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PAGE 1 1 PLASTIC HINGE BEHAVIOR OF REINFORCED CONCRETE AND ULTRA HIGH PERFORMANCE CONCRETE BEAM COLUMNS UNDER SEVERE AND SHORT DURATION DYNAMIC LOADS By TRICIA CALDWELL A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ENGINEERING UNIVERSITY OF FLORIDA 2011 PAGE 2 2 2011 Tricia Caldwell PAGE 3 3 To my family PAGE 4 4 ACKNOWLEDGMENTS I sincerely thank my advisors, Dr. Theodor Krauthammer and Dr. Serdar Astarlioglu for their guidance and assistance over the course of this research, as well as the Defense Threat Reduction Agency ( DTRA ) the generous sponsors of this project. Finally, I thank my par ents for their continued support of all of my efforts. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS .................................................................................................. 4 LIST OF TABLES ............................................................................................................ 7 LIST OF FIGURES .......................................................................................................... 9 LIST OF ABBREVIATIONS ........................................................................................... 13 ABSTRACT ................................................................................................................... 17 CHAPTER 1 INTRODUCTION .................................................................................................... 19 1.1 Problem Statement ........................................................................................... 19 1.2 Objective and Scope ......................................................................................... 21 1.3 Research Significance ...................................................................................... 22 2 LITERATURE REVIEW .......................................................................................... 23 2.1 Overview ........................................................................................................... 23 2.2 Structural Load and Response Analysis ........................................................... 23 2.2.1 Static and Dynamic Responses ............................................................... 23 2.2.2 Dynamic Analysis Methodology ............................................................... 24 2.2.2.1 Newmark beta method ................................................................... 26 2.2.2.2 Reaction forces .............................................................................. 27 2.2. 3 Blast Loads .............................................................................................. 28 2.3 Structural Analysis of Reinforced Concrete Columns ....................................... 30 2.3.1 Stress Strain Relationships for Reinforced Concrete Columns ............... 30 2.3.2 Dynamic Increase Factors ....................................................................... 32 2.3.3 Flexural Behavior and Moment Curvature Development ......................... 33 2.3.4 Diagonal and Direct Shear Behavior ....................................................... 34 2.3.5 Axial Behavior ......................................................................................... 35 2.3.6 Large Deformation Behavior .................................................................... 36 2.4 Plastic Hinge Formation .................................................................................... 38 2.4.1 Plasticity of Reinforced Concrete ............................................................ 39 2.4.2 Locations of Plastic Hinge Formation ...................................................... 40 2.4.3 Factor s of Plastic Hinge Length ............................................................... 41 2.4.4. Plastic Hinge Length............................................................................... 42 2.4.5 Moment Curvature and Plastic Hinges .................................................... 4 4 2.5 Ultra High Performance Concrete ..................................................................... 45 2.5.1 UHPC Defined ......................................................................................... 46 2.5.2 Evolution of UHPC ................................................................................... 48 2.5.3 Characterization of UHPC ....................................................................... 50 PAGE 6 6 2.5.3.1 Constituent materials and mixing methods .................................... 51 2.5.3.2 Mechanical properties and behavior .............................................. 54 2.5.3.3 Blast and protection research ........................................................ 55 2.5.4 UHPC Manufacturers .............................................................................. 57 2.5.4.1 Ductal technology ......................................................................... 58 2.5.4.2 CeraCem (formerly BSI) ................................................................. 59 2.5.4.3 Cor Tuf ........................................................................................... 59 2.5.5 UHPC Codes and Regulations ................................................................ 60 2.5.6 Applications of Ultra High Performance Concrete ................................... 63 2.6 Summary .......................................................................................................... 65 3 METHODOLOGY ................................................................................................... 80 3.1 Overview ........................................................................................................... 80 3.2 Plastic Hinge Development in DSAS ................................................................ 80 3.3 Experimental Validation Case ........................................................................... 85 3.4 Finite Element Analysis ..................................................................................... 88 3.5 Parametric Study .............................................................................................. 90 3.6 Summ ary .......................................................................................................... 91 4 ANALYSIS ............................................................................................................ 104 4.1 Overview ......................................................................................................... 104 4.2 Normal Strength Concrete Column ................................................................. 105 4.2.1 Simply Supported NSC Column ............................................................ 106 4.2.2 Fi xed NSC Column ................................................................................ 108 4.2.3 Axial Loads ............................................................................................ 110 4.3 Ultra High Performance Concrete Column ...................................................... 110 4.3.1 Simply Supported UHPC Column .......................................................... 112 4.3.2 Fixed UHPC Column ............................................................................. 113 4.3.3 Axial Loads ............................................................................................ 113 4.4 Contrast of Concrete Types ............................................................................ 114 4.5 Supplemental Behavior Study ......................................................................... 116 4.5.1 Plastic Hinge Length.............................................................................. 117 4.5.2 Shear Considerations ............................................................................ 117 4.5.3 Tension Membrane Behavior ................................................................ 118 4.6 Summary ........................................................................................................ 119 5 CONCLUSIONS AND RECOMMENDATIONS ..................................................... 141 5.1 Overview ......................................................................................................... 141 5.2 Conclusions .................................................................................................... 141 5.3 Recommendations .......................................................................................... 144 LIST OF REFERENCES ............................................................................................. 148 BIOGRAPHICAL SKETCH .......................................................................................... 153 PAGE 7 7 LIST OF TABLES Table page 2 1 Sample of UHPC mix proportions ....................................................................... 66 2 2 Percent replacement of Portland cement with silica fume .................................. 66 2 3 Static mechanical tests with increasing volume of steel fibers ........................... 66 2 4 Optimized UPHC mix proportions. ...................................................................... 67 2 5 Varied packing densities ..................................................................................... 67 2 6 Summary of CeraCem (structural premix) properties ......................................... 67 3 1 Plastic hinge length expressions and calculations for the test beam. ................. 92 3 2 Beam 1C and Beam 1H material properties. ...................................................... 92 3 3 DSAS peak responses per the inclusion of behavioral effects. ........................... 92 3 4 Strain rat e effect on DSAS output. ...................................................................... 93 3 5 Dynamic increase factors for beams 1C and 1H. ............................................... 93 3 6 Comparison of experimental and DSAS output for beams1C and 1H. ............... 93 3 7 Comparison of DSAS and ABAQUS output for Beam 1C. ................................. 93 3 8 Sample of parametric study blast trials. .............................................................. 93 4 1 Dynamic increase factors for the NSC column. ................................................ 120 4 2 Simply supported NSC column response. ........................................................ 120 4 3 Fixed support NSC column response. .............................................................. 120 4 4 NSC response to combinations of blast and axial loads. .................................. 120 4 5 DSAS input of UHPCs properties. ................................................................... 121 4 6 Simply supported UHPC column response. ..................................................... 121 4 7 Fixed support UHPC column response. ........................................................... 121 4 8 UHPC response to combinations of blast and axial loads. ............................... 122 4 9 Comparison of NSC and UHPC test columns DSAS peak response. ........... 122 PAGE 8 8 4 10 Comparison of NSC and U HPC test columns ABAQUS peak response. ...... 122 4 11 Effect on plastic hinge length expressions. ....................................................... 122 4 12 Effect of diagonal shear. ................................................................................... 123 4 13 Direct shear response. ..................................................................................... 123 4 14 Effect of tension membrane behavior. .............................................................. 123 PAGE 9 9 LIST OF FIGURES Figure page 2 1 Example of frame loading ................................................................................... 68 2 2 Equivalent systems. ............................................................................................ 68 2 3 Reaction force schematic for beam with arbitrary boundary conditions. ............. 69 2 4 Free air blast. ..................................................................................................... 69 2 5 Pressure time history for an idealized freeair blast wave .................................. 70 2 6 Simplified pressuretime history. ......................................................................... 70 2 7 Equivalent triangular pressuretime history. ....................................................... 71 2 8 Ideal concrete stress strain curve for uniaxial compression ............................... 71 2 9 Confined concrete stress str ain curve ................................................................ 72 2 10 Steel reinforcement stress strain curve .............................................................. 72 2 11 Concrete section with strain, stress, and force distributions. .............................. 73 2 12 Typical moment curvature diagram .................................................................... 73 2 13 Flexureshear cracking pattern ........................................................................... 74 2 14 Influence of shear mo del without web reinforcement ....................................... 74 2 15 Relationship between direct shear stress and shear slip .................................... 75 2 16 Compression and tension membrane behavior .................................................. 75 2 17 Ductile and brittle concrete behavior. ................................................................. 76 2 18 Curvature along a beam at ultimate moment ...................................................... 76 2 19 Plast ic hinge of a cantilever column ................................................................... 77 2 20 Comparison of plastic hinge length expressions ................................................. 77 2 21 Increasing brittleness with strength .................................................................... 78 2 22 Stress strain diagrams normal concrete vs. UHPFRC ..................................... 78 2 23 Production cost wit h respect to compressive strength ........................................ 79 PAGE 10 10 2 24 Tensile constitutive law of UHPFRC ................................................................... 79 3 1 Moment curvature diagram for test beam. .......................................................... 94 3 2 Loaddeflection curve for test beam with 19 nodes. ........................................... 94 3 3 Progressive deformed shape of test beam with 19 nodes. ................................. 95 3 4 Progressive rotation of test beam with 19 nodes. ............................................... 95 3 5 Progressive curvature of test beam with 19 nodes. ............................................ 96 3 6 Loaddeflection curve for test beam with 51 nodes. ........................................... 96 3 7 Progressive deformed shape of test beam with 51 nodes. ................................. 97 3 8 Progressive rotation of test beam with 51 nodes. ............................................... 97 3 9 Progressive curvature of test beam with 51 nodes. ............................................ 98 3 10 Zoomed progressive curvature of test bea m with 51 nodes. .............................. 98 3 11 Experimental setup for beams 1C and 1H. ......................................................... 99 3 12 Cross section of beam (Beam 1C has open stirrups; Beam 1H, closed). .......... 99 3 13 Dynamic loading of Beam 1C. .......................................................................... 100 3 14 Dynamic loading of Beam 1H. .......................................................................... 100 3 15 ABAQUS interface and Beam 1C schematic. ................................................... 101 3 16 Modified Hognestad curve for Beam 1Cs normal strength concrete. ............... 101 3 17 Beam 1C steel material model. ........................................................................ 102 3 18 Comparison of Beam 1Cs response. ............................................................... 102 3 19 Beam 1C rotation at peak response. ................................................................ 103 3 20 Beam 1C curvature at peak response. ............................................................. 103 4 1 Cross section of test column. ........................................................................... 124 4 2 Schematic of test column span. ........................................................................ 124 4 3 Moment curvature of NSC column. .................................................................. 125 4 4 Loaddeflection curves of NSC column. ........................................................... 125 PAGE 11 11 4 5 Progression of deformation of simply supported NSC column. ........................ 126 4 6 Progression of rotation of simply supported NSC column. ............................... 126 4 7 Progression of curvature of NSC simply supported column. ............................ 127 4 8 Curvature of simply supported NSC column post yield. ................................... 127 4 9 Example of ultimate deformati on via ABAQUS (trial NS2). ............................... 128 4 10 ABAQUS deflected shapes for simply supported NSC column. ....................... 128 4 11 ABAQUS rotations for simply supported NSC column. ..................................... 129 4 12 ABAQUS curvature for simply supported NSC column. ................................... 129 4 13 Progression of deformation of fixed NSC column. ............................................ 130 4 14 Progression of rotation of fixed NSC column. ................................................... 130 4 15 Progression of curvature of fixed NSC column. ................................................ 131 4 16 ABAQUS deflected shapes for fixed NSC column ........................................... 131 4 17 ABAQUS rotations for fixed NSC column. ........................................................ 132 4 18 ABAQUS curvatures for fixed NSC column. ..................................................... 132 4 19 UHPC material model. ...................................................................................... 133 4 20 Moment curvature diagram for UHPC column. ................................................. 133 4 21 Loaddeflection curves of UHPC column. ......................................................... 134 4 22 Progression of deformation of simply supported UHPC column. ...................... 134 4 23 Progression of rotation of simply supported UHPC column. ............................. 135 4 24 Progression of curvature of simply supported UHPC column. .......................... 135 4 25 ABAQUS deflected shapes of simply supported UHPC column. ...................... 136 4 26 ABAQUS rotations of simply supported UHPC column. ................................... 136 4 27 ABAQUS curvatures of simply supported UHPC column. ................................ 137 4 28 Progression of deformation of fixed UHPC column. ......................................... 137 4 29 Progression of rotation of fixed UHPC column. ................................................ 138 PAGE 12 12 4 30 Progression of curvature of fixed UHPC column. ............................................. 138 4 31 ABAQUS deflected shapes for fixed UHPC col umn. ........................................ 139 4 32 ABAQUS rotations for fixed UHPC column. ...................................................... 139 4 33 ABAQUS curvatures for fixed UHPC column. ................................................... 140 4 34 Comparison of NSC and UHPC constitutive models. ....................................... 140 PAGE 13 13 LIST OF ABBREVIATION S Ag Area of gross section As Area of steel bw Width of concrete section c Damping coefficient cNA Neutral axis depth d Effective depth of element Ec Concrete elastic modulus Es Steel elastic modulus fc Concrete stress fc Compressive strength of concrete fc Maximum strength of concrete fs Steel stress fsu Ultimate steel stress ftj Tensile strength of concrete matrix fy Yield stress F Force Fe Equivalent force FT Total force h Depth of element H Height of element ipos Positive phase impulse ILF Inertia load factor k Stiffness kd Neutral axis depth PAGE 14 14 Ke Equivalent stiffness KE Kinetic energy KEe Kinetic energy of equivalent system KL Load factor KM Mass factor lp Plastic hinge length L Length of element m Mass M Moment Me Equivalent mass MT Total mass Nu Axial load P Pressure Pd Downward pressure Pmax Peak pressure Pr Reflected pressure PS0 Incident pressure Pt Transverse pressure Pu Upward pressure Qi Reaction force per time step R Standoff distance Re Resistance function t Time Tneg Duration of negative phase Tpos Duration of positive phase PAGE 15 15 Tpos/ Equi valent duration of positive phase u System displacement System velocity System acceleration U Shock front velocity V Shear/reaction force w Point load w/c Water cement ratio w(x) Distributed load W TNT equivalent charge weight WE W ork WEe W ork of equivalent system x Location non structural element X Acceleration z Distance from the critical section to the point of contraflexure Z Scaled distance Blast decay coefficient Newmark Beta coefficient Newmark Beta coefficient i Load proportionality factor Shear slip Deflection e Elastic deflection p Plastic deflection o Strain at maximum concrete stress PAGE 16 16 c Concrete strain cm Strain at extreme compression fiber s Steel strain sh Steel hardening strain su Ultimate steel strain y Yield strai n Rotation between two points e Elastic rotation p Plastic rotation Modification factor relative to normal weight concrete Longitudinal reinforcement ratio Shear stress Curvature u Ultimate curvature y Yield curvature Shape function along element length Deflected shape function PAGE 17 17 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Engineering PLASTIC HINGE BEHAVIOR OF REINFORCED CONCRETE AND ULTRA HIGH PERFO RMANCE CONCRETE BEAM COLUMNS UNDER SEVERE AND SHORT DURATION DYNAMIC LOADS By Tricia Caldwell August 2011 Chair: Theodor Krauthammer Major: Civil Engineering Given the prospective threat of a blast or impact load causing severe damage to a structural element or system, it is critical to investigate and understand the dynamic behavior and potential failure modes of such member s. One method used to perform such an analysis in a computati onally efficient manner is the programming code Dynamic Structural Analysis Suite (DSAS). The presented studys intent i s to evaluate the incurrence of plastic hinges via DSAS and inspect the resultant behavior of both normal strength reinforced concrete and ultra high performance conc rete columns under severe loads. Ultra hig h performance concrete (UHPC) is an emerging engineering technology characterized by increased strength and durability compared with normal and high performance concretes. As a relatively new material, UHPC remains to be fully characterized, and to study the materials response under simulated loading conditions contributes to widening its use, especially with respect to protective applications. As a particularly vulnerable structural element under blast and impact loading s, columns are the specific interest of the study To introduc e the dynamic behavior of concrete columns, the analytical methods and models detailed in engineering literature PAGE 18 18 are first reviewed. The study then examines the process by which DSAS was adjust ed to approximate the plastic hinge formation and respective curvature along the column length, allowing for the evaluation of concrete columns behavioral response under various boundary and load conditions. The considered normal strength concrete and UHPC columns a re subsequently compared with the output from the finite element program ABAQUS for corresponding material model s as well as the simulated behavior of each other using DSAS Such comparisons respectively intend to validate the g enerated models and to demonstrat e the elevated properties of UHPC. PAGE 19 19 CHAPTER 1 INTRODUCTION 1.1 Problem Statement A cyclic relationship exists between the development of protective structures and the destructive forces used against them. While advancements are made on behalf of defense engineering, so too is technology dedicated toward the improvement of weaponry. The increasing severity of explosive devices warrants an examination and improvement of the techniques used to analyze severe blast and impact forces and prepare structural entities to withstand resultant loads. Though challenging, government and engineering agencies across the globe strive to meet the adaptability inherent to and required of protective technologies. It is recognized that a number of steps may be taken to enhance defense systems and procure an increased degree of safety. F or instance, in a recent report, the U.S. Army Engineer Research and Development Center (ERDC) emphasized the necessity of research and development of both advanced computational methods and structural materials to aid in disabling evolving threats (Roth et al. 2008) The time required of complex dynamic and finite element analyses is a critical facet, and it is therefore relevant to develop simplified a nd expedite, but nevertheless accurate, numerical methods for such work. In addition to thorough behavioral studies, the engineering of improved and/or new materials is essential to the evolution of protective structures and defense mechanisms. The consideration of blast and impact loads involves a detailed look at the failure modes of structural elements. Of particular interest is the vulnerability of reinforced concrete columns under explosive or other detrimental attacks. Generally, the linear elastic mo del of behavior for reinforced concrete is regarded as conservative and its PAGE 20 20 plasticity is ignored. Although this approach may be suitable for design purposes, a proper analysis would address the realistic aspects of the materials plastic range. The sectio n ( s) of a reinforced column reaching ultimate moment and incipient failure may exhibit the formation of a plastic hinge and an additional load carrying capacity. The inspection of these plastic rotations would be prudent and contribute to the efforts of advancing the analysis techniques of systems under severe dynamic loading. The Dynamic Structural Analysis Suite (DSAS) is a software program developed at the Center of Infrastructure Protection and Physical Security (CIPPS) and responds to the need of numer ical methods for modeling (Astarlioglu 2008) The program performs static and dynamic analyses of structural elements including reinforced concrete, steel, and masonry members. DSAS is specifically intended to analyze severe dynamic loads and it uses sing le degreeof freedom (SDOF) systems to simplify and expedite the process. With respect to reinforced concrete columns, the program considers combined axial and transverse loads as well as the effects of large deformation behavior ( Tran 2009; Morency 2010) The integration of plastic hinge formation in reinforced concrete columns would supplement DSAS and allow the program t o complete its analyses in a more realistic fashion than at present. In response to the need for stronger yet feasible structural materi als, ultra high performance concrete (UHPC) has emerged as a prominent research topic over the course of the past few decades. This new technology is characterized by increased strength and durability compared with normal or high performance concretes as w ell as an impressive resistance to blast and impact loadings. Taking advantage of such traits, the ERDC has implemented the use of UHPC in threat ening environments with the PAGE 21 21 creation of an armor panel and invested in the further research and development of the material ( Roth et al. 2008) As the employment of UHPC with respect to protective applications grows, it is pertinent to study the materials response under simulated impacts Alongside a full material characterization, the ability to model UHPC via pr ograms such as DSAS will assist in advanc ing its current technological state. 