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Numerical modelling of experimental data of reinforced concrete beam-to-column joints Baraka, Miljenko 1996

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NUMERICAL MODELLING OF EXPERIMENTAL DATA OF REINFORCED CONCRETE BEAM-TO-COLUMN JOINTS by Miljenko Baraka B.A.Sc, (Civil Engineering), The University of British Columbia, 1992 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF CIVIL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA December, 1996 © Miljenko Baraka In presenting this thesis in partial fulfilment of the - requirements for an advanced, degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of .QW\U £/V6(#E££/rf6 The University of British Columbia Vancouver, Canada Date J f e ' C , - 2 3 , WJb . DE-6 (2/88) ABSTRACT Since the 1970's, increasing attention has been given to seismic design in building codes with emphasis on ductility. Ductile behaviour in reinforced concrete moment resisting frames is important from the point of view of an energy dissipating mechanism. Modern design codes today have stringent guidelines on the design of the beam to column joint region in order to achieve ductile behaviour without brittle shear failure. There are many older buildings, however, that are deficient in strength and ductility with respect to seismic loading. Deficient structures such as these may be retrofitted by encasing the beam to column joint in a steel shell. Cycl ic testing of reinforced concrete beam and column sub-assemblies have proven that a very substantial increase in bending and shear strength can be achieved in the joint area by encasing the region with a steel tube and filling the cavity with cement grout. Failures were deflected from the joint area to adjacent members, which were intentionally weakened to form plastic hinges. Subsequent tests on the remaining joint specimens, which forced the failure mechanism into the joint region, provided strength and ductility data for the joint itself. Because experimental testing of scale models can be expensive and at times impractical for every situation that may arise in practice, a non-linear finite element program was written for the analysis of the joint area. The program utilizes plasticity based constitutive descriptions of the concrete and steel material models and intends to be able to predict the behaviour and peak values of the strength envelopes of the joints. Comparisons with available experimental results were encouraging insofar as the plastic behaviour of the concrete and steel were captured. Due to the complex nature of the problem the program was unable to accurately predict the maximum load carrying capacity and more research is required in "fine tuning" the material constitutive models and finite element models. Recommendations for continuing research are given. TABLE OF CONTENTS Page ABSTRACT ii LIST OF TABLES vii LIST OF ILLUSTRATIONS ix LIST OF SYMBOLS IN ORDER OF APPEARANCE xiv CHAPTER 11NTRODUCTION 1 CHAPTER 2 EXPERIMENTAL WORK 13 2.1 Phase 1: Beam and Column Subassembly Tests 13 2.2 Phase 2: Joint Tests 21 2.3 Concluding Remarks 24 CHAPTER 3 THE CONSTITUTIVE MODELS 43 3.1 Overview 43 3.2 Plasticity Based Constitutive models 44 3.3 A Perfectly Plastic Drucker-Prager Model 46 3.4Response in the Tension Regime, Method 1 49 3.5 Response in the Tension Regime, Method 2 51 3.6 A Hardening/Softening Drucker-Prager Model 54 Response in the Tension Regime 56 Hardening/Softening Modulus 59 3.7 Plastic Dilatancy and Passive Confinement 61 3.8 Summary of the parameters in the Drucker-Prager models 64 3.9 The Von-Mises Steel Constitutive Model 65 CHAPTER 4 THE FINITE ELEMENT SOLUTION 70 4.1 Overview . 70 4.2 The Weak Equilibrium Equations 72 4.3 The Numerical Procedure 75 Gaussian Integration 80 4.4 The Concrete Finite Element 81 4.5 The Steel Plate Finite Element 83 4.6 The Reinforcing Steel Element 86 The Solution Algorithm 89 CHAPTER 5 ANALYSES WITH THE PROGRAM 73 Overview 73 5.1 The Program APOSEC 73 5.2 Analyses with the Perfectly Plastic Model 74 One Element Under Compression 74 Drucker-Prager Iteration Sensitivity 76 Passive Confinement Analysis 79 5.3Analyses with the Hardening/Softening Model 87 One Element Under Compression 87 Passive Confinement Analysis 90 5.4Analysis of a Joint Phase 1 92 5.5 Analysis of a Joint Phase 2 96 5.6 Analysis of a Joint Phase 3 98 5.7Closing Commentary 100 CHAPTER 6 CONCLUSION 105 Summary 105 Concluding Remarks 105 Future Work 107 REFERENCES 109 APPENDIX A 133 v Some Mathematical Facilitations: 133 Procedure for determining CT^ 134 Perfectly Plastic Model 134 Hardening/Softening Model 135 APPENDIX B 138 Element Shape Functions 138 Concrete Element Shape Functions 138 Shape functions corresponding to eight noded prism 138 Shape functions corresponding to midside nodes 139 Shape functions for twenty noded prism 140 Shape Functions For the Membrane Element 141 APPENDIX C 142 The Three Point Gauss Quadrature Rule 142 APPENDIX D 143 The Concrete Element [B] matrix 143 APPENDIX E 145 The Membrane Element [B] Matrix 145 vi LIST OF TABLES Table Page Table 2.1. Strain gauge readings (micro strain) at 100 kN. actuator thrust (direction of the arrow in Figure 2.12) 32 Table 2.2. Strain gauge readings in micro strain at 92 kN (moment = 33kNm) actuator thrust 35 Table 2.3. Strain gauge readings in micro strain on 105 kN actuator thrust for second square specimen 39 Table 2.4. Maximum and minimum observed envelope moments of the joint test specimens 42 Table 3.1. Table of parameters defining the Plastic Modulus H 61 Table 3.2. Parameters in the Drucker-Prager model 64 Table 5.1. Stresses at the center Gauss point for the different numbers of iterations at load step 10 just as the Gauss points make the elasto-plastic transition 96 Table 5.2. Stresses at the center Gauss point for the different number of iterations for the last load step 96 Table 5.3. Yield surface tolerance to number of iterations at the last load step 97 Table 5.4. Material parameters in passive confinement test 99 Table 5.5. APOSEC arguments used to conduct passive confinement analysis 99 Table 5.6. Stresses for the first load step at middle plane of Gauss points for the above model. p= 0.5 102 Table 5.7. Coordinates of the Gauss points shown in figure 5.7 102 Table 5.8. Constitutive parameters for Case A 106 VII Table 5.9. Constitutive parameters for Case B 106 Table 5.10. Constitutive parameters for Case C 107 Table 5.11. Constitutive parameters for Case D 107 Table 5.12. Constitutive parameters for Case E 107 Figure 5.10. Average vertical stress vs. vertical strain for two tension tests on single concrete element with the hardening/softening Drucker-Prager model 107 Table 5.13. Steel material parameters for passive confinement test on the Hardening/Softening Drucker-Prager model 108 Table 5.14. Concrete material parameters for passive confinement test on the Hardening/Softening Drucker-Prager model 108 Table 5.15. Material parameters in joint analysis using the perfect plasticity Drucker-Prager constitutive model 113 Table 5.16. APOSEC arguments used to conduct joint analysis using the perfect plasticity Drucker-Prager concrete constitutive model.113 Table 5.17. Steel plate constitutive properties in Phase 2 116 Table 5.18. Concrete constitutive properties in Phase 2 116 Table 5.19. Concrete constitutive properties for Case A 118 Table 5.20. Concrete constitutive properties for Case B 118 viii LIST OF ILLUSTRATIONS Figure Page Figure 1.1. Shear Wall Building 2 Figure 1.2. Moment Resisting Frame Building 3 Figure 1.3. Moment resisting frame subjected to seismic forces 5 Figure 1.4. Beam forces 6 Figure 1.5. Column design forces 6 Figure 1.6. Joint shear failure on I-880 during the 1989 Loma Prieta earthquake 8 Figure 1.7. Schematic of the California State University parkade 9 Figure 1.8. Schematic of the California State University parkade after collapse 9 Figure 2.1. Typical reinforced concrete test specimens 15 Figure 2.2. Loading apparatus used by Hoffschild 16 Figure 2.3. Jacket geometry 17 Figure 2.4. Hysteresis curves of square specimens 18 Figure 2.5. Hysterisis curves of circular specimens 19 Figure 2.5. Beam failure outside retrofit region. (Hoffschild) 20 Figure 2.6. Circular retrofitted specimen with multiple gaps. (Hoffschild) .20 Figure 2.7. Salvage and repair sequence of Hoffschild's used specimens. 26 Figure 2.8. Plastic hinge formation inside steel jacket 27 Figure 2.9. Failure of joint region in Hoffschild's unretrofitted test [11] 28 Figure 2.10. Test rig for joint test 29 Figure 2.11. Strain gauges on circular specimen, top view 30 Figure 2.12. Strain gauges on circular specimen, side view 30 Figure 2.13. Strain gauges on circular specimen, bottom view 30 IX Figure 2.14. Strain gauges on circular specimen, frontal view 31 Figure 2.15. Hysteresis curve for the circular specimen 33 Figure 2.16. Photograph showing failure of the steel shell on the circular specimen 33 Figure 2.17. Strain gauge placement on first square specimen, top view. ..34 Figure 2.18. Strain Gauge placement on first square specimen, side view.34 Figure 2.19. Strain gauge placement on first square specimen, bottom view. 34 Figure 2.20. Strain gauge placement on first square specimen, back view.35 Figure 2.21. Hysteresis curve for first square specimen 36 Figure 2.22. Photograph showing tearing of the weld in the corners of the square tube 36 Figure 2.23. Corner stiffener placement on second square specimen 37 Figure 2.24. Strain gauge placement on second square specimen, side view 38 Figure 2.25. Strain gauge placement on second square specimen, front and back view 38 Figure 2.26. Hysteresis curves for second square specimen 39 Figure 2.27. Tearing of the steel shell around the stiffener elements 40 Figure 2.28. Comparison of beam and beam-column joint strengths(Hoffschild [11]) 41 Figure 3.1. Drucker-Prager yield surface in principal stress space 47 Figure 3.2. Drucker-Prager yield surface in meridional space 47 Figure 3.3. Modified yield function in meridional space 50 Figure 3.4. Cracked material element of dimensions dx and dy showing principal and global axes 51 Figure 3.5. Variation of cohesion with hardening/softening 54 x Figure 3.6. Variation of friction angle with hardening/softening 55 Figure 3.7. Proposed function of mobilized friction as a function of K 60 Figure 3.8. Variation of H as a function of K for the values of parameters shown in the above table 61 Figure 3.9. Plastic dilatancy as a crack sliding phenomenon 61 Figure 3.10. Plastic volumetric expansion vs. plastic distortion 63 Figure 3.11. Von-Mises yield surface in principal stress space 65 Figure 3.12. Uniaxial Stress-Strain curve with hardening 66 Figure 3.13. Illustration of local and global coordinate system in the steel 68 Figure 4.1. General body modeled by finite elements 72 Figure 4.2. Modified Newton-Raphson method for single degree of freedom 79 Figure 4.3. Concrete finite element 82 Figure 4.4. Plate Steel Finite Element 84 Figure 4.5. Reinforcing steel element and degrees of freedom 86 Figure 5.1. Single 230mmX230mmX230mm Concrete Element Under Cycled Displacement 93 Figure 5.2. Average vertical stress vs. strain for cycled response of a concrete element 94 Figure 5.3. Post yield stress sensitivity to the number of iterations per loadstep on a single concrete element. (3 = 0.5 95 Figure 5.4. Exploded view of composite model and elements used to test the passive confinement effect of the plate steel 98 Figure 5.5. Vertical stress vs. vertical strain at center Gauss point of the above model for various dilatancy values 100 xi Figure 5.6. Passive confining pressure at center Gauss point for various values of dilatancy 100 Figure 5.7. Middle layer of Gauss points 102 Figure 5.8. a,b,c,d,e. 3D plots of a z at middle layer of Gauss points in the concrete element for load steps 1, 5,10,15 and 20. p = 0.5. 105 Figure 5.9. Average vertical stress vs. vertical strain for various compression tests on single concrete element with the hardening/softening Drucker-Prager model 105 Figure 5.11. Brittle response in uniaxial tension 108 Figure 5.12. Average vertical stress vs. strain for the passive confinement test on a singe concrete element confined by four steel plates using the Hardening/Softening Drucker-Prager model 110 Figure 5.13. Average confining pressure for the passive confinement test using the Hardening/Softening Drucker-Prager model 110 Figure 5.14. Average hardening modulus for the passive confinement test on the Hardening/Softening Drucker-Prager model 111 Figure 5.15. Model 1: 4 bricks and 15 plates 112 Figure 5.16. Model 3: 40 bricks and 60 plates 112 Figure 5.17. Model 3: 20 Bricks and 36 plates 112 Figure 5.18. Force displacement plot for two finite element models of a square specimen 114 Figure 5.19. Separation of steel plate from concrete on the tension side of specimen 115 Figure 5.20. Comparison of responses between tensile stress release due to cracking and no tensile stress release due to cracking 116 Figure 5.21. Response of the 4 concrete element model in Phase 3 117 xii Figure 5.22. Response of the 20 concrete element model in Phase 3 118 Figure 5.23. Force displacement response of the Square Joint #2 with fins 121 Figure 5.24. Force displacement response of the Square Joint # 1 121 Figure 5.25. Cross-section of retrofitted specimen 122 Figure B.1. Eight noded prism 138 Figure B.2. Prism with only midside nodes 139 Figure C.3. The three point Gauss quadrature rule 142 Figure E.1. Local basis vectors at a typical node 146 LIST OF SYMBOLS {ds'} {dee} {dzP} dzex,dzey,dzez,dyxy,dyxz,dy^ dzp,dspy,dep,dypxy,dypxz,dypyz {da} [V] K dk H [D e p] IN ORDER OF APPEARANCE Differential in total strain vector in Global Cartesian coordinates. Differential in total strain vector in Local Cartesian coordinates. Differential in elastic strain vector in Global Cartesian coordinates. Differential in plastic strain vector in Global Cartesian coordinates. Components of differential elastic strain i n Global Cartesian coordinates. Components of differential plastic strain in Global Cartesian coordinates. Differential in stress vector in Global Cartesian coordinates. Differential in stress vector in Local Cartesian coordinates. Linear elastic constitutive matrix. Y ie ld function.. Hardening/softening parameter. Magnitude of the plastic strain increment. Hardening/Softening modulus. Plastic flow function. Elasto plastic matrix. Components of the stress tensor. Components of the deviator stress tensor. Kronecker delta. First invariant of stress. Hydrostatic pressure. Second invariant of the deviator stress tensor. Third invariant of the deviator stress tensor. xiv Square roof of the second invariant of stress. Angle measure of the third invariant. Differential in the tensor plastic strain. Differnetial in the deviator of the plastic strain. Differential in plastic volume expansion. Differntial in plastic distortion. Parameter in perfectly plastic Drucker-Prager yield function. Parameter in perfectly plastic Drucker-Prager yield function. Cohesion. Friction angle. Concrete strength in compression. Concrete strength in tension. Dilatancy factor. Parameter that locates the apex of the erfectly plastic Drucker-Prager yield function. Principal stress vector. Principal stress direction vectors. Components of the principal stress vector. Strain normal to a crack plane. Strain transformation matrix. Shear modulus. Youngs modulus. Poissons ratio. Rate of shear modulus reduction. Mobil ised cohesion. Mobil ised friction angle. Parameter in hardening/softening Drucker-Prager yield function. K Parameter in hardening/softening Drucker-Prager yield function. y* Mobilised dilatancy angle. <j>* Mobil ised friction angle. <t>cv Value of mobilised friction angle at which a transition from compaction to dilatancy occurs. a*A Parameter that locates the apex of the hardening/softening Drucker-Prager yield function. r ( a „ , , a * / ) , / c , A) Function relating damage in concrete to plastic distortion. Y Initial yield stress of steel in uniaxial tension. YU(K.) Y ie ld stress of steel in uniaxial tension. x,y,z Global Cartesian coordinates. x',y',z' Local Cartesian coordinates. {Ae} Finite increment in strain vector. [B] Strain-displacement compatibility matrix. {Aa} Vector of nodal displacement increments. {Aa} Finite increment in stress vector. {P} Vector of external forces. {AP} Increment in the vector of external forces. W Virtual work. {a} Virtual displacement. {5a} Corrective increment to displacement. e Euclidean norm. [K'] Tangent stiffness matrix. [K']q Initial tangent stiffness matrix. {AO} Unbalanced load vector. ^,ti,(^ Dimensionless coordinates. [J] Jacobian matrix. lx,m\,nx Direction cosines. xvi Global Cartesian base vectors. Local Cartesian base vectors. CHAPTER 1 INTRODUCTION During the past two to three decades, building codes in Canada, the United States and Japan have undergone significant changes by updating design rules to address the effects of earthquakes. Most of the current design rules regarding building member resistance have been established during the last two decades from experimental work, a great deal of which originated in New Zealand and North America. M u c h was also learned from the observation of failed buildings and bridges in past earthquakes, which prompted the adoption of modern design philosophies. Extensive research has been conducted to determine the response of buildings during earthquakes and to predict the forces generated in a structure as a result of seismic motion. In a simplified approach the basic premise of a seismic load is that of a lateral load, which is a percentage of the structure's total weight, distributed vertically according to the mass distribution and predicted accelerations. A case in point regarding the evolution of this lateral load in design codes can be made using Japan as an example [1]. The first time a seismic load was included in Japanese building codes was in 1924, which prescribed a load representing 10% of the building weight, applied uniformly over the height. In 1950 the seismic design coefficient was increased to 20%. After the Second World Conference on Earthquake Engineering held in Japan in 1964 the seismic coefficient was changed to 20% for buildings up to 16 m in height and increased by 1% for every 4 m increase in height. The distribution of lateral load was still uniform. After the 1971 San Fernando, California earthquake, the Ministry of Construction of Japan conducted a five year research program aimed at establishing a rational design procedure regarding seismic design. After the 1978 Miyagiken-oki earthquake, the findings of the five year research program were 1 incorporated into a new building code. The seismic force became dependent upon the soil conditions at the site, the building natural period and ductility. Furthermore, the code now required more sophisticated analysis methods such as dynamic and non-linear time history studies for buildings taller than 60 m. There are basically three major mechanisms by which a building resists the effects of lateral motion caused by an earthquake: • Shearwall type structures, which, as the name suggests transfer lateral loads to deep reinforced masonry or concrete walls, often in pairs and joined with lintel beams. These can be located throughout the structure and/or comprise the elevator shaft in the case of multistorey structures. Shearwalls are used in a wide variety of buildings ranging from single storey warehouses to multistorey buildings. Link Beams Figure 1.1. Shear Wall Building. • Diagonal braces are frequently used in simply connected frames to form an efficient lateral load resisting system. The braces, however, are a major impediment to the free flow of traffic. • Moment resisting frame type structures provide an open plan system comprising of interacting beams and columns. Lateral resistance of such a building is provided predominantly through bending action of the beams and columns, thus relying on rigid connections between members. Figure 1.2. Moment Resisting Frame Building. The beam-column joint which is a vital component of moment resisting frames is the focus of study in this thesis. Quite often it is impractical to design a structure to remain elastic during a major earthquake, as the overall cost of materials would be prohibitively high. When properly designed for controlled deformations, however, the structure is expected to behave inelastically at selectively chosen locations to allow for energy dissipation and thus damping of the system, without collapse of any vital parts. These locations may be the bases of shear walls in shear wall type buildings, the bases of columns in bridge bents or at the junctions of beam to column joints in tall frame-type buildings. The key issue is, however, that no fracture or instability occur during such plastic action. This has several implications, namely: 1. The design forces that a building is required to resist are reduced by the "ductility factor", due to the fact that ductile behavior dampens the structure and reduces its resonance during shaking. When a structure becomes inelastic it dissipates the energy of motion through hysteretic damping. This damping component significantly reduces the forces experienced by the building. 