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PAGE 1 1 EFFECT OF SHORT DURATION HIGH IMPULSE VARIABLE AXIAL AND TRANSVERSE LOAD S ON REINFORCED CONCRETE COLUMN By THIEN P HUOC TRAN A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2009 PAGE 2 2 2009 Thien Phuoc Tran PAGE 3 3 To my p arents b rother, sister, my wife and a ll my c hildren PAGE 4 4 ACKNOWLEDGMENTS I would like to thank my advisor, Dr. Theodor Krauthammer, for all his valuable advice s and guidance. I would also like to extend my appreciation to Dr. Serdar Astarlioglu for all his constructive ideas and his attentiveness to facilitate the completion of this thesis paper. I am grateful to the Canadian Armed Forces particularly the Defence Research and Developm ent Canada in Suffield for providing the opportunity for me to complete this postgraduate program. I would like to thank all my fri ends and colleagues in Canada and at the Center for Infrastructure Protection and Physical Security, University of Florida for all the supports during the last two years Lastly, I am indebted to my p arents, my brother, my sister, my wife and all my chil dren for all the sacrifices that they have made to provide the opportunity for my achievement PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS .................................................................................................................... 4 LIST OF TABLES ................................................................................................................................ 7 LIST OF FIGURES .............................................................................................................................. 8 LIST OF SYMBOLS AND ABBREVIATIONS ............................................................................. 12 ABSTRACT ........................................................................................................................................ 1 5 CHAPTER 1 INTRODUCTION ....................................................................................................................... 16 1.1 Problem Statement ............................................................................................................ 16 1.2 Objectives and Scope ........................................................................................................ 16 1.3 Research Significance ....................................................................................................... 17 2 LITERATURE REVIEW ........................................................................................................... 18 2.1 Introduction ....................................................................................................................... 18 2.2 Blast Loading on Structure ............................................................................................... 1 8 2.3 Structure and Its Equivalent System ................................................................................ 23 2.3.1 Introduction ........................................................................................................... 23 2. 3 2 Equivalent M ass .................................................................................................... 24 2. 3 3 Equivalent L oad .................................................................................................... 25 2. 3 4 Shape F unctions .................................................................................................... 27 2. 3 5 Resistance F unctions ............................................................................................ 27 2. 4 Static Analysis ................................................................................................................... 28 2. 4 .1 Flexural Behavior ................................................................................................ 28 2. 4 .2 Influence of S hear on F lexural R esponse ............................................................ 30 2. 4 .3 Influence of A xial F orce on S hear and F lexural R esponses .............................. 32 2. 4 .4 Direct S hear M ode of R esponse .......................................................................... 34 2. 5 Dynamic Analysis ............................................................................................................. 36 2. 5 .1 Newmark Beta M ethod ........................................................................................ 36 2. 5 .2 Dynamic R esistance F unctions ............................................................................ 37 2. 5 .3 Modified E quivalent P arameters for SDOF S ystem ........................................... 40 2 5 3.1 Mass factor ................................................................................................ 4 0 2 5 3.2 Load factor ................................................................................................ 41 2. 5 4 Dynamic Reaction s ............................................................................................... 4 1 2.6 Pressure Impulse (P I) Diagrams ..................................................................................... 4 3 2.6.1 Characteristics of P I Diagram s ........................................................................... 4 3 2.6.2 Derivation of P I Diagram s .................................................................................. 4 4 2.7 Summary ............................................................................................................................ 4 6 PAGE 6 6 3 METHODOLOGY ...................................................................................................................... 4 7 3.1 Introduction ....................................................................................................................... 4 7 3.2 Load Determination .......................................................................................................... 4 7 3.2.1 Overview of Structure .......................................................................................... 4 7 3.2.2 Effects of Transverse Loads ................................................................................. 5 0 3.2.3 Axial Loads ........................................................................................................... 5 0 3.3 Load Deformation Analysis .............................................................................................. 5 1 3.4 Computations of Dynamic Reactions, Shear and Flexural Responses .......................... 5 4 3.4.1 Computation of Dynamic Reactions for the Supported Mass ........................... 5 4 3.4.2 Computation of Shear and Flexural Responses on Columns ............................. 5 5 3.5 Summary ............................................................................................................................ 5 8 4 ANALYSIS ................................................................................................................................. 59 4.1 Introduction ....................................................................................................................... 59 4.2 Description of DSAS ........................................................................................................ 59 4.3 Validations with Experimental Data ................................................................................ 6 0 4. 3 .1 Experimental Data ................................................................................................ 6 0 4. 3 1.1 Material and physical properties .............................................................. 6 0 4. 3 1.2 Loading functions ..................................................................................... 6 1 4. 3 2 ABAQUS Validations .......................................................................................... 6 4 4. 3 3 DSAS Validations ................................................................................................ 73 4. 3 4 Comparison of Results from ABAQUS and DSAS to Experimental Data ....... 74 4.4 Validation for Beam Subject to Uniform Load Using ABAQUS and DSAS ............... 7 7 4. 5 Validation for Column Using ABAQUS and DSAS ...................................................... 79 4. 6 Summary ............................................................................................................................ 82 5 PARAMETRIC STUDY ............................................................................................................ 83 5.1 Description of Columns .................................................................................................... 83 5.2 Columns Subject to Transverse and Constant Axial Load ............................................. 84 5. 3 ......................... Columns Subject to Transverse, Constant and Variable Axial Loads 101 5. 4 Summary .......................................................................................................................... 106 6 CONCLUSIONS AND RECOMMENDATIONS ................................................................. 107 6 .1 Summary .......................................................................................................................... 1 07 6 .2 Conclusions ..................................................................................................................... 1 08 6 .3 Recommendations ........................................................................................................... 1 08 APPENDIX SAMPLE ABAQUS INPUT FILE BEAM 1 C .................................................... 1 10 LIST OF REFERENCES ................................................................................................................. 1 18 BIOGRAPHICAL SKETCH ........................................................................................................... 1 21 PAGE 7 7 LIST OF TABLES Table page 2 1 Definition of direct shear ...................................................................... 35 4 1 Concrete material properties ................................................................................................ .. 6 0 4 2 Steel reinforcements material properties .............................................................................. 6 0 4 3 Material model properties for beams in ABAQUS .............................................................. 67 4 4 Strain rate hardening and material enhancement factors ..................................................... 67 4 5 Comparison o n ABAQUS DSAS and e xperiment r esults .................................................. 7 4 5 1 Summary of columns physical and material properties ....................................................... 83 5 2 Constant axial load cases ....................................................................................................... 8 7 5 3 ......... Comparisons on displacements resulted from ABAQUS and DSAS for P1 and P2 88 5 4 ......... Compar isons on displacements resulted from ABAQUS and DSAS for P3 and P4 88 5 5 ................ Comparisons on displacements induced by constant and variable axial loads 1 01 PAGE 8 8 LIST OF FIGURES Figure page 2 1 Air burst explosion................................................................................................................. 19 2 2 Groundburst explosion ......................................................................................................... 19 2 3 Typical blast wave pressure time history graph ............................................................... 20 2 4 Partial and simplied time history graph. ............................................................................... 2 2 2 5 Real and typical equivalent SDOF system ........................................................................... 2 3 2 6 Mass diagram ......................................................................................................................... 24 2 7 Load diagram .......................................................................................................................... 25 2 8 Resistance functions ............................................................................................................... 28 2 9 Stress and strain diagram of a cross section ......................................................................... 29 2 10 Typical shear failure of a column .......................................................................................... 30 2 11 Flexure s hear interaction model v alley of d iagonal f ailure ......................................... 31 2 1 2 Flexure s hear interaction model ........................................................................................... 32 2 13 Direct shear slip relationship ................................................................................................ 34 2 14 Degrading stiffness method ................................................................................................ ... 38 2 15 Idealized hysteresis loops for reinforced concrete ............................................................... 38 2 16 Typical response of a SDOF system ..................................................................................... 39 2 17 Dynamic reactions for beam with arbitrary boundary conditions ....................................... 42 2 1 8 Typical p ressure i mpulse diagram ....................................................................................... 4 4 2 1 9 Search algorithm in developing P I diagram ....................................................................... 4 5 2 20 Typical P I diagram for multi failure modes ....................................................................... 4 6 3 1 Blast load s on structure .......................................................................................................... 4 8 3 2 Load diagram for supported mass ......................................................................................... 49 3 3 Load diagram for column ...................................................................................................... 49 PAGE 9 9 3 4 Newton Rhapson method ...................................................................................................... 5 1 3 5 Typical load deformation diagram of a strucutre ................................................................ 5 2 3 6 Spherical constant arc length criterion for SDOF system ................................................... 5 3 3 7 ........................................................................................................ S upported mass diagram 5 5 3 8 Column with axial load .......................................................................................................... 5 5 3 9 Flow chart for determining dynamic reactions based on failure mode ............................... 5 6 3 10 Newmark Beta method for computing displacement .......................................................... 5 7 4 1 Beam 1 C layout ..................................................................................................................... 6 0 4 2 Load function for beam 1 C ................................................................................................ .. 