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PAGE 1 ANALYZING BURIED REINFORCED CONCRETE STRUCTURES SUBJECTED TO GROUND SHOCK FROM UNDERGRO UND LOCALIZED EXPLOSIONS By NICHOLAS HENRIQUEZ A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2009 1 PAGE 2 2 2009 Nicholas Henriquez PAGE 3 3 To 1504 PAGE 4 4 ACKNOWLEDGMENTS I thank my chair and advisor Dr. Theodor Krauthammer for first introducing me to the study of protective structures, as well as for his guidance with this report. I would also like to thank Dr. Serdar Astarlioglu for all of his assistance with the creation of the program and suggestions for improvement. The financial support for this study provided by the U S Army Engineer Research and Development Center is gratefully acknowledged. I especially need to thank my family and friends for all of their support. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS .................................................................................................................... 4 LIST OF TABLES ................................................................................................................................ 7 LIST OF FIGURES .............................................................................................................................. 8 LIS T OF SYMBOLS .......................................................................................................................... 10 ABSTRACT ........................................................................................................................................ 15 CHAPTER 1 INTRODUCTION ....................................................................................................................... 16 1.1 Proble m Statement ................................................................................................................ 16 1.2 Objective and Scope ............................................................................................................. 17 1.3 Research Significance ........................................................................................................... 17 2 BACKGROUND AND LITERATURE REVIEW ................................................................... 18 2.1 Introduction ........................................................................................................................... 18 2.2 Single Degree of Freedom (SDOF) Systems ...................................................................... 18 2.3 Flexure in Reinforced Concrete Walls ................................................................................ 20 2.4 Direct Shear ........................................................................................................................... 27 2.5 Use of the N ewmark Beta Method for Integration ............................................................. 29 2.6 Underground Blasts .............................................................................................................. 30 2.6.1 Ground Shock ............................................................................................................. 30 2.6.2 Soil Arching ................................................................................................................ 33 2.7 One Dimensional Elastic Wave Behavior ........................................................................... 34 2.8 Summary................................................................................................................................ 37 3 METHODOLOGY ...................................................................................................................... 38 3.1 Introduction ........................................................................................................................... 38 3.2 Flexural Response ................................................................................................................. 38 3.3 Direct Shear ........................................................................................................................... 42 3.4 Load Function Creation ........................................................................................................ 43 3.4.1 Soil Layer Reflections and Transmi ssions ................................................................ 43 3.4.2 Converting Free Field Soil Pressure to an Interface Pressure ................................. 46 3.4.3 Calculating Average Pressure .................................................................................... 46 3.5 Thrust ..................................................................................................................................... 48 3.6 Program Flowcharts .............................................................................................................. 51 3.7 Summary................................................................................................................................ 51 PAGE 6 6 4 RESULTS AND DISCUSSION ................................................................................................ 55 4.1 Introduction ........................................................................................................................... 55 4.2 Box Validation Using Experiment al Data ........................................................................... 55 4.2.1 Box Resistance Models .............................................................................................. 56 4.2.2 Test Shots .................................................................................................................... 57 4. 2.3 Calculated Deflection Histories ................................................................................. 59 4.2.4 Results Comparison.................................................................................................... 63 4.3 Load Function Creation and Possible Improvement ........................................................... 65 4.4 Summary................................................................................................................................ 70 5 CONCLUSION AND RECOMMENDATIONS ...................................................................... 71 5.1 Summary................................................................................................................................ 71 5.2 Conclusions ........................................................................................................................... 72 5.3 Recommendations for Further Study ................................................................................... 72 APPENDIX KIGER AND ALBRITTON (1980) TESTS ........................................................... 74 Box Compositions ....................................................................................................................... 74 Soils .............................................................................................................................................. 75 Test Shots and Data Recording .................................................................................................. 76 Values Used in Computer Calculations ..................................................................................... 77 LIST OF REFERENCES ................................................................................................................... 82 BIOGRAPHICAL SKETCH ............................................................................................................. 84 PAGE 7 7 LIST OF TABLES Table page 4 1 Validation results .................................................................................................................... 64 4 2 Best matched values ............................................................................................................... 68 A 1 List of values used in computer calculations ........................................................................ 77 PAGE 8 8 LIST OF FIGURES Figure page 2 1 SDOF system .......................................................................................................................... 19 2 2 Load deflection diagram for an RC slab ............................................................................... 21 2 3 Assumed yield line and strip geometry (Park and Gamble 2000) ....................................... 22 2 4 Deflections and plastic hinges of a restrained strip (Park and Gamble 2000) .................... 22 2 5 Full slab thickness between plastic hinges (Park and Gamble 2000) ................................. 23 2 6 Thrust forces added to concrete strips .................................................................................. 26 2 7 Action of tension membrane forces (Park and Gamble 2000) ............................................ 27 2 8 Load deflection model for a slab (Krauthammer et al. 1986) ............................................. 27 2 9 Empirical model for shear stress slip relationship (Krauthammer et al. 1986) .................. 29 2 10 Sketch of an underground blast pressure time history ......................................................... 32 2 11 Ground shock coupling factor as a function of scaled depth (ESL TR 8757 1989) ........ 33 2 12 Change in direction after a wave transmission ..................................................................... 37 3 1 Unit width divided into layers and corresponding stress distributions ............................... 39 3 2 Variation of load and mass factors ........................................................................................ 41 3 3 Illustration of simplification used in reflected and transmitted wave calculations, including soil layer resizing and investigating each pressure wave as coming directly from its own charge ................................................................................................................ 45 3 4 Original design for a 10 x 10 rectangular grid across the entire wall ................................. 47 3 5 Modified square 10 x 10 grid nearest to the explosion ........................................................ 48 3 6 Distributed trapezoidal forces creating thrust loads ............................................................. 50 3 7 Program flowchart .................................................................................................................. 53 3 8 Load function flowchart ......................................................................................................... 54 4 1 Flexural resistance model for box 3C ................................................................................... 56 4 2 Direct shear resistance model for box 3C ............................................................................. 57 PAGE 9 9 4 3 Damage after shot 3C2 (Kiger and Albritton 1980) ............................................................ 58 4 4 Near failure damage after shot 3C3 (Kiger and Albritton 1980) ........................................ 58 4 5 Damage after shot 3D6 (Kiger and Albritton 1980) ............................................................ 59 4 6 Experimentally measured pressure time histories (Kiger and Albritton 1980) ................. 60 4 7 Calculated deflection time history for test shot 3C1 from DSAS using digitized loads ... 61 4 8 Calculated deflection time histor y for test shot 3C2 from DSAS using digitized loads ... 