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Out-of-plane dynamic stability of unreinforced masonry walls connected to flexible diaphragms Penner, Osmar 2014

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Out-of-Plane Dynamic Stability ofUnreinforced Masonry WallsConnected to Flexible DiaphragmsbyOsmar PennerB.A.Sc., The University of British Columbia, 2006A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Civil Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)February 2014c© Osmar Penner 2014AbstractThe vulnerability of unreinforced masonry (URM) buildings to out-of-planedamage and collapse has been clearly demonstrated in past earthquakes.Given sufficient anchorage to the diaphragms (a minimum-level retrofit),a URM wall subjected to out-of-plane inertial forces will likely develop ahorizontal crack at an intermediate height. This crack will cause the wallto behave as two semi-rigid bodies, which rock in the out-of-plane direction.Past studies have demonstrated that the out-of-plane stability of a URMwall connected to the diaphragms can be related to the height to thicknessratio (h/t) and the spectral acceleration at 1 s. However, treatment of theeffects of diaphragm flexibility and ground motion variability on out-of-planewall stability in studies to date has been limited.This dissertation presents an experimental and analytical study examin-ing the out-of-plane stability under seismic loading of URM walls connectedto flexible diaphragms. In the experimental phase, five full-scale unrein-forced solid clay brick wall specimens spanning one storey were subjectedto earthquake ground motions using a shake table. The top and bottom ofthe walls were connected to the shake table through coil springs, simulatingthe flexibility of the diaphragms. The apparatus allowed the wall supportsto undergo large absolute displacements, as well as out-of-phase top andbottom displacements, consistent with the expected performance of URMbuildings with timber diaphragms. Variables examined experimentally in-cluded diaphragm stiffness and wall height.An analytical rigid body model was validated against the experimentalresults, and it was demonstrated that the model was able to reproduce theobserved rocking behaviour with reasonable accuracy. The validated modelwas used to undertake a parametric study investigating the effects of nu-merous parameters on out-of-plane wall stability. Ground motion variabilitywas accounted for by using a large suite of motions. Based on the resultsof the modelling, an updated out-of-plane assessment procedure was pro-posed. The procedure, which could be incorporated into ASCE 41, providesreference curves of h/t vs. spectral acceleration at 1 s, along with correctionfactors for axial load, wall thickness, ground-level walls, and exposure.iiPrefaceThis dissertation is based on original work by the author, Osmar Penner.The concept and design of the test apparatus is my own. I carried out alarge portion of the construction of the test apparatus and the set-up ofthe tests, and supervised all testing. The set-up of the model in this study,including all code and data processing, is my work.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . xx1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Earthquake performance of URM buildings . . . . . . . . . 11.2 Research motivation and objectives . . . . . . . . . . . . . 61.3 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . 72 Background and Literature Review . . . . . . . . . . . . . 92.1 Building characterization . . . . . . . . . . . . . . . . . . . 92.1.1 In-plane system . . . . . . . . . . . . . . . . . . . . 102.1.2 Diaphragms . . . . . . . . . . . . . . . . . . . . . . 112.1.2.1 Idealization . . . . . . . . . . . . . . . . . 122.1.2.2 Stiffness range . . . . . . . . . . . . . . . . 152.2 Out-of-plane wall behaviour . . . . . . . . . . . . . . . . . 162.2.1 Experimental testing . . . . . . . . . . . . . . . . . 172.2.2 Analysis methods . . . . . . . . . . . . . . . . . . . 222.2.3 Current assessment standards . . . . . . . . . . . . 252.2.3.1 Rationale . . . . . . . . . . . . . . . . . . 252.2.3.2 Limits . . . . . . . . . . . . . . . . . . . . 272.2.3.3 Discussion . . . . . . . . . . . . . . . . . . 28ivTable of Contents3 Experimental Program . . . . . . . . . . . . . . . . . . . . . 313.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2 Wall specimens . . . . . . . . . . . . . . . . . . . . . . . . . 313.2.1 Material tests . . . . . . . . . . . . . . . . . . . . . 393.3 Experimental set-up . . . . . . . . . . . . . . . . . . . . . . 403.3.1 Shake table . . . . . . . . . . . . . . . . . . . . . . . 413.3.2 Test frame . . . . . . . . . . . . . . . . . . . . . . . 413.4 Instrumentation and data collection . . . . . . . . . . . . . 503.4.1 Wall instrumentation . . . . . . . . . . . . . . . . . 503.4.2 Shake table and test frame instrumentation . . . . . 513.4.3 Data collection and processing . . . . . . . . . . . . 513.4.4 Video recording . . . . . . . . . . . . . . . . . . . . 513.5 Ground motions . . . . . . . . . . . . . . . . . . . . . . . . 523.6 Shake table tests . . . . . . . . . . . . . . . . . . . . . . . . 543.7 Performance of test apparatus . . . . . . . . . . . . . . . . 573.7.1 In-plane response of test frame . . . . . . . . . . . . 573.7.2 Response of springs . . . . . . . . . . . . . . . . . . 594 Shake Table Test Results . . . . . . . . . . . . . . . . . . . . 614.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 614.2 Visual observations . . . . . . . . . . . . . . . . . . . . . . 614.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . 654.3.1 Cracked response summary . . . . . . . . . . . . . . 654.3.2 Displacement response . . . . . . . . . . . . . . . . 664.3.2.1 Static stability limits . . . . . . . . . . . . 684.3.2.2 Time history response . . . . . . . . . . . 724.3.2.3 Peak response . . . . . . . . . . . . . . . . 784.3.2.4 Rocking period . . . . . . . . . . . . . . . 834.3.3 Acceleration response . . . . . . . . . . . . . . . . . 854.3.3.1 Time history response . . . . . . . . . . . 874.3.3.2 Peak response . . . . . . . . . . . . . . . . 904.3.4 Force demands . . . . . . . . . . . . . . . . . . . . . 944.3.5 Hysteretic response . . . . . . . . . . . . . . . . . . 1004.3.5.1 Acceleration-displacement response . . . . 1004.3.5.2 Force-displacement response . . . . . . . . 1034.3.6 Cracking . . . . . . . . . . . . . . . . . . . . . . . . 1064.3.6.1 Cracking predictions . . . . . . . . . . . . 111vTable of Contents5 Validation of Analytical Model . . . . . . . . . . . . . . . . 1145.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1145.2 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1155.3 Model construction . . . . . . . . . . . . . . . . . . . . . . 1155.4 Modelled response . . . . . . . . . . . . . . . . . . . . . . . 1196 Parametric Study . . . . . . . . . . . . . . . . . . . . . . . . . 1246.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1246.2 Ground motions . . . . . . . . . . . . . . . . . . . . . . . . 1246.2.1 Background . . . . . . . . . . . . . . . . . . . . . . 1256.2.2 Ground motion selection . . . . . . . . . . . . . . . 1286.2.3 Ground motion characterization . . . . . . . . . . . 1296.3 Model configuration . . . . . . . . . . . . . . . . . . . . . . 1326.4 Study methodology . . . . . . . . . . . . . . . . . . . . . . 1366.5 Modelling results . . . . . . . . . . . . . . . . . . . . . . . . 1386.5.1 Reference configuration . . . . . . . . . . . . . . . . 1386.5.2 Phase 1a: Effect of diaphragm stiffness . . . . . . . 1436.5.2.1 Near-fault effects . . . . . . . . . . . . . . 1476.5.2.2 Spectral shape effects . . . . . . . . . . . . 1506.5.2.3 Variable top and bottom period . . . . . . 1546.5.2.4 Allowable spectrum . . . . . . . . . . . . . 1586.5.3 Phase 1b: Effect of other parameters . . . . . . . . 1606.5.3.1 Crack height . . . . . . . . . . . . . . . . . 1606.5.3.2 Slenderness ratio . . . . . . . . . . . . . . 1636.5.3.3 Thickness . . . . . . . . . . . . . . . . . . 1646.5.3.4 Spalling . . . . . . . . . . . . . . . . . . . 1666.5.3.5 Damping . . . . . . . . . . . . . . . . . . . 1676.5.3.6 Diaphragm mass ratio . . . . . . . . . . . 1696.5.3.7 Axial load . . . . . . . . . . . . . . . . . . 1706.5.4 Phase 2: Parametric combinations . . . . . . . . . . 1756.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1897 Recommendations for Assessment Guidelines . . . . . . . 1927.1 Considerations . . . . . . . . . . . . . . . . . . . . . . . . . 1927.2 Recommended assessment procedure . . . . . . . . . . . . . 2007.2.1 Base curves and classification of diaphragms . . . . 2017.2.2 Correction factors . . . . . . . . . . . . . . . . . . . 2027.2.3 Safe seismic hazard level . . . . . . . . . . . . . . . 2077.2.4 Anchorage demands . . . . . . . . . . . . . . . . . . 2077.3 Assessment examples . . . . . . . . . . . . . . . . . . . . . 209viTable of Contents7.4 Summary and additional considerations . . . . . . . . . . . 2158 Summary and Conclusions . . . . . . . . . . . . . . . . . . . 2168.1 Experimental phase . . . . . . . . . . . . . . . . . . . . . . 2168.2 Analytical phase . . . . . . . . . . . . . . . . . . . . . . . . 2178.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . 2198.4 Future research . . . . . . . . . . . . . . . . . . . . . . . . 220Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222AppendicesA Wall Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . 229B Materials Testing . . . . . . . . . . . . . . . . . . . . . . . . . 231B.1 Mortar compression . . . . . . . . . . . . . . . . . . . . . . 232B.2 Masonry compression . . . . . . . . . . . . . . . . . . . . . 236B.3 Masonry flexural tension . . . . . . . . . . . . . . . . . . . 239B.4 Brick compression . . . . . . . . . . . . . . . . . . . . . . . 243B.5 Brick absorption . . . . . . . . . . . . . . . . . . . . . . . . 244C Apparatus Sketches . . . . . . . . . . . . . . . . . . . . . . . 246D Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . 253E Photos of Shake Table Testing . . . . . . . . . . . . . . . . 256F Shake Table Test Results . . . . . . . . . . . . . . . . . . . . 271G Model Validation Results . . . . . . . . . . . . . . . . . . . . 326H Working Model 2D Code Listing . . . . . . . . . . . . . . . 357I Ground Motions . . . . . . . . . . . . . . . . . . . . . . . . . 385viiList of Tables2.1 Range of diaphragm periods . . . . . . . . . . . . . . . . . . 162.2 ASCE 41 special procedure h/t limits . . . . . . . . . . . . 283.1 Wall geometry . . . . . . . . . . . . . . . . . . . . . . . . . 323.2 Wall age . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.3 Mortar properties . . . . . . . . . . . . . . . . . . . . . . . . 393.4 Brick properties . . . . . . . . . . . . . . . . . . . . . . . . . 393.5 Masonry properties . . . . . . . . . . . . . . . . . . . . . . . 403.6 Spring test data . . . . . . . . . . . . . . . . . . . . . . . . . 473.7 Test protocol - wall FF-3 . . . . . . . . . . . . . . . . . . . 553.8 Test protocol - wall FR-3 . . . . . . . . . . . . . . . . . . . 553.9 Test protocol - wall FF-2 . . . . . . . . . . . . . . . . . . . 563.10 Test protocol - wall SS-3 . . . . . . . . . . . . . . . . . . . 563.11 Test protocol - wall RR-3 . . . . . . . . . . . . . . . . . . . 573.12 Peak carriage displacements, wall FF-3, uncracked . . . . . 604.1 Results summary . . . . . . . . . . . . . . . . . . . . . . . . 654.2 Stability limits . . . . . . . . . . . . . . . . . . . . . . . . . 704.3 Peak displacement response in highest stable run . . . . . . 784.4 Peak displacement response in collapse run . . . . . . . . . 784.5 Shortest observed rocking periods . . . . . . . . . . . . . . . 854.6 Peak acceleration response in highest stable run . . . . . . . 904.7 Peak acceleration response in collapse run . . . . . . . . . . 914.8 Peak force demands . . . . . . . . . . . . . . . . . . . . . . 974.9 Period, damping, and full-scale Sa values . . . . . . . . . . 984.10 Peak force demands . . . . . . . . . . . . . . . . . . . . . . 994.11 Peak normalized forces in cracking run . . . . . . . . . . . . 1094.12 Peak forces in cracking and cracked runs . . . . . . . . . . . 1114.13 Peak stresses in cracking run . . . . . . . . . . . . . . . . . 1136.1 Model parameters . . . . . . . . . . . . . . . . . . . . . . . 1356.2 Incrementation parameters . . . . . . . . . . . . . . . . . . 137viiiList of Tables6.3 Phase 1 runs . . . . . . . . . . . . . . . . . . . . . . . . . . 1376.4 Phase 2 runs . . . . . . . . . . . . . . . . . . . . . . . . . . 1386.5 Probability of crack height occurring . . . . . . . . . . . . . 1616.6 Probability of crack height occurring . . . . . . . . . . . . . 1767.1 Diaphragm classification . . . . . . . . . . . . . . . . . . . . 2017.2 Axial load base factor . . . . . . . . . . . . . . . . . . . . . 2047.3 Exposure factor . . . . . . . . . . . . . . . . . . . . . . . . . 206A.1 Wall dimensions . . . . . . . . . . . . . . . . . . . . . . . . 230B.1 Mortar compression, walls FF-3, FR-3, and FF-2 . . . . . . 232B.2 Mortar compression, walls SS-3 and RR-3 . . . . . . . . . . 234B.3 Dimensions of masonry prisms . . . . . . . . . . . . . . . . 237B.4 Prism compression, walls FF-3, FR-3, and FF-2 . . . . . . 237B.5 Prism compression, walls SS-3 and RR-3 . . . . . . . . . . 237B.6 Bond wrench test results, walls FF-3, FR-3, and FF-2 . . . 241B.7 Bond wrench test results, walls SS-3 and RR-3 . . . . . . . 242B.8 Brick compression test results, type A . . . . . . . . . . . . 243B.9 Brick compression test results, type B . . . . . . . . . . . . 244B.10 Absorption test results, type A . . . . . . . . . . . . . . . . 244B.11 Absorption test results, type B . . . . . . . . . . . . . . . . 245D.1 Instrumentation listing . . . . . . . . . . . . . . . . . . . . . 255I.1 Listing of ground motions . . . . . . . . . . . . . . . . . . . 386ixList of Figures1.1 Chimney failures . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Parapet failures . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Gable wall failures . . . . . . . . . . . . . . . . . . . . . . . 31.4 In-plane failures . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 Out-of-plane wall failures . . . . . . . . . . . . . . . . . . . 41.6 Wall-diaphragm anchorage . . . . . . . . . . . . . . . . . . . 52.1 Response of diaphragm and equivalent SDOF system . . . . 142.2 Test setup, ABK . . . . . . . . . . . . . . . . . . . . . . . . 182.3 Test setup, Doherty . . . . . . . . . . . . . . . . . . . . . . 192.4 Test setup, Simsir . . . . . . . . . . . . . . . . . . . . . . . 202.5 Test setup, Meisl . . . . . . . . . . . . . . . . . . . . . . . . 212.6 Quasi-static response of SDOF wall model . . . . . . . . . . 232.7 ASCE 41 diaphragm DCR zones . . . . . . . . . . . . . . . 262.8 ASCE 41 h/t limits . . . . . . . . . . . . . . . . . . . . . . . 293.1 Wall geometry . . . . . . . . . . . . . . . . . . . . . . . . . 323.2 Two-wythe header course . . . . . . . . . . . . . . . . . . . 343.3 Construction of three-wythe header course . . . . . . . . . . 343.4 Three-wythe header course after slushing of joints . . . . . 353.5 Pointing the joints . . . . . . . . . . . . . . . . . . . . . . . 353.6 Completed walls in the EERF . . . . . . . . . . . . . . . . . 363.7 Weighing a wall . . . . . . . . . . . . . . . . . . . . . . . . . 373.8 Lifting a wall into the test frame . . . . . . . . . . . . . . . 383.9 Overview of experimental set-up . . . . . . . . . . . . . . . 423.10 Model depiction of test apparatus . . . . . . . . . . . . . . . 433.11 Bottom of test frame . . . . . . . . . . . . . . . . . . . . . . 443.12 Top of test frame . . . . . . . . . . . . . . . . . . . . . . . . 453.13 Spring assembly . . . . . . . . . . . . . . . . . . . . . . . . . 463.14 Detail of bottom connection . . . . . . . . . . . . . . . . . . 483.15 Detail of top connection . . . . . . . . . . . . . . . . . . . . 493.16 Response spectra of recorded table motions . . . . . . . . . 53xList of Figures3.17 Table displacement time history . . . . . . . . . . . . . . . . 533.18 Table acceleration time history . . . . . . . . . . . . . . . . 543.19 Power spectral density of table acceleration . . . . . . . . . 583.20 Time history of acceleration at table and top of test frame . 593.21 Peak carriage displacements, wall FF-3, uncracked . . . . . 604.1 Typical crack configuration . . . . . . . . . . . . . . . . . . 624.2 Typical detail of fresh crack . . . . . . . . . . . . . . . . . . 624.3 Maximum observed spalling . . . . . . . . . . . . . . . . . . 624.4 Lower wall section at collapse . . . . . . . . . . . . . . . . . 644.5 Wall rocking before collapse . . . . . . . . . . . . . . . . . . 644.6 Displacement nomenclature . . . . . . . . . . . . . . . . . . 674.7 Typical displacement profiles of cracked wall . . . . . . . . . 674.8 Stability criteria . . . . . . . . . . . . . . . . . . . . . . . . 694.9 Instability rotations, without overburden load . . . . . . . . 714.10 Instability displacements, varying overburden load . . . . . 714.11 Displacement time histories, wall FF-2, run 10 . . . . . . . 744.12 Displacement profiles, wall FF-2, run 10 . . . . . . . . . . . 754.13 Rocking displacement time histories . . . . . . . . . . . . . 764.14 Instability ratio time histories . . . . . . . . . . . . . . . . . 774.15 Peak rocking displacements . . . . . . . . . . . . . . . . . . 794.16 Peak instability ratio . . . . . . . . . . . . . . . . . . . . . . 804.17 Peak top carriage displacements . . . . . . . . . . . . . . . . 814.18 Peak bottom carriage displacements . . . . . . . . . . . . . 814.19 Peak differential carriage displacements . . . . . . . . . . . 824.20 Peak wall segment angles . . . . . . . . . . . . . . . . . . . 824.21 Peak top carriage and rocking displacements vs. Sd . . . . . 834.22 Rocking period . . . . . . . . . . . . . . . . . . . . . . . . . 844.23 Typical acceleration profiles of cracked wall . . . . . . . . . 864.24 Acceleration time histories, wall SS-3, run 12 . . . . . . . . 884.25 Close-up of acceleration time histories, wall SS-3, run 12 . . 894.26 Acceleration profiles, wall SS-3, run 12 . . . . . . . . . . . . 904.27 Peak carriage accelerations . . . . . . . . . . . . . . . . . . 914.28 Peak wall accelerations at top and bottom . . . . . . . . . . 924.29 Ratio of peak top to bottom accelerations . . . . . . . . . . 934.30 Peak wall accelerations at crack . . . . . . . . . . . . . . . . 944.31 Force time histories, wall SS-3, run 12 . . . . . . . . . . . . 964.32 Components of acceleration profile . . . . . . . . . . . . . . 974.33 Peak force demands on cracked wall . . . . . . . . . . . . . 994.34 Acceleration hysteresis in highest stable runs . . . . . . . . 101xiList of Figures4.35 Acceleration hysteresis in collapse runs . . . . . . . . . . . . 1024.36 Force hysteresis in highest stable runs . . . . . . . . . . . . 1044.37 Force hysteresis in collapse runs . . . . . . . . . . . . . . . . 1054.38 Acceleration and force time histories, wall FF-3, run 7 . . . 1074.39 Acceleration profiles of wall before and after cracking . . . . 1084.40 Location of centroid of wall force prior to cracking . . . . . 1104.41 Stress time histories, wall FF-3, run 7 . . . . . . . . . . . . 1124.42 Location of peak tensile stress, wall FF-3, run 7 . . . . . . . 1135.1 Model of test setup in Working Model 2D . . . . . . . . . . 1165.2 Detail of crack in Working Model 2D . . . . . . . . . . . . . 1165.3 Modelled peak rocking displacements . . . . . . . . . . . . . 1205.4 Modelled displacement time histories, wall FF-2 . . . . . . 1215.5 Modelled displacement time histories, wall SS-3 . . . . . . . 1226.1 Sample illustration of ε calculation . . . . . . . . . . . . . . 1276.2 Source properties of far-field ground motions . . . . . . . . 1306.3 Response spectra for far-field ground motions - original . . 1316.4 Response spectra for far-field ground motions - normalized . 1316.5 Model configuration . . . . . . . . . . . . . . . . . . . . . . 1346.6 Collapse spectra, reference configuration . . . . . . . . . . . 1396.7 Reference fragility curve . . . . . . . . . . . . . . . . . . . . 1406.8 Coefficient of variation, for varying intensity measure . . . . 1426.9 Fragility curves, varying period . . . . . . . . . . . . . . . . 1436.10 Target points, varying period . . . . . . . . . . . . . . . . . 1446.11 Target points, varying period: effect of intensity measure . . 1456.12 Collapse spectra, varying period . . . . . . . . . . . . . . . 1466.13 Target points, varying period: with near-fault motions . . . 1486.14 Collapse spectra, pulse motions . . . . . . . . . . . . . . . . 1496.15 ln (Sacol) vs. ε regressions . . . . . . . . . . . . . . . . . . . 1526.16 R2 and slope of ln (Sacol) vs. ε regression . . . . . . . . . . 1536.17 Fragility curves, varying top period only, Tb = 1.0 s . . . . . 1556.18 Fragility curves, varying top period only, Tb = 0.5 s . . . . . 1556.19 Target points, varying top and bottom periods . . . . . . . 1566.20 Fragility curves, Ts = 0.5 and 1.0 s combinations . . . . . . 1576.21 Fragility curves, Ts = 1.0 and 1.25 s combinations . . . . . . 1576.22 Allowable spectrum . . . . . . . . . . . . . . . . . . . . . . . 1596.23 Fragility curves, varying crack height . . . . . . . . . . . . . 1626.24 Target points, varying crack height . . . . . . . . . . . . . . 1626.25 Fragility curves, varying slenderness . . . . . . . . . . . . . 163xiiList of Figures6.26 Target points, varying slenderness ratio . . . . . . . . . . . 1646.27 Fragility curves, varying thickness, Ts = 1.0 s . . . . . . . . 1656.28 Fragility curves, varying thickness, Ts = 0.5 s . . . . . . . . 1656.29 Relative collapse intensity, varying thickness . . . . . . . . . 1666.30 Fragility curves, varying spalling . . . . . . . . . . . . . . . 1676.31 Fragility curves, varying damping . . . . . . . . . . . . . . . 1686.32 Target points, varying damping . . . . . . . . . . . . . . . . 1686.33 Mass ratio definitions . . . . . . . . . . . . . . . . . . . . . 1696.34 Fragility curves, varying diaphragm mass ratio, T = 1.0 s . . 1706.35 Axial load application . . . . . . . . . . . . . . . . . . . . . 1716.36 Fragility curves, varying axial load, Case 2 . . . . . . . . . . 1726.37 Fragility curves, varying axial load, Case 3 . . . . . . . . . . 1726.38 Target points, varying P and boundary condition . . . . . . 1736.39 Stabilizing limits of axial load application . . . . . . . . . . 1746.40 Stabilizing effect of axial load application . . . . . . . . . . 1746.41 h/t curves, top storey, P = 0 . . . . . . . . . . . . . . . . . 1776.42 T curves, top storey, P = 0 . . . . . . . . . . . . . . . . . . 1786.43 h/t curves, top storey, P = 10 kN/m . . . . . . . . . . . . . 1806.44 T curves, top storey, P = 10 kN/m . . . . . . . . . . . . . . 1816.45 SGR due to P = 10 kN/m, as h/t . . . . . . . . . . . . . . . 1826.46 SGR due to P = 10 kN/m, as T . . . . . . . . . . . . . . . . 1836.47 SGR due to rigid base, at P = 0, as h/t . . . . . . . . . . . 1856.48 SGR due to rigid base, at P = 0, as T . . . . . . . . . . . . 1866.49 SGR due to rigid base, at P = 10 kN/m, as h/t . . . . . . . 1876.50 SGR due to rigid base, at P = 10 kN/m, as T . . . . . . . . 1887.1 Spectral displacement vs spectral acceleration . . . . . . . . 1937.2 Wall response along the length of a flexible diaphragm . . . 1947.3 Base curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 2037.4 Axial load correction factors . . . . . . . . . . . . . . . . . . 2047.5 Axial load correction factors, compared to model results . . 2057.6 Thickness correction factors, compared to model results . . 2067.7 Axial load correction factors, example 1 . . . . . . . . . . . 2107.8 Assessment curves, example 1 . . . . . . . . . . . . . . . . . 2117.9 Axial load correction factors, example 2 . . . . . . . . . . . 2127.10 Assessment curves, example 2 . . . . . . . . . . . . . . . . . 2127.11 Axial load correction factors, example 3 . . . . . . . . . . . 2147.12 Assessment curves, example 3 . . . . . . . . . . . . . . . . . 214B.1 Typical mortar cube compression failure . . . . . . . . . . . 232xiiiList of FiguresB.2 Typical masonry prism compression failure . . . . . . . . . 236B.3 Masonry prism compression tests . . . . . . . . . . . . . . . 238B.4 Bond wrench test apparatus . . . . . . . . . . . . . . . . . . 239B.5 Typical bond wrench failure . . . . . . . . . . . . . . . . . . 240B.6 Typical brick compression failure . . . . . . . . . . . . . . . 243E.1 Wall FF-3, cracking pattern . . . . . . . . . . . . . . . . . . 257E.2 Wall FR-3, cracking pattern . . . . . . . . . . . . . . . . . . 258E.3 Wall FF-2, cracking pattern . . . . . . . . . . . . . . . . . . 259E.4 Wall FF-2, detail of crack step . . . . . . . . . . . . . . . . 260E.5 Wall FF-2, detail of spalling . . . . . . . . . . . . . . . . . . 260E.6 Wall FF-2, apparatus in lowered position . . . . . . . . . . 261E.7 Wall SS-3, cracking pattern . . . . . . . . . . . . . . . . . . 262E.8 Wall RR-3, cracking pattern . . . . . . . . . . . . . . . . . . 263E.9 Wall RR-3, detail of spalling . . . . . . . . . . . . . . . . . . 264E.10 Wall FF-3, video frame captures . . . . . . . . . . . . . . . 265E.11 Wall FR-3, video frame captures . . . . . . . . . . . . . . . 266E.12 Wall FF-2, video frame captures . . . . . . . . . . . . . . . 267E.13 Wall SS-3, video frame captures . . . . . . . . . . . . . . . . 268E.14 Wall RR-3, video frame captures . . . . . . . . . . . . . . . 269E.15 High speed video frame captures, stable runs . . . . . . . . 270G.1 Modelled vs. tested displacements, wall FF-3, run 9 . . . . 327G.2 Modelled vs. tested displacements, wall FF-3, run 10 . . . . 328G.3 Modelled vs. tested displacements, wall FF-3, run 11 . . . . 329G.4 Modelled vs. tested displacements, wall FF-3, run 12 . . . . 330G.5 Modelled vs. tested displacements, wall FF-3, run 13 . . . . 331G.6 Modelled vs. tested displacements, wall FR-3, run 4 . . . . 332G.7 Modelled vs. tested displacements, wall FR-3, run 5 . . . . 333G.8 Modelled vs. tested displacements, wall FR-3, run 6 . . . . 334G.9 Modelled vs. tested displacements, wall FR-3, run 7 . . . . 335G.10 Modelled vs. tested displacements, wall FR-3, run 8 . . . . 336G.11 Modelled vs. tested displacements, wall FR-3, run 9 . . . . 337G.12 Modelled vs. tested displacements, wall FR-3, run 10 . . . . 338G.13 Modelled vs. tested displacements, wall FF-2, run 5 . . . . 339G.14 Modelled vs. tested displacements, wall FF-2, run 6 . . . . 340G.15 Modelled vs. tested displacements, wall FF-2, run 7 . . . . 341G.16 Modelled vs. tested displacements, wall FF-2, run 8 . . . . 342G.17 Modelled vs. tested displacements, wall FF-2, run 9 . . . . 343G.18 Modelled vs. tested displacements, wall FF-2, run 10 . . . . 344xivList of FiguresG.19 Modelled vs. tested displacements, wall FF-2, run 11 . . . . 345G.20 Modelled vs. tested displacements, wall SS-3, run 7 . . . . . 346G.21 Modelled vs. tested displacements, wall SS-3, run 8 . . . . . 347G.22 Modelled vs. tested displacements, wall SS-3, run 9 . . . . . 348G.23 Modelled vs. tested displacements, wall SS-3, run 10 . . . . 349G.24 Modelled vs. tested displacements, wall SS-3, run 11 . . . . 350G.25 Modelled vs. tested displacements, wall SS-3, run 12 . . . . 351G.26 Modelled vs. tested displacements, wall SS-3, run 13 . . . . 352G.27 Modelled vs. tested displacements, wall RR-3, run 4 . . . . 353G.28 Modelled vs. tested displacements, wall RR-3, run 5 . . . . 354G.29 Modelled vs. tested displacements, wall RR-3, run 6 . . . . 355G.30 Modelled vs. tested displacements, wall RR-3, run 7 . . . . 356xvList of SymbolsNote: the subscript t,b indicates two variants of that symbol, referring totop and bottom locations. This is indicated in the description as t/b.a Accelerationacrack Acceleration at crackat,b Acceleration at t/b of wallB Diaphragm depthCa Axial load correction factorC ′a Base parameter for axial load correction factorCe Exposure correction factorCg Ground level correction factorCt Thickness correction factorct,b t/b damping constantcv Coefficient of variationdb Displacement of top of bottom wall segment relative to base ofwallddiff Differential displacement between top and bottom carriagesdm Moment armdrelt,b Relative displacement of t/b carriagedrel Relative displacementdrock Rocking displacementdrocknorm Rocking displacement normalized to wall thicknessdrockthreshold Threshold at which wall is considered to have started rockingE Modulus of elasticitye Eccentricity of axial load OR base of natural logarithmFd Total force applied to diaphragmFIt,b Inertial force of t/b wall segmentxviList of SymbolsFt,b Horizontal reaction force at t/b of wallFv Vertical reaction force at base of wallFw Total force on wallf Frequencyf ′b Brick compressive strengthf ′fb Masonry flexural bond strengthf ′j Mortar compressive strengthf ′m Masonry compressive strengthfTmax Peak tensile stress in wallGd Diaphragm shear stiffnessG′deff Effective diaphragm shear stiffnessg Acceleration due to gravityh Wall heighthcr Height of crack relative to height of wallht,b Height of t/b wall segmentkt,b t/b spring constantL Diaphragm spanL˜ Generalized excitation factorMdt,b Mass of t/b diaphragmMW Moment magnitudeMw Mass of wallMwt,b Mass of t/b wall segmentm˜ Generalized massm(x) Mass per unit lengthmtrib Total tributary massP Axial load applied to wall (overburden) OR probabilityPcol Probability of collapsep Axial load per unit lengthR2 Coefficient of determinationRins Static instability ratioRMt,b t/b ratio of diaphragm mass to wall massrjb Joyner-Boore distance, to surface projectionxviiList of SymbolsSa Spectral accelerationS′a(1) Corrected level of allowable spectral acceleration for given h/tSab(1) Base level of allowable spectral acceleration for given h/tSacol Spectral acceleration at level causing collapseSd Spectral displacementSmajor Major ground motion scale incrementSminor Minor ground motion scale incrementSstart Ground motion scale at start of simulationSX1 Spectral acceleration at 1 s, in ASCE 41sh Horizontal depth of spallingsv Joint thickness at spalling locationT PeriodT1 Fundamental period of general structural systemTim Period used in intensity measureTrock Effective period of rocking excursionTs Fundamental period of a diaphragmTt,b Fundamental period of t/b diaphragmt Wall thickness OR timeu Displacement of structure relative to supportsueq Displacement of equivalent SDOF systemug Ground displacementut Absolute displacement of structureVS30 Average shear wave velocity in upper 30m of soilW Weight of wallWd Diaphragm tributary weightWt,b Weight of t/b wall segmentx Horizontal co-ordinate along a spany Vertical co-ordinate up a spany¯ Height to centroid of wall forceyTmax Height to point of maximum tensile stressz Generalized displacementxviiiList of SymbolsΓ˜ Generalized SDOF scale factor∆d Diaphragm deflection at mid-spanε Ground motion shape parameterζ Damping ratioζt,b t/b damping ratioθiL,R L/R rotation limit for top block under axial load case iθp Processed rotation angle of bottom wall segmentθt,b Rotation angle from vertical of t/b wall segmentµ Meanµdynamic Dynamic coefficient of frictionµstatic Static coefficient of frictionρ Densityσ Standard deviationψ Shape functionxixAcknowledgementsFirstly, I would like to thank my supervisor Ken Elwood for his supportthroughout this process — for providing guidance and direction, for alwaysmaking himself available to discuss ideas, for providing rapid and thoroughfeedback through the writing process, for boosting morale, and for continu-ously pushing me to do my best.I thank my committee members, Don Anderson and Carlos Ventura, fortheir time and for contributing their valuable insights. I thank SvetlanaBrzev for sharing her masonry expertise, and her students at BCIT for theirassistance in materials testing. Thank you also to Jason Ingham and hisresearch team at the University of Auckland, for sharing their work andtheir ideas.The technicians at UBC, Doug Hudniuk, Harald Schrempp, Bill Leung,and Scott Jackson, were instrumental to the success of the experimentalphase and their first-rate work was greatly appreciated. I would like toexpress a particular thanks to Doug, for the inordinate amount of time andeffort he expended on this project and for his many valuable ideas. I thankGeza Gergo for his help with the apparatus construction. Thank you alsoto the other staff and students who contributed to the project, and to myfellow students with whom I shared the lab.Thank you to Bill McEwen and J.P. LeBerg of the Masonry Institute ofBC, for providing the masonry materials, testing equipment, and technicalsupport throughout the experimental work. I also thank the masons for theconstruction of the wall specimens.The research was conducted with the financial support of the NaturalSciences and Engineering Research Council of Canada (NSERC) and of theCanadian Seismic Research Network (CSRN). I also thank my employer,BC Hydro, for providing me with time away and financial support.I am indebted to my fellow graduate students at UBC, for bouncing ideasaround, for sharing the burdens of graduate school, and for helping me toretain my sanity. Thank you in particular to my office mates and friends,Jeff, Stephen, Mathieu, Carla, and Nazli, whose company created a greatoffice atmosphere.xxAcknowledgementsI thank my parents, Arno and Liese, for providing me with the abilitiesand the means to even attempt such an undertaking, and for their incrediblesupport throughout the process. I also thank my brother, Marcio, for alwaysoffering encouragement, his ideas, and his time. Thank you also to myfriends, for providing many revitalizing breaks from the work.Finally, I thank my wife, Allison, for her incredible patience and tirelessencouragement. This project could not have been completed without her.xxiChapter 1IntroductionA significant stock of unreinforced masonry (URM) buildings is presentthroughout the world, including seismically active regions in Canada [Bru-neau and Lamontagne, 1994]. An extensive building survey of Victoria,British Columbia found URM buildings up to 3 stories to be the secondmost common building type in the city centre [Onur et al., 2005]. A surveyin Vancouver, BC found that URM accounts for the majority of high-seismic-vulnerability buildings [Eng, 2013]. Both of these cities are located in areasof moderate to high seismic hazard.URM buildings representative of those considered in this study were con-structed in the late 1800s to early 1900s, and typically have a simple, usuallyrectangular footprint with minimal vertical irregularities. Exterior walls areconstructed of clay brick and consist of multiple wythes (vertical sectionsof masonry one brick thick). Bricks could be laid in a number of possiblebond patterns, but the wythes are tied together intermittently by headers(bricks laid perpendicular to the wall, so that one brick spans two wythes).Floor and roof diaphragms are typically timber joists (or trusses) sheathedwith narrow (∼ 150mm) boards nailed to the joists. These diaphragmsare typically very flexible in-plane, and are often poorly connected to theperimeter walls. Characterization of buildings is presented in more detail inSection 2.1.1.1 Earthquake performance of URM buildingsURM buildings have performed consistently poorly in past earthquakes.Typical damage includes chimney, parapet, and gable failures, in-plane wallfailures due to sliding or diagonal shear, toe crushing, or rocking, and out-of-plane wall failures. Examples of typical failures in Christchurch, NewZealand, following the Darfield earthquake of 4 September 2010 are shownin Figures 1.1–1.5. Where URM buildings have completely collapsed (i.e.only a pile of rubble remains), it is typically not possible to determine afailure mode; however, many buildings have sustained significant damagewithout undergoing total collapse (e.g., Figure 1.5a).11.1. Earthquake performance of URM buildings(a) (b)Figure 1.1: Chimney failures (credit: D. Dizhur)(a) (b)Figure 1.2: Parapet failures (credit: D. Dizhur)21.1. Earthquake performance of URM buildings(a) (b)Figure 1.3: Gable wall failures (credit: D. Dizhur)(a) (b)Figure 1.4: In-plane failures (credit: D. Dizhur)31.1. Earthquake performance of URM buildings(a)(b) (c)Figure 1.5: Out-of-plane wall failures (credit: D. Dizhur)41.1. Earthquake performance of URM buildings(a) Intact (b) Failed (credit: D. Dizhur)Figure 1.6: Wall-diaphragm anchorageBruneau and Lamontagne [1994] reported on damage from numerousearthquakes in eastern Canada, and found that out-of-plane wall failure wasboth the most common failure mode as well as the mode presenting the great-est life safety hazard. Dizhur et al. [2011] likewise found that out-of-planewall collapse was the most commonly observed failure in URM buildings fol-lowing the 22nd February 2011 earthquake in Christchurch, New Zealand.Extensive in-plane damage was also observed, particularly shear cracking inpiers and spandrels, most often occurring in a stair step pattern along mor-tar joints. While such damage affects the lateral load-carrying capacity ofa building, gravity loads can remain supported even under large lateral dis-placements where cracks follow mortar joints. Shear cracks through brickswill result in much sooner loss of gravity support [Russell et al., 2013]. In-plane damage by itself thus often does not lead to collapse without excessivedeformations or accompanying out-of-plane failure.Vintage URM buildings often have limited or no lateral connections be-tween load-bearing walls and floor and roof diaphragms. This leads to wallsessentially falling off the building by tipping over when subjected to groundshaking, in what is termed a ‘cantilever’ out-of-plane failure. If walls are ad-equately anchored to diaphragms at each level (e.g., Figure 1.6a), this failuremode is prevented. Inadequate anchorage leads back to cantilever wall fail-ure, commonly leaving bare anchors protruding from the diaphragms (e.g.,Figure 1.6b). Given effective anchorage, out-of-plane failures occur by bend-ing of the wall between support points. Where limited vertical supports arepresent between floors, such as in long walls or in piers between windows,walls can fail in one-way vertical bending — spanning between adjacent dia-51.2. Research motivation and objectivesphragms. Where vertical supports are significant (e.g., cross walls, corners),walls can fail in two-way bending. All three failure modes (cantilever, one-way and two-way bending) have been observed in the field [Dizhur et al.,2011].Walls prone to cantilever failure are far less resilient — that is, ableto withstand earthquake ground motions without collapsing — than thoselimited to one- or two-way bending: Doherty et al. [2002] showed that thestability of a cantilever wall is approximately equivalent to that of a simply-supported wall four times more slender. The relatively low cost of installingwall-to-diaphragm anchors is therefore easily justified by the significant im-provements in out-of-plane stability provided [Sharif et al., 2007]. Accord-ingly, the present study focuses on URM walls that have been appropriatelyanchored to diaphragms, particularly on the least resilient case of one-waybending.1.2 Research motivation and objectivesDue to its recognized poor earthquake performance, URM is no longer per-mitted as a structural system in the construction of new buildings in regionsof notable seismicity in Canada [National Research Council of Canada, 2010].In contrast, mitigation of the life safety risk presented by existing URMbuildings in seismically active areas remains to date on a primarily vol-untary basis in most jurisdictions [Paxton et al., 2013]. A lack of ownerincentives for retrofit combined with an often high heritage building value(making demolition unlikely) highlight the need to maximize the efficiencyand cost-effectiveness of retrofit schemes. Sufficient knowledge to allow iden-tification of the most vulnerable components and systems is required to beable to prioritize retrofits and achieve the greatest reduction in seismic riskfor the limited available funding.The prevalence of out-of-plane wall failures, in combination with theassociated life safety risks, have resulted in the prevention of out-of-planefailure becoming a high priority in the retrofit of URM buildings. While out-of-plane wall response has been the subject of significant research effortsin the past three decades (see Section 2.2), the treatment of the effect ofdiaphragm flexibility on this response in the literature has been limited.In particular, there has been a lack of dynamic testing in which the effectof diaphragm flexibility has been addressed. Current assessment guidelines(see Section 2.2.3) do not adequately address the issue, and the current stateof knowledge does not provide a basis for comment on whether the existing61.3. Thesis outlineguidelines are conservative or unconservative in this respect. The presentstudy aims to make a significant contribution towards filling this gap inknowledge.The objectives of the study, each constituting a significant contributionto the body of knowledge relating to the seismic performance of URM walls,are as follows:• Obtain observations of the dynamic one-way out-of-plane bending re-sponse of URM walls connected to flexible supports over a variety ofsupport stiffnesses, including variable top and bottom support stiff-nesses.• Validate a modelling approach that can accurately simulate the ob-served dynamic response over the range of tested conditions.• Using the validated model, conduct simulations over a large range ofparameters to:– characterize wall response as a function of relevant parameters,and– investigate the effect of ground motion variability on the wallresponse.• Produce recommendations for a new out-of-plane wall assessment pro-cedure.1.3 Thesis outlineA multi-phase study including both experimental and analytical work wasundertaken to achieve the objectives listed above. This dissertation, docu-menting the study, is arranged as follows:Background and Literature Review, Chapter 2: Typical character-istics of URM buildings to be used in the study are established, includingthe idealization of walls and diaphragms, and the relevant ranges of pa-rameters describing them. Past research regarding out-of-plane response ofURM walls and the effect of diaphragm flexibility is reviewed. A summaryof current assessment standards is provided.Experimental Program, Chapter 3: Five URM wall specimens wereconstructed in the laboratory and dynamically tested on a shake table ina purpose-built apparatus. The apparatus included provisions to simulate71.3. Thesis outlinethe effect of flexible diaphragms at both the top and base of the wall, andallowed walls to be tested to collapse. Details of the specimen construction,the apparatus, the instrumentation, the shake table inputs, and the testingprocedure are presented.Shake Table Test Results, Chapter 4: The wall specimens were eachtested at incrementally greater amplitudes of input motion until collapsewas achieved. The displacement and acceleration responses of the wall, aswell as force demands, are examined in detail and discussed.Validation of Analytical Model, Chapter 5: A rigid body computermodel representing a cracked one-way out-of-plane wall spanning one storeywas set up using commercially-available software. The model was validatedusing the results of the dynamic testing presented in the previous chapter.Details of the model construction and inputs, and comparisons of measuredto predicted reponses are presented.Parametric Study, Chapter 6: The validated model was used to carryout an extensive parametric study. Over 300 different wall configurationswere modelled, collectively investigating the effects of eight parameters.Each configuration was run through incremental dynamic analysis with upto 100 ground motions, producing a total of over 220,000 runs. The studywas conducted in two phases. In the first phase, the effects of each modelparameter are first investigated relative to a reference configuration. Thesecond phase consists of a full combination matrix of a subset of primary pa-rameters: slenderness ratio, diaphragm period, axial load, and crack height.The results are interpreted in terms of fragility curves providing probabilityof collapse for a given intensity measure, and are compiled into a series ofslenderness ratio curves for varying risk levels.Recommendations for Assessment Guidelines, Chapter 7: A newprocedure for the seismic assessment of out-of-plane URM walls is proposed.The procedure accounts for diaphragm flexibility, axial load, exposure level,wall thickness, and whether a wall is at ground level or above. Considera-tions involved in the interpretation of the study results are discussed, andthe procedure is presented. The new procedure is applied to several examplescenarios and assessment results are compared with the current standard.8Chapter 2Background and LiteratureReviewThe following sections provide background and review of previous researchon the characterization of the most relevant components — in-plane wallsand diaphragms — of typical URM buildings, followed by a review of pre-vious research on the subject of out-of-plane URM wall response.2.1 Building characterizationDerakhshan [2010] addressed the issue of the effects of diaphragm flexibilityon out-of-plane wall response from a static testing approach. While theapproach to that study was different than that for the present one, thesubject matter is effectively the same. In his literature review, he notedthat the following characteristics had been shown to be of importance:• Overall shape and size of building, number of stories• Wall thickness• Wall slenderness• Diaphragm in-plane stiffnessRussell [2010] conducted a comprehensive assessment of New Zealand’sURM stock, grouping buildings into typologies and compiling typical charac-teristics for each. A comparably detailed typological study of URM buildingsin western Canada has not been conducted to date. A comparison can bedrawn, however, between URM construction in western Canada and in NewZealand due to the heavy British colonial influence in both areas. The citiesof Victoria, BC and Christchurch, NZ are considered briefly as examples.Both are the oldest cities in their respective regions, with Victoria beingincorporated in 1856 and Christchurch in 1862. URM construction in bothoccurred primarily between the 1880s and the 1930s. The URM stock in the92.1. Building characterizationcentral business districts consists mostly of two- and three-storey buildings,many in row arrangements.A small-scale survey carried out by Paxton et al. [2013]1 and discussionswith local engineering consultants point towards general consistency betweenURM construction practices in western Canada and in New Zealand. WhileRussell found top storeys commonly constructed with both 2-wythe and3-wythe walls, local engineers noted that 2-wythe walls were scarce, andthat 3-wythe walls were typical. In addition, the upper ranges of the typicalstorey heights listed by Russell were considered to be higher than is typicallyfound locally. Russell’s work, with the addition of local input, formed thebasis of the building characteristics considered in the current study.Focusing on local conditions, the default wall considered in this study isthree wythes thick, with an assumed thickness of 330mm. A 2-wythe wallis also considered, with an assumed thickness of 220mm. Slenderness ratiosof between 8 and 22 are considered, which translate to maximum heights of7.3m for 3-wythe walls, and 4.8m for 2-wythe walls.The excitation applied to the out-of-plane walls is the ground motion,filtered through the building’s in-plane system and the diaphragms. Thetreatment of these components is discussed in the sections below.2.1.1 In-plane systemThe in-plane walls for low-rise URM buildings were assumed to be verystiff compared to flexible diaphragms. As such, they were modelled asrigid, transferring the ground motion unaltered to the endpoints of eachdiaphragm. This assumption is consistent with past work on out-of-planeresponse of URM walls [ABK Joint Venture, 1981b, Doherty, 2000, Meisl,2006].One instrumented URM building with flexible diaphragms has been sub-jected to strong motion shaking. A 2-storey former firehouse in Gilroy, Cal-ifornia was subjected to peak ground accelerations up to 0.29 g during theLoma Prieta earthquake of 1989. Tena-Colunga and Abrams [1992] presentan analysis of the data, which shows that motions recorded in-plane at thetop of the central URM wall exhibited notable amplification at short pe-riods (below 0.5 s), but minimal amplification at periods longer than this.It is shown later in the study that the response of cracked walls is mostsensitive to long-period energy content (see Chapter 6). It is therefore rea-1The survey referenced here was not yet published at the time of writing, but is partof the same study discussed in the listed paper102.1. Building characterizationsonable to assume that short-period amplification would have minimal effecton out-of-plane wall response.Non-linear modelling of two, three, and five-storey URM buildings car-ried out by Knox [2012] showed significant amplification of absolute acceler-ations up the height of in-plane walls for the two and three-storey buildings,and minimal amplification in the five-storey building. Spectral analysis ofthe output motions was not published. Menon and Magenes [2011a,b] con-ducted a parametric study of URM buildings with rigid diaphragms using anon-linear model, which also showed the possibility of acceleration amplifica-tion up the height of the building. However, it was cautioned that buildingswith flexible diaphragms behave fundamentally differently and that furtherwork was necessary to address the topic.The topic of ground motion amplification up the height of URM buildingswith flexible diaphragms is one of great interest, but it is beyond the scopeof the present study. Available data suggests the amplification in low-risebuildings is likely concentrated at short periods, to which out-of-plane wallsare least sensitive. While the assumption of rigid in-plane walls is in thiscontext considered reasonable, efforts are made to present the results of thestudy in sufficient detail to allow re-interpretation in the light of possiblefuture work regarding the amplification issue.2.1.2 DiaphragmsFloor diaphragms in vintage URM buildings commonly consist of timbersheathing supported on timber framing. In smaller buildings, joists typicallyspan directly between load-bearing URM walls, and are either supported onthe ledge created by a change in the number of wythes between adjacentstories, or are embedded in cavities created in the walls for this purpose. Inlarger buildings, joists may be supported by heavier timber or steel beams,and by columns in large open plan areas. Sheathing arrangements vary, andinclude either straight sheathing (perpendicular to the joists) or diagonalsheathing (typically at 45◦ to the joists), applied in either one or two layers.While the in-plane stiffness of such diaphragms varies depending on theconfiguration, in general the stiffness is very low compared to alternativediaphragm systems such as concrete slabs. The matter of quantifying thestiffness of vintage timber diaphragms has been the subject of much uncer-tainty over the past decades. ASCE 41 [ASCE, 2014] contains suggestedstiffness parameters for various diaphragm configurations, but the sourceand accuracy of these values has been questionable, and they have not seenany recent updates.112.1. Building characterizationCohen [2001] constructed two half-scale, single-story, reinforced concreteblock buildings and subjected them to shake table testing. The roof dia-phragm of one specimen consisted of diagonal timber sheathing on timberjoists, the other of corrugated metal decking on open-web steel joists. Rock-ing behaviour of the out-of-plane walls was not observed since the walls werereinforced, but notably it was confirmed that the overall structural responsewas dominated by the deformation of the diaphragm rather than of the in-plane walls. Non-linear analyses [ABK Joint Venture, 1981b, Paquette andBruneau, 1999] have likewise shown that response of low-rise URM buildingsis dominated by the response of their flexible diaphragms. It is thereforeclear that the characterization of flexible diaphragm response is importantin the study of out-of-plane wall response.Testing of wood diaphragms began in the 1980s with a large experimentalprogram by ABK Joint Venture [1981a]. Additional testing, primarily withinthe last 10 years, has significantly improved understanding of the effects ofdiaphragm condition, configuration, perforations, interaction with walls, andorthotropic behaviour [Brignola et al., 2012, Giongo et al., 2013, Peraltaet al., 2004, Wilson, 2012]. Wilson et al. [2013] provides the most up-to-date compilation of research on the matter, including recommendations forstiffness assessment. That analysis put significant weight on the results ofGiongo et al. [2013], who conducted the first full-scale in-situ testing of theas-built diaphragms in a vintage building. The recommendations of Wilsonet al. [2013] were adopted in the present study.2.1.2.1 IdealizationWilson [2012] showed that the deformation response of wood sheathed dia-phragms is most aptly represented by a shear beam model. The stiffness ofa uniform shear beam is proportional to its depth; diaphragms can thereforebe characterized by the shear stiffness, Gd, which is independent of the plandimensions of the diaphragm. For a diaphragm with span L and depth Bloaded parabolically with a total load of Fd, the mid-span displacement isdefined by Wilson et al. [2013] as:∆d = 316 ·FdLGdB(2.1)Wilson et al. [2013] conclude that the shear stiffness varies with dia-phragm condition (defined as good, fair, or poor), orientation (perpendic-ular or parallel to joists) and joist continuity (continuous or discontinuousjoists). For the base case of single straight-sheathed diaphragms, loading122.1. Building characterizationparallel to joists produces the stiffest response (Gd = 225− 350 kN/m), fol-lowed by perpendicular to joists–continuous (Gd = 170 − 265 kN/m) andperpendicular to joists–discontinuous (Gd = 135− 210 kN/m). The highestvalue in this series, 350 kN/m, matches the value for straight-sheathed dia-phragms listed in ASCE 41. Wilson et al. [2013] recommend that for otherdiaphragm configurations, the relative multipliers derived from ASCE 41 areapplied to the new base-case values. The multipliers range up to a maximumof 9.0 for chorded diaphragms with double layered sheathing, producing amaximum base stiffness of 3150 kN/m. The listed stiffness values must fi-nally be modified to include the effects of diaphragm penetrations and theadded stiffness of boundary walls, producing the effective stiffness, G′deff .The stiffness values listed above are representative of secant stiffnesses at100mm displacement.The natural period of a diaphragm with a uniformly distributed tributaryweight, Wd, is given by Wilson et al. [2013] as:Ts = 0.7·√√√√WdLG′deffB(2.2)This relation is derived by assuming a quartic displacement shape func-tion (due to the parabolic loading recommended by ASCE 41), and calculat-ing the properties of the generalized single-degree-of-freedom system. Here,the shape function was:ψ (x) = 325L ·(x2 −x3L2 +x42L3)(2.3)The generalized mass, m˜ and generalized excitation factor, L˜ are cal-culated as functions of the total tributary mass of the diaphragm, mtrib[Chopra, 2007]:m˜ =∫ L0 m(x) [ψ (x)]2 dx = 39687875 ·mtrib (2.4)L˜ =∫ L0 m(x)ψ (x) dx =1625 ·mtrib (2.5)The above formulations are for a generalized co-ordinate, z, of translationat mid-span of the diaphragm. The response of the equivalent SDOF system,ueq, is not equal to the response at this co-ordinate, however. To obtain theresponse of the generalized co-ordinate, the SDOF response must be scaledby the factor:132.1. Building characterizationΓ˜ = L˜m˜ =315248 = 1.27 (2.6)Thus, the response at the generalized co-ordinate is:z(t) = Γ˜·ueq(t) = 1.27·ueq(t) (2.7)The response of the diaphragm at any point along its span, x, is:u(x, t) = ψ(x)·z(t) (2.8)The response along the length of the diaphragm and that of the equiv-alent SDOF system are shown in Figure 2.1. This plot illustrates how theequivalent SDOF system is representative of approximately 0.8 times thepeak diaphragm response. In this study, the equivalent SDOF response wasconsidered to be representative of the conditions around mid-span of thediaphragm. Using the mid-span response, z, as the input to out-of-planewall excitation would be conservative when also not accounting for two-waybending effects. Nevertheless, the reader is advised to keep this simplifica-tion in mind throughout the study.u(x)ueqz0 0.5 1.000.51.0xNormalizedresponseFigure 2.1: Response of diaphragm and equivalent SDOF systemThe displacement history of any point along the diaphragm relative toits endpoints can be determined using Equation (2.8). This relative dis-placement is the same as the relative displacement of the equivalent SDOFsystem when subjected to the ground motion at a different scale. The totaldisplacement, however, is the sum of the relative displacement and the dis-placement at the endpoints of the diaphragm (here assumed to be equal tothe ground displacement):142.1. Building characterizationut(x, t) = ug(t) + u(x, t) = ug(t) + ψ(x)·z(t) (2.9)The total displacement is thus the sum of the unscaled ground motionand the scaled SDOF response. This total response can not be reproducedby any SDOF system, except at the points where u(x) = ueq (Figure 2.1).While this was illustrated here for total displacements, the same holds fortotal velocities and total accelerations. Considering the equivalent SDOFresponse to be representative of the conditions around mid-span thereforeallows the simplification of directly modelling this response as an SDOFsystem. This greatly simplified the representation of the system both in theexperimental and analytical phases of the project.Two conditions were covered by the study: (1) the performance aroundmid-span, represented by the equivalent SDOF system, and (2) the perfor-mance near the ends of the diaphragm, represented by the rigid diaphragmmodel. While not directly addressed due to the limitations of the simplifiedmodels, the conditions at other points along the span can reasonably beassumed to be bracketed by these two conditions.2.1.2.2 Stiffness rangeThe period of a diaphragm system — defined by its stiffness and tributarymass — governs its seismic response, as opposed to its stiffness value alone.In this study, the period will therefore be used as an indicator of diaphragmstiffness.The representative range of diaphragm periods to be used in the studywas determined by considering a number of typical building configurations.The upper floor diaphragm in a two-storey building was used as the basecase. Walls were assumed to be 3-wythe with a thickness of 330mm. Arelatively heavy masonry density of 2100 kg/m3 was assumed, representativeof walls with few voids. Wall heights used were 4m in the storey below and3.6m in the storey above. A diaphragm self weight of 0.5 kPa was used (thetotal tributary weight is dominated by the wall weight, so results are notvery sensitive to the assumed diaphragm self weight). Four different buildingplans were considered: 6× 10m, 8× 14m, 16× 24m, and an elongated planof 8× 24m. Joists were assumed to span the short length of each building,meaning the short diaphragm span is loaded perpendicular to joists, and thelong span is loaded parallel to joists. Effects of penetrations and boundarywall added stiffness were ignored for this order-of-magnitude study, sincethe factors counter one another and the effect of either is moderate. Threestiffness values were used: the ‘poor’ and ‘good’ base case values for single152.2. Out-of-plane wall behaviourstraight sheathing, and those using the maximum multiplier of 9.0 on the‘good’ case. The calculated periods are listed in Table 2.1.Table 2.1: Range of diaphragm periodsPeriod (s)Single straight sheathing Two-layer, chordedPlan (m) Span Poor Good Good6 × 10 short 0.86 0.61 0.20long 1.41 1.13 0.388 × 14 short 0.99 0.70 0.23long 1.72 1.38 0.4616 × 24 short 1.57 1.12 0.37long 2.16 1.73 0.588 × 24 short 0.79 0.56 0.19long 2.95 2.37 0.79Minimum 0.79 0.56 0.19Maximum 2.95 2.37 0.79For buildings with single straight sheathed diaphragms — the most com-mon in vintage URM buildings — periods range between approximately 0.5and 3 s. For very stiff diaphragms, periods range between 0.2 and 0.8 s.Values outside these ranges are possible, due to lighter or heavier walls ordiaphragms and more or less elongated floor plans, but this range is repre-sentative of a majority of URM buildings.In this study, periods up to 2 s are considered; this range covers most typ-ical buildings. A 2 s period is already pushing the boundary of what wouldlikely be considered an acceptable diaphragm flexibility when assessing abuilding due to the large deformations involved. Diaphragms at periods be-yond this range (e.g., a large elongated building with single straight sheatheddiaphragm in poor condition) should likely be retrofitted regardless of thepredicted out-of-plane wall performance. Consequently, it is not of greatinterest to examine wall performance at very large periods.2.2 Out-of-plane wall behaviourAt its simplest, an out-of-plane wall can be idealized as a one-way verticallyspanning strip with a horizontal crack at some height within the span. Such162.2. Out-of-plane wall behavioura crack does not cause collapse at its initiation; instead, the cracked wall canform a stable out-of-plane rocking response about the crack location. Thisone-way idealization is representative of a real wall when two-way effectsare minimal (e.g., in a long wall away from cross walls and corners) andthe wall is adequately anchored to the diaphragms at both the top andbottom. Where two-way effects become significant, this idealization willproduce more conservative results, but nevertheless forms a good startingpoint for understanding the response.Past research has predominantly focused on the one-way model, whichhas remained overall poorly understood despite its apparent simplicity. Thefollowing sections provide a review of past experimental work and analyticalrepresentations of this idealized behaviour. Detailed review of the relevantliterature was provided most recently by Derakhshan [2010] and by Meisl[2006]. The following review serves to provide the reader with a brief back-ground on the subject, with a focus on dynamic testing.2.2.1 Experimental testingEarly experimental work on the topic of out-of-plane URM wall response wasrestricted to quasi-static testing [Anderson, 1984, West et al., 1977, Yokeland Dikkers, 1971]. Walls were supported at the top and base with varyingboundary conditions, and transverse load was applied slowly. Observationsincluded consistent horizontal cracking of wall panels near or slightly abovemid-height, and the stabilizing effect of axial load and its eccentricity.Dynamic testing began with the large-scale programme carried out byABK Joint Venture [1981b]. In this study, 22 wall specimens with differentoverburden loads and height to thickness (h/t) ratios were tested underdynamic loading. The tests were carried out using displacement-controlledactuators at both the top and bottom of the walls (Figure 2.2). The issueof diaphragm flexibility was addressed by estimating the input motions atthe top and bottom of walls using a computer model that consisted of anon-linear shear-deformable beam representing the diaphragm, and lumpedmasses on the beam representing the out-of-plane walls. The calculateddiaphragm response was then applied to the actuators. This design excludedthe possibility of observing the effects of interaction between out-of-planewall rocking and diaphragm flexibility. Cracking was observed near mid-height and at the base of walls, and it was noted that dynamically stablerocking was possible at relative mid-height displacements significantly inexcess of those at crack initiation.172.2. Out-of-plane wall behaviourActuatorActuators (×2)MechanicalheaderBaseplateon rollersURM wallBack stop LoadcellLoadcellFigure 2.2: Test setup, ABK182.2. Out-of-plane wall behaviourDoherty [2000] carried out shake table testing of half-scale (1.5m tall)50 and 110mm thick walls. These tests applied equal top and bottom inputmotions to the wall via a stiff frame on a shake table (Figure 2.3). Axialload was applied to the top of the wall by a series of springs. Notably,the upward displacement at the top of the wall resulting from out-of-planerocking increased the applied axial force (arching action), and so a constantaxial load could not be assumed at larger displacements with the thickerwalls.Cornicesupportat topof wallEarthquake simulator platformWall compressionloading rigStiffenedframeURMwallTimber propsto preventwall collapseFigure 2.3: Test setup, DohertyIn commentary on these tests, Griffith et al. [2004] suggested that spec-tral displacements seem to be a more direct and convenient parameter todefine seismic demand and capacity than accelerations. Furthermore, it wasnoted that for the rigid-support conditions in these tests, the static force-displacement relationship developed for the wall was a reasonable bound ofthe dynamic hysteretic behaviour. Testing also confirmed that wall rockingfrequency and damping are both displacement dependent.Simsir [2004] performed shake-table testing of an idealized URM building192.2. Out-of-plane wall behaviourto study the influence of diaphragm flexibility on out-of-plane URM wallperformance. The test was conducted at half-scale using hollow concretemasonry units. A high axial load was applied, representing a bottom storeywall in a multi-storey building. In-plane walls were reinforced and connectedto the top of the unreinforced out-of-plane walls using a flexible steel beam,representing a diaphragm (see schematic representation in Figure 2.4). Thewalls were observed to crack just above the base at very low input levels, andsubsequently underwent rocking about this crack. Due to the high axial load,rocking about a mid-height crack was not observed until additional masswas added to one wall; the wall collapsed in the same run. Consequently,limited information on the dynamic rocking response of walls was obtainedfrom these tests.URMwallShake tableAxial loadDiaphragmflexibilityIn-plane wall(reinforcedmasonry)Figure 2.4: Test setup, SimsirMeisl [2006] performed full-scale shake table testing of solid clay brickURM walls subjected to out-of-plane excitation, with approximately equalinput motions at the top and bottom of the walls (Figure 2.5). Two dif-ferent ground motions — one recorded on firm ground and one on softground, both during the 1989 Loma Prieta earthquake — were used asinput motions. Rigid-body rocking about cracks above mid-height was ob-served for each specimen. Cracking occurred above mid-height in all walls.It was found that the quality of the collar joint (whether mostly voids orfully slushed) made minimal difference in the rocking response of the walls.202.2. Out-of-plane wall behaviourThe tests provided an important dataset of full-scale rigid-body rocking towhich an analytical model could be calibrated, within the limitations ofrigid-diaphragm conditions.URMwallShake tableBraced frameKey atwall baseKey at top of wallTensionbracesFigure 2.5: Test setup, MeislDazio [2008] conducted shake table testing of 2.4m tall walls with thick-nesss between 125 and 200mm, varying the boundary conditions includingfixity, axial load, and axial load eccentricity. It was observed that the simplysupported case with no overburden was not always the most vulnerable con-dition; small axial loads applied at large eccentricity in some cases resultedin lower stability levels than in walls without overburden. Considerable rock-ing displacement capacity was observed in tests without overburden, but itwas noted that collapse occurred more suddenly with less rocking in wallsthat were axially loaded.Derakhshan [2010] conducted quasi-static air bag testing of full-scaleURM wall panels, including both panels built in the lab and in-situ testingof wall segments in vintage buildings. The in-situ testing demonstrated thatarching action due to vintage timber diaphragms is negligible, and that theforce-displacement characterization obtained in the lab was representative212.2. Out-of-plane wall behaviourof in-situ conditions.In testing of 5m square single-storey full-scale masonry houses [Cloughet al., 1990, Gülkan et al., 1990], it was observed that “typical single-storeymasonry houses are so rigid that they do not develop very complicated dy-namic response mechanisms during earthquakes”. Under strong excitation,out-of-plane walls were observed to crack near mid-height and underwentstable rocking about this crack. Notably, neither greater damage nor re-sponse limits were observed when applying three-axis table motions, com-pared with one- or two-axis motions.Paquette and Bruneau [2003] conducted pseudo-dynamic testing of a4 × 6m full-scale, single-storey URM building with a flexible timber dia-phragm. The base of the building was fixed, while a single actuator appliedan earthquake motion to the roof diaphragm at mid-span. Relative dia-phragm deformation was limited to 20mm, and the diaphragm remainedelastic. The test focused on in-plane performance, and out-of-plane wallrocking was not observed.Three 4 × 6m two-storey full-scale stone masonry houses with flexibletimber diaphragms were subjected to shake table testing by Magenes et al.[2010, 2012]. One specimen was unretrofitted, the second was retrofitted byadding wall-diaphragm connections, but without stiffening the diaphragms,and the third was retrofitted including significant diaphragm stiffening.Both retrofitted specimens withstood significantly higher levels of shakingthan the unretrofitted specimen. Comparisons between the two retrofittedbuildings suggested that the observed improvement of the seismic perfor-mance was related more to the wall-diaphragm connections, and less so tothe stiffening of the diaphragms.2.2.2 Analysis methodsUncracked URM walls behave elastically, and can be modelled accuratelyby simple methods. Given an assumed acceleration distribution, theoreticalcrack locations can be predicted by solving for the location of peak tensilestress. Derakhshan [2010] conducted a detailed analysis of wall crackingbehaviour for varying overburden and tensile strength (typically governedby flexural bond strength). He found that for a uniform applied load, thepredicted crack height changes appreciably only for very low values of tensilestrength (less than 0.1MPa), and is most sensitive under low overburden.Approaches to modelling cracked walls can be grouped into three broadcategories. The methods are summarized briefly here; Derakhshan [2010]provides a more detailed review.222.2. Out-of-plane wall behaviourComplex finite element models: The most detailed models involve thediscretization of the wall into masonry units and mortar joints. Approacheshave included a block-interface model [Martini, 1998], rigid bricks with co-hesionless friction joints [Felice and Giannini, 2001], lumped masses withfibre-element mortar joints [Simsir, 2004], and continuum modelling (non-discrete) [Hamed and Rabinovitch, 2008]. While some of these methodshave proven to be able to simulate the out-of-plane response of URM walls,including crack formation and rocking, they are invariably computationallydemanding, and consequently ill-suited for use in a large-scale parametricstudy.Stick models: Doherty et al. [2002] proposed a simplified representationof a cracked URM wall with rigid top and bottom supports as an equiva-lent single-degree-of-freedom (SDOF) model. A modal mass was calculatedbased on the assumed triangular displaced shape, and a triangular inertialforce distribution was assumed. A [total force]–[displacement at crack] rela-tionship was derived for rigid and deformable wall conditions (Figure 2.6).The deformable model, idealized as a tri-linear curve, included the effectsof (1) elastic deformation prior to rocking and (2) finite dimensions of pivotpoints due to mortar strength limits. Viscous damping was empirically ap-proximated by observing the decay of the free rocking motion. With mass,damping, and stiffness defined, non-linear analysis can be carried out usingthe model.ExperimentalIdealized tri-linearTheoretical rigid bodyDisplacementForceFigure 2.6: Quasi-static response of SDOF wall modelLinearized displacement-based procedures have been proposed based onthese curves, involving the definition of a secant stiffness and assumed damp-ing [Doherty et al., 2002, Priestly et al., 2007]. Simsir [2004] extended theSDOF model to two degrees of freedom by adding a flexible top diaphragm,232.2. Out-of-plane wall behaviourand Derakhshan [2010] extended it further to a two-storey model with flex-ible diaphragms.Rigid body models: Makris and Konstantinidis [2003] demonstrated thatthe response of a rocking system is fundamentally different from that of aregular SDOF oscillator, and recommended that “the response of one shouldnot be used to draw conclusion on the response of the other”. While therestoring force in an SDOF oscillator is due to the elasticity of the structure(k), that in a rocking block is due to gravity. An oscillator has a uniqueperiod, while a rocking block does not. In addition, even non-linear modelslike the tri-linear model in Figure 2.6 do not capture the change in pivotpoint location with rocking direction reversal in a cracked wall. Instead,the effects of this location change and its associated impact and energydissipation are approximated by viscous damping.In rigid body modelling, like that used by Makris and Konstantinidis,bodies are represented with finite geometry. Consequently, changes in pivotpoint locations in a cracked wall can be accurately captured, and energydissipation due to impacts is explicitly accounted for by a coefficient ofrestitution. A rocking wall can therefore be modelled more directly usingthis method, without the need for empirically-calibrated viscous dampingvalues.Konstantinidis and Makris [2005] validated the ability of commercially-available rigid body software Working Model 2D [Design Simulation Tech-nologies, Inc., 2010] to simulate the pure sliding and pure rocking responsesof a block as part of an investigation into the seismic performance of multi-drum columns. Meisl [2006] further demonstrated that the software couldadequately simulate the rocking response of one-way spanning URM wallswith rigid diaphragm boundary conditions.The stick-model approaches use workarounds to approximate the rock-ing response of a cracked wall, while rigid body modelling represents therocking response explicitly. The drawbacks of the rigid body approach are alack of a simple representation of initial elastic wall stiffness and finite mor-tar strength. However, these issues are relatively minor compared with thebenefit of explicit representation of rocking behaviour, and workaroundsare possible to adjust contact point geometry to simulate finite mortarstrength. Given that rigid body models are computationally efficient, andthat commercially-available software makes configuration and setup rela-tively simple, there is not a strong driver towards using a stick model ap-proach. Consequently, the rigid body approach will be used in this study(see Chapter 5).242.2. Out-of-plane wall behaviour2.2.3 Current assessment standardsThe most widely used current standard for assessment of out-of-plane wallstability is ASCE 41 [ASCE, 2014], which is based on the recommendationsof ABK Joint Venture [1984]. In ASCE 41, two options are presented forwall out-of-plane evaluation: (1) a simplified procedure, contained within themain body of the standard, and (2) a more detailed procedure, containedwithin a standalone special procedure for URM. The Canadian “Guidelinesfor Seismic Evaluation of Existing Buildings” [National Research Council ofCanada, 1993] also contains a special procedure for URM in which the out-of-plane provisions are effectively equivalent to those in the special procedurein ASCE 41. Differences are limited to notation and hazard definition; thisprocedure will therefore not be reviewed separately. Application of eitherof the special procedures is restricted to buildings with flexible diaphragmsat all levels above the base of the structure, a minimum of two lines ofwalls in each principal direction (except for single-storey buildings with anopen front on one side), and a maximum of six stories above the base of thestructure.2.2.3.1 RationaleBruneau [1994] provides an overview of the rationale based on which thisspecial procedure was developed. From their experimental data, ABK JointVenture produced non-linear regression curves of h/t as a function of theoverburden ratio (axial load divided by wall weight) and the square rootsum of squares (SRSS) of top and bottom peak input velocities, for fixed‘probabilities of survival’. It is critical to note that the data set on which theregressions were performed was very limited: only two ground motions wereused at each hazard level. Recent work by Baker [2011] suggests that con-siderably more ground motions are required to adequately address motion-to-motion variability.Maximum h/t values to satisfy a 98% probability of survival were estab-lished. Overburden ratios were taken as 0 for walls without stories above,and 0.5 for all other walls — these are conservative values. Input velocitiesat wall supports were taken as ground velocities amplified by the flexiblediaphragms.Amplification was assessed based on the diaphragm demand-to-capacityratio (DCR). Note that this is a strength parameter only, and diaphragmstiffness is not explicitly accounted for in the procedure. As defined by ABK,the DCR is hazard-independent: the demand is calculated based on dynamic252.2. Out-of-plane wall behaviourloading of the tributary diaphragm weight multiplied by 1.0 g. This demandwas assumed based on modelled diaphragm amplifications in the highestseismic hazard zone considered, with an effective peak acceleration (EPA) of0.4 g (refer to the UBC 1994 [International Conference of Building Officials,1994] regarding EPA). A maximum allowable DCR was incorporated, whichincreases with decreasing diaphragm span, reaching a maximum of 5.0 fordiaphragms with a span of less than ∼ 10m (see Figure 2.7). This DCRcorresponded to a peak diaphragm deformation of∼ 125mm in experimentaltesting of diaphragms at the highest hazard level.3120 1 2 3 4 5050100150DCRDiaphragmspan,L,betweenshearwalls(m)Figure 2.7: ASCE 41 diaphragm DCR zonesLower amplification factors were used for ‘softer’ diaphragms (higherDCR), resulting in more stringent h/t limits for diaphragms with higherstrength. The rationale here was that greater non-linear response in thediaphragm would result in lower amplification. These lower amplificationfactors were also allowed for long-span low-DCR diaphragms if adequatewood-framed cross walls spanning between diaphragms were present to pro-vide added damping. Amplifications were varied depending on the dia-262.2. Out-of-plane wall behaviourphragm DCR only at the highest hazard level; at the lower levels, amplifi-cation was assumed constant regardless of the diaphragm capacity.2.2.3.2 LimitsLimits on h/t were derived in this way for three levels of seismic hazard,with a peak ground velocity of 0.3m/s used for the highest hazard level.Where allowable h/t limits produced from the data were deemed too high,they were arbitrarily reduced. The recommended h/t limits were furthertightened in some cases, and reformulated based on spectral acceleration.These final limits were then incorporated in various guidelines, includingFEMA 273 [FEMA, 1997], forming an out-of-plane assessment procedurethat went effectively unchanged into the FEMA 356 prestandard [FEMA,2000] and the ASCE 41 standard [ASCE, 2014]. The procedure, as presentedin the ASCE 41 special procedure and in the main document, is brieflyoutlined below.Special procedure: Diaphragms are categorized into one of three pos-sible zones based on their DCR and span (see Figure 2.7): (1) long-spandiaphragms, (2) short-span diaphragms with high DCR, and (3) short-spandiaphragms with low DCR. Here, diaphragms are considered long-span be-ginning at a span of 55m — the majority of typical commercial URM build-ings will therefore be classified as short-span. The DCR at the transitionbetween (2) and (3) is between 2.5 and 2.8, depending on the diaphragmspan. In the adaptation of the original rationale to form this procedure, thecalculation of the DCR has been tied to the hazard: the demand is definedas 2.1SX1Wd, where SX1 is the design spectral acceleration at 1 s (2/3 ofthe MCE value specified in US codes), adjusted for site class, and Wd is theweight tributary to the diaphragm. Here, the factor of 2.1 appears to beback-calculated to produce results roughly consistent with ABK’s originalintentions for amplifications, which were only specified at a single hazardlevel.Limits on h/t are presented at three discrete hazard levels and four walllocation types, as shown in Table 2.2. Diaphragm classification affects theallowable h/t only at the highest hazard level, where distinction is madebetween values in columns A and B. The more lenient h/t limits in columnA may be used with long-span diaphragms that meet minimum cross-wallrequirements or with short-span, high-DCR diaphragms regardless of cross-walls. Other systems, including short-span, low-DCR diaphragms, must usethe more stringent limits in column B. For evaluation of earthquake-damaged272.2. Out-of-plane wall behaviourwalls, FEMA 306 [FEMA, 1998] stipulates damage-dependent λh/t factorsby which to multiply allowable h/t values; the factors range between 0.6 forheavy damage to 1.0 for insignificant damage.Table 2.2: ASCE 41 special procedure h/t limitsSX1 ≥ 0.40Wall Location 0.13 ≤ SX1 < 0.25 0.25 ≤ SX1 < 0.40 A[a] B[b]One-storey 20 16 16[c] 13Top storey 14 14 14[c] 9First storey 20 18 16 15All other walls 20 16 16 13[a] For long-span diaphragms (zone 1 in Figure 2.7) meeting minimum cross-wall require-ments, or for short-span, high-DCR diaphragms (zone 2) regardless of crosswalls[b] For systems not meeting requirements in [a][c] Minimum in-plane shear requirements apply to use these valuesSimplified procedure: As an alternative to the special procedure, a sim-plified procedure not requiring diaphragm assessment is presented as part ofthe main body of ASCE 41. Here, the assessment procedure is divided intothree performance levels: immediate occupancy (IO), life safety (LS), andcollapse prevention (CP). At the IO level, wall flexural cracking must beprevented. This is evaluated based on the masonry tensile strength and theseismic demands. Both the LS and CP performance levels allow cracking,but require that the walls remain dynamically stable. Stability must eitherbe evaluated by an analytical time-step integration model, or else the wallsmust meet the specified h/t limits (Figure 2.8). The h/t limits in the sim-plified procedure are the same as those in the special procedure using themore stringent column B requirements for all buildings, with slightly differ-ent breakpoints in SX1. These simplified limits will be used as a base casefor comparison of results in this dissertation. Note that the requirementsfor ‘all other walls’ are the same as for one-storey walls.2.2.3.3 DiscussionWhere applicable, the following discussion refers to both the special and sim-plified procedures. First, while there are maximum permissible h/t ratios,there is no upper limit on the hazard level. This is a remnant of seismichazard being classified in discrete zones at the time of the work of ABK282.2. Out-of-plane wall behaviourfirststoreyonestoreytopstorey9 13 14 15 16 18 2000.240.37h/tS a(1.0)(g)Figure 2.8: ASCE 41 h/t limitsJoint Venture, and is not reflective of the current state of hazard analysis.The fact that the same outcome is obtained from assessment of a given wallat SX1 = 0.40 g as at SX1 = 0.60 g or higher may be cause for concern.Characterization of wall stability with respect to seismic hazard is a majorobjective of this dissertation, and this topic is investigated in Chapter 6. Inaddition, in the simplified procedure, no guidance is provided on modellingwall stability, which effectively prevents that option from being exercised inpractice.Furthermore, the procedure accounts for important parameters only verycoarsely, implicitly, or not at all. For example, axial load can vary sig-nificantly among top-storey walls, depending on the size of parapets andwhether roof joists or trusses are bearing on the wall or running parallel toit, but the procedure does not distinguish between these cases. Diaphragmflexibility could vary among diaphragms with the same DCR, but this ef-fect is disregarded. In addition, the procedure assumes that wall stability isindependent of scale effects, accounting only for the h/t ratio, but not forvariation in wall thickness.Finally, the data set on which the ABK Joint Venture analysis was basedwas very limited. Ground motion variability was not accounted for, and theeffects of parameters other than those varied in the experimental work werenot considered. Recommended limits on h/t were arbitrarily reduced at var-292.2. Out-of-plane wall behaviourious stages prior to incorporation in assessment guidelines, with inadequatedocumentation.Aside from hazard definition, no revisions have been made to the out-of-plane portion of the procedure since its inception, despite significant researchprogress having been made since ABK Joint Venture [1984]. The limitationsof the original research and the following adaptations of its recommenda-tions create great uncertainty about the risk levels produced by the currentstandard. New research and modern computational power offer the abilityto re-examine the out-of-plane response of URM walls far more thoroughlythan before, with the potential to produce a new assessment procedure thatbetter defines and controls the associated seismic risks.30Chapter 3Experimental Program3.1 IntroductionFull-scale shake table tests were carried out on URM wall specimens usinga testing apparatus which allows for the simulation of flexible diaphragmboundary conditions. Five wall specimens were tested. This chapter de-scribes the test apparatus, wall specimens, data collection and testing pro-tocols. More details are provided in Appendix C.3.2 Wall specimensFive wall specimens were constructed by professional masons in the Earth-quake Engineering Research Facility (EERF) at the University of BritishColumbia (UBC). Specimens were intended to represent a portion of a top-storey wall in an early 1900s load-bearing URM building in British Columbia(see Section 2.1). Additionally, they were to be generally consistent withthose tested by Meisl [2006], which were representative of an early 1900sschool building in British Columbia.Wall dimensions are listed in Table 3.1 and typical dimensions illustratedin Figure 3.1. Detailed measurements of each specimen are provided inAppendix A. Four 3-wythe walls and one 2-wythe wall were constructed(Figures 3.2–3.6). American bond was used in all walls with a single headercourse at every sixth course. Specimens are named by diaphragm condition([F]lexible, [S]tiff, or [R]igid at top and bottom) and the number of wythesin the wall specimen. The 3-wythe walls are slightly shorter and thinnerthan the walls tested by Meisl, which were 4.2m tall, 0.33m thick, and1.5m long.Mortar was mixed on site by the masons in an electric mixer. To repre-sent the deterioration of the mortar in existing buildings, a Type O mortarmix (1:2:9 cement:lime:sand by volume) was selected due to its low compres-sive strength. Brick units were solid clay and measured 64mm × 89mm ×191mm. Brick units were placed dry to further minimize the bond strength.313.2. Wall specimensTable 3.1: Wall geometrySpecimenIDThickness(mm)Length(mm)Height(mm) h/tMass(kg)Density(kg/m3)FF-3 291 1509 3947 13.6 3627 2095FR-3 291 1500 3984 13.7 3614 2081FF-2 191 1504 2790 14.6 1739 2176SS-3 300 1518 3985 13.3 3833 2113RR-3 296 1513 3973 13.4 3768 2118Figure 3.1: Wall geometry323.2. Wall specimensWhile the workmanship in older buildings may be variable, this factor can bedifficult to quantify. To achieve reasonable consistency among specimens inthis regard, it was therefore decided to employ good construction practices -bricks were precisely placed and collar joints were slushed in all specimens.Meisl [2006] investigated the effect of varying the quality of the collar jointsand found that it had no significant effect on the out-of-plane response ofwalls.The walls were constructed and cured under dry conditions in the lab-oratory. The total construction time for any single wall varied between 5and 7 days. The walls were built in two phases: walls FF-3, FR-3 and FF-2were built simultaneously and then tested. Walls SS-3 and RR-3 were builtafter testing of the first three walls was complete. The age of the walls attesting varied between approximately 2 and 10 months (Table 3.2).Table 3.2: Wall ageAge at testingSpecimen ID Date completed Date tested Days MonthsFF-32011-06-212012-02-27 251 8.4FR-3 2012-04-04 288 9.6FF-2 2012-04-26 310 10.3RR-3 2012-04-24 2012-06-13 50 1.7SS-3 2012-07-11 78 2.6Each wall was built on the web surface of a steel wide-flange section.To move the wall from the place of construction onto the test apparatus,a steel lifting beam was placed on top of the wall and a threaded rod wasinstalled between the lifting beam and the base beam at each corner. Apiece of plywood was placed between the top of the wall and the liftingbeam to absorb surface irregularities. The threaded rods were tightened,placing compressive stress in the wall. The wall could then be lifted usingthe lab’s bridge crane. The wall was first weighed using a load cell placedbetween the lifting beam and the crane hook (Figure 3.7). It was then movedinto position on the testing apparatus (Figure 3.8). No cracking or otherdamage occurred during movement of the walls.333.2. Wall specimensFigure 3.2: Two-wythe header courseFigure 3.3: Construction of three-wythe header course343.2. Wall specimensFigure 3.4: Three-wythe header course after slushing of jointsFigure 3.5: Pointing the joints353.2. Wall specimensFigure 3.6: Completed walls in the EERF (shake table in foreground)363.2. Wall specimensLIFTINGBEAMLOAD CELLBASE BEAMWALLFigure 3.7: Weighing a wall373.2. Wall specimensFigure 3.8: Lifting a wall into the test frame383.2. Wall specimens3.2.1 Material testsMaterial samples were made during construction of the walls. Since the pur-pose of the materials testing was to estimate the as-tested properties of thewalls, the samples were cured in the laboratory under the same conditionsto which the walls were subject. Masonry prims were cured dry (exposed),while mortar cubes were cured in a sealed bag for the first 7 days, thenexposed to cure dry. Materials testing was carried out for each of the twophases: after shake table testing of walls FF-3, FR-3 and FF-2, and aftershake table testing of walls SS-3 and RR-3. Testing procedures specifiedby the indicated standards were followed with the exception of the curingconditions and curing times noted above. Test results are summarized inTables 3.3–3.5.Table 3.3: Mortar propertiesCompressive strength[a]Walls f ′j (MPa) cvFF-34.0 0.23FR-3FF-2SS-3 4.2 0.35RR-3[a] CAN/CSA A179-04 (R2009) [CSA,2009]Table 3.4: Brick propertiesCompressivestrength[a]Absorption[a]24-hr soak 5-hr boilBricks f ′b (MPa) cv (%) cv (%) cvType A 119 0.06 5.1 0.17 7.0 0.12Type B 157 0.07 4.8 0.03 6.6 0.02[a] CAN/CSA A82-06 (R2011) [CSA, 2011]Two sub-varieties of bricks with slightly differing appearance and prop-erties were used in the walls; type A bricks were more porous and had393.3. Experimental set-upTable 3.5: Masonry propertiesCompressive strength[a] Elastic modulus Flexural strength[b]Walls f ′m (MPa) cv E (MPa) cv f ′fb (MPa) cvFF-333 0.20 6.4×103 0.40 0.38 0.42FR-3FF-2SS-3 46 0.13 10.0×103 0.21 0.55 0.31RR-3[a] ASTM C1314 - 11a [ASTM, 2011][b] ASTM C1072 - 10 [ASTM, 2010]deeper colouring than type B bricks, which had a higher strength. Thetwo brick types were mixed in the construction of the walls. Both brickswere of extraordinarily high strength - the stronger type B bricks had amean compressive strength of 157MPa, based on the gross area of half-bricks with the load applied against the bedding faces. The bricks were forrefractory use, and were selected due to a lack of availability of solid regularface bricks. The high brick strength also produced a correspondingly highmasonry compression strength despite relatively weak mortar being used.In out-of-plane behaviour of URM walls, the flexural bond strength (testedvalues in Table 3.5) is the primary parameter influencing cracking load, andbrick strength is expected to have minimal impact on out-of-plane response.The greatest expected difference, should regular-strength bricks have beenused, would be greater spalling of the bricks at the crack.Batching and mixing of mortar on-site by the masons produced notablebatch-to-batch variability in properties. Typically three to four batches weremixed in a day; six mortar cubes were made from each of three batches daily,and eight masonry prisms were made from one batch daily. Note that the lowtarget strength of the Type O mix exaggerates the magnitude of cv for themortar compression strength. Further details can be found in Appendix B.3.3 Experimental set-upThe walls were tested on the single degree of freedom shake table in theEERF in a purpose-built test frame. The following sections detail the con-figuration of the experimental set-up.403.3. Experimental set-up3.3.1 Shake tableThe shake table (Figure 3.6) consists of a steel frame and steel top sheetapproximately 3m by 4m in plan. The top sheet was removed from the tablefor the duration of the testing to reduce force demands on the actuator. Thetable is supported on four double v-groove casters running on steel angletracks and is not restrained against uplift. The test frame was designed tominimize overturning demands and the table was monitored for uplift in theinitial tests; no uplift was detected.The table is controlled by a single hydraulic actuator housed in an open-ing in the centre of the shake table frame. The actuator has a maximumdisplacement of ±457mm and a static capacity of 298 kN. The hydraulicsystem is powered by a 5.7L/s pump supplying 20.7MPa, with a 454Laccumulator. The table is displacement controlled; hydraulic pressure iscontrolled by an MTS 458 servo-controller. The displacement waveform isinput to a PC, where it is converted to a voltage signal by the software Dasy-Lab. This command signal is sent to the servo-controller, which then sendsa signal to a MOOG hydraulic proportional servo valve, thereby regulatingthe hydraulic pressure. The table position is fed back to the servo-controllerby an analog MTS Temposonic displacement transducer.3.3.2 Test frameThe test frame imposed a simplified representation of flexible diaphragmboundary conditions on the wall. The simplifications that were made were afunction of construction and testing simplicity, in addition to ensuring thatthe test results could be modelled accurately in the analytical portion of thestudy. A description of pertinent features is provided in this section, andmore details can be found in the sketches in Appendix C.An overview photo of the test frame is shown in Figure 3.9. A 3-d ren-dering is provided in Figure 3.10, illustrating the various components byvarying colours. A stiff steel braced frame, representing the in-plane walls,was constructed on the shake table. The table motion was transferred tothe top of this frame with minimal amplification (see Section 3.7.1). The in-clusion of in-plane wall flexibility was outside the scope of the experimentalwork. It was instead assumed that the flexibility of the in-plane walls couldbe considered negligible compared to that of the flexible timber diaphragms.Top and bottom diaphragms were represented by rolling steel carriages con-nected to the frame by coil springs. The carriages were able to roll parallel tothe direction of motion of the shake table. Spring stiffnesses were selected413.3. Experimental set-upto achieve natural periods of vibration in the test setup representative offirst-mode in-plane behaviour of typical diaphragm-wall assemblies (refer toSection 2.1.2.2). The top and bottom of the wall were connected to therespective carriages (Figures 3.11–3.12).TOP COILSPRINGSBOTTOMCOIL SPRINGSTOP CARRIAGEBOTTOMCARRIAGEBRACEDFRAMEWALLSPECIMENWEBBINGFigure 3.9: Overview of experimental set-upEach carriage could be ‘locked out’ to simulate a rigid diaphragm con-dition. This was achieved by connecting the carriage to the braced framewith 19mm threaded steel rods. Four rods were used for each carriage423.3. Experimental set-upCoil springassemblyBracedframeTop carriage Hold-downShake tableWallspecimenBottomcarriageFigure 3.10: Model depiction of test apparatus433.3. Experimental set-upFigure 3.11: Bottom of test frame(one on each corner) and all rods were pre-tensioned to prevent any ‘slop’in this connection.Overturning of the top carriage was prevented by a hold-down assemblyconsisting of a steel channel spanning across the top carriage, with a pair ofangles welded to each end. The lower ends of the angles were bolted to thetop frame. A pair of polyurethane-coated steel casters were bolted under-neath the channel in line with the longitudinal beams of the top carriage.Each beam was also supported underneath by 2 v-groove casters mountedon the top frame near the ends of the carriage, thus creating a 3-point con-straint and preventing overturning. The bottom carriage did not require anoverturning constraint, since the weight of the wall specimen in the middleof the carriage provided sufficient overturning resistance.Coil springs were custom-fabricated by a local supplier2. A total offour spring assemblies were used (two for each carriage). Each assembly(Figure 3.13) consisted of a base housing, a shaft, two springs, end caps, anda carriage connection plate. The base housing was made of a square steelhollow structural section (HSS) roughly 450mm long, with 25mm plates2Dendoff Springs Ltd., Surrey, British Columbia443.3. Experimental set-upTOP CARRIAGE HOLD-DOWNTOP CARRIAGEFigure 3.12: Top of test frame453.3. Experimental set-upwelded into each end of the HSS. The plates were bored to accomodate nylonbushings. The shaft, a73mm round HSS, slid through these bushings. Thecoil springs were designed to fit over the shaft; one spring was placed oneach side of the base housing. The springs were then compressed to roughlyone-half of their maximum displacement and end caps were threaded ontothe shaft. Finally, a carriage connection plate (not shown in Figure 3.13)was threaded on one end of the shaft. This plate had two slotted holeswhich mated with holes in each carriage, and allowed for adjustment of thecarriage ‘neutral’ position prior to connecting the spring assemblies. Thebase housings of the spring assemblies were bolted to the braced frame.Pre-loading the springs to half-capacity resulted in a system with no ‘slop’when transitioning from positive to negative displacement. The shafts andsprings were greased, resulting in very little resistance to motion.Figure 3.13: Spring assemblyForce-displacement response of each spring strut assembly was measuredby bolting the assembly to a Tinius-Olsen screw-drive compression testingmachine and running the assembly through nearly the full design range ofmotion. All assemblies showed a force-displacement response that was nearlyperfectly linear over the entire measured range. Key parameters are shownin Table 3.6. Mean and coefficient of variation are shown for each pair ofassemblies (north and south sides).The base of the wall was supported by the bottom carriage, and the wallbase beam was bolted to the carriage. The connection between the walland the base beam is shown in Figure 3.14. A strip of ultra-high molecularweight polyethylene (UHMW) was fastened to each side of the wall at thebase using masonry screws. A steel bar with a stiff rubber spacer, also linedwith a UHMW strip, was snug-tightened against the wall on each side usingbolts in the flanges of the base beam. A shorter steel bar was then placedacross these wall restraint bars at each end of the wall and welded to both thelonger bars and the flanges of the base beam, thus fixing the wall restraintbars in place relative to the wall base beam. The contact surfaces betweenthe two pieces of UHMW were coated with grease to reduce friction. This463.3. Experimental set-upTable 3.6: Spring test data (per strut assembly)Travel (mm) Spring rateSystem LocationDesignmaximum Tested toMean(kN/m) cvFlexible Top 460 430 18.5 0.034Bottom 19.7 0.031Stiff Top 190 180 73.5 0.008Bottom 71.2 0.009connection effectively restrained the lateral displacement of the base of thewall relative to the carriage, but allowed the base of the wall to rotate andlift up with minimal resistance.A steel channel assembly was placed on the top of the wall (Figure 3.15).The channel’s long axis was oriented parallel to the long edge of the wall, andthree plates were welded underneath perpendicular to the channel. Slottedholes were cut in each end of the plates, and heavy angles were bolted to theplates from below. The inside faces of the angles were lined with rubber toabsorb surface irregularities. The angles were clamped tight against the walland then the bolts were tightened, thereby securing the channel assembly tothe wall. A 64mm steel pin was inserted into a bored rectangular blockbolted to each end of this channel, with the pin protruding beyond the edgeof the wall by approximately 200mm.A steel plate with a milled vertical slot was bolted underneath the topcarriage on each side, as shown in Figure 3.15. The pins at the top of thewall travelled within the vertical slots on these carriage plates, allowing thetop of the wall free rotation and vertical displacement while restraining thelateral displacement of the top of the wall relative to the carriage. The slotwas machined approximately 0.03mm wider than the diameter of the pinsto allow for imperfections in assembly and to prevent binding of the system.The bearing interface was thoroughly greased before each test. To minimizethe risk of damage to the interface, the pins were machined from re-purposedhigh-strength machine shafts and the slotted plates were milled of ASTMA514 steel. No damage or alignment error was observed during the testing.No provisions were made in the test frame for the application of over-burden load to the wall beyond that imposed by the steel channel assemblybolted to the top of the wall (approximately 179 kg). The simulations insteadconcentrated on the worst-case stability conditions found in upper-storey473.3. Experimental set-up(a) prior to restraintinstallationUHMW STRIPON WALLBASE BEAMBOTTOMCARRIAGE(b) with restraint inplace; unsecuredSTEEL BARRUBBER SPACERUHMW STRIPON WALLUHMW STRIPON RESTRAINT(c) with restraint inplace and securedFLANGE OFBASE BEAMSTEEL BAR STEEL BARWELDSWELDSFigure 3.14: Detail of bottom connection483.3. Experimental set-upSLOTTEDPLATEPINCLAMPINGANGLETOPCARRIAGEFigure 3.15: Detail of top connection493.4. Instrumentation and data collectionwalls with minimal tributary gravity loads.Testing was carried out until collapse of the walls was achieved. In aneffort to minimize damage to the test frame components and the instrumen-tation, nylon webbing was installed on either side of the wall to catch thefalling wall segments after collapse (see Figure 3.9). The channel assemblythat was clamped to the top of the wall was tethered to the top carriage,also using nylon webbing. Timber cover assemblies were built for the bottomcarriage, the lower test frame, and the bottom spring assemblies to blockfalling debris. All of these measures were installed so as not to interfere withthe response of any of the components during the testing.3.4 Instrumentation and data collectionData were recorded from 33 instrumentation channels measuring the re-sponse of the shake table, test frame, carriages, and the wall specimen.Instrumentation consisted of displacement transducers and accelerometers.A summary of the instrumentation setup is provided in this section; moredetails can be found in Appendix D.3.4.1 Wall instrumentationOne displacement transducer and one accelerometer were mounted at eachheader course on the walls. IC Sensors (ICS) model 3028 piezoresistivesilicon accelerometers with an output range of ±10 g were used. The ac-celerometers were housed in steel boxes, which were screwed to plywoodmounts glued to the wall. String potentiometers were used to measure dis-placements, and consisted of Celesco models PT101 and 632036 units. Theinstruments were mounted on a purpose-built timber instrumentation framemounted on the east wall of the building, and the strings were attached tohooks mounted on the walls. These units were thus measuring total dis-placement of the wall (including the shake table displacement).In addition to the header-course instrumentation, a tri-directional ac-celerometer, consisting of three ICS model 3026 units (±5 g), was mountedon the steel channel assembly at the top of the wall. Two linear voltage po-tentiometers (Novotechnik model TR100) were installed at the base of thewall on the south side to measure uplift of the edges of the wall relative tothe bottom carriage. Two of the same units measured vertical displacementof the pins at the top of the wall relative to the top carriage.503.4. Instrumentation and data collectionThe potentiometers at the base of the wall were removed for high-intensity tests to avoid damage during collapse of the wall. All other in-strumentation remained in place and active for all runs.3.4.2 Shake table and test frame instrumentationShake table displacement was measured by an MTS Temposonics 6 analogtransducer (model LPRCVU03601). Shake table acceleration was measuredby an ICS model 3026 accelerometer (±10 g).Top and bottom carriage accelerations were measured by ICS model 3028accelerometers (±10 g). Carriage displacements were measured relative tothe test frame (not including shake table displacements) using Celesco modelSP1-50 string potentiometers.The response of the top of the test frame was recorded to compare withthe shake table response. An ICS 3028 accelerometer (±10 g) and a CelescoSP1-50 string potentiometer (mounted on the instrumentation frame) wereused.3.4.3 Data collection and processingA computer-based data acquisition system using a National Instrument 16-bit PCI 6052E multi-function board and an SCXI signal conditioning chassiswith a SCXI 1520 module was used. This module has a programmable filter,which was set as a (1 kHz) low pass, 4th order Butterworth filter, and wasused for all instrumentation. The software DasyLab was used to acquire,control and store the data. Data were sampled at 200Hz (∆t = 0.005 s).High-frequency noise was removed from all data using zero-phase digi-tal filtering with the built-in Matlab filtfilt function, using a 4th-orderButterworth filter. This function carries out two filtering passes — forwardsand backwards — which in this case produces effectively an 8th-order filter.Only low-pass filtering was applied, with the cutoff frequency set at 25Hz.3.4.4 Video recordingRegular video (1280×720 resolution at 30 fps) was recorded from four anglesduring dynamic testing. Wide angle shots from floor level and from anelevated position, plus close-up shots of the top and base of the wall weremade.In addition, high-speed video was recorded using a Phantom v4.2 camera.The combination of resolution, recording time and frame rate of which thecamera is capable are limited by its 2GB internal memory; 100 fps and513.5. Ground motionsthe maximum resolution of 512 × 512 were selected, providing a recordingwindow of roughly 80 s. The high-speed video was shot normal to the southend of the wall, lined up with the centre of the wall, from an elevation ofroughly 2m.3.5 Ground motionsTwo ground motions were used as input to the shake table, with one mo-tion selected for significant long-period spectral response and the other fora dominant short-period spectral response. The long-period motion se-lected (CHHC1) was recorded during the 22 February 2011 earthquake inChristchurch, New Zealand at the Christchurch Hospital. The short-periodmotion selected (NGA0763) was recorded during the 18 October 1989 LomaPrieta earthquake at the Gavilan College in Gilroy, California. Accelerationresponse spectra as recorded on the shake table are shown in Figure 3.16along with design spectra for Seattle, WA, USA and Victoria, BC, Canada.Displacement and acceleration time histories of the two motions as recordedon the shake table are shown in Figures 3.17 and 3.18, respectively. Scalefactors are shown relative to the original motion as recorded during theearthquake, and reference the amplitude of the displacement time history.It can be observed that the displacement control of the shake table resultsin significant response amplification at the natural frequency of the hydraulicsystem, producing a large response peak at a period of about 0.10 to 0.15seconds. The effect of this amplification may be notable for runs in whichthe carriages were ‘locked out’; however, for runs in which the carriageswere driven through the springs, this amplification was filtered out due tothe much longer natural period of the spring-carriage-wall system.523.5. Ground motions0 0.5 1.0 1.5 2.0 2.5 3.000.51.01.52.0CHHC1@100%NGA0763@60%Victoria, BCSeattle, WAPeriod (s)Spectralacceleration(g)Figure 3.16: Response spectra of recorded table motions (5% damped elas-tic)0 5 10 15 20 25 30−200−1000100200CHHC1@100%NGA0763@60%Time (s)Displacement(mm)Figure 3.17: Table displacement time history533.6. Shake table tests0 5 10 15 20 25 30−0.6−0.4−0.200.20.40.6NGA0763@60% CHHC1@100%Time (s)Acceleration(g)Figure 3.18: Table acceleration time history3.6 Shake table testsThe mortar used in the construction of the test walls (Type O) is of signif-icantly lower strength than that used in modern structural masonry. How-ever, in particular the flexural bond strength of walls found in early 1900sbuildings may be weaker still than that of the test walls. It was thereforedecided not to rely on the cracking resistance of the test walls in assessingtheir dynamic stability on the shake table, but rather to assume that thewalls would experience cracking at very low levels of excitation.Initial runs for each uncracked wall were made with the desired dia-phragm conditions and the CHHC1 motion, to verify correct operation ofthe apparatus. To ensure that walls would remain stable after crack ini-tiation, allowing further tests to be carried out, cracking was initiated byrunning the NGA0763 motion with both top and bottom carriages lockedout (rigid diaphragm conditions). After cracking was achieved, the carriageconnections were adjusted for the desired diaphragm conditions, and subse-quent runs were made using the CHHC1 motion at increasing amplitude ofinput motion until collapse was observed. Shake table tests are summarizedin Tables 3.7–3.11.543.6. Shake table testsTable 3.7: Test protocol - wall FF-3Diaphragm state Recorded table responseRun Motion Scale Top Bottom PGA (g) PGD (mm)1CHHC110%flexible flexible0.05 182 30% 0.19 573 50% 0.28 964 70% 0.42 1355 80% 0.52 1556 100% 0.64 1947 NGA0763 50% rigid rigid 0.23 298 60% 0.32 369CHHC130%flexible flexible0.18 5710 50% 0.28 9611 70% 0.44 13512 80% 0.49 15513 100% 0.64 194Table 3.8: Test protocol - wall FR-3Diaphragm state Recorded table responseRun Motion Scale Top Bottom PGA (g) PGD (mm)1 CHHC1 50% flexible rigid 0.43 962 70% 0.59 1353 NGA0763 60% rigid rigid 0.28 354CHHC150%flexible rigid0.29 965 70% 0.43 1356 80% 0.41 1557 90% 0.45 1748 100% 0.53 1949 110% 0.49 21310 120% 0.70 233553.6. Shake table testsTable 3.9: Test protocol - wall FF-2Diaphragm state Recorded table responseRun Motion Scale Top Bottom PGA (g) PGD (mm)1 CHHC1 50% flexible flexible 0.36 962 80% 0.54 1353 NGA0763 60% rigid rigid 0.26 354 70% 0.35 415CHHC150%flexible flexible0.31 966 70% 0.45 1357 80% 0.53 1558 90% 0.58 1749 100% 0.67 19410 110% 0.69 21411 120% 0.73 233Table 3.10: Test protocol - wall SS-3Diaphragm state Recorded table responseRun Motion Scale Top Bottom PGA (g) PGD (mm)1CHHC130%stiff stiff0.20 572 40% 0.22 763 50% 0.28 964 60% 0.33 1165 NGA0763 50% rigid rigid 0.22 296 60% 0.27 357CHHC130%stiff stiff0.18 578 50% 0.27 969 60% 0.34 11610 65% 0.36 12511 70% 0.42 13512 75% 0.47 14513 80% 0.45 155563.7. Performance of test apparatusTable 3.11: Test protocol - wall RR-3Diaphragm state Recorded table responseRun Motion Scale Top Bottom PGA (g) PGD (mm)1NGA076350%rigid rigid0.25 292 60% 0.28 353 70% 0.47 424CHHC150%rigid rigid0.43 965 60% 0.53 1166 55% 0.44 1067 65% 0.77 1253.7 Performance of test apparatusTwo aspects of the performance of the test frame are briefly examined inthis section: the lateral stiffness of the frame in the in-plane direction, andthe response of the spring-carriage systems.3.7.1 In-plane response of test frameAccelerations were measured on the shake table as well as at the top of thetest frame during all testing. Run six from wall FF-3 was used as a casestudy to examine the frame response. This run was with an uncracked wallrunning the CHHC1 motion at 100% scale. A power spectral density plotof the table acceleration is shown in Figure 3.19a. This plot shows thatthe energy content of the table response is concentrated at frequencies lowerthan roughly 12Hz.The transmissibility of the test frame can be defined as the ratio of theabsolute values of the Fourier transform of the acceleration at the top of thetest