3 2. The location of the yield zones and mechanisms must be carefully chosen and detailed by the designer to avoid undesirable modes of failure such as shear in reinforced concrete members. 3. The overstrength of members must be considered to ensure that the energy dissipating elements are indeed the weakest. Adjacent members and connections must be able to resist the forces generated in the plastic element (this is called capacity design philosophy). Contemporary design codes deal with these items fairly well when new buildings are designed. There are many older concrete buildings, however, that have been designed to the old codes. In addition to inadequate force levels, they are also often deficient in detailing to allow for the formation of plastic hinges. To avoid the possible collapse of such buildings in future earthquakes retrofit methods must be explored. Unfortunately, design codes do not, as a rule, deal with retrofitting of structures. The work presented herein particularly concerns items 1,2 and 3. For a discussion on capacity design the reader is referred to the work by Paulay and Priestley [2]. The Canadian concrete design code C S A - A 2 3 . 3 , [3], addresses items 1 and 3 in the following manner: • Having calculated the total elastic base shear Ve = vSIFW, as given in the National Building Code of Canada (NBC) , the design base shear V is 0.6Ve/R, where R is the ductility factor of the structural system (for ductile moment resisting frames, R=4). This total shear is then distributed per every storey according to height and storey mass to obtain a set of equivalent static lateral loads, (Figure 1.3). The beams are required to resist the resulting factored moments (Mf) and overstrength shears (Vo) as shown in Figure 1.4, while the columns are required to resist both the overstrength moments (Mo) and overstrength shears (Vo) (see Figure 1.5 and equations 1.1 and 1.2). The overstrength shear is the shear that results from the overstrength moment gradient which corresponds to the resistance of the beam when the material resistance factors, <)>c and <|)s, are taken as unity and using fc and 1.25/y as the material strengths. The main reason for this procedure is to avoid shear failures which are known to be very brittle. There is a probability that the concrete and steel materials used in the construction may be stronger than the nominal values and the fact that steel strain-hardens when subjected to large strains as might be expected in a plastic hinge. The shear resistance of the beams and columns need not exceed the factored shear corresponding to seismic forces computed with a ductility factor R = l (since theoretically the largest forces a structure may experience are the elastic forces). The beams, being designed for a higher shear than the factored shear, are thus protected from a brittle shear failure. Figure 1.3. Moment resisting frame subjected to seismic forces. 5 Resulting factored beam moments Vf Factored shear forces resulting from above moments Mo+ Mo-Overstrength beam moments resulting from member plastic hinging Vo Overstrength shear forces resulting from above moments Figure 1.4. Beam forces. The column design moments and shear are given by: (1.1) (1.2) Where kj and &2 represent the rotational stiffness in double curvature bending of the upper and lower column at the joint in question. To achieve ductility in the beams, the concrete code requires closely spaced closed stirrups in the plastic hinge regions of the beams and closely spaced ties in the ends of the columns to confine the concrete. Furthermore, ties must be placed in the beam to column joint which is a region of compound forces and especially susceptible to shear failure. Many structures designed to codes more than twenty years old would typically lack such design details as hooped stirrups in beams and confining steel in columns. A classic failure involving the beam to column joint during the Loma Prieta earthquake was the collapse of a two kilometer stretch of the Nimi tz freeway in Oakland (highway 1-880) [4]. The structure was completed in 1957 and consisted of a double deck superstructure supported on piers as shown in Figure 1.6. 7 Joint Failure Figure 1.6. Joint shear failure on I-880 during the 1989 Loma Prieta earthquake. To simplify the design process, hinges were built into the piers at selected locations (Figure 1.6) to make the structure statically determinate and to minimize the effects of creep, shrinkage and temperature forces. Consequently, when the earthquake occurred, the base of the upper right column was subjected to high moment and shear forces. The high shear and moment, coupled with the fact that there was no confining steel in the joint and only minimal shear reinforcing (ties) in the column, resulted in a brittle joint and column failure. More recently, during the 1994 Northridge earthquake, failures involving shear in columns occurred in a parkade at the California State University [5]. The structure consisted of a ductile moment resisting perimeter frame, precast beams supported by corbels on cast-in- place gravity columns supporting a cast-in-place floor slab. A simple schematic representation of the structure is shown in Figure 1.7. The interior gravity columns were not detailed for ductility and subsequently suffered a brittle shear failure. 8 The loss of interior support caused the floor slabs to collapse and pull the outside perimeter frame inward (Figure 1.8). i-T. 7 Short columns Figure 1.7. Schematic of the California State University parkade. Shear failure in columns Figure 1.8. Schematic of the California State University parkade after collapse. Such deficient structures can be "retrofitted" by encasing the columns in grouted steel jackets as was done with many bridge columns on California's freeway bridges following the 1989 Loma Prieta earthquake. Such a steel jacket externally provides the concrete with confinement and ductility that would otherwise be provided by the transverse reinforcement in a contemporarily detailed member. M u c h experimental work has been done by such researchers as Mander and Park on the behaviour of concrete subjected to confining pressure [6], [7]. It can be shown that confined concrete exhibits an increased compressive strength and crushing strain, which is a key issue regarding the ductility capacity of a plastic hinge. 9 In the process of design, one models a reinforced concrete structure with a linear elastic computer analysis program and performs linear elastic member force analyses based on loading that is representative of the seismicity of the local area. The objective of the design process is to ensure that the member elastic forces calculated in the analyses be less than (R x Member Resistance). For conventional beams and columns the member resistances are easily obtainable from the charts and tables in design codes, while the ductility capacities (R factors) can be obtained from building codes that have published values for various structural systems. Alternatively one may resort to more sophisticated computer analysis programs such as D R A I N T A B S . The program D R A I N T A B S and many others have non-linear analysis capability; that is, they can perform a step by step time analysis response of a structure subjected to earthquake ground motion. In performing such analyses the solution algorithms of these programs take account of the changing beam and column member properties when these members yield. The output of the software consists of member forces and deformations tabulated versus time. From the output, the user may calculate member ductility demands (u=maximum deformation/elastic deformation) and compare against allowable values to ensure that |j, < R. A retrofitted structure, such as a reinforced concrete moment frame that has the beam to column joints encased in a steel shell w i l l have unknown values of strength and ductility capacity. Building codes do not deal with this situation, but the yield strength and ductility must be determined to asses the structure's capacity to seismic loading. To obtain the properties of the elements comprising such structures one may test scale model subassemblies in the laboratory. But obviously one cannot perform experiments for every situation that may arise in practice. From a practical standpoint, one would like experimental results to be predictable to some degree i f the results of the experiments are 10 to be used in design of members for use in practical structures, which often are of a larger scale and have more complex configurations than laboratory models. This thesis explores the modelling applications of beam-column joint subassemblies with a non-linear finite element based computer program in an attempt to predict the yield strengths and the post yield behaviour (ductility) of the joint. Such information is important for assessing the capacity of a structure with steel encased joints as a seismic retrofit. Furthermore, obtaining the information in an analytical way w i l l reduce the time and cost associated with experimental testing. Initially, carefully performed experimental testing w i l l be required for the calibration of the numerical models. Once calibrated, a computer program may be regarded as an experimental laboratory at the keyboard that, although not replacing the physical laboratory, could provide initial insight into the behaviour of laboratory models to facilitate the design of the experimental procedure itself. The program is based on non-associated plasticity concrete and steel constitutive models. The impetus for this work came from a plasticity based constitutive model of concrete recently developed by J. Jiang [8] and implemented in a finite element analysis computer program by J. Jiang and F . A . Mi rza [9] to determine the response of reinforced concrete slabs to loading. The computed load history results matched very closely those of experiments. Initially the concrete constitutive model by J. Jiang was obtained as a subroutine and attempted to be used in this program in conjunction with a plate element subroutine to model the steel encased joints. The author encountered numerical problems in the implementation, however. The fact that the constitutive model formulation is complex, and the author being unfamiliar with the intricate details of the subroutine, rendered the numerical problems insoluble. A n alternative approach was taken where a simpler constitutive formulation was implemented as a complete new subroutine in the computer program. Such an approach required the writing of a constitutive subroutine or 11 subroutines and program from scratch, but the effort expended was returned as familiarity with the resulting program, such that any corrections and modifications i f needed would be performed smoothly. In fact, the complexity of the application required several modifications during the development of the program. In the following chapter some previous experimental work on the ductile cyclic load behaviour of steel encased beam-column joint specimens are presented [11] along with recent experimental work dealing with an attempt to determine the strength of the beam-column joint area itself. In Chapter 3 two concrete constitutive models, one simple and the other complex, based on the Drucker-Prager yield criterion [10] are developed from concepts adopted from various literature on the subject. Model complexity is kept to a minimum by implementing only those aspects necessary to predict the strength envelope of cyclic load tests. In some work the cyclic behavior of concrete has been modeled [14]. The key features of the concrete constitutive model are that it be confining pressure sensitive and plastically dilatant. It is important to obtain accurate concrete behavior with respect to these properties for the following reasons. The properties of the concrete model are important when considering the composite action of the steel shell that provides the passive confinement. It is the tendency of a concrete volume element (and many geologic materials such as sand) to expand as it deforms plastically (this is plastic dilatancy and is not related to the Poisson's ratio effect which is an elastic phenomenon). The plastic expansion w i l l stretch the steel which w i l l in turn provide passive confinement. Passively confined concrete, as was stated previously, displays increased ductility and crushing strain. In Chapter 4 the aspects of incorporating the constitutive models in a finite element computer program are discussed. The predictions of the program are compared to some available load history experimental results in an attempt to compare the sophisticated with the simple constitutive models in Chapter 5. 12 CHAPTER 2 EXPERIMENTAL WORK The experimental work of this project was done in two phases. Hoffschild [11] investigated the effectiveness of steel encasing as a retrofit method for reinforced concrete frames with weak joints. The author continued the study by determining the behavior of the retrofitted joint region itself. A brief summary of Hoffschild's work is given here. 2.1 Phase 1: Beam and Column Subassembly Tests During 1990-1992 Thomas E . Hoffschild prepared a series of half-scale beam column reinforced concrete test specimens representing part of a 2 storey frame [11]. The reinforcement details used were designed to the codes of the early 1970's with the joint ties eliminated. The dimensions and reinforcement details of the reinforced concrete specimens are as shown in Figure 2.1. A l l the specimens were tested under cyclic loading in the loading apparatus shown in Figure 2.2, which consisted of a load controlled actuator that maintained a constant axial load on the column and a displacement controlled actuator that cycled the beam end up and down. Four specimens were prepared, two of which were pre-damaged before retrofitting. Two of the specimens were left undamaged. Later all were encased in steel jackets, two circular and two square, (Figure 2.3) so that two damaged specimens were available, one each with square and circular jackets, and two undamaged ones, one each with square and circular jacket. The specimens were labeled as R E T R O - C U , R E T R O - C D , R E T R O - S U and R E T R O - S D , 13 corresponding to circular undamaged (CU), circular damaged (CD), square undamaged (SU) and square damaged (SD). The purpose of damaging the beam-column specimens prior to applying the retrofit jackets was to determine the retrofit effectiveness on earthquake damaged structures. To limit the bending strength increase of the beams, the retrofit jackets were provided with small gaps as shown in Figure 2.3. For the last test, additional gaps were cut as shown in the photograph of Figure 2.7. A s it turned out, pre-damaging of the joint region had no effect on the performance of the specimens. The added strength and stiffness resulting from the retrofit significantly changed the damage locations, which often occurred just outside of the encasing as seen in Figure 2.6. Even with the gap acting as the weak link there was no assurance that the gapped region would fail. Furthermore, it was found that a single gap resulted in a very short plastic hinge with limited ductility. Consequently, three gaps were cut in the last specimen (Figure 2.7) to ensure proper performance of the retrofit. This specimen exhibited a markedly improved rotational ductility of 7 times yield as opposed to 5 for the specimen with only one gap. In all the tests, no damage was observed in the joint region itself. Load deformation curves of the four specimens are shown in Figures 2.4 and 2.5, which also indicate the location of failure. 14 10mm <p hoops Spacing 70mm Column: 190x190 3 x 10mm <p 2550 10mm <p stirrups Spacing 70mm 4 x 10mm <p Beam: 165 x 200 2 x 10mm <p 495 1200 1175 All dimensions In mm Figure 2.1. Typical reinforced concrete test specimens. beam actuator (displacement; controlled)! load cell specimen Figure 2.2. Loading apparatus used by Hoffschild. 16 Figure 2.3. Jacket geometry. HYSTERESIS CURVE - RETRO-SU HYSTERESIS CURVE - RETRO-CU -0.05 0 0.05 JOINT ROTATION a [rad] HYSTERESIS CURVE - RETRO-CD 0.1 -0.05 0 0.05 JOINT ROTATION a [rad] Figure 2.5. Hysterisis curves of circular specimens. 0.1 20 2.3 Phase 2: Joint Tests Although Hoffschild's experiments addressed the performance of the retrofit schemes as a whole, the performance of the beam to column connection itself had not been assessed. To avoid potential brittle behaviour of the frame, the joint area, which could have difficulties in confinement of the concrete, must be assured to have sufficient strength to deflect plastic hinges to adjoining members. Following capacity design procedures, it should be significantly stronger than the adjoining members to allow for unforeseen overstrength of these members. In May - August of 1993 the failed specimens of T. Hoffschild were salvaged and repaired. The salvage and repair program consisted of: 1. Removing the steel jacket from the damaged beam portion. 