61 4 9 Calculated deflection time history for test shot 3D1 from DSAS using digitized loads ... 62 4 10 Calculated deflection time history for test shot 3D2 from DSAS using digitized loads ... 62 4 11 Calculated deflection time history for test shot 3D6 from DSAS using digiti zed loads ... 63 4 12 Overlay of original calculated load and digitized experimental load for test 3C1 ............. 65 4 13 Overlay of original c alculated load and digitized experimental load for test 3C2 ............. 66 4 14 Overlay of original calculated load and digitized experimental load for test 3D1 ............ 66 4 15 Overlay of original calculated load and digitized experimental load for test 3D2 ............ 67 4 16 Overlay of original calculated load and digitized experimental load for te st 3D6 ............ 67 4 17 Overlay of best fit calculation and digitized experimental load from test 3D2 ................. 69 4 18 Comparison of calculated pr essure at the wall center and calculated average pressure .... 70 A 1 Typical concrete stress strain curve (Kiger and Albritton 1980) ........................................ 75 A 2 Steel stress strain curve (Kiger and Albritton 1980) ........................................................... 76 A 3 Box layouts (Kiger and Albritton 1980) ............................................................................... 79 A 4 Shot and instr umentation layouts, box 3C (Kiger and Albritton 1980) .............................. 80 A 5 Shot and instrumentation layouts, box 3D (Kiger and Albritton 1980) .............................. 81 PAGE 10 10 LIST OF SYMBOLS Ac Cross se ctional area of concrete Asb Area of reinforcement Ai Area of a layer C Damping c Elastic wave velocity c Seismic velocity c Neutral axis depth at Section 1 c Neutral axis depth at Section 2 Cc Concrete compressive force of section 1 Cc Concrete compress ive force of section 2 Cs Steel compressive force of section 1 Cs Steel compressive force of section 2 d Depth to steel layer d Soil layer thickness db Bar diameter E Modulus of elasticity Ec Concrete modulus of elasticity f Coupling factor fi S tress in a layer fs Tensile strength of reinforcement fy Yield strength of reinforcement fc Concrete cylinder strength Fcc Total compressive concrete force in a section Fi F orce in a layer PAGE 11 11 Fsc Total compressive steel force in a section Fst Total tensil e steel forc e in a section Ft Total force Fe(t) Equivalent forcing function h Slab thickness I Moment of inertia K0 Coefficient of lateral earth pressure KL Load factor Km Mass factor L Length of shorter dimension of box wall L Length of structure l Strip length Lx Le ngth of slab in the xdirection Ly Length of slab in the y direction m Unit mass Me Equivalent mass Mt Total mass mc Moment in a concrete layer ms Moment in a steel layer mu Total internal moment about the neutral axis N Thrust force n Attenuation coeffici ent nu Total membrane force P(t) Pressure function p(x) Pressure function PAGE 12 12 P0 Peak freefield pressure Pwall (t) Pressure function on the wall R Range R(x) Dynamic resistance function S Surround stiffness t Lateral movement of strip ta Load arrival time T S teel tensile force of section 1 T Steel tensile force of section 2 Tx Yield force per unit width in the x direction Ty Yield force per unit width in the y direction Iu Incident normal particle displacement Ru Refl ected normal particle displacement Tu Reflected normal particle displacement u Normal particle velocity Iu Incident normal particle velocity Ru Reflected normal particle v elocity Tu Reflected normal particle velocity V Shear force W Equivalent charge weight w Beam displacement w Distributed load on the strip x Displacement x Velocity PAGE 13 13 x Acceleration Newmark Beta integration constant Portion of slab length between central and support plastic hinges 1 Ratio of ACI stress block depth to neutral axis depth Central deflectio n of strip max Maximum deflection at wall center max Maximum shear slip z Thickness of concrete layer Axial strain c Concrete strain cu Concrete strain at failure i Strain in a layer s Steel strain Direction angle of an elastic wave End rotation Material density 0 Soil density vt Ratio of total reinforcement area to the area of the plane it crosses Normal stress I Incident normal stress R Reflected norm al stress T Transmitted normal stress e Elastic direct shear resistance PAGE 14 14 L Limiting direct shear capacity m Maximum direct shear resistance Frict ion angle of soil ) (x Shape function PAGE 15 15 Abstract of Thesis Presen ted to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science ANALYZING BURIED REINFORCED CON CRETE STRUCTURES SUBJECTED TO GROUND SHOCK FROM UNDERGRO UND LOCALIZED EXPLOSIONS By Nicholas Henriquez August 2009 Chair: Theodor Krauthammer Major: Civil Engineering Closein localized HE detonations pose a substantial risk to buried RC boxtype structures. This study investigated the relationships between the HE charge and its distance from an RC box wall, the existing soil la yers and their properties, th e directinduced ground shock transmitted through soil layers, the load distributi on on the structural wall, and the structural behavior. Previous experimental studies were examined and their results were compared with those obtained from the computer code Dynamic Structural Analysis Suite (DSAS) that was modified to handle such complicated conditions. The box structure was represented in DSAS by addressing the wall slab as a single degree of freedom system, while th e effects of adjacent structural components were incor porated into the resistance func tion for the wall. The spatial dynamic pressure distribution on the wall was pro cessed to derive an equivalent uniformlydistributed dynamic pressure on the wall to be us ed for the fully nonlinear structural analyses. PAGE 16 16 CHAPTER 1 INTRODUCTION 1.1 Problem Statement Having a military structure located underground achieves more than just concealment. Burying a structure allows the builders to make use of the grounds natural damping to absorb and dissipate the blast wave energy from a munitions explosion. Most commonly, these buried structu res take the form of a box, built using reinforced concrete. These types of concrete structures are common for defense against conventional and nuclear weapons. Should a buried box fail, it could result in the loss of human lives. Also, munitions and oth er supplies may be stored in these facilities, the loss of which might lead to a supply shortage. Analytical methods and computer programs which are meant to examine the effects of buried expl osions on buriedbox structures exist but they have drawbacks. More complex programs, such as finite element codes or hydrocodes, consider a high number of degrees of freed om and are computationally intensive. These programs may model the soil using finite elements, assuming a uniform soil type. Since the soil prop erties will be neither uniform nor able to be characterized by a simple material model the results that these programs give for the transmission of the blast wave may or may not be more accurate than using empirical equations, and the amount of time and m emory required to track all of the soil nodes can be excessive. A method to analyze quickly but accurately the effects of a buried blast on a buried box would be ide al for use during a preliminary design phase, as it would save time. Use of a s ingle d egre e of f reedom (SDOF) model would aid in achieving this goal, since calculations on such a model can be performed quickly. PAGE 17 17 1.2 Objective and Scope The objective of this work is to develop a n SDOF computational approach to analyze quickly and accurately the d ynamic response of a buried reinforced concrete (RC) structure to a buried explosives blast loads, using a rational resistance function and including different modes of response. Doing so will aid in the proper selection of ma terials and layout to protec t the structure against common or predicted explosions. Such factors include concrete strength and concrete thickness in the structures walls, steel reinforcement, and soil backfi ll for the structures location. The loads on the structure, its deflectio n, and its flexural and direct shear modes will be analyzed over the course of the explosion event. This study is limited to a localized buried explosion whose most severe loads would occur near the center of one of the boxs sides. T he effects of a blas t on the corners or roof of a box are not considered The load on the wall will be approximated as a uniformly distributed load. The side walls of the box structure will be treated as slabs with axial and lateral forces caused by the effects of the blast The use of up to three layers of soil will be allowed, with the box located in either of the two upper layers or spanning across both. The proposed methods will be compared with available test data. 1.3 Research Significance This work can provide a sim ple, accurate procedure to dynamically analyze a buried RC box structure subject ed to an underground blast loading. The method create s a time history of both the loads on the wall and the response of the center point of the wall, using a n SDOF computation al model that considers both flexure and direct shear behaviors. PAGE 18 18 CHAPTER 2 BACKGROUND AND LITER ATURE REVIEW 2.1 Introduction Burying a structure provides a good measure of protection against air blast. If an explosive device is able to penetrate into the ground before detonation, however, it can exert a much greater load on the structure and present a significant danger. An adequate thickness of concrete and reinforcing steel must be provided to protect against these threats. During the design phase of a n RC box, the possible threats are usually known or assumed. These threats can then be simplified to a design load on the boxes. With this information, the chosen box design can be evaluated by analyzing the relevant structural response modes. This study is focused on buried RC boxes whose outer side walls are loaded by a buried explosi on Section 2.2 of this review discuss es the use of a n SDOF system. In Sections 2.3 and 2.4, the two most likely structural response modes, flexure and direct shear, are discussed. A review of blast loading and the specifics of underground blasts are presented in Section 2.5. Reflection and transmission of elastic waves are discussed in Section 2.6. 