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UBC Theses and Dissertations

Thin lightly-reinforced concrete walls in concrete shear wall buildings Fenton, Robert Kyle 2020

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THIN LIGHTLY-REINFORCEDCONCRETE WALLS IN CONCRETESHEAR WALL BUILDINGSbyRobert Kyle FentonB.A.Sc., The University of British Columbia, 2015A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Civil Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2020c© Robert Kyle Fenton 2020The following individuals certify that they have read, and recommend to the Faculty of Graduate andPostdoctoral Studies for acceptance, the thesis entitled:Thin Lightly-Reinforced Concrete Walls in Concrete Shear Wall Buildingssubmitted by Robert Kyle Fenton in partial fulfillment of the requirements forthe degree of Master of Applied Sciencein Civil EngineeringExamining Committee:Perry Adebar, Ph.D., FCAE, P.Eng., Civil EngineeringSupervisorTony T.Y. Yang, Ph.D., P.Eng., Civil EngineeringAdditional ExamineriiAbstractThis thesis focuses on the behaviour of thin and lightly-reinforced concrete walls subjected to axial and lateralin-plane force and displacement demands within high-rise cantilever and coupled shear wall type buildings.Many older and some new high-rise buildings employ thin wall elements of this type as a part of the maingravity system. A brief case study of a fictitious sample building is used to identify some shortcomings ofthese elements in design practice.Current North American design codes employ two main design methods to determine the in-plane, uni-axial capacity of thin bearing walls. Results of past wall tests are aggregated and compared to the empiricalmethod and rational ”moment magnifier” method of slender wall design. Comparisons of the design methodsshow that significantly different design axial load capacities are possible within the same design code. Theresults of this comparison are used to derive a new empirical bearing wall design formula which bettercorresponds with the results and design input parameters of the rational ”moment magnifier” design method.Recent seismic events have also shown that these thin walls are subject to sudden compression failureswhen subjected large in plane lateral displacements. A database analysis of past wall tests is used to identifyparameters which influence the drift capacity of these elements, and a new empirical relationship of overallwall drift capacity based on shear height and compression zone slenderness is derived. The database resultsare used to identify several low drift capacity elements for further analysis. An analysis of several previouswall tests and non-linear finite element models is used to determine the sectional and global response charac-teristics of these members. The results of this test specimen analysis shows that thin and lightly-reinforcedwall elements show very little vertical spread of plasticity resulting in smaller than anticipated plastic hingelengths, however sectional analysis methods produce good estimates of overall sectional properties. Finally,a model of in-plane shear displacements based on measured average vertical strains in the plastic hinge zoneis validated for these types of elements.iiiLay SummaryThis research aims to better understand thin and lightly-reinforced concrete walls within shear wall typebuildings. This thesis presents advanced insight into several broad areas of thin and lightly-reinforced wallbehaviour, largely relying on aggregation of past research programs.A case study of a fictitious building representative of buildings in South-Western British Columbia ispresented and discussed. Next, a review of the most common design and analysis procedures within Canadiancodes is presented. A newly simplified design method for lightly reinforced bearing walls is presented andcompared with results of past tests. The final section focuses on the response of these members under seismicloading conditions. A database analysis of previous test specimens is used to identify wall tests with lowlevels of drift capacity for further analysis. Further analysis is applied to validate several proposed models.ivPrefaceThis thesis is original, unpublished, independent work by the author, R. Kyle Fenton.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.1 Demands on Gravity Load Walls Due to Lateral Response of Core Wall Buildings . . 21.1.2 Thin Lightly-Reinforced Bearing Walls . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.3 Thin Lightly-Reinforced Shear Walls . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Demands on Gravity Load Walls Due to Lateral Response of Cantilever and CoupledWall Buildings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Building Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Analysis Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 Wind Load Demands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4.1 Wind Demands According to NBCC . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4.1.1 NBCC Static Wind Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4.1.2 NBCC Dynamic Wind Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 62.4.1.3 FE Model Implementation of Wind Loads . . . . . . . . . . . . . . . . . . . 72.4.2 Results from NBCC Wind Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.5 Seismic Load Demands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.5.1 Seismic Demands According to NBCC . . . . . . . . . . . . . . . . . . . . . . . . . . 102.5.1.1 NBCC Seismic Analysis Procedure . . . . . . . . . . . . . . . . . . . . . . . 102.5.1.2 NBCC Equivalent Static Force Procedure . . . . . . . . . . . . . . . . . . . 102.5.1.3 Linear Modal Response Spectrum Analysis . . . . . . . . . . . . . . . . . . . 112.5.1.4 FE Model Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.5.2 Results from NBCC Seismic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 12vi2.6 Chapter Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Thin Lightly-Reinforced Bearing Walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2 Tests on Bearing Walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2.1 Bearing Wall Test History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2.1.1 Oberlender and Everard, 1977 . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2.1.2 Pillai and Parthasarathy, 1977 . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2.1.3 Saheb and Desayi, 1989 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2.1.4 Fragomeni, 1995 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2.1.5 Sanjayan, 2000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2.1.6 Doh and Fragomeni, 2005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2.1.7 Robinson, Palmeri, and Austin, 2013 . . . . . . . . . . . . . . . . . . . . . . 203.2.1.8 Huang, Hamed, Chang, and Foster, 2015 . . . . . . . . . . . . . . . . . . . . 213.2.2 Comparison of Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2.2.1 Effect of Concrete Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2.2.2 Effect of Number of Reinforcing Layers . . . . . . . . . . . . . . . . . . . . . 253.2.3 Conclusion of Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.3 Empirical Strength Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.3.1 Empirical Strength Equation History . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.3.1.1 Pre 1970s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.3.1.2 Kripanarayanan, 1977 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.3.1.3 Overlender and Everard, 1977 . . . . . . . . . . . . . . . . . . . . . . . . . . 283.3.1.4 Pillai and Parthasarathy, 1977 . . . . . . . . . . . . . . . . . . . . . . . . . . 283.3.1.5 Fragomeni, Mendis, and Grayson, 1994 . . . . . . . . . . . . . . . . . . . . . 283.3.1.6 Bartlett, Loov, and Allen, 2002 . . . . . . . . . . . . . . . . . . . . . . . . . 283.3.1.7 ACI 318 and CSA A23.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.3.2 Empirical Strength Equations Summary . . . . . . . . . . . . . . . . . . . . . . . . . 303.3.3 Comparison of Empirical Strength Equations with Experimental Results . . . . . . . 313.4 Rational Method for Strength of Bearing Walls . . . . . . . . . . . . . . . . . . . . . . . . . . 343.4.1 Moment Magnifier Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.4.1.1 MacGregor, Breen, and Pfrang, 1970 . . . . . . . . . . . . . . . . . . . . . . 353.4.1.2 Mirza, 1990 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.4.1.3 MacGregor, 1993 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.4.1.4 Khuntia and Ghosh, 2004 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.4.1.5 Mirza, 2006 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.4.2 Rational Method in CSA A23.3-14 and ACI 318-14 Standards . . . . . . . . . . . . . 403.4.2.1 CSA A23.3-14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.4.2.2 ACI 318-14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.4.3 Parameters Affecting Strength Estimations . . . . . . . . . . . . . . . . . . . . . . . . 423.4.3.1 Minimum Eccentricity and Limiting Factored Axial Resistance . . . . . . . 423.4.3.2 Flexural Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.4.3.3 Sustained Loading (Creep) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.4.3.4 Buckling Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44vii3.4.4 Comparison of Rational Method Strength Equation with Experimental Results . . . . 443.4.4.1 Comparison Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.4.4.2 Slender Interaction Diagram Generation . . . . . . . . . . . . . . . . . . . . 453.4.4.3 Presentation of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.4.4.4 Discussion of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.5 Comparison of Empirical and Rational Methods . . . . . . . . . . . . . . . . . . . . . . . . . 523.5.1 Comparison of Empirical and Rational Methods . . . . . . . . . . . . . . . . . . . . . 523.6 Simplified Rational Method for Strength of Lightly Reinforced Bearing Walls . . . . . . . . . 553.6.1 Derivation of CSA A23.3 Moment Magnifier Method for an Unreinforced RectangularCross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.6.2 Comparison of Closed Form Solution and Other Methods . . . . . . . . . . . . . . . . 583.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.7.1 Empirical Bearing Wall Design in CSA A23.3-14 . . . . . . . . . . . . . . . . . . . . . 603.7.1.1 General Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.7.1.2 Sustained Load Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.7.1.3 Higher Strength Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.7.1.4 Member Buckling Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.7.1.5 Effective Length Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.7.1.6 Use of the Empirical Method in Moderate Seismic Assessments . . . . . . . 633.7.2 Rational Method of Design in CSA A23.3-14 . . . . . . . . . . . . . . . . . . . . . . . 633.7.2.1 General Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.7.2.2 Effective Flexural Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.7.2.3 Sustained Load Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.7.2.4 Maximum Slenderness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.7.3 Simplified Rational Member Design Equation . . . . . . . . . . . . . . . . . . . . . . 654 Thin Lightly-Reinforced Shear Walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.2 UCLA RCWalls Database . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.2.2 Database Query Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.2.3 Detailed Analysis of Selective Test Results . . . . . . . . . . . . . . . . . . . . . . . . 704.2.4 Simple Empirical Models from Subset of UCLA Wall Database . . . . . . . . . . . . . 724.2.4.1 Abdullah and Wallace 2019 Model of Maximum Drift Capacity . . . . . . . 724.2.4.2 Proposed Modified Abdullah and Wallace 2019 Model of Maximum DriftCapacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.2.4.3 Proposed Empirical Model of Maximum Drift Capacity for Thin and LightlyReinforced Walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.2.4.4 Selection of UCLA Database Wall Specimens for Further Study . . . . . . . 784.3 Walls Tested at E´cole Polytechnique Fe´de´rale de Lausanne (EPFL) . . . . . . . . . . . . . . 794.3.1 Summary of Experimental Program at EPFL . . . . . . . . . . . . . . . . . . . . . . . 794.3.1.1 Testing Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.3.1.2 Description of Test Specimens . . . . . . . . . . . . . . . . . . . . . . . . . . 814.3.1.3 Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83viii4.3.1.4 Test Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.3.2 Summary of Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.3.2.1 Load-Deformation Response . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.3.2.2 Failure Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.3.2.3 Vertical Strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924.3.2.4 Curvatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.3.2.5 Shear Strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.3.2.6 Deformation Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . 974.3.2.7 Influence of Out-of-plane Demands . . . . . . . . . . . . . . . . . . . . . . . 1034.3.3 Analytical Methods for Predicting Wall Response . . . . . . . . . . . . . . . . . . . . 1044.3.3.1 Sectional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044.3.3.2 NLFE Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054.3.4 Flexural Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084.3.4.1 Observed and Predicted Flexural Capacity . . . . . . . . . . . . . . . . . . . 1094.3.4.2 Discussion on the Determination of Flexural Capacity . . . . . . . . . . . . 1114.3.5 Verification of NLFE Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1124.3.6 Horizontal Variation of Vertical Strains . . . . . . . . . . . . . . . . . . . . . . . . . . 1124.3.7 Influence of Test Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1154.3.8 Maximum Tension and Compression Strains at Wall Ends . . . . . . . . . . . . . . . 1164.3.9 Curvature Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1214.3.10 Shear Strain Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1214.3.11 Crack Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1264.4 Development of Proposed Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1294.4.1 Plastic Hinge Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1294.4.1.1 Description of Plastic Hinge Model Based on Uniformly Varying InelasticCurvatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1294.4.1.2 Description of Plastic Hinge Model Based on Linearly Varying Inelastic Cur-vatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1304.4.1.3 Estimates of Elastic Curvatures . . . . . . . . . . . . . . . . . . . . . . . . . 1314.4.1.4 Distributions of Inelastic Curvatures . . . . . . . . . . . . . . . . . . . . . . 1334.4.1.5 Lengths of Linearly Varying Inelastic Curvature . . . . . . . . . . . . . . . . 1334.4.1.6 Model of Plastic Hinge Length for Thin Lightly-Reinforced Concrete Walls . 1334.4.2 Shear Strain Estimation Based on Average Vertical Strain . . . . . . . . . . . . . . . 1384.4.2.1 Description of Shear Strain Model . . . . . . . . . . . . . . . . . . . . . . . . 1384.4.2.2 Shear Model Analysis Results . . . . . . . . . . . . . . . . . . . . . . . . . . 1394.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1435 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1445.1 Demands on Gravity Load Walls due to Lateral Response of Cantilever and Coupled WallBuildings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1445.2 Thin Lightly-Reinforced Bearing Walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1445.2.1 Empirical Bearing Wall Design in CSA A23.3-14 . . . . . . . . . . . . . . . . . . . . . 1455.2.2 Rational Method of Design in CSA A23.3-14 . . . . . . . . . . . . . . . . . . . . . . . 1455.2.3 Simplified Rational Member Design Equation . . . . . . . . . . . . . . . . . . . . . . 146ix5.3 Thin Lightly-Reinforced Shear Walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1475.3.1 UCLA RCWalls Database Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1475.3.2 Analysis of EPFL Walls and Proposed Models . . . . . . . . . . . . . . . . . . . . . . 1475.4 Opportunities for Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149AppendicesA Demands on Gravity Load Walls Due to Lateral Response of Cantilever and CoupledWall Buildings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152A.1 Example Building Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152A.1.1 Building Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152A.1.2 Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152A.2 NBCC Wind Demands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152A.2.1 Static Wind Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155A.2.2 Dynamic Wind Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164A.2.3 Comparison of Static and Dynamic Wind Loading . . . . . . . . . . . . . . . . . . . . 191B Thin Lightly-Reinforced Bearing Walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195B.1 Bearing Wall Test Specimen Geometry and Material Properties . . . . . . . . . . . . . . . . 196B.2 Bearing Wall Test Experimental Results and Predicted Capacity Ratios . . . . . . . . . . . . 200C Thin Lightly-Reinforced Shear Walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203C.1 UCLA RCWalls Selective Walls Database Query Results . . . . . . . . . . . . . . . . . . . . 204C.1.1 Specimen Test Setup Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204C.1.2 Specimen Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207C.1.3 Specimen Concrete and Reinforcing Material Properties . . . . . . . . . . . . . . . . . 210C.1.4 Specimen Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213C.2 EPFL Test Results for TW2, TW4, and TW5 . . . . . . . . . . . . . . . . . . . . . . . . . . 220C.2.1 Vertical Strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221C.2.1.1 TW2 Tension Strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221C.2.1.2 TW2 Compression Strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222C.2.1.3 TW4 Tension Strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223C.2.1.4 TW4 Compression Strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224C.2.1.5 TW5 Tension Strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225C.2.1.6 TW5 Compression Strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226C.2.2 Curvatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227C.2.2.1 TW2 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227C.2.2.2 TW4 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228C.2.2.3 TW5 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229C.2.3 Shear Strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230C.2.3.1 TW2 Shear Strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230C.2.3.2 TW4 Shear Strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231xC.2.3.3 TW5 Shear Strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232C.3 EPFL Test Setup Influence Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233C.3.1 NLFE Vertical Strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234C.3.1.1 TW1 NLFE Vertical Strains . . . . . . . . . . . . . . . . . . . . . . . . . . . 234C.3.1.2 TW2 NLFE Vertical Strains . . . . . . . . . . . . . . . . . . . . . . . . . . . 235C.3.2 NLFE Horizontal Variation of Vertical Strains . . . . . . . . . . . . . . . . . . . . . . 236C.3.2.1 TW1 NLFE Horizontal Variation of Vertical Strains . . . . . . . . . . . . . 236C.3.2.2 TW2 NLFE Horizontal Variation of Vertical Strains . . . . . . . . . . . . . 237C.3.3 NLFE Average Shear Strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238C.3.3.1 TW1 NLFE Average Shear Strains . . . . . . . . . . . . . . . . . . . . . . . 238C.3.3.2 TW2 NLFE Average Shear Strains . . . . . . . . . . . . . . . . . . . . . . . 239C.4 Development of Proposed Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240C.4.1 Inelastic Curvatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241C.4.1.1 TW2 Test Specimen Inelastic Curvature . . . . . . . . . . . . . . . . . . . . 241C.4.1.2 TW4 Test Specimen Inelastic Curvature . . . . . . . . . . . . . . . . . . . . 242C.4.1.3 TW5 Test Specimen Inelastic Curvature . . . . . . . . . . . . . . . . . . . . 243C.4.1.4 TW1 NLFE Inelastic Curvature . . . . . . . . . . . . . . . . . . . . . . . . . 244C.4.1.5 TW2 NLFE Inelastic Curvature . . . . . . . . . . . . . . . . . . . . . . . . . 245C.4.1.6 TW1 NLFE Linearly Varying Inelastic Curvature . . . . . . . . . . . . . . 246C.4.1.7 TW2 NLFE Linearly Varying Inelastic Curvature . . . . . . . . . . . . . . 247C.4.1.8 TW4 Linearly Varying Inelastic Curvature . . . . . . . . . . . . . . . . . . 248C.4.1.9 TW5 Linearly Varying Inelastic Curvature . . . . . . . . . . . . . . . . . . 249xiList of Tables2.1 Effective stiffnesses applied for example building wind load analysis finite element models. . . 82.2 Sample sectional forces on thin wall elements due to NBCC dynamic wind demands. . . . . . 102.3 NBCC spectral acceleration design spectrum used for the analysis of the example building. . 112.4 Effective stiffnesses applied to the example building seismic load analysis finite element models. 122.5 Sample sectional forces on thin wall elements due to NBCC dynamic seismic demands in theexample building. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.6 Sample thin wall inelastic rotational demands and capacities due to NBCC dynamic seismicdemands. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.1 Summary of eccentrically loaded bearing wall test parameters. . . . . . . . . . . . . . . . . . . 223.2 Summary of previously published and implemented bearing wall empirical axial load strengthequations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.3 Summary of statistical analysis of moment magnifier method estimation compared with testresults for varying estimates of flexural stiffness (n=111). . . . . . . . . . . . . . . . . . . . . 514.1 Summary of EPFL test specimen geometric and concrete material properties. . . . . . . . . . 824.2 Summary of EPFL test specimen reinforcing material properties. . . . . . . . . . . . . . . . . 824.3 Wall TW1 summary of displacement components. . . . . . . . . . . . . . . . . . . . . . . . . . 994.4 Wall TW1 summary of displacement component increases by load cycle. . . . . . . . . . . . . 1004.5 Wall TW2 summary of displacement components. . . . . . . . . . . . . . . . . . . . . . . . . . 1004.6 Wall TW2 summary of displacement component increases by load cycle. . . . . . . . . . . . . 1004.7 Wall TW4 summary of displacement components. . . . . . . . . . . . . . . . . . . . . . . . . . 1014.8 Wall TW4 summary of displacement component increases by load cycle. . . . . . . . . . . . . 1014.9 Wall TW5 summary of displacement components. . . . . . . . . . . . . . . . . . . . . . . . . . 1034.10 Wall TW5 summary of displacement component increases by load cycle. . . . . . . . . . . . . 1034.11 TW1 comparison of observed and predicted flexural capacities for in-plane bending. . . . . . 1094.12 TW2 comparison of observed and predicted flexural capacities for in-plane bending. . . . . . 1094.13 EPFL Test specimen estimates of elastic curvature based on plane sections analysis. . . . . . 1324.14 Summary of EPFL specimen linearly varying predicted yield moments, inelastic curvatures,plastic hinge lengths, spread of inelastic curvatures, and maximum compression strains. . . . 137A.1 NBCC 2005 static wind load calculations (E-W). . . . . . . . . . . . . . . . . . . . . . . . . . 157A.2 NBCC 2005 static wind load calculations (N-S). . . . . . . . . . . . . . . . . . . . . . . . . . . 161A.3 NBCC 2005 dynamic wind load calculations (E-W). . . . . . . . . . . . . . . . . . . . . . . . 172A.4 NBCC 2005 dynamic wind load calculations (N-S). . . . . . . . . . . . . . . . . . . . . . . . . 182A.5 Comparison of static and dynamic pressure distributions (E-W). . . . . . . . . . . . . . . . . 192xiiA.6 Comparison of static and dynamic pressure distributions (N-S). . . . . . . . . . . . . . . . . . 193A.7 Static and dynamic wind load distribution comparison. . . . . . . . . . . . . . . . . . . . . . . 194B.1 Summary of bearing wall test specimen geometry and material properties. . . . . . . . . . . . 196B.2 Summary of bearing wall test experimental results and predicted capacity ratios. . . . . . . . 200C.1 Summary of UCLA database query wall experimental test setup details. . . . . . . . . . . . . 204C.2 Summary of UCLA database query wall experimental test specimen geometry. . . . . . . . . 207C.3 Summary of UCLA database query wall experimental test specimen concrete and reinforcingmaterial properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210C.4 Summary of UCLA database query wall experimental test specimen results. . . . . . . . . . . 213xiiiList of Figures1.1 Photo of downtown Vancouver showing many examples of concrete core style shear wall highrise buildings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Case study building typical floor plan. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Example building NBCC static wind pressure distribution. . . . . . . . . . . . . . . . . . . . . 72.3 Example building NBCC dynamic wind pressure distribution. . . . . . . . . . . . . . . . . . . 82.4 Wind moment resistance distribution profiles for analysis model results, including influenceof the gravity frame system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5 Seismic moment resistance distribution profiles for analysis model including influence of thegravity frame system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.1 Test axial load capacity results vs. wall height to thickness ratio for e = t/6†. . . . . . . . . . 233.2 Test axial load capacity results vs. wall height to thickness ratio for e = t/6†, including testspecimen concrete cylinder strengths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.3 Test specimen axial load capacity results grouped by high and low concrete cylinder strengthsincluding linear regressions of each data set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.4 Test specimen axial load capacity results grouped by number of reinforcing layers. . . . . . . 263.5 Comparison of experimental data and historically applied empirical axial strength equations. 323.6 Ratio of test to CSA A23.3-14 Eqn. 14-1 predicted ultimate axial load capacity vs. wallslenderness h/t. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.7 Ratio of test to CSA A23.3-14 Eqn. 14-1 predicted ultimate axial load capacity vs. wallconcrete cylinder strength f ′c. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.8 Slender column or wall free body diagrams and typical axial load and moment interactiondiagram, (MacGregor 1971). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.9 Typical slender column or wall axial load and bending moment interaction diagrams, (Mac-Gregor 1971). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.10 Axial load and moment interaction diagram showing the limiting axial load resistance basedon minimum applied axial load eccentricity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.11 CSA A23.3-14 chapter 10 slender interaction diagrams for t = 203mm, ρv = 0.15% in 2 layers,25mm clear cover, and EI = 0.4EcIg. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.12 CSA A23.3-14 chapter 10 slender interaction diagrams for t = 203mm, ρv = 0.15% in 2 layers,25mm clear cover, and EI = 0.2EcIg +AsIst. . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.13 Comparison of predicted axial load capacity using CSA A23.3-14 moment magnifier methodand experimentally measured maximum axial load results for bearing walls. . . . . . . . . . . 473.14 Ratio of experimentally recorded maximum axial load to CSA A23.3-14 moment magnifierpredicted axial load capacity vs. member slenderness h/t using flexural stiffness EI = 0.4EcIg. 48xiv3.15 Ratio of experimentally recorded maximum axial load to CSA A23.3-14 moment magnifierpredicted axial load capacity vs. concrete cylinder strength f ′c using flexural stiffness EI =0.4EcIg. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.16 Ratio of experimentally recorded maximum axial load to CSA A23.3-14 moment magnifierpredicted axial load capacity vs. member slenderness h/t using flexural stiffness EI =0.2EcIg + EsIst. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.17 Ratio of experimentally recorded maximum axial load to CSA A23.3-14 moment magnifierpredicted axial load capacity vs. concrete cylinder strength f ′c using flexural stiffness EI =0.2EcIg + EsIst. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.18 Ratio of experimentally recorded maximum axial load to CSA A23.3-14 moment magnifierpredicted axial load capacity vs. member slenderness h/t using flexural stiffness from Equation3.38. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.19 Ratio of experimentally recorded maximum axial load to CSA A23.3-14 moment magnifierpredicted axial load capacity vs. concrete cylinder strength f ′c using flexural stiffness fromEquation 3.38 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.20 Comparison of experimentally observed axial load capacities and CSA A23.3-14 Chapter 10design axial strength procedures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.21 Comparison of CSA A23.3-14 Chapter 10 moment magnifier method and Chapter 14 Eqn.14-1 empirical predicted maximum axial load capacity. . . . . . . . . . . . . . . . . . . . . . . 543.22 Axial load and moment interaction diagrams of lightly reinforced bearing walls for varying ρvplaced in two layers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.23 Axial load and moment interaction diagrams of lightly reinforced bearing walls for varying ρvplaced in one centered layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.24 Maximum axial load vs. slenderness for lightly reinforced bearing walls using CSA A23.3-14moment magnifier equations for varying ρv in two layers with eccentricity e = 15 + 0.03t andempirical equation 14-1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.25 Maximum axial load vs. slenderness for lightly reinforced bearing walls using CSA A23.3-14moment magnifier equations for varying ρv in one centered layer with eccentricity e = 15+0.03tand empirical equation 14-1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.26 Maximum axial load vs. slenderness for lightly reinforced bearing walls using CSA A23.3-14 moment magnifier equations for varying ρv in two layers with eccentricity e = t/6 andempirical equation 14-1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.27 Maximum axial load vs. slenderness for lightly reinforced bearing walls using CSA A23.3-14moment magnifier equations for varying ρv in one centered layer with eccentricity e = t/6 andempirical equation 14-1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.1 Histograms of selective parameters and results from UCLA wall database result query. . . . . 694.2 UCLA database maximum drift capacity at shear height vs. axial load ratio P/(f ′cAg). . . . . 704.3 UCLA database maximum drift capacity at shear height vs. shear span ratio M/V lw. . . . . 714.4 UCLA database maximum drift capacity at shear height vs. shear stress ratio Vmax/Av√f ′cwith f ′c in MPa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.5 UCLA database maximum drift capacity at shear height vs. cross section slenderness lw/t. . 724.6 UCLA database maximum drift capacity at shear height vs. compression zone slendernessc/t, including linear trend. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73xv4.7 UCLA database maximum drift capacity at shear height vs. specimen height to length aspectratio hw/lw. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.8 UCLA database maximum drift capacity at shear height vs. specimen slenderness parameterλ = clw/t2, including linear trend. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.9 UCLA database maximum drift capacity at shear height vs. specimen slenderness parameterλ = clw/t2 including linear best fit trends, binned by shear stress ratios lower and greaterthan 0.25vmax/√f ′c. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.10 Abdullah model predicted vs. experimentally observed maximum drift capacity at shear height. 764.11 Comparison of UCLA database wall cross section slenderness lw/t vs. compression zoneslenderness c/t. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.12 UCLA database maximum drift at shear height vs. compression zone slenderness c/t, dividedinto groups of shear stress greater than and less than 0.25vmax/√f ′c. . . . . . . . . . . . . . . 774.13 UCLA database maximum drift at shear height vs. compression zone slenderness c/t, dividedinto groups of shear span ratio, M/V lw greater than and less than 2.0. . . . . . . . . . . . . . 784.14 Comparison of proposed model predicted vs. experimentally observed maximum drift capacityat shear height. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.15 UCLA database tests with observed maximum drift capacities at the shear height of less than1.0%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.16 EPFL test wall setup (adapted from (Almeida et al. 2017)). . . . . . . . . . . . . . . . . . . . 814.17 TW1 hysteretic response at 2 m above the base of the wall. . . . . . . . . . . . . . . . . . . . 854.18 TW1 hysteresis plot for bending with slender wall end in compression with load steps identified. 864.19 TW4 hysteresis plot for bending with slender wall end in compression with load steps identified. 874.20 TW2 hysteresis plot for bending with slender wall end in compression with load steps identified. 874.21 TW5 hysteresis plot for bending with slender wall end in compression with load steps identified. 884.22 TW1 slender wall end damage progression (Almeida et al. 2017). . . . . . . . . . . . . . . . . 894.23 TW2 slender wall end damage progression (Almeida et al. 2017). . . . . . . . . . . . . . . . . 894.24 TW4 slender wall end damage progression (Almeida et al. 2017). . . . . . . . . . . . . . . . . 904.25 TW5 slender wall end damage progression (Almeida et al. 2017). . . . . . . . . . . . . . . . . 914.26 TW1 observed tension strains at 116 mm inside the extreme fibre for multiple load steps. . . 934.27 TW1 observed compression strains at the extreme fibre for multiple load steps. . . . . . . . . 944.28 TW1 observed curvature for bending with tension induced in the wall flange at various loadstep maximum drifts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964.29 TW1 observed shear strains for bending with tension induced in the wall flange at variousload step maximum drifts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 984.30 EPFL observed displacements vs. sum of shear and flexure component estimations. . . . . . . 994.31 Displacement components at load cycle maximum drifts. . . . . . . . . . . . . . . . . . . . . . 1024.32 TW1 comparison of NLFE predicted and observed load-deformation response measured at 2mabove the base of the wall. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1134.33 TW2 comparison of NLFE predicted and observed load-deformation response measured at 2mabove the base of the wall. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1134.34 TW1 and TW2 comparison of NLFE and plane sections vertical strains at the wall base atultimate flexural capacity and displacement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1144.35 TW1 comparison of NLFE predicted and observed tensions strains near the flanged wall end. 117xvi4.36 TW1 comparison of NLFE predicted and observed compression strains at the 80 mm thick end.1184.37 TW2 comparison of NLFE predicted and observed tensions strains near the flanged wall end. 1194.38 TW2 comparison of NLFE predicted and observed compression strains at the 80 mm thick end.1204.39 TW1 comparison of NLFE predicted and observed total curvature distribution. . . . . . . . . 1224.40 TW2 comparison of NLFE predicted and observed total curvature distribution. . . . . . . . . 1234.41 TW1 comparison of NLFE predicted and observed shear strains and displacements. . . . . . . 1244.42 TW2 comparison of NLFE predicted and observed shear strains and displacements. . . . . . . 1254.43 TW1 comparison of NLFE predicted and observed crack patterns. . . . . . . . . . . . . . . . 1274.44 TW2 comparison of NLFE predicted and observed crack patterns. . . . . . . . . . . . . . . . 1284.45 Plastic hinge model based on uniform inelastic curvature. . . . . . . . . . . . . . . . . . . . . 1294.46 Plastic hinge model based on linear inelastic curvature. . . . . . . . . . . . . . . . . . . . . . 1314.47 TW1 observed distributions of inelastic curvature for load-steps 17 to 29. . . . . . . . . . . . 1344.48 TW1 model of linearly varying inelastic curvature based on equal area and first moment ofarea in comparison to length of reinforcement yielding. . . . . . . . . . . . . . . . . . . . . . . 1354.49 TW2 model of linearly varying inelastic curvature based on equal area and first moment ofarea in comparison to length of reinforcement yielding. . . . . . . . . . . . . . . . . . . . . . . 1364.50 EPFL test specimen observed shear strains at individual measurement elevations in compari-son to estimated average vertical strains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1404.51 EPFL test specimen observed average shear strains in comparison to estimated average verticalstrains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1414.52 NLFE predicted average shear strain in comparison to estimate based on average vertical strain.142A.1 Example building typical floor plan. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153A.3 Static wind load applied at story (E-W). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159A.4 Static wind load cumulative applied load (E-W). . . . . . . . . . . . . . . . . . . . . . . . . . 159A.5 Static wind load applied at story (N-S). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163A.6 Static wind load cumulative applied load (N-S). . . . . . . . . . . . . . . . . . . . . . . . . . . 163A.2 NBCC partial wind loading cases taken from NBCC 2005 structural commentaries. . . . . . . 164A.7 Dynamic wind load applied at story (E-W). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175A.8 Dynamic wind load cumulative applied load (E-W). . . . . . . . . . . . . . . . . . . . . . . . 175A.9 Dynamic wind load applied at story (N-S). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185A.10 Dynamic wind load cumulative applied load (N-S). . . . . . . . . . . . . . . . . . . . . . . . . 185A.11 Comparison of static and dynamic pressure distributions (E-W). . . . . . . . . . . . . . . . . 192A.12 Comparison of static and dynamic pressure distributions (N-S). . . . . . . . . . . . . . . . . . 193C.1 TW2 observed tension strains at 162 mm inside the extreme fibre for multiple load steps. . . 221C.2 TW2 observed compression strains at 68 mm inside the extreme fibre for multiple load steps. 222C.3 TW4 observed tension strains at 162 mm inside the extreme fibre for multiple load steps. . . 223C.4 TW4 observed compression strains at the extreme fibre for multiple load steps. . . . . . . . . 224C.5 TW5 observed tension strains at 162 mm inside the extreme fibre for multiple load steps. . . 225C.6 TW5 observed compression strains at 68 mm inside the extreme fibre for multiple load steps. 226C.7 TW2 observed curvature for multiple load steps. . . . . . . . . . . . . . . . . . . . . . . . . . 227C.8 TW4 observed curvature for multiple load steps. . . . . . . . . . . . . . . . . . . . . . . . . . 228C.9 TW5 observed curvature for multiple load steps. . . . . . . . . . . . . . . . . . . . . . . . . . 229xviiC.10 TW2 observed shear strains for multiple load steps. . . . . . . . . . . . . . . . . . . . . . . . . 230C.11 TW4 observed shear strains for multiple load steps. . . . . . . . . . . . . . . . . . . . . . . . . 231C.12 TW5 observed shear strains for multiple load steps. . . . . . . . . . . . . . . . . . . . . . . . . 232C.13 TW1 predicted vertical strains for the test setup and full height NLFE models. . . . . . . . . 234C.14 TW2 predicted vertical strains for the test setup and full height NLFE models. . . . . . . . . 235C.15 TW1 predicted horizontal variation of vertical strains for the test setup and full height NLFEmodels at the base of the wall panel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236C.16 TW2 predicted horizontal variation of vertical strains for the test setup and full height NLFEmodels at the base of the wall panel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237C.17 TW1 predicted average shear strains for the test setup and full height NLFE models. . . . . . 238C.18 TW2 predicted average shear strains for the test setup and full height NLFE models. . . . . . 239C.19 TW2 test specimen estimated distribution of inelastic curvature for load steps 15 to 19. . . . 241C.20 TW4 test specimen estimated distribution of inelastic curvature for load steps 26 to 48. . . . 242C.21 TW5 test specimen estimated distribution of inelastic curvature for load steps 32 to 38. . . . 243C.22 TW1 NLFE and test specimen estimated inelastic curvature. . . . . . . . . . . . . . . . . . . 244C.23 TW2 NLFE and test specimen estimated inelastic curvature. . . . . . . . . . . . . . . . . . . 245C.24 TW1 NLFE Model of linearly varying inelastic curvature based on equal area and first momentof area in comparison to length of reinforcement yielding. . . . . . . . . . . . . . . . . . . . . 246C.25 TW2 NLFE Model of linearly varying inelastic curvature based on equal area and first momentof area in comparison to length of reinforcement yielding. . . . . . . . . . . . . . . . . . . . . 247C.26 TW4 Model of linearly varying inelastic curvature based on equal area and first moment ofarea in comparison to length of reinforcement yielding. . . . . . . . . . . . . . . . . . . . . . . 248C.27 TW5 Model of linearly varying inelastic curvature based on equal area and first moment ofarea in comparison to length of reinforcement yielding. . . . . . . . . . . . . . . . . . . . . . . 249xviiiAcknowledgementsI would like to thank my wife Lindsay for her patience and support during my studies. I could not havecompleted this endeavour without her.Thanks to Perry Adebar for the many hours of discussions, generous funding, and overall guidance incompleting this work.xixChapter 1IntroductionReinforced concrete building design in Western Canada relies heavily on the use of cantilever wall stylelateral force resisting systems. This style of building typically consists of a core made up of an assembly ofcantilever and coupled concrete walls which serve to resist lateral forces on the building, and a distinctlyseparate gravity load resisting system made up of concrete columns, walls, beams, and slabs. The lateralforce resisting system’s (LFRS) primary purpose is to not simply resist lateral forces on the building, but toprotect the gravity force resisting system (GFRS) to ensure that it is able to retain the ability to carry thegravity loads present in the building when lateral deformations due to wind, seismic events, or other sourcesare imparted on the structure. In the simplest cases, this style of building follows a fairly simplistic designand analysis philosophy which is structurally efficient and economical. In practice, the simplistic nature ofthis style of building is easily complicated due to non-structural design constraints, and often the distinctionbetween LFRS and GFRS is less apparent. A common complication is the presence of reinforced concretewalls which are not part of the main core wall assembly. These walls are often mainly used as a part of theGFRS, however due to their often high in-plane lateral stiffness they can also be subjected to significantlateral forces. This thesis explores several aspects of the codified design and analysis of these wall elements.Figure 1.1: Photo of downtown Vancouver showing many examples of concrete core style shear wall highrise buildings.As previously mentioned, the current design philosophy in most cases is to think of the LFRS and GFRSas distinctly separate systems with design checks to the gravity system under lateral loading applied separatefrom the main LFRS analysis. This approach is typically robust when applied to buildings made up of a corewall assembly LFRS and a GFRS made up of concrete columns with small plan dimensions in comparisonto the core wall assembly. As the length of the column elements become longer with respect to the maincore, they begin to resemble what would be described as a wall. In buildings of this style, there havebeen documented events of the thin and lightly-reinforced walls outside the core being heavily damaged1after seismic events (Adebar 2013; Elwood 2013). These wall elements are often observed to have failed incompression dominated modes as a result of the high axial and lateral demands they undergo in seismicevents. CSA A23.3-14 (CSA Group 2014) contains guidance on the level of displacement which gravityload members must be able to undergo. Despite the codified requirements, the behaviour of wall elementssubjected to in-plane loading has been largely focused on heavily reinforced, ductile members. However,the presence of thin and lightly-reinforced wall elements persists in some new, and certainly in many olderbuildings.In addition to the issues of in-plane lateral design and analysis of thin and lightly-reinforced wall elements,CSA A23.3-14 allows several methods of design and analysis for axially loaded bearing walls. Two of thesecodified methods which have been widely applied, have been shown to produce conflicting predictions ofcapacity (Bartlett, Loov, and Allen 2002). Since accurate predictions of capacity are not only requiredto ensure safe, but also economical designs, the opportunity for more in-depth understanding exists. Thisthesis explores these various behaviours of thin and lightly-reinforced concrete walls within shear wall stylebuildings.1.1 Thesis Overview1.1.1 Demands on Gravity Load Walls Due to Lateral Response of Core WallBuildingsThe first section of this thesis presents a case study of a fictitious 35 storey reinforced concrete high risebuilding similar to those built in South-Western British Columbia. The study building represents a verytypical style of construction in the area, and includes a several thin and lightly reinforced wall elementsoutside of the main core. The main lateral force resisting system consists of a single ductile reinforcedconcrete core with cantilever shear walls in one direction, and coupled walls along the orthogonal axisdirection. The gravity load system is made up of a combination of concrete columns and bearing walls. Thiscase study exemplifies some of the issues to be discussed in detail in the subsequent chapters.A wind and seismic analysis of the building is included to determine the types of demands present inthe various thin wall elements throughout the building. The seismic analysis consists of a linear responsespectrum analysis. This type of dynamic seismic analysis was selected to represent standard industry practice.The results of the case study are used to frame the importance of the topics covered in subsequent chapters.This case study is not intended to be a rigorous presentation of the dynamic behaviour of the case building.1.1.2 Thin Lightly-Reinforced Bearing WallsAs informed by the results of the case study building in the preceding chapter, this chapter delves into thetreatment of thin and lightly-reinforced bearing walls within Canadian reinforced concrete design codes. Thewall elements in this section are subjected to monotonic, uni-axial, in-plane eccentric loading only.There are two main methods of analysis employed in Canadian concrete codes. The first is a semi-empirical analysis which has been in use since before the 1950’s, and the second is the more robust “momentmagnifier” method of design, more commonly employed in columns. Past test results from various researchprograms are aggregated and the results are compared with predictions of axial load capacity based onthe two design code analysis methods. The comparisons are used to inform whether one design method ispreferable over the other.2In addition to the comparison of the two analysis methods in general, a comparison of the variousparameters which influence the axial load capacity estimate is made, with the relative merits of each discussedin detail. As a result of the comparisons made, modifications to the semi-empirical analysis method arepresented. A new closed form solution of the moment magnifier method is derived based on an unreinforcedwall element, and is shown to produce good results in comparison to the aggregated test results.A final discussion on code methods for bearing wall analysis is included, with various specific relevantdiscussion topics such as sustained loading effects, high strength concrete, member buckling, effective length,and member slenderness.1.1.3 Thin Lightly-Reinforced Shear WallsTo better understand the behaviour of thin and lightly-reinforced walls subjected to lateral demands, thischapter investigates the behaviour of these elements both in aggregate, and on a case specific basis.The aggregate analysis of these members is based on study of test results identified using a newly builtdatabase of wall tests kindly provided by colleagues at the University of California Los Angeles (UCLA). TheULCA RCWalls database is a clearing house of wall test results from researchers all over the world. Usingthe results from the database, parameters which are important to the overall displacement capacity of thinand lightly-reinforced walls is identified. The results of this analysis are then compared with a previouslydeveloped model of estimated overall drift capacity, and in addition a new predictive model of total driftcapacity is derived and presented for specific application to thin and lightly-reinforced concrete walls.As a result of the ULCA RCWalls database analysis in the leading section of this chapter, several partic-ularly poor performing test results are identified for further analysis to determine the important parameterswhich result in such low levels of drift capacity. Of the walls identified, one set of test results from the E´colePolytechnique Fe´de´rale de Lausanne (EPFL) present an opportunity for further analysis. The researchers atEPFL made a special emphasis on documenting the raw data from their heavily instrumented test specimens,with an aim that other researchers would be able perform their own analyses on the test specimens. As themain focus of the EPFL tests was geared towards out-of-plane buckling phenomena, this thesis includes anin depth summary of the results of several of the test specimens, with a focus on the test observed failuremodes, flexural strains, curvatures, and out-of-plane behaviour.Non linear finite element models (NLFE) of the EPFL test specimens are developed to provide furtherinsight into the behaviour of these elements, and to determine if the test set up influenced inelastic verticalspread of plasticity. The results as observed from the EPFL tests and the NLFE models are used to validatea proposed plastic hinge model based on the spread of inelastic curvatures, and shear strain model based onaverage vertical strains in walls.3Chapter 2Demands on Gravity Load Walls Dueto Lateral Response of Cantilever andCoupled Wall Buildings2.1 IntroductionRecent seismic events have shown that buildings which employ the use of thin wall elements as part oftheir gravity load system may be prone to failures due to the lateral and gravity load demands impartedduring seismic events (Adebar 2013; Saatcioglu et al. 2013; Elwood 2013). The style of design which relieson distributed walls as part of the gravity and lateral load design is commonly observed in older high risebuildings in the Vancouver area (Yathon, Adebar, and Elwood 2017), however has fallen out of favour in thepast 20 years as buildings codes have evolved. Increased emphasis on torsional behaviour, problems withflexural ductility, and increased design seismic demands have all contributed to a paradigm shift away fromthe use thin wall elements in high rise buildings. Despite these changes some designers have still opted forthe use of thin walls within cantilever wall style buildings due to architectural constraints, pressure fromdevelopers to maximize floor areas and to preserve exterior views, and to reduce slab spans. To betterunderstand the role which thin wall elements play in the response of cantilever and coupled wall buildings,this case study looks at how thin walls influence the ultimate behaviour of a high rise building subjected towind and seismic lateral loads. In general, serviceability requirements are not considered in this case study,with ultimate strength design and inelastic displacement capacity being the main focus.To determine the demands on thin wall elements in cantilever and coupled wall type buildings, relativelysimple methods of analysis using linear models, and reduced section properties are typically employed bydesigners in practice. It is acknowledged that the industry is trending towards performance based designfor high profile and more complex tall buildings, however the majority of typical high rise buildings arestill designed using traditional linear methods. To better align the results of this thesis toward practicalapplications, the industry standard software ETABS (Computers and Structures Inc. 2016) has been chosenfor analysis of the example building’s global structural analysis. The demands resulting from the analysisin this chapter are intended to provide insight into the behaviour of thin wall elements within a core wallstructure, and are used simply to frame subsequent sections of this thesis in an established context.2.2 Building DescriptionThe case study building represents a design column layout more typical of buildings constructed prior to1980, however is based on the design of more recently completed structures in the Vancouver area. Newer4core wall buildings in south west British Columbia typically rely on the ductile core alone to resist lateralforce demands, with the gravity system consisting solely of gravity force resisting column elements. Thesample building in this study employs the use of thin wall elements as part of the gravity frame system,however due to their in plane stiffness they are expected to significantly influence the response of the buildingunder lateral demands, and attract large in plane lateral shear forces.The fictitious case study building is a 35-story residential tower assumed to be in South West BritishColumbia. The main tower section is approximately 300 feet tall. The building consists of a central corewith coupled shear walls in one direction, and cantilever walls in the perpendicular direction. The maingravity load resisting system consists mainly of concrete columns, however also includes the use of bearingwall type elements in both direction. The floors plates are typically 7 1/2 inch conventionally reinforced flatplate slab. Foundations are a combination of strip and spread footings on soil classified as seismic site class“C”. The typical tower floor plan is shown in Figure 2.1. A more detailed plan is included in Appendix A.1.Figure 2.1: Case study building typical floor plan.The structure uses assumed specified concrete strengths ranging from 30 MPa to 55 MPa, with the higherstrengths represented in the gravity and lateral frame elements in lower portions of the building. Reinforcingis assumed as standard 400 MPa yield stress.2.3 Analysis Model DescriptionThe analysis model applied to the case study building is implemented in the industry standard softwarepackage ETABS (Computers and Structures Inc. 2016). The analysis model is constructed in 3D using a5combination of frame and shell elements. The core was made up of entirely of shell elements with appropriatestiffness modifiers which will be discussed in a subsequent section. For the dynamic analysis, the buildingmass and its distribution is automatically generated by the model elements. The typical tower floors havein-plane diaphragm multi-point constraints applied to increase computational efficiency.2.4 Wind Load Demands2.4.1 Wind Demands According to NBCCThe NBCC defines two analysis methods of determining wind loads on the primary structural system of abuilding. The static method which is based on an assumed pressure distribution and a relatively rigid build-ing, and the dynamic procedure which includes