(hax(i), 'CameraViewAngleMode' ), 'manual' ); % test if we are viewing in 2d mode or a 3d view is2d = all(bsxfun(@eq, [az,el], views2d), 2); if iszoomed && ~any(is2d) error( 'TIGHTFIG:haszoomed3d' 'Cannot make figures containing zoomed 3D axes tight.' ) end end % we will move all the axes down and to the left by the amount % necessary to just show the bottom and leftmost axes and labels etc. moveleft = min(pos(:,1) ti(:,1)); movedown = min(pos(:,2) ti(:,2)); % we will also alter the height and width of the figure to just % encompass the topmost and rightmost axes and lables figwidth = max(pos(:,1) + pos(:,3) + ti(:,3) moveleft); figheight = max(pos(:,2) + pos(:,4) + ti(:,4) movedown); % move all the axes for i = 1:numel(hax) set(hax(i), 'Position' [pos(i,1:2) [moveleft,movedown], pos(i,3:4)]); end

PAGE 102

102 origfigunits = get(hfig, 'Units' ); set(hfig, 'Units' 'centimeters' ); % change the size of the figure figpos = get(hfig, 'Position' ); set(hfig, 'Position' [figpos(1), figpos(2), figwidth, figheight]); % change the size of the paper set(hfig, 'PaperUnits' 'centimeters' ); set(hfig, 'PaperSize' [figwidth, figheight]); set(hfig 'PaperPositionMode' 'manual' ); set(hfig, 'PaperPosition' ,[0 0 figwidth figheight]); % reset to original units for axes and figure if ~iscell(origaxunits) origaxunits = {origaxunits}; end for i = 1:numel(hax) set(hax(i), 'Units' origaxunits{i}); end set(hfig, 'Units' origfigunits); % set(hfig, 'WindowStyle', origwindowstyle); end function Hd = BandStop(Pass1,Stop1,Stop2,Pass2,Frequency) %BANDSTOP Returns a discrete time filter object. % % MATLAB Code % Generated by MATLAB(R) 7.13 and the Signal Processing Toolbox 6.16. % % Generated on: 23 Oct 2013 09:02:06 % % Butterworth Bandstop filter designed using FDESIGN.BANDSTOP. % All frequency values are in Hz. Fs = Frequency; % Sampling Frequency Fpass1 = Pass1; % First Passband Frequency Fstop1 = Stop1; % First Stopband Frequency Fstop2 = Stop2; % Second Stopband Frequency Fpass2 = Pass2; % Second Passband Frequency Apass1 = 0. 5; % First Passband Ripple (dB) Astop = 60; % Stopband Attenuation (dB) Apass2 = 1; % Second Passband Ripple (dB)

PAGE 103

103 match = 'stopband' ; % Band to match exactly % Construct an FDESIGN object and call its BUTTER method. h = fdesign.bandstop(Fpass1, Fstop1, Fstop2, Fpass2, Apass1, Astop, ... Apass2, Fs); Hd = design(h, 'butter' 'MatchExactly' match); % [EOF] FFT Code clc; clear % Load vs Time % figure( 'units' 'normalized' 'outerposition' ,[0 0 1 1]); load( 'CT1S 1 1 D 2(P,DvT).mat' ); startload=157577; endload=185590; P=Load(startload:endload,:); t5=Time(startload:endload,:); N2=2^16; P1 = zeros(N2,1); [N3,nd] = size(P); P1(1:N3)=P(1:N3); % P2= smooth(P,50); subplot(2,4,1) plot(t5,P2) title( 'CT1S 1 1 D 2 P(t)' ) xlabel( 'Time (sec)' ) ylabel( 'Load (kip)' ) % FFT of Load vs Time dt2=0.000001; tmax2 = dt2*(N2 1); t2=0:dt2:tmax2; df2= 1/tmax2; fmax2 = 1/dt2; f2 = 0:df2:fmax2; % Pw = (1/N2)*fft(P1); % FFT of Load vs Time % subplot(2,4,5) plot(f2',abs(Pw)) title( 'Load(w)' ) xlabel( 'Frequency (Hz)' ) ylabel( 'Amplitude (load*sec)' ) axis([0 2500 0 10]) % startslip=159098+100; endslip=178698+100; lengthplot=(endslip startslip)/200; U2=zeros(lengthplot,1); t1=zeros(lengthplot,1);

