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2-D non-linear seismic analysis of one-storey eccentric precast concrete buildings Parmar, Surinder Singh 1987-12-31

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2-D NON-LINEAR SEISMIC ANALYSIS OF ONE-STOREY ECCENTRIC PRECAST CONCRETE BUILDINGS by Surinder Singh Parmar M.A.Sc, PANAJAB UNIVERSITY CHANDIGARH, INDIA, 1984 A THESIS SUBMITTED IN PARTIAL FULFILLEMNT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES (CIVIL ENGINEERING DEPARTMENT) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA May 1987 ©Surinder Parmar, 1987 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date Oct is, mi DE-6(3/81) Abstract Investigations into the behaviour of precast buildings under earthquake loading have shown that the connections are likely to be the weakest link in a pre-cast structure, and the stability of the structure under earthquake loading depends upon the strength & stability of these connections. A 2-dimensional non-linear dynamic analysis of a one storey box-type pre-cast buildings is presented. The shear walls in the buildings are modelled by linear springs, the properties of which depend upon the connections connecting the rigid panels of the shear walls. To check the effectiveness of the NBCC code design, computer studies have been made on a box-type building statically designed for different eccentricities. The strength of the shear walls was calculated assuming that each panel was a cantilever fixed at the base with dowel bars providing the flexural steel. To make the building survive a major earthquake, we need dowel connections that can take 5mm to 6mm elongation which can be easily accomodated. Studies have also shown that under the action of an earthquake, the response of a highly unsymmetric building will not be very different from that of a symmetric building as long as the building is properly designed using the NBCC code provisions for earthquake loading. It has also been shown that the NBCC code design eccentricity equation is somewhat conservative in calculating the design eccentricity and that a small change in the stiffness of walls perpendicular to the direction of earthquake has little effect on the response of the structures. ii Contents Page Abstract ii Table of Contents iiList of Figures vi List of Tables x Acknowledgement xiii 1 INTRODUCTION 1 1.1 General1.2 Categories of Pre-cast Concret Construction 1 1.3 Earthquake Response of Buildings 2 1.4 Scope and Objective of Present study 5 1.5 Thesis Layout 5 2 CYCLIC TESTS ON CONNECTIONS 6 2.1 Introduction 6 2.2 Types of connections 7 2.3 Dry Connections 8 iii 2.3.1 Bolted Connections 8 2.3.2 Welded Connections 13 2.4 Wet Connections 35 2.4.1 Platform type Horizontal Connection 32.4.2 Dowel Connection 38 2.5 Hysteresis Model for Vertical Connection 47 2.6 Hysteresis Model for Horizontal Connection 55 DYNAMIC ANALYSIS 60 3.1 Introduction3.2 Idealization Technique 61 3.3 2-D Mathematical Model 5 3.4 Wall Displacements 67 3.5 Time Step Integration 8 3.6 Panel Stiffness 71 3.6.1 Case 0 : Both the foundation springs in contact with the panel 2 3.6.2 Case 1 : Only right foundation spring in contact with the panel 73.6.3 Case - 1 : Only left foundation spring in contact with the panel • • 75 3.6.4 Case 2 : Neither foundation spring in contact with the panel 75 3.7 Wall Stiffness 73.8 Time-Step Solution Technique 78 3.9 Damping 81 3.10 Energy Balance 2 Static Design of the Buildings 85 iv 4.1 General 85 4.2 Stiffness Method 89 4.2.1 Shortcomings of the Stiffness Method 96 4.3 Strength Method 94.3.1 Shortcomings of the Strength Method 103 5 Results of Dynamic Analysis 104 5.1 Effect of non-symmetry in the buildings 105.2 Structural Data 106 5.3 Discussion of Results Ill 5.3.1 Effect of different methods of static design Ill 5.3.2 Effect of different E/Q data on the dynamic analysis 121 5.3.3 Effect of geometry of the building 127 5.3.4 E/Q direction 139 5.3.5 Effect of design eccentricity 142 5.3.6 Effect of variation of stiffness of wall 3 & 4 152 6 Discussion, Conclusions and Future Studies 157 6.1 Introduction 156.2 Discussion of results 158 6.3 Conclusions 161 6.4 Future studies 2 Bibliography 163 Appendix A 7 A.l Time Periods of the Buildings Considered in the Present Studies . . 167 v List of Figures 1.1 Storage Tank Analogy for the Seismic Response of Buildings by Clough [9] 4 2.1 Wall to Floor connections, from Ref. [14] 9 2.2 Floor to Floor connection, from Ref. [14] 10 2.3 Wall to Wall Horizontal connection, from Ref. [14] 10 2.4 Limited Slip Bolted connections, from Ref. [24] 11 2.5 Load-Deformation Response of Limited Slip Bolted (LSB) Joints, from Ref. [24] 12 2.6 Typical Headed Stud Connection showing two common stud configurations and the ease with which minor imperfections and misalignments may be accomodated, from Ref. [23] 14 2.7 Details of Specimens reported by Spencer & Neille [31] 15 2.8 Hysteretic Behaviours of Welded Headed Stud Connections reported by Spencer &; Neille [31] 15 2.9 Alternate Embedment 19 2.10 Details of Connections reported by Aswad [2] 20 2.11 Cyclic Behaviour of Connections Anchored With Steel Bars, by Spencer [34] 24 2.12 Models Developed by Spencer [34] 28 2.13 Cyclic Tests on Thin Panel Embedded Rebar Connections reported by Kallros [17] 30 vi 2.14 Split Pipe Connection by Saxena [29] 34 2.15 Typical platform-type horizontal connection 36 2.16 Coefficients of friction from tests on horizontal joints by Hanson [12] 32.17 Shear-displacement hysteretic loops from horizontal tests by Han son [12] 6 2.18 3/32 Scale Model Interior Horizontal Joint used by Harris & Abbound [13] 37 2.19 Comparison of Coefficients of Friction by Harris & Abbound [13] 37 2.20 Specimens Tested by Dimitrov & Georgiev [10] 39 2.21 Load-Deflection Curves of Specimens in Fig. [2.20] by Dimitrov & Georgiev [10] 32.22 Test Specimen - Phase 1, by Hawkins & Lin [15] 41 2.23 Load-displacement curve, # 10 straight bar, Bamboo-type defor mations, Phase 1 by Hawkins & Lin [15] 41 2.24 Test Specimens - Phase 2, by Hawkins & Lin [15] 43 2.25 Load-displacement curve, # 10 straight bar, Bamboo-type defor mations, Phase 2 by Hawkins & Lin [15] 44 2.26 Test Specimen - Phase 3, by Hawkins & Lin [15] 46 2.27 Load-displacement curve, # 6 straight bar, Bamboo-type defor mations, Phase 1 by Hawkins & Lin [15] 46 2.28 Elasto-plastic Model 48 2.29 Degrading Connector Models, by Muller & Becker [21] 48 2.30 Trilinear Hysteresis Loops used for both Stress-Strain and Moment-Curvature Relationships for Studs, by Neille [23] .... 49 2.31 Shear-slip Relationships for Shear-friction element, by Schricker & Powell [30] 50 vii 2.32 Hysteretic Model : Primary Curve, model proposed by Tong [36] 52 2.33 Hysteretic Model : Cyclic behaviour before yielding, model proposed by Tong [36] 52 2.34 Hysteretic Model : Cyclic behaviour after yielding, model proposed by Tong [36] 53 2.35 Joint Connector Characteristics proposed by Aswad [3] 56 2.36 Bar Force-Pull Out Model for Cyclic Loading, proposed by Hawkins & Lin [15] 57 2.37 Hysteretic Model for the Dowel Connection, by Tong [36] ... 58 3.1 Idealization of Shear Wall 64 3.2 Three Degrees of Freedom Model 66 3.3 Linear Variation of Acceleration 9 3.4 Free body diagrams of an interior panel corresponding to Case 0 73 3.5 Free body diagrams of an interior panel corresponding to Case 1 and Case -1 74 3.6 Free body diagrams of an interior panel corresponding to Case 2 76 3.7 Shear Wall System 77 3.8 Solution Strategies 80 3.9 Strain Energy from a F-A curve 83 4.1 One Storey Buildings Analysed in the Present Studies 86 4.2 Assumed Line of Action of Equivalent Static Force 91 5.1 Panel Connections 107 5.2 Earthquake Accelerograms used in the present studies 109 5.3 Resisting forces on the building 112 5.4 Comparison of static & dynamic analysis results of a square building designed by the strength and stiffness methods 115 viii 5.5 Comparison of dynamic analysis results of a square building using different earthquake accelerograms for dynamic analysis 122 5.6 Comparison of static Sz dynamic analysis results for square and rect angular buildings using NBCC recommended equivalent static design approach with strength method of design 128 5.7 Comparison of static & dynamic analysis results for square and rect angular buildings using NBCC recommended equivalent static design approach with design eccentricity equal to the actual eccentricity of the building and using the strength method of design 135 5.8 Comparison of dynamic analysis results of a rectangular building for the earthquake in X & Y directions 140 5.9 Comparison of static & dynamic analysis results for a rectangular building designed using NBCC recommended equivalent static design approach for various design eccentricities with the strength method of design 143 5.10 Comparison of static & dynamic analysis results for a square build ing designed using NBCC recommended equivalent static design ap proach for various design eccentricities with the strength method of design 148 5.11 Comparison of dynamic analysis results of a square building stati cally designed using strength method of design and using different stiffnesses of walls 3 & 4 in the dynamic analysis 153 ix List of Tables 4.1 Forces in various walls of a square building, Fig. [4.1a], for E/Q in X-direction, using stiffness method of design 94 4.2 Forces in various walls of a square building, Fig. [4.1a], for E/Q in Y-direction, using stiffness method of design 94 4.3 Maximum Panel Forces governing the design of various walls in a Square Building, Fig. [4.1a], using stiffness method of design .... 95 4.4 Code recommended Design Eccentricity & Torsional Moment for Square Building with Stiffness of each wall proportional to Wall Force in Table [4.3] 95 4.5 Forces in various walls of a square building, Fig. [4.1a], for E/Q in X-direction using stiffness method of design with Stiffness of each wall proportional to Wall Force in Table [4.3] 96 4.6 Forces in various walls of a square building, Fig. [4.1a], for E/Q in Y-direction, using stiffness method of design with Stiffness of each wall proportional to Wall Force in Table [4.3] 97 4.7 Maximum Panel Forces governing the design of various walls in a Square Building, Fig. [4.1a], using stiffness method of design with Stiffness of each wall proportional to Wall Force in Table [4.3] .... 97 4.8 Forces in various walls of a rectangular building, Fig. [4.1b], using stiffness method of design 98 x 4.9 Forces in various walls of a Square Building, Fig. [4.1a], for E/Q acting in X-direction, using Strength Method of design 101 4.10 Forces in various walls of a Square Building, Fig. [4.1a], for E/Q acting in Y-direction, using Strength Method of design 102 4.11 Maximum Panel Forces governing the design of various walls in a Square Building, Fig. [4.1a], using Strength Method of design. . . . 102 5.1 List of the dynamic analyses made in this studies 105 A.l Time Periods ( in sec ) of the Buildings in Figs. [4.1a] and [4.1b], de signed by the Strength Method, for displacements in the X direction 167 xi DEDICATED TO MY FAMILY xii Acknowledgement I would like to express my sincere appreciation and gratitude to my advisor, Dr. R. A. Spencer, for his invaluable assistance and guidance throughout the duration of the research period and the writing of this thesis. The advice given by coadviser Dr. D. L. Anderson during the research is gratefully acknowledged. Finally, I would like to thank the U.B.C. Civil Engineering Department staff, my friends and my wife for their assistance, advice, and encourage ment in completing this thesis. xiii Chapter 1 INTRODUCTION 1.1 General : Immediately after the second world war, an immense necessity for residence facil ities was felt all over the world especially in Euorpe where the war had left its mark in every big and small town. The classical methods of construction were not adequate to meet this construction demand. Therefore prefabricated construction was introduced in which members like beams, columns, walls etc. were fabricated in the plants and then assembled at the construction site. Using standard precast building components for construction offers a much greater speed in construction. This decrease in the total time of construction has enabled the concrete industry to compete effectively with the steel and timber in dustries which have a major advantage over in-situ concrete in this field. The use of precast members is widely prevalent everywhere in the world espe cially in non-seismic regions. But in seismic regions this industry is still lagging behind because of a lack of information on the performance of precast buildings and the joints used for connecting the precast elements under the earthquake loads. 1.2 Categories of Pre-cast Concret Construction : Precast buildings are of two types, 'jointed' and 'monolithic', with widely different structural properties [9]. In monolithic construction, precast elements are joined by 1 well reinforced connections posessing continuity of stiffness, strength and ductility comparable to well designed cast-in place concrete. In jointed construction the precast elements are connected to each other by relatively small connections while the inter element boundaries behave as zones of reduced stiffness, strength and ductility. While the existing code requirements on strength and ductility can be used for the design of monolithic precast structures, code requirements for jointed structures are not well established. In a jointed precast structure, joints open and close as energy is input into the structure. If sufficient energy is input to open up the joints, connections between the precast elements will yield because they are weaker than the jointed element. These connections start acting as the localized sites for energy dissipation. As this yielding is confined to small volumes only, the energy dissipation capacity of a jointed structure will be less than that of a monolithic structure detailed as conventional cast-in situ concrete. On the other hand, in a jointed structure as connections starts yielding before the monolithic structure, the structure will exibit a non-linear force displacement relationship even though the concrete and steel stresses in the elements are within elastic limit. This non-linear response of a jointed structure should result in reduced seismic energy input into the structure. This shows that a jointed structure behaves quite differently from a monolithic structure in terms of energy input as well as energy output. 1.3 Earthquake Response of Buildings : Earthquakes and their ability to do damage can best be described in terms of energy. When a fault ruptures, elastic strain energy is released which sets the neighbouring earth strata into motion. The earth strata act as a filter for this wave, filtering the high frequency waves. Thus at a point some distance away from the epicentre only the low frequency waves are present. 2 As the waves pass underneath a structure, some energy is transferred into the structure, the amount of which depends upon characteristics of the structure and the duration of ground shaking. It takes some time to build up energy in the structure i.e. we need a finite duration of ground shaking underneath the structure. The more receptive structures are those with linearly elastic stiffness properties, and with natural vibration frequencies similar to the frequencies of the strongest ground shaking. Structures with non-linear force displacement relationship do not have well defined natural frequencies. As these structures begin to respond to harmonic loads, their effective period of vibration changes, thus forcing the structure away from resonance, reducing futher energy build up. The behaviour of structures under earthquake loads has been demonstated as a 'Storage Tank Analogy for the Absorption and Dissipation of Seismic Energy' by Clough [9]. He represents the structure by a storage tank Fig. [l.l]. The elastic strength of the structure is represented by the height of an over flow nozzel above the base. Two large pipes conduct energy into and away from the main tank. An inlet pipe is fitted with a valve which is wide open for a linear elastic structure and partially closed for a non-linear structure. For a strong ground motion of finite duration, response will build up until the structure's elastic limit is attained i.e. the storage tank is fully filled. Any additional input of energy will lead to spill over into the smaller reservoir tank. If the structure is brittle ( i.e. the size of the smaller reservoir is very small ), the reservoir will soon be filled and start spilling over i.e. the structure will fail. But if the structure is ductile, the smaller tank can manage to take some of the additional energy i.e. avoiding failure. When the ground shaking stops, elastic strain and kinetic energy stored in the structure are dissipated by damping and the structure comes to rest. The contents of the overflow tank represents plastic work done on the structure which usually means structural damage. 3 NEGATIVE AND POSITIVE WORK RESULT FROM THE INTERACTION OF BASE SHEAR AND GROUND VELOCITY INELASTIC ELASTIC RESPONSE RESPONSE (Damage) Figure 1.1: Storage Tank Analogy for the Seismic Response of Buildings by Clough [9] 1.4 Scope and Objective of Present study : This thesis is the continuation of the work done by William Tong [36], who worked on the one dimensional earthquake response of precast jointed box type structures. In this study the two dimensional response of a one storey precast box type building is studied. The building walls are made up of standard double-tees connected together by connections. To study the effect of eccentricity, panels in one of the walls are changed while keeping the other three walls same all the times. Non-linear connection behaviour is modelled from the results of test performed on isolated precast connections under cyclic loading. The main objectives of this study are : 1. To evaluate the validity of various static design methods for unsymmetrical buildings. 2. To evaluate the performace of unsymmetrical one storey box type precast structures designed according to building code requirements. 3. Based on analytical results, to make recommendations on any necessary changes in the building code requirements for the design of jointed structures. 1.5 Thesis Layout : This thesis begins with a discussion of cyclic tests performed on isolated connections used for connecting the precast units. In the following chapter, detailed modelling of the one storey structure is studied. Results of the dynamic analysis of the buildings designed according to code requirements are shown in the next chapter. In the final chapter conclusions of the present study are presented. 