1.2 Objective and Scope The culmination of the conducted research is twofold: the evaluation of DSASs ability to recognize and account for the formation of plastic hinges in concrete columns and the analytical comparison of normal strength concrete (NSC) against the performance of UHPC as a structural material. In completing this effort, a literature review was o rganized to present the pertinent information supporting each function. To fulfill the goals presented above and enhance the structural analysis software, the following items a re accomplished and accordingly reported. Dynamic analysis techniques, such as those for blast and impact loads, a re reviewed, and the current methods and algorithms by which DSAS operates a re summarized. The theoretical background of plastic hinge formation, experimental proceedings on hinge leng th development, and the resultant impact on column curvature a re presented. Dually, a full overview of ultra high performance concrete as a relatively new material is organized to establish its usability. An algorithm i s verifie d to signify the development of plastic hinges at the critical sections of reinforced concrete columns based on t he establish ment of the hinge length. This algorithm i s then integrated into the current DSAS programming. The comparison of DSAS with the finite element software ABAQUS ( SIMULIA 2010) is verified with respect to material models. Validation of the plastic hinge algorithm for NSC and UHPC is conducted. ABAQUS is used to run simulations to be compared with the DSAS output as part of a parametric study PAGE 22 22 1.3 Research Significance The research and algorithm development proposed enhances DSAS and allow s it to broaden its structural analysis capabilities The verified program is subsequently capable of expediently evaluating the plastic hinge formation in reinforced concr ete columns under severe dynamic loading as well as analyzing the behavior of ultra high performance concrete employed as structural elements un der similar conditions. With t he introduction of plastic hinges the program perform s a more realistic analysis of column behavior and its resistance to failure, and the addition of UHPC updates the DSAS system and includes the relatively new, but technological ly available, material. PAGE 23 23 C HAPTER 2 LITERATURE REVIEW 2.1 Overview The literature review provides a compreh ensive background of the structural analysis of reinforced normal strength and ultra high performance concrete beam columns under short d uration blast loads. Section 2.2 reviews methods of dynamic analysis and the assessment of pressures exerted by blasts or explosions. Section 2.3 discusses the behavioral response of reinforced concrete columns, while section 2.4 examines the impact of plastic hinge formation on concret e behavior. Finally, section 2.5 provides a thorough background on the development of ul tra high performance concrete and an introduction to the materials analytical behavior. 2.2 Structural Load and Response Analysis To design or analyze a structural element, information must be known about the loads to which it is or may be s ubjected. For example, Figure 21 illustrates a potential loading scheme of a frame. In the diagram Pt(t), Pd(t), and Pu(t) represent pressure loads that vary over time. This section reviews the techniques used to assess a materials response to such dynamic loads. In p articular, the ability to approximate the pressure emitted from a known explosive or blasting device is examined. 2.2.1 Static and Dynamic Responses A static analysis requires only displacement dependent forces to be considered, whereas dynamic analyses include velocity and accelerationdependent forces. All structural systems realistically behave in a dynamic sense, however it is often reasonable to ignore the negligible effect of the time dependent forces and conduct a static analysis using stiffness rela tionships. Such is not the case for timedependent PAGE 24 24 blast or impact forces as the damping and inertial responses of a structural element become significant. 2.2.2 Dynamic Analysis Methodology Dynamic systems may be modeled either as singledegreeof freedom or multi degreeof freedom (MDOF) problems. Given the additional computational energy inherent of MDOF systems (which may have infinite degrees of freedom ), methods for simplifying them to equivalent SDOF systems are typically sought and successfully used to expedite the analysis process (Biggs 1964) For a SDOF system, the equation of motion is expressed as Equation 21 where m is the systems mass, c is the damping coefficient, k is the stiffness, F(t) is the force varied over time, and , and u are the system acceleration, velocity, and displacement, respectively. S imilar ly, E quation 2 2 is used to represent the equivalent SDOF system of a more complex system, where m k, and F(t) are replaced by Me, Ke, and Fe(t), or th e equivalent system parameters. + + = ( ) (2 1) + + = ( ) (2 2) Figure 22 displays the representation of a beam element by its equivalent SDOF system. The equivalent system is chosen so that its displacement corresponds to that of a designated point along the length of the real system. The simply supported beam of Fig ure 22 is shown to have experienced deformation, the shape of which is expressed by the function Th is shape function of a structural member is used in the evaluation of the equivalent SDOF parameters. The ku and Keu terms of the equations of motion apply to linear elastic systems. For nonlinear systems the equation of motion is typically expressed by Equation 23 PAGE 25 25 where Re(u) is known as the resistance function. The stiffness term is no longer proportional to displacement for a nonlinear system. The resistance function, defined as the restoring force in the spring, therefore becomes dependent on the loading path of the system and its material properties rather than simply its displacement. + + ( ) = ( ) (2 3) Equivalent mass. The equivalent mass of a system is determined by equating the kinetic energies of the real and equivalent structures. The velocit y function is estimated using the shape function by Equation 24. The kinetic energy of the real system is given by Equation 25, within which the velocity function may be replaced by Equation 24 to produce Equation 26. The equivalent systems kinetic energy is given by Equation 27. Finally, equating the kinetic energy equations and solving for the equivalent mass leads to Equation 28. Another significant value used in dynamic analysis is the mass factor ( KM), a ratio of the equivalent mass to the systems total mass ( MT), or Equation 29. ( ) = ( ) ( ) (2 4) = ( ) ( ) (2 5) = ( ) ( ) ( ) (2 6) = ( ) (2 7) = ( ) ( ) (2 8) = / (2 9) Equivalent load. The equivalent load of a system is determined by equating the work done by the external load on the real and equivalent structures. Like the velocity function for the equivalent mass, the displacement function is estimated using the shape function which resul t s in E quation 2 10. The work done by the external load on the real PAGE 26 26 system is found by integrating the product of the load and displacement across the length of a member via Equation 211. The substitution of Equation 2 10 for the displacement function res ults in Equation 212. For a uniformly distributed load w(t) may be substituted for w(x,t) and pulled out of the integral since the load would not vary across the length of the member. The work done on the equivalent system is given by Equation 213. Equat ing the works performed by the external load, the equivalent force is given by Equation 214. The load factor ( KL) may then be found as the ratio of the equivalent load to the total load ( FT) or Equation 215. ( ) = ( ) ( ) (2 10) = ( ) ( ) (2 11) = ( ) ( ) ( ) (2 12) = ( ) (2 13) = ( ) ( ) (2 14) = / (2 15) Equivalent parameters per time step. For every time step t he mass and load factors, KM and KL, will be linearly interpolated by the displacement at each timestep i The generalized method for determining either parameter (simply represented by K ) is demonstrated by Equation 2 16, where ui < u < ui+1. = + ( ) (2 16) 2.2.2.1 Newmark b eta method While a number of accepted methods exist for solving the equation of motion for a linear or nonlinear system subjected to dynamic loading, the technique considered within this study i s a specific case of the Newmark b eta m ethod referred to as the linear PAGE 27 27 acceleration method. This technique involves the direct integration of the equation of motion and is considered more applicable to nonlinear systems compared with other techniques. Along with the equation of motion, the two incremental timestep e quation s of Equations 217 and 218 are to be used, where the coefficie nts and are to be taken as 1/2 and 1/6, respectively. = + ( 1 ) + (2 17) = + + ( )+ ( ) (2 18) The initial values for ui and i should be known, and a timestep value ( ) should be selected that establishes an accurate and stable system. The solution is considered stable if a timestep less than / times the natural period is utilized. However, a smaller time step may be required to reach an appropriate degree of accuracy. The general procedure for determining the system behavior at each timestep is as follows: Compute i using the equation of motion. Estimate a value for i+1. Compute ui+1 and i+1 using E quations 2 17 and 2 18. Compute i+1 from the equation of motion. Verify convergence of i+1 or iterate until convergence is established. 2.2.2.2 Reaction f orces The support r eactions of a dynamically loaded element may be analyzed following the method prescribed by Krauthammer et al. (1990). This technique expands Biggs s (1964) assumption that the inertia force distribution corresponds to the deflected shape function so as to account for structures not behaviorally perfectly elastic or plastic. Considering the beam element illustrated in Figure 23 with a length L dynamic load Q(t) and deflected shape function the procedure for determining the reaction forces is as follows : PAGE 28 28 Calculate the reactions ( Q1 and Q2) at the supports for each timestep ( i ) and report the load proportionality factors ( 1 and 2) by Equations 219 and 220. = / (2 19) = / (2 20) Calculate the inertia load factor (ILF) by integrating across the beam length via Equation 221. = ( ) (2 21) Estimate the inertia proportionality factors 1i and 2i by linear beam theory per time step. Calculate the e nd reactions per timestep by Equations 222 and 223, where m is the beams mass and Xi is the acceleration. = ( ) + (2 22) = ( ) + (2 23) As for the equivalent mass and load factors, the load and inertial proportionality factors are linearly interpolated between each timestep via Equation 2 24, where may represent any of 1, 2, 1, or 2, and u is the system displacement. = + ( ) (2 24) 2.2.3 Blast Loads A blast or explosion is a sudden release of energy emitted as a shock or pressure wave. Blasts may be idealized as either freeair or ground bursts, the difference being the resultant wave propagation and pressure functions. Figure 24 illustrates a freeair blast. For a ground burst t he shock wave immediately reflects off the rigid ground surface whereas from Figure 24 it is evident that a secondary reflected wave is generated for a freeair burst. The overall effect of a blast is demonstrated by the pressuretime history of Figure 25 At an arrival time after the blast is detonated the structure (or point) of interes t experiences a spike above the ambient pressure to a PAGE 29 29 peak overpressure. As the shock wave passes, the pressure declines and drops below the ambient pressure, effectively creating a vacuum. The pressuretime history is easily divided into positive and negative phases based on whether the pressure is above or below the ambient pressure (Smith and Rose 2002) The pressuretime history of a blast with respect to a specific point is also represented in Figure 26. The chart assumes that the arrival time is the start time and demonstrates the difference between the incident pressure PS 0 and the reflected pressure Pr. The overpressure Pr is the result the shock waves reflection off of an encountered obstacle in addition to the incident pressure. The pressuretime history of a blast may be modeled by Equation 2 25, the modified Friendlander equation, where Pmax is the peak pressure, Tpos is the duration of the positive phase, and is the blast decay coefficient. Due to the blast loads short duration, a systems dynamic response is dependent upon the pressure impulse. The positive impulse load is found by integrating the pressure over time by Equation 226. ( ) = 1 / (2 25) = ( ) (2 26) If the peak pressure and impulse are known values, the positive pressuretime history may be simplified by a triangular loading history. As Figure 27 details, the duration of the triangular positive phase could then be estimated by Equation 227. /= 2 / (2 27) Any number of materials may be contrived into explosives, but for ease of analysis explosive materials are typically equivocated to TNT. The size of an explosive charge may be expressed by its TNT equivalency ( W ) or the weight of TNT needed to produce PAGE 30 30 an identical effect. Blast analyses are further simplified by the use of the scaled distance parameter ( Z ). This parameter is dimensional and relates the standoff distance from an explosion ( R ) to the charge weight ( W ) by Equation 228. = / (2 28) Curves have been prepared for the determination of a number of blast wave parameters from the scaled distance values. The U.S. Department of Defense for instance, assembled charts for idealized spherical and hemispherical TNT equivalent explosions (Unified Facilities Criteria 2008) Using the scaled distance functions reported include the peak positive incident and normal reflected pressures, positive incident impulse, time of arrival, duration of the positive phase, shock front velocity, and positive phase wavelength. 2.3 Structural Analysis of Reinforced Concrete Columns This portion of the literature review summarizes the general characteristics examined during the structural analysis of a reinforced concrete column. First the stressstrain relationships for concrete and steel reinforcement as well as their potential dynamic increase factors are presented. The flexural, shear, and axial behaviors of reinforced concrete elements are then reviewed, while a detailing of l arge deformation behavior, including the P delta effect and the Euler buckling model, concludes the overview o f anal ytical behavior 2.3.1 Stress Strain Relationships for Reinforced Concrete Columns The behavior of a reinforced concrete column is dependent on the stress strain properties of its constitutive materials. The relationships between stress and strain for each of concrete and steel reinforcement are influenced by a number of parameters and the curves presented on their behalf are idealized representations. PAGE 31 31 The modified Hognestad curve for concrete is displayed in Figure 2 8 This stressstrain relationship assumes the concrete section is unconfined and uniaxially loaded in compression. The modified Hognestad curve consists of two functions between the concrete stress ( fc) and strain ( c). As the stress approaches its maximum value ( fc ) 0, its relationship with the strain is represented by the parabolic function of Equation 229. After achieving the maximum strength, the stress of a reinforced concrete member is assumed to linearly decrease in relation to str ain. = (2 29) The reference materials of Park and Paulay (1975) and Krauthammer and Shahriar (1988) address the changes in stress strain behavior for instances of biaxial or triaxial compressive behavior or for concrete confined by circul ar spirals or rectangular hoops, all instances outside of the assumptions of the modified Hognestad curve. Figure 29 exemplifies a factored adaptation of the stress train curve for a rect angular beam or column given the inclusion of confinement. As depicted, confinement improves the strength of a reinforced concrete member as well as the ductility through its increasing of the experienced lateral pressure. The strength of concrete in tensi on is only a small fraction of that displayed in compression. The tensile strength is represented by the stress at which the concrete fc and fc. The behavior may be idealized as linear using the elastic modulus observed in the compressive range. A sample stress strain relationship for steel reinforcement is shown in Figure 210. This reinforcement curve assumes that the steel bar s are loaded monotonically in PAGE 32 32 tension. The function between steel stress ( fs) and steel strain ( s) consists of four distinct regions. The first region is characterized by the linear elastic relationship of Equation 230 until the steel yields, where Es is the modulus of elasticity for the steel. A yield plateau is then observed until the strain sh is reached. The third reg ion (strainhardening) is mark ed by a second range of increasing stress This region spans between the yield and ultimate ( fsu) stresses and the strain hardening and ultimate ( su) strains Park and Paulay (1975) modeled the region by Equation 231, where the variable r is represented by Equation 232. Though not illustrated in Figure 210, a fter achieving the ultimate steel stress and strain, the final region of reinforcement s curve consists of the stress decreasing until fracture occurs. = (2 30) = ( ) ( ) +( ) ( ) ( ) (2 31) = / ( 30 30 + 1 ) 60 ( ) 1 / [ 15 ( )] (2 32) 2.3.2 Dynamic Increase Factors When loads are i ntroduc ed to a structural system in a dynamic sense it has been observed that engineering materials such as concrete and steel display strengths in excess of those statically reported and held a s standard (Dusenberry 2010) The degree by which an apparent strength is heightened is specifi ed by its dynamic increase factor (DIF). DIFs have typically been linked to the strain rate imposed during loading, the value of which may exceed 100 s1 for blast loads. Numerous experimental studies have been conducted and are ongoing to develop DIF curves for various materials models, and a preliminary compilation of those for concrete has been assembled by Malvar and Crawford (1998). PAGE 33 33 The high strain rate work outlined by Malvar and Crawford primarily address es the DIFs of concrete in tension, but also briefly touches upon the effect held on concrete in compression and reinforc ing steel. In tension, the concrete DIF has been shown to potentially reach 2.0 at a strain rate of 1 s1 and to exceed 6.0 at a rate of 100 s1. For the higher spectrum of strain rates, the compressive concrete DIF may reach 2.0, and the DIF for steel reinforcement, 1.5. 2.3.3 Flexural Behavior and Moment Curvature Development The flexural behavior of reinforced concrete is typically reflected by a structural elements moment curvature diagram. A cross sectional view of a concrete beam or column, along with its respective strain, stress, and force diagrams, may be configured as exemplified in Figure 21 1 Given the stress strain relationships for concrete and steel rei nforcement (as previously explicated) and using the methods of strain compatibility and equilibrium, the moment curvature curve for a concrete element may be prepared like tha t of Figure 2 12. An elements curvature is its rotation per unit length as well as its strain profile gradient. Based on the diagram of Figure 211, the curvature ( ) and moment ( M ) may be derived from trigonometry and the summation of moments about the neutral axis, respectively by Equations 2 33 and 234. The variables cm and kd represent the strain at the extreme compression fiber and the neutral axis depth, respectively. As part of the moment equation, fsi, Asi, and di are the strength, area, and depth of each longitudinal steel bar. The mean stress factor ( ) is derived from the area under the stress strain curve, while the centroid factor ( ) is related to the first moment of area about the origin of area under the stress strain curve. These factors are expressed more specifically by Equations 235 and 236. = / (2 33) PAGE 34 34 = ( ) + (2 34) = / ( ) (2 35) = 1 / (2 36) To use E quations 2 33 through 236, the strain at the extreme compression fiber is varied (thereby varying curvature), and the depth of the neutral axis is determined by equilibrium of the internal forces. The given equations a pply to rectangular concrete sections, though the prescribed process would be similar for irregular section areas. With respect to flexural behavior, reinforced concrete may potentially fail by either the crushing of concrete in compression or the fracture of longitudinal reinforcement in tension. Both possibilities are to be accounted for when establishing the ultimate moment curvature point. 