Strength, Stiffness, and Stability ofSolid Continua: Gravity LoadingScenario on Cross-Laminated TimberbyP.K.M. MoniruzzamanB.Sc., Bangladesh University of Engineering and Technology, 2009M.A.Sc., The University of British Columbia, 2013A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Forestry)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)November 2019c© P.K.M. Moniruzzaman, 2019The following individuals certify that they have read, and recommend to the Faculty ofGraduate and Postdoctoral Studies for acceptance, the dissertation entitled:Strength, Stiffness, and Stability of Solid Continua: Gravity Loading Scenarioon Cross-Laminated Timbersubmitted by P.K.M. Moniruzzaman in partial fulfillment of the requirements forthe degree of Doctor of Philosophyin ForestryExamining Committee:Prof. Frank Lam, Department of Wood Science, UBCSupervisorProf. Ricardo O. Foschi, Department of Civil Engineering, UBCSupervisory Committee MemberProf. Stavros Avramidis, Department of Wood Science, UBCSupervisory Committee MemberProf. Tsung Yuan (Tony) Yang, Department of Civil Engineering, UBCUniversity ExaminerProf. W.D. Liam Finn, Department of Civil Engineering, UBCUniversity ExaminerAdditional Supervisory Committee Members:Prof. Terje Haukaas, Department of Civil Engineering, UBCSupervisory Committee MemberiiAbstractGiven the variety of wood species available, understanding of cross-laminated timber (CLT)materials is by no means complete. This dissertation serves to advance the state-of-the-artin understanding the material and structural response of the CLT system and developingengineering tools for modelling and predicting such responses. The investigation consistedof an experimental study, numerical study and reliability analysis. The hypothesis beingtested is that the cross-layers have some contributions towards the CLT’s behaviour underthe axial compression load.In this context, to evaluate the physical and mechanical properties of CLT-lamella(sawnlumber), testing was done on the small-scale (specimens’ length ≤ 250 mm) clear woodand wood contains defects specimens. Then, a medium-scale (495 mm ≤ specimens’ length≤ 1000 mm) 3-, 5-, 7- and 9-layer CLT columns and a full-scale (specimens’ length ≥ 2400mm) 3- and 5-layer CLT elements have been tested. In addition, to characterize the stiff-ness (modulus of elasticity) of CLT materials, we employed three types of testing, namely,compression test, flexural test, and transverse vibration test.A numerical study is then employed. In order to compute the strength and stiffness ofmedium-scale CLT composite, we developed a nonlinear material model, namely, Subrou-tine for Orthotropic Materials’ Elasticity & Rate-independent Plasticity (SOME&RIP),and implemented into ANSYS as an UserMat library. In addition, a finite element tool,namely, Analysis of Universal Beam-Columns (AnUBC), considering the material andstructural nonlinearities for the stability analysis of full-scale CLT structures is developedin MATLAB. Finally, reliability analysis is carried out considering the sources of uncertain-ties that can be resulted from production, construction, material and loading conditions.Results show that characteristic strengths of the medium-scale 3-, 5-, 7-, and 9-layer CLTspecimen groups are 42%, 21%, 64% and 65% higher than the code specified strength,respectively. Moreover, characteristic stiffness is approximately the same as its code’scounterpart. Following the reliability analysis, we conclude that for utilizing CLT capacityefficiently and economically, using the characteristic properties and a performance factor of0.9 instead the current practice value of 0.8 is recommended in the CSA O86 code designequation.iiiLay SummaryThe lamination effects in cross-laminated timber (CLT) system is difficult to quantify sinceit depends on many factors such as layer thickness, number of layers, edge gap or lumberproperties. To minimize the knowledge gap, the primary objective of this dissertation isto investigate the effects of cross-layers on the CLT’s performance under axial compressionloads.In this context, we conducted experiments on different layers of CLT including clear woodand wood contains defects specimens. Then, numerical study is employed. Knowing thatthe structural performance of CLT elements relates to many sources of uncertainty, includ-ing wood properties, product configurations and loading behaviour. Thus, we performedreliability analysis to evaluate the probability of failure of a CLT element under a randomload.The study found that the design capacity of CLT structures using the characteristic prop-erties that were established in the experimental and numerical studies can satisfy the safetylevel in the existing CSA O86 code.ivPrefaceThis research project was originally proposed by Professor Frank Lam, and completelyconducted by P.K.M. Moniruzzaman. Five scientific articles are under preparation for sub-mitting into journals from this dissertation, in which P.K.M. Moniruzzaman is the leadauthor. P.K.M. Moniruzzaman was responsible for designing and performing the experi-mental tests, developing the numerical models, writing the scripts and programs, analyzingthe results and preparing the manuscripts. Professor Frank Lam provided guidance for thetest design, model development, and comments and revisions for the papers.List of potential publications related to this thesis:• Moniruzzaman, P. and Lam, F. “Axial Compression Behaviour of Clear Wood, Woodcontains defects, and Multi-scale Cross-Laminated Timber Composites”. (To be sub-mitted).• Moniruzzaman, P. and Lam, F. “An Orthotropic Elasticity and Rate-IndepedentPlasticity Model for Multi-layer Cross-Laminated Timber Composite”. (To be sub-mitted).• Moniruzzaman, P. and Lam, F. “Analytical Homogenization Method Guided for De-termining the Elastic Constants of Wood”. (To be submitted).• Moniruzzaman, P.and Lam, F. “Corotational Formulation for Locking-free Beam-Column Element for Multi-layer Solid Continua”. (To be submitted).• Moniruzzaman, P. and Lam, F. “Structural Reliability Analysis of Beam-ColumnAction in Cross-Laminated Timber Structures”. (To be submitted).vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xxviiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xxviii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Cross-laminated timber (CLT) . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Manufacturing specification . . . . . . . . . . . . . . . . . . . . . . . 21.1.1.1 Lumber laminations . . . . . . . . . . . . . . . . . . . . . . 31.1.1.2 Lamination sizes . . . . . . . . . . . . . . . . . . . . . . . . 31.1.1.3 Adhesives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.2 Application and research . . . . . . . . . . . . . . . . . . . . . . . . 41.1.3 Research opportunity . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 State of the art in CLT wall under in-plane gravity loading . . . . . . . . . 51.2.1 Previous studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3 Research questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3.1 Code assumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3.2 Shear contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3.3 Stressed volume effect . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3.4 Nonlinearities and uncertainties . . . . . . . . . . . . . . . . . . . . . 10vi1.4 Proposed research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.5 Research contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.6 Navigation of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Strength of CLT: Experimental . . . . . . . . . . . . . . . . . . . . . . . . . 132.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Small-scale: clear wood (CW) . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2.1 Clear wood parallel (CW0) . . . . . . . . . . . . . . . . . . . . . . . 172.2.1.1 Test setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2.1.2 Failure modes . . . . . . . . . . . . . . . . . . . . . . . . . 192.2.1.3 Mechanical properties . . . . . . . . . . . . . . . . . . . . . 202.2.2 Clear wood perpendicular (CW90) . . . . . . . . . . . . . . . . . . . 222.2.2.1 Test setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2.2.2 Failure modes . . . . . . . . . . . . . . . . . . . . . . . . . 232.2.2.3 Mechanical properties . . . . . . . . . . . . . . . . . . . . . 232.3 Small-scale: Wood contains defects (DW) . . . . . . . . . . . . . . . . . . . 242.3.1 Test setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.3.2 Failure modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.3.3 Mechanical properties . . . . . . . . . . . . . . . . . . . . . . . . . . 252.4 Medium-scale: 3-layer CLT (CLT3C) . . . . . . . . . . . . . . . . . . . . . . 272.4.1 Test setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.4.2 Failure modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.4.3 Mechanical properties . . . . . . . . . . . . . . . . . . . . . . . . . . 282.5 Medium-scale: 5-layer CLT (CLT5C) . . . . . . . . . . . . . . . . . . . . . . 302.5.1 Test setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.5.2 Failure modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.5.3 Mechanical properties . . . . . . . . . . . . . . . . . . . . . . . . . . 312.6 Medium-scale: 7-layer CLT (CLT7C) . . . . . . . . . . . . . . . . . . . . . . 322.6.1 Test setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.6.2 Failure modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.6.3 Mechanical properties . . . . . . . . . . . . . . . . . . . . . . . . . . 342.7 Medium-scale: 9-layer CLT (CLT9C) . . . . . . . . . . . . . . . . . . . . . . 352.7.1 Test setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.7.2 Failure modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.7.3 Mechanical properties . . . . . . . . . . . . . . . . . . . . . . . . . . 362.8 Testing data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.8.1 Sampling quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.8.1.1 Content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.8.1.2 Confidence interval . . . . . . . . . . . . . . . . . . . . . . 392.8.2 Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.8.3 Central tendency and variability . . . . . . . . . . . . . . . . . . . . 41vii2.8.4 Distribution function . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.9 Uniaxial compressive strength . . . . . . . . . . . . . . . . . . . . . . . . . . 462.9.1 Test measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.9.2 Unifying test measurement . . . . . . . . . . . . . . . . . . . . . . . 472.9.3 Adjusting test measurement . . . . . . . . . . . . . . . . . . . . . . . 482.9.3.1 Adjustment factor and results . . . . . . . . . . . . . . . . 512.10 Characteristic strength and CSA code strength . . . . . . . . . . . . . . . . 572.10.1 Strength variation reason . . . . . . . . . . . . . . . . . . . . . . . . 582.11 Analytical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602.12 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623 Strength of CLT: Numerical . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.2 SOME&RIP implementation: archetype model of FEM . . . . . . . . . . . 653.2.1 Meshing, loading and boundary condition . . . . . . . . . . . . . . . 663.2.2 Glue-line idealization . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.2.3 UserMat subroutine . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.2.4 Solution strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.3 Model calibration and verification . . . . . . . . . . . . . . . . . . . . . . . . 693.4 Stress analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723.4.1 Behaviour of lamellae . . . . . . . . . . . . . . . . . . . . . . . . . . 733.4.2 Behaviour of lamellae-interface . . . . . . . . . . . . . . . . . . . . . 763.4.3 Failure mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774 Stiffness of CLT: Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . 794.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.2 Small-scale and medium-scale specimens . . . . . . . . . . . . . . . . . . . . 804.3 Full-scale: 3- and 5-layer CLT (CLT3B and CLT5B) . . . . . . . . . . . . . 814.3.1 Test setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.3.1.1 Dynamic modulus . . . . . . . . . . . . . . . . . . . . . . . 814.3.1.2 Static bending modulus . . . . . . . . . . . . . . . . . . . . 824.3.2 Mechanical properties . . . . . . . . . . . . . . . . . . . . . . . . . . 844.4 Testing data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.4.1 Sampling quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.4.2 Central tendency and variability . . . . . . . . . . . . . . . . . . . . 854.4.3 Distribution function . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.5 MOE results of the specimens . . . . . . . . . . . . . . . . . . . . . . . . . . 864.5.1 Test measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.5.2 Unifying test measurement . . . . . . . . . . . . . . . . . . . . . . . 884.5.3 Adjusting test measurements . . . . . . . . . . . . . . . . . . . . . . 89viii4.6 CLT characteristic stiffness and the CSA code stiffness . . . . . . . . . . . . 934.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955 Stiffness of CLT: Numerical . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975.2 Stiffness: archetype model of FEM . . . . . . . . . . . . . . . . . . . . . . . 975.2.1 Medium-scale: CLT3C, CLT5C, CLT7C and CLT9C . . . . . . . . . 975.2.2 Full-scale: CLT3B and CLT5B . . . . . . . . . . . . . . . . . . . . . 985.3 Model calibration and verification . . . . . . . . . . . . . . . . . . . . . . . . 995.3.1 Medium-scale: CLT3C, CLT5C, CLT7C and CLT9C . . . . . . . . . 995.3.2 Full-scale: CLT3B and CLT5B . . . . . . . . . . . . . . . . . . . . . 1005.4 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.4.1 Medium-scale: CLT3C, CLT5C, CLT7C and CLT9C . . . . . . . . . 1015.4.2 Full-scale: CLT3B and CLT5B . . . . . . . . . . . . . . . . . . . . . 1015.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1036 Stability of CLT: Experimental . . . . . . . . . . . . . . . . . . . . . . . . . 1046.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046.2 Full-scale: 3- and 5-layer CLT (CLT3B and CLT5B) . . . . . . . . . . . . . 1056.2.1 Test setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1056.2.2 Failure modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076.2.3 Mechanical properties . . . . . . . . . . . . . . . . . . . . . . . . . . 1096.2.3.1 Full-scale: 3-layer CLT (CLT3B) . . . . . . . . . . . . . . . 1096.2.3.2 Full-scale: 5-layer CLT (CLT5B) . . . . . . . . . . . . . . . 1106.3 Testing data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1116.3.1 Sampling quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1116.3.2 Central tendency and variability . . . . . . . . . . . . . . . . . . . . 1126.3.3 Distribution function . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136.4 Stability capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136.4.1 Test measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136.4.2 Unifying test measurement . . . . . . . . . . . . . . . . . . . . . . . 1146.4.3 Adjusting test measurement . . . . . . . . . . . . . . . . . . . . . . . 1156.5 Characteristic capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1166.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1177 Stability of CLT: Numerical . . . . . . . . . . . . . . . . . . . . . . . . . . . 1187.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1187.2 AnUBC development: multi-layer fibre element . . . . . . . . . . . . . . . . 1187.2.1 Displacement formulation algorithm . . . . . . . . . . . . . . . . . . 1207.2.2 Force formulation algorithm . . . . . . . . . . . . . . . . . . . . . . . 1217.2.3 Implementation result . . . . . . . . . . . . . . . . . . . . . . . . . . 122ix7.3 AnUBC development: solution control . . . . . . . . . . . . . . . . . . . . . 1247.3.1 Load control algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 1257.3.2 Displacement control algorithm . . . . . . . . . . . . . . . . . . . . . 1267.3.3 Implementation results . . . . . . . . . . . . . . . . . . . . . . . . . . 1277.4 AnUBC development: shear deformation . . . . . . . . . . . . . . . . . . . . 1287.4.1 Finite element formulation . . . . . . . . . . . . . . . . . . . . . . . 1317.4.1.1 Hermite Interpolation Function for w(x) . . . . . . . . . . 1317.4.1.2 Linear interpolation functions for θs(x) . . . . . . . . . . . 1327.4.1.3 Governing equations . . . . . . . . . . . . . . . . . . . . . . 1327.4.2 Implementation results . . . . . . . . . . . . . . . . . . . . . . . . . . 1337.5 AnUBC development: out-of-straightness . . . . . . . . . . . . . . . . . . . 1347.5.1 Angle of rotation of the co-rotating frame . . . . . . . . . . . . . . . 1357.5.2 Relation between global and local variables . . . . . . . . . . . . . . 1367.5.3 Consistent tangent stiffness matrix . . . . . . . . . . . . . . . . . . . 1377.5.4 Implementation results . . . . . . . . . . . . . . . . . . . . . . . . . . 1397.6 AnUBC development: support restraints . . . . . . . . . . . . . . . . . . . . 1407.6.1 Implementation results . . . . . . . . . . . . . . . . . . . . . . . . . . 1427.7 Stability of CLT: FE model calibration . . . . . . . . . . . . . . . . . . . . . 1437.7.1 Material model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1437.7.1.1 Size and stress distribution effects . . . . . . . . . . . . . . 1447.7.2 Quantifying initial imperfection . . . . . . . . . . . . . . . . . . . . . 1447.7.2.1 Southwell approach . . . . . . . . . . . . . . . . . . . . . . 1447.7.2.2 Probabilistic approach . . . . . . . . . . . . . . . . . . . . . 1477.7.2.3 FEM approach . . . . . . . . . . . . . . . . . . . . . . . . . 1487.7.3 Spring constant calibration . . . . . . . . . . . . . . . . . . . . . . . 1497.8 Stability of CLT: FE model verification . . . . . . . . . . . . . . . . . . . . 1507.8.1 Stochastic material data . . . . . . . . . . . . . . . . . . . . . . . . . 1507.8.2 FE model verification results . . . . . . . . . . . . . . . . . . . . . . 1517.9 Stability of CLT: P-M interaction capacity . . . . . . . . . . . . . . . . . . . 1527.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1548 Structural capacity of CLT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1558.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1558.2 Experimental methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1568.2.1 Capacity by experiments . . . . . . . . . . . . . . . . . . . . . . . . . 1568.2.2 Capacity by Soutwell plot . . . . . . . . . . . . . . . . . . . . . . . . 1578.3 Theoretical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1588.3.1 Capacity by elastic buckling theory . . . . . . . . . . . . . . . . . . . 1598.3.2 Capacity by inelastic Engesser’s theory . . . . . . . . . . . . . . . . 1608.4 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1618.4.1 Capacity by FEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161x8.4.2 Capacity by residual stress effects . . . . . . . . . . . . . . . . . . . 1628.5 Empirical method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1658.5.1 Capacity by CSA O86 code . . . . . . . . . . . . . . . . . . . . . . . 1658.5.2 Slenderness factor, Kc . . . . . . . . . . . . . . . . . . . . . . . . . . 1658.5.3 Effective length factor, Ke . . . . . . . . . . . . . . . . . . . . . . . . 1658.5.3.1 Ke by different methods . . . . . . . . . . . . . . . . . . . . 1678.6 Capacity comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1698.7 CSA code capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1708.7.1 Sensibility of the fitting parameter . . . . . . . . . . . . . . . . . . . 1718.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1749 Structural reliability of CLT . . . . . . . . . . . . . . . . . . . . . . . . . . . 1759.1 Reliability analysis approach . . . . . . . . . . . . . . . . . . . . . . . . . . 1759.1.1 FORM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1769.1.2 Importance sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . 1779.1.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1779.2 Reliability analysis: limit state function . . . . . . . . . . . . . . . . . . . . 1789.2.1 Design equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1799.2.1.1 P-M no-interaction . . . . . . . . . . . . . . . . . . . . . . 1809.2.1.2 P-M linear-interaction . . . . . . . . . . . . . . . . . . . . . 1809.2.1.3 P-M parabolic-interaction . . . . . . . . . . . . . . . . . . . 1819.3 Reliability analysis: resistance variables . . . . . . . . . . . . . . . . . . . . 1819.3.1 Probability distribution . . . . . . . . . . . . . . . . . . . . . . . . . 1829.3.2 Response surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1839.4 Reliability analysis: demand variables . . . . . . . . . . . . . . . . . . . . . 1879.4.1 Probability distribution . . . . . . . . . . . . . . . . . . . . . . . . . 1879.5 Reliability analysis results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1879.5.1 P-M no-interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1899.5.2 P-M linear-interaction . . . . . . . . . . . . . . . . . . . . . . . . . . 1919.5.3 P-M parabolic-interaction . . . . . . . . . . . . . . . . . . . . . . . . 1939.5.4 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1959.6 Reliability-based design capacity . . . . . . . . . . . . . . . . . . . . . . . . 1979.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19910 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20010.1 Summary of the study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20010.1.1 Strength of CLT structures . . . . . . . . . . . . . . . . . . . . . . . 20010.1.2 Stiffness of CLT structures . . . . . . . . . . . . . . . . . . . . . . . 20110.1.3 Stability of CLT structures . . . . . . . . . . . . . . . . . . . . . . . 20110.1.4 Structural capacity of CLT structures . . . . . . . . . . . . . . . . . 20210.1.5 Structural reliability of CLT structures . . . . . . . . . . . . . . . . . 203xi10.2 Impact of the study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20310.2.1 Example problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20310.2.2 Solution of the problem . . . . . . . . . . . . . . . . . . . . . . . . . 20410.2.3 Comments on the solution . . . . . . . . . . . . . . . . . . . . . . . . 20810.3 Future works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210Appendix A First Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220A.1 Physical and mechanical properties of all specimens . . . . . . . . . . . . . 220A.2 Correlation table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224Appendix B Second Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . 225B.1 The general development of SOME&RIP subroutine . . . . . . . . . . . . . 