2.2 Single Degree of Freedom (SDOF) Systems For both simplicity and spee d of calculations, it is advantageous to analyze a wall of the buried box structure as an SDOF system. This type of system is an approximation of reality, where a nearly infinite number of degrees of freedom exists An SDOF system involves motion in only one direction, which, in this case, is the walls movement. Figure 2 1 shows an idealized damped SDOF system. PAGE 19 19 Figure 2 1. SDOF s ystem In an SDOF system, t here is only one mass, resistance function and damper, and this mass is acted upon by a single forcing function. Each of these is the SDOF equivalent of the total mass, force, etc. The degree of freedom is the vertical displacement, x Me is the equivalent mass, Fe(t) the equivalent forcing function, C the damping, and R (x) the resistance function Often, it is possible to combine all the existing masses, springs, and dampers into this kind of simple case. By converting a more complicated system into an SDOF system, calculations are si mplified. To be useful the displ acement term must correspond to the portion of the element that deflects the most, such as the midpoint on a si mple beam, or the center portion of a wall or slab. The motion of a n SDOF system (with damping) is defined by the following forcing function: ) ( ) (. ..x R x C x M t Fe e (2 1) where the first derivative of the displacement term x is velocity and the second derivative is acceleration. In this case, the forcing function is created by the pressure wave in the ground. A conversion is re quired to calculate the SDOF equivalents of the real values of the total mass and forcing function The total and equivalent mass es of the system can be calculated using the following equations (Biggs 1964): R (x) C M e F e (t) ) ( ), ( ), ( t x t x t x PAGE 20 20 L tdx x m M ) ( (2 2 ) L edx x m M ) (2 (2 3 ) where Mt is the total mass, Me the equivalent mass, L the length, m the unit mass and (x) the shape function. T he mass factor, KM, is defined as the ratio of the eq uivale nt mass to the total mass: t e MM M K (2 4 ) T he total and equivalent loading function s and load factor can be found with the following equations: L tdx x p F ) ( (2 5) L ed x x p F ) ( ) ( (2 6 ) t e LF F K (2 7 ) w here Ft is the total load, Fe the equivalent load, KL the load factor, and p(x) the pressure function. T ables of value of mass and load factors for structural elements with different support conditi ons at elastic, p lastic, or elastoplastic states are found in Biggs (1964). 2.3 Flexure in Reinforced Concrete Walls For the purpose of analysis, it is possible to treat the side walls of the buried box as laterally restrained RC slabs. The se slabs have two possible failure modes. The first, flexure, is discussed in this section. The second, direct shear, is discussed in the next section. PAGE 21 21 Figure 2 2 shows the resistance function of a laterally restrained RC slab. The yield line pattern, w hich is further discussed below, develops between points A and B. According to Johansens yield line theory, the slab should have yielded when it first reached a load equal to the load at point C. However, the slab experiences an enhanced strength at B d ue to compressive membrane forces, caused by the lateral restraint. After peaking at point B, if load is still applied, there is a reduction in the compression membrane forces until point C As point C is reached cracks in the concrete extend all the way through the slabs depth. T he compressive membrane forces in the concrete become tensile membrane forces. T he tensile load near the slabs center is carried by the steel reinforcing, strengthened by the concrete pieces still bonded to it. The slab can t hen carry an increasing load while continuing to deflect, until failure occurs at point D. Depending on the amount of steel reinforcing, it is possible that this failure load may be even greater than the load at point B. Figur e 2 2 Load deflection diagram for an RC slab Due to the composition and support conditions of a n RC slab, when a uniform load is applied, the s lab wants to rotate about all of its supports. This results in a 45 yield pattern, Central Deflection Applied Uniform Load A B C D PAGE 22 22 wh ich can be seen in Figu re 2 3. For analysis, the slab can be divided into individually evaluated unit width strips in both the x and y directions Each of the strips shown in Figure 2 3 can be analyzed as a beam with proper boundary conditions, using the plastic deformation explained in Park and Gamble (2000). The boundary conditions restrain rotation and vertical translation; however, minimal horizontal translation is allowed. In order for there to be a rotation at the end of the beams, plastic hinges must be formed. This i s illustrated in Figure 2 4. Figure 2 3 Assumed y ield l ine and s trip g eometry (Park and Gamble 2000) Figure 2 4 Deflections and plastic hinges of a restrained strip (Park and Gamble 2000) PAGE 23 23 The original length of the beam is l and the lateral moveme nt is t The central deflection is and the length between the center and end plastic hinges is Th e lateral movement t allows for the formation of the compression membrane forces. The locations of the plastic hinges are symmetric about the beam's c enter. The segments between the plastic hinges are assumed to be straight. For there to be a plastic hinge, the steel must have yielded and the concrete must have reached its maximum strength. Physical reality differs from this simplified diagram due to the slabs depth. This can be seen in Figure 2 5. Although the beam portions are assumed to remain straight, geometric problems arise as portions of the slab overlap with other segments and the support s Figure 2 5 Full slab thickness between plast ic hinges (Park and Gamble 2000) From the geometry and force equilibrium in Figure 2 5, the following equations can be developed: ) 2 ( 2 22L t L h c c (2 8 ) T C C T C Cs c s c (2 9 ) PAGE 24 24 w here c and c are the neutral axis depths for secti ons 1 and 2, respectively, h is the slab thickness, Cc and Cc are the concrete compressive forces, Cs and Cs are the steel compressive forces, and T and T are the steel tensile forces. Using the ACI equivalent rectangular stress block assumption, t he co mpressive forces of the concrete can be calculated as c f Cc c 185 0 (2 10) w here fc is the concretes cylinder strength and 1 is the ratio of the depth of the ACI stress block to the depth of the neutral axis. The load central deflection relationship can then be determined from the following equation from Park and Gamble (2000), which is derived using virtual work principles and the moments caused by the previous forces: 13 242 y x yI I wl 1 1 2 1 '2 0 47 0 141 0 188 0 85 0 y x cI I h f 1 5 0 1 5 1 2 16 3 5 3 161 1 2 1 2 y x y x y x y xI I h I I I h I 2 2 2 4 14 3 1 16sx sx x x c y y x x y y xC C T T f I I h I I I 2 '8 3x sx sx sy sy y y y xd h C C C C T TI I 8 4 8 3' 'h d h I I C C h d T Ty y x sy sy x x x PAGE 25 25 8 4'h h d I I T Ty y x y y (2 11) where: x x x xI t 2' (2 1 2 ) a nd y y y yI t 2' (2 1 3 ) In these equations, I is the moment of inertia in the xor y direction, d the depth to the tension steel layer, w the distributed load on the strip and l the strip length. External thru st applied to the outsides of the slab can enhance the compressi on membrane portion of the load deflection diagram (Krauthammer 1984). Calculation of this thrust is discussed in Section 3.5. The new thrust force, N, can then be included in the deflection diagrams and equilibrium equations above as shown in Figure 2 6 Using these equations, the new neutral axis can be found. Calculations using the new thrust force and neutral axis can be found in Section 3.2. This procedure is used for calculating point s between B and C on the diagram. At point C, the compressive membrane forces have reached zero. T ensile membrane forces begin to develop. Figure 2 7 illustrates h ow these f orces act An equation was derived (Park and Gamble 2000) to calculate the rela tionship between load and deflection in this section of the diagram : ,... 5 3 1 3 2 / ) 1 ( 3 22 cosh / 1 1 ) 1 ( 4n x y y x n y yT T L L n n T L w (2 1 4 ) where Tx and Ty yield forces per unit width in the x and y directions. PAGE 26 26 Figure 2 6 Thrust forces added to concrete strips These aforementi oned equations require there to be plastic deformations, and, therefore, large deflections. Consequently, the relationships in the early portion of the load and deflection diagrams are not addressed. A model for this segment was proposed by Krauthammer e t al. (1986). Between points A and B, a quadratic function is fit. Straight lines are then used to model the portions between both point s B and C and points C and D. A drawing of this model is shown in Figure 2 8. This model uses Park and Gambles (2000) assertions that the maximum load is reached at a deflection equal to half the slab thickness, and that the compressive membrane forces end at a deflection equal to the complete slab thickness. The accuracy of this model was verified through comparison s with experimental data. PAGE 27 27 Figure 2 7 Action of tension membrane forces (Park and Gamble 2000) Figure 2 8 Load deflection model for a slab (Krauthammer et al. 1986) 2.4 Direct Shear D irect shear failure occurs through an excessive slipping along the slab's supports. When the concrete box structure fails in direc t shear, it does so in less than a few millis econds after the loads arrival, without time to develop a significant flexural respons e. F or this reason, the direct shear response can be uncoupled from flexural response in calculations (Krauthammer et al. Deflection Load w max 0.5h h A B C D Quadratic function Linear function Linear function PAGE 28 28 1986) Direct shear failure is of s ig nificant concern when dealing with blast loads, due to the ir impulsive nature An empirical model is used to determine the wall s response to direct shear. An earlier model developed by Hawkins (1972) was enhanced in Krauthammer et al. (1986) to take into account compression and rate effects. This was done by increasing the original model by a factor o f 1.4. Th e original and enhanced models are shown in Figure 2 9 The highest shear strength of the wall occurs at B and exists through C. Failure due to direc t shear occurs at E, where maximum displacement is reached. The values of the graph points c ome from the following equations: '157 0 165c ef (2 1 5 ) '8 0 8c y vt c mf f f (2 1 6 ) c s sb LA f A'85 0 (2 1 7 ) 120 1 0 2maxxe (2 1 8 ) b cd f x'86 2 900 (2 1 9 ) where vt is the ratio of total reinforcement area to the area of the plane which it crosses, db the bar diameter, Asb the total reinforcement area, Ac the concrete cross sectional area, fs the re inforcements tensile strength, fy the .reinforcements yield strength, and vt the ratio of total reinforcement area to the area of the plane it crosses. PAGE 29 29 Figure 2 9 Empirical model for shear stress slip relationship (Krauthammer et al. 1986) 2.5 Use of the Newmark Beta Method for Integration Even when solving simplified equations of motion, it is useful to employ a numerical evaluation method to more easily calculate the dynamic response and find a closedform solution. The Newmark Beta method (Newmark et al. 1962) has been chosen for use in direct integration of the equations of motion in both the flexure and direct shear cases. The method is summarized below. 1 ) The equation to be used in this case is (2 1), the motion of an SDOF sys tem: ) ( ) (. ..x R x C x M t Fe e 2 ) The values of x .x and ..x are known at the initial time, t = ta. The values of Fe should be known at every time, t PAGE 30 30 3 ) Let ti+1 = ti t where t is the time step. 4 ) A value of 1 .. ix must be assumed. 5 ) Compute the values 2 ) (1 .. .. 1 .t x x x xi i i i (2 20) and 2 1 .. 2 .. 1) ( ) ( ) 21 ( t x t x t x x xi i i i i (2 21) 1 0 6 ) In this case, a value of 1/6 was used for which corresponds to a parabolic variation. 7 ) By inputting these new values into the original equation of motion, (2 1), compute a new value for 1 .. ix 8 ) Repeat steps 5 and 7 with the new values of 1 .. ix until a convergent value is reached. 9 ) Repeat the process for the next time step. 10) The method starts at time t = 0 the time when the load is first applied. The system is initially at rest, so 0. ix x and m F x ) 0 ( 0 ... 