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PAGE 1 1 LARGE DEFLECTION BEHAVIOR EFFECT IN REINFORCED CONCRETE COLUMN UNDER SEVERE DYNAMIC SHORT DURATION LOAD By DAVE MORENCY A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORID A IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2010 PAGE 2 2 2010 Dave Morency PAGE 3 3 To my girlfriend, family and friends PAGE 4 4 ACKNOWLEDGMENTS I would like to express my sincere gr atitude to Dr. Theodor Krauthammer, my research advisor, for his precious advice an d direction during the completion of this research. I would also like to thank Dr. Serdar Astarlioglu for his constructive advice and support which greatly facilitated the completi on of this research. I am thankful for Canadian Armed Forces, mo re particularly 1 ESU (1st Engineer Support Unit) for giving me the opportunity to complete graduate studi es in structural engineering. My final gratitude goes to my family, friends and loved one who supported me during the completion of this study. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDG MENTS .................................................................................................. 4LIST OF TABLES ............................................................................................................ 8LIST OF FIGURES .......................................................................................................... 9LIST OF ABBR EVIATIONS ........................................................................................... 13ABSTRACT ................................................................................................................... 18 CHA PTER 1 INTRODUC TION .................................................................................................... 20Problem St atement ................................................................................................. 20Objective a nd Scope ............................................................................................... 22Research Si gnificance ............................................................................................ 232 LITERATURE REVIEW .......................................................................................... 25Introduc tion ............................................................................................................. 25Dynamic Structural A nalysis Suit e (DSAS) ............................................................. 25Blast Load Calc ulation ............................................................................................ 26Blast Load Calculations for Simple Structures .................................................. 26Blast Load Calculations fo r Complex St ructures .............................................. 27Structural Analysis .................................................................................................. 28Dynamic A nalysis ............................................................................................. 29Shape func tion ........................................................................................... 30Equivalent mass calculat ion ....................................................................... 30Equivalent load calculation ........................................................................ 32Mass factor and load factor ca lculation: DS AS approach .......................... 34Numerical integration of equivalent SDOF ................................................. 34Dynamic r eactions ..................................................................................... 36Resistance F unction ......................................................................................... 37Flexural Behavior ............................................................................................. 39Diagonal Shear Behavior .................................................................................. 44Direct Sh ear ..................................................................................................... 47Axial Behavior and Anal ysis ............................................................................. 47Rate Effect ........................................................................................................ 49Large Defo rmation ............................................................................................ 50PDelta e ffect ............................................................................................. 51Development of resistance function curve in DSAS ................................... 53 PAGE 6 6 Euler Bu ckling .................................................................................................. 57Dynamic analysis: Reinforce concrete globa l buckling .............................. 58Dynamic analysis: Reinforce concre te compression longitudinal rebar buckling .................................................................................................. 60Membrane B ehavior ......................................................................................... 62Compressive membra ne calcul ation .......................................................... 64Tension membrane calculat ion .................................................................. 70LoadImpulse Diagram s ................................................................................... 75Summary ................................................................................................................ 763 METHODOLOGY ................................................................................................. 102Introduc tion ........................................................................................................... 102Structure Ov erview ............................................................................................... 102Dynamic A nalysis ................................................................................................. 103Tension Membr ane Analysis .......................................................................... 103PDelta E ffect ................................................................................................. 105Buckling .......................................................................................................... 106Column bu ckling ...................................................................................... 106Buckling of compressive l ongitudinal rein forcement ................................ 106Flexural Failure and Transit ion to Tensi on Memb rane ................................... 107Column buckling transition to tension memb rane behavior ...................... 107Flexural failure transition into tension membrane ..................................... 108Transition into tension membrane due to excessive deformation ............ 108DSAS Overall Dynamic Analysis Algorithm .................................................... 109Summary .............................................................................................................. 1094 ANALYSI S ............................................................................................................ 125Introduc tion ........................................................................................................... 125Material Model Validation ...................................................................................... 125Column Model Validat ion ...................................................................................... 128Modified DSAS Validation ..................................................................................... 129Tension Membr ane Validat ion .............................................................................. 130Slender Co lumn .................................................................................................... 134Summary .............................................................................................................. 1355 DISCUSSION AND RECOMMENDATIONS ......................................................... 160Introduc tion ........................................................................................................... 160Limitations ............................................................................................................. 160Parametric study ................................................................................................... 163Column Parametric Study ............................................................................... 164PI Diagr am .................................................................................................... 165Future Development/Recommendations ............................................................... 166Conclusi on ............................................................................................................ 168 PAGE 7 7 LIST OF RE FERENCES ............................................................................................. 180BIOGRAPHICAL SKETCH .......................................................................................... 184 PAGE 8 8 LIST OF TABLES Table page 31 Reinforced concrete column param eters. ......................................................... 12442 Concrete damaged plasticity model parameters. .............................................. 15543 Steel properties of beam C for ABAQUS, US unit with stress in ksi. ................ 15644 Reinforced concrete column under no axial l oad result s. ................................. 15645 Reinforced concrete colu mn under no ax ial load. ............................................. 15746 Twelve foot reinforced concrete column under 500 kip axial load. ................... 15747 Twelve foot reinforced concrete column under 800 kip axial load. ................... 15748 Twelve foot reinforced concrete column under 1500 kip axial load. ................. 15849 Comparison of ABAQUS results wit h DSAS V3.0 witho ut axial load. ............... 158410 Twelve foot reinforced concrete column under 800 kip axial load con't. ........... 158411 Twelve foot reinforced concrete co lumn under 1500 kip axial load con't. ......... 159412 Twentyfour foot reinforced concrete column results. ....................................... 15951 Twelve foot reinforced concrete column analyzed with various longitudinal reinforcement size rebar. .................................................................................. 179 PAGE 9 9 LIST OF FIGURES Figure page 11 Blast load combination on inte rnal structura l components. ................................. 2421 Blast environment from air burst. ......................................................................... 7822 Free field pressu retime va riation ....................................................................... 78 23 One degree of freedom system representation of a real structu ral element. ...... 7924 Deformati on prof ile. ............................................................................................ 7925 Equivalent load diagr am. .................................................................................... 8026 Determination of dynamic reac tion for a beam with arbitrary boundary condition s. .......................................................................................................... 8027 Resistance functions example for a re inforce concrete material with fixed boundaries conditions obtai n with DSAS V3.0. .................................................. 8128 Resistance disp lacement model. ........................................................................ 8229 Stress and strain distributions across reinforced conc rete section ..................... 83210 Unconfined concrete stress strain curve. ............................................................ 84211 Confined concrete stress strain curve. ............................................................... 85212 Steel Stressstra in curve model. ......................................................................... 86213 General model for shear influence on beams without web reinforcement. ......... 87214 Shear redu ction m odel. ...................................................................................... 88215 Direct shear resistance envelop and revers al l oads. .......................................... 89216 Influence of axial forc e on section response. ...................................................... 90217 Flexural resistanc e function envelope. ............................................................... 91218 PDelta effect on pinned column. ........................................................................ 92219 SnapThrough buckling. ..................................................................................... 93220 Beam element. ................................................................................................... 93221 Membrane behavior trans ition beam/column. ..................................................... 94 PAGE 10 10 222 General flexural resistance functions including compression membrane and tension membr ane behavio r. .............................................................................. 95223 Free body diagram of deformed slab strip. ......................................................... 95224 Compressive memb rane geometri c model. ........................................................ 96225 Compressive me mbrane sect ion. ....................................................................... 96226 Compressive membrane externe section forces. ................................................ 97227 Tension membrane geom etric defo rmation. ....................................................... 97228 Cable/truss deformation. .................................................................................... 97229 Pure tension memb rane modifi cation. ................................................................ 98230 Resistance function modificati on path for stabi lity failure. .................................. 99231 DSAS combined resistance function ex ample with shifted tension membrane. 100232 Typical res ponse func tions. .............................................................................. 10131 Reinforced concre te column model. ................................................................. 11032 Tension membrane algor ithm. .......................................................................... 11134 Pure tension membrane resistanc e function calculat ion example. ................... 11235 DSAS combined resist ance function example. ................................................. 11336 Buckling che ck algorit hm. ................................................................................. 11437 Eulerbuckling function obtained from DSAS DPLOT linked software. ............. 11538 Flexural resistance functions only. .................................................................... 11639 Resistance functions for differ ent axial load on same column. ......................... 117310 Combined resistance function for constant 800 kips axial load. ....................... 118311 Resistance function transition into tensile me mbrane behav ior. ....................... 119312 Flexural resistance function of RC column with no axial load. .......................... 120313 DSAS simplified column analysis flowchar t chart 1. ......................................... 121314 DSAS simplified column analysis flowchar t chart 2. ......................................... 122315 DSAS simplified column analysis flowchar t chart 3. ......................................... 123 PAGE 11 11 41 Beam C meshing. ............................................................................................. 13642 Beam C load ti me history. ................................................................................ 13743 Displacement time history beam C ABAQUS vs. experiment. .......................... 13844 Beam C experiment time history co mparison. .................................................. 13945 DSAS V3.0 vs. ABAQUS V6.8 an alysis with triangul ar blas t load .................... 14046 DSAS V3.0 vs ABAQUS V6.8 ana lysis with triangul ar blast load ..................... 14147 Reinforced concrete column model in AB AQUS V6.8. ..................................... 14248 Flexural time history analysis com parison modified DSAS vs. ABAQUS V6.8 14349 Interaction diagram for 12 ft long colu mn model. .............................................. 144410 Support displacement time history for different constant axial load cases. ....... 145411 Flexural resistance function compar ison of local buckling with a constant axial load of 500 ki ps. ....................................................................................... 146412 Principal maximum stress, case 1500 kip axial load, Pr=5262, t = 0.0014272 1....................................................................................................... 147413 Steel reinforcement axial stress at midspan, case 1500 ki p axial load, Pr = 5262 psi, t = 0.00142721 se c. .......................................................................... 148414 Principal minimum stress, case 1500 kip axial load, Pr=5262 psi, t = 0.00142721 se c. ............................................................................................... 149415 Flexural time histor y example case with 800 ki ps constant axial load, triangular blast load with Pr = 2922 psi and t = 1.28 msec. .............................. 150416 Flexural time histor y example case with 1500 ki ps constant axial load, triangular blast load with Pr = 2922 psi and t = 1.28 msec. .............................. 151417 Flexural time histor y example case with 1500 ki ps constant axial load, triangular blast load with Pr = 4757 psi and t = 1.39 msec. .............................. 152418 TwentyFour foot column maximum general principal stress results case: Pr = 2922 psi, t = 0.00128337 se c. ....................................................................... 153419 TwentyFour foot column maximum general principal stress results case: Pr = 3400 psi, t = 0.00128337 sec, direct s hear failure reported by DSAS. .......... 15451 Twelve foot reinforced concrete column combined resistance function comparis on. ...................................................................................................... 170 PAGE 12 12 52 Interaction diagram 12 ft RC co lumn with different rebar sizes......................... 17153 LoadImpulse diagram 12 ft RC column under 800 kips axial load. .................. 17254 LoadImpulse diagram 12 ft RC column under 1500 kips axial load. ................ 17355 Experimental setup. ......................................................................................... 17356 Failure mode of the sect ion. ............................................................................. 17457 Experimental resi stance func tion. ..................................................................... 17458 Reinforced concrete column under going tension membrane behavior picture 1. ...................................................................................................................... 17559 Reinforced concrete column under going tension membrane behavior picture 2. ...................................................................................................................... 176510 Reinforced concrete column under going tension membrane behavior picture 3. ...................................................................................................................... 177511 Reinforced concrete column under going tension membrane behavior picture 4. ...................................................................................................................... 