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Ductility demands of filtered earthquakes on reinforced concrete frames Kuan, Steven Yet Wui 1993

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DUCTILITY DEMANDS OFFILTERED EARTHQUAKES ONREINFORCED CONCRETE FRAMESbySTEVEN YET WUI KUANB.A.Sc.(Honours), The University of British Columbia, 1986M.Eng., The University of British Columbia, 1990A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF CIVIL ENGINEERINGWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAApril 1993© Steven Yet Wui Kuan, 1993In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature) Department of ^EnaineerilThe University of British ColumbiaVancouver, CanadaDate ^April 28 , iq93DE-6 (2/88)AbstractThe Mexico City 1985 and the Loma Prieta 1989 earthquakes demonstrated the destructive natureof seismic waves filtered through soft soils. Key characteristics of filtered ground motions are generallyamplified acceleration, increased predominant period, and lengthened duration of strong shaking.Filtering of seismic waves through the soft soils in the Fraser River delta in British Columbia wasinvestigated using the two-dimensional, dynamic analysis program SHAKE. It was found that the peakacceleration and the predominant period of the bedrock vibration was increased by about fifty percent.Surface ground motion amplification similar to the strong motion that occurred in Mexico City in 1985only occurred when the bedrock motion was of low intensity and vibrated predominantly at the inherentnatural period of the soil layers, two conditions which may be brought on by large earthquakes at a largeepicentral distance.Filtered earthquakes can result in increasing displacements with decreasing yield strength of astructure, in contrast with the usually accepted equal-displacement criterion. In other words, in a filteredearthquake, the inelastic displacement response curve, which is a line connecting the maximum responseof a given structure with different yield strengths in a lateral load-deflection plot, is, for most structures,outward-sloping rather than vertical, implying that higher strengths are required to maintain a certainductility. These results were indicated by the response of single-degree-of-freedom, elasto-plastic systemsin time-step computer analyses. They can also be predicted reasonably accurately by the R-ii equation,a relationship derived from energy principles applied to simple elasto-plastic vibrating systems anddeveloped in this thesis. The strength and ductility demand analyses using the single-degree-of-freedomsystems and the R-A equation further revealed that the response of a structure in any earthquake, filteredor unfiltered, is influenced mainly by the relationship of the fundamental structural period to thepredominant period of the ground motion, or by the slope of the velocity response spectrum. Thesefindings point to possible adjustments to the seismic design philosophy of the building code for design ofstructures for filtered earthquakes.Inelastic, time-step analyses of reinforced concrete frames and shear walls, designed in accordancewith the Canadian codes, showed that filtered earthquakes caused more cycles into the inelastic range thanunfiltered earthquakes, probably as a direct result of the longer duration of strong shaking of the filteredground motion. Also, it was calculated that code-designed frames would respond to acceptable ductilitylevels in the filtered version of the design earthquake for the Fraser River delta.Based on results of the computer analyses of prototype six-storey frames, cyclic loads wereapplied to five large-scale, reinforced concrete exterior beam-column-slab subassemblies designed anddetailed in accordance with the codes. The tests indicated that structures would behave well up to thedeflection limits that they were designed for, with only moderate damage and with no influence fromdifferences in the loading history, but larger displacements could cause severe structural damage and couldlead to a significant loss of load-carrying capacity. Most importantly, it was shown that maintaining theintegrity of the joint core is required to develop the required strength and stiffness of a reinforced concreteframe in a code-level seismic event and that joints designed to satisfy the National Building Code ofCanada suffered considerable damage. In order to behave well in a filtered surface ground motion, manystructures need to be designed for higher strengths or for larger ductility capacities when compared tostructures on rock subjected to the same earthquake.ivTable of ContentsPageAbstract ^Table of Contents ^  ivList of Tables  xifiList of Figures ^  xvAcknowledgement Dedication ^  xxix1^Introduction ^  11.1 Earthquakes  11.2 Filtered Earthquakes ^  21.3 Earthquake Parameters Having Influence on Structural Response ^ 31.4 Past Occurrences of Filtered Earthquakes ^  51.4.1 Mexico Earthquake 1985 ^  51.4.2 Loma Prieta Earthquake 1989  71.5 Fraser River Delta and Richmond, B.C. ^ 91.6 Purpose ^  91.7 Scope  101.8 Objectives ^  102^Filtered Earthquakes in the Fraser River Delta ^  112.1 Geography of Fraser River Delta ^  112.2 Geology of Fraser River Delta  112.3 Seismicity of Fraser River Delta ^  122.4 Review of Seismic Studies of Fraser River Delta Soils ^  142.5 Dynamic Analyses of Fraser River Delta Soils  162.5.1 Computer Program SHAKE ^  16V2.5.2 Modelling of Richmond Soils ^  162.5.3 Bedrock Motions ^  172.5.4 Surface Ground Motions ^  182.6 Remarks ^  19I^Analytical Studies of Simple Elasto-Plastic Systems ^  203^Elasto-Plastic Representation for Seismic Response of Structures ^ 213.1 Idealized Seismic Response of Elasto-Plastic Systems ^  213.2 Ductility, Force Reduction Factor, and Inelastic Displacement Response Curves . . ^ 223.3 Concept of Equal-Displacement Response ^  233.4 Application in Seismic Design Code  243.5 Elastic and Inelastic Response Spectra ^  243.6 Change in Fundamental Period of Vibration  254^Strength and Ductility Demands of Filtered Earthquakes ^  264.1 Computer Program, Test Structure and Procedure  264.2 Earthquake Records Studied ^  274.3 Results of Analyses ^  274.4 Expected Results Based on Equal-Displacement Criterion ^  284.5 Ductility Demands ^  294.6 Strength Demands  294.7 Inelastic Displacement Response Curves ^  314.8 Summary of Results and Conclusions  325^Strength and Ductility Demands in Relation to Predominant Periods of Earthquakes . 345.1 Background ^  345.2 Past Observations on Effects of Predominant Period ^  355.3 Observations from Present Strength and Ductility Demand Analyses ^ 365.4 Characteristic Periods from Acceleration and Velocity Response Spectra ^ 37vi5.5 Summary ^  396^Relationship Between Strength Requirements and Ductility Demands of Earthquakes . 406.1 Introduction ^  406.2 Energy Concepts in Seismic Response of Structures ^  406.3 Concept of Equal-Energy Response ^  436.4 R-p Equation: Relationship Between Strength, Ductility, and Spectral Velocities^436.5 Significance of the R-p Equation ^  446.6 Shift in Fundamental Period and Equivalent Elastic Structure ^  466.7 Comparison Between Calculated and Actual R ^  476.8 Modified R-p Equation ^  486.9 Shift in Structural Damping  496.10 Comparison Between Calculated and Actual R with Damping Shifts ^ 526.11 Summary ^  547^Strength and Ductility Demands in Relation to Velocity Spectrum ^ 558^Relevance of R-p Equation to Seismic Design Code ^  588.1 Seismic Design of Structures in Canada ^  588.1.1 General Procedure ^  588.1.2 Seismic Response Factor  598.1.3 Expected Response of Structures ^  608.1.4 Foundation Factor ^  608.2 Combined Effects of Amplification and Period Shift ^  618.3 Effects of Filtered Earthquakes in Richmond^  618.4 Recommended Design of Structures for Filtered Earthquakes ^  62Analytical Studies of Reinforced Concrete Structures ^  649^Dynamic Analyses of Reinforced Concrete Frames Designed for R of 4 ^ 659.1 Purpose and Procedure ^  65vii9.2 Design of Study Frame ^  659.3 Member Properties of Study Frame ^  679.4 Computer Program ^  689.5 Modelling of Study Frame  689.6 Structural Properties of Study Frame ^  699.6.1 Elastic Lateral Stiffness  709.6.2 Lateral Roof Deflection at Adjusted Code Yield Load ^  709.6.3 Natural Frequencies ^  719.6.4 Damping ^  729.7 Dynamic, Inelastic Response of R4 Frame ^  729.7.1 Response to Richmond Surface Motions  739.7.2 Response to Earthquakes Scaled to 0.21g ^  739.7.3 Response to Earthquakes Scaled to 0.21m/s  749.7.4 Response to Earthquakes Scaled for Ductility of 4 ^  759.7.5 Selected Response of R4 Frame at Ductility of 4  769.8 Remarks ^  7610 Dynamic Analyses of Reinforced Concrete Frames Designed for F of 2 ^ 7810.1 Purpose and Procedure ^  7810.2 Design of Study Frame  7810.3 Member Properties of Study Frame ^  7910.4 Computer Program and Modelling of Study Frame ^  7910.5 Structure Properties of Study Frame ^  7910.6 Dynamic, Inelastic Response of F2 Frame  8010.7 Remarks ^  8011 Dynamic Analyses of Reinforced Concrete Nominal Shear Walls ^ 8211.1 Introduction ^  82viii11.2 Past Research on Shear Walls ^  8311.3 Analysis Procedure ^  8311.4 Design of Study Shear Wall Structure ^  8411.5 Member Properties of Study Shear Wall Structure ^  8411.6 Computer Program and Modelling of Study Shear Wall Structure ^ 8511.7 Structural Properties of Study Shear Wall Structure ^  8611.7.1 Elastic Lateral Stiffness ^  8611.7.2 Lateral Roof Deflection at Adjusted Code Yield Load ^  8611.7.3 Natural Frequencies ^  8611.7.4 Damping ^  8711.8 Dynamic, Inelastic Response of Study Shear Wall Structure ^  8711.8.1 Response to Earthquakes Scaled to 0.21g ^  8711.8.2 Response to Earthquakes Scaled for Ductility of 2 ^  8811.8.3 Selected Response of Shear Wall at Ductility of 2  8911.9 Remarks ^  8912 Static-Load-To-Collapse Analyses for Study Frames and Shear Wall ^ 9012.1 Purpose ^  9012.2 Procedure  9012.3 Computer Program and Modelling of Structures ^  9112.4 Lateral Load-Deflection Plots for Study Structures  9112.5 Yield Points of Study Frames ^  9412.6 Remarks ^  9513 Examination of Dynamic Displacement Response with Load-Deflection Plots ^ 9613.1 Introduction ^  9613.2 Response of R4 Frame, F2 Frame, and Shear Wall Structure ^  9613.3 Discussions of Elastic Response ^  97ix13.4 Discussions of Inelastic Response ^  9713.5 Examination Using Inelastic Displacement Response Curves ^  9814 Examination of Local Curvature Ductility in R4 Frame ^  10014.1 Introduction ^  10014.2 Plastic Hinge Length ^  10114.3 Determination of Local Curvature Ductility from Tip Displacement Ductility . . . . 10214.4 Plastic Hinge Rotations from Static-Load-to-Collapse Analyses ^ 10314.5 Plastic Hinge Rotations from Dynamic Analyses ^  10414.6 Local Curvature Ductility - Tip Displacement Ductility Relationship for R4 Frame^10414.7 Summary ^  10515 Examination of Response in Plastic Hinges in Study Structures ^ 10615.1 Introduction ^  10615.2 Plastic Hinge Response of R4 Frame ^  10715.3 Plastic Hinge Response of Study Shear Wall Structure ^  10815.4 Remarks ^  10816 Examination of Base Shears, Storey Shears, and Lateral Forces ^ 10916.1 Introduction ^  10916.2 Dynamic Base Shears in Study Structures ^  10916.3 Storey Shears and Lateral Forces in Elastic Frames ^  110III Experimental Studies Using Reinforced Concrete Exterior Beam-ColumnSubassemblies ^  11317 Description of Code-Response Tests ^  11417.1 Purpose ^  11417.2 Method of Study and Test Set-Up ^  11417.3 Test Specimens ^  11517.4 Components of Beam Tip Deflection ^  11717.5 Scope of Experimental Study ^  117x17.6 Testing Procedure and Loading Programs ^  11917.7 Detailing of Specimens ^  12117.8 Instrumentation and Data Acquisition ^  12117.9 Fabrication of Specimens ^  12317.9.1 Reinforcing Steel  12317.9.2 Concrete ^  12417.10 Repair of Specimen BC1 ^  12517.11 Past Research on Reinforced Concrete Exterior Beam-Column Joints ^ 12517.12 Testing ^  12818 Damage to Specimens ^  12918.1 Overall Damage Pattern ^  12918.2 Beam Damage ^  13018.3 Slab Damage  13118.4 Transverse Beam Damage ^  13218.5 Column Damage ^  13218.6 Joint Damage  13318.7 Summary ^  13319 Overall Hysteretic Behaviour of Specimens ^  13519.1 Testing Frame Deflection ^  13519.2 Beam Tip Load-Deflection Behaviour ^  13619.2.1 Applied Load Versus Global Deflection ^  13619.2.2 Applied Load Versus Relative Deflection  13719.2.3 Overall Behaviour ^  13719.3 Comparison with Theoretical Behaviour ^  13819.4 Comparison with Previous Experimental Results ^  13919.5 Yield Loads and Deflections ^  140xi19.6 Displacement Ductility ^  14119.7 Yield Excursions  14119.8 Rotation of Upper Column ^  14220 Examination of Yield Behaviour of Specimens ^  14520.1 Flexural Strength Ratios ^  14520.2 Effective Slab Width in Tension ^  14820.3 Yielding of Main Beam ^  15020.4 Flexural Response of Column  15220.5 Response of Joint Core ^  15320.6 Summary ^  15621^Large-Deflection Test  15821.1 Purpose ^  15821.2 Test Specimen  15821.3 Loading Program ^  15921.4 Instrumentation and Data Acquisition ^  15921.5 Testing ^  16021.6 Hysteretic Behaviour ^  16121.7 Yield Behaviour  16221.8 Damage to Specimen ^  16521.9 Failure Mode of Specimen  16521.10 Summary ^  16722 Comments on Experimental Study ^  16822.1 Effects of Filtered Earthquakes  16822.1.1 Effects of Loading History ^  16822.1.2 Effects of Large Ductility Demand ^  16922.1.3 Response of Stronger Structures  170xii22.2 General Comments ^  17022.2.1 Effective Slab Width ^  17122.2.2 Strong Column-Weak Beam ^  17122.2.3 Integrity of Joint Core  171Conclusions ^  173List of References  400Appendix A^Calculation of Quasi-Static Seismic Loads for Design of R4 Frame . . . 408Appendix B^Calculation of Quasi-Static Seismic Loads for Design of Study Shear WallStructure ^  410Appendix C^Response of Beam-Column Subassemblies Repaired with InjectedEpoxy ^  413List of TablesPageTable 1.1 Ground Motion Classification ^  175Table 1.2 Parameters Having Influence on Seismic Response of Structures ^ 175Table 2.1 Peak Accelerations and Predominant Periods of Bedrock and Surface Ground Motionsin Richmond, B.C. ^ 175Table 4.1 Earthquake Records Used in Strength and Ductility Demands Analyses ^ 176Table 5.1^Characteristic Periods of Study Earthquakes ^  176Table 9.1 Member Properties of R4 Frame ^  177Table 9.2 Structure Properties of R4 Frame  177Table 9.3 Response of R4 Frame to Richmond Ground Motions ^  178Table 9.4 Response of R4 Frame to Earthquakes Scaled to 0.21g  178Table 9.5 Response of R4 Frame to Earthquakes Scaled to 0.21 m/s ^  178Table 9.6 Response of R4 Frame to Earthquakes Scaled for Ductility of 4  179Table 10.1 Member Properties of F2 Frame ^  179Table 10.2 Structure Properties of F2 Frame  179Table 10.3 Response of F2 Frame to Study Filtered Earthquakes ^  180Table 11.1 Section Properties of Study Shear Wall Structure  180Table 11.2 Structure Properties of Study Shear Wall Structure ^  180Table 11.3 Response of Study Shear Wall Structure to Earthquakes Scaled to 0.21g ^ 181Table 11.4 Response of Study Shear Wall Structure to Earthquakes Scaled for Ductility of 2 . 181Table 14.1 Local Curvature Ductility from Tip Displacement Ductility for R4 Frame Based onPark and Paulay's Formula ^  181Table 14.2 Plastic Hinge Rotation at Critical Section in R4 Frame from Static Analysis . . . . 182Table 14.3 Plastic Hinge Rotation at Critical Section in R4 Frame from Dynamic Analyses . . 182Table 14.4 Comparison of Local Curvature and Tip Displacement Ductilities for R4 Frame . . 182xivTable 15.1 Plastic Hinge Response at Critical Section in R4 Frame Deflecting to Tip Ductilityof 4 in Study Earthquakes ^  183Table 15.2 Plastic Hinge Response at Critical Section in Study Shear Wall Structure Deflectingto Tip Ductility of 2 in Study Earthquakes ^  184Table 16.1 Earthquake Records for Storey Shears Analyses  184Table 17.1 Control Deflection Parameters for Tests BC1 to BC4 ^  185Table 17.2 Actual Positions of Reinforcement in Specimens  186Table 17.3 Measured Material Properties of Specimens ^  187Table 19.1 Flexibility of Testing Frame ^  188Table 19.2 Comparison between Specimen BC! and Ehsani and Wight Specimen ^ 189Table 19.3 Comparison between Specimen BC1 and Paultre and Mitchell Specimen ^ 190Table 19.4 Measured Yield Loads and Yield Deflections of Specimens ^  191Table 19.5 Comparison between Theoretical and Measured Yield Loads  191Table 19.6 Negative Beam Tip Displacement Ductilities of Specimens ^  192Table 19.7 Positive Beam Tip Displacement Ductilities of Specimens  193Table 19.8 Yield Excursions of Specimens ^  194Table 19.9 Flexibility of Upper Column in Bending ^  194Table 20.1 Axial Loads in Upper Column in Specimens  195Table 20.2 Balanced Axial Loads and Moments of Column in Specimens ^ 195Table 20.3 Flexural Strength Ratios of Specimens ^  196Table 20.4 Flexural Strength Ratios at Maximum Negative Beam Tip Loads ^ 197Table 20.5 Joint Shear Stress in Specimens ^  197Table 20.6 Response of Middle Transverse Reinforcement in Joint Core ^  198Table 20.7 First-Crack Load for Joint from Beam Bar Hook Strains  198Table A.1 Distribution of Design Base Shear for R4 Frame ^  409Table B.1 Distribution of Design Base Shear for Study Shear Wall ^  411XVList of FiguresPageFigure 1.1 (a) Accelerogram, (b) Absolute Acceleration Response Spectrum,and (c) Relative Velocity Response Spectrum for Taft S69E 1952Ground Motion   199Figure 1.2^(a) Accelerogram, (b) Absolute Acceleration Response Spectrum,and (c) Relative Velocity Response Spectrum for El Centro EW 1940Ground Motion ^  200Figure 1.3^Epicentre of September 19, 1985 Mexico Earthquake ^  201Figure 1.4^Accelerograms for La Villita, UNAM, and SCT in the 1985 MexicoEarthquake ^  202Figure 1.5^Soil Zones of Mexico City ^  202Figure 1.6^Mexico City SCT EW 1985 Ground Motion Accelerogram ^ 203Figure 1.7^(a) Absolute Acceleration and (b) Relative Velocity Response Spectrafor Mexico City SCT EW 1985 Ground Motion ^  204Figure 1.8^Epicentre of October 17, 1989 Loma Prieta Earthquake and Recorded PeakHorizontal and Vertical Ground Accelerations (in units of g) ^ 205Figure 1.9^Accelerograms and Acceleration Response Spectra at Yerba Buena Islandand Treasure Island Stations in the 1989 Loma Prieta Earthquake ^ 206Figure 2.1^Location Map of Fraser River Delta and Richmond, B.C.  207Figure 2.2^Soil Types in Fraser Delta ^  208Figure 2.3^Typical Soil Profiles in Fraser Delta ^  209Figure 2.4^Seismicity in Western Canada ^  210Figure 2.5^Plate Tectonics along West Coast of Canada ^  211Figure 2.6^Amplification Curves for Richmond Soils  212Figure 2.7^Acceleration Response Spectra (5% Damping) for Artificial SurfaceMotions in Richmond, B.0^  213Figure 2.8^Idriss' Amplification Curve for Soft Soils ^  213Figure 2.9^Richmond Soil Profile Model with Variations of Shear Modulusand Damping for SHAKE Analysis ^  214xviFigure 4.2Figure 4.3Figure 4.4Figure 4.5Figure 4.6Figure 4.7Accelerograms for (a) Taft S69E 1952 Scaled to 0.21gand (b) Filtered Taft S69E ^  215Accelerograms for (a) Caltech EW 1971 Scaled to 0.21gand (b) Filtered Caltech EW ^  216Accelerograms for (a) El Centro EW 1940 Scaled to 0.21gand (b) Filtered El Centro EW ^  217Accelerograms for (a) Modified Taft and (b) Artificial RichmondGround Motions ^  218(a) Absolute Acceleration and (b) Relative Velocity Response Spectrafor Taft (0.21g) and Filtered Taft Motions ^  219(a) Absolute Acceleration and (b) Relative Velocity Response Spectrafor Caltech (0.21g) and Filtered Caltech Motions ^  220(a) Absolute Acceleration and (b) Relative Velocity Response Spectrafor El Centro EW (0.21g) and Filtered El Centro EW Motions ^ 221(a) Absolute Acceleration and (b) Relative Velocity Response Spectrafor Modified Taft and Artificial Richmond Motions  ^222Acceleration Response Spectra for Taft S69E 1952, El Centro EW 1940,Mexico City SCT EW 1985, and Artificial Richmond Ground Motions ^ 223Idealized Bilinear Response of Structures under Static Lateral Loads ^ 224Idealized Response of Structures under Dynamic Lateral Loads ^ 224Definition of Force Reduction Factor ^  225Definition of Inelastic Displacement Response Curve ^  225Equal-Displacement Seismic Response Assumption of Structures ^ 226(a) Accelerogram, (b) Absolute Acceleration Response Spectrum,and (c) Relative Velocity Response Spectrum for Mexico City CUIP EW 1985Ground Motion ^  227Ductility Demands of Taft S69E 1952 (0.21g) ^  228Ductility Demands of Filtered Taft^  228Ductility Demands of Caltech EW 1971 (0.21g) ^  229Ductility Demands of Filtered Caltech ^  229Ductility Demands of El Centro EW 1940 (0.21g) ^  230Ductility Demands of Filtered El Centro EW  230Figure 2.10Figure 2.11Figure 2.12Figure 2.13Figure 2.14Figure 2.15Figure 2.16Figure 2.17Figure 2.18Figure 3.1Figure 3.2Figure 3.3Figure 3.4Figure 3.5Figure 4.1Figure 4.8Figure 4.9Figure 4.10Figure 4.11Figure 4.12Figure 4.13Figure 4.14Figure 4.15Figure 4.16Figure 4.17Figure 4.18Figure 4.19Figure 4.20Figure 4.21Figure 4.22Figure 4.23Figure 4.24Figure 4.25Figure 4.26Figure 4.27Figure 4.28Figure 4.29Figure 4.30Figure 4.31Figure 4.32Figure 4.33xviiDuctility Demands of Modified Taft ^  231Ductility Demands of Artificial Richmond  231Ductility Demands of Mexico City CUIP EW 1985 ^  232Ductility Demands of Mexico City SCT EW 1985  232Force Reduction Factors for Taft S69E 1952 (0.21g) ^  233Force Reduction Factors for Filtered Taft ^  233Force Reduction Factors for Caltech EW 1971 (0.21g) ^  234Force Reduction Factors for Filtered Caltech ^  234Force Reduction Factors for El Centro EW 1940 (0.21g) ^  235Force Reduction Factors for Filtered El Centro EW  235Force Reduction Factors for Modified Taft^  236Force Reduction Factors for Artificial Richmond  236Force Reduction Factors for Mexico City CUIP EW 1985 ^ 237Force Reduction Factors for Mexico City SCT EW 1985  237Inelastic Acceleration Response Spectra for Taft S69E 1952 (0.21g) ^ 238Inelastic Acceleration Response Spectra for Filtered Taft ^  238Inelastic Acceleration Response Spectra for Caltech EW 1971 (0.21g) ^ 239Inelastic Acceleration Response Spectra for Filtered Caltech ^ 239Inelastic Acceleration Response Spectra for El Centro EW 1940 (0.21g) ^ 240Inelastic Acceleration Response Spectra for Filtered El Centro EW ^ 240Inelastic Acceleration Response Spectra for Modified Taft ^ 241Inelastic Acceleration Response Spectra for Artificial Richmond ^ 241Inelastic Acceleration Response Spectra for Mexico City CUIP EW 1985 . . . ^ 242Inelastic Acceleration Response Spectra for Mexico City SCT EW 1985 ^ 242Inelastic Displacement Response Curves for Taft S69E 1952 (0.21g) ^ 243Inelastic Displacement Response Curves for Filtered Taft ^  244Figure 4.34Figure 4.35Figure 4.36Figure 4.37Figure 4.38Figure 4.39Figure 4.40Figure 4.41Figure 4.42Figure 5.1Figure 6.1Figure 6.2Figure 6.3Figure 6.4Figure 6.5Figure 6.6Figure 6.7Figure 6.8Figure 6.9Figure 6.10Figure 6.11Figure 6.12Figure 6.13xviiiInelastic Displacement Response Curves for Caltech EW 1971 (0.21g) ^ 245Inelastic Displacement Response Curves for Filtered Caltech ^ 246^Inelastic Displacement Response Curves for El Centro EW 1940 (0.21g) .   247Inelastic Displacement Response Curves for Filtered El Centro EW ^ 248Inelastic Displacement Response Curves for Modified Taft ^ 249Inelastic Displacement Response Curves for Artificial Richmond ^ 250Inelastic Displacement Response Curves for Mexico City CUIP EW 1985 . .^251Inelastic Displacement Response Curves for Mexico City SCT EVV 1985 . .^252Strength and Ductility Demands of Outward-SlopingInelastic Displacement Response ^  253Relationship Between Acceleration Response Spectrum andInelastic Displacement Response ^  254Example of Single-Mass Vibrating System ^  255Strain Energy in (a) Elastic and (b) Inelastic Structures ^  255Equal-Energy Seismic Response of Structures ^  256Results of R-A. Equation for Various Ratios of Spectral Velocities ^ 256Equivalent Elastic Structure ^  257Period Shift as Function of Ductility ^  257Comparison Between Calculated (R-p. equation) and Actual (time-step analyses)R for all Earthquakes with Period Shift only ^  258Comparison Between Calculated (R-i equation) and Actual (time-step analyses)R for Ductility of 4 for Taft S69E 1952 (0.21g) ^  258Modification Factor for the R-A Equation  259Definition of Secant Slope in Velocity Response Spectrum ^ 260Energy Dissipated in Elasto-Plastic Structures ^  260Damping as Function of Ductility for Initial Damping of 0.05 ^ 261Comparison Between Calculated and Actual R for Ductilities of (a) Two,(b) Three, and (c) Four for Taft S69E 1952 (0.21g) ^  262xixFigure 6.14^Comparison Between Calculated and Actual R for Ductilities of (a) Two,(b) Three, and (c) Four for Filtered Taft ^  263Figure 6.15^Comparison Between Calculated and Actual R for Ductilities of (a) Two,(b) Three, and (c) Four for Caltech EW 1971 (0.21g) ^  264Figure 6.16^Comparison Between Calculated and Actual R for Ductilities of (a) Two,(b) Three, and (c) Four for Filtered Caltech ^  265Figure 6.17^Comparison Between Calculated and Actual R for Ductilities of (a) Two,(b) Three, and (c) Four for El Centro EW 1940 (0.21g) ^  266Figure 6.18^Comparison Between Calculated and Actual R for Ductilities of (a) Two,(b) Three, and (c) Four for Filtered El Centro EW^  267Figure 6.19^Comparison Between Calculated and Actual R for Ductilities of (a) Two,(b) Three, and (c) Four for Modified Taft ^  268Figure 6.20^Comparison Between Calculated and Actual R for Ductilities of (a) Two,(b) Three, and (c) Four for Artificial Richmond ^  269Figure 6.21^Comparison Between Calculated and Actual R for Ductilities of (a) Two,(b) Three, and (c) Four for Mexico City CUIP EW 1985 ^  270Figure 6.22^Comparison Between Calculated and Actual R for Ductilities of (a) Two,(b) Three, and (c) Four for Mexico City SCT EW 1985 ^  271Figure 6.23^Comparison Between Calculated and Actual R with Period and DampingShifts for All Study Earthquakes ^  272Figure 6.24^Influence of Spectral Velocity Ratios on Results of R-p. Equation ^ 272Figure 6.25^Comparison Between Calculated and Actual with Iwan Period and DampingShifts for All Study Earthquakes ^  273Figure 6.26^Influence of Spectral Velocity Ratios on Results of R-p. Equationwith Iwan Period and Damping Shifts ^  273Figure 8.1^Code Seismic Response Factor  274Figure 8.2^Code Design Acceleration Response Spectra ^  274Figure 8.3^Effects of Changes in Response Spectrum on Seismic Response ^ 275Figure 8.4^Comparison Between Code Design Spectrum and Acceleration ResponseSpectrum for Artificial Richmond Ground Motion ^  276Figure 8.5^Comparison Between Code Design Spectrum and Acceleration ResponseSpectrum for Mexico City SCT EW 1985 Ground Motion ^ 277Figure 9.1^6-Storey, 3-Bay Study Frame ^  278xx^Figure 9.2^Detailing of Members in R4 Frame ^  279Figure 9.3^Moment-Curvature Relationships for Beam Sections in R4 Frame ^ 280Figure 9.4^Moment-Axial Load Interaction Diagram for Column Sections in R4 Fram0 . ^ 281Figure 9.5^Takeda Model for Stiffness-Degrading Moment-Rotation Relationship ^ 281Figure 9.6^Modelling of Study Frame ^  281Figure 9.7^(a) Accelerogram, (b) Absolute Acceleration Response Spectrum,and (c) Relative Velocity Response Spectrum for El Centro NS 1940Ground Motion ^  282Figure 9.8^(a) Accelerogram, (b) Absolute Acceleration Response Spectrum,and (c) Relative Velocity Response Spectrum for Olympia N86E 1948Ground Motion ^  283Figure 9.9^Displacement Time Histories for R4 Frame at Ductility of 4in Taft 569E 1952 ^  284Figure 9.10^Displacement Time Histories for R4 Frame at Ductility of 4in El Centro NS 1940 ^  284Figure 9.11^Displacement Time Histories for R4 Frame at Ductility of 4in El Centro EW 1940 ^  285Figure 9.12^Displacement Time Histories for R4 Frame at Ductility of 4in Olympia N86E 1948 ^  285Figure 9.13^Displacement Time Histories for R4 Frame at Ductility of 4in Artificial Richmond ^  286Figure 9.14^Displacement Time Histories for R4 Frame at Ductility of 4in Mexico City SCT EW 1985 ^  286Figure 9.15^Locations of Plastic Hinges in R4 Frame at Ductility of 4in Study Earthquakes ^  287Figure 9.16^Moment Time History for Exterior End of First-Floor Beam in R4 Frameat Ductility of 4 in Taft S69E 1952 ^  288Figure 9.17^Moment Time History for Exterior End of First-Floor Beam in R4 Frameat Ductility of 4 in El Centro NS 1940 ^  288Figure 9.18^Moment Time History for Exterior End of First-Floor Beam in R4 Frameat Ductility of 4 in El Centro EW 1944^  289Figure 9.19^Moment Time History for Exterior End of First-Floor Beam in R4 Frameat Ductility of 4 in Olympia N86E 1948 ^  289xxiFigure 10.6Figure 10.7Figure 11.1Figure 11.2Figure 11.3Figure 11.4Moment Time History for Exterior End of First-Floor Beam in R4 Frameat Ductility of 4 in Artificial Richmond ^  290Moment Time History for Exterior End of First-Floor Beam in R4 Frameat Ductility of 4 in Mexico City SCT EW 1985 ^  290Moment-Hinge Rotation Response for Exterior End of First-Floor Beamin R4 Frame at Ductility of 4 in Taft S69E 1952 ^  291Moment-Hinge Rotation Response for Exterior End of First-Floor Beamin R4 Frame at Ductility of 4 in El Centro NS 1940 ^  291Moment-Hinge Rotation Response for Exterior End of First-Floor Beamin R4 Frame at Ductility of 4 in El Centro EVV 1940 ^  292Moment-Hinge Rotation Response for Exterior End of First-Floor Beamin R4 Frame at Ductility of 4 in Olympia N86E 1948 ^  292Moment-Hinge Rotation Response for Exterior End of First-Floor Beamin R4 Frame at Ductility of 4 in Artificial Richmond ^  293Moment-Hinge Rotation Response for Exterior End of First-Floor Beamin R4 Frame at Ductility of 4 in Mexico City SCT EW 1985 ^ 293Detailing of Members in F2 Frame ^  294Moment-Curvature Relationships for Beam Sections in F2 Frame ^ 295Moment-Axial Load Interaction Diagrams for Column Sections in F2 Frame . ^ 295Displacement Time Histories for F2 Frame in Mexico City SCT EW 1985 . . ^ 296Moment Time History for Exterior End of First-Floor Beam in F2 Framein Mexico City SCT EW 1985 ^  296Moment-Hinge Rotation Response for Exterior End of First-Floor Beamin F2 Frame in Mexico City SCT EW 1985 ^  297Locations of Plastic Hinges in F2 Frame in Study Earthquakes ^ 298Study Shear Wall Structure ^  299Detailing of Members in Study Shear Wall Structure ^  300Moment-Curvature Relationships for Shear Wall Sectionsin Study Shear Wall Structure ^  300Moment-Axial Load Interaction Diagram for Columns Sectionsin Study Shear Wall Structure ^  301Figure 9.20Figure 9.21Figure 9.22Figure 9.23Figure 9.24Figure 9.25Figure 9.26Figure 9.27Figure 10.1Figure 10.2Figure 10.3Figure 10.4Figure 10.5Figure 11.5^Modelling of Study Shear Wall Structure ^  301xxiiFigure 11.6^Displacement Time Histories for Study Shear Wall Structureat Ductility of 2 in Taft S69E 1952 ^  302Figure 11.7^Displacement Time Histories for Study Shear Wall Structureat Ductility of 2 in El Centro NS 1940 ^  302Figure 11.8^Displacement Time Histories for Study Shear Wall Structureat Ductility of 2 in El Centro EW 1940^  303Figure 11.9^Displacement Time Histories for Study Shear Wall Structureat Ductility of 2 in Artificial Richmond ^  303Figure 11.10 Displacement Time Histories for Study Shear Wall Structureat Ductility of 2 in Mexico City SCT EW 1985 ^  304Figure 11.11 Base Moment Time History for Study Shear Wall Structureat Ductility of 2 in Taft S69E 1952 ^  304Figure 11.12 Base Moment Time History for Study Shear Wall Structureat Ductility of 2 in El Centro NS 1940 ^  305Figure 11.13 Base Moment Time History for Study Shear Wall Structureat Ductility of 2 in El Centro EW 1940^  305Figure 11.14 Base Moment Time History for Study Shear Wall Structureat Ductility of 2 in Artificial Richmond ^  306Figure 11.15 Base Moment Time History for Study Shear Wall Structureat Ductility of 2 in Mexico City SCT EW 1985 ^  306Figure 11.16 Base Shear Time History for Study Shear Wall Structureat Ductility of 2 in Taft S69E 1952 ^  307Figure 11.17 Base Shear Time History for Study Shear Wall Structureat Ductility of 2 in El Centro NS 1940 ^  307Figure 11.18 Base Shear Time History for Study Shear Wall Structureat Ductility of 2 in El Centro EW 1940^  308Figure 11.19 Base Shear Time History for Study Shear Wall Structureat Ductility of 2 in Artificial Richmond ^  308Figure 11.20 Base Shear Time History for Study Shear Wall Structureat Ductility of 2 in Mexico City SCT EW 1985 ^  309Figure 11.21 Moment-Hinge Rotation Response at Base of Shear Wallat Ductility of 2 in Taft S69E 1952 ^  309Figure 11.22 Moment-Hinge Rotation Response at Base of Shear Wallat Ductility of 2 in El Centro NS 1940 ^  310Figure 11.23Figure 11.24Figure 11.25Figure 12.1Figure 12.2Figure 12.3Figure 12.4Figure 12.5Figure 12.6Figure 13.1Figure 13.2Figure 13.3Figure 13.4Figure 13.5Figure 13.6Figure 13.7Figure 13.8Figure 14.1Figure 15.1Figure 16.1Figure 16.2Figure 16.3Moment-Hinge Rotation Response at Base of Shear Wallat Ductility of 2 in El Centro EW 1940^  310Moment-Hinge Rotation Response at Base of Shear Wallat Ductility of 2 in Artificial Richmond ^  311Moment-Hinge Rotation Response at Base of Shear Wallat Ductility of 2 in Mexico City SCT ENV 1985 ^  311Modelling of Plastic Hinges in Static-Load-to-Collapse Analysis ^ 312Static-Load-to-Collapse Response of R4 Frame ^  313Static-Load-to-Collapse Response of F2 Frame  314Static-Load-to-Collapse Response of Study Shear Wall Structure ^ 315Moments in Design Beam Section in R4 Frame ^  316Definitions for Yield Displacement of Structure  317Response of Inelastic R4 Frame in Study Earthquakes Scaled to 0.21g ^ 318Response of Inelastic R4 Frame in Study Earthquakes Scaled to 0.21 m/s . . . ^ 318Response of Elastic and Inelastic R4 Frame in Study EarthquakesScaled for Ductility of Four ^  319Response of Inelastic F2 Frame in Study Earthquakes ^  319Response of Inelastic Shear Wall Structure in Study EarthquakesScaled to 0.21g ^  320Response of Elastic and Inelastic Shear Wall Structurein Study Earthquakes Scaled for Ductility of Two ^  320Simplified Inelastic Displacement Response Curves for R4 Framein Study Earthquakes Scaled for Ductility of Four ^  321Simplified Inelastic Displacement Response Curves for Shear WallStructure in Study Earthquakes Scaled for Ductility of Two ^ 321Relationship Between Ultimate Tip Displacement and Plastic Hinge Rotation^322Definitions of Accumulated Primary and Secondary Plastic Hinge Rotations^32320-Storey, 3-Bay Study Frame for Storey Shear Analysis ^  324Results of Storey Shear Analysis for R4 Frame ^  325Results of Storey Shear Analysis for 20-Storey Frame  326xxivFigure 17.1^Representative Exterior Beam-Column Subassembly ^  327Figure 17.2^Deflected Shape of Exterior Beam-Column Joint in a) Real Structuresand b) Laboratory Tests ^  327Figure 17.3^Overall Test Set-Up  328Figure 17.4^Bending Moments in Exterior First-Floor Beam of Prototype Framea) Under Gravity Loads only and b) When Beam Ends Yield ^Figure 17.5^Dimensions of Test Specimens ^Figure 17.6^Components of Beam Tip Deflection Figure 17.7^Scope of Experimental Study ^Figure 17.8^Loading Program for Test BC1 Figure 17.9^Loading Program for Test BC2 ^Figure 17.10 Loading Program for Test BC3 Figure 17.11 Loading Program for Test BC4 ^Figure 17.12 Detailing of Specimens BC! to BC3 Figure 17.13Figure 17.14 Locations of Strain Gauges in Specimens ^Detailing of Specimen BC4329330331332333334335336337338339Figure 17.15Figure 17.16Figure 17.17Figure 17.18Figure 17.19Figure 18.1Figure 18.2Figure 18.3Figure 18.4Figure 18.5Figure 18.6a) Instrumentation Around Joint Core of Specimenand b) Parameters Measured ^Reinforcement in Joint Core of Specimens ^Typical Stress-Strain Curve for Reinforcing Steel in Specimens ^Formwork ^Beam of Epoxy-Repaired Specimen BC1 ^Typical Damage Pattern in Specimens Beam Damage in Specimens a) BC2, b) BC3, and c) BC4 ^Propagation of Negative Cracks in Beam of Specimen BC3 Typical Slab Damage Showing Transverse and Longitudinal Cracks ^Typical Torsional Cracks in Transverse Beams ^Typical Upper Column Damage ^340341342343343344345346347348349xxvTypical Lower Column Damage ^  350Typical Damage to Back of Joint Core  351Splitting of Beam-Column Specimen ^  352Deflection of Testing Frame in Tests a) BC2, b) BC3, and c) BC4 ^ 353Beam Tip Load-Global Deflection Relationship for Specimen BC! ^ 354Beam Tip Load-Global Deflection Relationship for Specimen BC2 ^ 355Beam Tip Load-Global Deflection Relationship for Specimen BC3 ^ 356Beam Tip Load-Global Deflection Relationship for Specimen BC4 ^ 357Beam Tip Load-Relative Deflection Relationship for Specimen BC1 ^ 358Beam Tip Load-Relative Deflection Relationship for Specimen BC2 ^ 358Beam Tip Load-Relative Deflection Relationship for Specimen BC3 ^ 359Beam Tip Load-Relative Deflection Relationship for Specimen BC4 ^ 359Computed Moment-Rotation Response of End of Exterior Beam of PrototypeFrame in a) El Centro EW, b) Taft S69E, and c) Mexico City SCT EWGround Motions ^  360Beam Tip Load-Deflection Relationship for Ehsani and Wight Specimen ^ 361Beam Tip Load-Deflection Relationship for Paultre and Mitchell Specimen . . ^ 361a) Entire Response and b) Initial Response of Rotationof Upper Column in Tests BC1 to BC3 ^  362Deflected Shape of Joint Core with Gap at Base of Upper Column ^ 363Actions at Joint Core under Negative Beam Tip Loads ^  364Axial Load in Upper Column in Test BC1 ^  365Axial Load in Upper Column in Test BC2  365Axial Load in Upper Column in Test BC3 ^  366Axial Load in Upper Column in Test BC4  366Effective Slab Width for Specimens ^  367Strains in Slab Reinforcement on Left Side of Specimen BC3 ^ 368Strains in Middle Slab Reinforcing Bar in Specimens a) BC2 and b) BC4 . . . . 369Figure 18.7Figure 18.8Figure 18.9Figure 19.1Figure 19.2Figure 19.3Figure 19.4Figure 19.5Figure 19.6Figure 19.7Figure 19.8Figure 19.9Figure 19.10Figure 19.11Figure 19.12Figure 19.13Figure 19.14Figure 20.1Figure 20.2Figure 20.3Figure 20.4Figure 20.5Figure 20.6Figure 20.7Figure 20.8xxviFigure 20.9Figure 20.10Figure 20.11Figure 20.12Figure 20.13Figure 20.14Figure 20.15Figure 20.16Figure 20.17Figure 20.18Figure 20.19Figure 20.20Figure 20.21Figure 20.22Figure 20.23Figure 20.24Figure 20.25Figure 21.1Figure 21.2Figure 21.3Figure 21.4Figure 21.5Figure 21.6Strains in Outer Slab Reinforcing Bars in Specimens BC2, BC3, and BC4 . . ^ 370Strains in Beam Longitudinal Reinforcement in Specimen BC2^ 371Strains in Beam Longitudinal Reinforcement in Specimen BC3 ^ 372Strains in Beam Longitudinal Reinforcement in Specimen BC4^ 373Measured Moment-Curvature Relationship for Beam Section at Joint Facein Specimen BC3 ^  374Rotation of Main Beam in Specimen BC2 ^  375Rotation of Main Beam in Specimen BC3  376Rotation of Main Beam in Specimen BC4 ^  377Strains in Column Longitudinal Reinforcement in Specimen BC2 ^ 378Strains in Column Longitudinal Reinforcement in Specimen BC3 ^ 379Strains in Column Longitudinal Reinforcement in Specimen BC4 ^ 380Strain Gauges in Joint Core ^  381Strains in Joint Core Transverse Reinforcementin Specimens BC2, BC3, and BC4 ^  382Strains in Embedment Length of Beam Top Longitudinal Reinforcementin Specimens BC2, BC3, and BC4 ^  383Strains in Embedment Length of Beam Bottom Longitudinal Reinforcementin Specimen BC3 ^  384Strains in Hooks of Beam Longitudinal Reinforcementin Specimens BC2, BC3, and BC4 ^  385Strains in Hook of Beam Longitudinal ReinforcementUnder Initial Loadings in Specimens BC3 and BC4 ^  386Loading Program for Test BC5 ^  387Beam Tip Load-Global Deflection Relationship for Specimen BC5 ^ 388Beam Tip Load-Relative Deflection Relationship for Specimen BC5 ^ 389Deflection of Testing Frame in Test BC5 ^  389Rotation of Upper Column a) Before and b) After Bang in Test BC5 ^ 390Strains in Beam Longitudinal Reinforcement in Specimen BC5^ 391Figure 21.7Figure 21.8Figure 21.9Figure 21.10Figure 21.11Figure 21.12Figure 21.13Figure 21.14Figure 21.15Figure A.1Figure B.1Figure C.1Figure C.2Figure C.3Figure C.4xxviiStrains in Column Longitudinal Reinforcement in Specimen BC5 ^ 392Strains in Joint Core Transverse Reinforcement in Specimen BC5 ^ 393Strains in Embedment Length of Beam Bottom Longitudinal Reinforcementin Specimen BC5 ^  393a) Entire Response and b) Initial Response of Strains in Hook of BeamLongitudinal Reinforcement in Specimen BC5 ^  394Joint Core Shear Deformation in Specimen BC5  395Damage to Beam of Specimen BC5 ^  396Back of Specimen BC5 a) Before and b) After the Bang^  397Rear View of Specimen BC5 without Back Concrete Cover  398Side View of Specimen BC5 in Last Cycle of Loading ^  399Quasi-Static Seismic Loads for Design of R4 Frame  409Quasi-Static Seismic Loads for Design of Study Shear Wall Structure ^ 412a) Beam and b) Back of Specimen BC3-Z ^  416Beam Tip Load-Global Deflection Relationship for Specimen BC3-Z ^ 417Damage to Main Beam of Specimens BC1 and BC3-Z ^  418Initial Joint Shear Deformations in Tests BC3 and BC3-Z  419AcknowledgementMany people have contributed to this research and their assistance is greatly appreciated.I would like to acknowledge the funding provided by Natural Sciences and Engineering ResearchCouncil of Canada for this project.I appreciate very much the help from Bernie Merkli, without whom my experiments would stillbe on the drawing board; Tamunoiyala Koko, who endured the boring task of recording test data; andBryan Katzensteiner, who is a good partner in tackling projects, venting gripes, and sharing laughs.I would like to thank Dr. Don Anderson who kindly provided, without hesitation, financialsupport, supervision, and critical reviews in the final one and a half years of my study.Very special thanks go to Dr. Noel Nathan for giving me valuable guidance, support, andknowledge; for having patience with and trust in me for all these years; and for being a great teacher,mentor, and friend to me, even after he retired!Most of all, I am forever grateful to my parents, brothers, and sister, who all did not complainone bit about my continuous education at the university. Especially, I thank Mom and Dad for all theirlove and sacrifices; Troy, for lending me his computer; George, for letting me use the car all the time;and Caroline, for allowing me to move upstairs.Finally, I would like to thank Jenny for making my life "bearable".To my daddy and mommy,whose love, care, and supportare immeasurable and unmatchable.1Chapter 1Introduction1.1 EarthquakesEarthquake ground motions are usually measured by sensitive instruments which record theacceleration of the ground. The time histories of the accelerations, called accelerograms, provide valuableinformation about the ground motion, such as maximum acceleration, duration of strong shaking, andpredominant period of vibration. These parameters are useful for many engineering studies. Examplesof recorded acceleration time histories are shown in Figures 1.1a and 1.2a. These records provide agraphical representation of one component of the ground motion at a point; they are unique because eachearthquake has its own set of characteristics. No two earthquakes are exactly the same since there aremany parameters and control variables involved.Another method of displaying the characteristics of an earthquake is the response spectrum. Itis a plot of the maximum earthquake response attained by vibrating systems of various periods anddamping. Response spectra of system acceleration, velocity and displacement are normally useful indesign. Examples of acceleration and velocity response spectra are shown in Figures 1.1 and 1.2. Sincethe vibration period of a rigid system is zero, its acceleration for this period will be equal to the maximumacceleration of the earthquake. When the natural frequency of a structure is similar to the predominantfrequency of the ground shaking, a quasi-resonance situation will develop, leading to a peak in theresponse spectrum. Therefore, an acceleration response spectrum is very useful in demonstrating clearlythe maximum acceleration and the predominant period of vibration of a ground motion.Earthquakes can be classified into two main groups and four subgroups as listed in Table 1.1.The descriptions used in this classification are reflected in the characteristics of their recorded groundmotions, or of their associated accelerograms. Available records and observations indicate that vibrations2close to the epicentre are usually of the Group I type and that the Group II, or harmonic, ground motionsare the results of linear filtering, by distance or local soil conditions, of Group I seismic waves.1.2 Filtered EarthquakesAs the seismic waves move away from the focus, their nature is altered by the geology andtopography through which they pass. Filtering is the screening of energy over a certain band ofoscillation frequencies in the seismic waves. The energy decays due to kinetic action and damping in therocks. The amount and nature of the decrease depends on the properties of the earthquake and thegeology, and it has been shown that the energy decays exponentially with the product of the epicentraldistance and vibration frequency [Bolt 1973]. Therefore, the attenuation of high frequency waves is fasterthan that of low frequency ones. This means that ground motion velocity attenuates more slowly withdistance than acceleration. The consequences of filtering over a relatively large epicentral distance, then,are lower energy level, lower maximum acceleration, and more periodic behaviour in the oscillations.Furthermore, the periodic nature would lengthen the duration of the strongest portion of the groundmotion.The preceding remarks refer to the basic motion of the bedrock; at a particular site, the motionis then propagated upward through the local soil profile. In this case, the characteristics of the earthquakeare altered by the presence of layers of soft soils. Past observations have indicated that the amplitude ofvibration on soft soil is larger than that on hard ground in the same earthquake [Davis, Reitan, andPestrong 19761. The filtering process can be pictured simply as shear waves propagating from thebedrock to the free ground surface through horizontal soil layers which act like a multi-storey structurehaving a specific stiffness and damping. The mass of soil vibrates to produce a motion at the free surfacethat, relative to the bedrock, has the peak ground acceleration amplified, the predominant period ofvibration shifted to a higher value, and the duration of strong shaking lengthened; these are the keycharacteristics of a ground motion filtered by soft soils. Similar to the situation of filtering by distance,3this filtered free surface motion also has a lower peak acceleration-to-velocity ratio and a more prominentperiodic behaviour; but it often has increased maximum acceleration relative to the bedrock motion.In this thesis, ground motions which have been filtered through soft soil will be studied. Fromhere on, the term "filtered earthquake" or "filtered ground motion" shall mean ground shaking that hasbeen altered by local soft soil conditions or by distance.1.3 Earthquake Parameters Having Influence on Structural ResponseThe parameters which are known to have major effects on the seismic response of structures arelisted in Table 1.2. It should be noted that all the key characteristics affected by filtering of groundmotions --- amplified acceleration, long duration, and lower frequency --- are included in the list.The forces in the structure are, of course, proportional to the acceleration of the masses so thatit is natural to expect the ground acceleration to play an important role. Therefore, the peak groundacceleration is a simple guide for damage potential, and indeed it is a very popular factor used in seismicdesign of structures.However, analytical studies and field observations have consistently indicated that peak groundacceleration correlates very poorly with actual structural response and damage and so may not be a goodmeasure of the ground motion for design purposes and for seismic risk mapping [Hall, McCabe, andZahrah 1984; Applied Technology Council 1982; Sarma and Yang 1987; Housner 1975]. Also, it is verydifficult to predict correctly the maximum acceleration which a future earthquake will attain. Themaximum acceleration usually occurs in large, high-frequency pulses to which most structures do not havetime to respond fully; tall structures having periods longer than 0.5 second tend to respond in relationto velocity rather than acceleration. Characterizing ground motions by the peak acceleration implies thatother relevant parameters are similar, which is not the case. Nevertheless, the single value of the peakground acceleration provides a simple indicator for engineering purposes.4The effect of duration on structures is not as direct as acceleration. It has been shown that, whentwo ground motions have the same maximum amplitude, the one with the longer duration has strongerdestructive potential against structures [Nagahashi 1980]. The manner in which duration causes greaterdamage is by increasing the number of hysteresis cycles in the structural response and thus imposing ademand for large energy-absorption capacity on the structure [Hall, McCabe, and Zahrah 1984]. Also,in the case of quasi-resonant vibration of a structure, the longer duration of ground shaking would allowthe resonant effects to build up for increased response.Since the peak ground acceleration alone is a poor indicator of damage potential and the durationof the ground motion can be significant, it has been suggested [Applied Technology Council 1982] thata more representative measure of earthquake severity would include the peak ground acceleration or otheramplitude value and at least one of the following parameters: frequency content, duration of strongshaking, and sustained level of vibration. Unfortunately, there is still no single established definition forthe duration of strong shaking of a ground motion. Although many researchers have come up withdifferent criteria for it [Bolt 1973; Housner 1975; Vanmarcke and Lai 1980; McCann and Shah 1979;Shahabi and Mostaghel 1984], the complexity of earthquake vibrations creates great difficulty indetermining the exact segment of a ground motion responsible for causing damage. The number of cyclessustained above a specific amplitude may be a better measure of the ground motion severity since it alsoincorporates, in a way, the duration. This characteristic is reflected later on as the number of yieldreversals in the response of a structure. The number of yield reversals is defined as the number of timesa structure yields in opposite directions consecutively; the number of yield excursions points out howmany times a structure is in the yield state, and its value has been shown to increase with increasingduration [Hall, McCabe, and Zahrah 1984]. So, the duration and the sustained level of strong shakingare two significant ground motion parameters affecting structural damage potential. However, theirinclusion in seismic design procedure has so far only been suggested and has not yet been accepted widelyin practice.5In short, each of the earthquake characteristics of amplitude, duration, and frequency is veryinfluential with respect to the response of and the damage in a structure. Since all three characteristicscan be made worse in filtered earthquakes, concerns for the safety of structures under this type of motionnaturally arise and are intensified by actual experience of filtered ground motions in the past several years.1.4 Past Occurrences of Filtered EarthquakesIn recent years, the damaging effects of filtered earthquakes were revealed by two seismic eventsin which tremendous amounts of damage occurred in structures founded on soft soil. The first is the 1985Mexico earthquake and the second is the 1989 Loma Prieta earthquake in California.1.4.1 Mexico Earthquake 1985On September 19, 1985 at 1:17 PM, a strong earthquake, which was measured at magnitude 8.1on the Richter scale, occurred in the Michoacan area on the west coast of Mexico. The location of theepicentre is shown in Figure 1.3. The focus was at a depth of 18 km in a fault where the Cocos Platesubducts under the North American Plate. The length and width of the fault rupture were estimated tobe 170 km and 50 km, respectively [Anderson et al. 19871.The most interesting aspect of this earthquake is the different types of ground motion observedat various locations. Near the epicentre, a moderately strong vibration was recorded in the town of LaVillita. It had a maximum acceleration of 0.12g and a recorded duration of about 60 seconds (see Figure1.4). No prevailing period of vibration was present. In Mexico City at a distance of 350 km from theepicentre, the ground motion was completely different. At the UNAM station in the hill zone of the city,a relatively weaker shaking was observed. The maximum acceleration was 0.04 g only, but the durationhad been lengthened to 180 seconds, and an oscillation period of two seconds predominated in thevibration. These characteristics are the direct results of filtering of the seismic waves over the large6epicentral distance. At station SCT in the lake zone of the city, however, a much stronger ground motionoccurred because of the local soil conditions. The accelerograms for La Villita, UNAM, and SCT areshown in Figure 1.4.Mexico City is located in the mountainous interior region of the country. Part of the city issituated on an old lake bed composed of mixed sand, clay, and silt. These soft soils have high watercontent (100-400%), low undrained shear strength (25 to 78 kPa), and low shear wave velocity (40 m/s)[Romo and Seed 1987; Whitman 1987]. Most importantly, the soil layers are able to exhibit nearly linearbehaviour in response to large dynamic strains [Whitman 19871. As shown in Figure 1.5, the city isdivided into three major areas reflecting the different soil conditions: hill zone, transition zone, and lakezone. The soft clayey soil had already demonstrated its capability of magnifying bedrock motions fromtwo earthquakes in 1957 and 1979 [Kobayashi et al. 1987].In the 1985 event, the seismic waves arrived at Mexico City with a predominant period ofoscillation of two seconds. The inherent natural period of the soil layers in some regions is also twoseconds. Combined with further filtering through the soil layers, a periodic ground motion having a peakacceleration of 0.16g, a pronounced predominant period of 2.0 seconds, and a long overall duration ofover 180 seconds with about 45 seconds of strong shaking was produced at the station SCT. Themaximum amplitude of ground movement at SCT was about 400 mm The accelerogram along the east-west direction at SCT is drawn in Figure 1.6 and its response spectra are plotted in Figure 1.7; it maybe seen that the peak value of acceleration response was about 1.0g.As a result of the soft-soil filtering, the damage suffered in Mexico City was excessive. The totaldamage was estimated at $4 billion US [Sozen and Lopez 1987]. The hardest-hit area was the inner coreof the city where the soil thickness is between 30 and 48 meters [Romo and Seed 1987]. Of the 53,368structures in this region, 757 (or 14%) were damaged, including 133 collapsed, 33 partially collapsed,and 271 severely damaged [Borja-Navarette et al. 1987]. A large number of failures occurred in buildings6 to 15 storeys high and in frame-type structures [Borja-Navarette et al. 1987]. The fundamental periodof affected structures range from 0.5 to 2.0 seconds and the damping coefficient was about 5% [Kobayashi7et al. 1987]. In addition, reinforced concrete structures were the most severely affected. The dominantmodes of failure include shear or eccentric compression failure of columns, shear cracking of beams andribs of waffle slabs, punching failure, failure of beam-column joints, foundation rotation and settlements,and pounding [Meli 1987]. Surprisingly, there were not many instances of flexural hinging at beam ends[Meli 1987]. The patterns of damage distribution were very similar for the earthquakes in 1985, 1979,and 1959. No liquefaction was reported.1.4.2 Loma Prieta Earthquake 1989On October 17, 1989 at 5:04 PM, a 45-km section of the San Andreas fault slipped in the LomaPrieta region just north of Santa Cruz (see Figure 1.8). The resulting strong earthquake registered 7.0on the Richter scale. Strong ground movements were felt in a banded region stretching northwest andsoutheast along the fault.Like the 1985 Mexico earthquake, local site effects played a significant role in causing damagein this earthquake. In the epicentral region, large peak vertical accelerations in addition to largehorizontal ground surface accelerations were recorded. But both components of acceleration attenuatedquickly with distance; values recorded at various stations in the affected region are shown in Figure 1.8.However, several sites in the San Francisco Bay area at a distance of 50 to 100 km from the rupture zonehad relatively larger horizontal accelerations than other sites at the same distance. A study of the geologyof the Bay Area reveals that these high accelerations occurred in zones containing soft soils. This indicatesclearly the results of amplification of the bedrock acceleration by the local soft soils.In broad terms, the geology of the Bay Area can be categorized into 3 units: bedrock, alluvium,and Bay mud. The alluvium is made up of silty clay, sand, and gravel, and has a moisture content of lessthan 40%. The shear wave velocity at the surface of this soil reaches about 200 m/s. The Bay mudconsists mostly of recent deposits (8000 years and younger) of soft plastic carbonaceous clay, silt, andminor sand intrusions. The material is very loose and its moisture content can exceed 50%. The deposit8thickness in some areas is about 40 m and the shear wave velocity in the mud ranges from 90 to 130 m/s[Rainer et al. 1990].In the central San Francisco region, peak ground horizontal accelerations on rock were about0.06g to 0.12g; however, these accelerations were amplified to about 0.16 to 0.33g on soft, Bayshoresites, giving an amplification factor of about two to three. An excellent illustration of the influence ofthe Bay mud on ground shaking characteristics is provided by a comparison of the motions recorded ata station founded on rock on Yerba Buena Island and a station founded on soft soil on Treasure Island.The acceleration time histories recorded at these two stations and their response spectra are shown inFigure 1.9 in which the motion at Yerba Buena Island is considered to be the bedrock motion underTreasure Island. Both islands are located in San Francisco Bay and are about 70 kilometres from theepicentre.The larger ground surface acceleration along with a shift in the frequency of