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ANALYSIS OF SHALLOW BURIED REINFORCED CONCRETE BOX STRUCTURES SUBJECTED TO AIRBLAST LOADS By KAY HYANG CHEE A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2008 2008 Kay Hyang Chee To my lovely wife ACKNOWLEDGMENTS I thank my advisor, Dr Theodor Krauthammer, for his advice and guidance. I want to express my appreciation to Dr Serdar Astarlioglu for his valuable suggestions and help in the programming aspects. I am grateful to the Defence Science and Technology Agency, Singapore, for the postgraduate scholarship. I thank all of my friends at the Center for Infrastructure Protection and Physical Security, University of Florida and Maguire Village, for a great stay and experience in Gainesville. Last of all, I thank my dear wife, Hiong Suan, for all she has done for me and her willingness to sacrifice many things over the last two years. TABLE OF CONTENTS page A CK N O W LED G M EN T S ................................................................. ........... ............. ..... L IST O F T A B L E S .......................................................................... 7 LIST OF FIGURES .................................. .. ..... ..... ................. .8 LIST OF SYM BOLS .................. ................................ ............ 12 A B S T R A C T ............ ................... ............................................................ 16 CHAPTER 1 IN TR OD U CTION .......................................................................... .. ... .... 17 1.1 Problem Statem ent .................. ................................ ........ .. ............ 17 1.2 Objectives and Scope .................................................. .. ................. 18 1.3 R research Significance ..................................................... ... .. ............ 18 2 BACKGROUND AND LITERATURE REVIEW .............................................................19 2 .1 In tro du ctio n ...................................... ................................................. 19 2 .2 B last L o ad s ......................................................... .............. ................ 19 2.2.1 A irblast from H igh Explosive.................................... .................................... 20 2.2.2 N nuclear D devices .................................. ... .. ..... ............ 21 2 .3 E effects on B uried Structures .............................................................. .....................22 2.3.1 Soil Arching Effect .......................... ........ ... ...... ............ 22 2.4 Dynamic Structural Behavior and Analysis................................ ...................... 24 2.4.1 SingleDegreeofFreedom (SDOF) System................. ............................27 2.4.2 Transformation Factors for Equivalent SDOF...............................................28 2.4.3 Numerical Integration (NewmarkBeta method) ...........................................29 2.5 Behavior of Reinforced Concrete Slabs..................................................... ............. 30 2.5.1 Flexural Behavior: Johansen's Yield Line Theory .......................................30 2.5.2 Flexural Behavior: M embrane Action ................................... .................31 2.5.3 Slab Com pressive M em brane ........................................ ....................... 33 2.5.4 Slab Tensile M em brane.............................................. ........................... 38 2.5.5 Reinforced Concrete Slab Flexural M odel.......................................................39 2.5.6 D irect Shear B behavior ......................................................... .............. 40 2.5.7 Hawkins Shear M odel ................... ................................. 41 2.5.8 Dynamic Resistance Function and Response........................................43 2.6 PressureImpulse Diagrams and their Application ....................................... .......... 46 2.6.1 Characteristics of PI D iagram .............................................. ............... 46 2.6.2 Numerical Approach to PI Diagram ..... .......... ...................................... 48 2 .6.3 M multiple F failure M odes......................................................................... ..... 48 2 .7 S u m m ary ............................................................................ 5 0 3 M ETH OD OLO G Y .............. .......................................... .......................... 51 3 .1 In tro d u ctio n .............................................................................................................. 5 1 3.2 Flexural M ode ................... .......................................................................... ........51 3.2.1 Externally Applied Thrust.................. ..... ...... ..... ............... 51 3.2.2 Numerical Approach for Resistance Curve Calculation .................................54 3.2.3 Variation of Mass and Load Factor...........................................................58 3.3 Soil Structural Interaction ...................................................................................... 60 3.3.1 Influence of Parameters on Soil Arching Effect .............. ..... .............. 60 3.3.2 Effect on SDOF Load and Mass Factor................................................61 3.4 Direct Shear Mode .................... ........................... ......65 3.4.1 R resistance Curve .......... .. .................................. ...... ............ ...... .... 65 3.4.2 Shear M ass and Load Factors ........................................ ........................ 67 3.5 Shear F failure M ode for Slab ........................................ ..........................................68 3 .6 P program F low chart.......... .................................................................... .......... ....... .. 70 3 .7 S u m m a ry .................................................................................................................. 7 0 4 R E SU L TS A N D D ISCU SSIO N S............................................................. .........................73 4.1 Introduction .... .......... ......... ............................................. 73 4.2 V alidation w ith Experim ental D ata.......................................... ........................... 73 4.2.1 Test FH1 ...................... ..................... ............... .........74 4.2.2 Test FH2 ............... ....... ...................... ................. 77 4.2.3 Test FH3 ......... ......... ............. ............... .........81 4 .2 .4 T e st F H 4 ............. ........... .................. ........................................8 4 4.2.5 Test FH5 ............... ....... ...................... ............... ........ 88 4.2.6 Test FH6 ............... ....... ...................... ................. 92 4 .2 .7 S u m m a ry ...................................................................................................... 9 6 4.3 A ssessm ent by PI D iagram s ................................................. ........ 96 4 .4 Su m m ary ...............................................................10 0 5 CONCLUSIONS AND RECOMMENDATIONS .......... .............................101 5 .1 S u m m a ry ................................................................................................................ 1 0 1 5.2 C conclusions .................................................................................................................. 10 1 5.3 Recommendations for Future Study ........................................ ... ........ 102 APPENDIX EXPERIMENT TEST ON SHALLOW BURIED FLAT ROOF S T R U C T U R E S .............................................................................103 LIST OF REFERENCES ......... ......... ................................ 113 BIOGRAPHICAL SKETCH ........................................................................... ....... ........ ... ........115 6 LIST OF TABLES Table page 41 Summary of results ................................ .. ... ..... .................. 96 A Param eters for Foam H est tests......... .................. ................................... ............... 106 A 2 Test FH 1 input param eters....... ......... .......... .......... ....................... ............... 107 A 3 Test FH 2 input param eters....... ......... .......... .......... ....................... ............... 108 A 4 Test FH 3 input param eters....... ......... .......... .......... ....................... ............... 109 A 5 Test FH 4 input param eters .................................................................. ............... 110 A 6 T est FH 5 input param eters ......... ........................................................... ..................... A 7 Test FH 6 input param eters .................................................................. ............... 112 LIST OF FIGURES Figure page 21 B last pressuretim e curve.......................................................................... ................... 2 1 22 Soil arching demonstrated by trap door experiment ......................................................23 23 Soil arching as function of depth of burial................................. ........................ .......... 24 24 Uniform beam subject to arbitrary load ................................................. ...... ......... 25 25 Yield lines development in a uniformly loaded simply supported slab .............................31 26 Loaddeflection curve for twoway RC slab with laterally restrained edges..................... 33 27 Assumed yield line pattern for uniformly loaded slab with restrained edges ..................34 28 Plastic hinges of a restrained strip .................................. .....................................34 29 Portion of strip betw een plastic hinges ........................................ ......................... 35 210 Conditions at positive moment yield section ....................................... ...............37 211 Uniformly loaded plastic tensile membrane. ........................................ ............... 39 211 Flexural resistance m odel for slab. ............................................ ............................ 40 213 Slab in direct shear failure m ode ............................................................ .....................4 1 214 Hawkins model for direct shear stressslip relationship ......................................... 42 215 Equivalent SDOF systems for structural element...........................................................44 216 D ynam ic flexural resistance functions................................................................... ......45 217 Dynamic direct shear resistance function. .............................................. ............... 46 218 Typical response spectra and PI diagram ........................................ ...... ............... 47 219 Search algorithm for PI diagram ............................................. ............................. 49 220 PressureImpulse diagram with two failure modes............................................................49 31 M odel for externally applied thrust........................................................ ............... 52 32 Calculation of externally applied thrust................................ ................... ...... ........ 53 33 Stress and strain distributions across reinforced concrete section............... .................54 34 Restrained strip with external thrust ............................................................................ 55 35 Portion of strip between plastic hinges with external thrust............................................55 36 Variation of load and m ass factor ..................................................... ...................58 37 Variation of soil arching factor with friction angle and burial depth .............................61 39 V aviation of XL.................................... ......................................................... 63 310 Variation of kM ....................................................... ........ ....... .......... 65 311 D irect shear m odel for tw ow ay slab ........................................ ........................... 66 312 Direct shear resistance curve for twoway slab ...................................... ............... 67 313 Deformed shape for direct shear response................................ ................................. 68 314 Slab in shear failure m ode......................................................................... ...................69 315 Resistance curve for slab with shear failure mode.................................. ...............71 316 Program flowchart ...................................... .. ...... .... .... .. ............72 41 Post test view of FH 1 ................... ............... .............. .......... 75 42 FH 1 flexural displacement time history................................................................. 75 43 FH 1 flexural resistance function.............................................. .............................. 76 44 FH1 direct shear displacement time history............................... .............. 76 45 FH 1 direct shear resistance function........................................... ........................... 77 46 Post test view of FH2 ................. ............................... .......... 78 47 FH2 flexural displacement time history................................................................. 79 48 FH 2 flexural resistance function...................... ...................... ................. ............... 79 49 FH2 direct shear displacement time history............................... .............. 80 410 FH 2 direct shear resistance function........................................... ........................... 80 411 Post test view of FH3 ................... ............................. .......... 82 412 FH3 direct shear displacement time history............................... .............. 82 413 FH 3 direct shear resistance function........................................... ........................... 83 414 FH3 flexural displacement time history................................................................. 83 415 FH 3 flexural resistance function................................................ ............................ 84 416 Post test view of FH 4 ............... ............................ ............ ........... .......... 85 417 FH4 flexural displacement time history................................................................. 86 418 FH 4 flexural resistance function...................... ...................... ................. ............... 86 419 FH 4 direct shear displaced ent tim e history................................................. ................. .... 87 420 FH 4 direct shear resistance function........................................... ........................... 87 421 Post test view of FH 5 ..................... .. ....................................................... 88 422 FH5 direct shear displacement time history............................... .............. 90 423 FH 5 direct shear resistance function........................................... ........................... 90 424 FH5 flexural displacement time history............................................................91 425 FH 5 displaced ent tim e history ................................................. ............................. 91 426 F H 5 resistance function .......................................................................... .....................92 427 Post test view of FH6 ................. ............................... .......... 93 428 FH6 flexural displacement time history....... ................ ...................................94 429 FH 6 flexural resistance function...................... ...................... ................. ............... 94 430 FH6 direct shear displacement time history............................... .............. 95 431 FH 6 direct shear resistance function........................................... ........................... 95 432 FH 1 PressureIm pulse diagram ............................................................... .....................97 433 FH 2 PressureIm pulse diagram ...................... .... ......... ........................ ............... 97 434 FH 3 PressureIm pulse diagram ............................................................... .....................98 435 FH 4 PressureIm pulse diagram .................................. ........................ ............... 98 436 FH 5 PressureIm pulse diagram ...................... .... ......... ........................ ............... 99 437 FH 6 PressureIm pulse diagram ...................... .... ......... ........................ ............... 99 A Experiment test configuration for FH3 ........................... ....... ............................... 104 A2 Construction dimensions and details of FH ...................................... .......................105 A3 Construction dimensions and details of FH5 ............................... .............105 LIST OF SYMBOLS A, Concrete crosssectional area Asb Area of reinforcement c Neutral axis depth at Section 1 c' Neutral axis depth at Section 2 C', Concrete compressive force at SectionI Cc Concrete compressive force at Section 2 C's Steel compressive force at SectionI Cs Steel compressive force at Section 2 Ca Soil arching ratio C Damping E Elastic modulus Ec Concrete elastic modulus fc Concrete cylinder strength fs Tensile strength of the reinforcement fy Yield strength of steel reinforcement Fe Equivalent force F(x,t) Arbitrary distributed force H Depth of burial h Thickness of slab I Moment of inertia I Impulse K Spring stiffness Ke Equivalent stiffness Ke Elastic stiffness of direct shear degreeoffreedom Ku Negative stiffness at segment CD of Hawkins model KM Mass factor KL Load factor Ko Coefficient of static lateral earth pressure Kr Generalized (modal) stiffness for the rh mode L Length of structure Lx Long span of slab Ly Long span of slab m Unit mass mu' Resisting moment at Section 1 mu Resisting moment at Section 2 M Lumped mass Me Equivalent mass Mr Generalized (modal) mass N External thrust nu Membrane force p(t) Pressure function PB Average pressure acting on structure Ps Uniform pressure acting on soil surface Po Peak load Po Atmospheric pressure Pr Generalized (modal) force of the rh mode Pso Peak pressure q Displacement of the selected representative point q Velocity of the selected representative point q Acceleration of the selected representative point R Dynamic resistance function R1 Residual function 1 R2 Residual function 2 S Surround stiffness ta Time of arrival to Positive phase duration to Negative phase duration t Strip outward lateral movement t Time td Loading function time duration T' Steel tensile force at Section 1 T Steel tensile force at Section 2 Tx Yield force of reinforcement per unit width in the xdirection Ty Yield force of reinforcement per unit width in the ydirection T Kinetic energy Tn Natural period V Shear force V Potential energy w Beam displacement w Uniform load per unit area W Width of structure x, Displacement x, Velocity x, Acceleration Xmax Maximum displacement /8 NewmarkBeta integration constant /,i Ratio of depth of the equivalent ACI stress block to neutralaxis depth E Axial strain ES, Steel strain sC, Concrete strain Ec, Concrete ultimate strain at failure 0 Virtual rotation 3 Slab central displacement S Angle of internal friction O(x) Shape function r (x) Normal vibration modes for the beam AL Ratio of load factors A, Ratio of mass factors p, Ratio of total reinforcement area to the area of plane that it crosses ,q,. Virtual displacement At Time step interval Amax Maximum shear slip r, Direct shear resistance (elastic) TL Limiting direct shear capacity r, Maximum direct shear resistance 0' Natural circular frequency E Damping ratio Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science ANALYSIS OF SHALLOW BURIED REINFORCED CONCRETE BOX STRUCTURES SUBJECTED TO AIRBLAST LOADS By Kay Hyang Chee May 2008 Chair: Theodor Krauthammer Major: Civil Engineering A numerical method for the dynamic analysis of shallowburied reinforced concrete box type structures subjected to air blast loadings is presented in this study. The proposed method was based on the SingleDegreeofFreedom (SDOF) approach, where two loosely coupled SDOF systems were considered to take into account the flexural and direct shear mode of structural response. The effects of compression and tension membrane in reinforced concrete slabs and soilstructural interaction were considered in the study. The resistance functions for the structure for each structural response mode were generated and used in the dynamic analysis. The issue of soilstructure interaction and relationship with structural behavior, in terms of soil arching effect was examined in more detail. A rational model was proposed to incorporate the soil arching effect and a varying SDOF equivalent load and mass factors in the dynamic analysis. The algorithm was implemented in a computer program. Results of the study were validated using available experimental data from a number of buried reinforced concrete boxes that were tested by other investigators. CHAPTER 1 INTRODUCTION 1.1 Problem Statement Recognizing the potential of soil as a protective medium long ago, man has built underground and buried structures to provide shelter and refuge from his potential enemy. By doing so, the soil's performance in the provision of protection to the structure is harnessed through its inertia effects and its ability to diffuse load and dissipate energy. Commonly, reinforced concrete is used as the construction material in protective engineering applications due to its suitable properties and economical value. In addition, most underground or buried structures can usually be approximated as boxtype structures. Therefore, in protective engineering applications, the design and evaluation of the performance of a buried reinforced concrete box structures under transient severe loading is an important consideration. Since the behavior of buried reinforced concrete box structure under the effects of intense pressure pulses applied to the soil surface is of interest, the structural element located closest to the applied load will strongly affect the performance of the entire structure. Therefore, the behavior of a buried structure can be adequately represented by the response of the roof slab which form part of the rectangular box structure. The design of structural elements under transient loading requires dynamic analysis to be carried out to determine the response characteristics, such as the displacement time history and reaction forces. Analytical studies have been performed previously by various researchers using finite element codes or singledegreeoffreedom (SDOF) models. The SDOF approach is comparatively a simple tool, but with an accurate prediction of the structural behavior, it is a useful tool in preliminary design or parametric studies and can be combined with more advanced analytical techniques to reduce the total computational time and cost. 1.2 Objectives and Scope This study aims to develop a reliable, simple and accurate analytical approach and numerical procedure to perform dynamic analysis for the design and evaluation of buried reinforced concrete box structure against airblast loads. The procedure will consider nonlinear resistance mechanisms for reinforced concrete slab structure in the flexural and direct shear mode of behavior. The scope of this study is limited to reinforced concrete slabs subjected to transient uniformly distributed airblast pressure load on the soil surface. This study includes modification of the resistance function to consider inplane compressive force due to internal membrane effect and external thrust due to wave propagation through the soil. The SDOF equivalent load and mass factors are also varied with respect to the slab response regime. This study also includes the evaluation of the dynamic soil arching effect and its corresponding effect on the SDOF land and mass factors. The proposed procedure is to be validated with available experimental test data in order to evaluate its accuracy. 1.3 Research Significance This study can offer a methodology for a reliable, simple and accurate dynamic analysis engine to study the behavior of a shallowburied reinforced concrete box structure. The approach includes rational and physicsbased resistance functions, taking into account the effect of soil structural interaction (e.g. wave propagation and soil arching effect) and improves on the approach to approximate the real continuous system into an equivalent SDOF system in order to provide an accurate numerical result. CHAPTER 2 BACKGROUND AND LITERATURE REVIEW 2.1 Introduction Since humans recognized the potential of soil as a protective medium, underground structure have been constructed to provide shelter and refuge from their enemy. In order to design an underground structure or to evaluate its structural performance, the first requirement is to characterize the expected performance under the specified design loads. With the necessary loading applied, the relevant mode of structural behavior has to be modeled accurately and the response results from dynamic analysis can be used for design purpose. If the main design concern is a specific response limit state (rotation, failure, etc.), a suitable computational aid to use is a PressureImpulse diagram. In this study, the form of loading considered is from blast and the corresponding response of reinforced concrete box structures roof slabs. A brief introduction to blast loads and its effects on buried structures are presented in Sections 2.2 and 2.3. Blast loadings are transient and the response is highly dependent on the peak load and duration. The dynamic structural behavior and analysis are reviewed in Section 2.4. Section 2.5 focuses on the different structural response mode for reinforced concrete slabs and their corresponding resistance model under static and dynamic loading. Lastly, the background of pressureimpulse diagrams and their applications are presented in Section 2.6. 2.2 Blast Loads Generally, the most common sources of explosions and blast loads are derived from either chemical (High Explosive, HE) or nuclear materials. The environment created by an explosion consists of the following effects: * Airblast * Groundshock * Ejecta * Fragments * Fire, thermal and chemical (nuclear explosions only) * Radiation (nuclear explosions only) * Electromagnetic pulse (EMP) (nuclear explosions only) 2.2.1 Airblast from High Explosive In an open air High Explosive (HE) explosion, the reaction generates gases at very high pressure and temperature, causing a violent expansion of these explosive gases and the surrounding air is forced out of the volume it occupies. The shock front is essentially vertical, reflecting the sudden rise in pressure due to the explosion. It is a layer of compressed air, forms in front of the explosive gases. As the gases expand and cool, their pressure decreases. The pressure of the blast wave front also falls with increasing distance from source. Eventually, the pressure falls a little below atmospheric pressure because the momentum of the moving gas molecules. The gases are overexpanded near the explosion location and a reversal of flow towards the source occurs. This is the negative phase which is characterized by a pressure lower than the ambient air pressure. Eventually, the pressure and temperature of the gases returns to equilibrium (Smith and Hetherington 1994, Tedesco et al. 1998, Krauthammer 2008). An idealized pressuretime function for an airblast shockwave in free air is shown in Figure 21, where Po is the atmospheric pressure; Pso is the peak pressure; ta is the time of arrival; to is the positive phase duration and to is the negative phase duration. The impulse of the blast wave is defined as the area under the pressure time curve and can be calculated simply by t +t I = p(t)dt (21) ta Pressure Pso  Negative Phase  to Ambient, Po time ta to Positive Phase Figure 21. Blast pressuretime curve. 2.2.2 Nuclear Devices A nuclear device can deliver its explosive effects from air burst, surface burst or shallow burst. When the nuclear device is exploded at an altitude below 100,000 ft, about 50% of the released energy will result in blast and shock (ASCE 1985). For high altitude bursts above 100,000 ft, they generate only strong EMP effects (which should be considered in facility design) but are of little interest to the structural engineer. The characteristics of the blast pressure wave in a nuclear explosion are similar to those of a high explosive (HE) explosion and are a function of the weapon yield, the height of burst and the distance from the burst. There is also dynamic pressure which results from the mass flow behind the shock front. The dynamic pressure is a function of the gas density and the flow velocity. As with the HE explosion, there is both a positive and negative phase for the over pressure and for the dynamic pressure (Krauthammer 2008). 2.3 Effects on Buried Structures Although buried structures offered protection from aerially delivered weapons and airborne blast effects, but these structures can be vulnerable to the transient stresses propagated through the soil and rock in which they have been constructed (Smith and Hetherington 1994). Other types of loadings such as buried charges or structural penetration are also important but they are not considered in this study. A shock wave will be induced in the soil when an air blast is applied to the free soil surface (soilair interface). The shock wave travels downwards in the vertical direction until meeting with the structure. Based on data from soil stress gages and interface pressure gages in nuclear tests, the shock front can be considered as quite planar (Krauthammer et al. 1986). The possible modulation in the interface pressures will appear as a result of wave reflections and soilstructure interaction effects, including soilarching. The resultant pressure time history may not be uniformly distributed over the roof slab of the buried structure. 2.3.1 Soil Arching Effect Loads acting on the buried structure are influenced by the interaction between the structure and the surrounding soil. An effect of such interaction is soil arching and it is defined as the ability of a soil to transfer loads through a system of shear stresses from one location to another in response to a relative displacement between the locations. A stiffer structure in the soil tends to attract more loads, while stress will be diverted from or around buried structures that are less stiff (Kiger 1988). Soil arching occurs when there is a relative motion between structure and soil. The classical approach for computing soil arching is by the use of the "trapdoor" mechanism (see Figure 22), as discussed in Terzaghi and Peck (1948). There are two types of soil arching, namely: * Passive arching: The structure moves away from the loading soil, and the soil cannot follow it due to shear resistance. * Active arching: The structure is pushed into the soil. c d ," '. '. "'(a) Apparatus for S":investigating arching in layer of sand above Shielding trap door in '' :! horizontal platform. o' b Y (b) Pressure on platform and trap door before and after slight lowering of (b) r f r door. Figure 22. Soil arching demonstrated by trap door experiment (Terzaghi and Peck 1948). The soil arching ratio Ca is defined as the ratio of the average pressure on the unsupported clear span of the structure to the applied surface structure. For a shallow burial depth, the arching ratio is given by Equation 22 (ASCE 1985 and Kiger 1988): SPB_ 2Kotan().(W + L)H Ca PS WL (22) (22) where PB is the average pressure acting on structure; Ps is the uniform pressure acting on soil surface; Ko is the coefficient of static lateral earth pressure; q is the angle of internal friction in the soil and W, L, H are the width, length and depth of burial of the structure respectively. Typical arching factors for rectangular and arch structures are shown in Figure 23. Besides reducing the average pressure acting on the buried structure, the actual pressure distribution is also no longer uniform due to soil arching. With a responding roof slab, the pressure at the center will be smaller while towards the edge the pressure is much higher. A parabolic pressure distribution can be assumed (Kiger 1988). I) BURED RECTAMBU i SWUCTURE URIED ARCH OR CVICREFt 9 B,4  X 0.6 0S Ko A* K,.# Pa"a K*08 1( .2 SD COHESIVE SOl COHESIVE SOIL 0 0 750O6L 0 7.'6 O76 0 L AL OWL 0 kL 0s 0 'BI ibU DEPTH OF BURIAL Figure 23. Soil arching as function of depth of burial (ASCE 1985). 