225B.1.1 Tensor basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226B.1.2 State variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227B.1.3 Yield condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227B.1.4 Flow rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228B.1.5 Plastic evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229B.2 The algorithmic development of SOME&RIP subroutine . . . . . . . . . . . 230B.2.1 Evaluation of consistency condition . . . . . . . . . . . . . . . . . . . 233B.2.2 Update of stress and state variables . . . . . . . . . . . . . . . . . . 234B.2.3 Consistent tangent moduli . . . . . . . . . . . . . . . . . . . . . . . . 235B.2.4 Failure index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237B.2.5 Computation flowchart . . . . . . . . . . . . . . . . . . . . . . . . . . 237Appendix C Third Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239C.1 Reliability analysis results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239xiiList of Tables1.1 A study by [Hor13] on CLT behaviour under axial compression loading . . . 82.1 Overview of the test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2 Number and geometric properties of all specimen groups . . . . . . . . . . . 172.3 Statistical data of the properties of the CW0 specimen group . . . . . . . . 222.4 Statistical data of the properties of the CW90 specimen group . . . . . . . . 242.5 Statistical data of the properties of the DW specimen groups . . . . . . . . 262.6 Statistical data of the properties of the CLT3C specimen groups . . . . . . 292.7 Statistical data of the properties of the CLT5C specimen groups . . . . . . 312.8 Statistical data of the properties of the CLT7C specimen group . . . . . . . 342.9 Statistical data of the properties of the CLT9C specimen group . . . . . . . 362.10 Estimation of the distribution parameters . . . . . . . . . . . . . . . . . . . 452.11 Compressive strength without adjustment . . . . . . . . . . . . . . . . . . . 472.12 Effective compressive strength without adjustment . . . . . . . . . . . . . . 482.13 Density of small- and medium-scale specimens . . . . . . . . . . . . . . . . . 492.14 Density of full-scale specimens . . . . . . . . . . . . . . . . . . . . . . . . . 502.15 Moisture content of small- and medium-scale specimens . . . . . . . . . . . 512.16 Moisture content of full-scale specimens . . . . . . . . . . . . . . . . . . . . 512.17 Effective compressive strength adjusted with the moisture content . . . . . 522.18 Effective compressive strength adjusted with the moisture content & sizefactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552.19 Characteristic compressive strength of small- & medium-scale specimen groups 562.20 Value of the properties used in the Equation 2.5 . . . . . . . . . . . . . . . 612.21 Characteristic compressive strength of CLT specimens with its code coun-terpart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.1 Material properties data for SOME&RIP . . . . . . . . . . . . . . . . . . . 683.2 Compressive strength and MOE results from test and FEM analysis . . . . 713.3 Characteristic strength of medium-scale specimens from test & FEM analysis 724.1 Overview of the test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.2 MOE test results of small- and medium-scale specimens . . . . . . . . . . . 80xiii4.3 Dynamic MOE test results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.4 Static MOE test results of full-scale specimens . . . . . . . . . . . . . . . . 844.5 Statistical data of MOE of small- & medium-scale specimens without ad-justment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.6 Statistical data of MOE of full-scale specimens without adjustment . . . . . 874.7 Statistical data of the effective MOE of small- and medium-scale specimenswithout adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.8 Statistical data of the effective MOE of full-scale specimens without adjust-ment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.9 Statistical data of the effective MOE of small- and medium-scale specimensadjusted with sample number . . . . . . . . . . . . . . . . . . . . . . . . . . 914.10 Statistical data of the effective MOE of small- and medium-scale specimensadjusted with sample number and moisture content . . . . . . . . . . . . . . 924.11 Statistical data of the effective MOE of full-scale specimens adjusted withsample number and moisture content . . . . . . . . . . . . . . . . . . . . . . 934.12 MOE values compared to the CSA code . . . . . . . . . . . . . . . . . . . . 955.1 Material properties of full-scale specimens . . . . . . . . . . . . . . . . . . . 985.2 FEM calibration results of full-scale specimens . . . . . . . . . . . . . . . . 1015.3 MOE results of medium-scale specimens from test and FEM analysis . . . . 1015.4 MOE results of full-scale specimens from test and FEM analysis . . . . . . 1036.1 Overview of the test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046.2 Loading capacity of CLT3B groups . . . . . . . . . . . . . . . . . . . . . . . 1106.3 Loading capacity of CLT5B . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106.4 Statistical data of the effective load carrying capacity without adjustment . 1146.5 Statistical data of adjusted capacity of full-scale specimens . . . . . . . . . 1157.1 Features in the finite element analysis programs . . . . . . . . . . . . . . . . 1197.2 Fundamental equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1197.3 Material properties data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1447.4 Quantifying the initial imperfection by different methods and spring constant1467.5 Sensitivity of initial deflection . . . . . . . . . . . . . . . . . . . . . . . . . . 1487.6 Stochastic parameters of material properties . . . . . . . . . . . . . . . . . . 1507.7 FEM and test results of the load carrying capacity of full-scale CLT specimens1518.1 Design case overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1558.2 Statistical data of characteristic capacity of 3- and 5-layer CLT specimens . 1578.3 Capacity of full-scale specimens by test, Southwell plot and Euler equation 1588.4 Statistical data of capacity of 3- and 5-layer CLT specimens by test and FEM 1628.5 Effective length factor obtained by different methods . . . . . . . . . . . . . 1698.6 Comparison of load carrying capacity obtained by different methods . . . . 169xiv8.7 Capacity comparison with different values of the performance factor parameter1718.8 Capacity comparison with different values of the curve fitting parameter . . 1739.1 Random variables’ distribution parameters . . . . . . . . . . . . . . . . . . . 1829.2 Response surface model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1869.3 Loading distribution parameters . . . . . . . . . . . . . . . . . . . . . . . . 1879.4 Reliability indices considering Dead loadLive load of 0.25 and P-M no-interaction . . . 1919.5 Reliability indices considering Dead loadLive load of 0.25 and P-M linear-interaction . 1939.6 Reliability indices considering Dead loadLive load of 0.25 and P-M parabolic-interaction1959.7 Reliability indices considering different design equations . . . . . . . . . . . 196A.1 Experimental and FEM results of all specimens . . . . . . . . . . . . . . . . 220A.2 Pearson correlation coefficient table for all specimen groups . . . . . . . . . 224C.1 Reliability indices of full-scale CLT specimens using P-M no-interaction . . 239C.2 Reliability indices of full-scale CLT specimens using P-M linear-interaction 240C.3 Reliability indices of full-scale CLT specimens using P-M parabolic-interaction241xvList of Figures1.1 A CLT system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 A typical wall element under load . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Load carrying capacity curve of a typical CLT member based on [CSA16]| . 61.4 Design procedure of CLT under compression loading according to [CLT11]| . 72.1 Naming system of the specimens notation . . . . . . . . . . . . . . . . . . . 152.2 All specimen groups for the examination of the mechanical properties: (leftto right) CLT9C, CLT7C, CLT5B, CLT5C1, CLT5C2, CLT3B4, CLT3B3,CLT3B2, CLT3B1, CLT3C2, CLT3C1, DW1, DW2, CW0, and CW90; loadingplates and LVDTs are attached with the specimens; cross-section of eachspecimen is shown underneath respectively . . . . . . . . . . . . . . . . . . . 162.3 Clear wood samples: (a) cross-section, and (b) side view . . . . . . . . . . . 182.4 Test setup of CW0 specimen group: (a) real, (b) rendering . . . . . . . . . . 192.5 Representative macro-level failure modes of CW0 specimen group: (a) crush-ing, (b) end rolling, (c) crushing and wedge splitting, and (d) shearing . . . 202.6 Test results of the clear wood specimen groups: (a) CW0, and (b) CW90 . . 212.7 Test setup of CW90 specimen group: (a) real, (b) rendering . . . . . . . . . 222.8 Representative failure modes of CW90 specimen group: (a) wood crushingfrom the side view, and (b) wood crushing on the cross-section . . . . . . . 232.9 Test setup of DW specimen groups: (a, b) real duo, and (c, d) rendering duo 242.10 Representative failure modes of DW specimen groups: (a) crushing due toknots on the width side, and (b) crushing on the thickness side . . . . . . . 252.11 Test results of DW specimen groups: (a) DW1 and (b) DW2 . . . . . . . . . 262.12 Test setup of CLT3C groups: (a, b) real duo, and (c,d) rendering duo . . . 272.13 Representative failure modes of CLT3C groups: (a) fibre buckling followedby wood crushing, (b) wood crushing commencement on the unconfined-parallel layers, (c) wood crushing due to knots, and (d) wood crushing nearto the end location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.14 Test results of the CLT3C specimen groups: (a) CLT3C1 and (b) CLT3C2 . 292.15 Test setup of CLT5C groups: (a, b) real duo, and (c, d) rendering duo . . . 30xvi2.16 Representative failure modes of CLT5C groups: (a) fibre buckling followedby wood crushing and wood-adhesive interface splitting, (b) wood crushingcommencement on the confined- and unconfined-parallel layers, (c) crushingfailure followed by lamella splitting, and (d) wood crushing due to knots onthe width side . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.17 Test results of CLT5C specimen groups: (a) CLT5C1 and (b) CLT5C2 . . . 322.18 Test setup of CLT7C group: (a) real, and (b) rendering . . . . . . . . . . . 332.19 Representative failure modes of CLT7C group: (a) wood crushing com-mencement on the parallel layers, (b) fibre buckling followed by wood crush-ing and wood-adhesive interface splitting, (c) shearing type crushing failureat the mid-height, and (d) wood crushing due to knots on the width side . . 332.20 Test results of the specimen groups: (a) CLT7C, and (b) CLT9C . . . . . . 342.21 Test setup of CLT9C group: (a) real, and (b) rendering . . . . . . . . . . . 352.22 Representative failure modes of the CLT9C group: (a) wood crushing com-mencement on the confined- and unconfined-parallel layers, (b) wood crush-ing followed by lamella splitting, (c) crushing failure at different location onthe lamellae, and (d) wood crushing due to knots on the width side . . . . . 362.23 Compression test results of small- and medium-scale specimen groups: (a)spectrum, and (b) average line . . . . . . . . . . . . . . . . . . . . . . . . . 372.24 Sphere with the 95% probability and data points of the standardized strength,stiffness (MOE) and density parameters of all specimen groups . . . . . . . 392.25 Confidence ellipse and data points of the normalized strength and stiffnessparameters of CLT3C1 group . . . . . . . . . . . . . . . . . . . . . . . . . . 402.26 Representation of measuring the correlation between the strength (str), stiff-ness (MOE) and density (den) parameters of all samples . . . . . . . . . . . 412.27 Representation of measuring the central tendency and data variability of thestrength parameter of the CLT3C1 group . . . . . . . . . . . . . . . . . . . 422.28 Mirrored histogram and box-and-whisker plot of the strength data points ofCLT3C specimen groups with the test realizations . . . . . . . . . . . . . . 432.29 Measuring the symmetry and tail-strength of strength data points of CLT3C1specimen group relative to a normal distribution . . . . . . . . . . . . . . . 442.30 Plots for the strength data points of CLT3C1 specimen group consideringfour different distribution types: 2P-Weibull, lognormal, gamma and normal:(a) Probability density, (b) Cumulative density, (c) P-P plot, and (d) Q-Qplot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.31 Probability density function and cumulative distribution function of uniaxialcompressive strength of small- and medium-scale specimens: (a) CLT3C1group, and (b) small- and medium-scale specimen groups . . . . . . . . . . 462.32 Compressive strength of small- & medium-scale specimens without adjustment 472.33 Effective uniaxial compressive strength of small- and medium-scale speci-mens without adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48xvii2.34 Density of all specimen groups . . . . . . . . . . . . . . . . . . . . . . . . . 492.35 Moisture content of all specimen groups . . . . . . . . . . . . . . . . . . . . 502.36 Moisture content adjustment factor in the ASTM standard . . . . . . . . . 512.37 Effective uniaxial compressive strength of small- and medium-scale speci-mens adjusted with the moisture content . . . . . . . . . . . . . . . . . . . . 522.38 Size adjustment factor in the ASTM standard for the strength parameter . 532.39 Size adjustment factor in the CSA code for the strength parameter . . . . . 542.40 Effective uniaxial compressive strength of small- and medium-scale speci-mens adjusted with the moisture content and size factors . . . . . . . . . . 542.41 Duration-of-load adjustment factor in the ASTM standard . . . . . . . . . . 552.42 Effective uniaxial compressive strength of small- and medium-scale speci-mens adjusted with the moisture content, size, safety and duration-of-loadfactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562.43 Adjusted compressive strength of small- and medium-scale specimen groups 572.44 Characteristic strength of CLT obtained from tests with its code counterpart:(a) actual value, and (b) normalized value with the code specified strength . 582.45 Slenderness ratio of CLT specimens . . . . . . . . . . . . . . . . . . . . . . . 582.46 Characteristic strength, and percentages of cross-layers, unconfined-parallellayers and confined-parallel layers of CLT: (a) actual value, and (b) normal-ized value with the CLT3C group . . . . . . . . . . . . . . . . . . . . . . . . 592.47 Trend plot of the normalized characteristic strength, and percentages ofcross-layers, unconfined-parallel layers and confined-parallel layers of CLT . 602.48 Characteristic compressive strength of CLT specimens from test and analyt-ical model with its code counterpart: (a) actual value, (b) normalized valuewith the test result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.1 A representative archetype model of FEM of the medium-scale specimengroups: (a) meshing, lamella orientation with gap and boundary conditionof a CLT7C specimen, and (b) contact surfaces of a CLT9C specimen . . . 663.2 FE model calibration of the medium-scale specimen groups: (a) CLT3C1,(b) CLT5C2, (c) CLT7C, and (d) CLT9C . . . . . . . . . . . . . . . . . . . 703.3 Compressive strength of the medium-scale CLT specimens from test, analyt-ical and FEM: (a) actual value, and (b) normalized value with the CLT3Cgroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713.4 Von-Mises stress (√3J2) contour diagram of the medium-scale specimens atthe ultimate state: (a) CLT3C1, (b) CLT5C2, (c) CLT7C, and (d) CLT9C . 733.5 Von-Mises stress (√3J2) contour diagram of a typical CLT9C specimen withthe maximum and minimum value at the ultimate state: (a) an unconfined-parallel layer (1st sequence) and a confined-parallel layer (7st sequence), and(b) two cross-layers (2nd and 8th sequence) . . . . . . . . . . . . . . . . . . . 74xviii3.6 Von-Mises stress (√3J2) of a confined- and unconfined-parallel layers ofCLT9C specimen at the ultimate limit state: (a) location of the probesalong the length of the specimen at various width section, and (b) stressdistribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753.7 Von-Mises stress (√3J2) of a confined- and unconfined-parallel layers ofCLT9C specimen at the ultimate limit state: (a) location of the probesalong the width of the specimen at various length section, and (b) stressdistribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753.8 Nodal solution (average from the contact element’s integration points) at theultimate state of a CLT5C2 specimen: (a) contact stress, and (b) octahedralnormal stress (13I1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763.9 Stress ratio contour diagram with the maximum and minimum value at theultimate state: (a) an unconfined-parallel layer (1st sequence) of a CLT3C1specimen, (b) an unconfined-parallel layer (1st sequence) and a confined-parallel layer (3rd sequence) of a CLT5C2 specimen, (c) an unconfined-parallel layer (1st sequence) and a confined-parallel layer (5th sequence) ofa CLT7C specimen, and (d) an unconfined-parallel layer (1st sequence) anda confined-parallel layer (7th sequence) of a CLT9C specimen . . . . . . . . 774.1 Transverse vibration test setup . . . . . . . . . . . . . . . . . . . . . . . . . 814.2 A representative test setup of a specimen under third-point loading: (a) atypical CLT5BET specimen, (b) a typical CLT3B1FA specimen, (c) renderingof a CLT5BEA specimen, and (d) rendering of a CLT5BFA specimen . . . . 834.3 Test results of the specimen groups: (a) CLT3B1, and (b) CLT5B . . . . . 844.4 Representation of measuring the central tendency and data variability of thestiffness parameter of the CLT3B2D group . . . . . . . . . . . . . . . . . . . 854.5 Probability density function and cumulative distribution function of the stiff-ness parameter: (a) CLT3B2D group, and (b) all specimen groups . . . . . . 864.6 Stiffness results of all specimen groups without adjustment . . . . . . . . . 874.7 Effective stiffness results of all specimen groups without adjustment . . . . 884.8 Traceplot of the simulated MOE values by Bayesian analysis using Markovchain Monte Carlo sampling technique for the CLT7C group: (a) meanvalue, (b) standard deviation value, (c) probability density of mean, and (d)probability density of standard deviation . . . . . . . . . . . . . . . . . . . . 904.9 Effective stiffness results of all specimen groups adjusted with sample number 904.10 Moisture content adjustment factor in the ASTM standard for the stiffnessparameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.11 Effective stiffness results of all specimen groups adjusted with sample num-ber and moisture content . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924.12 Adjusted stiffness results of all specimen groups . . . . . . . . . . . . . . . . 944.13 Characteristic MOE compared with the code specified value . . . . . . . . . 94xix5.1 A typical archetype model of FEM with the mesh, elements, loading &boundary condition of the CLT3B1EA, CLT3B1ET , CLT3B1FA, CLT3B1FT ,CLT5BEA, CLT5BET , CLT5BFA, CLT5BFT , CLT3B2, CLT3B3, and CLT3B4specimen groups: (a) CLT3B1EA, (b) CLT3B1FA, (c) CLT5BEA, and (d)CLT5BFA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.2 Simulated deformation contour results (magnifying scale: 15) of the full-scalespecimen groups with the maximum and minimum value at the final solutionstep: (a) CLT3B1EA, (b) CLT3B1FA, (c) CLT5BEA, and (d) CLT5BFA . . 1005.3 Stiffness in the longitudinal direction of full-scale specimens from FEM sim-ulations and test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1026.1 Test setup of full-scale CLT specimen groups: (a) a CLT3B1 specimen underaxial compression loading, (b) a CLT5B specimen with the lateral support,(c) side-view of a CLT5B specimen, (d) a CLT3B1 specimen with the lateralsupport that restrains buckling in the vertical direction, (e) rolling supportcondition at the loading end of a specimen, (f) fixed-in-vertical but free-in-lateral support condition at mid-span, and (g) fixed support condition atthe far end of a specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066.2 Rendering of a representative test setup of full-scale CLT specimens: (a)top view a CLT3B1 specimen under axial compression loading with the endsupport conditions and location of LVDTs, and (b) side view of a typicalspecimen with the lateral supports that restrain buckling in the verticaldirection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076.3 Representative failure modes of full-scale 3-layer CLT specimens: (a) typicallateral deformation of a CLT3B4 specimen, and (b) buckling followed bywood crushing failure at the mid-span of a CLT3B3 specimen . . . . . . . . 1086.4 Representative failure modes of the CLT5B group: (a, b, c) buckling followedby splitting at the wood-adhesive interface, (d) buckling followed by woodcrushing and splitting of a lamella, and (e) buckling deformation of a specimen1086.5 Test results of the CLT3B1 specimens: (a) load-displacement curve, and(b) lateral displacement along the length of the specimens at 60% of theultimate load carrying capacity level . . . . . . . . . . . . . . . . . . . . . . 1096.6 Test results of the CLT5B group: (a) load-displacement curve, and (b) lat-eral displacement along the length of the specimens at 60% of the ultimateload carrying capacity level . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106.7 Test results of the CLT3B1, CLT3B2, CLT3B3,CLT3B4, and CLT5B speci-men groups: (a) spectrum of the load-displacement curves, and (b) averageline of the lateral displacement along the length of each specimen group at60% of their ultimate load carrying capacity level . . . . . . . . . . . . . . . 1116.8 Representation of measuring the central tendency and data variability of thestability capacity parameter of the CLT3B2 group . . . . . . . . . . . . . . 112xx6.9 Probability density function and cumulative distribution function of the sta-bility capacity parameter of the CLT3B1, CLT3B2, CLT3B3,CLT3B4, andCLT5B specimen groups: (a) CLT3B2 specimen, and (b) full-scale specimengroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136.10 Load carrying capacity of full-scale specimen groups without adjustment . . 1146.11 Effective (with the same width of 184 mm) load carrying capacity of full-scale specimen groups without adjustment . . . . . . . . . . . . . . . . . . . 1156.12 Effective load carrying capacity of all specimen groups adjusted with themoisture content and duration-of-loading . . . . . . . . . . . . . . . . . . . 1166.13 Characteristic capacity of full-scale specimen groups . . . . . . . . . . . . . 1167.1 Definition of variables for the FEM formulation . . . . . . . . . . . . . . . . 1197.2 Displacement formulation algorithm for the AnUBC program . . . . . . . . 1207.3 Force formulation algorithm for the AnUBC program . . . . . . . . . . . . . 