178 PAGE 13 13 LIST OF ABBREVIATIONS Ac Area of concrete Aci Area of concrete at each layer Ag Area of gross section Asi Area of steel at each layer c Neutral axis depth C Damping coefficient Cc Compressive concrete force at midspan Cc Compressive concrete force at support cm Section neutral axis depth at midspan cs Section neutral ax is depth at support Cs Compressive steel force at midspan Cs Compressive steel force at support D Nominal diameter of longitu dinal compression reinforcement (inches) D Nominal diameter of hoops (inches) dsi Depth of steel layer Ec Elastic modulus of concrete (psi) Enhc Concrete enhancement factor in compression Enhct Concrete enhancement factor in tension Enhs Steel enhancement factor Es Steel modulus of elasticity F Force fci Stress of concrete layer PAGE 14 14 Fci Concrete layer forces Fe Equivalent Force Fn Function fs Steel stress fsi Stress of steel layer Fsi Steel layer forces Ft Total force fu Ultimate steel stress fy Steel yield stress fy Yield stress of the hoops/stirrups h Concrete section depth H Average core dimensions of the confined concrete compression zone measure to the outside of stirrups (inches) I Moment of inertia i Increment variable ILF Inertia load factor is Impulse is Negative Impulse k Effective length factor Ke Equivalent Stiffness KEb Total kinetic energy of beam KEe Total kinetic energy of equivalent system KL Load factor Km Mass factor PAGE 15 15 l Length lu Unsupported length m Mass M1 Column end moment at support 1 M2 Column end moment at support 2 mci Concrete layer moment Me Equivalent mass Mfl Ultimate moment capac ity due only to flexure Mnm Moment at midspan Mns Moment at support msi Steel layer moment Mt Total mass Mu Ultimate moment c apacity due only to shear N Membrane axial force Next External axial load P Pressure/axial load Pr Resistance function load Pmax Maximum pressure Pso Incident pressure Po Atmospheric Temperature ta Time of arrival to Time of positive Phase to Time of negative phase PAGE 16 16 q Transverse load Q1i Static reaction at support 1 at time step i Q2 Static reaction at support 2 at time step i Qi Load at time step i R Distance from a reference explosion r Radius of gyration Rm Resistance function s Spacing of hoops (inches) S Surrounding support stiffness SRF Shear reduction factor t Section lateral displacement at support TF Trust factor Ts Tensile steel force at midspan Ts Tensile steel force at support u Displacement u Velocity u Acceleration ue Equivalent displacement eu Equivalent velocity eu Equivalent acceleration W Equivalent TNT weight w Deflection WEb External Work in beam PAGE 17 17 WEe External Work of equivalent system x Position Z Scaled distance zi Depth of concrete layer NewmarkBeta constant Increment space Deflection Total Strain a Axial strain h/2 Reinforced concrete strain at h/2 p Strain due to creep and shrinkage s Steel strain sh Hardening steel strain su Ultimate steel strain Infinity Steel ratio r Ratio of the confining steel volume to the confined concrete core volume per unit length of the element in compression zone Longitudinal compression reinforcement ratio Web reinforcement ratio Curvature Stress Inertia proportionality factor PAGE 18 18 Abstract of Thesis Pres ented to the Graduate School of the University of Florida in Partial Fulf illment of the Requirements for t he Degree of Master of Science LARGE DEFLECTION BEHAVIOR EFFECT IN REINFORCED CONCRETE COLUMN UNDER SEVERE DYNAMIC SHORT DURATION LOAD By Dave Morency May 2010 Chair: Theodore Krauthammer Major: Civil Engineering Reinforced concrete columns may undergo la rge deformations when subjected to blast load. In addition to static loads that impose flexural, axial and shear effects, a combination of blastinduced transient transverse and axial loads due to blast loads may induce large deformations and lead to geom etric instabilities. Furthermore, if the flexural deformation becomes very large, the structural mem ber might shift to a tension membrane response. One needs to addres s all these phenomena when conducting a dynamic analysis. Advanced explicit nonlin ear dynamic finite element codes may perform such analysis but require large numer ical resources and very long execution times. Consequently, advanced simplified an d accurate computational tools are required to address these complicated behaviors with much more limited computational resources, in support of design activities and/or rapid incident and/or damage assessments. Therefore, a new numerical computational capabilit y was developed for the computer code DSAS to address those needs. The new approach can account for transverse and axial responses, sec ondary moments when undergoing large deformations, geometric instabilities, and the transition into a tension membrane. The PAGE 19 19 present thesis explains this new approach de veloped, explains how it was validated and presents numerical results for reinforc ed concrete columns undergoing explosioninduced large deformations. Conclusions and recommendations are also included. PAGE 20 20 CHAPTER 1 INTRODUCTION Problem Statement In recent years, many studies hav e been conducted by governmental and nongovernmental organizations across the worl d such as the Engineer Research and Development Center U.S. Army, Defens e Research Development Canada and the Center of Infrastructure Protection and Physic al Security University of Florida, in an attempt to better understand the effect of blast, fragmentation and impact loads on buildings in order to better des ign against specific threats. P ublications such as Unified Facilities Criteria 334002 (U .S Department of Defense, 2008) ASCE Structural Design for Physical Security (ASCE, 1999), and ASCE Guideline for Blast Resistance Buildings in Petrochemical Facilities (ASCE, 1997) ar e few example of public ations produced in light of the development accomplishments in th is field and are available to engineers to better design both military and civilian struct ures. In spite of t hese guidelines, much more work still remains to be done to better understand structural behavior of a variety of different structures, stru ctural components and structural materials when subjected to shortduration severe dynamic loads. Exam ples of such required works include reinforced concrete or steel connections, co mposite sections and reinforced concrete columns undergoing large deformation. Of all components in a structure, columns ma y be the most critical ones as they carry the most amount of the structural component. In the past, they were generally not designed to sustain large lateral dynamic lo ads such as blast loading and, for that reason, are very susceptible to terrorist attacks. As an example, in 1995, part of the Alfred P. Murrah Federal Building in Okl ahoma City collapsed after three of the four PAGE 21 21 north face columns were destroyed by a blas t, resulting in 168 casualties and several injured citizens (Bangash et al. 2006). Fo r this reason, the need to better understand column behavior quickly became clear in the fi eld of progressive co llapse. At the design stages, a finite element analysis may be the most widely used approach to verify a design for a specific threat as it offers great capabilities, but it often requires advanced knowledge of finite elements and copious am ounts of time. On the other end, design guidelines may be expedient to design against s pecific threats but also come with a cost as they should not be applied to all cases a nd may not yield the most cost efficient design as conservative measures may be built into them. When it comes to a realtime possible threat on a building and counter m easures that are i mmediately needed, both approaches are usually of no use to first re spondent engineers. In such scenarios, a unified approach yielding accurate results in a restrained timeframe is desired and has now been under development for some years. In the case of reinforced concrete elem ents, research and development has made great progress in recent years for the behavior of several cases such as reinforced concrete slabs, beams and columns, i.e. Ross (1983), Krauthammer et al. (1988), Shanaa (1991), Krauthammer, T., Schoedel, R., Shanaa, H. (2002), Krauthammer et al. (2004), Tran (2009), and many more. Algorithms have been developed to quickly analyze structural components using advan ced single degreeof freedom system (SDOF) approach coupled with nonlinear mate rial behavior models and different modes of failure such as flexural and direct shear Improvement in the an alysis of the capacity of reinforced concrete columns subjected to both lateral forces and axial forces were recently accomplished by Tran et al. (2009). PAGE 22 22 In his research, Tran et al. (2009) devel oped an algorithm to analyze reinforced concrete column structures as seen in Figure 11 subjected to both a variable lateral and variable axial load. By decoupling t he column and the beam elements of the structure in to two distinct single degree of freedom systems, it was possible to analyze both elements at each time step and take into account variable load interaction on each element, changing with time. Parametric stud ies conducted during the research with similar models using finite element softw are and DSAS lead to the conclusion that improvement is still required for the analysis of reinforced concrete columns when undergoing large deformation. As an example, tension memb rane effects in columns have yet to be implemented in such algorithm although studies show it would improve significantly the section capacity prediction. In addition, second order moments, also known as PDelta effects, need to be incorpor ated in such analysis as they increase the flexural stress of the columns. Objective and Scope The primary objective of this research was to develop an expedient and accurate numerical algorithm to analyze the response of reinforced concrete columns subjected to severe short dynamic loads such as blast and impact loads. The following are the steps of this research. Develop a flexural resistance function ca lculation algorithm for columns, including tension membrane behavior, PDelta e ffects and Euler buckling, and adapt the algorithm for variable axial load when present. Validate algorithm using finite element numerical software ABAQUS by comparing similar analysis between DSAS and ABAQUS. Conduct parametric studies using DSAS to fully understand the effect of large deformation and tension membrane action in reinforced concrete columns under severe short dynamic loads. PAGE 23 23 Although protective struct ures are designed against blas t and fragmentation, this study only focused on the behavior of reinfo rced concrete columns under severe, shortduration dynamic loads, or, more precisely, blast loads. The effect of fragmentation load was dismissed due to the need for more resear ch and development in this field of study to fully capture its effect on the stru cture and properly model its behavior using adequate numerical software. Research Significance The algorithm developed in this study was incorporated into the computer code DSAS developed at the Center for Infrastructure Protection and Physical Security at the University of Florida and will provide the means to generate more realistic reinforced concrete column resistance functions. As a re sult of this research, the computer code DSAS has now the capability to account fo r large deformations, geometric instability and tension membrane effects in addition to the other phenomena already taken into account by the program when conducting dy namic analysis of reinforced concrete section. The research is discusse d in detail in the next chapters. PAGE 24 24 Figure 11. Blast load combination on inter nal structural compone nts. [Reprinted with permission from Tran T.P. 2009. Effect of short durati on high impulse variable axial and transverse loads on reinforced concrete column. M.S. dissertation (Page 48, Figure 31). University of Florida, Gainesville, Florida.] PAGE 25 25 CHAPTER 2 LITERATURE REVIEW Introduction Chapter 2 provides a review of recent studies and developments in the field of protective structures based on a number of technical reports, books and publications related to the scope of this research. The first topic discussed is an overview of the Dynamic Structural Analysis Suite (DSAS) software which will be intensively used for this research. Blast load calculations fo r simple and complex structures are then discussed followed by structural analysis theoretical models rele vant to this research, to conclude with discussion on loadimpulse diagrams. Dynamic Structural Analysis Suite (DSAS) DSAS has been under continuous developm ent under the direction of Dr. Theodore Krauthammer since the early 1980s. The program is designed to perform a static and dynamic analysis of a wide range of structural elements such as reinforced concrete beams/column/slabs, masonry walls steel components, and others, for point load, uniform load, single degree of freedom load, blast load and impact. The program has the ability to generate moment curv ature for beams and columns as well as resistance functions and pressure/load impulse diagrams of all elements it analyzes. The program uses advanced single degree of freedom (SDOF) and nonlinear material behavior allowing for fast analysis, ma king it efficient and expedient. As this chapter reviews theoretical concepts, references will be made to the software to ensure good theoretical backg round comprehension of the program for subjects related to this research and how the concepts are incorporated into DSAS. PAGE 26 26 One may refer to the DSAS user manual (Ast arlioglu, 2008) for a complete description of the software. Blast Load Calculation Blast Load Calculations for Simple Structures A shock wave is caused by a sudden violent release of energy as a result of an explosion. There are many different sources of explosions such as chemical and nuclear, but the principle sources of interest for this research are the chemical explosive materials available to terrorism, such as TNT, capable of damaging or destroying civilian and military structures. Upon arrival of a blast wave on a structure, many factors may affect the load generated on the structure due to the shock wave propagation such as the shape of the explosive, obstacles en counter during the wave propagation phase, and the distance of the explosion to the st ructure. For ease of load calculation and research purposes, some assumptions are made for the calculation of the loads. First, the shape of the explosive is considered perf ectly spherical, the explosion occurs far enough to assume uniformly distributed load on the structure, and no obstacles are encountered during wave propagati on before time of arrival. Figure 21 shows a typical blast wave path to a structure. The calculation of the l oad on the st ructure may be performed using a scaling approach represented by Equatio n 21. The explosive char ge needs to be transformed into an equivalent TNT charge in the case wh en TNT is not the primary explosive used. UFC 334002, (Department of Defense, 2008) contains charts that have been developed to simplify the calculation. 3/1 W R Z (21) PAGE 27 27 where W represents the equivalent TNT Weight (lbs/Kg) R represents the radial di stance from centre of expl osion to target (ft/m) Z represents the scaled distance The free field pressure time variation presented in Figure 22 may be used to generate the load on a structure and one may use appropriate charts provided by UFC334002 to obtain the different parameters. The modifi ed Friedlander equation, Equation 22, may then be used to compute the pressure on t he structure as a function of time. / max1)(t pose T t PtP (22) where represents the decay coefficient. The im pulse may then be calculated using Eqation 23 by integrati ng the area under the curve. oatt sdttPi0)( (23) In literature, equations more complex than the Friedlander equation having better agreement with experimental results do exist, but the extra amount of computational time required to perform these calculations does not justify the small increase in accuracy and therefore are not c onsidered for this research. Blast Load Calculations for Complex Structures In reality, when a blast wave hits a stru cture, the load will ac t on the structural elements in various ways depending on the type of structure, the layout of its structural elements, its material properti es, and its architectural compos ition. The blast wave will PAGE 28 28 penetrate through doors, windows and various openi ngs. Vertical structural elements will be affected by a transverse pressure while horizontal components will be affected by a vertical and transverse pressure. Figure 11 shows a simple example of how blast load may act on the internal members. The same approach discus sed in the previous section with the appropriate chart may be used to combine all types of loads a structural element is subjected to as a function of time. Structural Analysis To properly analyze a reinforced concrete structure under dynamic load, proper constitutive models must be used. Over the past century, reinforced concrete has been studied very intensively and many book, paper s and technical reports covering a wide range of the subject matter are available i. e. Ross (1983), Krautha mmer et al. (1988), Shanaa (1991), Krauthammer, T., Schoedel, R., Shanaa, H. (2002), Krauthammer et al. (2004), Park and Gamble (2000), Park (1975) etc. The following se ction discusses the structural analysis approach use for this research and reviews the recent developments in the field of protective structures with r egards to reinforced concrete analysis relevant to this research. First, a review of dy namic analysis is presented which covers the notion of equivalent single degree of freedom system, follo wed by the notion of a resistance function used for nonlinear materi al behavior in a dynamic analysis. Flexural behavior is then discussed along with constituti ve material models of interest for reinforced concrete columns. Different failure mechanism for reinforced concrete is then discussed, including shear and axial behavior to conclude with a review of large deformation and phenomenon such as memb rane behavior which may be present during large deformation of rein forced concrete elements. PAGE 29 29 Dynamic Analysis Dynamic analysis may be conducted for one degree of freedom problems which require simple computational effort or for multiple degree of freedom problems requiring more computational effort. The general equat ion of motion remains the same in both cases and is represented by Equation 24. )()( tFuRucum (24) where m represents the mass which gener ates the inertia force, c represents the damping coefficient used to calculate the damping force and R represents the resistance force or stiffness of the struct ural element. Usually, the more degrees of freedom used, the more accurate the representation of the actual parameters. However, for a specific location of interest, it is possible to use an equivalent single degree of freedom system approach. In the case of a column having a distri buted mass, distributed varying forces, and a varying stiffness, a complete dynamic analysis may become cumbersome computationally since the column can be di vided in an infinitesimal number of degrees of freedom. For a simple problem such as evaluating the localized deflection of a structural element, an equiva lent system may be used to accurately calculate the structural response at a desired locati on using a single degr ee of freedom system (SDOF) having an equivalent stiffness Ke, mass Me and load Fe. The equation of motion may then be expressed by Equat ion 25 for a linear elastic system )( tFuKucuMe e e (25) and Equation 26 for nonlinear system )( tFRucuMee e (26) PAGE 30 30 The equivalent system parameters are calc ulated such that the equivalent SDOF system displacement at the degree of freedom lo cation is equal to the real structure at the same point in time and both systems assume the same displaced shape. As Biggs (1964) pointed out, stresses and forces in the equivalent system must be based on displaced shapes, as they are not directly equi valent to the same quantities in the real structure. Since we are dealing with reinfo rced concrete material having non linear material properties in the elastic and nonelastic domain, the general academic stiffness term K will be replaced by the notion of resistance function discussed later in this section. The following procedure to obtain the equiva lent SDOF parameters was presented by Biggs (1964) for an elastic or perfectly plastic domain and did not account for the transition phase in between. Furthermore, it assumed ideal boun dary conditions. Krauthammer et al. (1988) improved this c oncept by developing a procedure based on Biggs (1964) to account for the transition between elastic behaviors to inelastic behaviors valid for all boundary conditions and is discussed in this section. Shape function As shown in Figure 24, when an element of a st ructure undergoes deformation it assumes a deformation profile that can ma thematically be represented by a shape function (x). The shape function can then be used to derive a mathematical model using laws of physics to calculate the equival ent SDOF terms required for the equivalent system. It may also be used to calc ulate the reaction at the support. Equivalent mass calculation The equivalent mass may be calculated using an energy balance solution by balancing the kinetic ener gy in both systems. PAGE 31 31 The total kinetic energy of the beam is calculated using Equation 27 L bdxtxuxm KE0 2),()( 2 1 (27) Where the displacement velocity is calculated using Equation 28 )()(),(' tuxtxu (28) The kinetic energy in the equivalent system may then be obtained using Equation 29 2)( 2 1 tuMKEee e (29) Then Me is solved for by equating Equation 27 and Equation 29 2 0 2)( ),()( tu dxtxuxm Me L e (210) One may the use Equation 211 )()( tutue (211) and substituting Equation 211 into Equati on 210 to obtained a new equivalent mass relationship expressed by Equation 212 L edxxxmM0 2)()( (212) The mass factor Km representing the rati o of the equivalent mass to total mass, can then be calculated by Equation 213 t e mM M K (213) In a dynamic analysis, the equivalent mass and equivalent mass factor will be affected at each time step by the change in displacement prof ile. To account for this, the PAGE 32 32 linear interpolation presented by Equation 214 was used by Krauthammer et al. (1988) to calculate the equivalent mass factor at each time step. i i i mi im mi mKK KK )1( )1( (214) Where )1( i i Once hinge formation starts to occur, the equivalent mass will remain constant, since the inelastic deform ed shape function is assumed to be effected only minorly (Krauthammer et al. 1988). DSAS uses a simplified approach where it generates an equivalent mass function curve based on displacement Me( ) which reduces computational time. Equivalent load calculation Following the same approach as that of the equivalent mass, the equivalent load Fe is then derived by balancing the exter nal work done on the real system to the equivalent system. The work done on the real syst em is calculated using Equation 215 L n i ii btutFdxtxutxwWE0 1)()(),(),( (215) Where the displacement u(x) c an be substituted by Equation 216 )()(),( tuxtxu (216) To obtain Equation 217 L i iiiii btuxtxFdxtuxtxwWE0)()((),( ))()((),( (217) The work done on the equivalent system can be expressed by Equation 218 )( tuFWEeee (218) PAGE 33 33 Where Fe can be obtained by balancing the work done on both systems to be equal