5 Chapter 2 CYCLIC TESTS ON CONNECTIONS 2.1 Introduction Connections are one of the most important parts of a precast structure because they are the weakest link in the structure. Test results on full scale structures have shown that connections start yielding even when the steel in the precast panels is in the elastic range. This is because the strength of a connection is much less than that of the surrounding panel. And once the connection starts yielding, its strength may start decreasing with the cyclic load leading to its failure. The design philosphy for the design of connections is : • to remain elastic for small earthquakes. • to be able to take inelastic deformations without structural collapse. Connections can be used to dissipate energy if they show stable elasto-plastic be haviour. A word of caution here is that if connections are used as energy dissipators, extreme caution must be taken to ensure their integrity, even as they degrade, espe cially the connections upon which the ultimate stability of the structure depends. Thus in some connections e.g. in the diaphragm, it is desirable for connections to resist earthquake without yielding. According to Spencer [32], these relatively strong and rigid connections designed for earthquake loading can lead to certain 6 problems under service loads. Hence, a designer has to forsee these effects and try to eliminate them by proper detailing and design. To reduce both erection and production costs, connections should be able to take some variation of size and mis-alignment of panel edges & faces etc. However the tolerance must not be so large as to cause aesthetically objectionable features in the completed structure, nor to cause functional problems. Joints should be detailed so as to provide adequate clearances between components of joints. Such characteristics help speed the work of erection crew by facilitating the placing of grout or welding and tightening of bolts etc. Spencer and Tong [36] have shown that shear walls with overdesigned connec tions are not always the best solution for earthquake loading, because overdesigned connections mean higher probable strengths of the walls, which implies that wall to roof connections have to be stronger to take the increased load transferred to them by strong walls. Good quality control of the steel that goes into the connections is necessary. Kallros [17] has shown that the yield and ultimate strength of two rebars he got from two different suppliers had much higher values than the recommended strength for that steel. He also found that the use of high yield strength bars for the connections will lead to brittle failure because these bars have less ductility. 2.2 Types of connections : Precast connections are classified as • Dry Connections • Wet Connections 7 2.3 Dry Connections These connections utilize mechanical anchors such as bolts or welded metal to trans fer the load. Shear is transferred across the connection through the bearing of steel shapes; shear of the connecting elements; shear of the welds or bolts or through friction between bolted plates. Various types of dry connections available are : 2.3.1 Bolted Connections In these connections steel embedments are bolted together to form the connec tion. The advantage of these connections is that they can be quickly assembled and erected. The primary disadvantage is that close tolerances are required for placement of the connectors and its receptacle when they are embedded in concrete members. One example of bolted connections which has behaved well under the cyclic load ing is Drescon-Concordia System [14]. Connections are generally composed of steel inserts embeded in concrete panel. These are held in place by stud-welded anchor bars which are used to develop the required strength by shear friction. During the erection when two such inserts are placed adjacent to each other, a third element ( a steel make up piece ) is friction bolted into the embeded inserts. The makeup pieces are pre-slotted to take up manufacturing and erection tolerances. These connections can be used as wall to floor connection Fig. [2.1], floor to floor connection Fig. [2.2], wall to wall horizontal connection Fig. [2.3]. Tests made by the National Bureau of Standards in Gaithersburg, Maryland indicate that these friction bolted connections have a capacity for energy dissipation and that the capacity of the connections is not impaired for cyclic loading tests beyond the ultimate slip load. A slightly modified form of the above connections is the Limited Slip Bolted Joint [ LSB ] developed by Pall & Marsh [24]. The wall to wall vertical connection 8 1 • Exterior beoring woll porta! 2. Floor panel 3. Cladding 4; Wall top insert 5. Wall bottom insert ' 6. Floor insert 7. Connection make-up pieces 8. Friction bolted connections-' (a) Exterior Wall 0 © 1. WALL PANEL ' 2. FLOOR.. PANEL 3. WALL INSERT A. FLOOR INSERT S. METAL SHIMS Q. CONNECTION MAKE-UP PIECES 7. FRICTION BOLTEO CONNECTIONS Interior Wall Figure 2.1: Wall to Floor connections, from Ref. l14^ 9 Figure 2.2: Floor to Floor connection, from Ref. [14] Figure 2.3: Wall to Wall Horizontal connection, from Ref. [14] 10 weld «fter erection connecting ^ anchors INSERT recu washers insert CONNECTING PLATE ^:-;:;::::ay. [fl i pF r i—^ .connecting pl^, .bolts (ASTM A^s) |—insert ,rm, 21 pane) 1 i SECTION (a) Wall to Wall Joint nuts welded to insert -wall panels Connecting, angles -connecting angles insert SECTION SECTION (b) Corner Wall to Wall Joint gure 2.4: Limited Slip Bolted connections, from Ref. [24] 11 (a) Mill Scale '• (b) Sand Blasted (c) Inorganic Zinc-Rich Paint 1-* • « I «0 • . . . \ :. • * ,4 2 • .; '_ .. 20 40 A (mm) (d) Metalized • • 40 10 t ( 4 2 3 2 4 S 1 A (mm) 10 '40 • (e) Brake Lining Pads (f) Polyethylene Coating 120-Z100-c >eo-< o ml 1. 2. 3. 4. 5. 6. Mill Scale Grit Blasted Zinc-Rich Paint Metalized j Sand Blasted Brake Lining Pads 6 Expected shear failure of bolt 60 40-20 It— 4 ,6 8 DISPLACEMENT, mm 10. 12 Figure 2.5: Load-Deformation Response of Limited Slip Bolted (LSB) Joints, from Ref. [24] 12 is shown in Fig. [2.4]. The connection plate is bolted in position during erection but is finally welded on one side to prevent the rotation of the plate when slipping occurs. These joints are designed not to slip for service loads, but are allowed to slip during severe seismic excitation and so they will not be grouted but sealed by other appropriate means. The static and dynamic tests on LSB connections; having different slipping surface treatment were conducted by Pall & Marsh [24], Load deformation curves and hysteresis loops; using 12.7mm (1/2 inch ) diameter high strength bolts ( ASTM - A325 ), are shown in Fig. [2.5]. The load deformation characteristics are elasto-plastic upto the point of slipping and after that these are plastic. If slipping of the bolts exceeds the hole clearance then it becomes elastic again upto the load causing failure in the bolt. The best behaviour is shown by brake lining pads inserted between steel plates with mill scale surfaces. The joint exhibited a constant repeatable slip load and nearly elasto-plastic behaviour with negligible degradation. A sand blasted steel surface was the second choice. Thus it is possible to achieve the appropriate strength and energy dissipation by an appropriate choice of joint surface and clearance of slotted holes. 2.3.2 Welded Connections In these connections steel embedments in the precast panels are welded together with or without a steel piece used to make welding of two connections easier. Research has been done on the cyclic behaviour of different types of welded connections. These include : • Headed Stud Connection. • Embeded Rebar Connection. • Split Pipe Connection. 13 Figure 2.6: Typical Headed Stud Connection showing two common stud configurations and the ease with which minor imperfections and misalignments may be accomodated, from Ref. [23] 14 CONNECTION DETAILS OF STUOS I PANEL REINFORCEMENT Al A2.A3 Bl B2 B3 £3-/i2*i%tilO*LONS—^ ~T=W 6' 1 an Figure 2.7: Details of Specimens reported by Spencer & Neille [31] YWIuturr u*r tUBUTT UlflT W±0 tmtLOH. (a) Load-deflection loops (connection A2) showing stable behaviour after 35 cycles of loading. Groups of curves have been seperated for greater clarity Figure 2.8: Hysteretic Behaviours of Welded Headed Stud Connections reported by Spencer & Neille 31 ^ 15 v 40-tJ == » Mi ' ItMIUTV LD4I | v£> ^ (b) Resuhs f()r Specimens Aj & A3 » IEFIECTI0N (In) (c) Load-deflection loops 20 (connection Bl). Rapid increase in the applied deflection led to .30 failure after relatively few cycles. •YIE10 ENVELOPE (d) Load-deflection loops " (connection B2). The result of a stud failure can be -oo seen in Cycle 19. Figure 2.8 Continued 16 Headed Stud Connection A typical headed stud connection is shown in Fig. [2.6]. It is clear from the fig ure that minor imperfections and misalignments can be accomodated very easily in these connections. Spencer & Neille [31] have done work on the behaviour of headed stud connections under cyclic loading. Details of the tests are shown in Fig. [2.7]. Connection Al was loaded monotonically to failure while the other five connections were tested for cyclic loading at frequencies in the range of .01 to .02 Hz. Load deflection curves for the connections are shown in Fig. [2.8]. The following conclusions were made by Spencer & Neille [31]. : • The PCI procedure for calculating the ultimate design strength of these con nections under static loading gives conservative results. • The strength of the connection in the first cycle of loading up to yield will be approximately the same as the strength in monotonic loading. • If cyclic loading is continued above the stability limit the strength of the con nections will fall with increasing number of cycles and yield strength envelop will tend to approach the stability limit. • The deflections reached before failure were seven to twenty times the theoret ical elastic deflection corresponding to design ultimate strength. • These connections, if properly designed and detailed appear to be suitable for use in earthquake-resistant box type buildings. Neille [23] showed that three factors contibuted to the capacity of these connections; friction between the faceplate and concrete; bearing of the end faceplate and con crete; and bearing and bending of studs. He showed that friction forces decreased rapidly under cyclic loading and were negligible compared to other forces in the connections. Existence of these three mechanisms through which the connection 17 transfers shear forces to surrounding concrete tends to contradict the shear friction analogy which is often used in the design of these connections. It was found that strength of the connection is directly dependent upon the strength of concrete in which it is cast, while the shear friction theory does not contain concrete strength as variable. Embedded Rebar Connection An alternate method for embedding the steel sections in precast members is to replace the headed stud with welded reinforcement as shown in Fig. [2.9]. A major advantage of these connections is the large surface area available for welding which reduces the weld stress and chances of brittle weld failure. A brief summary of the rebar connections tested for cyclic loading is given below : Aswad [2] did some low cyclic tests on connections shown in Fig. [2.10]. The tests were done on 21* * 8' panels. Two types of connections; connections for double-tee ( 2" flange thickness ) and connections for precast wall panel ( 6" thick ); were tested. Load was statically applied. In the majority of cases the load was cycled three times between ± P0j where Po is the value smaller than the actual maximum load and the connection was brought to failure. Thus these tests do not represent the real cyclic behaviour of the connections. The load-displacement curves are shown in Fig. [2.10]. These curves do not show much deterioration in strength and stiffness because the cyclic tests remained essentially in the elastic range. The following conclusions have been stated by Aswad [2] : • When force applied to the plate was simple shear, ultimate capacities were higher than ones listed in Ref. [1] by up to 100 % in some cases. The extra capacity was higher whenever the connection included a | thick plate which increased the bearing portion of capacity. • Ductility of the standard plates tested was good to excellent except for D-34 18 re 2.9: Alternate Embedment 19 Figure 2.10: Details of Connections reported by Aswad [2] 20 D-40 Tie Cyclic Shear Force vs. Relative Slip for D-40/D-40 ties (c) MI'ZS.I Cyclic Shear Force vs. Relative Slip for P-9/P-9 (connections previously loaded in Shear to 15.0 K) Figure 2.10 : Continued 21 PLATES P-3 (TOP) AND P-8 (BOTTOM) SHEAR PORCEVS. SLIP FOR P-8/P-3 COMBINATION Figure 2.10: Continued 22 with straight rebars, although its actual capacity was superior to the recom mended one. • Rebar length was sufficient to develop ultimate capacity. • Plates subjected to cyclic tests showed major deterioration in stiffness after three cycles. • Pull out forces normal to the panel surface substantially decrease ultimate capacity in shear. Moderate in-plane pull-out forces acting simultaneously with shear do not noticeably affect ultimate capacity of precast connections although they reduce the connection stiffness. • Size of field weld plates is critical in wall panel connections if premature weld failure is to be avoided. Spencer [34] has done cyclic shear load tests on connections anchored into 4** 2**5.5" concrete panels. The load-displacement curves for the tests are shown in Fig. [2.11]. The following conclusions were made by Spencer [34] : • Loading cycles in the elastic range do not reduce the strength of the connec tions. • The nominal strength of the connections can be found using models shown in Fig. [2.12]. • The strength of the connections, with the rebar running into the connection at 45°, falls to about 50 % of the nominal strength under cyclic loading into inelastic range. • The strength of connections, with rebar running at 90°, falls to less than 50 % of the nominal strength under cyclic loading into the inelastic range. These connections were not recommended for use in situations where they might be 23 Figure 2.11: Cyclic Behaviour of Connections Anchored With Steel Bars, by Spencer [34] 24 25 (f) . p.. Figure 2.11 : Continued 26 Figure 2.11: Continued 27 (a) Embedded bent rebar | Fb (b) Embedded rebar welded to an angle Fi, — 4.4 * fc * d* lP and V = 2 * Fb (c) Embedded rebar at 90 degrees Figure 2.12: Models Developed by Spencer [34] 28 loaded past their elastic limit when the potential loss of strength is considered on the design. • The connection with a recess in the panel edge and straight embedded bars at 45° appears to perform best under simulated earthquake loading. • Panel thickness and concrete quality can have a marked effect on the behaviour of connections. Another investigation has been done by Kallros [17] on thin panel embedded rebar connections. Six different types of connections were tested. All the connections were loaded into the elastic range and loading continued until the connection failed. The failure of the connections occured either by bar failure or by spalling of the concrete. The bar failure generally occured at very small deflections. For the connections that failed by spalling, once the connection started cracking, the stiffness gradually decreased with increased cycles into inelastic range and width of the load-deflection curve also increased. Connections that were welded straight to reinforcing mesh did not behave well as compared to other connections. Connections with lower yield strength rebar ( rebar # 1 ) behaved better than rebar # 2, with higher yield strength. The connections and load-deflection curves are shown in Fig. [2.13]. The following conclusions have been made by Kallros [17] : • Good quality control of steel that goes into connections is necessary. Brittle failure of connections, which is more likely with high yield strength bars, is not desirable. • A thicker flange will behave better during earthquake loading since there is more cover around the reinforcement. A thicker flange is also less fragile during transport. • The design strength of connections should be taken as 50 % of expected max-29 Cyclto Wit or cenntetlon __ BC — 1 (a) Rebar # 1 at 45 degrees Cyclic tail of connection -it -a -i Displacement (mm) S m y -I (b) Rebar # 1 bent to 45 degrees and welded to an angle Cyclic teat of connection T**t f« ai.ar.tfM 2 -1 < Displacement (mm) '% Cyclic U«t of conntetion -14 •>*• Displacement (mm) Cyclic Ust of tonntcllon O -1* . i • i=?-^— vr, O e-J -« 1 • Displacement (mm) Cyclic teit of conntetion T*«4 It W.**.IM4 Ik If =^—,—,—.—. 'f Displacement (mm) Displacement (mm) Figure 2.13: Cyclic Tests on Thin Panel Embedded Rebar Connections reported by Kallros f]7| 30 Cyclic tt»l of c*»nn«cllon lot ft It »» •• i \ 1 (c) Rebar # 2 at 45 degrees with a short recess Cyclic Uat of connection Displacement (mm) Cyclic Ust of connection 7\ *-~3l—" —— IT • V T* S -IV Ml -l» -« "« • ' Displacement (mm) Displacement (mm) Cyclic Uit of connection M fll H.MJW* s s 1 (d) Rebar # 2 at 45 degrees with a recess in a thicker slab Cyclic Ust of connection t«l |U tLM.IMf o Displacement (mm) Cyclic teat of connntlen »••* fl* M.lttM Kb. » •» y T3 • 10 -O -a . -J - • rr \\\ \ r — n-•M 'Iff Displacement (mm) Displacement (mm) Figure 2.13 : Continued 31 I -\-(e) Reversed angle welded to reinforcing mesh A \-J4U II •• 1-Cyclic Ult of connection TMl II* CLMLlfti Cyclic Ult sf connection Till lit If H.1M4 -« -« I I Displacement (mm) -tt -I -4 I « I Displacement (mm) ESS 1553 n ri I «00 mm Angle welded to reinforcing mesh Cyclic test of connection Tut ft UAIM Cyclic test of connection Tail lis la^niraa -la -ie >• H l « w Displacement (mm) Displacement (mm) Figure 2.13 : Continued 32 imum capacity of the connections. Spacing between connections can be re duced in order to reduce the load on each connection. • Loading cycles in the elastic range do not seem to reduce the capacity of the connections. • The connections with a recess in the panel edge and embeded bars at 45° appear to perform best under simulated loading. These connections can also be easily attached to an adjacent connection. • Connections in which an angle is welded directly to the reinforcing mesh are not recommended since they damage the reinforcing steel and cause severe cracking in flange itself. • The models used to predict the connection strength are adequate for predicting the lower limit of the connection strength. Split Pipe Connection Saxena [29] replaced the rigid bar or plate normally used to connect adjacent pieces with a ductile steel connector ( a steel pipe with longitudinal slit). The connection is shown in Fig. [2.14]. The following conclusions were made by Saxena [29] : • The split pipe connection is able to accomodate the relative movement be tween panels due to shrinkage and temperature changes. • The split pipe connection limits the forces that develop during dynamic load ing which leaves the panels largely undamaged. • Slight inaccuracies in the dimensions of the precast panels during casting can be easily accommodated when pipe is welded in its position. 33 PLAN Figure 2.14: Split Pipe Connection by Saxena [29 34 2.4 Wet Connections These are cast in place connections using reinforced or unreinforced concrete to form a junction between members. The strength and performance of these connections dependend upon precast and cast-in-place concrete; the amount of transverse and longitudinal steel; the tensile and compressive forces acting on the connections and the surface prepration of precast panels. Shear is trasferred mainly through shear friction mechanism. Some tests have been made to test the performance of wet connections under cyclic loading. A few of these tests are summarised below : 2.4.1 Platform type Horizontal Connection This connection is most commonly used in America as a horizontal joint between vertical panels. The cross-section of a typical platform joint is shown in Fig. [2.15]. For these connections shear transferred by friction is equal to a coefficient of friction times the compressive force acting on the joint. If the induced shear force exceeds this value then slip will occur in the joint. This shows that the coefficient of friction is important in the design of these joints. Joahl and Hanson [16] have tested these connections under monotonic loading and Hanson [12] has tested these connections under cyclic loading. These connections transfer shear through coulomb friction due to normal stress, and through shear friction due to clamping and dowel action of vertical wall reinforcement passing through the joint. It was found that the coefficient of friction effecting slip of these connections showed a very low value varying from 0.2 to 0.4 ( see Fig. [2.16] ). The load-displacement curve, Fig. [2.17b], was found to be elasto-plastic. When a longitudinal reinforcing bar crossed the joint, cyclic reversing loading caused spalling and damage of concrete around the bar; while the resistance of the bar to distortion caused to post-elastic-slip stiffness to develop. When longitudinal reinforcement was enclosed within a soft layer of padding, concrete spalling was avoided and the hysteresis loop shown in Fig. [2.17a] 35 DRY PACK VERTICAL CONTINUITY REINFORCING PRECAST PLANK GROUT BEARING STRIP WALL PANEL Figure 2.15: Typical platform-type horizontal connection 0.5 r 3. 