1217.4 Moment-curvature-thrust results of a beam-column structural member . . . 1227.5 Simulation results by varying the number of elements parameter of a beam-column structural member: (a) moment-rotation curves, (b) curvature alongthe length of the member, and (c) trend of the computational time required 1237.6 Simulation results by varying the number of integration points parameterof a beam-column structural member: (a) moment-rotation curves, (b) cur-vature along the length of the member, and (c) trend of the computationaltime required . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1247.7 Solution strategy for getting the convergence points: (a) load control algo-rithm, and (b) displacement control algorithm . . . . . . . . . . . . . . . . . 1277.8 Simulation results by varying the solution control algorithm of a beam-column structural member: (a) load-displacement curves, (b) displacementalong the length of the member, (c) curvature along the length of the mem-ber, and (d) computational time required . . . . . . . . . . . . . . . . . . . 1287.9 Rendering of the bending and shear kinematics of a typical shear deformablebeam under asymmetrical bending and shear loading . . . . . . . . . . . . . 1297.10 Simulation results considering with- and without-shear deformation of abeam-column structural members having different lengths: (a) displace-ments along length of the members, (b) curvatures along length of the mem-bers, and (c) trend of shear deformation contribution of the members . . . . 1347.11 Rendering of the initial and current configuration for a typical beam elementin the corotating frame: (a) ignoring flexural deformation, (b) includingflexural deformation, and (c) having a small movement from the currentconfiguration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136xxi7.12 Simulation results considering with- and without-corotational algorithm ofa beam-column structural member: (a) load-displacement curves, (b) dis-placement along length of the member, (c) curvature along length of themember, and (d) computational time required . . . . . . . . . . . . . . . . . 1397.13 Simulation results by varying the out-of-straightness parameter of a beam-column structural member: (a) load-displacement curves, (b) curvaturealong length of the member, and (c) computational time required . . . . . . 1407.14 Simulation results by varying the end support conditions of a beam-columnstructural member: (a) load-displacement curves, (b) displacement alonglength of the member, and (c) curvature along length of the member . . . . 1427.15 Schematic material model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1447.16 Southwell plot: (a) a typical specimen, (b) all specimens of CLT3B4 group . 1467.17 Probabilistic characteristics of the out-of-straightness variable: (a) actualvalue, (b) standardized value . . . . . . . . . . . . . . . . . . . . . . . . . . 1487.18 Spring constant calibration result of CLT3B4 specimen group: (a) simulationresults at various spring constants with the test observation, and (b) errormeasurement at each run resulting the optimal value . . . . . . . . . . . . . 1507.19 The simulated ultimate load carrying capacity results of the full-scale CLTspecimen groups with the test observations, and the CDF curves . . . . . . 1517.20 2nd order elastic load carrying capacity in relation to slenderness ratio andloading eccentricity of a typical CLT5B specimen . . . . . . . . . . . . . . . 1527.21 2nd order plastic load carrying capacity in relation to slenderness ratio andloading eccentricity of a typical CLT5B specimen . . . . . . . . . . . . . . . 1537.22 Elastic and plastic range load carrying capacity of a typical CLT5B spec-imen, and the contour of P-M interaction curves from the plastic analysisresult . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1538.1 Characteristic ultimate load carrying capacity of 3- and 5-layer CLT specimens1568.2 Capacities of CLT3B4 group by Southwell plot, experiments and Euler equa-tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1578.3 Capacities of full-scale specimen groups by Southwell plot with the test results1588.4 Capacity curves of CLT3B1 group considering flexural and torsional buckling 1598.5 Tangent modulus curve of CLT3C1 group with its stress-strain curve . . . . 1608.6 Capacity curves of CLT3B1 group considering tangent and double modulus 1618.7 Load carrying capacity of 3- and 5-layer CLT specimens by test and FEM . 1618.8 Residual stress due to drying: (a) above fibre saturation point (FSP) through-out, (b) shell below FSP and core above FSP; core moisture migrates out-ward to the shell, (c) below FSP throughout; eventually reaches uniformlylow equilibrium moisture content . . . . . . . . . . . . . . . . . . . . . . . . 1628.9 Finding compression capacity considering residual stress algorithm . . . . . 163xxii8.10 Residual stress of a typical CLT3B1 specimen: (a) linear distribution, (b)linearlayer wise distribution, and (c) parabolic distribution, (d) schematic profile1648.11 Capacity curves of CLT3B1 groups considering residual stress distribution . 1658.12 Effective length factor: (a) varying with slenderness ratio, and (b) varyingwith spring constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1668.13 Effective length factor in relation with spring constant and slenderness ratio 1678.14 Finding effective length factor for the compression member . . . . . . . . . 1688.15 Load carrying capacity curves of the 3-layer CLT with the test observations 1708.16 Load carrying capacity curves of the 5-layer CLT with the test observations 1718.17 Load carrying capacity curves of the 3-layer CLT with the test observationsand sensitivity of the curve fitting parameter . . . . . . . . . . . . . . . . . 1728.18 Load carrying capacity curves of the 5-layer CLT with the test observationsand sensitivity of the curve fitting parameter . . . . . . . . . . . . . . . . . 1739.1 Sampling algorithm: (a) Monte Carlo, (b) importance sampling . . . . . . . 1779.2 Methodology of reliability study . . . . . . . . . . . . . . . . . . . . . . . . 1789.3 Typical beam-column layout under loading . . . . . . . . . . . . . . . . . . 1799.4 Parametric results from the FEM simulations with the test observations . . 1829.5 Plot of leverage points for checking the fitness of data for developing thecapacity regression model of a typical CLT3B2 group: (a) relation with out-of-straightness, and (b) relation with spring constant . . . . . . . . . . . . . 1839.6 Plots for checking the quality of capacity regression model of a typicalCLT3B2 specimen group: (a) regression model result with the ± 2 stan-dard error, (b) Cook’s distance, and (c) influence point plot . . . . . . . . . 1849.7 Regression model results of 3- and 5-layer CLT specimen groups: (a) loadcarrying capacity, (b) standard deviation . . . . . . . . . . . . . . . . . . . . 1859.8 Response surface of CLT3B2 specimen group with the input variables . . . 1869.9 Graphical representation of the random variables: (a) PDFs of response anddemand variables, and (b) histogram of their joint probability density . . . 1889.10 Graphical representation of the reliability index (β) with the joint proba-bility distribution and contour diagram in the standard normal space, andthree criteria for the limit state function evaluation (g > 0 denotes safe, g=0denotes neutral, and g < 0 denotes failure) . . . . . . . . . . . . . . . . . . 1899.11 Reliability analysis results of 3- and 5-layer CLT specimens at six differentlocations in Canada considering P-M no-interaction in the design equation:(a) using code specified material properties, and (b) using characteristicmaterial properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1909.12 Reliability analysis results of 3- and 5-layer CLT specimens at six differ-ent locations in Canada considering P-M linear-interaction in the designequation: (a) using code specified material properties, and (b) using char-acteristic material properties . . . . . . . . . . . . . . . . . . . . . . . . . . 192xxiii9.13 Reliability analysis results of 3- and 5-layer CLT specimens at six differ-ent locations in Canada considering P-M parabolic-interaction in the designequation: (a) using code specified material properties, and (b) using char-acteristic material properties . . . . . . . . . . . . . . . . . . . . . . . . . . 1949.14 The minimum reliability analysis results of 3- and 5-layer CLT specimensconsidering P-M no-interaction, P-M linear-interaction and P-M parabolic-interaction in the design equation: (a) using code specified material proper-ties, and (b) using characteristic material properties . . . . . . . . . . . . . 1969.15 Bounds of the reliability analysis results of the 3- and 5-layer CLT speci-mens considering P-M no-interaction in the design equation: (a) using codespecified material properties, and (b) using characteristic material properties 1979.16 Load carrying capacity curves of 3-layer CLT using the code specified andcharacteristic material properties, and considering the sensitivity of the per-formance factor parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1989.17 Load carrying capacity curves of 5-layer CLT specimens using the code spec-ified and characteristic material properties, and considering the sensitivityof the performance factor parameter . . . . . . . . . . . . . . . . . . . . . . 19810.1 An example of a CLT wall used in the foundation level . . . . . . . . . . . . 204xxivList of SymbolsAeff Effective areaCc,eff Effective slenderness ratioCs Spring constantdp Ratio of PdPdndq Ratio of QdQdnDi Initial out-of-straightnesse Load eccentricityen Nominal load eccentricityE Elastic modulusEd Descending elastic modulusEt Tangent modulusE0 Initial elastic modulusE05 5th percentile MOEFb Bending strengthFc Compressive strengthFt Tensile strengthG Shear modulusKc Slenderness factorKc,eff Effective slenderness factorKe Effective length factorKSE Service condition factorKT Treatment factorKZc,eff Effective size factorlp Ratio of PlPlnlq Ratio of QlQlnMu Ultimate bending capacityN Fitting parameterP Axial loadPd Dead load component of the total axial load PPdn Nominal axial dead loadPEuler Euler capacityPl Live load component of the total axial load PxxvPln Nominal axial live loadPr Factored capacityPu Ultimate capacity if the load P acted aloneQ Lateral loadQd Dead load component of the total lateral load QQdn Nominal lateral dead loadQl Live load component of the total lateral load QQln Nominal lateral live loadSeff Effective section modulusαdLoad factors for dead loadsαlLoad factors for live loadsβ Reliability indexγ Ratio of Dead loadnominalLive loadnominalγp Ratio of PdnPlnγq Ratio of QdnQlnζd Geometric factor for dead loadζl Geometric factor for live loadφb Performance factor for bendingφc Performance factor for compressionf Ratio of Lateral loadAxial loadxxviAcknowledgementsProf. Frank Lam: I want to tell you something, Prof. For writing this, I have travelledback and forth inside me since I met you. I saw where I went wrong, I saw where Istopped, I saw the moment when I lost argument. I experienced things I never imagined Iwould experience in my Ph.D. journey. And all because you were by my side, guiding me,cultivating my research skills, fostering my personal growth even though you might nothave been aware that you were. On basis of that, I can call you the walking encyclopedia.I thank you for all that you gave me.I desire to show my gratitude toward my supervisory committee members, Prof. EmeritusRicardo O. Foschi, Prof. Stavros Avramidis and Prof. Terje Haukaas who guided me timeto time on the research topic. To Prof. Andrea Frangi, Prof. W.D. Liam Finn, and Prof.Tsung Yuan (Tony) Yang, I would like to thank-you, most sincerely, for taking the timeto serve my PhD defence examining committee. I am so much grateful to Prof. Foschifor his time to enlighten me on various topics. I was fortunate to get his insight on mydissertation and to formuate my academic acumen under his guidance.I also extend my thank-you to George Li and Chao (Tom) Zhang, who helped me so muchthroughout my study. I also appreciate my excellent TA students for their inquisitionalmind to ask me so many questions; my course teachers who taught me various state-of-the-art contents very passionately; my university for a great academic environment; departmentstuffs for responding necessary demands so quickly; Network for Engineered Wood-basedBuilding Systems (NEWBuildS) and Sustainable Building Science Program (SBSP) forfunding; and donors who supported me though fellowships and scholarships.I enjoyed being a part of Timber Engineering and Applied Mechanics (TEAM) group andthank every member of this group. I also thank my known Bangladeshi community atVancouver and Kelowna for arranging plenty of social gatherings and giving me such afeeling like a family-away-family. Finally, I express my deepest gratitude to my family andfriends for believing in my strength and giving me the moral supports I need. As always,my Ratnagorva1 mother (Mst. Mazeda Begum) is one of my biggest support systems. Iget inspiration from her, she turned failure into struggle that led to success. I love you‘Amma’. Furthermore, almost at the end of my Ph.D. journey, it was a truly blessing forme to meet with the change-of-wind, my wife, Ananna. Thank you all.1Ratnagorva mothers who have given birth to at least 3 children who are graduates and have establishedthemselves in life by their own talents and achievements.xxviiDedicationAmma: Mazeda - Abba: AzizKaka: Bari - Vai: SohelSheulypa, Luckfa, Nicepa, Mukta, Koki and AnannaxxviiiChapter 1IntroductionWhile solid sawn material in its natural state is relatively weak building material[Din00]|,the properties and performance of wood can be modified to a significant degree by mod-ern processing methods[OW15]|, resulting in engineered wood products which are stronger,more rigid, more consistent, more dimensionally stable and more efficient than traditionalsolid sawn material [Kin17]|. However, given the variety of wood species available, under-standing of the intrinsic and extrinsic characteristics of these engineered wood productsremain critical to the successful design of timber buildings.In this Chapter, we describe the background of an engineered wood product, namely, cross-laminated timber (CLT), and our motivation to characterize its strength, stiffness, andstability issues, in particular under the axial compression loading scenario. The overviewof CLT research is presented in section 1.1. Then, state of the art of CLT wall behavioursubjected to in-plane gravity loading is described in section 1.2. Research questions thatare addressed in this study are laid out in section 1.3. The research methodology and con-tribution are pointed out in section 1.4 and section 1.5, respectively. Finally, the navigationof the chapters of this dissertation is presented in section 1.6.1.1 Cross-laminated timber (CLT)The selection of a construction material is based on the 3S metrics, namely, Strength, Ser-viceability and Savings for estimating the required building performance and subsequentlyfor deciding the structural design and policy development of the construction industry[AJ02]|. The addition of one more ‘S’ metric, Sustainability makes the material − wood asa precursor among the conventional construction materials such as concrete, steel and glassdue to its low embedded energy characteristic [SWH14]|. Until recently, modern massivewooden tall structures are rare because of the relative weakness of the conventional woodmaterials [GT17]|. This scenario is starting to change with a new engineered wood product,namely, cross laminated timber (CLT). This product has expanded the possibilities of us-ing wood as a structural products in mid- and high-rise residential buildings, as well as forindustrial and commercial construction [Jon17]|. A CLT system is portrayed in Figure 1.1.It shows a typical manufacturing specification, lamellae species and dimension of a panel,and component assemble in a building.1CLT offers simple structural assemblies needed for making cost-effective buildings, as wellas a strong combination of benefits such as easy and fast installation, reduced weight andwaste, improved thermal and sound insulating qualities and assured design versatility andstructural integrity [OW15]|. In one example, a 9−story building saved an estimated 22weeks of construction time by using CLT instead of concrete [Jon17]|. Essentially, thesecharacteristics put forward CLT to be a strong competitor of conventional building materi-als such as concrete and steel − particularly in multiple-storey buildings [GT17]|. The firstwork with CLT was found in Switzerland and Austria in the early to mid−1990s [Kin17]|.After flourishing in Europe over the past couple of decades, recently North American build-ing designers are welcoming CLT to their community due to proving its superiority in termsof the design flexibility, structural integrity, economic aspect, environmental performanceand sustainability. The first North American non-residential CLT structure was a belltower completed in 2010 in North Carolina [Cre11]|.An amount of 200g/m2Polyurethane (PUR) orMelamine Urea Formalde-hyde (MUF) glue is ap-plied on lumber’s wide sur-face in orthogonal lay-ers under a pressure of0.8MPa for about 2 hours+ =(a) A CLT panelNo.3/StudNo.2/No.145 ∼400mmUpto5m16 ∼18m(b) A CLT wall system (c) A CLT wall-floor assembleFigure 1.1: A CLT system1.1.1 Manufacturing specificationCLT is manufactured in panels that have an odd number of softwood plank layers laidon top of each other at right angles and glued together under pressure. In general, thecommercially available CLT is produced from balanced combinations of orthogonal layerswhere all laminations oriented in the same direction are made of the same structural sawnlumber of the same grade and species combinations. The CLT manufacturing specificationcomplied from [ANS12; Piz94; Piz89; CLT17; Str18; WDM10]|is described in the followingsubsection.21.1.1.1 Lumber laminationsThe lumber in the outer layers of the CLT panel for a wall and a floor or roof system areusually oriented parallel to vertical loads and parallel to major span direction, respectively.This type of orientation is adopted to maximize the CLT system resistance.[CSA16]|limitsthe materials used in CLT panels to structural sawn lumber. The minimum grades of lum-ber in the longitudinal direction of the CLT panel (the major strength direction) permittedare visual graded2 #1 or #23. The minimum grade in the transverse (minor strength direc-tion) is visual graded #3 sawn lumber (Figure 1.1). [ANS12]|limits the moisture contentof the lumber laminations at the time of CLT manufacturing to 12 ± 3%.1.1.1.2 Lamination sizes[ANS12]|requires a minimum net width of lamination of 1.75 times the lamination thicknessin the major strength direction. For the minor strength direction, if the laminations arenot edge glued, the net width of a lamination is not to be less than 3.5 times the lamina-tion thickness unless the planner shear and creep are evaluated by testing to demonstrateperformance. Typically, the thickness of a single layer lumber can be 19 mm or 25 mmor 38 mm up to 50 mm (Figure 1.1). During manufacturing, the lamellae are stackedtogether with a standard approved adhesive on their wide faces and, sometimes, on thenarrow faces as well [Piz89; CLT11]|. Thus, the total thickness and width of a CLT panelranges from 0.045 m to 0.4 m and 0.5 m to 5 m, respectively, whereas length can be upto 18 m. The size and form of CLT panels is limited by production, transportation anderection possibilities [CLT11]|.1.1.1.3 AdhesivesTypical structural adhesives, such as, one-component polyurethane (PUR), emulsion poly-mer isocyanate (EPI), and phenolic types such as phenol-resorcinol formaldehyde (PRF),are used in bonding CLT lamellae. Edge gluing of wood components within the CLTlayers is not a common practise among North American manufacturers due to the addedmanufacturing cost [CLT11]|.2Visual grading is accomplished from an examination of all four faces and the ends of the piece, in whichthe location as well as the size and nature of the knots and other features appearing on the surfaces areevaluated over the entire length of the piece.3According to the Standard grading rules by the National Lumber Grades Authority, visually gradedCanadian structural lumber can be put into four categories: Structural Light Framing, Structural Joistsand Planks, Light Framing, and Studs. This classification is based on the end use applications and membersize. [WDM10]|grades CLT lamella (Structural Joists and Planks whose smaller dimension is 38 to 89 mm,and larger dimension is 114 mm or more), as Select Structural, #1, #2, and #3 based on the quality.31.1.2 Application and researchSuperior to conventional wood construction, CLT as a plate element has reasonable strengthproperties in both directions in-plane due to its lay-up matrix. Therefore, CLT panels arenormally used as a structural floor, roof and wall element of a multi-story buildings. Un-like masonry, which limits the building’s height and leads to heavy and material-intensiveconstruction, a study by [Tec09] indicated 12−storey buildings possible with CLT, using135 mm internal wall, 125 mm external wall and 125 mm thick floor panels. Furthermore,buildings with increased timber content relative to steel and concrete are generally lighter,which alleviates pressure on foundations and means that savings can be made by reducingtheir size. In fact, to avoid over-specification of the panels, CLT is best applied to large-scale medium- and high-rise projects.Along with increasing production demand globally [Cre11]|, research is advancing on CLTsystem, targeting to ensure a CLT structure which would be safe and serviceable to use,economical to build and maintain, and satisfactorily performs its intended function. Forexample, in component level, based on experimental studies, different analytical and em-pirical design equations have been adopted in the codes and standards for determining themechanical properties of CLT. The procedures are reported in [Eur04; CLT11; CSA16]|.Moreover, design procedure for some connection systems in CLT (e.g. bolted connection,dowel-type connection) have been developed and adopted in the design manuals. In systemlevel, researchers demonstrated that CLT wall panels possess adequate seismic resistancecapacity when they assembled with appropriate connection [CLT11; Cec08; PPK02]|. Theload duration behaviour of CLT has also been studied by researchers [CLT11; Li15]. Aseries of small and full scale fire experiments have been conducted by several researchers toget the fire resistance behaviour of CLT, such as [CLT11; LL13]|. In addition, some workhas been done on the vibration and acoustic performance of CLT system, such as [CLT11;Log17]|.1.1.3 Research opportunityTo