as shown in Equation 219 L i iiiii eetuxtxFdxtuxtxwtuF0)()((),( ))()((),()( (219) To finally obtain Equation 220 )( )()((),( ))()((),(0tu tuxtxFdxtuxtxw Fe L i iiiii e (220) And since )()( tutue (221) Equation 220 may be simp lified to Equation 221 L i iiii extxFdxxtxwF0)((),()(),( (222) The load factor KL representing the ratio of the equi valent load to total load may be calculated with Equation 223 t e LF F K (223) Krauthammer et al. (1988) used the same approach as was used for the equivalent mass factor to compute the equivalent load factor at every time step during the dynamic analysis using Equation 224 i ii Li iL LiLuu uu KK KK )1( )1( (224) Where )1( i iuuu Again, a simplified approach was implemented in DSAS where it generates a load factor function curve based on displacement Re(u), which reduces computational time. PAGE 34 34 Mass factor and load factor ca lculation: DSAS approach As mentioned in the previous section, DSAS uses a modified approach based on the previous theory to evaluat e the load factor and the mass fa ctor at every time step of the analysis and does so by using finite beam elements. Based on material properties, section properties and the curvature relationship, DSAS will generate an equivalent resistance function for a given section (dis placement versus load) using Equation 225 i e Nnodes i i mid i j i j i eR d d fF (225) where j represents the node location and i the load increment. Using the same approach, it will generate the equivalent mass using Equation 226 Nnodes i i mid i j i j i ed d M M2 (226) Finally, it will generate the loadi ng function using Equation 227 )( )( )( ),( tw uw uF tuFe e (227) where w represents the distributed static load applied on the element that would cause the control displacement, u Numerical integration of equivalent SDOF Various approaches may be used to numerica lly integrate the equation of motion. The method is usually chosen to minimize computational time and provide better accuracy for a given type of problem. The met hod used for the scope of this research is a special case of the NewmarkBeta Met hod. This method is an implicit method meaning the equation of motion is satisfied at time t + t unlike an explicit method, where the equation of motion is satisfied at time t For multi degree of freedom system, PAGE 35 35 an implicit method may have been computationa lly inefficient requiring a full stiffness matrix inversion which would be very ine fficient for a nonlinear resistance function. However, this method has proven to be very efficient and stable for a single degree of freedom system. The method uses Equat ion 228 and Equation 229 to compute velocities and displacement: ttt tttuu t uu 2 (228) 2 22 1 tutu tuuutt t tttt (229) Where is taken as 1/6. The following are t he steps the method uses to solve the equation of motion. Compute tu using the equation of motion and known tu and tu Estimatettu, usually chosen as tu or zero for fi rst time step. Compute ttu and ttu using Equation 228 and Equation 229 Compute ttu using the equation of motion and previously computed ttu and ttu. Iterate the process using newly computed ttu until ttu satisfy the convergence tolerance. Once convergence is satisfied, continue to next time step and repeat process. The value of time step must also be carefully chosen to ensure accuracy and stability. For the NewmarkBeta method, typical values when dealing with short impulsive load are one tenth of the natural period or one twel fth of the pulse duration. This takes into account the effect of plasti city on the system period. The main criterion of importance is the time step must be sm all enough to properly capture the loading PAGE 36 36 time history since blast occurs in a very s hort duration of time. Therefore, it is not uncommon to use a time step of 0.0001 sec ond in the case of blast loading. To computationally improve the calculation time, one can use a time step of 0.0001 second during loading phase and then use a bigger time step during free vibration phase. Dynamic reactions The following procedure to compute reacti ons at the supports were developed by Krauthammer et al. (1988) to overcome the limit ations of the elastic perfectly plastic material model previously used in literature such as Biggs (1964), which are also only applicable to a limited number of load cases (Krauthammer et al. 1988). The distribution of the inertia force is assumed to be equal to the deformed shape function as seen in Figure 26. For each load step in the loaddeflection relationship, compute the reaction and proportionality factor corresponding to each support using Equation 230 and Equation 231. i i iQ Q1 1 (230) i i iQ Q2 2 (231) Where iQ1 and iQ2 represent the static reaction at load time step i iQ represent the load at time step i i1 and i2 represent the load pr oportionality factors. Compute the inertial load factor ILF at every load step i using the following relationship. PAGE 37 37 L dxx ILFL i i0)( (232) By using the principle of linear beam theory, inertia pr oportionality factors 1i and 2i are approximated for every load time step. At this point, the inertia proportionality factor can onl y be approximated due to the fact that the magnitude of the inertia load is unknown, theref ore the required iter ative procedure to evaluate these factors prec isely cannot be performed. Using Equation 233 and Equation 234, compute the end reactions of the element )'())((1 11uMILFtQ Vti ii (233) )'())((2 2 2uMILFtQ Vti ii (234) Where )(tQ represents the forcing function tM represents the ma ss of the element u represents the acceleration Equation 235 is used to calculate 1i, 2i, 1i and 2i at every time step )(1 1 i ii ii iuu uu (235) Where 1 i iuuu and urepresents the dynamic displacem ent for a specific time step and is simply the generic name for load and inertia proportionality factors. Resistance Function The resistance function is by definition the restoring force Rm exercised by an element to regain its in itial condition when su bjected to a load. It is important to note that the resistance function of an element will be different for each different type of deformation. For example, the resistanc e function for an axial deformation on a PAGE 38 38 concrete column will not be the same as fo r a flexural deformation on the same column. Also, in the case of reinforced concrete, t he resistance function due to axial load will not be the same in tension and in compression. Figure 27 shows an example of a resistance function obtained us ing the software DSAS V3.0 for a reinforced concrete section. A detailed discussion on the softwar e DSAS may be found in a later section. Resistance function R(u) allows for the dynamic equilibrium equation for a SDOF system to be written as pr esented by Equation 236 )()(tFuRucum (236) The maximum resistance is the total load having the given distribution which the element could support static ally. The stiffness is numerically equal to the total load of the same distribution which would cause a unit deflection at the point wher e the deflection is equal to that of the equivalent system. (Biggs, 1964) Using Biggs definitions quoted above, t he equivalent SDOF equation of motion may be expressed with Equation 237 )()( tFtRKuCuMe L e (237) where )( tFe is the equivalent force as a function of time eM is the equivalent mass )(t euM is the equivalent mass as a function of displacement C is the damping coefficient parameter KL is the load factor Using the previously describe definition by (Biggs, 1964), one may assume Equation 238 LRKK (238) PAGE 39 39 where Kr and KL represent the resistance factor and load factor. In the case of dynamic analysis, one must al so consider the effect of load reversal into the resistance function model. Many m odels may be found in t he literature but the model used in this research was develo ped by Krauthammer et al. (1990) based on Sozen (1974) models and is seen in Figure 28. Line AA represents the loading and unloading in the linear domain. For instance, when the element is loaded, displacement will start going from O to A. If A is not reached, the unl oading will occur elastically following line A to A and will oscillate until dampi ng brings the structure to rest at O. If yielding is reached, the beam is said to have reached yielding and plastic deformation will start occurring. If point C is reached, the beam is said to have failed in flexure. Otherwise, the beam will unload following points B, D, E, F, G, D and E. The beam will then come to rest with a plastic defo rmation when all energy has been dissipated through damping. Flexural Behavior Flexural behavior of reinforced concrete is probably the subject mostly documented in the literature since most des igns are controlled by flexure, (Park and Pauley, 1975, MacGregor, 2009, Park and Gamble, 2000, McCormack and Brown, 2009). The concept of a mom ent curvature relationship has been extensively used to represent the flexural behavior of reinforced concrete. The moment curvature function is generated using strain compatibility and equi librium. The following assumptions have been used to develop the numerical algorithm embedded into DSAS. First, multiaxial stresses are ignored and therefore, only uniaxial stresses are considered in the concrete beam. Second, plane sections re main plane before and after bending based on Bernoullis principle. Finally, stressstrain curves for concrete and steel are known. PAGE 40 40 Procedure may be found in literature where tension is ignored since the concrete under tension stress quickly loses its effect when cr acks are formed in the tensile region. In this case, the tension is not ignored since strain rate may have a significant effect on the tension strength of the concrete when subjected to blast or other loading cases. A more detailed discussion on strain rate effect is presented in a later section. Figure 29 shows a stress and strain diagram of a reinforced concrete cross section under flexural stress. The confined concrete is separated from the unconfined concrete and the cross section is divided into layers. By using equilibrium and compatibility, the stress strain curve may be obtained b y increm enting the curvature from 0 to failure of the secti on using the following steps: Increment Assume strain distribution Calculate strain i for each layer of the section Calculate each stress i as a function strain i using materials constitutive model Calculate moment and forces Iterate the process until equilibrium Equation 239, is satisfied 0 PFx (239) Repeat process for new value of until ultimate strain has been reach. The constitutive material models used to generate the moment curvature diagram for confined and unconfined concrete under high intensity load and steel have been developed by Krauthammer et al (1988) and are presented in Figure 210, Figure 211 and Figure 212. PAGE 41 41 For the unconfined concrete stress strain model the model is divided into three stages represented by Equation 240, Equat ion 241 and Equation 242 for different strain values: From 002.00 c 2002.0002.0 2 'c c ccff (240) From 004.0 002.0 c 002.0'1' c ccZff (241) Where Z represents the line slope From 004.0 c 0 cf (242) For the confined concrete stress strain model, Figure 211, the model is also divided into three stages for di fferent values of strain and represented by Equation 243, Equation 244 and Equation 248: From oc 0 02 1 c oc o o c oc cfK E K f E f (243) From K co 3.0 1 8.01'o o c cZ fKf (244) Where PAGE 42 42 c yr of f s h 734.0 1005.00024.0 (245) c y rf f D D h s K 245.010091.01 (246) 002.0 1000' '002.03 4 3 5.0 c c rf f s h Z (247) From cK 3.0 c cfKf '3.0 (248) And, Ec is the elastic modulus of concrete (psi) H is the average core dimension of the confined concrete compression zone, measured to the outside of stirrups (inches) r is the ratio of the confining steel volume to the confined concrete core volume per unit length of the element in compression zone. is the longitudinal compression reinforcement ratio fy is the yield stress of the hoops s is the spacing of the hoops (inches) D is the nominal diamet er of the hoops (inches) D is the nominal diameter of the longitudinal compressi on reinforcement (inches) It is important to note the parameters K, Z and o will vary as the location of the neutral axis changes and therefore, as DSAS increments the section curvature, a new confined stress strain curve is generated for t he calculation of the moment curvature. PAGE 43 43 Furthermore, different cross section will have different stress strain curves unique to them. The stressstrain model used for steel is also taken from Krauthammer et al. (1988) and is shown in Figure 212 and is separated in 3 stages: elastic using Equation 249, perfectly plastic using Equati on 250 and hardening using Equation 251. From A to B: ys yssEf (249) From B to C: shsy ysff (250) From C to D: shs 21302 60 2 60 2 r m m ffshs shs shs ys (251) where 2 215 160130 r r r f f my u (252) is the difference from Ultimat e strain to strain hardening Es is the steel modulus of elasticity s is the steel strain sh is the strain where steel hardening begins su is the ultimate strain of steel fs is the steel stress fu is the ultimate steel stress PAGE 44 44 fy is the steel yield stress It is important to note that when a reinforced c oncrete section fails in flexure, it is a combination of both flexural behavior and diagonal shear behavior and not only a flexure failure. Adjustments are required to account for diagonal shear behavior in the dynamic analysis and are discussed next. Diagonal Shear Behavior Diagonal shear failure occurs on reinforced concrete sections where flexural stress and shear stress are acting toget her with a significant amount of stress creating cracks perpendicular to the principle tensile stresse s along the member causing brittle failure, thus the need to provide web reinforcement. For the case of this study, a shear reduction factor will be used based on Krauthammer et al. (1988) findings that di agonal shear must not be neglected when computing the deflection of beams influenc ed by shear, especially for deep beams and slender beams. The shear reduction concept was first dev eloped by Krauthammer et al. (1979) and later modified by Krauthammer et al. (1988) to better account for the effect of diagonal shear in deep and sl ender beams. Previous res earch conducted by Kani (1966) and Leoanhardt (1964) on beams without web reinforcement indicated the shear span to effective depth ratio a/d was governing the strengt h of simply supported rectangular reinforced concrete beams, where a is the distance from the load to the support and d is the distance from the compression ex ternal fiber to th e first layer of steel in tension. Two different ty pe of failure were observed based on a/d Shear compression failure: 5.2/0.1 da Diagonal failure: 7/5.2 da PAGE 45 45 Kani (1966) proposed a model for the in fluence of shear on beams having no web reinforcement as shown in Figure 213. Lines P1 and P3 r epresent the limit where there is no need for reduction of moment capac ity and line P2 represent the minimum moment capacity. Krauthammer et al. (1979) represented the influence of shear on the section capacity for fl exure using Equation 253 fl uM M SRF (253) Mu being the ultimate moment capacity due to shear and flexure and Mfl the ultimate moment capacity due only to flexure. To obtain the SRF Krauthammer et al. (1979) developed the relationship presented by Equation 254, Equation 255 and Equation 256 based on Kani (1966) experimental results to describe the minimum SRF ratio as a function of 0.1)(:%65.00 m fl uM M (254) )0065.0(6.360.1)(:%88.1%65.0 m fl uM M (255) 6.0)(:%88.2%88.1 m fl uM M (256) Using Equation 254, Equation 255 and Equa tion 25, Krauthammer et al. (1988) developed mathematical models to account for shear influence on reinforced concrete moment capacity of a given section with and without web reinforcement and for both underreinforced and overreinforced sections. Figure 214 shows the model developed by Krauthammer et al. (1988). PAGE 46 46 For the case of this study, the important fa ctor to take from the Krauthammer et al. (1988) investigation is the modification of the model when web reinforcements are present. Krauthammer et al. (1988) developed the relationship presented by Equation 257 to compute the minimum mom ent capacity ratio, point P2 on Figure 213. )tan()(0.1)()(' m fl u m fl u m fl uM M M M M M (257) Where ')(m fl uM M is the modified minimum moment c apacity ratio accounting for the effect of transverse reinforcement and is taken as the angle of diagonal compression strut at ultimate and is calculated using Equation 258 and Equation 259 for different type of beams. Deep rectangular beam: 08.4)/(72.2* da (258) Slender rectangular beam: 22.7)/(06.3*da (259) is the web reinforcement ratio, calc ulated using the fo llowing relationship c yf f ""* (260) is the web reinforcement ratio yf" is the yield stress of the stirrup cf is the compressi ve strength of concrete This approach was embedded into DSAS. The computed moment is multiplied by SRF and the curvature is divided by SRF. PAGE 47 47 Direct Shear Direct shear failure occurs at a lo cation of high concentration of shear forces such as support and point load loca tions. In this ca se, cracks form almost perpendicular to the beam. (Krauthammer et al. 2002). In the case of direct shear failure, Krauthammer et al. (1986) developed a shear resistance function model based on an empi rical model develop by Mattock and Hawkins (1972) and Murtha and Holland (1982) to include the effect of load reversal and is represented in Figure 215. To account for direct shear failure in a dynamic column analysis one may use Krauthammer et al. (1988) approach to verify both direct shear failure and f lexural failure at each time step of the analysis. The method uses two singledegreeoffreedom systems running simultaneously, one for direct s hear analysis and the other for flexural analysis. This method is already implement ed in DSAS and has been intensively used in previous research conducted at the Cent er for Infrastructure Protection and Physical Security. A similar approach is also used to evaluate variable axial load on a column from adjacent components and is discussed in the next section. Axial Behavior and Analysis When axial forces are acting on a concrete beam or column, t heir effect may be taken directly into account in the moment cu rvature relationship since all forces acting on the section must be in equilibrium. Typi cally, the presence of compressive axial load increases the ultimate moment of the section; however it also reduces its ductility. If the axial force is less than the balanced axial fo rce, the moment capacity will be increased, and if the axial force in compression exceeds the balance point, the moment capacity will be reduced. For most case s, compressive axial forces will also improve the shear resistance capacity of the section. PAGE 48 48 Holmquist and Krauthammer (1984) proposed a model to account for shear capacity enhancement due to compressive axial force. They proposed the thrust factor TF, calculated using Equation 261, )/(0005.01gAP TF (261) which multiplies the stress axis in the forc e slip model, when compressive axial loads are present. When a variable axial force is acting on a column, it will result in a variable resistance function. Figure 216 from Tran ( 2009) shows the influence different values of axial forces may have on a section when su bjected to both lateral and axial forces. To evaluat e the resulting axial load on the column due to upward and downward pressures acting on the horizontal element attached to the column as seen in Figure 11, one may used a SDOF system to analyze the response of the horizontal element and transfer the calculated reaction at the suppor t to the column. It is then possible to use a SDOF to evaluate the column response to the load combination. Such analyses must be conducted simultaneously at each time step as both member reacti ons to the load affect one another, especiall y if one member is to fail. Also, within one element, it may be required to have more than one DOF to evaluate each possible failure mode. In order to account for the variable axial forces on the column and to solve for the dynamic SDOF equation equilibrium, one may gener ate a series of resistance functions within an upper and lower bound envelop for the flexural resistance function as seen in Figure 217. Theses curves may then be used to interpolate the resistance value at a given deflection for the variable axial forces. The upper bound would represent the resistance function for the ma ximum compressive axial forces generated by the loading PAGE 49 49 function and the lower bound would represent the maximum axial tension forces or minimum axial compressive force. A search algorithm may then be used at each time step to solve for the resistance function poi nt within the envelope during the dynamic analysis. The aim of generating a set of curves to interpolat e from is to minimize the computational time by serving as a starti ng point to solve for the resistance function during the flexural dynamic analysis. At the present time, D SAS is regenerating a complete resistance curve ever y time the axial forces vari es which becomes very time consuming when the axial load varies continuously during the analysis. Rate Effect Researchers such as Evan (1942), Wast ein (1953) and Soroushian et al. (1986) have reported in literature an incr ease in strength material for material subjected to high loading rates (Shanna, 1991). Shanna (1991) repor ted two techniques used in analysis, One of these is a dynamic enhancement factor based on st raining rate to increase material properties used in the derivation of momentcurvature