NUMERICAL MODELLING OF EXPERIMENTAL DATA OF REINFORCED CONCRETE BEAM-TO-COLUMN JOINTS by Miljenko Baraka B . A . S c , (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 degree at the thesis in University of partial fulfilment of British Columbia, I agree that the freely available for reference and study. I further copying of department this thesis for or by his or the - requirements representatives. an advanced, Library shall make it agree that permission for extensive scholarly purposes may be granted her for It is by the understood that head of copying my 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 DE-6 (2/88) Jfe'C, -23, WJb . 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 o f view o f an energy dissipating mechanism. Modern design codes today have stringent guidelines on the design o f 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. C y c l i c testing o f reinforced concrete beam and column sub-assemblies have proven that a very substantial increase i n bending and shear strength can be achieved i n 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 o f 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 o f the joint area. The program utilizes plasticity based constitutive descriptions o f the concrete and steel material models and intends to be able to predict the behaviour and peak values o f the strength envelopes o f the joints. Comparisons with available experimental results were encouraging insofar as the plastic behaviour o f the concrete and steel were captured. Due to the complex nature o f the problem the program was unable to accurately predict the maximum load carrying capacity and more research is required i n "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 4.1 Overview 70 . 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 CHAPTER 5 ANALYSES WITH THE PROGRAM 89 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 CHAPTER 6 CONCLUSION 100 105 Summary 105 Concluding Remarks 105 Future Work 107 REFERENCES 109 APPENDIX A 133 v Some Mathematical Facilitations: Procedure for determining 133 CT^ 134 Perfectly Plastic Model 134 Hardening/Softening Model 135 APPENDIX B Element Shape Functions 138 138 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 The Three Point Gauss Quadrature Rule APPENDIX D The Concrete Element [B] matrix APPENDIX E The Membrane Element [B] Matrix 142 142 143 143 145 145 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 Table 5.4. Material parameters in passive confinement test 97 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 Figure 1.7. Schematic of the California State University parkade 8 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 Figure 2.23. Corner stiffener placement on second square specimen 36 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 Figure 3.5. Variation of cohesion with hardening/softening 51 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 at middle layer of Gauss points in the z 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 Figure 5.11. Brittle response in uniaxial tension 105 108 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 Figure 5.21. Response of the 4 concrete element model in Phase 3 116 117 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 SYMBOLSIN ORDER OF APPEARANCE Differential in total strain vector i n Global Cartesian coordinates. {ds'} Differential in total strain vector i n L o c a l Cartesian coordinates. {de } Differential in elastic strain vector i n Global Cartesian coordinates. {dzP} Differential i n plastic strain vector i n Global Cartesian coordinates. e dz ,dz ,dz ,dy ,dy ,dy^ e e x e y z xy xz Components o f differential elastic strain i n Global Cartesian coordinates. dz ,ds ,de ,dy ,dy ,dy p p p y p p xy p xz yz Components o f differential plastic strain i n Global Cartesian coordinates. {da} Differential in stress vector i n Global Cartesian coordinates. Differential in stress vector in L o c a l Cartesian coordinates. [V] Linear elastic constitutive matrix. Y i e l d function.. K Hardening/softening parameter. dk Magnitude o f the plastic strain increment. H Hardening/Softening modulus. Plastic flow function. [D ] ep Elasto plastic matrix. Components o f the stress tensor. Components o f the deviator stress tensor. Kronecker delta. First invariant o f stress. Hydrostatic pressure. Second invariant o f the deviator stress tensor. Third invariant o f the deviator stress tensor. xiv Square roof o f the second invariant o f stress. Angle measure o f the third invariant. Differential in the tensor plastic strain. Differnetial in the deviator o f the plastic strain. Differential i n plastic volume expansion. Differntial in plastic distortion. Parameter i n perfectly plastic DruckerPrager yield function. Parameter i n perfectly plastic DruckerPrager yield function. Cohesion. Friction angle. Concrete strength in compression. Concrete strength i n tension. Dilatancy factor. Parameter that locates the apex o f the erfectly plastic Drucker-Prager yield function. Principal stress vector. Principal stress direction vectors. Components o f the principal stress vector. Strain normal to a crack plane. Strain transformation matrix. Shear modulus. Youngs modulus. Poissons ratio. Rate o f shear modulus reduction. Mobilised cohesion. Mobilised friction angle. Parameter i n hardening/softening DruckerPrager yield function. K Parameter i n hardening/softening DruckerPrager yield function. y* Mobilised dilatancy angle. <j>* Mobilised friction angle. <t> Value o f mobilised friction angle at which a transition from compaction to dilatancy occurs. a* Parameter that locates the apex o f the hardening/softening Drucker-Prager yield function. cv A r ( a „ , , a * , / , A) Function relating damage i n concrete to plastic distortion. Y Initial yield stress o f steel in uniaxial / ) c tension. Y (K.) Y i e l d stress o f steel i n uniaxial tension. x,y,z Global Cartesian coordinates. x',y',z' Local Cartesian coordinates. {Ae} Finite increment i n strain vector. [B] Strain-displacement compatibility matrix. {Aa} Vector o f nodal displacement increments. {Aa} Finite increment i n stress vector. {P} Vector o f external forces. {AP} Increment i n the vector o f external forces. W Virtual work. {a} Virtual displacement. {5a} Corrective increment to displacement. e Euclidean norm. [K'] Tangent stiffness matrix. U [K'] Initial tangent stiffness matrix. {AO} Unbalanced load vector. ^,ti,(^ Dimensionless coordinates. [J] Jacobian matrix. q l ,m\,n x x Direction cosines. xvi Global Cartesian base vectors. Local Cartesian base vectors. CHAPTER 1 INTRODUCTION During the past two to three decades, building codes i n Canada, the United States and Japan have undergone significant changes by updating design rules to address the effects o f earthquakes. Most o f the current design rules regarding building member resistance have been established during the last two decades from experimental work, a great deal o f which originated i n N e w Zealand and North America. M u c h was also learned from the observation o f failed buildings and bridges in past earthquakes, which prompted the adoption o f modern design philosophies. Extensive research has been conducted to determine the response o f buildings during earthquakes and to predict the forces generated i n a structure as a result o f seismic motion. In a simplified approach the basic premise o f a seismic load is that o f a lateral load, which is a percentage o f the structure's total weight, distributed vertically according to the mass distribution and predicted accelerations. A case in point regarding the evolution o f this lateral load i n design codes can be made using Japan as an example [1]. The first time a seismic load was included i n Japanese building codes was in 1924, which prescribed a load representing building weight, applied uniformly over the height. 10% o f the In 1950 the seismic design coefficient was increased to 20%. After the Second W o r l d Conference on Earthquake Engineering held i n Japan in 1964 the seismic coefficient was changed to 20% for buildings up to 16 m i n height and increased by 1% for every 4 m increase i n height. The distribution o f lateral load was still uniform. After the 1971 San Fernando, California earthquake, the Ministry o f Construction o f 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 o f 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 o f 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 i n pairs and joined with lintel beams. These can be located throughout the structure and/or comprise the elevator shaft i n the case o f multistorey structures. Shearwalls are used in a wide variety o f buildings ranging from single storey warehouses to multistorey buildings. Link B e a m s 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 o f traffic. • Moment resisting frame type structures provide an open plan system comprising o f interacting beams and columns. Lateral resistance o f such a building is provided predominantly through bending action o f 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 o f moment resisting frames is the focus o f study i n this thesis. Quite often it is impractical to design a structure to remain elastic during a major earthquake, as the overall cost o f 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 o f the system, without collapse o f any vital parts. These locations may be the bases o f shear walls i n shear wall type buildings, the bases o f columns i n bridge bents or at the junctions o f 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 o f motion through hysteretic damping. This damping component significantly reduces the forces experienced by the building. 3 2. The location o f the yield zones and mechanisms must be carefully chosen and detailed by the designer to avoid undesirable modes o f failure such as shear in reinforced concrete members. 3. The overstrength o f 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 i n detailing to allow for the formation o f plastic hinges. To avoid the possible collapse o f such buildings in future earthquakes retrofit methods must be explored. Unfortunately, design codes do not, as a rule, deal with retrofitting o f 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 i n the following manner: • Having calculated the total elastic base shear V e = v S I F W , as given i n the National Building Code o f Canada ( N B C ) , the design base shear V is 0.6Ve/R, where R is the ductility factor o f 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 o f equivalent static lateral loads, (Figure 1.3). The beams are required to resist the resulting factored moments (Mf) and overstrength shears (Vo) as shown i n 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 o f the beam when the material resistance factors, <)> and <|), are taken as unity and using f c s c 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 i n 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 i n a plastic hinge. The shear resistance o f 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 i n double curvature bending o f the upper and lower column at the joint in question. To achieve ductility i n the beams, the concrete code requires closely spaced closed stirrups i n the plastic hinge regions o f the beams and closely spaced ties in the ends o f the columns to confine the concrete. Furthermore, ties must be placed i n the beam to column joint which is a region o f compound forces and especially susceptible to shear failure. M a n y 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 L o m a Prieta earthquake was the collapse o f a two kilometer stretch o f the N i m i t z freeway i n Oakland (highway 1-880) [4]. The structure was completed in 1957 and consisted o f 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 o f creep, shrinkage and temperature forces. Consequently, when the earthquake occurred, the base o f 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 i n the joint and only minimal shear reinforcing (ties) in the column, resulted i n a brittle joint and column failure. More recently, during the 1994 Northridge earthquake, failures involving shear i n columns occurred i n a parkade at the California State University [5]. The structure consisted o f 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 o f the structure is shown i n Figure 1.7. The interior gravity columns were not detailed for ductility and subsequently suffered a brittle shear failure. 8 The loss o f 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 i n grouted steel jackets as was done with many bridge columns on California's freeway bridges following the 1989 L o m a 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 o f 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 o f a plastic hinge. 