PAGE 104

104 for i=1:lengthplot; U2(i)=Slip(startslip+200*i,1); t1(i)=Time(startslip+200*i,1); end U=U2; dt1= 0.000001*200; subplot(2,4,2) plot(t1,U) title( 'Slip vs Time' ) xlabel( 'Time (sec)' ) ylabel( 'Slip (in)' ) % % Differentiate % V=zeros(length(U),1); for i = 2:length(U); V(i)=(U(i) U(i 1))/(t1(i ) t1(i 1)); end subplot(2,4,3) plot(t1,V) title( 'Velocity vs Time' ) xlabel( 'Time (sec)' ) ylabel( 'Velocity (in/sec)' ) A=zeros(length(U),1); for i = 2:length(U); A(i)=(V(i) V(i 1))/(t1(i) t1(i 1)); end subplot( 2,4,4) plot(t1,A) title( 'Acceleration vs Time' ) xlabel( 'Time (sec)' ) ylabel( 'Acceleration (in/sec^2)' ) % % FFT of Displacement vs Time % N=2^8; U1 = zeros(N,1); [N1,nd] = size(U); U1(1:N1)=U(1:N1); % tmax = dt1*(N 1); t=0:dt1:tmax; df= 1/tmax; fmax = 1/dt1; f = 0:df:fmax; % Uw = (1/N)*fft(U1); subplot(2,4,6) plot(f',abs(Uw)) title( 'Slip(w)' ) xlabel( 'Frequency (Hz)' ) ylabel( 'Amplitude (in*sec)' ) axis([0 fmax/2 0 0.2]) % % FFT of Veleocity vs Time

PAGE 105

105 % V1 = zeros(N,1); [N1,nd] = size(V); V1(1:N1)=V(1:N1); Vw = (1/N)*fft(V1); % subplot(2,4,7) plot(f',abs(Vw)) title( 'Velocity(w)' ) xlabel( 'Frequency (Hz)' ) ylab el( 'Amplitude (in/sec*sec)' ) axis([0 fmax/2 0 8]) % % FFR of Acceleration vs Time % A1 = zeros(N,1); [N1,nd] = size(A); A1(1:N1)=A(1:N1); Aw = (1/N)*fft(A1); subplot(2,4,8) plot(f',abs(Aw)) title( 'Acceleration(w)' ) xlabel( 'Frequency (Hz)' ) ylabel( 'Amplitude (in/sec^2*sec)' ) axis([0 fmax/2 0 8e4])

PAGE 106

106 APPENDIX B TEST SPECIMENS Table B 1 Push off specimen geometry Specimen Size L (in) h (in) d (in) b (in) Gap (in) Shear Plane bd (in 2 ) 1 11.5 25 5.5 10 1 55 1A 11.5 27 5.5 11 1.5 60.5 Table B 2. Push off specimen reinforcement and compressive strengths. Specimen f`c (ksi) Reinforcement Number of Stirrups Stirrup Size NC 1A 0 5.3 0.0% 0 NA NSC 1 1 5.3 0.8% 4 #3 NSC 1 2 5.3 1.6% 8 #3 CT1 1A 0 29 0.0% 0 NA CT1S 1 1 29 0.8% 4 #3 CT1S 1 2 29 1.6% 8 #3 CT2 1A 0 29 0.0% 0 NA CT2S 1 1 29 0.8% 4 #3 CT2S 1 2 29 1.6% 8 #3