relationships, diagonal shear and direct shear relationships. Another technique used is to directly apply the enhancement factor to the resistance functi on by multiplying the shear and flexure capacity of the section by the enhancement factor (Shaana, 1991). One may use one of the following rate to develop the enhancement factor as they are correlated, the loading rate, the stress rate or the strain rate. Since the strain is the controlling parameter in the co mputation of the moment curv ature relationship, it is much easier computationally to use the st rain rate as the independent parameter. Shanna (1991) and Krauthammer et al. (2002) offer a more detailed discussion on rate effect and proposed the use of the Sorous hian and Obaseki (1986) model for the enhancement factor. PAGE 50 50 For steel material, the enhancement factor presented by Equation 262 may be used. )( )( )( )(staticf dynamicf staticf dynamicf Enhu u y y s (262) y y y sf t f f Enh)/(log)05.065.0(2.11.310 (263) And for concrete materials, the enhancement fa ctor presented by Equation 264 may be used. )(' )(' staticf dynamicf Enhc c c (264) For sec//6.15inchinchE t )/(log03.014.110t Enhc (265) For sec//6.15inchinchE t )/(log03.038.110t Enhc (266) Finally for concrete in tens ion, Equation 267 may be used ) 0164.0exp(086.2E Enhct (267) Where, )/(log0.710t E (268) Large Deformation The behavior of structural elements under large deformation is well known in literature and is often referred to as second order analysis. In such analyses, the effect of PDelta is taken into account and the EulerB uckling is verified at each time step of PAGE 51 51 the analysis. The following review the notion of PDelta effect and a finite element method that implicitly a ccounts for its effect. PDelta effect Columns may be classified into two categories: short columns and slender columns. The difference lies in the influenc e of axial force combined with lateral deflection on section strength capacity. Short columns are classified as such as the strength relies solely on st rength of material and sectio n geometry and would fail before enough lateral deflection occurs for a PDelta effect to be significant on the section capacity. In the case of a slender column, t he eccentricity of an axial load may create a significant moment amplification that cannot be ignored which results in a reduction of column strength capacity. There are two types of second order mome nts that are required to be taken into account in a Panalysis. The first type of moment is induced by a deflection to the central cords causing a load eccentricity as seen in Figure 218. The second type occurs when the supports move away from t he cord, creating an extra eccentricity for the applied load to the central cord. For the case of this study, only the first type will be considered as supports are not expected to move significantly compared to the center of the column when subjected to a blast load. MacGregor (2009) in its slender column se ction discuss ed in details the different paths of failure a column may take using a reinforced concrete interaction diagram (Figure 128 of the reference). He demonstr ated a linear correlation for a short column and a second order correlation for a slender column. The linear correlation of the short column is due to the fact that it fails before Pmoment has a significant influence on the section. The second order correlation is due precisely to the second moment PAGE 52 52 introduced as the slender columns deflect. ACI 31808 provide slenderness ratios for braced and unbraced members where slenderness effects may be neglected for ease of design. Non braced members: 22 r lku (269) Braced members: 40)/(123421 MM r lku (270) Where K is the effective length factor lu is the unsupported length r is the radius of gyration of cross section M1/M2 is the ration of end moment and 21MM Those slenderness ratios were developed for static analysis and may not apply for the case of severe short duration dynamic loads. Therefore, one shall take into consideration PDelta effect for all type of columns regardless of their slenderness ratio when dealing with blast loading. Two types of failure may occur for slender column: material failure and stability failure. Stability failure occurs when P M (271) Therefore, in a dynamic analysis, stability fail ure shall be verified at each time step of the dynamic analysis. PAGE 53 53 In order to design for second moment effect, the moment magnifier method presented by MacGregor (2009) combined with a firstorder analysis approach may be used. For the case of this study an exac t approach solution using finite element application and numerical proced ure to develop the resistance curve in DSAS was used based on the following discussion and fu rther discussed in Chapter 3. Development of resistance function curve in DSAS To generate the resistance curve one may use a finite beam element approach that implicitly accounts for the axial forces in its formulation. The following is a short review of the DSAS finite beam element formulation procedure used to generate the resistance function of an element followed by a discussion of a modified finite beam element formulation. DSAS procedure in steps: Generate the moment curv ature of the reinforced c oncrete cross section. Divide the column into multiple beam elements with 2 degrees of freedom per nodes as seen in Figure 220. Using a displacement control approach implicitly through the cylindrical arclength method developed by Crisfield M.A (1981) and equival ent parameters, increment displacement at mid span and obtain rotation and displacement of all nodes due to midspan displacement. Obtain Curvature at each node using a straindisplacement matrix and Equation 272 and Equation 273. UBEE (272) UB (273) Recover Forces at each node using Equation 274 and obtain the stiffness term from the moment curvat ure diagram for each node. UKf (274) Repeat steps by incrementi ng displacement until failure PAGE 54 54 The cylindrical arclength method m entioned above is an iterative solution developed by Crisfield M.A (1981) to capture snapthrough buckling in a finite element analysis. When a load control analysis is perfo rmed, the load deflection path will follow OAC in Figure 219, creating a dynamic snapthrough which is not truly representative of the element behavior. In order to truly capture the element behavior, a deflection control analysis is required. Path OABC of Figure 219 shows the true load deflection path of an element. To properly capture this path using finite elements, a solution allowing tracing negative slope is required. To do so, Crisfiel d (1981) used a cylindric al arclength iterative approach wh ere the finite governing differential equation is checked along the arc until convergence occurs. A more detailed discussion of this method may be found in Crisfield (1981). To account for the axial forces, Smith and Griffiths (1998) presented the following modification to the EulerBernoulli beam el ement. The governing differential equation is modified and is presented by Equation 275 q x w P x w IE 2 2 4 4 (275) Where E is the elastic modulus of the section I is the moment of in ertia of the section P is the axial load (compression > 0) w is the deflection x is the position on the beam q is the transverse load per unit length carry by the element. PAGE 55 55 The Galerkin method and finite element discr etisation was used to solve the matrix Equation 276 for the axial force cont ribution in the system equilibrium. L j iw w dx x N x N P0 2 2 1 1 (276) Where i,j = 1,2,3,4 And lead to the stiffness matrix Equation 277 2 2 1 143 3 3 36 3 36 343 3 36 3 36 w w L L L L L L L L P (277) Then the equilibrium equat ion becomes Equation 278. f w w L L L L L L L L P L L L L LLLL L L L L LLLL IE 2 2 1 1 2 2 2 32 3 2 2 23 2 343 3 3 36 3 36 343 3 36 3 36 4626 612612 2646 612612 (278) For a displacement control approach, this method may be modify in step 5 in the presented procedure to account for the extra curvature created by the PDelta effect using Equation 279 and Equation 280. 1 1 111IE u P (279) 2 2 222IE u P (280) PAGE 56 56 Where the stiffness I E may be approximated using the mo ment curvature diagram for the unmodified curvature and implying a suffi cient number of elements are present. It is important to note there haves been many different finite element models developed over the years better suited fo r large deformation problems than the one presented here. For example, Bathe et al. (2005) presented two formulations for large strain and large deformation, the total fo rmulation and the rate formulation. The total formulations are based on kinematic quantities defined with respect to an adopted reference confi guration and on incremental solution procedure and the rate formulations are based on integration of constitutive relations involving stress and stra in rate. (Bathe et al. 2005). The modified EulerBernoulli beam elem ent is an approximate method based on the theory of small rotation and deflection that may be used for reinforced concrete elements as they are expected to fall into tension membrane behavior while being in the small deflection/deformation range. Krauthammer et al. (1988) proved that small deflection theory and large defle ction theory hold true for defle ction up to 15% of the beam length in the case of simply supported beams. Furthermore, it was also shown by Kraut hammer et al. (1984), Krauthammer et al. (1986), and Park and Pauley (1975) that comp ressive membrane effe cts for slabs start developing at a deflection of about 0.5*h where h is the slab thickness. It was also shown that tensile membranes fully devel op in between a deflect ion of h and 2*h. Krauthammer et al. (2003) also showed tension membrane effects in deep beams point C in Figure 222, start occurring at about 0.17*h. In the case of columns, recent studies conducted by Tran et al. (2009) demons trated that an increase of axial forces on columns resulted in a smaller deflection at failure which indicated transition into a PAGE 57 57 tension membrane behavior would occur at a smaller deflection than without axial forces. All the above demonstrated that the use of large deformation finite elements for reinforced concrete columns is not necessary since by the time the central deflection reaches 15%, slender columns and even short columns will have already transited into tension membrane. Therefore, the a modified EulerBernoulli beam element is expected to provide excellent results in the ca se of reinforced concrete sections. Euler Buckling The Euler Buckling represents a failure mode of a structural element subjected to a compressive stress exceeding its capac ity. The theory was developed by Euler Bernoulli using its beam theory formulation and goes as follows. Using Equation 281 for a simply support ed column with only an axial load applied on it M dx du IE (281) where the moment may be calculated using Equation 282 uPM (282) the solution for displacement, u, will ta ke the form presented by Equation 283 xCxCu cos sin2 1 (283) where IE P (284) The constants C1 and C2 may then be solved using boundary conditions. For this case one may show that C2 is 0 for x =0 and C1 is not zero for x = L Therefore if C1 is not PAGE 58 58 zero but the boundary condition states the displa cement must be zero, it implies that the variable in the sin(var) must be zero. Thus Equation 285 nL (285) where n represents the mode shape of t he deflection function leading minimum P for n = 1 Therefore using Equation 284 and Equation 285 one may solv e Equation 286 for the critical load Pcr 2 2L IE Pcr (286) Many reference books provide the Euler buckling equation for different possible cases. However, it is impor tant to note this formulation is based on the assumption of small angle deformation, linear elastic mate rial, no yielding before buckling and the presence of only pure axial load. This is, howev er, not true in reality but in most cases, calculating the Euler buckling represents safety verification in design and an upper limit for analysis. For the case of reinforced concrete, there are two different types of Euler buckling that may be verified. The first one is the global buckling where the entire section is in compressive buckling and the second one is where buckling occurs in the longitudinal reinforcement between stirrups. The following section discusses the two different types during a dynamic analysis. Dynamic analysis: Reinforce concrete global buckling When a column is subjected to a blast load ing or any lateral l oad, the deflection of the column will be due to the combined effect s of both axial and lateral load. Therefore PAGE 59 59 during a dynamic analysis, it is possible to approximate the Euler buckling load at each time step by using the effective stiffness of the column. This approach may be facilitated by using a finite element code since curv ature may be obtained at each node and the stiffness may then be obtained for each node us ing the moment curvature diagram of the reinforced concrete section. Therefor e, for a one degree of freedom system, the effective stiffness is simply the average of all stiffness. Another method that has been used in literature is the energy balanced method. To use this approach in a single degree of freedom dynamic anal ysis, the axial load must also be expressed in terms of equi valent axial load using an energy balance solution and a shape function. In the equation of motion, the axial load only affects the stiffness and when combined with the stiffness fo r the flexural stiff ness, a new natural frequency is formed at each time step as any la teral displacement or axial load variation will have an effect on the system stiffness. If one considers Kf to be the equivalent flexural stiffness and Ka to be the equivalent axial stiffness, the natural frequency of the system may then be expressed using Equation 287 e af eM KK (287) when the axial load acts in compression, the natural frequency would be decreased. From Equation 287, it is possibl e to solve for the value of Pcr using Equation 288 0 afKK (288) to yield Equation 289 PAGE 60 60 dxx dxxxk Pcr 2 2)( )()( (289) A more detailed discussion on the derivatio n of the previous equation may be found in Tedesco et al. (1999). By using Equation 289, it is possible to form a graph of Pcr versus central displacement that may be used by advanced dynamic analysis software such as DSAS to verify the buckling load at each time step of the analysis. Dynamic analysis: Reinforce concrete compression longitudinal rebar buckling Buckling may occur in elements subjected to compressive forces within a composite section such as reinforced concrete when conditions for such phenomenon are present. Test experiments conducted by Ya mashiro and Siess (1962) on reinforced concrete beam members subjected to bending, shear and axial load have revealed the presence of local buckling of the compression longitudinal steel reinforcement on several test specimens. There were two ty pe of compression failure observed, steel compression failure and concre te compression failure. It was, however, observed that when a beam failed due to concrete comp ression failure, the longitudinal steel had buckled as well. It was also observed that co ncrete compression failure was occurring in lightly reinforced concrete specimens. In the case of reinforced concrete columns, the amount of longitudinal steel reinforcement is generally much larger compare to reinforced concrete beam and one may expect a steel compression failure to occur before concrete compression failure. When using a finite element approach such has the one used by DSAS, concrete compression failure is easily captured using the moment curvature diagram and finite elements as concrete will failed in layers. To capture the steel compression failure, PAGE 61 61 Yamashiro and Siess (1962) proposed using the Euler Buckling formulation for longitudinal steel in compression. The unsuppor ted length is then taken as the stirrups spacing and the formulation is solve for a fixedfi xed case in view of th e fact that the test specimen demonstrated that all rebar bu ckling occurred between two adjacent ties having the shape of a fixedfix ed buckling case. Furthermore, it was also observed that the buckling of the steel never occurred downward or upward but rather horizontally. Equation 290 is the relationship used by Yamashiro and Siess (1962) to account for the possibility of rebar buckling in between two ties in terms of buckling stress as the strain in the compression steel and t he stress strain curve were known. 2 2)/( rs E ft cr (290) where Et is the tangent modulus of the steel s is the spacing of the ties r represents the radius of compression bars At the time, the computational results di d not quite match the experiment results as some beams continue to provide flexural strength after the buckling of longitudinal reinforcement. The assumption by Yamash iro and Siess (1962) was that the beam would fully fail if the longitudinal steel buckles. The reason why the beam continued to provide flexural strength may be explained by the fact the stress in the beam were redistributed to the remaining of the sect ion after the compression longitudinal rebar buckled. This limitation may however be overco me by the use of a finite element code and the moment curvature diagram for a given section. PAGE 62 62 For a dynamic analysis using a finite elemen t code, it is possible to modify the steel stress strain curve re lationship in the compression range to account for the Euler buckling. One may solve for the stress or the strain at which Euler buckling occurs using the following procedure Increment stress (or increment strain) solve for the secant stiffness calculate the Euler buckling stress corresponding to the secant stiffness compare the calculated Euler buckling st ress to the actual incremental stress Increment until Euler buckling stress is equal to the incremental stress Using this approach, it is possible to capt ure buckling in longitudinal steel element and continue the analysis has the stress distribution in the failed region of the cross section would simply be redistri buted to the unfailed region. Membrane Behavior When a reinforced concrete section goes into large deformation, the section may experience an increase in flexural capacity due to compressive membrane action which may be followed by tension membrane acti on caused by catenary action of steel reinforcement when conditions fo r such effects are present. In that case, the membrane behavior start occurring once reinforced conc rete stops behaving in an elastic manner and falls into a nonlinear material behavior. Pr inciple cracks form at the centre of the element and continue to increase as the beam /column deflects. Once the beam is fully cracked, only the steel continues to provi de strength until all steel layers have failed. Figure 221 shows a simplified transition progression int o compressive membrane followed by tension membrane. PAGE 63 63 For general design purposes, the membrane e ffect is generally not taken into account as the serviceability issue does not allow for large deformation to occur. The effect was although studied in literature for limited cases su ch as slabs under large deformations (Park 1964, Park and Gamble, 2000 and Guice et al. 1989) and resistance slabs under fire (Bailey, 2000). T he subject is also of interest in the field of progressive collapse as it may provide sufficient time fo r a building to be evacuated before complete collapse of the building occurs or in some cases may actually prevent the collapse of the building. Figure 222 shows a general loaddeflection/resistancefunction at midspan of a fully fixed two way rectangular reinforc ed concrete slab. The membrane effect development of a slab section or a beam /column section is based on the same principles. The steel reinforcement must be continuous along the entire length of the element and be well anchored int o the boundar y to allow for compression and tensile membranes to develop. A continuous sectio n would behave the same as long as the steel is not discontinued and is long enough to have proper length development for the steel beyond the continuous supports. To develop a compressive membrane, the boundaries have to be stiffed enough to allow for outward reaction as the section deforms as seen in Figure 223. Only then would the compressive membrane have a signific ant impact on the section flexural strength capacity. Park and Gamble (2000) showed that compressive membrane forces in slabs may improve by 1.5 to 2 times the ultimate capacity of both oneway and twoway slabs. The section behaves in compressive membrane up to point A of Figure 222. As the section continues to deflect, transition into tensile membrane occurs, as seen in PAGE 64 64 segment AB of Figure 222. In that region, the beam is nearly fully cracked and concrete no longer provides sufficient str ength for the section flexural capacit y to continue to increase. Segment BC of Figure 222 represents the tensile membrane region, which shows an increase in section capac ity as the deflection increases At that point, the section is fully cracked and only t he steel provides flexural strength to the section. The steel will continue to carry the load until it fails. This phase is characterized by tensile membrane. Compressive membrane calculation Compressive membrane calculation t heory has been developed for slabs by various authors. The following is a review of one way slab compressive membrane calculation based on Guice et al. (1989) and Park (1964). One may also use this theory to approximate compressive membrane behavior of column, as long as good judgment is used, as the deformation profile for oneway slabs used to develop the theory differ from the deformation pr ofile of columns. Using geometry of deformation as seen in Figure 224, one may derive Equation 291 x c chx txm s )tan()tan( )cos( (291) where represents the total stra in, shrinkage and creep strains t represents the lateral movement of one support cm represents the neutral axis depth at midspan cs represents the neutral axis depth at support x represents the horizon tal projected length of deformed element PAGE 65 65 Guice et al. (1989) rewrote Equation 291 using trigonometric identities and small angle deformation to obtain Equation 292 txx hccms 22 (292) Equation 292 may then be rea rranged to yield Equation 293 x tx hccms 22 (293) Park and Gamble (2000) developed another expression to solve for cs and cm using Figure 223 using Equation 294. )tan( )1()tan( )sec( 215.0 m scl ch tl l (294) which may be simplified to Equation 295 )sin( )21(5.0cos ) 2 sin(2 tl l l cchcs (295) to yield Equation 296 l tl hccms2 222 (296) using small angle approximation and the trigonom etry identities presented by Equation 297 and 298 l ) 2 sin(2)sin( (297) and 1)cos( (298) For oneway slabs, may be taken as 1 and the variable l may be substituted for x to obtain a similar expression to Guice et al. (1989) presented by Equation 299. PAGE 66 66 x tx hccms2 222 (299) The difference between both expression li es in small angle approximation and trigonometry identities where Guice et al. (1 989) and Park and Gamble (2000) relied on different simplification approaches. By comp aring Equation 299 and Equation 292, one can see that the difference relies on the strain where the Park and Gamble (2000) equation uses half the strain displacement used by Guice et al. (1989). Experimental investigation of Guice et al (1989) theoretical model to calculate the compressive membrane capacity of several slabs r eported an excellent correlation with the experimental ultimate deflection. A mean value of analysis predictions over experimental result of 0. 