vibration causedextensive damage in the soft-soil zones. They were responsible for many collapses of structures such asthe 3- and 4- storey apartment buildings in the Marina District in San Francisco, the upperdeck of theelevated Highway 880 in Alameda, and the upperdeck of the Bay Bridge. Structures which were severelydamaged by the earthquake were unreinforced masonry structures, and buildings that had a "soft storey"on the ground floor. Only a few older reinforced concrete structures in the epicentral region weredamaged. New structures and structures located on firm ground suffered very little damage [EERC 1989].Unlike the 1985 Mexico City earthquake, liquefaction was widespread in this event, and much of thestructural damage was actually liquefaction related.Even though the heavily damaged areas make up less than one percent of the total area affectedby the earthquake, it was estimated that more than 105000 homes, 500 apartment buildings, and 3500businesses had been damaged by the earthquake [Seed et al. 1990]. Damage directly attributable to theearthquake amounted to nearly ten billion dollars, making it the most costly single natural disaster inUnited States history.91.5 Fraser River Delta and Richmond, B.C.The earthquakes in Mexico City and San Francisco have revealed the destructive nature of filteredearthquakes. Consequently, one might be concerned about the possibility of this type of ground motionoccurring again elsewhere in a region which is known to be capable of filtering seismic waves. One suchregion is the Fraser River delta, within which lies the municipality of Richmond, in the province of BritishColumbia in Canada.Located adjacent to the city of Vancouver, the municipality of Richmond is founded on the largestisland in the delta of the Fraser River. The composition of this island is mainly layers of alluvial depositsof loose sand and clay underlain by till and bedrock. Studies of the soil have indicated its ability to filterearthquake waves and its susceptibility to liquefaction [Wallis 1979; Byrne and Anderson 1983, 1987].Unfortunately, Richmond is located in a seismically active part of the country. Strong earthquakes ofmagnitude in the order of 7 to 8 have occurred in the past and an earthquake of this magnitude, or evenhigher, can be expected to hit the area again [Rogers 1988].1.6 PurposeWith its rapid urbanization and the special properties of its underlying soils, the municipality ofRichmond may suffer substantial damage during a strong earthquake. The tragic observations made inthe 1985 Mexico City and 1989 Loma Prieta earthquakes, and a concern for the behaviour of engineeringstructures in Richmond, form the motive for this research.The purpose of this research is to understand better the behaviour of structures in filteredearthquakes, in particular the response of reinforced concrete structures subjected to high levels ofsustained, strong ground shaking.101.7 ScopeIn this research, the filtering effects of the Richmond soils to generate strong surface groundmotions will be examined first. Then, the influence of the filtered earthquakes on the ductilityrequirements for single-degree-of-freedom systems will be studied. To compliment the studies on thesingle-degree-of-freedom systems, analytical studies on 6-storey reinforced concrete frames and 8-storeyreinforced concrete shear wall structures subjected to typical and filtered earthquakes will also beperformed. The results from the analytical studies on the response of frame structures will then be appliedto cyclic-loading tests on large-scale specimens of reinforced concrete exterior beam-column joints.1.8 ObjectivesIt is hoped that this thesis will improve understanding of the following:1) possibility of a severe filtered earthquake occurring in Richmond2) response of structures located in Richmond to the design earthquake3) ductility demands of filtered earthquakes4) response of reinforced concrete frames in filtered earthquakes5) behaviour of reinforced concrete structures designed according to Canadian codes6) extent of damage in beam-column joints during filtered earthquakes7)^behaviour of exterior reinforced concrete beam-column joints with slab and transversebeams11Chapter 2Filtered Earthquakes in the Fraser River Delta2.1 Geography of Fraser River DeltaThe Fraser River Delta is located in the southwest corner of the province of British Columbia inCanada. It covers a plan area of about 350 sq. km . adjacent to the city of Vancouver. The Fraser Riversplits into two main channels and four secondary channels before reaching the Strait of Georgia. Anumber of islands were formed among these channels and the two largest ones --- Lulu Island and SeaIsland --- make up the municipality of Richmond (see Figure 2.1). The municipality is very flat withnearly three-quarters of the total land area only 4 to 6.5 feet above sea level [Blunden 1975]. To provideprotection against flooding from the surging river and sea, dykes were built around this municipalitywhich is now a fast-growing suburb with high concentrations of residential, commercial, and industrialbuildings.2.2 Geology of Fraser River DeltaThe Fraser River delta began to form westward when ice retreated from the Vancouver area about8000 years ago [Blunden 1975]. The river has since been depositing large amounts of sand, silt, and clay.As a result of slow subsidence and plant growth and decay, most of the upstream part of the delta hasturned into peat bogs, the surfaces of which are free of sand. The thickness of the peat layer reaches 8meters in some areas. Figure 2.2 shows the extent of the peat and the alluvial deposits in the delta.The alluvial deposit is generally very deep with an average thickness of about 250 meters abovethe bedrock. A north-south section near the western edge of the delta is shown in Figure 2.3. Typically,the Richmond soil deposit is comprised of four layers [Byrne, Yan, and Lee 1990]:121) a thin discontinuous layer of surficial deposits of clays, silts, and peats up to 8 m inthickness, underlain by2) a sand and silty sand stratum 20 to 45 m in thickness, underlain by3) a silt-clay stratum 100 to 300 m in thickness, underlain by4) a glacial till stratum 90 to 600 m in thickness, underlain by5)^bedrock.Three profiles along the north-south cross section are also drawn in Figure 2.3. The water table is usuallyvery close to the top surface. The soils are very loose materials with low N values obtained from conepenetration tests. Compared to the soft clay in Mexico City, the Fraser River delta soils have a lowerwater content and a higher shear wave velocity.2.3 Seismicity of Fraser River DeltaAs part of the west coast of Canada, the Fraser River delta lies in an area of strong seismicity.The seismicity of British Columbia is shown in Figure 2.4. Two zones of high concentration ofearthquakes near the delta can be clearly seen. These areas are the Georgia Strait/Puget Sound zone tothe south and the offshore belt off the west coast of Vancouver Island to the west. The causes ofearthquakes in these zones can be explained by plate tectonics along the coast of Canada and the UnitedStates. Located off this coast is the boundary between the continental America plate and the oceanicPacific plate (see Figure 2.5). It runs from northern California to about the 60-degree parallel in Alaska.However, south of the Queen Charlottes Islands in British Columbia, the two large plates are separatedby three very small oceanic plates. The movements of these small plates are responsible for creating thetwo earthquake source zones. But the causes of seismic events in these two zones are different.In the Puget Sound area, the earthquakes are the direct results of the Juan de Fuca platesubducting the America plate. The subduction process builds up strains in the overlying plate at anaverage depth of about 25 km below the Puget Sound. Release of this strain creates earthquakes; the13largest earthquake recorded in this source zone is the 1949 Olympia earthquake registered at 7.0 on theRichter scale.The causes of earthquakes in the zone off the coast of Vancouver Island are, on the other hand,spreading of the seafloor and sliding of the large Pacific and America plates against each other. It hasbeen noted that the subduction of the Explorer plate may have stopped [Rogers 1988]. So almost all theearthquake epicentres lie along the Explorer Ridge separating the Pacific and Explorer plates and alongthe sliding boundary between the Pacific and America plates, as can be seen in Figure 2.4. However,some subduction earthquakes can still occur, and the low angle of subduction extends the earthquake zoneeastward across Vancouver Island. A large earthquake of magnitude 7.4 occurred in 1946 on the east sideof the island in the vicinity of the town of Courtenay.Even though the two source zones are very active seismically, there is an absence of thrustearthquakes on the subduction interface, in historic time at least. But it has been argued by Rogers [1988]that a large thrust earthquake is possible in western Canada and may occur in the very near future. Heconcluded that an earthquake of magnitude 9.1 with a return period of 400 years may occur in the Juande Fuca plate and an earthquake of magnitude 8.7 with a return period of only 91 years may occur in theExplorer plate. In his study, he cited as an example the Mexico 1985 event in which the rupture mayhave been the first in at least 200 years in the relatively inactive region of the subduction zone.There also has been no occurrence of strong earthquakes in the Fraser River Delta in recordedhistory. The largest earthquake which had any effects on the Richmond soil was the 1946 Courtenayearthquake during which "runways at Vancouver Airport rolled in waves" [Blunden 1975]. However, nodamage was reported. In 1971, seismographs were triggered by an earthquake centred at Pender Islandbut again the surface ground motions were not intense. However, evidence of the occurrence of a strongearthquake in Richmond in the past was recently found in the soil layers at an excavation site in thatmunicipality [Naesgaard, Sy, and Clague 1992], where several vertical sand dykes rising through the siltyclay were observed. These are signs of liquefaction during an earthquake.14Using statistical analyses of earthquakes that have occurred in and around Canada, andconsidering the geology and tectonics of the various regions, the National Building Code of Canada[National Research Council of Canada 1985, 1990] has established seismic zones across the entirecountry. The zones are characterized by the peak ground acceleration and the peak ground velocity thathave a ten percent chance of being exceeded in fifty years or a probability of 0.0021 of being exceededon an annual basis. For the Vancouver area, the "design" peak ground acceleration is 0.21g. This valuecorresponds to a magnitude of seven for a probable earthquake originating from the Georgia Strait/PugetSound source zone and is characteristic of the expected bedrock motion underneath Richmond.2.4 Review of Seismic Studies of Fraser River Delta SoilsWith the presence of loose soils and the high probability of earthquakes, the response of the soildeposits in the Fraser River delta during an earthquake is of interest and a concern to engineers. Theconcern includes liquefaction and filtering of seismic waves, but only the effects of filtering will beconsidered in this thesis.Dynamic analyses of the Richmond soils to study filtering effects have been carried out over manyyears. In 1979, Wallis performed computer analyses to investigate potential seismic ground motions inthe Fraser River delta. Using design earthquakes as input bedrock motions, he showed that the periodof vibration would be shifted to a higher value after the seismic waves passed through the soil deposits.But the maximum acceleration was amplified for small-magnitude bedrock motions only; for large-magnitude bedrock motions, the maximum acceleration at the top decreased! Similar results wereobtained by Byrne and Anderson in their report to the municipality of Richmond in 1983 [Byrne andAnderson 1983]. In addition, both Wallis and Byrne and Anderson found that liquefaction is possible inthe loose saturated sand layer to a depth of about ten meters. With new soil data and a revised buildingcode, Byrne and Anderson updated their report in 1987, but the results on the amplification of the peak15acceleration were similar to previous findings. Their results are duplicated in Figure 2.6. They alsoshowed that much of the high amplification occurs in the top silt layer of three meters in thickness.The results in Figure 2.6 indicate the difference in the behaviour of the Richmond soil and theMexico City soil; amplification in Mexico City was considerably higher. However, one important factorin the event in Mexico City was not considered in the above analyses on the Richmond soils, namely thefiltering of the bedrock motion (by distance) so that it had a predominant period approximately equal tothe natural period of the soft overlying layer. This was a major factor in the severe damage suffered inMexico City. This point was subsequently considered in a study by Finn and Nichols [1988]. Using thesame computer program and a similar soil profile as in the Blyne and Anderson's study, they inputvarious scaled acceleration records as bedrock vibrations; in particular, one motion was obtained fromscaling the Pasadena 1971 record to a low amplitude of 0.035g and to a predominant period of 2.5seconds. The resulting surface motions were expressed in the form of acceleration response spectra andthese are shown in Figure 2.7. The results show that a severe surface ground motion similar to that whichoccurred on the clay deposits in Mexico City can occur in Richmond. This severe vibration is the endproduct of filtering the scaled Pasadena record.After the Finn and Nichols' study, the 1989 Loma Prieta earthquake occurred and provided newinformation on the amplification of soft soils. The main observation of that event is that highamplifications were obtained even with moderately high accelerations in the bedrock. Based on thisobservation, a new amplification curve for soft soils was derived by Idriss [1990] and this is shown inFigure 2.8. With the new soil data, several groups of researchers have repeated the analytical studies onthe Richmond soils [Lo et al. 1991; Sy et al. 1991; Byrne, Yan, and Lee 1991]. They found that abedrock motion of 0.2g would be amplified to a surface motion of 0.3g, a factor of 1.5. This is inagreement with the new amplification curve. Their findings have been included in a guideline drafted bya special task force to deal with seismic design in the Fraser River delta [ Byrne et al. 1991].162.5 Dynamic Analyses of Fraser River Delta SoilsThe concern for a severe filtered ground motion in Richmond is justified by Finn and Nichols'results in which a ground motion similar to that in Mexico City was obtained. With the availability ofnew soil data following the 1989 Loma Ptieta earthquake, the analysis for the Richmond soils should berepeated. Also, the Fraser River delta lacks a historical record of strong ground motions. Therefore,more investigations of strong motion for Richmond would be beneficial for further studies in this thesis.To obtain a set of strong motions records for Richmond, various earthquake records were inputas base motions to a typical set of soil layers to obtain the resulting motions at the free surface on top.2.5.1 Computer Program SHAKEThe computer program SHAKE [Schnabel et al. 1972] developed at the University of Californiain Berkeley was used for the analysis. This is the same program that was used by the researchersmentioned earlier. In this program, the soil deposits are idealized as perfectly horizontal, two-dimensionalsoil layers with specified properties such as damping and shear modulus for each layer. An accelerogramis input as bedrock motion and the vibration at the top of any layer can be determined based on shearwave analysis.2.5.2 Modelling of Richmond SoilsThe soil profile used in the program to represent a typical Fraser River delta soil is shown inFigure 2.9. It is actually a representative profile for Sea Island, which is one of the islands in Richmondand is the site of the Vancouver International Airport. The variations of maximum shear modulus anddamping with depth are also shown in the figure.17A Fourier analysis performed by the SHAKE program on this soil profile gives the natural periodof the soil deposit as 2.9 seconds. The same period was obtained by Sy et al. [1991] for a similar profile.2.5.3 Bedrock MotionsFour bedrock motions were used to study the effects of filtering through the Richmond soils. Thefirst three are the Taft S69E 1952, the Caltech EW 1971, and the El Centro EW 1940 ground motions.All records were scaled to give a maximum acceleration of 0.21g, the level of earthquake expected tooccur in the Fraser River delta. The scaled accelerograms are shown in Figures 2.10a, 2.11a, and 2.12a.These records have been used by many researchers for seismic design and analyses of structures inVancouver and the Fraser River delta [Byrne and Anderson 1987; Taylor et al. 1983; Khalil and Bush1987; Finn and Nichols 1988] because of the close match between the characteristics of these motions andof the code design earthquake.The fourth motion used is also the Taft S69E 1952 record but it was scaled down to 0.04g andalso scaled to give a higher predominant period of 2.9 seconds, the inherent natural period of therepresentative soil deposit as modelled, by increasing the time interval in the acceleration record. Theresulting bedrock motion comes out to be about six times longer in duration than the original, unscaledTaft motion. This scaling procedure follows that of Finn and Nichols in their analysis. The method ofscaling the time domain of a recorded accelerogram for input bedrock motion has also been used by Romoand Seed [1987] in their study of Mexico City ground motions. The maximum acceleration of 0.04g wasselected for three reasons. First, this is exactly the maximum ground acceleration obtained in the hillzone in Mexico City. Thus the results can be compared to the event in Mexico City. Second, based onattenuation studies done by Hasegawa et al. [1981] for western Canada, the peak ground acceleration of0.04g corresponds to an epicentral distance of 500 km. This epicentral distance would place the sourcezone around Queen Charlotte Islands, where an active fault exists and where earthquakes of magnitudegreater than 7 are likely to occur. Third, the lower bound value for the peak ground acceleration in zone181, the lowest seismic zone in the code, is taken to be 0.04g. So, this scaled input bedrock motion canbe thought of as the attenuated waves of a large magnitude earthquake centred some 500 km away. Thisis similar to the situation in Mexico City. The accelerogram for this low-amplitude vibration, which shallbe called the "Modified Taft" motion, is shown in Figure 2.13a.2.5.4 Surface Ground MotionsEach of the four vibrations was input as a bedrock motion into the SHAKE program and theresulting surface ground motions are those shown in Figures 2.10 to 2.13. Actually, these vibrations wereobtained at the bottom of the surface clay crust at a depth of three meters. Byrne and Anderson [1987]showed that this top crust can exaggerate the amplification of the acceleration, and so the present studyexamines the vibrations at the bottom of this layer. This procedure is still reasonable because thefoundation level of a structure can be considered to be located at this elevation.In the accelerograms in Figures 2.10 to 2.13, a change in the frequency content of eachearthquake is very apparent; the oscillation frequency can be seen to be much lower in the surfacemotions. For a clearer examination of the effects of filtering, the acceleration and velocity responsespectra of the bedrock and surface motions were determined and are shown in Figures 2.14 to 2.17. Thespectra are for a damping of five percent of critical. For the three "design" earthquakes, amplificationfactors of 1.18 to 1.45 for the peak ground acceleration at zero period are obtained, and the shift in thepeak response period varies from a factor of 1.20 to 1.89. As for the low-level bedrock shaking in theModified Taft motion, the soil deposit was sent into resonance, producing a response spectrum similarto that of the Mexico City record and to that obtained by Finn and Nichols. A high accelerationamplification factor of 2.68 is obtained for this motion. Table 2.1 lists the peak ground accelerations andthe predominant periods of the bedrock and surface vibrations. The predominant period of each groundmotion is considered to be the period at which the largest peak in the elastic acceleration responsespectrum occurs.19For comparison purposes, the response spectra for the strong surface motion in resonance inRichmond, the Mexico City SCT EW record, the Taft S69E 1952 record (0.21g), and the El Centro EW1940 record (0.21g) are plotted together in Figure 2.18. As expected, the spectral response is lower forRichmond than for Mexico City SCT. The Richmond soil with its higher content of sand and lower watercontent cannot amplify the bedrock ground motion as much as the clayey, high-water content/low dampingsoil in Mexico City.The surface ground motion in resonance will be used extensively in the remainder of this researchto study the effects of filtered earthquakes. It will be designated as the "Artificial Richmond" earthquake.By no means is this motion the definite ground vibration to be experienced in Richmond. It is simply astrong motion that can occur in the area; it is a possible one, not the most probable, the most severe, northe design one.2.6 RemarksFrom the SHAKE analyses on the typical Richmond soil profile, it can be concluded thatamplification of acceleration and shift in predominant period can occur in the Fraser River delta. For acode design earthquake, the increases in the acceleration and the predominant period are close to fiftypercent. However, the acceleration amplification is higher as the magnitude of the bedrock motiondecreases. When the low-magnitude vibration is able to resonate the soil deposit, a surface motion similarto that in Mexico City in 1985 could occur in the Fraser River delta. Thus, the damage potential of thisfiltered earthquake may be high and the predominant period of the incoming seismic waves or bedrockmotion is therefore an important factor in determining the severity of filtering through soft soils.20Part IAnalytical Studies of Simple Elasto-Plastic SystemsBefore the effects of filtered earthquakes on reinforced concrete structures are studied, theresponse of single-degree-of-freedom systems to this type of ground motion is examined first. In thisexamination, single-degree-of-freedom (SDOF) systems with elastic-perfectly plastic load-deflectionbehaviour will be subjected to several filtered and unfiltered earthquakes in dynamic, inelastic analyses.Then comparisons between the demands of the various ground motions on the response of the systems willbe made. This investigation will provide a general picture of the behaviour of structures in filteredearthquakes as well as in the typical, unfiltered motions; the SDOF systems are used here as a generalmodel of any structure, since structural response is generally dominated by the first mode. The resultsexhibited by the SDOF systems will be useful for and applicable to the analytical studies on multi-storeys,multi-degrees-of-freedom reinforced concrete structures later in this thesis.The results of this investigation will be analyzed using the maximum displacement response ofthe systems, or specifically, the amounts of post-yield ductility demanded in the earthquakes. The useof ductility to indicate the extent of damage suffered in a structural member is popular [Banon, Biggs, andIrvine 1981]. This idea can be extended to relate the displacement ductility of an entire structure, basedon the centre of mass or the location of resultant inertia forces, to the damage it experiences.A section defining several concepts used in studies of seismic response of structures will bepresented first.21Chapter 3Elasto-Plastic Representation for Seismic Response of StructuresIn this chapter, several definitions and concepts used in earthquake engineering are presented.These will be used extensively for analyzing the results in this thesis.3.1 Idealized Seismic Response of Elasto-Plastic SystemsUnder increasing static lateral loads, the response of a structure can be idealized by a bilinearcurve in a lateral load-lateral deflection plot as shown in Figure 3.1. This type of behaviour is calledelastic-perfectly plastic or elasto-plastic. The first part of the curve represents elastic response in whichthe force is directly proportional to the displacement. The proportionality constant is the elastic structuralstiffness which is a function of the geometry, the composition, and the state of the structure, and it canbe determined reasonably accurately. When any part of the structure begins to behave inelastically, thestructural state is then changed and the load-deflection relationship becomes different. In an elasto-plasticbehaviour, the displacement becomes independent of the applied lateral load once the state of the structureis changed. The level at which the plastic behaviour occurs is called the yield level or the yield strength.In earthquake engineering, the displacement which is commonly used is either the displacement of the topof the structure or the displacement of the centre of mass.During seismic response, the direction of the applied load is reversed, and so the response of thestructure traces out trapezoidal loops of various sizes at different locations in the load-deflection plot asshown in Figure 3.2. It can often be assumed that the yield strengths in both directions are the same andthat the unloading and reloading stiffnesses are identical to the original elastic stiffness.For most engineering purposes, only the maximum response of the structure is of interest. So,to demonstrate the behaviour of a structure in an earthquake, the complex array of loops for dynamic22response is replaced by a simple version of the one-directional bilinear load-deflection plot, like thatshown in Figure 3.1. The end point of the bilinear curve will be the ultimate displacement reached duringthe entire duration of the ground motion. This simple load-deflection plot is very useful, as thecharacteristics of the earthquake controls only the point of maximum response; the slope of the elasticsegment and the yield strength are both properties of the structure.3.2 Ductility, Force Reduction Factor, and Inelastic Displacement Response CurvesBased on the one-directional, elasto-plastic representation for seismic response of structures, theimportant concepts of ductility, force reduction factor, and inelastic displacement response envelopes aredefined.Commonly designated as p., the displacement ductility is defined as the ratio of the ultimatedisplacement to the displacement at the yield point of the structure. Based on Figure 3.1,(3.1)Awhere Au and Ay are the ultimate and yield displacements, respectively. Structures which can exhibitconsiderable amounts of post-yield deflections before collapse are referred to as ductile structures.The force reduction factor relates the largest lateral load carried by an elastic structure to that byan inelastic one. Consider a structure having a particular stiffness but infinite strength. Its elasto-plasticresponse curve for an earthquake would be a straight line, like line OAB in Figure 3.3. The plasticportion of this curve is actually zero. The maximum shear force induced in the structure during the entireearthquake is Pe . If another structure having the same stiffness but a lower and finite yield strength issubjected to the same earthquake, its response is the bilinear curve OAC in Figure 3.3, with the yieldlevel P being the largest force developed. The force reduction factor, R, for the latter structure is thendefined as the ratio of the maximum force in its elastic counterpart divided by its yield strength, or23(3.2)The idea used in defining the force reduction factor can be expanded further to develop thedefinition for an inelastic displacement response curve. If a structure has its strength varied from infinitydownwards, and at each yield level it is subjected to the same earthquake, different ultimatedisplacements, like those in Figure 3.4, will be produced. Each curve has its own end point of ultimatedisplacement. An inelastic displacement response curve (IDRC) is a line joining the end points of thebilinear response curves for the different yield levels of a structure (see Figure 3.4). The IDRC is a veryuseful tool because it relates the displacement response of a structure in an earthquake to the yieldstrength. Any point along this curve can be treated as an indication of the ductility demand of theearthquake given the yield strength, or vice versa, the amount of strength required to reach a certaindisplacement ductility.3.3 Concept of Equal-Displacement ResponseThe effects of varying the yield strength of a structure subjected to earthquake ground motionshave been studied by Veletsos and Newmark [1960]. By applying the El Centro NS 1940 ground motionto a single-degree-of-freedom inverted pendulum system with elasto-plastic behaviour, they found that,at any yield level, the maximum displacement is always very close to the displacement of the structurewhich remains elastic. In other words, no matter how strong a structure of a certain stiffness is, itsmaximum displacement under one earthquake loading always reaches approximately the same value. Interms of the load-deflection relationship, the ductility developed in an earthquake is equal to the forcereduction factor for the structure, or the inelastic displacement response curve (IDRC) is vertical (seeFigure 3.5).243.4 Application in Seismic Design CodeThe concept of equal-displacement response has become the basis for most building codes for theearthquake-resistant design of structures. A probable design earthquake is found for a region based onprevious records and statistics. Using this earthquake, the maximum lateral force (the "elastic" force) thatcan be developed in the structure is determined. This maximum force is then reduced by an appropriateforce reduction factor to arrive at the quasi-static earthquake force that is to be applied to the designstructure in an elastic analysis. This structure will then be expected to achieve the ductility allowed for(that is, the same value as the force reduction factor) when it is subjected to a ground motion matchingthe design one. A ductility factor of 4 is most commonly used for ductile frame structures.3.5 Elastic and Inelastic Response SpectraAs mentioned earlier in Chapter 1, an elastic acceleration response spectrum can be determinedfor an earthquake by noting the maximum accelerations attained in oscillating systems of various naturalfrequencies. For a single-mass system, the maximum acceleration when multiplied by the mass gives themaximum force. This force, such as Pe in the load-deflection plot in Figure 3.3, is the maximum lateralload induced in an elastic structure. Therefore, the load-deflection plot reflects the response spectrum.The same principle can be applied to the inelastic system to generate inelastic accelerationresponse spectra. The strength of the system is adjusted until a desired ductility is reached. By dividingthe yield force by the mass and repeating the procedure at various periods, an inelastic accelerationresponse spectrum for the given ductility is produced. However, the common approach to generateinelastic acceleration response spectra for an earthquake is to scale down the elastic spectrum. This isbased on the equal-displacement criterion in which the force reduction factor is identical to the ductility.253.6 Change in Fundamental Period of VibrationThe elastic stiffness under lateral loads is an inherent property of a structure. Graphically, it isthe slope of the initial straight-line response in a load-deflection plot. But when the structure behavesinelastically, its state is changed and consequently its properties, including its stiffness, change also.More specifically, yielding would soften the structure and thus reduces its stiffness and its vibrationfrequency. So the maximum inelastic response actually corresponds to a structure with a fundamentalperiod higher than the original value. For simplicity, in this thesis, this response will be associated withthe initial vibration period of the structure.26Chapter 4Strength and Ductility Demands of Filtered EarthquakesThe strength and ductility demands resulting from filtered and unfiltered earthquakes acting onthe SDOF systems are investigated in this chapter. The results are presented and analyzed based on thedefinitions and concepts explained in the previous chapter.4.1 Computer Program, Test Structure and ProcedureThe inelastic response of single-degree-of-freedom systems subjected to filtered and unfilteredearthquakes was determined in order to examine structural strength and ductility demands. The computerprogram DRAIN-2D [Kanaan and Powell 1975], which is a two-dimensional, time-step analysis program,was used for this purpose.The single-degree-of-freedom system used was a simple spring-mass-dashpot system having theelasto-plastic load-deflection behaviour shown in Figure 3.2 and a damping value of 5% of critical. Thestiffness and the yield strength of the system were the only variables in the model structure. Changingthe stiffness directly changes the fundamental period of the structure.The study structure was subjected to a number of earthquakes and, for each earthquake, themaximum displacement of the structure was obtained for various stiffness and yield strength combinations.In the study, the stiffness is expressed in terms of the fundamental period and the yield strength is givenas a force reduction factor. Forty different periods between 0.1 second and 5.0 seconds were used witheach earthquake. To examine the ductility demand of an earthquake, the yield strength of the studystructure at each period was reduced downwards by three force reduction factors --- 1.3, 2.0, and 4.0 --- and the maximum displacement and the corresponding displacement ductility at each yield level wasnoted. To examine the strength demand of an earthquake, the displacement ductility response of the study27structure at each period was set at 3 different values --- 2, 3, and 4 --- and the yield strengths requiredto achieve these ductilities were determined. In total, 160 values were obtained for each earthquake ineach of the strength- and ductility-demand analyses.4.2 Earthquake Records StudiedA total of ten earthquakes were used in this study. They include the four input bedrock motionsand their respective filtered surface motions used for the seismic response studies of the Richmond soils,presented earlier in Chapter 2, and two motions recorded in Mexico City in 1985. One is the east-westcomponent of the motion recorded on hard ground at the station CUIP in Mexico City, and the other isthe east-west component of the motion recorded on the soft clays at station SCT. The latter record isassumed to be the results of filtering the former. The accelerogram and the response spectra for the CHIPrecord are shown in Figure 4.1. The strength and ductility demands of the study structures resulting fromthe five pairs of unfiltered and filtered earthquakes were evaluated using DRAIN-2D. These pairedmotions and their maximum accelerations are listed in Table 4.1. It should be noted again that theModified Taft and the Mexico City CUIP represent earthquake motions which have been filtered throughlong distances.4.3 Results of AnalysesFor each ground motion record, relationships between yield strength, natural period of thestructure, and the ductility demand were obtained. These results are displayed in several different waysin Figures 4.2 to 4.31.In Figures 4.2 to 4.11, the ductility demand is plotted against the period for three values of forcereduction factor R. In Figures 4.12 to 4.21, the reverse is performed; R is plotted against period for28selected values of ductility demand. In Figures 4.22 to 4.31, the results are displayed as inelasticresponse spectra, which will be shown to contain the same information.In Figures 4.2 to 4.21, the results for the elastic structure (R=1 and A=1) are not plotted, as theyare by definition equal to unity. For clarity, the spectra for ductility of 3 are not shown in Figures 4.22to 4.31.4.4 Expected Results Based on Equal-Displacement CriterionBefore discussing the results presented in Figures 4.2 to 4.31, it is of interest to consider the formthat each type of plot should have according to the equal-displacement criterion described in Section 3.3.If the equal-displacement criterion is valid, straight horizontal lines with constant values ofequal to the R values and straight horizontal lines with constant values of R equal to the values areexpected in Figures 4.2 to 4.21. This follows since the elastic structure is expected to have a linear force-displacement relationship, so that the yield force and the yield displacement are both obtained by dividingthe corresponding elastic values by R. Then, if the ultimate displacement is equal to the elastic value,the ductility demand is equal to R. The vertical dotted lines in Figures 4.2 to 4.21 represent thepredominant periods of the ground motions.For the inelastic acceleration response spectra in Figures 4.22 to 4.31, the curves shown for eachearthquake should have the same shape and be separated by constant spacings if the equal-displacementcriterion is valid. That is, the lower curves in each figure should be equal to the top one divided by thegiven ductility demand.It has been mentioned [Park and Paulay 1975] that the equal-displacement response does not applyto structures with very short period. In fact, it should be noted that, when the period of the structure iszero, the force reduction factor equals to one for any yield strength in the structure.294.5 Ductility DemandsThe ductility demands of the earthquakes studied are revealed by the results in Figures 4.2 to4.11. In these figures, the curves show very high ductility demands at low periods, as expected, sinceit is known that the equal-displacement criterion is not valid in this range. Each curve appears to oscillatearound a ductility value equal to the force reduction factor at medium to high periods. The curves arefar from smooth although the oscillation amplitudes seem to decrease with decreasing force reductionfactor; the curves for R=1.3 are the smoothest and those for R of 4 are very jagged. This is to beexpected, as the response of the weaker (higher R value) structures is largely in the inelastic range.Comparing the plots for a filtered motion with those for the corresponding unfiltered earthquakeshows that the effect of filtering is to move the beginning of the roughly horizontal portion of the curvesto higher periods; the shift is somewhat more pronounced in the case of R=4. Indeed, it is found thatthis beginning point is invariably just to the left of the predominant period of the ground motion, whichis marked in each of the Figures 4.2 to 4.11 by a vertical dotted line. This is clearly apparent in Figures4.9 and 4.11, for the Artificial Richmond and Mexico SCT EW records. Thus filtered earthquakes giverise to higher ductility demands in structures with lower natural periods. It should be noted that,compared to the other bedrock motions, the high ductility demands persist to longer periods for theModified Taft and Mexico City CUIP records (see Figures 4.8 to 4.10) since these are filtered bydistance.4.6 Strength DemandsThe strength demands of the earthquakes studied are revealed by the results in Figures 4.12 to4.31. As described above, Figures 4.12 to 4.21 show R values plotted against period for three differentductility demands: 2.0, 3.0, and 4.0. (It is to be noted that R is inversely proportional to the strengthdemand.) It is recalled that the equal-displacement criterion would lead to horizontal ductility demand30plots centred around R values which are equal to the ductility values; this is, very roughly, the case forall the unfiltered records (especially the Taft S69E 1952 earthquake).For each of the filtered earthquakes, it is noted that the three lines remain closely spaced at shortperiods. This means that these structures must all be designed for high strengths (approaching the elasticvalue at R=1) regardless of the available ductility. As the natural period of the structure becomes longerthan the predominant period of the ground motion, however, the R values can be seen to rise above thecorresponding values for the parent unfiltered motion, indicating that the strength demands are lower inthe filtered case. Again, this observation is most noticeable in the cases of the Mexico City SCT EW andthe Artificial Richmond ground motions since these have the longer predominant periods.The inelastic response spectra in Figures 4.22 to 4.31 (these spectra are not as accurate as thoseof Chapters 1 and 2 since fewer points were used in their generation) are compared to the elasticspectrum, which is the curve for a ductility of 1. The inelastic curves are expected to have the sameshape as the elastic spectrum, with the acceleration values scaled down by the ductility. For the Taft,Caltech, and El Centro EW unfiltered earthquakes, this is roughly the case.However, in the response spectra for the Mexico City SCT and Artificial Richmond records(Figures 4.29 and 4.31), the inelastic spectra are clearly not a constant ratio of the elastic ones. At lowperiods, the entire family of curves are close together, and only after the predominant period of theearthquake (recognizable by the peaking of the elastic response spectrum) is exceeded do the curvesapproach the expected proportional spacing. The results for these two ground motions show that areasonably accurate inelastic acceleration spectrum cannot always be obtained by simply scaling the elasticspectrum. This also implies that the shapes of the elastic and inelastic spectra for the filtered groundmotions are not similar. The most noticeable change when inelastic behaviour is considered is thedisappearance of the elastic peak response at the predominant frequency of the ground vibration. Evenat a ductility as low as two, the inelastic behaviour is able to suppress the resonant amplification effect.This "smoothing out" of the peak has been noted by Whitman [1977], and it indicates an important point:that the high lateral acceleration expected to occur in structures at the predominant period of the ground31motion will not be of great concern if structures are allowed to undergo plastic deformations during theearthquake.When the shape of the inelastic response spectrum for a filtered earthquake deviates from that ofthe elastic spectrum, it takes on the shape of an inelastic spectrum for an unfiltered earthquake; thespectral accelerations are large at low periods and they decrease as the period increases. However, inthe filtered spectra, the low end accelerations are higher, and they persist into a higher period range. Thatis to say, lower period structures must be designed for higher accelerations in the case of filteredearthquakes --- the same conclusion as was reached earlier from the results of the strength-demand study.4.7 Inelastic Displacement Response CurvesThe preceding results can be displayed in yet another way which is, perhaps, more powerful andmeaningful than those used above. Figures 4.32 to 4.41 show, for each earthquake, the inelasticdisplacement response curves for SDOF systems of different periods. An inelastic displacement responsecurve plots the maximum displacements for a series of SDOF structures having the same natural periodbut different yield strengths. For another group of structures with a different natural period, a differentcurve is obtained.In view of the equal-displacement hypothesis, the inelastic displacement response curves wouldbe expected to be vertical straight lines at A/Ae = 1, where A is the maximum inelastic displacement andAe is the corresponding elastic response. If they slope down towards the right ("outward"), it means theductility demands are greater than expected, or that greater strengths are required to give the requiredductility demands. If they slope down towards the left ("inward"), it means the ductility demands arelower than expected, or that lower strengths are required to give the required ductility demands.In Figures 4.32 to 4.41, normalized inelastic displacement response curves for twenty differentinitial periods are shown for each of the earthquakes studied. Some of the twenty curves in each plot areidentified.32It can be seen that, for the unfiltered earthquakes, the envelopes do approximate vertical lines,except at very low periods or very low strengths. For the filtered earthquakes of Figures 4.33, 4.35, and4.38 to 4.41, there is a much more pronounced tendency for lower periods to be outward-sloping, but thelonger periods appear to be more inward-sloping than the corresponding unfiltered cases. For theArtificial Richmond and Mexico City SCT EW records, which are doubly filtered --- by distance and bysoft local soils --- these effects are particularly strong. The filtered El Centro EW 1940 record (Figure4.37) shows a slightly different pattern, in that longer periods also tend to have increased outward slopes,and there is a complete absence of inward sloping cases. The reasons for these observations will bediscussed later.4.8 Summary of Results and ConclusionsIn summary, the major differences in inelastic response between unfiltered and filteredearthquakes are:1) Low-period structures which are designed to the same force reduction factors have tomeet higher ductility demands if the earthquake is filtered.2) Low-period structures designed for given ductility demands require lower force reductionfactors (ie. higher strengths) for filtered earthquakes.3)^The inelastic displacement response curves generally slope outward at lower periods forfiltered earthquakes.The three conclusions all imply the same thing, as can be demonstrated more clearly by Figure4.42, showing two idealized inelastic displacement response curves.Consider a filtered earthquake and an unfiltered earthquake giving the same elastic response atA for a certain structure. The filtered earthquake would produce the outward-sloping IDRC marked AB,while the unfiltered motion would give the vertical curve AC. When the strength of the structure isreduced to P 1 , the structure will deflect to the ultimate point at C in the unfiltered earthquake but to a33larger value at B in the filtered motion. Therefore, higher ductility is needed in the filtered groundmotion. When the point C is assumed to correspond to an arbitrary ductility of 4 for the structure withthe yield strength P 1 , a line indicating the locations at which ductility of 4 is reached in the plot fordifferent yield strengths would extend from the origin and pass through the point C. This line is dottedin Figure 4.42, and it intersects the sloping IDRC for the filtered earthquake at point D. It shows thatductility of 4 is obtained in the filtered earthquake only if the structural strength is increased to P2 , asstated by the second conclusion above. This simplified plot demonstrates very clearly the three majorfindings on the strength and ductility demands of filtered earthquakes.34Chapter 5Strength and Ductility Demands in Relation to Predominant Periods of Earthquakes5.1 BackgroundIt has been mentioned in Chapter 1 that the frequency content of an earthquake is an importantparameter influencing structural response. The significance of this parameter lies in the possibility thata structure will respond in quasi-resonance when the structural period matches the predominant period ofthe ground motion. Vibrating in resonance produces large deformations and forces that are undesirablefor any structure. This is indicated by a peak in an elastic acceleration response spectrum. Major damagecan be avoided if adequate strength is provided for structures responding in resonance or if thefundamental vibration frequency of the structure can be altered to a value different from the predominantground vibration frequency. The second option involves a decision to choose between a larger and asmaller frequency value. In the situation illustrated by the acceleration spectrum in Figure 5.1 for threestructures with identical mass and infinite strength, the structures with periods Ta and Te can be chosenso they will be subjected to equal lateral forces that are smaller than the force developed in the structurewith period Tb. But which one of structures A and C is better? The answer to the question isstraightforward when both structures are strong enough to behave elastically during the entire groundmotion. In this case, they would experience the same level of lateral loading, as governed by the elasticacceleration response spectrum. But their difference in stiffness would affect their absolute deflections,with structure C deflecting more. However, for reasons of economy and practicality, seismic designcodes allow yielding in structures provided the ductility demand is satisfied, and most structures designedaccording to the code will not remain in the elastic stage during a design level earthquake. This meansthat the fundamental period of vibration of a structure will change during the earthquake, as explainedearlier in Section 3.6. Because of this, the answer to the above question becomes unclear.35The fundamental period of vibration always increases with the onset of yielding, so that it mayapproach or move away from the predominant frequency of the ground motion, depending on the valueof the initial fundamental period. Thus the response of structures A and C in Figure 5.1 will be differentunder inelastic actions.5.2 Past Observations on Effects of Predominant PeriodIn a concurrent study similar to this research, Qi and Moehle [1991] at the University ofCalifornia at Berkeley concluded that "the displacement response of SDOF systems subjected toearthquake ground motions can be characterized in two period ranges, divided by the characteristic periodof the ground motion." They defined the characteristic period to be the predominant period of the groundshaking. Furthermore, Qi and Moehle recognized the occurrence of equal-displacement response forperiods higher than the characteristic period, and they noticed the dependency of the maximum inelasticdisplacement response on both the strength and stiffness of the system for periods lower than thepredominant period.From their analytical studies on isolated, reinforced concrete shear walls, Derecho et al. [1978]noted that the frequency characteristics of ground shaking has a significant effect on the inelastic responseof structures. They realized that, for structures with extensive yielding, an earthquake with an ascendingacceleration response spectrum produces greater damage than an earthquake of the same intensity but witha descending spectrum. Ascending means that the spectral acceleration increases as the period increases,while descending means decreasing spectral value with increasing period.Since an acceleration response spectrum usually slopes upwards before the predominant periodbecause of the existence of a peak in the spectrum at that period, the results of Qi and Moehle and ofDerecho et al. carry indirectly the same meaning --- the response of an inelastic structure with a periodlower than the predominant period of ground vibration can be different from and worse than that of astructure with a period on the other side of the predominant period.365.3 Observations from Present Strength and Ductility Demand AnalysesThe predominant periods of the ten ground motions studied in the strength and ductility demandanalyses have been indicated in Chapters 2 and 4. These are identified in the ductility-demand plots inFigures 4.2 to 4.11 by vertical, dotted lines and in the plots of the inelastic displacement response curvesin Figures 4.32 to 4.41 by circling the single-letter label for the inelastic displacement response curve forthe structural period closest to the predominant period. From these results, it can be seen that thepredominant period of an earthquake is a key parameter influencing the response of the SDOF system andthat the response follows the behaviour indicated by Qi and Moehle and Derecho et al. For periodshigher than the predominant period, the ductility demand and the force reduction factor are approximatelythe same, and the inelastic displacement response curves are vertical. That is to say, the equal-displacement criterion applies. For periods lower than the predominant period, the strength reductionfactor is lower than the ductility, and the curves slope outwards; the equal-displacement criterion doesnot apply, and the strength requirement is higher than implied by that criterion. But, in addition to thesetwo types of response, it can also be observed that, close to the predominant period, the strength reductionfactor is actually lower than the ductility, and the curves slope inwards.An examination of the inelastic displacement response curves for the filtered motions of MexicoCity SCT and Artificial Richmond (Figures 4.41 and 4.39, respectively) reveals a pattern in theorientations of the curves. At very low periods, the curves slope gently outwards from the elasticresponse point towards the large-displacement end of the plot. As the period increases, the curves beginto sweep towards the vertical position. The vertical position is attained at a structural period slightlylower than the predominant value. As the structural period approaches the predominant period, the curvescontinue its sweep and swing towards the elastic displacement response curve, which is a line joining theorigin of the plot and the elastic response point. The closest approach to this line is reached at thepredominant period. Finally, when the structural period is large (exceeds the predominant period), thecurves move back to the vertical position and remain there for equal-displacement response.37From these observations, three types of seismic response can be identified based on therelationship of the fundamental period of vibration of the structure to the predominant period of theground motion (see Figure 5.1). The three types of response are characterized by outward-sloping (A),inward-sloping (B), and vertical (C) inelastic displacement response curves for periods lower than, closeto, and higher than the predominant period of the ground motion, respectively. Note that this suggestsa third category to be added to the two detected by Qi and Moehle. The range of applicability for eachtype of response has yet to be determined, but the inelastic displacement response curves for Mexico CitySCT and Artificial Richmond earthquakes show that the upper bound period for the range of outward-sloping curves is about 0.5 second below the predominant period. This is an interesting observationbecause it points out that structures having periods less than 1.5 seconds would be forced to larger thanexpected deformations in the Mexico City earthquake. Based on the code assumption that the fundamentalperiod of a frame-type building is one-tenth of the number of storeys, this observation implies thatbuildings up to fifteen storeys in height would exhibit very large amounts of ductility. Indeed, most ofthe severely damaged or collapsed structures in Mexico City in 1985 were observed to be frame-typebuildings between six- and fifteen-storey in height. [Borja-Navarette et al. 198715.4 Characteristic Periods from Acceleration and Velocity Response SpectraA typical elastic acceleration response spectrum for an unfiltered earthquake generally exhibitsa peak at a rather low period. The ground motion has a wide range of period content, but most of theearthquake energy is generally concentrated in the lower periods. Filtering the seismic waves tends toshift the energy peak towards a higher period and to make it sharper, as the band of dominant periods isnarrowed and moved to a higher value.The period at which the peak in an acceleration response spectrum occurs, for either a filteredor unfiltered earthquake, is a critical characteristic of the ground motion; following the nomenclatureused by Qi and Moehle [1991], this value will be called the characteristic period. Its importance lies in38the fact that it is the predominant period of the ground vibration and that the acceleration responsespectrum is ascending before this period and descending afterwards. The characteristic periods obtainedfrom the acceleration spectra for the ten study earthquakes are listed in Table 5.1. These are identicalto the values tabulated in Table 2.1, but an additional value in brackets is included for the filtered Caltechand El Centro EW earthquake in Table 5.1. This bracketed value is the period at which a second peakhaving a spectral value very close to that of the largest peak occurs in the spectrum. Beyond this point,the spectrum is entirely descending.The typical shape of an elastic velocity response spectrum for an unfiltered earthquake resemblesa bilinear plot with the curve ascending quite sharply at low periods and then gradually levelling off athigher periods. This shape can be seen in the velocity spectra for the Taft, Caltech, and El Centro EWmotions (Figures 1.1c, 2.15b, and 1.2c, respectively). Even the spectra for the filtered motions show thisgeneral shape, except that the point at which the slope of the spectrum begins to change is at a higherperiod, and after this point the spectrum is more downward-sloping than horizontal. The peaks in thespectra for the Mexico City SCT and the Artificial Richmond motions (Figures 1.7b and 2.17b,respectively) are so large and sharp that they distort the shape and make the velocity spectra look liketheir respective acceleration spectra. The ground motions of Mexico City CUIP is actually a filteredmotion and so its velocity spectrum has a descending portion at long periods.A characteristic period can be identified for the velocity response spectrum in the same way asbefore. It is taken to be the value at which the ascending branch ends; for filtered motions, this wouldalso correspond to the period at which the highest peak in the spectrum occurs. Again, the spectrum isascending before this period and descending or horizontal afterwards. The characteristic periods fromthe velocity spectra for the ten study earthquakes are listed in Table 5.1.A comparison between the two sets of periods in Table 5.1 shows that, for all earthquakes exceptthe El Centro EW earthquake and its filtered motion, the characteristic period for a velocity spectrum isthe same as or very close to the value for an acceleration spectrum. For the filtered Caltech motion, aclose match is obtained when the period for the second peak is used. An examination of the velocity39spectra for the two versions of the El Centro EW earthquake (Figure 2.16b) reveals that both contain asmall peak at a lower characteristic period, which is close to that given by the acceleration spectrum. Theperiods for these peaks are shown in square brackets in Table 5.1. But, in each velocity spectrum, thespectral velocities continue to rise after this lower characteristic period (in square brackets) and reachglobal peak values at a much higher period. This is an important observation, as will be explained laterin another chapter.From the study of the characteristic periods for the ten study earthquakes, it is concluded that thecharacteristic periods obtained from acceleration and velocity spectra for an earthquake can generally beregarded as equal, although a difference between the two values can occur. The characteristic periodfrom an acceleration response spectrum is chosen in this thesis as the critical parameter defining theexpected structural response because, based on the characteristic periods for the El Centro EW motions,the period defined by the acceleration spectrum is seen to be lower than the corresponding velocityspectrum period; both the acceleration and velocity response spectra would be ascending before thecharacteristic period defined in this manner.5.5 SummaryBased on the type of response observed for the SDOF systems examined, the strength andductility demands are seen to be dependent on the relationship of the structural period to the predominantperiod of the ground vibration. High strength and ductility demands are required for structures withperiods lower than the characteristic period. Therefore, the major effects of filtered earthquakes onstructural response come from the shift in the predominant period to a higher value which exposes a largerrange of periods to outward-sloping inelastic displacement response curves and thus to larger ductilitydemands. The shift in predominant period is only one of the three key characteristics of filteredearthquakes. The effects of the other two characteristics of large acceleration and long duration will bedealt with later in this thesis.40Chapter 6Relationship Between Strength Requirements and Ductility Demands of Earthquakes6.1 IntroductionIt was concluded in the previous chapter that the orientation (outward-sloping, inward-sloping,or vertical) of the inelastic displacement response curve is related to the predominant period of theearthquake. Outward-sloping inelastic displacement response curves were observed to occur in filteredearthquakes and they are undesirable because of their high strength or ductility demands. Unfortunately,the results from the previous analyses do not indicate quantitatively the conditions under which the variousorientations of the curve occur. A significant sweeping pattern was noted in Chapter 5 in the curves forthe Mexico City SCT and the Artificial Richmond earthquakes, but no explanation was offered as to whysloping curves occur or even why equal-displacement response occurs.This chapter attempts to establish a general rule for the inelastic response of structures which willexplain the existence of sloping inelastic displacement response curves and their orientation and determinethe range of periods for each of the three types of curves. This will be done by applying basic energyprinciples to the idealized bilinear seismic response curves. The validity will be checked by comparingpredicted results with those obtained analytically.6.2 Energy Concepts in Seismic Response of StructuresIn this section, energy principles from basic dynamics and applied mechanics will be used inevaluating the seismic response of structures. The forms of energy to be considered are potential energyand kinetic energy, and the law of conservation of energy will be used to relate these two quantities.41Consider a general single-mass system in free undamped vibration (for example, the pendulumin Figure 6.1). At either end of an oscillation where the deflection is the greatest, the system has themaximum potential energy. When the mass passes through its original undeflected position, the potentialenergy is transformed entirely into kinetic energy. By conservation of energy, the maximum kineticenergy would be equal to the maximum potential energy.Now consider a structure vibrating elastically under seismic loading. Although it is not in freevibration, in the sense that the ground imparts energy to it during each cycle, the maximum energy ofdeformation (strain energy) is present at the point of maximum displacement, and the maximum kineticenergy must be approximately equal to this amount.From applied mechanics, the area under a load-deflection curve is the potential energy stored ina deflected system. This principle can be applied to the idealized seismic response curve. For an elasticstructure, the seismic response is represented by a straight line in a lateral load-deflection plot. Themaximum potential energy in the structure is the triangular area under the curve, as shown in Figure 6.2a.If the stiffness of the structure is k, and the maximum deflection and lateral force experienced are e andPe , respectively, then the maximum potential energy (PE) is given by(6.1)It should be recalled that the end point of the idealized seismic response curve indicates the maximumresponse during the entire earthquake. Therefore, the energy given by the area under the curve is onlyattained at the time of maximum response.By the principle of conservation of energy, the maximum kinetic energy KEmax is the same asthe maximum potential energy, provided there is no damping or additional input of energy during the timeit takes to go from the position of maximum kinetic energy to that of maximum potential energy. Thatis,42KEmax — PE^ (6.2)The maximum kinetic energy in an oscillating structure can also be written asKE.