2.3.4 Diagonal and Direct Shear Behavior Two forms of shear are apparent in reinforced concrete elements: diagonal shear and direct shear. Diagonal shear occurs in coincidence with flexure. As cracks form in the tensile region of a concrete beam or column, the shear stress acts in addition to the flexural stresses to propagate secondary cracks in a diagonal direction. This behavior is illustrated in Figure 21 3 The effect of diagonal shear on the flexural behavior and failure of concrete has been determined to be a function of the longitudinal steel reinforcement ratio ( ) as well as the shear span to effective depth ratio ( a/d ). For structural elements without web reinforcement, Figure 21 4 models the trend of the shear reduction factor (SRF) by which the ultimate moment is decreased to account for diagonal shear. The SRF is plotted as the ratio Mu /Mfl, where Mu is t he ultimate moment due to the combi ned effort of flexure and shear and Mfl is the moment due solely to PAGE 35 35 flexural. To further counteract diagonal shear failure, stirrups may be placed vertically or diagonally in the concrete element to provide web reinforcem ent. Direct shear is recognized by cracking perpendicular to a concrete beam or columns axis. Such behavior is typically caused by a concentrated shear force located at a support or under a point load and assumed to act independently of flexural behavior. Therefore, individual analyses may be performed for each of the flexural and direct shear failures. Figure 21 5 charts the relationship between direct shear stress and shear slip as elaborated upon by Krauthammer et al. (2002). The model consists of five regions of either linear or c onstant stress slip relationships. To appropriate the di agram for dynamic loading cases Krauthammer et al. (1988) provides a shear resistance envelop that accounts for and models the effect of load reversal. 2.3.5 Axial Behavio r The column of Figure 21 is both transversely and axially loaded by the dynamic pressures. The diagram represents a typical loading scenario, but thus far only transverse, flexure inducing loads have been accounted for behaviorally. To consider an axial load working alone on a column, the structural member exhibits a compressive axial strength ( Nu) based on the relative areas and strengths of the concrete and reinforcing steel as depicted by Equation 237. Given steels superior tensile strength, the tensile axial strength of the column is based solely on the area of reinforcement. = 0 85 + (2 37) The consideration of an axial loads effect on a columns flexural and shear behavior has been documented in the ACI 3 18 08 code. For the case of flexure combined with axial compression, the experienced moment is altered as a function of PAGE 36 36 the load, column depth, and tension reinforcement depth. Also note that the P delta effect may become predominant given axial loads (see large deformation behavior) The effect of axial load s on diagonal shear behavior is separated into whether the loads act in tension or compression. If subjected to axial tension, the shear strength ( Vc) may be calculated by Equation 238, where Nu is negative for tensile forces. For axial compression however, the shear strength is to be taken by Equation 239 with Nu /Ag expressed in units of psi. The maximum shear under thi s condition is to be capped at the value given by Equation 240. For Equati ons 2 38 through 240, Ag, bw, and fc are defined as the gross concrete area, the concrete section width, the modification factor relative to normal weight concrete, and the concrete compressive strength, respectively. = 2 1 + (2 38) = 2 1 + (2 39) 3 5 1 + ( /500 ) (2 40) 2.3.6 Large Deformation Behavior The large deformation behavior of a reinforced concrete column includes the columns response to deflectiondependent mechanisms such as the P delta effect, Euler buckling, and compression and tension membrane modeling This section highlights the critical aspects of these behaviors, but for a more deta iled evaluation of each, Morencys (2010) report may be referenced. An eccentrically loaded column will deflect laterally under an axial load. The resultant deflection will generate a secondary moment related to the axial load, an occurrence known as the P delta effect. The delta may be represented by either or PAGE 37 37 based on the deformation reference point. The P moment is produced by a columns deflection away from its initial axis, while the P deflection of a columns joint(s). This behavior is also highly influenced by a columns classificati on as either short or slender. Short columns are less likely to deform enough for the secondary moment to generate and impact the whole of the structural elements behavior. Slender columns are also more likely to exhibit failure by buckling. Buckling of slender reinforced concrete columns is based on the differential equation for an axially loaded elastic column or Equation 241, where EI is the concretes flexural rigidity and P is the applied axial load. F or Equation 242 Euler derived the critical axial load ( Pcr) where n is the mode shape, and l is the unsupported member length. The Euler buckling load ( PE) corresponds to the first modal shape, as given by Equation 243. = (2 41) = ( ) / (2 42) = ( ) / (2 43) The Euler buckling form of failure is subdivided into two categories: global, compressive buckling of a section and buckling of the longitudinal r einforcement in the areas between stirrups. For the case of global buckling, Equation 243 may be directly applied to the concrete element. For localized buckling of the longitudinal steel reinforcement, the critical steel stress ( fcr) may be approximated in terms of the steels tangent modulus ( Et), the tie spacing ( s), and the radius of the reinforcement ( rs) by the relationship given in Equation 244. = ( ) /( /) (2 44) PAGE 38 38 A final result of large deformations in reinforced concrete elements is that the elements may exhibit nonlinear material behavior and the development of compression and tension membranes. This behavior has been commonly documented for the case of reinforced concrete slabs, though the general theory may be expanded to include members such as beams or columns. Post initial concrete cracking, a members flexural strength may be increased by the interaction of a membrane form ed along its compressive zone. Figure 21 6 displays a simplified diagram of a beams deflection, cracking, and formation of a compressive membrane. The illustration dually documents the beams ability to act with a tension membrane. After full cracking of the section, only the longitudinal reinforcement remains to generate the entire members strength. This tension membrane is often modeled by steel cable theory, as the reinforcement alone carries the flexural load until its own failure. The analysis of compression and tension membranes is a significant contribution to the modeling of collapse mecha nisms under blast and impact loadings. The procedure for such analysis has been chronicled by Morency (2010) and adapted into a methodological algorithm. 2.4 Plastic Hinge Formation The consideration of plastic hinge formation with respect to columns or other structural elements typically lowers the conservatism behind approximating member behavior. The occurrence of inelastic curvature allows for additional load to be carried after the c ritical section has sustained its ultimate moment. Predicting the exact behavior of plastic hinges though is difficult and largely based on experimental evidence. Research devoted to the topic has seen varied results, particularly in relation to the calcul ation of plastic hinge length. It has also predominately focused on hinge PAGE 39 39 development in solely flexural members rather than in columns enduring both flexural and axial loads. Nevertheless, analyzing the large deformations characteristic of plastic hingin g and the final collapse mechanisms of reinforced concrete columns is critical of dynamic loading applications. This se ction compiles the information relevant to the plasticity of reinforced concrete columns. Topics to be discussed include t he formation an d locations of plastic hinges factors influenc ing and methods used to approximate hinge length, and the relationship between plastic hinge length and a columns moment curvature response. 2.4.1 Plasticity of Reinforced Concrete A concrete element, such as a beam or column, exhibits one of two behaviors upon reaching its maximum load or capacity, brittleness or ductility. Brittle failure occurs at the ultimate load and usually without significant forewarning. This mode of collapse m ay be avoided through selective composition of the concrete and/or the addition of steel reinforcement. Figure 2 1 7 illustrates the flexural response for both behaviors. If ductile, the member will continue to deflect under a load without a sudden, abrupt failure. Plastic hinges exercise concretes ductility. Rather than fail upon sustaining its ultimate capacity, a beam or column may experience the continued absorption of energy. The allowance to exceed the elastic deform ation and generate moment redistrib ution is a trait of the inelastic, or plastic, range of concrete behavior. The resultant large inelastic curvatures are typically braced over cracking in the concretes tension zone by steel reinforcement. At the location(s) deemed a critical section of a structural element, the possibility of plastic hinge formation exists. A hinge results in the inelastic rotation and moment redistribution of a member. Figure 218 displays the effect of a plastic hinge on a PAGE 40 40 cantilever beams curvature: i n (a) the reinforc ed beam is shown with a tip load and cracking along its tensile region, in (b) the corresponding moment diagram is drawn, and in (c) t he curvature diagram details the actual and idealized curvatures of the beam (indicated by thick and dashed lines, respect ively) as the plastic hinge spreads across a portion of it. The shaded region denotes the inelastic behavior and hinge rotation, the area of which is translated into a rectangular area of a certain length to simplify and idealize the diagr am. Over this specific distance ( the plastic hinge length (lp), curvature is assumed to jump from approaching the yield curvature ( y) to the ultimate ( u). The effect of cracking is also illustrated, as the curvature realistically peaks at each crack. Plastic hinges are often ignored, partially due to the difficulty inherent in their prediction. Although the spread of a hinge is not easily approx imated given the number of factors affecting length, the locations where a hinge may form are easily discernable. 2.4.2 Locations of Plastic Hinge Formation As stated previously plastic hinges form in the region where a member reaches its maximum moment. Inelastic rotation of the element occurs and the moment continues to build along its remainder. Moment is essentially redistributed given a plastic hinge. The locality of plastic moments is dependent on a structural members system of support and loading pattern. Fi gure 2 19 demonstrates the relationship between the experienced moment and the hinge formation f or a cantilever column. For a cantilever column loaded at its tip, the maximum moment develops at its base, as does the hinge. If a beam is simply supported however the location of the maximum moment, and therefore plastic hinge, will spread outward from its midpoint. If the end supports are fixed, then plastic hinges will potentially occur at the supports while the ultimate PAGE 41 41 moment may also be subsequently reached at its midpoint forming a third hinge. Hinge location is predictable given the type of support and loading scheme. 2.4.3 Factors of Plastic Hinge Length In early studies of plastic hinge formation in concrete members beams were the primary focus of research. The hinge lengths were typically consider ed functions of concrete strength, tension reinforcement, moment gradient, and beam depth. Past research reflect s a lack of consensus regarding the influence of axial load on column hinges. This conflict was addressed by Bae and Bayrak (2008), who resolved to determine the factors specifically affecting columns. In their report Bae and Bay r ak list ed the following factors as holding influence on the length of a plastic hinge: axial load, moment gradient, shear stress, type and quantity of reinforcement, concrete strength, and confinement in the hinge area. Of these characteri stics, three were examined intently by Bae and Bayrak, namely axial load, the shear spandepth ratio, and amount of longitudinal reinforcement. Bae and Bayrak discovered axial load to in general, have a positive trending affect on the hinge length of a co lumn F or a small axial load, the plastic hinge length is not substantially affected. This case of having an axial load ratio below 0.2 was cause for some experimenters deeming axial load insignificant. L arger applied axial loads have substant ial influence over hinge length however, with Bae and Bayrak demonstrating that the hinge length ratio (plastic hinge length to member depth) increased from approximately 0.65 to 1.15 when the axial load ratio (acting axial load to ultimate axial load) was increased fr om 0.4 to 0.8 This trend is also emphasized by the relationship between hinge length, shear span depth ratio ( L / h ) and axial load. The L / h ratio linearly relates to the length, and as the axial load approaches capacity the trend PAGE 42 42 steepens. Finally, Bae and Bayrak relate d hinge length to longitudinal reinforcement. The greater the ratio of steel area to gross concrete area, the longer the plastic hinge region stretches. Increasing the steel ratio from 2% to 10% approximately triples the hinge length f or any given axial load scenario. 2.4.4. Plastic Hinge Length The following discussion summarizes the research conducted and empirical expressions form ulated for plastic hinge length as well as the applicability of such expressions to columns. Baker (Park and Paulay 1975). For unconfi ned concrete Baker proposed E quation 2 45 for the plastic hinge length where k1, k2, and k3 are factors for the type of steel employed, the relationship between axial compressive force and axial compressive strength with bending moment of a member, and the strength of concrete, respectively, z represents the distance from the critical section to the point of contraflexure, and d is the members effective depth. E quation 2 45 was based on testing the variables of concrete strength, tensile reinforcement, concentrated loads, and axial loads. Baker indicated that for a normal range of spanto depth and z / d ratios found in practice, lp lies between 0.4 d and 2.4d Later work by Baker considere d concrete confined by transverse steel and resulted in Equation 246 for hinge length, where c is the neutral axis depth at the ultimate moment. = ( / ) / (2 45) = 0 8 ( / ) (2 46) Corley (Bae and Bayrak 2008). Corley studied simply support beams and proposed that the plastic hinge length may be given by Equation 247. Equation 247 PAGE 43 43 was generated after considering the concrete confinement, size effects, moment gradient, and reinforcement, and the variables of d and z are t o be reported in inches. = 0 5 + 0 2 ( / ) (2 47) Mattock (Park and Paulay 1975). After a pproximating data trends Mattock simplified Corleys expression to Equation 248. = 0 5 + 0 05 (2 48) Sawyer (Park and Paulay 1975). The formula for the equivalent plastic hinge length as proposed by Sawyer is given by Equation 249. Sawyers expression is based on a number of assumptions regarding a members moment distribution. The maximum moment is assumed the ultimate, the ratio of yield to ultimate moment is assumed to be 0.85, and the yield region is assumed to spread d /4 past where the being moment is reduced to the yielding moment. = 0 25 + 0 075 (2 49) Paulay and Priestly (Bae and Bayrak 2008). Paulay and Priestly included the influence of steel reinforcement via bar size and yield strength into their equation for the equivalent plastic hinge length of a concrete member, resulting in Equation 250. In Equation 250 L is the member length, db is the reinforcing bar diameter, and the steel yield strength ( fy) is to be reported in units of ksi. = 0 08 + 0 15 (2 50) Bae and Bayrak (2008). Bae and Bayrak generated the chart represented in Figure 220 to compare the equivalent hinge length predictions of some of the empirical equations reviewed above. Amongst these expressions are the constant values approximated by Shelkh and Khoury and Park et al. for all shear spandepth ratios. The PAGE 44 44 variation between the equations is evident. It i s also interesting to note that for several of the equations proposed z and d (the distance from the critical section to the point of contraflex u r e and effective beam depth, respectively) are the contributing variables. From their experimental and parametric studies, Bae and Bayrak also produced a new equation for the plastic hinge length in a concrete column. This expression is given by Equation 251, where h is the depth of the column. All three factors examined during the parametric studies are i ncorporated into Equation 251, including axial load level, shear spandepth ratio, and amount of longitudinal reinforcement. A least squares analysis technique was used for determination of each parameters coefficient and the procurement of Equation 251 / = 0 3 ( /) + 3 ( /) ( / ) + 0 25 0 25 (2 51) 2.4.5 Moment Curvature and Plastic Hinges The rotation and deflection of a structural element may be determined from its curvature. The rotation ( ) between two points is the result of integrating curvature along the member via Equ ation 252, where dx is an i ncremental length of the member, while the deflection ( ) of the member is expressed by Equation 253. = (2 52) = (2 53) Referring to Figure 218 the rotation between two points, or the integration of curvature, is equivalent to the area under the curvature graph. This area may be considered to have two distinct sections, specifically the elastic and plastic regions. Therefore, the rotation or def lectio n of a member may be calculated by adding the contribution of each region as represented by Equations 254 and 255. PAGE 45 45 = + (2 54) = + (2 55) For elastic behavior, the curvature may be substituted for by the ratio M / EI with the rotation given by Equation 256. In Equation 256 M is the moment and EI is the elastic flexural rigidity at a given location of a member. The plastic rotation may be idealized, like that in Figure 219, by Equation 257. = ( / ) (2 56) = (2 57) Through the use of strain compatibility and small angle approximation (as Figure 2 11 represent s), the curvature may be represented by E quation 2 58, where kd is the neutral axis depth for a strain Th e plastic rotation may thereby be taken as Equation 2 59, where c and cNA denote the strain and neutral axis depth at the ultimate moment, and cm and kd represent the strain and neutral axis depth at the yield moment. = / (2 58) = [( /) ( / ) ] (2 59) 2.5 Ultra High Performance Concrete An emerging technology, ultra high performance concrete is predominately characterized by an increased strength and durability in comparison with normal concrete or high performance concrete (HPC). The remainder of the literature review provides a comprehensive background o n this structural material. First, a broad description and the developmental history of UHPC are presented, followed by an examination of the material c haracterization of UHPC. A description of the UHPC manufacturing industry is narrated, noting the products marketed as Ductal, CeraCem, PAGE 46 46 and Cor Tuf and f inally review s of the status of regulatory codes and the completed and potential applications of UHPC are conducted. 2.5.1 UHPC Defined The American Concrete Institute (ACI) depicts high performance concrete as meeting special combinations of performance and uniformity requirements that cannot always be achieved routinely using conventional constituent materials and normal mixing, placing, and curing practices ( FHWA 2005) This definition also encompass es ultra high performance concrete as it alludes to a material functioning at a level above that of normal concr ete. To differentiate the term ultra high from high, a limit compressive strength may be used. UHPC is commonly classified by a strength of, or exceeding, 150 MPa (22 ksi). To wholly define a concretes performance level, a number of material properties must be evaluated, or specifications met. The Federal Highway Administration ( FHWA), for example, uses freezethaw durability, scaling resistance, abrasion resistance, chloride ion penetration, alk alisilica reactivity, sulfate resistance compressive st rength, modulus of elasticity, shrinkage, and creep as criteria for characterizing HPC ( FHWA 2005) While research is underway to document similar standards for UHPC, the comparatively new material is characterized by fewer and more general traits namely strength, durability, and ductility (Astarlioglu et al. 2010) UHPC is marked by high durability in addition to an impressive strength These qualities are attributed to a low porosity which coincides with a denser packing of th e constitu ent molecules. A sample composition of UHPC is provided in Table 21 The concrete characteristically excludes the use of coarse aggregates and classically consists of Portl and cement, silica fume, a superplasticizer, and steel fibers. The materials collaborate on a microparticle level (the average size being 2.5 mm for fine PAGE 47 47 aggregate and 0.2 m for silica fume) and induce a densification of the concrete (Rong et al. 2009) Sand and/or other fine aggregates are combined with Portland cement to create a more homogeneous and compact mixture than if large aggregate s were utilized. In addition to being pozzolanic (reactive with calcium hydroxide to enhance concrete strength), silica fume particles fill void spaces left in the cement, thereby reducing porosity ( Shah and Weiss 19 98) Superplasticizers further promote the densification process as water reducing admixtures. They are employed to increase the composite mixtures fluidity and allow a simultaneous decrease in the water cement ratio a value inversely proportional to strength. The high compressive strength of UHPC is attributed to its mixture proportions and methods, factors dually resulting in an increased brittleness compared with normal concrete. Figure 221 represents the brittle behavior of UHPC via its stressstr ain diagram. After reaching the peak load (peak stress), the decrease in load carrying capacity is steeper for UHPC and HPC than for the normal concrete. The near vertical slope of the UHPC stress strain curve (post peak capacity) illustrates the possibili ty of an unpredicted, rapid failure of the material. To lessen the severity of such a failure and increase ductility, micro fibers, either of steel and/ or another synthetic material, may be added to the cement mix, creating a composite material commonly known as ultra high performance fiber reinf orced concrete (UHPFRC). Steel fibers strengthen the cement bonding and positively affect the cracking pattern of concrete. The fibers cause a number of microcracks to form in the place of large single cracks, impr oving both ductility and tensile strength. Embedded fibers may also l imit the need for mild steel reinforcement such as stirrups used for shear or confinement reinforcement. PAGE 48 48 2.5.2 Evolution of UHPC The development of UHPC began in the 1960s a century after W.B. Wilkinson patented reinforced concrete in 1854. At that time, the strongest concrete had a compressive strength in the range of 60 to 80 MPa and a water to cement ( w/c ) ratio of 0.30 ( Buitelaar 2004) Given a firm understanding of normal concrete, r esearchers hoped to exploit the recognized inverse relationship between strength and the w/c ratio. Concrete with a compressive strength in excess of 800 MPa was subsequently created. The material was prepared following a strict procedure however, utilizing high pressure and thermal treatment in a laboratory to drive its w/c ratio down ( Schmidt and Fehling 2004) Though relevant to improve concrete strength, such a specialized treatment does not prove overly construction or cost effective. Under the leadership of Hans Henrik Bache in Denmark, the idea that the ultimate strength of concrete depended on more than the selection of cementitious materials was tested. It was discovered through experimentation that strength i s also dependent on the compaction of the