2.4 Dynamic Structural Behavior and Analysis Structural behavior under time dependent loading can be obtained by dynamic analysis. The dynamic equilibrium of a system can be described by the equation of motion. An important result from the equation is the displacement time history of the structure subjected to a time varying load (Tedesco et al. 1998). All structures are in reality distributed mass and stiffness systems and are referred to as distributed systems, or continuous systems. Each system consists of an infinite number of degrees of freedom and can be considered as a discrete small element connected by springs to all other elements. The governing equations for continuous system can be expressed in partial differential equations and analytical or closedform solutions can be obtained only for relatively simple continuous systems with welldefined boundary conditions. Using an example of a uniform beam subjected to an arbitrary distributed force F(x,t) as shown in Fig 24, the equation of motion for the system is given by 04W 02W El a4 na F(x,t) (23) ax4 t2j F(xt) Fdx I M+ MM dx EI, m I V + dx x w(xOt) L 8 Figure 24. Uniform beam subject to arbitrary load. The normal vibration modes for the beam ,r (x) must satisfy the boundary conditions and a4 El a4x nwo, b(x) 0 (24) where or (x) represents the normal vibration mode shape for the rh mode. The normal modes are orthogonal functions that must satisfy the mass orthogonality relationship L f for r s SmnOr(x) (x)dx= fo (25) o r for r = s where the generalized (modal) mass is given by L M,=r mq(x)dx (26) The general solution in terms of the normal modes r (x) and normal coordinates qr(t), w(x,t)= r (x) q (t) (27) r=1 Establishing the kinetic and potential energy gives T = mw2(xt)dx = 1 M, q (28) 2 2 r L 2W(X, 2 ri V = El t)dx = Kr q2 (29) 0 I a2 I r= r where Kr is the generalized (modal) stiffness for the rth mode. Finally, the generalized (modal) force of the rth mode Pr must be determined from the work done by the applied force F(x,t) acting through the virtual displacement 6q,. Therefore L c L 8W= F(x,t) r ,(x)q dx = qr F(x,t)4r (x)dx (210) 0 d=1 =1 0 L and P, =F(x,t)Or(x)dx (211) 0 Substituting the above expressions for T, V and Pr into the Lagrange's equations d OT OT cV S  + = P, (212) dt q q, 2q, d rt c aq cqr aqr the equation of motion in normal coordinates is Mrijr + Krqr = Pr (213) Equation 213 represents the uncoupled equations of motion for r = 1,2,3,..., o . For approximation to the continuous system of a real structure, only a few of the lower modes have responses of any significance for practical purposes, and in some cases only the fundamental mode is of importance. Depending on the chosen approximation, the real system can be considered as either a multidegreeoffreedom (MDOF) system or a singledegreeof freedom (SDOF) system. 2.4.1 SingleDegreeofFreedom (SDOF) System As mentioned in the previous section, in reality all structural system consists of an infinite number of degrees of freedom. An infinite number of independent spatial coordinates are necessary to completely define the geometric location of all the masses and stiffness of a structure (Tedesco et al. 1998). However, it is frequently possible to approximate the real system to a single degree of freedom having equivalent parameters of load, mass and stiffness where the fundamental mode of response is significant. It is advantageous to model the structure as a SingleDegreeofFreedom (SDOF) as this approximate method permit rapid analysis of complex structures with reasonable accuracy (Biggs 1964). SDOF formulation gives designer valuable information on the dynamic characteristics of the system and they are usually used in preparation of detailed analysis using more advanced methods (Krauthammer 1998). The equivalent system is selected so that the deflection of the concentrated mass is the same as that for some significant representative point on the structure, e.g. midspan of beam or center of slab. Since the time scale is not altered, the response of the equivalent system, defined in terms of displacement and time, will be exactly the same as the chosen representative point. As presented in the previous section, the equation of motion for a structure system (Equation 2 13) can be simplified for the SDOF system (inclusive of damping) as M,q + Cq + K,q = F, (214) where Me, C, Ke and Fe are the equivalent mass, damping, stiffness and force; q, q, q is the displacement, velocity and acceleration value of the selected representative point. The constants of the equivalent system are evaluated on the basis of an assumed shape function for the deflected structure. When the total load, mass, resistance and stiffness of the real structure are multiplied by the corresponding "transformation factors", we obtain the parameters for the equivalent singledegreeoffreedom system. 2.4.2 Transformation Factors for Equivalent SDOF To convert an actual continuous structure into an equivalent singledegreeoffreedom system, the equivalent parameters of the system like the equivalent mass and equivalent loading and resistance function have to be evaluated. Biggs (1964) used transformation factors, denoted by K, to convert the real system into the equivalent system. The equivalent mass of a SDOF system for a structure with continuous mass can be determined by Equation 215 and the mass factor, KM, is defined as the ratio of the equivalent mass to the actual total mass of the structure (Biggs 1964). M, =nm2 (x)dx (215) L M Jm2 (x)dx KM L (216) Mt Mt The equivalent force on the SDOF system for distributed loads can be found by Equation 217 and the load factor, KL, is defined as the ratio of the equivalent to actual total force F,= p(x)q(x)dx (217) L f p(x) 0(x) dx KL = L (218) Ft Ft where m is the unit mass, p(x) is the distributed loading acting on structure and O(x) is the assumed shape function on which the equivalent system is based. Biggs (1964) had tabulated transformation factors for beams and slabs with various types of support conditions. For a fixed end beam or oneway slab, the load factor varies from 0.50 for the plastic case to 0.64 for the elasticplastic case and 0.53 for the elastic case. For the mass factor, it varies from 0.33 for plastic case to 0.50 for the elasticplastic case and 0.41 for the elastic case. 2.4.3 Numerical Integration (NewmarkBeta method) The analytical, or closedform, solution of the equation of motion can be cumbersome even for relatively simple excitations. Therefore, for most practical problems, numerical evaluation technique is employed to obtain the dynamic response (Tedesco et al. 1998). In this study, the NewmarkBeta method is used in the direct integration of the equation of motion and is briefly summarized below (Newmark and Rosenblueth 1971): A. For an equivalent SDOF system, the equation of motion is as follows: mi + ck + kx = F(t) (219) B. Let the values ofxi, , and Y, be known at time t = ti. Let ti+l = ti + At, where At is the time step interval. Assume a value of5 ,. At C. Compute the value 1 = x, + (5, + Y,) (220) 2 1 D. Compute the value x,, = x, + At + (2 p)Y, (At)2 + p6+, (At)2 (221) 2 E. Compute a new approximation to Y, using equation of motion (Equation 219) F. Repeat steps B to D beginning with the newly computed Y+,, until a satisfactory degree of convergence is attained. G. Step B is consistent with a straight line approximation to Y in the interval considered. IfP = 1/4, the method is consistent with a straight line variation of x in the same interval (constant average acceleration). If/ = 1/6, method corresponds to a parabolic variation. In this study, it was set as / = 1/6. H. The numerical method starts at t=0, the time instant when the load is applied. The initial condition is that the mass is at rest. 0 F(t = 0 = o = 0. m This numerical integration method is unconditionally stable. However, a proper value of the time step interval must be chosen to ensure sufficient accuracy. The time step is dependent on the natural period of the system (Tn) and the loading function time duration (td). According to Bathe (1996) and Clough and Penzien (1993), the chosen time step is given by Tt At < Min( d) (222) 10 12 2.5 Behavior of Reinforced Concrete Slabs A reinforced concrete box effectively is made up a number of reinforced concrete slabs. Therefore, it is essential to first understand the load resisting mechanism of reinforced concrete slabs. This section discusses the two possible failure mechanisms, namely the flexural and direct shear mode, and their respective resistancedisplacement functions. Sections 2.5.1 and 2.5.2 discuss the structural behavior of reinforced concrete slabs in flexure using yield line theory and consideration of actual membrane actions respectively. Sections 2.5.3 and 2.5.4 focus on the compressive membrane and tensile membrane behavior. Section 2.5.5 presents a rational flexural model for slabs which was proposed by Krauthammer et al. (1986). Section 2.5.6 discusses the direct shear mode of failure for slabs and the Hawkins model for direct shear failure is presented in Section 2.5.7. Lastly, the dynamic resistance functions and response is covered in Section 2.5.8. 2.5.1 Flexural Behavior: Johansen's Yield Line Theory The Johansen's yield line theory is a method for the limit analysis of reinforced concrete slabs. The ultimate load of the slab is calculated by postulating a collapse mechanism that is compatible with the boundary conditions (Park and Gamble 2000). The moments at the plastic hinge lines are the ultimate moments of resistance of the sections, and the ultimate load is determined using the principle of virtual work or equations of equilibrium. A yield line refers to a line in the plane of the slab about which plastic rotation occurs and across which the reinforcing bars are yielding. When a slab is progressively loaded to failure, yielding of the tension steel occurs at section of maximum moment with a large change in section curvature while the moment remains almost constant at the ultimate moment of resistance. As the load is increased, the yield line propagates from the point at which yielding originated until eventually the yield lines have formed in sufficient numbers to divide the slab into segments that can form a collapse mechanism. The positions of the yield lines developed are governed by the arrangement of reinforcement, boundary conditions and loading (Park and Gamble 2000). The development of the yield line pattern for a uniformly loaded simply supported rectangular slab is shown in Figure 25. First Yielding Further development of yield lines Collapsed mechanism formed Figure 25. Yield lines development in a uniformly loaded simply supported slab (Park and Gamble 2000). 2.5.2 Flexural Behavior: Membrane Action The resistance of reinforced concrete slabs computed using the traditional approach (e.g. ACI 2005) of oneway or twoway slabs formulation will be adequate for normal design applications, but they are not accurate representations of the actual capacity of the slabs. Often under the extreme loading conditions in the blast and shock environment, compressive and tensile membrane actions in slabs can enhance the ultimate structural capacity of the slab. Typical design serviceability requirements like deflections and cracking need not be enforced in some field of applications such as protective design, since moderate to severe degree of damage may be acceptable under such extreme loading. Figure 26 shows the typical loadcentral deflection curve of a uniformly loaded twoway rectangular slab with laterally restrained edges (Park and Gamble 2000). When the applied load is increased from point A to B, although the initial resistance is developed by conventional two way slab mechanism, a compression membrane mechanism sets in with the corresponding increase in the central deflection due to the restraint of the outward of movement of the slab edges. The induced compressive membrane force in the slabs results in an enhancement of the flexural strength. Tests have demonstrated that the ultimate load may be significantly (about two to eight times) higher than that given by the Johansen's yield line theory (Section 2.5.1), particularly if the boundary restraint is stiff, high spandepth ratio and small reinforcement steel ratio (Park and Gamble 2000). With continued loading and increase of the deflection beyond point B, the load carried by the slab decreases rapidly because of a reduction in the compressive membrane force. As point C is approached, the membrane forces in the central region of the slab change from compressive to tensile. Beyond point C, with an increased loading, the effect of restrained edges sets in and allows the slab reinforcement to act as a plastic tensile membrane with fulldepth cracking of the concrete over the central region of the slab due to the large stretch of the slab surface. The slab continues to carry further load with an increase in deflection until point D when the reinforcement fractures. Tests have indicated that for heavily reinforced slabs the load at point D can exceed the ultimate load at point B. Therefore, in many cases tensile membrane action also provides a useful means of preventing catastrophic failure at ultimate conditions. Load Yield line (Johansen) Yield line Elastic Elastic Central Deflection Figure 26. Loaddeflection curve for twoway RC slab with laterally restrained edges. 2.5.3 Slab Compressive Membrane The compressive membrane behavior of slabs covers two ranges of deflections, point A to B and point B to C (as shown in Figure 27). As the load is increased from A to B, the slab behavior is initially elastic, combined with inelastic behavior at critical sections at higher loads. Yield line pattern for the slab is fully developed at point B. As deflection increases from B to C, the deformation is mainly caused by plastic rotation at the yield lines. Therefore, the slab is deforming as a mechanism in the range BC. Plastic theory can be developed first for a restrained strip and then extended to a twoway slab. For a twoway slab, it can be assumed to be composed of strips running in the x and y directions. The strips have the same thickness as the slab. The xdirection strips contain only x direction steel and the ydirection strips contain only ydirection steel. The yield line pattern of the slab is as shown in Figure 27. Then yield sections of the strips lie on the yield lines and have the same deflection as the actual slab. The corer yield lines are simplified to be at 450 to the edges (Park and Gamble 2000). This simplification of assuming corner lines at 450 results in not more than 3% error in theoretical ultimate load for slabs with all edges fixed against restraint (Park and Gamble 2000). Each of these strips can be analyzed using the plastic theory presented in Park and Gamble (2000). A fixedend strip with plastic hinges developed is shown in Figure 28. This strip is initially of length L and is fully restrained against rotation and vertical translation at the ends. The ends of the strip are considered to be partially restrained against lateral displacement, and the outward lateral movement at the other end is t. A Ix I~~  Negativemoment yield line  Positivemoment yield line xDirection strips  or I Y4eld sections (b) Figure 27. Assumed yield line pattern for uniformly loaded slab with restrained edges (Park and Gamble 2000). A) Actual slab. B) Systems of strips. 2 ^ ^  t^U 3 L Figure 28. Plastic hinges of a restrained strip (Park and Gamble, 2000). Compressive membrane action is dependent on the restriction of small lateral displacement, and the behavior of the strip is sensitive to any lateral displacements that may occur. The lateral displacement t may be calculated from the movement of the boundary system under the action of the membrane force (Park and Gamble 2000). The strip shown in Figure 28 is considered to have symmetrically positioned plastic hinges. The symmetry assumption leads to the necessary assumptions that the top steel at opposite ends must be equal and the bottom steel is constant along the length. The top and bottom steel may be different. It is assumed that at each plastic hinge that the tension steel has yielded and the concrete has reached its compressive strength The portions of the strip between the plastic hinges are assumed to remain straight. The sum of the elastic, creep and shrinkage axial strain, e, will be constant. The change in dimensions of end Section 12 due to and t is shown in Figure 29. BL+ 0.5sd1 2O)L+ t V A ottorm steel Figure 29. Portion of strip between plastic hinges (Park and Gamble 2000). Based on the geometry of the deformations, the compatibility equation can be written as g L 2t c'+c= h (s+) (223) 2 23 L where c' and c are the neutral axis depths at yield Sections 1 and 2 respectively, and h is the thickness of the strip. For equilibrium, the membrane forces acting on Sections 1 and 2 of the strip are equal, C; + C T'= C, + C, T (224) where C'c and Cc are the concrete compressive forces, C's and Cs are the steel compressive forces and T' and T are the steel tensile forces, acting on crosssections 1 and 2 respectively. The concrete compressive forces can be written for a unit width strip as C; =0.85 f'/1c' (225) C = 0.85f'/ c (226) where f, is the concrete cylinder strength and /, is the ratio of the depth of the equivalent ACI rectangular stress block to the neutralaxis depth (ACI 2005). Using Equations 224, 225 and 226, rT'TC' +c c c = (227) 0.85f'f/ Solving simultaneously Equations 26 and 210, the neutral axis depths are given as h L2 2t T'TC' +Cs c'= (E + T) + (228) 2 4 23 L 1.7f'/, h 3 L2 +2t T' T C' +C 2 4 23 L 1.7f/1s Figure 210 the shows the conditions at a positivemoment yield section of unit width. The stress resultants at the section Co, Cs and T are statically equivalent to the membrane force nu, acting at mid depth, and the resisting moment mu, summed about the mid depth axis. Therefore, for a unit width strip, n = C, + C, T = 0.85f/'1c + C, T (230) m, =0.85f/'fc(0.5h 0.5/,1c)+ C, (0.5h d') + T(d 0.5h) (231) where c is given by Equation 228. For a negative moment yield section, mu is given by an equation similar to Equation 231, and for equilibrium nu = nu. Unit width 0 0 O Cross section Elevation 0,85/; y  r 4 Neutral < c ax is Strain Stress internal distribution d istribu tion actions Figure 210. Conditions at positive moment yield section (Park and Gamble 2000). Considering end sections 12 or 34 of the strip, the sum of the moments of the stress resultants at the yield sections about an axis at middepth at one end can be written as h 81 8L2 2t g2 81 (1 )+ (f, 3)+ (/, 1)(sE+ )+ (2 2 2 4 43 L 8h 2 m' +m,, n,,5 = 0.85 fc',1h2 2 4 (2 + L(1 )(E + 2t) 8 4(2 + 2t 4h 2 L 16hg2 L 1 h 3 34 (T'TC +C,)2 +(C +C,)( ) 3.4fc' 2 2 hd +(T'+ T)(d + 2 2 (232) If Sections 12 or 34 of the strip is given a virtual rotation 0, the virtual work done at the yield sections is given by (m' +m, n,,)0 (233) By equating the work done (Equation 233) to the work done by the loading on the strip, an equation relating the deflection of the strip to the load carried can then be obtained (Park and Gamble 2000). 2.5.4 Slab Tensile Membrane Towards the end of the compressive membrane action range, the large stretch of the slab surface causes the cracks at the central region of the slab to penetrate across the whole thickness of the slab depth, and the load is entirely carried by the reinforcing bars acting as a tensile membrane through caternary action. With further deflection (beyond point C in Figure 26), the region of tensile membrane action gradually spreads throughout the slab, and the load carried increases until the steel reinforcement starts to fracture at point D. Figure 211 shows the forces acting on a uniformly loaded plastic tensile membrane for a rectangular slab. In Park and Gamble (2000), a linear relationship between load and deflection (Equation 2 34) for a uniformly loaded slab is given as an approximation for the tensile membrane region of point CD of Figure 26. L 3(234) Ty 4. ()n1)/2 I 1/cosh nx . 1,5, n3 2LY ,T where w is the uniform load per unit area, 6 is the central deflection, Lx and Ly are the long and short span of the slab, Tx and Ty are the yield force of the reinforcement per unit width in the x and ydirections respectively. Ty dx TY dx 1 Y' f Sw dx +dy ir rdy I T rdy  l dy y 2 Membrane Forces acting on element Figure 211. Uniformly loaded plastic tensile membrane. 2.5.5 Reinforced Concrete Slab Flexural Model The plastic theory and its assumptions in the Section 2.5.3 are only applicable at large deflections. Therefore, the initial portion of the load deflection relationship will not be representative at deflections when the slab is still within the elastic or elasticplastic regime. Krauthammer (1984) and Krauthammer et al. (1986) proposed a rational model to rectify the shortcoming. A second order polynomial is fitted to segment AB and a straight line is fitted to segment BC. To better describe the tensile membrane resistance, a straight line which is not required to pass through the origin is proposed for segment CD. The proposed model is shown in Figure 212. Park and Gamble (2000) reported that a good estimate of the ultimate load of the slab would be obtained at a central deflection of about half the slab thickness (actually half the effective depth). The flexural model as shown in Figure 212 is completed with approximations that the displacements at point B and point C at 0.5h and h respectively, where h is the slab thickness. Krauthammer et al (1986) showed that this approach was able to represent accurately based on comparison against experimental test data. This approach was modified by Frye (2002) to consider the differences for slender, intermediate and deep slabs. For slender slabs, the corresponding central deflections at point B and C are 0.5h and 1.0h. For deep slabs, the corresponding central deflections at point B and C are 0.07h and 0.17h. For intermediate slabs, the displacement and resistance is linearly interpolated between the slender and deep models. Load B D max  Linear function Quadratic function A I I 0.5 h Deflection Figure 211. Flexural resistance model for slab (Krauthammer 1984). 2.5.6 Direct Shear Behavior Kiger and Getchell (19801982) and Slawson (1984) both reported that reinforced concrete slabs exhibited another type of behavior under severe and rapid loading. Beside failure in flexural mode, some slabs failed in a direct shear mode. A photograph of a test specimen which failed in a direct shear is shown in Figure 213. This type of shear failure is characterized by slipping and large displacement along the vertical interface shear plane (Krauthammer et al. 1986). The shear failure produced a vertical failure plane at the edge of roof and both the top and bottom steel exhibited necking and were severed nearly flush with the failure plane (Crawford et al. 1983). Direct shear failure will occur at the very early stage of the loading, usually a fraction of a millisecond, and before any significant dynamic flexural response can be observed. Similarly, once the slab survived the initial loading phase without failure in direct shear mode, it was observed that a flexural mode of failure will dominate which will occur at a much later time. ir1 Figure 213. Slab in direct shear failure mode (Slawson 1984). Crawford et al. (1999) consider this direct shear mode of failure as an important element in the blast effects design process. This mode is associated with geometric or load discontinuity, but not with flexure, and is caused by the high shear inertia forces which do not exist under static or slow dynamic loads. 2.5.7 Hawkins Shear Model The direct shear model used in this study is based on a model proposed by Hawkins (1972). The model describes the static interface shear transfer in RC members with well anchored main reinforcement in the absence of axial forces. Krauthammer et al. (1986) modified the model to account for the effects of compression and rate effects by applying an enhancement factor of 1.4 (see Figure 214). This same approach is used for this present study. Shear Stress S ENHANCED  ORIGINAL Tm B C A' / / K, Z D' E' e IA TL   D E Ke A1 A, A3 Shear Slip Amx Figure 214. Hawkins model for direct shear stressslip relationship (Krauthammer et al. 1986). A detailed description of the model is given below. Region OA: The response is assumed elastic and the slope, Ke, is defined by the shear resistance, ze, for a slip of 0.004 inch. The resistance is given by the following expression r, = 165+0.157f' (235) where both ,, and f, are in psi. The response should be taken to be elastic and not greater than r /2. Region AB: The slope of the curve decrease continuity with increasing displacements until a maximum strength, Tm is reached at a slip of 0.012 inch. The maximum strength is given by m = 8' +0.8p"tf (236) where both rm, f, and fy are in psi, p,, is the ratio of total reinforcement area to the area of plane that it crosses and fy is the yield strength of the reinforcement. Region BC: The shear capacity remains constant with increasing slips. Point C corresponds to a slip of 0.024 inch. Region CD: The slope of the curve is negative, constant and independent of the amount of reinforcement crossing the shear plane. The slope is given by K = 2000 + 0.75f" (237) Region DE: The capacity remains essentially constant until failure occurs at a slip ofAmax For a well anchored bar, the slip for failure in inches is given by Amx =2 12 (238) max 2120 where 900 x 9 (239) 2.86 c and db is the bar diameter (in inch). The limiting shear capacity, r, is given by 0.85Ashfs L (240) Ac where Asb is the area of reinforcement, fs is the tensile strength of the reinforcement and A, is the crosssectional area. 2.5.8 Dynamic Resistance Function and Response The governing equation of motion for the equivalent SDOF system (see Fig 215) is given by the following (Krauthammer et al. 1990). R F, (t) Flexural Y(t) + 2g9' xk(t) + (241) Me Me Direct M,\her y(t) + 2 y(t) + R V(t) (242) Me Me where x(t), k(t) and x(t) are the flexural displacement, velocity and acceleration respectively; y(t), y(t)andy(t) are the direct shear slip, velocity and acceleration respectively; Me is the equivalent mass; R is the dynamic resistance function; 0' is the natural circular frequency; is the damping ratio; Fe(t) is the equivalent forcing function; and V(t) is the dynamic shear force x, y P (t) V(t) I V(t) A Pe(t) V(t) Me M, x(t), x(t), x(t) I M y(t),J y(t), y(t) M R C R, C, B C Figure 215. Equivalent SDOF systems for structural element (Krauthammer et al. 1990). A) Continuous structural system. B) Flexural Response. C) Direct Shear Response. The modeling of unloadingreloading paths is important in the analysis of a nonlinear dynamic behavior. For an elastic perfectly plastic resistance function, the typical loading and unloading path is as shown in Figure 216A (Tedesco et al. 1998). Krauthammer et al. (1990) proposed a more realistic hysteretic loadingunloading looping as shown in Figure 216B. As the loading acting on the slab increases, the resistancedisplacement follows the resistance curve in the corresponding direction, (from Point O to Point A to Point B). If flexural failure (Point C) is not reached and unloading occurs, positive unloading path is assumed to follow a straight line BD, which has a stiffness value equals to the initial stiffness. Beyond Point D where negative unloading occurs, the unloading path is assumed to follow the straight line DB', where B' is a mirror image of the point of the last maximum displacement attained at Point B. Reloading path (e.g. EF) is assumed to remain parallel to BD and then traces towards Point B. If the reloading exceeds the displacement at Point B, it will reload along the resistance curve and Point B will march forward. The procedure will repeat for the next unloading cycle with a new position of Points B and B' The unloadingreloading paths will affect the amount of internal damping from the hysteretic energy dissipation. For the direct shear mode of behavior, the unloadingreloading of the resistance curve follows the same procedures for the flexural resistance curve. Resistance, R Resistance, R A C Rm  displacement F D displacement S E B' C' Rmax A B Figure 216. Dynamic flexural resistance functions. A) Typical elastic perfectly plastic function. B) Hysteretic loadingunloading. Shear Stress E' C D Shear slp Figure 217. Dynamic direct shear resistance function. 2.6 PressureImpulse Diagrams and their Application For structural dynamic analysis and design, it is often the final states (e.g. maximum displacement and stresses) of utmost relevance, instead of a detailed response time history of the structure. Pressureimpulse (PI) diagrams are characteristic curves that describe the behavior of a structural component under different time dependent loads. These diagrams, often known as "isodamage curves", were developed to aid the assessment of structure against blast (May and Smith 1995). Detailed studies on pressureimpulse diagrams for beam and slab have been covered by Soh and Krauthammer (2004) and Ng (2004). Therefore, only a brief introduction on pressureimpulse diagrams and their applications will be covered. 2.6.1 Characteristics of PI Diagram A typical response spectrum for an undamped, perfectly elastic SDOF system is shown in Figure 218(a). In this figure, Xmax is the maximum displacement, M is the lumped mass, K is the spring stiffness, Po is the peak load, td is the load duration and T is the natural period. By defining a different set of axes, the same response spectrum can be transformed into a PI diagram, Figure 218(b). The response spectrum focuses the influence of scaled time on the system response, while the PI diagram shows the combination of peak load and impulse for a given damage level (Soh and Krauthammer 2004). With a damage level defined, the PI curve indicates the combination of pressure and impulse values that will cause the specified damage. The curve divides the PI diagram into two regions which indicate either failure or nonfailure cases. Pressure and impulse points falling to the right and above the threshold curve indicates failure in excess of the specified damage level criterion. To the left and below the curve indicates no failure is induced. p(t) M ^Ax K x_ Initial p  Tangent PK Impulsive Regime Dynamic Regime Quasistatic Regime 0 ;<   Impulsive Dynamic Quasistatic  Regime Regime Regime T xma A B Figure 218. Typical response spectra and PI diagram (Soh and Krauthammer 2004). A) Shock Spectrum. B) PI diagram. In structural dynamics, there is a strong relationship between the structural response and the ratio of the load duration to the natural period of the structure (Biggs 1964, Clough and Penzien 1993). This relationship can be categorized into the impulsive, dynamic and quasistatic regimes. As seen in Figure 218, the PI diagram better differentiates the impulsive and quasi static regimes, in the form of vertical and horizontal asymptotes. 2.6.2 Numerical Approach to PI Diagram Closed form solutions of PI diagram can be obtained for idealized structures subjected to a simplified load pulse (Ng 2004). However, in order to allow complex nonlinear resistance functions and complex loading conditions to be considered, a numerical approach to generate the PI diagram must be adopted. PI diagram can be generated numerically by performing many single dynamic analysis. Each result from a dynamic analysis will determine whether the pressure and impulse combination is in the failure or nonfailure region. With sufficient runs, a threshold curve can be plotted. Since it is computationally expensive to run all possible pressure and impulse combinations, an efficient search algorithm must be employed to locate the required threshold points (Krauthammer et al. 2008). Soh and Krauthammer (2004) and Ng (2004) developed numerical procedures to numerically generate PI diagram. Blasko et al. (2007) developed a more efficient search algorithm. The procedure uses a single radial search direction, originating from a pivot point (Ip, Pp) which is located in the failure zone of the PI diagram (see Figure 219). Iterations using Bisection method are carried to generate the threshold curve. This approach can be applied effectively to any structural system for which a resistance function can be defined. The pressure impulse diagrams which will be presented in the later part of this study are generated numerically using this same approach. 2.6.3 Multiple Failure Modes In general, the response and failure for most structures can occur in more than one mode. Although flexure is usually the predominant mode, but under certain circumstances, failure may occur in other mode (e.g. direct shear). If there exists two a single failure modes, the PI diagram will consists of two threshold curves, each representing a failure mode. The true threshold curve will therefore be represented by the lower bound of the two curves as shown by the dotted line (see Figure 220). With the two threshold curves plotted, it is possible to identify the actual failure mode by plotting the pressureimpulse combination and examining which region the point is located. Figure 219. Search algorithm for PI diagram (Blasko et al. 2007). A) Establish pivot point. B) Data pivot search Pressure A Failure in Mode 2 Failure in Mode 1 & 2 * Mode 2 (Direct Sh Safe \\ Failure in Mode 1 Mode 1 (Flexure)  ear) Impulse Figure 220. PressureImpulse diagram with two failure modes. 