6 2 4 3 Load function for beam 1 G ................................................................................................ .. 6 2 4 4 Load function for beam 1 H ................................................................................................ .. 6 3 4 5 Load function for beam 1 I .................................................................................................... 6 3 4 6 Load function for beam 1 J ................................................................................................ ... 6 4 4 7 Typical modeling of experimental beam using ABAQUS .................................................. 6 5 4 8 .......... Graphical presentation of parameters in the M odified Drucker Prager/ Cap M odel 6 6 4 9 Stress strain relationship tension reinforcements b eam 1 C ......................................... 68 4 10 Stress strain relationship compression reinforcement b eam 1 C ................................ .. 68 4 11 Stress strain relationship tension reinforcements b eam 1 G ......................................... 69 4 12 Stress strain relationship compression reinforcement b eam 1 G ................................ 69 4 1 3 Stress strain relationship tension reinforcements b eam 1 H ......................................... 70 4 14 Stress strain relationship compression reinforcement b eam 1 H ................................ 70 4 15 Stress strain relationship tension reinforcements b eam 1 I .......................................... 71 4 16 Stress strain relationship compression reinforcement b eam 1 I ................................ ... 71 4 17 Stress strain relationship tens ion reinforcements b eam 1 J .......................................... 72 4 18 Stress strain relationship compression reinforcement b eam 1 J ................................ ... 72 PAGE 10 10 4 19 Typical DSAS data entry screen ........................................................................................... 7 3 4 20 Comparison of d isplacement t ime history for beam 1 C ..................................................... 75 4 21 Comparison of d i splacement t ime history for beam 1 G .................................................... 7 5 4 22 Comparison of d ispla cement t ime history for beam 1 H .................................................... 76 4 23 Comparison of d isplacement t ime history for beam 1 I ...................................................... 7 6 4 24 Comparison of d isplacement t ime history for beam 1 J ...................................................... 77 4 25 Loading function for 500 pounds of T rinitrotoluene (T NT ) at 20 ft ................................ .. 78 4 2 6 Displacement t ime history of beam 1 C ............................................................................... 7 8 4 2 7 Typical column layout in ABAQUS ..................................................................................... 80 4 2 8 Loads and boundary conditions for column in ABAQUS ................................................... 8 0 4 2 9 Longitudinal and transverse steel re inforcement layout in ABAQUS ................................ 81 4 30 Displacement t ime history column subject to blast load .................................................. 8 1 5 1 Axial moment interaction diagram confined conrete DSAS v er s us CRSI .................. 8 4 5 2 Stress strain relationship tension and compression reinforcements column 1 ............. 8 5 5 3 Stress strain relationship tension and compression reinforcements column 2 ............. 86 5 4 Stress strain relationship tension and compression reinforcements column 3 ............. 86 5 5 Stress strain relationship tension and compression reinforcements column 4 ............. 87 5 6 Axial moment interaction diagram 8 No. 7 RC confined ............................................. 9 0 5 7 Displacement time history diagram 8 No. 7 RC confined ............................................ 9 1 5 8 Flexure resistance diagram 8 No. 7 RC confined ................................................. 9 1 5 9 Moment curvature diagram 8 No. 7 RC confined ......................................................... 9 2 5 10 Pressure impuluse diagram 8 No. 7 RC confined .......................................................... 9 2 5 11 Axial moment interaction diagram 8 No. 10 RC confined ........................................... 9 3 5 1 2 Displacement time history diagram 8 No. 10 RC confined .......................................... 9 3 5 1 3 Flexure resistance diagram 8 No. 10 RC confined ........................................................ 9 4 PAGE 11 11 5 14 Moment curvature diagram 8 No. 1 0 RC confined ....................................................... 9 4 5 1 5 Pressure impuluse diagram 8 No. 1 0 RC confined........................................................ 95 5 1 6 Axial moment interaction diagram 12 No. 1 1 RC confined ......................................... 95 5 1 7 Displacement time history diagram 12 No. 1 1 RC P = 0 to 570 kips confined.......... 96 5 1 8 Displacement time history diagram 12 No. 11 RC P > 570 kips confined ................. 96 5 19 Flexure resistance diagram 12 No. 1 1 RC confined ...................................................... 97 5 20 Moment cur vature diagram 12 No. 1 1 RC confined ..................................................... 97 5 21 Pressure impuluse diagram 12 No. 1 1 RC confined...................................................... 98 5 22 Axial moment interaction diagram 4 No. 14 RC confined ........................................... 98 5 23 .......................................... Displacement time history diagram 4 No. 14 RC confined 99 5 24 Flexure resistance diagram 4 No. 14 RC confined ........................................................ 99 5 25 Moment curvature diagram 4 No. 14 RC confined ..................................................... 1 00 5 26 Pressure impuluse diagram 4 No. 14 RC confined...................................................... 100 5 27 Variable axial load profile ................................................................................................... 1 02 5 28 Displacement time history 8 No. 7 RC P ............................................. 1 02 5 29 Displacement time history 8 No. 7 RC P > Pbal + Pvar ............................................. 1 03 5 30 Displacement time history 8 No. 10 RC P ........................................... 1 03 5 31 Displacement time history 8 No. 10 RC P > Pbal + Pvar ........................................... 1 04 5 32 Displacement time history 12 No. 11 RC P ......................................... 1 04 5 33 Displacement time history 12 No. 11 RC P > Pbal + Pvar ......................................... 1 05 5 34 Displacement time history 4 No. 14 RC P + Pvar ........................................... 1 05 5 35 Displacement time history 4 No. 14 RC P > Pbal + Pvar ........................................... 1 06 PAGE 12 12 LIST OF SYMBOLS AND ABBREVIATIONS a Distance from s upport to the l oad a Speed o f s ound 0 A Cross section gross area g ATM Atmosp heric pressure d Depth from the top of concrete to the layer of reinforced bars f Specified compressive strength c f Yield strength y F(t) Load time function h Depth of concrete cross section I Positive incident impulse s I Positive normal reflected impulse r K Tangent stiffness matrix K Coefficient matrix K Equivalent stiffness e K Load factor L K Mass factor M L Wavelength of positive phase w m Mass M Moment M Equivalent mass e M Ultimate moment due to pure flexure fl M Nominal flexural strength n PAGE 13 13 M Ultimate moment due to shear and flexure u M Total mass t N Factored axial force normal to cross section u P Equivalent l oad e P d Downward l oad (t) P Reflected p ressure r P Incident p ressure so P Actual total load t P u Upward l oad (t) q Dynamic p ressure s R Stand off distance G R Load v ector R Maximum plastic limit load m R(t) Resistance f unction SRF Shear r eduction f actor t Time of arrival of blast wave A t Positive duration of positive phase pos U Shock front velocity V Nominal shear strength c w Width of concrete beam cross section W T rinitrotoluene (TNT) e quivalent c harge w eight Angle of i ncident Maximum c ompression s train cm PAGE 14 14 Displacement Incremental d isplacement Incremental l oad Inertia and l oad p roportionality f actors Load m ultiplier Curvature Assume s hape f unction Reinforcement r atio Air density behind the shock front s Density of air beyond the blast wave at atmospheric pressure 0 Shape f unction Stress Maximum s hear s tress m PAGE 15 15 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science EFFECT OF SHORT DURATION HIGH IMPULSE VARIABLE AXIAL AND TRANSVERSE LOAD S ON REINFORCED CONCRETE COLUMN By Thien Phuoc Tran May 2009 Chair: Theodor Krauthammer Major: Civil Engineering Previous studies were conducted on the deformations of reinforced concrete columns induced by blast load that combined both axial and transverse loading components. Most of those studies assumed that the response of the mass supported by the column in its axial direction developed much slower compared to that in the lateral movement. Thus, the load transferred from the supported mass to the colu mn in its axial direction could be treated as a static load. Moreover, when comparing the vertical displacement with the lateral displacement of the column it was assumed that the former was much small er, and therefore was negligible Consequently, the failure of the column was assumed to be governed by the flexure caused by transverse loads. While this may be true, the effect of variable axial load s may still be an important factor in determining the failure of the column. Thus, the above simplified a ssumption should be re examined to determine the actual effect of variable axial loads on the behavior of a column. PAGE 16 16 CHAPTER 1 INTRODUCTION 1 1 Problem Statement Progressive collapse of a building is normally caused by an abrupt failure of one or more structural bearing members such as beams or columns. Therefore, the endurance of these members under short duration but highly impulsive loads is crucial for the survivability of the building. While beams are normally subject to transverse loads, columns are always exposed to both transverse and axial loads. In practice, it is assumed that failure of a column is normally caused by transverse rather than axial loads. This may not be accurate, particularly in the case where a structure is subjected to sho rt duration but highly impulsive loads such as blast loads. While the failure of the column will most likely be induced by the transverse loads, the effect of variable axial loads should also be considered as a contributing factor. The column resistance m ay be reduced due to the variable axial load s under the same material and physical properties, a nd the column may fail sooner. On the other hand, the alterations in directions and the eccentricity of the variable axial loads over the time period may act as an enhancement factor to the strength of the column, thus preventing it from failing in the early stage. The discussion on e ccentricity is, however, not a in the scope of this work and therefore is not included in this study. 1 2 Objective and Scope The objective of this research is to determine the actual effect of variable axial loads on the column, allowing them to be properly a ccounted for during the design stage of structure subjected to blast loads. PAGE 17 17 This research will: Develop a Single Degree of Freedom (SDOF) algorithm for axial and transverse loads on a reinforced concrete column and implement it within the Dynamic Structu re Analysis Suite (DSAS) Version 2.0 (Center for Infrastructure and Physical Security, University of Florida (CIPPS, UF)) Model the columns in ABAQUS Version 6.8 1 (Dassault Systmes, 2008) using the same physical and material properties Validate the a bove software applications with available experimental data. Validate analytically between the two software applications. Conduct parametric study using DSAS Version 2.0 based on the above results 1 3 Research Significance The outcome of this research will v erify the assumption stated in the problem statement. By knowing the significant effect of axial loads, building structures in high threat environment will have a better chance to resist and endure when subject to blast effects. On the other hand, for st ructure that may not be subject to these conditions, excluding the effect of variable axial loads will reduce the cost and the runtime required for the analysis and design. PAGE 18 18 CHAPTER 2 LITERATURE REVIEW 2 1 Introduction In the past few decades, there are number s of books, publications, and technical papers explaining and discussing the effects of blast load on structures, particularly on the combined effects of flexure, axial, diagonal and direct shears (Biggs 1964, Murtha and Holland 1982, Baker et al. 1983, Kr authammer et al. 1988). Properties and behavior of blast when exerted on a structure were experimentally studied and recorded. Equations for calculations of all necessary parameters were either derived mathematically or empirically. The results were wel l explained and summarized in tables and charts allowing the users to expediently obtain pertinent information for using in the design of structures (Department of the Army, 1990). Although fragmentation is normally associated with blast, it is not within the scope of this work. Therefore, t he following sections in this chapter will provide a brief summary of these studies with respect to blast only as well as the required information to be used in this research. 