concrete and the porosity established after the hardening process (Buitelaar 2004). Results reported in 1970 of Baches research displayed the compressive strength increasing from approximately 60 MPa to 290 MPa when the porosity ratio w as decreased from 0.6 to 0.4. The use of vibration or pressure to compact cement paste i s able to substantially increase the compressive strength. Through the 1970s alternate methods for decreasing porosity were investig ated. Superplasticizers emerged as e ffective admixtures for improving the workability of concrete in the presence of lower water contents. They are generally sufficient in quantities of 12% of the cement weight, and an excessive amount will induce concrete segregation. T heir development eased the dispersion of particles throughout PAGE 49 49 cementitious mixtures and led to a rise in the interest of fine particles, particularly the microsilica by products of silicon and ferrosilicon production ( Buitelaar 2004) Over a hundred times smaller than typical cement particles, silica f ume acts as a void filler and compac tor. Its reactivity with calcium hydroxide in the cement paste creates calcium silicate hydrate and dually enhances the molecular bo nding and strength. The Danish research t eam led by Bache developed a superplasticizer and microsilica infused cement paste named D.S.P By 1980 t he material saw practical application in the security field, employed in the construction of vaults and other protective structures ( Schmidt and Fehling 2004). Alongside a further restriction in fine aggregate grain size (< 1 mm in diameter), the cement mixtures implementing the use of reactive constituents became know n as reacti ve powder concretes (RPC). The term ultra high performance concrete was est ablished to reference a boarder scope of concretes, encompassing RPCs and other formulations reaching a compressive strength of 150 MPa ( Schmidt and Fehling 2004). As for fiber reinforcement, it became a modern development w ith the addition of asbestos fib ers to cement paste in the early 1900s. Steel fibers, in particular, were first proposed in reports published by Romualdi in 196364 and have been researched and refined for over 45 years. Other well established, though less common, reinforcements include polypropylene and glass fibers ( Brandt 2008) The advantages of each material var y Steel fibers increase fracture energy (ductility) and compr essive and tensile strength s and lower concretes tendency toward cracking. Polypropylene fibers limit early shri nkage and microcracking but improv e fire resistance, while g lass fibers reduce the internal stresses present in early age concrete ( Orgass and Klug 2004). PAGE 50 50 Ultra high performance fiber reinforced concrete has become a prominent research topic over the past two decades. As of the mid1990s, multiple engineering and production companies have ventured into producing particular varieties of UHPFRC. Among the earliest companies were Lafarge and Eiffage, developing their concretes Ductal and BSI (now CeraCem), r espectively. The library of data collect ed, including research of its constituent materials, reinforcement, characterization, construction method, etc., has since begun to compil e on an international level. Two international symposiums have been held in Kassel, Germany (in 2004 and 2008) with specific interest in UHPC. The proceedings from each symposium have been published and cover presentations on topics such as raw materials, fiber reinforcement, early age behavior, structural behavior, material modeling and predicting, impact and blast effect, and worldwide applications. R elevant data and information have also been gathered by several engineering com mittees into sets of preliminary design recommendations. T he current research and findings on behalf of UHPC are subsequently reviewed in greater detail. 2.5.3 Characterization of UHPC The analysis of ultra high performance concretes material properties has been conducted across the globe by researchers seeking to fully understand their structural advantages and disadvantages. An overview of the material characteristics distinctive to UHPC ensues, and it is divided into three areas of focus. First, the effects of raw material proportions and production methods on material behavior are examined. The typical m echanical properties of UHPC are then summarized, followed by a discussion of the research pertinent to the blast and impact resistance of UHPC. PAGE 51 51 Important to the design proc ess and structural use of UHPC, the general compressive and tensile behaviors of UHPC are illustrated in Figure 222. The stress strain diagrams are representative of the employment of fiber reinforcement. Displayed are the general ranges for compressive and tensile strengths (typically 150 200 MPa and 20 30 MPa, respectively), as well as a comparison between the behaviors of UHPC and normal strength concrete. Not only does UHPC substantially surpass normal concrete in strength, but it also exhibits a fiber enhanced ductility. 2.5.3.1 Constituent materials and mixing methods The proportio ning of raw materials and the mixing techniques utilized in concrete preparation are studied in order to generate product optimization. Raw materials such as silica fume and fiber reinforcement contribute to UHPCs functionality provided they are employed in effective quantities while specialized compaction and mixing methods improve the strength and durability properties of UHPC mixtures. However, for optimization purposes it is important to recognize that these noted benefits may not always prove effici ent coupled with time or cost constraints. For high strength concretes the cost of production is generally proportional to the strength developed. Figure 22 3 presents this relationship as determined by the Norwegian Defence Estates Agency (Markeset 2002) The costs are normalized to the concrete strength of 45 MPa. At a point, the increase in concrete strength fails to outweigh the additional incurred cost. Silica f ume. Silica fume is added to a mixture as a percentage or ratio of the cement present. A study of silica fumes effect on the mechanical properties of UHPC was conducted by Jayakumar (2004) the results of which are organized in Table 2 2 The percentage of silica fume reported is relative to the density of ordinary Portland cement. The case o f no silica fume associates with a cement density of 500 kg/m3, and PAGE 52 52 the case of 5.0% silica fume corresponds to 23.8 kg/m3 of silica fume and 476.2 kg/m3 of cement. For each of the evaluated strengths, the optimum employment was at 7.5%, though the tensile and flexural strengths displayed limited improvement with silica fume. Steel fiber reinforcement and curing time. Fiber reinforcement is added to UHPC mixes to enhance the overall ductility of the resultant concrete. Testing performed by Rong et al. (2009) examined the effect of percent volume of steel fiber reinforcement on the compressive strength, flexural strength, and toughness of UHPC. Three volume fractions (relative to the entire cementitious mixture) of steel fibers were tested: 0%, 3%, and 4%, an d they are compared in Table 23. An increase in the quantity of steel fibers saw an increase in all values presented, particularly the toughness index. As expected of concrete, strength accrues over time, and the compressive and flexural strengths rise per an extended curing time. The benefit of employing fibers was particularly observed by a jump in flexural strength between 0% and 3% fiber content from approximately 20 MPa to 65 MPa at 180 curing days The bonding inherent between the steel fibers and ce ment matrix enhances the concretes tensile region during flexure. Optimum mixture. Park et al. (2008) completed similar laboratory testing for each UHPC raw material and generated the sample configuration depicted in Table 2 4 Densification and heat treatment. Teichmann and Schmidt (2004) analyzed the effect of packing density on the durability properties of UHPC. Normal, high performance, and ultra high performance concretes were differentiated by relative packing density values of 0.68, 0.71, and 0.76. The experimental outcomes are exemplified in Table 25 The percentages of porosity and capillary pores decrease with PAGE 53 53 an increase in the d ensity. As a form of transportation through concrete, capillary pores promote chemical int rusion, and UHPC is evinced to provide improved defense against harmful attacks. O ther durability t raits evaluated were the permeability coefficient, water absorption, and chloride penetration depth of each material. The permeability coefficient dropped fr om 6.7E 17 m2 for normal concrete to 0.08E 17 m2 for UHPC, and a corresponding trend w as viewed for the water absorption. Monitoring the chloride diffusion via a quick migration test, the penetration depth of chloride into the concrete was substantially reduced by the densification (measured at 2.3 cm, 0.7, and 0.1 cm for normal, high performance, and ultra high performance concrete). While all three materials were cured under water, a second batch of UHPC was cured under a two day heat treatment of 90C fo r comparison purposes Th is variation between curing te chniques was reflected by a difference in compressive strengths at 28 days. The heat treatment was observed to raise the UHPCs strength from 162 MPa to 213 MPa. Also, t he permeability coefficient dropped to 0.01E 17 m2 (oneeighth of that for normally cured UHPC). Pressure treatment. The relationship between strength and density was also emphasized by Schachinger et al. s (2004) test ing the eff ect of pressure treatment on cement mixtures. Multiple techniques for removing the air void content of UHPC were examined, one of which was pressurization. A mixer with vacuum capabilities was used, and the change in pressure from 1000 mbar to 50 mbar decreased the air void content from approximately 4% (of volume) to less than 1%. T he specialized treatment was demonstrated to raise the compressive strength of UHPC by nearly 100 MPa a substantial increase. PAGE 54 54 2.5.3.2 Mechanical properties and behavior Structural materials are typically defined by their mechanical properties, and the characteristics of UHPC to be examined include the compressive, flexural, and tensile behaviors, shrinkage and creep, and the effect imposed by strain rate. Compressive behavior. In addition to the stress strain representation of compressive strength in Figure 222 and the effects held on it by various production methods, the biaxial compression behavior of UHPC has been investigated by Curbach and Speck (2008). The difference between uniaxial and bi axial behavior was exhibited by Curbach and Speck via comparison of their resultant stress strain diagrams The confinement caused by dually loaded axes decreases the brittle behavior of the UHPC, expressed by an extension of the stressstrain diagram. Flexural behavior. The flexural behavior of UHPC, compared with other types of concrete such as high strength concrete, fiber reinforced concrete, and engineered cementitious composite (ECC), was investigated by Millard et al. (2009) The impressive strength of UHPC may be credited to the use of fiber reinforcement and the cracking pattern it influences. Instead of a single large flexural crack forming, a number of microcracks develop in the concrete. Millard et al. also prepared load deflection curves which correspond to the energy absorption possible of the concretes, as their potential is indicated by the area under a curve. Based on such curves, the fiber reinforced UHPC had the greatest energy absorption potential of the concretes examined. Tensile behavior. Ultra high performance concretes tensile strength is also greatly shaped by its employment of fiber reinforcement. Habel and Gauvreau (2008) present ed the general behavior divid ing it in two areas: strain hardening and strain softening. Differentiated b y their manner of energy dissipation, strain hardenings loss is PAGE 55 55 considered volumetric and strain softenings loss is via localized cracking Habel and Gauvreau represented t he hardening and softening behaviors by a stress strain curve and the relationship between stress and crack width, respectively. Shrinkage and creep. The shrinkage and creep of UHPC under various curing methods was tested by Graybeal (2005). Although steam treatment speeds the process and the ultimate shrinkage is attained very early, t he air cured UHPC experiences less shrinkage overall. Nearly the opposite is true for creep, as the steam cured concrete experiences approximately onefourth of the ultimate creep of the air cured concrete Strain rate. The primary intent of Rong et al.s ( 2009) research was to investigate the dynamic compression behavior of UHPC. Testing was carried out on specimens varied by their volume percentage of steel fibers (again, 0%, 3% or 4%). For each UHPC and fiber configuration multiple strain rates were appl ied The stressstrain relationships were prepared by Rong et al. to represent strain rates from 25.9/s to 93.4/s. Not only does the increase in strain rate amplify the peak stress attained, but it also contributes to an increase in the ultimate strain. Th e sensitivity to strain rate inherent of UHPC was clear from the experiments performed by Rong et al 2.5.3.3 Blast and protection research The enhanced material properties of ultra high performance concr ete have been shown to provide notable def ense against dynamic loading of a structural sys te m, and i ts attributes have influenced specific interest in its use as protection from impact and blast forces. The durabilit y and ductility of UHPC impart resistance against aggressive environments and its incr eased strength allows for thinner members, which in turn reduces cost and construction needs. Blast research has been performed by Cavill PAGE 56 56 (2005) and Wu et al. (2009), while penetration testing has been conducted by Markeset (2002), the discussion and resul ts of which follow. The endurance of UHPC panels against blast loading was documented by Cavill (2005) after testing was conducted in 2004. The experiment at Woomera consisted of subjecting plain and pretensioned panels at three varying distances (30, 40, and 50 meters) to a six ton TNT equivalent of Hexolite. The panels had thick nesses of 50mm, 75mm, and 100mm and compressive strengths of approximately 180 MPa. The blasting resulted in minimal damages. Fractures occurred in the plain panels and the pretens ioned panels subjected to the highest pressures (relative to their thickness), and the largest final deformatio n observed was 150 mm. The thickest and farthest placed panels appeared entirely undamaged. For all cases, spalling and fragmentation did not occur. The testing was deemed successful as all panels endured the exerted pressures, with note being given to their thinness. Supporting the Woomera conclusions, Wu et al. (2009) studied a number of different types of concrete slabs under blasting. Two UHPC slabs were investigated, one with steel reinforcement and one without it. Each slab measured 2000 mm x 1000 mm x 100 mm, and their compressive and tensile strengths were recorded at 151.6 MPa and 30.2 MPa, respectively. The charge was suspended from a pipe and centered over a slab constrained by a steel frame. The slab lacking reinforcement had a scaled distance of 0.5 m/kg1/3 (standoff distance of 0.75 m and 3.433 kg of explosiv e), while the reinforced slab had a scaled distance of 0.37 m/kg1/3 (distanced at 1 m with 20.101 kg of explosive). After blasting, the plain slab experienced flexural cracking at its center and quarter points, as well as a final deflection of 4.1 mm. The reinforced slab, PAGE 57 57 subjected to the much larger quantity of explosive, was displayed post blast to have crushed at midslab and the LVDT recording the deflection had been destroyed. Though the reinforced UHPC slab experienced considerable damage, it was deem ed superior to the others by Wu et al. based on its predicted energy absorption capacity. Finally, Markeset (2002) studied the relationship between compressive strength and penetration depth of UHP C materials. To determine the protective abilities of UHPC, the concrete samples were subjected to 45 kg, 152 mm caliber projectiles, with the average impact velocity being about 500 m/s. The experimental results were based on a penetration depth relative to a compressive strength of 30 MPa, and the penetration depth was observed to decrease by approximately half with an increase to 150 MPa Protection is evidenced to substantially improve given the use of UHPC over normal concrete. Markeset also compared t he penetration results with the cost effectiveness of UHPC production. Referencing Figure 2 23, the increased cost of creating 200 MPa concrete over 150 MPa does not appear to result in an equivalent return in penetration depth (Markeset 2002) Cost feasib ility plays a significant role in the consideration of a materials use in construction projects 2.5.4 UHPC Manufacturers The continued development and optimization of ultra high performance concrete will ease its production and distribution over time. Currently, few UHPC mixtures or products have been commercialized. Research practices call for UHPCs production either on an individual laboratory basis or through one of the readily available manufacturers. Three examples of the industrialization of UHPC are Ductal, CeraCem, and Cor Tuf. Ductal and CeraCem were each developed in France over the past few PAGE 58 58 decades, while Cor Tuf emerged more recently in the United States. A review of these UHPC products and their research data follows. 2.5.4.1 Ductal technology Ductal is an ultra high performance fiber reinforced concrete product originating from the efforts of three companies: LaFarge (construction materials manufacturer) Bouygues (civil engineering contractor) and Rhodia (chemical materials manuf acturer) in response to the development of r eactive powder concrete in the 1990s. The term d uctal refers to the enhanced durability that is generated by a reduction in defects like micro crac king and/or pore spaces (Acker and Behloul 2004) The standar d mixes reflect the typical UHPC composition, with particle sizes ranging between 0.1 m and 600 m, but variations in fiber reinforcement exist. The fibers used may either be steel, an organic material, or a combination of the two. The parent companies ga thered research data from over twenty different laboratory facilities and universities testing their products The results supported the enhancement of mechanical and durability behaviors expected. Indicators of the materials heightened properties are an average w/c ratio of 0.18, an average density of 2 450 kg/m3, and a 2% 6% porosity ( four to five times lower than that of normal concrete). The use of steel fibers ( which demonstrat e a tensile strength of 2 000 MPa and Youngs modulus of 200, 000 MPa) provides additional insight to the concrete strengths recorded. The research reports include ranges for the compressive strength, flexural st rength, and fracture energy measured: 174 186 MPa, 25 39 MPa, and 20,000 30, 000 J/m2, respectively ( Cavill 2005). A cker and Behloul (2004) present the compressive and bending behaviors of Ductal. The ir graphical representations also compare the Ductal stress strain and PAGE 59 59 stressdisplacement diagrams with those of ordinary concrete. The ductility apparent prior to reaching the peak bending stress is due to microcracking o f the concrete (Acker and Behloul 2004). The formation of multiple small cracks provides the beams with greater support than if a central major crack were to develop. Ductal materials may or may not be thermally treated. In either case, the total concrete shrinkage is comparatively minimal; heat treatment simply expedites the process. With heat treatment, a concern need not exist with respect to the members shrinkage after placement. A similar benefit occurs for creep. Heat treatment minimizes the creep coefficient to nearly 0.2, as opposed to approximately 0.8 without treatment. 2.5.4.2 CeraCem (formerly BSI) CeraCem was developed by Eiffage, a contracting and consulting company based in France, in partnership with Sika Regional Technology. CeraCem is classified as an ultra high performance fiber reinforced concrete, as well as a self compacting concrete. CeraCem consists of three comm ercial premixes each of which applies to either the structural, aesthetic, or grouting and anchoring fields (Abdelrazig 2008) Property values reported by Eiffage for the structural premix (commonly called BFM Millau) are listed in Table 26 CeraCem was o riginally marketed as BSI (Bton Spcial Industriel), the name under which several engineering projects were carried out. The majority of its projects have been localized in France, the most prominent of which was the Millau viaduct toll gate constructed i n 2004 and discussed later alongside other UHPC applications 2.5.4.3 Cor Tuf Cor Tuf was developed by the Geotechnical and Structural Laboratory (GSL) of the U.S. Army Corp of Engineers Engineer Research and Development Center. It was PAGE 60 60 created with a fixed material composition to regulate the UHPC testing performed. In 2009, the ERDC published a detailed report on Cor Tuf: Laboratory Characterization of Cor Tuf With and Without Steel Fibers The report is the primary literature available on the Cor Tuf mat erial and describes the ultra high performance concrete as a reactive powder concrete (RPC) with concrete cylinder strengths in the range of 190 to 244 MPa. It documents the results of six mechanical tests completed for both concrete with and without steel fibers, including hydrostatic compression, unconfined compression, triaxial compression, unconfined direct pull, uniaxial strain, and uniaxial strain/constant volume strain loading tests. The triaxial compression tests performed characterize the brittle v ersus ductile behavior of concrete. A collection of the data from a total of twelve such tests i s provided by the ERDC is graphical form The principal stress difference versus axial strain graph compares six different confining pressures that range from 1 0 MPa to 300 MPa. Below a pressure of 100 MPa, strain softening is observed by a drop in the stress after reaching its peak. The 200 MPa and 300 MPa confining pressures however promoted strainhardening and an increased grow th in strength (Williams et al. 2009). 