provisions for determining any resonant dynamic response.The static method is typically sufficient for low and mid rise structures, while the dynamic procedure appliesto slender or tall structures. NBCC specifies the use of the dynamic procedure when the height to effectivewidth of a building exceeds 4.0, or the overall height is greater than 120 meters. The building in this studyhas height to effective width ratios of approximately 3.3 and 4.6 in each direction. Since the slenderness ofthe structure requires the use of the dynamic procedure in the direction with slenderness of 4.6, and thedifference in final design loading based on dynamic versus static procedure is not known, as an exercise bothdesign pressure distributions is calculated for comparison. The final design wind pressure distribution to beused in the analysis is based on the dynamic procedure.2.4.1.1 NBCC Static Wind AnalysisWind loads applied to the main structural system in the NBCC are derived from static pressure distributionsapplied to the windward and leeward faces of the building in various combinations as to produce the mostsevere effect. The main underlying assumption in the use of the static procedure is that the structure isassumed to be relatively rigid, resulting in a system which is insensitive to dynamic effects. This is generallytrue for low and mid rise buildings. Figure 2.2 shows the static pressure distribution, detailed calculationsand loading effects are outlined in Appendix A.2.2.4.1.2 NBCC Dynamic Wind AnalysisThe NBCC dynamic wind pressure distribution applied to tall or slender structures. It’s application isgenerally similar to the static approach, however provides inclusions for the dynamic effects of wind andstructure interaction. The dynamic approach alters the wind gust factor, Cg to account for various effectsincluding turbulence, size effects, resonant frequencies, critical damping ratio, and wind speed variationsover the height of the structure. The determination of gust effect factor is much more involved than thestatic approach of Cg = 2.0, however often produces higher wind loads due to the dynamic properties of thestructure.In addition to gust effects, the exposure factor Ce also differs in it’s application, and it is responsiblefor producing different windward face pressure distributions. The dynamic exposure factor produces higherwind pressures at higher elevations for rough exposures, to represent the decreasing sheltering effects aselevation rises. This has the effect of producing higher overturning moments than the static method, evenwhen total wind loads stay constant.601020304050607080901000.00 0.25 0.50 0.75 1.00 1.25Height Above Ground [m]ULS Pressure [kPa]Windward LeewardFigure 2.2: Example building NBCC static wind pressure distribution.The dynamic pressure distribution for the example building is shown in Figure 2.3, and does not differsignificantly in the X or Y direction which is purely coincidental. The gust factor typically increases asfundamental period in the along wind direction increases, however decreases as effective width decreases.The gust effect factors in the X and Y directions of the case study building are the same due to counteractivewidth and fundamental frequency parameters. The applied dynamic wind pressure distribution is based onthe dynamic characteristics of a model including the main core and gravity frame system, and is applied toall models regardless of if the gravity frame is omitted or not.2.4.1.3 FE Model Implementation of Wind LoadsThe wind pressure distributions as outlined in the previous section are implemented in ETABS to determinethe strength demands on various critical structural elements in the building. Two models were producedfor comparison of wind effects in this thesis. The first model includes the main core of the building only,neglecting any contribution from the gravity load system against lateral wind loads. The second includesthe entire gravity system in the model. The core only model is used as the benchmark model, from whichany reductions or deviations of demands in the second full frame model can be compared. The deviations indemands are then known to be due to the inclusion of the frame system. In practice, it is common practicefor designers to include any resistance provided by the gravity frame against wind lateral loading.Since the the model to be implemented in ETABS is linear, reduced section properties are employed tocapture any material non-linearities which would occur. The reduced stiffnesses applied to the wind analysismodel are as suggested by CSA A23.3-14 (CSA Group 2014), and are shown in Table 2.1.The ETABS models has been developed to use a combination of shell and beam elements depending onthe desired structural member to be implemented. Typically, walls and slab elements are shells, and smallercolumns and the coupling beams are beam elements. All joints are assumed to be fully fixed due to being701020304050607080901000.00 0.25 0.50 0.75 1.00 1.25Height Above Ground [m]ULS Pressure [kPa]Windward LeewardFigure 2.3: Example building NBCC dynamic wind pressure distribution.Table 2.1: Effective stiffnesses applied for example building wind load analysis finite element models.Member Type Effective StiffnessWall Piers in Core Ie = 0.75Ig, Ae = 0.75AgDiagonally Reinforced Coupling Beams Ie = 0.35Ig, Ave = 0.40AgOut-of-plane Bending of Slabs Ie = 0.20IgColumns Ie = 0.70Ig, Ae = 0.70AgThin Shear Walls Outside Core Ie = 0.75Ig, Ae = 0.75Agcast in place, and any joint softening is assumed to be captured as part of the reduced section propertiespresented previously.All FE models are taken as pinned at the level of the footings, and in general the below grade parkingstructure was not included in the analysis. Where the below grade floor diaphragm elements providedstiffness to the lateral force resisting system, it is assumed that the diaphragm not deformable and pinned inthe horizontal directions to simplify the analysis. Due to this simplifying analysis, any below grade responseof the structure is not necessarily indicative of the actual response of the structure. Since the focus of thisstudy is the behaviour of the above grade lateral force resisting system, this simplifying assumption is deemedto be appropriate.2.4.2 Results from NBCC Wind AnalysisThis section presents the demands on the lateral load resisting systems as determined from each modeldeveloped. Results presented are the distribution of overturning moment resistance between the main corewalls and the gravity frame system, as well as sample thin wall sectional forces which produce the governingworst case combined design forces.The distribution of member sectional forces is influenced by inclusion of the gravity frame system in8the structural model. To determine the contribution of the gravity frame system in resisting lateral windpressures, a brief analysis of the overturning moment resistance distribution between members is presented.The core only model moment resistance is analysed in terms of proportion of moment resisted by coupling inthe wall piers and sectional moment resistance of the wall piers. The core and gravity frame model momentresistance is analysed in terms of proportion of moment resisted by the core walls, the thin wall elementsalong grid lines C or 4, and the rest of the gravity system. Figure 2.4 shows the results of the momentresistance analysis including the gravity frame system contributions.0.0010.0020.0030.0040.0050.0060.0070.0080.0090.00100.00-50000 50000 150000 250000Height Above Grade [m]Moment Resisted [kN-m]Core AloneThin Walls Along GL 4Gravity FrameTotal System(a) X-Direction (Coupled Wall Direction)0.0010.0020.0030.0040.0050.0060.0070.0080.0090.00100.00-50000 50000 150000 250000Height Above Grade [m]Moment Resisted [kN-m]Core WallsGravity FrameTotal System(b) Y-Direction (Cantilever Wall Direction)Figure 2.4: Wind moment resistance distribution profiles for analysis model results, including influence ofthe gravity frame system.Sectional forces including gravity load demands are presented in Table 2.2 below. In addition to thesectional force levels, the normalized values based on concrete strength and cross section are presented forreference.9Table 2.2: Sample sectional forces on thin wall elements due to NBCC dynamic wind demands.Member Axial Force, P [kN] Shear Force, V [kN] Bending Moment, M [kN-m]Normalized Pf ′cbtV√f ′cbtMf ′cbt2Wall C3(X-Dir) 9857 kN 581 kN 5781 kN-mNormalized 0.215 0.085 0.025Wall C1(Y-Dir) 17325 kN 2125 kN 7517 kN-mNormalized 0.240 0.198 0.016As shown in Figure 2.4a, the gravity frame system contributes roughly 36% of the overturning resistancein the coupled wall direction (X-Direction) is provided by the gravity frame system, with the thin wallsalong GL 4 making up around a third of that resistance. This results in the sectional forces as shown inTable 2.2 which shows that the overturning and shear force demands are not particularly high, especiallyconsidering an axial load ratio of 21%, which serves to aid the thin wall element’s ability to resist moments.These results show that wind forces in the coupled wall direction are not a cause of significant concern withrespect to lateral wind demands.Moment resistance distributions for the cantilever wall direction (Y-Direction) are presented in Figure2.4b, and show that the gravity frame system demands in this direction make up a significantly lowercontribution to overturning resistance. However as shown in Table 2.2, sectional moment demands are quitelow, however shear demands on this element are quite high. This result shows that while the stiff coredominates overturning moment resistance, shear demands may remain high in these type of elements.It is acknowledged that these simplistic results do not tell the whole story with respect to wind responsein this structure. This analysis shows that, as expected, the long, thin wall elements in this structure doindeed make a significant contribution to wind demands. It is common practice in some design scenarios tosize the main core to resist all lateral wind demands, and apply a separate analysis with inclusion of thegravity frame to help control deflections.2.5 Seismic Load Demands2.5.1 Seismic Demands According to NBCC2.5.1.1 NBCC Seismic Analysis ProcedureThe NBCC requires that deflections and loading due to seismic ground motions be considered in the design ofa building structure. Depending on site seismic hazard, the dynamic response characteristics of the structure,and the complexity of the structure, the method of analysis prescribed by the NBCC varies considerably. Therequired method of analysis varies from the most rudimentary Equivalent Static Force Procedure (ESFP),to more complex dynamic analysis procedures.Since the structure in this study exceeds a height of 60 meters, the NBCC requires the use of a dynamicanalysis procedure. To align the results of this study’s analysis with those typically implemented in designpractice, the most commonly applied dynamic analysis procedure is used.2.5.1.2 NBCC Equivalent Static Force ProcedureRegardless of the dynamic analysis required by NBCC, an ESFP analysis must be performed to provide abaseline estimate of the maximum base forces in the structure. The basis of the ESFP is the assumption10that the main features of the dynamic response of the structure can be represented by a single fundamentalmode response. The result of the ESFP is the lateral earthquake design force, V which is defined as,V =S(Ta)MvIEWRdRo(2.1)where S(Ta) is the spectral acceleration at the fundamental period Ta of the structure, Mv is a factor toaccount for higher modes in long period buildings, IE is structure importance factor, W is the seismic weight,Rd is a ductility force modification, and Ro is an overstrength force modification. The structural systemductility and overstrength force modifications are defined in the NBCC for this structure as 4.0 and 1.7 forductile coupled walls in the east-west direction, and 3.5 and 1.6 for ductile shear walls in the north-southdirection.2.5.1.3 Linear Modal Response Spectrum AnalysisThe dynamic analysis procedure to be applied in this study was selected to align closely with typical designpractice. In practice, the most common dynamic analysis applied is a Linear Modal Response SpectrumAnalysis (RSA). This medium complexity analysis procedure measures the contribution of the natural modesof vibration to determine a likely seismic response of the elastic structure.To determine the modal response of the structure for analysis, the idealized multiple degree of freedom(MDOF) system is typically implemented in commercial software. The results of the modal analysis are thenapplied to a design spectrum which is used to determine the single mode maximum pseudo-acceleration anddeformation. The modal results of the MDOF system applied to the design spectrum provide the distributionof forces and deflections for that modal response. From there, the peak response of each mode is determinedand combined using a modal combination method. The combined modal deformation and element forceresults are then used in the design of the structure.The design spectrum applied is determined by the NBCC. For this study the design spectrum is appliedand defined as shown in Table 2.3.Table 2.3: NBCC spectral acceleration design spectrum used for the analysis of the example building.Spectral Period [s] Spectral Acceleration [g]0.2 1.00.5 0.691.0 0.332.0 0.17PGA 0.512.5.1.4 FE Model ImplementationKeeping in line with applying an analysis approach which is commonly applied in professional practice, twoanalysis models are developed. The first model is that which is typically used in professional practice for thedesign of the main LFRS. This first ”core only” model for the study structure in this chapter is the mainvertical circulation core which consists of the 3 main elevator shafts, the 2 exit stairs, and small commonarea. This model represents the level of complexity typically applied in design practice.The second model developed is a full representation of the gravity and lateral force resisting systems.While a model of this complexity is often generated for use in the design of the gravity system, many11designers opt to reduce computational complexity by using the previously described ”core only” model forlateral analysis. These two models will used to compare the sectional force demands on the lateral andgravity force resisting system elements.Since the method of analysis is a simple linear RSA, effective member stiffnesses are applied to ap-proximate system non-linearity. For the this analysis, the CSA A23.3-14 (CSA Group 2014) recommendedeffective stiffnesses are applied as shown below in Table 2.4.Table 2.4: Effective stiffnesses applied to the example building seismic load analysis finite element models.Member Type Effective StiffnessBeams Ie = 0.40IgColumns Ie = 0.60IgCoupling Beams Ave = 0.45Ag, Ie = 0.25IgSlab Frame Elements Ie = 0.20IgWalls Axe = 0.70Ag, Ie = 0.70Ig2.5.2 Results from NBCC Seismic AnalysisThis section includes the results of the two separate model linear response spectrum analyses. Similar to thewind analysis results presented above, overturning moment resistance of the main core and gravity framesystem are quantified over the height of the structure in Figure 2.5, and sample sectional forces on thethin wall elements are presented in Table 2.5. In addition, a simple check of the thin wall element inelasticrotational capacity and demands are presented in Table 2.6.The modal analysis results identified a fundamental period of 4.45s in the coupled wall direction. Thefirst 2 modes are dominated by behaviour in orthogonal directions, and the third mode is torsional. 90%mass participation of all modes is achieved in the first 15 modes.Table 2.5: Sample sectional forces on thin wall elements due to NBCC dynamic seismic demands in theexample building.Member Axial Force, P [kN] Shear Force, V [kN] Bending Moment, M [kN-m]Normalized Pf ′cbtV√f ′cbtMf ′cbt2Wall C3(X-Dir) 9084 kN 381 kN 3401 kN-mNormalized 0.198 0.056 0.015Wall C1(Y-Dir) 16338 kN 3075 kN 10062 kN-mNormalized 0.240 0.286 0.022As shown in Figure 2.5a, the gravity frame system contributes roughly 25% of the overturning resistancein the coupled wall direction (X-Direction) is provided by the gravity frame system, with the thin wallsalong GL 4 making up around a third of the total resistance. This results in the sectional forces as shownin Table 2.2 which shows that the overturning and shear force demands are not particularly high, especiallyconsidering an axial load ratio of 20%, which serves to aid the thin wall element’s ability to resist moments.However as will be shown in the inelastic rotational demand results, this is not indicative of the thin wallelement’s ability to undergo deformations.Moment resistance distributions for the cantilever wall direction (Y-Direction) are presented in Figure2.5b, and show that the gravity frame system demands in this direction make up a low contribution tooverturning resistance. As shown in Table 2.2, sectional moment demands are moderate, however similar120.0010.0020.0030.0040.0050.0060.0070.0080.0090.00100.00-50000 0 50000 100000 150000 200000Height Above Grade [m]Moment Resisted [kN-m]Core AloneThin Walls Along GL 4Gravity FrameTotal System(a) X-Direction0.0010.0020.0030.0040.0050.0060.0070.0080.0090.00100.00-50000 0 50000 100000 150000 200000Height Above Grade [m]Moment Resisted [kN-m]Core WallsGravity FrameTotal System(b) Y-DirectionFigure 2.5: Seismic moment resistance distribution profiles for analysis model including influence of thegravity frame system.13to the wind analysis, shear demands on this element are quite high at 28%. This result again shows thatwhile the stiff core dominates overturning moment resistance, shear demands may be high in these type ofelements.Up until this point, the analysis results have shown that the thin wall elements have certainly contributedto the overall behaviour of the structure under seismic response, however none of the thin wall force demandshave been shown to be of particular concern. Herein lies the fundamental error many designers makewhen considering the analysis results of a simple linear response spectrum analysis. The modal responseindicates that these force demands occur with a corresponding distribution of deformations, which will belargely controlled by the more structurally dominant core walls. In the case of this structure, the responsespectrum analysis results indicate total displacements of approximately 1000 mm in the coupled wall direction(X-Direction), and 700 mm in the cantilever wall direction (Y-Direction). Since the it is not possible todecouple the displacements of the main core and the thin wall elements, these walls must also undergo thesedisplacements.To quantify the effects of inelastic rotations the core walls will undergo in the plastic hinge region of thestructure, CSA A23.3-14 (CSA Group 2014) applies a method of calculating inelastic rotational demands incantilever walls taken to be,θid =∆fRdRo −∆fγwhw − 0.5lw (2.2)where θid is the inelastic rotational demand in the plastic hinge region, ∆f is the design displacement atthe top of the building, RdRo are the ductility and overstrength force reductions, gammaw is the designoverstrength of the core walls, hw is the height of the structure, and lw is the length of the cantilever wall.A slightly modified version of this equation is applied to coupled wall piers.To compare the calculated inelastic rotational demands, it is necessary to determine the thin wall elementinelastic rotational capacity. It can be shown that inelastic rotations in an axially loaded member subjectedto in-plane shear demands can be related to the compression zone depth, and in-lieu of a direct inelasticrotational comparison, A23.3-14 applies the following limit of compression zone depth,cmax ≤ cu2θid + 0.004lw (2.3)where cmax is the maximum allowable compression zone depth of the wall, cu is the maximum designcompression strain capacity of the concrete, lw is the length of the wall element, and θid is the previouslycalculated inelastic rotational demand of the system.The maximum compression zone depth is then compared with the anticipated compression zone depthof the member from a simple sectional moment-curvature analysis. The results of this analysis for thethin wall elements along grid line 4 and grid line C are shown in Table 2.6. As shown in Table 2.6, theTable 2.6: Sample thin wall inelastic rotational demands and capacities due to NBCC dynamic seismicdemands.Member Inelastic Rotational Demand Maximum Compression Depth D/Cθid [radians] cmax [mm]Wall GL 4(X-Dir) 0.0116 659 2.61Wall GL C(Y-Dir) 0.0050 1595 1.57maximum compression zone depth for the thin walls in the coupled wall direction (X-Direction) is exceededby over 2 times. Similarly, the thin walls along GL C in the cantilever wall direction (Y-Direction) have a14compression zone which exceeds the maximum by roughly 50%. What this result shows is that regardless ofthe distribution forces which the response spectrum analysis purports to present to the designer, the abilityfor these thin wall members to actually undergo the deformations imposed by the system is not possible.2.6 Chapter ConclusionThis chapter presented the results of a fictitious case study of structure similar to some built within the last10 years in the Vancouver area of British Columbia. As was shown, the behaviour of the core wall structureunder lateral wind and seismic loading was influenced by the presence of several thin wall elements which area part of the main gravity resisting system. The sectional forces on the thin wall elements under wind andseismic loading were shown to be significant, with walls in the coupled direction shown to resist up to 15%of the total overturning moments in a combination of bending and unintended outrigger effect. In contrastto the seemingly manageable ultimate force levels in the thin walls, a simple analysis of inelastic rotationaldemands and capacities of these elements shows that the ability for these thin wall elements to undergodeformations of the full system are of paramount importance. The thin walls undergo significant levels ofinelastic demand, far in excess of what they are able to withstand. It is expected that providing increasedamounts of reinforcing would aid in increasing the inelastic rotational demands, however this would alsoserve to stiffen the walls, ultimately increasing force demands.This case study, while not intended to be a rigorous presentation of the dynamic behaviour of thestructure, shows how a simple force based linear analysis method is alone inadequate for the design of thegravity system in a shear wall core type building. This case study represents a small glimpse into thecomplexity of the behaviour of thin wall elements within shear wall core type buildings. The subsequentchapters of this thesis expand on the ability for thin and lightly-reinforced walls to undergo lateral in-planedisplacements and in-plane axial forces in shear wall buildings.15Chapter 3Thin Lightly-Reinforced BearingWalls3.1 IntroductionReinforced concrete bearing walls are vertical load carrying elements which carry axial loads parallel to theirvertical axis, and are subjected to low levels of in plane forces. Axial loads distributed over the horizontal crosssection of the wall are inherently distributed unevenly resulting in a vertical load vector which is eccentricwith respect to the length and thickness of the wall. Eccentricities parallel to the horizontal longitudinalaxis of the wall are generally small with respect to the overall length and are ignored. Eccentricities ofthe vertical load vector transverse to the horizontal longitudinal axis are often substantial with respect tothe thickness of the wall that its inclusion in the analysis is warranted. The eccentric vertical load resultsin an applied moment generated out-of-plane in single curvature for non-sway systems such as shear wallbuildings. As the height to thickness ratio (slenderness) of the wall increases, out-of-plane moments resultin deflections which produce non-linear second order moments (commonly known as P − δ effects). Typicalbearing wall elements are of a height to thickness ratio which requires that these second order effects betaken into consideration.Second order effects due to P − δ in bearing walls can be addressed in several common ways. The mostrigorous method involves non-linear structural analysis techniques which solve the system deflections underthe loads considered and produce a corresponding updated loading scheme under the deformed configuration.This type of second order analysis is generally considered a very detailed analysis method which can produceprecise results. The downside of this type of analysis is that this accuracy comes at the cost of increasedcomplexity and computational time. This type of analysis is generally considered infeasible for the vastmajority of design cases where the applied loads, material properties, and often the structural geometry areuncertain. This thesis does not consider this type of complex analysis.As an alternative to a full second order analysis method, concrete bearing walls have been constructedunder laboratory conditions and tested to failure to determine their ultimate load carrying capacity. Labora-tory tests have resulted in semi-empirical models being developed which marry mechanics based approacheswith observed results. Examples of these methods are employed in many modern design codes such as CSAA23.3-14 (CSA Group 2014) and ACI 318-14 (ACI Committee 318 2014). These methods are generallyaccepted to produce good results under specific conditions, however are less certain in their applicability asparameters move away from the most basic situations.In addition to empirical types of analysis for bearing walls, a common design approach is to perform afirst-order linear analysis and amplify any applied out-of-plane moments based on a procedure commonlyknown as the “Moment Magnifier Method”. This method of analysis forms the basis of slender memberdesign in many design standards including CSA A23.3-14 and ACI 318-14. The Moment Magnifier Method16is well established, and studies have been performed which validate its use for columns and wall type elements.However, the method is not without its pitfalls. The Moment Magnifier Method uses a member’s criticalbuckling load in its magnification and is thus based on the flexural stiffness of the member. Since reinforcedconcrete is materially non-linear, this flexural stiffness varies substantially as loading conditions change.This chapter investigates the CSA A23.3-14 empirical and moment magnifier methods of estimatingmaximum axial load applied to slender wall elements. Both methods are compared with the results over100 laboratory tests of bearing wall elements loaded with eccentric axial loads. This study then performs acomparison of the two methods with the aim of determining if they produce similar estimates of ultimateaxial load capacity when applied in isolation from each other.Once a preferred method of analysis is established to produce appropriate estimates of ultimate axialload capacity, this chapter presents a closed-form solution for a special case of the Moment Magnifier methodfor use in the design and analysis of lightly reinforced slender bearing walls. The method is presented asan alternative to both the empirical and full Moment Magnifier procedure. The alternative method is thenvalidated for lightly reinforced bearing wall elements.Other topics relevant to the behaviour of slender and eccentric axial loaded bearing walls are also discussedas they become relevant. A brief historical overview of bearing wall test and design procedures is presented.Parameters which may effect behaviour of bearing walls are investigated such as concrete strength, verticalreinforcing placement, vertical reinforcement amount, minimum applied axial load eccentricity, estimates offlexural stiffness, sustained loads, effective lengths, slenderness, and member buckling factors.3.2 Tests on Bearing WallsThis section presents and compares experimental testing schemes and results of bearing wall elements sub-jected to eccentric monotonic uniaxial loads. Typically the tests were devised to determine the relationbetween wall slenderness1 and maximum axial load carrying capacity. Some experimental programs in-cluded other varying parameters and are discussed as they arise in each specific case. The test results whichare identified to be appropriate for comparison are then aggregated to determine the general influence ofvarying parameters, and for use in comparison to established design and analysis methods. Table 3.1 in-cludes a summary of the test parameters from each author, and Appendix B.1 includes a full list of specimengeometric and material properties.3.2.1 Bearing Wall Test HistoryBearing wall axial load testing mainly began in the early 1970s with an aim to validate existing empiricaldesign methods in use at the time. The experimental test programs are discussed chronologically from thedate of their publication. A literature review of experimental wall test programs identified the existence walltests prior to the 1970s (Leabu 1959; Seddon 1956), however they focus on research topics more generallyrelated to bearing wall behaviour and typically neglect in depth investigation of ultimate load carryingcapacity. As such, research prior to approximately 1970 is not included for discussion in this thesis.1Wall height to thickness ratio (h/t) is typically used as a proxy for wall slenderness (KL/r). Where no distinction isrequired, the term slenderness may represent either definition in this thesis173.2.1.1 Oberlender and Everard, 1977Oberlender and Everard’s testing program performed from 1971 to 1973 was the first large series of testsperformed on reinforced concrete load-bearing walls (Oberlender 1973; Oberlender and Everard 1977). 54individual wall panels were tested to failure to determine the ultimate strength and structural behavior ofthe panel elements. Of the 54 panels, 27 were loaded eccentrically, with the remaining specimens loadedconcentrically. Test results were then compared with ACI Building Code (ACI Committee 318 1971) designmethods which were in use at the time.Test specimens were approximately half-scale with cross-sectional dimensions of 76.2 mm (3 inches) thickand 609.6 mm (24 inches) in length. Heights were varied from 609.6 mm to 2133.6 mm (2 ft to 7 ft),producing height to thickness ratios ranging from 8 to 28 or slenderness ratios of approximately 27 to 93.The height to thickness values were selected to represent a typical range of element dimensions used inconventional building construction design. In addition to variations in wall slenderness, the amount andplacement of reinforcing was varied. Vertical reinforcing ratios (As/Ag) of 0.33% and 0.47% were placed intwo layers with a cover of 15.875 mm (5/8 in) or 9.525 mm (3/8 in) to determine the effects of reinforcingamount and placement on behaviour and ultimate axial load capacity. Concrete cylinder strengths rangedfrom 28 MPa to 42 MPa (4000 psi to 6000 psi), which represented a common range of design strengthsin use at the time. The actual concrete strengths of each specimen was not recorded, and all results arebased on an average concrete strength of a group of walls of the same height. Load eccentricity effects werestudied by having two different loading configurations. Half of the specimens were loaded concentricallywith a uniformly distributed load over the full cross-sectional area, with the remaining half loaded with theuniform loading applied at and eccentricity of t/6 of 12.7 mm (0.5 in) from the centreline of the section.The maximum applied eccentric load of the tests represents the minimum assumed value used for empiricalwall design in the ACI Building Code. The wall support conditions were set up to be essentially pinnedto produce a column effective length factor of K = 1.0. The loading and support configurations of theOberlender and Everard tests were typically used in other future test programs, with variations in elementgeometry or reinforcing being selected as target test parameters.Some typical observations found that walls with low height to thickness ratios (h/t = 8) failed due tobearing stress failures where the loads were applied. Load eccentricity had no effect on the failure mode ofthese squat members. Concentrically loaded members with values of h/t = 12 failed in multiple locationsalong their heights, with eccentric members of the same slenderness failing at the ends. Concentrically loadedmembers of h/t = 16 failed in multiple locations along their height, with horizontal cracking becoming moreapparent than in less slender members. Eccentrically loaded members with h/t = 16 and greater, all failed bybuckling and collapsed at ultimate load levels. All remaining concentrically loaded members with h/t = 20and greater failed horizontally at the centreline of the wall. Of all the specimens tested, only a singleconcentrically loaded member was found to have deflected in double curvature, all other members werefound to deflect in single curvature under the applied loads.3.2.1.2 Pillai and Parthasarathy, 1977Pillai and Parathasarathy’s 1977 paper (Pillai and Parthasarathy 1977) was both a comparison and vali-dation of several analytical wall design methods, and a presentation of Parathasarathy’s 1973 MSc thesisexperimental testing work at Calicut University in Calicut, India.22Parathasarathy’s MSc thesis was not available from Calicut University’s collection or archives at the time of writing thisthesis18Parathasarathy’s thesis work involved ultimate axial load testing of 18 wall specimens. The wall specimenheight to thickness ratios (h/t) varied from 5 to 30, and vertical reinforcing ratios As/Ag of 0.00%, 0.15%,and 0.30% were used. Reinforcing was placed in a single layer at the centre of the cross section of the wallelements. Field cured concrete cylinder strengths ranging from 15 MPa to 31 MPa were used in the tests. Inkeeping with the premise of a “reasonably concentric” loading scheme, the wall elements were loading withan eccentricity of t/6. The wall support conditions were assumed to be pinned (K = 1.0), and the testingapparatus was designed to provide a negligible amount of wall end fixity.Results of the testing program showed that elements with low height to thickness ratios (h/t) failed bycracking or splitting of the members near the supports. More slender elements (h/t > 20) showed formationof horizontal tensile cracking near mid-height of the members at ultimate load levels.3.2.1.3 Saheb and Desayi, 1989Saheb and Desayi’s work was based on testing of 24 approximately full sized reinforced concrete wall panelsin one-way action (Saheb and Desayi 1989). Tests were performed as part of Saheb’s 1985 PhD thesis3. Thetest program was performed parametrically, where parameters were selected and varied to determine theinfluence of slenderness ratio, aspect ratio, vertical reinforcing, and horizontal reinforcing on ultimate loadof the wall elements. In addition to the presentation of test results, a comparison of past test results andrecommendations was performed.Wall testing was broken into six groups of four wall specimens, with each group representing a differentparameter to be tested. The first group varied in aspect ratio with panels ranging in height to length from0.67 to 2.00. The aspect ratio group used a height to thickness ratio (h/t) of 12, concrete cylinder strength of17.9 MPa, vertical reinforcing ratio of 0.173%, and a horizontal reinforcing ratio of 0.199%. The second groupvaried in height to thickness ratio from 9 to 27. The height to thickness ratio tests used an aspect ratio of 1.5,a concrete cylinder strength of 17.3 MPa, a vertical reinforcing ratio of 0.165%, and a horizontal reinforcingratio of 0.199%. The third group varied in vertical reinforcing ratio with panels ranging from 0.173% to0.845%. This group was the first of two groups of varying vertical reinforcing ratio, each performed with adifferent slenderness ratio and aspect ratio. The first vertical reinforcing tests had a height to thickness ratioof 12, an aspect ratio of 0.67, concrete cylinder strength of 20.1 MPa, and a horizontal reinforcing ratio of0.199%. The fourth group again varied in vertical reinforcing, however with differing values of slendernessand aspect ratio. The fourth group had an aspect ratio of 1.50, a height to thickness ratio of 24, concretecylinder strength of 18.3 MPa, and a horizontal reinforcing ratio of 0.199%. The fifth group was the firstof two groups to vary in horizontal reinforcing ratio, at two different values of height to thickness ratio andaspect ratio. The fifth group varied in horizontal reinforcing ratio from 0.199% to 0.507%. Other parametersfor group five were a height to thickness of 12, an aspect ratio of 0.67, cylinder strength of 19.6 MPa, and avertical reinforcing ratio of 0.173%. The final group again varied in horizontal reinforcing ratio from 0.199%to 0.507%. Other parameters were a height to thickness ratio of 24, aspect ratio of 1.50, concrete cylinderstrength 16.1 MPa, and a vertical reinforcing ratio of 0.176%. All reinforcing was placed in two layers witha clear cover of 10 mm.The wall testing apparatus was devised to be a hinge at both bearing ends, with the axial load appliedwith a constant eccentricity of t/6. Loading was monotonically increased in phases to failure. The phasingand overall duration of loading is not specified. Failure modes for low height to thickness ratios (h/t < 19)3Saheb’s MSc thesis was not available from the Indian Institute of Science collection or archives at the time of writing thisthesis19were typically by crushing at the ends of the specimen. More slender members (h/t > 18) typically failed inbending at approximately mid-height.3.2.1.4 Fragomeni, 1995Fragomeni’s work (Fragomeni 1995) was focused on the behaviour of slender concrete walls. 20 wall specimenswere constructed and tested to failure. Member thickness varied from 35 mm to 50 mm, wall length from200 mm to 500 mm, height to thickness ratios from 12 to 25, and reinforcing ratios from 0.17% to 0.86%in one and two layers. This paper was not available at the time of writing this thesis, and all values are asreported by Doh and Fragomeni (2005).3.2.1.5 Sanjayan, 2000Sanjayan’s testing (Sanjayan 2000) included 4 approximately quarter scaled wall elements all with heightto thickness ratios of 40. The walls included a variety of reinforcing including wire mesh, conventionalreinforcing, fibre reinforcing, and a combination of wire and conventional reinforcing. The first specimenused wire mesh with a vertical reinforcing ratio of 0.99% and a concrete cylinder strength of 58.5 MPa.The second used a combination of wire mesh with supplementary conventional reinforcing for a verticalreinforcing ratio of 2.80% and a concrete cylinder strength of 59.0 MPa. The third specimen was reinforcedwith wire mesh and a vertical reinforcing ratio of 0.21% and a concrete cylinder strength of 59.0 MPa. Thefinal specimen used only fibre steel reinforcing with an approximate reinforcing ratio of 1.22% and a concretecylinder strength of 60.5 MPa4.Test specimens were loaded with a constant eccentricity of t/2 (25 mm), with the line of action parallelwith the exterior edge of the wall specimen. The end conditions were set up in such a way that they wereeffectively hinged (K = 1.0). All failure modes were by buckling of the member at mid-height.3.2.1.6 Doh and Fragomeni, 2005Doh and Fragomeni’s work (Doh and Fragomeni 2005) was focused on both two way and one way concretepanel action. 18 approximately 1/5 scale wall specimens were tested, with 6 of those being one way tests.Only the 6 one way tests are described here for brevity. Concrete cylinder strengths varied from 35.7 MPa to78.2 MPa with each set of 2 walls meant to represent normal and high strength concrete mixes respectively.All specimens were reinforced with steel wire mesh reinforcing at a vertical reinforcing ratio of 0.31%. Heightto thickness ratios were varied from 30 to 40 for these tests. The testing apparatus was set up to applythe axial load with a constant eccentricity of t/6 and the end supports were devised to be effectively hinged(K = 1.0). All test specimens were observed to have failed at mid-height of the specimen in a bucklingaction.3.2.1.7 Robinson, Palmeri, and Austin, 2013Robinson tested sixteen eccentrically loaded wall panels in one-way action (Robinson, Palmeri, and Austin2013). Two of the tests were loaded concentrically, and are not included for comparison in this thesis. Theremaining fourteen walls were loaded with an applied eccentricity of either t/3 or t/6. Concrete cylinderstrengths varied from 49.1 MPa to 53.2 MPa. All specimens were reinforced with 8mm wire reinforcing4It is unclear whether the concrete cylinder corresponding to the fibre reinforced concrete specimen included fibre reinforcing,however this specimen is omitted from the comparison data, as fibre reinforced behaviour is not a part of this study20placed at the centre of the section, with a yield strength which is not noted in the paper. Since the axialload capacity is not sensitive to the reinforcing yield strength, an assumed value of 450 MPa is applied foranalysis purposes in this thesis. Test specimen height to thickness ratios varied from 25 to 30. The tests wereset up in such a way that the end supports were effectively pinned, resulting in an assumed effective lengthfactor of K = 1.0. Some specimens were pre-cracked at mid height to assess sensitivity of the panel bucklingcapacity for cracked sections. All specimens were observed to have failed at mid height in a buckling action.3.2.1.8 Huang, Hamed, Chang, and Foster, 2015Huang tested eight wall panels to observe the effects of reinforcement ratio and arrangement, load eccentricity,and slenderness ratio on the ultimate axial load capacity (Huang, Chang, and Foster 2015). The walls wereloaded with eccentricities of t/12, t/6, and t/3. Panels were reinforced with wire mesh in two layers (except asingle member with one layer), with a yield strength of approximately 550 MPa, and reinforcing ratios from0.164% to 0.592%. Concrete cylinder strength was quite high at 81.4 MPa. Height to thickness ratios werevaried from 17 to 27 for these specimens. The tests were set up in such a way that the end supports wereeffectively pinned, resulting in an assumed effective length factor of K = 1.0. All specimens were observed tohave failed at mid height in a buckling action, with some members fracturing into two pieces upon failure atpeak axial load. The member with the highest reinforcing ratio 0.592% was noted to have not fractured atpeak load, however it is uncertain whether or not this would have been the case had the loading apparatusbeen directly applied load, as opposed to a displacement controlled apparatus.21Table 3.1: Summary of eccentrically loaded bearing wall test parameters.ResearcherNumberofTestsSpecimen Dimensions Reinforcing Details ConcreteCylinderStrength[MPa]Thickness,h[mm]Length,L[mm]Height,h[mm]Height toThicknessh/tVerticalReinf.Ratio, ρv#ofLayersBarDiameter[mm]ClearCover[mm]Oberlender and Everard 1977 27 76 610 610-2130 8-280.33%0.47%2 4-5 10-15 28-42Pillai and Parthasarathy 19773 40 400 1200 300.00%0.15%0.30%1 Unknown Centered 16-323 48 500 1200 253 60 560 1200 209 80 700 400-1200 5-15Saheb and Desayi 1989 24 50 300-900 450-1350 9-270.17%to0.86%2 2-5 10 20-26Fragomeni 1995 20 35-50 200-500 420-1000 12-250.17%to0.86%1-2 Unknown10Centered33-67Sanjayan 2000 3 50 1500 2000 400.21%to2.80%1 5-12 Centered 59-61Doh and Fragomeni 2005 6 40 1000-1600 1000-1600 30-40 0.31% 1 4 Centered 36-78Robinson, Palmeri, and Austin 2013 14 100 500 2500-3000 25-30 0.50% 1 8 Centered 49-52Huang, Chang, and Foster 2015 8 100-160 460 2700 17-270.16%to0.59%1-2 525Centered81Note: A complete list of individual test specimen geometric and material properties is included in Appendix B.1223.2.2 Comparison of Experimental ResultsEach experimental program selected in this comparison has similar testing parameters and conditions and areappropriate for comparison. All tests of interest were loaded with an eccentricity of t/6 with the exceptionof Sanjayan’s tests which were loaded at t/2. Sanjayan’s test results are included for qualitative comparison,however are omitted for any quantitative results presented in this section.Results of the various test programs have been aggregated for comparison and analysis with respectto the ultimate axial load capacity of monotonically loaded, uniaxial, eccentrically loaded wall elements.In each case, the wall elements are of an appropriate range of test scale, loading configuration, concretestrength, reinforcing type, and overall form. Where any important deviation occurs with a given data set,it is appropriately noted.The test results as extracted from work by the authors described in the previous section is shown inFigure 3.1. This figure presents the maximum axial load achieved by each wall specimen normalized by theproduct of the reported concrete strength and the gross cross-sectional area, plotted as a function of wallheight to thickness ratio which is defined as the quotient of wall height and wall thickness.Figure 3.1: Test axial load capacity results vs. wall height to thickness ratio for e = t/6†.† Applied eccentricity for Sanjayan (2000) was t/2The plotted results show that as wall slenderness is increased, a corresponding drop in maximum axialload occurs. This result is well documented in the literature, and is the expected behaviour of a typicalbearing wall member undergoing an eccentric uniaxial applied load based on a simple mechanics basedapproach. A fairly distinct lower bound of axial load strength is observed with upward scatter present acrossthe entire range of slenderness. For test specimens with height to thickness ratios of approximately 30 andabove, the maximum axial load capacity appears to decrease at a lower rate than for more squat specimens.This is important to note, since the effect of slenderness on ultimate axial load capacity is not linear over23the entire range of test results.3.2.2.1 Effect of Concrete StrengthA wide range of concrete cylinder strengths are present in the test results, therefore a comparison of thebehaviour with respect to cylinder strength is noted. Past studies have shown that higher strength concretewalls produce lower normalised axial load capacities (Doh and Fragomeni 2005; Fragomeni, Mendis, andGrayson 1994; Robinson, Palmeri, and Austin 2013; Fragomeni 1995). Figure 3.2 shows the test resultssorted into quartiles of concrete cylinder strength. The figure shows that a wide array of cylinder strengthsare represented over the range of slenderness values. The highest strength tests tended to correspond to moreslender wall specimens. From this figure, a possible connection between concrete strength and maximumaxial load ratio may be present. A statistical analysis of the general trend of the effect of concrete strengthon the ultimate axial load level has been completed.Figure 3.2: Test axial load capacity results vs. wall height to thickness ratio for e = t/6†, including testspecimen concrete cylinder strengths.By grouping the test results into bins by high or low concrete strength, and performing a least squareslinear regression on each set, the general trend of each data set can be assessed. Cylinder strengths above40 MPa are considered as higher strength, and lower than 40 MPa as lower strength. This level is selectedso each bin has an appropriate amount of tests to make a comparison, and strengths above 40 MPa areless common for typical construction practice for bearing walls. For tests of slenderness of h/t ≈ 10, allmembers were made of lower strength concrete. Similarly, tests on members with slenderness greater thanh/t ≈ 35, correspond to higher strength specimens. Both extreme slenderness values have been omittedfrom the comparison data set since they include limited ranges of concrete strength. While this rudimentaryanalysis is not statistically rigorous, it will provide some insight into any effect concrete strength may have24on ultimate axial load level for bearing walls.Figure 3.3 shows the results of the linear regressions for the lower and higher concrete cylinder strengths.From this analysis, there appears to be some relation between the strength of concrete and the ultimate axialload capacity of the wall element. The plot shows that higher strength concretes produce lower ultimateaxial loads, and lower strengths producing relatively higher strengths. This trend can only be assumed tobe valid over the range of slenderness selected for comparison.Figure 3.3: Test specimen axial load capacity results grouped by high and low concrete cylinder strengthsincluding linear regressions of each data set.This short study confirms the presence of concrete cylinder strength effects on ultimate axial load capacityover a range of slenderness of 10 < h/t ≤ 30. Higher strength concrete cylinder strengths are shownqualitatively to have produced lower normalized ultimate axial load capacities. Producing a model for axialstrength reductions as a result of increasing concrete cylinder strength may be possible, however is notthe focus of this research. High strength effects will be discussed within the context of established designstandard strength estimation methods in subsequent sections of this thesis.3.2.2.2 Effect of Number of Reinforcing LayersAnother potentially significant factor in the ultimate axial load level of slender concrete bearing walls is thereinforcing configuration, however it has been suggested in past studies (Kripanarayanan 1977) that for lowamounts of vertical reinforcing the location of reinforcing within the cross section may have little effect onthe overall capacity of a wall element.Figure 3.4 shows the test results sorted by single or double layer reinforcing, with the aim of identifyingany relation between reinforcing location and ultimate axial load capacity. As is evident, there is limitedevidence to support any difference in ultimate axial load capacity for slender wall elements based on the25configuration of the vertical reinforcing.Figure 3.4: Test specimen axial load capacity results grouped by number of reinforcing layers.3.2.3 Conclusion of Experimental ResultsIn total, 111 test specimens of similar loading configuration, from 8 separate studies were identified. Thesetests results represent over 45 years of study on the topic of eccentrically loaded bearing wall behaviour.The results of these tests will form the basis of comparison for the proceeding sections on the topic ofeccentric monotonic uniaxial loaded walls. These results will be used to gauge the effectiveness existingdesign standards to estimate the maximum capacity of bearing wall elements.It is acknowledged that the variety of test programs were attempting to identify the effect of severaldifferent parameters on ultimate axial load capacity. The presence of many confounding variables such asreinforcing ratio, test specimen dimensions, test scale, type and placement of reinforcing, and other intangibledifferences makes determining the explicit cause of higher or lower ultimate axial load capacities difficult.However the main purpose of aggregating the test data is to determine the lower bound and general trend ofaxial load capacity, for comparison to existing analysis methods available. With this purpose in mind, thetest results are acceptable as presented.3.3 Empirical Strength EquationsThis section includes a brief historical outline of empirical ultimate axial strength equations and a comparisonof those equations to the bearing wall test data presented in the previous section. A summary of the historicalempirical strength equations is presented and several of the functions are compared to the experimental26results. Finally, an in depth comparison of the CSA A23.3-14 empirical design equation is compared to theexperimental data and general trends are discussed.3.3.1 Empirical Strength Equation History3.3.1.1 Pre 1970sVarious forms of empirical strength equations exist prior to the 1970’s research into bearing walls began.Given that prior to the 1970s designers were predominantly using a variety of stress based design procedures,a comparison to the newer more generic force based equations was not performed. It is known that theempirical design equations date back to at least the 1928 ACI design code (ACI Committee 318 1928).These early codes lacked the documentation of modern codes, and the development of the equations withinis cumbersome to trace. No effort was made to identify or document specific design equations in use priorto 1970.3.3.1.2 Kripanarayanan, 1977Kripanarayanan’s 1977 paper (Kripanarayanan 1977) explored the historical aspects of the ACI BuildingCode empirical wall equation, and performed a theoretical examination of ultimate capacity wall elementsunder axial loads with varying height to thickness ratios h/t.The author presents a brief history of empirical wall design starting with early allowable stress designmethods from as early as 1908. Various limits based on allowable stress design were used up until in 1971when the ACI Building Code introduced a strength design method approach using the equation,Pu = 0.55φf′cAg[1.0− (lu/40h)2] (3.1)One provision of using this equation as defined was that the applied loads must be “reasonably concentric”,which was defined as to be within the middle third of the cross-section.Kripanarayanan identified the empirical wall design equation to be a product of two individual functions,F1 which is a function of eccentricity and is a value less than 1.0, and F2 which is a function of height tothickness ratio and is assumed to be a negative parabolic function.The author used a computer based parametric analysis approach to determine theoretical load carryingcapacity of the wall elements under axial load at the defined eccentricity of t/6. Three values of verticalreinforcing ratio were used in the analysis 0.00%, 0.15%, and 1.00%. In addition, three thicknesses of wallsections were considered, 203.2 mm (8 in), 254.0 mm (10 in), and 304.8 mm (12 in).A modified empirical design equation was proposed of,Pu = 0.55φf′cbh[1.0− (klu/32h)2] (3.2)which was calibrated to closely match with the lower bound of the ACI Building Code rational analysisapproach for columns. Kripanarayanan found that the proposed equation was better suited to match Ober-lender’s 1973 PhD thesis data for a wider range of slenderness values, which the 1971 ACI Chapter 14equation had overestimated. Kripanarayanan also identified that wall elements would not see a significantincrease in axial load carrying capacity until vertical reinforcing ratios were in the range of 1.00%.273.3.1.3 Overlender and Everard, 1977In the same paper which presented the bearing wall test results (Oberlender and Everard 1977), the author’salso provided insight into the ongoing use of the empirical design equations within the context of their results.The authors suggested that the ACI Building Code adopt the following empirical equation for use in Chapter14,Pu = 0.60φf′cbh[1− (l/30h)2] (3.3)The proposed equation was developed to closely match the lower bound of the eccentrically loaded specimentest results.In addition to the test program, the proposed empirical equation was compared with the existing ACIBuilding Code moment magnifier column design provisions. The authors found that the Chapter 10 procedureproduced results which closely matched those of the test program and even went so far as to recommend theChapter 10 design method as the preferred method of design, even though it produced conservative resultsfor small h/t values (8 and 12). It was also suggested that all wall design procedures and provisions beconsolidated within Chapter 13 of the ACI Building Code, with the two methods of design be correlated andany limitations of the empirical method stated therein.3.3.1.4 Pillai and Parthasarathy, 1977In the same paper which presented the bearing wall test results (Pillai and Parthasarathy 1977), the author’salso provided insight into the ongoing use of the empirical design equations within the context of their results.The authors found that the ultimate axial load capacity estimated by all of theoretical analysis methodsused were conservative in comparison to test results. An equation was presented to replace the ACI 318-71Chapter 14 empirical approach,Pu = 0.57φf′cbt[1.0− (h/50t)2] (3.4)Ultimately the author’s suggested that the ACI 318-71 Chapter 14 empirical design approach was overlyconservative, and