^M V nax^ (6.3)where M is the total mass of the structure and vmax is the maximum velocity attained by the structureduring the entire earthquake. Substituting Equations 6.1 and 6.3 into 6.2 gives— k A, — M v2^21^2^1^2^ (6.4)which states symbolically the law of conservation of energy for an undamped, elastic vibrating system.The energy principles applied to an elastic structure under seismic loadings can also be used todetermine the energy in an inelastic structure. The only difference for an inelastic structure is that itsidealized response curve is no longer a straight line but a bilinear curve. Despite that, the strain energyis still given by the area under the curve. For an elasto-plastic structure as shown in Figure 6.2b, themaximum strain energy in it isPE — Py (A.—Ay ) + k Ay^1 ^2^(6.5)with k being the elastic stiffness, Py being the yield strength of the structure, Ay being the deflection atfirst yield, and Au being the maximum displacement reached in the earthquake. The maximum kineticenergy for an inelastic system remains as calculated by Equation 6.3. This energy will be transformedinto strain energy before the system comes to rest at the ultimate displacement level, thus giving(6.6)436.3 Concept of Equal-Energy ResponseThese concepts of potential energy stored in an elastic and an inelastic system have been used byBlume [1970] and Park and Paulay [1975] to relate the displacement ductility to the force reduction factorin an earthquake. They consider a structure with a bilinear response as indicated in Figure 6.3. If themaximum kinetic energy acquired by this structure is not affected by its yield level, the area under thedisplacement curve OBC to the ultimate displacement C must be equal to the area under the unyieldingelastic curve (OA). Thus, from Equations 6.1 and 6.5,1^2^ 1^2— k^Py^) + k Ay2(6.7)Recalling that R = Pe/Py = / L5,Ay  and A = AuLtly, this leads toR (6.8)which gives rise to the dotted line in Figure 6.3. Blume suggested that this line defines the probableupper limit for seismic response of structures; however, this is based on the assumption that themaximum kinetic energy in a yielding structure would never be greater than that in an elastic structure.6.4 R-p, Equation: Relationship Between Strength, Ductility, and Spectral VelocitiesIn the previous section, the strain energies of elastic and inelastic systems were equated based onthe idealized seismic response curves like the ones shown in Figure 6.3. However, it was assumed thatthe kinetic energy was the same for the two systems; in fact, the yield level affects the equivalentstiffness of the system, and thus changes the kinetic energy the system acquires from the ground motion.This effect will now be taken into account.Consider two structures of equal initial stiffness and equal mass being subjected to an earthquakeloading. If one has an infinite strength and the other has a finite strength, then the law of conservationof energy leads to Equations 6.4 and 6.6, repeated here:A2,2A2 + 2 A (A -A y )^viy^u 2Ve (6.11)441^2^1^22— k A , — M ve2(6.9)12P (A -A ) + —k A 2 -1  M v.Y^mi^Yi^2^Yi^2^I(6.10)where the subscripts "e" and "i" stand for elastic structure and inelastic structure, respectively. Thesubscript "max" is dropped from the velocity term for clarity. The response of the two structures can berelated by dividing the equation for the inelastic one into that for the elastic one. This results in a singlerelationship containing all four energy quantities, which, by putting Py = laiy, can be simplified toBy definition from Chapter 3, Pe=k,k, Py=kAy, R=Pe/Py, and =Au/Ay. Substituting thesedefinitions into Equation 6.11 and rearranging terms result in a very compact equation:R^8/211 - 1^ (6.12)viEquation 6.12 will be designated as the "R-p. equation". It is a useful relationship because it relates theresponse of the elastic and inelastic structures. Note the restriction that the value for the ductility mustalways be greater than one.6.5 Significance of the R-p. EquationThe R-it equation has some important implications. First, it leads directly to an inelasticdisplacement response curve, given the ratio of ye/v1. A given it value implies a particular R valuethrough the equation and also gives the ultimate deflection if the yield deflection is known. The resultsgiven by the R-p. equation for various ratios of ve/vi are shown in Figure 6.4. This is the same type of45plot as those in Chapter 4. The restrictions on R and A should be noted; both must be greater than one.It can be observed that some curves lie in the range of A/Ae greater than one and some are in the rangeof less than one. The curve closest to being a vertical line at il/Ae of one is that for ve/vi=1.2.Therefore, the R-p equation shows, in a way, that inelastic displacement response curves can slopeoutwards (increasing A/Ae for increasing R) and inwards (decreasing A/ile for increasing R) besidesvertically downwards.Second, the R-p. equation also reveals when each of the outward-sloping, inward-sloping, andvertical IDRC will occur. From Figure 6.4, it can be seen that the ratio ve/vi is the key parameterinfluencing their slope. As mentioned above, the velocity ratio of 1.2 gives an almost vertical inelasticdisplacement response curve. Thus, it can be concluded that the IDRC slopes outwards for ye/v1 less than1.2 and slopes inwards for ve/vi greater than 1.2.To understand better the condition under which each type of envelope occurs, it is necessary tounderstand the meaning of the velocity ratio. As defined in the previous section, the terms ye and vi arethe maximum velocities attained by an elastic and inelastic structure, respectively, in an earthquake. Inrelation to a velocity spectrum, ye would be the elastic spectral velocity for a period Te, and vi would bethe value for the inelastic one with effective period Ti. But since both structures are assumed to have thesame initial elastic stiffness, the period Ti would be larger than Te because yielding inevitably softens thestructure and lengthens the period. Therefore, ve/vi is the ratio of the velocity at the initial period to thatat a higher period. Then three possible situations can arise; vi can be greater than, less than, or equalto ve. When vi is greater, it means that the velocity spectrum slopes upwards between these two periods;when vi is smaller, then the velocity spectrum slopes downwards; finally, equal spectral velocity valueswould signify a horizontal spectrum. For these three situations respectively, the force reduction factorwould be less than, greater than, or equal to the square root of 2p-1. That is, if vi is equal to ye, thenthe velocity ratio is one and the R-p, equation becomes the equal-energy relationship, Equation 6.8,presented in Section 6.3. It was mentioned earlier that this equation was regarded by Blume [1970] as46a probable upper bound for seismic response of structures. Obviously, based on the results shown inFigure 6.4, this is unlikely to be the case.Finally, the third implication is that the force reduction factors obtained for different ductilitiesby inelastic, dynamic computer analyses can be represented by the R-p. equation once the ratio of thespectral velocities is known. Therefore, the equation can be used to determine inelastic accelerationresponse spectra based on known elastic acceleration and velocity response spectra, if the period shift canbe determined.6.6 Shift in Fundamental Period and Equivalent Elastic StructureIn determining the velocity ratio, the two spectral velocity values for the structure at its initialand final period have to be known. The velocity response spectrum for a given ground motion can beeasily determined, as can be the initial period of the structure. But the effective period of the inelasticsystem is difficult to determine. So far it is known that the fundamental period for an inelastic systemis greater than that for an elastic one, but the amount of increase is uncertain.In this section, the vibration period of an inelastic structure is determined based on the idealizedelasto-plastic structural response curve and the concept of an equivalent elastic structure. An equivalentelastic structure is defined as an elastic structure such that its maximum strain energy and its ultimatedisplacement are identical to those of the inelastic structure in question. The lateral load-deflection curvefor this system is thus a straight line and its relation to that of the inelastic system represented is shownin Figure 6.5. The spectral velocity of this equivalent elastic system is assumed to give the maximumvelocity of the inelastic system. Based on Equation 6.1, the strain energy of the equivalent elasticstructure is1^2PE = k A .—2 eq(6.13)47where k is the equivalent elastic stiffness. The potential energy for the inelastic structure is given byeqEquation 6.5. Knowing that the elastic stiffness of a structure is related to the period byk - 47E2M^(6.14)T2where M is the mass and T is the fundamental period, equating the two potential energies would give-   Te^ (6.15)211 — 1Te is the fundamental period of the elastic structure, while Ti is considered to be that of the inelasticstructure. It can be seen that the ratio of the periods Ti/Te is a function of the ductility only. This is inagreement with a period-shift equation based on an empirical study by Iwan [1980], who comparedinelastic displacements of single-degree-of-freedom systems of different hysteretic behaviour to elasticspectral displacements, and given by the equationT.- 1 + 0.121 (p.-1)"39Te(6.16)The two functions are compared in Figure 6.6. The difference between these two functions increases asthe ductility value goes up, but it is only about 11% at ductility of 4.Therefore, Equation 6.15 can be used to calculate Ti once the displacement ductility is known,and then the spectral velocity vi of the equivalent elastic structure can be determined.6.7 Comparison Between Calculated and Actual RThe validity of the R-A equation is checked in this section by comparing the force reductionfactors calculated by the R-p. equation (known from now on as the "calculated" R) to those found by thedynamic analyses in Chapter 4 (known from now on as the "actual "R). The comparison is made usingthe same ten earthquakes used earlier in the strength and ductility demands analyses and the same48ductilities (2, 3, and 4) and periods (0.1 to 5.0 seconds) as considered in Chapter 4. Calculated forcereduction factors were obtained from the R-p equation by first determining the change in period at eachductility level, and then determining the spectral velocity values at the various periods. A damping valueof five percent critical was used in the calculation of the spectral velocities. This amount of damping wasselected because the elastic spectra for all the earthquakes studied were determined based on this value.Figure 6.7 is a plot of the calculated force reduction factors against the actual ones for all earthquakesand all ductility levels. It can be seen that the comparison is good at low values of R, but at high values,the calculated factors are lower than what they are supposed to be.6.8 Modified R-pc EquationThe comparison between calculated and actual force reduction factors shows that the R-A equationdoes not give the correct values in many cases. This is expected since the simple R-it equation does notentirely account for the complex nature of an earthquake and of a vibrating inelastic system. Also, theeffects of hysteretic damping on the response are not considered.Shown in Figure 6.8 is a comparison between the calculated and actual R values for the Taft S69E1952 ground motion for ductility of four. The figure shows that the calculated force reduction factors arein most cases lower than the actual values. However, the patterns of the behaviour of the two plots inthe figure can be seen to be similar for periods up to two seconds. Therefore it is possible that a slightmodification to the R-A equation may improve the estimation.An investigation of a possible modification was made based on force reduction factors at ductilitylevels of two and four for three earthquakes --- Taft S69E 1952, Mexico City SCT EW 1985, and anotherartificial, strong filtered motion in Richmond, similar to the Artificial Richmond vibration determined inChapter 2. Figure 6.9 is a plot of the ratio of actual to calculated reduction factors against the secantslope of the velocity response spectra. The secant slope is defined as the amount of change in the spectralvelocity over the change in the fundamental period from Te to Ti (see Figure 6.10). It is given by49— vem =  ^ (6.17)- TeFigure 6.9 shows that fN, which is the ratio of the actual to calculated R, is a function of both the ductilityand the secant slope, but only for positive slopes. Thus, a modification to the R-p, equation is proposedasR^21L-1^ (6.18)viwhere fN is the modification factor which essentially accounts for the energy loss to hysteretic damping.After best-fit, straight lines are drawn through the data points in the figure, the modification factor forductility of four is given by1 , m 0 (6.19)INwhile that for ductility of two is given by1 + 0.01602m , m > 01 , m 0 (6.20)IN { 1 + 0.00467m , m > 0with m in units of centimetre per second squared. As shown in Figure 6.9, most of the values for m liebelow 200 cm/s/s; the modification factor for m of 200 cm/s/s is 4.20 for ductility of 4 and 1.93 forductility of 2. This modification has been reported by Kuan and Nathan [1991].6.9 Shift in Structural DampingBesides being a function of the structural period, spectral response is also a function of dampingin the structure. Unlike stiffness, damping increases when yielding occurs. Consequently, the spectral50velocity value for the inelastic structure, vi , should be calculated based on the damping in the inelasticstructure. The problem again is to determine the amount of change in the structural damping.For an inelastic structure with an ultimate displacement Au under seismic loading, the strainenergy has been shown to be the area under the bilinear curve. If the structure goes through one completecycle to the ultimate displacement in each direction (see Figure 6.11), the energy dissipated in this cyclewill be given by the area enclosed in the load-deflection loop. This type of energy dissipation is attributedto hysteretic damping. The amount of damping in an oscillating system is obtained from [dough andPenzien 1975](6.21)where is the ratio of the equivalent amount of viscous damping in the system to the critical damping,and Wd and Ws are the energy dissipated in a cycle and the strain energy, respectively. The strain energyin the structure at the ultimate displacement Au would be the maximum potential energy in the system andso is given by Equation (6.5). Based on Figure 6.11, the area enclosed in the loop isenclosed area = 4 Py (A. —A)^ (6.22)Substituting Equation 6.5 for Ws and Equation 6.22 for Wd, the equivalent viscous damping in thestructure reaching displacement Au is2 IA —1^ (6.23)it 211 — 1where is the ductility and is equal toThe value given by Equation 6.23 is actually the additional amount of viscous damping added bythe hysteretic damping to the initial viscous damping in the system. The result is considered as a"change" in the damping in the system since the equation does not contain the initial damping value.Also, Equation 6.23 gives the damping change to the ultimate displacement. It is recalled that the amountof period shift in an inelastic structure is determined based on the concept of an equivalent elastic51structure. The stiffness of this equivalent system is somewhere between the initial elastic stiffness of theoriginal structure and the stiffness corresponding to the ultimate inelastic deflection (see Figure 6.5).Assuming that there is a relationship between the amount of damping change and the amount of stiffnesschange (that is, period shift) for an inelastic structure, then, when the equivalent elastic structure is usedagain to model the inelastic system, the actual damping change would be related to the amount of stiffnesschange for the equivalent elastic structure and would be less than the damping change at the ultimatedeflection as given by Equation 6.23.The ratio of the stiffness changes based on Figure 6.5 is determined and the result iskeg — k^11_1kg,^k^p.Then, by substituting for k, the ratio of the damping values can be written simply as,(6.24)(6.25)The assumption behind Equation 6.25 seems reasonable because both the period shift given by Equation6.15 and the damping shift given by Equation 6.25 are functions of ductility only. Substituting Equation6.23 into 6.25 givesp, —1 12 p,— 1 4 +eq^IC 211 —1(6.26)This equation gives the damping value of an inelastic structure reaching a certain ductility with a knowninitial damping value, . With an initial damping of five percent of critical damping (ie, =0.05), therelationship between and it is plotted in Figure 6.12.Damping change in inelastic systems has also been examined by other researchers. Usingstatistical studies on analytical results of the response of SDOF systems, Iwan [1980] proposed thefollowing damping change equation:52eq -^+ 0.058701-1)°3n^(6.27)Based on the response of reinforced concrete beams and frames, Gulkan and Sozen [1974] give anequation for damping change:eq = E + 0.20 1 - 1^ (6.28)1firThese two relationships, with initial damping of 0.05, are also plotted in Figure 6.12 for comparison.This figure shows that the damping changes given by the latter two expressions are very close to eachother and that Equation 6.26 gives a damping change that is significantly higher than the others. At aductility of four, the damping change given by Equation 6.26 is about 57% higher than that given byIwan. The appropriateness of Equations 6.26 and 6.27 for the R-p. equation will be discussed in the nextsection when the calculated force reduction factors are compared to the actual values.6.10 Comparison Between Calculated and Actual R with Damping ShiftsIn Section 6.7, the force reduction factors calculated by the R-it equation considering period shiftonly were compared to the actual factors obtained from dynamic analyses. Now, with the amount ofchanges in damping also being known, new comparisons are made in this section based on calculated forcereduction factors which have considered damping shifts as well as period shifts.Using the R-p. equation and Equations 6.15 and 6.26 for the period and damping shifts,respectively, force reduction factors at ductilities of 2, 3, and 4 for each of the ten study earthquakes weredetermined and are compared to the actual values in Figures 6.13 to 6.22. In each plot, the agreementis fairly reasonable (in earthquake engineering studies) over the entire range of periods examined; eventhe local peaks and valleys in the plots are reasonably matched with some discrepancies only at sharppeaks. Figure 6.23 is a plot of all the calculated force reduction factors against their respective actualvalues. It can be seen from the figure that a reasonable agreement is obtained, even at high values of R.53To examine the effects of the velocity ratio on the calculated factors, the ratio of the calculated to actualR is plotted against the spectral velocity ratio NT/ye in Figure 6.24. Except at low velocity ratios, the ratioof the force reduction factors centre around the value of one for most values of vi/ve, indicating theindependence of the results of the R-it equation with respect to the velocity ratio. This is contrary to theidea of the dependence of the calculated force reduction factors on the secant slope of the responsespectrum.If the period and damping shifts given by Iwan (Equations 6.16 and 6.27) were used inconjunction with the equation to determine force reduction factors, then the results shown in Figures6.25 and 6.26 would be obtained. Comparing these two figures to Figures 6.23 and 6.24 shows that thereis not a great deal of difference between the two sets of figures, meaning that the two different sets ofperiod- and damping-shift relationships produce similar results. However, for actual R greater than five,the calculated factors based on the Iwan expressions can be seen to be generally lower than the actualvalues, whereas the R values based on Equations 6.15 and 6.26 are generally in better agreement in thisrange. Therefore, the period and damping shifts given by Equations 6.15 and 6.26 are preferable for usewith the R-A equation over the extended range of force reduction factors.One further observation to be noted is that the difference between the calculated factors based onthe two sets of shift relationships is mainly contributed by the difference in the amount of dampingchange. This is because the period shifts given by Equation 6.15 and the Iwan expression (Equation 6.26)are fairly close to each other as shown in Section 6.6. The effects of damping shift on the calculated Rcan be shown by comparing the plots in Figure 6.7 (with Equation 6.15 period shift only), Figure 6.23(with Equation 6.15 period shift and Equation 6.26 damping shift) and Figure 6.25 (with Iwan period anddamping shifts). It can be seen that Figure 6.7 and 6.25 look very much alike; the amount of dampingshift given by Iwan did not significantly affect the force reduction factors calculated without dampingchange. But the damping shift given by Equation 6.26 improved the comparison between the calculatedand actual R values, as shown in Figure 6.25. This observation justifies the use of the relatively largerdamping change that is produced by Equation 6.26.54In conclusion, the R-tt equation is a reasonable formula for calculating the force reduction factorwhen the changes in period and damping as determined by Equations 6.15 and 6.26, respectively, areconsidered in the determination of the spectral velocity of the inelastic structure.6.11 SummaryIn this chapter, a relationship was developed which explains the general seismic response ofstructures. Called the R-p. equation and given by Equation 6.12, this relationship determines the strengthrequired to achieve a certain ductility response in an earthquake using values of the spectral velocity ofthe earthquake. The spectral velocities are determined with the use of Equation 6.15 for the period shiftand Equation 6.26 for the damping shift. Using the inelastic displacement response curves as indicatorsof the type of seismic response, the R-A equation shows why, when, and how a certain response isobtained in an earthquake, whether it is filtered or not.The R-it equation suggests that the response of a structure or the orientation of the inelasticdisplacement response curve is governed by the velocity spectrum. More specifically, the controllingparameter is the slope of the velocity spectrum, that is, whether the spectrum is upwards, downwards,or horizontal in the range of the period shift.55Chapter 7Strength and Ductility Demands in Relation to Velocity SpectrumIn Chapter 5, the relationship of the fundamental structural period to the predominant period ofthe ground motion was observed to be a determining factor for the type of inelastic displacement responsecurve that will occur in an earthquake. In the Chapter 6, the controlling parameter on the orientation ofan IDRC was extended by the R-it equation to include the slope of the elastic velocity response spectrum.In this chapter, the influence of the slope of the velocity spectrum will be examined further by relatingthe inelastic displacement response curves obtained from the strength- and ductility- demand analyses inChapter 4 to the velocity spectra of the respective earthquakes. It is hoped that this examination willexplain the observations made earlier on the behaviour of the curves.A recapitulation of the general conclusions and observations on the behaviour of the inelasticdisplacement response curves in Chapter 4 is as follows:1) In unfiltered earthquakes, the curves are mostly vertical with only a few curves sloping outwardbut none sloping inward.2) In filtered earthquakes, outward- and inward-sloping curves occur frequently.The curves for the filtered and unfiltered motions of the El Centro EW record were exceptions to theseobservations. For the unfiltered motion, the curves were more outward-sloping than vertical; for thefiltered motion, there was no inward-sloping curves. The reasons for these behaviour will be revealedby investigating the slopes of the velocity spectra.In the previous chapter, a slightly downward-sloping velocity spectrum (that is, decreasingvelocity with increasing period) with a ve/vi ratio of 1.2 was considered to be the dividing spectrum forthe three types of inelastic displacement response curves. In general then, the occurrence of outward-sloping, inward-sloping, and vertical inelastic displacement response curves can be taken to correspondapproximately to ascending, descending, and horizontal velocity response spectrum, respectively. This56generalization is useful for analysis and explanation purposes, and it also helps in the understanding ofthe concept in a general sense.First, the occurrence of outward-sloping curves in filtered earthquakes is examined. Each of thevelocity response spectra for the five study filtered motions can be noticed to contain a long, upward-sloping part at the beginning at low periods, especially in the spectra for the Mexico City SCT (Figure1.7b) and the Artificial Richmond (Figure 2.17b) earthquakes. Furthermore, these spectra slopedownwards beyond the characteristic periods. These velocity spectrum characteristics explain theabundance of both outward- and inward-sloping inelastic displacement response curves in the filteredearthquakes. The behaviour of the response curves for the Modified Taft and the Mexico City CUIPmotions follow the same pattern since both earthquakes are actually filtered motions. Outward-slopingand inward-sloping curves occurred in the Mexico City CUIP earthquake because its velocity spectrum(Figure 4.1c) is an ascending and then descending curve. No inward-sloping curves occurred in theModified Taft earthquake because its spectrum (Figure 2.17b) levels off to a horizontal line, rather thana descending one, after a long ascending portion.The velocity spectra for the Taft and Caltech motions (Figures 2.14b and 2.15b) are horizontalfor nearly all periods and so their curves were observed to be mostly vertical. These two spectra alsohave an ascending initial portion, but the range of periods for each portion is so short that outward-slopingcurves cannot be produced.The reasons for the unusual behaviour of the response curves for the filtered and unfiltered ElCentro EW motions become clear once their velocity response spectra (Figure 2.16b) are examined. Theoccurrence of outward-sloping curves in the unfiltered motion comes from the fact that the spectralvelocity continues to increase for a long range of periods. The absence of inward-sloping curves in thefiltered motion is a result of the spectrum being horizontal at high periods, which is different from thedescending characteristic in other filtered motions.Therefore, the results on the behaviour of the inelastic displacement response curves in the studyearthquakes can be explained by the characteristics of the velocity spectrum. It should be emphasized that57the velocity response spectrum, and not the acceleration response spectrum, is the controlling parameter.A good example is given by the results for the filtered El Centro ENV ground motion. As mentioned inChapter 5, the acceleration and velocity spectra for this earthquake give two very different characteristicperiods, with the one from the acceleration spectrum being lower. In the range of periods between thesetwo characteristic values, the acceleration spectrum is sloping downwards, signifying vertical curves, andthe velocity spectrum is sloping upwards, signifying outward-sloping curves. The outward orientationof the curves for these periods, as shown in Figure 4.38, demonstrates that the velocity spectrum indeedcontrols the response of structures. This also illustrates that the type of inelastic displacement responsecurves to be expected should be based on the characteristic period from the velocity spectrum.The discussion so far has been concentrated on the characteristics of velocity spectra. However,acceleration response spectra rather than velocity response spectra are widely used by engineers. Since,in most earthquakes, the characteristic periods from acceleration and velocity spectra are roughly thesame, acceleration spectra generally can give good insights into the response of structures. But it shouldbe cautioned that, when the two characteristic periods from the two types of spectrum are not the same,the velocity spectrum should be used since it has a more direct effect on structural response.Finally, it should be noted that, based on the R-ii equation and on the discussions above, theconcept of the slope of the velocity spectrum influencing the orientation of the inelastic displacementresponse curves is really applicable to all earthquakes, filtered and unfiltered. The large differences inbehaviour seen between the curves for the two types of ground motion are mainly attributed to thedifferences in their characteristic periods. Typical, unfiltered earthquakes tend to have very lowcharacteristic periods, and so the effects of the upward-sloping spectrum are significant to only a verysmall range of structures; on the other hand, the characteristic periods of filtered earthquakes are shiftedto higher values than those associated with unfiltered earthquakes, which means the existence of a long,ascending spectrum.58Chapter 8Relevance of R-pc Equation to Seismic Design CodeIn Chapters 4 to 7, the effects of filtered earthquakes were examined through the use of theinelastic displacement response curves. In this section, the significance of the R-u equation and of theslope of an inelastic displacement response curve to the seismic design code will be discussed.8.1 Seismic Design of Structures in Canada8.1.1 General ProcedureThe seismic design of structures in Canada is generally based on static analyses. In theseanalyses, factored quasi-static lateral loads are applied to the structures, and the design forces ordeflections are obtained. The sum of the quasi-static lateral loads applied to a structure, or the totaldesign base shear, is given in the National Building Code of Canada (NBCC) [National Research Councilof Canada 1990a] asV U V^(8.1)V is the lateral seismic force to be applied to the base of the structure, and V, is called the equivalentelastic force. U is a factor which represents the level of protection in the structure and is taken to be aconstant value of 0.6. R is the force reduction factor which is structure-dependent and reflects the abilityof the structure to behave well inelastically. The only term which is dependent on the characteristics ofthe earthquake is V,.598.1.2 Seismic Response FactorThe equivalent elastic force V. is given by the Code asVe — vSIFW (8.2)v is the zonal horizontal ground velocity expressed as a ratio to 1 m/s and it reflects the magnitude of thedesign earthquake expected for the area. S is the seismic response factor which is a function of periodof the structure. I is the importance factor emphasizing the necessity of the structure to survive theearthquake. F is the foundation factor which is dependent on the soil conditions and will be discussedfurther later in this chapter. W is the total weight of the structure.For Vancouver, v is given as 0.2 m/s. If I and F are taken to be one, and W is expressed as Mgwith M being the total mass of the structure and g the acceleration due to gravity, then the equivalentelastic force for Vancouver can be written asVe — M (0.2 Sg) (8.3)This is the basic definition of a force and the term in the brackets is the acceleration value expressed ing. Therefore, the factor S must reflect the characteristics of the design earthquake. Indeed, the codegives the seismic response factor as a function of the natural period of the structure and of the ratio ofpeak horizontal ground acceleration Z. to peak horizontal ground velocity Zv , as shown in Figure 8.1.For Vancouver, the values for Z. and Zv are four, and thus the ratio Za/Zv is one. Figure 8.2 shows theseismic response factor for Za/Zv =1 after it has been multiplied by 0.2g. It can be seen that the curveresembles an elastic acceleration response spectrum for a typical, unfiltered earthquake --- large valuesat low periods and decreasing values towards higher periods. Since the V. will be multiplied by the factorU in Equation 8.1. It is easier to group U and V. together for explanation purposes. The seismicresponse factor curve with the U factor included is also shown in Figure 8.2.608.1.3 Expected Response of StructuresAs explained earlier, the actual design force level or the yield strength of the structure is less thanthat given by the solid curve in Figure 8.2 to allow for inelastic structural behaviour. In fact, it is 1/Rtimes the value of the term UVe , as dictated by Equation 8.1. Based on the equal-displacement principle,the code implies that the structure designed for this yield level will exhibit a level of ductility equal invalue to the force reduction factor. Graphically, the implication of the code on the seismic design ofstructures is shown by the vertical inelastic displacement response curve in Figure 3.5. It should be notedthat the lateral load-deflection diagram in Figure 3.5 is expressed in absolute values of forces anddisplacements and not in relative values of force reduction factors and ductilities.8.1.4 Foundation FactorThe code recognizes the fact that the soft soils can amplify the seismic waves travelling throughthem and thus the foundation factor F is included in the calculation of the equivalent elastic force for thedesign of a structure. A maximum value of two is suggested for F by the NBCC 1990 for very loose orsoft materials greater than fifteen meters in depth. There is a limit, however, to the product term FS.For Za less than or equal to , this product term need not exceed a value of three. For the Vancouverarea, where za= zy, the seismic response factor curve, when modified by a foundation factor of 2, is thatshown in Figure 8.2. It should be noted that the foundation factor simply shifts most of the seismicresponse curve upwards; the original, downward-sloping shape of the curve is retained. Also, theconcept of equal-displacement response is still implied by the curves in Figure 8.2.618.2 Combined Effects of Amplification and Period ShiftThe effects of the combination of amplification and period shift can be understood when theeffects of each characteristic based on the observations made so far in this thesis are put together. Fora structural design based on a descending spectrum, an amplified earthquake would just shift the spectrumupwards and thus would cause larger deflections, as shown by the load-deflection response in Figure 8.3a.In another situation, when only a change in the slope of the response spectrum occurs as a result of a shiftin the predominant period of the ground motion without a change in the force level, the response of thesame structure would also be forced to a higher ductility, as governed by the outward-sloping inelasticdisplacement response curve in Figure 8.3b. Now, when the two situations are combined, the adverseeffects of the two characteristics acting together are apparent (see Figure 8.3c). The inelasticdisplacement response curves for the first two situations are included in this case for comparison. Toachieve the same ductility as the original intended value, a lower R value or a higher yield strength isneeded. This strength may even be larger than the level obtained with the foundation factor included.8.3 Effects of Filtered Earthquakes in RichmondRealizing the severe effects of amplification and period shift acting together, it is interesting tocompare the Artificial Richmond earthquake generated in Chapter 2 to the code design spectrum forRichmond. The elastic design response spectrum for Richmond is determined using a foundation factorof two. This spectrum, along with the acceleration response spectrum for the Artificial Richmond groundmotion, is drawn in Figure 8.4. Also shown in the figure are their respective inelastic spectra for aductility of four. It can be seen that, even though the elastic spectra for the two earthquakes are verydifferent, their inelastic spectra are similar in shape. In fact, the spectral values match each other veryclosely over the entire range of periods shown. This shows that structures designed in accordance withthe code for a ductility of four will be required to exhibit only this level of ductility as a result of this62ground motion. In other words, these structures would survive the Artificial Richmond earthquake. Itshould be noted that this comparison by no means evaluates the adequacy of structural design inRichmond; it simply indicates the performance of structures in this one particular earthquake. It isinteresting also to examine how well a structure designed for Richmond would resist the Mexico City1985 earthquake loading. As in the above comparison with the Artificial Richmond earthquake, theresponse spectra for the code and the Mexico City SCT motion are drawn together in Figure 8.5. It canbe seen that the inelastic spectrum for a ductility of four is well below the inelastic spectrum for the sameductility for the Mexico City earthquake. This means that structures designed for ductility of 4 would beforced to much larger ductilities and thus severe damage or even collapse could be expected.8.4 Recommended Design of Structures for Filtered EarthquakesThe previous section emphasized that both an amplification of the earthquake intensity and achange in the slope of the earthquake response spectrum would lead to increased ductility demands in astructure. In these situations, a large yield strength is required for response to a reasonable level ofinelastic deflections. Retaining the use of the equal-displacement assumption, the foundation factor givenby the code more or less accounts for the higher earthquake intensity only. Another factor is thus neededto account for the effects of the sloping inelastic displacement response curves.From Chapter 6, it was shown that the orientation of an inelastic displacement response curvecould be approximated by the R-A equation. The use of this equation adds new dimensions to seismicdesign of structures. First, it implies that the design spectrum can be quite different from the typical,downward-sloping spectrum generally used for design. Second, as a consequence of the first idea, theR-A equation leads to values of R and which are not equal to each other. Third and most important ofall, with the inclusion of the spectral velocities, whose values are dependent on the structural period, theR-it equation implies that the force reduction factor itself is period-dependent. Period-dependent forcereduction factors have been proposed by Tso and Naumoski [1991] and Moehle and Aschheim [1993].63Both suggested a relatively lower and linearly varying factor for low structural periods; for highstructural periods, the values of R and j are identical. In particular, Moehle and Aschheim proposed theforce reduction factor be varied linearly from one at a period of zero to the value of the ductility at aperiod corresponding to the predominant period of the expected ground motion, while Tso and Naumoskiproposed a range of periods between 0 and 0.5 second for the linearly varying R. Furthermore, Tso andNaumoski found that using their suggested factor would decrease the ductility demand in the short periodrange to nearly the level given by the equal-displacement criterion. The results of this thesis, especiallythe IDRC for the study filtered earthquakes, indicate that, indeed, R should be varied from one to theductility value over low structural periods. The R-A equation determines automatically which type ofresponse, namely outward-sloping, inward-sloping, or vertical inelastic displacement response curve, isapplicable to the design period (that is, the design structure). However, a time history or a velocityspectrum of the design earthquake is not always available, which will prevent the use of the R-/L equation.In this case, the predominant period of ground vibration, which is often available for sites with soft soils,can be used to separate the outward-sloping and vertical inelastic displacement response curves, as impliedby Moehle and Aschheim's proposed adjustment to R and indicated by the results of this thesis. If thepredominant period of the design earthquake is not known, then the cut-off point can be assumed to be0.5 second, as used by Tso and Naumoski and by the design code in the seismic response factor curve.The following procedure is recommended for the seismic design of structures, especially forstructures excited by filtered earthquakes. First, the desired level of ductility response for the structureis set, and its structural properties, such as stiffness and damping, are determined. Then, when thevelocity response spectrum of the design earthquake is known, the R-A equation can be used to determinethe level of lateral forces to be applied to the structure. In other words, the value of R to be used withthe base shear equation (Equation 8.1) is determined. It should be noted that the design force level is stillrelative to the spectral value given by the elastic acceleration spectrum, that is, to the term Ve inEquations 8.1 and 8.2. Thus, the foundation factor F can still be used to reflect the greater intensity ofa filtered ground motion.64Part IIAnalytical Studies of Reinforced Concrete StructuresIn Part I, the effects of filtered earthquakes on structural response were illustrated by means ofsingle-degree-of-freedom, elasto-plastic systems. Also, the impact of these effects on the seismic designof structures was discussed. The response of typical, multi-storey frames subjected to filtered earthquakesand designed in accordance with the codes will be examined in this part of the thesis through dynamiccomputer analyses. The results from these analyses will be used as the basis for experimental studies tobe performed later.Shear wall structures will also be studied in this part of the thesis to provide further informationon the effects of filtered earthquakes on code-designed structures and to offer a basis of comparison forthe results of the analysis of the frames. The emphasis of the examination, however, will still be on thebehaviour of the code-designed frames.65Chapter 9Dynamic Analyses of Reinforced Concrete Frames Designed for R of 49.1 Purpose and ProcedureIn the 1985 Mexico City earthquake, most of the collapsed structures were reinforced concreteframe buildings between six and fifteen storeys in height. It is of interest to examine the behaviour ofthis type of structure in filtered earthquakes.Properly detailed reinforced concrete frames are able to dissipate energy through inelastic action.A displacement ductility of four is usually assumed, so that, based on the concept of equal-displacementresponse, a structure only needs to be designed for a force reduction factor of four.In this analysis, a multi-storey reinforced concrete frame will be designed in accordance with thecodes for a force reduction factor of four. Then the structure will be modelled in a plane-frame computerprogram, and its response in filtered and unfiltered earthquakes will be examined. The response to benoted include ultimate roof displacements, maximum base shears, and plastic hinge rotations. The lateralstiffness and natural frequencies of the structure will also be determined.9.2 Design of Study FrameFor the analysis, a typical 6-storey reinforced concrete frame in a multi-frame building wasselected. The number of storeys is within the range of building heights of the collapsed frames in MexicoCity in 1985. The geometry of the frame follows the 6-storey, 3-bay frame used in a design example inthe Concrete Design Handbook published by the Canadian Portland Cement Association [1985].However, a three-quarter-scale version of this frame was studied in this analysis. Figure 9.1 shows thegeometry and the dimensions of the study frame. It is desirable to correlate analytical and experimental66studies and therefore the smaller scale was chosen in order to meet the limitations imposed by the sizeconstraints of the experiments undertaken with this study; however, the gravity loadings on the framewere taken as those given in the Handbook. Dead loads on the structure include self-weight of thereinforced concrete, partition loadings, and weight of insulation; live loads were chosen to be thoseestimated for office occupancy. Finally, the frame was assumed to be located on firm ground inVancouver.The design of the study frame followed the rules set out by the 1985 edition of the NationalBuilding Code of Canada [National Research Council of Canada 1985] and the 1984 metric version of theCSA-CAN3-A23.3 concrete code [Canadian Standards Association 1984]. The former was used toprovide the design earthquake loadings, and the latter was used to determine the detailing of the framemembers. (The 1985 edition of the National Building Code was used because the study was performedbefore the publication of the 1990 edition. Even though the method for calculating the design earthquakeloading was revised in the latest edition, the factored end results given by the two editions are the same.)To arrive at the design forces in the members, a static analysis on the study frame was performed.In this static analysis, the stiffness of the beams and columns was taken to be fifty and eighty percent ofthe gross moment of inertia of the sections, respectively. (More elaborate properties were used insubsequent analyses.) The strength of the concrete was assumed to be 30 MPa and the estimated yieldstrength of the reinforcing steel was 400 MPa. The dead and live loads were those specified above, andthe quasi-static seismic loads were those calculated in Appendix A. The corresponding load factors werealso applied. Based on the forces determined in the static analysis, the beams, columns, and slabs weredesigned and detailed. The criterion of strong column-weak beam was strictly followed. The final designfor the first floor of the frame is shown in Figure 9.2. For simplicity, the dimensions and detailing ofthe members were kept the same for all floors.Since it was designed for a force reduction factor of four, this study frame will be referred to as"R4 frame" in this thesis.679.3 Member Properties of Study FrameFrom the final design of the frame, five critical sections in the beams and columns are identifiedand shown in Figure 9.2. The transformed gross area and the transformed cracked moments of inertiafor these sections were calculated and are listed in Table 9.1. The values for the beams are based onrectangular cross sections for negative bending and on T-beam cross sections for positive bending.The flexural strength or resistance of each of the five sections was also determined. Indetermining the member strength, the unfactorekl material properties of the concrete and reinforcing steelwere used. The concrete was considered to have a stress-strain relationship under compression given as- 4'12(  ec0.002(9.1)fc and ec are the compressive stress and strain in the concrete, respectively, and fc' is the maximumcompressive stress for the material which was assumed to be 30 MPa. The strain at this maximum pointis 0.002, and an ultimate strain of 0.003 was assumed. The elastic modulus was considered to be5000Vfc' MPA and the modulus of rupture to be O.6%/f' MPa. A bilinear, elasto-plastic stress-strainrelationship was assumed for the reinforcing steel having a yield point at a stress of 400 MPa and a strainof 0.002, corresponding to an elastic modulus of 200,000 MPa. The moment resistance of each sectionwas calculated based on a linear strain distribution over the depth of the member.For the beams, the flexural resistance at each critical section is given in terms of a moment-curvature diagram with the effects of axial load ignored. This diagram is approximated by establishingthree points signifying first crack in the concrete, first yield in the tensile steel, and crushing of theconcrete. The moment-curvature relationships under positive and negative bending for the beam sectionsare shown in Figure 9.3. The yield moments can be extracted from the plots, and they are tabulated inTable 9.1. For the columns, however, the effects of axial loads on the sections cannot be ignored, andtheir resistances are expressed in axial load-moment interaction diagrams. Figure 9.4 contains theinteraction diagrams for the two column sections. The dotted lines in this figure mark out the moments68at first yield of the tensile steel. The yield moments and section properties at zero axial load for the twosections are listed in Table 9.1.9.4 Computer ProgramIn this analysis, the computer program DRAIN-2D developed at the University of California atBerkeley was used to determine the response of the frame under earthquake loadings. This program hasalready been used in Chapter 4 to determine the strength and ductility demands on the single-degree-of-freedom systems. Two capabilities of the program are important to this analysis. One is the capabilityto perform a static analysis on the structure before the time-step analysis. This is ideal for the applicationof gravity loads on the structure. The results of the static analysis are carried over into the dynamicsession. Another capability of the program is the availability of different types of elements to modelvarious members of a structure. Beam-column elements and beam elements with degrading stiffness areavailable. The yield strength of a beam-column element is governed by a moment-axial load interactiondiagram and the post-yield action in the plastic hinge in this element follows the elasto-plastic moment-rotation relationship. The beam element with degrading stiffness has a moment-rotation relationship asshown in Figure 9.5. This relationship has zero strain hardening stiffness and follows the rules proposedby Takeda et al.