0.4 2 o r-o cr Lu ll. UJ o u 0.3 0.2 0.1 A A O A A 1.0 2.0 NORMAL STRESS (ksi) 3.0 Figure 2.16: Coefficients of friction from tests on horizontal joints by Hanson [12] 36 I*— 3/* D*e*U of St««I Tabulur CoonceCor Figure 2.18: 3/32 Scale Model Interior Horizontal Joint used by Harris & Abbound [131 MPa 2.0 10 IS e e u o •a - a < In. (7) • Jones-bonded (7) • rrene (10) • CaaCon and Krlt (6) • Uanaon (9) P This Study 1000 2000 Normal SI '« • • . pal 300O Figure 2.19: Comparison of Coefficients of Friction by Harris & Abbound [13] 37 was obtained. Another study on cyclic behaviour of interior American type horizontal joints has been done by Harris & Abbound [13], using ^ scaled models of the joint shown in Fig. [2.18]. They found a much higher value of the coefficient of friction, Fig. [2.19], as compared to that from Hanson [12]. 2.4.2 Dowel Connection Dowel connections can be used as horizontal connections between wall panels and footings. The strength of these connections depends upon the area of cross section and on the embededment length and the developed bond of the dowel bars. As in the platform type connections, the co-efficient of friction is an important parameter in the design of these connections. One disadvantage of these connections is that any adjustment after the initial set of grout may destroy the dowel bond thus reducing the connection strength. Dowel connections may be keyed to increase the shear resistance through interlocking of keys Fig. [2.20]. Once keys fail, the residual shear strength depends upon the friction and strength of the dowels. The only experimental data available on the cyclic test on dowel connection is from Dimitrov &; Georgiev [10] who did tests on two full sized panels connected to a rigid base by trapezoidal dowels. The reinforcement bars passing through the dowels are welded and the cavity is closed with fine aggregates. The concrete strength in the panels was 20 MPa. The dowel reinforcement in the first specimen; ( W\ in Fig. [2.20] ) consisted of two longitudinal bars each of 16 mm dia. with a design steel stress of 210 MPa and yield stress of 300 MPa. The second specimen ( Wi in Fig. [2.20] ) had longitudinal reinforcement of two 22 mm dia. bars with a design steel stress of 375 MPa and a yield stress of 500 MPa. Normal forces on the panels; 2 * 270 kN on Wi and 2 * 275 on W2; were provided by hydraulic jacks to simulate the vertical loading on the panel. Reversed cyclic loading was applied along the top 38 3U0 1.-WM J.- IMkN 3.- 180W* I.- »kN 5. - 20OkM 6. - 260.280** Figure 2.20: Specimens Tested by Dimitrov & Georgiev [10] P-0Acm-20kN Zi. - 1cm - 1mm P - 0,35cm - 20kN x\ - 1cm - 1mm Figure 2.21: Load-Deflection Curves of Specimens in Fig. [2.20] by Dimitrov & Georgiev [io] \ 39 edge of the panel. The loading sequence followed the expression : P = 0, IH, 0, -IH, 0, 2H, — 0, jH, 0, -jH, 0 -in which j is the number corresponding to loading cycle H is the load amplitude increment assumed constant at 20 kN for each cycle. The curves in Fig. [2.21] are obtained by connecting the peak values of the dis placement at each load cycle which are obtained from the hysteresis loops of the specimens. The main objective of these tests was to check the ductility of the panel with dowel connections. From the curves in Fig. [2.21], it can be concluded that strength and stiffness of the panel does not degrade too much as the cyclic displacements go into the inelastic range. For a dowel connection if it is assumed that the connection fails due to failure of dowel bond because of the slip between dowel bar and the surrounding concrete, leading to pull out type failure; then the experimental data available on bond char acteristics of reinforcing bars for seismic loading can be used in the modelling of these connections. One such study has been done by Hawinks & Lin [15]. They tested three types of test specimens Figs. [2.22, 2.24, 2.26]. Load-displacement curves are shown in Figs. [2.23, 2.25, 2.27]. The dotted line on these curves is for connections tested under monotonic loading. A brief description of conclusions reached by Hawkins & Lin [15] is given below : • The displacements for first yielding were very similar for all specimens and equal to 40 * IO-3 inch ( 1.016 mm ). • The characteristics of the loading history have a marked effect on rate of bond deterioration and mode of failure. 40 io-r -i to .EZZZZL — r —j r—• tactlta A4 sir" 0 TZZZZZ z btaM 1*1 la J. -All reinforinclng bars are Grade 40. -Shaded areas are supports. Figure 2.22: Test Specimen - Phase 1, by Hawkins k Lin [15] to TTTTi __-monotonic curve MO no i •Utlt* «1t». I. i It'1 <«**. zz I A A A A A A A ICnlal (Law MittO) Cm mif (a) /„ = 48 Ksi, fe = 3100 psi Figure 2.23: Load-displacement curve, # 10 straight bar, Bamboo-type defor mations, Phase 1 by Hawkins & Lin [15] 41 laHlaa (b) fv = 48 Ksi, fe = 3200 psi (c) /„ = 48 Ksi, fc = 4200 psi Figure 2.23 : Continued 42 Figure 2.24: Test Specimens - Phase 2, by Hawkins & Lin [15] 43 lfO.flOO ua.tDD to.000 K.000 «caoo f to-sco 0 8 -0.900 -40.000 -40.000 -mjxD -100.000 -IS-CCO' (a) /, = 48 Ksi, fc = 2640 psi JXB • 400-000 KS .000 (b) /„ = 60 Ksi, fc = 3760 psi Figure 2.25: Load-displacement curve, # 10 straight bar, Bamboo-type defor mations, Phase 2 by Hawkins & Lin '.15; 44 (c) /„ = 60 Ksi, fe = 4100 psi (d) /„ = 60 Ksi, fe = 5120 psi Figure 2.25 : Continued 45 twin* Uavrata Figure 2.26: Test Specimen - Phase 3, by Hawkins & Lin [15' 60.000 40.000 20.000 r--20.000 r -40-ooor -60.000 -flC.o:3 -203.000 -100.000 Cyclt. Ductility Mlatory 0 100.000 203-000 0lSHJCDl£NT XIO"1 IN. 300.000 430.000 (a) /„ = 67 Ksi, fc = 3400 psi Figure 2 27: Load-displacement curve, #6 straight bar, Bamboo-type deforma tions, Phase 1 by Hawkins & Lin |15| 46 • The surface geometry of the bar had a significant effect on rate of bond dete rioration. The rate increased as the ratio of lug spacing to bar diameter in creased. Bars with different surface geometries and similar load displacement characteristics for monotonic loading had significantly different characteristics for reversed cyclic loading. • For reversed cyclic loading, the response for specimens with 180° hooks was much poorer than for specimens with straight bars because once slip pen etrated to the hook, the motions of the hook broke the connection. The 90° connection maintains good characteristics for tensile loading considerably longer than for compressive loading but even then its characteristics are not nearly as good as those for specimens with straight bars. • The grade of bar had less effect than general form of its stress-strain charac teristics. • The strength of concrete had a marked effect on load displacement curves. The displacement corresponding to maximum load capacity increased in direct proportion to the concrete compressive strength. • Additional hoop reinforcement markedly improved the load-displacement char acteristics for bars terminating with 90° hooks but had little effect on the characteristics of straight bars. 2.5 Hysteresis Model for Vertical Connection A number of mathematical models have been proposed for the connections between vertical wall panels. As there hasn't been a complete comprehensive study on a particular type of connection which would have taken into account all the vari ables involved all these hysteresis models are approximate to same degree. A brief discussion on these models is given below. 47 Figure 2.30: Trilinear Hysteresis Loops used for both Stress-Strain and Moment-Curvature Relationships for Studs, by Neille [23] (c) Stabalized Hysteresis Loops After (d) Strength Envelop Stiffness Degradation Figure 2.31: Shear-slip Relationships for Shear-friction element, by Schricker & Powell [30] 50 The simplest of the hysteresis models used for precast connections is the elasto-plastic model Fig. [2.28]. This model has been used by Ashwad [3] to model his rebar type vertical connection. A similar model has been used by Pall & Marsh [24] to model the LSB joint. Looking at the hysteresis loops obtained by Pall & Marsh Fig. [2.5], one finds that elasto-plastic model is valid for the LSB joint, but for a rebar type connection the validity of this model is still questionable because of its inability to model degrading strength and stiffness of the connection. Neille [23] has used a trilinear load-deformation model, Fig. [2.30], for welded headed stud connections. The test results for load decrement vs. associated maxi mum deflection in one cycle were represented by a parabola which has been incorpo rated in the model to model the strength and stiffness degradation in the connection. Muller & Becker [21] used the degrading model in Fig. [2.29a] to represent the test data obtained by Neille [23]. They proposed another model, Fig. [2.29b] to represent a highly degrading connector. Schricker V. & Powell G. [30] have developed a number of elements to model several types of joint behaviour for the inelastic analysis of pre-cast buildings. Each element is assigned a simple force-displacement relationship Fig. [2.31]. Complex relationships for the behaviour of various joints were obtained by placing two or more elements in parallel in a single joint. Tong [36] has used a simple degrading model with strain hardening to model the rebar connection. The model given by Tong can be used to model vertical connections between wall panels ( no vertical connections have been included in the buildings studied in this thesis but the computer program has the capability to model these connections ). A brief description of this model is given below. The model consists of a primary curve Fig. [2.32] defined by four linear seg ments ( each segment is assigned an arbitrary number ). The initial stiffness of the connection is ki which remains constant until the elastic limit point ( fe , de) is 51 M = b*f> Displacement Figure 2.32: Hysteretic Model : Primary Curve, model proposed by Tong [36] Displacement Figure 2.33: Hysteretic Model proposed by Tong [36] Cyclic behaviour before yielding, model 52 -f, tOB; to B: Relative Displacement -3 Figure 2.34: Hysteretic Model : Cyclic behaviour after yielding, model proposed by Tong [36] reached. The ascending branch representing strain hardening starts after the yield point ( fy , dy ) is reached. The ascending branch stiffness is k& defined as kz = a* k\ 0 < a < 1 Max. rise of the ascending branch is given by A/=6*fc3 0 < 6 < 1 Descending branch stiffness is defined by /c4 = —c * /c3 0 < c < 1 where a, b, & c are constants. The hysteresis model cyclic behaviour before and after yielding are shown in Fig. [2.33] & Fig. [2.34] respectively. There is no strength degradation for cyclic behaviour before yielding. Once the yielding has taken place, the strength degra dation was defined during load reversals from ascending and descending branches. Everytime unloading takes place from these branches, a reduced yield limit is de fined for the next cycle by multiplying the load at reversal by a factor <f> which is given by <p = Rn where R = 0.5 d\ fy Two other constants f3 & a are defined, which when multiplied by slope of ascending and descending branches, give the slopes of new ascending and descending branches. (3 = Rr 0 <r <1 a = ^- 0<5<1 where n, r & s are constants. 54 2.6 Hysteresis Model for Horizontal Connection A number of models have been proposed for horizontal connections commonly used in industry. A brief discussion on these models is given below : Aswad [3] used P-8 and P-3, ( Fig. [2.9]) in combination as a horizontal joint and he modelled this connection with an elasto-plastic model shown in Fig. [2.35]. From the experimental results for tests on platform-type horizontal connection Fig. [2.17], it seems that the elasto-plastic model without any strength or stiffness degradation can be used to model this connection. There have been few other models proposed for the horizontal connections commonly used in Europe Ref. [38,39], but all the above mentioned models have been used for wall to wall horizontal connections. A model for a dowel type horizontal connection between wall panel and foundation, has been proposed by Tong [36]. His model is for the force-displacement characteristics of the dowels. He has based his model on the indirect study done by Hawkins & Lin [15]. Hawkins & Lin proposed the model given by thick unbroken lines in Fig. [2.36] while the broken lines in the figure represent the test results from which this model is derived. The model proposed by Tong is given in Fig. [2.37] and it is a further simplification of the one in Fig. [2.36]. For the present studies the model given by Tong [36] has been used to model the horizontal connection. A brief description of the model is given below : The bilinear model has a horizontal post-yielding branch. On unloading the curve stiffness is ki, the same as the initial stiffness. When the load is reversed, the unloading stiffness is obtained by considering the slope of a line connecting the load reversal point to the yield point, or, if the connection has already yielded in the new direction, the point on post yielding branch corresponding to the maximum deformation experienced. Hysteresis energy dissipation for deformations smaller than yield is not considered and no strength degradation is included. The validity of this model can be questioned, but considering the approximate nature of the 55 Shear Force Shear Force CASE I CASE II (a) Vertical Joint Connector Characteristics Shear Force • Shear Force dljpL Axial Spring ac Edjej (truii eleaent) (b) Horizontal Joint Connector Characteristics Figure 2.35: Joint Connector Characteristics proposed by Aswad (3] 56 Figure 2.36: Bar Force-Pull Out Model for Cyclic Loading, proposed by Hawkins & Lin [15] 57 58 modelling process and conservative nature of this model as compared to one by Hawkins &; Lin [15], the proposed model is adequate for present studies. In general, for all the connections used by the precast industry, it can be said that the proposed models are approximate to varying degrees because of the lack of experimental data and analytical models available for the connections. For those connections for which some data is available, not all the variables have been taken into account, e.g. in the studies done in Ref [29,34], the effect of load normal to the connection and the effect of vertical load on the panel on the behaviour of the connection have not been taken into account. 59 Chapter 3 DYNAMIC ANALYSIS 3.1 Introduction : The usual Code approach for designing buildings for earthquake loads; i.e. to calculate the equivalent static loads on the structure; does not explicitly consider the dynamic characteristics of the structure. One way to investigate the influence of the dynamic characteristics of the building on earthquake response is by using the 'response spectrum design approach'. The response spectrum used corresponds to a probability of exceedence of seismic ground motion of 10 percent in 50 years ( .0021 per annum). The Supplement to NBCC - 85 [35], commentary J - 39, recommends a simplified response spectrum technique for dynamic analysis. This approach ensures that the dynamic analysis will not result in lower base shears than those prescribed by the code because the code values are already minimum values commensurate with the acceptable level of safety. Another method of dynamic analysis is to find the response of the building over the total duration of a known ground motion using either a linear or non-linear analytical model with the time-history of the earthquake as the input data. This approach is expensive and is generally used only to check the performance of the final design of complex structures. 60 3.2 Idealization Technique : A number of structural idealization techniques are available to investigate the be haviour of panel structures [5]. A Beam Modelling technique is used for shear wall structures in which the roof or walls of the structure have no in plane rotation and the lateral rigidity of the walls is uniform. The mass is lumped at the roof and floor levels. The horizontal joints are modelled as beams. Brankov and Sacha-niski [4] have used this approach in the dynamic analysis of multi degree of freedom panelized structures. Shear Medium theory has been widely used in the area of coupled shear walls and shear wall buildings. It allows not only a straight forward analysis of the overall response of the walls joined by coupling beams but has also been used successfully in shear wall buildings to reduce the number of degrees of freedom involved. In this method the coupling beams are replaced by a continuous shear medium. The basic assumptions for this theory are : • Discrete coupling elements are evenly smeared over the joint length. • The lateral deflections of the individual walls are equal. • For the individual walls the classical beam theory holds i.e. plane cross sec tions remain plane after bending. Muller and Becker [21] used this theory for precast walls coupled by vertical con nections. For grouted connections, the replacement of coupling elements by a con tinuous shear medium is exact. For mechanical connections this replacement of coupling element is not exact. In the Frame Analogy the structure is idealized as an equivalent frame having wide columns with rigid arms and connected by joints. The panels are modelled as beam columns with flexural, axial and shear deformations taken into account. 61 Constant stiffness is specified for the panels as they remain in the elastic range. This procedure is efficient in the three dimensional computer analysis of shear wall structures. Pall & Marsh [24] used this technique to model a large panel wall with Limited-slip bolts as vertical connections. Most of the work done on the computer analysis of panelled structures has used the Finite Element for modelling because of this method's ability to model diverse geometries and material properties. The whole structure is discritized into small elements connected to each other only at the nodes. The response of the structure depends upon the response of the nodes. With some modifications, the existing computer programs [28] available for the earthquake analysis of the ordinary reinforced structures can be used to find the response of the panelled structures. Usually the panels are assumed to remain elastic while the connections exibit non linear characteristics. The size of the problem is reduced by eliminating all nodes other than that of the connections, by static condensation. The present study is based on an idealization technique used by Tong [36]. In this technique the panels are assumed to be rigid and non-linear hysteretic models are used for the connections. The nonlinear hysteretic loop for the dowel connection is shown in Fig. [2.37]. This technique is based on the study done by Becker & Llorente [6] and Hanson [12] on the study of wall to wall pedestal type horizontal connections. They found that for a friction coefficient of 0.4 there is very little shear slip along the horizontal joint and rocking of the wall panels governs the response. Fintel, Schutz & Iqbal [11] report that the co-efficient of friction specified by design codes may vary from 0.2 - 0.8. CAN3-A23.3-M84 [8] recommends a coefficient friction of 0.5, thus for the wall panel with dowel joints as horizontal connections it can be safely assumed that rocking of the panels will govern the response and there will be no slippage in the joints. 