2.6 Underground Blasts 2.6.1 Ground Shock A blast taking place below the ground surface behaves differently than a blast in the open air. In an explosive event occurring in the open air, the explosi on pushes air away, creating a vacuum. On ce the pressure is gone, air flows back into this vacuum, creating a negative pressure phase. This does not occur in soils. Instead, the blast pushes on the soil and creates a crater, which may eventually be filled by soil due to gravity effects T here i s no negative PAGE 31 31 pressure phase. An underground explosion usually generates a greater stress and has a longer duration than if that same explosion were to occur in air (ESL TR 87 57 1989). Therefore its impulse will be much greater. When detonation occu rs the intense pressure wave caused by expanding ga s ses at areas close to the blast creates stress waves in the soil and crushes air voids present in the soil creatin g a crater or cavity. Initially, these gasses are very hot, but they cool as this new exp anding soil cavity is being formed. As the gas s es cool, their volume decreases, resulting in relief or unloading waves, similar to the suction pressures in an air blast. Since the soil through which these relief waves travel has been densified by the ini tial stress waves, they travel faster than the initial waves did, and eventually overtake them. In doing so, they attenuate the intensity of the shock front. To account for this in calculations, an attenuation coefficient is included, though only a rough estimation. Soils where this attenuation occurs more quickly have a low relative density or a large percentage of air voids. Conversely, soils with a high relative density or a low percentage of air voids will attenuate the ground s hock much more slowly Saturation in soils can also affect the shock transmission. Water can fill air voids and increase a soil's density. In cohesive soils, as saturation approaches 100 percent, the peak pressure and stress transmissions begin to behave like they would in w ater. In saturated granular soils with low relative densities, it is possible for the pr essure wave to collapse the soil skeleton, liquefying the sand. These types of granular soils are not recommended for use in the construction of buried facilities (ES L TR 8757 1989). Since a pressure wave expands s pherically after an explosion, the pressure at any point is proportional to a ratio of the range of this point to the cube root of the charge weight. This is PAGE 32 32 known as the scaled distance, and the validity of its use in buried explosives has been proven using 35 years of explosion data. Using this scaled distance and attenuation coefficient, an equation was created for t he calculation of the peak free field pressure in the soil at a gi ven distance from the explosion: nW R fc P 3 1 0 0144 160 (2 22) Here, P0 is the peak free field pressure (psi), c the seismic velocity of the soil (ft/s), 0 the soil density (lb/ft3), R the range (ft), W the charge weight (equi valent weight in lbs of C4), n th e atte nuation coefficient (unitless), and f (unitless) the coupling factor A basic sketch of an underground blast pressure time history is shown in Figure 2 1 0 Figure 2 10. Sketch of an underground blast pressure time histo ry After the arrival of the pressure wave, the pressure at any given time can be calculated from the following equations: at te P t P 0) ( (2 23) Time P 0 t a Pressure PAGE 33 33 c R ta (2 24) R c (2 25) wh ere, ta is the arrival time (seconds). The relevant duration of the blast load recommended for use in these equations is approximately four times the arrival time. The use of a linear rise is recommended in place of an instantaneous rise to the peak pressure at its arrival time The duration of this recommended linear rise is one tenth of the arrival time (ESL TR 8757 1989). The coupling factor f reflects how much of the blasts energy has been coupled into the soi l, as opposed to bei ng lost into the air, etc. at the grounds surface. This value can be interpreted off of the graph in Figure 2 1 1 Figure 2 11. Ground shock coupling factor as a function of scaled depth (ESL TR 8757 1989) 2.6.2 Soil Arching Soil arching is a process in which the shear strength properties of a soil can favorably redistribute a load applied to a buried structure (ASCE 1985). When a uniform load is applied to PAGE 34 34 a face of a buried structure, and it responds flexurally, the soil may passively redistribute t he load away from the center of the wall. Through strong shear forces, the load is directed away from the retreating wall and s e n t towards the stationary portions of the system. This includes the outer walls of the box and surrounding soil, where the loa d will be less damaging to the structure. The effects of soil arching will not be including in this study for two reasons. First, it is often ignored for purposes of a conservative design approach (ASCE 1985). Second, soil arching is a process where a un iform load becomes redistributed; the load from an underground blast is not uniformly applied to the structure. The blast will be spherical, resulting in higher loads at the central portion of the wall and decreasing loads toward the outer edges. It is possible that this load distribution will counteract t he effects of soil arching. 2.7 One Dimensional Elastic Wave Behavior E lastic wave behavior is the simplest option for use in predicting the pressure wave s actions at material interfaces, including betw een soil layers, at the soil surface, and at the boundaries between the box and soil. This is an approximation, however, because the attenuated pressure waves may behave as elastic waves, but the pressure waves near a buried explosion do not. The material recovers back to its undisturbed state once an elastic wave has passed ; no plastic defo rmation has occurred. The propagation of an elastic wave is defined by the dynamic equilibrium equation: ma F (2 26) Inputting the forces, m asses, and acceleration in a small, one yields the following equation: x x A t u x A 2 2 (2 27) PAGE 35 35 where is the solids density, A the cross sectional area of the volume being investigated, u the particle v elocity, stress, x the direction of travel, and t time. Simplifying this equation and applying Hookes Law yields: 2 2 2 2x u E t u (2 28) where E is the materials modulus of elasticity. By simplifying this equation, the elastic wave velo city in any solid can be found: E c (2 29) where c is the wave velocity. Each material has its own elastic wave velocity For example, the elastic wave velocity in concrete is around 10,000 ft/s In soils, elastic wave velocit y is more commonly referred to as seismic velocity. When an elastic wave traveling through one medium encounters the boundary with another medium, including air, a portion of the stress wave will be transmitted into this new medium, and a portion will be r eflected back into the original medium. At these interfaces, there is assumed to be continuity in both normal stress and displacement, resulting in an equality of velocity. The interface is assumed to remain stationary, and the materials are assumed to r emain in contact. From these assumptions, t he following equilibrium equations can be applied at the boundary: T R I (2 30) T R Iu u u (2 31) T R Iu u u (2 32) PAGE 36 36 w here is the normal stress, u is the normal particle velocity, and u the normal particle displacement. I, R and T indicate incident, reflected, and transmitted, respectively. Stress and particle velocity are related by the following equ ation: u c (2 3 3 ) Using these equations, equations to calculate the values for actual stresses transmitted and reflected can be derived : 2 2 1 1 2 22 c c cI T (2 3 4 ) 2 2 1 1 1 1 2 2c c c cI R (2 3 5 ) w here 1 and 2 indicate the initial and newly encountered media, respectively (Tedesco 1999) Depending on the material properties, it is possible for the reflected wave to have a different sign than the incident wave. As the density of the second medium approaches zero, as is assumed with air, the reflected wave approaches the full intensity of the incident wave, but with the opposite sign. As the density of the second medium approaches infinity, the reflected wave approaches the full intensity of the in cident wave, with the same sign (Tedesco 1999). It is possible that the wave itself will not be traveling in a direction normal to the boundary. In this case it is important to be able to calculate the direction of the new waves. The reflected w ave will rebound at the same angle as the incident angle. The transmitted wave will have the same normal wave velocity as the incident wave. Having different elastic wave velocities, they will have different directions of travel. This is illustrated in Figure 2 1 2 Using trigonometry these angles can be found from the following equation: 2 2 1 1cos cos c c (2 3 6 ) PAGE 37 37 Figure 2 1 2 Change in direction after a wave transmission 2.8 Summary In this chapter, t he behavior of the wall of a concrete box as well as methods for estimating this behavior using SDOF systems, was first discussed This discussion focused on flexure and direct shear behavior, the most likely failure modes for the wall of a buried conc rete box subjected to the effects of a buried HE explosive. The numerical method for integration was then discussed. The effects of a buried explosive on soil were also presented as well as the behavior of elastic wave propagation. The background presen ted in this chapter sets the foundation for the methodology discussed in the following chapter. c 1 1 c 2 2 Boundary PAGE 38 38 CHAPTER 3 METHODOLOGY 3.1 Introduction Using the material behaviors discussed in the previous chapter, methods for calculating loads on the structure and the s tructural response can be formulated. This chapter discusses the methodology used in creating the resistance functions for both flexural and direct shear responses for the wall of a n RC box, in Sections 3.2 and 3.3, respectively. The methods used in the creation of the load and the thrust c reated by the load are discussed in Sections 3.4 and 3.5, respectively. F low chart s of the procedure s as included in DSAS are located in Section 3.6. 