2 2 Blast load ing on Structures When explosive detonates, it generates a sudden, violent release of energy. After the arrival time, tA following the detonation, the pressure peaks at its highest value, Ps o above the atmospheric pressure (1 ATM = 14.7 psi). There are two types of blast bursts with e ach type generates different effect These are air burst and ground burst If the charge is air burst, depending of the angle of attack, the Pso is further increased by the reflection off from the ground as shown in Fig. 2 1. If the charge is groundbu rst, then the Pso is at its maximum as the reflection of the blast wave occurs immediately as shown in Fig. 2 2. PAGE 19 19 Figure 2 1. Air b urst e xplosion (TM5 1300, 1990) Figure 2 2 Groundb urst e xplosion (TM51300, 1990) PAGE 20 20 As the shock wave moves through air, it is followed by an air pressure pocket which travels at a slower speed. This air pressure, a function of time, is known as the dynamic pressure q(t). As shown in Fig. 2 3, two stages are formed within a very short d uration. An outward burst of the blast wave immediately raises the pressure over the ambient atmospheric pressure. This is also known as the positive phase. Shortly after that, this pressure drops below the ambient atmospheric pressure (negative phase) as the distance increases. Figure 2 3. Typical b last w ave p ressure t ime h istory g raph Where: Pso P Incident pressure, r I Reflected pressure, s/W1/3 I Scaled unit positive incident impulse, r/W1/3 t Scaled unit positive normal reflected impulse, A/W1/3 t Scaled time of arrival of blast wave, pos/W1/3 U Shock front velocity, Scaled positive duration of positive phase, Lw/W1/3 Ps0 P0P(t) t tA t0 t0 1 ATM TPOSTNEG Area iS Ps0 PrPr Reflected Pressure Incident Pressure Scaled wavelength of positive phase. PAGE 21 21 A scaled distance, Z, expressed in terms of stand off distance, RG 3W R ZG and the Trinitrotoluene ( TNT ) equivalent charge weight, W, is used as a common factor to determine all the above parameters. The scaled distance, Z, is calculated as follow: (2 1) Depending on the category of explosion, air burst (spherical) or ground burst (hemispherical), data for the listed parameters can be obtained from the charts provided in the US Army TM 5 1300 (1990). Calculations for these parameters can be found in numerous references. Brode (1955) introduced the calcu lations for the over pressure in unit of bars: 1 7 63 ZPso bar for 10 soP bars (2 2) 019 0 85 5 455 1 975 03 2 Z Z Z Pso bars for 10 1 0 soP bars (2 3) This formula was later refined by Newmark and Hansen (1961) for ground burst explosion: 2 1 3 393 6784 R W R W Pso bars (2 4) s, and dynamic pressure, qs 0 0 07 7 6 a P P P Uso were introduced by Rankine and Hugoniot (1870): (2 5) 0 0 07 7 6 P P P Pso so s (2 6) ) 7 ( 2 50 2P P P qso so s (2 7) PAGE 22 22 Where a0 0 so so so rP P P P P P 0 07 4 7 2 is the density of air beyond the blast wave at atmospheric pressure. In the case where reflection occurs, then: (2 8) To compute the value of pressure, P, along the curve, one can use the Modified Friedlander equation: t pose T t P t P 1 ) (max (2 9) The impulse, the area under the positive portion of the blast wave time history graph as shown in Fig. 2 3, can be calculated by integrating Eq uatio n 2 9 posT sdt t P i0) ( (2 10) However, approximation can be made for Fig. 2 3 by considering only the positive portion of the curve as a triangle. It can be simplified and re drawn as shown in Fig. 2 4 A B Figure 2 4. Partial and s implified T ime H istory g raph A) p artial blast wave B) simplified blast wave P(t) t P max i S T pos P(t) t T pos P max i S PAGE 23 23 T he relation of impulse with respect to a blast pressure and time can then be approximated based on Fig. 4 B: pos sT P i max2 1 (2 11) Since is and Pmax can be determine d from E quations 2 4 and 2 10, the Tpos2 3 Structure and Its Equivalent System can be computed by manipulating Equation 2 11. 2 3 1 Introduction For most structure, there will be an infinite number of degrees of freedom. It will be cumbersome and inefficient to analyze the structure in this form, not to mention the unfeasibility at times. Therefore, it is often possible to reduce the system to a single degree of freedom (SDOF) system to simp lify the process. To achieve this, equivalent parameters for the SDOF system such as mass, Me, stiffness, Ke, load, FeA and load time function F(t) need to be setup. Although the response of the equivalent system, in term of forces and stress, are not the same as that of the actual system; the deformation and time, however, are the same in both systems. As such, selection of the equivalent system should be based on the criteria that the deflection of the concentrated mass is the same as that for a signifi cant point on the actual structure as shown in Fig. 2 5 B Figure 2 5 Real s ystem and t ypical e quivalent SDOF s ystem A) Actual structural member. B) Equivalent SDOF system C K e M e W u L L w(x) u PAGE 24 24 2 3 2 Equivalent M ass As shown in Fig. 2 6 m(x) is the continuous mass of the element u is the deflection at mid span, u shape function of the element. Figure 2 6 Mass d iagram Assuming that the displacement, u, and the velocity u can be appr oximated as: ) ( ) ( ) (t u x t x u ) ( ) ( ) (t u x t x u ( 2 12) (2 13) Then the kinetic energy of the distributed mass system can be: dx t x u x m KEL 0 2)) ( ( ) ( 2 1 dx t u x x m KEL 0 2 2)) ( ( )) ( ( ) ( 2 1 (2 14) (2 15) And the kinetic energy of the equivalent system is 2)) ( ( 2 1 t u M KEe e (2 16) If Me is selected so that the kinetic energy of the real system is the same as that of the equivalent system, then by equating and simplifying the right hand sides of Eq uatio ns 2 1 5 and 2 16, Me m(x) u u(x) x L L (x) for the distributed mass system can be obtained and expressed as below: or PAGE 25 25 L edx x x m M0 2) ( ) ( (2 17) Similar process can be applied for the lumped mass system, which will yield: )] ( [2 1 i n i i ex m M (2 18) Thus the mass factor KM is the ratio of equivalent load, Me, to actual total load, Mt t e MM M K (2 19) 2 3 3 Equivalent Load Using the parameters shown in Fig. 2 7 as well as defining w(x) as the distributed load and Pi Figure 2 7 Load d iagram The work done by the external load, WE, on the real system must be equal to the work done by the external load, WE as the concentrated load at location i, the equivalent load can also be determined following similar approach. e, on the equivalent system. Thus by defining, equating and simplifying the expressions for WE and WEe, the equivalent load, Pe P1P2Pn w(x) u u(x) x1 x2 xn L L (x) for the distributed load system can be expressed as follow. PAGE 26 2 6 Ldx t x u t x w WE0) ( ) ( (2 20) Where u(x, t) is as previously 2(x) represents the assumedshape function for the distributed load. Since this is a uniformly distributed load 2 ) ( ) ( ) ( ) (2t w t w x t x w (x) = 1. The expression for w(x, t) becomes ( 2 21) Therefore, Ldx t u x t w WE0) ( ) ( ) ( (2 22) Or Ldx x t u t w WE0) ( ) ( ) ( (2 23) The work done by the equivalent external load on the equivalent system will be: ) ( t u P WEe e ( 2 2 4 ) Equating and modifying the right hand sides of the expressions for the work done on the real system, WE, and the work done on the equivalent system, WEe, Pe L edx x t w P0) ( ) ( for the distributed load system can be obtained: (2 25) T he same process can be repeated again for the concentrated load case; of which: )) ( (1 i n i i ex P P (2 2 6 ) Thus the load factor KL is the ratio of equivalent load, Pe, to actual total load, Pt t e LP P K (2 27) PAGE 27 27 2 3 4 Shape F unctions As shown in Section 2.3.4, to obtain the equivalent mass and equivalent load, it is required to use the assumedfactor s Biggs (1964) also provided some assumedshape functions for simply supported beams and one way slabs under various types of loading conditions. The da ta is, however, only applicable for either elastic or fully plastic and nothing in between. E mployment of these assumed shape functions will introduce some small errors and will be limited to the outlined load cases. Therefore, to correct these errors an d eliminate these constraints, modifications to the employment of these shape functions will be required. Various approaches were done in the past with the attempt to overcome the above mentioned limitations. Summary of some of these approaches will be p rovided in the later section. 2 3 5 Resistance F unctions In general, when an external load is exerted on a column, the column tends to produce a resistance force trying to reinstate it to its original position. Biggs (1964) suggested, as shown in Fig. 2 8 A the three possible shapes corresponding to the three categories of materials: brittle, ductile, and plain concrete or instable structures. Simplification is made for most structures by using the bilinear functions in computing the resistant factor as shown i n Fig. 2 8 B ; where Rm is the maximum plastic limit load that the beam could support statically. Thus in the linear elastic range, the resistance factor, KR, is the same as the load factor, KL, due the fact that the deflection is the same for both systems Since this does not apply nonlinear situation, further development will be required and will be discussed in another section. PAGE 28 28 A B Figure 2 8 Resistance f unctions A) Actual. B) S implify (Biggs, 1964) T he general equation of motion and its applicable form for a linear elastic system can be expressed as shown in Equations 2 28 and 2 29, repectively. ) ( t F u k u c u m ( 2 28 ) ) ( ) ( t P t R K u C u Me L e ( 2 29 ) Where R(t) is the resistance function that replaces the product of the spring stiffness and the displacement for an elastic beam and is defined as the restoring force in the spring, and the maximum resistance is the ultimate load the beam can carry under static conditions for an inelastic beam (Krauthammer et al., 1988). 2 4 St atic Analysis 2 4 1 Flexural Behavior Figure 2 9 shows a typical stress and strain diagram of a cross section in the plastic deformation state under axial load. The axial force P, at any instance t can be computed from the blast pressure p(t) as outlined in Eq uatio n 2 9. R Brittle Ductile Plain / Instable R k 1 elRm PAGE 29 29 Figure 2 9 Stress and strain diagram of the cross section Moment curvature relationship can be derived using the computed values of stresses and strains at any given point on h/2 as the strain value at the mid depth of the z z hh z ) 2 ( ) tan() 2 ( can be calculated using similar triangle. Thus: (2 30) A ssuming small angle is used the above equation can be re written as: z hh z22 (2 31) From the constitutive relation of stress and strain, stress at any point on the cross section can be expressed as a function of strain at the corresponding point. z zf ( 2 32 ) T he associated moment can then be defined as: dz zh Mh z 02 (2 33) Eq uation s 2 31 and 2 33 define the Moment Curvature relationship for the cross section. h b dsbh/2 P Strain, Stress, cmc dst z zh/2 PAGE 30 30 2 4 2 Influence of S hear on F lexural R esponse Aside from flexure failure, shear failure is also a factor that needs to be considered. Two types of shear failures commonly known are diagonal shear and direct shear. Fig. 2 10 shows a typical shear failure of a column. Figure 2 10. Typical s hear f ailure of a c olumn (MacGregor and Wight 2009 ) To account for shear effect in the design for members that are subject to shear and flexure only, ACI 31805 use s Eq uatio n 2 34. However, further knowledge is required to fully understand the actual behavior and the interaction between shear and flexure. d b f Vw c c '2 (2 34) Where: Vc f' the nominal shear strength. cNumerous studies were conducted to examine the influence of shear on flexural response (Kani, 1966, Placas and Regan, 1971, Haddadin et al., 1971, Bazant and Kim (1984), Ahamad the compressive strength. PAGE 31 31 and Lue., 1987, Krauthammer et al., 1988). It was determined that the failure due to shear effective depth ratio, a /d ; where a is the distance from the support to the load and d is depth taken from the top of concrete beam to the first layer of reinforcing bars. Fig s 2 11 and 2 12 show two models proposed by Kani (1966) and Ahamad et al. (1987) respectively; where Mu/Mfl Figure 2 11. Flexure s hear i nteraction m odel v alley of d iagonal f ailure (Kani, 1966) is the ratio between the ultimate moment due to shear and flexure and the ultimate moment due to pure flexure. This ratio is known as the shear reduction factor (SRF). PAGE 32 32 Figure 2 12. Flexure s hear i nteraction m odel (A hmad et al., 1987) These two models were evaluated and further developed and modified by Krauthammer et al. (1988) and Russo et al., (1991 and 1997). Detailed discussion on the developments by these two authors is not part of the scope of this work and ca n be found in Krauthammer et al. (2004). Based on the result found, approach by Krauthammer et al. (1988) concluded in the closer match with the experimental data presented by Kani (1966). Hence, this research paper will adopt this approach. This means that to take the shear effect into account more correctly the moment will be multiplied by the SRF and the curvature will be divided by the SRF (Krauthammer et al., 2004). This, in turn, produces a more accurate result for both moment and curvature. 2 4 3 Infl uence of A xial F orce on S hear and F lexural R esponses The presence of axial load enhances the moment capacity of the cross section as well as delays cracks from occurring (Krauthammer et al 1988). This is because the compressive load increases the normal stress and reduces effect of the principal tensile stress. To maintain equilibrium, the sum of forces must still be zero. Thus: PAGE 33 33 h zP dz Fx00 (2 35) The current code, ACI 31805, accounts for the axial load effect on flexure and shear by employing the equations for nominal shear strength, Vc, and nominal flexural strength, Mn d b f A N Vw c g u c '2000 1 2 Eq uatio ns 2 36 and 2 37 represent the lower and upper bound of the shear strength respectively: (2 36) g u w c cA N d b f V 500 1 5 3' (2 37) 8 4 d h N M Mu u n (2 3 8 ) Where: Vc : nominal shear strength provided by concrete. Nu : factored axial force normal to cross section. Ag : section gross area. fc : specified concrete compressive strength. bw : width of the section. d: is the effective depth which is from the top compression fiber to the centroid of the longitudinal tension reinforcement. Mattock and Wangs studies (1984) conducted a study and found that Equatio n s 2 36 and 2 37 were too conservative. The recommended r eplacements for Eq uations 2 36 and 2 37, respectively, are: '3 1 2c g u c cf A N f V (2 3 9 ) PAGE 34 34 g u c cA N f V 3 0 5 3' (2 40) 2 4 4 Direct S hear M ode of R esponse There have not been many studies done on the direct shear effect. In addition to the above mentioned factors, Park and Paulay (1975) also suggested that the transferring of high shear stress across a weak section where cracks have formed was another facto r that indicated the significant effects of direct shear on flexural members. The failure due to direct shear was further proven by the dynamic tests that were conducted by Kiger et al. (1984) and Slawson (1984). Hawkins (1974) proposed an empirical mode l on the behavior of shear stress versus slip; which did not include the effect of compression loads. This relationship was later modified by the Krauthammer et al. (1988) to include the effects of load reversals. Figure 2 13. D irect s hear slip r elationship (Krauthammer et al. 2002) Summary of the slip range and the associated descriptions for each segment in Fig. 2 13 is shown in Table 2 1. O A B C D E mLe1234maxShear Stress, Slip, Ke Ku PAGE 35 35 Table 2 1 Definition of d irect s hear s relationship (Hawkins, 1974) Segment Slip Description 3, in OA 0 4 Elastic response. Positive slope e eK 2 157 0 165' m c ef (2 41) AB 4 12 Slope decreases but remains positive. m '35 0 8 0 8c y vt c mf f f (2 42) Where vt g s vtA A is the ratio between the area of reinforcement crossing the shear plane and the gross area, fy steel yield strength BC 12 24 m remains constant. CD Constant decrease slope. Independent of reinforcement crossing the shear plane. '75 0 2000c uf K (2 43) DE > 24 Shear capacity remains constant. Deformation at E varies with the level of damage. Krauthammer et al. (2002) went further to define the upper limits of segments CD and DE. Eq uatio n s 2 42 and 2 43 show the upper limits of D and E respectively: g s s LA f A'85 0 (2 44) 60 1max xe Where b cd f x'86 2 900 (2 45) PAGE 36 36 Based on the proven fact that axial load increases the strength of the beam in a sense that it deters the cracks from extending into the compression block, it is safe to assume that the maximum stress will also increase. Hence, it is proposed that with the effect of axial load, the m (Eq uatio n 2 42) will increase its value by 1 + (Nu / (2000*Ag)). Thus the equation m '35 0 8 0 2000 1 8c y vt c g u mf f f A N can be re written as below. This point still needs to be proven as part of this research. (2 46) 2 5 Dynamic Analysis 2 5 1 Newmark Beta M ethod For a nonlinear system, a different approach will be required to obtain the solut ion for the equation of motion ( Eq uatio n 2 28). There are numbers of methods that can be used to for computing the displacements, some of which are the acceleration method (implicit method) and the central difference method (explicit method). The main difference between these two is the time at which the equation of motion is satisfied. The former method satisfies the equation of motion at time ti+1 and the latter one is at time ti Knowing y For this research, the linear acceleration (implicit) method will be used. It is basically a special case of Newmark Beta Method is as follow: i iy and at time ti iy (Eq uatio n 2 12), compute for Estimate 1 iy Compute 1 iy and yi+1 i i i iy y t y y ) 1 (1 1 (2 47) PAGE 37 37 i i i i iy y t y t y y 2 11 2 1 (2 48) Using the equation of motion Eq uatio n 2 28, and the above computed values of 1 iy and yi+1, 1 iy compute for Check if convergence is satisfied. If so, move to the next time step; if not, use the newly computed value of 1 iy for the next iteration. ss than T 3 where T is the natural period of the system. A flowchart for this method will be presented in the next chapter. 