96 with standard deviation of 0.14 and a variance of 1.9% were reported (Guice et al. 1989). In order to solve for a uniqu e solution more equations ar e required to solve for the unknown terms introduced by Equation 292. By using horizontal equilibrium of each force components in the reinforced concrete section in Figure 225, one may use Equation 2100 TCCTCCscssc''' (2100) or simply use Equation 2101 midspan portF Fsup (2101) It is now possible to use an algorithm as discussed in the flexural behavior section to solve for each forces Cs, Cc, Ts, Cs, Cc and Ts in a more accurate manner by dividing the section in multiple layer as seen in Figure 29. The forces may then be expressed using Equation 2102, Equation 2103, Equation 2104 and Equatio n 2105. PAGE 67 67 sisisiAfF (2102) ciciciAfF (2103) ) 2 (si sisid h Fm (2104) ) 2 (i ciciz h Fm (2105) where Fsi represents the steel layer forces Fci represents the concrete layer forces Asi represents the area of steel at each layer Aci represents the area of concrete at each layer dsi represents the depth of each steel layer zi represents the depth of each concrete layer msi represents the steel layer moment mci represents the concrete layer moment fsi represents the steel stress fci represents the concrete stress Using a stressstrain relationship with previously defined material models fsi and fci may be expressed as a func tion of neutral axis depth cs or cm, Equation 2106, depending on side of interest. )(si siFf (2106) where si may be expressed using Equation 2107 cd ci cu si (2107) PAGE 68 68 Using the same approach, one may obtained the stress of concrete as a function of central axis depth us ing Equation 2108 cz c Ffi cu ci (2108) The total section forces and moment may then be expressed using Equation 2109 and Equation 2110 ii ci si nmmmM (2109) ii ci siFFN (2110) By looking at the external forces equilib rium of the reinforced concrete element, one may derive Equation 2111 for the sum of ex ternal moment on the element as seen in Guice et al. (1989) and using Figure 226 2 x FNMMe nm ns (2111) Where Mns may be calculated using the section forc es equilibrium at the support using the same approach as for Mnm. The final equations required to obtain a unique solution needs to account for the axial thrust in order to have a complete picture of membrane behavior in the section equilibrium model. The magnitude of the thrust may be affect ed by various factors such as support movement, axial strain deformati on and creep. To account for these, Guice and al (1989) developed a relationship which accounts for the strain due to axial deformation, the elastic shortening strain and the creep and shrinkage strain. Equation 2112 neglects the effect of longitudinal reinforcement on the axial stiffness. p cAcE N (2112) PAGE 69 69 where a represents the axial strain Ec represents the concrete elastic modulus Ac represents the gr oss section area p represents the creep and shrinkage strain If lateral displacement occurs at the co lumn support it may be taken into account using Equation 2113 and assuming t he displacement is elastic. S N t (2113) Where t represents lateral di splacement at support N represents the compressive axial forces at support S represents the surrounding support stiffness. then Equation 299 may be written as xS N AcE Nx hccp c ms 22 (2114) Creep and shrinkage prior to t he loading of the section may be ignored since data on the matter are often not available in pr actice and may also be ignored during blast loading due to short load duration. Therefore Equati on 2114 may be written as Equation 2115 xS N AcE Nx hccc ms 22 (2115) The formulation may be further modified to account for external axial forces as shown in Equation 2116 PAGE 70 70 xS NN AcE Nx hccext c ms 22 (2116) where Next represents the external axial forces taken as negative for compression and positive for tension. In the case of most columns, no lateral displacement or separation is expected to occur at the support since columns support la rge loads, making the connection very stiff in that regard. Therefore, one may consider the connection stiffness to be infinite and rewrite Equation 2116 to obtain Equation 2117 AcE Nx hccc ms 22 (2117) Although the external axial te rm disappears from Equation 2116, it remains part of the equilibrium of the external forces and Equation 2111 may be adjusted to account for the PDelta effect created by the extra axial load to obtain Equation 2118. 2 )( x FNNMMe ext nm ns (2118) cs and cm may now be computed using Equation 2114 or Equation 2116 depending on the cases for any displacement With cs and cm known, one may calculate all internal forces in the concrete section element and finally, one may use Equation 2117 to solve the load carried by the section. Tension membrane calculation The tension membrane forces may be computed by applying the general cable theory to the steel reinforcem ent. The following is a review of this approach and only considers the case of a uniformly distri buted load on the section. Under uniformly PAGE 71 71 distributed load, the section def lection profile is expected to take the form of a parabola as seen in Figure 227 where the def lected curve may be described by Equation 2119 2xay (2119) To solve for the parameter a in the parabola equation, one may use a truss approach where the cable is divided into an infinite number of truss link together supporting the distributed load as seen in Figure 228. At any loca tion on the curve, the axial force ds may be represented into vertical and horizontal component force v and t Equation 2120 is then derived using similar triangle. dy dx v t (2120) The shear force at any point may then be expressed by Equation 2121 and Equation 2122 dx dy tv (2121) and w dx yd tdv 2 2 (2122) where w is taken as the load per unit length. By integrating Equation 2122, one obtain Equation 2123 Cxw dx dy t (2123) and since at x = 0 the deflection is at a maximum, the constant C become 0. Integrating once more, one obtain Equation 2124 C xw yt 22 (2124) PAGE 72 72 and at x = L /2 y = u one obtained E quation 2125 for C 82L wutC (2125) then Equation 2124 becom es Equation 2126 8 22 2L wut xw yt (2126) One may sum the moments at one end of t he parabola for the equilibrium of half the parabola to obtained Equation 2127 82L wut (2127) where in this case t is the horizontal force at x = 0 w here there is no vertical force. Then substituting Equation 2127 into Equation 2126 one obtain Equation 2128 8822 2 2L w L w xw yt (2128) and solve for Equation 2129 t xw y 22 (2129) The term a in the parabola Equati on 2119 may be taken expressed using Equation 2130 t w a 2 (2130) and using Equation 2127, one may substitute w in Equation 2130 to obtain Equation 2131 24 L u a (2131) PAGE 73 73 The steel strain may then be calculated usin g the arclength formula to solve for the change in length. Therefore, using symmetr y and solving one half of the parabola, one obtain Equation 2132 dx dx dy LL arc 2/ 0 21 (2132) one may recall from Equation 2119 xa dx dy 2 (2133) Then solving the integral equat ion to obtain Equation 2134 22 221 4 4 ) 1ln( La L a aLLa Larc (2134) The total strain may then be computed Equation 2135 and Equation 2136 1 2 2 L L L LLarc arc (2135) 1 2 1 2 ) 1ln(22 22 La La aLLa (2136) By defining La (2137) and 21 (2138) Equation 2136 becomes Equation 2139 1 )ln( 2 1 (2139) PAGE 74 74 The strain is then expressed as a function of central deflection u Using stress strain compatibility, one may solve the load deflection curve for any steel configuration in a reinforced concrete section. Park and Gamble (2000) compared theoret ical model calculations of tension membrane load deflection curve versus expe rimental data for different slabs. The results demonstrated a need to adjust the theoretical ca lculation of the tension membrane to the experimental tension me mbrane. The theoretical membrane curve needs to be shifted up and merged to the end of the load deflection curve of the compressive membrane theoretical curve for appropriate results. Krauthammer (1986) developed the Equation 2140 for tension membrane behavior into one wayslabs that account for such shift. 2)(8 )( L uTM uwn (2140) where w represents the appl y distributed load Mn represents the nominal section moment T represents the tens ion in the steel u represents the midspan deflection. Since columns and beams are literally oneway slabs, one may use Equation 2140 to adjust the pure tension membrane calcul ated using the cable theory. The tension membrane behavior would then begin at a pressu re value calculated using Equation 2141 28 L Mn (2141) PAGE 75 75 and the shift would be done following a perpe ndicular path to the pure membrane behavior as shown in Figure 229. As a column deflects under a severe short dynamic load, it is possible for the column to stop behaving as a beam/column and behav e as a tension membrane if the axial load is taken over by other elements of the structure. This is possible for the cases where the column would fail in flexure or due to a stability issue yielding to sudden buckling of the section. In the case of buckling a sudden change into tension membrane behavior would occur. If such a scenario occu rs, the resistance function of the section must be modified to account for the tension membrane behavior. Figure 230 shows an example of such transition into tension membrane and Figure 231 shows a column resistance function generated using the new version of DSAS with shifted tension membrane resistance function. It i s important to note the maximum resistance function in tension membrane may be limited by an upper limit base on steel development length or pulling capacity of the connection. LoadImpulse Diagrams The PressureImpulse (PI ) diagrams or more accurately referred to as LoadImpulse Diagrams, were developed to dete rmine levels of damage on a structure and have also been used in the past to evaluate human response to shock wave generated by an explosion. After World War II, they became widely used in the field of protective structure engineering. As seen in Figure 232, the PI diagram may be divided into three different regimes. The impulsiv e regime is charac terized by short load duration where the maximal structural response is not reac hed before the load duration is over. The dynamic regime is characterized by the maximum response being reached close to the PAGE 76 76 end of the loading regime. Lastly, the quasistatic regime is characterized by a structure having reached its maximum response befor e the applied load is removed. Figure 232 also shows one of the advantages of using the PI diagram to better differentiate the impulsive regime and quasistatic regime by mean of vertical and horizontal asymptotes. The points on the PI curve repr esent the combination of pressure or load and impulse that would ca use failure based on a predefine damage or failure criteria. If a combination of load and impulse is lo cated on the left and below the curve, the structure wi ll not exceed t he predetermined acceptable damage level or failure criteria. The predetermined acceptable damage level or failure criteria will only be exceed for a combination of load and impulse located above and on the right side of the curve. For simple problems, the PI diagram may be generated using closed form solutions. For other cases, the energy bal anced method may be used to establish the quasistatic and impulsive asymptote and t he dynamic range may be approximated by fitting a curve based on available data. More details on these methods may be found in literature such as Krauthammer (2008). For more complex loading scenario, a numerical solution must be employed to generat e the PI curve. Blasko et al. (2007) developed a numerical solution that was inco rporated into DSAS and will be used in the scope of this research and. A detailed descrip tion of his method may be found in Blasko et al. (2007). Summary This chapter presented various theoretical m odels from various fields of study that is used in Chapter 3 to develop a new comput ational tool to analyze reinforced concrete column subjected to blast load and undergoi ng large deformation. An introduction to PAGE 77 77 explosive load calculation was presented fo llowed by a review of dynamic analysis procedures for single degree of freedom system. A review of different concepts used in the field of protective stru cture for reinforced concrete material was also conducted before discussing large deformation behavior of reinforced concrete and loadimpulse diagram. PAGE 78 78 Figure 21. Blast environment from air burst. [Adapted from Kr authammer, T. 2008. Modern Protective Structure (Page 72, Figure 35). CRC Press, Boca Raton, Florida.] Figure 22. Free field pressuretime variation. Pso = Incident over pressure, Po = Atmospheric pressure, ta = Time of arrival, to = Positive phase duration, to = Negative phase duration, is = Positive phase impulse, is = Negative phase impulse. [Adapted from Krauthammer, T. 2008. Modern Protective Structure (Page 68, Figure 31). CRC Press, Boca Raton, Florida.] PAGE 79 79 Figure 23. One degree of freedom system representation of a real structural element. Figure 24. Deformation profile. w(x) u L Me Ke ce ue SDOF System Real System PAGE 80 80 Figure 25. Equiva lent load diagram. Figure 26. Determination of dynamic reaction for a beam with arbitrary boundary conditions. [Adapted from Krautha mmer, T., Shahriar, S. 1988. A Computational Method for Evaluati ng Modular Prefabricated Structural Element for Rapid Construction of Faci lities, Barriers, and Revetments to Resist Modern Conventional Weapons Effects. Rep. No. ESLTR8760 (Page 122, Figure 44). Engineering & Services Laboratory Air Force Engineering & Services Center, Tyndall Air Force Base.] w(x) u(x) L (x) u F1 Me Ke ce ue Fe F2 Fn PAGE 81 81 Figure 27. Resistance functions example for a reinforced concrete material with fixed boundaries conditions obtai n with DSAS V3.0. PAGE 82 82 Figure 28. Resistance displacement model [Adapted from Krauthammer, T., Shahriar, S. 1988. A Computational Method fo r Evaluating Modular Prefabricated Structural Element for Rapid Constr uction of Facilities, Barriers, and Revetments to Resist Modern Conventi onal Weapons Effects. Rep. No. ESLTR8760 (Page 103, Figure 38). Engi neering & Services Laboratory Air Force Engineering & Services Center, Tyndall Air Force Base.] PAGE 83 83 Figure 29. Stress and strain distributions across reinforced concrete section. A) Reinforced concrete section. B) Strain distribution. C) Concrete and steel stress. [Adapted from Chee, K.H, Krauthammer, T., Astarlioglu, S. 2008. Analysis Of Shallow Buried Reinforced Concrete Box Structures Subjected To Air Blast Loads. Rep. No. CIPPSTR0022008 (Page 38, Figure 33) Center for Infrastructure Protection and Physica l Security, University of Florida. Gainesville, Florida.] PAGE 84 84 Figure 210. Unconfined concrete stress st rain curve. [Adapted from Krauthammer, T., Shahriar, S. 1988. A Computationa l Method for Evaluating Modular Prefabricated Structural Element for Rapid Construction of Facilities, Barriers, and Revetments to Resist Modern Convent ional Weapons Effects. Rep. No. ESLTR8760 (Page 9, Figure 2). Engineering & Services Laboratory Air Force Engineering & Services Center, Tyndall Air Force Base.] PAGE 85 85 Figure 211. Confined concrete stress stra in curve. [Adapted fr om Krauthammer, T., Shahriar, S. 1988. A Computationa l Method for Evaluating Modular Prefabricated Structural Element for Rapid Construction of Facilities, Barriers, and Revetments to Resist Modern Convent ional Weapons Effects. Rep. No. ESLTR8760 (Page 11, Figure 3). Engineering & Services Laboratory Air Force Engineering & Services Center, Tyndall Air Force Base.] PAGE 86 86 Figure 212. Steel Stressstrain curve model. [Adapted from Krauthammer, T., Shahriar, S. 1988. A Computationa l Method for Evaluating Modular Prefabricated Structural Element for Rapid Construction of Facilities, Barriers, and Revetments to Resist Modern Convent ional Weapons Effects. Rep. No. ESLTR8760 (Page 11, Figure 3). Engineering & Services Laboratory Air Force Engineering & Services Center, Tyndall Air Force Base.] PAGE 87 87 Figure 213. General model fo r shear influence on beams without web reinforcement. [Adapted from Krauthammer, T. 2008. Modern Protective Structure (Page 229, Figure 526). CRC Press, Boca Raton, Florida.] PAGE 88 88 Figure 214. Shear reduction model. [Adapted from Krauthammer, T., Shahriar, S. 1988. A Computational Method for Evaluati ng Modular Prefabricated Structural Element for Rapid Construction of Faci lities, Barriers, and Revetments to Resist Modern Conventional Weapons Effects. Rep. No. ESLTR8760 (Page 37, Figure 12). Engineering & Services Laboratory Air Force Engineering & Services Center, Tyndall Air Force Base.] PAGE 89 89 Figure 215. Direct shear resistance envelop and reversal loads. [Adapted from Krauthammer, T., Shahriar, S. 1988. A Computational Me thod for Evaluating Modular Prefabricated Struct ural Element for Rapid C onstruction of Facilities, Barriers, and Revetments to Resist Modern Conventional Weapons Effects. Rep. No. ESLTR8760 (Page 126, Figur e 46). Engineering & Services Laboratory Air Force Engineering & Servic es Center, Tyndall Air Force Base.] PAGE 90 90 Figure 216. Influence of axial force on sect ion response. [Reprinted with permission from Tran T.P. 2009. Effect of short duration high im pulse variable axial and transverse loads on reinforced concrete column. M.S. dissertation (Page 91, Figure 57). University of Fl orida, Gainesville, Florida.] PAGE 91 91 Figure 217. Flexural resistance function envelope. PAGE 92 92 Figure 218. PDelta effect on pinned column. PAGE 93 93 Figure 219. SnapThrough buckling. Figure 220. Beam element. PAGE 94 94 Figure 221. Membrane behavio r transition beam/column. PAGE 95 95 Figure 222. General flexural resistance functions including compression membrane and tension membrane behavior. Figure 223. Free body diagram of deform ed slab strip. [Adapted from Park, R., Gamble, W.L. 2000. Reinforced Concrete Slabs. (Page 642, Figure 12.4 ) John Wiley & Sons, Inc. NewYork, NewYork.] PAGE 96 96 Figure 224. Compressive membrane geometric model. [Adapted from Guice, L.K., Slawson, R., Rhomberg, E.J.1989. Me mbrane analysis of flat plate slabs. (Page 88, Figure 5) ACI Structural Journal, Vol 86(1), 8392.] Figure 225. Compressive membrane section. [Adapted from Guice, L.K., Slawson, R., Rhomberg, E.J.1989. Membrane analysis of flat plate slabs. (Page 88, Figure 6) ACI Structural Journal, Vol 86(1), 8392.] CsCc Ts Cs Cc Ts h d d d d c c xx e x + t PAGE 97 97 Figure 226. Compressive membrane externe se ction forces. [Adapted from Guice, L.K., Slawson, R., Rhomberg, E.J.1989. Me mbrane analysis of flat plate slabs. (Page 88, Figure 5) ACI Structural Journal, Vol 86(1), 8392.] Figure 227. Tension memb rane geometric deformation. Figure 228. Cable/truss deformation. Mnm Mns N N w PAGE 98 98 Figure 229. Pure tension membrane modification. Deflection Load Pure Tension Membrane Modified Tension Membrane O PAGE 99 99 Figure 230. Resistance function modifi cation path for stability failure. PAGE 100 100 Figure 231. DSAS combined resistance f unction example with shifted tension membrane. PAGE 101 101 Figure 232. Typical response functions. [Reprinted with permission from Tran T.P. 2009. Effect of short duration high im pulse variable axial and transverse loads on reinforced concrete column. M.S. dissertation (Page 44, Figure 218). University of Florida, Gainesville, Florida.] PAGE 102 102 CHAPTER 3 METHODOLOGY Introduction This chapter presents the enhanced development of RC column analysis implemented into the nonlinear SDOF dynamic analysis procedure in DSAS to account for the different large deformation behav ior phenomenon that ma y