2.5.5 UHPC Codes and Regulations Though ultra high performance concrete developed from normal concrete, their properties are too fundamentally different to group them together as a single material. Correspondingly, their regulat ions of their use mus t not be assumed to overlap with guidelines prepared for normal concrete or HPC to be recycled on UHPCs behalf. The next step toward generating a broader use of UHPC is establishing standards to which the material can be held. As the breadth of UHPC literature has expanded, civil engineering societies have begun organizing the findings and creating a foundation for PAGE 61 61 future codes. UHPC is still relatively young with respect to its material characterization, and according ly, the completion of such regulatory codes will likely not occur before the material has sufficiently aged. To date though, societies in France and Japan have each published a set of recommendations, with similar work being conducted in other nations. The first complete set of guidelines prepared on behalf of ultra high performance fiber reinforced concrete was the French Association for Civil Engineerings (AFGC) Interim Recommendations of 2002. The recommendations classify UHPC as a cement based product with a compressive strength in the range of 150 MPa to 250 MPa (approximately equivalent to 22 ksi to 36 ksi), and it elicits the use of steel fiber reinforcement to achieve ductile behavior under tension and, if possible, to dispense with the need for passive (nonprestressed) reinforcement. AFGCs Interim Recommendations is divided into three sections: characterization, design and analysis, and durability of UHPFRC. In Part 1 the relationship between UHPFRCs characterization and the behavioral contribution of its fiber reinforcement is highlighted. Considered are the effects imposed on tensile strength and the concretes geometry during placement. The diagram displayed in Figure 22 4 represents the elastic and post cracking stages defin ing the tensile behavior of UHPFRC. The elastic stage is surpassed upon reaching of the cements tensile strength ( ft j) and the transfer of strength to that of the composite structure. The document notes that the diagram is representative of the strainhardening case. Part 1 of the Interim Recommendations also serves as a provision for laboratory testing, including instruction for compressive, direct tensile, and flexural tensile strength PAGE 62 62 tests, and for the routine checking of the concretes design, mixing, and placement. Th e document emphasizes the usefulness of evaluating each batch of UHPFRC on an individual basis and provides property coe fficients for use when they cannot be directly determined. Values defined are as follows: Static modulus of elasticity, 55 MPa Poissons ratio, 0.2 Thermal expansion coefficient, 1.1x105 m/m/ C Shrinkage, 55 m/m Creep coefficient, 0.2 (with heat treatment) or 0.8 (without heat treatment) Part 2 delves into the design and analysis of UHPFRC members and is largely based on the French BPEL and BAEL codes (which respectively address the limit state designs of prestressed concrete structures and reinforced concrete structures). Some excerpts are pulled directly from one of the previous codes, while others are slightly adapted to the UHPFRC mat erial. The recommen dations third and final section, Part 3 evaluates potential threats (biological or chemical agents) to the concretes sustainability and its potential to endure any such damage. Observations of early UHPFRC applications demonstrate the concretes suitability to aggressive environments and structural longevity. Also, though not typically considered a trait of durability, the section addresses fire resistance and its connection to the materials bearing capacity. The Japanese Society for Civil Engineers (JSCE) released its Recommendations for Design and Construction of Ultra High Strength Fiber Reinforced Concrete Structures, Draft in 2006. The recommendations classify UHPFRC by a compressive strength exceeding 150 MPa, similar to AFGCs g uidelines, and a tensile strength greater than 5 MPa. JSCE specifically highlights two ways by which the document deviates from normal concrete standards First, in reference to the tensile strength PAGE 63 63 apparent in UHPFRC without conventional steel reinforcement, methods for safety and serviceability evaluations are provided. The recommendations also limit the required durability testing and set the materials standard lifespan at 100 years (under typical environmental conditions) due to the enhan ced densit y and its resultant durability 2.5.6 Applications of Ultra High Performance Concrete Structural materials are selected for use based on the evaluation of four primary characteristics: strength, workability, durability, and affordability (Tang 2004) As exemplified, UHPC excels in the areas of strength and durability, but its widespread employment has been inhibited by both its workability and affordability. Though the addition of superplasticizers lessens the c onstraint inherently placed upon wor kability (fluidity), it remains a challenge to readil y adapt established construction methods to the more specialized practices required. This same needed specialization leaves the c ost of UHPC above that of other comparable structural materials. The lack of widespread availability and increased cost encourage developers to seek new, advantageous applications for UHPC. Even given its relative youth with respect to research and development, UHPC has been successfully employed in a number of completed project s based on its benefits of high durability, high strength, and architectural aesthetics. Examples of these three aspects are subsequently discussed. Introduced in the recap of UHPCs history, the earliest UHPC applications involved the use of the D.S.P. cement created in Denmark. The applications primarily focused on exploiting durability and pertained to protective structures and wear resistance. For instance, scoop feeders were built from UHPC for the c ement mill of Aalborg, Denmark and cavities caused by erosion in the Kinuza (Denmark) and Raul Leoni (Venezuela) dams were corrected with UHPC. The material performed well under testing devised to PAGE 64 64 qualify barrier materials and began employment in security structures such as vaul ts or AT Ms worldwide (Buitelaar 2004). Its continued high performance has warranted its use against blast and impact loads and in the construction of larger protective structures. For example, the ERDC has worked in conjunction with the United States Gypsum Company to develop and supply armor material ready for military use (Roth et al. 2008). Larger scale applications have included the construction of pedestrian and traffic bridges, as well as a runway expansion at the Haneda Airport (Tokyo, Japan). The more commo n bridge designs consist of precast and prestressed UHPC beams and girders. Examples of such include the Sherbrooke footbridge constructed in 1997 (the first prestressed hybrid bridge) and the Sakat a Mirai footbridge constructed in 2002. One of the most documented bridge applications has been the Grtnerplatz Bridge in Kassel, Germany. The bridge is a hybrid of UHPC and structural steel, the first of its kind, and consists of six spans totaling a distance of 132 meters. UHPC is well suited for bridge constr uction due to its combination of inherent high strength and use of fiber reinforcement. Its increase in strength over nor mal concrete allows for thinner and ultimately lighter precast elements. Another facet of research has focused on designing new beam and girder specifications to optimize UHPCs particular attributes. The steel fiber reinforcement on the other hand, is instrumental in replacing mild steel. The improved flexural strength due to the fibers, especially in tension, dually improves beam shear capacity. The capacity is increased enough to void the need of mild reinforcement, and without any stirrups or other passive steel, constructi on costs may be reduced (Acker and Behloul 2004). PAGE 65 65 An example of the architectural use of UHPC is the toll gate c onstructed for the Millau viaduct in France. The toll also serves to demonstrate use of the CeraCem product. Completed in 2004, its arched roof consists of 53 concrete segments joined longitudinally with prestressing rods and spans a total of 98 meters. UH PCs properties support a reduction in the structures depth, without which the implementation of concrete would likely not have been feasible. Utilizing thin members trims the overall weight of the structure (beneficial for the structural support) and produces an aesthetically pleasing design. 2.6 Summary This chapter prepared the literature review for the structural and dynamic analysis of normal strength and ultra high performance concrete columns subjected to severe blast and impact loads. Section 2.2 w as devoted to processing an elements response to dynamic loads and modeling the pressure exerted from explosives. Section 2.3 summarized the modes of failure and material behavior of reinforced concrete columns. The primary focus of the development of plastic hinges in reinforced concrete columns and the material of ultra high performance concrete were discussed in depth in sections 2.4 and 2.5, respectively PAGE 66 66 Table 2 1. Sample of UHPC mix proportions. [Adapted from Habel, K. and Gauvreau, P. 2008. Response to ultrahigh performance fiber reinforced concrete (UHPFRC) to impact and static loading. (Page 939, Table 1) Cement & Concrete Composites 30.] Constituent Type Weight (kg/m 3 ) Cement Portland cement 967 Silica fume White, specific surface: 15 18 m 2 /g 251 Silica sand Size < 0.5 mm 675 Steel fibers Straight (length/diameter: 10 mm/0.2 mm) 430 Superplasticizer Polycarboxylate 35 W ater 244 Table 22 Percent replacement of Portland cement with silica fume. [Adapted from Jayakumar, K. 2004. Role of Silica fume Concrete in Concrete Technology. (Page 169, Table 3). Proceedings of the International Symposium on Ultra High Performance Concrete. Kassel Ger many ] Strengths 0.0% 2.5% 5.0% 7.5% 10.0% 12.5% Compressive @ 90 days (MPa) 72.75 74.50 79.75 89.25 81.50 76.25 Split tensile @ 28 days (MPa) 4.46 4.52 4.65 5.47 4.71 4.52 Flexural @ 28 days (MPa) 6.42 6.44 6.90 7.63 6.51 6.36 Youngs modulus @ 28 days (GPa) 37.17 38.26 38.65 43.73 39.13 38.41 Table 23. Static mechanical tests with increasing volume of steel fibers. [Adapted from Rong Z. et al. 2009. Dynamic compression behavior of ultrahigh performance cement based composites. (Page 517, Table 2). International Journal of Impact Engineering 37 ] UHPC t ype Compressive strength (MPa) Peak value of strain (x 10 3 ) Elastic m odulus (GPa) Toughness i ndex c5 c10 c30 0 % fiber 143 2.817 54.7 2.43 2.43 2.43 3 % fiber 186 3.857 57.3 53.59 5.08 5.57 4 % fiber 204 4.165 57.9 4.57 6.32 7.39 PAGE 67 67 Table 2 4. Optimized UPHC mix proportions. [Adapted from Park, J.J. et al. 2008. Influence on the Ingredients of the Compressive Strength of UHPC as a Fundamental Study to Optimize the Mixing Proportion. (Page 111 Figure 10). Proceedings of the Second International Symposium on Ultra High Performance Concrete, Kassel, Germany. ] Mixutr e c omponent Proportion Sand 1.1 Cement 1.0 Filling Powder 0.3 Water 0.25 Silica Fume 0.25 Superplasticizer 0.016 Steel fiber (% of volume) 2.0 Table 25 Varied packing densities. [Adapted from Teichmann, T. and Schmidt, M. 2004. Influence of the packing density of fine particles on structure, strength and durability of UHPC. (Page 318, Table 3). Proceedings of the International Symposium on Ultra High Performance Concrete. Kassel, Germany ] Property NC C35 HPC C100 UHPC C200 Total porosity ( vol. % ) 15.0 8.3 6.0 Capillary pores ( vol. % ) 8.3 5.2 1.5 Table 2 6. Summary of CeraCem (structural premix) properties. [Adapted from Abdelrazig, B. 2008. Properties & Applications of CeraCem Ultra High Performance Self Compacting Concr ete. (Page 220, Table 1). Proceedings of the International Conference on Construction and Building Technology. Kuala Lumpur, Malaysia. ] Property Measured Value Compressive strength @ 2 days 122 MPa Compressive strength @ 28 days 199 MPa 3 point flexural strength @ 28 days 30 MPa 4 point flexural strength @ 28 days 29 MPa Tensile strength @ 28 days 8 MPa Modulus of elasticity @ 28 days 71 GPa Total shrinkage @ 1 year 725 microstrain Total shrinkage (with SRA) @ 1 year <500 microstrain PAGE 68 68 Figure 21 Example of frame loading [Adapted from Tran, B A 2009. Effect of short durationhigh impulse variable axial and transverse loads o n reinforced concrete column. MS thesis (Page 48, Figure 31). University of Florida, Gainesville, Florida.] Figure 22. Equivalent systems. A) Loaded beam element B) Equivalent SDOF system. [Adapted from Morency, D. 2010. Large deflection behavior effect in reinforced concrete columns under severe dynamic short duration load. MS thesis. (Page 79, Figure 23). University of Florida, Gainesville, Florida.] L H Pu(t) Pd(t) Pt(t) w(x) L u w u Me c KeB A PAGE 69 69 Figure 23 Reaction force schematic for beam with arbitrary boundary conditions. [Adapted from Krauthammer, T. and Shahriar, S. 1988. A Computational Method for Evaluating M odular Prefabricated Structural Element for Rapid Construction of Facilities, Barriers, and Revetments to Resist Modern Conventional Weapons Effects. ESL TR 8760. (Page 122, Figure 44). Engineering & Services Laboratory Air Force Engineering & Services Center, Florida .] Figure 24 Free air blast. [Adapted from Unified Facilities Criteria 2008. Structures to Resist the Effects of Accidental Explosions. UFC 334002. (Page 87, Figure 2 12) U.S. Department of Defense, Washington D.C. ] Q(t) 2 i1 i (t) 2Q(t) 1Q(t) M2M1 H Blast Standoff Distance Point of Interest Angle of Incidence Incident Wave Reflected Wave Path of Triple Point Mach Front PAGE 70 70 Figure 25. Pressure time history for an idealized freeair blast wave. Figure 26. Simplified pressuretime history Peak Pressure Positive Impulse Negative Impulse Positive Phase Duration Negative Phase Duration Arrival TimeTime Time Reflected Pressure Incident Pressure TposTnegPrPS 0 PAGE 71 71 Figure 27 Equivalent triangular pressuretime history Figure 28. Ideal concrete stress strain curve for uniaxial compression. t Tpos P(t) Pmaxipos t TposP(t) Pmaxipos Strain, c0.15fc fc 0 Linear Parabolic u PAGE 72 72 Figure 29. Confined concrete stress strain curve. [ Adapted from Krauthammer, T. and Shahriar, S. 1988. A Computational Method for Evaluating Modular Prefabricated Structural Element for Rapid Construction of Facilities, Barriers, and Revetments to Resist Modern Conventional Weapons Effects. ESL TR 8760. (Page 11 Figure 3 ) Engineering & Services Laboratory Air Force Engineering & Services Center, Florida.] Figure 210. Steel reinforcement stress strain curve Concrete Strain fc/ fc 01.2 0 8 0.4 Strain, sfsu Stress, fssh Linear su yfy PAGE 73 73 Figure 211. Concrete section with strain, stress, and force distributions. [Adapted from Park R. and Paulay T. 1975. Reinforced Concrete Structures. (Page 201, Figure 6.5 ) John Wiley & Sons Inc., New York, New York. ] Figure 212. Typical m oment curvature diagram [ Adapted from Park R. and Paulay T. 1975. Reinforced Concrete Structures. (Page 198, Figure 6.3 ). John Wiley & Sons Inc., New York, New York. ] b kd s1s 2cms 3s 4 Elevation Section Strain Stress hNeutral Axisfs2fs 1fs4fs 3 Internal Forces External Forces S4S3S2S1CcP M h/ 2 kd Moment, M Curvature, First crack First yield of steel PAGE 74 74 Figure 213. Flexureshear cracking pattern. [ Adapted from Krauthammer, T. and Shahriar, S. 1988. A Computational Method for Evaluating Modular Prefabricated Structural Element for Rapid Construction of Facilities, Barriers, and Revetments to Resist Modern Conventional Weapons Effects. ESL TR 8760. (Page 27 Figure 8 ). Engineering & Services Laboratory Air Force Engineering & Services Center, Florida.] Figure 214. Influence of shear model without web reinforcement [ Adapted from Krauthammer, T. and Shahriar, S. 1988. A Computational Method for Evaluating Modular Prefabricated Structural Element for Rapid Construction of Facilities, Barriers, and Revetments to Resist Modern Conventional Weapons Effects. ESL TR 8760. (Page 35 Figure 11). Engineering & Services Laboratory Air Force Engineering & Services Center, Florida .] Initiation of cracking Flexureshear crack Secondary cracking 100% 90 80 70 60 50 40 30 20 10 0 0 1 2 3 4 5 6 8 7P2P3P1 a/d Mu / Mfl = 0 5 % = 2 80% = 1 88% = 0.8% PAGE 75 75 Fig ure 215. Relationship between direct shear stress and shear slip. [ Adapted from Tran, B.A. 2009. Effect of short durationhigh impulse variable axial and transverse loads on reinforced concrete column. M.S. thesis (Page 34, Figure 2 13). University of F lorida, Gainesville, Florida.] Figure 216. Compression and tension membrane behavior [ Adapted from Morency, D. 2010. Large deflection behavior effect in reinforced concrete columns under severe dynamic short duration load. M.S. thesis. (Page 94 Figure 2 21). University of Florida, Gainesville, Florida.] Shear Stress, Slip, maxe12max43L Beam/column initial configuration Compressive membrane range Tension membrane range PAGE 76 76 Figure 217. Ductile and brittle concrete behavior. Figure 218. Curvature along a beam at ultimate moment [ Adapted from Park R. and Paulay T. 1975. Reinforced Concrete Structures (Page 243 Figure 6.26). John Wiley & Sons Inc., New York, New York. ] Ductile Behavior Brittle Behavior Deflection Load Mucrack M uA) B) C) uy lp PAGE 77 77 Figure 219. Plastic hinge of a cantilever column [ Adapted from Park R. and Paulay T. 1975. Reinforced Concrete Structures. (Page 245 Figure 6.28). John Wiley & Sons Inc., New York, New York. ] Figure 220. Comparison of plastic hinge length expressions. [Adapted from Bae, S. and Bayrak O. 2008. Plastic Hinge Length of Reinforced Concrete Columns (Page 292, Figure 2). ACI Structural Journal 105 (3). ] Cantilever Moment Diagram Curvature DistributionL Mulpuy 12 10 8 6 4 2 0 0 0 2 0 4 0.6 0.8 1.0 1 2 1 4 Shear SpanDepth Ratio (L/h) Sheikh and Khoury Park et al. Paulay & Priestly Baker Mattock Corley 0 5 Po0 3 Po0 0 Po #9 bar #7 bar PAGE 78 78 Figure 221. Increasing brittleness with strength. [Adapted from Shah, S.P. and Weiss W.J. 1998. Ultra High Performance Concrete: A Look to the Future. (Figure 4 ). ACI Special Proceedings from the Paul Zia Symposium Atlanta, Georgia. ] Figure 222. Stress strain diagrams normal concrete vs. UHPFRC. [Adapted from Wu C. et al. 2009. Blast testing of ultra high performance fibre and FRP retrofitted concrete slabs. (Page 2061, Figure 1 ). Engineering Structures 31. ] Strain (mm/mm) 0 0025 0 0050 0 0075 0.0100 200 150 50 100 0Ultra High Strength Concrete High Strength Concrete Normal Strength Concrete Compressive strain0 005 0.010 100 200 0Ductility plateau NSC UHPFRC Tensile strain0 005 0 010 15 30 0Ductility plateau NSC UHPFRC Gradual softening PAGE 79 79 Figure 223. Production cost with respect to compressive strength [Adapted from Markeset G. 2002 Ultra High Performance Concrete is Ideal for Protective Structures. HighPerformance Concrete. (Page 137 Figure 9 ). Performance and Quality of Concrete Structures Proceedings, Third International Conference. ] Figure 224. Tensile constitutive law of UHPFRC. [Adapted from Association Franaise de Gnie Civil 2002. Btons fibrs ultrahautes performances Recommandations provisoires (Page 17, Figure 1.2 ). Association Franaise de Gnie Civil. France. ] 45 65 85 105 125 145 165 185 205 225 1 3 5 7 9 11 13 Compressive Strength ( MPa ) ftj ( wi) wiwCrack opening0Elastic strain PAGE 80 80 CHAPTER 3 METHODOLOGY 3.1 Overview The fo cus of this study is to examine the Dynamic Structural Analysis Suites ability to properly model the formation of plastic hinges in concrete columns T he engineering materials of interest include both normal strength and ultra high performance concrete, and b y realistically modeling their hinge and curvature behavior advanced insight may be gathered with respect to the failure mechanisms of such concrete systems. Though previous studies have sought to analyze plastic hinging in NSC columns, much of the UHPC focus has been on developing its basic functionality. The brevity of analysis available is ther efore expanded by this effort, and whether UHPC exhibits similar plastic behavior to NSC is demonstrated This chapter elaborat es on the process followed to complete the study. Section 3.2 measures DSAS V3.2.1s (Astarlioglu 2008) capabilities in terms of monitoring the hinge formation over time and describes the program adjustments made to more acutely account f or the mechanism. Section 3.3 establishes the Feldman and Siess (1958) experimental beams used to verify DSASs compatibility with material input a nd behavioral analyses while section 3.4 overviews the employment of the finite element code A BAQUS/Explicit V6.10 (SIMULIA 2010) to further validate the DSAS results Finally, section 3.5 prepares the parametric study and section 3.6 summarizes of the i mplemented methodology and the prominent analytical issues to be discussed. 3.2 Plastic Hinge Development in DSAS DSAS performs a simplified dynamic analysis of concrete columns subjected to severe and short duration loading. Aligned with the methods described in sections 2.2 PAGE 81 81 and 2.3, t he program considers the combined incurrence of axial and transverse loads as well as the effects of large deformation behavior on representative singledegreeof freedom systems. Standard data for output includes the in dication of flexural and/or shear failure, the deformationtime response at the critical point of interest, and the corresp ondent pressureimpulse diagram Though the basic output provi des the maximum response at the critical point via a SDOF analysis, the program is also capable of track ing the relationship between load and deflection of designat ed elements along the columns length. The record of a columns deformation ultimately allows for the rotation and curvature along its length to be monitored, and an evaluat ion of the displacement, rotation, and curvature output provided by the originating DSAS program follows To explore the ideas outlined and analyze DSASs abilities, a standard simply supported reinforced concrete test beam ( similar to Beam 1C d iscussed in section 3.3) is subjected to a varying point load at its midspan. The moment curvature relationship of the NSC beam is contingent upon its cross sectional properties and is provided in Figure 31 D ependent on this relationship, the loaddeflec tion curve of Figure 32 is generated by DSAS. This curve is relative to the beams critical location of maximum response, i.e. the point at midspan. For each data point on the loaddeflection curve, DSAS dually records the displacement at nodes along the beams span and thereby enables the creation of a deformed beam schematic. DSAS assumes the size of the elements between nodes to be half the beams depth ( h ). With the motivation to limit and thereby expedite calculations, the fewe st number of nodes and elements that can accurately account for the structural members shape is desired. The appropriateness of PAGE 82 82 using 0.5 h as the approximate elemental length is to be determined through a later comparison with a finer mesh. Since the test beams cri tical location corresponds to midspan, an element that expands 0.5 h to either of its side s exists and is taken to be the plastic hinge, for a total plastic hinge length of h The remainder of the beam is divided into equally sized elements of approximately 0.5 h ( a whole number of elements is established) Note that f or this particular beam case, the total number of nodes, including one midhinge at midspan, sums to nineteen Each node represent s a point at which displacement and rotation values are reported. Although the curvature is not directly provided through DSAS, a simple calculat ion is performed based on the rotation change per element length. Five points along the loaddeflection curve of Figure 3 2 are highlighted and correspond to the beams tracked points. Figures 33 through 35 respectively display the deformation, rotation, and curvature across the beam span per the five designated point s of Figure 3 2. The beam s shape becomes m ore triangular with the growth of the midspan displacement as expected, and a similar trend is observed through the sharpening of the rotation curves slope across the plastic hinge element. Regarding Figure 35, the curvature is calculated as the change i n curvature per element and is therefore specified at elemental midpoints. Introduced in the literature review, t he curvature across a plastic hinge is theoretically idealized as constant and may be recognized by a drop to or below yield on either of its side s. From the moment curvature relationship, the yield curvature for th e beam case is approximately 0.00 03 in1. The yield curvature is mark ed by a dotted line in Figure 35, and the chart demonstrates that PAGE 83 83 the yield condition is exceeded outside of the hinge element. This observation indicates the need to reevaluate the DSAS programming for hinge generation. Given the 0.5 h (6 inch) length of each element, the beam meshing could plausibly be further refined. It is logical that a more defined picture would be found by refining the mesh, but doing so would also increase the computational cost. For comparison, the experimental setup was run via DSAS for a nodal count of fifty one. The alternate DSAS configuration adjusts the elemental lengths to conform to a specified number of elements and does not account for a designated hinge element. As completed for th e previous mesh density, Figure 36 shows the beams loaddeflection relationship, while Figure s 3 7 through 39 display the deflected shapes, rotations and curvatures across the beam span. Due to the increase in the number of elements, Figure 310 is provided to show a zoomed view of the curvature. Figure 310 is intended to emphas ize the yield curvature and more clearly illustrate the plastic hinge development. Contrasting the two beam cases highlights the effect of increasing the number of elements along the beam. Figure 35 demonstrates that the concrete length surpassing the yield curvature increases as t he midspan deflection increases and that the inferred hinge would exceed its length h The hinge length estimated for the first beam case from the curvature output is on the order of 30 inches. This approximation is considerably rough however due to the elements themselves being large and the curvature point s distantly spread apart. To focus on the second case depicted in Figures 39 and 3 10, the yielded area fluctuates as the midspan deformation grows, and the ultimate hinge length is approximated at 10 inches. It is clear that the number of elements present affects the clarity and insight on the beam behavior. PAGE 84 84 The second beam case (51 nodes) illustrates the need to implement a properly sized plastic hinge element in order to obt ain a realistic ultimate curvature. The curvature across a hinge ought to be roughly constant, and as the critical sections element decreases in size, the rotation change across it becomes too drastic. The ultimate curvature from Figure 31 is approximately 0.028 in1, and given that the same moment curvature relationship applies to the two cases, the ultimate curvature should not be exceeded as it is in Figure 39. A discussion on the calculation of a satisfactory plastic hinge length and the corresponding DSAS modifications made follows To determine the plastic hinge length typical of reinforced concrete beam columns, the equations presented in the literature review are em ployed. Table 31 presents the propos ed hinge length equations and resultant calculations f or the test beam. Included are the expressions issu ed by Corley, Mattock, Sawyer, Paulay and Priestly, and Sheikh and Khoury (Bae and Bayrak 2008) The Bae and Bayrak (2008) expression is omitted beca use of its minimal estim at e without the presence of a known axial load. Due to a number of studies relating hinge length to either beam height or tension steel depth, a hinge length dependent on effective depth is addit ionally c onsidered. For columns the confined concrete area is the area providing primary support against the severe loading inherent of blasts or impacts. Therefore, this effective depth of resistance is understood as the true column depth. The assessment of the two DSAS cases and their comparison with the Table 31 expressions indicates that the effective depth equation may be a reasonable representati on of hinge length. Though Corley and Mattocks expressions equate to similar values for the length, the discernable difficulty in calculating the z parameter (the distance from critical point to contraflexure) per PAGE 85 85 combination of boundary and loading conditions elicits a desire for simplified calculations. The effective depth provides the wanted, and a comparable result for a decreased computational cost. The effective depth equation is therefore chosen as the modified DSAS hinge length to better represent the concretes behavior. To implement the derivation and use of this value, the algorithm t hat establishes the elemental lengths used by DSAS is slightly adjusted. Because the critical region(s) from which the plastic hinge(s) spreads is based on user defined boundary conditions, the full algorithm considers each boundary pairing (simplefixed, free fixed, etc.) separately. For each boundary set of the original program, an element the length of the beam columns depth ( h ) is centered on the critical section, whereas for the modified algorithm, an element the size of the effective depth is substit uted at the critical section. The elemental length outside of the hinge region is also adjusted between the original and modified algorithms. Although the length remains initially set at 0.5h it is made an adaptable parameter in order to ease the refinement process. It appears that the increase in nodes only supplemented the adherence to yield curvature outside of the hinge region for the last points on the loaddeflection curve, and the larger element sizes are thereby similarly applicable. 