2. Removing bits of broken concrete from the beam. 3. Welding new reinforcing bars to replace broken ones. 4. Welding the steel jacket that was removed in step 1 to the beam stub. 5. Fi l l ing the j acket with fresh concrete. The previous five steps are illustrated below in figure 2.8. The three specimens that were salvaged in this manner were one circular and two square jacketed specimens. The second circular jacketed specimen had been tested previously in a pilot study by a group of students as part of a term project [12]. During this test an attempt was made to fail the joint region. The test procedure and experimental rig resembled the one shown in Figure 2.11 with the exception that the hold down points were not symmetrically placed about the centerline of the joint, which eventually resulted in a failure occurring in the column. Since the objective of the test (to fail the joint region) was not realized, the results of the test were inconclusive. A postmortem analysis of the specimen revealed the formation of a flexural plastic hinge inside the jacket (Figure 2.9). If one compares this with Hoffschild's pre-retrofit tests, Figure 2.10, it is clear that the failure mechanism changed from shear in the joint panel to a flexural hinge in the beam. Hoffchild's and the students' 21 experiments are not directly comparable, however, since Hoffschild had a larger moment to shear ratio, whereas the students had a relatively short lever arm and a larger shear to moment ratio. Even with the larger shear experienced by the specimen in the student's test, the steel jacket prevented a shear failure of the joint. Following the repair of the salvaged specimens, a test rig was devised that held the specimen down securely and prevented a failure outside of the joint region. Shown in Figure 2.11 is the apparatus used to test the joint region of the beam to column connection. The specimen was held securely down by four threaded rods that were bolted into the concrete floor of the testing laboratory. A steel channel was placed across the member at each end to which the other ends of the threaded rods were bolted as shown. The loading of the specimen occurred by applying a cyclic displacement through a servo-controlled hydraulic actuator at the position indicated in the figure. A Linear Variable Differential Transformer (L .V.D.T . ) was mounted 152 mm above the face of the specimen for the measurement of displacements of the beam relative to the column. Rotations of the joint are defined as the measured displacement at the (L .V .D .T . ) divided by 152 mm, whereas the moment arm is referred to point A in Figure 2.11. Initially, when the first specimen was tested, the load displacement curve was carefully observed on a computer screen that plotted the signals from the data acquisition equipment in real time. A n actuator displacement of 2 mm forwards and backwards was chosen to plot the first hysteresis curve. A t this point it was not known what the range of displacement would be required to the point of failure. A l l of the other hysteresis loops were loaded based on progressive 1mm increments so that the point where strength degradation occurs is not missed. Having observed the hysteresis curves for the first specimen, it was decided that all subsequent load tests would be based on 1 mm progressive increments of the actuator. 22 The first specimen to be tested was the circular one. Strain gauges were placed on the jacket for the recording of strains with a computer controlled data acquisition system. The placement of the strain gauges is shown in Figures 2.12 to 2.15. In Table 2.1 are shown the readings of the respective strain gauges at a thrust of 100 k N on the actuator. Referring to Figure 2.11, a thrust force would be to the left in the picture. Enough data was acquired to plot the moment vs. rotation hysteresis loops (Figure 2.16). The placement of both the circular and square specimens was such that the actuator always formed a lever arm of about 500 mm with respect to point A . I f the yield rotation is taken to be 0.02 radians (which is approximately the point at which some form of yield plateau was reached as shown in Figure 2.16) and the ultimate rotation to be 0.065 radians, the joint itself exhibited a rotational ductility ratio of about 3.25. While the ductility of the joint region itself is not of major concern in capacity design procedures, the peak strength being greater than that of the adjacent members is more critical. A comparison of peak strengths of the joints and beams as conducted by Baraka and Hoffschild are made at the end of this chapter. Eventually the weld connecting the beam tube to the column tube failed, which is exhibited as a drastic drop-off in strength (Figure 2.16) after a rotation of about 0.065 radians. The next specimen to be tested was the first of the square specimens. Shown in Figures 2.18 to 2.22 are the placements of the strain gauges, the strain gauge readings at a thrust of 100 k N and the corresponding hysteresis curves. A s with the circular specimen, the square specimen failed by tearing of the weld line along the tube connections. The failure occurred at a rotation of about 0.04 radians, with the hysteresis loops showing significant pinching at higher rotations. In both specimens tested the weak links were the welds connecting the beam and column jacket. In the case of the square specimen, tearing of the weld initiated in the corner of the tube where the highest stress concentrations occurred. To improve the performance of the square retrofit, four steel fin 23 plates (of dimensions 6.4 X 38 X 229 mm) were welded on to the second square specimen to reinforce the corners (Figure 2.24). In this test, strain gauges were concentrated in the highly stressed regions around the reinforcement fins (Figure 2.25 to Figure 2.26). It is seen that the hysteresis loops in Figure 2.27 of the corner reinforced specimen display a remarkable improvement in ductility, almost as much as the circular encased specimen. The post failure inspection of the failed specimen (Figure 2.28) indicated that the tear in the steel jacket went around the steel stiffeners as opposed to the weld seam failure observed in the other tests. 2.4 Concluding Remarks Shown in Figure 2.29 is a comparison of the beam strengths as observed by Hoffschild [11] and the strengths of the beam to column joints as obtained by Baraka. Generally, the joint is significantly stronger than the moment required to develop the ductile plastic hinges in Hoffschild's tests. This is a desirable performance feature of the retrofit scheme as it applies to capacity design philosophy. The steel jacketing method was shown to be an effective technique for improving the strength and ductility of inadequately reinforced concrete member joints. Careful attention must be paid, however, to the design of a retrofit scheme by considering the possibility of a shear failure outside the retrofitted region due to higher moment gradients resulting from flexural overstrength. The observed improvement in ductility which was achieved by adding steel stiffener elements, indicates that careful detailing of beam column joints is needed to avoid premature failures. This also may lead to further simplifications by using methods that are easy to implement in the field such as by building a retrofit cage around a beam-column joint composed of steel angles and bars as suggested by Alcoccer [13]. 24 A finite element analysis technique wi l l be explored next that incorporates a hydrostatic pressure sensitive concrete constitutive model to predict the behaviour of steel encased concrete joints. 25 Steps 1 and 2. Weld Steps 3 and 4. Step 5 Figure 2.8. Salvage and repair sequence of Hoffschild's used specimens. 26 27 Figure 2.10. Failure of joint region in Hoffschild's unretrofitted test [11]. 28 •I L.V.D.T. [ J152mm Actuator 496mm I Threaded Rod Threaded Rod 960 mm Figure 2.11. Test rig for joint test. 29 PLACEMENT OF STRAIN GAUGES ON CIRCULAR SPECIMEN End of tube Figure 2.12. Strain gauges on circular specimen, top view. Center Line Front #1 #6 #5 #2 124mm < T Actuator #12 0 #11 200mm End of tube #13 c=> #18 \\<=i-#15 u #17 60° arc Center Line Figure 2.13. Strain gauges on circular specimen, side view. 160mm End of tube #20 ,#19 #21 [ ,#22 Figure 2.14. Strain gauges on circular specimen, bottom view. 30 124mm Figure 2.15. Strain gauges on circular specimen, frontal view. 31 Table 2.1. Strain gauge readings (micro strain) at 100 kN. actuator thrust (direction of the arrow in Figure 2.13). Gauge #1 -33.6 Gauge #2 16.8 Gauge #3 39.1 Gauge #4 -117.4 Gauge #5 151 Gauge #6 -307.6 Gauge #7 447.4 Gauge #8 33.6 Gauge #9 251.6 Gauge #10 33.6 Gauge #11 151 Gauge#12 100.7 Gauge #13 55.9 Gauge #14 486.5 Gauge #15 -89.5 Gauge #16 190.2 Gauge #17 -106.2 Gauge #18 123 Gauge #19 -167.8 Gauge #20 83.9 Gauge #21 61.5 Gauge #22 -167.8 32 200 -0.1 -0.05 0 0.05 0.1 Rotation (radians) Figure 2.16. Hysteresis curve for the circular specimen. Figure 2.17. Photograph showing failure of the steel shell on the circular specimen. 33 PLACEMENT OF STRAIN GAUGES ON FIRST SQUARE SPECIMEN Figure 2.18. Strain gauge placement on first square specimen, top view. Front #8 60mn+ 40i Actuator Back 356mm Figure 2.19. Strain Gauge placement on first square specimen, side view. Figure 2.20. Strain gauge placement on first square specimen, bottom view. 34 Figure 2.21. Strain gauge p 110mm 125mni #10 r~ '—l-#9 lOOmrrl acement on first square specimen, back view. Table 2.2. Strain gauge readings in micro strain at 92 kN (moment = 33kNm) actuator thrust. Gauge #1 57.24 Gauge #2 -332.02 Gauge #3 -5.72 Gauge #4 51.53 Gauge #5 out of order Gauge #6 68.7 Gauge #7 40.08 Gauge #8 68.69 Gauge #9 154.56 Gauge #10 240.43 35 Figure 2.22. Hysteresis curve for first square specimen. Figure 2.23. Photograph showing tearing of the weld in the corners of the square tube. Figure 2.24. Corner stiffener placement on second square specimen. STRAIN GAUGE PLACEMENTS ON SECOND SQUARE SPECIMEN Actuator Fronti Back 356mm #1 #3 #4 i i #2 Figure 2.25. Strain gauge placement on second square specimen, side view. #5 #7 #6 n n n #8 #9 Figure 2.26. Strain gauge placement on second square specimen, front and back view. 38 Table 2.3. Strain gauge readings in micro strain on 105 kN actuator thrust for second square specimen. Gauge #1 -27.96 • Gauge #2 33.55 Gauge #3 -11.18 Gauge #4 -50.32 Gauge #5 -419.39 Gauge #6 -542.41 Gauge #7 -738.13 Gauge #8 492.31 Gauge #9 45.8 125 Rotation (radians) Figure 2.27. Hysteresis curves for second square specimen. 39 Figure 2.28. Tearing of the steel shell around the stiffener elements. 40 MOMENT RESISTANCES OF RETROFIT SCHEMES RETRO-CU 39.6 kNm 56.4 kNm 44.3 kNm RETROFITTED JOINT STRENGTHS SQUARE1 SQUARE2 97.6 kNm I I 104.6 kNm 106.3 kNm I I 118.6 kNm CIRCULAR 139.0 kNm 156.0 kNm Figure 2.29. Comparison of beam and beam-column joint strengths (Hoffschild [11]). Table 2.4. Maximum and minimum observed envelope moments of the joint test specimens. MINIMUM MAXIMUM Circular -139 k N m 156 k N m Square #1 -106.3 k N m 97.6 k N m Square #2 -104.6 k N m 118.6 k N m 42 CHAPTER 3 THE CONSTITUTIVE MODELS 3.1 Overview The previous two chapters have explored the problem of earthquake loading on structures and the retrofitting of beam to column joints in deficient moment resisting frames for increased shear strength and ductility. The basic problem is the brittle nature o f concrete failure when insufficient shear and confinement reinforcement is provided. When attention is being paid to provide concrete confinement by either closed stirrups or by encasement in a steel jacket, an increase in concrete strength and ultimate strain can be achieved. It remains a very difficult task, however, to predict the behavior of a steel encased beam to column joint. To avoid the need for testing a joint prototype for every possible configuration, an analytical procedure would present an acceptable method for studying the influence of changing certain design parameters. A non-linear finite element model was developed as part of this project. A n important part of the model, however, is the choice of appropriate material models. In this chapter a set of constitutive models, described in the context of continuum mechanics and rate independent non-associated plasticity theory 1, are presented for the eventual incorporation into a finite element based computer program. The main focus is on the selection of a concrete constitutive model because its behavior under compound stress and strain is very complex, especially in comparison with steel. A significant To account for rate dependent effects one would resort to viscoplasticity or endochronic theory [17]. 43 difference, for example, is the fact that the plasticity models capture the increased concrete strength due to confining pressure. More complex models [8] even simulate the ultimate brittle crushing failure of concrete as a function of confining pressure, which in the continuum mechanics context is represented by the first invariant of stress or the hydrostatic pressure. Increased strength and strain range to the point of crushing are important features of concrete in a beam column joint required to undergo plastic hinging. The characteristics of three plasticity based constitutive models are investigated for incorporation in the program: two concrete models are based on the Drucker-Prager yield criterion, considering non-associated plasticity, while an associated plasticity Von-Mises yield criterion model is adopted for the steel [10]. 3.2 Plasticity Based Constitutive models In the constitutive modelling of materials exhibiting plastic deformations the assumption is made that the total strain increment at a point is composed of an elastic strain component and a plastic strain component, which in the case of general three dimensional engineering representation both have six components of strain: {de} = {d£e} + {d£p} {*} = d€x dep' deey dspy del dzpz dfxy dypxy' dfxz dypxz dl"yz (3.1 a,b) The elastic component, {dse}, is what gives rise to the stress increment in the material at the point in question: {<fo} = [De]{<fee} (3.2) 44 where the matrix [De] is the constitutive matrix of linear elasticity. Also central to the notion of associated and non-associated plasticity are the yield criterion and plastic flow rule: F({CT},K) = 0 yield criterion (3.3) = eft. j^ j plastic flow rule (3.4) The plastic flow rule states that the strain increment is proportional to the gradient of a scalar function, V|/({CT}), called the plastic potential or plastic flow function. The scalar parameter dk represents the magnitude of the plastic strain increment and is given by f V k * i dk = dF_ da (3.5) The yield function is generally dependent on some hardening/softening parameter, K, which represents the evolution of the yield surface F with accumulated plastic strains sP. The above results are used to derive the elastoplastic constitutive matrix which relates the stress increment to the total strain increment [10]: {da} = [l)ep]{de} (3.6) [wp] = [D-] da [da [D'] 8F_ da M | 3 + H [jyp] = [jy] When yielded When elastic (3.7) The parameter H is known as the hardening/softening modulus in the literature and is related to the yield function F and the hardening/softening parameter K as follows: dF 5 K H = 9 K dk (3.8) 45 Departing from the previous engineering representations of stress and strain, a three dimensional tensorial representation wi l l be employed from here forward. To facilitate the work here the following stress tensors and stress invariants are defined: The deviator of the stress tensor: su = a ! / - ^ 6 / / a * * ( 3 - 9 ) The first invariant of stress: 1 = (3.10) The hydrostatic pressure: 1 (3.11) The second invariant of the deviator stress J2 1 - 2 s o s o (3.12) The third invariant of the deviator: J 3 1 — 2 Si.iS.ikSki (3.13) Square root of the second invariant: a • = Jj2 (3.14) Angle measure of the third invariant: 9 = s in - 1 { 3V3 J3) I 2 * 3 J ; - 7 c / 6 < e < 7 i / 6 (3.15) Where 8^ is the Kronecker delta, which is defined as 5^ = 0 for tej and 5^ = 1 for i=j. Although there are six invariants defined, only three are independent (/,J 2,J 3) or ( a m , a ,6). In addition the following plastic strain tensors and plastic strain measures are defined: Increment of plastic strain: de? (3.16) Deviator of the increment in the plastic strain: de[] = dzP. - ^ Syde^ (3-17) The first invariant of plastic strain. (Measure of plastic volume change): dz^^de^. (3.18) The second invariant of the deviator of plastic strains. (Measure of plastic distortions): dep = ^[dePjdePj) (3-19) 3.3 A Perfectly Plastic Drucker-Prager Model A perfectly plastic model, (H= 0 in equation 3.8), to be implemented for concrete is based on the simple hydrostatic pressure sensitive Drucker-Prager yield criterion which is an approximation of the Mohr-Coulomb yield criterion: 46 F(aiJ) = 3aam + G-K = Q (3.20) The yield function represented by equation 3.20 can be visualized as a cone in principal stress space as shown in Figure 3.1 or as a straight line in the meridional space in Figure 3.2. The parameters K and a can vary as a function of the plastic strains in concrete exhibiting strain hardening and softening. In this simple model the K and a shall hold constant values, hence a perfect plasticity Drucker-Prager model is represented above. (3.21) -30 -10 10 30 Figure 3.2. Drucker-Prager yield surface in meridional space. 47 The parameters <j) and c are the friction angle and cohesion in the Mohr-Coulomb failure envelope for the material. The model as stated requires only two parameters to define it, either the friction angle, and cohesion, c, or the uniaxial tensile and compressive stresses at failure, fc and ft. Given any of the two parameters the remaining two may be calculated as: (3.22) _ 2ccos(<j>) _ 2ccos(c|)) Jc ~ ^ : 7TT ' / / 1 - sin((j)) or (j) = sin f ' c - f , f c + f , . l + sin((j)) (3.23) These relationships are readily determined from the geometric relationship of the plot of the strength envelope in compression and of the Mohr circle representing the state of stress of uniaxial compression. The gradient of the yield function is in engineering notation: do J c T l s„ y dF 1 1 1 — • « > 3 0 2 a 0 0 2 t Volumetric Deviatoric (3-24) Component Component In the plastic modelling of geologic materials and concrete the plastic flow function, is chosen to be non-associated with the yield function, F. The amount of plastic flow for concrete is non-associated with respect to the volumetric expansion. The form of the yield function gradient can then be stated as follows: 48 d\\i So 7 ay 1 5a„.3 Y ' s * ' 1 sy 1 < 0 1 • + 2 a > 2x 0 0 2 V where: Volumetric Deviatoric Component Component dxv = p - 5 F (3.25) (3.26) The parameter p has the interpretation of a plastic dilatancy factor (Bazant [15]) since it has the effect of increasing or decreasing the sensitivity of the total plastic strain increment to the volumetric component. The importance of plastic dilatancy in concrete w i l l be discussed later. For the case where (3 = 1 the plasticity is associated in which case {dF I da} = {dy I da}. For a real material the factor P has to be determined experimentally but in general a value less than unity wi l l give realistic results. 3.4 Response in the Tension Regime, Method 1 While the yield function of equation 3.20 may be calibrated to provide realistic results in compression, in tension it w i l l give erroneous results. Values of 10 M P a and 30° for the parameters c and (j) respectively wi l l yield fc = 34.6 M P a and ft= 11.55 M P a . A value of compressive stress of 34.6 M P a is not unreasonable, while a high value of 11.55 M P a for tensile strength disagrees with experience. To provide acceptable results in tension the yield surface needs to be described in two parts in the following way; in the compression domain (am < 0) by equation 3.20, while in tension ( am >= 0) by a connecting surface that makes a smooth transition from the function of equation 3.20 down to an apex that is independent of the parameters a and K. The modified yield function is the following equation: 49 F(atJ) = {3aom-K) 1- 5L» 2 -K 1-2 M ^A J F ( a , ) = 3aa„ , + 5 - ^ = 0 30 a + a = 0 for a,„ > 0 for a„, < 0 (3.27) Figure 3.3. Modified yield function in meridional space. A l l o f the equations of the previous section are still valid with the exception of the expression for 8F/dam which takes the form: dF do.,, = 3a 1- -2{3aam-K)^f + 2K ( \ ( \ 2 C\„ o"„, in m — m ^2 dF 3a for a„, > 0 (3.28) for <ym < 0 (3.29) The parameter <JA locates the apex of the yield surface in the tension domain (<jm > 0). Generally the parameter is dependent on the concrete tensile strength,^, and cohesion, c, a procedure for determining this parameter is given in Appendix A . The modified yield function is shown in Figure 3.3 for the following values of constitutive constants: (j) = 30°, c = 10 M P a , CJ4 = 5 M P a . This framework allows for the specification of the response in compression via parameters c and (> and the response in tension via the parameter aA(cft). Alternatively it sometimes may be useful to calculate (j) from c a n d / c by rearranging the first expression in equation 3.22: 50 <|> = sin -Ac1 (3.30) 3.5 Response in the Tension Regime, Method 2 A n alternative method to model the response in tension is to release the tension in the direction of maximum principal stress and thus model the formation of a crack. To incorporate the tensile stress release in the direction of maximum principal tension the initial stress "vector {a}initiai is transformed into the principal stress vector {a}' at every load step: [T(v„v 2 ,v 3)]~ {o}.B.