occur when a RC column is subjected to blast loads. A structur al overview of the pr oblem is presented to familiarize the reader with the analytical problem addressed in this research. A discussion on the different improvements made to the dynamic analysis solution is also presented. Simplified algorithm s are provided, as well as example calculations from both, previous and improved, analysis methods Finally, the chapter concludes with the overall general analysis algorithm that incorporates all discussed dynamic analysis theory and procedures. Structure Overview The following discussion sets the pr oblem parameters and presents the assumptions made in this study. The structur es of interest are reinforced concrete columns subjected to blast loads. The support conditions for the columns are considered to be fixed at both ends. It is assumed that all longit udinal reinforcements are continuous through the entire span of th e column, and well anchored into the supports such that no pullout of steel rein forcement could occur allowing for tension membrane behavior to be developed. It is al so assumed that once flexural failure occurs, the axial load on the column will be taken by other elements of the structure allowing for tension membrane behavior to occur. Figure 31 illustrates the structural problem that satisfies t he abovestated conditions. PAGE 103 103 When a blast load hits the structure, it will generate a time varying pressure/load that may be characterized by F(t). When c onditions are present for the blast wave to pursue its way into the building through various openings, upward and downward loads, characterized by Pu(t) and Pd(t), may occur on horizont al structural elements transferring their loads to t he reinforced concrete columns. Under such conditions, various phenomena may occur, and need to be addressed during the analysis: The first phenomenon of interest is t he possibility that the RC columns undergo large deformation and transit into a tens ion membrane behavior. This case would only occur if the axial load acting on the columns is taken by other structural elements once the column fails in flexure. The second phenomenon of interest is t he possibility of second order moment to arise, also referred to as PDelta effect. The third phenomenon is the possibility of buckling to occur due to the weakening of the section stiffness as t he column deflects coupling with an axial load that may increase due to the dynamic axial load. Chapter 2 reviewed different structural behavior analysis theories that were used to develop different theoretical models and al gorithm procedures that were incorporated into DSAS to account for the different modes of behavior and are presented in this section. Dynamic Analysis This section lays down the different algorithms and dynamic analysis approaches used to develop the overall improved dyna mic analysis algorithm procedure to address the different phenomena discussed in the last section. Tension Membrane Analysis The dynamic analysis of the reinforced concrete column subjected to blasts loads is conducted using the advanced nonlinear SDOF dynamic analysis software DSAS and uses a variety of the conc epts discussed in Chapter 2. Since the analysis is time PAGE 104 104 dependant and used resistance functions to calc ulate the effective SDOF equivalent stiffness of the column at each time step for various deflection, it is possible to develop a resistance function for the tension memb rane behavior to replace the resistance function of the flexural behavio r once flexural failure occurs and continue the dynamic analysis until tension membrane failure occurs or dynamic behavior stops. Figure 32 illustrates the main steps in the algorit hm develop ed and incorporated into DSAS to create the tension membrane resistance functi on based on the cable theory presented in Chapter 2. Figure 33 and Table 31 set the param eters, as an example, for the pure tension membrane resistance func tion illustrated in Figure 34. The large displacement is mainly due to the fact the maximum strain consi der ed to be possible for the steel is 15% and since the length of the steel re inforcement is long in this example, it allows for a large central deflection before failure occurs. During the dynamic analysis, the flexural resistance function is first generated and the maximum moment capacity of the secti on is obtained from t he moment curvature diagram. It is then possible to apply Equati on 2141 to shift the tension membrane base on the discussion of Chapter 2 to obtai n the resistance function shown in Figure 35. It is important to note here both the flexur al and tension resistance functions have been combined together to show a combinat ion of both. In the dynamic analysis however, once transition occurs into tensi on membrane behavior, there is no going back into a flexural behavior mode as the concrete no longer provides strength to the section. Therefore, the hysteresis loop will be conducted on the tension membrane resistance function only. One may realize the total displacement on the shifted curve is less than PAGE 105 105 the previously generated tension membrane resistance curve. The reason is the maximum allowable stress for the steel cannot increase when the membrane is shifted up and therefore has to fail at an earlier central displacement. The equivalent parameters in the tensi on membrane range must be evaluated to complete the equivalent SDOF equation of motion. To do so, the same approach discussed in Chapter 2 is used. The equival ent mass and equivalent load are computed using the Equation 31, Equation 32, Equation 33 and Equation 34 L edxxxmM0 2)()( (31) Nnodes i i mid i j i j i ed d M M2 (32) L edxxtxwF0)(),( (33) Nnodes i i mid i j i j i ed d fF (34) Since the membrane deflection take the form of a parabola, the load factor and mass factor remain constant throughout the analysis and the so lution yield Equation 35 and 36. Total eMLM 15 8 (35) )( 3 2 xwLFe (36) PDelta Effect To account for PDelta effect, the fini te element beam formulation was simply modified using Equation 279 and Equation 280 to account for the axial effect when PAGE 106 106 computing the nodes rotations. This method formulation may be used for both slender and short columns with and without axial forces. Buckling As discussed earlier, there are two types of buckling that may occur in a reinforced concrete section, the first of which being th e overall buckling of the column and second being buckling of the compressive longitudi nal reinforcement. Th e following discusses both types and how they were handled in the dynamic analysis Column buckling The Euler Buckling theory is introduced in the dynamic analysis as a check allowing the user using the program DSAS to be aware of the possibi lity of such stability failures to occur. When the Euler buckling is included in the dynamic analysis and such failure occurs, transition into tension membrane will suddenly occur. The theory presented in Chapter 2 to calculate the Eule r buckling during a single degree of freedom system was adapted and incorporated into the Software DSAS to generate a function of the critical buckling load versus the central displacement that may be used at each time step to check for buckling. Figure 36 represents t he algorithm that wa s incorporated in DSAS to create such function. The incorporat ion of this algorithm was quite simple as the program used finite beam elements that provide dis placement and rotation at each node. An example of the resultant Eu lerBuckling function is presented in Figure 37 for a 12 ft reinforced concrete column that will be introduce d in more details in the next chapter. Buckling of compressive longitudinal reinforcement In the case of local buckling, the steel stress strain curve was modified in the compressive range to have failure occur onc e the Euler Buckling stress was reached in PAGE 107 107 between rebars. This modifi ed the moment curvature di agram of the section and therefore modified the flexural resistance function of the section. Since the resistance function is generated by means of finite elements, a localized failure in a rebar element does not imply total failure of the struct ural element and therefore the analysis is continued until overall failure of the section occurs. Figure 38 shows flexural resistance functions examples of a given sec tion with and without compressive longitudinal reinforcement buckling. Flexural Failure and Transi tion to Tension Membrane Different conditions were incorporated in to the dynamic analysis program for the transition into tension membrane. Thos e conditions are based on geometric behavior and theoretical models. Column buckling transition to tension membrane behavior Different scenarios may trigger a stability fa ilure as the column central deflection increase such as an increase in axial load, a diminution of the section effective stiffness under a given axial load or a combination of both. During the dynam ic analysis, if the axial load increases, the flexural resistance function is regenerated to account for the stiffness changes due to the axial load. Figure 39 is an example of two different flexural resistance functions for two different axial loads. These resistance functions also include the effect of local buckling as discuss ed previously and the shear reduction factor. An increase in the axia l load results into push of t he flexural resistance function to the left allowing less deflection capac ity and transverse load carrying capacity. Figure 310 is the resistance function fo r the case of 800 kip axial load as s hown previously but it includes the tension membrane resistance curve. Note this section is heavily reinforced with # 11 rebar having an ultimate strain of 15% PAGE 108 108 As one may observed by looking at Figure 310, if a stability failure is to occur and the column is to transit into tension me mbrane behavior, the only capacity the section would have is the steel carrying capacity. Ther efore, as the sectio n loses its flexural strength capacity, the resist ance functions suddenly falls onto the tension membrane resistance function and continues to deflect following the tension membrane resistance function path. Figure 311 represents the general case for such transition. Flexural fa ilure transition into tension membrane The most obvious transition into tension membrane is due to flexural failure. The program DSAS already takes into account the di fferent modes of flexural failure such as material failure and diagonal shear failure. Th e only concern here is the transition into tension membrane behavior. As one may have observed from Figure 310, transition into tension membrane shall occur at the end of the flexural resistance curve, w hich corresponds to the deflection where the concrete no longer provides resistance to the section. Therefore, as the flexural resistance suddenly drops to zero, the tension membrane resistance function curve is in tercepted and the program continues the dynamic analysis using the tension memb rane resistance curve. To capture the interception point, a simple algorithm wa s programmed that si mply compares both curves. Transition into tension memb rane due to excessive deformation For any given element acting as a beam, column or slab, if the c entral deflection is greater than the section depth, the entire section will then fall into tension. This type of behavior, however, may not be fully captured by the current finite element procedure as it is using beam element. Figure 312 is an example of DSAS V3.0 limitations as the resistance function goes to about twice the section depth which, for this case, was 16 PAGE 109 109 inches. Note that this resistance function do es not account for local buckling and has no axial load on it. Obviously, a column carrying no axial load does not exist in reality. This is more or less an extreme case. However, to ensur e transition occurs due to the geometric limitations of the section, a simple condition was introduced in the algorithm that simply transitions into tension membrane if the central deflection becom es greater than the section depth. DSAS Overall Dynamic Analysis Algorithm Figure 313, Figure 314, and Figure 315, are the simplified program flowchart of the improved DSAS procedure for the analysis of reinforced concrete columns subjected to blast loading. Summary The dynamic analysis procedure and algorithms for the analysis of reinforced concrete columns subjected to blast loads were described in this Chapter. First, the structural components of interest were discussed along with the assumptions used for the problem analysis. Various improvement s were discussed to capture different possible behaviors reinforced concrete colu mns with fully fixedfixed supports could undertake. Those improvements of the currently used dynamic analysis procedure make for a more complete algorithm/program that provides a strong analysis capacity of reinforced concrete columns undergoing large deformation. In the following chapter, the developed program will be validated using finite element software and a past study of reinforced concrete behavior subjected to blast loads. PAGE 110 110 Figure 31. Reinforced concrete column model. PAGE 111 111 Figure 32. Tension membrane algorithm. 33. RC column tension memb rane calculation example. PAGE 112 112 Figure 34. Pure tension membrane re sistance function calculation example. PAGE 113 113 Figure 35. DSAS combined resistance function example. PAGE 114 114 Figure 36. Buckling check algorithm. PAGE 115 115 Figure 37. Eulerbuckling function obt ained from DSAS DPLOT linked software. PAGE 116 116 Figure 38. Flexural resistance functions only. PAGE 117 117 Figure 39. Resistance functions for di fferent axial load on same column. PAGE 118 118 Figure 310. Combined resistance functi on for constant 800 kips axial load. PAGE 119 119 Figure 311. Resistance function transit ion into tensile membrane behavior. Deflection (U) Resistance (W) Pure tension membrane Modified tension membrane calculation O Stability failure PAGE 120 120 Figure 312. Flexural resistance functi on of RC column with no axial load. PAGE 121 121 Figure 313. DSAS simplified column analysis flowchart chart 1. PAGE 122 122 Figure 314. DSAS simplified column analysis flowchart chart 2. PAGE 123 123 Figure 315. DSAS simplified column analysis flowchart chart 3. PAGE 124 124 Table 31. Reinforced concrete column parameters. Dimension Length (ft) Height (in) Width (in) 26 16 16 Reinforcement Number of bars Designation Depth (in) Area (in2) 2 US #6 2.5 0.44 2 US #6 8 0.44 2 US #6 13.5 0.44 Materials properties f'c (psi) 4000 f y (psi) 60000 y 0.002 fu (psi) 75000 u 0.15 ff (psi) 75000 f 0.15 PAGE 125 125 CHAPTER 4 ANALYSIS Introduction The objectives validation of the research and the confirmation of the methodology presented in Chapter 3 were conducted using the finite element software ABAQUS/Explicit V6.8 (Dassault Systemes, 2008). The first step was to validate the material model using experimental data obtained from Feldman and Siess (1958) experiment on reinforced concrete beams subjected to impact loads. The second step was to apply the material model to both DS AS V3.0 and ABAQUS V6.8 and finally use the developed model to validat e the tension membrane analys is algorithm by analyzing a reinforced concrete column under different loading conditions. Material Model Validation The reinforced concrete model used for re inforced concrete was developed using experimental data from Feldman and Sie ss (1958) on reinforced concrete beams subjected to impact load. For ABAQUS V6.8, the concrete damaged plasticity (CDP) model was used for the concrete material as it is suitabl e for concrete subjected to dynamic and cyclic loading such as blast l oading. The following citation was directly taken from ABAQUS V6.8 theory manual to describe the model. The concrete damaged plasticity model is primarily intended to provide a general capability for the analysis of c oncrete structures under cyclic and/or dynamic loading. The model is also suitable for the analysis of other quasibrittle materials, such as rock, mortar and cerami cs. Under low confining pressures, concrete behaves in a brittle manner; the main failure mechanisms are cracking in tension and crushing in compression. The brittle behavior of concrete disapp ears when the confining pressure is sufficiently large to prevent crack propagation. In these circumstances failure is driven by the consoli dation and collapse of the concrete microporous microstructure, leading to a macroscopic response that resembles that of a ductile material with work hardening. (ABAQUS V6.8 User Theory Manual 2008) PAGE 126 126 The model parameters used for analysis we re derived by Tan Loo Yong at CIPPS as part of his M.S. thesis titled Characte rizing a Reinforced Concrete Connection for Progressive Collapse Assessment, to be pub lished in May 2010. He validated his concrete model by analyzing the different beams from the expe riment conducted by Feldman and Siess (1958). Dynamic increase fa ctors for both concre te and steel were applied on the materials stressstrain curve to properly capture the dynamic effect onto the material behavior. Table 41 presents typical values used for such factors. Table 42 presents the CDP parameter s developed to m odel beam C from Feldman and Siess (1958) expe riment. The steel stress st rain material model was based on the experiment data but was trans formed into true stress and logarithmic strain as required by ABAQUS V6.8 explicit. Table 43 presents the different steel material parameter s for Beam C. Figure 41 shows the develo ped model in ABAQUS V6.8 The element types used were cubic 8 node reduced integr ation with hour glass control (C3D8R) for the concrete element and 2D Timoshenko bea m element for the steel elements (B31). Finally, gravity loads were applied along with a point load to simulate the impact load located at the center of the beam hav ing the load time history presented in Figure 42. Figure 43 illustrates the displacement ti me history of Beam C exp eriment data versus ABAQUS V6.8 results. The maxi mum displacement obtained from ABAQUS V6.8 is 3.03 inches versus 3.01 inches from the experiment, which represents a difference of 0.66%. The results obtained from ABAQUS V6.8 proved to be representative of the experiment for the peak displacement va lue. However, the results obtained from PAGE 127 127 ABAQUS V6.8 past the peak displacement va ries from the experiment as ABAQUS V6.8 lacks the ability to properly capture the hysteresis behavior of reinforced concrete models in free vibration mode. This limitation has been reported in previous studies conducted using the ABAQUS software for impact and blast loads and will be disregarded for the remainder of the validation. However, since structural damages are related to the peak support rotation, one can compare the peak displacements. When pushing the Timoshenko beam element into large deformation under higher load, it was found that t he Timoshenko formulation was unable to handle large axial deformation. A Timoshenko hybrid element fo rmulation to handle such case is available in ABAQUS/Standard V6.8 but unfortunately not in ABAQUS/Explicit V6.8. Therefore, the Timoshenko beam elements were replac ed by truss elements and the material properties for the steel were modified to refl ect proper dynamic increase factor for truss elements. Figure 44 shows the results obtained for Beam C using truss elements compared with Timos henko beam elements, D SAS V3.0 and the experiment results. The maximum displacement obtained using truss elements is 3.0 and represents a difference of 0.33% compared to experimental data. The same beam was then subjected to a triangular blast load having a peak reflected pressure of 1181 psi and load duration of 0.00132375 sec. The results are shown in Figure 45 for DSAS V3.0 and ABAQUS V6.8. The peak displacement value obtained from ABAQUS V6.8 is 1.72 inches vers us 1.66 inches obtained from DSAS 3.0 for a difference of 3.33%. The same ex ercise was conducted with an increased triangular load function having a peak reflect ed pressure of 1777 psi and a load duration of 1.28 msec. The results are shown in Figure 46. The results obtained for this case PAGE 128 128 demonstrated a difference of 1.60% for a ma ximum displacement of 3.36 inches for DSAS V3.0 and 3.41 inches for ABAQUS V6.8. The material model developed for ABAQUS V6.8 and DSAS V3.0 based on the Feldman and Siess (1958) experi ment proved to yield accura te results for beam cases. Column Model Validation The model was further validated against a reinforced concrete column having no axial load subjected to various combinations of triangular blast load. The column design was chosen based on previous research conduc ted by Thien Tran (2008) on reinforced columns subjected to blast. The column cross section was 16X16 inches and the column free span length was 12 ft. 8 # 11 longi tudinal rebar was placed at 1.5 inches, 8 inches and 14.5 inches deep leaving 1.5 inc hes of cover all around. The longitudinal reinforcements were closed with stirrups sp aced at 12 inches. The column continued six inches at each end to properly model boundar y conditions. A steel plate was emplaced at the top and bottom of the column to properly distribute the axial point load onto the surface of the column. The supports were fi xed in all directions and the longitudinal reinforcement ends were also fixed in the ve rtical direction to ensure full development capacity of the reinforcement. Figure 47 shows a general view of ABAQUS V6.8 model and Table 44 shows the peak dis placement results obtained for all different loading cases from DSAS V3.0 and includes the perc ent difference from ABQUS V6.8 