3.2 Flexural Response As mentioned in Section 2.3, modifications need to be made to values on the load deflection diagram (Figure 2 2) to account for the compression membrane forces enhanced by the external thrust. To determine this new flexural resistance, calculations begin by dividing unit width s of the concrete slab in to a series of layers as illustrated in Figure 3 1. The stresses in each layer are then determined individually, using the chosen stress strain relationships: the Hognestad model (MacGregor and Wight 2005) for concrete, and the Krauthammer and Hall (19 82) model for steel. Once the resulting total force and moment about the neutral axis, nu and mu, respectively, have been calculated, the walls deflection can be found by equating the internal work done by these values to the external work done by the blas t load This process is used for points B and C, where the deflections have been assumed as equal to one half the slabs depth and the full slab depth, respectively. Then a linear function can be fit between the two points. The value at point B should b e a local maximum, and compression membrane forces should no longer be present at point C. PAGE 39 39 Figure 3 1 Unit width divided into layers and corresponding stress distributions As evident from the figure, nu and mu are functions o f the neutral axis depth, c These apply to the central portion of the slab, between the middle plastic hinges. The same holds true for the corresponding forces nu and mu, which are functions of their neutral axis depth, c The s t rains in the concrete and steel at each of their layers can be calculated from the following equation, derived from the geometry of the strain distribution: i cu id c c (3 1) where i is the steel and/or concrete strain in the layer, cu the ultimate concrete strain at its failure, di the depth to the layer, and c the neutral axis depth. The stresses in each of these layers, fi, can then be determined f rom the materials stress stra in diagrams. These strains can then be converted into forces, and the forces can be used to calculate the total section moment and axial force, as shown in the following equations: i i iA f F (3 2 ) layers i i i ud h F m# 12 (3 3 ) Unit width of RC slab Compression steel n u m u h Linear strain distribution Strain distribution sj Neutral axis cu f ci c d j d i i f sj f si Concrete and steel stresses Tension steel PAGE 40 40 layers i i uF n# 1 (3 4 ) where Fi is the steel and /or concrete layer force, Ai the area of the layer, h the slab thickness, mu the total section moment, and nu the total section axial force. Based on deformation geometry, the strain and move ment portion of equation (2 8) can be determined as follows: total u c uLS N n hE n L t 2 2 (3 5 ) where Ec is the elastic modulus of the concrete, S the surround stiffness, and N the thrust force. By su bstituting equation (3 9) into equation (2 8), it can b e written as a function of the neutral axis depths c and c as can the equilibrium of membrane forces: 0 2 2 2 ) (2 1 LS N n hE n L h c c c c Ru c u (3 6 ) 0 ) (2 u un n c c R (3 7 ) Through iteration, values of c and c at any displacement value, can be found from equations (3 6 ) and (3 7 ). These values of c and c can then be used in equations (3 3 ) and (3 4 ) to find the total values of nu, mu, and mu in each strip, for use in virtual work calculations. As mentioned in Chapter 2, in order to use an SDOF system approximation, factors need to be applied to the total load and mass. Since a range of factors are listed in Biggs (1964), dependent on the behavior of the material (elastic, plastic, etc), it was decided to use different factors throug hout the course of the load deflection diagram. Initially, at point A, the behavior is elastic. Between point s A and B the factors are varied linearly to elasticplastic first, and then to plastic at B. From B to C, the values go from plastic to the te nsion membrane values. These tension membrane values are then used from point C through D. This is illustrated in Figure 3 2. PAGE 41 41 Figure 3 2 Variation of l oad and m ass f actors The factors for elastic, plastic and elastoplastic portions can be found in Biggs (1964), but the tension membrane factors must be calculated using the methods discuss in Section 2.2. During the tension membrane portion of its response, a beam with a uniform load applied is assumed to have a parabolic def ormed shape, defined by the following equation: 2 24 4 ) ( L x x L x for L x 0 (3 8 ) Inputting this value into equations (2 4 ) and (2 7 ): L dx L x x L mL dx x m M M KL L t e M 0 2 2 2 0 24 4 ) ( (3 9 ) 533 0 15 8 MK L x p dx L x x L x p L x p dx x x p F F KL L t e L) ( 4 4 ) ( ) ( ) ( ) (0 2 2 0 (3 1 0 ) Deflection Load and Mass Factors 0.5h h A B C D Compression Membrane Portion Transition Portion Tension Membrane Portion Elastic Elastoplastic Plastic Tension Membrane PAGE 42 42 667 0 3 2 LK These equations correspond to the unit width (treated as a beam) o f a one way slab. The same approach can be used for ca lculating the factors for a twoway slab. 3.3 Direct Shear T he basic concepts and load deflection curve s used in direct shear calculations were discussed in Chapter 2. These concepts apply to one way slabs, but modifications must be made for the ir use with two way slabs. The x and y directions can be looked at separately, as shown in the following equations: ) ( ) ( t V w R w C w Mex x x x x ex (3 1 1 ) ) ( ) ( t V w R w C w Mey y y y y ey (3 1 2 ) where w is the degree of freedom, i.e. the slip of the slab, and the forcing function is V a shear force. Since the slab is assumed to not flex, it can be treated as a single moving mass. Therefore, the x and y directions both experience the same slab displacement w Equations (31 1 ) and (3 1 2 ) can then be combined as follows: )) ( ) ( ( ) ( ) ( t V t V w R wR w C C w M My x y x y X ey ex ) (, ,t V R w C w Mtotal e total total total e (3 1 3 ) In two way slab s the resistance can be as sumed as the sum of the resistances in the x and y directions. In addition due to the simple deformed shape, the load and mass factors can be taken as 1.0. The equivalent mass is the total slab mass, and the equivalent load is the total shear load on t he slab. T he resistance is the sum of the resistance s around the support perimeter PAGE 43 43 3.4 Load Function Creation To treat the wall of the RC box as a n SDOF system, a single forcing function must be applied. Since an underground blast does not create a unif orm load on the wall, an average force from t he whole wall must be created. In this section, first the blast wave reflections and transmissions are discussed then the conversion from free field pressures t o a wall surface pressure is explained. Lastly, t he techniques used in creating an average pressure on the wall are discussed. 3.4.1 Soil Layer Reflections and Transmissions The existence of dense soil layers can increase the load on the wall by reflecting additional pressure waves back towards the wall as discussed in Section 2.7. Waves reflecting from the grounds surface or a less dense layer may also send a negative pressure wave, reducing the load on the wall. All of these additional reflected pressures must be accounted for when calculating the t otal load on the wall. The values of these pressure waves can be calculated using a combination of the pressure equations found in Section 2.6 and the reflection and transmission coefficients found in Section 2.7. First, the maximum pressure is calculated using equation (2 22), this is then multiplied by the reflection and/or transmission equations (2 3 4 ) and (2 3 5 ) as necessary, dependent upon the number of times the wave has been reflected or transmitted before reaching the wall. Equations (2 23) through (2 25) can then be used in conjunction with this newly modified peak pressure to develop a pressure time history of the reflected wave. The soil weight and wave speed to be used in these calculations are the density and wave speed of the soil in which t he wall is located since th e s e will be the final speed of the wave as it reaches the wall and the density of the soil that will be pressing on the wall. PAGE 44 44 It may be difficult to calculate the range of a wave which has been transmitted through an additional layer because, as shown in Figure 2 1 2 the wave will change direction. The true range is the sum of the total distance s traveled by the wave, so these changes in direction must be accounted for. In order to simplify this, the soil layers can be artifici ally stretched or compressed so that the same wave speed can be used in calculations. The wave will still have the same arrival time, but there will be no change in direction at the interfaces, allowing for a simpler calculation of range. The equation us ed to artificially change a layers depth is shown below: layer current layer blast newc c d d_ (3 1 4 ) where d is the layer depth. This layer resizing is illustrated in Figure 3 3 Another way to simplify the reflection and transmission calculations is to consi der each as its own direct pressure wave coming from a new source rather than as a reflection emanating from the original explosive charge. These new charges are located directly above or below the charge in such a way that their direct pressure wave wou ld travel for the same amount of time as the reflected/transmitted waves, but never change direction. The reflection and transmission coefficients still apply. In relation to the original charge itself, only the pressure time history of the direct pressu re wave would be considered. All of these individual pressure time histories can then be summed to get a total pressure time history at a point. Th e use of these new charges is illustrated in Figure 3 3 PAGE 45 45 Figure 3 3 Illustration of simplification used in reflected and transmitted wave calculations, including soil layer resizing and investigating each pressure wave as coming directly from its own charge New Charge Air Air Box Box S oil 1 Soil 1 (Resized) Soil 2 Charge Soil 2 Soil 3 Soil 3 Charge Pressures Pressure PAGE 46 46 3.4.2 Converting Free Field Soil Pressure to an Interface Pressure Th e equations presented in Section 2.6 and expanded upon in other portions of this paper only calculate the free field pressures in soil. This is different from the pressure felt on a surface. In order to find this surface pressure, elastic wave reflection is again used. The pressure on the wall is the sum of the free field pressure and the force of reflecting the wave back. Once all the free field pressures, both direct and from reflections, have been calculated, the surface pressure can be calculated us ing the following equations: ) ( ) 1 ( ) ( t P r t Pwall (3 15) so soil concrete concrete so soil concrete concretec c c c r (3 16) where Pwall(t) is the pressure on the wall and P(t) is the original total of all free field pressures. Equation (3 16) is a modi fication of equation (2 35), specifically for use with soils and concrete. 3.4.3 Calculating Average Pressure The pressure equations discussed earlier can only calculate a pressure time history at one single point. Taking only the pressure on the center point of the wall would overestimate the walls loading; an average pressure on the walls s urface must be created for use in the SDOF calculations. It was originally determined to divide the entire wall into a grid of one hundred rectangles, ten rectangles vertically by ten horizontally. This is illustrated in Figure 3 4. Ten was chosen to allow for simple divisions. The time dependent pressure equations could then be found at the center of each rectangle By averaging these pressures and multiplying by the area of the rectangle, a n average force could be obtained. This average force could then be converted to a uniform load on the wall. However, since the pressure equations are continuous in respect to time, a finite number of times would need to be u sed in order to have values to average. PAGE 47 47 Figure 3 4 Original design for a 10 x 10 rectangular grid across the entire wall The duration of the entire loading on the structure begin s at the start of the rise of the first pressure to reach the wall. It en d s at the end of the duration of the last pressure to reach the wall. As mentioned in Section 2.6, the duration of one pressure time history was estimated as four times its arrival time. Th e overall duration was then divided into a number of time steps. Then, at each of step, the time w as put into the one hundred rectangle s pressure equations, and a total pressure was found. It was discovered that using the average pressure over the entire wall resulted in a greatly underestimated load. Since the outer portion of the box would take not feel any pressure until much later, many rectangular areas were contributing a zero pressure to the average while the walls center, the most important section and the section most affected by flexure, was experiencing it s greatest load. It was decided that