[1970]; it is not influenced by axial loads in the member.9.5 Modelling of Study FrameThe modelling of the structure was governed by the DRAIN-2D program. The study frame wasmodelled as a plane frame consisting of line elements and nodes. The line elements were located alongthe centrelines of the members, and the nodes were located at beam-to-column intersections and at theintersection of the primary and secondary beams (see Figure 9.1). Rigid end zones, called endeccentricities, were put into the appropriate beam and column members to account for the relatively stiffer69regions within the beam-column joints. The base of the structure was assumed to be fixed at groundlevel. This line-element frame is shown in Figure 9.6.The properties of the members followed those given in Section 9.3. The beam members weremodelled as beam elements with degrading stiffness, while the column members were modelled as beam-column elements. Each exterior or interior beam was composed of three members (see Figure 9.6). Forthe two outer members, the cross-sectional area was taken as the average of the two areas for Section A-Ashown in Table 9.1, and their moment of inertia was the value shown for negative bending. For thecentre member in the exterior or interior beam, the average area was again used but the moment of inertiawas the value shown for positive bending. The Takeda model for the stiffness-degradation moment-rotation relationship was used. Even though the slab was not actually modelled, it was considered to bea rigid diaphragm so that all the nodes in one floor have identical horizontal displacements.The mass of the structure consisted of unfactored dead loads at each level with twenty-five percentof the snow load included at the roof. The mass in each floor was distributed proportionally to the fourbeam-column joint nodes. Only translational mass in the horizontal direction was considered.Static loads were also applied to the structure. These are factored, vertical gravity loads whichare a combination of 1.25 times the dead loads and 1.05 times the live loads. The loads were representedby concentrated forces at each node of the frame (see Figure 9.6).P-A effects were also considered in the analyses to account for the nonlinear effects resulting fromthe large displacements of the structure. The influence of these effects was approximated by utilizing aconstant geometric stiffness, which is based on the axial forces in the columns under gravity loads.9.6 Structural Properties of Study FrameIn Section 9.3, various section properties of the members of the study frame were calculated.In this section, properties of the entire structure are determined. These properties are listed in Table 9.2and their derivations are explained in the following subsections.709.6.1 Elastic Lateral StiffnessThe stiffness of the structure under lateral loadings can be determined by applying a set of lateralloads to the structure and noting its deflection at the top. One horizontal load was applied to each floor,and these loads formed a distribution which is similar in shape to that specified by the National BuildingCode (see Appendix A). Their sum is equal to the total base shear under the frame. Then, the lateraldisplacement of the roof was determined from the static analysis part of the DRAIN-2D program. Fora total base shear of 100 kN, the displacement at the top is 26.6 mm, giving a lateral stiffness of 3.76kN/ram. This is the elastic stiffness because there are no plastic hinges formed. This property can beused in the study of elastic response of the frame.9.6.2 Lateral Roof Deflection at Adjusted Code Yield LoadOne interesting response for a code-designed structure is the lateral deflection of the roof at thecode yield level. This value is important because it is used in the definition of ductility. It can bedetermined from the elastic lateral stiffness calculated above, but the code yield load has to be found first.In Appendix A, the design total base shear for the frame and the distribution of this force overthe height of the building are determined But this base shear is given as a design value; the actual valuein the real structure will be higher when all the load and resistance factors are considered. This has beenexplained by Mital et al. [1987]. They pointed out that the base shear expected under a real structure inthe design earthquake should bev — *IX Q va^4) (9.2)71where V is the calculated, unfactored design base shear and Va is the actual base shear which will becalled the "adjusted code yield load" in this thesis. aQ, 1,G, and are the earthquake load factor, the loadcombination factor, and the combined material resistance factor, respectively. Using an earthquake loadfactor of 1.5 (for V calculated in accordance with the 1985 design code), a load combination factor of 0.7(corresponding to the combination of live loads and earthquake loads that was used as the governingcombination in the design of the study frame), and a material resistance factor of 0.67 for reinforcedconcrete, the expected base shear capacity of the frame is 164 kN, which is 1.58 times higher than theunfactored design base shear. Dividing the actual base shear by the elastic lateral stiffness then gives thelateral deflection at the adjusted code yield load as 43 6 mm.9.6.3 Natural FrequenciesThe fundamental vibration frequency of the study frame can be determined from a free-vibrationanalysis of the structure in the dynamic part of DRAIN-2D. In the analysis, the structure was subjectedto a weak impulse and was then allowed to vibrate freely for a long duration. The result of this simpleanalysis reveals the fundamental period of the study frame to be 1.48 sec.This value was verified by modelling the structure in a computer program called DYNA [Law1978]. This program was developed at the University of British Columbia to perform dynamic, modalanalysis of plane structures. The modelling of the structure in this program is almost the same as inDRAIN-2D. However, DYNA cannot model the rigid end zone at the beam-column joints so that theshorter lengths between end eccentricities were used. One advantage of using DYNA is that it cancalculate the frequencies for higher modes of vibration. For the study frame, this program determinedthe periods for the first three modes of vibration to be 1.42 sec, 0.45 sec, and 0.25 sec.The fundamental periods given by the two programs are quite close to each other, but both area lot higher than the code value of 0.6 sec indicated in Appendix A. More precisely, the periodscalculated in this section are about two times higher! One reason for the higher fundamental period in72this study is that the stiffnesses of the members in the computer model were based on cracked,transformed cross sections rather than on the relatively larger uncracked, gross sections which would givea larger stiffness.For this research, the fundamental period of the study frame of 1.42 second was chosen.9.6.4 DampingA damping of five percent of critical was assumed to exist in the structure in the dynamicanalyses. This is a reasonable value for reinforced concrete structures and is the same value used for theresponse spectra of the various earthquakes shown earlier in this thesis. For the modelled structure inDRAIN-2D, damping was given as mass-dependent and stiffness-dependent damping. This combinationresulted in 5 percent of critical damping in each of the first two modes of vibration and 7.5 percent in thethird mode.9.7 Dynamic, Inelastic Response of R4 FrameInelastic response of the R4 frame was computed by the DRAIN-2D program for various scaledearthquake loadings. In some cases, elastic analyses were also carried out. Four groups of study wereperformed with the scale of the earthquake being the variable among the studies. In all dynamic analyses,a time step of 0.01 second was used. This time interval is small compared to the fundamental period ofthe frame (1.42 seconds). In each elastic or inelastic analysis, the maximum lateral displacement of theroof, the maximum total base shear under the frame, and their respective time of occurrence were noted.739.7.1 Response to Richmond Surface MotionsThe first examination was carried out for the eight bedrock and surface ground motions that werestudied in Chapter 2 of this thesis for the municipality of Richmond. It should be recalled that three ofthe four bedrock motions are considered representatives of the design earthquake for the Vancouver areawith a maximum ground acceleration of 0.21 g. The inelastic responses of the R4 frame in the eightground motions are listed in Table 9.3. The results show clearly that the response for each of the surfacefiltered motions is higher than the response for the respective original bedrock motion. The filteredmotions for the three 0.21g "design" earthquakes produce maximum roof displacements of the frame that,on average, are 3.3 times higher than the corresponding average displacement induced by the bedrockmotions. The adverse effect produced by filtering is even greater for the low-intensity Modified Taftmotion (8.6 times). Therefore, as indicated by the response of the single-degree-of-freedom systems,greater ductility demands are imposed on structures by filtered earthquakes.It is interesting to evaluate the response of the R4 frame under the loading of the actual MexicoCity SCT EW ground motion. When a DRAIN-2D inelastic analysis of the frame was performed for theearthquake, the execution of the program was terminated during the strong motion portion of the inputbecause the displacement limit set in the program was exceeded. The value of this limit is greater thanthe total height of the frame. Therefore, the R4 frame would have collapsed under the loading of theunsealed Mexico City earthquake.9.7.2 Response to Earthquakes Scaled to 0.21gAs noted earlier, the peak horizontal ground acceleration for the design earthquake for Vancouveris given as 0.21 g. In the previous section, the response of the R4 frame in the Taft and El Centro EWground motions scaled to this maximum acceleration level was found. In this study, four moreacceleration records were scaled for 0.21 g. Two of them are the El Centro NS 1940 record and the74Olympia N86E 1949 record, and their accelerogram and elastic response spectra are shown in Figures 9.7and 9.8. These records, along with the Taft and El Centro EW motions, represent typical, unfilteredearthquakes. The other two earthquakes are the Mexico City SCT EW record and the Artificial Richmondrecord. Both of these events have been studied in the previous section and represent motions that havebeen filtered by distance and soft soils. Thus, a total of six earthquakes were scaled to 0.21 g and therequired scale factors are listed in Table 9.4. Table 9.4 also shows the inelastic response of the R4 frameto these scaled earthquakes. The results show that the responses to the four unfiltered earthquakes areclose to each other but are quite different from the responses corresponding to the filtered motions. Inthe typical, "design" earthquakes, the maximum deflections reached are only about 0.48% of the totalheight of the building, which is acceptable in terms of amount of damage expected. In the filteredearthquakes, however, the frame deflects to a level that is beyond the large displacement limit set (greaterthan the total height of the building) in the computer program. In this situation, the execution of theprogram is terminated automatically, and the frame is assumed to have collapsed.9.7.3 Response to Earthquakes Scaled to 0.21m/sIt was noted earlier in Chapter 1 that the peak ground acceleration can be a poor indicator of thedamage potential of an earthquake, so there is justification for using another reference parameter toevaluate intensity of ground motion. In the National Building Code of Canada 1990, the design base shearof a structure is directly proportional to the peak horizontal ground velocity (see Equations 8.1 and 8.2);also, it has been found that the response of long-period structures is affected by velocity more than byacceleration. Therefore, the six accelerograms which were scaled earlier for 0.21 g were scaled to givea maximum velocity of 0.21 m/s, the velocity value given for the design earthquake expected inVancouver. The scale factors required for the six study earthquakes are shown in Table 9.5 along withthe inelastic response of the frame in these scaled earthquakes.75Unlike the response to ground motions of 0.21g, the response to the earthquakes scaled to 0.21m/s shows similar values between the maximum deflections reached in filtered earthquakes and those inunfiltered earthquakes. No excessive deflection in the frame is observed, and even the deflection attainedin the Mexico City SCT earthquake is exactly the same as that in the Taft earthquake. On average, themaximum roof deflection is only 0.53% of the building height, which is about the same as that for theresponse to the unfiltered earthquakes scaled to 0.21g.9.7.4 Response to Earthquakes Scaled for Ductility of 4Since the study frame was designed for a force reduction factor of four, it should be capable ofundergoing inelastic deflections up to a displacement ductility of four. Earlier in Section 9.6.2, the lateraldeflection at the adjusted code yield load for the study frame was determined. Responding to a ductilityof four means that the ultimate displacement of the study frame should be four times this yield deflection,or 175 mm For each of the six earthquakes used in the previous two sections, the record was scaled untilthe desired ultimate tip displacement was reached in the inelastic R4 frame. After the required scalefactors were known, elastic analyses of the study frame were also performed using the scaledaccelerograms. The results of this study are summarized in Table 9.6.Table 9.6 shows that the scale factors needed by the unfiltered earthquakes to produce a ductilityof four are larger than those required by the filtered earthquakes. In particular, a factor of 2.35 had tobe applied to the Taft motion, while the Mexico City SCT was reduced to only 53% of its originalintensity.The difference in the scale factors for the two types of ground motion is reflected in the resultsof the elastic analyses. Both the maximum base shears and the maximum deflections are larger inunfiltered earthquakes than in filtered ones, even for the long-period structure under study.769.7.5 Selected Response of R4 Frame at Ductility of 4In addition to the values of maximum tip displacement and maximum base shear, other responsesof the study frame deflecting to a ductility of four were also noted. Shown in Figures 9.9 to 9.14 are thetime histories of the roof and first floor displacements in each of the scaled earthquakes. No significantdifferences are observed for the displacement time histories under the various earthquake loadings. Theframe has a permanent offset at the end of each earthquake.The response at a critical section of the frame where very large inelastic actions occur is also ofgreat interest. Figure 9.15 shows the locations where plastic hinges formed in the study frame beamswhen it is deflected to ductility of four under the six study earthquakes. This figure indicates that theplastic hinging is limited to the beams in the lower floors only. But, in general, the frames in theunfiltered earthquakes are observed to have more plastic hinges than the frames in the filtered motions.The results of the dynamic analyses show that the section which exhibits the most inelastic action in eachearthquake is the exterior end of the first-floor exterior beam. The bending moment time histories andthe moment-hinge rotation relationships at the critical section on the right side of the frame in the sixearthquakes were evaluated. The former are shown in Figures 9.16 to 9.21 and the latter are drawn inFigures 9.22 to 9.27. These results will be examined later in Chapter 15.9.8 RemarksThe design principles for the R4 frame implies that the structure would deflect to a displacementductility of four in the design earthquake. This ductility limit corresponds to a roof displacement of 175mm or a drift ratio of 1.0%. Particularly for the unsealed Taft ground motion, which has characteristics(Amax =0.18 g and Za/Z,=1) close to those of the design earthquake, the response of the R4 frameshould correspond approximately to the design ductility. But, when the loadings of the four unfilteredearthquakes scaled to 0.21 g were applied to the frame, an average drift ratio of only 0.48% was77obtained, which means that the design ductility level would not be reached in the design earthquake. Thisresult is also indirectly indicated by the large scale factors needed by these unfiltered earthquakes to reacha ductility of four. The response to earthquakes scaled to 0.21 m/s is also lower than the expected designlevel. In fact, they are almost identical to the results in the analyses for earthquakes scaled to 0.21 g.Therefore, it can be concluded that reinforced concrete frames designed in accordance with the codeswould survive typical, design earthquakes with minor damage only. The reason for the relatively lowdeflection response will be explained in Chapter 13. However, when the design earthquakes were filteredthrough soft soils, the response of the R4 frame increased threefold to an average drift ratio of 1.4%,which is about 1.5 times the design ductility level. As a result, for such earthquake inputs, the R4 framewould be required to exhibit ductilities that are larger than its design ductility. The ductility level istremendously high in the case of the actual Mexico City earthquake loading. On the other hand, the R4frame deflects to a drift ratio of 0.8% in the Artificial Richmond earthquake. That is, the R4 framewould survive the Artificial Richmond earthquake but would suffer severe damage in the actual MexicoCity earthquake. This is the same result as indicated by Figures 8.5 and 8.6, in which the code inelasticdesign spectrum was seen to lie below the inelastic spectrum for the Mexico City ground motion but tomatch almost perfectly the elastic spectrum for the Artificial Richmond earthquake.It can be argued that subjecting the R4 frame to the strong, filtered motions of Mexico City andArtificial Richmond would not be fair to a frame designed for firm ground conditions. However, itshould be remembered that the focus of the analyses was to demonstrate the effects of ground motionswhich have different characteristics than the expected "design" earthquakes.Although not indicated in Tables 9.3 to 9.6, it was found that the times at which the maximumroof deflection and the maximum base shear occurred during each earthquake were usually close to eachother.78Chapter 10Dynamic Analyses of Reinforced Concrete Frames Designed for F of 210.1 Purpose and ProcedureIn the previous chapter, the study frame was assumed to be situated on hard ground in Vancouver.Thus, the value of the foundation factor in the calculation of the design base shear was one. If thestructure is assumed to be situated on a thick layer of soft soils, as in Richmond, then the foundationfactor will be two and the design base shear will be doubled. All other values in the determination of thebase shear remain unchanged.In this chapter, the dynamic, inelastic response of a frame designed for a force reduction factorof four and a foundation factor of two is examined. The purposes are to examine how well framesdesigned with the higher foundation factor behave in the study filtered earthquakes and to provide somecomparisons for the results of the analyses of the R4 frame. The emphasis is on the response of frameslocated in Richmond to surface motions expected for that area. It was shown earlier in the previouschapter that a ductile frame designed for a foundation factor of one would collapse under the Mexico Cityearthquake. It is interesting to see what difference a stronger frame makes to the response under the samesituation. The analysis procedure is similar to that followed in Chapter 9; the structure is modelled inthe DRAIN-2D program and then subjected to different ground motions.10.2 Design of Study FrameThe study frame used in this analysis was the same as the R4 frame in Chapter 9, except that thedetailing was different because of the higher strengths required. Again, the design of the structurefollowed the specifications given by the 1985 edition of the National Building Code of Canada and the791984 edition of the CSA CAN3-A23.3 reinforced concrete code. Since the frame geometry anddimensions were the same as before, the dead loads and the live loads also remained unchanged. Butdesigning the structure for a foundation factor of two means that the design base shear would be twicethe value used for the R4 frame. Design forces in the frame based on the larger lateral loads applied weredetermined; and the resulting frame design is shown in Figure 10.1. Once again, the detailing in eachfloor was kept the same over the entire height of the building.103 Member Properties of Study FrameFor the design shown in Figure 10.1, the areas and the moments of inertia of the sections in themembers were determined. These values are shown in Table 10.1. The moment-curvature curves forthe beam sections and the axial load-moment interaction diagrams for the column sections are shown inFigure 10.2 and Figure 10.3, respectively. The procedures to arrive at these properties have beenexplained in Section 9.3.10.4 Computer Program and Modelling of Study FrameThe computer program DRAIN-2D was used for the analysis, and the modelling of the structurewas the same as for the R4 frame. The differences in this frame, though, were in the values for themember properties of cross-sectional areas, stiffness, and yield moments of the members. The valuesshown in Table 10.1 were used.10.5 Structure Properties of Study FrameFrom the determination of the structure properties of the R4 frame in the previous chapter, it wasobserved that the properties were highly dependent upon the stiffness of the frame members. With new80stiffnesses for the members, the elastic lateral stiffness, the lateral deflection at code yield load, the naturalfrequencies, and the damping were determined for the F2 frame following the procedures used earlier inSection 9.6. The results are summarized in Table 10.2.10.6 Dynamic, Inelastic Response of F2 FrameThe dynamic analysis of the F2 frame was carried out using the surface motions in Richmond asobtained in Chapter 2 and the Mexico City SCT motion. The motions were not scaled. Subjecting theF2 frame to the Richmond surface motions provides an indication of the actual response of structuresdesigned for this municipality, since these motions are the end products of filtering the design earthquakesfor the region. The response to the Artificial Richmond ground motion is of great interest.As in the analysis of the R4 frame, the maximum roof displacement and the maximum base shearof the F2 frame in each earthquake were determined. The results of the dynamic analyses of the F2 frameare tabulated in Table 10.3. It should be noted that all the maximum base shears are larger than theadjusted code yield load of 328 kN.In addition to maximum tip displacement and maximum base shear, other responses were alsoobtained for the F2 frame from the analysis of the structure for the unscaled Mexico City SCT EW groundmotion. Figure 10.4 and Figure 10.5 are the time histories of the roof and first floor displacements andthe time history of the moment at the end of the exterior beam, respectively. The moment-rotationrelationship for this latter location is shown in Figure 10.6. Finally, the locations of the plastic hingesin the frame resulting from the various unsealed ground motions are indicated in Figure 10.7.10.7 RemarksTable 10.3 shows a wide variation of response for the unscaled ground motions. The resultsrange from a maximum deflection of 97 mm in the Artificial Richmond earthquake, to over twice this81value in the Filtered Taft motion (216 mm), and to collapse under the Filtered El Centro EW earthquakeloadings. This is the same pattern of response observed for the R4 frame when subjected to the Richmondfiltered motions, as indicated in Section 9.7.1; the maximum roof displacements of the R4 frame in theArtificial Richmond, Filtered Taft, and Filtered El Centro EW ground motions are 138 mm, 206 mm, and407 mm, respectively. Also, the response of the F2 frame can be seen to be close to but slightly largerthan that of the R4 frame in the three filtered design earthquakes; on the other hand, the F2 framedeflected to a lower level in the Artificial Richmond ground motion.From the previous section, the roof deflection at the adjusted code yield load was calculated tobe 65 mm. An ultimate displacement of 260 mm would then correspond to a ductility of four for the F2frame. As for the R4 frame, the maximum roof deflections reached in the unsealed earthquakes for theF2 frame are all quite low compared to the ductility-of-four displacement, with the exception of theFiltered El Centro EW earthquake. A possible explanation for the response in the Filtered El Centro ENVmotion may be the steep, upward-sloping velocity spectrum in the period range covered by the inelasticF2 frame, which induces large ductility demands in the frame. But more importantly, the results showthat the six-storey frame would survive the loading of the actual Mexico City ground motion when it isdesigned for a higher strength.82Chapter 11Dynamic Analyses of Reinforced Concrete Nominal Shear Walls11.1 IntroductionIn the previous chapters, analyses were performed on reinforced concrete frames in which lateralloads are resisted by beams and columns. However, most buildings in the Vancouver area, especially tallapartments or offices, are usually of shear wall-type construction. Shear wall is a general term forstructural elements such as elevator shafts, stairwells, central core units, and exterior or interior walls thatresist lateral loads [ACI Committee 442 1971]. These elements have a much greater lateral stiffness thanthat of columns because of their large widths and thicknesses. Since they provide rigidity for resistinglateral loads and can carry large shear forces, reinforced concrete shear walls are often introduced intomulti-storey frame buildings to reduce deformations and minimize damage to non-structural componentsunder service loadings. Under severe earthquake loadings, they can provide sufficient strength and energyabsorption and dissipation capacities to prevent collapse and loss of life [Vallenas, Bertero, and Popov1979]. A dual structural system, known as frame-wall system, consisting of a shear wall connected inseries to a frame can offer considerable advantages over ductile moment-resisting space frames.The term "shear wall" is misleading since the critical mode of resistance is rarely associated withshear [Park and Paulay 1975]. A reinforced concrete shear wall in a multi-storey building is really adeep, slender cantilever beam that deforms in a bending mode [Blume, Newmark, and Corning 1968; ACI1971]. High bending moments, in addition to high shear forces and large compression forces from gravityloads, can occur in a shear wall during an earthquake.In the strong earthquakes in Mexico City and the San Francisco Bay area, shear wall structureswere observed to respond well with only minor damage in the walls [EERC 1989]. Nevertheless, it isstill of interest to examine the behaviour of shear walls in filtered earthquakes because of the possible83occurrence of the combination of large moments, shear forces, and axial forces at the bases of the walls,together with many excursions into yielding. The base of a shear wall is a critical region where plasticdeformations are expected to occur. Therefore, the effects of filtered earthquakes on the force anddeformation parameters at the bases of shear walls need to be examined.11.2 Past Research on Shear WallsIn the mid-1970's, a comprehensive research program on dynamic analyses of isolated structuralwalls was carried out by the Portland Cement Association for the National Science Foundation [Derechoet al. 1977, 1978]. The program contained extensive analytical studies on the characteristics of strongground motions and on the response of isolated shear walls to these motions. Using the stiffness-degrading model in the computer program DRAIN-2D, the researchers at the Portland Cement Associationfound that the frequency characteristic and the duration of the ground shaking affect the response of shearwalls significantly. In particular, they realized that, for structures going through extensive plasticdeformations and thus significant increases in their effective period of vibration, an accelerogram withan ascending spectrum can be expected to produce greater deformations than another accelerogram of thesame intensity but with a descending spectrum. Their "ascending" accelerogram was that for the ElCentro EW 1940 record. They also found that long duration of shaking increases the number of inelasticoscillations which a structure will undergo.11.3 Analysis ProcedureIn this study, a simple, nominal shear wall structure is designed and then modelled in a dynamic,nonlinear computer analysis program. The behaviour of this structure during filtered and unfilteredearthquakes is examined with attention directed towards the moment, shear, and plastic deformation atthe base of the wall and the lateral displacement at the roof level.8411.4 Design of Study Shear Wall StructureAn eight-storey, single-shear wall building was chosen as the study structure. The building layouthas the shear wall centrally located with four columns at the corners. The shear wall is rectangular inshape and stretches across the entire 6000-mm width of the building. Its thickness is 200 mm and hasa height to width ratio of 5. The column dimensions are 400 mm by 400 mm. The floors of this buildingwere assumed to be flat slabs with no supporting beams. The layout and dimensions of this studystructure are given in Figure 11.1. The shear wall is located at the centre of the building to carry largeaxial loads, as most walls do. Because the building is symmetric in plan, it is not subjected to torsion.(Accidental torsion will be excluded.) The absence of frames on either side of the wall also eliminatesthe problem of frame-wall coupling during earthquakes. It is assumed that the structure deforms alongthe plane of the wall with no out-of-plane movements. The study structure was considered to be locatedin Vancouver, and was designed in accordance with the National Building Code of Canada 1990 and theCSA CAN3-A23 .3-M84 concrete code for a force reduction factor of two. With the critical design loadcombination of 85% of the dead loads and 100% of the seismic loads, the wall and columns weredesigned and the results are shown in Figure 11.2. The calculation of the design seismic loads is shownin Appendix B. The planar wall has concentrated reinforcement of four No.20 bars at each end and twolayers of No.10 bars at 400-mm spacing in between. The columns contain the minimum two percentreinforcement as permitted by the code. The detailing shown was kept constant throughout the entireheight of the building.11.5 Member Properties of Study Shear Wall StructureBearing gravity loads and resisting lateral loads, the study shear wall is subjected to both axialloads and bending moments. But because the axial load varies along the height of the building, the sectionproperties of the wall change between storeys. Based on unfactored strengths of concrete and reinforcing85steel, the moment-curvature relationships at the various axial load levels were obtained and are drawn inFigure 11.3. From these relationships, the stiffness (cracked El), yield strength, and post-yield strainhardening stiffness of the section at each level were determined and are tabulated in Table 11.1. Theunfactored axial load-moment interaction curve for the column section is shown in Figure 11.4. Thecracked El of the column section was used as the stiffness and was calculated to be 0.3638 x 109kN.mm2 For simplicity, the stiffness of the column was kept constant over the entire height of thebuilding. The floor slabs were assumed to act as rigid diaphragms, constraining the lateral displacementsof the wall and columns at each storey to be the same.11.6 Computer Program and Modelling of Study Shear Wall StructureDynamic, nonlinear analyses of the study shear wall structure were performed using the computerprogram DRAIN-TABS [Guendelman-Israel and Powell 1977]. This program works on the sameprinciples and with almost the same subroutines as DRAIN-2D, but the former can analyze plane framesoriented in three-dimensional space. In this program, the shear wall is modelled as a single column (seeFigure 11.5) consisting of elements that exhibit degrading stiffness in their moment-rotation relationships.The moment-rotation relationship at each axial load level follows the rules proposed by Takeda et al[1970] and has the strain-hardening stiffness calculated in the previous section. Static axial loadscorresponding to 85% of the design dead loads were applied to the shear walls before the start of thedynamic loading. The corner columns were modelled as regular beam-column elements with a yieldsurface similar to the axial load-moment interaction diagram shown previously. The slabs were assumedto be rigid diaphragms. Mass- and stiffness-dependent damping were considered, and the damping valuesused correspond to five percent of critical in each of the first two modes. The structure was restrictedto deform in the plane of the shear wall.8611.7 Structural Properties of Study Shear Wall Structure11.7.1 Elastic Lateral StiffnessUsing the static analysis part of the DRAIN-TABS program, the lateral roof deflection of thestructure subjected to a set of lateral loads was determined. The lateral loads have a distribution identicalto the quasi-static seismic loads distribution specified by the code for the structure (see Appendix B). Theloads were applied at the floor levels and the additional concentrated force at the roof was included.When the sum of these lateral loads equal to 100 kN, the deflection at the top was found to be 28.6 mmand there were no plastic hinges in the structure. This gives an elastic lateral stiffness of 3.50 kNimm.11.7.2 Lateral Roof Deflection at Adjusted Code Yield LoadThe design base shear for the structure was calculated to be 459 IN. With an earthquake loadfactor of one and with the load combination factor and the material resistance factor having the same valueof 0.7, the adjusted code yield load for the structure remained as 459 kN. From the elastic lateralstiffness determined above, the deflection at this adjusted base shear is 131 mm.11.7.3 Natural FrequenciesThe vibrating frequencies of the study shear wall structure were determined using the computerprogram PITSA [Tam 1985]. Like the program DYNA, PITSA is a modal-analysis program developedat the University of British Columbia. The first three periods of the building were calculated to be 1.75sec, 0.22 sec, and 0.15 sec. A free vibration test of the structure was also carried out using the programDRAIN-TABS. In the test, a weak impulse was applied to the structure and then the free vibration of thestructure was tracked. The free vibration period was found to be the same as that from PITSA.8711.7.4 DampingIn DRAIN-TABS, damping can be assigned to the structure as mass-dependent and/or stiffness-dependent damping. In this analysis, both types were used and values for them were calculated to give5% critical damping in each of the first two modes of vibration and 13% in the third mode.11.8 Dynamic, Inelastic Response of Study Shear Wall StructureThe set of dynamic analyses carried out for the shear wall structure is similar to that used withthe R4 frame. The study shear wall structure was analyzed under filtered and unfiltered earthquakeloadings to examine its elastic and inelastic response to these motions. The time step used in the analyseswas 0.01 second. Five strong ground motions were used in this study. They are the records of Taft S69E1952, El Centro NS 1940, El Centro EW 1940, Mexico City SCT EW 1985, and the Artificial Richmondmotion. The first three are considered to be typical, unfiltered motions, while the last two are strongfiltered ground motions. Derecho et al. [1978] have noted that the El Centro EW 1940 motion has anascending response spectrum, and the previous chapters have shown the unusual results given by thisearthquake. This earthquake can be regarded as having a ground motion which is between the unfilteredand the filtered motions. For each earthquake, the maximum lateral displacement of the top of the shearwall and the maximum base shear of only the shear wall were noted. The base shears under the columnswere not considered because these forces are very small compared to the wall base shear and so can beregarded as negligible.11.8.1 Response to Earthquakes Scaled to 0.21gThe design earthquake for Vancouver is expected to have a peak ground acceleration of 0.21g.For the five study earthquakes selected, the scale factors required to attain this maximum acceleration88value in these motions are listed in Table 9.3. Under these scaled earthquakes, the inelastic responsesof the study shear wall structure were found and are listed in Table 11.3. Similar to the results for theR4 frame subjected to earthquakes scaled to 0.21g, the results in Table 11.3 show that the shear wallstructure would collapse in the filtered earthquakes, but that the average maximum roof deflection in theunfiltered earthquakes would be about 0.5% of the total building height. Unlike the R4 frame, in whichthe maximum base shear is confined to the yield value, the shear wall model has strain hardening stiffnesswhich allows the base shear to grow with the increasing deflections. Therefore, very large shears wereobtained in the analyses using the filtered motions.11.8.2 Response to Earthquakes Scaled for Ductility of 2Since the shear wall was designed for nominal ductility, it was expected to reach a displacementductility of two in earthquakes. Therefore, the five ground motions were scaled to a level such that thestudy shear wall structure would be forced to deflect inelastically to a ultimate displacement of 260 mmwhich is twice the roof deflection at the adjusted code yield load. The scale factors required are shownin Table 11.4 along with the maximum base shears obtained in the structure in the five earthquakes.Table 11.4 also lists the elastic response of the shear wall structure under these same earthquake loadings.Again, the results obtained here for the shear wall structure are similar to those for the R4 frame. Highscale factors are needed by the unfiltered earthquakes to force the shear wall to displace to a ductility oftwo, while low factors are sufficient for the filtered motions. However, there is one major differencebetween the results for the two structures. In the inelastic analysis of the shear wall structures, themaximum base shear is larger in an unfiltered earthquake than in a filtered earthquake; for the R4 frame,the maximum base shears are identical because of no strain hardening.One point to be noted is that, in the Taft ground motion scaled by the factor shown in Table 11.4,the lateral deflection of the shear wall actually increases suddenly to very large values near the end of theearthquake (see Figure 11.6). The deflection for ductility of two is reached at the beginning of the89earthquake. This large increase in the deflection at the end of the event is considered to be an errorproduced by the computer program, because the intensity of the ground motion at this time is too low forsuch a large response and no resonant effect was observed in the response of the structure.11.8.3 Selected Response of Shear Wall at Ductility of 2Results other than maximum base shear and maximum tip displacement were also obtained fromthe DRAIN-TABS analyses for the shear wall structure deflecting to a ductility of two. Figures 11.6 to11.10 show the time histories of the roof and first floor displacements in the five earthquakes. Thesudden increase in deflection in the Taft motion as mentioned earlier can be seen clearly in Figure 11.6.In fact, the shear wall was observed to have very large deflections at the end of most of the earthquakesstudied. The time histories of the base shear and the base moment were also noted and are presented inFigures 11.11 to 11.20. The base moment in the inelastic study structure is also the moment at thecritical section, because the shear wall has only one plastic hinge which occurs at the base. Moment-hingerotation responses at this location in the study earthquakes are shown in Figures 11.21 to 11.25.11.9 RemarksThe overall results obtained for the shear wall structure are very similar to the results for the R4frame. In general, like the R4 frame, a nominal shear wall structure designed in accordance with thecodes would behave well in "design" earthquakes but would be forced to excessively large displacementsin filtered ground motions.90Chapter 12Static-Load-To-Collapse Analyses for Study Frames and Shear Wall12.1 PurposeIn the dynamic analyses of the study frames and shear wall structure, several properties such aselastic lateral stiffness, yield deflection, and natural frequencies of the structure were determined.However, there is one more property that is an important characteristic of a structure; it is the behaviourunder increasing static lateral loads, or the lateral load-lateral deflection relationship. Yielding inindividual members of the structure would be considered in this relationship.Earlier in Chapter 3, the load-deflection relationship under static, lateral loads for a simple,elasto-plastic structure was shown to be a bilinear curve with a definite yield point and a horizontalsegment after this point. Subsequently, dynamic response of the single-degree-of-freedom systems wasexpressed in terms of this curve. This bilinear curve is also used by the seismic design codes as a basisfor the design philosophy.In this chapter, the static lateral load-deflection relationships for the two study frames and thestudy shear wall structure are determined. The objectives are to examine the behaviour of these structuresunder increasing static loads and to see how well they are represented by the idealized bilinear curves.12.2 ProcedureThe basic step in obtaining the lateral load-deflection relationship is to apply an increasing butstatic base shear to the structure. The base shear is represented by a set of lateral forces over the entireheight of the structure, and its distribution is that specified by the National Building Code of Canada (seeAppendices A and B). The base shear is increased in small increments and the lateral displacement of91the roof at each load level is noted. When the yield moment at a section in a member is reached, a plastichinge is inserted at that location. This reduces the overall stiffness of the structure so that the load-deflection behaviour over the next load step becomes different. The process of increasing the base shearand introducing plastic hinges is repeated until excessive deflections occur in the structure. The resultsof the analyses are then presented in load-deflection plots.12.3 Computer Program and Modelling of StructuresThe static-analysis portion of the DRAIN-2D and DRAIN-TABS programs were used to determinethe lateral deflections of the frames and shear wall, respectively. The model for each structure was thesame as that used in the respective dynamic analysis. In order to model plastic hinges, a different modelof the end of a member was needed. The end eccentricity was replaced by a very stiff, short member;in addition, the intersection of this rigid member and the main member was represented by two separatenodes instead of one. Since they are common to the two members, these nodes initially have the sametranslations and rotation initially. However, when the yield moment at this section is reached, the rotationconstraint is relaxed (see Figure 12.1), and a pair of equal but opposite external moments are applied atthese nodes. The value of these moments is equal to the yield moment and this value was kept constantthroughout the analysis. Therefore, the effect of strain-hardening was not modelled.12.4 Lateral Load-Deflection Plots for Study StructuresThe results of the static-load-to-collapse analyses for the three study structures are shown inFigures 12.2 to 12.4. Also shown in each figure are the locations of the plastic hinges formed and thesequence in which they occurred.From a comparison of the load-deflection plots for the three study structures, a significantdifference among the shapes of these plots is observed. For the shear wall structure, the lateral load-92deflection curve resembles very closely a bilinear, elasto-plastic relationship; for both frames, however,their behaviour under increasing lateral loads are more curvilinear with no definite yield point. The typeof load-deflection behaviour displayed seems to be dependent upon the pattern of plastic hinges in thestructure. In the response of the frame, a distinct yield point is absent because a large number ofredundancies are present in the structural system and occurrences of yielding at these redundancies arenot simultaneous. A reasonable mechanism only came into effect in the frame when all the column basesbecame inelastic. Figures 12.2 and 12.3 show that almost all of the plastic hinges in the frames occurredin the bottom half of the building. This is consistent with the plastic hinging patterns in the R4 frame inthe dynamic, inelastic analyses. The absence of yielding in the upper storeys can be attributed to the factthat it is impractical to design all members to be exactly at the yield strengths required for seismic effects.It is to be noted that column hinging, aside from that at the bases, occurred in the F2 frame while noneoccurred in the R4 frame, even though both frames were designed to satisfy the strong column-weak beamcriterion. One reason for this difference is that the sum of the yield strengths of the columns above andbelow a joint is relatively closer to the sum of the yield strengths of the beams framing into the joint inthe F2 frame than in the R4 frame. Therefore, the possibility of yielding occurring in the columns underlarger lateral loads on the structure after the beams have yielded is higher in the F2 frame. The static-load-to-collapse analysis for the shear wall structure shows that only one plastic hinge is formed in thisbuilding. For a shear wall, a plastic hinge at the base is all that is needed to produce a mechanism. Thisexplains the near perfect elasto-plastic behaviour in the lateral load-deflection plot of the shear wallstructure. Clearly, the idealized bilinear curve used by the codes is more appropriate for representing theresponse of a shear wall structure than a frame.One interesting point to be examined is the relationship of the strength obtained from the static-load-to-collapse analysis to the design yield load given by the seismic design code or, in this study, theadjusted code yield load. Two dotted lines denoting the adjusted code yield load and the load at theformation of the first plastic hinge are shown with each of the lateral load-deflection plots for the threestructures. The results indicate that the load at first yield for the R4 and F2 frames are 45% and 25%,93respectively, higher than their adjusted code yield load, while the two loads for the shear wall areapproximately the same. The reason for this lies in the design of the structure. In the normal designprocedure, a static analysis of the structure under the code quasi-static lateral loads is carried out to arriveat the design forces in the members, and then the sections are designed based on the requirement that thefactored resistances have to be larger than or equal to the factored design forces. This procedure wasfollowed in the design of the study structures. However, there is no restriction in the code as to howmuch larger the factored resistance can be. For the R4 frame, the factored design bending moment forthe end of the exterior beam was determined to be 167 kN.m, but a section with a factored flexuralresistance of 184 k.N.m was selected; the section resistance is therefore larger by 10%. This means thatthe actual lateral loads required to cause yielding in that section are higher than the design value. But theincrease in lateral loads necessary to overcome this extra strength is not the same as the amount of theextra strength, due to the presence of gravity loads. Figure 12.5 shows the linear relationship betweenthe applied total base shear and the moment at the exterior end of the exterior beam for the R4 frameunder elastic behaviour. The moment at zero base shear is caused by the initial static gravity loads. Thefactored design force and the factored resistance provided at this section are indicated, and it can be seenthat the a 10% increase in the moment translates to a 43% increase in the base shear. If the materialfactor is neglected from the resistance value, thus giving the so-called adjusted moment resistance, thenthe base shear required for first yielding is 237 kN, or an increase of 117% from the initial factored baseshear. This yield base shear value of 237 kN agrees well with the first-yield load obtained from the static-load-to-collapse analysis. The reasons for the close match between the adjusted code yield load and theload at first yield for the shear wall are that, in this case, the wall has a factored resistance almostidentical to the factored design moment at the base, and the moment from the gravity loads is zero.In addition to the large differences between the adjusted code yield load and the load at first yield,the load-deflection plots for the frames also show that these structures can continue to resist a lot morelateral load after first yield. Again, this can be attributed to the non-simultaneous formation of the plastic94hinges in these structures, and the difference between the lateral load at first yield and the actual level ofthe yield plateau can be regarded as an inherent overstrength.12.5 Yield Points of Study FramesIt was indicated in the previous section that the lateral load-deflection response for the shear wallstructure shows a perfect bilinear behaviour while the response for the two study frames are morecurvilinear in behaviour, containing no apparent yield points. Furthermore, the frames were observedto remain elastic well after the adjusted code yield load is reached. But, in a study of the seismicresponse of a structure, a yield point is required because it serves as a basis for the calculation of thedisplacement ductility. It is therefore necessary to establish the yield points for the R4 and F2 frames.In a report on ductility evaluation, Park [1991] pointed out four different definitions used byresearchers to obtain the yield displacement for a structure with no definite yield point. Based on a load-deflection diagram, the four possible yield displacements are:"1)^the displacement when yielding first occurs,2) the yield displacement of the equivalent elasto-plastic system with the same elasticstiffness and ultimate load as the real system,3) the yield displacement of the equivalent elasto-plastic system with the same energyabsorption as the real system, and4)^the yield displacement of the equivalent elasto-plastic system with reduced stiffness foundas the secant stiffness at 75% of the ultimate lateral load of the real system."These four definitions are shown in Figure 12.6. Park stated that the yield displacement obtained by thelast definition is the most realistic for reinforced concrete systems as it accounts for the reduction instiffness resulting from cracking.For this thesis, however, the yield point is considered to be the response at the formation of thefirst plastic hinge in the structure. This was chosen because the corresponding yield strength is closer to95the expected yield strength given by the design code. The code implies that a structure would developall its plastic hinges simultaneously when the design yield strength is reached. But, as shown in Figures12.2 and 12.3, there is overstrength in both study frames. The overstrength is the result of the designphilosophy of designing the section with the largest moment under the factored code seismic loadings firstand then making the designs of all levels of the building identical. So the ultimate load is significantlylarger than the expected yield capacity, and the use of this load to define the yield strength of thestructure, as indicated by Park's last three definitions, would result in large overestimates. In addition,a yield strength at the ultimate load of a structure with overstrength would signify a greater yielddeflection compared to the yield deflection corresponding to the formation of the first plastic hinge or tothe elastic deflection corresponding to the adjusted code yield load.12.6 RemarksIt was shown in Section 12.4 that the first plastic hinge in the R4 frame occurred at a lateral loadlarger than the adjusted code yield load and that the frame could continue to carry more lateral loads afterfirst yield. Furthermore, it was decided in Section 12.5 that the point of first hinging is considered to bethe yield point of the study frame. These effects lead to the conclusion that, for the R4 frame, theresponse at the adjusted code yield load is still in the elastic range and the 175-mm lateral roof deflection,as indicated in Section 9.7.4, would not correspond to the design ductility (four for the R4 frame).However, the emphasis of this thesis is on the responses of frames subjected to design earthquakes, whichmeans that the deflection of interest is still four times (for the R4 frame) the deflection at the adjustedcode yield load, regardless of the state and behaviour of the structure.96Chapter 13Examination of Dynamic Displacement Response with Load-Deflection Plots13.1 IntroductionWhen using the lateral load-deflection plots to represent dynamic seismic response of structures,the end point of the curve signifies the maximum displacement reached by the vibrating structure. In thedynamic analyses of the frames and shear wall structure, maximum elastic and inelastic displacements invarious earthquakes were obtained. In this chapter, these results from the dynamic analyses are examinedwith respect to the lateral load-deflection behaviour obtained in the previous chapter. The procedure isto plot the dynamic results as data pairs of maximum base shear and displacement on the correspondingload-deflection plot. In this examination, the maximum base shear and maximum displacement areassumed to occur at the same time even though they may not.13.2 Response of R4 Frame, F2 Frame, and Shear Wall StructureThe results of the elastic and inelastic dynamic analyses of the three study structures are plottedin Figures 13.1 to 13.6. The first three figures show the results for the R4 frame subjected to earthquakesscaled to 0.21g, to 0.21m/s, and for displacement ductility of four. Figure 13.4 contains the results forthe F2 frame, and Figures 13.5 and 13.6 are for the shear wall structure subjected to earthquakes scaledto 0.21g and for displacement ductility of two, respectively. In each figure, the results are plotted againstthe corresponding lateral load-deflection curve obtained in the previous chapter. Also drawn in each ofthe figures is a dashed line, which is an extension of the straight, initial portion of the load-deflection plot,to mark out the elastic response of an infinitely strong structure.9713.3 Discussions of Elastic ResponseThe elastic dynamic responses of the study structures are examined first. In Figures 13.3 and13.6, all the dynamic results do not fall exactly on the elastic lines (dashed) derived from static-load-to-collapse analyses. This points out quickly that the lateral load distributions used in the static-load-to-collapse analyses are not a good representation of the actual seismic load distributions. As a matter offact, all the response points lie above the elastic line, signifying that the load distribution is more of auniform type than an approximately upper triangular type, since the former produces smaller deflectionsthan the latter under the same total lateral load [Chamey and Bertero 1982]. In other words, second modeeffects are significant in the response. However, it can be seen in the plots for the R4 frame and theshear wall structure that the response for the Mexico City SCT and the Artificial Richmond filteredmotions are closer to the elastic line. Since the lateral load distributions given by the code reflect theloadings for the fundamental response of the structure, the results mean that the response of a structurein filtered earthquakes is in the fundamental mode while higher modes can be assumed to be containedin the response in typical, unfiltered earthquakes.13.4 Discussions of Inelastic ResponseThe inelastic response can be examined with respect to the total base shear or the lateral roofdisplacement.In terms of total base shear, the inelastic dynamic analysis results, like the elastic response, alsodo not fall exactly on the load-deflection curves obtained from static-load-to-collapse analyses. However,while it is observed that the dynamic results lie close to the static-load-to-collapse results for the twoframes, this agreement is not valid for the shear wall. Also, the good agreement for the frames appliesto a large range of deflections. The reason for the poor match for the shear wall is that strain hardening98at the plastic hinge is considered in the dynamic analyses but not in the static-load-to-collapse analyses.Therefore, the shear wall modelled in the dynamic analyses can resist much larger shear forces.In terms of lateral roof displacement, an interesting observation can be seen for the inelasticresults. As mentioned earlier in Chapter 9 and 11, the deflection responses of the study structures toearthquakes scaled to the design level of 0.21g are, on average, about half of the design deflection inaccordance with the code. This design deflection is identified in each of Figures 13.1 to 13.6. Thesefigures reveal that most of the inelastic responses are only slightly larger than the respective deflectionsat first yield. This is especially the case for the R4 frame responding to unfiltered earthquakes of 0.21g,the R4 frame responding to all earthquakes of 0.21m/s, and the study shear wall structure responding tounfiltered earthquakes of 0.21g. Therefore, very large inelastic actions in these structures are not implied.13.5 Examination Using Inelastic Displacement Response CurvesIn the previous two sections, the elastic and inelastic responses were examined separately. In thissection, they are examined together with the use of the Inelastic Displacement Response Curve. Each ofthe static load-deflection curves in Figures 13.1 to 13.6 can be represented by an idealized bilinear seismicresponse curve such as that shown in Figure 3.1, even though the yield plateau may be significantlyhigher than the level of first yield in the structure. So when the points signifying elastic and inelasticresponses for a particular earthquake are plotted on the load-deflection diagram and then joined by astraight line, a segment of the Inelastic Displacement Response Curve is obtained. This was done for theductility-of-four response of the R4 frame and for the ductility-of-two response of the shear wall structure.The resulting response curves are shown in Figure 13.7 for the frame and in Figure 13.8 for the shearwall. From these two figures, it can be seen that the curves for the unfiltered earthquakes of Taft, ElCentro EW, El Centro NS, and Olympia are almost vertical. The filtered Artificial Richmond groundmotion produces curves which slope outwards for both the frame and the shear wall. But