frame divided by that of the table acceleration. This transmissibilityfunction is plotted as a function of the frequency in Figure 3.19b. Here itcan be observed that in the range of significant energy content (f < 12Hz),the transmissibility is very close to 1. This indicates that the majority ofthe input motion energy content is transferred to the top of the frame withminimal amplification, meaning also that the natural frequency of the testframe under those test conditions is greater than 12Hz. A rudimentaryinterpretation of the transmissibility plot suggests that the fundamental573.7. Performance of test apparatus10−1010−810−610−4(a)PSD0 5 10 15 20 25 300510(b)Frequency (Hz)TransmissibilityFigure 3.19: Power spectral density of table acceleration583.7. Performance of test apparatusfrequency of the test frame is likely around 21Hz. While the transmissibilityat this frequency is high (> 10), the energy content of the input is very low.A two-second portion of the acceleration time histories at the top ofthe test frame and at the shake table is shown in Figure 3.20. This plotconfirms that there is negligible phase lag between the two locations, andthat amplification is minor. Consequently, the response of the test frameis expected to have minimal effect on the response of the wall, and it isreasonable to simplify the system as having equal top and bottom inputs.9 9.2 9.4 9.6 9.8 10 10.2 10.4 10.6 10.8 11−0.500.5 TableTop of frameTime (s)Acceleration(g)Figure 3.20: Time history of acceleration at table and top of test frame3.7.2 Response of springsThe static response of the springs as measured prior to installation on thetest frame was nearly perfectly linear. The response of the assembled wall-carriage-spring system in the test frame was also close to linear when excitedat large displacement amplitudes, but exhibited some ‘stiction’ at low lev-els of excitation. The response characteristics of the system are examinedusing the first six tests of wall FF-3. In these tests the wall was uncracked,which resulted in in-phase top and bottom carriage displacements; the en-tire system is thus analogous to an SDOF oscillator. Maximum carriagedisplacements from these runs are shown in Table 3.12 and Figure 3.21.The additional resistance in the system evident at low motion amplitudescould be attributed to several factors, including misalignment between eachpair of spring shafts connected to a carriage, misalignment between thespring shafts and the carriage rails, and/or rolling resistance of the casters593.7. Performance of test apparatuson the carriage. This non-linearity in the spring-carriage systems shouldbe of minimal importance, particularly for collapse and near-collapse tests,which occur at moderate to high amplitudes.Table 3.12: Peak carriage displacements, wall FF-3, uncrackedMaximum carriage displacement (mm)Test Motion scale Top Bottom Average1 10% 0.7 0.5 12 30% 18.2 17.0 183 50% 69.9 70.6 704 70% 130.5 129.3 1305 80% 170.9 172.0 1716 100% 246.5 249.8 2480 50 100 150 200 2500.00.20.40.60.81.0Displacement (mm)MotionscaleFigure 3.21: Peak carriage displacements, wall FF-3, uncracked. Referenceline drawn through 100% scale point60Chapter 4Shake Table Test Results4.1 IntroductionThis chapter presents the results of dynamic shake table testing. Generalvisual observations from wall tests are presented, followed by a detailedanalysis and discussion of numerical test data. Pertinent response historyresults for each run are provided in Appendix F.4.2 Visual observationsEach of the five wall specimens developed a single horizontal crack near mid-height during the tests in which the carriages were locked out (Figure 4.1;refer to Tables 3.7–3.11 for test sequence). In every case, the crack occurredat the brick-mortar interface, but the crack location varied between thespecimens. In wall RR-3, the crack was located at a height of 0.74 timesthe wall height, while in the other four specimens the crack height variedbetween 0.47 and 0.55 times the wall height. Cracks occurred both at headercourses and at common courses. In Walls FF-3, FR-3, and SS-3, the crackwas located in a single horizontal plane across the entire wall section. InWalls FF-2 and RR-3, the crack stepped down by one course. Even aftersustained rocking in later runs, all cracks consistently closed up withouthorizontal offset and with minimal spalling of mortar or brick.Prior to undergoing significant rigid-body rocking, cracks were visuallynearly imperceptible (for example see Figure 4.2). Walls FF-2 and RR-3 un-derwent the greatest amount of rigid body rocking, and correspondingly alsosustained the most spalling damage at the crack. The maximum amount ofspalling that was observed prior to collapse is shown in Figure 4.3. The otherwall specimens exhibited less damage than shown here. Additional photosshowing the cracking patterns in each wall are included in Appendix E.The extent of spalling at the base of the wall could not be observed dueto the configuration of the base restraint. Every wall underwent rockingabout the [wall]-[base beam] interface; no cracking was observed within the614.2. Visual observationsFigure 4.1: Typicalcrack configurationFigure 4.2: Typicaldetail of fresh crackFigure 4.3:Maximum observedspalling624.2. Visual observationswall near the base. A video frame of wall FF-3 in the process of collapse(Figure 4.4) illustrates how the lower section of the wall remained intact andthe wall was able to rock unrestrained as intended.At sufficiently high input motion intensities, each wall specimen even-tually underwent rigid-body rocking. Since only a single crack was formedin each wall, the rocking was clearly defined as occurring between two bod-ies: the lower wall block, from the base beam to the horizontal crack, andthe upper wall block, from the crack to the top pin connection. This isillustrated in a video frame of wall FR-3 just before collapse (Figure 4.5).634.2. Visual observationsFigure 4.4:Lower wallsection atcollapseFigure 4.5: Wallrocking beforecollapse644.3. Numerical results4.3 Numerical resultsThis section presents recorded and derived data obtained from the shake-table testing. Response of the cracked walls are presented first, includingrocking behaviour, wall stability limits, carriage response, and force de-mands. Cracking response of the walls is presented subsequently in Sec-tion 4.3.6.4.3.1 Cracked response summaryThe crack height, system period, and motion scales at collapse and in therun prior to collapse for each specimen are listed in Table 4.1. The systemperiod is defined independently for the top and bottom of each wall, andshould be interpreted as an indicator of the stiffness of each support ratherthan a true period of response, since in a multi-degree-of-freedom (MDOF)system like this one a period can only be defined for a modal response, andthe modes in this sytem link both the top and bottom responses. In additionthe height of the crack affects the distribution of tributary mass to the topand bottom diaphragms, and once the wall is cracked it no longer exhibitsa periodic response.For consistency and simplicity, each of these periods is calculated usingexactly half the total wall mass plus the mass of the respective carriagesystem; it is thus independent of the crack height. The wall height used tonormalize the crack height is taken as the distance from the base of the wallto the centerline of the top pin.Table 4.1: Results summaryPeriod (sec) Motion scaleWallNormalizedcrack height[a] Top BottomHigheststable runCollapserunFF-3 0.47 1.59 1.59 80% 100%FR-3 0.55 1.59 0 110% 120%FF-2 0.49 1.24 1.24 110% 120%SS-3 0.51 0.83 0.83 75% 80%RR-3 0.74 0 0 65% 65%[a] Normalized with respect to wall height654.3. Numerical results4.3.2 Displacement responsePrior to cracking, the wall behaves as a single rigid body. The out-of-plane elastic stiffness of the wall at this point is high, and the limitations ofthe string pot instrumentation setup prevented reliable measurement of theelastic bending displacements (the instruments could not resolve beyond a2–3mm precision). Consequently, no further analysis regarding the elasticbehaviour of the wall was included here.Once cracked, the wall behaves as two rigid bodies. In the test specimens,the combination of surface friction and potential interlock effect due to a non-planar crack surface effectively prevented any notable sliding at the crackinterface. For analysis purposes, the two wall segments were thus assumedto be linked at the crack location. The relative displacement of the wall wasassumed to be equal to that of the carriage at the base and the top of thewall, which is consistent with the observed performance of the wall-carriageconnections during testing.Displacements of the carriages were measured relative to the table, whiledisplacements of the wall headers were measured relative to the ground (ab-solute). The nomenclature is illustrated in Figure 4.6. To obtain relativedisplacements of the wall, drel, the table displacement was subtracted fromthe measured wall displacements. Relative displacements are zeroed at theinitial at-rest positions of the carriages. Top and bottom carriage displace-ments, drelt and drelb , remain as measured — relative to the shake table.The differential carriage displacement, ddiff , is the difference between dreltand drelb . Rocking displacement, drock, is defined as the difference betweenthe measured horizontal displacement of the wall at the crack height andthe straight-line interpolation between the top and bottom of the wall atthe same height. The normalized rocking displacement, drocknorm , is simplythe rocking displacement divided by the wall thickness.A typical displacement profile of a cracked wall with flexible diaphragmconditions is shown in Figure 4.7a. The displacement at the crack wasestimated as the mean of the straight-line extrapolated values from the topand bottom wall segments. The lines used in this process were defined as theleast-squares fit subject to being forced through the carriage displacementvalue, since the carriage displacements were subject to less measurementnoise than the wall displacements. This method minimizes the effect oferrors for any single string pot measurement. Slopes of the segments werecalculated from these best-fit lines. The corresponding rocking displacementprofile is shown in Figure 4.7b.664.3. Numerical resultsdrelbdrockdrelddiffdreltθtθbFigure 4.6: Displacement nomenclature−100 −50 001234drel (mm)Height(m)Measured Calculated(a) Relative displacement−100 −50 001234drock (mm)Height(m)Measured Calculated(b) Rocking displacementFigure 4.7: Typical displacement profiles of cracked wall674.3. Numerical results4.3.2.1 Static stability limitsIt is convenient to describe a stability limit to aid in understanding how‘near to collapse’ a wall specimen comes in a shake table run. The stabilityof a single rigid rocking block depends on the time variation of the forcebalance and the body’s position. For a given ground motion, stability is abinary outcome of the whole run that can only be determined at the endof the run: either the block collapsed, or it did not. It is thus not possibleto determine the actual stability at a given timestep, since whether or notthe block will reach a collapsed state depends on the details of the appliedmotion following that timestep.For example, a rocking block may have reached a significant rotation atsome particular instant during a ground motion. Proceeding forwards fromthis time, a ground motion pulse might follow that would push the blocktowards the upright position, stabilizing it. Alternatively, a pulse in theopposite direction might follow, which would result in the block fully col-lapsing. At the original time instant considered, therefore, it is not possibleto directly quantify the stability of the block. This is different from, say, aductile column being subjected to a ground motion, where at any instantone can objectively calculate that the column is at x % of yield strength —regardless of the time variation of the ground motion after that instant.It is possible, however, to define a static stability limit for a body thatcan be used as an indicator of approximately how ‘near to collapse’ a bodycomes. Without dynamic effects (horizontal reactions at contact points andinertial forces), the stability can be described in terms of whether the forcebalance pushes the body towards a stable (in the case of a wall — vertical) oran unstable (tipped over) position. At any time step in a dynamic scenario,a body’s force and position conditions can be compared to this static limit.While this criterion will not accurately predict the stability outcome of adynamic run, it can provide a good idea of whether the body was on theverge of collapsing or was relatively stable.Two rocking mode shapes are possible for a wall with a single crack andat least one flexible diaphragm (Figure 4.8). The concepts here have beenadapted from Derakhshan et al. [2014]. Assuming that the wall-diaphragmanchorage remains intact and that slip at the crack interface is negligible,the top wall segment is unconditionally stable and collapse of the wall canbe determined by assessing the stability of the bottom segment. The upperportion of the figure shows the whole wall section, including dynamic effects,while the lower portion shows just the bottom wall segment simplified to thestatic conditions. Under static conditions, no inertial load is acting on the684.3. Numerical resultswall, and it is assumed that the horizontal force transferred between thetwo wall segments at the crack interface and the horizontal force impartedby the base restraint are both negligible. At the peaks of large rockingexcursions (near collapse, or at collapse initiation), accelerations at the crackare relatively small and the static simplification may not be too far fromreality (e.g., Figure 4.35).hthbtθtθbθbWbWbWtWt + PPFtFbFvFvdbdbθtθbθbWbWbWtWt + PPFtFbFvFvdbdbFIbFItFIbFItCase 1 Case 2Bottom segment:(static conditions)Figure 4.8: Stability criteriaIn Case 1, θtθb < 1, and the points of contact at the crack and at the baseof the wall are on opposite sides of the bottom wall segment. The weight ofthe top wall segment (Wt) and the overburden load (P ) both act as restoringforces against overturning of the bottom segment while θb remains below theinstability threshold.694.3. Numerical resultsIn Case 2, θtθb > 1, and the points of contact at the crack and at thebase of the wall are on the same side of the bottom wall segment. Here, Wtand P both act as destabilizing forces for any value of θb, and consequentlythe instability threshold rotation is significantly smaller for Case 2 than forCase 1.For both cases, the static instability threshold rotation can be calculatedby a taking the sum of the moments acting on the lower wall segment aboutthe point of contact at the base. It should be noted that in this model,the eccentricity of the overburden load does not affect the static instabilityrotation of the bottom segment — but it does still affect the dynamics ofthe wall behaviour, which determine when and how the conditions nearinstability are reached. The instability threshold was calculated in terms ofboth rotations and displacements for each wall specimen (Table 4.2). Smallangle approximations were used in these calculations; for angles up to 15◦,the maximum resulting error is roughly 2%.Table 4.2: Stability limitsCase 1 Case 2θb1 db1 (mm) θb2 db2 (mm)thbθb1 ·hb = t t2hb ·Wb12Wb + (Wt + P ) θb2 ·hb < tFF-3 9.1◦ 291 2.6◦ 83FR-3 7.6◦ 291 2.7◦ 103FF-2 8.0◦ 191 2.3◦ 54SS-3 8.5◦ 300 2.7◦ 96RR-3 5.8◦ 296 3.1◦ 160The instability rotations for generic wall configurations with no overbur-den load are shown in Figure 4.9. The instability rotations decrease withincreasing slenderness ratio. The instability conditions are also plotted asnormalized displacements (displacement of bottom block at crack divided bywall thickness), for varying levels of overburden (Figure 4.10). Plotting thenormalized displacement results in the same lines for all slenderness ratios.The overburden load in this plot is indicated as normalized to the total wallweight, W . Note that while the overburden load does not affect the Case1 static stability threshold. The addition of overburden loads significantlyreduce the stability thresholds for Case 2.704.3. Numerical results0 0.2 0.4 0.6 0.8 104812h/t = 8h/t = 8h/t = 14h/t = 14h/t = 20h/t = 20Typical crackheightsCase 1Case 2Normalized crack heightθ ins(◦)Figure 4.9: Instability rotations, without overburden load0 0.2 0.4 0.6 0.8 100.20.40.60.81.0 Case 1P/W= 0P/W= 0.2P/W= 0.5P/W = 1.0Typicalcrack heightsCase 2Normalized crack heightd ins/tFigure 4.10: Normalized instability displacements, varying overburden load714.3. Numerical results4.3.2.2 Time history responseDisplacement time history results of run 10 of wall FF-2 are shown in Fig-ure 4.11. This run illustrates sustained rocking behaviour. Three timeinstants of interest are indicated by the dotted vertical lines. The dis-placement profiles of the wall at these instants are shown in Figure 4.12.Figures 4.11a–4.11c simply show the previously-described displacement pa-rameters. Figure 4.11d shows the processed bottom rotation, θp, which isdefined as:θp = |θb| ·0, if |θt − θb| < θthreshold — no rocking−1, if θtθb < 1 — Case 1 rocking1, if θtθb > 1 — Case 2 rocking(4.1)When rocking is negligible (θthreshold = 0.1◦ was used to accomodateerrors in the calculated top and bottom rotations), the processed rotationis zero. Otherwise, the sign indicates the type of rocking (Case 1 or 2)as opposed to the direction of rotation. Sudden changes in sign thereforeresult when the top block is rotating relative to the bottom block and thepoint of contact at the crack switches sides (an impact occurs at this time).The static instability rotations (see Table 4.2) for both rocking cases aresuperimposed on this plot, providing a subjective idea of how near to collapsethe wall is.The instability ratio, Rins, provides a normalized representation of theprocessed rotation, where the instability limit is reached at either −1 (Case1 rocking) or 1 (Case 2 rocking). It is shown in Figure 4.11e, and is definedas:Rins =0, if |θt − θb| < θthreshold — no rockingθpθb1, if θtθb < 1 — Case 1 rockingθpθb2, if θtθb > 1 — Case 2 rocking(4.2)It can be observed that the rocking displacement becomes zero when thetop and bottom rotations are equal, at which point the relative displacementat the crack falls between the top and bottom relative displacements. Thelargest rocking displacement occurs concurrently with the largest differencein rotations (e.g., at time C, Figure 4.12c). However, the processed bottomrotation, and thus the instability ratio, do not necessarily peak at thesesame instances. The instability ratio is directly related to the bottom blockrotation; it can reach large values even when rocking is minimal (e.g., at time724.3. Numerical resultsB, Figure 4.12b). The top block needs to rotate only slightly relative to thebottom block to place all its weight on one edge of the bottom block, andthus produce one of the rocking cases of Figure 4.8. This slight differencein rotation can arise while both blocks were already at a significant rotation(along with differential diaphragm displacement). Conversely, large rockingdisplacements can occur while the instability ratio is small, if the bottomblock is near vertical while the top block experiences a large rotation (e.g.,at time A, Figure 4.12a).The normalized rocking displacement is plotted for two runs for eachwall in Figure 4.13: the highest stable run, and the run causing collapse.Significant differences are apparent among the different specimens. WallRR-3 exhibited almost no rocking, even in the collapse run. Wall FF-3showed limited rocking in the highest stable run, but this could be partiallydue to the large difference in scale between the last two runs for this wall(80% → 100%). Wall RR-3 underwent a long series of gradually decayingrocking excursions in the highest stable run, something not seen in any ofthe specimens with flexible diaphragms.The instability ratio is plotted in Figure 4.14 for the same two runsfor each wall (highest stable and collapse). As expected, wall RR-3 (Fig-ure 4.14e) exhibits only Case 1 rocking (negative Rins values). Case 2 rock-ing requires some degree of diaphragm flexibility, which was not present inthis configuration. The next-stiffest configuration, wall SS-3 (Figure 4.14d),showed a few instances of Case 2 rocking, but of short duration and small am-plitude. The walls with the more flexible diaphragms showed more frequentinstances of Case 2 rocking with longer durations and larger amplitudes. Ofthese, wall FR-3 (Figure 4.14b) showed the largest amplitudes in the Case2 direction, but these peaks were typically abruptly terminated, indicatingthat the amplitude of rocking was very minor and the crack kept closing up,all while the rotations of both blocks were large due to the large differentialdiaphragm displacements.In each of the five wall configurations, collapse occurred as Case 1 rocking(negativeRins), despite significant Case 2 rocking sometimes occurring in theprevious run or earlier in the collapse run. Unless the diaphragm stiffnessis extremely soft — much softer than in any of the tests — collapse willalways occur as Case 1 rocking. Instances of Case 2 rocking can contributemomentum to the bottom block that will push it towards collapse, but thetop block will always flip back into the Case 1 direction before an actualcollapse will occur (e.g., the collapse sequence of wall FR-3, to a minordegree). To accommodate a Case 2 collapse, the differential diaphragmdisplacement would need to be exceedingly large. Such a condition may be734.3. Numerical results-2000200(a)d rel(mm)Top Bottom Crack-1000100(b)d rock(mm)Crack-404(c)θ(◦)Top Bottom−8−40A B CCase 2Case 1(d)θ p(◦)Bottom (processed) Stability limits6 8 10 12 14 16 18-101Case 2Case 1(e)Time (s)RinsInstability ratio Stability limitsFigure 4.11: Displacement time histories, wall FF-2, run 10744.3. Numerical results−100 −50 00123drel (mm)Height(m)Measured Calculated(a) Time A−100 −50 0drel (mm)(b) Time B−150−100 −50 0drel (mm)(c) Time CFigure 4.12: Displacement profiles, wall FF-2, run 10possible for some building configurations (e.g., a long, narrow, unretrofitteddiaphragm), but for most practical purposes collapse can be assumed tooccur as Case 1 rocking.754.3. Numerical results-101(a)FF-3d rock normHighest stable run Collapse run-101(b)FR-3d rock norm-101(c)FF-2d rock norm-101(d)SS-3d rock norm6 8 10 12 14 16 18-101(e)Time (s)RR-3d rock normFigure 4.13: Rocking time histories, highest stable and collapse runs764.3. Numerical results-101Case 2Case 1(a)FF-3RinsHighest stable run Collapse run Stability limits-101Case 2Case 1(b)FR-3Rins-101Case 2Case 1(c)FF-2Rins-101Case 2Case 1(d)SS-3Rins6 8 10 12 14 16 18-101Case 2Case 1(e)Time (s)RR-3RinsFigure 4.14: Instability ratio time histories, highest stable and collapse runs774.3. Numerical results4.3.2.3 Peak responseFor each specimen, peak values of the displacement parameters observedin the most intense stable run are listed in Table 4.3. The peak rockingdisplacements occur at the crack height. Peak values for the collapse runare listed in Table 4.4. None of the peak carriage displacement values occuras a result of the wall collapsing — they occur before collapse in every case.Table 4.3: Peak displacement response in highest stable rundrelt(mm)drelb(mm)ddiff(mm)drock [a](mm)drocknorm [a](—)θt[a](◦)θb[a](◦)FF-3 157 180 73 46 0.16 2.3 0.4FR-3 206 0 206 14 0.05 2.8 3.2FF-2 102 142 74 89 0.47 4.7 2.4SS-3 61 91 52 182 0.61 5.3 5.0RR-3 0 0 0 132 0.45 6.7 2.6[a] Values calculated based on linear fit of measured displacements on each wallsegmentTable 4.4: Peak displacement response in collapse rundrelt(mm)drelb(mm)ddiff(mm)FF-3 182 273 144FR-3 227 0 227FF-2 130 169 110SS-3 65 84 48RR-3 0 0 0In Figures 4.15–4.20, peak displacement parameters are plotted againstthe intensity of the ground motion in each run. Peak values of some ofthese parameters can not be defined when the wall collapses, and arbitraryresponse values have been selected to indicate collapse in these cases (asnoted on the plots).Significant rocking without collapse was observed in four of the five spec-imens (Figure 4.15). Wall FR-3, with rigid bottom diaphragm condition,underwent limited rocking in all runs prior to the collapse run despite largedisplacements of the top carriage. Of the remaining walls, FF-3 displayed784.3. Numerical resultsthe least rocking in the run prior to collapse, but presumably this is in largepart because the change in intensity was larger between the last two runsfor this wall than for the others (20% vs. 5-10%).0 0.2 0.4 0.6 0.8 1.0020%40%60%80%100%120%00.20.40.60.81.0FR-3FF-2FF-3SS-3RR-3CollapsedrocknormMotionscaleS a(1.0)(g)Figure 4.15: Peak rocking displacementsInstability ratios reached in each run are shown in Figure 4.16. Asdiscussed previously (page 73), collapse is reached in the Case 1 directionfor each specimen. Significant ratios in the Case 2 direction were reachedby walls FR-3 and FF-2, and small ratios by walls FF-3 and SS-3. WallSS-3 showed decreasing Case 2 ratios in the highest two runs, with thelargest Case 2 ratio reached in the third-highest run. All of these ratioswere relatively small, however, and differences such as these can easily arisedue to the chaotic nature of the wall’s rocking response. Recall that fora given instability ratio, θb in Case 1 will be significantly larger than inCase 2. Correspondingly, it can be noted that frequently Case 2 ratios ina particular run are larger than those for Case 1. Case 2 ratios are largestfor wall FR-3 by a significant margin, which is due to the nature of theflexible/rigid diaphragm configuration forcing large differential diaphragmdisplacements, leading to large rotations of the bottom wall segment. Anysmall amount of rocking at this point can produce a large instability ratioof either case.Peak displacements of top and bottom carriages are shown in Figures 4.17and 4.18, and differential displacements in Figure 4.19. Peak top and bot-794.3. Numerical results-1.0 -0.5 0 0.5 1.0020%40%60%80%100%120%00.20.40.60.81.0FR-3FF-2FF-3SS-3RR-3Stability limit Stability limitCase 2Case 1RinsMotionscaleS a(1.0)(g)Figure 4.16: Peak instability ratiotom carriage displacements follow roughly linear trends with respect to themotion intensity, but the extrapolated fit lines would intersect the intensityscale at a value greater than zero. This can be largely attributed to the non-linear resistance discussed in Section 3.7.2, though some additional effectsare likely due to the interaction of the rocking wall with the diaphragm.Peak differential carriage displacements are significantly larger than the dif-ference between peak top and bottom displacements, indicating that themotion of the carriages consistently ends up out of phase, and thus thepeak differential displacement does not necessarily coincide with the peaksof either the top or the bottom.Peak angles of rotation of the top and bottom wall segments are shownin Figure 4.20. At a given instant for a particular wall, the distributionbetween top and bottom angles depends on the top and bottom diaphragmdisplacements and the rocking displacement, and additionally on the relativecrack height. A crack higher on the wall will produce larger rotations in thetop segment and smaller rotations in the bottom segment (e.g., wall RR-3,with a crack at 0.74h). Walls FF-3 and FF-2 also showed larger rotationsin the top segment than in the bottom one, despite the cracks in thesewalls being near mid-height. A contributing factor may have been somedegree of rotational restraint imparted by the bottom connection, althoughthe connection was designed to minimize any such restraint. Wall SS-3804.3. Numerical results0 50 100 150 200 250020%40%60%80%100%120%00.20.40.60.81.0FR-3FF-2FF-3SS-3RR-3dt (mm)MotionscaleS a(1.0)(g)Figure 4.17: Peak top carriage displacements0 50 100 150 200 250020%40%60%80%100%120%00.20.40.60.81.0FR-3FF-2FF-3SS-3RR-3db (mm)MotionscaleS a(1.0)(g)Figure 4.18: Peak bottom carriage displacements814.3. Numerical results0 50 100 150 200 250020%40%60%80%100%120%00.20.40.60.81.0FR-3FF-2FF-3SS-3RR-3ddiff (mm)MotionscaleS a(1.0)(g)Figure 4.19: Peak differential carriage displacements10 8 6 4 2 0 2 4 6 8 10020%40%60%80%100%120%00.20.40.60.81.0FR-3FF-2FF-3SS-3RR-3CollapseCollapseBottom segmentTop segmentθ (◦ from vertical)MotionscaleS a(1.0)(g)Figure 4.20: Peak wall segment angles824.3. Numerical resultsproduced fairly similar peak rotations in top and bottom segments, andwall FR-3 produced bottom rotations that were slightly, but consistently,larger than top rotations. Both of these walls were also cracked near mid-height. The variability in these trends among the different walls suggeststhat the bottom connection rotational restraint was likely not an importanteffect, but rather that the characteristics of the motion, the exact crackheight, and the diaphragm response were the primary factors of influence.The rocking displacement at the crack is compared to the top carriagedisplacement in Figure 4.21. The spectral displacement at 1 s is indicatedon the y-axis on the right side. The rocking displacements are shown by theheavy lines, while the top carriage displacements are shown by the lighterweight lines of the corresponding colours. This plot illustrates that whilethe carriage (i.e. diaphragm) displacements roughly follow the spectral dis-placement linearly, the rocking response is fundamentally different.0 50 100 150 200 250 300020%40%60%80%100%120%050100150200FR-3FF-2 FF-3SS-3RR-3drock, drelt (mm)MotionscaleS d(1.0)(mm)drock dtFigure 4.21: Peak top carriage and rocking displacements vs. Sd4.3.2.4 Rocking periodMakris and Konstantinidis [2003] showed that the rocking response of a sim-ple rectangular block subjected to ground shaking can not be characterizedby a single degree of freedom system with a fixed period. Griffith et al. [2004]834.3. Numerical resultsfurther determined that the rocking frequency of a cracked wall is displace-ment dependent. Neither Griffith et al. nor Makris and Konstantinidis hadconsidered the effect of flexible supports in their work.The ‘period’ of the rocking response of the walls in the current testswas evaluated by considering each rocking excursion separately. One rock-ing excursion was defined as the response between adjacent points at whichdrock = 0. The period of an excursion was defined as two times the du-ration of the excursion, with one excursion approximating a half-cycle sinepulse. The amplitude of the excursion was recorded as the peak rockingdisplacement. Amplitudes of less than 5mm were eliminated due to thelimited precision of the measurements from which the rocking displacementwas derived. The highest stable run from each wall was considered, with theexception of wall RR-3, due the lack of rocking observed. The observed peri-ods are plotted against the corresponding normalized rocking displacementsin Figure 4.22.0 0.2 0.4 0.600.51.01.52.02.53.0drocknormT rock(s)FF-2 FF-3 SS-3 RR-3Figure 4.22: Rocking periodThe rocking response of all walls follows generally the same trend: theminimum observed period increases as the rocking amplitude increases. Fora given rocking amplitude (at moderate values), wall RR-3 exhibits theoverall shortest rocking periods, while the walls with flexible diaphragmsexhibit generally longer periods. Wall RR-3 also produces the most consis-844.3. Numerical resultstently sine-like rocking oscillations; the shape of the rocking excursions ofwalls with flexible diphragms are more irregular, and one or more directionreversal cycles may be contained within a single rocking excursion. Thiscorresponds with the increased scatter present in Figure 4.22 for these wallsrelative to wall RR-3.In general, one can conclude that at smaller rocking amplitudes, themore flexible diaphragms have the capacity to allow longer rocking periods,but also create greater variability in the rocking period. As the rockingexcursions become larger, the effect of diaphragm flexibility becomes lessimportant, implying that the period at these points becomes more a func-tion of the wall characteristics rather than of the support conditions. Theapproximate shortest observed periods are listed in Table 4.5.Table 4.5: Shortest observed rocking periodsdrocknorm Trock (s)0.1 0.50.2 0.80.3 1.04.3.3 Acceleration responseAn uncracked wall, idealized as rigid, undergoing dynamic excitation willexhibit a linear acceleration profile. If the excitations at the top and bottomof the wall happen to be in-phase and equal, the wall will exhibit a uniformacceleration profile. However, such idealized conditions can not be producedin a real structure (nor in a real test apparatus), and the difference in topand bottom excitations will produce a linearly-varying profile. Under theseconditions, the acceleration profile is uniform only for brief instances as theslope of the linear profile changes sign.Since a real wall has a finite elastic stiffness, the acceleration will deviatefrom this idealized linear profile and will instead assume a curved profile ap-proximating a quadratic shape. Uncracked walls in this study that were runwith flexible diaphragms — producing gentle direction reversals and thusminimal impact effects at the top pin connection — exhibited only smallcurvatures of the acceleration profile. With rigid diaphragm conditions, themore sudden direction reversals produced more pronounced profile curva-tures (see Section 4.3.6).854.3. Numerical results0 0.1 0.2 0.301234a (g)Height(m)Measured Calculated(a) Uncracked−1 −0.5 0a (g)(b) Cracked−0.5 0 0.5a (g)(c) Cracked, with dis-continuityFigure 4.23: Typical acceleration profiles of cracked wallOnce the wall is cracked, the acceleration profile becomes triangular,with local peaks at the crack height and at the top and bottom of the wall.The free boundary condition created by the crack prevents the formationof significant bending moments in the wall. Each segment of the crackedacceleration profile is thus very close to linear.Typical acceleration profiles of cracked and uncracked walls with flexiblediaphragm conditions are shown in Figures 4.23a and 4.23b. The accelera-tion at the crack was estimated as the mean of the straight-line extrapolatedvalues from the top and bottom wall segments. The lines used in this pro-cess were defined as the least-squares fit through the wall sensors only. Thecarriage accelerations are shown on the profile plots, but were not consid-ered in the line fitting since accelerations were not transferred continuouslythrough the wall-carriage connections. In the top connection, the tolerancebetween the pin and the slot allowed for a small amount of ‘slop’, while inthe bottom connection, the rubber spacers provided some flexibility. Con-sequently, the accelerations extrapolated from the wall sensors to the topor bottom of the wall are often significantly different than the accelerationsmeasured on the carriages. It is also common for the acceleration profile toexhibit a discontinuity at the crack height. Figure 4.23c shows an exampleprofile with discontinuities at all three locations, while within each of thewall segments, the measured accelerations are nearly perfectly linear.864.3. Numerical results4.3.3.1 Time history responseAcceleration time history results of run 10 of wall SS-3 are shown in Fig-ure 4.24, and the shaded time is shown in greater detail in Figure 4.25. Thisrun illustrates a large rocking excursion and the associated impact when thecrack closes up. Three time instants of interest are indicated by the verticaldashed lines, with the acceleration profiles at these instants shown in Fig-ure 4.26. Figures 4.24a and 4.24b show the relative displacements and therocking displacement, respectively, while Figure 4.24c shows the accelera-tions of the top and bottom carriages and the calculated mean accelerationat the crack.In general, the acceleration time history during rocking is characterizedby periods of relatively smoothly varying accelerations during rocking ex-cursions followed by periods of rapid variation following the impacts causedby the crack closing up. Peak accelerations at the crack and at the carriagesoccur at impact times.At the peak of the large rocking excursion (time A, Figure 4.25), rel-ative velocities of the carriages and of the crack are small, and the crackacceleration is in the opposite direction of the carriage accelerations — theyare accelerating towards closing up the crack (Figure 4.26a). The acceler-ations slowly converge as the crack closes up, until roughly 0.015 s beforethe impact point (time B, and Figure 4.26b), at which point the acceler-ation profile is fairly uniform. The crack is now moving in the negativedirection with considerable velocity, while the carriages are moving slightlyin the positive direction. As the crack now closes, the wall at crack heightquickly picks up positive acceleration (slowing its negative travel) while thecarriages pick up negative acceleration. Peak acceleration is reached roughly0.015 s after the crack has closed (time C, and Figure 4.26c). The rockingresponse of the wall has therefore dragged the carriages ‘along for the ride’,demonstrating that significant two-way interaction can occur between wallsand diaphragms under the right conditions.In this case, the carriages were lighter than the wall, facilitating thistwo-way interaction. If the carriages had been much heavier than the wall,the carriage motions would have approached that predicted for SDOF oscil-lators, and the wall would have been ‘along for the ride’ instead. It is likelythat in this case, there would have been a larger rocking excursion followingthe main excursion, as the cracked wall would snap through between thecarriages while they would carry on moving in their original (opposite) di-rection. One might intuitively expect that this characteristic would reducethe overall stability of the system.874.3. Numerical results-2000200(a)d rel(mm)Top Bottom Crack−2000200(b)d rock(mm)Crack6 7 8 9 10 11 12 13 14−101(c)Time (s)a(g)Top carriage Bottom carriage CrackFigure 4.24: Acceleration and displacement time histories, wall SS-3, run12884.3. Numerical results−2000200(a)d rel(mm)Top Bottom Crack−2000200A B C(b)d rock(mm)Crack10.8 11 11.2 11.4 11.6 11.8 12−101(c)Time (s)a(g)Top carriage Bottom carriage CrackFigure 4.25: Close-up of acceleration and displacement time histories, wallSS-3, run 12894.3. Numerical results−1 0 101234a (g)Height(m)Measured Calculated(a) Time A, at maxi-mum rocking−1 0 1a (g)(b) Time B, prior toimpact−1 0 1a (g)(c) Time C, after im-pactFigure 4.26: Acceleration profiles, wall SS-3, run 124.3.3.2 Peak responseFor each specimen, peak values of the accelerations observed in the mostintense stable run are listed in Table 4.6. Peak values for the collapse runare listed in Table 4.7; these values do not include the impacts caused bycollapse of the wall.Table 4.6: Peak acceleration response in highest stable run (g)Bottom Crack TopCarriage Wall[a] Bottom[a] Mean[a] Top[a] Carriage Wall[a]FF-3 0.57 0.56 0.55 0.56 0.58 0.34 0.33FR-3 0.52 0.55 0.42 0.42 0.42 0.56 0.66FF-2 0.57 0.56 0.93 0.96 0.99 0.38 0.42SS-3 1.21 1.33 1.35 1.35 1.35 1.10 0.78RR-3 0.54 0.57 0.48 0.46 0.46 0.73 0.77[a] Values calculated based on linear fit of measured accelerations on each wall segmentPeak accelerations measured on the carriages for each run are shown inFigure 4.27. Peak accelerations calculated for the top and base of the wallbased on a linear fit to the measured accelerations on the wall are shown904.3. Numerical resultsTable 4.7: Peak acceleration response in collapse run (g)Bottom Crack TopCarriage Wall[a] Bottom[a] Mean[a] Top[a] Carriage Wall[a]FF-3 1.33 1.04 1.71 1.79 1.87 0.79 0.57FR-3 0.65 0.67 0.58 0.55 0.52 0.60 0.74FF-2 0.52 0.53 0.95 0.95 0.94 0.49 0.49SS-3 1.01 0.99 1.92 1.80 1.67 0.93 0.90RR-3 0.66 0.70 0.54 0.51 0.56 0.86 0.82[a] Values calculated based on linear fit of measured accelerations on each wall segmentin Figure 4.28. The ratios between peak top and bottom accelerations forboth the wall and carriage values are shown in Figure 4.29. The peak meanaccelerations calculated at the crack height are shown in Figure 4.30.1.0 0.5 0 0.5 1.0020%40%60%80%100%120%00.20.40.60.81.0FR-3 FF-2FF-3SS-3RR-3TopBottoma (g)MotionscaleS a(1.0)(g)Figure 4.27: Peak carriage accelerationsIn Figures 4.27 and 4.28, it is evident that some specimens producedmonotonic increases of peak accelerations with increasing motion intensity.However, in several specimens, at least one of the top or bottom accelerationsproduced a reversal of the direction near the collapse run. Recalling thatpeak accelerations are produced upon impact after rocking excursions, these914.3. Numerical results1.0 0.5 0 0.5 1.0020%40%60%80%100%120%00.20.40.60.81.0FR-3 FF-2FF-3SS-3RR-3TopBottoma (g)MotionscaleS a(1.0)(g)Figure 4.28: Peak wall accelerations at top and bottom (extrapolated)trend reversals would be expected for some scenarios. For wall SS-3, forexample, the peak acceleration in run 12 (the highest stable run) is producedby the impact shown in Figure 4.25. In run 13 (the collapse run), this samerocking excursion becomes large enough to continue into collapse, withoutproducing that particular impact, thus resulting in a lower peak accelerationfor the whole run. In other specimens, collapse can result from new rockingexcursions (e.g., a larger follow-through excursion), and the accelerationsproduced by particular large impacts can remain present in the collapserun.Wall FF-2 produced the lowest peak accelerations despite undergoingsignificant rocking. This is likely in large part due to the significantly lowerwall mass (less than 50% of the mass of the 3-wythe walls), meaning theresponse of the system was more heavily dominated by the mass of thecarriages. Furthermore, the combination of the reduced mass and thick-ness would have resulted in lower impact energy from a given magnitude ofnormalized rocking.Wall FR-3 produced larger peak accelerations at low motion intensitiesthan FF-3, but the peak accelerations of FR-3 increased only moderatelywith increasing motion intensity, as would be expected given the very mini-mal rocking response of this wall in all runs. In contrast, wall FF-3 producedlarge increases in the runs where rocking became significant. The larger ini-924.3. Numerical resultstial peak accelerations of FR-3 could be due to the change in effective periodof the rigid-base system vs. the fully flexible system, in combination with apossibly higher response at the base due to the rigid connection.1.5 1.0 0.5 0 0.5 1.0 1.5020%40%60%80%100%120%00.20.40.60.81.0FR-3 FF-2FF-3SS-3RR-3WallCarriageat/abMotionscaleS a(1.0)(g)Figure 4.29: Ratio of peak top to bottom accelerationsThe ratios of the peak top to peak bottom accelerations varied signifi-cantly among the various specimens and runs. The ratios calculated fromthe carriage response were generally very to those calculated from the wallresponse. Wall RR-3 was the only specimen to exhibit ratios in a consistentdirection — larger top accelerations; all other walls produced ratios that fellon either side of 1.0 depending on the run or the measurement location.Peak crack accelerations increased mainly monotonically for walls FF-3,FF-2, and SS-3, and remained fairly constant for walls FR-3 and RR-3(Figure 4.30). The lack of large increases for FR-3 are again likely dueto the lack of large rocking impacts. In contrast, wall RR-3 was subjectto large rocking impacts without significant increases in crack accelerations.The fixed diaphragm conditions appear to ‘damp out’ the acceleration peaksby preventing the ‘flicking’ effect at snap-through that is allowed by theflexible diaphragms. With a fixed diaphragm, a wall segment pivots aboutthe diaphragm connection, while with a flexible diaphragm, the effectivepivot is located within the wall segment, allowing snap-through to occurmore violently by accelerating both ends of the wall segment in oppositedirections.934.3. Numerical results0 0.5 1.0 1.5020%40%60%80%100%120%00.20.40.60.81.0FR-3 FF-2FF-3SS-3RR-3a (g)MotionscaleS a(1.0)(g)Figure 4.30: Peak wall accelerations at crack (mean of extrapolated values)4.3.4 Force demandsForce demands imposed on the wall-to-diaphragm connections are of interestto engineers involved in building assessment and retrofit design. In thissection, the forces imposed by the wall on the carriages during the shaketable testing are examined. Two approaches were used to calculate the totalforce demands on the connections.In the first approach, the demand at a single connection (top or bottom)was calculated by subtracting the inertial force of the carriage (obtained fromthe accelerometer on the carriage) from the force in the springs (obtainedfrom the measured spring stiffness and spring displacement).The second approach allows the calculation of the total connection de-mand (sum of top and bottom). The total force on the wall was calculatedby multiplying the acceleration measured on the wall at each header courseby the wall mass tributary to the elevation of that course. Data from eithereight (for the 4m high walls) or six (for the 2.8m high wall) accelerometerswere used in this calculation. The two methods should produce equivalentresults; comparing the results serves as a check on these calculations and onthe instrumentation.Results are plotted for run 12 of wall SS-3 as an example in Figure 4.31.Time histories of each component calculated in the first method are shown944.3. Numerical resultsfor the top and bottom in Figures 4.31c and 4.31d, and the total demandis compared for the two methods in Figure 4.31e. The two methods pro-duce nearly identical results, with the forces calculated from the carriagesexhibiting a slight time lag behind those calculated from the wall. The timelag is smallest for large amplitude excursions; for small oscillations both thelag and difference in peak magnitudes increase, with the forces calculatedfrom the carriages having a smaller amplitude than those measured on thewall.It is important to note that the acceleration profile of a cracked wallis the sum of several components, as illustrated in Figure 4.32 (adaptedfrom Meisl [2006]). Since the total force is the integration of this profilealong with the mass density over the height, it is a high-level quantity, andconsequently many details about the response can be hidden within it. Twodrastically different acceleration profiles can produce the same total force.For example, a uniform profile equal to zero everywhere on the wall and aprofile with large accelerations at the crack and at each carriage, where thecrack acceleration has the opposite sign as that at the carriages, can bothproduce a total force that is equal to zero even though the wall responseis very different at those two times. Plots in which only the total force isshown should therefore be interpreted cautiously.The forces can be calculated from the carriages only for flexible dia-phragm configurations. When a carriage is locked into the rigid mode, thestiffness is too high and the displacements are too low to reliably measurethe spring force. To compare consistent demands among all five wall speci-mens, only the forces calculated from the wall inertia should be used. Peakdemands from both sources are listed for the highest stable run and for thecollapse run in Table 4.8. Here, total forces are normalized to the total wallweight, while individual connection forces are normalized to 50% of the wallweight — representative of the way in which tributary wall weights wouldbe assigned in the assessment of a real building.The maximum recorded normalized force for a cracked wall recordedfrom any source, in any run, is 0.61 g. This includes forces caused by im-pact when cracks close up, and includes the collapse run (up to a rockingdisplacement of just over one wall thickness). Considering the wide range ofboundary conditions encompassed by these results, it is reasonable to con-clude that connection force demands in cracked, one-way rocking walls withno overburden are unlikely to exceed 0.6 g. However, these conditions formclose to a lower bound on wall stability, and changing boundary conditionsto improve stability could certainly result in higher demands if walls areexcited to correspondingly higher intensities.954.3. Numerical results−2000200(a)d rel(mm)Top Bottom Crack−2000200(b)d rock(mm)Crack−10010(c)F t(kN)Carriage inertia Spring force Connection force−10010(d)F b(kN)Carriage inertia Spring force Connection force6 7 8 9 10 11 12 13 14−10010(e)Time (s)F w(kN)Total: wall Total: connectionFigure 4.31: Force time histories, wall SS-3, run 12964.3. Numerical results± ± ± =GroundmotionTopflexibilityBottomflexibilityRigid bodyrockingTotalaccelerationFigure 4.32: Components of acceleration profileTable 4.8: Peak normalized force demands (g)Highest stable run Collapse runWall Connection Wall ConnectionWall Run Total Total Top Bot. Run Total Total Top Bot.FF-3 12 0.26 0.28 0.23 0.38 13 0.58 0.47 0.41 0.53FR-3 9 0.31 — 0.31 — 10 0.33 — 0.33 —FF-2 10 0.35 0.36 0.25 0.49 11 0.39 0.38 0.35 0.55SS-3 12 0.44 0.42 0.34 0.52 13 0.58 0.50 0.39 0.61RR-3 5 0.30 — — — 7 0.33 — — —Maximum: 0.44 0.42 0.34 0.52 0.58 0.50 0.41 0.61Total wall forces are normalized to total wall weightTop and bottom connection forces are each normalized to one-half of the wall weight974.3. Numerical resultsPeak force demands were compared with those predicted using the spec-tral acceleration of the shake table motion at the diaphragm period. Damp-ing values obtained from calibration of the analytical model were used (referto Section 5.3). Periods, damping ratios, and Sa values at 100% scale areshown in Table 4.9. Using these parameters, the force demands listed inTable 4.8 are compared to the corresponding scaled Sa values in Table 4.10.The results indicate that the spectral acceleration is in general an reason-ably good approximate predictor of demands. However, forces significantlyhigher than those predicted by this method were observed, particularly forthe bottom connection demand, for which several specimens developed forcesup to 1.6 times higher than those predicted from Sa. Total force demandsup to 1.7 times higher than the Sa predictions were also recorded. Thesehigh demands appear to be caused by the impacts induced when the crackcloses up after large rocking excursions — refer to the illustration of force-displacement hysteresis in Figure 4.37.Table 4.9: Period, damping, and full-scale Sa valuesWall Ts (s) ζ Sa(Ts)100% (g)FF-3 1.59 0.08 0.35FR-3 1.59 0.08 0.35FF-2 1.24 0.12 0.31SS-3 0.83 0.08 0.49RR-3 0 0.08 0.50[a][a] PGAPeak total force demands (from wall inertia) for each run are plottedin Figure 4.33. In general, trends for each wall are fairly linear, with some‘softening’ possible towards the collapse run (i.e. the point for the collapserun falls at lower force than the linear extrapolation from the other runswould suggest). Wall FF-3 showed the greatest deviation from linearity inthe collapse run, while in other walls the effect was less notable. The speci-mens with the flexible springs (FF-3, FR-3, and FF-2 ) showed a generallylower rate of change of force with respect to motion intensity than the stiffand rigid specimens, with FR-3 showing the lowest rate by a significantmargin.984.3. Numerical resultsTable 4.10: Ratio of peak normalized force demands to Sa(Ts)Highest stable run Collapse runWall Connection Wall ConnectionWall Run Total Total Top Bot. Run Total Total Top Bot.FF-3 12 0.94 1.03 0.84 1.39 13 1.69 1.36 1.17 1.54FR-3 9 0.81 — 0.82 — 10 0.78 — 0.80 —FF-2 10 1.04 1.08 0.75 1.46 11 1.05 1.04 0.94 1.50SS-3 12 1.20 1.15 0.93 1.42 13 1.49 1.28 1.00 1.56RR-3 5 1.00 — — — 7 1.01 — — —Maximum: 1.20 1.15 0.93 1.46 1.69 1.36 1.17 1.56Total wall forces are normalized to total wall weightTop and bottom connection forces are each normalized to one-half of the wall weight0 0.2 0.4 0.6020%40%60%80%100%120%00.20.40.60.81.0FR-3FF-2 FF-3SS-3RR-3Fw (g)MotionscaleS a(1.0)(g)Figure 4.33: Peak force demands on cracked wall994.3. Numerical results4.3.5 Hysteretic responseThe hysteretic response of the rocking displacement at the crack height isexamined in this section with respect to two parameters: the acceleration atthe crack, and the total force acting on the wall. As mentioned previously,it should be recalled that the rocking displacement is calculated from thestring pot measurements. The configuration of the string pots created someinaccuracies in the measurements, and consequently the measured rockingdisplacement is unreliable below roughly 8mm, i.e. drocknorm = 0.025 for3-wythe walls and drocknorm = 0.04 for the 2-wythe wall. As a result, theuncracked stiffness of the walls could not be measured from the shake tabletests. The rocking response is, however, adequately captured.4.3.5.1 Acceleration-displacement responseFigure 4.34 shows the crack acceleration vs. rocking displacement hystereticresponse for the highest stable run for each wall. Note that to better illus-trate the response within each run, the x- and y-scales of the plots vary. Ingeneral, each hysteresis plot displays two zones: with and without rocking.Prior to a significant rocking cycle being initiated, the response is basicallyelastic, with a stiff slope. During a rocking excursion, the crack accelerationremains relatively constant while the rocking displacement increases. Os-cillations of the acceleration occur during rocking, typically on the leadingportion of the excursion following an impact from the previous excursion.In Figure 4.34a, a large double oscillation occurs on the main rocking excur-sion (towards negative rocking displacement) that results in a full reversalof the crack acceleration while the rocking displacement changes minimally.These oscillations transferred into the carriages, which also experienced asign change of accelerations.Figure 4.35 shows the same plots for the collapse run for each wall.The x-scale (rocking displacement) is consistent in all these plots at ±1,which is an approximation of the static instability condition (the exact staticinstability limit is determined by the angle of the bottom wall segment,as discussed in Section 4.3.2.1, but by the time the rocking displacementreaches ±1, the bottom block is usually near this rotation limit). In eachspecimen, the crack acceleration decays approximately towards zero as therocking displacement increases towards one. This decay occurs smoothly insome cases (e.g., Figure 4.37c) and with large oscillations in others (e.g.,Figure 4.37d). In Figure 4.37e, the acceleration decays through zero beforethe rocking displacement reaches 1.1004.3. Numerical results−0.2 0 0.2−0.500.5drocknorma crack(g)(a) FF-3, run 12−0.1 0 0.1−0.500.5drocknorm(b) FR-3, run 9−0.5 0 0.5−101drocknorma crack(g)(c) FF-2, run 10−0.5 0 0.5−101drocknorm(d) SS-3, run 12−0.5 0 0.5−0.500.5drocknorma crack(g)(e) RR-3, run 5Figure 4.34: Acceleration hysteresis in highest stable runs1014.3. Numerical results−1 0 1−101drocknorma crack(g)(a) FF-3, run 13−1 0 1−0.500.5drocknorm(b) FR-3, run 10−1 0 1−101drocknorma crack(g)(c) FF-2, run 11−1 0 1−101drocknorm(d) SS-3, run 13−1 0 1−0.500.5drocknorma crack(g)(e) RR-3, run 7Figure 4.35: Acceleration hysteresis in collapse runs1024.3. Numerical resultsIn each of the plots in Figures 4.34 and 4.35, the dashed lines indicateaccelerations of ±0.35 g. While these lines serve as a reference for the verticalscale, in general, these accelerations also bracket the ‘mean’ accelerations(moving time-averaged, discounting the oscillatory peaks caused by impacts)occurring during rocking excursions. Meisl [2006] termed this the ‘effectiverocking acceleration’, and found that it was fairly consistent among all of histests. He found the mean effective rocking acceleration to be 0.57 g, whichis considerably higher than the 0.35 g noted for the current tests. It shouldbe noted that estimation of this parameter is subjective. Meisl’s test setupwas reasonably equivalent to that used in wall RR-3 in the current testing,as were his typical crack heights. However, wall RR-3 — for which 0.35 gwould likely be a high estimate — showed perhaps the largest difference ineffective rocking acceleration from Meisl’s values. A significant differencebetween the tests was the ground motion used, suggesting that perhaps thedetails of the ground motion time history may affect the effective rockingacceleration at the crack, despite Meisl noting similar accelerations for bothground motions that he ran.4.3.5.2 Force-displacement responseFigure 4.36 shows the total wall force vs. rocking displacement hystereticresponse for the highest stable run for each wall. Note that in this case,the y-scale is constant among all runs, while the x-scale varies. Comparedto Figure 4.34, oscillations and extreme values in the force response plotsare generally less intense than in the crack acceleration plots. For example,the large double oscillation of wall FF-3 visible in the acceleration plot ofFigure 4.34a is much less significant in terms of the total force on the wall asshown in Figure 4.36a. On the other hand, for wall RR-3, the accelerationand force plots look reasonably similar in terms of oscillations and extremevalues (Figures 4.34e and 4.36e).This is consistent with the concept presented earlier that when the car-riages are more easily influenced by the wall response (more flexible and/orrelatively lighter carriages), the cracked wall snaps through the neutral po-sition harder, creating larger accelerations at the crack and the base. Theselarge acceleration spikes can occur without much change in the total forceon the wall, since they occur in opposing directions. With rigid diaphragms,the snap through is more subdued, and the accelerations are more reflectiveof the total force on the wall.Figure 4.37 shows the same plots for the collapse run for each wall. In theacceleration plots, the collapse runs were marked by at least one significant1034.3. Numerical results−0.2 0 0.2−0.500.5drocknormF w(g)(a) FF-3, run 12−0.1 0 0.1−0.500.5drocknorm(b) FR-3, run 9−0.5 0 0.5−0.500.5drocknormF w(g)(c) FF-2, run 10−0.5 0 0.5−0.500.5drocknorm(d) SS-3, run 12−0.5 0 0.5−0.500.5drocknormF w(g)(e) RR-3, run 5Figure 4.36: Force hysteresis in highest stable runs1044.3. Numerical results−1 0 1−0.500.5drocknormF w(g)(a) FF-3, run 13−1 0 1−0.500.5drocknorm(b) FR-3, run 10−1 0 1−0.500.5drocknormF w(g)(c) FF-2, run 11−1 0 1−0.500.5drocknorm(d) SS-3, run 13−1 0 1−0.500.5drocknormF w(g)(e) RR-3, run 7Figure 4.37: Force hysteresis in collapse runs1054.3. Numerical resultslarge pulse, typically much larger than in the highest stable run. In theforce plots, the differences between the collapse and highest stable runs forany given wall are more subtle, and peak forces are not much larger — notethat the y-scales in Figure 4.37 are consistent with those in Figure 4.36.In each of the plots in Figures 4.34 and 4.35, the dashed lines indicatea total wall inertia of ±0.35 g — equivalent to the entire wall accelerat-ing at the value of the dashed lines in the acceleration plots. The dashedforce lines bracket the mean rocking force similarly to the way in whichthe effective rocking acceleration was bracketed. Meisl [2006] noted thatmultiplying the wall mass by the effective rocking acceleration at the crackproduced a reasonable estimate of the maximum total force on the crackedwall. This appears to be roughly valid for the results of the current testing,with the caveat that the maximum forces can exceed this estimate due tothe greater propensity for larger force oscillations with flexible diaphragmconfigurations.4.3.6 CrackingCracking was initiated in each wall by subjecting it to the NGA0763 motionwith both carriages locked into the rigid mode. Confirming visually whetheror not a run had initiated cracking was difficult in some specimens, and sincedata viewing capabilities during testing were limited, an additional run ofthe same motion at slightly increased intensity was then carried out to ensurethat a crack had in fact formed.The precise time at which the crack formed was estimated during post-processing of the data. The cracking runs typically produced very smallrocking displacements, and the limited precision of the string potentiometersdid not provide sufficient resolution of the rocking displacement to pinpointthe time of crack formation. Instead, the shape of the wall’s accelerationprofile at each time step was used to evaluate whether the wall was crackedor uncracked. An uncracked wall generally exhibits a roughly quadraticacceleration profile as opposed to the bilinear profile of a cracked wall. Ateach time step, a quadratic curve was fit to the acceleration profile usinglinear regression. The coefficient of determination, R2, was calculated forthe regression at each time step. This coefficient was used as an indicatorof ‘how cracked’ the acceleration profile was at any given time.The time histories of selected accelerations and of total force on thewall are shown in Figures 4.38a and 4.38b. Since the wall is not crackedfor a portion of this run, the recorded acceleration at header 4 (just belowthe crack) is shown rather than the mean of the linearly extrapolated ac-1064.3. Numerical results−0.500.5(a)a(g)Top carriage Bottom carriage Header 4−10010(b)F w(kN)Total3 4 5 60.91.0A BCrackedUncracked(c)Time (s)R2Quadratic fitFigure 4.38: Acceleration and force time histories, wall FF-3, run 71074.3. Numerical resultscelerations. Figure 4.38c shows the time variation of R2 (filtered to removehigh-frequency variations). Two times of interest are indicated by the dottedvertical lines.Prior to time A, R2 remains very close to 1, with the exception of isolatedspikes where the coefficient drops — typically as the acceleration profilestraightens and the curvature switches direction. After time B, R2 remainsconsistently below 1, with the exception of reaching values near 1 brieflyas the profile nears the curvature switch and the bilinearity becomes lesspronounced. The differences in the R2 time history may appear subtle atfirst glance, but the transition is distinct and was consistently identifiablefor each wall specimen. The acceleration profile at time A is shown inFigure 4.39a along with the fitted quadratic curve, and the profile at timeB is shown in Figure 4.39b along with the fitted bilinear curve.0 0.2 0.401234a (g)Height(m)Measured Calculated(a) Uncracked (Time A)−0.4 −0.3 −0.2 −0.1 0a (g)(b) Cracked (Time B)Figure 4.39: Acceleration profiles of wall before and after crackingIn addition to the R2 transition at cracking, the acceleration at header 4transitions from being generally in-phase with the carriage accelerations tosignificantly out-of-phase. This is illustrated in Figure 4.38a, and was alsoconsistently identified in each specimen. The R2 method, however, allowsthe precise time of transition to be identified more readily, and consequentlywas used as the primary indicator — verified by checking individual accel-eration profile shapes.The peak forces occurring in the cracking runs (normalized to wall1084.3. Numerical resultsweight) for each specimen are summarized in Table 4.11. Time A in Fig-ure 4.38 is representative of the data under ‘crack initiation’. In each speci-men, the force peak at crack initiation was smaller than the maximum forceof the entire run. In addition, the maximum force of the entire run occurredbefore the crack initiation time in every specimen. This suggests that cracksare formed progressively — some damage is caused during the large initialforce peaks, but not enough to propagate the crack through the entire wallthickness. Subsequent force peaks, though smaller, cause further damageuntil eventually the crack has propagated through the wall.Table 4.11: Peak normalized forces in cracking runCrack initiation Entire runWall Run Scale Time[a](s) Force[b](g) Time (s) Force (g)FF-3 7 50% 4.32 0.39 3.515 0.40FR-3 3 60% 5.18 0.36 4.385 0.52FF-2 4 70% 4.12 0.58 3.385 0.58SS-3 6 60% 5.03 0.30 4.540 0.49RR-3 2 60% 4.55 0.49 4.400 0.50[a] The latest time at which the wall could definitively be identified as not cracked.In each specimen, a cracked profile was confirmed no more than 0.1 s after this time.