<fa/ = {d} (3.31) The matrix T is given by equation 4.39 and v j , V2 and V3 are the direction vectors (with respect to the global Cartesian base vectors) of the principal axes. Here primes, ('), are used to indicate quantities in the principal directions. Figure 3.4 illustrates the above concepts more clearly. The stress o"j represents the maximum principal stress and CT3 represents the minimum principal stress (sign convention is tension positive). 51 y Global Cartesian coordinates. Figure 3.4. Cracked material element of dimensions dx and dy showing principal and global axes. If G 3 exceeds the concrete tensile strength ft stress is released as follows: 0 { C T } ^ = i a L f a / - [ T ] (3.32) The left-hand side of the above equation indicates a new value for the stress vector after cracking has occurred. The linear elasticity matrix is made anisotropic because a crack plane has formed and tensile stresses cannot be resisted normal to it. Furthermore, it is also proposed that the shear stiffness in the plane normal to the crack plane be reduced [16]. The two phenomena are implemented in the constitutive models as follows: 1. The third row and column of [De] are zeroed out. 2. The strain normal to the crack plane, s „ c r , is calculated from the strains referred to the global coordinate system, {e}, by the expression: *w = I X e , - (3-33) 52 The shear moduli G 1 3 and G23 are reduced linearly from their initial uncracked values by the expression: a,=G?,(l-Be.„) (3.34) G23 = G^-Bsncr) ; (\-Bzncr)>0 The parameter B is conveniently referred to as the rate of shear modulus reduction. The resulting elasticity matrix takes the form: £ ( l - v ) Ev ( l + v ) ( l - 2 v ) ( l + v ) ( l - 2 v ) Ev E(\-v) ( l + v ) ( l - 2 v ) ( l + v ) ( l - 2 v ) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 G„ 0 0 0 0 0 G & O - B O 0 0 0 0 0 0 0 G°23(l-Bsncr) (3.35) The resulting matrix, [De]', which is only valid in the principal stress space must be transformed into the global coordinate system: The subscript 1 on [D e]j is used to indicate that one plane of cracks has occurred. The above process can be continued for the next set of cracks by determining the direction vectors o f the next crack plane, zeroing the third row and column of [D e]j and transforming it into [D e] 2: Two sets of crack planes are supported in the computer program that was written with a minimum angle between crack planes arbitrarily chosen to be 30°. This method of tensile response is applicable to both the perfectly plastic and hardening/softening models. 53 3.6 A Hardening/Softening Drucker-Prager Model The next logical step is to incorporate a hardening/softening rule into the Drucker-Prager model. In equation 3.21, instead of the cohesion and friction angle remaining constant throughout the load history, the evolution law as suggested by Hinton et. al. [16] for the Mohr-Coulomb yield function is adopted: c = cexp sin(|)* = 2 KE 7 K + S sin(j) for K < S ( 7 (3.38) (3.39) sine))* = sin<|) for K > s / The parameters a and K in the Drucker-Prager yield condition now become: <X*(K) = -2sin(() 6c* cos<j)* (3.40) V 3 ( 3 - s i n f ) ' " V3(3-sin<|>*) The parameters c* and <)>* have the interpretation of mobilized cohesion and mobilized friction angle while c and <j) represent their ultimate values. The parameters s c and sy-need to be determined experimentally. The graphs of the two functions in equations 3.38 and 3.39 are shown below in Figures 3.5 and 3.6. Figure 3.5. Variation of cohesion with hardening/softening. 54 0.00000 0.50000 1.00000 1.50000 K / S y 2.00000 Figure 3.6. Variation of friction angle with hardening/softening. The cohesion starts at the maximum value and gradually diminishes as the plastic strains accumulate. The friction on the other hand starts at zero and eventually is fully mobilized. To determine the form of the plastic flow function, we use the same arguments as in section 3.4 in modifying the 8FI d<3M term by the dilatancy factor, (3; that the rate of plastic volume production must be non-associated with the yield function, F, and hence d\\i I dam * dF 15am. However, in this case where hardening and softening taking place a new parameter a'(K) is defined : 2siny*(K) CC (K): (3.41) V 3 ( 3 - s i n y * ( K ) ) The parameter Y*(K) is conveniently referred to as the mobilized dilatancy angle which is in general independent of the mobilized friction angle, (J)*(K). Hinton and Owen [16] define the mobilized dilatancy angle to be a function of the mobilized friction angle as: sin<j)* -sin<j)cv siny (3.42) l-sin(|>*sin(|>C), The value of <|>cv represents the value of the mobilized friction angle <|>* at which a transition from compaction to dilatancy takes place. 55 Response in the Tension Regime If Method 1 is chosen to model the response in the tension regime then the yield function is similar in form to the one for the perfectly plastic model, (equation 3.27), with the exception that the parameters that define it have evolution laws: a becomes O * ( K ), K becomes K*(K) etc. F ( a , ) = ( 3 a * a , „ - / : * ) 1 -o\<0O -K' 1-( v M K ) + a = 0 for a„, >0 (3.43) F(Og) = 3a*a„, + a - K* = 0 for om < 0 (3.44) The gradient of the yield function and plastic potential are still given by equations 3.24 and 3.25 with the exception of dF I dam and d\\i 1d<3M which become: dF (K) = 3a* 1- o„ v M K ) y - 2 ( 3 a a M - ^ ) ^ - + 2^ * a ^(K) ( \ 2 ( \ 1 < ^ ( K ) J , a * ( K ) , dF 5a„ (K) = 3 a for a,„ >0 (3.45) for a„, < 0 (3.46) d\\i (K) = 3a' 1- - 2 ( 3 a ' a m - r ) ^ - + 2 ^ a*5i(K) * 2 / \ 9\(/ & 7 (K) = 3GC' v a ^ ( K ) y for a,„ >0 (3.47) for a,„ < 0 (3.48) Here it is proposed that a*A be a function of the. hardening/softening parameter, K . The evolution law that is chosen for aA is the same one as for the cohesion: (3.49) The above formulation wi l l ensure that the apex of the yield surface moves toward the origin of the a and o~„, space as the cohesion vanishes. The non-associativeness is 56 achieved by replacing a* in the expression for dF I dam with a ' to obtain the expression for dy I dam. The hardening/softening parameter, K , in the context of the present model represents the amount of damage in the material. A definition for it is somewhat of an open question, but basing it on the second invariant of the deviator of plastic strains seems appropriate for geologic materials and concrete: dKocde" (3.50) The parameter dzp, as was stated in section 3.2, is a measure of plastic distortions. The amount of plastic hardening or softening must be sensitive to the confining pressure. A s the hydrostatic pressure increases the rate of damage in the material slows hence K must be inversely proportional to the hydrostatic pressure. Also as the hydrostatic pressure becomes tensile near the apex of the yield surface, the rate of damage must increase at an ever increasing rate. The approach adopted here is similar to that used by Jiang and M i r z a [8]. The damage in the material must be a function of plastic distortions and the stress level. The functional form that is chosen is as shown below: dK = (3.51) r(.om,<yA,fc,A) The form of the function f that w i l l give the appropriate behaviour is: r ( a , „ , a ^ , / c , ^ ) = F A O V fc J f - \ 1-for cr „ < 0 for G,„ > 0 (3.52) V A * ^ ( K ) . Where A is a constant and fc is the specified concrete strength in uniaxial compression. The negative sign is placed in front of am because the concrete specified strength is a positive number and the hydrostatic pressure is negative i f compressive. From the definition of the plastic distortion of equation 3.19 and from the second term in equation 3.25, then: 57 de" =dk 1 dev dev (3.53) The symbol, dev, means "the deviatoric component o f . The product under the square root sign is shown to be 1 / 4l in Appendix A . The expression for dx. therefore takes the form: Or alternatively: dK = dK dk 1 1 (3.54) (3.55) dk r(am,aA,fe,A)J2 If Method 2 is chosen to represent the response in tension then the cases of functions in the compression domain (am < 0) would be used for both am < 0 and am > 0 with the exception that tensile stresses in the direction of maximum principal stress would be released at the concrete tensile stress in accordance with Method 2. 58 Hardening/Softening Modulus The hardening/softening modulus H was previously defined as H = -(dF I 9K)(5K / dk). If the expression for H is expanded by the chain rule of differentiation and substituting the expression for dK /dk then: H- 1 r(°m,GA,fc,A)j2 dF da d<j>* + dF_^dKdc_+ dK d f ^ da <9<j) DK dK dc 8K 5<j) 8K The partial derivatives in the square brackets are readily determined as: 8F da SF dK da a f d f dK dK dc d^ dK dK 3a„ -1 2cos(j)* • + 2sin(|)*cos(|)* V 3 ( 3 - s i n f ) " V3(3-sin(|>*) 2 4 - 2 -KS, A/KS7(K + S / ) (K + B / ) sin(() COS(j) for K < s f = 0 for K > s 6cos(j)* V3(3-sinc|)*) = -2c exp ' K ' -6c*sin<j)* 6c* cos 2^* V3 (3 - sin f ) + V3 (3 - sin (j>* ) 2 It is the author's opinion that the expression for mobilized friction in equation 3.39 is unsuitable because the derivative introduces a singularity in the expression for H. Consequently for the case where the material is not yet yielded (no accumulated plastic strains, K = 0), the hardening/softening parameter holds an infinite value and plastic deformations could never begin to take place: (3.56) (3.57) (3.58) (3.59) (3.60) (3.61) (3.62) (3.63) 59 l im d.K = dk "~ r(am,aA,fe,A)J2 r(am,GA,fc,A)>f2 $} M{f + H = 0 (3.64) A s an alternative to equation 3.39 the Author proposes the following expression which introduces no singularities in H.\ sinij) = ( \ 2 o K K K 3 3 + £ / I 8 / J sin<)) for K < ef for K > 8 , (3.65) sin<t>* = sin<() ^ ^ ^ 0 / The bracketed term [] is shown below in Figure 3.7 for visual comparison to Figure 3.6. K / E / Figure 3.7. Proposed function of mobilized friction as a function of K. A n d the derivative d$*/dK is: r - "i sin(j) 3 K K 6 — T + 3 — r COS(() 7 for K < 8 , (3.66) = 0 for K > E , 8K ^ A s a case in point the hardening/softening modulus H is plotted in figure 3.8 for the following values of parameters that it is dependent on. 60 Table 3.1. Table of parameters defining the Plastic Modulus H. -10 M P a Zc 0.0005 *f 0.0015 C 6 M P a + 23° Figure 3.8. Variation of H as a function of K for the values of parameters shown in the above table. 3.7 Plastic Dilatancy and Passive Confinement Dilatancy in concrete comes from the sliding of crack surfaces past each other [15]. The phenomenon can be illustrated with the aid of the following figure: Figure 3.9. Plastic dilatancy as a crack sliding phenomenon. 61 Such behaviour is not observed in steel and hence the steel constitutive model allows for plastic shear deformation under constant volume as w i l l be shown in a later section. Passive confinement of concrete by steel is initiated by the concrete plastic volume expansion which is associated with plastic deformation. Therefore, when modelling the confinement effect of steel on concrete in the plastic range one must try to reproduce the plastic dilatant behaviour of concrete as accurately as possible. The dilatant behaviour offered by the two Drucker-Prager concrete constitutive models is examined here. From the first term in 3.25 and the definition in equation 3.18 the volumetric plastic strain increment can be derived, dzp = dk(d\\i 13cm) / 3, while the plastic distortion increment was already implied to be dzp = ^ dejjdej} = dX IV2. The ratio of the two is the rate of plastic expansion to plastic distortion. For the case where the Drucker-Prager model is perfectly plastic the ratio is: d^_ = < = V 2 j f y = V 2 p _ a ^ _ ( 3 6 7 ) dzp dK 3 d<Jm 3 dam A n d for the case where the hardening/softening rule was introduced: < = < = ^ ( K ) (3.68) dzp dK 3 dam A s a comparison of the above two equations the quantities and K are plotted below in Figure 3.10 for the case of ^ = 45°, zf= 0.01, §cv = 20°, p = 0.5, um = -3 M P a . 62 -0.0005 |> -0.001 hardening/softening 0.002 0.004 0.006 0.008 0.01 Figure 3.10. Plastic volumetric expansion vs. plastic distortion. From the above figure it can be seen that for the case of perfect plasticity (d\\) I dam = const) the plastic volumetric strain accumulates linearly at a slope of V2f3a. For the case where hardening and softening take place (d\\i I dam- SIJ/(K) / 9 a m ) the plastic volumetric strains are initially negative (the material contracts) and then a smooth transition occurs to dilatancy later in the load history. 63 3.8 Summary of the parameters in the Drucker-Prager models The parameters that comprise the Drucker-Prager models are listed below in tabular format. Table 3.2. Parameters in the Drucker-Prager model. Parameter Perfectly Plastic Model Hardening/ Softening Model Description Yes Yes Peak value of Friction Angle c Yes Yes Peak value of Cohesion P Yes N o Dilatancy Factor <|>*(K,SF) N o Yes Mobi l ized Friction Angle C*(K,S C ) N o Yes Mobi l ized Cohesion N o Yes Mobi l ized Dilatancy Angle K N o Yes Plastic Damage Parameter e f N o Yes Parameter in equation 3.39 e c N o Yes Parameter in equation 3.38 <t>cv N o Yes Parameter in equation 3.42 A N o Yes Parameter in equation 3.52 B Yes Yes Rate of shear modulus reduction ( used in Method 2 only) 64 3.9 The Von-Mises Steel Constitutive Model The steel constitutive model is based on the Von-Mises yield criterion which is briefly stated as: F ( a / y ) = V 3 o - r „ ( K ) = 0 (3.69) a.3 A a l Figure 3.11. Von-Mises yield surface in principal stress space. The parameter Yu is the yield stress in uniaxial tension which is in general dependent on the hardening/softening parameter H. The parameter H can be interpreted from an uniaxial stress test as the ratio of the change in yield stress to the change in plastic strain increment [10]: H = ^ - (3.70) 65 In Figure 3.12 the parameter His constant and is given as: H = ^ (3.71) \-E,IE where Et is the post yield tangent modulus and E is the elastic modulus. For the case where the steel is perfectly plastic H = 0. Since we are dealing with an associated plasticity model the gradient of the yield function and the gradient of the plastic flow function are the same: 5a dF_ 5a V3 2 a s. 2x 2x 2x xy (3.72) Deviatoric Component It should be remarked that unlike the Drucker-Prager model the steel model is hydrostatic pressure insensitive since equation 3.69 does not contain a term involving the first 66 invariant of stress. The parameter K in the Von-Mises model has the interpretation of the amount of plastic work: dK = {a}T {dsp} = {of dlf^pi (3.73) The evolution of the yield surface can be derived from Figure 3.12 as follows: If Yu represents the present uniaxial yield stress then the amount of plastic work, dK, is: dK = Yudsp (3.74) The change in the uniaxial yield stress is from equations 3.70 and 3.74: dtc dYu = Hdepu = Hy (3.75) u When one substitutes the general expression for the plastic work increment in equation 3.73 the change in the uniaxial yield stress becomes: Which is integrated to give: r^+S^+lfy^JX (3.77) A l l of the variables in the integrand are functions of X. The magnitude of the plastic strain increment, dX, is defined in equation 3.5 in terms of the total strain increment, and thus the evolution law for the Von-Mises yield surface is established. The steel plate confining concrete joints acts as a membrane that has negligible stresses on the surfaces that have the z axis as their normal. The concept is illustrated in Figure 3.13 where the local coordinate frame, x',y',z' is shown at a point in relation to the global frame x,y,z. The stresses at a point in the membrane have only three components; dox', day* and dxxy\ 67 membrane {do., 1 day, dxxy (3.78) 0 0 Figure 3.13. Illustration of local and global coordinate system in the steel plate. The strains, however, are dependent on the global deformation of the membrane and are generally stated as: membrane del, dz", deey, dzpy, dz\, dzp, dyc,y dypy dyl; dypx,. dypyz. (3.79) 68 The three-dimensional elastic constitutive matrix that properly relates the membrane stresses and the elastic component of strains is an expanded version of the plane stress constitutive matrix which is given as: [D-] L J met \ membrane 1 - v 2 Ev 1 - v 2 E 1 - v 2 0 0 0 symmetric 0 0 0 G 0 0 0 0 0 0 0 0 0 0 0 (3.80) Wi th the exception of the above stated differences, all of the concepts developed in the preceding sections are equally applicable to the membrane constitutive model. 69 C H A P T E R 4 T H E FINITE E L E M E N T SOLUTION 4.1 Overview The previous chapter dealt with concrete and steel constitutive relationships at an infinitesimal point and within infinitesimal increments of strain. This chapter presents, in the context of finite element analysis, a numerical procedure for the solution of problems o f finite size and deformation. The finite element numerical procedure may be thought of as a bridge between the theoretical material representation of Chapter 3 and the physical world. In the solution of solid mechanics problems, one typically uses solid finite elements that have eight or more boundary nodes. In this thesis an element consisting of twenty nodes is used to model the concrete. The steel plate element is modeled as a membrane surface with eight nodes. Eight nodes are required to ensure compatibility of deformations when the membrane element is attached to the concrete element, which has eight nodes on any given side. Geometric generality is maintained by using the isoparametric formulation, [18], for both the concrete element and steel membrane. Displacements at the boundary nodes are interpolated through each of the element surfaces and volumes by compatible shape functions. The strain increments, which are linear functions of the gradients in the displacement increments, are calculated at a number of finite points in the finite element called the Gauss Points. The Gauss Points are points within the element where quantities such as stress or elements of the compatibility matrix, [B], are sampled for integration by 70 the Gauss Quadrature integration rule, [19]. The Gauss Quadrature integration rule assumes a priori that the sampled quantities are the values of a truncated Taylor series of a predetermined order. The integration rule that is chosen here w i l l exactly integrate quantities inside elements that have straight sides for the associated shape functions. This is not true for elements that have curved sides. The integration w i l l be approximate, but w i l l in general converge as the number of elements is increased. The stress-strain relationships presented in Chapter 3 are used to calculate the corresponding stress increments from the strain increment at each Gauss Point. The integration of the stress increment is accomplished by subdividing the strain increment into smaller sub increments for the purpose of obtaining more accurate stress increments. The stress at a Gauss point at the end of a number of cumulative load increments is a function of the strain history at that point. In a computer solution this has the implication that the material constitutive parameters must be stored in memory for each Gauss point and each element. A s a result, it was found that the memory requirements can be quite demanding for models consisting of more than twenty concrete elements. In the solution of non-linear constitutive problems it is often necessary to find a solution to the finite element nodal displacements by a series of iterations. The iteration procedure utilized here is the modified Newton-Raphson method [10]. In the adoption of this method, a set of known displacements are applied at the controlled nodes while the free nodes are assigned initial zero values of displacement. A n unbalanced load is calculated (which represents the non zero value of virtual work) by integrating the stress increments at the Gauss Points resulting from the associated displacement increments that are applied at the controlled nodes. From the unbalanced load, a corrective set of displacements for the free nodes is calculated and consecutively added to the previous 71 values. Hence the unbalanced load is used to "steer" the procedure toward a solution. The steps are repeated for all load increments. 