9 In the process o f 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 o f the seismicity o f the local area. The objective o f the design process is to ensure that the member elastic forces calculated i n 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 i n 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 o f a structure subjected to earthquake ground motion. In performing such analyses the solution algorithms o f these programs take account o f the changing beam and column member properties when these members yield. The output of the software consists o f 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 i n a steel shell w i l l have unknown values o f 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 o f the elements comprising such structures one may test scale model subassemblies i n 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 o f the experiments are 10 to be used i n design o f members for use in practical structures, which often are o f a larger scale and have more complex configurations than laboratory models. This thesis explores the modelling applications o f 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) o f the joint. Such information is important for assessing the capacity o f 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 o f 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 o f laboratory models to facilitate the design o f 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 o f concrete recently developed by J. Jiang [8] and implemented in a finite element analysis computer program by J. Jiang and F . A . M i r z a [9] to determine the response o f reinforced concrete slabs to loading. experiments. The computed load history results matched very closely those o f Initially the concrete constitutive model by J. Jiang was obtained as a subroutine and attempted to be used in this program i n 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 o f 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 i n the computer program. Such an approach required the writing o f 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 o f the application required several modifications during the development o f the program. In the following chapter some previous experimental work on the ductile cyclic load behaviour o f steel encased beam-column joint specimens are presented [11] along with recent experimental work dealing with an attempt to determine the strength o f the beamcolumn 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. M o d e l 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 o f concrete has been modeled [14]. The key features o f 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 o f the concrete model are important when considering the composite action o f the steel shell that provides the passive confinement. It is the tendency o f 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 i n turn provide passive confinement. Passively confined concrete, as was stated previously, displays increased ductility and crushing strain. In Chapter 4 the aspects o f incorporating the constitutive models i n a finite element computer program are discussed. The predictions o f the program are compared to some available load history experimental results in an attempt to compare the sophisticated with the simple constitutive models i n Chapter 5. 12 CHAPTER 2 EXPERIMENTAL WORK The experimental work o f this project was done in two phases. Hoffschild [11] investigated the effectiveness o f steel encasing as a retrofit method for reinforced concrete frames with weak joints. The author continued the study by determining the behavior o f the retrofitted joint region itself. A brief summary o f Hoffschild's work is given here. 2.1 Phase 1: Beam and Column Subassembly Tests During 1990-1992 Thomas E . Hoffschild prepared a series o f half-scale beam column reinforced concrete test specimens representing part o f a 2 storey frame [11]. The reinforcement details used were designed to the codes o f the early 1970's with the joint ties eliminated. The dimensions and reinforcement details o f the reinforced concrete specimens are as shown i n Figure 2.1. A l l the specimens were tested under cyclic loading in the loading apparatus shown in Figure 2.2, which consisted o f 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 o f which were pre-damaged before retrofitting. T w o o f the specimens were left undamaged. Later all were encased i n 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 ( C U ) , circular damaged ( C D ) , square undamaged (SU) and square damaged (SD). The purpose o f 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 o f the beams, the retrofit jackets were provided with small gaps as shown i n Figure 2.3. For the last test, additional gaps were cut as shown i n the photograph o f Figure 2.7. A s it turned out, pre-damaging o f the joint region had no effect on the performance o f the specimens. The added strength and stiffness resulting from the retrofit significantly changed the damage locations, which often occurred just outside o f 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 i n a very short plastic hinge with limited ductility. Consequently, three gaps were cut in the last specimen (Figure 2.7) to ensure proper performance o f the retrofit. This specimen exhibited a markedly improved rotational ductility o f 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 o f the four specimens are shown in Figures 2.4 and 2.5, which also indicate the location o f failure. 14 10mm <p hoops Spacing 70mm Column: 190x190 3 x 10mm <p 10mm <p stirrups Spacing 70mm 4 x 10mm Beam: 165 x 200 <p 2 x 10mm <p 2550 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 0.1 JOINT ROTATION a [rad] HYSTERESIS CURVE - RETRO-CD -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 o f the retrofit schemes as a whole, the performance o f the beam to column connection itself had not been assessed. To avoid potential brittle behaviour o f the frame, the joint area, which could have difficulties in confinement o f 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 o f these members. In M a y - August o f 1993 the failed specimens o f 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 o f 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. Filling the j acket with fresh concrete. The previous five steps are illustrated below i n figure 2.8. The three specimens that were salvaged i n this manner were one circular and two square jacketed specimens. The second circular jacketed specimen had been tested previously i n a pilot study by a group o f students as part o f 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 i n Figure 2.11 with the exception that the hold down points were not symmetrically placed about the centerline o f the joint, which eventually resulted in a failure occurring i n the column. Since the objective o f the test (to fail the joint region) was not realized, the results o f the test were inconclusive. A postmortem analysis o f the specimen revealed the formation o f 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 i n the joint panel to a flexural hinge i n 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 i n the student's test, the steel jacket prevented a shear failure o f the joint. Following the repair o f the salvaged specimens, a test rig was devised that held the specimen down securely and prevented a failure outside o f the joint region. Shown i n Figure 2.11 is the apparatus used to test the joint region o f the beam to column connection. The specimen was held securely down by four threaded rods that were bolted into the concrete floor o f the testing laboratory. A steel channel was placed across the member at each end to which the other ends o f the threaded rods were bolted as shown. The loading o f the specimen occurred by applying a cyclic displacement through a servocontrolled hydraulic actuator at the position indicated in the figure. A Linear Variable Differential Transformer ( L . V . D . T . ) was mounted 152 m m above the face o f the specimen for the measurement o f displacements o f the beam relative to the column. Rotations o f the joint are defined as the measured displacement at the ( L . V . D . T . ) divided by 152 m m , whereas the moment arm is referred to point A i n 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 i n real time. A n actuator displacement o f 2 m m forwards and backwards was chosen to plot the first hysteresis curve. A t this point it was not known what the range o f displacement would be required to the point o f failure. A l l o f 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 m m progressive increments o f the actuator. 22 The first specimen to be tested was the circular one. Strain gauges were placed on the jacket for the recording o f strains with a computer controlled data acquisition system. The placement o f the strain gauges is shown in Figures 2.12 to 2.15. In Table 2.1 are shown the readings o f the respective strain gauges at a thrust o f 100 k N on the actuator. Referring to Figure 2.11, a thrust force would be to the left i n the picture. Enough data was acquired to plot the moment vs. rotation hysteresis loops (Figure 2.16). The placement o f both the circular and square specimens was such that the actuator always formed a lever arm o f about 500 m m 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 o f 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 o f about 3.25. While the ductility o f the joint region itself is not o f major concern i n capacity design procedures, the peak strength being greater than that o f the adjacent members is more critical. A comparison o f peak strengths o f the joints and beams as conducted by Baraka and Hoffschild are made at the end o f 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 o f about 0.065 radians. The next specimen to be tested was the first o f the square specimens. Shown in Figures 2.18 to 2.22 are the placements o f the strain gauges, the strain gauge readings at a thrust o f 100 k N and the corresponding hysteresis curves. A s with the circular specimen, the square specimen failed by tearing o f the weld line along the tube connections. The failure occurred at a rotation o f 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 o f the square specimen, tearing o f the weld initiated in the corner o f the tube where the highest stress concentrations occurred. To improve the performance o f 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 o f the corner reinforced specimen display a remarkable improvement i n ductility, almost as much as the circular encased specimen. The post failure inspection o f the failed specimen (Figure 2.28) indicated that the tear i n the steel jacket went around the steel stiffeners as opposed to the weld seam failure observed i n the other tests. 2.4 Concluding Remarks Shown i n Figure 2.29 is a comparison o f the beam strengths as observed by Hoffschild [11] and the strengths o f 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 i n Hoffschild's tests. This is a desirable performance feature o f 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 o f inadequately reinforced concrete member joints. Careful attention must be paid, however, to the design o f a retrofit scheme by considering the possibility o f a shear failure outside the retrofitted region due to higher moment gradients resulting from flexural overstrength. The observed improvement i n ductility which was achieved by adding steel stiffener elements, indicates that careful detailing o f beam column joints is needed to avoid premature failures. This also may lead to further simplifications by using methods that are easy to implement i n the field such as by building a retrofit cage around a beam-column joint composed o f steel angles and bars as suggested by Alcoccer [13]. 24 A finite element analysis technique w i l l be explored next that incorporates a hydrostatic pressure sensitive concrete constitutive model to predict the behaviour o f 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 Actuator L.V.D.T. 496mm [ J152mm •I 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 124mm Front #1 #2 Actuator <T #6 #5 200mm #12 0 End of tube #11 60° arc Center Line #13 c=> #15 #18 \\<=iu #17 Figure 2.13. Strain gauges on circular specimen, side view. 160mm End of tube ,#19 #20 ,#22 #21 [ 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. Actuator Front Back 356mm #8 60mn+ 40i 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 110mm #10 125mni r~'—l#9 lOOmrrl Figure 2.21. Strain gauge p 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 o f 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 #1 Back #4 #3 i i 356mm #2 Figure 2.25. Strain gauge placement on second square specimen, side view. #8 #9 #5 #7 #6 n n n 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 CIRCULAR 97.6 kNm I I 104.6 kNm 139.0 kNm 106.3 kNm I I 118.6 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 o f earthquake loading on structures and the retrofitting o f 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 i n a steel jacket, an increase i n concrete strength and ultimate strain can be achieved. It remains a very difficult task, however, to predict the behavior o f 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 o f changing certain design parameters. A non-linear finite element model was developed as part o f this project. A n important part o f the model, however, is the choice o f appropriate material models. In this chapter a set o f constitutive models, described i n the context o f 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 o f 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 o f concrete as a function o f confining pressure, which i n the continuum mechanics context is represented by the first invariant o f stress or the hydrostatic pressure. Increased strength and strain range to the point o f crushing are important features o f concrete in a beam column joint required to undergo plastic hinging. The characteristics o f three plasticity based constitutive models are investigated for incorporation i n 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 o f materials exhibiting plastic deformations the assumption is made that the total strain increment at a point is composed o f an elastic strain component and a plastic strain component, which i n the case o f general three dimensional engineering representation both have six components o f strain: {de} = {d£ } + {d£ } e p d€ de ' p x de ds e {*} = p y y del dz df dy ' df dy p z (3.1 a,b) p xy xy p xz xz dl" yz The elastic component, {ds }, is what gives rise to the stress increment i n the material at e the point i n question: {<fo} = [D ]{<fe } e e (3.2) 44 where the matrix [D ] is the constitutive matrix o f linear elasticity. A l s o central to the e notion o f associated and non-associated plasticity are the yield criterion and plastic flow rule: F({CT},K) = 0 = eft. j^j yield criterion (3.3) plastic flow rule (3.4) The plastic flow rule states that the strain increment is proportional to the gradient o f a scalar function, V|/({CT}), called the plastic potential or plastic flow function. The scalar parameter dk represents the magnitude o f the plastic strain increment and is given by dk = f Vk*i (3.5) dF_ da The yield function is generally dependent on some hardening/softening parameter, K , which represents the evolution o f 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) ]{de} (3.6) ep [D-] [w ] p = 8F_ da [jy ] = [jy] p da [da M | 3 [D'] When yielded + H When elastic (3.7) The parameter H is known as the hardening/softening modulus i n 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 o f stress and strain, a three dimensional tensorial representation w i l l be employed from here forward. To facilitate the work here the following stress tensors and stress invariants are defined: The deviator o f the stress tensor: The first invariant o f stress: u s = a !