the adjusted equation suggested may better represent the ultimate axial load capacity ofwall elements.3.3.1.5 Fragomeni, Mendis, and Grayson, 1994This paper focussed on bearing wall design of members in one and two-way action, with an emphasis onthe development of the ACI 318 empirical equation with regard to one-way action (Fragomeni, Mendis,and Grayson 1994). A history of the ACI 318 empirical equation was presented with a comparison of thevarious forms. No further developments to the existing equation in use at the time of publication was made,however it was identified that the basis of the empirical equation was limited to a specific set of parametersand conditions, and that designers should exercise caution in its use. In conclusion it was suggested thatthe buckling failure of bearing wall elements was poorly understood, and a significant gap in research waspresent.3.3.1.6 Bartlett, Loov, and Allen, 2002Bartlett, Loov, and Allen’s 2002 paper (Bartlett, Loov, and Allen 2002) deals specifically with bearing walldesign in the context of Canadian design codes. The paper outlines and compares wall design methodsin CSA A23.3-94. The authors identify the empirical design equation procedure can produce significantly28higher maximum design axial loads when compared with the moment magnifier method for compressionmember design in A23.3.Additional to comparing bearing wall design methods in CSA A23.3, this paper provides a mechanicsbased approach to derivation of the empirical bearing wall design equation. By considering the deflectedshape of a pin ended simple wall, with a load applied at one third of the thickness of the wall at both ends,the deflection reduces the depth of mid-height internal compression block which resists the external loads.Equilibrium of the external load applied at t/6 and the internal compression block resultant force resultsin the following expression,a2+ ∆ =h3(3.5)where a is the width of the equivalent rectangular stress block. Mid height deflection of the wall can berelated to the curvature of the wall through the moment-area theorems with constant curvature as,∆ =13θmax(lc2)2(3.6)where θmax is the maximum curvature at mid-height and is related to the failure strain and neutral axis ofthe concrete section through the expression,θmax =cβ1a(3.7)where the equivalent stress block factor β1 = 0.97 − 0.0025f ′c. By combining the three presented equationsand noting that the depth of the compression stress block is is proportional to the applied axial load asP/Po, and simplifying, the following expression results,12(PPo)2− 13PPo+13cβ14(lch)2= 0 (3.8)Solving for the positive root of this expression through the use the first two terms of Taylor expansion andreducing to a single term results in the expression,PPo≈ 23[1− 38cβ1(lch)2](3.9)Finally assuming a maximum concrete failure strain of c = 0.0035 and an equivalent compression blockfactor β1 = 0.92 corresponding to 20 MPa concrete, the equation simplifies to,PPo=23[1−(lc29h)2](3.10)This final expression is the same as the empirical bearing wall equation, with the derived version havinga slightly more conservative slenderness term. It could be easily shown that assuming a slightly lowermaximum concrete compression strain of c = 0.0029, the derived equation would be practically identical tothe empirical equations in CSA A23.3-14 and ACI 318-14. This mechanics based derivation shows that theempirical equation is not simply a best fit solution.In addition to the mechanics based derivation of the empirical design equation, the authors also identifythe absence of sustained loading effects, more commonly known as creep, in the empirical equation. The29inclusion of creep and other effects which are required to be considered when using the general compressionmember design procedure in CSA A23.3-14, can result in higher axial load capacities resulting from theempirical method. Despite the higher apparent axial loads predicted by the empirical equations, the authorsmake a comparison with the ACI 318 design equation and show that CSA consistently predicts lower valuesthan the ACI version, and therefore need not be adjusted.3.3.1.7 ACI 318 and CSA A23.3Use of the bearing wall empirical axial strength equation in the current general form began with ACI 318-71(ACI Committee 318 1971) and with CSA A23.3 development occurring in tandem. Minor changes to themaximum achievable load, and slenderness effects have been made as time went on as new research to bearingwalls was performed, as presented in the previous section.Currently, ACI 318-14 uses the following expression as the empirical method of bearing wall design,PN = 0.55f′cAg[1−(klu32h)2](3.11)This equation is defined as valid for members which are subject to axial load with out of plane flexure whichare loaded within the middle third of the thickness of a rectangular solid wall. The effective length factor kis defined as 0.8, 1.0, or 2.0 for walls with rotations restrained at at least one end, unrestrained at both ends(pinned), and walls unbraced against lateral rotation, respectively.Likewise, CSA A23.3-14 uses the following expression (A23.3-14 Equation 14-1) as the empirical methodof bearing wall design,Pr =23α1φcf′cAg[1−(khu32t)2](3.12)This equation is defined as valid for solid rectangular cross-sections, which the principal moments actingabout a horizontal axis parallel to the plane of the wall, with the resultant of the all loads located within themiddle third of the of the thickness. Additionally the wall must be supported against lateral displacementalong the top and bottom edges. The effective length factor k is defined as 0.8 for walls restrained againstrotation at one or both ends, and as 1.0 for walls unrestrained against rotations at both ends.Both ACI 318-14 (ACI Committee 318 2014) and A23.3-14 (CSA Group 2014) have no additional restric-tions on the use of the empirical equations outside those defined generally for all wall elements, regardlessof the method of design or analysis.3.3.2 Empirical Strength Equations SummaryA summary of the various empirical design equations presented in the literature and those in use in ACI318 and CSA A23.3 are shown in Table 3.2. In addition to the mathematical formulations, Figure 3.5 showsa plot of the various design equations compared with the results of experimental testing presented in theprevious section. A discussion of the the test results in comparison to the CSA A23.3-14 design equation ispresented in the succeeding section.30Table 3.2: Summary of previously published and implemented bearing wall empirical axial load strengthequations.Equation Ultimate Axial Load ExpressionOberlender and Everard, 1973 Pu = 0.60f′cAg[1− (h/30t)2]Pillai and Parthasarathy, 1977 Pu = 0.57f′cAg[1− (h/50t)2]Kripanarayanan, 1977 Pu = 0.55f′cAg[1− (h/32t)2]Saheb and Desay, 1989 Pu = 0.55[Agf′c + (fy − f ′c)Asv][1− (h/32t)2]ζwhere ζ = [1.20− (h/10L)]for h/L < 2.0Pu = 0.55[Agf′c + (fy − f ′c)Asv][1− (h/32t)2]for h/L ≥ 2.0ACI 318-71 to 77 Pu = 0.55f′cAg[1− (h/40t)2]ACI 318-77(1980) to 318-14 Pu = 0.55f′cAg[1− (kh/32t)2]CSA A23.3-94 to 14 Pu = 0.67α1f′cAg[1− (kh/32t)2]3.3.3 Comparison of Empirical Strength Equations with Experimental ResultsThis section includes a presentation of historical design equations used in current and historical design codes,and equations presented or recommended in the literature.As shown in Figure 3.5 for the majority of cases, the empirical equations provide reasonable estimatesof the ultimate axial load at low slenderness values, with none of the equations exceeding test results forh/t ≤ 12. This suggests that the maximum ultimate axial load is consistently overestimated by all of theequations in use. Similarly, with the exception of ACI 318-71, the empirical equation appears to provide alow estimate of the strength of the wall. The empirical equation as used in ACI 318-14 and CSA A23.3-14provides a lower bound estimate of the ultimate axial load of a bearing wall which is loaded within themiddle third of its thickness.To aid in determining the applicability of Equation 3.12 used in CSA A23.3-14, Figure 3.6 presents theratio of the theoretical ultimate axial load as predicted by Equation 3.12, to the results of each test specimenpresented in the previous section. The plot includes only wall test results for specimens with h/t ≤ 30, asthis is the upper limit of wall slenderness defined in CSA A23.3-14. Additionally, the use of Equation 3.12for walls with h/t > 32 would be irrational, as the equation is simply a downward opening parabola whichintercepts the slenderness axis at a value of h/t = 32 (or h/t = 40 in the case of K = 0.8), and its use beyondwould produce negative predictions of strength. It is possible, however that the empirical equation could beuseful over the intermediate 25 > h/t > 30 range of slenderness.Figure 3.6 shows that Equation 3.12 can produce unconservative estimates of ultimate axial load in aminority of cases for h/t > 12. As h/t increases beyond the CSA A23.3-14 Chapter 14 upper limit of 25,increasingly conservative estimates are produced as expected, with the exception of a single member with asevere over estimation of ultimate load capacity predicted.The previous section of this thesis has established a relationship between strength of concrete and ultimateaxial load capacity, therefore it is rational to present these results in the context of cylinder strength. Figure3.19 shows the ratios of experimental strength to theoretical strength for Equation 3.12 plotted againstthe recorded cylinder strength. As suspected, as cylinder strength is increased a corresponding decrease inthe ratio of predicted to test strength is observed. The lowest ratio of predicted to test strength was 0.82observed in a specimen from Fragomeni (1995), with a cylinder strength of f ′c = 60MPa. This shows thatthe use of Equation 3.12 may be unconservative when applied to members with concrete cylinder strengths31Figure 3.5: Comparison of experimental data and historically applied empirical axial strength equations.32Figure 3.6: Ratio of test to CSA A23.3-14 Eqn. 14-1 predicted ultimate axial load capacity vs. wallslenderness h/t.Figure 3.7: Ratio of test to CSA A23.3-14 Eqn. 14-1 predicted ultimate axial load capacity vs. wall concretecylinder strength f ′c.33higher than ≈ 40MPa.It is noted that three test results for lower concrete strengths were also observed to have failed at alower value than predicted5, however the specimen’s predicted strengths were within 10% of observed values,and may represent statistical outliers. Specific details of all three of the aforementioned lower strength testresults are published in PhD works which were not available at the time of writing this thesis.It is interesting to note that in practice, the CSA A23.3-14 empirical equation 3.12 would be implementedincluding the effect of end restraint, with a effective length factor of k = 0.8 applied in the majority of typicalcases. The addition of this condition results in modest increases to the ultimate axial load capacity over therange of slenderness which is allowable by the ACI and CSA design standards, which is h/t ≤ 25 in mostcases. Considering that none of the tests have been performed on elements with end restraints applied, thisincrease is justified only through a basic mechanics assumption. It is unknown whether the end restraintassumptions are borne out in actual wall elements. Additionally in cases where rotational end restraints ofthe wall element could be subjected to softening though cyclic motion, such as imposed displacement dueto seismic ground motions, a resulting reduction in ultimate load carrying capacity may occur at higherslenderness levels. The effect would increase as the slenderness of the wall element increased, as the effectivelength effects only the h/t in any of the empirical equations presented.Given that few restrictions are made on the use of either the ACI 318-14 or CSA A23.3-14 equations, it iseasy to imagine cases where the in practice use of either equation could depart from the test conditions. Forexample, possible reductions in ultimate axial load capacity due to sustained loading (creep), initial allowableconstruction imperfections, concrete strength, variations in applied load eccentricity, vertical reinforcingratio, and reinforcing placement may occur. It is apparent that the empirical method, while useful inthe generic case, does not encapsulate many of the other considerations present in the design of concretestructures.3.4 Rational Method for Strength of Bearing WallsAs an alternative to the empirical equation method of bearing wall design presented in the previous section,both ACI 318-14 and CSA A23.3-14 allow the use of a rational procedure for slender compression memberdesign. This section contains details of the rational method for slender compression member design as isimplemented in ACI 318-14 and CSA A23.3-14 design codes, and their implementations within the contextof slender bearing walls. Both design codes relevant to this discussion use the moment magnifier method forslender member design and a brief presentation of the method is included. A comparison of the CSA A23.3-14 moment magnifier with respect to the test results presented in previous sections is performed. In addition,topics relevant to axial load design of bearing walls in general are presented and discussed. Topics includeminimum eccentricity, maximum axial load resistance reductions, flexural stiffness estimations, sustainedload effects, and critical buckling load factors.3.4.1 Moment Magnifier MethodThe general procedure for slender compression member design and analysis in ACI 318-14 and CSA A23.3-14uses the moment magnifier method, which is an approximate elastic first order analysis method developedfor slender columns. The moment magnifier method was fist introduced into North American practice as5Saheb and Desayi (1989) specimens WAR-3 and WAR-4, and Fragomeni (1995) specimen 7b. See Appendix B.2 for details.34a part of ACI 318-70, and was the result of perceived shortcomings in the reduction factor method in useprior to 1970. This section includes a brief literature review of the moment magnifier method for design ofslender compression members, and relevant studies regarding the estimation of flexural stiffness used whichis important when the method is applied to slender bearing walls.3.4.1.1 MacGregor, Breen, and Pfrang, 1970The moment magnifier method for slender column design was first introduced in a 1971 paper by MacGregor,Breen, and Pfrang (MacGregor, Breen, and Pfrang 1971). In their proposal they provide an outline of slendercolumn behaviour, the major variables affecting strength, and an overview of the proposed general designprocedure. A summary of the contents of that paper is provided here, as it forms the basis of the rationalcolumn analysis and design method in modern North American standards.Columns can be generally divided into two categories. Short-columns, denoted by their ability to resistcombined axial and bending forces with the full expected capacity of the member cross section. And slender-columns, which exhibit reductions in member resistance due to second-order deformations. Using thesedefinitions as the basis, a column of fixed slenderness ratio may be both a short and slender memberdepending on the applied loading and restraints. As shown in Figure 3.8, the maximum moment occursat Section A-A as a result of the eccentricity e and the deflection ∆. From this configuration two failuremodes are possible. First the applied axial load P and moment M in the deformed position may exceed thecross sectional capacity of the column, indicated as a “material failure” in Figure 3.8(c). Alternatively if thecolumn is very slender, the deflection may reach a point such that the value of δM/δP is zero. This type offailure is denoted in Figure 3.8(c) as a “stability failure”. Typically, a material failure may occur in bracedor non-sway systems, and a stability failure will occur in an unbraced or sway frame.Representing the column axial load and moment interaction diagram with varying slenderness valuesresults in a family of curves which represent the slender column response of the member. Shown in Figure3.9(a), the slender column response can be represented as the short column response with the momentmagnified by the additional moment due to deflection of the column. To determine the slender columncurve, the intersection of the interaction envelope and the slender column axial is found, shown as point Bin Figure 3.9(a). The slender failure intersection point is then projected back to the short column line offixed eccentricity to generate a single point on the slender interaction diagram for a given slenderness. Thisis repeated for varying eccentricities and slenderness values to create the family of slender curves for themember of interest. An example of the final group of slender axial load and moment interaction diagrams isas shown in Figure 3.9(b).Several factors affect the strength of a slender column, with the major effects resulting from the degree ofrotational end restraint, the degree of lateral restraint, the ratio of end eccentricities, the ratio of axial loadlevel to cylinder strength P/f ′c, and the level of sustained loads (creep). Knowing these factors, the momentmagnifier method is presented as a good approximation of the second order effects.To approximate the maximum moment in an elastic beam-column in single curvature using the momentmagnifier method the following equation is presented,Mmax = Mo +P∆o1− (P/Pc) (3.13)where Mo and ∆o are the first order moment and deflections respectively, and Pc is the critical axial load35Figure 3.8: Slender column or wall free body diagrams and typical axial load and moment interactiondiagram, (MacGregor 1971).Figure 3.9: Typical slender column or wall axial load and bending moment interaction diagrams, (MacGregor1971).36more commonly known as the Euler buckling load. It can be shown that equation 3.13 can be reduced to,Mmax =Mo1− (P/Pc) (3.14)for a column deflecting in single curvature, since the maximum moment and deflection occur at the sameplace. To account for unequal end moments, the concept of “equivalent uniform moments” is introduced.The equivalent moment factor Cm is developed to produce the same column strength as the actual momentsand is the ratio of smaller end moment to larger end moment. Equation 3.14 with the inclusion of theequivalent uniform moment factor is,Mmax =CmMo1− (P/Pc) ≥Mo (3.15)In practice for a reinforced concrete member, the axial load P resulting from a first-order analysis is used,and the corresponding maximum moment at that axial load level is Mmax given by Equation 3.15.The moment magnifier method employs the critical buckling load Pc as a main measure of the influenceof the applied axial load on the slenderness effects of the member. The critical buckling load is defined as,Pc =pi2EIl2(3.16)where the flexural stiffness(EI) applied in practice, is an approximate equivalent stiffness which takes ma-terial and cross sectional non linearities into consideration. The author suggests the following estimates offlexural stiffness are appropriate for analysis of column type members,EI =(0.2EcIg + EsIse)(1 + βd)(3.17)for heavily reinforced members (ρv ≈ 8%), orEI =0.4EcIg(1 + βd)(3.18)for lightly reinforced members (ρv ≈ 1%). Both estimations for flexural stiffness were selected for use becausethey represent lower bound estimates of stiffness in most cases.This forms the basis of the moment magnifier method as originally outlined. Modifications are alsopossible for the inclusion of frame sway effects. Sway effects are defined as lateral displacements of themember ends with respect to one another, as would occur in a frame type structure. This thesis is focussedon the behaviour of wall elements, which typically occur in shear wall type buildings which are not expectedto have a frame type of sway response.3.4.1.2 Mirza, 1990Mirza’s review (Mirza 1990) of the ACI Building Code provides an overview of the moment magnifierapproach to compression member design. The main focus of the study is to provide insight into the influenceof flexural stiffness (EI), and appropriately measure the functionality of the existing estimations used in themoment magnifier method. Mirza used a theoretical nonlinear analysis of approximately 9500 configurationsof slender columns, and compared the results to those estimated using existing equations.37Mirza identified a 305 x 305 mm (12 x 12 in) as the smallest practical cross section which would be usedin design, and used this cross section as the basis of his analysis. The effects of varying concrete cylinderstrength f ′c, vertical reinforcing yield strength fy, concrete cover, length to thickness ratio, load eccentricityto thickness ratio, and vertical reinforcement arrangement, were taken into consideration.A statistical analysis of the difference to flexural stiffness estimation compared with the nonlinear analysisresults is presented. It was found that both equations 3.17 and 3.18, produce stiffness ratios (EI Theoretical/ EI Estimated) which average very close to unity. However the coefficient of variation of stiffness ratios wasvery high (0.33 and 0.38 for equations 3.17 and 3.18 respectively). It was determined this deviation was dueto both design equations missing various factors which affect the flexural stiffness.The study performed multiple regression analyses to identify the slenderness ratio and end eccentricityratio caused the greatest discrepancies of flexural stiffness. From the regression analyses, it was possibleto develop new proposed design equations for flexural stiffness which includes the effects of slenderness andeccentricity. The equations proposed were,EI =αEcIg + EsIs(1 + βd)(3.19)where,α = (0.27 + 0.003l/h− 0.3e/h)EcIg + EsIs ≥ 0 (3.20)or alternatively,α = (0.3− 0.3e/h)EcIg + EsIs ≥ 0 (3.21)where βd is greater than or equal to zero, l is the unsupported height of the member, and e is the largerend eccentricity. Equations 3.17 and 3.18 are subject to the following limits: f ′c ≤ 6000 psi, ρv ≥ 1%, ande/h ≥ 0.1. Both equations were not implemented in ACI 318, however they provide valuable insight into theapplicability and usefulness of the commonly used estimations for flexural stiffness in design codes.3.4.1.3 MacGregor, 1993MacGregor was involved in the original implementation of the moment magnifier method in North Amer-ican design codes (MacGregor, Breen, and Pfrang 1971). In this follow up publication (MacGregor 1993),MacGregor presents some simplifications and provisions for second-order analysis in the ACI design codes.In the context of bearing wall ultimate load design, the main addition of this research is the inclusion of anew proposed slenderness limit for non-sway frames. Previous versions of slenderness limit were found tohave been derived inappropriately, and a new limit of,klur≤ 25− 10 (M1/M2)√Puf ′cAg(3.22)has been derived. The new slenderness limit shown as Equation 3.22 corresponds to a lower bound of momentmagnifier factor of 1.05. In other words, a member is not defined as a slender member unless the effect ofmagnified moment is at least 5% greater than the short column response.383.4.1.4 Khuntia and Ghosh, 2004In a pair of companion papers (Khuntia and Ghosh 2004a; Khuntia and Ghosh 2004b) which explored theinfluence of flexural stiffness in the ACI 318-02 implementation of the moment magnifier method, Kuntiaand Ghosh provided both an analytical approach, and experimental verification of the approximations im-plemented in practice. The analytical approach discussed the factors which generally influence the flexuralstiffness of a reinforced concrete compression member, including reinforcement ratio, axial load ratio, ec-centricity ratio, and compressive strength of concrete. Considering these factors, a parametric study wasperformed which focussed on column type members. The main result of the study was that the flexuralstiffness estimations in Equations 3.17 and 3.18, produced overly conservative magnified moments whencompared with those observed from more advanced non-linear analysis. Ultimately the authors present thefollowing expressions for estimation of flexural stiffness for typical columns,EIe = EcIg (0.80 + 25ρg)(1− eh− 0.5PuPo)≤ EcIg ≥ EcIbeam (3.23)where ρg is the vertical reinforcing ratio, e/h is the load eccentricity to thickness ratio, and Pu/Po is theaxial load ratio. The authors found that Equation 3.23 produced a mean analytical to predicted ratio offlexural stiffness of 1.24 with a standard deviation of 0.15, with only 2 of the 50 values of the ratio marginallyless than 1.0. Low levels of axial load ratio produced the most conservative results.Flexural stiffness estimated by Equation 3.23 was compared in the companion paper which focused onexperimental verification of the proposed equation. Three investigations with a total of nine sets of test datawere identified as relevant for the case of a column in a nonsway frame under single curvature bending. Thestudy found that Equation 3.23 produced results which matched well to the test results, and the typicalACI 318-02 procedure which in all cases used Equation 3.17 as an estimate for flexural stiffness, typicallyproduced more conservative results. A final recommendation to implement Equation 3.23 into subsequentversions of ACI 318 was suggested. This method of flexural stiffness estimation is provided as an alternativemethod in ACI 318-14.3.4.1.5 Mirza, 2006This study (Mirza 2006) undertakes a meta-analysis of 25 past investigations for a total of 354 physical tests,with the aim of validation of the moment magnifier method of slender column design in CSA A23.3-94. Theanalysis compared the maximum axial load of the tests to the predicted maximum axial load using the CSAA23.3-94 implementation of the moment magnifier method with flexural stiffness estimated using equation3.18. In addition, an alternative estimation of flexural stiffness proposed by a study by Mirza (Mirza 1989)is compared. The alternative estimation of stiffness is,EI = [(0.3− 0.3e/h)EcIg + EsIrs] ≥ EsIrs (3.24)The study finds that the CSA A23.3-94 method using Equation 3.18 produces a mean value of theoreticalto test maximum axial load of 1.08 at a standard deviation of 0.17. The alternative Equation 3.24, improvesthe results slightly to a mean of 1.07 at a standard deviation of 0.14. According to the study, the modestincrease in accuracy warrants the use of equation 3.24 in future editions of CSA A23.3. The study does notinclude considerations with regard to axial load ratio, which has been shown to have a significant effect oneffective flexural stiffness estimations (Khuntia and Ghosh 2004a; Khuntia and Ghosh 2004b).393.4.2 Rational Method in CSA A23.3-14 and ACI 318-14 StandardsThis section includes an overview of slender compression member design using the moment magnifiermethod as implemented in CSA A23.3-14 and ACI 318-14 in the context of its application to bearing wallaxial load design.3.4.2.1 CSA A23.3-14Design of bearing walls in Canada is generally governed by CSA A23.3-14 Design of Concrete Structures(CSA Group 2014). Chapter 14 of the standard defines the requirements for wall type elements. A wall isdefined in CSA A23.3-14 as “a vertical element in which the horizontal length, lw, is at least six times thethickness, t, and at least one-third the clear height of the element.” Further, the definition of a bearing wallis specified as a wall which supports,a) factored in-plane vertical loads exceeding 0.04f ′cAgb) weak axis moments about a horizontal axis in the plane of the wall; andc) the shear forces necessary to equilibrate the moments specified in Item b)Structural design of bearing walls is governed by Clause 14.2 “Structural design of bearing walls”. ThisClause provides designers with two avenues of design of these members, by using a general approach governedby Clauses 7, 10, and 11, and by using the empirical design equation as defined in Clause 14.2.2.CSA A23.3-14 provisions of Chapters 7, 10, and 11 are required to be followed in the general design ofbearing wall type elements. Contained within Chapter 10 are the design requirements for axial and flexureload design, including provisions for combined axial and flexure analysis.Typically for vertical load carrying elements vertical reinforcing is required to be fully tied, howeverbecause of Clause 10.10.4.c which defines a maximum axial load resistance for “other walls”, the standardacknowledges the existence of a vertical load carrying wall element within Chapter which does not containtransverse reinforcing ties. Clause 14.1.8.7 allows vertical distributed reinforcement to be untied in wallelements if the area of vertical reinforcing is less than 0.005Ag, and the reinforcing used is 20M or smaller.This feature is what allows the design of untied wall type elements under the provisions of Chapter 10.As a part of Chapter 10, all compression members must be checked for slenderness effects under Clause10.13 “Slenderness effects - General”. Since this study is concerned with reinforced concrete shear wallstructures, the slenderness effects of a “Non-sway frame” system will typically govern the design of anycompression members in the structure. Designation of a system as a non-sway frame is covered in Clause10.14.4 of the standard, and gauges whether or not the structure is resisting lateral load effects through theresistance of a stiff lateral load resisting system such as shear walls, or if the gravity load resisting framedefoemations contribute greatly to lateral resistance. Typically this check can be performed by inspection(CSA Group 2014).Under Clause 10.15 “Slenderness effects - Non-sway frames”, slenderness effects may be neglected forcompression members which satisfy,klur≤ 25− 10(M1/M2)√Pf/(f ′cAg)(3.25)which can be reduced further due to the assumption of any flexure in the bearing wall occurring in singlecurvature with equal end moments due to applied minimum eccentricity, therefore M1/M2 = 1.0, which40results in the reduced equation,klur≤ 15√Pf/(f ′cAg)(3.26)Once the slenderness trigger of Clause 10.15.2 has been exceeded (Equations 3.25 and 3.26), a secondorder analysis taking into consideration the effects of member curvature is required. CSA A23.3-14 usesa moment magnifier procedure, which amplifies the applied moments to account second order effects incompression members. In the case where there is no or low magnitude applied end moments, a minimummoment of M2 = Pf (15mm+ 0.03h) is to be applied (minimum eccentricity of e = 15mm+ 0.03h.The amplified moment based on member curvature is defined in Clause 10.15.3.1 as,Mc =CmM21− PfφmPc≥M2 (3.27)where the member resistance factor is defined as,φm = 0.75 (3.28)the critical axial load (Euler buckling load) is,Pc =pi2EI(klu)2(3.29)and the member flexural stiffness can be approximated by either,EI =0.2EcIg + EsIst1 + βd(3.30)orEI =0.4EcIg1 + βd(3.31)Also, a factor which relates the actual member moment diagram to an equivalent uniform moment diagramis defined as,Cm = 0.6 + 0.4M1M2≥ 0.4 (3.32)The modulus of elasticity of concrete applicable to the estimates of flexural stiffness in Equations 3.29,3.30, and 3.31 are defined in CSA A23.3-14 as,Ec = (3300√f ′c + 6900)( γc2300)1.5(3.33)where γc is the unit weight of the concrete. CSA A23.3-14 also provides an alternative estimation of elasticmodulus for normal density concrete with compressive strength between 20 and 40 MPa which is,Ec = 4500√f ′c (3.34)Both estimates of elastic modulus are based on the average of the secant modulus for a stress of 0.40f ′cdetermined for similar concrete in accordance with ASTM C469.When applied to a bearing wall element assumed to be in a non sway frame system with the only momentarising from equal end eccentricities, the ratio M1/M2 is positive and equal to unity. For this special case,41Equation 3.32 reduces to unity.The term βd in Equations 3.31 and 3.30 accounts for member creep and for non-sway frames is definedas the ratio of the maximum sustained axial load to the maximum factored axial load associated with thesame load combination (CSA Group 2014). For a typical residential building, it has been suggested thatβd ≈ 0.6 (MacGregor and Bartlett 2000), however an analysis of a typical reinforced concrete shear wallbuilding in the Vancouver area has shown that the value of β may reach as high as 0.9 for some elements.Further details are provided on this study of member sustained load in a subsequent section.3.4.2.2 ACI 318-14Design of bearing walls in the United States is generally governed by ACI 318-14 Building Code Requirementsfor Structural Concrete (ACI Committee 318 2014). Chapter 11 of the standard defines the requirementsfor wall type elements. A wall is defined in ACI 318-14 as “a vertical element designed to resist axial load,lateral load, or both, with a horizontal length-to-thickness ratio greater than 3, used to enclose or separatespaces.”Structural strength design of bearing walls is governed by Clause 11.4 “Required Strength”, and Clause11.5 “Design strength”. This clause provides designers with two avenues of design of these members, byusing a general approach governed by Chapter 6 and Clause 22.4, and by using the empirical design equationas defined in Clause 11.5.3.In general, the provisions of ACI 318-14 are identical to those presented for CSA A23.3-14. One majorexception is an alternative measure of approximate member moment of inertia to be used in the momentmagnifier method. The alternative effective moment of inertia is based off the previously discussed work ofKhuntia and Ghosh (2004a) and Khuntia and Ghosh (2004b) and is presented as,I = Ig (0.80 + 25ρg)(1− eh− 0.5PuPo)(3.35)and is restricted to be greater than 0.25Ig and less than 0.875Ig. This alternative moment of inertia isdefined to only be valid for column and wall type elements.3.4.3 Parameters Affecting Strength EstimationsThis section includes a brief discussion of the various parameters which have been identified to effect axialstrength estimations of bearing wall elements. Parameters include minimum eccentricity, estimations offlexural stiffness, sustained loading (creep), buckling factor, and end restraint.3.4.3.1 Minimum Eccentricity and Limiting Factored Axial ResistanceThe general procedure for short compression member axial load design in ACI 318-14 and CSA A23.3-14applies a limit to the theoretical maximum axial load which may be applied. The maximum axial load is inplace to represent the effect on axial strength which an applied moment would have. This relation can beshown graphically on an axial load and moment interaction diagram. Figure 3.10 shows the relation betweenmaximum axial load, Pmax and the minimum eccentricity, emin.The minimum applied moment is intended to account for eccentricities not quantified in the analysis,and the possibility that concrete strength at high loads may be reduced to less than f ′c (CSA Group 2014).The levels of Pr,max were calibrated to roughly correspond to the minimum column eccentricities used in42MrPrPr,maxeminFigure 3.10: Axial load and moment interaction diagram showing the limiting axial load resistance based onminimum applied axial load eccentricity.design prior to the introduction of the maximum axial load level approach (ACI Committee 318 1977; Neville1980). The minimum assumed column design eccentricities are 10% and 5% of the member dimension in thedirection of interest for normal and spirally reinforced members respectively.In the design of slender members with the moment magnifier method, a minimum moment to determinethe slenderness effects for members loaded without applied moments is required. The minimum moment toapply for slender members is set at Pf (15+0.03h) for CSA A23.3-14 and Pu(0.6+0.03h) for ACI 318-14. Thelevel of minimum moments are unchanged since the introduction of the moment magnifier method in designstandards. This minimum moment was established to roughly correspond with the minimum eccentricityemployed to determine the level of Pr,max. No work has been identified which rigorously establishes thevalidity of either the levels of minimum eccentricity for wall elements.3.4.3.2 Flexural StiffnessAn important aspect of the moment magnifier method is the estimate of flexural stiffness used to determinethe critical axial load. More flexible members will have more severe second order slenderness effects, thanstiffer members. Since the flexural stiffness of reinforced concrete is non-linear in material response andcracked section geometry, an equivalent linear elastic approximation is typically used in practice. Theequivalent flexural stiffness for a reinforced concrete compression member in CSA A23.3-14 and ACI 318-14is approximated as,EI =0.2EcIg + EsIst1 + βd(3.36)orEI =0.4EcIg1 + βd(3.37)where Ec is the tangent modulus of elasticity, Es is the modulus of elasticity of the reinforcing steel, Igis gross section moment of inertia, Ist is the moment of inertia of the reinforcing, and βd is a reductionmodifier for sustained member load. ACI 318-14 also allows the use of an alternative flexural stiffness basedon Khuntia and Ghosh (2004a) and Khuntia and Ghosh (2004b) of,EI =EcI1 + βd(3.38)43where I is defined as,0.35Ig ≤(0.80 + 25AstAg)(1− MuPuh− 0.5PuPo)≤ 0.875Ig (3.39)where Ast is the total cross sectional area of reinforcing, Ag is the gross cross section of the member, Mu isapplied moment, Pu is the applied axial load, h is the section thickness, and Po is the maximum axial loadresistance of the member.The estimation of flexural stiffness greatly effects the predicted behaviour of wall and column elements.An accurate estimation of flexural stiffness is perhaps the single most important parameter to be determinedin the design and analysis of slender members.3.4.3.3 Sustained Loading (Creep)Sustained loading effects, more commonly known as creep, are the increase in deformations which occur afterinitial elastic deformations. Creep is addressed in the moment magnifier method through a flexural stiffnessreduction factor based on the ratio of sustained load to total load level. The reduction factor βd as shownin equations 3.36, 3.37, and 3.38, acts to reduce the equivalent effective flexural stiffness.A ratio of sustained factored dead load to factored total load of βd = 0.6 has been suggested as reasonable.However an analysis of a 35 storey residential tower in the Metro Vancouver area has shown that somemembers may experience factored sustained load ratios as high as βd = 0.9. This high value of βd representsa typical vertical load carrying element in high rise residential constructions, where already light live loadsare reduced based on floor area and dead loads predominate the majority of the design load. High axialload ratios are especially prevalent in wall type members due to code allowable live load reductions whichare based on the total floor area which a member carries. Since walls, by virtue of their length, are able tocarry much higher total floor areas than typical columns.In practice, the effect of member sustained load on slender bearing wall elements is not well known.Factors such as level of sustained load, initial curvature due to construction imperfections, and reinforcingratio are all known to influence the sustained load effect on members in general, however no studies whichoutline the effects for slender bearing walls are known to have been completed.3.4.3.4 Buckling FactorAs part of the moment magnifier’s implementation in design codes, the critical buckling load used whichis part of Equation 3.27 is reduced to represent the uncertainty associated with understrength of a singlemember in isolation. Both ACI 318-14 and CSA A23.3-14 apply a reduction of φm = 0.75 to the criticalload for both sway and non-sway frames. The reduction factor is based on a study of approximately 9500configurations of a 305 mm x 305 mm column (Mirza 1990). Critical buckling load of wall type membershas not been explicitly studied.3.4.4 Comparison of Rational Method Strength Equation with ExperimentalResultsWith the framework established for rational strength analysis of slender compression members both generallyand as implemented in North American design codes, a comparison of the rational method to experimental44results is possible. This section applies the CSA A23.3-14 moment magnifier method for use as a comparisonto the experimental strength results presented in the opening section of this chapter.3.4.4.1 Comparison ParametersTo appropriately compare the rational strength analysis method to the experimental results as presentedearlier in this chapter, the rational method parameters must match those of the experimental programs.Since the experimental data is based on members of various cross sectional area, amount of reinforcing, andother parameters, each test specimen must be evaluated. Where unknown parameters exist, tests have eitherbeen excluded or reasonable estimates have been used in place. Appendix B.1 includes a detailed list of thetest parameters for each individual specimen.3.4.4.2 Slender Interaction Diagram GenerationFor the comparison, slender interaction diagrams are generated to compare the observed maximum axialload level of the experimental tests. Implementing the A23.3-14 procedure for slender members in performedin a MATLAB program which first generates the CSA A23.3-14 out-of-plane interaction diagram for the wallsection of interest. Next the magnified moments are generated along a fixed eccentricity over the range ofaxial load levels in the member. By calculating the intersection of the interaction failure envelope and themagnified moment trajectory, the maximum axial load for the magnified applied moment is determined. Togenerate the complete final slender interaction diagram, the previously determined maximum axial loads aretraced back to the unmagnified eccentric line along the moment axis. These intersections form the basis ofthe slender interaction diagrams for each level of member slenderness investigated. Figures 3.11, and 3.12show the slender interaction diagrams generated for a 203mm wall section with 0.15% vertical reinforcingin 2 layers. The maximum axial load for a given slenderness is recorded and plotted for comparison toexperimental results and empirical equations in the next section.3.4.4.3 Presentation of ResultsThe previous section outlined the generation of slender interaction diagrams for a typical lightly reinforcedthin wall section. This section presents a comparison of the experimental test results and the CSA A23.3-14implementation of the moment magnifier method. Figure 3.13 shows the normalized test results plottedagainst a typical maximum axial to slenderness relation. This shows that the data tends to follow the typicalshape of the estimated curve.To rigorously evaluate the CSA A23.3-14 moment magnifier procedure when applied to slender bearingwall elements in non-sway frames, a comparison of each test result to a predicted strength is performed.Figures 3.16, 3.14, and 3.18 show the computed ratios of maximum theoretical axial load capacity basedon the CSA A23.3-14 moment magnifier method versus that of each test result. Each plot represents adifferent estimation of stiffness, including Equations 3.36 and 3.37. In addition, a comparison plot whichutilizes the ACI 318-14 alternative stiffness of Equation 3.38 is presented to determine its potential utilityapplied to bearing wall elements. The figures presented include the maximum slenderness cut-off levels asdefined in CSA A23.3-14 Clauses 10.13.2 and 14.1.7.1. Some ambiguity exists within CSA A23.3-14 as towhich slenderness level governs for wall designed using the moment magnifier methods, however they areboth included for comparison.45Figure 3.11: CSA A23.3-14 chapter 10 slender interaction diagrams for t = 203mm, ρv = 0.15% in 2 layers,25mm clear cover, and EI = 0.4EcIg.Figure 3.12: CSA A23.3-14 chapter 10 slender interaction diagrams for t = 203mm, ρv = 0.15% in 2 layers,25mm clear cover, and EI = 0.2EcIg +AsIst.46Figure 3.13: Comparison of predicted axial load capacity using CSA A23.3-14 moment magnifier methodand experimentally measured maximum axial load results for bearing walls.Concrete cylinder strength has been established to play an important role in the behaviour of slenderaxially loaded members. It is therefore important to determine if any under estimation of strength using themoment magnifier method is due to concrete strength. Plots of the theoretical calculated maximum axialload to test results ratio are again presented, however are plotted versus the measured concrete cylinderstrength. Figures 3.17, 3.15, and 3.19 show the results of this analysis. Again, the three figures presentedrepresent the two CSA A23.3-14 estimations of flexural stiffness (Equations 3.36 and 3.37), as well as theACI 318-14 alternative (Equation 3.38). Specimens are identified where they exceed the minimum CSAA23.3-14 minimum slenderness limits of h/t = 25 and h/t = 30.3.4.4.4 Discussion of ResultsAs shown in Figures 3.14 and 3.15, Equation 3.31 produces estimates of strength which generally match thoseof the test results. Several predictions fall below the test results, however the margin of error is small. Onehigh slenderness test appears to produce the worst case failure, however it occurs at a level of slendernesswhich is far outside the allowable range defined by CSA A23.3-14.Figures 3.16 and 3.17 show the results when Equation 3.30 is used as the estimate for flexural stiffness.This estimation is the most conservative, and produces over estimations which appear to become increasinglypoor as slenderness is increased.Figures 3.18 and 3.19 show the analysis results when the alternative ACI 318-14 estimation of flexuralstiffness is applied (Equation 3.38). This method appears to produce results with much lower standarddeviation than the two CSA A23.3-14 methods of flexural stiffness. The lower level of variance appears tocome at the cost of reduced conservatism, as many members’ strengths are over estimated.47Figure 3.14: Ratio of experimentally recorded maximum axial load to CSA A23.3-14 moment magnifierpredicted axial load capacity vs. member slenderness h/t using flexural stiffness EI = 0.4EcIg.Figure 3.15: Ratio of experimentally recorded maximum axial load to CSA A23.3-14 moment magnifierpredicted axial load capacity vs. concrete cylinder strength f ′c using flexural stiffness EI = 0.4EcIg.48Figure 3.16: Ratio of experimentally recorded maximum axial load to CSA A23.3-14 moment magnifierpredicted axial load capacity vs. member slenderness h/t using flexural stiffness EI = 0.2EcIg + EsIst.Figure 3.17: Ratio of experimentally recorded maximum axial load to CSA A23.3-14 moment magnifierpredicted axial load capacity vs. concrete cylinder strength f ′c using flexural stiffness EI = 0.2EcIg +EsIst.49Figure 3.18: Ratio of experimentally recorded maximum axial load to CSA A23.3-14 moment magnifierpredicted axial load capacity vs. member slenderness h/t using flexural stiffness from Equation 3.38.Figure 3.19: Ratio of experimentally recorded maximum axial load to CSA A23.3-14 moment magnifierpredicted axial load capacity vs. concrete cylinder strength f ′c using flexural stiffness from Equation 3.3850The numerical data presented in Figures 3.14 to 3.19 are included in Appendix B.2. A summary of thestatistical properties of the analysis completed is included in Table 3.3.Table 3.3: Summary of statistical analysis of moment magnifier method estimation compared with testresults for varying estimates of flexural stiffness (n=111).FlexuralStiffnessEstimationMin. Mean Max.Std.Dev.NumberFailed% FailedMeanFailedStd.Dev.FailedEI = 0.4EcIg 0.84 1.72 4.29 0.53 4 3.6% 0.92 0.05EI = 0.2EcIg+EsIst1.07 2.72 7.59 1.13 0 0.0% N/A N/AEquation 3.38 0.58 1.23 2.84 0.32 25 22.5% 0.90 0.09From this analysis, it appears that Equation 3.36 produces overly conservative estimates of maximumaxial load capacity of lightly reinforced, slender bearing wall elements. On average it produced estimateswhich were over 2.5 times lower than observed and estimates varied considerably, with the most conservativeestimate producing an estimate of strength over 7.5 times lower than observed. For a data set this large, itis expected to observe some under estimates of capacity due to random variations.Equation 3.37 produced good results with only 4 failures out of 111 specimens observed, resulting in afailure rate of 3.6%. Of the under estimates of strength observed, the worst case was an over estimation ofaxial load capacity of appropriately 20%. While this represents a relatively severe under estimation, it isof note that the this specimen was extremely slender at h/t = 40 and was made of a fairly high strengthconcrete at f ′c ≈ 60MPa. The outstanding 3 failed specimens were of a much more modest level of underestimated strength in the range of 5-6%, well within the traditional material strength reductions applied byφc and φs in CSA A23.3-14.The final estimation of flexural stiffness of Equation 3.38, which is allowed in ACI 318-14, produced25 failures which made up 22.5% of the test specimens. While at first glance this number appears veryhigh, it is important to remember that these are nominal estimates of strength and had load and materialresistance factors been applied, the number of over estimates of axial load capacity would have been greatlyreduced. Also of importance is that the 2 worst case over estimates of strength (0.58 and 0.69) wereobserved in members with high slenderness (h/t of 40 and 30 respectively). While one the latter specimenwould perhaps be allowable in CSA A23.3-14, it still would have represented a marginal failure had loadand material strength reductions been imposed. The worst case member would not have been allowed byCSA A23.3-14 at its slenderness ratio without further second-order analysis methods implemented. Thisalternative estimate of flexural stiffness appears to provide the most accurate estimate for use with themoment magnifier, however comes at the cost of reduced conservatism. Given that this method producessuch a small level of variance in its results, it may represent an economical design method for bearing wallelements and warrants consideration for adoption by CSA A23.3-14. It’s applicability for use in column typemembers has been established (Khuntia and Ghosh 2004a; Khuntia and Ghosh 2004b), and it is already inuse for design in ACI 318-14.In summary, the CSA A23.3-14 moment magnifier method produces estimates of maximum axial loadcapacity which appropriately estimate the lower bound of experimental test results from lightly reinforced,slender, eccentrically loaded bearing wall elements. Flexural stiffness estimated using equation 3.30 tends toproduce more conservative estimates than 3.31, which is as expected due to the small influence on stiffness51that the reinforcing provides for lightly reinforced elements. The alternative ACI 318-14 estimate of flexuralstiffness has also been shown to produce the most accurate estimates of ultimate axial load capacity forbearing wall type elements, however comes at the cost of conservatism. Future modifications to Equation3.38 may result in estimates of bearing wall axial strength which maximize the economy of elements of thistype.3.5 Comparison of Empirical and Rational MethodsThis section presents a comparison of the CSA A23.3-14 rational and empirical design methods for bearingwall design with test results as presented in previous sections. This is followed by a comparison of the designmethods based on varying parameters used in the CSA A23.3-14 implemented moment magnifier procedure.The wall member parameters for comparison purposes are a thickness of 203 mm, length of 1218 mm,concrete cylinder strength of 30 MPa, and a vertical reinforcing ratio of 0.15%, and two layers of reinforcing.3.5.1 Comparison of Empirical and Rational MethodsTo compare the empirical and moment magnifier methods to past test results, the parameters of the rationalmethod are set to match those assumed in the empirical equation and test set up. As was noted in earliersections, the empirical equation assumes the axial load to be applied within the middle third of the section,resulting in a maximum eccentricity of e = t/6. In addition, the CSA A23.3-14 moment magnifier methodapplies a reduction to critical axial load Pc equal to φm = 0.75. To appropriately compare the two designmethods, the critical axial load reduction factor is set as φm = 1.0 since the empirical method does notinclude reductions for individual member effects for critical load. Also any reductions to flexural stiffness inthe moment magnifier method due to sustained axial load are neglected. These parameters allow the twomethods and the test results to be compared with little difference to the assumed parameters in each case.Figure 3.20 shows the plot of each design method when compared with the test results.It is evident that the rational and empirical methods produce similar estimations of ultimate maximumaxial load for approximately h/t < 24 when Equation 3.31 is used as the estimation for flexural stiffness.The flexural stiffness estimation in Equation 3.31 has produced what seems to be a more accurate responsewhen compared with the test results and the empirical equation over much of the domain. This result wasexpected as has been discussed, and Equation 3.31 is suggested for use in lightly reinforced bearing walls. Incontrast, applying Equation 3.30 to the moment magnifier method results in a lower estimation of flexuralstiffness, and by extension a lower ultimate axial load once second order effects become a driving factor inthe response of the member.At high slenderness levels, the moment magnifier method continues to predict axial load carrying capacity,whereas the empirical method by virtue of it’s formulation reduces axial load capacity to zero. This behaviouris important to make note of, however in practice the use of such slender members is atypical. For exampleCSA A23.3-14 limits the use of the moment magnifier method to klu/r ≤ 100, however wall thickness ofbearing walls is limited to hu/25. So walls are restricted from exceeding an h/t of 25. Interestingly, therestriction of hu/25 does not take effective length into consideration, which may have a beneficial effect onthe stability of the wall element.The empirical method and the rational method using Equation 3.31 as the estimate for flexural stiffnessprovide good agreement with the lower bound of the test results over the domain of interest. The rational52Figure 3.20: Comparison of experimentally observed axial load capacities and CSA A23.3-14 Chapter 10design axial strength procedures.method with Equation 3.30 applied as the estimate of flexural stiffness is overly conservative when secondorder effects are considered. This is as discussed in the previous section.It is important to note that the rational moment magnifier method relies on several assumptions tobe made and the adjustment of the parameters results in changes to the overall response of the member.Since the moment magnifier method requires that sustained load effects and full member stability effects beconsidered. By varying these parameters individually, the effect of each can be quantified.First the baseline case which uses the same parameters as presented for the comparison with the testresults is presented in uppermost plot of Figure 3.21. The plot presents the ratio of rational method strengthestimation to the empirical equation with the empirical method providing the datum. The empirical methodand the rational method with Equation 3.31 closely match, with the rational method with Equation 3.30producing an estimate of strength which is 34.4% lower than the empirical method in the worst case.Next the effect of sustained load and the buckling member resistance factor on the rational method isexplored. A typical residential building may have a typical ratio of factored sustained axial dead load tofactored total load of βd = 0.6 (MacGregor and Bartlett 2000), however as previously discussed may be ashigh as βd = 0.9 for modern residential construction. The member resistance factor reduces the critical loadwhich increases the moment magnifier effect in general. CSA A23.3-14 specifies the member resistance factorto be φm = 0.75. The sustained load and buckling resistance factor have the effect of reducing the flexuralstiffness of the member. The central plot of Figure 3.21 shows the effect of sustained load on the rationalmethod axial load capacity.The combination of sustained load and buckling resistance factor serves to reduce the maximum axialload capacity using Equation 3.17 and 3.18 by up to approximately 65.6% and 42.7% respectively. Thisshows that the inclusion of sustained