PAGE 107

107 Figure B 1. Push off specimen geometry

PAGE 108

108 APPENDIX C EXPERIMENTAL TEST SUMMARY Table C 1. Summary of static test results. Specimen f`c (ksi) Shear Plane (in 2 ) Peak Load (kips) max (ksi) Slip at Peak (in) Avg Residual (kips) avg Residual (ksi) Slip at Ultimate Failure (in) NC 1A 0 S 1 1 5 60.5 41.2 0.68 0.018 0.018 NC 1A 0 S 2 1 5 60.5 43.4 0.72 0.029 0.029 NSC 1 1 S 1 3 5 55 59.9 1.09 0.054 30.5 0.55 NSC 1 1 S 2 5 55 62.1 1.13 0.057 NSC 1 2 S 1 3 5 55 NSC 1 2 S 2 3 5 55 NSC 1 2 S 3 3 5 55 NSC 1 2 S 4 5 55 71.1 1.29 0.088 28.0 0.51 NSC 1 2 S 5 3 5 55 75.3 1.37 CT1 1A 0 S 1 1,2 29 60.5 187.4 3.1 0.106 0.106 CT1 1A 0 S 2 1,4 29 60.5 206.7 3.42 0.067 7.5 0.12 0.067 CT1 1A 0 S 3 1 29 60.5 181.0 2.99 0.123 7.1 0.12 0.124 CT1S 1 1 S 1 1,4 29 55 213.8 3.89 0.090 55.5 1.01 0.650 CT1S 1 1 S 2 1,4 29 55 200.1 3.64 0.086 48.4 0.88 0.820 CT1S 1 1 S 3 1 29 55 208.0 3.78 0.093 40.3 0.73 0.560 CT1S 1 2 S 1 29 55 235.5 4.28 0.134 83.2 1.51 0.620 CT1S 1 2 S 2 29 55 222.0 4.04 0.114 97.3 1.77 0.670 CT2 1A 0 S 1 1 29 60.5 70.7 1.17 0.033 0.033 CT2 1A 0 S 2 1 29 60.5 76.2 1.26 CT2S 1 1 S 1 4 29 55 137.0 2.49 0.074 55.4 1.01 CT2S 1 1 S 2 29 55 121.3 2.21 0.050 70.4 1.28 CT2S 1 1 S 3 29 55 155.9 2.83 0.087 64.8 1.18 CT2S 1 2 S 1 3 29 55 152.4 2.77 CT2S 1 2 S 2 29 55 156.0 2.84 0.078 66.9 1.22 CT2S 1 2 S 3 5 29 55 160.2 2.91 79.8 1.45 CT2S 1 2 S 4 29 55 167.0 3.04 0.123 83.5 1.52 1 Complete failure of specimen 2 Test stopped early, did not obtain full residual strength 3 Top of block failed instead of intended shear plane 4 Test stopped after initial crack formed throughout shear plan. Testing restarted to obtain residual strength. 5 Displacement date lost due to detached tab or laser blocked by debris.

PAGE 109

109 Table C 2. Summary of impact test results. Specimen f`c ( ksi) Hammer Mass (lb) Height (in) Shear Plane (in 2 ) Approx Peak Load (Kip) (ksi) Slip at Ultimate Failure (in) Measured Velocity (mm/s) NC 1A 0 D 1 1 5 750 12 60.5 72 1.19 1.5 2427 NC 1A 0 D 2 1 5 750 12 60.5 78 1.29 1.5 2130 NSC 1 1 D 1 5 5715 12 55 93 1.69 1 1974 NSC 1 1 D 2 5 5715 12 55 88 1.60 1 2011 NSC 1 1 D 3 4 5 5715 12 55 86 1.56 1894 NSC 1 2 D 1 3 5 5715 12 55 101 1.84 1 1823 NSC 1 2 D 2 3 5 5715 12 55 95 1.73 1 1728 CT1 1A 0 D 1 29 5715 24 60.5 265 4.38 1.5 2234 CT1 1A 0 D 2 29 5715 18 60.5 252 4.17 1.5 2775 CT1S 1 1 D 1 29 5715 30 55 285 5.18 1 3060 CT1S 1 1 D 2 29 5715 30 55 285 5.18 1 3160 CT1S 1 2 D 1 1 2 29 5715 18 55 74 1.35 CT1S 1 2 D 1 2 2 29 5715 18 55 270 4.91 CT1S 1 2 D 1 3 2 29 750 102 55 0.225 5223 CT1S 1 2 D 1 4 2 29 750 120 55 CT1S 1 2 D 1 5 2 29 750 138 55 CT1S 1 2 D 2 1 29 5715 24 55 320 5.82 0.393 2867 CT1S 1 2 D 2 2 29 5715 24 55 120 2.18 1 2813 CT1S 1 2 D 3 1 29 5715 30 55 300 5.45 0.437 3402 CT1S 1 2 D 3 2 29 5715 24 55 145 2.64 1 2735 CT1S 1 2 D 4 1 29 5715 45 55 297 5.40 0.640 4046 CT1S 1 2 D 4 2 29 5715 24 55 222 4.04 1 2696 CT1S 1 2 D 5 29 5715 24 55 346 6.30 1 3723 CT2 1A 0 D 1 1 1 29 750 11 60.5 145 2.40 0.113 2164 CT2 1A 0 D 1 2 1,4 29 750 18 60.5 2960 CT2 1A 0 D 2 1 29 750 18 60.5 114 1.88 1.5 2513 CT2 1A 0 D 3 1 29 5715 6 60.5 127 2.10 1.5 1675 CT2 1A 0 D 4 1 29 5715 6 60.5 103 1.71 1.5 1675 CT2S 1 1 D 1 29 5715 24 55 189 3.44 1 2770 CT2S 1 1 D 2 29 5715 18 55 154 2.80 1 2454 CT2S 1 1 D 3 29 5715 18 55 160 2.91 1 2457 CT2S 1 2 D 1 29 5715 30 55 208 3.78 1 3249 CT2S 1 2 D 2 29 5715 30 55 202 3.67 1 3149 1Complete failure of specimen 2Trial test to make sure set up is OK 3Top of block failed instead of intended shear plane 4Displacement data lost due to detached tab or laser blocked by debris.