/-^ // ** 6 3 1= The second invariant o f the deviator stress J - (3.11) 1 (3.12) o o s 2 s 2 1 The third invariant o f the deviator: J Square root o f the second invariant: a •= Jj 3 — 2 (3.13) i.i .i S S kSki (3.14) 2 sin -1 9 (3.10) 1 The hydrostatic pressure: Angle measure o f the third invariant: 9 = ( - ) a { I 3V3 J) 3 ; -7c/6<e<7i/6 (3.15) * J 2 3 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 ,J ) or ( a , a ,6). 2 3 m In addition the following plastic strain tensors and plastic strain measures are defined: Increment o f plastic strain: de? (3.16) Deviator o f the increment in the plastic strain: de[] = dz . - ^ Syde^ P (3-17) The first invariant o f plastic strain. (Measure o f plastic volume change): dz^^de^. (3.18) The second invariant o f the deviator o f plastic strains. (Measure o f plastic distortions): de p = ^[de jde j) P P (3-19) 3.3 A Perfectly Plastic Drucker-Prager Model A perfectly plastic model, (H= 0 i n equation 3.8), to be implemented for concrete is based on the simple hydrostatic pressure sensitive Drucker-Prager yield criterion which is an approximation o f the Mohr-Coulomb yield criterion: 46 F(a ) = 3aa + G-K iJ m =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 i n Figure 3.2. The parameters K and a can vary as a function o f the plastic strains i n 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 i n the Mohr-Coulomb failure envelope for the material. The model as stated requires only two parameters to define it, either the friction angle, stresses at failure, f c and cohesion, c, or the uniaxial tensile and compressive and f . t Given any o f the two parameters the remaining two may be calculated as: Jc _ 2ccos(<j>) _ 2ccos(c|)) ~ ^ : 7TT ' / / 1 - sin((j)) l + sin((j)) (j) = sin or f'c-f, (3.22) (3.23) fc+f,. These relationships are readily determined from the geometric relationship o f the plot o f the strength envelope in compression and o f the M o h r circle representing the state o f stress o f uniaxial compression. The gradient o f the yield function is in engineering notation: T s„ y l dF 1 1 1 3 0 2a —• do J c « > 0 0 2t Volumetric Deviatoric Component Component (3-24) In the plastic modelling o f geologic materials and concrete the plastic flow function, is chosen to be non-associated with the yield function, F. The amount o f plastic flow for concrete is non-associated with respect to the volumetric expansion. The form o f the yield function gradient can then be stated as follows: 48 Y ' 1 So y 1 < •+ 5a„.3 0 7 ' s ay 1 1 d\\i * s 2a 2x > 0 0 Volumetric V Deviatoric Component Component 2 (3.25) where: v dx = p- 5 F (3.26) The parameter p has the interpretation o f a plastic dilatancy factor (Bazant [15]) since it has the effect o f increasing or decreasing the sensitivity o f the total plastic strain increment to the volumetric component. The importance o f plastic dilatancy i n concrete w i l l be discussed later. For the case where (3 = 1 the plasticity is associated i n which case {dF I da} = {dy I da}. For a real material the factor P has to be determined experimentally but i n general a value less than unity w i l l give realistic results. 3.4 Response in the Tension Regime, Method 1 While the yield function o f equation 3.20 may be calibrated to provide realistic results in compression, in tension it w i l l give erroneous results. Values o f 10 M P a and 30° for the parameters c and (j) respectively w i l l yield f c = 34.6 M P a and f = t 11.55 M P a . A value o f compressive stress o f 34.6 M P a is not unreasonable, while a high value o f 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 i n the following way; i n the compression domain (a m < 0) by equation 3.20, while i n tension ( a m >= 0) by a connecting surface that makes a smooth transition from the function o f equation 3.20 down to an apex that is independent o f the parameters a and K. The modified yield function is the following equation: 49 F(a ) = tJ {3ao -K) 1m 5L» 2 2 -K 1^A J M +a =0 for a,„ > 0 F ( a , ) = 3aa„, + 5 - ^ = 0 for a„, < 0 (3.27) 30 a Figure 3.3. Modified yield function in meridional space. A l l o f the equations o f the previous section are still valid with the exception o f the expression for 8F/da m which takes the form: ( dF = 3 a 1do.,, -2{3aa -K)^f + 2K m C\„ in ^2 \( \2 m — o"„, m for a„, > 0 (3.28) dF 3a for <y < 0 (3.29) m The parameter <J locates the apex o f the yield surface i n the tension domain (<j A m Generally the parameter > 0). is dependent on the concrete tensile strength,^, and cohesion, c, a procedure for determining this parameter is given i n Appendix A . The modified yield function is shown in Figure 3.3 for the following values o f constitutive constants: (j) = 30°, c = 10 M P a , CJ4 = 5 M P a . This framework allows for the specification o f the response i n compression v i a parameters c and ((> and the response i n tension v i a the parameter a (cf ). A t Alternatively it sometimes may be useful to calculate (j) from c a n d / c by rearranging the first expression i n equation 3.22: 50 -Ac 1 <> | = sin (3.30) 3.5 Response in the Tension Regime, Method 2 A n alternative method to model the response i n tension is to release the tension i n the direction o f maximum principal stress and thus model the formation o f a crack. To incorporate the tensile stress release i n the direction o f maximum principal tension the initial stress "vector {a} i is transformed into the principal stress vector {a}' at every initia load step: [T(v„v ,v )]~ {o}. . = {d} 2 3 (3.31) B <fa/ 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) o f 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 CT 3 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 exceeds the concrete tensile strength f stress is released as follows: 3 t 0 { C T }^=i L a /-[ ] T f a (3.32) The left-hand side o f 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 [D ] are zeroed out. e 2. The strain normal to the crack plane, s „ , is calculated from the strains referred to the c r 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.„) G = G^-Bs ) 23 ncr ; (3.34) (\-Bz )>0 ncr The parameter B is conveniently referred to as the rate o f shear modulus reduction. The resulting elasticity matrix takes the form: £(l-v) Ev 0 0 0 0 0 0 0 0 0 0 0 0 0 G & O - B O 0 0 0 (l + v ) ( l - 2 v ) Ev (l + v)(l-2v) E(\-v) (l + v ) ( l - 2 v ) 0 (l + v)(l-2v) 0 0 0 0 G„ 0 0 0 0 0 0 G° (l-Bs ) 23 ncr (3.35) The resulting matrix, [D ]', which is only valid i n the principal stress space must be e transformed into the global coordinate system: The subscript 1 on [D ]j is used to indicate that one plane o f cracks has occurred. The e above process can be continued for the next set o f cracks by determining the direction vectors o f the next crack plane, zeroing the third r o w and column o f [ D ] j and e transforming it into [D ] : e 2 Two sets o f crack planes are supported i n the computer program that was written with a minimum angle between crack planes arbitrarily chosen to be 30°. This method o f 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 DruckerPrager model. In equation 3.21, instead o f 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: (3.38) c = cexp KE sin(|)* = 2 K + 7 sin(j) S 7 sine))* = sin<|) for K < S for K ( (3.39) > s/ The parameters a and K i n the Drucker-Prager yield condition now become: <X*(K) = - 2sin(() V3(3-sinf) 6c* cos<j)* ' " (3.40) V3(3-sin<|>*) The parameters c* and <)>* have the interpretation o f mobilized cohesion and mobilized friction angle while c and <j) represent their ultimate values. The parameters s and sy-need c to be determined experimentally. The graphs o f 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 2.00000 K/Sy 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. mobilized. The friction on the other hand starts at zero and eventually is fully To determine the form o f the plastic flow function, we use the same arguments as in section 3.4 i n modifying the 8FI d<3 term by the dilatancy factor, (3; that M the rate o f plastic volume production must be non-associated with the yield function, F, and hence d\\i I da m * dF 15a . m However, i n this case where hardening and softening taking place a new parameter a'(K) is defined : CC(K): 2siny*(K) V3(3-siny*(K)) (3.41) The parameter Y*(K) is conveniently referred to as the mobilized dilatancy angle which is in general independent o f the mobilized friction angle, (J)*(K). Hinton and O w e n [16] define the mobilized dilatancy angle to be a function o f the mobilized friction angle as: siny sin<j)* -sin<j) cv l-sin(|>*sin(|> , (3.42) C) The value o f <|> represents the value o f the mobilized friction angle <> | * at which a cv transition from compaction to dilatancy takes place. 55 Response in the Tension Regime If Method 1 is chosen to model the response i n the tension regime then the yield function is similar i n 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 , ) = (3a*a,„-/:*) 1- ( 1- -K' o\<0O + a =0 vMK) for a„, > 0 F(Og) = 3a*a„, + a - K* = 0 for o m (3.43) <0 (3.44) The gradient o f the yield function and plastic potential are still given by equations 3.24 and 3.25 with the exception o f dF I da and d\\i 1d<3 which become: ( \ 2 ( \ dF o„ (K) = 3a* 1- 2 ( 3 a a - ^ ) ^ - 2^* a ^(K) ,a*(K), vM )y 1<^( )J m M M + K K dF 5a„ d\\i (K) = 3a' 1- -2(3a'a -r)^m + (K) = 2^ 3a for a,„ > 0 (3.45) for a„, < 0 (3.46) a*5i(K) v ^ ( ) y a 9\(/ &7 ( K ) = 3GC' K * 2 / \ for a,„ > 0 (3.47) for a,„ < 0 (3.48) Here it is proposed that a* be a function o f the. hardening/softening parameter, K. The A evolution law that is chosen for a A is the same one as for the cohesion: (3.49) The above formulation w i l l ensure that the apex o f the yield surface moves toward the origin o f the a and o~„, space as the cohesion vanishes. The non-associativeness is 56 achieved by replacing a * i n the expression for dF I da with a ' to obtain the expression m for dy I da . m The hardening/softening parameter, K , i n the context o f the present model represents the amount o f damage i n the material. A definition for it is somewhat o f an open question, but basing it on the second invariant o f the deviator o f plastic strains seems appropriate for geologic materials and concrete: dKocde" (3.50) The parameter dz , as was stated i n section 3.2, is a measure o f plastic distortions. The p amount o f plastic hardening or softening must be sensitive to the confining pressure. A s the hydrostatic pressure increases the rate o f damage i n the material slows hence K must be inversely proportional to the hydrostatic pressure. A l s o as the hydrostatic pressure becomes tensile near the apex o f the yield surface, the rate o f 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 i n the material must be a function o f plastic distortions and the stress level. The functional form that is chosen is as shown below: d = (3.51) K r(.o ,<y ,f ,A) m A c The form o f the function f that w i l l give the appropriate behaviour is: F r(a,„,a^,/ ,^) = A c V O for cr „ < 0 fc J f - for 1V (3.52) \ A G,„ > 0 *^(K). Where A is a constant and f is the specified concrete strength i n uniaxial compression. c The negative sign is placed i n front o f a m because the concrete specified strength is a positive number and the hydrostatic pressure is negative i f compressive. F r o m the definition o f the plastic distortion o f equation 3.19 and from the second term i n equation 3.25, then: 57 de" =dk dev dev (3.53) 1 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: dK = dk 1 (3.54) Or alternatively: dK dk 1 (3.55) r(a ,a ,f ,A)J2 m A e If Method 2 is chosen to represent the response i n tension then the cases o f functions i n the compression domain (a m < 0) would be used for both a m < 0 and a m > 0 with the exception that tensile stresses i n the direction o f maximum principal stress would be released at the concrete tensile stress in accordance with Method 2. 58 Hardening/Softening Modulus The hardening/softening H = -(dF I 9K)(5K / dk). modulus H was previously defined as If the expression for H is expanded by the chain rule o f differentiation and substituting the expression for dK /dk then: 1 dF da d<j>* (° ,G ,f ,A)j2 da <9<j) DK Hr m A c + dF_^dKdc_ dK dK 5<j) 8K + dc 8K df^ (3.56) The partial derivatives in the square brackets are readily determined as: 8F 3a„ da SF -1 dK da 2cos(j)* 2sin(|)*cos(|)* •+ V 3 ( 3 - s i n f ) " V3(3-sin(|>*) af A df dK dK dc d^ dK dK /KS7(K + S ) / -2- for K 6cos(j)* < s f (3.60) for K > s (3.61) V3(3-sinc|)*) ' K ' exp -6c*sin<j)* V3 (3 - sin f ) (3.59) COS(j) / = 0 = -2c (3.58) 2 sin(() 4 KS, (K + B ) (3.57) (3.62) 6c* cos ^* 2 + V3 (3 - sin (j>* (3.63) ) 2 It is the author's opinion that the expression for mobilized friction i n equation 3.39 is unsuitable because the derivative introduces a singularity i n 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: 59 l i m d.K = "~ dk m A $} Mf{ =0 r(a ,a ,f ,A)J2 e r(a ,G ,f ,A)>f2 m A c +H (3.64) A s an alternative to equation 3.39 the Author proposes the following expression which introduces no singularities in H.