4.2 The Weak Equilibrium Equations To begin the finite element numerical procedure the weak form of the equilibrium equations for a general body (or continuum) modeled by finite elements is required. Consider such a body as shown in Figure 4.1 to which an incremental displacement is applied at one or more of the nodes. Controlled displacements are chosen as the independent variables instead of load increments to avoid instability problems during the stages when the load-deflection curve has a very low or even negative gradient. The same loading procedure is often also applied in the laboratory to avoid sudden collapse of a specimen. Figure 4.1. General body modeled by finite elements. If the incremental displacements in the continuum are denoted by Au,Av and Aw in the x, y and z directions respectively, then the continuum displacements are discretized in terms 72 of the element nodal displacements Aaxi, Aayi and Aazi via a suitably chosen set of compatible shape functions Ffxy£): Au^Aa^x^z) Av = YJ^yFiv(x,y,z) (4.1a,b,c) i Aw - ^Aa^F^ (x,y,z) ; i = 1...number of nodes The functions F^xj^z) are at this point general. Shape functions specific to different elements are described in Appendix B . The strains and stresses in the continuum are defined as: {As} = [B]{Aa} (4.2) {E+AS} {AG} = J[D*(s)]{&} (4.3) {e} It should be noted that the [B] matrix contains only derivatives of the shape functions with respect to the coordinate variables x, y and z. The nodal degrees of freedom can be conveniently split into three components, namely those degrees of freedom that are free to move, those that have prescribed boundary conditions and those that have controlled displacements applied to them ({$)free, {^boundary {^control)- The nodal forces can also be split up into their corresponding components: free, boundary and control. If one applies a displacement increment in the set of displacement control degrees of freedom { A a } c o n / r o / a set of equilibrium stresses, {Aa(Aa c o n ( T O / )} , is produced in the continuum and a set of contact loads, {AP(Aa c o n , r o / ) } , at the displacement controlled degrees of freedom and support points. The weak form of the equilibrium equations is formulated by the method of virtual work using variations in displacements and strains that are compatible with the kinematic boundary conditions. Since the total virtual work, W + AW, is zero, therefore: 73 I {a + Aa}T{e}dV - {P + AP}r{a} =W+AW=0 V (4.4 a,b) j" {a + Aaf[B]{a}jF - JP + APf{a} = W + AW = 0 v where virtual quantities are denoted by a tilde (~). Since, in general, the body of Figure 4.1 has been loaded prior to the application of the displacement increment, {Aa} c o n W ,the quantities {o-(acojMTO/)} and {p(acon,ro/)} are the internal stresses and external loads prior to the application of the displacement increment which are, in general, functions of the total displacement, {%\controb applied at the displacement controlled points. The quantities W and AW are the virtual work associated with the state prior to the application of the displacement increment and the increment in virtual work associated with the displacement increment. The state prior and after the application of the load increment are equilibrium states, therefore equation (4.4b) can be split up into two equations, each of which is theoretically zero: J { ° } r [ B ] ( a } ^ - { P } r { a } = ^ = 0 (4.5) V J {Aa}r[B]{a} dV - { A P f {a} = AW = 0 (4.6) V It is stated here that the above two equations are each theoretically zero because, as w i l l be seen later in this chapter, in numerical applications they are only approximately zero. In the Newton Raphson iteration procedure the above equations w i l l contain a residual that must be carried into successive load steps. Equations 4.5 and 4.6 can be further simplified i f one considers the fact that the only externally applied forces are at the displacement controlled degrees of freedom. A t the nodes corresponding to boundary conditions, the contact forces do not contribute to the virtual work since those degrees of freedom cannot be varied. The displacement controlled degrees of freedom are prescribed for the load increment (analogous to kinematic boundary conditions) and 74 cannot be varied either, thus the second terms in equations 4.5 and 4.6, which represent the virtual work of external loads, are zero: L J free {P}r{5} = ' control ^ ^ ^ boundary f ^ (4.7) And : {AP} 7 '{a} = { A P U = {O} {AP}, ' control to, = 0 (4.8) The weak form of the equilibrium equations are therefore: l{<y}T[B]{a}dV = W = 0 v \{A<j}T[B]{a}dV = AW=0 (4.9) (4.10) N o w , since it is argued that the variation in the nodal displacements, {a} is arbitrary, the following terms in equations 4.9 and 4.10 must be identically equal to the zero vector: j{oY[B]dV = {0}r (4.11) j{AG}T[B]dV = {0}7 (4.12) Again it is stressed that in a numerical application the right hand side of equations 4.11 and 4.12 w i l l only be approximately zero. 4.3 The Numerical Procedure The finite element solution of problems dealing with non-linear constitutive relationships must be carried out in a stepwise iterative fashion. The displacements must be stepped so as to permit the updating of material constitutive properties at every step, since, in general, they are load history dependent. The increments in the displacements 75 must be iterated by the modified Newton-Raphson method [10] at every load step to satisfy zero total virtual work, as shown in equations 4.9 and 4.10. To begin the numerical loading simulation, taking as an example the structure of Figure 4.1, a set of displacements, {Ad} , is applied at the nodes corresponding to points of load application, {Aa} . The increment in the entire displacement vector may then be represented mathematically as follows {Aa} = {Aa}, {Aa} boundary control L J free ^ ^boundary {Ad}, * control N o w there is a corresponding set of displacements, { A d } / r e e , of the free nodes associated with the { A d } c o n W displacements that w i l l produce a set of stresses, { A a } , which make equation 4.12 equal to zero. The procedure for finding the displacements of the free nodes upon the application of displacements to the controlled nodes is the next topic of discussion. The displacement increment, { A d } c o n m j / , produces a set of strains in the body and a resulting set of stresses via equations 4.2 and 4.3. The integral of equation 4.3 is evaluated as a simple sum: {e+As} { A a } = J[D*(e)]{dfe}«-i:2;[D*(e + //JVAE)]{As} (4.13) /=i where N is the number of subincrements. Having obtained {Aa} from the above equation, the integral in equation 4.12 is evaluated and is found not to equal the zero vector: J{Aa} r[B]</K = {AO}%{0} r (4.14) v Integrals such as the one above as well as all volume integrals are evaluated by the Gauss quadrature integration rule. A discussion on the Gauss quadrature integration rule w i l l follow after discussing how the {Ad} displacements are obtained. The above integral 76 is not equal to the zero vector because, before the {Ad} / r e e values are known, equilibrium is not satisfied, which is represented by the unbalanced force vector, {AO}. The method of obtaining the {Ad} / r e e increment is the modified Newton-Raphson iteration procedure [10]. A corrective increment, {5a}]5 is computed in the nodal displacements and added to {Aa} of the load step: {5a}1 = -[K'] o{A0} (4.15) {Aa}, = {Aa} + {5a}1 (4.16) The subscript, 1, indicates that this is the first iteration of a series of corrections. The tangent stiffness matrix, [k']q, corresponds to the boundary conditions of the problem such that: (4.17) Next the new strains, stresses and unbalanced load associated with {Aa}, are computed from equations 4.2, 4.13 and 4.14: {As},=[B]{Aa}, (4.18) {e+M N {Aa},= J [ D * ( e ) ] { & } « - 2 [ D v ( e + //^AE1)]{Ae}1 (4.19) {e} / _ 1 J{Aa};'[B]^ = {A(D};' (4.20) f 1 L J free ' ( 5 D U " {5a}, =• {^*^ ^ boundary • = • ^ ^-^^^ control 1 Having obtained the unbalanced load from iteration 1, the next corrective increment, {8a}2, can be computed and added to the total: { S a ^ - f K ' J j A O } , (4.21) {Aa}2={Aa} + {5a} ]+{5a}2 (4.22) The procedure can be continued indefinitely with each successive {Aa}M being a better approximation than {Aa} which wi l l be indicated in the Euclidean norm: 77 VL " J " <p - y J " - ' 1 J »- ' (4.23) In a converging process, the successive corrections and unbalanced loads get smaller, but w i l l never be zero. It should be noted that: { ^ ^ { S d J ^ . (4.24) The modified Newton-Raphson method uses the initial stiffness matrix throughout the iterations, (Figure 4.2) instead of the updated one at every iteration point. This procedure takes longer to converge but is convenient from the point of view that the updated method would involve recalculating the stiffness matrix more frequently. The updated tangent stiffness matrix is based on the elasto-plastic constitutive matrix, [DeP]. This matrix is non-symmetric for the case of non-associated plasticity and hence the updated tangent stiffness matrix is non-symmetric as well : [ K ' ] = J [ B f [ D e p ] [ B ] r f r (4.25) v The initial stiffness matrix, however, is based on the elastic constitutive matrix, which is symmetric, and is given as: [ K l 0 = j [ B ] > e ] [ B K (4.26) v One can visualize the modified Newton-Raphson procedure with the aid of Figure 4.2 for the case with a single free degree of freedom , Aafr e e , and one displacement controlled degree of freedom A a c o n t r o j . In Figure 4.2 the unbalanced load is on the vertical axis, which is a function of both the controlled displacement and free degree of freedom, A( | ) (Aa c o n t r o i ,Aa f r e e ) . For a load step A a c o n t r o ] is fixed, while the corresponding displacement in the free variable, Aafr e e , that makes A(|>=0, is found by a series of iterations. Each increment in the displacement variable 8a is added to the cumulative total for the load step and eventually the result should converge. 78 A a i J + 1 = Aa-ui + baiA 6 a i , j = - ^ A ( t ) i j (4.27) From Figure 4.2 it can be visualized that this method may fail when a displacement increment is chosen that straddles the peak load point. This iteration procedure w i l l fail to converge as it crosses over to the unloading part of the load-deflection curve. For this case more sphisticated iteration procedures have to be used. 79 Gaussian Integration In general the stress increment {Aa(x,y,z)} of equation 4.13 is a continuous function of position inside the elements and continuum. To apply the Gauss quadrature integration rule, the stress increment is determined at a set of prescribed points, (x0j,z^), in the elements, called the Gauss points: {C+AE} {B} ^ (4.28) where TV is the number of sub increments used in computing the stress increment from the associated strain increment. The indices (ij,k) in parenthesis are not tensorial indices as used in Chapter 3, refer to the Gauss point numbers in each element. Simply put, the quantity {ACT(/7.A)} is the engineering stress increment evaluated at the Gauss points which have coordinates (x^z^). Having obtained { A a ^ ^ J from the above equation, the integral in equation 4.14 is evaluated by the Gauss quadrature rule in each element sub volume and their contributions added to the total. The Gauss quarature rule is normally applied to a cubic element of dimensions 2,2,2 with the dimensionless element coordinate axes Z,,r\,C, at the center of the element. Subsequently in applications to an element of general dimensions in x, y and z space the Jacobian [J] matrix needs to be determined that transforms a volume element in £,,r|,C, to x, y and z space: dx dy dz [J] = d\ 8^ 8^ dx dy dz dr\ dr\ dr\ dx dy dz ~d~i a ; dc; (4.29) Thus the volume differential dV\s: dV = det([ j])d^dC,dr\ (4.30) 80 A n d 4.14 becomes: i i i | {Aa} r [B]JF = JJJ{Aa} r [B]det([ j ] )^^T 1 (4.31) V -1-1-1 Thus applying a three point Gauss quadrature rule the above is evaluated as: j ] j { A a } r [ B ] d e t ( [ j ] ) « ^ = Z Z Z^^{A C T(u.*)}[B('J.*)] d e t( J0.y.*)) = * W (4.32) The coefficients Wj are weight coefficients and are given for the three point integration rule in Appendix C. For a complete discussion on isoparametric mapping, shape functions and Gaussian integration the reader is referred to [18] and [19]. From here forward, for clarity, integrals as in equation 4.14 shall be shown as: but are understood to be evaluated by the Gauss quadrature integration rule without an explicit expression as in equation 4.32. 4.4 The Concrete Finite Element The finite element used to model the concrete is a prism of twenty nodes, as described in reference 18, with three degrees of freedom per node for a total of sixty degrees of freedom. The representation of the element is more conveniently expressed in dimensionless coordinates ^, n and C, as shown in Figure 4.3. The cube shown in dimensionless coordinates has dimensions of 2, 2 and 2 with center at (0,0,0). Although an element of eight nodes would have been sufficient for the model, the choice of an element of twenty nodes was influenced by two reasons: 1. The twenty noded element has higher order shape functions associated with it than an eight noded type, consequently a model based on the twenty noded elements converges faster. V 81 2. Having three nodes on every side allows the element to be warped and curved in the Cartesian x, y and z space, hence allowing more accurate modelling of circular specimens with relatively few elements. 5 Figure 4.3. Concrete finite element. The shape functions of the element in Figure 4.3 are derived in accordance with the methods given in [18] and are shown in Appendix D with the [B] matrix . The Cartesian coordinates in the element are related to the dimensionless coordinates by the isoparametric mapping: i and the displacements in the element are given by. 82 i v = Y,ayiNi(^0 (4-34) w = Nt(^,r\,Q ; i = 0... number o f nodes i 4.5 The Steel Plate Finite Element The steel plate finite element is a membrane that has no bending stiffness or shear stiffness out of plane because stresses and strains are defined only in the middle surface. Since the laboratory specimens were both circular and square, the element that is being used in the computer program must be a membrane of general shape for flexibility. Later the results are applied to a flat membrane, since only the square specimen was chosen to be modelled because of it's simpler geometry. The results that follow are a variation on a general shell element described in reference [20], with eight nodes and three degrees of freedom per node for a total of twenty-four degrees of freedom. It is necessary to use a membrane with eight nodes for compatibility with the concrete finite element that has eight nodes on any particular side of the cube. The properties of the concrete element presented in the previous section can be applied directly in the context of section 4.2, whereas the plate finite element requires some special details. Figure 4.4 shows the membrane shell, used to model the steel casing, with the curvilinear coordinates n and C, shown in relation to the global Cartesian coordinates x, y and z. A t any point on the middle surface of the shell a local Cartesian frame (x',y',z') can be defined that has base vectors S j , e 2 and e 3 associated with it. The base vectors of the global Cartesian frame are referred to as I , J and K . It is important to establish the relationships between the local basis vectors in the membrane, that are a function of position in the membrane coordinates (^,r\,Q, and the global basis 83 for the purpose of defining the transformation laws of stresses and strains between the coordinate systems (x',y',z') and (x,y,z). U p p e r Surfac Lower Surface M i d d l e Surface ^=0 Global Cartesian Coordinates Figure 4.4. Plate Steel Finite Element. In this case the stiffness matrix for the element is not given by equation (4.26) but by the expression: [ K ' 1 = [ [ B ] 7 ' [ T 1 > " 1 [ T j B ] 7 ' dV (4.35) L Jo J L •'membrane!- sj L Imembrane'- -"- •'membrane x ' V The transformation matrix [T E ] is a function of the direction cosines between the base vectors in the local frame x', y', z' and the global frame x, y, z. The details of obtaining the ^>\membrane matrix can be found in Appendix E . The expression [ T J ^ [ D e ] m e w £ r a 7 i e [ T E ] has the interpretation of transforming the constitutive matrix from 84 the local coordinate system to the global one. The elastic constitutive matrix, Vbe]membrane > *s m e anisotropic expanded version of the plane stress constitutive matrix given in Chapter 3. The strains or stresses in the global Cartesian frame (x,y,z) can be transformed to and from the membrane local frame (x',y',z') by: {e'} = [TE]{e} (4.36) {o} = [T.]V} (4-37) {&} = [TjT{<j} (4.38) where the symbol [ ] r means the inverse of the transpose and the primes are used to indicate quantities referred to (x',y,'z') local Cartesian coordinates. l\ m\ n\ lxmx mxnx nxlx ll m\ n\ l2m2 m2n2 n2l2 l\ m\ n\ l3m3 m3n3 n3l3 2lxl2 2mlm2 2nxn2 hmi + hm\ mxn2 + m2nx nxl2 + n2lx 2/ 2 / 3 2m2m3 2n2n3 l2m3 + l3m2 m2n3 + m3n2 n2l3 + n3l2 2l3lx 2m3mx 2n3nx l3mx + lxm3 m3nx + mxn3 n3lx+nxl3 (4.39) where lv mx and n\ are the components relative to the global Cartesian frame of the z'th base vector e;-. Many of the above concepts may be found in reference [20]. The usage of the element in the context of section 4.2 is as follows: • From the nodal displacement of the current load step and iteration, the global strains are computed at the Gauss points and the transformation of equation 4.36 is applied to obtain the strains in the membrane local coordinates: {A8'} = [T e ] [BL m A r a n e {Aa} (4.40) • Next, the local membrane stress increments at the Gauss points are obtained from equation 4.13: 85 {E'+AE'} N { A a ' j = f [»Zmhrane(£')]{ds'} * ± - ^ [ D Z m h r + / / JVAs')]{As'} (4.41) M . ^ 1 = 1 • The element contribution to the unbalanced load is computed via equations 4.14 and 4.37 { A O } r = J [ [ T j r { A a ' } ] r [ B L m A _ ^ = \{^'}\\}[B]memhrandV (4.42) ^membrane ^membrane 4.6 The Reinforcing Steel Element The reinforcing steel is modeled by a bar element of three nodes that has only one stress and strain component along its axis. In keeping with the previous discussion of local and global coordinate systems, the element is defined to have its own local coordinate system (x',y',z') of arbitrary orientation to the global one (x,y,z). Figure 4.5 illustrates the point. I Local I Figure 4.5. Reinforcing steel element and degrees of freedom. To maintain displacement compatibility with the concrete elements the reinforcing steel element must have three nodes (ie. quadratic shape functions). The only local coordinate of importance is x' since the element has only one stress and strain defined along its' axis. The geometry of the element is defined by isoparametric mapping, with only one 86 dimensionless coordinate, as shown in Figure 4.5. The shape functions for the reinforcing element shall be referred to as LfQ and the coordinates of the nodes as XJ, y\ and zf. i y = Y,yM® (4-43) i The nodal base vector corresponding to the x' local coordinate is easily derived as: u = I(x 3 - x , ) + J(y 3 - v,) + K ( z 3 - z , ) _u (4.44 a,b) lul The axial strain in the local coordinate system is given in terms of the global strains as: s r t . = k 1 } 7 ' { e } = {e} r { , i ; i } (4.45) {°} = K h ^ (4-46) where the term { T E l } r is the top row of matrix [ T 8 ] . The initial tangent stiffness matrix for the element and unbalanced load vector contribution are: [ K ' ] o = AE' j [ B f { T e l } { T £ l f [B]dl (4.47) 0 { A O f = ^ j ^ { A a } r [ B f { T E l } { T E l } r [ B > / / (4.48) 0 The matrix [B] for the element is given as: 87 (4.49) dx Z X ' A 5 ( 4 5 0 ) f - y V (4'51) f - y V (4-52) The term £ ^ is the uniaxial tangent modulus (either E when the material is elastic or Et when plastic) and Ef is the initial tangent slope ,E. The term Ef? is inside the integral since it is a material property dependent on position inside the element (not all Gauss points may be yielded). 