[b] The local force peak occurring at approximately the noted time.The wall forces listed in Table 4.11 are equal to the sum of the top andbottom connection demands, which are representative of the demands onwall-to-diaphragm anchors. The distribution of these total forces betweenthe top and bottom locations could not be measured directly with carriageslocked out. However, individual connection demands can be approximatedby assuming the wall to be simply supported (i.e. no moment support atthe connections), and then calculating the reactions from equilibrium basedon the distribution of the forces. Here, the force distribution at each timestep was assumed to be proportional to the quadratic fit of the accelerationprofile, denoted here as f(y), calculated earlier. The height of the centroidof the force, y¯, was calculated as:y¯ =∫ h0 y ·f(y) dy∫ h0 f(y) dy(4.3)1094.3. Numerical resultswhere y is the height from the base of the wall. From equilibrium, theindividual connection forces can then be approximated as:Ft = y¯h ·Fw (4.4)Fb =(1− y¯h)·Fw (4.5)The relative height of the force centroid is plotted against the normalizedtotal wall force, Fnorm, in Figure 4.40. Note that numerical inaccuracies withthe quadratic fitting at very low force levels sometimes resulted in a centroidcalculated as out of the wall height range; these values were then capped toeither top or bottom of the wall. At large force levels, the force centroid wasconsistently located slightly above mid-height of the wall (on average roughlyat 55% of the wall height). The total force demand can be interpreted tobe allocated to the top and bottom connections as between 50%/50% and60% top/40% bottom. It is critical to note that these allocations are basedon the observed values from this particular test apparatus only, and they donot necessarily reflect conditions that may be encountered in real buildings.0 0.2 0.4 0.600.51.0Fw (g)y¯/hFF-3 FR-3 FF-2 SS-3 RR-3Figure 4.40: Location of centroid of wall force prior to crackingThe total wall forces in the cracking run are compared with those mea-sured in the two highest cracked runs for each wall in Table 4.12. Maximum1104.3. Numerical resultsforces for each wall are shown in bold. Maximum forces attained in the high-est stable rocking runs were lower in each case than those attained duringthe cracking runs. During collapse runs, force levels in walls FF-3 and SS-3exceeded those attained during the respective cracking runs. In all cases,maximum wall forces recorded were between 0.5 and 0.6 g.Table 4.12: Peak normalized forces in cracking run and cracked runs (g)Cracking run Cracked runsWall Initiation Entire run Highest stable CollapseFF-3 0.39 0.40 0.26 0.58FR-3 0.36 0.52 0.31 0.33FF-2 0.58 0.58 0.35 0.39SS-3 0.30 0.49 0.44 0.58RR-3 0.49 0.50 0.30 0.334.3.6.1 Cracking predictionsFlexural cracks like those observed in the test specimens are initiated whenthe tensile stress in the material exceeds the tensile strength. The flexuraltensile strength of the masonry was tested using the bond wrench method(Table 3.5). The nature of this test method and the low tensile strength ofthe masonry produced large variability in the results, but the test nonethe-less provides a rough idea of the expected tensile strength.The wall in the test setup can be simplified as a simply-supported, one-way spanning beam. When undergoing excitation on the shake table, thewall’s inertia produces a distributed horizontal load on the wall, creating abending moment, and thus compressive and tensile stress. In addition, theself-weight of the wall and the top beam assembly create axial compressivestress, which increases from the top of the wall towards the base. The totalstress at any location in the wall is the sum of the axial and bending stresses.To calculate the bending stresses in an uncracked wall at a given timestep, a horizontal load distribution was assumed that had the same quadraticshape as the curve that was fit through the acceleration profile (as in Fig-ure 4.39a), but was scaled to produce the total load previously calculatedfrom the tributary weight method. Top and bottom horizontal reactionswere calculated from equilibrium, and the shear and moment equations wereobtained by integrating the load equation. Finally, a distribution for the1114.3. Numerical resultsmaximum net tensile stress was obtained by calculating the flexural tensilestress using half the wall thickness as the distance from the neutral axis tothe tensile fiber and then adding the axial compression. The time historiesof the peak tensile stress in the wall and the location of this peak stress areshown in Figure 4.41.−10010(a)F w(kN)Total00.20.4f ′fb: mean f ′fb: mean - 1σ(b)f Tmax(MPa)Peak tensile stress in wall3 4 5 600.51.0A BCrackedUncracked(c)Time (s)y Tmax/hLocation of peak tensile stressFigure 4.41: Stress time histories, wall FF-3, run 7The location of the peak stress varies significantly, and is plotted as afunction of the stress in Figure 4.42, for all measurement points occurringprior to cracking (time A). The location of the peak tensile stress consistentlydecreases as the stress increases. At the largest stresses (those that wouldinitiate cracking), the peak stress occurs at approximately 55% of the wallheight. This trend, including the location at high stresses, was consistentamong all specimens. This indicates that the variability in crack heightamong specimens (Table 4.1) can likely be attributed to the details of eachwall’s construction — the strength of each joint in the wall will not be the1124.3. Numerical resultssame due to construction variability and errors — rather than to a significantdifference in demands imposed upon the wall.0 0.1 0.2 0.300.51.0fTmax (MPa)y Tmax/hFigure 4.42: Location of peak tensile stress, wall FF-3, run 7Calculated tensile stresses in the cracking run for each specimen arecompared with the flexural tensile strength measured by bond wrench testingin Table 4.13. Shake table results are within one standard deviation ofthe mean strength predicted by the static tests. While the bond wrenchresults for walls SS-3 and RR-3 indicated a slightly higher strength thanfor the other three walls, the calculated stresses during dynamic testing werefairly consistent among all of the specimens. These results are reasonable,considering the variability inherent in the bond wrench testing method aswell as in the wall construction and the dynamic load application.Table 4.13: Peak stresses in cracking runBond wrench Shake tablef ′fb (MPa) cv fTmax (MPa) #σ from meanFF-30.38 0.420.31 −0.4FR-3 0.41 0.2FF-2 0.36 −0.1SS-3 0.55 0.31 0.38 −1.0RR-3 0.39 −1.0113Chapter 5Validation of AnalyticalModel5.1 IntroductionThe shake table tests conducted as part of the experimental program wereable to examine a limited number and range of variables, and by them-selves can not provide a comprehensive understanding of the out-of-planeresponse of URM walls with flexible diaphragms. Since shake table testingis prohibitively time- and resource-intensive, it is necessary to develop ananalytical model which can reliably approximate the results of the physicaltesting.As discussed in Section 2.2.2, two main simplistic approaches have beenconsidered for modelling the rocking behaviour of a cracked URM wall:1. A non-linear elastic system consisting of stick models for the wall seg-ments connected by a rotational spring at the crack, producing a tri-linear force-displacement response and using variable Rayleigh damp-ing [Doherty, 2000], and2. A rigid body model consisting of wall segments with a defined thick-ness, with rocking explicitly simulated using body geometry and im-pact mechanics [Meisl, 2006].Makris and Konstantinidis [2003] demonstrated that the response of arocking system is fundamentally different from that of a regular SDOF os-cillator, and recommended that “the response of one should not be used todraw conclusion on the response of the other”. Since the out-of-plane re-sponse of URM walls has been extensively shown in laboratory tests to bedominated by rocking, the rigid body approach was selected for the analyt-ical phase of the project.In this chapter, an analytical rigid body model is validated to the shaketable test results of Chapter 4. Modelling methodology and details arepresented, followed by validation results.1145.2. Software5.2 SoftwareThe commercially available software Working Model 2D [Design SimulationTechnologies, Inc., 2010] was used to simulate the response of the wall-diaphragm system. Konstantinidis and Makris [2005] validated the abilityof this software to simulate the pure sliding and pure rocking responses ofa block as part of an investigation into the seismic performance of multi-drum columns. Meisl [2006] further demonstrated that the software couldadequately simulate the rocking response of one-way spanning URM wallswith rigid diaphragm boundary conditions.Working Model 2D (WM) carries out time history analysis on a sys-tem of rigid bodies and constraints. A problem is time-discretized suchthat the program can compute motions and forces, while ensuring that theconstraints are satisfied. The motion of bodies is governed by differentialequations, which are solved in WM using the explicit Kutta-Merson (5th-order Runge-Kutta) integration method. The critical aspect of rigid bodyanalysis is the treatment of contact between bodies. Sliding along a contactinterface is treated with Coulomb friction. In addition, bodies are monitoredcontinuously to detect collisions, and impact forces are calculated using animpulse-based model and the coefficient of restitution [Design SimulationTechnologies, Inc., 2010].5.3 Model constructionModels were created using scripts written in Working Model Basic [Knowl-edge Revolution, 1995], which is an offshoot of Visual Basic developed forWorking Model 2D. The use of scripting rather than creating the models inthe graphical interface provided much greater capabilities for parametriza-tion, which were later used extensively, as well as allowing for greater preci-sion and detail in the definition of body geometry and constraint placement.The general configuration of the model is shown in Figure 5.1. The wallis modelled as two rigid bodies stacked one on top of the other, restingon a frictionless base, which represents the shake table. A rigid frame ex-tends up from this base, representing the test frame. Spring-damper unitsconnect this rigid frame to the top and bottom carriages. Each carriage isconstrained to travel only horizontally using a square slot restraint.Each wall segment was modelled with uniformly distributed mass, suchthat combined weight of the segments corresponds to the measured weight ofthe wall. WM calculates the corresponding mass moment of inertia. Spalling1155.3. Model constructionTop wallsegmentBottom wallsegmentTop carriageRigidframeSpring-damperunitsBottomcarriageFrictionlessbaseDisplacement-controlledactuatorFixed anchorblockCrackFigure 5.1: Model of test setup in Working Model 2DActiveconstraintInactiveconstraintStruts (partof bottomwall segment)Figure 5.2: Detail of crack in Working Model 2D (size of chamfer exagger-ated for illustration purposes)1165.3. Model constructionat the rocking interfaces (at the base and at the crack) was represented byassigning a 45◦ chamfer to the appropriate corners of the wall body. Atthe crack, the chamfer was set at 2mm in all walls, consistent with theminimal amount of spalling that was visible during testing. The amount ofspalling at the base could not be reliably assessed during testing due to thevisual obstruction created by the bottom connection. Greater spalling of themortar at the base than at the crack was expected due to the higher gravityloads and the fact that the base mortar was bearing on the steel beam. Itwas decided that a value of 10mm at the base was reasonable, and it wasfound that this value produced good calibration results.A triangular body, representing the top beam assembly, is connected tothe top of the top wall segment by a rigid link. A pin node is placed atthe peak of this triangle, connected to a vertical slot constraint in the topcarriage. This modelled connection thus accurately simulates the responseof the test setup by allowing rotation and vertical travel in the slot, whilemating the horizontal displacement of the top of the wall to that of the topcarriage.The crack interface between the top and bottom wall segments provedchallenging to model. In initial attempts, the blocks were modelled as simplyresting on top of one another, with reasonable expected friction properties(µstatic = 0.75, µdynamic = 0.75). However, this resulted in the blockssometimes experiencing large sliding instabilities during rocking, when nosliding was observed in the tests. This issue was resolved by constrainingthe blocks to one another horizontally (Figure 5.2).Thin struts were added to the bottom wall segment, such that the topof each strut was at the same elevation as the crack. The additional masscontributed by the struts is negligible due to their small size. A workaroundfor a rigid link was connected from the end of each strut to the point ofcontact on the top wall segment when undergoing rocking (i.e. the insideof the chamfer). This link consists of a separator and a rope connected tothe same nodes. The separator only acts as a constraint when its endpointsattempt to move closer together than its initial length, while the rope onlyacts as constraint when its endpoints attempt to move farther apart thanits initial length. The initial length for the separator was set at the startingdistance between the two nodes minus a tolerance dimension, and that ofthe rope was set at the starting distance plus the same tolerance dimen-sion. The tolerance dimension was determined by trial and error. It wasfound that using a value of 0.2mm, when the program’s integrator, overlapand assembly errors were also set to this same value, produced consistentstability. While this link configuration creates some slack in the connection1175.3. Model construction(compared to a true rigid link), the effects of this slack were found to beminimal with the small selected value of the tolerance dimension.When the wall is undergoing rocking, there is only one contact pointbetween the two wall segments at any time. To allow rotation of the topsegment relative to the bottom segment, the link at the side of contact mustbe active, while the link at the other side must be inactive. This was achievedby setting the ‘active when’ property of the links to refer to the directionof relative rotation of the two wall segments. A tolerance of ±10−7 rad wasallowed during which links on both sides would be simultaneously active.The base of the bottom wall segment rests on the frictionless body rep-resenting the shake table. The bottom carriage is connected horizontallyto the bottom wall segment using the same links as at the crack interface.These links are connected to the wall segment at a height of 40mm abovethe base (representing the middle of the rubber spacer in the test setup),and connected to the bottom carriage such that the links are horizontal inthe at-rest position. The ‘active when’ property of these links referred tothe absolute rotation of the bottom segment, and required no tolerance al-lowance. This link configuration simulates the test conditions accurately,and produces a different response than a slotted pin joint would.Elasticity is implemented in WM using the coefficient of restitution,which is the ratio of the relative velocities of collided objects immediatelybefore and after a collision. This is a property of a collision, but a coef-ficient is assigned to each body in WM. The coefficient of restitution in acollision is defined by WM as the lower value of the constants given to thetwo bodies involved in the collision. Meisl [2006] found that a small amountof elasticity was appropriate to simulate the rocking response of URM walls,and used values between 0.02 and 0.023. A value of 0.02 was assigned tothe wall segments in this study. Due to the more constrained nature of thebase connection as opposed to the totally free crack interface, a value of 0was used for the shake table body. Overall, the model response was foundto be minimally sensitive to the exact values of the coefficient of restitution,provided that they were in this general range.The carriages were connected to the rigid frame with spring-damperunits. The spring constants were set equal to the sum of those measured foreach spring assembly. The damping values were empirically calibrated toachieve a good fit to the measured carriage response for the larger oscilla-tions. The ‘stiction’ present in the system during low amplitude oscillationscould not be captured by the viscous damping model, and this aspect wasnot further pursued in the modelling due to its low importance. The damp-ing ratio for each carriage was defined in terms of the total spring constant1185.4. Modelled responsefor that carriage and the mass of the carriage plus half of the wall mass. Adamping ratio of 12% was used for wall FF-2, and 8% was used for all otherwalls. Actual damping constants were then calculated by the WM script.Since the viscous damping in the model accounts for multiple sources ofdamping in the test setup (e.g., wheel friction, carriage rail misalignment,friction on the springs, spring assembly misalignment), and some of thesesources of real damping can be dependent on the test specimen size andweight, it was decided that was little basis for applying the same dampingconstant for both the large and small walls. Consequently, the dampingratio was tuned independently for each of the two wall sizes.The input motion in the model is applied to the shake table body bya displacement-controlled actuator connected to a fixed anchor block. Theas-recorded table motions, low-pass filtered as described in Section 3.4.3,were fed into the actuator. The flexibility of the test frame was disregarded,and the rigid frame in the model applied the same input motion to both thetop and bottom springs.5.4 Modelled responseEach run on the shake table in which the CHHC1 ground motion was appliedto a cracked wall was simulated using the model. Since wall stability isfundamentally a displacement-governed problem, the displacement responseis of primary interest in the validation of the model. In particular, therocking response of the wall is the critical output, for which a prerequisiteis accurate modelling of the response of the diaphragms.The peak normalized rocking displacements shown in Figure 4.15 arereplicated in Figure 5.3, with the modelled results overlaid in blue. Notethat for some walls — FR-3, SS-3, and RR-3 — the model did not producea collapse outcome for the motion that caused collapse on the shake table.In these cases, further simulations were carried out by scaling up the finaltest motion until collapse was achieved in the model. The motion scale wasincremented by scale factors of 1% at a time. In each case, collapse wasproduced in the model at scale factors no more than 3% greater than thescale factor used on the shake table.The model approximates the general trend of rocking displacements vs.motion intensity reasonably well. The collapse scale is very well simulated;the largest discrepancy is the premature simulated collapse of wall FF-2 ata scale of 110% vs the observed collapse at a scale of 120%. The minimalrocking of wall FR-3 prior to the collapse run is well represented. There are1195.4. Modelled response0 0.2 0.4 0.6 0.8 1.0020%40%60%80%100%120%00.20.40.60.81.0FR-3FF-2FF-3SS-3RR-3CollapseBAdrocknormMotionscaleS a(1.0)(g)Test ModelFigure 5.3: Modelled vs. tested peak rocking displacementssome discrepancies in the peak amounts of rocking observed in non-collapseruns, but significantly higher accuracy in the modelling of this parametercan not be expected given the nature of the rocking response.A more detailed look at the model’s performance can be obtained byexamining the time history output. Selected pairs of modelled and testedpoints are highlighted in Figure 5.3, labeled A and B. Points A illustrate lowmagnitude rocking from wall FF-2, and points B illustrate high magnituderocking from wall SS-3. Time history comparisons are shown for points Ain Figure 5.4 and points B in Figure 5.5.Figure 5.4 illustrates the ability of the rocking model to accurately repro-duce the non-periodic rocking response. The time variation of the rockingmotion is matched very well by the model, though the magnitudes of someof the rocking excursion peaks are smaller in the model than recorded inthe test. In addition, the response of both the top and bottom carriages arematched exceedingly well for the entire duration of the run.The larger rocking displacements shown in Figure 5.5 are likewise re-produced well until the peak of the largest excursion is reached. Here, themodel (which is running at 80% scale vs. the test at 75% scale) overshootsthe rocking displacement slightly on this cycle. Since this happens to beon a very large rocking excursion (at a significant instability factor), this1205.4. Modelled response−50050d rock(mm)Crack: Test Model−1000100d rel(mm)Top carriage: Test Model6 8 10 12 14 16 18−1000100Time (s)d rel(mm)Bottom carriage: Test ModelFigure 5.4: Modelled vs. tested displacement time histories, wall FF-2, run9 (modelled) vs. run 9 (tested)1215.4. Modelled response−2000200d rock(mm)Crack: Test Model−1000100d rel(mm)Top carriage: Test Model6 8 10 12 14 16 18−1000100Time (s)d rel(mm)Bottom carriage: Test ModelFigure 5.5: Modelled vs. tested displacement time histories, wall SS-3, run13 (modelled) vs. run 12 (tested)1225.4. Modelled responseovershoot significantly elongates the duration (‘period’) of that rocking cy-cle in addition to increasing the peak rocking displacement. Consequently,when the wall returns to the closed position (no rocking displacement) inthe model, the diaphragms are at significantly different positions than at theearlier point in the test. The wall thus reacts quite differently at this point,following through with a large excursion in the opposing direction ratherthan damping out the rocking rapidly as in the test. Due to the strong in-teraction of this heavy wall with the carriages, the simulated response of thecarriages is also thrown off significantly after this point, and the modelledresponse actually ends up nearly out-of-phase with the measured response.This example illustrates the particular difficulty associated with repli-cating a measured response with a model for a non-periodic response likerocking: slight variances in a single excursion can throw off the remainder ofthe simulated time history by very large amounts. It is therefore importantto note that consistent and exact matching of an entire run’s time historyis an overly ambitious goal that serves little practical purpose. The inputsfor the current model were calibrated roughly and consistently among wallspecimens. Certainly further tweaking of the inputs specifically for each wallmight lead to some gains in the accuracy of a particular simulation, but thereis no need for this. In its current state, the model clearly reproduces withreasonable accuracy and consistency the following:• the general time variation of the rocking response,• peak rocking magnitudes, and• motion scales causing wall collapse.The model is thus adequate as a predictor of wall performance for othergeometric configurations and ground motions, where the goal is to assesstrends in wall stability as these variables change. Validation results for eachrun are provided in Appendix G.123Chapter 6Parametric Study6.1 IntroductionThe experimental phase of the project was restricted to examining a verylimited number of combinations of ground motions, wall geometries, andboundary conditions. The out-of-plane response of walls must be exploredunder a significantly larger range of these variables in order to allow gener-alized conclusions to be drawn. To fulfil this need, a parametric study wasconducted using the analytical model described in Chapter 5. The effectsof relevant variables were examined by running time history analyses with alarge suite of ground motions for each configuration of interest. The thresh-old of out-of-plane wall collapse is evaluated for each run, and results arecompiled and interpreted. This chapter describes the ground motions used,the model configuration, the modelling procedure, and the results.6.2 Ground motionsThe response of a non-linear system can be sensitive to the details of thetime variation of the ground motion. These details of a motion’s responsehistory can not be precisely described by simple intensity measures, nor canthe response of a non-linear system be predicted based only on such intensitymeasures. It is therefore necessary to run a model of a system through theentire history of a ground motion to determine the system’s response.The inability to predict the system response based on intensity measuresmakes the selection of individual ground motions a gamble — it is not pos-sible to determine before running an analysis whether a particular groundmotion will result in below or above average system response for a givenintensity measure. To obtain an accurate picture of the system response,it is necessary to account for motion-to-motion variability. In practice, fora particular project this variability might be estimated using a relativelysmall number of motions (e.g., seven). In this exploratory study, however,it is desired to take a detailed look at motion-to-motion variability and to1246.2. Ground motionsinterpret the findings. A larger set of motions is needed to achieve thesegoals, and the selection of motions must be carefully considered. This sec-tion describes the rationale for the selection of ground motions, followed bya summary of pertinent characteristics of the selected set of motions.6.2.1 BackgroundGround motion selection methodology continues to be a contentious issueamong researchers and practicing engineers. As discussed above, it is diffi-cult to characterize the time-variable response details of motions using scalarintensity measures. The response spectrum achieves it adequately for linearsingle-degree-of-freedom (SDOF) systems — for these systems, it suffices toread the spectral value at the system’s natural period of vibration to de-termine precisely its peak response. For systems with non-linearity and/ormultiple degrees of freedom (MDOF), a simple solution does not exist.MDOF systems have (typically) different periods of vibration for eachmode of response, with the fundamental period being the longest one. Thevalues at multiple periods on the response spectrum are therefore relevant inestimating the response of such a system. Consider as an example two mo-tions which have the same spectral value at a MDOF system’s fundamentalperiod. At shorter periods, the spectrum of motion A is larger than that ofmotion B. The modal periods of the system’s higher modes will fall withinthis shorter period range. One can therefore reasonably predict that motionA should result in a larger system demands than motion B. The charac-teristic that differs between these two motions can be termed the spectralshape.In the present study, the base system is a cracked wall connected at thetop and bottom to flexible diaphragms. In its most basic representation, thissystem consists of three degrees of freedom: the lateral displacement of thetop of the wall, the crack location, and the bottom of the wall. Given thatthe top and bottom diaphragm stiffnesses may not necessarily be the same,and that the rocking response does not even have a characteristic period, itis clear that MDOF effects may be significant. It was therefore decided toconsider the effect of spectral shape in the analysis.To account for this effect, Baker and Cornell [2005] proposed a vector-valued ground motion intensity measure consisting of spectral acceleration(Sa) and a shape parameter, ε. Both Sa and ε are period-dependent. Sincethe concept of ε is significantly newer than that of response spectra, it isbriefly explained here.1256.2. Ground motionsFor a given ground motion record, ε is calculated by comparing the elas-tic response spectrum of that record with the predicted response spectrumfor the same magnitude, distance, site conditions, etc. Several ground mo-tion attenuation relations are available in the literature; that proposed byBoore et al. [1997] was used in this study. The other relations are generallyconsistent with this one, and for the purposes of this study the differencescan be expected to be of little consequence. This relation requires as inputthe period of interest (T ), the type of fault (normal or reverse), and MW ,rjb, and VS30 . It returns the predicted mean and standard deviation of thespectral acceleration at that period. Given both the recorded and predictedspectra, ε is then calculated as the number of standard deviations that therecorded spectral acceleration falls from the mean predicted spectral accel-eration (where the spectral accelerations are lognormally distributed):ε = lnSa(T )− µˆlnSa(T )σˆlnSa(T ) (6.1)The process is illustrated for a sample motion in Figure 6.1. Figure 6.1ashows the mean and ±1σ of the prediction from the ground motion attenua-tion relation, in addition to the spectrum from the recorded ground motion.Figure 6.1b shows the calculated ε values at each period. Periods wherethe recorded spectrum exceeds the mean prediction have positive ε values,and vice versa. Ground motions will typically not have consistent ε valuesover the entire period range, and thus a high ε value at a particular periodwill tend to indicate a peaked shape of the response spectrum around thatperiod, whereas a low ε indicates a valley.1266.2. Ground motionsµˆSaµˆSa + 1σˆSaµˆSa − 1σˆSaRecorded motion00.51.0(a)S a(g)ε = 1.32ε = −0.930 0.5 1.0 1.5 2.0−2−1012(b)T (sec)εFigure 6.1: Sample illustration of ε calculation1276.2. Ground motions6.2.2 Ground motion selectionFor site specific studies, such as those typically carried out in practice for aparticular project at a single location, recent advances have greatly improvedthe basis for ground motion selection. Probabilistic seismic hazard analysis(PSHA) allows the deaggregation of the hazard, which provides insight intothe main sources of contribution and their characteristics. At each period,a target ε value can be derived by considering the ε values of the dominantcontributions to the hazard. A conditional mean spectrum (CMS) can bedeveloped for a period of interest [Baker, 2011]. The CMS can then beused as a reference shape for matching ground motion spectra. For a simplestructure whose response is dominated by the fundamental period, a singleCMS at that period might be used; for a more complex structure multipleCMS’s may be used to examine several characteristic periods.In a non-site specific study, for example where one might aim to drawconclusions about the performance of a class of structures, of which spec-imens could be located anywhere, the CMS method can not be directlyapplied. This is due primarily to two factors: target ε values depend on thehazard deaggregation, which cannot be generalized to different sites, and aCMS must be constructed around a single period of interest, which can notbe defined over a class of structures with variable configurations. Haseltonet al. [2011] suggested a procedure whereby the effects of the spectral shapecould be incorporated in the application of the results of a generalized study.The procedure is summarized as follows (adapted from Haselton et al.):1. Select a general far-field ground-motion set without regard to ε valuesof the motions2. Calculate the collapse capacity by conducting a nonlinear incrementaldynamic analysis (refer to Vamvatsikos and Cornell [2002])3. Select a period of interest, and perform a linear regression analysisbetween the collapse capacity of each record and the ε(T1) of eachrecord4. Adjust the collapse capacity distribution, by using the regression rela-tionship, to be consistent with the target ε(T1) for the site and hazardlevel of interest.5. Repeat steps 3 and 4 for different periods as necessaryIn step (1), the far-field motion set could be selected based on any num-ber of criteria (e.g., magnitude, distance, peak response parameters, etc.).1286.2. Ground motionsIn the absence of any well-defined justification for choosing selection criteriafor a generalized study, the use of a standardized ground motion set car-ries some appeal. It offers the benefit of consistency with other research inwhich it is used, including the possibility of retroactively applying findingsfrom future research using this motion set to conclusions from the currentproject.The ground motion set prescribed by FEMA P695 [FEMA, 2009] for usein quantifying seismic performance factors was selected for the current study.It includes subsets of far-field and near-fault motions, with the near-faultmotions further split into pulse and non-pulse motions. The set includesboth horizontal components from 50 records, for a total of 100 motions. Ofthese, 22 records (44 motions) are classified as far-field, 14 (28) as near-faultpulse, and 14 (28) as near-fault non-pulse. Portions of the study were runusing the full set of motions, though the analysis focuses primarily on thefar-field set.6.2.3 Ground motion characterizationThe motions in the far-field set had moment magnitudes (MW ) between6.5 and 7.62. Shear-wave velocities in the upper 30m of soil (VS30) variedbetween 192 and 724m/s, and Joyner-Boore distances (rjb) varied between 7and 26 km (see Figure 6.2). This distance measure represents the horizontaldistance from the station to a point on the earth’s surface that lies directlyabove the rupture [Boore et al., 1997].The effects of spectral shape were examined for the far-field set as partof this study. In order to do so, the ε values for each ground motion werecalculated. The original (unscaled) response spectra of the far-field motionsare shown in Figure 6.3a, with mean and ±1σ curves overlaid in blue. Fig-ure 6.3b shows the ε calculated for each period for each motion. Note thaton average, the motions have a slight positive-ε bias, with µε ≈ 0.5 for shortand mid-range periods. The dip between periods of 0 and 0.1 s is due tothe interpolation between the listed coefficients in Boore et al., and can beignored — values in this range were not used in the study.Scaling of ground motions is critical for procedures such as the incre-mental dynamic analysis (IDA) prescribed in FEMA P695, in which thewhole set of motions is uniformly incremented at each step in the analysis.A variation of this procedure, in which each ground motion is incrementedseparately, was used in the current study. This renders the reference scal-ing of the ground motions irrelevant, except for the convenience aspect ofobtaining similar scale factors at various performance points among differ-1296.2. Ground motions0102030(a)r jb(km)6 6.5 7 7.5 80200400600800(b)MWV S30(m/s)Figure 6.2: Magnitude, distance, and shear wave velocity of far-field groundmotions1306.2. Ground motions012(a)S a(g)0 0.5 1.0 1.5 2.0−3−2−10123(b)T (sec)εFigure 6.3: Response spectra for far-field ground motions - original0 0.5 1.0 1.5 2.0012T (sec)S a(g)Figure 6.4: Response spectra for far-field ground motions - normalized1316.3. Model configurationent motions. To achieve this convenience, the motions were normalized bythe same method discussed in Appendix A of FEMA P695 — namely, byscaling the geometric mean of the peak ground velocities (PGV) of the twocomponents of each record to a constant value, such that the average of themean PGV did not change for the set of motions. The normalized spectraare shown in Figure 6.4. All motion scales referenced in this chapter arerelative to these normalized motions. The full set of motions is tabulated inAppendix I.6.3 Model configurationThe functional configuration of the model is illustrated in Figure 6.5. TheWorking Model Basic code from a sample script creating the model for asingle configuration is listed in Appendix H. The details of the model aregenerally the same as those described in Chapter 5, with a few changes:• The vertical offset of the wall-diaphragm connections relative to thetop and bottom of the wall was changed to zero. The offsets used inChapter 5 were intended only to replicate the conditions of the testrig. The height between supports is now equal to the exact height ofthe wall.• The definition of the chamfers at the crack and at the base of thewall was changed to accommodate separate vertical and horizontaldimensions. The vertical dimension was set to represent one mortarbed thickness, while the horizontal dimension is a parameter withinthe study. The change to a fixed vertical dimension is expected tohave minimal impact on the results.• Provision for an applied overburden load was added to the model.Two methods of application were investigated, indicated by details (a)and (b) in Figure 6.5. In detail (a), the load is applied to a blockresting on top of the wall. The block is frictionless and constrainedfrom rotation. Its horizontal displacement is matched to that of thetop diaphragm, but it is free to move vertically. The eccentricities e1and e2 define the extents of the block. As the top wall segment rotatesthrough vertical, the point of contact at which the overburden load isapplied changes from one side of the block to the other. This detail isrepresentative of the load being applied by either a wall in the storeyabove, a parapet, or joists. In detail (b), the overburden is applied as1326.3. Model configurationa force at a fixed eccentricity. The location of the force remains thesame regardless of the rotation of the top wall segment. This detailcould be representative of a post-tensioning retrofit.The input parameters used in a given configuration determine the valuesof the derived parameters. Input and derived parameters are defined in theupper and lower sections, respectively, of Table 6.1. When an input param-eter is changed between two configurations, the remaining input parametersretain constant values, and the derived parameters are recalculated. Forexample, if changing the slenderness ratio, the thickness remains the samebut the height is recalculated. The new height results in a new wall volume,which for the same density results in greater wall mass. The stiffness thenchanges to retain the same period. With the changed stiffness and mass,the damping constant changes to retain the same damping ratio.It is worth revisiting the definition of the ‘diaphragm period’ (Tb or Tt,or more generally the system period, Ts). A cracked wall connected to flex-ible diaphragms combines three responses: the vibration of two diaphragmsystems and the rocking response of the wall. The diaphragm period, asdefined here, is a reference indicator of diaphragm stiffness, and is an ap-proximation of the initial period of vibration of a diaphragm connected touncracked walls. As observed in the testing, the rocking motion of a crackedwall does not have a distinct period (Section 4.3.2.4). Furthermore, the ad-ditional degree of freedom created by the crack changes the effective massof the diaphragm system, thereby changing its effective period relative tothat calculated by simple tributary mass. While the diaphragm period is aconvenient and intuitive way to characterize such a system, it is importantto keep note of its limitations.1336.3. Model configurationkbcbktctMDbMDtthbhthPeshsve1 e2Ground motioninput(a)(b)MWtMWbRigid frameFigure 6.5: Model configuration1346.3. Model configurationTable 6.1: Model parametersParameterReferencevalue Description/FormulationInputt 330mm wall thicknessh/t 11 slenderness ratioL 1.0m wall lengthρ 2100 kg/m3 wall densityhcr 0.6 relative crack heightsv 12mm joint thicknesssh 10mm spall depthζt, ζb 0.05 damping ratio, top & bottomTt, Tb 1.0 s period, top & bottomRMt , RMb 3.0 mass ratio, top & bottomp 0 kN/m axial load per unit lengthDerivedh 3.63m t·(h/t)ht 1.452m h·(1− hcr)hb 2.178m h·hcrMw 2516 kg ρ·h·t·LMwt 1006 kg Mw ·(1− hcr)Mwb 1510 kg Mw ·hcrMdt ,Mdb 3773 kg RMt,b ·Mw2kt, kb 199 kN/m( 2piTt,b)2·(Mw2 +Mdt,b)ct, cb 3160Ns/m ζt,b · 2√kt,b ·(Mw2 +Mdt,b)P 0 kN p·L1356.4. Study methodology6.4 Study methodologyThe parametric study was grouped into two phases: (1) investigating theeffect of each single parameter relative to a reference configuration, and (2)running the full matrix of parametric combinations of a subset of primaryparameters. Phase 1 was split into two sub-phases: phase 1a investigatedthe effect of diaphragm flexibility, including ground motion characteristics,while phase 1b investigated the effects of other parameters. The intent ofphase 1 is to provide a rudimentary look at effects of individual parameters:the subsection for each parameter is a figurative cross-section of the effect ofthat parameter on the reference configuration. The results are intended tobe applied only conceptually, and care should be taken not to lose contextof any numerical values shown in plots. Phase 2 is intended to provide a fulldataset to be used as a basis for developing recommendations for assessment.Here, conclusions drawn in phase 1 are incorporated, and numerical valuesare intended for direct application. The analysis procedure for a singleconfiguration was the same in both phases; only the selection of parametersused to create the configurations varied.The normalized ground motions described in Section 6.2.3 were used inan IDA. It is difficult to pin down a fundamental period for a given configu-ration of the model, since there could be variable diaphragm periods at topand bottom, in addition to the non-periodic response of the rocking wall. Atraditional IDA in which the whole set of motions is uniformly incrementedafter being scaled at the fundamental period of the structure would thusbe difficult to both define and interpret. To circumvent these issues, eachground motion was incremented separately, and the results aggregated inpost-processing. This renders the reference scale of each motion irrelevant.Each motion was incremented (↑) and decremented (↓) as follows. Here,S refers to motion scale — relative to the normalized motions (Figure 6.4)— and the parameter values are listed in Table 6.2.1. Run at Sstart2. ↑ by Smajor until a run reaches drockthreshold3. ↓ by Smajor once4. ↑ by Sminor until a run reaches collapse5. ↓ by 12 ·Sminor onceThe range of parameter values considered in phase 1 is listed in Table 6.3,divided into phases 1a (top section) and 1b (bottom section). Parameters1366.4. Study methodologyTable 6.2: Incrementation parametersParameter ValueSstart 0.50Smajor 0.40Sminor 0.10drockthreshold 0.10·twere varied one at a time, i.e. each configuration is the same as the referenceconfiguration with the exception of one parameter only. The selection ofreference values and ranges for the various parameters are discussed in theapplicable subsections in Sections 6.5.2 and 6.5.3. Where parameters arespecified for both top and bottom locations, they were changed such thatthe top and bottom values were the same, expect as specifically noted. Whilethe primary reference period was Ts = 1.0 s, all the runs listed in Table 6.3were repeated for a secondary reference period of Ts = 0.5 s. The totalnumber of configurations tested in this phase was 73. The full set of 100ground motions was run for each of these configurations.Table 6.3: Phase 1 runsParameterReferencevalue Modelled values1a Tt, Tb 1.0 s 0.0 0.2 0.5 0.75 1.0 1.25 1.5 2.0Tt only[a] 1.0 s 0.0 0.2 0.5 0.75 1.0 1.25 1.5 2.01bt[b] 330mm 110 220 330h/t 11 4 8 11 14 18 22hcr 0.6 0.4 0.5 0.6 0.7 0.8sh 10mm 5 10 15ζt, ζb 0.05 0.03 0.05 0.07 0.10RMt , RMb 3.0 0.3 1.0 3.0 10.0p 0 kN/m 0 2.5 10 25 50[a] For constant values of Tb = 1.0 s and Tb = 0.5 s[b] For all combinations of h/t = 11 and 22, and Ts = 0, 0.5, and 1.0 sThe range of parameter values considered in phase 2 is listed in Ta-ble 6.4. Each combination of parameters was run, resulting in a total of210 unique configurations (though several of these configurations duplicate1376.5. Modelling resultsa configuration from phase 1). The full set of 100 ground motions was runfor those configurations with no overburden load (P = 0), while for thoseconfigurations with overburden load, only the far-field set of motions wasused in an effort to reduce computational time3.Table 6.4: Phase 2 runsParameterReferencevalue Modelled valuesh/t 11 8 11 14 18 22hcr 0.6 0.5 0.6 0.7Tt, Tb[a] 1.00 s 0.00 0.50 1.00 2.00p 0 kN/m 0 10[a] For Tb = Tt and for constant values of Tb = 0 s6.5 Modelling resultsThis section presents the most pertinent results of the parametric modelling.Visualizing the data from roughly 200,000 runs of a multi-degree-of-freedommodel will inevitably involve a significant number of plots. To aid the reader,the plots in this section have been designed to maintain consistency in style,size, and limits as much as possible, and reference curves that remain thesame between various plots are shown in black. Unless indicated otherwise,all plots refer to the far-field motion subset only. Concepts are introducedfor the reference configuration, followed by the presentation of results forphase 1 configurations, and finally results for phase 2.6.5.1 Reference configurationFor any single configuration (in this case, the reference configuration), eachground motion was incrementally scaled until collapse of the wall was ob-served (see Section 6.4). The primary interest in the analysis of the resultslies in quantifying the intensity of the motions that result in collapse. Oncethe analysis is complete, each motion’s scale at the lowest level causing col-lapse is known, and it remains to describe the intensity of these motions.Many possible intensity measures could be used to achieve this — e.g.,3A computer with an Intel i7 4770K CPU and 16GB RAM was able to run the modelabout 20% faster than real-time1386.5. Modelling resultspeak ground acceleration/velocity/displacement, spectral acceleration/ve-locity/displacement at a period of choice, Arias intensity, etc. As a startingpoint, the acceleration response spectrum of each ground motion at thescale causing collapse was compiled. The distribution of spectral values ateach period increment was evaluated, and the mean4, 10th percentile, and90th percentile of these spectral values at collapse are shown in Figure 6.6.These spectral curves will subsequently be referred to collectively as collapsespectra.Distribution of Sa at T = 1.0 s90th percentilemean10th percentile0 0.5 1.0 1.5 2.000.51.01.5T (s)S a(T)(g)Figure 6.6: Collapse spectra, reference configurationTo allow comparison of results across various configurations, the sameintensity measure should be used consistently throughout the analysis. Ini-tially, Sa(1.0 s) is selected as the intensity measure, with the choice of in-tensity measure discussed subsequently. The ordinate at T = 1.0 s on theacceleration spectrum at collapse of each ground motion is noted. A set ofSa values is thus obtained — one for each ground motion. The distributionof these values can then be plotted as an empirical CDF, and a lognormalCDF can be fitted to the data, as shown in Figure 6.7. These curves rep-resent the vertical distribution of points at T = 1.0 s in Figure 6.6. The4Spectral values at collapse are assumed to be lognormally distributed. When referringto spectral values, the term mean shall be taken to signify the geometric mean, which isequivalent to the median in the case of a lognormal distribution.1396.5. Modelling resultsintensity measure at a probability of collapse of interest (in this case, illus-trated as 10%) can then be obtained from the fitted CDF. Here, we considerthe term probability of collapse, Pcol, to be synonymous with the proportionof ground motions causing collapse (which is what is actually plotted). It istherefore a conditional probability, conditioned on the parameters involvedin the particular plot. In this case, it is conditioned on the spectral shapedistribution of the far-field motions, among other factors (e.g., crack height,slenderness ratio, etc.).ReferenceconfigurationPcol = 10%0 0.2 0.4 0.6 0.8 1.0 1.200.20.40.60.81.0Sa(1.0) (g)Proportionofmotionscausingcollapse Empirical Lognormal fitFigure 6.7: Reference fragility curveThe choice of Sa(1.0 s) as intensity measure is reasonable for a systemwith diaphragm periods of Ts = 1.0 s. For systems with different diaphragmperiods, or systems in which the top and bottom periods differ, the choiceof measure is not clear. One approach to quantifying the appropriateness ofan intensity measure is to consider the variability produced in the fragilitycurve using that intensity measure. Tightly distributed results — exhibitingless variance (steeper fragility curves) — are preferable to results exhibitinglarger variance, in the sense that it is undesirable to use an intensity measurethat introduces additional variance simply because the measure is unrelatedto the response quantity of interest. A low variance suggests that the selectedmeasure is in fact closely related to the response quantity. The coefficient ofvariation (cv) is a convenient indicator of this variance, since it is normalized1406.5. Modelling resultsto the mean of the distribution. For a lognormal distribution, it can becalculated as:cv =√eσ2 − 1 (6.2)Here, σ is the standard deviation in log space. For a given configuration,one can calculate cv for any possible choice of intensity measure. Consid-ering spectral acceleration as the measure of choice, cv can be calculatedfor every possible period selection. These results are plotted for the refer-ence configuration (Ts = 1.0 s) in Figure 6.8a, where the x-axis indicates theperiod, Tim, for the intensity measure, Sa(Tim). It