\ ( o K K sinij) = 3 3 £ \2 K + I /J sin<)) for K < e f 8 / sin<t>* = sin<() ^ K ^> ^ for (3.65) 8 /, 0 The bracketed term [] is shown below i n Figure 3.7 for visual comparison to Figure 3.6. K / E / Figure 3.7. Proposed function of mobilized friction as a function of K. A n d the derivative d$*/dK is: r 8K 3 - "i K K 6—T + 3 — r sin(j) 7 for K < 8, COS(() =0 (3.66) for K > E, ^ A s a case i n point the hardening/softening modulus H is plotted i n figure 3.8 for the following values o f 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 MPa + 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 i n concrete comes from the sliding o f crack surfaces past each other [15]. The phenomenon can be illustrated with the aid o f the following figure: Figure 3.9. Plastic dilatancy as a crack sliding phenomenon. 61 Such behaviour is not observed i n steel and hence the steel constitutive model allows for plastic shear deformation under constant volume as w i l l be shown i n a later section. Passive confinement o f concrete by steel is initiated by the concrete plastic volume expansion which is associated with plastic deformation. Therefore, when modelling the confinement effect o f steel on concrete i n the plastic range one must try to reproduce the plastic dilatant behaviour o f 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 i n equation 3.18 the volumetric plastic strain increment can be derived, dz = dk(d\\i 13c ) / 3, while the plastic p m distortion increment was already implied to be dz p = ^dejjdej} = dX IV2. The ratio o f the two is the rate o f plastic expansion to plastic distortion. For the case where the DruckerPrager model is perfectly plastic the ratio is: d^_ < = dz = V2jfy dK p V2 _a^_ = p 3 d<J m 3 ( 3 6 7 ) da m A n d for the case where the hardening/softening rule was introduced: < = < dz = dK p ^ ( 3 K da m A s a comparison o f the above two equations the quantities Figure 3.10 for the case o f ^ = 45°, z = 0.01, § f cv (3.68) ) and K are plotted below in = 20°, p = 0.5, u m = -3 M P a . 62 hardening/softening -0.0005 |> 0.002 0.004 0.006 0.008 0.01 -0.001 Figure 3.10. Plastic volumetric expansion vs. plastic distortion. From the (d\\) I da m above figure it can be seen that for the case o f perfect plasticity = const) the plastic volumetric strain accumulates linearly at a slope o f V2f3a. For the case where hardening and softening take place (d\\i I da m SIJ/(K) / 9 a ) the m 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 i n tabular format. Table 3.2. Parameters in the Drucker-Prager model. Parameter Perfectly Hardening/ Plastic Softening Model Model Yes Yes Description Peak value of Friction Angle c Yes Yes Peak value o f Cohesion P Yes No Dilatancy Factor <|>*(K,S ) No Yes M o b i l i z e d Friction Angle C*(K,S ) No Yes M o b i l i z e d Cohesion No Yes Mobilized F C Dilatancy Angle K No Yes Plastic Damage Parameter e f No Yes Parameter i n equation 3.39 e c No Yes Parameter i n equation 3.38 <t>cv No Yes Parameter i n equation 3.42 A No Yes Parameter i n equation 3.52 B Yes Yes Rate of reduction shear ( modulus 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 ) = V 3 o - r „ ( K ) = 0 / y (3.69) a.3 A al Figure 3.11. Von-Mises yield surface in principal stress space. The parameter Y is the yield stress i n uniaxial tension which is i n general dependent on u the hardening/softening parameter H. The parameter H can be interpreted from an uniaxial stress test as the ratio o f the change i n yield stress to the change i n 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 E is the post yield tangent modulus and E is the elastic modulus. t where the steel is perfectly plastic H = 0. For the case Since we are dealing with an associated plasticity model the gradient o f the yield function and the gradient o f the plastic flow function are the same: 5a dF_ V3 s. 5a 2a 2x xy 2x 2x Deviatoric (3.72) 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 o f stress. The parameter K in the Von-Mises model has the interpretation o f the amount o f plastic work: dK = {a} {ds } T = {of p dlf^pi (3.73) The evolution o f the yield surface can be derived from Figure 3.12 as follows: If Y u represents the present uniaxial yield stress then the amount o f plastic work, dK, is: dK = Y ds (3.74) p u The change i n the uniaxial yield stress is from equations 3.70 and 3.74: dtc dY = Hde = Hy p u u (3.75) u When one substitutes the general expression for the plastic work increment in equation 3.73 the change i n the uniaxial yield stress becomes: W h i c h is integrated to give: r^+S^+lfy^JX (3.77) A l l o f the variables in the integrand are functions o f X. The magnitude o f the plastic strain increment, dX, is defined in equation 3.5 in terms o f 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; do ', day* and dx y\ x x 67 {do., 1 da , y membrane (3.78) dx xy 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 o f the membrane and are generally stated as: del, dz", de , dz , dz\, dz , e y membrane p y p dy , dy dyl; dy ,. p c y (3.79) y p x dy . p yz 68 The three-dimensional elastic constitutive matrix that properly relates the membrane stresses and the elastic component o f strains is an expanded version o f the plane stress constitutive matrix which is given as: Ev 1-v 2 1-v E 2 1-v 2 0 0 0 0 0 0 0 0 0 0 0 0 symmetric G 0 0 0 0 [D-] L membrane J\met 0 (3.80) W i t h the exception o f the above stated differences, all o f the concepts developed in the preceding sections are equally applicable to the membrane constitutive model. 69 CHAPTER 4 T H E FINITE E L E M E N T S O L U T I O N 4.1 Overview The previous chapter dealt with concrete and steel constitutive relationships at an infinitesimal point and within infinitesimal increments o f strain. This chapter presents, i n the context o f finite element analysis, a numerical procedure for the solution o f problems o f finite size and deformation. The finite element numerical procedure may be thought o f as a bridge between the theoretical material representation o f Chapter 3 and the physical world. In the solution o f solid mechanics problems, one typically uses solid finite elements that have eight or more boundary nodes. In this thesis an element consisting o f 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 o f 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 o f the element surfaces and volumes by compatible shape functions. The strain increments, which are linear functions o f the gradients i n the displacement increments, are calculated at a number o f finite points i n the finite element called the Gauss Points. The Gauss Points are points within the element where quantities such as stress or elements o f 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 o f a truncated Taylor series o f 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 o f 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 o f the stress increment is accomplished by subdividing the strain increment into smaller sub increments for the purpose o f obtaining more accurate stress increments. The stress at a Gauss point at the end o f a number o f cumulative load increments is a function o f 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 o f more than twenty concrete elements. In the solution o f non-linear constitutive problems it is often necessary to find a solution to the finite element nodal displacements by a series o f iterations. The iteration procedure utilized here is the modified Newton-Raphson method [10]. In the adoption o f this method, a set o f known displacements are applied at the controlled nodes while the free nodes are assigned initial zero values o f displacement. A n unbalanced load is calculated (which represents the non zero value o f 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 o f 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 o f 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 o f the nodes. Controlled displacements are chosen as the independent variables instead o f 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 o f 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 i n the x, y and z directions respectively, then the continuum displacements are discretized i n terms 72 o f the element nodal displacements Aa , Aa xi compatible shape functions and Aa yi zi v i a a suitably chosen set o f Ffxy£): Au^Aa^x^z) Av = Y ^ F (x,y,z) (4.1a,b,c) i Aw - ^Aa^F^ (x,y,z) ; i = 1...number o f nodes v J y i The functions F ^ x j ^ z ) are at this point general. Shape functions specific to different elements are described i n Appendix B . The strains and stresses i n 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 o f the shape functions with respect to the coordinate variables x, y and z. The nodal degrees o f freedom can be conveniently split into three components, namely those degrees o f freedom that are free to move, those that have prescribed boundary conditions and those that have controlled displacements applied to them ({$)f , {^boundary {^control)- The nodal forces can also ree be split up into their corresponding components: free, boundary and control. I f one applies a displacement increment in the set o f displacement control degrees o f freedom {Aa} con/ro/ a set o f equilibrium stresses, { A a ( A a con(TO/ ) } , is produced i n the continuum and a set o f contact loads, { A P ( A a , ) } , at the displacement controlled degrees o f freedom con ro/ and support points. The weak form o f the equilibrium equations is formulated by the method o f 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} {e}dV - {P + AP} {a} =W+AW=0 T r (4.4 a,b) V j" {a + A a f [ B ] { a } j F - JP + A P f { a } = W + AW = 0 v where virtual quantities are denoted by a tilde (~). Since, i n general, the body o f Figure 4.1 has been loaded prior to the application o f the displacement increment, { A a } quantities {o-(a conW ,the )} and {p(a , )} are the internal stresses and external loads prior to cojMTO/ con ro/ the application o f the displacement increment which are, in general, functions o f the total displacement, {%\ ob applied at the displacement controlled points. The quantities W contr and AW are the virtual work associated with the state prior to the application o f the displacement increment and the increment i n virtual work associated with the displacement increment. The state prior and after the application o f the load increment are equilibrium states, therefore equation (4.4b) can be split up into two equations, each of which is theoretically zero: J{°} [B](a}^-{P} {a} =^ = 0 V (4.5) J {Aa} [B]{a} dV - { A P f {a} = AW = 0 V (4.6) r r r It is stated here that the above two equations are each theoretically zero because, as w i l l be seen later i n this chapter, i n 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 o f freedom. A t the nodes corresponding to boundary conditions, the contact forces do not contribute to the virtual work since those degrees o f freedom cannot be varied. The displacement controlled degrees o f freedom are prescribed for the load increment (analogous to kinematic boundary conditions) and 74 cannot be varied either, thus the second terms i n equations 4.5 and 4.6, which represent the virtual work o f external loads, are zero: L {P} {5} = free J r ^ ^ ^ boundary f ^ (4.7) ' control And: to, { A P U = {O} {AP} '{a} = 7 =0 (4.8) {AP},' control The weak form o f the equilibrium equations are therefore: l{<y} [B]{a}dV = W = 0 T (4.9) v \{A<j} [B]{a}dV = AW=0 T (4.10) N o w , since it is argued that the variation i n the nodal displacements, {a} is arbitrary, the following terms i n equations 4.9 and 4.10 must be identically equal to the zero vector: j{oY[B]dV = {0} (4.11) r j{AG} [B]dV = {0} T 7 (4.12) A g a i n it is stressed that i n a numerical application the right hand side o f equations 4.11 and 4.12 w i l l only be approximately zero. 4.3 The Numerical Procedure The finite element solution o f problems dealing with non-linear constitutive relationships must be carried out i n a stepwise iterative fashion. The displacements must be stepped so as to permit the updating o f material constitutive properties at every step, since, i n general, they are load history dependent. The increments i n 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 o f Figure 4.1, a set o f displacements, {Ad} points o f load application, { A a } , is applied at the nodes corresponding to . The increment i n the entire displacement vector may then be represented mathematically as follows L {Aa} = {Aa}, boundary {Aa} control c o n W free ^ ^boundary {Ad},* control N o w there is a corresponding set o f displacements, { A d } with the { A d } J /ree , o f the free nodes associated displacements that w i l l produce a set o f stresses, { A a } , which make equation 4.12 equal to zero. The procedure for finding the displacements o f the free nodes upon the application o f displacements to the controlled nodes is the next topic o f discussion. The displacement increment, { A d } c o n m j / , produces a set o f strains i n the body and a resulting set o f stresses v i a equations 4.2 and 4.3. The integral o f equation 4.3 is evaluated as a simple sum: {e+As} {Aa}= J[D*(e)]{dfe}«-i:2;[D*(e + //JVAE)]{As} /=i where N is the number o f subincrements. Having obtained (4.13) { A a } from the above equation, the integral i n equation 4.12 is evaluated and is found not to equal the zero vector: J{Aa} [B]</K = {AO}%{0} r 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} /ree values are known, equilibrium is not satisfied, which is represented by the unbalanced force vector, {AO}. The method o f obtaining the {Ad} increment is the modified Newton-Raphson iteration procedure /ree [10]. A corrective increment, {5a} is computed i n the nodal displacements and added to ]5 {Aa} o f the load step: {5a} 1= -[K'] {A0} (4.15) o {Aa}, = {Aa} + {5a} (4.16) 1 The subscript, 1, indicates that this is the first iteration o f a series o f corrections. The tangent stiffness matrix, [k'] , corresponds to the boundary conditions o f the problem q such that: f {5a}, =• '( 1 free {^*^ ^boundary • L 5 D U" J ^ ^-^^^ control (4.17) = • 1 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}, {e+M {Aa},= (4.18) N J [ D * ( e ) ] { & } « - 2 [ D ( e + //^AE )]{Ae} (4.19) v 1 {e} / _ 1 1 J{Aa};'[B]^ = {A(D};' (4.20) Having obtained the unbalanced load from iteration 1, the next corrective increment, {8a} , can be computed and added to the total: 2 {Sa^-fK'JjAO}, (4.21) {Aa} ={Aa} + {5a} {5a} 2 ]+ (4.22) 2 The procedure can be continued indefinitely with each successive {Aa} being a better M approximation than {Aa} which w i 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 ^ . The modified Newton-Raphson method uses (4.24) the initial stiffness matrix throughout the iterations, (Figure 4.2) instead o f the updated one at every iteration point. This procedure takes longer to converge but is convenient from the point o f 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, [D P]. e This matrix is non-symmetric for the case o f non-associated plasticity and hence the updated tangent stiffness matrix is non-symmetric as well: [K']= J [ B f [D ][B]rfr v (4.25) ep The initial stiffness matrix, however, is based on the elastic constitutive matrix, which is symmetric, and is given as: [Kl 0 = j[B]> ][BK e (4.26) v One can visualize the modified Newton-Raphson procedure with the aid o f Figure 4.2 for the case with a single free degree o f freedom , Aafr , and one displacement ee controlled degree o f 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 o f both the controlled displacement and free degree o f freedom, A ( | ) ( A a contro i , A a f ) . For a load step A a ree c o n t r o ] is fixed, while the corresponding displacement in the free variable, Aafr , that makes A(|>=0, is found by a series o f ee 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 Aa i J + 1 = Aa- + ba ui iA (4.27) 6 a i,j = - ^ A ( t ij ) 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 o f 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)} o f equation 4.13 is a continuous function o f position inside the elements and continuum. T o apply the Gauss quadrature integration rule, the stress increment is determined at a set o f prescribed points, (x0j,z^), in the elements, called the Gauss points: {C+AE} {B} ^ (4.28) where TV is the number o f sub increments used i n computing the stress increment from the associated strain increment. The indices (ij,k) in parenthesis are not tensorial indices as used i n Chapter 3, refer to the Gauss point numbers i n each element. Simply put, the quantity {ACT . } is the engineering stress increment evaluated at the Gauss points (/7 A) which have coordinates (x^z^). Having obtained { A a ^ ^ J from the above equation, the integral i n equation 4.14 is evaluated by the Gauss quadrature rule i n each element sub volume and their contributions added to the total. The Gauss quarature rule is normally applied to a cubic element o f dimensions 2,2,2 with the dimensionless element coordinate axes Z,,r\,C, at the center o f the element. Subsequently i n applications to an element o f general dimensions i n 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: [J] = dx dy dz d\ dx 8^ 8^ dy dz dr\ dr\ dr\ dx dy dz ~d~i a ; (4.29) dc; 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} [B]JF = JJJ{Aa} [B]det([j])^^T r r V 1 (4.31) -1-1-1 Thus applying a three point Gauss quadrature rule the above is evaluated as: j ] j { A a } [ B ] d e t ( [ j ] ) « ^ = Z Z Z^^{ r ACT (u.*)}[ ('J.*)] ( 0.y.*)) = B det J * 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: V 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 o f twenty nodes, as described i n reference 18, with three degrees o f freedom per node for a total o f sixty degrees o f freedom. The representation o f the element is more conveniently expressed in dimensionless coordinates ^, n and C, as shown in Figure 4.3. The cube shown i n dimensionless coordinates has dimensions o f 2, 2 and 2 with center at (0,0,0). Although an element o f eight nodes would have been sufficient for the model, the choice o f an element o f 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. 81 2. Having three nodes on every side allows the element to be warped and curved i n the Cartesian x, y and z space, hence allowing more accurate modelling o f circular specimens with relatively few elements. 5 Figure 4.3. Concrete finite element. The shape functions o f the element in Figure 4.3 are derived in accordance with the methods given i n [18] and are shown in Appendix D with the [B] matrix . The Cartesian coordinates i n 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, a i(^0 (4-34) N yi w = N (^,r\,Q t ; 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 o f 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 i n the computer program must be a membrane o f general shape for flexibility. Later the results are applied to a flat membrane, since only the square specimen was chosen to be modelled because o f 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 o f freedom per node for a total o f twenty-four degrees o f 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 o f the cube. The properties o f the concrete element presented in the previous section can be applied directly i n the context o f 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 i n relation to the global Cartesian coordinates x, y and z. A t any point on the middle surface o f the shell a local Cartesian frame (x',y',z') can be defined that has base vectors S j , e and e 2 3 associated with it. The base vectors o f the global Cartesian frame are referred to as I , J and K . It is important to establish the relationships between the local basis vectors i n the membrane, that are a function o f position in the membrane coordinates (^,r\,Q, and the global basis 83 for the purpose o f defining the transformation laws o f 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 L Jo J L V [T1>"1 •'membrane!- sj L [TjB] ' Imembrane'- 7 -"- dV •'membrane (4.35) x ' The transformation matrix [ T ] is a function o f the direction cosines between the E base vectors i n the local frame x', y', z' and the global frame x, y, z. The details o f obtaining the ^>\ brane mem [TJ^[D ] e m e w £ r a 7 i e matrix can be found i n Appendix E . The expression [ T ] has the interpretation o f transforming the constitutive matrix from E 84 the local coordinate system to the global one. Vb ]membrane > * e sm e The elastic constitutive matrix, anisotropic expanded version o f the plane stress constitutive matrix given i n 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'} = [T ]{e} (4.36) {o} = [T.]V} (4-37) {&} = [Tj {<j} (4.38) E T where the symbol [ ] m e a n s the inverse o f the transpose and the primes are used to r indicate quantities referred to (x',y,'z') local Cartesian coordinates. l\ m\ n\ lm mn nl ll m\ n\ lm mn nl l\ m\ n\ lm 2l l x 2m m 2 2/ / 2 l 3 2l l 3 x 2 2m m 2 3 2m m 3 x x 2 3 x x 2 2 x x 2 2 mn 3 3 x 2 33 n 3 l (4.39) 2n n hi + h\ mn + mn 2n n lm + lm mn + mn nl + nl lm + lm mn + mn n l +n l x 2 2 3 2n n 3 x m 2 3 m 3 x 3 x 2 3 x 2 3 2 3 x 2 3 x x 2 3 nl + nl x 2 3 2 2 3 x 3 x x 2 3 where l m and n\ are the components relative to the global Cartesian frame o f the z'th v x base vector e -. Many o f the above concepts may be found i n reference [20]. The usage ; of the element i n the context o f section 4.2 is as follows: • From the nodal displacement o f the current load step and iteration, the global strains are computed at the Gauss points and the transformation o f equation 4.36 is applied to obtain the strains in the membrane local coordinates: {A8'} = [ T e ] [ B L • mArane {Aa} (4.40) Next, the local membrane stress increments at the Gauss points are obtained from equation 4.13: 85 {E'+AE'} N f [»Z {Aa'j = (£')]{ds'} mhrane M • *± - ^ [ D Z . ^ m h r + / / JVAs')]{As'} (4.41) 1 = 1 The element contribution to the unbalanced load is computed v i a equations 4.14 and 4.37 J[[Tj {Aa'}]r[BLmA_^= {AO} = r r ^membrane \{^'}\\}[B] dV (4.42) memhran ^membrane 4.6 The Reinforcing Steel Element The reinforcing steel is modeled by a bar element o f three nodes that has only one stress and strain component along its axis. In keeping with the previous discussion o f local and global coordinate systems, the element is defined to have its o w n local coordinate system (x',y',z') o f 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 o f importance is x' since the element has only one stress and strain defined along its' axis. The geometry o f the element is defined by isoparametric mapping, with only one 86 dimensionless coordinate, as shown i n Figure 4.5. reinforcing element shall be referred to as LfQ The shape functions for the and the coordinates o f the nodes as XJ, y\ and zf. i = Y,yM® y (- ) 4 43 i The nodal base vector corresponding to the x' local coordinate is easily derived as: u = I ( x - x , ) + J(y - v,) + K ( z - z , ) 3 3 3 _u (4.44 a,b) lul The axial strain i n the local coordinate system is given in terms o f the global strains as: s . = k } ' { e } = {e} r { , i; i } (4.45) {°} = K h ^ (4-46) 7 r t where the term { T 1 } is the top row o f matrix [ T ] . The initial tangent stiffness matrix r E l 8 for the element and unbalanced load vector contribution are: [ K ' ] = AE' j [ B f { T } { T f [B]dl o e l (4.47) £ l 0 {AOf = ^j^{Aa} [Bf{T }{T } [B>// r r E l E l (4.48) 0 The matrix [B] for the element is given as: 87 (4.49) dx Z ' A X 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 E t 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 i n 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. A p p l y displacement increment for the present load step, i, i n {Aa}. ^"control Iteration Loop To Iterate Out the Unbalanced Load 2. Calculate the strain increments at the Gauss points i n all o f the elements from: 3. Compute the stress increment at all o f the Gauss points from equation (4.3): {Ao} =2;[D*( )l{AE} u 4. (4.54) e Determine the load unbalance { A O } , for the load step from (4.9) using Gaussian integration. { A O } . . = {AO},., + | { A a } / [ B ] d V r ; * {o} (4.55 a,b) T v The term { A O } . , is the residual unbalanced load left over from the previous load step. Adding {AO}. l to the unbalanced load o f the current load step w i l l minimize the total unbalanced load, { o } , at the end o f 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 {5a}.. L J '.Jfree I H , - (4.56 a,b) (»•}„ ., = M »J control l W e note that the only degrees o f freedom that have corrections applied to them are those that are free. 6. G o back to step 2 with the updated values o f nodal displacements and continue until the corrections i n 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 i n nodal displacements to the total displacement from previous load stepjaWaj^+lAa}.^.. 8. A t all the Gauss points o f all finite elements compute the increment i n the hardening parameter AK; and add to the previous value. 8F AK = H : AI H da [D<] d\\i 8K 3K 5a [B]{Aa}i,last j (4.58) + H 9. Update constitutive parameters i n 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 o f the shape functions and possibly to detect computer coding errors. The elements used i n 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 i n detail. Finally, three models o f a joint, the largest one consisting o f forty concrete and sixty steel plate elements, were used in an attempt to replicate the load deflection curve that was obtained i n the laboratory for one o f the square specimens. The results o f 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 o f individual displacements or a total displacement that is 91 divided into a prescribed number o f equal steps. The output from A P O S E C contains the contact forces at the point o f application o f the displacements. More specifically, the contact forces are the non-zero elements o f the unbalanced load vector for the load step: {0} {AO}, = (5.1) ^-^^ -J boundary conditions {Ac()} J The non-zero components ( displacement control correspond to boundary displacement control is applied. conditions and points where In our case the contact forces at the point o f displacement application are extracted and the cumulative total written to an output file along with the total displacements o f 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 o f 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 o f the first test o f the Drucker-Prager model was to obtain a qualitative assessment o f the behavior o f the elements under cycled loads. displacement Although cyclic simulations were not conducted on the joint models themselves, it is always possible i n a complex model to have local loading and unloading o f Gauss points even though the global displacement loading was monotonic. Hence an observation o f the loading and unloading behaviour o f the constitutive model is prudent. In Figure 5.1 is shown a single concrete element, based on the perfectly plastic Drucker-Prager constitutive model o f 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 i n the horizontal plane. The type o f loading used is consistent with that o f infinitely stiff and frictionless plates. T w o sub increments per displacement step were used to allow for the more frequent updating o f material constitutive parameters for improved accuracy o f the response. The number o f sub increments in this case was two. CYCLIC LOAD Figure 5.1. Single 230mmX230mmX230mm Concrete Element Under Cycled Displacement. The response o f the model is depicted