62 The rigid panel is assumed to rock about its base corners only. The foundation underneath the panel is replaced by two elastic springs, one at each corner of the panel. These springs are not connected to the panel i.e. the panel can lift off a foundation spring. Due to rocking of the panel, there can be three possible situations ( see Fig. [3.1]) : • Case 0 : both the foundation springs are in contact with the wall panel. • Case 1 or -1 : only one foundation spring is in contact with the panel. • Case 2 : neither foundation spring is in contact with the panel. The lateral stiffness and shear force for each case for an interior panel is given in section [3.6]. 63 4 f f f * i ' ! FS f J un n f 4 Dowel at right Dowel at left Foundation spring at right Foundation spring at left (a) Idealization of Shear Wall Case-1 Case 0 Vertical connection Wall Panel Fixed Base Case 1 Case 2 (b) Transition between Different Cases Figure 3.1: Idealization of Shear Wall 64 3.3 2-D Mathematical Model A one storey shear wall building is modelled by the mathematical model shown in Fig. [3.2]. The model has three degrees of freedom; 2 translations and a rotation at the CG. of mass. The following notation is used in the derivation of the equation of motion for the building under earthquake loading : L : building dimension in the direction of ground motion B : building dimension perpendicular to the direction of ground motion M : mass of the building J : rotational moment of inertia about the CG. of the building kwi : lateral stiffness of wall i found as shown in Section [3.7] and i varies from 1 to 4 xg : ground acceleration acting on the structure Cx : damping in X-direction Cy : damping in Y-direction Cg : rotational damping x : relative displacement of the building CG. in X-direction Fig. [3.2] y : relative displacement of the building CG. in Y-direction Fig. [3.2] 9 : relative building rotation about CG. of the building Fig. [3.2] x : relative velocity of the building CG. in X-direction Fig. [3.2] y : relative velocity of the building CG. in Y-direction Fig. [3.2] 9 : relative rotational velocity about CG. of the building x, : relative acceleration of the building CG. in X-direction y : relative acceleration of the building CG. in Y-direction 6 : relative rotational acceleration about CG. of the building The equations of motion for each of freedom ( for building CG. is at center of the building ) are : Mi + Cxx + (kwi + kw2)x + (kwx — kw2)B/29 -Mi, My + Cyy + (kw3 + kw4)y + (kw3 - kw4)L/20 0 J0 + CB9 + ((x + B/26)kw1 -{x- B/29)kw2)B/2+ ((y + L/29)kws - (y - L/29)kw4)L/2 0 65 66 In general : [M]{X} + [C]{X} + [K]{X} = {F} (3.1) where M C = K = {X} = MOO 0 MO 0 0 J cx 0 0 cy 0 0 kwi (kwi — x y 0 0 Ce + kw2 0 (kwi - kw2)B/2 0 ^3 + kw4 (kws — kw^)L/2 kw2)B/2 (kw3- kw4)L/2 (kwt + kw2)B2/4+ (kw3 + kw4)L2/4 {X} {X} {F} x y 0 y e -MXg 0 0 3.4 Wall Displacements : x, y & 0 are the displacements at the centre of mass. If xwl...4 are the displacements of Wall 1...4 . Then XWi = x + B ~2 * 9 xw2 = x — B 2 * 9 xw3 = y + L 2 * 9 XWi - y- L 2 * e 67 3.5 Time Step Integration : In incremental form eqn.(3.1) can be written as : [ M}{ AX } + [ C ]{ AX } + [ K ]{ AX } = { AF } Variables with subscript 1 are for time t and Variables with subscript 2 are for time t + At. Assuming a linear variation in acceleration ( see Fig. [3.3] ) we get : {X2} = {Xi} + {S} * At Integrating eqn.(3.3) : Ar2 {X2} = {X,.} + {X^ * At + {Si-Integrating eqn.(3.4) : - At2 At3 {X2} = {X^ + {Xi} * At + {Xi}— + {S} — From eqn.(3.5) : From eqn.(3.3) & (3.6) : {AX} = {X2}-{Xi} = ({X2} - {Xi})-^ - {^} ± - 3 * {Xi} But {AX} = {X2} - {Xi} Thus {AX} = ^{AX} - -^{X,} - 3 * {Xx} 68 <u Assumed MA L—Ax—J Rei. Disp. (a) Assumed Constant Stiffness over AX o o < 0) Actual (unknown) Mr -At-Assumed Time (b) Assumed Linear Variation of Acceleration Figure 3.3: Linear Variation of Acceleration 60 And {AX} = {X2} - {X,} Q . A/ .. —{AX} - 3 * {Xj - -^-{XJ (3.9) Putting eqn. (3.7), (3.8), (3.9) in eqn. (3.2) we get [M] ( ^{AX} - j-t{Xl} - 3 * {Xx} ) + ^ . At [C](^{AX}-3{X1}-|{X1}) + [K]{AX} = {AF} (3.10) Rearranging various terms or where and [Knet] {AX} = {Fnet} {AX} = [Knet]"1 {FneJ (3.11) [Kn.] = [K] + £j[M] + £[C] (3.12) [Fnet] = {AF} + [M] ( ^{X,} + 3{X1} ) + [C}(3{X1 + ^(Xl}) (3.13) Knowing the values of various variables at time-step t, and knowing the incremental change in the the ground acceleration from time t to t + At, we can find { AX} from eqns. (3.11), (3.12) & (3.13) . Then use eqn. (3.7), (3.8) & (3.9) to find the incremental change in displacement, acceleration & velocity. 70 3.6 Panel Stiffness The following notations are used in the derivation of the stiffness of the panel for various possible panel positions : a : Width of panel b : Height of panel c : Height of panel upto roof level s : Separation between two panels e : Distance of dowel from the nearer panel face w : Weight of panel x : Horizontal wall displacement j/i : Deformation of the left foundation spring y2 : Deformation of the right foundation spring zx : Defomation of the left dowel springs measured from level 2 zi : Defomation of the right dowel springs measured from level 2 v : Relative shear displacement between the vertical connections 6 : Rotation of the panel about one of the corners &3 : Vertical spring stiffness fc4 : Left dowel spring stiffness k$ : Right dowel spring stiffness kg : Foundation spring stiffness Fi : Horizontal force in the panel at the roof level /s : Vertical spring force /4 : Left dowel spring force fs : Right dowel spring force /9i : Left foundation spring stiffness /92 : Right foundation spring stiffness Nv : Number of vertical connections on one face of the panel Level 1 : Undeformed position of foundation Level 2 : Deformed position of foudation under the dead load of the panel Level 3 : Final Deformed position of the foundation Note : &3, £4, k^, are found from the hysteresis loops of the respective connections and may vary for each time-step. 71 3.6.1 Case 0 : Both the foundation springs in contact with the panel The free body diagram for the panel in Case 0 is shown in Fig. [3.4]. Level 1 is the original surface of the foundation. Level 2 represents the initial deformation due to dead load of the panel, while Level 3 represents the total deformation due to the rotation of the panel. The deformations of the dowels are measured from Level 2. The stiffnesses of various springs and other structural data is given in section [5.2]. The stiffness of the panel for this case is given by the eqn.(3.14) ( Ref. [36], Page 111 ) : _ , l_1In, , E{2k4ks + k4k9 + k5kg)  Ki ~ [2c> + a\x\][Dks+ k4 + k5 + kQ ^(sgnx) + G] (3.14) and .. the shear force Fi is given by : 2Nvcfs + c{a - 2e)(/4 - /5) + (a2fc9 - wb)x Fi = 2c2 + a\x\ (3'15) where D = 6a(a + s) (3.16) E = (a-e)2 (3.17G - a2 k9 - wb (3.18) 3.6.2 Case 1 : Only right foundation spring in contact with the panel The free body diagram for the panel in Case 1 is shown in Fig. [3.5a]. The stiffness of the panel for this case is given by the eqn.(3.19) ( Refer. [36], Page 114 ) : _ f 1 nn), , 2[Ek4k5 + (o - e)2fc4fc9 + e2A;5A;9]  Ki ~ l2c2 + ax""** + k4 + k5 + k9 - aFi - wb] (3.19) and .. the shear force is given by : 72 (b) Negative displacement (a) Positive displacement Figure 3.4: Free body diagrams of an interior panel corresponding to Case 0 73 -X f»1 f$ f, f5 f,2 (b) Case -1 (a) Case 1 Figure 3.5: Free body diagrams of an interior panel corresponding to Case 1 and Case -1 74 2Nvcf3 + 2c(a - e) /4 + 2ce/5 + acw - wbx F< - + ^ (3.20) where D & E are constants given by eqn.(3.16) & (3.17). 3.6.3 Case - 1 : Only left foundation spring in contact with the panel The free body diagram for the panel in Case -1 is shown in Fig. [3.5b]. The stiffness of the panel for this case is given by the eqn.(3.21) ( Refer. [36], Page 115 ) : _ f_J_lfn, , 2[Ek4k5 + e2fc4fc9 + (a - e)akBkg]  Ki ~ [2c* - ax l[Dk3 + k4 + h + k9 - aFi - wb] (3.21) and .. the shear force Fi is given by : 2Nvcfs-2cef4-2c(a-e)f5-acw-wbx Fi = 2c' - ax (3-22) where D &; E are constants given by eqn.(3.16) & (3.17). 3.6.4 Case 2 : Neither foundation spring in contact with the panel The free body diagram for the panel in Case 2 is shown in Fig. [3.6] The stiffness of the panel for this case is given by the eqn.(3.23) ( Refer. [36], Page 115 ) : jv- _ r 1 ]In, , 2Ek4k5 Ki ~ l2c2-(a-2e)|x|Jl^A:3+ k4 + k5 " (a - 2e) (Fi) {sgnx) - wb] (3.23) and .. the shear force F{ is given by : _ 2iV„c/3 + 2c(a - 2e)/4 + c(a - 2e)w - wbx Fi ~ 2c'-(a-2<0|*| (3-24) where D , E & G are constants given by eqn.(3.16), (3.17) & (3.18). 3.7 Wall Stiffness The stiffness of a wall is the sum of the stiffnesses of all the panels in the wall. A wall is made up of exterior and interior panels as shown in Fig. [3.7]. Interior panels can 75 original foundation surface f. (b) Negative displacement (a) Positive displacement Figure 3.6: Free body diagrams of an interior panel corresponding to Case 2 76 77 have vertical connections on both of their faces while the exterior panels can have vertical connections only on one face. The exterior panels are needed at the ends as well as at the openings in the wall. The eqns. (3.14), (3.19), (3.21) and (3.23) give the stiffness of interior panels only. The exterior panel stiffness is derived from the interior panel stiffness ( Ref. [36] Appendix A ). Stiffness of 'n' exterior panels is equal to the stiffness of | modified interior panels . In the modified interior panel weight, dowel spring stiffness & yield strength and the foundation spring stiffness are obtained by multiplying the respective exterior panel properties by a factor of 2. Thus the stiffness of a wall with n\ interior and n2 exterior panels is given by : Kwall = Ml Kint + 1*2 Kext where : Kint & Kext are obtained using eqns. (3.14) (3.19) (3.21) or (3.23) using proper values of the variables Note that if the panels do not have any vertical connections, then the exterior panel stiffness will be same as the interior panel stiffness ( as is the case in the present studies ). 3.8 Time-Step Solution Technique The earthquake excitation record is subdivided into small time-steps and the re sponse of the structure is found at each time-step. This allows one to follow the exact force-deflection curve of the structure. As we know that stiffness of each wall panel varies from one stage to another as the wall displacement changes, so the force-deflection curve keeps on changing. To follow the exact force-deflection curve; we would have to use a very small time-step, and for each time-step we would have to use an iteration procedure ( Newton - Raphlson or some other similar tech nique ) so that the solution proceeds from event ( i.e. stiffness change ) to event 78 ( see Fig. [3.8b] ). There should be no imbalance of load at the end of any time-step and the solution should follow the exact load-deflection curve. This procedure will be very expensive. A second technique is to find the unbalanced force at the end of each time-step and add this unbalanced force R to the incremental force AR for the next time-step ( see Fig. [3.8a] ). After a few load reversals we would be far off the true load displacement curve and the energy input would not equal energy output ( i.e. we loose energy balance ). The approach used here is a combination of the above two approaches. Events (i.e. stiffness changes and hence load sub-steps ) are recognised for certain elements only, namely those in which large stiffness changes, and hence large equilibrium errors, can occur. These specified events occurs when the rocking of panels change from one equilibrium position to another ( see Fig. [3.1b] ]. All other stiffness changes are ignored and any resulting errors are taken into account by applying an equilibrium correction, i.e. for each time step we follow the second approach. Then if we find that within some time step, say from time ' t ' to ' t + At ', the stiffness of a panel has changed from one position of equilibrium to another, we come back to find, with a given tolerance, the time t'; time at which the change occured. We then find the response of the structure at time t' and use the new stiffness until the panel position changes again. If within a time-step a number of panels change their positions, then first we find the least time t'x at which one panel is going to change its position; find the response of the structure at this time t[ then use the new stiffness for that panel and find the next time t'2 at which the next panel will change its position ( see Fig. [3.8c] ), and so on. In other words we try to find the times where the panel stiffnesses change, as exactly as possible. 79 (a) Equilibrium Correction (b) Event to Event 0 oo Figure 3.8: Solution Strategies 3.9 Damping : Damping is an inherent property of the system, and is a mechanism by which the mechanical energy of the system, kinetic and potential energy, is transferred to other forms of energy such as heat etc. The mechanics of this transformation or dissipation are quite complex. This mechanism can be accounted for by viscous damping approach described below : Viscous Damping : In this approach the damping force is assumed to be propor tional to the velocity of the system . The coefficient of damping is given by : C = 2^KeM (3.25) where Ke = Elatic stiffness of the system M = Mass of the system £ = Damping ratio of the system as % of critical damping Experimental results have shown that damping forces in the buildings are nearly independent of the test frequency, while the viscous damping procedure has a fre quency dependent damping. However there is very little literature available on damping in panelized structures. During an earthquake analysis, most of the en ergy is dissipated by F-A hysteresis loops and the energy dissiapated by damping is only a small fraction of the total energy dissipated, so the assumption of viscous damping is often made regardless of the dissipative characteristics of the system be cause it leads to relatively simpler mathematical analysis. For the present studies, damping ratio £ is taken as 1 %. 81 3.10 Energy Balance : When the ground underneath a structure moves, energy is either input into the structure or fed back into the ground. The net energy input is given by : Einput = Base Shear Force * Ground Displacement = Mass * Abs. Acceleration * Ground Displacement For time step analysis this equation can be written as ^2Einp = jT{M*xt)Az (3.26) where M = Mass of the building xt = absolute acceleration of the building Ax = incremental displacement of the building at time t As the structure moves relative to the ground the stored energy is dissipated or transformed into other forms of energy. Various mechanisms through which the energy is dissipated or stored are : (i) Energy stored as strain energy : The total cumulative strain energy stored and/or dissipated at any instant is given by area under the F-A curve, calculated with due regard for the signs of F and A. When the structure comes to rest, the net cumulative strain energy is the energy dissipated by hysteresis during the earthquake excitation. For example in Fig. [3.9], if the slope ol is the elastic stiffness, the energy stored at point 1 is the area olao and at point 2 is ol2bo which is made up of the area ol24o which represents dissipative or heat energy, and area 2b42 which is strain energy. From point 2 to 3 as the structure unloads elastically the strain energy reduces and at point 3 the strain energy is given by the area 3c43. Now if the 82 LOAD A 2 ^ DISP. Figure 3.9: Strain Energy from a F-A curve 83 structure comes to rest at point 6, there is no strain energy and the area ol23456 of the F-A diagram gives the total energy dissipated by hysteresis. (ii) Energy dissipated by damping : This is the velocity proportional energy dis sipated due to the damping of the system. The governing equation is : = C*x*x (3.27) where C = Coefficient of damping x = Displacent of the structure relative to ground x = Velocity of the structure relative to ground (iii) Instantaneous kinetic energy : The governing eqn. for energy stored ( or dissipated ) as kinetic energy is : EK.E. = (3.28) where xabB is the abosolute velocity of the structure Thus the energy balance equation is given by : Einput = Net stored strain energy + ^2 dissipated hysteresis energy + ^2 dissipated viscous energy + kinetic energy At the end of each time step energy input should be equal to energy output. During the time-step analysis, energy balance is checked at the end of each time-step. Because of the approximations in the iteration scheme, ( see section [3.8] ) there will be some error in the energy balance at the end of a time-step. The maximum energy balance error allowed was 5 % at any time during the dynamic analysis. 84 Chapter 4 Static Design of the Buildings 4.1 General The loading on a structure resulting from seismic ground motion effects is quite complex. Design codes attempt to reduce this complex situation into a simpler set of equivalent static lateral forces. In a one storey building a single force is assumed to act at the roof level. The building is designed to resist this lateral force by developing forces in walls and other building components. In this chapter, two methods which are used to calculate the forces acting on the walls are discussed. The first method of design is based on the stiffness of each wall. This method is referred to as the Stiffness Method . This method has been used in most of the design handbooks [25] to calculate the forces in various walls. An alternate method based on the strength of the walls is also presented. This method is referred to as the Strength Method. Two buildings, a square building of plan area 38.4m * 38.4m, Fig. [4.1a], and a rectangular building of plan area 38.4m*76.8m, Fig. [4.1b], were designed according to the NBCC [22] recommendations, using both the strength and stiffness method techniques to calculate the forces in various walls. The roof diaphragm of each building is considered to be rigid and the mass is uniformly distributed ( i.e. the centre of mass ( CM. ) is at the centre of the building ). Other structural data 85 o m t-Parapet I Roof o Floor —S I f Footing o •38.4 m -WALL 3 WALL 4 (a) Square Building 76.8 m — WALL 3 WALL 4 (b) Rectangular Building 2400 155 V 100 Elevation 400 ' 1200 -JL120 (c) Wall Panel Figure 4.1: One Storey Buildings Analysed in the Present Studies £6 common to both the buildings is given below : Height of the building = 5.4m Height of the parapet wall = 0.75m Width of the panel = 2.4m Thickness of the panel = 100mm Weight of each panel per unit area = 3.29 kN/m2 Separation between panels -• 0.01m For both the buildings, there are no openings in walls 2, 3 and 4. Openings in wall 1 are created by removing full panels i.e. the number of panels in wall 1 is varied from 0 to 16 to study the effects of different eccentricities. It is assumed that the number of panels in wall 1 is fixed by architectural requirements and not by strength requirements. A brief description of the NBCC [22] recommendations for calculating the lat eral force & torsional design moment acting on a building under the influence of an earthquake, section [4.1.9] in Ref. [22], is given below. a) Mimimum specified lateral seismic force V = v.S.K.I.F.W where v — zonal velocity ratio S = seismic response factor for the structure K = numerical coefficient that reflects the ductility and other properties of the structure I = importance factor F — foundation factor (4.1) 87 W = dead load, plus 25% of design snow load, plus 60% of the storage load b) Torsional moment in the horizontal plane of the building, about the centre of rigidity is Mtx = (Ft + ^F^e, (4.2) i=x where Mtx = torsional moment at level x. Ft — portion of V concentrated at the top of the structure Fx = lateral force applied at level x ex = design eccentricity at level x Design eccentricity, ex, is computed by one of the following equations, using whichever produces the greater stresses for a given wall: ex = 1.5e + 0.lA, (4.3) ex = 0.5e-0.lD„ (4.4) where e = distance between the location of the resultant of all equivalent static forces applied at and above the level being considered and the centre of rigidity at the level being considered. Dn = plan dimension of the building in the direction of computed eccentricity. The buildings being considered are assumed to be in zone 5 of the NBCC [35]. The buildings have been designed for earthquake acting in the X direction and have been checked for the earthquake in the Y direction. The buildings have been designed according to NBCC [22] requirements for equivalent static loads using both the 88 stiffness and strength methods. In the following paragraphs a brief description of the two methods of design is given. 