3.3 Experiment al Validation Case To standardize the column study beams experimented upon by Feldman and Siess (1958) are examined. The beams marked as 1c and h by Feldman and Siess are subsequently referred to as Beam 1C and Beam 1H. The verification of Beam 1C and Beam 1H s behaviors offers a basis for the parametric studys constitutive material model s. The documented B eam 1C case is also r un to configure the agreement between the DSAS analysis and the use of the finite element code PAGE 86 86 ABAQUS. W ork has previously been completed to model the Feldman and Siess beams via DSAS for comparison wit h the experiment al output (Tran 2009; Morency 2010), but this current effort confirms the latest versions compatibility and continues on to extract the beams curvature data fr om DSAS. The geometry and experimental setup for the Feldman and Siess beams are diagrammed in Figures 3 11 and 312 and their concrete and reinforcing steel material properties are detailed in Table 32. The measurements and properties of beam s 1C and 1H are drawn from Table 2 of Feldman and Siess (1958). The steel layers measure 1.5 and 10 inches from the top of the beam, while the open stirrups of Beam 1C and the closed stirrups of Beam 1H maintain approximately 0.75 inches of cover. For the material properties not expressly provided, the model employs assumed standard values in DSAS such as an ultimate steel strength of 90,000 psi. An important item to note is the beams increased depth at midspan, a column stub upon which the point load is exerted. At its current status DSAS does not allot for a varying depth along a members span, and the results must be considered as missing the effects of th e 6 inch by 12 inch column stub. Aside from the material, boundary, and loading input, DSAS also requires the selection of beam behavioral effects, including diagonal shear, compression buckling, and strain rate. Table 33 compares how each of these considerations when acting alone, affects the peak deformations of the two beams and indicates whether failure occurs. Noting Beam 1C s maximum deflection of 3.0 inches during the experiment, it is evident that diagonal shear does not play a significant role in its case while strain rate does. Beam 1H on the other hand experimentally peaked at 8.9 inches, and as PAGE 87 87 demonstrated in Table 33, diagonal shear is deemed to work in conjunction with the estimated strain rate. As Figure 311 depicts, the beams experience a varying point load at midspan. The loadings for beams 1C and 1H, severe and short in duration, are plott ed in Figures 3 1 3 and 31 4 respectively. Due to their impulsive nature, the application of dynamic increase factors (DIF s) to the material strengths is supported. Feldman and Siess report approximated initial strain rates in their Table 4, upon which val ues for the DIFs may be based; however, the internal DSAS estimation results in alternate strain rate and DIF values Table 34 compare s the use of either reference. The employed DIFs derive from the consideration of both strain rate determinations and are listed in Table 35 The results of the Feldman and Siess investigation are detailed in Table 36 and compared against those of DSAS for Beam 1C and Beam 1H. Both DSAS cases account for strain rate effects, while only Beam 1H accounts for diagonal shear. T able 3 6 denotes compatibility between the experimental and analytical output and establish es DSASs functionality. As configured, the differences between the reported maximum response of beams 1C and 1H are 0 3 % and 2.3%, respectively. Also considering D SASs ability to capture the elastic plastic behavior of concrete beams, the difference in the permanent deformations for B eam 1C is 6 8 % while that for B eam 1H is 5.6%. Although several assumptions are built into the model s, including the dependence on Feldman and Siesss static tests to detail the material strengths and the variances in the ultimate steel strength and column stub, the magnitudes and range s of deflections reported are within engineering reason and appropriate. PAGE 88 88 3.4 Finite Element Analysis The DSAS programming is tested and validated against the finite element software ABAQUS to certify their correlation. Finite element analysis (FEA) provides a thoroughly detailed look at a structural system s operation, including and not limited to deformation and stress distribution over time. FEA is based on approximating solutions to the partial differential equations relevant of material behavior, and it s accuracy can be as great as that of the data put into it (i.e. the ability t o incorporate nonlinearity depends on how well the phenomena is understood) Although a finite element solution can be exceptionally refined and complex its use incurs a cost in computational time, and this is where the expediency of DSAS becomes largely beneficial This section details the ABAQUS validation of the Feldman and Siess B eam 1C including the standard assumptions and practices of the coding system, in order to establish the material models employed for the parametric studys columns. Within t he ABAQUS input file of B eam 1C, the beam geometry adheres to Figure 3 11. The simple supports are configured as rollers, and the midspan plane is fixed in the axial direction to ensure symmetry restrictions. The ABAQUS schematic also lacks the existence o f the beams increased depth at midspan since the results are desired for comparison with DSAS, not the experimental data. As shown in Figure 311, the beam does extend 7 in. past the supports in an effort to more realistically capture the boundary conditions and stresses at work. The point load is centered at midspan with a few adjacent nodes set as rigid to avoid severe concrete discontinuity at impact, and gravity is dually enacted. Figure 31 5 illustrates th e beam schematic via the ABAQUS interf ace. Lastly, the mesh size of 0.5 inches is a result of running trials to simultaneously capture convergence and promote minimal computation time. PAGE 89 89 T he material model s are entered in to ABAQUS in two separate regimes: the linear elastic and the plastic. T he normal strength concrete is modeled by the modified Hognestad curve and tension stiffening as displayed in Figure 316 while the steel reinforcement model is an adjusted Hsu (1993) curve for steel embedded in concrete per Figure 317 (the curve is compare d to the strain hardening steel model of section 2.3) Though the yield point is based on the Hsu model, the remainder of the model is linearized to reach the ultimate stress. Note that the material models are illustrated without the application of DIFs N onlinear inelastic behavior in ABAQUS requires the specification of true stress and plastic strain experienced post yielding and the input must be aptly adjusted. The NSC behavior is more specifically detailed by the concrete damaged plasticity (CDP) m odel, the parameters of which derive from previous studies (Morency 2010). The dilation angle, eccentricity, fbo/fco, K, and viscosity parameters are respectively set to 40, 0.1, 1.16, 0.6667, and 0 (SIMULIA 2010) The CDP model is designed to account for the lessening of elastic stiffness through cyclic loading in the plastic regimes, and it is therefore well suited for the impos ed dynamic load. The Beam 1C ABAQUS output is compared in Table 37 a gainst the DSAS results. Overall correlation exists between the maximum response data gathered. D emonstrated by th e case however is ABAQUSs negligible depiction of the beams rebound after achieving its peak def lection Fi gure 31 8 d isplay s the lack of congruency. A reasonable explanation for the discrepancy is the lack of specifics known about the concretes damage criteria. Once again, the modeling is largel y theoretical and results mu st be approached from this perspective. The contrast credit s DSAS with the ability to more succinctly evaluate a members el astic and plastic deformations PAGE 90 90 The rotation and curvature data at the peak response are compared in Figures 319 and 320, respectively. Note that the depicted beam length is 120 inches whil e the span is 106 inches and that the ABAQUS nodal points are positioned every four inches along the tension rebar. Given the symmetry of the analysis, the rotation data was drawn from ABAQUS for half of the span and then inversely replicated for the secon d half span. Excepting for the ABAQUS scatter ing the graphs communicate similar behavioral descriptions and a similar procedure for relaying deformation specifics is thereby implemented throughout the parametric study for a more indepth analysis 3.5 Parametric Study The congruency shown to exist between DSAS and ABAQUS for the dynamically loaded beam s provides the foundation for conducting a parametric study. The study expands to incorporate UHPC as a second investigated material model, in addition to the reinforced NSC formerly addressed. The investigation also varies the boundary and loading conditions imposed on the columns for assessment T he basic experimental procedure is now introduced with the analysis of results processed in Ch apter 4. In term s of the boundary conditions, simply supported and fixedend columns are the primary focus. These forms of support are common of columns situated within protective structures. The ir con sideration encompasses a broader scope of conditions (fixed free, simpl e fixed, etc.) as they represent the least and most complex hinge behaviors of a single span column. For a simply supported column, a single plastic hinge is expected to develop at a critical location (analogous to that of the maximum momen t) along the col umn span, as shown by the validating case. For a fixedend column, three plastic hinges are expected to evolve in sequence of maximum moment magnitude: two at the supports and one at a critical location along its length. PAGE 91 91 Various instances of blast and axial loads are also imposed on the four column types (combinations of NSC, UHPC, simple supports, and fixed supports) The blast pressures are assumed to be uniformly distributed across the column span and linearly decrease through time as per the triangular loading of Figure 27. Table 38 provides a sample of the blast parameters imposed for the trial run s, including the reflected pressures and impulse s to which the columns are subjected and the resultant positive duration time s for proper configur ation The blast pressure and impulse magnitudes are intended to correlate with the work performed by Morency (2010) and derive from the blast parameter curves proposed by the Unified Facilities Criteria (2008) The study primarily focuses on the set of blast s run in the absence of axial load, while additional trial trials are placed under axial loads to vary between 750 and 1750 kips for the NSC column and 1750 and 5750 kips for the UHPC column. The parametric study aims to evaluate the appropriateness of DSASs analysis. Since the plastic hinge is represented by a single element, a constant curvature is inherently imposed across it. However, the region outside of the hinge is not expected to subs tantially exceed the yield curvature and requires examination. The range of blast pressures is intended to mark different points along the columns loaddeflection curve, thereby promoting various deformation and curvature stages and the creation of evolut ion diagram s. Finally, the employed hinge length calculation is analyzed and any necessary recommendations for its correction are made. 3.6 Summary The methodology followed for the analysis of the plastic hinge formation in normal strength concrete and ul tra high performance concrete columns has been presented throughout this chapter The verification of DSASs and ABAQUS s compatibility with PAGE 92 92 the experimental data of Feldman and Siess for Beam 1C and Beam 1H establishes a basis for the parametric study. Wi th the acceptance of the material model for normal reinforced concrete the next chapter develops the UHPC material model and narrows on the effects held by varying boundary conditions and combinations of blast and axial loads on the hinge formation. The parametric studys analysis also convey s the modeling assumptions made and their respective impact. Table 31. Plastic hinge length expressions and calculations for the test b eam. Expression Equation Length (in) Corley 8.4 Mattock 0.5d + 0.05z 7.7 Sawyer 0.25d + 0.075z 6.6 Paulay and Priestly 0.08L + 0.15d b f y 14.7 Sheikh and Khoury 1.0h 12.0 Effective Depth d d 8.5 Table 32. Beam 1C and Beam 1H material properties. Property Beam 1C Beam 1H Concrete f c (psi) 6000 5775 E c (ksi) 4000 4450 f t (psi) 733 935 Steel f y (psi) 46080 47170 E s (ksi) 29520 34900 y (in/in) 0.0016 0.0014 o (in/in) 0.0144 0.0150 T able 33. DSAS peak responses per the inclusion of behavior al effects Effect Beam 1C (in) Beam 1H (in) None 7.40 7.65 Fail Diagonal Shear 9.42 Fail 8.65 Fail Compression Buckling 6.69 Fail 6.74 F ail Estimated Strain Rate 2.83 3.93 Diagonal Shear & Estimated Strain Rate 6.61 8.70 PAGE 93 93 Table 34. Strain rate effect on DSAS output. Effect M aximum d eflect ion 1C (in) F inal d eflect ion 1C (in) Maximum d eflect ion 1H (in) F inal d eflect ion 1H (in) None 7.40 5.52 8.65 Fail 8.65 DSAS Strain Rate Estimate ( 1 4 3 in/in) 2.83 2.18 8.70 7.92 Feldman & Siess Strain Rate Estimate ( 0. 3 in/in) 3.47 2.82 9.08 Fail 9.08 Table 35. Dynamic increase factors for beams 1C and 1H Beam Concrete in c om pression, f c Concrete in t ension f t Yield s teel, s trength, f y Ultimate s teel s trength, f u 1C 1.3 2.5 1.2 1.2 1H 1.3 3.0 1.2 1.2 Table 36. Comparison of experimental and DSAS output for beams1C and 1H. Beam Exp. m ax imum d eflection (in) DSAS m ax imum d eflection (in) Exp. p ermanent d eflection (in) DSAS p ermanent d eflection (in) 1C 3.0 3.01 2.2 2.35 1H 8.9 8.70 7.5 7.92 Table 37. Comparison of DSAS and ABAQUS output for Beam 1C. Deflection DSAS (in) ABAQUS (in) % Difference Maximum 3.01 3.15 4 4 % Final 2.35 2. 96 20 6 % Table 38. Sample of parametric study blast trials. Trial No. Reflected p ressure (psi) Reflected i mpulse (psi ms) Duration t ime (ms) 1 1777 1134 1.276 2 2922 1875 1.283 3 3602 2369 1.315 4 4208 2844 1.352 5 4757 3305 1.390 PAGE 94 94 Figure 31. Moment curvature diagram for test beam. Figure 3 2. Loaddeflection curve for test beam with 19 nodes. 0 10 20 30 40 50 60 70 0 0.005 0.01 0.015 0.02 0.025 0.03Moment (ft kips) Curvature (in1) 0 5 10 15 20 25 30 35 0 2 4 6 8 10 12Load (kips) Displacement (in) Pt. 1 Pt. 2 Pt. 3 Pt. 4 Pt. 5 PAGE 95 95 Figure 33. Progressive deformed shape of test beam with 19 nodes. Figure 34. Progressive rotation of test b eam with 19 nodes. 0 2 4 6 8 10 12 0 10 20 30 40 50 60 70 80 90 100Displacement (in) Coordinate (in) Pt. 1 Pt. 2 Pt. 3 Pt. 4 Pt. 5 0.25 0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2 0.25 0 10 20 30 40 50 60 70 80 90 100Rotation (rad) Coordinate (in) Pt. 1 Pt. 2 Pt. 3 Pt. 4 Pt. 5 PAGE 96 96 Figure 35. Progressive curvature of test b eam with 19 nodes. Figure 36. Loaddeflection curve for test b eam with 51 nodes. 0.03 0.025 0.02 0.015 0.01 0.005 0 0 10 20 30 40 50 60 70 80 90 100Curvature (1/in) Coordinate (in) Pt. 1 Pt. 2 Pt. 3 Pt. 4 Pt. 5 0 5 10 15 20 25 30 35 0 2 4 6 8 10 12 14Load (kips) Displacement (in) Pt. 1 Pt. 2 Pt. 5 Pt. 4 Pt. 3 PAGE 97 97 Figure 37. Progressive deformed shape of test b eam with 51 nodes. Figure 38. Progressive rotation of test b eam with 51 nodes. 0 2 4 6 8 10 12 0 10 20 30 40 50 60 70 80 90 100Displacement (in) Coordinate (in) Pt. 1 Pt. 2 Pt. 3 Pt. 4 Pt. 5 0.25 0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2 0.25 0 10 20 30 40 50 60 70 80 90 100Rotation (rad) Coordinate (in) Pt. 1 Pt. 2 Pt. 3 Pt. 4 Pt. 5 PAGE 98 98 Figure 39. Progressive curvature of test b eam with 51 nodes. Figure 310. Zoomed progressive curvature of test b eam with 51 nodes. 0.1 0.08 0.06 0.04 0.02 0 0 10 20 30 40 50 60 70 80 90 100 Curvature (1/in) Coordinate (in) Pt. 1 Pt. 2 Pt. 3 Pt. 4 Pt. 5 0.006 0.005 0.004 0.003 0.002 0.001 0.000 20 30 40 50 60 70 80 90Curvature (1/in) Coordinate (in) Pt. 1 Pt. 2 Pt. 3 Pt. 4 Pt. 5 PAGE 99 99 Figure 311. Experimental setup for b eam s 1C and 1H. Figure 312. Cross section of b eam (Beam 1C has open stirrups ; Beam 1H, closed ) L = 106 in. No. 3 stirrups @ 7 in. 2 No. 6 bars 2 No. 7 bars Applied Force 1.5 in. 6.0 in. 2.0 in. 12.0 in. #6 rebar #3 stirrup #7 rebar PAGE 100 100 Figure 31 3 Dynamic loading of Beam 1C. Figure 31 4 Dynamic loading of Beam 1H 5 0 5 10 15 20 25 30 35 0 0.02 0.04 0.06 0.08 0.1 0.12Load (kips) Time (s) 5 0 5 10 15 20 25 30 35 40 0 0.02 0.04 0.06 0.08 0.1 0.12Load (kips) Time (s) PAGE 101 101 Figure 315. ABAQUS interface and Beam 1C schematic. Figure 316. Modified Hognestad curve for Beam 1C s normal strength concrete. 2000 0 2000 4000 6000 0.004 0.002 1E18 0.002 0.004Stress (psi) Strain (in/in) Compression Tension PAGE 102 102 Figure 317. Beam 1C steel material model. Figure 318. Comparison of Beam 1C s response 0 30000 60000 90000 0 0.03 0.06 0.09 0.12 0.15Stress (psi) Strain (in/in) Hsu (See section 2.3) 0 0.5 1 1.5 2 2.5 3 3.5 0 0.02 0.04 0.06 0.08 0.1 0.12Deflection (in) Time (s) Experiment DSAS ABAQUS PAGE 103 103 Figure 319. Beam 1C rotation at peak response. Figure 320. Beam 1C curvature at peak response. 0.08 0.06 0.04 0.02 0 0.02 0.04 0.06 0.08 0 12 24 36 48 60 72 84 96 108 120Rotation (rad) Coordinate (in) DSAS ABAQUS 0.012 0.009 0.006 0.003 0.000 0.003 0 12 24 36 48 60 72 84 96 108 120Curvature (in1) Coordinate (in) DSAS ABAQUS PAGE 104 104 CHAPTER 4 ANALYSIS 4.1 Overview Following the demonstration of DSASs ability to realistically model the response of reinforced concrete members subjected to severe dynamic loads, the study narrows its focus onto concrete columns. Just as for the previous Beam 1C work, DSAS V3.2.1 (Astarlioglu 2008) and ABAQUS/Explicit V6.10 (SIMULIA 2010) are the analysis programs used to complete the parametric study outlined in section 3.5. Altho ugh the two programs have been comparatively validated from a maximum deformation standpoint, the curvature schemes they project are now more critically evaluated. The presented analysis is theoretical and seeks to establish a reasonable method for processing the inelastic behavior of NSC and UHPC. Furthermore, in contrast to the extensive documentation of NSCs behavior is the lack of an equally well defined UHPC model and the study implements an experiment al model derived from multiple proposals The ongoing characterization of UHPC and the various mixture proportions and construction techniques that may be us ed for its creation leave the proposed UHPC constitutive models subject to a degree of variability This chapter serves to study the formation of plastic hinges in normal strength and ultra high performance concrete columns and to compare the two material s inelastic and large deformation characteristics. Section 4.2 establishes the NSC columns geometric and material properties and addresses the evolution of its deformation and curvature under a series of increasing blast loads for both simple and fixed b oundary conditions. Section 4.3 introduces the UHPC constitutive material model employed and conducts th e same column inspection of UHPC as for the NSC and section 4.4 PAGE 105 105 processes the comparison and contrast of the two concrete types. Section 4.5 then devel ops a number of additional studies to supplement the background and relevance of the parametric study. For instance, the appropriateness of the plastic hinge length calculation implemented in DSAS as well as the effects of diagonal shear and tension membra ne behaviors are considered. Finally, section 4.6 briefly summarizes the key outcomes of the performed analys e s. 4.2 Normal Strength Concrete Column First examined is a normal strength, reinforced concrete column subjected to the Table 38 series of blast loads. Figure 41 illustrates the columns 16 inch square crosssection with eight #7 reinforcing bars confined by #3 ties s paced at 12 inches. The column spans 12 feet and overhangs t he supports by 6 inches (used for the ABAQUS modeling) as depicted in Figure 42. The concrete and steel pro perties are drawn from those of Beam 1C listed in Table 3 2. Note that adherence to the ACI31808 ( American Concrete Institute 2008) code is maintai ned for the structural integrity of a column constructed under precast conditions. The overall column geometry and blast loads also resemble those employed by Morencys (2010) investigation in order to expand the breadth of data gathered for potential comparison. The moment curvature diagram (from DSAS) for the NSC column is presented in Figure 43. As a characteristic of the member cross section, the moment curvature diagram is unaffected by the combination of boundary and uniform loading conditions implem ented. The displayed curve includes the consideration of diagonal shear as well as tension membrane behavior and the observed yield and ultimate curvatures are approximately 0.0005 in1 and 0.03 in1, respectively. T he curvatures induced on a NSC PAGE 106 106 column w ith simple supports are subsequently detailed, followed by the alteration of the boundary conditions to fixed supports. To model the column in ABAQUS, the loads time step i s set to invoke the explicit mode for dynamic analysis. The concrete i s created as a homogeneous soli d while the steel reinforcement is generated by beam elements, and the dynamic increase factors applied to the material strengths a re appropriated based on DSASs estimated strain rate per trial. The employed DIFs are catalogued in Table 4 1 for reference. By analysis, a mesh density of one inch was ultimately deemed computationa l l y efficient. 