i n Figure 5.2. A t first the element experiences uniaxial compression under the concrete tangent modulus o f 30000 M P a , then the material flows plastically at a stress o f 30 M P a . In the elongation cycle it unloads elastically and is brought into tension. In the context o f plasticity, tensile failure is depicted as yielding with.no stress release as would occur across a crack surface. The tensile strength o f 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 i n Figure 5.2 exhibits a m i l d slope o f 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 i n detail i n the next section. Drucker-Prager Iteration Sensitivity The concrete element o f Figure 5.1 was next subjected to a series o f monotonic loadings to determine the accuracy o f the post yield stress prediction o f the perfectly plastic Drucker-Prager constitutive model when the number o f iterations per load step are changed. The material properties were as follows: Elastic modulus E = 15000 M P a . Compressive strength f = 30 M P a . Cohesion c = 3.2404 M P a . c c 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 o f the Euclidean norm and checks it against a user specified value. It also checks to see that the maximum number o f 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 o f the previous criteria are met. The maximum number o f iterations was set at 15, 30, 60 and 100 for each test respectively while the target value o f the Euclidean norm was set at exactly 0. Setting the target value o f the Euclidean norm at exactly 0 forces the program to iterate to the maximum number o f iterations each time, since the target value is i n practice unreachable. The results o f 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 ( M P a ) 30 20 10 -0 0 0.001 0.002 0.003 S 0.004 0.005 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 o f about 0.002, after which 95 one would expect the stress curve to stay flat since i n the loading test simulations there was no confining pressure. Actually, a small amount o f error i n the confining pressure is picked up in the numerical computation sequence as shown i n Table 5.1 and Table 5.2 under the rows labeled a and a-y. x The error in the confining pressure gradually diminishes as the number o f iterations per load step is increased but the effects o f the error on the vertical stress a z can be quite significant as shown below. Conversely, this test could have been carried out by using a very large number o f iterations, say 1000000, and selecting different target values o f 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 a n = 15 -8.589e-02 -8.589e-02 n = 30 -6.066e-02 -6.066e-02 °z x CT n = 60 -3.025e-02 n = 100 -1.197e-02 -1.197e-02 -3.184e+01 -3.130e+01 -3.025e-02 -3.065e+01 CT -9.834e-16 -9.205e-16 -1.232e-15 CT xv xz -7.885e-16 -8.900e-16 -8.319e-16 -8.438e-16 -8.125e-16 vz +5.163e-16 +4.326e-16 +6.673e-16 +6.053e-16 Y CT -3.026e+01 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 a -1.162e+00 -8.209e-01 -4.094e-01 -1.619e-01 -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 xv xz +8.480e-16 +8.000e-16 -2.156e-15 +5.120e-16 -1.356e-16 -2.503e-17 +8.605e-16 -5.235e-16 VZ +1.200e-15 +1.144e-15 +1.658e-15 +2.137e-16 v a a CT A s another check on the robustness o f the algorithm, a check can be made to see how close the stresses for the last load step o f 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) o f the Mohr-Coulomb failure envelope i n compression is: f /*i 30 - 4 x 3 . 2 4 0 42 A , -4<r2\ § = sin = 65.62° = sin 30 +4x3.2404 2 2 2 v (5.3) y from which we can determine the parameters a and K o f the Drucker-Prager model as: ^_ 2 sine)) ~ V3(3-sin<t)) a = 0.5034 _ ' 6ccos(|) ~ V3(3-sinc(>) (5.4) ; ^ = 2.2179 The Drucker-Prager yield condition is reiterated here as: F(a ) = 3a<s + a-K IJ m =0 (5.5) The following table shows i n the last row the value o f 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. a n = 15 n = 30 -19.078 31.0314 0.0019 -16.410 27.0021 0.0017 If the row labeled F(GJJ) n = 60 -13.196 n=100 -11.265 22.1475 0.0010 19.230 0.00014 o f 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 o f iterations is increased. Passive Confinement Analysis The next model to be analyzed is shown i n Figure 5.4: A concrete cube surrounded by four steel plates o f 3 m m thickness undergoing uniaxial compression. The loading simulation consisted o f a uniformly distributed displacement applied to the top nodes for a series o f twenty load steps. The bottom nodes were restrained vertically while the cube was allowed to expand i n 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 o f the test was to determine the. effect o f the dilatancy factor p on the passive confinement ability o f 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 0.3 0.2 o> Y i e l d stress in uniaxial tension. 300 M P a N.A. Et Y i e l d tangent modulus. t Plate thickness. OMPa N.A. 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 N.A. 0.5 Poissons ratio Dilatancy factor Table 5.5. APOSEC arguments used to conduct passive confinement analysis. Parameter Value Number o f load steps Maximum iterations Number increments 20 no. of of 15 sub 2 Target Euclidean N o r m Total displacement 0.01 1.00 m m The results o f the analysis are given in Figures 5.5, 5.6, 5.8 and Table 5.6. Figure 5.5 is a plot o f CT (the vertical stress) at the center Gauss point for the various values o f z dilatancy factor p. It can be seen that i n the post yield section o f the plot the vertical compressive stress <J continues to increase due to confining pressure. Z The accuracy o f 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 o f the passive confinement modelling to the choice o f dilatancy factor. O f interest here is the confining pressure <3 on C = - ( x v ) / 2 which is shown in Figure 5.6. The negative sign indicates a + a 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 o f 0.2 and 0.3 respectively, the steel tends to expand more than the concrete in the elastic range thus producing primarily tensile stresses i n the concrete. In the inelastic range, however, the concrete tends to dilate considerably more than the steel shell due to sliding o f crack surfaces over each other, thus stretching the plate steel and developing positive confining pressure. The sudden sharp dip i n confining pressure just before the concrete makes the elasto-plastic transition can be attributed to the modelling process. - Beta = 0 Beta = — 0.25 —A—-Beta = 0.5 — • - - Beta = 0.75 0.001 0.002 0.003 0.004 0.005 Figure 5.5. Vertical stress vs. vertical strain at center Gauss point of the above model for various dilatancy values. 100 m-Beta = 0 Beta = 0.25 - A - Beta = 0.5 Beta = 0.75 (MPa) 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 o f Gauss points for the first load step. Figure 5.8 is a three dimensional graphical depiction o f the stresses for several load steps starting in the elastic range and progressing into the inelastic range over the layer o f the element. Table 5.7 gives their (xj,yj) coordinates relative to the coordinate axes shown i n Figure 5.7. The first column o f Table 5.6 gives the Gauss point number (1-27) and the corresponding indices on the (xj,yj) coordinates. Although the behavior o f a concrete element encased by four steel plates is difficult to analyze by hand methods, an inspection o f the stresses in Table 5.6 provides a qualitative assessment o f the behavior o f the steel and concrete model. To begin with, the in-plane stresses o x and Oy are greatest at the Gauss points that are near the corners o f the concrete element, which is correct since they are nearest to the highly confined corners o f the steel shell. There are no out o f plane shear stresses u and Oy as would be expected from uniaxial loading. The i n plane shear stress a y is greatest at a corner Gauss Point and negligible at the other xz X points. One would Z expect that at a corner point the direction o f the principal stresses would be 4 5 ° . This assumption may be verified for Gauss point #1 for the first load step by calculating the principal angle v i a a simple formula that is found i n many Mechanics textbooks: 101 tan29 = ^ (a,-a,)/2 0.1082 = , P-°) c = 00 (0.1162-0.1162) .-. 20 = 90° 0 = 45° A l s o one would expect the vertical stress to be smallest in magnitude near the corners and greatest in magnitude at the center o f the concrete element while the concrete is behaving elastically, since it is subject to negative confining pressure that is greatest i n 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 x y Sigma x z Sigma y z Gauss Pt. 1 +1.162e-01 +1.162e-01 -3.214e+00 +1.082e-01 +4.706e-12 +6.962e-12 i=lj=l Gauss Pt 4 +4.407e-02 +9.816e-02 -3.232e+00 +1.124e-13 +1.127e-ll +8.326e-12 i=lj=2 Gauss Pt. 7 +1.162e-01 +1.162e-01 -3.214e+00 -1.082e-01 +2.156e-ll +9.301e-12 i=l,j=3 Gauss Pt. 10 +9.816e-02 +4.407e-02 -3.232e+00 +1.816e-13 +6.031e-12 +3.335e-12 i=2j=l Gauss Pt. 13 +2.604e-02 +2.604e-02 -3.250e+00 +1.81 le-13 +1.186e-ll +2.852e-12 Gauss Pt. 16 i=2,j=3 +9.816e-02 +4.407e-02 -3.232e+00 +1.808e-13 +1.892e-ll +1.685e-12 Gauss Pt. 19 +1.162e-01 + 1.162e-01 -3.214e+00 -1.082e-01 +9.325e-12 +7.892e-12 i=3J=l Gauss Pt. 22 i=3,j=2 +4.407e-02 +9.816e-02 -3.232e+00 +2.499e-13 +1.890e-ll +6.947e-12 Gauss Pt 25 +1.162e-01 +1.162e-01 -3.214e+00 + 1.082e-01 +2.719e-ll +5.021e-12 i=2,j=2 i=3J=3 y m • 7 • 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. i= 1 i =2 i =3 j = l (25.92,25.92) j =2 (25.92,115) j =3 (25.92,204.08) (115,25.92) (204.08,25.92) (115,115) (204.08,115) (115,204.08) (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 i n 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 i n 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 a at middle layer of Gauss points in the concrete element for load steps 1, 5, 10, 15 and 20. p = 0.5. z 105 5.3 Analyses with the Hardening/Softening Model A series o f tests were performed on the finite element o f Figure 5.1 using the Hardening/Softening Drucker-Prager model in place o f the perfectly plastic one. Many parameters were introduced i n 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 o f 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 , averaged over all o f the Gauss points vs. the vertical strain, s , for three z z different cases o f 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 , it w i l l be recalled from Chapter 3, controls the rate o f softening i n the post c 106 yield region. Generally a larger value o f this parameter tends to delay the softening effect, which is a trend also observed in Figure 5.9. In Case C the parameters (p , s cv c and were changed to 0.1, 0.035 and 0.005 respectively over their values in Case B . The parameter cp affects the plastic expansion cv to distortion ratio. A lower value o f 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 o f hardening i n 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 o f Figure 5.9 tends to indicate that the response shows a peak of 22.5 M P a over the initial yield value o f 20 M P a indicating that hardening took place to reach the 22.5 M P a value. Logically, decreasing and perhaps increasing s w i l l tend to c 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 f = 30 MPa f = NA v = 0.2 A = 0.3 s = 0.0075 <p =17.2° Sty = 0.0025 c f c w (j) = 22.62°1 c = 10 MPa Table 5.9. Constitutive parameters for Case B. E = 27000 MPa f = 30 MPa f = NA v = 0.2 A = 0.3 cp =17.2° s = 0.015 e,f= 0.0025 f c cv < > | = 22.62 f Calculated from <>| = sin - i c c = 10 MPa 01 r'2 /; +4c 2 2\ 2 107 Table 5.10. Constitutive parameters for Case C. E = 27000 MPa f = 30 MPa v = 0.2 i = NA c A = 0.3 <>| = 2 2 . 6 2 ° t (p =5.73° s c l / 1 c = 0.035 = 0.005 c = 10 MPa The next two tests were carried out i n tension on the same model for the values o f constitutive constants shown i n Tables 5.11 and 5.12. Cases D and E was A i n the function Y{<5 ,G ,f ,A) M computed stress strain response. A The only parameter to vary i n and generally had no effect on the c Shown i n Figure 5.10 are the responses i n tension averaged over all Gauss points in the element for both Cases D and E . Table 5.11. Constitutive parameters for Case D. f = 30 MPa f = 2 MPa v = 0.2 E = 27000 MPa c f A = 0.3 <p = 5 . 7 3 ° $ = 22.62° c= 10 MPa cv s c = 0.035 = 0.005 Table 5.12. Constitutive parameters for Case E. f = 30 MPa v = 0.2 f = 2 MPa E = 27000 MPa c f 4 = 0.15 <Pcv = 5-73° 4 = 22.62° c= 10 MPa e = 0.035 c 5^ = 0.005 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 o f Chapter 3 utilized to release the stresses i n the direction o f principal stress. The resulting brittle response is as shown i n 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 o f tests were performed on the single concrete element surrounded by four steel plates o f Figure 5.4 using the Hardening/Softening Drucker-Prager constitutive model for the following set o f 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 = 300 MPa E = 0 MPa v = 0.3 t = Varies t v Table 5.14. Concrete material parameters for passive confinement test on the Hardening/Softening Drucker-Prager model. E = 27000 MPa f = 30 MPa f = 2 MPa v = 0.2 A = 0.3 e = 0.02 Vcv = 5-73° s = 0.01 < > | = 22.62° c = 10 MPa B= NA c f c f 109 A l l the constitutive parameters were kept constant while the thickness o f the steel shell was changed from 1 mm, 2 m m and to 4 m m in three tests referred to as Case A , Case B and Case C i n Figures 5.12 to 5.14. Figure 5.12 shows the vertical stress, a , i n the z concrete element averaged over all the Gauss points while Figures 5.13 and 5.14 show the variation i n confining pressure and the hardening modulus, H, respectively for the three tests. There is a prominent discontinuity i n Figures 5.13 and 5.14. The discontinuity is caused by the hardening modulus, H, which is a discontinuous function o f 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 i n Figure 5.12 but it is somewhat subtle. The discontinuity in Figure 5.12 is the change o f 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 o f Figure 5.14 are missing. These computed points were omitted from the figure because they were characterized by values that deviated by an order o f magnitude from the points preceding them. From observations o f the corresponding values o f the Euclidean N o r m these points correspond to convergence failures o f the Newton-Raphson iteration procedure which can occur i n 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 i n the corresponding regions. It seems apparent that one should look at the smoothness o f the hardening modulus curve to determine i f the computed response is sound. 