is clear that the lowestvariance is produced when a period of Tim = 1.0 s is selected (i.e. the in-tensity measure becomes Sa(1.0 s)). The variance dips sharply from bothshorter and longer periods to reach its minimum where Tim = Ts.The same analysis was conducted for configurations with different systemperiods (such that Tt = Tb within each configuration, denoted simply as Ts).The cv curves for each configuration are overlaid onto the curve from thereference configuration in Figure 6.8b. Several noteworthy features can beobserved. First, all configurations with system periods of 0.75 s or longerfeature similar sharp minimums of cv at Tim = Ts, and the minimum cvvalues are similar for each of these configurations. For configurations withshorter periods, however, there is no well-defined minimum of cv at anyTim. Second, configurations with system periods of 1.25 s or longer exhibitconsiderably larger variance for small Tim than those systems with periodsof 1.0 s or shorter.Examining the trends in this plot suggests that the choice of Tim = 1.0 sproduces close to the lowest overall variance among systems with periodsvarying between 0 and 2 s. Long period systems are subject to the highestvariance with this selection. For the systems with periods of 0, 0.2, and 0.5 s,this selection actually yields close to the lowest possible variance - betterthan at Tim = Ts.Spectral acceleration is a universal design parameter readily availableto practicing engineers, which makes it a logical choice for selection as anintensity measure. In addition, Sa(1.0 s) specifically is typically a key bench-mark within a design spectrum. It is also the parameter used by the existingassessment procedure for out-of-plane stability of URM walls in ASCE 41[ASCE, 2014]. Combined with the results of Figure 6.8b discussed previ-ously, these factors form a strong case for the selection of Sa(1.0 s) as theintensity measure of choice, and it will be used for most aspects in theremainder of this chapter.1416.5. Modelling resultsReference configuration00.51.0(a)c vTs = 0Ts = 0.2 sTs = 0.5 sTs = 0.75 sTs = 1.0 sTs = 1.25 sTs = 1.5 sTs = 2.0 s0 0.5 1.0 1.5 2.0 2.5 3.000.51.0(b)Tim (s)c vFigure 6.8: Coefficient of variation, for varying intensity measure choice1426.5. Modelling results6.5.2 Phase 1a: Effect of diaphragm stiffnessConfigurations with periods between 0 (rigid) and 2 s were considered. Fra-gility curves are plotted in Figure 6.9. At first glance, it is difficult to spotany trends in this plot. It is also notable that there is a wide range ofvariance among these fragility curves; this is expected, considering that allthe curves are all plotted for the same intensity measure. The variance withchoice of intensity measure was illustrated earlier in Figure 6.8.Ts = 0Ts = 0.2 sTs = 0.5 sTs = 1.0 sTs = 1.25 sT s=0.75sT s=1.5s Ts =2.0 sPcol = 10%0 0.2 0.4 0.6 0.8 1.0 1.200.20.40.60.81.0Sa(1.0) (g)ProportionofmotionscausingcollapseFigure 6.9: Fragility curves, varying periodFigure 6.10 shows constant Pcol points for 10% and 50% levels, whichclarifies the fragility plot. At a given period, the difference between the twocurves indicates the variance of the fragility curve for that period. Varianceis smallest at Ts = Tim = 1.0 s, is larger but reasonably consistent forshorter periods, and increases more significantly towards longer periods —consistent with Figure 6.8.Due to the discrepancies in variance among the various periods, the 50%curve is perhaps most representative of the ‘real’ trend in this figure. Wallsin systems where Ts ≤ 0.2 s are significantly more resilient than those withlonger periods. Resilience increases once more at periods beyond 1.25 s.Stability is lowest at Ts = 0.75 s, but not much lower than at other pointsbetween 0.5 and 1.25 s.1436.5. Modelling resultsPcol = 10%Pcol= 50%0 0.5 1.0 1.5 2.000.20.40.6Ts (s)S a(1.0)(g)atvariousP colFigure 6.10: Target points, varying periodTo confirm whether this trend is only a product of the choice of inten-sity measure, the same plot was repeated for Pcol = 10% and 50% usingspectral acceleration at various Tim as alternate intensity measures (Fig-ure 6.11). Each curve on these plots is for a different intensity measure.The curves are shifted vertically corresponding roughly to the shape of themean ground motion spectrum: mid-range periods produce the highest Savalues. For Pcol = 50% (Figure 6.11a), the trend is consistent among allintensity measures. For Pcol = 10% (Figure 6.11b), local upward spikes inSa are visible for the longer periods where Ts = Tim, since the variance dipsat these points. Aside from these differences, however, the trend is gener-ally consistent among all the curves: short periods perform best, mid-rangeperiods perform worst, and some improvement occurs towards the longestperiods — regardless of the intensity measure used.As a complementary look at the issue, Figure 6.12 shows the collapsespectra for each of the configurations overlaid on the results from the ref-erence configuration (Ts = 1.0 s, shaded in grey). For clarity, only two orthree additional curves are shown on each plot. These plots illustrate therelative performance of walls with varying system periods perhaps least am-biguously. The fact that the shape of the mean curves varies little amongall the configurations emphasizes that a single intensity measure is perfectlyadequate to describe the relative performance of the different configurations1446.5. Modelling resultsTim = 0.00 (red)Tim= 0.20Tim= 0.50Tim= 0.75Tim =1.00 (black)Tim = 1.25Tim = 1.50Tim = 2.0000.20.40.60.81.01.21.4(a)S a(Tim)(g)atP col=50%Tim = 0.00 Tim= 0.20Tim =0.50Tim =0.75Tim = 1.00Tim = 1.25Tim = 1.50Tim = 2.000 0.5 1.0 1.5 2.000.20.40.60.8(b)Ts (s)S a(Tim)(g)atP col=10%Figure 6.11: Target points, varying period: effect of intensity measure1456.5. Modelling resultsTs = 0.0 sTs = 0.2 sShort periods00.51.01.5(a) Pcol = 50%Pcol = 10%, 90%S a(T)(g)Ts = 1.0 sTs = 0.5 sTs = 0.75 sMid-range periods00.51.01.5(b)S a(T)(g)Ts = 1.25 sTs = 1.5 sTs = 2.0 sLong periods0 0.5 1.0 1.5 2.000.51.01.5(c)T (s)S a(T)(g)Figure 6.12: Collapse spectra, varying period1466.5. Modelling resultsif the mean curves are used as the basis, and that it could almost equallywell be any intensity measure — consistent with the findings in Figure 6.11a.The variability in the variance among the configurations complicatesmatters when one is concerned with points on the tails of the distributions.At the T = 1.0 s ordinate in Figure 6.12b, for example, the 10% points forTs = 0.5 and 0.75 s configurations fall significantly below the 10% point forthe Ts = 1.0 s configuration. This is despite the fact that there is littledifference between the mean curves.This illustrates a drawback of using a single intensity measure in theanalysis of results. The variability in the variance exists because the Savalues that are plotted are from the spectra of real ground motions, which aresubject to significant peaks and valleys throughout the period range. WhenTs 6= Tim, the mean collapse spectrum implicitly relates the mean shape ofthe spectra at Ts to that at Tim. This translates well when comparing theresults with a uniform hazard spectrum, which is an aggregation of hazardsevaluated independently at each period. At low Pcol, however, the collapsespectrum is implicitly picking out motions that have relative peaks at Timand relative valleys at Ts. This is technically correct, but care must be takenif these results are interpreted in the context of a comparison to a uniformhazard spectrum. If taken at face value, low Pcol values at Ts 6= Tim mayproduce conservative assessments of wall stability.When forming assessment criteria based on such results, some judge-ment should be exercised in the interpretation of the low Pcol values. Anapproximation of ‘design’ Pcol = 10% values based on Figure 6.10 might beto take the Pcol = 50% curve and shift it down until it hits the Pcol = 10%value at T = 1.0 s. This would preserve the mean relative responses betweensystems with different periods while accounting for the lower collapse prob-ability based on the best matched response at Ts = Tim. This issue will beexplored further in the following sections as the effects of ground motioncharacteristics are investigated.6.5.2.1 Near-fault effectsThe findings of the analysis are conditioned on the characteristics, includingthe spectral shapes, of the ground motions used in the study. It is of someinterest to examine the effects of near-fault motions. In addition to the far-field records, the FEMA P695 ground motion set includes 56 ground motionsfrom 28 near-fault records: 14 of these are characterized as pulse motions,while 14 are characterized as non-pulse. For each record, the rotated com-ponents (fault-normal and fault-parallel) were used. It is important to note1476.5. Modelling resultsthat each of these subsets has a considerably smaller sample size (14) thanthe far-field set of 44 motions.Far-fieldPulseNon-pulsePcol = 10%Pcol = 50%0 0.5 1.0 1.5 2.000.20.40.6Ts (s)S a(1.0)(g)atvariousP colFault-normal Fault-parallel } for near-fault motionsFigure 6.13: Target points, varying period: with near-fault motionsThe plot of Figure 6.10 is repeated in Figure 6.13 with additional curvesshowing the results for the near-fault motions. In general, the trends iden-tified earlier hold true for all ground motion types: short periods performbest, mid-range periods perform worst, and some improvement occurs to-wards the longest periods. For the most part, the far-field motions resultin the least stable systems, with the exception that pulse motions are theleast stable at periods of 1.5 s or longer. Non-pulse motions produce curvesthat are very similar to the far-field motions. Pulse motions, on the otherhand, exhibit a nearly flat response for periods of 0.75 s or longer, with lessof the increase in stability with long periods seen in the other motions. Thisis consistent with pulse motions generally containing relatively more energyat long periods compared with non-pulse (including far-field) motions.The differences in response are examined further through the plotting ofthe collapse spectra in Figure 6.14. These plots show the collapse spectra forthe pulse motions overlaid on the far-field spectra (shaded in grey), for thereference configuration, with Ts = 1.0 s (Figure 6.14a), and the configurationwith Ts = 0.2 s (Figure 6.14b). Figure 6.14a shows that the pulse motionshave noticeably flatter collapse spectra on average than the far-field motions.1486.5. Modelling resultsTs = 1.0 sFault-parallelFault-normal00.51.01.5(a) Pcol = 50%Pcol = 10%, 90%S a(T)(g)Far-fieldTs = 0.2 sFault-parallelFault-normal0 0.5 1.0 1.5 2.000.51.01.5(b)T (s)S a(T)(g)Figure 6.14: Collapse spectra, pulse motions1496.5. Modelling resultsThis is reflective not only of the ‘collapse’ spectra, but rather the shape ofthe mean curve here is also simply an indication of the shape of the meanresponse spectra of the motions. It is thus evident that the pulse motionsdo, in fact, have significantly more energy content at long periods relativeto short periods than the far-field motions. The near-fault spectra are veryclose to the far-field spectra around the ordinate of T = 1.0 s. At shorterperiods, the near-fault spectra are thus significantly lower than the far-fieldones as a result of the flatter shape.Figure 6.14b shows that for the short-period configuration, the near-faultspectra are again lower than the far-field spectra for short periods, and arevery similar in the mid-range periods. The near-fault spectra actually creephigher than the far-field ones in the longer periods. It so happens that atthe ordinate of T = 1.0 s, the near-fault spectra are already a little higherthan the far-field ones, thus creating the results at T = 0.2 s of Figure 6.13.Perhaps the most noteworthy observation from this examination is thatit seems to indicate fairly definitively that the wall response for systems withstiff diaphragms depends to a greater extent on the spectral values at longerperiods than it does on the spectral values at the period of the diaphragm.For ground motions with significantly differing spectral shapes, the config-uration with Ts = 0.2 s reached collapse at similar values of Sa(1.0 s), butat very different values of Sa(0.2 s) (Figure 6.14b). This behaviour is con-sistent with the long periods observed in testing for large-amplitude rockingexcursions (Section 4.3.2.4), for both rigid and flexible diaphragms. For stiffsystems, it thus appears that the original diaphragm period is of limitedrelevance once rocking occurs. This supports the use of a single intensitymeasure, at least for systems with relatively stiff diaphragms, and moreoverconfirms that Sa(1.0 s) is in fact a reasonable choice.6.5.2.2 Spectral shape effectsThe previous section established that near-fault pulse motions exhibit dif-ferent characteristics than far-field motions, and that this has a notableeffect on out-of-plane wall stability under certain conditions. In particular,the flatter shape of the response spectra of the pulse motions produces pre-dictable differences in spectral values at collapse once it is noted that thewall response is heavily dependent on the spectral acceleration at longerperiods. In this section, the effect of the spectral shape of the far-field mo-tions themselves is examined. Here, the spectral shape is quantified by theε parameter, which is dependent on the period (ε values were plotted forfar-field motions in Figure 6.3b).1506.5. Modelling resultsFor a given configuration, a regression analysis can be done for Sacol vs.ε. Typically, in such a procedure, Sacol is taken at the fundamental periodof the structure [Haselton et al., 2011]; here it is therefore taken at Ts,which is dependent on the configuration. In keeping with the assumption oflognormally distributed Sa values, a linear regression is done between lnSaand ε. This results in a curved regression line in non-log space. Regressionplots for selected periods are shown in Figure 6.15.A particular regression can be summarized by the slope of the fit line andthe coefficient of determination, R2. These values are illustrated for eachperiod in Figure 6.16. R2 represents the proportion of the variability in theregressand (lnSa) that is explained by the regressor (ε). Since the regressionis linear in the log space, the slope varies in the real space; a reference‘slope’ is specified as the change in Sa between ε = 0 and ε = 1. This slopeis illustrated in both absolute (Figure 6.16b) and relative (Figure 6.16c)terms.The regression has minimal significance at periods of T = 0.75 s orgreater, with R2 exceeding 0.1 only at T = 0.2 and 0.5 s. These resultsare consistent with a system that is primarily sensitive to long-period con-tent, as explained below.In general, a large ε at one period indicates that the spectrum will tendto be peaked around that period, and will be relatively lower at other pe-riods (e.g., the significant peak around T = 1.2 s in Figure 6.1) [Baker andCornell, 2005]. Consider a hypothetical system that is sensitive to the spec-tral acceleration at 1 s, but not to that at other periods. All ground motionswould cause collapse at roughly the same Sa(1 s) values, regardless of theirspectral values around 0.2 s. At the collapse scale, the motions that havelarge Sa(0.2 s) values will then also have large relative Sa(0.2 s)/Sa(1 s) val-ues. This means that these motions will also tend to have large ε valuesaround 0.2 s. Consequently, there would be a direct correlation betweenε(0.2 s) and Sacol(0.2 s).At the same time, the set of collapse motions would include a full rangeof ε at 1 s, which are indicative of whether the spectrum of a particularmotion is relatively high or low at 1 s. Since the motions all caused collapseat roughly the same Sa(1 s) values, there would be little correlation betweenε(1 s) and Sacol(1 s). Both of these predictions for this hypothetical systemare observed in Figures 6.15 and 6.16 for short period systems. Notably,the correlation is roughly equally weak for all long period systems. Thisindicates that there is not one particular period to which all the systems aresensitive, but that for long periods, the system is sensitive at its own period.These observations are furthermore consistent with what was observed in the1516.5. Modelling results−3 −2 −1 0 1 2 300.51.01.52.0S acol(Ts)(a) Ts = 0 s−3 −2 −1 0 1 2 300.51.01.52.02.53.0(b) Ts = 0.2 s−3 −2 −1 0 1 2 300.51.01.52.0S acol(Ts)(c) Ts = 0.5 s−3 −2 −1 0 1 2 300.51.01.52.0(d) Ts = 0.75 s−3 −2 −1 0 1 2 300.51.01.52.0εS acol(Ts)(e) Ts = 1.0 s−3 −2 −1 0 1 2 300.51.01.52.0ε(f) Ts = 1.5 sFigure 6.15: ln (Sacol) vs. ε regressions1526.5. Modelling results00.20.4(a)R200.20.4(b)∆S a(ε=0→1)(g)0 0.5 1.0 1.5 2.000.20.4(c)T , Ts (s)∆S a(ε=0→1)/Sa(ε=0)Figure 6.16: R2 and slope of ln (Sacol) vs. ε regression1536.5. Modelling resultscoefficient of variation plot in Figure 6.8. There, it was noted that for short-period configurations, the spectral acceleration around T = 1.0 s was asgood or better of an indicator of collapse than the system period, while forlong period systems the spectral acceleration at Ts was the best indicator.It can be concluded that the effect of spectral shape is negligible whenappropriate intensity measures are used, given that the rocking walls aresensitive to long-period energy content. If one were to use Tim = Ts atshort periods, the results should be adjusted for spectral shape — but thiswould simply be a convoluted method of returning to the spectral contentat the longer periods. It is more appropriate to use the long period intensitymeasures in the first place.6.5.2.3 Variable top and bottom periodThe previous sections have all dealt with configurations in which the top andbottom diaphragm periods are the same. In reality, diaphragms are likelyto vary in stiffness, particularly in the top storey where one diaphragm is aroof while the other is a floor. The top and bottom inputs into the out-of-plane wall can thus be out of phase, which has the potential to change theresponse significantly. In this section, configurations with different periodsat the top and bottom of the wall are examined. The top period is variedwhile the bottom period remains constant. Bottom periods of Tb = 0.5 and1.0 s were considered.Fragility curves are shown in Figure 6.17 for Tb = 1.0 s and in Figure 6.18for Tb = 0.5 s. When compared with Figure 6.9, in which both periods varysimultaneously, it is immediately apparent that changing only the top periodhas a lesser effect on the results than changing both periods. Constant Pcolpoints for these three plots are shown in Figure 6.19, which illustrates thedifferences well.The variance of the fragility curves remains more consistent when onlychanging the top period than when changing both periods. Notably, theparticularly small variance observed at Ts = 1.0 s (resulting from the choiceof Sa(1.0 s) as the intensity measure) persists even as Tt is decreased to 0.2 s.This suggests that between the top and bottom periods, the bottom periodis the more dominant in terms of the effect on out-of-plane wall stability.It was noted in Section 4.3.2.1 that wall collapse is dictated only by thestability of the bottom wall segment. In a cracked wall, the motion of thetop diaphragm is only indirectly transmitted to the critical bottom segment,while the bottom diaphragm is directly connected to it. In this regard, the1546.5. Modelling resultsTt = 0.2 sTt = 0.5 sTt = 0.75 sTt = 1.0 sTt = 1.25 sTt = 1.5 sTt = 2.0 sPcol = 10%0 0.2 0.4 0.6 0.8 1.0 1.200.20.40.60.81.0Sa(1.0) (g)ProportionofmotionscausingcollapseFigure 6.17: Fragility curves, varying top period only, Tb = 1.0 sTt = 0.2 sTt = 0.5 sTt = 0.75 sTt = 1.0 sTt = 1.25 sTt = 1.5 sTt = 2.0 sPcol = 10%0 0.2 0.4 0.6 0.8 1.0 1.200.20.40.60.81.0Sa(1.0) (g)ProportionofmotionscausingcollapseFigure 6.18: Fragility curves, varying top period only, Tb = 0.5 s1556.5. Modelling resultsTb =TtTb = 1.0 sTb = 0.5 s0 0.5 1.0 1.5 2.000.20.40.6Tt (s)S a(1.0)(g)atvariousP colPcol=50% Pcol = 10%Figure 6.19: Target points, varying top and bottom periodsgreater observed influence of the bottom diaphragm stiffness is consistentwith expectations.Selected fragility curves are isolated to illustrate these effects in greaterdetail. Figure 6.20 shows all four combinations of periods of 0.5 and 1.0 sdistributed between the top and bottom of the wall. The four curves allshow different variances, but intersect at roughly the same location nearPcol = 45%. The configurations with equal top and bottom periods bracketthe responses, with the varying top and bottom periods falling in between.Beginning with a reference of an equal period curve, changing the bottomperiod pushes the new curve further towards the second equal period curvethan does changing the top period. In this particular case, the curve for Tt =0.5 s/Tb = 1.0 s has changed very little relative to that of Tt = 1.0 s/Tb =1.0 s, particularly for low Pcol.Figure 6.21 shows periods of 1.0 and 1.25 s. Here, changing the top periodfrom 1.0 to 1.25 s pushes the fragility curve close to that for Tt = 1.25 s/Tb =1.25 s, particularly at low Pcol. While the earlier observations suggested thatthe bottom period appears to be the generally more dominant of the two, itis important to note that the top period can still have a large effect on wallstability.In general, it can be concluded that the out-of-phase action produced1566.5. Modelling resultsTs =0.5 s0.5 sTs =1.0 s0.5 sTs =0.5 s1.0 sTs =1.0 s1.0 sPcol = 10%note: periods denoted as Ts =TtTb0 0.2 0.4 0.6 0.8 1.0 1.200.20.40.60.81.0Sa(1.0) (g)ProportionofmotionscausingcollapseFigure 6.20: Fragility curves, Ts = 0.5 and 1.0 s combinationsTs =1.25 s1.25 sTs =1.0 s1.0 sTs =1.25 s1.0 sPcol = 10%note: periods denoted as Ts =TtTb0 0.2 0.4 0.6 0.8 1.0 1.200.20.40.60.81.0Sa(1.0) (g)ProportionofmotionscausingcollapseFigure 6.21: Fragility curves, Ts = 1.0 and 1.25 s combinations1576.5. Modelling resultsby diaphragms of different stiffness at the top and base of a wall will resultin out-of-plane wall stability in between that of configurations when bothdiaphragms are equal to one of these periods. It is critical to clarify that theterm stability, in this sense, refers to the aggregated collapse levels of a suf-ficiently large suite of ground motions — i.e. the entire fragility curve. Theconclusion does not necessarily hold for any single ground motion, where thestability at various periods is dependent on the distribution of peaks andvalleys in the response spectrum. On fragility curves of systems with differ-ent periods, one can not assume that each ground motion falls at the samelocation on each curve — in fact, it is unlikely. Comparisons of these analysisresults with the test results must therefore be made cautiously. Because thetests used only a single ground motion, the fact that wall FR-3 performedbetter than both FF-3 and RR-3 can still be considered consistent with theanalysis results.Typically there is significant uncertainty in the assessment of diaphragmperiods, and it would not be prudent to rely on a calculated difference inperiod to produce an increase in assessed wall stability. For a given wall, itwould be reasonable to use the least stable of the two calculated periods atthe top and bottom when conducting an assessment. Accordingly, configu-rations with different top and bottom periods are not considered further inthis study.6.5.2.4 Allowable spectrumAll analysis discussed up to this point has involved the comparison of thevarious configurations at some single intensity measure. In particular forthe simple model in question, and for configurations which have top andbottom periods equal, an alternative approach could be to tie the choiceof intensity measure to the system configuration — i.e. set Tim = Ts foreach configuration. Plotting constant Pcol curves in such a way produceswhat could be termed an ‘allowable spectrum’. The Pcol = 10, 50, and 90%curves are plotted in Figure 6.22. Here, the allowable spectrum for the Pcolof choice is analogous to a capacity.This plot illustrates the period dependency of the rocking wall system.At short periods (e.g., Ts = 0.2 s), the variance is very large. As discussed inSections 6.5.2.1 and 6.5.2.2, this is symptomatic of a system that is sensitiveto long-period content. At long periods, the variance is consistently small— for such periods, this ’allowable spectrum’ approach holds appeal.If desired, the ε regression procedure (see Section 6.5.2.2) could be usedto create ε-specific allowable spectra. To do so, the Sacol value for each1586.5. Modelling resultsPcol =50%Pcol = 10%Pcol =90%0 0.5 1.0 1.5 2.000.51.01.5T , Ts (s)S a(T)(g)Figure 6.22: Allowable spectrumground motion for each configuration would be adjusted according to theregression curve and the actual and target ε values. This creates a newempirical distribution for Sacol for each configuration; from this point, theprocess for creating the allowable spectrum is the same. Such a curve wouldhave reduced variance at Ts = 0.2 s, and potentially a significant shift invalues at this period depending on the target ε. These effects would also benotable, though reduced, for Ts = 0.5 s. At other periods, differences wouldbe minimal.The value of using the allowable spectrum approach should be carefullyevaluated. At short periods, it was already concluded that using Tim = Tsis a poor choice since the rocking response of the wall is not sensitive tocontent at short periods. For periods longer than 1 s, additional considera-tions relating to effects of large diaphragm displacements will likely becomesignificant (e.g., damage at wall corners due to horizontal bending deforma-tions). While these effects are beyond the scope of this study, it would beprudent to err on the conservative side when deciding allowable limits forlong-period systems. Given that long-period systems have shown the sameor better resiliency as systems with Ts = 1 s, it is perhaps not necessaryto quantify this additional resiliency by using the allowable spectrum, butrather to conservatively assign the same limits for long period systems asfor those at Ts = 1 s. With this intention in mind, the allowable spectrum1596.5. Modelling resultsmethod is not pursued in the remainder of this study.6.5.3 Phase 1b: Effect of other parametersThis section covers the single-parameter variations from the reference wall,as listed in Table 6.3. Results are presented and discussed for each parameterin turn. Recall that each of the parameter variations was run for the defaultreference configuration with Ts = 1.0 s as well as for the secondary referenceconfiguration with Ts = 0.5 s. Results in this section are plotted by defaultfor configurations with Ts = 1.0 s, unless otherwise indicated. Results atTs = 0.5 s were examined to ensure that trends are consistent, but for brevityare not included here.6.5.3.1 Crack heightThe parameters examined in preceding and subsequent sections are tied tophysical properties of the structure. They are subject primarily to epistemicuncertainty: ‘correct’ values of each parameter exist, but there is uncertaintyin determining them due to measurement difficulties, poor representation inthe model, etc. The crack location, on the other hand, is subject to aleatoricuncertainty: it does not have a ‘correct’ value at the assessment stage, sincewalls are likely uncracked. While the theoretical location of the crack can becalculated as the location of maximum stress (with some assumptions aboutacceleration distributions), the crack may well form at a different locationdue to spatial variability in materials and workmanship in the wall. This wasdemonstrated in the experimental phase of this project (see Section 4.3.6).The crack location is thus aptly considered as a random variable. Onecould reasonably assume that the expectation of this variable should fall atthe predicted location of maximum stress, with variance on either side ofthis value due to the construction details of the wall. As a basis to determinethis variance, results from three sets of experimental tests were used: ABKJoint Venture [1981b], Meisl et al. [2007], and the tests from the currentproject. Sharif et al. [2007] compiled the crack locations from the first twosets and calculated a mean relative crack height of 0.63 with a standarddeviation of 0.07. The current testing produced a mean crack height of 0.55with a standard deviation of 0.11. The location of maximum observed stressin the current tests was at a relative height of roughly 0.55.Considering all of these results, it is reasonable to assume normally-distributed crack heights with a mean of 0.60 and a standard deviation of0.10. Crack heights of 0.4, 0.5, 0.6, 0.7, and 0.8 were considered, correspond-1606.5. Modelling resultsing to the mean ±2σ. To represent the assumed normal distribution withthese discrete values, the distribution in Table 6.5 was assigned.Table 6.5: Probability of crack height occurringx P (hcr = x)0.4 0.070.5 0.240.6 0.380.7 0.240.8 0.07Total 1.00The fragility curve for each crack height is obtained from the analysis(Figure 6.23). At each Sa value, we can then calculate the total probabilityof collapse as:Pcol = ∑i[Pcol| (hcr = xi)] · [P (hcr = xi)] (6.3)Plotting this result for each Sa value yields the thick grey line in Fig-ure 6.23, which is the total fragility curve. Note that the total curve hasgreater variance than any of the curves for single crack locations, because ithas incorporated the additional uncertainty associated with the crack heightdistribution, whereas the individual curves are conditional on a particularcrack height.The constant Pcol values are plotted in Figure 6.24. A cubic function fitsthe data well within the domain considered. For this particular case, therelation is approximately:Sa = −1.37 (hcr)3 + 3.67 (hcr)2 − 3.39 (hcr) + 1.31 (6.4)1616.5. Modelling resultshcr = 0.6hcr = 0.7hcr = 0.8h cr=0.4h cr=0.5TotalPcol = 10%0 0.2 0.4 0.6 0.8 1.000.20.40.60.81.0Sa(1.0) (g)ProportionofmotionscausingcollapseFigure 6.23: Fragility curves, varying crack height0 0.2 0.4 0.6 0.8 1.000.10.20.30.40.50.60.7hcrS a(1.0)(g)atP col=10%Figure 6.24: Target points, varying crack height1626.5. Modelling results6.5.3.2 Slenderness ratioWalls of constant thickness and varying height were evaluated, where slen-derness ratios varied between 4 and 22. Fragility curves are plotted in Fig-ure 6.25. More slender (taller) walls consistently exhibited lower resiliencethan less slender (shorter) walls. Variance is fairly consistent among thedistributions, with the exception of the curve at h/t = 4, which shows no-tably more variance than the others. Plotting the constant Pcol values (Fig-ure 6.26) produces a smooth trend that is well defined by a power relation(the line shown). For this particular case, the relation is approximately:Sa = 2.77·(ht)−0.9 (6.5)h/t = 14h/t = 18h/t = 22h/t =4h/t =8h/t =11Pcol = 10%0 0.2 0.4 0.6 0.8 1.0 1.200.20.40.60.81.0Sa(1.0) (g)ProportionofmotionscausingcollapseFigure 6.25: Fragility curves, varying slenderness1636.5. Modelling results0 4 8 12 16 2000.20.40.60.8h/tS a(1.0)(g)atP col=10%Figure 6.26: Target points, varying slenderness ratio6.5.3.3 ThicknessWalls with the same slenderness ratio but different thicknesses (and thereforeheights) were evaluated — effectively, one wall was a scaled version of theother, with the exception of the spalling geometry, which remained fixedat its absolute dimensions. The thicknesses were representative of typicalone-, two-, and three-wythe walls: 110, 220 and 330mm, respectively. Thethinner walls were evaluated at each combination of h/t = 11 and 22, andTs = 0, 0.5, and 1.0 s.The fragility curves for Ts = 0.5 and 1.0 s are plotted in Figures 6.27and 6.28, both at h/t = 11. Relative collapse intensities of two-wythe vs.three-wythe and one-wythe vs. three-wythe walls are shown in Figure 6.29for the various system periods and h/t ratios. It is clear that thinner wallsexhibited consistently lower stability than the thicker walls. The differencewas most pronounced at Ts = 0.5 s, less so at Ts = 0, and the least so atTs = 1.0 s. The effect showed minimal variation with h/t.Makris and Konstantinidis [2003] conducted 2-dimensional analyses onrocking of simple rectangular blocks subjected to ground motions. Theynoted that for a given slenderness ratio, blocks exhibited roughly monotoni-cally decreasing peak rocking rotations as size increased — i.e. larger blockswere more stable than smaller blocks of the same proportions. The results1646.5. Modelling resultst =220mmt =330mmt =110mmPcol = 10%0 0.2 0.4 0.6 0.8 1.000.20.40.60.81.0Sa(1.0) (g)ProportionofmotionscausingcollapseFigure 6.27: Fragility curves, varying thickness, Ts = 1.0 st =220mmt =330mmt =110mmPcol = 10%0 0.2 0.4 0.6 0.8 1.000.20.40.60.81.0Sa(1.0) (g)ProportionofmotionscausingcollapseFigure 6.28: Fragility curves, varying thickness, Ts = 0.5 s1656.5. Modelling resultst = 110mm, h/t = 11t = 110mm, h/t = 22t = 220mm, h/t = 11t = 220mm, h/t = 220 0.5 1.000.20.40.60.81.0Ts (s)S acol/S acol 3−wytheatP col=10%Figure 6.29: Relative collapse intensity, one- and two-wythe walls vs. three-wythe wallsof this section are consistent with those findings, and the effect is of consid-erable importance, particularly with short- to moderate-period systems.6.5.3.4 SpallingThe horizontal size of the chamfer representing mortar spalling (sh, seeFigure 6.5) at the crack location was varied between 5 and 15mm. Thefragility curves are shown in Figure 6.30. Greater spalling produced lessresilient walls, as would be expected, but the magnitude of the influencewas negligible within the considered range.1666.5. Modelling resultssh = 5mmsh = 10mmsh = 15mmPcol = 10%0 0.2 0.4 0.6 0.8 1.000.20.40.60.81.0Sa(1.0) (g)ProportionofmotionscausingcollapseFigure 6.30: Fragility curves, varying spalling6.5.3.5 DampingDamping values of 3, 5, 7, and 10% were considered; fragility curves are plot-ted in Figure 6.31. Variance remains fairly consistent among all curves, withwalls becoming notably more resilient at higher damping values. ConstantPcol values are plotted in Figure 6.32.ASCE 41 allows the use of 10% damping for buildings with wood dia-phragms and cross walls that interconnect the diaphragm levels at a max-imum spacing of 12m transverse to the direction of motion, however therationale behind this value is not known. Wilson [2012] conducted full-scale dynamic testing on wood diaphragms representative of those in typicalURM buildings, in which he found that “the results provide no evidenceagainst the 5% inherent damping that is typically assumed for dynamicallyresponding structures, and this value is therefore recommended for timberfloor diaphragms”. Consequently, 5% damping was adopted as the referencevalue for the remainder of the study.1676.5. Modelling resultsζ = 3%ζ = 5%ζ = 7%ζ = 10%Pcol = 10%0 0.2 0.4 0.6 0.8 1.000.20.40.60.81.0Sa(1.0) (g)ProportionofmotionscausingcollapseFigure 6.31: Fragility curves, varying damping0 0.02 0.04 0.06 0.08 0.1000.10.20.30.40.50.60.7ζS a(1.0)(g)atP col=10%Figure 6.32: Target points, varying damping1686.5. Modelling results6.5.3.6 Diaphragm mass ratioThe diaphragm mass ratio in the model is defined as the mass of one dia-phragm divided by half the total wall mass. While this is unambiguous, itis less clear what this ratio should represent in a prototype building. Theratio is intended to capture the relative weight of translation-only elementsvs. rocking elements. Two methods for calculating the mass ratio were used,illustrated in Figure 6.33 for the diaphragm at the base of an upper-storeywall. The elements contributing to the diaphragm mass are hatched in red,while those contributing to the wall mass are hatched in blue.Method 1 Method 2MwMdMwMdFigure 6.33: Mass ratio definitionsIn method 1, the tributary wall mass — half of the storey height — bothabove and below the diaphragm is counted as wall mass. However, the wallin the storey below will be more stable than the one above due to the largeroverburden, and it is therefore less likely to undergo rocking. If it is not un-dergoing rocking, then it is more aptly modelled as a lumped mass attachedto the diaphragm. In this case, method 2 may be a better approximation,where the tributary wall mass below the diaphragm is counted towards thediaphragm mass.The range of possible ratios was examined for a sample prototype build-ing 10× 20m in plan. Light (2-wythe, 3m tall) and heavy (3-wythe, 4.5mtall) walls were considered. Diaphragm loads were considered as either bare1696.5. Modelling resultsfloor (0.5 kPa), floor with partitions (1.0 kPa), or floor with partitions and50% live load for office occupancy (2.2 kPa). Each of these combinationswas considered in both the long and short span directions of the diaphragm,and for calculation methods 1 and 2.For method 1, ratios fell between 0.2 and 3.2, while for method 2 theyfell between 1.3 and 8.6. A reference ratio of RM = 3 was used, and ratios of0.5, 1, 3, and 10 were modelled. Fragility curves are shown in Figure 6.34.The plot demonstrates that in general, lighter diaphragms result in moreresilient walls. Intuitively, this makes sense — a rocking wall will be ableto ‘push back’ against a lighter diaphragm more readily than on a heavierone while undergoing rocking. The differences are notable, but not large,particularly at low probabilities of collapse. At Pcol = 10%, Sa values forRM = 10 and for RM = 1 are within 7% of that at RM = 3, while forRM = 0.5, Sa is 18% higher.RM = 0.5RM = 1RM = 3RM = 10Pcol = 10%0 0.2 0.4 0.6 0.8 1.000.20.40.60.81.0Sa(1.0) (g)ProportionofmotionscausingcollapseFigure 6.34: Fragility curves, varying diaphragm mass ratio, T = 1.0 s6.5.3.7 Axial loadThe details of how the axial load is applied result in very significant dif-ferences in its effect. The two types of load application considered wereintroduced in details (a) and (b) of Figure 6.5. Five specific load applica-tions were considered in this section (illustrated specifically in Figure 6.35):1706.5. Modelling results• An applied force at a fixed location: (1) at the wall centerline, and (2)at 0.4·t away from the centerline• A load applied to a block that remains horizontal, where the blockstarts at one edge of the wall below and extends across the wall thick-ness by (3) one wythe, (4) two wythes, and (5) three wythesCase 1 Case 2 Case 3 Case 4 Case 5Figure 6.35: Axial load applicationOverburden loads consist of live loads plus dead loads from roofs, floors,walls, and parapets. Since overburden increases stability, it is conservativeto err on the low side when estimating loads. Live loads were therefore notconsidered in this analysis. At the lower bound, the overburden approacheszero — e.g., a top-storey (or single-storey) wall running parallel to roof joists,with a low or non-existent parapet. Heavily-loaded upper storey walls maysee overburden in the range of 15 kN/m with a large parapet and a roof load.Walls in the 2nd-from-top storey may see overburden in the range of 20 to50 kN/m, when including a parapet, roof and floor loads, and a top storeywall.Fragility curves for Case 2 are shown in Figure 6.36, and for Case 3in Figure 6.37. Constant Pcol values are plotted in Figure 6.38 for all 5cases at all axial load levels, including the best-fit straight lines (constrainedthrough the point at 0 load). The greatest stability gains for a given load areproduced by Case 5 (3-wythe block), followed by Case 4 (2-wythe block).Case 3 (1-wythe block) produced nearly the same effect as Case 1 (concentricfixed force).1716.5. Modelling resultsP=0P=2.5 kN/mP=10kN/mP=25kN/mP=50kN/mPcol = 10%0 0.2 0.4 0.6 0.8 1.0 1.200.20.40.60.81.0Sa(1.0) (g)ProportionofmotionscausingcollapseFigure 6.36: Fragility curves, varying axial load, Case 2P=0P=10kN/mP=50kN/mPcol = 10%0 0.2 0.4 0.6 0.8 1.0 1.200.20.40.60.81.0Sa(1.0) (g)ProportionofmotionscausingcollapseFigure 6.37: Fragility curves, varying axial load, Case 31726.5. Modelling results(2) ecc = 0.4·t, fixed(1) ecc= 0, fixed(3) joist pocket on 3-wythe(4)2-wythe on 3-wythe(5)3-wytheon 3-wythe0 10 20 30 40 5000.51.01.52.0P (kN/m)S a(1.0)(g)atP col=10%Figure 6.38: Target points, varying P and boundary conditionThe results illustrate that the effects of axial load are the result of acombination of the effects in each of the two possible rocking directions. Inevery case, the axial load is stabilizing until the rotation of the top wallsegment becomes such that the contact point between the top and bottomwall segments is directly below the point of action of the axial load on thetop block. These rotation limits are illustrated in Figure 6.39 for Cases 1and 2. In Case 1, the limit is the same in both directions (θ1L = θ1R),while in Case 2, the rotation limit is very small in one direction (θ2L < θ1L),but larger than the Case 1 limit in the other direction (θ2R > θ1R). Forrigid diaphragm conditions, Doherty et al. [2002] related the static force-displacement curves of Case 1 and Case 5 systems to those of equivalentcantilever walls with suitably adjusted geometry. In this study, the axialload application is modelled explicitly for all cases.While the top wall segment has not exceeded the rotation limit, thestabilization occurs as a result of the moment applied by the axial loadabout the point of contact between the wall segments. The moment arm,dm, is the horizontal distance between (1) the point of application of the axialforce and (2) the point of contact between the wall segments (Figure 6.40).The moment arm is greatest at the onset of a rocking excursion, as the crackjust begins to open. Beyond this point, any additional rotation of the top1736.5. Modelling resultsθ1L θ1Rθ2Lθ2RCase 1 Case 2Figure 6.39: Stabilizing limits of axial load applicationwall segment only reduces the moment arm, and thus proportionally reducesthe stabilizing moment.dmFigure 6.40: Stabilizing effect of axial load applicationIn Case 2, the eccentric force applies a large restoring moment in onedirection, but only a minimal one in the opposite direction — even at onsetof cracking. In the latter direction the wall is thus able to rock easily, andonce it rocks even slightly, the axial force is now destabilizing the upperblock (i.e. accelerating the rotation of the upper segment). The relativenet effect of stabilization vs. destabilization appears to be dependent on themagnitude of the axial load, resulting in progressively smaller stability gains1746.5. Modelling resultsper unit axial force as the axial force increases (Figure 6.38).Case 3 appears to shows a milder version of this behaviour: in Fig-ure 6.38, the slope from 0 to 50 kN/m is 16% lower than the slope from0 to 10 kN/m. Cases 1, 4, and 5 — all of which are more symmetricalthan Cases 2 and 3 — produce notably more linear results. Though thesample size is small, it would seem that in general, more symmetrical loadapplication leads to more linear stability gains, while more asymmetricalload application leads to lower stability gains for higher loads than for lowerloads.6.5.4 Phase 2: Parametric combinationsThis section provides the results of the Phase 2 runs, which consisted of afull combination matrix of selected parameters — crack height, slendernessratio, period, and axial load (Table 6.4). The results were processed into aseries of constant Pcol points, arranged in parallel plots as a function of h/tand diaphragm period. Both sets of plots are included in this section sinceeach illustrates different trends. On h/t plots, the best-fit power function isdrawn for each data series. Each plot type is presented as a series of fourplots, one for each Pcol value of 5%, 10%, 20%, and 50%. For reference,ASCE 41 allowable slenderness limits (see Section 2.2.3) for top storey andone storey walls are included in the h/t plots. Recall that these limits donot distinguish between diaphragm stiffnesses nor axial load levels.As discussed in Sections 6.5.2.1 and 6.5.2.2, the effects of variations inspectral shape, including due to near-fault motions, is minimal when evalu-ating stability as a function of Sa(1.0 s). Consequently, results are presentedfor far-field motions only in this section.Crack height was considered as a random variable, and the differentcrack heights were combined into a total probability curve using an assumeddistribution (see Section 6.5.3.1). The distribution of Table 6.5 was modifiedto include only the three crack heights evaluated in Phase 2 (0.5, 0.6, 0.7)as opposed to the five used in Phase 1 (0.4, 0.5, 0.6, 0.7, 0.8). This wasachieved by assigning the probabilities of occurrence for the 0.4 and 0.8crack heights to the 0.5 and 0.7 crack heights, respectively (Table 6.6).A comparison of the total probability curves for these two distributionsshowed negligible differences for low to moderate Pcol, with the modifiedcurve returning somewhat higher Pcol values for Pcol > 0.7. Since there islittle interest in the behaviour at high Pcol values, the modified curve wasconsidered an acceptable approximation of the original.1756.5. Modelling resultsTable 6.6: Probability of crack height occurringP (hcr = x)x Phase 1 Phase 20.4 0.07 —0.5 0.24 0.310.6 0.38 0.380.7 0.24 0.310.8 0.07 —Total 1.00 1.00Base case: Results are shown for the base case of no overburden, withTb = Tt. Slenderness ratio and period plots are provided in Figures 6.41 and6.42, respectively. Both of these plots illustrate that the magnitude of theeffect of the diaphragm period is heavily dependent on the slenderness ratio,with the effect being more important for less slender walls. In particular forthe lower Pcol values, it is evident that when h/t reaches 22, there is minimaldifference in allowable Sa(1.0) between all the diaphragm periods. At lowPcol, it is also clear that the Ts = 0 and 0.2 s configurations consistentlyresult in the highest stability. At higher Pcol, the Ts = 2 s configurationreaches the same level of stability as the stiff systems, with the Ts = 0.5 and1.0 s configurations remaining at notably lower stability levels.Overburden: Results are shown for the runs otherwise equivalent to thebase case runs, excepting the application of an overburden of P = 10 kN/m.The overburden was applied entirely as joist pocket loading (Case 3 in Fig-ure 6.35). Slenderness ratio and period plots are provided in Figures 6.43and 6.44, respectively. To aid in comparing the results to the base case,plots of the ratio of Sa with overburden to Sa for the base case at constantPcol are shown, also as functions of h/t and Ts, in Figures 6.45 and 6.46.This ratio will generally be denoted as the ‘stability gain ratio’ (SGR), notlimited to overburden effects only. Linear regressions are plotted for theh/t curves. These plots demonstrate that, like the effect of period, the sta-bilizing effect of overburden depends heavily on h/t, with greater relativestabilty increases occurring for smaller h/t.ABK Joint Venture [1981b] considered overburden as a ratio of overbur-den to wall self-weight (P/W ). Here, the wall weight increased linearly withh/t, as the height increased and the thickness remained constant, while the1766.5. Modelling results8 12 16 20 2400.20.40.60.81.0S a(1.0)(g)top storey one storey } ASCE41 limits(a) Pcol = 5%8 12 16 20 2400.20.40.60.81.0Ts = 0 s Ts = 0.2 s Ts = 0.5 s Ts = 1 s Ts = 2 s(b) Pcol = 10%8 12 16 20 2400.20.40.60.81.0h/tS a(1.0)(g)(c) Pcol = 20%8 12 16 20 2400.20.40.60.81.0h/t(d) Pcol = 50%Figure 6.41: h/t curves, top storey, P = 01776.5. Modelling results0 0.5 1.0 1.5 2.000.20.40.60.81.0S a(1.0)(g)(a) Pcol = 5%0 0.5 1.0 1.5 2.000.20.40.60.81.0h/t = 8 h/t = 11 h/t = 14 h/t = 18 h/t = 22(b) Pcol = 10%0 0.5 1.0 1.5 2.000.20.40.60.81.0Ts (s)S a(1.0)(g)(c) Pcol = 20%0 0.5 1.0 1.5 2.000.20.40.60.81.0Ts (s)(d) Pcol = 50%Figure 6.42: T curves, top storey, P = 01786.5. Modelling resultsapplied overburden was constant. Using a constant overburden and the ac-tual wall weights provides a better illustration of the effect of increasing wallheight alone in a building. In addition, it is clear that the effects can not beexplained simply using P/W rather than P . Consider Figure 6.45b as anexample. Here, the wall self-weight doubles from h/t = 11 to 22. For thissame range, the stability gain ratio goes from roughly 1.7 to 1.2 at Ts = 1 s,while for Ts = 0.5 s, the ratio goes from 1.2 to 1.0.This also highlights a very defined trend best illustrated in Figure 6.46,namely that there is a very sharp dip in the stability gain around Ts = 0.5 s.The dip is most severe at low Pcol. The points at Ts = 0.2 s are already onthe descending arm of this dip with considerably lower stability gain thanthe points at Ts = 0, while the Ts = 1 s points are effectively out of the dipalready. Presumably, the inclusion of additional period increments around0.5 s would have rounded out the sharpness of the dip seen in this plot, butit would nevertheless still form a very notable feature. The combination ofthe low stability gains at Ts = 0.5 s and the decreasing effect of overburdenwith h/t actually result in effectively no stability gain at h/t = 22 at thisperiod for low Pcol.It is difficult to intuitively explain all of these trends. In general, itwould be expected that the effect of constant overburden would decreasewith increasing wall height. The stabilizing moment provided by the over-burden does not change, as neither the available moment arm — relatedto the constant wall thickness — nor the load are changing. In contrast,the destabilizing loads are changing: there is more wall mass loaded acrossa taller span, both of which result in lower stability. The variation withperiod is less easily explained; it is possible that a reduction in the ‘effectiverocking period’ of the wall due to the additional axial load results in an in-creased rocking response around a period of 0.5 s. With a diaphragm periodnear this value, the resulting amplification could counteract a portion of thestability gain from the axial load. Barring more detailed investigation intothe matter, which is outside the scope of this study, the results producedhere should serve as a caution to allow only conservative stability gains dueto axial load.Bottom vs. top storey: Runs otherwise equivalent to each of the previouscases, excepting that the bottom diaphragm period was set to rigid (T = 0 s),were also considered. This is representative of a wall in a one-storey building,where the base of the wall is on a foundation and the top is connected to aflexible diaphragm. These cases were examined for the longer periods only(0.5, 1, and 2 s), since minimal difference was expected from fixing the base1796.5. Modelling results8 12 16 20 2400.20.40.60.81.0S a(1.0)(g)top storey one storey } ASCE41 limits(a) Pcol = 5%8 12 16 20 2400.20.40.60.81.0Ts = 0 s Ts = 0.2 s Ts = 0.5 s Ts = 1 s Ts = 2 s(b) Pcol = 10%8 12 16 20 2400.20.40.60.81.0h/tS a(1.0)(g)(c) Pcol = 20%8 12 16 20 2400.20.40.60.81.0h/t(d) Pcol = 50%Figure 6.43: h/t curves, top storey, P = 10 kN/m1806.5. Modelling results0 0.5 1.0 1.5 2.000.20.40.60.81.0S a(1.0)(g)(a) Pcol = 5%0 0.5 1.0 1.5 2.000.20.40.60.81.0h/t = 8 h/t = 11 h/t = 14 h/t = 18 h/t = 22(b) Pcol = 10%0 0.5 1.0 1.5 2.000.20.40.60.81.0Ts (s)S a(1.0)(g)(c) Pcol = 20%0 0.5 1.0 1.5 2.000.20.40.60.81.0Ts (s)(d) Pcol = 50%Figure 6.44: T curves, top storey, P = 10 kN/m1816.5. Modelling results8 12 16 20 241.01.21.41.61.82.0S a(1.0) P=10/Sa(1.0)P=0(a) Pcol = 5%8 12 16 20 241.01.21.41.61.82.0Ts = 0 s Ts = 0.2 s Ts = 0.5 s Ts = 1 s Ts = 2 s(b) Pcol = 10%8 12 16 20 241.01.21.41.61.82.0h/tS a(1.0) P=10/Sa(1.0)P=0(c) Pcol = 20%8 12 16 20 241.01.21.41.61.82.0h/t(d) Pcol = 50%Figure 6.45: SGR due to P = 10 kN/m, as h/t1826.5. Modelling results0 0.5 1.0 1.5 2.01.01.21.41.61.82.0S a(1.0) P=10/Sa(1.0)P=0(a) Pcol = 5%0 0.5 1.0 1.5 2.01.01.21.41.61.82.0h/t = 8 h/t = 11 h/t = 14 h/t = 18 h/t = 22(b) Pcol = 10%0 0.5 1.0 1.5 2.01.01.21.41.61.82.0Ts(s)S a(1.0) P=10/Sa(1.0)P=0(c) Pcol = 20%0 0.5 1.0 1.5 2.01.01.21.41.61.82.0Ts(s)(d) Pcol = 50%Figure 6.46: SGR due to P = 10 kN/m, as T1836.5. Modelling resultsin a short-period configuration. For brevity, the results are shown only asrelative stability increases. Figures 6.47 and 6.48 show the results comparingthe configurations with no overburden, while Figures 6.49 and 6.50 comparethe results with overburden.It can be noted that the slenderness ratio has minimal influence overall onthe effect of fixing the bottom diaphragm. The period has notable influence,however, with the greatest stability gains occurring at Ts = 0.5 s. For themost part, fixing the bottom diaphragm results in mild stability gains. Theexception is that at Ts = 2 s for larger Pcol, it actually results in slightstability decreases. This is consistent with earlier observations of systemswith mixed top and bottom periods exhibiting stability in between that ateither period (see Section 6.5.2.3), and the fact that the stability at Ts = 2 sis actually higher than at Ts = 0 s in some cases (see Figures 6.42d and6.44d).1846.5. Modelling results8 12 16 20 240.81.01.21.41.61.8S a(1.0) Tb=0/S a(1.0) Tb=T t(a) Pcol = 5%8 12 16 20 240.81.01.21.41.61.8Ts = 0.5 s Ts = 1 s Ts = 2 s(b) Pcol = 10%8 12 16 20 240.81.01.21.41.61.8h/tS a(1.0) Tb=0/S a(1.0) Tb=T t(c) Pcol = 20%8 12 16 20 240.81.01.21.41.61.8h/t(d) Pcol = 50%Figure 6.47: SGR due to rigid bottom diaphragm, at P = 0, as h/t1856.5. Modelling results0 0.5 1.0 1.5 2.00.81.01.21.41.61.8S a(1.0) Tb=0/S a(1.0) Tb=T t(a) Pcol = 5%0 0.5 1.0 1.5 2.00.81.01.21.41.61.8h/t = 8 h/t = 11 h/t = 14 h/t = 18 h/t = 22(b) Pcol = 10%0 0.5 1.0 1.5 2.00.81.01.21.41.61.8Ts(s)S a(1.0) Tb=0/S a(1.0) Tb=T t(c) Pcol = 20%0 0.5 1.0 1.5 2.00.81.01.21.41.61.8Ts(s)(d) Pcol = 50%Figure 6.48: SGR due to rigid bottom diaphragm, at P = 0, as T1866.5. Modelling results8 12 16 20 240.81.01.21.41.61.8S a(1.0) Tb=0/S a(1.0) Tb=T t(a) Pcol = 5%8 12 16 20 240.81.01.21.41.61.8Ts = 0.5 s Ts = 1 s Ts = 2 s(b) Pcol = 10%8 12 16 20 240.81.01.21.41.61.8h/tS a(1.0) Tb=0/S a(1.0) Tb=T t(c) Pcol = 20%8 12 16 20 240.81.01.21.41.61.8h/t(d) Pcol = 50%Figure 6.49: SGR due to rigid bottom diaphragm, at P = 10 kN/m, as h/t1876.5. Modelling results0 0.5 1.0 1.5 2.00.81.01.21.41.61.8S a(1.0) Tb=0/S a(1.0) Tb=T t(a) Pcol = 5%0 0.5 1.0 1.5 2.00.81.01.21.41.61.8h/t = 8 h/t = 11 h/t = 14 h/t = 18 h/t = 22(b) Pcol = 10%0 0.5 1.0 1.5 2.00.81.01.21.41.61.8Ts(s)S a(1.0) Tb=0/S a(1.0) Tb=T t(c) Pcol = 20%0 0.5 1.0 1.5 2.00.81.01.21.41.61.8Ts(s)(d) Pcol = 50%Figure 6.50: SGR due to rigid bottom diaphragm, at P = 10 kN/m, as T1886.6. Summary6.6 SummaryThis section provides a summary of key observations from the various phasesof the parametric modelling. These observations will form the basis for therecommended assessment procedure in Chapter 7. As mentioned earlier,the concept of stability in these observations refers to the aggregation ofresults for many ground motions. Results for any single ground motion donot necessarily obey these trends; for this reason, experimental results werenot compared with fragility curves. The primary purpose of the testingwas validation of the model, and this was successfully achieved (refer toChapter 5).Diaphragm period and ground motions• Rigid and very stiff diaphragms (Ts < 0.2 s) resulted in the most stablewalls, while diaphragms with moderate periods (0.5 s < Ts < 1.25 s)resulted in the least stable walls. Stability increased at periods longerthan 1.25 s.• Wall stability is more dependent on ground motion content at longperiods than on that at short periods. For Ts < 0.5 s, Sa(1 s) wasan equally good or better predictor of collapse than Sa(Ts). ForTs > 0.75 s, Sa(Ts) was the best predictor, but using Sa(1 s) producedreasonable results.• Using Sa(1 s) as the intensity measure for long-period systems addsvariance to the results, since this method implicitly relates Sa(Ts) toSa(1 s). This additional variance becomes apparent at low (or high)Pcol levels. For long period systems, trends with respect to varying Tsare best described by the median results (Pcol = 50%).• Near-fault non-pulse motions produced results very similar to thoseof far-field motions. Near-fault pulse motions showed significantly lessstability improvements at long periods than far-field motions.• The spectral shape factor, ε, of far-field motions is insignificant ifcharacterizing stability by Sa(1 s) or at longer periods. The spectralshape would only become significant if describing short-period systemsby Sa(Ts). In this case, correcting outcomes for ε would serve as aworkaround to relate Sa(Ts) to Sa(1 s).• Systems in which the diaphragm period differs between the top andbottom of the wall generally exhibit stability in between that of con-1896.6. Summaryfigurations when both diaphragms are equal to either one of theseperiods.• The magnitude of the effect of diaphragm period is heavily dependenton the slenderness ratio, with more slender walls being less sensitiveto differences in diaphragm periods.