in the case of99the Mexico City SCT EW earthquake, the curve for the frame slopes outwards while that for the shearwall slopes inwards.The difference in the response in the Mexico City earthquake can be explained by what has beenlearned so far regarding the response of structures in relation to the predominant period of an earthquake.As calculated earlier, the fundamental periods for the R4 frame and the shear wall structure are 1.42seconds and 1.75 seconds, respectively. The fundamental period of the shear wall is closer to thepredominant period of the ground motion which is 2.0 seconds. In fact, the period is close enough to fallin the range of periods giving inward-sloping inelastic displacement response curves. This concept hasbeen explained in Chapter 5.Therefore, it is concluded that the responses of the frame and the shear wall somewhat follow thebehaviour of structures represented by the elasto-plastic single-degree-of-freedom systems.100Chapter 14Examination of Local Curvature Ductility in R4 Frame14.1 IntroductionCurvature at a section of a flexural member is defined as the change in rotation over the lengthof a very short segment about the section. Yield curvature, (AY' is attained when the bending moment atthe section reaches the yield value. From strength of materials, it can be calculated by(14.1)where M is the yield moment and El is the flexural rigidity of the member. Once the short segment hasyielded and thus becomes a plastic hinge, the curvature can continue to increase with very little increasein the moment. This has been shown by the moment-curvature relationships for the various sections inthe R4 and F2 frames in Chapters 9 and 10. Curvature ductility, /to, is defined as the ratio of the totalcurvature, (Au, to the curvature at yield. That is,(1),1-14,^4)), (14.2)By definition, the plastic hinge rotation is the amount of increase in curvature beyond yieldmultiplied by the length of the hinge segment. In terms of curvature ductility, the plastic hinge rotationis given byOp - (V+ - 1) 4)y lp^ (14.3)101where Op is the plastic hinge rotation and lp is the length of the plastic hinge. This equation shows thedirect relationship between plastic hinge rotation and curvature ductility. Since the amount of plastichinge rotation has been described as a good indicator of damage in a flexural member, the curvatureductility can also be used as a damage index.14.2 Plastic Hinge LengthIn both the dynamic analysis and the static-load-to-collapse analysis in the earlier chapters, aplastic hinge in the model frame was assumed to be a point with no length. But in a real structure, itwould have a finite length. The actual length of a plastic hinge is very difficult to determine and dependson many loading parameters and member properties. Several expressions for it have been indicated anddiscussed by Park and Paulay [1975]. One expression was proposed by Mattock [1967] in his discussionof an expression given by Corley [1966]; it has a simple form1p - 0.5d + 0.05z^ (14.4)where d is the effective depth of the beam and z is the distance from the critical section to the nearestpoint of zero moment in the beam.For the end section of the exterior beam in the R4 frame, the effective depth is 384 mm undernegative bending. The value of z was obtained by examining the variation of moments in the exteriorbeam having one end yielded in negative bending under different static lateral loads. Based on the modelused in the static-load-to-collapse analysis, the distance z for the R4 frame was determined to be 1663mm. Therefore, using these values and Equation 14.4, the plastic hinge length at the exterior end of theexterior beam in the R4 frame was calculated to be 275 mm10214.3 Determination of Local Curvature Ductility from Tip Displacement DuctilitySo far in this thesis, the seismic response has been examined with reference to ductility based onthe lateral displacement of the top of a structure. Known as the tip displacement ductility, this was thetype of ductility employed in the strength and ductility demand analyses of single-degree-of-freedomsystems and in the dynamic analyses of the frames and shear wall. However, it is known that the globalresponse is different from the local response within a structure, and so the curvature ductility exhibitedat one section of a member is different from the global tip displacement ductility of the structure. Itwould be useful if the local curvature ductility can be determined based on a known tip displacementductility. Since the plastic hinge rotation is directly related to the curvature ductility, the amount ofplastic hinge rotation at a local section can be used to develop this relationship.Park and Paulay [1975] gave a formula relating the lateral displacement of the top of a frame andthe beam plastic hinge rotation. Based on terms defined in Figure 14.1, this formula isr^lbAu Ay + epb (14.5)where pb is identical to 9 defined in Section 14.1 and r is the number of storeys. The frame is assumedto have developed a perfect beam sidesway mechanism like the one shown in Figure 14.1. Withdefinitions given by Equations 14.1, 14.2, and 14.3 and with the tip displacement ductility JL equal toAu/Ay , the above formula can be rewritten as1 El114,^(11-1) A + 1 (14.6)r lc lb lp MyFor the study R4 frame, r =6, lc = 2738 mm, 1=6750 mm, lb =6412 mm, lp =275 mm,EI=2.717x1010 IN mm2, My =216.5 IN.m, and Ay =64 mm The values for E, I, and My are the sameas those used in the dynamic analyses in Chapter 9, and the moment of inertia is based on the cracked,transformed section of the beam member. Equation 14.6 then becomes103p. — 1.8716 (^— 1) + 1^ (14.7)Table 14.1 lists the curvature ductilities for various lateral roof deflections or tip displacement ductilities.Park and Paulay found that a tip displacement of 4 gives a curvature ductility of 16.7 for theirstudy frame with five storeys. Table 14.1 shows that this value of curvature ductility at a tip displacementductility of four was not obtained for the study R4 frame. The reasons for the discrepancy lie in thevalues used for the distance between the two plastic hinges in the beam, lb, and the yield displacementof the structure, Ay. For the R4 frame, lb is almost equal to the entire length of the beam, but Park andPaulay assumed lb to be only two-thirds of the total beam length. They also made many assumptions inthe calculation of Ay, including simultaneous yielding of all critical sections to produce a beam sideswaymechanism. In the present study, the yield deflection of the R4 frame is taken to be the displacementat first yield in the static-load-to-collapse analysis.14.4 Plastic Hinge Rotations from Static-Load-to-Collapse AnalysesWhen the Park and Paulay formula for local curvature ductility was applied to the R4 frame, theresults were found to be different from those obtained by the two researchers for their study frame. Toexamine further the level of curvature ductility in the study R4 frame, the results from the static-load-to-collapse analysis are studied.In the static-load-to-collapse analysis, a plastic hinge is modelled by two nodes at the samelocation but with different rotations. Therefore, the difference between the rotations of these nodes is theplastic hinge rotation. For the plastic hinge at the end of the first-floor beam on the loaded edge of theR4 frame, the amount of plastic hinge rotation at this location was determined at four levels of roofdeflections --- 175 mm, 250 mm, 350 mm, and 450 mm --- using the procedure for the static-load-to-collapse analysis. (The plastic hinge on the loaded edge is chosen for the large negative flexural hingingthat occurs there.) The total base shear was increased in small increments to reach the desired roofdeflections. The results are shown in Table 14.2. At a deflection level of 450 mm, the frame was very104close to a mechanism, making it very sensitive to small increases in the lateral loads. As a result, thisdeflection could not be obtained by this procedure; therefore, the value of the plastic hinge rotation wasdetermined by extrapolating from the results at the other three deflection levels.14.5 Plastic Hinge Rotations from Dynamic AnalysesThe DRAIN-2D program also indicates the maximum plastic hinge rotations in its output.Inelastic responses in the Taft 1952 and the Mexico City 1985 earthquakes are studied. For eachearthquake, the accelerogram input was scaled to develop roof deflections of 175 mm, 250 mm, 350 mm,and 450 mm. The results of these dynamic analyses are tabulated in Table 14.3. These values are themaximum accumulated primary hinge rotations computed by the DRAIN-2D program.Table 14.3 indicates that the two earthquakes give approximately the same plastic hinge rotationsat the same lateral roof deflection level. Therefore, the characteristics of the ground motion do notsignificantly influence the relationship between the tip displacement and the amount of local plastichinging. The relationship is dependent on the state of the structure.When the dynamic results in Table 14.3 are compared to the static results in Table 14.2, it canbe seen that they are very close to each other. Thus a static analysis can be used to give the plastic hingerotation directly without having to perform the dynamic analysis, provided the roof deflection is known.14.6 Local Curvature Ductility - Tip Displacement Ductility Relationship for R4 FrameThe previous two sections have shown that, given the tip deflection, the plastic hinge rotation canbe determined from a static-load-to-collapse analysis. In turn, when the hinge rotation at a section isknown, the curvature ductility exhibited at that location can be calculated from Equation 14.3 with knownvalues of the yield curvature and the plastic hinge length.105The plastic hinge length for the exterior beam of the study frame has already been determinedin Section 14.2; it was calculated to be 275 mm. The yield curvature can be calculated by Equation14.1. For the exterior beam in the R4 frame, with My = 216.5 kN.m and El = 2.717x1010 kN mm2,the yield curvature is 0.7968x1(16 mm-1. Based on Equation 14.3, the curvature ductilities at the end ofthe first-floor exterior beam at various levels of roof deflections were calculated and are shown in Table14.4. This table also lists the tip displacement ductilities for the different deflection levels. These weredetermined based on the deflection at first yield of the structure. For a tip displacement ductility of four,a local curvature ductility of 12.6 was obtained. This is close to the value of 16.7 calculated by Park andPaulay for their study frame.14.7 SummaryThe amount of curvature ductility exhibited at the critical section in the first-floor exterior beamin the R4 frame is a direct function of the amount of tip displacement ductility. The curvature ductilityis calculated after the plastic hinge rotation in that section is determined from a static-load-to-collapseanalysis. The rotation from a static analysis is about the same as that from a dynamic analysis. Eventhough a perfect beam sidesway mechanism is not achieved in the R4 frame, the relationship between thelocal curvature ductility and the tip displacement ductility compares well with that given by Park andPaulay for their perfectly hinged frame.106Chapter 15Examination of Response in Plastic Hinges in Study Structures15.1 IntroductionIn the previous chapter, it was found that the static-load-to-collapse analysis can indicate theamount of hinge rotation at a yielded section in a structure when the maximum tip deflection of thestructure is known. In this chapter, the plastic hinge rotation will be examined further through moment-hinge rotation relationships at the critical yielded exterior end of the first-floor exterior beam in the R4frame and at the base of the wall in the study shear wall structure. The moment-hinge rotationrelationships for the two study structures at the design ductility levels have already been shown inChapters 9 and 11. These were obtained from dynamic analyses using different earthquakes. Therefore,the response at a plastic hinge under unfiltered and filtered earthquake loadings can be compared.There are two reasons to look at the response at a plastic hinge. One is that plastic hinges areresponsible for the dissipation of earthquake energy input into the structure through hysteretic action. Theother reason is that, as mentioned earlier, the damage in a structural member can be evaluated by theaccumulated plastic hinge rotation.In the computer programs DRAIN-2D and DRAIN-TABS, the accumulated plastic hinge rotationis separated into primary and secondary rotations. The definitions of these rotations, based on the Takedamodel for the moment-rotation relationship of a yielded section, are shown in Figure 15.1. Recall thatthe Takeda model was used for the beams and shear walls in the dynamic analyses. In this model, thehinge rotation in a new yield excursion would begin at the previous maximum rotation attained.Therefore, the accumulated primary plastic hinge rotation is the same as the maximum hinge rotation.In the following sections, the response at the critical plastic hinge in each of the two structureswill be analyzed in terms of the number of cycles of inelastic behaviour and the amount of accumulated107plastic hinge rotation. When not specified, the term accumulated plastic hinge rotation shall mean the sumof the primary and secondary rotations.15.2 Plastic Hinge Response of R4 FrameThe response at the exterior end of the first-floor beam in the R4 frame deflecting to tip ductilityof four has been shown in Section 9.7.6. In particular, the moment-hinge rotation response at thislocation in various earthquakes was shown in Figures 9.22 to 9.27. From these figures, the number ofyield excursion and the accumulated plastic hinge rotations are obtained, and these values are tabulatedin Table 15.1. Also listed in this table is the accumulated primary plastic hinge rotation at the exteriorend of the beam on the other side of the frame. The results shown in Chapter 9 and in the second tofourth columns in Table 15.1 were taken from the same side of the frame. The reason for showing theresults for the two sides is that the plastic hinges at these locations behave differently. As shown in thetable, one end has approximately the same hinge rotations under positive and negative bending, and theother end has almost all hinging in negative flexure only. It can be observed that the large rotations undernegative bending have approximately the same value in the study earthquakes. This is expected becausethe ultimate roof deflections of the frame in all the earthquakes are identical. The previous chapter hasshown that the relationship between the tip displacement and the local plastic hinge rotation is independentof the type of ground motion applied to the frame. However, from Table 15.1, the accumulatedsecondary plastic hinge rotation is observed to be higher in filtered earthquakes than in unfilteredearthquakes. It is important to note that the filtered earthquakes force the beam section to yield moretimes than the unfiltered earthquakes.10815.3 Plastic Hinge Response of Study Shear Wall StructureFor the shear wall structure, there is only one plastic hinge in the structure and the response ofthis hinge at the base of the wall corresponding to a design tip displacement ductility of two is studied.The moment-hinge rotation responses have been plotted in Figures 11.21 to 11.25. Again, the responseis evaluated in terms of the number of cycles and the accumulated plastic hinge rotations. As shown inTable 15.2, the results indicate responses that are similar to that of the R4 frame. The number of cyclesand the accumulated plastic hinge rotations are larger in the filtered earthquakes; the maximumaccumulated primary plastic hinge rotation is about the same in all earthquakes. In addition, it can beobserved that the amounts of hinging in positive and negative flexure are approximately equal in filteredmotions but they are lopsided in typical, unfiltered earthquakes.15.4 RemarksFrom the examination of the plastic hinge response in the R4 frame and the study shear wallstructure, it can be concluded that the behaviour of a structure in a filtered earthquake is different fromthat in an unfiltered earthquake even though the same maximum tip displacements and the same maximumplastic hinge rotations are reached in the two types of ground shaking. In reaching a certain tip deflectionin a filtered earthquake, a structure has to go through more cycles of inelastic response and equaldeflections on both sides of the neutral position, the consequence of which is larger accumulated hingerotations. When the accumulated plastic hinge rotation is used as a measure of damage, a large amountof damage can be expected in a filtered earthquake.It was mentioned in Chapter 1 that longer duration of ground shaking causes a larger number ofhysteretic cycles and demands larger energy-absorption capacity in a structure [Hall, McCabe, and Zahrah1984]. The dynamic analyses of the study frames and shear wall demonstrate that this is indeed thesituation in filtered earthquakes, which have long durations of shaking.109Chapter 16Examination of Base Shears, Storey Shears, and Lateral Forces16.1 IntroductionIn Chapter 13, the dynamic response of the inelastic study structures were investigated bypresenting each response as a set of maximum lateral roof displacement and maximum total base shear.The emphasis, however, was on the maximum displacement; in this chapter, the total base shears areexamined.In addition, the manner in which the base shear is distributed along the height of an elasticstructure in filtered earthquakes is also studied. During the 1985 Mexico City event, many reinforcedconcrete buildings had structural failures in the top storeys [Borja-Navarette et al. 1987; Mitchell et al.1986; Mitchell 1987]. Although the reason for these failures is thought to be the excessive gravity loadsin the top storeys as a result of drastic changes in the type of occupancy, the idea of a different and severetype of lateral loading brought on by the filtered motion has been raised. Therefore, the storey shearsdeveloped in a structure during filtered earthquakes are examined.16.2 Dynamic Base Shears in Study StructuresThe maximum base shears obtained for the inelastic study frames and shear walls subjected todifferent earthquakes have been reported earlier in Chapters 9 to 11. Specifically, the response of the R4frame at a tip displacement ductility of four (Table 9.6) and of the shear wall structure at a tipdisplacement ductility of two (Table 11.4) were looked at.From Table 9.6, it can be seen that the maximum base shears in the frame structure for the sixearthquakes for a ductility of four are nearly equal. But the results given in Table 11.4 show that the base110shears in the shear wall structure are much lower in filtered earthquakes (about 600 kN) than in unfilteredearthquakes (800 to 1200 kN). The difference in the response of the two structures arises because theresponse of the shorter 6-storey frame is governed mainly by shear whereas the taller 20-storey framebehaves mostly in flexure. Since the base moments are the same in all earthquakes for a shear wallyielded at its base with no strain hardening, the higher base shears under the shear wall in unfilteredearthquakes indicate indirectly that the shear to moment ratio at the base of the wall is higher in theseearthquakes. These results can be attributed to second mode effects. The presence of the second modewould change the lateral load distribution and would lower the point of application of the total lateralforce. The shorter lever arm implies that a larger lateral force is needed for a given base moment.Therefore, it can be concluded that filtered earthquakes would produce approximately an upper triangularlateral-load distribution connected with response in the fundamental mode.16.3 Storey Shears and Lateral Forces in Elastic FramesIn the examination above, the base shears were obtained for earthquakes scaled for identical tipdisplacement ductilities. If the structure was to remain elastic, the base shears in these scaled earthquakeswould differ from each other, as demonstrated by the differently sloping inelastic displacement responsecurves noted in Part I. In this section, the distribution of base shear over the height of an elastic structureis examined using earthquakes scaled for identical elastic base shears.Two structures were analyzed for their elastic response. One is the R4 frame designed in Chapter9, and the other is a 20-storey, 3-bay frame studied by Tsai and Popov [1988] and shown in Figure 16.1.The former represents a short building while the latter, with a height to width ratio of five, represents atall building. These two structures were modelled in the DRAIN-2D computer program and weresubjected to the five earthquakes --- namely the Taft, El Centro NS, El Centro EW, Mexico City SCT,and Artificial Richmond motions --- that were used in the dynamic analyses of the study shear wallstructure, The earthquakes were scaled to produce a certain base shear under each structure. For the111R4 frame, the desired base shear was 658 kN, which is four times the adjusted code yield load for theframe and represents the shear force for an elastic R4 frame in the design earthquake for Vancouver. Thescale factors required by the earthquakes to give the desired base shear under the frame are listed in Table16.1. For the 20-storey frame, the base shear corresponding to an elastic structure deflecting to a tipductility of four is 683 kN. Table 16.1 also lists the scale factors for the earthquakes to reach this valueunder the structure.Using the DRAIN-2D program, the response of the two frames under elastic behaviour in eachearthquake were obtained and are shown in Figures 16.2 and 16.3. The results are expressed in termsof the envelope of storey shear for the entire duration of the earthquake and the profile of lateral seismicforces at the time of the maximum base shear. In addition, from the storey shear envelope, anotherprofile of lateral forces can be obtained by calculating the difference between the storey shears in twoadjacent levels over the height of the building. In fact, this distribution from the storey shear envelopeis a reflection of the quasi-static load distribution specified by the design code. By reversing theprocedure, then, the storey shear envelope specified by the code can be obtained from the specified lateralload distribution. The code lateral load distribution and its corresponding storey shear envelope for eachframe are shown as dotted lines in Figures 16.2 and 16.3. It is to be remembered that the codedistribution is based on the assumption of fundamental-mode response in the structure.The plots in Figures 16.2 and 16.3 show that the code lateral force distribution is matched in onlya few earthquakes. Moderately reasonable agreement is obtained for the 20-storey frame in the twofiltered earthquakes of Mexico City SCT and Artificial Richmond. These results can be explained bynoting the periods for the first two modes of vibration of the frames and their elastic spectral accelerationsin the earthquakes. The fundamental period of the 6-storey frame was determined to be 1.42 seconds,and its second-mode period was found to be 0.45 second. For the 20-storey frame, the first- and second-mode periods are 2.67 seconds and 0.91 second, respectively. In the three unfiltered earthquakes of Taft,El Centro NS, and El Centro EW, the elastic spectral acceleration for the second-mode is noted to beconsiderably larger than that for the fundamental mode for both the 6-storey and the 20-storey frames.112Therefore, second-mode effects would occur in each frame in the unfiltered earthquakes, and this isclearly shown by the lateral force distributions at the time of maximum base shear in these earthquakes,especially in El Centro NS for the 20-storey frame.The dependency of the response of the two frames on the relationship of the vibrational periodsto the spectral acceleration is even more apparent when the structures are subjected to filtered earthquakes.For the 6-storey frame, its spectral acceleration in the second-mode is very close to that in the first-modein both the Mexico City and the Artificial Richmond earthquakes. The response of the second modewould therefore be significant for the 6-storey frame in these two earthquakes, and this is reflected by thealmost uniform lateral load distributions shown in Figure 16.2. For the 20-storey frame, on the otherhand, the spectral acceleration for the second mode is very much lower than the first-mode, and this frameessentially responded in its fundamental mode in the two filtered ground motions, as shown in Figure16.3. Consequently, the lateral force distribution resembles the code distribution which is based onfundamental response. However, the response of the 20-storey frame in the Mexico City earthquake doesnot show the large concentrated load at the top that is specified by the code.In conclusion, a response in the fundamental mode of vibration is more likely in filteredearthquakes than in unfiltered earthquakes because the shift in the predominant period of the groundmotion exposes a larger range of structures to an ascending elastic acceleration response spectrum, whichallows the fundamental mode of response to dominate. This means that the code lateral force distribution,which is based on fundamental-mode response, would represent the seismic loading better in filteredearthquakes. But, as shown by the response to filtered earthquakes in Figures 16.2 and 16.3, themaximum storey shears in the top storeys are lower than the code storey shears. Therefore, thesuggestion of a damaging type of lateral loading produced by a filtered earthquake as the cause of the top-storey collapses in the 1985 Mexico City event is not justified. The real cause seems to be the highgravity loads brought on by changes in the type of occupancy in the upper storeys.113Part IIIExperimental Studies Using Reinforced ConcreteExterior Beam-Column SubassembliesIn Parts I and II of this thesis, analytical studies were performed to examine the effects of filteredearthquakes on the response of reinforced concrete structures. The examination is carried out further inthis part of the thesis through tests on large-scale specimens.Two sets of tests were performed. The first set, called the "code-response tests", consisted offour specimens designated BC!, BC2, BC3, and BC4; the second set consisted of only one specimen,BC5, and is called the "large-deflection test". The second set differs from the first one mainly by theamount of displacement imposed on the specimen, and the discussion of its results will be separate fromthe discussions on results of the other tests. However, tables showing the test results will group themeasurements and data from the two sets together.The entire operation of the experimental program, from fabricating the specimens to the actualtesting, was conducted in the Structures Laboratory in the Department of Civil Engineering at TheUniversity of British Columbia.114Chapter 17Description of Code-Response Tests17.1 PurposeEarlier in this thesis, the analytical studies on the seismic response of single-degree-of-freedomsystems and of reinforced concrete structures showed that, for structures with periods up to thepredominant period of the ground motion, which is typical of many buildings or bridges, higher ductility(or higher strength for equal ductility response) and more cycles of inelastic displacements are demandedby filtered earthquakes. These responses in reinforced concrete structures were examined numericallyusing computer analyses. The effects of the demands of filtered earthquakes were determined physicallythrough tests on models of reinforced concrete structures and the results are reported in this part of thethesis. The purpose of the experiments was to look at the extent of damage in structures during filteredand unfiltered earthquakes. The emphasis was placed on what might actually happen to structures thatare designed in accordance with the codes. It was hoped that the results of the experiments woulddemonstrate the significance of the effects of filtered earthquakes and the need to design for them.17.2 Method of Study and Test Set-UpOnly reinforced concrete frames were tested in the experimental study. Reinforced concrete shearwall structures were not selected because shear walls behave like deep, flexural beams and the set-up totest them is more complex and expensive. Also, it has been mentioned in Chapter 1 that more frame-typebuildings than shear wall structures suffered severe damage in the Mexico City 1985 earthquake.Specifically, the six-storey, 3/4-scale frames studied in Part II were used in the experimental study. Thetests, however, were not performed on entire reinforced concrete frames, but rather on frame components115only. The component tested is an exterior beam-column joint. The joint and adjacent beam section wasselected because that is where inelastic action is expected to occur, and an exterior joint was chosenbecause the set-up to test such a specimen is simpler and cheaper than that for an interior one.The joint tested represents an exterior joint in the first-floor of the six-storey reinforced concreteframe, and it includes the slab and the transverse beams framing into the joint core. A drawing showingthe shape of the specimen is shown in Figure 17.1. In a real reinforced concrete frame building, thedeflected shape of an exterior joint enclosed by inflection points in the members is that shown in Figure17.2a. Interstorey drift causes relative, lateral displacements between the top and bottom ends of thecolumn with the beam end remaining at the same level. However, for large-scale testing, it is commonto hold the column ends against translation and to allow the beam end to deflect under applied verticalloads as shown in Figure 17.2b. The same method was followed in this study. The overall test set-upis shown in Figure 17.3. The specimen sits vertically within the testing frame and is held in placethrough three pin connections. The testing frame consists of three parallel steel bents each made up oftwo wide-flange columns and two connecting channels across the top. One end bent is braced with tworound tubes and is responsible for holding the specimen using a lever arm extending from the connectingchannels of the bent. The top of the column of the specimen is connected to this lever arm and thebottom of the column is pinned to the strong floor. The other two bents support two hydraulic loadingactuators, one for the column axial load and the other for the beam tip load. Axial load in the columnis actually applied through the lever arm at the top of the specimen. The free end of the main beam isconnected directly to the actuator for the application of slow, cyclic loads. Both actuators can producea maximum load of 450 lcN and can provide a maximum displacement of 150 mm173 Test SpecimensThe general size of the specimen, or specifically the lengths of the main beam and column, isdetermined by the locations of zero moments in the members of the six-storey, three-bay frame, as based116on Figure 17.2a. Under the code quasi-static lateral loads, the inflection point in exterior columns abovethe first floor is found to be approximately at mid-height. For the first-storey column, the inflection pointis located approximately two-thirds down the length from the first floor; but the actual distance to theinflection point from the joint remains the same as in the other columns because this column is relativelylonger. So the distance between two adjacent inflection points in the exterior column would be equal tothe storey height of the building. For the test specimen, the length of the column between the top andbottom pins was set equal to the storey height of 2743 mm.Under the initial factored gravity loads, the variation of static bending moment in the first-floorexterior beam of the study frame is that shown in Figure 17.4a. The point of zero moment close to theexterior column is at a distance of 1250 mm from the centreline of the column But when the exteriorend of the beam yields under negative bending and the interior end yields under positive loading, theinflection point shifts to a location 1835 mm from the column centreline, as shown in Figure 17.4b. Theactual length of the main beam in the test specimen, however, is governed by the limitations in the loadingactuator and by the dimensions of the testing frame. To allow for the proper positioning of the loadingactuator and, more critically, to keep the beam end deflections within the limit of travel of the actuatorpiston, the length of the main beam is set at 1524 mm from the centreline of the column, which is inbetween the values for the two possible locations of the inflection point. It should be noted that, if theexterior end yields under positive bending, there will be no inflection point in the exterior-half of thebeam. However, this case was not used in the determination of the length of the beam because the resultsof the dynamic analyses of the R4 frame indicated that the response of the exterior end of the first-floorbeam was more critical and frequent in negative flexure.The transverse dimension of the specimen is governed by the widths of the bents, and themaximum allowable width of the specimen is 1422 mm. The overall dimensions of the specimen areshown in Figure 17.5.11717.4 Components of Beam Tip DeflectionFor an exterior beam-column joint specimen tested as shown in Figure 17.2b, the deflection ofthe beam tip is an important parameter in its response. Vertical displacements at the end of the mainbeam (Av in Figure 17.2b) can be caused by various actions in the specimens. The major actions includebeam bending, column bending, beam hinging, column hinging, shear deformation of the joint core, andreinforcing bars slipping. Their influence on the beam tip deflection is illustrated in Figure 17.6.However, in this experimental study, the deflection of the testing frame was sufficiently large tocontribute to the beam tip deflection. Since the specimen was pinned to the strong floor, any lateralmovement of the testing frame would cause a rigid body rotation of the beam-column subassembly aboutits hinge at the bottom of the column, as shown in Figure 17.6. The absolute, overall deflection of thebeam tip in space, including the contribution from the deflection of the testing frame, will be called the"global beam tip deflection" in this research. If the bending of the column is not considered for adeflected joint such as that shown in Figure 17.2b, then the displacement at the end of the beam will becalled the "relative beam tip deflection". That is, the relative beam tip deflection is the global beam tipdeflection minus the contributions from the testing frame deflection and column flexure.17.5 Scope of Experimental StudyFor the purpose of examining the actual response of structures designed in accordance with thecodes, the experimental study looked at the response at the deflection level specified by the codes. Thescope of the tests can be better understood using idealized lateral load-deflection plots such as those shownin Figure 17.7. As shown in Part II of this thesis, a lateral roof displacement of 175 mm corresponds toa tip displacement ductility of 4 for the study R4 frame, and dynamic, inelastic analyses of the frameresponding to this deflection limit under various earthquake loadings were examined. In particular,earthquake records of El Centro EW 1940, Taft S69E 1952, and Mexico City SCT EW 1985 were scaled118to produce the response to this ductility level which is specified by the code for the design of the frame.Graphically, Curve 1 in Figure 17.7 represents the response of the R4 frame under lateral loads, withPoint A being the response at the code deflection, and the three scaled earthquakes are represented by thetwo linear inelastic displacement response curves reaching point A (lines Q1 and Q2). It was decided tosimulate the loadings of these three sealed earthquakes in the experimental study to examine the extentof damage at the code-deflection level and to see if there is any difference between the response underthe filtered and unfiltered earthquakes.Even though the maximum response in the two types of earthquakes are the same, the loadinghistories to attain them are different. In Part II, the moment-hinge rotation response for the first-floorbeam in the R4 frame showed that the response in El Centro EW has the same ultimate hinge rotation asthat in Mexico City but the number of cycles is less in the former; also the amount of hinge rotation islower in Taft than in either El Centro EW or Mexico City.It was also found in Part II that the R4 frame would "collapse" under the actual, unsealed MexicoCity earthquake loading, but that the stronger F2 frame would survive this shaking with a maximumlateral roof displacement of 188 mm, which is very close to the code deflection for the R4 frame. Thisscenario is represented in Figure 17.7 by the dotted inelastic displacement response curve Q3, which givesa large-deflection response (point B) for the R4 frame but a lower level of response (point C) for thestronger structure whose response under lateral loads is given by curve 2. Hence a test was alsoperformed to investigate this case.In summary, four tests were carried out to examine the response at the deflection level specifiedby the code --- three for the R4 frame in the scaled El Centro EW, Taft, and Mexico City SCTearthquakes with designations of BC1, BC2, and BC3, respectively, and one for the F2 frame in theunsealed Mexico City SCT EW earthquake with a designation of BC4. One specimen was fabricated foreach test.11917.6 Testing Procedure and Loading ProgramsThe specimens were loaded by displacing the end of the main beam with slow, cyclic loads whileholding the ends of the column in place through the pinned connections. Since the intention behind theexperiments was to try to simulate as closely as possible the response of an exterior beam-column jointduring a seismic event, the actions applied to the joint should reflect the computed time history of theresponse at the end of the first-floor exterior beam for a selected earthquake. The response quantitiesused in controlling the experiments were the bending moment and the plastic hinge rotation that wereobtained from the dynamic analyses. The basic testing procedure then was to apply these bendingmoments to the joint and impose certain amounts of plastic hinge rotation in the beam when the yieldmoment was reached. Thus the tests were run under load control for part of the time and deflectioncontrol during other times.The loading programs for tests BC1 to BC4 are shown in Figures 17.8 to 17.11. These programsreflect the time histories of the bending moment for the R4 frame in the El Centro EW, Taft, and MexicoCity earthquakes and for the F2 frame in the Mexico City earthquake. The moment time histories havebeen shown in Figures 9.16, 9.18, 9.21, and 10.5, and the high-frequency contents in these momenthistories had been discarded to produce the loading programs. The existence of a large number of small-intensity cycles before the strong portion in the two programs for the Mexico City earthquake loadingsshould be noted. In each program, the bending moment is that in the beam at the face of the joint andits values are given as ratios to the yield moment. With a fixed distance between the joint face and theloading point at the beam tip, the moment would be directly proportional to the applied vertical load.Therefore, the loading programs can also be regarded as time histories of the vertical load at the end ofthe beam. This load was monitored during the tests.Similarly, the amount of plastic hinge rotation to be imposed on the specimen when the yieldmoment is reached was converted to a value that can be easily controlled and measured in the tests. Bymultiplying the rotation by the length of the beam between the plastic hinge and the beam tip, the plastic120hinge rotation was converted into a vertical linear displacement at the beam tip. This beam tip deflectionwould be the relative beam tip deflection as defined in Section 17.4 and was monitored during the tests.The amounts of hinge rotation and the corresponding beam tip deflections in the four tests are shown inTable 17.1. These are also shown in the loading programs in Figures 17.8 to 17.11 as small plateaus ata moment ratio of one.Since the actual yield moments in the specimens would not be expected to be exactly the sameas the theoretical values, the actual yield points were established during the progress of the tests. Forthose cycles before yield, the applied moments were based on the theoretical value; after the actual yieldmoment was determined during the test, the loads in subsequent cycles were based on the new value. Theplastic hinge rotations obtained from the dynamic analyses were also adjusted for the overstrength in theflexural resistance of the beam by multiplying its values shown in Table 17.1 by the overstrength factor,the ratio of the measured strength to the theoretical value. This scaling procedure would increase thelevels of the earthquake loading; however, the intention was to adjust the intensity of the earthquake sothat the ductility response of the structure would be the same as designed or obtained in the dynamicanalyses (which is a ductility of 4). It is also realized that the stiffness of the specimen might also bedifferent from the design value and that the difference would affect the amounts of plastic hinge rotationto be imposed on the specimen.Before the cyclic loads at the beam tip were applied, an axial load of about 800 kN was appliedto the column through the top loading lever arm. This level of loading corresponds to that in the first-storey exterior column in the R4 and F2 frames under factored gravity loads. Once the desired axial loadwas reached, the actuator was set to deflection control; the displacement of its piston was fixed for theentire loading program. Therefore, the force in the column fluctuated during the tests as the beam tipload was varied.12117.7 Detailing of SpecimensSince specimens BC!, BC2, and BC3 represent an exterior joint of the R4 frame, their detailingfollows the design of the R4 frame as shown in Figure 9.2. For specimen BC4, the detailing follows thedesign of the F2 frame as shown in Figure 10.1. The actual detailing of the specimens, however, variedslightly from the design. Rather than being a combination of hoops and stirrups at different spacings, allthe transverse reinforcement in the beam were hoops spaced equally apart along the length of the beam,thereby extending the zone detailed for plastic hinging. Also, the longitudinal reinforcement in the slabwas placed at 230-mm spacing, instead of the 400-mm spacing indicated in the design, to look at theircontribution to the beam flexural strength. In each specimen, the column longitudinal reinforcementextended all the way to the bottom of the column and was welded to a base plate. At the top of thecolumn and at the end of the main beam, the longitudinal reinforcement was terminated 50 mm from theconcrete surface of the members. The detailing for specimens BC1, BC2, and BC3 is shown in Figure17.12, and that for specimen BC4 is shown in Figure 17.13.17.8 Instrumentation and Data AcquisitionSeveral types of measuring device, such as strain gauges mounted on reinforcing bars,displacement transducers measuring deflection of various parts of the specimen, and load cells measuringthe loads applied by the two hydraulic actuators, were used in the tests.Thirty strain gauges were mounted on reinforcing bars at different locations in each specimen.The locations are shown in Figure 17.14. The strain gauges have a gauge length of 6.35 mm, a widthof 3.05 mm, and a resistance of 350 ohms. Their maximum elongation is 5% of the gauge length. Atthe point where a strain gauge was to be located, the reinforcing bar had to be machined and smoothedto provide a flat surface for mounting the device. The depth of the machining was about one-third of thebar diameter, resulting in a loss of almost 30% of the cross-sectional area. The strain gauge was glued122onto the machined surface, and wires were soldered to its terminals. Finally, a coating of water-resistantcompound was spread over the entire gauge for protection. All strain gauges were installed on individualbars before the reinforcing bars were assembled together. The cables leading from the gauges wereallowed to pass through the concrete in the specimen and emerged at the top of the slab.Several linear displacement transducers were used to measure various significant displacementsthat would affect the response (that is, the beam tip deflection) of the beam-column subassembly. Thesedisplacements have been identified in Section 17.4. A linear displacement transducer was installed insidethe beam tip load actuator to measure the stroke of the loading piston or the global deflection of the beamtip. Another transducer was located at the top of the testing frame to measure the lateral displacementof the top hinge of the specimen and thus to indicate the rigidity of the testing frame. An aluminum rig,having a horizontal arm extending to the end of the beam and a vertical one to the lower column, wasmounted on the side of the base of the upper column just above the slab (see Figure 17.15a), and it wasused to measure three displacements. First, since it was attached to the upper column, it rotated with theupper column during the test and the rotation, Oc in Figure 17.15b, was measured directly by a rotationtransducer mounted on the rig. Secondly, also because of its rotation with the base of the upper column,a linear displacement transducer mounted at the end of the horizontal arm of the aluminum rig wouldmeasure the relative beam tip deflection, Ai. in Figure 17.15b. Thirdly, a transducer was mounted on thevertical arm of the rig to attempt to measure the shear deformation of the joint core, Ai in Figure 17.15b.Six more displacement transducers were used to measure the rotations at three sections of the beamlocated at 200-mm spacings from the face of the column An L-shaped frame was attached to the beamat each section (see Figure 17.15a) whose rotation can be determined by measuring the vertical deflectionsof two points on the short leg of the L-shaped frame using the two displacement transducers.In summary, 43 different measurements (from 30 strain gauges, 11 displacement transducers, and2 load cells) were recorded simultaneously in each test. Of the 43 measurements, 39 were recorded andstored in a computer by the OPTILOG data acquisition system which scanned the devices at a constanttime interval. In tests BC1 and BC2, the time interval for collection of data was 15 seconds; in tests BC3123and BC4, it was 30 seconds. The three measurements not read by OFTILOG --- beam tip load, force forcolumn axial load, and upper column rotation --- were recorded manually by two assistants at the samescan interval as the other devices. In addition, for every scan, the computer printed out values for sixselected devices: the linear transducers measuring the relative beam tip deflection and the joint sheardeformation, and four strain gauges in the top and bottom beam reinforcement. Furthermore, the load-deflection relationship of the beam tip was plotted continuously in each test, which helped to indicate theyield points of the specimen. Yielding was considered to have occurred when the load-deflection curvebegan to deviate from its linear behaviour. This was confirmed by examining the strains in thereinforcing bars at the face of the joint in the tests.17.9 Fabrication of Specimens17.9.1 Reinforcing SteelAll reinforcing steel for the specimens were obtained from a local supplier and were shipped cutand bent. The reinforcement cages were assembled after all the strain gauges were installed. Figure17.16 is a picture of the reinforcement in the joint core of the specimens. It can be seen that the jointcore is very congested with reinforcing steel. Also, because of the way the bars were bent, many of thedesired spacings and positions of the reinforcement specified in the design drawings could not beachieved. This resulted in larger covers for the hooks of the beam longitudinal reinforcement. In otherwords, the anchorage lengths for these bars were shorter than the design values. For each specimenexcept BC 1, the positions of the reinforcing bars in the formwork were noted before the concrete waspoured. Table 17.2 indicates the actual positions of the main beam and column reinforcement at sectionsadjacent to the joint core and compares them to the design values. It should be noted that all the columnlongitudinal bars were closer to the centre in the actual specimens than in the design. Only two sets ofsteel were used for the four specimens; specimens BC1 to BC3 contained steel from the same delivery.124Tensile strengths of the reinforcing steel were obtained from tests on short samples. The average tensilestrength for each bar size of each set is shown in Table 17.3. A typical stress-strain curve for thereinforcing steel is shown in Figure 17.17.17.9.2 ConcreteThe formwork for the specimens was built in the Structures Laboratory at The University ofBritish Columbia. Figure 17.18 is a picture showing the formwork, with reinforcement in place, readyto receive concrete. The concrete for the specimens were ordered ready-mixed from a local supplier.The materials received had high slump values ranging from 75 to 130 mm and a maximum aggregate sizeof 10 mm. All the specimens were formed vertically and completed in one continuous pour with theexception of specimen BC1 which had the upper column cast a day later. All four specimens were caston different days.Compressive strengths of the concrete were obtained from tests on standard 150 mm by 300 mmcylinders cast during pouring of the concrete and "field-cured" beside the beam-column specimens. Thecylinders were broken at the day of the test of their corresponding beam-column subassembly. Theaverage compressive strengths of the concrete for the four beam-column specimens are shown in Table17.3, and they are all very close to the specified strength of 30 MPa which was used in the analyticalstudies of the reinforced concrete frames. Stress-strain curves for the concrete cylinders were notobtained.As a result of a stiffer mix and the lack of proper vibration, specimen BC4 had a largehoneycomb cavity in the lower column. Opened to the surface and as wide as half the dimension of thecolumn, the hole was patched up using an aggregate-free grout. Fortunately, the defect was in a locationnot of direct concern in the test.12517.10 Repair of Specimen BC1During setting up of test BC!, a sudden drop in hydraulic pressure in the beam-load actuatoroccurred after the three hinges of the specimen were connected. This resulted in an impulsive load pullingthe end of the beam upwards, and it caused some damage to the bottom of the beam and the top of thecolumn Since the damage was minor, it was decided that the cracks in the beam would be repaired byinjected epoxy and the column top by sandy grout. The repair work was carried out by an outsidecontractor specializing in repairs of concrete structures. Figure 17.19 shows the beam of specimen BC1after the repair. Further information and discussions on the epoxy injection procedure and its effects onthe behaviour of the joint are presented in Appendix C.17.11 Past Research on Reinforced Concrete Exterior Beam-Column JointsExperimental studies of reinforced concrete exterior beam-column joints have been going on forthe past twenty-five years. The first study was carried out by the Portland Cement Association andreported by Hanson and Connor [1967]. This famous study involved tests on full-scale specimens ofsimple exterior beam-column connections ( ie. no slabs and no transverse beams) under slow, reversedloadings. It concluded that "properly designed and detailed cast-in-place reinforced concrete frames canresist moderate earthquakes without damage and severe earthquakes without loss of strength". Later,based on their own cyclic-loading tests on simple exterior beam-column connections, Park and Paulaymade important recommendations for the design of joint reinforcement and anchorage of beamreinforcement and presented comprehensive discussions on the response of beam-column joints underseismic loadings [Park and Paulay 1973, 1975; Park, Paulay, and Priestley 1978]. Then, in the mid-1970's, a group of tests on reinforced concrete beam-column joints was performed [Lee, Wight, andHanson 1977; Townsend and Hanson 1977; Uzumeri 1977; Bertero and Popov 19771 and the results werepublished in special publication SP-53 of the American Concrete Institute (ACI) [1977].126One common parameter studied in the works published in ACI SP-53 was the loading history,and some of the major conclusions on its effects on the response of reinforced concrete beam-columnjoints are the following:1) "Most of the cracking occurred during the first cycle of loading. Subsequent cycles at the samedisplacement level and at the higher displacement level produced little additional joint cracking."[Lee, Wight, and Hanson 1977]2) "Large displacement levels of hinge rotation causes the hysteretic behaviour of the connections todeteriorate more rapidly." [Townsend and Hanson 1977]3) "A large number of cycles of inelastic loading at a relatively low amplitude has no significantreduction of moment capacity of the connection for larger amplitude of loading." [Townsend andHanson 1977]4) "Loading history does not affect the strength but seriously affects the stiffness of beam-columnsubassemblies." [Uzumeri 1977]5) "Provided that the joint steel has not yielded, the additional stiffness deteriorations of a sub-assemblyin subsequent cycles is insignificant when the displacements which the sub-assembly are subjected toare smaller than the immediately preceding maximum displacement." [Uzumeri 1977]6) "If the maximum deformation of a member in either direction is increased, the initial stiffness andenergy dissipation per cycle degrade during the following cycle." [Bertero and Popov 1977]7) "The hysteretic behaviour of a structure appears to be very sensitive to the history of excitations.Thus, the application of repeated reversal loading cycles in which the peak values of the load aregradually increased, might not be necessarily a 'conservative' method of testing behaviour. Thestructure might show considerably less energy dissipation capacity, and even less maximum strength,if it is loaded to or near its ultimate resistance during the first cycles." [Bertero and Popov 1977]These observations are relevant to the present study of the effects of filtered earthquakes because it hasbeen shown earlier in this research that the manner in which a structure reaches a certain deflection levelis considerably different in a filtered earthquake than in an unfiltered earthquake.127All the experiments mentioned so far involved simple beam-column connections that did notcontain transverse beams nor slabs. But in most frame-type buildings, the slab and transverse beamswould be cast monolithically with the main lateral-load resisting elements. Beginning in the mid-1980's,many tests --- most of them full-scale --- have been performed on specimens of reinforced concrete beam-column joints containing transverse beams and slab. These include tests performed at The University ofTexas at Austin as part of the U.S.