110 25 —m— Case A —A— Case B 20 - — • — Case C 15 - °z (MPa)™ 5 o • f i 0 1 0.001 1 0.002 1 0.003 1 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 o f section 3.3 to analyze a T-joint with square tubing is that shown i n Figure 5.15. It consists o f 4 concrete elements o f the dimensions shown surrounded by 15 steel plate elements o f 3 m m thickness and no reinforcing steel elements. The loading simulation consisted o f applying a total displacement o f 6 m m 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 0.3 0.2 300 M P a N.A. OMPa N.A. 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 N.A. 0.5 Poission's ratio Y i e l d stress in uniaxial tension. Et Y i e l d tangent modulus. t Plate thickness. Dilatancy factor * This is representative o f the secant modulus o f concrete i n uniaxial compression up to a strain o f 0.002 corresponding to a peak stress o f 30 M p a [21]: 30MPa = 15000MPa 0.002 114 Table 5.16. APOSEC arguments used to conduct joint analysis using the perfect plasticity Drucker-Prager concrete constitutive model. Parameter Value Number o f load steps M a x i m u m iterations Number of increments 30 15 sub Target Euclidean N o r m Total displacement 5 0.005 6.00 m m 1000 0 1 2 3 4 Displacement (mm) 5 Figure 5.18. Force displacement plot for two finite element models of a square specimen. The second model o f Figure 5.16 consists o f 40 concrete elements and 60 plate elements. It also differs from the first model in the boundary conditions. The vertical restraints i n the second model are applied on the midside nodes o f a set o f bare concrete elements at each end o f the model, which is more consistent with laboratory conditions during the physical tests. The force displacement curves for the two models are plotted i n Figure 5.18. The general trend appears to be a softening o f the response as the number o f elements is increased although the plot for the second model o f the joint still overpredicts the strength o f 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 o f 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 o f Figure 5.17. The models o f 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 o f the specimen. This phenomenon is illustrated i n the following diagram: Plate steel peels away on tension side 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 o f force that does not contribute to the moment resistance o f the beam stub. This phenomenon can be modeled in the finite element program by defining an independent set o f 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 i n Figure 5.19) to possibly model the bond slip characteristics o f the steel-grout interface. A complication arises when using nodes for the steel shell 116 that are independent o f 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 o f constitutive constants: Table 5.17. Steel plate constitutive properties in Phase 2. E = 200000 MPa CT = 300 MPa E = 0 MPa v = 0.3 f = 3 mm f v Table 5.18. Concrete constitutive properties in phase 2. E = 27000 MPa f = 30 MPa f = 2 MPa v = 0.2 A = 0.2 s = 0.005 s = 0.01 <Pcv = 7° (j, = 22.62 c= 10 MPa B = See text * c t c f 01 * This is not P o f the perfectly plastic Drucker-Prager model, it is the parameter B introduced in Chapter 3 as the rate o f shear modulus reduction. A . Here the model i n Figure 5.17 o f 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 MPa. C . Same as B except the shear reduction rate was used as B = 10000 M P a . 117 0 0.5 1 1.5 Displacement (mm) 2 2.5 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 o f about 2 m m as the response became unstable. Debonding the nodes on the tension side results i n very little decrease i n the predicted strength (Case A vs. Case B ) . A l s o shear stiffness degradation has very little effect (Case C ) . 5.6 Analysis of a Joint Phase 3 The next phase o f joint finite element analyses utilized the Hardening/Softening Drucker-Prager model with Method 1 as the option for the response i n tension. Two cases were examined each for the 4-concrete-finite-element-model o f Figure 5.15 and the 20-concrete-fmite-element-model of Figure 5.17. The concrete constitutive parameters for each o f the cases are shown in Tables 5.19 and 5.20 while the steel plate constitutive parameter values were kept the same as i n Phase 2. 118 — • — Case A 800 • d * T —m— . Case B j£r 600 - if 400 < n 200 0 1 2 ,f 0 h- 4 1 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 f = 30 MPa f = 2 MPa v = 0.2 A = 0.2 c p = 11.46° s = 0.01 6f = 0.02 <j, = 22.62° c = 10 MPa B = See text * c t cv c 1 Table 5.20. Concrete constitutive properties for Case B. E = 27000 MPa f = 30 MPa f = 2 MPa v = 0.2 A = 0.2 c p = 17.19° s = 0.01 e = 0.02 ()> = 22.62° c= 10 MPa B = See text * c t cv c f 1 119 Figures 5.21 and 5.22 show the responses to the displacement loading simulations o f the two finite element models as the parameter cp was varied. The parameter cp cv cv is the value o f 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 i n the finite element procedure as the confinement effect o f 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 o f (p . Also the 20 element concrete model became unstable after a cv displacement o f about 1.5 m m and 2 m m for the two cases o f <p . cv 5.7 Closing Commentary A t the beginning o f this chapter simple analyses were conducted on a single concrete element and later on a concrete element surrounded by steel plates, with the objectives o f determining the behavior o f the constitutive models and the sensitivity o f 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 o f a finite element analysis provided a reasonable framework with which to model the behavior o f steel encased concrete . Phase 1 o f the joint loading simulation utilized the perfectly plastic DruckerPrager concrete constitutive model. The two models, (4 concrete elements and 40 concrete elements), predicted yield strengths o f 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 i n 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 o f 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 i n subsequent test comparisons. In Phase 2 the Hardening/Softening Drucker-Prager concrete constitutive model was used with the brittle cracking option o f Method 2 i n Chapter 3 utilized to fine tune 2 the response i n 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 i n the load history, but interestingly enough, the load displacement curves i n Phase 2 show a yield "knee" at about 350 k N which is on par with the 40 element model o f Phase 1. Phase 3 o f 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 i n conjunction with variations i n the parameter cp to determine the confinement effect on the response. The cv 4 element model predicted yield points at about 600 k N and peak strengths o f 700 k N and 900 k N . The 4 element model was stable enough to simulate the post yield region o f the load deflection curve. This is the only combination o f joint finite element and concrete constitutive models that permitted the observation o f the post yield branch. The 20 element finite element model became unstable right after the yield plateau i n Figure 5.22. The lowest predicted peak strength was about 350 k N which is about the value o f the yield knee i n Phases 1 and 2. Curiously the shape o f the load deflection curve i n Figure 5.22 does resemble the shape o f the laboratory measured response envelopes o f 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 o f the two square joints are presented below in Figures 5.23 and 5.24 i n force displacement units for comparison. 250 Displacement (mm) Figure 5.24. Force displacement response of the Square Joint # 1. Part o f the problem associated with trying to predict the peak strength o f 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 i n subsequent testing. Such behavior may result i n the concrete not contributing at all to the response o f 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 o f concrete i n tension to the capacity o f the above section and making the crude assumption that the neutral axis passes through the geometric center o f the gross 3 section, one can do the following analysis assuming that all o f the steel has yielded. The reinforcing and plate steel properties can be found in the appendix o f Hoffschild's thesis [11]: 1. Contribution o f reinforcing bars to moment capacity: momentarm y d = 2 0 0 - 2 x 1 5 - 2 x 1 0 - 1 0 = 140 m m yield strength f = 700 M P a y steel area A = 200 m m 2 s Mr, = AJ jd y = 200 x 10" x 700 x 140 k N m 6 Mr, =19.6 k N m 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. 123 2. Contribution o f the steel shell on the top and bottom to moment capacity: moment arm jd = 230 m m yield strength f = 267 M P a y steel area A = 230 x 3 = 690 m m 2 s i M r = Afyjd = 690 x 10" x 267 6 x 230 kNm Mr = 42.4 k N m 2 3. Contribution o f the steel shell on the sides to moment capacity: moment arm jd = 2 3 0 - 0.5 x 230 = 115 m m yield strength f = 267 M P a y steel area A = 0.5 x 2 x 230 x 3 = 690 m m 2 s Mr, = AJ jd y = 6 9 0 x 1 0 x 267 x 115 k N m - 6 Mr = 21.2 k N m 3 4. The total is: M = Mr + Mr + Mr, T x 2 M = 19.6 + 42.4 + 21.2 T M j . = 83.2 k N m If we compare the above calculation with the measured moment capacity o f the square retrofit without fins. In the positive quadrant (see Figures 5.23) we have a peak force o f about 200 k N . The moment arm to the face o f 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 o f steel jacketing was chosen as the topic o f experimentation i n the work presented here. Following the tests performed by Hoffschild, who observed failures i n the members outside o f 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 i n Chapter 3 to represent the behavior o f passively confined concrete with the goal o f predicting the load deflection response o f such beam column joints. Chapter 4 presents the details o f the incorporation o f the constitutive models into a nonlinear finite element based computer program. Chapter 5 contains details o f some simple analyses that were performed on single concrete elements confined by four steel plates, followed by models o f the beam to column joints consisting o f 4, 20 and 40 concrete elements. The simple models served to simulate showed the behavior under passive confinement. The models o f the joints, although unable to predict the peak strength as measured i n the laboratory, were able to reproduce the shape o f the strength envelope as observed. 125 Concluding Remarks The method o f steel jacketing inadequately reinforced beam column joints was experimentally shown to significantly improve the strength and ductile response o f 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 o f plastic hinging to provide confinement, this is not always the case i n practice. A steel jacket retrofit has been shown to be an effective means o f remediating such deficiencies i n reinforced concrete members and joints. Careful attention must be paid, however, to the design o f a retrofit scheme, considering the possibility o f shear failure outside the retrofitted region due to severe moment gradients resulting from flexural overstrength and shortening o f the weak sections. The overstrength forces may be reduced by providing gaps in the retrofit jacket in the region o f the plastic hinge. Central to the concept o f capacity design, i n 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 o f stress concentrations are reinforced on the steel jacket, the joint region also exhibits a very ductile response. The theoretical analysis and modelling o f steel encased joints led to development o f a nonlinear material finite element based computer program. the The computer program features non linearity i n plasticity based material constitutive models, which range from the very complex to the very simple. During the development o f the plasticity models, many constitutive parameters had to be introduced that would be difficult to quantify precisely without accurate experimental data. The constitutive modelling o f the concrete in the joints was focused on the plastic nature o f confined 126 concrete and its brittle response in tension. Intuitively it is argued that stresses representing both conditions are present during the loading o f such joints. It was shown, however, that the concrete may not contribute significantly to the strength o f such a joint. The predicted strength is relatively insensitive to the parameters governing the response of the concrete i n tension while changes i n parameters governing the response in compression such as cp produce significant differences in the predicted response. cv Also, differences between the two Drucker-Prager plasticity models have a significant impact on the post yield response o f the finite element models. The observed strength o f 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 i n the predicted response crops up i n the models involving more elements which is a not uncommon i n finite element procedures involving nonlinear material characteristics. The unstable nature is a function o f the constitutive model used and the number o f 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 