for both beam and truss elements models. Overall, the average percent difference is 13.37% for truss element cases versus 17.92% from Timoshenko beam element case s. However, when ignoring the smaller load case, the average percentage diffe rence for truss and beam elements is respectively 6.26% and 17.89%. The truss element model proved to yield more PAGE 129 129 accurate results than the beam element model and also proved to be capable of handling large deformation, and as a results, higher loading cases. Modified DSAS Validation To validate the modification done to the software DSAS, the previously discussed cases were tested using the modified version of DSAS to ensure the program would yield the same results as DSAS V3.0 for ca ses where modification done to improved the program were not expected to be engaged during an analysis. Table 45 represents the results obtained from both programs. Under low impulsive load, the results are exactly the same as expected as neither compressive longitudinal reinforcement buckling nor tension membrane behav ior influence the response. However, under the hi gher impulsive load of 1875 psimsec, the tension membrane was engaged because the comp ression longitudinal reinforcement buckling was not taken into a ccount, yielding a higher tensi on membrane shift due to a higher nominal moment on the moment curv ature diagram. Finally, as the impulsive load was increased, direct shear failure occurred. Although tension membrane is captured, it has no significance as the column has already failed. This will not be the case under the presence of axial load since t he axial load increases the direct shear capacity of the section while reducing the ma ximum lateral deflection in flexure. This case was more or less a beam case with the configuration of a column used for validation purposes only. Figure 48 is an example of flexural time history analysis to compare the modi fied DSAS 3.0 response versus ABAQUS 6.8 using truss element. PAGE 130 130 Both programs yield very close peak disp lacements; however, one may see the peak displacements occurring at different times. This is mainly due to the fact both programs used different approaches to calculat e the stiffness of the column during the dynamic analysis which yielded different response perio ds. However, both analyses yield acceptable results from a design stand point as the main interests are the peak displacement and reactions. Tension Membrane Validation To validate the modified version of DSAS in tension membrane range, the previous column was subjected to several seri es of blast loads while being subjected to a constant axial load. For each series const ant axial loads were chosen to be located below balance point, at balance point and a bove balance point based on the interaction diagram of the column presented in Figure 49. Two analytical steps were used in an attempt to capture the effect of the axial load on the column behavior during the ABAQUS 6.8 analysis. The first step consisted of capturing the displacement time history of the support under only axial loading. The second step was to reproduce the effect of t he axial load on the column by imposing a support displacement time history as of step one. This allowed t he stress due to the axial load to develop into the column prior of being subjected to a blast load and also respects the assumption that the supports ar e fixed when the column is subjected to transverse load and undergoing tension membrane behavior. The first axial load imposed on the first seri es of lateral loads is 500 kip. Because the analysis is conducted using ABAQUS Exp lic it, the axial load was gradually applied from 0 to 0.1 sec in order to reduce the anal ysis time for the support to come to rest. It was found that a total time of 0.15 sec was suffi cient to properly capt ure the effect of the PAGE 131 131 axial load on the column. Figure 410 represents the support displacement time history for the three different axial l oads imposed on the column. The results obtained from ABAQUS V6.8 an d DSAS for a reinforced concrete column under 500 kips axial load subjected to different loading cases are shown in Table 46. The results obtained from the modified progr am DSAS for the case of 500 kips axial load demonstrated t he column would not fail in fl exure and would not undergo tension membrane behavior under an impulsive load of 1134 and 1354 psimsec. For the case of 1875 psimsec, the modified DSAS program demonstr ated the column would undergo flexural failure followed by tension membrane behavior. ABAQUS V6.8 however demonstrated all cases would undergo tension membrane behavior based on the strain in the compression longitudinal reinforcement at midspan going from compression to tension. The reason is t hat the effect of the axial load quickly disappears as the column deflects under a transverse load due to the boundary conditions remaining fixed. Under real conditions, the data obtained for the 500 Kips axial loads would have been closer to the mo dified DSAS results as axial load taken below balanced point increase the flexural capac ity of the column. It is interesting to note DSAS V3.0 analysis of the impulse case of 1875 psimsec did not yield a flexural failure. However, when including the local bu ckling effect and ignoring the possibility of tension membrane, the modified version of DSAS yield a flexural failure. This is demonstrated in Figure 411 which represents t he flexural resistance function of both cases. In the case where longitudinal rein forcement buckling is included, the maximum permissible displacement is 3.99 inches and therefore demonstrated the column would fail in flexure and undergo tension membrane behavior. PAGE 132 132 In the case of an axial load of 800 kip and 1500 kip, Table 47 and Table 48, all cases demonstrated flexural failure and t ension membrane behavior in the modified DSAS program. ABAQUS V6.8, under the pr escribed analysis condition, also demonstrated tension membrane behavior for a ll cases. However, these results are somewhat biased, as discussed earlier, by the fact the axial l oad does not remain constant up to flexural failure during the analysis onto the longitudinal reinforcement. Under higher lateral loads, t he effect of the ax ial load becomes less important and this may be observed by looking at the percent diffe rence from all the prev ious cases, as the central deflection increases, the percent difference from both progr ams improves. This point is further enforced when compari ng the results obtained from ABAQUS V6.8 versus Modified DSAS where no axial load is present as demonstrated in Table 49 where the percent difference for all cases are within 15% and mo stly within 1 0%. The last discussion only proved that the approach used in ABAQUS V6.8 lacks the capacity to properly capture the effect of the axial load up to flexural failure but this limitation could be overcome by increasing the axial load. Therefore, the column was subjected to higher lateral load where the ax ial load is expected to have less effect on the overall analysis and the results are pr esented in the Table 410 and Table 411 for the case of 800 kips and 1500 kips. The 500 kips cases are ignored as the column would failed in direct shear failure prior to reaching those central deflection levels. The results obtained from the last analysis proved to be much more conclusive to evaluate the tension membrane behavior. In the case of 800 kips axial load, although the percent difference was in the range of 16%, the result s proved to be consistent under different load cases. For the case of 1500 kips axial load, the percent difference PAGE 133 133 varies but are overall is within 10%. T he suspected reason for percent difference variation in the case of 1500 kip is the need to better understand the sh ifting condition of the tension membrane resistance function. The results obtained are overall remarkable when considering one program us es over fifty thousand elements while the other only uses a few and also considering the complexity of ABAQUS versus DSAS. Direct shear failure was observed for the 800 kips axial load during the modified DSAS program run. Even though DSAS detects a direct shear failure, it performs a full flexural analysis. The results obtained may still be used for comparison. It was however not the intent to prove DSAS capability of handling direct shear failure and no further discussion on the subject will be made as part of the results analysis section of this dissertation. One thing to notes however, an increase in axial load increases the direct shear capacity of a section which explains why direct shear failure was not detected during the 1500 kips axial load analysis under the same loading conditions as the 800 kips axial load. Figure 412 shows the deflected shape of t he columns for the last reported cases. The maxim um principal stress are reported but are not representative of the maximum stress in the steel rebar as truss elem ents are being used which may only report S11 (axial) stress. In this case, the maximum prin cipal stresses are loca ted at the interface of the reinforcement and the steel plate on top of the column is used to uniformly distribute the axial stress due to axial load. However, Figure 413 shows the axial stress time history of steel rebar at midspan. It demonstrates compression reinforcement goes from co mpression stress to tension stress an d the maximum tensile stresses are below ultimate capacity. The transition into t ension membrane behavior of the compressive PAGE 134 134 reinforcement occurs at 0.1535 sec of the analysis corresponding to a displacement of 2.82 in. This represents a transition displace ment of 17.63% of t he section depth (h) which is comparable to transition point of deep beam as expected based on Krauthammer et al. (2003) where 17% was observed in deep beam. Finally, Figure 414 shows the minimum principal stress where t he concrete is shown to be crushed and in tension. To conclude on the validation of the 12ft column, Figure 415, Figure 416 and Figure 417 are example of various time re sponses of central displacement obtained from both program for different ca ses. Slender Column To confirm the validation of the SDOF al gorithm ability to handle large deformation behavior into a reinforced concrete column with a fixed boundary condition, the previous column was expended 12 feet, ma king it fall into a slender column classification. All other properties remained the same. The colu mn was subjected to an axial load of 1200 kips and the same approach discussed earlier was used in ABAQUS V6.8 to mimic the effect of the axial load. Table 412 are the results obtained from both programs. The results obtained are in good agreemen t. Under higher load, the percent difference increases a little above 10%. Howeve r, after increasing the load until direct shear failure occurs, the percent difference is 14.6% which is an acceptable range. Figure 418 and Figure 419 shows the deformed s hape of the column and the S11 (axial) stress at th e maximum displacem ent step for the last two cases analyzed. It demonstrates both cases have remaining capac ity in the steel to develop tension membrane behavior. PAGE 135 135 Summary This chapter presented the different steps and approaches used to validate the modification made to the dynamic analysi s procedure embedded into the software DSAS for the analysis of reinforced concre te columns having fixed boundary conditions and the capacity to undergo tension membrane behavior under large deformation. First, a material model was developed for both DSAS V3.1 and ABAQUS V6.8 based on past experimental data on reinforced concrete beams subjected to impact load. The material model was then used to model a reinforced concrete column in ABAQUS V6.8 and DSAS 3.1 and again tested and validated under se vere short dynamic transverse load. Finally, the modification made to the pr ogram to account for large deformation and tension membrane behavior were validated. The results obtained were conclusive and both ABAQUS and the modified DSAS yielded results within 15% difference which at this point may be considered acceptable. However, it is im portant to understand limitations and assumptions made for the validation and will be discussed in the next chapter as part of the discussion and recommendations. PAGE 136 136 Figure 41. Beam C meshing. PAGE 137 137 Figure 42. Beam C load time history. PAGE 138 138 Figure 43. Displacement time hi story beam C ABAQUS vs. experiment. PAGE 139 139 Figure 44. Beam C experiment time history comparison. PAGE 140 140 Figure 45. DSAS V3.0 vs. ABAQUS V6.8 analysis with triangular blast load Pr = 1183 psi and t = 1.32 msec. PAGE 141 141 Figure 46. DSAS V3.0 vs ABAQUS V6.8 ana lysis with triangular blast load Pr = 1777 psi and t = 1.28 msec. PAGE 142 142 Figure 47. Reinforced concrete column model in ABAQUS V6.8. A) General model. B) Steel reinforcement cage. PAGE 143 143 Figure 48. Flexural time history analysis comparison modified DSAS vs. ABAQUS V6.8 for triangular blast l oad Pr=1777 psi and t = 0.00127630. PAGE 144 144 Figure 49. Interaction diagram for 12 ft long column model obtained with Modified DSAS DPLOT linked software. PAGE 145 145 Figure 410. Support displacement time history for different constant axial load cases. PAGE 146 146 Figure 411. Flexural resistance function com parison of local buckling with a constant axial load of 500 kips. PAGE 147 147 Figure 412. Principal maximum stress, case 1500 kip axial load, Pr=5262, t = 0.00142721. PAGE 148 148 Figure 413. Steel reinforcement axial stress at midspan, case 1500 kip axial load, Pr = 5262 psi, t = 0.00142721 sec. PAGE 149 149 Figure 414. Principal mini mum stress, case 1500 kip ax ial load, Pr=5262 psi, t = 0.00142721 sec. PAGE 150 150 Figure 415. Flexural time hi story example case with 800 kips constant axial load, triangular blast load with Pr = 2922 psi and t = 1.28 msec. PAGE 151 151 Figure 416. Flexural time hi story example case with 1500 kips constant axial load, triangular blast load with Pr = 2922 psi and t = 1.28 msec. PAGE 152 152 Figure 417. Flexural time hi story example case with 1500 kips constant axial load, triangular blast load with Pr = 4757 psi and t = 1.39 msec. PAGE 153 153 Figure 418. TwentyFour foot column maxi mum general principal stress results case: Pr = 2922 psi, t = 0.00128337 sec. PAGE 154 154 Figure 419. TwentyFour foot column maxi mum general principal stress results case: Pr = 3400 psi, t = 0.00128337 sec, dire ct shear failure reported by DSAS. PAGE 155 155 Table 41. Reinforced concrete design dynamic increase factors. Type of Stress Far design range Close design range Steel reinforcement Concrete Steel reinforcement Concrete fdy/fy fdu/fu f'dc/f'c fdy/fy fdu/fu f'dc/f'c Bending 1.17 1.05 1.19 1.230 1.050 1.250 Diagonal tension 1.00 1.00 1.100 1.000 1.000 Direct shear 1.10 1.00 1.10 1.100 1.000 1.100 Bond 1.17 1.05 1.00 1.230 1.050 1.000 Compression 1.10 1.12 1.130 1.100 Adapted from Krauthammer, T. Hinman, E. Rittenhouse, T. Structural syst emBehavior and design philosophy.(Page 324, Figure 3.1) Structural Design for Physical SecurityState of the practice rep, ASCE, Reston, Virginia. Table 42. Concrete damaged pl asticity model parameters. Mass density 2.21E07 Lbf*s 2 /in Young modulus 4816 ksi Poisson's ratio 0.2 Concrete damaged plasticity parameters Dilatation angle Eccentricity fb0/fc0 k 30 0.1 1.16 0.666 Compressive behavior Yield stress (ksi) Inelastic strain Damage parameter Inelastic strain 5.355 0 0 0 6.24636 0.0007031 0 0.000703 6.426 0.0010674 0 0.001067 5.88124 0.0018797 0.084774 0.00188 4.24697 0.0029182 0.339097 0.002918 Tensile behavior Yield stress (ksi) Cracking strain Damage parameter Cracking strain 0.872 0 0 0 0.437 0.0009054 0.499427 0.000905 0.001 0.0018108 0.99 0.001811 PAGE 156 156 Table 43. Steel properties of beam C for ABAQUS, US unit with stress in ksi. Steel general properties Mass density 7.3386E07 Lbf*s2/in Young's modulus 29520 ksi Poisson ration 0.3 Tension steel Increased stress Nominal strain True stress (ksi) True strain Plastic strain 51.414 0.00174 51.504 0.00174 0 75 0.15000 86.250 0.13976 0.137 Compression steel Increased stress Nominal strain True stress (ksi) True strain Plastic strain 52.106 0.00177 52.198 0.00176 0 75 0.15000 86.250 0.13976 0.137 Stirrups Increased stress Nominal strain True stress (ksi) True strain Plastic strain 47.129 0.00160 47.204 0.00160 0 75 0.15000 86.250 0.13976 0.137 Table 44. Reinforced concrete co lumn under no axial load results. Triangular load parameters DSAS V3.0 max disp. (in) FEA results ABAQUS V6.8 % difference Reflected pressure (psi) Incident reflected impulse (psimsec) Time duration (msec) No SRF With SRF Using truss element average midspan disp. (in) Using beam element Average midspan disp. (in) DSAS VS truss element DSAS VS beam element 1183 783 1.324 0.60 0.99 0.68 0.84 44.3 18.0 1777 1134 1.276 1.08 1.77 1.76 1.61 0.5 9.6 2134 1354 1.269 1.47 2.39 2.51 2.27 4.9 5.3 2922 1875 1.283 2.57 4.18 4.71 5.02 11.2 16.6 3602 2369 1.315 3.85* 6.25* 6.82 10.44 8.4 40.1 *Direct shear failure. PAGE 157 157 Table 45. Reinforced concrete column under no axial load. Triangular load parameters DSAS V3.0 max disp. (in) Modified DSAS Including tension membrane behavior max disp. (in) FEA Results ABAQUS V6.8 midspan disp. (in) Reflecte d pressure (psi) Incident reflecte d impulse (psimsec) Time duration (msec) No SRF With SRF With SRF & no rebar buckling With SRF & rebar buckling Using truss element average Using beam element Average 1183 783 1.324 0.60 0.99 0.99 0.99 0.68 0.84 1777 1134 1.276 1.08 1.77 1.77 1.77 1.76 1.61 2134 1354 1.269 1.47 2.39 2.39 2.39 2.51 2.27 2922 1875 1.283 2.57 4.18 4.191 4.18 4.71 5.02 3602 2369 1.315 3.852 6.252 6.182 6.27 6.82 10.44 1Tension membrane engaged, 2Direct shear failure. Table 46. Twelve foot reinforced conc rete column under 500 kip axial load. Triangular load parameter s DSAS results FEA results Results comparison Reflected pressure (psi) Incident reflected impulse (psimsec) Time duration (msec) DSAS V3.0 max disp. (in) Modified DSAS V 3.0 max disp. (in) ABAQUS V6.8 average midspan disp. (in) % difference 1777 1134 1.276 1.41 1.41 1.72 18.25 2134 1354 1.269 2.05 2.05 2.52 18.87 2922 1875 1.283 4.05 4.031 4.69 14.05 1Tension membrane engaged. Table 47. Twelve foot reinforced conc rete column under 800 kip axial load. Triangular load parameter s DSAS results FEA results Results comparison Reflected pressure (psi) Incident reflected impulse (psimsec) Time duration (msec) DSAS V3.0 max disp. (in) Modified DSAS V 3.0 max disp. (in) ABAQUS V6.8 average midspan disp. (in) % difference 1777 1134 1.276 1.333 1.551 1.90 18.81 2134 1354 1.269 1.333 2.351 2.56 8.23 2922 1875 1.283 1.333 4.221 4.68 9.71 1Tension membrane engaged, 2Direct shear failure, 3Flexural failure. PAGE 158 158 Table 48. Twelve foot reinforced c oncrete column under 1500 kip axial load. Triangular load parameter s DSAS results FEA results Results comparison Reflected pressure (psi) Incident reflected impulse (psimsec) Time duration (msec) DSAS V3.0 max disp. (in) Modified DSAS V 3.0 max disp. (in) ABAQUS V6.8 average midspan disp. (in) % difference 1777 1134 1.276 0.783 2.501 2.04 22.08 2134 1354 1.269 0.783 3.341 2.55 30.84 2922 1875 1.283 0.783 5.131 4.86 5.39 1Tension membrane engaged, 2Direct shear failure, 3Flexural failure. Table 49. Comparison of ABAQUS resu lts with DSAS V3.0 without axial load. Reflected pressure (psi) ABAQUS V6.8 with 500 kip axial load max disp. (in) ABAQUS V6.8 with 800 kip axial load max disp. (in) ABAQUS V6.8 with 1500 kip axial load max disp. (in) Modified DSAS V 3.0 (with SRF & rebar buckling) no axial load % difference 500 kip case % difference 800 kip case % difference 1500 kip case 1777 1.72 1.90 2.04 1.77 2.37 7.30 13.67 2134 2.52 2.56 2.55 2.39 5.25 6.47 6.41 2922 4.69 4.68 4.86 4.18 10.76 10.60 14.00 Table 410. Twelve foot reinforced conc rete column under 800 kip axial load con't. Triangular load parameter s DSAS results FEA results Results comparison Reflected pressure (psi) Incident reflected impulse (psimsec) Time duration (msec) DSAS V3.0 max disp. (in) Modified DSAS V 3.0 max disp. (in) ABAQUS V6.8 average midspan disp. (in) % difference 4208 2844 1.352 1.323 7.591 8.98 15.45 4757 3305 1.390 1.323 9.451 11.36 16.75 5262 3755 1.427 1.323 11.312 13.58 16.70 1Tension membrane engaged, 2Direct shear failure, 3Flexural failure. PAGE 159 159 Table 411. Twelve foot reinforced concre te column under 1500 ki p axial load con't. Triangular load parameter s DSAS results FEA results Results comparison Reflected pressure (psi) Incident reflected impulse (psimsec) Time duration (msec) DSAS V3.0 max disp. (in) Modified DSAS V 3.0 max disp. (in) ABAQUS V6.8 average midspan disp. (in) % difference 4208 2844 1.352 0.773 8.601 9.10 5.51 4757 3305 1.390 0.783 10.481 10.33 1.47 5262 3755 1.427 0.783 12.261 13.48 9.03 1Tension membrane engaged, 2Direct shear failure, 3Flexural failure. Table 412. Twentyfour foot rein forced concrete column results. Triangular load parameter s DSAS results FEA results Results comparison Reflected pressure (psi) Incident reflected impulse (psimsec) Time duration (msec) DSAS V3.0 max disp. (in) Modified DSAS V 3.0 max disp. (in) ABAQUS V6.8 Average midspan disp. (in) % difference 1777 1134 1.276 0.533 6.441 6.74 4.44 2134 1354 1.269 0.523 8.231 9.02 8.80 2922 1875 1.283 0.493 12.731 14.76 13.75 3400 2182 1.284 0.493 15.562 18.22 14.59 1Tension membrane engaged, 2Direct shear failure, 3Flexural failure. PAGE 160 160 CHAPTER 5 DISCUSSION AND RECOMMENDATIONS Introduction In this chapter, the observations made as part of the validation of the tension membrane are further discussed. First, a di scussion on the limitat ion of the analysis procedure based on