2.7 Summary The background of the effects of blast loads on buried structures and the dynamic structural behavior and analysis were presented in this chapter. Pressureimpulse diagrams and their application were also briefly discussed. The different mode of behavior for reinforced concrete slabs under loading was discussed in greater detail in this chapter. This chapter is the basis for the methodology to derive the numerical method for the dynamic analysis of shallowburied reinforced concrete boxtype structures subjected to air blast loadings in Chapter 3. CHAPTER 3 METHODOLOGY 3.1 Introduction For dynamic analysis of any structures, the material and constitutive models of the structure must first be derived in order to describe the relationship between the dynamic resistance (e.g. bending moment, shear forces) and the structure response (e.g. displacement). Then a suitable dynamic structural model (e.g. continuous system, multi degree of freedom, single degree of freedom) can be chosen to accurately represent the mechanical characteristics of the structure. This chapter covers the methodology to generate the resistance functions for reinforced concrete slabs and the buried box structure in both the flexural mode (Section 3.2) and direct shear mode (Section 3.4). The issue of soil arching and the required modifications to the load and mass factors are discussed in details in Section 3.4. The effect of shear failure on the slab resistance is also discussed in Section 3.5. Lastly, a flowchart of the procedure is presented in Section 3.6. 3.2 Flexural Mode As described in Section 2.4.1, the center point of the slab is chosen as the reference point for the singledegreeoffreedom system and the slab response is in accordance with the resistance function described in Section 2.5.4. However, for a buried box structure, several modifications have to be introduced into the approach described in Section 2.5.2 in order to account for certain conditions that will affect the system behavior. 3.2.1 Externally Applied Thrust The peak structural resistance in the compression membrane mode can be enhanced if there is an external inplane compressive force being applied to the slab (Krauthammer 1984). The inplane compressive force may exist in the form of prestressing force (Meamarian et al, 1994) or due to horizontal component of the vertical forces for a box structure buried in soil (Krauthammer et al. 1986). For the box type structures under consideration, the inplane compressive forces are generated by the horizontal component of the pressure pulse that propagates vertically through the soil from the air blast load on the surface (see Figure 31). These compressive forces vary with time and therefore have to be calculated at every time step of the analysis. Pv(t) Nroof Roof Nroof Ph(t) Ph(t) Floor Floor Nfloor Figure 31. Model for externally applied thrust (Krauthammer et al. 1986). In this study, the following procedure is implemented: 1. The wall is subdivided into n number of layers (see Figure 32). The depth at the middepth of each of the n layers is calculated and denoted as h,. The time of arrival at each layer can be calculated by considering the seismic wave velocity of the soil. 2. The vertical pressure pulse in the soil is traced at each time step and the horizontal stress component is converted into point loads acting the each of the n layers. 3. The horizontal stresses generated by the propagating pressure pulse in the soil can be computed as a ratio of the vertical stress. The ratio is the coefficient of static lateral earth pressure, Ko. The ratio is dependent on the effective friction angle of the soil and it can be estimated as about 1.0 for clay and about 0.5 for sand (Krauthammer et al. 1986). 4. The compressive thrust at the roof and floor can then be calculated from the points load acting at each layer. The compressive thrust, Nx and Ny, will be used in the formulation for the calculation of the enhanced membrane peak resistance. 5. When the whole vertical pressure pulse propagates beyond the floor level of the buried box structure, the compressive thrust (due to static earth pressure) remains constant and therefore need not be reevaluated again at the further time steps. Pv(t) Nroof i i i=2 Roof h, Ph(t) /  Wall divided into n layers Floor Nfloor Figure 32. Calculation of externally applied thrust. 3.2.2 Numerical Approach for Resistance Curve Calculation Following the plastic theory described in Section 2.5, some modifications were introduced in order to calculate the slab resistance at point B and point C (see Figure 25) more accurately. An externally applied thrust is included in the formulation. Instead of using the approximation of using the ACI stress block for the concrete compressive stress, the approach in this study is to divide the concrete into layers parallel to the neutral axis and the stresses and forces for all layers are determined based on the appropriate stressstrain relationship chosen (see Figure 33). Compressive steel Ecu .diI z si i/ fE ch ======== nu Neutral Saxis Tensile unit width linear strain distribution RC Slab Strain Distribution Concrete and steel Stress Figure 33. Stress and strain distributions across reinforced concrete section. In this study, the concrete stressstrain relationship was based on the Hognestad model (MacGregor and Wight 2005) whereas for the reinforcing steel, the stressstrain model by Park and Paulay (1975) was used. In Figures 34 and 35, the restrained strip with plastic hinges is applied with an external thrust. Using similar compatibility and equilibrium equations presented in Section 2.5.2, an iterative procedure is implemented in order to find the neutral axis depths that will be satisfied for each strip displacement. With the neutral axis depth, the corresponding axial forces and moment in the strips can then be calculated. The slab resistance at point B and C of the resistance curve can then be determined based on the strips forces and moment. Plastic hinges / N 2 3 SL L  1 LE tI<  L 2t Figure 34. Restrained strip with external thrust. BL+ 0,5E(1 20)L+ t Figure 35. Portion of strip between plastic hinges with external thrust. Based on the strain distribution and stress strain relationship (Figure 33), the concrete and steel stress for a unit width of the slab at yield Section 2 (Figure 35) can be expressed as a function of the neutral axis depth, c. f, = Fn(es,) n= c (c d) (31) fe n(c, lFn c c (3 where s,, and cc are the steel and concrete strain, s is the ultimate concrete strain at failure, d, is the depth of reinforcing steel, z, is the depth of concrete layer, c is the neutral axis depth and Fn() represent the material stressstrain function. The forces and moment for the steel and concrete can be expressed as F,, =f, A,, (3 F, = jcAz (3 m,, = F,,(h/2ds,) (3 m, = F (h/2 z,) (3 where Fsi and Fci are the steel and concrete layer force, msi and mci are the steel and concrete layer moment about midheight of slab section, Asi is the steel area, Az is the concrete layer thickness, h is the slab thickness. The total section moment, mu, and axial force, nu, can be calculated as: steel layers concrete layers m = ms, + im, (3 steel layers concrete layers n = Fs, + F, (3 3) 4) 5) 6) 7) 8) Similarly, for yield Section 1 (Figure 35), m,' and n,' can also be expressed as a function of the neutral axis depth, c'. Based on the geometry of the deformations, the compatibility equation can be written as g h L2 2t 9) c' + c = h (E +2t) (39) 2 28 L 2t n 2 (n N) and (E + )= + = Etotal (310) L h.Ec L.S where t is the outward lateral movement of the slab, Ec is the concrete elastic modulus and S is the surround stiffness. For equilibrium, the membrane forces at Section 1 and 2 are equal n, =nu (311) Equations 310 and 311 can be rearranged and written as a function in c and c': R h+ fL2 n 2(n N) R,(c')= c'+ch+ ( + N)=0 (312) S2 23 h.Ec L.S R2(c,c')= nu n = 0 (313) For any displacement value, 3, the values of the neutral axis depth c and c' can be determined by solving Equations 312 and 313. In this study, NewtonRaphson iteration (Zienkiewicz and Taylor 2005) is used to solve the two equations numerically. With values of the neutral axis depth c and c', the values of mu, mu' and nu for each strip can be calculated using Equations 37 and 38. By equating the internal virtual work done (Equation 2.16) to the external virtual work done by the loading on the strip, the load carried by the strip can be obtained for any assumed displacement value of the strip. As presented in Section 2.6.2, the whole slab is assumed to be divided into xdirection and ydirection strips. With an assumed deflection value at the center of the slab, the strip deflection can be calculated and the force and moments in each strip are solved using the above procedure. The maximum slab resistance can then be determined using the appropriate displacement value at Point B on the slab resistance curve. The resistance at Point C on the resistance curve is also calculated using the same procedure. The only difference is that the membrane forces are set as zero. The resistance at Point C corresponds to the Johansen's yield line load. 3.2.3 Variation of Mass and Load Factor As presented in Section 2.4.2, to convert a continuous structure into an equivalent single degreeoffreedom system, the equivalent parameters like the equivalent mass and equivalent loading and resistance function have to be evaluated using the appropriate mass and load factor. Instead of applying a constant mass and load factor for analysis, a variation of the load and mass factor is proposed. As shown in Figure 36, the variation of the factors corresponds to the resistance curve for the slab. In the compressive membrane region of the resistance curve, the factors will varied from the elastic value at point A to the elasticplastic value and then to the fully plastic value at point B, where the resistance is at the maximum. For the region represented by Point C to Point D, the factors used are the tension membrane value. A linear variation is assumed for the factors in between Points A to D. Load Factor Mass Factor Tension membrane Elasticplastic Elastic Plastic B Compression Transition Tension membrane membrane A I I 0.5h h Deflection Figure 36. Variation of load and mass factor. As mentioned in Section 2.4.2, the load and mass factors for beams and slabs under various loading and support conditions for different range of response can be found in Biggs (1964). The values for the load and mass factors for the tension membrane can also be found using the same approach. Taking a fixed end beam or oneway slab under uniform load as an example, the load and mass factor for the tension membrane can be calculated using Equations 2 16 and 218. For the beam (or oneway slab) of length L in tension membrane, under a uniform applied load, the deformed shape can be assumed as a parabola. Therefore, the deformed shape function can be written as 4 4x2 A(x) = x for 0 From Equations 216 and 218, m f M2 ()dx 4 x 4x22 dx KM M (315) Mt m.L L 8 K = =0.533 15 pf (t)qO(x)dx p(t)Lf 4x 4x dx KL = o (316) F, p(t).L p(t).L 2 KL = = 0.667 3 Using the same approach, once the deflected shape for twoway slabs have been assumed, the load and mass factors can be obtained by integration over the entire slab surface. 3.3 Soil Structural Interaction As presented in Section 2.3.1, one of important effect caused by the interaction between the structure and the surrounding soil is soil arching. Buried reinforced concrete box structures are usually much stiffer than their surrounding soil medium and will tend to attract load. Experimental data also showed that the pressure acting on the flexible center of the roof slab is significantly less than the applied overpressure acting on the free soil surface. The roof edges are relatively stiffer since they are supported by the walls and the pressure acting there are much higher. The pressure distribution is therefore not uniform. The load reduction due to soil arching is accounted for by using the soil arching factor, Ca, as given by Equation 22. The load distribution effect has to be accounted for by adjusting the load and mass factors. 3.3.1 Influence of Parameters on Soil Arching Effect The soil arching factor, Ca, represent the ratio of the average pressure acting on the roof slab to the applied surface pressure. Looking at Equation 22, we can see that the factors that will influence the value of the arching factor includes, the friction angle q!, the span ratio of the roof slab and the depth of burial of the roof slab. Bowles (1996) provides some representative values for the angle of friction for different types of soil. Friction angle for cohesionless soils vary from 200 for loose silty sand to about 460 for dense sand. For cohesive soils such as clay, the friction angle is appreciably smaller than cohesionless soil. For saturated clayey soil with very small shear strength, the friction angle can be assumed as zero and there will be no soil arching effect. The depth of burial for the roof slab is expressed as a ratio of the short span of the slab and in this study for shallow buried box structures, the ratio of burial depth to short span is limited to a maximum value of 1.0. With regards to the slab span ratio, the arching ratio will be the maximum for a square slab, given the same soil properties and burial depth. The variation of the soil arching factor (for a slab span ratio =1) with respect to the friction angle and burial depth is shown in Fig 37. From Figure 37, one can assume conservatively that the soil arching factor Ca will varies from a value of 0.3 (maximum soil arching effect) to a value of 1.0 (no soil arching) for different combination of parameters, including the friction angle, burial depth and span ratio. Variation of Soil Arching Factor (Span ratio =1) 1 .1 0.90.2L 0.8 0.7 0. 6 L 0.6L 0. 8L 0.5 burial depth=l. OL 0.4 0.3 0 5 10 15 20 25 30 35 40 45 Friction Angle Figure 37. Variation of soil arching factor with friction angle and burial depth. 3.3.2 Effect on SDOF Load and Mass Factor As presented in Section 2.4.2, the load factor and mass factor can be calculated based on the Equations 216 to 218. For a uniform loading case, Biggs (1964) had tabulated these factors for beam and slab with different support conditions. Soil arching effect changes the load distribution and the uniform pressure applied on the soil surface is no longer uniform at the buried slab level. It is assumed that the pressure distribution due to soil arching now follows a parabolic shape (Figure 38). The pressure at the centre of the slab or beam is assumed to be a factor/f of the uniform pressure given by Ca p(t), where p(t) is the applied pressure at the soil surface. Factor/ will vary between 1 and 0, whereby value of 1 represent there is no soil arching effect; and value of 0 represent the presence of a maximum soil arching. Factor/ will give an indication on the relative strength of the soil arching effect, based on the soil and geometric properties given in Section 3.3.1. p(t) Parabolic Pressure Distribution Depth of Burial {. Uniform Pressure Distribution U*Ca*p(t) f*1* ,___C__ P p*Ct*p(t)) 1./9 L/2 .Figure 38. Parabolic pressure distribution under soil arching. Based on the assumed parabolic pressure distribution and Equation 218, one can calculate the new load factor, KL, which will account for the effect of nonuniform pressure distribution acting on the slab in the equivalent SDOF system. The ratio AL defined by: KL where K'L is the load factor with soil arching parabolic pressure; and KL is the normal load factor under uniform load. The variation of ratio AL is plotted in Figure 39 for the four different cases, namely the elastic, elasticplastic, plastic and tension membrane. The plot shows how the load factor changes with the soil arching factor. A linear function between the soil arching factor, Ca, and the ratio AL can be derived based on the plot. This function was used in the computer code to adjust the load factors used in the dynamic analysis based on the calculated arching factor value. 1 0.9 0.8 4. 0. 7 1. elastic 2L 2. elasticplastic 3. plastic 1. 4. tension 0.5 0.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 soil arching factor soil arching factor Ca = 0.3 Ca = 1.0 Figure 39. Variation of AL. Based on Figure 39, AL Elas,, =0.816C, +0.1837 (318A) AL Elashcplash =0.612C, +0.3877 (318B) AL Plash =0.714C, +0.2857 (318C) AL Tenson =0.571C, +0.4286 (318D) Similarly, the mass factor has also to be adjusted due to soil arching effect. The mass factor for the structural slab will remain the same since soil arching does not actually affect it. Instead, the weight of the soil overburden which is acting on the slab will also be affected in the same manner described above. Due to active soil arching, more weight of the soil will be acting at the edges where the slab is stiffer and lesser weight on the center. Therefore, with a strong soil arching effect, it can be expected that less soil mass is active in the system response. Using the same parabolic distribution profile for the soil mass, one can calculate the new soil mass factor, KM, which will account for the soil arching effect on soil mass participation in the equivalent SDOF system. The ratio A, defined by: AM (319) KM where K'M is the mass factor with soil arching parabolic profile; and KM is the normal soil mass factor with uniform mass distribution. The variation of ratio AM is plotted in Figure 310 for the four different cases, namely the elastic, elasticplastic, plastic and tension membrane. The plot shows how the soil mass factor changes with the soil arching factor. A linear function between the soil arching factor, Ca, and the ratio AM can be derived based on the plot. This function is used in the computer code to adjust the soil mass factors based on the calculated arching factor value. Based on Figure 310, AM Elash, =1.039Ca 0.039 (320A) AM Elashclas =0.863Co +0.137 (320B) AM Plashe =1.0 C (320C) A ren,,on =0.816C +0.1837 (320D) Cz 4. 0.6 4 1. elastic 2. elasticplastic XM r 3. plastic .4 4. tension 0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 soil arching factor soil arching factor Ca = 0.3 Ca = 1.0 Figure 310. Variation of A . 3.4 Direct Shear Mode In the dynamic response for reinforced concrete slabs, beside the usual dominant flexural mode, it had been experimentally observed that slabs when subjected to severe impulsive loading have also shown to exhibit direct shear mode of failure. 3.4.1 Resistance Curve As presented in Section 2.5.7, the direct shear model used in this study is based on a model proposed by Hawkins (1972) and modified by Krauthammer et al. (1986). For reinforced concrete beam or oneway slab, the formulation presented in Section 2.5.6 can be used to generate the direct shear resistance curve. For a twoway slab where the longitudinal reinforcing steel in the xdirection and ydirection are different, the following approach is adopted to find the effective resistance curve. The governing direct shear SDOF equation of motion can be written as Mex ,x (t) + C., w (t) + R, = V, (t) (321) Mey i, (t) + Cy wy (t) + Ry = V, (t) (322) where Mex and Mey are the equivalent mass in x and ydirection; Cx and Cy are the damping, Rx and Ry are the direct shear resistance; Vx(t) and Vy(t) are the dynamic shear force; and w(t)andiw(t) are the direct shear slip velocity and acceleration. As shown in Figure 311, it is assumed that the reinforcement is the same at the supports for each x and ydirection. Since the direct shear failure occurs at the very early stage of the loading, one can assume that the flexural mode of deformation is not significant and the whole slab displaced downwards as a rigid body motion. Therefore, Equations 321 and 322 can be simplified to w, (0 = (0t) = Wy (t); (t)= w (t) = Wy (t); (t) = ix (t) = wy (t) (Mex + AM,) w, (t) + (C + C, ) s (t) + (Rx + R( ) = (Vx (t) + V, (t)) AM, (t)+ Cs (t)+ R = V (t) (323) Simultaneous failure in ystrips xdirection ydirection r Simull. ... ..... _ . ...... M failure i, '. 1 1np '    \   \\\\\\\\ S t (ty s WFt) Ry31 Cy Figure 311. Direct shear model for twoway slab. Therefore, the equivalent resistance for a twoway slab can be obtained by considering the resistance in the xdirection and ydirection and adding them together (see Figure 312). The equivalent mass and loading for the singledegreeoffreedom system in direct shear was calculated by applying the appropriate transformation factors, using the same procedures presented in Section 2.4.2. The shear mass and load factors are presented in the next section. Shear Stress Resistance R, = Rx+Ry Resistance R, /Resistance R \ SFailure in y I /Failure in x Shear Slip max (xdir) max (ydir) Figure 312. Direct shear resistance curve for twoway slab. 3.4.2 Shear Mass and Load Factors The equivalent shear mass and load factors are computed based on the assumed mode and deformed shape of the slab under direct shear failure mode. Since the assumption for the slab under flexural mode deformation is a symmetrical plastic hinge formation, it requires that the steel reinforcement at the either sides of the support is the same. Therefore, simultaneous shear failure at the supports will occur and the deformed shape is as shown in Figure 313. Based on the deformed shape, a shear mass factor of 1.0 can be taken for the structural slab. In addition, for a buried box structure, the mass of the soil overburden has to be considered as well. Since the direct shear failure (if it was to occur) happened very early in the loading stage and the entire roof slab was pushed into the box, followed by the soil overburden, soil arching effect can be assumed be neglected in direct shear mode. Therefore, the entire mass of the soil overburden can be assumed be effective. Simultaneous xdirection failure in ystrips ydirection Simultaneous failure in xstrips Deformed Shape (and distribution of inertia force) Figure 313. Deformed shape for direct shear response. Using the same assumed deformed shape, the shear load and resistance factor can also be taken as 1.0, where the total resistance and loading for the entire slab is used in the SDOF equation of motion. 3.5 Shear Failure Mode for Slab The flexural and direct shear modes of behavior of reinforced concrete slab have been considered in the previous Sections. Shear is generally not critical when slabs carry distributed loads and supported by walls or beams since the maximum shear force per unit length is relatively small (Park and Gamble 2000). However, shear failure can become critical when the span to effective depth ratio is small and the corresponding flexural resistance due to membrane action increases. A photograph of a test specimen tested by Slawson (1984) which failed in a shear mode is shown in Figure 314. The shear failure occurs near the wall support where the shear stress level is the highest. Figure 314. Slab in shear failure mode (Slawson 1984). In order to model the shear failure mode, it was proposed to implement a simplified modification to the flexural resistance curve as shown in Figure 315. When the applied load is increased from point A, the initial resistance will follow the path towards point B, which is representative of the flexure resistance in the compressive membrane zone. With continued loading and if the shear strength of the slab section is lower than the maximum flexural resistance, wmax, the slab will failed in shear at point B'. With the increase of load and deflection beyond point B', the slab will continue to deform in the flexure mode towards the Johansen's yield line load (point C). Beyond point C, the slab will behave in the tension membrane mode until the slab reinforcement failed at point D. The nominal shear strength of a reinforced concrete section is given by: V, = V, + V (324) where Vn is the nominal shear strength, Vc and Vs are the shear strength provided by concrete and steel reinforcement respectively. Nawy (2000) states that for a deep beam or oneway slab section, where the clear span to effective depth ratio is less than 5, the shear resisting force of concrete and shear reinforcement can be calculated using the following expressions. V = 2f b wd (325) A, l+l,/d A, ll, 11,,d S AI+ d+AhI d fyd (326) s. 12 s, 12 where bw is the width; d is the effective depth; f, is the concrete strength; fy is the steel yield strength; In is the clear span of beam/slab; Av is the total area of vertical reinforcement; Avh is the total area of horizontal reinforcement; Sv is the horizontal spacing of the vertical reinforcement and Sh is the vertical spacing of the horizontal reinforcement. 3.6 Program Flowchart The flowchart of the proposed procedure to generate the resistance function and solving the equation of motion for the required dynamic response is shown in Figure 316. The approach consists of two SDOF systems for evaluating the flexural response and the direct shear response separately. For the flexural mode or response, the resistance function has to be recalculated at each time step since the maximum resistance is dependent on the wall force for a buried box structure. For the plotting of PressureImpulse diagram, this program will be used to solve the system for each pressureimpulse iteration run in order to generate the threshold curve, following the process described in Section 2.6.2. 3.7 Summary This chapter presented the methodology to generate the resistance function for buried box structure for both flexural and direct shear mode of behavior. A variation of the load and mass factor with respect to the resistance curve and the modification on these factors due to soil arching effect were discussed. The modification to consider shear failure mode was also presented. The proposed methodology was implemented in a computer language and the numerical analysis results generated will be presented in Chapter 4. Load B D Flexural ma  / Flexural mode /C Shear B'/ strength,ws Transition A Johansen's Load Shear Failure AB Ac Deflection Figure 315. Resistance curve for slab with shear failure mode. Generate Flexural Resistance Function Generate Direct Shear Resistance Function Direct Shear Response Using NewmarkBeta Integration Figure 316. Program flowchart. CHAPTER 4 RESULTS AND DISCUSSIONS 4.1 Introduction The proposed procedure discussed in Chapter 3 was implemented in a computer programming language to test its viability. The results generated using the procedure are compared with the available data from past experimental work carried out on shallow buried reinforced concrete box structures subjected to airblast loading. This validation of the numerical procedure will be presented in Section 4.2. In Section 4.3, the assessment on the same set of experimental results using PressureImpulse diagrams was demonstrated. 4.2 Validation with Experimental Data The capability of the proposed numerical procedure to perform dynamic analysis for reinforced concrete box structure subjected to airblast loads is presented. Experimental data from the tests conducted by Kiger and Getchell (1980) was used to validate the proposed methodology presented in Chapter 3. The experimental test series conducted by Kiger and Getchell (1980) consisted of a total number of seven tests on shallow buried reinforced concrete box structures subjected to airblast loads. The airblast loads was a simulation of a distant nuclear explosion with a sharp rise time and a uniform pressure over the top of the soil surface. Details of the experiment are given in the Appendix. The test series consisted of a total of 7 cases, whereby six of them are single bay rectangular box structures and another one a multibay box structure. Dynamic analysis was conducted and the flexural and direct shear response time history and resistance functions were generated for each test. The forcing function was obtained from the airblast gauge on the free surface and the damping ratio used was 20% for flexural analysis and 5% for direct shear analysis. A high damping ratio for the flexural mode was used to consider the significant energy dissipation due to soilstructure interaction (Krauthammer et al. 1986). The numerical results were compared against the measured test data and the comparison is presented in the following sections. 4.2.1 Test FH1 Test FH1 was conducted in a sand (noncohesive) backfill at a depth of burial (DOB) equal to 50% of the short clean span. The reinforced concrete box had wall, floor and roof thickness of 5.6 inches, giving the roof slab a span to effective depth ratio of 10. The structure had one percent principal reinforcing steel in each face, with a concrete strength of 7000psi. The test charge density was 0.9 lb/ft3 and produced a peak pressure of 2400psi. The roof and floor of the structure suffered cracking and some permanent deflections, but there was no structural failure. The top surface had longitudinal cracks, located roughly above the inside walls, almost the entire length of the structure. The roof had a maximum permanent deflection at its midspan of about 0.44 inch. A photograph of the box slab after the sand backfill was excavated is shown in Figure 41. The displacement time history for the flexural degreeoffreedom is plotted in Figure 42. The numerical results show that the permanent displacement at the center of the roof slab is about 0.5 inch, which compares well with the experimental result. The numerical analysis also indicated that the slab flexural response is still within the compression membrane mode. The flexural resistance function for FH1 is shown in Figure 43. The numerical analysis shows that the roof slab did not fail in the direct shear mode, which is consistent with the experimental observation. The displacement time history and resistance function for the direct shear degreeoffreedom are plotted in Figures 44 and 45. 4' *' f. ,1 Figure 41. Post test view ofFH1. 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.005 0.01 0.015 0.02 Time 0.025 0.03 0.035 0.04 0.045 Figure 42. FH1 flexural displacement time history. C 600000 500000 400000 300000 200000 100000 0 100000 Displacement (in) Figure 43. FH1 flexural resistance function. 0.008 0.007  0.006       0.006  0.005  0.004   0.003  0.002 00.005  0 0 0.005 0.01 0.015 0.02 Time (s) Figure 44. FH1 direct shear displacement time history. 0.025 8000000 7000000   6000000  5000000  4000000 S3000000  1 2000000   1000000   0 0.01 0.02 0.03 0.04 0.)5 1000000 1000000  2000000 Displacement (in) Figure 45. FH1 direct shear resistance function. 4.2.2 Test FH2 Test FH2 was conducted in a sand (noncohesive) backfill at a depth of burial (DOB) equal to 50% of the short clean span. The reinforced concrete box had wall, floor and roof thickness of 5.6 inches, giving the roof slab a span to effective depth ratio of 10. The structure had one percent principal reinforcing steel in each face, with a concrete strength of 5200psi. The test charge density was 2.7 lb/ft3 and produced a peak pressure of 5250 psi. The test bed had a distinct, elongated depression above the top of the structure. Excavation of the test bed revealed that the roof of the structure suffered complete failure. Post test examination indicated that the roof had been sheared off at the wall supports. The principal steel reinforcing bars, except a few that were not broken near the corners of the end wall, were necked down and broken at the wall supports. An inspection of the reinforcement bars near the center of the roof slab did not indicate the occurrence of significant flexure behavior. A photograph of the box slab after the sand backfill was excavated is shown in Figure 46. ..'i Figure 46. Post test view of FH2. The displacement time history and resistance function for both the flexural and direct shear degreeoffreedom are plotted in Figure 47, 48, 49 and 410. The numerical results show that the roof slab failed in direct shear mode first, when the applied loading exceeded the direct shear resistance of the entire slab. The roof slab is also expected to fail in flexural mode in the numerical results. However, since the direct shear failure occurs at about 1 millisecond after the arrival of the loading, the slab will shear off the wall support and do not have enough time to go into the flexure response mode. The numerical prediction is therefore consistent with the experiment observation. r, . .. m '" Fiue4.Ps ts iw fF2 Th islcmettiehstr ndrsitne ucion fo oh tefeua nddrcha dereofeeomar pote i Fgue 7 48 49 nd40.Th nmrialreuls ho ta the oofslabfaied i diect hea mod fist, henthe ppled ladig exeedd th diect hea resistance~~~~~~~~~~~~~~~~~ ofteetr sa.Terofsa sas epce ofali lxra oei h numrial eslts Hwevrsine he irct her filre ccrs t bou 1miliscon aterth arrival~~ ~ ~ ~~~~~~~~~~~5 oftelaig h lbwl ha f h al upr n onthv nuhtm og into~' th feuersosmoeThnueiaprdconis hrefr ossetwt h experment bservtion 14 12 10 S8 _2 Z. 6 4 2 0 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 Time (s) Figure 47. FH2 flexural displacement time history. 2000000 1800000 1600000 1400000 11200000 S1000000 800000 600000 400000 200000 0 0 2 4 6 8 10 Displacement (in) 12 14 16 Figure 48. FH2 flexural resistance function. 0.2  o.15 S0.1 0.05 0 0 F 7000000 6000000 5000000 '4000000 Z 3000000 2000000 1000000 0 0.05 0.1 0.15 0.2 Displacement (in) Figure 410. FH2 direct shear resistance function. 0.002 0.004 0.006 0.008 Time (s) igure 49. FH2 direct shear displacement time history. 