instead of taking an average pressure over the entire wall, the new force would come from a square area of the wall nearest to the charge. This square would be as large as the height of the box, and have the charge loc ated at its center. In this way, the portions PAGE 48 48 of the box experiencing very little pressure, which also experience the least deflection, w ould not distort the averages The square layout is illustrated in Figure 3 5. Figure 3 5 Modified square 10 x 10 grid nearest to the explosion 3.5 Thrust As mention ed in Section 2.3, the external thrust must be taken into account when creating the walls resistance functions. By increasing compression membrane forces, these axial forces can increase the ultimate lo ad that a wall or slab can resist. A method had to be formed to calculate these thrusts, having c alculated the forcing function. This thrust is created by the pressure wave as it travels across the surfaces of the box (the sides, top, and base ) and appli es a load to all of these faces. These loads are then transmitted to the ends of these side s, where they create axial forces on the adjoining walls. To calculate the valu es of these thrust forces, unit width strips at the boxs roof, base, and one side ar e used. As the pressure wave travels across one of these strips, the load it applies is dependent upon the coefficient of lateral earth pressure, a soil property that determines how much force applied to a soil in one direction will be felt in perpendicul ar directions T he PAGE 49 49 pressure waves strength continues to dissipate due to the increase in range from the explosive, as discussed in Section 2.6. The coefficient of lateral earth pressure is so named because it was originally used to determine what kind o f lateral force, due only to a soils weight, would be applied to a vertical wall meant to hold back a volume of soil. It can be calculated from the following equation: sin 10 K (3 17) where K0 is the coefficient of lateral earth pr essure and is the soils friction angle. The thrust on the wall is calculated from the reactions of the strips in the ad j oining members assuming the strips are simply supported. The load on the face of the structure at each time s tep over the entire explosive event is already known before the thrust function is calculated. At each time step, the following procedure is performed to calculate the thrust: 1 ) The load applied to the wall facing the blast is multiplied by K0 and appl ied a t the very end of the unit width strip. 2 ) Any load points already on the unit width strip are moved down a distance equal to the time step multiplied by the soils seismic velocity. They are also decreased in magnitude by this increase in range. Should any of these points move beyond the end of the box, they are ignored. 3 ) The areas between these load points are taken as trapezoidal distributed loads. Using the areas of these trapezoids, the locations of their centroids, and the summation of moments, the re action forces are calculated. 4 ) The inverse of the reaction force at the end of the strip corresponding to the wall being analyzed is the thrust force. A portion of this thrust method is illustrated in Figure 3 6 PAGE 50 50 Figure 3 6 Di stributed trapezoidal forces creating thrust loads The roof, base, and one side wall are each analyzed separately due to the possibility of there being different soils present, and, therefore, the possibility of different coefficients of lateral earth pres sure and seismic velocities. For example, the box may have been built or placed on a denser soil and then backfilled with a less dense material, resulting in a different soi l along the boxs base. For this reason, two separate resistance functions are al so used, one for the horizontal direction and one for the vertical. For the horizontal direction, the thrust on one side Equivalent Load at Support Box Load Traveling Across Box Blast Load on Wall Face PAGE 51 5 1 of the box was used in calculating the resistance function; for the vertical, an average thrust from the roof and base was used. 3.6 P rogram Flowchart s A flowchart outlining the order of the procedures performed by DSAS to analyze the effects of an explosive on the wall of a buried RC box structure is shown in Figure 3 7 Figure 3 8 provides a flowchart expanding upon the load function generation of the program. In the main program, a fter all parameters have been input, the load function is calculated first. From this, the thrust can then be calculated. Two SDOF systems are then run to analyz e the wall response: one for calculating fle xural response and one for calculating direct shear response. For flexural response calculations, the resistance function must be recalculated at each time step due to the change of the thrust force The load function generating portion of the program begins by artificially stretching the soil layers so as to use only the seismic velocity of the layer in which the box is located. The square portion of the wall nearest to the charge is then divided into a ten by ten grid, with the location of the center of each grid box known. Additional charges, representing the source locations of reflected and transmitted waves are then created. The arrival and departure time s for the load on the entire box wall are then calculated. At each time step between the arri val and departure time s the loads on each rectangle are calculated and averaged to create the average load on the box. 3.7 Summary In this chapter, the methodology used in creating both the loads and the resistance functions was discussed, allowing for the calculation of the buried RC box walls reaction to an underground blast. These methods were coded into a portion of the computer program DSAS PAGE 52 52 and tested against existing experimental data. The results of these comparisons can be found in the next chap ter. PAGE 53 53 Figure 3 7 Program flowchart Flexural Response Calculated with Newmark Beta Method Start Data Input Generate Load Function Generate Thrust Function Failure I ncrease Time Step Output End Generate Flexural Resistance Function Direct Shear Response Calculated with Newmark Beta Method Failure Increase Time Step Generate Direct Shear Resistance Function Yes Yes No No PAGE 54 54 Figure 3 8 Load function flowchart Start Retrieve Data Stretch Soil Layers Create 10x10 Square Grid on Wall Generate New Charges Representing Reflected and Transmitted Waves Calculate Arrival and Departure Times Calculate Total Pressure o n Each Grid Point Calculate Average Pressure Increase Time Step End Pass Load Function to Main Program PAGE 55 55 CHAPTER 4 RESULTS AND DISCUSSI ON 4.1 Introduction The procedures proposed in Chapter 3 were coded into the computer program Dynamic Structural Analysis Suite (DSAS) in order to validate their results. Pressure and damage data from an existing experiment performed by Kiger and Albritton (1980) subjecting buried boxes to buried explosives, was used to validate the proposed met hods for generating resistance models for a box. This pressure data was also used to determine the validity of the proposed methods for calculating loads. Intermediate portions of pressure calculations were compared with those from th e existing computer program ConW ep (Hyde 1992) to verify that they indicated similar results. 4.2 Box Validation Using Experimental Data The Kiger and Albritton (1980) tests involved the burial of two box structures, known as 3C and 3D. A number of charges of an equivalent weight were buried and detonated at predetermined points around either of these boxes. The size of the explosives used is detailed in their report. More details on the boxes site, and test conditions can be found in the Appendix. Pressure time histories on the box surface were recorded for five of these detonations, referred to as shots, two from box 3C and three from box 3D. The make up of these boxes, along with their recorded load functions, was put into DSAS to test the flexure and direct shear re sistance functions. Damping ratios of 20% for flexure and 5% for direct shear were used. The higher damping ratio in the flexural case was meant to account for energy dissipation caused by soil structure interaction (Krauthammer et al. 1986). Calculated displacements were then equated to a possible damage level and compared to the observed damage. The means of doing so, as well as the data comparison can be found in the next sections. PAGE 56 56 4.2.1 Box Resistance Models From the properties entered, DSAS calcul ated equivalent resistance models for each box. Examples of the flexural and direct shear resistance models that were generated by the program are shown in Figures 4 1 and 4 2. Information on the dimensions, material properties, and rebar layouts used in both boxes is located in the Appendix. Displacement (in)Pressure (psi)Flexural ResistanceBox 3C 0 2 4 6 8 10 12 0 100 200 300 400 500 Figure 4 1 Flexural resistance model for box 3C PAGE 57 57 Displacement (in)Pressure (psi)Direct Shear ResistanceBox 3C 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0 300 600 900 1200 1500 1800 Figure 4 2 Direct shear resistance model for box 3C 4.2.2 Test Shots All test shots shown here were located at the center of one of the longer sides of the boxe s, at varying distances. Shot 3C1 was placed eight feet away from the wall center of box 3C. No structural damage was sustained. Shot 3C2 was located six feet from the wall center of box 3C. Moderate cracking was observed at the center of the wall sectio n with cracks radiating longitudinally along the wall. The damage is shown in Figure 4 3. Shot 3C3 was located four fee t from the wall center opposite from the previous shots, so that an undamaged wall could be used. T his portion of the wall however, d id not have a pressure gage. Since the distance was identical to shot 3D6, the load function from 3D6 is used in place of the unavailable date for 3C3 in this study. Problems presented by this substitution are explained in Section 4.2.4. This test resul ted in a deflection in the wall of approximately 10.5 inches, with breaching assumed to be imminent. Researchers believed that this near failure response mode was flexure. The damage is shown in Figure 4 4. PAGE 58 58 Shot 3D1 was located eight feet from the center of the long wall of box 3D. No damage was observed. Shot 3D2 was located six feet from the center of the long wall of box 3D. No damage was observed. Shot 3D6 was located four feet from the center of the long wall. It produced minor longitudinal cracks The damage is shown in Figure 4 5. Figure 4 3 Damage after shot 3C2 (Kiger and Albritton 1980) Figure 4 4 Near failure damage after shot 3C3 (Kiger and Albritton 1980) PAGE 59 59 Figure 4 5 Damage after shot 3D6 (Kiger and Albritton 1980) The recorded pressure time histories for the test shots are shown in Figure 4 6. 