results obtained is conduct ed. Second, a parametric study of the modified DSAS program is carried out to i dentify key issues that may help someone design a RC column capable of undergoing tension membrane behavior. Finally, recommendations are made for future development. Limitations Various limitations were encountered duri ng the validation proc edures that were only discussed briefly in the previous chapter ; more specifically these limitations are regarding the effect of axial load on the colu mn. The first limitation discussed here is the lack of ability to properly capture the effect of axial load on the overall column during the dynamic analysis in the software ABAQUS V6.8. The main purpose of columns is obviously to carry large axial forces and as such, they are usually not designed to sustain large lateral loads. On the other hand, when buildings are designed to be able to sustain the loss of a column, the adjacent components are not designed to take over the axial load on the lost column before flexural fa ilure occurs. The realit y is, if the structure is to overtake the axial load onto the column prior to flexural failure, no one can predict accurately the exact time and the exact vertic al support displacement at flexural failure using a finite element code or any other anal ysis tool available today to conduct such analysis. Furthermore, the axial load time hi story on the column going from fully loaded to zero would also be impossible to predict and yet very important to be taken into PAGE 161 161 account. The fact is, for any given design able to sustain the loss of a column, flexural failure would occur before significant axial lo ad variation would have an effect on the column response and the problem needs to be treated as such. This was however the limitation encountered with t he software ABAQUS V6.8 where it was not possible to model both a constant axial load onto the column and going into tension membrane at flexural failure. An induced support disp lacement approach was then used in an attempt to mimic the effect of a constant axial load onto the column but was very limited as the effect of the axial load was quickly dissipated as lateral deflection occurred. It was however found that axial load effect becam e less important as the transverse load increased and the model was then validated under higher load. In the same matter, DSAS also lacks the ab ility to properly capture the effect of support displacement during an analysis. It assumes fixed boundary conditions at the support and the axial effect if taken into account when DSAS numerically developed the moment curvature relationship prior to c onducting the dynamic analysis. One may argue a validation approach using ABAQUS V6.8 would have been to first conduct an analysis of the column under both axial and transverse load, capture t he time displacement time history of the support until flexural failure o ccur and mimic the effect of the axial load by imposing the same displacement time hist ory at the support. Th is would have been ideally the best route to take. However, because DSAS does not have the ability to capture the effect of support displacement for the analysis of column, this approach was not preferable as the results from ABAQ US would have yielded much higher central displacement and there would have been many difficulties capturing the exact time of flexural failure. Note that in this case, the displacement time history of the support due PAGE 162 162 to both axial and transverse load would hav e been much higher than the displacement time history due only to axial load which was not significant enough to be taken into account during the validation in Chapter 4. So not only was the validation limited by the uses of ABAQUS, it was also limited by the capacity of DSAS. Nonetheless, DSAS proved to have the capability to yield very good results under the conditions it was design for and in the case of a significant support displacement, DSAS had the ability to predict within an acceptable range if the column would actually undergo tension membrane or fail in direct shear failure. When conducting a dynamic analysis, one mu st take into account the physical limitation of the overall stru cture and not only the capacit y of the column to undergo tension membrane behavior. If a column is subjected to an explosion strong enough to bring the column into a tension membr ane state, one must c onsider not only the damage done to the column but also to t he area surrounding the column. For example, while the concrete of the column is likely to be crushed into pieces, concrete within the steel reinforcement may remain trapped inside the steel cage or not based on the stirrup spacing and overall reinforcement conf iguration. This will have an effect on the central displacement of the column as the loss of mass leads to the loss of inertia force. This problem may be overcome by minimizing the stirrup spacing or through the use of FRP to wrap the column. Another important factor is the capacity of the connection to resist blast load. One must considered the damage that may occur at the fixed connection itself and how it may affect t he column capacity to develop tension membrane behavior. For example, the longit udinal rebar nearest to the blast may be completely cut due to fragments while the r ebar farthest from the blast load remains PAGE 163 163 intact. In that case te nsion membrane could develo ped but with less resistance capacity. Finally, the SDOF approach itself has it s own assumptions and limitations that need to be considered. One of these is t he assumed shape function of the tension membrane to take the form of a parabola. Th is shape function represents the ideal case where once the column undergoes tension me mbrane, the concrete does not provide movement restriction to the st eel rebar. However, this may not be the case as concrete is likely to remain caged in the rebar and change the shape of the deflection. Also, the failure criteria of the steel is uniaxial, and therefore does not consider the stress from lateral load on the steel. On the other end, once concrete is crushed, it is impossible to predict the exact lateral stress on the rebar. However, as seen in the validation and further discussed in the next section, direct shear failure would occur before full tension membrane capacity may be developed. Overall, the improvements made to the SDOF procedure as part of this thesis provide an excellent tool for analysis within certain limitati ons. Those limitations could be better addressed by conducting experiments and will be discussed with more depth in a later section. Nonetheless, good agr eement was obtained from both programs during the validation which per mits a further investigati on of the benefit tension membrane behavior may bring to a column under blast loads. Therefore, the improved software is put to work in the next section to identify key benefits and better understand the tension membrane behavioral mode wit h different columns configurations. Parametric study The following section conducts paramet ric studies to better understand the behavioral mode of reinforced concrete columns undergoing large deformation. First a PAGE 164 164 parametric study to better underst and the effect of direct s hear failure in the overall analysis is conducted for the 12ft column used in the previous section having different longitudinal rebar sizes. Second, pressurei mpulse diagram of the 12 ft columns are presented to demonstrate the benefit of tension membrane capabilities. Column Parametric Study The first key issue for a column to be able to undergo tension membrane behavior under blast load is to have the ability to resi st direct shear failure. Different factors will affect the direct shear capac ity of a section but the main one to be considered in the case where tension membrane behavior is desi red is the axial load and the size of longitudinal reinforcement. Therefore the 12 foot column designed as part of the validation was modified to compare the effect of using different longitudinal reinforcement rebars sizes. The columns we re subjected to various loading scenarios and the results are presented in Table 51. The different resistance functions for each columns analyzed under a 500 kips axial load are presented in Fi gure 51 represents. Note t he resistances functions are combined for demonstration purposes, once t ension membranes occur, only tension membrane resistance functions remain for th e dynamic analysis. The results highlighted in orange in Table 51 represent the c entral displacements obtained under tension membrane behavior. The results obtained clear ly demonstrate the tension membrane capacity is limited by the direct shear failure capacity of the section under blast. In other words, direct shear failure is expected to occur before a column may reach its full capacity in tension membrane and fails as such As the rebar size decreases, the direct shear capacity decreases as well as the tension membrane capacity. However, as the PAGE 165 165 axial load is increased, the direct shear capacity is increased which in some cases provides extra strength to the column to develop higher tensi on membrane capacity. When comparing the extra str ength provided by the tension membrane, the results are impressive. For the worst case loading of 5262 psi reflected pressure under an axial load of 1500 kip and #11 reinforcement, the results show an increased transverse load capacity of 715% when compared with the 735.9 ps i reflected pressure case that did not develop tension membrane behavior. Figure 52 represents the inte raction diagram of the column analyzed as part of the paramet ric study. Note that no design reduction factors are applied. PI Diagram In this section, the loadimpulse diagram of the 12 ft RC column is investigated for a constant axial load of 800 kip and 1500 kip. Figure 53 and 54 are the loadimpulse diagram obtained including direct shear and flexural behavior from both modified and DSAS V3.0. Both cases, 800 kip and 1500 kip axial load, demonstrated t he benefit of the tension membrane action of the RC column. The modified DSAS yield a flexural loadimpulse curve much higher than DSAS V3.0 which only includes flexural behavior. The results obtained also demonstrated in the dynamic response range of high explosive, the direct shear failure domi nated the failure mechanism up to impulsive load response 7285 psimsec and 7253 psimsec respectively for 800 kip and 1500 kip axial load cases. To bring the RC column to fail in tension membrane without failing in direct shear, higher impulsive load are required. This may be of intere st for a building to resist enhanced blast weapons such as thermobaric explosives. PAGE 166 166 Future Development/Recommendations The analytical procedure and algorithm devel oped in this research to conduct a single degree of freedom analysis of RC column under blast induced transient and axial loads demonstrated considerable progress in the capability to conduct large deformation analysis. The following discussion identifies key development issues that shall be considered in the future to further develop the capacity of capturing the tension membrane behavior of reinforced concrete columns as well as better capturing large deformation behavior during the analysis. The first recommendation is to conduct fi eld experiments to validate the approach presented in this research. During the ex periment, key issues other than tension membrane behavior shall be considered. The first issue of great interest is to capture and study the spalling and the capacity of th e steel reinforcement cage to retain concrete from flying out and becoming hazards. This would provide better insight into what percentages of loads are actually tr ansferred onto the steel once the concrete crushed and spalled and if they occur before or after transi ting to tension membrane. The second would be to capture the damage occurring at the support to evaluate the capacity of the steel to carry tension me mbrane behavior and develop a reduction factor for the resistance function or steel capac ity based on the findings. The third one would be to evaluate local damages that may occur to the steel rebar and develop a reduction factor if needed. Finally, a recommendation is made to capt ure the effect of support displacement versus fully fixed while undergo ing tension membrane behavior in order to develop a more realistic model for DSAS. The second recommendation is regardi ng software application of ABAQUS. It would be of great interest to develop a rein forced concrete model in ABAQUS/Explicit fit PAGE 167 167 for blast that actually models the spalli ng of concrete. Such a model would better account for the change of inertia forc es due to loss of concrete mass. The final recommendation is regarding the software applications of DSAS. Obviously, based on the first recommendations, t he findings should be incorporated into DSAS as part of future studies conducted at CIPPS. However, one aspect that was not discussed is the capacity of capturing the effect of compression membrane behavior that was discussed in Chapter 2 that may have a significant effect on the column response. This effect was not observed during th is research as it was not of interest but previous studies on reinforced concrete slabs demonstrated compression membranes may have significant effect on the section capacity and shall considered. A study on Timoshenko beam elements is currently underway at CIPPS that should allow capturing such effect in DSAS. During this research, only distributed l oads were investigated. However, the development of tension membrane behavior is also possible under different type of load such as point load. Jun and Hai, (2010) in vestigated experimentally the effect of compressive membrane behavior and tens ion membrane behavior in RC Beam subjected to point load. Figure 55 and Figure 56 are pictures obtained from Jun and Hai (2010) that demonstrate the experiment setup. T he experimental resistance function obtained and shown in Figure 57 clearly demonstrat ed the capacity of the RC beam to ful ly develop the tension me mbrane behavior. The maximum central displacement obtained is about 24 inches whic h is in the displacement range modified DSAS would yield for comparable section le ngth. The failure mechanism at the end support observed during the experiment also corresponds to the failure mechanism of PAGE 168 168 tension membrane behavior embedded into DS AS based on the cable theory. The displacement profile of the ent ire beam at failure is not c onclusive with the shape of the cable theory as parts of t he RC beam/column remain uncracked. This may be due to the fact this was more or less a static test and no spalling occurred as it is expected to occur when subjected to blast load. Ther efore more investigation of RC beam/column subjected to impact loads is required to pr operly understand the disp lacement profile of the RC beam in the tension membrane range in order to develop a tension membrane resistance function algorithm to handle point loads due to impact. The pictures shown in Figure 58 to Fi gure 511, were obtained from previous tests conducted by the Defense Threat Redu ction Agency on RC columns. They show a reinforced concrete columns undergoing tens ion membrane behavior when subjected to a blast load. These pictures are good visual examples of fully developed catenary action and further justify the recommendations made in this section. Conclusion A new computational capability was dev eloped for DSAS to address geometric instability and the possible transition into the tension membrane behavioral mode. This significantly extends the capability of D SAS to analyze reinforced concrete columns subjected to explosive load. However, further research and experiment on the matter is required. For example, it is assumed the stru cture would free the colu mn of its axial load once it falls into tension membrane. However, this generally implies that a vertical displacement of the column supports w ould occur which needs to be capture in the dynamic analysis. When hardened structure are to be designed to sustain large loading capacity and survive the loss of a column component, one s hall consider the design of such columns PAGE 169 169 as to obtain extra lateral capacity by dev eloping a tension membrane behavior. This study demonstrated the significant late ral strength one may gain due to tension membrane behavior. Such lateral support, al though the element may not carry axial load, continues to provide obstacle fr agment, to secondary blast and secondary fragment as well as continuing to support any lateral load a column may be designed to support. However, the cost will quickly increase as larger diameter of steel rebar is going to be required and it is likely to make for an over designed reinforced concrete column and contradict the design code. Ho wever, because it is designed for tension membrane behavior and so the remaining of t he structure may take on the axial load, the issue of sudden failure is not an issue in this case. The effect of direct shear failures also plays a signi ficant role as it was demonstrated during the validatio n and parametric study that t he column is likely to fail in direct shear prior to reach full tens ion membrane capacity. However, the study showed important increase in flexural capac ity due to tension membrane that proved to provide a significant flexural capacity compared with a column having no tension membrane capacity. Finally, the overall study pr ovided important improvements to rapidly and efficiently analyze reinforced concrete columns subjected to blast while yielding accurate values. PAGE 170 170 Figure 51. Twelve foot reinforced conc rete column combined resistance function comparison. PAGE 171 171 Figure 52. Interaction diagram 12 ft RC column with different rebar sizes. PAGE 172 172 Figure 53. LoadImpulse diagram 12 ft RC column under 800 kips axial load. PAGE 173 173 Figure 54. LoadImpulse diagram 12 ft RC column under 1500 kips axial load. Figure 55. Experimental setup. [Reprinted with permission from Jun, Y., Hai, T.K. 2010. Progressive Collapse Resistance of RC BeamColumn Subassemblages. (Page 3, Figure 3). Proc. 3rd Int. Conf. on Design and Analysis of Protective Structure (DAPS 2010), Singapore, 1012 May, 2010.] PAGE 174 174 Figure 56. Failure mode of the section. [R eprinted with permission from Jun, Y., Hai, T.K. 2010. Progressive Coll apse Resistance of RC BeamColumn Subassemblages. (Page 3, Figure 4). Proc. 3rd In t. Conf. on Design and Analysis of Protective Structure (DAPS2010), Singapore, 1012 May, 2010.] Figure 57. Experimental resi stance function. [Reprinted wit h permission from Jun, Y., Hai, T.K. 2010. Progressi ve Collapse Resistance of RC BeamColumn Subassemblages. (Page 4, Figure 5). Proc. 3rd Int. Conf. on Design and Analysis of Protective Structure (DAPS 2010), Singapore, 1012 May, 2010.] PAGE 175 175 Figure 58. Reinforced concrete column undergoing tension membrane behavior picture 1. [Picture provided by the De fense Threat Reduction Agency] PAGE 176 176 Figure 59. Reinforced concrete column undergoing tension membrane behavior picture 2. [Picture provided by the Defense Threat Reduction Agency] PAGE 177 177 Figure 510. Reinforced concrete column undergoing tension membrane behavior picture 3. [Picture provided by t he Defense Threat Reduction Agency] PAGE 178 178 Figure 511. Reinforced concrete colu mn undergoing tension membrane behavior picture 4. [Picture provided by t he Defense Threat Reduction Agency] PAGE 179 179 Table 51. Twelve foot reinforced concre te column analyzed with various longitudinal reinforcement size rebar. Triangular Load Parameters #7 Longitudinal Reinforcement Reflected pressure (psi) Incident reflected impulse (psimsec) Time duration (msec) 500 kip axial load disp. (in) 800 kip axial load disp. (in) 1500 kip axial load disp. (in) 735.9 526.6 1.431 0.40 0.39 0.801 1183 783 1.324 0.82 0.941 1.721 1777 1134 1.276 2.141 2.121 3.341 2134 1354 1.269 3.141 3.021 4.401 2922 1875 1.283 5.682 5.372 7.201 4208 2844 1.352 12.952 4757 3305 1.390 5262 3755 1.427 Triangular Load Parameters #9 Longitudinal Reinforcement Reflected pressure (psi) Incident reflected impulse (psimsec) Time duration (msec) 500 kip axial load disp. (in) 800 kip axial load disp. (in) 1500 kip axial load disp. (in) 735.9 526.6 1.431 0.45 0.43 0.781 1183 783 1.324 0.85 0.90 1.611 1777 1134 1.276 1.841 2.121 3.071 2134 1354 1.269 2.831 3.011 3.991 2922 1875 1.283 5.131 5.151 6.281 4208 2844 1.352 9.862 9.682 11.061 4757 3305 1.390 13.221 5262 3755 1.427 15.152 Triangular Load Parameters #11 Longitudinal Reinforcement Reflected pressure (psi) Incident reflected impulse (psimsec) Time duration (msec) 500 kip axial load disp. (in) 800 kip axial load disp. (in) 1500 kip axial load disp. 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MomentRotation Characteristics of Reinforced Concrete members Subjected to Bending, Shear and Axial Load. Rep. No. SRS260 Department of Civil Eng., University of Illinois, Urbana, IL. PAGE 184 184 BIOGRAPHICAL SKETCH In June 2000, Dave Morency enrolled in the Canadian Regular Officer Training Plan (ROTP) where he attended the Royal Military College of Canada and graduated with a bachelor degree in Civil Engineering in 2005. He then served as a combat engineer officer and completed one tour of duty in Afghanistan. In 2008, he was selected by the Canadian army to return to school and complete graduate studies with a specialization in Force Protection. He was then accepted to attend the University of Florida where he studied under the direction of Professor Dr Theodor Krauthammer as a structural engineer graduate student. 