4.2.3 Test FH3 Test FH3 was conducted in a clay backfill at a depth of burial (DOB) equal to 50% of the short clean span. The clay backfill were wetted during the test to ensure the backfill would be a low shear strength material. The reinforced concrete box had wall, floor and roof thickness of 5.6 inches, giving the roof slab a span to effective depth ratio of 10. The structure had one percent principal reinforcing steel in each face, with a concrete strength of 7900psi. The test charge density was 0.9 lb/ft3 and produced a peak pressure of 2650psi. After excavation, post test examination indicated that the roof slab responded primarily in the flexure mode, with a permanent center deflection of about 6 inches. Passive deflection gage recorded a maximum transient deflection of about 7 inches, indicating a rebound of about 1 inch after removal of load. Extensive longitudinal direction cracks were observed and they were concentrated along the edge of the wall supports and center area of the roof. This observation corresponded with a flexural response with three hinges forming at the two supports and at the center. The inside of the roof was extensively cracked longitudinally down the center with the concrete broken off and the reinforcing bars exposed. The exposed principal bars in the roof center were all necked down and some were broken. A photograph of the damaged box structure after excavation is shown in Figure 411. The displacement time history and resistance function for the direct shear degreeof freedom are plotted in Figure 412 and 413. The numerical results show that the roof slab did not fail in direct shear mode, same as the experimental observation. The displacement time history for the flexural degreeoffreedom is plotted in Figure 414. The numerical results show that the permanent displacement at the center of the roof slab is about 5.9 inch, with a maximum transient deflection of 6.7 inches. This compares well with the experimental result. Looking at the resistance function shown in Figure 415, it indicates that the slab underwent extensive deformation into the tensile membrane region. These numerical predictions are consistent with the experiment observations and measurement. Figure 411. Post test view of FH3. 0.12 0.1 .0.08 S0.06 .s 40.04 0.02 0 0 0.002 0.004 0.006 0.008 Time (s) Figure 412. FH3 direct shear displacement time history. 7000000 6000000 5000000 4000000 = Z 3000000 2000000 1000000 0 0.05 0.1 0.15 Displacement (in) Figure 413. FH3 direct shear resistance function. 0 0.01 0.02 0.03 0.04 Time (s) 0.05 0.06 0.07 0.08 Figure 414. FH3 flexural displacement time history. 1400000 1200000 i 1200000 . 1000000  800000 8 600000  S400000   200000 / 2 00000    0 1 2 4 5 7 200000 Displacement (in) Figure 415. FH3 flexural resistance function. 4.2.4 Test FH4 Test FH4 was conducted in a sand backfill at a shallow depth of burial (DOB) equal to 20% of the short clean span. The reinforced concrete box had wall, floor and roof thickness of 5.6 inches, giving the roof slab a span to effective depth ratio of 10. The structure had one percent principal reinforcing steel in each face, with a concrete strength of 6700psi. The test charge density was 0.9 lb/ft3 and produced a peak pressure of 3000psi. After excavation, post test examination indicated that the structure suffered severe damage with the roof slab on the verge of collapse. The measured permanent center deflection was about 12.5 inches. Passive deflection gage recorded a maximum transient deflection of about 13.5 inches, indicating a rebound of about 1 inch after removal of load. About 6 feet along one edge of the roof slab had failed along a vertical failure surface directly over the supporting wall and the steel reinforcement were necked down and broken. On the inside of the roof, much of the concrete were spalled off and cracks went through the slab in some locations. A photograph of the damaged box structure after excavation is shown in Figure 416. The displacement time history and resistance function for both the flexural and direct shear degreeoffreedom are plotted in Figure 417, 418, 419 and 420. The numerical results show that the roof slab did not fail in direct shear mode, same as the experimental observation. The numerical results for flexure show that the permanent displacement at the center of the roof slab is about 11.4 inch, which compares reasonably well with the experimental result. The numerical analysis also indicated that the slab underwent extensive flexural deformation into the tensile membrane region, very close to the calculated failure point at about 14.6 inches of deformation. The numerical prediction is consistent with the experiment observation. Figure 416. Post test view of FH4. 0.01 0.02 0.03 0.04 0.05 Time (s) Figure 417. FH4 flexural displacement time history. Displacement (in) Figure 418. FH4 flexural resistance function. 0.06 2500000 2000000 .1500000 1000000 500000 0 500000 0.2 0.15 , 0.1 0.05 0 0 0.002 0.004 0.006 0.008 0.01 Time (s) Figure 419. FH4 direct shear displacement time history. 7000000 6000000  5000000 5000000     4000000 4000000 , I 4  . 3000000  2000000  1000000 0 0 0.05 0.1 0.15 0.2 Displacement (in) 0.25 0.3 0.35 Figure 420. FH4 direct shear resistance function. 0.012 4.2.5 Test FH5 Test FH5 was conducted in a sand backfill at a shallow depth of burial (DOB) equal to 20% of the short clean span. The reinforced concrete box had wall, floor and roof thickness of 13.5 inches, giving the roof slab had a span to effective depth ratio of 4. The structure had 1.5 percent principal reinforcing steel in each face, with a concrete strength of 6000psi. The test charge density was 3.6 lb/ft3 and produced a peak pressure of 18,000psi. Post test observation indicated that the sand backfill area above the buried structure was slightly depressed, with the sand backfill being sheared and making an outline of the structure perimeter. After excavation, it was observed that the structure suffered moderate damage with a permanent deflection of about 3.1 inches at the roof center. The roof slab was found to have failed in shear, with the cracked and deformed area near the supporting side walls. The center area is relatively flat. On the inside of the roof, the concrete had severely spalled off and the reinforcing bars were bent near the edges indicating a shear deflection. A photograph of the damaged box structure after excavation is shown in Figure 421. Figure 421. Post test view of FH5. The displacement time history and resistance function for the direct shear degreeof freedom are plotted in Figure 422 and 423. The numerical results show that the roof slab did not fail in direct shear mode, same as the experimental observation. For the flexural response, the predicted displacement at the center of the roof slab center is about 0.71 inch (see Figure 424), which is much smaller than deflection measured in the experiment. The roof is a deep slab with a span to effective depth ratio of 4, therefore the slab is very stiff in the flexural response mode. However, based on the post test observation, the failure mode is in shear rather than flexure. Therefore it is not surprising that the numerical prediction and experimental data did not match up. To overcome this shortcoming, a simplified approach to include the shear effect into the flexural resistance function was presented in Section 3.5. The new displacement time history and resistance function are plotted in Figure 425 and 426. The revised maximum permanent deflection at the center of the roof slab is calculated to be about 3.4 inches, and the resistance function shows that the slab had failed in shear before the slab cold reached its maximum flexural capacity in the compression membrane mode. After failing in shear, the slab continues to deflect under load until the applied loading completed and the slab response reached equilibrium. With the proposed modification to the flexure resistance function to consider shear strength effect, the numerical prediction compares very well with experiment measurement and is consistent with the experiment observation. 0.05 .0.04 0.03 _2 40.02 0.01 0 0 0.002 0.004 0.006 0.008 Time (s) Figure 422. FH5 direct shear displacement time history. 18000000 16000000  14000000      14000000 12000000 10000000  8000000 S6000000 4000000 2000000  0 S0.05 0 1 0.15 02 2000000 ( Displacement (in) Figure 423. FH5 direct shear resistance function. 0 0.005 0.01 0.015 0.02 0.025 Time (s) Figure 424. FH5 flexural displacement time history. 3.5 t 2.5 4 1.5 4 0.5 4 0 0.005 0.015 Time (s) 0.02 0.025 Figure 425. FH5 displacement time history. 0.8 0.7 j0.6 S0.5 0.4 0.3 0.2 0.1 0 permanent displacement=3.4" LL 18000000 16000000   14000000 flexural resistance function 12000000 : 10000000 I /Failed in Shear = 8000000  8000000   \  '^^.' 6000000  4000000 Test FH5 resistance curve 2000000 2000000   ...    0 2000000 2f4 6 10 12 14 1 2000000 Displacement (in) Figure 426. FH5 resistance function. 4.2.6 Test FH6 Test FH6 was conducted in a clay backfill at a depth of burial (DOB) equal to 50% of the short clean span. The reinforced concrete box had wall, floor and roof thickness of 5.6 inches, giving the roof slab had a span to effective depth ratio of 10. The structure had one percent principal reinforcing steel in each face, with a concrete strength of 6800psi. The test charge density was 1.8 lb/ft3 and produced a peak pressure of 8,320psi. A photograph of the box slab after partial excavated is shown in Figure 427. Severe structural damage occurred and the roof collapsed completely. The lateral earth pressure pushed the side walls inward after roof collapse. The numerical results show that the roof slab failed in both direct shear mode and flexural mode. When the applied loading exceeded the direct shear resistance of the entire slab, the slab will shear off the wall support and the failure occurs at about 1 millisecond after the arrival of the loading. The numerical prediction is therefore consistent with the experiment observation. The displacement time history and resistance function for both the flexural and direct shear degree offreedom are plotted in Figure 428, 429, 430 and 431. ... .. .. O LR^ ^!'l Figure 427. Post test view of FH6. 16 14 12 10 S8 Z. 6 4 2 0 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 Figure 428. FH6 flexural displacement time history. 3000000 2500000 ,2000000 2 1 1500000 1000000 500000 0 0 2 4 6 8 10 12 14 16 Displacement (in) Figure 429. FH6 flexural resistance function. 0.2 0.15 " 0.1 0.05 0 0 0.002 0.004 0.006 0.008 Time (s) Figure 430. FH6 direct shear displacement time history. 7000000 6000000 5000000 4000000  SIFailed . 3000000  2000000  1000000 0 0 0.05 0.1 0.15 0.2 Displacement (in) 0.25 0.3 0.35 Figure 431. FH6 direct shear resistance function. 4.2.7 Summary A summary of all the six tests present in the above sections is presented in Table 41. Table 41. Summary of results Experiment Average Measured Numerical Computed Failure Time Test Structural Peak Permanent Structural Permanent A2 Direct of Behavior/ nBehavior/ Shear No. Behavior / Pressure Deflection Behavior /Deflection A Slip Failure Failure Failure \ Shp Fa(psi) (in) A Failure (in) A (msec) e de () Mode (in) FH1 Flexure 2400 0.44 Flexure 0.50 1.14   Direct Direct FH2 Diect 5200 Collapsed Diret Collapsed 1.00 0.20 1.1 Shear Shear FH3 Flexure 2650 6 Flexure 5.9 0.98   FH4 Flexure 3000 12.5 Flexure 11.4 0.91   Flexure / FH5 Shear 18000 3.1 Flexure3.4 1.09   Shear Direct Direct FH6 Drect 8320 Collapsed Drect Collapsed 1.00 0.23 1.2 Shear Shear 4.3 Assessment by PI Diagrams For all the test cases presented in Section 4.2, PressureImpulse (PI) diagrams were generated. The experiment measured pressure and corresponding impulse value was plotted on the same PI graph. The assessment of the structural response using the PI diagram were compared with the post test observations. The PI diagrams are shown in Figure 432 to 437. For tests FH1, FH3, FH4 and FH5, the experimental pressureimpulse point lies on the left side of both the flexure and direct shear mode threshold curve, and this agrees with the observations of the experiment whereby the structure did not suffer any complete failure. For test FH2 and FH6, experiment observations indicated that the roof slab suffered completed failure in the test. The same result is obtained from the assessment using PI diagram whereby the experiment pressureimpulse data point lies on the right side of the threshold curves. 10 20 30 40 50 60 Impulse (psimsec) Figure 432. FH1 PressureImpulse diagram. 10 20 30 40 50 60 Impulse (psimsec) Figure 433. FH2 PressureImpulse diagram. 6000 5000 4000 S numerical direct shear 3 3 numerical flexural g 3000 A experimental FH3 2000 1000 0  0 10 20 30 40 50 60 70 80 Impulse (psimsec) Figure 434. FH3 PressureImpulse diagram. 8000  7000 6000  5000 numerical direct shear 4000 numerical flexural g 4000 SA experimental FH4 3000 2000 1000 0 10 20 30 40 50 60 70 80 Impulse (psimsec) Figure 435. FH4 PressureImpulse diagram. 40000 35000 30000 25000 numerical direct shear Snumerical flexural only g 20000 A experimental FH5 S A numerical flexureshear 15000 10000 5000 0 0 50 100 150 200 250 300 Impulse (psimsec) Figure 436. FH5 PressureImpulse diagram. 14000 12000 10000 SI ~ numerical direct shear S80 numerical flexural SA experimental FH6 6000 4000 2000 0 5 10 15 20 25 30 35 40 Impulse (psimsec) Figure 437. FH6 PressureImpulse diagram. 4.4 Summary Based on the numerical approach presented in Chapter 3, the dynamic response for buried box structures are generated and the numerical results compared very well with the experimental data. The approach also correctly predicted the correct mode of failure by examining the response time history and resistance functions. PressureImpulse diagrams are also generated for all the test cases considered. The assessment of structure behavior using PI diagram also predicted the correct result when compared with experiment data. CHAPTER 5 CONCLUSIONS AND RECOMMENDATIONS 5.1 Summary A numerical method for the dynamic analysis of shallowburied reinforced concrete box type structures subjected to air blast loadings was presented in this study. The proposed method adopted the SingleDegreeofFreedom (SDOF) approach, where two loosely coupled SDOF systems were considered to model the flexural and direct shear mode of structural response. An introduction to blast loads and the effects on buried structures was presented in Chapter 2. Dynamic structural behavior and analysis for real continuous system and the approximation into an equivalent SDOF system was reviewed. A review on the structural response mode for reinforced concrete slabs and the resistance model under static and dynamic loading as well as the background and applications of pressureimpulse diagrams was presented. The proposed methodology to generate the resistance functions for reinforced concrete slabs of the buried box structure in both the flexural mode and direct shear mode was presented in Chapter 3. The issue of soil arching and the required modifications to the load and mass factors for the equivalent SDOF system was also discussed and incorporated into the proposed methodology. For the consideration of deep slab behavior, the flexural resistance function was incorporated with a modification to capture the possibility of shear failure in the slab. The proposed procedure was implemented in a computer programming language and the results were validated using experimental data from a number of explosive tests on buried reinforced concrete boxes. The numerical results compared very well with experimental data. 5.2 Conclusions Based on the results from this present study, the following conclusions can be drawn. S The proposed methodology can be employed for the approximate analyses of reinforced concrete slabs, and structural systems that are composed of such elements. * Singledegreeoffreedom analyses which are based on rational models for structural behavior mechanism have been validated, with good accuracy and consistency. * The use of pressureimpulse diagram enables a quick and accurate assessment of the likely performance of the structure, by comparison of the location of the pressure and impulse point with respect to the flexural and direct shear threshold curves plotted on the PI diagram. * The proposed variation of the load and mass transformation factors for SDOF system enable a closer match of the factors with respect to the actual response regime of the structure under different loading combination. * Dynamic soil arching effect reduced the load acting on the buried roof slab and changed the load distribution. The load and mass transformation factors must be modified with an appropriate reduction factor in order to reflect the soil arching effect and to obtain an accurate numerical result. * Shear failure on the slab was not captured with the original proposed flexural and direct shear mode SDOF systems. A simplified approach to consider shear failure on the flexural resistance function was able to better model behavior for slab which failed in shear. 5.3 Recommendations for Future Study Based on the results and observations, the following recommendations for future research are proposed. * The current methodology assumed that the loading on the slab is assumed to be a uniformly distributed airblast load on the soil surface. Further consideration of non uniform loads that may be caused by localized HE (high explosives) explosions is recommended. * Direct shear mode of response was based on the Hawkins model and a single enhancement factor is applied to account for the effects of compression and rate effects. A detailed study of the significance of the possible variation of the enhancement factor is recommended. * Shear failure mode is only considered in this study in a simplified way. A separate response model for tension shear may be considered and its significance can be studied in greater detail. * To study the possible interaction between the roof, wall and floor elements and their effects on the structural response of the box structure under blast loads. APPENDIX EXPERIMENT TEST ON SHALLOW BURIED FLAT ROOF STRUCTURES The series of experiment tests carried out by the United States Army Engineer Waterways Experiment Station, Structures Laboratory (Kiger and Getchell 1980) are used in Chapter 4 as validation data. The details of the experiment carried out are given in this appendix. The tests were carried out to obtain structural response data in a simulated nuclear overpressure environment at the severe damage level for a buried reinforced concrete structure. Six quarterscale models of rectangular, single bay reinforced concrete box structures with inside dimensions of 4 feet high, 4 feet wide and 16 feet long, were tested with simulated nuclear blast. The box structures were designed to model one bay from a rectangular multibay structure with span to effective depth ratios of between 4 and 10. The box structure was tested in a shallowburied configuration using a HEST (High Explosive Simulation Technique) test which can simulate the peak pressure and duration characteristic of the overpressure generated in a nuclear detonation. The charge cavity was composed of conventional high explosives and plastic foam for which the tests were designated Foam Hest (FH). The test involves distributing a high explosive over a relatively large surface area and covering the explosive with a soil overburden to momentarily confine the blast. The test configuration for FH3 is shown in Figure A1. The material properties such as the concrete and steel strength, steel ratio and charge density were varied for all the six tests. The depth of burial for the tests was varied, with a ratio of depth of burial to the short clear span between 0.2 to 0.5. The parameters of the six cases presented in Chapter 4 are tabulated in Table A1. Construction details for FH1 and FH5 are shown in Figure A2 and Figure A3 respectively. Parameters used in the numerical procedure are summarized in Table A2, A3, A4, A5, A6 and A7. 0.9 M (3 FT) 1  NATIVE SOIL OVERBURDEN SAND SACKFILL 0. M (0 FT) Figure A1. Experiment test configuration for FH3. A) Elevation. B) Plan view. CPL. STIRRUPS EACH S/d IM END WALLS I STIRRUP EACH ' LOCATION STIRRUP DETAIL FMOR TR RAERS P) RiB84R ENTRANCE COVER METAL,. 10" THICK HO 7/;i'BUR FOR OPErING ENTRANCE DETAILS AN VIEW TM/CAL Ao. 4 REBAR 4rd"C 2 NO. 3 REBAR. STIRRUPS 4" O.C' EACH WAY SEE DETAIL  TCA.AL H0 J REBAR' 4"0C fACn EO WALL  NO. 3 REBAR STIRRUPS 4" O.C EACH WAY SEE DETAIL SECTION AA SIDE VIEW RECTANGULAR BOX STRUCTURE (L/d I0) Figure A2. Construction dimensions and details of FH1 4.11 rL r~ ~ .     I. UNLE JOTHIEROIE NOTE, TOLERANCE ON ALL r LENi.CNi ST ACTED f IN INCHES is 1/*4' TOLER ANCE ON ALL DIMENSIONS STATED IN FEET IS I IN. 2. ALL RESARS WILL BE ASTM A61568 GRADE 60. 3. CONCRETE STRENGTH 5000 PSI tN 28 DAYS. PLAN VIEW TYPICAL Pr. 3 Ji;;f.6A 5" O.C EACh FACE  LTYPI.'LNC iVRE BAR, 5.5" *;.C EACh uA e SECTION AA SIDE VIEW RECTANGULAR BOX STRUCTURE t L/D = 4) STRUCTURAL DETAILS AND DIMENSIONS Figure A3. Construction dimensions and details of FH5. STIRRUP DETAIL 13.5' i  Table A1. Parameters for FoamHest tests Parameters FH1 FH2 FH3 FH4 FH5 FH6 Height (ft) 4 4 4 4 4 4 Ls(ft) 4 4 4 4 4 4 LL (ft) 16 16 16 16 16 16 Thickness (in) 5.6 5.6 5.6 5.6 13.5 5.6 Spaneffective depth ratio 10 10 10 10 4 10 DOB (ft) 2 2 2 0.8 0.8 2 DOBLs ratio 0.5 0.5 0.5 0.2 0.2 0.2 Concrete strength (psi) 7000 7600 7900 6700 6000 6800 Fy (psi) 75,000 57,000 57,000 65,000 69,000 65,000 Percentage steel (%) 1 1 1 1 1.5 1 Soil Type Sand Sand Clay Sand Sand Clay Charge density (pcf) 0.9 2.7 0.9 0.9 3.6 1.8 Table A2. Test FH1 input parameters Parameters Value Height 4 ft Lx 16 ft Ly 4ft Slab Thickness 5.6 in Concrete cylinder strength 7000 psi Concrete density 0.0868 lb/in3 Xdirection Reinforcement area 0.11 in2 / 4 in spacing Reinforcement bar diameter 0.375 Reinforcement yield strength 75,000 psi Reinforcement ultimate strain 0.21 Depth of top reinforcement 0.8 in Depth of top reinforcement 4.8 in Ydirection Reinforcement area 0.20 in2 / 4 in spacing Reinforcement bar diameter 0.50 Reinforcement yield strength 75,000 psi Reinforcement ultimate strain 0.21 Depth of top reinforcement 0.8 in Depth of top reinforcement 4.8 in Depth of burial (ft) 2 Soil Type Sand Soil density (pci) 0.061 Friction angle 0q 35.5 Coefficient of static lateral earth pressure Ko 0.5 Soil wave velocity (in/s) 18,000 Damping (Flexure / Direct shear) 20% / 5% Pressure Time History FH1 2500 2000  i.1500  1000 500  0 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 Time (s) Table A3. Test FH2 input parameters Parameters Value Height 4 ft Lx 16 ft Ly_ 4ft Slab Thickness 5.6 in Concrete cylinder strength 7600 psi Concrete density 0.0868 lb/in3 Xdirection Reinforcement area 0.11 in2 / 4 in spacing Reinforcement bar diameter 0.375 Reinforcement yield strength 57,000 psi Reinforcement ultimate strain 0.21 Depth of top reinforcement 0.8 in Depth of top reinforcement 4.8 in Ydirection Reinforcement area 0.20 in2 / 4 in spacing Reinforcement bar diameter 0.50 Reinforcement yield strength 57,000 psi Reinforcement ultimate strain 0.21 Depth of top reinforcement 0.8 in Depth of top reinforcement 4.8 in Depth of burial (ft) 2 Soil Type Sand Soil density (pci) 0.0613 Friction angle qS 35.5 Coefficient of static lateral earth pressure Ko 0.5 Soil wave velocity (in/s) 18,000 Damping (Flexure / Direct shear) 20% / 5% Pressure Time History FH2 6000 5000  3000    4000 t3000 2000 1000 0 0 0.005 0.01 0.015 0.02 Time (s) Table A4. Test FH3 input parameters Parameters Value Height 4 ft Lx 16 ft Ly_ 4ft Slab Thickness 5.6 in Concrete cylinder strength 7900 psi Concrete density 0.0868 lb/in3 Xdirection Reinforcement area 0.11 in2 / 4 in spacing Reinforcement bar diameter 0.375 Reinforcement yield strength 57,000 psi Reinforcement ultimate strain 0.21 Depth of top reinforcement 0.8 in Depth of top reinforcement 4.8 in Ydirection Reinforcement area 0.20 in2 / 4 in spacing Reinforcement bar diameter 0.50 Reinforcement yield strength 57,000 psi Reinforcement ultimate strain 0.21 Depth of top reinforcement 0.8 in Depth of top reinforcement 4.8 in Depth of burial (ft) 2 Soil Type Clay Soil density (pci) 0.0714 Friction angle qS 0 Coefficient of static lateral earth pressure Ko 1.0 Soil wave velocity (in/s) 24,000 Damping (Flexure / Direct shear) 20% / 5% Pressure Time History FH3 3000 2500  2000  1500  1000  500  0 0 0.005 0.01 0.015 0.02 0.025 0.03 Time (s) Table A5. Test FH4 input parameters Parameters Value Height 4 ft Lx 16 ft Ly_ 4ft Slab Thickness 5.6 in Concrete cylinder strength 6700 psi Concrete density 0.0868 lb/in3 Xdirection Reinforcement area 0.11 in2 / 4 in spacing Reinforcement bar diameter 0.375 Reinforcement yield strength 65,000 psi Reinforcement ultimate strain 0.21 Depth of top reinforcement 0.8 in Depth of top reinforcement 4.8 in Ydirection Reinforcement area 0.20 in2 / 4 in spacing Reinforcement bar diameter 0.50 Reinforcement yield strength 65,000 psi Reinforcement ultimate strain 0.21 Depth of top reinforcement 0.8 in Depth of top reinforcement 4.8 in Depth of burial (ft) 0.8 Soil Type Sand Soil density (pci) 0.0612 Friction angle qS 35.5 Coefficient of static lateral earth pressure Ko 0.5 Soil wave velocity (in/s) 18,000 Damping (Flexure / Direct shear) 20% / 5% Pressure Time History FH4 3000 2500  2000  1500  1000  500  0 0 0.005 0.01 0.015 0.02 0.025 Time (s) Table A6. Test FH5 input parameters Parameters Value Height 4 ft Lx 16 ft Ly_ 4ft Slab Thickness 13.5 in Concrete cylinder strength 6000 psi Concrete density 0.0868 lb/in3 Xdirection Reinforcement area 0.11 in2 / 5 in spacing Reinforcement bar diameter 0.375 Reinforcement yield strength 69,000 psi Reinforcement ultimate strain 0.21 Depth of top reinforcement 1.5 in Depth of top reinforcement 12.0 in Ydirection Reinforcement area 1.0 in2 / 5.5 in spacing Reinforcement bar diameter 0.50 Reinforcement yield strength 69,000 psi Reinforcement ultimate strain 0.21 Depth of top reinforcement 1.5 in Depth of top reinforcement 12.0 in Depth of burial (ft) 0.8 Soil Type Sand Soil density (pci) 0.0623 Friction angle qS 35.5 Coefficient of static lateral earth pressure Ko 0.5 Soil wave velocity (in/s) 18,000 Damping (Flexure / Direct shear) 20% / 5% Pressure Time History FH5 18000 16000  14000   12000  "10000 L L S8000    6000  4000 2000 r r 0 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 Time (s) Table A7. Test FH6 input parameters Parameters Value Height 4 ft Lx 16 ft Ly_ 4ft Slab Thickness 5.6 in Concrete cylinder strength 6800 psi Concrete density 0.0868 lb/in3 Xdirection Reinforcement area 0.11 in2 / 4 in spacing Reinforcement bar diameter 0.375 Reinforcement yield strength 65,000 psi Reinforcement ultimate strain 0.21 Depth of top reinforcement 0.8 in Depth of top reinforcement 4.8 in Ydirection Reinforcement area 0.20 in2 / 4 in spacing Reinforcement bar diameter 0.50 Reinforcement yield strength 65,000 psi Reinforcement ultimate strain 0.21 Depth of top reinforcement 0.8 in Depth of top reinforcement 4.8 in Depth of burial (ft) 2 Soil Type Clay Soil density (pci) 0.0701 Friction angle qS 0 Coefficient of static lateral earth pressure Ko 1.0 Soil wave velocity (in/s) 24,000 Damping (Flexure / Direct shear) 20% / 5% Pressure Time History FH6 9000 8000  7000  6000  '5000  S4000  3000  2000  1000 i 0 0 0.005 0.01 0.015 0.02 Time (s) LIST OF REFERENCES ACI Committee 318. (2005). Building code requirementsfor structural concrete (ACI 31805) and commentary (ACI 318R05), Farmington Hills, Mich, American Concrete Institute. ASCE. (1985). Design of Structures to Resist Nuclear Weapon Effects, ASCE Manuals and Reports on Engineering Practice No. 42, ASCE. Bathe, K.J. (1996). Finite element procedures, Englewood Cliffs, N.J. Biggs, J. M. (1964). Introduction to structural dynamics, McGrawHill, New York.. Blasko, J.R., Krauthammer, T., and Astarlioglu, S. (2007). "Pressureimpulse diagrams for structural elements subjected to dynamic loads." Technical report PTCTR0022007, University Park, PA: Protective Technology Center, The Pennsylvania State University. Bowles, J.E. (1996). Foundation Analysis andDesign, 5th Edition, McGrawHill. Clough, R. W., and Penzien, J. (1993). Dynamics of structures, McGrawHill, New York. Crawford, J. E., Holland, T.J., Mendoza, P.J. and Murtha, R. (1983). "A failure methodology based on shear deformation." Fourth ASCE Engineering Mechanics Division Specialty Conference. Purdue University, Lafayette, IN. Crawford, J. E., Krauthammer, T., Karagozian, J. and Hinman, E. (1999). "Structural components Analysis and design examples." Structural design for physical security: state of the practice. Chapter 4, ASCE, SEI, Reston, Va. Department of Army (1986). Fundamentals ofprotective design for conventional weapons, Technical Manual No. 58551, Headquarters Department of the Army, Washington, D.C.. Frye, M. (2002). Relationships between slender and deep reinforced concrete slabs subjected to shortduration dynamic loading, MS Thesis, The Pennsylvania State University, PA. Hawkins, N.M. (1974). "The strength of stud shear connections." Civil Engineering Transactions, IE, Australia, 3945. Kiger, S.A. (1988). "Ultimate capacity of earthcovered slab." J Struct. Eng., ASCE, 114(10), 23432356. Kiger, S.A., and Getchell, J.V. (19801982). "Vulnerability of shallowburied flat roof structures." 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"Analysis and design of laterally restrained structural concrete oneway members." ACI Structural Journal, 91(6), 719725. Nawy, E.G. (2000). Reinforced concrete: a fundamental approach, PrenticeHall, N.J. Newmark, N. M., and Rosenblueth, E. (1971). Fundamentals of earthquake engineering, PrenticeHall, Englewood Cliffs, N.J. Ng, P.H. (2004). Pressure impulse diagrams for reinforced concrete slabs, MS Thesis, The Pennsylvania State University, PA. Park, R., and Gamble, W.L. (2000). Reinforced concrete slabs, Wiley, New York. Park, R., and Paulay, T. (1975). Reinforced concrete structures, Wiley, New York. Slawson, T. R. (1984). "Dynamic shear failure of shallowburied flatroofed reinforced concrete structures subjected to blast loading." Technical Report SL847, U.S. Army Engineer Waterways Experiment Station, Vicksburg, Miss. Smith, P.D., and Hetherington, J.G. (1994). Blast and ballistic loading of structures, ButterworthHeinemann, Oxford, Boston. Soh, T.B., and Krauthammer, T. (2004). "Loadimpulse diagrams of reinforced concrete beams subjected to concentrated transient loading." Technical report PTCTR0062004. University Park, PA: Protective Technology Center, The Pennsylvania State University. Tedesco, J. W., McDougal, W. G., and Ross, C. A. (1999). Structural dynamics: theory and applications, Addison Wesley Longman, Menlo Park, Calif. Terzaghi, K., and Peck, R. B. (1949). Soil mechanics in engineering practice, Wiley, New York. Zienkiewicz, O. C., and Taylor, R. L. (2005). Thefinite element method for solid and structural mechanics, Elsevier ButterworthHeinemann, Amsterdam. BIOGRAPHICAL SKETCH Kay Hyang Chee was born in Singapore in 1973. He attended secondary school and junior college in Singapore. After obtaining his GCE "A" level in 1991, he served his national service in the army from 1991 to 1993. He began his undergraduate studies in Civil Engineering at the National University of Singapore in July 1993. He graduated in July 1997 with his Bachelor of Engineering degree in civil engineering. He continued with his graduate studies at the National University of Singapore and obtained his Master of Engineering degree in 1999. In August 1999, he joined the Defence Science and Technology Agency, Singapore, as a project engineer. In 2006, he was awarded a postgraduate scholarship to pursue a master's degree in civil engineering, focusing on protective engineering at the University of Florida. PAGE 1 1 ANALYSIS OF SHALLOW BURIED REIN FORCED CONCRETE BOX STRUCTURES SUBJECTED TO AIRBLAST LOADS By KAY HYANG CHEE 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 2008 PAGE 2 2 2008 Kay Hyang Chee PAGE 3 3 To my lovely wife PAGE 4 4 ACKNOWLEDGMENTS I thank m y advisor, Dr Theodor Krauthammer, for his advice and guidance. I want to express my appreciation to Dr Serdar Astarliogl u for his valuable suggestions and help in the programming aspects. I am grateful to the Defence Science and Technology Agency, Singapore, for the postgraduate scholarship. I thank all of my friends at the Center for Infrastructure Protection and Physical Security, University of Florida and Maguire Village, for a great stay and experience in Gainesville. Last of all, I thank my d ear wife, Hiong Suan, for all she has done for me and her willingness to sacrifice many things over the last two years. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ............................................................................................................... 4 LIST OF TABLES ...........................................................................................................................7 LIST OF FIGURES .........................................................................................................................8 LIST OF SYMBOLS .....................................................................................................................12 ABSTRACT ...................................................................................................................... .............16 CHAP TER 1 INTRODUCTION .................................................................................................................. 17 1.1 Problem Statement ........................................................................................................ 