-Japan Cooperative Project on seismic research on reinforced concretebuildings [Joglekar et al. 1985] and a series of coordinated tests that were performed collaboratively byresearchers from U.S., Japan, New Zealand, and China to bring together the earthquake-resistant designof reinforced concrete frame joints in these countries [American Concrete Institute 1991].The presence of the unloaded transverse beams was found to be helpful in developing confinementin the joint and in eliminating the pull-out of the main beam longitudinal reinforcement [Ehsani and Wight1984, 1985], but large torsional damage in these members at the column face occurred at high deflectionlevels [Joglekar et al. 1985]. One major influence of the slab on the response of exterior beam-columnjoint is the increase in the flexural strength of the main beam under negative loadings (top in tension).Defined as the slab width within which the longitudinal reinforcement in the slab help to resist negativebending of the main beam, the effective slab width in the tests was found to be larger than that given bythe ACI for T-beam analyses. By testing specimens with different slab widths, Durrani and Zerbe [1987]arrived at an estimate for the effective slab width. The influence of the transverse beams on the effectiveslab width has also been noted [Paultre and Mitchell 1987]. Another effect, unfortunately undesirable,of the actions in the slab is the introduction of additional shear stresses to the joint core. The results ofthe past studies demonstrated the significance of including slab and transverse beams in an exterior beam-column joint specimen because these elements can affect the behaviour of the most important part of thesubassembly, which is the joint core. Furthermore, the tests by Joglekar et al. [1985] showed that thebehaviour of exterior beam-column joints with slab and transverse beams under continued cyclic loadingwas excellent up to storey drift levels estimated to correspond to the maximum deflection level imposedon the prototype structure. Also, the specimens of Paultre and Mitchell which were designed and detailed128in accordance with the latest Canadian codes for a force reduction factor of four (that is, ductilestructures) performed very well under cyclic loading, but those specimens designed for lower factors anddetailed accordingly showed poor performances.Finally, it should be noted that all the tests mentioned in this discussion have been performedusing cyclic loading of increasing displacement ductility. The total number of cycles, the number ofcycles at a particular ductility level, and the maximum ductility were varied from one test to another.17.12 TestingThe four specimens were tested in the order they are numbered: BC!, BC2, BC3, and BC4. Eachtest was carried out using very slow loading rates. In addition, the applied beam tip load was held atcertain load points during the tests to allow cracks to be marked with felt pens and photographs to betaken. The acquisition of the data, however, was performed uninterruptedly at the preset time intervalsof 15 seconds for tests BC1 and BC2 and 30 seconds for tests BC3 and BC4. There are a few, butimportant, points to be noted regarding the data collection in the tests:i) In test BC 1, the recording of data by the OPTILOG system was accidentally turned off at the startof the test. So no measurements were recorded on computer disks; only the six readings printed by thecomputer are available. These readings are for the relative beam tip deflection, the joint sheardeformation, and strain measurements at four locations in the beam longitudinal reinforcing bars.ii) In test BC4, the transducer for measuring the rotation of the upper column was not mountedbecause it was broken before the test. Also, the linear transducer for measuring the joint sheardeformation did not function properly during the test, and so no reasonable reading was obtained for thismeasurement.iii) Strain gauges are quite fragile, and the chances of them being damaged before and during the testswere high. On average, about 70 percent of the thirty strain gauges mounted actually produced reasonableresults in each test.129Chapter 18Damage to SpecimensIn this chapter, the damage observed in different parts of a specimen are discussed. Only thegeneral visual observations are presented, and correlations between the extent of damage and thedisplacement ductility reached for the damage will not be made.18.1 Overall Damage PatternIn general, the four specimens were not severely damaged, with cracking being the most commonform of damage. Some spalling did occur, but that usually occurred at the back of the joint and columnand at the bottom of the beam at the joint face. No reinforcing bar was observed to be broken or buckled.There was no major failure in terms of complete breakage of the members of the specimens. All fourspecimens remained intact at the end of the tests, leaving no difficulty in the removal of the specimensfrom the testing frame. The damage received in these tests was relatively minor when compared to theheavy damage, which included crushing and spalling of the concrete and buckling of the reinforcing bars,observed in the field or in other tests of beam-column subassemblies. The most significant observationfrom tests BC1 to BC4 is that the damage to the four specimens are similar. Figure 18.1 shows thetypical pattern of damage in the beam-column subassemblies. Damage to each member of the beam-column subassembly will be explained in detail in the following sections of this chapter. Since all fourspecimens showed similar damage, their results will be considered together in the discussion.13018.2 Beam DamageThe damage in the main beam consisted almost entirely of cracking of the concrete. Flexural andshear cracks were observed. Figures 18.2a to 18.2c show the final damage in the portion of the mainbeam adjacent to the joint core in specimens BC2 to BC4. Three types of cracks can be seen to haveoccurred and their occurrence is dependent on the position along the beam.At the face of the joint, a vertical flexural crack over the entire depth of the beam can be seen.This crack was expected because the bending moment at that section was the largest along the beam. Itappeared initially at very low loadings as two cracks at the top and bottom, and they eventually joinedup, covering the whole beam depth, by the time yielding in both directions of loading were reached.During positive loading, this crack opened up into a wide gap at the bottom of the beam. To give someperspective, this gap was measured to be about 4 mm wide at a beam tip displacement ductility of 2 andgrew rapidly to about 10 mm at a ductility of 2.5. This large gap closed completely though when theloading at the beam tip was reversed to produce negative bending in the beam. Minor spalling wasobserved at the bottom of the beam near the joint in specimen BC3 only. The crack extended all the waythrough the slab and appeared on the top of the slab adjacent to the column. Contrary to the large gapthat occurred at the bottom of the beam under positive loads, a wide gap did not occur at the top of theslab during negative loading. The reason for this phenomenon is not known exactly but the large widthof the slab may be a contributing factor.Away from the face of the joint, cracks of the flexural-shear type occurred from both top andbottom. These cracks began as vertical flexural cracks at low loads and then propagated into inclinedshear cracks at larger loads. Both negative and positive shear cracks propagated inwards towards the jointcore. This type of crack occurred within a distance of about one and a half times the effective depth ofthe beam from the joint face. Beyond the flexural-shear crack zone, mostly cracks extending from thetop occurred; these were pure shear cracks inclined at 45 degrees towards the joint core. In general, all131the cracks from the top extended downwards to about three-quarters the depth of the beam, and the cracksfrom the bottom extended upwards to only half the depth of the beam.The propagation of flexural-shear cracks and pure shear cracks in the main beam during the testswas noted. For example, Figure 18.3 is a drawing showing the negative cracks only in the first-half ofthe main beam in specimen BC3. The hash marks in this figure indicate the extent of the cracks at theend of a load cycle before the beam tip load was reversed. The numbers beside the hash markscorrespond to the load peak numbers given in the loading program. From this figure, it can be seen thatthe cracks grew with the loading but then stopped almost completely after load peak 54. Beyond thispoint, the additional damage was relatively minor even though the load reached the same level severaltimes. Similar results were observed in the other tests. Therefore, virtually all the damage in the mainbeam occurred in the loadings up to a certain load cycle, and larger deflections of the specimen insubsequent cycles produced no further significant damage. This cycle after which propagation of cracksceases actually corresponds to the first-yield of the beam-column subassembly.18.3 Slab DamageIn each specimen, the damage observed in the slab was mainly transverse cracks running acrossthe entire width of the slab and penetrating through the thickness. These cracks can be regarded as partof the negative flexural cracks in the main beam. The first crack appeared at the section adjacent to thecolumn; subsequent cracks were formed parallel to the first one at fairly equal spacings. All the cracksin the slab, including the one at the column face, were very narrow. Typical damage to the top of theslab is shown in Figure 18.4.In addition to these transverse cracks, two longitudinal cracks were observed at locations abovethe sides of the main beam (see Figure 18.4). These two cracks might be caused by the relative verticaldisplacement between the main beam and the slab.132Also, a crack around the base of the upper column was observed. This crack wrapped aroundthe inside half of the perimeter of the column and then branched into inclined cracks in the slab, one oneither side of the column, extending to the back edge of the slab (see Figure 18.1). These inclined cracksjoined up with the cracks in the transverse beams.18.4 Transverse Beam DamageIn-plane forces in the slab induce torsion in the unloaded transverse beams, and indeed torsionalcracks inclined at 45 degrees to the beam axis were observed. These cracks were more prominent on theback side of the transverse beams. As shown in Figure 18.5, two pairs of torsional cracks were observed.One set began at the column face and below the mid-height of the transverse beams. The other set beganat the top of the transverse beams near the column face and was connected to the crack that went aroundthe perimeter of the upper column on the top of the slab as described in the previous section. The latterset was longer, and the damage presented by it was more noticeable. Both sets of cracks wrapped aroundthe bottom, outside edge of the transverse beams and they then turned towards the column again at 45degrees at the bottom of the beams. The extension of the larger torsional crack on the underside of thetransverse beams eventually joined up with the vertical flexural crack in the main beam at the joint face.18.5 Column DamageMost of the damage in the column of the subassembly occurred in the upper column. Typically,a long and wide crack extended upwards near the back of the upper column, as shown in Figure 18.6,and almost reached the top of the specimen. It cut through the width of the column and was located inthe plane of the back surface of the column reinforcement cage. Its width was noticed to increase withthe deflection level of the beam-column subassembly. Generally speaking then, the back cover in thelower half of the upper column can be considered to have spalled off. On the front side of the upper133column, however, there were only a few flexural cracks and the cracking zone covered about the bottomtwo-thirds of the column.In contrast to the heavy damage in the upper column, the damage in the lower column was verylight (see Figure 18.7). Flexural cracks occurred on both front and back sides. A longitudinal cracksimilar to the one in the upper column could be seen extending downwards from the joint on the back sideof the lower column. But this crack was relatively short and did not open up very much. The lighterdamage to the lower column can be attributed to the higher axial load in this member, making it strongerin flexure.18.6 Joint DamageBecause of the presence of the transverse beams, visual observations of the damage in the jointwere very limited. Only damage at the back of the joint was visible. Similar to the results in the uppercolumn, the major damage in the joint was that of spalling of the back concrete cover. At early stagesof loading, cracks appeared along the edges where the transverse beams frame into the joint region, asshown in Figure 18.8. At higher deflection levels, two horizontal cracks formed along the top and bottomof the joint at the back, thereby enabling the back side concrete cover to break away from the joint core.After the tests, the loose cover was removed manually, and the concrete confined within the reinforcementin the joint core did not show signs of severe damage.18.7 SummaryIn conclusion, the level of damage suffered by the four beam-column joints in the tests can beconsidered to be moderate. The most significant damage occurred at the back of the upper column andjoint where the concrete cover was spalled off.134As a result of the torsional cracks joining up with the cracks in the slab and the main beam, thebeam-column subassembly can be pictured to be splitting into two large portions, as shown in Figure 18.9.Under downward loads applied at the beam tip, the main beam, along with the slab and large portions ofthe transverse beams, twisted out of the column, to which two small "wings" that belong to the transversebeams remained attached. This mode of damage has been noted also by Durrani and Zerbe [1987] in theirexperiments on exterior reinforced concrete beam-column joints. Indeed, a large out-of-plane offset alongthe major inclined cracks on the back side of the transverse beams could be seen in the tests, especiallyBC3 and BC4. This offset in specimen BC3 was about 5 mm at negative displacement ductility of about2, and it increased to about 28 mm at a ductility of 3.Once again, it should be noted that the same damage pattern was observed in all four tests, eventhough the specimens were loaded to different ductilities by different loading histories.135Chapter 19Overall Hysteretic Behaviour of SpecimensIn this chapter, the hysteretic behaviour exhibited by the four specimens through the beam tipload-deflection response is examined. Only the overall hysteretic behaviour such as the shape of thehysteresis loops, the locations of the yield points, and the amount of ductility attained are discussed;detailed examinations of the yield behaviour of individual members of the subassembly will be presentedin the next chapter.19.1 Testing Frame DeflectionAs mentioned in Section 17.4, a part of the global beam tip deflection is contributed by thedeflection of the testing frame. Hence, the actual response of the beam-column subassemblies only, asrepresented by Lli in Figures 17.2b, would be smaller than the measured global beam tip deflection. Thissection examines the beam tip deflection component contributed by the deflection of the testing frame inthe tests.A linear transducer was mounted on a fixed, vertical pole to measure the absolute, lateraldisplacement of the loading lever arm on top of the specimen, and the measurements are plotted againstthe applied loads at the beam tip in tests BC2 to BC4 in Figure 19.1. (The measurements of thistransducer in test Bel were not recorded.)It can be seen from Figure 19.1 that the load-deflection response of the testing frame wasreasonably linear in each test. Perfectly linear behaviour was not achieved because of possible factorssuch as lateral restraint from the top loading actuator and twisting of the members of the testing frame.Nevertheless, a linear relationship between the testing frame deflection and the applied beam tip load canbe considered and the "flexibility" of the testing frame can be determined. Based on geometry, the136horizontal deflection at the top of the specimen is directly proportional to the vertical deflection at thebeam tip and hence the flexibility can be expressed in terms of the beam tip deflection, which would thenindicate the component of the beam tip deflection that is contributed by the testing frame deflection. The"flexibility" of the testing frame, given in terms of beam tip deflection per unit of applied beam tip load,for each test is listed in Table 19.1. For test BC1, its value is assumed to be the average of the valuesfor tests BC2 and BC3. It can be noted from this table that the flexibilities in tests BC2, BC3, and BC4differ, with the flexibility in test BC4 being twice that in BC2. Unfortunately, no explanation can befound for the larger value obtained in test BC4.To study the beam tip deflection of the specimens without the influence of the testing framedeflection, the component contributed by the deflection of the testing frame should be deducted from themeasured global beam tip deflection. However, values of the global beam tip deflection were notrecorded nor stored in the computer during the tests and only a plot of these values as a function of theapplied beam tip load was made by an X-Y plotter. Therefore, the load axis in the load-global deflectionplot should be rotated about the origin to account for the contribution of the testing frame deflection.19.2 Beam Tip Load-Deflection BehaviourTwo relationships can be noted for the load and the deflection of the beam tip. One is betweenthe load and the global deflection, and the other is between the load and the relative deflection.19.2.1 Applied Load versus Global DeflectionThe beam tip load-global deflection relationship for each specimen was recorded by an X-Yplotter which continuously measured the load applied by the hydraulic actuator and the displacement ofthe piston of the actuator. The load-global deflection plots for the four specimens are shown in Figures19.2 to 19.5. The scale of the plots are made the same for easy comparison. For specimen BC4, the137limit of the plotter was reached just before the maximum downward displacement of the beam tip,resulting in a vertical line at the end of each of the three largest cycles. Estimates of the response at theends of these cycles are drawn in the plot as dashed lines. An inclined load axis is also drawn in eachplot to indicate the deflections without the contributions from the deflection of the testing frame.19.2.2 Applied Load versus Relative DeflectionThe beam tip load-relative deflection relationship for each specimen can be obtained by plottingthe applied load against the measurement of the linear displacement transducer mounted between the beamtip and the end of the aluminum frame extending from the base of the upper column. Since the load anddeflection were measured at the preset fifteen- or thirty-second interval, the plots for this relationshipwould not be as smooth as the continuous plots for the load-global deflection relationship. The beam tipload-relative deflection plots for the four specimens are shown in Figures 19.6 to 19.9.19.2.3 Overall BehaviourThe beam tip load-global deflection relationships shown in Figures 19.2 to 19.5 demonstrate thatall four specimens have similar general behaviour with stable hysteretic response.There are several common characteristics among the plots. First, the strength in the negativeloading direction is larger than that in the positive direction. This non-symmetry in strengths is expectedsince the beam was designed to be stronger in negative flexure. Second, as dictated by the loadingprograms, there are more large loadings in the negative direction than in the positive direction. But moreimportantly, the beam-column subassemblies are observed to be able to maintain their positive andnegative yield strengths up to the maximum deflections reached. Third, the most noticeable feature ofthe hysteretic behaviour of the specimens is the pinching of the hysteresis loops. This characteristicoccurred in both directions of loading. A common cause of pinching is the loss of stiffness as a result138of large inelastic strains in the tensile reinforcement in the beam, which necessitate closing of wide cracksupon reloading. But pinching can also signify the undesirable result of shear failure of the joint. Thefailure modes of the specimens and the cause of the pinching will be examined in the next chapter. Eventhough the load-deflection curves are pinched near the zero-load level, the reloading curve heads towardsthe maximum response attained in the previous cycle, but softens as the maximum load is approached.19.3 Comparison with Theoretical BehaviourAs previously stated, the loads used in the tests of the beam-column subassemblies were basedon the results of the dynamic computer analyses of the study six-storey frames. In the analytical study,the flexural behaviour of the members of the frames were assumed to follow the Takeda model [1970]with perfectly plastic yield behaviour, degradation of stiffness in relation to the amount of yielddisplacement, and return to the previous maximum response on reloading (see Figure 9.5). In Figure19.10, the moment-rotation response of the end of the first-floor exterior beam in the R4 study frame inthe El Centro EW, Taft, and Mexico City SCT EW ground motions are shown. The major differencebetween the analytical and experimental response is that pinching exists in the experimental results. Thisindicates that the amount of energy dissipated in one cycle by the specimens is less than that assumed inthe modelled structure, which means that the specimens should be subjected to larger deflections if thesame energy was to be dissipated. Analytical models incorporating pinching have been used by someresearchers [Kitoyama, Otani, and Aoyama 1991; Paultre and Mitchell 1987] and would represent a betterresponse of the specimens in this study. However, the occurrence of pinching as significant as thatobserved was not expected before the tests.13919.4 Comparison with Previous Experimental ResultsAs mentioned in Chapter 17, many experimental studies on reinforced concrete exterior beam-column joints have been performed. Ehsani and Wight [1984, 1985] conducted cyclic loading tests usinga set-up similar to that in the present research. Furthermore, by coincidence, the dimensions and detailingof one of their specimens are almost the same as those of specimens BC1 to BC3. Table 19.2 comparesthe properties of the specimens in the two studies. It shows that parameters such as the ratio of the beamto column length, the cross-sectional dimensions of the beam and column, and the reinforcement ratiosare almost the same between the two specimens. When subjected to a loading program of increasingdisplacement ductility, the Ehsani and Wight specimen produced the beam tip load-deflection responseshown in Figure 19.11. It can be seen that the overall general behaviour of this beam-column specimenis very much like that of specimens BC1 to BC4; the figure even displays the pinching of the hysteresisloops. Therefore, the hysteretic behaviour of the specimens in this research can be considered to bereasonable since similar responses have been obtained for a specimen of nearly identical properties.However, the load-deflection response in the test by Ehsani and Wight and in the present studyare considerably different from the results obtained by Paultre and Mitchell [1987] in their tests onexterior beam-column subassemblies that represent the six-storey frame given in the CPCA ConcreteDesign Handbook. Figure 19.12 is a reproduction of the beam tip load-deflection relationship of aspecimen which was designed and detailed in accordance with the Canadian codes for a ductile behaviour(K=0.7), and Table 19.3 compares the properties of this specimen to specimen BC1. Figure 19.12 showsthat hysteresis loops having only minor pinching in one direction were obtained by Paultre and Mitchell.Two factors may account for the discrepancy in the hysteretic behaviour obtained in this researchand the Paultre and Mitchell test. Even though specimens in both tests are representative of an exteriorjoint in the six-storey frame given in the CPCA Concrete Design Handbook, the dimensions of thespecimens in the present tests were reduced; a larger specimen could provide a better anchorage zonefor the longitudinal reinforcement of the main beam, which would prevent pull-outs and slippage of the140bars in the joint. But the main factor is the smaller amount of top reinforcement in the beam in thePaultre and Mitchell specimen (0.6% compared to 1.4% in specimens BC! to BC3). This, in combinationwith the larger plan area of the joint core, would reduce the shear stresses and thus the damage to thecritical joint core. This in turn would help to maintain the integrity and stiffness of the joint core, leadingto fatter hysteresis loops.19.5 Yield Loads and DeflectionsYield points in both directions of loading were established during the tests, as required by theloading programs. To determine the actual yield points of a specimen, both the load-global deflection plotand the load-relative deflection plot were examined for a significant change in slope in the initial part ofthe curves. The results are identified in the beam tip load-global deflection plots in Figures 19.2 to 19.5,and the load and global deflection of the beam tip at these points are listed in Table 19.4.In Table 19.5, the yield loads obtained from the load-deflection plots are compared to the valuesrecalculated using measured material properties and actual positions of the reinforcement. One slabreinforcing bar on either side of the beam is included in the theoretical strength. It can be seen that thetheoretical and measured yield loads agree with each other very well in each test. On average, the actualyield load is only about 7% higher than the theoretical value. However, it should be noted that the yieldmoments used earlier in the analytical studies for the beam of the prototype frame are considerably lowerthan the measured and the nominal (recalculated) strengths shown in this section; the negative andpositive yield loads corresponding to these yield moments for specimen BC1 to BC3 (that is, the R4frame) are 160 lcIsl and 68 IN, respectively, while those for specimen BC4 (F2 frame) are 210 lds1 and112 1c/s1.14119.6 Displacement DuctilityOnce the yield points are identified, the amount of displacement ductility attained in each test canbe determined. The ductility was calculated based on the beam tip deflection without the componentcontributed by the deflection of the testing frame, that is, based on the rotated load axis in the load-globaldeflection plots. The levels of displacement ductility are indicated by hash marks drawn in the beam tipload-global deflection plots in Figures 19.2 to 19.5. Table 19.6 and Table 19.7 list the negative andpositive displacement ductilities, respectively, at significant cycles in each test. It can be seen that themaximum negative displacement ductility reached is about 4.4 in test BC1, 2.5 in BC2, 3.4 in BC3, and2.4 also in BC4. The maximum positive displacement ductilities for specimens BC1 to BC4 are allslightly above 2 with the exception of 2.9 in test BC3.Tables 19.6 and 19.7 also give the cumulative displacement ductility at each cycle. Thecumulative displacement ductility is the sum of all the ductilities reached up to and including the deflectionin question. It is a reasonable indicator of the number of cycles of yield excursion experienced by ayielding member. For test BC3, which represents response in a filtered earthquake, the cumulativedisplacement ductilities can be seen to be higher than those of the tests BC1 and BC2.19.7 Yield ExcursionsAs stated earlier, the loading history in a filtered earthquake is different from that in a typical,unfiltered earthquake. Table 19.8 shows values for the number of yield excursion in each direction. Intest BC3, the specimen was loaded to negative yield six times compared to only twice in tests BC1 andBC2. This characteristic regarding loading history has been revealed in Parts I and II and is demonstratedhere again in the experimental results because of the correlation of the loading programs to the analyticalresults.14219.8 Rotation of Upper ColumnThe relative beam tip deflection was measured relative to the base of the upper column.Therefore, an examination of the behaviour of the upper column during the tests should provide someinformation on the measurements of the relative beam tip deflection.A rotation transducer mounted on the aluminum rig attached to the upper column was used tomeasure the rotation of the base of the upper column. Since this transducer measured the absolute andnot the relative rotation of the base of the upper column, a part of the measurement was caused by thedeflection of the testing frame. After the contribution of the testing frame deflection is deducted from theabsolute deflection, the resulting rotations of the base of the upper column during the duration of testsBC1 to BC3 are plotted in Figure 19.13a. The rotation is plotted against the bending moment at thesection which is calculated from the beam tip load based on equilibrium. There are no results forspecimen BC4 because the transducer was not mounted. The results for specimen BC1 are available eventhough the computer was not storing data in that test since the rotation measurements were recordedmanually . However, for this specimen, the estimated testing frame flexibility given in Section 19.4 wasused to obtain the adjusted moment-rotation plot. The results in Figure 19.13a show two types ofbehaviour for the column rotation in each test. The relationship between the rotation and the bendingmoment during initial loadings is linear, but the response becomes very erratic later on. The linearresponse, when separated out from the plots, are shown in Figure 19.13b.The linear relationship between the moment and rotation is a measure of the flexibility of thecolumn in bending. A proportionality constant between the two parameters is determined from Figure19.13b for specimens BC1, BC2 and BC3, and the values are listed in Table 19.9. The results show thatthe value in test BC1 is many times larger than that in test BC2 or BC3. Using the actual materialproperties and reinforcement positions, the average value of the theoretical flexibility of the column forspecimens BC1 to BC3, based on a pin-ended member subjected to a concentrated moment at one end,is calculated to be 0.113 x 10-7 rad/kN-mm when an uncracked section is used and 0.400 x i0 rad/kN-143mm when a cracked section is used. Comparing these numbers to the measured column flexibilities ofspecimens BC! to BC3 shows that the measured values for tests BC2 and BC3 are close to the flexibilitybased on an uncracked cross section while the value for test BC1 is close to that based on the crackedcross sections. Therefore, the column in specimens BC2 and BC3 can be assumed to be uncracked duringinitial loadings applied at the beam tip and the column in specimen BC1 can be assumed to be crackedby the accidental load occurred during the test set-up.As shown in Figure 19.13a, the linear behaviour between the rotation and moment does not lastfor the entire duration of the tests. But inelastic behaviour of the column, in terms of yielding of itslongitudinal reinforcement, is not expected because the beam-column subassembly was designed so thatthe columns were stronger than the beams. If plastic hinging does occur at the base of the upper column,then the rotation will increase with only slight increase in the load, which is the typical yield behaviourof materials or structural members. The non-linear behaviour in the moment-rotation response for thecolumn in the test specimen, however, does not follow this normal yield behaviour. In fact, the rotationin the column actually reverses at the end of the linear response range. Arrows drawn to show the pathof the moment-rotation curves indicate this clearly in Figure 19.13b. On the other hand, the beam tipcontinues to deflect further under the same loading.The explanation for this behaviour of the column rotation can be found by observing the damagein the upper column. A picture of the upper column in specimen BC3 under a negative (downward) beamtip loading was shown in Figure 18.6. In this picture, a crack can be seen to have opened at the base ofthe upper column and it seems to run underneath the member, separating the upper column from the beamand slab. Diagrammatically, Figure 19.14 shows the relatively smaller rotation for the base of the uppercolumn when a gap develops at that section under negative loadings of the beam-column subassembly.The figure also indicates the shear deformation of the joint core that may be associated with theoccurrence of the gap. Therefore, it can be assumed that, under large negative beam tip loads, the sheardeformation of the joint core generates large tensile strains on the inner half of the base of the uppercolumn, causing a crack to develop there and subsequent reversal of the upper column rotation. The144reverse rotation of the upper column is possible if the longitudinal reinforcement in the column is regardedto be slipping through the concrete in the column. Under negative beam tip loadings, the inner portionof the upper column would be under tension, causing loss of bond around the longitudinal reinforcementand subsequent slipping through the concrete. Therefore, it is concluded that the reversal in the rotationof the upper column is caused by the gap opening at the base on top of the slab, leading to acorresponding loss of bond around the column reinforcement.As explained in Section 17.6, the loading procedure at yield of the specimen was to displace thespecimen through a beam tip deflection corresponding to a certain level of plastic hinge rotation at thejoint end of the main beam; the relative beam tip deflection was used as the controlling parameter.However, the reversal of the upper column rotation would contribute an additional amount to the valueof the relative beam tip deflection. Therefore, the level of deflection reflecting the hinging in the mainbeam would be lower than intended or planned.145Chapter 20Examination of Yield Behaviour of SpecimensIn the previous chapter, the response of the specimens was discussed in terms of the overallhysteretic behaviour. In this chapter, the at-yield and post-yield behaviour of the beam-columnsubassemblies and their individual members will be examined. The discussion will be concentrated onthe response under negative beam tip loads because the specimens were loaded in the negative directionmore frequently and because negative flexure (tension at the top) at the joint face was the critical modeof response in the design of the exterior beams of the prototype frame. Under negative loadings of thebeam-column subassembly, the various actions around the joint core would be those shown in Figure 20.1for a typical exterior reinforced concrete beam-column joint. Identifying these actions is useful for anunderstanding of the behaviour and response of the subassemblies in the tests.20.1 Flexural Strength RatiosThe post-yield behaviour of the beam-column subassembly depends a great deal on the yieldmechanism that is formed in the subassembly which, in turn, is governed by the strengths of the members.In the design of earthquake-resistant structures, the strong column-weak beam philosophy is followed toforce plastic hinging to occur in the beams and not in the columns The design code ensures this for anexterior joint by requiring the sum of the nominal strengths of the column at sections above and belowthe joint to be at least 10% greater than the probable strength of the beam at the joint face. In the designof the six-storey study frame test specimens, the code requirement of strong column-weak beam wassatisfied. Therefore, only the main beam is expected to yield in the tests of the beam-columnsubassemblies.146To examine the actual yielding mechanism in the exterior beam-column subassemblies tested, theratio of flexural yield strength of the column to that of the beam, which is called the flexural strengthratio, is studied. A ratio of less than one means that the column would develop plastic hinging before thebeam would. In this study, the flexural strength ratios are based on the nominal values of the yieldstrength which are calculated using the measured material properties and actual reinforcement positions.The strength of the column is dependent on the axial load in the member. Since the column ineach subassembly was loaded axially by a hydraulic actuator set at displacement-control (that is,displacement of the piston held fixed during the entire test), the axial load in the column varied asdifferent loads were applied at the beam tip. Figures 20.2 to 20.5 show the variations of the column axialforce with the beam tip loads in the four tests. All the columns were in compression at all times. Theinitial, average, and maximum axial loads in the upper column in each specimen are listed in Table 20.1.The table also shows the axial load that was in the upper column when first-yield was reached in eachdirection of loading. For comparison, the axial load and moment at the balanced point for the columnare calculated and shown in Table 20.2. It can be seen that in each specimen the average axial load inthe upper column is near the balanced load.In calculating the yield strengths of the upper and lower columns, the axial loads associated withthe first yield of the subassembly were used. Depending on the direction of loading, the axial load in theupper column is one of those shown in the last two columns in Table 20.1, and the corresponding axialload in the lower column is the sum of the upper column load and the beam tip load. The flexural yieldstrengths of the columns subjected to these axial loads are listed in Table 20.3. It should be noted thatthe strengths of the upper and lower column in each specimen are very close to each other. Thecalculated yield strengths of the main beam in each specimen are also shown in Table 20.3. The flexuralstrength ratios for negative and positive loadings are determined for each specimen and are tabulated inTable 20.3. It can be seen that the ratios are greater than 1.3 for negative loading and greater than 2.0for positive loading. Hence, negative deflection of the beam-column subassembly is the critical directionof loading.147The specimen set-up is statically determinate; therefore, the moment at the base of the uppercolumn is determinate and is directly proportional to the vertical load at the beam tip. The bendingmoment in the beam at the face of the joint is also proportional to this load. Based on the geometry ofthe specimen, the moment at the base of the upper column is 42% of that in the beam at the joint face fora given vertical beam tip load. Table 20.3 shows that the flexural strength of the upper column is morethan half of the beam strength in each specimen. (This explanation also applies to the lower column.)This fact, combined with the observation that the flexural strength ratios are more than one, indicates thatthe main beam would develop plastic hinging first, as expected from satisfying the strong column-weakbeam criterion.When strain hardening comes into effect in the yielded tensile reinforcement in the beam, the loadcarrying capacity of the main beam should increase; this is confirmed by the beam tip load-deflectionplots, in which the beam tip load can be seen to increase after yield. The effect of the higher momentin the beam at the joint face is to decrease the value of the flexural strength ratio. Table 20.4 shows themaximum negative beam tip loads attained in the tests, the corresponding axial loads in the upper andlower columns, and the flexural strength ratio based on these new loads. The flexural strength ratios arenow closer to, but still greater than, one. In Section 19.5, the actual yield strengths of the beam-columnsubassemblies were shown to be higher than the theoretical yield strength of the main beam. But thedifference is only several percent and so its effects on the flexural strength ratios calculated in Table 20.3would be minorTherefore, in summary, the calculated flexural strength ratios indicate that the column is strongerthan the beam in flexure during the entire test of each specimen, which means that only the main beamshould develop plastic hinging. The occurrence of hinging in the main beam and the column will beexamined in later sections.14820.2 Effective Slab Width in TensionOnly one reinforcing bar in the slab on either side of the main beam was assumed to contributein the calculation of the negative flexural yield strength of the beam for the determination of the flexuralstrength ratios in the previous section, . However, as stated earlier, the actual yield strength of the mainbeam was slightly higher than the theoretical value. Therefore, it is likely that more slab reinforcementis effective in resisting the negative bending moments in the main beam.Based on their tests on exterior beam-column joints with various slab widths, Durrani and Zerbe[1987] found that the effective slab width, defined as the width over which the slab reinforcementparticipates in resisting negative bending moments in the main beam, can be taken as the column widthplus two times the height of the transverse beam. When this effective slab width is applied to the beam-column subassemblies studied in this research, two bars on other side of the main beam should beconsidered in providing flexural resistance (see Figure 20.6).The participation of slab reinforcement in the tests can be examined by looking at the strains inthe reinforcing bars in the slab. In each specimen, all six longitudinal bars in the slab were instrumentedwith strain gauges at a section adjacent to the joint face. However, only the set of strain measurementsfor the left side of the slab in specimen BC3 is complete. Figure 20.7 shows these strains measured undersix different negative loads, one of which is the observed negative yield load. The yield strain obtainedfrom tensile tests of these bars is indicated in the figure by a dotted line. The results indicate that the barclosest to the beam yielded before the yield load of the beam-column subassembly was reached. Themiddle bars eventually reached yield when the beam tip load was increased by about 10% above the initialyield. These responses of the slab reinforcement were confirmed by measurements of the strain in themiddle reinforcing bar in specimens BC2 and BC4; these are plotted against the applied beam tip loadin Figure 20.8. In this figure, yielding is indicated very clearly by the flattening of the curve and occursat a beam tip load that is about 10% higher than the load at first yield of the subassembly, indicated inthe figure by a vertical dotted line. This amount of increase in the beam tip load for the middle149reinforcing bar to yield agrees well with the 13% increase in the theoretical flexural yield strength of themain beam when two bars, rather than one, on either side of the beam are considered. However, thestrains in the middle reinforcing bars were quite high at the yield point of the subassembly when comparedto the strains in the inner reinforcing bars. Therefore, it can be concluded that two bars on either sideof the main beam contribute to the negative flexural resistance of the beam. The additional strengthcontributed by the two middle reinforcing bars in the slab would account for the average 7% differencebetween the theoretical and the measured yield strengths as indicated in Section 19.5.Figure 20.9 shows the strains in the outermost reinforcing bars in the slab during the entireduration of tests BC2 to BC4. The strains are given in terms of the yield value and are plotted againstthe beam tip load. The results demonstrate that these bars in specimens BC2 and BC3 were not yieldedin the tests and the maximum strain attained was only about half of the yield value. In specimen BC4,however, the strains in the outer bars were relatively larger, with gauge SRR reaching the yield strain.But the strains at the first-yield of the specimen were only about 50% of the yield value. Therefore, itcan be concluded that the outermost reinforcing bars in the slab did not contribute significantly to theflexural yield strength of the beam in the tests and would contribute only a small amount at largerloadings.In summary, the effective slab width in tension in the beam-column subassembly under negativeloadings is in agreement with the width proposed by Durrani and Zerbe. Two bars on either side of themain beam participated in resisting negative moments in the beam-column subassembly of this study.However, only the two bars adjacent to the main beam will have yielded when yield is first reached inthe beam tensile reinforcement; the other two reinforcing bars within the effective slab width would yieldwhen the moment in the beam is increased by about a further 10%.15020.3 Yielding of Main BeamBased on the design of the study frame and on the flexural strength ratios determined in Section20.1, plastic hinging is expected to occur in the main beam. In each specimen, strain gauges weremounted on one top and one bottom reinforcing bar in the beam to indicate yielding. Four sections,designated as sections C, D, E, and F (see Figure 20.10), were monitored by a pair of top and bottomgauges at each section. The first section, C, is at the joint face, and the next three sections, D, E, andF, are located 2, 4, and 8 times the hoop spacing, respectively, from it. The distance to section E fromthe joint face, which is about 380 mm, is approximately equal to the effective beam depth.In Figures 20.10 to 20.12, the measured strains in the longitudinal beam reinforcement are plottedagainst the beam tip load for specimens BC2 to BC4; only those gauges that produced reasonable resultsare shown. Many of these gauges, though, malfunctioned at the start of the large loadings in the tests,and the malfunctioning is indicated by a horizontal line in the plot. In these figures, positive strain meanstensile strain and positive beam tip load signifies upward loading at the beam tip; positive strains weredeveloped in the top reinforcement during negative loading.Also drawn in each plot in Figures 20.10 to 20.12 is a dotted line indicating the measured tensileyield strain of the reinforcing steel. It can be seen that the top reinforcement at section C in eachspecimen first reached the yield strain at a negative load that is very close to the yield load indicated bythe beam tip load-deflection plot. Similar results are observed for the bottom reinforcement at section C.Therefore, yielding of the beam-column subassembly in either direction of loading is associated withyielding in the main beam at the joint face.As the displacement and load of the beam-column subassembly are increased, yielding of thebeam reinforcement is expected to spread down the length of the beam for a distance of at least oneeffective beam depth, that is, to section E. However, as shown in Figures 20.10 to 20.12, the topreinforcement at section D in specimens BC2 and BC3 did not reach yield in the tests; unfortunately, thegauge on the top reinforcement at section D in specimen BC4 was not operational. Furthermore, no data151were available from the gauge on the top reinforcement at section E in all the specimens. On the otherhand, the gauges on the bottom reinforcement at section D seem to have reached yield in the cycle of thelargest positive beam tip load in all specimens. No conclusive results can be obtained from the strainmeasurements at section E. Therefore, based on these observations, the yielding in the main beam spreadbeyond section D at the bottom of the beam only and it never reached this section at the top. All thegauges at section F in the specimens showed strains lower than the yield value, which is expected sincethis section is two times the effective beam depth away from the joint face.The strain measurements and the onset of yielding can be checked by calculating the curvatureat the joint face using the data from the pair of gauges at the section and then comparing the result withthe theoretical yield curvature. Only the results in specimen BC3 are complete for such examination.Figure 20.13 plots the curvature calculated from the measured strains against the bending moment at thejoint face for specimen BC3. The theoretical negative yield curvature for the beam (9.49 x 10 -6 mm-1)is also shown in the figure as a dotted line. It can be seen that the measured moment-curvature curvebegins to flatten at the theoretical yield curvature value. This provides confidence that the strainmeasurements shown in Figure 20.10 to 20.12 are proper, and the onset of yielding in the main beam canbe indicated by the strains measured in the longitudinal reinforcement.Curvature in the main beam can also be obtained using the beam rotation measured at threesections at 200-mm spacing from the joint face (sections 1, 2, and 3 in Figure 20.14). These measuredrotations for specimens BC2 to BC4 are plotted against the applied beam tip load in Figures 20.14 to20.16. The plots show the same type of hysteretic behaviour as their respective beam tip load-deflectionrelationships. Curvature is obtained when the difference between the rotations of adjacent sections isdivided by the 200-mm spacing, and this curvature would be the average value over the 200-mm lengthbetween sections. However, the computed curvatures for both sections in each specimen show very highvalues with negative curvatures at least two times larger than the theoretical values. These largecurvatures are contrary to the results indicated by the measured strains in the beam longitudinalreinforcement which demonstrated that yielding in the top reinforcement did not extend past section 2 (that152is, section E based on the designations used for the strain gauges) in the beam. Thus the measurementsare discredited, with the conclusion that the method used is ineffective for obtaining accurate curvatureestimates. Factors such as bending of the L-shaped measuring frames on which the transducers weremounted and bending of the measuring rods of the transducers might have affected the measurements.20.4 Flexural Response of ColumnFour strain gauges were mounted on the column longitudinal reinforcement at sections just aboveand below the joint region (see Figure 20.17). These are identified by their positions relative to the sideview of the beam-column subassembly as top-inside (CTI), top-outside (CTO), bottom-inside (CBI), andbottom-outside (CBO) gauges. Unfortunately, the top-inside gauge in every specimen was damaged beforethe test. Strain measurements for the remaining part of the tests after first yield are plotted in Figures20.17 to 20.19. Again, erroneous results produced by malfunctioning of the gauges have been removedfrom the plots. Gauges CTI and CB0 would be in tension under negative beam tip loads; gauges CTOand CBI would be so under positive loads.Figure 20.17 shows that the upper and lower columns in Specimen BC2 never yielded. Thisconclusion cannot be made for specimens BC3 because the two gauges that were operational at the startof the test malfunctioned early, before the cycles of large loadings. But gauge CBO was operational longenough to show that the tensile yield in the outer longitudinal reinforcement in the bottom column waswell below the yield value when the negative first-yield of the main beam occurred (see Figure 20.18).Similarly, the strain at gauge CBO in specimen BC4 was also lower than the yield value at the negativefirst-yield of the subassembly (see Figure 20.19). These low strains confirm that the column was strongerthan the beam in the subassembly under negative beam tip loadings and support the results indicated fromthe examination of the flexural strength ratio . The strength ratios for positive loadings were even higherthan those for negative loadings, but gauge CBI in specimen BC4 shows possible yielding of the columnlongitudinal reinforcement at the largest positive beam tip load (see Figure 20.19); unfortunately, the153gauge malfunctioned just as the yield strain was reached. No explanation can be made for these largevalues of strain for the inside edge of the lower column in specimen BC4.20.5 Response of Joint CoreThe examination so far has indicated that yielding was expected to occur, and did occur, in themain beam of the beam-column specimens. However, the spread of the yield zone from the joint facedown the length of the beam was limited. The lack of severe damage observed in the beam and the largepinching displayed in the beam tip load-deflection response indicate that the joint core played a criticalrole in the response. It is known that the inelastic behaviour of the joint core is not very effective indissipating energy and, therefore, it is important to examine the response of the joint core of the beam-column specimens in the tests.In an exterior beam-column joint, the joint core is subjected to horizontal shear forces arisingfrom the transfer of shear forces in the column and axial forces in the beam reinforcement. Undernegative loads, the difference between the shear in the column (V' in Figure 20.1) and the tension forcein the top reinforcement of the beam (T) is the shear force acting on the joint core. For the discussionin this section, the tensile force in the top beam reinforcement is taken at 7% above the yield value, toaccount for the effects of strain hardening, and includes the participation of two reinforcing bars in theslab on each side of the main beam. The amount of strain hardening considered corresponds to the strainsat five times the yield strain. The tensile force in each specimen is listed in the first column of Table20.5. In the second column of the table, the maximum shear force in the upper column of the specimensin each test is shown. This is calculated from the maximum negative beam tip load by equilibriumconsiderations. The difference between the values in the first and second columns of Table 20.5, then,is the maximum shear that occurred in the joint core of the specimen, and this is shown in the thirdcolumn of the table. This maximum shear force is converted into stress in the fourth column of the table154by dividing it by the plan area of the joint. The shear stress is also expressed in terms of its ratio to thesquare root of the concrete compressive strength, as shown in the last column of Table 20.5.It should be noted that the participation of the slab reinforcement in resisting negative momentsincreases the shear force in the joint core by causing additional shear stress through the torsional forcesthat the slab reinforcement introduces in the transverse beams, rather than by increasing the total tensileforce at the top of the beam. The shear stress based on the latter procedure, however, is larger and thusconservative.Table 20.5 shows that the joint shear stresses in specimens BC!, BC2, and BC3 are about 1.2q,while that in specimen BC4 is higher at 1.6Vfc,. All values are close to but lower than the unfactoreddesign shear strength for an exterior joint, which is given as 1.8Vfc, by the Canadian reinforced concretecode, and so failure of the joint core was avoided in the tests. However, it has been suggested that thejoint shear stress should be limited to 1.0Vfc, in order to prevent excessive damage in the core region[Ehsani and Wight 1984], and to 9 MPa in order to avoid brittle diagonal compression failure [Paulay andPriestley 1992]. Based on these criteria, the joint cores in this experimental study can be considered tobe severely damaged, especially the joint core of specimen BC4, which experienced a relatively largeshear stress of 9 MPa.A strain gauge (HJC) was mounted at the centre of the joint core on a long leg of the transversereinforcement enclosing the inner longitudinal bars of the column (see Figure 20.20), and it measuredstrains in the direction along the axis of the main beam. Measurements of this strain gauge in tests BC2to BC4 are plotted against the applied beam tip load in Figure 20.21. The measurement for specimenBC1 was not recorded. This figure shows that the strain in the hoop did not vary significantly duringinitial low loadings of the specimens, but that it suddenly showed a softening with permanent large tensilestrains. This response agrees with the occurrence of diagonal cracks in the joint core. The beam tip loadsat which cracking of the joint core occurred in specimens BC3 and BC4 are shown in Table 20.6. Noconclusive results can be obtained from the response of specimen BC2. Comparing to the load at firstnegative yield, the first-crack load is lower, which indicates that the diagonal cracks in the joint core155developed before yielding of the main beam. Theoretical cracking stress of the joint core was determinedassuming that cracks occur when the principal tensile stress reaches 0.33Vfc,, which corresponds to anegative beam tip load of 131 kN for the four specimens. The measured first-crack loads are on average18% lower than the theoretical value.The measured yield strain of the reinforcing steel is indicated in each load-strain plot in Figure20.21 by a dotted line. It can be seen that the shear reinforcement in the middle of the joint core yieldedin each test. The load at which the joint shear reinforcement yielded is also shown in Table 20.6, andit is just slightly larger than the first-yield load of the subassembly as given in Table 19.4. Therefore,the beam-column subassemblies suffered shear yielding of the joint core immediately after the main beamyielded. Shear yielding means that the damage in the joint core was high, as indicated by the shearstresses computed above.Another indication of the extent of damage to the joint core is the penetration of yield in the beamreinforcement into the anchorage zone. Strain gauges were mounted on one top and one bottomreinforcing bar at a location corresponding to the centre of the column. Another gauge was mounted onthe hook of the bar at a location corresponding to the mid-height of the joint (see Figure 20.20). Figure20.22 shows the results obtained from the gauge in the embedment length of the top bar only (BTB) inspecimens BC2 to BC4. The load-strain response in Figure 20.22 indicates that the section of the beamtop reinforcement at the centre of the column yielded in test BC4. In specimen BC3, the maximum strainwas close to the yield value, and thus it can be expected that the section in specimen BC3 also yieldedeven though the gauge malfunctioned just before the yield value. This yield penetration into theembedment length of the beam reinforcing bar is a direct result of the increased damage in the joint core;extensive cracking in this region would decrease the bond between the bar and its surrounding concrete,allowing yield to penetrate from the joint face. Yield penetration to the centre of the column did notoccur in specimen BC2. The gauges on the bottom bars did not function in tests BC2 and BC4.However, this gauge indicates that yielding occurred in test BC3, as shown in Figure 20.23. Although156yield penetrating into the embedment zone signifies deterioration of the integrity of the concrete in thejoint core, it indicates that there is still adequate anchorage for the reinforcement.The strains in the hooks of the bars in each specimen are shown in Figure 20.24. It can be seenthat, with the exception of gauge BBA in specimens BC3 and BC4, these strains remained below the yieldvalue during the entire test. The strains under initial loadings are separated out from the entire responseand are drawn in Figure 20.25. This figure shows that the strains remained constant at a negative value,presumably due to the column axial load, during small initial loadings and then suddenly increased intension at a certain load. This is the same behaviour displayed by the response of the transversereinforcement in the middle of the joint core and can be attributed again to the development of diagonalshear cracks. The first-crack loads indicated by the strains in the hooks of the top and bottom beam barsin specimens BC3 and BC4 are listed in Table 20.7. The two loads for specimen BC4 correspond to thefirst-crack load obtained earlier using the response of the joint hoop. The results for specimen BC3,however, are higher but a lot closer to the theoretical value calculated above.In each test, the joint core was also instrumented with a linear transducer in order to measure itsshear deformation. However, this measuring device was mounted off the rig that was attached to the baseof the upper column. As revealed earlier, the response of the upper