4.2 Stiffness Method The centre of rigidity is assumed to be on the line of action of the resultants of the resisting forces acting when the building is given a lateral displacement with no rotation. In the stiffness method, the resisting force contributed by each element is assumed to be proportional to its stiffness. If it is assumed that the stiffness of each wall is proportional to its number of panels i.e. all the walls have similar panels ( or panel connections ), then the location of the centre of rigidity of the building can be calculated as follows : ecr = L/2 - Ni/(Ni + N2) * L where ecr = distance from the geometric centre of the roof diaphragm to the centre of rigidity. iV,- = number of panels in wall i and L = building dimension perpendicular to the direction of the E/Q The equivalent static force applied to the building is assumed to act through the centre of mass. If the centre of mass and the geometric centre of the building are coincident ( i.e. mass is uniformly distributed in the roof diaphragm ), then e = ecr = L/2 - 7Vi/(/Vi + N2)*L (4.5) An example calculation of the resisting forces in the walls for the square building, Fig. [4.1a], with 16 panels in walls 2, 3 & 4 and 2 panels in wall 1, when a force is applied through the CM., is given below : 89 From eqn.(4.5), eccentricity e is e = 38.4/2-2/(2 + 16) * 38.4 m = 14.93 m As noted above, the code requires the consideration of two design eccentricities, tx ( see eqn. (4.3) and (4.4) ). The building must be designed to resist the lateral force V and the torsional moment V * ex. For the design eccentricity, both the code equations; eqn.(4.3) & eqn.(4.4) are used. The resisting force system must provide a resultant lateral force ( = V ) acting through the CR. and a resisting moment M = V*ex. The force V is therefore assumed to be applied to the building as shown in Fig [4.2]. For each wall, the maximum resultant force found using one of the two design eccentricities is taken as the design force. Design eccentricity e<j from eqn.(4.3) : ed = 1.5*e + 0.1*L m = 1.5 * 14.93 + 0.1 * 38.4 m = 25.94 m The lateral force V is calculated using eqn. [4.1]. The seismic response factor of the structure ' S ' used in the eqn. [4.1] corresponds to fundamental period of the buildings determined by the formula 0.09/in/-/Dl ( section [4.1.9.1 (6)] NBCC [22]) where hn = height of the building in metres = 5.4 m Ds = the dimension of the building in the direction of the = applied forces (in metres ) For the building being discussed here, the fundamental period is 0.08 sec and the corresponding seismic response factor S from NBCC [22] ( Table [4.1.9.A] ) is 0.44 90 (a) cs = 1.5e + 0.lDn D„ -H (b) es = 0.5e - 0.lDn Figure 4.2: Assumed Line of Action of Equivalent Static Force 91 ( the actual fundamental periods of the statically designed buildings are shown in the Appendix [A] ). The torsional moment M, for V = 1071 kN is M = V * ed kN-m = 1071 * 25.94 kN-m = 28124 kN-m Design eccentricity t& from eqn.(4.4) : ed = 0.5 * e — 0.1 * L m = 0.5 * 14.93 - 0.1 * 38.4 m = 3.62 m and corresponding torsional moment M is M - V * ed kN-m = 1071 * 3.62 kN-m = 3877 kN-m Now if Fvi = Force in wall i due to lateral force V Fti = Force in wall i due to torsional moment M Walls 1 and 2 take the lateral force V in proportion to their stiffness . Thus forces in various walls due to the lateral force V are : FVl = V*N1/{N1 + N2) kN = 1071 * 2/(2 + 16) kN 92 = 119 kN Fv, = V *N2/[N1 + N2) kN = 1071 * 16/(2 + 16) kN = 952 kN FV3 = 0 kN F„4 = 0 kN Force on wall i due to torsional moment M is given by : t=i where : r,- = Stiffness of wall i d{ = Distance of wall i from CR. of the building i.e. all the walls contribute in resisting the torsional moment. The forces acting on various walls for the earthquake acting in the X-direction are given in Table [4.1]. The calculations are repeated for the earthquake acting in the other direction. The design eccentricity and torsional moment for the earthquake acting in the Y direc tion are given below and the forces in various walls are given in Table [4.2]. V = Specified lateral force on the building = 1071 kN e = Actual eccentricity of the building = 0.0 m ( from eqn(4.5) ) ed = Design eccentricity of the building = 3.84 m ( from eqn.(4.3) ) = -3.84 m ( from eqn.(4.4) ) 93 Wall No. Fv Ft (kN) Force c >n wall Max. Force Panel No. of kN from from from from on wall Force panels eqn(4.3) eqn(4.4) eqn(4.3) eqn(4.4) (kN) (kN) (1) (2) (3) (4) (5) (6) = (3)+(4) (7) = (S)+(5) (8)=max. of (6) & (7) (9) = (8)/(2) 1 2 119 133 19 252 138 252 126 2 16 952 -133 -19 819 933 933 58.5 3 16 0 -598 -83 -598 -83 598 37.3 4 16 0 598 83 598 83 598 37.3 Table 4.1: Forces in various walls of a square building, Fig. [4.1a], for E/Q in X-direction, using stiffness method of design M = Torsional Moment on the building = 4113 kN-m ( using eqn.(4.3) ) = 4113 kN-m ( using eqn.(4.4) ) Panel design is governed by the maximum of the panel forces ( Col (9) ) in Wall No. Fv Ft (kN) Force on wall Max. Force Panel No. of kN from from from from on wall Force panels eqn(4.3) eqn(4.4) eqn(4.3) eqn (4.4) (kN) (kN) (1) (2) (3) (4) (5) (6) = (3)+(4) (7) = (3)+(5) (8)=max. of (6) & (7) (9) = (8)/(2) 1 2 0 20 -20 20 -20 20 10 2 16 0 -20 20 -20 20 20 1.25 3 16 536 -88 88 448 624 624 39. 4 16 536 +88 -88 624 448 624 39. Table 4.2: Forces in various walls of a square building, Fig. [4.1a], for E/Q in Y-direction, using stiffness method of design Tables [4.1] & [4.2]. It was assumed that the stiffness of each wall was proportional to its number of panels but the maximum force for each wall ( Col 4 in Table [4.3] ) is not proportional to the number of panels in that wall. The force on each panel in wall 1 ( the weaker wall ) is much higher than that for each panel in the other 94 Wall No. Maximum Wall Maximum Panel No. of Force Force panels kN kN (1) (2) (3) = Max. of Col (8) in Tables 4.1 & 4.2 (4) = Max. of Col (9) in Tables 4.1 &; 4.2 1 2 252 126 2 16 933 58.5 3 16 624 39 4 16 624 39 Table 4.3: Maximum Panel Forces governing the design of various walls in a Square Building, Fig. [4.1a], using stiffness method of design Wall No. e ( m ) e<j & M for the E/Q ed & M for the E/Q No. of V for the E/Q in in X dir. in Y dir. panels (kN) X dir. Y dir. eqn(4.3) eqn(4.4) eqn(4.3) eqn(4.4) 1 2 252 ed = ed = ed = ed = 2 16 933 20.39 1.68 3.84 -3.84 1071 11.03 0.0 3 16 624 M = M = M = M = 4 16 624 21837 1796 4113 -4113 Table 4.4: Code recommended Design Eccentricity & Torsional Moment for Square Building with Stiffness of each wall proportional to Wall Force in Table [4.3] 95 walls. In general we would expect panel stiffness to be affected by panel strength. If panel stiffness is assumed linearly proportional to the panel strength, as given in Table [4.3] and the calculations are repeated then the weaker wall ( wall 1 ) seems to attract more force per panel than the other walls because of the higher stiffness panels required for this wall. See Tables [4.4]...[4.7] for the forces in the walls. ( Note that load factors must still be applied to the panel forces when the panels are design ). Wall No. Fv Ft (kN) Force c >n wall Max. Force Panel No. of kN from from from from on wall Force panels eqn(4.3) eqn(4.4) eqn(4.3) eqn(4.4) (kN) (kN) (1) (2) (3) (4) (5) (6) = (3)+(4) (7) = (3)+(5) (8) =max. of (6) & (7) (9) = (8)/(2) 1 2 228 221 18 449 246 449 224.5 2 16 843 -221 -18 622 825 825 51.6 3 16 0 -348 -29 -348 -29 348 21.8 4 16 0 348 29 348 29 348 21.8 Table 4.5: Forces in various walls of a square building, Fig. [4.1a], for E/Q in X-direction using stiffness method of design with Stiffness of each wall proportional to Wall Force in Table [4.3] 4.2.1 Shortcomings of the Stiffness Method For the example shown above, if it is assumed that all the panels have the same stiffness regardless of their strength then the walls can be designed by providing the panel strengths found from the panel forces in Table [4.3]. But for a highly eccentric building, the required panel strength for the weaker wall ( wall 1 ) might become so high as to make the panel design impractical. This is shown in the following example of a highly eccentric rectangular building. The rectangular building, Fig. [4.1b] has 32 panels in walls 3 & 4,16 panels in wall 2 96 Wall No. Fv Ft 'kN) Force c >n wall Max. Force Panel No. of kN from from from from on wall Force panels eqn (4.3) eqn(4.4) eqn(4.3) eqn(4.4) (kN) (kN) (1) (2) (3) (4) (5) (6) = (7) = (8)=max. of (9) = (3)+(4) (3)+(5) (6) & (7) (8)/(2) 1 2 0 42 -42 42 -42 42 21 2 16 0 -42 42 -42 42 42 2.6 3 16 536 -66 66 470 602 602 37.6 4 16 536 +66 -66 602 470 602 37.6 Table 4.6: Forces in various walls of a square building, Fig. [4.1a], for E/Q in Y-direction, using stiffness method of design with Stiffness of each wall proportional to Wall Force in Table [4.3] Wall No. Maximum Wall Maximum Panel No. of Force Force panels kN kN (1) (2) (3) = Max. of Col (8) in Tables 4.5 & 4.6 (4) = Max. of Col (9) in Tables 4.5 & 4.6 1 2 449 224.5 2 16 825 51.6 3 16 602 37.6 4 16 602 37.6 Table 4.7: Maximum Panel Forces governing the design of various walls in a Square Building, Fig. [4.1a], using stiffness method of design with Stiffness of each wall proportional to Wall Force in Table [4.3] 97 and 2 panels in wall 1. V — Specified lateral force on the building = 2058 kN e = Actual eccentricity of the building = 29.87 m ( from eqn(4.5) ) ed — Design eccentricity of the building = 52.5 m ( from eqn.(4.3) ) = 7.25 m ( from eqn.(4.4) ) M — Torsional Moment on the building = 108025 kN-m ( using eqn.(4.3) ) = 14930 kN-m ( using eqn.(4.4) ) Forces in various walls for the rectangular building are given in Table [4.8]. Wall No. Fv Ft (kN) Force < >n wall Max. Force Panel No. of kN from from from from on wall Force panels eqn(4.3) eqn(4.4) eqn(4.3) eqn(4.4) (kN) (kN) (1) (2) (3) (4) (5) (6) = (3)+(4) (7) = (3)+(5) (8)=max. of (6) & (7) (9) = (8)/(2) 1 2 229 432 60 661 289 661 330.5 2 16 1829 -432 -60 1397 1769 1769 110.6 3 32 0 -1948 -269 -1948 -269 1948 60.8 4 32 0 1948 269 1948 269 1948 60.8 Table 4.8: Forces in various walls of a rectangular building, Fig. [4.1b], using stiffness method of design The panel force ( Table [4.8], Col (9) ) for wall 1 is 330.5 kN. It would be difficult to design panels to resist this force. The panel force for walls 3 and 4 is much less than that for the other walls and if it is assumed that the panels in walls 3 and 98 4 have lower stiffnesses than those in walls 1 and 2, the situation becomes worse because in that case the stronger walls attract more force than given in Table [4.8], thus ending up with much higher strength requirement for the panels in wall 1. This would make the design even more difficult. 4.3 Strength Method This method is based on the idea that when a building is resisting a severe earth quake the design should be based on the assumption that all the walls would yield. The location of the centre of rigidity of the building ( CR.) is therefore calculated from the yield strength of the walls. er = L/2-F1/{F1 + F2)*L where Fi = yield strength of wall i and er = distance of the CR. from the geometric centre of the building If walls 1 and 2 have similar panels, then the strength of these walls will be propor tional to their number of panels. If it is assumed that mass is uniformly distributed, then the eccentricity e is e = L/2- Nx/[Ni + N2) * L i.e. same as in the stiffness method, eqn. (4.5). The square building, Fig. [4.1a] in section [4.2] has been redesigned using the strength method for calculating the forces in the walls. The calculations for the forces in various walls are given below : From eqn.(4.5), eccentricity e is e = 38.4/2 - 2/(2 + 16) * 38.4 m = 14.93 m 99 In this method, the centre of rigidity ( CR. ) of the building is calculated from the strength of walls 1 and 2 thus the resultant force resisting motion in the direction of the E/Q, acts through the CR. Therefore moment of forces in walls 1 and 2 about the CR. is zero. The design moment must therefore be resisted entirely by walls 3 &; 4. As before, the NBCC [22] equations (4.3) and (4.4) are used to calculate the design eccentricity. But as walls 1 and 2 do not contribute anything to the resistance of this torsional moment; eqn.(4.3) is the only equation that needs to be used in the calculation of the applied torsional moment because this moment is always greater than that from eqn.(4.4). Design eccentricity e<j from eqn.(4.3) Design moment M is M = V * ed kN-m = 1071 * 25.94 kN-m = 28124 kN-m Walls 1 &; 2 resist lateral force V while Walls 3 & 4 resist moment M. Thus forces in various walls due to the lateral force V are : 1.5*e + 0.1*L m 1.5 * 14.93 + 0.1 * 38.4 m 25.94 m V * Ni/iNt + N2) 1071 * 2/(2 + 16) kN 119 kN V * Nt/iNi + N2) 1071 * 16/(2 + 16) kN 100 = 952 kN FV3 = 0 kN FVi = 0 kN Force on wall i due to torsional moment M is given by : Ftl = 0 kN Fta = 0 kN Fta = M/L = -28124/38.4 kN = -732.8 kN Fti = -Fti = +732.8 kN Forces in various walls, for E/Q acting in X-direction are given in Table [4.9]. All Wall No. Fv Ft Max. Force Panel No. of Panels (kN) (kN) on wall ( kN ) Force ( kN ) (1) (2) (3) (4) (5) = (3) + (4) (6) = (5)/(2) 1 2 119 0 119 59.5 2 16 952 0 952 59.5 3 16 0 -733 -733 45.8 4 16 0 733 733 45.8 Table 4.9: Forces in various walls of a Square Building, Fig. [4.1a], for E/Q acting in X-direction, using Strength Method of design walls have to be checked for the E/Q acting in the other direction i.e in the direction of wall 3 or 4 (see Table [4.10] ). The largest force in each panel will govern the design ( Table [4.11] ). 101 Wall No. Fv Ft Max. Force Panel No. of Panels (kN) (kN) on wall ( kN ) Force ( kN ) (1) (2) (3) (4) (5) = (3)+(4) (6) = (5)/(2) 1 2 0 107 107 6.69 2 16 0 -107 -107 6.69 3 16 536 0 536 33.5 4 16 536 0 536 33.5 Table 4.10: Forces in various walls of a Square Building, Fig. [4.1a], for E/Q acting in Y-direction, using Strength Method of design Wall No. Maximum Wall Maximum Panel No. of Force Force panels kN kN (1) (2) (3) = Max. of Col (8) (4) = Max. of Col (9) in Tables 4.8 & 4.9 in Tables 4.8 & 4.9 1 2 119 59.5 2 16 952 59.5 3 16 733 45.8 4 16 733 45.8 Table 4.11: Maximum Panel Forces governing the design of various walls in a Square Building, Fig. [4.1a], using Strength Method of design. 102 4.3.1 Shortcomings of the Strength Method As explained in the above paragraph, with this method it was assumed that the walls in the direction of the earthquake do not take part in resisting the torsional moment. To resist the torsional moment, walls which are not parallel to the direction of the earthquake are needed. For the example given in section [4.2], the strength method cannot be used if there are no panels in walls 3 and 4; because then there are no walls to take the torsional moment. Another shortcoming of this method may be the increased ductility demand in the weaker wall because the strength of the weaker wall designed by this method is less than that designed by the stiffness method. This may lead to an increase in the eccentricity if the walls tend to decrease in strength over several cycles above the yield level. 103 Chapter 5 Results of Dynamic Analysis 5.1 Effect of non-symmetry in the buildings If the center of mass and center of resistance of a building do not coincide, a rota tional couple will act on the building during an earthquake. It is not easy to develop a design method using equivalent static lateral forces that properly considers their effects. To study the adequacy or inadequacy of the code design, a number of dy namic analyses were made for different variables such as eathquake data, building geometry etc. ( see Table [5.1] ). In each of the these cases, the panels in walls 2, 3 & 4 were not changed while the panels in wall 1 were varied from 0 to 16 i.e. moving from a highly unsymmetrical building with no panels in wall 1 to a symmetrical building with wall 1 &; wall 2 having the same number of panels. All the panels were designed as cantilevers with dowel bars at the base acting as flexural steel. To reduce the time required for dynamic analysis, the panels were the same in every wall. All dowels in a given wall were the same, but different walls could have different dowels. In the static design, the dowels used were not limited to standard dowel sizes available in the industry, but it was assumed that any dowel area which would give adequate resisting force could be used, i.e. no wall had any surplus strength beyond that required in the design. For example say the required factored panel strength is 128.5 kN which can be provided by using two dowel connections each of area 888 mm2. If we used 104 Building Design Method E/Q E/Q Direction Static Design Eccentricity Stiffness Stiffness Strength El Centro X r3 & r4 Square Strength El Centro X e r3 & r4 Strength El Centro San Fernando Taft X ed r8 & r4 Strength El Centro X Y ed r3 & r4 Rectangular Strength El Centro X 0.95ed 0.85ed 0.75ed 0.65ed r3 & r4 Strength El Centro X ed 0.5 r3 & 0.5 r4 0.2 r3 &; 0.2 r4 Note :- r3 & r4 are the stiffnesses of walls 3 & 4 respectively, derived from the panel connections required in the static design with assumption that the stiffness of each wall is proportional to the number of panels in the wall. Note :- Dynamic analysis has always been done using the actual eccentricity of the building. Table 5.1: List of the dynamic analyses made in this studies 105 dowels that are available in market then we would have to use a dowel of area of 1000 ram2 which would increase the panel strength as well as the wall strength beyond that required by design. As a consequence we would not be able to check the adequacy of the static design. 