• Ground floor walls in systems with Ts > 0.5 s, in which the bottom‘diaphragm’ is rigid, exhibited on the order of 10% higher stabilitythan walls in which the bottom diaphragm has the same stiffness asthe top diaphragm. This stability gain was lowest for very long periodsystems (Ts = 2 s), where at high Pcol there was actually a slightstability decrease.Other factors• Stability increases with decreasing crack height. The magnitude of theeffect on stability is largest at lower crack height — i.e. the differencein stability between relative heights of 0.4 and 0.5 is greater than thatbetween heights of 0.7 and 0.8.• Stability increases with decreasing slenderness ratio, h/t. The relationof Sacol to h/t is described well by a power function.• Stability increases with increasing wall thickness, for constant h/t.Two-wythe walls had Sacol on the order of 10–30% lower than that ofthree-wythe walls of the same h/t, with larger differences at Ts = 0.5 sthan at Ts = 1.0 s.• The effect of the amount of spalling at the crack was minimal for valuesbetween 5 and 15mm.• Stability increases with increasing damping ratios.• Stability increases as the relative mass of diaphragms to walls de-creases.• Stability increases with increasing axial load. The magnitude of theeffect is heavily dependent on the manner in which the load is applied.More symmetrical load applications result in close to linear stabilitygains with increasing axial load, while more eccentric load applicationsresult in progressively lower gains as the axial load increases.1906.6. Summary• The stabilizing effects of axial load are highly dependent on the slen-derness ratio of the wall and on the period. More slender walls gainless stability for a given axial load than do less slender walls. In ad-dition, systems with Ts = 0.5 s exhibited significantly lower stabilitygains than systems at other periods.191Chapter 7Recommendations forAssessment GuidelinesMoving from the results of the parametric analysis to an assessment pro-cedure involves simplifications and the subjective evaluation of numerousfactors. In this section, principal factors requiring consideration are dis-cussed, and suggestions for an updated assessment procedure are offeredbased on the results of the parametric study, the experimental study, andthe evaluation of these factors.7.1 ConsiderationsKey factors that should be considered in the implementation of the studyresults into assessment guidelines are briefly discussed. Most of these topicswould be of interest for future research.1. Variance in Sa(1 s) at collapseAmong systems with varying periods, it was shown that comparingSa(1 s) values at low Pcol can obscure the mean trend, particularlyfor long period systems, because the variance in results changes withsystem period. For shorter period systems, there is an inherent largevariance in the collapse spectra at all periods; consequently, the largevariance in Sa(1 s) is more representative of the actual trends. Thisis a continuously varying phenomenon, however, and there is no well-defined cutoff period at which the effect starts or stops being relevant.For long periods (Ts > 1 s), it would be appropriate to use the Pcol =50% values as the indicator of the relative stability between periods.The lower Pcol curves could be approximated by shifting the 50% curvethrough the appropriate Pcol points at Ts = 1 s, which has the smallvariance representative of the long period systems.2. Diaphragm displacements1927.1. ConsiderationsResults in previous sections have been presented in terms of spectralaccelerations, in large part because this is the most common design cri-terion used in practice. When dealing with long-period diaphragms,it is important to maintain an awareness of what sort of deforma-tions a given spectral acceleration creates. Spectral displacements areplotted as a function of spectral acceleration in Figure 7.1 for threeperiods: 0.5, 1, and 2 s. The thickness of a typical 3-wythe URM wallis also indicated for reference. The displacements shown are thosecorresponding to a SDOF system; peak diaphragm displacement atmid-span would be approximately 27% larger (see Equation (2.7) andFigure 2.1).t3−wytheT = 0.5 sT = 1 sT =2 s0 0.2 0.4 0.6 0.80200400600800Sa(T ) (g)S d(mm)Figure 7.1: Spectral displacement vs spectral accelerationDiaphragm deformations at moderate Sa values are less than roughlyhalf a wall thickness for systems with diaphragm periods up to roughly1 s, assuming that the linear response of a system with a secant stiff-ness is representative of the actual response of the diaphragm. (e.g.,at Sa(1 s) = 0.6 g, Sd(1 s) ≈ 150mm). At periods much beyond this,deformations increase rapidly, since Sd is proportional to T 2. Large de-formations, particularly if occurring in buildings with short diaphragmspans, can cause issues not directly related to out-of-plane wall failure,such as damage at corners and cross walls and wall cracking due tohorizontal bending.The results presented in the parametric study indicate that the largedisplacements associated with long-period systems do not create ahigher risk of collapse within the limitations of the one-way rockingmodel. In light of the potential issues associated with large diaphragm1937.1. Considerationsdeformations other than out-of-plane wall failure, it would be prudentto make assessment guidelines increasingly conservative as diaphragmperiods increase. This consideration must be balanced against thefindings of Point (1), which suggested that stability gains at long pe-riods are actually more significant than is implied by results at lowPcol.3. Response along the diaphragmAs discussed in Section 2.1.2 and mentioned in Point (2), the dia-phragm response varies significantly along its length. Disregardingtwo-way effects, one could conservatively imagine an out-of-plane wallalong the span of a diaphragm as a series of tall and narrow one-way spanning strips. This concept is illustrated in Figure 7.2, whichshows a single wall with window perforations connected to flexiblediaphragms at top and bottom. The wall strips in the illustration arecracked and undergoing rocking.FlexiblediaphragmOne-way URMwall stripFigure 7.2: Wall response along the length of a flexible diaphragmThe strips near the ends of the diaphragm would be subject to theequivalent of a rigid diaphragm input, while the strips in the middlewould be subject to a relative diaphragm response greater than that ofthe equivalent SDOF model used in this study, which considered onlythe strips at particular points near mid-span (see Figure 2.1). Notonly would the inputs to the wall strips at the other points along thespan be different, but the resulting differences in wall rocking response1947.1. Considerationswould also affect the response of the rest of the diaphragm. Neither ofthese effects was accounted for within the scope of the present study.Given that the parametric study showed that stiff and rigid diaphragmsystems resulted in the highest out-of-plane stability, it is unlikely thatpoints of intermediate response between the equivalent SDOF systemand the rigid system would produce lower stabilities. However, it ispossible that the mid-span points would result in lower stability thanthe SDOF system modelled due to the larger diaphragm diaphragmdeformations at this location. Whether additional conservativenessbeyond the modelled results is necessary to account for this effectmay be a matter of opinion, but the effects of two-way bending (notmodelled here) would in many cases provide some reserve resistance.4. Two-way bending and wall geometryGriffith et al. [2007] conducted cyclic air bag testing of two-way sup-ported URM wall panels. They showed that the addition of verticalsupports at the ends of wall panels resulted in significant additionalreserve displacement capacity beyond that available through one-waybending. In this same study, however, it was noted that a large pro-portion of vertical cracking was due to line failure (cracking throughbricks) rather than stepped failure (cracking at the brick–mortar in-terface). While “stepped cracks can possess significant reserve post-cracking moment capacity due to the torsional resistance from frictionacting on the bed-joints”, line cracks can not. In the most extremecases, full-height line cracks can effectively completely negate two-waybending effects.Full-scale dynamic testing of two-way supported wall panels has notbeen carried out to date, and the existing work does not clearly con-clude how much additional dynamic stability is achieved by two-waysupports vs. one-way supports. Two-way effects appear unlikely tobe detrimental to dynamic stability, however the demonstrated poten-tial for line cracking necessitates caution in incorporating any benefitsfrom two-way bending into assessment guidelines. Due to these issues,two-way effects were ignored for the time being in the derivation ofguidelines.In addition to two-way effects, other geometry issues can complicatewall assessment. Most notably, gable end walls have continuously vary-ing heights along their length, and the relationship between their dy-namic stability and that of a constant-height wall is unclear. Use of1957.1. Considerationsthe peak gable height is simple, but likely over-conservative. The useof some intermediate height is likely reasonable, but the selection ofthis height has not received adequate treatment in the literature. Bothof these topics are of interest for future research.5. Amplification up the buildingAs discussed in Section 2.1.1, there is very limited research available onthe amplification of ground motions up the height of URM buildings.ABK Joint Venture [1981b] ignored the issue entirely by rationalizingthat between foundation rocking and non-linearity of URM in-planeresponse it was unlikely that accelerations would be greater at the topof a building than at the base. However, one available instrumentationrecord in a real building revealed notable amplification [Tena-Colungaand Abrams, 1992], particularly at short periods, and recent modelling[Knox, 2012], while not exhaustive, certainly suggested that some am-plification may be likely.The present study demonstrated that out-of-plane wall stability — forboth rigid and flexible diaphragm systems — is more dependent onlong-period input motion content than short-period content. Shortperiod amplification of input motions would therefore not be expectedto cause significant changes in wall stability, and the use of the resultsof the current study, which used unamplified ground motions as inputto diaphragms, should be reasonable. The subject should definitely beaddressed in greater detail by future work; new efforts on this frontare currently getting under way [Paxton, 2014].6. Arching actionDerakhshan [2010] found during in-situ airbag testing of URM wallsin vintage buildings that the effect of arching action provided by tim-ber roof diaphragms was negligible. Other construction details, likeconcrete ring beams, can provide notable arching action effects thatcould substantially increase the wall stability. While investigationslike the aforementioned have shown significant strength increases dueto arching, the effect has not been demonstrated in dynamic testing.An approximation could be rationalized based on the effects of axialload demonstrated in the present study by calculating the verticalstiffness of the support. It would be expected that the effect wouldbe less than that due to an axial load corresponding to the maximumforce predicted due to the arching, since arching resistance is only1967.1. Considerationsmobilized as the wall rocking displacement increases. Alternatively,the effect could be incorporated into future modelling as a spring-loaded support block placed on top of the wall. These considerationsare beyond the scope of the current study, however. Excluding theeffect is conservative in all cases, and for flexible timber diaphragmsit is in fact accurate.7. Masonry strengthMeisl et al. [2007] showed that the quality of collar joints in multi-wythe walls had little effect on their out-of-plane response during dy-namic testing. Type O mortar was used in both those tests and inthe current tests. Vintage URM construction may exhibit consider-ably weaker mortar than that used during testing (e.g., Lumantarna[2012]), in terms of both flexural bond strength and mortar compres-sive strength.While wall segments in the current tests sustained negligible damageoutside of the characteristic horizontal cracks, it is not clear how wellvintage masonry with very low strength mortar would hold togetherunder sustained rocking behaviour. It would be of interest to carryout dynamic testing with mortars of varying strengths to observe thedifference in degradation during out-of-plane rocking. The currentmodelling did, however, include the effects of moderate amounts ofspalling at the crack location. While it is out of the scope of thisstudy, it may be reasonable to include limits on acceptable mortarstrength in out-of-plane assessment guidelines, below which reductionfactors would be applied to allowable Sa values.Additionally, masonry strength can be a consideration for anchoragedesign. A recent anchor-testing program conducted in New Zealandshowed anchor capacity to be dependent on masonry strength, butalso that properly-installed anchors can still function adequately inlow-strength walls [Dizhur, 2012]. Provisions for adequate anchoringin low-strength walls should be carefully considered along with thosefor out-of-plane stability.8. Damping and non-linearityDamping in flexible timber diaphragms in URM buildings has receivedlimited study to date. At this point, the available evidence suggeststhat 5% should be reasonable. However, the parametric study showedthat damping does in fact have a moderately significant effect on out-1977.1. Considerationsof-plane stability. Further testing regarding this issue, particularly in-situ, would improve confidence in this assumption. Based on availabledata, it seems unlikely that 5% would be unconservative, and so thestudy results using this value should be appropriate for use.Diaphragm non-linearity was not accounted for in this study. Thestiffness recommendations on which the period ranges were based wereapproximating secant stiffnesses at 100mm deformation. This is a sig-nificant deformation, and tested response characteristics [Wilson, 2012]would suggest that stiffness degradation beyond this point is likely tobe minor. In addition, the reduced input accelerations that wouldresult from non-linearity would be unlikely to reduce wall stability.Nevertheless, it would be of interest to incorporate non-linearity intofurther parametric modelling, and particularly to examine the effectsof the potential for increased diaphragm displacements.9. In-plane damagePerhaps the most significant simplification in the current study (andin most previous work on the topic) is that it considers out-of-planewall response independently of in-plane response. In reality, all wallsin a building are both ‘out-of-plane walls’ and ‘in-plane walls’ simul-taneously when subjected to an actual earthquake. It could be arguedthat this issue is not of significant importance since in-plane damagetends to be concentrated in lower floors where in-plane demands ac-cumulate, while out-of-plane failures tend to occur in the top floor,where axial load is lowest.The effect of in-plane damage on out-of-plane dynamic stability re-mains one of great interest, and it has not been addressed specifically.Whole buildings have been tested, but in such tests it is difficult todraw conclusions on this issue in particular. Clough et al. [1990],Gülkan et al. [1990] noted in shake table testing of one-storey ma-sonry houses that minimal differences in response were observed whentesting with three-axis input versus one-axis input, but these observa-tions can hardly be considered conclusive. Dynamic testing similar tothat done in this study, but with wall specimens subjected to diagonalcracking prior to the test would be helpful.10. Vertical accelerationsAn issue that has received limited treatment across structural engi-neering is the effect of vertical accelerations. Historically, it has been1987.1. Considerationsneglected due to two factors: (1) vertical motions are typically dom-inated by high-frequency content, and as such they input less energyinto a structure as well as attenuate more rapidly with distance fromthe source, and (2) structures are typically designed with a high safetyfactor against gravity (vertical) loads, in addition to which is unlikelythat the gravity system would be stressed by full snow and occupancyloads when an earthquake occurs. While the bases for these factorsare accurate, the conclusion that vertical accelerations are insignifi-cant is overly simplistic for a number of reasons in typical modernconstruction (see Papazoglou and Elnashai [1996]).In the particular case of out-of-plane stability of URM walls, the issueis worthy of investigation, but is out of the scope of this study. Thefact that out-of-plane response depends heavily on the axial force onthe wall suggests that vertical effects could be important. On the otherhand, the high-frequency nature of the vertical excitation means thatseveral full cycles of vertical motion may occur during a single rockingexcursion, which may in the end result in minimal overall difference.At this point, the issue is unresolved, and merits further attention.11. Acceptable risk levelIt is common for retrofit legislation to allow compliance with a lowerhazard level for existing buildings than for new construction, to avoidretrofits becoming prohibitively expensive and thus not happening atall — the argument is that some retrofit is better than no retrofit. Forexample, the “bolts plus” provision in San Francisco’s ordinance 225-92 of 1992 mandated retrofits of URM buildings, but allowed buildingsmeeting certain occupancy and configuration restrictions to satisfyretrofit requirements simply by installing wall-to-diaphragm connec-tions and satisfying wall h/t limits [Paxton et al., 2013]. Both ASCE41 and New Zealand’s equivalent document, NZSEE [2006], allow theselection of performance targets from a range of options. While highperformance is encouraged, in New Zealand the minimum national re-quirement is only that existing buildings undergoing retrofit satisfyearthquake demands of at least 33% of the design level for new build-ings. Clearly, buildings retrofitted to a lower hazard level will still posea higher risk to the public than new buildings, but this is consideredacceptable given the alternative (no retrofit).It is proposed that variation in risk levels could similarly be consideredamong different specimens of the same building, and within a building1997.2. Recommended assessment procedureitself, based on an assessment of the exposure. In the context of out-of-plane wall failure, the greatest life safety risk is due to wall debrisfalling onto an occupied street or sidewalk, rather than precipitatingtotal building collapse. Ingham and Griffith [2011] noted during apost-earthquake survey of URM damage in Christchurch that wallsand gables are significantly more likely to fall outwards from a buildingthan inwards due to restraint from the diaphragms, resulting in a largerlife safety risk for passers-by than for building occupants.Consider as an example the top storey of a two-storey commercialURM building, with the façade located on a busy pedestrian streetand roof joists supported on the front and back walls. Out-of-planefailure of the front wall in the daytime has the potential to causesignificant loss of life, both for pedestrians on the ground in front ofthe building, and for occupants of the top storey, since the roof issupported on this wall. In comparison, out-of-plane failure of a sidewall has much lower potential to cause loss of life, since the area below(say, an alley) is unlikely to be occupied, and this wall is not integralto the support of the roof.While the conservative (and simple) solution would be to apply thesame assessment standards to both of these walls, in the context ofa limited retrofit budget, a greater overall risk reduction would beachieved by applying a higher standard to the high-risk front wall anda lower standard to the low-risk side wall. To this end, it is proposed toadd an ‘exposure factor’ to the out-of-plane wall assessment procedure(see Section 7.2).7.2 Recommended assessment procedureIn this section, an out-of-plane wall assessment procedure is proposed as anupdate to the current procedure in ASCE 41. As discussed in Section 7.1,many considerations are involved in the transformation of study results toassessment procedure. The subjective nature of most of these considerationsmeans that significant judgement is involved in defining an assessment pro-cedure, and there will likely be differences of opinion on these issues amongreaders. The proposed procedure in this section should be viewed as a rea-sonable starting point for further discussion among members of the relevantstandards committees and the practising community in general.The recommended procedure is summarized as follows, with subsequentsections providing more details regarding each step.2007.2. Recommended assessment procedure• Classify diaphragms as stiff or flexible• Obtain the corresponding base curve of Sab(1) vs. h/t• Obtain a correction factor for axial load, Ca• Obtain a correction factor for wall thickness, Ct• Obtain a correction factor for exposure level, Ce• Obtain a correction factor for ground level walls, Cg• Compute the final relationship of S′a(1) = Ca ·Ct ·Ce ·Cg ·Sab(1)7.2.1 Base curves and classification of diaphragmsThe parametric study showed that very stiff and rigid diaphragms resulted inthe most stable walls, while mid-range periods resulted in considerably lowerstability. Very long periods showed some improvement over mid-range peri-ods. It is recommended that diaphragm flexibility classification be limitedto two categories: stiff and flexible. Allowing more detailed classification offlexible diaphragms is not recommended for two reasons: (1) there is signif-icant uncertainty in assessing diaphragm periods, and (2) allowing benefitsfor longer periods is not prudent due to the other issues that may arisewith the larger deformations to which long-period systems are subject (referto Point (2) in Section 7.1). The recommended classifications are listed inTable 7.1.Table 7.1: Diaphragm classificationClassification Period (s)stiff < 0.2flexible > 0.2The base curves for each case are defined for no axial load, at Pcol = 10%,in 3-wythe walls in upper stories. For each classification, the base curves areselected to approximate the results for the most conservative periods: (T =0.2 s for stiff systems, and T = 0.5 s for flexible systems). The recommendedrelationships are listed in Equation (7.1), and the curves are plotted againstthe relevant model data (from Figure 6.41b) and against the current ASCE41 limits in Figures 7.3a and 7.3b, respectively. The base curve for stiffdiaphragms is reasonably consistent with current ASCE 41 limits for top2017.2. Recommended assessment procedurestorey walls, at low h/t, and with the limits for one storey walls, at higherh/t. The base curve for flexible diaphragms is notably more conservativethan current limits, in accordance with the findings of this study.Sab(1) =4·(ht)−1 stiff diaphragms32 ·(ht)− 34 flexible diaphragms(7.1)7.2.2 Correction factorsThe base curves must be adjusted to account for the effects of axial load,wall thickness, exposure level, and whether walls are at ground level or inan upper storey. Correction factors for each of these effects are presented inthis section.Axial load: Modelling results indicated that stability gains due to axialload decreased with increasing h/t (see Figure 6.45). Results varied signifi-cantly for different periods, and again the most conservative points were atT = 0.2 s for stiff systems, and T = 0.5 s for flexible systems. These resultsare all based on joist pocket loading (onto the outer wythe of a 3-wythewall). Figure 6.38 suggested that for such a load case, the relative stabil-ity gains decrease with increasing load. Furthermore, it is prudent not torely too heavily on axial load gains due to possible countering effects fromvertical accelerations.The following formulation is recommended for the axial load correctionfactor, Ca:Ca =1 + C ′a ·( P10) ht < 81 + C ′a ·( P10)(1− 112(ht − 8))8 ≤ ht ≤ 201 ht > 20(7.2)Here, P is in kN/m and C ′a is defined in Table 7.2. While the modellingresults suggest that these relationships should remain valid even at large P ,it is recommended to implement a cap at P = 20 kN/m to avoid excessivestability gains. Ca values are plotted for P = 10 and 20 kN/m in Figure 7.4.Notably, very slender walls are not allocated stability benefits from axialload.2027.2. Recommended assessment procedureBase curve: stiffBase curve: flexible8 12 16 20 2400.20.40.60.8S a(1.0)(g)T = 0 s T = 0.2 s T = 0.5 s T = 1 s T = 2 s } at Pcol = 10%(a) Compared with model datatopstoreyonestoreyBase curve: stiffBase curve: flexible8 12 16 20 2400.20.40.60.8h/tS a(1.0)(g)(b) Compared with current ASCE 41 limitsFigure 7.3: Base curves, for 3-wythe wall, no axial load, high exposure,upper storey2037.2. Recommended assessment procedureTable 7.2: Axial load base factorClassification C ′astiff 0.5flexible 0.2The suggested curves are compared to the modelled stability gains atPcol = 10% (Figure 6.45b) in Figure 7.5. The stability gains at other Pcolvalues were similar (refer to Figure 6.45). Figure 7.5 illustrates that therecommended axial load correction factors are generally conservative, whichis prudent when not considering the effects of vertical accelerations.stiff, P = 10 kN/mflexible, P = 10 kN/mstiff, P=20 kN/mflexible, P = 20 kN/m8 12 16 20 241.01.21.41.61.82.0h/tC aFigure 7.4: Axial load correction factorsWall thickness: Past research and modelling results indicate that for agiven aspect ratio, rocking stability is greater for larger bodies than forsmaller ones. In section Section 6.5.3.3 it was shown that the relative col-lapse intensity of thinner walls shows some variability with period, but isaffected minimally by h/t. Differences between Ts = 0 and 0.5 s are moder-ate, and it could be expected that results for Ts = 0.2 s, though not mod-elled here, would fall somewhere in between those values. For simplicity, itis recommended that the thickness correction factor be defined as period-2047.2. Recommended assessment procedurestiff, P = 10 kN/mflexible, P = 10 kN/m8 12 16 20 241.01.21.41.61.82.0h/tC aorS a(1.0) P=10/Sa(1.0)P=0Ts = 0 s Ts = 0.2 s Ts = 0.5 s Ts = 1 s Ts = 2 sFigure 7.5: Axial load correction factors, compared to model results atPcol = 10%independent for all diaphragms. The results of Figure 6.29 are repeatedwith the recommended correction factor, calculated using Equation (7.3),overlaid in Figure 7.6.Ct = 0.2 + 52 ·t ≤ 1.0, t in m (7.3)Exposure: As discussed in Section 7.1, it would be of interest to accountfor varying levels of risk corresponding to higher or lower exposure condi-tions. An exposure factor, Ce, is proposed that would account for variationsfrom the base curve case of Pcol = 10% according to the assessed exposurecaused by a particular wall. The assessment should take into account thewall’s role in the support of the structure’s gravity system, and also thelikelihood of occupants being located in the impact zone in the case of wallfailure. In regards to the latter, it has been shown to be far more commonfor debris from out-of-plane wall failures to fall outward than inward [Ing-ham and Griffith, 2011]; as such, assessment emphasis should be placed onthe exposure outside the building. The base case of Pcol = 10% is deemedto be a reasonable risk level for default high-risk conditions. It is recom-2057.2. Recommended assessment procedure0 100 200 300 40000.20.40.60.81.0t (mm)C torS acol/S acol 3−wytheTs = 0 s Ts = 0.5 s Ts = 1.0 sFigure 7.6: Thickness correction factors, compared to model results atPcol = 10%mended to define Ce as approximating the differences between Pcol valuesfrom parametric studies, as listed in Table 7.3. These values were derivedby comparing results within the subfigures of Figures 6.41 and 6.43.Table 7.3: Exposure factorCeExposure Pcol Stiff Flexiblevery high 5% 0.9 0.9high 10% 1.0 1.0low 20% 1.15 1.1very low 50% 1.5 1.25The definition of what constitutes each level of exposure should be care-fully considered. In particular, the ‘very low’ case defined here should beused only under stringent conditions. Note that the study used as a basisfor these results considered only the uncertainty in the wall response. Acomplete risk study should include the uncertainty in both the hazard and2067.2. Recommended assessment procedurethe exposure before any decisions are made regarding adopting specific fac-tors. This is a highly subjective matter that merits further discussion, butthe numbers are included here for completeness.Ground level: In flexible diaphragm systems, modelling results showedmild stability gains for walls in which the base was connected to a rigiddiaphragm (e.g., for a one-storey building) vs. those connected to flexiblediaphragms at top and bottom (upper storey walls) — see Figures 6.47–6.50.While these results showed a slight decrease in stability at very long periods(2 s), the simplifications made in the definition of the base curves left reservecapacity at these periods, making it reasonable to apply constant stabilitygains at all periods. These gains can be accounted for by applying a groundlevel correction factor, Cg, defined as follows:Cg ={1.0 stiff diaphragms1.1 flexible diaphragms (7.4)7.2.3 Safe seismic hazard levelThe ASCE 41 special procedure specifies that out-of-plane stability neednot be evaluated for sites at which SX1 ≤ 0.133. This lower bound is re-evaluated using the new procedure. Conservative conditions are chosen torepresent worst-case walls: t = 200mm represents two-wythe walls, andwalls are assumed to be in the top storey with no axial load and very highexposure. This produces Ct = 0.7, Ce = 0.9, and Ca = Cg = 1.0. Flexiblediaphragm conditions are used. At h/t = 26, this produces an allowableS′a(1) of 0.08 g.It can be concluded that at sites with a hazard level of Sa(1 s) ≤ 0.08 g,out-of-plane stability need not be evaluated for URM walls that meet all ofthe following criteria:• have a thickness of at least 200mm,• are adequately anchored to the diaphragms at all levels, and• are within reasonable h/t bounds bearing in mind other stability issues.7.2.4 Anchorage demandsDesign forces for wall-to-diaphragm anchors are currently specified in theASCE 41 special procedure as the maximum of (a) 2.1 ·SX1 ·W , where W2077.2. Recommended assessment procedureis the weight of the wall, or (b) 2.9 kN/m. The amplification factor of 2.1is derived from ABK Joint Venture’s original conclusions regarding the am-plification of the ground motion caused by the flexible diaphragms, whichresulted in the recommendation of designing anchorage for a demand of 1.0times the wall weight in a seismic hazard zone with an effective peak accel-eration of 0.4 g. This recommendation was reformulated in terms of SX1,and the amplification factor adjusted to suit. The investigation of anchor-age demands was outside the scope of the analytical study, but observationscan be made regarding the anchorage demands observed in the experimentalphase of the project.In shake table testing of cracked walls, it was observed that peak con-nection (i.e. anchorage) demands in runs causing collapse (but prior to thetime of collapse) reached levels up to 1.6 times greater than those predictedfrom the spectral acceleration of the applied motion at the period of thediaphragm system. Total wall forces (which would be taken up by the com-bination of top and bottom anchorage) reached levels up to 1.7 times greaterthan predicted. In the highest stable cracked runs, connection demands wereup to 1.5 times predicted levels, and total wall forces were up to 1.2 timespredicted levels. These large force peaks occurred due to impact when thewall crack closed up after a large rocking excursion. Individual connectiondemands were consistently largest at the bottom — the peak top connectiondemand observed in all testing was only 1.2 times greater than predicted,compared with 1.6 for the bottom connection.When considering the shape of a typical uniform hazard spectrum, thespecified force of 2.1 ·SX1 ·W would be sufficiently large to meet all thedemands incurred in the testing, which included diaphragm periods of ap-proximately 0, 0.8, 1.2, and 1.6 s. Here, the amplification factor is coveringthe demands created by (1) uneven distribution of forces between top andbottom connections, and (2) force peaks caused by impacts during rocking.Neither of these matches the original rationale for the amplification fac-tor, but by chance the existing factor happens to reasonable for the groundmotion considered.In walls that undergo limited or no rocking, the force amplification wouldbe lower due a reduction in the above-mentioned issues. The peak forcedemands would in this case be more aptly described by the unamplifiedspectral acceleration at the period of the diaphragm. Without impact effects,peak anchorage demands will be limited by the strength of the diaphragm.In particular where periods are in the high-amplification range (e.g., under0.5 s), using full elastic spectral acceleration may be over-conservative for atimber diaphragm due to non-linear effects.2087.3. Assessment examplesAnchorage should be capacity-designed — that is, designed so that othercomponents are force-limiting, and that the anchorage will not fail under anycircumstances. The two components to which anchorage is connected aretimber diaphragms, whose highly ductile nature makes them good candi-dates as force limiters, and URM walls, which have some ‘ductility’ capacitydue to rocking. The experimental tests in this study have demonstrated thatanchorage force demands can be amplified if URM rocking is mobilized.The existing demands specified by the ASCE 41 special procedure arereasonable for URM walls connected to long-period diaphragms. It is rec-ommended that further work be conducted regarding anchorage demands,particularly for diaphragms with periods in the high-amplification spectralregion. Retrofitted diaphragms in small to moderate buildings could berepresentative of such periods. Design demands might be formulated as anamplified spectral acceleration at the diaphragm period, with possible re-duction allowed to an amplified upper-bound diaphragm capacity (e.g., referto Wilson et al. [2013] for diaphragm properties). In addition, a value thatis possibly close to the current requirement could serve as a lower bound toprevent demand reductions for long-period diaphragms. The temptation toreduce anchorage demands should be avoided — they are arguably the mostcritical component in a URM retrofit, and relatively inexpensive.7.3 Assessment examplesIn this section, assessment curves are produced for several example build-ings and compared with applicable results from ASCE 41. Wall height isconsidered as a variable. Masonry density is assumed to be 1800 kg/m3.Wall thickness is assumed to be 110mm per wythe.Example 1: A small one-storey building, 6 × 8m in plan, with roof joistsspanning the short dimension is considered. Walls are three wythes thick,and a two-wythe parapet, 1m high, is on the front wall, which is 6m long.No significant parapet is present along the sides and back of the building.The vintage straight-sheathed wooden diaphragm is classified as flexible inboth directions. The side walls are classified as high risk because they aresupporting the roof structure. The front and rear walls are also classifiedas high risk due to the exposure of sidewalk and parking areas in front andbehind the building. The self-weight of the roof diaphragm is assumed tobe 0.5 kN/m2.The axial load on the walls is calculated as follows:2097.3. Assessment examples• Front wall: P = 0.22m·1.0m·17.7 kN/m3 = 3.9 kN/m• Side walls: P = 3m·0.5 kN/m2 = 1.5 kN/m• Rear wall: P = 0The axial load correction factor, Ca, is calculated from Equation (7.2),for flexible diaphragm conditions. The factor is plotted as a function of h/tfor each wall in Figure 7.7.front wallside wallsrear wall8 12 16 20 241.01.1h/tC aFigure 7.7: Axial load correction factors, example 1From Equation (7.3), the thickness factor is determined to be Ct = 1.0for all walls since they are three wythes thick.From Table 7.3, the exposure factor is determined to be Ce = 1.0 for allwalls based on the risk assessment.From Equation (7.4), the ground level factor is determined to be Cg = 1.1for all walls since they are at ground level.The final assessment curve is computed for each wall as S′a(1) = Ca ·Ct ·Ce·Cg ·Sab(1). The curves are shown in Figure 7.8, along with the limits forwalls in one-storey buildings from ASCE 41.In this example, the differences in axial load among the various wallsare moderate, and thus there is minimal effect on the respective assessmentcurves. Of note is the significant discrepancy between the recommendedprocedure and the current standard, depending on h/t. In the vicinity ofh/t = 16 (equivalent to a wall height of 5.3m for a thickness of 0.33mm),the two methods produce very similar results, and for more slender walls thedifferences are relatively minor. For less slender walls, the new procedure issignificantly more conservative.2107.3. Assessment examplesonestoreyfront wallside wallrear wall8 12 16 20 2400.20.40.6h/tS a(1.0)(g)Figure 7.8: Assessment curves, example 1Example 2: The front walls of a large two-storey commercial building areto be assessed. The building has three-wythe walls in both levels, and a1.5m tall parapet on the front. The building is 16 × 24m in plan, withvintage single-sheathed wooden diaphragms classified as flexible. The frontwalls are bearing a 2.5m tributary width of diaphragm weight. The wallshave been assessed as high risk.For the purpose of calculating axial load, the top storey wall will beassumed to have a height of 3.5m. The axial load on the walls is calculatedas follows:• Top storey:P = 0.22m·1.5m·17.7 kN/m3 + 2.5m·0.5 kN/m2 = 7.1 kN/m• Bottom storey:P = 7.1 kN/m + 0.33m·3.5m·17.7 kN/m3+ 2.5m·0.5 kN/m2 = 28.8 kN/mThe axial load correction factor, Ca, is calculated from Equation (7.2),for flexible diaphragm conditions. The factor is plotted as a function of h/tfor each wall in Figure 7.9. Note that while P = 28.8 kN/m for the bottomstorey wall, a maximum contribution of P = 20 kN/m is permitted in thecalculation of Ca.2117.3. Assessment examplestop storeybottomstorey8 12 16 20 241.01.21.4h/tC aFigure 7.9: Axial load correction factors, example 2From Equation (7.3), the thickness factor is determined to be Ct = 1.0for both walls since they are three wythes thick.From Table 7.3, the exposure factor is determined to be Ce = 1.0 forboth walls based on the risk assessment.From Equation (7.4), the ground level factor is determined to be Cg = 1.0for the top storey wall, and Cg = 1.1 for the first storey wall.The final assessment curve is computed for each wall as S′a(1) = Ca ·Ct ·Ce ·Cg ·Sab(1). The curves are shown in Figure 7.10, along with the limitsfor walls in multi-storey buildings from ASCE 41.firststoreytopstoreytop storeybottom storey8 12 16 20 2400.20.40.6h/tS a(1.0)(g)Figure 7.10: Assessment curves, example 22127.3. Assessment examplesIn this case, the assessment curves show significant differences betweenthe two walls. While the bottom storey wall is assessed to withstand sub-stantially higher intensity ground motions than the top storey wall, bothdiffer significantly from the current limits. In each case, the new curvesare similar to the most conservative points on the ASCE 41 limits: aroundh/t = 9 for the top storey, and around h/t = 18 for the first storey. At otherh/t values, the new curves are more conservative than the current limits.Example 3: The building from example 2 is retrofitted by pouring a100mm thick concrete slab on top of the existing wooden diaphragms, cre-ating effectively rigid diaphragm conditions. The total self-weight of thenew diaphragms is assumed to be 2.5 kN/m2.The axial load on the walls, including the additional diaphragm loads,is calculated as follows:• Top storey:P = 0.22m·1.5m·17.7 kN/m3 + 2.5m·2.5 kN/m2 = 12.1 kN/m• Bottom storey:P = 12.1 kN/m + 0.33m·3.5m·17.7 kN/m3+ 2.5m·2.5 kN/m2 = 38.8 kN/mThe axial load correction factor, Ca, is calculated from Equation (7.2),for stiff diaphragm conditions. The factor is plotted as a function of h/tfor each wall in Figure 7.9. Note that while P = 38.8 kN/m for the bottomstorey wall, a maximum contribution of P = 20 kN/m is permitted in thecalculation of Ca.The other factors, Ct, Ce, and Cg, remain unchanged from Example 2.The final assessment curve is computed for each wall as S′a(1) = Ca ·Ct ·Ce ·Cg ·Sab(1). The curves are shown in Figure 7.12, along with the limitsfor walls in multi-storey buildings from ASCE 41.Comparing Figure 7.12 to Figure 7.10, it is apparent that the retrofittedbuilding is allowed significantly higher demands on out-of-plane walls. Inthe top storey, the new curve compares well with the least conservativepoint in the current standard, while in the bottom storey, the new curve isrepresentative of the middle range of the old standard.2137.3. Assessment examplestop storeybottomstorey8 12 16 20 241.01.21.41.61.82.0h/tC aFigure 7.11: Axial load correction factors, example 3firststoreytopstoreytop storeybottom storey8 12 16 20 2400.20.40.6h/tS a(1.0)(g)Figure 7.12: Assessment curves, example 32147.4. Summary and additional considerations7.4 Summary and additional considerationsThe new assessment method recommended in this study is a significantchange from the procedure in the current ASCE 41 standard. For flexiblediaphragm systems, the new method tends to be more conservative than thecurrent standard. In some cases, the differences are minor, while in othercases they are significant. For stiff diaphragm systems, the new methodtends to be consistent with or slightly less conservative than the currentstandard.The new method has been specified based on the conclusions drawn froma large-scale parametric model. Key aspects of the model, including the wallresponse under varying diaphragm flexibility, were calibrated to the resultsof shake table testing. While there remains significant room for refinementand additional research, the method presented here offers significant im-provements over the current method in terms of risk consistency amongvarying wall configurations, and by providing a thorough rationale for itsdetails based in analytical and experimental work.Several additional considerations are noteworthy in the process of imple-menting potential changes to the current assessment standard. The currentASCE 41 limits cut off allowable h/t values at maximum values. Resultsfrom the current study do not objectively support an h/t cut-off based purelyon the stability of the one-way rocking model. While wall heights and h/tvalues are already effectively limited by construction practices, general pru-dence and engineering judgement may favour applying hard limits in theassessment procedure.In addition, the current limits do not impose a maximum Sa limit. Forthe sake of prudence, it may be of interest to cap the base curves at someSa limit. Again, the results of this study do not objectively support thedefinition of such a limit, but cutting the curves off horizontally at the Savalues corresponding to h/t = 8 would be a reasonable starting point.Finally, the definition of base curves and correction factors in these rec-ommendations erred towards conservativeness, most prominently so in flex-ible diaphragm portions. The notable variance in modelling results withvarying system period (refer to Section 6.5.4) resulted in systems with pe-riods of 0.5 s typically governing the definition of parameters. Particularlyin regards to the axial load effect, these values were significantly more con-servative than at other periods. The subjectivity inherent in the choice ofparameter values may well allow for more relaxed selections in certain cases.215Chapter 8Summary and ConclusionsThis study has investigated the out-of-plane seismic performance of URMwalls, with a specific focus on the effect of diaphragm flexibility. The studyconsisted of experimental and analytical phases, ultimately leading to rec-ommendations for an improved out-of-plane seismic assessment procedure.A review of past research showed that the timber floor and roof dia-phragms typical of vintage URM buildings have very low stiffness, withfundamental periods of vibration of typical diaphragm-wall systems rang-ing between 0.5 and 3 s. Ground motions applied to buildings are filteredthrough relatively stiff in-plane walls into the ends of diaphragms, then fil-tered through the diaphragms into the attached out-of-plane walls. Flexiblediaphragms therefore have the potential to significantly affect out-of-planewall performance relative to walls connected to rigid diaphragms. This mat-ter has received very limited treatment in past research, and no dynamictests specifically addressing this effect have been conducted.8.1 Experimental phaseA dynamic testing programme was developed and conducted at the Univer-sity of British Columbia’s Earthquake Engineering Research Facility. Fivefull-scale unreinforced solid clay brick wall specimens spanning one storeywere subjected to earthquake ground motions on a shake table. The top andbottom of the walls were connected to the shake table through coil springs,simulating the flexibility of the diaphragms. The apparatus allowed the wallsupports to undergo large absolute displacements, as well as out-of-phasetop and bottom displacements, consistent with the expected performanceof URM buildings with timber diaphragms. Variables examined includeddiaphragm stiffness and wall height.Walls were cracked by applying a ground motion under rigid diaphragmconditions, then tested to failure with flexible diaphragm conditions by in-crementally increasing the intensity of the applied ground motion. All wallssustained horizontal cracks at an intermediate height, with four out of fivewalls cracking between 0.47 and 0.55 times the wall height, and one wall2168.2. Analytical phasecracking at 0.74 times the wall height. Analysis of peak tensile stresses inwalls at crack initiation showed good agreement with flexural tensile strengthvalues obtained from bond wrench testing.The ground motion intensity at collapse varied widely among the wallswith different boundary conditions. The lowest level causing collapse was60% of the as-recorded amplitude, while the highest was at 120%. In gen-eral, the more flexible diaphragms reached higher motion intensities. Thespecimen tested to failure with rigid diaphragms collapsed at the lowest in-tensity level, but also had sustained a crack height significantly higher thanthe other walls — a factor that reduces stability.Force demands on connections at the top and base of walls in stableruns were observed to be up to 1.5 times those predicted from the spectralacceleration of the input ground motion at the period of the diaphragm-wallsystems. In runs causing collapse, total force demands on the wall were upto 1.7 times higher than predicted. Demands on the bottom connection wereconsistently larger than on the top connection. These observations empha-size that rocking cannot be relied upon to reduce anchorage demands, butrather that the impacts caused by rocking can amplify anchorage demands.Extensive stable rocking was observed in some specimens, while in othersthe amount of rocking was very limited. The specimen with rigid diaphragmconditions exhibited the most rocking cycles in the run prior to collapse. Thespecimen with a flexible top diaphragm and rigid bottom diaphragm exhib-ited effectively no rocking until collapse, yet withstood the highest intensityof ground motion. This observation emphasizes that it is not possible to re-liably tell how near to collapse a wall came in any single run by the degree ofrocking experienced. In real buildings following an earthquake, it is difficultto determine how much rocking a wall experienced in the first place sincehorizontal cracks can be difficult to detect, doubly complicating the matter.8.2 Analytical phaseThe primary purpose of the tests was to provide a dataset for validation of ananalytical model. A rigid body rocking model, first proposed by Meisl [2006]for use in modelling out-of-plane URM wall response, was created in thesoftware Working Model 2D and successfully validated against the test data.The model was able to reproduce the time-variation of the chaotic, non-linear rocking behaviour of the cracked walls, and also reproduce the motionintensity at collapse for each wall with reasonable accuracy. The ability ofthe rigid body model to explicitly represent the true rocking behaviour of2178.2. Analytical phasethe wall is a significant benefit over approaches in previous research (e.g.,Doherty et al. [2002]).Conclusions regarding the relative stability of walls with different bound-ary conditions cannot be drawn from the tests using a single ground motion.The highly non-linear rocking response of the wall is very sensitive to thedetails of the time variation of the applied excitation, and these detailscannot adequately be quantified by simple intensity measures. To addressthis issue, and to investigate a wider range of parameters not feasible toinvestigate on the shake table, the analytical model was used to conduct aparametric study on out-of-plane performance. The study used the suiteof 100 ground motions prescribed by FEMA P695 to examine motion-to-motion variability, and investigated the effect of numerous variables, mostnotably including the diaphragm period. Motion-to-motion variability con-sistently followed a lognormal distribution closely, and was significant undersome conditions, highlighting the importance of a sufficiently large suite ofmotions in such a study.It was determined that the spectral acceleration at a period of 1 s wasthe best predictor of collapse for walls connected to most diaphragms withperiods of less than 1 s. For periods longer than this, the spectral acceler-ation at the diaphragm period was the best predictor. In the interest ofsimplicity and consistency with the current ASCE 41 standard, Sa(1) wasused as the intensity measure for all results.In general, short-period systems (T ≤ 0.2 s) were the most resilient tocollapse. Periods between 0.5 and 1 s resulted in the lowest resilience levels,while some increase in resilience occurred at periods longer than this. Wheretop and bottom periods were different, results typically ended up in betweenthose bracketed by the cases where both periods were equal to either thetop or the bottom period.The addition of overburden load had a stabilizing effect on wall response,with the magnitude of the effect heavily dependent on the manner in whichthe load was applied. Simulating a joist-pocket load produced roughly thesame stability increase as by applying the same load fixed in the centre ofthe wall, despite the eccentricity of the joist pocket loading. For a givenload level, the relative stability increases varied with period and h/t, withstability increases lowest at a period of 0.5 s, and consistently decreasingwith increasing h/t.The relative crack height had a significant effect on stability, with highercracks resulting in less stable walls. As the crack height approaches the topof the wall, its behaviour will approach that of a cantilever failure mode.Since the height of cracking can not be controlled, however, it was treated2188.3. Contributionsas a random variable in the parametric study and explicitly accounted forin the resulting probabilities of collapse.The results of the parametric study were compiled and interpreted toform recommendations for an improved assessment procedure. Like thecurrent ASCE 41 procedure, the new procedure is based on h/t vs. Sa(1)curves. Two base curves are provided: one for stiff diaphragms and onefor flexible ones. Due to the high uncertainty involved in assessing thestiffness of an existing diaphragm, it is not recommended to allow assessmentbenefit based on the precise period of the diaphragm; all flexible diaphragmsare thus grouped together and a reasonably conservative curve is used forthe group. Correction factors are provided to account for axial load, wallthickness, and walls at ground level. Notably, the concept of acceptingvariable risk is proposed, by including an additional correction factor totolerate higher risks of collapse depending on the assessed exposure of aparticular wall.The new procedure was compared with the existing simplified ASCE 41procedure for several example scenarios for the common case of three-wythewalls. It was found that the new procedure was generally more conservativethan the existing one for flexible diaphragms, most prominently so whencomparing results for the first storey of multi-storey buildings. For stiffdiaphragms, the new procedure was generally on par with, or more lenientthan the existing procedure, particularly for top-storey walls subjected tosignificant axial loads. For two-wythe walls, the thickness correction factorwould make results more conservative than those for three-wythe walls.8.3 ContributionsThe study documented in this dissertation has made significant contribu-tions to the body of knowledge relating to the seismic performance of URMwalls by completing the objectives of Section 1.2. The dynamic one-wayout-of-plane bending response of URM walls connected to flexible supportswas observed for the first time over a variety of support stiffnesses, includ-ing variable top and bottom support stiffnesses. An analytical model wasvalidated to the experimental data, and it was demonstrated that the modelcould accurately simulate the dynamic response of the walls over the rangeof tested conditions. A large-scale parametric study, consisting of over 300wall configurations and over 200,000 individual simulations, was conductedusing the model, and insights into the effects of eight parameters, as well asground motion variability, were obtained. The results of the modelling were2198.4. Future researchcompiled and interpreted to produce a new out-of-plane wall assessmentprocedure that offers significant improvements over current methods.8.4 Future researchThe need for further research in specific areas has been discussed at variouspoints in this thesis, notably in Section 7.1. The most pertinent needs arebriefly listed below.• The effect of two-way bending should be investigated by dynamic test-ing. Data is needed specifically comparing two-way bending to one-way stability for otherwise equivalent walls. This could be added to theassessment procedure as an additional correction factor, as warranted.• The effect of arching action should be investigated. This could likelybe achieved analytically by extending the current model’s axial loadcapabilities to model various top support vertical stiffnesses.• The effect of varying wall thickness should be investigated more thor-oughly than was possible in this study. Ideally, the entire second phaseof the parametric study would be re-run at several thickness levels.• The effect of different wall geometries should be investigated. In par-ticular, the response of gable walls should be compared with regu-lar one-way spanning walls, and additional correction factors or otherguidelines could be added to the assessment procedure to account forgeometrical differences.• The effect of varying response along the span of a flexible diaphragmshould be investigated. The current SDOF idealization represents onlyone point on the diaphragm. Ideally, a 3-dimensional model wouldinclude one-way spanning strips at short intervals along the diaphragmspan, all excited simultaneously.• Practical limits on diaphragm flexibility should be developed. Thestability of one-way spanning walls does not by itself form a basis forsuggesting any limits, but other considerations like damage to cornersand non-structural components may warrant displacement limits.• The effect of diaphragm non-linearity should be investigated. Thiscan be done analytically, possibly by extending the current model,2208.4. Future researchbut implementation of complicated non-linearity in Working Model2D may prove challenging.• Amplification up the height of URM buildings needs to be adequatelycharacterized, and incorporated into out-of-plane recommendations.• A lower bound on masonry strength should be established, below whichout-of-plane rocking can not be relied upon as a stable mechanism.With very low-strength walls, it is possible that wall-diaphragm an-chorage may be unreliable, and additionally that walls may fall apartduring rocking. Further dynamic testing may be warranted to developconfidence on this matter.• The effect of in-plane damage on out-of-plane response should be in-vestigated. Dynamic tests that are suitably controlled to provide ref-erence to existing one-way tests could be conducted with pre-damagedwalls.