load and the buckling resistance factor in the rational method has a53Figure 3.21: Comparison of CSA A23.3-14 Chapter 10 moment magnifier method and Chapter 14 Eqn. 14-1empirical predicted maximum axial load capacity.54significant effect on the maximum load estimate which is not captured by the empirical method.Up to this point comparisons have been made which represent the two CSA A23.3-14 methods beingapplied to a situation where the minimum applied eccentricity is set as equal at e = t/6, however the minimumeccentricity for each method when applied in CSA A23.3-14 is not equal. The Chapter 14 empirical bearingwall method employs an assumed maximum eccentricity of t/6, the Chapter 10 rational method requiresthat slender members have an end moment of 15mm + 0.03t be applied. Since when using the rationalmethod for simple bearing wall members it would be illogical to apply a minimum eccentricity greater thanthe minimum specified if it was assumed that none existed, a comparison of the two methods at each of theirrespective minimum eccentricities is reasonable.Since the effects of sustained load βd, and member resistance factor φm have a significant effect on themoment magnifier method in CSA A23.3-14, any comparison made should include the use of both parameters.The lower plot in Figure 3.21 shows the comparison including the effects of sustained load βd = 0.6 andmember resistance factor φm = 0.75.When sustained load effects are considered (βd = 0.6), and the buckling member resistance factor (φm =0.75) is applied, the disparity in maximum axial load estimations is apparent. As shown in the bottom ofFigure 3.21 the moment magnifier produces maximum axial loads up to 30.9% lower when Equation 3.31 isapplied, and 59.8% lower when Equation 3.30 is applied.From this comparison it is clear that a significant disparity between the rational moment magnifiermethod and the empirical equation method is applied in practice in CSA A23.3-14 for bearing walls. Adesigner who elects to use one method over the other, may be restricted in the maximum axial load capacitywhich is allowed. As shown in the previous section, the moment magnifier method is assumed to producegood estimates of ultimate axial load when compared with test results, and the empirical method producinghigher estimates would be unsafe. Since the empirical design method has been developed based on shortterm testing, its application for long term load applications is dubious, and certainly no apparent effort hasbeen made for the inclusion of member stability effects.One caveat to the results shown in Figure 3.21 is that in CSA A23.3-14, untied compression memberswith thicknesses less than 300mm are restricted to maximum axial ratios less than 0.75Pro, and for the caseof the 203 mm (8 inch) member used in this example, the axial load is limited to 0.61Pro. The inclusion ofthe maximum axial load restriction restricts any gain in apparent strength from the rational method whencompared with the empirical equation to a negligible amount for low slenderness levels. This result is curioussince the assumed eccentricity is different in each case, however the maximum axial load is the roughly equal.This represents a significant inconsistency between the application of the two design methods.Clearly the use of the empirical method of analysis is sufficient under the correct circumstances, howevergaps in CSA A23.3-14 present the opportunity for misuse. In the worst case of a bearing wall element carryingsubstantial long term dead loads, the disparity between the two methods is severe, with the empirical methodseverely over estimating the axial load carrying capacity of the member.3.6 Simplified Rational Method for Strength of LightlyReinforced Bearing WallsIn previous sections the rational method for bearing wall design was presented and contrasted with theempirical solution based on CSA A23.3-14. Applying the rational moment magnifier method requires devel-opment of member specific moment-interaction diagrams, and amplifications to applied moments to arrive55at an allowable maximum axial load capacity. This section presents a simplified special case closed formsolution of the moment magnifier method applied to lightly reinforced bearing wall members. The inputsto the method are outlined and compared with solutions based on the fully implemented CSA A23.3-14moment magnifier and empirical methods.3.6.1 Derivation of CSA A23.3 Moment Magnifier Method for anUnreinforced Rectangular Cross SectionAs previously discussed, the moment magnifier method implemented in CSA A23.3-14 serves to determinethe intersection of a non linear loading trajectory (the magnified moment), and an axial-moment resistanceinteraction diagram for a specific member cross section and reinforcing configuration. One major downsideof the method is the complexity of determining a solution, and many steps are required. For simple crosssections with light reinforcing, this process represents a significant design task. To improve on this a simplifiedmethod such as the empirical method can be used, however as shown in the previous section the empiricalequation is inaccurate in determining the maximum axial load capacity of a reinforced concrete member. Analternative expression is to derived in the following.As mentioned, implementing the moment magnifier method depends on the axial-moment interactiondiagram of the member cross section to be determined. Since for many applications, reinforced concretebearing walls contain design standard minimum vertical reinforcing, these walls are lightly reinforced. In thecase of CSA A23.3-14, ρv,min = 0.0015Ag. For lightly reinforced cross sections the axial-moment resistanceinteraction diagram approaches that of an unreinforced cross section. That is a section which contains noreinforcing steel and resists applied axial and bending moments entirely through the internal compressionstress block. The axial-moment resistance interaction diagram for an unreinforced rectangular section, usingan equivalent rectangular internal stress block can be shown to be,Mr =Prt2(1− Prα1f ′ctlw)(3.40)To determine the validity of applying this simplified interaction diagram to the solution of the problem,a comparison to sections with varying amounts of reinforcing is shown in Figure 3.22 for two layers ofreinforcing, and Figure 3.23 for a single layer at the center of the section.As shown in Figures 3.22 and 3.23, Equation 3.40 (Shown in red) produces close estimates for lightlyreinforced sections. This is especially true for the single layer reinforced section, which reduces to the un-reinforced section moment capacity at P/(φcf′cAg) ≈ 0.5. The double layer reinforced section producesincreasingly poor estimates of maximum moment capacity as the amount of reinforcing in the section in-creases. At the CSA A23.3-14 specified applied eccentricities of t/6 and 15+0.03t, the difference in maximumaxial and moment resistances is minimal, especially for low levels of reinforcing. This simple analysis showsthat for low levels of reinforcing, Equation 3.40 is appropriate to estimate the axial-moment interactiondiagram for a slender concrete wall with a rectangular cross section.To determine a simple expression for maximum axial load capacity for a slender bearing wall memberbased on the CSA A23.3-14 moment magnifier method, the simplified axial-moment interaction of Equation3.40 is equated with the magnified moment load equation,Mf =MsCm1− PfφmPcr(3.41)56Figure 3.22: Axial load and moment interaction diagrams of lightly reinforced bearing walls for varying ρvplaced in two layers.Figure 3.23: Axial load and moment interaction diagrams of lightly reinforced bearing walls for varying ρvplaced in one centered layer.57where the inputs are as described in Section 3.4.2.By setting Equations 3.40 and 3.41 equal as Mf = Mr and solving for axial load P , the result is a quarticequation of which one root is the desired solution. By inspection a trivial solution exists at the origin, whichreduces the solution of the remaining roots to a simple quadratic. Solving for the roots and simplifyingresults in the following desired root of,Pr =A2−√(A2)2−B(1− 2et)(3.42)where,A = φmPcr + α1f′cAgB = φmPcrα1f′cAgThis solution represents the maximum axial load capacity Pr of an unreinforced concrete bearing wall,based on a CSA A23.3-14 Chapter 10 moment magnifier load amplification to account for second order effects.Next a comparison of the closed form solution to the full CSA A23.3-14 chapter 10 moment magnifier andChapter 14 empirical methods is presented.3.6.2 Comparison of Closed Form Solution and Other MethodsThe closed from solution of the A23.3-14 moment magnifier method as presented in Equation 3.42 is used tosolve for a maximum allowable axial load level for a given slenderness h/t, and other A23.3-14 modificationparameters such as φm and βd. For comparison, the closed form method (CSA A23.3-14 Chapter 10 momentmagnifier method with ρv = 0.00) is plotted alongside the results of a CSA A23.3-14 Chapter 10 momentmagnifier method with increasing vertical reinforcing ratios, and also the Chapter 14 Equation 14-1 empiricalcurve. To represent typical practice, the comparison uses the following inputs:t = 203 mm Wall Thicknesslw = 1000 mm Wall Lengthf ′c = 30 MPa Specified Compressive Strength of Concretefy = 400 MPa Specified Yield Strength of ReinforcementEI = 0.4EcIg Flexural Stiffnessφm = 0.75 Member Resistance Factorβd = 0.00 Creep Factore = 15 + 0.03t and t/6 Applied EccentricityThe results of the comparison for applied eccentricity of e = 15 + 0.03t (A23.3 Chapter 10) are shown inFigures 3.24 and 3.25, and for eccentricity of e = t/6(A23.3 Chapter 14) in Figures 3.26 and 3.27.Figure 3.24 shows that when Chapter 10 minimum eccentricity is applied to a section with two layers ofreinforcement at each face, the unreinforced section only produces significantly higher allowable axial loadsfor reinforcing ratios of ρv ≈ 0.0050. As slenderness is increased, the effect of any reinforcing is significantlydiminished as the maximum axial loads for the reinforced sections approaches the unreinforced case. TheA23.3-14 Chapter 14 Equation 14-1 (Equation 3.12) is shown to compare how the different applied minimumeccentricities affects the outcome of the analysis. In this case, the Chapter 14 empirical method producesroughly the same maximum axial load as the Chapter 10 solutions in the range of 15 < h/t < 25, however58Figure 3.24: Maximum axial load vs. slenderness for lightly reinforced bearing walls using CSA A23.3-14moment magnifier equations for varying ρv in two layers with eccentricity e = 15 + 0.03t and empiricalequation 14-1.Figure 3.25: Maximum axial load vs. slenderness for lightly reinforced bearing walls using CSA A23.3-14moment magnifier equations for varying ρv in one centered layer with eccentricity e = 15+0.03t and empiricalequation 14-1.59the Chapter 14 equation assumes a higher level of eccentricity of e = t/6. This result does not agree withexperimental results and is discussed in previous sections of this thesis.Figure 3.25 again shows the results as presented previously, however for a section with a single layer ofreinforcing centred in the wall. The result is similar to the double layer reinforcing results, however theeffect of reinforcing is diminished even further, especially for slenderness values in the 15 to 20 range. Thisis explained by the neutral bending axis of the wall and the reinforcing occurring close to the same depthwithin the cross section and the strains induced in the reinforcing being small as a result. This analysisshows that the configuration of the reinforcing is important to the behaviour of the member, and that acentrally placed wall reinforcing provides little or no added resistance to a member of a certain slenderness.Figures 3.26 and 3.27 show the maximum axial load to slenderness curves for walls subjected to aminimum eccentricity of e = t/6 as is assumed for the empirical method of design. As can been seen for bothdouble and single layer reinforcing, the effect of added minimum reinforcing to the cross section provide aminimal strength increase. In the case of a single layer with height to thickness of approximately 10 to 20,the reinforcing has essentially no effect. Again this is due to the location of neutral axis at the failure statecoinciding closely to the location of the reinforcing.Figures 3.24 to 3.27 validate that the simplified moment magnifier method derived in Equation 3.42is appropriate for use in lightly reinforced members and provides a significant reduction to the amount ofeffort input for the accurate design of a slender reinforced concrete wall member. The method also has theadvantage of the inclusion of sustained loading and individual member buckling reductions.3.7 Conclusion3.7.1 Empirical Bearing Wall Design in CSA A23.3-143.7.1.1 General CommentsEmpirical bearing wall design by the CSA A23.3-14 method has been established to be a generally usefultool for estimating short term, monotonic axial load capacity of lightly reinforced, and modestly slender wallmembers (ρv ≈ 0.002 and h/t ≤ 25). This has been establish through comparison of analysis estimates tomultiple past experimental studies of wall elements loaded within the middle third of their cross section.The term “empirical” has been shown by Bartlett, Loov, and Allen (2002) to somewhat of a misnomer.While the empirical design equation has been established based on the results of experimental studies, thebehaviour is not simply a best fit to available data. The method has is rooted in well established principles ofstructural mechanics. While this feature adds to its credibility for use as a design and analysis tool, it doesnot come without its shortcomings which include the absence of sustained load effects, high strength concretestrength reductions, no member buckling factor applied, and little experimental validation of effective lengthfactors.3.7.1.2 Sustained Load EffectsPerhaps the most significant drawback of the CSA A23.3-14 empirical bearing wall design method, is theomission of sustained load effects. The well known and widely studied effects which sustained loading hason reinforced concrete members is an essential consideration in the design of any concrete structure whichis intended to perform suitably beyond a short period of time. Over time as sustained loads are imparted,60Figure 3.26: Maximum axial load vs. slenderness for lightly reinforced bearing walls using CSA A23.3-14moment magnifier equations for varying ρv in two layers with eccentricity e = t/6 and empirical equation14-1.Figure 3.27: Maximum axial load vs. slenderness for lightly reinforced bearing walls using CSA A23.3-14moment magnifier equations for varying ρv in one centered layer with eccentricity e = t/6 and empiricalequation 14-1.61permanent deformations increase which have a significant effect on the response of a reinforced concretemember.By comparing the empirical design method with another well established method of analysis, the momentmagnifier method, it was shown that the empirical method may produce estimations of axial load which canbe in the range of 150% to 250% higher than those estimated by the moment magnifier method under thesame conditions. This is shown in Figure 3.20.The effect of sustained loads is of particular concern in members with high ratios of dead to total loadsβd. Mid and high rise residential buildings may have sustained load ratios βd as high as 0.90 in many cases.This is due in part to the use of live load reductions based on floor area carried by a member, and also therelatively low design live loads of residential structures when compared with structure self weight.As such, the use of the empirical design method in CSA A23.3-14 Equation 14-1 (Equation 3.12 in thisthesis), is not recommended for use in elements which are expected to be affected in any way by sustainedloads. The disparity between estimated design axial loads when using the two design methods in CSAA23.3-14 is identified as an area of future improvement, as it creates the opportunity for unsafe designs, andprovides little incentive for a designer to apply a more rigorous design solution, as the moment magnifierwould ultimately result in a lower design capacity under most conditions.3.7.1.3 Higher Strength ConcreteAs was shown in Figure 3.7, as concrete cylinder strengths increase above f ′c ≈ 40MPa, a reduction innormalized axial load capacity occurs. This effect has been documented in previous studies (Fragomeni1995; Doh and Fragomeni 2005), and is unaccounted for in CSA A23.3-14. In the worst case CSA A23.3-14allowable scenario (h/t ≤ 25), a member’s capacity was over estimated by approximately 20%. Given thesmall sample size of members with f ′c > 40MPa (n = 17), the failure rate of 35% for members in this categoryis of particular concern. Given that the CSA A23.3-14 does not account for this reduction in estimated axialload capacity, its use for concrete strength of f ′c > 40MPa is not recommended.3.7.1.4 Member Buckling FactorThe moment magnifier method, which has been established in thesis to provide acceptable results of axialload estimation, employs the use of the critical buckling load, or Euler buckling load in is estimate of second-order effects. There is an inherent uncertainty in the level of critical buckling load due to the estimatesof flexural stiffness and concrete strength. To account for the potential of a reduced critical axial load, amember buckling factor φm = 0.75 is employed. This factor has the effect of reducing the axial load capacityof a slender member analysed using the moment magnifier method. The empirical method of design in CSAA23.3-14 does not have an equivalent analogue for which full member effects are quantified.It was shown in the uppermost plot of Figure 3.20 that the empirical method closely approximates themoment magnifier when a critical buckling factor of φm = 1.0 is applied. Since this would not be the case inpractice, this results in yet another avenue of strength over estimation when the empirical method is applied.For this reason it is suggested that the empirical method is recalibrated to better fit with the results of themoment magnifier method when a member buckling factor of φm = 0.75 is applied.623.7.1.5 Effective Length FactorNo studies have been found which investigate the influence of effective length K on the capacity of axiallyloaded bearing wall elements. Elements with larger cross sectional dimensions such as typical columns areexpected to provide significant end restraint when cast in place, resulting in a reduced effective length. Sincebearing wall elements have by definition an elongated profile, their ability to restrict end rotations about theweak axis may be impacted. One aggravating factor to this may be cycling of loads through seismic actions.In addition, no tests have been identified which validate the effective length factors applied to bearing wallelements undergoing weak axis bending.In practice, Equation 3.12 is applied in many cases with K = 0.8, assuming the ends are fixed againstrotations, however the level of restraint provided by connection of a thin wall element to a thin slab elementis questionable. Since the reduction of effective length represents a significant increase in axial load capacityin some cases, caution is advised when applying effective length factors when thin elements are connected.3.7.1.6 Use of the Empirical Method in Moderate Seismic AssessmentsGiven the inherent uncertainty in the seismic response estimation of a concrete structure when comparedwith estimates of sectional and member strength, the use of Equation 3.12 in estimating the axial loadcapacity of a bearing wall element warrants some consideration. Even though the empirical equation hasbeen shown to provide poor estimations of axial load capacity in some cases, when short term uniaxial loadsare applied it can produce reasonable estimates of member capacity when small out-of-plane eccentricitiesare present.When tasked with performing a seismic assessment of an existing structure, often with little or uncertaininformation, and few resources provided, quick an simple estimates of member capacity are in high demand.While the empirical method as it is employed for design of new members in CSA A23.3-14 is less than an idealtool, it does have the advantage of being both somewhat accurate for short term loading, and is extremelysimple and fast to employ. As a tool for mid level seismic estimations, where uncertainty is inherently highand economic resources are low, the empirical method can provide valuable insight into the behaviour ofbearing wall elements under uniaxial seismically induced loading.3.7.2 Rational Method of Design in CSA A23.3-143.7.2.1 General CommentsThe moment magnifier method of slender compression member design in CSA A23.3-14 provides a rationalmethod to estimate second-order effects without the use advanced analysis tools. It also has the advantage ofbeing able to produce slender interaction diagrams such as shown in Figures 3.11 and 3.12. These diagramsprovide a useful tool for preliminary and detailed designs when time or budget constraints apply.Estimations of axial load capacity using the moment magnifier method have been shown to generallyproduce good estimations of axial load capacity, however depending on the assumptions made may produceoverly conservative results. The method was validated against 111 test specimens or various slenderness,concrete strength, and levels of reinforcing. The results of this analysis are as shown in Figure 3.14 to 3.19.633.7.2.2 Effective Flexural StiffnessOne of the main factors affecting the estimation of axially loaded members using the moment magnifiermethod is the estimate of flexural stiffness used in the analysis. Deflections along the length of the memberare determined according to an equivalent flexural stiffness which encapsulates the varied response of theactual member.Three estimate of flexural stiffness were investigated Equations 3.36 and 3.37 which are employed byCSA A23.3-14, and Equation 3.38 which is an alternative allowed by ACI 318-14.Equation 3.36 was shown to provide a conservative estimate of strength, however was certainly overlyconservative in many cases, especially at higher slenderness levels. Equation 3.37 produced good resultsacross the range of slenderness, however some over estimates of axial load capacity were observed, yet theworst case was for an extremely slender member h/t = 40 which would not qualify to be designed using thismethod under provisions of CSA A23.3-14. The final alternative Equation 3.38 produced the best resultsbased on overall member behaviour, however produced somewhat unconservative estimates in some cases,particularly for higher slenderness levels.As a result of this analysis, Equation 3.36 is not recommended for use for lightly reinforced slender bearingwall elements, as it produces an overly conservative estimate of capacity which results in an unwarrantedand inefficient use of resources. Equation 3.37 is recommended for use in typical members of this type, andhas been shown to produce reasonable estimates of axial load capacity without producing overly conservativeresults. The alternative Equation 3.38 does provide an excellent overall prediction of member behaviour,with the lowest observed variance of the the three methods. However its use is not recommended at thistime, as it is uncertain if the level of failure produces an acceptable level of risk when applied to a muchlarger number of members.3.7.2.3 Sustained Load EffectsAs was noted for empirical method of axial load estimation, sustained load effects result in significant effectson the behaviour of slender bearing wall elements. Levels of sustained load to total load in residential highrise construction can be upwards of 90% and the effects are not insignificant. The moment magnifier methodapproach reduces the effective flexural stiffness to account for sustained loading effects, however the lack ofexperimental data to confirm this behaviour creates some uncertainty. Since few or no axial load failures ofmembers under sustained loads have been observed in practice, it is cautiously assumed that the method isat least satisfactory, if not conservative. More investigation into this phenomenon may produce more refinedmethods of sustained load effect quantification, however for now it would seem there is little risk in it’sapplication within the CSA A23.3-14 moment magnifier method.3.7.2.4 Maximum SlendernessOne curious result of this research is the ambiguity between the CSA A23.3-14 Clause 10.13.2 and Clause14.1.7.1 maximum slenderness limits. Chapter 14 indicates a maximum slenderness of h/t > 25, howeverdirects the user to the provisions of Chapter 10 for the detailed design of the element, which specifies amaximum slenderness ratio of 100, or h/t = 30 for a rectangular cross sections. No guidance is provided as towhich provision governs, and a prudent user would perhaps err on the side of conservatism. However, coherentinterpretation of standards and codes is important for uniform application of the provisions within. Betterguidance on slenderness limits for members and when those limits apply may reduce user misapplication.643.7.3 Simplified Rational Member Design EquationA simplified equation to determine the maximum axial load capacity of a lightly reinforced concrete wallmember derived from the axial-moment interaction of an unreinforced member, and the magnified momentusing the moment magnifier method is presented as Equation 3.42. This new equation provides a significantlyimproved method of member design based on a well established design approach. It was shown that theequation is valid for lightly reinforced members with little or no loss to the maximum axial load capacity, sincelight reinforcing provides little strength gain to the member. An addition to the reduced effort and increasedaccuracy of Equation 3.42 when compared to the full moment magnifier method and the empirical equation,no loss of sustained loading effect or reduced member buckling capacity as implemented in CSA A23.3-14 ispresent. The newly derived equation represents a significant improvement to the empirical equation whichthis thesis has confirmed can produce unconservative and unreliable estimates of maximum axial load.65Chapter 4Thin Lightly-Reinforced Shear Walls4.1 IntroductionAs discussed earlier in this thesis, thin and lightly-reinforced concrete walls have, in some cases, exhibitedpoor behaviour when subject to in-plane lateral displacement demands due to seismic forces. A selectionof these elements have shown to fail in a brittle compression dominated mode, even under relatively lowaxial loads. In some cases the compression failures modes are accompanied by out of plane buckling of thecompression zone, buckling of reinforcing, fracture of reinforcing, or a combination of all of these modes. Tobetter understand the behaviour of thin and lightly-reinforced shear walls, this chapter presents a study onthe effect of induced in-plane lateral displacements on flexural dominated wall elements. To this aim, thischapter is divided into three main sections.The first section presents the results and analysis on a sample of test walls extracted from a databasegenerously provided by researchers at the University of California, Los Angeles. These test results are used toidentify tests on thin and lightly reinforced wall elements and provide insight to the factors which affect theirdrift capacity. A small study of several typically significant factors affecting the response of walls includingaxial load, shear span, shear stress, slenderness, and depth of compression is presented. The results of thedatabase analysis and the study of their parameters affecting drift capacity is then applied to an existingempirical model to predict drift capacity of walls generally. Finally, a new empirical model of drift capacityis presented to predict the drift capacity of thin and lightly-reinforced shear walls based on some basicparameters.Using the results of the ULCA database analysis, several tests are identified to have particularly low levelsof overall drift capacity. Included in this group of tests are a series of walls tested at E´cole PolytechniqueFe´de´rale de Lausanne (EPFL). The researchers involved in this series of tests, which focuses on the outof-plane behaviour of thin walls, did an excellent job documenting and recording their data for public use.Using the results of these well instrumented tests, a study of the walls is performed to assess many elementsof their response to lateral loading. The study includes analysis of the load-deformation response, failuremodes, vertical strains, shear strains, and curvatures of the test walls.After presenting and discussing the results of the EPFL tests in the context of this study, non linearfinite element (NLFE) models of a selection of the test members is performed. The results of test walls isused to validate and calibrate the NLFE models. The NLFE models are then used in the development ofplastic hinge and shear strain models for thin and lightly-reinforced walls. A plastic hinge models based ondistributions linearly varying inelastic curvatures is validated. In addition a model used to estimate shearstrains based on average vertical strains is validated for use in elements of the type used in the EPFL tests.This chapter serves as a resource for the identification and discussion of many of the characteristics whichhave been proposed to affect thin and lightly-reinforced shear walls subject to seismic in-plane lateral loads.664.2 UCLA RCWalls Database4.2.1 BackgroundTest programs for many reinforced concrete elements have been completed at a wide variety of researchinstitutions, with their results presented in the literature with varying degrees of completeness, often withimportant information missing. These published test results form the body of knowledge which is thebasis for understanding the intricacies of reinforced concrete behaviour. Traditionally, the results of testprograms are held in periodicals with comparisons between various studies completed as the requirements ofan individual researcher’s needs dictate. A distinct downside of single study results being held in isolation,is the dissemination of test outcomes to other researchers is very limited.In an attempt further understanding of reinforced concrete structural walls subjected to lateral defor-mations, researchers at UCLA have developed the ULCA-RCWalls Database (Abdullah and Wallace 2018;Abdullah and Wallace 2019). The database contains the detailed results of over 1000 wall specimens, andis able to be searched and filtered to easily gain access to the results of many tests previously only availablethrough meticulous scouring of numerous sources.On the topic of thin and lightly reinforced shear walls, many studies have been completed with a varietyof research outcomes in mind. The ultimate goal of this study is to develop models to accurately assess thedeformation capacity of this class of wall, along with various other ancillary conclusions along the way. Withthis aim in mind, it is necessary to identify which test results can be used to provide productive insight. TheULCA researchers have graciously allowed the use of the RCWalls Database to analyse the results of manymore test results than would have been possible by the traditional method. The results of this databaseanalysis are used to draw conclusions on the most important parameters affecting deformation capacity ofthin and lightly reinforced shear walls. In addition, the results of the database analysis are used to selectcandidate test specimens identified as important for further study.4.2.2 Database Query ParametersFor this study, walls will be filtered from the UCLA Database to create a subset of test specimen resultswhich have properties typical of “thin” members. With this aim in mind, the subset of data to be used inthis analysis was filtered according to the following criteria:1. Main element thickness t < 200 mm2. Axial load ratio Pf ′cAg> 0.033. Cross section aspect ratio lw/t > 64. Flexural failure modes only5. Quasi-static, cyclic loading programs6. No boundary zone confining reinforcementFrom the criteria set above, 42 individual test specimens were identified in the database. The detaileddatabase results as extracted are shown in Appendix C.1.The database results contain three main data groups. The first is the test specimens and setup informa-tion, including the specimen geometry, material properties, loading protocol, and reinforcing configuration.67The second is the results of the tests as described in the published materials, for example the backbonecurves, damage details, and failure modes. The final grouping of data is analytical results which are derivedfrom the test results, such as the depth of the neutral axis, flexural strength, and curvature.The parameters of interest in this study have been identified as:• Axial load ratio, Pf ′cAg• Shear span ratio, MV lw• Shear stress ratio, VmaxAv√f ′c• Cross section slenderness, lwt• Compression zone slenderness, ct• Test panel aspect ratio, hwlw• Top drift capacity, ∆maxhwFor this study, the definition of top drift capacity is based on the values presented in the UCLA Database.The database defines two cases of maximum observed top displacement, ∆max. The first is for specimenswhich have been tested to the full height of their shear span, M/V lw, which is simply the displacement atthe top of the test specimen. The second definition applies to test specimens which are set up in a way tohave a shear height which is greater than the test specimen height. This effect is achieved by applying anadditional top moment to the test specimen. In this case the UCLA Database makes a prediction of the topdrift based on the recorded drift at the top of the specimen, increased to reflect the member rotation at thetop of the specimen and the additional flexural displacements present above the top, up to the theoreticalshear height. The ULCA Database curators use the recorded material and geometric properties to make anestimate of the flexural stiffness of the test specimen.Figure 4.1 shows the histograms of number of specimens with respect to each parameter. Typically, thereis a good number of samples representing a wide range of parameter results. Some notable exceptions are:axial load ratios in the 15% to 25% range, shear stress ratios less than 0.10 (MPa), cross section slendernessbetween 12.5 and 15.0, and compression zone slenderness of greater than 4. The samples sizes available areexpected to provide good estimates to determine the general trend of parameters which influence the driftcapacity of the test specimens.The range of axial loads as shown in Figure 4.2 represent a wide range of potential load levels. The gapin data between 15% and 25% isn’t expected to lower the quality of results, as the lower number of samplesis bounded by many higher and lower results. Since increased axial load leads to compression dominatedfailures for members with combined axial and moment loads, it is expected that this parameter serves as agood indicator of member drift capacity.The range of shear span ratios represented in the data set is shown in Figure 4.3. Members with lowershear span ratios begin to exhibit begin to exhibit shear dominated responses through the formation ofcompression struts, and tension ties. Members with higher shear span ratios are expected to behave in aflexural dominated mode. The members with responses defined by flexure are expected to have higher driftcapacities owing to the larger deformations typically possible through flexural deformations.68(a) Axial Load Ratio, Puf ′cAg(b) Shear Span Ratio, MV lw(c) Shear Stress Ratio, VmaxAv√f ′c[MPa] (d) Cross Section Slenderness, lwt(e) Compression Zone Slenderness, ct (f) Top Drift Capacity,∆tophwFigure 4.1: Histograms of selective parameters and results from UCLA wall database result query.694.2.3 Detailed Analysis of Selective Test ResultsFigures 4.2 to 4.7 present the results of the UCLA RCWalls Database query for the various parameters ofinterest. All results are plotted with respect to the maximum drift based on the actual or theoretical shearheight as discussed in the previous section.Figure 4.2 presents the influence of axial load on the selected test results. It appears that the generaltrend is for the axial load to have little effect on global drift capacity, however the maximum attainabledrift may be reduced as axial loads increase. The maximum levels recorded clearly decline as axial loads areincreased, however the presence of many lower recorded drifts tends to weaken the effect of axial load forthe entire dataset.0%1%2%3%0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4Maximum Drift Capacity at Shear HeightAxial Load Ratio, P/(f'cAg)Figure 4.2: UCLA database maximum drift capacity at shear height vs. axial load ratio P/(f ′cAg).Figure 4.3 presents the influence of shear span ratio for the database query. As shear span is increased,there is a strong upwards trend in displacement capacity. This effect is generally as expected, since asshear span increases, the wall’s flexural response becomes dominant over it’s shear response, leading to largedisplacements.Figure 4.4 presents the influence of shear stress on the global drift capacity of the member. Since shearforces serve to increase demand on the vertical reinforcing in reinforced concrete member undergoing flexure,it is expected that the level of applied shear will generally serve to decrease ultimate displacements. Thisappears to hold true for the ULCA Database query walls, however the effect is scattered and only offers aweak downward trend as shear stress is increased.Figure 4.5 presents the influence of cross section slenderness on the global drift capacity. It is evidentthat as a member is elongated with respect to its thickness, it’s maximum global drift is reduced. Thiseffect is as expected, since as the length of the member is increased, the internal moment gradient at asection is decreased, which tends leads to compression zones which are distributed farther into the section700%1%2%3%1 2 3 4 5 6Maximum Drift Capacity at Shear HeightShear Span Ratio, M/VlwFigure 4.3: UCLA database maximum drift capacity at shear height vs. shear span ratio M/V lw.0%1%2%3%0.0 0.1 0.2 0.3 0.4 0.5 0.6Maximum Drift Capacity at Shear HeightShear Stress Ratio, Vmax/Av√f'cFigure 4.4: UCLA database maximum drift capacity at shear height vs. shear stress ratio Vmax/Av√f ′c withf ′c in MPa.71longitudinally.0%1%2%3%5 10 15 20 25 30 35Maximum Drift Capacity at Shear HeightCross Section Slenderness, lw/tFigure 4.5: UCLA database maximum drift capacity at shear height vs. cross section slenderness lw/t.Figure 4.6 presents the influence of the compression zone slenderness on the global drift capacity of themember. As shown, when the compression zone length to thickness ratio increases, the is a well definedcorresponding drop in global drift capacity. This effect has been observed in other studies, and is generallyas expected (Abdullah and Wallace 2019).Figure 4.7 shows there is little influence of the specimen height to length ratio on the global drift capacityof the member. This may be in part due to the variety of test set ups which are included in the databaseresults.4.2.4 Simple Empirical Models from Subset of UCLA Wall DatabaseThis section presents several models to predict the global drift capacity of a thin and lightly reinforced wallmember, based on the member geometry, and the distribution of internal and external forces. The firstmodel discussed is one proposed by Abdullah and Wallace 2019, the second is the Abdullah model modifiedto fit the ULCA database of this study, and the final is an original proposed model based on the analysisresults of the UCLA database from the previous section.4.2.4.1 Abdullah and Wallace 2019 Model of Maximum Drift CapacityA study by Abdullah and Wallace 2019 has proposed that for walls with special boundary elements whichadhere to the requirements of ACI 318-14 special structural walls, the drift capacity is primarily a functionof neutral axis depth to compression zone width, c/t, wall length to compression zone width, lw/t, and shear720%1%2%3%0 1 2 3 4 5 6Maximum Drift Capacity at Shear HeightCompression Zone Slenderness, c/tFigure 4.6: UCLA database maximum drift capacity at shear height vs. compression zone slenderness c/t,including linear trend.0%1%2%3%10 15 20 25 30Maximum Drift Capacity at Shear HeightTest Panel Aspect Ratio, hw/lwFigure 4.7: UCLA database maximum drift capacity at shear height vs. specimen height to length aspectratio hw/lw.73stress ratio, Vmax/(Av√f ′c). Their research presents the proposed linear relationship,δchw(%) = 3.85− λα− vmax0.83√f ′c(4.1)where δc/hw is the maximum global drift capacity, λ is the parameter clw/t2, and vmax is the maximumapplied shear stress Vmax/Av. The parameter α is based on the configuration of boundary reinforcing andis 60 when overlapping hoops are used, and 45 when a single hoop with supplemental cross ties is used. Thefirst term of the proposed model is mean maximum drift capacity based on a best fit linear relation betweenobserved maximum global drift capacity and slenderness parameter λ. The model was shown to have goodsuccess in predicting the global drift capacity of the type of walls which were the focus of the study.4.2.4.2 Proposed Modified Abdullah and Wallace 2019 Model of Maximum Drift CapacityAdjustments can be made to Abdullah’s linear model to apply it to the wall specimen dataset of this study.The first term is changed to reflect the maximum mean drift capacity for a linear model of maximum driftcapacity versus slenderness parameter, λ. As shown in Figure 4.8 is 2.08. Also from the same figure, thereduction in drift capacity is simply the slope of the linear best fit, equal to 1/60.01230 20 40 60 80 100 120 140 160 180Maximum Top Drift Capacity at Specimen Shear Height [%]Slenderness Parameter, λ = clw/t2Figure 4.8: UCLA database maximum drift capacity at shear height vs. specimen slenderness parameterλ = clw/t2, including linear trend.Next, the parameter for shear stress is to be determined. Specimen shear stresses lower than 0.25vmax/√f ′care taken as the low stress bin, and those above the higher stress bin. The results of the two specimen resultsbinned by high and low shear stress are shown in Figure 4.9. As shown, the average drop associated to anincrease in shear stress is approximately 0.65%. It is noted that unlike was observed for the specimensstudied by Abdullah, the specimens included here do not exhibit the same rate of drift capacity change74for each shear stress bin. Since the application of the Abdullah model to the dataset in this study is forillustrative purposes only, the effect is ignored. The average quantification of the effect of shear for thismodel is determined to be vmax/0.60√f ′c.01230 20 40 60 80 100 120 140 160Maximum Top Drift Capacity at Specimen Shear Height [%]Slenderness Parameter, λ = clw/t2vmax/vmax/ ≥ 0.25Figure 4.9: UCLA database maximum drift capacity at shear height vs. specimen slenderness parameterλ = clw/t2 including linear best fit trends, binned by shear stress ratios lower and greater than 0.25vmax/√f ′c.The resulting modified Abdullah model is taken as,δchw(%) = 2.08− λ75− vmax0.60√f ′c(4.2)This model as applied to the dataset of thin and lightly reinforced walls in this study results in a mean valueof predicted to experimental maximum drift capacity at shear height of 0.70 with a variance of 0.26. Thepredicted versus experimental results are shown in Figure 4.10. It is clear that the Abdullah model approachis overly conservative over a wide range, and the adjustment terms result in negative predictions for somespecimens.4.2.4.3 Proposed Empirical Model of Maximum Drift Capacity for Thin and LightlyReinforced WallsAn observation of the effect of shear stress on drift capacity which was ignored for the model is that memberswith very high slenderness parameters, the effect of shear appears to have diminishing effects. In addition,unlike the set of walls applied to the Abdullah tests, the thin and lightly reinforced wall drift capacity isless strongly correlated to cross section slenderness, lw/t, as shown in Figure 4.5. By plotting the specimencross section slenderness against the compression zone slenderness (Figure 4.11), it is shown that memberswith slender compression zones typically also have slender cross sections. This effect is explained by the750.00.51.01.52.02.53.0-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0Abdullah Model Predicited Maximum Drift at Shear Height [%]Experimental Maximum Drift at Shear Height [%]Figure 4.10: Abdullah model predicted vs. experimentally observed maximum drift capacity at shear height.compression zone depth at maximum being in effect a function of the length of the wall. With this in mind itis natural to attempt predictions based on compression zone slenderness alone, as its relation to drift capacityappears to be better suited than the slenderness parameter, λ, or the cross section slenderness alone, lw/t.As discussed, Abdullah’s study found that the maximum level of drift is sensitive to the shear stresspresent in the wall. By separating the UCLA database results into groups of specimens with recorded shearstresses less than and greater than 0.25vmax/√f ′c, any effect on the base linear relation between drift capacityand compression zone slenderness is observed. The results of this analysis are shown in Figure 4.12. For thethin and lightly reinforced walls included in this study, increasing shear stress appears to have little, if anyoverall effect on the maximum drift capacity at the shear height of the specimen. As such, shear effects areomitted from the model to be developed.As shown in Figure 4.3, increasing shear span ratio may play a role in improving the maximum driftcapacity of members of this type. To determine the effect of shear span ratio on the proposed model, theUCLA database results are separated into bins of shear spans less than and greater than 2.0. Shown inFigure 4.13, the shear span ratio has a distinct influence on the test data. An increase of avergae shearspan ratio of 1.50 to 2.82 results in a corresponding increase to maximum drift capacity at the shear heightof approximately 0.43%, as is evidenced by the offset of linear relationships. To quantify this effect in themodel, a linear relation between the two average values is developed and added to the initial linear relationas was shown in Figure 4.6.The basic linear model taken from the relationship between maximum drift capacity at shear height andcompression zone slenderness, as shown in Figure 4.6 is determined to be,δhw(%) = 2.85− c1.92t(4.3)76048121620242832360 1 2 3 4 5 6Cross Section Slenderness, lw/tCompression Zone Slenderness, c/tFigure 4.11: Comparison of UCLA database wall cross section slenderness lw/t vs. compression zone slen-derness c/t.0.0%0.5%1.0%1.5%2.0%2.5%3.0%0 1 2 3 4 5 6Maximum Drift Capacity at Shear HeightCompression Zone Slenderness, c/tvmax/vmax/Figure 4.12: UCLA database maximum drift at shear height vs. compression zone slenderness c/t, dividedinto groups of shear stress greater than and less than 0.25vmax/√f ′c.770.0%0.5%1.0%1.5%2.0%2.5%3.0%3.5%0 1 2 3 4 5 6Maximum Drift Capacity at Shear HeightCompression Zone Slenderness, c/tM/Vlw ≥ 2.0M/Vlw < 2.0Figure 4.13: UCLA database maximum drift at shear height vs. compression zone slenderness c/t, dividedinto groups of shear span ratio, M/V lw greater than and less than 2.0.The accompanying model of shear span ratio with respect to the linear model of compression zone slendernessversus maximum drift capacity is determined to be,∆δhw(%) =M3.05V lw− 0.71 (4.4)By combining the two models, the resulting proposed linear model to predict the maximum drift capacityat the shear height δ/hw with respect to the compression zone slenderness, c/t, and the shear span ratio,M/V lw is,δchw(%) = 2.14 +M3.05V lw− c1.92t(4.5)Applying the proposed linear model to the walls in the test database results in a mean predicted maximumdrift capacity versus experimentally observed maximum drift capacity of 1.02, with a variation of 0.04. Thepredicted results are shown plotted against the observed results in Figure 4.14. This comparison shows thatthe model produces acceptable estimates of the maximum drift capacity at the shear height for thin andlightly reinforced shear walls.4.2.4.4 Selection of UCLA Database Wall Specimens for Further StudyFrom the UCLA RCWalls database analysis and the proposed model which was developed in the previoussection, several test specimens are identified for further analysis. The walls with the lowest observed driftcapacities (less than 1.0%) are the results of three test programs. These tests are identified in Figure 4.15.In addition to the low observed drift capacities, the proposed model predicted maximum drift at the shear780.0%0.5%1.0%1.5%2.0%2.5%3.0%3.5%4.0%0.0% 1.0% 2.0% 3.0% 4.0%Proposed Model Predicited Maximum Drift at Shear HeightExperimental Maximum Drift at Shear HeightFigure 4.14: Comparison of proposed model predicted vs. experimentally observed maximum drift capacityat shear height.height for these members was higher than observed for 6 out of 8 of them, with an average of a 10%, and ashigh as 47% over prediction.Of the tests identified the studies by Tomazevic 1995, and Ho 2006, provide few details regarding theobserved low drift capacities. In addition these studies provide little test data to perform supplementaryanalysis. In contrast the study performed by Almeida et al. 2017 at E´cole Polytechnique Fe´de´rale deLausanne (EPFL), provides the full data set for 5 heavily instrumented test specimens. Of these specimensthere are two pairs of complimentary walls, with varying thickness, reinforcing configurations, and loadingconfigurations. These four test specimens are of interest to this study, and offer an excellent opportunity tofurther explore the topic of thin and lightly reinforced compression dominated concrete walls subjected toin-plane and out-of-plane demands.4.3 Walls Tested at E´cole Polytechnique Fe´de´rale de Lausanne(EPFL)4.3.1 Summary of Experimental Program at EPFL4.3.1.1 Testing ApproachThe EPFL test program was devised to study out-of-plane and lap splice failure modes. The wall specimenswere proportioned to represent designs typical to lost cost Columbian mid and high rise residential buildings(TW1 and TW4), and Swiss construction practices prevalent in the 1950s to 1970s (TW2, TW3, and TW5).Wall specimen TW3 which focused on lap splice failure is outside the scope of this study and further790.0%0.5%1.0%1.5%2.0%2.5%3.0%0 1 2 3 4 5 6Maximum Drift Capacity at Shear HeightCompression Zone Slenderness, c/tTomazevic et al., 1995Almeida et al., 2014Ho, 2006Figure 4.15: UCLA database tests with observed maximum drift capacities at the shear height of less than1.0%.description of this specimen will be omitted. The test walls were proportioned to be rectangular with asmall flange at one end. Detailed descriptions of the test specimens is included in a subsequent section.The EPFL test program included two loading configurations, both quasi-static cyclic tests. The first setof tests (TW1, TW2) imposed the specimens to in-plane deformations only to determine baseline results,and the second set of tests (TW4, TW5) included both in-plane and out-of-plane imposed deformations.The test walls were constructed and loaded in such a way as to represent the lower portion of the wall only,with actuators at the top of the walls imposing additional axial forces, and bending moments to simulate adistribution of forces which would be present in a taller specimen. A diagram of the test set up is shownin Figure 4.16. A detailed description of the test set up and loading program is included in a subsequentsection.Since the main goal of the EPFL testing program was to determine out-of-plane behaviour of the testwalls, the specimens were heavily instrumented. Strains at the ends of the wall specimens were measuredwith conventional linearly variable displacement transducers (LVDTs). The LVDT data was supplementedwith an optical triangulation system which used a camera system to track an array of LED sensors located onone side of the test specimens. In addition, a digital image correlation (DIC) system was applied to the faceopposite to the LED optical triangulation system. The DIC system uses cameras to track the movements of aspecially applied speckle pattern on the face of the wall. A more detailed description of the instrumentationused in this study is included in a subsequent section.The results of the EPFL tests have been used to validate models of out-of-plane displacements of wallelements due to residual tension strains which develop through cyclic loading (Rosso, Almeida, and Beyer2016). The analysis of the EPFL tests by Rosso focused on models of the out-of-plane deformations present80 Horizontal actuator(Load 1 / Stroke 1)Flange edge actuator(Load 2 / Stroke 2)Web edge actuator(Load 3 / Stroke 3)2,200 mm3,600 mmTest SpecimenTop Beam3,600 mm2,000 mmLoading BeamVertical ActuatorsStiffener BeamTest SpecimenWall PanelTest SpecimenFoundationHorizontalActuatorStrong FloorReaction Wall2,700 mm400 mm420 mmFigure 4.16: EPFL test wall setup (adapted from (Almeida et al. 2017)).in thin members, and found that imposed out-of-plane displacements at the top of the wall had a significanteffect on observed out-of-plane instability of the wall boundary elements.The availability of the very detailed EPFL test data as provided by Almeida et al. 2017 offers furtheropportunity to study the behaviour of these test specimens. As the study on drift capacity using the resultsof the UCLA RCWalls database has indicated, the EPFL test specimens represent excellent candidates forfurther investigation. Since the work of previous researchers has focused on the localized effects observed inthe tests, this study will aim to garner insight on the global behaviour of these slender and lightly reinforcedwalls.4.3.1.2 Description of Test SpecimensTwo wall types were developed for the EPFL tests. Walls TW1 and TW4 represent one pair of matchingspecimens, and TW2 and TW5 present the other matching pair. All walls were 2000 mm in height and 2700mm in length. Table 4.1 lists the geometric and reinforcing parameters of the four specimens of interest.TW1 and TW4 Walls TW1 and TW4 represent the more slender of the two types, with a thickness of80 mm, and a 440 mm wide and 80 mm thick flange at one end. The vertical reinforcing was a single layerof 6 mm bars spaced at 240 mm, with 3 additional 16 mm bars at each end, spaced at 100 mm. The 6 mmvertical reinforcing was lap spliced at the base with a 350 mm length, and the extra 16 mm vertical barswere fully developed into the top and bottom supports. The horizontal reinforcing consisted of 6 mm barsplaced in a single layer spaced at 200 mm. The material properties of walls TW1 and TW4 are as shown inTable 4.1.81Table 4.1: Summary of EPFL test specimen geometric and concrete material properties.Property Units Test SpecimenGeometry and Loading TW1 TW4 TW2 TW5Clear Height mm 2000 2000Length mm 2700 2700Thickness mm 80 120Shear Span mm 10,029 3150 7350Shear Span Ratio 3.70 1.17 2.72Axial Load Ratio 4.3% 3.3% 3.2% 4.8%Loading Configuration (In-Plane/Out-of-Plane) IP OOP IP OOPVertical Reinforcing ContentWeb Vertical Reinforcing Diameter mm 6 6Boundary Vertical Reinforcing Diameter mm 16 6Total Vertical Reinforcing Ratio 0.67% 0.51%Web Vertical Reinforcing Ratio 0.15% 0.49%Web Boundary Vertical Reinforcing Ratio 2.63% 0.50%Flange Boundary Vertical Reinforcing Ratio 0.98% 0.64%Horizontal Reinforcing ContentHorizontal Reinforcing Diameter mm 6 6Horizontal Reinforcing Ratio 0.18% 0.36%Concrete Material PropertiesConcrete Cylinder Strength MPa 28.8 31.2 50.7 33.6Concrete Tensile Strength MPa 2.2 1.5 2.1 1.7Concrete Initial Tangent Modulus of Elasticity MPa 25,300 29,200 31,800 31,700Table 4.2: Summary of EPFL test specimen reinforcing material properties.Property Units Test Specimen6 mm Reinforcing TW1 TW4 TW2 TW5Yield Strength MPa 460Strength at Onset of Hardening MPa No Yield PlateauUltimate Strength MPa 625Yield Strain mm/m 2.5Strain at Onset of Hardening mm/m No Yield PlateauUltimate Tensile Strain mm/m 99Modulus of Elasticity MPa 184,00016 mm ReinforcingYield Strength MPa 565 515 -Strength at Onset of Hardening MPa 565 515 -Ultimate Strength MPa 650 618 -Yield Strain mm/m 2.7 3.2 -Strain at Onset of Hardening mm/m 27 29 -Ultimate Tensile Strain mm/m 141 127 -Modulus of Elasticity MPa 208,150 200,00082TW2 and TW5 Walls TW2 and TW5 represent the less slender of the two types, with a thickness of 120mm, and a 440 mm wide and 120 mm thick flange at one end. The vertical reinforcing was two layers of 6mm bars spaced at 95 mm. The 6 mm vertical reinforcing was not spliced, and was fully developed into thetop and bottom supports. The horizontal reinforcing consisted of 6 mm bars placed in two layers spaced at130 mm. The material properties of walls TW2 and TW5 are as shown in Table 4.1.The reinforcing properties as described above represent the average values of several individual specimenstested for each diameter of reinforcing. All parameters described with the exception of ultimate strain, werecharacterised to be fairly uniform across the tests performed.Concrete material strength was measured with a series of concrete cylinder tests which were field curedto accurately represent the in-situ conditions. Concrete tensile strength was measured using a double-punchtest. The concrete elastic modulus was measured using the relevant Swiss standard procedure, with the testcylinders being loaded 3 times to a pre determined strain level, with the third and final cycle taken as themeasured elastic tangent modulus.4.3.1.3 InstrumentationThe EPFL specimens were instrumented with vertical LVDTs placed at one or both of the wall ends, a gridof optical triangulation LEDs on one face, and a digital image correlation system with an applied specklepattern on one face. For the purpose of the analysis in this thesis, the primary instrumentation relied uponis the optical triangulation LEDs, with the vertical LVDT instruments providing validation of opticallymeasured results when available.The optical triangulation grid consists of a field of infrared light emitting diodes (LEDs), which are trackedby a camera which records the relative three dimensional position of each LED as the test is performed.Accuracy of measurements is improved by the use of two separate cameras which each cover approximatelyhalf of the length of the wall. All LED grids applied to each specimen included LEDs located on thefoundation to measure relative movements and boundary effects of the wall base. Additionally the top beamhas LEDs to measure any wall top boundary effects.Specimens TW1 and TW4 include a grid of 255 LEDs for optical triangulation in a roughly 240mmx 200mm grid, with additional sensors at the foundation, top beam, and lap splice locations. SpecimensTW2 and TW5 include a grid of 492 LEDs for optical triangulation in a roughly 95mm x 130mm grid, withadditional sensors at the foundation, and top beam.4.3.1.4 Test ProcedureEach test specimen was subjected to either in-plane or a combination of in-plane and out-of-plane displace-ments applied at mid-height of the top beam of the specimen. In addition, vertical actuators located aboveeach end of the top beam served to apply axial load and additional moment to increase the shear height of thewall specimens. This configuration was devised to the reduce the height of the test specimen for practicality,while retaining the height of wall expected to undergo the plastic deformations. Each specimen was testedover several days, with the load actuators remaining engaged continuously until the test was complete, orwhen a system safety shut down was required.Specimen TW1 was subjected to in-plane loads only, with two fully-reversed load cycles applied at eachdrift level (four total load stages per target drift level). Drift levels are 0.05%, 0.10%, 0.15%, 0.25%, 0.35%,0.50%, 0.75%, and 1.00% were applied.83Specimen TW2 was subjected to in-plane loads only, with one fully-reversed load cycle applied at eachdrift level (two total load stages per target drift level). Drift levels are 0.05%, 0.10%, 0.15%, 0.25%, 0.35%,0.50%, 0.75%, 1.00%, and 3.00% were applied.Specimen TW4 was subjected to in-plane and out-of-plane loads, which is applied in a “clover leaf”pattern. At drifts below 0.50%, only a half clover leaf is applied. The in-plane drifts applied were 0.05%,0.10%, 0.15%, 0.25%, 0.35%, 0.50%, and 0.75%. Out-of-plane drifts applied were 0.05%, 0.10%, 0.15%,0.25%, 0.35%, 0.50%, 0.50%, 0.75%, and 0.75%.Specimen TW5 was subjected to in-plane and out-of-plane loads, which is applied in a half “clover leaf”pattern. The in-plane drifts applied were 0.05%, 0.10%, 0.15%, 0.25%, 0.35%, 0.50%, 0.75%, 1.00%, and1.50%. Out-of-plane drifts applied were 0.05%, 0.10%, 0.15%, 0.25%, 0.35%, 0.50%, 0.75%, 1.00%, and1.50%.4.3.2 Summary of Test ResultsThis study focuses on the behaviour of slender, lightly reinforced compression dominated flexural wallssubjected to in-plane forces. The results from the wall tests are compression and tension strains at the endsof the specimen, distribution of elastic and inelastic curvatures over the height, distribution of shear strains,and deformation components. Each of these values, and their quantification methodologies are presentedand described in the subsequent sections. All results shown are for bending which induces compression inthe slender wall end and tension in the flange end. Results for bending in induced in the opposite directionare identified as they are relevant to the outcome of this study, and are clearly identified.4.3.2.1 Load-Deformation ResponseLoad-deformation response is a useful result of any shear wall test program. The EPFL walls are instrumentedin a way which allows for precise analysis of the load-deformation characteristics of the test specimens. Thissection presents the load-deformation results of the EPFL tests.In this section, the drift reported is taken from the optical sensor data, and is taken for the section ofwall spanning from just below the base crack, to just below the top of the wall. In other words, the reporteddrift is taken for the wall panel including base effects, but neglecting top boundary effects. The influence ofthe base crack on the deformation response of the test is discussed in later sections.The definition of drift capacity used in this study is based on the maximum observed top displacement,∆max, and the clear height of the specimen hw as,δmax =∆maxhw(4.6)The maximum top displacement is defined as the displacement at a well defined sudden loss of load carryingcapacity, or an 80% drop in load carrying capacity. In later sections, estimates of the drift capacity at theshear span (height) are made in an attempt to make estimates of the drift capacity of a full height wall.Estimates of displacement ductility of the various test specimens are also presented. Displacement duc-tility is based on the displacement at yield, ∆y and the maximum top displacement, ∆max, and is definedas,δmax =∆max∆y(4.7)84Where a well defined yield point is not apparent, yield is taken as the displacement which corresponds to aload of 85% of the maximum observed load capacity.Figure 4.17 shows the full hysteretic response for the specimen TW1. This full response is included as anexample of the full cyclic response of the wall members. The focus of this study is the behaviour of the EPFLtest specimens as they are loaded in such a way where compression is induced in the thin end, and tensionin the flanged end. Only the quadrant of the cyclic response for this direction of loading will be included inthe subsequent results. Further to the presentation of the load-deformation results for a single direction ofloading, the load steps (typically half cycles of loading) are identified only for the load step which inducestension in the flange of the test specimen. The context of the discussion to follow will dictate whether theload step (identified as LS#) is referring to the full or half cycle of interest. Where the distinction betweenthe a full and half cycle of loading is of exceptional importance, the specific load step of interest will beexplicitly identified.