pave the way for using CLT elements in a structure, it requires to be proven satisfactorybehaviour under all external forces. In a building, walls are important vertical structuralelements that carry in-plane gravity and lateral loads (e.g. earthquake) as well as out-of-plane lateral loads (e.g. wind). The possibility of using CLT with high aspect ratiosas wall and column elements requires particular attention to the study their structuralstability under in-plane gravity loadings. In this context, the inertia forces caused bygravity loadings from all kinds of dead and live loads need to be transferred by the wallelements to the supporting floors or foundation. Over this load path, the strength of thelumber and in-plane stiffness will affect the load distribution among the wall systems, whichneeds to be considered by the design procedures.4Evidence found that the current design practise— [CSA16]|overestimates the CLT capac-ity by a significant amount [Hor13]|. Therefore, appropriate characteristic values of thelumbers in the codes for estimating the in-plane behaviour of different CLT configurationsneeds to be re-examined. Moreover, the current code’s practise of neglecting the crosslayers contribution in the design equation required to evaluate based on experimental andnumerical evidence. Research studies on the in-plane bending rigidity followed by bucklingstrength of CLT wall systems including uncertainty have not been conducted yet.1.2 State of the art in CLT wall under in-plane gravityloadingShort columnLong columnFigure 1.2: A typical wall element under loadWhen a timber slender member is loadedin the gravity direction, the fibres whichare in compression have a tendencyto buckle as the compression stressreaches a certain critical value. Sincea wall is relatively stiff in the hor-izontal direction (i.e. along length orwidth), it is not likely that buck-ling4 would occur in that direction.This leaves the weaker direction (i.e.along thickness) and the plane per-pendicular to it, as the only possi-ble path of instability. Since thesource of wall instability is not influ-enced by its non-buckling direction (i.e.along length or width); therefore, it canbe considered as a column strip (Fig-ure 1.2).The performance of a column element depends on its slenderness ratio5. The failure modesof short and long columns are shown in Figure 1.2.For the short columns, its design is gov-erned by the compression strength of the column parallel-to-load-direction. For the longcolumns, its design is controlled by Euler buckling i.e. the modulus of elasticity (MOE) ofthe element. Here, the element resists out-of-plane buckling primarily through out-of-planebending stiffness. For the intermediate columns, the design is governed by both compres-sion strength of the column parallel-to-load-direction and MOE.4Buckling is a mode of failure generally resulting from structural instability due to compressive actionon the structural element involved.5Slenderness ratio is the ratio of the length of a structural element and the least radius of gyration ofits cross section.5In Canada, an analytical design procedure for a CLT member under axial compressionload is given in [CLT11]| based on [CSA16]|. With the assumption of neglecting cross-layers carrying the load followed by stiffness calculated by shear analogy method6, theeffective compressive resistance (Pr,eff ) and effective Euler buckling capacity (PE,eff ) canbe determined as given in Figure 1.4. The ultimate load carrying capacity and Eulerbuckling capacity of a typical 3-layer CLT following the design procedure described inFigure 1.4 and considering the lumber properties tabulated in [CSA16]| are depicted inFigure 1.3. It shows, the stubby CLT columns (slenderness ratio < 5) are simply designedfor the ratio of Pr,effAeff to be less than the compression-parallel-to-grain strength of theparallel layers. The slender CLT columns (generally 25 < slenderness ratio < 50) aredesigned based upon the Euler buckling capacity (PE,eff ). For the intermediate CLTcolumns, a reasonable parabolic equation is used solely on curve fitting. For the Canadiancode practise ([CSA16]|), the Cubic-Rankine- Gordon (CRG) formula is adopted for gettinga continuous capacity curve ranges from stubby to slender columns as shown in Figure 1.3.The unified focus of the thesis is to examine the suitability of this capacity curve. Thefollowing subsection describes the previous studies done along the line of this work.01002003004005000 10 20 30 40 50Effective slenderness ratioUltimate Capacity (kN)Code interpretationEuler curveFigure 1.3: Load carrying capacity curve of a typical CLT member based on [CSA16]|6Adopted from [Eur04], effective stiffness of the CLT system can be determined by three approaches:mechanically jointed beams theory (Gamma method), composite theory (K method), and shear analogymethod. Gamma method merely provides an exact solution only for simply supported single span memberswith uniformly distributed load distribution. K method does not take into account the shear deformationof the cross layers. Shear analogy method does consider the shear deformation and is not restricted to amaximum of layers or any specific load distribution.6InputsLumberpropertiesWall con-figuration CLT configuration Modification factors- Width [b]- Height [L]- End fixation[Ke]- No of ply- Ply thickness [hi]- Ply orientation[‖ / ⊥]- Species- Grade- MOE[E,E05]- Strength[fc]- System effect [KH ]- Load duration [KD]- Strength reduction [φ]- Lumber treatment [KT ]- Curve shape parameter [N ]- Service condition for fc [KSc]- Service condition for MOE [KSE ]Effective lengthLe = KeLEffective areaAeff = bheff= b∑i(‖)hiEffective stiffness(EI)eff = (EI)A + (EI)B=n∑i=1Eibihi312 +n∑i=1EiAizi2E⊥ =E‖30Effective slenderness ratioCc,eff =Le√12reff= Le√12√Ieff√Aeff= Le√12√(EI)effEmean√AeffFactored strengthFc = fc(KDKHKSckT )Effective size factorKZc,eff = 6.3(heffL)−0.13 ≤ 1.3Effective slenderness factorKc,eff =(1 +FcKZc,effC3c,effNE05KSEKT)−1Factored capacityPr,eff = φAeffFcKZc,effKc,effEuler bucklingPE,eff =pi2E05KSEKT IeffLe2Figure 1.4: Design procedure of CLT under compression loading according to [CLT11]|71.2.1 Previous studiesThe study on the CLT wall subjected to in-plane gravity loading is very limited in theliterature. An investigation on the stability of CLT intermediate columns has been doneby [Hor13]|. Stripes of CLT panels as intermediate length members were tested. Moreover,a short column compression test was done to examine the influence of transverse-layers tothe stability behaviour. The study parameters and findings are stated in Table 1.1.The study reported that the characteristic compression strength of CLT elements are lowerthan the code specified strength by a factor of 1.68 which led to a lower capacity forthe intermediate columns. This shows that either the strength of the cross laminae arecontributing to the overall compression strength of the CLT column or a possibility thatthe cross laminae are contributing a laminating effects that can lead to a higher specifiedcompressive strength parallel-to-grain for lumbers in CLT.Table 1.1: A study by [Hor13] on CLT behaviour under axial compression loadingGroupSlendernessratio(Le/√12reff )Length[mm]Avg. MOE forwholeX-section[MPa]Avg.peakload[kN ]Compressionstrength ||to grain[MPa]FailureS 4.1 495 10601 405 31.38?(19.31†,11.5‡)CompressionandsplittingA 13 3000 8995 332B 17 3750 9486 273C 21 4500 9290 220?5th percentile exp. value, †Characteristic value from exp., ‡[CSA16] value for the S-P-F #2 lumberAlong with the experimental studies, [Hor13]|did numerical modelling of the intermediateCLT elements (Group A, B and C) to estimate the compressive capacity. However, thenumerical modelling did not consider material orthotropic behaviour and individual layersset-up in different directions. Therefore, there was no knowledge to which extent andhow the cross-layers influence the stability behaviour of CLT system with different layupor structural configurations. Moreover, the critical buckling load of CLT elements can beaffected by variation of materials properties, structural properties, initial imperfection, loadeccentricity, and external loads. Therefore, reliability analysis, which is a natural choiceto evaluate the probability of buckling failure, has not been done in the study. To have abetter understanding on the CLT stability behaviour, research questions are discussed inthe following section.81.3 Research questionsTo minimize the knowledge gap found in the previous studies, we address the followingresearch questions in our study.1.3.1 Code assumptionAll different technical approvals, codes and standards, such as [Eur04]|and [CLT11]|, arebased on the same basic assumption: the load-bearing members of CLT are the parallel-to-loading-direction layers only. This seems to be a realistic estimation only for the columnshaving high slenderness ratios because the MOE parallel-to-grain is 20 to 30 times higherthan the MOE perpendicular-to-grain, hence load transfer is mainly governed by the par-allel layers. This is even more valid, when it is taken into account that there can be gapsbetween the lumbers perpendicular-to-load-direction. On the other hand, these assump-tions cannot explain the intermediate column test results obtained by [Hor13]|. Therefore,we aim to answer the question: What extent of neglecting the cross layers contributions isvalid up to?1.3.2 Shear contributionAs a result of stacking of the crosswise layers in CLT panels, the load bearing behaviour ofthis planar element is affected by the material itself and by its constructive anisotropy. Forthe short columns, shear deformation may affect the strength. Whereas, for the intermedi-ate and long CLT columns, shear deformations can be neglected in the longitudinal layersand should be included for the cross layers. Thus, the study attempts to get an answer of:What is the impact of shear contribution on the CLT stiffness behaviour?1.3.3 Stressed volume effectCLT strength can be affected by size and stress distribution profiles, which are essentiallycaused by the inhomogeneity of wood material. The short columns are usually loaded bya force that is constant along the length of the members and develop compression wrinklesat higher load (Figure 1.2). This enables compression failures to have some load sharingwithin the weakest cross-section, leading to a ductile failure manner [Mad92; IWD11].Hence, size effects are less pronounced for the short columns. On the other hand, for theintermediate and long columns, since buckling would create a bending force on the outerlayers weaker plane, the geometric imperfections such as local slope of grain, knots andout-of-straightness of the wall would affect the buckling strength. Also since the eventualfailure of intermediate CLT columns was observed to be brittle fracture, size or stressedvolume effect on the strength properties of wood should be considered. Hence, we wouldlike to incorporate a research question: How to include size and stress distributions effectson the CLT wall stability behaviour?91.3.4 Nonlinearities and uncertaintiesSignificant axial compressive force in a member, particularly for the intermediate and longcolumns, reduces its bending stiffness [TG70]|. This change in stiffness due to axial forceis called geometric stiffness, and is often referred to a ‘P −∆’ effects. Since the member,in general, can be subjected to both elastic and inelastic behaviour, this effect should betaken into consideration. On the other hand, the uncertainty associated with the materials,constructions of the panels and loading conditions should be perused in the design process.Therefore, the study finds an answer for: How to quantify CLT performance consideringnonlinearities and uncertainties?1.4 Proposed researchFocusing on previously stated research questions, the aim of this study is to investigate thestability behaviour of a CLT elements subjected to in-plane gravity loading by adoptingthree-fold approach: experimental, numerical and probabilistic approach.• Experimental study: Quantification of the impact of lamination in CLT systemis difficult, since it depends on many factors such as layer thickness, number of lay-ers, edge gap or lumber properties. Due to having a number of controlling factorsassociated with CLT product configuration, a good number of testing should be con-ducted for better understanding its behaviour and presenting the accurate analyticalformulation. In the experimental study, the CLT stability behaviour is quantifiedfor the given material and system properties. In this context, CLTs having 3-, 5-,7- and 9-layer are tested under axial compression load. Moreover, clear wood andwood contains defects were tested to seek the mechanical properties of the lumber. Interms of sizing variation, the test specimens have been classified into three scales—small-scale, medium-scale, and full-scale. The small-, medium-, and full-scale speci-mens are defined as those whose maximum dimension can be ≤250 mm, ≤1000 mmand ≥2400 mm, respectively. After conducting the experiments, the test results havebeen used for the purpose of validation of the developed numerical models and forscaling the limit state capacity of the CLT system.• Numerical study: The numerical study aims to develop technical evidence of sizeand stress effects on the performance, as well as how the stress would be distributedin the layers for the short and intermediate length columns. Developing a calibratednumerical model, once verified with the test results, offers a more general and lesscostly alternative. [Hor13]|’s study unveiled that the design equation in [CSA16]|givesa conservative capacity compared to the test results. Thereby, the basic assumptionof neglecting cross-layers in the codes is questioned. To debugging the root of thisdiscrepancy, the numerical study is employed to examine the impact of cross-layerson the CLT performance considering parameters, such as, number of layers, layerthickness, and lumber properties.10• Reliability analysis: In the reliability study, we put an effort to find a value for theperformance factor (φc) used in the [CSA16]|design equation for compression capac-ity of CLT structures which will satisfy both criteria, achieving uniformity in safetyacross materials and without unduly penalizing previous design practise. The levelof variation of uncertainties makes it almost impossible to conduct reliability-basedperformance analysis with experimental studies [FFY89]|. Therefore, numerical sim-ulation tools calibrated against experimental results are used for reliability analysis.1.5 Research contributionsThis study aim to create the following contribution in the field:• Acquire the test data of the mechanical properties of CLT elements subjected toin-plane gravity loading.• Evaluate CLT short column behaviour and predict its short-term compressive strength.• Examine the stress distribution in the CLT layers.• Measure the stiffness (MOE) of the CLT system under different loading conditions.• Quantify the cross layer contribution to the load carrying capacity of CLT system.• Develop a finite element method (FEM) simulation environment capable of stabilityanalysis of CLT elements considering material and geometric nonlinearities.• Measure the effects of material and construction uncertainties on the CLT stabilitybehaviour under a random load.1.6 Navigation of the thesisChapter 1 describes the introduction of CLT, problem statement, research questions,methodology, and objectives of the study.Chapter 2 delineates the experimental details and strength result of three types ofmaterials— clear wood, wood contains defects and small-scale CLT elements.Chapter 3 writes about the numerical modelling techniques and analytical model for de-termining the compressive strength of CLT system.Chapter 4 reports the experimental setup and stiffness (modulus of elasticity) results ofthree types of materials— clear wood, wood with defect, and CLT elements.11Chapter 5 covers the development of a finite element method model to compute the mod-ulus of elasticity of CLT material.Chapter 6 addresses the experimental setup and results of full-scale CLT specimens underaxial compression test.Chapter 7 explains the development of a numerical program to compute the ultimate loadcarrying capacity of CLT structure by idealizing the component as a beam-column element.Chapter 8 presents the compression capacity of CLT structures having different slender-ness ratios.Chapter 9 inscribes the reliability analysis and reliability-based design capacity of CLTstructures.Chapter 10 summaries the strength, stiffness and stability behaviour of CLT structuresunder axial compression loads.12Chapter 2Strength of CLT: ExperimentalIn this chapter, we describe the experimental setup and compressive strength resultsof three types of materials- clear wood (CW), wood contains defects (DW), and cross-laminated timber (CLT). Here, CW is referred as the sample which is defect free, and DWis denoted as the specimen which contains defects in terms of knots and other disturbance,resembling the characteristics of dimensional lumber. In terms of sizing variation, the spec-imens have been classified into three scales- small-scale, medium-scale and full-scale. Wedefine the domain of small-scale as a function of dimensions of the specimens, having one ofthe cross-sectional (plane perpendicular to the load axis) dimensions as low as 25 mm andlength as high as 250 mm. In the similar fashion, we define the domain of medium-scaleas a function of dimensions of the specimens, having one of the cross-sectional dimensionsas low as 99 mm and length between 495 mm to 1000 mm. Third, specimens having adimension more than 2400 mm is refereed to full-scale specimens.The test setup and testing condition, failure modes, and mechanical properties of eachspecimen group7 are presented from section 2.2 to section 2.7. Then, testing data analysisand data diagnosis are described in the section 2.8. In order to unify the testing results ofdifferent specimen groups, the adjustment of the experimental results to a specific condi-tion8 and statistical representative properties are laid out in section 2.9. The comparisonof nominal properties with its code’s counterpart is pointed out in section 2.10. Finally,the analytical model is presented in section 2.11.2.1 OverviewOur primary objective is to investigate the effects of cross-layers9 on the CLT’s compressivestrength. The hypothesis being tested is that the cross-layers have some contributions to-ward the compressive strength in CLT. According to the rule of mixture, the contributionsof cross-layers can be quantified by its relative volume fraction occupied in CLT composite.7Clear wood parallel-to-loading; Clear wood perpendicular-to-loading; Wood with defects; 3-, 5-, 7-, and9-layer CLT.8Adjustment due to size, moisture content, and duration of load.9Cross-layers are the layers whose longitudinal grain lies perpendicular to the loading axis.13In the experimental phase, the testing method was identical to each other across the mate-rials type. The ultimate load carrying capacity was considered as the independent variable.The experimental group consisted of 3-, 5-, 7-, and 9-layer CLT. In addition, we experi-mented the clear wood and sawn lumber (wood containing defects) under the same testingprotocol. Furthermore, clear wood specimens were tested under loading parallel and per-pendicular to the grain direction. Two-fold objectives are satisfied by testing the CW andDW specimens; first, is to examine whether there are any difference in the failure modebetween multi-layer specimens (i.e. CLT) and mono-layer specimens (i.e. CW and DW),having the same slenderness ratio, and second, is to develop a database for material’s inputparameters for the numerical modelling of CLT structure. Table 2.1 states the informationregarding specimens scale, species identity, testing standards and loading protocol. Allspecimens were made out of visual graded S-P-F10 #2 or better graded materials.Table 2.1: Overview of the testKeywords DescriptionScaling Small-scale:Clear wood‖ (CW0)Clear wood⊥ (CW90)Wood contains defects (DW)Medium-scale:CLT0−90−011 (CLT3C)CLT0−90−0−90−0 (CLT5C)CLT0−90−0−90−0−90−0 (CLT7C)CLT0−90−0−90−0−90−0−90−0 (CLT9C)Species S-P-F #2, Visual gradingV2M1.1Standards ASTM: D4933-99, D2915-10ASTM: D143-14, D198-15, D4761-13, D4442-15ASTM: D2555-15, D245-06, D1990-14, EN 408-07Loading Displacement controlledMonotonic conservative loadingIn terms of nomenclature, the 5-parts naming system, namely, ‘VWXY Z ’ of the specimengroup is elaborated in Figure 2.1. Here, the metaphor, ‘VWXY Z ’ is referred to the type ofmaterial, number of CLT layers, type of experiment, grain-to-loading-direction or experi-ment set and stiffness measuring method, respectively.10Spruce-Pine-Fir, stamp identification by S-P-F, species included in the combination are Spruce (allspecies except coast Sitka spruce), Jack pine, Lodgepole pine, Balsam fir, Alpine fir.11The orientation of lamella under loading is designated by 0 and 90 which signify that the lamella’slongitudinal direction is parallel and perpendicular to the loading axis, respectively.14VWXY ZType of material:Clear wood (CW),Wood contains defects (DW),Cross-laminated timber (CLT).(optional, if applicatble) Number of CLT layers:3,5,7,9.(optional, if applicable) Type of experiments:B for long-member buckling test,C and empty field for short-member compression test.(optional, if applicable) Loading direction or set of experiments:Y=0 or 90 for the direction of applied loading,otherwise, sequence of the experiments more than one set.