17 1.2 Objectives and Scope ....................................................................................................18 1.3 Research Significance ...................................................................................................18 2 BACKGROUND AND LITER AT URE REVIEW ................................................................ 19 2.1 Introduction ...................................................................................................................19 2.2 Blast Loads ....................................................................................................................19 2.2.1 Airblast from High Explosive ...........................................................................20 2.2.2 Nuclear Devices ................................................................................................ 21 2.3 Effects on Buried Structures .........................................................................................22 2.3.1 Soil Arching Effect ...........................................................................................22 2.4 Dynamic Structural Behavior and Analysis ..................................................................24 2.4.1 SingleDegreeofFr eedom (SDOF) System ..................................................... 27 2.4.2 Transformation Factors for Equivalent SDOF .................................................. 28 2.4.3 Numerical Integration (Newm arkBeta method) .............................................. 29 2.5 Behavior of Reinforced Concrete Slabs ........................................................................ 30 2.5.1 Flexural Behavior: Johans ens Yield Line Theory ........................................... 30 2.5.2 Flexural Behavior: Membrane Action ..............................................................31 2.5.3 Slab Compressive Membrane ........................................................................... 33 2.5.4 Slab Tensile Membrane.....................................................................................38 2.5.5 Reinforced Concrete Slab Flexural Model ........................................................ 39 2.5.6 Direct Shear Behavior .......................................................................................40 2.5.7 Hawkins Shear Model .......................................................................................41 2.5.8 Dynamic Resistance Function and Response .................................................... 43 2.6 PressureIm pulse Diagrams and their Application .......................................................46 2.6.1 Characteristics of PI Diagram ..........................................................................46 2.6.2 Numerical Approach to PI Diagram ................................................................ 48 2.6.3 Multiple Failure Modes ..................................................................................... 48 2.7 Summary .................................................................................................................. .....50 PAGE 6 6 3 METHODOLOGY ................................................................................................................. 51 3.1 Introduction ...................................................................................................................51 3.2 Flexural Mode ...............................................................................................................51 3.2.1 Externally Applied Thrust ................................................................................. 51 3.2.2 Numerical Approach for Re sistance Curve Calculation ................................... 54 3.2.3 Variation of Mass and Load Factor ................................................................... 58 3.3 Soil Structural Interaction ............................................................................................. 60 3.3.1 Influence of Parameters on Soil Arching Effect ............................................... 60 3.3.2 Effect on SDOF Load and Mass Factor ............................................................61 3.4 Direct Shear Mode ........................................................................................................65 3.4.1 Resistance Curve ............................................................................................... 65 3.4.2 Shear Mass and Load Factors ........................................................................... 67 3.5 Shear Failure Mode for Slab ......................................................................................... 68 3.6 Program Flowchart ........................................................................................................ 70 3.7 Summary .................................................................................................................. .....70 4 RESULTS AND DISCUSSIONS........................................................................................... 73 4.1 Introduction ...................................................................................................................73 4.2 Validation with Experimental Data ............................................................................... 73 4.2.1 Test FH1 ............................................................................................................74 4.2.2 Test FH2 ............................................................................................................77 4.2.3 Test FH3 ............................................................................................................81 4.2.4 Test FH4 ............................................................................................................84 4.2.5 Test FH5 ............................................................................................................88 4.2.6 Test FH6 ............................................................................................................92 4.2.7 Summary ........................................................................................................... 96 4.3 Assessment by PI Diagrams ........................................................................................ 96 4.4 Summary .................................................................................................................. ...100 5 CONCLUSIONS AND RECOMMENDATIONS ............................................................... 101 5.1 Summary .................................................................................................................. ...101 5.2 Conclusions .................................................................................................................101 5.3 Recommendations for Future Study ...........................................................................102 APPENDIX EXPERIMENT TEST ON SHALLOW BURIED FLAT ROOF STRUCTURES ....................................................................................................................103 LIST OF REFERENCES .............................................................................................................113 BIOGRAPHICAL SKETCH .......................................................................................................115 PAGE 7 7 LIST OF TABLES Table page 41 Summary of results ............................................................................................................96 A1 Parameters for FoamHest tests .........................................................................................106 A2 Test FH1 input parameters ............................................................................................... 107 A3 Test FH2 input parameters ............................................................................................... 108 A4 Test FH3 input parameters ............................................................................................... 109 A5 Test FH4 input parameters ............................................................................................... 110 A6 Test FH5 input parameters ............................................................................................... 111 A7 Test FH6 input parameters ............................................................................................... 112 PAGE 8 8 LIST OF FIGURES Figure page 21 Blast pressuretime curve ................................................................................................. ..21 22 Soil arching demonstrated by trap door experim ent .......................................................... 23 23 Soil arching as functi on of depth of burial. ........................................................................24 24 Uniform beam subject to arbitrary load. ............................................................................25 25 Yield lines development in a unifo rmly loaded simply supported slab ............................. 31 26 Loaddeflection curve for twoway RC sl ab with laterally restrained edges. .................... 33 27 Assumed yield line pattern for uniformly loaded slab with restrained edges .................... 34 28 Plastic hinges of a restrained strip .....................................................................................34 29 Portion of strip be tween plastic hinges ..............................................................................35 210 Conditions at positive moment yield section .....................................................................37 211 Uniformly loaded plastic tensile membrane. ..................................................................... 39 211 Flexural resistance model for slab. .................................................................................... 40 213 Slab in direct shear failure mode ....................................................................................... 41 214 Hawkins model for direct sh ear stressslip relationship .................................................... 42 215 Equivalent SDOF system s for structural elem ent .............................................................. 44 216 Dynamic flexural resistance functions ...............................................................................45 217 Dynamic direct shear resistance function. .........................................................................46 218 Typical response spectra and PI diagram ......................................................................... 47 219 Search algorithm for PI diagram ...................................................................................... 49 220 PressureImpulse diagram with two failure modes ............................................................ 49 31 Model for externally applied thrust .................................................................................... 52 32 Calculation of extern ally applied thrust ............................................................................. 53 33 Stress and strain distributions ac ross reinforced concrete section ..................................... 54 PAGE 9 9 34 Restrained strip with external thrust .................................................................................. 55 35 Portion of strip between plasti c hinges with external thrust .............................................. 55 36 Variation of load and mass factor ...................................................................................... 58 37 Variation of soil arching factor with f riction angle and burial depth ................................61 39 Variation of L....................................................................................................................63 310 Variation of M ...................................................................................................................65 311 Direct shear mode l for twoway slab ................................................................................. 66 312 Direct shear resistance curve for twoway slab .................................................................67 313 Deformed shape for direct shear response ......................................................................... 68 314 Slab in shear failure mode ..................................................................................................69 315 Resistance curve for slab with shear failure mode ............................................................. 71 316 Program flowchart ........................................................................................................ .....72 41 Post test view of FH1 ..................................................................................................... ....75 42 FH1 flexural displacement time history ............................................................................. 75 43 FH1 flexural resistance function ........................................................................................76 44 FH1 direct shear disp lacem ent time history....................................................................... 76 45 FH1 direct shear resistance function ..................................................................................77 46 Post test view of FH2 ..................................................................................................... ....78 47 FH2 flexural displacement time history ............................................................................. 79 48 FH2 flexural resistance function ........................................................................................79 49 FH2 direct shear disp lacem ent time history....................................................................... 80 410 FH2 direct shear resistance function ..................................................................................80 411 Post test view of FH3 .................................................................................................... .....82 412 FH3 direct shear disp lacem ent time history....................................................................... 82 413 FH3 direct shear resistance function ..................................................................................83 PAGE 10 10 414 FH3 flexural displacement time history ............................................................................. 83 415 FH3 flexural resistance function ........................................................................................84 416 Post test view of FH4 .................................................................................................... .....85 417 FH4 flexural displacement time history ............................................................................. 86 418 FH4 flexural resistance function ........................................................................................86 419 FH4 direct shear disp lacem ent time history....................................................................... 87 420 FH4 direct shear resistance function ..................................................................................87 421 Post test view of FH5 .................................................................................................... .....88 422 FH5 direct shear disp lacem ent time history....................................................................... 90 423 FH5 direct shear resistance function ..................................................................................90 424 FH5 flexural displacement time history ............................................................................. 91 425 FH5 displacement time history .......................................................................................... 91 426 FH5 resistance function .....................................................................................................92 427 Post test view of FH6 .................................................................................................... .....93 428 FH6 flexural displacement time history ............................................................................. 94 429 FH6 flexural resistance function ........................................................................................94 430 FH6 direct shear disp lacem ent time history....................................................................... 95 431 FH6 direct shear resistance function ..................................................................................95 432 FH1 PressureImpulse diagram .......................................................................................... 97 433 FH2 PressureImpulse diagram .......................................................................................... 97 434 FH3 PressureImpulse diagram .......................................................................................... 98 435 FH4 PressureImpulse diagram .......................................................................................... 98 436 FH5 PressureImpulse diagram .......................................................................................... 99 437 FH6 PressureImpulse diagram .......................................................................................... 99 A1 Experiment test c onfiguration for FH3 ............................................................................104 PAGE 11 11 A2 Construction dimensions and details of FH1 ................................................................... 105 A3 Construction dimensions and details of FH5 ................................................................... 105 PAGE 12 12 LIST OF SYMBOLS Ac Concrete crosssectional area Asb Area of reinforcement c Neutral axis depth at Section 1 c Neutral axis depth at Section 2 Cc Concrete compressive force at Section1 Cc Concrete compressive force at Section 2 Cs Steel compressive force at Section1 Cs Steel compressive force at Section 2 Ca Soil arching ratio C Damping E Elastic modulus Ec Concrete elastic modulus fc Concrete cylinder strength fs Tensile strength of the reinforcement fy Yield strength of steel reinforcement Fe Equivalent force F(x,t) Arbitrary distributed force H Depth of burial h Thickness of slab I Moment of inertia I Impulse K Spring stiffness Ke Equivalent stiffness Ke Elastic stiffness of dire ct shear degreeoffreedom PAGE 13 13 Ku Negative stiffness at segment CD of Hawkins model KM Mass factor KL Load factor Ko Coefficient of static lateral earth pressure Kr Generalized (modal) stiffness for the rth mode L Length of structure Lx Long span of slab Ly Long span of slab m Unit mass mu Resisting moment at Section 1 mu Resisting moment at Section 2 M Lumped mass Me Equivalent mass Mr Generalized (modal) mass N External thrust nu Membrane force p(t) Pressure function PB Average pressure acting on structure PS Uniform pressure acting on soil surface P0 Peak load Po Atmospheric pressure Pr Generalized (modal) force of the rth mode Pso Peak pressure q Displacement of the selected representative point q Velocity of the select ed representative point PAGE 14 14 q Acceleration of the selected representative point R Dynamic resistance function R1 Residual function 1 R2 Residual function 2 S Surround stiffness ta Time of arrival to Positive phase duration to Negative phase duration t Strip outward lateral movement t Time td Loading function time duration T Steel tensile fo rce at Section 1 T Steel tensile force at Section 2 Tx Yield force of reinforcement per unit width in the xdirection Ty Yield force of reinforcement per unit width in the ydirection T Kinetic energy Tn Natural period V Shear force V Potential energy w Beam displacement w Uniform load per unit area W Width of structure ix Displacement ix Velocity ix Acceleration PAGE 15 15 xmax Maximum displacement NewmarkBeta integration constant 1 Ratio of depth of the equivalent AC I stress block to neutralaxis depth Axial strain si Steel strain ci Concrete strain cu Concrete ultimate strain at failure Virtual rotation Slab central displacement Angle of internal friction )( x Shape function )( xr Normal vibration modes for the beam L Ratio of load factors M Ratio of mass factors vt Ratio of total reinforcement area to the area of plane that it crosses rq Virtual displacement t Time step interval max Maximum shear slip e Direct shear resistance (elastic) L Limiting direct shear capacity m Maximum direct shear resistance Natural circular frequency Damping ratio PAGE 16 16 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 ANALYSIS OF SHALLOW BURIED REIN FORCED CONCRETE BOX STRUCTURES SUBJECTED TO AIRBLAST LOADS By Kay Hyang Chee May 2008 Chair: Theodor Krauthammer Major: Civil Engineering A numerical method for the dynamic analysis of shallowburied reinforced concrete boxtype structures subjected to ai r blast loadings is presented in this study. The proposed method was based on the SingleDegreeofFreedom (S DOF) approach, where two loosely coupled SDOF systems were considered to take into account the flexural and direct shear mode of structural response. The effects of compression and tension membrane in reinforced concrete slabs and soilstructural interac tion were considered in the stu dy. The resistance functions for the structure for each structural response mode we re generated and used in the dynamic analysis. The issue of soilstructure interaction and relatio nship with structural behavior, in terms of soil arching effect was examined in more detail A rational model was proposed to incorporate the soil arching effect and a varying SDOF e quivalent load and mass factors in the dynamic analysis. The algorithm was implemented in a computer program. Results of the study were validated using available experimental data from a number of buried reinforced concrete boxes that were tested by other investigators. PAGE 17 17 CHAPTER 1 INTRODUCTION 1.1 Problem Statement Recognizing the potential of soil as a protective medium long ago, man has built underground and buried structures to provide shelter and refuge from his potential enemy. By doing so, the soils performance in the provision of protection to the structure is harnessed through its inertia effects and its ability to di ffuse load and dissipate energy. Commonly, reinforced concrete is used as the construction material in pr otective engineering applications due to its suitable properties and economical value. In addition, most underground or buried structures can usually be appr oximated as boxtype structures Therefore, in protective engineering applications, the de sign and evaluation of the perfor mance of a buried reinforced concrete box structures under transient severe loading is an important consideration. Since the behavior of buried reinforced conc rete box structure under th e effects of intense pressure pulses applied to the soil surface is of in terest, the structural element located closest to the applied load will strongly affect the performance of the entire structure. Therefore, the behavior of a buried structure can be adequately represented by the response of the roof slab which form part of the r ectangular box structure. The design of structural elements under transi ent loading requires dynamic analysis to be carried out to determine the response characterist ics, such as the displacement time history and reaction forces. Analytical stud ies have been performed previously by various researchers using finite element codes or singl edegreeoffreedom (SDOF) mode ls. The SDOF approach is comparatively a simple tool, but with an accurate prediction of the structural behavior, it is a useful tool in preliminary design or parametric studies and can be combined with more advanced analytical techniques to reduce the to tal computational time and cost. PAGE 18 18 1.2 Objectives and Scope This study aim s to develop a reliable, simp le and accurate analytical approach and numerical procedure to perform dynamic analys is for the design and evaluation of buried reinforced concrete box structure against airblast loads. The procedure will consider nonlinear resistance mechanisms for reinforced concrete sl ab structure in the flexural and direct shear mode of behavior. The scope of this study is limited to reinfor ced concrete slabs s ubjected to transient uniformly distributed airblast pressure load on th e soil surface. This study includes modification of the resistance function to consider inplane co mpressive force due to internal membrane effect and external thrust due to wave propagation th rough the soil. The SDOF equivalent load and mass factors are also varied with respect to the slab response re gime. This study also includes the evaluation of the dynamic soil arching effect and its corresponding effect on the SDOF land and mass factors. The proposed procedure is to be validated with availa ble experimental test data in order to evaluate its accuracy. 1.3 Research Significance This study can offer a methodology f or a reliable, simple and accurate dynamic analysis engine to study the behavior of a shallowburied reinforced concrete box structure. The approach includes rational and phys icsbased resistance f unctions, taking into account the effect of soil structural inte raction (e.g. wave propagation and soil arching effect) and improves on the approach to approximate the real continuous system into an equivalent SDOF system in order to provide an accurate numerical result. PAGE 19 19 CHAPTER 2 BACKGROUND AND LITERATURE REVIEW 2.1 Introduction Since hum ans recognized the potential of soil as a protective medium, underground structure have been constructed to provide shel ter and refuge from their enemy. In order to design an underground structure or to evaluate its structural performance, the first requirement is to characterize the expected performance under th e specified design loads. With the necessary loading applied, the relevant mode of structural behavior has to be modeled accurately and the response results from dynamic analysis can be used for design purpose. If the main design concern is a specific response limit state (rotation, failure, etc.), a suitable computational aid to use is a PressureImpulse diagram. In this study, the form of loading consider ed is from blast and the corresponding response of reinforced concrete box structures roof slabs. A brief introduction to bl ast loads and its effects on buried structures are presented in Sections 2.2 and 2.3. Blast loadings are transient and the response is highly dependent on the peak load and duration. The dynamic structural behavior and analysis are reviewed in S ection 2.4. Section 2.5 focuses on the different structural response mode for reinforced concrete slabs and thei r corresponding resistance model under static and dynamic loading. Lastly, the background of pressu reimpulse diagrams and their applications are presented in Section 2.6. 2.2 Blast Loads Generally, the m ost common sources of explosio ns and blast loads are derived from either chemical (High Explosive, HE) or nuclear materi als. The environment created by an explosion consists of the following effects: Airblast Groundshock PAGE 20 20 Ejecta Fragments Fire, thermal and chemical (nuclear explosions only) Radiation (nuclear explosions only) Electromagnetic pulse (EMP) (nuclear explosions only) 2.2.1 Airblast from High Explosive In an open air High Explosive (HE) explosion, the reaction generates gases at very high pressure and temperature, causing a violent expansion of these explosive gases and the surrounding air is forced out of th e volume it occupies. The shock front is essentially vertical, reflecting the sudden rise in pressu re due to the explosion. It is a layer of compressed air, forms in front of the explosive gases. As the gases expand and cool, their pressure decreases. The pressure of the blast wave front also falls with increasing distance from source. Eventually, the pressure falls a little below atmospheric pre ssure because the moment um of the moving gas molecules. The gases are overexpanded near th e explosion location and a reversal of flow towards the source occurs. This is the negative phase which is characterized by a pressure lower than the ambient air pressure. Eventually, the pressure and temp erature of the gases returns to equilibrium (Smith and Hetherington 1994, Tedesco et al. 1998, Krauthammer 2008). An idealized pressuretime function for an ai rblast shockwave in free air is shown in Figure 21, where Po is the atmospheric pressure; Pso is the peak pressure; ta is the time of arrival; to is the positive phase duration and to is the negative phase duration. The impulse of the blast wave is defined as the area under the pressure time curve and can be calculated simply by tt ta adttpI) ( (21) PAGE 21 21 Figure 21. Blast pressuretime curve. 2.2.2 Nuclear Devices A nuclear device can d eliver its explosive eff ects from air burst, surface burst or shallow burst. When the nuclear device is exploded at an altitude below 100,000 ft, about 50% of the released energy will result in blast and shoc k (ASCE 1985). For high altitude bursts above 100,000 ft, they generate only strong EMP effects (whi ch should be considered in facility design) but are of little interest to the structural engineer. The characteristics of the blast pressure wave in a nuclear explosion are similar to those of a high explosive (HE) explosion and are a function of the weapon yield, the height of burst and the distance from the burst. There is also dyna mic pressure which resu lts from the mass flow behind the shock front. The dynamic pressure is a function of the gas density and the flow velocity. As with the HE explosion, there is both a positive and ne gative phase for the overpressure and for the dynamic pressure (Krauthammer 2008). Pressure time Pso ta to to Ambient, Po Positive Phase Negative Phase PAGE 22 22 2.3 Effects on Buried Structures Although buried structures offered protection fr om aerially delivered weapons and airborne blast effects, but these structur es can be vulnerable to the tr ansient stresses propagated through the soil and rock in which they have been constructed (Sm ith and Hetherington 1994). Other types of loadings such as buried charges or struct ural penetration are also important but they are not considered in this study. A shock wave will be induced in the soil when an air blast is applied to the free soil surface (soilair interface). The shock wave travels downw ards in the vertical direction until meeting with the structure. Based on data from soil stress gages and interface pressure gages in nuclear tests, the shock front can be considered as quite planar (Krauthammer et al. 1986). The possible modulation in the interface pressure s will appear as a result of wave reflections and soilstructure interaction effects, including soilarching. The resultant pressure time history may not be uniformly distributed over the roof slab of the buried structure. 2.3.1 Soil Arching Effect Loads acting on the buried structure are influenc ed by the interaction between the structure and the surrounding soil. An effect of such inter action is soil arching an d it is defined as the ability of a soil to transfer loads through a syst em of shear stresses from one location to another in response to a relative displacem ent between the locations. A sti ffer structure in the soil tends to attract more loads, while stress will be dive rted from or around buried structures that are less stiff (Kiger 1988). Soil arching occurs when there is a relative motion between structure and soil. The classical approach for computing soil arching is by the use of the trapdoor mechanism (see Figure 22), as discussed in Te rzaghi and Peck (1948). There are two types of soil arching, namely: PAGE 23 23 Passive arching: The structure moves away from the loading soil, and the soil cannot follow it due to shear resistance. Active arching: The structure is pushed into the soil. Figure 22. Soil arching demonstrated by trap door experiment (Terzaghi and Peck 1948). The soil arching ratio Ca is defined as the ratio of the average pressure on the unsupported clear span of the structure to the applied surface structure. For a shallow burial depth, the arching ratio is given by Equati on 22 (ASCE 1985 and Kiger 1988): C a P B P S exp 2 K o tan WL ()H WL (22) where PB is the average pressure acting on structure; PS is the uniform pressure acting on soil surface; Ko is the coefficient of static lateral earth pressure; is the angle of internal friction in the soil and W, L, H are the width, length and depth of burial of the structure respectively. Typical arching factors for rectangular and arch structures are shown in Figure 23. Besides reducing the average pressure acting on the buried structure, the actual pressure distribution is also no longer uniform due to soil arching. With a responding roof slab, the (b) Pressure on platform and trap door before and after slight lowering of door. (a) Apparatus for investigating arching in layer of sand above yielding trap door in horizontal platform. PAGE 24 24 pressure at the center will be smaller while towa rds the edge the pressure is much higher. A parabolic pressure distribution can be assumed (Kiger 1988). Figure 23. Soil arching as functi on of depth of burial (ASCE 1985). 