5.2 Structural Data The two buildings used in these studies are shown in Fig. [4.1]. Building and panel dimensions are given in section [4.1] and the properties of the connections used in the studies are discussed below. See Appendix A for the time periods of the square and rectangular buildings designed by the strength method using NBCC design ec centricity. Panel to Panel Connections :-In the design of the buildings, it was assumed that there are no vertical connec tions between the wall panels because in the initial runs it was found that the dowel connections are more effective in taking the lateral shear force than the vertical connections. However both the computer program for the dynamic analysis and the static design program in LOTUS - 123 has the capability of including panel to panel connection data. The static design program has been written to consider connections which use embedded rebars, as shown in Fig. [5.1a]. Panel to Foundation Connections :-These connections consist of two dowels, one embedded in each stem as shown in Fig. [5.1b]. As explained in section [5.1], dowels used in the design are not limited to those available in the market, but may have any area. Initial stiffness of the dowels was based on the study done by Hawkins & Lin [15] who found that the dis-lacement at the loaded end of the embedded bars under axial loads was 1.066 mm 106 Section •Man V = y/2*Ab*fv *<f>, (a) Panel to Panel Connection Wall Panel 0.6m O Q 7W 2.4m 0.6m I O Q 1 1 CO d (b) Dowel Connection Figure 5.1: Panel Connections 107 at the onset of yielding for all the cases. Thus for the present studies it is assumed that all dowel connections have 1.066 mm displacement at yield point regardless of dowel size i.e. for any dowel of area As : Yield force Fy = 1.25 fy * A, N Stiffness kdow = F„/(1.066 * IO-3) N/m where l-25/„ is the probable strength of the steel. After yielding the load deflection curve is parallel to the deflection axis. Foundation Springs :-The foundation underneath the panel is modelled by two linear elastic springs. The stiffness of the foundation is found from the equation k = (A* Ec )/L where A is the bearing area of the flange of the panel on the foundation, Ec is the modulus of elasticity of concrete and L is the depth of the foudation equal to 1000 mm. The approximate value of k used in the studies is 6.0 * 109 N/m. Damping :-In all the computer runs, viscous damping of approximately 1% of critical damp ing has been used. Earthquake Records :-Three earthquakes have been used for these studies; El Centro - S00E 1940, San Fernando - S62E recorded at 2011 Zonal Ave. and Taft - N21E. Time vs. ground acceleration plots for first 20 seconds of these earthquakes are shown in Fig. [5.2a...c]. The El Centro E/Q has a peak acceleration of 3.417m/sec2 ( 0.35g ) which is quite close to 0.32g, the acceleration requirement for zone 5 from Ref. [35]. Ground accelerations of the other two earthquakes were linearly scaled so as to have a peak acceleration of 3.417ra/sec2. 108 GROUND ACCELERATION for ELCENTRO - 1940 E/Q Maximum acceleration = 3.417 m/sec/secj.;. - i i i i i i III 0 2 4 6 8 10 12 14 16 18 20 Time - sec (a) Accelerogram for the El Centro E/Q Figure 5.2: Earthquake Accelerograms used in the present studies 109 GROUND ACCELERATION for SANFERNANDO - S62E E/Q o -800 I i i I I I I I I i 1 0 2 4 6 8 10 12 14 16 18 20 Time - sec (b) Accelerogram for the San Fernando E/Q GROUND ACCELERATION for TAFT E/Q 8 10 12 Time - sec (c) Accelerogram for the Taft E/Q Fig. 5.2: Continued 110 Both El Centro & Taft have ratios of acceleration to velocity ( a/v - units m/sec ) approximately equal to 1, while San Fernando has a/v equal to 0.48. 5.3 Discussion of Results The results have been divided into the following subsections to discuss the effects of various parameters involved. ( Note that dynamic analysis has always been done using the actual eccentricity of the building ). • Effect of different methods of static design. • Effect of different E/Q data in the dynamic analysis. • Effect of the geomtry of the building. • E/Q direction • Effect of change in the design eccentricity of the buildings in the static design (to calculate the torsional moment) with dynamic analysis done using actual eccentricity of the buildings. • Effect of the variation of stiffnesses of walls 3 &i 4. 5.3.1 Effect of different methods of static design As explained earlier in section [4.1], two methods, a strength method and a stiffness method, have been used in the static design of the buildings shown in Fig. [4.1]. To enable one to measure the effect of the two methods of design, Strength ratios ( SR ) and Moment ratios ( MR ) are defined which relate the actual design strength of the walls to certain forces required from the static design. These ratios are defined to be, with reference to Fig. [5.3] : th R t" SR Strength of the walls in the X-direction Code recommended lateral force (factored) 111 .J < WALL 3 - - —it CM. OR. w .J WALL 4 Figure 5.3: Resisting forces on the building 112 Moment Ratio MR = Moment Strength of walls for the E/Q in X-dir. Larger moment used for the design in the X-dir. (factored) Fw\ * ii — FW2 * l2 + Fw3 * h + Fw4 * ^4 Mfx The wall strengths are calculated from considering the earthquake requirements in both the X and Y directions, and thus the strengths may be larger than required if only the X direction was considered. Considering the SR the wall strengths Fwi and FW2 are not affected by the earthquake in the Y direction, and thus for the strength method of design SR is equal to one for all cases. For the stiffness method of design, part of the wall strength is used to resist torsion, and thus in most cases the lateral strength exceeds the lateral force and so SR is greater than 1. The MR considers the torsional capacity of the walls at the same time as the X direction Vf is resisted, that is, it is not the pure torsional capacity of the walls. This is why the forces FwX and Fw2 are shown in the same direction in Fig. [5.3]. For most cases considered the Fws and Fw4 capacities are governed by the earthquake capacities in the Y direction and for these cases the MR will be greater than unity. It will be shown that displacement of the CM. tends to fall as the SR increases, and rotation 6 tends to fall as the MR increases. It will also be shown that buildings designed by the stiffness method generally have higher SR and MR than the same building designed by the strength method The strength ratio SR and moment ratio MR for the square building ( Fig. [4.1a]) with varying degrees of eccentricity, are shown in Figs.[5.4a..b]. To investigate the 113 effect of different methods on the response of the buildings, the buildings were sub jected to the El Centro earthquake. Comparison of various variables e.g. building displacement x at CM., rotation 0 etc. obtained from the dynamic analysis have been shown in Fig. [5.4 c.l]. The static design results show that the stiffness method is more conservative than the strength method for all the eccentricities except when the number of panels in wall 1 is 0 ( i.e. Ni ), for which two methods give the same results. Dynamic analysis results confirm that the stiffness method is conservative. Fig [5.4c] shows the variation of building displacement at CM. as Ni (the number of panels in wall 1 ) is varied from 0 to 16. Both design methods give similar results for Ni = 0. Then for the stiffness method, displacement decreases as Ni is increased. The average value of this displacement at CM. for the strength method is slightly higher than for the stiffness method as Ni is varied from 0 to 16. The reason for this is the high strength of walls in the direction of the earthquake with the stiff ness method approach ( see Fig [5.4a] ). The maximum difference in the building displacement at CM. for the two methods is 25% which is obtained at Ni = 4 when the wall designed by the stiffness method had extra strength of 15% in X direction as compared to that for strength method Fig [5.4a]. Variation of rotation 0 for the two methods is shown in Fig. [5.4d]. Curves for both the methods are similar except that the strength method gives slightly higher results than stiffness method which is predictable because the moment ratio, Fig. [5.4b], for the former is lower. The maximum value of 0 for all buildings is 0.28 * 10-3 rad. which is very small and can be easily accomodated. The decrease in rotation as Ni increases is related to the moment ratio. As long as the moment ratio is equal to 1 there is only a small change in 6 with increase in N\ showing that static design methods give reasonable results as the eccentricity varies. And as moment ratio starts increasing, the extra strength in walls 3 & 4 causes 0 to 114 1.30 in CO 1.25-1.20-1.15-1.10 cm c <t> 1.05 cn 1A 0.95 •• Square Building with Panels Nl, 16. 16, 16 Static Design using Code Design Ecc. Legend A Strength Method X Stiffness Method jC... ..: ; ;.. ^"-> / -1 1 I I I 2 4 6 8 10 Nl - Panels in Wall 1 (a) Strength Ratio 12 -T-14 16 6 5.5-5 4.5 4 3.5 2 CD B o S 3 2.5 2 1.5 1 Square Building with Panels Nl, 16, 16, 16 Static Design using Code Design Ecc. Legend A Strength Method X Stiffness Method -A" 2 •X' -A" 4 6 8 10 Nl - Panels in Wall 1 (b) Moment Ratio Figure 5.4: Comparison of static & dynamic analysis results of a square building designed by the strength and stiffness methods 115 X cj (0 no C OQ 55-50-45-40-35-30-25-20-15-10-5-Square Building with Panels Nl, 16, 16, 16 Dynamic Analysis using El Centro E/Q X Legend A Strength Methud x Stiffness Method i 1 l I 4 6 8 10 Nl - Panels in Wall 1 (c) Building Displacement at CM. 12 14 16; 0.35-Square Building with Panels Nl, 16, 16, 16 Dynamic Analysis using El Centro E/Q Legend A Strength Method x Stiffness Method 8 10 Panels in Wall 1 -x— 14 16 (d) Rotation 0 Fig. 5.4: Continued 116 e 60-50-a i> 6 30 o 20' Square Building with Panels Nl, 16, 16, 16 Dynamic Analysis using El Centro E/Q Legend A Strength Method X Stiffness Method 1 1 6 8 10 Nl - Panels in Wall 1 12 14 16 (e) Wall 1 Displacement c £ O "E, w O 60 50-40-Square Building with Panels Nl, 16, 16, 16 Dynamic Analysis using El Centro E/Q Nl - Panels in Wall 1 (f) Wall 2 Displacement Fig. 5.4: Continued 117 20 16 12-1 8-Square Building with Panels Nl, 16, 16, 16 Dynamic Analysis using El Centro E/Q •X" T" 2 Legend A Strength Method x Stiffness Method —T" 12 6 8 10 Nl - Panels in Wall 1 14 18 (g) Max. Dowel Displacement I O 3 CL G 2? (U C Cd 0.6 0.5-0.4-0.3 0.2 0.1 0.0 V Square Building with Panels Nl, 16, 16, 16 Dynamic Analysis using El Centro E/Q •A" •X' 3^ .•x« Legend A Strength Method x Stiffness Method -T-8 "T— 10 —r— 12 14 Nl - Panels in Wall 1 (h) Energy Input Fig. 5.4: Continued 16 118 o (1) 1 o o < <rj S 550-525-500-A 475-450-425-Square Building with Panels Nl. 16, 16, 16 Dynamic Analysis using El Centro E/Q -r 2 Legend A Strength Method X Square Method -T* 4 6 8 10 Nl - Panels in Wall 1 12 —T— 14 16 (i) Max. Absolute Velocity o <L> 1 i ci o < < X (0 S 4600 4400 4200 4000 3800 3600 3400 3200 3000 2800 Square Building with Panels Nl, 16, 16, 16 Dynamic Analysis using El Centro E/Q / •x« t 6 8 10 12 Nl - Panels in Wall 1 (j) Max. Absolute Acceleration Fig. 5.4: Continued 14 """"^ A -A : A Legend A Strength Method X Stiffness Method 16 119 6.75-6 5.25-4.50 3.75 3 2.25 1.50 0.75 0 Square Building with Panels Nl, 16, 16, 16 Dynamic Analysis using El Centro E/Q Legend A Strength Method X Stiffness Method Nl - Panels in Wall 1 (k) Max. Absolute Rotational Velocity Nl - Panels in Wall 1 (1) Max. Absolute Rotational Acceleration Fig. 5.4: Continued 120 decrease sharply. Curves for displacements of wall 1, wall 2 are similar to building displacement confirming that the rotation 0 is small and has only a slight effect on the wall displacements. Maximum dowel displacement is 13mm which is well within the capability of the dowel connections. Other variables such as energy input Einp, maximum velocity x, maximum rotational velocity 0, and maximum rotational ac celeration 0 are similar for both the methods, but maximum acceleration 'x is quite different for the two methods. This can be explained by the fact that the stiffness method results in stronger walls and hence in higher forces and accelerations. 5.3.2 Effect of different E/Q data on the dynamic analysis In this section dynamic analysis results for a square building Fig [4.1a], designed by the strength method, using ground acceleration data for the three earthquakes, El Centro [5.2a], San Fernando [5.2b] & Taft [5.2c], are discussed and shown in Figs. [5.5a...j]. The building displacement at the CM. ( x ) is quite similar for all the three earthquakes. For each earthquake the value of x does not vary significantly as Ni increases from 0 to 16. Curves for rotation 0 for the three earthquakes are very close with the maximum value being 0.28 * IO-3 rad. which is very small. Curves for displacement of wall 1, wall 2 and maximum dowel displacement are similar to building displacement thus confirming that the smal rotations have only a small effect on the wall displacement. Curves for maximum acceleration x, maximum rotational velocity 0 and max imum rotational acceleration 0 are also similar for these earthquakes. Maximum velocity x is similar for the El Centro & the Taft earthquakes while for the San Fer nando earthquake it is almost double. This is because the ^ ratios for El Centro and Taft earthquakes are approximately equal to 1 while for the San Fernando earth-121 60 50-Square Building with Panels Nl, 16, 16, 16 Static Design by Strength Method  <K 20-10-Legend - E/Q El Centro X E/Q - San Fernando Q E/Q - Ta rt  -T 1 1—1 1 1 r-4 6 8 10 12 14 Nl - Panels in Wall 1 T" 2 16 (a) Building Displacement at CM. Square Building with Panels Nl, 16, 16, 16 Static Design by Strength Method  Legend A E/Q - El Centro  x E/Q ~ San Fernando • E/Q - Taft  ••MM • • • mmmmmm mmm •••••MM 6 8 10 Nl - Panels in Wall 1 —r-14 16 (b) Rotation 6 Figure 5.5: Comparison of dynamic analysis results of a square building using different earthquake accelerograms for dynamic analysis 122 e X 60 50 Square Building with Panels Nl, 16, 16, 16 Static Design by Strength Method ..a -X I •A Legend A E/Q - El Centro X E/Q - San Fernando D E/Q - Taft 6 8 10 Nl - Panels in Wall 1 12 14 16 (c) Wall 1 Displacement 6 X c: e o "a, w a 0 60 50-30-20 Square Building with Panels Nl, 16, 16.16 Static Design by Strength Method - ^^^^**ml\wm~ = —A. j- * i | t Legend A E/Q - El Centro X E/q - San Fernando • E/Q - Taft •MM • • • MM* • • • HHWi 1 "1 T" v : * • -1 p. i i j— 6 8 10 Nl - Panels in Wall 1 (d) Wall 2 Displacement Fig. 5.5: Continued 12 14 16 123 20 Square Building with Panels Nl, 16, 16, 16 Static Design by Strength Method Legend A E/Q - El Centro X E/Q - San Fernando • E/Q - Taft  12 —I— 14 "T" 2 6 8 10 Nl - Panels in Wall 1 16 (e) Max. Dowel Displacement 6 i O a a. c: c 0.8 0.7 H 0.6 0.5 Square Building with Panels Nl, 16. 16, 16 Static Design by Strength Method  ...-7 0*... 0.4-j 0.3 0.2 OH 0.0 -0.1 2 •X' !•••""' j ~ " " D—MM • • • MMMli • • • -T-4 T" 8 Legend A E/Q - El Centro x E/Q - San Fernando a E/Q - Taft MMM mmm MMMMM * • * MMM 10 12 14 16 Nl - Panels in Wall 1 (f) Energy Input Fig. 5.5: Continued 124 ej o o OT JO < X S 1100 1000 900 Square Building with Panels Nl, 16, 16, 16 Static Design by Strength Method 800 £ 700-600-500-A" • oo -200 Legend A E/Q - El Centro X E/Q - San Fernando O E/Q - Taft MHMI * • • mmmmmmm mm m MMMM 10 12 14 16 Nl - Panels in Wall 1 (g) Max. Absolute Velocity o <u ^. cr a; 1 6 o o <: OT < X CCS s 4000-3800 I 3600-3400-3200-3000-Square Building with Panels Nl, 16,16, 16 Static Design by Strength Method  Legend A E/Q - El Centro X E/Q - San Fernando • E/Q - Taft  MM • • • MMMM « • • MMMM i i i 1 r— 4 6 8 10 12 Nl - Panels in Wall 1 (h) Max. Absolute Acceleration Fig. 5.5: Continued 14 16 125 0 2 4 6 8 10 12 14 16 Nl - Panels in Wall 1 (i) Max. Absolute Rotational Velocity o 0) (fl u n o l O I <D ci o < C o o 350-300 Square Building with Panels Nl, 16. 16. 16 Static Design by Strength Method Nl - Panels in Wall 1 (j) Max. Absolute Rotational Acceleration Fig. 5.5: Continued 126 quake this ratio is 0.48. As the acceleration (a) for all the earthquakes was made same for the dynamic analyses, the velocity (v) for the San Fernando earthquake is almost doubled. 5.3.3 Effect of geometry of the building In this section dynamic analysis results for a square building Fig. [4.1a] and a rectangular building Fig. [4.1b], statically designed by the strength method, are compared. The moment ratio ( MR ) is shown in Fig. [5.6a]. The El Centro earthquake acting in the X direction has been used in the dynamic analysis. Results for the two buildings are shown in Fig. [5.6b...k]. Building displacement at the CM. ( x ) for the two buildings is shown in Fig. [5.6b]. The maximum difference for the two buildings is about 15%. If for each building an average value of displacement x is taken as Ni varies from 0 to 16, then two buildings will have very close average displacement values showing that geometry of the building has very little effect on the building displacement at the CM. Variation of rotation 6 is shown in Fig. [5.6c], which shows that the value of 8 for the square building decreases very sharply for Nx > 2 while 6 for the rectangular building does not decrease until Ni > 10, and after that it decreases sharply. This phenomenon can be explained from Fig. [5.6a], which shows that the moment ratio MR for the square building starts increasing from unity for Nx > 4 while for the rectangular building MR does not increase until Ni > 10. For MR values near 1 the rotation varies around 0.25 * 10-3 radians. It is interesting to note that when MR reaches 2.5, the rotation essentially vanishes. This again shows that the code equation for calculating the design moment is adequate for a wide range of eccentricities and for different geometries. Displacements of wall 1, wall 2 & maximum dowel displacements for the two buildings are quite close to each other and 127 (0 cc c e o 5.5 5-4.5-4-3.5 3-2.5-2 1.5-J Square Building with Panels Nl, 16, 16,16 Rectangular Building with Panels Nl, 16, 32, 32 Static Design by Strength Method with Design Ecc. = ed = 1.5e + 0.11 Legend A Square Building x Rectangular Building Nl - Panels in Wall 1 (a) Moment Ratio Figure 5.6: Comparison of static & dynamic analysis results for square and rectan gular buildings using NBCC recommended equivalent static design approach with strength method of design 128 50-40-20-10-Square Building with Panels Nl, 16, 16, 16 Rectangular Building with Panels Nl, 16. 32, 32 Static Design by Strength Method Dynamic Analysis using El Centro E/Q  Legend A Square Building X Rectangular Building —r— 10 14 40-35-| 30-6 12 Nl - Panels in Wall 1 (b) Building Displacement at C.M. 16 Square Building with Panels Nl. 16, 16. 16 Rectangular Building with Panels Nl, 16, 32, 32 Static Design by Strength Method Dynamic Analysis using El Centro E/Q Legend A Square Building X Rectangular Building T" 2 6 —1— 10 Nl - Panels in Wall 1 (c) Rotation 0 Fig. 5.6: Continued 129 £ £ x i a £ QJ o jd "a. w JO 60 50-40-A 30-20-10 Square Building with Panels Nl, 16, 16, 16 Rectangular Building with Panels Nl, 16, 32, 32 Static Design by Strength Method Dynamic Analysis using El Centro E/Q  6 8 10 12 14 Nl - Panels in Wall 1 Legend A Square Building X Rectangular Building 16 (d) Wall 1 Displacement £ £ CO X C £ o co H W Q CO 2 60-50-40-Square Building with Panels Nl, 16, 16, 16 Rectangular Building with Panels Nl, 16, 32. 32 Static Design by Strength Method Dynamic Analysis using El Centro E/Q 30-20-104 0--10-Legend A Square Building X Rectangular Building 6 8 -Tr io T-12 —T-14 Nl - Panels in Wall 1 16 (e) Wall 2 Displacement Fig. 5.6: Continued 130 25 B B o Q X Square Building with Panels Nl, 16, 16, 16 Rectangular Building with Panels Nl. 16, 32. 32 Static Design by Strength Method Dynamic Analysis using El Centro E/Q 10-5-Legend A Square Building X Rectangular Building , , , , , p. 6 8 10 12 Nl - Panels in Wall 1 (f) Max. Dowel Displacement 14 16 B i oo O d a. Ctf) <U C W 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 Square Building with Panels Nl, 16, 16. 16 Rectangular Building with Panels Nl. 16. 32, 32 Static Design by Strength Method Dynamic Analysis using El Centro E/Q  *x-Legend £. Square Building X Rectangular Building * • : i • i i i i i i 1 6 8 10 Nl - Panels in Wall 1 12 14 16 (g) Energy Input Fig. 5.6: Continued 131 600-i o cu Square Building with Panels Nl. 16, 16, 16 Rectangular Building with Panels Nl, 16, 32. 32 Static Design by Strength Method Dynamici Analysis using El Centro E/Q 6 8 10 Nl - Panels in Wall 1 (h) Max. Absolute Velocity o <u s O o < GO < X 4600 4400 Square Building with Panels Nl, 16, 16, 16 Rectangular Building with Panels Nl, 16, 32. 