-1.25% -0.75% -0.25% 0.25% 0.75% 1.25%-2000-1500-1000-5000500100015002000-25 -20 -15 -10 -5 0 5 10 15 20 25-200-150-100-50050100150200Drift at 2m Above Wall BaseBase Moment [kN-m]Displacement at 2m Above Wall Base [mm]Applied Shear [kN]Compression in the FlangeTension in the FlangeFigure 4.17: TW1 hysteretic response at 2 m above the base of the wall.As shown in Figure 4.18, TW1 achieved a drift capacity of 0.75% at LS27. Two stable cycles at a driftof 0.75% were performed with little strength loss observed. The last cycle to 1.0% drift showed significantstrength degradation prior to a drift of 0.75%, suggesting that failure was likely to have occurred soon aftera drift of 0.75% was achieved during the previous load cycles. A well defined yield point at 0.29% wasobserved, which results in an approximate displacement ductility of 2.4.The load-deformation response for member TW4 is shown in Figure 4.20. TW4 achieved a drift capacityof 0.71%. One stable cycle at a drift of 0.71% was observed. The final cycle planned to impose a drift of1.0%, and only an 11% loss of strength was observed. As the final cycle approached 0.75% drift, a sudden85LS3,5LS7,9LS11,13LS15,17LS19 LS21 LS23LS25LS27LS29LS310.00% 0.25% 0.50% 0.75% 1.00% 1.25%0204060801001201401601802000 5 10 15 20 250500100015002000Drift at 2m Above Wall BaseBase Shear [kN]Displacement at 2m Above Wall Base [mm]Base Moment [kN-m]Figure 4.18: TW1 hysteresis plot for bending with slender wall end in compression with load steps identified.complete loss of strength was recorded. Similar to wall TW1, a loss of strength is likely to have occurredhad the previous cycle of 0.75% been continued to failure. Similarly to wall TW1, TW4 was observed tohave a well defined yield point at approximately 0.31% drift. The displacement ductility of TW4 is 2.3.As shown in Figure 4.20, TW2 achieved a drift capacity of 0.90%. One stable drift cycle of 0.75% wasobserved. The final cycle was planned to impose a drift of 1.0%, however significant loss of strength wasobserved at a drift of 0.90%. This specimen had a less well defined yield point, and yield is taken as thedrift at 85% of the ultimate load, and was observed to be 0.14%. The preceding drift values at ultimate andyield result in a displacement ductility of 6.4.As shown in Figure 4.21, TW5 achieved an ultimate drift capacity of 0.84%. The final cycle had a plannedimposed drift of 1.0%, however a complete loss of strength was observed at 0.84%. Similar to wall TW5,a well defined yield point was not observed, therefore yield is again taken to be at the drift correspondingto 85% of the maximum observed load. The yield drift is taken to be 0.17%, resulting in a displacementductility of 4.9.The more slender specimens TW1 and TW4 had ultimate drift capacities (0.75% and 0.71%) which were17% and 15% lower than those observed in walls TW2 and TW5 (0.90% and 0.84%). Perhaps of greaterconcern than low drift capacity is the low displacement ductility observed in specimens TW1 and TW4(2.4 and 2.3) when compared with those of TW2 and TW5 (6.4 and 4.9). The disparity in drift capacityand displacement ductility between walls TW2 and TW5 may be attributed to the difference in shear span,imposed out-of-plane displacement, or some other phenomenon.86LS2LS8LS14LS20LS26 LS32LS42LS48LS580.00% 0.25% 0.50% 0.75%0204060801001201401601800 5 10 15020040060080010001200140016001800Drift at 2m Above Wall BaseApplied Shear [kN]Displacement at 2m Above Wall Base [mm]Base Moment [kN-m]Figure 4.19: TW4 hysteresis plot for bending with slender wall end in compression with load steps identified.LS5LS7LS9LS11 LS13 LS15 LS17LS190.00% 0.25% 0.50% 0.75% 1.00% 1.25%01002003004005006007008000 5 10 15 20 2505001000150020002500Drift at 2m Above Wall BaseBase Shear [kN]Displacement at 2m Above Wall Base [mm]Base Moment [kN-m]Figure 4.20: TW2 hysteresis plot for bending with slender wall end in compression with load steps identified.87LS2LS8LS14LS20 LS26 LS32 LS38LS440.00% 0.25% 0.50% 0.75% 1.00%050010001500200025000 5 10 15 20050100150200250300350Drift at 2m Above Base of WallApplied Shear [kN]Displacement at 2m Above Base of Wall [mm]Base Moment [kN-m]Figure 4.21: TW5 hysteresis plot for bending with slender wall end in compression with load steps identified.4.3.2.2 Failure ModesThis section explores the dominant failure modes of the test walls, along with the general progression ofdamage observed in late stage load cycles. The results of this section are purely qualitative observationsbased on photos and videos as provided in the data set assembled by the EPFL researchers (Almeida et al.2017), and observations of the load-deformation response presented in the previous section.For specimen TW1, the EPFL researchers reported an in-plane failure caused by damage induced dueby out-of-plane deformations (Rosso, Almeida, and Beyer 2016). In the video of the test, it is evident thatlarge out-of-plane displacements were induced near mid-height of the specimen as a result of residual tensionstrains present in the slender end of the wall. The residual tension strains allow out-of-plane curvature tobe induced in the wall prior to the closing of cracks in the subsequent load step. The closing of the crackscorresponds with a recovery of the out-of-plane displacements, and as a result the member continues alongits planned loading program with no visible out-of-plane deformations. Simply put, as the wall experienceshigher levels of drift than in previous cycles, the wall is not subjected to increased out-of-plane deformationsat the or near the maximum drift.As shown in Figure 4.18, wall TW1’s last three cycles are LS27, LS29, and LS31. Photos of the conditionof the wall toe are shown in Figure 4.22, which illustrate the progression of damage to the toe of the wall.Figure 4.22a shows that for LS27, the first cycle of drift to 0.75%, the wall toe is beginning to show signsthat the concrete is nearing its compression strain capacity. The damage does not appear to be concentratedtowards a single edge of the wall section, and is relatively symmetric with vertical cracking and slight bulgingevident at both sides of the member. Figure 4.22b shows the damage induced by the next compression loadstep, LS29 (the photo was taken at LS30, however the damage is indicative of that induced in LS29). As88(a) LS27 (b) LS29 (c) LS31Figure 4.22: TW1 slender wall end damage progression (Almeida et al. 2017).shown in Figure 4.18, LS29 was the second of two cycles to approximately 0.75% drift, and the specimenexhibited a slight reduction in load carrying capacity. The slight spalling of the concrete at LS29 suggeststhat the reduction in load carrying capacity at this stage is due to damage induced in previous cycles. Finallyin Figure 4.22c, the wall toe has experienced significant damage, with large portions of the concrete spalledaway, along with significant buckling of the vertical reinforcing. This level of damage is accompanied by alarge decrease of in-plane load carrying capacity as shown in Figure 4.18.The description of the progression of late stage damage to the wall toe indicates that the wall TW1failed in concrete compression at the toe of the wall. As shown in the photos of LS27 and LS29 (Figures4.22a, and 4.22b), and corroborated in the video recordings of the test, the buckling of the reinforcing wasprecipitated by the toe of the wall reaching its compression strain capacity and spalling away, leaving thevertical reinforcing to be subjected to large compression strains with little or no lateral support. The typeof failure observed in TW1 is typically described as brittle, as the member exhibited a rapid loss of loadcarrying capacity, with little or no residual capacity near ultimate drift levels.Wall TW2’s failure mode was described by the EPFL researchers as extensive crushing at the slenderwall end, which affected the load carrying capacity of the member. Unlike the thinner wall TW1, wall TW2did not show significant levels of out-of-plane displacements, and damage was not ascribed to the effects ofresidual tension strains during subsequent compression load cycles.(a) LS18 (b) LS20Figure 4.23: TW2 slender wall end damage progression (Almeida et al. 2017).89Figure 4.20 shows the last two cycles of TW2 are LS17 and LS19. The photos in Figure 4.23 show thetoe of TW2 for LS18 and LS20, which represent the damage induced by the previous load cycles LS17 andLS19. The damage induced during LS17 shown in Figure 4.23a indicates the wall has reached its concretecompression strain capacity, with a small amount of spalling visible. At this stage, the load carrying capacityhas not significantly decreased, as is shown in Figure 4.20. Figure 4.23b shows that the subsequent loadcycle has significantly damaged the compression zone of the wall, with the fractured reinforcing occurringupon reloading to LS20. This damage corresponds to a large and rapid decrease in load carrying capacityas shown in Figure 4.20. In addition to the photos shown, the EPFL researchers report that little or noout-of-plane displacement was visible throughout the test.The late cycle damage to wall TW2 indicates that the wall failed in compression when the toe reachedits strain limit. The failure observed is decidedly brittle, as the member did not show significant indicationof failure prior to complete loss of load carrying capacity at ultimate load and drift levels.Wall TW4, which is the companion specimen to TW1, was described by EPFL researchers to fail inconcrete crushing and spalling, which was promoted partially by buckling of the vertical reinforcing. Unlikespecimen TW1, wall TW4 was subjected to imposed out-of-plane displacement load cycles, which may havecontributed to increased damage to the compression zone prior to failure indicated by loss of load carryingcapacity.(a) LS48 (b) LS54Figure 4.24: TW4 slender wall end damage progression (Almeida et al. 2017).Wall TW4’s hysteretic response shown in Figure 4.19 shows that failure occurred over the load stagesLS48 and LS54. Figure 4.24 shows the wall toe at these final load stages. LS48 shown in Figure 4.24a showsthat the wall toe has sustained some crushing damage. Unlike its partner specimen TW1, wall TW4 appearsto have sustained compression zone damage in an asymmetric fashion. The left side of the specimen hasclearly sustained more damage. This asymmetry of damage is possibly caused by the imposed out-of-planedeformations from previous cycles. The next in-plane load cycle LS54 is shown in Figure 4.24b, and showsthe damage from the previous cycle has progressed to a complete failure of the compression zone. Similarly towall TW1, it is likely that the buckling observed in the vertical reinforcing is a by-product of the compressionfailure of the concrete, and is not a root cause of the member failure. More simply, the reinforcing does notshow any indication that it has a tendency to buckle prior to loss of section due to significant crushing and90spalling.The failure observed in wall TW4 indicates that imposed out-of-plane deformations may serve to increasedamage to the compression zone. That said, the failure mode appears to to remain the same as the base case,with the compression zone concrete reaching its strain capacity, followed by a rapid decrease in load carryingcapacity. The asymmetry of the damage may indicate more a complex distribution of strains throughoutthe compression zone of the wall, however the ultimate resulting failure mode does not appear significantlyaltered.The EPFL researchers describe the failure of wall TW5 as concrete crushing and spalling at the slenderend. They went on to state that the imposed out-of-plane deformations did not significantly affect theresponse of the member. It is noted that the wall TW5 and its companion TW2 were not identical specimens,with the two members having different shear spans as noted in the member descriptions.(a) LS38 (b) LS44Figure 4.25: TW5 slender wall end damage progression (Almeida et al. 2017).Figure 4.21 shows that the late stage load steps of interest are LS38 and LS44. Figure 4.25 shows thedamage of the wall toe at LS38 and LS44. LS38 shown in Figure 4.25a shows that the compression zone isshowing significant signs that it is at or near to its compression strain limit. Unlike as was seen in membersTW1 and TW4, the effect of imposed out-of-plane deformations appears to show little effect on the damageto the thicker wall specimen. Damage to the toe at LS38 does not appear to be significantly concentratedto a single side of the member, indicating relatively uniform compression strains across the thickness of thecompression zone. Figure 4.25b showing LS44, indicates a significant failure of the compression zone, and issupplemented by the loss of load carrying capacity shown in Figure 4.21. The photo of LS44 shows bucklingof the vertical reinforcing, however similar to the wall TW2, video evidence indicates that the buckling issimply a by-product of the compression failure of the wall.The failure observed in member TW5 indicates that the wall failed in a brittle, concrete compressiontype mode. The late cycle damage indicates that the out-of-plane deformations did not significantly increasedamage to the slender end of the wall. None of the wall specimens were observed to have lost their abilityto carry the axial loads induced during the tests.The significant findings of the qualitative analysis of the failure modes of specimens TW1, TW2, TW4,and TW5 are as follows:911. All specimens failed in a brittle mode caused by the slender end of the wall reaching their concretecompressive strain limit.2. All specimens were observed to have a rapid onset of failure once the compression zone was compro-mised, with little or no residual load carrying capacity past the ultimate measured level of drift.3. The failure mode of the more slender members TW1 and TW4, with imposed out-of-plane deformationsindicate increased damage to the compression zone which results in a non-uniform distribution of strainsacross the thickness of the compression zone, possibly leading to a change of failure mode.4. The failure mode of the thicker walls TW2 and TW5 appear to be relatively unaffected by imposedout-of-plane deformations.5. None of the specimens lost axial load carrying capacity in combination with lateral failure.4.3.2.3 Vertical StrainsThis section presents the observed vertical tension and compression strains. The vertical strains are deter-mined using the optical sensor data. The optical sensor records the position of the sensor in an x,y,z format.To determine the vertical strains the initial vertical distance between two sensors in a column is recorded as,dinitial = yupper − ylower (4.8)This initial vertical spacing of sensors is taken as the gauge length over which the strain will be measured.Upon loading the final vertical positions of the sensors is used to determine a final distance between thesensors, dfinal. The final measured vertical strain is then determined as,εv =dfinal − dinitialdinitial(4.9)For this study, the strains recorded are taken at the maximum drift levels observed for each load step ofinterest. Tension strains are taken at the first column of sensors located inside the flange of the wall specimen,since the flange sensors are located on the flange edge out-of-plane and may be affected by shear lag effects.The compression strains are taken at the first column of sensors on the slender end of the wall specimen. Anexample of the recorded tension strains for specimen TW1 is shown in Figure 4.26, and compression strainsin Figure 4.27. The strain value presented is shown at the elevation that represents the center of the verticalgauge length for which it was measured.As is shown in Figure 4.26, the distribution of tension strains for member TW1 show that strains increaseas the drift level in each load cycle increases, however few increases in tension strains are observed for loadcycles with repeated levels of drift. Another interesting feature is the presence of low observed tension strainsat 100 mm above the base of the wall. Photos of the wall crack pattern indicate that a wedge of uncrackedconcrete formed near the base, which explains the low strain values. This indicates that vertical strainsconcentrated in cracks above and below the uncracked portion near the base. The large recorded tensionstrains at the base crack are attributed to the effects of strain penetration, where the reinforcing strainextends some distance into the support foundations, such that the strain values recorded are taken over areduced gauge length than is present in the test.As shown in Figure 4.27, the distribution of compression strains at the toe of specimen TW1 indicatesthat strains are increasing towards the base of the member as expected. The base compression strains for9201000 20 40 60 80 100 120 140 160Tension Strain [mm/m]02004006008001000120014001600180020000 2 4 6 8 10 12 14Elevation [mm]LS19LS21LS23LS25LS27Figure 4.26: TW1 observed tension strains at 116 mm inside the extreme fibre for multiple load steps.930200400600800100012001400160018002000-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0Elevation [mm]LS19LS21LS23LS25LS270100-20.0 -15.0 -10.0 -5.0 0.0Compression Strain [mm/m]180019002000-30.0 -25.0 -20.0 -15.0 -10.0 -5.0 0.0Figure 4.27: TW1 observed compression strains at the extreme fibre for multiple load steps.94load cycles with repeated levels of drift (LS19 to LS21 and LS23 to LS25) indicate that damage to thecompression zone is occurring with each repeated cycle, as strains increase without the presence of higherdrifts. The base strain for LS27 is taken over a longer gauge length than previous load steps, as the sensor wascompromised due to damage to the compression zone. The observed compression strains support qualitativeobservations of failure mode of TW1, with failure occurring due to damage to the compression zone of thespecimen.The results for specimens TW2, TW4, and TW4 are included in Appendix C.2.1.4.3.2.4 CurvaturesFor this study curvature is determined using the vertical strains presented in the previous section. Thecurvature is taken as,φ =εt − εclh(4.10)where εt and εc are the measured vertical tension and compression strain respectively, and lh is the horizontaldistance between the measured tension and compression strains. The validity of this definition of curvature isinvestigated in a subsequent section to determine if integration of curvatures results in accurate estimationsof flexural displacements.As shown in 4.28, the curvatures for TW1 indicate the presence of inelastic curvatures. Near the top ofthe wall, curvatures show little increase for cycle to cycle, with lower portions of the wall having much largerincreases indicative of plasticity in the member. The distribution of curvature has many of the propertiesof both the observed tension and compression strains presented in the previous sections, which indicatesthat the behaviour is not largely driven by a single strain component. As was observed in the compressionstrain results, small increases to curvature are present for load steps with repeated drifts. On the tensionside, the uncracked portion near the base is clearly represented in the results. As discussed in the verticalstrain results, the phenomenon of strain penetration at the base of the wall results in higher than expectedcurvature level, and the implications of this will be discussed in subsequent sections.The curvature distributions for specimens TW2, TW4, and TW5 are included in Appendix C.2.2.4.3.2.5 Shear StrainsFor this study shear strains are determined modifying the method described by Hiraishi 1983 which accountsfor the change in curvature over the measured height. The modifications to the method described areimplemented to account for differences in horizontal and vertical positions between the sensors, which arenot typically present when measuring a full panel section. The changes to the method simply replacethe horizontal and vertical panel dimensions, with the average vertical and horizontal heights between thesensors. The modified method determines the shear deformations as,∆s =d1 + d22lh1 + 2lh2(δ1 − δ2)− (α− 0.5) θhv (4.11)where d is the initial diagonal length between sensors, lh1 and lh2 are the horizontal distances betweensensors, δ1 and δ2 are the deformed lengths of diagonals between sensors, α is a parameter to account forvariations in curvature over the height, θ is the difference in rotations between the top and bottom set ofsensors, and hv is the average vertical height between the sensors. This method was typically developedto determine the average vertical strain over the full height of a test specimen. In this study the average9502004006008001000120014001600180020000 1 2 3 4 5 6 7 8Elevation [mm]LS17LS19LS21LS23LS25LS27LS290501001502000 10 20 30 40 50Elevation [mm]Curvature [rad/km]Figure 4.28: TW1 observed curvature for bending with tension induced in the wall flange at various loadstep maximum drifts.96curvature was determined over much shorter height intervals. As such the change in rotations observed overthe height of each interval was very low, and occurred over a much smaller height than over a full panelheight. This results in the last term of Equation 4.11 being very small in comparison to the first two, andcan thus be omitted in many cases.Figure 4.29 presents the shear strain distribution for wall TW1 for load steps 19 to 27. As shown, theshear strains at the base of the wall are the largest measured, and indicate sliding of the base at the boundary.In addition, the shear strains near the base of the wall correspond with the location low tension strains andcurvatures. Above the base, shear strains are relatively constant over the height, with redistributions evidentfor load cycles of repeated drift. This indicates that shear strains are less stable for cycles of repeated driftlevels, unlike as observed for curvatures, which showed little change for cycles of repeated drift levels.The shear strain distributions for walls TW2, TW4, and TW5 are included in Appendix C.2.3.4.3.2.6 Deformation ComponentsBy using the measured curvatures and shear strains, it is possible to make estimates of the deformationcomponents of the test specimens. Flexural displacements are determined by integrating curvatures over theheight of the member with respect to the top. Since the relative contribution of the base crack to overallflexural displacements is of interest, the flexural displacements are determined for the those due to baserotation only, and those due to the remainder of the member above the level of the base crack. Curvaturesare integrated with respect to the top of the wall panel, located at 2000 mm above the base of the wall.Curvatures are integrated using the moment-area theorem as follows,∆f =∫yφ(y)dy (4.12)where φ(y) is the distribution of curvature over the height of the member. For the case of discrete curvaturemeasurements can be expressed as,∆f ≈∑φihiyi (4.13)where φi is the observed curvature at yi, and hi is height over which the curvature at yi occurs. Thedistance yi is taken the distance between the centroid of the curvature interval φi and the location wheredisplacements are to be determined.The shear strains determined in the previous section are used to determine the contributions due to shearand base sliding. Shear displacements and base sliding deformations are determined by taking the measuredshear strains and multiplying by the height over which they are measured. All shear strains at the base ofthe wall are described as base sliding in this study, as no distinction can be made between shear displacementand base sliding effects due to the lack of instrumentation.To validate the estimated flexural and shear displacements, the estimates are summed and a comparisonto the total measured displacement is made. Figure 4.30 shows the estimated flexural and shear displacementsversus the total measured displacement for all specimens over various load cycles. For specimen TW1, onlyload cycles for the first level of imposed drift are shown, not the second repeated cycle of drift. As is shown,the estimations of shear and flexural displacements result in accurate estimations of total displacement.This brief analysis serves provide validation of the estimates of flexural and shear deformation of the testspecimens.Table 4.3 and presents the displacement components for specimen TW1. Table 4.4 presents the displace-97180019002000-3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.502004006008001000120014001600180020000.0 0.5 1.0 1.5 2.0 2.5Elevation [mm]LS19LS21LS23LS25LS2701000 2 4 6 8 10 12 14 16Shear Strain [mm/m]Figure 4.29: TW1 observed shear strains for bending with tension induced in the wall flange at various loadstep maximum drifts.9802468101214160 2 4 6 8 10 12 14 16Estimated Shear + Flexural Displacement [mm]Measured Total Displacement [mm]Figure 4.30: EPFL observed displacements vs. sum of shear and flexure component estimations.ment component increases by load cycle. As shown, the largest proportion of displacement at each load stepis through panel flexure, however as the base rotation increases it’s proportion increases. Interestingly, whilebase sliding accounts for a very low proportion of the overall displacement, shear displacements in the panelaccount for approximately 17% to 22% of the total displacement. This indicates that panel shear plays asignificant role in the displacement capacity of these specimens. Similarly, base rotations proportions of 37%at ultimate indicate that base rotations play a significant role in the displacement capacity of the member.As the level of imposed drift increases, panel flexure and base rotations appear to increase relatively uni-formly. Increases in panel shear appear to be growing as drift is increased, and base sliding appears to bedeclining.Table 4.3: Wall TW1 summary of displacement components.MeasuredDisplacements Percentage of[mm] Total DisplacementLoad Step 19 23 27 19 23 27Panel Flexure 2.9 4.3 6.0 49% 46% 40%Panel Shear 1.3 1.7 2.6 22% 18% 17%Base Rotation 1.4 2.9 5.6 23% 30% 37%Base Sliding 0.1 0.3 0.5 2% 3% 3%Unaccounted 0.3 0.2 0.4 4% 2% 3%Sum 6.0 9.4 15.2 100% 100% 100%Table 4.5 and presents the displacement components for specimen TW2. Table 4.6 presents the dis-99Table 4.4: Wall TW1 summary of displacement component increases by load cycle.Displacement IncreaseBy Load Step Percentage Increase of[mm] Component by Load StepLoad Step 19 to 23 23 to 27 19 to 23 23 to 27Panel Flexure 1.4 1.7 41% 29%Panel Shear 0.3 1.0 10% 17%Base Rotation 1.4 2.7 43% 46%Base Sliding 0.2 0.2 5% 4%Unaccounted 0.0 0.2 -1% 4%Sum 3.4 5.9 56% 62%placement component increases by load cycle. As shown, in load step 15 panel flexure dominates the totaldisplacement at 76%. As the steps increase, the proportion of displacement due to flexure decreases drasti-cally to under 20% of the total. Base rotations begin to rapidly increase growing by 54% from load step 15 to17 and 380% from load cycle 17 to 19. Similar as observed in wall TW1, shear displacements steadily makeup approximately 20% of the total observed displacement. It is noted that load step 19 was observed toelicit a drop in moment carrying capacity of the member. From these results, the failure of the wall sectionoccurred at the base of the wall, as was observed and noted in previous sections.Table 4.5: Wall TW2 summary of displacement components.MeasuredDisplacements Percentage of[mm] Total DisplacementLoad Step 15 17 19 15 17 19Panel Flexure 7.5 8.1 3.2 76% 54% 19%Panel Shear 1.9 3.2 3.6 19% 21% 22%Base Rotation 0.6 3.3 9.0 6% 22% 55%Base Sliding 0.2 0.2 0.0 2% 2% 0%Unaccounted -0.4 0.0 0.5 -4% 0% 3%Sum 9.8 14.9 16.4 100% 100% 100%Table 4.6: Wall TW2 summary of displacement component increases by load cycle.Displacement IncreaseBy Load Step Percentage Increase of[mm] Component by Load StepLoad Step 15 to 17 17 to 19 15 to 17 17 to 19Panel Flexure 0.6 -4.9 11% -328%Panel Shear 1.3 0.5 26% 31%Base Rotation 2.7 5.7 54% 380%Base Sliding 0.1 -0.2 1% -16%Unaccounted 0.4 0.5 8% 33%Sum 5.1 1.5 52% 10%Table 4.7 and presents the displacement components for specimen TW4. Table 4.8 presents the dis-placement component increases by load cycle. Since TW4 is the companion element to wall TW1, somecomparisons are noted in addition to the general observations. As shown, the proportion of displacement100due to panel shear are held relatively constant at around 26% of the observed total. Panel flexure representsroughly half of the total observed displacement, showing a declining trend as displacements increase. WallTW2’s base rotations make up less than 25% of the total, unlike the nearly 40% as observed in wall TW1.Additionally, displacements due to increases of base rotations tend to decline as total displacement increases.Table 4.7: Wall TW4 summary of displacement components.MeasuredDisplacements Percentage of[mm] Total DisplacementLoad Step 26 32 48 26 32 48Panel Flexure 3.1 4.2 6.6 52% 46% 48%Panel Shear 1.6 2.2 3.6 27% 24% 26%Base Rotation 1.1 2.4 3.3 18% 27% 24%Base Sliding 0.0 0.1 0.2 0% 1% 2%Unaccounted 0.1 0.1 0.0 2% 1% 0%Sum 6.0 9.0 13.6 100% 100% 100%Table 4.8: Wall TW4 summary of displacement component increases by load cycle.Displacement IncreaseBy Load Step Percentage Increase of[mm] Component by Load StepLoad Step 26 to 32 32 to 48 26 to 32 32 to 48Panel Flexure 1.1 2.4 36% 51%Panel Shear 0.6 1.4 19% 29%Base Rotation 1.3 0.9 43% 19%Base Sliding 0.1 0.1 4% 2%Unaccounted -0.1 -0.1 -2% -2%Sum 3.0 4.6 50% 51%Table 4.9 presents the displacement components for specimen TW5. Table 4.10 presents the displacementcomponent increases by load cycle. Unlike the previous three specimens, the results of the final two loadcycles are presented for wall TW5. As shown, panel flexure at failure made up 73% of the total displacement,due to a decline in measured shear displacements. Unlike the previous three members, base rotations makeup less than 15% of the total observed displacements. The lower base rotations observed in wall TW5 whencompared with those of TW2 are likely due to the higher shear height of the member which serves to decreasethe moment gradient in the element, resulting in less concentrated moments at the base of the member.The displacement components at load cycle maximum drifts for all members is shown graphically inFigure 4.31. The results of the displacement component analysis shows that for all of the members, the totaldisplacement is driven largely by panel flexure, panel shear, and base rotation. Each main displacementcomponent accounts for roughly 50%, 25% and 25% respectively, with some variation to these proportionsobserved. As total displacement is increased, base rotations increase more rapidly than panel flexure resultingin generally increasing proportions of base rotations. Increases of shear displacement proportions appear tobe relatively stable as total displacement rises.1010246810121416LS19 LS23 LS27Displacement [mm]Load StepUnaccountedPanel FlexurePanel ShearBase RotationBase Sliding(a) Specimen TW10246810121416LS32 LS42 LS48Displacement [mm]Load StepUnaccountedPanel FlexurePanel ShearBase RotationBase Sliding(b) Specimen TW4-20246810121416LS15 LS17 LS19Displacement [mm]Load Step(c) Specimen TW2-20246810121416LS32 LS38Displacement [mm]Load StepUnaccountedPanel FlexurePanel ShearBase RotationBase Sliding(d) Specimen TW5Figure 4.31: Displacement components at load cycle maximum drifts.102Table 4.9: Wall TW5 summary of displacement components.MeasuredDisplacements Percentage of[mm] Total DisplacementLoad Step 32 38 32 33Panel Flexure 4.4 9.3 48% 73%Panel Shear 2.3 1.5 25% 12%Base Rotation 1.6 1.7 17% 13%Base Sliding 0.0 0.0 0% 0%Unaccounted 0.4 0.1 4% 1%Sum 9.1 12.7 100% 100%Table 4.10: Wall TW5 summary of displacement component increases by load cycle.Displacement IncreaseBy Load Step Percentage Increase of[mm] Component by Load StepLoad Step 33 to 38 32 to 38Panel Flexure 5.0 137%Panel Shear -0.8 -22%Base Rotation 0.1 3%Base Sliding 0.0 0%Unaccounted -0.2 -6%Sum 3.6 40%4.3.2.7 Influence of Out-of-plane DemandsThe EPFL tests represent an opportunity to to determine the effect, if any, of imposed out-of-plane demandson slender and lightly reinforced wall elements. By imposing cycles of out-of-plane drift to two of the testspecimens (TW4 and TW5), any effects on the in-plane behaviour can be determined by comparison to thespecimens loaded solely in-plane (TW1 and TW2).TW1 and TW4 represent the more slender of the two types of specimen. As discussed in Section 4.3.2.1,the drift capacity of TW4 was slightly lower than that observed in wall TW1. This result may indicatean influence due to out-of-plane loading, however the difference is not overly dramatic, and in addition theobserved displacement ductility was only slightly lower (2.3 versus 2.4). Of note however, is that memberTW4 was not subjected to repeated cycles of in-plane displacement.The observed failure modes of TW1 and TW4, as presented in Section 4.3.2.2, were largely similar withboth determined to have failed due to excessive compression strain demands at the slender toe of both walls.As indicated was the asymmetry of compression zone damage in member TW4 which may have been due tothe presence of out-of-plane cycle, which served to favour damage to a single side of the wall.The vertical strains, curvatures, and shear strains measured in both members did not indicate any largeaberrations between the two loading configurations. Strains were observed to be relatively uniform for boththe compression and tension extremes, and the distributions of curvatures were not significantly different.The shear strains in both members were observed to be large at the base and uniformly distributed alongthe height.Walls TW1 and TW4 indicate that out-of-plane demands may lead to increased, asymmetric damage to103the compression zone of the wall. This conclusion is supported by the analysis of EPFL researchers whofocused on the out-of-plane response of the test specimens (Rosso et al. 2018). The presence of amplified out-of-plane deformations due to imposed out-of-plane deformations at the wall top, does appear to correspondto a higher level of damage to he compression zone of the wall, however the implications of this additionaldamage appear to be benign. The drift capacity of the wall was only modestly impacted and the sectionaldemands were not significantly influenced.TW2 and TW5 represent the less slender of the two types of specimen. As discussed in Section 4.3.2.1,the drift capacity of TW5 was slightly higher than that observed in wall TW2. This result is counter-intuitive,as the out-of-plane demands are expected to lower the drift capacity. This suggests that the difference indrift capacity is due to differences in material properties. A supporting argument for the difference due tomaterial properties is the difference between the observed displacement ductilities of 6.4 for TW2 and 4.9for TW5. This may suggest that the out-of-plane demands are having some effect on the drift capacity.The observed failure modes of TW2 and TW5, as presented in Section 4.3.2.2, were largely similar withboth determined to have failed due to excessive compression strain demands at the slender toe of bothwalls. Asymmetry of compression strains induced across the compression zone of wall TW5 may be due toout-of-plane demands, and may have served to damage a single side of the wall.The vertical strains, curvatures, and shear strains measured in both members were largely dissimilar dueto the difference in induced shear span in the members. Aside from the difference in height of inelasticcurvatures, the maximum measured curvatures were of a similar order.Walls TW2 and TW5 indicate that out-of-plane demands may lead to increased, asymmetric damage tothe compression zone of the wall. This conclusion is supported by the analysis of EPFL researchers whofocused on the out-of-plane response of the test specimens (Rosso et al. 2018). The presence of amplified out-of-plane deformations due to imposed out-of-plane deformations at the wall top, does appear to correspondto a higher level of damage to he compression zone of the wall, however the implications of this additionaldamage appear to be benign. The drift capacity of the wall was only modestly impacted and the sectionaldemands were not significantly influenced.4.3.3 Analytical Methods for Predicting Wall ResponseA wide array of analysis tools exist, with an equally wide array of complexities. The simplest methods includeempirically derived solutions based on simple observations, which are useful for low complexity problems. Themost complex analysis methods are general-purpose, non-linear, finite element(NLFE) computer programs,which require a in-depth understanding of the physical mechanics involved to successfully derive solutions.While NLFE analysis represents the gold-standard of solutions available, the time and effort required forimplementation can easily surpass what is reasonable in everyday practice. The sectional analysis methodoffers a practical middle ground, it has had wide success and is by far the most widely implemented methodof analysis. This study will use both sectional and NLFE analysis methods to gain insight into the behaviourof thin, lightly reinforced concrete walls.4.3.3.1 Sectional AnalysisAs mentioned, sectional analysis forms the standard basis for member design and analysis for most rein-forced concrete structures. The main assumption made in sectional design is that strains which arise from104shear, bending, and axial forces act in such a way that their distribution is linear across the depth of themember. This assumption is often described as “plane sections remain plane”. This assumption provides thecompatibility necessary to determine a solution based on internal equilibrium and the material properties.Plane sections analysis represents the simplest method of determining flexural displacements in reinforcedconcrete elements.This study uses the program Response2000 (R2k) (Bentz 2016) to implement the sectional analysismethod. This software is selected for its ease of use, ability to implement advanced material models, and thewide variety of results available. To perform analyses using R2k, many model options must be made. Forthis study, R2k is used to make estimations of flexural displacements only. The reinforcing material modelis implemented directly with the parameters as listed in Table 4.2. The concrete model assumed uses theproperties listed in Table 4.1, and applies them to a parabolic concrete stress-strain model, a tension stiffeningmodel as described by Bentz 2000, and a compression softening model as described by Vecchio and Collins1986. The maximum concrete compressive strain in the plane sections models is taken as εc,max = 0.0035to reflect the standard of practice under Canadian design standards. No shear loads are applied in the R2kmodels, as only the flexural response of the member is of interest.4.3.3.2 NLFE AnalysisNon-linear finite element (NLFE) analysis is a method of analysis which has been applied to concretestructures for many years. The non-linear aspect is derived from the ability to apply non-linear materials orgeometry. Finite element refers to the underlying mathematical model which solves the static or dynamicinternal and external equilibrium of the model. Many general purpose and speciality NLFE analysis packagesexist, however many are too general or too cumbersome to implement with ease. With this in mind, theprogram VecTor2 (VT2) (Vecchio 2019) was selected as the NLFE software of choice for this study.Developed by researchers the University of Toronto, VT2 is based on the Modified Compression FieldTheory (Vecchio and Collins 1986) and the Disturbed Stress Field Model (Vecchio 2000), which are analyticalmodels used in prediction of the response of reinforced concrete subjected to in-plane normal and shearstresses. The formulations applied in VT2 represent a state-of-the-art model for shear behaviour of reinforcedconcrete structures. The plane-sections assumption is not held, and shear deformations are included in theanalysis. The method of analysis used in VT2 is two-dimensional, however some out-of-plane effects withrespect to material properties are captured by the VT2 analysis.Pre-peak Concrete Compression A main input of the NLFE analysis in VT2 is the selection of thevarious concrete material models to be applied. The concrete material model implemented in VT2 for thisstudy is based an amalgamation of several models. The pre-peak compression model applied is the parabolicHognestad model described as,fci = fp(2(εciεp)−(εciεp)2)for 0 > εci ≥ εp (4.14)where the concrete stress (fc) and strain (εc) are negative values (compression). The initial elastic tangentstiffness used to determine the strain at peak stress is defined as,Ec =2fp|εp| (4.15)105This concrete compression stress-strain relation is typically accepted for concretes with cylinder strengthsof f ′c < 40 MPa. It is noted that specimen TW2’s cylinder strength exceeds this recommendation, howevermodifying a single specimens model is not deemed as appropriate for the needs of this study.Post-peak Concrete Compression The concrete post-peak response is important for near failure re-sponse of the overall member. A modified Park-Kent model is implemented in VT2 which accounts for anyconfinement present in the member. This modification may seem superfluous for the members describedin the EPFL test program, however some level of confinement is provided near the toe of the wall, wherethe change in cross section from the wall to the foundation serves to restrict transverse strains in the wallsection, resulting in a confined behaviour. The model is as follows,fci = − (fp + Zmfp (εci − εp)) < 0 or − 0.2fp for εci < εp < 0 (4.16)where Zm is a factor which accounts for changes based on concrete cylinder strength, transverse principalstresses, and the compressive strain at peak stress.Concrete Compression Softening Compression softening is the loss of compressive strength and stiffnessdue to transverse cracking and tension strains. The Vecchio 1992-A model is implemented in VT2 and allowsreductions to both strength and stiffness. This model makes reductions to the concrete cylinder strength,and the strain at peak stress through the factor βd. The factor is defined as,βd =11 + CsCd≤ 1 (4.17)where Cs and Cd are factors which account for the presence of shear slip and the ratio between elementprincipal compression and tension strains respectively.Concrete Tension Stiffening Tension stiffening accounts for the variations in reinforced concrete stiffnessat, and away from crack locations. At crack locations, the tensile stresses are resisted predominately by thereinforcing steel. Away from cracks, the bond between concrete and reinforcing allows the two materials toshare stresses, resulting in higher stiffness than away from cracks. Tension stiffening is implemented in theVT2 models using the Modified Bentz 2003 model. This model is described as,fc1 =f ′t1 +√ctεc1for εc1 > ε′t (4.18)where ct is a factor which accounts for the reinforcing ratio, reinforcing diameter, inclination of the principalaxis, and the inclination of the reinforcing.Concrete Tension Softening Concrete tension softening is the post-cracking tensile stresses which arepresent in plain concrete. As plain concrete is loaded in tension, tension stresses do not suddenly dissipateafter the maximum stress is attained, however a gradual reduction in stress occurs as strains increase post-peak. This phenomenon is important to the response of lightly-reinforced concrete elements which mayexhibit brittle failure modes. The model applied in the VT2 analysis is the non-linear Hordijk model. Thismodel uses a degradation of concrete strength in tension based on an empirically derived relation which isbased mainly on the ratio of crack width to ultimate crack width, and a fracture energy parameter.106Concrete Confined Strength As previously mentioned, concrete confinement is not provided by rein-forcing in the EPFL VT2 models, however the top loading beam and bottom foundation serve to restricthorizontal strains in the top and bottom rows of wall model elements. This restriction of strains results ina lateral compressive stress developing at the toe of the wall which produces a confining effect. To capturethis effect, VT2 implements the Kupfer/Richart model which uses the difference in normal lateral stressesin an element to modify the concrete cylinder strength and strain at peak stress.Concrete Cracking Criterion Concrete cracking strength, fcr differs from tensile strength and varieswith member size, compressive strength, and the stress state of the member. In general, the cracking strengthis reduced by the presence of transverse compression stresses. VT2 uses a stress based Mohr-Coulomb modelto determine the concrete cracking strength. This model uses a Mohr’s circle of stress to defined the stressstate at which cracking will occur. VT2 assumes the internal angle of friction, φ to be 37 degrees and fromthis it is possible to determine the ultimate cracking stress (fcru) which occurs when the transverse normalcompression stress fc3 = 0. To then determine the cracking stress, the Mohr’s circle is again solved forthe state in which the failure envelope is tangent to the Mohr’s circle with the given value of transversecompressive stress. This solution is defined as,fcr = fcru(1 +fc3f ′c)for 0.20f ′t ≤ fcr ≤ f ′t (4.19)Once the cracking strength is computed using this solution, the cracking strain is computed using the linear-elastic relationship εcr = fcr/Ec.Concrete Crack Stress Criterion The shear stress at a crack is determined using the basic method asdefined in the DSFM/MCFT. VT2 offers the ability to use a more advanced crack check criteria, however itis not warranted for the purpose of this study. Shear stresses at a crack arise through aggregate interlock,and diminish as normal tension stresses are increased.Concrete Crack Width Check As crack widths increase, the ability of local compressive stresses to betransmitted across a crack is reduced. To account for this phenomenon, a crack check is performed in VT2to limit the compressive stresses. The limiting crack width applied in this study is the 40% of the aggregatesize. Once the crack width w is calculated to be larger than the limiting value wl, a reduction of averagecompressive stress-strain response is performed with the factor,βcr = 1− (w − wl) /3 ≥ 0 (4.20)Since the aggregate size is not noted in the data published by the EPFL researchers, the VT2 defaultmaximum aggregate size of 10 mm has been applied for this study.Concrete Crack Slip Calculation Shear strains along the crack face result in increased shear strains ofa reinforced concrete element. VT2 allows this effect to explicitly included as it is forms a part of the DSFM.For this study, the Walraven approach to determining crack slip is implemented. This method relates thelocal shear stresses at the crack vci, to the level of crack slip applied to the DSFM. The Walraven modeluses a model based on aggregate interlock to make the estimate of crack slip.107Reinforcement Stress-Strain The reinforcement stress strain model in VT2 is applied using the materialproperties as presented in Table 4.2, with some modifications made to calibrate the models to the testresults. The model applies a initial linear-elastic relation, followed by a yield plateau, and a non-linear strainhardening segment until fracture strain is attained. The VT2 reinforcing model is defined as,fs =Esεs for εs ≤ εyfy for εy < εs ≤ εshfu + (fy − fu)(εu−εsεu−εsh)4for εy < εs ≤ εsh0 for εu < εs(4.21)The model is applied for both compression and tension loading cases.Reinforcement Dowel Action Resistance to shear forces through the reinforcing bars which cross acrack, as the crack slips transversely to the reinforcing axis is known as dowel action. Dowel action contributesto the shear strength and the post-peak ductility of reinforced concrete members which have low amountsof transverse reinforcing. The model applied in VT2 for this study is the Tassios model, which is an elastic-plastic force-displacement relationship based on the amount of shear slip, δs along the element crack. Themodel dowel force is taken as,Vd = EsIzγ3δs ≤ Vdu (4.22)where Vdu is the ultimate dowel force which corresponds to plastic hinging of the reinforcing and crushingof the concrete at the dowel interface. The shear resistance offered by the dowels is converted to a smearedarea of shear reinforcing based on the reinforcing ratio relevant to the direction of loading.Reinforcement Buckling Bar buckling occurs when the reinforcing is strained in compression to thepoint where lateral support no longer provides stability to the axially loaded bar. The method of modellingthis behaviour in VT2 is a reduction of post-yield reinforcing compressive strength calculated mainly usingthe unsupported length ratio as the main input parameter. For the purpose of this study, it is assumed thatthe unsupported length is based on the full clear height of the test member, or 2000 mm for the EPFL tests.Analysis Model Options Several generic analysis model parameters are required to be selected in theVT2 model. For this study, strain rate effects are not considered, since the loading scheme in the testswas not uniform with respect to strain rate. Geometric non-linearity is considered, with care taken inconstruction of the analysis model to appropriately apply loads as they were in the test program. Crackspacing is determined according to the 1978 CEB-FIP model.4.3.4 Flexural CapacityThe maximum flexural moment capacity of a reinforced concrete section is a function of its material prop-erties, concrete geometry, and reinforcement configuration. In this section comparisons are made betweenthe observed, plane sections predicted, and NLFE predicted flexural capacities of specimens TW1 and TW2.The effect of shear on the NLFE predicted level of flexural capacity is studied. Finally, the effect of theassumed reinforcing stress-strain relation is considered.1084.3.4.1 Observed and Predicted Flexural CapacityThe observed and predicted flexural capacities are presented in Table 4.11 wall TW1, and 4.12 for wall TW2.Each table includes results for the various analyses performed in both in-plane directions of applied loading.Table 4.11: TW1 comparison of observed and predicted flexural capacities for in-plane bending.