2 5 2 Dynamic Resistance Functions The dynamic resistance responses of a structure depend on a number of factors. These include the stability nonlinearity, the geometric nonlinearity, and material nonlinearity of the structure. When an external load that acts on a structure is less than the yield load of the structure, the response of the structure is still in the elas tic range. Thus, the resistance force or the internal force is the product of structure stiffness and the displacement (Fint = R = k*u). However, when the external load is greater than the yield load, the resistance force or the internal force is no long er linear and becomes a function of displacement (Fint = R = fn(u)). Figs 2 14 and 215 represent t wo models that were proposed by Clough et al. (1966) and Sozen (1974) that illustrated the nonlinear behaviors of a structure. Both of these models were por trayed by a number of bilinear resistance functions, of which each segment represented a loading stage. The order of loading sequence is as shown in alphabetical order. PAGE 38 38 Figure 2 14. D egrading s tiffness m odel (Clough, 1966) Figure 2 15. Idealized h ysteresis l oops for reinforced c oncrete (Sozen, 1974) Based on these two models, Krauthammer et al. (1988) proposed a piecewise multi linear curve model (Fig. 2 16) that included all the above mentioned nonlinearities and portrayed both elastic and inelastic response of a beam with the following assumptions: a b h / Yi g c o d e f M/MU1 1 1 a, b, c, ... is the loading sequence PAGE 39 39 a Beam is symmetrically reinforced. For unsymmetrical reinforced beam, two R will be required for both negative and negative loadings. b Maximum displacement is reached during the first po sitive cycle, which is valid only for blast or impact loads. max) is less than the y). Within this range, the beam will behave elastically and oscillation will occur al ong line A A. This oscillation will eventually come to rest once external damping max y, the beam will be in the inelastic range, at which point permanent deformation will take place. The order of points A, B, D, E, F, G, D, and E in the figure below (quadrants I and IV) describes the sequence of loading and plastic ) of the beam when all the energy is completely dissipated. Point C represents the flexure failure of the beam. Figure 2 16. Typical response of a SDOF s ystem (Krauthammer et al., 1988) PAGE 40 40 2 5 3 Modified Equivalent Parameters for SDOF System. As mentioned in Section 2.3, the mass factor and the load factor derived by Biggs (1964) were based on the assumed shape functions that were either for elastic or fully plastic and nothing in between. Moreover, these assumedshape functions were also derived for specific cases of end supports and loading conditions. With a better insight on the be haviors of a structure member in both elastic and inelastic range based on the above model proposed by Krauthammer et al. (1988), these factors can be re evaluated to account for the behaviors of the member in the transition stage between elastic and inela stic. The procedures to compute mass factor and load factor were extracted from the above mentioned reference. 2 5 3 1 Mass factor The mass factor can now be computed along the loaddeflection curve by taking the integration over the length of the beam for each load step j. Using Fig. 2 6 and all previously defined variables, the equivalent mass and the mass factor at load step j can be expressed as below. n i j i j i L j ex m dx x x m Mj1 2 0 2) ( ) ( ) ( (2 4 9) t e MM M Kj j (2 50) Thus the mass factor, KM j j j M M M Mj J jK K K K ) 1 () 1 ( for each time step can be computed: (2 51) j Reference to Fig. 2 16, K(j+1) M is computed for every time step until it reaches point B, which is the maximum inelastic displacement. Beyond that, it will remain constant based on the PAGE 41 41 assumption that the shape functions do not significantly change after the formation of the plastic hi nges. 2 5 3 2 Load factor The procedure to re evaluate the load factor for the transition stage between elastic and plastic is similar to that of the mass factor. Hence, using Fig. 2 7, for each load step j with shape (j,i) ) () (i j Lx Kj where i is the location to be evaluated, the load factor for each load step can be expressed as: ( 2 52 ) Thus the load factor for each time step will be ) () 1 () 1 (j j j L L L Lj J jK K K K (2 53) j (j+1)2 5 4 Dynamic Reactions Biggs (1964) provided a series of tables of equations for the dynamic reactions of various loading scenarios with different boundary conditions. As mentioned earlier, t he limitations of these equations are that they are only applicable to either perfectly elastic or plastic structures and are bounded by specific load cases. Computations of the dynamic reactions based on these equations will introduce inaccurate results for the purpose of this paper. Therefore, another approach is required. Krauthammer e t al. (1988) introduced a procedure that could be used to neutralize the above limitations. This procedure was based on the assumption that the distribution of the inertia forces is identical to the deformed shape function of the beam as shown in Fig. 2 17. PAGE 42 42 Figure 2 17. Dynamic r eactions for b eam with a rbitrary b oundary c onditions (Krauthammer et al., 1988) Summary of the procedure is as follow (Krauthammer et al., 1988): a For each load step i, obtaining the reactions at each end of the of the element and compute i i iQ Q /1 1 ( 2 54 ) i i iQ Q/2 2 ( 2 5 5 ) Where: Q1i and Q2i Q are the static reactions at load step i of end 1 and 2. i is the load at step i. 1i 2i are the load proportionality factors at end 1 and 2. b For every load step, compute the Inertial Load Factor, IFL: dx x L ILFL i i)) ( ( 10 (2 5 6 ) Where: IFL is the load factor associated with the distribution of the inertial forces. shape function at load step i. c 1i 2i at every load step i. These factors are approximated using the principles of the linear beam theory. d Compute the dynamic reaction at each end of the element for each time step: 1i' 2iQ(t) 1Q(t) 2Q(t) (x) M1 M2 PAGE 43 43 X M ILF t Q Vt i i 1 1 1) ( ( 2 57 ) X M ILF t Q Vt i i 2 2 2) ( ( 2 5 8 ) Where: Q(t) is the forcing function Mt is the mass of the beam X is the acceleration 1i, 2i i 1 and i 2 specific displacement at every time step by using the following linear interpolation equation: i i i i i i 1 1 for 1 i i (2 5 9 ) 2 6 PressureImpulse (P I) Diagrams 2 6 1 Characteristics of P I Diagrams P I diagrams are graphical tools used to determine the potential damage of a structure caused by dynamic loads. Detailed descriptions on P I diagrams can be found in numerous references (Krauthammer, 2008). There are three distinguished regions on the P I curve, as shown in Fig. 2 18. These are the Impulsive Loading Region, the Quasi Static Loading Region and the Dynamic Loading Region. In addition, there are two asymptotes. The Impu lse Asymptote is tangent to the Impulsive Loading Region and the Pressure Asymptote is tangent to the Quasi Static Loading Region. In the Impulsive Loading Region, the response time of the structure is much longer than the duration of the loading. Hence, before the structure can experience any permanent deformation, the load is already dissipated. In the Dynamic Loading Region, the duration for both loading and natural period is approximately the same. The PAGE 44 44 response of the structure in this region depends on the loading history. In the Quasi Static Loading Region, the loading duration is much longer than the natural period. Therefore, the structure experiences maximum deformation before the load completely dissolves. (Smith and Hetherington, 1994) Figure 2 18. Typical p ressure i mpulse diagram 2 6 2 Derivation of P I Diagrams Although approach to develop a P I diagram for a c omplex non linear structure could be possible, it would be however, extremely cumbersome. Hence, numerical approach should be used. The P I diagram is derived from the results of numerous single dynamic analyses where the computed threshold points are used to plot the P I curve. Since the process to obtain these threshold points is intensive in term of computational time, an effective and efficient search algorithm is required. Impulse, psimsecPressure, psi 0 2 4 6 8 10 12 14 16 18 20 100 0 100 200 300 400 500 Impulsive Loading Region Quasi Static Loading Region Pressure Asymptote Impulse Asymptote Dynamic Loading Region PAGE 45 45 Blasko et al. (2007) developed a good search engine where a single radial s earch direction was originated from a n arbitrary pivot point that was located in the failure zone of the P I diagram as shown in Fig. 2 19. The iteration process continued where another arbitrary point between the point in the safe zone and the first assumed point was evaluated. The same procedure was repeated until all the threshold points were successfully acquired This process can be completed for structure that may also experience more than one failure modes such as shear and flexure. The result s from both of these failure modes can then be plotted together for use in evaluating the structure to determine the likelihood of the mode of failure as shown in Fig. 2 20. Figure 2 19. Search algorithm in developing P I diagram (Blasko et al., 2007) PAGE 46 46 Figure 2 20. Typical P I diagram for m ulti failure modes (Chee, 2008) 2 7 Summary In this chapter, the property of explosive blast and the behavior of a beam under the influence of transverse and axial loads due to blast pressure were briefly reviewed along with the possible failure modes due to shear and flexure. The effect of axial load on these modes of failures was also considered. Moreover, the transformation from an actual structure to an equivalent SDOF system through the use o f equivalent parameters such as mass factor, load factor and the dynamic reactions that were based on the employment of the assumed shaped functions was discussed. Since this problem includes nonlinearity, closedform approach would be very cumbersome and inefficient. Hence, direct integration techniques both implicit and explicit were considered. A short summary on the Pressure Impulse diagram and how it can be used to quickly determine the failure of a structure was provided T he brief discussions on each topic in this chapter provide adequate source of information to form a basis for the analysis that will be discussed in the next chapter. Pressure Impulse Mode 1 (Flexure) Mode 2 (Direct Shear) Failure in Mode 1 & 2 Failure in Mode 1 Failure in Mode 2 Safe PAGE 47 47 CHAPTER 3 METHODOLOGY 3 1 Introduction This chapter provide s an overview of the structure being considered as well as outlines the approaches used in determining the effects of variable and constant axial loads on a reinforced concrete column that is also subject ed to transverse loads. Assumptions and simplificatio ns used in the implementation steps will also be included in the appropriate sections of this chapter. 3 2 Load Determination 3 2 1 Overview of Structure In either case of air or ground blast as shown in Figs. 2 1 or 2 2, once the first wave of blast strikes the bu ilding, it will destroy architectural items such as windows and doors. This creates openings in the structure allowing subsequent blast waves to act as internal pressures in the outwards and upwards directions against walls, floor and roof of the building. The structure considered in this case is as shown in Fig. 3 1 of which only the column is being investigated. It should be noted that the loading diagram caused by both exterior and interior pressure is much more complicated than shown. For simplicity purpose, only loads that have significant effects on the column are being considered. In addition, although the arrival time tA, of the blast load will be different for the column and the supported mass ; for ease of computations, it is assumed that tA is the same for the above mentioned structures Moreover, to reduce the computer runtime, the end boundary conditions for the column will be reduced to simply support condition rather than fixedfixed condition. PAGE 48 48 Figure 3 1. Blast loads on structure T he column of the above structure will be subject to flexure, diagonal shear and direct shear induced by: a Lateral dynamic loads F(t). b Downward loads, Pdc Upward loads, P (t). uThe equivalent system for the above structure will be a multi degree of freedom (MDOF) system under the influence of transverse loads and axial loads. Solving for the above structure using multi degree of freedom approach may be inept. Thus the system wi ll be de coupled into two independent members that will eventually be reduced to two equivalent single degree of freedom systems. The first member to be considered is the supported mass. As shown in Fig. 3 2 P(t) is the net load resulting from the di fferential pressure between upward and downward pressure. R (t). 