32 Static Design by Strength Method Dynamic Analysis using El Centro E/Q (i) Max. Absolute Acceleration Fig. 5.6: Continued 132 o (ti C o (0 o Oi 6.75-6-5.25-Square Building with Panels Nl. 16. 16. 16 Rectangular Building with Panels Nl. 16, 32, 32 Static Design by Strength Method Dynamic Analysis using El Centro E/Q 1.50- Legend A Square Building X Rectangular Building 6 8 10 Nl - Panels in Wall 1 12 -1— 14 -A 16 (j) Max. Absolute Rotational Velocity o v «j u o o I o I d o < C o 300 -r Z50- • O Square Building with Panels Nl, 16, 16, 16 Rectangular Building with Panels Nl, 16, 32, 32 Static Design by Strength Method Dynamic Analysis using El Centro E/Q 200-150-100 a eo 6 8 10 Nl - Panels in Wall 1 (k) Max. Absolute Rotational Acceleration Fig. 5.6: Continued 133 have a shape that is similar to the shape of the curves for the building displacement at the CM. The energy input for the rectangular building is higher because it has a higher mass than the square building. Maximum velocity x is almost constant for the two buildings while the curve for rotational velocity has a shape that is similar to that for rotation 0. For rotational acceleration the trend is similar but the curves are flatter. Static Design with Design Ecc. = e Fig. [5.7a...g] shows the static and dynamic analysis results for a square and a rectangular building designed using the strength method and taking the design eccentricity equal to the actual eccentricity of the building i.e. no extra eccentricity for calculating the design moment. Fig. [5.7a] shows that for the square building the moment ratio MR becomes greater than 1 for Ni > 2 i.e. for Ni > 1 the design of walls 3 & 4 is governed by the earthquake in the Y direction. ( Note that the larger moment used in calculating MR is calculated using actual eccentricity e and not the code design eccentricity e<j ). Fig. [5.7b] shows the variation of building displacement at CM. ( x ) for the two buildings. The displacements are quite similar as Ni is varied from 0 to 16. The reason for this is that the strength ratio SR of the walls in the direction of the earthquake (i.e. parallel to X direction ) is the same. Maximum rotation 9 for both the buildings is quite high at Ni = 0. For the square building 0 decreases quite rapidly to a value comparable to that obtained for the building designed for code eccentricity ( Fig. [5.6c] ); 9 for the rectangular building is approximately 0.7* IO-3 rad. when MR = 1 and then decreases rapidly as the value of MR increases. For both the buildings the maximum value of 9 is very high; which means that displacements of wall 1 and wall 2 are greatly influenced by rotation and these 134 17 15 OS 13-l o 11-(0 9-OS en 7-e o 5-3-1-4 0 Square Building with Panels Nl, 16, 16, 16 Rectangular Building with Panels Nl, 16, 32, 32 Static Design by Strength Method with Design Ecc. = e = actual ecc. of the building Legend A Square Building X Rectangular Buildirig 8 10 Nl - Panels in Wall 1 (a) Moment Ratio Figure 5.7: Comparison of static & dynamic analysis results for square and rectan gular buildings using NBCC recommended equivalent static design approach with design eccentricity equal to the actual eccentricity of the building and using the strength method of design 135 60 50-40-B B I CJ CO CU w Q 20-c 3 CQ Square Building with Panels Nl, 16, 16. 16 Rectangular Building with Panels Nl. 16. 32, 32 Dynamic Analysis using El Centro E/Q Static Design by Strength Method with Design Ecc. = e = actual ecc. of the building ^.rr..r-m.mxm^mmh L...7!^ •m^mmm- *XM ^X' ^ Legend A Square Building x Rectangular Building 6 8 10 Nl - Panels in Wall 1 12 14 16 X) cd u o o I O CD 0.8*, 0.6 o "fO 0.4 O (b) Building Displacement at CM. Square Building with Panels Nl. 16. 16. 16 Rectangular Building with Panels Nl, 16. 32. 32 Dynamic Analysis using El Centro E/Q Static Design by Strength Method with Design Ecc. = e = actual ecc. of the building Legend A Square Building X Rectangular Building _ .i^^^^^^A^, X i ! ! \ i ! - i i'i 6 8 10 12 14 16 Nl - Panels in Wall 1 (c) Rotation 0 Fig. 5.7: Continued 136 B 6 X I 6 cu o "a. 90 80 H 70 60 T ••x« Square Building with Panels Nl, 16, 16. 16 Rectangular Building with Panels Nl, 16, 32, 32 Dynamic Analysis using El Centro E/Q Static Design by Strength Method with Design Ecc. = e = actual ecc. of the building 30 20 •X« """-—A ^ Legend A Square Building X Rectangular Building 6 "T" 8 -r— 10 i 12 —T— 14 18 Nl - Panels in Wall 1 (d) Wall 1 Displacement 60 6 6 50-Cv? X 1 40-c e 30-0) o a. Ix 20-w Q CO 10-A =e 0 + Square Building with Panels Nl, 16, 16,16 Rectangular Building with Panels Nl, 16, 32, 32 Dynamic Analysis using El Centro E/Q Static Design by Strength Method with Design Ecc. = e = actual ecc. of the building Legend A Square Building X Rectangular Buildint 6 8 10 Nl - Panels in Wall 1 12 14 16 (e) Wall 2 Displacement Fig. 5.7: Continued 137 £ £ 30 25-•£ O a x I 20 Q 15 T Square Building with Panels Nl, 16. 16. 16 Rectangular Building with Panels Nl, 16, 32, 32 Dynamic Analysis using El Centro E/Q Static Design by Strength Method with Design Ecc. = e - actual ecc. of the building 10-;..: ; i —„^ j \^ j | Legend A Square Building X Rectangular Building (f) 6 8 10 Nl - Panels in Wall 1 Max. Dowel Displacement 12 14 16 0.8 | 0.7 «, 0.6 O 0.5 Square Building with Panels Nl, 16, 16, 16 Rectangular Building with Panels Nl, 16, 32, 32 Dynamic Analysis using El Centro E/Q Static Design by Strength Method with Design Ecc. = e = actual ecc. of the building^ ^x Legend A Square Building x Rectangular Building 6 8 10 Nl - Panels in Wall 1 12 -r-14 16 (g) Energy Input Fig. 5.7: Continued 138 displacements are too large to be handled by typical pre-cast connections. Looking at the wall and dowel displacements for this case one concludes that the rectangular building, which has a higher eccentricity e because of its dimensions, is effected to a greater extend than the square building. This conclusion is in agreement with the code design equation eqn. [4.3] which shows the necessity of larger design eccentricity for the building with high eccentricity e. The dynamic analysis results confirm that the actual eccentricity of the building alone is insufficient for calculating the design moment. 5.3.4 E/Q direction NBCC [22] recommends that a building has to be designed to withstand the earth quake in two mutually perpendicular directions. In the preceeding paragraphs we have dicussed various results for the earthquake acting in the X direction on a building designed for earthquake in the X &; Y directions. To check the adequacy of the building design to withstand the earthquake in the Y direction, a comparison of the various results obtained from dynamic analysis of the rectangular building ( see Fig. [4.1b] ) subjected to the El Centro earthquake acting in X & Y direc tion is given in Fig. [5.8a...c] ( actual eccentricity in the X & Y directions used in the dynamic analysis ). The results show that all the variables for the earthquake in Y direction are lower than those for the X direction. The difference is much more for lower values of Ni. This occurs because for these values of Ni the design of walls 3 & 4 is goverened by moment requirements in the X direction (i.e. MR = 1). For higher values of Ni ( Ni > 10 ), the design of walls 3 & 4 is governed by force requirements in the Y direction ( i.e. MR > 1 in Fig. [5.56] ). The values of the re sults for the earthquake in both the directions become comparable, although the Y direction results for the displacements and energy input are slightly lower than the X direction results. This shows that the building design is safe for the earthquake 139 B B x I U .*-> ca a, CO c 3 45 40 35 30-25 20 H 15 10 H 5 Rectangular Building with Panels Nl. 16. 32. 32 Static Design by Strength Method Dynamic Analysis using El Centro E/Q Legend L\ E/Q in X - direction X E/Q In Y - direction T 6 8 10 Nl - Panels in Wall 1 -T— 12 —P-14 16 (a) Building Displacement at CM. Figure 5.8: Comparison of dynamic analysis results of a rectangular building for the earthquake in X & Y directions 140 18 16 e I 12-Cu w 10 Q Rectangular Building with Panels Nl. 16. 32, 32 Static Design by Strength Method Dynamic Analysis using El Centro E/Q "•"••A. | A ^] e o Q x cd 2 i ! l/l / _ : i : /. Legend A E/Q in X - direction / X E/Q in Y - direction i i i 6 8 10 Nl - Panels in Wall 1 12 14 16 (b) Max. Dowel Displacement 0.8-6 0.7-i 2 Rectangular Building with Panels Nl, 16, 32. 32 Static Design by Strength Method Dynamic Analysis using El Centro E/Q 6 8 10 Nl - Panels in Wall 1 (c) Energy Input Fig. 5.8: Continued 141 in the Y direction. The critical direction for dynamic response will always will be the direction where e is greater. 5.3.5 Effect of design eccentricity Rectangular Building :-The NBCC [22], in its equivalent static design approach, has given two equations, eqn.(4.3) & eqn.(4.4) to calculate the design eccentricity of the building, although as explained earlier in section [4.3], only eqn.(4.3) is needed for calculating the design eccentricity when the building is designed by the strength method. The following section presents a comparison of results obtained from the dynamic analysis, which always use the actual eccentricity of the building, for buildings using different design eccentricities in the static design. Fig.[5.9a] shows the moment ratio for different static designs. Fig. [5.9b...g] shows the variation of response with Nx for a rectan gular building. The El Centro earthquake has been used for the dynamic analysis. ed is the code design eccentricity given by eqn. [4.3]. Variation of building displacement at the CM. ( x ), with Ni for various design eccentricities is given in Fig. [5.9b]. Displacement x does not vary significantly for various design eccentricities because the walls have the same strength in the direction of the earthquake for all the design eccentricities. Rotation 0 also does not vary greatly as design eccentricity is changed from l.Oej to 0.856,*. Then from 0.85e<j to 0.75ed there is 100 % increase in the value of 0 for low values of N\ and as the eccentricity is further decreased from 0.75ej to 0.65e<j there is another increase of 100 % in the value of 0. Wall 1 & wall 2 displacements also show similar trends of very small variation for design eccentric ities of 1.0e<i, 0.95e<i & 0.85e<i with displacements increasing quickly after that as the eccentricity is further reduced. From this we conclude that the code equation to calculate the design eccentricity, eqn. [4.3] is conservative and reducing ed by a 142 Rectangular Building with Panels Nl. 16, 32. 32 Static Design by Strength Method Code Ecc. = ed = 1.5e + 0.11 e = actual eccentricity of the building Legend X a a a x Ecc. = 1.0 * ed Ecc. = .95 * ed Ecc. = .85 ' ed Ecc. = .75* ed Ecc. = .65 * ed Ecc. = e 6 8 10 Nl - Panels in Wall 1 (a) Moment Ratio Figure 5.9: 'Comparison of static & dynamic analysis results for a rectangular build ing designed using NBCC recommended equivalent static design approach for vari ous design eccentricities with the strength method of design 143 t B B x i s cj CO a. w 00 c 3 CQ 60 50 40 30 20 10 0 -10 -20 -30 -40 Rectangular Building with Panels Nl, 16. 32. 32 Static Design by Strength Method Code Ecc. = ed = 1.5e + 0.11 e = actual eccentricity of the building Legend A Ecc. = -1.0 * ed X Ecc. = .95 * ed • Ecc. = .85 ' ed MM • • • MM m H Ecc. = .75 * ed MMMMMM • MMMMM K Ecc. = e 6 8 10 Nl - Panels in Wall 1 12 14 16 (b) Building Displacement at CM. 1.75 TJ (0 *- 1.50 O 125 1 I CD C o 02 O 0.75 4-> CO Rectangular Building with Panels Nl, 16, 32. 32 Static Design by Strength Method Code Ecc. = ed = 1.5e + 0.11 e =» actual eccentricity of the building  "••Tr K~^X|;., ; xp^^**£«!i 6 8 10 Nl - Panels in Wall 1 12 (c) Rotation 6 Fig. 5.9: Continued 144 a B c £ so 4 CO 120 110 100-90-80 70 60 50 40 Rectangular Building with Panels Nl, 16, 32. 32 Static Design by Strength Method Code Ecc. = ed = 1.5e + 0.11 e = actual eccentricity of the building 6 8 10 Nl - Panels in Wall 1 Legend A Ecc. = 1.0 ' ed X Ecc. = .95 * ed • Ecc. = .85 * ed B Ecc. = .75 * ed H Ecc. = .65 * ed ••lllllilliiiillil X Ecc. = e (d) Wall 1 Displacement B B X C <D B CD O co "a. V) Q 0 40 20-y ***«^>.» 10 -f _ , n ^i-'x-~W -10--20 Rectangular Building with Panels Nl. 16, 32. 32 Static Design by Strength Method Code Ecc. = ed = 1.5e + 0.11 e = actual eccentricity of the building  Ecc. = 1.0 * ed X Ecc. = .95 * ed - • Ecc. = .85 * ed Ecc. = .75 ' ed H Ecc. = .65 ' ed X Ecc. = e 1 I 1 • 1 1 2 4 6 8 10 Nl - Panels in Wall 1 I 12 —r-14 16 (e) Wall 2 Displacement Fig. 5.9: Continued 145 B B Cu 5 a o a x (0 S a 15 10 5 ft! 0 -5 -10 -15H -20 Rectangular Building with Panels Nl. 18, 32, 32 Static Design by Strength Method Code Ecc. = ed = 1.5e + 0.11 e =: actual eccentricity of the building T" 4 Legend A Ecc. = 1.0 * ed X Ecc. = .95 * ed • Ecc. = .85 ' ed MM mum «WB • B Ecc. = .75 m mm • ed E Ecc. = .65 * ed X Ecc. = e 6 8 10 Nl - Panels in Wall 1 12 14 16 (f) Max. Dowel Displacement B i CO O 0.7-0.6-0.5 I 0.4 a, 0.3 c to 02 U 01 • 0.0 Rectangular Building with Panels Nl, 16, 32, 32 Static Design by Strength Method Code Ecc. = ed = 1.5e + 0.11 e = actual eccentricity of the building  Legend A Ecc. = 1.0 • ed X Ecc. = .95 * ed • Ecc. = .85 * ed • • • * mm m a Ecc. = .75 ' ed B Ecc. = .65 " ed X Ecc. = e 6 T* 8 -T-10 —r-12 14 Nl - Panels in Wall 1 (g) Energy Input Fig. 5.9: Continued 16 146 factor of 0.85 would not significantly increase the response of the buildings analysed here. Square Building:-Fig.[5.10a] shows the moment ratio for a squre building shown in Fig. [4.1a] statically designed using the strength method of design, with design eccentricity equal to the code recommended design eccentricity ed and the actual eccentricity e of the building. Fig. [5.10b...g] shows the comparison of the dynamic analysis results of the building. Dynamic analysis has been done using the El Centro earthquake acting in the X direction. From Fig. [5.10a], the moment ratio MR for the square building with eccentricity equal to the actual eccentricity e, is > 1 at Ni > 2. For this reason there is a sharp decrease in rotation 6 for the building with Ecc. = e as Ni increases from 0. For the moment ratio of 1.5 ( at Ni = 4 in Fig. [5.10a] ) i.e. walls 3 & 4 capable of resisting a moment 50 % higher than that obtained from the design eccentricity Ecc. = e, 6 reduces to values comparable to those obtained using Ecc. = e<j. The displacements of wall 1, wall 2 and maximum dowel displacements show similar trends. As the results of the wall displacements for Ni = 3 are very close for the two cases, it could be said that the code design eccentricity e<j could be reduced by a factor of ( 30933/38076 ) 0.812 for this case ( at Ni = 3 the moment strength of the building with Ecc. = td is 38076 kN-m and with Ecc. = e it is 30933 kN-m..) The square building reduction factor of 0.812 is quite close to the reduction factor of 0.85 obtained for the rectangular building. Thus it is safe to conclude that the NBCC [22] gives a conservative equation to find the design eccentricity and a reduction factor of 0.85 could be applied to the code design eccentricity equation, at least for buildings similar to those considered here. 147 15-11-Q> 13 s I o as .>-> c <u B o Square Building with Panels Nl, 16, 16, 16 Static Design by Strength Method Code Design Ecc. = ed = 1.5e + 0.11 where e = actual eccentricity of the building / m .. / f e a Nl - Panels in Wall 1 (a) Moment Ratio Figure 5.10: Comparison of static & dynamic analysis results for a square building designed using NBCC recommended equivalent static design approach for various design eccentricities with the strength method of design 148 1 £ £ X I S cj c 3 DQ CD C o • -kJ (0 -4-) o ct: 65 55-45 Square Building with Panels Nl. 16, 16, 16 Dynamic Analysis using El Centro E/Q Static Design by Strength Method Code Ecc. = ed = 1.5e + 0.11 where e = actual eccentricity of the building 251 15-Legend A Ecc = ed x Ecc = e 6 8 10 12 14 16 Nl - Panels in Wall 1 (b) Building Displacement at CM. 1.2-(TJ . U. 1 o O 0 8 iV 0.6 0.4 0.2 A \ \ Square Building with Panels Nl, 16, 16, 16 Dynamic Analysis using El Centro E/Q Static Design by Strength Method Code Ecc. = ed = 1.5e + 0.11 where e = actual eccentricity of the building \ V Legend A Ecc.= ed Nl - Panels in Wall 1 (c) Rotation 0 Fig. 5.10: Continued 149 E B x i c e D O JS Cu CO . —^ 80 70 40 30 20-10 Square Building with Panels Nl, 16. 16. 16 Dynamic Analysis using El Centro E/Q Static Design by Strength Method Code Ecc. = ed = 1.5e + 0.11 where e = actual eccentricity of the building : Legend A Ecc. ~ ed X Ecc. 1 T —I 6 8 10 Nl - Panels in Wall 1 —r~ 12 14 16 (d) Wall 1 Displacement I X c 6 CD O jd "a, w Q CM 60-50 Square Building with Panels Nl. 16, 16. 16 Dynamic Analysis using El Centro E/Q Static Design by Strength Method Code Ecc. = ed = 1.5e + 0.11 where e = actual eccentricity of the building 8 10 Nl - Panels in Wall 1 16 (e) Wall 2 Displacement Fig. 5.10: Continued 150 25-6 fi 20-Square Building with Panels Nl, 16, 16, 16 Dynamic Analysis using El Centro E/Q Static Design by Strength Method Code Ecc. = ed = 1.5e + 0.11 where e = actual eccentricity of the building Nl - Panels in Wall 1 (f) Max. Dowel Displacement 0.50 fi 0.45 I 0.40 CO O I 0.35-d 2- 0.30 c Square Building with Panels Nl, 16, 16, 16 Dynamic Analysis using El Centro E/Q Static Design by Strength Method Code Ecc. = ed = 1.5e + 0.11 where e = actual eccentricity of the building &0 0-25 rS 020 0.15 -r 2 Legend A Ecc. = ed X Ecc.= e 6 10 T-12 —J— 14 16 Nl - Panels in Wall 1 (g) Energy Input Fig. 5.10: Continued 151 5.3.6 Effect of variation of stiffness of wall 3 & 4 As explained earlier in section [4.2] for the stiffness method of design the force in each wall is distributed according to the its stiffness. To check the effect of wall stiffness on response, the rectangular building in Fig. [4.1b] was designed by the strength method. The stiffness of walls 3 &; 4 was then varied when the dynamic analysis was done for the earthquake in X direction. The stiffness of walls 3 & 4 ( K3 & K4 ) was reduced by factors of 1.0, 0.5 & 0.2 while the strength was kept constant. The results of the response of the building for the X direction earthquake are shown in Fig. [5.11a...f]. The building displacement at CM. ( see Fig. [5.11a]) does not vary significantly as the stiffness of walls 3 &; 4 is reduced from Ks to K3/2. For low values of Ni building displacement at CM. ( x ) increases by about 100% as the stiffness of walls 3 & 4 is further reduced to K3/5. For high values of Nx, there is not too much change even when stiffness varies from K3/I to K3/5. Rotation 6 shows a change of 100% as stiffness is reduced from Ks/1 to K3/2. 0 increases by another 100% as the stiffness is reduced to Ks/5. For low eccentricities i.e. high Ni, the variation is less drastic. Curves for maximum wall displacement and dowel displacement show a variation similar to building displacement at CM. This shows that reduction in stiffnesses of walls 3 & 4 by one half has no significant effect on the response of the buildings for all values of Nx. For low eccentricities, even a further reduction in the stiffness does not effect the response drastically. Thus we conclude that the stiffness of walls 3 & 4 does not play a major role in the response of the structure for low eccentricities. For high eccentricities, the reduction in the stiffnesses of walls 3 & 4 from Ks/2 to Ks/5 effects the response. The variation of stiffness from K3/I to K3/5 is quite large whereas the corresponding variation in the response is not that large. We can assume that small variations in panel connection stiffness from one wall to another which may arise due to different strength panel 152 £ £ X cj ->-> CO tx CO C -a '3 CQ 65 55 45 35 H 25 15 5 • - v-\ _ • Rectangular Building with Panels Nl, 16, 32, 32 Dynamic Analysis using El Centro E/Q Stiffness of walls 3 & 4 is fraction of actual stiffness 4 6 8 10 Nl - Panels in Wall 1 (a) Building Displacement at CM. 12 14 Legend A K3/1 & K4/1 X K3/2 & K4/2 O K3/5 & K4/5 IIII 16 M T -a i_ CO o I O I CD C o -•J o 1-0.9-0.8 0.7 0.6-0.5-0.4-Rectangular Building with Panels Nl, 16, 32, 32 Dynamic Analysis using El Centro E/Q Stiffness of walls 3 & 4 is fraction of actual stiffness Legend A K3/1 & K4/1 6 8 10 Nl - Panels in Wall 1 (b) Rotation 0 Figure 5.11: Comparison of dynamic analysis results of a square building statically designed using strength method of design and using different stiffnesses of walls 3 & 4 in the dynamic analysis 153 6 B X a OJ B OJ o JO Cu co 90 80 A^. 