• The effect of vertical accelerations should be investigated. 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Journal of the Structural Division, 97(5):1593–1609.228Appendix AWall DimensionsNotes: This section contains detailed measurements for each wall specimen.229Appendix A. Wall DimensionsTable A.1: Wall dimensionsFF-3 FR-3 FF-2 SS-3 RR-3Elevations (mm)Base 0 0 0 0 0SP1 427 425 432 417 417SP2 887 877 879 873 862SP3 1347 1326 1327 1316 1307SP4 1800 1773 1782 1762 1752SP5 2259 2224 2230 2275 2191SP6 2720 2682 2606 2795 2626SP7 3167 3135 — 3232 3140SP8 3622 3654 — 3738 3584Top 3947 3984 2790 3985 3973Crack 1838 2186 1365 2019 2920Width — East (mm)H1 1512 1499 1507 1510 1512H2 1513 1505 1504 1515 1511H3 1516 1504 1504 1521 1518H4 1520 1505 1502 1517 1513H5 1513 1500 1504 1520 1515H6 1514 1499 1505 1522 1514H7 1512 1503 — 1520 1515H8 1511 1500 — 1520 1513Width — West (mm)H1 1515 1498 1502 1515 1509H2 1517 1502 1503 1514 1511H3 1417 1504 1505 1518 1514H4 1520 1501 1503 1518 1512H5 1517 1496 1505 1518 1511H6 1515 1496 1505 1517 1516H7 1514 1497 — 1519 1519H8 1516 1495 — 1518 1511Thickness — North (mm)H1 292 290 190 300 291H2 292 291 192 299 292H3 290 290 191 300 295H4 294 291 189 304 294H5 291 289 190 303 296H6 291 290 191 302 300H7 290 291 — 299 299H8 290 288 — 299 297Thickness — South (mm)H1 289 289 191 298 295H2 291 292 192 298 298H3 290 290 190 297 297H4 290 292 190 299 295H5 289 292 191 300 295H6 292 289 189 299 301H7 290 292 — 303 295H8 290 292 — 300 293Weight (kN) 35.58 35.45 17.06 37.60 36.96Mass (kg) 3627 3614 1739 3833 3768Density (kg/m3) 2095 2081 2176 2113 2118230Appendix BMaterials TestingNotes: This section contains detailed testing data for mortar compression, masonry prismcompression, masonry bond wrench, brick compression, and brick absorption testing carriedout as a portion of the experimental study.231B.1. Mortar compressionB.1 Mortar compressionA total of 102 mortar cubes were tested in compression: 66 specimens created duringconstruction of the first three walls (FF-3, FR-3, FF-2 ) and 36 specimens created duringconstruction of the last two walls (SS-3, RR-3 ). Testing was carried out in accordance withCAN/CSA A179-04 (R2009) [CSA, 2009] on the compression testing machine in the UBCmaterials lab. A typical compression failure from the testing is shown in Figure B.1.Figure B.1: Typical mortar cube compression failureCubes measured 50mm on each edge. Due to the relatively small number of batches andlarge variability within individual batches, mean and cv were calculated simply from the listof all successful tests, without regard for batches. Test results are shown in Tables B.1 andB.2. Due to the large number of batches and small variability within individual batches,mean and cv were calculated from the list of batch means, thereby preventing bias due tovarying numbers of samples per batch.Table B.1: Mortar cube compression results, walls FF-3, FR-3, and FF-2Fmax fmax mean of cv ofDate cast Batch Date tested Age Specimen (kN) (MPa) fmax fmax13-Jun-2011 1 02-May-2012 324 1 9.7 3.88 3.56 8.1%2 8.3 3.323 8.7 3.4814-Jun-2011 4 02-May-2012 323 1 8.6 3.44 3.03 12.0%2 7.2 2.883 6.9 2.7615-Jun-2011 1 04-May-2012 324 1 14.6 5.84 5.75 6.2%2 13.5 5.403 13.1 5.24continued on next page. . .232B.1. Mortar compressionFmax fmax mean of cv ofDate cast Batch Date tested Age Specimen (kN) (MPa) fmax fmax4 15 6.005 15.4 6.166 14.7 5.8815-Jun-2011 2 04-May-2012 324 1 10.5 4.20 4.42 13.7%2 10 4.003 9.4 3.764 13.3 5.325 12.5 5.006 10.6 4.2415-Jun-2011 3 04-May-2012 324 1 12.3 4.92 5.07 7.4%2 11.8 4.723 11.9 4.764 14 5.605 12.4 4.966 13.7 5.4817-Jun-2011 1 04-May-2012 322 1 8.2 3.28 3.55 5.2%2 8.7 3.483 8.6 3.444 9.4 3.765 9.3 3.726 9.1 3.6417-Jun-2011 2 04-May-2012 322 1 9.2 3.68 4.19 6.9%2 10.8 4.323 11 4.404 10.1 4.045 10.7 4.286 11.1 4.4417-Jun-2011 3 04-May-2012 322 1 9.2 3.68 4.19 6.9%2 10.8 4.323 11 4.404 10.1 4.045 10.7 4.286 11.1 4.4420-Jun-2011 1 04-May-2012 319 1 6.7 2.68 3.20 16.2%2 7 2.803 6.8 2.724 9.4 3.765 9.3 3.726 8.8 3.5220-Jun-2011 2 04-May-2012 319 1 8.6 3.44 3.73 7.4%2 8.7 3.483 8.9 3.564 9.9 3.965 10.3 4.126 9.5 3.8021-Jun-2011 1 04-May-2012 318 1 11.3 4.52 4.41 2.1%2 11.1 4.443 11.2 4.484 10.7 4.285 11 4.406 10.8 4.3221-Jun-2011 2 04-May-2012 318 1 5.9 2.36 2.56 5.5%2 6.2 2.48continued on next page. . .233B.1. Mortar compressionFmax fmax mean of cv ofDate cast Batch Date tested Age Specimen (kN) (MPa) fmax fmax3 6.3 2.524 6.4 2.565 6.8 2.726 6.8 2.72Mean of batch means: 3.97 22.5%Min. of batch means: 2.56Max. of batch means: 5.75Table B.2: Mortar cube compression results, walls SS-3 and RR-3Fmax fmax mean of cv ofDate cast Batch Date tested Age Specimen (kN) (MPa) fmax fmax19-Apr-2012 1 21-Jun-2012 63 1 12.5 5.00 5.28 4.7%2 13.4 5.363 13.7 5.4819-Apr-2012 2 21-Jun-2012 63 1 6.5 2.60 2.60 6.2%2 6.9 2.763 6.1 2.4419-Apr-2012 3 21-Jun-2012 63 1 18.6 7.44 7.52 3.3%2 18.3 7.323 19.5 7.8020-Apr-2012 1 21-Jun-2012 62 1 11.4 4.56 4.33 6.8%2 11.1 4.443 10 4.0020-Apr-2012 2 21-Jun-2012 62 1 13.2 5.28 5.27 2.7%2 13.5 5.403 12.8 5.1220-Apr-2012 3 21-Jun-2012 62 1 11.5 4.60 4.96 9.3%2 13.7 5.483 12 4.8023-Apr-2012 1 21-Jun-2012 59 1 5.9 2.36 2.31 2.6%2 5.6 2.243 5.8 2.3223-Apr-2012 2 21-Jun-2012 59 1 8 3.20 3.05 4.2%2 7.5 3.003 7.4 2.9623-Apr-2012 3 21-Jun-2012 59 1 8.4 3.36 3.41 4.9%2 8.2 3.283 9 3.6024-Apr-2012 1 21-Jun-2012 58 1 9.2 3.68 3.40 7.3%2 8.3 3.323 8 3.2024-Apr-2012 2 21-Jun-2012 58 1 11.3 4.52 4.73 4.3%2 12.3 4.923 11.9 4.7624-Apr-2012 3 21-Jun-2012 58 1 7.7 3.08 3.04 3.5%2 7.8 3.123 7.3 2.92continued on next page. . .234B.1. Mortar compressionFmax fmax mean of cv ofDate cast Batch Date tested Age Specimen (kN) (MPa) fmax fmaxMean of batch means: 4.16 35.7%Min. of batch means: 2.31Max. of batch means: 7.52235B.2. Masonry compressionB.2 Masonry compressionA total of 10 masonry prisms were tested in compression: six specimens (1–6) created duringconstruction of the first three walls (FF-3, FR-3, FF-2 ) and four specimens (A–D) createdduring construction of the last two walls (SS-3, RR-3 ). Testing was carried out in accor-dance with ASTM C1314 - 11a [ASTM, 2011] on the Baldwin testing machine in the UBCstructures lab. Prior to testing, all prisms were capped top and bottom with hydrostone,cast against a precision-ground steel plate to ensure total flatness of the bearing surface.Specimens 1–6 were tested on 25 May 2012, and specimens A–D were tested on 16 July2012. A typical compression failure from the testing is shown in Figure B.2. Displacementswere measured by two linear potentiometers, one on each side of the loading plate; therecorded displacement was taken as the average of the two readings, thus compensating fortilt of the loading plate. Load and displacement readings were low-pass filtered to reducenoise.Figure B.2: Typical masonry prism compression failurePrism dimensions are listed in Table B.3. In accordance with ASTM C1314, the maxi-mum gross stress, fmax is calculated by dividing the peak load, Fmax, by the area, A. Thegross stress is then multiplied by a correction factor for slenderness, Ch/t, which is specifiedin ASTM C1314, to obtain the masonry compressive strength, f ′m. The elastic modulus,E, is calculated based on the change in gross stress, df , and the change in strain, d, be-tween the points on the force-displacement curve at 5% and 33% of peak load. Results aresummarized in Tables B.4 and B.5. Force-displacement curves are shown in Figure B.3.236B.2. Masonry compressionTable B.3: Dimensions of masonry prisms (mm, mm2)1 2 3 4 5 6 A B C DL1 190 191 190 190 189 189 190 189 191 190L2 191 190 190 191 190 190 191 190 191 191L3 190 187 191 190 190 189 192 190 191 191L4 192 187 192 191 190 189 191 190 190 191W1 89 88 89 89 88 89 90 89 87 89W2 89 89 90 90 88 88 89 89 88 89W3 88 87 89 89 89 88 89 89 87 90W4 88 86 89 89 89 88 89 89 88 89H1 299 295 296 299 291 292 291 287 288 289H2 299 295 296 299 291 292 291 287 288 289H3 299 295 296 299 291 292 291 287 288 289H4 299 295 296 299 291 292 291 287 288 289Lmean 190.8 188.8 190.8 190.5 189.8 189.3 191.0 189.8 190.8 190.8Wmean 88.5 87.5 89.3 89.3 88.5 88.3 89.3 89.0 87.5 89.3Hmean 299.0 295.0 296.0 299.0 291.0 292.0 291.0 287.0 288.0 289.0h/t 3.38 3.37 3.32 3.35 3.29 3.31 3.26 3.22 3.29 3.24A 16881 16516 17024 17002 16793 16701 17047 16888 16691 17024Table B.4: Prism compression test results, walls FF-3, FR-3, and FF-2df d E Pmax fmax Ch/t f ′mSpecimen (MPa) — (MPa) (kN) (MPa) — (MPa)1 5.583 0.00105 5302 340 20.1 1.100 22.12 7.836 0.00108 7257 462 28.0 1.100 30.73 9.466 0.00093 10145 579 34.0 1.095 37.24 8.850 0.00114 7788 538 31.7 1.098 34.85 8.605 0.00191 4512 520 30.9 1.093 33.86 10.687 0.00341 3138 638 38.2 1.095 41.8mean 8.504 0.00159 6357 513 30.5 1.097 33.4cv 0.20 0.60 0.40 0.20 0.20 0.00 0.20Table B.5: Prism compression test results, walls SS-3 and RR-3df d E Pmax fmax Ch/t f ′mSpecimen (MPa) — (MPa) (kN) (MPa) — (MPa)A 10.707 0.00098 10912 650 38.1 1.091 41.6B 13.381 0.00137 9772 810 47.9 1.088 52.2C 12.290 0.00170 7230 733 43.9 1.093 48.0D 10.264 0.00085 12117 621 36.5 1.089 39.7mean 11.660 0.00122 10008 703 41.6 1.090 45.4cv 0.12 0.32 0.21 0.12 0.13 0.00 0.13237B.2. Masonry compression0 2 4 60200400600800δ (mm)P(kN)(a) Walls FF-3, FR-3, and FF-20 2 4 60200400600800δ (mm)P(kN)(b) Walls SS-3 and RR-3Figure B.3: Masonry prism compression tests238B.3. Masonry flexural tensionB.3 Masonry flexural tensionA total of 9 batches of masonry prisms, each consisting of three prisms four bricks high,were subjected to bond wrench testing. Each batch was sampled from a different day ofconstruction of the walls; five batches (1, 3, 4, 5, 6) were created during construction of thefirst three walls (FF-3, FR-3, FF-2 ) and four batches (A–D) were created during construc-tion of the last two walls (SS-3, RR-3 ). Batch 2 from the first set of walls was damagedand was not suitable for testing. Bond wrench testing was carried out in accordance withASTM C1072 - 10 [ASTM, 2010], with an apparatus constructed as per the drawings inthat standard. The force was applied to the apparatus by the Tinius Olsen testing machinein the UBC structures lab, and was measured digitally using a load cell. The setup is shownin Figure B.4.(a) End view (b) Side viewFigure B.4: Bond wrench test apparatusMasonry prisms were extremely fragile due to the low tensile bond strength, and numer-ous prisms were broken accidentally during handling. A number of courses broke at very lowloads; it is likely that these were subjected to damage in preparation for the test, and theseresults were therefore neglected. Nearly all specimens broke cleanly at the brick-mortarinterface (Figure B.5), while a few specimens had portions of the failure surface within themortar bed.ASTM C1072 specifies the formula to calculate the flexural tensile strength, here denotedf ′fb, as follows:f ′fb = 6 (PL+ PlLl)bd2 −P + Plbd (B.1)where the parameters are defined as follows (where applicable, values specific to the appa-ratus and bricks used are indicated):239B.3. Masonry flexural tensionFigure B.5: Typical bond wrench failuref ′fb = — gross area flexural tensile strengthP = — maximum applied loadPl = 159N weight of loading arm including one brickL = 369.5mm distance from center of prism to loading pointLl = −13.5mm distance from center of prism to centroid of loading armb = 191mm cross-sectional width of the mortar-bedded aread = 89mm cross-sectional depth of the mortar-bedded areaA few of the mortar beds had notable void spaces in the tension zone. Where it wasdeemed significant, the estimated void ratio in the tension zone was estimated by eye, andthe strength was multiplied by (1−Rvoid)−1 to adjust for this effect. Due to the relativelysmall number of batches and large variability within individual batches, mean and cv werecalculated simply from the list of all successful tests, without regard for batches.240B.3. Masonry flexural tensionTable B.6: Bond wrench test results, walls FF-3, FR-3, and FF-2Age P f ′fb mean of cv ofMix date Batch Test date (days) ID (N) use? Rvoid (MPa) f ′fb f ′fb13-Jun-2011 1 08-May-2012 330 1 – 1 40.5 no 5% 0.256 0.15– 2 233.1 yes 0% 0.310– 3 156.6 yes 10% 0.2252 – 1 no 0%– 2 178.0 yes 10% 0.258– 3 176.6 yes 0% 0.2313 – 1 107.2 no 0%– 2 no 0%– 3 31.1 no 0%15-Jun-2011 3 08-May-2012 328 1 – 1 231.8 yes 0% 0.308 0.317 0.29– 2 214.0 yes 5% 0.298– 3 190.0 yes 5% 0.2622 – 1 31.1 no 0%– 2 214.4 yes 0% 0.284– 3 219.8 yes 0% 0.2913 – 1 383.5 yes 0% 0.522– 2 no 0%– 3 195.3 yes 0% 0.25717-Jun-2011 4 08-May-2012 326 1 – 1 261.2 yes 0% 0.349 0.299 0.20– 2 261.2 yes 0% 0.349– 3 256.3 yes 0% 0.3432 – 1 266.5 yes 0% 0.357– 2 196.2 yes 0% 0.258– 3 202.0 yes 5% 0.2803 – 1 190.4 yes 0% 0.250– 2 133.5 yes 5% 0.179– 3 242.0 yes 0% 0.32320-Jun-2011 5 08-May-2012 323 1 – 1 235.4 yes 0% 0.313 0.314 0.11– 2 232.7 yes 0% 0.309– 3 223.8 yes 0% 0.2972 – 1 193.5 yes 0% 0.254– 2 251.4 yes 5% 0.353– 3 276.3 yes 0% 0.3713 – 1 218.4 yes 0% 0.289– 2 211.8 yes 5% 0.295– 3 254.9 yes 0% 0.34121-Jun-2011 6 08-May-2012 322 1 – 1 446.7 yes 0% 0.610 0.621 0.20– 2 480.5 yes 0% 0.658– 3 383.5 yes 20% 0.6522 – 1 574.8 yes 0% 0.791– 2 519.6 yes 0% 0.713– 3 358.1 yes 0% 0.4863 – 1 526.3 yes 0% 0.722– 2 286.5 yes 5% 0.405– 3 404.0 yes 0% 0.550Mean of all samples: 0.378 0.42241B.3. Masonry flexural tensionTable B.7: Bond wrench test results, walls SS-3 and RR-3Age P f ′fb mean of cv ofMix date Batch Test date (days) ID (N) use? Rvoid (MPa) f ′fb f ′fb19-Apr-2012 A 20-Jun-2012 62 1 – 1 555.2 yes 0% 0.763 0.791 0.23– 2 no 0%– 3 311.4 yes 0% 0.4202 – 1 618.9 yes 0% 0.853– 2 610.4 yes 0% 0.841– 3 544.6 yes 0% 0.7483 – 1 708.7 yes 0% 0.979– 2 678.5 yes 0% 0.936– 3 159.3 no 0%–20-Apr-2012 B 20-Jun-2012 61 1 – 1 483.2 yes 5% 0.697 0.519 0.17– 2 425.8 yes 0% 0.581– 3 347.9 yes 0% 0.4712 – 1 400.0 yes 0% 0.545– 2 303.4 yes 0% 0.409– 3 357.3 yes 0% 0.4853 – 1 413.3 yes 0% 0.563– 2 361.7 yes 0% 0.491– 3 319.9 yes 0% 0.432–23-Apr-2012 C 20-Jun-2012 58 1 – 1 287.4 yes 0% 0.386 0.411 0.24– 2 221.1 yes 5% 0.309– 3 396.4 yes 0% 0.5402 – 1 177.1 yes 0% 0.231– 2 293.6 yes 0% 0.395– 3 283.8 yes 0% 0.3813 – 1 344.8 yes 0% 0.467– 2 370.2 yes 0% 0.503– 3 361.7 yes 0% 0.491–24-Apr-2012 D 20-Jun-2012 57 1 – 1 412.0 yes 0% 0.562 0.539 0.13– 2 349.7 yes 0% 0.474– 3 no 0%2 – 1 359.9 yes 0% 0.488– 2 463.6 yes 0% 0.634– 3 392.8 yes 0% 0.5353 – 1 469.8 yes 0% 0.643– 2 353.7 yes 0% 0.480– 3 366.6 yes 0% 0.498Mean of all samples: 0.552 0.31242B.4. Brick compressionB.4 Brick compressionA total of 10 half-brick specimens were tested under compression, according to CAN/CSAA82-06 (R2011) [CSA, 2011]. Bricks were saw-cut to form the half-brick specimens. Twotypes of bricks were mixed through construction of the wall specimens: type A being morerough-faced, and type B being more smooth-faced. Each specimen was capped with hydro-stone on top and bottom prior to testing to ensure flat bearing surfaces. Specimens weretested in the compression machine in the concrete lab at BCIT. As specified in CAN/CSAA82-06, bricks were tested normal to their bedding face (i.e. in the direction of lowestprofile). A typical compression failure is shown in Figure B.6. Dimensions and test resultsare shown in Table B.8.Figure B.6: Typical brick compression failureTable B.8: Brick compression test results, type AL W H A P f ′bID (mm) (mm) (mm) (mm2) (kN) (MPa)A6 89.2 95.8 66.5 8544 1063 124A7 90.3 93.8 64.9 8468 895 106A8 90.0 95.8 66.7 8618 1052 122A9 91.8 92.5 65.0 8483 1017 120A10 90.2 95.2 66.5 8582 1042 121Mean: 119cv : 0.06243B.5. Brick absorptionTable B.9: Brick compression test results, type BL W H A P f ′bID (mm) (mm) (mm) (mm2) (kN) (MPa)B6 89.6 93.4 66.4 8371 1263 151B7 90.3 94.1 66.5 8498 1257 148B8 93.4 90.9 67.5 8490 1251 147B9 95.7 89.9 66.2 8605 1417 165B10 91.2 90.6 64.7 8256 1430 173Mean: 157cv : 0.07B.5 Brick absorptionA total of 10 half-brick specimens, five each of types A and B (refer to Section B.4), weretested for absorption, according to CAN/CSA A82-06 (R2011) [CSA, 2011]. Both 24-hourimmersion and 5-hour boiling absorption were measured. Results are shown in Tables B.10and B.11.Table B.10: Absorption test results, type AMass (g) Absorption (%)Immersed Saturated 5-hour 24-hour 5-hourID Initial Dry in water surface dry boil soak boilA1 1135.1 1134.7 693 1207 1228 6.37 8.22A2 1148.3 1139.6 687 1200 1222 5.30 7.23A3 1100.0 1098.5 642 1146 1167 4.32 6.24A4 1109.5 1102.7 661 1160 1181 5.20 7.10A5 1189.7 1188.6 710 1240 1262 4.32 6.18Mean: 5.10 6.99cv : 0.166 0.120244B.5. Brick absorptionTable B.11: Absorption test results, type BMass (g) Absorption (%)Immersed Saturated 5-hour 24-hour 5-hourID Initial Dry in water surface dry boil soak boilB1 1181.2 1179.9 692 1236 1257 4.75 6.53B2 1131.4 1129.0 675 1181 1202 4.61 6.47B3 1189.5 1188.2 713 1245 1266 4.78 6.55B4 1135.8 1135.1 681 1191 1211 4.92 6.69B5 1120.1 1118.7 670 1173 1195 4.85 6.82Mean: 4.78 6.61cv : 0.025 0.021245Appendix CApparatus SketchesNotes: This section contains sketches detailing the functional configuration of the testapparatus. Overall dimensions and primary member sizes are indicated. All bolts used inprimary connections are ASTM A325, and are 1 in unless otherwise indicated. All bolts,except those clamping the wall base connection, were tightened by turn-of-nut to satisfyslip-critical requirements. In general, bolting capacity far exceeded design forces and noslip was noted during any point of the testing. Welded connections — those not shown asbolted — are typically 6mm fillets all around.246Appendix C. Apparatus SketchesISOMETRIC247Appendix C. Apparatus Sketches4273 (VARIES)3306HSS102x102x6.4(TYP)C250x23 (TYP)4 - 3/4”3947 (VARIES)2606290 (VARIES)1501149A AB B1.5” THREADEDROD THROUGHTABLE (TYP.)PLANELEVATION - SIDEC C248Appendix C. Apparatus SketchesL76x76x6.4 (TYP)1500 (VARIES)212515741650ELEVATION - END249Appendix C. Apparatus Sketches DETAIL A SCALE 1 : 20BSCALE 1 : 50SECTION A-A W460x52C250x23 (TYP)L102x102x6.4 (TYP)C200x17 (TYP)18501800ASCALE 1 : 50SECTION B-B L102x102x6.4 (TYP)W150x30 (TYP)C250x23 (TYP) 1733 2125DETAIL B SCALE 1 : 20250Appendix C. Apparatus Sketches DCSECTION C-C SCALE 1 : 5018001450 89DETAIL C SCALE 1 : 1072SCALE 1 : 10DETAIL D TO FLANGE OFNUT WELDEDW-SECTIONSPACERRUBBERSTEEL BARJAM NUT251Appendix C. Apparatus Sketches DETAIL - BOTTOM CARRIAGE (WALL HIDDEN)DETAIL - TOP CONNECTION252Appendix DInstrumentationNotes: This section contains a sketch of the instrumentation layout and a full instrumen-tation listing.253Appendix D. InstrumentationSP-H3SP-H2SP-H8SP-H7SP-H6SP-H5SP-H4SP-H1SP-BOT.CARRIAGEH1H3H4H5H6ACC-TOP.WALLH7ACC-H8 H8DISPL-TABLEACCEL-TABLEACC-H1H2ACC-H2ACC-H3ACC-H5ACC-H6SP-TOP.CARRIAGEACC-H7LP W CARRIAGEACC-H4LP W TABLELP-TOP.PINLP E TABLEACC-BOT.CARRIAGEACC-TOP.FRAMEE (+X)LP-BOT.WALL.EWLP-BOT.WALL.W(+Z)SP-TOP.FRAMEACC-TOP.CARRIAGELP E CARRIAGE254Appendix D. InstrumentationTable D.1: Instrumentation listing# Ch. Designation ID Mount Reference Brand Model1 8-1 LP E TABLE LP5 Table Floor Duncan 606 R6k2 8-2 LP W TABLE LP6 Table Floor Duncan 606 R6k3 8-3 LP E CARRIAGE LP11 Bot. car. Table Duncan 612 R12k4 8-4 LP W CARRIAGE LP12 Bot. car. Table Duncan 612 R12k5 9-1 SP-H1 SP53 Wall H1 Ref. fr. Celesco 632036 Rev. B6 9-2 SP-H2 SP13 Wall H2 Ref. fr. Celesco PT1017 9-3 SP-H3 SP15 Wall H3 Ref. fr. Celesco PT1018 9-4 SP-H4 SP16 Wall H4 Ref. fr. Celesco PT1019 9-5 SP-H5 SP17 Wall H5 Ref. fr. Celesco PT10110 9-6 SP-H6 SP18 Wall H6 Ref. fr. Celesco PT10111 9-7 SP-H7 SP19 Wall H7 Ref. fr. Celesco PT10112 9-8 SP-H8 SP20 Wall H8 Ref. fr. Celesco PT10113 10-1 SP-TOP.CARRIAGE SP49 Top car. Top fr. Celesco SP1-5014 10-2 SP-BOT.CARRIAGE SP51 Bot. car. Bot. fr. Celesco SP1-5015 10-3 SP-TOP.FRAME SP52 Top fr. Ref. fr. Celesco SP1-5016 10-4 LP-TOP.PIN.N LP7 Top car. Top pin Novotechnik TR10017 10-5 LP-TOP.PIN.S LP8 Top car. Top pin Novotechnik TR10018 10-6 LP-BOT.WALL.E LP9 Wall bot. Bot. beam Novotechnik TR10019 10-7 LP-BOT.WALL.W LP10 Wall bot. Bot. beam Novotechnik TR10020 11-1 ACC-H1 A1D-1 Wall H1 — ICS 302821 11-2 ACC-H2 A1D-2 Wall H2 — ICS 302822 11-3 ACC-H3 A1D-3 Wall H3 — ICS 302823 11-4 ACC-H4 A1D-4 Wall H4 — ICS 302824 11-5 ACC-H5 A1D-5 Wall H5 — ICS 302825 11-6 ACC-H6 A1D-6 Wall H6 — ICS 302826 11-7 ACC-H7 A1D-7 Wall H7 — ICS 302827 11-8 ACC-H8 A1D-8 Wall H8 — ICS 302828 12-1 ACC-TOP.CARRIAGE A1D-9 Top car. — ICS 302829 12-2 ACC-BOT.CARRIAGE A1D-14 Bot. car. — ICS 302830 12-3 ACC-TOP.WALL.X A3D-3X Top beam — ICS 302631 12-4 ACC-TOP.WALL.Y A3D-3Y Top beam — ICS 302632 12-5 ACC-TOP.WALL.Z A3D-3Z Top beam — ICS 302633 12-6 ACC-TOP.FRAME A1D-10 Top fr. — ICS 302834 1-1 TABLEDISP — Floor Table MTS Temposonic 6Analog35 1-2 TABLEACCEL — Table — ICS 3028255Appendix EPhotos of Shake Table TestingNotes: This section contains photos and video frame captures taken during shake tabletesting. The photos illustrate the crack configurations in walls, and damage (where appli-cable). Observed damage was sparse, and limited to minor spalling at crack locations insome walls only. Video frame captures are provided for each wall at one point in the higheststable run (typically illustrating the point of greatest rocking or carriage displacement) andin the collapse run, immediately prior to collapse. Finally, high speed video frame capturesare provided for each wall in the highest stable run, illustrating the maximum displacedshape in profile.256Appendix E. Photos of Shake Table Testing(a) East face(b) West faceFigure E.1: Wall FF-3, cracking pattern257Appendix E. Photos of Shake Table Testing(a) East face(b) West faceFigure E.2: Wall FR-3, cracking pattern258Appendix E. Photos of Shake Table Testing(a) East face(b) West faceFigure E.3: Wall FF-2, cracking pattern259Appendix E. Photos of Shake Table TestingFigure E.4: Wall FF-2, detail of crack stepFigure E.5: Wall FF-2, detail of spalling260Appendix E. Photos of Shake Table TestingFigure E.6: Wall FF-2, apparatus in lowered position261Appendix E. Photos of Shake Table Testing(a) East face(b) West faceFigure E.7: Wall SS-3, cracking pattern262Appendix E. Photos of Shake Table Testing(a) East face(b) West faceFigure E.8: Wall RR-3, cracking pattern263Appendix E. Photos of Shake Table TestingFigure E.9: Wall RR-3, detail of spalling264Appendix E. Photos of Shake Table Testing(a) Run 12: stable(b) Run 13: at collapseFigure E.10: Wall FF-3, video frame captures265Appendix E. Photos of Shake Table Testing(a) Run 9: stable(b) Run 10: at collapseFigure E.11: Wall FR-3, video frame captures266Appendix E. Photos of Shake Table Testing(a) Run 10: stable(b) Run 11: at collapseFigure E.12: Wall FF-2, video frame captures267Appendix E. Photos of Shake Table Testing(a) Run 12: stable(b) Run 13: at collapseFigure E.13: Wall SS-3, video frame captures268Appendix E. Photos of Shake Table Testing(a) Run 5: stable(b) Run 7: at collapseFigure E.14: Wall RR-3, video frame captures269Appendix E. Photos of Shake Table Testing(a) Wall FF-3, run 12 (b) Wall FR-3, run 9(c) Wall FF-2, run 10 (d) Wall SS-3, run 12(e) Wall RR-3, run 5Figure E.15: High speed video frame captures, stable runs270Appendix FShake Table Test ResultsNotes: This section contains a summary page of plotted data for each shake table run.There are four time series plots:• Relative displacement (relative to the shake table) of the top and bottom carriagesand the crack location• Rocking displacement of the crack location• Acceleration of the top and bottom carriages and the crack location• The total force on the wall, calculated from the measured wall accelerationsProfiles up the height of the wall are shown for the first three time series at one selectedtime step. The time step chosen was typically at the peak force, for uncracked walls, orat the peak of the largest rocking excursion, where rocking was notable. In runs in whichinitiated cracking, the profiles are shown at the last time step at which it was confirmedthat the wall was uncracked (i.e. immediately before cracking). In the bottom right is aplot of [total force]–[rocking displacement at the crack] hysteresis.It is important to note that the scales of the plots vary among the different runs so asto best illustrate the results in each case. In addition, the displacement measurements inruns with significant table or carriage motion are subject to at least ± several mm error.The rocking displacements in particular should therefore be interpreted carefully whereamplitudes are small. The rocking profile is a good indicator of the relative importance ofthe displacement measurement error for a given run.271Appendix F. Shake Table Test Results6 8 10 12 14 16 18 20-10-50510drel(mm)6 8 10 12 14 16 18 20-10-50510drock(mm)6 8 10 12 14 16 18 20-0.0500.05a(g)6 8 10 12 14 16 18 20-101F(kN)Time (sec)-10 0 1001234drel (mm)Height(m)-10 0 1001234drock (mm)Height(m)-0.05 0 0.0501234a (g)Height(m)-10 0 10-101F(kN)drock (mm)Time = 10.405 sWall: FF-3 Run: 1 Motion: CHHC1@10% PGA: 0.05 g PGD: 18mmTop carriage Bottom carriage Crack272Appendix F. Shake Table Test Results6 8 10 12 14 16 18 20-20-1001020drel(mm)6 8 10 12 14 16 18 20-10-50510drock(mm)6 8 10 12 14 16 18 20-0.0500.05a(g)6 8 10 12 14 16 18 20-2-1012F(kN)Time (sec)-10 0 1001234drel (mm)Height(m)-10 0 1001234drock (mm)Height(m)-0.06 0 0.0601234a (g)Height(m)-10 0 10-2-1012F(kN)drock (mm)Time = 10.955 sWall: FF-3 Run: 2 Motion: CHHC1@30% PGA: 0.19 g PGD: 57mmTop carriage Bottom carriage Crack273Appendix F. Shake Table Test Results6 8 10 12 14 16 18 20-50050drel(mm)6 8 10 12 14 16 18 20-10-50510drock(mm)6 8 10 12 14 16 18 20-0.100.1a(g)6 8 10 12 14 16 18 20-4-2024F(kN)Time (sec)-70 0 7001234drel (mm)Height(m)-10 0 1001234drock (mm)Height(m)-0.1 0 0.101234a (g)Height(m)-10 0 10-4-2024F(kN)drock (mm)Time = 11.090 sWall: FF-3 Run: 3 Motion: CHHC1@50% PGA: 0.27 g PGD: 96mmTop carriage Bottom carriage Crack274Appendix F. Shake Table Test Results6 8 10 12 14 16 18 20-100-50050100drel(mm)6 8 10 12 14 16 18 20-10-50510drock(mm)6 8 10 12 14 16 18 20-0.200.2a(g)6 8 10 12 14 16 18 20-505F(kN)Time (sec)-100 0 10001234drel (mm)Height(m)-10 0 1001234drock (mm)Height(m)-0.2 0 0.201234a (g)Height(m)-10 0 10-505F(kN)drock (mm)Time = 11.115 sWall: FF-3 Run: 4 Motion: CHHC1@70% PGA: 0.41 g PGD: 135mmTop carriage Bottom carriage Crack275Appendix F. Shake Table Test Results6 8 10 12 14 16 18 20-1000100drel(mm)6 8 10 12 14 16 18 20-10-50510drock(mm)6 8 10 12 14 16 18 20-0.200.2a(g)6 8 10 12 14 16 18 20-10-50510F(kN)Time (sec)-100 0 10001234drel (mm)Height(m)-10 0 1001234drock (mm)Height(m)-0.3 0 0.301234a (g)Height(m)-10 0 10-10-50510F(kN)drock (mm)Time = 11.140 sWall: FF-3 Run: 5 Motion: CHHC1@80% PGA: 0.51 g PGD: 155mmTop carriage Bottom carriage Crack276Appendix F. Shake Table Test Results6 8 10 12 14 16 18 20-200-1000100200drel(mm)6 8 10 12 14 16 18 20-10010drock(mm)6 8 10 12 14 16 18 20-0.4-0.200.20.4a(g)6 8 10 12 14 16 18 20-10-50510F(kN)Time (sec)-200 0 20001234drel (mm)Height(m)-10 0 1001234drock (mm)Height(m)-0.4 0 0.401234a (g)Height(m)-10 0 10-10-50510F(kN)drock (mm)Time = 11.175 sWall: FF-3 Run: 6 Motion: CHHC1@100% PGA: 0.64 g PGD: 194mmTop carriage Bottom carriage Crack277Appendix F. Shake Table Test Results1 2 3 4 5 6 7-10-50510drel(mm)1 2 3 4 5 6 7-10-50510drock(mm)1 2 3 4 5 6 7-0.500.5a(g)1 2 3 4 5 6 7-10010F(kN)Time (sec)-10 0 1001234drel (mm)Height(m)-10 0 1001234drock (mm)Height(m)-0.4 0 0.401234a (g)Height(m)-10 0 10-10010F(kN)drock (mm)Time = 4.320 sWall: FF-3 Run: 7 Motion: NGA0763@50% PGA: 0.24 g PGD: 29mmTop carriage Bottom carriage Crack278Appendix F. Shake Table Test Results1 2 3 4 5 6 7-10-50510drel(mm)1 2 3 4 5 6 7-10-50510drock(mm)1 2 3 4 5 6 7-0.500.5a(g)1 2 3 4 5 6 7-10-50510F(kN)Time (sec)-10 0 1001234drel (mm)Height(m)-10 0 1001234drock (mm)Height(m)-0.3 0 0.301234a (g)Height(m)-10 0 10-10-50510F(kN)drock (mm)Time = 4.170 sWall: FF-3 Run: 8 Motion: NGA0763@60% PGA: 0.31 g PGD: 36mmTop carriage Bottom carriage Crack279Appendix F. Shake Table Test Results6 8 10 12 14 16 18 20-10010drel(mm)6 8 10 12 14 16 18 20-10-50510drock(mm)6 8 10 12 14 16 18 20-0.1-0.0500.050.1a(g)6 8 10 12 14 16 18 20-2-1012F(kN)Time (sec)-10 0 1001234drel (mm)Height(m)-10 0 1001234drock (mm)Height(m)-0.07 0 0.0701234a (g)Height(m)-10 0 10-2-1012F(kN)drock (mm)Time = 10.955 sWall: FF-3 Run: 9 Motion: CHHC1@30% PGA: 0.18 g PGD: 57mmTop carriage Bottom carriage Crack280Appendix F. Shake Table Test Results6 8 10 12 14 16 18 20-50050drel(mm)6 8 10 12 14 16 18 20-10-50510drock(mm)6 8 10 12 14 16 18 20-0.100.1a(g)6 8 10 12 14 16 18 20-4-2024F(kN)Time (sec)-70 0 7001234drel (mm)Height(m)-10 0 1001234drock (mm)Height(m)-0.1 0 0.101234a (g)Height(m)-10 0 10-4-2024F(kN)drock (mm)Time = 11.110 sWall: FF-3 Run: 10 Motion: CHHC1@50% PGA: 0.28 g PGD: 96mmTop carriage Bottom carriage Crack281Appendix F. Shake Table Test Results6 8 10 12 14 16 18 20-100-50050100drel(mm)6 8 10 12 14 16 18 20-10010drock(mm)6 8 10 12 14 16 18 20-0.200.2a(g)6 8 10 12 14 16 18 20-505F(kN)Time (sec)-100 0 10001234drel (mm)Height(m)-10 0 1001234drock (mm)Height(m)-0.1 0 0.101234a (g)Height(m)-10 0 10-505F(kN)drock (mm)Time = 12.275 sWall: FF-3 Run: 11 Motion: CHHC1@70% PGA: 0.43 g PGD: 135mmTop carriage Bottom carriage Crack282Appendix F. Shake Table Test Results6 8 10 12 14 16 18 20-200-1000100200drel(mm)6 8 10 12 14 16 18 20-40-2002040drock(mm)6 8 10 12 14 16 18 20-0.500.5a(g)6 8 10 12 14 16 18 20-505F(kN)Time (sec)-100 0 10001234drel (mm)Height(m)-40 0 4001234drock (mm)Height(m)-0.3 0 0.301234a (g)Height(m)-40 0 40-505F(kN)drock (mm)Time = 12.145 sWall: FF-3 Run: 12 Motion: CHHC1@80% PGA: 0.50 g PGD: 155mmTop carriage Bottom carriage Crack283Appendix F. Shake Table Test Results6 8 10 12 14 16 18 20-400-2000200400drel(mm)6 8 10 12 14 16 18 20-2000200drock(mm)6 8 10 12 14 16 18 20-101a(g)6 8 10 12 14 16 18 20-20-1001020F(kN)Time (sec)-200 0 20001234drel (mm)Height(m)-200 0 20001234drock (mm)Height(m)-0.3 0 0.301234a (g)Height(m)-300 0 300-20-1001020F(kN)drock (mm)Time = 11.555 sWall: FF-3 Run: 13 Motion: CHHC1@100% PGA: 0.63 g PGD: 194mmTop carriage Bottom carriage Crack284Appendix F. Shake Table Test Results6 8 10 12 14 16 18 20-50050drel(mm)6 8 10 12 14 16 18 20-10-50510drock(mm)6 8 10 12 14 16 18 20-0.4-0.200.20.4a(g)6 8 10 12 14 16 18 20-505F(kN)Time (sec)-70 0 7001234drel (mm)Height(m)-10 0 1001234drock (mm)Height(m)-0.1 0 0.101234a (g)Height(m)-10 0 10-505F(kN)drock (mm)Time = 10.910 sWall: FR-3 Run: 1 Motion: CHHC1@50% PGA: 0.43 g PGD: 96mmTop carriage Bottom carriage Crack285Appendix F. Shake Table Test Results6 8 10 12 14 16 18 20-1000100drel(mm)6 8 10 12 14 16 18 20-10-50510drock(mm)6 8 10 12 14 16 18 20-0.500.5a(g)6 8 10 12 14 16 18 20-10-50510F(kN)Time (sec)-100 0 10001234drel (mm)Height(m)-10 0 1001234drock (mm)Height(m)-0.3 0 0.301234a (g)Height(m)-10 0 10-10-50510F(kN)drock (mm)Time = 10.985 sWall: FR-3 Run: 2 Motion: CHHC1@70% PGA: 0.57 g PGD: 135mmTop carriage Bottom carriage Crack286Appendix F. Shake Table Test Results1 2 3 4 5 6 7-10-50510drel(mm)1 2 3 4 5 6 7-10-50510drock(mm)1 2 3 4 5 6 7-0.500.5a(g)1 2 3 4 5 6 7-10010F(kN)Time (sec)-10 0 1001234drel (mm)Height(m)-10 0 1001234drock (mm)Height(m)-0.4 0 0.401234a (g)Height(m)-10 0 10-10010F(kN)drock (mm)Time = 5.180 sWall: FR-3 Run: 3 Motion: NGA0763@60% PGA: 0.27 g PGD: 35mmTop carriage Bottom carriage Crack287Appendix F. Shake Table Test Results6 8 10 12 14 16 18 20-50050drel(mm)6 8 10 12 14 16 18 20-10-50510drock(mm)6 8 10 12 14 16 18 20-0.4-0.200.20.4a(g)6 8 10 12 14 16 18 20-505F(kN)Time (sec)-80 0 8001234drel (mm)Height(m)-10 0 1001234drock (mm)Height(m)-0.2 0 0.201234a (g)Height(m)-10 0 10-505F(kN)drock (mm)Time = 10.870 sWall: FR-3 Run: 4 Motion: CHHC1@50% PGA: 0.30 g PGD: 96mmTop carriage Bottom carriage Crack288Appendix F. Shake Table Test Results6 8 10 12 14 16 18 20-1000100drel(mm)6 8 10 12 14 16 18 20-10-50510drock(mm)6 8 10 12 14 16 18 20-0.4-0.200.20.4a(g)6 8 10 12 14 16 18 20-505F(kN)Time (sec)-50 0 5001234drel (mm)Height(m)-10 0 1001234drock (mm)Height(m)-0.3 0 0.301234a (g)Height(m)-10 0 10-505F(kN)drock (mm)Time = 9.340 sWall: FR-3 Run: 5 Motion: CHHC1@70% PGA: 0.44 g PGD: 135mmTop carriage Bottom carriage Crack289Appendix F. Shake Table Test Results6 8 10 12 14 16 18 20-1000100drel(mm)6 8 10 12 14 16 18 20-10-50510drock(mm)6 8 10 12 14 16 18 20-0.4-0.200.20.4a(g)6 8 10 12 14 16 18 20-505F(kN)Time (sec)-100 0 10001234drel (mm)Height(m)-10 0 1001234drock (mm)Height(m)-0.2 0 0.201234a (g)Height(m)-10 0 10-505F(kN)drock (mm)Time = 10.850 sWall: FR-3 Run: 6 Motion: CHHC1@80% PGA: 0.43 g PGD: 155mmTop carriage Bottom carriage Crack290Appendix F. Shake Table Test Results6 8 10 12 14 16 18 20-1000100drel(mm)6 8 10 12 14 16 18 20-10-50510drock(mm)6 8 10 12 14 16 18 20-0.500.5a(g)6 8 10 12 14 16 18 20-10-50510F(kN)Time (sec)-50 0 5001234drel (mm)Height(m)-10 0 1001234drock (mm)Height(m)-0.3 0 0.301234a (g)Height(m)-10 0 10-10-50510F(kN)drock (mm)Time = 9.380 sWall: FR-3 Run: 7 Motion: CHHC1@90% PGA: 0.44 g PGD: 174mmTop carriage Bottom carriage Crack291Appendix F. Shake Table Test Results6 8 10 12 14 16 18 20-1000100drel(mm)6 8 10 12 14 16 18 20-10-50510drock(mm)6 8 10 12 14 16 18 20-0.500.5a(g)6 8 10 12 14 16 18 20-10-50510F(kN)Time (sec)-10 0 1001234drel (mm)Height(m)-10 0 1001234drock (mm)Height(m)-0.1 0 0.101234a (g)Height(m)-10 0 10-10-50510F(kN)drock (mm)Time = 10.620 sWall: FR-3 Run: 8 Motion: CHHC1@100% PGA: 0.51 g PGD: 194mmTop carriage Bottom carriage Crack292Appendix F. Shake Table Test Results6 8 10 12 14 16 18 20-200-1000100200drel(mm)6 8 10 12 14 16 18 20-10010drock(mm)6 8 10 12 14 16 18 20-0.500.5a(g)6 8 10 12 14 16 18 20-10-50510F(kN)Time (sec)-100 0 10001234drel (mm)Height(m)-10 0 1001234drock (mm)Height(m)-0.3 0 0.301234a (g)Height(m)-10 0 10-10-50510F(kN)drock (mm)Time = 13.875 sWall: FR-3 Run: 9 Motion: CHHC1@110% PGA: 0.50 g PGD: 213mmTop carriage Bottom carriage Crack293Appendix F. Shake Table Test Results6 8 10 12 14 16 18 20-2000200drel(mm)6 8 10 12 14 16 18 20-2000200drock(mm)6 8 10 12 14 16 18 20-0.500.5a(g)6 8 10 12 14 16 18 20-10-50510F(kN)Time (sec)-100 0 10001234drel (mm)Height(m)-50 0 5001234drock (mm)Height(m)-0.2 0 0.201234a (g)Height(m)-300 0 300-10-50510F(kN)drock (mm)Time = 14.030 sWall: FR-3 Run: 10 Motion: CHHC1@120% PGA: 0.61 g PGD: 233mmTop carriage Bottom carriage Crack294Appendix F. Shake Table Test Results6 8 10 12 14 16 18 20-40-2002040drel(mm)6 8 10 12 14 16 18 20-10-50510drock(mm)6 8 10 12 14 16 18 20-0.100.1a(g)6 8 10 12 14 16 18 20-2-1012F(kN)Time (sec)-30 0 3001234drel (mm)Height(m)-10 0 1001234drock (mm)Height(m)-0.1 0 0.101234a (g)Height(m)-10 0 10-2-1012F(kN)drock (mm)Time = 10.940 sWall: FF-2 Run: 1 Motion: CHHC1@50% PGA: 0.36 g PGD: 96mmTop carriage Bottom carriage Crack295Appendix F. Shake Table Test Results6 8 10 12 14 16 18 20-100-50050100drel(mm)6 8 10 12 14 16 18 20-10-50510drock(mm)6 8 10 12 14 16 18 20-0.2-0.100.10.2a(g)6 8 10 12 14 16 18 20-505F(kN)Time (sec)-100 0 10001234drel (mm)Height(m)-10 0 1001234drock (mm)Height(m)-0.2 0 0.201234a (g)Height(m)-10 0 10-505F(kN)drock (mm)Time = 10.975 sWall: FF-2 Run: 2 Motion: CHHC1@80% PGA: 0.53 g PGD: 155mmTop carriage Bottom carriage Crack296Appendix F. Shake Table Test Results1 2 3 4 5 6 7-10-50510drel(mm)1 2 3 4 5 6 7-10-50510drock(mm)1 2 3 4 5 6 7-0.500.5a(g)1 2 3 4 5 6 7-505F(kN)Time (sec)-10 0 1001234drel (mm)Height(m)-10 0 1001234drock (mm)Height(m)-0.6 0 0.601234a (g)Height(m)-10 0 10-505F(kN)drock (mm)Time = 3.375 sWall: FF-2 Run: 3 Motion: NGA0763@60% PGA: 0.27 g PGD: 35mmTop carriage Bottom carriage Crack297Appendix F. Shake Table Test Results1 2 3 4 5 6 7-10-50510drel(mm)1 2 3 4 5 6 7-10-50510drock(mm)1 2 3 4 5 6 7-0.500.5a(g)1 2 3 4 5 6 7-10-50510F(kN)Time (sec)-10 0 1001234drel (mm)Height(m)-10 0 1001234drock (mm)Height(m)-0.8 0 0.801234a (g)Height(m)-10 0 10-10-50510F(kN)drock (mm)Time = 4.120 sWall: FF-2 Run: 4 Motion: NGA0763@70% PGA: 0.36 g PGD: 41mmTop carriage Bottom carriage Crack298Appendix F. Shake Table Test Results6 8 10 12 14 16 18 20-40-2002040drel(mm)6 8 10 12 14 16 18 20-10-50510drock(mm)6 8 10 12 14 16 18 20-0.100.1a(g)6 8 10 12 14 16 18 20-2-1012F(kN)Time (sec)-30 0 3001234drel (mm)Height(m)-10 0 1001234drock (mm)Height(m)-0.1 0 0.101234a (g)Height(m)-10 0 10-2-1012F(kN)drock (mm)Time = 10.930 sWall: FF-2 Run: 5 Motion: CHHC1@50% PGA: 0.31 g PGD: 96mmTop carriage Bottom carriage Crack299Appendix F. Shake Table Test Results6 8 10 12 14 16 18 20-50050drel(mm)6 8 10 12 14 16 18 20-10-50510drock(mm)6 8 10 12 14 16 18 20-0.2-0.100.10.2a(g)6 8 10 12 14 16 18 20-202F(kN)Time (sec)-80 0 8001234drel (mm)Height(m)-10 0 1001234drock (mm)Height(m)-0.2 0 0.201234a (g)Height(m)-10 0 10-202F(kN)drock (mm)Time = 11.035 sWall: FF-2 Run: 6 Motion: CHHC1@70% PGA: 0.45 g PGD: 135mmTop carriage Bottom carriage Crack300Appendix F. Shake Table Test Results6 8 10 12 14 16 18 20-100-50050100drel(mm)6 8 10 12 14 16 18 20-10-50510drock(mm)6 8 10 12 14 16 18 20-0.200.2a(g)6 8 10 12 14 16 18 20-4-2024F(kN)Time (sec)-50 0 5001234drel (mm)Height(m)-10 0 1001234drock (mm)Height(m)-0.1 0 0.101234a (g)Height(m)-10 0 10-4-2024F(kN)drock (mm)Time = 11.585 sWall: FF-2 Run: 7 Motion: CHHC1@80% PGA: 0.53 g PGD: 155mmTop carriage Bottom carriage Crack301Appendix F. Shake Table Test Results6 8 10 12 14 16 18 20-100-50050100drel(mm)6 8 10 12 14 16 18 20-20-1001020drock(mm)6 8 10 12 14 16 18 20-0.4-0.200.20.4a(g)6 8 10 12 14 16 18 20-505F(kN)Time (sec)-60 0 6001234drel (mm)Height(m)-10 0 1001234drock (mm)Height(m)-0.3 0 0.301234a (g)Height(m)-10 0 10-505F(kN)drock (mm)Time = 11.570 sWall: FF-2 Run: 8 Motion: CHHC1@90% PGA: 0.59 g PGD: 174mmTop carriage Bottom carriage Crack302Appendix F. Shake Table Test Results6 8 10 12 14 16 18 20-1000100drel(mm)6 8 10 12 14 16 18 20-20020drock(mm)6 8 10 12 14 16 18 20-0.500.5a(g)6 8 10 12 14 16 18 20-505F(kN)Time (sec)-100 0 10001234drel (mm)Height(m)-30 0 3001234drock (mm)Height(m)-0.3 0 0.301234a (g)Height(m)-30 0 30-505F(kN)drock (mm)Time = 11.110 sWall: FF-2 Run: 9 Motion: CHHC1@100% PGA: 0.66 g PGD: 194mmTop carriage Bottom carriage Crack303Appendix F. Shake Table Test Results6 8 10 12 14 16 18 20-1000100drel(mm)6 8 10 12 14 16 18 20-50050drock(mm)6 8 10 12 14 16 18 20-0.500.5a(g)6 8 10 12 14 16 18 20-505F(kN)Time (sec)-100 0 10001234drel (mm)Height(m)-80 0 8001234drock (mm)Height(m)-0.3 0 0.301234a (g)Height(m)-80 0 80-505F(kN)drock (mm)Time = 14.640 sWall: FF-2 Run: 10 Motion: CHHC1@110% PGA: 0.69 g PGD: 214mmTop carriage Bottom carriage Crack304Appendix F. Shake Table Test Results6 8 10 12 14 16 18 20-200-1000100200drel(mm)6 8 10 12 14 16 18 20-200-1000100200drock(mm)6 8 10 12 14 16 18 20-0.500.5a(g)6 8 10 12 14 16 18 20-505F(kN)Time (sec)-100 0 10001234drel (mm)Height(m)-90 0 9001234drock (mm)Height(m)-0.3 0 0.301234a (g)Height(m)-200 0 200-505F(kN)drock (mm)Time = 9.245 sWall: FF-2 Run: 11 Motion: CHHC1@120% PGA: 0.74 g PGD: 233mmTop carriage Bottom carriage Crack305Appendix F. Shake Table Test Results1 2 3 4 5 6 7-10-50510drel(mm)1 2 3 4 5 6 7-10-50510drock(mm)1 2 3 4 5 6 7-0.500.5a(g)1 2 3 4 5 6 7-10010F(kN)Time (sec)-10 0 1001234drel (mm)Height(m)-10 0 1001234drock (mm)Height(m)-0.4 0 0.401234a (g)Height(m)-10 0 10-10010F(kN)drock (mm)Time = 4.305 sWall: RR-3 Run: 1 Motion: NGA0763@50% PGA: 0.25 g PGD: 29mmTop carriage Bottom carriage Crack306Appendix F. Shake Table Test Results1 2 3 4 5 6 7-10-50510drel(mm)1 2 3 4 5 6 7-10-50510drock(mm)1 2 3 4 5 6 7-0.500.5a(g)1 2 3 4 5 6 7-10010F(kN)Time (sec)-10 0 1001234drel (mm)Height(m)-10 0 1001234drock (mm)Height(m)-0.6 0 0.601234a (g)Height(m)-10 0 10-10010F(kN)drock (mm)Time = 4.550 sWall: RR-3 Run: 2 Motion: NGA0763@60% PGA: 0.27 g PGD: 35mmTop carriage Bottom carriage Crack307Appendix F. Shake Table Test Results1 2 3 4 5 6 7-10-50510drel(mm)1 2 3 4 5 6 7-10-50510drock(mm)1 2 3 4 5 6 7-0.500.5a(g)1 2 3 4 5 6 7-10-50510F(kN)Time (sec)-10 0 1001234drel (mm)Height(m)-10 0 1001234drock (mm)Height(m)-0.2 0 0.201234a (g)Height(m)-10 0 10-10-50510F(kN)drock (mm)Time = 4.260 sWall: RR-3 Run: 3 Motion: NGA0763@70% PGA: 0.47 g PGD: 42mmTop carriage Bottom carriage Crack308Appendix F. Shake Table Test Results6 8 10 12 14 16 18 20-10010drel(mm)6 8 10 12 14 16 18 20-10010drock(mm)6 8 10 12 14 16 18 20-0.500.5a(g)6 8 10 12 14 16 18 20-10-50510F(kN)Time (sec)-10 0 1001234drel (mm)Height(m)-10 0 1001234drock (mm)Height(m)-0.2 0 0.201234a (g)Height(m)-10 0 10-10-50510F(kN)drock (mm)Time = 11.465 sWall: RR-3 Run: 4 Motion: CHHC1@50% PGA: 0.43 g PGD: 96mmTop carriage Bottom carriage Crack309Appendix F. Shake Table Test Results6 8 10 12 14 16 18 20-50050drel(mm)6 8 10 12 14 16 18 20-50050drock(mm)6 8 10 12 14 16 18 20-0.500.5a(g)6 8 10 12 14 16 18 20-10-50510F(kN)Time (sec)-60 0 6001234drel (mm)Height(m)-60 0 6001234drock (mm)Height(m)-0.3 0 0.301234a (g)Height(m)-50 0 50-10-50510F(kN)drock (mm)Time = 9.590 sWall: RR-3 Run: 6 Motion: CHHC1@55% PGA: 0.44 g PGD: 106mmTop carriage Bottom carriage Crack310Appendix F. Shake Table Test Results6 8 10 12 14 16 18 20-100-50050100drel(mm)6 8 10 12 14 16 18 20-100-50050100drock(mm)6 8 10 12 14 16 18 20-0.500.5a(g)6 8 10 12 14 16 18 20-10-50510F(kN)Time (sec)-100 0 10001234drel (mm)Height(m)-100 0 10001234drock (mm)Height(m)-0.2 0 0.201234a (g)Height(m)-100 0 100-10-50510F(kN)drock (mm)Time = 10.995 sWall: RR-3 Run: 5 Motion: CHHC1@60% PGA: 0.52 g PGD: 116mmTop carriage Bottom carriage Crack311Appendix F. Shake Table Test Results6 8 10 12 14 16 18 20-2000200drel(mm)6 8 10 12 14 16 18 20-2000200drock(mm)6 8 10 12 14 16 18 20-0.500.5a(g)6 8 10 12 14 16 18 20-10-50510F(kN)Time (sec)-80 0 8001234drel (mm)Height(m)-80 0 8001234drock (mm)Height(m)-0.2 0 0.201234a (g)Height(m)-300 0 300-10-50510F(kN)drock (mm)Time = 10.235 sWall: RR-3 Run: 7 Motion: CHHC1@65% PGA: 0.62 g PGD: 125mmTop carriage Bottom carriage Crack312Appendix F. Shake Table Test Results6 8 10 12 14 16 18 20-10010drel(mm)6 8 10 12 14 16 18 20-10-50510drock(mm)6 8 10 12 14 16 18 20-0.1-0.0500.050.1a(g)6 8 10 12 14 16 18 20-4-2024F(kN)Time (sec)-10 0 1001234drel (mm)Height(m)-10 0 1001234drock (mm)Height(m)-0.1 0 0.101234a (g)Height(m)-10 0 10-4-2024F(kN)drock (mm)Time = 10.720 sWall: SS-3 Run: 1 Motion: CHHC1@30% PGA: 0.19 g PGD: 57mmTop carriage Bottom carriage Crack313Appendix F. Shake Table Test Results6 8 10 12 14 16 18 20-20-1001020drel(mm)6 8 10 12 14 16 18 20-10-50510drock(mm)6 8 10 12 14 16 18 20-0.100.1a(g)6 8 10 12 14 16 18 20-505F(kN)Time (sec)-20 0 2001234drel (mm)Height(m)-10 0 1001234drock (mm)Height(m)-0.1 0 0.101234a (g)Height(m)-10 0 10-505F(kN)drock (mm)Time = 10.770 sWall: SS-3 Run: 2 Motion: CHHC1@40% PGA: 0.22 g PGD: 76mmTop carriage Bottom carriage Crack314Appendix F. Shake Table Test Results6 8 10 12 14 16 18 20-40-2002040drel(mm)6 8 10 12 14 16 18 20-10-50510drock(mm)6 8 10 12 14 16 18 20-0.2-0.100.10.2a(g)6 8 10 12 14 16 18 20-505F(kN)Time (sec)-30 0 3001234drel (mm)Height(m)-10 0 1001234drock (mm)Height(m)-0.2 0 0.201234a (g)Height(m)-10 0 10-505F(kN)drock (mm)Time = 12.790 sWall: SS-3 Run: 3 Motion: CHHC1@50% PGA: 0.27 g PGD: 96mmTop carriage Bottom carriage Crack315Appendix F. Shake Table Test Results6 8 10 12 14 16 18 20-50050drel(mm)6 8 10 12 14 16 18 20-10-50510drock(mm)6 8 10 12 14 16 18 20-0.200.2a(g)6 8 10 12 14 16 18 20-10-50510F(kN)Time (sec)-50 0 5001234drel (mm)Height(m)-10 0 1001234drock (mm)Height(m)-0.3 0 0.301234a (g)Height(m)-10 0 10-10-50510F(kN)drock (mm)Time = 12.825 sWall: SS-3 Run: 4 Motion: CHHC1@60% PGA: 0.33 g PGD: 116mmTop carriage Bottom carriage Crack316Appendix F. Shake Table Test Results1 2 3 4 5 6 7-10-50510drel(mm)1 2 3 4 5 6 7-10-50510drock(mm)1 2 3 4 5 6 7-0.500.5a(g)1 2 3 4 5 6 7-10-50510F(kN)Time (sec)-10 0 1001234drel (mm)Height(m)-10 0 1001234drock (mm)Height(m)-0.4 0 0.401234a (g)Height(m)-10 0 10-10-50510F(kN)drock (mm)Time = 4.445 sWall: SS-3 Run: 5 Motion: NGA0763@50% PGA: 0.23 g PGD: 29mmTop carriage Bottom carriage Crack317Appendix F. Shake Table Test Results1 2 3 4 5 6 7-10-50510drel(mm)1 2 3 4 5 6 7-10-50510drock(mm)1 2 3 4 5 6 7-0.500.5a(g)1 2 3 4 5 6 7-10010F(kN)Time (sec)-10 0 1001234drel (mm)Height(m)-10 0 1001234drock (mm)Height(m)-0.3 0 0.301234a (g)Height(m)-10 0 10-10010F(kN)drock (mm)Time = 5.030 sWall: SS-3 Run: 6 Motion: NGA0763@60% PGA: 0.27 g PGD: 35mmTop carriage Bottom carriage Crack318Appendix F. Shake Table Test Results6 8 10 12 14 16 18 20-10010drel(mm)6 8 10 12 14 16 18 20-10-50510drock(mm)6 8 10 12 14 16 18 20-0.1-0.0500.050.1a(g)6 8 10 12 14 16 18 20-4-2024F(kN)Time (sec)-10 0 1001234drel (mm)Height(m)-10 0 1001234drock (mm)Height(m)-0.1 0 0.101234a (g)Height(m)-10 0 10-4-2024F(kN)drock (mm)Time = 10.740 sWall: SS-3 Run: 7 Motion: CHHC1@30% PGA: 0.18 g PGD: 57mmTop carriage Bottom carriage Crack319Appendix F. Shake Table Test Results6 8 10 12 14 16 18 20-40-2002040drel(mm)6 8 10 12 14 16 18 20-10-50510drock(mm)6 8 10 12 14 16 18 20-0.2-0.100.10.2a(g)6 8 10 12 14 16 18 20-505F(kN)Time (sec)-20 0 2001234drel (mm)Height(m)-10 0 1001234drock (mm)Height(m)-0.1 0 0.101234a (g)Height(m)-10 0 10-505F(kN)drock (mm)Time = 10.905 sWall: SS-3 Run: 8 Motion: CHHC1@50% PGA: 0.26 g PGD: 96mmTop carriage Bottom carriage Crack320Appendix F. Shake Table Test Results6 8 10 12 14 16 18 20-50050drel(mm)6 8 10 12 14 16 18 20-20020drock(mm)6 8 10 12 14 16 18 20-0.4-0.200.20.4a(g)6 8 10 12 14 16 18 20-10-50510F(kN)Time (sec)-60 0 6001234drel (mm)Height(m)-20 0 2001234drock (mm)Height(m)-0.3 0 0.301234a (g)Height(m)-20 0 20-10-50510F(kN)drock (mm)Time = 12.920 sWall: SS-3 Run: 9 Motion: CHHC1@60% PGA: 0.33 g PGD: 116mmTop carriage Bottom carriage Crack321Appendix F. Shake Table Test Results6 8 10 12 14 16 18 20-100-50050100drel(mm)6 8 10 12 14 16 18 20-50050drock(mm)6 8 10 12 14 16 18 20-1-0.500.51a(g)6 8 10 12 14 16 18 20-10-50510F(kN)Time (sec)-90 0 9001234drel (mm)Height(m)-70 0 7001234drock (mm)Height(m)-0.3 0 0.301234a (g)Height(m)-60 0 60-10-50510F(kN)drock (mm)Time = 13.070 sWall: SS-3 Run: 10 Motion: CHHC1@65% PGA: 0.36 g PGD: 125mmTop carriage Bottom carriage Crack322Appendix F. Shake Table Test Results6 8 10 12 14 16 18 20-1000100drel(mm)6 8 10 12 14 16 18 20-1000100drock(mm)6 8 10 12 14 16 18 20-1-0.500.51a(g)6 8 10 12 14 16 18 20-10010F(kN)Time (sec)-100 0 10001234drel (mm)Height(m)-100 0 10001234drock (mm)Height(m)-0.3 0 0.301234a (g)Height(m)-100 0 100-10010F(kN)drock (mm)Time = 11.035 sWall: SS-3 Run: 11 Motion: CHHC1@70% PGA: 0.41 g PGD: 135mmTop carriage Bottom carriage Crack323Appendix F. Shake Table Test Results6 8 10 12 14 16 18 20-1000100drel(mm)6 8 10 12 14 16 18 20-1000100drock(mm)6 8 10 12 14 16 18 20-1-0.500.51a(g)6 8 10 12 14 16 18 20-10010F(kN)Time (sec)-100 0 10001234drel (mm)Height(m)-100 0 10001234drock (mm)Height(m)-0.4 0 0.401234a (g)Height(m)-100 0 100-10010F(kN)drock (mm)Time = 11.045 sWall: SS-3 Run: 12 Motion: CHHC1@75% PGA: 0.45 g PGD: 145mmTop carriage Bottom carriage Crack324Appendix F. Shake Table Test Results6 8 10 12 14 16 18 20-2000200drel(mm)6 8 10 12 14 16 18 20-2000200drock(mm)6 8 10 12 14 16 18 20-101a(g)6 8 10 12 14 16 18 20-20-1001020F(kN)Time (sec)-90 0 9001234drel (mm)Height(m)-100 0 10001234drock (mm)Height(m)-0.6 0 0.601234a (g)Height(m)-300 0 300-20-1001020F(kN)drock (mm)Time = 10.600 sWall: SS-3 Run: 13 Motion: CHHC1@80% PGA: 0.46 g PGD: 155mmTop carriage Bottom carriage Crack325Appendix GModel Validation ResultsNotes: This section compares output from Working Model 2D with results recorded duringshake table testing. Rocking displacement at the crack and relative displacements of thetop and bottom carriages are plotted for each run.326Appendix G. Model Validation Results−20020d rock(mm)Crack: Test Model−50050d rel(mm)Top carriage: Test Model6 7 8 9 10 11 12 13 14 15 16 17 18−50050Time (s)d rel(mm)Bottom carriage: Test ModelFigure G.1: Modelled vs. tested displacements, wall FF-3, run 9327Appendix G. Model Validation Results−20020d rock(mm)Crack: Test Model−1000100d rel(mm)Top carriage: Test Model6 7 8 9 10 11 12 13 14 15 16 17 18−1000100Time (s)d rel(mm)Bottom carriage: Test ModelFigure G.2: Modelled vs. tested displacements, wall FF-3, run 10328Appendix G. Model Validation Results−20020d rock(mm)Crack: Test Model−1500150d rel(mm)Top carriage: Test Model6 7 8 9 10 11 12 13 14 15 16 17 18−1500150Time (s)d rel(mm)Bottom carriage: Test ModelFigure G.3: Modelled vs. tested displacements, wall FF-3, run 11329Appendix G. Model Validation Results−50050d rock(mm)Crack: Test Model−2000200d rel(mm)Top carriage: Test Model6 7 8 9 10 11 12 13 14 15 16 17 18−2000200Time (s)d rel(mm)Bottom carriage: Test ModelFigure G.4: Modelled vs. tested displacements, wall FF-3, run 12330Appendix G. Model Validation Results−3000300d rock(mm)Crack: Test Model−3000300d rel(mm)Top carriage: Test Model6 7 8 9 10 11 12 13 14 15 16 17 18−3000300Time (s)d rel(mm)Bottom carriage: Test ModelFigure G.5: Modelled vs. tested displacements, wall FF-3, run 13331Appendix G. Model Validation Results−20020d rock(mm)Crack: Test Model−1000100d rel(mm)Top carriage: Test Model6 7 8 9 10 11 12 13 14 15 16 17 18−20020Time (s)d rel(mm)Bottom carriage: Test ModelFigure G.6: Modelled vs. tested displacements, wall FR-3, run 4332Appendix G. Model Validation Results−20020d rock(mm)Crack: Test Model−1500150d rel(mm)Top carriage: Test Model6 7 8 9 10 11 12 13 14 15 16 17 18−20020Time (s)d rel(mm)Bottom carriage: Test ModelFigure G.7: Modelled vs. tested displacements, wall FR-3, run 5333Appendix G. Model Validation Results−20020d rock(mm)Crack: Test Model−1500150d rel(mm)Top carriage: Test Model6 7 8 9 10 11 12 13 14 15 16 17 18−20020Time (s)d rel(mm)Bottom carriage: Test ModelFigure G.8: Modelled vs. tested displacements, wall FR-3, run 6334Appendix G. Model Validation Results−20020d rock(mm)Crack: Test Model−1500150d rel(mm)Top carriage: Test Model6 7 8 9 10 11 12 13 14 15 16 17 18−20020Time (s)d rel(mm)Bottom carriage: Test ModelFigure G.9: Modelled vs. tested displacements, wall FR-3, run 7335Appendix G. Model Validation Results−20020d rock(mm)Crack: Test Model−2000200d rel(mm)Top carriage: Test Model6 7 8 9 10 11 12 13 14 15 16 17 18−20020Time (s)d rel(mm)Bottom carriage: Test ModelFigure G.10: Modelled vs. tested displacements, wall FR-3, run 8336Appendix G. Model Validation Results−30030d rock(mm)Crack: Test Model−2000200d rel(mm)Top carriage: Test Model6 7 8 9 10 11 12 13 14 15 16 17 18−20020Time (s)d rel(mm)Bottom carriage: Test ModelFigure G.11: Modelled vs. tested displacements, wall FR-3, run 9337Appendix G. Model Validation Results−3000300d rock(mm)Crack: Test Model−2000200d rel(mm)Top carriage: Test Model6 7 8 9 10 11 12 13 14 15 16 17 18−20020Time (s)d rel(mm)Bottom carriage: Test ModelFigure G.12: Modelled vs. tested displacements, wall FR-3, run 10338Appendix G. Model Validation Results−20020d rock(mm)Crack: Test Model−50050d rel(mm)Top carriage: Test Model6 7 8 9 10 11 12 13 14 15 16 17 18−50050Time (s)d rel(mm)Bottom carriage: Test ModelFigure G.13: Modelled vs. tested displacements, wall FF-2, run 5339Appendix G. Model Validation Results−20020d rock(mm)Crack: Test Model−80080d rel(mm)Top carriage: Test Model6 7 8 9 10 11 12 13 14 15 16 17 18−80080Time (s)d rel(mm)Bottom carriage: Test ModelFigure G.14: Modelled vs. tested displacements, wall FF-2, run 6340Appendix G. Model Validation Results−20020d rock(mm)Crack: Test Model−1000100d rel(mm)Top carriage: Test Model6 7 8 9 10 11 12 13 14 15 16 17 18−1000100Time (s)d rel(mm)Bottom carriage: Test ModelFigure G.15: Modelled vs. tested displacements, wall FF-2, run 7341Appendix G. Model Validation Results−20020d rock(mm)Crack: Test Model−1000100d rel(mm)Top carriage: Test Model6 7 8 9 10 11 12 13 14 15 16 17 18−1000100Time (s)d rel(mm)Bottom carriage: Test ModelFigure G.16: Modelled vs. tested displacements, wall FF-2, run 8342Appendix G. Model Validation Results−30030d rock(mm)Crack: Test Model−1500150d rel(mm)Top carriage: Test Model6 7 8 9 10 11 12 13 14 15 16 17 18−1500150Time (s)d rel(mm)Bottom carriage: Test ModelFigure G.17: Modelled vs. tested displacements, wall FF-2, run 9343Appendix G. Model Validation Results−50050d rock(mm)Crack: Test Model−1500150d rel(mm)Top carriage: Test Model6 7 8 9 10 11 12 13 14 15 16 17 18−1500150Time (s)d rel(mm)Bottom carriage: Test ModelFigure G.18: Modelled vs. tested displacements, wall FF-2, run 10344Appendix G. Model Validation Results−2000200d rock(mm)Crack: Test Model−1500150d rel(mm)Top carriage: Test Model6 7 8 9 10 11 12 13 14 15 16 17 18−1500150Time (s)d rel(mm)Bottom carriage: Test ModelFigure G.19: Modelled vs. tested displacements, wall FF-2, run 11345Appendix G. Model Validation Results−20020d rock(mm)Crack: Test Model−30030d rel(mm)Top carriage: Test Model6 7 8 9 10 11 12 13 14 15 16 17 18−30030Time (s)d rel(mm)Bottom carriage: Test ModelFigure G.20: Modelled vs. tested displacements, wall SS-3, run 7346Appendix G. Model Validation Results−20020d rock(mm)Crack: Test Model−50050d rel(mm)Top carriage: Test Model6 7 8 9 10 11 12 13 14 15 16 17 18−50050Time (s)d rel(mm)Bottom carriage: Test ModelFigure G.21: Modelled vs. tested displacements, wall SS-3, run 8347Appendix G. Model Validation Results−30030d rock(mm)Crack: Test Model−70070d rel(mm)Top carriage: Test Model6 7 8 9 10 11 12 13 14 15 16 17 18−70070Time (s)d rel(mm)Bottom carriage: Test ModelFigure G.22: Modelled vs. tested displacements, wall SS-3, run 9348Appendix G. Model Validation Results−70070d rock(mm)Crack: Test Model−70070d rel(mm)Top carriage: Test Model6 7 8 9 10 11 12 13 14 15 16 17 18−70070Time (s)d rel(mm)Bottom carriage: Test ModelFigure G.23: Modelled vs. tested displacements, wall SS-3, run 10349Appendix G. Model Validation Results−1500150d rock(mm)Crack: Test Model−80080d rel(mm)Top carriage: Test Model6 7 8 9 10 11 12 13 14 15 16 17 18−80080Time (s)d rel(mm)Bottom carriage: Test ModelFigure G.24: Modelled vs. tested displacements, wall SS-3, run 11350Appendix G. Model Validation Results−1500150d rock(mm)Crack: Test Model−80080d rel(mm)Top carriage: Test Model6 7 8 9 10 11 12 13 14 15 16 17 18−80080Time (s)d rel(mm)Bottom carriage: Test ModelFigure G.25: Modelled vs. tested displacements, wall SS-3, run 12351Appendix G. Model Validation Results−3000300d rock(mm)Crack: Test Model−80080d rel(mm)Top carriage: Test Model6 7 8 9 10 11 12 13 14 15 16 17 18−80080Time (s)d rel(mm)Bottom carriage: Test ModelFigure G.26: Modelled vs. tested displacements, wall SS-3, run 13352Appendix G. Model Validation Results−20020d rock(mm)Crack: Test Model−20020d rel(mm)Top carriage: Test Model6 7 8 9 10 11 12 13 14 15 16 17 18−20020Time (s)d rel(mm)Bottom carriage: Test ModelFigure G.27: Modelled vs. tested displacements, wall RR-3, run 4353Appendix G. Model Validation Results−1000100d rock(mm)Crack: Test Model−20020d rel(mm)Top carriage: Test Model6 7 8 9 10 11 12 13 14 15 16 17 18−20020Time (s)d rel(mm)Bottom carriage: Test ModelFigure G.28: Modelled vs. tested displacements, wall RR-3, run 5354Appendix G. Model Validation Results−50050d rock(mm)Crack: Test Model−20020d rel(mm)Top carriage: Test Model6 7 8 9 10 11 12 13 14 15 16 17 18−20020Time (s)d rel(mm)Bottom carriage: Test ModelFigure G.29: Modelled vs. tested displacements, wall RR-3, run 6355Appendix G. Model Validation Results−3000300d rock(mm)Crack: Test Model−20020d rel(mm)Top carriage: Test Model6 7 8 9 10 11 12 13 14 15 16 17 18−20020Time (s)d rel(mm)Bottom carriage: Test ModelFigure G.30: Modelled vs. tested displacements, wall RR-3, run 7356Appendix HWorking Model 2D Code ListingNotes: This section contains the complete code listing used to construct the reference casemodel from Chapter 6.357Appendix H. Working Model 2D Code Listing1 Sub Single_Motion(motion_filename as String, ground_motion_folder as String)23 ’**********************************4 ’**** DECLARE SCRIPT VARIABLES ****5 ’**********************************67 ’** Parameter inputs **89 Dim wall_height as double10 Dim wall_thickness as double11 Dim wall_length as double12 Dim wall_density as double13 Dim wall_mass as double14 Dim crack_height_rel as double15 Dim spall_chamfer_crack as double16 Dim spall_chamfer_base as double17 Dim joint_thickness as double1819 Dim coef_friction_static as double20 Dim coef_friction_kinetic as double21 Dim coef_restitution as double22 Dim coef_restitution_table as double2324 Dim damping_ratio_top as double25 Dim damping_ratio_bot as double26 Dim coef_damping_bot as double27 Dim coef_damping_top as double28 Dim rot_K_top as double29 Dim rot_K_bot as double30 Dim trans_K_top as double31 Dim trans_K_bot as double32 Dim diaphragm_mass_top as double33 Dim carriage_mass_top as double34 Dim beam_mass_top as double35 Dim diaphragm_mass_bot as double3637 Dim rigid_diaphragm_top as integer38 Dim rigid_diaphragm_bot as integer3940 Dim axial_load as double41 Dim axial_load_eccentricity as double4243 Dim cracked as integer44 Dim top_diaphragm_locked as integer45 Dim bot_diaphragm_locked as integer464748 ’** Model inputs **4950 Dim base_constraint_slack as double51 Dim crack_constraint_slack as double52 Dim frame_horiz_offset as double53 Dim diaphragm_length as double54 Dim bot_diaphragm_vert_offset as double55 Dim top_diaphragm_vert_offset as double56358Appendix H. Working Model 2D Code Listing57 Dim time_step as double58 Dim overlap_error as double59 Dim sig_figs as integer606162 ’** File handling variables **6364 ’Dim ground_motion_folder as string ’ removed when written as function instead of sub65 ’Dim motion_filename as string ’ removed when written as function instead of sub66 Dim output_folder as string67 Dim config_ID as string68 Dim save_name as string69 Dim GM_prefix as string70 Dim GM_ID as string71 Dim GM_duration as integer72 