(optional, if applicable) Stiffness measuring way:Edge-wise apparent (EA),Edge-wise true (ET),Flat-wise apparent (FA),Flat-wise true (FT),Dynamic transverse vibration (D).Figure 2.1: Naming system of the specimens notationThe dimension of all specimen groups are depicted in Figure 2.2. The longitudinal perspec-tive of the specimen groups relative to a standard human height is shown for the bettervisual clarification. Moreover, loading plate, the number and location of linear variabledisplacement transducer (LVDT), and gauge length12 are shown in the figure. The dimen-sions of specimens and the number of specimens of each group are stated in Table 2.2. Notethat full-scale specimen groups which are discussed in Chapters 4 and 6 are also includedin Figure 2.2 and Table 2.2.The lengths of the specimens of small- and medium-scale specimen groups were chosen suchthat uniformly distributed compressive force are existed throughout the specimen duringloading; hence no flexure occurred. To meet this requirement, the length of the specimenswas greater than 10 times the least radius of gyration, but less than 17 times the leastradius of gyration of the cross-section of the specimen.12The distance along the specimen upon which extension calculations are made.15170 mm102 mm300 mm495 mm180 mm99 mm700 mm200 mm32 mm89 mm250 mm100 mm89 mm230 mm30 mm100 mm25 mm25 mm50 mm100 mm32 mm50 mm 150 mm2400 mm3000 mm3750 mm180 mm180 mm4500 mm170 mm102 mm99 mm99 mm180 mm99 mm800 mm170 mm170 mm200 mm495 mm180 mm170 mm200 mm2400 mm170 mm200 mm250 mm306 mm1000 mm306 mm1000 mm238 mm400 mm400 mmFigure 2.2: All specimen groups for the examination of the mechanical properties: (leftto right) CLT9C, CLT7C, CLT5B, CLT5C1, CLT5C2, CLT3B4, CLT3B3,CLT3B2, CLT3B1, CLT3C2, CLT3C1, DW1, DW2, CW0, and CW90; load-ing plates and LVDTs are attached with the specimens; cross-section of eachspecimen is shown underneath respectivelyExperiments were conducted at the Timber Engineering and Applied Mechanics (TEAM)Laboratory and Civil Engineering Laboratory at the University of British Columbia, Van-couver, Canada. The measurement and recording of load and deflection are accomplishedusing electronic equipment and computerized apparatus available in the laboratories. Theaccuracy of LVDTs was ± 2 mm. Throughout the testing period the room temperatureand relative humidity in the lab was 230C and 45%, respectively, with a minor variation.The physical and mechanical properties of all specimens are presented in A.16Table 2.2: Number and geometric properties of all specimen groupsScale GroupNo. ofspeci-mensSlendernessratio,Le√12 reffCross section[width(mm)× thickness(mm)]Totallength[mm]Lumbergrading SpeciesSmallCW0 35 2.6 25 × 25 100 #2 & Btr S-P-FCW90 33 - 50 × 150 32 #2 & Btr S-P-FDW1 28 5.1 89 × 32 250 #2 & Btr S-P-FDW2 30 5.0 89 × 30 230 #2 & Btr S-P-FMediumCLT3C1 28 4.1 180 × 99 495 #2 & Btr S-P-FCLT3C2 28 5.7 170 × 102 700 #2 & Btr S-P-FCLT5C1 26 4.1 170 × 170 700 #2 & Btr S-P-FCLT5C2 32 2.5 180 × 170 495 #2 & Btr S-P-FCLT7C 05 2.4 306 × 238 1000 #2 & Btr S-P-FCLT9C 05 1.9 253 × 306 1000 #2 & Btr S-P-FFullCLT3B1 20 12.60 170 × 102 2400 #2 & Btr S-P-FCLT3B2 28 13.16 180 × 99 3000 #2 & Btr S-P-FCLT3B3 28 16.50 180 × 99 3750 #2 & Btr S-P-FCLT3B4 28 19.76 180 × 99 4500 #2 & Btr S-P-FCLT5B 20 08.75 200 × 170 2400 #2 & Btr S-P-F2.2 Small-scale: clear wood (CW)To evaluate the physical and mechanical properties of CLT-lamella (sawn lumber), inthe first step, testing was done on small clear wood specimens to obtain the fundamentalinformation of wood species for properties scaling and numerical modelling. The specimensof CW0 and CW90 were obtained from cutting the outside lamella of CLT3C and CLT5Cspecimen groups. Then, we smoothed the surfaces of the lamella and eliminated knots,defects and sloping grain to make a representative ‘clear wood’ state.2.2.1 Clear wood parallel (CW0)2.2.1.1 Test setupThe testing setup and dimension of a typical specimen are depicted in Figure 2.4. In thefigure, the location of LVDTs and gauge length are also specified. A total number of 35specimens were tested. The species grading information and geometric properties of thespecimen are listed in Table 2.2. Figure 2.3 shows the photographs of cross-section, andthe radial and tangential surface of the specimens. It resembles the material variability interms of grain and sawing variation of the lamella.17(a) Cross-section(b) Side viewFigure 2.3: Clear wood samples: (a) cross-section, and (b) side viewWe considered the testing standard adopted by [BLJ94]|, where the small clear wood re-sults of several Canadian species are enlisted. Therefore, following the ASTM D143, a 25× 25-mm cross-section for compression parallel-to-grain test has been used in this study.We tested the specimens under the monotonic displacement controlled compressive loadingcondition. The load was applied through a metal bearing plate, placed across the uppersurface of the specimens. Top platen of the testing machine was equipped with a sphericalbearing to obtain uniform distribution of load over the ends of the specimen. We choosedisplacement controlled loading rate as 0.3% of the nominal length of the specimen. Theloading rate was within ± 25% limit specified in the specification for the testing standard.During testing we monitored and recorded the loading from the hydraulic actuator andthe displacement at several location along the longitudinal axis (parallel-to-loading) of thespecimen using LVDTs. Load-compression readings was continued until the end of theexperiment after proportional limit is well passed. The test was continued until eitherthe load dropped to 80% of the peak load or the strain reached 5% (small deformationassumption). Thus, the full load-deflection trajectory of each specimens was obtained.The actual height, width, and length were recorded at three different location of the spec-18imens. The specimens were weighed immediately before the test, and after the specimendried in oven to quantify the mass density13 and moisture content, MC14. The measure-ments of compression was taken over a central gauge length by LVDTs. Moreover, thelocation and description of failure15 were recorded.(a) Real (b) Rendering32 mm89 mm250 mm100 mm89 mm230 mm30 mm100 mm25 mm25 mm50 mm100 mm32 mm50 mm 150 mmA LVDTFigure 2.4: Test setup of CW0 specimen group: (a) real, (b) rendering2.2.1.2 Failure modesCompression failure can be described as a slow yielding process in which there is a progres-sive development of structural change. At a lower order of magnification, the sequence ofirreversible anatomical changes leading to failure originates in the fibre wall at that pointwhere the longitudinal cell is displaced vertically to accommodate the horizontally runningray [SWH14]|. Compression deformation assumes the form of a small kink in the microfib-rillar structure [BJ93]|. The number and distribution of kinks is dependent on temperatureand moisture content [Din00]|. Increasing moisture content results in the production ofmore kinks, although each is smaller in size than its ‘dry’ counterpart. Increasing temper-ature results in a similar wider distribution of the kinks.At a given temperature and moisture content, as stress and strain increase the kinks be-come more prominent and increase in number, generally in a preferred lateral direction,horizontally on the longitudinal-radial plane and at an angle to the vertical axis of from450 to 600 on the longitudinal-tangential plane [Mar67]|. These lines of deformation, gen-erally called a wrinkle and comprising numerous kinks, continue to develop in width and13The volumetric mass density, of a substance is its mass per unit volume at a moisture content.14MC=Massinitial −Massoven-drying at 1050CMassoven-drying at 1050C15Failure is defined by observing the development of cracks in the surface.19length. At failure, these wrinkles was observed by eye on the face of the block of timber(Figure 2.5) which appeared to be associated with longitudinal strain of 0.33% [AJ99]|.At this stage there was considerable buckling of the cell wall and delamination within it,usually between the S1 and S2 layers [Din00]|.In a visible eye-scale, the observed wood failure modes, namely, crushing, end rolling, wedgesplitting and shearing are shown in Figure 2.5-a, b, c, and d, respectively. None of thespecimen was failed in splitting16 mode of failure. Crushing failure mode (Figure 2.5a)describes a pattern in which the crushed fibres near the end lie in a plane approximatelyparallel to the end surface. End rolling failure mode (Figure 2.5b) occurred because thespecimen was at an elevated moisture content or because the end surfaces were improperlycut [BJ93]|. Wedge splitting failure mode (Figure 2.5c) was easily identified by the ‘Y’shape of the failure line at the surface of the specimen. Failure mode caused by shearingat an angle to the loaded surface (Figure 2.5d) was associated with a plane of weakness.(a) (b) (c) (d)Figure 2.5: Representative macro-level failure modes of CW0 specimen group: (a) crushing,(b) end rolling, (c) crushing and wedge splitting, and (d) shearing2.2.1.3 Mechanical propertiesDuring testing, under a controlled displacement (δ), a transducer connected in series withthe specimen provided an electronic reading of the load P (δ). Then, the engineeringmeasures of stress and strain were determined from the measured load and deflection usingthe original specimen cross-sectional area A0 and length L0 as PA0 andδL0, respectively. Agraphical measurement of mechanical properties in terms of engineering stress-strain curvefor CW0 specimens is shown in Figure 2.6a.16Splitting along the length of the sample may be the result of internal defects in the specimen.20In the early portion of the curve, the material obeyed Hooke’s law17 to a reasonable approx-imation, so that stress was proportional to strain with the constant of proportionality beingnamed as the modulus of elasticity (MOE). Varying with specimens, the obey of Hooke’slaw appeared at a stress up to 25% to 60% of the ultimate failing stress (Figure 2.6a).As strain was increased, specimens eventually deviated from this linear proportionality,the point of departure being termed as the proportional limit18. Then the stress neededto increase the strain beyond the proportional limit continued to rise; the phenomenon istermed as strain hardening. However, the rate of strain hardening would then diminishedup to a point, and labelled as the ultimate compression strength. Beyond this point, thematerial appeared to strain soften, so that each increment of additional strain required asmaller stress.0 1 2 3051015202530354045500 2 4 600.511.522.533.544.55(a) CW0 (b) CW90A specimen curve6Average curve?Slope = MOEffStrengthA specimen curve6Average curve6Slope at 1% strainffSlope ‖ 1% strain-slopeffA = StrengthB = 0.4 × StrengthC = 0.1 × StrengthABCFigure 2.6: Test results of the clear wood specimen groups: (a) CW0, and (b) CW9017Within certain limits, the force required to stretch an elastic object is directly proportional to itsextension.18A closely related term is the yield stress— the stress needed to induce plastic deformation in thespecimen.21The results in terms of average volumetric mass density, MOE, and ultimate compressivestrength of the specimens as a value of 430 kg/m3, 15000 MPa and 36.2 MPa, respectively,are shown in the inset of Figure 2.6a. Moreover, a descriptive statistics of the properties,such as density, moisture content, MOE and strength are presented in Table 2.3. It showsthe coefficient of variation of strength and MOE is 9.1% and 34.0%, respectively.Table 2.3: Statistical data of the properties of the CW0 specimen groupProperties? Min Max Average SD COV [%] 5th percentileDensity [kg/m3] 390 563 431 33 7.70 377MC [%] 12.08 19.17 13.78 1.94 14.12 10.58MOE [MPa] 6219 20038 11000 3736 33.97 4854Strength [MPa] 29.30 40.82 36.19 3.29 9.08 30.79?Sample size = 352.2.2 Clear wood perpendicular (CW90)2.2.2.1 Test setupIn general, the entire surface of the cross-layers experiences the application of gravityloading in a practical case of CLT structure. Therefore, a ‘pure block test’ method, EN408-07, is adopted in this study. The testing setup and dimension of a typical specimen aredepicted in Figure 2.7. A total number of 33 specimens were tested. The species gradinginformation and geometric properties of the specimen are listed in Table 2.2. The load wasapplied continuously using the displacement control method, so that the maximum loadwas reached either within 5 ± 2 minutes or the strain reached 9%.(a) Real (b) Rendering32 mm89 mm250 mm100 mm89 mm230 mm30 mm100 mm25 mm25 mm50 mm100 mm32 mm50 mm 150 mmFigure 2.7: Test setup of CW90 specimen group: (a) real, (b) rendering222.2.2.2 Failure modesIt was observed that when a load was applied perpendicular to the grains, the thin walledtubes were affected laterally and squeezed together with the increase of compression stressesand started to collapse. This behaviour continued until all the fibres were fully crushed.When all fibres were crushed together the loading capacity was once again increased; there-fore, it was difficult to define a failure level19. The estimation procedure of the compressivestrength perpendicular to grain is stated in the next subsection.Three basic patterns of failure depending on the orientation of the growth rings in relationto the direction of the applied load was observed as shown in Figure 2.8. Compression inthe radial direction produced crushing failure in the earlywood zone. Compression in thetangential direction resulted in buckling of the growth rings. In addition, loading at anangle to the growth rings generated shear failure.(a) (b)Figure 2.8: Representative failure modes of CW90 specimen group: (a) wood crushing fromthe side view, and (b) wood crushing on the cross-section2.2.2.3 Mechanical propertiesThe compressive strength perpendicular to grain was estimated from the tests performedaccording to EN 408:2010. Firstly, the load-deformation curve, shown in Figure 2.6b wasdrawn. From this curve an estimated value for the ultimate compressive strength, pointA in Figure 2.6b was decided. Secondly, the value of (0.4 × strength) and the value of(0.1 × strength) was calculated and their value levels were marked onto the curve as pointB and C, respectively. Then, a line-1 was drawn through these two points as shown inFigure 2.6b. Another line-2 was then drawn parallel to line-1, with the deformation atthe initial load equal to 1% strain as shown in Figure 2.6b. The value of the compressivestrength then labelled as the stress value that corresponds to the intersection of line-2 andthe load-deformation curve of test results.The results in terms of average volumetric mass density, MOE, and ultimate compressivestrength of specimens as a value of 525 kg/m3, 300 MPa and 3.5 MPa, respectively, are19Defining a failure level, international discussion has been often controlled by the question, whethercompression strain perpendicular to grain leads to dangerous loss of strength or only to a reduction ofserviceability limit state [ALH01; LLP09]|.23shown in the inset of Figure 2.6b. Moreover, a descriptive statistics of the properties, suchas density, moisture content, MOE and strength are presented in Table 2.4. It shows thecoefficient of variation of strength and MOE is 18.4% and 17.9%, respectively.Table 2.4: Statistical data of the properties of the CW90 specimen groupProperties? Min Max Average SD COV [%] 5th percentileDensity [kg/m3] 435 597 525 43 8.26 454MC [%] 13.58 15.12 14.19 0.36 2.54 13.59MOE [MPa] 103 206 146 26 17.86 103Strength [MPa] 2.34 4.70 3.48 0.64 18.37 2.43?Sample size = 332.3 Small-scale: Wood contains defects (DW)2.3.1 Test setupThe testing setup and dimension of a typical specimen are depicted in Figure 2.9. Two setsof specimens, namely, DW1 and DW2 having number of samples of 28 and 30, respectivelywere tested. The specimens of DW1 and DW2 were obtained from the outside lamellaof CLT3C and CLT5C specimen groups, respectively. The species grading informationand geometric properties of the specimen are listed in Table 2.2. Testing was conductedaccording to the ASTM D198-15 standard. The load was applied continuously using thedisplacement control method, so that the maximum load was reached either within 4 min-utes or the strain reached 1.5%.(a) DW1 (b) DW2 (c) DW1 (d) DW232 mm89 mm250 mm100 mm89 mm230 mm30 mm100 mm25 mm25 mm50 mm100 mm32 mm50 mm 150 mm32 mm89 mm250 mm100 mm89 mm230 mm30 mm100 mm25 mm25 mm50 mm100 mm32 mm50 mm 150 mmALVDTFigure 2.9: Test setup of DW specimen groups: (a, b) real duo, and (c, d) rendering duo242.3.2 Failure modesTypical failure modes are shown in Figure 2.10. Similar to the CW0 specimens group, fibremicrobuckling creating a shearing angle, followed by wood crushing was observed. Woodwrinkles started near the defects. The location, size, distribution, nature, soundness, andfirmness of knots20, deviations (slope) of grains21, and presence of shakes22, checks23, andsplits24 controlled the failure initiation point and wrinkles propagation plane. In general,it was observed that knots in clusters were more influential than knots of a similar sizewhich were evenly distributed; large knots were much more critical than small knots.(a) (b)Figure 2.10: Representative failure modes of DW specimen groups: (a) crushing due toknots on the width side, and (b) crushing on the thickness side2.3.3 Mechanical propertiesThe mechanical properties of the specimen group are depicted in Figure 2.11. All specimenstress-strain curves are drawn. Moreover, the average curve is plotted and mechanicalproperties are computed based on this curve. The average density, MOE and strength20Knots are imperfection that appear as a roughly circular ‘solid’ where the direction of the wood (graindirection) is up to 90 degrees different from the grain direction of the regular wood.21Grain refers to the pattern resulting from the longitudinal arrangement of wood fibres.22Shakes are a defined lengthwise separation of the wood along the grain, usually occurring between orthrough the rings of annual growth.23Checks are a separation of the wood normally occurring across or through the rings of annual growthand usually as a result of drying stresses.24Splits are a separation of the wood through the piece to the opposite surface or to an adjoining surfacedue to the tearing apart of the wood cells.25values are shown in the inset of Figure 2.11. For an instance, the average volumetric massdensity, MOE, and ultimate compressive strength (point A in the figure) of DW1 specimengroup are obtained as a value of 490 kg/m3, 10500 MPa and 39.6 MPa, respectively.Moreover, descriptive statistics of the properties are presented in Table 2.5. It shows thecoefficient of variation of strength and MOE of DW1 specimen group is 12.0% and 19.3%,respectively.0 0.5 1 1.5051015202530354045500 0.5 1 1.505101520253035404550(a) DW1 (b) DW2A = StrengthASlope = MOEffA specimen curveffAverage curveffA = StrengthASlope = MOEffA specimen curveffAverage curveffFigure 2.11: Test results of DW specimen groups: (a) DW1 and (b) DW2Table 2.5: Statistical data of the properties of the DW specimen groupsProperties Min Max Average SD COV [%] 5th percentileDW1? DW2† DW1 DW2 DW1 DW2 DW1 DW2 DW1 DW2 DW1 DW2Density [kg/m3] 425 351 560 525 487 475 36 42 7.38 8.77 428 406MC [%] 7.08 5.09 8.75 8.04 7.92 7.46 0.36 0.51 4.55 6.83 7.33 6.62MOE [MPa] 5909 7514 14866 17891 10513 11843 2028 2469 19.29 20.85 7177 7781Strength [MPa] 30.44 30.71 46.70 50.41 40.55 43.05 4.86 6.07 11.99 14.11 32.55 33.06?Sample size = 28, †Sample size = 30262.4 Medium-scale: 3-layer CLT (CLT3C)2.4.1 Test setupThe testing setup and dimension of a typical specimen are depicted in Figure 2.12. Thespecimens of CLT3C1 and CLT3C2 were obtained from a 3 × 3 m CLT panel. The speciesgrading information and geometric properties of the specimen are listed in Table 2.2.Two sets of specimens, namely, CLT3C1 and CLT3C2 having number of samples of 28 and30, respectively were tested. The first set, CLT3C1, was obtained from a panel supplied byStructurelam, a mass timber manufacturer company in Canada. Structurelam labelled theCLT panel as V2M1.1 class, which composed all parallel (longitudinal/major) and cross(tangential/minor) layers from #2& Btr visually graded lumbers. It was a 3-layer CLT,with two outer layers of 32 mm and one inner layer of 35 mm, i.e. the panels had a totalthickness of 99 mm. The second set, CLT3C2, was produced at TEAM lab according tothe American National Standard ANSI / APA PRG 320-2012. It was a 3-layer CLT, withtwo outer and one cross-layers of 34 mm each, resulted in total thickness of 102 mm. Noneof the lumber in both sets was edge glued, which led to having gaps in cross-layers. Testingfollowed the testing standard—ASTM D198-15. The load was applied continuously usingthe displacement control method, so that the maximum load was reached either within 10minutes or the strain reached at 1.5%.(a) CLT3C1 (b) CLT3C2 (c) CLT3C1 (d) CLT3C2170 mm102 mm300 mm495 mm180 mm99 mm700 mm200 mm170 mm102 mm300 mm495 mm180 mm99 mm700 mm200 mmA LVDTFigure 2.12: Test setup of CLT3C groups: (a, b) real duo, and (c,d) rendering duo272.4.2 Failure modesFailure modes are shown in Figure 2.13. It shows wood wrinkles formed near the knots.Also, it specifies the failure location and wrinkle line on the surface of the layers. Similar tothe CW0 specimen group, fibre microbuckling creating a shearing angle, followed by woodcrushing was observed. Similar to the failure modes of the DW specimen group, woodcrushing started on unconfined-parallel layers due to having knots and other disturbances.Here, the ‘unconfined’ parallel layers means the parallel layers where one out of its two L-Rplanes is in contact with the cross-layers, whereas, the ‘confined’ parallel layers means theparallel layers where both of its L-R planes are in contact with the cross-layers. After thepeak load drops about 6%, cracks propagated from parallel to cross-layers; this transitionmainly happened closer to a gap.(a) (b) (c) (d)Figure 2.13: Representative failure modes of CLT3C groups: (a) fibre buckling followed bywood crushing, (b) wood crushing commencement on the unconfined-parallellayers, (c) wood crushing due to knots, and (d) wood crushing near to the endlocation2.4.3 Mechanical propertiesThe mechanical properties of the specimen group are depicted in Figure 2.14 with thestress strain curves of all specimens shown. Moreover, the average curve is plotted andmechanical properties of the specimen group are computed based on this curve. For anexample, the average volumetric mass density, MOE, and ultimate compressive strength(point A in the figure) of CLT3C1 specimens group as a value of 420 kg/m3, 7000 MPa and22.7 MPa, respectively, are obtained and shown in the inset of Figure 2.14a. Moreover,28descriptive statistics of the properties, such as density, moisture content, MOE and strengthare presented in Table 2.6. It shows the coefficient of variation of strength and MOEproperties of CLT3C1 is 8.6% and 13.4%, respectively. Moreover, it states the compressivestrength of CLT3C1 is lower than that of CLT3C2.(a) CLT3C1 (b) CLT3C2A = StrengthASlope = MOEffA specimen curveffAverage curveffA = StrengthASlope = MOEffA specimen curveffAverage curveffFigure 2.14: Test results of the CLT3C specimen groups: (a) CLT3C1 and (b) CLT3C2Table 2.6: Statistical data of the properties of the CLT3C specimen groupsProperties Min Max Average SD COV [%] 5th percentileCLT3C1? CLT3C2† CLT3C1 CLT3C2 CLT3C1 CLT3C2 CLT3C1 CLT3C2 CLT3C1 CLT3C2 CLT3C1 CLT3C2Density [kg/m3] 386 455 469 511 422 484 21 14 4.93 2.95 388 461MC [%] 10.60 8.22 12.20 9.65 11.42 8.98 0.51 0.33 4.49 3.66 10.57 8.44MOE [MPa] 5351 5666 9521 10143 6961 7875 930 1292 13.36 16.40 5432 5750Strength [MPa] 20.19 21.12 28.48 35.49 22.71 28.43 1.95 3.30 8.57 11.61 19.51 23.00?Sample size = 28, †Sample size = 28292.5 Medium-scale: 5-layer CLT (CLT5C)2.5.1 Test setupThe testing setup and dimension of a typical specimen are depicted in Figure 2.15. Two setsof specimens, namely, CLT5C1 and CLT5C2 having sample size of 28 and 30, respectivelywere tested. The specimens of CLT5C1 and CLT5C2 were obtained from the 3 × 3 mV2M1.1 graded CLT panel supplied by Structurelam. It was a 5-layer CLT, with twounconfined-parallel layers, one confined-parallel layer and two cross-layers of 34 mm each,resulted in total thickness of 170 mm. The species grading information and geometricproperties of the specimen are listed in Table 2.2. Similar to the CLT3C specimen group,the same testing standard and loading rate was followed to conduct the experiment.