4.2.1 Simpl y Support ed NSC Column For this study, the simplesupport condition is defined by pin and roller boundary conditions. The simply supported NSC column described is subjected to each of five b lasts ( labeled trials NS1 through NS5) and the DSAS max imum response results are listed in Table 42 T he column deflection s range between 2.7 and 12.7 inches, well into the large def ormation regime. The de flected shape and curvature data drawn from the DSAS output are based on the load deflection curve of Figure 44 which is respective to the point of maximum response, or the midspan point. The progressions of the co lumns deformation, rotation, and curvature as the blast pressure increases are displayed in Figures 4 5 through 4 7, respectively. The columns effective depth equates to 13 inches and this value is employed by DSAS as the plastic hinge length. Since the maximum moment generated by a uniform pressure load occurs at midspan, it is also the expected hinge location Figure 47 chronicles the spread of the curvature over the series of blasts. The NS1 blast serves to push the column past yielding and at this point, the length of the columns span surpassing the yield curvature is nearly 50 inches (25 inches to either side of midspan) PAGE 107 107 which substantially violates the predicted plastic hinge length. Based on the NS1 and NS2 curves, it may be plausible tha t DSAS errs in perceiving the proper curvature across the beam, the mesh is too rough to represent the behavior, or the plastic hinge does not expressly generate as theorized. However, because even the NS1 blast significantly exceeds yield, the column curv atures for small deformations are examined. In Figure 48, the four charted points represent the midspans deflection increasing from 0.96 to 1.12 inches ( deformations just post yield) The curvature is shown to lock at a distance of 15 inches to either si de of midspan as the curvature across the hinge element rises. This behavior reflects that expected of a plastic hinge, and although the designated hinge still exceeds the predicted 13 inches, it is possible that the hinge spreads this distance on either s ide of midspan or that, again, the mesh may be too large to properly perceive the tightened change in rotation about midspan. Now transitioning back to the large deformation curvature patterns represented in Figure 47, although yield is exceeded for the N S4 and NS5 trials, there exists an obvious jump in curvature upon reaching 0.00 3 in1. If this area is to be characterized as the plastic hinge, its length is roughly equal to the one present at post yield (Figure 4 8) The trend observed may also be relat ed back to the cross sectionss moment curvature, where an increase in the supportable moment is demonstrated after the initial yielding plateau (approximately at 0.002 in1). In order to decipher whether this re generation of the later hinge is appropriate, the ABAQUS results are consulted. Table 42 lists the ABAQUS peak responses alongside those from DSAS for the blast trial s. The compatibility of the results varies in accuracy ; the occurrence of which is discussed in section 4.4. Figure 49 provides an ABAQUS illustration of the NSC PAGE 108 108 column after subjection to the 2922 psi blast load, and Figure 410 chronicles the ABAQUS deflected shape for the NS1, NS2, and NS3 cases. While the NS4 and NS5 cases resemble exaggerated deflected shapes of NS3 they are not shown for simplicity and clarity of the figure. Note that the deformations presented correspond to the final state of the middepth rebar and not the peak response. For the lower blast pressures the column deforms as expected and assumes a somewhat triangular shape. For the NS3 and nonplotted NS4 and NS5 cases however, the column is shown to deflect under a severe impact with high rotations nearer to the boundaries. T o inspect the hinge formation of the more triangular deformed shape, the r otations of trials NS1 and NS2 are observed in Figure 4 11. The ABAQUS output is from a collection of nodes along the columns rebar spaced at one inch and demonstrates an incurred variability Shifting of the critical section slightly away from the midspa n point is observed because of the allowance of axial deformation and the enactment of gravity Due to the simple supports, small vertical deformations misalign the nodal references. The small, incremental nod e approach to curvature results in the NS1 curv ature plot of Figure 412 with a rough estimate plot accompanying it (derived from manually smoothing the rotation curve) Given the difficulty in deciphering the ABAQUS curvature from its plot, the hinge length is estimated on the rotation curve. A near constant change in rotation occurs 12 inch es to either side of midspan which reiterates the extended hinge length observed in DSAS. Unlike DSAS however, the ABAQUS curve shows a greater adherence to the yield curvature. 4.2.2 Fixed NSC Column By fixing the columns supports, three plastic hinges are expected to form as opposed to a single one at the critical location. Due to the moment distribution induced PAGE 109 109 by a uniform pressure load, a hinge first form s at e ach boundary and then a third hinge g enerates at midspan (the location of maximum moment along the column span). To conceptualize this occurrence, the response data of cases NF1 through NF5 are extracted from DSAS and ABAQUS. Table 43 compares the peak midspan deflections recorded by the analytical programs. The load deflection curve of the fixed column is shown in Figure 44 for differentiation with that for the simply supported column. At a certain point, the fixed support curve is observed to realign with the simple support curve, plausibl y due to the loss of rotation stiffness at the supports. T he progressions of deformation, rotation, and curvature for trials NF1 to NF5 are displayed in Figures 413 through 415. Indicated by the sharp changes in rotation in Figure 414 and the peaking cu rvatures of Figure 415, it is evident that three hinges are present as expected. Outside of the hinge elements located at the supports, t he curvature abides to yield. The dictated hinge length of 13 inches may even be larger than necessary to represent th e plasticity at the fixed ends. The midspan plastic hinge behaves similarly to its simple support counterpart. A significant spread of the columns mid section deviates from the yield curvature; however, the prominent change in rotation occurs over a more compressed distance. The shorter hinge length (as compared with that at midspan for the simple NSC column) supports the introduction of the z parameter ( distance from the critical section to the point of contraflexure) to plastic hinge length expressions. Figures 41 6 through 4 1 8 respectively further the inspect ion of deformation, rotation and curvature by diagramming the ABAQUS data for trials NF1, NF2, and NF3. The larger pressure loads demonstrate the same impact effect of a sharp increase in PAGE 110 110 deflection nearer to the boundaries, and this response leaves trial NF1 to best illustrate the hinge indicators. Visualized at the supports of trial NF1 in Figure 41 7 the change in rotation transpires over approximately 7 inches, thereby highlighting the potential overestimation of the hinge length at this section in DSAS. At midspan, the yielded portion of the column is maintained within 12 inches to either of its side, and this observation again supports that the hinge spreads one length in each directi on from the critical points. 4.2.3 Axial Loads Realistically columns serve to support axial loads; transverse, flexure inducing loads via blast ing or impact are incidental. To reflect upon how the inclusion of an a xial load impacts the peak response of th e addressed columns, varying forces are implement ed in combination with a few of the simple and fixed column blast loads. The selected axial loads are 750, 1250, and 1750 kips, with the latter being just shy of the columns maximum allowable compressive fo rce. The spread of the axial magnitudes is intended to reasonably capture the fluctuation in column behavior between no axial load and the ultimate load. Table 44 looks at the result of adding an axial load to three of the blast s, name ly the 1777 psi, 3608 psi, and 4757 psi case s The 750 and 1250 kip loads contribute to the columns deformation resistance, while the 1750 kip load incurs further damage. To compare the boundary conditions i n turn, the simple column proves more subjective to the lower loads while the fixed column is more affected by the 1750 kip load. 4.3 Ultra High Performance Concrete Column To compare ultra high performance concrete with the normal strength concrete, a column setup similar to that shown in Figures 4 1 and 42 is modele d. Aside from the PAGE 111 111 alteration in the material con crete, the only difference between the NSC and UHPC columns is the UHPCs lack of transverse ties. The common use and addition of steel fibers to the UHPCs mixture is intend ed to provide enough strength to r educe or elim inate the need of transverse reinforcement. The UHPC material model is based on the compilation of strength models proposed by various research excursions, including Acker and Behloul (2004), Habel et al. (2004), Spasojevic et al. (2008), and Fehling et al. ( 2004 ). The constitutive model employed by this study is recognized as the standard in DSAS and is shown in Figure 4 19. In the compressi ve regime, the strength is model by three linear segments: an elastic region, a level plateau, and a decrease in strength to the ultimate strain. The UHPC dually exhibits impressive strength in tension and displays ductility well into a tension stiffening region. The strength properties are reiterated in T able 45 alongside an outline of the rebar propert ies drawn from the NSC study. Dynamic increase factors are only applied to the steels material model and the factors used correspond to those listed in Table 41. For ABAQUS and its consideration of the CDP model, the dilatation angle has been inversely related to the compressive strength of concrete, and it is therefore decreased from the value employed to model the NSC. A dilation angle of 35 degrees was determined to best represent the UHPC durability, and all other CDP parameters remain the same as for the NSC. The result of the UHPC columns cross sectional configuration is diagrammed in Figure 420 as the moment curvature diagram. This relationship may be compared to that for the NSC column related in Figure 43. From Figure 420 it is noted that the yield curvature is approximately 0.0005 in1, while the ultimate curvature is nearly 0.015 in1. PAGE 112 112 4.3.1 Simpl y Support ed UHPC Column The described UHPC column is first situated with simple supports ( a pin and roller mechanism) and subjected to the blast lo ad sequence. The resultant peak deflections of trials US1 through US5 are detailed in Table 4 6, including both the DSAS and ABAQUS output. Based on the moment curvature relationship and stiffness imposed by the boundary conditions, the loaddeflection cur ve in Figure 421 is assembled. The deformation, rotation, and curvature charts respectively included in Figures 422 through 424 delineate the first four blast trials, while trial US5 is absent due to DSASs loaddeflection curves cutoff at about 9.8 inches. The evolution of the plastic hinge remains evident in Figure 424 even lacking the depiction of trial US5. The ultimate hinge is shown to extend 14 inches to either side of midspan, and the curvature outside of the hinge element clearly increases from zero toward the yield curvature. The ABAQUS files for the simply support ed UHPC column illustrate a deviating behavior from the norm. Figures 4 25, 4 26, and 427 contain the deflected shape, rotation, and curvature schematics, respectively. The crit ical section is shown to be misplaced from midspan by nearly 5 inches with uncharacteristic rotations occurring for most trials along the bottom half of the span. The boundary conditions were established as simple, but since axial deformations are permitted and the bottom support allows the freer rotation, the data is offset Also, the enactment of gravity is expected to somewhat exaggerate the axial deformation effect. Still, spread of the hinge is demonstrated about the spans critical section. The curvat ure diagram for US1 supports the DSAS hinge estimate by illustrating the spike in curvature to occur between 12 inches on either side of the critical point while the remainder of the span maintains unyielded curvature. PAGE 113 113 4.3.2 Fixed UHPC Column Upon fixing the columns boundary conditions the response behavior is examined for the formation of three distinct hinges. The DSAS output is a function of the loaddeflection curve drawn in Figure 421, and the respective deflected shapes, rotations, and curvatures for the blast loaded UHPC column (trials UF1 through UF5) are pronounced in Figures 4 28, 4 29, and 4 30 respectively On the evolution of curvature graph (Figure 430) the hinged sections are clearly defined. The hinge at midspan is observed to grow from 12 inches to 28 inches (or to 14 inch span lengths on either of its sides) and matches the reported simple support behavior Meanwhile, the elemental hinge at the support provides a sufficient length for the significant change in curvature to occur. Also observed for the fixed NSC column, the boundary hinge may be shorter than projected by the 13.0 inch element. The comparison of peak responses attained from both DSAS and ABAQUS are listed in Table 47. The ABAQUS data expresses the sever e mid column impact ion of trials UF2 through UF5 by the chronicled deformations of Figure 431. Prominently focusing on trial UF1, the Figure 432 rotation curve proves to be fairly smooth and results in the perceivable hinge lengths designated in Figure 433. The UF1 curvature plot shows a 6 inch hinge alongside each of the supports and a third 24 inch hinge equally spread at midspan. 4.3.3 Axial Loads The employed UHPC has nearly four times the compressive strength of the NSC tested, and the axial load supportable by the UHPC column is thereby approximately quadruple that of the NSC column. In contrast to axial loads ranging between 750 and 1750 kips t he UHPC blast trials are coupled with axial loads on the order of 1750, PAGE 114 114 3750, and 5750 kips. Table 48 serves to demonstrate the axial load effect under these specified conditions The column behaves similarly by either boundary condition; the two smaller axial loads aid in significantly decreasing the peak response, while the 5750 kip load forces slightly greater deformations on the column. 4.4 Contrast of Concrete Types The dynamic analysis of n ormal strength and ultra high performance concrete columns wi th both simple and fixed boundary conditions ha s been presented in sections 4.2 and 4.3 and are now compared against one another First, Figure 4 34 demonstrates the difference in the concretes constitutive models. UHPCs compressive strength is about four times that o f NSC and boasts nearly double the ultimate strain. In tension the UHPC is again superior to NSC, in part due to the typical employment of fiber reinforcement The ductility of the fiber r einforced UHPC is also evidenced by its t ensile ultimate strain. For the given geometry, a comparis on of the NSC and UHPC columns moment curvature diagrams may be made between Figures 4 3 and 420. The two materials possess a similar yield curvature, but the UHPC has double the moment capacity at that instance. Before comparing the maximum response data of the NSC and UHPC columns collected the perspectives from which the analyses were completed must be understood. For example, diagonal shear and tension membrane behaviors were applied (in DSAS) to both concrete types; the specific effects of which are discussed in section 4.5. As demonstrated by the Feldman and Siess (1958) beam cases, the estimate of diagonal shear may or may not be applicable to each column under the NSC model. Also, whether t he same model for the shear reduction factor extends to the heightened properties of UHPC is not verified. The consideration of tension membrane PAGE 115 115 action is more easily translated to the UHPC case on the other hand, because of its basis on the longitudinal reinforcement s operation after the concrete cracks. Finally, in order to judge the peak responses off of one another, it is recognized that the UHPC column has no transverse reinforcement or DIFs applied to its co ncrete strength. While the steel fiber enhancement of UHPC is built into its material model and has been demonstrated to diminish the need for transverse reinforcement, the UHPC column is nevertheless without the confinement applied to the NSC column. T hou gh UHPC has shown a relationship between dynamic strength and strain rate experimentally (Rebentrost and Wight 2009), studies equivalent to those of NSC to determine and specify a consistent DIF relationship have not yet been established. The UHPC column i s therefore subjected to the blasts loads without the inclusion of dynamic increase factors unlike the NSC column. The DSAS comparison between the simple and fixed NSC and UHPC columns peak responses is detailed in Table 49 while that for ABAQUS appear s in Table 410 Per DSAS, the use of UHPC on average generates a 16% reduction in deflection for the case of simple supports and a 25% reduction for fixed boundaries. Should DIFs be applied to the UHPC or should it be determined that diagonal shear is less effective in the UHPC c olumn, the already significant reductions m ight exceptionally increase. T he formation of plastic hinges varies in representation between DSAS and ABAQUS. Granted that the computational arrangement of DSAS would not generate the fluctuations observed from the ABAQUS oneinch nodes, but even excepting for this variability, the impacted shape ABAQUS configures for the blasts larger than 1777 psi deviates from the shape proposed by DSAS. DSAS proves that after yielding the PAGE 116 116 column bec omes more triangular in nature, linearly shaped between the hinge regions. The deformation in ABAQUS however virtually introduces additional hingelike segments between the support and midspan for large blast pressure s or possibly extends the support hin ge to a much larger length. With respect to differences in the plastic hinge formation of NSC versus UHPC, ABAQUS demonstrates no observable discrepancy while DSAS elicits one important distinction. It was repeatedly shown that the support hinge in fixed columns has a length of or less than the effective depth. ABAQUS supports that the length may be nearly half the effective depth, as most 1777 psi cases promote a 6 inch hinge. At midspan the hinge area consistently extends between 12 and 14 inches toward either boundary. These hinge lengths apply to NSC and UHPC alike. The materials notably differ in their surpassing of yield curvature along the column span outside of the hinge region. UHPC displays a behavior that theory supports: a plastic hinge is noted by a rise in curvature from yield to ultimate. The NSC violates this principle by DSAS calculations however as the yielded segment of the column is nearly half of the span. 4.5 Supplemental Behavior Study A great number of factors affect the flexural b ehavior of columns subjected to severe dynamic loads. For instance, it was specified that both diagonal shear and tension membrane behavior were considered in the DSAS response calculations of the test columns. In order to more effectively characterize the demonstrated deformations, additional topics under the branch of inelastic concrete behavior are addressed. First, the appropriateness of the selected plastic hinge length formulation with respect to the proposed hinge length expressions is evaluated, and t hen the individual effects of shear and large deformations are examined by way of DSAS PAGE 117 117 4.5.1 Plastic Hinge Length The preceding discussion of simply supported and fixed NSC and UHPC columns assumed a plastic hinge length equal to the columns effective depth in DSAS. Although this estimation proved to be reasonable for the distance of severe rotation changes to each side of a critical section, as based on the ABAQUS diagrams, it has not yet be gauged against the proposed expressions of Corley, Mattock, e tc (Bae and Bayrak 2008) A brief comparative analysis of the assumed length is therefore conducted against the analytical findings specified in the literature. The proposed plastic hinge length expressions of section 2.4 employ factors such as section d epth, rebar depth, rebar diameter, steel yield strength, and distance from the critical section to the point of contr aflexure in their calculations. The result of using other applicable equations for predicting the hinge length of this studys test column is found in the second column of Table 411 While the effective depth of 13.0 inches ranges within the spread of the other expressions, it also corresponds well with the others average. To determine the degree to which the effective depth remains within an appropriate range under different circumstances Table 411 also details the outcome of altering various param eters of the control test column. The parameter changes includ e lengthening the column ( to 14 feet), providing greater rebar cover ( to 3 inches ), and compacting the column depth ( to 12 inches). Overall, several of the hinge length equations depend upon similar factors, i.e. functions of the columns depth, and allow the effective depth to correspond well with the changes in column geometry. 4.5.2 Shear Considerations Two forms of shear behavior act within concrete columns that are, and must be appropriately treated as separate facets : diagonal and direct shear. PAGE 118 118 Diagonal shear works in conjunction with flexure and has been observed to weaken the resistance of concrete m embers. Referenced in section 2.3, diagonal shear supplements crack propagation to produce an inarguable effect on the failure criteria of beam and column members. An estimate of this eff ect is proposed by Krauthammer and Shahriar (1988) via a shear reduction factor. The SRF is applied to a members moment capacity, effectively shifting the moment curvature diagram downwards The f actor is based on the reinforcement and shear span to depth ratios, and since the longitudinal reinforcement remains unchanged through the various NSC and UHPC column trials, the SRF remains constant. In each of the presented cases the SRF employed i s 0.77, and the Table 412 exemplifies the effect held by this factor on the columns peak response. Direct shear operates as a separate failure mode, juxtaposed to the flexural failure focused upon in the study. Direct shear is not ed by cracking perpendicular to a members axis typical in areas of high concentrated sh ear force, i.e. under the supports or point loads DSAS considers the inc urrence of direct shear and its peak behavior for several of the blast trials presented in Table 4 13. It is observed that for all trials, except the UHPC subjected to a 1777 psi p r e ssure direct shear failure is indicated. 4.5.3 Tension Membrane Behavior After a structural member sufficiently crack s to dissipate the strength contribution of the concrete the steel reinforcement begins to act as a tension membrane The study formerly conducted by Morency (2010) chronicled the adaption of DSAS to account for tension membrane action in reinforced concrete columns under severe dynamic loads. Table 414 now compares the e ffects of including the tension membrane strengt h in computations for the NSC and UHPC columns. For the simply supported spans, as the PAGE 119 119 deformation grows the counteraction of the membrane logically increases as well. The ductility and durability of UHPC is also observed by the lesser effect held by the m embrane action compared with its impact on the NSC columns. For fixed columns, the membrane does not enact for the smaller blast loads, but it begins to contribute greater support as the blasts more severely increase the column deformation. 