Dim wall_ID as string73 Dim version_ID as integer747576 ’** Internal variables **7778 Dim bot_block_height as double79 Dim bot_block_mass as double80 Dim bot_block_x_pos as double81 Dim bot_block_y_pos as double8283 Dim top_block_height as double84 Dim top_block_mass as double85 Dim top_block_x_pos as double86 Dim top_block_y_pos as double8788 Dim frame_height as double8990 Dim GM_table_inputID as integer91 Dim GM_scale_input_inputID as integer92 Dim top_block_inputID as integer93 Dim bot_block_inputID as integer9495 Dim crack_height_rel_eff as double9697 Dim num_time_steps_to_run as long98 Dim num_time_steps_to_run_checked as long99100 Dim GM_scale as double101 Dim GM_scale_start as double102 Dim GM_scale_minor_inc as double103 Dim GM_scale_major_inc as double104 Dim GM_scale_to_not_repeat as double105 Dim GM_scale_to_not_repeat_collapsed as double106107 Dim collapse_check as integer108 Dim final_increment as integer109 Dim rocking_check_current as integer110 Dim rocking_check_previous as integer111 Dim rocking_threshold as double112 Dim increment_number as integer359Appendix H. Working Model 2D Code Listing113114115116 ’**********************************117 ’**** DECLARE OBJECT VARIABLES ****118 ’**********************************119120 ’** Document **121122 Dim Doc as WMDocument123124 ’** Bodies **125126 Dim bot_block as WMBody127 Dim top_block as WMBody128 Dim table as WMBody129 Dim left_frame as WMBody130 Dim right_frame as WMBody131 Dim top_diaphragm as WMBody132 Dim bot_diaphragm as WMBody133 Dim actuator_anchor as WMBody134 Dim top_pin as WMBody135 Dim left_crack_block as WMBody136 Dim right_crack_block as WMBody137138 Dim trigger1_base as WMBody139 Dim trigger1_stop as WMBody140 Dim trigger1_bullet as WMBody141142 ’** Slots **143144 Dim table_slot as WMConstraint145 Dim top_slot as WMConstraint146 Dim bot_diaphragm_slot as WMConstraint147 Dim top_diaphragm_slot as WMConstraint148149 ’** Joints **150151 Dim left_frame_joint as WMConstraint152 Dim right_frame_joint as WMConstraint153 Dim top_pin_joint as WMConstraint154 Dim left_crack_block_joint as WMConstraint155 Dim right_crack_block_joint as WMConstraint156 Dim uncracked_wall_joint as WMConstraint157 Dim trigger1_joint as WMConstraint158159 ’** Links **160161 Dim left_bot_rope as WMConstraint162 Dim right_bot_rope as WMConstraint163 Dim left_bot_separator as WMConstraint164 Dim right_bot_separator as WMConstraint165166 Dim left_crack_rope as WMConstraint167 Dim right_crack_rope as WMConstraint168 Dim left_crack_separator as WMConstraint360Appendix H. Working Model 2D Code Listing169 Dim right_crack_separator as WMConstraint170171 Dim bot_spring as WMConstraint172 Dim top_spring as WMConstraint173174 Dim left_bot_diaphragm_rope as WMConstraint175 Dim right_bot_diaphragm_rope as WMConstraint176 Dim left_top_diaphragm_rope as WMConstraint177 Dim right_top_diaphragm_rope as WMConstraint178179 ’** Actuator **180181 Dim actuator_constraint as WMConstraint182183 ’** Inputs **184185 Dim GM_table as WMInput186 Dim GM_scale_input as WMInput187188 ’** Points **189190 Dim anchor_point as WMPoint191 Dim wall_base_measuring_point as WMPoint192 Dim wall_crack_bot_measuring_point as WMPoint193 Dim wall_crack_top_measuring_point as WMPoint194 Dim table_measuring_point as WMPoint195 Dim trigger1_anchor_point as WMPoint196197 ’** Meters **198199 Dim meter_left_rocking_constraints as WMOutput200 Dim meter_right_rocking_constraints as WMOutput201 Dim meter_rel_displacements as WMOutput202 Dim meter_accelerations as WMOutput203 Dim meter_table as WMOutput204205 ’** Forces **206207 Dim trigger1_force as WMConstraint208 Dim axial_load_force as WMConstraint209210211 ’****************212 ’**** INPUTS ****213 ’****************214215 ’%%% START OF EXCEL GENERATED FILE %%%216217 ’** Configuration inputs **218219 config_ID = 100220 version_ID = 014221222 ’** Wall-specific parameters **223224 wall_height = 3.630 ’m361Appendix H. Working Model 2D Code Listing225 wall_thickness = 0.330 ’m226 wall_length = 1.000 ’m227 wall_density = 2100 ’kg/m3228 crack_height_rel = 0.600 ’unitless229 joint_thickness = 0.012 ’m230 spall_chamfer_crack = 0.010 ’m231 spall_chamfer_base = 0.010 ’m232233 damping_ratio_top = 0.050 ’unitless234 damping_ratio_bot = 0.050 ’unitless235 trans_K_top = 198623 ’N/m236 trans_K_bot = 198623 ’N/m237238 cracked = 1 ’binary (1 = cracked, 0 = uncracked)239 top_diaphragm_locked = 0 ’binary (1 = locked, 0 = flexible)240 bot_diaphragm_locked = 0 ’binary (1 = locked, 0 = flexible)241242 time_step = 0.005 ’sec243244245246 ’** Common parameters **247248 coef_friction_static = 0.75 ’unitless249 coef_friction_kinetic = 0.70 ’unitless250 coef_restitution = 0.02 ’unitless251 rot_K_top = 0 ’N/rad252 rot_K_bot = 0 ’N/rad253 carriage_mass_top = 3773 ’kg254 beam_mass_top = 0.1 ’kg255 diaphragm_mass_bot = 3773 ’kg256257 axial_load = 0 ’N258 axial_load_eccentricity = 0 ’m259260 ’** Model inputs **261262 base_constraint_slack = 0.0002 ’m263 crack_constraint_slack = 0.0002 ’m264 frame_horiz_offset = 4.00 ’m265 diaphragm_length = 0.80 ’m266 bot_diaphragm_vert_offset = 0.000 ’m267 top_diaphragm_vert_offset = -0.001 ’m268269 overlap_error = 0.0002 ’m270 sig_figs = 8 ’integer271272 ’** File handling inputs **273274 file_prefix = "c100v014"275 ground_motion_folder = "D:\Google Drive\Thesis\Analysis\Ground Motions\FEMA P695\Resampled Motions\"276 output_folder = "D:\My Documents\Thesis\WM2D Runs\c100v014\"277278 ’** Ground motion incrementing inputs **279280 GM_scale_start = 0.50362Appendix H. Working Model 2D Code Listing281 GM_scale_minor_inc = 0.10282 GM_scale_major_inc = 0.40283284 rocking_threshold = 0.1285286287 ’%%% END OF EXCEL-GENERATED FILE %%%288289 ’*************************************290 ’**** MODEL VARIABLE CALCULATIONS ****291 ’*************************************292293 bot_diaphragm_vert_offset = 0 ’override the bottom vertical offset to 0 (new in V12)294295 diaphragm_mass_top = carriage_mass_top + beam_mass_top296297 bot_block_height = wall_height*crack_height_rel298 top_block_height = wall_height - bot_block_height299300 bot_block_x_pos = 0301 top_block_x_pos = 0302303 bot_block_y_pos = bot_block_height/2304 top_block_y_pos = bot_block_height + top_block_height/2305306 wall_mass = (wall_height * wall_thickness * wall_length) * wall_density307 bot_block_mass = wall_mass * crack_height_rel308 top_block_mass = wall_mass * (1 - crack_height_rel)309310 frame_height = wall_height + top_diaphragm_vert_offset311312 crack_height_rel_eff = crack_height_rel * wall_height / frame_height313314 coef_damping_bot = damping_ratio_bot * (2*(trans_K_bot*(wall_mass/2+diaphragm_mass_bot))^0.5)315 coef_damping_top = damping_ratio_top * (2*(trans_K_top*(wall_mass/2+diaphragm_mass_top))^0.5)316317318 ’****************************319 ’**** MODEL CONSTRUCTION ****320 ’****************************321322 ’Set Doc = WM.ActiveDocument323 Set Doc = WM.Open("D:\Google Drive\Thesis\Analysis\WM2D\Startup File\startupC1.wm2d")324325 ’**** Delete everything before starting ****326327 Doc.SelectAll True328 Doc.Delete329 Doc.SelectAll False330331 ’**** Wall bodies ****332333 Set bot_block = Doc.NewBody("polygon")334 bot_block.name = "bot_block"335 Set top_block = Doc.NewBody("polygon")336 top_block.name = "top_block"363Appendix H. Working Model 2D Code Listing337338 bot_block.PX.Value = bot_block_x_pos: bot_block.PY.Value = bot_block_y_pos339 top_block.PX.Value = top_block_x_pos: top_block.PY.Value = top_block_y_pos340341 ’**** Bottom block vertex definition ****342343 If spall_chamfer_crack = 0 And spall_chamfer_base = 0 Then344345 bot_block.AddVertex 1, -wall_thickness/2, -bot_block_height/2 ’bottom left corner346347 bot_block.AddVertex 2, -wall_thickness/2, bot_block_height/2 - 0.20 ’bottom of left spike348 bot_block.AddVertex 3, -wall_thickness/2 - 0.20, bot_block_height/2 ’outside of left spike349 bot_block.AddVertex 4, -wall_thickness/2, bot_block_height/2 - 0.195 ’top of left spike350351 bot_block.AddVertex 5, -wall_thickness/2, bot_block_height/2 ’top left corner352 bot_block.AddVertex 6, wall_thickness/2, bot_block_height/2 ’top right corner353354 bot_block.AddVertex 7, wall_thickness/2, bot_block_height/2 - 0.195 ’top of right spike355 bot_block.AddVertex 8, wall_thickness/2 + 0.20, bot_block_height/2 ’outside of left spike356 bot_block.AddVertex 9, wall_thickness/2, bot_block_height/2 - 0.20 ’bottom of left spike357358 bot_block.AddVertex 10, wall_thickness/2, -bot_block_height/2 ’bottom right corner359360 bot_block.DeleteVertex 13 ’delete three default vertices361 bot_block.DeleteVertex 12362 bot_block.DeleteVertex 11363364 ElseIf spall_chamfer_crack <> 0 And spall_chamfer_base = 0 Then365366 bot_block.AddVertex 1, -wall_thickness/2, -bot_block_height/2 ’bottom left corner367368 bot_block.AddVertex 2, -wall_thickness/2, bot_block_height/2 - 0.20 ’bottom of left spike369 bot_block.AddVertex 3, -wall_thickness/2 - 0.20, bot_block_height/2 ’outside of left spike370 bot_block.AddVertex 4, -wall_thickness/2, bot_block_height/2 - 0.195 ’top of left spike371372 bot_block.AddVertex 5, -wall_thickness/2, bot_block_height/2 - joint_thickness/2 ’top leftOUTSIDE corner373 bot_block.AddVertex 6, -wall_thickness/2 + spall_chamfer_crack, bot_block_height/2 ’top leftINSIDE corner374 bot_block.AddVertex 7, wall_thickness/2 - spall_chamfer_crack, bot_block_height/2 ’top rightINSIDE corner375 bot_block.AddVertex 8, wall_thickness/2, bot_block_height/2 - joint_thickness/2 ’top rightOUTSIDE corner376377 bot_block.AddVertex 9, wall_thickness/2, bot_block_height/2 - 0.195 ’top of right spike378 bot_block.AddVertex 10, wall_thickness/2 + 0.20, bot_block_height/2 ’outside of left spike379 bot_block.AddVertex 11, wall_thickness/2, bot_block_height/2 - 0.20 ’bottom of left spike380381 bot_block.AddVertex 12, wall_thickness/2, -bot_block_height/2 ’bottom right corner382383 bot_block.DeleteVertex 15 ’delete three default vertices384 bot_block.DeleteVertex 14385 bot_block.DeleteVertex 13386387 ElseIf spall_chamfer_crack = 0 And spall_chamfer_base <> 0 Then388364Appendix H. Working Model 2D Code Listing389 bot_block.AddVertex 1, -wall_thickness/2 + spall_chamfer_base, -bot_block_height/2 ’bottomleft INSIDE corner390 bot_block.AddVertex 2, -wall_thickness/2, -bot_block_height/2 + joint_thickness/2 ’bottom leftOUTSIDE corner391392 bot_block.AddVertex 3, -wall_thickness/2, bot_block_height/2 - 0.20 ’bottom of left spike393 bot_block.AddVertex 4, -wall_thickness/2 - 0.20, bot_block_height/2 ’outside of left spike394 bot_block.AddVertex 5, -wall_thickness/2, bot_block_height/2 - 0.195 ’top of left spike395396 bot_block.AddVertex 6, -wall_thickness/2, bot_block_height/2 ’top left corner397 bot_block.AddVertex 7, wall_thickness/2, bot_block_height/2 ’top right corner398399 bot_block.AddVertex 8, wall_thickness/2, bot_block_height/2 - 0.195 ’top of right spike400 bot_block.AddVertex 9, wall_thickness/2 + 0.20, bot_block_height/2 ’outside of left spike401 bot_block.AddVertex 10, wall_thickness/2, bot_block_height/2 - 0.20 ’bottom of left spike402403 bot_block.AddVertex 11, wall_thickness/2, -bot_block_height/2 + joint_thickness/2 ’bottomright OUTSIDE corner404 bot_block.AddVertex 12, wall_thickness/2 - spall_chamfer_base, -bot_block_height/2 ’bottomright INSIDE corner405406 bot_block.DeleteVertex 15 ’delete three default vertices407 bot_block.DeleteVertex 14408 bot_block.DeleteVertex 13409410 Else411412 bot_block.AddVertex 1, -wall_thickness/2 + spall_chamfer_base, -bot_block_height/2 ’bottomleft INSIDE corner413 bot_block.AddVertex 2, -wall_thickness/2, -bot_block_height/2 + joint_thickness/2 ’bottom leftOUTSIDE corner414415 bot_block.AddVertex 3, -wall_thickness/2, bot_block_height/2 - 0.20 ’bottom of left spike416 bot_block.AddVertex 4, -wall_thickness/2 - 0.20, bot_block_height/2 ’outside of left spike417 bot_block.AddVertex 5, -wall_thickness/2, bot_block_height/2 - 0.195 ’top of left spike418419 bot_block.AddVertex 6, -wall_thickness/2, bot_block_height/2 - joint_thickness/2 ’top leftOUTSIDE corner420 bot_block.AddVertex 7, -wall_thickness/2 + spall_chamfer_crack, bot_block_height/2 ’top leftINSIDE corner421 bot_block.AddVertex 8, wall_thickness/2 - spall_chamfer_crack, bot_block_height/2 ’top rightINSIDE corner422 bot_block.AddVertex 9, wall_thickness/2, bot_block_height/2 - joint_thickness/2 ’top rightOUTSIDE corner423424 bot_block.AddVertex 10, wall_thickness/2, bot_block_height/2 - 0.195 ’top of right spike425 bot_block.AddVertex 11, wall_thickness/2 + 0.20, bot_block_height/2 ’outside of left spike426 bot_block.AddVertex 12, wall_thickness/2, bot_block_height/2 - 0.20 ’bottom of left spike427428429 bot_block.AddVertex 13, wall_thickness/2, -bot_block_height/2 + joint_thickness/2 ’bottomright OUTSIDE corner430 bot_block.AddVertex 14, wall_thickness/2 - spall_chamfer_base, -bot_block_height/2 ’bottomright INSIDE corner431432 bot_block.DeleteVertex 17 ’delete three default vertices365Appendix H. Working Model 2D Code Listing433 bot_block.DeleteVertex 16434 bot_block.DeleteVertex 15435436 End If437438439 ’**** Top block vertex definition ****440441 top_block.AddVertex 1, wall_thickness/2, top_block_height/2 ’top right corner442443 If spall_chamfer_crack = 0 Then444445 top_block.AddVertex 2, wall_thickness/2, -top_block_height/2 ’bottom right corner446 top_block.AddVertex 3, -wall_thickness/2, -top_block_height/2 ’bottom left corner447 top_block.AddVertex 4, -wall_thickness/2, top_block_height/2 ’top left corner448449 top_block.DeleteVertex 7 ’delete three default vertices450 top_block.DeleteVertex 6451 top_block.DeleteVertex 5452453 Else454455 top_block.AddVertex 2, wall_thickness/2, -top_block_height/2 + joint_thickness/2 ’bottom rightOUTSIDE corner456 top_block.AddVertex 3, wall_thickness/2 - spall_chamfer_crack, -top_block_height/2 ’bottom rightINSIDE corner457 top_block.AddVertex 4, -wall_thickness/2 + spall_chamfer_crack, -top_block_height/2 ’bottom leftINSIDE corner458 top_block.AddVertex 5, -wall_thickness/2, -top_block_height/2 + joint_thickness/2 ’bottom leftOUTSIDE corner459 top_block.AddVertex 6, -wall_thickness/2, top_block_height/2 ’top left corner460461 top_block.DeleteVertex 9 ’delete three default vertices462 top_block.DeleteVertex 8463 top_block.DeleteVertex 7464465 End If466467 ’**** Assign body ID values for wall blocks ****468469 top_block_inputID = top_block.ID470 bot_block_inputID = bot_block.ID471472 ’**** Fixed joint between wall blocks (if uncracked) ****473474 If cracked = 0 Then475476 Set uncracked_wall_joint = Doc.NewConstraint("SquarePin")477 uncracked_wall_joint.name = "uncracked_wall_joint"478 Set uncracked_wall_joint.Point(1).Body = top_block479 uncracked_wall_joint.Point(1).PX.Value = -0.02 ’noticed instability when this fixed joint waslocated in the center of the wall body, seems to be OK when moved off center480 uncracked_wall_joint.Point(1).PY.Value = -top_block_height/2481 Set uncracked_wall_joint.Point(2).Body = bot_block482 uncracked_wall_joint.Point(2).PX.Value = -0.02483 uncracked_wall_joint.Point(2).PY.Value = bot_block_height/2366Appendix H. Working Model 2D Code Listing484485 top_block.PY.Value = top_block_y_pos ’put the top block back to to the correct height, for somereason it moves when defining the above constraint486487 End If488489 ’**** Table ****490491 Set table = Doc.NewBody("rectangle")492 table.name = "table"493494 table.PX.Value = 0495 table.PY.Value = -0.10496497 table.width.value = 4.00498 table.height.value = 0.20499500 table.staticfriction.value = 0501 table.kineticfriction.value = 0502503 ’**** Table constraint ****504505 Set table_slot = Doc.NewConstraint("KeyedHSlot")506507 Set table_slot.Point(2).Body = table ’point 2 is the point508 table_slot.Point(2).PX.Value = -2509 table_slot.Point(2).PY.Value = -.1510511 table_slot.Point(1).PY.Value = -.2 ’point 1 is the slot - this MUST be defined after point 2 forsome reason512513 table_slot.name = "table_slot"514 table_slot.Point(2).name = "table_slot_pt"515 table_slot.Point(1).name = "table_slot_bk_pt"516517 ’**** Vertical frame segments ****518519 Set left_frame = Doc.NewBody("rectangle")520 left_frame.name = "left_frame"521 left_frame.PX.Value = -frame_horiz_offset522 left_frame.PY.Value = frame_height/2523 left_frame.width.value = 0.10524 left_frame.height.value = frame_height525526 Set right_frame = Doc.NewBody("rectangle")527 right_frame.name = "right_frame"528 right_frame.PX.Value = frame_horiz_offset529 right_frame.PY.Value = frame_height/2530 right_frame.width.value = 0.10531 right_frame.height.value = frame_height532533 ’**** Vertical frame fixed joints ****534535 Set left_frame_joint = Doc.NewConstraint("SquarePin")536 Set left_frame_joint.Point(2).Body = left_frame537 left_frame_joint.Point(2).PX.Value = 0.00367Appendix H. Working Model 2D Code Listing538 left_frame_joint.Point(2).PY.Value = -frame_height/2539 Set left_frame_joint.Point(1).Body = table540 left_frame_joint.Point(1).PX.Value = -frame_horiz_offset541 left_frame_joint.Point(1).PY.Value = 0.10542543 left_frame_joint.name = "left_frame_joint"544 left_frame_joint.Point(2).name = "left_frame_joint_pt2"545 left_frame_joint.Point(1).name = "left_frame_joint_pt1"546547 Set right_frame_joint = Doc.NewConstraint("SquarePin")548 Set right_frame_joint.Point(2).Body = right_frame549 right_frame_joint.Point(2).PX.Value = 0.00550 right_frame_joint.Point(2).PY.Value = -frame_height/2551 Set right_frame_joint.Point(1).Body = table552 right_frame_joint.Point(1).PX.Value = frame_horiz_offset553 right_frame_joint.Point(1).PY.Value = 0.10554555 right_frame_joint.name = "right_frame_joint"556 right_frame_joint.Point(2).name = "right_frame_joint_pt2"557 right_frame_joint.Point(1).name = "right_frame_joint_pt1"558559 table.PX.Value = 0 ’put the table back to 0, for some reason it moves when defining the aboveconstraints560561 ’**** Diaphragms ****562563 Set bot_diaphragm = Doc.NewBody("rectangle")564 bot_diaphragm.name = "bot_diaphragm"565 bot_diaphragm.PX.Value = 0566 bot_diaphragm.PY.Value = bot_diaphragm_vert_offset567 bot_diaphragm.width.value = diaphragm_length568 bot_diaphragm.height.value = 0.050569570 Set top_diaphragm = Doc.NewBody("rectangle")571 top_diaphragm.name = "top_diaphragm"572 top_diaphragm.PX.Value = 0573 top_diaphragm.PY.Value = wall_height + top_diaphragm_vert_offset574 top_diaphragm.width.value = diaphragm_length575 top_diaphragm.height.value = 0.050576577 ’**** Top pin connection ****578579 Set top_pin = Doc.NewBody("polygon")580 top_pin.name = "top_beam"581582 top_pin.PX.Value = 0: top_pin.PY.Value = wall_height583584 top_pin.AddVertex 1, wall_thickness/2, 0 ’bottom right corner585 top_pin.AddVertex 2, -wall_thickness/2, 0 ’bottom left corner586 top_pin.AddVertex 3, 0, top_diaphragm_vert_offset ’top corner587588 top_pin.DeleteVertex 6 ’delete three default vertices589 top_pin.DeleteVertex 5590 top_pin.DeleteVertex 4591592 ’**** Top pin fixed joint ****368Appendix H. Working Model 2D Code Listing593594 Set top_pin_joint = Doc.NewConstraint("SquarePin")595 Set top_pin_joint.Point(1).Body = top_block596 top_pin_joint.Point(1).PX.Value = 0597 top_pin_joint.Point(1).PY.Value = top_block_height/2598 Set top_pin_joint.Point(2).Body = top_pin599 top_pin_joint.Point(2).PX.Value = 0.00600 top_pin_joint.Point(2).PY.Value = 0601602 top_pin_joint.name = "top_beam_fixed_joint"603 top_pin_joint.Point(1).name = "top_beam_fixed_joint_pt1"604 top_pin_joint.Point(2).name = "top_beam_fixed_joint_pt2"605606 top_block.PY.Value = top_block_y_pos ’put the top block back to to the correct height, for somereason it moves when defining the above constraint607608 ’**** Slot for top pin ****609610 Set top_slot = Doc.NewConstraint("VSlot")611612 Set top_slot.Point(2).Body = top_pin ’point 2 is the point613 top_slot.Point(2).PX.Value = 0614 top_slot.Point(2).PY.Value = top_diaphragm_vert_offset615616 Set top_slot.Point(1).Body = top_diaphragm617 top_slot.Point(1).PX.Value = 0 ’point 1 is the slot - this MUST be defined after point 2 forsome reason618619 top_slot.name = "top_pin_slot"620 top_slot.Point(2).name = "top_pin_slot_pt"621 top_slot.Point(1).name = "top_pin_slot_bkpt"622623 ’**** Lateral restraints at base of the wall (to diaphragm) ****624625 Set left_bot_rope = Doc.NewConstraint("Rope")626 left_bot_rope.Point(2).PX.Value = -wall_thickness/2 + spall_chamfer_base627 left_bot_rope.Point(2).PY.Value = -bot_block_height/2 + bot_diaphragm_vert_offset628 Set left_bot_rope.Point(2).Body = bot_block ’set the body after defining the relative co-ordinates on that body, then it avoids moving the body629 left_bot_rope.Point(1).PX.Value = -diaphragm_length/2630 left_bot_rope.Point(1).PY.Value = 0631 Set left_bot_rope.Point(1).Body = bot_diaphragm632 left_bot_rope.length.value = diaphragm_length/2 - wall_thickness/2 + base_constraint_slack +spall_chamfer_base633634635 Set right_bot_rope = Doc.NewConstraint("Rope")636 right_bot_rope.Point(2).PX.Value = wall_thickness/2 - spall_chamfer_base637 right_bot_rope.Point(2).PY.Value = -bot_block_height/2 + bot_diaphragm_vert_offset638 Set right_bot_rope.Point(2).Body = bot_block639 right_bot_rope.Point(1).PX.Value = diaphragm_length/2640 right_bot_rope.Point(1).PY.Value = 0641 Set right_bot_rope.Point(1).Body = bot_diaphragm642 right_bot_rope.length.value = diaphragm_length/2 - wall_thickness/2 + base_constraint_slack +spall_chamfer_base643369Appendix H. Working Model 2D Code Listing644645646 Set left_bot_separator = Doc.NewConstraint("Separator")647 left_bot_separator.length.value = diaphragm_length/2 - wall_thickness/2 - base_constraint_slack648 left_bot_separator.Point(2).PX.Value = -wall_thickness/2 + spall_chamfer_base649 left_bot_separator.Point(2).PY.Value = -bot_block_height/2 + bot_diaphragm_vert_offset650 Set left_bot_separator.Point(2).Body = bot_block651 left_bot_separator.Point(1).PX.Value = -diaphragm_length/2652 left_bot_separator.Point(1).PY.Value = 0653 Set left_bot_separator.Point(1).Body = bot_diaphragm654 left_bot_separator.length.value = diaphragm_length/2 - wall_thickness/2 - base_constraint_slack +spall_chamfer_base655656657 Set right_bot_separator = Doc.NewConstraint("Separator")658 right_bot_separator.length.value = diaphragm_length/2 - wall_thickness/2 - base_constraint_slack659 right_bot_separator.Point(2).PX.Value = wall_thickness/2 - spall_chamfer_base660 right_bot_separator.Point(2).PY.Value = -bot_block_height/2 + bot_diaphragm_vert_offset661 Set right_bot_separator.Point(2).Body = bot_block662 right_bot_separator.Point(1).PX.Value = diaphragm_length/2663 right_bot_separator.Point(1).PY.Value = 0664 Set right_bot_separator.Point(1).Body = bot_diaphragm665 right_bot_separator.length.value = diaphragm_length/2 - wall_thickness/2 - base_constraint_slack +spall_chamfer_base666667 left_bot_rope.name = "left_bot_rope"668 left_bot_rope.Point(2).name = "left_bot_rope_pt2"669 left_bot_rope.Point(1).name = "left_bot_rope_pt1"670671 right_bot_rope.name = "right_bot_rope"672 right_bot_rope.Point(2).name = "right_bot_rope_pt2"673 right_bot_rope.Point(1).name = "right_bot_rope_pt1"674675 left_bot_separator.name = "left_bot_separator"676 left_bot_separator.Point(2).name = "left_bot_separator_pt2"677 left_bot_separator.Point(1).name = "left_bot_separator_pt1"678679 right_bot_separator.name = "right_bot_separator"680 right_bot_separator.Point(2).name = "right_bot_separator_pt2"681 right_bot_separator.Point(1).name = "right_bot_separator_pt1"682683 ’**** Lateral restraints at crack (between wall blocks) ****684685 Set left_crack_rope = Doc.NewConstraint("Rope")686 left_crack_rope.length.value = 10 ’make rope temporarily long so it doesn’t move other bodiesduring definition687 left_crack_rope.Point(2).PX.Value = -wall_thickness/2 + spall_chamfer_crack688 left_crack_rope.Point(2).PY.Value = -top_block_height/2689 Set left_crack_rope.Point(2).Body = top_block690 left_crack_rope.Point(1).PX.Value = -wall_thickness/2 - 0.20691 left_crack_rope.Point(1).PY.Value = bot_block_height/2692 Set left_crack_rope.Point(1).Body = bot_block693 top_block.PX.Value = top_block_x_pos: top_block.PY.Value = top_block_y_pos694 left_crack_rope.length.value = 0.20 + spall_chamfer_crack + crack_constraint_slack695696 Set right_crack_rope = Doc.NewConstraint("Rope")370Appendix H. Working Model 2D Code Listing697 right_crack_rope.length.value = 10 ’make rope temporarily long so it doesn’t move other bodiesduring definition698 right_crack_rope.Point(2).PX.Value = wall_thickness/2 - spall_chamfer_crack699 right_crack_rope.Point(2).PY.Value = -top_block_height/2700 Set right_crack_rope.Point(2).Body = top_block701 right_crack_rope.Point(1).PX.Value = wall_thickness/2 + 0.20702 right_crack_rope.Point(1).PY.Value = bot_block_height/2703 Set right_crack_rope.Point(1).Body = bot_block704 top_block.PX.Value = top_block_x_pos: top_block.PY.Value = top_block_y_pos705 right_crack_rope.length.value = 0.20 + spall_chamfer_crack + crack_constraint_slack706707708 Set left_crack_separator = Doc.NewConstraint("Separator")709 left_crack_separator.length.value = 0.01 ’make separator temporarily short so it doesn’t moveother bodies during definition710 left_crack_separator.Point(2).PX.Value = -wall_thickness/2 + spall_chamfer_crack711 left_crack_separator.Point(2).PY.Value = -top_block_height/2712 Set left_crack_separator.Point(2).Body = top_block713 left_crack_separator.Point(1).PX.Value = -wall_thickness/2 - 0.20714 left_crack_separator.Point(1).PY.Value = bot_block_height/2715 Set left_crack_separator.Point(1).Body = bot_block716 top_block.PX.Value = top_block_x_pos: top_block.PY.Value = top_block_y_pos717 left_crack_separator.length.value = 0.20 + spall_chamfer_crack - crack_constraint_slack718719 Set right_crack_separator = Doc.NewConstraint("Separator")720 right_crack_separator.length.value = 0.01 ’make separator temporarily short so it doesn’t moveother bodies during definition721 right_crack_separator.Point(2).PX.Value = wall_thickness/2 - spall_chamfer_crack722 right_crack_separator.Point(2).PY.Value = -top_block_height/2723 Set right_crack_separator.Point(2).Body = top_block724 right_crack_separator.Point(1).PX.Value = wall_thickness/2 + 0.20725 right_crack_separator.Point(1).PY.Value = bot_block_height/2726 Set right_crack_separator.Point(1).Body = bot_block727 top_block.PX.Value = top_block_x_pos: top_block.PY.Value = top_block_y_pos728 right_crack_separator.length.value = 0.20 + spall_chamfer_crack - crack_constraint_slack729730 left_crack_rope.name = "left_crack_rope"731 left_crack_rope.Point(2).name = "left_crack_rope_pt2"732 left_crack_rope.Point(1).name = "left_crack_rope_pt1"733734 right_crack_rope.name = "right_crack_rope"735 right_crack_rope.Point(2).name = "right_crack_rope_pt2"736 right_crack_rope.Point(1).name = "right_crack_rope_pt1"737738 left_crack_separator.name = "left_crack_separator"739 left_crack_separator.Point(2).name = "left_crack_separator_pt2"740 left_crack_separator.Point(1).name = "left_crack_separator_pt1"741742 right_crack_separator.name = "right_crack_separator"743 right_crack_separator.Point(2).name = "right_crack_separator_pt2"744 right_crack_separator.Point(1).name = "right_crack_separator_pt1"745746 ’**** Set "Active When" conditions for constraints at crack and at base of wall ****747748 left_crack_rope.AlwaysActive = False749 left_crack_separator.AlwaysActive = False371Appendix H. Working Model 2D Code Listing750 right_crack_rope.AlwaysActive = False751 right_crack_separator.AlwaysActive = False752 left_bot_rope.AlwaysActive = False753 left_bot_separator.AlwaysActive = False754 right_bot_rope.AlwaysActive = False755 right_bot_separator.AlwaysActive = False756757 left_crack_rope.ActiveWhen.Formula = "Body["+ str$(top_block_inputID) +"].p.r-Body["+ str$(bot_block_inputID) +"].p.r >= -0.0000001"758 left_crack_separator.ActiveWhen.Formula = "Body["+ str$(top_block_inputID) +"].p.r-Body["+ str$(bot_block_inputID) +"].p.r >= -0.0000001"759 right_crack_rope.ActiveWhen.Formula = "Body["+ str$(top_block_inputID) +"].p.r-Body["+ str$(bot_block_inputID) +"].p.r <= 0.0000001"760 right_crack_separator.ActiveWhen.Formula = "Body["+ str$(top_block_inputID) +"].p.r-Body["+ str$(bot_block_inputID) +"].p.r <= 0.0000001"761 left_bot_rope.ActiveWhen.Formula = "Body["+ str$(bot_block_inputID) +"].p.r >= 0"762 left_bot_separator.ActiveWhen.Formula = "Body["+ str$(bot_block_inputID) +"].p.r >= 0"763 right_bot_rope.ActiveWhen.Formula = "Body["+ str$(bot_block_inputID) +"].p.r <= 0"764 right_bot_separator.ActiveWhen.Formula = "Body["+ str$(bot_block_inputID) +"].p.r <= 0"765766767768 ’**** Bottom diaphragm spring or rigid link ****769770 If bot_diaphragm_locked = 0 Then771772 Set bot_spring = Doc.NewConstraint("SpringDamper")773 bot_spring.Point(1).PX.Value = 0774 bot_spring.Point(1).PY.Value = -frame_height/2 + bot_diaphragm_vert_offset775 Set bot_spring.Point(1).Body = left_frame776 bot_spring.Point(2).PX.Value = -diaphragm_length/2777 Set bot_spring.Point(2).Body = bot_diaphragm778 bot_spring.Point(2).PY.Value = 0779 bot_spring.length.value = frame_horiz_offset - diaphragm_length/2780781 bot_spring.name = "bot_spring"782 bot_spring.Point(2).name = "bot_spring_pt2"783 bot_spring.Point(1).name = "bot_spring_pt1"784785 bot_spring.K.value = trans_K_bot786 bot_spring.damperK.value = coef_damping_bot787788 Else789790 Set bot_spring = Doc.NewConstraint("Rod")791 bot_spring.Point(1).PX.Value = 0792 bot_spring.Point(1).PY.Value = -frame_height/2 + bot_diaphragm_vert_offset793 Set bot_spring.Point(1).Body = left_frame794 bot_spring.Point(2).PX.Value = -diaphragm_length/2795 Set bot_spring.Point(2).Body = bot_diaphragm796 bot_spring.Point(2).PY.Value = 0797 bot_spring.length.value = frame_horiz_offset - diaphragm_length/2798799 bot_spring.name = "bot_rod"800 bot_spring.Point(2).name = "bot_rod_pt2"801 bot_spring.Point(1).name = "bot_rod_pt1"372Appendix H. Working Model 2D Code Listing802803 End If804805806 bot_diaphragm.PX.Value = 0807 top_diaphragm.PX.Value = 0808809 ’**** Diaphragm slot constraints ****810811 Set bot_diaphragm_slot = Doc.NewConstraint("KeyedHSlot")812 Set bot_diaphragm_slot.Point(2).Body = bot_diaphragm ’point 2 is the point813 bot_diaphragm_slot.Point(2).PX.Value = diaphragm_length/2814 bot_diaphragm_slot.Point(1).PY.Value = bot_diaphragm_vert_offset ’point 1 is the slot - this MUSTbe defined after point 2 for some reason815816817 Set top_diaphragm_slot = Doc.NewConstraint("KeyedHSlot")818 Set top_diaphragm_slot.Point(2).Body = top_diaphragm ’point 2 is the point819 top_diaphragm_slot.Point(2).PX.Value = diaphragm_length/2820 top_diaphragm_slot.Point(1).PY.Value = frame_height ’point 1 is the slot - this MUST be definedafter point 2 for some reason821822 bot_diaphragm_slot.name = "bot_diaphragm_slot"823 bot_diaphragm_slot.Point(2).name = "bot_diaphragm_slot_pt"824 bot_diaphragm_slot.Point(1).name = "bot_diaphragm_slot_bkpt"825826 top_diaphragm_slot.name = "top_diaphragm_slot"827 top_diaphragm_slot.Point(2).name = "top_diaphragm_slot_pt"828 top_diaphragm_slot.Point(1).name = "top_diaphragm_slot_bkpt"829830831 ’**** Top diaphragm spring or rigid link ****832833834 If top_diaphragm_locked = 0 Then835836 Set top_spring = Doc.NewConstraint("SpringDamper")837 top_spring.Point(1).PX.Value = 0838 top_spring.Point(1).PY.Value = frame_height/2839 Set top_spring.Point(1).Body = left_frame840 top_spring.Point(2).PX.Value = -diaphragm_length/2841 Set top_spring.Point(2).Body = top_diaphragm842 top_spring.Point(2).PY.Value = 0843 top_spring.length.value = frame_horiz_offset - diaphragm_length/2844845 top_spring.name = "top_spring"846 top_spring.Point(2).name = "top_spring_pt2"847 top_spring.Point(1).name = "top_spring_pt1"848849 top_spring.K.value = trans_K_top850 top_spring.damperK.value = coef_damping_top851852 Else853854 Set top_spring = Doc.NewConstraint("Rod")855373Appendix H. Working Model 2D Code Listing856 top_spring.length.value = frame_horiz_offset - diaphragm_length/2857858 top_spring.Point(1).PX.Value = 0859 top_spring.Point(1).PY.Value = frame_height/2860861 top_spring.Point(2).PX.Value = -diaphragm_length/2862 top_spring.Point(2).PY.Value = 0863864 Set top_spring.Point(1).Body = left_frame865 Set top_spring.Point(2).Body = top_diaphragm866867 top_spring.name = "top_rod"868 top_spring.Point(2).name = "top_rod_pt2"869 top_spring.Point(1).name = "top_rod_pt1"870871 End If872873874 bot_diaphragm.PX.Value = 0875 top_diaphragm.PX.Value = 0876877 table.PX.value = 0878 top_block.PX.value = 0879 bot_block.PX.value = 0880881 ’**** Axial load ****882883 Set axial_load_force = Doc.NewConstraint("Force")884 axial_load_force.AlwaysActive = True885 Set axial_load_force.Point(1).Body = top_block886 axial_load_force.Point(1).px.value = axial_load_eccentricity887 axial_load_force.Point(1).py.value = top_block_height/2888 axial_load_force.Point(1).name = "axial_load_pt"889890 axial_load_force.FX.Value = 0891 axial_load_force.FY.Value = -axial_load892 axial_load_force.name = "axial_load"893894 ’**** Actuator anchor block ****895896 Set actuator_anchor = Doc.NewBody("rectangle")897 actuator_anchor.PX.Value = 10898 actuator_anchor.PY.Value = -0.10899 actuator_anchor.width.value = 1.00900 actuator_anchor.height.value = 1.00901902 Set anchor_point = Doc.NewPoint("Anchor")903 Set anchor_point.Body = actuator_anchor904905 actuator_anchor.name = "actuator_anchor"906 anchor_point.name = "actuator_anchor_pt"907908 ’**** Actuator ****909910 Set actuator_constraint = Doc.NewConstraint("Actuator")911 actuator_constraint.Point(1).PX.Value = -0.50374Appendix H. Working Model 2D Code Listing912 actuator_constraint.Point(1).PY.Value = 0913 Set actuator_constraint.Point(1).Body = actuator_anchor914 actuator_constraint.Point(2).PX.Value = 2915 Set actuator_constraint.Point(2).Body = table916 actuator_constraint.Point(2).PY.Value = 0917 actuator_constraint.ActuatorType = "Length"918919 actuator_constraint.name = "actuator"920 actuator_constraint.Point(2).name = "actuator_pt2"921 actuator_constraint.Point(1).name = "actuator_pt1"922923 ’**** Measuring points on bodies ****924925 Set wall_base_measuring_point = Doc.NewPoint("Point")926 wall_base_measuring_point.name = "wall_base_measuring_point"927 Set wall_crack_bot_measuring_point = Doc.NewPoint("Point")928 wall_crack_bot_measuring_point.name = "wall_crack_bot_measuring_point"929 Set wall_crack_top_measuring_point = Doc.NewPoint("Point")930 wall_crack_top_measuring_point.name = "wall_crack_top_measuring_point"931 Set table_measuring_point = Doc.NewPoint("Point")932 table_measuring_point.name = "table_measuring_point"933934 Set wall_base_measuring_point.Body = bot_block935 wall_base_measuring_point.py.value = -bot_block_height/2936937 Set wall_crack_bot_measuring_point.Body = bot_block938 wall_crack_bot_measuring_point.py.value = bot_block_height/2939940 Set wall_crack_top_measuring_point.Body = top_block941 wall_crack_top_measuring_point.py.value = -top_block_height/2942943 Set table_measuring_point.Body = table944 ’table_measuring_point.py.value = -top_block_height/2945946947 ’**** GM data table ****948949 Set GM_table = Doc.NewInput()950 GM_table.Format = "Table"951 GM_table.X = 20 ’position of the text box from top left corner of screen952 GM_table.Y = 20953 GM_table.Name = "Ground Motion"954 GM_table_inputID = GM_table.ID ’get the input ID number (for reference in formlas)955 GM_table.TimeColumn = 1956 GM_table.DataColumn = 2957958 ’**** GM scale box ****959960 Set GM_scale_input = Doc.NewInput()961 GM_scale_input.Format = "TextBox"962 GM_scale_input.X = 40 ’position of the text box from top left corner of screen963 GM_scale_input.Y = 70964 GM_scale_input.Name = "Ground Motion Scale"965 GM_scale_input.Min = 0966 GM_scale_input.Max = 100967 GM_scale_input_inputID = GM_scale_input.ID ’get the input ID number (for reference in formlas)375Appendix H. Working Model 2D Code Listing968969 ’**** Set actuator formula to reference the ground motion table and ground motion scale ****970971 actuator_constraint.field.formula = "input[" + str$(GM_table_inputID) + "]*input[" + str$(GM_scale_input_inputID) + "] + 7.50"972973 ’**** Set collision properties ****974975 ’Deselect everything976977 Doc.SelectAll False978979 ’Select pairs of objects and turn off collisions980981 Doc.Select bot_diaphragm982 Doc.Select bot_block983 Doc.Collide False984 Doc.SelectAll False985986 Doc.Select bot_diaphragm987 Doc.Select table988 Doc.Collide False989 Doc.SelectAll False990991 Doc.Select top_diaphragm992 Doc.Select top_block993 Doc.Collide False994 Doc.SelectAll False995996 Doc.Select top_diaphragm997 Doc.Select top_pin998 Doc.Collide False999 Doc.SelectAll False10001001 Doc.Select actuator_anchor1002 Doc.Select right_frame1003 Doc.Collide False1004 Doc.SelectAll False10051006 Doc.Select actuator_anchor1007 Doc.Select left_frame1008 Doc.Collide False1009 Doc.SelectAll False10101011 Doc.Select actuator_anchor1012 Doc.Select table1013 Doc.Collide False1014 Doc.SelectAll False10151016 ’**** Assign object parameters ****10171018 bot_block.elasticity.value = coef_restitution1019 bot_block.staticfriction.value = coef_friction_static1020 bot_block.kineticfriction.value = coef_friction_kinetic1021 bot_block.mass.value = bot_block_mass1022376Appendix H. Working Model 2D Code Listing1023 top_block.elasticity.value = coef_restitution1024 top_block.staticfriction.value = coef_friction_static1025 top_block.kineticfriction.value = coef_friction_kinetic1026 top_block.mass.value = top_block_mass10271028 top_pin.mass.value = beam_mass_top1029 top_diaphragm.mass.value = carriage_mass_top1030 bot_diaphragm.mass.value = diaphragm_mass_bot1031103210331034 table.elasticity.value = coef_restitution_table103510361037 ’**************************1038 ’**** MODEL PROPERTIES ****1039 ’**************************10401041 Doc.SimulationMode = "accurate"1042 Doc.AutoAnimationStep = False1043 Doc.VariableIntegrationStep = False1044 Doc.AutoOverlapError = True1045 Doc.AutoIntegratorError = False1046 Doc.WarnInaccurate = False1047 Doc.WarnInconsistent = False1048 Doc.WarnOverlap = False1049 Doc.WarnRedundant = False1050 Doc.AutoSignificantDigits = False1051 Doc.AutoAssemblyError = True10521053 Doc.IntegrationStep = time_step1054 Doc.IntegratorError = overlap_error1055 ’Doc.OverlapError = overlap_error1056 Doc.SignificantDigits = sig_figs1057 Doc.AnimationStep = time_step105810591060 ’****************1061 ’**** METERS ****1062 ’****************10631064 ’**** Left side rocking constraints ****10651066 ’Set meter_left_rocking_constraints = Doc.NewOutput()1067 ’meter_left_rocking_constraints.X = 2001068 ’meter_left_rocking_constraints.Y = 201069 ’meter_left_rocking_constraints.Name = "Left Side Constraints"1070 ’meter_left_rocking_constraints.Height = 2001071 ’meter_left_rocking_constraints.Width = 3001072 ’meter_left_rocking_constraints.Column(0).Label = "Time"1073 ’meter_left_rocking_constraints.Column(0).Cell.Formula = "time"1074 ’meter_left_rocking_constraints.Column(1).Label = "Crack rope length"1075 ’meter_left_rocking_constraints.Column(1).Cell.Formula = "Constraint["+ str$(left_crack_rope.ID)+"].length"1076 ’meter_left_rocking_constraints.Column(2).Label = "Crack separator length"377Appendix H. Working Model 2D Code Listing1077 ’meter_left_rocking_constraints.Column(2).Cell.Formula = "Constraint["+ str$(left_crack_separator.ID) +"].length"1078 ’meter_left_rocking_constraints.Column(3).Label = "Base rope length"1079 ’meter_left_rocking_constraints.Column(3).Cell.Formula = "Constraint["+ str$(left_bot_rope.ID) +"].length"1080 ’meter_left_rocking_constraints.Column(4).Label = "Base separator length"1081 ’meter_left_rocking_constraints.Column(4).Cell.Formula = "Constraint["+ str$(left_bot_separator.ID)+"].length"10821083 ’**** Right side rocking constraints ****10841085 ’Set meter_right_rocking_constraints = Doc.NewOutput()1086 ’meter_right_rocking_constraints.X = 6001087 ’meter_right_rocking_constraints.Y = 201088 ’meter_right_rocking_constraints.Name = "Right Side Constraints"1089 ’meter_right_rocking_constraints.Height = 2001090 ’meter_right_rocking_constraints.Width = 3001091 ’meter_right_rocking_constraints.Column(0).Label = "Time"1092 ’meter_right_rocking_constraints.Column(0).Cell.Formula = "time"1093 ’meter_right_rocking_constraints.Column(1).Label = "Crack rope length"1094 ’meter_right_rocking_constraints.Column(1).Cell.Formula = "Constraint["+ str$(right_crack_rope.ID)+"].length"1095 ’meter_right_rocking_constraints.Column(2).Label = "Crack separator length"1096 ’meter_right_rocking_constraints.Column(2).Cell.Formula = "Constraint["+ str$(right_crack_separator.ID) +"].length"1097 ’meter_right_rocking_constraints.Column(3).Label = "Base rope length"1098 ’meter_right_rocking_constraints.Column(3).Cell.Formula = "Constraint["+ str$(right_bot_rope.ID)+"].length"1099 ’meter_right_rocking_constraints.Column(4).Label = "Base separator length"1100 ’meter_right_rocking_constraints.Column(4).Cell.Formula = "Constraint["+ str$(right_bot_separator.ID) +"].length"11011102 ’**** Relative displacements ****11031104 ’Define some local string variables here for formulas (all x-displacements)11051106 dim table_for as string1107 dim top_for as string1108 dim crackfor as string1109 dim bot_for as string1110 dim rock_for as string11111112 table_for = "(Body["+ str$(table.ID) +"].p.x)"1113 ’top_for = "(Point["+ str$(top_slot.Point(2).ID) +"].p.x - " + table_for + ")"1114 top_for = "(Body["+ str$(top_diaphragm.ID) +"].p.x - " + table_for + ")"1115 crack_for = "((Point["+ str$(wall_crack_top_measuring_point.ID) +"].p.x + " + "Point["+ str$(wall_crack_bot_measuring_point.ID) +"].p.x)/2 - " + table_for + ")"1116 ’bot_for = "(Point["+ str$(wall_base_measuring_point.ID) +"].p.x - " + table_for + ")"1117 bot_for = "(Body["+ str$(bot_diaphragm.ID) +"].p.x - " + table_for + ")"1118 rock_for = crack_for +"- ("+ bot_for +"+("+ top_for +"-"+ bot_for +")*"+str$(crack_height_rel_eff) +")"11191120 ’msgbox rock_for112111221123 Set meter_rel_displacements = Doc.NewOutput()378Appendix H. Working Model 2D Code Listing1124 meter_rel_displacements.X = 6001125 meter_rel_displacements.Y = 3001126 meter_rel_displacements.Name = "Relative Displacements (to table)"1127 ’meter_rel_displacements.Height = 2001128 meter_rel_displacements.Width = 3001129 meter_rel_displacements.Column(0).Label = "Time"1130 meter_rel_displacements.Column(0).Cell.Formula = "time"1131 meter_rel_displacements.Column(1).Label = "Top diaphragm"1132 meter_rel_displacements.Column(1).Cell.Formula = top_for1133 meter_rel_displacements.Column(2).Label = "Crack (mean)"1134 meter_rel_displacements.Column(2).Cell.Formula = crack_for1135 meter_rel_displacements.Column(3).Label = "Bottom diaphragm"1136 meter_rel_displacements.Column(3).Cell.Formula = bot_for1137 meter_rel_displacements.Column(4).Label = "Rocking displacement"1138 meter_rel_displacements.Column(4).Cell.Formula = rock_for11391140 ’**** Accelerations ****11411142 Set meter_accelerations = Doc.NewOutput()1143 meter_accelerations.X = 2001144 meter_accelerations.Y = 3001145 meter_accelerations.Name = "Accelerations"1146 ’meter_accelerations.Height = 2001147 meter_accelerations.Width = 3001148 meter_accelerations.Column(0).Label = "Time"1149 meter_accelerations.Column(0).Cell.Formula = "time"1150 meter_accelerations.Column(1).Label = "Top diaphragm"1151 meter_accelerations.Column(1).Cell.Formula = "Point["+ str$(top_slot.Point(2).ID) +"].a.x"1152 meter_accelerations.Column(2).Label = "Crack (mean)"1153 meter_accelerations.Column(2).Cell.Formula = "(Point["+ str$(wall_crack_top_measuring_point.ID) +"].a.x + " + "Point["+ str$(wall_crack_bot_measuring_point.ID) +"].a.x)/2"1154 meter_accelerations.Column(3).Label = "Bottom diaphragm"1155 meter_accelerations.Column(3).Cell.Formula = "Point["+ str$(wall_base_measuring_point.ID) +"].a.x"115611571158 ’**** Table ****11591160 Set meter_table = Doc.NewOutput()1161 meter_table.X = 2001162 meter_table.Y = 1501163 meter_table.Name = "Table"1164 ’meter_table.Height = 2001165 meter_table.Width = 3001166 meter_table.Column(0).Label = "Time"1167 meter_table.Column(0).Cell.Formula = "time"1168 meter_table.Column(1).Label = "Table Accel"1169 meter_table.Column(1).Cell.Formula = "Point["+ str$(table_measuring_point.ID) +"].a.x"1170 meter_table.Column(2).Label = "Table Displ"1171 meter_table.Column(2).Cell.Formula = "Point["+ str$(table_measuring_point.ID) +"].p.x"117211731174 ’******************1175 ’**** TRIGGERS ****1176 ’******************11771178 ’**** Trigger 1 (for rocking threshold) ****379Appendix H. Working Model 2D Code Listing11791180 Set trigger1_base = Doc.NewBody("polygon")1181 Set trigger1_bullet = Doc.NewBody("rectangle")1182 trigger1_base.name = "trigger1_base"1183 trigger1_bullet.name = "trigger1_bullet"118411851186 trigger1_base.PX.Value = 01187 trigger1_base.PY.Value = -1.0011881189 trigger1_base.addVertex 1, -0.10,01190 trigger1_base.addVertex 2, -0.10,0.11191 trigger1_base.addVertex 3, 0.3,0.11192 trigger1_base.addVertex 4, 0.3,0.31193 trigger1_base.addVertex 5, 0.4,0.31194 trigger1_base.addVertex 6, 0.4,011951196 trigger1_base.DeleteVertex 9 ’delete three default vertices1197 trigger1_base.DeleteVertex 81198 trigger1_base.DeleteVertex 711991200 Set trigger1_anchor_point = Doc.NewPoint("Anchor")1201 Set trigger1_anchor_point.Body = trigger1_base1202 trigger1_anchor_point.name = "trigger1_anchor_pt"12031204 ’Projectile12051206 trigger1_bullet.PX.Value = 01207 trigger1_bullet.PY.Value = -0.801208 trigger1_bullet.width.value = 0.201209 trigger1_bullet.height.value = 0.2012101211 trigger1_base.elasticity.value = 01212 trigger1_bullet.elasticity.value = 0121312141215 ’trigger1_bullet.staticfriction.value = 0.31216 ’trigger1_bullet.kineticfriction.value = 0.312171218 ’Set the trigger to activate when rocking displacement exceeds X% of wall thickness12191220 Set trigger1_force = Doc.NewConstraint("Force")1221 trigger1_force.AlwaysActive = False1222 trigger1_force.ActiveWhen.Formula = "abs(Output["+ str$(meter_rel_displacements.ID) +"].y4) >= " &wall_thickness * rocking_threshold1223 Set trigger1_force.Point(1).Body = trigger1_bullet1224 trigger1_force.FX.Value = 5001225 trigger1_force.FY.Value = 01226 trigger1_force.name = "trigger1_force"1227 trigger1_force.Point(1).name = "trigger1_force_pt"12281229123012311232 ’*******************1233 ’**** EXECUTION ****380Appendix H. Working Model 2D Code Listing1234 ’*******************1235123612371238123912401241124212431244 ’**** Read the ground motion ****12451246 ’MsgBox "Ground motion file: " & motion_filename12471248 ’**** Read the ground motion into the data table in the model ****1249 GM_table.ReadTable ground_motion_folder & motion_filename12501251 ’**** Read ground motion prefix, ID, and duration from filename ****1252 ’GM_prefix = Left(motion_filename,7)1253 ’MsgBox GM_prefix1254 GM_ID = Left(motion_filename,17)1255 ’MsgBox GM_ID1256 GM_duration = CInt(Left(Right(motion_filename,9),3))1257 ’MsgBox GM_duration12581259 ’**** Calculate number of time steps to run to match motion duration ****1260 num_time_steps_to_run = CLng(GM_duration/time_step)1261 ’MsgBox num_time_steps_to_run12621263 ’**** Clear and re-assign pause controls ****12641265 While Doc.PauseControlCount > 01266 Doc.DeletePauseControl 11267 Wend12681269 Doc.NewPauseControl1270 Doc.NewPauseControl1271 Doc.PauseControl(1).Formula = "time > " & str$(GM_duration+2)1272 Doc.PauseControl(2).Formula = "Point["+ str$(top_slot.Point(2).ID) +"].p.y < " + str$(wall_height - 0.5)1273 Doc.SetPauseControlType 1, "stop"1274 Doc.SetPauseControlType 2, "stop"12751276 ’############ START LOOP THROUGH GROUND MOTION SCALES ############’12771278 GM_scale = GM_scale_start1279 GM_scale_input.Value = GM_scale1280 ’GM_scale_input.Value = 1.0012811282 rocking_check_previous = 01283 rocking_check_current = 01284 collapse_check = 01285 increment_number = 11286 final_increment = 01287 GM_scale_to_not_repeat = 01288 GM_scale_to_not_repeat_collapsed = 0381Appendix H. Working Model 2D Code Listing12891290 ’Exit Sub12911292 While collapse_check <= 112931294 ’**** Assemble output file name ****1295 save_name = file_prefix & " - " & GM_ID & " - " & Format$(GM_scale,"0.00")1296 ’MsgBox "Filename: " & save_name12971298 ’**** Export meter data for calculated number of frames ****1299 Doc.ExportStartFrame = 013001301 ’MsgBox "Number of time steps to run: " & num_time_steps_to_run13021303 If num_time_steps_to_run > 32767 Then1304 num_time_steps_to_run = 327671305 End If13061307 ’MsgBox "Number of time steps to run: " & num_time_steps_to_run1308 Doc.ExportStopFrame = num_time_steps_to_run1309 ’Doc.ExportStopFrame = 130001310 Doc.ExportMeterData output_folder & save_name & ".txt"13111312 ’**** Check if rocking occurred ****1313 If trigger1_bullet.PX.Value > 0.0001 Then1314 ’msgbox "Rocking detected"1315 rocking_check_current = 11316 End If131713181319 ’**** Check if run ended due to collapse or time expiry ****1320 ’msgbox "final position: " & Format$(top_pin.PY.Value, "0.00") & " initial position: " &Format$(wall_height, "0.00")1321 If top_pin.PY.Value < (wall_height - 0.3) Then1322 ’msgbox "Wall collapse"1323 collapse_check = 11324 Else1325 ’msgbox "Time expiry"1326 End If13271328 ’**** If this is the final run after previous collapse, then exit the while loop ****1329 If final_increment = 1 Then1330 Goto End_GM_Loop1331 End If13321333 ’**** Reset for next run ****1334 Doc.Reset13351336 ’**** Increment ground motion scale by appropriate amount ****1337 If rocking_check_current = 0 Then1338 GM_scale = GM_scale + GM_scale_major_inc ’*** if no rocking has ever been detected, then do a big increment13391340 ElseIf rocking_check_current = 1 Then ’*** if rocking has been detected now,13411342 If rocking_check_previous = 1 Then ’*** if rocking has been detected before,382Appendix H. Working Model 2D Code Listing1343 If collapse_check = 0 Then1344 GM_scale = GM_scale + GM_scale_minor_inc ’*** and the wall has notcollapsed yet, then keep going with small increments13451346 ’Msgbox "GM_scale: " & GM_scale & " to skip: " &GM_scale_to_not_repeat & " to skip (collapse): " &GM_scale_to_not_repeat_collapsed1347 ’Msgbox GM_scale - GM_scale_to_not_repeat_collapsed13481349 If Abs(GM_scale - GM_scale_to_not_repeat)<0.00001 Then1350 GM_scale = GM_scale + GM_scale_minor_inc ’*** if we have alreadyhit this GM scale before (with the major increments) thendon’t repeat it (skip to next minor inc up)1351 End If13521353 If Abs(GM_scale - GM_scale_to_not_repeat_collapsed)<0.00001 Then1354 GM_scale = GM_scale - GM_scale_minor_inc/2 ’*** and the wall hadpreviously collapsed at this increment, then go back halfof one small increment and make next one the last increment1355 final_increment = 11356 End If1357 Else1358 GM_scale = GM_scale - GM_scale_minor_inc/2 ’*** and the wall hascollapsed, then go back half of one small increment and make nextone the last increment1359 final_increment = 113601361 End If13621363 Else ’*** if this is the first time rocking has been detected,13641365 ’*** make a note of current GM scale so we don’t repeat it with the smallincrements1366 If collapse_check = 0 Then1367 GM_scale_to_not_repeat = GM_scale ’*** if it has NOT collapsed1368 Else1369 GM_scale_to_not_repeat_collapsed = GM_scale ’*** if it has collapsed1370 End If13711372 ’*** go back in increment1373 If increment_number = 1 Then1374 GM_scale = GM_scale - GM_scale_major_inc ’*** and if this is the firstrun, then go back one full big increment (whether wall hascollapsed or not)1375 Else1376 GM_scale = GM_scale - GM_scale_major_inc + GM_scale_minor_inc ’*** andif this is not the first run, then go back to one smallincrement above the previous large one1377 End If13781379 collapse_check = 0 ’*** reset collapse check to 0 (in case the wall hadcollapsed - we want to reach first collapse with small increments)13801381 End If1382 End If1383383Appendix H. Working Model 2D Code Listing1384 GM_scale_input.Value = GM_scale138513861387 ’**** Set previous rocking check to current value ****1388 rocking_check_previous = rocking_check_current13891390 ’**** Update increment number ****1391 increment_number = increment_number + 113921393 ’**** Update collapse_check_previous ****1394 If collapse_check = 1 Then1395 collapse_check_previous = 11396 End If13971398 Wend13991400 End_GM_Loop:1401 ’############ END LOOP THROUGH GROUND MOTION SCALES ############’14021403140414051406140714081409 ’GM_scale_input.Value = 1.00141014111412141314141415 End Sub14161417 ’%%% END OF FILE %%%384Appendix IGround MotionsNotes: This section contains a listing of the ground motions used in the parametric studyof Chapter 6. The accelerograms were downloaded from the PEER database (http://peer.berkeley.edu/peer_ground_motion_database). The motions and normalizationfactors used match those prescribed by FEMA P695 [FEMA, 2009].385Appendix I. Ground MotionsTable I.1: Listing of ground motionsNGA filename Study ID NGA # Component Set Normalization factorNGA_no_953_MUL009.AT2 001 0953 1 FF 0.65NGA_no_953_MUL279.AT2 002 0953 2 FF 0.65NGA_no_960_LOS000.AT2 003 0960 1 FF 0.83NGA_no_960_LOS270.AT2 004 0960 2 FF 0.83NGA_no_1602_BOL000.AT2 005 1602 1 FF 0.63NGA_no_1602_BOL090.AT2 006 1602 2 FF 0.63NGA_no_1787_HEC000.AT2 007 1787 1 FF 1.09NGA_no_1787_HEC090.AT2 008 1787 2 FF 1.09NGA_no_169_H-DLT262.AT2 009 0169 1 FF 1.31NGA_no_169_H-DLT352.AT2 010 0169 2 FF 1.31NGA_no_174_H-E11140.AT2 011 0174 1 FF 1.01NGA_no_174_H-E11230.AT2 012 0174 2 FF 1.01NGA_no_1111_NIS000.AT2 013 1111 1 FF 1.03NGA_no_1111_NIS090.AT2 014 1111 2 FF 1.03NGA_no_1116_SHI000.AT2 015 1116 1 FF 1.10NGA_no_1116_SHI090.AT2 016 1116 2 FF 1.10NGA_no_1158_DZC180.AT2 017 1158 1 FF 0.69NGA_no_1158_DZC270.AT2 018 1158 2 FF 0.69NGA_no_1148_ARC000.AT2 019 1148 1 FF 1.36NGA_no_1148_ARC090.AT2 020 1148 2 FF 1.36NGA_no_900_YER270.AT2 021 0900 1 FF 0.99NGA_no_900_YER360.AT2 022 0900 2 FF 0.99NGA_no_848_CLW-LN.AT2 023 0848 1 FF 1.15NGA_no_848_CLW-TR.AT2 024 0848 2 FF 1.15NGA_no_752_CAP000.AT2 025 0752 1 FF 1.09NGA_no_752_CAP090.AT2 026 0752 2 FF 1.09NGA_no_767_G03000.AT2 027 0767 1 FF 0.88NGA_no_767_G03090.AT2 028 0767 2 FF 0.88NGA_no_1633_ABBAR–L.AT2 029 1633 1 FF 0.79NGA_no_1633_ABBAR–T.AT2 030 1633 2 FF 0.79NGA_no_721_B-ICC000.AT2 031 0721 1 FF 0.87NGA_no_721_B-ICC090.AT2 032 0721 2 FF 0.87NGA_no_725_B-POE270.AT2 033 0725 1 FF 1.17NGA_no_725_B-POE360.AT2 034 0725 2 FF 1.17NGA_no_829_RIO270.AT2 035 0829 1 FF 0.82NGA_no_829_RIO360.AT2 036 0829 2 FF 0.82NGA_no_1244_CHY101-E.AT2 037 1244 1 FF 0.41NGA_no_1244_CHY101-N.AT2 038 1244 2 FF 0.41NGA_no_1485_TCU045-E.AT2 039 1485 1 FF 0.96NGA_no_1485_TCU045-N.AT2 040 1485 2 FF 0.96NGA_no_68_PEL090.AT2 041 0068 1 FF 2.10NGA_no_68_PEL180.AT2 042 0068 2 FF 2.10NGA_no_125_A-TMZ000.AT2 043 0125 1 FF 1.44NGA_no_125_A-TMZ270.AT2 044 0125 2 FF 1.44NGA_181IMPVALL.H-E06_FN.acc 045 0181 1 PL 0.90NGA_181IMPVALL.H-E06_FP.acc 046 0181 2 PL 0.90NGA_182IMPVALL.H-E07_FN.acc 047 0182 1 PL 0.96NGA_182IMPVALL.H-E07_FP.acc 048 0182 2 PL 0.96NGA_292ITALY.A-STU_FN.acc 049 0292 1 PL 1.72continued on next page. . .386Appendix I. Ground MotionsNGA filename Study ID NGA # Component Set Normalization factorNGA_292ITALY.A-STU_FP.acc 050 0292 2 PL 1.72NGA_723SUPERST.B-PTS_FN.acc 051 0723 1 PL 1.04NGA_723SUPERST.B-PTS_FP.acc 052 0723 2 PL 1.04NGA_802LOMAP.STG_FN.acc 053 0802 1 PL 1.63NGA_802LOMAP.STG_FP.acc 054 0802 2 PL 1.63NGA_821ERZIKAN.ERZ_FN.acc 055 0821 1 PL 1.09NGA_821ERZIKAN.ERZ_FP.acc 056 0821 2 PL 1.09NGA_828CAPEMEND.PET_FN.acc 057 0828 1 PL 1.08NGA_828CAPEMEND.PET_FP.acc 058 0828 2 PL 1.08NGA_879LANDERS.LCN_FN.acc 059 0879 1 PL 0.77NGA_879LANDERS.LCN_FP.acc 060 0879 2 PL 0.77NGA_1063NORTHR.RRS_FN.acc 061 1063 1 PL 0.69NGA_1063NORTHR.RRS_FP.acc 062 1063 2 PL 0.69NGA_1086NORTHR.SYL_FN.acc 063 1086 1 PL 0.80NGA_1086NORTHR.SYL_FP.acc 064 1086 2 PL 0.80NGA_1165KOCAELI.IZT_FN.acc 065 1165 1 PL 2.79NGA_1165KOCAELI.IZT_FP.acc 066 1165 2 PL 2.79NGA_1503CHICHI.TCU065_FN.acc 067 1503 1 PL 0.74NGA_1503CHICHI.TCU065_FP.acc 068 1503 2 PL 0.74NGA_1529CHICHI.TCU102_FN.acc 069 1529 1 PL 0.86NGA_1529CHICHI.TCU102_FP.acc 070 1529 2 PL 0.86NGA_1605DUZCE.DZC_FN.acc 071 1605 1 PL 1.08NGA_1605DUZCE.DZC_FP.acc 072 1605 2 PL 1.08NGA_126GAZLI.GAZ_FN.acc 073 0126 1 NP 0.86NGA_126GAZLI.GAZ_FP.acc 074 0126 2 NP 0.86NGA_160IMPVALL.H-BCR_FN.acc 075 0160 1 NP 1.13NGA_160IMPVALL.H-BCR_FP.acc 076 0160 2 NP 1.13NGA_165IMPVALL.H-CHI_FN.acc 077 0165 1 NP 1.99NGA_165IMPVALL.H-CHI_FP.acc 078 0165 2 NP 1.99NGA_495NAHANNI.S1_FN.acc 079 0495 1 NP 1.27NGA_495NAHANNI.S1_FP.acc 080 0495 2 NP 1.27NGA_496NAHANNI.S2_FN.acc 081 0496 1 NP 1.95NGA_496NAHANNI.S2_FP.acc 082 0496 2 NP 1.95NGA_741LOMAP.BRN_FN.acc 083 0741 1 NP 1.15NGA_741LOMAP.BRN_FP.acc 084 0741 2 NP 1.15NGA_753LOMAP.CLS_FN.acc 085 0753 1 NP 1.17NGA_753LOMAP.CLS_FP.acc 086 0753 2 NP 1.17NGA_825CAPEMEND.CPM_FN.acc 087 0825 1 NP 0.66NGA_825CAPEMEND.CPM_FP.acc 088 0825 2 NP 0.66NGA_1004NORTHR.0637_FN.acc 089 1004 1 NP 0.77NGA_1004NORTHR.0637_FP.acc 090 1004 2 NP 0.77NGA_1048NORTHR.STC_FN.acc 091 1048 1 NP 1.18NGA_1048NORTHR.STC_FP.acc 092 1048 2 NP 1.18NGA_1176KOCAELI.YPT_FN.acc 093 1176 1 NP 0.90NGA_1176KOCAELI.YPT_FP.acc 094 1176 2 NP 0.90NGA_1504CHICHI.TCU067_FN.acc 095 1504 1 NP 0.78NGA_1504CHICHI.TCU067_FP.acc 096 1504 2 NP 0.78NGA_1517CHICHI.TCU084_FN.acc 097 1517 1 NP 0.62NGA_1517CHICHI.TCU084_FP.acc 098 1517 2 NP 0.62NGA_2114DENALI.ps10_FN.acc 099 2114 1 NP 0.57NGA_2114DENALI.ps10_FP.acc 100 2114 2 NP 0.57387

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