column was adversely affected bya gap opening up at the base of the member. Therefore, the measurement of the joint shear deformationwas also affected by this movement and the true values of the shear deformation were not obtained. Onlythe results obtained during the small initial loadings of the tests can be considered reasonable, but theydid not provide any useful information.20.6 SummaryFrom an examination of the response of the beam, column, and joint core of the beam-columnsubassemblies, it can be concluded that specimens BC! to BC4 performed adequately in the cyclic-loadingtests; the maximum loads were maintained and the hysteretic behaviour was stable.157For all four specimens, the main beam developed plastic hinging while the column steel remainedbelow yield, as predicted by the flexural strength ratios. The resistance of the main beam should alsoinclude contributions from the slab reinforcement; in this experimental study, two slab reinforcing barson either side of the beam were effective in providing flexural resistance.Although yielding of the longitudinal reinforcement in the main beam causes pinching in the load-deflection response of the beam-column subassembly, the extensive pinching that occurred in the tests wasmainly caused by damage to the joint core. As indicated by the strains in the transverse reinforcementin the middle of the joint core, the joint core yielded in shear. The high damage to this region alsoreduced the bond of the beam reinforcement, allowing bond slip to occur; slipping of the columnlongitudinal reinforcement was also observed. Slipping of reinforcement also contributed to theoccurrence of pinching in the tests.In summary, the sequence of significant events in the inelastic response of the beam-columnsubassembly under negative loading is as follows. At the end of the elastic response, the reinforcementat the top of the main beam and in the slab yielded. At this stage, diagonal cracks had occurred in thejoint core. The upper column then developed a splitting crack near the back face, and a gap was formedat the base of the upper column, resulting in reversal of the column rotation. Finally, after furtherreversed loadings, increased damage in the joint reduced the bond around the beam bars, allowing yieldpenetration into the embedment zone and reducing the stiffness of the beam-column connection.158Chapter 21Large-Deflection Test21.1 PurposeAs explained in Chapter 17, tests BC1 to BC3 are considered to be investigations of the responseof code-designed reinforced concrete frames deflecting to the design level specified by the codes. Forthe prototype frames studied, the design deflection level was a tip displacement ductility of 4. Asdemonstrated in the analytical studies of the six-storey frames, considerable overstrength exists in a framedesigned in accordance with the codes and this reduces the amount of inelastic action or ductility in theframe responding to the design earthquake. However, studies of the strength and ductility demands ofearthquakes in Part I revealed that a higher ductility would be imposed by a filtered ground motion havingan intensity of the design earthquake. Therefore, a test of the beam-column subassembly to largerdeformations was carried out. This test was intended to demonstrate the effects of large deflections onthe response of the beam-column subassemblies, based on the large ductility requirement of a filteredearthquake, and to reveal the response to the actual ductility level of the code, based on the effects of theoverstrength.21.2 Test SpecimenFor the large-deflection test, the response of the R4 frame was selected to be examined, and thusthe beam-column subassembly was the same as specimens BC1 to BC3. The dimensions and the detailingwere exactly the same as those shown in Figure 17.15 and 17.12, respectively. The actual positions ofthe reinforcement are indicated in Table 17.2, and the measured properties of the materials are listed inTable 17.3. It should be noted that the compressive strength of the concrete of this specimen was only15983% of the specified strength or the strength of specimens BC1 to BC4; the yield strength of thereinforcement, on the other hand, was about 10% higher than the specified value and that in the otherspecimens. This specimen is designated as BC5.21.3 Loading ProgramIn tests BC! to BC4, the results from dynamic analyses of the study frames were used in settingup the loading programs (see Section 17.7). The same procedure of scaling some earthquake records toachieve a certain displacement ductility level in the structure was not used to set up the loading programfor specimen BC5, since the response of the study R4 frame to large ductilities was found to be quiteunstable, creating difficulties in achieving the desired level of response. As the emphasis of the test wassimply to load the beam-column joint to larger displacements, simple cyclic loadings with increasingdisplacement ductility were used instead. The loading program for test BC5, which is expressed in termsof the beam tip displacement ductility, is shown in Figure 21.1. This type of loading program iscommonly used by researchers in tests of beam-column joints. The maximum ductility level intended was7, which is higher than the level of about 4 reached in tests BC1 to BC4. It was estimated that thisductility level for the beam tip displacement of the beam-column subassembly would correspond to theactual design system ductility level, which is 4 in this case for the study R4 frame. The numbering ofthe cycles or load points is also shown in the loading program in Figure 21.1 and is based on the levelof ductility intended to be reached in the test.21.4 Instrumentation and Data AcquisitionThe instrumentation in test BC5 is, in most parts, the same as that in tests BC1 to BC4. Thirtystrain gauges were mounted on reinforcing bars in the specimen at locations identified in Figure 17.14.But since large deformations of the specimen were expected, strain gauges having a maximum elongation160of 20% of the gauge length, compared to only 5% for those gauges used earlier, were employed. Mostof the linear displacement transducers and their location remained unchanged from the other four tests.The only changes were that two sections of the beam (sections 1 and 2) instead of three were instrumentedfor rotation measurements and one more transducer was added to measure the shear deformation of thejoint core. This shear deformation is reflected by the relative lateral displacement between two rods thatwere embedded into the core region from the back of the joint at the levels of the longitudinalreinforcement in the main beam. This second measurement of the joint shear deformation was set upbecause it was found in the earlier tests that the transducer mounted on the aluminum rig attached to theupper column did not produce correct results. Unlike tests BC1 to BC4, in which measurements of thelinear rotation transducer and of the load cells on the hydraulic actuators were recorded manually, allmeasurements in test BC5 were recorded and stored by a computer through the OPTILOG data acquisitionsystem.21.5 TestingDuring the set-up of the test, the loading lever arm on top of the specimen was accidentallydamaged by an unexpected downward load from the top hydraulic actuator. The arm buckled under theload, and its original stiffness could not be restored by subsequent repairs. Therefore, it was decided toreduce the level of the initial axial load in the column to about 500 kN from 800 kN.After the axial load in the column was applied, the test was carried out using a slow loading rate,and the loading was held constant at the end of each cycle for cracks to be marked and pictures to betaken. As in test BC1, the recording of the measurements was accidentally shut off during the test. Thishappened in the middle segment of the test between load points 3C + and 4 - , but fortunately, the mishapwas noticed early and the recording was resumed immediately. So data for only a small portion of thetest are not available.161When the specimen was being loaded to load point 6 - , a loud bang was heard and a sudden dropin the beam tip load occurred. Nevertheless, the test was continued to the end of the loading program.21.6 Hysteretic BehaviourThe beam tip load-deflection response of specimen BC5 based on the global beam tip deflectionis shown in Figure 21.2 and that based on the relative beam tip deflection is given in Figure 21.3. Again,the load axis in the load-global deflection plot was rotated to account for the effects of the deflection ofthe testing frame, the measurement of which in test BC5 is shown in Figure 21.4. The testing frameflexibility is given in Table 19.3. The early hysteretic behaviour of specimen BC5 can be seen to besimilar to those displayed by specimens BC1 to BC4. Pinching of the hysteresis loops is significant, andits onset can be observed in the cycle 3B. The most significant difference observed in the hystereticbehaviour of specimen BC5 is the loss of load-carrying capacity in the negative loading direction in thelatter stages of the test. The capacity in the positive direction, however, was maintained in the entire test.The negative strength of the beam-column subassembly seems to have decreased after the occurrence ofthe loud bang which shows up as a notch in the load-deflection curve. Discussion of the bang and thesubsequent loss of load-carrying capacity will be made in the next two sections.The yield points of the specimen are indicated in the beam tip load-global deflection plot in Figure21.2, and the beam tip load and deflection at these points are listed in Table 19.4. The measured yieldloads are compared to the theoretical values, calculated using actual material properties and positions ofreinforcement, in Table 19.5. The measured positive strength is 12% lower than the theoretical value butthe measured negative strength is only 5% higher.The negative yield point found from the load-deflection plots is also higher than the yield pointestablished during the test; this point is indicated in Figure 21.2 for comparison. The consequence ofunderestimating the yield point is lower ductilities imposed on the beam-column subassembly in thenegative loading direction. The actual ductility at each cycle, with the testing frame deflection component162deducted, is shown in Tables 19.5 and 19.6. The maximum ductility reached is 5.5 in the negativedirection and 7.2 in the positive direction. Therefore, the planned ductility of seven was achieved onlyunder positive loadings. Loading in the negative direction was not continued to the planned ductility levelbecause the load carried by the subassembly at the end of the last cycle was already one-half of theultimate load (see Figure 21.2). This amount of strength loss indicates severe damage to the specimenand is usually used as the criterion to terminate cyclic-loading tests of beam-column subassemblies.The cumulative displacement ductilities for specimen BC5 are also shown in Tables 19.6 and19.7. For negative loading, the value is close to that for specimen BC3; but for positive loading, thecumulative ductility is many times larger than that for specimens BC1 to BC4 because both the numberof yield excursion and the displacement ductility in the positive loading direction in test BC5 are relativelylarger. The number of times yield was reached in test BC5 is shown in Table 19.8.21.7 Yield BehaviourAs mentioned in Section 21.2, the compressive strength of the concrete was lower and the yieldstrength of the reinforcing steel was higher in specimen BC5. This variation in the material propertieswould affect the nominal strengths of the beam and column, and with a lower axial load in the column,the flexural strength ratio of the specimen would be reduced.The axial load and moment at the balanced point for the column in specimen BC5 are shown inTable 20.2. The balanced moment does not differ much from that for specimens BC1 to BC3, but thebalanced axial load is a lot lower. Fortunately, the initial axial load in the column was reduced becauseof the damage to the loading lever arm, as indicated in Section 21.5, and the resulting level of axial loadended up mostly around the balanced point, thereby utilizing the higher flexural capacities of the columnThe initial, average, and maximum axial load in the upper column in the test are shown in Table 20.1.The table also shows the column axial loads at the positive and negative yield points of the beam-columnsubassembly.163The bending flexibility of the upper column under initial loadings of the beam-columnsubassembly was also determined from the initial portion of the moment-rotation response shown in Figure21.5a. The calculated value is given in Table 19.9 and it is found to be almost identical to the theoreticalflexibility based on a cracked cross section. It can be concluded that the upper column in specimen BC5was cracked in the early stages of the test, and this can be attributed to the relatively low axial loadapplied to the columnThe flexural yield strengths of the upper column, lower column, and main beam were calculatedbased on the actual material properties and reinforcement positions and are shown in Table 20.3. Thetheoretical flexural strength ratio under negative loadings is determined to be only 1.20, which is 80%of the ratios in specimens BC1 to BC3. But the value is still greater than one, which means that initialyielding would occur in the main beam. However, when the applied moment in the beam is increasedto the level given by the ultimate beam tip load achieved in the test, the flexural strength ratio is reducedto 0.90, as shown in Table 20.4. This signifies that the column of specimen BC5 yielded at large beamtip loads in the test.Figure 21.6 shows the strains in the longitudinal reinforcement in the main beam. It can be seenclearly that the top reinforcement at the joint face reached yield at the measured yield load. Furthermore,the figure indicates that section D of the main beam also yielded at about the same load, but it occurredin the cycle following the one that caused first-yield at section C. Yielding in the main beam did notreach section E until late in the loading program and never reached section F.Figure 21.7 shows the strains in the longitudinal reinforcement in the column at sections aboveand below the joint. Again, the top-inside gauge did not function in the test. The other three gaugesfunctioned properly at the start of the test but they malfunctioned later on, producing erratic response inthe load-strain plots. Before malfunctioning, the strain gauges did not show yielding of the columnreinforcement. Therefore, strain measurements for the column longitudinal reinforcement cannot verifythe occurrence of yielding in the column.164Yielding of the joint core, as observed in the other four tests, was indicated by the response ofthe strain gauge on the middle horizontal transverse reinforcement bar. The strains in this bar are shownin Figure 21.8. As expected for shear reinforcement embedded in concrete that has diagonal cracks intwo directions, the joint hoop gains tensile strain under either positive or negative loads. Similar to theresponse obtained in the other tests, the strain in the joint hoop did not respond to the initial applied beamtip load, but shows a large tensile strain after the joint core cracked. The first-crack load and the yieldload for the joint core in test BC5 are estimated and compared to the results in the other four tests inTable 20.6.The high level of damage in the joint core is indirectly indicated by the penetration of yield intothe embedment zone for the beam longitudinal reinforcement, as shown in Figure 21.9. The yieldpenetration was possible because, again, the hooks of the bars was able to provide adequate anchorage.The strain measurements for the hooks also indicate the load at which diagonal cracks developed in thejoint core and can be seen in Figure 21.10b as the load at which the strain suddenly changes direction.This load is listed in Table 20.7 and is close to that found in tests BC2 and BC3. All these values of thefirst-crack load from the strain measurements of the joint shear reinforcement and the hooks of the mainbeam reinforcement are only 8% higher than the theoretical value of 103 kN.With the lower compressive strength of the concrete and the higher tensile strength of thereinforcing steel, the shear stress level in the joint was relatively high and the maximum value approachedthe theoretical strength of 1.8Vfc, given by the reinforced concrete code (see Table 20.5). Therefore, thejoint core region can be considered to have suffered severe damage.The joint shear deformation measured using the two rods embedded in the joint core is shown inFigure 21.11. The deformation can be seen to remain at near zero until a negative beam tip load of about230 kN. This load is larger than the first-crack load indicated by the joint transverse reinforcement butclose to the yield load indicated by this reinforcement. Therefore, this indicates that the sheardeformation (and thus the damage) becomes significant when the joint shear reinforcement yields.16521.8 Damage to SpecimenThe pattern of damage in specimen BC5 is similar to those in specimens BC1 to BC4. However,the extent of the damage seems to be higher in this specimen. Figure 21.12 shows the damaged specimenunder a positive beam tip load near the end of the test. The flexural cracks at the bottom of the mainbeam in this test can be seen to be wider than the cracks in specimens BC1 to BC4. The damage to theback of this specimen is also more severe than those in the other specimens, and it greatly affected theresponse of the beam-column subassembly, initiating failure of the specimen, as will be discussed in thefollowing section.21.9 Failure Mode of SpecimenAt the lower ductility levels, specimen BC5 behaved in a similar manner as the other four beam-column specimens, in which yielding of the main beam and the joint core occurred. However, asindicated by the beam tip load-deflection plot, specimen BC5 can be considered to have failed in the sensethat the load-carrying capacity of the subassembly was observed to drop off dramatically in the last fewcycles after the bang.It was initially thought that the bang was caused by fracture of one of the reinforcement bars.However, the reinforcement exposed by the spalling of the concrete cover at the back of the specimenshowed no signs of fracture. Also, the concrete cover at critical sections of the beam and column werehammered off after the test but no broken reinforcing bars could be found. Therefore, it has to beconcluded that the bang was not caused by fracture of any of the reinforcing bars.The cause of the loud bang was most probably the release of the strain in the steel testing frameas a consequence of the sudden drop in the applied load at the beam tip. The drop in load was most likelycaused by a sudden extension of the crack in the concrete cover at the back of the column and joint core.Figure 21.13 shows the back of the specimen moments before and after the bang. It can be seen that,166before the bang, a long and wide crack had already developed near the back of the upper column and jointand that, after the bang, the crack extended down into the lower column, thus causing the entire backcover to be separated from the specimen. The cover did not fall off the specimen completely at the bangbecause the rods which were inserted into the joint core were holding it up; it was removed by handafterwards. Figure 21.14 shows the rear view of the specimen without the back cover. The idea of theloud bang being associated with the loss of the back cover is supported by an abrupt increase in the strainin the outer longitudinal reinforcement in the column below the joint at the time of the bang (see gaugeCBO in Figure 21.8).As shown in Figure 20.1, the outer edge of the upper column is in compression under negativebeam tip loads. But when the back concrete cover of the upper column is absent, the transfer of thecompressive force to the diagonal concrete struts in the joint core would shift the neutral axis in thesection of the column towards the centre. The result is a reduction in the flexural capacity of the column.Using the value of the axial load in the upper column observed at the time of the bang, the momentresistance for a column section without the cover on the compression side is computed to be 141 kN-m,a 14% decrease from the resistance for a section with the cover. This is in agreement with the behaviourin the beam tip load-deflection response in which the negative beam tip load dropped by 17% betweencycles immediately before and after the bang.The observed amount of decrease in the flexural strength of the upper column would reduce theflexural strength ratio of the specimen at that stage such that the upper column would develop plastichinging at the joint face. Although yielding of the column could not be demonstrated by themeasurements of the strains in the longitudinal reinforcement, the highly distorted geometry of thespecimen, as shown in Figure 21.15, indicates that yielding in the upper column might have occurred.The picture in Figure 21.15 is a side view of the specimen loaded at the end of the second last cycle ofloading, and it clearly shows a kink between the upper column and the beam-slab unit; yielding of themember and also slippage of the column bars would be associated with such large local deformation. Theerratic moment-rotation response of the upper column in the last few cycles of loading, as shown in167Figure 21.5, also points to a different behaviour of the beam-column subassembly in the end stages of thetest. Furthermore, Figure 21.15 shows that the main beam and the joint core region displaced forwardconsiderably after the bang had occurred. The large forward displacement would induce additionalbending moments into the upper column under negative loading of the subassembly and consequentlyreduced the available strength of the column to resist the primary moments induced by the beam tip load.Further cycling of the specimen deteriorated the integrity of the joint core more and perhapsdeteriorated the integrity of the concrete confined within the reinforcement in the upper column also,reducing the flexural capacity of the column even more. This, combined with the increased forwardmovement of the beam-slab assembly, caused further reduction in the load-carrying capacity of the beam-column subassembly in the last three cycles of loading. In the last cycle, the negative beam tip loadcarried was only 54% of the maximum attained.21.10 SummaryIn summary, specimen BC5 was fabricated identical to specimen BC1 to BC3 but was loaded tolarger deflections under cyclic loading of increasing beam tip displacement ductility. The maximumductility reached was 5.5 in the negative direction and 7.2 in the positive direction. The specimen showedsimilar early yield behaviour as the other specimens in this experimental study --- the main beam yieldedat the joint face and significant pinching occurred during load reversal because of distress in the jointcore. However, this specimen yielded in the column when the main beam capacity increased as a resultof strain hardening and other effects. The yielding was possible because of the weaker concrete andstronger reinforcing steel and was enhanced by the loss of the back concrete cover of the joint core. Thelatter reduced significantly the flexural strength of the upper column The occurrence of yielding in thecolumn created a mechanism in the beam-column subassembly which caused large deflections anddrastically reduced the load-carrying capacity of the subassembly.168Chapter 22Comments on Experimental StudySome lessons learned from the results of the five cyclic-loading tests on exterior beam-columnsubassemblies are presented in this chapter.22.1 Effects of Mitered EarthquakesIn summary, five beam-column subassemblies were tested to look at the response of reinforcedconcrete frames detailed in accordance with the code. Tests BC1 and BC2 looked at the response of acode-designed structure when it responds to the code-response level of 4) in unfiltered earthquakes ofEl Centro EW and Taft, respectively. Test BC5 examined what happens to the same structure when itis loaded to higher ductility levels, say, in a filtered earthquake. Tests BC3 and BC4 examined twomodifications that were made which influenced the structure's response to filtered earthquakes, inparticular, to the Mexico City SCT ground motion; test BC3 studied the case of reducing the intensityof the ground motion such that the code-response level of the structure is reached, and test BC4 studiedthe case of increasing the strength of the structure to reduce the ductility demands in the Mexico Cityearthquake. The influence of two main effects of filtered earthquakes --- larger number of strong cyclesand larger ductility demand --- on reinforced concrete frames were revealed by the five tests.22.1.1 Effects of Loading HistorySince tests BC1 to BC3 represent response of structures to the same level of structural ductilitybut under different earthquake loadings, their results would reveal the effects of loading history. It wasobserved that the damage in the three specimens were only moderate, which means that the damage169suffered by a ductile moment-resisting frame designed in accordance with the code would be onlymoderate in the design earthquake. The extent of damage was such that the structure could be repairedand put back into service again, as proven by the response of the specimen repaired with injected epoxy(see Appendix C).It should be emphasized that the pattern and extent of damage among specimens BC!, BC2, andBC3 was similar. The response of the joint core is the factor responsible for the similar damagedisplayed. It was shown in Chapter 18 that the propagation of cracks in the beam ceased after yieldingoccurred at the joint face. The reason for it is connected to the onset of inelastic response in the jointcore. Furthermore, the damage to the joint core is hidden by the beams framing into the region on threesides, which made examination and thus comparison of the damage to the joint core very difficult.Comparison of the beam tip load-deflection behaviour for the three specimens is also not effectivein revealing differences among the tests because all the specimens were loaded differently and the loadingprograms contained many cycles of small loading. These small-load cycles do not seem to have anynoticeable adverse effects on the response of the specimen. But the occurrence of pinching is a commonfeature of the hysteretic response of the subassemblies. Therefore, the results of tests BC1, BC2, andBC3 demonstrate that there are no significant effects caused by differences in their loading history,particularly in the number of strong cycles, for structural response up to the ductility level specified bythe code.22.1.2 Effects of Large Ductility DemandComparing the results of tests BC!, BC2, and BC3 to the result of test BC5 reveal the effects oflarge ductility demands on a code-designed structure. When loaded to a much higher ductility level inthe positive loading direction, specimen BC5 developed wider flexural cracks at the bottom of the mainbeam, but its strength did not degrade. But, in the negative direction, the strength did degrade after abeam tip ductility of 4, and the damage was even more severe when loss of back cover of the specimen170occurred. Furthermore, near the end of the test under negative loading, the specimen was observed tohave undergone a significant change in its deflected shape, which was not observed in the other tests.This was caused by the unwanted yielding in the column, which was heightened by the spalling of theback concrete cover of the joint core. As shown in Figure 21.11, the joint core shear deformation inspecimen BC5 increased continuously as the specimen was subjected to more and larger cycles. Theexcessive shear deformation in the joint core caused the region to expand, which helped to break off theback cover. Therefore, the most significant consequence of responding to large ductilities is thedeterioration of the integrity of the joint core which leads to spalling of the cover in the joint core andto reduction in the load-carrying capacity.22.1.3 Response of Stronger StructureSpecimen BC4 was designed to be stronger than the other four specimens and its prototype framewas found to be able to withstand the unsealed Mexico City earthquake with moderately low displacementductility demand. The response of the beam-column subassembly shows that satisfactory behaviour wasexhibited; the hysteretic behaviour and the extent of cracking were both similar to those of the weakerspecimens. These results were obtained despite the relatively large joint shear stress caused by the greateramount of top reinforcement in the main beam. The lower ductility level in this test probably helped tomaintain the satisfactory performance of this specimen. Therefore, providing stronger beams and columnsmay be necessary to resist strong filtered earthquakes, but the effects of the additional reinforcement onthe joint response should be checked to ensure that shear failure of the joint core will not occur.22.2 General CommentsBesides revealing the effects of filtered earthquakes, the cyclic-loading tests in this experimentalstudy also point out several important issues regarding the seismic response of exterior reinforced concrete171beam-column subassemblies and also, indirectly, the seismic response of ductile reinforced concreteframes.22.2.1 Effective Slab WidthThe test specimens under negative beam tip loads showed that two reinforcing bars on each sideof the main beam contributed significantly to the negative flexural resistance of the beam. The slab widthwhich contains these reinforcing bars corresponds to the effective slab width proposed by Durrani andZerbe [1987]. The results suggest that the participation of the slab reinforcement should be included indetermining the flexural strength of the beam, and that the effective width proposed by Durrani and Zerbeis reasonable for use in design of ductile members.22.2.2 Strong Column-Weak BeamWith strength loss of about 50% following yielding in the upper column, test BC5 reinforces theimportance of having strong columns and weak beams in a ductile moment-resisting space frame.22.2.3 Integrity of Joint CoreMost important of all, the tests demonstrate the significance of the response of the joint core tothe behaviour of a reinforced concrete frame.The joint core in each specimen in this study yielded in shear under negative beam tip loads afterthe main beam had yielded in flexure. Consequently, the response of the joint core dominated theresponse of the entire beam-column subassembly under negative loadings after the initial beam yield; theassociated high level of damage in the core region caused large pinching in the hysteretic behaviour of172the subassembly, and the high level of damage normally shown in the main beam by tests on beam-columnsubassemblies did not occur in this study.Under positive loadings, the beam-column subassemblies behaved very well, as indicated by theresponse of specimen BC5. Since the level of shear stress in the joint core under positive beam tip loadswas only about half of that under negative loads, it can be assumed that the response of the joint corewould be satisfactory for this level of shear stress. As stated in Section 20.5, the limit of the joint shearstress to prevent excessive core damage has been given as 1.0■./fc' by Fhsani and Wight. The results ofthe tests in this study, therefore, support this finding. Also, Paulay and Priestley [1992] stated that thearea of joint horizontal reinforcement should be at least 1.25 times the area of top reinforcement in thebeam for good performance of a beam-column joint. This, too, is supported by the tests since the ratioof the area of the above-mentioned reinforcement was only 0.86 in specimens BC!, BC2, BC3, and BC5and 0.60 in specimen BC4, and all specimens showed undesirable large shear deformations in the jointcore. It should be noted that large joint shear deformations help to break off the back cover, whichreduces the column flexural strength; severely cracked joint cores result in costlier and less effectiverepairs.Since the joint core was designed in accordance with the reinforced concrete code, the test resultsindicate that large damage in, and yielding of, the joint core should be expected for structures similar tothe prototype frame studied. Shear yielding of the joint can be prevented by placing more shearreinforcement in the joint core (as according to Paulay and Priestley), but the ease of construction shouldalso be considered in the design of a joint. As shown in Figure 17.16, the reinforcement cage in the jointcore of the specimen in this study was already very congested with the three layers of transversereinforcement as specified by the design. It should be noted that the shear in the upper column in the testspecimen is lower than that in the design, which means higher shear stresses and greater damage wouldoccur in the joint of the tests. However, compared to the shear in the upper column, the tensile force inthe main beam at the joint face (see Figure 20.1 and Table 20.5) is usually a lot larger and so it wouldgovern and reflect the level of shear stresses in the joint core.173ConclusionsThe following conclusions can be drawn from this research:1) The inelastic response of a structure to an earthquake is greatly affected by the slope of thevelocity spectrum over the range of change of the fundamental period that occurs after yielding.2) Three types of inelastic response can be detected, characterized by the inelastic displacementresponse curve as defined in Chapter 3. This curve may be outward-sloping, inward-sloping, orvertical, depending on the slope of the velocity spectrum, which, in turn, depends upon therelationship of the fundamental period of the structure to the predominant period of theearthquake. If the structure period is below the predominant period of the earthquake, thevelocity spectrum will be upward-sloping and the response curve will be outward-sloping; if thestructure period is close to the predominant period, the velocity spectrum will have a downwardslope and the response curve will be inward-sloping; finally, if the structure period is higher thanthe predominant period, the velocity spectrum will be horizontal and the response curve will bevertical.3)^The strength and ductility demands on structures vary with the three classes of inelastic response.Particularly, in the case of an outward-sloping inelastic displacement response curve, higherductility is demanded for a given strength or higher strength is required to reach a desired levelof ductility. Since outward-sloping response curves are associated with upward-sloping velocityspectra, the value of the force reduction factor, R, should be reduced when the fundamentalperiod of the structure is below the predominant period of the earthquake.1744) The effect discussed in paragraph 3 above is in addition to the amplification of spectral responsewhich may occur in filtering of seismic waves by local soft soils and is reflected in the foundationfactor, F, of the code.5) The strength and ductility demands on structures can be estimated quite reasonably by the R- 11.equation developed in Chapter 6, which accounts for the effective-period shift and the change indamping.6) A further consequence of filtering of the ground motion by soft soil layers is that the number ofcycle of strong shaking is increased; provided the ductility demands are kept low, this researchsuggests that the increased number of cycles may not have a detrimental effect. At higherductility demands, however, there was some evidence to suggest that performance may deterioratewith increased cycles.7) Regular reinforced concrete frames designed in accordance with the codes will generally have aconsiderable overstrength due to the piecemeal development of a mechanism as plastic hinges areformed, and to other effects. In consequence of this overstrength factor, ductile reinforcedconcrete frames designed for the Vancouver area will perform quite well when the designearthquake is filtered by the soft soils of the Fraser River delta, although moderate damage canbe expected in exterior beam-column connections. In general, however, a reduction in the Rvalue should be considered for short-period buildings subject to filtered ground motions.8)^The joint core regions in exterior beam-column connections are very important to the stability andstrength of a reinforced concrete frame. The results of this investigation tend to confirm theassertions by other researchers that the code requirements for joint design (1.8%/fe, as the nominalshear strength of a joint) may be somewhat unconservative.175Table 1.1 Ground Motion Classification (after Newmark and Rosenblueth 1971)Main Group SubgroupI.^Impulsive A.B.C.Single ShockModerately long, extremely irregular motionGround motion involving large-scale, permanentdeformations of the groundII. Harmonic D. Long ground motion exhibiting pronounced prevailingperiod of vibrationTable 1.2 Parameters Having Influences on Seismic Response of Structures(after Hall, McCabe, and Zahrah 1984)Ground Motion Parameters Structural ParametersAmplitudeShape / ConfigurationFrequencyDurationNatural FrequencyDampingMaterial ResistanceTable 2.1^Peak Accelerations and Predominant Periods of Bedrock and SurfaceGround Motions in Richmond, B.C.InputMotionPeak Acceleration (g) Predominant Period (sec)Bedrock Surface AmplificationFactorBedrock Surface ShiftFactorTaft 0.21 0.304 1.45 0.45 0.85 1.89Caltech 0.21 0.287 1.37 0.25 0.37 1.48El Centro EVV 0.21 0.248 1.18 0.50 0.60 1.20Modified Taft' 0.04 0.107 2.68 2.80 2.80 1.00'Obtained by time-scaling and amplitude-scaling of the Taft S69E 1952 recordTable 4.1 Earthquake Records Used in Strength and Ductility Demands AnalysesUnfiltered Earthquakes Filtered EarthquakesTaft S69E 1952Caltech EW 1971El Centro EVV 1940Modified Taft S69E 1952bMexico City CUIP EW 1985Arimx=0.21gA.=0.21gAmax =0.21gAmax =0.04gAmx=0.04gFiltered Taft S69E 1952a^A.=0.304gFiltered Caltech EW 1971a^Amax =0.287gFiltered El Centro EW 1940a Amax=0.248gArtificial Richmond'^A.=0. 107gMexico City SCT EW 1985^Amax=0.171g'Filtered through Richmond Soils (see Ch.2)bObtained by time-scaling and amplitude-scaling of Taft S69E 1952 recordTable 5.1 Characteristic Periods of Study EarthquakesEarthquake Characteristic Period (sec)Acceleration Spectrum Velocity SpectrumTaft 0.45 0.45Caltech 0.25 0.27El Centro EW 0.50 2.10 [0.60]bMexico City CU1P 0.92 (2.00)a 2.20Modified Taft 2.80 2.80Filtered Taft 0.85 0.85Filtered Caltech 0.37 (0.77)" 0.77Filtered El Centro EW 0.60 (1.25)a 3.20 [1•25]bMexico City SCT 2.00 2.00Artificial Richmond 2.80 2.90'Period for second peak of comparable spectral valuebPeriod for first prominent peak in the spectrum176Table 9.1 Member Properties of R4 FrameSections SectionNo .aAiR(MM2)ICR,T4(X 10'mm4)MY(1c.N.m)Ends of All Beams A-A +262116 +0.5287 + 92.4-263377 -0.9921 -216.5Centre of Exterior Beam B-B +259595 +0.9181 +149.3-260855 -0.6294 -115.5Centre of Interior Beam C-C +257074 +0.5764 + 90.8-258334 -0.6250 -115.7Exterior Column D-D 129370 0.3400 98Interior Column E-E 163317 0.5514 161aSee Figure 9.2.Note: Positive sign for positive bending; negative sign for negative bending.Table 9.2 Structure Properties of R4 FrameElastic lateral stiffness = 3.76 kNimmAdjusted code yield load ---- 164 INRoof deflection at adjusted code yield load --=-- 43.6 mmFundamental vibration period = 1.42 secSecond-mode vibration period = 0.45 secThird-mode vibration period = 0.25 secFundamental-mode damping = 5% critical177178Table 9.3 Response of R4 Frame to Richmond Ground MotionsEarthquakes MaximumGroundAccelerationMaximumDeflection(mm)MaximumBase Shear(kN)Surface Filtered Taft 0.304 g 206 473Bedrock Taft S69E 1952 0.210 g 77 386Surface Filtered Caltech 0.287 g 128 355Bedrock Caltech EW 1971 0.210 g 39 152Surface Filtered El Centro EW 0.248 g 407 556Bedrock El Centro EW 1940 0.210 g 105 404Surface Artificial Richmond 0.107 g 138 447Bedrock Modified Taft 0.040 g 16 85Table 9.4 Response of R4 Frame to Earthquakes Scaled to 0.21gEarthquake Scale Factor Maximum Deflection(mm)Maximum Base Shear(kN)Taft S69E 1.1709 77 386El Centro NS 0.6029 72 325El Centro EW 0.9803 105 404Olympia N86E 0.7501 76 328Mexico City SCT EW 1.2268 ___a 583Artificial Richmond 1.9539 ___a 583isp acement limit in program >^mg^t exTable 9.5 Response of R4 Frame to Earthquakes Scaled to 0.21 m/sEarthquake Scale Factor Maximum Deflection(mm)Maximum Base Shear(kN)Taft S69E 1.1855 77 389El Centro NS 0.6278 75 336El Centro EW 0.5688 71 288Olympia N86E 1.2288 109 410Mexico City SCT EW 0.3471 77 323Artificial Richmond 0.3596 48 206Table 9.6 Response of R4 Frame to Earthquakes Scaled for Ductility of 4Earthquake Scale Factor Inelastic ElasticAmax(mm)Vmax(cN)Amax(mm)Vmax(N)Taft S69E 2.3500 175 485 170 933El Centro NS 1.3835 175 448 165 764El Centro EW 1.5235 175 473 191 773Olympia N86E 2.0505 175 463 209 919Mexico City SCT EW 0.5305 175 469 116 505Artificial Richmond 1.0850 175 473 143 613Table 10.1 Member Properties of F2 FrameSection SectionNo •aATR(Mm2)I CRTR(X 109 MM4)MY(IN.m)Ends of All Beams A-A +268419 +0.9266 +152.1-269680 -1.2015 -284.6Centre of Exterior Beam B-B +268419 +1.5081 +276.4-269680 -0.8547 -166.8Centre of Interior Beam C-C +264637 +1.1616 +200.6-265898 -0.8562 -167.8Exterior Column D-D 136936 0.4513 139Interior Column E-E 173401 0.8420 198ee rigure lu.Note: Positive sign for positive bending; negative sign for negative bending.Table 10.2 Structure Properties of F2 FrameElastic lateral stiffness = 5.05 IN/mm^Adjusted code yield load = 328^INRoof Deflection at adjusted code yield load = 65^mmFundamental vibration period = 1.21^secSecond mode vibration period = 0.39^secThird mode vibration period = 0.22^secFundamental mode damping = 5% critical179180Table 10.3 Response of F2 Frame to Study Filtered EarthquakesEarthquakes ScaleFactorPeakGroundAccelerationMaximumRoof Deflection(mm)MaximumBase Shear(r-N)Filtered Taft 1.00 0.304 g 216 647Filtered Caltech 1.00 0.287 g 154 553Filtered El Centro EW 1.00 0.248 g ___a 676Artificial Richmond 1.00 0.107 g 97 518Mexico City SCT EVV 1.00 0.171 g 188 643"Displacement limit in program (>1^mg e ) excTable 11.1 Section Properties of Study Shear Wall StructureStorey Axial Loads(N)El(x1012kN-mm2)My(kN.m)EIsha8 264 12.28 5035 0.013067 569 13.70 5732 0.014736 872 15.21 6502 0.016705 1176 16.59 7221 0.017714 1481 17.84 7896 0.019773 1785 19.07 8573 0.022732 2088 20.33 9287 0.023341 2397 21.58 10011 0.02586'Strain hardeningfiTfuess expressed as fractions oTable 11.2 Structure Properties of Study Shear Wall Structure^Elastic lateral stiffness^=^3.50kN/mmAdjusted code yield load = 459^kNRoof deflection at adjusted code yield load^=^131^mmFundamental vibration period =^1.75^secSecond mode vibration period = 0.22^secThird mode vibration period = 0.15^secFundamental mode damping = 5% criticalTable 11.3 Response of Study Shear Wall Structure to Earthquakes Scaled to 0.21gEarthquake Scale Factor Amax(mm)Vmax(N)Taft S69E 1.1709 152 808El Centro NS 0.6029 124 648El Centro EW 0.9803 174 727Mexico City SCT EW 1.2265 ___aArtificial Richmond 1.9539 __ a ___baDeflection limit in the program exceededbLarge shear due to strain hardening and large deflectionTable 11.4 Response of Study Shear Wall Structure to Earthquakes Scaled for Ductility of 2Earthquake Scale Inelastic ElasticAmax(mm)'max(1c.N)Amax(mm)Vmax(kN)Taft S69E 2.933 260 1199 461 2461El Centro NS 1.390 260 1313 285 1499El Centro EW 1.431 260 809 221 1070Mexico City SCT EW 0.417 260 612 296 1077Artificial Richmond 0.738 260 600 162 641Table 14.1^Local Curvature Ductility from Tip Displacement Ductility for R4 FrameBased on Park and Paulay's FormulaRoof Deflection(mm)Tip DisplacementDuctilityLocal CurvatureDuctility175 2.7 4.2250 3.9 6.5350 5.5 9.4450 7.0 12.3181Table 14.2 Plastic Hinge Rotation at Critical Section' in R4 Frame from Static AnalysisRoof Deflection(mm)Base Shear(N)Plastic Hinge Rotation(rad)175 422.5 0.0158250 446.0 0.0254350 450.6 0.0382450 ___C 0.049b'Exterior, loaded end of exterior beam in first floorbObtained from extrapolation of the other three rotations'Not availableTable 14.3 Plastic Hinge Rotation at Critical Section' in R4 Frame from Dynamic AnalysesRoof Deflection(mm)Taft Mexico City SCTScale Factor 0P(rad)Scale Factor 0P(rad)175 2.350 0.0130 0.5305 0.0184250 3.010 0.0230 0.6560 0.0296350 3.567 0.0369 0.6756 0.0437450 4.150 0.0521 0.6847 0.0624'Exterior end of exterior beam in first floorTable 14.4 Comparison of Local Curvature and Tip Displacement Ductilities for R4 FrameRoof Deflection(mm)Tip DisplacementDuctility'Local CurvatureDuctility"175 2.7 8.2250 3.9 12.6350 5.5 18.5450 7.0 23.4"Yield displacement of 64 mmbExterior end of first-floor exterior beam182183Table 15.1^Plastic Hinge Response at Critical Section' in R4 FrameDeflecting to Tip Ductility of 4 in Study EarthquakesEarthquake Scale No. ofTimesReachingYieldbAccumulated PlasticHinge Rotation'(rad)Right End Left EndPrimary Secondary PrimaryTaft S69E 2.3500 4 0.0043 0.0896 0.005053-0.0130 -0.0854 -0.012118El Centro NS 1.3835 4 0.0081 0.0277 0.0004-0.0081 -0.0320 -0.0162El Centro EW 1.5235 2 0.0022 0.1195 0.0103-0.0179 -0.1180 -0.0099Olympia N86E 2.0505 3 0.0080 0.0726 0.0019-0.0094 -0.0732 -0.0153Mexico City SCT EW 0.5305 7 0.0057 0.1220 0.0108-0.0184 -0.1237 -0.0134Artificial Richmond 1.0850 8 0.0120 0.1992 0.0008-0.0083 -0.2037 -0.0195'Exterior end of exterior beam in first floorbPositive and negative'Positive value for positive bending.184Table 15.2^Plastic Hinge Response at Critical Section' in Study Shear Wall StructureDeflecting to Tip Ductility of 2 in Study EarthquakesEarthquake Scale No. ofTimesReachingYield'Accumulated PlasticHinge Rotationc(rad)Primary SecondaryTaft S69E 2.933 3 0.0012 0.0246-0.0040 -0.0235El Centro NS 1.390 4 0.0010 0.0146-0.0036 -0.0131El Centro EW 1.431 5 0.0034 0.0205-0.0011 -0.0209Mexico City SCT EW 0.417 7 0.0038 0.0325-0.0035 -0.0351Artificial Richmond 0.738 6 0.0041 0.0444-0.0039 -0.0452'Base of shear wallbPositive and negativecPositive value for positive bendingTable 16.1 Earthquake Records for Storey-Shear AnalysesEarthquake 6-Storey Frame 20-Storey FrameScale PeakAccelerationScale PeakAccelerationUnfilteredTaft 569E 1.9867 0.356g 0.7301 0.131gEl Centro NS 1.2523 0.436g 0.5398 0.188gEl Centro EW 1.2907 0.276g 0.6309 0.135gFilteredMexico City SCT EW 0.8834 0.151g 0.2557 0.044gArtificial Richmond 1.1298 0.121g 1.6203 0.174gTable 17.1 Control Deflection Parameters for Tests BC1 to BC4Test YieldExcursionaBeam HingeRotation(rad)Beam TipDeflectionb(mm)Negative 1 0.0178 24.3BC1Positive 1 0.0022 3.0Negative 1 0.0048 6.6BC2 2 0.0081 11.0Positive 1 0.0018 2.42 0.0025 3.4Negative 1 0.0008 1.1BC3 2 0.0010 9.53 0.0033 4.54 0.0066 8.95 0.0061 83Positive 1 0.0056 7.6Negative 1 0.0009 1.3BC4 2 0.0049 6.63 0.0131 17.8Positive 1 0.0003 0.52 0.0098 13.33 0.0013 1.8aNegative for downward displacement of beam tipbCalculated by multiplying the hinge rotation by the beam length185- ^dci  dal dcY fdcaTable 17.2 Actual Positions of Reinforcement' in SpecimensSpecimenBeam Columnd1(nun)d2(mm)d3(mm)di(mm)da(mm)da(mm)dc4(mm)BC1b -- -- -- -- -- -- --BC2 381 329 54 67 131 183 266BC3 373 341 61 76 140 194 273BC4 380 333 48 68 147 196 267BC5 389 354 76 67 142 204 272Design:BC1,BC2,BC3,BC5391 350 59 59 132 206 279Design:BC4391 350 59 61 133 205 277'Average positions of bar layersbNot measured1863• • •kTable 17.3 Measured Material Properties of SpecimensSpecimen 1^Concrete' Reinforcing SteelbCompressiveStrength(MPa)Bar Size YieldStrength(MPa)UltimateStrength(MPa)ElasticModulus(GPa)YieldStrain'BC1 33.1BC2 29.5No.10 448 700 184 0.0024BC3 30.0 No.15 446 675 196 0.0023BC4 30.7 No. 10 435 702 200 0.0022No. 15 463 717 209 0.0022No. 20 458 721 184 0.0025BC5 25.5 No. 10 476 673 198 0.0024No. 15 502 781 186 0.0025'Average of two field-cured 150 mm by 300 nun cylindersbAverage of three samples'Calculated from measured yield strength and measured elastic modulusTable 19.1 Flexibility of Testing FrameTest Testing Frame Flexibility'(mmilds1)BC1 0.023bBC2 0.020BC3 0.026BC4 0.037BC5 0.030'Expressed in terms of vertical beam tip deflection per unit of applied beam tip loadbAverage of BC2 and BC3 values188Table 19.2 Comparison between Specimen BC1 and Ehsani and Wight Specimen■^Property 1 Specimen BC1^.Ehsani and Wight SpecimenLb^(mm) 1355 1067Lc (mm) 2743 2210Lb / 1.,, 0.49 0.48he^(mm) 338 340hb (mm) 450 480hs^(mm) 83 102hb / I.,b 0.34 0.45bb^(mm) 305 300bs (mm) 1422 1016Pc 0.021 0.020Pb 0.014 0.011Pb' 0.005 0.009hc / dbb 21.1 15.3hb / d 25.2PC = column reinforcement ratioPb = beam top reinforcement ratioPb' = beam bottom reinforcement ratiodbb = diameter of beam longitudinal reinforcementdin = diameter of column longitudinal reinforcementLb b8L ch bVbb189L cbbhs;^ri ^I 6H b^IbsTable 19.3 Comparison between Specimen BC1 and Paultre and Mitchell SpecimenI^Property I Specimen BC1 I Paultre and Mitchell SpecimenLb^(mm) 1355 2000Lc (mm) 2743 3650Lb / Lc 0.49 0.48hc^(mm) 338 450hb (mm) 450 600h.^(mm) 83 110hb / Lb 0.34 0.45bb^(mm) 305 400bs (mm) 1422 1900Pc 0.021 0.012Pb 0.014 0.006Pb' 0.005 0.006hc / dbb 21.1 23.1hb / d 30.8Pc = column reinforcement ratioPb = beam top reinforcement ratioPb' = beam bottom reinforcement ratiodbb = diameter of beam longitudinal reinforcementd ^diameter of column longitudinal reinforcement190hcTable 19.4 Measured Yield Loads and Yield Deflections of SpecimensSpecimen^ ,^Negative Loading^II^Positive LoadingYield Loada(kN)Yield Deflectionb(mm)Yield Loada(kN)Yield Deflectionb(mm)BC1 198 23 81 9BC2 183 20 84 12BC3 187 25 89 12BC4 236 32 130 20.5BC5 208 32 71 15aExpressed in terms of beam tip loadbGlobal beam tip deflectionTable 19.5 Comparison between Theoretical and Measured Yield LoadsTest Negative Yield Loada (kN) Positive Yield Loada (kN)Theoreticalb Measured' Measured Theoreticalb Measured' MeasuredTheoretical TheoreticalBC1 172d 198 1.15 rd 81 1.05BC2 173 183 1.06 78 84 1.08BC3 171 187 1.09 77 89 1.16BC4 235 236 1.00 135 130 0.96BC5 198 208 1.05 81 71 0.88aExpressed in terms of beam tip loadbCalculated using actual reinforcement positions and measured material properties'As determined from beam tip load-deflection plotsdEstimated using BC2 reinforcement positions191Table 19.6 Negative Beam Tip Displacement Ductilities of SpecimensSpecimen AdjustedYield Deflection'(mm)YieldExcursionGlobalDeflection,^(mm)Ductility CumulativeDisplacementDuctility1 29 1.3 1.3BC1 19 2 78 3.8 5.13 88.5 4.4 9.5BC2 16 1 31 1.7 1.72 44.5 2.5 4.21 33 1.4 1.42 47.5 2.1 3.5BC3 20 3 53 2.4 5.94 59.5 2.7 8.65 64.5 3.0 11.66 74 3.4 15.01 39 1.3 1.3BC4 23 2 44.5 1.5 2.83 65 2.4 5.21 64 2.2 2.22 85.5 3.0 5.2BC5 26 3 107 3.8 9.04 128.5 4.7 13.75 149.5 5.5 19.2aC,omponent of testing frame deflection deducted192Table 19.7 Positive Beam Tip Displacement Ductilities of SpecimensSpecimen AdjustedYield Deflection'(mm)YieldExcursionGlobalDeflection(mm)Ductility CumulativeDisplacementDuctilityBC1 7 1 16.5 2.1 2.1BC2 10 1 24.5 23 2.3BC3 10 1 26.5 2.4 2.42 31 2.9 5.31 21 1.0 1.0BC4 16 2 37.5 2.0 3.03 42 2.3 5.31 44 3.2 3.22 58.5 4.3 7.5BC5 13 3 64.5 4.7 12.24 81.5 6.0 18.25 96.5 7.2 25.4aComponent of testing frame deflection deducted193Table 19.8 Yield Excursions of SpecimensSpecimen No. ofNegative YieldNo. ofPositive YieldBC1 2 1BC2 2 2BC3 6 2BC4 3 3BC5 10 10Table 19.9 Flexibility of Upper Column in BendingTest Measured Column Flexibilitya(rad/IN-mm)BC1 0.646 x 10-7BC2 0.192 x 10-7B C3 0.167 x 10-7B C4 ____bB C5 0.427 x 10-7a Relationship between rotation and moment at base of upper column without testingframe deflection componentb Not measured194Table 20.1 Measured Axial Loads in Upper Column of SpecimensSpecimen Axial Load in Upper Column (kN)Initial Average Maximum AtNegativeYieldAtPositiveYieldBC1 796 730 842 764 656BC2 797 782 856 812 771BC3 802 836 1023 942 901BC4 771 798 969 896 749BC5 480 490 647 537 454Table 20.2 Balanced Axial Load and Moment of Column in Specimens'Specimen BalancedAxial Load(IN)BalancedMoment(kN-m)BC1 912b 175bBC2 912 175BC3 921 177BC4 784 210BC5 632 164'Based on measured material properties and reinforcement positionsbEstimated using BC2 values195Table 20.3 Flexural Strength Ratio in SpecimensSpecimenFlexural Strength (1(11-m)UpperColumnLowerColumnUpper+LowerBeama FlexuralStrengthRatioI Negative Beam Tip LoadingBC1 169b 175b 344 234b 1.47BC2 172 175 347 234 1.48BC3 176 175 351 232 1.51BC4 208 205 413 318 1.30BC5 160 163 323 269 1.20Positive Beam Tip Loading IBC1 163b 159b 322 105b 3.07BC2 170 164 334 105 3.18BC3 177 175 352 104 3.38BC4 208 204 412 182 2.26BC5 157 148 305 110 2.77aBased on one slab reinforcing bar on either side of beambEstimated using BC2 reinforcement positions196Table 20.4 Flexural Strength Ratio at Maximum Negative Beam Tip LoadSpecimen Flexural Strength (kN-m) FlexuralStrengthRatioUpperColumnLowerColumnUpper+LowerBeamaBC1 172b 175b 347 339b 1.02BC2 173 175 348 295 1.18BC3 176 174 350 290 1.21BC4 207 204 411 358 1.15BC5 164 162 326 363 0.90aCalculated by multiplying maximum beam tip load by length of beambEstimated using BC2 reinforcement positionsTable 20.5 Joint Shear Stress in SpecimensSpecimen TY(kN)V'max(kN)V.J(kN)T.J(MPa)T- / Vf ,j^cBC1 860 139 721 6.3 1.1BC2 860 121 739 6.5 1.2BC3 860 119 741 6.5 1.2BC4 1177 147 1030 9.0 1.6BC5 956 149 807 7.1 1.4• Tensile force at top of main beamCalculated assuming participation of two slab bars on either side of beamand strain hardening to 7% above the yield strengthV 'max =^Shear force in upper column at maximum negative beam tip loadV.•^Maximum joint shear = T - V 'maxT.^Maximum joint shear stress197Table 20.6 Response of Middle Transverse Reinforcement in Joint CoreSpecimen First-Crack Load'(kN)Yield Load'(kN)BC1 ___b bBC2 ___c 206BC3 109 210BC4 112 236BC5 111 258'Expressed in terms of beam tip loadbData not recordedcNo conclusive result observedTable 20.7 First-Crack Load for Joint from Beam Bar Hook StrainSpecimen First-Crack Load' (kN)Top Bar Bottom BarBC1 ___b bBC2 ___c cBC3 134 134BC4 116 116BC5 111 111'Expressed in terms of beam tip loadbData not recorded'No conclusive result observed198199(a)6020^40Time (sec)0.30.2Tii0.1co-0.2(b)0.8 0.7 -0.6 -0.5 -0.4 -0.30.2 _I0.1 -0.0(c)1.0 ^0.8-"cf7 0.6-—E-c;)'0.2 -5% DampingI11^2^3Period (sec)5% Damping4^50.0 oi 1^2^3^4Period (sec)Figure 1.1^(a) Accelerogram, (b) Absolute Acceleration Response Spectrum, and(c) Relative Velocity Response Spectrum for Taft S69E 1952 Ground Motion50.30.2CD0.1cC(a) t;^oo8 -0.10a-0.220^'^40Time (sec)-0.30 605% Damping,1^2^3Period (sec)1.0 0.90.8 -0.7 -...,T6' 0.6E(C)^0.5>cn 0.40.30.20.1.0 0^1^2^3Period (sec)5% Damping4^5Figure 1.2^(a) Accelerogram, (b) Absolute Acceleration Response Spectrum, and(c) Relative Velocity Response Spectrum for El Centro EW 1940 Ground Motion200(b)^0.8^0.7 -0.6 -0.5 -0.4 -0.3 -I0.220.1^0.0 ^0 4^5Colima^F ^'Colima^Jalisco^r''Guerrero^a leto de Campos^ +^ +^ l- - =^3.8,141,89) --)‘).-.‘...".,- '^/ Zihuatanejo(99,156,101)^ /C^, Oaxacaoyuca C600 i)76,/,Ce/7 _ .,',...1^( 156,112,81) (•^, .,Pa^1E211200:coo) (40,34,18)El Ocotito (49,54,21)^)^_c.Sip, LaVillito(125,122,58) '1La UnionL_^(166,148,129))^4._.‘+ Michoacan^+^-,^.:,^4-^>c . ^ ---,^.r- (Mexico''' \1k95 ,) 6El Suchil^6( 53,59,60)^eXaltianguis (25,18,20)^/..,:99,78,50) El Cayaeo.^ 'Las Mesas (22,18,19)(41,48,24) • La yenta (18,21,16)Pe^ Acapulco^..Cerro de Piedra (26,15,12)r'r0^ c,-d-s-1.^MorelosTeaca)cOT,(49,24,27) s...J-k^Puebla• / -Mexico' City_,D.F.'Mexico.00 Cif/650^100 km00,7• Accelerograph* EpicenterC-.103°W^102°W^101 °W^100°W^99°WFigure 1.3 Epicenter of September 19, 1985 Mexico Earthquake [Anderson et al. 1986]202-411 04".‘441**44411"f4tIAA44dLA VILLITAWYtorio,3%UNAM, MEXICO CITY-,NA041r\-.0,(A10%SCT, MEXICO CITYNSEWNSEWNSEW1^t io^)^1^1^I0 10^30 40 50 60SecondsFigure 1.4^Accelerograms for La Villita, UNAM, and SCT in the 1985 Mexico Earthquake[after Mitchell 1987]Figure 1.5^Soil Zones of Mexico City [reproduced from Mitchell 1987]-0.171 g20^40^60^80^100Time (sec)0.30.2la)0.1I.171^...-1,..,-..,- "p...,a)-a-)00 -0.1<-02-0.3 ^0 120Figure 1.6 Mexico City SCT EW 1985 Ground Motion Accelerogram204(e)^4.0 ^3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 -0.0 ^05% Damping2^3^5Period (sec)5% Damping1^2^3i i^4Period (sec)(a)1.2 ^1.1 -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 0.1 -0.0 ^05Figure 1.7^(a) Absolute Acceleration and (b) Relative Velocity Response Spectra forMexico City SCT EW 1985 Ground MotionBerkeley (0.0. 0.02)Emeryville (0.26. 006)Oakland (0.\ 9.0.07)(0.26, 0.16)San FranciscoSouth San Francisco(0.33. 0.05)an JoseEpicentre [0.64,0.470Santa Crt;:..•(0.47,0.40)Watsonville (039. 0.66)Scale 0^5^10 milestimesimesra0^10^20 kmMontereyRichmond205Figure 1.8^Epicentre of October 17, 1989 Loma Prieta Earthquake and Recorded Peak Horizontaland Vertical Ground Accelerations (in units of g) [Mitchell et al. 1990]50 10^15Ton - Seconds20^25^300.200.200.00.20Depth - Fest^ Ground Surface0 -or-7w- -4-11-0- AV A,Loose Sandy3045WM. -n*"4"" Argarm0.00.20100140MAX 0.07 gIt MAX . 0.18 gLoan Slay SandYoung Bay MudDense Fine Sand and Silty Sandwith alternating layers ofswtdy claySttlf to Hard Silly daynth °cantonalsandy and gravely seams2060 5 10^1571me-Seconds20^25^300.80 . 0 0— Max. Surface Accel.i. 0.16 gDamping^5 ftSurface- - - - Rock--------------1 2^3^4PERIOD, sFigure 1.9^Accelerograms and Acceleration Response Spectra at Yerba Buena Island and TreasureIsland Stations in the 1989 Loma Prieta Earthquake [after Seed et al. 1990]...DELTAVANCOUVERRICHMONDLulu is.^..0 400kmBritishColumbiaCANADAU.S.A.FRASER RIVER0^5^10 kmI^I^,^I,Figure 2.1^Location Map of Fraser River Delta and Richmond, B.C.TIDAL FLATDYKED DELTA TOPPEAT BOGUPLANDS (earlier deposits)FERRYATERMINALBOUNDARYBAYPT. ROBERTSPENINSULA208Figure 2.2^Soil Types in Fraser Delta [reproduced from Luternauer and Finn 1983]VANCOUVERSEA ISLANDFRASER RIVER DELTA DEPOSITS ^•LAC1AL DEPOITSTERTIARY BEDROCKLULU ISLAND0^1^2 3 kmISTRAITOFGEORGIAE=3 FRASER DELTAUPLANDSUrn- 1 000Uicr-100 2-200ELI-300 —J209SOUTH NORTH — -400SECTION E-FLEGENDLJ SAND[0:11^SILTEEO^TILLSITE A^SITE B SITE C130-75-175—-.... •- ■•■^ <▪ 0^...., ••▪ •■■E Ne E le E080 p,^80 (.,^0 cyCC IX CCFigure 2.3^'Typical Soil Profiles in Fraser Delta [reproduced from Sy et al. 1991]210Figure 2.4^Seismicity in Western Canada [reproduced from Milne et al. 1978]OUE•CHAPLOT if'^WilsonKnolls\44\,Empocs,,, \Plate ,A:1st \\Nootk•F•ullZOn•BRITISHA COLUMBIAso"VictoriaAWASHINGTONEgolowRWIlsSorancoF.Z.OwenCharlotteFaunt? Juan do--Fuca PlatoAA^\coluntsi.A us^ tu^ astA QOREGON gCaps^ABlanco 111fa•^i• .4c^•eCape^AMendocino0' 4CALIFORNIA iSan Andros's^\.41.