(a) Bending inducing tension in the flange.Shear % of Moment % of[kN] Observed [kN-m] ObservedObserved 173 1739Plane Sections(R2k) 0 0% 1601 92%Plane Sections(R2k) True Stress-Strain 0 0% 1629 94%NLFE (VT2) 159 92% 1621 93%NLFE (VT2) Reduced Shear 53 31% 1621 93%NLFE (VT2) True Stress-Strain 164 95% 1672 96%(b) Bending inducing compression in the flange.Shear % of Moment % of[kN] Observed [kN-m] ObservedObserved 154 1534Plane Sections(R2k) 0 0% 1449 94%Plane Sections(R2k) True Stress-Strain 0 0% 1458 95%NLFE (VT2) 153 99% 1459 95%NLFE (VT2) Reduced Shear 50 33% 1439 94%NLFE (VT2) True Stress-Strain 156 101% 1493 97%Table 4.12: TW2 comparison of observed and predicted flexural capacities for in-plane bending.(a) Bending inducing tension in the flange.Shear % of Moment % of[kN] Observed [kN-m] ObservedObserved 758 2368Plane Sections(R2k) 0 0% 2168 92%Plane Sections(R2k) True Stress-Strain 0 0% 2131 90%NLFE (VT2) 678 90% 2173 92%NLFE (VT2) Reduced Shear 212 28% 2176 92%NLFE (VT2) True Stress-Strain 702 93% 2244 95%(b) Bending inducing compression in the flange.Shear % of Moment % of[kN] Observed [kN-m] ObservedObserved 689 2190Plane Sections(R2k) 0 0% 1797 82%Plane Sections(R2k) True Stress-Strain 0 0% 1762 80%NLFE (VT2) 656 95% 1916 87%NLFE (VT2) Reduced Shear 203 30% 1909 87%NLFE (VT2) True Stress-Strain 680 99% 1993 91%Specimen TW1 achieved a maximum flexural capacity of 1739 kN-m for bending with the flange intension, and 1534 kN-m for the flange in compression. The plane sections analysis resulted in predictionswhich were 0.4% lower when tension is induced in the flange, and 3.5% higher when compression is inducedin the flange. This result is of interest since the plane sections analysis is a pure flexural response only, withno shear effects applied to the model. The NLFE prediction of wall TW1 was 6.8% lower than observed when109tension is induced in the flange, and 4.9% lower when compression is induced in the flange. To determinewhether or not the underestimation of the NFLE estimate is due to shear effects in the model, a secondNLFE prediction of wall TW1 was performed with the applied shear reduced to approximately 30% of thatobserved from the tests. These model showed no appreciable increase in flexural capacity.Specimen TW2 achieved a maximum flexural capacity of 2368 kN-m for bending with the flange intension, and 2190 kN-m for the flange in compression. The plane sections analysis resulted in predictionswhich were 0.6% lower when tension is induced in the flange, and 9.6% lower when compression is induced inthe flange. As with specimen TW1, this result is of interest since the plane sections analysis is a pure flexuralresponse only, with no shear effects applied to the model. The NLFE prediction of wall TW2 was 8.2% lowerthan observed when tension is induced in the flange, and 12.5% lower when compression is induced in theflange. To determine whether the reduced NLFE flexural capacity is a result of the shear model appliedin VT2, models with roughly 30% of the test observed shear were analysed. As was observed in specimenTW1, the reduced shear models of TW2 resulted in no major difference to the predicted maximum flexuralcapacity.The probable cause of the plane sections under-predictions is likely the result of the plane sectionsassumption itself. As will be shown in a proceeding section, vertical strains along the length of these wallsis not predicted to remain plane due to their lengths. The plane sections assumption may under estimatevertical strains near the neutral axis, leading to reduced vertical reinforcing strains and resulting internalstresses. This would result in lower internal resultant reinforcing force, and by extension the internal moment.Since the NLFE model does not apply a plane sections assumption, it would not be the cause of theunder-predictions for these models. Since the NLFE analysis uses an analysis model which includes sheareffects, it may be that this model is serving to influence the flexural capacity in such a way that it is reduced.To investigate this effect, identical NLFE models of TW1 and TW2 are implemented, however significantlyreduced levels of shear are applied. The result of this analysis is that a 70% reduction in shear to the NLFEmodels results in no appreciable difference to the ultimate flexural capacity of these two specimens.With shear effects ruled out in leading to reductions of flexural capacity, other factors must be hypoth-esised. One potential issue is the validity of the reinforcing stress-strain relationship at high strains. TheEPFL material properties presented in their study are the result of tests performed on several test speci-mens, so naturally some variation is present. However further study of the reinforcing test properties doesnot result in variations which would lead to large observed differences at the full member level.Variation in reinforcing material properties aside, the definition of the stress-strain relation at highstrains may be at issue. As is typical of tests performed on reinforcing specimens, the resulting stress-strain relationship is the engineering relationship. That is to say the stress-strain relationship is based on aconstant cross-sectional area. This relationship is known to be appropriate at low levels of strains, howeveras strains exceed their elastic limit, the validity of the relationship begins to degrade. For conventional linearanalysis models, which are not based on the deformed geometry of the elements, the engineering relationshipis appropriate. However, for non-linear analyses which calculate stresses based on the deformed elementgeometry, errors in the estimation of stress at high strains may occur.To better predict the member internal stress state when the analysis is heavily into the materially non-linear portion of the analysis, the true stress-strain relation should be applied which takes into considerationthe internal stress-state of the deformed geometry.110The true stress-strain relationship is known to follow the relationship:σt = σe (1 + εe) (4.23)εt = ln (1 + εe) (4.24)By applying this relationship to the material properties presented in the EPFL study and implementingit into the NLFE models for TW1 and TW2, it is found that the maximum flexural capacity is raised to 96%and 97% of the test observed levels for member TW1, and 95% and 91% for TW2. As shown, the result of theimproved stress-strain model is observed levels of flexural capacity which are all within 5% of the observedvalues, with the exception of TW2 bending to induce compression in the flange. The maximum flexuralcapacity for member TW2 bending for compression induced in the flange is noted to be highly sensitiveto the fracture strain of the reinforcing, since the flexural capacity was positively increasing at the point offailure, which indicated the specimen’s ultimate flexural capacity is governed by bar fracture. The EPFL testresults corroborate this failure mode as the governing case. Limitations on customization of the reinforcingstress-strain model in the NLFE models limit the ability to allow for residual fracture strain capacity paststrain at ultimate stress. This limitation is also expected to contribute to the slightly lower than predictedflexural capacity, as it limits the ability for tension reinforcing to exceed the strain at ultimate stress priorto fracture occurring.4.3.4.2 Discussion on the Determination of Flexural CapacityPredictions of the flexural capacity of t-shaped sections by several different methods has shown variousdegrees of accuracy (Orakcal and Wallace 2006; Pantazopoulou, Moehle, and Shahrooz 1988). For this studyit is hypothesised that the inaccuracy of both the plane sections and NLFE predictions stems from variationsof the reported material properties and modelling limitations. One known limitation of the NLFE model isrigorous modelling of the wall base to foundation interface. Large vertical strains are known to have beenpresent in the test specimens (as shown in the test results section), it is likely that reinforcing tension strainswere also very high. The higher than predicted reinforcing tension strains would result in increased levels ofstrain hardening to occur, resulting in higher tension stresses than predicted by the NLFE models.To address the irregularities in the predictions of the NLFE model, the decision was made to make minoradjustments to the reinforcing material properties to better reflect the observed flexural capacity, so longas the changes do not significantly alter the overall behaviour of the member. Since it is assumed that thevertical reinforcing was able to attain a larger amount of strain hardening than the NLFE predicted, anadjustment to the yield strength of the reinforcing to allow for a higher strength levels was applied. For wallTW1, the 16 mm reinforcing was adjusted to be elastic-plastic, with a yield strength equal to the recordedultimate strength. This adjustment resulted in a higher attained maximum flexural capacity recorded to be1751 kN-m, which is 6.9% higher than observed in the test. Wall TW2 was also adjusted to reach a higherflexural capacity by changing the 6 mm reinforcing in the flange only to an elastic-plastic material whichyields at the ultimate recorded stress. After the adjustment was made, the TW2 NLFE model had a flexuralcapacity of 2406 kN-m, which is 1.6% greater than observed in the test.The adjustments made to the reinforcing properties result in NLFE predictions which are capable ofattaining the flexural capacity observed in the test specimens. This result suggests that the large base crackobserved in the tests did allow for a higher than predicted level of strain hardening to occur in the verticalreinforcing. Also since the plane sections prediction had success in determining the flexural capacity, a brief111study of the effect of horizontal variation of vertical strains is discussed in a subsequent section.4.3.5 Verification of NLFE ModelA goal of this study is to use the results of the NLFE models to supplement the test results, the remainingflexural capacity discrepancy is adjusted to better match the results of the tests. This is accomplishedby adjusting the reinforcing and concrete material properties to create a better fit to the observed load-displacement data. This so called “perfect” fit model will be applied for the remainder of this study.For members TW1 and TW2, the NLFE model was observed to be slightly stiffer than observed. This islikely due to the difference in strain rates between the concrete material tests and the full-scale wall tests.There is a known proportional relation between strain rate and observed stiffness. It was observed that theconcrete cylinder tests were subjected to a strain rate which was approximately 10 times greater than inthe wall tests. To account for this effect, the concrete model was softened to better reflect the as testedconditions.Specimen TW1’s concrete cylinder initial tangent stiffness was decreased from 25,300 MPa to 22,500MPa, a reduction of 11%. Specimen TW2’s concrete cylinder initial tangent was decreased from 31,800 MPato 27,000 MPa, a reduction of 15%. The higher reduction necessary for TW2 is due in part to the inaccuracyof the parabolic concrete model applied to this member with a measured cylinder strength of f’c = 50.7MPa.As discussed in the previous section the flexural capacity of the NLFE models underpredicts those ob-served in the tests. To account for this, the vertical reinforcing which undergoes tension strains, has its yieldstress increased to bring the prediction of flexural capacity up to the observed levels.For member TW1 the 16mm and 6mm reinforcing stresses over the range of plastic strains is raised by5%. For member TW2 the 6mm reinforcing stresses over the range of plastic strains is raised by 15%.The load-deformation response for the panel section of the wall alone for members TW1 and TW2 areshown in Figures 4.32 and 4.33. The moment presented is at the base of the wall, and the deformation isthe lateral displacement at 2 m above the base of the wall.As shown both NLFE models produce good estimates of the load-deformation response. In particular, thelevel of drift observed at ultimate is closely predicted. The TW1 NLFE load-deformation prediction slightlyover estimates the post-cracking, pre-yield stiffness, however produces close member stiffness predictionsfor the uncracked and post-yield conditions. The TW2 NLFE load-deformation prediction produces goodestimates of stiffness, however the response near yield, and the post-yield stiffness is slightly over-estimated.These comparisons of NLFE prediction to the observed test results is evidence that the predictions areaccurately capturing the member load-deformation behaviour.4.3.6 Horizontal Variation of Vertical StrainsThis section investigates the influence of the plane sections assumption on the prediction of maximum flexuralcapacity and displacement.Figure 4.34 shows presents a comparison of the horizontal variation of vertical strains for the planesections and NLFE model predictions at the base of the wall at maximum flexural capacity and deformation.The distribution of vertical strains predicted by the NLFE model for member TW1 clearly shows thatthe web reinforcing undergoes an increased level of strain when compared with the more heavily reinforced1120200400600800100012001400160018000 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Base Moment [kN-m]Horizontal Displacement of Wall Panel [mm]Observed Cyclic ResponseObserved Cyclic BackboneNLFE PredictionFigure 4.32: TW1 comparison of NLFE predicted and observed load-deformation response measured at 2mabove the base of the wall.050010001500200025000 2 4 6 8 10 12 14 16Base Moment [kN-m]Horizontal Displacement of Panel [mm]Observed Cyclic ResponseObserved Cyclic BackboneNLFE PredictionFigure 4.33: TW2 comparison of NLFE predicted and observed load-deformation response measured at 2mabove the base of the wall.113-10-5051015202530350 500 1000 1500 2000 2500Vertical Strain [mm/m]Horizontal Position Along Wall [mm]TW1 NLFE ModelTW2 NLFE ModelTW1 Plane Sections ModelTW2 Plane Sections ModelFigure 4.34: TW1 and TW2 comparison of NLFE and plane sections vertical strains at the wall base atultimate flexural capacity and displacement.flange end zone. This increased strain in the web region serves to increase the internal stress in the webbeyond what would be possible from the plane sections prediction.The distribution of vertical strains by the NLFE model for member TW2 shows that vertical strainsextend uniformly into the web over approximately 1900 mm. The TW1 plane sections and NLFE modelpredicts similar levels of vertical strains in the extreme tension end of the element. The TW2 plane sectionsand NLFE model predicts higher levels of vertical strains in the extreme tension end of the element. Thisdistribution of strains suggests that the plane sections response should produce somewhat accurate, if notconservative estimates of maximum member curvature based on the strains at the extreme member ends.Given that the plane sections predicted curvature produces a somewhat accurate estimate. It follows thatif the prediction over the height of the member is accurate as well, the plane sections curvatures integratedover the would produce similar estimates of flexural displacements.The difference in distribution of vertical strains between the plane sections and NLFE model predictionshelps to explain the differences in flexural capacity of the two methods. The plane sections prediction under-predicts vertical strains in the web, leading to lower levels of stress in the vertical reinforcing here, leadingto a lower internal resultant tension force. Since the majority of the reinforcing in tension exceeds the yieldstrain of the reinforcing for both the NLFE and plane sections models, the difference in flexural capacity isdue to the level of strain hardening which can occur.In addition to lower stresses due to reduced strain hardening effects in the plane sections model, higherstrains are observed near the neutral axis of the NFLE model, resulting in even further increased levelsresultant tension force available. Both of these effects help to explain the lower observed flexural capacityobserved in the plane sections model when compared with the NLFE estimations.114The presence of a “local” curvature is also noted from the results of this analysis. Each NLFE modeldistribution of vertical strains shows a length of wall with an apparent increased curvature toward thecompression end of the wall. This difference in curvature from the tension zone leads to an increasedcurvature gradient, which may affect the compression zone stability.The curvature of member TW1 based on the NLFE extreme fibre strains is 9.82 rad/km, compared with9.81 rad/km as predicted from the plane sections analysis. However, this is compared with a “local” curvaturein the compression zone of 33.51 rad/km. The curvature of member TW2 based on the NFLE extreme fibrestrains is 8.20 rad/km, compared with 7.73 rad/km as predicted from the plane sections analysis. However,this is compared with a “local” curvature in the compression zone of 27.90 rad/km. For member TW1and TW2, the maximum observed compression zone “local” curvature is 3.41 and 3.40 times the apparentcurvature measured from the extreme fibres.4.3.7 Influence of Test SetupSince the EPFL test walls were developed to model the bottom portion of a theoretically taller wall anyeffect this setup may have on the observed behavioural properties such as distributions of vertical and shearstrains is unknown. This section include a brief study on the influence of the test setup used in the EPFLtest program for walls TW1 and TW2. The main behavioural properties to be studied are the horizontaldistribution of vertical strains, vertical distributions of vertical compression and tension strains over theheight of the member, and the vertical distribution of average shear strains.The “full height” models of TW1 and TW2 are assumed to be the height of the theoretical shear span ofeach member, with a constant cross-section and a single lateral load applied at the shear height. The resultsof extreme fibre vertical strains, horizontal variation of vertical strains at the base, and average shear strainsare included in Appendix C.3.The result of the analysis shows that the vertical strains predicted by the full height model showsthat compression strains are largely unchanged over the 2m lower portion of the wall. The tension side,however appears to show some small differences in their distribution and magnitude. The full height NLFEmodel vertical tension strains are slightly more concentrated towards the base of the member, however themagnitude appears largely unchanged. Overall, the full height NLFE model appears to drive vertical strainsapproximately 100mm closer to the base when compared with the results of the test setup NLFE models.In addition to the extreme fibre vertical strain distributions, the horizontal variation of vertical strainsat the base of each model is compared. For both walls TW1 and TW2 the horizontal variation is slightlyaltered, however the results are not dissimilar enough to warrant further analysis.The final test setup influence to be investigated is the distribution of average shear strains over the heightof each NLFE model prediction. Wall TW2 shows little difference in the overall distribution and magnitudesof shear strains. The full height NLFE model of wall TW1 predicts higher shear strains than the test setupprediction at approximately 1000mm and above. The result of the discrepancy is that test setup NLFEmodel may under-predict shear displacements by over the 2m section of the wall by approximately 38%,when compared with the full height NLFE model.The result of this comparison is that the EPFL test setup may restrict the level of shear displacementsin the upper half of the wall if the distribution of strains is such that large inelastic strains are present. Theobserved differences in tension strains are minimal and any resulting effect of the test setup are unlikely toresult in largely dissimilar behaviour when compared with a full height member.1154.3.8 Maximum Tension and Compression Strains at Wall EndsThis section presents the results of the NLFE and plane sections model predicted maximum tension andcompression strains at the wall ends in comparison to the EPFL test observed results.Figure 4.35 shows the distribution of tension strains over the bottom 2m of wall TW1. The NLFEprediction produces a good prediction for the upper half of the wall, however the estimate for the lowerhalf is less well represented. The plane sections prediction of vertical tension strains deviates significantlyover the lower half of the wall, with observed strains under-predicited by as much approximately 100%.The maximum tension strains predicted at the base of the wall in the NLFE and plane sections models isquite close, as was discussed in previous sections. Both predictions in comparison to the observed resultsunder-predict the base vertical tension strain, however as discussed the observed tension strains at the baseof the wall suffer from measurement errors due to strain penetration and the presence of a large base crack.As shown, when the average of the two observed base vertical strains is taken, the result is that the NLFEonly under estimates strains by approximately 25%.Moving to the compression end of wall TW1, the distribution of NLFE and plane sections predictedvertical strains in comparison to the observed test results is shown in Figure 4.36. The compression strainpredictions over the upper 1700mm are very close to the test observed results. The NLFE prediction atthe bottom 300 mm above the base of the wall is significantly lower than observed, however similar to thevertical tension strains, the observed test results at the base vary significantly based on the gage length usedin the determination of strains. In comparison to the NLFE model results, the plane sections analysis underestimates the maximum compression strains.Figure 4.37 presents the NLFE and plane section predictions in comparison to the test observed resultsfor extreme vertical tension strains in wall TW2. As shown the NLFE model predicted tension strainsproduce good estimates of the vertical tension strains, however similar to specimen TW1 the tension strainsat the base of the wall are under-predicted. Unlike the results of wall TW1, the plane sections predictionof vertical tension strains is significantly lower from 200 mm to 1400 mm above the base of the wall. Theplane sections prediction at the base of the wall is higher than that observed in the NLFE model.The NLFE and plane sections predicted extreme compression strains in comparison to the test observedresults are presented in Figure 4.38. Both the NLFE and plane sections models slightly over-predict thecompression strains from approximately 200 mm to 900 mm above the base. The plane sections modelprediction at the base of the wall is significantly lower than that of the NLFE and observed results. TheNLFE model prediction near the base produces a reasonable estimate of the maximum observed.Given the relative simplicity of the NLFE and plane sections analysis methods, both appear to producereasonable estimates of the vertical tension and compression strains. In the case of the NLFE model, had amore complex model of the wall base and foundation interaction been implemented, it is plausible a morerefined result of the ultimate tension and compression strains is possible. With regard to the plane sectionsmodel, the vertical strains are under-predicted just above the base up the level of first applied inelasticmoment. Given that there is a relatively complex internal distribution of forces near the base of the wall,it is unlikely that the plane sections predictions of vertical strains near the base of the wall will produceexcellent estimates. However the use of reasonable maximum values of tension and compression strains inpractice allow for the method to be of significant value. The gains made by the NLFE model in the predictionof vertical strains appears to be limited by the complexity of the modelling assumptions near the base crackof the wall. Granted, very precise estimates of strain levels would be possible, however since the vast majorityof design cases are based on assumed material limitations, the refinements are of little practical use. The11602004006008001000120014001600180020000 5 10 15 20 25 30 35Elevation [mm]Tension Strain [mm/m]NLFE PredictionPlane Sections PredictionObserved Gauge LengthObserved StrainsTwo Strains Averaged Near Wall BaseBase Tension Strain Continuesto 150 mm/mFigure 4.35: TW1 comparison of NLFE predicted and observed tensions strains near the flanged wall end.11702004006008001000120014001600180020000 1 2 3 4 5 6 7 8Elevation [mm]Compression Strain [mm/m]NLFE PredictionPlane Sections PredicitonObserved Gauge LengthObserved Compression StrainsFigure 4.36: TW1 comparison of NLFE predicted and observed compression strains at the 80 mm thick end.1180200400600800100012001400160018002000-5 5 15 25 35 45 55 65 75Elevation [mm]Tension Strain [mm/m]NLFE PredictionPlane Sections PredictionObserved Gauge LengthObserved StrainsTwo Strains Averaged Near Wall BaseBase Tension Strain Continues to 184 mm/mFigure 4.37: TW2 comparison of NLFE predicted and observed tensions strains near the flanged wall end.11902004006008001000120014001600180020000 1 2 3 4 5 6 7 8 9 10Elevation [mm]Compression Strain [mm/m]NLFE PredictionPlane Sections PredictionObserved Gauge LengthObserved Compression StrainsFigure 4.38: TW2 comparison of NLFE predicted and observed compression strains at the 80 mm thick end.120NLFE predictions of vertical strains above the base crack produce good estimates which are useful for designpractice, and research applications.4.3.9 Curvature DistributionsThis section includes a presentation of NLFE and plane sections model predictions of curvature with com-parisons made to TW1 and TW2 test results. The definition of curvature is as discussed in previous sectionsand is based on the strains in the extreme tension and compression fibres of the test or model.Figure 4.39 presents the NLFE and plane sections predictions for wall TW1. As shown the NLFEestimates of curvature match closely over the entire height of the member. This observation is of interest,as in the previous section it was shown that the estimates of vertical strains at the extreme fibres near thebase of the wall are less accurate than the results observed for curvatures. This comparison is made basedon an average the two measured curvatures nearest the base of the wall. The plane sections predictionpresented shows that while the base maximum observed curvature estimate is near that of the NLFE modeland observed results, the estimate of curvature just above this section is under-estimated.Figure 4.40 presents the NLFE and plane sections predictions for wall TW2. Unlike as observed for wallTW1, the NLFE produces somewhat worse estimates of curvatures near the base of the wall, however thedistribution above the base is somewhat better. The irregular distribution of curvatures observed over theheight of member TW2 isn’t well represented by the NLFE model, however the average behaviour is capturedby the model. The plane sections model produces a somewhat better estimate of maximum observed basecurvature, however the estimate of curvature from 200 mm to 1400 mm above the base is significantlyunderestimated.The NLFE and plane sections predicitons of both walls TW1 and TW2 appear to produce relativelyaccurate estimates of curvatures over the height of the member, with the exception plane sections predictionsover the inelastic portion of the member. As was shown in the analysis of vertical strains in the previoussection, the under-predicted curvatures are driven by inaccurate estimates of vertical tension strains due tothe plane sections assumption. The result of this analysis identifies this as a major short coming of applyingthe plane sections assumption to walls of this type. In contrast to this negative result, since the plane sectionsestimates of curvature at the base of the wall produce reasonably accurate estimates, a relation between basemaximum curvature and the distribution may be possible.4.3.10 Shear Strain DistributionsThis section presents the NLFE predicted average shear strains and displacement profiles in comparison tothe EPFL test wall results. The results of this section are used to validate the NLFE shear model whencompared with the test results.Figure 4.41a presents the shear strain distribution for wall TW1. The NLFE model generally predictsshear strains of an appropriate magnitude, however the distribution of the strains somewhat differs fromthose observed in the test. As the NLFE model does not include complex modelling of the base crack, thelarge observed shear strain associated with base sliding effects is not captured. The resulting displacementsresulting from the shear strains presented are shown in Figure 4.41b. The NLFE model provides reasonablepredictions of shear displacements over the height of the wall. When accounting for the large 1 mm basesliding shear, the NLFE model quite accurately matches that observed in the test.Figure 4.42a presents the shear strain distribution for wall TW2. The NLFE model provides as less12102004006008001000120014001600180020000 2 4 6 8 10 12 14 16Elevation [mm]Total Curvature [mm/m]NLFE PredictionPlane Sections PredictionObserved Gauge LengthObserved Total CurvatureAverage of Bottom 387 mm of PredicitonFigure 4.39: TW1 comparison of NLFE predicted and observed total curvature distribution.12202004006008001000120014001600180020000 2 4 6 8 10 12 14 16 18Elevation [mm]Total Curvature [mm/m]NLFE PredictionPlane Sections PredictionObserved Gauge LengthObserved Total CurvatureFigure 4.40: TW2 comparison of NLFE predicted and observed total curvature distribution.12302004006008001000120014001600180020000 1 2 3 4 5Elevation [mm]Average Shear Strain [mm/m]ObservedNLFE PredictionBase Shear Strain Extends to 14 mm/m(a) Shear Strain Distribution02004006008001000120014001600180020000 1 2 3 4Elevation [mm]Shear Displacement [mm]ObservedNLFE Prediction(b) Shear Displacement DistributionFigure 4.41: TW1 comparison of NLFE predicted and observed shear strains and displacements.124irregular distribution than observed in the test specimen, however provides reasonable estimates magnitudesof shear strains over much of the height of the member. The resulting displacements from the distributionsof shear strains is shown in Figure 4.42b. The distribution of NLFE predicted shear displacements over thelower 600 mm matches well with those observed in the EPFL tests. NLFE predicted displacements over theupper 1400 mm are over-estimated when compared with those observed in the test wall.0200400600800100012001400160018002000-1 0 1 2 3 4 5Elevation [mm]Average Shear Strain [mm/m]ObservedNLFE PredictionBase Shear Strain Extends to 10.7 mm/m(a) Shear Strain Distribution02004006008001000120014001600180020000 1 2 3 4 5Elevation [mm]Shear Displacement [mm]ObservedNLFE Prediction(b) Shear Displacement DistributionFigure 4.42: TW2 comparison of NLFE predicted and observed shear strains and displacements.Some minor NLFE prediction errors associated with base sliding and areas of low curvature were observed,however overall the model provides reasonably accurate results for predicting the shear displacements andstrains of the thin and lightly reinforced walls. As was shown in the results of the EPFL members, sheardisplacements can account for upwards of 30% of the total displacement of members of this type. Howeverit is noted that had a full height member been tested, the proportion of shear displacement at full height125would be reduced as rotations near the base are amplified higher up in the member.4.3.11 Crack PatternsThis section presents the NLFE predicted crack patterns in comparison to those observed in the EPFL testwalls TW1 and TW2. The NLFE model does not explicitly produce a predicted crack pattern, so variousresults are aggregated and interpreted to produce a likely crack pattern. The number of horizontal cracksdeveloped at any given vertical section of wall is determined by the reported element crack spacing at thatvertical section. As the crack propagate from the extreme tension fibre into the web of the wall, the initiallyhorizontal crack is rotated based on the NLFE reported element crack direction. Where two cracks meet,they are assumed to develop into a single crack. Starting at the extreme tension fibre, the initial number ofcracks in the flange area is determined by dividing the height of wall by the reported crack spacing.Figure 4.43 presents the observed and NLFE interpreted prediction of crack pattern for wall TW1. Asshown in the upper Figure 4.43a, horizontal cracks initially begin horizontally in the flange area, and as theypropagate into the web they become increasingly diagonal to a maximum of a 45 degree angle. Since theEPFL tests were cyclic, when cracks reach those developed in a previous reverse cycle tend to run along thesecracks until they diverge and begin propagate in the natural direction. This effect of cracks running alongpreviously created cracks tends to allow cracks to propagate further horizontally than they may have, hadnot reverse cycles been introduced. The NLFE interpreted prediction of cracks is shown the lower Figure4.43b. Beginning at the flange, the observed wall had approximately 13 cracks develop, compared with 11main cracks in the NLFE prediction. Continuing into the web, the NLFE prediction has 5 large cracksdevelop significantly into the web, which is similar to the those observed in the test wall. The upper twolarge cracks in the NLFE model extend into the web more horizontally than in the test wall. The test wall’supper most crack develops into a roughly 45 degree shear crack.Figure 4.44 presents the observed and NLFE interpreted prediction of crack pattern for wall TW2. Theupper Figure 4.44a shows the crack pattern as observed in the EPFL test wall. As shown, the main flexuralcracks developed in the flange extend roughly two-thirds of the height of the wall. The cracks then propagateat a roughly 30 degree angle in the web. As observed in wall TW1, when the tension cracks meet thosedeveloped during a previous reverse loading cycle, the cracks propagate along the existing cracks for a shortdistance, until diverging in their natural direction. Figure 4.44b presents the NLFE interpreted predictionfor wall TW2. The number of main flexural cracks predicted by the NLFE model is 8, compared withapproximately 9 in the test wall. The propagation of cracks into the web predicted by the NLFE model is ata slightly flatter angle for the uppermost cracks, however it is noted that the cracked area predicted by theNLFE model was of a similar shape to that observed in the test, with the discrepancy due to the method ofinterpretation used.Overall, the NLFE prediction produces quite accurate predictions of crack patterns. Given that theNLFE model loading was monotonic, compared with the cyclic loading program used in the EPFL tests,it is expected that had the NLFE program been applied cyclically the results would only improve. Thissections serves to further validate the results of the NLFE program for use with thin and lightly reinforcedwalls subjected to in-plane loads.126(a) Observed During Test(b) NLFE Interpreted PredictionFigure 4.43: TW1 comparison of NLFE predicted and observed crack patterns.127(a) Observed During Test(b) NLFE Interpreted PredictionFigure 4.44: TW2 comparison of NLFE predicted and observed crack patterns.1284.4 Development of Proposed ModelsAs was presented in the previous section, while the EPFL tests on thin and lightly-reinforced shear walls wereobserved to deform primarily in flexure, shear deformations can account for roughly 30% of the total observeddeformation near the base of the wall. This result suggests that classical models of flexural deformationsalone may poorly predict drift capacities near the base of these types of walls. This section presents twoproposed models which are used to predict the flexural and shear capacities of thin and lightly-reinforcedconcrete walls.The first model applies a classical plastic hinge analysis for the analysis of flexural deformations. Thesecond model uses the relationship between average element vertical strains and shear strains to produceestimates of shear deformations in members of this type.4.4.1 Plastic Hinge ModelsThis section will introduce the concept of a classical plastic hinge model based on a region of uniformlyvarying inelastic curvature used to estimate first mode flexural displacements in flexural concrete walls.Once the classical plastic hinge concept is established, an alternative formulation of classical plastic hingemodel is established on the basis of a a region linearly varying inelastic curvature. This refined hinge modelis proposed for analysis of thin and lightly-reinforced concrete walls which are shown to exhibit little verticalspread of plasticity resulting in little4.4.1.1 Description of Plastic Hinge Model Based on Uniformly Varying InelasticCurvaturesHF ∆f=∆y+∆iθi12 lpφy φiφlpIdealizedElastic CurvatureUniformlyVaryingInelasticCurvatureActual CurvatureDistributionFigure 4.45: Plastic hinge model based on uniform inelastic curvature.129As a part of the seismic design of a concrete wall, it is important to assess the flexural displacementcapacity in relation to the flexural displacement demand. Inelastic flexural displacement demands are de-termined through non-linear or equivalent linear elastic analysis methods. The total displacement at anypoint over the height of the wall is made up of three displacement components, the flexural displacements∆f , the shear displacements ∆v, and the shear slip displacements ∆s. The sum of the three displacementcomponents results in the total displacement at the level of interest as,∆ = ∆f + ∆v + ∆s (4.25)Inelastic flexural displacement ∆f capacities are analysed on based on the formation of an inelastic(plastic) region near the base of the wall where inelastic curvatures φi dominate over elastic curvatures φe.The total flexural displacement at a given level of the wall is then given by,∆f = ∆ef + ∆if (4.26)where ∆ef and ∆if are the flexural displacements due to elastic and inelastic deformations respectively.Application of the described model of flexural displacements is typically performed on the basis of as-suming that inelastic displacements occur due to a region of uniform inelastic curvature near the base ofthe wall commonly referred to as the plastic hinge length lp. The inelastic curvature is determined from thedifference between the total measured curvature φ and the yield curvature φy,φi = φ− φy (4.27)Inelastic rotations can then be determined by integrating the inelastic curvatures over the plastic hingelength as,θi = φilp (4.28)By considering plastic hinge rotations to be located at the height of the centroid of inelastic curvatures, theinelastic displacements are expressed as,∆i = θi(H − lp2)= (φ− φy) lp(H − lp2)(4.29)The final expression for flexural displacements is given as,∆f = ∆ef + (φ− φy) lp(H − lp2)(4.30)This model is dependant on the assumption of uniformly varying inelastic curvatures near the base of thewall over the plastic hinge length. This section will investigate the assumption of uniformly varying inelasticcurvatures compared to an alternative region of linearly varying inelastic curvatures.4.4.1.2 Description of Plastic Hinge Model Based on Linearly Varying Inelastic CurvaturesAn alternative to the formulation of the plastic hinge model presented in the previous section has beensuggested (Bohl 2006). This alternative model is based on linear variation of inelastic curvatures as shownin Figure 4.46.130HF ∆f=∆y+∆iθi13 l∗pφy φiφl∗pIdealizedElastic CurvatureLinearlyVaryingInelasticCurvatureActual CurvatureDistributionFigure 4.46: Plastic hinge model based on linear inelastic curvature.By applying the same methodology as the uniformly varying inelastic curvature model, inelastic displace-ments can be shown to be,∆i = θi(H − l∗p3)= (φ− φy)l∗p2(H − l∗p3)(4.31)Comparing this result to an equivalent uniform plastic hinge model, the resulting estimate of inelasticdisplacements is similar for walls with small plastic hinge length to wall height ratios. For walls with largerplastic hinge length to wall height ratios, the difference between the two models becomes larger with theuniform model producing higher estimates inelastic displacements than the linear model.For a 10m tall wall, the resulting difference in inelastic displacements varies from less than 1% for aplastic hinge length to wall height ratio of 0.05, to over 12% for a ratio of 0.5. This result shows that as wallsbecome more squat, care must be exercised to ensure that an appropriate plastic hinge based on realisticdistributions of inelastic curvature is applied.Many researchers have used uniform distributions of inelastic curvatures to develop models of plastichinge lengths, while others have used actual curvature distributions. Since the actual extent of plasticcurvatures is roughly twice the plastic hinge length of the uniform inelastic model, and the distribution ofinelastic curvatures is approximately linear, the models presented in the subsequent sections are based onlinearly varying inelastic curvatures.4.4.1.3 Estimates of Elastic CurvaturesTo determine the inelastic contribution of curvature and elastic portion must be subtracted from the totaldistribution over the height of the member. The estimate of elastic curvature can have a significant influenceon the magnitude of inelastic curvature, as well as the apparent height over which plasticity occurs. With131this in mind, this section presents a brief discussion on the selection of elastic curvature estimate used in theremainder of this study.The limit of elastic curvature depends on the definition of the onset of plasticity in the section. As curva-ture increases in a member which is subjected to flexural displacements, the tensile demands on longitudinalreinforcing increase elastically until the onset of plastic deformations at the yield limit. The point of overallsection yielding is dependant on several factors including the method of analysis.Using a plane sections analysis, the most basic and widely applied definition of the elastic curvaturelimit in a reinforced concrete section is often taken as the point at which reinforcing first begins to yield,beginning with the most extreme bars and progressing inward towards the compression zone as curvaturesare increased. Typically this point of first yielding represents a well defined point on a moment-curvatureinteraction diagram of the section where the slope begins to fall and the overall sectional response begins tosignificantly soften.Depending on the reinforcing configuration over the cross-section of the member, reinforcing may beginto yield at the most extreme end of the member, however this may not coincide with the point of apparentoverall section yielding as observed on the moment-curvature diagram. This type of behaviour can beobserved in members with low boundary reinforcing ratios in comparison to the distributed reinforcing ratio.In this case, the overall section yield point may not occur until a larger proportion of distributed longitudinalreinforcing has reached the yield limit.Given the differences noted, the definition of elastic curvature limit which is most appropriate to applyis dependant on the application in question.As a more in-depth analysis method, NLFE model estimates of elastic curvature can be more difficult todefine depending on the reinforcing configuration and non-linear properties of the reinforcing and concrete.Similar to the plane sections estimate, a member with no zones of concentrated reinforcing and uniformreinforcing properties will begin to yield longitudinal reinforcing at the most extreme bars first. Howeverfor more complex reinforcing configurations, such as shear walls with areas of concentrated zone reinforcingat the ends and comparatively small amounts of distributed reinforcing, the first onset of longitudinal barreinforcing may occur in the web section prior to the most extreme bars. Depending on the relative amountsof reinforcing present in each zone, yielding may occur in distributed reinforcing much sooner than is apparenton a plot of the overall section moment-curvature behaviour. This type of behaviour present as a result ofthe NLFE analysis complicates the definition of the elastic curvature limit when applied.In the context of this study, it was shown that the EPFL test members typically exhibited an apparentelastic curvature limit which is more easily defined using a plane section estimate of elastic curvature basedon first yield of the extreme reinforcing bars. This point of first bar yielding also corresponded well withthe point of apparent overall section yield as observed from the results of the plane-sections analyses of theEPFL test members. The plane sections elastic curvature estimates are shown below in Table 4.13.Table 4.13: EPFL Test specimen estimates of elastic curvature based on plane sections analysis.Specimen Name Plane Sections Estimate of Elastic Curvature [rad/km]TW1 1.10TW2 0.83TW4 1.05TW5 0.981324.4.1.4 Distributions of Inelastic CurvaturesEstimates of inelastic curvatures in the EPFL test specimens and the NLFE results are taken by subtract-ing the elastic curvature limit discussed in the previous section, from the total measured distributions ofcurvatures. An example plot of the inelastic curvatures as observed in the EPFL test wall TW1 is shownin Figure 4.47. The results for walls TW2, TW4, TW5 including those determined by NLFE analysis forspecimens TW1 and TW4 are included in Appendix C.4.1.4.4.1.5 Lengths of Linearly Varying Inelastic CurvatureThe model of linearly varying inelastic curvature is applied to the observed distributions of inelastic curva-tures by creating an equivalent linear distribution. Equivalence is based on producing a model which hasboth equal area and first moment of area as the observed results. This equal area and first moment of areamethod ensures the equivalent model of linearly varying curvature has the same displacement characteristicsas the test specimen.To assess the vertical spread of plasticity present in the EPFL test members and NLFE estimates, theequivalent linear model results are presented alongside the estimate of plastic hinge length estimate basedon reinforcement yielding,l∗p = z(1− MyMmax)(4.32)By comparing these two lengths, an estimate of the vertical spread of plasticity is made.As shown in Figure 4.48, EPFL test specimen TW1 shows no spread of plasticity over its height. Inelasticcurvatures for this specimen are restricted to the height over which reinforcement is expected to yield, evenwith the formation of large diagonal shear cracks.In contrast to these results, the results of the thicker wall TW2 are shown in Figure 4.49. Unlike specimenTW1, this wall shows significant spread of vertical plasticity beyond the height over which reinforcement isexpected to yield.Similar results for the NLFE estimates of specimens TW1 and TW2, and the test results for TW4 andTW5 are included in Appendix C.3.3.4.4.1.6 Model of Plastic Hinge Length for Thin Lightly-Reinforced Concrete WallsThe results of the EPFL test specimens in comparison to the model of linearly varying inelastic curvaturesare summarized in Table 4.14.13302004006008001000120014001600180020000 1 2 3 4 5 6 7 8Elevation [mm]LS17LS19LS21LS23LS25LS27LS290501001502000 10 20 30 40 50Elevation [mm]Inelastic Curvature [rad/km]Figure 4.47: TW1 observed distributions of inelastic curvature for load-steps 17 to 29.13402004006008001000120014001600180020000 2 4 6 8 10 12 14 16Elevation [mm]Inelastic Curvature [rad/km]Observed Gauge LengthObserved Inelastic CurvatureLinear Model - Equal Area and Moment of AreaLinearly Varying Curvature due to Reinforcement YieldingFigure 4.48: TW1 model of linearly varying inelastic curvature based on equal area and first moment of areain comparison to length of reinforcement yielding.13502004006008001000120014001600180020000 2 4 6 8 10 12 14 16Elevation [mm]Inelastic Curvature [rad/km]Observed Gauge LengthObserved Inelastic CurvatureLinear Model - Equal Area and Moment of AreaLinearly Varying Curvature due to Reinforcement YieldingFigure 4.49: TW2 model of linearly varying inelastic curvature based on equal area and first moment of areain comparison to length of reinforcement yielding.136Table 4.14: Summary of EPFL specimen linearly varying predicted yield moments, inelastic curvatures, plastic hinge lengths, spread of inelasticcurvatures, and maximum compression strains.Specimen Name TW1 TW1 TW2 TW2 TW4 TW5NLFE NLFESpecimen DetailsWall Thickness [mm] 80 80 120 120 80 120Height to Thickness Ratio 25.0 25.0 16.7 16.7 25.0 16.7LoadingLoad Stage 27 Max. 17 Max. 48 38Maximum Bending Moment Applied at Base [kN-m] 1728.7 1728.7 2367.7 2367.7 1650.1 2260.1Applied Shear Force [kN] 172.4 172.4 757.1 757.1 165.0 307.4Shear Height M/V [mm] 10029 10029 3127 3127 10000 7354Yield MomentPredicted My [kN-m] 1505 1505 1950 1950 1392 1844.71Height from Base where Bending Moment at Yield [mm] 1753 1753 620 620 1455 1092Predicted CurvaturesPredicted Curvature at Reinforcement Yield [rad/km] 1.10 1.10 0.83 0.83 1.05 0.98Predicted Inelastic Curvature Capacity, φi,pred [rad/km] 8.20 8.20 7.56 7.56 9.40 5.69Linear ModelHeight of Linearly Varying Inelastic Curvature, l∗p[mm] 1583 1441 1153 1428 1297 1689Maximum Inelastic Curvature, φi [rad/km] 7.74 7.62 6.65 7.30 10.71 5.44φi/φi,pred 0.94 0.93 0.88 0.97 1.14 0.96Maximum Measured Inelastic Curvature[kN-m] 9.00 8.97 8.47 7.41 4.03 7.98Spread of InelasticCurvatureHeight of Inelastic Curvatures Above My, Hi [mm] -170 -312 533 809 -158 597Hi/Lw -0.06 -0.12 0.20 0.30 -0.06 0.22Max. CompressionStrainPredicted φc,max at φi,max 0.0024 0.0024 0.0045 0.0045 0.0021 0.0035Maximum Measured Compression Strain 0.0026 0.0026 0.0059 0.0059 0.0027 0.0063137As shown, specimens TW1 and TW4 exhibit no apparent vertical spread of inelastic curvature, withmaximum compression strains which are similar to those predicted by plane sections estimates. By com-parison, the thicker members TW2 and TW5 have a distinct spread of vertical plasticity beyond the yieldheight. The spread of inelastic curvatures for the two EPFL test specimens TW2 and TW5 were 20% and22% of the wall length, and the NLFE model of TW2 was 30% of the wall length. Plane sections predictionsof maximum compression strains for the thicker members was typically good, with the exception of specimenTW5 which reached notably higher than predicted maximum compression strain.The model of linearly varying inelastic curvatures in comparison to the test results shows that the thinnerEPFL test specimens TW1 and TW4 show little tendency for spread of inelastic curvatures to occur. Thisis contrasted by the thicker specimens TW2 and TW5 distinctly observed vertical spreading of plasticityabove the yield height of the walls. This result illustrates that thin and lightly reinforced wall membersmay not form a well defined plastic hinge region due to spread of vertical plasticity at the base of the wall.This lack of vertical spread of plasticity contributes to lower than expected overall top of wall displacementcapacity through reduction of the overall area enclosed by the inelastic curvature distribution at the base ofthe wall. A simple lower bound estimate of the distribution of inelastic curvatures based on the height ofreinforcement yielding is suggested for members of this type.4.4.2 Shear Strain Estimation Based on Average Vertical StrainBased on observations from the EPFL test walls and the UCLA database analysis, many thin and lightly-reinforced walls have been shown to have low levels of measured overall displacement capacity. Since dis-placement ductility of flexural walls is typically developed through inelastic flexural hinging at the base, asthe measured level of flexural ductility decreases, shear strains may begin to contribute a proportionallyhigher amount to overall displacements.Typically, as the wall height to length aspect ratio increases, the overall displacements at the top of thewall begin to quickly be dominated by rotations induced in the plastic region which develops near the base.For thicker, more heavily reinforced walls which are able to develop high levels of displacement ductility,shear strains make up very little of the overall displacement. In contrast to the case of wall with high flexuralductility, thin and lightly-reinforced members subject such as those in the EPFL test program, have lowlevels of flexural ductility. This low flexural ductility allows the usually low proportion of shear strains tobecome more substantial.To account for the shear strains present in flexural dominated walls, it has been proposed that the averagevertical strains can be used to determine shear strains (Bazargani and Adebar 2015). This section presentsthe method of using average vertical strains to perform shear strain analyses, and uses the analysis results ofthe EPFL walls TW1, TW2, TW4, and TW5 from the previous section to validate the utility of the modelwhen applied to thin and lightly reinforced concrete walls.4.4.2.1 Description of Shear Strain ModelBy considering a simple wall element with uniformly spaced vertical and horizontal reinforcing subject toa tensile uniaxial vertical normal strain εv and a shear strain γvh, it has been shown that this element isable to represent the flexural hinge region of a concrete wall (Bazargani and Adebar 2015). This model onlyrequires an estimate of average shear angle θ and vertical tensile strain to establish an estimate of shear138strains by using a plane strain transformation. This transformation is defined as,γvh = εv (− tan 2θ) (4.33)Previous research has suggested that an average principal strain angle of θ = 75 degrees be applied. Byapplying this to the simplified strain model reduces to the expression,γ¯vh = 0.58ε¯v (4.34)where γ¯vh is the average shear strain, and ε¯v is the average vertical strain. The subsequent section will usethe results of the EPFL test walls to validate the use of this expression for thin and lightly-reinforced walls.4.4.2.2 Shear Model Analysis ResultsThe simple shear model was applied to the results of the EPFL test walls TW1, TW2, TW3, and TW4.Figure 4.50 presents the model predicted shear strain estimates versus the measured average vertical strain atthe individual measurement elevations. For discrete measurements at each gage length, the model producespoor results. This indicates that the shear model assumption that several cracks are present is importantto the applicability of the model. Figure 4.51 presents the EPFL test specimen results for average membershear strain over their full heights, versus the estimated average vertical strain at mid-height of the testspecimens. As shown the model produces good estimates of average shear strain. This shows that overmultiple crack locations, the average shear strains are well correlated with average vertical strains. Theresults of the EPFL tests shown serve to validate the simple shear strain model based on average verticalstrain of these thin and lightly-reinforced walls. However the results for individual measurements over theheight of the test members do not produce good results for this model. This results may be due to thecoarseness of strain measurements or some measurement gage lengths not crossing crack locations. Thiseffect will be investigated by investigating the results of the NLFE models for walls TW1 and TW2.Further to the analysis of the EPFL test specimen results, the results of the NLFE model for wallsTW1 and TW2 are presented in Figure 4.52. These plots show the distribution of average shear strain overthe height of each wall in comparison to shear model using the estimated average vertical strain at eachelevation from a plane sections assumption of strains across the length of the wall. As shown in Figure4.52a, the results for wall TW1 show that the average predicted shear strains near the base of the wall areoverestimated. Conversely, in higher elevations the shear strain response is under predicted in comparison tothe NLFE model results. The former results is due to the presence of high vertical strains predicted in theNLFE model near the base of the wall, with low predicted shear strains due to boundary effects. The underpredicted results higher in the wall are due to the presence of high vertical strains in the lightly reinforcedweb section which are not captured by the plane sections assumption of vertical strains over the length ofthe wall.Figure 4.52b presents the results over the member height for the TW2 NLFE model. As shown, thismember produces much better results over the height of the wall. This better prediction is due to themember’s uniform vertical reinforcing which does not produce overly irregular vertical strains over the lengthof the wall. In addition, the more closely spaced cracks in wall TW2 better match the shear strain modelassumption of several uniform cracks over the biaxial shear element.1390.0000.0020.0040.0060.0080.0100.000 0.002 0.004 0.006 0.008 0.010γεTW1TW4TW2TW51.0:1.00.6:1.0Figure 4.50: EPFL test specimen observed shear strains at individual measurement elevations in comparisonto estimated average vertical strains.1400.00000.00050.00100.00150.00200.00250.00300.00350.00400.