4. 6 Summary An analytical study of the severe deformation of normal strength and ultra high performance concrete columns has been presented. The primary focus was to demonstrate the theory of plastic hinge formation through the computational DSAS V3.2.1 and ABAQUS V610 programs. Sections 4.2 and 4.3 evaluated the behaviors of NSC and UHPC columns respectively, exploring the single and multiple hinge generations inherent of simply supported and fixed columns. A thorough comparison of the two concrete types then proceeded in section 4.4, followed by the reemphasis of particular inelastic behaviors, including the calculation of a columns plastic hinge length, in section 4.5. PAGE 120 120 Table 41. Dynamic increase factors for the NSC column. Blast t rial Concrete in compression, f c Concrete in tension, f t Reinforcing steel, f y and f u 1 1.29 2.64 1.20 2 1.31 2.82 1.21 3 1.32 2.90 1.22 4 1.32 2.97 1.22 5 1.33 3.03 1.23 Table 42. Simply supported NSC column response. Trial Uniform p ressure (psi) DSAS (in) ABAQUS (in) Percent d ifference (%) NS1 1777 2.69 2.20 22.2 NS2 2922 5.72 5.72 0.0 NS3 3602 8.03 8.79 8.6 NS4 4208 10.35 13.71 24.5 NS5 4757 12.64 19.15 34.0 Table 43. Fixed support NSC column response. Trial Uniform p ressure (psi) DSAS (in) ABAQUS (in) Percent d ifference (%) NF1 1777 1.82 1.09 67.0 NF2 2922 4.28 3.70 15.7 NF3 3602 6.50 6.72 3.3 NF4 4208 9.16 10.79 15.1 NF5 4757 11.84 15.30 22.6 Table 44. NSC response to combinations of blast and axial loads. Trial Uniform p ressure (psi) 0 kip (in) 750 kip (in) 1250 kip (in) 1750 kip (in) NS1 1777 2.69 1.59 1.76 3.15 NS3 3602 8.03 5.55 6.22 8.65 NS5 4757 12.64 9.37 9.89 13.24 NF1 1777 1.82 1.43 1.76 3.15 NF3 3602 6.50 5.68 6.13 8.68 NF5 4757 11.84 9.88 10.25 13.34 PAGE 121 121 Table 45. DSAS input of UHPCs properties. Concrete p roperties Values Steel p roperties Values Compression Comp/Tension E c ( k si) 5950 E s (ksi) 29520 y ,c1 (in/in) 0.004 y (in/in ) 0.00156 c2 (in/in) 0.006 f y (psi) 0.0144 f max ( p si) 23800 sh (in/in ) 90000 u (in/in) 0.01 u (in/in ) 0.15 f u (psi) 20300 f u (psi) 0.15 Tension y (in/in) 0.0003 f y (psi) 1300 2 (in/in) 0.0015 f 2 (psi) 1600 3 (in/in) 0.0055 f 3 (psi) 362 u (in/in) 0.0125 Table 46. Simply supported UHPC column response. Trial Uniform p ressure (psi) DSAS (in) ABAQUS (in) Percent d ifference (%) U S1 1777 2.24 1.97 13.7 U S2 2922 4.70 4.96 5.2 U S3 3602 6.78 7.93 14.5 U S4 4208 8.89 10.87 18.2 U S5 4757 11.09 17.73 37.5 Table 47. Fixed support UHPC column response. Trial Uniform p ressure (psi) DSAS (in) ABAQUS (in) Percent d ifference (%) U F1 1777 1.42 0.79 80.0 U F2 2922 3.22 1.93 66.8 U F3 3602 4.78 4.37 9.4 U F4 4208 6.69 8.40 20.4 U F5 4757 9.29 12.08 23.1 PAGE 122 122 Table 48. UHPC respons e to combinations of blast and axial loads. Trial Uniform p ressure (psi) 0 kip (in) 1750 kip (in) 3750 kip (in) 5750 kip (in) US1 1777 2.24 1.08 1.00 2.29 US3 3602 6.78 3.12 3.05 7.20 US5 4757 11.09 7.00 5.33 11.66 UF1 1777 1.42 0.55 0.73 2.22 UF3 3602 4.78 2.72 2.89 7.29 UF5 4757 9.29 5.25 5.40 11.83 Table 49. Comparison of NSC and UHPC test columns DSAS peak response. Blast trial Simple NSC (in) Simple UHPC (in) Fixed NSC (in) Fixed UHPC (in) 1 2.69 2.24 1.82 1.42 2 5.72 4.70 4.28 3.22 3 8.03 6.78 6.50 4.78 4 10.35 8.89 9.16 6.69 5 12.64 11.09 11.84 9.29 Table 410. Comparison of NSC and UHPC test columns ABAQUS peak response. Blast trial Simple NSC (in) Simple UHPC (in) Fixed NSC (in) Fixed UHPC (in) 1 2.20 1.97 1.09 0.79 2 5.72 4.96 3.70 1.93 3 8.79 7.93 6.72 4.37 4 13.71 10.87 10.79 8.40 5 19.15 17.73 15.30 12.08 Table 411. Effect on plastic hinge length expressions. Expression Control: NSC c olumn Increase span to 14 feet Increase rebar cover to 3 in. Decrease column depth to 12 in. Corley 11.0 11.7 10.5 9.7 Mattock 10.9 11.5 10.1 8.9 Sawyer 9.0 9.9 8.7 8.0 Paulay & Priestly 17.6 19.5 17.6 17.6 Sheikh & Khour y 16.0 16.0 16.0 12.0 Average (of above) 12.9 13.7 12.6 11.2 Effective Depth 13.0 13.0 10.0 9.0 PAGE 123 123 Table 412. Effect of diagonal shear. Trial With SRF (in) Without SRF (in) Trial With SRF (in) Without SRF (in) NS1 2.69 2.17 US1 2.24 1.77 NS4 10.35 9.10 US4 8.89 7.59 NF1 1.82 1.40 UF1 1.42 1.12 NF4 9.16 7.20 UF4 6.69 5.84 Table 413. Direct shear response. Column t ype 1777 psi (in) 2922 psi (in) 3608 psi (in) 4208 psi (in) 4757 psi (in) Simple NSC 0.45 Fail 0.44 Fail 0.44 Fail 0.42 Fail 0.43 Fail Fixed NSC 0.45 Fail 0.44 Fail 0.44 Fail 0.43 Fail 0.43 Fail Simple UHPC 0.67 1.05 Fail 1.05 Fail 1.05 Fail 1.05 Fail Fixed UHPC 0.67 1.05 Fail 1.05 Fail 1.04 Fail 1.05 Fail Table 414. Effect of tension membrane behavior Normal Strength Concrete Ultra High Performance Concrete Trial Consider TM (in) Without TM (in) Trial Consider TM (in) Without TM (in) NS1 2.69 3.61 US1 2.24 2.55 NS2 5.72 8.24 US2 4.70 5.80 NS3 8.03 12.44 US3 6.78 8.70 NS4 10.35 17.30 US4 8.89 10.34 Fail NS5 12.64 22.74 US5 11.09 13.74 Fail NF1 1.82 1.40 UF1 1.42 1.42 NF2 4.28 4.28 UF2 3.22 3.22 NF3 6.50 6.50 UF3 4.78 4.78 NF4 9.16 9.08 UF4 6.69 6.97 NF5 11.84 12.06 UF5 9.29 11.12 PAGE 124 124 Figure 41 Cross section of test column. Figure 42 Schematic of test column span. b = 16 in. h = 16 in. # 7 rebar # 3 tie 1 5 in. 12 ft. # 7 rebar # 3 ties @ 12 in. PAGE 125 125 Figure 43 Moment curvature of NSC c olumn. Figure 44 Loaddeflection curves of NSC column. 0 50 100 150 200 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035Moment (ft kips) Curvature (in1) 0 20 40 60 80 100 0 4 8 12 16 20Pressure (psi) Displacement (in) Simple Fixed PAGE 126 126 Figure 45 Progression of deformation of simply supported NSC column. Figure 46 Progression of rotation of simply supported NSC c olumn. 0 2 4 6 8 10 12 14 0 12 24 36 48 60 72 84 96 108 120 132 144Displacement (in) Coordinate (in) 1777 psi 2922 psi 3602 psi 4208 psi 4757 psi 0.25 0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2 0.25 0 12 24 36 48 60 72 84 96 108 120 132 144Rotation (rad) Coordinate (in) 1777 psi 2922 psi 3602 psi 4208 psi 4757 psi PAGE 127 127 Figure 47 Progression of curvature of NSC simply supported c olumn. Figure 48 Curvature of simply supported NSC column post yield 0.015 0.012 0.009 0.006 0.003 0.000 0 12 24 36 48 60 72 84 96 108 120 132 144Curvature (in1) Coordinate (in) 1777 psi 2922 psi 3602 psi 4208 psi 4757 psi 0.0010 0.0008 0.0006 0.0004 0.0002 0.0000 0 12 24 36 48 60 72 84 96 108 120 132 144Curvature (in1) Coordinate (in) Pt. 1 Pt. 2 Pt. 3 Pt. 4Hinge Excess of Y ield PAGE 128 128 Figure 49 Example of ultimate deformation via ABAQUS (trial NS2) Figure 410. ABAQUS deflected shapes for simply supported NSC c olumn. 0 2 4 6 8 10 0 12 24 36 48 60 72 84 96 108 120 132 144 156Displacement (in) Coordinate(in) 1777 psi 2922 psi 3602 psi PAGE 129 129 Figure 411. ABAQUS rotations for simply supported NSC c olumn. Figure 412. ABAQUS curvature for simply supported NSC c olumn. 0.15 0.1 0.05 0 0.05 0.1 0.15 0 12 24 36 48 60 72 84 96 108 120 132 144 156Rotation (rad) Coordinate (in) 1777 psi 2922 psi 0.0050 0.0025 0.0000 0.0025 0.0050 0 12 24 36 48 60 72 84 96 108 120 132 144 156Curvature (in1) Coordinate (in) 1777 psi Rough Approximation 1777 psi Hinge PAGE 130 130 Figure 413. Progression of deformation of fixed NSC column. Figure 414. Progression of rotation of fixed NSC column. 0 2 4 6 8 10 12 0 12 24 36 48 60 72 84 96 108 120 132 144Displacement (in) Coordinate (in) 1777 psi 2922 psi 3602 psi 4208 psi 4757 psi 0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2 0 12 24 36 48 60 72 84 96 108 120 132 144Rotation (rad) Coordinate (in) 1777 psi 2922 psi 3602 psi 4208 psi 4757 psi PAGE 131 131 Figure 415. Progression of curvature of fixed NSC column. Figure 416. ABAQUS deflected shapes for fixed NSC column. 0.025 0.020 0.015 0.010 0.005 0.000 0.005 0.010 0.015 0.020 0 12 24 36 48 60 72 84 96 108 120 132 144Curvature (in1) Coordinate (in) 1777 psi 2922 psi 3602 psi 4208 psi 4757 psi 0 2 4 6 8 0 12 24 36 48 60 72 84 96 108 120 132 144 156Displacement (in) Coordinate (in) 1777 psi 2922 psi 3602 psi Excess of Y ield PAGE 132 132 Figure 417. ABAQUS rotations for fixed NSC column. Figure 418. ABAQUS curvatures for fixed NSC c olumn. 0.1 0.05 0 0.05 0.1 0 12 24 36 48 60 72 84 96 108 120 132 144 156Rotation (rad) Coordinate (in) 1777 psi 2922 psi 3602 psi 0.010 0.005 0.000 0.005 0.010 0 12 24 36 48 60 72 84 96 108 120 132 144 156Curvature (in1) Coordinate (in) 1777 psi 2922 psi 3602 psi 1777 psi Hinge Hinge Hinge PAGE 133 133 Figure 419. UHPC material model. Figure 420. Moment curvature diagram for UHPC c olumn. 5000 0 5000 10000 15000 20000 25000 0.015 0.01 0.005 0 0.005 0.01 0.015Stress (psi) Strain (in/in) 0 50 100 150 200 250 0 0.003 0.006 0.009 0.012 0.015Moment (ft kips) Curvature (in1) Compression Tensi on PAGE 134 1 34 Figure 421. Loaddeflection curves of UHPC c olumn. Figure 422. Progression of deformation of simply supported UHPC c olumn. 0 30 60 90 120 150 0 2 4 6 8 10Pressure (psi) Displacement (in) Fixed Simple 0 2 4 6 8 10 12 0 12 24 36 48 60 72 84 96 108 120 132 144Displacement (in) Coordinate (in) 1777 psi 2922 psi 3602 psi 4208 psi 4757 psi PAGE 135 135 Figure 423. Progression of rotation of simply supported UHPC c olumn. Figure 424. Progression of curvature of simply supported UHPC c olumn. 0.15 0.1 0.05 0 0.05 0.1 0.15 0 12 24 36 48 60 72 84 96 108 120 132 144Rotation (rad) Coordinate (in) 1777 psi 2922 psi 3602 psi 4208 psi 0.015 0.010 0.005 0.000 0 12 24 36 48 60 72 84 96 108 120 132 144Curvature (in1) Coordinate (in) 1777 psi 2922 psi 3602 psi 4208 psi PAGE 136 136 Figure 425. ABAQUS deflected shapes of simply supported UHPC c olumn. Figure 426. ABAQUS rotations of simply supported UHPC c olumn. 0 2 4 6 8 0 12 24 36 48 60 72 84 96 108 120 132 144 156Displacement (in) Coordinate (in) 1777 psi 2922 psi 3602 psi 0.15 0.1 0.05 0 0.05 0.1 0.15 0 12 24 36 48 60 72 84 96 108 120 132 144 156Rotation (rad) Coordinate (in) 1777 psi 2922 psi 3602 psi PAGE 137 137 Figure 427. ABAQUS curvatures of simply supported UHPC c olumn. Figure 428. Progression of deformation of fixed UHPC c olumn. 0.015 0.005 0.005 0.015 0 12 24 36 48 60 72 84 96 108 120 132 144 156Curvature (in1) Coordinate (in) 1777 psi 2922 psi 0 2 4 6 8 10 0 12 24 36 48 60 72 84 96 108 120 132 144Displacement (in) Coordinate (in) 1777 psi 2922 psi 3602 psi 4208 psi 4757 psi 1777 psi Hinge PAGE 138 138 Figure 429. Progression of rotation of fixed UHPC c olumn. Figure 430. Progression of curvature of fixed UHPC c olumn. 0.15 0.1 0.05 0 0.05 0.1 0.15 0 12 24 36 48 60 72 84 96 108 120 132 144Rotation (rad) Coordinate (in) 1777 psi 2922 psi 3602 psi 4208 psi 4757 psi 0.020 0.015 0.010 0.005 0.000 0.005 0.010 0.015 0 12 24 36 48 60 72 84 96 108 120 132 144Curvature (in1) Coordinate (in) 1777 psi 2922 psi 3602 psi 4208 psi 4757 psi PAGE 139 139 Figure 431. ABAQUS deflected shapes for fixed UHPC c olumn. Figure 432. ABAQUS rotations for fixed UHPC c olumn. 0 1 2 3 4 5 0 12 24 36 48 60 72 84 96 108 120 132 144 156Displacement (in) Coordinate (in) 1777 psi 2922 psi 3602 psi 0.1 0.05 0 0.05 0.1 0 12 24 36 48 60 72 84 96 108 120 132 144 156Rotation (rad) Coordinate (in) 1777 psi 2922 psi 3602 psi PAGE 140 140 Figure 433. ABAQUS curvatures for fixed UHPC c olumn. Figure 434. Comparison of NSC and UHPC constitutive models 0.002 0.001 0.000 0.001 0.002 0 12 24 36 48 60 72 84 96 108 120 132 144 156Curvature (in1) Coordinate (in) 1777 psi 3602 psi 5000 0 5000 10000 15000 20000 25000 0.01 0.005 0 0.005 0.01Stress (psi) Strain (in/in) NSC UHPC 1777 psi Hinge Hinge Hinge PAGE 141 141 CHAPTER 5 CONCLUSIONS AND RECOMMENDATIONS 5.1 Overview The preceding chapters examined the plastic hinge behavior relevant of normal strength and ultra high performance concretes. Chapter 1 introduced the motivation behind and the general focus of th e study, while Chapter 2 provided the necessary background for conducting structural analyses of NSC concrete and an overview of UHPCs material pr operties and applications. Chapter 3 then organized the methodological steps by which the study was carried out, including the adaptation of DSASs programming and the techniques employed for study validation. Finally, Chapter 4 processed the parametric st udys results regarding the plastic hinge formation of concrete columns with both simple and fixed supports. This chapter in turn encapsulates the presented work as a whole and remarks on the conducted study retrospectively. In addition to an abbreviated summary of activities, this chapter elaborates on the progress made in plastic hinge and UHPC analyses and demonstrates the need for continued experimentation in the fields of interest. Section 5.2 narrates the conclusions that can be drawn from the analys is of Chapters 3 and 4, and section 5.3 details a number of recommendations made for consideration with future endeavors. The suggestions reference the analytical tools akin to concrete members as well as the investigation of plastic hinge formation in str uctural entities. 5.2 Conclusions The validation stage of the work that dealt with the Feldman and Siess (1958) beams 1C and 1H was shown to be successful. T hough several assumptions were PAGE 142 142 made for the DSAS input parameters, the programs ability to simula te the beams maximum responses was refined within engineering reason. In addition to representing the experimentally displayed behavior, the Beam 1C case exemplified correlation between DSAS and ABAQUS. T he overall product of the validation was the NSC an d reinforcing steel material models and the verification of properly plot ting the rotation and curvature data of a concrete beam. Regarding the peak response analysis of the NSC and UHPC columns, the greatest compatibility observed between DSAS and ABAQUS was most often for the middle range of blast pressures (2922 psi and 3602 psi). The divergence of the small and excessively large deformations is invariably, in part, a product of relating the constitutive material models between programs, especially consi dering ABAQUSs use of the concrete damaged plasticity model to represent the behavioral inelasticity. Individually, the two programs expressed consistent comparisons between the improved strength of using UHPC over NSC. The plastic hinge modeling proved generally consistent with the overriding theory and demonstrated respectable correlation between DSAS and the finite element modeling of ABAQUS. Originally, the hinge element introduced to DSAS was intended to take the place of the entire hinge and spread a distance of the effective depth Upon inspection however, t he hinges formed within the columns span were consistently observed to expand two effective depth lengths in total (or a hinge length in either direction away from the critical section). Though this result was not the projected behavior, the implementation of the effective depth element eased the graphical representation of curvature for DSASs larger mesh density It also sufficiently PAGE 143 143 encapsulated the hinge generated at the boundaries, which proved in ABAQUS to be of a length less than effective depth. The large hinge element present at the boundaries in DSAS also weakens the depiction of curvature change when represented graphically. The study of reinforced concrete columns clearly demonstrates the significance of plastic hinge behavior in its contribution to column failure. For fixed columns, a fully formed hinge releases the boundary constraints and enables the member to continue to act as though it were a simple support system. The final hinge to develop in a fixed column, or the only one to do so in simply supported column, provides the ultimate failure mechanism around which the column will invariably collapse under extreme forces. Ultimately the contras t between the NSC and UHPC columns hig hlighted UHPCs dominant strength and endurance. The advanced ductility of UHPC was to be expected, but the material also proved quite resistant to rotation outside of the critical sections. The NSC on the other hand exhibited excessive span yielding under the severe uniform pressures, at least via DSASs configuration. This discrepancy introduces the plausibility of a second tier plastic hinge as demonstrated by Figure 47 and 48 Though not validated by the ABAQUS output, extensive experimental data this area would assist in the clarification. The deviation from theory may also plausibly delineate from the large meshing sized used in an attempt to capture the behavior or an error in shape data processing. Finally, while processing the additional behav ior studies of the diagonal shear, direct shear, and tension membrane behaviors, it is noted that these models and computational considerations are not necessarily translatable between NSC and UHPC. PAGE 144 144 Without proper examination of UHPC through experimentatio n, the shear reduction factor employed is functionally meaningless, and it was only included to provide a resemblance between the concrete types and limit the altered variables within the column study. A discrepancy also exists with the prediction of UHPC s failure amongst the lack of tension membrane consideration. Unless the transverse reinforcement is the originating cause, the UHPC should display equal ability to the NSC in terms of ductility and large deformations. The designation of failure for the US 4 and US5 trials in the absence of tension membrane behavior emphasizes a need for further development of the DSAS UHPC analytical capabilities. 5.3 Recommendations After completion of the validation and parametric studies, it is prudent to reflect upon and critique the explicated methodology and results as well as to express how they would best be implemented or adapted This section addresses the limitations of the parametric study performed and presents recommendations for future s tudies relative to th e inelastic hinge behavior of NSC and UHPC columns. The congruency between the programming codes used for the dynamic analyses was foremost limited by the extent to which the material models are known. Although the NSC column properties derived from Beam 1C, which demonstrated a strong relationship between DSAS and ABAQUS, the effect of uniform pressures on the column were not equally compatible. The two code models were intended to correspond to one another, but discrepancies in the plastic material model s may have resulted, especially with respect to describing the concrete plasticity model. The CDP parameters were based on previous studies and standard ABAQUS assumptions, but their true value definitions are not directly known. Also, although parametric studies have formerly PAGE 145 145 been conducted to hone on the most suitable CDP model for the Beam 1C concrete, the same background is not provided for UHPC. Instead, various dilation angles between 15 and 40 degrees were tested for the overall most reasonable compatibility with the DSAS output. The dilation angle was ultimately changed from 40 degrees (for the NSC model) to 35 degrees to more appropriately suit the UHPC. The parametric study of plastic hinge formation was also limited by the bre adths of column supports considered and the forms of loading to which they were subjected. The supports considered were idealized and therefore simplified. In reality, column supports are not precisely simple or fixed and demonstrate varying grades of stiffness per their d egrees of freedom. Given that hinges form at the locations of highest concentrated moment, whether one is to form at a particular support or the length it would assume is dependent upon the supports true disposition. With respect to loadings, uniform span loa ds were the type primarily considered. The validation case briefly introduced the effects of a concentrated load, but no further load cases were pursued. For uniform loads, the critical point of maximum moment (not including the supports) corresponds t o midspan which was appropriately identified by DSAS. Whether the program properly develops hinges along the span not at midspan, as would occur under a linearly distributed load for example, was not expressly tested. Though brief checks of the program wer e made to verify its functionality such as placing a point load at the quarter span, etc., extensive studies in this fashion were not completed or compared with an alternate source (ABAQUS or experimental data). To remedy the narrated study limitations, t he recommendations made are geared toward the study of plastic hinge behavior as well as the continued development of PAGE 146 146 UHPCs characterization. It may be beneficial to run and thoroughly analyze more theoretical cases, such as the hinging of circular concre te columns, nonuniform loads, or alter ed UHPC behavioral models Yet, based on the research and analytical focus of this paper, the all encompassing recommendation is to verify the work through actual experimentation. Addressing the topic of reinfor ced concrete columns Bae and Bayrak (2008) presented findings on plastic hinges in cantilever s, and t heir exposition proposed a new expression for hinge lengths under severe axial loads Their experimentations d id not address varied boundary conditions or eff ects of critical transverse loads however So while this case provides an example of work being conducted in the inelastic deformation regime of normal strength concrete, additional realistic data in this area would significantly benefit the analytical study. Finally, t he effort of describing concretes complete behavior, regardless of type, is unending. To further capture the most realistic behavior of concrete structures, experimentation is required. T he programming capabilities of both DSAS and ABAQUS are deterred when there is little support of their processed outcomes. Again, without a firm understanding of concretes behavior in the inelastic regime, which is not to be modeled by a single, simple theory, the study of large deformations exponentially increases in difficulty Recognizing UHPCs relative youth and high material variability, the scarcity of a uni versal ly available behavioral model is to be expected. UHPC research is being conducted on a worldwide scale, yet the specific i nterest in large deformations or plastic hinging has not necessarily been thoroughly explored. It has been demonstrated that PAGE 147 147 the shear and tension membrane models used through the parametric study may not necessarily be replicated between normal strength a nd ultra high performance concretes. These areas of behavior provide just a few examples of those remaining to be further explored with respect UHPC. To better situate UHPCs use in protective structures, subjecting UHPC columns to blast and severe impact loads is crucial Such experimentation on UHPC beams is being advanced at CIPPS, and the results will enable and l ead to a more defined picture of the inelastic modes of UHPC structural members PAGE 148 148 LIST OF REFERENCES Abdelrazig, B. (2008). Properties & a pplications of CeraCem u ltra h igh p erformance self compacting concrete. Proceedings of the International Conference on Construction and Building Technology Kuala Lumpur, Malaysia, 217 226. Acker, P. and Behloul, M. (2004). Ductal t echnology: a l arge spectrum of p roperties, a w ide r ange of a pplications. Proceedings of the International Symposium on Ultra High Performance Concrete, Kassel, Germany, 11 23. American Concrete Institute (ACI). (2008). Bui lding code r equirements for s tructural concrete . ACI 318 08, ACI 318R 08, Farmington Hills, MI. Association Franaise de Gnie Civil (2002). 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PAGE 153 153 BIOGRAPHICAL SKETCH Tricia Caldwell received her Bachelor of Science degree in civil engineering from University of Florida in December of 2009. The following semester she continued studying at UF in pursuit of her Master of Engineering with emphasis in the area of structural engineering. Tricia worked with the Center for Infrastructure Protection and Physical Security (CIPPS) to complete her graduate research and thesis under the guidance of Drs. Theodor Krauthammer and S erdar Astarlioglu. Her research focused on the computational methods of analyzing normal strength and ultra high performance concrete columns under blast loading. 