(a) CLT5C1 (b) CLT5C2 (c) CLT5C1 (d) CLT5C2800 mm170 mm170 mm200 mm495 mm180 mm170 mm200 mm800 mm170 mm170 mm200 mm495 mm180 mm170 mm200 mmALVDTFigure 2.15: Test setup of CLT5C groups: (a, b) real duo, and (c, d) rendering duo2.5.2 Failure modesFailure modes are shown in Figure 2.16. It shows wood wrinkles form near the knots.Similar to CW specimen groups, fibre microbuckling creating a shearing angle, followedby wood crushing was observed. Similar to the CLT3C specimen group, wood crushingstarted on unconfined-parallel layers due to having knots and other disturbances. Afterthe peak load drops about 8%, cracks propagated from parallel layers to cross-layers; thistransition mainly happened closer to a gap.30(a) (b) (c) (d)Figure 2.16: Representative failure modes of CLT5C groups: (a) fibre buckling followedby wood crushing and wood-adhesive interface splitting, (b) wood crushingcommencement on the confined- and unconfined-parallel layers, (c) crushingfailure followed by lamella splitting, and (d) wood crushing due to knots onthe width side2.5.3 Mechanical propertiesThe mechanical properties of the specimen group are depicted in Figure 2.17 and descrip-tive statistics of the properties, such as density, moisture content, MOE and strength arepresented in Table 2.7. The average volumetric mass density, MOE, and ultimate com-pressive strength (point A in the figure) of CLT5C1 as a value of 460 kg/m3, 6500 MPaand 17.7 MPa, respectively, are obtained and shown in the inset of Figure 2.17a. More-over, it shows the coefficient of variation of strength and MOE of CLT5C1 is 6.4% and28.2%, respectively. The calculated compressive strength of CLT5C1 is lower than that ofCLT5C2.Table 2.7: Statistical data of the properties of the CLT5C specimen groupsProperties Min Max Average SD COV [%] 5th percentileCLT5C1? CLT5C2† CLT5C1 CLT5C2 CLT5C1 CLT5C2 CLT5C1 CLT5C2 CLT5C1 CLT5C2 CLT5C1 CLT5C2Density [kg/m3] 426 419 501 499 464 455 19 21 4.05 4.52 433 421MC [%] 10.90 8.95 13.90 11.12 12.42 9.93 0.79 0.50 6.40 4.99 11.11 9.11MOE [MPa] 4668 5035 14367 8451 6377 6244 1801 876 28.24 14.04 3415 4803Strength [MPa] 15.29 16.97 19.58 24.96 17.73 20.82 1.13 1.98 6.39 9.53 15.86 17.55?Sample size = 26, †Sample size = 3231(a) CLT5C1 (b) CLT5C2A = StrengthASlope = MOEffA specimen curveffAverage curveffA = StrengthASlope = MOEffA specimen curveffAverage curveffFigure 2.17: Test results of CLT5C specimen groups: (a) CLT5C1 and (b) CLT5C22.6 Medium-scale: 7-layer CLT (CLT7C)2.6.1 Test setupThe testing setup and dimension of a typical specimen are depicted in Figure 2.18. 5specimens were tested. The specimens of CLT7C were obtained from V2M1.1 graded CLTpanel supplied by Structurelam. It was a 7-layer CLT, with two unconfined-parallel layers,two confined-parallel layers and three cross-layers of 34 mm each, resulted in total thicknessof 238 mm. The species grading information and geometric properties of the specimen arelisted in Table 2.2. The load was applied continuously using the displacement controlmethod, so that the maximum load was reached either within 20 minutes or the strainreached at 1.5%.32(a) Real (b) Rendering306 mm1000 mm238 mm400 mmA LVDTFigure 2.18: Test setup of CLT7C group: (a) real, and (b) rendering2.6.2 Failure modesFailure modes are shown in Figure 2.19. Similar to the CLT3C and CLT5C specimens, woodcrushing started on unconfined-parallel layers due to having knots and other disturbances.After the peak load drops about 5%, cracks propagated from parallel layers to cross-layers.(a) (b) (c) (d)Figure 2.19: Representative failure modes of CLT7C group: (a) wood crushing commence-ment on the parallel layers, (b) fibre buckling followed by wood crushing andwood-adhesive interface splitting, (c) shearing type crushing failure at themid-height, and (d) wood crushing due to knots on the width side332.6.3 Mechanical propertiesThe mechanical properties of the specimen group are depicted in Figure 2.20a. The averagevolumetric mass density, MOE, and ultimate compressive strength (point A in the figure)of CLT7C as a value of 545 kg/m3, 4500 MPa and 20.1 MPa, respectively, are obtainedand shown in the inset of Figure 2.20a. Moreover, descriptive statistics of the properties,such as density, moisture content, MOE and strength are presented in Table 2.8. It showsthe coefficient of variation of strength and MOE is 3.7% and 15.6%, respectively.0 0.5 1 1.505101520250 0.5 1 1.50510152025(a) CLT7C (b) CLT9CA = StrengthASlope = MOEffA specimen curveffAverage curveffA = StrengthASlope = MOEffA specimen curveffAverage curveffFigure 2.20: Test results of the specimen groups: (a) CLT7C, and (b) CLT9CTable 2.8: Statistical data of the properties of the CLT7C specimen groupProperties? Min Max Average SD COV [%] 5th percentileDensity [kg/m3] 496 652 545 63 11.48 442MC [%] 11.09 11.98 11.57 0.42 3.65 10.88MOE [MPa] 6590 9704 8282 1291 15.59 6158Strength [MPa] 19.35 20.85 20.07 0.74 3.67 18.85?Sample size = 05342.7 Medium-scale: 9-layer CLT (CLT9C)2.7.1 Test setupTypical test set up of CLT9C group is shown in Figure 2.21. In order to distribute theload uniformly a 30 mm thick steel plate was put between the machine head and specimen.The displacement-controlled loading was applied by Baldo Machine. The specimen sizewas determined by machine’s hydraulic capacity and taking into consideration about thematerial failure by keeping slenderness ratio sufficient low. The species grading informationand geometric properties of the specimen are listed in Table 2.2.Three LVDTs were mounted on three faces of CLT (2 on wide surface and 1 on thicknessside). The gauge length was considered as 400 mm. Five specimens were tested. Thespecimens of CLT9C were obtained from the 9-layer V2M1.1 graded CLT panel suppliedby Structurelam. It was a 9-layer CLT, with two unconfined-parallel layers, three confined-parallel layers and four cross-layers of 34 mm each, resulted in total thickness of 306 mm.The load was applied continuously using the displacement control method, so that themaximum load was reached either within 24 minutes or the strain reached at 1.5%.(a) Real (b) Rendering250 mm306 mm1000 mm400 mmA LVDTFigure 2.21: Test setup of CLT9C group: (a) real, and (b) rendering352.7.2 Failure modesTypical failure modes of CLT9C group are shown in Figure 2.22. It shows wood crushingfailure of the parallel layers. In the post-ultimate phase, with increasing the displacement-controlled loading, lamella splitting occurs. From sample to sample, wood crushed notedin various location, for example, near the knots, and near the support of the parallel layers.(a) (b) (c) (d)Figure 2.22: Representative failure modes of the CLT9C group: (a) wood crushing com-mencement on the confined- and unconfined-parallel layers, (b) wood crushingfollowed by lamella splitting, (c) crushing failure at different location on thelamellae, and (d) wood crushing due to knots on the width side2.7.3 Mechanical propertiesThe mechanical properties of the specimen group is depicted in Figure 2.20b. Moreover,descriptive statistics of the properties, such as density, moisture content, MOE and strengthare presented in Table 2.9. It shows the coefficient of variation of strength and MOE is3.9% and 16.4%, respectively. The average volumetric mass density, MOE, and ultimatecompressive strength (point A in the figure) as a value of 503 kg/m3, 4000 MPa and 20.4MPa, respectively, are obtained and shown in the inset of Figure 2.20b.Table 2.9: Statistical data of the properties of the CLT9C specimen groupProperties? Min Max Average SD COV [%] 5th percentileDensity [kg/m3] 486 529 503 17 3.42 475MC [%] 10.91 12.08 11.37 0.43 3.81 10.66MOE [MPa] 6120 9453 7533 1235 16.39 5502Strength [MPa] 19.36 21.15 20.35 0.78 3.86 19.06?Sample size = 05362.8 Testing data analysisThe spectrum and average line of all specimen groups’ stress-strain curve are portrayedin Figure 2.23-a, -b, respectively. It shows that at the average level, apart from thecompression-perpendicular capacity, the maximum and minimum parallel-to-grain com-pressive capacities belong to the DW and CLT5C group, respectively. The CW0 andCW90 groups, however, show the maximum and minimum MOE value, respectively. Themechanical properties of all specimens are stated in A. The following subsection examinesthe test data quality and quantity to represent the characteristic properties of the specimengroups.DW2DW1CW0CW90CLT3C2CLT9CCLT7CCLT5C1CLT3C1CLT5C2(a) Spectrum (b) Average lineFigure 2.23: Compression test results of small- and medium-scale specimen groups: (a)spectrum, and (b) average line372.8.1 Sampling qualityThe number of samples of each specimen group is examined first whether it was a sufficientnumber to represent the characteristic properties. Selection of a sample size depends uponthe properties to be estimated, the actual variation in properties occurring in the popula-tion, and the precision with which the property is to be estimated [Mon04]|.The ASTM standard specified a statistical methodology for evaluating the mean andnear minimum property25 estimates. The two approaches, namely, parametric and non-parametric are suggested to follow. A nonparametric approach requires fewer assumptionsand a parametric approach assumes a known distribution of the underlying population. Todetermine sample size based on a non-paramteric tolerance limit, the desired content andassociated confidence level must be selected.We choose a content of 95% and a confidence level of 75% to evaluate the mechanicalproperties of each group. In this case, a sample size of 28 is required to estimate the5% nonparametric tolerance limit value. The content and confidence level of the data isexamined in the following subsection.2.8.1.1 ContentIn order to compare between groups, the test results from all groups are standardized26 first.Sphere with the 95% probability and data points of the standardized strength, stiffness(MOE) and density parameters (univariate Gaussian distribution case) of all samples isshown in Figure 2.24. It shows data for all small- and medium-scale specimen groups aresufficient to measure the mechanical properties. It is to be noted that data from full-scalespecimen groups which described in the Chapters 4 and 6 are also plotted here.25The MOE and strength properties are considered as the mean property and near minimum property,respectively.26Standardization transforms data to have zero mean and unit variance; we adopted the following formulafor data standardization: Datanew =Dataold −DatameanDatamax − Totoal no. of data38-220220 0-2-2-3 -2 -1 0 1 2 3-3-2-10123-3 -2 -1 0 1 2 3-3-2-10123-3 -2 -1 0 1 2 3-3-2-10123Figure 2.24: Sphere with the 95% probability and data points of the standardized strength,stiffness (MOE) and density parameters of all specimen groups2.8.1.2 Confidence intervalIn order to scale all numeric values in the range of 0 to 1, data was normalized27 first. Then75% confidence ellipse28 for a set of 2D normally distributed data samples is computed.27Normalization was done using the formula: Datanew =Dataold −DataminDatamax −Datamin .28For plotting the error ellipse, the length of the axes are defined by the standard deviations SDx andSDy of the data such that the equation of the error ellipse becomes: (DataxSDx )2 + (DataySDy)2 = s, where sdefines the scale of the ellipse. We choose s, such that the scale of the resulting ellipse represents a chosenconfidence level. For example, a 75% confidence level corresponds to s = 2.1459.39Figure 2.25 shows a typical 75% confidence ellipse with the normalized mean (zero) of thestrength and MOE properties of the CLT3C1 specimen group. The vectors shown by pinkand green arrows in Figure 2.25, are the eigenvectors of the covariance matrix29 of thedata, whereas the length of the vectors corresponds to the eigenvalues. The eigenvaluestherefore represent the spread of the data in the direction of the eigenvectors. Figure 2.25shows that the largest variance is in the direction of the X-axis (MOE parameter), whereasthe smallest variance lies in the direction of the Y-axis (strength parameter) and thereis relatively high correlation exist between these two parameters. The next subsectiondetermines the magnitude of the correlation.-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3-0.3-0.2-0.10.00.10.20.37Figure 2.25: Confidence ellipse and data points of the normalized strength and stiffnessparameters of CLT3C1 group2.8.2 CorrelationThe Pearson correlation coefficient is calculated to measure the linear correlation betweentwo variables. The correlation is defined by the covariance of the two variables divided bythe product of their standard deviations. The correlation coefficient is plotted in Figure 2.26whereas the values are given in A. It has a value between +1 and -1, where 1 is total positivelinear correlation, 0 is no linear correlation, and -1 is total negative linear correlation.For example, for the CLT3C1 specimen group, it shows that strength is correlated withMOE by 0.78; whereas density is correlated with strength and stiffness by -0.31 and -0.24,respectively. The correlation values between the parameters are used later in the reliabilityanalysis.29The matrix that linearly transformed original data to normalized data.401str1MOE1den2str2MOE2den1str1MOE1den2str2MOE2denstrMOEdenstrMOEden1str1MOE1den2str2MOE2den0str0MOE0den90str90MOE90den1str1MOE1den2str2MOE2den3str3MOE3den4str4MOE4denstrMOEdenCorrelationcoefficientstr = strength; den= densityFigure 2.26: Representation of measuring the correlation between the strength (str), stiff-ness (MOE) and density (den) parameters of all samples2.8.3 Central tendency and variabilityThe central tendency and variability of data are examined to do reliability analysis later onby plotting data in different ways. For example, Figure 2.27 represents the visual inspectionof the four underlying assumptions30 of the univariate statistical analyses of the strength30If the 4 underlying assumptions holds, the four plots will have a characteristic appearance. First, if thefixed location assumption holds, then the run sequence plot will be flat and non-drifting. Second, if thefixed variation assumption holds, then the vertical spread in the run sequence plot will be the approximatelythe same over the entire horizontal axis. Third, if the randomness assumption holds, then the lag plot willbe structureless and random. Fourth, if the fixed normal distribution holds, then the histogram will bebell-shaped, and the normal probability plot will be linear.41parameter of the CLT3C1 specimen group having n observational values. In the four sub-panels, the run sequence plot (strength[n] vs n), a lag plot (strength[n] vs strength[n-1]), ahistogram, and a normal probability plot are shown. Within these axes, the mean value ofstrength is drawn as a blue straight line. In addition, a 5th panel shows a box-and-whiskerplot of strength. The box-and-whisker plot shows the median (red line), mean and SD (inblue), the 25th and 75th percentile (the box), and outliers (‘+’ symbol), if any. The whiskersare shown as the lowest value within 1.5 times the inter-quartile range (IQR) of the lowerquartile, and the highest value within 1.5 IQR of the upper quartile. Raw data are plottedin grey colour. The plots in Figure 2.27 show that strength parameter is randomly variedalong the run sequence and indicates the non-normal distribution pattern. Also, samplingand run sequence of the tests are fairly reasonable. The probability distribution parameteris sought out in the next subsection and the variability properties are used later in thereliability analysis.0 10 2028303234363828 30 32 34 36 3828303234363828 30 32 34 36 3800.050.10.150.20.250.30.3528 30 32 34 36 38-4-3-2-1012342829303132333435363738Figure 2.27: Representation of measuring the central tendency and data variability of thestrength parameter of the CLT3C1 group422.8.4 Distribution functionFinding a suitable distribution function, data’s variability and normality conditions areexamined. While a box plot only shows summary statistics such as mean/median andinterquartile ranges; to examine the full distribution pattern, the test data are plotted as aviolin plot31. For example, for the CLT3C1 and CLT3C2 specimen groups plots are shownin Figure 2.28. Wider sections of the violin plot of the CLT3C2 group represent a higherprobability than the skinnier sections of CLT3C1 group. In addition, it can be seen thatCLT3C2 group has higher median and higher interquartile range than the that of CLT3C1group indicated by white dot and thick grey bar in the figure, respectively.CLT3C161820222426161820222426CLT3C1CLT3C2Strength(MPa)Strength(MPa)Figure 2.28: Mirrored histogram and box-and-whisker plot of the strength data points ofCLT3C specimen groups with the test realizationsThe symmetry and tail strength of the data relative to a normal distribution are diagnosednext. For example, Figure 2.29 shows how closely the skewness32 and kurtosis33 of theCLT3C1 specimen group would match the skewness and kurtosis of several commonlyused distributions. It shows the kurtosis and skewness of the CLT3C1 specimen group are4.5 and 1.3, respectively, which indicates it belongs to a lack of symmetric heavy-taileddistribution. This information helps to find suitable probability distribution later on.31Violin plot is a box-and-whisker plot with a rotated kernel density plot on each side.32Skewness is a measure of symmetry.33Kurtosis is a measure of whether the data are heavy-tailed or light-tailed relative to a normal distribu-tion.43l0 1 2 3 4 5Cullen and Frey graphsquare of skewnesskurtosis11987654321 l Observationl bootstrapped valuesTheoretical distributionsnormaluniformexponentiallogisticbetalognormalgamma(Weibull is close to gamma and lognormal)llllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllFigure 2.29: Measuring the symmetry and tail-strength of strength data points of CLT3C1specimen group relative to a normal distributionThen, the probability distribution parameters are sought out for the specimen groups. Forinstance, the probability density, cumulative density, P-P34 and Q-Q35 plots for the com-pressive strength of CLT3C1 are showing in Figure 2.30 considering several distributions,such as, 2P-Weibull, lognormal, gamma and normal distributions.For estimating the relative quality of the distribution parameters, we relied upon scoresof goodness-of-fit of the distributions based on two criteria, namely, Akaike’s informationcriterion and Bayesian information criterion36. The best model in the group comparedis the one that minimizes these scores, in both cases. In terms of lower tail goodness-of-fitness, lognormal distribution seems most appropriate as shown in Figure 2.30 and scoresstated in Table 2.10.34A P-P plot compares the empirical cumulative distribution function of a data set with a specifiedtheoretical cumulative distribution function.35A Q-Q plot compares the quantiles of a data distribution with the quantiles of a standardized theoreticaldistribution from a specified family of distributions.36The Akaike information criterion (AIC) tells nothing about the absolute quality of a model, does notdepend directly on sample size, only provides the quality relative to other models. Thus, if all the candidatemodels fit poorly, AIC will not give any warning of that. The difference between the Bayesian informationcriterion and the AIC is the greater penalty imposed for the number of parameters by the former than thelatter.440.00.10.20.315 20 25 30Strength [MPa]Probability densityWeibulllognormalgammanormalllllllllllllllllllllllllllll0.000.250.500.751.0015 20 25 30Strength [MPa]CDFWeibulllognormalgammanormall llll lllllllllllllllllllllllll llllllllllllllllll lll ll lllllllllll lllllll1520253015 20 25 30Theoretical quantilesEmpirical quantilesllllWeibulllognormalgammanormallllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll0.000.250.500.751.000.00 0.25 0.50 0.75 1.00Theoretical probabilitiesEmpirical probabilitiesllllWeibulllognormalgammanormal(a) (b)(c) (d)Figure 2.30: Plots for the strength data points of CLT3C1 specimen group considering fourdifferent distribution types: 2P-Weibull, lognormal, gamma and normal: (a)Probability density, (b) Cumulative density, (c) P-P plot, and (d) Q-Q plotTable 2.10: Estimation of the distribution parametersGoodness-of-fit criteria DistributionNormal 2P-Weibull Gamma LognormalAkaike’s Information Criterion 126.65 128.24 118.22 117.55Bayesian Information Criterion 129.71 130.90 120.88 120.22Knowing the best fitted distribution, along with specimen data, Figure 2.31 is present-ing the smooth cumulative distribution function (CDF) and probability density function(PDF)37 curves of all specimen group considering the lognormal probability distribution.The adjustment of the test results to a specific condition is described in the subsequentsection.37The probability density function (PDF) is defined as the first derivative of the cumulative distributionfunction. The cumulative distribution function (CDF) is defined as follows: Fx(x) = the total sum of allprobability function corresponding to values less than or equal to x.4520 25 30 35 40 45 5000.10.20.30.40.50.60.70.80.91(a) (b)0 5 10 15 20 25 30 35 40 45 5000.10.20.30.40.50.60.70.80.91Figure 2.31: Probability density function and cumulative distribution function of uniax-ial compressive strength of small- and medium-scale specimens: (a) CLT3C1group, and (b) small- and medium-scale specimen groups2.9 Uniaxial compressive strengthThe uniaxial compressive strength38 of CW, DW and CLT is presented in this section.First, test data for all groups are presented; then, adjustment of the test data to a specificcondition is done.2.9.1 Test measurementThe compressive strength of each group are laid out in Figure 2.32 by box plot with whiskerat ± 1.5 of interquartile range and identifying the outliers which lie beyond the whiskerfence. The jitter of specimen results of each group are also shown by black dots in thefigure respectively. Moreover, descriptive statistics of the compression parallel-to-grainstrength of the specimen groups are presented in Table 2.11. Roughly, the compressivestrength parallel-to-grain ranges from 17 to 51 MPa. The reason of this wide range is thatstress39 is calculated upon total area. For instance, the strength for CW and DW showshigher value than the counterpart of CLT specimens. In CW and DW cases, the graindirection of the entire loaded surface is in-line with the load direction. In CLT, however,the strength of the cross-layers contributed little to the overall capacity of the CLT member.This resulted in lower