2.4 Dynamic Structural Behavior and Analysis Structural behavior under tim e dependent lo ading can be obtained by dynamic analysis. The dynamic equilibrium of a system can be desc ribed by the equation of motion. An important result from the equation is the displacement tim e history of the structure subjected to a timevarying load (Tedesco et al. 1998). All structures are in reality distributed mass and stiffness systems and are referred to as distributed systems, or continuous systems. E ach system consists of an infinite number of degrees of freedom and can be considered as a di screte small element connected by springs to all other elements. The governing equations for co ntinuous system can be expressed in partial differential equations and analytic al or closedform solutions can be obtained only for relatively simple continuous systems with welldefined boundary conditions. PAGE 25 25 Using an example of a uniform beam subj ected to an arbitrary distributed force F(x,t) as shown in Fig 24, the equation of motion for the system is given by ),(2 2 4 4txF t w m x w EI (23) Figure 24. Uniform beam s ubject to arbitrary load. The normal vibration modes for the beam )( xr must satisfy the boundary conditions and 0)( )(2 4 4 xm x x EIrr r (24) where )( xr represents the normal vibr ation mode shape for the rth mode. The normal modes are orthogonal functions th at must satisfy the mass orthogonality relationship L r srsrfor srfor M dxxxm00 )()( (25) where the generalized (modal) mass is given by L r rdxxmM0 2)( (26) The general solution in terms of the normal modes )( xr and normal coordinates qr(t), 1)()(),(r rrtqxtxw (27) L EI, m F(x,t) x w(x,t) dx dx x V V V dx x M M M wmdx)(Fdx PAGE 26 26 Establishing the kinetic and potential energy gives 1 2 0 22 1 ),( 2 1r rr LqMdxtxwmT (28) 1 2 0 2 2 2),(r rr LqKdx x txw EIV (29) where Kr is the generalized (modal) stiffness for the rth mode. Finally, the generalized (modal) force of the rth mode Pr must be determined from the work done by the applied force F(x,t) acting through the virtual displacement rq Therefore L r L r r r r rdxxtxFqdxqxtxFW0 1 0 1)(),( )(),( (210) and L r rdxxtxFP0)(),( (211) Substituting the above expressions for T, V and Pr into the Lagranges equations r rr rP q V q T q T dt d 2 (212) the equation of motion in normal coordinates is rrrrrPqKqM (213) Equation 213 represents the unc oupled equations of motion for ,...,3,2,1r. For approximation to the continuous system of a real structure, only a few of the lower modes have responses of any significance for practical purposes, and in some cases only the fundamental mode is of importance. Depending on the chosen approximation, the real system can be considered as either a multidegreeoff reedom (MDOF) system or a singledegreeoffreedom (SDOF) system. PAGE 27 27 2.4.1 SingleDegreeofFreedom (SDOF) System As mentioned in the previous sec tion, in reality all st ructural system consists of an infinite number of degrees of freedom. An infinite number of independent spatial coordinates are necessary to completely define the geometric location of all the masses and stiffness of a structure (Tedesco et al. 1998). Ho wever, it is frequently possible to approximate the real system to a single degree of freedom ha ving equivalent parameters of load, mass and stiffness where the fundamental mode of response is significant. It is advantageous to model the structure as a SingleDegreeofFreedom (SDOF) as this approximate method perm it rapid analysis of complex structures with reasonable accuracy (Biggs 1964) SDOF formulation gives designer valuable information on the dynamic characteristics of the sy stem and they are usually used in preparation of detailed analysis using more a dvanced methods (Krauthammer 1998). The equivalent system is selected so that the deflection of the concentrated mass is the same as that for some significant representative point on the structure, e.g. midspan of beam or center of slab. Since the time scale is not altered, the response of the equivalent system, defined in terms of displacement and time, will be exactly the same as the chosen representative point. As presented in the previous section, the equa tion of motion for a structure system (Equation 213) can be simplified for the SDOF sy stem (inclusive of damping) as ee eFqKqCqM (214) where Me, C, Ke and Fe are the equivalent mass, damping, stiffness and force; qqq ,, is the displacement, velocity and acceleration value of the sele cted representative point. The constants of the equivalent system are evaluated on the basis of an assumed shape function for the deflected structure. When the to tal load, mass, resistance and stiffness of the PAGE 28 28 real structure are multiplied by the correspond ing transformation factors, we obtain the parameters for the equivalent si ngledegreeoffreedom system. 2.4.2 Transformation Factor s for Equivalent SDOF To convert an actual continuous structure in to an equivalent si ngledegreeoffreedom system, the equivalent parameters of the system like the equivalent mass and equivalent loading and resistance function have to be evaluated. Biggs (1964) used transformation factors, denoted by K, to convert the real system into the equivalent system. The equivalent mass of a SDOF system fo r a structure with continuous mass can be determined by Equation 215 and the mass factor, KM, is defined as the ratio of the equivalent mass to the actual total mass of the structure (Biggs 1964). L edxxmM )(2 (215) t L t e MM dxxm M M K )(2 (216) The equivalent force on the SDOF system fo r distributed loads can be found by Equation 217 and the load factor, KL, is defined as the ratio of the equivalent to actual total force L edxxxpF )()( (217) t L t e LF dxxxp F F K )()( (218) where m is the unit mass, p(x) is the distributed loading acting on structure and )( x is the assumed shape function on which the equivalent system is based. Biggs (1964) had tabulated transformation fact ors for beams and slabs with various types of support conditions. For a fixed end beam or on eway slab, the load factor varies from 0.50 for PAGE 29 29 the plastic case to 0.64 for the elasticplastic case and 0.53 for the elastic case. For the mass factor, it varies from 0.33 for plastic case to 0.50 for the elasticplastic case and 0.41 for the elastic case. 2.4.3 Numerical Integration (NewmarkBeta method) The analytical, or closedform, solution of the equation of motion can be cumbersome even for relatively simple excitations. Therefore, fo r most practical proble ms, numerical evaluation technique is employed to obtain the d ynamic response (Tedesco et al. 1998). In this study, the NewmarkBeta method is used in the direct integra tion of the equation of motion and is briefly summarized belo w (Newmark and Rosenblueth 1971): A. For an equivalent SDOF system, th e equation of motion is as follows: )(tFkxxcxm (219) B. Let the values of xi, ixand ix be known at time t = ti. Let ti+1 = ti + t where t is the time step interval. Assume a value of1ix. C. Compute the value 2 )(1 1t xxxxiiii (220) D. Compute the value 2 1 2 1)()() 2 1 ( txtxtxxxi i iii (221) E. Compute a new approximation to 1 ix using equation of motion (Equation 219) F. Repeat steps B to D beginning with the newly computed1 ix until a satisfactory degree of convergence is attained. G. Step B is consistent with a straight line approximation to xin the interval considered. If 41 the method is consistent with a straight line variation of xin the same interval (constant average acceleration). If 61 method corresponds to a parabolic variati on. In this study, it was set as 61 H. The numerical method starts at t=0, the time instant when the load is applied. The initial condition is that the mass is at rest. m tF x )0(0 000 xx PAGE 30 30 This numerical integration method is unconditionally stable. However, a proper value of the time step interval must be chosen to ensu re sufficient accuracy. The time step is dependent on the natural period of the system (Tn) and the loading function time duration (td). According to Bathe (1996) and Clough and Penzien (1993) the chosen time step is given by ) 12 10 (dntT Mint (222) 2.5 Behavior of Reinforced Concrete Slabs A reinforced concrete box effectively is made up a number of reinforced concrete slabs. Therefore, it is essential to first understand the load resisting mechanism of reinforced concrete slabs. This section discusses th e two possible failure mechanisms, namely the flexural and direct shear mode, and their respective re sistancedisplacement functions. Sections 2.5.1 and 2.5.2 discuss the structural behavior of reinforced concrete slabs in flexure using yield line theory and consideration of actual membrane actions respectively. Sections 2.5.3 and 2.5.4 focus on the compressive membrane and tensile membrane behavior. Section 2.5.5 presents a rational flexural model for slabs which was proposed by Krauthammer et al. (1986). Section 2.5.6 discusse s the direct shear mode of failure for slabs and the Hawkins model for direct shear failure is presented in Section 2.5.7. Lastly, the dynamic resistance functions and response is c overed in Section 2.5.8. 2.5.1 Flexural Behavior: Johansens Yield Line Theory The Johansens yield line theo ry is a method for the limit anal ysis of reinforced concrete slabs. The ultimate load of the slab is calcula ted by postulating a collapse mechanism that is compatible with the boundary conditions (Park and Gamble 2000). The moments at the plastic hinge lines are the ultimate moments of resistance of the sections, and the ultimate load is determined using the principle of virtual work or equations of equilibrium. PAGE 31 31 A yield line refers to a line in the plane of the slab about which plastic rotation occurs and across which the reinforcing bars are yielding. When a slab is progressively loaded to failure, yielding of the tension steel o ccurs at section of maximum moment with a large change in section curvature while the moment remains almost constant at the ultimate moment of resistance. As the load is increased, the yield line propagates from th e point at which yielding originated until eventually the yield lines have fo rmed in sufficient numbers to divide the slab into segments that can form a collapse mechanis m. The positions of the yield lines developed are governed by the arrangement of reinforcem ent, boundary conditions and loading (Park and Gamble 2000). The development of the yield li ne pattern for a uniformly loaded simply supported rectangular slab is shown in Figure 25. Figure 25. Yield lines devel opment in a uniformly loaded si mply supported slab (Park and Gamble 2000). 2.5.2 Flexural Behavior: Membrane Action The resistance of reinforced concrete slabs co mputed using the tradit ional approach (e.g. ACI 2005) of oneway or twoway slabs form ulation will be adequate for normal design applications, but they are not accurate representations of the actual capacity of the slabs. Often under the extreme loading conditi ons in the blast and shock environment, compressive and tensile membrane actions in slabs can enhance th e ultimate structural capacity of the slab. Typical design serviceability requirements like defl ections and cracking need not be enforced in First Yielding Further development of yield lines Collapsed mechanism formed PAGE 32 32 some field of applications such as protective design, since moderate to severe degree of damage may be acceptable under such extreme loading. Figure 26 shows the typical lo adcentral deflection curve of a uniformly loaded twoway rectangular slab with laterally restrained edges (Park and Gamble 2000). When the applied load is increased from point A to B, although the initial re sistance is developed by conventional twoway slab mechanism, a compression membrane mechanism sets in with the corresponding increase in the central deflecti on due to the restraint of the out ward of movement of the slab edges. The induced compressive membrane force in the slabs results in an enhancement of the flexural strength. Tests have demonstrated that the ultimate lo ad may be significantly (about two to eight times) higher than that given by the J ohansens yield line theory (Section 2.5.1), particularly if the boundary restra int is stiff, high spandepth ratio and small reinforcement steel ratio (Park and Gamble 2000). With continued loading and increase of the de flection beyond point B, the load carried by the slab decreases rapidly because of a reduction in the compressive membrane force. As point C is approached, the membrane forces in the central region of the slab change from compressive to tensile. Beyond point C, with an increased loading, th e effect of restrained edges sets in and allows the slab reinforcement to act as a plastic tensile membrane with fulldepth cracking of the concrete over the central region of the slab due to the large stretch of the slab surface. The slab continues to carry further load with an increase in deflection until point D when the reinforcement fractures. Tests have indicated that for heavily reinforced slabs the load at point D can exceed the ultimate load at point B. Theref ore, in many cases tensile membrane action also provides a useful means of preventing catastrophic failure at ultimate conditions. PAGE 33 33 Figure 26. Loaddeflection curve for twowa y RC slab with laterally restrained edges. 2.5.3 Slab Compressive Membrane The compressive membrane behavior of slabs co vers two ranges of deflections, point A to B and point B to C (as shown in Figure 27). As the load is increased from A to B, the slab behavior is initially elastic, combin ed with inelastic behavior at crit ical sections at higher loads. Yield line pattern for the slab is fully developed at point B. As deflection increases from B to C, the deformation is mainly caused by plastic rotati on at the yield lines. Therefore, the slab is deforming as a mechanism in the range BC. Plastic theory can be developed first for a restrained strip and then extended to a twoway slab. For a twoway slab, it can be assumed to be composed of strips running in the xand ydirections. The strips have the same thickness as the slab. The xdirection strips contain only xdirection steel and the ydirection stri ps contain only ydirection steel. The yield line pattern of the slab is as shown in Figure 27. Then yield sections of the strips lie on the yield lines and ha ve the same deflection as the actua l slab. The corner yield lines Central Deflection Applied Uniform Load A B C D X Yield line ( Johansen ) Yield line Elastic PAGE 34 34 are simplified to be at 45 to the edges (Park and Gamble 2000) This simplification of assuming corner lines at 45 results in not more than 3% error in th eoretical ultimate load for slabs with all edges fixed against restraint (Park and Gamble 2000). Each of these strips can be analyzed using th e plastic theory presented in Park and Gamble (2000). A fixedend strip with plastic hinges deve loped is shown in Figure 28. This strip is initially of length L and is fully restrained against rotation and vertical translation at the ends. The ends of the strip are considered to be part ially restrained against lateral displacement, and the outward lateral movement at the other end is t Figure 27. Assumed yiel d line pattern for uniformly loaded sl ab with restrained edges (Park and Gamble 2000). A) Actual slab. B) Systems of strips. Figure 28. Plastic hinges of a re strained strip (Park and Gamble, 2000). L L L PAGE 35 35 Compressive membrane action is dependent on the restriction of small lateral displacement, and the behavior of the strip is sensitive to any lateral displacements that may occur. The lateral displacement t may be calcu lated from the movement of the boundary system under the action of the membrane force (Park and Gamble 2000). The strip shown in Figure 28 is considered to have symmetrically positioned plastic hinges. The symmetry assumption leads to the necessary assumptions that the top steel at opposite ends must be equal and the bottom steel is constant along the length. The top and bottom steel may be different. It is assumed that at each plastic hinge that the tension steel has yielded and the concrete has re ached its compressive strength The portions of the strip between the plastic hi nges are assumed to remain straight. The sum of the elastic, creep and shrinkage axial strain, will be constant. The change in dimensions of end Section 12 due to and t is shown in Figure 29. Figure 29. Portion of strip between plastic hinges (Park and Gamble 2000). Based on the geometry of the deformations, th e compatibility equation can be written as ) 2 ( 222L tL hcc (223) L L L PAGE 36 36 where c and c are the neutral axis depths at yi eld Sections 1 and 2 respectively, and h is the thickness of the strip. For equilibrium, the membrane forces acting on Sections 1 and 2 of the strip are equal, TCCTCCsc sc (224) where 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 acting on crosssections 1 and 2 respectively. The concrete compressive forces can be written for a unit width strip as cfCc c 185.0 (225) cfCc c185.0 (226) where fc is the concrete cylinder strength and 1 is the ratio of the depth of the equivalent ACI rectangular stress block to th e neutralaxis depth (ACI 2005). Using Equations 224, 225 and 226, 185.0c ssf CCTT cc (227) Solving simultaneously Equations 26 and 210, the neutral axis depths are given as 1 27.1 ) 2 ( 242 c ssf CCTT L tLh c (228) 1 27.1 ) 2 ( 242 c ssf CCTT L tLh c (229) Figure 210 the shows the conditions at a positivemoment yield section of unit width. The stress resultants at the section Cc, Cs and T are statically equivalent to the membrane force nu, acting at mid depth, and the resisting moment mu, summed about the mid depth axis. Therefore, for a unit width strip, PAGE 37 37 TCcfTCCns c scu 185.0 (230) )5.0()5.0()5.05.0(85.01 1hdTdhCchcfms c u (231) where c is given by Equation 228. Fo r a negative moment yield section, mu is given by an equation similar to Equation 231, and for equilibrium nu = nu. Figure 210. Conditions at positive moment yield sect ion (Park and Gamble 2000). Considering end sections 12 or 34 of the st rip, the sum of the moments of the stress resultants at the yield sections about an axis at middepth at one end can be written as ) 22 )(( ) 22 )(() ( 4.3 1 ) 2 ( 16 ) 2 )( 2 1( 4 ) 2 2( 8 ) 2 )(1( 4 )3( 4 ) 2 1( 2 85.02 2 2 42 1 1 2 1 2 1 2 1 1 1 h dTT d h CCCCTT f L t h L L t h L hL t L h hfnmmss ss c c uuu (232) PAGE 38 38 If Sections 12 or 34 of the st rip is given a virtual rotation the virtual work done at the yield sections is given by ) (uuunmm (233) By equating the work done (Equation 233) to the work done by the loading on the strip, an equation relating the deflection of the strip to the load carried can then be obtained (Park and Gamble 2000). 2.5.4 Slab Tensile Membrane Towards the end of the compressive membrane action range, the large stretch of the slab surface causes the cracks at the central region of th e slab to penetrate across the whole thickness of the slab depth, and the load is entirely ca rried by the reinforcing bars acting as a tensile membrane through caternary action. With further deflection (beyond point C in Figure 26), the region of tensile membrane action gradually spre ads throughout the slab, and the load carried increases until the steel reinforcement starts to fr acture at point D. Figure 211 shows the forces acting on a uniformly loaded plastic tens ile membrane for a rectangular slab. In Park and Gamble (2000), a linear relati onship between load and deflection (Equation 234) for a uniformly loaded slab is given as an approximation for the tensile membrane region of point CD of Figure 26. ,...5,3,1 3 2/)1( 3 22 cosh/11 )1( 4n x y y x n y yT T L Ln n T Lw (234) where w is the uniform load per unit area, is the central deflection, Lx and Ly are the long and short span of the slab, Tx and Ty are the yield force of the reinforcement per unit width in the xand ydirections respectively. PAGE 39 39 Figure 211. Uniformly loaded plastic tensile membrane. 2.5.5 Reinforced Concrete Slab Flexural Model The plastic theory and its assumptions in th e Section 2.5.3 are only applicable at large deflections. Therefore, the initial portion of the load deflection relationship will not be representative at deflections when the slab is st ill within the elastic or elasticplastic regime. Krauthammer (1984) and Krauthammer et al. (1986) proposed a rationa l model to rectify the shortcoming. A second order polynomial is fitt ed to segment AB and a straight line is fitted to segment BC. To better describe the tensile membrane resistance, a straight line which is not required to pass through the origin is proposed fo r segment CD. The proposed model is shown in Figure 212. Park and Gamble (2000) reported that a good es timate of the ultimate load of the slab would be obtained at a central deflection of a bout half the slab thickness (actually half the effective depth). The flexural model as shown in Figure 212 is completed with approximations that the displacements at point B and point C at 0.5h and h respectively, where h is the slab thickness. Krauthammer et al (1986) showed that this approach was able to represent accurately based on comparison against e xperimental test data. PAGE 40 40 This approach was modified by Frye (2002) to consider the differences for slender, intermediate and deep slabs. For slender slabs, the correspondi ng central deflections at point B and C are 0.5h and 1.0h. For deep slabs, the corresponding central deflec tions at point B and C are 0.07h and 0.17h. For intermediate slabs, the displacement and resistance is linearly interpolated between the slender and deep models. Figure 211. Flexural resistance model for slab (Krauthammer 1984). 2.5.6 Direct Shear Behavior Kiger and Getchell (19801982) a nd Slawson (1984) both reported that reinforced concrete slabs exhibited another type of behavior under severe and rapid loading. Beside failure in flexural mode, some slabs failed in a direct shear mode. A photograph of a test specimen which failed in a direct shear is shown in Figure 213. This type of shear failure is characteri zed by slipping and larg e displacement along the vertical interface shear plane (Krauthammer et al 1986). The shear failure produced a vertical failure plane at the edge of r oof and both the top and bottom st eel exhibited necking and were severed nearly flush with the failure plane (Crawford et al. 1983). Direct shear failure will occur Deflection Load wmax 0.5 h A B C D Quadratic function Linear function PAGE 41 41 at the very early stage of the loading, usua lly a fraction of a millisecond, and before any significant dynamic flexural response can be obser ved. Similarly, once the slab survived the initial loading phase without failure in direct sh ear mode, it was observed that a flexural mode of failure will dominate which will occur at a much later time. Figure 213. Slab in direct shear failure mode (Slawson 1984). Crawford et al. (1999) consider this direct shear mode of fail ure as an important element in the blast effects design process. This mode is associated with geometric or load discontinuity, but not with flexure, and is cause d by the high shear inertia forces which do not exist under static or slow dynamic loads. 2.5.7 Hawkins Shear Model The direct shear model used in this study is based on a model proposed by Hawkins (1972). The model describes the static inte rface shear transfer in RC members with wellanchored main reinforcement in the absence of ax ial forces. Krauthammer et al. (1986) modified the model to account for the effects of compressi on and rate effects by applying an enhancement factor of 1.4 (see Figure 214) This same approach is used for this present study. PAGE 42 42 Figure 214. Hawkins model for direct shear stresssl ip relationship (Krauthammer et al. 1986). A detailed description of the model is given below. Region OA: The response is assumed elastic and the slope, Ke, is defined by the shear resistance, e for a slip of 0.004 inch. The resistan ce is given by the following expression c ef 157.0165 (235) where both m and fc are in psi. The response should be ta ken to be elastic and not greater than 2/m Region AB: The slope of the curve decrease con tinuity with increasing displacements until a maximum strength, tm is reached at a slip of 0.012 in ch. The maximum strength is given by yvt c mff 8.08 (236) Shear Stress Shear Slip ENHANCED ORIGINAL E D C B E A 0 Ku D C B A Ke m e L 1 2 3max PAGE 43 43 where bothm fc and fy are in psi,vt is the ratio of total reinfor cement area to the area of plane that it crosses and fy is the yield strength of the reinforcement. Region BC: The shear capacity remains constant with increasing slips. Point C corresponds to a slip of 0.024 inch. Region CD: The slope of the curve is negative, cons tant and independent of the amount of reinforcement crossing the shear plane. The slope is given by c uf K 75.0 2000 (237) Region DE: The capacity remains essentially constant until failure occurs at a slip ofmax For a well anchored bar, the slip for failure in inches is given by 120 1 2max xe (238) where b cd f x 86.2 900 (239) and db is the bar diameter (in inch). The limiting shear capacity, L is given by c ssb LA fA 85.0 (240) where Asb is the area of reinforcement, fs is the tensile strength of the reinforcement and Ac is the crosssectional area. 2.5.8 Dynamic Resistance Function and Response The governing equation of motion for the equiva lent SDOF system (see Fig 215) is given by the following (Krauthammer et al. 1990). PAGE 44 44 Flexural e e eM tF M R txtx )( )('2)( (241) Direct Shear e eM tV M R tyty )( )('2)( (242) where ) ()(),( txandtxtxare the flexural displacement, velo city and acceleration respectively; ) ()(),( tyandtytyare the direct shear slip, veloci ty and acceleration respectively; Me is the equivalent mass; R is the dynamic resistance function; is the natural circular frequency; is the damping ratio; Fe(t) is the equivalent forcing function; and V(t) is the dynamic shear force Figure 215. Equivalent SDOF syst ems for structural element (Krauthammer et al. 1990). A) Continuous structural syst em. B) Flexural Response. C) Direct Shear Response. The modeling of unloadingreloading paths is important in the anal ysis of a nonlinear dynamic behavior. For an elastic perfectly plas tic resistance function, the typical loading and unloading path is as shown in Figure 216A (Ted esco et al. 1998). Kr authammer et al. (1990) proposed a more realistic hysteretic loadingunlo ading looping as shown in Figure 216B. R C Me R s C s Pe(t) M s )(),(),( txtxtx)(),(),( tytyty V(t) C A V(t) V(t) P(t) x, y z B PAGE 45 45 As the loading acting on the slab increas es, the resistancedisplacement follows the resistance curve in the correspondin g direction, (from Point O to Point A to Point B). If flexural failure (Point C) is not reached and unloadi ng occurs, positive unloadi ng path is assumed to follow a straight line BD, which has a stiffness va lue equals to the initial stiffness. Beyond Point D where negative unloading occurs, the unloading path is assumed to follow the straight line DB, where B is a mirror image of the point of the last maximum displacement attained at Point B. Reloading path (e.g. EF) is assumed to rema in parallel to BD and then traces towards Point B. If the reloading exceeds the displacement at Point B, it will reload along the resistance curve and Point B will march forward. The procedure w ill repeat for the next unloading cycle with a new position of Points B and B The unloading reloading paths will affect the amount of internal damping from the hystere tic energy dissipation. For the di rect shear mode of behavior, the unloadingreloading of the resistance curve follows the same procedures for the flexural resistance curve. Figure 216. Dynamic flexural re sistance functions. A) Typical elas tic perfectly plastic function. B) Hysteretic loadingunloading. displacement Resistance, R O A B C C B D E F G Rmax Resistance, R Rmax displacement A B PAGE 46 46 Figure 217. Dynamic direct shear resistance function. 2.6 PressureImpulse Diagrams and their Application For structural dynamic analysis and design, it is often the final states (e.g. maximum displacement and stresses) of utmost relevance, instead of a detail ed response time history of the structure. Pressureimpulse (PI) diagrams are char acteristic curves that de scribe the behavior of a structural component under diffe rent time dependent loads. These diagrams, often known as isodamage curves, were devel oped to aid the assessment of st ructure against blast (May and Smith 1995). Detailed studies on pressureimpulse diagrams for beam and slab have been covered by Soh and Krauthammer (2004) and Ng (2004). Therefore, only a brief introduction on pressureimpulse diagrams and their applications will be covered. 2.6.1 Characteristics of PI Diagram A typical response spectrum for an undamped, perfectly elastic SDOF system is shown in Figure 218(a). In this figure, xmax is the maximum displacement, M is the lumped mass, K is the spring stiffness, P0 is the peak load, td is the load duration and T is the natural period. By defining a different set of axes, the same re sponse spectrum can be transformed into a PI diagram, Figure 218(b). The response spectrum focuses the influence of scaled time on the Shear slip Shear Stress O A B C D F E A E B C PAGE 47 47 system response, while the PI diagram shows the combination of peak load and impulse for a given damage level (Soh and Krauthammer 2004). With a damage level defined, the PI curve indicates the combination of pressure and impulse values that will cause the specified damage. The curve divides the PI diagram into two regions which indicate either fa ilure or nonfailure cases. Pressu re and impulse points falling to the right and above the threshold curve indicates failure in excess of the specified damage level criterion. To the left and below the curve indicates no failure is induced. Figure 218. Typical response spectra and P I diagram (Soh and Krauthammer 2004). A) Shock Spectrum. B) PI diagram. In structural dynamics, there is a strong relationship between the structural response and the ratio of the load duration to the natural period of the structure (Biggs 1964, Clough and Penzien 1993). This relationship can be categorized into the im pulsive, dynamic and quasistatic regimes. As seen in Figure 218, the PI di agram better differentiates the impulsive and quasistatic regimes, in the form of vertical and horizontal asymptotes. A B PAGE 48 48 2.6.2 Numerical Approach to PI Diagram Closed form solutions of PI diagram can be obtained for idealized structures subjected to a simplified load pulse (Ng 2004). However, in order to allow comp lex nonlinear resistance functions and complex loading cond itions to be considered, a numeri cal approach to generate the PI diagram must be adopted. PI diagram can be generated numeri cally by performing many single dynamic analysis. Each result from a dynamic analysis will determine whether the pressure and impulse combination is in the failur e or nonfailure region. With sufficient runs, a threshold curve can be plotted. Since it is comp utationally expensive to run all possible pressure and impulse combinations, an efficient search al gorithm must be employed to locate the required threshold points (Krauthammer et al. 2008). Soh and Krauthammer (2004) and Ng (2004) developed numerical procedures to numerically generate PI diagram. Blasko et al. (2007) developed a more efficient search algorithm. The procedure uses a single radial se arch direction, origina ting from a pivot point (Ip, Pp) which is located in the failure zone of the P I diagram (see Figure 219). Iterations using Bisection method are carried to ge nerate the threshold curve. This approach can be applied effectively to any structural system for which a re sistance function can be defined. The pressureimpulse diagrams which will be presented in the later part of this study are generated numerically using this same approach. 2.6.3 Multiple Failure Modes In general, the response and failure for most st ructures can occur in more than one mode. Although flexure is usually the predominant mode but under certain circumstances, failure may occur in other mode (e.g. direct shear). If there exists two a singl e failure modes, the PI diagram will consists of two threshold curves each representing a failure mode. The true threshold curve will therefore be represented by the lower bound of the two curves as shown by PAGE 49 49 the dotted line (see Figure 220). With the two thre shold curves plotted, it is possible to identify the actual failure mode by plotting the pre ssureimpulse combination and examining which region the point is located. Figure 219. Search algorithm for PI diagram (B lasko et al. 2007). A) Establish pivot point. B) Data pivot search Pressure Impulse Mode 1 (Flexure) Mode 2 (Direct Shear) Failure in Mode 1 & 2 Failure in Mode 1 Failure in Mode 2 Safe Figure 220. PressureImpulse diagram with two failure modes. A B PAGE 50 50 2.7 Summary The background of the effects of blast loads on buried structures and the dynamic structural behavior and analysis were presented in this chapter. Pressureimpulse diagrams and their application were also briefly discussed. The different mode of behavior for reinforced concrete slabs under loading was discussed in greater detail in this chapter. This chapter is the basis for the methodol ogy to derive the numerical method for the dynamic analysis of shallowburied reinforced concrete boxtype structures subjected to air blast loadings in Chapter 3. PAGE 51 51 CHAPTER 3 METHODOLOGY 3.1 Introduction For dynamic analysis of any structures, the material and constitutive models of the structure must first be derived in order to describe the relationsh ip between the dynamic resistance (e.g. bending moment, shear forces) and the structure response (e.g. displacement). Then a suitable dynamic structural model (e.g continuous system, multi degree of freedom, single degree of freedom) can be chosen to accurately represent the mechan ical characteristics of the structure. This chapter covers the methodology to genera te the resistance functions for reinforced concrete slabs and the buried box structure in bot h the flexural mode (Section 3.2) and direct shear mode (Section 3.4). The issue of soil arch ing and the required modifications to the load and mass factors are discussed in details in Section 3.4. The effect of shear failure on the slab resistance is also discussed in Section 3.5. La stly, a flowchart of the procedure is presented in Section 3.6. 