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.0040γεEPFL Test Walls1.0:1.00.6:1.0Figure 4.51: EPFL test specimen observed average shear strains in comparison to estimated average verticalstrains.1410100020003000400050006000700080009000100000 1 2 3 4 5Elevation [mm]Average Shear Strain [mm/m]NLFE ModelAverage VerticalStrain Model(a) TW1 NLFE Results0500100015002000250030000 1 2 3 4Elevation [mm]Average Shear Strain [mm/m]NLFE ModelAverage VerticalStrain Model(b) TW2 NLFE ResultsFigure 4.52: NLFE predicted average shear strain in comparison to estimate based on average vertical strain.1424.5 ConclusionThis chapter presented an in-depth look at the in-plane behaviour of thin and lightly-reinforced concreteshear walls. Using the UCLA RCWalls database, parameters which affect the global drift capacity of thesetypes of walls were identified. This analysis resulted in the development of a refined model of wall globaldrift capacity with respect to shear span and compression zone slenderness. Using the results of the databaseanalysis, a set of test specimens with particularly low levels of recorded global drift capacity was selected forfurther analysis. This set of wall specimens tested at EPFL presented a unique opportunity to apply testspecimen results to a different application than the original testing program research topic. The availabilityof high quality test data allowed an in-depth analysis of the load-deformation response, failure modes, verticalstrains, shear strains, and curvatures in these specimens. A NLFE model was then calibrated to the EPFLspecimens, and the results of which were applied to validate models of inelastic curvature and shear strains.The UCLA RCWalls database identified that thin and lightly-reinforced wall drift capacity is mainlyaffected by compression zone slenderness, c/t, and shear span ratio, M/V lw. This result is of interest as itwas expected that walls with high shear stresses or axial loads would result in low observed drift capacity.The results of this analysis were then applied to a simple linear model of maximum drift capacity based oncompression zone slenderness and shear span ratio. The model was shown to have good results in predictingmaximum drift capacities of the test specimens with little observed scatter.The EPFL wall test results identified that out-of-plane buckling failures may have occurred after theformation of a compression strain limit failure, which bears the conclusion that while out-of-plane effectsmay contribute to cyclical degradation of the compression zone, the ultimate failure is not destabilizationof the wall panel out-of-plane. This result is supported by the presence of high strains measured in thecompression zone, as well as observed in test video. When the NLFE element models were developed, thefailure mode was also a compression zone strain failure.Analysis of EPFL wall shear and vertical strains showed that plastic hinge zone horizontal deformationsconsist of significant amounts of wall base rotation, panel flexure, and panel shear. These three componentsmust all be accounted for in any estimation of displacement in the plastic hinge zone of this type of wall.NLFE and plane sections models of the selected EPFL wall specimens were well calibrated with the testresults and the model results were used to augment the observed data in validating a plastic hinge modelfor thin and lightly reinforced shear walls. It was shown that even though vertical strains were not linearlydistributed across the length of the wall were not linearly distributed, the plane sections analysis resultedin good estimations of overall member curvature capacity. The NLFE results were able to confirm that thetest set up was unlikely to have caused any reduction in spread of vertical plasticity in the test specimens.Using models of linearly and uniformly varying inelastic curvature, it was shown that these thin andlightly-reinforced walls exhibit very little vertical spread of inelastic curvatures. This behaviour results in aplastic hinge length which is based mainly on the height over which the extreme fibre reinforcing is yieldingwith additional vertical reinforcing strain demand due to shear stress playing a smaller role than in moreheavily reinforced members. Since accurate estimations of plastic hinge length are important in many linearand non-linear analysis situations, this result is of significant importance.Finally a model of estimated shear strains based on average vertical strains was validated using the resultsof the EPFL NLFE models. These results showed that this model which is typically applied to more heavilyreinforced walls also holds valid in thin and lightly-reinforced elements.143Chapter 5ConclusionThe use of lateral force resisting systems consisting of a main core of cantilever and coupled walls is ubiquitousin the design of high rise buildings in Canada. Understanding how thin and lightly-reinforced walls influence,and are influenced by these structural systems is of utmost importance to practising engineers, researchers,local governments, and building end users. Ensuring our codified design procedures are robust and consistent,and exploring new design and analysis tools, serves to attain the goal of ensuring appropriate levels ofstructural safety are achieved. This thesis has presented insight on a broad range of topics related to thinand lightly-reinforced concrete bearing walls.5.1 Demands on Gravity Load Walls due to Lateral Response ofCantilever and Coupled Wall BuildingsThe first main chapter of this thesis was dedicated to framing some of the issues with thin and lightly-reinforced within the context of a typical structure representative of those in South Western British Columbia.To this aim a fictitious building with some typical features observed in new and old high rise buildings inBritish Columbia was analysed. As was shown, the behaviour of the core wall structure under lateralwind and seismic loading was influenced by the presence of several thin wall elements which are a part ofthe main gravity resisting system. The sectional forces on the thin wall elements under wind and seismicloading were shown to be significant, with walls in the coupled direction shown to resist up to 15% of thetotal overturning moments in a combination of bending and unintended outrigger effect. In contrast to theseemingly manageable ultimate force levels in the thin walls, a simple analysis of inelastic rotational demandsand capacities of these elements shows that the ability for these thin wall elements to undergo deformationsof the full system are of paramount importance. The thin walls undergo significant levels of inelastic demand,far in excess of what they are able to withstand.5.2 Thin Lightly-Reinforced Bearing WallsThe next subject of study in this thesis was the treatment of uni-axially loaded thin and lightly-reinforcedbearing walls in Canadian design codes. Uni-axial bearing wall test results were aggregated and comparisonsto code based rational and empirical design methods were made. It was shown that the rational momentmagnifier approach to design and analysis of bearing walls was well correlated with test results. Unconserva-tive discrepancies were found when applying the CSA A23.3-14 empirical design equation. Specifically, thelack of consideration of sustained loading effects, concrete strength, member buckling factors, and effectivelengths served to provide higher estimated capacity results when compared with the more rigorous momentmagnifier method. With an aim at improving uniformity across the various allowable code based design andanalysis methods, a new simplified empirical bearing wall equation was derived based on the behaviour of144an unreinforced wall section. This new design equation was shown to provide similar axial load carryingcapacity results to the moment magnifier method for lightly reinforced wall sections.5.2.1 Empirical Bearing Wall Design in CSA A23.3-14Empirical bearing wall design by the CSA A23.3-14 method has been established to be a generally usefultool for estimating short term, monotonic axial load capacity of lightly reinforced, and modestly slender wallmembers (ρv ≈ 0.002 and h/t ≤ 25). Perhaps the most significant drawback of the CSA A23.3-14 empiricalbearing wall design method, is the omission of sustained load effects. By comparing the empirical designmethod with another well established method of analysis, the moment magnifier method, it was shown thatthe empirical method may produce estimations of axial load which can be in the range of 150% to 250%higher than those estimated by the moment magnifier method under the same conditions. The effect ofsustained loads is of particular concern in members with high ratios of dead to total loads βd. Mid and highrise residential buildings may have sustained load ratios βd as high as 0.90 in many cases. This is due inpart to the use of live load reductions based on floor area carried by a member, and also the relatively lowdesign live loads of residential structures when compared with structure self weight.It was also shown that as concrete cylinder strengths increase above f ′c ≈ 40MPa, a reduction in nor-malized axial load capacity occurs. In the worst case CSA A23.3-14 allowable scenario (h/t ≤ 25), amember’s capacity was over estimated by approximately 20%. Given the small sample size of members withf ′c > 40MPa (n = 17), the failure rate of 35% for members in this category is of particular concern.There is an inherent uncertainty in the level of critical buckling load due to the estimates of flexuralstiffness and concrete strength. To account for the potential of a reduced critical axial load, a memberbuckling factor φm = 0.75 is employed. It was shown that the empirical method closely approximates themoment magnifier when a critical buckling factor of φm = 1.0 is applied. Since this would not be the case inpractice, this results in yet another avenue of strength over estimation when the empirical method is applied.For this reason it is suggested that the empirical method is recalibrated to better fit with the results of themoment magnifier method when a member buckling factor of φm = 0.75 is applied.5.2.2 Rational Method of Design in CSA A23.3-14The moment magnifier method of slender compression member design in CSA A23.3-14 provides a rationalmethod to estimate second-order effects without the use advanced analysis tools. Estimations of axial loadcapacity using the moment magnifier method have been shown to generally produce good estimations ofaxial load capacity, however depending on the assumptions made may produce overly conservative results.The method was validated against 111 test specimens or various slenderness, concrete strength, and levelsof reinforcing.One of the main factors affecting the estimation of axially loaded members using the moment magnifiermethod is the estimate of flexural stiffness used in the analysis. Deflections along the length of the member aredetermined according to an equivalent flexural stiffness which encapsulates the varied response of the actualmember. Equation 3.36 was shown to provide a conservative estimate of strength, however was certainlyoverly conservative in many cases, especially at higher slenderness levels. Equation 3.37 produced goodresults across the range of slenderness, however some over estimates of axial load capacity were observed.Equation 3.38 produced the best results based on overall member behaviour, however produced somewhatunconservative estimates in some cases, particularly for higher slenderness levels. As a result of this analysis,145Equation 3.36 is not recommended for use for lightly reinforced slender bearing wall elements, as it producesan overly conservative estimate of capacity which results in an unwarranted and inefficient use of resources.Equation 3.37 is recommended for use in typical members of this type, and has been shown to producereasonable estimates of axial load capacity without producing overly conservative results. The alternativeEquation 3.38 does provide an excellent overall prediction of member behaviour, with the lowest observedvariance of the the three methods. However its use is not recommended at this time, as it is uncertain if thelevel of failure produces an acceptable level of risk when applied to a much larger number of members.As was noted for empirical method of axial load estimation, sustained load effects result in significanteffects on the behaviour of slender bearing wall elements. Levels of sustained load to total load in residentialhigh rise construction can be upwards of 90% and the effects are not insignificant. The moment magnifiermethod approach reduces the effective flexural stiffness to account for sustained loading effects, however thelack of experimental data to confirm this behaviour creates some uncertainty. Since few or no axial loadfailures of members under sustained loads have been observed in practice, it is cautiously assumed that themethod is at least satisfactory, if not conservative.It was also shown that ambiguity exists between the CSA A23.3-14 Clause 10.13.2 and Clause 14.1.7.1maximum slenderness limits. Chapter 14 indicates a maximum slenderness of h/t > 25, however directsthe user to the provisions of Chapter 10 for the detailed design of the element, which specifies a maximumslenderness ratio of 100, or h/t = 30 for a rectangular cross sections. Better guidance on slenderness limitsfor members and when those limits apply may reduce user misapplication.5.2.3 Simplified Rational Member Design EquationA simplified equation to determine the maximum axial load capacity of a lightly reinforced concrete wallmember derived from the axial-moment interaction of an unreinforced member, and the magnified momentusing the moment magnifier method is presented as,Pr =A2−√(A2)2−B(1− 2et)(5.1)where,A = φmPcr + α1f′cAgB = φmPcrα1f′cAgThis new equation provides a significantly improved method of member design based on a well establisheddesign approach. It was shown that the equation is valid for lightly reinforced members with little or no lossto the maximum axial load capacity, since light reinforcing provides little strength gain to the member. Anaddition to the reduced effort and increased accuracy of the new equation when compared to the full momentmagnifier method and the empirical equation, no loss of sustained loading effect or reduced member bucklingcapacity as implemented in CSA A23.3-14 is present. The newly derived equation represents a significantimprovement to the empirical equation which this thesis has confirmed can produce unconservative andunreliable estimates of maximum axial load.1465.3 Thin Lightly-Reinforced Shear WallsThe focus of the thesis then shifted to address the more complex response of thin and lightly-reinforced wallssubjected to in-plane shear and bending demands.5.3.1 UCLA RCWalls Database AnalysisA database analysis of past test results was used to estimate the aggregate behaviour of this style of wall.It was found that overall member drift capacities less than 1% were often observed. A parametric analysisof the database test results was performed and it was shown that overall wall drift capacity was moststrongly influenced by wall shear span MV lw , and compression zone slendernessct . These results were appliedto the derivation of a simple empirical equation which predicts wall overall drift capacity based on theaforementioned parameters.The UCLA RCWalls database identified that thin and lightly-reinforced wall drift capacity is mainlyaffected by compression zone slenderness, c/t, and shear span ratio, M/V lw. This result is of interest as itwas expected that walls with high shear stresses or axial loads would result in low observed drift capacity.The results of this analysis were then applied to a simple linear model of maximum drift capacity based oncompression zone slenderness and shear span ratio,δchw(%) = 3.85− λα− vmax0.83√f ′c(5.2)The model was shown to have good results in predicting maximum drift capacities of the test specimenswith little observed scatter.5.3.2 Analysis of EPFL Walls and Proposed ModelsIt was shown that the displacement capacity of these flexurally dominated walls included significant sheardisplacements and base rotations. Very little vertical spread of plasticity was present, and inelastic verticalstrains were mostly constrained within the expected height of reinforcing yield based on simple sectionalflexural analysis. NLFE and plane sections models of the selected EPFL wall specimens were well calibratedwith the test results and the model results were used to augment the observed data in validating a plastichinge model for thin and lightly reinforced shear walls. It was shown that even though vertical strains werenot linearly distributed across the length of the wall, the plane sections analysis resulted in good estimationsof overall member curvature capacity.Using models of linearly and uniformly varying inelastic curvature, it was shown that these thin andlightly-reinforced walls exhibit very little vertical spread of inelastic curvatures. The observed behaviourresults in a plastic hinge length which is based mainly on the height over which the extreme fibre reinforcingis yielding with additional vertical reinforcing strain demand due to shear stress playing a smaller role thanin more heavily reinforced members. Since accurate estimations of plastic hinge length are important inmany linear and non-linear analysis situations, this result is of significant importance. Analysis of EPFLwall shear and vertical strains showed that plastic hinge zone horizontal deformations consist of significantamounts of wall base rotation, panel flexure, and panel shear. These three components must all be accountedfor in any estimation of displacement in the plastic hinge zone of this type of wall.Finally a model of estimated shear strains based on average vertical strains was validated using the results147of the EPFL NLFE models. These results showed that this model which is typically applied to more heavilyreinforced walls also holds valid in thin and lightly-reinforced elements.5.4 Opportunities for Future ResearchSeveral opportunities for future research were identified as a result of the various research topics presented:• The effect of sustained eccentric axial loads on thin and lightly-reinforced bearing wall elements.• Validation of effective length factors applied to the analysis uniaxial load carrying capacity of thin andlightly-reinforced bearing wall elements.• Instability of slender compression zones in thin and lightly-reinforced walls subjected to combinedin-plane shear and significant axial loads.• Plastic hinge lengths in thin and lightly-reinforced members with low levels of vertical spread of plas-ticity in walls with significant observed diagonal shear cracking.• Influence of base crack formation on overall drift of thin and lightly-reinforced walls subjected toin-plane shear.148BibliographyAbdullah, Saman A. and John W. 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In: Earthquake Spectra.151Appendix ADemands on Gravity Load Walls Dueto Lateral Response of Cantilever andCoupled Wall BuildingsA.1 Example Building DescriptionA.1.1 Building GeometryThis section includes the typical floor layout used in the case study presented in Chapter 2. The typicalcolumn layout is based on real buildings in the Vancouver, BC area to reflect real world design configurations.Figure A.1 shows the typical floor plan of the example building.A.1.2 Material PropertiesThe material properties applied to the analysis model are as follows:Specified concrete cylinder strengths:Wall elements 30 MPa to 55 MPa Column elements 35 MPa to 55 MPa Typical slabs 35 MPa Transferslabs 45 MPaConcrete modulus of elasticity is defined according to the CSA A23.3-14 standard as E = 4500√f ′c.Reinforcing steel yield strength, fy = 400MPa Reinforcing steel modulus of elasticity, Es = 200000MPaA.2 NBCC Wind DemandsStatic and dynamic wind demands are based on NBCC 2005 provisions. The final wind demands used foranalysis are those resulting from the dynamic analysis procedure. The results of the static and dynamic loadestimation methods are included in the following sections. Results include design pressure distributions,resulting story shears, overturning moments, cumulative storey shears, cumulative overturning moments,and relevant intermediate calculations.NBCC 2005 requires that wind pressure distributions be determined based on the following formula,p = IwqCeCgCp (A.1)where Iw is the importance factor for wind loads, q is the reference velocity pressure, Ce is the exposurefactor, Cg is the gust effect factor, and Cp is the external pressure coefficient.The importance factors for wind loads for normal buildings are 1.0 and 0.75 for ultimate limit state(ULS),and serviceability limit states(SLS), respectively. The ULS factors are presented for the determination of the15227.00 ft23.02 ft12.51 ft 12.51 ft23.02 ft29.19 ft78.07 ft112.06 ft14.33 ft 23.10 ft 23.10 ft 14.35 ft1.60 ft 1.57 ft5.84 ft11.34 ft20.41 ft36.87 ft20.41 ft11.34 ft5.84 ft4.78 ft 5.39 ft 8.59 ft3.33 ft4.91 ft2.51 ftFigure A.1: Example building typical floor plan.153design pressure distribution, and the 0.75 SLS reduction is applied as required during model implementationof the wind pressure.The reference velocity pressure for this building is 0.47 kPa based on the assumed location in the GreaterVancouver area. Reference velocity pressures are based off of historical and predicted 1/50 year maximumhourly wind events which are published as part of the NBCC.Exposure factors are based off of the changes in wind speed due to height, and roughness of surroundingterrain. The static and dynamic procedures differ in their determination of this factor. Each factor used willbe presented in their respective sections.Gust factors attempt to reduce several different effects into a single modification. Effects include randomfluctuations caused by turbulence of incoming wind, fluctuations induced by the wake of the structure itself,inertial forces from the motion of the structure in response to fluctuating wind forces, and aerodynamic forcesdue to alterations in the airflow caused by the motion of the structure itself (aero-elastic effects). The gustfactor accounts for any resonant effects of inertial forces caused by excitation close to a natural vibrationfrequency. Normally smaller buildings which are not expected to have significant resonant component can beanalysed using the static procedure, while taller and more slender structures require more in depth analysisusing the dynamic method. The determination of the gust factor for the static and dynamic methods willbe presented in each respective following section as required.The final factor required is the external pressure coefficient. This factor represents a modification to thewind pressure distribution based the results of scale model wind tunnel testing. The static and dynamicprocedures differ in their determination of this factor. Each factor used will be presented in their respectivesections.Applying the equation for external pressure results in a pressure distribution which varies with heightover the structure on the windward side, and an accompanying leeward constant suction which acts intandem. From this final pressure distribution, the overall shear and overturning moment demands can beeasily determined. When implemented in a structural analysis software program, an accurate estimate ofthe demands on the building can be easily determined. The results of both such analyses for the building ofinterest are presented in the following sections.The external pressure coefficient used in the static and dynamic methods is based on an assumed pressuredistribution which varies by ratio of height to width in the direction of wind applied. The windward coefficientis determined by,Cp =0.6 for H/D < 0.250.27(H/D + 2) for 0.25 < H/D < 10.8 for H/D ≥ 1(A.2)and the leeward coefficient by,Cp =0.3 for H/D < 0.250.27(H/D + 0.88) for 0.25 < H/D < 10.5 for H/D ≥ 1(A.3)In addition to full pressure distributions determined from either the static or dynamic analysis, partialloading effects are required to be analysed. The required partial loading cases are as shown in Figure A.2.154A.2.1 Static Wind CalculationsAs discussed in the previous section, the static wind demands are based on determining the external pressureusing the equation presented. As has been discussed, the building in this study requires the implementation ofthe dynamic analysis method, however it is of interest to also determine the static distribution for comparison.The exposure factor Ce determined for use in the static method is based on the surrounding terrain ofthe building site. For this building the terrain is typically flat with mostly low rise buildings, with a fewscatted high rise also present. This type of surrounding area represents “rough terrain” as described in theNBCC 2005 commentaries. The rough terrain exposure factor in this case is defined as Ce = 0.7(h/12)0.3,but not less than 0.7 which is applied at each level for the windward case, and at the reference height forthe leeward case.The gust effect factor for the static case is simply defined for the building as a whole as Cg = 2.0.The results of the static procedure wind loading are as shown in the following.155Static Wind Loading - NBCC 2005 Description: East West Direction Static Wind Loading (X Direction)Building Height Above Grade, H 100.26 m 328.84 ftLeeward Side Reference Height 50.13 m 164.42 ftBuilding Dimension in Y-Dir, Dy 30.15 m 98.89 ftCross Wind Dimension in Y-Dir, Wy 21.57 m 70.74 fth/Dy 3.3 3.3Cp Windward Side 0.80 0.80Cp Leeward Side 0.50 0.501/50 Year Reference Pressure, q50 0.47 kPa 9.8 psfULS Importance Factor, Iw 1.00SLS Importance Factor, Iw 0.75Leeward Exposure Factor, Ce 1.07 Assumed rough terrain per 4.1.7.5.5Gust Effect factor, Cg 2.0 Structure as a whole per 4.1.7.1.5.6External Wind PressureCpCgULS Pe [kPa]ULS Pe [psf]Y-Dir Leeward 1.0 0.51 10.6ZoneExternal156LevelFloor to Floor [ft]Elevation [ft]Height Above Ground [ft]Height Above Ground [m]Wind Height [m] Width [ft] Width [m]43 0 633.84 328.84 100.26 1.51 29.17 8.8942 9.92 623.93 318.93 97.23 2.92 29.17 8.8941 9.24 614.69 309.69 94.42 2.99 29.17 8.8940 10.40 604.29 299.29 91.25 3.08 104.00 31.7139 9.82 594.47 289.47 88.25 2.82 112.00 34.1538 8.67 585.80 280.80 85.61 2.64 112.00 34.1537 8.67 577.14 272.14 82.97 2.64 112.00 34.1536 8.67 568.47 263.47 80.33 2.64 112.00 34.1535 8.67 559.80 254.80 77.68 2.64 112.00 34.1533 8.67 551.14 246.14 75.04 2.64 112.00 34.1532 8.67 542.47 237.47 72.40 2.64 112.00 34.1531 8.67 533.80 228.80 69.76 2.64 112.00 34.1530 8.67 525.14 220.14 67.11 2.64 112.00 34.1529 8.67 516.47 211.47 64.47 2.64 112.00 34.1528 8.67 507.80 202.80 61.83 2.64 112.00 34.1527 8.67 499.14 194.14 59.19 2.64 112.00 34.1526 8.67 490.47 185.47 56.55 2.64 112.00 34.1525 8.67 481.80 176.80 53.90 2.64 112.00 34.1523 8.67 473.14 168.14 51.26 2.64 112.00 34.1522 8.67 464.47 159.47 48.62 2.64 112.00 34.1521 8.67 455.80 150.80 45.98 2.64 112.00 34.1520 8.67 447.14 142.14 43.33 2.64 112.00 34.1519 8.67 438.47 133.47 40.69 2.64 112.00 34.1518 8.67 429.80 124.80 38.05 2.64 112.00 34.1517 8.67 421.14 116.14 35.41 2.64 112.00 34.1516 8.67 412.47 107.47 32.76 2.64 112.00 34.1515 8.67 403.80 98.80 30.12 2.64 112.00 34.1512 8.67 395.14 90.14 27.48 2.64 112.00 34.1511 8.67 386.47 81.47 24.84 2.64 112.00 34.1510 8.67 377.80 72.80 22.20 2.64 112.00 34.159 8.67 369.14 64.14 19.55 2.64 112.00 34.158 8.67 360.47 55.47 16.91 2.64 112.00 34.157 8.67 351.80 46.80 14.27 2.64 112.00 34.156 8.67 343.14 38.14 11.63 2.64 112.00 34.155 8.67 334.47 29.47 8.98 2.64 112.00 34.153 8.67 325.80 20.80 6.34 2.64 112.00 34.152 8.67 317.14 12.14 3.70 3.17 112.00 34.151 12.14 305.00 0.00 0.00 1.85 112.00 34.15157LevelWindward Exposure Factor, CeULS Windward Pressure, [kPa]ULS Leeward Pressure, [kPa]ULS Total Storey Pressure, [kPa]ULS Applied Wind Load at Level, [kN]ULS Cumulative Story Wind Load, [kN]ULS Overturning Moment at Level, [kN-m]ULS Cumulative Overturning Moment, [kN-m]43 1.32 1.00 0.51 1.50 20.17 20 2022 202242 1.31 0.99 0.51 1.49 38.73 59 3766 578841 1.30 0.98 0.51 1.48 39.47 98 3726 951440 1.29 0.97 0.51 1.47 143.92 242 13132 2264639 1.27 0.96 0.51 1.46 140.80 383 12426 3507338 1.26 0.95 0.51 1.45 131.21 514 11233 4630637 1.25 0.94 0.51 1.45 130.41 645 10820 5712636 1.24 0.93 0.51 1.44 129.59 774 10410 6753635 1.23 0.92 0.51 1.43 128.75 903 10002 7753833 1.21 0.91 0.51 1.42 127.90 1031 9597 8713532 1.20 0.90 0.51 1.41 127.02 1158 9196 9633131 1.19 0.89 0.51 1.40 126.11 1284 8797 10512830 1.17 0.88 0.51 1.39 125.18 1409 8402 11353029 1.16 0.87 0.51 1.38 124.23 1534 8009 12153928 1.14 0.86 0.51 1.37 123.25 1657 7621 12916027 1.13 0.85 0.51 1.35 122.24 1779 7235 13639526 1.11 0.84 0.51 1.34 121.20 1900 6853 14324825 1.10 0.83 0.51 1.33 120.12 2020 6475 14972323 1.08 0.81 0.51 1.32 119.00 2139 6100 15582322 1.07 0.80 0.51 1.31 117.85 2257 5729 16155221 1.05 0.79 0.51 1.29 116.64 2374 5363 16691520 1.03 0.77 0.51 1.28 115.39 2489 5000 17191619 1.01 0.76 0.51 1.26 114.09 2603 4642 17655818 0.99 0.74 0.51 1.25 112.72 2716 4289 18084717 0.97 0.73 0.51 1.23 111.29 2827 3940 18478816 0.95 0.71 0.51 1.22 109.78 2937 3597 18838515 0.92 0.69 0.51 1.20 108.18 3045 3259 19164312 0.90 0.67 0.51 1.18 106.48 3152 2926 19456911 0.87 0.65 0.51 1.16 104.66 3256 2600 19716910 0.84 0.63 0.51 1.14 102.70 3359 2279 1994489 0.81 0.61 0.51 1.11 100.57 3460 1966 2014158 0.78 0.58 0.51 1.09 98.22 3558 1661 2030767 0.74 0.55 0.51 1.06 95.61 3653 1364 2044406 0.70 0.53 0.51 1.03 93.08 3747 1082 2055225 0.70 0.53 0.51 1.03 93.08 3840 836 2063583 0.70 0.53 0.51 1.03 93.08 3933 590 2069492 0.70 0.53 0.51 1.03 111.70 4044 413 2073621 0.70 0.53 0.51 1.03 65.16 4110 0 20736215801020304050607080901000 50 100 150 200Height Above Ground [m]Story Wind Force [kN]ULS Applied Wind Load at Level, [kN]01020304050607080901000 1000 2000 3000 4000 5000Height Above Ground [m]Cumulative Story Force [kN]ULS Cumulative Story Wind Load, [kN]159Static Wind Loading - NBCC 2005 Description: North South Direction Static Wind Loading (Y Direction)Building Height Above Grade, H 100.26 m 328.84 ftLeeward Side Reference Height 50.13 m 164.42 ftBuilding Dimension in Y-Dir, Dy 21.57 m 70.74 ftCross Wind Dimension in Y-Dir, Wy 30.15 m 98.89 fth/Dy 4.6 4.6Cp Windward Side 0.80 0.80Cp Leeward Side 0.50 0.501/50 Year Reference Pressure, q50 0.47 kPa 9.8 psfULS Importance Factor, Iw 1.00SLS Importance Factor, Iw 0.75Leeward Exposure Factor, Ce 1.07 Assumed rough terrain per 4.1.7.5.5Gust Effect factor, Cg 2.0 Structure as a whole per 4.1.7.1.5.6External Wind PressureCpCgULS Pe [kPa]ULS Pe [psf]Y-Dir Leeward 1.0 0.51 10.6ZoneExternal160LevelFloor to Floor [ft]Elevation [ft]Height Above Ground [ft]Height Above Ground [m]Wind Height [m] Width [ft] Width [m]43 0 633.84 328.84 100.26 1.51 27.00 8.2342 9.92 623.93 318.93 97.23 2.92 27.00 8.2341 9.24 614.69 309.69 94.42 2.99 27.00 8.2340 10.40 604.29 299.29 91.25 3.08 78.00 23.7839 9.82 594.47 289.47 88.25 2.82 78.00 23.7838 8.67 585.80 280.80 85.61 2.64 78.00 23.7837 8.67 577.14 272.14 82.97 2.64 78.00 23.7836 8.67 568.47 263.47 80.33 2.64 78.00 23.7835 8.67 559.80 254.80 77.68 2.64 78.00 23.7833 8.67 551.14 246.14 75.04 2.64 78.00 23.7832 8.67 542.47 237.47 72.40 2.64 78.00 23.7831 8.67 533.80 228.80 69.76 2.64 78.00 23.7830 8.67 525.14 220.14 67.11 2.64 78.00 23.7829 8.67 516.47 211.47 64.47 2.64 78.00 23.7828 8.67 507.80 202.80 61.83 2.64 78.00 23.7827 8.67 499.14 194.14 59.19 2.64 78.00 23.7826 8.67 490.47 185.47 56.55 2.64 78.00 23.7825 8.67 481.80 176.80 53.90 2.64 78.00 23.7823 8.67 473.14 168.14 51.26 2.64 78.00 23.7822 8.67 464.47 159.47 48.62 2.64 78.00 23.7821 8.67 455.80 150.80 45.98 2.64 78.00 23.7820 8.67 447.14 142.14 43.33 2.64 78.00 23.7819 8.67 438.47 133.47 40.69 2.64 78.00 23.7818 8.67 429.80 124.80 38.05 2.64 78.00 23.7817 8.67 421.14 116.14 35.41 2.64 78.00 23.7816 8.67 412.47 107.47 32.76 2.64 78.00 23.7815 8.67 403.80 98.80 30.12 2.64 78.00 23.7812 8.67 395.14 90.14 27.48 2.64 78.00 23.7811 8.67 386.47 81.47 24.84 2.64 78.00 23.7810 8.67 377.80 72.80 22.20 2.64 88.58 27.019 8.67 369.14 64.14 19.55 2.64 88.58 27.018 8.67 360.47 55.47 16.91 2.64 88.58 27.017 8.67 351.80 46.80 14.27 2.64 88.58 27.016 8.67 343.14 38.14 11.63 2.64 88.58 27.015 8.67 334.47 29.47 8.98 2.64 88.58 27.013 8.67 325.80 20.80 6.34 2.64 88.58 27.012 8.67 317.14 12.14 3.70 3.17 88.58 27.011 12.14 305.00 0.00 0.00 1.85 92.00 28.05161LevelWindward Exposure Factor, CeULS Windward Pressure, [kPa]ULS Leeward Pressure, [kPa]ULS Total Storey Pressure, [kPa]ULS Applied Wind Load at Level, [kN]ULS Cumulative Story Wind Load, [kN]ULS Overturning Moment at Level, [kN-m]ULS Cumulative Overturning Moment, [kN-m]43 1.32 1.00 0.51 1.50 18.67 19 1872 187242 1.31 0.99 0.51 1.49 35.85 55 3486 535741 1.30 0.98 0.51 1.48 36.53 91 3449 880640 1.29 0.97 0.51 1.47 107.94 199 9849 1865539 1.27 0.96 0.51 1.46 98.06 297 8654 2731038 1.26 0.95 0.51 1.45 91.38 388 7823 3513337 1.25 0.94 0.51 1.45 90.82 479 7535 4266836 1.24 0.93 0.51 1.44 90.25 570 7250 4991835 1.23 0.92 0.51 1.43 89.67 659 6966 5688433 1.21 0.91 0.51 1.42 89.07 748 6684 6356832 1.20 0.90 0.51 1.41 88.46 837 6404 6997231 1.19 0.89 0.51 1.40 87.83 925 6127 7609830 1.17 0.88 0.51 1.39 87.18 1012 5851 8195029 1.16 0.87 0.51 1.38 86.52 1098 5578 8752828 1.14 0.86 0.51 1.37 85.83 1184 5307 9283527 1.13 0.85 0.51 1.35 85.13 1269 5039 9787326 1.11 0.84 0.51 1.34 84.40 1354 4773 10264625 1.10 0.83 0.51 1.33 83.65 1437 4509 10715523 1.08 0.81 0.51 1.32 82.88 1520 4248 11140422 1.07 0.80 0.51 1.31 82.07 1602 3990 11539421 1.05 0.79 0.51 1.29 81.23 1683 3735 11912920 1.03 0.77 0.51 1.28 80.36 1764 3482 12261119 1.01 0.76 0.51 1.26 79.45 1843 3233 12584418 0.99 0.74 0.51 1.25 78.50 1922 2987 12883117 0.97 0.73 0.51 1.23 77.50 1999 2744 13157516 0.95 0.71 0.51 1.22 76.45 2076 2505 13408015 0.92 0.69 0.51 1.20 75.34 2151 2269 13635012 0.90 0.67 0.51 1.18 74.15 2225 2038 13838811 0.87 0.65 0.51 1.16 72.89 2298 1810 14019810 0.84 0.63 0.51 1.14 81.22 2379 1803 1420019 0.81 0.61 0.51 1.11 79.54 2459 1555 1435568 0.78 0.58 0.51 1.09 77.68 2537 1314 1448707 0.74 0.55 0.51 1.06 75.62 2612 1079 1459496 0.70 0.53 0.51 1.03 73.61 2686 856 1468055 0.70 0.53 0.51 1.03 73.61 2759 661 1474663 0.70 0.53 0.51 1.03 73.61 2833 467 1479332 0.70 0.53 0.51 1.03 88.34 2921 327 1482601 0.70 0.53 0.51 1.03 53.53 2975 0 14826016201020304050607080901000 50 100 150Height Above Ground [m]Story Wind Force [kN]ULS Applied Wind Load at Level, [kN]01020304050607080901000 1000 2000 3000 4000Height Above Ground [m]Cumulative Story Force [kN]ULS Cumulative Story Wind Load, [kN]163PwPwEHPlCase A: Full wind pressure applied in both directions separatelyPw—tPw-------- ► —f Pl orI I P l i 'PtCase B: Case A wind pressure applied only on parts of wall faces0.75pyvMMi .Pl o r 0.75pw 0.75PlTT+il0.75PlCase C: 75% of full wind pressure applied in both directions simultaneously0.38pw 0-75pwiwwUiliiJ. ^ ' ' ' ' ' '  11111 —^0.75pw ' 0.38pw^0.75Pl0.38pL0.38pl 075p^Case D: 50% of Case C wind load removed from part of projected areaEG00942AFigure A.2: NBCC partial wind loading cases taken from NBCC 2005 structural commentaries.A.2.2 Dynamic Wind CalculationsAs with the case of the static wind distribution, the dynamic procedure is based on the equation for externalwind pressure with alternative values of exposure, gust, and external pressure factors.The exposure factor Ce determined for use in the dynamic method is based on the surrounding terrainof the building site. For this building the terrain is typically flat with mostly low rise buildings, with a fewscatted high rise also present. This type of surrounding area represents Exposure B (rough exposure) asdescribed in the NBCC 2005 commentaries. The rough terrain exposure factor in this case is defined asCe = 0.5(h/12.7)0.50 for 0.5 ≤ Ce ≤ 2.5 (A.4)which is applied at each level for the windward case, and at the reference height for the leeward case.The gust effect factor applied for the dynamic method is much more in depth than applied in the staticmethod. A brief overview of the dynamic wind gust factor is included here, with details specific to the studybuilding included in the following calculations. The dynamic gust factor is based on a statistical method fordetermining the peak loading effect defined as follows,Wp = µ+ gpσ (A.5)where µ = mean loading effect, gp = statistical peak factor for the loading effect, and σ = root mean squareof the loading effect. When rearranged the following expression for the gust effect factor is obtained,Cg =Wpµ= 1 + gp(σµ)(A.6)164The coefficient of variation σ/µ can be expressed by the following,σ/µ =√KCeH(B +sFβ)(A.7)whereK = factor related to surface roughness of the surrounding terrain= 0.10 for Exposure BCeH = exposure factor at the top of the building, Equation A.4B = background turbulence factor as defined beloww = effective width of the windward face of the buildingH = height of the windward face of the buildings = size reduction factor as a function of w/h and the reduced frequency fnDH/VHfnD = natural frequency of vibration in the along-wind directionVH = mean wind speed at the top of the structure as defined belowF = gust energy ratio at the natural frequency of the structure as defined belowβ = critical damping ratio in the along-wind directionThe background turbulence factor is a function of the width and height of the structure and is determinedby evaluating the following expression,B =43914/H∫ [11 + xH457][11 + xw122][x(1 + x2)4/3]dx (A.8)The size reduction factor is a function of height, width, and the reduced frequency of the structure and isexpressed by,s =pi3[11 + 8fnH3VH][11 + 10fnwVH](A.9)The gust energy ratio is a function of the wave number fn/VH , and is expressed by,F =x20(1 + x20)4/3(A.10)wherex20 = 1220fn/VH (A.11)The peak factor is a function of the average fluctuation rate, /nu and is expressed by,gp =√2 loge νT +0.577√2 loge νT(A.12)165where T = 3600s The mean wind speed at the top of the structure, VH is given by,VH = V¯√CeH (A.13)where V¯ = 39.2√q is the reference wind speed at a height of 10 m. Finally the average fluctuation rate canbe estimated by the following expression,v = fn√sFsF + βB(A.14)Since the dynamic method of determining wind loading is based on the dynamic properties of the structureitself, two main parameters must be known or estimated. First the critical damping ratio must be estimated.The critical damping ratio β for concrete frames with respect to wind loading is suggested to be estimatedat 2%. Second the fundamental periods in each direction of the structure must be determined. The periodscan be estimated using various approximate equations, or determined based on a modal analysis of thestructure. Since the study building is implemented in ETABS for later analysis, running a modal analysis isthe most accurate method. Since there are two configurations of the structure to be analysed, one includingthe gravity frame, and one omitting its contributions, both cases must be considered. The resulting pressuredistributions for each dynamic wind model are shown in the following.Calculations for determining the gust effect factor in each primary wind loading direction of the studybuilding was implemented in Mathcad and is shown in the following.166Dynamic wind gust factor - Windward X-Direction - Core and Gravity Period≔H 100.26 m Building height≔w 30.2 m Windward effective width≔β 0.02 Damping ratio≔Tn 4.452 s Fundamental period in direction of interest≔fnD =Tn-1 0.225 Hz Fundamental frequency in direction of interest≔K 0.1 Exposure A - K = 0.08Exposure B - K = 0.10Exposure C - K = 0.14≔q 0.47 kPa Reference velocity ≔V10m =⋅⋅39.2‾‾‾‾‾‾‾⋅q kPa-1 m s-1 26.874 ⋅m s-1 Mean wind at 10m elevationExposure factor at the top of the buildingExposure B (Rough terrain)≔CeH =0.5⎛⎜⎝―――H12.7 m⎞⎟⎠0.501.405≔VH =⋅V10m ‾‾‾‾CeH 31.853 ⋅m s-1 Mean wind at top of structure≔B =―43⌠⎮⎮⎮⎮⌡d0―――914mH⎛⎜⎜⎜⎝――――1+1 ―――⋅x H457 m⎞⎟⎟⎟⎠⎛⎜⎜⎜⎝――――1+1 ―――⋅x w122 m⎞⎟⎟⎟⎠⎛⎜⎜⎜⎝――――x⎛⎝ +1 x2 ⎞⎠―43⎞⎟⎟⎟⎠x 0.757 Background turbulence factor≔s =―pi3⎛⎜⎜⎜⎝―――――1+1 ――――⋅⋅8 fnD H⋅3 VH⎞⎟⎟⎟⎠⎛⎜⎜⎜⎝―――――1+1 ――――⋅⋅10 fnD wVH⎞⎟⎟⎟⎠0.116 Size reduction factor≔x0 =⋅1220 m ――fnDVH8.603 Gust energy ratio≔F =――――x02⎛⎝ +1 x02 ⎞⎠―430.234≔COV =‾‾‾‾‾‾‾‾‾‾‾‾‾――KCeH⎛⎜⎝+B ――⋅s Fβ⎞⎟⎠0.388 Coefficient of variation of gust factor≔v =⋅fnD‾‾‾‾‾‾‾‾‾‾――――⋅s F+⋅s F ⋅β B0.18 ―1sAverage fluctuation rate≔T 3600 s≔gp =+‾‾‾‾‾‾‾‾‾⋅2 ln (( ⋅v T)) ―――――0.577‾‾‾‾‾‾‾‾‾⋅2 ln (( ⋅v T))3.759 Peak gust factor≔Cg =+1 ⋅gp COV 2.46 External Gust Effect Factor, Cg167Dynamic wind gust factor - Windward Y-Direction - Core and Gravity Period≔H 100.26 m Building height≔w 21.6 m Windward effective width≔β 0.02 Damping ratio≔Tn 3.861 s Fundamental period in direction of interest≔fnD =Tn-1 0.259 Hz Fundamental frequency in direction of interest≔K 0.1 Exposure A - K = 0.08Exposure B - K = 0.10Exposure C - K = 0.14≔q 0.47 kPa Reference velocity ≔V10m =⋅⋅39.2‾‾‾‾‾‾‾⋅q kPa-1 m s-1 26.874 ⋅m s-1 Mean wind at 10m elevationExposure factor at the top of the buildingExposure B (Rough terrain)≔CeH =0.5⎛⎜⎝―――H12.7 m⎞⎟⎠0.501.405≔VH =⋅V10m ‾‾‾‾CeH 31.853 ⋅m s-1 Mean wind at top of structure≔B =―43⌠⎮⎮⎮⎮⌡d0―――914mH⎛⎜⎜⎜⎝――――1+1 ―――⋅x H457 m⎞⎟⎟⎟⎠⎛⎜⎜⎜⎝――――1+1 ―――⋅x w122 m⎞⎟⎟⎟⎠⎛⎜⎜⎜⎝――――x⎛⎝ +1 x2 ⎞⎠―43⎞⎟⎟⎟⎠x 0.818 Background turbulence factor≔s =―pi3⎛⎜⎜⎜⎝―――――1+1 ――――⋅⋅8 fnD H⋅3 VH⎞⎟⎟⎟⎠⎛⎜⎜⎜⎝―――――1+1 ――――⋅⋅10 fnD wVH⎞⎟⎟⎟⎠0.12 Size reduction factor≔x0 =⋅1220 m ――fnDVH9.92 Gust energy ratio≔F =――――x02⎛⎝ +1 x02 ⎞⎠―430.214≔COV =‾‾‾‾‾‾‾‾‾‾‾‾‾――KCeH⎛⎜⎝+B ――⋅s Fβ⎞⎟⎠0.386 Coefficient of variation of gust factor≔v =⋅fnD‾‾‾‾‾‾‾‾‾‾――――⋅s F+⋅s F ⋅β B0.202 ―1sAverage fluctuation rate≔T 3600 s≔gp =+‾‾‾‾‾‾‾‾‾⋅2 ln (( ⋅v T)) ―――――0.577‾‾‾‾‾‾‾‾‾⋅2 ln (( ⋅v T))3.79 Peak gust factor≔Cg =+1 ⋅gp COV 2.46 External Gust Effect Factor, Cg168Dynamic wind gust factor - Windward X-Direction - Core Only Period≔H 100.26 m Building height≔w 30.2 m Windward effective width≔β 0.02 Damping ratio≔Tn 5.27 s Fundamental period in direction of interest≔fnD =Tn-1 0.19 Hz Fundamental frequency in direction of interest≔K 0.1 Exposure A - K = 0.08Exposure B - K = 0.10Exposure C - K = 0.14≔q 0.47 kPa Reference velocity ≔V10m =⋅⋅39.2‾‾‾‾‾‾‾⋅q kPa-1 m s-1 26.874 ⋅m s-1 Mean wind at 10m elevationExposure factor at the top of the buildingExposure B (Rough terrain)≔CeH =0.5⎛⎜⎝―――H12.7 m⎞⎟⎠0.501.405≔VH =⋅V10m ‾‾‾‾CeH 31.853 ⋅m s-1 Mean wind at top of structure≔B =―43⌠⎮⎮⎮⎮⌡d0―――914mH⎛⎜⎜⎜⎝――――1+1 ―――⋅x H457 m⎞⎟⎟⎟⎠⎛⎜⎜⎜⎝――――1+1 ―――⋅x w122 m⎞⎟⎟⎟⎠⎛⎜⎜⎜⎝――――x⎛⎝ +1 x2 ⎞⎠―43⎞⎟⎟⎟⎠x 0.757 Background turbulence factor≔s =―pi3⎛⎜⎜⎜⎝―――――1+1 ――――⋅⋅8 fnD H⋅3 VH⎞⎟⎟⎟⎠⎛⎜⎜⎜⎝―――――1+1 ――――⋅⋅10 fnD wVH⎞⎟⎟⎟⎠0.144 Size reduction factor≔x0 =⋅1220 m ――fnDVH7.268 Gust energy ratio≔F =――――x02⎛⎝ +1 x02 ⎞⎠―430.26≔COV =‾‾‾‾‾‾‾‾‾‾‾‾‾――KCeH⎛⎜⎝+B ――⋅s Fβ⎞⎟⎠0.433 Coefficient of variation of gust factor≔v =⋅fnD‾‾‾‾‾‾‾‾‾‾――――⋅s F+⋅s F ⋅β B0.16 ―1sAverage fluctuation rate≔T 3600 s≔gp =+‾‾‾‾‾‾‾‾‾⋅2 ln (( ⋅v T)) ―――――0.577‾‾‾‾‾‾‾‾‾⋅2 ln (( ⋅v T))3.728 Peak gust factor≔Cg =+1 ⋅gp COV 2.61 External Gust Effect Factor, Cg169Dynamic wind gust factor - Windward Y-Direction - Core Only Period≔H 100.26 m Building height≔w 21.6 m Windward effective width≔β 0.02 Damping ratio≔Tn 4.271 s Fundamental period in direction of interest≔fnD =Tn-1 0.234 Hz Fundamental frequency in direction of interest≔K 0.1 Exposure A - K = 0.08Exposure B - K = 0.10Exposure C - K = 0.14≔q 0.47 kPa Reference velocity ≔V10m =⋅⋅39.2‾‾‾‾‾‾‾⋅q kPa-1 m s-1 26.874 ⋅m s-1 Mean wind at 10m elevationExposure factor at the top of the buildingExposure B (Rough terrain)≔CeH =0.5⎛⎜⎝―――H12.7 m⎞⎟⎠0.501.405≔VH =⋅V10m ‾‾‾‾CeH 31.853 ⋅m s-1 Mean wind at top of structure≔B =―43⌠⎮⎮⎮⎮⌡d0―――914mH⎛⎜⎜⎜⎝――――1+1 ―――⋅x H457 m⎞⎟⎟⎟⎠⎛⎜⎜⎜⎝――――1+1 ―――⋅x w122 m⎞⎟⎟⎟⎠⎛⎜⎜⎜⎝――――x⎛⎝ +1 x2 ⎞⎠―43⎞⎟⎟⎟⎠x 0.818 Background turbulence factor≔s =―pi3⎛⎜⎜⎜⎝―――――1+1 ――――⋅⋅8 fnD H⋅3 VH⎞⎟⎟⎟⎠⎛⎜⎜⎜⎝―――――1+1 ――――⋅⋅10 fnD wVH⎞⎟⎟⎟⎠0.136 Size reduction factor≔x0 =⋅1220 m ――fnDVH8.968 Gust energy ratio≔F =――――x02⎛⎝ +1 x02 ⎞⎠―430.228≔COV =‾‾‾‾‾‾‾‾‾‾‾‾‾――KCeH⎛⎜⎝+B ――⋅s Fβ⎞⎟⎠0.411 Coefficient of variation of gust factor≔v =⋅fnD‾‾‾‾‾‾‾‾‾‾――――⋅s F+⋅s F ⋅β B0.19 ―1sAverage fluctuation rate≔T 3600 s≔gp =+‾‾‾‾‾‾‾‾‾⋅2 ln (( ⋅v T)) ―――――0.577‾‾‾‾‾‾‾‾‾⋅2 ln (( ⋅v T))3.772 Peak gust factor≔Cg =+1 ⋅gp COV 2.55 External Gust Effect Factor, Cg170Dynamic Wind Loading - NBCC 2005 Description: East West Direction Dynamic Wind Loading (X Direction)Core and Gravity Load System Period Applied1/50 Year Reference Pressure, q50 0.47 kPaULS Importance Factor, Iw 1.00SLS Importance Factor, Iw 0.75Cg, Rough 2.46 See separate calculations for detailsCp Windward 0.8H/D 3.3Effective Width 30.2 mH Leeward 50.1 mH/D 1.7Ce Leeward 0.99Cp Leeward 0.50.738 kip-ft / kN-m0.225 kip / kN171LevelFloor to Floor [ft] Elevation [ft]Height Above Ground [ft]Height Above Ground [m]Wind Height [m] Width [ft] Width [m]43 0 633.84 328.84 100.26 1.51 29.2 8.8942 9.92 623.93 318.93 97.23 2.92 29.2 8.8941 9.24 614.69 309.69 94.42 2.99 29.2 8.8940 10.40 604.29 299.29 91.25 3.08 104.0 31.7139 9.82 594.47 289.47 88.25 2.82 112.0 34.1538 8.67 585.80 280.80 85.61 2.64 112.0 34.1537 8.67 577.14 272.14 82.97 2.64 112.0 34.1536 8.67 568.47 263.47 80.33 2.64 112.0 34.1535 8.67 559.80 254.80 77.68 2.64 112.0 34.1533 8.67 551.14 246.14 75.04 2.64 112.0 34.1532 8.67 542.47 237.47 72.40 2.64 112.0 34.1531 8.67 533.80 228.80 69.76 2.64 112.0 34.1530 8.67 525.14 220.14 67.11 2.64 112.0 34.1529 8.67 516.47 211.47 64.47 2.64 112.0 34.1528 8.67 507.80 202.80 61.83 2.64 112.0 34.1527 8.67 499.14 194.14 59.19 2.64 112.0 34.1526 8.67 490.47 185.47 56.55 2.64 112.0 34.1525 8.67 481.80 176.80 53.90 2.64 112.0 34.1523 8.67 473.14 168.14 51.26 2.64 112.0 34.1522 8.67 464.47 159.47 48.62 2.64 112.0 34.1521 8.67 455.80 150.80 45.98 2.64 112.0 34.1520 8.67 447.14 142.14 43.33 2.64 112.0 34.1519 8.67 438.47 133.47 40.69 2.64 112.0 34.1518 8.67 429.80 124.80 38.05 2.64 112.0 34.1517 8.67 421.14 116.14 35.41 2.64 112.0 34.1516 8.67 412.47 107.47 32.76 2.64 112.0 34.1515 8.67 403.80 98.80 30.12 2.64 112.0 34.1512 8.67 395.14 90.14 27.48 2.64 112.0 34.1511 8.67 386.47 81.47 24.84 2.64 112.0 34.1510 8.67 377.80 72.80 22.20 2.64 112.0 34.159 8.67 369.14 64.14 19.55 2.64 112.0 34.158 8.67 360.47 55.47 16.91 2.64 112.0 34.157 8.67 351.80 46.80 14.27 2.64 112.0 34.156 8.67 343.14 38.14 11.63 2.64 112.0 34.155 8.67 334.47 29.47 8.98 2.64 112.0 34.153 8.67 325.80 20.80 6.34 2.64 112.0 34.152 8.67 317.14 12.14 3.70 3.17 112.0 34.151 12.14 305 0 0.00 1.85 112.0 34.15sum 1900172Level Wi x hi [m2] Ce, RoughDynamic Windward p [kPa]Windward p SLS [kPa]Dynamic Leeward p  [kPa]Leeward p SLS [kPa]Total Dynamic Pressure [kPa]43 892 1.40 1.30 0.97 0.57 0.43 1.8742 865 1.38 1.28 0.96 0.57 0.43 1.8541 840 1.36 1.26 0.95 0.57 0.43 1.8440 2893 1.34 1.24 0.93 0.57 0.43 1.8139 3014 1.32 1.22 0.91 0.57 0.43 1.7938 2923 1.30 1.20 0.90 0.57 0.43 1.7837 2833 1.28 1.18 0.89 0.57 0.43 1.7636 2743 1.26 1.16 0.87 0.57 0.43 1.7435 2653 1.24 1.14 0.86 0.57 0.43 1.7233 2562 1.22 1.12 0.84 0.57 0.43 1.7032 2472 1.19 1.10 0.83 0.57 0.43 1.6831 2382 1.17 1.08 0.81 0.57 0.43 1.6630 2292 1.15 1.06 0.80 0.57 0.43 1.6429 2201 1.13 1.04 0.78 0.57 0.43 1.6228 2111 1.10 1.02 0.77 0.57 0.43 1.5927 2021 1.08 1.00 0.75 0.57 0.43 1.5726 1931 1.06 0.98 0.73 0.57 0.43 1.5525 1841 1.03 0.95 0.71 0.57 0.43 1.5323 1750 1.00 0.93 0.70 0.57 0.43 1.5022 1660 0.98 0.90 0.68 0.57 0.43 1.4821 1570 0.95 0.88 0.66 0.57 0.43 1.4520 1480 0.92 0.85 0.64 0.57 0.43 1.4319 1389 0.89 0.83 0.62 0.57 0.43 1.4018 1299 0.87 0.80 0.60 0.57 0.43 1.3717 1209 0.83 0.77 0.58 0.57 0.43 1.3516 1119 0.80 0.74 0.56 0.57 0.43 1.3215 1029 0.77 0.71 0.53 0.57 0.43 1.2912 938 0.74 0.68 0.51 0.57 0.43 1.2511 848 0.70 0.65 0.49 0.57 0.43 1.2210 758 0.66 0.61 0.46 0.57 0.43 1.199 668 0.62 0.57 0.43 0.57 0.43 1.158 577 0.58 0.53 0.40 0.57 0.43 1.117 487 0.53 0.49 0.37 0.57 0.43 1.066 397 0.50 0.46 0.35 0.57 0.43 1.045 307 0.50 0.46 0.35 0.57 0.43 1.043 217 0.50 0.46 0.35 0.57 0.43 1.042 126 0.50 0.46 0.35 0.57 0.43 1.041 0 0.50 0.46 0.35 0.57 0.43 1.04sum 57297173LevelULS Storey Force [kN]ULS Cumulative Shear [kN]ULS Overturning Moment [kN-m]Cumulative Overturning Moment [kN-m]43 25.2 25 2525 252542 48.1 73 4681 720741 48.9 122 4613 1182040 177.3 299 16175 2799539 172.6 472 15233 4322738 160.1 632 13710 5693837 158.5 791 13148 7008536 156.8 947 12591 8267735 155.0 1102 12042 9471933 153.2 1256 11499 10621832 151.4 1407 10964 11718231 149.6 1557 10436 12761830 147.7 1704 9915 13753329 145.8 1850 9402 14693528 143.9 1994 8896 15583127 141.9 2136 8398 16423026 139.9 2276 7908 17213825 137.8 2414 7427 17956523 135.6 2549 6953 18651822 133.5 2683 6488 19300621 131.2 2814 6032 19903920 128.9 2943 5585 20462419 126.5 3069 5148 20977218 124.0 3193 4720 21449117 121.5 3315 4301 21879316 118.8 3434 3894 22268615 116.1 3550 3496 22618312 113.2 3663 3111 22929311 110.2 3773 2736 23203010 107.0 3880 2374 2344049 103.6 3984 2026 2364308 100.0 4084 1690 2381207 96.0 4180 1370 2394906 93.5 4273 1088 2405785 93.5 4367 840 2414183 93.5 4460 593 2420122 112.3 4573 415 2424271 65.5 4638 0 24242717401020304050607080901000 100 200Height Above Ground [m]Story Wind Force [kN]ULS Storey Force [kN]01020304050607080901000 1000 2000 3000 4000 5000Height Above Ground [m]Cumulative Story Force [kN]ULS Cumulative Shear [kN]175Dynamic Wind Loading - NBCC 2005 Description: East West Direction Dynamic Wind Loading (X Direction)Core Only System Period Applied1/50 Year Reference Pressure, q50 0.47 kPaULS Importance Factor, Iw 1.00SLS Importance Factor, Iw 0.75Cg, Rough 2.61 See separate calculations for detailsCp Windward 0.8H/D 3.3Effective Width 30.2 mH Leeward 50.1 mH/D 1.7Ce Leeward 0.99Cp Leeward 0.50.738 kip-ft / kN-m0.225 kip / kN176LevelFloor to Floor [ft] Elevation [ft]Height Above Ground [ft]Height Above Ground [m]Wind Height [m] Width [ft] Width [m]43 0 633.84 328.84 100.26 1.51 29.2 8.8942 9.92 623.93 318.93 97.23 2.92 29.2 8.8941 9.24 614.69 309.69 94.42 2.99 29.2 8.8940 10.40 604.29 299.29 91.25 3.08 104.0 31.7139 9.82 594.47 289.47 88.25 2.82 112.0 34.1538 8.67 585.80 280.80 85.61 2.64 112.0 34.1537 8.67 577.14 272.14 82.97 2.64 112.0 34.1536 8.67 568.47 263.47 80.33 2.64 112.0 34.1535 8.67 559.80 254.80 77.68 2.64 112.0 34.1533 8.67 551.14 246.14 75.04 2.64 112.0 34.1532 8.67 542.47 237.47 72.40 2.64 112.0 34.1531 8.67 533.80 228.80 69.76 2.64 112.0 34.1530 8.67 525.14 220.14 67.11 2.64 112.0 34.1529 8.67 516.47 211.47 64.47 2.64 112.0 34.1528 8.67 507.80 202.80 61.83 2.64 112.0 34.1527 8.67 499.14 194.14 59.19 2.64 112.0 34.1526 8.67 490.47 185.47 56.55 2.64 112.0 34.1525 8.67 481.80 176.80 53.90 2.64 112.0 34.1523 8.67 473.14 168.14 51.26 2.64 112.0 34.1522 8.67 464.47 159.47 48.62 2.64 112.0 34.1521 8.67 455.80 150.80 45.98 2.64 112.0 34.1520 8.67 447.14 142.14 43.33 2.64 112.0 34.1519 8.67 438.47 133.47 40.69 2.64 112.0 34.1518 8.67 429.80 124.80 38.05 2.64 112.0 34.1517 8.67 421.14 116.14 35.41 2.64 112.0 34.1516 8.67 412.47 107.47 32.76 2.64 112.0 34.1515 8.67 403.80 98.80 30.12 2.64 112.0 34.1512 8.67 395.14 90.14 27.48 2.64 112.0 34.1511 8.67 386.47 81.47 24.84 2.64 112.0 34.1510 8.67 377.80 72.80 22.20 2.64 112.0 34.159 8.67 369.14 64.14 19.55 2.64 112.0 34.158 8.67 360.47 55.47 16.91 2.64 112.0 34.157 8.67 351.80 46.80 14.27 2.64 112.0 34.156 8.67 343.14 38.14 11.63 2.64 112.0 34.155 8.67 334.47 29.47 8.98 2.64 112.0 34.153 8.67 325.80 20.80 6.34 2.64 112.0 34.152 8.67 317.14 12.14 3.70 3.17 112.0 34.151 12.14 305 0 0.00 1.85 112.0 34.15sum 1900177Level Wi x hi [m2] Ce, RoughDynamic Windward p [kPa]Windward p SLS [kPa]Dynamic Leeward p  [kPa]Leeward p SLS [kPa]Total Dynamic Pressure [kPa]43 892 1.40 1.38 1.03 0.61 0.46 1.9942 865 1.38 1.36 1.02 0.61 0.46 1.9741 840 1.36 1.34 1.00 0.61 0.46 1.9540 2893 1.34 1.32 0.99 0.61 0.46 1.9239 3014 1.32 1.29 0.97 0.61 0.46 1.9038 2923 1.30 1.27 0.96 0.61 0.46 1.8837 2833 1.28 1.25 0.94 0.61 0.46 1.8636 2743 1.26 1.23 0.93 0.61 0.46 1.8435 2653 1.24 1.21 0.91 0.61 0.46 1.8233 2562 1.22 1.19 0.89 0.61 0.46 1.8032 2472 1.19 1.17 0.88 0.61 0.46 1.7831 2382 1.17 1.15 0.86 0.61 0.46 1.7630 2292 1.15 1.13 0.85 0.61 0.46 1.7429 2201 1.13 1.11 0.83 0.61 0.46 1.7128 2111 1.10 1.08 0.81 0.61 0.46 1.6927 2021 1.08 1.06 0.79 0.61 0.46 1.6726 1931 1.06 1.04 0.78 0.61 0.46 1.6425 1841 1.03 1.01 0.76 0.61 0.46 1.6223 1750 1.00 0.99 0.74 0.61 0.46 1.6022 1660 0.98 0.96 0.72 0.61 0.46 1.5721 1570 0.95 0.93 0.70 0.61 0.46 1.5420 1480 0.92 0.91 0.68 0.61 0.46 1.5219 1389 0.89 0.88 0.66 0.61 0.46 1.4918 1299 0.87 0.85 0.64 0.61 0.46 1.4617 1209 0.83 0.82 0.61 0.61 0.46 1.4316 1119 0.80 0.79 0.59 0.61 0.46 1.4015 1029 0.77 0.76 0.57 0.61 0.46 1.3612 938 0.74 0.72 0.54 0.61 0.46 1.3311 848 0.70 0.69 0.51 0.61 0.46 1.3010 758 0.66 0.65 0.49 0.61 0.46 1.269 668 0.62 0.61 0.46 0.61 0.46 1.228 577 0.58 0.57 0.42 0.61 0.46 1.187 487 0.53 0.52 0.39 0.61 0.46 1.136 397 0.50 0.49 0.37 0.61 0.46 1.105 307 0.50 0.49 0.37 0.61 0.46 1.103 217 0.50 0.49 0.37 0.61 0.46 1.102 126 0.50 0.49 0.37 0.61 0.46 1.101 0 0.50 0.49 0.37 0.61 0.46 1.10sum 57297178LevelULS Storey Force [kN]ULS Cumulative Shear [kN]ULS Overturning Moment [kN-m]Cumulative Overturning Moment [kN-m]43 26.7 27 2679 267942 51.1 78 4967 764641 51.8 130 4894 1254040 188.1 318 17161 2970239 183.1 501 16161 4586338 169.9 671 14546 6041037 168.1 839 13949 7435936 166.3 1005 13359 8771835 164.5 1170 12776 10049433 162.6 1332 12201 11269532 160.7 1493 11633 12432731 158.7 1652 11072 13540030 156.7 1808 10520 14592029 154.7 1963 9975 15589528 152.7 2116 9439 16533327 150.5 2266 8910 17424426 148.4 2415 8391 18263425 146.2 2561 7879 19051423 143.9 2705 7377 19789122 141.6 2846 6884 20477521 139.2 2986 6400 21117520 136.7 3122 5926 21710119 134.2 3257 5462 22256318 131.6 3388 5007 22757017 128.9 3517 4564 23213416 126.1 3643 4131 23626515 123.2 3766 3710 23997412 120.1 3886 3300 24327511 116.9 4003 2903 24617810 113.5 4117 2519 2486979 109.9 4227 2149 2508468 106.1 4333 1794 2526407 101.9 4435 1454 2540946 99.2 4534 1154 2552475 99.2 4633 892 2561393 99.2 4732 629 2567682 119.1 4851 441 2572091 69.5 4921 0 25720917901020304050607080901000 100 200Height Above Ground [m]Story Wind Force [kN]ULS Storey Force [kN]01020304050607080901000 2000 4000 6000Height Above Ground [m]Cumulative Story Force [kN]ULS Cumulative Shear [kN]180Dynamic Wind Loading - NBCC 2005 Description: North South Direction Dynamic Wind Loading (Y Direction)Core and Gravity Load System Period Applied1/50 Year Reference Pressure, q50 0.47 kPaULS Importance Factor, Iw 1.00SLS Importance Factor, Iw 0.75Cg, Rough 2.46 See separate