88 The Solution Algorithm The solution procedure may be summarized as follows in nine steps. The bold subscripts i and j refer to the load step and iteration number respectively. The upper case delta, A, represents the load step increment and the lower case delta, 8, represents the corrective increment corresponding to the iteration. 1. Apply displacement increment for the present load step, i, in {Aa}. ^"control Iteration Loop To Iterate Out the Unbalanced Load 2. Calculate the strain increments at the Gauss points in all of the elements from: 3. Compute the stress increment at all of the Gauss points from equation (4.3): {Ao}u=2;[D*(e)l{AE} (4.54) 4. Determine the load unbalance {AO}, for the load step from (4.9) using Gaussian integration. { A O } . . r = {AO},., + |{Aa} ; / [B]dV * {o}T (4.55 a,b) v The term { A O } . , is the residual unbalanced load left over from the previous load step. Adding {AO}. l to the unbalanced load of the current load step w i l l minimize the total unbalanced load, { o } , at the end of the last load step (equation 4.11 w i l l be minimized). The term { A O } . . is interpreted as the unbalanced force vector for load step i, iteration j . 5. While the load imbalance is not zero determine the correction increment to the nodal displacements {Aa}. from the initial tangent stiffness matrix and unbalanced load for the current iteration: 89 I H , -{5a}.. L J '.Jfree (»•}„ ., = M l»J control (4.56 a,b) We note that the only degrees of freedom that have corrections applied to them are those that are free. 6. Go back to step 2 with the updated values of nodal displacements and continue until the corrections in the displacements become sufficiently small. This is done by calculating the Euclidean norm and comparing it to a prescribed tolerance: < specified tolerance (4.57) End of Iteration Loop 7. A d d increment in nodal displacements to the total displacement from previous load s t e p j a W a j ^ + l A a } . ^ . . 8. A t all the Gauss points of all finite elements compute the increment in the hardening parameter AK; and add to the previous value. AK : = H AI H 8F da [D<] 8K 3K d\\i 5a + H [B]{Aa} i,last j (4.58) 9. Update constitutive parameters in the yield criteria and plastic flow rules that depend on K , calculate [j)ep\+ and go back to step 1 to apply next load step. 90 CHAPTER 5 ANALYSES WITH THE PROGRAM Overview The algorithms that were presented in the previous chapter are here brought together into a working program called A P O S E C . A P O S E C is an acronym for Analysis Program O f Steel Encased Concrete. It was intended to serve as a research tool and as such is relatively primitive in its design, as compared to commercial analysis programs. Several finite element models were used to test the program. The tests performed on the models include linear elastic analysis patch tests on element pairs to test the integrity of the shape functions and possibly to detect computer coding errors. The elements used in the computer program did pass the patch tests and detailed results are not presented. The more interesting tests were the plastic analysis uniaxial compression and tension tests that w i l l be discussed in detail. Finally, three models of a joint, the largest one consisting of forty concrete and sixty steel plate elements, were used in an attempt to replicate the load deflection curve that was obtained in the laboratory for one of the square specimens. The results of the analysis are inconclusive at this stage but the computer program does open the door for more research. 5.1 The Program APOSEC The program A P O S E C reads as input a data file that describes the geometry and element types used in the model. In a separate file the displacement data is contained, which could be either a number of individual displacements or a total displacement that is 91 divided into a prescribed number of equal steps. The output from A P O S E C contains the contact forces at the point of application of the displacements. More specifically, the contact forces are the non-zero elements of the unbalanced load vector for the load step: {AO}, = {0} ^-^^ -J boundary conditions {Ac()}( (5.1) J displacement control The non-zero components correspond to boundary conditions and points where displacement control is applied. In our case the contact forces at the point of displacement application are extracted and the cumulative total written to an output file along with the total displacements of the nodes: The output file is a text file that can be imported into a spreadsheet for plotting and other post-processing. In addition, a stress file, consisting of the stresses at various select elements is output at the Gauss points. 5.2 Analyses with the Perfectly Plastic Model One Element Under Compression The intent of the first test of the Drucker-Prager model was to obtain a qualitative assessment of the behavior of the elements under cycled loads. Although cyclic displacement simulations were not conducted on the joint models themselves, it is always possible in a complex model to have local loading and unloading of Gauss points even though the global displacement loading was monotonic. Hence an observation of the loading and unloading behaviour of the constitutive model is prudent. In Figure 5.1 is shown a single concrete element, based on the perfectly plastic Drucker-Prager constitutive model of section 3.4 under a cycled displacement. Displacement increments were applied to the top nodes, the bottom nodes were restrained vertically but both top 92 and bottom nodes were free to expand in the horizontal plane. The type of loading used is consistent with that of infinitely stiff and frictionless plates. Two sub increments per displacement step were used to allow for the more frequent updating of material constitutive parameters for improved accuracy of the response. The number of sub increments in this case was two. CYCLIC LOAD Figure 5.1. Single 230mmX230mmX230mm Concrete Element Under Cycled Displacement. The response of the model is depicted in Figure 5.2. A t first the element experiences uniaxial compression under the concrete tangent modulus of 30000 M P a , then the material flows plastically at a stress of 30 M P a . In the elongation cycle it unloads elastically and is brought into tension. In the context of plasticity, tensile failure is depicted as yielding with.no stress release as would occur across a crack surface. The tensile strength of the concrete here is 1.4 M P a and the P factor is 0.5. 93 (MPa) Figure 5.2. Average vertical stress vs. strain for cycled response of a concrete element. The stress-strain cycle in Figure 5.2 exhibits a mi ld slope of the yield plateau i n compression. This can be explained by the fact that the concrete constitutive model is hydrostatic pressure sensitive. When the Gauss points make the transition from the elastic to the plastic regime, a slight error is picked up in the stresses that, i f not corrected in the Modified Newton Raphson procedure, w i l l give rise to errors in the stress components and hence to the post yield branch which is sensitive to the hydrostatic pressure component. This phenomenon w i l l be examined in detail in the next section. Drucker-Prager Iteration Sensitivity The concrete element of Figure 5.1 was next subjected to a series of monotonic loadings to determine the accuracy of the post yield stress prediction of the perfectly plastic Drucker-Prager constitutive model when the number of iterations per load step are changed. The material properties were as follows: Elastic modulus Ec = 15000 M P a . Compressive strength fc = 30 M P a . Cohesion c = 3.2404 M P a . 94 Poisson's ratio v = 0.2. Dilatancy factor P = 0.5. The program A P O S E C checks two criteria for terminating its iteration procedure. It computes the value of the Euclidean norm and checks it against a user specified value. It also checks to see that the maximum number of iterations, which is also specified by the user, is not exceeded. The program terminates the iteration procedure and goes on to the next load step when either of the previous criteria are met. The maximum number of iterations was set at 15, 30, 60 and 100 for each test respectively while the target value of the Euclidean norm was set at exactly 0. Setting the target value of the Euclidean norm at exactly 0 forces the program to iterate to the maximum number of iterations each time, since the target value is in practice unreachable. The results of these analyses are shown in Figure 5.3, Table 5.1 and Table 5.2. Figure 5.3 shows the average vertical stress on the element vs. strain. 60 50 40 (MPa) 30 20 10 -0 0 0.001 0.002 0.003 0.004 0.005 S z Figure 5.3. Post yield stress sensitivity to the number of iterations per load step on a single concrete element, p = 0.5 Table 5.1 shows the stresses at the center Gauss point just as it makes the transition from elastic to plastic behavior and Figure 5.2 shows the stresses at the center Gauss point for the last load step. The concrete becomes plastic after a strain of about 0.002, after which 95 one would expect the stress curve to stay flat since in the loading test simulations there was no confining pressure. Actually, a small amount of error in the confining pressure is picked up in the numerical computation sequence as shown in Table 5.1 and Table 5.2 under the rows labeled ax and a-y. The error in the confining pressure gradually diminishes as the number of iterations per load step is increased but the effects of the error on the vertical stress az can be quite significant as shown below. Conversely, this test could have been carried out by using a very large number of iterations, say 1000000, and selecting different target values of the Euclidean norm, for example 0.05, 0.01, 0.005 etc. Table 5.1. Stresses at the center Gauss point for the different numbers of iterations at load step 10 just as the Gauss points make the elasto-plastic transition. Stress n = 15 n = 30 n = 60 n = 100 CTx -8.589e-02 -6.066e-02 -3.025e-02 -1.197e-02 aY -8.589e-02 -6.066e-02 -3.025e-02 -1.197e-02 °z -3.184e+01 -3.130e+01 -3.065e+01 -3.026e+01 CTxv -7.885e-16 -9.834e-16 -9.205e-16 -1.232e-15 CTxz -8.900e-16 -8.319e-16 -8.438e-16 -8.125e-16 CTvz +5.163e-16 +4.326e-16 +6.673e-16 +6.053e-16 Table 5.2. Stresses at the center Gauss point for the different number of iterations for the last load step. Stress n = 15 n = 30 n = 60 n = 100 ° x -1.162e+00 -8.209e-01 -4.094e-01 -1.619e-01 a v -1.162e+00 -8.209e-01 -4.094e-01 -1.619e-01 ° z -5.491e+01 -4.759e+01 -3.877e+01 -3.347e+01 a x v +8.480e-16 +8.000e-16 -2.156e-15 +5.120e-16 a x z -1.356e-16 -2.503e-17 +8.605e-16 -5.235e-16 CTVZ +1.200e-15 +1.144e-15 +1.658e-15 +2.137e-16 A s another check on the robustness of the algorithm, a check can be made to see how close the stresses for the last load step of Table 5.2 lie on the Drucker-Prager yield surface. In perfect plasticity the yield surface does not evolve, we know all the 96 parameters that define it for the entire load history. The corresponding material constant (J) of the Mohr-Coulomb failure envelope in compression is: f *i § = sin 2\ / , -4<r = sin 2 A 30 2 - 4 x 3 . 2 4 0 4 v 3 0 2 + 4 x 3 . 2 4 0 4 2 y = 65.62° (5.3) from which we can determine the parameters a and K of the Drucker-Prager model as: ^ _ 2 sine)) _ 6ccos(|) ~ V3(3-sin<t)) ' ~ V3(3-sinc(>) (5.4) a = 0.5034 ; ^ = 2.2179 The Drucker-Prager yield condition is reiterated here as: F(aIJ) = 3a<sm + a-K = 0 (5.5) The following table shows in the last row the value of equation 5.5 at the center Gauss point for the four different iteration criteria at the last load step. Table 5.3. Yield surface tolerance to number of iterations at the last load step. n = 15 n = 30 n = 60 n = 1 0 0 -19.078 -16.410 -13.196 -11.265 a 31.0314 27.0021 22.1475 19.230 0.0019 0.0017 0.0010 0.00014 If the row labeled F(GJJ) of Table 5.3 is examined one can see that the stresses at the last load step lie closer and closer to the yield surface as the number of iterations is increased. Passive Confinement Analysis The next model to be analyzed is shown in Figure 5.4: A concrete cube surrounded by four steel plates of 3 mm thickness undergoing uniaxial compression. The loading simulation consisted of a uniformly distributed displacement applied to the top nodes for a series of twenty load steps. The bottom nodes were restrained vertically while the cube was allowed to expand in the x, y plane. A t this stage the concrete and 97 steel are connected to each other and compression, shear or tension forces can be transmitted at the nodes. Uniformly Applied Displacement Figure 5.4. Exploded view of composite model and elements used to test the passive confinement effect of the plate steel. The purpose of the test was to determine the. effect of the dilatancy factor p on the passive confinement ability of the plate steel. The following table summarizes the material properties and program parameters used to conduct the analysis. 98 Table 5.4. Material parameters in passive confinement test. Steel Concrete Parameter E elastic modulus 200000 M P a 15000 M P a V Poissons ratio 0.3 0.2 o> Yie ld stress in uniaxial tension. 300 M P a N . A . Et Y i e l d tangent modulus. O M P a N . A . t Plate thickness. 0.003 m N . A . fc Compressive strength N . A . 30 M P a ft Tensile strength N . A . 1.4 M P a P Dilatancy factor N . A . 0.5 Table 5.5. APOSEC arguments used to conduct passive confinement analysis. Parameter Value Number of load steps 20 Maximum no. of iterations 15 Number of sub increments 2 Target Euclidean Norm 0.01 Total displacement 1.00 mm The results of the analysis are given in Figures 5.5, 5.6, 5.8 and Table 5.6. Figure 5.5 is a plot of CTz (the vertical stress) at the center Gauss point for the various values of dilatancy factor p. It can be seen that in the post yield section of the plot the vertical compressive stress <JZ continues to increase due to confining pressure. The accuracy of the absolute stress values may be somewhat lacking since 15 iterations are not sufficient to produce accurate results on the confining pressure as was shown in the previous 99 section . However, the purpose here is to explore the sensitivity of the passive confinement modelling to the choice of dilatancy factor. O f interest here is the confining pressure <3Con = - ( a x + a v ) / 2 which is shown in Figure 5.6. The negative sign indicates compressive stress. Initially, while the steel and concrete are both elastic (up to strain 0.0016) there is negative confinement, as shown in Figure 5.6. Since the two materials concrete and steel have Poisson's ratios of 0.2 and 0.3 respectively, the steel tends to expand more than the concrete in the elastic range thus producing primarily tensile stresses in the concrete. In the inelastic range, however, the concrete tends to dilate considerably more than the steel shell due to sliding of crack surfaces over each other, thus stretching the plate steel and developing positive confining pressure. The sudden sharp dip in confining pressure just before the concrete makes the elasto-plastic transition can be attributed to the modelling process. 0.001 0.002 0.003 0.004 - Beta = 0 — - Beta = 0.25 —A— -Beta = 0.5 — • - - Beta = 0.75 0.005 Figure 5.5. Vertical stress vs. vertical strain at center Gauss point of the above model for various dilatancy values. 100 (MPa) m-Beta = 0 Beta = 0.25 - A - Beta = 0.5 Beta = 0.75 0.005 Figure 5.6. Passive confining pressure at center Gauss point for various values of dilatancy. Table 5.6 gives the stresses at the middle layer of Gauss points for the first load step. Figure 5.8 is a three dimensional graphical depiction of the stresses for several load steps starting in the elastic range and progressing into the inelastic range over the layer of the element. Table 5.7 gives their (xj,yj) coordinates relative to the coordinate axes shown in Figure 5.7. The first column of Table 5.6 gives the Gauss point number (1-27) and the corresponding indices on the (xj,yj) coordinates. Although the behavior of a concrete element encased by four steel plates is difficult to analyze by hand methods, an inspection of the stresses in Table 5.6 provides a qualitative assessment of the behavior of the steel and concrete model. To begin with, the in-plane stresses ox and Oy are greatest at the Gauss points that are near the corners of the concrete element, which is correct since they are nearest to the highly confined corners of the steel shell. There are no out of plane shear stresses uxz and OyZ as would be expected from uniaxial loading. The in plane shear stress aXy is greatest at a corner Gauss Point and negligible at the other points. One would expect that at a corner point the direction of the principal stresses would be 45°. This assumption may be verified for Gauss point #1 for the first load step by calculating the principal angle via a simple formula that is found in many Mechanics textbooks: 101 tan29 = ^ ( a , - a , ) / 2 0.1082 , c = = 00 P-°) (0.1162-0.1162) .-. 20 = 90° 0 = 45° Also one would expect the vertical stress to be smallest in magnitude near the corners and greatest in magnitude at the center of the concrete element while the concrete is behaving elastically, since it is subject to negative confining pressure that is greatest in magnitude near the corners. 102 Table 5.6. Stresses for the first load step at middle plane of Gauss points for the above model. p= 0.5 Sigma x Sigma y Sigma z Sigma xy Sigma xz Sigma yz Gauss Pt. 1 i = l j = l +1.162e-01 +1.162e-01 -3.214e+00 +1.082e-01 +4.706e-12 +6.962e-12 Gauss Pt 4 i = l j = 2 +4.407e-02 +9.816e-02 -3.232e+00 +1.124e-13 +1.127e- l l +8.326e-12 Gauss Pt. 7 i=l,j=3 +1.162e-01 +1.162e-01 -3.214e+00 -1.082e-01 +2.156e- l l +9.301e-12 Gauss Pt. 10 i=2 j= l +9.816e-02 +4.407e-02 -3.232e+00 +1.816e-13 +6.031e-12 +3.335e-12 Gauss Pt. 13 i=2,j=2 +2.604e-02 +2.604e-02 -3.250e+00 +1.81 le-13 +1.186e- l l +2.852e-12 Gauss Pt. 16 i=2,j=3 +9.816e-02 +4.407e-02 -3.232e+00 +1.808e-13 +1.892e- l l +1.685e-12 Gauss Pt. 19 i=3J=l +1.162e-01 + 1.162e-01 -3.214e+00 -1.082e-01 +9.325e-12 +7.892e-12 Gauss Pt. 22 i=3,j=2 +4.407e-02 +9.816e-02 -3.232e+00 +2.499e-13 +1.890e- l l +6.947e-12 Gauss Pt 25 i=3J=3 +1.162e-01 +1.162e-01 -3.214e+00 + 1.082e-01 +2.719e- l l +5.021e-12 y • 7 m 16 • 25 • 4 • 13 • 22 r • 10 • 19 X Figure 5.7. Middle layer of Gauss points. Table 5.7. Coordinates of the Gauss points shown in figure 5.7. j = l j = 2 j = 3 i = 1 (25.92,25.92) (25.92,115) (25.92,204.08) i = 2 (115,25.92) (115,115) (115,204.08) i = 3 (204.08,25.92) (204.08,115) (204.08,204.08) 103 Figure 5.8 depicts the compressive vertical stress (plotted as compression positive vs. the Gauss point indices i,j) at five different load steps. In Figures 5.8 a, b and c the concrete is acting in the elastic range with the compressive stresses being least at the corners, giving the stress plot a convex appearance. The concrete makes the plastic transition in Figure 5.8 d and subsequently the confining pressure increases at the corners relative to the center section, thus giving the stress plot a concave appearance. 104 53.35 53.25 Load Step 20 Figure 5.8. a,b,c,d,e. 3D plots of az at middle layer of Gauss points in the concrete element for load steps 1, 5, 10, 15 and 20. p = 0.5. 