CLT compression strength (based on total area) compared to thecompression strengths of CW and DW. The following subsections describe the strength ofdifferent groups from the same level of perspective.38Strength is defined by the ultimate stress point.39Stress is referred to first Piola-Kirchhoff (PK1) stress.46lllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllllllllllllll llll lll ll l lllllllllllllllllllllllllllllllllllllllll lllll01020304050CLT3C1CLT3C2CLT5C1CLT5C2CLT7CCLT9CCW0CW90DW1DW2SpecimenCompressive Strength (MPa)s Outlier· Specimen resultFigure 2.32: Compressive strength of small- & medium-scale specimens without adjustmentTable 2.11: Compressive strength without adjustmentEstimations SpecimensCLT3C1 CLT3C2 CLT5C1 CLT5C2 CLT7C CLT9C CW0 DW1 DW2Min [MPa] 20.19 21.12 15.29 16.97 19.35 19.36 29.30 30.44 30.71Max [MPa] 28.48 35.49 19.58 24.96 20.85 21.15 40.82 46.70 50.41Average [MPa] 22.71 28.43 17.73 20.82 20.07 20.35 36.19 40.55 43.05SD [MPa] 1.95 3.30 1.13 1.98 0.74 0.78 3.29 4.86 6.07COV [%] 8.57 11.61 6.39 9.53 3.67 3.86 9.08 11.99 14.115th P. [MPa] 19.51 23.00 15.86 17.55 18.85 19.06 30.79 32.55 33.06CSA code specified compression parallel strength of a SPF#2 dimension lumber is 11.5 MPa.2.9.2 Unifying test measurementIn order to compare the strength of the different groups, strength is calculated based onthe effective cross-sectional area40. Therefore, at this level, CLT’s strength is comparableto the CW and DW groups. The effective strength of each group and code specifiedcompression strength is plotted in Figure 2.33. Moreover, descriptive statistics of theeffective compression parallel-to-grain strength of the specimen groups are presented in40Effective cross-section area is defined as the area considering only parallel layers; thus cross-layers inCLT are neglected.47Table 2.12. It shows the strength of DW group is higher than that of the CW0 group; itis due to data have yet been adjusted to a standard moisture content. The next sectiondescribes the adjustment of the data.lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll ll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll lll lllllllllllllllllllllll l lll ll ll lllllllllllllllllllllllllllllllllllllllllllllll01020304050CLT3C1CLT3C2CLT5C1CLT5C2CLT7CCLT9CCW0CW90DW1DW2SpecimenCompressive Strength eff (MPa)Code specified compression parallel strengths Outlier· Specimen resultFigure 2.33: Effective uniaxial compressive strength of small- and medium-scale specimenswithout adjustmentTable 2.12: Effective compressive strength without adjustmentEstimations SpecimensCLT3C1 CLT3C2 CLT5C1 CLT5C2 CLT7C CLT9C CW0 DW1 DW2Min [MPa] 31.23 31.68 25.49 28.29 33.86 34.84 29.30 30.44 30.71Max [MPa] 44.06 53.24 32.63 41.60 36.49 38.07 40.82 46.70 50.41Average [MPa] 35.13 42.64 29.54 34.69 35.11 36.64 36.19 40.55 43.05SD [MPa] 3.01 4.95 1.89 3.31 1.29 1.41 3.29 4.86 6.07COV [%] 8.57 11.61 6.39 9.53 3.67 3.86 9.08 11.99 14.115th P. [MPa] 30.17 34.50 26.44 29.25 32.99 34.31 30.79 32.55 33.06CSA code specified compression parallel strength of a SPF#2 solid lumber is 11.5 MPa.2.9.3 Adjusting test measurementThe test results are adjusted with moisture content, size, duration of loads and safetyfactors. During adjustment it is followed the ASTM standards and CSA codes. To see48the variation of characteristic property of the specimens, the test data for density of eachgroup is presented in Figure 2.34 as a violin plot and embedded box and whisker plot. Theplot is included the sample groups which are used for full-scale stiffness and stability testsas described in Chapters 4 and 6. The density results of each specimen group are listed inTable 2.13 and Table 2.14. It shows average density of the groups ranges from 415 to 540kg/m3. This variation is due to different species and moisture conditions. The test dataare not adjusted with the density variation rather it is adjusted with the moisture content.llllllllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllllllllllllll lllllll lllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll lllll llllllllllllllllll lllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllll lllll ll llllllllllllllllllllllllllllllllllllllll ll lllllllll llllllllllllllllllll200400600800CLT3B2DCLT3B3DCLT3B4DCLT3B1EACLT3B1ETCLT3B1FACLT3B1FTCLT3C1CLT3C2CLT5BEACLT5BETCLT5BFACLT5BFTCLT5C1CLT5C2CLT7CCLT9CCW0CW90DW1DW2SpecimenDensity (kg/m3 )s Outlier· Specimen resultFigure 2.34: Density of all specimen groupsTable 2.13: Density of small- and medium-scale specimensEstimations SpecimensCLT3C1 CLT3C2 CLT5C1 CLT5C2 CLT7C CLT9C CW0 CW90 DW1 DW2Min [kg/m3] 372 455 426 419 442 432 390 435 425 351Max [kg/m3] 469 511 501 499 652 562 563 597 560 525Average [kg/m3] 420 484 464 456 541 493 431 525 487 475SD [kg/m3] 25 14 19 21 46 35 33 43 36 42COV [%] 5.98 2.95 4.05 4.62 8.58 7.04 7.70 8.26 7.38 8.775th P. [kg/m3] 379 461 433 421 464 436 377 454 428 40649Table 2.14: Density of full-scale specimensEstimations SpecimensCLT3B1EA CLT3B1ET CLT3B1FA CLT3B1FT CLT3B2D CLT3B3D CLT3B4D CLT5BEA CLT5BET CLT5BFA CLT5BFTMin [kg/m3] 462 455 455 455 365 360 372 425 419 419 419Max [kg/m3] 508 511 511 511 451 487 452 483 499 499 499Average [kg/m3] 484 485 481 486 417 418 419 456 454 456 453SD [kg/m3] 13 14 14 16 22 32 21 20 22 22 20COV [%] 2.78 2.94 2.86 3.19 5.32 7.57 5.02 4.29 4.82 4.91 4.515th P. [kg/m3] 462 461 459 460 381 366 385 424 418 419 419The test data for moisture content of each group is presented in Figure 2.35. The plotis included the sample groups which are used for full-scale stiffness and stability test, asdescribed in Chapters 4 and 6. Moreover, the moisture content results of each specimengroup are listed in Table 2.15 and Table 2.16. The lowest and highest moisture content arerecorded for the defect and clear wood group with an average value of 7.5% and 14.2%,respectively. The reason for high value for CW is that the samples were placed in theclimate chamber with setup temperature and humidity as 200c and 65%, respectively, formore than 2 weeks. Whereas, for DW case, test was done as received conditions withoutfurther climatic conditioning. Oven dry method is used to calculate moisture content ofeach specimens except CLT3B2D, CLT3B3D, CLT3B4D groups; the moisture contents ofthese groups were measured by [Hor13]|with a moisture meter.llllllllllllllllllll lllllllllll llllllllllllllllllllllllllllllllllll llllllll lllllllll llllllllllllllllll lllllllllllllllllllllllllllll lll llllllllllllllllllllllllllllllllllllllllllll llll llllllllllllll lllllllllllllllllllllllllllllllllllllllll lllllllllllllll lllllllllllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllll llllll llllllllll lllllllllllll lllllllllllllllllllllll llllllllllllllllllllllll lll llll llllllllllll l05101520CLT3B2DCLT3B3DCLT3B4DCLT3B1EACLT3B1ETCLT3B1FACLT3B1FTCLT3C1CLT3C2CLT5BEACLT5BETCLT5BFACLT5BFTCLT5C1CLT5C2CLT7CCLT9CCW0CW90DW1DW2SpecimenMC (%)s Outlier· Specimen resultFigure 2.35: Moisture content of all specimen groups50Table 2.15: Moisture content of small- and medium-scale specimensEstimations SpecimensCLT3C1 CLT3C2 CLT5C1 CLT5C2 CLT7C CLT9C CW0 CW90 DW1 DW2Min [%] 10.60 8.22 10.90 8.95 11.09 10.91 12.08 13.58 7.08 5.09Max [%] 12.20 9.65 13.90 11.07 11.98 12.08 19.17 15.12 8.75 8.04Average [%] 11.42 8.98 12.42 9.86 11.61 11.35 13.78 14.19 7.92 7.46SD [%] 0.51 0.33 0.79 0.46 0.36 0.38 1.94 0.36 0.36 0.51COV [%] 4.49 3.66 6.40 4.66 3.13 3.30 14.12 2.54 4.55 6.835th P. [%] 10.57 8.44 11.11 9.10 11.01 10.73 10.58 13.59 7.33 6.62Table 2.16: Moisture content of full-scale specimensEstimations SpecimensCLT3B1EA CLT3B1ET CLT3B1FA CLT3B1FT CLT3B2D CLT3B3D CLT3B4D CLT5BEA CLT5BET CLT5BFA CLT5BFTMin [%] 8.22 8.40 8.40 8.22 10.60 10.20 10.20 8.95 8.95 8.95 9.33Max [%] 9.65 9.55 9.55 9.65 12.60 13.50 14.10 11.12 11.07 11.07 11.12Average [%] 9.00 9.00 9.02 8.91 11.60 12.03 12.71 9.88 9.91 9.87 10.00SD [%] 0.33 0.30 0.29 0.32 0.69 0.95 0.96 0.51 0.52 0.51 0.47COV [%] 3.71 3.36 3.25 3.61 5.94 7.86 7.53 5.17 5.28 5.20 4.715th P. [%] 8.45 8.50 8.54 8.38 10.46 10.48 11.14 9.04 9.05 9.03 9.222.9.3.1 Adjustment factor and resultsFor the strength parallel-to-grain (σ‖) parameter, the adjustment factor for moisture con-tent (MC) is calculated from Equation 2.1 according to the ASTM standard. The adjust-ment equations are valid for MC values between 10 and 23%. We set our target MC is15%. The graphical representation of this adjustment factor is depicted in Figure 2.36. Itshows that at a given MC, the adjustment factor varies with the strength.σ‖new = σ‖old +σ‖old − 140034−MCold (MCold −MCnew) (2.1)where σ‖ in psi and greater than 1400 psi, and MC in %.10152025303540455010 11 12 13 14 15 16 17 18 19 20 21 22 23MC (%)0.60.70.80.911.11.2fMCFigure 2.36: Moisture content adjustment factor in the ASTM standard51The test data for strength adjusted with the moisture content at 15% level and codespecified strength value are presented in Figure 2.37. Moreover, descriptive statistics ofthe adjusted compression parallel-to-grain strength of the specimen groups are presented inTable 2.17. For all group of specimens, the experimental near-minimum strengths are morethan double the code values. Additional adjustments are needed to account for durationof load and safety so that the code values can be compared with the experimental data.lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllllll lllllllll llllllllllllllllllllll lll llllllllllllllllllllllllllll llllllllllllllllllllllllllll01020304050CLT3C1CLT3C2CLT5C1CLT5C2CLT7CCLT9CCW0CW90DW1DW2SpecimenCompressive Strength effadj (MPa)Code specified compression parallel strengths Outlier· Specimen resultFigure 2.37: Effective uniaxial compressive strength of small- and medium-scale specimensadjusted with the moisture contentTable 2.17: Effective compressive strength adjusted with the moisture contentEstimations SpecimensCLT3C1 CLT3C2 CLT5C1 CLT5C2 CLT7C CLT9C CW0 DW1 DW2Min [MPa] 27.55 26.48 22.85 24.30 29.73 30.67 27.88 24.78 24.90Max [MPa] 38.84 42.91 30.08 34.49 32.78 33.39 45.17 36.38 39.13Average [MPa] 31.11 34.72 27.19 29.41 31.23 32.30 34.74 32.16 33.58SD [MPa] 2.70 3.90 1.83 2.56 1.18 1.06 3.46 3.48 4.45COV [%] 8.67 11.22 6.74 8.69 3.78 3.27 9.96 10.83 13.255th P. [MPa] 26.67 28.31 24.18 25.21 29.29 30.57 29.05 26.43 26.26CSA code specified compression parallel strength of a SPF#2 solid lumber is 11.5 MPa.52Then, the compressive strength test results are adjusted with the characteristic size (38 ×184 × 3658 mm at 15% MC) according to the ASTM standard using the Equation 2.2.The graphical representation of the factor is given in Figure 2.38. It shows that the factorfor the compressive strength parallel-to-grain only depends on the width variation.σ‖new = σ‖old(WidtholdWidthnew)0.13(LengtholdLengthnew)0(ThicknessoldThicknessnew)0(2.2)where σ‖ in psi and dimensions are in inch.010020030040050060070080090010000 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000.80.850.90.9511.051.11.151.21.251.3Figure 2.38: Size adjustment factor in the ASTM standard for the strength parameterNote that the CSA code has a size adjustment factor for determining the compressiveresistance parallel-to-grain, as written in Equation 2.3. The graphical representation ofthe factor is given in Figure 2.39. It shows that the factor for the compressive strengthparallel-to-grain decrease with increasing the size of a specimen by a power factor. Unlikethe ASTM standard where size factor only depends on the width of a specimen, the sizefactor in the CSA code depends on the length and a dimension associated with minor axisdirection (axis with the lowest moment of inertia). The size adjustment factor by CSAcode is used in calculating the capacity of the specimens which described in Chapters 6and 7.Size factor,KZc = 6.3(Dimensionbuckling direction × Length)−0.13 (2.3)where dimensions are in mm and size factor value is limited to 1.3.53010020030040050060070080090010000 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000.80.850.90.9511.051.11.151.21.251.3KZcFigure 2.39: Size adjustment factor in the CSA code for the strength parameterThe specimens results under moisture content and size adjusted are presented in Fig-ure 2.40. Moreover, descriptive statistics of the adjusted compression parallel-to-grainstrength of the specimen groups are presented in Table 2.18. With this adjustment itshows the defect wood strength is lower than the clear wood strength. Also, we observedata variation for defect wood was slightly larger than the clear wood counterpart.lllllllll lllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllll llllllllll llll lllllllllllllllllllllllll lllll l lllllllllllllllll llllllllllllllllllllllllllll01020304050CLT3C1CLT3C2CLT5C1CLT5C2CLT7CCLT9CCW0CW90DW1DW2SpecimenCompressive Strength effadj (MPa)Code specified compression parallel strengthr Outlier· Specimen resultFigure 2.40: Effective uniaxial compressive strength of small- and medium-scale specimensadjusted with the moisture content and size factors54Table 2.18: Effective compressive strength adjusted with the moisture content & size factorsEstimations SpecimensCLT3C1 CLT3C2 CLT5C1 CLT5C2 CLT7C CLT9C CW0 DW1 DW2Min [MPa] 27.47 26.21 22.62 24.23 31.75 32.01 27.88 22.55 22.66Max [MPa] 38.72 42.47 29.77 34.39 34.98 34.81 45.17 33.10 35.60Average [MPa] 31.02 34.37 26.91 29.33 33.35 33.69 34.74 29.26 30.55SD [MPa] 2.69 3.86 1.81 2.55 1.25 1.09 3.46 3.17 4.05COV [%] 8.67 11.22 6.74 8.69 3.75 3.23 9.96 10.83 13.255th P. [MPa] 26.60 28.02 23.93 25.14 31.29 31.89 29.05 24.05 23.89CSA code specified compression parallel strength of a SPF#2 solid lumber is 11.5 MPa.Another factor, the duration-of-load adjustment factor for the strength parameter is de-picted in Figure 2.41. Following the Wood Design Manual [WDM10]|, we choose durationof load factor as 1.25. At this level, the factor incorporate snow and occupancy loads.The duration of loads is anticipated to exceed short-term condition41 and underperformlong-term condition42. The load duration factor can be calculated as follows:KD = 1.0− 0.5 log(PLPS)≥ 0.65 (2.4)where PL and PS are the specified long-term load and specified standard-term load basedon load combination.0.00.51.01.52.010 min7 days2 months1 year10 yearsTimeLoad duration factorFigure 2.41: Duration-of-load adjustment factor in the ASTM standard41Loads is expected to last less than 7 days.42Load is expected to last as long a period of time as the dead loads themselves.55Finally, Figure 2.42 shows the adjusted strength of small- and medium-scale specimengroups with the CSA code specified strength. The descriptive statistics of the adjustedcompression parallel-to-grain strength of the specimen groups are stated in Table 2.19.The results are adjusted with the moisture content at 15% level, characteristic size of aspecimen of 38 mm × 184 mm × 3658 mm, safety and duration-of-load factors specified inthe ASTM standard. The data variation also depicts in the Figure 2.42. It shows that thecharacteristics strength of all specimen groups are outperformed the code specified value.llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllll lllllllllllll llllllllllllllllllllllllll llllllllllllllllllllllllllllllllll llllllllllllllllllll llllllll lll llllllllllllllllllllllllllllllllllllllllllll0510152025CLT3C1CLT3C2CLT5C1CLT5C2CLT7CCLT9CCW0CW90DW1DW2SpecimenCompressive Strength specifiedTest (MPa)Code specified compression parallel strengths Outlier· Specimen resultFigure 2.42: Effective uniaxial compressive strength of small- and medium-scale specimensadjusted with the moisture content, size, safety and duration-of-load factorsTable 2.19: Characteristic compressive strength of small- & medium-scale specimen groupsEstimations SpecimensCLT3C1 CLT3C2 CLT5C1 CLT5C2 CLT7C CLT9C CW0 DW1 DW1Min [MPa] 16.91 16.13 13.92 14.91 19.54 19.70 17.16 13.88 13.94Max [MPa] 23.83 26.14 18.32 21.16 21.53 21.42 27.80 20.37 21.91Average [MPa] 19.09 21.15 16.56 18.05 20.52 20.73 21.38 18.01 18.80SD [MPa] 1.66 2.37 1.12 1.57 0.77 0.67 2.13 1.95 2.49COV [%] 8.67 11.22 6.74 8.69 3.75 3.23 9.96 10.83 13.255th P. [MPa] 16.37 17.24 14.73 15.47 19.26 19.63 17.88 14.80 14.70CSA code specified compression parallel strength of a SPF#2 solid lumber is 11.5 MPa.562.10 Characteristic strength and CSA code strengthUsing the adjustment factors derived earlier, the adjusted compression capacity of all spec-imen groups and its code counterpart are shown in Figure 2.43, and descriptive statisticsare stated in Table 2.19. The strength of CW and DW groups is higher than the codespecified strength due to code adaption based on ‘in-grade’ testing specimens.lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllll lllllllllllll llllllllllllllllllllllllllllllllllllll llllllllllllllllllllll llllllllllllllllll lllllll llllll llllllllllllllllllllllllllllllllllllllllllll0510152025CLT3C1CLT3C2CLT5C1CLT5C2CLT7CCLT9CCW0CW90DW1DW2SpecimenCompressive Strength specifiedTest (MPa)Code specified compression parallel strengths Outlier· Specimen resultFigure 2.43: Adjusted compressive strength of small- and medium-scale specimen groupsNow, the characteristic strength43 of effective CLT specimens and CSA code specifiedstrength for solid lumber is shown in Figure 2.44a. The strength for CLT3C44, CLT5C ,CLT7C and CLT9C specimen groups is obtained as 16.6 MPa, 14.9 MPa, 19.3 MPa, and19.6 MPa, respectively. Figure 2.44b represents the groups’ normalized strength withrespect to the code specified strength. It shows strengths of the CLT3C , CLT5C , CLT7Cand CLT9C specimen groups are 42%, 21%, 64% and 65% higher than the code specifiedstrength, respectively. The variation reason is discussed in the following subsection.43Characteristic strength is a lower minimum estimated value of the test data having adjusted to stan-dardized conditions of temperature, moisture content and characteristic size.44For comparison, the group having two sets are averaged and then compared.57CLT3C CLT5C CLT7C CLT9C024681011.5121416182022CLT3C CLT5C CLT7C CLT9C00.20.40.60.811.21.41.61.8(a) Actual (b) NormalizedFigure 2.44: Characteristic strength of CLT obtained from tests with its code counterpart:(a) actual value, and (b) normalized value with the code specified strength2.10.1 Strength variation reasonTo identify the reason behind thevariation between characteristic andcode specified strength, first we ex-amine the slenderness ratio of eachspecimen group whether it has anyinfluence on the failure mode. Theslenderness ratio of each group isrepresented in Figure 2.45. It showsthe slenderness ratio lies in between2.9 to 4.8. This low slenderness ra-tio justify the material failure modeof all medium-scale CLT specimenswhich observed in the tests. There-fore, we take out the slenderness ra-tio as a governing factor for strengthvariation reason of the groups.CLT3C CLT5C CLT7C CLT9C00.511.522.533.544.55Figure 2.45: Slenderness ratio of CLT specimensThen, we examine the volume fraction of cross and parallel layers in CLT. Each groups’58strength and volume fraction of layers are presented in Figure 2.46-a. Moreover, Fig-ure 2.46-b shows the normalized values of all groups with respect to the values of CLT3Cgroup; noted, the normalization for the confined-parallel layers parameter is done with re-spect to the value of CLT5C due to having no confined-parallel layer in the CLT3C specimengroup. Figure 2.46 shows the percentage of cross-layers in CLT3C and CLT9C specimengroups is 33% and 8%, respectively; the fraction of unconfined-parallel layers in these twospecimen groups is 67% and 22%, respectively, whereas, the confined-parallel layers is 0%and 44%, respectively.It is to be noted that the strength and stiffness (MOE) of cross-layers are almost 15 and120 ,respectively, of its parallel layers counterpart. Since loading was displacement controlled,with a constant compression loading, cross-layers experienced very low disturbance andhence contribute negligibly to the overall capacity to the CLT composite system. More-over, during testing no damage was observed in the cross-layers at the ultimate limit state.This phenomenon guides us to identify the governing contributors for the CLT compos-ite strength. As a consequence, we propose that the volume fraction of confined- andunconfined-parallel layers make a difference in CLT composite strength. In the next sub-section, an analytical model is presented to calculate the CLT strength.CLT3C CLT5C CLT7C CLT9C010203040506070010203040506070CLT3C CLT5C CLT7C CLT9C00.20.40.60.811.21.41.61.800.20.40.60.811.21.41.61.8(a) (b)Figure 2.46: Characteristic strength, and percentages of cross-layers, unconfined-parallellayers and confined-parallel layers of CLT: (a) actual value, and (b) normalizedvalue with the CLT3C group592.11 Analytical modelTo develop an analytical model of CLT strength, the trend lines of layers’ percentagesare plotted first. The trend lines of normalized strength and normalized volume fractionof layers are presented in Figure 3.3. It shows, strength drops from CLT3C to CLT5Cgently, then increases from CLT5C to CLT7C sharply and again increases from CLT7Cto CLT9C steadily. The cross-layers’ percentage increases from CLT3C to CLT9C mildly.The confined-parallel layers’ percentage increases from CLT5C to CLT9C steeply. Theunconfined-parallel layers’ percentage drops from CLT3C to CLT9C gradually.CLT3C CLT5C CLT7C CLT9C00.20.40.60.811.21.41.61.800.20.40.60.811.21.41.61.8Figure 2.47: Trend plot of the normalized characteristic strength, and percentages of cross-layers, unconfined-parallel layers and confined-parallel layers of CLTBased on the trends noticed in Figure 3.3, we developed the Equation 2.5 for determiningthe effective strength of CLT having confined- and unconfined-parallel layers. The Equa-tion 2.5 depends on mainly two variables. One is confining factor on the strength, α, andanother one is volume fraction of confined- and unconfined-parallel layers. We find theconfining factor for unconfined-parallel layers from the testing of CW and DW specimengroups and confining factor for confined-parallel layers from the simulations of the verifiedfinite element (FE) models (FE modelling is described in Chapter 3). The values of theparameters used in the Equation 2.5 are stated in Table 2.20.60fCLTeffc =αUPL × fCSAc‖A−B ×(tUPLtPL)−n CLT with only unconfined-parallel layers (UPL)αCPL × fCSAc‖A−B ×(tCPLtPL)−m CLT with only confined-parallel layers (CPL)αUPL × fCSAc‖A−B ×(tUPLtPL)−n + αCPL × fCSAc‖3− 2×(tCPLtPL)−m CLT with both parallel layers (PL)(2.5)where A,B, n, andm are the fitting parameters, fCLTeffc is the calculated effective strengthof the CLT composite in MPa, fCSAc‖ is the CSA code specified strength for the parallellayers in MPa, and tUPL , tCPL , and tPL are the total thickness of the unconfined-parallellayers, confined-parallel layers and total parallel layers, respectively.Table 2.20: Value of the properties used in the Equation 2.5Property ValueαUPL 1.44αCPL 1.60A 3B 2n 2m 0.12The results obtained by the developed analytical model are stated in Table 2.21. Fig-ure 2.48a