calculations for detailsCp Windward 0.8H/D 4.6Effective Width 21.6 mH Leeward 50.1 mH/D 2.3Ce Leeward 0.99Cp Leeward 0.50.738 kip-ft / kN-m0.225 kip / kN181LevelFloor to Floor [ft] Elevation [ft]Height Above Ground [ft]Height Above Ground [m]Wind Height [m] Width [ft] Width [m]43 0 633.84 328.84 100.26 1.51 27.0 8.2342 9.92 623.93 318.93 97.23 2.92 27.0 8.2341 9.24 614.69 309.69 94.42 2.99 27.0 8.2340 10.40 604.29 299.29 91.25 3.08 78.0 23.7839 9.82 594.47 289.47 88.25 2.82 78.0 23.7838 8.67 585.80 280.80 85.61 2.64 78.0 23.7837 8.67 577.14 272.14 82.97 2.64 78.0 23.7836 8.67 568.47 263.47 80.33 2.64 78.0 23.7835 8.67 559.80 254.80 77.68 2.64 78.0 23.7833 8.67 551.14 246.14 75.04 2.64 78.0 23.7832 8.67 542.47 237.47 72.40 2.64 78.0 23.7831 8.67 533.80 228.80 69.76 2.64 78.0 23.7830 8.67 525.14 220.14 67.11 2.64 78.0 23.7829 8.67 516.47 211.47 64.47 2.64 78.0 23.7828 8.67 507.80 202.80 61.83 2.64 78.0 23.7827 8.67 499.14 194.14 59.19 2.64 78.0 23.7826 8.67 490.47 185.47 56.55 2.64 78.0 23.7825 8.67 481.80 176.80 53.90 2.64 78.0 23.7823 8.67 473.14 168.14 51.26 2.64 78.0 23.7822 8.67 464.47 159.47 48.62 2.64 78.0 23.7821 8.67 455.80 150.80 45.98 2.64 78.0 23.7820 8.67 447.14 142.14 43.33 2.64 78.0 23.7819 8.67 438.47 133.47 40.69 2.64 78.0 23.7818 8.67 429.80 124.80 38.05 2.64 78.0 23.7817 8.67 421.14 116.14 35.41 2.64 78.0 23.7816 8.67 412.47 107.47 32.76 2.64 78.0 23.7815 8.67 403.80 98.80 30.12 2.64 78.0 23.7812 8.67 395.14 90.14 27.48 2.64 78.0 23.7811 8.67 386.47 81.47 24.84 2.64 78.0 23.7810 8.67 377.80 72.80 22.20 2.64 88.6 27.019 8.67 369.14 64.14 19.55 2.64 88.6 27.018 8.67 360.47 55.47 16.91 2.64 88.6 27.017 8.67 351.80 46.80 14.27 2.64 88.6 27.016 8.67 343.14 38.14 11.63 2.64 88.6 27.015 8.67 334.47 29.47 8.98 2.64 88.6 27.013 8.67 325.80 20.80 6.34 2.64 88.6 27.012 8.67 317.14 12.14 3.70 3.17 88.6 27.011 12.14 305 0 0.00 1.85 92.0 28.05sum 1900182Level Wi x hi [m2] Ce, RoughDynamic Windward p [kPa]Windward p SLS [kPa]Dynamic Leeward p  [kPa]Leeward p SLS [kPa]Total Dynamic Pressure [kPa]43 825 1.40 1.30 0.97 0.57 0.43 1.8742 800 1.38 1.28 0.96 0.57 0.43 1.8541 777 1.36 1.26 0.95 0.57 0.43 1.8440 2170 1.34 1.24 0.93 0.57 0.43 1.8139 2099 1.32 1.22 0.91 0.57 0.43 1.7938 2036 1.30 1.20 0.90 0.57 0.43 1.7837 1973 1.28 1.18 0.89 0.57 0.43 1.7636 1910 1.26 1.16 0.87 0.57 0.43 1.7435 1847 1.24 1.14 0.86 0.57 0.43 1.7233 1785 1.22 1.12 0.84 0.57 0.43 1.7032 1722 1.19 1.10 0.83 0.57 0.43 1.6831 1659 1.17 1.08 0.81 0.57 0.43 1.6630 1596 1.15 1.06 0.80 0.57 0.43 1.6429 1533 1.13 1.04 0.78 0.57 0.43 1.6228 1470 1.10 1.02 0.77 0.57 0.43 1.5927 1408 1.08 1.00 0.75 0.57 0.43 1.5726 1345 1.06 0.98 0.73 0.57 0.43 1.5525 1282 1.03 0.95 0.71 0.57 0.43 1.5323 1219 1.00 0.93 0.70 0.57 0.43 1.5022 1156 0.98 0.90 0.68 0.57 0.43 1.4821 1093 0.95 0.88 0.66 0.57 0.43 1.4520 1031 0.92 0.85 0.64 0.57 0.43 1.4319 968 0.89 0.83 0.62 0.57 0.43 1.4018 905 0.87 0.80 0.60 0.57 0.43 1.3717 842 0.83 0.77 0.58 0.57 0.43 1.3516 779 0.80 0.74 0.56 0.57 0.43 1.3215 716 0.77 0.71 0.53 0.57 0.43 1.2912 653 0.74 0.68 0.51 0.57 0.43 1.2511 591 0.70 0.65 0.49 0.57 0.43 1.2210 599 0.66 0.61 0.46 0.57 0.43 1.199 528 0.62 0.57 0.43 0.57 0.43 1.158 457 0.58 0.53 0.40 0.57 0.43 1.117 385 0.53 0.49 0.37 0.57 0.43 1.066 314 0.50 0.46 0.35 0.57 0.43 1.045 243 0.50 0.46 0.35 0.57 0.43 1.043 171 0.50 0.46 0.35 0.57 0.43 1.042 100 0.50 0.46 0.35 0.57 0.43 1.041 0 0.50 0.46 0.35 0.57 0.43 1.04sum 40987183LevelULS Storey Force [kN]ULS Cumulative Shear [kN]ULS Overturning Moment [kN-m]Cumulative Overturning Moment [kN-m]43 23.3 23 2338 233842 44.6 68 4333 667141 45.2 113 4269 1094040 133.0 246 12131 2307239 120.2 366 10608 3368038 111.5 478 9548 4322837 110.4 588 9156 5238536 109.2 697 8769 6115435 108.0 805 8386 6954033 106.7 912 8009 7754932 105.5 1017 7636 8518431 104.2 1122 7268 9245230 102.9 1225 6905 9935829 101.6 1326 6548 10590528 100.2 1426 6196 11210127 98.8 1525 5849 11795026 97.4 1623 5508 12345725 96.0 1718 5172 12862923 94.5 1813 4842 13347222 92.9 1906 4519 13799021 91.4 1997 4201 14219220 89.8 2087 3890 14608119 88.1 2175 3585 14966618 86.4 2262 3287 15295317 84.6 2346 2996 15594916 82.8 2429 2712 15866015 80.8 2510 2435 16109512 78.8 2589 2166 16326211 76.7 2665 1906 16516710 84.6 2750 1878 1670459 81.9 2832 1602 1686478 79.1 2911 1337 1699847 76.0 2987 1084 1710686 74.0 3061 860 1719285 74.0 3135 665 1725933 74.0 3209 469 1730622 88.8 3298 328 1733911 53.8 3351 0 17339118401020304050607080901000 50 100 150Height Above Ground [m]Story Wind Force [kN]ULS Storey Force [kN]01020304050607080901000 1000 2000 3000 4000Height Above Ground [m]Cumulative Story Force [kN]ULS Cumulative Shear [kN]185Dynamic Wind Loading - NBCC 2005 Description: North South Direction Dynamic Wind Loading (Y Direction)Core Only System Period Applied1/50 Year Reference Pressure, q50 0.47 kPaULS Importance Factor, Iw 1.00SLS Importance Factor, Iw 0.75Cg, Rough 2.55 See separate calculations for detailsCp Windward 0.8H/D 4.6Effective Width 21.6 mH Leeward 50.1 mH/D 2.3Ce Leeward 0.99Cp Leeward 0.50.738 kip-ft / kN-m0.225 kip / kN186LevelFloor to Floor [ft] Elevation [ft]Height Above Ground [ft]Height Above Ground [m]Wind Height [m] Width [ft] Width [m]43 0 633.84 328.84 100.26 1.51 27.0 8.2342 9.92 623.93 318.93 97.23 2.92 27.0 8.2341 9.24 614.69 309.69 94.42 2.99 27.0 8.2340 10.40 604.29 299.29 91.25 3.08 78.0 23.7839 9.82 594.47 289.47 88.25 2.82 78.0 23.7838 8.67 585.80 280.80 85.61 2.64 78.0 23.7837 8.67 577.14 272.14 82.97 2.64 78.0 23.7836 8.67 568.47 263.47 80.33 2.64 78.0 23.7835 8.67 559.80 254.80 77.68 2.64 78.0 23.7833 8.67 551.14 246.14 75.04 2.64 78.0 23.7832 8.67 542.47 237.47 72.40 2.64 78.0 23.7831 8.67 533.80 228.80 69.76 2.64 78.0 23.7830 8.67 525.14 220.14 67.11 2.64 78.0 23.7829 8.67 516.47 211.47 64.47 2.64 78.0 23.7828 8.67 507.80 202.80 61.83 2.64 78.0 23.7827 8.67 499.14 194.14 59.19 2.64 78.0 23.7826 8.67 490.47 185.47 56.55 2.64 78.0 23.7825 8.67 481.80 176.80 53.90 2.64 78.0 23.7823 8.67 473.14 168.14 51.26 2.64 78.0 23.7822 8.67 464.47 159.47 48.62 2.64 78.0 23.7821 8.67 455.80 150.80 45.98 2.64 78.0 23.7820 8.67 447.14 142.14 43.33 2.64 78.0 23.7819 8.67 438.47 133.47 40.69 2.64 78.0 23.7818 8.67 429.80 124.80 38.05 2.64 78.0 23.7817 8.67 421.14 116.14 35.41 2.64 78.0 23.7816 8.67 412.47 107.47 32.76 2.64 78.0 23.7815 8.67 403.80 98.80 30.12 2.64 78.0 23.7812 8.67 395.14 90.14 27.48 2.64 78.0 23.7811 8.67 386.47 81.47 24.84 2.64 78.0 23.7810 8.67 377.80 72.80 22.20 2.64 88.6 27.019 8.67 369.14 64.14 19.55 2.64 88.6 27.018 8.67 360.47 55.47 16.91 2.64 88.6 27.017 8.67 351.80 46.80 14.27 2.64 88.6 27.016 8.67 343.14 38.14 11.63 2.64 88.6 27.015 8.67 334.47 29.47 8.98 2.64 88.6 27.013 8.67 325.80 20.80 6.34 2.64 88.6 27.012 8.67 317.14 12.14 3.70 3.17 88.6 27.011 12.14 305 0 0.00 1.85 92.0 28.05sum 1900187Level Wi x hi [m2] Ce, RoughULS Dynamic Windward p [kPa]Windward p SLS [kPa]ULS Dynamic Leeward p  [kPa]Leeward p SLS [kPa]Total Dynamic Pressure [kPa]43 825 1.40 1.35 1.01 0.60 0.45 1.9442 800 1.38 1.33 0.99 0.60 0.45 1.4441 777 1.36 1.31 0.98 0.60 0.45 1.4340 2170 1.34 1.29 0.96 0.60 0.45 1.4139 2099 1.32 1.26 0.95 0.60 0.45 1.3938 2036 1.30 1.24 0.93 0.60 0.45 1.3837 1973 1.28 1.23 0.92 0.60 0.45 1.3736 1910 1.26 1.21 0.90 0.60 0.45 1.3535 1847 1.24 1.19 0.89 0.60 0.45 1.3433 1785 1.22 1.17 0.87 0.60 0.45 1.3232 1722 1.19 1.14 0.86 0.60 0.45 1.3031 1659 1.17 1.12 0.84 0.60 0.45 1.2930 1596 1.15 1.10 0.83 0.60 0.45 1.2729 1533 1.13 1.08 0.81 0.60 0.45 1.2628 1470 1.10 1.06 0.79 0.60 0.45 1.2427 1408 1.08 1.03 0.78 0.60 0.45 1.2226 1345 1.06 1.01 0.76 0.60 0.45 1.2125 1282 1.03 0.99 0.74 0.60 0.45 1.1923 1219 1.00 0.96 0.72 0.60 0.45 1.1722 1156 0.98 0.94 0.70 0.60 0.45 1.1521 1093 0.95 0.91 0.68 0.60 0.45 1.1320 1031 0.92 0.89 0.66 0.60 0.45 1.1119 968 0.89 0.86 0.64 0.60 0.45 1.0918 905 0.87 0.83 0.62 0.60 0.45 1.0717 842 0.83 0.80 0.60 0.60 0.45 1.0516 779 0.80 0.77 0.58 0.60 0.45 1.0215 716 0.77 0.74 0.55 0.60 0.45 1.0012 653 0.74 0.71 0.53 0.60 0.45 0.9811 591 0.70 0.67 0.50 0.60 0.45 0.9510 599 0.66 0.63 0.48 0.60 0.45 0.929 528 0.62 0.59 0.45 0.60 0.45 0.898 457 0.58 0.55 0.41 0.60 0.45 0.867 385 0.53 0.51 0.38 0.60 0.45 0.836 314 0.50 0.48 0.36 0.60 0.45 0.815 243 0.50 0.48 0.36 0.60 0.45 0.813 171 0.50 0.48 0.36 0.60 0.45 0.812 100 0.50 0.48 0.36 0.60 0.45 0.811 0 0.50 0.48 0.36 0.60 0.45 0.81sum 40987188LevelULS Storey Force [kN]ULS Cumulative Shear [kN]ULS Overturning Moment [kN-m]Cumulative Overturning Moment [kN-m]43 24.2 24 2423 242342 46.2 70 4492 691541 46.9 117 4426 1134140 137.8 255 12575 2391639 124.6 380 10997 3491238 115.6 495 9898 4481037 114.4 610 9491 5430136 113.2 723 9090 6339135 111.9 835 8693 7208433 110.6 945 8302 8038632 109.3 1055 7915 8830131 108.0 1163 7534 9583530 106.7 1269 7158 10299329 105.3 1375 6787 10978028 103.9 1478 6422 11620227 102.4 1581 6063 12226526 101.0 1682 5709 12797425 99.5 1781 5361 13333523 97.9 1879 5020 13835522 96.3 1976 4684 14303921 94.7 2070 4355 14739420 93.0 2163 4032 15142619 91.3 2255 3716 15514218 89.5 2344 3407 15854917 87.7 2432 3105 16165416 85.8 2518 2811 16446515 83.8 2602 2524 16698912 81.7 2683 2246 16923511 79.5 2763 1975 17121010 87.7 2850 1947 1731579 84.9 2935 1661 1748178 82.0 3017 1386 1762037 78.7 3096 1124 1773276 76.7 3173 892 1782185 76.7 3249 689 1789073 76.7 3326 486 1793942 92.0 3418 341 1797341 55.8 3474 0 17973418901020304050607080901000 50 100 150Height Above Ground [m]Story Wind Force [kN]ULS Storey Force [kN]01020304050607080901000 1000 2000 3000 4000Height Above Ground [m]Cumulative Story Force [kN]ULS Cumulative Shear [kN]190A.2.3 Comparison of Static and Dynamic Wind LoadingThe static and dynamic wind analysis methods produce different pressure distributions as was shown in theprevious section. A brief comparison of the two methods is included to outline the importance of applyingthe correct method for tall or slender structures. Included in the comparison is any change in wind effectsbased on the dynamic properties of the structure based on including or excluding the gravity frame systemas part of the analysis. The results of this analysis are included in the following pages.The analysis shows that the dynamic method produces 13% to 20% base shears and 17% to 24% higherbase overturning moments when applied to the study building. This proves that neglecting to account fordynamic wind effects produces unconservative loading distributions. The higher overturning moment is ofparticular concern, as capacity design for seismic loads ensures that member capacity is controlled by flexure,not shear. If the flexural demands due to wind are 24% higher, unintended plastic behaviour of elasticallydesigned members may result.The comparison of wind effects based on inclusion or exclusion of the gravity frame system tends to havea modest effect for the study building. Inclusion of the core serves to increase the stiffness of the buildingand subsequently lowers any dynamic response due to wind loading. This reduction is a linear scaling basedon the gust factor, so in the case of base shear and overturning moment increases in the 4% to 7% range areobserved. Since the focus of the study is to compare the demands on the gravity frame, the loading for thecore and gravity frame period will be applied in subsequent models and analysis. Should the need arise tocompare results to the higher pressure distribution of the core only, it is simply a linear scaling of the lowerload level.191Comparison of NBCC 2005 Static and Dynamic Wind Loading MethodsX-Direction (Coupled Walls)LevelHeight Above Grade [m]Static Wind Pressure [kPa]Dynamic Wind Pressure Core and Gravity [kPa]Dynamic Wind Pressure Core Only [kPa]43 100.26 1.50 1.87 1.9942 97.23 1.49 1.85 1.9741 94.42 1.48 1.84 1.9540 91.25 1.47 1.81 1.9239 88.25 1.46 1.79 1.9038 85.61 1.45 1.78 1.8837 82.97 1.45 1.76 1.8636 80.33 1.44 1.74 1.8435 77.68 1.43 1.72 1.8233 75.04 1.42 1.70 1.8032 72.40 1.41 1.68 1.7831 69.76 1.40 1.66 1.7630 67.11 1.39 1.64 1.7429 64.47 1.38 1.62 1.7128 61.83 1.37 1.59 1.6927 59.19 1.35 1.57 1.6726 56.55 1.34 1.55 1.6425 53.90 1.33 1.53 1.6223 51.26 1.32 1.50 1.6022 48.62 1.31 1.48 1.5721 45.98 1.29 1.45 1.5420 43.33 1.28 1.43 1.5219 40.69 1.26 1.40 1.4918 38.05 1.25 1.37 1.4617 35.41 1.23 1.35 1.4316 32.76 1.22 1.32 1.4015 30.12 1.20 1.29 1.3612 27.48 1.18 1.25 1.3311 24.84 1.16 1.22 1.3010 22.20 1.14 1.19 1.269 19.55 1.11 1.15 1.228 16.91 1.09 1.11 1.187 14.27 1.06 1.06 1.136 11.63 1.03 1.04 1.105 8.98 1.03 1.04 1.103 6.34 1.03 1.04 1.102 3.70 1.03 1.04 1.101 0.00 1.03 1.04 1.1001020304050607080901000.00 0.50 1.00 1.50 2.00Height Above Grade [m]Pressure [kPa]X-Direction Wind Pressure DistributionsStatic Wind Pressure [kPa]Dynamic Wind Pressure Core andGravity [kPa]Dynamic Wind Pressure Core Only [kPa]192Y-Direction (Cantilever Walls)LevelHeight Above Grade [m]Static Wind Pressure [kPa]Dynamic Wind Pressure Core and Gravity [kPa]Dynamic Wind Pressure Core Only [kPa]43 100.26 1.50 1.87 1.9442 97.23 1.49 1.85 1.4441 94.42 1.48 1.84 1.4340 91.25 1.47 1.81 1.4139 88.25 1.46 1.79 1.3938 85.61 1.45 1.78 1.3837 82.97 1.45 1.76 1.3736 80.33 1.44 1.74 1.3535 77.68 1.43 1.72 1.3433 75.04 1.42 1.70 1.3232 72.40 1.41 1.68 1.3031 69.76 1.40 1.66 1.2930 67.11 1.39 1.64 1.2729 64.47 1.38 1.62 1.2628 61.83 1.37 1.59 1.2427 59.19 1.35 1.57 1.2226 56.55 1.34 1.55 1.2125 53.90 1.33 1.53 1.1923 51.26 1.32 1.50 1.1722 48.62 1.31 1.48 1.1521 45.98 1.29 1.45 1.1320 43.33 1.28 1.43 1.1119 40.69 1.26 1.40 1.0918 38.05 1.25 1.37 1.0717 35.41 1.23 1.35 1.0516 32.76 1.22 1.32 1.0215 30.12 1.20 1.29 1.0012 27.48 1.18 1.25 0.9811 24.84 1.16 1.22 0.9510 22.20 1.14 1.19 0.929 19.55 1.11 1.15 0.898 16.91 1.09 1.11 0.867 14.27 1.06 1.06 0.836 11.63 1.03 1.04 0.815 8.98 1.03 1.04 0.813 6.34 1.03 1.04 0.812 3.70 1.03 1.04 0.811 0.00 1.03 1.04 0.8101020304050607080901000.00 0.50 1.00 1.50 2.00Height Above Grade [m]Pressure [kPa]Y-Direction Wind Pressure DistributionsStatic Wind Pressure [kPa]Dynamic Wind Pressure Core andGravity [kPa]Dynamic Wind Pressure Core Only [kPa]193Static and Dynamic Load Distribution ComparisonAnalysis TypeBase Shear [kN]Base Moment [kN-m]Increase of Dynamic to Static ShearIncrease of Dynamic to Static MomentStatic X 4110 207362 N/A N/ADynamic Core & Gravity X 4638 242427 1.13 1.17Dynamic Core Only Y 4921 257209 1.20 1.24Static Y 2975 148260 N/A N/ADynamic Core & Gravity Y 3351 173391 1.13 1.17Dynamic Core Only Y 3474 179734 1.17 1.21Dynamic method produces 13% - 20% higher load effects for study buildingDynamic method is to be applied based on height to width slendernessAll recorded values are based on ULS criteria194Appendix BThin Lightly-Reinforced BearingWalls195B.1 Bearing Wall Test Specimen Geometry and Material PropertiesTable B.1: Summary of bearing wall test specimen geometry and material properties.Researcher Specimen Wall Wall Wall Slenderness Load Cylinder Reinforcing Reinforcing Reinforcing ReinforcingName Length Height Thickness Eccentricity Strength Yield Layers Placement RatioL h t h/t e f ′c fy ρv[mm] [mm] [mm] [mm] [MPa] [MPa] [mm]Oberlender, 1977 A-2-1-M 610 610 76 8 13 27.1 518 2 18 0.00331Oberlender, 1977 A-2-2-M 610 610 76 8 13 35.2 435 2 18 0.00467Oberlender, 1977 A-3-1-M 610 914 76 12 13 24.9 518 2 18 0.00331Oberlender, 1977 A-3-2-M 610 914 76 12 13 25.7 435 2 18 0.00467Oberlender, 1977 A-4-1-M 610 1219 76 16 13 27.7 518 2 18 0.00331Oberlender, 1977 A-4-2-M 610 1219 76 16 13 34.7 435 2 18 0.00467Oberlender, 1977 A-5-1-M 610 1524 76 20 13 41.1 518 2 18 0.00331Oberlender, 1977 A-5-2-M 610 1524 76 20 13 40.3 435 2 18 0.00467Oberlender, 1977 A-6-1-M 610 1829 76 24 13 45.1 518 2 18 0.00331Oberlender, 1977 A-6-2-M 610 1829 76 24 13 47.7 435 2 18 0.00467Oberlender, 1977 A-7-1-M 610 2134 76 28 13 37.9 518 2 18 0.00331Oberlender, 1977 A-7-2-M 610 2134 76 28 13 42.2 435 2 18 0.00467Oberlender, 1977 B-2-1-M 610 610 76 8 13 29.3 518 2 18 0.00331Oberlender, 1977 B-2-2-M 610 610 76 8 13 31.9 435 2 18 0.00467Oberlender, 1977 B-3-1-M 610 914 76 12 13 26.6 518 2 18 0.00331Oberlender, 1977 B-3-1-MC 610 914 76 12 13 29.9 518 2 12 0.00331Oberlender, 1977 B-3-2-M 610 914 76 12 13 23.9 435 2 18 0.00467Oberlender, 1977 B-4-1-M 610 1219 76 16 13 38.8 518 2 18 0.00331Oberlender, 1977 B-4-2-M 610 1219 76 16 13 33.4 435 2 18 0.00467Oberlender, 1977 B-5-1-M 610 1524 76 20 13 29.7 518 2 18 0.00331Oberlender, 1977 B-5-1-MC 610 1524 76 20 13 13.4 518 2 12 0.00331Oberlender, 1977 B-5-2-M 610 1524 76 20 13 37.7 435 2 18 0.00467Oberlender, 1977 B-6-1-M 610 1829 76 24 13 48.5 518 2 18 0.00331Oberlender, 1977 B-6-2-M 610 1829 76 24 13 44.0 435 2 18 0.00467Oberlender, 1977 B-7-1-M 610 2134 76 28 13 42.4 518 2 18 0.00331Continued on Next Page196Researcher Specimen Wall Wall Wall Slenderness Load Cylinder Reinforcing Reinforcing Reinforcing ReinforcingName Length Height Thickness Eccentricity Strength Yield Layers Placement RatioL h t h/t e f ′c fy ρv[mm] [mm] [mm] [mm] [MPa] [MPa] [mm]Oberlender, 1977 B-7-1-MC 610 2134 76 28 13 41.6 518 2 12 0.00331Oberlender, 1977 B-7-2-M 610 2134 76 28 13 45.5 435 2 18 0.00467Pillai, 1977 A1 400 1200 40 30 7 25.0 274 1 20 0.00156Pillai, 1977 A2 500 1200 48 25 8 25.0 233 1 24 0.00150Pillai, 1977 A3 550 1200 60 20 10 20.8 233 1 30 0.00153Pillai, 1977 A4 700 1200 80 15 13 15.6 347 1 40 0.00150Pillai, 1977 A5 700 800 80 10 13 20.8 347 1 40 0.00150Pillai, 1977 A6 700 400 80 5 13 15.6 347 1 40 0.00150Pillai, 1977 B1 400 1200 40 30 7 24.3 233 1 20 0.00300Pillai, 1977 B2 500 1200 48 25 8 24.3 233 1 24 0.00300Pillai, 1977 B3 560 1200 60 20 10 31.1 233 1 30 0.00301Pillai, 1977 B4 700 1200 80 15 13 22.8 347 1 40 0.00300Pillai, 1977 B5 700 800 80 10 13 22.8 347 1 40 0.00300Pillai, 1977 B6 700 400 80 5 13 15.6 347 1 40 0.00300Saheb, 1989 WAR-1 900 600 50 12 8 22.3 297 2 11 0.00173Saheb, 1989 WAR-2 600 600 50 12 8 22.3 297 2 11 0.00173Saheb, 1989 WAR-3 400 600 50 12 8 22.3 297 2 11 0.00173Saheb, 1989 WAR-4 300 600 50 12 8 22.3 297 2 11 0.00173Saheb, 1989 WSR-1 300 450 50 9 8 21.7 297 2 11 0.00165Saheb, 1989 WSR-2 400 600 50 12 8 21.7 297 2 11 0.00165Saheb, 1989 WSR-3 600 900 50 18 8 21.7 297 2 11 0.00165Saheb, 1989 WSR-4 900 1350 50 27 8 21.7 297 2 11 0.00165Saheb, 1989 WSTV-2 900 600 50 12 8 25.2 297 2 12 0.00173Saheb, 1989 WSTV-3 900 600 50 12 8 25.2 286 2 12 0.00331Saheb, 1989 WSTV-4 900 600 50 12 8 25.2 581 2 13 0.00528Saheb, 1989 WSTV-5 800 1200 50 24 8 22.8 570 2 11 0.00845Saheb, 1989 WSTV-6 800 1200 50 24 8 22.8 297 2 12 0.00177Saheb, 1989 WSTV-7 800 1200 50 24 8 22.8 286 2 12 0.00335Saheb, 1989 WSTV-8 800 1200 50 24 8 22.8 581 2 13 0.00528Continued on Next Page197Researcher Specimen Wall Wall Wall Slenderness Load Cylinder Reinforcing Reinforcing Reinforcing ReinforcingName Length Height Thickness Eccentricity Strength Yield Layers Placement RatioL h t h/t e f ′c fy ρv[mm] [mm] [mm] [mm] [MPa] [MPa] [mm]Saheb, 1989 WSTH-2 900 600 50 12 8 24.5 297 2 11 0.00856Saheb, 1989 WSTH-3 900 600 50 12 8 24.5 297 2 11 0.00173Saheb, 1989 WSTH-4 900 600 50 12 8 24.5 297 2 11 0.00173Saheb, 1989 WSTH-6 800 1200 50 24 8 20.2 297 2 11 0.00176Saheb, 1989 WSTH-7 800 1200 50 24 8 20.2 297 2 11 0.00176Saheb, 1989 WSTH-8 800 1200 50 24 8 20.2 297 2 11 0.00176Sanjayan, 2000 1 1500 2000 50 40 25 58.5 518 1 25 0.00990Sanjayan, 2000 2 1500 2000 50 40 25 59.0 450 1 25 0.02800Sanjayan, 2000 3 1500 2000 50 40 25 59.0 506 1 25 0.00210Doh, 2005 OWNS2 1200 1200 40 30 7 35.7 450 1 20 0.00310Doh, 2005 OWNS3 1400 1400 40 35 7 52.0 450 1 20 0.00310Doh, 2005 OWNS4 1600 1600 40 40 7 51.0 450 1 20 0.00310Doh, 2005 OWHS2 1200 1200 40 30 7 78.2 450 1 20 0.00310Doh, 2005 OWHS3 1400 1400 40 35 7 63.0 450 1 20 0.00310Doh, 2005 OWHS4 1600 1600 40 40 7 75.9 450 1 20 0.00310Fragomeni, 1995 1a 200 1000 50 20 8 40.7 450 1 20 0.00310Fragomeni, 1995 2a 300 1000 50 20 8 42.4 450 1 20 0.00310Fragomeni, 1995 3a 200 1000 40 25 7 37.1 450 1 20 0.00310Fragomeni, 1995 4a 300 1000 40 25 7 35.7 450 1 20 0.00310Fragomeni, 1995 5a 500 1000 40 25 7 35.7 450 1 20 0.00310Fragomeni, 1995 6a 200 600 40 15 7 38.3 450 1 20 0.00310Fragomeni, 1995 7a 200 600 40 15 7 32.9 450 1 20 0.00310Fragomeni, 1995 8a 210 420 35 12 6 39.6 450 1 20 0.00310Fragomeni, 1995 9a 200 1000 50 20 8 34.2 450 1 20 0.00310Fragomeni, 1995 10a 300 1000 50 20 8 34.6 450 1 20 0.00310Fragomeni, 1995 1b 200 1000 50 20 8 58.9 450 1 20 0.00310Fragomeni, 1995 2b 300 1000 50 20 8 65.4 450 1 20 0.00310Fragomeni, 1995 3b 200 1000 40 25 7 54.0 450 1 20 0.00310Fragomeni, 1995 4b 300 1000 40 25 7 54.0 450 1 20 0.00310Continued on Next Page198Researcher Specimen Wall Wall Wall Slenderness Load Cylinder Reinforcing Reinforcing Reinforcing ReinforcingName Length Height Thickness Eccentricity Strength Yield Layers Placement RatioL h t h/t e f ′c fy ρv[mm] [mm] [mm] [mm] [MPa] [MPa] [mm]Fragomeni, 1995 5b 500 1000 40 25 7 59.7 450 1 20 0.00310Fragomeni, 1995 6b 200 600 40 15 7 67.4 450 1 20 0.00310Fragomeni, 1995 7b 200 600 40 15 7 45.1 450 1 20 0.00310Fragomeni, 1995 8b 210 420 35 12 6 67.4 450 1 20 0.00310Fragomeni, 1995 9b 200 1000 50 20 8 60.0 450 1 20 0.00310Fragomeni, 1995 10b 300 1000 50 20 8 60.7 450 1 20 0.00310Huang, 2014 ST1 460 2700 100 27 17 81.4 550 2 20 0.00233Huang, 2014 ST2 460 2700 100 27 17 81.4 550 1 50 0.00233Huang, 2014 ST3 460 2700 100 27 7 81.4 550 2 20 0.00233Huang, 2014 ST4 460 2700 100 27 36 81.4 550 2 20 0.00233Huang, 2014 ST5 460 2700 130 21 23 81.4 550 2 20 0.00284Huang, 2014 ST6 460 2700 160 17 33 81.4 550 2 20 0.00292Huang, 2014 ST7 460 2700 100 27 18 81.4 550 2 20 0.00164Huang, 2014 ST8 460 2700 100 27 17 81.4 550 2 20 0.00592Robinson, 2013 3 500 3000 100 30 5 53.2 550 1 50 0.00502Robinson, 2013 4 500 3000 100 30 5 53.2 550 1 50 0.00502Robinson, 2013 5 500 3000 100 30 17 49.1 550 1 50 0.00502Robinson, 2013 6 500 3000 100 30 17 49.1 550 1 50 0.00502Robinson, 2013 7 500 2500 100 25 17 51.5 550 1 50 0.00502Robinson, 2013 8 500 2500 100 25 17 51.5 550 1 50 0.00502Robinson, 2013 9 500 2800 100 28 17 52.4 550 1 50 0.00502Robinson, 2013 10 500 2800 100 28 17 52.4 550 1 50 0.00502Robinson, 2013 11 500 3000 100 30 17 51.6 550 1 50 0.00502Robinson, 2013 12 500 3000 100 30 17 51.6 550 1 50 0.00502Robinson, 2013 13 500 3000 100 30 17 51.6 550 1 50 0.00502Robinson, 2013 14 500 3000 100 30 17 51.6 550 1 50 0.00502Robinson, 2013 15 500 3000 100 30 33 52.4 550 1 50 0.00502Robinson, 2013 16 500 3000 100 30 33 52.4 550 1 50 0.00502199B.2 Bearing Wall Test Experimental Results and PredictedCapacity RatiosTable B.2: Summary of bearing wall test experimental results and predicted capacity ratios.Experimental Results Prediction to Experimental Strength RatioResearcher Specimen Ultimate Normalized Equation Rational Rational ACI 318-14Name Axial Ultimate 14-1 Method Method AlternativeLoad Axial Load EI = 0.2EcIg StiffnessPu Pu/(f ′cAg) +AstIst EI = 0.4EcIg[kN]Oberlender, 1977 A-2-1-M 1272 1.01 2.00 2.10 1.94 1.86Oberlender, 1977 A-2-2-M 979 0.60 1.20 1.29 1.18 1.12Oberlender, 1977 A-3-1-M 912 0.79 1.70 2.04 1.69 1.52Oberlender, 1977 A-3-2-M 818 0.68 1.47 1.73 1.45 1.30Oberlender, 1977 A-4-1-M 925 0.72 1.78 2.72 1.93 1.57Oberlender, 1977 A-4-2-M 787 0.49 1.22 2.00 1.40 1.10Oberlender, 1977 A-5-1-M 765 0.40 1.25 2.69 1.69 1.16Oberlender, 1977 A-5-2-M 850 0.45 1.41 2.87 1.85 1.26Oberlender, 1977 A-6-1-M 649 0.31 1.36 2.99 1.82 1.15Oberlender, 1977 A-6-2-M 565 0.26 1.12 2.43 1.51 0.93Oberlender, 1977 A-7-1-M 583 0.33 † 3.72 2.25 1.40Oberlender, 1977 A-7-2-M 449 0.23 † 2.62 1.63 0.98Oberlender, 1977 B-2-1-M 1041 0.77 1.52 1.62 1.49 1.42Oberlender, 1977 B-2-2-M 881 0.59 1.18 1.25 1.15 1.10Oberlender, 1977 B-3-1-M 854 0.69 1.49 1.83 1.50 1.35Oberlender, 1977 B-3-1-MC 916 0.66 1.43 1.73 1.45 1.30Oberlender, 1977 B-3-2-M 698 0.63 1.35 1.54 1.30 1.18Oberlender, 1977 B-4-1-M 952 0.53 1.34 2.41 1.62 1.26Oberlender, 1977 B-4-2-M 765 0.49 1.23 1.98 1.40 1.10Oberlender, 1977 B-5-1-M 858 0.62 1.90 3.43 2.22 1.59Oberlender, 1977 B-5-1-MC 356 0.57 1.69 1.76 1.35 1.11Oberlender, 1977 B-5-2-M 885 0.51 1.57 3.07 2.00 1.37Oberlender, 1977 B-6-1-M 636 0.28 1.25 2.85 1.72 1.09Oberlender, 1977 B-6-2-M 552 0.27 1.18 2.44 1.53 0.95Oberlender, 1977 B-7-1-M 485 0.25 † 2.97 1.79 1.10Oberlender, 1977 B-7-1-MC 414 0.21 † 2.42 1.53 0.93Oberlender, 1977 B-7-2-M 489 0.23 † 2.78 1.72 1.03Pillai, 1977 A1 229 0.57 † 6.42 3.57 2.36Pillai, 1977 A2 367 0.61 2.90 5.01 2.87 2.01Pillai, 1977 A3 382 0.56 1.68 2.85 1.81 1.40Pillai, 1977 A4 392 0.45 1.04 1.39 1.05 0.93Pillai, 1977 A5 932 0.80 1.62 1.92 1.65 1.55Pillai, 1977 A6 647 0.74 1.38 1.32 1.32 1.32Pillai, 1977 B1 282 0.73 † 7.59 4.29 2.84Pillai, 1977 B2 402 0.69 3.26 5.31 3.14 2.18Continued on Next Page200Researcher Specimen Ultimate Normalized Equation Rational Rational ACI 318-14Name Axial Ultimate 14-1 Method Method AlternativeLoad Axial Load EI = 0.2EcIg StiffnessPu Pu/(f ′cAg) +AstIst EI = 0.4EcIg[kN]Pillai, 1977 B3 616 0.59 1.81 3.66 2.25 1.63Pillai, 1977 B4 883 0.69 1.63 2.49 1.77 1.49Pillai, 1977 B5 971 0.76 1.55 1.87 1.58 1.48Pillai, 1977 B6 559 0.64 1.19 1.13 1.13 1.13Saheb, 1989 WAR-1 484 0.48 1.03 1.25 1.03 0.94Saheb, 1989 WAR-2 315 0.47 1.00 1.22 1.01 0.92Saheb, 1989 WAR-3 198 0.44 0.95 1.15 0.95 0.87Saheb, 1989 WAR-4 147 0.44 0.94 1.14 0.94 0.86Saheb, 1989 WSR-1 214 0.66 1.31 1.42 1.29 1.23Saheb, 1989 WSR-2 254 0.59 1.25 1.51 1.25 1.15Saheb, 1989 WSR-3 299 0.46 1.23 1.93 1.29 1.04Saheb, 1989 WSR-4 374 0.38 † 3.09 1.85 1.24Saheb, 1989 WSTV-2 535 0.47 1.02 1.29 1.04 0.94Saheb, 1989 WSTV-3 584 0.52 1.11 1.34 1.11 1.00Saheb, 1989 WSTV-4 704 0.62 1.34 1.54 1.30 1.16Saheb, 1989 WSTV-5 339 0.37 1.56 1.82 1.35 0.87Saheb, 1989 WSTV-6 399 0.44 1.83 2.97 1.81 1.26Saheb, 1989 WSTV-7 463 0.51 2.13 3.19 2.03 1.39Saheb, 1989 WSTV-8 503 0.55 2.32 3.21 2.13 1.43Saheb, 1989 WSTH-2 538 0.49 1.05 1.07 0.95 0.85Saheb, 1989 WSTH-3 538 0.49 1.05 1.31 1.07 0.97Saheb, 1989 WSTH-4 538 0.49 1.05 1.31 1.07 0.97Saheb, 1989 WSTH-6 349 0.43 1.81 2.71 1.68 1.18Saheb, 1989 WSTH-7 344 0.43 1.78 2.67 1.65 1.17Saheb, 1989 WSTH-8 349 0.43 1.81 2.71 1.68 1.18Sanjayan, 2000 1 238 0.05 † 1.93‡ 1.21‡ 1.02Sanjayan, 2000 2 202 0.05 † 1.39‡ 0.84‡ 0.58Sanjayan, 2000 3 212 0.05 † 2.70‡ 1.81‡ 1.75Doh, 2005 OWNS2 253 0.15 † 1.96 1.09 0.69Doh, 2005 OWNS3 427 0.15 † 3.24‡ 1.76‡ 1.05Doh, 2005 OWNS4 442 0.14 † 3.75‡ 2.03‡ 1.20Doh, 2005 OWHS2 483 0.13 † 2.79 1.51 0.90Doh, 2005 OWHS3 442 0.13 † 3.12‡ 1.69‡ 1.00Doh, 2005 OWHS4 456 0.09 † 3.34‡ 1.80‡ 1.06Fragomeni, 1995 1a 162 0.40 1.24 2.85 1.71 1.20Fragomeni, 1995 2a 232 0.36 1.14 2.67 1.60 1.12Fragomeni, 1995 3a 100 0.34 1.63 3.35 1.89 1.26Fragomeni, 1995 4a 199 0.46 2.24 4.51 2.56 1.71Fragomeni, 1995 5a 201 0.28 1.36 2.74 1.55 1.04Fragomeni, 1995 6a 163 0.53 1.29 2.40 1.59 1.27Fragomeni, 1995 7a 111 0.42 1.01 1.78 1.20 0.97Continued on Next Page201Researcher Specimen Ultimate Normalized Equation Rational Rational ACI 318-14Name Axial Ultimate 14-1 Method Method AlternativeLoad Axial Load EI = 0.2EcIg StiffnessPu Pu/(f ′cAg) +AstIst EI = 0.4EcIg[kN]Fragomeni, 1995 8a 158 0.54 1.20 1.88 1.37 1.18Fragomeni, 1995 9a 148 0.43 1.33 2.80 1.72 1.23Fragomeni, 1995 10a 230 0.44 1.37 2.88 1.77 1.27Fragomeni, 1995 1b 187 0.32 1.03 2.82 1.63 1.10Fragomeni, 1995 2b 264 0.27 0.88 2.54 1.46 0.97Fragomeni, 1995 3b 168 0.39 1.94 4.86 2.69 1.72Fragomeni, 1995 4b 217 0.33 1.67 4.19 2.32 1.48Fragomeni, 1995 5b 269 0.23 1.14 3.00 1.66 1.05Fragomeni, 1995 6b 178 0.33 0.85 1.98 1.23 0.92Fragomeni, 1995 7b 132 0.37 0.90 1.79 1.17 0.91Fragomeni, 1995 8b 233 0.47 1.10 2.05 1.39 1.14Fragomeni, 1995 9b 151 0.25 0.82 2.26 1.31 0.88Fragomeni, 1995 10b 265 0.29 0.94 2.63 1.52 1.02Huang, 2014 ST1 795 0.21 † 3.73 2.14 1.30Huang, 2014 ST2 804 0.21 † 4.04 2.18 1.34Huang, 2014 ST3 1274 0.34 † 4.87 2.67 1.42Huang, 2014 ST4 297 0.08 † 2.22‡ 1.47‡ 1.20Huang, 2014 ST5 1427 0.29 1.04 3.20 1.94 1.25Huang, 2014 ST6 1882 0.31 0.90 2.58‡ 1.67‡ 1.19Huang, 2014 ST7 846 0.23 † 4.22 2.38 1.49Huang, 2014 ST8 839 0.22 † 3.42 2.16 1.23Robinson, 2013 3 672 0.25 † 3.43 1.77 0.86Robinson, 2013 4 725 0.27 † 3.70 1.90 0.93Robinson, 2013 5 595 0.24 † 3.80 2.11 1.27Robinson, 2013 6 557 0.23 † 3.56 1.97 1.19Robinson, 2013 7 871 0.34 1.69 3.99 2.25 1.42Robinson, 2013 8 858 0.33 1.66 3.93 2.21 1.40Robinson, 2013 9 692 0.26 † 3.83 2.13 1.30Robinson, 2013 10 683 0.26 † 3.78 2.10 1.28Robinson, 2013 11 582 0.23 † 3.65 2.02 1.21Robinson, 2013 12 597 0.23 † 3.74 2.07 1.24Robinson, 2013 13 572 0.22 † 3.59 1.99 1.19Robinson, 2013 14 568 0.22 † 3.56 1.97 1.18Robinson, 2013 15 322 0.12 † 2.59‡ 1.59‡ 1.20Robinson, 2013 16 336 0.13 † 2.70‡ 1.66‡ 1.25†Slenderness h/t exceeds CSA A23.3-14 §14.1.7.1 limit of 25‡Slenderness h/t exceeds CSA A23.3-14 §10.13.2 limit of 30202Appendix CThin Lightly-Reinforced Shear Walls203C.1 UCLA RCWalls Selective Walls Database Query ResultsC.1.1 Specimen Test Setup DetailsTable C.1: Summary of UCLA database query wall experimental test setup details.GENERAL INFORMATION TESTING SETUP INFORMATIONLateral Loading Test SetupAuthor(s) Specimen Vertical Axial Axial Total No. of Height Unsupported Height of Test ShearName Reinforcing Load Load No. of Repeated of Lateral Panel Displacement Configuration‖ SpanLayers Ratio Cycles Cycles Load Height Measurement Ratio[kN] [%] [mm] [mm] [mm]Alarcon 2013 W1 1 287 14.98 16.5 2 1750 1600 1750 1 2.50Alarcon 2013 W2 1 479 24.97 14 2 1750 1600 1750 1 2.50Alarcon 2013 W3 1 672 35.02 12.5 2 1750 1600 1750 1 2.50Almeida et al., 2014 TW1 1 303 4.30 15 2 2210 2000 2210 2 3.70Almeida et al., 2014 TW2 2 590 3.21 8.5 1 2210 2000 2210 2 1.17Almeida et al., 2014 TW3 2 597 3.41 10 1 2210 2000 2210 2 1.17Almeida et al., 2014 TW4 1 253 3.31 8 1 2210 2000 2210 2 3.70Almeida et al., 2014 TW5 2 590 4.85 8.5 1 2210 2000 2210 2 2.72Altheeb 2016 Wall1 2 190 5.00 28 2 2650 2700 1 2.94Altheeb 2016 Wall2 2 187 5.00 26 2 2650 2700 1 2.94Albidah 2016 Wall3 2 230 4.99 2 2650 2700 1 2.94Carvajal and Pollner 1983 M-2 1 118 8.38 12 1 1550 3100 1550 1 3.10Carvajal and Pollner 1983 M-3 1 118 8.35 13.5 1 1550 3100 1550 1 3.10Carvajal and Pollner 1983 M-4 1 118 9.22 6.5 1 1550 3100 1550 1 3.10Carvajal and Pollner 1983 M-5 1 118 8.21 9.5 1 1550 3100 1550 1 3.10Carvajal and Pollner 1983 M-6 1 118 11.35 5.5 1 1550 3100 1550 1 3.10Ireland 2007 W1 2 150 3.10 20 2 1500 1350 1500 1 1.47Matsui et al., 2014 WF 1 343 8.00 11 2 2500 1985 2 1.50Yamakawa et al., 1993 RCW-NN 2 126 7.84 1140 1140 1 1.43Tomazevic et al., 1995 SW00N1 2 140 8.75 3 1785 1400 1450 1 2.37Tomazevic et al., 1995 SW00N2 2 279 17.50 3 1785 1400 1450 1 2.37Continued on Next Page204Lateral Loading Test SetupAuthor(s) Specimen Vertical Axial Axial Total No. of Height Unsupported Height of Test ShearName Reinforcing Load Load No. of Repeated of Lateral Panel Displacement Configuration∗∗ SpanLayers Ratio Cycles Cycles Load Height Measurement Ratio[kN] [%] [mm] [mm] [mm]Ho 2006 N-1.0 2 500 13.71 9 2 1350 1140 1350 1 1.13Ho 2006 N-1.5-B 1 500 12.14 7 2 1950 1740 1950 1 1.63Ho 2006 M-1.0-T 2 500 11.05 12.5 2 1350 1140 1350 1 1.13Ho and Kuang 2008 U1.5 2 500 13.71 6.5 2 1950 1740 1950 1 1.63Ho and Kuang 2008 C1.0 1 500 12.14 8 2 1350 1140 1350 1 1.13Marihuen 2014 W4 1 216 15.02 13 2 1750 1600 1750 1 2.50Marihuen 2014 W5 1 287 14.96 15 2 1330 1180 1330 1 1.90Marihuen 2014 W6 2 287 14.96 16.5 2 1750 1600 1750 1 2.50Marihuen 2014 W7 1 287 14.96 16.5 2 1750 1600 1750 1 2.50Ogura et al., 2014 NSW4 1 458 15.02 2 2100 2100 1 2.00Zhang et al., 2010 SW-1 2 550 19.56 59 3 1750 1600 1 2.06Deng et al., 2012 1 2 950 54.98 14 2 2925 2925 2925 1 3.90Deng et al., 2012 2 2 605 35.01 14 2 2925 2925 2925 1 3.90Deng et al., 2012 3 2 1140 54.98 12.5 2 2925 2925 2925 1 3.25Deng et al., 2012 4 2 725 34.96 13 2 2925 2925 2925 1 3.25Peng et al., 2013 1 2 605 35.01 2 2925 2925 2925 1 3.25Peng et al., 2013 2 2 950 54.98 2 2925 2925 2925 1 3.25Wang et al., 2011 W-1A 2 640 12.66 3 2025 1850 2025 1 1.45Wang et al., 2011 W-1B 2 640 12.66 26 3 2025 1850 2025 1 1.45Wang et al., 2015 W-1A 2 640 11.37 21 3 2025 1850 2025 1 1.45Wang et al., 2015 W-1B 2 640 11.37 3 2025 1850 2025 1 1.45Lu et al., 2017 C1 2 290 3.59 31 3 2963 2800 2650 1 2.00Lu et al., 2017 C2 2 290 4.00 31 3 2963 2800 2650 2 4.00Lu et al., 2017 C3 2 290 3.81 31 3 2963 2800 2650 2 6.00Dazio et al., 1999 WSH4 1 695 5.66 8 2 4560 1700 4560 1 2.28Baek et al., 2018 FW2 1 1080 29.09 2 3375 3375 1 2.70Chun et al., 2013 MW1 2 768 9.30 27.5 3 3700 2800 2800 2 2.51Zhu and Guo 2013 XJ1 2 600 9.87 9 1 1700 1700 1 1.70Continued on Next Page205Lateral Loading Test SetupAuthor(s) Specimen Vertical Axial Axial Total No. of Height Unsupported Height of Test ShearName Reinforcing Load Load No. of Repeated of Lateral Panel Displacement Configuration∗∗ SpanLayers Ratio Cycles Cycles Load Height Measurement Ratio[kN] [%] [mm] [mm] [mm]Oh 1998 HRI-W6 1 828 8.71 21.5 3 3000 2000 3000 1 2.00Wang et al., 2012 W-1 2 600 5.56 1 2800 2800 1 1.40Villalobos 2015 W-MC-N 1 890 9.57 24 3 3315 3048 3315 1 2.18Villalobos 2014 W-60-N 1 890 9.57 27.5 3 3315 3048 3315 1 2.18Villalobos 2015 W-60-N2 1 890 8.77 22 3 3315 3048 3315 1 2.18∗∗1 = Lateral Load Only Applied; 2 = Lateral Load and Top Moment Applied206C.1.2 Specimen GeometryTable C.2: Summary of UCLA database query wall experimental test specimen geometry.GENERAL INFORMATION SPECIMEN GEOMETRYAuthor(s) Specimen Flanged Clear Length Wall Flange Geometry Gross Aspect Cross-SectionName Cross-Section Wall Of Web Flange Flange Area Ratio AspectHeight, Hw Wall, Lw Thickness, tw Length, Lf Thickness, tf Ag Hw/Lw Ratio, Lw/tw[mm] [mm] [mm] [mm] [mm] [mm] [mm]Alarcon 2013 W1 No 1600 700 100 70000 2.29 7.0Alarcon 2013 W2 No 1600 700 100 70000 2.29 7.0Alarcon 2013 W3 No 1600 700 100 70000 2.29 7.0Almeida et al., 2014 TW1 Yes 2000 2700 80 440 80 244800 0.74 33.8Almeida et al., 2014 TW2 Yes 2000 2700 120 440 120 362400 0.74 22.5Almeida et al., 2014 TW3 Yes 2000 2700 120 440 120 362400 0.74 22.5Almeida et al., 2014 TW4 Yes 2000 2700 80 440 80 244800 0.74 33.8Almeida et al., 2014 TW5 Yes 2000 2700 120 440 120 362400 0.74 22.5Altheeb 2016 Wall1 No 2750 900 120 108000 3.06 7.5Altheeb 2016 Wall2 No 2750 900 120 108000 3.06 7.5Albidah 2016 Wall3 No 2750 900 120 108000 3.06 7.5Carvajal and Pollner 1983 M-2 No 1550 500 100 50000 3.10 5.0Carvajal and Pollner 1983 M-3 No 1550 500 100 50000 3.10 5.0Carvajal and Pollner 1983 M-4 No 1550 500 100 50000 3.10 5.0Carvajal and Pollner 1983 M-5 No 1550 500 100 50000 3.10 5.0Carvajal and Pollner 1983 M-6 No 1550 500 100 50000 3.10 5.0Ireland 2007 W1 No 1350 1020 125 127500 1.32 8.2Matsui et al., 2014 WF No 1600 1600 80 128000 1.00 20.0Yamakawa et al., 1993 RCW-NN No 950 800 80 64000 1.19 10.0Tomazevic et al., 1995 SW00N1 No 1400 970 50 48500 1.44 19.4Tomazevic et al., 1995 SW00N2 No 1400 970 50 48500 1.44 19.4Ho 2006 N-1.0 No 1200 1200 100 120000 1.00 12.0Ho 2006 N-1.5-B No 1800 1200 100 120000 1.50 12.0Ho 2006 M-1.0-T No 1200 1200 100 120000 1.00 12.0Continued on Next Page207Author(s) Specimen Flanged Clear Length Wall Flange Geometry Gross Aspect Cross-SectionName Cross-Section Wall Of Web Flange Flange Area Ratio AspectHeight, Hw Wall, Lw Thickness, tw Length, Lf Thickness, tf Ag Hw/Lw Ratio, Lw/tw[mm] [mm] [mm] [mm] [mm] [mm] [mm]Ho and Kuang 2008 U1.5 No 1800 1200 100 120000 1.50 12.0Ho and Kuang 2008 C1.0 No 1200 1200 100 120000 1.00 12.0Marihuen 2014 W4 No 1600 700 75 52500 2.29 9.3Marihuen 2014 W5 No 1180 700 100 70000 1.69 7.0Marihuen 2014 W6 No 1600 700 100 70000 2.29 7.0Marihuen 2014 W7 No 1600 700 100 70000 2.29 7.0Ogura et al., 2014 NSW4 No 2100 1050 120 126000 2.00 8.8Zhang et al., 2010 SW-1 No 1600 850 125 106250 1.88 6.8Deng et al., 2012 1 No 2800 750 160 120000 3.73 4.7Deng et al., 2012 2 No 2800 750 160 120000 3.73 4.7Deng et al., 2012 3 No 2800 900 160 144000 3.11 5.6Deng et al., 2012 4 No 2800 900 160 144000 3.11 5.6Peng et al., 2013 1 No 2800 750 160 120000 3.73 4.7Peng et al., 2013 2 No 2800 750 160 120000 3.73 4.7Wang et al., 2011 W-1A No 1850 1400 160 224000 1.32 8.8Wang et al., 2011 W-1B No 1850 1400 160 224000 1.32 8.8Wang et al., 2015 W-1A No 1850 1400 160 224000 1.32 8.8Wang et al., 2015 W-1B No 1850 1400 160 224000 1.32 8.8Lu et al., 2017 C1 No 2800 1400 150 210000 2.00 9.3Lu et al., 2017 C2 No 2800 1400 150 210000 2.00 9.3Lu et al., 2017 C3 No 2800 1400 150 210000 2.00 9.3Dazio et al., 1999 WSH4 No 4030 2000 150 300000 2.02 13.3Baek et al., 2018 FW2 No 3125 1250 180 225000 2.50 6.9Chun et al., 2013 MW1 No 2800 1600 200 320000 1.75 8.0Zhu and Guo 2013 XJ1 No 1800 1000 200 200000 1.80 5.0Oh 1998 HRI-W6 No 2000 1500 200 300000 1.33 7.5Wang et al., 2012 W-1 No 2800 2000 200 400000 1.40 10.0Villalobos 2015 W-MC-N No 3658 1524 203 309982 2.40 7.5Villalobos 2014 W-60-N No 3658 1524 203 309982 2.40 7.5Continued on Next Page208Author(s) Specimen Flanged Clear Length Wall Flange Geometry Gross Aspect Cross-SectionName Cross-Section Wall Of Web Flange Flange Area Ratio AspectHeight, Hw Wall, Lw Thickness, tw Length, Lf Thickness, tf Ag Hw/Lw Ratio, Lw/tw[mm] [mm] [mm] [mm] [mm] [mm] [mm]Villalobos 2015 W-60-N2 No 3658 1524 203 309982 2.40 7.5209C.1.3 Specimen Concrete and Reinforcing Material PropertiesTable C.3: Summary of UCLA database query wall experimental test specimen concrete and reinforcing material properties.GENERAL INFORMATION CONCRETE AND REINFORCING MATERIALSAuthor(s) Specimen Measured Web Vertical Reinforcing Web Horizontal Reinforcing Boundary Vertical ReinforcingName Cylinder Reinforcing Yield Ultimate Reinforcing Yield Ultimate Reinforcing Yield UltimateStrength Ratio Strength Strength Ratio Strength Strength Ratio Strength Strengthf ′c ρv fy fu ρv fy fu ρv fy fu[MPa] [%] [MPa] [MPa] [%] [MPa] [MPa] [%] [MPa] [MPa]Alarcon 2013 W1 27.4 0.74 446 599 0.44 609 668 469 676 3.14Alarcon 2013 W2 27.4 0.74 446 599 0.44 609 668 469 676 3.14Alarcon 2013 W3 27.4 0.74 446 599 0.44 609 668 469 676 3.14Almeida et al., 2014 TW1 28.8 0.15 460 625 0.18 460 625 565 650 2.63Almeida et al., 2014 TW2 50.7 0.50 460 625 0.36 460 625 0.50Almeida et al., 2014 TW3 48.3 0.50 460 625 0.36 460 625 0.50Almeida et al., 2014 TW4 31.2 0.15 460 625 0.18 460 625 512 618 2.63Almeida et al., 2014 TW5 33.6 0.50 460 625 0.36 460 625 0.50Altheeb 2016 Wall1 35.19 0.32 500 720 0.33 500 720 0.32Altheeb 2016 Wall2 34.66 0.73 500 720 0.33 500 720 0.73Albidah 2016 Wall3 42.71 0.90 500 720 0.33 500 720 0.90Carvajal and Pollner 1983 M-2 28.15 0.25 589 0.25 589 412 1.49Carvajal and Pollner 1983 M-3 28.25 0.25 589 0.25 589 412 1.49Carvajal and Pollner 1983 M-4 25.6 0.25 589 0.25 589 451 0.84Carvajal and Pollner 1983 M-5 28.74 0.25 589 0.25 589 451 0.84Carvajal and Pollner 1983 M-6 20.8 0.25 589 0.25 589 451 0.84Ireland 2007 W1 38 0.42 316 415 0.24 376 460 0.42Matsui et al., 2014 WF 33.5 0.26 359 474 0.26 359 474 355 516 2.49Yamakawa et al., 1993 RCW-NN 25.1 0.79 438 446 0.79 438 446 0.79Tomazevic et al., 1995 SW00N1 32.88 0.24 478 531 0.24 478 531 0.24Tomazevic et al., 1995 SW00N2 32.88 0.26 478 531 0.24 478 531 0.26Ho 2006 N-1.0 30.4 0.87 520 615 1.05 520 615 0.87Ho 2006 N-1.5-B 34.32 0.52 520 615 1.05 520 615 520 615 1.96Continued on Next Page210Author(s) Specimen Measured Web Vertical Reinforcing Web Horizontal Reinforcing Boundary Vertical ReinforcingName Cylinder Reinforcing Yield Ultimate Reinforcing Yield Ultimate Reinforcing Yield UltimateStrength Ratio Strength Strength Ratio Strength Strength Ratio Strength Strengthf ′c ρv fy fu ρv fy fu ρv fy fu[MPa] [%] [MPa] [MPa] [%] [MPa] [MPa] [%] [MPa] [MPa]Ho 2006 M-1.0-T 37.7 0.87 520 615 1.05 520 615 0.87Ho and Kuang 2008 U1.5 30.4 0.87 520 615 1.05 520 615 0.87Ho and Kuang 2008 C1.0 34.32 0.52 520 615 1.05 520 615 520 615 1.96Marihuen 2014 W4 27.4 0.99 446 599 0.46 524 576 469 676 3.43Marihuen 2014 W5 27.4 0.74 446 599 0.44 609 668 469 676 3.14Marihuen 2014 W6 27.4 1.36 446 599 0.44 609 668 469 676 1.36Marihuen 2014 W7 27.4 0.74 446 599 0.44 609 668 469 676 3.14Ogura et al., 2014 NSW4 24.2 0.24 547 484 0.48 547 484 360 527 1.03Zhang et al., 2010 SW-1 26.46 0.40 392 479 0.43 392 479 0.40Deng et al., 2012 1 14.4 0.14 259 0.14 259 0.14Deng et al., 2012 2 14.4 0.14 259 0.14 259 0.14Deng et al., 2012 3 14.4 0.13 259 0.07 259 0.13Deng et al., 2012 4 14.4 0.13 259 0.07 259 0.13Peng et al., 2013 1 14.4 0.10 259 0.14 259 0.10Peng et al., 2013 2 14.4 0.10 259 0.14 259 0.10Wang et al., 2011 W-1A 22.56 0.49 362 477 0.49 362 477 0.49Wang et al., 2011 W-1B 22.56 0.49 362 477 0.49 362 477 0.49Wang et al., 2015 W-1A 25.12 0.49 575 595 0.49 575 595 0.49Wang et al., 2015 W-1B 25.12 0.49 575 595 0.49 575 595 0.49Lu et al., 2017 C1 38.5 0.47 300 409 0.25 301 462 0.47Lu et al., 2017 C2 34.5 0.47 300 409 0.25 301 462 0.47Lu et al., 2017 C3 36.2 0.47 300 409 0.25 301 462 0.47Dazio et al., 1999 WSH4 40.9 0.54 584 714 0.25 519 559 576 675 1.74Baek et al., 2018 FW2 16.5 0.18 540 0.20 540 555 0.80Chun et al., 2013 MW1 25.8 0.51 506 624 0.31 506 624 0.51Zhu and Guo 2013 XJ1 30.4 0.75 492 656 0.28 317 460 0.75Oh 1998 HRI-W6 31.7 0.32 329 435 0.29 329 435 355 600 1.29Wang et al., 2012 W-1 27 0.57 335 0.57 335 0.57Continued on Next Page211Author(s) Specimen Measured Web Vertical Reinforcing Web Horizontal Reinforcing Boundary Vertical ReinforcingName Cylinder Reinforcing Yield Ultimate Reinforcing Yield Ultimate Reinforcing Yield UltimateStrength Ratio Strength Strength Ratio Strength Strength Ratio Strength Strengthf ′c ρv fy fu ρv fy fu ρv fy fu[MPa] [%] [MPa] [MPa] [%] [MPa] [MPa] [%] [MPa] [MPa]Villalobos 2015 W-MC-N 30 0.41 434 634 0.55 531 683 462 655 4.91Villalobos 2014 W-60-N 30 0.41 434 634 0.55 531 683 462 655 4.91Villalobos 2015 W-60-N2 32.75 0.41 434 634 0.55 455 710 469 669 4.91212C.1.4 Specimen Test ResultsTable C.4: Summary of UCLA database query wall experimental test specimen results.GENERAL INFORMATION TEST SPECIMEN RESULTSAuthor(s) Specimen Measured Base Shear and Top Displacement Boundary Vertical ReinforcingName Top Drift Failure Neutral AxisCracking Yielding Maximum Ultimate Residual Collapse Capacity Mode Depth, c[mm,kN] [mm,kN] [mm,kN] [mm,kN] [mm,kN] [mm,kN] [%] [mm]Alarcon 2013 W17.2 11.3 31.5 43.1 48.0 48.02.46 Concrete Crushing 18986.5 106.0 144.3 115.4 80.0 0.0Alarcon 2013 W24.0 11.2 31.0 31.0 31.0 31.01.77 Concrete Crushing 25568.0 124.0 161.0 153.5 0.0 0.0Alarcon 2013 W34.2 11.0 24.0 26.3 26.3 26.31.50 Concrete Crushing 32268.0 131.0 165.0 146.5 0.0 0.0Almeida et al., 2014 TW11.0 7.3 16.5 16.5 22.0 22.00.75 Lateral Instability 35777.5 167.5 172.5 138.0 0.0 0.0Almeida et al., 2014 TW21.5 6.0 17.0 20.0 23.0 23.00.90 Concrete Crushing 325400.0 700.0 755.0 604.0 500.0 0.0Almeida et al., 2014 TW32.0 8.0 16.0 21.0 33.0 33.00.95 Bar Fracture 341480.0 713.0 750.0 600.0 280.0 0.0Almeida et al., 2014 TW41.0 7.8 17.0 17.0 17.0 17.00.77 Lateral Instability 34373.0 164.0 173.0 138.0 0.0 0.0Almeida et al., 2014 TW51.0 5.5 16.7 19.2 22.0 22.00.87 Concrete Crushing 403189.0 275.0 307.0 246.0 180.0 0.0Continued on Next Page213Author(s) Specimen Measured Base Shear and Top Displacement Boundary Vertical ReinforcingName Top Drift Failure Neutral AxisCracking Yielding Maximum Ultimate Residual Collapse Capacity Mode Depth, c[mm,kN] [mm,kN] [mm,kN] [mm,kN] [mm,kN] [mm,kN] [%] [mm]Altheeb 2016 Wall16.0 14.0 39.0 60.0 85.0 124.02.22 Bar Fracture 10844.0 66.0 68.0 63.0 28.0 28.0Altheeb 2016 Wall25.0 17.0 43.0 77.0 95.0 118.02.85 Concrete Crushing 15255.0 88.0 95.0 85.0 16.0 16.0Albidah 2016 Wall35.0 21.5 32.5 58.0 88.0 100.02.15 Concrete Crushing 16677.0 125.0 134.4 107.5 12.0 12.0Carvajal and Pollner 1983 M-22.0 9.0 12.5 64.04.13 Bar Buckling 8219.6 36.5 38.8 34.6Carvajal and Pollner 1983 M-32.0 7.6 9.0 50.03.23 Bar Buckling 8219.6 34.4 36.4 35.8Carvajal and Pollner 1983 M-42.3 7.0 8.0 65.04.19 Bar Buckling 8719.6 29.4 30.3 25.0Carvajal and Pollner 1983 M-52.0 10.0 13.5 51.23.30 Bar Buckling 8011.8 28.0 29.0 23.5Carvajal and Pollner 1983 M-62.0 8.0 9.8 34.02.19 Bar Buckling 10216.2 28.5 31.0 23.4Ireland 2007 W10.8 3.3 15.8 31.0 38.0 45.02.07 Bar Fracture 6646.0 100.0 110.0 88.0 85.0 70.0Continued on Next Page214Author(s) Specimen Measured Base Shear and Top Displacement Boundary Vertical ReinforcingName Top Drift Failure Neutral AxisCracking Yielding Maximum Ultimate Residual Collapse Capacity Mode Depth, c[mm,kN] [mm,kN] [mm,kN] [mm,kN] [mm,kN] [mm,kN] [%] [mm]Matsui et al., 2014 WF1.0 5.7 13.5 19.9 26.81.15 Concrete Crushing 241120.0 213.0 230.0 224.0 95.0Yamakawa et al., 1993 RCW-NN2.7 10.3 13.61.19 Bar Fracture 169100.0 112.0 90.0Tomazevic et al., 1995 SW00N10.7 3.6 8.4 19.21.32 Bar Fracture 15125.6 36.8 39.0 29.3Tomazevic et al., 1995 SW00N21.2 3.0 6.7 8.50.59 Concrete Crushing 25840.0 53.7 62.5 60.6Ho 2006 N-1.00.7 4.6 10.0 11.8 16.20.87 Concrete Crushing 333142.0 257.0 359.0 287.0 160.0Ho 2006 N-1.5-B2.5 9.0 11.2 17.0 18.60.87 Lateral Instability 318140.0 271.0 288.0 230.4 150.0Ho 2006 M-1.0-T0.7 3.6 8.3 14.0 16.81.04 Concrete Crushing 291179.5 297.0 378.0 302.0 215.0Ho and Kuang 2008 U1.51.8 9.3 14.3 19.5 21.31.00 Concrete Crushing 333123.0 252.0 277.0 222.0 172.0Ho and Kuang 2008 C1.01.8 6.9 12.8 14.5 14.51.07 Concrete Crushing 318196.0 362.0 412.0 330.0 215.0Continued on Next Page215Author(s) Specimen Measured Base Shear and Top Displacement Boundary Vertical ReinforcingName Top Drift Failure Neutral AxisCracking Yielding Maximum Ultimate Residual Collapse Capacity Mode Depth, c[mm,kN] [mm,kN] [mm,kN] [mm,kN] [mm,kN] [mm,kN] [%] [mm]Marihuen 2014 W45.8 12.8 24.5 29.3 31.5 31.51.67 Concrete Crushing 19950.0 90.0 113.0 90.4 0.0 0.0Marihuen 2014 W53.5 9.7 20.4 25.5 25.5 25.51.92 Concrete Crushing 18989.0 167.0 191.0 185.0 0.0 0.0Marihuen 2014 W67.7 13.1 28.2 36.4 36.4 36.42.08 Bar Buckling 21287.5 116.0 138.0 110.0 0.0 0.0Marihuen 2014 W75.4 13.1 30.1 40.0 44.0 47.72.29 Bar Buckling 18972.5 116.0 140.0 129.0 75.0 75.0Ogura et al., 2014 NSW47.0 30.2 42.0 42.0 42.02.00 Concrete Crushing 234153.0 187.0 183.0 0.0 0.0Zhang et al., 2010 SW-10.8 3.8 17.0 35.02.19 Bar Buckling 23630.0 78.0 110.0 90.0Deng et al., 20121 5.0 9.5 23.0 53.01.81 Concrete Crushing 50975.2 91.0 85.0 75.0Deng et al., 20122 3.0 6.5 12.5 36.01.23 Concrete Crushing 33243.6 56.0 59.2 47.0Deng et al., 20123 3.8 9.0 13.0 34.51.18 Concrete Crushing 61183.3 112.0 118.6 94.4Continued on Next Page216Author(s) Specimen Measured Base Shear and Top Displacement Boundary Vertical ReinforcingName Top Drift Failure Neutral AxisCracking Yielding Maximum Ultimate Residual Collapse Capacity Mode Depth, c[mm,kN] [mm,kN] [mm,kN] [mm,kN] [mm,kN] [mm,kN] [%] [mm]Deng et al., 20124 3.0 13.6 22.0 46.01.57 Concrete Crushing 39948.1 95.0 99.0 79.0Peng et al., 20131 2.5 6.0 10.0 54.01.85 Concrete Crushing 33251.0 60.0 64.0 55.0Peng et al., 20132 3.2 8.0 23.0 55.01.88 Concrete Crushing 51258.0 70.0 74.0 70.0Wang et al., 2011 W-1A2.1 7.2 13.2 28.21.39 Bar Fracture 317112.0 224.0 271.1 225.0Wang et al., 2011 W-1B2.0 7.4 24.7 30.81.52 Bar Fracture 317100.0 207.0 290.0 244.0Wang et al., 2015 W-1A2.5 7.0 17.5 24.11.19 Bar Fracture 328180.0 237.0 399.5 315.0Wang et al., 2015 W-1B3.5 10.3 24.6 34.81.72 Bar Fracture 328180.0 321.0 418.4 355.6Lu et al., 2017 C13.5 8.0 40.2 70.02.64 Bar Fracture 139118.8 153.9 173.6 158.0Lu et al., 2017 C22.7 11.1 41.8 70.82.64 Bar Fracture 14558.0 77.9 89.1 80.0Continued on Next Page217Author(s) Specimen Measured Base Shear and Top Displacement Boundary Vertical ReinforcingName Top Drift Failure Neutral AxisCracking Yielding Maximum Ultimate Residual Collapse Capacity Mode Depth, c[mm,kN] [mm,kN] [mm,kN] [mm,kN] [mm,kN] [mm,kN] [%] [mm]Lu et al., 2017 C34.4 9.5 42.1 71.22.81 Bar Fracture 13636.3 47.6 56.5 51.8Dazio et al., 1999 WSH43.3 27.0 46.5 73.0 77.01.60 Bar Buckling 313196.0 425.0 443.0 354.0 335.0Baek et al., 2018 FW216.5 28.7 60.0 62.0 62.01.78 Concrete Crushing 485215.0 231.0 184.8 0.0 0.0Chun et al., 2013 MW13.8 20.3 40.9 56.0 70.0 70.02.00 Concrete Crushing 341136.0 262.5 282.6 267.2 133.0 133.0Zhu and Guo 2013 XJ14.0 16.0 50.0 80.04.71 Concrete Crushing 182120.0 260.0 305.0 255.0Oh 1998 HRI-W61.6 13.9 31.6 64.4 66.0 67.52.15 Bar Buckling 236167.6 306.7 363.6 290.0 250.0 250.0Wang et al., 2012 W-115.5 17.3 38.81.39 Bar Fracture 313597.0 600.0 532.0Villalobos 2015 W-MC-N1.7 18.6 49.7 80.0 80.0 80.02.41 Bar Buckling 325134.0 587.0 719.0 575.0 0.0 0.0Villalobos 2014 W-60-N1.7 18.9 49.7 66.3 66.5 99.52.00 Bar Buckling 325187.0 614.0 729.0 584.0 330.0 330.0Continued on Next Page218Author(s) Specimen Measured Base Shear and Top Displacement Boundary Vertical ReinforcingName Top Drift Failure Neutral AxisCracking Yielding Maximum Ultimate Residual Collapse Capacity Mode Depth, c[mm,kN] [mm,kN] [mm,kN] [mm,kN] [mm,kN] [mm,kN] [%] [mm]Villalobos 2015 W-60-N22.0 18.9 49.7 66.3 66.3 83.02.00 Bar Buckling 325200.0 614.0 754.0 603.0 400.0 400.0219C.2 EPFL Test Results for TW2, TW4, and TW5This appendix includes various supplemental results of the EPFL test specimens TW2, TW4, and TW5.220C.2.1 Vertical StrainsC.2.1.1 TW2 Tension Strains01000 50 100 150 200Tension Strain [mm/m]02004006008001000120014001600180020000 2 4 6 8 10 12 14 16 18 20 22 24Elevation [mm]LS15LS17LS19Figure C.1: TW2 observed tension strains at 162 mm inside the extreme fibre for multiple load steps.221C.2.1.2 TW2 Compression Strains050100-70 -60 -50 -40 -30 -20 -10 0Compression Strain [mm/m]0200400600800100012001400160018002000-10 -8 -6 -4 -2 0 2 4Elevation [mm]LS15LS17LS19Figure C.2: TW2 observed compression strains at 68 mm inside the extreme fibre for multiple load steps.222C.2.1.3 TW4 Tension Strains02004006008001000120014001600180020000 2 4 6 8 10 12Elevation [mm]LS26LS32LS42LS4801000 20 40 60 80 100 120Tension Strain [mm/m]18002000-70 -60 -50 -40 -30 -20 -10 0Figure C.3: TW4 observed tension strains at 162 mm inside the extreme fibre for multiple load steps.223C.2.1.4 TW4 Compression Strains0200400600800100012001400160018002000-3 -2.5 -2 -1.5 -1 -0.5 0Elevation [mm]LS26LS32LS42LS480-20.0 -15.0 -10.0 -5.0 0.0Compression Strain [mm/m]180019002000-80.0 -70.0 -60.0 -50.0 -40.0 -30.0 -20.0 -10.0 0.0Figure C.4: TW4 observed compression strains at the extreme fibre for multiple load steps.224C.2.1.5 TW5 Tension Strains02004006008001000120014001600180020000 2 4 6 8 10 12 14 16 18Elevation [mm]LS32LS3801000 10 20 30 40 50 60 70Tension Strain [mm/m]180020000 10 20 30 40 50Figure C.5: TW5 observed tension strains at 162 mm inside the extreme fibre for multiple load steps.225C.2.1.6 TW5 Compression Strains18002000-4 -2 0 2 4 6 8 100200-35 -30 -25 -20 -15 -10 -5 0Compression Strain [mm/m]0200400600800100012001400160018002000-6.5 -6 -5.5 -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0Elevation [mm]LS32LS38Figure C.6: TW5 observed compression strains at 68 mm inside the extreme fibre for multiple load steps.226C.2.2 CurvaturesC.2.2.1 TW2 Curvature02004006008001000120014001600180020000 2 4 6 8 10 12Elevation [mm]LS15LS17LS1901000 20 40 60 80 100 120Curvature [rad/km]Figure C.7: TW2 observed curvature for multiple load steps.227C.2.2.2 TW4 Curvature02004006008001000120014001600180020000 1 2 3 4 5 6Elevation [mm]LS26LS32LS42LS4801000 10 20 30 40 50Curvature [rad/km]Figure C.8: TW4 observed curvature for multiple load steps.228C.2.2.3 TW5 Curvature01000 5 10 15 20 25 30 35 40Curvature [rad/km]1800190020000 50 100 150 200 250 30002004006008001000120014001600180020000 2 4 6 8 10Elevation [mm]LS32LS38Figure C.9: TW5 observed curvature for multiple load steps.229C.2.3 Shear StrainsC.2.3.1 TW2 Shear Strains02004006008001000120014001600180020000 2 4 6 8 10Elevation [mm]Shear Strain [mm/m]LS15LS17LS19Figure C.10: TW2 observed shear strains for multiple load steps.230C.2.3.2 TW4 Shear Strains02004006008001000120014001600180020000 2 4 6Elevation [mm]Shear Strain [mm/m]LS26LS32LS42LS48Figure C.11: TW4 observed shear strains for multiple load steps.231C.2.3.3 TW5 Shear Strains02004006008001000120014001600180020000 1 2 3 4 5Elevation [mm]Shear Strain [mm/m]LS32LS38Figure C.12: TW5 observed shear strains for multiple load steps.232C.3 EPFL Test Setup Influence ResultsThis section includes figures supporting the study on test setup influence on various parameters.233C.3.1 NLFE Vertical StrainsC.3.1.1 TW1 NLFE Vertical Strains0200400600800100012001400160018002000-5 0 5 10 15 20 25Elevation [mm]Vertical Strain [mm/m]Test Setup Tension StrainsTest Setup Compression StrainsFull Height Tension StrainsFull Height Compression StrainsFigure C.13: TW1 predicted vertical strains for the test setup and full height NLFE models.234C.3.1.2 TW2 NLFE Vertical Strains0200400600800100012001400160018002000-10 -5 0 5 10 15 20Elevation [mm]Vertical Strain [mm/m]Test Setup Tension StrainsTest Setup Compression StrainsFull Height Tension StrainsFull Height Compression StrainsFigure C.14: TW2 predicted vertical strains for the test setup and full height NLFE models.235C.3.2 NLFE Horizontal Variation of Vertical StrainsC.3.2.1 TW1 NLFE Horizontal Variation of Vertical Strains-5051015202530350 500 1000 1500 2000 2500Base Element Vertical Strain [mm/m]Horizontal Position [mm]Test Setup ModelFull Height ModelFigure C.15: TW1 predicted horizontal variation of vertical strains for the test setup and full height NLFEmodels at the base of the wall panel.236C.3.2.2 TW2 NLFE Horizontal Variation of Vertical Strains-10-5051015200 500 1000 1500 2000 2500Base Element Vertical Straain [mm/m]Horizontal Position [mm]Test Setup ModelFull Height ModelFigure C.16: TW2 predicted horizontal variation of vertical strains for the test setup and full height NLFEmodels at the base of the wall panel.237C.3.3 NLFE Average Shear StrainsC.3.3.1 TW1 NLFE Average Shear Strains02004006008001000120014001600180020000 0.5 1 1.5 2 2.5 3 3.5 4 4.5Elevation [mm]Average Shear Strain [mm/m]Test Setup ModelFull Height ModelFigure C.17: TW1 predicted average shear strains for the test setup and full height NLFE models.238C.3.3.2 TW2 NLFE Average Shear Strains02004006008001000120014001600180020000 0.5 1 1.5 2 2.5 3 3.5 4Elevation [mm]Average Shear Strain [mm/m]Test Setup ModelFull Height ModelFigure C.18: TW2 predicted average shear strains for the test setup and full height NLFE models.239C.4 Development of Proposed ModelThis appendix contains various figures associated with the development of the proposed plastic hinge modelbased on the EPFL wall specimens.240C.4.1 Inelastic CurvaturesC.4.1.1 TW2 Test Specimen Inelastic Curvature01000 20 40 60 80 100 120Inelastic Curvature [rad/km]02004006008001000120014001600180020000 2 4 6 8 10 12Elevation [mm]LS15LS17LS19Figure C.19: TW2 test specimen estimated distribution of inelastic curvature for load steps 15 to 19.241C.4.1.2 TW4 Test Specimen Inelastic Curvature02004006008001000120014001600180020000 1 2 3 4 5 6Elevation [mm]LS26LS32LS42LS4801000 10 20 30 40 50Inelastic Curvature [rad/km]Figure C.20: TW4 test specimen estimated distribution of inelastic curvature for load steps 26 to 48.242C.4.1.3 TW5 Test Specimen Inelastic Curvature01000 5 10 15 20 25 30 35 40Curvature [rad/km]1800190020000 50 100 150 200 250 30002004006008001000120014001600180020000 2 4 6 8 10Elevation [mm]LS32LS38Figure C.21: TW5 test specimen estimated distribution of inelastic curvature for load steps 32 to 38.243C.4.1.4 TW1 NLFE Inelastic Curvature02004006008001000120014001600180020000 2 4 6 8 10 12 14 16Elevation [mm]Inelastic Curvature [rad/km]Observed Gauge LengthObserved Inelastic CurvatureNLFE PredictionFigure C.22: TW1 NLFE and test specimen estimated inelastic curvature.244C.4.1.5 TW2 NLFE Inelastic Curvature02004006008001000120014001600180020000 2 4 6 8 10 12 14 16Elevation [mm]Inelastic Curvature [rad/km]Observed Gauge LengthObserved Inelastic CurvatureNLFE PredictionFigure C.23: TW2 NLFE and test specimen estimated inelastic curvature.245C.4.1.6 TW1 NLFE Linearly Varying Inelastic Curvature02004006008001000120014001600180020000 2 4 6 8 10 12 14 16Elevation [mm]Inelastic Curvature [rad/km]NLFE PredictionLinear Model - Equal Area and Moment of AreaLinearly Varying Curvature due to Reinforcement YieldingFigure C.24: TW1 NLFE Model of linearly varying inelastic curvature based on equal area and first momentof area in comparison to length of reinforcement yielding.246C.4.1.7 TW2 NLFE Linearly Varying Inelastic Curvature02004006008001000120014001600180020000 2 4 6 8 10 12 14 16Elevation [mm]Inelastic Curvature [rad/km]NLFE PredictionLinear Model - Equal Area and Moment of AreaLinearly Varying Curvature due to Reinforcement YieldingFigure C.25: TW2 NLFE Model of linearly varying inelastic curvature based on equal area and first momentof area in comparison to length of reinforcement yielding.247C.4.1.8 TW4 Linearly Varying Inelastic Curvature02004006008001000120014001600180020000 2 4 6 8 10 12 14 16Elevation [mm]Inelastic Curvature [rad/km]Observed Inelastic CurvatureLinear Model - Equal Area and Moment of AreaLinearly Varying Curvature due to Reinforcement YieldingFigure C.26: TW4 Model of linearly varying inelastic curvature based on equal area and first moment ofarea in comparison to length of reinforcement yielding.248C.4.1.9 TW5 Linearly Varying Inelastic Curvature02004006008001000120014001600180020000 2 4 6 8 10 12 14 16Elevation [mm]Inelastic Curvature [rad/km]Observed Inelastic CurvatureLinear Model - Equal Area and Moment of AreaLinearly Varying Curvature due to Reinforcement YieldingFigure C.27: TW5 Model of linearly varying inelastic curvature based on equal area and first moment ofarea in comparison to length of reinforcement yielding.249

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