o f finite elements i n a model w i l l tend to produce instability i n 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 o f constructing a steel shell or cage around a joint i n an actual building i n the field. But more to the point would be the prefabrication o f complete moment resisting frames utilizing steel encased joints that could be installed as a redundant system i n moment resisting frames that do not have 127 contemporary details and hence are brittle. The installation o f a prefabricated ductile moment resisting frame into a system that has limited ductility is an area that is not addressed by building codes i n Canada. Research into the behavior o f the two systems would prove to be very beneficial. The numerical research here has focused on the representation from the continuum level o f beam column joints involving models o f many degrees o f freedom with results that represent reality qualitatively i f not quantitatively. From the work i n Chapter 5 it was found that the accurate representation o f 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 o f numerical research should focus on simpler beam and column models (possibly stick representations) with the aim o f calibrating them from laboratory experiments and using them in the modeling o f entire frames. A n interesting application o f such a model would be to attempt to predict the response o f 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 o f Steel Buildings: Lessons F r o m The 1995 Hyogoken-Nanbu Earthquake", Canadian Journal o f C i v i l Engineering, August 14, 1995 T. Paulay, M . J . N . Priestley, "Seismic Design o f Reinforced Concrete and Masonry Buildings", John Wiley and Sons, 1992 Canadian Standards Association, CAN3-A23.3-M92 "Design of Concrete Structures for Buildings" Michel Bruneau, "Preliminary report o f structural damage from the L o m a Prieta (San Francisco) earthquake o f 1989 and pertinence to Canadian structural engineering practice", Canadian Journal o f 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 i v . Eng. 22: 361-377 (1995) 129 J.B. Mander, M.J.N. Priestley, R. Park, "Theoretical Stress-Strain M o d e l For Confined Concrete", Journal o f Structural Engineering, V o l . 114, N o . 8, August 1988. J.B. Mander, M . J . N . Priestley, R. Park, "Observed Stress-Strain Behaviour o f Confined Concrete", Journal o f Structural Engineering, V o l . 114, N o . 8, August 1988. S. Pietruszczak, J.Jiang and F . A . Mirza, " A n Elastoplastic Constitutive M o d e l 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 o f 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 M c G r a w H i l l . Thomas E. Hoffschild, "Retrofitting Beam-to-Column Joints for Improved Seismic Performance", M . A . S c . thesis, Dept. o f 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. o f C i v i l Engineering, University o f British Columbia, A p r i l 1993. 130 (13) Sergio M . Alcoccer, " R C Frame Connections Rehabilitated by Jacketing", Journal o f Structural Engineering, V o l 119, N o . 5, M a y 1993. (14) J . Jiang and S. Pietruszczak, "Modeling o f Concrete Response to Fluctuating Load", International Journal for Numerical and Analytical Methods i n Geomechanics, V o l . 13, 171-181, 1989. (15) Zdenek P. Bazant, "Advanced Topics in Inelasticity and Failure o f 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 o f Inelasticity and Failure o f Concrete", Mechanics Division, Proceedings Journal o f the Engineering o f the American Society o f C i v i l Engineers, V o l . 102, N o . E M 4 , August, 1976. (18) O. C . Zienkiewicz and R. L . Taylor, "The Finite Element Method", Fourth Eddition, Volume 1, 1989 M c G r a w H i l l . (19) Press, Flannery, Teukolsky, Vetterling, "Numerical Recipes, The A r t o f Scientific Computing", Chapter 4, 1989 Cambridge University Press. (20) R. D. Cook, D. S. Malkus, M . E . Plesha, "Concepts A n d Applications o f 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 o f 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 dev d\\j v5a</y Here we set out to prove this result. To begin, it is useful to rewrite the plastic flow function i n tensor notation as follows: d\\f d\\j 1 5.,+- 1 (A.1) Recognizing that the deviatoric component o f the flow function is the second term i n the above equation: 1 dev I (A.2) 2 ^ And: dev ( ^ \ d\\i dev __J 1_ (A.3) ^A/*^~ ^ffi 2 Substituting the definition for J : 2 (A.4) = —S,:S:: IJ 1 IJ 1 <2JJ,2jj' SuSu V 4J, 2J, (A.5) 133 Procedure for determining G . A Perfectly Plastic Model The procedure for calibrating the yield function o f section 3.4 i n the tension regime is as follows. The yield function is presented here for clarity: ( F(o ) = ij {3ao -K) 1 m om \2" ( 1- -K O"m ^A) F(rj,) = 3aa m \2 +a = 0 \*A +a - ^ = 0 for a,„ > 0 J forCT„,< 0 (A.6) 2 sine)) a = K = V3(3-sin((>) 6ccosc() V3(3-sin<|>) (A.7) (A.8) Imagine performing an experiment where a concrete specimen is subjected to a state o f uniaxial tension. A t the point o f tensile failure, the stress tensor, a^-, has the following values: 'f t o o" 0 0 0 0 0 0 (A.9) A n d the deviator is: 3 ' 7 0 0 In the tension regime (a m 0 0 - i / 3 0 0 (A.10) 3 >0) at the point o f yielding the following conditions are satisfied: 134 (A.11) a... = f, (A.12) Substituting the above into the yield function results in: (3aa -K) 1- m A. + A K 1- V3 \ °AJ 3 = 0 (A. 13) The equation A . 13 is a non linear equation dependent only on the material parameters c,f t and <> | . G i v e n the material parameters c,f and t the parameter a A can be found by a series o f trial and error iterations. Hardening/Softening Model The procedure for calibrating the yield function o f section 3.6 i n the tension regime follows similarly as for the yield function o f section 3.4. t F(a,) = (3a*a -^) +a = 0 m F(o ) = 3 a o + a - J i r = 0 (y B for a„, > 0 (A. 14) for a.„ < 0 (A. 15) ( A . 16) In the tension regime ( a OT >0) at the point o f yielding the following conditions are satisfied: a = K = c=c (A. 17) fy = 0 ( A . 18) 2smfy* \/3(3-sin(j)*) 0 (A. 19) 6c* cos fy" _ 2 A/3(3-sin(j)*) ~ V 3 ' - (A.20) 135 (A.22) a.,. = (A.23) °A=V A Substituting the above into the yield function results in: 2c V3 1- + 2c V3 A 1- ' A * \ 3 ° A j K AJ 3(J A V3 = 0 (A.24) The equation A . 2 4 is a non linear equation dependent only on the material parameters c and f , which can be solved by trial and error iteration as A . 13. t A s an example the following table gives solutions for the parameter cr for various values o f c andf . Values A t may be interpolated linearly for values o f c and f not given. t 136 Table A.1. Table of in MPa for various values of c and f in the Hardening/Softening model. t ' c = 5 c = 10 c = 15 MPa MPa MPa ft =2 M P a 0.70273 0.68399 0.67806 /, = 4MPa 1.49071 1.40545 1.38013 f =6M?a 2.39046 2.16931 2.39046 t 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 o f shape functions. One set o f 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 o o o # = i(i+0(i-Ti)(i+0 7 9 o N = I(i - 0 ( i - TIXI+0 5 N = UI+0(i+TIXI - 0 N = UI- 0 ( i - TIXI - 0 JV = 1 ( I + 0 ( I - T I X I - 0 13 I5 o The other set o f 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 N M 2 N» N M 6 4<' 4<> 4<>-4 )(l-Tl)(l + 0 -4 )(i 2 + T 1 )(i c;) + 2 -T! )(IH)(I C;) 2 + < 4c - 4 ) ( l + T,)(l-0 N -T, )(l-ai-0 M 4c 2 J N M JV 14 N M IV iV 2 17 N M JV 4c 4c 4c -^)(i-^)(i -T! )(IH)(I-C;) 16 19 -; )(i 2 + n)(i ^) + ) + T1 -; )(l-^)(l-ri) 2 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 N = 2 N M 2 N = N 3 N = 4 - ^ N 3 N M 2 - ^ N M 4 - ^ M 4 N = N -^N -^N -^Nf{ M 9 9 l M 0 l 6 ^ 1 1 = ^ 1 1 - ^ 1 0 - ^ 1 2 - ^ 1 8 N = N M N = < l2 H X 2 N = N% N = N{f i6 l7 *is = AT iAT v 2 0 < - J\T -- i VKT M M 2 0 Shape Functions For the Membrane Element 1/ M x ~ 4 _ 1/ M 2 M 3 4 5 6 7 ~ 2 ~ 4 1/ 8 3 (-1,1) 2 (0,1) 1 (1,1) l-TI )(l-0 l-OU-Tl) (-1,0) 8(1,0) (0,0) i-^ )(i-r,) 2 l 1/ M M ~ 4 _ 1/ M 2 2 ~ 2^ 1/ M 1 r, i-^ )(i+r,) _ 1 / l - 0 ( l + r|) ~ 4 _ 1/ M 1 + 0 ( 1 + Tl) 1 + 0(1-Tl) i - n ) d H ) 2 ~ 2^ (-1.-1) 6 (-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=\ ^=0.555555555555556 W 2 (CI) =0.888888888888889 W = 0.555555555555556 The three point Gaussian integration rule w i l l integrate a fifth order polynomial exactly. 3 142 APPENDIX D The Concrete Element [B] matrix The Cartesian coordinates i n 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 w = N :(£,,ri,Q t (D.2) ; i = 0... number o f nodes The variable x , y and z are the coordinates o f the nodes i n the Cartesian space. Equation t t t D . l is more conveniently expressed in matrix form as: a a u 0 <V 0 w 0 0 0 •- N 0 • •• 0 N • •• 0 x N 0 0 a. 0 0 N J3x60 a N X a yN a (D.3) 143 From which the strains are easily derived as: to = [B]M (DA) The [B] matrix is implied from the following: r su ] 3x '\,x *yy Bv By 0 C dw Bz < pxy > = • 811 _ i _ Bv By ~ Bx r 0 >,y N 0 20,* 0 0 0 0 20,y 0 0 0 N.20,2 N.20,y N.20,x 0 N.20,z 0 N.20,* 0 N.20,; N.2Q,y 0 > N, 1,2 0 0 Bu _ i _ 3w Bz " " dx + Bv , Bw Bz ~ ~ By l 0 0 lo- (D.5) a. a yn a. In the case where the shape functions are given i n terms o f dimensionless coordinates, their derivatives with respect to the Cartesian coordinates x, y and z are not known explicitly but can be found v i a the Jacobian o f the transformation D . 1; [J] dx dy dz dx dy_ &> dz dr\ dx dr\ dy dr\ dz aw,. i i aw. i__ I* dr\ dN 2>, x— 1 t 2>< dN: I ft, cW.z_ aw. dNi dr\ dN I 5>, (D.6) ( i 'dN t \ dx dN: < dy dN — I d N i ] dN: (D.7) - t dz , > dr\ dN t 144 APPENDIX E The Membrane Element [B] Matrix A n appropriate set o f shape functions, (call them M,- to distinguish them from the concrete element shape functions N ), for the membrane are only a function o f the middle surface { coordinates, which define the membrane geometry: r ~\ X X (E.l) I > , ( $ , T I > yi middle surface middle surface Thus i f the coordinates o f 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. 1 = I> r TV V; 1 = I> a$ / JY dM,. i L yi + KT i (E.2) t V x, + J> dM r 01 Where the symbol, j x, + 1 , 9M. - l i dM i f * \ an , means the function that precedes it is evaluated at node j. From which the third basis vector is defined as the cross product o f u and v scaled to unity: Uj '3; X Uj X V j V (E.3) j The remaining basis vectors are defined as: U; ; e. 2 - e 3j x e ly (E.4 a,b) Ui The basis vectors are computed at all o f the nodes comprising the finite element and stored as variables i n 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 i a the shape functions: 145 (E.5) '2/ l 3j e In a numerical computer program application, the process o f 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 o f the curvilinear coordinates (£,,r|,Q: i x m ' middle surface 3i (E.6) rh.. The terms / , TW and « are the components (direction cosines) o f the third basis vector i n 3 3 3 the global Cartesian coordinates. Equation E.6 also allows us to determine the Jacobian o f the transformation that shall be needed to define the strains i n the element: 146 dz' dM, dM, ~&i &> dx dy di dz d^ dM, dk dM, an dy dr\ dz an dr\ dx [J]= dy &\ dx v ( t )dM ^ \ i r v ( i dM ?l ' 2^J^nz + _5C (E.7) A s with the case o f 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 v i a the shape functions: a a u V w ~M 0 0 •- M 0 M\ 0 • •• 0 0 0 M, • • 0 X > = 8 0 M 8 0 0 a. 0 My 3x24 a a y* a8 z 24x1 (E.8) W e note that the displacements are not a function o f the depth through the membrane, ( Q , since we are assuming that any displacement gradients through the depth o f a membrane are negligibly small. Following the procedure as i n Appendix D , the strains with respect to the global Cartesian frame are defined by: F r o m the definition o f engineering strains and equation E.8, the [B] brane mem matrix can be implied from the following equation: 147
- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- UBC Theses and Dissertations /
- Numerical modelling of experimental data of reinforced...
Open Collections
UBC Theses and Dissertations
Featured Collection
UBC Theses and Dissertations
Numerical modelling of experimental data of reinforced concrete beam-to-column joints Baraka, Miljenko 1996
pdf
Page Metadata
Item Metadata
Title | Numerical modelling of experimental data of reinforced concrete beam-to-column joints |
Creator |
Baraka, Miljenko |
Date Issued | 1996 |
Description | 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. Cyclic 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. |
Extent | 7959540 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
File Format | application/pdf |
Language | eng |
Date Available | 2009-03-07 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0050384 |
URI | http://hdl.handle.net/2429/5725 |
Degree |
Master of Applied Science - MASc |
Program |
Civil Engineering |
Affiliation |
Applied Science, Faculty of Civil Engineering, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 1997-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
Download
- Media
- 831-ubc_1997-0046.pdf [ 7.59MB ]
- Metadata
- JSON: 831-1.0050384.json
- JSON-LD: 831-1.0050384-ld.json
- RDF/XML (Pretty): 831-1.0050384-rdf.xml
- RDF/JSON: 831-1.0050384-rdf.json
- Turtle: 831-1.0050384-turtle.txt
- N-Triples: 831-1.0050384-rdf-ntriples.txt
- Original Record: 831-1.0050384-source.json
- Full Text
- 831-1.0050384-fulltext.txt
- Citation
- 831-1.0050384.ris
Full Text
Cite
Citation Scheme:
Usage Statistics
Share
Embed
Customize your widget with the following options, then copy and paste the code below into the HTML
of your page to embed this item in your website.
<div id="ubcOpenCollectionsWidgetDisplay">
<script id="ubcOpenCollectionsWidget"
src="{[{embed.src}]}"
data-item="{[{embed.item}]}"
data-collection="{[{embed.collection}]}"
data-metadata="{[{embed.showMetadata}]}"
data-width="{[{embed.width}]}"
async >
</script>
</div>
Our image viewer uses the IIIF 2.0 standard.
To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0050384/manifest