1(t) and R2 F(t) H H Pd(t) Pu(t) L L load. PAGE 49 49 A Figure 3 2 Load d iagram for s upported m ass A) Load diagram, B) Free body diagram The second member to be considered is the column; which is the main focus of this research. Loads that act on this member include th e transverse load F(t), the internal moment M, the weight of the structure above, the variable axial load s which are the dynamic reaction s resulting from the first member, R(t), and the self weight of the column. It should be noted that in both cases, me mbers are subject to transverse loads that could induce failure by either shear or flexure. Thus, consideration of the effects of transverse load is required. A B Figure 3 3 Load diagram for c olumn A) Initial stage, B) Deformed stage R(t) M Plastic Hinge 3 Locations (Typ) M H H F(t) R(t) H H F(t) M M P(t) L L L L P(t) PAGE 50 50 3 2 2 Effect of Transverse Loads. Previous studies on box type buried reinforced concrete structure subject to blast load (Kiger et al., 1984, Slawson et al., 1984, and Ross, T.J. 1983 and 1985) indicated that when element of the studied structure failed due to shear, the flexural response was negligible. On the other hand, when element of the studied structure failed due to flexure, the element was able to withstand the shear forces at the early stage. Based on the result, Krauthammer et al. (1988) suggested the employment of two separate SDOF systems for evaluating flexural and direct shear responses of a beam. Verification against the failur e criterion of the computed results from these two SDOFs was conducted at the end of each time step to determine the mode of failure. As mentioned above, the failure of the two members could also be induced by either flexure or direct shear. Therefore, t his research paper will employ the approach suggested by Krauthammer et al. (1988). This means that each of the two members described above will have two SDOF systems. The failure mode of the first member will be used as a governing factor in the computa tion for the second member. In other word, the dynamic reaction s resulting from the failure mode in the first member, by direct shear or flexure, will be used as the variable axial load s acting on the second member. 3 2 3 Axial Loads As indicated above, two typ es of axial loads will be used in the modeling and the required computations. These are constant and variable axial loads. The constant axial loads are assumed to be induced by the weight of the supported mass over the period of time Application of v ar ious magnitudes of constant axial load s to the column will be considered for comparison purpose Variable axial loads are derived from the computations of the dynamic reactions caused by the effect of the blast load on the supported mass. PAGE 51 51 3 3 Load Deformation Analysis Within the elastic range, the stress distribution of a cross section at any point along the beam will remain linear. As the load increases, the resulting moment also increases until it passes the yield point; when the stress distribut ion of the cross section becomes nonlinear and a plastic hinge is formed. Since the load deformation relationship is no longer linear, equations derived for the linear elastic are no longer valid. As such, a different method is required to trace the nonl inear path of the beam behavior. One of t he most commonly u sed methods wa s the Newton Rhapson method (Fig. 3 4 ); where for any load displacement function, the displacement value at point B could be determined based on the known value at point A. The aim of each iteration, i, wa s to reduce the out of i i ) 1 () 1 ( i i BR u Ki to a satisfactorily small value. Thus: ( 3 1 ) Where: i the iteration number K tangent stiffness matrix i incremental displacement at ith iteration (i 1) incremental load at (i 1)th iteration Figure 3 4 Newton Rhapson m ethod (Bathe, 1996) uAuBDisplacement, u u1u2 KB 1 Load RARB u1 u2 RB FB RB FB KB 1A A 1 1 PAGE 52 52 It was, however, found that this method d id not converge for zero slopes. Thus for typical load deformation diagram as shown in Fig. 3 5 this method w ould not yield the best res ult as it might not be able to pass the post buckling response point. Figure 3 5 Typical l oad d eformation d iagram of a s tructure (Bathe, 1996) To account for this issue and to allow for the post buckling response of a structure, the Cylindrical Arc Length method developed by Crisfield M.A. (1981) is used This procedure allows the tracing to be possible even when the slope of loaddeflection curve is negative. The main difference between this method and the Newto n Rhapson method is the assumption that the load vector, R varies proportionally during the response calculation and the use of the load multiplier; which will need to be determined. In short, Newton Rhapson is a forced controlled method; whereas, Cylind rical Arc Length is a displacement controlled method. Using Fig. 3 6 the algorithm of this procedure can be described as follow with detailed discussion can be found in the reference (Cook et al., 1974): Displacement Load Large load increments Small load increments Load decreases Postbuckling response PAGE 53 53 Figure 3 6 Spherical c onstant a rc l ength c riterion for SDOF s ystem (Cook et al., 1974) a The governing finite element equations for n equations in (n+1) unknowns 0) ( ) ( t t t tF R ( 3 2 ) Or ) 1 ( ) ( ) 1 ( ) ( i t t i i t t iF R u K ( 3 3 ) Where: or decreased. R is the reference load vector for n DOFs of the FEA model. It can contain any loading on the structure but is constant throughout the response calculati on. F is the vector of n nodal point forces corresponding to the element stresses at time i is the iteration order. K is the coefficient matrix. b Additional equation requires for determining the unknowns vector of displacement i i Displacement, u Load RA RB l PAGE 54 54 0 ) ( i iu f ( 3 4 ) Where: t i t t iu u u ) ( ( 3 5 ) t i t t i ) ( ( 3 6 ) c Effective constraint equation is given by the spherical constant arc length criterion 2 2) ( l u ui T i i (3 7 ) Where ui is the total increment of displacement within each load step for the ith iteration. i is the total increment in the load multiplier for the ith iteration. l is the arc length for the time step. 3 4 Computati ons of Dynamic Reactions, Shear and Flexural Responses As mentioned in Sect. 3.2.1, t wo separate SDOF systems will be used for the supported mass and the column respectively. Within the SDOF system for the supported mass, a sub set of two SDOF systems will be used to determine the mode of failure. The governing dynamic reactions of the supported mass produced either by shear or flexure will be used as the time dependant axial load s a pplied to the column. 3 4 1 Computation of Dynamic Reactions for the Supported Mass The equivalent SDOF system and the associated freebody diagram, as shown in Fig. 3 7, can be used to illustrate all the pertinent loads used in the computations The complete process in determining the mode of failure of a structure as well as obtaining the required parameters such as acceleration, velocity, displacement and resistance associated with the failure mode is shown in Fig 3 9. Figure 3 10 presents the Newmark Beta method, which is a part of the PAGE 55 55 complete process. Both of Figs 3 9 and 310 are applicable to Section 3.4.1 and Section 3.4.2. The results obtained from the complete process, particularly the dynamic reactions, will be used for the necessary computat ions in the next section A B Figure 3 7 S upported mass diagram A) SDOF for supported mass B) Free body diagram 3 4 2 Computation of Shear and Flexural Responses on Column. The column is subjected to three different types of loads. First ly it is experienc ed by a transverse load that could cause failure in either flexure or shear. Secondly, it is exerted by the dead load of the structure directly above it as well as its own weight These are considered to be the static loads. Finally, it is also experienced by the variable axial loads which are the dynamic reactions resulted from the supported mass being subjected to the blast load. A B Figure 3 8 Column with a xial load. A) Equivalent SDOF system B) Free body diagram R F (t) R A (t) C*x' M Fe *x" F(t) C K F M Fe F(t) x P(t) R A (t) C*y' M Ae *y" y C K A M Ae P(t) PAGE 56 56 No yes Input Parameters, Pertinent Data Compute Velocity, Displacement, Acceleration, Mass and Load Factors, Resistance START Initialize All Variables Initial Time Increment by t Last Time Step? Compute Applied Force (for beam and column) Convergence? Compute Plastic Displacement Direct Shear Analysis? Compute Shear Force Compute Shear Velocity, Displacement, Shear Resistance and Acceleration Convergence? Shear Failure? Outputs STOP Flexural Failure? No yes No No No yes Process to compute listed variables using NewmarkBeta Method Process to compute listed variables using NewmarkBeta Method Figure 3 9 Flow c hart for d etermining dy namic r eactions based on f ailure m ode PAGE 57 57 Yes Yes START Known Initial Conditions: Select Time Step h Estimate Output No Start Counter Increment Counter Counter 1 STOP Fe, Me, Re Compute 0u m u u t F u u u / ) ( 20 2 0 0 ) 1 ( iu ) 1 ( i trialu u 2 1 ) 1 (2 1 h u uh u u ui i i i i ) 2 ( ) () 1 ( ) 1 (h u u u ui i i i ) 1 ( 2 ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 (2 i i e e e i i i iu u M R M u u t F u 0) 1 ( trial iu u u u u 0 0, u u Figure 3 10. N ewmark Beta m ethod for c omputing d isplacement PAGE 58 58 3 5 Summary This chapter provided the layout of the structure as well as outlined the assumptions and limitations applied to the column being considered. It described the loads exerted on each member of the structure and the computation process to transfer the applicable loads from one member to another The effect of transverse load on the column was also discussed. Approach on load deformation analysis was outlined along with the approach on the computations of the dynamic reactions, shear and flexural response of the equivalent SDOF system. PAGE 59 59 CHAPTER 4 ANALYSIS 4.1 Introduction Numerous steps were taken in order to confirm the objective. Firstly, validations for the ABAQUS Version 6.8 1 (Da ssault Systmes, 2008) and the Dynamic Structure Analysis Suite (DSAS) Version 2.0 (CIPPS, 2008)) were required to ensure that they could produce reliable results. This was completed by comparing the results obtaining from these two applications against k nown data of a series of experiment on reinforced concrete beams. Secondly, the validation was completed on a reinforced concrete column. However, since there was no experimental data available for the column, these two software applications were validat ed analytically using a standard size column with the same material properties of the experimental beam. Lastly, a series of columns of the same physical and material properties but different reinforcements configurations was arbitrarily picked from the C oncrete Reinforcing Steel Institute Design Handbook (CRSI, 2002) for further analysis in the parametric study to determine the effect of axial loads on the P I relationships. This was followed by the analysis on the effects of variable axial loads on the columns. The computer code DSAS was modified to address the effects of axial force on RC columns, and it was used to derive the corresponding P I curves. 4.2 Description of DSAS DSAS (CIPPS 2008) is a comprehensive software suite developed for the analysi s and assessment of structural members subject to severe dynamic loads such as blast and impact. The primary analysis engine in DSAS is based on an advanced single degree of freedom (SDOF) formulation and is capable of developing fully nonlinear resistan ce functions for reinforced concrete, steel, masonry and other members with diverse end conditions using force or displacement controlled solution procedures (Krauthammer et al 1990 and 2003, DSAS User PAGE 60 60 Manual, 2008). The moment curvature relationships f or RC that are derived by DSAS are based on layered section analysis with fully nonlinear material models for steel and confined and/or unconfined concrete. The resistance function is based on a displacement controlled solution approach, and the Direct Sh ear function uses the Hawkins model. The present study enabled the development of an enhanced version of DSAS that allows for constant gravity loads to be specified and modifications can be made to account for dynamic variations in axial force. Moreover, DSAS is also capable to conduct Physics based P I analysis and produce the P I diagrams. 4.3 Validations with Experimental Data 4 3 1 Experimental Data 4 3 1 1 Material and p hysical p roperties Data of five beams from the experiment conducted by Feldman and Siess (1958) were used as part of the validation. Detailed layouts of steel reinforcements for all beams are shown in Fig. 4 2. The main difference in the physical configuration between beam 1 C and the other four beams was the transverse reinforcements. Beam 1 C used opened stirrups and the other two beams used closed stirrups. Other properties of the beams are described in Tables 4 1 and 4 2. Figure 4 1 Beam 1 C l ayout (Feldma n and Siess, 1958) PAGE 61 61 Table 4 1 Concrete material properties b = 6 in, h = 12 in, d = 10 in, d = 2.0 in, span =106 in Beam Compressive Strength, f c Modulus of Elasticity, Ec, k si k si Rupture Strength, f r k si 1 C 5.67 4292.1 0.9 1 G 6.21 4491.8 1.0 1 H 6.15 4450 0.935 1 I 6.50 4730 0.85 1 J 6.09 3890 0.935 Table 4 2 Steel reinforcements material p roperties Tension reinforcement = 2 #7 bars, compression reinforcement = 2 #6 bars, stirrups = 16 #3 bars at 7 in on center Beam Compression r einforcements Tension r einforcements f y E Ksi s Ksi y in/in sh f in/in y E Ksi s Ksi y in/in sh in/in 1 C 46.70 46.08 29520 0.0016 0.0144 1 G 48.30 47.75 1 H 47.61 32280 0.0015 0.015 47.17 34900 0.0014 0.0125 1 I 47.95 47.00 32600 0.0014 0.015 1 J 48.86 29560 0.0016 0.012 47.42 4 3 1 2 Loading functions Loading functions for all beams were reproduced by extracting the points from the loading graphs provided in the experimental report. For beam 1G, due to the failure of test recording equipment, the actual loads were not properly recorded. Hence, it had to be estimated. Two main criteria were based on when conducting this process. Firstly, the load profile was assumed to follow the shape of the sum of the reactions curve. Secondly, it must take into account the inertia effect of the beam under loading. Loading for beam 1 G was stopped at 0.072 sec when the wooden stop was hit. For beam 1 J, it should be noted that it was subject ed to two separate sets of impact loads. Similar to the loading situation for beam 1 G, the first set of loads stopped at 0.072 sec and resulted in a displacement of 0.5 in. It was then subjected to the second set of loads which was stopped at approximately 0.068 sec. For simplicity, these two sets of loads were combined in one simulation. Figs 4 2 to 4 6 show the loading functions for the beams. PAGE 62 62 Figure 4 2 Load f unction for b eam 1 C (Feldm an et al., 1958) Figure 4 3 Load f unction for b eam 1 G (Feldman et al., 1958) Time (msec)Load (kips) 0 10 20 30 40 50 60 70 80 90 100 110 120 5 0 5 10 15 20 25 30 35 Time (msec)Load (kips) 0 10 20 30 40 50 60 70 80 90 100 110 120 0 5 10 15 20 25 30 35 40 Loads, Kips (Actual) Loads, Kips (Estimated) PAGE 63 63 Figure 4 4 Load f unction for b eam 1 H (Feldman et al., 1958) Figure 4 5 Load f unction for b eam 1 I (Feldman et al., 1958) Time (msec)Load (kips) 0 10 20 30 40 50 60 70 80 90 100 110 120 5 0 5 10 15 20 25 30 35 40 Time (msec)Load (kips) 0 20 40 60 80 100 120 0 10 20 30 40 Stop at 72 msec PAGE 64 64 Figure 4 6 Load f unction for b eam 1 J (Feldman et al., 1958) 4 3 2 ABAQUS Validations All five beams were modeled with ABAQUS Version 6.8.1 (Dassault Systmes, 2008) A lthough various mesh sizes were evaluated, it was found that a cubical mesh size of 1 in ch yielded the most effective and economical results in terms of computer runtime and the accuracy of the outcomes. Hence, this mesh size and the solid element type C3D8I w ere used in the modeling of all the beams. Various types of material models in ABAQUS were also explored for the modeling of the concrete beam. It was determined that the Modified Drucker Prager/C ap Model was the most suitable one to be used in th is case for numerous reasons. Firstly, it could respond to large stress reversals in the cap region Secondly, it provided