4.2.3 Calculated Deflection Histories The loads shown above were digitized and input into DSAS, where they were applied to the proper structures. Figures 4 7 through 4 11 show the calcula ted deflection time histories. A comparison with actual tests results is shown in the following section. It should be noted that these are the pressures on the center of the wall, which are the greatest pressures felt anywhere with the configurations in t his experiment. Therefore, using them as the loading function for the SDOF calculations overestimates the average pressure on the wall. Since no other pressure measurements on the wall are available, however, no more can be done without trying to assume an unverifiable factor to decrease the load. PAGE 60 60 Figure 4 6 Experimentally measured pressure time histories (Kiger and Albritton 1980) PAGE 61 61 Time (sec)Displacement (in)Flexural TimeHistory3C1 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Figure 4 7 Calculated deflection time history for test shot 3C1 from DSAS using digitized loads Time (sec)Displacement (in)Flexural TimeHistory3C2 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 0.25 0.5 0.75 1 1.25 1.5 1.75 Figure 4 8 Calculated deflection time history for test shot 3C2 from DSAS using digitized loads PAGE 62 62 Time (sec)Displacement (in)Flexural TimeHistory3D1 0 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2 0.225 0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Figure 4 9 Calculated deflection time history for test shot 3D1 from DSAS using digitized loads Time (sec)Displacement (in)Flexural TimeHistory3D2 0 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2 0.225 0.2 0 0.2 0.4 0.6 0.8 Figure 4 10. Calculated deflection time histor y for test shot 3D2 from DSAS using digitized loads PAGE 63 63 Time (sec)Displacement (in)Flexural TimeHistory3D6 0 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2 0.225 1 0.5 0 0.5 1 1.5 2 Figure 4 11. Calculated deflection time history for test shot 3D6 from DSAS using digitized loads 4.2.4 Results Comparison The maximum wall deflections from the output were used to calculate the walls end rotations using the afore mentioned 45 yield lines and the trigonometric equation: L 5 0 tanmax1 (4 1) w here L is the length of the shorter dimension of the box wall, max the maximum deflection, and the angle of rotation. Using the following criteria for damage based on the end rotation of a slab found in UFC 3 34002: 2 0 Light damage 6 0 Moderate damage 12 6 Severe damage PAGE 64 64 expected damage could be determined. Calculation of expected damage was then compared against the recorded damage observations from the test report (Kiger and Albritton 1980) to validate the methods used in DSAS. This information is pres ented in Table 4 1. Table 4 1. Validation results Shot Calculated Deflection (in) L (in) Calculated Calculated Damage Level Observed Damage Level (from tests) Calculated Final Deflection (in) 3C1 1.22 59.2 2.36 Moderate No Damage 0.20 3C2 1.60 59.2 3.09 Moderate Moderate Cracking 0.53 3C3* 9.02 59.2 17.95 Beyond Severe Breaching imminent, permanent deflection of 10.5 inches 5.16 3D1 0.40 74.0 0.62 Light No Damage 0.00 3D2 0.73 74.0 1.13 Light No Damage 0.00 3D6 1.55 74.0 2.40 Moderate Minor Cracking 0.02 *Shot 3C3 did not have a recorded pressure time history. Since the char ge used was located at a similar distance to the one used in 3D6, that pressure time history was used for this comparison. From this table, it can be seen that, in regards to the five tests with actual pressure time histories, the program calculated a simi lar level of damage to that seen in the experiment. None of the boxes failed in direct shear according to the program and according to the experiment. Since no data for loading was available for a case with direct shear failure, the effectiveness of this portion of the program could not be determined. Special consideration should be made for test 3C3. The deflection did not match what was observed; however, a true pressure time history was not used. The loading used from shot 3D6 would have been similar to its actual load, but, as can be seen from the other tests, explosives at the same distances from the two boxes will not result in the same pressures. It should be noted that in preliminary tests using a less precise version the 3D6 loading where the first spike trough, and second spike were assumed as just one spike the program showed the PAGE 65 65 box failing at a deflection of just over 10.5 inches. As it exists now, there was still more than a severe amount of damage. 4. 3 Load Function Creation and Possib le Improvement As described in Chapter 2, semi empirical equations exist for the calculation of the free field pressures in soil. Th ese equations were coded into DSAS The results from DSAS matched well with those of an existing DOS version of the progra m ConWep With validation of the first portion of the load function completed, additional modifications, discussed in Sections 3.4 and 3.5 were added. Upon completion of the load function generating portion of DSAS the results calculated using the exper imental set up were compared with the pressure time histories measured during the Kiger and A l britton (1980) experiment. The values used in the calculations can be found at the end of the Appendix. Overlays of these results are shown in Figure s 4 12 thro ugh 4 16. Time (sec)Pressure (psi) 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0 50 100 150 200 250 300 350 400 450 Calculated Measured Figure 4 12. Overlay of original calculated load and digitized experimental load for test 3C1 PAGE 66 66 Time (sec)Pressure (psi) 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0 200 400 600 800 1000 1200 Calculated Measured Figure 4 13. Overlay of original calculated load and digitized experimental load for test 3C2 Time (sec)Pressure (psi) 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0 50 100 150 200 250 300 350 400 450 500 Calculated Measured Figure 4 14. Overlay of original calculated load and digitized experimental load for test 3D1 PAGE 67 67 Time (sec)Pressure (psi) 0 0.0025 0.005 0.0075 0.01 0.0125 0.015 0.0175 0.02 0.0225 0 200 400 600 800 1000 1200 Calculated Measured Figure 4 15. Overlay of original calculated load and digitized experimental load for test 3D2 Time (sec)Pressure (psi) 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0 500 1000 1500 2000 2500 3000 3500 4000 Calculated Measured Figure 4 16. Overlay of original calculated load and digitized experimental load for test 3D6 PAGE 68 68 In the pressure e quations, the wave is assumed to be traveling at a speed equal to the soils seismic velocity. However, the recorded pressure time histories show that this is not the case. The arrival times shown on these graphs indicate a wave which has traveled at an average speed much lower than the seismic velocity, as little as three or four times slower. From the pressure equations, it can be seen that a wave traveling at a slower speed will exert less pressure. Since this pressure wave is not elastic and must us e up energy by permanently crushing and moving soil, it would make sense that it would not travel at the same speed as waves use d to measure seismic velocities (ESL TR 8757 1989). The design man ual recommends a rise time of approximately 10% of the arriva l time. The test results indicate a much larger rise time. The rises shown are between 22% and 24% of the arrival time. This more than doubling of the rise time can have a large impact on the overall impulse. To achieve better correlation between the an alytical and experimental loads the individual values used in the pressure calculations were modified for each case until the best match could be found. These best matched values are shown in Table 4 2. Table 4 2 Best matched values Test c (ft/s) n Den sity (lbs/ft^3) Rise Time (% of arrival time) Decay Factor Difference in Peak Pressure (psi) (Compared to Using Measured Pressures) Difference in Max Deflection (in) Difference in Permanent Deflection (in) 3C1 1400 3 112 45 e 21 0.15 0.104 3D1 750 2.75 112 20 e 14 0.06 0.006 3C2 600 3 112 15 e 49 0.068 0.007 3D2 1050 3.25 112 20 e 47 0.054 0.002 3D6 2100 3 112 10 30 90 0.14 0.013 PAGE 69 69 W hile these value changes are a great improvement over the existing method, they are still somewhat inaccurat e. A lthough t he change in rise time can be easily adopted into the methodology, a way to calculate these average wave speeds using only the soil and charge data has not been determined. An example comparison of one of these best fit values and the recorded data is sho wn in Figure 4 17. Time (sec)Pressure (psi) 0 0.003 0.006 0.009 0.012 0.015 0.018 0.021 0.024 0.027 0 50 100 150 200 250 300 350 400 Calculated Measured Figure 4 17. Overlay of best fit calculation and digitized experimental load from test 3D2 Because the only pressure data recorded during the experiments occurred at the center point of the wall, the only valid comparison is one whose pressure is calculated by DSAS only at the center point of the wall. The calculated average loads on the walls surface could not be validated against any real world data, since none is available. Figure 4 18 shows the difference between the calculated average load on the walls face and the central load. These loading functions come from the original calculated load from shot 3C1. PAGE 70 70 Time (sec)Pressure (psi)Comparison of Calculated Central and Average PressuresOriginal Pressure Calculations of Shot 3C1 0 0.002 0.004 0.006 0.008 0.01 0.012 0 100 200 300 400 500 Pressure at Center Average Pressure Figure 4 1 8 Comparison of calculated pressure at the wall center and calculated average pressure 4.4 Summary In this chapter, the methods proposed in Chapter 3 were compared with experimental results. The results obtained from DSAS were in good agreement with the behavior observed in the experiments when the pressure time histories obtained in the actual experiments were used as input for DSAS. However, when the loading was generated using the approach described in this chapter, the correlation between the pressures generated by DSAS and the pressures that were measured in the experiments was poor. Modifications were sug gested, which helped create a closer match to the test data. Although an improvement, these methods are still not entirely accurate, and a method to calculate these new functions using only soil and charge data was unable to be created. PAGE 71 71 CHAPTER 5 CONCL USION AND RECOMMENDA TIONS 5.1 Summary A numerical method for analyzing the response of a n RC box type structure subjected to loading from a buried high energy explosive was developed in this study. This method uses a pair of SDOF models to model the flexure and direct shear modes of response on a selected box wall. An attempt was made to calculate the forcing function created by a buried explosive. A method using existing, semi empirical equations did not compare well with experimental results. Changes to these existing equations were investigated for improvements. While improvements could be made, a pattern that could be used to modify the equations for every case could not be found. Background information on RC boxes and underground blast loading was presented in Chapter 2. The assumptions used in adapting a wall into a SDOF system were discussed. Possible failure modes, namely flexure and direct shear, were presented. The methods for calculating free field soil pressures using elastic wave reflecti ons and transmissions were described. The proposed methodology for creating the two SDOF systems was discussed in Chapter 3. Methods for creating flexural and direct shear resistance functions were presen ted. Modifications to the free field soil equation s and wave behavior in order to generate a single loading function on the wall were also discussed. Experimental data was used to validate the direct shear and flexural response models, once their methods had been coded into the computer program DSAS Val idation for flexure was found, but no experimental data containing direct shear behavior could be found to validate the direct shear methods. An attempt was made to validate the methodology used in creating a PAGE 72 72 load function, but a good comparison to experi mental data was not found. Further improvements were made to the load function calculations for each individual case, but a pattern that could be used to develop an improved method for every case could not be found. 