105 5.3 Analyses with the Hardening/Softening Model A series of tests were performed on the finite element of Figure 5.1 using the Hardening/Softening Drucker-Prager model in place of the perfectly plastic one. Many parameters were introduced in Chapter 3 to define the hardening/softening model that are not verified by experimentation. In this work one may gain qualitative insight into the behavior of the predicted responses as the model parameters are varied. One Element Under Compression A t first, three compression tests were conducted. Figure 5.9 shows the vertical stress, G z , averaged over all of the Gauss points vs. the vertical strain, s z, for three different cases of constitutive constants shown in Tables 5.8 to 5.10. 25 0 0.001 0.002 0.003 0.004 0.005 Figure 5.9. Average vertical stress vs. vertical strain for various compression tests on single concrete element with the hardening/softening Drucker-Prager model. In the first two tests, Case A and B , the variable chosen to be varied was s c . The parameter E c , it w i l l be recalled from Chapter 3, controls the rate of softening in the post 106 yield region. Generally a larger value of this parameter tends to delay the softening effect, which is a trend also observed in Figure 5.9. In Case C the parameters (p c v, s c and were changed to 0.1, 0.035 and 0.005 respectively over their values in Case B . The parameter cp c v affects the plastic expansion to distortion ratio. A lower value of this parameter was observed to improve the stability of the computed post yield response and generally should not influence the computed stress strain values. Increasing s c and s§, ( it w i l l be recalled from Chapter 3 that controls the rate of hardening in the mobilized friction angle), w i l l tend to delay the softening and cause the hardening effect to occur more gradually in the post yield response. The observation of Figure 5.9 tends to indicate that the response shows a peak of 22.5 M P a over the initial yield value of 20 M P a indicating that hardening took place to reach the 22.5 M P a value. Logically, decreasing and perhaps increasing s c w i l l tend to produce a peak near 30 M P a (the compressive strength used in the model). Table 5.8. Constitutive parameters for Case A. E = 27000 MPa fc = 30 MPa ff = NA v = 0.2 A = 0.3 <pw=17.2° s c = 0.0075 Sty = 0.0025 (j) = 22.62°1 c = 10 MPa Table 5.9. Constitutive parameters for Case B. E = 27000 MPa fc = 30 MPa ff = NA v = 0.2 A = 0.3 cp c v=17.2° s c = 0.015 e,f= 0.0025 <|> = 22.62 0 1 c = 10 MPa f r'2 Calculated from <|> = sin - i / ; 2 + 4 c 2 2\ 107 Table 5.10. Constitutive parameters for Case C. E = 27000 MPa fc = 30 MPa it= NA v = 0.2 A = 0.3 ( p c l / = 5 . 7 3 ° s c = 0.035 = 0.005 <|> = 2 2 . 6 2 ° 1 c = 10 MPa The next two tests were carried out in tension on the same model for the values of constitutive constants shown in Tables 5.11 and 5.12. The only parameter to vary in Cases D and E was A in the function Y{<5M,GA,fc,A) and generally had no effect on the computed stress strain response. Shown in Figure 5.10 are the responses in tension averaged over all Gauss points in the element for both Cases D and E . Table 5.11. Constitutive parameters for Case D. E = 27000 MPa fc = 30 MPa ff = 2 MPa v = 0.2 A = 0.3 <pcv = 5 .73° s c = 0.035 = 0.005 $ = 2 2 . 6 2 ° c= 10 MPa Table 5.12. Constitutive parameters for Case E. E = 27000 MPa fc = 30 MPa ff = 2 MPa v = 0.2 4 = 0.15 <Pcv = 5-73° e c = 0.035 5^ = 0.005 4 = 2 2 . 6 2 ° c= 10 MPa 2.5 0 0.0001 0.0002 0.0003 0.0004 0.0005 Figure 5.10. Average vertical stress vs. vertical strain for two tension tests on single concrete element with the hardening/softening Drucker-Prager model. 108 Alternatively the tension test was carried out with Method 2 of Chapter 3 utilized to release the stresses in the direction of principal stress. The resulting brittle response is as shown in Figure 5.11 0 0.00005 0.0001 0.00015 0.0002 0.00025 Vertical Strain Figure 5.11. Brittle response in uniaxial tension. Passive Confinement Analysis The next series of tests were performed on the single concrete element surrounded by four steel plates of Figure 5.4 using the Hardening/Softening Drucker-Prager constitutive model for the following set of constitutive constants for steel and concrete. Table 5.13. Steel material parameters for passive confinement test on the Hardening/Softening Drucker-Prager model. E = 200000 MPa a v = 300 MPa Et = 0 MPa v = 0.3 t = Varies Table 5.14. Concrete material parameters for passive confinement test on the Hardening/Softening Drucker-Prager model. E = 27000 MPa f c = 30 MPa ff = 2 MPa v = 0.2 A = 0.3 Vcv = 5-73° s c = 0.01 e f = 0.02 <|> = 22.62° c = 10 MPa B= NA 109 A l l the constitutive parameters were kept constant while the thickness of the steel shell was changed from 1 mm, 2 mm and to 4 mm in three tests referred to as Case A , Case B and Case C in Figures 5.12 to 5.14. Figure 5.12 shows the vertical stress, az, in the concrete element averaged over all the Gauss points while Figures 5.13 and 5.14 show the variation in confining pressure and the hardening modulus, H, respectively for the three tests. There is a prominent discontinuity in Figures 5.13 and 5.14. The discontinuity is caused by the hardening modulus, H, which is a discontinuous function of deformation. When the Gauss points are not yielded, the hardening modulus is zero, when the Gauss points yield, it takes on a finite value. The discontinuity is also present in Figure 5.12 but it is somewhat subtle. The discontinuity in Figure 5.12 is the change of the stress strain curve from the linear portion to the curved plateau. It w i l l be noticed that some points after the discontinuity on the curves of Figure 5.14 are missing. These computed points were omitted from the figure because they were characterized by values that deviated by an order of magnitude from the points preceding them. From observations of the corresponding values of the Euclidean Norm these points correspond to convergence failures of the Newton-Raphson iteration procedure which can occur in finite element procedures utilizing plasticity based constitutive models [22]. Interestingly enough, Figure 5.12 and 5.13 did not show any outliers in the computed responses but they do have some jumps and kinks in the corresponding regions. It seems apparent that one should look at the smoothness of the hardening modulus curve to determine i f the computed response is sound. 110 25 —m— Case A 20 - —A— Case B — • — Case C 15 -° z ( M P a ) ™ -5 o • f i 1 1 1 1 0 0.001 0.002 0.003 0.004 0.005 Figure 5.12. Average vertical stress vs. strain for the passive confinement test on a singe concrete element confined by four steel plates using the Hardening/Softening Drucker-Prager model. Figure 5.13. Average confining pressure for the passive confinement test using the Hardening/Softening Drucker-Prager model. 111 -1200 J I Figure 5.14. Average hardening modulus for the passive confinement test on the Hardening/Softening Drucker-Prager model. 5.4 Analysis of a Joint Phase 1 The first model utilizing the perfectly plastic Drucker-Prager concrete model of section 3.3 to analyze a T-joint with square tubing is that shown in Figure 5.15. It consists of 4 concrete elements of the dimensions shown surrounded by 15 steel plate elements of 3 mm thickness and no reinforcing steel elements. The loading simulation consisted of applying a total displacement of 6 mm over 30 steps at the position indicated in Figure 5.15. 112 360 mm a 496 mm Center Line 690 mm Figure 5.15. Model 1: 4 bricks and 15 plates ' 960 mm Figure 5.16. Model 3: 40 bricks and 60 plates. 113 690 mm Figure 5.17. Model 3: 20 Bricks and 36 plates. Table 5.15. Material parameters in joint analysis using the perfect plasticity Drucker-Prager constitutive model. Steel Concrete Parameter E elastic modulus 200000 M P a 15000 M P a * V Poission's ratio 0.3 0.2 Yie ld stress in uniaxial tension. 300 M P a N . A . Et Yie ld tangent modulus. O M P a N . A . t Plate thickness. 0.003 m N . A . fc Compressive strength N . A . 30 M P a ft Tensile strength N . A . 1.4 M P a P Dilatancy factor N . A . 0.5 * This is representative of the secant modulus of concrete in uniaxial compression up to a strain of 0.002 corresponding to a peak stress of 30 M p a [21]: 30MPa 0.002 = 15000MPa 114 Table 5.16. APOSEC arguments used to conduct joint analysis using the perfect plasticity Drucker-Prager concrete constitutive model. Parameter Value Number of load steps 30 Maximum iterations 15 Number of sub 5 increments Target Euclidean Norm 0.005 Total displacement 6.00 mm 1000 0 1 2 3 4 5 Displacement (mm) Figure 5.18. Force displacement plot for two finite element models of a square specimen. The second model of Figure 5.16 consists of 40 concrete elements and 60 plate elements. It also differs from the first model in the boundary conditions. The vertical restraints in the second model are applied on the midside nodes of a set of bare concrete elements at each end of the model, which is more consistent with laboratory conditions during the physical tests. The force displacement curves for the two models are plotted in Figure 5.18. The general trend appears to be a softening of the response as the number of elements is increased although the plot for the second model of the joint still overpredicts the strength of the square joint with fins by about 120 % and the joint without fins by 170 %. 115 5.5 Analysis of a Joint Phase 2 The first series of tests, utilizing the Hardening/Softening Drucker-Prager model with Method 2 as the option for tensile response, on a joint were performed on the finite element model of Figure 5.17. The models of the joints presented in Figures 5.15 and 5.16 had the steel plates perfectly bonded to the concrete elements. In the laboratory it was observed, however, that the steel plate separated from the concrete joint at the corners on the tension side of the specimen. This phenomenon is illustrated in the following diagram: Figure 5.19. Separation of steel plate from concrete on the tension side of specimen. When the steel plate separates at the beam column interface, the load capacity should decrease since the plate now has a horizontal component of force that does not contribute to the moment resistance of the beam stub. This phenomenon can be modeled in the finite element program by defining an independent set of nodes at the beam column interface on which to connect the plates and another set on which to connect the concrete elements. The nodes would initially share the same coordinates but are free to move independently during the load history. The steel shell finite elements were debonded at selected nodes (as shown in Figure 5.19) to possibly model the bond slip characteristics of the steel-grout interface. A complication arises when using nodes for the steel shell Plate steel peels away on tension side 116 that are independent of the concrete nodes, since nodes belonging to the steel shell are able to move into the concrete elements during loading. This is physically impossible, but i f we consider the fact that the steel shell can loose strength by buckling in compression then the physical interpretation may be a reasonable compromise. Three cases were examined for the following values of constitutive constants: Table 5.17. Steel plate constitutive properties in Phase 2. E = 200000 MPa CTv = 300 MPa E f = 0 MPa v = 0.3 f = 3 mm Table 5.18. Concrete constitutive properties in phase 2. E = 27000 MPa fc = 30 MPa ft = 2 MPa v = 0.2 A = 0.2 <Pcv = 7° s c = 0.005 s f = 0.01 (j, = 22.62 0 1 c= 10 MPa B = See text * * This is not P of the perfectly plastic Drucker-Prager model, it is the parameter B introduced in Chapter 3 as the rate of shear modulus reduction. A . Here the model in Figure 5.17 of 20 brick elements was used with the steel shell fully bonded to the concrete. The tension due to cracking was released with B = 0 M P a (no shear stiffness degradation). B . The 20 brick model was used with selected nodes on the steel shell on the tension side debonded from the concrete. The tension due to cracking was released with B = 0 M P a . C . Same as B except the shear reduction rate was used as B = 10000 M P a . 117 0 0.5 1 1.5 2 2.5 Displacement (mm) Figure 5.20. Comparison of responses between tensile stress release due to cracking and no tensile stress release due to cracking. In every case presented above the analysis failed to converge after a displacement of about 2 mm as the response became unstable. Debonding the nodes on the tension side results in very little decrease in the predicted strength (Case A vs. Case B) . A l so shear stiffness degradation has very little effect (Case C). 5.6 Analysis of a Joint Phase 3 The next phase of joint finite element analyses utilized the Hardening/Softening Drucker-Prager model with Method 1 as the option for the response in tension. Two cases were examined each for the 4-concrete-finite-element-model of Figure 5.15 and the 20-concrete-fmite-element-model of Figure 5.17. The concrete constitutive parameters for each of the cases are shown in Tables 5.19 and 5.20 while the steel plate constitutive parameter values were kept the same as in Phase 2. 118 < — • — Case A 800 • d * T n . —m— Case B 600 - j£r 400 if 200 0 , f 1 h - 1 0 2 4 6 8 Displacement (mm) Figure 5.21. Response of the 4 concrete element model in Phase 3. 0 1 2 3 Displacement (mm) Figure 5.22. Response of the 20 concrete element model in Phase 3. Table 5.19. Concrete constitutive properties for Case A. E = 27000 MPa fc = 30 MPa ft = 2 MPa v = 0.2 A = 0.2 cp c v = 11.46° s c = 0.01 6f = 0.02 <j, = 22.62°1 c = 10 MPa B = See text * Table 5.20. Concrete constitutive properties for Case B. E = 27000 MPa fc = 30 MPa ft = 2 MPa v = 0.2 A = 0.2 cp c v = 17.19° s c = 0.01 e f = 0.02 ()> = 22.62°1 c= 10 MPa B = See text * 119 Figures 5.21 and 5.22 show the responses to the displacement loading simulations of the two finite element models as the parameter cp c v was varied. The parameter cp c v is the value of the mobilized friction angle at which a transition from plastic compaction to dilatancy takes place. One would expect a higher value in this parameter to reduce the peak predicted strength in the finite element procedure as the confinement effect of the steel shell would be less pronounced. The 20 concrete element model appears to obey this intuition while the 4 element model does not, it predicted a higher peak strength for the higher value of (p c v . Also the 20 element concrete model became unstable after a displacement of about 1.5 mm and 2 mm for the two cases of <pcv. 5.7 Closing Commentary A t the beginning of this chapter simple analyses were conducted on a single concrete element and later on a concrete element surrounded by steel plates, with the objectives of determining the behavior of the constitutive models and the sensitivity of the predicted responses to material constitutive parameter variations and as well to program argument variations. The one element analysis indicated that the concrete and steel constitutive models used in the context of a finite element analysis provided a reasonable framework with which to model the behavior o f steel encased concrete . Phase 1 of the joint loading simulation utilized the perfectly plastic Drucker-Prager concrete constitutive model. The two models, (4 concrete elements and 40 concrete elements), predicted yield strengths of about 350 k N and 450 k N respectively. Consistent with finite element theory the 40 concrete element model predicted a softer initial and post yield response. Since softening did not take place in the perfectly plastic constitutive models, the predicted responses were stable. The increased computing time and complexity associated with the 40 concrete element model did not justify its use. In fact the results obtained from it did not differ by an order of magnitude over the 4 120 concrete element model. Consequently, a compromise between the 4 element and 40 element model was devised; the 20 concrete element model. The 20 concrete element model was used with the 4 concrete element model in subsequent test comparisons. In Phase 2 the Hardening/Softening Drucker-Prager concrete constitutive model was used with the brittle cracking option of Method 2 2 in Chapter 3 utilized to fine tune the response in tension. Other features were used such as attempting to debond the steel shell from the concrete. The predicted responses using this method showed instability later in the load history, but interestingly enough, the load displacement curves in Phase 2 show a yield "knee" at about 350 k N which is on par with the 40 element model of Phase 1. Phase 3 of the joint loading simulations utilized the Hardening/Softening Drucker-Prager model with Method 1 as the option for the modified response in tension. The 4 element and 20 element finite element models were used in conjunction with variations in the parameter cp c v to determine the confinement effect on the response. The 4 element model predicted yield points at about 600 k N and peak strengths of 700 k N and 900 k N . The 4 element model was stable enough to simulate the post yield region of the load deflection curve. This is the only combination of joint finite element and concrete constitutive models that permitted the observation of the post yield branch. The 20 element finite element model became unstable right after the yield plateau in Figure 5.22. The lowest predicted peak strength was about 350 k N which is about the value of the yield knee in Phases 1 and 2. Curiously the shape of the load deflection curve in Figure 5.22 does resemble the shape of the laboratory measured response envelopes of the Although in Chapter 3 the brittle stress release method is called Method 2, in the development of the finite element computer program it was the first method to be used. Hence in the modelling cases presented here in Chapter 5, those that use Method 2 come before those that use Method 1. 121 square specimens. The response envelopes of the two square joints are presented below in Figures 5.23 and 5.24 in force displacement units for comparison. 