3.2 Flexural Mode As described in Section 2.4.1, the center point of the slab is chosen as the reference point for the singledegreeoffreedom system and the slab response is in accordance with the resistance function described in Section 2.5.4. However, for a buried box structure, several modifications have to be introduced into the ap proach described in Section 2.5.2 in order to account for certain conditions that will affect the system behavior. 3.2.1 Externally Applied Thrust The peak structural resistance in the comp ression membrane mode can be enhanced if there is an external inplane compressive for ce being applied to the slab (Krauthammer 1984). PAGE 52 52 The inplane compressive force may exist in th e form of prestressing force (Meamarian et al, 1994) or due to horizontal component of the vert ical forces for a box structure buried in soil (Krauthammer et al. 1986). For the box type structures under considerati on, the inplane compressive forces are generated by the horizontal component of the pr essure pulse that propagates vertically through the soil from the air blast load on the surface (s ee Figure 31). These compressive forces vary with time and therefore have to be calculated at every time step of the analysis. Roof Floor NroofNfloorPv(t) Ph(t) Ph(t) NroofNfloor Figure 31. Model for externally applied thrust (Krauthammer et al. 1986). In this study, the following procedure is implemented: 1. The wall is subdivided into n number of layers (see Figure 32). The depth at the middepth of each of the n layers is calculated and denoted as hi. The time of arrival at each layer can be calculated by considering the seismic wave velocity of the soil. 2. The vertical pressure pulse in the soil is traced at each time step and the horizontal stress component is converted into point loads acting the each of the n layers. PAGE 53 53 3. The horizontal stresses generated by the propa gating pressure pulse in the soil can be computed as a ratio of the vertical stress. The ratio is the coefficient of static lateral earth pressure, K0. The ratio is dependent on the effective friction angle of the soil and it can be estimated as about 1.0 for cl ay and about 0.5 for sand (Krauthammer et al. 1986). 4. The compressive thrust at the roof and floor can then be calculated from the points load acting at each layer. The compressive thrust, Nx and Ny, will be used in the formulation for the calculation of the enhanced membrane peak resistance. 5. When the whole vertical pressure pulse propagates beyond the floor level of the buried box structure, the compressive thrust (due to static earth pressure) remains constant and therefore need not be reeva luated again at the further time steps. Figure 32. Calculation of externally applied thrust. Roof Floor NroofNfloo r Pv(t) Ph(t) Ph_i Wall divided into n layers i =1 i =2 i =n hi PAGE 54 54 3.2.2 Numerical Approach for Resistance Curve Calculation Following the plastic theory described in Sectio n 2.5, some modifications were introduced in order to calculate the slab re sistance at point B and point C (see Figure 25) more accurately. An externally applied thrust is included in the formulation. Instead of using the approximation of using the ACI stress block for th e concrete compressive stress, th e approach in this study is to divide the concrete into layers parallel to the ne utral axis and the stresses and forces for all layers are determined based on the appropriate stress strain relationship chosen (see Figure 33). Figure 33. Stress and stra in distributions across reinforced concrete section. In this study, the concrete stressstrain relationship was based on the Hognestad model (MacGregor and Wight 2005) whereas for the reinfo rcing steel, the stressstrain model by Park and Paulay (1975) was used. In Figures 34 and 35, the restrained strip with plastic hinges is applied with an external thrust. Using similar compatibility and equilibr ium equations presented in Section 2.5.2, an iterative procedure is implemented in order to fi nd the neutral axis depths that will be satisfied for each strip displacement. With the neutral axis depth, the corresponding axial forces and did j h nu mu Strain Distribution Concrete and steel Stress fsifs j fci c zi ciRC Slab Tensile steel Compressive steel Neutral axis unit width linear strain distribution sicus j PAGE 55 55 moment in the strips can then be calculated. The slab resistance at point B and C of the resistance curve can then be determined based on the strips forces and moment. L L L mu mu nu nu N N Figure 34. Restrained st rip with external thrust. L N L L N Figure 35. Portion of strip between plastic hinges with external thrust. Based on the strain distribution and stress strain relationship (Figure 33), the concrete and steel stress for a unit width of the slab at yiel d Section 2 (Figure 35) can be expressed as a function of the neutral axis depth, c. )()(i cu si sidc c Fn Fnf (31) )()(i cu ci cizc c Fn Fnf (3where si and ci are the steel and concrete strain, cu is the ultimate concrete strain at failure, di PAGE 56 56 is the depth of reinforcing steel, zi is the depth of concrete layer, c is the neutral axis depth and Fn( ) represent the material stressstrain function. The forces and moment for the steel and concrete can be expressed as sisisiAfF (33) zfFcici (34) )2/(si sisidhFm (35) )2/(i cicizhFm (36) where Fsi and Fci are the steel and concrete layer force, msi and mci are the steel and concrete layer moment about midheight of slab section, Asi is the steel area, z is the concrete layer thickness, h is the slab thickness. The total section moment, mu, and axial force, nu, can be calculated as: layers concrete i ci layerssteel i si um mm (37) layers concrete i ci layerssteel i si uF Fn (38) Similarly, for yield Section 1 (Figure 35), mu and nu can also be expressed as a function of the neutral axis depth, c. Based on the geometry of the deformations, th e compatibility equation can be written as ) 2 ( 222L tL hcc (39) and total u c uSL Nn Eh n L t )(2 ) 2 ( (310) where t is the outward lateral movement of the slab, Ec is the concrete elastic modulus and S is the surround stiffness. PAGE 57 57 For equilibrium, the membrane for ces at Section 1 and 2 are equal uunn (311) Equations 310 and 311 can be rearranged and written as a function in c and c: 0) )(2 ( 22 ),(2 1 SL Nn Eh n L hccccRu c u (312) 0 ),(2 uunnccR (313) For any displacement value, the values of the neutral axis depth c and c can be determined by solving Equations 312 and 313. In this study, NewtonRaphson iteration (Zienkiewicz and Taylor 2005) is used to solve the two equations numerically. With values of the neutral axis depth c and c, the values of mu, mu and nu for each strip can be calculated using Equations 37 and 38. By equating the internal virtual work done (E quation 2.16) to the ex ternal virtual work done by the loading on the strip, the load carried by th e strip can be obtained for any assumed displacement value of the strip. As presented in Section 2.6.2, the whole slab is assumed to be divided into xdirection and ydirection strips. With an assu med deflection value at the center of the slab, the strip deflection can be calculated and the force and moments in ea ch strip are solved using the above procedure. The maximum slab resistance can then be determ ined using the appropriate displacement value at Point B on the slab resistance curve. The resistance at Point C on the resistance curve is also calculated using the same procedure. The only difference is that the membra ne forces are set as zero. The resistance at Point C corresponds to the Johansens yield line load. PAGE 58 58 3.2.3 Variation of Mass and Load Factor As presented in Section 2.4.2, to convert a c ontinuous structure into an equivalent singledegreeoffreedom system, the equivalent parameters like the equivalent mass and equivalent loading and resistance function ha ve to be evaluated using the a ppropriate mass and load factor. Instead of applying a constant mass and load fact or for analysis, a variation of the load and mass factor is proposed. As shown in Figure 36, the variation of the factors corresponds to the resistance curve for the slab. In the compressive membrane region of the resistance curve, the factors will varied from the elas tic value at point A to the elasticplastic value and then to the fully plastic value at point B, where the resistance is at the maximum. For the region represented by Point C to Point D, the factors used are the tension membrane value. A linear variation is assumed for the factors in between Points A to D. Figure 36. Variation of load and mass factor. Load Factor Mass Facto r Deflection 0.5h h A B C D Compression membrane Transition Tension membrane Elastic Elasticplastic Plastic Tension membrane PAGE 59 59 As mentioned in Section 2.4.2, the load and mass factors for beams and slabs under various loading and support conditions for diff erent range of response can be found in Biggs (1964). The values for the load and mass factor s for the tension membrane can also be found using the same approach. Taking a fixed end be am or oneway slab under uniform load as an example, the load and mass factor for the tensio n membrane can be calcul ated using Equations 216 and 218. For the beam (or oneway slab) of length L in tension membrane, under a uniform applied load, the deformed shape can be assumed as a parabola. Therefore, the deformed shape function can be written as 2 244 )( L x x L x Lxfor 0 (314) From Equations 216 and 218, L dx L x x L Lm dxxm M M KL L t e M 0 2 2 2 0 244 )( (315) 533.0 15 8MK Ltp dx L x x L tp Ltp dxxtp F F KL L t e L.)( 44 )( ).( )()(0 2 2 0 (316) 667.0 3 2 LK Using the same approach, once the deflected shape for twoway slabs have been assumed, the load and mass factors can be obtained by integration over the en tire slab surface. PAGE 60 60 3.3 Soil Structural Interaction As presented in Section 2.3.1, one of impor tant effect caused by th e interaction between the structure and the surrounding soil is soil archi ng. Buried reinforced concrete box structures are usually much stiffer than their surrounding soil medium and will tend to attract load. Experimental data also showed that the pressure acting on the flexib le center of th e roof slab is significantly less than the applied overpressure acting on the free soil surface. The roof edges are relatively stiffer since they are supported by the walls and the pressure acting there are much higher. The pressure distribut ion is therefore not uniform. The load reduction due to soil arching is acc ounted for by using the soil arching factor, Ca, as given by Equation 22. The lo ad distribution effect has to be accounted for by adjusting the load and mass factors. 3.3.1 Influence of Parameters on Soil Arching Effect The soil arching factor, Ca, represent the ratio of the aver age pressure acting on the roof slab to the applied surface pressure. Looking at Equation 22, we can see that the factors that will influence the value of the arching factor includes, the friction angle the span ratio of the roof slab and the depth of burial of the roof slab. Bowles (1996) provides some repr esentative values for the angle of friction for different types of soil. Friction angle for cohesionless soils vary from 20 for loose silty sand to about 46 for dense sand. For cohesive soils such as clay the friction angle is appreciably smaller than cohesionless soil. For saturated clayey soil with very small shear strength, the friction angle can be assumed as zero and there wi ll be no soil arching effect. The depth of burial for the roof slab is expresse d as a ratio of the shor t span of the slab and in this study for shallow buried box structures, the ratio of burial de pth to short span is limited to PAGE 61 61 a maximum value of 1.0. With regards to the slab span ratio, the arching ratio will be the maximum for a square slab, given the same soil properties and burial depth. The variation of the soil arching factor (for a slab span ratio =1) with respect to the friction angle and burial depth is shown in Fig 37. From Figure 37, one can assume conservatively that the soil arching factor Ca will varies from a value of 0.3 (maximum soil arching effect) to a value of 1.0 (no soil arching) for different combinati on of parameters, including the friction angle, burial depth and span ratio. 051015202530354045 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 Variation of Soil Arching Factor (Span ratio = 1)Friction AngleSoil Arching Factor0.2L 0.4L 0.6L 0.8L burial depth=1.0L Figure 37. Variation of so il arching factor with fric tion angle and burial depth. 3.3.2 Effect on SDOF Load and Mass Factor As presented in Section 2.4.2, the load fact or and mass factor can be calculated based on the Equations 216 to 218. For a uniform loadi ng case, Biggs (1964) had tabulated these factors for beam and slab with di fferent support conditions. Soil arching effect changes the load distribut ion and the uniform pressure applied on the soil surface is no longer uniform at the buried slab level. It is assumed that the pressure PAGE 62 62 distribution due to soil arching now follows a parabolic shape (Fi gure 38). The pressure at the centre of the slab or beam is assumed to be a factor of the uniform pressure given by Ca p(t), where p(t) is the applied pressu re at the soil surface. Factor will vary between 1 and 0, whereby value of 1 represent there is no soil arch ing effect; and value of 0 represent the presence of a maximum soil arching. Factor will give an indication on the relative strength of the soil arching effect, based on the soil and geom etric properties given in Section 3.3.1. .Figure 38. Parabolic pressure distribution under soil arching. Based on the assumed parabolic pressure dist ribution and Equation 218, one can calculate the new load factor, KL, which will account for the effect of nonuniform pressure distribution acting on the slab in the equiva lent SDOF system. The ratioL defined by: L L LK K' (317) where KL is the load factor with soil arching parabolic pressure; and KL is the normal load factor under uniform load. L/2 L/2 Ca*p(t) *Ca*p(t) Parabolic Pressure Distribution Uniform Pressure Distribution Depth of Burial p(t) *Ca*p(t) PAGE 63 63 The variation of ratioL is plotted in Figure 39 for the four different cases, namely the elastic, elasticplastic, plastic and tension me mbrane. The plot shows how the load factor changes with the soil arching factor. A lin ear function between the soil arching factor, Ca, and the ratioL can be derived based on the plot. This f unction was used in the computer code to adjust the load factors used in the dynamic analysis based on the calculated arching factor value. 00.10.20.30.40.50.60.70.80.91 0.4 0.5 0.6 0.7 0.8 0.9 1 betalamda4. 1. elastic 2. 2. elasticplastic 3. 3. plastic 4. tension 1. soil arching factor Ca = 0.3 soil arching factor Ca = 1.0 Figure 39. Variation ofL Based on Figure 39, 1837.0816.0_a ElasticLC (318A) 3877.0612.0_a stic ElasticPla LC (318B) 2857.0714.0_a PlasticLC (318C) 4286.0571.0_a TensionLC (318D) L PAGE 64 64 Similarly, the mass factor has also to be ad justed due to soil arching effect. The mass factor for the structural slab will remain the sa me since soil arching does not actually affect it. Instead, the weight of the soil overburden which is acting on the slab will al so be affected in the same manner described above. Due to active soil arching, more weight of the soil will be acting at the edges where the slab is stiffer and lesser weight on the cen ter. Therefore, with a strong soil arching effect, it can be e xpected that less soil mass is active in the system response. Using the same parabolic distribution profile for the soil ma ss, one can calculate the new soil mass factor, KM, which will account for the soil archi ng effect on soil mass participation in the equivalent SDOF system. The ratioM defined by: M M MK K' (319) where KM is the mass factor with soil arching parabolic profile; and KM is the normal soil mass factor with unifor m mass distribution. The variation of ratioM is plotted in Figure 310 for the four different cases, namely the elastic, elasticplastic, plastic and tension memb rane. The plot shows how the soil mass factor changes with the soil arching factor. A lin ear function between the soil arching factor, Ca, and the ratioM can be derived based on the plot. This f unction is used in th e computer code to adjust the soil mass factors based on th e calculated arching factor value. Based on Figure 310, 039.0039.1_a ElasticMC (320A) 137.0863.0_a stic ElasticPla MC (320B) a PlasticMC 0.1_ (320C) 1837.0816.0_a TensionMC (320D) PAGE 65 65 00.10.20.30.40.50.60.70.80.91 0.2 0.4 0.6 0.8 1 betalamda_M4. 1. elastic 2. 3. 2. elasticplastic 1. 3. plastic 4. tension soil arching factor Ca = 0.3 soil arching factor Ca = 1.0 Figure 310. Variation ofM 3.4 Direct Shear Mode In the dynamic response for reinforced concrete slabs, beside the usua l dominant flexural mode, it had been experimentally observed that sl abs when subjected to severe impulsive loading have also shown to exhibit di rect shear mode of failure. 3.4.1 Resistance Curve As presented in Section 2.5.7, the direct shea r model used in this study is based on a model proposed by Hawkins (1972) and modi fied by Krauthammer et al. (1986). For reinforced concrete beam or oneway slab the formulation presented in Section 2.5.6 can be used to generate the di rect shear resistance curve. For a twoway slab where the longitudinal reinforcing steel in the xdirec tion and ydirection are different, the following approach is adopted to find th e effective resistance curve. M PAGE 66 66 The governing direct shear SDOF equa tion of motion can be written as )()()( tVRtwCtwMxx xx xex (321) )()()( tVRtwCtwMyy yy yey (322) where Mex and Mey are the equivalent mass in xand ydirection; Cx and Cy are the damping, Rx and Ry are the direct shear resistance; Vx(t) and Vy(t) are the dynamic shear force; and )()( twandtw are the direct shear slip velocity and acceleration. As shown in Figure 311, it is assumed that th e reinforcement is the same at the supports for each xand ydirection. Since the direct shear failure occurs at the very early stage of the loading, one can assume that the flexural mode of deformation is not si gnificant and the whole slab displaced downwards as a rigid body motion. Therefore, Equations 321 and 322 can be simplified to )()()(;)()()(;)()()( twtwtwtwtwtwtwtwtwy x s y x s y x s ))()(()()()()()( tVtVRRtwCCtwMMy x yx syX sey ex )()()( tVRtwCtwMss sS ss (323) Figure 311. Direct shea r model for twoway slab. xdirection ydirection Ry Rx Cy Cx w y (t) wx(t) V y (t) V x (t) Simultaneous failure in ystrips Simultaneous failure in xstrips Rs C w(t) M PAGE 67 67 Therefore, the equivalent resistance for a twoway slab can be obtained by considering the resistance in the xdirection and ydirection and adding them together (see Figure 312). The equivalent mass and loading for the singledegreeoffreedom system in direct shear was calculated by applying the appropriate transf ormation factors, using the same procedures presented in Section 2.4.2. The shear mass and load factors are presented in the next section. Figure 312. Direct shear resistance curve for twoway slab. 3.4.2 Shear Mass and Load Factors The equivalent shear mass and load factors are computed based on the assumed mode and deformed shape of the slab under direct shear fa ilure mode. Since the assumption for the slab under flexural mode deformation is a symmetrical plastic hinge formation, it requires that the steel reinforcement at th e either sides of the support is the same. Therefore, simultaneous shear failure at the supports will occur and the de formed shape is as shown in Figure 313. Based on the deformed shape, a shear mass factor of 1.0 can be taken for the structural slab. In addition, for a buried box structure, th e mass of the soil overburde n has to be considered Shear Stress Shear Slip 0 Resistance R x Resistance R y Resistance R s = R x +R y Dmax (xdir) Dmax (ydir) Failure in x Failure in y PAGE 68 68 as well. Since the direct shear failure (if it was to occur) happened very early in the loading stage and the entire roof slab was pushed into the box, followed by the soil overburden, soil arching effect can be assumed be neglected in di rect shear mode. Theref ore, the entire mass of the soil overburden can be assumed be effective. Figure 313. Deformed shape for direct shear response. Using the same assumed deformed shape, the sh ear load and resistance factor can also be taken as 1.0, where the total resistance and load ing for the entire slab is used in the SDOF equation of motion. 3.5 Shear Failure Mode for Slab The flexural and direct shear modes of behavi or of reinforced concrete slab have been considered in the previous Sections. Shear is ge nerally not critical when slabs carry distributed loads and supported by walls or beams since th e maximum shear force per unit length is relatively small (Park and Gamble 2000). However, shear failure can become critical when the span to effective depth ratio is small and the corresponding flexural resistance due to membrane action increases. A photograph of a test specimen tested by Slawson (1 984) which failed in a shear mode is shown in Figure 314. The shear failure occurs near the wall support where the shear stress level is the highest. xdirection ydirection Simultaneous failure in ystrips Simultaneous failure in xstrips Deformed Shape (and distribution of inertia force) PAGE 69 69 Figure 314. Slab in shear failure mode (Slawson 1984). In order to model the shear failure mode, it was proposed to implement a simplified modification to the flexural resistance curve as shown in Figure 315. When the applied load is increased from point A, the initial resistance wi ll follow the path towards point B, which is representative of the flexure resistance in th e compressive membrane zone. With continued loading and if the shear strength of the slab section is lower than the maximum flexural resistance, wmax, the slab will failed in shear at point B With the increase of load and deflection beyond point B, the slab will continue to deform in the flexure mode towards the Johansens yield line load (point C). Beyond point C, the sl ab will behave in the tension membrane mode until the slab reinforcement failed at point D. The nominal shear strength of a reinfo rced concrete section is given by: scnVVV (324) where Vn is the nominal shear strength, Vc and Vs are the shear strength provided by concrete and steel reinforcement respectively. PAGE 70 70 Nawy (2000) states that for a d eep beam or oneway slab sec tion, where the clear span to effective depth ratio is less than 5, the shear resisting force of concrete and shear reinforcement can be calculated using the following expressions. dbfVwc c '2 (325) df dl s Adl s A Vy n h vh n v v s 12 11 12 1 (326) where bw is the width; d is the effective depth; fc is the concrete strength; fy is the steel yield strength; ln is the clear span of beam/slab; Av is the total area of vertical reinforcement; Avh is the total area of horizontal reinforcement; sv is the horizontal spacing of the vertical reinforcement and sh is the vertical spacing of the horizontal reinforcement. 3.6 Program Flowchart The flowchart of the proposed procedure to generate the resistance function and solving the equation of motion for the required dynamic response is shown in Figure 316. The approach consists of two SDOF systems for evaluating the fle xural response and the direct shear response separately. For the flexur al mode or response, th e resistance function has to be recalculated at each time step since th e maximum resistance is dependent on the wall force for a buried box structure. For the plotting of PressureImpulse diagram, this program will be used to solve the system for each pressureimpulse iteration run in order to generate the threshold curve, following the process described in Section 2.6.2. 3.7 Summary This chapter presented the methodology to gene rate the resistance function for buried box structure for both flexural and di rect shear mode of behavior. A variation of the load and mass PAGE 71 71 factor with respect to the resistance curve and the modification on these factors due to soil arching effect were discussed. The modification to consider shear failure mode was also presented. The proposed methodology was implemented in a computer language and the numerical analysis results generated will be presented in Chapter 4. Figure 315. Resistance curve for slab with shear failure mode. Deflection Load Flexural wmax BA B C D C Shear strength,ws Johansens Load Shear Failure TransitionB Flexural mode PAGE 72 72 Start Input Data Flexural Response Using NewmarkBeta Integration Direct Shear Response Using NewmarkBeta Integration Failure Failure Increment In Time Step Vertical Wave Propagation Calculate Wall Force Output End Yes No Yes No Generate Direct Shear Resistance Function Generate Flexural Resistance Function Increment In Time Step Figure 316. Program flowchart. PAGE 73 73 CHAPTER 4 RESULTS AND DISCUSSIONS 4.1 Introduction The proposed procedure discussed in Chapter 3 was implemented in a computer programming language to test its viability. The results generated using the procedure are compared with the available data from past experimental work carried out on shallow buried reinforced concrete box structures subjected to airblast loading. This validation of the numerical procedure will be presented in Section 4.2. In Section 4.3, the assessment on the same set of experimental results using PressureI mpulse diagrams was demonstrated. 4.2 Validation with Experimental Data The capability of the proposed numerical procedure to perform dynamic analysis for reinforced concrete box structure subjected to airblast loads is presented. Experimental data from the tests conducted by Kiger and Getchell (1980) was used to validate the proposed methodology presented in Chapter 3. The experimental test series conducted by Kiger and Getchell (1980) consisted of a total number of seven tests on shallow buried reinforc ed concrete box structures subjected to airblast loads. The airblast loads was a simulation of a distant nuclear explosion with a sharp rise time and a uniform pressure over the top of the soil surf ace. Details of the experiment are given in the Appendix. The test series consisted of a total of 7 cases, whereby six of them are single bay rectangular box structures and another one a mu ltibay box structure. Dynamic analysis was conducted and the flexural and direct shear resp onse time history and resistance functions were generated for each test. The forcing function was obtained from the airblast gauge on the free surface and the damping ratio used was 20% for flexural analysis and 5% for direct shear PAGE 74 74 analysis. A high damping ratio for the flexural m ode was used to consider the significant energy dissipation due to soilstructure interaction (Krauthammer et al 1986). The numerical results were compared against the measured test data and the comparison is presented in the following sections. 4.2.1 Test FH1 Test FH1 was conducted in a sand (noncohesive) backfill at a depth of burial (DOB) equal to 50% of the short clean span. The reinforced concrete box had wall, floo r and roof thickness of 5.6 inches, giving the roof slab a span to effective depth ratio of 10. The structure had one percent principal reinforcing steel in each face, with a concrete strength of 7000psi. The test charge density was 0.9 lb/ft3 and produced a peak pressure of 2400psi. The roof and floor of the structure suffered cracking and some permanent deflections, but there was no structural failure. The top surf ace had longitudinal cracks, located roughly above the inside walls, almost the entire length of the structure. The roof had a maximum permanent deflection at its midspan of about 0.44 inch. A photograph of the box slab after the sand backfill was excavated is shown in Figure 41. The displacement time history for the flexural degreeoffreedom is pl otted in Figure 42. The numerical results show that the permanent di splacement at the center of the roof slab is about 0.5 inch, which compares well with the expe rimental result. The numerical analysis also indicated that the slab flexural response is still within the compression membrane mode. The flexural resistance function for FH1 is shown in Figure 43. The numerical analysis shows that the roof slab did not fail in the di rect shear mode, which is consistent with the experimental observation. The displacement time history and resistance function for the direct shear degreeoffree dom are plotted in Figures 44 and 45. PAGE 75 75 Figure 41. Post test view of FH1. Flexural Displacement Time History FH1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 00.0050.010.0150.020.0250.030.0350.040.045 Time (s)Displacement (in) permanent displ = 0.50" Figure 42. FH1 flexural displacement time history. PAGE 76 76 Flexural Resistance Function FH1 100000 0 100000 200000 300000 400000 500000 600000 00.10.20.30.40.50.60.70.8 Displacement (in)Resistance (lbs) Figure 43. FH1 flexur al resistance function. Direct Shear Displacement Time History FH10 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0 0.0050.010.0150.020.025Time (s)Displacement (in) Figure 44. FH1 direct shear displacement time history. PAGE 77 77 Direct Shear Resistance Function FH12000000 1000000 0 1000000 2000000 3000000 4000000 5000000 6000000 7000000 8000000 0 0.010.020.030.040.05Displacement (in)Resistance (lbs) Figure 45. FH1 direct shear resistance function. 