5.2 Conclusions The following conclusions could be made from this study: Provided the actual load time history, the SDOF analysis engine in DSAS can produce accurate results. The existing methodology for calculating pressure in soils caused by a below ground detonation is not accurat e. Two rea sons for this can be verified from experimental data: on average, the blast wave appears to be traveling at a rate much slower than the soils seismic velocity and rise time is greater than 10%. Furthermore, the change in soil density due to compaction b y the pressure wave and the nonlinear propagation of a pressure wave also need to be considered for a more accurate analysis. 5.3 Recommendations for Further Study Based on the knowledge gained from the work completed, the following recommendations for fur ther study can be made: This study was focused on the central part of the walls of a buried box; possible research into the responses of corners where the walls or the walls and roof meet, is worth pursuing. More research into properly calculating the loading functions caused by a buried explosive should be performed, including a look into how changing any of the different variables will change the pressure time history. Specifically, research into calculating the waves ac tual propagation speed is importa nt. Also, a look into how to calculate a more appropriate rise time would be helpful. The possibility of soil arching a ffecting the load should be studied further. As existing methods to simplify soil arching effects are based on the assumption of a unif ormly distributed load, they cannot be used in their current form. Since available experimental data does not include the average load on the wall, v alidation of the methodology used in creating an average loading function for SDOF systems could not be com pleted When this data is obtained, validation should be performed. Additional experiments need to be performed with real buried structures, especially experiments where direct shear failure is likely to occur and experiments using a variety PAGE 73 73 of soil back fills, with extensive soil data collected both before and after the explosion, so that these methods for calculating loading and response functions can be further verified. These experiments should also include pressure gages located throughout the wall, not just on the walls center, as well as gage s to measure the wall s deflection. PAGE 74 74 A PPENDIX KIGER AND ALBRITTON (1980) TESTS The experimental tests used in the validation work are further explained in this section. This includes detailed information on the box compositions, the soil, the test shots and data recorded in each, and the data used for the test calculations. The reason these last values are included is that, at times, a large range of possible values is given in the report. At other times, som e information is not included at all. Therefore, the values used in the actual input files need to be listed. Box Compositions Two boxes were used in these experiments, known as Structure 3C and 3D. The dimensions and layouts of the boxes were different, but the properties of the materials used were the same. The compressive strength of the concrete in Structure 3C was 6,595 psi. In 3D it was 6,613 psi. The No. 4 bars used had an average yield stress of 76,000 psi and an ultimate stress of 124,000 psi. For the No. 6 bars, the yield stress was 71,000 psi and the ultimate stress was 128,000 psi. The typical stress strain curves are shown in Figures A 1 and A 2. For both boxes, the interior dimensions were 4 feet high by 4 feet wide by 16 feet long. Stru cture 3C had a wall thickness of 5.6 inches. This includes the thickness of the roof and floor. The wall thickness in structure 3D was 13 inches. The transverse reinforcement of Structure 3C consisted of No. 4 bars spaced at 4 inches on center in both fa ces and all four sides. L ongitudinal reinforcement consisted of No. 3 bars spaced at 4 inches on center in both faces and all four sides. S hear reinforcement consisted of No. 3 bar shear stirrups spaced at 4 inches on center. PAGE 75 75 Figure A 1 Typical concr ete stress strain curve (Kiger and Albritton 1980) The transverse reinforcement of Structure 3D consisted of No. 6 bars spaced at 4 inches on center in both faces and all four sides. L ongitudinal reinforcement consisted of No. 3 bars spaced at 4 inches on center in both faces and all four sides. S hear reinforcement consisted of No. 3 bar shear stirrups spaced at 4 inches on center. Sketches of these layouts can be seen in Figure A 3. Soils At the test site, there were two major regions of soils, although a silty sand was encountered at 10 feet. No thickness or seismic velocity is given for this silty sand. The top layer of soil is a clayey silty sand. It has a wave velocity of 1345 to 1360 ft/s. This layer exists for a depth of 3 feet. The second layer is a red or tan sandy clay. It has a wave velocity of 2360 PAGE 76 76 to 2590 ft/s. This soil la yer exists f ro m a depth of 3 feet to 26 feet. However, on the test day, the water table was located at a depth of 24 feet. Figure A 2 Steel stress strain curve (Kiger and Albritton 1980) Test Shots and Data Recording Although many test shot s were performed on the structures, only six of these shots were used in this study. There were numerous gages set up throughout the boxes, but only one pressure gage per box. These pressure gages were located at the center of one of the long walls PAGE 77 77 of each box. Also located at the center of the long walls, but on the interior side, was an accelerometer. The resea rchers tried to integrate the acceler ometer data to calculate the wall deflections, but the results do not appear to be accurate. Only five test shots were placed adjacent to the wall with the pressure sensor. These were shots 1 and 2 on Structure 3C (3C1 and 3C 2) and shots 1, 2, and 6 on St ructure 3D (3D1, 3D2, and 3D 6). Shot 3 on Structure 3C (3C 3) was also located on a long box wall, but not on the one with the pressure sensor. The researchers had decided that the second shot had done too much damage to the wall and the use of a fresh wa ll was necessary to obtain good data. Diagrams of instrumentation and shot locations can be seen in Figures A 4 and A 5. Values Used in Computer Calculations Table A 1 lists the input values used in the original calculations with the test date. These val ues were used for both boxes unless otherwise noted. Table A 1 List of values used in computer calculations Item Value Unit Interior Length X 203.2 in Interior Length Y 59.2 in Interior Length Z 59.2 in Burial Depth 24 in Wall, Floor, Roof Thickness es, Box 3C 5.6 in Wall, Floor, Roof Thicknesses, Box 3 D 13 in Concrete f'c 7500 psi Steel Yield 75000 psi Steel Ultimate 90000 psi Steel Strain Hardening 0.00275 in/in Steel Ultimate Strain 0.12 in/in Steel Failure Strain 0.15 in/in Wall Rebar Z, B ox 3C #4 bar # Wall Rebar Z, Box 3D #6 bar # Wall Rebar X #3 bar # Rebar Spacing (all) 4 in Outer Rebar Depth 0.8 in Inner Rebar Depth 4.8 in PAGE 78 78 Table A 1 Continued. Item Value Unit Wave Reflection from Surface yes First Soil Layer Thickness 36 in First Soil Layer Unit Weight 110 lb/ft 3 First Soil Layer Seismic Velocity 1350 ft/s First Soil Layer Attenuation Coefficient 3 First Soil Layer Friction Angle 30 degrees Second Soil Layer Thickness 252 in Second Soil Layer Unit Weight 112 lb/ft 3 Sec ond Soil Layer Seismic Velocity 24 50 ft/s Second Soil Layer Attenuation Coefficient 3 Second Soil Layer Friction Angle 30 degrees Third Soil Layer Thickness 300 in Third Soil Layer Unit Weight 125 lb/ft 3 Third Soil Layer Seismic Velocity 2450 ft/s T hird Soil Layer Attenuation Coefficient 3 Third Soil Layer Friction Angle 30 degrees Flexural Damping 20 % Direct Shear 5 % PAGE 79 79 Figure A 3 Box layouts (Kiger and Albritton 1980) PAGE 80 80 Figure A 4 Shot and instrumentation la youts, box 3C (Kiger and Albritton 1980) PAGE 81 81 Figure A 5 Shot and instrumentation layouts, box 3D (Kiger and Albritton 1980) PAGE 82 82 LIST OF REFERENCES Astarlioglu, S., and Krauthammer, T. Dynamic Structural Analysis Suite (DSAS). Center for Infrastructure Protection and Physical Security, University of Florida, 2009. Biggs, John M. Introduction to Structural Dynamics New York: McGraw Hill, 1964. "Design of Structures to Resist Nuclear Weapon 1985, ASCE Manuals and Reports on Engi neering Practice No. 42, ASCE Hyde, D., ConWep Application of TM5 8551 Structural Mechanics Division, Structures Laboratory, USAE Waterways Experiment Station, Vicksburg, Mississippi, 1992. Kiger, S. A., and Albritton, G.E., Response of Buried Hardened Box Structures to the Effects of Localized Explosions, U.S. Army Engineer Waterways Experiments Station, Technical Report SL 801, March 1980. Krauthammer, T. Modern Protective Structures CRC Press, 2008. Krauthammer, T., et al., 1986 Modified SDOF Analysis of R. C. BoxType Structures Journal of Structural Engineering, Vol. 112, No. 4, pgs 726744 Krauthammer, T. and W.J. Hall, 1982. Modified Analysis of Reinforced Concrete Beams Proceedings of the ASCE Journal of the Structural Division, Vol. 108, No. 2, pgs 457474 Krauthammer, T. and Mehul Parikh, 2005 Structural Response Under Localized Dynamic Loads Proceedings of Second Symposium on the Interaction of NonNuclear Munitions with Structures pgs. 52 55 MacGregor, J.G. and J.K. Wight Rei nforced Concrete: Mechanics and Design Upper Saddle River, N.J.: Prentice Hall 2005. Newmark, N., et al., 1962 A Method of Computation for Structural Dynamics American Society of Civil Engineers Transactions Vol. 127, Part 1, pgs 601 630 Park, Robert and Thomas Paulay. Reinforced Concrete Structures New York: John Wiley & Sons, Inc., 1975. Park, Robert and William L. Gamble. Reinforced Concrete Slabs New York: John Wiley & Sons, Inc., 2000. Parikh, Mehul and T. Krauthammer, 1987 Behavior of Burie d Reinforced Concrete Boxes Under the Effects of Localized HE Detonations Structural Engineering Report ST 87 02, University of Minnesota, Department of Civil and Mineral Engineering Institute of Technology PAGE 83 83 Protective Construction Design Manual 1989, E SL TR 8757, U.S. Air Force Engineering and Services Center, Engineering and Services Laboratory, Tyndall Air Force Base, Florida. "Structures to Resist the E ffects of Accidental Explosions 2008, UFC 3 34002 Tedesco, Joseph W. et al. Structural Dynamic s: Theory and Applications California: Addison Wesley, 1999. Terzaghi, K. and R.B. Peck. Soil Mechanics in Engineering New York: Wiley, 1949. PAGE 84 84 BIOGRAPHICAL SKETCH Nick Henriquez was born in Tampa, Florida in 1984. He stayed in Tampa, where he graduat ed from Jesuit High School in 2003. Nick enrolled at the University of Florida in 2003, completing a Bachelor of Science degree in civil engineering in 2007. During his time as an undergraduate, he became a member of Sigma Nu fraternity. In 2008, at the University of Florida, he began the pursuit of a Master of Science degree in civil engineering with an emphasis in protective structures. While seeking this degree, Nick has worked as a research assistant at UFs Center for Infrastructure Protection and Physical Security (CIPPS). 