250 Displacement (mm) Figure 5.24. Force displacement response of the Square Joint # 1. Part of the problem associated with trying to predict the peak strength of previously damaged specimens that have been subjected to cyclic loading is that the reinforcing steel may have stretched to a point that cracks have permanently opened in the concrete and which did not close in subsequent testing. Such behavior may result in the concrete not contributing at all to the response of the member as the following simple analysis for the square reinforced concrete specimen, without the steel fins welded to the side, w i l l show. 122 230 mm 10 mm Reinforcing Bars 230 mni Grout filled gap 200 mm It 15 mm cover (approximate) 3 mm Steel Shell Figure 5.25. Cross-section of retrofitted specimen. If elementary beam theory is used to analyze the above cross-section, ignoring the contribution of concrete in tension to the capacity of the above section and making the crude assumption that the neutral axis passes through the geometric center3 o f the gross section, one can do the following analysis assuming that all of the steel has yielded. The reinforcing and plate steel properties can be found in the appendix of Hoffschild's thesis In most practical reinforced concrete beams, the neutral axis is very shallow at the state where all steel has yielded. A reasonable estimate is at 25% of the depth of the section. [11]: 1. Contribution of reinforcing bars to moment capacity: momentarm yd = 2 0 0 - 2 x 1 5 - 2 x 10 -10 = 140 mm yield strength fy = 700 M P a steel area As = 200 mm 2 Mr, = AJyjd = 200 x 10"6 x 700 x 140 k N m Mr, =19.6 k N m 123 2. Contribution of the steel shell on the top and bottom to moment capacity: moment arm jd = 230 mm yield strength fy = 267 M P a steel area As = 230 x 3 = 690 mm 2 M r i = Afyjd = 690 x 10"6 x 267 x 230 k N m Mr2 = 42.4 k N m 3. Contribution of the steel shell on the sides to moment capacity: moment arm jd = 2 3 0 - 0.5 x 230 = 115 mm yield strength fy = 267 M P a steel area As = 0.5 x 2 x 230 x 3 = 690 mm 2 Mr, = AJyjd = 6 9 0 x 1 0 - 6 x 267 x 115 k N m Mr3= 21.2 k N m 4. The total is: MT = Mrx + Mr2 + Mr, MT = 19.6 + 42.4 + 21.2 M j . = 83.2 k N m If we compare the above calculation with the measured moment capacity of the square retrofit without fins. In the positive quadrant (see Figures 5.23) we have a peak force of about 200 k N . The moment arm to the face of the steel shell on the column is 0.456m -l/2*0.230m = 0.341m (see Figure 2.10). The resulting moment capacity is 200kN*0.341m = 68.2kNm. 124 CHAPTER 6 CONCLUSION Summary In addressing the seismic retrofit o f deficient beam to column concrete joints, the method of steel jacketing was chosen as the topic of experimentation in the work presented here. Following the tests performed by Hoffschild, who observed failures in the members outside of the beam column joint region, a procedure was set up to test the joint area itself for strength and behavior when subjected to cyclic loading. Following tests on the joint region itself, intricate concrete and steel constitutive models were developed in Chapter 3 to represent the behavior of passively confined concrete with the goal of predicting the load deflection response of such beam column joints. Chapter 4 presents the details of the incorporation of the constitutive models into a nonlinear finite element based computer program. Chapter 5 contains details of some simple analyses that were performed on single concrete elements confined by four steel plates, followed by models of the beam to column joints consisting of 4, 20 and 40 concrete elements. The simple models served to simulate showed the behavior under passive confinement. The models of the joints, although unable to predict the peak strength as measured in the laboratory, were able to reproduce the shape of the strength envelope as observed. 125 Concluding Remarks The method of steel jacketing inadequately reinforced beam column joints was experimentally shown to significantly improve the strength and ductile response of such joints during cyclic loading. This is because confined concrete exhibits increased strength and an expanded strain range before failure. Whereas modern codes call for closely spaced stirrups within beam to column joints and in regions of plastic hinging to provide confinement, this is not always the case in practice. A steel jacket retrofit has been shown to be an effective means of remediating such deficiencies in reinforced concrete members and joints. Careful attention must be paid, however, to the design of a retrofit scheme, considering the possibility of shear failure outside the retrofitted region due to severe moment gradients resulting from flexural overstrength and shortening of the weak sections. The overstrength forces may be reduced by providing gaps in the retrofit jacket in the region of the plastic hinge. Central to the concept of capacity design, in the design or retrofit o f beam column joints, is that the joint region itself be stronger than the adjacent beam, which was found to be the case from test comparisons made in Chapter 2 with Hoffschild's work. When regions of stress concentrations are reinforced on the steel jacket, the joint region also exhibits a very ductile response. The theoretical analysis and modelling of steel encased joints led to the development of a nonlinear material finite element based computer program. The computer program features non linearity in plasticity based material constitutive models, which range from the very complex to the very simple. During the development of the plasticity models, many constitutive parameters had to be introduced that would be difficult to quantify precisely without accurate experimental data. The constitutive modelling of the concrete in the joints was focused on the plastic nature of confined 126 concrete and its brittle response in tension. Intuitively it is argued that stresses representing both conditions are present during the loading of such joints. It was shown, however, that the concrete may not contribute significantly to the strength of such a joint. The predicted strength is relatively insensitive to the parameters governing the response of the concrete in tension while changes in parameters governing the response in compression such as cp c v produce significant differences in the predicted response. Also , differences between the two Drucker-Prager plasticity models have a significant impact on the post yield response of the finite element models. The observed strength of the joints was difficult to reproduce in the finite element models. It was observed that models involving many elements tend to produce a softer response and hence lower peak strengths than models involving few elements. A n instability in the predicted response crops up in the models involving more elements which is a not uncommon in finite element procedures involving nonlinear material characteristics. The unstable nature is a function of the constitutive model used and the number of elements. Generally the constitutive models exhibiting a softening response w i l l tend to produce an unstable response while the simple perfectly plastic Drucker-Prager model does not. A n increase in the number of finite elements in a model w i l l tend to produce instability in the Hardening/Softening Drucker-Prager model while fewer elements may not. Future Work A s recommendations for further research it would be advisable to continue the experimental research involving reinforcement method of constructing a steel shell or cage around a joint in an actual building in the field. But more to the point would be the prefabrication of complete moment resisting frames utilizing steel encased joints that could be installed as a redundant system in moment resisting frames that do not have 127 contemporary details and hence are brittle. The installation of a prefabricated ductile moment resisting frame into a system that has limited ductility is an area that is not addressed by building codes in Canada. Research into the behavior of the two systems would prove to be very beneficial. The numerical research here has focused on the representation from the continuum level of beam column joints involving models of many degrees of freedom with results that represent reality qualitatively i f not quantitatively. From the work in Chapter 5 it was found that the accurate representation of the concrete plastic dilatant behavior is critical in obtaining realistic force displacement responses. More work needs to be directed into this area. The next phase of numerical research should focus on simpler beam and column models (possibly stick representations) with the aim of calibrating them from laboratory experiments and using them in the modeling of entire frames. A n interesting application of such a model would be to attempt to predict the response of ductile and non ductile systems working together as outlined above for future experimental work. 128 REFERENCES R. Tremblay, M . Bruneau, M . Nakashima, H . G . L . Prion, A. Filiatrault, R. DeVall, "Seismic Design of Steel Buildings: Lessons From The 1995 Hyogoken-Nanbu Earthquake", Canadian Journal of C i v i l Engineering, August 14, 1995 T. Paulay, M.J.N. Priestley, "Seismic Design of Reinforced Concrete and Masonry Buildings", John Wiley and Sons, 1992 Canadian Standards Association, C A N 3 - A 2 3 . 3 - M 9 2 "Design of Concrete Structures for Buildings" Michel Bruneau, "Preliminary report of structural damage from the Loma Prieta (San Francisco) earthquake of 1989 and pertinence to Canadian structural engineering practice", Canadian Journal of C i v i l Engineering, A p r i l 1990 D. Mitchell, R .H. DeVall, M . Saatcioglu, R. Simpson, R. Tinawi, R. Tremblay, "Damage to concrete structures due to the 1994 Northridge earthquake", Can. J. C iv . Eng. 22: 361-377 (1995) 129 J.B. Mander, M.J.N. Priestley, R. Park, "Theoretical Stress-Strain Model For Confined Concrete", Journal of Structural Engineering, V o l . 114, N o . 8, August 1988. J.B. Mander, M.J.N. Priestley, R. Park, "Observed Stress-Strain Behaviour of Confined Concrete", Journal of Structural Engineering, V o l . 114, N o . 8, August 1988. S. Pietruszczak, J.Jiang and F.A. Mirza, " A n Elastoplastic Constitutive Model For Concrete", Int. J. Solids Structures V o l . 24 N o . 7 pp. 705-722, 1988. J . Jiang and F.A. Mirza, "Nonlinear Analysis of Reinforced Concrete Slabs by a Discrete Finite Element Approach", unpublished manuscript. O. C. Zienkievvicz and R. L . Taylor, "The Finite Element Method", Fourth Eddition, Volume 2, Chapter7, 1991 McGraw H i l l . Thomas E . Hoffschild, "Retrofitting Beam-to-Column Joints for Improved Seismic Performance", M . A . S c . thesis, Dept. of C i v i l Eng., University of British Columbia, Vancouver, B . C . , Canada, 1990. G. Lee, P. Behrouzi, A. Sabounchi, A. Mirza-Soleimani, K . Vorell, L . Erven, C I V I L 321 Laboratory Report, Dept. of C i v i l Engineering, University of British Columbia, A p r i l 1993. 130 (13) Sergio M . Alcoccer, " R C Frame Connections Rehabilitated by Jacketing", Journal of Structural Engineering, V o l 119, N o . 5, M a y 1993. (14) J . Jiang and S. Pietruszczak, "Modeling of Concrete Response to Fluctuating Load", International Journal for Numerical and Analytical Methods in Geomechanics, V o l . 13, 171-181, 1989. (15) Zdenek P. Bazant, "Advanced Topics in Inelasticity and Failure of Concrete", Swedish Cement and Concrete Research Institute, 1977 (16) Ernest Hinton and Roger Owen, "Computational Modeling of Reinforced Concrete Structures", Pineridge Press, 1986 (17) Zdenek P. Bazant and Parameshwara D. Bhat, "Endochronic Theory of Inelasticity and Failure of Concrete", Journal of the Engineering Mechanics Division, Proceedings of the American Society of C i v i l Engineers, V o l . 102, No . E M 4 , August, 1976. (18) O. C. Zienkiewicz and R. L . Taylor, "The Finite Element Method", Fourth Eddition, Volume 1, 1989 McGraw H i l l . (19) Press, Flannery, Teukolsky, Vetterling, "Numerical Recipes, The Art of Scientific Computing", Chapter 4, 1989 Cambridge University Press. (20) R. D. Cook, D. S. Malkus, M . E . Plesha, "Concepts A n d Applications of Finite Element Analysis", John Wiley & Sons, 1989. 131 (21) Collins & Mitchell, "Prestressed Concrete Basics", 1987 Canadian Prestressed Concrete Institute (22) Z . P. Bazant, L . Cedolin, "Stability of Structures", Chapter 10, 1991 Oxford University Press, Inc. 132 APPENDIX A Some Mathematical Facilitations: In Chapter 3 it was stated that the quantity: i dev v5a</y dev d\\j Here we set out to prove this result. To begin, it is useful to rewrite the plastic flow function in tensor notation as follows: d\\f d\\j 1 5.,+-1 (A.1) Recognizing that the deviatoric component of the flow function is the second term in the above equation: dev I 1 2 ^ And: ( ^ \ dev d\\i dev __J 1 _ ^A/*^~ 2^ffi Substituting the definition for J2: = —S,:S:: IJ IJ (A.2) (A.3) (A.4) 1 1 <2JJ,2jj'SuSu V 4 J , 2 J , (A.5) 133 Procedure for determining G A . Perfectly Plastic Model The procedure for calibrating the yield function of section 3.4 in the tension regime is as follows. The yield function is presented here for clarity: F(oij) = {3aom-K) ( \ 2" ( \ 2 1- o 1- O" m -K m ^A) \*A J F ( r j , ) = 3 a a m + a - ^ = 0 + a = 0 for a,„ > 0 for CT„, < 0 (A.6) a = K = 2 sine)) V3(3-sin((>) 6ccosc() (A.7) (A.8) V3(3-sin<|>) Imagine performing an experiment where a concrete specimen is subjected to a state of uniaxial tension. A t the point of tensile failure, the stress tensor, a^-, has the following values: 'ft o o" 0 0 0 0 0 0 (A.9) A n d the deviator is: 3 7 ' 0 - i / 3 0 0 0 3 0 0 (A.10) In the tension regime (am >0) at the point of yielding the following conditions are satisfied: 134 a... = f, Substituting the above into the yield function results in: (3aam-K) 1- K 1-\3°AJ A. + A = 0 V3 (A.11) (A.12) (A. 13) The equation A . 13 is a non linear equation dependent only on the material parameters c,ft and <|>. Given the material parameters c,ft and the parameter aA can be found by a series of trial and error iterations. Hardening/Softening Model The procedure for calibrating the yield function of section 3.6 in the tension regime follows similarly as for the yield function of section 3.4. F ( a , ) = ( 3 a * a m - ^ ) t F(o ( y) = 3 a o B + a - J i r = 0 for a„, > 0 for a.„ < 0 + a = 0 (A. 14) (A. 15) (A. 16) In the tension regime (a O T >0) at the point of yielding the following conditions are satisfied: c=c (A. 17) (A. 18) a = fy =0 2smfy* 0 K = \/3(3-sin(j)*) 6c* cos fy" _ 2 A/3(3-sin(j)*) ~ V 3 - ' (A. 19) (A.20) 135 a.,. = °A=VA Substituting the above into the yield function results in: 2c V3 1- + 2c V3 1- ' A * \ 3 ° A j A K3(JAJ A V3 = 0 (A.22) (A.23) (A.24) The equation A.24 is a non linear equation dependent only on the material parameters c and ft, which can be solved by trial and error iteration as A . 13. A s an example the following table gives solutions for the parameter crA for various values of c andft. Values may be interpolated linearly for values of c and ft not given. 136 Table A.1. Table of in MPa for various values of c and ft in the Hardening/Softening model. ' c = 5 c = 10 c = 15 M P a M P a M P a ft =2 M P a 0.70273 0.68399 0.67806 / , = 4 M P a 1.49071 1.40545 1.38013 ft=6M?a 2.39046 2.16931 2.39046 137 APPENDIX B Element Shape Functions Concrete Element Shape Functions The shape functions for the twenty noded concrete element are formed from two families /•V of shape functions. One set of shape functions iVy corresponds to the eight noded prism as shown bellow: Figure B.1. Eight noded prism. Shape functions corresponding to eight noded prism % = \ (i+OU+T,)(I+0 N9 = UI+0(i+TIXI - 0 o o o o N5 = I(i - 0 ( i - TIXI+0 N13 = UI- 0 ( i - TIXI - 0 #7 = i(i+0(i-Ti)(i+0 JV I 5 = 1 ( I + 0 ( I - T I X I - 0 o The other set of shape functions Nf* is defined from a prism with only midside nodes. 138 19 17 14 16 Figure B.2. Prism with only midside nodes. Shape functions corresponding to midside nodes N2M 4<' - 4 2 ) ( i + T 1 ) ( i + c ; ) N» 4<> N6M 4<> -42)(l-Tl)(l + 0 - T ! 2 ) ( I H ) ( I + C ; ) < 4c - 4 2 ) ( l + T , ) ( l -0 NM 4c - T , J ) ( l - a i - 0 NM J V14 NM I V16 4c - T ! 2 ) ( I H ) ( I - C ; ) iV17 4c - ; 2 ) ( i + n ) ( i + ^ ) NM 4c -^)(i-^)(i + T1) J V19 - ; 2 ) ( l - ^ ) ( l - r i ) 4c - ? ) (1 + O0-Tl) The shape functions for the composite element are formed by appropriately combining the A^- and • shape functions so that the composite set satisfies unity value at their corresponding nodes and zero at other nodes. 139 Shape functions for twenty noded prism N2 = N2M N3 = N 3 - ^ N 2 M - ^ N 4 M - ^ N4 = N4M N9 = N9-^NlM0-^NlM6-^Nf{ ^ 1 1 = ^ 1 1 - ^ 1 0 - ^ 1 2 - ^ 1 8 Nl2 = NXM2 NH = < Ni6 = N% Nl7 = N{f *is = < AT - J\TM AT - KTM i v 2 0 - i V 2 0 Shape Functions For the Membrane Element Mx 1 / ~ 4 1 + 0 ( 1 + Tl) M2 _ 1/ i - ^ 2 ) ( i + r , ) M3 _ 1 / ~ 4 l - 0 ( l + r|) M4 _ 1/ ~ 2 ^ l - T I 2 ) ( l - 0 M5 1 / ~ 4 l - O U - T l ) M6 _ 1/ ~ 2 l i - ^ 2 ) ( i - r , ) M7 1/ ~ 4 1 + 0(1-Tl) M 8 1/ ~ 2^ i - n 2 ) d H ) 3 (-1,1) 1 r, 2 (0,1) (-1,0) (0,0) (-1.-1) 6 (-1,0) 1 (1,1) 8(1,0) 7 (-1.D 141 APPENDIX C The Three Point Gauss Quadrature Rule / 4 -0.774596669241483 0 0.774596669241483 Figure C.3. The three point Gauss quadrature rule. 1 3 -1 i=\ ^ = 0 . 5 5 5 5 5 5 5 5 5 5 5 5 5 5 6 ( C I ) W2 =0.888888888888889 W3 = 0.555555555555556 The three point Gaussian integration rule w i l l integrate a fifth order polynomial exactly. 142 APPENDIX D The Concrete Element [B] matrix The Cartesian coordinates in the Concrete finite element are related to the dimensionless coordinates by the isoparametric mapping: (D.l) i A n d the displacements in the element are given by. i v = ^ / , ( ^ , Q (D.2) w = Nt:(£,,ri,Q ; i = 0... number o f nodes The variable xt, yt and zt are the coordinates of the nodes in the Cartesian space. Equation D . l is more conveniently expressed in matrix form as: u 0 0 • - NN 0 0 < V 0 0 • •• 0 0 w 0 0 Nx • •• 0 0 N J3x60 a a a. a XN a yN a (D.3) 143 From which the strains are easily derived as: to = [B]M The [B] matrix is implied from the following: (DA) r su ] 3x *yy Bv By C dw < p > = • Bz 811 _ i _ Bv > xy By ~r Bx Bu _ i _ 3w Bz "+" dx Bv , Bw Bz ~l~ By '\,x 0 0 N, 0 0 N>,y 0 0 0 0 1,2 0 lo-20,* 0 0 0 20,y 0 0 0 N. N. 20,y N. N. 20,z 0 20,x 0 20,2 0 N. 20,* N. 20,; N. 2Q,y a. a yn a. (D.5) In the case where the shape functions are given in terms of dimensionless coordinates, their derivatives with respect to the Cartesian coordinates x, y and z are not known explicitly but can be found via the Jacobian of the transformation D . 1; [J] dx dy dz &> dx dy_ dz dr\ dr\ dr\ dx dy dz i I * 2>, i aw,. 2>< dN: I ft, aw. cW.- aw. i__ dr\ z_ dr\ dNt x—1 dNi 5>, i dN( I (D.6) 'dNt \ d N i ] dx dN: < — -dy dNt dN: > dr\ dNt I dz , (D.7) 144 APPENDIX E The Membrane Element [B] Matrix A n appropriate set of shape functions, (call them M,- to distinguish them from the concrete element shape functions N{), for the membrane are only a function of the middle surface coordinates, which define the membrane geometry: r ~\ X I > , ( $ , T I > middle surface X yi (E . l ) middle surface Thus i f the coordinates of the nodes are known in x, y and z space we can define two tangent vectors to the middle surface, u and v evaluated at each node j: u. = I > 1 1 r a$ - T V dMi V; = I > L 1 r 01 dM,. x, + JY / i t V dMi x, + J> l-, i * \ yi + KT 9 M . f an (E.2) Where the symbol, j , means the function that precedes it is evaluated at node j. From which the third basis vector is defined as the cross product of u and v scaled to unity: ' 3 ; Uj X V j The remaining basis vectors are defined as: U; Ui Uj X V j ; e2. - e3j x e l y (E.3) (E.4 a,b) The basis vectors are computed at all of the nodes comprising the finite element and stored as variables in the computer program. Once the nodal basis vectors are known we may find the basis vector for any coordinates (^,r)), such as at the Gauss points, v ia the shape functions: 145 '2/ (E.5) l e 3 j In a numerical computer program application, the process of finding the basis vectors by equation E.5 from their nodal values is faster than using equations E.2-E.4 every time they are needed. Figure E.1. Local basis vectors at a typical node. N o w that the basis vectors are defined, we may define the global Cartesian coordinates (x,y,z) o f any point as a function of the curvilinear coordinates (£,,r|,Q: xi m 3 i rh.. (E.6) ' middle surface The terms / 3 , TW3 and « 3 are the components (direction cosines) of the third basis vector in the global Cartesian coordinates. Equation E.6 also allows us to determine the Jacobian of the transformation that shall be needed to define the strains in the element: 146 [J]= dx dy dz' ~&i &> di dx dy dz &\ an dr\ dx dy dz _5C dM, d^ dM, an dM, dk dM, dr\ v ( tr )dMi v ( ^ \dMi ? l z ' + 2 ^ J ^ n -(E.7) A s with the case of the concrete element we define the three continuum variables u, v and w, that are the displacements with respect to the global basis vectors I, J and K to be interpolated from their nodal values via the shape functions: u ~MX 0 0 • - M8 0 0 V > = 0 M\ 0 • •• 0 M 8 0 w 0 0 M , • • 0 0 My 3x24 a a a. a a y* a z8 24x1 (E.8) We note that the displacements are not a function of the depth through the membrane, (Q, since we are assuming that any displacement gradients through the depth of a membrane are negligibly small. Following the procedure as in Appendix D , the strains with respect to the global Cartesian frame are defined by: From the definition of engineering strains and equation E.8, the [B]membrane matrix can be implied from the following equation: 147 

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