- FaultPLATEPACIFICliondocino AteanciscoSouthFlat.GordaOVERRIDING BUTNO "SUBDUCTION'.OVERRIDING PLUS'SUBDUCTION'AMERICAPLATEWINONABLOCKEXPLORERPLATEJUAN DEFUCA PLATE211Figure 2.5^Plate Tectonics along West Coast of Canada [reproduced from Rogers 1988]PRESENT STUDY D=0.0MSEE)COLDERWALLISBYRNE k ANDERSON, 198 2MDCICOPENDER ISLAND(1976)PRESENT STUDY D=3.0M0.1^0.15^0.2^0.25^0.3^0.35^0.4BEDROCK ACCELERATION (G)0.0506212Figure 2.6^Amplification Curves for Richmond Soils[reproduced from Byrne and Anderson 1987]— Taft 0.2g as Input• Pasadena 0.035g Peak Perlod=2.5--- CALE NOOE 0.2g as Input• CUIP N9OW 0.035g as Input0.6 ^10.5 —r- OA5cr)-0▪ -• 0.30g 0.2.17;0.14J°• 0.00.0Median relationshiprecommended for use —in empirical correlations0.1^0.2^0.3^0.4^0.5^0.6Acceleration on Rock Sites — g1989 Loma Prieto11111111111111111111141111111985 Mexico City1 1 1 12^3^4^5Period (s)Figure 2.7^Acceleration Response Spectra (5% Damping) for Artificial Surface Motionsin Richmond, B.C. [reproduced from Finn and Nichols 1988]Figure 2.8^Idriss' Amplification Curve for Soft Soils [after Idriss 1990]213214Depth, m0CLAY K0: 0.6^31^I^ y =17.4KN/m3SATSAND Ko= 0.5^171 ^= 20KN/m3SATSILT Ko= 0.6y =18KN/m3ISATTILL Ko=0.7y =21KN/m3^SATGnu , MPa0 100 200 300 400 500 6001^00D^Cimox, e0 10 20 30 I^I5791 114202530507095125150195240ROCKOMPaFigure 2.9^Richmond Soil Profile Model with Variations of Shear Modulus and Dampingfor SHAKE Analysis [reproduced from Byrne and Anderson 1987]0.21 g40301100.4  0.3 -0.2 -0.1 -0-0.1 --0.2 --0.3 --0.4  0i20Time (sec)(b)0.4  0.3 -0.2 -0.1 -0  -0.1 --0.2 --0.3 --0.4.6I i-0.304 g,^110^20Time (sec)o 4030215Figure 2.10^Accelerograms for (a) Taft S69E 1952 Scaled to 0.21gand (b) Filtered Taft S69E0.40.30.20.10-0.1-0.2-0.3-0.40(a)40 ^0.4 ^0.3 -0.2 -0.1 -0 --0.1 --0.2 --0.3 -^-0.4 ^00.287 g\k(b) , •...., -.../,/ \ „,,..,^,/,44t..,, 'N.110,20Time (sec)30 40Figure 2.11^Accelerograms for (a) Caltech EW 1971 Scaled to 0.21gand (b) Filtered Caltech EW2160.21 g1(a)30 40110^0.4^0.3 -0.2 -0.1 -0-0.1 --0.2 --0.3 -^-0.4 ^0i20Time (sec)A,\lh1 00,,20Time (sec)0.4 0.3 -0.2 -0.1 -0-0.1 --0.2 --0.3 --0.40.248g30prAIV-A"T\40(b)ikMp,p,217Figure 2.12^Accelerograms for (a) El Centro EW 1940 Scaled to 0.21gand (b) Filtered El Centro EW0.30.2-0.2-0.3_-0.04 gI' ,N--218(a) 0< -0. 1td_a)a)C.)U0.1 -(b)o 20 40 60Time (sec)0.30.2 -0.107 gI^,iAP20^40^60^80^100^120Time (sec).NAAAA-0.2 --0.3 ^0Figure 2.13^Accelerograms for (a) Modified Taft and (b) Artificial Richmond Ground Motions-1.0 --0.8 - --acri^0.6 -co Filtered Taft_ „0.4 -1.25% Damping219, ,^‘Taft••,..._,___,,,",,- ---------------- --------------- -------i^i2 3^4^5Period (sec)5% DampingFiltered Taft1^2^3i 1Period (sec)Figure 2.14^(a) Absolute Acceleration and (b) Relative Velocity Response Spectrafor Taft (0.21g) and Filtered Taft Motions0.20.0—E".>C/)^2.0 ^1.8 -1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 -0.0 o 4^5 ^1.2 1^1.1 i1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.20.1 -0.0 05% Damping(a) „Caltech1Filtered Caltech/ii2^3Period (sec)4 5Filtered Caltech-- ^Caltech(b)'aE>co2.0 ^1 .8 -1 .6 -1.4 -1 .2 -1.0 -0.8 -0.6 -0.4 -0.2 -5% Damping2^3Period (sec)0.01 5220Figure 2.15^(a) Absolute Acceleration and (b) Relative Velocity Response Spectrafor Caltech (0.21g) and Filtered Caltech Motions221 ^1.2 ^1.1 -1.0 -0.9 -0.8 -0.7 --(350.6 -vscn 0.5 -0.4 -0.3 -0.20.1 -^0.0 ^(a)Filtered El Centro EWEl Centro EW5% Damping0 2^3^5Period (sec)2.0 ^1.8 -1 .6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 -0.05% DampingFiltered El Centro EW(b)2^3^4^5Period (sec)Figure 2.16^(a) Absolute Acceleration and (b) Relative Velocity Response Spectrafor El Centro EW (0.21g) and Filtered El Centro EW Motions2220.1 -0.005% DampingArtificial Richmond- _ --^------ _ - _ _ -1^2^3^14Period (sec)(a)1.2 ^1.1 -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 - .........--Modified Taft-----•____----Th55% DampingArtificial RichmondModified Taft,i2^3Period (sec)1I4^5(b)^3.0 ^2.8 -2.6 -2.4 -2.2 -2.0 -1 .8 -1 .6 -1 .4 -1.2 -1 .0 -0.8 -0.6 -0.4 -0.2 -0.0 0Figure 2.17^(a) Absolute Acceleration and (b) Relative Velocity Response Spectrafor Modified Taft and Artificial Richmond Motions2231.2 ^1.1 -1 .0 -0.9 -0.8 -0.7 - Taftt:n^1It „0.6 -^• 0co .0cn • 1 ; El Centro EW0.5- t; tv'!IV,0.4- /'^. ^1^-- .,f^i ,,....^.,,, ^•,/ r,q0.20.3 -5% DampingMexico City SCTRichmond0.1 -^----------------------------0.01^2^3^ 5Period (sec)Figure 2.18^Acceleration Response Spectra for Taft S69E 1952, El Centro EW 1940,Mexico City SCT EW 1985, and Artificial Richmond Ground Motions 224k = elastic stiffnessPy = yield strengthAy = yield displacementAu = ultimate displacementAu11 = AyI^ I^•Ay AuDisplacement AFigure 3.1^Idealized Bilinear Response of Structures under Static Lateral LoadsFigure 3.2^Idealized Response of Structures under Dynamic Lateral Loads 225PePeR = PyCAy^Ae^AuLateral DisplacementFigure 3.3^Definition of Force Reduction FactorAPeInelastic DisplacementResponse Curve (IDRC)AeLateral DisplacementFigure 3.4^Definition of Inelastic Displacement Response CurveAe Ile^Ae4 2 Lateral Displacement226Figure 3.5^Equal-Displacement Seismic Response Assumption of Structures11 i^1/1\v'pe\t'vq\:A4v\,,-0.04 g11111110^20^30^40Time (sec)^0.5^0.4 -(Z 0.3 -Eei0.1 -^0.0 ^0 1^2^3^;IPeriod (sec)(c)5227^0.10^0.08:...... 0.06 -0.04 :s 0.02-(a)^ts^o ^.t; -0.02'8 -0.04 -<-0.06:-0.08:-0.10 00.20-0.16 -(b)_-830.12 -cocf) 0.080.040.00 ^05% Damping1^2^3^4^5Period (sec)(a) Accelerogram, (b) Absolute Acceleration Response Spectrum,and (c) Relative Velocity Response Spectrumfor Mexico City CUIP EW 1985 Ground MotionFigure 4.122815 ^14 -13 -12 -11 -10 -: ^9 -1 8 -7 -o 6 -5 -4 -3 -2 -1 -0 o R=4R=2‘`,...-,........-.......---...--,.....-R71.32^3^4^5Period (sec)1Figure 4.2^Ductility Demands of Taft S69E 1952 (0.21g)15 ^14 -13 -12 -11 -10 -P 9 -0m^7 -CD^6 -5 -4 -3 -2 -1 -R=4R=2R=1.3Period (sec)1 4 5Figure 4.3^Ductility Demands of Filtered Taft1514 -13 -12 -11 -10 -:'^9-g^7 -0 6-5-4-3-2-1 R=1.315 ^14131211 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -0 0 1R=1.3121^3Period (sec)4 522902^3^:4Period (sec)0^1 5Figure 4.4^Ductility Demands of Caltech EW 1971 (0.21g)Figure 4.5^Ductility Demands of Filtered CaltechR=4R=2R=1.31^2^3^:4^5Period (sec)23015 ^14 -13 -12 -11 -10 -:**^9 -c.)=^7 -o 6 -5 -4 -3 -2 -1 -0 -^0Figure 4.615^14 -13 -12 -11 -10 -9 -17,^8 -(2^7 -0 6 -5 -4 -3 -2 -1 -0 0R=41R=1.312^3Period (sec)1.^5Ductility Demands of El Centro EW 1940 (0.21g)Figure 4.7^Ductility Demands of Filtered El Centro EW51 41514 -13 -12 -11 -10 -:'^9 -..7,'^8-S'^7 -0 6 -5 -4 -3 -2 -1 -00R=1.32^3Period (sec)415 ^14 -13 -12 -11 -10 ->.^9 -.8 -46=^7 -o 6 -5 -4 -3 -2 -1 -0 i2Period (sec)Figure 4.8^Ductility Demands of Modified TaftFigure 4.9^Ductility Demands of Artificial Richmond231151413 -12 -11 -10 -:' 9 -tg 8 -7 -0 6 -5 -4 -3 -2 -1 -0 o 1^2^3Period (sec)4^5R=4R=1.3R=4R=2R=1.312^3Period (sec)4^5Figure 4.10^Ductility Demands of Mexico City CUIP EW 198515 ^1413 -12 -11 -10 -:'^9 -:T.^8 -S)^7 -o 6 -5 -4 -3 -2 -1 -0 0Figure 4.11^Ductility Demands of Mexico City SCT EW 1985232,0^2^ 4^5Period (sec)Figure 4.12^Force Reduction Factors for Taft S69E 1952 (0.21g)233Period (sec)Figure 4.13^Force Reduction Factors for Filtered Taft4 5CC2,^iPeriod (sec)1Figure 4.14^Force Reduction Factors for Caltech EW 1971 (0.21g)2^3^4^5Period (sec)234Figure 4.15^Force Reduction Factors for Filtered CaltechCC2■Period (sec)5IF4_-_,2^3Period (sec)4^51Figure 4.16^Force Reduction Factors for El Centro EW 1940 (0.21g)235Figure 4.17^Force Reduction Factors for Filtered El Centro EW101^2^,3^4Period (sec)0 5Figure 4.18^Force Reduction Factors for Modified Taft26,32Period (sec)Figure 4.19^Force Reduction Factors for Artificial Richmond042365 -4 -CC^3 --237CC2^3^5Period (sec)Figure 4.20^Force Reduction Factors for Mexico City CUIP EW 19852^3Period (sec)4^5Figure 4.21^Force Reduction Factors for Mexico City SCT EW 19850.80.7 -0.6 -2380.01-,_2^^3i^4^5Period (sec)0.5 -ai 0.4 -CO0.3 -0.20.1 - p=2Figure 4.22^Inelastic Acceleration Response Spectra for Taft S69E 1952 (0.21g)1.2 ^1.1 -1.0 -0.9 -0.8 ---c53 0.7 -0.6 -asco 0.5 -0.4 -0.30.2 -0.1 -0.0  0p=1p=2p=4:4Period (sec)5Figure 4.23^Inelastic Acceleration Response Spectra for Filtered Taft0.8 10.7 i0.6 -0.5 -0.4 -0.3 -p=12^3^5Period (sec)0.0Figure 4.24^Inelastic Acceleration Response Spectra for Caltech EW 1971 (0.21g) 1.0 10.9 -I0.8 -0.7 -0.6 -0.5 -0.40.3^p=20.2-^p=4^=0.1 -0.0^o^1'6asu)i2Period (sec)50.2p=20.1p=4239Figure 4.25^Inelastic Acceleration Response Spectra for Filtered Caltech0.80.70.60.50.20.10.02^3^5Period (sec)Figure 4.26^Inelastic Acceleration Response Spectra for El Centro EW 1940 (0.21g)1.0  0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 -0.0 I2^3^4Period (sec)Figure 4.27^Inelastic Acceleration Response Spectra for Filtered El Centro EW5p=2p=4p=12404 5:1^2.3Period (sec)0.200.18 -0.16 -0.14 -, 0.12-alin0.10 -co 0.08-0.06-0.040.02 -0.000 i^2^3,Period (sec)4^5Figure 4.28^Inelastic Acceleration Response Spectra for Modified TaftFigure 4.29^Inelastic Acceleration Response Spectra for Artificial Richmond2412420.200.180.160.14, 0.12os0.10ccsCO 0.080.060.040.020.0021^2^I3^4Period (sec)0 5Figure 4.30^Inelastic Acceleration Response Spectra for Mexico City CUIP EW 19851.2  1.1 -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 -0.0 o^1g=i22^3Period (sec)4 5Figure 4.31^Inelastic Acceleration Response Spectra for Mexico City SCT EW 1985///1 1 1 1 " I 15.0A=4A/^BCDE0.2 sec F 1.2 sec0.4 sec G 1.4 sec0.6 sec H 1.6 sec0.8 sec I 1.8 sec1.0 sec J 2.0 seci..c=2////0.25 —1/ /^DI H J C^E()/0.00 1^I^IIIi0.0 1.0A / Ae//////1 . 00 —0.75 —0.50 —Figure 4.32^Inelastic Displacement Response Curves for Taft S69E 1952 (0.21g)k4=4A^O. 2 sec^F^1.2 sec/^B^0.4 sec G^1.4 secC^0. 6 sec^H^2.0 secD^0,8 sec I^2.2 secE^1.0 sec^J^3. 0 sec///A / Ae/Figure 4.33^Inelastic Displacement Response Curves for Filtered Taft^ `iA 0.2 sec H 1.6 secAz=4 B O. 4 sec I 2. 2 secC 0.6 sec J 2.6 sec/ D 0.8 sec K 3,2 secE 1.0 sec L sec3.6/ F 1.2 sec M 3.8 secG 1.4 sec N 4.0 seci Ili 1 ilii 1 1 1 I0.75 —0.50 —0.25 —////0.00 k 1^1^II^11 I 1^iil^i^Iiii^11.00 —0.0 1.0^2.0 3.0 4.0^5.0^6.0Figure 4.34^Inelastic Displacement Response Curves for Caltech EW 1971 (0.21g)///1.2=4^A^0.2 secB^O. 4 sec/^C^0 . 6 s e cD^0.8 secE^1 . 0 secF^1 . 2 secF^1.2 secG^2.0 secH^3.4 secI 3.8 secJ^4.0 secL^3,2 sec///a.0.25 —1^I^I^I^1^I^I^F^IIIIIIII111111111IIIII0.0 1.0 2.0^3.0^4.0^5.0^6 . 0A / Ae//a!-,,1 .00 —0.75 —0.50 —0.00Figure 4.35^Inelastic Displacement Response Curves for Filtered Caltech^tg/A^0.2 sec8^0.4 secC^0.8 secD^1.0 secE^1.2 secF^1.2 sec1F^1.4 secG^2.0 secH^2.4 secI^3.2 secJ^3.8 secL^3.2 secp=1 /4=2////0.50 —I1/0.25 — / /1//0/ I JGHEC////1.00 —0.75 —P ,^i^1^1^I^i^i^1^1^I^I^ililliiiiiiIIIIIII0.0^1.0 2.0 3 . 0^4.0^5 . 0^6.0A / AeFigure 4.36^Inelastic Displacement Response Curves for El Centro EW 1940 (0.21g)A 0.2 sec IB 0.4 sec JAz=4 C O. 6 sec KD 0.8 sec L/ E 1.0 sec MF 1.2 sec N/ G 1.4 sec 0H 1.6 sec/1 1.8 sec2.0 sec2.2 sec2.4 sec2.6 sec2.8 sec4.0 sec/Figure 4.37^Inelastic Displacement Response Curves for Filtered El Centro EWih=1^p,=2^ ih=4^D^O. 8 sec^M^2. 6 secE^1.0 sec N^2.8 sec/^F^1.2 sec^0^3.0 secG^1.4 sec P^3.2 sec/ H^1.6 sec Q^3.4 secI^1.8 sec R^4. 0 sec1 .00 —0.75 —0.50 —0 OP M LIK^ GIA^0.2 sec^J^2.0 secB^0.4 sec K^2.2 secC^O. 6 sec L^2. 4 sec0.00 1^0.0 1.0^2.0^3.0^4.0^5.0/ AeINJFigure 4.38^Inelastic Displacement Response Curves for Modified Taft%.06.04.0^5.00.0^1 .0^2.0^3.00.2 sec I 1.8 secO. 4 sec J sec2. 0O. 6 sec K 2.2 sec0.8 sec L 2.6 sec1.0 sec M 2.8 sec1.2 sec N 3.0 sec1.4 sec 0 3.2 sec1.6 sec P 3.4 sec1 .00 — p= 1 D/^EF/ GHABp=4^C0.75 —0.50 —DH^G1^I^li^II^T^i^1^I^r^il^ii1ii^-r^iill^I6.0A / AeFigure 4.39^Inelastic Displacement Response Curves for Artificial Richmondi.h=4^A^O. 2 sec^F^2. 0 secB^0.4 sec G^2.2 sec/^C^O. 6 sec^H^2. 6 secD^0.8 sec I^3.0 sec/ E^1.4 sec^J^3.4 sec/^F^1.2 sec L^3.2 sec1.00 —/0.75 —0.50 —0.25 —C1^1^1^I^I^1^1^I^I^I^i^I^i^i^1^11^Iiil^11111^III]0.0 1.0 2.0 3.0 4.0 5.0 6.0A / Ae0.00Figure 4.40^Inelastic Displacement Response Curves for Mexico City CUIP EW 19851.0^2.0^3.0^4.0^5.0IIII^I^1^-F^I^I^I^II^I^I^1^1^I^1^1^I^1^1^I^1^1^I^1^I6.0O. 2 sec G 1.^4 sec0.4 sec H 1,6 sec0.6 sec I 2.0 sec0.8 sec J 2.4 sec1. 0 sec K 3. 0 sec1.2 sec L 3.2 sec1 .00 —p.=1B/ CD/ EFA=4^A0.75 —0.50 —A / AeFigure 4.41^Inelastic Displacement Response Curves for Mexico City SCT EW 1985I253 P eP 2P1 ..•••••••...•••••.••••••'•.•'..•.••••.••...-^c0Lateral DisplacementFigure 4.42^Strength and Ductility Demands of Outward-SlopingInelastic Displacement Response Curvesc0a)a)C.)C.)<(Lt+6a)a_CD254TA^TB^TcPeriodP / PeA /A eFigure 5.1^Relationship Between Acceleration Response Spectrum andInelastic Displacement Response Curvesv = 0255max KE^max PEFigure 6.1^Example of Single-Mass Vibrating System///, = strain energy(b)Figure 6.2^Strain Energy in (a) Elastic and (b) Inelastic StructuresEqual-Energy Response• C..s%A256Pe_VG = 2 -=1.2Vi32- 0.75V= 0.5NP-4^5^eAy^A e^AuDisplacement AFigure 6.3^Equal-Energy Seismic Response of StructuresR^1^1.00 —^1.33^0.752^0.50 -^4^0.25 —co^0.00 ^0Figure 6.4^Results of R-p Equation for Various Ratios of Spectral VelocitiesArea OABD = Area OCD,-------AyDisplacement AAuFigure 6.5^Equivalent Elastic Structure2573.02.82.62.42.22.01.81.61.41 .21 .00.80.60.40.20.00^1^2^3^4^5^6^7^8^9Ductility1 0Figure 6.6^Period Shift as Function of Ductility3Period (sec)5.55 -4.5 -43.53 -2.52 -1.5 -1 -0.50 1Actual (Time Step)Calculated (R-p Equation)4^52580^3^6^9^12^15Actual Force Reduction Factor, RaFigure 6.7^Comparison Between Calculated (R-11 Equation) and Actual (Time-Step Analyses) Rfor all Earthquakes with Period Shift onlyFigure 6.8^Comparison Between Calculated (R-11 Equation) and Actual (Time-Step Analyses) Rfor Ductility of 4 for Taft S69E 1952 (0.21g)08O 11 = 2+ p = 4o\.,-400^-200^0^200^400 600N=1 /0o7z.1■■0 6.5as^5U-0• 40:4=^310Secant Slope m (cm/s/s)259Figure 6.9^Modification Factor for the R-ii EquationSVI - SVeT. - T1^esecant slope m —260= energy dissipatedin the cycle\\\ = strain energyTe^TiPeriodFigure 6.10^Definition of Secant Slope in Velocity Response SpectrumFigure 6.11^Energy Dissipated in Elasto-Plastic Structures0-re^.300.28• 0.260.24oTes 0.22'c^0.200.18• 0.160^0.14 -1--;fZ 0.120.1 00)^0.0871-^0.060.04ctS^0.02Ci0.002^3^4^;^6Ductility10 7 98 10261PresentGulkan and SozenlwanFigure 6.12^Damping as Function of Ductility for Initial Damping of 0.05Period (sec)1^2^3^:4Period (sec)5(b)iivi Atrikillp— Actual.--• Calculated41 54 ^3.5 -3 -2.5 -(a)^CC^2 -1.5 -1 -0.5 -0 o 587 -6 -5 -(C) cC 4 -3 -2 -1 -0 o,2^3Period (sec)262Figure 6.13^Comparison Between Calculated (R -p, Equation) and Actual (Time-Step) R forDuctilities of (a) Two, (b) Three, and (c) Four for Taft 569E 1952 (0.21g)— Actual•—• Calculated(c)41 510 9 -8 -7 -6 -ct 5 -4 -3 -2 -1 -0 o 2^3Period (sec)Period (sec)263(b)1 2^3^4. .Period (sec)5Figure 6.14^Comparison Between Calculated (R- 11 Equation) and Actual (Time-Step) R forDuctilities of (a) Two, (b) Three, and (c) Four for Filtered Taft v4k,— Actual•---• Calculated1 4 554 ^3.53 -2.5 -CC 2 -1.5 -1 -0.5 -0 o 6Period (sec)Period (sec)(a)(b)4 500^1^ 2^3Period (sec)(c)264Figure 6.15^Comparison Between Calculated (R- 11 Equation) and Actual (Time-Step) R forDuctilities of (a) Two, (b) Three, and (c) Four for Caltech EW 1971 (0.21g)1^3Period (sec)587 -6 -5 -ct^4 -3 -2 -1 -00 — ActualNI--N Calculated1 6Period (sec)565 -4 -CC^3 -2 -1 -0 o^1^2 3Period (sec)— ActualCalculated5265(b)(c)Figure 6.16^Comparison Between Calculated (R-p, Equation) and Actual (Time-Step) R forDuctilities of (a) Two, (b) Three, and (c) Four for Filtered Caltech(a)514 ^3.5 -3 -2.5 -cc^2 -1.5 -1 -0.5 -0 ^0 2^3^4Period (sec)1^2^e^:4Period (sec)5266(b)(c)765cc^4 -32 -_1-^ — Actual•---• Calculated00 1^2^3^4Period (sec)5Figure 6.17^Comparison Between Calculated (R-11 Equation) and Actual (Time-Step) R forDuctilities of (a) Two, (b) Three, and (c) Four for El Centro EW 1940 (0.21g)267(a)514 ^3.5 -3 -2.5 -CC^2 -1.5 -1 -0.5 -0 o 2^3^4,Period (sec)— Actual.-----• Calculated1^2^3Period (sec)4^5(c)Figure 6.18^Comparison Between Calculated (R-IL Equation) and Actual (Time-Step) R forDuctilities of (a) Two, (b) Three, and (c) Four for Filtered El Centro EW0Period (sec)54 ^3.5 -3 -2.5 -cc^2 -1.5 -1 -0.5 -056 ^5 -4 -cc^3 -2 -1 -0 o 1 2 3,^4Period (sec)268(a)(b)(c)i^i^6^4^5Period (sec)Figure 6.19^Comparison Between Calculated (R-p, Equation) and Actual (Time-Step) R forDuctilities of (a) Two, (b) Three, and (c) Four for Modified Taft76 -5 -(b)— Actual•—• Calculated0 1^2^3^4,Period (sec)(c) Period (sec)51^2^2^4^5Period (sec)269Figure 6.20^Comparison Between Calculated (R-11 Equation) and Actual (Time-Step) R forDuctilities of (a) Two, (b) Three, and (c) Four for Artificial Richmond 43.5 -3 -2.5 -CC^2 -1.5 -1 -0.5 -0 o270(a) if it ft 4*— Actual■--• Calculated4 5Period (sec)(b)544 5765cc 43210 oi^i^6Period (sec),2^3Period (sec)(c)1,— Actual•---• CalculatedFigure 6.21^Comparison Between Calculated (R-11 Equation) and Actual (Time-Step) R forDuctilities of (a) Two, (b) Three, and (c) Four for Mexico City CUIP EW 1985— Actual•--• Calculatedi^4^5Period (sec)9 ^8 -7 -6 -5 -4 -3 -2 -1 -0 o 5Period (sec)271(c)516 ^15 -14 -13 -12 -11 -10 -9 -X 8 -7 -6 -5 -4 -3 -2 -1 -0 o 2^3,Period (sec)Figure 6.22^Comparison Between Calculated (R-1.1 Equation) and Actual (Time-Step) R forDuctilities of (a) Two, (b) Three, and (c) Four for Mexico City SCT EW 19852720^3^6^9^12^15Actual Force Reduction Factor, RaFigure 6.23^Comparison Between Calculated (R-p, Equation) and Actual (Time-Step) R with Periodand Damping Shifts for All Study Earthquakes0^1^2^3^4Vi/VeFigure 6.24^Influence of Spectral Velocity Ratios on Results of R-ji Equation2730^3^6^9^12^15Actual Force Reduction Factor, RaFigure 6.25^Comparison Between Calculated and Actual with Iwan Period and Damping Shiftsfor All Study Earthquakes0^1^2^3^4Vi/VeFigure 6.26^Influence of Spectral Velocity Ratios on Results of R-p, Equationwith Iwan Period and Damping Shifts4.23.02.11.00 1.25274a> L,-a = V-5 . 1.5/ FT'-I^I^I 0.25^0,50^0.75Period, T, s0IVancouver Za=Zv Unit Mass\ Ve....• __ U*Ve''....^-- -. ,.... F=2---- — _ ..... _ _ ..... _ ..... ....F=1^•••• .................................................. •U*Ve • ..... • ••••• ••••■..Figure 8.1^Code Seismic Response Factor [reproduced from NBCC 199011f^I^2^I^3^f^4Period (see)Figure 8.2^Code Design Acceleration Response Spectra-6)Coco0.8 ^0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 -0 oSa seSsV275Se2 L.- Ve2VsSa se —VSs — VsIs^ A5T^(a)^AIs^ A sT^(b)^ASe2 —Sa se —Ss _VIs^ AsT^(c)^AFigure 8.3^Effects of Changes in Response Spectrum on Seismic Responsep=1p=10.2 — • ..■ •... ... •.. •■ •., .. ■0.7276 RichmondNBCC 19905% Damping0.5 —0.6 —..............................0.0 I^I^Io^1 2 3Period (sec)0.1 p=44^5Figure 8.4^Comparison Between Code Design Spectrum and Acceleration Response Spectrumfor Artificial Richmond Ground Motion1.11.0 -0.9 -0.8 -0.7 -CD 0.6 -CIS^0.5 -cn0.4 -0.3 -0.2 -0^1^2^3 4^5Period (sec)Figure 8.5^Comparison Between Code Design Spectrum and Acceleration Response Spectrumfor Mexico City SCT EW 1985 Ground Motion277278I s^11^11IIII^II675011^11^i^81^11^18^11st II t^11^11^II ts^III!^II^II^II^II^II 11^IIII 11 IIII^I11II II II^I.i 1^II^11It^II^I,is /1 II^II^I^11^1^II^11^11II^is^II It^it^II^11^11 II IIII^11 II^11 II II^11II 11^11^II^i!^ti^11^11^II^IIis^II II^11^st^if II__1L—__JL --^IL__ ^4500 I^675011^g^11^g^II^-- -1r —^II^11^11^11^11^"^:1^Int^11^1111^11^1:^11^11^"^=j11:1,  ==-111r1-^ f,1  ---411r-L irli^ ==liii"":"H'ILIII^11^II^11^i^it^II^is11^II 11 II^II^ii II 11 IIII^111^II^II^II^II11^II 11 11^II^II^I '^II^11II II^II it !!I I71 r-nr-^II ^II^1^11^II^I s^II^11si II II^II^11^11 it^11 IIII^11^11IIIs^11IIII II^IfII II^II^II^II^11^if 1111 I!^11^11 II 11ts^II^II si^sl I,^is^11II^II II if II^II II^1111^11^II^II^II^II^II^1111^II^111^11^II^II II11^14^II^11^II^II II^11„ ,^ „^ -II^II^it 11^IIII II^II^is II11^II^11^so^II 81If^II 11 If^II^itPLANAll dimensions in mm..A• /AA.^//../^ AELEVATIONFigure 9.1^6-Storey, 3-Bay Study Frame.-(g)2010.00Hoops @ 95 Hoops @ 95H Hoops @ 95 U-Stirrups 0 170 U-Stirrups @ 170U-Stirrups @ 807- No.15^3 - No.15^7 -No.15U-Stirrups @ 801150A 7- No.153- No.150.0c 0.I3- No.153- No.153- No.15A 1280 „..15 - No.15^3 - No.1512654  212- No.20Section B-B^ Section C-Ca. • AP • \ • /.1Section E-ESection D-D 375 x 375 • •^•338 x 33812- No.15Section A-AAll dimensions in mmSlab Reinforcement: No.10 @ 250300S. asstYP.All hoops and stirrups are No.10 CNDigpFigure 9.2^Detailing of Members in R4 Frame60005000 0.4934)4000300000200079- 1000—1000200^ 400Moment (kN.m)— Ultimate- - - Yield—2000Figure 9.4^Moment-Axial Load Interaction Diagram for Column Sections in R4 Frame.-144o)280300e"-■^250200150a)100 — Ext. & Int. Beam End Section50^ - — Exterior Beam Mid-Section- - - - Interior Beam Mid-Section0.0E+0^5.0E-5^1.0E-4^1.5E-4^2.0E-4^2.5E-4Curvature (mm-1)Figure 9.3^Moment-Curvature Relationships for Beam Sections in R4 Frame+Figure 9.5^Takeda Model for Stiffness-Degrading Moment-Rotation RelationshipDL and LL11.0111111.1111.111.1111111111.11.111.IMEM.111111111.1111.1111j nodes rigid endzones281iiiiFigure 9.6^Modelling of Study Frame5% Damping(b)4 51^1.2^1.00.80.610.4 -0.2 -^0.0 ^0 2^3Period (sec)5282 0.4 ^0.3 -- ^0.2-C 0.1o" tTi^00To? -0.1'"^-0.2--0.3 --0.4 ^00.348 g(a)20^'^40^60Time (sec)11.61.41.2re 1.0(C)^-ff 0.8>C/)060.40.20.00 2^3^:4Period (sec)Figure 9.7^(a) Accelerogram, (b) Absolute Acceleration Response Spectrum,and (c) Relative Velocity Response Spectrum for El Centro NS 1940 Ground Motion(a)60^0.3^0.2 -0.1 -^o^1-0.1--0.2--0.30-0.280 g20^.^40Time (sec)1.00.8 -(c)0.2 -0.0^0 1^2^3 4 52831 .05% Damping0.8 -(b)0.2 -1^2^3^4Period (sec)Period (sec)0.0 ^0 5Figure 9.8^(a) Accelerogram, (b) Absolute Acceleration Response Spectrum,and (c) Relative Velocity Response Spectrum for Olympia N86E 1948 Ground Motion30 401.0200150 —100 -Ecu^50 -0E8-50•4) -100 --150 -RootFirst Floor-2000^ 20^30Time (sec)Figure 9.9^Displacement Time Histories for R4 Frame at Ductility of 4 in Taft S69E 195240200150 —?E 10050 -Ec.)^0co-50--150 --2000 20Time (sec)284Figure 9.10^Displacement Time Histories for R4 Frame at Ductility of 4 in El Centro NS 19404030E 100EC)^50 -EC)c.)^0coa.co3-150 --2000I^I10^20Time (sec)40110 300?E 1 0 0 -• Ee 50 -Ee0coG.toTo:a) -100 -5-150 --200 1I^I20Time (sec)200^ RoofFirst Floor150 —285Figure 9.11^Displacement Time Histories for R4 Frame at Ductility of 4 in El Centro EW 1940Figure 9.12^Displacement Time Histories for R4 Frame at Ductility of 4 in Olympia N86E 194812080^100"40Time (sec)100e 50 -Eb• -50 --is-100--150-286-2000 20^tio^doTime (sec)80^100^120Figure 9.13^Displacement Time Histories for R4 Frame at Ductility of 4 in Artificial RichmondFigure 9.14^Displacement Time Histories for R4 Frame at Ductility of 4in Mexico City SCT EW 1985 MOIPININ1111111814•111MiNilIMMIMMMIINIMIONNOIMTaft^El Centro EW^El Centro NS• = Plastic HingeMIONNIIIIIIIIIIMIDOSIMMIIIMlliMIIIIINMINIPSIMINIMIOlympia^Mexico City SCT EW^RichmondFigure 9.15^Locations of Plastic Hinges in R4 Frame at Ductility of 4 in Study Earthquakes288200m+yi100 --200 -IA;110^20.^.Time (sec)-3000I t30 40Figure 9.16^Moment Time History for Exterior End of First-Floor Beam in R4 Frameat Ductility of 4 in Taft S69E 1952200m;1!100 -ItM -Y-100- I-200 --30010^I^I20^30Time (sec)Figure 9.17^Moment Time History for Exterior End of First-Floor Beam in R4 Frameat Ductility of 4 in El Centro NS 19400 40289200100 -^ M1+1-100-^ I^\ tiii 14 I \A \AAN\-200 - _,M-y-300 0^10^20^30^40Time (sec)Figure 9.18^Moment Time History for Exterior End of First-Floor Beam in R4 Frameat Ductility of 4 in El Centro EW 1940200100 -^I -100--200 -^i ^1^c,^I20Time (sec)Figure 9.19^Moment Time History for Exterior End of First-Floor Beam in R4 Frameat Ductility of 4 in Olympia N86E 1948-3000^10 30^40••!IM+Y100 -200M-Y1^1^T^1^I20^40 60 Eso^160Time (sec)Figure 9.20^Moment Time History for Exterior End of First-Floor Beam in R4 Frameat Ductility of 4 in Artificial Richmond-100 --200 --300 ^0 12020041146AKPK,,AK 4-300^0 20^40^60^sio^100Time (sec)Figure 9.21^Moment Time History for Exterior End of First-Floor Beam in R4 Frameat Ductility of 4 in Mexico City SCT EW 1985120290I100 --100--200 --0.02^-0.01^0^0.01^0.02Rotation (rad.)Figure 9.22^Moment-Hinge Rotation Response for Exterior End of First-Floor Beam in R4 Frameat Ductility of 4 in Taft S69E 1952291200 100 --200 --300-0.01 0.101-0.02 0Rotation (rad.)0.02Figure 9.23^Moment-Hinge Rotation Response for Exterior End of First-Floor Beam in R4 Frameat Ductility of 4 in El Centro NS 1940200100 --200 --3000.01 0.02-0.02 0Rotation (rad.)200100-3000Rotation (rad.)Figure 9.24^Moment-Hinge Rotation Response for Exterior End of First-Floor Beam in R4 Frameat Ductility of 4 in El Centro EW 1940292-0.02 -0.01 0.01 0.02Figure 9.25^Moment-Hinge Rotation Response for Exterior End of First-Floor Beam in R4 Frameat Ductility of 4 in Olympia N86E 1948100 --300-200 --0.01-0.022002930.02Figure 9.26^Moment-Hinge Rotation Response for Exterior End of First-Floor Beam in R4 Frameat Ductility of 4 in Artificial Richmond200-3000Rotation (rad.)-0.02 -0.01 0.01 0.02Figure 9.27^Moment-Hinge Rotation Response for Exterior End of First-Floor Beam in R4 Frameat Ductility of 4 in Mexico City SCT EW 198510 -No.155 -No.15E- c7- No.155 - No.15800, Section E-E375 x 3754-No.252-No.202-No.204-No.25Hoops @ 95 U-Stirrups @ 170 Hoops @ 95 Hoops @ 95 U-Stirrups @ 170U-Stirrups1000@ 80- No.1510B5 - No.15U-Stirrups10-@ 801150NMIIIIIIIANo.15AnimNNE•5.U.SINNENEMaiNNEUI11111UI5- No.15A^1280All dimensionsSlab Reinforcement:g^10 - No.15in mmNo.10 @ 2505- No.151EMUNNENONSEIB ENENo.2012- 8- No.254- No.20ENEUNE11.11 • OEMEEI...OROSection A-A• 41-•-s-so •Section B-B^ Section C-C• 4,‘• • ••A^•'sIk•••.•4 Section D-D338 x 338300 I— VP- • • ••12-No.20 4 Ali hoops and stirrups are No.10ii, il •Figure 10.1^Detailing of Members in E2 Frame1.2E-41.0E-42.0E-50.0E+0 8.0E-54.0E-5^6.0E-5350300250------------------------------------------------------w^1501 00— Ext. & Int. Beam End Section- — Exterior Beam Mid-Section• -^Interior Beam Mid-Section20050Curvature (mm-1)Figure 10.2^Moment-Curvature Relationships for Beam Sections in F2 Frame29560005000—V 4000300000_J 2000--x2 1000—1000—2000— Ultimate- - - Yield400Moment (kN.m)Figure 10.3^Moment-Axial Load Interaction Diagrams for Column Sections in F2 FrameRoofFirst Floor300I 200 -E.5 los -0E0N-6.cobm. -100-. ^-200--3000 20^40^I^80^I^80Time (sec)1^III ^100^120300 ^200 -100 --.E"....4E.a) -100 --200 --300 --400 ^0II^F^I20^40^60Time (sec)IA;80^100^120fv1+YFigure 10.4^Displacement Time Histories for F2 Frame in Mexico City SCT EW 1985296Figure 10.5^Moment Time History for Exterior End of First-Floor Beam in F2 Framein Mexico City SCT EW 1985Figure 10.6^Moment-Hinge Rotation Response for Exterior End of First-Floor Beam in F2 Framein Mexico City SCT EW 1985297Filtered Taft^Filtered Caltech^Filtered El Centro EWArtificial Richmond^Mexico City SCT EWFigure 10.7 Location of Plastic Hinges in F2 Frame in Study Earthquakes12000[^200 mm thick]floor slab400 x 400columns6000 x 200shearwall ^i2991V5800 5800•^60008000Floor PlanAll dimensions in mmH60000H3000026400228001920015600120008400 ico48000Elevation view of shearwallFigure 11.1^Study Shear Wall Structure4-No20I2 layers of No.10 @ 400 mm 4-No20300is^ :^ :^:^I s ISHEARWALL6000 X 200 MM8-No.15• •^•• •• •^•COLUMN400x 400 mmFigure 11.2^Detailing of Members in Study Shear Wall Structure.---..1500012500ELevel^1Level 2z—Y10000 Level 3Level 4■..-.0' Level 5Level 6.4.-Ca)7500 Level 7Level 8Eom5000250000.000000^0.000004^0.000008^0.000012Curvature (mm-1)Figure 11.3^Moment-Curvature Relationships for Shear Wall Sectionsin Study Shear Wall Structure_Y00^200^300Moment (kN.m)••• •• •• ••• •• •• I ^• •Figure 11.4^Moment-Axial Load Interaction Diagram for Columns Sectionsin Study Shear Wall Structure301^Column End Frame^Shearwall Frame^Column End Frame^Figure 11.5^Modelling of Study Shear Wall Structure-Es300200100YDTo- -100-200-300403010^20Time (sec)0302RoofFirst Floor--300RoofFirst Floor200 -5Figure 11.6^Displacement Time Histories for Study Shear Wall Structureat Ductility of 2 in Taft S69E 1952-200 --30020^40Time (sec)0 110Figure 11.7^Displacement Time Histories for Study Shear Wall Structureat Ductility of 2 in El Centro NS 194030110 20Time (sec)0300RoofFirst Floor2-200 --30040Figure 11.8^Displacement Time Histories for Study Shear Wall Structureat Ductility of 2 in El Centro EW 1940303300 RoofFirst Floor200 --200 -20^4I0^80^80^100^120Time (sec)Figure 11.9^Displacement Time Histories for Study Shear Wall Structureat Ductility of 2 in Artificial Richmond-3000310 40110300RoofFirst Floor200 --200 -20^I^4'0^60^I^80^100^120Time (sec)Figure 11.10^Displacement Time Histories for Study Shear Wall Structureat Ductility of 2 in Mexico City SCT EW 1985-30016000 ^12000 -8000 -4000:Z6e.o -4000 --8000 --12000 --16000 0 20Time (sec)304Figure 11.11^Base Moment Time History for Study Shear Wall Structureat Ductility of 2 in Taft S69E 19528000 --"g 4000 -Eo -4000 -2-8000 -M -Y-16000 I^,^0^10 20^30Time (sec)^Figure 11.12^Base Moment Time History for Study Shear Wall Structureat Ductility of 2 in El Centro NS 194040Figure 11.13^Base Moment Time History for Study Shear Wall Structureat Ductility of 2 in El Centro EW 1940+MYMY,3010 40^16000^12000 -_8000 -_--..- 4000 -Z4E,^0a) -Eo -4000 -2^,-8000 -_-12000--16000 ^0I^I^i20Time (sec)30512000 -_8000 ----... 4000 -Z..x^_+al^_E0 -4000 -2^_-8000 --12000-16000 ^12000 -8000 --12000 -4^I0 60^'^8100^20Time (sec)M-y-16000100 12016000306IIi1II1it IIM-y-160000 12020^I^,::^I0 80^1^80^'^100Time (sec)Figure 11.14^Base Moment Time History for Study Shear Wall Structureat Ductility of 2 in Artificial RichmondFigure 11.15^Base Moment Time History for Study Shear Wall Structureat Ductility of 2 in Mexico City SCT EW 198510 30 40^1600^1200 -800 -1 400 -...Acri^0.0a).cCO -400--800 -_-1200 --1600 0iII20Time (sec)16001200800bc 13^0.0a).cCI) -400-800-1200-1600307Figure 11.16^Base Shear Time History for Study Shear Wall Structureat Ductility of 2 in Taft S69E 1952Figure 11.17^Base Shear Time History for Study Shear Wall Structureat Ductility of 2 in El Centro NS 1940o 1016001200 -_800 --800 --1200 -_■^■^■20Time (sec)-1600 do 402-' 400 -_cis^0.0a) -.cCO -400-Figure 11.18^Base Shear Time History for Study Shear Wall Structureat Ductility of 2 in El Centro EW 19401600308_1200 -800 --I 400 -Cfs 0.0^APAAAra)^-.cu) -400 -1_-800 ---1200 --0^I^20^Ie\,,AMt\I40^60^80^100^120Time (sec)-16001^ tA\Figure 11.19^Base Shear Time History for Study Shear Wall Structureat Ductility of 2 in Artificial Richmond3091600 ^1200 -800 --800 --1200 -i-1600 ^0 20^40^60^80^100^120Time (sec)Figure 11.20^Base Shear Time History for Study Shear Wall Structureat Ductility of 2 in Mexico City SCT EW 1985-0.005^-0.003^-0.001^0.001^0.003^0.005Rotation (rad.)Figure 11.21^Moment-Hinge Rotation Response at Base of Shear Wallat Ductility of 2 in Taft S69E 19521600012000 -8000 _40002 -4°°° --8000-12000-16000-0.005 -0.003^-0.001^0.001Rotation (rad.)'^0.003 0.005Figure 11.22^Moment-Hinge Rotation Response at Base of Shear Wallat Ductility of 2 in El Centro NS 194031016000-0.003^-0.001^0.001^0.003Rotation (rad.)12000 -80004000-8000-12000 --16000 ^-0.005 0.005Figure 11.23^Moment-Hinge Rotation Response at Base of Shear Wallat Ductility of 2 in El Centro EW 1940-0.001^0.001Rotation (rad.)-0.003 0.003 0.005Figure 11.24^Moment-Hinge Rotation Response at Base of Shear Wallat Ductility of 2 in Artificial Richmond16000311_12000 -_8000 -_"Es 4000 --t^o_§2 -4000i_-8000--12000-_-16000 ^-0.005 -0.003^-0.001^0.001^0.003Rotation (rad.)0.005Figure 11.25^Moment-Hinge Rotation Response at Base of Shear Wallat Ductility of 2 in Mexico City SCT EW 1985yielded section_Physical Model^Computer ModelFigure 12.1 Modelling of Plastic Hinges in Static-Load-to-Collapse Analysis--_--_-_----^.._...,^_400—L_aw_c^V)^300 —a)w0^CO^200—_-7C-50 100—I--Load at First Hinge = 237 kNAdjusted Code Yield Load = 164 kN600—-^-500—__ •■•■•11..-IIIIIIIIIIIIIIIIIIIIIIIIIIIII0^100^200^300^400^500^600Lateral Roof Displacement (mm)Figure 12.2 Static-Load-to-Collapse Response of R4 Frame 800—-••■••141.■■■■•■••eThZ^H....Y 600 —...._.L._0a)^_..cV)^400 —a)cnaco _200——00I-- _-__-_--Load at First Hinge = 410 kNAdjusted Code Yield Load = 328 kN1^I^IIIIIIIIIIIIIIIIIIIIIIIIIII10 100^200^300^400^500^600Lateral Roof Displacement (mm)Figure 12.3 Static-Load-to-Collapse Response of F2 Frameif Strain Hardening ConsiderYield Plateauedlusted Code Yield Load = 459 kNLoad at First Hinge = 456 kN600—^--.--. 500—z_.Y•••—....•400—&._aa)v)^300—_a)0C3^_CO 200—_a-4-^ -0 1 0 0 -1--__^-_----_0i^1 I^I^1^I^I100IIII1200IIii^1300III^r^1400I^I^I^I^I^I-^I^I500 600Lateral Roof Displacement (mm)Figure 12.4 Static-Load-to-Collapse Response of Study Shearwall Structure--E.-..VC0t 250a)Cl)Ccm.....coa)C3trl 200ta)Eo20)C150Ca)cca)>fi0)a)Z 1 00Adjusted Moment Resistance = 216.5Adjusted Design Moment = 196.5Factored Moment Resistance = 184 Factored Design Moment = 167 IIIIco0v00a0^ 100^ 200^ 300Total Lateral Load (kN)Figure 12.5 Moments in Design Beam Section in R4 FrameAy DisplacementAyEqual AreasDisplacementUltimate LoadAy DisplacementAy DisplacementHuBased on Equivalent Elasto-Plastic YieldBased on First YieldUltimate LoadFirst YieldingBased on Equivalent Elasto-Plastic^Based on Reduced StiffnessEnergy Absorption^Equivalent Elasto-Plastic YieldFigure 12.6^Definitions for Yield Displacement of Structure [after Park 1991]100 200 300 400 500 600Figure 13.1^Response of Inelastic R4 Frame in Study Earthquakes Scaled to 0.21gLoad at First Hinge = 237 kNAdjusted Code Yield Load = 164 kNELCNSSCTELCEW//Olympia lTaft • /Richmond/600500400300200100600 / SCT * --0-Richmond * --0*-318, 500Z_x.....—•^ /400^ELCEW *Taft * /s-£a)^Olympia /^E-C ELCNSV)^300^ id,N.—U) ^ JL ^Load at First Hinge = 237 kNco0^ ccu 200^ 0^:.= ^Adjusted Code Yield Load = 164 kN^1_ 0 lir0^ 7-6a° 100i-- ccnIn0a 100 200^300 400^500 600Lateral Roof Displacement (mm)Lateral Roof Displacement (mm)Figure 13.2^Response of Inelastic R4 Frame in Study Earthquakes Scaled to 0.21 m/sTaft x^xOlympia /^ELCEW x Elastic^ELCNS x x /^* InelasticLoad at First Hinge = 237 kNAdfusted Code Yield Load = 164 kNC••011111r-111111111111111,111miRichmond x/ ; ESCT /'VI 00319100^200^300^400^500^600Lateral Roof Displacement (mm)Figure 13.3^Response of Elastic and Inelastic R4 Frame in Study EarthquakesScaled for Ductility of Four800 —Filtered El Centro EY/600 —_Mexico City/ SCT * * Filtered TaftRichmond */ Filtered Ca techLoad at First Hinge = 410 kN400 —200 —Adiusfed Code Yield Load = 328 kN100^200^300^400^500^600Lateral Roof Displacement (mm)Figure 13.4^Response of Inelastic F2 Frame in Study Earthquakes00-^1500 —a)^-U- -)a)cn^1000 —3201000 —* Taft750* ELCEWSCT *ELCN,Richmond *500— ----^--------------------------------250 —-0 1I^I^I^I^10^100^200 300 400^500^600Lateral Roof Displacement (mm)Figure 13.5^Response of Inelastic Shear Wall Structure in Study Earthquakes Scaled to 0.21g2500 — Taftx^Elastic*^Inelastic_Y2000 —_ELCNS* ELCNS* TaftELCEW^SCT>• ELCEWRichmond^SCTRichmondStrain Hardening Yield Plateaucri500 —_Design Deflection = 260 mm0I^i^1^I I^ 1^100 200^300^400^500^600Lateral Roof Displacement (mm)Figure 13.6^Response of Elastic and Inelastic Shear Wall Structurein Study Earthquakes Scaled for Ductility of Two1000—-^-Z_le^750 —...._.Taft^. x Olympiax ElasticELCNS x xI ELCEW * Inelastic__..—*,z 2000:..-■-_321-_^_^Richmond x_^500 —^SCT_--250 —----Itill^I^I1^1I^I^I^III;^li200^300 400 500^600Lateral Roof Displacement (mm)Figure 13.7^Simplified Inelastic Displacement Response Curves for R4 Framein Study Earthquakes Scaled for Ductility of Four2500 —_ TaffELCNSzxELCNS *Taft^*ELCEW x SCT *\ELCEW *x Elastic*^Inelastic_1500 —_-_1000 —-_-_500 —-Richmond x * SCTRichmond __o 1^IIIIIIIIIIIIIIIIiiIIIIIII;^IIo^100^200^300^400^500^600Lateral Roof Displacement (mm)Figure 13.8^Simplified Inelastic Displacement Response Curves for Shear Wall Structurein Study Earthquakes Scaled for Ductility of TwoBendingmoment• PlastichingeC\<---- I --->i-<--- I ---->.322Au = A + r1,1b 9Y^I^PbOp(Figure 14.1^Relationship Between Ultimate Tip Displacement and Plastic Hinge Rotation[from Reinforced Concrete Structures by R. Park and T. Paulay. Copyright ® JohnWiley & Sons, Inc., 1975. Reprinted by permission of John Wiley & Sons, Inc.]accumulated positive primary rotation^.^Zaccumulated negativen i  primary rotation^=^Ii3^accumulated positive secondary rotation .^Estaccumulated negative secondary rotation .^Esi323Figure 15.1^Definitions of Accumulated Primary and Secondary Plastic Hinge RotationsMassesRoof:^125.3 kNFloor: 333.6 kNW27 x84i W3C x99W3 C x99co'Cr1—W3C x99W3C x99W3Cx116W3Cx116W3Cx116CVCO4-NeictW3C x116W3C 116W3Cx116W3Cx116CVCo5^c'W3Cx116W3Cx116,c'W3Cx116cor.-1-rc)W3Cx116W3Cx116W3Cx116cor.1-wN...,.CV5W3Cx1161( T,^X0Ci)CVCO.r4^00COcf)@0).1-.W27x843241...^3 @ 6100^...1Figure 16.1^20-Storey, 3-Bay Study Frame for Storey Shear AnalysisLateral ForceDistribution attime of MaximumBase ShearLateral ForceDistribution fromStorey Shear^Storey ShearEnvelope Envelope519513415013395217411527587607658217194116so2051Taft951221281241088113213110010810780132263363471578658El Centro NS14817415610955161461711491076025148322478587642658El Centro EW( 93115119118113100951141181181131009520932744555865811313212611497761132453714855826581121331261149776ArtificialRichmondMexico City SCTFigure 16.2^Results of Storey Shear Analysis for R4 Frame325326Lateral ForceDistribution fromStorey ShearEnvelopeLateral ForceDistribution attime of MaximumBase Shear8236344633i 39Taft 257368114516El Centro NS3036Storey ShearEnvelope82303470606683114385428532683 9842759El Centro EW98356457535683Mexico City SCT464538261549230434588683812704626136834944372615ArtificialRichmond76433727-1 18144372710Figure 16.3^Results of Storey Shear Analysis for 20-Storey Frame—I.-f N + P^ t N + P(a)^ (b)H `\I test specimen11ELEVATIONFigure 17.1^Representative Exterior Beam-Column Subassembly[after Paultre et al. 19891Figure 17.2^Deflected Shape of Exterior Beam-Column Jointin a) Real Structures and b) Laboratory Tests327Left BentCross ChannelsSteel TubeBraceMiddle BentHydraulicActuatorjv— SpecimenLoad CellLoadingLever ArmHydraulicActuatorLoad Cell0Right BentcrStrong FloorFigure 17.3^Overall Test Set-Up2250 mmI. AN2081 mm —2081 mm....(a)-82 kN-m-122 kN-m(b)92 kN-m-216.5 kN-m6750 mm1835 mm1666 mm  ....76 kN-m163 kN-m.25 DL + 1.5 LL329Figure 174^Bending Moments in Exterior First-Floor Beam of Prototype Framea) Under Gravity Loads only and b) When Beam Ends Yield1524g1 00 ir^oLococv oLo•zr300 1508co80-2r-coco542 542300Loc.1o,_3381825330All dimensions in mm.Figure 17.5^Dimensions of Test Specimens•••••Beam Flexure^Column Flexure•••••••Beam Hinging^Column Hinging••••••• _ — -—••••••Joint Shear Deformation Bar Slip^ Frame DeflectionFigure 17.6^Components of Beam Tip DeflectionIDRC for Scaled El Centro EW and Taft S69EIDRC for Unscaled Mexico City SCT EWIDRC for Scaled Mexico City SCT EW•Deflection332Figure 17.7^Scope of Experimental Study59-"E-E0.037Test BC1El Centro EW 1940Selected load point numbers are indicated.( ) = Yield deflection at beam tip-39I^I291\I1 1241361^6 14Figure 17.8^Loading Program for Test BC1^ (,)(...)co6358• ■IrTest BC2Taft S69E 1952Selected load point numbers are indicated.( ) = Yield deflection at beam tip1\153927120^3040I45150I55•••• —6^10-.--.^EE Ecc::^0.(c) ——Figure 17.9^Loading Program for Test BC21.61.41.21.00.80.60.40.20-0.2-0.4-0.6-0.8-1.0-1.2-1.4-1.61 si3332 38105^11789I i127133f^9890112^132128-49--E'EcgN.6155V_54^62^68 70 72?^--f^-"E'^--' --'E E^E EE— u1^ig.^al cl— CD -4-^CO CO.._Test BC3Mexico City SCT EW 1985Selected load point numbers are indicated.( ) = Yield deflection at beam tip-69 71 73^—I ■ 82Figure 17.10^Loading Program for Test BC3BC4E^-E--^Mexico City SCT EW 1985E^ EC,).III cn^cq Selected load point numbers are indicated.—( ) = Yield deflection at beam tip63^71^7359319230785264 70^72cl^Co.coFigure 17.11^Loading Program for Test BC4• •No.10 Hoops @ 85 mm5 - No.15 with 10.5" hook2- No.15 with 10.5" hook3- No.15 with 10.5" hookNo.10 Hoops @ 95 mmNo.10 Hoops @ 85 mmAll cover: 40 mmunless indicated otherwiseMain Beam SectionTransverse Beam SectionColumn Section3 - No.15No.10 Hoops @ 85 mm25 mmtop coverNo.10 Hoops @ 140 mm 2-No.1512- No.15No.10 @225 mm with r hookNo.10 @ 400 mm56 mm covertop and bottom• • •337Figure 17.12^Detailing of Specimens BC1 to BC3No.10 Hoops @ 85 mm5- No.15 with 10.5" hook5 - No.15 with 10.5" hook5- No.15 with 10.5" hookNo.10 Hoops @95 mm— No.10 Hoops @ 85 mm12- No.20No.10 Hoops @ 85 mm25 mmtop coverNo.10 Hoops @ 140 mm3 -No.152- No.15e • •338All cover: 40 mmunless indicated otherwise•IriThNo.10 @ 225 mm with r hookNo.10 @ 400 mmMain Beam Section56 mm covertop and bottomTransverse Beam SectionColumn SectionFigure 17.13^Detailing of Specimen BC4M c25262728293020,2122,237,8,9,10,11,12NHInNIINMINIIIIINIn ,ligirimwmp-mwmmemommommi1IN E11111111111in munallhaiIIMIPillMEIME111111NINHINMNMIRNIN11•111. STA^16. TTL2. BTB 17. TTR3. BTC^18. TBL4. BTD 19. TBR5. BTE^20. CTO6. BTF 21. CB07. BBA^22. CTI8. BBB 23. Cal9. BBC^24. HJC10. BBD 25. SLL11. BBE^26. SLM12. BBF 27. SLR13. BM C^28. SRL14. HBC 29. SRM^15. HBE^30. SRR339IllINNwilmompsifilYwomwm111111111111EMI1011.U.of- 1,2,3,4,5,6• • ^•13‘r,Figure 17.14 Locations of Strain Gauges in Specimens(a)^ (b)Figure 17.15^a) Instrumentation Around Joint Core of Specimen and b) Parameters Measured341Figure 17.16^Reinforcement in Joint Core of SpecimensFigure 17.17^Typical Stress-Strain Curve for Reinforcing Steel in SpecimensFigure 17.18^Formwork343Figure 17.19^Beam of Epoxy-Repaired Specimen BC1(a)(b)(c)345Figure 18.2^Beam Damage in Specimens a) 13L2, b) BC3, and c) BC4344Figure 18.1^Typical Damage Pattern in SpecimensFigure 18.3^Propagation of Negative Cracks in Beam of Specimen BC3347Figure 18.4 Typical Slab Damage Showing Transverse and Longitudinal Cracks348Figure I^I wical Torsional Cracks in Transverse Beams349Figure 18.6^Typical Upper Column DamageFigure 18.7^Typical Lower Column Damage351Figure 18.8^Typical Damage to Back of Joint CoreTorsional CrocksFailure Surface352Figure 18.9^Splitting of Beam-Column Specimen [reproduced from Durrani and Zerbe 1987](b) a.I— -100Erococo -200Vco^o0...1-300-20 -16 -12^-8^-4^0^4^8Testing Frame Deflection (mm)200100,2002 100AC-0(a)^3^0aI= -100I -200353-300-20 -16 -12^-8^-4^0^4^8Testing Frame Deflection (mm) 2002 100—vV8 o_..1a.•:-- -100Ecowcci -200-300(c)-20 -16 -12^-8^-4^0 4^8Testing Frame Deflection (mm)Figure 19.1^Deflection of Testing Frame in Tests a) BC2, b) BC3, and c) BC4Load Axis without Testing Frame Deflection0^Yield PointDeflection (mm)-100^-75 25^50-100Specimen BC1El Centro EW 1940-200I^I^I^I^1^1I^1^-30010 ‘t co cv-1-II^II^II^II^'II.^iFigure 19.2^Beam Tip Load-Global Deflection Relationship for Specimen BC1CV^Co)^V'II^II II IIz200Load Axis Without Testing Frame Deflection0^Yield PointDeflection (mm)-100^-75^-50 25^50-100Specimen BC2Taft 569E 1952-200-300Figure 19.3^Beam Tip Load-Global Deflection Relationship for Specimen BC2CM^Cr)II^II^II^II^IIZ..^Z.^Z.11111Load Axis Without Testing Frame Deflection0^Yield Point/4.11111°.'1111;..4 :  I_ ,..-"I%#"1 /,/ /// I// // si7iiII1--- -1001I1,^Specimen BC311 Mexico City SCT EW 1985idt 1r,^11+ — -2001Ist111I^I^1^1^ II 14 - — -300t() •cr ce) cv 1II^II^II^il^'W^yDeflection (mm)-100^-75 25^50Figure 19.4^Beam Tip Load-Global Deflection Relationship for Specimen BC3II^IIZ. Z.dlollill'ill'i^f' '111. 111,1'.'1 1:111"11 1111 11,111i.illi ril'11111 11, 11111''' ''1,1"1111111,1'','1111111:'1 1 1 11 1 11111^;,011:11,1:'1)111,111111,411110111,1^11,1111,1111, ;I^irl^, ^ 2501'1' 'V101101(1111 111111111All ill 0%11 I :I j; 11` 11 II /I '1 ) IiI111111/ 41' 111,11P ;11 11 ;Ill 1111 // 111,41;111 11111 111,11p- -2000z 200Load Axis Without Testing Frame Deflection113Yield Point0..J 100I rDeflection (mm)1^-100^-75 -50 50-100Specimen BC4Mexico City SCT EW 1985--300Figure 19.5^Beam Tip Load-Global Deflection Relationship for Specimen BC4'^20^I^,t[)-300 111111111-100^-80^-60^-40^-20Specimen BC1200100 --200 -...Specimen BC2A ';;)7/Ifil,Wi- , l_111111111 1111100 -80 -60 -40 -20^0^20 40200100-200-300Deflection (mm)Figure 19.6^Beam Tip Load-Relative Deflection Relationship for Specimen BC1358Deflection (mm)Figure 19.7^Beam Tip Load-Relative Deflection Relationship for Specimen BC24020-300-100 -80^-60^-40^-20^0200Specimen BC3100--200 __200Specimen BC4100 --200 --300^ ,^-.^-100^-80^-60^-40^-20Deflection (mm)I 20 I 40Deflection (mm)Figure 19.8^Beam Tip Load-Relative Deflection Relationship for Specimen BC3359Figure 19.9^Beam Tip Load-Relative Deflection Relationship for Specimen BC4100 -(b)-100--200-(a)3600.02-002^-0.01^0^0.01Beam End Rotation (rad)2000.02-300^-002^-0.01^0^0.01Beam End Rotation (rad) (c)0.02-002^-0.01^0^0.01Beam End Rotation (rad)Figure 19.10^Computed Moment-Rotation Response of End of Exterior Beam of Prototype Framein a) El Centro EW, b) Taft S69E, and c) Mexico City SCT EW Ground Motions-100Deflection (mm)-50first yieldingo general yielding400300200100- 1 00-200-300Load (kN)Figure 19.11^Beam Tip Load-Deflection Relationship for Ehsani and Wight Specimen[reproduced from Ehsani and Wight 19851-100^-50^0^50^100^150^200tip deflection (mm)Figure 19.12^Beam Tip Load-Deflection Relationship for Paultre and Mitchell Specimen[reproduced from Paultre and Mitchell 19871361-100 •,■IIM50?1 01-5°BC1-100-0.006 -0.004 -0.002^0^0.002 0.004Rotation (rad)50-100-0.002^0^0.002^0.004^0.006Rotation (rad)1BC2To Load Point 6 OnlyV1 01-50BC150To Load Point 54 Only^BC3To Load Point 6 OnlyBC2-100-0.002^-0.001^0^0.001^0.002Rotation (rad)5050A-100-0.002^-0.001^0^0.001Rotation (rad)V-0.002(b)362-0.002^0^0.002^0.004^0.006Rotation (rad)(a)-100-0.006 -0.004 -0.002^0^0.002 0.004Rotation (rad)Figure 19.13^a) Entire Response and b) Initial Response of Rotation of Upper Columnin Tests BC1 to BC3Rotation ofBase of Upper ColumnAfter Crack DevelopedExpectedRotation ofBase of Upper Column363Figure 19.14^Deflected Shape of Joint Core with Gap at Base of Upper ColumnForces actingon Panel Zone1•1■•••••■■■•VSteel andBond ForcesFigure 20.1^Actions at Joint Core under Negative Beam Tip Loads1200.xZ 1100-1000 -tscts0-J 900800 -F 700 -5'5 600 -Crs 500 -a0 400 -300-300^-2000^-100^0^100Beam Tip Load (kN)200Figure 20.2^Axial Load in Upper Column in Test BC112001100 -1000 -900 -800 -700 -600 -500 -400 -_300-300 200-200^-100^100Beam Tip Load (kN)365Figure 20.3^Axial Load in Upper Column in Test BC21200Z 1100 -..v-ocoo...,^900 -Ws.•^800 -c 700 -E=-5 600 -0staaD 400 -3003661000 -500 -t-300^-200^-100^0^100^200Beam Tip Load (kN)Figure 20.4 Axial Load in Upper Column in Test BC3Z_v-ociso_JCu..cEm7508a.aD1200 1100 -1000 -900 -,800 -700 -600 -500 -400 -300-300 -200^-100^o^100Beam Tip Load (kN)200Figure 20.5 Axial Load in Upper Column in Test BC41422 mmDurrani and Zerbe1238 mmCSA CAN3-A23.3 Clause 21.4.2.2795 mm•No.10 bars^ 50 mm^230 mm^230 mm300 mmFigure 20.6^Effective Slab Width for Specimens50 mmFigure 20.7^Strains in Slab Reinforcement on Left Side of Specimen BC3• • •P .--- 214 kNP = 187 kN (Py)P = 177 kNP = 145 kNP = 109 kNP= 60 kN54321o--_-_-200To Load Point 10 Only-300To Load Point 78 Only369(a)4000^8000^12000MicrostrainSLMYield Strain(b)200z 100V48^o__Iarz -100ES -200co-300o^4000^8000Microstrain12000SAMFigure 20.8^Strains in Middle Slab Reinforcing Bar in Specimens a) BC2 and b) BC4BC2 -0.5^0^0.5^1Ratio to Yield Strain-1^-0.5^0^0.5^1Ratio to Yield StrainBC42-'-IC"0CD0-J0-I-COcpIn200100o-100-200-300Gauge SLL Gauge SRR,-, 200z-'1--`  100-02 0-J.2- -1 00g _2000m-300- -0.5^0^0.5^1Ratio to Yield Strain-0.5^0^0.5Ratio to Yield Strain,..., 200z100DSI^0a- . -1001--(V, -200CD131 -300-1 -0.5^0^0.5Ratio to Yield Strain-0.5^0^0.5Ratio to Yield StrainBC3 1200100o-100-200-3002'_vI,co0-J0-I-COCDCOSLL SRRFigure 20.9^Strains in Outer Slab Reinforcing Bars in Specimens BC2, BC3, and BC440000^2000MicrostrainO 2000Microstrain4000O 2000Microstrain4000Gauge BTDGauge BTC200• 1000iL2- -1 00• -2009900200100.1^02- - 100E2 -200310002001003o. -100,co -200-3°-9000Gauge BTFGauge BTENot AvailableBTC^BTEBTD„V. •BTFBBD^BBFBBC^BBEYield StrainGauge BBF200• 100^To Load Point 9 OnlyB 0iz -100,noil -2009000^0^2000 4000MicrostrainGauge BBC Gauge BBD Gauge BBE200100-1^09- -1oo1—ES -200-92000To Load 'folnt 9 OnlyNot Available Not AvailableO 2000 4000MicrostrainFigure 20.10^Strains in Beam Longitudinal Reinforcement in Specimen BC2200-Y 100 -seTo Load Point 60 Only-o -o0 co-9- -100I— -9--E2 -200 2-300^-2000 2000Microstrain0^2000Microstrain2001000-100-200300-2000Not Available Not AvailableTo LoadTo LoadTo Load Point 71 OnlyPoint 69 Only Point 73 Only2000Microstrain2000Microstrain2000Microstrain2000Microstrain200-Y 1000-Ji=a- -1002 -200co-300-2000200-se 100-000 0-J-1001-E2 -200-300-2000200100-00i= -1002 -200300-2000200100-oo°3^0.1002 -200-300-2000Gauge BTC Gauge BTD Gauge BTE Gauge BTFBTC^BTEBTD••‘,....---• >^*^*BBD BBFBBC^BBEBTFYield StrainGauge BBC Gauge BBD Gauge BBE Gauge BBFFigure 20.11^Strains in Beam Longitudinal Reinforcement in Specimen BC3Gauge BTFGauge BTEGauge BTDGauge BTCBBFBBDBBC^BBEGauge BBFGauge BBEGauge BBC^ Gauge BBDBTC^BTEBTD0,--'•'\.VEMIrr-• >l< 4,^*BTF-300^-2000^o^2000 4000 • wooMicrostrain2001000,^0-JISL -100cg -200-300-2000200-Y• 10068 0- -100-200-300-2000^290^• too•0 ^oo.1—Eai -200.-300^-2000200 ^• too•60• 0 ^a- .iz -100.g• 200-0^2000 4000 6000Microstrain0^2000 4000 6000MicrostrainFigure 20.12^Strains in Beam Longitudinal Reinforcement in Specimen BC4To Load Point 64 Only2001000- -1001—E-200-300-2000200-Ne too-J2- -100I -200-3®-2000Not Available Not Available0^2000 4000 6000Microstrain0^2000 4000 6000MicrostrainYield StrainTo LoadiPoint 71 Only2000 4000 6000MicrostrainTo Load Point 63 Only200100E.'.R2dV0To Load Point 54 Only.1Curvature^(x10-6 nnn-11)Figure 20.13 Measured Moment-Curvature Relationship for Beam Sectionat Joint Face in Specimen BC3374-200 --300-16^-12^-8^-4^0^4 8^12^16200 2002' 1 00-13CO^00-J0.-100ci -200200loo_v-0-J0--1 00-200-300-0.06-300-0.06-300-0.06 003 003 -0.03^0Rotation (rad)-0.03^0Rotation (rad)-0.03^0Rotation (rad)Section 1^Section 2^Section 3200 mm 200 mm 200 mm-P1^2^3Figure 20.14^Rotation of Main Beam in Specimen BC2Section 1 Section 2 Section 3200200 2002' 100..%-0al^0o-Jai= -100oq(2 -200-300-0.06 003-300-0.06 003-300--0.06 003 -0.03^0Rotation (rad)-0.03^0Rotation (rad)-0.03^0Rotation (rad)200 mm 200 mm 200 mm 1^2^3Figure 20.15^Rotation of Main Beam in Specimen BC3Section 1^Section 2^Section 32002- looI 0aF-100E1-200 200100l 0aII -100E1-200200Z' 100I 0aP -1001-200-300-0.06400-0.06400-0.06-0.03^0Rotation (rad)003 -0.03^0Rotation (rad)003 -0.03^0Rotalion (rad)003200 mm 200 mm 200 mm2^3 1-PFigure 20.16 Rotation of Main Beam in Specimen BC4200-1000^1000Misrostrain-3004000 3000200-1000^-1000MicrostrainGauge CTO-300-3000 -1000^1000Microstrain3000CTO^c-n• • •XCB0^CBIYield StrainGauge CBO Gauge CBIFigure 20.17 Strains in Column Longitudinal Reinforcement in Specimen BC2Gauge CTO290To Load Point 69 OnlyCTIX' 100-V•al^00-10.P -100COa)m -200•• • •-300-3000 CBI-1000^1000Microstrain3000CTOCB0200Yield StrainGauge CBO Gauge CBITo Load Point 62 Only3000• 100-V-0CO^00-Ja1-7: -100Easa)m -200-300-3000k,-1000^1000MicrostrainNot AvailableFigure 20.18^Strains in Column Longitudinal Reinforcement in Specimen BC3Gauge CTO200To Load Point 64 Only2. 10000-1i:: -100m -200-300-3000 -1000^1000Microstrain3000CTO^CTICB0^CBIYield StrainGauge CB0 Gauge CBI200 To Load Point 73 Only2's 100I— -100L200inoco0-JCL-100a)m -200-300-3000-300 ^-3000 -1000^1000Microstrain-1000^1000Microstrain3000Figure 20.19^Strains in Column Longitudinal Reinforcement in Specimen BC4BIBBTABBABBB\..:(1....C,/r.v V- HJC381a^> Direction of StrainHJCFigure 20.20^Strain Gauges in Joint Core10 OnlyTo Load Point382200100BC2:31 o,ca.I— -100E1 -200-300200To Load Point 68 Only2. 100 -_leBC32002. 100—V-0^o8_1aI— -100%123 -200-300BC4To Load Point 64 Only0^2000^4000Microstrain-300 ,2000Microstrain40000^2000Microstrain4000Figure 20.21^Strains in Joint Core Transverse Reinforcement in Specimens BC2, BC3, and BC4BTBTo Load Point 72 OnlyBC23832002 100 1— -100co -200-3002002 1000I— -1008CC) -200-300-1000^1000^3000^5000MicrostrainBC3To Load Point 62 Only-1000^1000^3000^5000MicrostrainYield StrainBC4200100o-100613co -200-300-1000^1000^3000^5000MicrostrainFigure 20.22 Strains in Embedment Length of Beam Top Longitudinal Reinforcementin Specimens BC2, BC3, and BC41000^2000^3000MicrostrainTo Load Point 69 Only„/,r,,,Ity..AiiriIf\,C..c03..bco-13TD5--300-10002002- loo00-J0-P-100asa)t -200BBBFigure 20.23^Strains in Embedment Length of Beam Bottom Longitudinal Reinforcement in Specimen BC3200• 1000ap -100 -EO -200 -co-300 ^-3000^-1000^1000^3000Microstrain200 ^.1 100p. -100 -P3O -200 -co-300 ^-3000 -1000^1000^3000MicrostrainBC2To Load Point 72 Only200 ^_1 100 --oto`/ap. -100 -E2 -200 --300 ^-3000 -1000^1000^3000Microstrain-1000^1000MicrostrainBC4 200To Load Point 63 Only0p -100O -200-300-3000 3000Gauge BTA Gauge BBA385BC3200200100.§^0p• -100g -200_Ne 100-300^,^.-3000^-1000^1000^3000Microstrain-300-3000 3000-1000^1000MicrostrainYield StrainFigure 20.24^Strains in Hooks of Beam Longitudinal Reinforcementin Specimens BC2, BC3, and BC4BTABBTo Load Point 48 OnlyTo LoadPoint 48 OnlyBC4-300 -100 100 300 500Microstrain-300 -100^100^300^500Microstrain2001-4- 1000.-o-100a) -200co-300-500200i!-100ri -200-300-500Gauge BTA Gauge BBA386200100200To Load Point 40 Only To Load Point 40 OnlyBC3 2 0;2-.100g -200F.:•-100-200-300-500 -300 -100^100Microstrain300 500-300-500 -300 -100^100^300^500MicrostrainBTABBFigure 20.25^Strains in Hook of Beam Longitudinal Reinforcement Under Initial Loadingsin Specimens BC3 and BC47A- 7B- 7C-Figure 21.1^Loading Program for Test BC57A+ 7B+ 7C+1Ig^IIcoZ.z.Cr)Z.- - - - - Load Axis Without Testing Frame Deflection0^Yield Point0^Yield Point Estimated During Test 3A+ 4+ 5+ 6+ 7A+-200Deflection (mm)-150-futi'd ' ..,tr' o'f'.rolltir . .opil l .tl'l 46,1111:04 -.;Ifiniii i ifj111:1:1111 1,11n141": 1 :1:1:1' :1/4aic ^.41,-'VAIr ip!iii ,00 1rfi li,/..,, f il0101 ,17?,:41:i 11r r11; ;11111.1:1111011111P/_, / ... iiioifililif 1 1..,..1^itiv.firciiii./14"^1 - -^---,-Iiimpli.:digiolliiii.;,-.,;;;'.iossid,;;Poriiiiii0-1 ;,".04,11Hilip.P.:0000.1051.11:fialtdot005.^ 1100,Tfolohroliptirsiti,ril lifAP..1.40iiidinii!,:dioirmilAiiiiiimpny'Aexitti AI/ip,iggiTis;fli;riitivillioNalliiipcti.rdtwirei 17lluilloillvAinumPriii rill 11tiliwy,LI.;iioariiiihoi.,ditilAiiiii):0o /, iiiitoorioniw Alp 11 r7A-6-5-^4-^3A-I-200-300-100Specimen BC5100 150U)^V'^CO^cvII II II II^IIFigure 21.2^Beam Tip Load-Global Deflection Relationship for Specimen BC54020389F-octs^oo.._1aI— -100Easa)c0-200-100^-80^-60^-40^-20^0Deflection (mm)Figure 21.3^Beam Tip Load-Relative Deflection Relationship for Specimen BC52002- 100-act%^0o_1aI— -100Ecd0.)CO -200-300 _8-20^-16^-12^-8^-4^0^4Testing Frame Deflection (mm)Figure 21.4^Deflection of Testing Frame in Test BC5390(a) Before Bang500a)0 -50-100-0.006^-0.002^0.002^0.006^0.01^0.014^0.018Rotation (rad)-0.006^-0.002^0.002^0.006^0.01^0.014^0.018Rotation (rad)Figure 21.5^Rotation of Upper Column a) Before and b) After Bang in Test BC5Gauge BTC Gauge BTD Gauge BTFGauge BTEGauge BBC Gauge BBD Gauge BBE40000^2000Microstrain200000^10000MicrostrainBTFBTC^BTEBTD200• 1002 0-J•C'I" -10012,• -200300200• 100g 01-1- -100E-200-300O 10000Microstrain20000200100-§ 0rz -100-200-300-2000*BBD^BBFBBC^BBEFigure 21.6^Strains in Beam Longitudinal Reinforcement in Specimen BC5Not Available200▪ 100• 021= -100 .• -20010000^.i15.000Microstrain-300^200^K. 100,0 ^-100 •fg• -200-300 ^-2000 2000^4000Microstrain200• 100co• 0-Ji= -100• -200-300-2000 0^2000MicrostrainYield StrainGauge BBF200Gauge CTO2- 100 --V(00P -100 -Em -200 --300 ^-3000CTO CTI• ••CB0 CBIGauge CBIGauge CB0-1000^1000 3000MicrostrainMicrostrainMicrostrainTo Load Point 3C+ OnlyYield StrainFigure 21.7^Strains in Column Longitudinal Reinforcement in Specimen BC5To Load Point 4+ Only1000—10.i:= -1000CO -200-300-30003000-300-3000 -1000^10002001000-1000To Load Point 3C+ Only-300-2000-200-1002001000I/ l^l\ A\0^2000 4000 6000 8000 10000 12000BBB393MicrostrainFigure 21.8^Strains in Joint Core Transverse Reinforcement in Specimen BC52001000a-100-200-300CrTo Load Point 38+ Onlyco05:-2000 0^2000 4000 6000 8000 10000 12000MicrostrainFigure 21.9^Strains in Embedment Length of Beam Bottom Longitudinal Reinforcementin Specimen BC5394(a)EntireResponseGauge BTA^Gauge BBATo Load Point C+ Only200100 --23-100 -Caa) -200 -co-300 ^-30002001003I._ -100 -E2 -200 --300 ^-3000 3000•^.^.-1000 1000Microstra in-1600^1000^3000MicrostrainTo Load Point 1- Only 200(b)^1±; 100InitialResponse ^-100g -200200100iz -1002 -200-300  ^-300--500 -300 -100^100^300^500 -500 -300 -100^100^Microstrain Microstrain300 500To Load Point 1- OnlyYield StrainBTABBFigure 21.10^a) Entire Response and b) Initial Response of Strains in Hook of BeamLongitudinal Reinforcement in Specimen BC5200To Load Point 6- Only100 -ZY-0Co0-J.c.).1— -100 -ECoa)co-200 -Bang-0.06^-0.04^-0.02^0^0.02^0.04Joint Core Shear Deformation e (rad)-300 ^-008395Figure 21.11^Joint Core Shear Deformation in Specimen BC5396Figure 21.12^Damage to Beam of Specimen BC5Back ol Specimen BC5 a) Belore and b) After the BangFigure 21.134.5:04I -4111-rIf)^._4111117111r--(a)V(b)398Figure 21.14^Rear View of Specimen BC5 without Back Concrete Cover399Figure 21.15^Side View of Specimen BC5 in Last Cycle of LoadingList of ReferencesAmerican Concrete Institute 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Skokie, Ill.: Portland Cement Association.402Derecho, A.T., et al. 1978.Structural walls in earthquake-resistant buildings. Dynamic analysis of isolated structural walls.Parametric studies. Skokie, Ill.: Portland Cement Association.Durrani, A.J., and Zerbe, H.E. 1987.Seismic resistance of R/C exterior connections with floor slab. Journal of the Struc. Div., Proc.ASCE 113, no.8 (Aug.): 1850-1864.Earthquake Engineering Research Centre (EERC). 1989.Preliminary report on the seismological and engineering aspects of the October 17, 1989 SantaCruz (Loma Prieta) earthquake. Report No. UCB/EERC-89/14. Earthquake EngineeringResearch Centre, University of California, Berkeley.Ehsani, M.R., and Wight, J.K. 1984.Reinforced concrete beam-to-column connections subjected to earthquake-type loading.Proc. Eighth World Conference on Earthquake Engineering 6:421-428.Ehsani, M.R., and Wight, J.K. 1985.Effect of transverse beams and slab on behaviour of reinforced concrete beam-to-columnconnections. AC! 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