70-60-50-40 30 0 -10 Rectangular Building with Panels Nl, 16, 32, 32 Dynamic Analysis using El Centro E/Q Stiffness of walls 3 & 4 is fraction of actual stiffness •TJ—..... 6 8 10 Nl - Panels in Wall 1 12 14 t I— " —— —vs- i_k . Legend A K3/1 & K4/1 X K3/2 & K4/2 1 1 1 1 1 1 Q K3/5 & K4/5 1 16 (c) Wall 1 Displacement B B CM X 50-45 40-ft-Rectangular Building with Panels Nl, 16, 32, 32 Dynamic Analysis using El Centro E/Q Stiffness of walls 3 & 4 is fraction of actual stiffness 2 Legend A K3/1 & K4/1 X K3/2 & K4/2 • K3/5 & K4/5 I I I 6 8 10 Nl - Panels in Wall 1 12 14 16 (d) Wall 2 Displacement Fig. 5.11: Continued 154 30 £ £ co CD o Q X (0 25-20-15-i 10--5 Rectangular Building with Panels Nl, 16, 32, 32 Dynamic Analysis using El Centro E/Q Stiffness of walls 3 & 4 is fraction of actual stiffnes Legend A K3/1 8c K4/1 X K3/2 & K4/2 • K3/5 & K4/5 6 8 10 Nl - Panels in Wall 1 12 14 16 1.1 1 0.9 0.8 0.7 0.6-0.5-(e) Max. Dowel Displacement Rectangular Building with Panels Nl, 16, 32, 32 Dynamic Analysis using El Centro E/Q Stiffness of walls 3 & 4 is fraction of actual stiffness Legend A K3/1 & K4/1 X K3/2 & K4/2 a K3/5 & K4/5 -r-12 16 Nl - Panels in Wall 1 (f) Energy Input Fig. 5.11: Continued 155 connections will not greatly effect the response. It has been shown that small variations in the panel connection stiffness will not significantly effect the results. This variation in the stiffness could be caused by using different strength panel connections in different walls. For a building designed by the stiffness method, it is safe to assume the stiffness of each wall proportional to the number of panels in that wall instead of calculating the stiffness of each panel connection and then calculating the stiffness of the walls. 156 Chapter 6 Discussion, Conclusions and Future Studies 6.1 Introduction This investigation was made to study the behaviour of eccentric precast concrete buildings designed according to the seismic provisions of the NBCC [22]. Two different approaches were taken in applying the static lateral load provisions of the NBCC [22]. These are refered to in chapter 4 as the strength method and the stiffness method of design. Results obtained from the dynamic analysis have been presented in chapter 5. The NBCC-1985 [22] contained major changes in the provisions for seismic loads on eccentric buildings. The torsional design moment was increased to take into con sideration the effects of dynamic magnification [7]. The previous recommendation for doubling the computed torsional effects when eccentricity exceeds 0.25D was eliminated because it was found to be unnecessarily conservative [37]. According to the NBCC [35], the peak horizontal seismic ground acceleration expected in zone 5 is 0.32g. Thus the El Centro 1940 earthquake, which has a peak acceleration of 0.35g, can be considered as a typical earthquake for earthquake-resistant design in zone 5. The CPCI has also modified its recommended design method for calculating the 157 strength of one storey precast panel buildings [20] because the previous method did not properly consider the strength of the lateral force resisting elements. The design example given in the CPCI handbook calculates the strength of a shear wall constructed from isolated wall panels by considering the probable & design flexural strength resulting from the dowels at the base of each panel. 6.2 Discussion of results Section [5.3] discusses the effects of changes in various parameters on eccentric buildings. The overall objective of this study was to answer the following questions : • Are the requirements for considering torsion in the NBCC-1985 [22] appro priate for the design of buildings of the type considered here ? • What recommendations can be made to improve the earthquake design of these buildings ? Guidelines to help answer the first question are given in Commentry J.l of Ref. [35], which states that : Structures designed in conformance with NBCC provisions should be able to resist moderate earthquakes without significant damage and ma jor earthquake without collapse. For the purpose of this section, collapse is defined as the state at which exit of the occupants from the building becomes impossible because of the failure of the primary structure. Although the terms moderate and major earthquakes are not explicitily defined, NBCC [35] has given recommendations for the maximum acceleration with a certain probability of exceedance that can expected in any part of the country. So for zone 5, an earthquake with peak acceleration of 0.32g can be taken as a major earthquake. In this study 3 different earthquakes with peak acceleration scaled to 0.35g have been used in the dynamic analysis ( see section [5.3] ). 158 Figs. [5.4c...1] [5.5a...j] & [5.6b...k] show the effect of various parameters such as building geometry, earthquake data, method of design etc., on the response of the buildings for various eccentricities. The maximum value of wall displacement for a building designed according to the NBCC recommendations using the strength method of design is 50 mm which is obtained for the Taft earthquake with JV"i=4 in Fig. [5.5c]. The average value of maximum wall displacement for various ec centricities for various parameters is between 35 to 40 mm. This is larger than the NBCC [35] recommended inter storey drift limitation of 0.005 times the storey height or 27mm in the present studies. But this NBCC drift limitation has been rec ommended assuming monolithic behaviour of the buildings in which a large volume of materials yield; thus the inelastic deformation is comparable to elastic deforma tion ( 3 times maximum as recommended by NBCC [35] ). But in a jointed pre-cast structure only a small volume of materials yield thus resulting in large inelastic deformations. At failure these can be 8 or more times the elastic deformations de pending upon the connection type. So for this type of structure we can accept much higher deformations than recommended by NBCC and these should not result in collapse. Thus for the type of the structure being dicussed here, a displacement of the order of 35 to 40 mm is quite reasonable. The average value of dowel displacements ( see Fig. [5.4g, 5.5e &; 5.6f] ) for buildings designed according to NBCC recommendations, is between 10 to 12 mm i.e. each side of the connection will be elongated by 5 to 6 mm. This elogation is well within the capability of the dowel connections. From Fig. [5.4d] [5.5b] & [5.6c] it is clear that the rotation 6 for the buildings designed according to NBCC recommendations does not vary significantly as long as the moment ratio MR for that case is equal to 1 i.e. as long as the design of the walls perpendicular to the earthquake direction, for the strength method of design, is done by the moment requirements for the earthquake in X direction. 159 The maximum, value of 0 for these cases is 0.28 * IO-3 radians which is very small. The effect of this small 0 on the wall displacements is very small compared to the building displacement at the center of mass. Also the maximum values of rotational velocity 0 and rotational acceleration 0 are very small and have only a very small effect on the overall response of the buildings. For buildings designed using the calculated building eccentricity with no magnification as the design eccentricity, the wall displacements and the rotation 0 are very large as shown in Figs. [5.7b..g], for builings with large eccentricity. This shows that building eccentricity alone is not adequate for the design of all buildings. The code design eccentricity eqn. (4.3) appears to be adequate for a wide range of eccentricities. Section 4.1.9.1 (13) of NBCC [35] states that total shear in the horizontal plane shall be distributed to various elements of the lateral force- resistant system accord ing to their rigidities found from a rational analysis; i.e. NBCC recommends the use of stiffness method approach explained in section [4.2] for finding the forces in the shear walls. Fig. [5.4c...1] show the response of the square building using both the stiffness method and the strength method of design. These figures show that the response of the building for the two methods of design is quite close. It has also been explained in setion [4.2.1] that the stiffness method may not be too practical for the highly eccentric precast buildings ( rectangular building Fig. [4.1 b] ) being considered in the present studies while the proposed strength method of design is ap plicable for all the eccentricities. This suggests that the proposed strength method is better suited for the design of pre-cast buildings of the type being considered in the present studies. Another advantage of the strength method is that we need only one design eccentricity equation ( eqn. (4.3) ) instead of the two equations, eqns.(4.3) & (4.4), recommended by NBCC [22], thus reducing the mathematical computations. 160 3 Conclusions 1. Design of connections based solely on their monotonic strength is inappro priate. Connections should exibit stable hysteresis loops without strength degradation to effectively dissipate the energy. 2. The CPCI suggested method for designing this type of building gives satis factory results. 3. The results of the dynamic analysis indicate that eccentric one storey box type precast buildings designed according to the NBCC recommendations will survive a major earthquake. 4. The strength method for calculating the forces in the walls is more effective than the code recommended stiffness method because the stiffness method tends to result in a requirement for higher strength connections in the weaker walls. For a highly eccentric building it becomes very difficult to provide these high strength connections. 5. The NBCC recommendations for building design are suitable for highly un-symmetrical as well as symmetrical buildings of the type being considered here. 6. For buildings of the type being considered here, it appears that the design eccentricity from eqn.(4.3) could be reduced by 15% without any significant changes in the overall response of the buildings. 7. A small reduction in the stiffness of the walls perpendicular to the direction of the earthquake does not effect the response of the buildings. 8. For a building designed by the stiffness method, the stiffness of the walls can be taken as proportional to their number of panels regardless of the different 161 strength panel connections used in the walls without significantly affecting the dynamic response. 6.4 Future studies A few of the areas which need to studied futher are given below : 1. Very little experimental data is available on the behaviour of the panel con nections, especially dowel connections, under cyclic loading. This work would increase our understanding on the seismic performance of the connections and thus of pre-cast structures as a whole. 2. Large scale tests on pre-cast buildings could be done to allow calibration of the results of computer studies. 3. The computer program should be extended to model an elastic roof diaphragm i.e. rigid roof panels connected by elastic springs. 4. The computer program should be modified to handle multi-storey buildings with different methods of construction. 162 Bibliography [l] Anon, Standard Plate Book; Published by Stanley Structures, Denver, June 1977. [2] Ashwad A.; 'Selected Precast Connections : Low - cycle behaviour and strength'; Proc. 3rd Canadian Conference on Earthquake Engineering; vol 2, Montreal 1979. [3] Ashawd A.; 'Seismic Design and In-plane Behaviour of Single Panel Cou pled Walls'; Proc. Workshop on Design of Prefabricated Concrete Buildings for Earthquake Loads; conducted by Applied Technology Council, Berkeley, April 1981. [4] Brankov G. & Sachaniski S.; 'Response of Large Panel Buildings for Earthquake Excitation in Non-elastic state'; Proc. 6th World Conference on Earthquake Engineering, New Delhi, India, Jan 1977. [5] Becker J.M., Llorente C, 'Seismic Design of Precast Concrete Panel Buildings'; Proceedings, Workshop on Earthquake Resistant Reinforced Concrete Building Construction, University of California, Berkeley, June 1977. [6] Becker J.M., Llorente C, 'Seismic Response of a Simple Precast Concrete Panel Walls'; Proceedings, Second U.S. National Conference on Earthquake Engineering, Stanford, California, 1979. [7] Bustamante J.I. &z Rosenblueth E.; 'Building Code Provisions on Torsional Oscillations'; Proc. 2nd World Conference on Earthquake Engineering, Japan 1960, vol 2, pp 879-892. [8] CANS - A2S.S - M84 - Design of Concrete Structures for Buildings, Canadian Standard Association (CSA), 178 Rexdale Boulevard Rexdale Ontario, Canada. [9] Clough D.P.; Design of Connections for Precast Prestressed Concrete Buildings for the effects of earthquake; Prestressed Concrete Institute, Chicago, Illinois, March 1985. [10] Dimitrov M.L. & Georgier G.B.; 'Determination of Hysteresis Curves in the Connections of Large - Panel Buildings'; Proc. 8"* European Conference on Earthquake Engineering; Lisbon 1986, vol 4. 163 [11] Fintel M., Schultz D., and Iqbal M.; 'PCA Report No. 2 : Philosophy of Structural Response of Normal & Abnormal Loads'; Design and Construction of Large Panel Structures ; office of Policy Development & Research, Deptt. of Housing and Urban Development, Washington D.C.; Aug 1976. [12] Hanson N.W.; 'Seismic tests of Horizontal Joints'; Design and Construction of Large Panel Structures, Supplement Report C; office of Policy Development & Research, Deptt. of Housing and Urban development, Washington D.C.; Jan 1979, NTIS No. PB 79-22701. [13] Harris H.G. & Abboud B.E.; 'Cyclic Shear Behaviour of Horizontal Joints in Precast Concrete Large Panel Buildings'; Proc. Workshop on Design of Prefabricated Concrete Buildings for Earthquake Loads, April 1981, pp 403-438. [14] Hawkins N.W.; 'State of the art report on Seismic Resistance of Prestressed and Precast Structures, Part 2 - Precast Concrete'; PCI Journal, Jan-Feb 1978, vol 23, No. 1, pp 40-58. [15] Hawkins N.M. h Lin I.J.; 'Bond Characteristics of Reinforcing Bars for Seismic Loading'; Proc. 3rd Canadian Conference on Earthquake Engineering; vol 2, Montreal 1979. [16] Joahl L.S. & Hanson N.W.; 'Horizontal Joints Tests'; Design and Construction of Large Panel Structures, Supplement Report B; office of Policy Development & Research, Deptt. of Housing and Urban Development, Washington D.C.; Jan 1979, NTIS No. PB 79- 22701. [17] Kallros M.; An Experimental Investigation of the Behaviour of Connections in Thin Precast Concrete Panels under Earthquake loading; M.A.Sc. Thesis, University of British Columbia, April 1987. [18] Llorente C, Becker J.M. & Roessel J.M.; 'The effects of Non- linear Inelastic Connection Behaviour of Precast Panelized Shear Walls'; Reinforced Concrete Structures Subjected to Wind & Earthquake Forces, ACI Publication, SP - 63, American Concrete Association, Detroit, Michigan, 1980. [19] Martin L.D. & Korkosz W.J.; Connections for Precast Prestressed Concrete Buildings, including earthquake resistance; Technical Report No. 2; Prestressed Concrete Institute, Chicago, Illinois, March 1982. [20] Metric Design Manual, Canadian Prestressed Concrete Institute, Ottowa, 2nd Edition, 1986. [21] Muller P. & Becker J.M.; 'Seismic Characteristics of Composite Precast Walls'; Proc. 3rd Canadian Conference on Earthquake Engineering; vol 2, Montreal 1979. 164 National Building Code of Canada 1985, Associate committee on the National Building Code, National Research Council of Canada. Neille D.S.; Behaviour of Headed Stud connections for Precast Concrete Panels under Monotonic and Cycled Shear Loading; Ph.D. Thesis, University of British Columbia, May 1977. Pall A.S. & Marsh C; 'Seismic Response of Large Panel Structures using Limited-Slip bolted Joints'; Proc. 3rd Canadian Conference on Earthquake En gineering; vol 2, Montreal 1979. Phillips W.R. & Sheppard D.A. Plant cast Precast and Prestressed Concrete, a design guide. Poller E., Tso W.K., Heidebrecht A.C.; 'Analysis of Shear Walls in Large Panel Construction'; Canadian Journal of Civil Engineering, vol 2, No. 3, Sept 1975. Popov E.P.; 'Mechanical Characteristics and Bond of Reinforcing Steel under Seismic Conditions'; Workshop on Earthquake - Resistant Reinforced Concrete Building Construction; ( ERCBC ), University of California, Berkeley, July 1977, pp 658-682. Powell G.H., Drain - 2D Users Guide, Report No. UCB/EERC - 73/22, Earth quake Engineering Research Centre, University of California, Berkeley, Oct 1973. Saxena R.P.; An Experimental Investigation of a Precast Connection; M.A.Sc. Thesis, University of British Columbia, Sept 1983. Schricker V. & Powell G. et. Al., Inelastic seismic analysis of large panel build ings; Califonia University Berkeley, Earthquake Engineering Research Centre, Sept. 1980, NTIS No. PB 81-154338. Spencer R.A. & Neille D.S.; 'Cyclic Tests of Welded Headed Stud connections'; PCI Journal; May-June 1976, vol 15, No. 1, pp 67-78. Spencer R.A.; 'Serviceability of Precast Diaphragms Sc Connections designed for Earthquake Resistant Structures'; Proc. Workshop on Design of Prefabri cated Concrete Buildings for Earthquake Loads, April 1981, pp 551-573. Spencer R.A. Si Tong W.K.T.; 'Design of a One-Storey Precast Concrete Build ing for earthquake loading'; Proc.8"* World Conference on Earthquake Engi neering; San Francisco 1984, vol 5, pp 653- 660. Spencer R.A.; 'Earthquake Resistant Connections for Low Rise Precast Con crete Buildings'; Seminar on Precast Concrete Construction in Seismic zones; Tokyo, October 27-31, 1986, pp 61-81. Supplement to the National Building Code of Canada 1985, Associate commit tee on the National Building Code, National Research Council of Canada. 165 [36] Tong W.K.T.; Non-Linear Seismic Analysis of a One-Storey Precast Concrete Building; M.A.Sc. Thesis, University of British Columbia, Febuary 1984. [37] Tso W.K. & Dempsey K.M.; 'Seismic Torsional Provisions for Dynamic Ec centricity'; Journal of Earthquake Engineering and Structural Dynamics, vol 8, 1980, pp 275-289. [38] Velkov Miodrag; 'Large Panel Systems in Yugoslavia : Design, Construction and Research for Improvement of Practice & Elaboration of Codes'; Proc. Workshop on Design of Prefabricated Concrete Buildings for Earthquake Loads; conducted by Applied Technology Council, Berkeley, April 1981. [39] Zamolo Mihaela; 'Evaluation of a Large - Panel Building System for Seismic Zones - Experimental Part'; Proc. 8th European Conference on Earthquake Engineering; Lisbon 1986, vol 5. [40] Zeck U.I.; Joints in Large Panel Precast Concrete Structures; NTIS No. PB-252852/9GI by U.S. Deptt. of Commerce Jan 1976. 166 Appendix A Time Periods of the Buildings Considered in the Present Studies Number of Panels in Wall 1 Ni Square Building with Panels Nu 16, 16, 16 Rectangular Building with Panels Nu 16, 32, 32 0 0.29 0.265 4 0.29 0.272 8 0.302 0.281 12 0.31 0.2897 16 0.311 0.293 Table A.l: Fundamental Periods (in sec ) of the Buildings in Figs. [4.1a] and [4.1b], designed by the Strength Method Note : The stiffness of the walls in the direction of the earthquake is determined from eqn. (3.19) ( corresponding to Case 1 ). 167 

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