4.2.2 Test FH2 Test FH2 was conducted in a sand (noncohesive) backfill at a depth of burial (DOB) equal to 50% of the short clean span. The reinforced concrete box had wall, floo r and roof thickness of 5.6 inches, giving the roof slab a span to effective depth ratio of 10. The structure had one percent principal reinforcing steel in each face, with a concrete strength of 5200psi. The test charge density was 2.7 lb/ft3 and produced a peak pressure of 5250 psi. The test bed had a distinct, el ongated depression above the top of the structure. Excavation of the test bed revealed that the roof of the structure suffered complete failure. Post test examination indicated that the roof had been sheared off at the wa ll supports. The principal steel reinforcing bars, except a few that were not broken near the corners of the end wall, were necked down and broken at the wall supports. An inspection of the reinforcement ba rs near the center of the roof slab did not indicate th e occurrence of significant flexure behavior. A photograph of the box slab after the sand backfill was ex cavated is shown in Figure 46. PAGE 78 78 Figure 46. Post test view of FH2. The displacement time history and resistance f unction for both the flexur al and direct shear degreeoffreedom are plotted in Figure 47, 48, 49 and 410. The numerical results show that the roof slab failed in direct shear mode first, when the applied loading exceeded the direct shear resistance of the entire slab. The roof slab is also expected to fail in flexural mode in the numerical results. However, since the direct shear failure occurs at about 1 millisecond after the arrival of the loading, the sl ab will shear off the wall support and do not have enough time to go into the flexure response mode. The numerical prediction is therefore consistent with the experiment observation. PAGE 79 79 Flexural Displacement Time History FH2 0 2 4 6 8 10 12 14 16 00.0020.0040.0060.0080.010.0120.014 Time (s)Displacement (in) Failed Figure 47. FH2 flexural displacement time history. Flexural Resistance Function FH2 0 200000 400000 600000 800000 1000000 1200000 1400000 1600000 1800000 2000000 0246810121416 Displacement (in)Resistance (lbs) Failed Figure 48. FH2 flexur al resistance function. PAGE 80 80 Direct Shear Displacement Time History FH20 0.05 0.1 0.15 0.2 0.25 0 0.0020.0040.0060.0080.01Time (s)Displacement (in) Failed Figure 49. FH2 direct shear displacement time history. Direct Shear Resistance Function FH20 1000000 2000000 3000000 4000000 5000000 6000000 7000000 0 0.050.1 0.150.20.25Displacement (in)Resistance (lbs) Failed Figure 410. FH2 direct shear resistance function. PAGE 81 81 4.2.3 Test FH3 Test FH3 was conducted in a clay backfill at a depth of burial (DOB) equal to 50% of the short clean span. The clay backfill were wetted dur ing the test to ensure the backfill would be a low shear strength material. The reinforced concrete box had wall, floor and roof thickness of 5.6 inches, giving the roof slab a span to effective depth ratio of 10. The structure had one percent principal reinforcing steel in each face, with a concrete strength of 7900psi. The test charge density was 0.9 lb/ft3 and produced a peak pressure of 2650psi. After excavation, post test examination indicated that the roof slab responded primarily in the flexure mode, with a permanent center deflecti on of about 6 inches. Passive deflection gage recorded a maximum transient deflection of abou t 7 inches, indicating a rebound of about 1 inch after removal of load. Extensive longitudinal direction cracks were obs erved and they were concentrated along the edge of the wall supports and center area of the roof. This observation corresponded with a flexural re sponse with three hinges forming at the two supports and at the center. The inside of the roof was extensively cracked longitu dinally down the center with the concrete broken off and the reinforcing bars expo sed. The exposed principal bars in the roof center were all necked down and some were br oken. A photograph of the damaged box structure after excavation is s hown in Figure 411. The displacement time history and resistan ce function for the direct shear degreeoffreedom are plotted in Figure 412 and 413. The numerical results show th at the roof slab did not fail in direct shear mode, same as the experiment al observation. The displacement time history for the flexural degreeoffreedom is pl otted in Figure 414. The numerical results show that the permanent di splacement at the center of the roof slab is about 5.9 inch, with a maximum transient deflection of 6.7 inches. This compares well with the experimental result. Looking at the resistance fu nction shown in Figure 415, it indicates that the PAGE 82 82 slab underwent extensive deformation into the tensile membrane region. These numerical predictions are consistent with the experiment observations and measurement. Figure 411. Post test view of FH3. Direct Shear Displacement Time History FH30 0.02 0.04 0.06 0.08 0.1 0.12 0 0.0020.0040.0060.0080.01Time (s)Displacement (in) Figure 412. FH3 direct shear displacement time history. PAGE 83 83 Direct Shear Resistance Function FH30 1000000 2000000 3000000 4000000 5000000 6000000 7000000 0 0.05 0.1 0.15 0.2Displacement (in)Resistance (lbs) Figure 413. FH3 direct shear resistance function. Flexural Displacement Time History FH3 0 1 2 3 4 5 6 7 8 00.010.020.030.040.050.060.070.08 Time (s)Displacement (in) permanent displacement=5.9" Figure 414. FH3 flexural displacement time history. PAGE 84 84 Flexural Resistance Function FH2 200000 0 200000 400000 600000 800000 1000000 1200000 1400000 012345678 Displacement (in)Resistance (lbs) Figure 415. FH3 flexural resistance function. 4.2.4 Test FH4 Test FH4 was conducted in a sand backfill at a shallow depth of burial (DOB) equal to 20% of the short clean span. The reinforced c oncrete box had wall, floor and roof thickness of 5.6 inches, giving the roof slab a span to effective depth ratio of 10. The structure had one percent principal reinforcing steel in each face, with a concrete strength of 6700psi. The test charge density was 0.9 lb/ft3 and produced a peak pressure of 3000psi. After excavation, post test examination indicated that the structure suffered severe damage with the roof slab on the verge of collapse. The measured permanent center deflection was about 12.5 inches. Passive deflection gage recorded a maximum transient deflection of about 13.5 inches, indicating a rebound of about 1 inch after removal of load About 6 feet along one edge of the roof slab had failed along a vertical failure surface directly over the supporting wall and the steel reinforcement were necked down and broke n. On the inside of the roof, much of the PAGE 85 85 concrete were spalled off and cracks went throug h the slab in some locations. A photograph of the damaged box structure after excavation is shown in Figure 416. The displacement time history and resistance f unction for both the flexur al and direct shear degreeoffreedom are plotted in Figure 417, 418, 419 and 420. The numerical results show that the roof slab did not fail in direct shear mode, same as the experimental observation. The numerical results for flexure show that the permanent displacement at the center of the roof slab is about 11.4 inch, which compares reasonably well with the experimental result. The numerical analysis also indicated that the slab underwent extensive flexural deformation into the tensile membrane region, very close to th e calculated failure point at about 14.6 inches of deformation. The numerical prediction is consiste nt with the experiment observation. Figure 416. Post test view of FH4. PAGE 86 86 Flexural Displacement Time History FH4 0 2 4 6 8 10 12 14 00.010.020.030.040.050.06 Time (s)Displacement (in) permanent displacement=11.4" Figure 417. FH4 flexural displacement time history. Flexural Resistance Function FH4 500000 0 500000 1000000 1500000 2000000 2500000 02468101214 Displacement (in)Resistance (lbs) Figure 418. FH4 flexural resistance function. PAGE 87 87 Direct Shear Displacement Time History FH40 0.05 0.1 0.15 0.2 0.25 00.0020.0040.0060.0080.010.012Time (s)Displacement (in) Figure 419. FH4 direct shear displacement time history. Direct Shear Resistance Function FH40 1000000 2000000 3000000 4000000 5000000 6000000 7000000 00.050.10.150.20.250.30.35Displacement (in)Resistance (lbs) Figure 420. FH4 direct shear resistance function. PAGE 88 88 4.2.5 Test FH5 Test FH5 was conducted in a sand backfill at a shallow depth of burial (DOB) equal to 20% of the short clean span. The reinforced c oncrete box had wall, floor and roof thickness of 13.5 inches, giving the roof slab had a span to ef fective depth ratio of 4. The structure had 1.5 percent principal reinforcing steel in each face, with a concrete strength of 6000psi. The test charge density was 3.6 lb/ft3 and produced a peak pressure of 18,000psi. Post test observation indicated that the sa nd backfill area above the buried structure was slightly depressed, with the sand backfill bei ng sheared and making an outline of the structure perimeter. After excavation, it was observed that the structure su ffered moderate damage with a permanent deflection of about 3.1 inches at the r oof center. The roof slab was found to have failed in shear, with the cracked and deformed area near the supporting side walls. The center area is relatively flat. On the inside of the r oof, the concrete had severely spalled off and the reinforcing bars were bent near the edges indicating a shear deflection. A photograph of the damaged box structure after excav ation is shown in Figure 421. Figure 421. Post test view of FH5. PAGE 89 89 The displacement time history and resistan ce function for the direct shear degreeoffreedom are plotted in Figure 422 and 423. The numerical results show th at the roof slab did not fail in direct shear mode, same as the experiment al observation. For the flexural response, the pr edicted displacement at the cente r of the roof slab center is about 0.71 inch (see Figure 424), which is much smaller than deflection measured in the experiment. The roof is a deep slab with a span to effective depth ratio of 4, therefore the slab is very stiff in the flexural response mode. Howeve r, based on the post test observation, the failure mode is in shear rather than flexure. Therefore it is not surprising that the numerical prediction and experimental data did not match up. To overcome this shortcoming, a simplified appr oach to include the shear effect into the flexural resistance function was presented in S ection 3.5. The new displacement time history and resistance function ar e plotted in Figure 425 and 426. The revised maximum permanent deflection at the center of the roof slab is calc ulated to be about 3.4 inches, and the resistance function shows that the slab had failed in shear before the slab cold reached its maximum flexural capacity in the compression membrane mode. After failing in shear, the slab continues to deflect under load until the applied loading completed and the slab response reached equilibrium. With the proposed modification to the flexure resistance func tion to consider shear strength effect, the numerical prediction compares very well with experiment measurement and is consistent with the experiment observation. PAGE 90 90 Direct Shear Displacement Time History FH50 0.01 0.02 0.03 0.04 0.05 0.06 0 0.0020.0040.0060.0080.01Time (s)Displacement (in) Figure 422. FH5 direct shear displacement time history. Direct Shear Resistance Function FH52000000 0 2000000 4000000 6000000 8000000 10000000 12000000 14000000 16000000 18000000 00.050.10.150.20.25Displacement (in)Resistance (lbs) Figure 423. FH5 direct shear resistance function. PAGE 91 91 Flexural Displacement Time History FH5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 00.0050.010.0150.020.0250.03 Time (s)Displacement (in) permanent displacement=0.71" Figure 424. FH5 flexural displacement time history. Flexural Displacement Time History FH5 0 0.5 1 1.5 2 2.5 3 3.5 4 00.0050.010.0150.020.0250.03 Time (s)Displacement (in) permanent displacement=3.4" Figure 425. FH5 displacement time history. PAGE 92 92 Flexural Resistance Function FH5 2000000 0 2000000 4000000 6000000 8000000 10000000 12000000 14000000 16000000 18000000 0246810121416 Displacement (in)Resistance (lbs) flexural resistance function Test FH5 resistance curve Failed in Shear Figure 426. FH5 resistance function. 4.2.6 Test FH6 Test FH6 was conducted in a clay backfill at a depth of burial (DOB) equal to 50% of the short clean span. The reinforced concrete box had wall, floor and roof thickness of 5.6 inches, giving the roof slab had a span to effective de pth ratio of 10. The structure had one percent principal reinforcing steel in each face, with a concrete strength of 6800psi. The test charge density was 1.8 lb/ft3 and produced a peak pressure of 8,320psi. A photograph of the box slab after partial exca vated is shown in Figure 427. Severe structural damage occurred and the roof collapsed completely. The lateral earth pressure pushed the side walls inward after roof collapse. The numerical results show that the roof slab failed in both direct shear mode and flexural mode. When the applied loading exceeded the direct shear resistance of the entire slab, the slab will shear off the wall support and the failure occurs at about 1 millisecond af ter the arrival of the loading. The numerical prediction is therefore consistent with the experiment observation. The PAGE 93 93 displacement time history and resistance function fo r both the flexural and direct shear degreeoffreedom are plotted in Figure 428, 429, 430 and 431. Figure 427. Post test view of FH6. PAGE 94 94 Flexural Displacement Time History FH6 0 2 4 6 8 10 12 14 16 00.0020.0040.0060.0080.010.0120.014 Time (s)Displacement (in) Failed Figure 428. FH6 flexural displacement time history. Flexural Resistance Function FH6 0 500000 1000000 1500000 2000000 2500000 3000000 0246810121416 Displacement (in)Resistance (lbs) Failed Figure 429. FH6 flexural resistance function. PAGE 95 95 Direct Shear Displacement Time History FH60 0.05 0.1 0.15 0.2 0.25 0 0.0020.0040.0060.0080.01Time (s)Displacement (in) Failed Figure 430. FH6 direct shear displacement time history. Direct Shear Resistance Function FH60 1000000 2000000 3000000 4000000 5000000 6000000 7000000 00.050.10.150.20.250.30.35Displacement (in)Resistance (lbs) Failed Figure 431. FH6 direct shear resistance function. PAGE 96 96 4.2.7 Summary A summary of all the six tests present in the a bove sections is presen ted in Table 41. Table 41. Summ ary of results Test No. Experiment Structural Behavior / Failure Mode Average Peak Pressure (psi) Measured Permanent Deflection (in) 1 Numerical Structural Behavior / Failure Mode Computed Permanent Deflection (in) 2 1 2 Failure Direct Shear Slip (in) Time of Failure (msec) FH1 Flexure 2400 0.44 Flexure 0.50 1.14 FH2 Direct Shear 5200 Collapsed Direct Shear Collapsed 1.00 0.20 1.1 FH3 Flexure 2650 6 Flexure 5.9 0.98 FH4 Flexure 3000 12.5 Flexure 11.4 0.91 FH5 Shear 18000 3.1 Flexure / Shear 3.4 1.09 FH6 Direct Shear 8320 Collapsed Direct Shear Collapsed 1.00 0.23 1.2 4.3 Assessment by PI Diagrams For all the test cases presented in Section 4.2, PressureImpulse (PI) diagrams were generated. The experiment measured pressure and corresponding impulse value was plotted on the same PI graph. The assessment of the st ructural response using the PI diagram were compared with the post test observations. The PI diagrams are shown in Figure 432 to 437. For tests FH1, FH3, FH4 and FH5, the experime ntal pressureimpulse point lies on the left side of both the flexure and di rect shear mode threshold curve, and this agrees with the observations of the experiment whereby the st ructure did not suffer any complete failure. For test FH2 and FH6, experiment observati ons indicated that the roof slab suffered completed failure in the test. The same result is obtained from the assess ment using PI diagram whereby the experiment pressureimpulse data poin t lies on the right side of the threshold curves. PAGE 97 97 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0102030405060 Impulse (psimsec)Pressure (psi) numerical direct shear numerical flexural experimental FH1PI Diagram for Test FH1 Figure 432. FH1 PressureImpulse diagram. 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 01 02 03 04 05 06 0 Impulse (psimsec)Pressure (psi) numerical direct shear numerical flexural experimental FH2PI Diagram for Test FH2 Figure 433. FH2 PressureImpulse diagram. PAGE 98 98 0 1000 2000 3000 4000 5000 6000 01020304050607080 Impulse (psimsec)Pressure (psi) numerical direct shear numerical flexural experimental FH3PI Diagram for Test FH3 Figure 434. FH3 PressureImpulse diagram. 0 1000 2000 3000 4000 5000 6000 7000 8000 01020304050607080 Impulse (psimsec)Pressure (psi) numerical direct shear numerical flexural experimental FH4PI Diagram for Test FH4 Figure 435. FH4 PressureImpulse diagram. PAGE 99 99 0 5000 10000 15000 20000 25000 30000 35000 40000 0 50 100 150 200 250 300 Impulse (psimsec)Pressure (psi) numerical direct shear numerical flexural only experimental FH5 numerical flexureshearPI Diagram for Test FH5 Figure 436. FH5 PressureImpulse diagram. 0 2000 4000 6000 8000 10000 12000 14000 0510152025303540 Impulse (psimsec)Pressure (psi) numerical direct shear numerical flexural experimental FH6PI Diagram for Test FH6 Figure 437. FH6 PressureImpulse diagram. PAGE 100 100 4.4 Summary Based on the numerical approach presented in Chapter 3, the dynamic response for buried box structures are generated and th e numerical results compared ve ry well with the experimental data. The approach also correctly predicted the correct mode of failure by examining the response time history and resistance functions. PressureImpulse di agrams are also generated for all the test cases considered. The assessment of structure behavior using PI diagram also predicted the correct result when compared with experiment data. PAGE 101 101 CHAPTER 5 CONCLUSIONS AND RECOMMENDATIONS 5.1 Summary A numerical method for the dynamic analysis of shallowburied reinforced concrete boxtype structures subjected to air blast loadings was presented in this study. The proposed method adopted the SingleDegreeofF reedom (SDOF) approach, wher e two loosely coupled SDOF systems were considered to model the flexural and direct shear mode of structural response. An introduction to blast load s and the effects on buried st ructures was presented in Chapter 2. Dynamic structural behavior and analysis for real continuous system and the approximation into an equivalent SDOF system was reviewed. A review on the structural response mode for reinforced concrete slabs a nd the resistance model under static and dynamic loading as well as the background and applications of pressureimpulse diagrams was presented. The proposed methodology to generate the resistance functions for reinforced concrete slabs of the buried box structure in both the flex ural mode and direct shear mode was presented in Chapter 3. The issue of soil arching and the required modifications to the load and mass factors for the equivalent SDOF system was also discussed and incorporated into the proposed methodology. For the consideration of deep slab behavior, the flexural resistance function was incorporated with a modification to capture the possibility of shear failure in the slab. The proposed procedure was implemented in a computer programming language and the results were validated using experimental data from a number of explosive tests on buried reinforced concrete boxes. The numerical results compared very well with experimental data. 5.2 Conclusions Based on the results from this present study, the following conclusions can be drawn. The proposed methodology can be employed for the approximate analyses of reinforced concrete slabs, and structural systems that are composed of such elements. PAGE 102 102 Singledegreeoffreedom analyses which are based on rational models for structural behavior mechanism have been validated, with good accuracy and consistency. The use of pressureimpulse diagram enables a quick and accurate assessment of the likely performance of the structure, by comparison of the location of the pressure and impulse point with respect to the fle xural and direct shear threshold curves plotted on the PI diagram. The proposed variation of the load and ma ss transformation factors for SDOF system enable a closer match of the factors with respect to the ac tual response regime of the structure under different loading combination. Dynamic soil arching effect reduced the load acting on the buried roof slab and changed the load distribution. The load and mass transformation factors must be modified with an appropriate reduction factor in order to reflect the soil arching effect and to obtain an accurate numerical result. Shear failure on the slab was not captured with the original proposed flexural and direct shear mode SDOF systems. A simplified approa ch to consider shear failure on the flexural resistance function was able to better model behavior for slab which failed in shear. 5.3 Recommendations for Future Study Based on the results and observations, the fo llowing recommendations for future research are proposed. The current methodology assumed that the lo ading on the slab is assumed to be a uniformly distributed airblast load on the soil surface. Further consideration of nonuniform loads that may be caused by locali zed HE (high explosives) explosions is recommended. Direct shear mode of response was based on the Hawkins model and a single enhancement factor is applied to account for the effects of compression and rate effects. A detailed study of the significance of th e possible variation of th e enhancement factor is recommended. Shear failure mode is only considered in this study in a simplified way. A separate response model for tension shear may be consid ered and its significance can be studied in greater detail. To study the possible interaction between the roof, wall and floor elements and their effects on the structural response of the box structure under blast loads. PAGE 103 103 APPENDIX EXPERIMENT TEST ON SHALLOW BU RIED FLAT ROOF STRUCT URES The series of experiment tests carried out by the United States Army Engineer Waterways Experiment Station, Structures Laboratory (Kiger and Getchell 1980) are used in Chapter 4 as validation data. The details of the experime nt carried out are given in this appendix. The tests were carried out to obtain struct ural response data in a simulated nuclear overpressure environment at the severe damage leve l for a buried reinforced concrete structure. Six quarterscale models of rectan gular, single bay reinforced conc rete box structures with inside dimensions of 4 feet high, 4 feet wide and 16 feet long, were tested with simulated nuclear blast. The box structures were designed to model one bay from a recta ngular multibay structure with span to effective depth ratios of between 4 and 10. The box structure was tested in a shallowburied configur ation using a HEST (High Explosive Simulation Technique) test which can simulate the peak pressure and duration characteristic of the overpressure generated in a nuclear detonation. The charge cavity was composed of conventional high explosives and pl astic foam for which the tests were designated Foam Hest (FH). The test involves distributing a high explosive over a relatively large surface area and covering the explosive with a soil overbu rden to momentarily confine the blast. The test configuration for FH3 is shown in Figure A1. The material properties such as the concrete and steel strength, st eel ratio and charge density were varied for all the six tests. The de pth of burial for the tests was varied, with a ratio of depth of burial to th e short clear span between 0.2 to 0.5. The parameters of the six cases presented in Chapter 4 are tabulated in Table A1. Construction details for FH1 and FH5 are shown in Figure A2 and Figure A3 respectively. Parameters used in the numerical procedure ar e summarized in Table A2, A3, A4, A5, A6 and A7. PAGE 104 104 Figure A1. Experiment test configuration for FH3. A) Elevation. B) Plan view. A B PAGE 105 105 Figure A2. Construction dime nsions and details of FH1. Figure A3. Construction dime nsions and details of FH5. PAGE 106 106 Table A1. Parameters for FoamHest tests Parameters FH1 FH2 FH3 FH4 FH5 FH6 Height (ft) 4 4 4 4 4 4 Ls (ft) 4 4 4 4 4 4 LL (ft) 16 16 16 16 16 16 Thickness (in) 5.6 5.6 5.6 5.6 13.5 5.6 Spaneffective depth ratio 10 10 10 10 4 10 DOB (ft) 2 2 2 0.8 0.8 2 DOBLs ratio 0.5 0.5 0.5 0.2 0.2 0.2 Concrete strength (psi) 7000 7600 7900 6700 6000 6800 Fy (psi) 75,000 57,000 57,000 65,000 69,000 65,000 Percentage steel (%) 1 1 1 1 1.5 1 Soil Type Sand Sand Clay Sand Sand Clay Charge density (pcf) 0.9 2.7 0.9 0.9 3.6 1.8 PAGE 107 107 Table A2. Test FH1 input parameters Parameters Value Height 4 ft Lx 16 ft L y 4 ft Slab Thickness 5.6 in Concrete cylinder strength 7000 psi Concrete density 0.0868 lb/in3 Xdirection Reinforcement area 0.11 in2 / 4 in spacing Reinforcement bar diameter 0.375 Reinforcement yield strength 75,000 psi Reinforcement ultimate strain 0.21 Depth of top reinforcement 0.8 in Depth of top reinforcement 4.8 in Ydirection Reinforcement area 0.20 in2 / 4 in spacing Reinforcement bar diameter 0.50 Reinforcement yield strength 75,000 psi Reinforcement ultimate strain 0.21 Depth of top reinforcement 0.8 in Depth of top reinforcement 4.8 in Depth of burial (ft) 2 Soil Type Sand Soil density (pci) 0.061 Friction angle 35.5 Coefficient of static lateral earth pressure Ko0.5 Soil wave velocity (in/s) 18,000 Damping (Flexure / Dire ct shear) 20% / 5% Pressure Time History FH10 500 1000 1500 2000 2500 00.0020.0040.0060.0080.010.0120.014Time (s)Pressure (psi) PAGE 108 108 Table A3. Test FH2 input parameters Parameters Value Height 4 ft Lx 16 ft L y 4 ft Slab Thickness 5.6 in Concrete cylinder strength 7600 psi Concrete density 0.0868 lb/in3 Xdirection Reinforcement area 0.11 in2 / 4 in spacing Reinforcement bar diameter 0.375 Reinforcement yield strength 57,000 psi Reinforcement ultimate strain 0.21 Depth of top reinforcement 0.8 in Depth of top reinforcement 4.8 in Ydirection Reinforcement area 0.20 in2 / 4 in spacing Reinforcement bar diameter 0.50 Reinforcement yield strength 57,000 psi Reinforcement ultimate strain 0.21 Depth of top reinforcement 0.8 in Depth of top reinforcement 4.8 in Depth of burial (ft) 2 Soil Type Sand Soil density (pci) 0.0613 Friction angle 35.5 Coefficient of static lateral earth pressure Ko0.5 Soil wave velocity (in/s) 18,000 Damping (Flexure / Dire ct shear) 20% / 5% Pressure Time History FH20 1000 2000 3000 4000 5000 6000 0 0.005 0.01 0.015 0.02 Time (s)Pressure (psi) PAGE 109 109 Table A4. Test FH3 input parameters Parameters Value Height 4 ft Lx 16 ft L y 4 ft Slab Thickness 5.6 in Concrete cylinder strength 7900 psi Concrete density 0.0868 lb/in3 Xdirection Reinforcement area 0.11 in2 / 4 in spacing Reinforcement bar diameter 0.375 Reinforcement yield strength 57,000 psi Reinforcement ultimate strain 0.21 Depth of top reinforcement 0.8 in Depth of top reinforcement 4.8 in Ydirection Reinforcement area 0.20 in2 / 4 in spacing Reinforcement bar diameter 0.50 Reinforcement yield strength 57,000 psi Reinforcement ultimate strain 0.21 Depth of top reinforcement 0.8 in Depth of top reinforcement 4.8 in Depth of burial (ft) 2 Soil Type Clay Soil density (pci) 0.0714 Friction angle 0 Coefficient of static lateral earth pressure Ko1.0 Soil wave velocity (in/s) 24,000 Damping (Flexure / Dire ct shear) 20% / 5% Pressure Time History FH30 500 1000 1500 2000 2500 3000 00.0050.010.0150.020.0250.03Time (s)Pressure (psi) PAGE 110 110 Table A5. Test FH4 input parameters Parameters Value Height 4 ft Lx 16 ft L y 4 ft Slab Thickness 5.6 in Concrete cylinder strength 6700 psi Concrete density 0.0868 lb/in3 Xdirection Reinforcement area 0.11 in2 / 4 in spacing Reinforcement bar diameter 0.375 Reinforcement yield strength 65,000 psi Reinforcement ultimate strain 0.21 Depth of top reinforcement 0.8 in Depth of top reinforcement 4.8 in Ydirection Reinforcement area 0.20 in2 / 4 in spacing Reinforcement bar diameter 0.50 Reinforcement yield strength 65,000 psi Reinforcement ultimate strain 0.21 Depth of top reinforcement 0.8 in Depth of top reinforcement 4.8 in Depth of burial (ft) 0.8 Soil Type Sand Soil density (pci) 0.0612 Friction angle 35.5 Coefficient of static lateral earth pressure Ko0.5 Soil wave velocity (in/s) 18,000 Damping (Flexure / Dire ct shear) 20% / 5% Pressure Time History FH40 500 1000 1500 2000 2500 3000 00.0050.010.0150.020.025 Time (s)Pressure (psi) PAGE 111 111 Table A6. Test FH5 input parameters Parameters Value Height 4 ft Lx 16 ft L y 4 ft Slab Thickness 13.5 in Concrete cylinder strength 6000 psi Concrete density 0.0868 lb/in3 Xdirection Reinforcement area 0.11 in2 / 5 in spacing Reinforcement bar diameter 0.375 Reinforcement yield strength 69,000 psi Reinforcement ultimate strain 0.21 Depth of top reinforcement 1.5 in Depth of top reinforcement 12.0 in Ydirection Reinforcement area 1.0 in2 / 5.5 in spacing Reinforcement bar diameter 0.50 Reinforcement yield strength 69,000 psi Reinforcement ultimate strain 0.21 Depth of top reinforcement 1.5 in Depth of top reinforcement 12.0 in Depth of burial (ft) 0.8 Soil Type Sand Soil density (pci) 0.0623 Friction angle 35.5 Coefficient of static lateral earth pressure Ko0.5 Soil wave velocity (in/s) 18,000 Damping (Flexure / Dire ct shear) 20% / 5% Pressure Time History FH50 2000 4000 6000 8000 10000 12000 14000 16000 18000 00.0020.0040.0060.0080.010.0120.014Time (s)Pressure (psi) PAGE 112 112 Table A7. Test FH6 input parameters Parameters Value Height 4 ft Lx 16 ft L y 4 ft Slab Thickness 5.6 in Concrete cylinder strength 6800 psi Concrete density 0.0868 lb/in3 Xdirection Reinforcement area 0.11 in2 / 4 in spacing Reinforcement bar diameter 0.375 Reinforcement yield strength 65,000 psi Reinforcement ultimate strain 0.21 Depth of top reinforcement 0.8 in Depth of top reinforcement 4.8 in Ydirection Reinforcement area 0.20 in2 / 4 in spacing Reinforcement bar diameter 0.50 Reinforcement yield strength 65,000 psi Reinforcement ultimate strain 0.21 Depth of top reinforcement 0.8 in Depth of top reinforcement 4.8 in Depth of burial (ft) 2 Soil Type Clay Soil density (pci) 0.0701 Friction angle 0 Coefficient of static lateral earth pressure Ko1.0 Soil wave velocity (in/s) 24,000 Damping (Flexure / Dire ct shear) 20% / 5% Pressure Time History FH60 1000 2000 3000 4000 5000 6000 7000 8000 9000 0 0.005 0.01 0.015 0.02 Time (s)Pressure (psi) PAGE 113 113 LIST OF REFERENCES ACI Comm ittee 318. (2005). Building code requirements for st ructural concrete (ACI 31805) and commentary (ACI 318R05), Farmington Hills, Mich, American Concrete Institute. ASCE. (1985). Design of Structures to Re sist Nuclear Weapon Effects ASCE Manuals and Reports on Engineering Practice No. 42, ASCE. Bathe, K.J. (1996). Finite element procedures, Englewood Cliffs, N.J. Biggs, J. M. (1964). Introduction to structural dynamics, McGrawHill, New York.. Blasko, J.R., Krauthammer, T., and Astarliogl u, S. (2007). Pressureimpulse diagrams for structural elements subjected to dynamic loads. Technical report PTCTR0022007, University Park, PA: Protective Technology Center, The Pennsylvania State University. Bowles, J.E. (1996). Foundation Analysis and Design, 5th Edition, McGrawHill. Clough, R. W., and Penzien, J. (1993). Dynamics of structures, McGrawHill, New York. Crawford, J. E., Holland, T.J., Mendoza, P.J. and Murtha, R. (1983). A failure methodology based on shear deformation. Fourth ASCE Engineering M echanics Division Specialty Conference Purdue University, Lafayette, IN. Crawford, J. E., Krauthammer, T., Karagozia n, J. and Hinman, E. (1999). Structural components Analysis and design examples. Structural design for physical security: state of the practice Chapter 4, ASCE, SEI, Reston, Va. Department of Army (1986). Fundamentals of protective design for conventional weapons, Technical Manual No. 58551, Headquarters Depa rtment of the Army, Washington, D.C.. Frye, M. (2002). Relationships between slender and deep reinforced concrete slabs subjected to shortduration dynamic loading, MS Thesis, The Pennsylvania State University, PA. Hawkins, N.M. (1974). The streng th of stud shear connections. Civil Engineering Transactions IE, Australia, 3945. Kiger, S.A. (1988). Ultimate cap acity of earthcovered slab. J. Struct. Eng., ASCE, 114(10), 23432356. Kiger, S.A., and Getchell, J.V. (19801982). Vul nerability of shallowburied flat roof structures. Technical Report SL807 five parts, U.S. Army Engineer Waterways Experiment Station, Vicksburg, Miss. Krauthammer, T. (1984). Shallowburied RC box type structures. J. Struct. Eng,, ASCE, 110(3), 637651. Krauthammer, T., Bazeos, N., and Holmquist, T. J. (1986). Modified SDOF analysis of RC boxtype structures. J. Struct. Eng. ASCE 112 (4), 726. PAGE 114 114 Krauthammer, T., Shaana, H.M., and AssaadiLamouki, A. (1994). Response of reinforced concrete structural elements to severe impulsive loads. Computer & Structures 53 (1), 119. Krauthammer, T. (2008). Modern protective structures, Civil and environmental engineering, 22. New York, CRC. Mays, G., and Smith, P.D. (1995). Blast effects on buildings: design of buildings to optimize resistance to blast loading T. Telford, London. MacGregor, J. G., and Wight, J. K. (2005). Reinforced concrete: mechanics and design, Prentice Hall, Upper Saddle River, N.J. Meamarian, N., Krauthammer, T., and O'Fallon, J. (1994). Analysis and design of laterally restrained structural concrete oneway members. ACI Structural Journal 91(6), 719725. Nawy, E.G. (2000). Reinforced concrete: a fundamental approach PrenticeHall, N.J. Newmark, N. M., and Rosenblueth, E. (1971). Fundamentals of earthquake engineering PrenticeHall, Englewood Cliffs, N.J. Ng, P.H. (2004). Pressureimpulse diagrams for reinforced concrete slabs MS Thesis, The Pennsylvania State University, PA. Park, R., and Gamble, W.L. (2000). Reinforced concrete slabs Wiley, New York. Park, R., and Paulay, T. (1975). Reinforced concrete structures Wiley, New York. Slawson, T. R. (1984). Dynamic shear failure of shallowburied flatroofed reinforced concrete structures subjected to blast loading. Technical Report SL847 U.S. Army Engineer Waterways Experiment Sta tion, Vicksburg, Miss. Smith, P.D., and Hetherington, J.G. (1994). Blast and ballistic l oading of structures ButterworthHeinemann, Oxford, Boston. Soh, T.B., and Krauthammer, T. (2004). Loadimpulse diagrams of reinforced concrete beams subjected to concentrated transient loading. Technical report PTCTR0062004 University Park, PA: Protective Technology Center, The Pennsylvania State University. Tedesco, J. W., McDougal, W. G., and Ross, C. A. (1999). Structural dynamics: theory and applications, Addison Wesley Longman, Menlo Park, Calif. Terzaghi, K., and Peck, R. B. (1949). Soil mechanics in engineering practice, Wiley, New York. Zienkiewicz, O. C., a nd Taylor, R. L. (2005). The finite element method for solid and structural mechanics, Elsevier Butt erworthHeinemann, Amsterdam. PAGE 115 115 BIOGRAPHICAL SKETCH Kay Hyang Chee was born in Singapore in 1973. He attended secondary school and junior colleg e in Singapore. After obtaining his GCE A level in 1991, he serv ed his national service in the army from 1991 to 1993. He began his undergraduate studies in Civil Engineering at the National University of Singapore in July 1993. He graduated in July 1997 with his Bachelor of Engineering degree in civil engineering. He continued with his gr aduate studies at the National University of Singapore and obtained his Master of Engineering degree in 1999. In August 1999, he joined the Defence Scie nce and Technology Agency, Singapore, as a project engineer. 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