the required inelastic hardening mechanism to account for the plastic compactions as well as the necessary controlling of the mater ial expansion when yielding in shear (ABAQUS Analysis Users Manual) With Time (msec)Load (kips) 0 20 40 60 80 100 120 140 160 180 0 5 10 15 20 25 30 35 40 45 First stop @ 72 msec First stop @ 68 msec PAGE 65 65 regards to the longitudinal and transverse reinforcements, two different element types were used. Beam element type B31, w as used to model the compression rein forcements S urface element type SFM3D4R, w as used for the modeling of tension and transverse reinforcements. This was done because the surfaceelement type required less work to model compared to the beam element type while still yielding the same result. Bond slip betwe en concrete and steel reinforcements was taken into account using the built in capability of ABAQU S known as the embedded region constraint. Fig. 4 7 shows a typical ABAQUS (Dassault Systmes, 2008) model for all the above mentioned beams. Figure 4 7 Typical modeling of experimental beams using ABAQUS Six parameters were required for the Modified Drucker Prager/ Cap M odel These were the transition surface radius ( ) and the flow stress ratio (K). Fig. 4 8 PAGE 66 66 provides the graphical illustration of the se parameters. Detailed explanations and derivations of these parameters can be found in the S ection 19.3.2 of the ABAQUS Ana lysis Users Manual ABAQUS Version 6.8.1 Figure 4 8 Graphic al presentation of parameters in the M odified D rucker Prager/ Cap M odel (ABAQUS a nalysis u sers m anual) Default values for most of the parameters were used with the exception of two paramete rs that had significant effect s on the result s These were the material cohesion and the angle of friction. Computation of the material cohesion was based on the following procedure: = (4 1) = 0 7 200 (4 2) = 4 (4 3) With regards to the angle of friction, for normal concrete, it should be taken about 37 degree However, i t was found that th is value of the angle of friction and those values produced from Eq uatio ns 4 1 to 4 3 for the material cohesion only provided a good st arting point. The calculated values of these two parameters for each beam still required adjustment s in order to PAGE 67 67 obtain the desired results. Numerous trials were made with different values of material cohesion and angle of friction in attempt to produce the outcomes that were comparable to the experimental results Table 4 3 provides the s ummary of the best values of the parameters used for all five beams Table 4 3. Material model properties for beams in ABAQUS Beam Material Cohesion, d Angle of Fricti on, Cap Eccentricity, R Initial Yd Surf Pos, Trans Surf Rad, Flow Stress Ratio, K 1 C 0.85 60 0.5 0.003 0.01 0 1 G 0.87 53 0.5 0.003 0.01 0 1 H 0.85 60 0.5 0.003 0.01 0 1 I 1.00 59 0.5 0.003 0.01 0 1 J 1.07 47.7 0.5 0.003 0.01 0 With regards to the material property of steel, Hsu theory (Hsu, 1993) on stress strain relationship of mild steel was applied taking into account the strain rate hardening for both concrete and steel reinforcements. Detailed explanations on the computation al procedure can be found in Chapter 7 of the above mentioned reference. The strain rate hardening and the enhancement factors for all five beams, which were required for the computations, were obtained from the previous research (Shana a 1991), and were implemented in the calculations of the stress strain relationships of concrete and steel reinforcements. Using the data provided in Table 4 4, the stress and strain relationships for all five beams were calculated The results are illus trated in Figs. 4 9 to 4 1 8 respectively. Table 4 4. Strain rate hardening and material enhancement factors (Shana a et al. 1991) Beam Strain rate, in/in/sec Strain rate enhancement factors Compression Reinforcements Tension Reinforcements Compression Concrete Tension Concrete 1 C 0.28 1.23 1.23 1.36 2.20 1 G 0.46 1.24 1.24 1.37 2.36 1 H 0.46 1.24 1.24 1.37 2.36 1 I 0.46 1.24 1.24 1.37 2.36 1 J 0.27 1.23 1.23 1.36 2.22 PAGE 68 68 Figure 4 9 Stress strain relationship tension reinforcements b eam 1 C Figure 4 10. Stress strain relationship compression reinforcements b eam 1 C Strain (in/in)Stress (ksi) 0 0.004 0.008 0.012 0.016 0.02 0 15 30 45 60 75 Strain, (in/in)Stress, (ksi) 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0 15 30 45 60 75 PAGE 69 69 Figure 4 1 1 Stress strain relationship tension reinforcements b eam 1 G Figure 4 1 2 Stress strain relationship compression reinforcements b eam 1 G Strain (in/in)Stress (ksi) 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0 15 30 45 60 75 Strain (in/in)Stress (ksi) 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0 15 30 45 60 75 PAGE 70 70 Figure 4 1 3 Stress strain relationship tension reinforcements b eam 1 H Figure 4 1 4 Stress strain relationship compression reinforcements b eam 1 H Strain (in/in)Stress (ksi) 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0 15 30 45 60 75 Strain (in/in)Stress (ksi) 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0 15 30 45 60 75 PAGE 71 71 Figure 4 1 5 Stress strain relationship tension reinforcements b eam 1 I Figure 4 1 6 Stress strain relationship compression reinforcements b eam 1 I Strain (in/in)Stress (ksi) 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0 15 30 45 60 75 Strain (in/in)Stress (ksi) 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0 15 30 45 60 75 PAGE 72 72 Figure 4 1 7 Stress strain relationship tension reinforcements b eam 1 J Figure 4 1 8 Stress strain relationship compression reinforcements b eam 1 J Strain (in/in)Stress (ksi) 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0 15 30 45 60 75 Strain (in/in)Stress (ksi) 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0 15 30 45 60 75 PAGE 73 73 Since ABAQUS Explicit required the use of true stress and logarithmic strain, the above calculated stress strain values were then converted using the two equations provided in the ABAQUS manual (2008) prior to input the data into the ABAQUS program: = ( 1 + ) (4 4) = ln ( 1 + ) (4 5) 4 3 3 DSAS Validations Validation using DSAS was a much simpler task. This software application required the inputs of physical and material properties of the beam as well as load time history. Strain hardening could either be applied manually to the material property prior to input or through the built in switch. For this validation, a n average st rain rate of 0.3 was used for all beam s Figure 4 1 9 Typical DSAS data entry screen PAGE 74 74 4 3 4 Comparison of Results from ABAQUS and DSAS to Experiment al Data Validation results for the displacements at the mid span of all the beams from ABAQUS (Dassault Systmes, 2008) and DSAS (CIPPS, UF) are summarized in Table 4 5 and illustrated in Figs. 4 20 to 4 2 4 It should be noted that the unloading portion in the ABAQUS result did not match up well compared to those of experimental and DSAS results. This was because hysteresis damping was not captured by the concrete material model in A BAQUS when modeling the beams. In general, most of the differences in the results compared to the experiment were in the range within 10%, which was acceptable. With regards to beam 1 G, the material model parameters for ABAQUS required some minor adjustment. It was found that these parameters, particularly the material cohesion and the angle of friction, were quite sensitive. A small adj ustment would change the result significantly. For beam 1 I, the recording instrument did not properly recorded the load s during the experiment. Therefore, the loading function had to be estimated based on two criteria. It must equal to the sum of the d ynamic reactions less the inertia effect due to the weight of the beam. It must also follow the shape of the experimental loading function. For beam 1 J, the load was accidentally applied and stopped. The damaged beam was then re subjected to another se t of loading. Table 4 5. Comparison of ABAQUS and DSAS to Experiment Results Beam Midspan Displacement, in (Experiment) Midspan Displacement, in (ABAQUS) % Difference Midspan Displacement, in (DSAS) % Difference 1 C 3.01 2.9 3.65% 3.07 1.99% 1 G 4.14 4.23 2.17% 4.02 2.90% 1 H 8.86 8.77 1.02% 8.2 7.45% 1 I 10.57 9.94 5.96% 9.55 9.65% 1 J 1st Set 0.95 1.21 27.37% 0.84 11.58% 1 J 2nd Set 10.17 10.6 4.23% 8.54 16.03% PAGE 75 75 Figure 4 20. Comparison of d isplacement t ime h istory for b eam 1 C Figure 4 2 1 Comparison of d isplacement t ime h istory for b eam 1 G Time (sec)Displacement (in) 0 0.02 0.04 0.06 0.08 0.1 0.12 1 0 1 2 3 4 Experiment DSAS ABAQUS Time (sec)Displacement (in) 0 0.02 0.04 0.06 0.08 0.1 0.12 0 1 2 3 4 5 Experiment DSAS ABAQUS PAGE 76 76 Figure 4 2 2 Comparison of d isplacement t ime h istory for b eam 1 H Figure 4 2 3 Comparison of d isplacement t ime history for beam 1 I Time (sec)Displacement (in) 0 0.02 0.04 0.06 0.08 0.1 0.12 0 1 2 3 4 5 6 7 8 9 10 Experiment DSAS ABAQUS Time (sec)Displacement (in) 0 0.02 0.04 0.06 0.08 0.1 0.12 1 0 1 2 3 4 5 6 7 8 9 10 11 Experiment DSAS ABAQUS Hit wooden stop at 0.072 sec PAGE 77 77 Figure 4 2 4 Comparison of d isplacement t ime history for beam 1 J 4. 4 Validation for Beam Subject to Uniform Load Using ABAQUS and DSAS Since data for beam under uniform load was not available, beam 1 C was also used to further validat e for the beam case under uniform loading. In this case, Conventional Weapons Effects (CONWEP, US Army Engineer Waterways Experiment Station, 1992) was used to derive the loading function for a blast load of 500 pounds of TNT at a distance of 20 feet. This loading function, as shown in Fig. 4 2 5, was used throughout the study in this paper. The results computed from ABAQUS and DSAS for beam 1 C under unifor m pressure load is shown in Fig. 4 2 6. A difference of 0.03 in ch or 2. 3 % for the peak displacement exists between the two outcomes, which is acceptable. As noted earlier, there is a more significant difference for the residual displacement that is due to the inability of the ABAQUS material model to capture the hysteretic behavior. Time (sec)Displacement (in) 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 1 0 1 2 3 4 5 6 7 8 9 10 11 Experiment DSAS ABAQUS 1 st stop @ 0.072 sec 2 nd stop @ 0.068 sec PAGE 78 78 Figure 4 2 5 Loading function for 500 pounds of T rinitrotoluene (TNT) at 20 f ee t Figure 4 2 6 D isplaceme nt t ime h istory of b eam 1 C under b last l oad Time (msec)Pressure (psi) 2 4 6 8 10 12 14 0 200 400 600 800 1000 1200 Time (sec)Displacement (in) 0 0.02 0.04 0.06 0.08 0.1 0.12 0.25 0 0.25 0.5 0.75 1 1.25 1.5 ABAQUS DSAS PAGE 79 79 4. 5 Validation for Columns Using ABAQUS and DSAS A 16 in x 16 in x 144 in reinforced concrete column was generated in ABAQUS (Dassault Systmes, 2008) and DSAS (CIPPS, 2008). The column is subject to the above mentioned loading function. Similarly to the beam models, a cub ical grid of 1 in ch was found to be the most effective and most economical size for the modeling of the column. Material properties were also taken to be within the range of those of the beam s in th e previous step Spacing for transverse reinforcements w as set at 12 in ches on center. Longitudinal reinforcements consisted of 8 No. 7 and were placed into 3 layers with a minimum concrete cover of 1.5 inches all around. For ABAQUS to work properly, both ends of the column were extended by 6 inches and were used as the end supports. The span length of the column was therefore, still at 12 feet. Typi cal layout of the column, loads, boundary conditions and reinforcement layout s are shown in Fig s 4 27 to 4 29, respectively A summary of the results for the column subjected to transverse load only is shown in Fig. 4 30. It should be noted that the difference between DSAS and ABAQUS was 0. 75 in ch or 35.8% for the maximum displacement at the mid span of the column This was most likely caused by the following factors: Th e material model parameters in ABAQUS required some minor adjustment T h e current version of DSAS did not include the effect of the shear reduction factor (S RF ) in the computations The column was modeled in ABAQUS using three dimensional solid element while the same column was modeled in DSAS using beam element. PAGE 80 80 Figure 4 2 7 Typical column layout in ABAQUS Figure 4 2 8 Loads and b oundary c onditions for c olumn in ABAQUS PAGE 81 81 Figure 4 2 9 Longitudinal and t ransverse s teel r einforcement l ayout in ABAQUS Figure 4 30. Displacement t ime h istory c olumn s ubject to b last l oad Time (sec)Displacement (in) 0 0.02 0.04 0.06 0.08 0.1 0 1 2 3 P=0 ABAQUS P=0 DSAS PAGE 82 82 4.6 Summary This chapter described the steps taken in the determination of the axial loads effects on a column. First, in order to establish the proper material mod el to be used and the accuracy of the software applications, validations of ABAQUS Version 6.81 (Dassault Systmes, 2008) and the Dynamic Structure Analysis Suite (DSAS) Version 2.0 (CIPPS, 2008) were completed against past experimental data During this step, discussion on the development of the steel stress strain relationship using Hsu T heory and how it was incorporated in the required computations were also included. This was followed by an analytical validation of the above mentioned s oftware applications on a beam and a column that are subjected to the same blast load. The results from both experimental and analytical validations indicated that either software application could be used for the parametric study. PAGE 83 83 CHAPTER 5 PARAMETRIC STUDY 5 1 Description of Columns The parametric study was based on four confined reinforced concrete columns with the same dimensions stated in Chapter 4, Section 4.5 but with various configurations of steel reinforcements. These columns were arbitrarily sel ected from the CRSI Design Handbook (2002) along with the sizes and spacing of the longitudinal and transverse reinforcements. Spacing of the transverse reinforcements was at 12 in ches on center for all configurations. A summary of the column material us ed is shown in Table 5 1. Table 5 1. Summary of c olumns p hysical and material p roperties fc = 4000 psi, fy = 60000 psi, Es = 29000 ksi, 16 x 16 x 144 Column Bars Stirrups s', in M max P k ips ft max k ips 1 8 #7 #3 1.88 12 323.7 172 0 2 8 #10 #3 4.88 12 450.7 2140 3 12 #11 #4 7.31 12 598.8 2820 4 4 #14 #4 3.52 12 474.5 2050 With the outcome from the above validations, both experimentally and analytically, DSAS and ABAQUS can now be used for the necessary computations. However, since experimental data on column was not available, one additional step was required to ensure the consistency of the result. An unconfined reinforced concrete column of 16 inch x 16 inch x 144 inch consisted of 8 No. 7 grade 60 steel reinforcements was used fo r this confirmation. The result obtained from DSAS was compared with the pre defined data provided in the CRSI Design Handbook (2002). The compression strength of the concrete, fc, was 4000 psi. The yield strength of the steel, fy, was 60000 psi. The layout of steel reinforcement was based on the recommendation outlined in Table 3 1 of the CRSI. A n axial load moment interaction diagram for the axial loads, P, acting on this column was generated from the data computed by DSAS. The result, shown in Fig 4 29, PAGE 84 84 indicated that the axial load and moment capacity values obtained from DSAS