PAGE 110

110 Table C 3. Approximate ratio of dynamic to static shear strength for UHPC. Specimen Average Peak Stress (ksi) Ratio (D/S) Shear Dynamic Static NC 1A 0 0 1.24 0.70 1.77 NC 1 1 0.8 1.62 1.11 1.46 NC 1 2 1.6 1.78 1.33 1.34 CT1 1A 0 0 4.27 3.17 1.35 CT1 1 1 0.8 5.18 3.77 1.37 CT1 1 2 1.6 5.74 4.16 1.38 CT2 1A 0 0 2.13 1.21 1.75 CT2 1 1 0.8 3.05 2.51 1.21 CT2 1 2 1.6 3.73 2.89 1.29

PAGE 111

111 LIST OF REFERENCES [1] R. J. Comparin R. Singh. Non linear frequency respon se characteristics of an impact pair. Journal of Sound and Vibration ; 1989 [2] H. P. Olesen and R. B. Randall A Guide to Mech anical Impedance and Structural R esponse Techniqu es. Nrum Denmark : B RUEL &KJAER Inc.; 1982. [3] T. Krauthammera, S. Astarlioglua, J. Blaskoa, T.B. Sohb, P.H. Ng Pressure impulse diagrams for the behavior assessment of structural components International Journal o f Impact Engineering 2008; 35 771 783 [4] Humar JR. Dynamics of structures. Lisse, Exton, PA: A.A. Balkema Pulishers; 2002 [5] R. French M. Stone, T. Krauthammer, J Kim Assessment of Direct Shear Behavior in Normal Strength and Ultra High Performance Concrete. Technical Report CIPPS TR 008 2013, Gainesville, FL: Center for Ifnrastructure Protection and Physical Security, University of Florida ; 2013 [6] J.W. Ted esco., W.G. Mcdougal, C.A. Ross. Structural Dynamics: Theory and application Addison Wesley; 1999 [7] Wight, J. K. Reinforced Concrete: Mechanics and Design 5th Edition Pearson Prentice Hall. ; 2005 [8] Hofbeck, J. A., Ibrahim, I O., and Mattock, A. H. ). Shear T ransfer in Reinforced Concrete. ACI Journal ; 1969 66(13), 119 128. [9] enhancement of blast resistant ultra high performance fibre reinforced concrete International Journal of Impact Engineering Elsevier Ltd ; 2010 ; 37(4), 405 413. [10] Rong, Z., Sun, W., and Zhang, Y. Dynamic compression behavior of ultra high International Journal of Impact Engineering Elsevier Ltd, ; 2010 ; 37(5), 515 520

PAGE 112

112 BIOGRAPHICAL SKETCH Jaeyoon Kim was born in 1984 in Gwangju, South Korea. In 2002, he attended the Chonnam National University graduating in 2008 with a Bachelor of Civi l Engineering. After being employed as a construction manager at Kumho Industrial Co., Ltd. and Hyundai Cons truction & Engineering Co., Ltd, h e began attending the University of Florida in 2012 to work on his Master of Science with a focus on impact and pr otective technology for structural engineering under Dr. Theodor Krauthammer. He graduated with a certificate in Security Engineering from CIPPS.