closely matched those outlined in the CRSI (2002). Figure 5 1. Axial m oment interaction diagram unconfined concrete DSAS v er s us CRSI 5 2 Columns Subject to Transverse and Constant Axial Load Since using ABAQUS Version 6.8 1 (Dassault Systmes, 2008) to produce the required axial load moment interaction (P M) diagrams the flexure resistance (P d) diagrams and the pressure impulse (P I) diagrams would be time consuming and cumbersome, only DSAS Version 2.0 (CIPPS, 2008) w as used to generate these diagrams for the columns listed in Table 5 1 However, to produce time displacement history diagrams for the above columns, both DSAS and ABAQUS were used. Similar to the beam cases, Hsu theory (Hsu, 1993) on stress strain relationship of mild steel was applied to the computations of nominal stress and strain values for concrete and steel Moment (ftkips)Axial Load (kips) 0 30 60 90 120 150 180 210 240 270 500 250 0 250 500 750 1000 1250 1500 DSAS CRSI PAGE 85 85 reinforcements. A value of 0.3 was used as the strain rate hardening for both concrete and steel reinforcements. Equation s 4 4 and 45 were then used to convert the computed nominal stress and strain values to the true stress and logarithm strain valu es prior to input into ABAQUS. Fig ure s 5 2 to 5 5 plotted the computed nominal stress strain r elationships of the steel reinforcements used in the columns. Since the tension and the compression steel reinforcements were assumed to have the same material properties for the four c olumns listed in Table 5 1 the computations were the same for both ty pes of steel and only one plot was used for each type of steel reinforcements. Figure 5 2. Stress strain relationship tension and compression reinforcements c olumn 1 Strain (in/in)Stress (ksi) 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0 20 40 60 80 100 PAGE 86 86 Figure 5 3. Stress strain relationship tension and compression reinforceme nts c olumn 2 Figure 5 4. Stress strain relationship tension and compression reinforcements c olumn 3 Strain (in/in)Stress (ksi) 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0 20 40 60 80 100 Strain (in/in)Stress (ksi) 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0 20 40 60 80 100 PAGE 87 87 Figure 5 5. Stress strain relationship tension and compression reinforcements c olumn 4 Four different load magnitudes as shown in Table 5 2, were arbitrary chosen to be used in the analyses for each column. In all cases, one load magnitude at zero, one below balance load, one at balance load and one above balance load were used in both DSAS and ABAQUS Summary of the results obtained from these two software applications are provided in Tables 5 3 and 5 4. As mentioned earlier, the differences in the results between DSAS and ABAQUS were due to the fact that the current version of DSAS did not include the SRF in the com putations. Table 5 2. Summary of load cases used in the analyses Column Rebar P bal P kips 1 P kips 2 P kips 3 P kips 4 kips 1 8 No. 7 560 0 250 560 1000 2 8 No. 10 560 0 250 560 1000 3 12 No. 11 570 0 250 570 1500 4 4 No. 14 530 0 250 530 1000 Strain (in/in)Stress (ksi) 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0 20 40 60 80 100 PAGE 88 88 Table 5 3. Comparisons on displacements result ed from ABAQUS and DSAS for P1 and P2 Load ABAQUS DSAS % Difference ABAQUS DSAS % Difference Column P1 P2 1 2.84 2.09 26.37 2.40 1.55 35.30 2 1.89 1.36 28.05 1.73 1.16 32.92 3 2.79 1.07 61.72 2.48 0.99 60.00 4 2.00 1.40 30.16 1.74 1.16 33.59 Table 5 4. Comparisons on displacements result ed from ABAQUS and DSAS for P3 and P4 Load ABAQUS DSAS % Difference ABAQUS DSAS % Difference Column P3 P4 1 2.28 0.99 56.45 2.66 0.68 74.32 2 1.88 1.12 40.49 1.81 0.85 53.38 3 2.30 0.95 58.87 24.61 1.27 94.85 4 1.59 1.09 31.62 1.78 0.90 49.70 Fig ure s 5 6 to 5 26 illustrate the four different approaches in presenting the outcomes of the study. The first approach showed the time displacement history of the columns computed from both ABAQUS and DSAS. Since this version of DSAS did not include the shear reduction factor (SRF) in the computations, the results produced by DSAS were, therefore, lower than those produced by A BAQUS. However, interpretation from these results yielded some common points on the behavior of the column. When being subjected to the transverse load only, t he column experienced a larger displacement at mid span compared to those obtained when axial l oad were exerted on the column. For all columns a s the axial load s P, increased the difference between the displacements in comparing to zero axial load became larger until P reached the balance load, Pbal. As P surpassed Pbal, th e difference in displ acement became smaller. This was an indication that the applications of axial loads actually enhanced the flexure resistance of the columns In the case of Column 2, the displacement at balance load was the same with that at zero axial load. This was due to the fact that the yielding of steel reinforcements and the crushing of concrete occurred early and Column 2 reached the plastic stage PAGE 89 89 T wo other approaches used the Moment Curvature and Flexure Resistance relationships to demonstrate column behavi or from the elastic to plastic range based on the analysis of loads and deformations. The results showed the changes in column stiffness for various magnitudes of constant axial loads, P. Within the elastic and elasto plastic range, the stiffness of the column decreased slowly as P increased The plastic range of the column depend ed on its ductility In th is range the lower P were, the larger the displacements the column could undertake. This was probably due to the formation of the hinge at the colum n mid span that allowed the column to behave in such a way. As P increased, the plastic range of the column became much shorter and failure took place sharply. Again, the time between the formations of the hinges at the column mid span and at the two end support s occurred much faster during this stage and therefore caused the column failure. It should also be noted that the area under the curve decreased as P increased and t he rate of change also in tensifie d significantly once P approached closer to Pba l. This implied that the strain energy, represented by the area under the Flexure Resistance curve, dissipated much faster once the first hinge was formed. Moreover, for P much less than PbalThe Pressure Impulse ( P I) diagram wa s the last approach for analyzing the results. Unlike the other three methods, the P I diagrams predicted the points of failure of the column P actually enhanced the moment capacity and strengthened the column. Hence the displacement due to the transverse load was actually less than that of P = 0 kips Once the balance point was surpassed P became the contributing factor to the column failure. From Fig. 5 8 it should also be noted that the final lateral displacement of the column was approximately 19 inches for P = 0, while Fig. 5 7 showed the column displacement of 2.1 inches (DSAS) or 2.85 inches (ABAQUS) for the same load This was because Fig 5 8 also included the failure of the column in the tension membrane mode. PAGE 90 90 based on each magnitude of load, P and impulse, I, that the column experienced over the time period. The Impulsive Loading region for P < Pbal had higher tolerance than that of P > Pbal. This implied that the strain energy decreased as P approached Pbal Figure 5 6 Axial m oment interaction diag ram 8 No. 7 RC c onfined As P increased, the Dynamic Load ing region became shorter and the columns exhibited longer Quasi Static Loading Regions; where the column deformation became larger and permanent damage could take place. Hence, the columns achieved failure. Moment (ftkips)Axial Force (kips) 0 50 100 150 200 250 300 350 600 400 200 0 200 400 600 800 1000 1200 1400 1600 1800 2000 P4 = 1000 kips > Pb P3 = Pb = 560 kips P2 = 250 kips < Pb P1 = 0 PAGE 91 91 Figure 5 7 Displacement t ime history diagram 8 No. 7 RC c onfined Figure 5 8 Flexure r esistance diagram 8 No. 7 RC c onfined Time (sec)Displacement (in) 0 0.02 0.04 0.06 0.08 0.1 0 0.5 1 1.5 2 2.5 3 P=250 kips ABAQUS P=560 kips ABAQUS P=1000 kips ABAQUS P=0 kips DSAS P=250 kips DSAS P=560 kips DSAS P=1000 kips DSAS P=0 kips ABAQUS Displacement (in)Pressure (psi) 0 2 4 6 8 10 12 14 16 18 20 0 20 40 60 80 100 P1 = 0 P2 = 250 kips < Pb P3 = Pb = 560 kips P4 = 1000 kips > Pb PAGE 92 92 Figure 5 9 Moment c urvature diagram 8 No. 7 RC c onfined Figure 5 10. Pressure i mpulse diagram 8 No. 7 RC c onfined Curvature (1/in)Moment (ftkips) 0 0.003 0.006 0.009 0.012 0.015 0.018 0.021 0.024 0 50 100 150 200 250 300 350 P1= 0 P2 = 250 kips < Pb P3 = Pb = 560 kips P4 = 1000 kips > Pb Impulse (psisec)Pressure (psi) 0 2 4 6 8 10 12 14 16 18 20 0 100 200 300 400 500 600 700 800 900 1000 Direct Shear P = 0 P = 250 kips < Pb P = Pb = 560 kips P = 1000 kips > Pb PAGE 93 93 Figure 5 11. Axial m oment interaction diagram 8 No. 10 RC c onfined Figure 5 12. Displacement t ime history diagram 8 No. 10 RC c onfined Moment (ftkips)Axial Force (kips) 0 50 100 150 200 250 300 350 400 450 1200 800 400 0 400 800 1200 1600 2000 P = 0 P = 250 kips < Pb P = Pb = 560 kips P = 1000 kips > Pb Time (sec)Displacement (in) 0 0.02 0.04 0.06 0.08 0.1 0 0.5 1 1.5 2 P=250 kips ABAQUS P=560 kips ABAQUS P=1000 kips ABAQUS P=0 kips DSAS P=250 kips DSAS P=560 kips DSAS P=1000 kips DSAS P=0 kips ABAQUS PAGE 94 94 Figure 5 13. Flexure r esistance diagram 8 No. 10 RC c onfined Figure 5 1 4 Moment c urvature diagram 8 No. 10 RC c onfined Displacement (in)Pressure (psi) 0 2 4 6 8 10 12 14 16 18 20 22 24 0 20 40 60 80 100 120 140 160 P = 0 P = 250 kips < Pb P = Pb = 560 kips P = 1000 kips > Pb Curvature (1/in)Moment (ftkips) 0 0.003 0.006 0.009 0.012 0.015 0.018 0.021 0.024 0 50 100 150 200 250 300 350 400 450 500 P = 0 P = 250 kips < Pb P = Pb = 560 kips P = 1000 kips > Pb PAGE 95 95 Figure 5 1 5 Pressure i mpulse diagram 8 No. 10 RC c onfined Figure 5 1 6 Axial m oment interaction diagram 12 No. 11 RC c onfined Impulse (psisec)Pressure (psi) 0 3 6 9 12 15 18 21 24 27 30 0 100 200 300 400 500 600 700 800 900 1000 Direct Shear P = 0 P = 250 kips < Pb P = Pb = 560 kips P = 1000 kips > Pb Moment (ftkips)Axial Force (kips) 0 50 100 150 200 250 300 350 400 450 500 550 600 2000 1600 1200 800 400 0 400 800 1200 1600 2000 2400 2800 P = 0 P = 250 kips < Pb P = 1500 kips > Pb PAGE 96 96 Figure 5 1 7 Displacement t ime history diagram 12 No. 11 RC P = 0 to 570 kips c onfined Figure 5 1 8 Displacement t ime history diagram 12 No. 11 RC c onfined Time (sec)Displacement (in) 0 0.02 0.04 0.06 0.08 0.1 0.5 0 0.5 1 1.5 2 2.5 3 P=250 kips ABAQUS P=570 kips ABAQUS P=0 kips DSAS P=250 kips DSAS P=570 kips DSAS P=0 kips ABAQUS Time (sec)Displacement (in) 0 0.02 0.04 0.06 0.08 0.1 0 5 10 15 20 25 P=1000 kips ABAQUS P=1500 kips ABAQUS P=1500 kips DSAS P=0 kips ABAQUS PAGE 97 97 Figure 5 1 9 Flexure r esistance diagram 12 No. 11 RC c onfined Figure 5 20. Moment c urvature diagram 12 No. 11 RC c onfined Displacement (in)Pressure (psi) 0 2 4 6 8 10 12 14 16 18 20 22 24 26 0 20 40 60 80 100 120 140 160 180 200 P = 0 P = 250 kips < Pb P = Pb = 570 kips P = 1500 kips > Pb Curvature (1/in)Moment (ftkips) 0 0.003 0.006 0.009 0.012 0.015 0.018 0.021 0.024 0.027 0.03 0 100 200 300 400 500 600 700 800 P = 0 P = 250 kips < Pb P = Pb = 570 kips P = 1500 kips > Pb PAGE 98 98 Figure 5 21. Pressure i mpulse diagram 12 No. 11 RC c onfined Figure 5 22. Axial m oment interaction diagram 4 No. 14 RC c onfined Impulse (psisec)Pressure (psi) 0 3 6 9 12 15 18 21 24 27 30 0 100 200 300 400 500 600 700 800 900 1000 Direct Shear P = 0 P = 250 kips < Pb P = Pb = 570 kips P = 1500 kips > Pb Moment (ftkips)Axial Force (kips) 0 50 100 150 200 250 300 350 400 450 500 1000 600 200 200 600 1000 1400 1800 2200 P = 0 P = 250 kips < Pb P = Pb = 530 kips P = 1000 kips > Pb PAGE 99 99 Figure 5 23. Displacement t ime history diagram 4 No. 1 4 RC c onfined Figure 5 2 4 Flexure r esistance diagram 4 No. 14 RC c onfined Time (sec)Displacement (in) 0 0.02 0.04 0.06 0.08 0.1 0 0.5 1 1.5 2 2.5 P=250 kips ABAQUS P=530 kips ABAQUS P=1000 kips ABAQUS P=0 kips DSAS P=250 kips DSAS P=530 kips DSAS P=1000 kips DSAS P=0 kips ABAQUS Displacement (in)Pressure (psi) 0 2 4 6 8 10 12 14 16 18 20 0 20 40 60 80 100 120 140 160 P = 0 P = 250 kips < Pb P = Pb = 530 kips P = 1000 kips > Pb PAGE 100 100 Figure 5 2 5 Moment c urvature diagram 4 No. 14 RC c onfined Figure 5 2 6 Pressure i mpulse diagram 4 No. 14 RC c onfined Curvature (1/in)Moment (ftkips) 0 0.003 0.006 0.009 0.012 0.015 0 50 100 150 200 250 300 350 400 450 500 550 600 P = 0 P = 250 kips < Pb P = Pb = 530 kips P = 1000 kips > Pb Impulse (psisec)Pressure (psi) 0 2 4 6 8 10 12 14 16 18 20 0 100 200 300 400 500 600 700 800 900 1000 Direct Shear P = 0 P = 250 kips < Pb P = Pb = 530 kips P = 1000 kips > Pb PAGE 101 101 5 3 Columns Subject to Transverse Constant and Variable Axial Loads With the known effects of constant axial loads P the same columns were then subjected to the variable axial loads Pvar, to determine the significance of their effects As indicated in Chapter 4, Section 4.4, during the course of t he required validations, beam 1 C was subjected to the same blast load as the column. The dynamic reactions at the two supports generated by this blast load were then used as the variable axial loads that acted on the columns. To reduce the computer runt ime, it was assumed that the arrival time of the blast load for the beam was the same as that of the column. Hence, in ABAQUS, the variable axial loads were activated at the same time as the variable transverse loads. The p rofile of the variable axial lo ads and the results of the application s of the variable axial loads on the columns are shown in Fig s 5 2 7 to 5 3 5 From Table 5 5, i t should be noted that for Column 3, at load P4 the column did not completely fail until it reached 24.61 inches. However, upon the exertion PvarTable 5 5. Comparisons on displacements induced by constant and variable axial loads the column actually failed at 4.46 inches. Load P2 P2 + P % Increase var P3 P3 + P % Increase var P4 P4 + P % Increase var Column 1 2.40 2.62 8.68 2.28 4.90 53.37 2.66 24.57 89.15 2 1.73 1.82 4.61 1.88 2.19 14.30 1.81 23.08 92.14 3 2.48 2.54 2.28 2.30 3.32 30.77 24.61 4.46 451.89 4 1.74 1.78 2.54 1.59 2.14 25.62 1.78 18.78 90.50 In all cases, for P bal, by exerti ng the v ariable axial loads, Pvar in addition to the constant axial loads, P, the effects on the column behavior were reversed Within this range, while the applications of P enhanced the flexure resistance of the columns and reduced the peak displacement at the column mid span, the applications of Pvar actually reduced the flexural resistance and increased the displacement at the same location As P reached Pbal, the effect of additional Pvar became more significant and the columns failed at sooner PAGE 102 102 Figure 5 2 7 Variable a xial l oad p rofile Figure 5 2 8 Displacement t ime h istory 8 No. 7 RC P bal + P Time (sec)Dynamic Reaction (kips) 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 100 0 100 200 300 400 var Time (sec)Displacement (in) 0 0.02 0.04 0.06 0.08 0.1 0 1 2 3 4 5 P=250 kips P=560 kips P=250 kips + Pvar P=560 kips + Pvar P=0 kips PAGE 103 103 Figure 5 2 9 Displacement time h istory 8 No. 7 RC P > Pbal + P Figure 5 30. Displacement t ime h istory 8 No. 10 RC P Pvar bal + P Time (sec)Displacement (in) 0 0.02 0.04 0.06 0.08 0.1 0 5 10 15 20 25 P=1000 kips P=1000 kips + Pvar P=0 kips var Time (sec)Displacement (in) 0 0.02 0.04 0.06 0.08 0.1 0 0.5 1 1.5 2 2.5 P=250 kips P=560 kips P=250 kips + Pvar P=560 kips + Pvar P=0 kips PAGE 104 104 Figure 5 31. Displacement time h istory 8 No. 10 RC P > Pbal + P Figure 5 32. Displacement t ime h istory 12 No. 1 1 RC P Pvar bal + P Time (sec)Displacement (in) 0 0.02 0.04 0.06 0.08 0.1 0 5 10 15 20 25 P=1000 kips P=1000 kips + Pvar P=0 kips var Time (sec)Displacement (in) 0 0.02 0.04 0.06 0.08 0.1 0 0.5 1 1.5 2 2.5 3 3.5 P=250 kips P=570 kips P=250 kips + Pvar P=570 kips + Pvar P=0 kips PAGE 105 105 Figure 5 33. Displacement t ime h istory 12 No. 11 RC P > Pbal + P Figure 5 3 4 Displacement t ime h istory 4 No. 14 RC P var bal + P Time (sec)Displacement (in) 0 0.02 0.04 0.06 0.08 0.1 0 5 10 15 20 25 30 P=1000 kips P=1500 kips P=1000 kips + Pvar P=1500 kips + Pvar P=0 kips var Time (sec)Displacement (in) 0 0.02 0.04 0.06 0.08 0.1 0 0.5 1 1.5 2 2.5 P=250 kips P=530 kips P=250 kips + Pvar P=530 kips + Pvar P=0 kips PAGE 106 106 Figure 5 3 5 Displacement t ime h istory 4 No. 14 RC P > Pbal + P 5 4 Summary var Parametric study on four reinforced concrete columns with different steel configurations was conducted in this chapter. The study was completed in t wo stages. In the first stage, only constant axial loads were applied to the columns that were subjected t o variable transverse load. Four different methods were used to illustrate the outcomes of the study along with the associated interpretation on the column behavior in each method. These methods were flexure resistance diagrams, moment curvature diagrams and pressure impulse diagrams. In the second stage, variable axial loads were applied to the columns in addition to the constant axial loads. The displacement time histor ies associated with the loads for each column obtained from both stages were compa red to determine the effects of the variable axial loads on the columns. Time (sec)Displacement (in) 0 0.02 0.04 0.06 0.08 0.1 0 4 8 12 16 20 P=1000 kips P=1000 + Pvar P=0 kips PAGE 107 107 CHAPTER 6 CONCLUSIONS AND RECOMMENDATIONS 6 1 Summary A study on the effects of short durationhighimpulsive loads such as blast load on a reinforced concrete column was presented in this paper. A summary on the computation of blast load and its effect were provided in Chapter 2. A review on a structure and its equivalent SDOF system along with the associated shape and resistance functions were also included in this chapter This wa s followed by the discussion of t he beh aviors of flexure and shear and approaches to obtain the solutions for the equation of motion of the equivalent SDOF system Chapter 3 discussed the load determination and the load deformation analysis approach using Newmark Beta m ethod and the computations of the dynamic reactions. Chapter 4 included the implementation of the above discussion using ABAQUS Version 6.8 1 (Dassault Systmes, 2008) and the Dynamic Structure Analysis Suite (DSAS) Version 2. 0 (CIPPS, 2008)). A series of steps were required to validate the two software application s. Upon the completion of the validations, a parametric study on column behaviors under the influence of short duration high impulsive transverse and axial loads was conducted as shown in Chapter 5. Four approaches were used to present the effects of variable transverse and axial loads on a column. The first approach used time displacement history to determine the effects of these above mentioned loads on the columns. T wo other approaches included the use of moment curvature and load deflection relationships to describe the columns behaviors in the elastic, elasto plastic, and fully plastic ranges. Pressure impulse relationship was the last approach used in analyzi ng the results PAGE 108 108 6 2 Conclusions Under normal loading conditions, a reinforced concrete column of a structure can be designed to support a given mass. However, consideration for maximum support capacity for the column should be taken into account when the st ructure is likely to be subject ed to short duration high impulse variable axial and transverse loads as this will significantly diminish the l oading capacity of the column. For a column with low and normal ductility, under the influence of short duration t ransverse loads, the column fails as the load resulted from the supported mass is greater than the balance load of the column. It should be noted that for loads less than the balance load, loads resulted from the supported mass act as an enhancement facto r to the strength of the column. However, chance of a column failure increases as the loads of the supported mass surpass the balance load. For column with high ductility, the column lasts longer even when its balance load capacity is surpassed to a cert ain load magnitude where failure of the column occurs. The probability of the shear failure of a column depends on the magnitudes of the transverse loads. The higher the transverse loads are, the higher the probability the column can fail in shear failure mode. In this study, the failure of the column was governed by flexure. Four methods of analyzing the performance of a structure were used in this study. While the employment of time displacemen t history diagram, load deflection diagram and moment curvature diagram provide a more thorough analysis P I diagram proves to be the most expedient one to in determining the state of the structure. ABAQUS Version 6.8 1 (Dassault Systmes, 2008) and the D ynamic Structure Analysis Suite (DSAS) Version 2.0 (CIPPS, 2008)) were heavily used in the study of this paper. Although DSAS is a noncommercial software application its portability, speed and ability to PAGE 109 109 provide accurate results prove that it is a valuable tool to be used for a quick prediction of a structure behavior. On the other hand, while being a commercial software application ABAQUS shows that it is too cumbersome and lacks of flexibility The properties of the material model used in ABAQUS are too sensitive. As such, a slight change in these properties may produce different results. In addition, the level of public technical supports such as the internet forum for ABAQUS is also found to be limited compared to that of other finite element sof tware application s. 6 3 Recommendations The following recommendations can be deduced from the above results and observations: T he results from this study should be verified by actual experiments using the same boundary conditions F urther study should be conducted on the column behaviors in the tension membrane stat e of the steel reinforcements. Reinforced concrete columns with different size configurations, strengths and boundary conditions can be used to further validate the concl usions Another finite element software application should be used for the modeling of the columns and the results obtained can be compared with those from ABAQUS. S hear reduction factor should be included in the computations in the next version of DSAS. PAGE 110 110 APPENDIX SAMPLE ABAQUS INPUT FILE BEAM 1 C *Heading ** Job name: Beam_C1_R17_9 Model name: Beam_C1_R17_9 ** Generated by: Abaqus/CAE Version 6.8 1 *Preprint, echo=NO, model=NO, history=NO, contact=NO ** ** PARTS ** *Part, name=BEAM_C1 1 *Node ** PAGE 111 111 0.375 0.,0.,1. ** Section: Section 4 SF_BARSBOTTOM *Surface Section, elset=SF_BARSBOTTOM *Rebar Layer BARS_BOTTOM, 0.60132, 3.125, STEEL, 0., 1 ** Section: Section 5 SF_STIRRUPS *Surface Section, elset=SF_STIRRUPS *Rebar Layer STIRRUPS, 0.11045, 6.625, STEEL_TRANS, 90., 1 ** Section: Section 6 STEEL_PLATE *Solid Section, elset=STEEL_PLATE, material=STEEL_PLATE 1., *End Part ** ** ** ASSEMBLY ** *Assembly, name=Assembly ** *Instance, name=BEAM_C1 1, part=BEAM_C1 1 *End Instance ** *Elset, elset=__PICKEDSURF8_S4, internal, instance=BEAM_C11, generate 7, 504, 7 *Nset, nset=SET 1, instance=BEAM_C1 1 53, *Elset, elset=SF_BARSBOTTOM, instance=BEAM_C1 1, generate 9381, 9498, 1 *Elset, elset=SF_STIRRUPS, instan ce=BEAM_C1 1, generate 9499, 12254, 1 *Elset, elset=__PICKEDSURF13_S4, internal, instance=BEAM_C1 1, generate 7, 504, 7 *Nset, nset=_PICKEDSET16, internal, instance=BEAM_C1 1 18, 19, 201, 202, 203, 204, 205 *Nset, nset=_PICKEDSET18, inter nal, instance=BEAM_C1 1 11, 12, 124, 125, 126, 127, 128 *Elset, elset=__PICKEDSURF26_S4, internal, instance=BEAM_C1 1, generate 7, 504, 7 *Elset, elset=_M12, internal, instance=BEAM_C1 1, generate 9381, 9498, 1 *Elset, elset=_M13, internal, instance=BEAM_C1 1, generate 9499, 12254, 1 *Elset, elset=_M14, internal, instance=BEAM_C1 1, generate 9263, 9380, 1 *Elset, elset=_M15, internal, instance=BEAM_C1 1, generate PAGE 112 112 9145, 9262, 1 *Elset, elset=_M16, internal, instance=BEAM_C 1 1, generate 9381, 9498, 1 *Elset, elset=_M17, internal, instance=BEAM_C1 1, generate 9499, 12254, 1 *Elset, elset=_M18, internal, instance=BEAM_C1 1, generate 9263, 9380, 1 *Elset, elset=_M19, internal, instance=BEAM_C1 1, generate 9145, 9262, 1 *Elset, elset=_M20, internal, instance=BEAM_C1 1, generate 9381, 9498, 1 *Elset, elset=_M21, internal, instance=BEAM_C1 1, generate 9499, 12254, 1 *Elset, elset=_M22, internal, instance=BEAM_C1 1, generate 9263, 9380, 1 *Elset, elset=_M23, internal, instance=BEAM_C1 1, generate 9145, 9262, 1 *Nset, nset=STRAIN_GAUGE, instance=BEAM_C11 2822, 2829, 2843, 2850 *Nset, nset=LEFT_RF, instance=BEAM_C1 1 18, 19, 201, 202, 203, 204, 205 *Nset, nset=RIGHT_RF, instanc e=BEAM_C1 1 11, 12, 124, 125, 126, 127, 128 *Nset, nset=_PickedSet38, internal, instance=BEAM_C11 12023, *Nset, nset=MIDSPAN_GAUGE_3, instance=BEAM_C1 1 3151, ** Constraint: BOTTOMBARS *Embedded Element _M20 ** Constraint: STIRRUPS *Embedded Element _M21 ** Constraint: TOPLEFT *Embedded Element _M22 ** Constraint: TOPRIGHT *Embedded Element _M23 *End Assembly *Amplitude, name=IMPACT2 0., 0., 0.003, 5., 0.004375, 10., 0.00625, 15. 0.006875, 20., 0.01125, 25., 0.01375, 30., 0.014, 31.2 0.0163, 29.2, 0.0175, 33.8, 0.02, 29.2, 0.022, 31.6 0.0225, 29.2, 0.025, 29., 0.027, 30.6, 0.027625, 28.6 0.031, 28.5, 0.032, 28.8, 0.034, 29.5, 0.03625, 28.8 PAGE 113 113 0.037, 28., 0.03825, 28.05, 0.0385, 28., 0.04181, 25. 0.04375, 22., 0.045, 20., 0.0475, 17., 0.04875, 15. 0.05, 14.5, 0.0525, 12., 0.05375, 10., 0.055, 7. 0.05625, 5., 0.0569, 4.8, 0.058, 5.5, 0.06, 3.8 0.063, 0., 0.064, 0.3, 0.066, 0., 0.072, 0. 0.075, 0.5, 0.07625, 0.1, 0.07725, 0.15, 0.08, 0.6 0.081, 0.7, 0.0875, 0., 0.09, 0.25, 0.0925, 0.25 0.094, 0.4, 0.0975, 0., 0.099, 0., 0.1025, 0.2 0.105, 0.2, 0.1075, 0.25, 0.11125, 0.1, 0.11375, 0.15 0.11625, 0., 0.11875, 0.05, 0.12, 0. ** ** MATERIALS *Material, name=CONCRETE *Cap Plasticity 0.85, 60., 0.5, 0.003, 0.01, 0. *Cap Hardening 6.,0. *Density 2.27e 07, *Elastic 5005.37, 0.2 *Material, name=STEEL *Density 7.33024e 07, *Elastic 36309.6, 0.3 *Plastic 49., 0. 50.6802, 0.000602224 51.6936, 0.00157182 52.709, 0.00254037 53.7253, 0.0035079 54.7445, 0.00447436 55.7656, 0.00543978 56.7887, 0.00640416 57.8137, 0.0073675 58.8406, 0.00832981 59.8694, 0.00929108 60.9001, 0.0102513 61.9328, 0. 0112105 62.9664, 0.0121688 64.0029, 0.0131259 65.0413, 0.0140821 66.0816, 0.0150372 67.1239, 0.0159913 PAGE 114 114 68.168, 0.0169443 69.2141, 0.0178964 *Material, name=STEEL_COMP *Density 7.33024e 07, *Elastic 36309.6, 0.3 *Plastic 48.5, 0. 50.5028, 0.000607109 51.5933, 0.00157458 52.6869, 0.00254098 53.7826, 0.00350632 54.8803, 0.00447062 55.9801, 0.00543387 57.082, 0.00639608 58.186, 0.00735725 59.2921, 0.00831737 60.4002, 0.00927646 61.5104, 0. 0102345 62.6226, 0.0111915 63.737, 0.0121475 64.8534, 0.0131025 65.9719, 0.0140564 67.0925, 0.0150093 68.2152, 0.0159612 69.3399, 0.0169121 70.4667, 0.0178619 *Material, name=STEEL_PLATE *Density 7.33024e 07, *Elastic 36000., 0.3 *Plastic 75.,0. *Material, name=STEEL_TRANS *Density 7.33024e 07, *Elastic 36309.6, 0.3 *Plastic 49., 0. 50.6802, 0.000602224 51.6936, 0.00157182 52.709, 0.00254037 53.7253, 0.0035079 PAGE 115 115 54.7445, 0.00447436 55.7656, 0.00543978 56.7887, 0.00640416 57.8137, 0.0073675 58.8406, 0.00832981 59.8694, 0.00929108 60.9001, 0.0102513 61.9328, 0.0112105 62.9664, 0.0121688 64.0029, 0.0131259 65.0413, 0.0140821 66.0816, 0.0150372 67.1239, 0.0159913 68.168, 0.0169443 69.2141, 0.0178964 ** ** ** STEP: BeamLoad ** *Step, name=BeamLoad *Dynamic, Explicit 0.12 *Bulk Viscosity 0.06, 0.12 ** ** BOUNDARY CONDITIONS ** ** Name: Disp BC 1 Type: Displacement/Rotation *Boundary, amplitude=IMPACT2 _PICKEDSET16, 1, 1 ** Name: Disp BC 2 Type: Displacement/Rotation *Boundary, amplitude=IMPACT2 _PICKEDSET16, 2, 2 ** Name: Disp BC 3 Type: Displacement/Rotation *Boundary, amplitude=IMPACT2 _PICKEDSET16, 3, 3 ** Name: Disp BC 4 Type: Displacement/Rotation *Boundary, amplitude=IMPACT2 _PICKEDSET16, 4, 4 ** Name: Disp BC 5 Type: Displacement/Rotation *Boundary, amplitude=IMPACT2 _PICKEDSET16, 5, 5 ** Name: Disp BC 6 Type: Displacement/Rotation *Boundary, amplitude=IMPACT2 _PICKEDSET18, 1, 1 ** Name: Disp BC 7 Type: Displacement/Rotation PAGE 116 116 *Boundary, amplitude=IMPACT2 _PICKEDSET18, 2, 2 ** Name: Disp BC 8 Type: Displacement/Rotation *Boundary, amplitude=IMPACT2 _PICKEDSET18, 3, 3 ** Name: Disp BC 9 Type: Displacement/Rotation *Boundary, amplitude=IMPACT2 _PICKEDSET18, 4, 4 ** Name: Disp BC 10 Type: Displacement/Rotation *Boundary, amplitude=IMPACT2 _PICKEDSET18, 5, 5 ** ** LOADS ** ** Name: GRAVITY BM Type: Gravity *Dload, amplitude=IMPACT2 BEAM_C1 1.BEAM_C1, GRAV, 386.1, 0., 1., 0. ** Name: GRAVITY STEELPLATE Type: Gravity *Dload BEAM_C1 1.STEEL_PLATE, GRAV, 386.1, 0., 1., 0. ** Name: Load 7 Type: Concentrated force *Cload, amplitude=IMPACT2 _PickedSet38, 2, 1. ** ** OUTPUT REQUESTS ** *Restart, write, number interval=1, time marks=NO ** ** FIELD OUTPUT: F Output 1 ** *Output, field *Node Output, nset=MIDSPAN_GAUGE_3 A, RF, U, V ** ** FIELD OUTPUT: F Output 2 ** *Node Output, nset=STRAIN_GAUGE A, U, V ** ** HISTORY OUTPUT: DRF_L EFT ** *Output, history *Node Output, nset=LEFT_RF CF2, CM3, RF1, RF2, RF3, RM1, RM2, RM3 ** ** HISTORY OUTPUT: DEF_MIDSPN_3 PAGE 117 117 ** *Node Output, nset=MIDSPAN_GAUGE_3 A1, A2, A3, AR, AR1, AR2, AR3, U1 U2, U3, UR, UR1, UR2, UR3, V1, V2 V3, VR, VR1, VR2, VR3 ** ** HISTORY OUTPUT: DRF_RIGHT ** *Node Output, nset=RIGHT_RF CF2, CM3, RF1, RF2, RF3, RM1, RM2, RM3 ** ** HISTORY OUTPUT: STRAIN ** *Node Output, nset=STRAIN_GAUGE A1, A2, A3, AR1, AR2, AR3, U1, U2 U3, UR3, V1, V2, V3, VR1, VR2, VR3 *End Step PAGE 118 118 LIS T OF REFERENCES ACI 31805, 2005. 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PAGE 121 121 BIOGRAPHICAL SKETCH Thien Phuoc Tran was born in Hue, Viet Nam in 1964. He escaped from the Vietnamese Communist Regime and arrived in Canada as a political refugee in 1980. He started his undergraduate studies in the c ivil e ngineering program at the University of Alberta, Canada in 1983. He graduated from this program and obtained his Bachelor of Science in e ngineering in 1988. He worked as a structural and construction engineer until 1997 when he joine d the Canadian Armed Forces as a Combat Engineer. He served two oversea tours. Upon his returning from the tour in Afghanistan in 2006, he was awarded a postgraduate scholarship to pursue a masters degree in c ivil e ngineering at the University of Florid a, specializing in the field of Force Protection. He received his Master of Science from the University of Florida in the spring of 2009. 