shows the effective strength of CLT by test and analytical model. The codespecified strength is also marked on the plot. Moreover, the results of the analytical modelwere normalized with respect to the test result, and is presented in Figure 2.48b. Themaximum variation between the test and prediction was found to be 7%. Therefore, it canbe seen that the analytical model predicts the test result reasonably well.61CLT3C CLT5C CLT7C CLT9C0510152025CLT3C CLT5C CLT7C CLT9C00.10.20.30.40.50.60.70.80.911.11.211.5(a) (b)Figure 2.48: Characteristic compressive strength of CLT specimens from test and analyticalmodel with its code counterpart: (a) actual value, (b) normalized value withthe test resultTable 2.21: Characteristic compressive strength of CLT specimens with its code counterpartSpecimenStrength RatioCode Test ModelAnalytical TestModelAnalytical[MPa] [MPa] [MPa]CLT3C 11.5 16.6 16.6 1.00CLT5C 11.5 14.9 14.8 1.01CLT7C 11.5 19.3 18.1 1.07CLT9C 11.5 19.6 18.8 1.052.12 ConclusionIn this chapter, the experimental setup and findings of small- and medium-scale specimensunder axial compression loading is presented. The experimental group consisted of medium-scale 3-, 5-, 7-, and 9-layer CLT members. In addition, to evaluate the physical andmechanical properties of CLT-lamella (sawn lumber), we tested small-scale clear woodand wood contains defects specimens. These fundamental information about small-scalewood species are used in properties scaling and numerical modelling of medium-scale CLTspecimens.62Test results in terms of dry density, moisture content, MOE and strength were recordedfor each specimen. Sampling quality and quantity of all specimen groups are examined.Confidence interval and correlation of all properties are identified. In addition, centraltendency, data variability, and probability distribution function of strength parameter ofsmall- and medium-scale specimens are determined. The compressive strength test resultsare adjusted with the moisture content, size, safety and duration-of-load factors.The study found that the characteristic strengths of 3-, 5-, 7-, and 9-layer CLT are 42%,21%, 64%, and 65% higher than the code specified strength, respectively. To identify thereason behind the variation between the characteristic and code specified strength of CLT,we examined its slenderness ratio, and volume fraction of cross and parallel layers. Wediscretized the parallel layers by unconfined- and confined-parallel layers. By unconfined-parallel layers we refer the exterior parallel layers, that means only one of its surfaces isglued with cross-layers’ surfaces. Whereas, any parallel layers which lie in between cross-layers is defined as confined-parallel layers, here, two of its surfaces are glued by respectivecross-layers’ surfaces.The slenderness ratio of medium-scale CLT specimens lies in between 2.9 to 4.8. Thislow slenderness ratio justify the material failure mode of all medium-scale CLT specimenswhich observed in the tests. Then, the study found evidence that the volume fractionof confined- and unconfined-parallel layers make a difference in CLT composite strength.Consequently, an empirical model is proposed for determining the effective strength of CLThaving confined- and unconfined-parallel layers. The maximum variation between the testobservations and analytical model predictions was found to be 7%.63Chapter 3Strength of CLT: NumericalWood is a fibrous orthotropic45 material [BJ93]|. As cross-laminated timber (CLT), madeout of wood, is a relatively new structural material in North American, design provisionsand design properties have been conservatively established that may not utilize the max-imum strength capacity of the materials. The experimental studies presented in Chapter2 have confirmed the validation of this hypothesis. Improvements in this area can beachieved with a comprehensive stress analysis of the material up to its limit state. In thischapter, we describe the implementation of an orthotropic elasticity and rate-independentplasticity model to compute the limit state behaviour of CLT material. The developmentof the constitutive model is described in B. The implementation of the subroutine intoANSYS Mechanical APDL [ANS17] is elaborated in section 3.2. Then, the calibration andverification of the finite element (FE) model are laid out in section 3.3. Finally, the stressanalysis of lamellae is pointed out in section 3.4.3.1 OverviewThe constitutive models developed for the behaviour simulation of simple isotropic mate-rials are not suitable for the analysis of composite materials due to the strong anisotropybehaviour [CH88]|. The representation by a single orthotropic material having propertiesof the whole set has not been satisfactory either [Cha10; Oll+95]|. There exist differentformulations for anisotropic materials presenting a non-linear constitutive response [Hil50;San72]|. These theories are based on threshold functions of discontinuity (yield functions)and anisotropic plastic potentials.In the past, numerous anisotropic criteria for initial yield (e.g. [Men68; SGL02]|) or failure(e.g. [TW71; Hil79; Gib+89; SK13]|) have been developed and extensively tested, in partic-ular for composite materials. However, limited attention was given to the characterizationof strongly anisotropic elastic-plastic behaviour. In this stream, the prediction of ultimatestrength for the mixed-mode tension/shear stress fields has been reported in the litera-tures by fracture mechanics [San03; CH88; Bar81]|. However, difficulties in determining thefracture toughness constants and lack of information available on more complex criteria45In fact, in a natural state, wood is an anisotropic material but in the commercial form, it can beconsidered as orthotropic in the longitudinal, radial and tangential directions.64for three-dimensional mixed-mode stress cases make this method less workable for wood[SLG03; Tal94]|. In addition, fracture mechanics approach also needs the assumption ofexistence of crack and in the case of axial compression its applicability can be questioned.Alternatively, tensor polynomial criteria such as the Tsai-Wu criterion can be used to re-late ultimate strengths in tension, compression and shear to the current state of stress intwo-dimensional models of wood composites [Hel+05; Clo95]|. In three-dimensions, how-ever, the determination of the interaction terms required for this model can be difficult orimpossible if all stress combinations are considered [Yu04]|. A comparatively simple buteffective method, stress based criteria are typically used for wood [PL01; HEM03; Abr08;MP04; Mal11]|.While numerical modelling is essential to conduct stress analysis, the linear elastic or-thotropic model normally used for wood has been proven to be insufficient for the predic-tion of ultimate strength in wood [Bar81; SK13; ML06]|. Therefore, to make the modelnon-linear and keep computational time reasonable, behaviour in compression can be mod-elled with the anisotropic plasticity model in conjunction with elastic orthotropic part.Consequently, in order to compute the ultimate strength of CLT composite, we developan ANSYS UserMat subroutine46, namely, Subroutine for Orthotropic Materials’ Elastic-ity & Rate-independent Plasticity (SOME&RIP), based on [Ree82; Hil47; PL98; SH98;NT73; Met12; DJS17]|which outline an anisotropic finite deformation plasticity model .The model is rate-type and rate-independent47 based on the Green-Naghdi finite plastic-ity theory, and the model is valid at a temperature while neglecting the recovery, creepand thermal phenomena. The development of the constitutive model as a subroutine isdescribed in B.3.2 SOME&RIP implementation: archetype model of FEMTo implement the SOME&RIP, we choose a commercially available finite element method(FEM) program, ANSYS mechanical APDL which implicitly solves the system of equationsinteractively using a Newton-Raphson procedure of a boundary-value problem on a domainΩ as follows [Bat82]|:RAui = −∫Ωeσij∂NA∂xjdv +∫ΩebiNAdv (3.1)and using the corresponding tangents, KABuiuk = −∂RAui∂uBk=∫Ωe∂NA∂xj∂σij∂kl∂NB∂xldv (3.2)46The UserMat subroutine is a user-programmable feature in ANSYS for developing a custom, constitutivemodel to define the stress-strain behaviour of a material to any analysis involving mechanical behaviour.47The elastic-plastic stress-strain relationship is established in its time derivative but the constitutiveparameters are independent of the strain rate65where bi is the externally-applied body force; NA is the shape functions with the indexA = 1, 2, ... denoting the nodes of the element; and uAi the nodal displacements. A typ-ical archetype model of FEM of short CLT member is presented in Figure 3.1. It showsthe mesh, boundary condition and contact surface of a specimen. The model buildingtechniques are described in the following subsection.1XYZ(a) (b)Figure 3.1: A representative archetype model of FEM of the medium-scale specimengroups: (a) meshing, lamella orientation with gap and boundary conditionof a CLT7C specimen, and (b) contact surfaces of a CLT9C specimen3.2.1 Meshing, loading and boundary conditionTo create the most appropriate mesh several attempts were made by changing the meshsize of the elements. The mesh quality was examined by checking the orthogonal quality48and aspect ratio. In the final model, the values of orthogonal quality and aspect ratio ofan element were 0.316667 and 3.157895, respectively, which were within the recommendedvalues by ANSYS guidelines49. On the other hand, a displacement value at each node ofthe top and bottom surfaces are used as the loading and boundary conditions50. Enteringa zero for a component prevents deformation in that direction. On the loading surface,displacement is allowed by a specified value (e.g. 10 mm in the longitudinal direction)whereas, on the boundary surface, displacement was restrained in all directions. Figure 3.1ashows the meshing, loading and boundary conditions of a typical model.48Orthogonal quality is computed for cells using the vector from the cell centroid to each of its faces, thecorresponding face area vector, and the vector from the cell centroid to the centroids of each of the adjacentcells. The worst cells will have an orthogonal quality closer to 0, with the best cells closer to 1.49The recommended value for the orthogonal quality is ≥ 0.05 and aspect ratio is ≤ 20.50Boundary conditions are the known values of the degrees of freedom on the boundary.663.2.2 Glue-line idealizationInitially, glue-lines of CLT panels are idealized as zero length spring51 element with stiffnessof 9800, 4900, 1 × 109N/m in the X, Y and Z direction, respectively. Alternatively, asurface-to-surface contact elements are used for comparing the results. These contactelements use a ‘target surface’ and a ‘contact surface’ to form a contact pair. The targetsurface is modelled with TARGE170 and the contact surface is modelled with CONTA174element. In the model, the target surfaces are considered as the parallel layer’s surfacesand contact surfaces are considered as the cross-layers surfaces due to their relative rigidity.For saving computational effort and avoiding convergence difficulties, zero length springtechnique is disregarded and surface-to-surface contact element is adopted for the analysis.3.2.3 UserMat subroutineThe task of an UserMat subroutine is to calculate the stress σij and material tangent∂σij/∂ij (Equation 3.2) at a given time so that ANSYS Mechanical APDL may performits Newton-Raphson iteration. User-defined materials can be used with the elements thathave a displacement-based element formulation. We used SOLID186 element formulationto implement the SOME&RIP UserMat subroutine. The subroutine is called at everymaterial integration point of the elements during the solution phase. The program passesin stresses, strains, and state variable values at the beginning of the time increment andstrain increment at the current increment, then updates the stresses and state variables tothe appropriate values at the end of the time increment. Input values and the number ofstate variables for UserMat are specified via the TB command as stated in Table 3.1. Itshows 37 parameters are used to define the material properties. Note that, the 6 parametersto define the plastic kinematic (back stress) behaviour does not interfere the model undermonotonic static loading. In addition, 12 parameters to define the failure criterion does notintervene the constitutive model; these parameters act in the post-processing calculation.Therefore, in total 19 parameters effectively contribute to the constitutive model understatic loading scenario. On the other hand, there are 14 state variables are assigned foreach lamella consist of 6 independent variables associated with the symmetric generalizedLagrangian plastic strain tensor EP , the isotropic hardening variable κ, the 6 terms ofkinematic hardening α and the failure index IF .51Zero length spring has properties that the same mass will be in equilibrium at any position. This isequivalent to saying that the restoring force is zero, or the period is infinite.67Table 3.1: Material properties data for SOME&RIP! Units are in N,mm! Elastic orthotropic definition! EX EY EZ NUXY NUYZ NUXZ GXY GYZ GXZ! Failure criteria definition! FXt FXc FYt FYc FZt FZc FXY FYZ FXZ c4 c5 c6! Plastic orthotropic definition! α1 α2 α3 α4 α5 γ1! Plastic kinematic (back stress) definition! h1 h2 h3 h4 h5 j1! Work hardening definition! σ0y H ∆σ βˆ! Material 1 is unconfined-parallel layersTB,USER,1,1,37, ! 37 properties for material 1 (Unconfined-parallel layers)TBTEMP,0 ! 1 temperatureTBDATA, ,12000,400,400,0.347,0.469,0.316TBDATA, ,750,75,750,43.8,27.81,2.949TBDATA, ,3.79,3.763,3.05,8.54,8.54,7TBDATA, ,-1,-1,-1, 0.5,1.5,1.5TBDATA, ,1.5,1.5,0.1,10,1.5,0.0TBDATA, ,0.0,0.0,1.6,31,10,15TBDATA, ,1000TB,STAT,1, ,14 ! 14 state variable for material 1TBDATA, ,0,0,0,0,0,0 ! Initialize the state variable to zeroTBDATA, ,0,0,0,0,0,0TBDATA, ,0,0, , , ,! Material 2 is confined-parallel layersTB,USER,2,1,37, ! 37 properties for material 2 (Confined-parallel layers)TBTEMP,0 ! 1 temperatureTBDATA, ,12000,400,400,0.347,0.469,0.316TBDATA, ,750,75,750,43.8,30.591,2.9667TBDATA, ,3.79,3.7807,3.05,8.54,8.54,7TBDATA, ,-1,-1,-1,0.5,1.5,1.5TBDATA, ,1.5,1.5,0.1,11.1,1.5,0.0TBDATA, ,0.0,0.0,1.6,34.1,100,20TBDATA, ,1000TB,STAT,2, ,14 ! 14 state variable for material 2TBDATA, ,0,0,0,0,0,0 ! Initialize the state variable to zeroTBDATA, ,0,0,0,0,0,0TBDATA, ,0,0, , , ,! Material 3 is cross layerTB,USER,3,1,37, ! 37 properties for material 3 (Cross-layer)TBTEMP,0 ! 1 temperatureTBDATA, ,400,1200,400,0.033,0.347,0.469TBDATA, ,750,75,750,3.0086,3.79,48.6TBDATA, ,30.7,3.6854,3.05,8.54,7,8.54TBDATA, ,-1,-1,-1,0.5,1.5,1.5TBDATA, ,1.5,1.5,0.1,1.5,11.1,0.0TBDATA, ,0.0,0.0,1.6,3.1,10,2TBDATA, ,100TB,STAT,3, ,14 ! 14 state variable for material 3TBDATA, ,0,0,0,0,0,0 ! Initialize the state variable to zeroTBDATA, ,0,0,0,0,0,0TBDATA, ,0,0, , , ,683.2.4 Solution strategyA nonlinear static analysis is performed in ANSYS Mechanical APDL platform. We acti-vated the large deformation feature in the solution strategy by the command NLGEOM,ON.In addition, the load was subdivided into a series of load increments (e.g. 0.1 mm) whichwas applied over several load steps. The program uses the Newton-Raphson approach tosolve nonlinear problems. Before each solution, the Newton-Raphson method evaluatesthe out-of-balance load vector, which is the difference between the restoring forces (theloads corresponding to the element stresses) and the applied loads. The program thenperforms a linear solution, using the out-of-balance loads, and checks for convergence. Ifconvergence criteria are not satisfied, the out-of-balance load vector is reevaluated, thestiffness matrix is updated, and a new solution is obtained. This iterative procedure con-tinues until the problem converges. A number of convergence-enhancement and recoveryfeatures, such as line search, automatic load stepping, and bisection, were activated at atime to help the problem to converge. If convergence was not achieved, then the programattempted to solve with a smaller load increment. In addition, in some nonlinear staticanalyses, using the Newton-Raphson method alone, the tangent stiffness matrix becamesingular (or non-unique), caused severe convergence difficulties. For such situations, analternative iteration scheme, namely, the arc-length method52 was activated to help avoidnon-convergence issues.Results from a nonlinear static analysis consist mainly of displacements, stresses, strains,reaction forces and state variables. The computational results were recorded in an outputfile by the command OUTRES,ALL,ALL whereas, the state variables were stored by thecommand OUTRES,SVAR,ALL. Finally load-history responses were plotted using POST2653,whereas, results for the desired load step and substep were read by POST154.3.3 Model calibration and verificationThe material properties, in particular, compression strength, of the confined- and unconfined-parallel layers are calibrated to match the test results of the CLT3C1 and CLT5C2 group atthe average level. After calibrating the confined- and unconfined-parallel layers’ materialproperties, the rest of the specimen groups were simulated with the calibrated materialproperties. The calibrated material properties are listed in Table 3.1. It shows the com-pression parallel to grain strength of the confined parallel layer is 30.59 MPa which is 10%higher than that of the unconfined parallel layer. The input values of the unconfined paral-lel layers comes from the small-scale test. The values of the elastic moduli kept unchanged52The arc-length method causes the Newton-Raphson equilibrium iterations to converge along an arc,thereby often prevent divergence, even when the slope of the load-deflection curve becomes zero or negative.53POST26 is the time-history postprocessor.54POST1 is the general postprocessor.69for the both layers. However, the nonlinearity of the stress-strain curve differ to each other.It is implemented by defining different work hardening parameters. The model calibrationand verification results are shown in Figure 3.2. Stress and strain are calculated basedon the reaction force over total cross-sectional area and nodal displacements over a gaugelength, respectively. The average results obtained by FEM and tests are listed in Table 3.2.It shows, the average strength of CLT3C, CLT5C, CLT7C and CLT9C specimen groupsby FEM simulations are 25.40 MPa, 21.30 MPa, 20.20 MPa, and 21.80 MPa, respectively,whereas, the MOE of the groups are 7285 MPa, 6500 MPa, 8696 MPa, and 7530 MPa,respectively.(a) (b)(c) (d)A specimen curve from testA specimen curve from testA specimen curve from testA specimen curve from testFigure 3.2: FE model calibration of the medium-scale specimen groups: (a) CLT3C1, (b)CLT5C2, (c) CLT7C, and (d) CLT9C70Table 3.2: Compressive strength and MOE results from test and FEM analysisSpecimenStrength MOETest ModelFEM TestModelFEMTest ModelFEM TestModelFEM[MPa] [MPa] [MPa] [MPa]CLT3C 25.57 25.40 1.01 7418 7285 1.02CLT5C 19.27 21.30 0.90 6310 6500 0.97CLT7C 20.07 20.20 0.99 8282 8696 0.95CLT9C 20.35 21.80 0.93 7533 8663 0.87Considering the same variability measured in tests and following the lognormal distribution,the 5th percentile strength of each group’s FEM results is calculated. Adjusting the 5thpercentile FEM results with the adjustment factors derives in Chapter 2, the adjustedresults are compared with the test and analytical results that were described in Chapter 2.Figure 3.3 and Table 3.3 are showing the comparison between results obtained by differentmethods. The maximum deviation of FEM results from test and analytical results isobtained as 7% and 13%, respectively. In general, FEM results are more conservative thanits test and analytical counterparts. Based on the good agreement between FEM, analytical& test results, a comprehensive stress analysis is presented in the following section.CLT3C CLT5C CLT7C CLT9C0510152025CLT3C CLT5C CLT7C CLT9C00.10.20.30.40.50.60.70.80.911.11.21.311.5(a) (b)Figure 3.3: Compressive strength of the medium-scale CLT specimens from test, analyticaland FEM: (a) actual value, and (b) normalized value with the CLT3C group71Table 3.3: Characteristic strength of medium-scale specimens from test & FEM analysisSpecimenStrength RatioCode Test ModelAnalytical ModelFEM TestModelAnalyticalTestModelFEM[MPa] [MPa] [MPa] [MPa]CLT3C 11.5 16.6 16.6 16.4 1.00 1.01CLT5C 11.5 14.9 14.8 15.2 1.01 0.98CLT7C 11.5 19.3 18.1 17.1 1.07 1.13CLT9C 11.5 19.6 18.8 18.3 1.05 1.073.4 Stress analysisTo represent any arbitrary three-dimensional stress state of an element as a single positivestress value, we articulate von-Mises stress55. Although von-Mises stress is not an accuraterepresentation for wood materials but here we choose it due to its simplicity. The von-Mises stress contour diagrams of all specimen groups at the final solution step are shown inFigure 3.4. The final step resembles the experimental observation as shown in Figure 3.2.Then, the von-Mises stresses of the specimen groups are extracted. Figure 3.4 shows, theconfined-parallel layers have experienced higher stress level than the unconfined-parallellayers. The maximum von-Mises stresses of a parallel layer of CLT3C1, CLT5C2, CLT7Cand CLT9C specimen groups are calculated as 31.6 MPa, 32.5 MPa, 33.2 MPa, and 41.0MPa, respectively.The von-Mises stress of 9-layer CLT are higher than the other counterparts due to higherpresence of the confined-parallel layers’ portion of 33% compared to the unconfined-parallellayers’ portion of 22%. Then, the strength of CLT considering whole cross-section is calcu-lated as von-Mises stress× AeffAtotal . Therefore, the average strengths of the CLT3C1, CLT5C2,CLT7C and CLT9C specimen groups are found as 21.1 MPa, 19.5 MPa, 19.0 MPa, 22.8MPa, respectively.55von-Mises stress is a value used to determine if a given material will reach a failure definition and canbe related to the principal stresses as follows:σvon−Mises =√3 J2 =√[(σ1 − σ2)2 + (σ2 − σ3)2 + (σ3 − σ1)22].721MN MXXYZCLT3 .0503113.554637.0589510.563314.067617.571921.076224.580628.084931.5892NODAL SOLUTIONSTEP=37SUB =3TIME=37SEQV (AVG)DMX =23.1924SMN =.050311SMX =31.58921MNMXXYZCLT5 .3933433.962857.5323511.101914.671418.240921.810425.379928.949432.5189NODAL SOLUTIONSTEP=8SUB =5TIME=7.00048SEQV (AVG)DMX =13.3846SMN =.393343SMX =32.51891MNMXXYZCLT7 .0309883.721147.4112911.101414.791618.481722.171925.86229.552233.2424NODAL SOLUTIONSTEP=22SUB =999999TIME=22SEQV (AVG)DMX =3.47245SMN =.030988SMX =33.24241MNMXXYZCLT9 .8281045.295749.7633714.23118.698623.166327.633932.101536.569241.0368NODAL SOLUTIONSTEP=1SUB =18TIME=5.39756SEQV (AVG)DMX =15.2752SMN =.828104SMX =41.0368(a) (b)(c) (d)Figure 3.4: Von-Mises stress (√3J2) contour diagram of the medium-scale specimens atthe ultimate state: (a) CLT3C1, (b) CLT5C2, (c) CLT7C, and (d) CLT9C3.4.1 Behaviour of lamellaeFigure 3.5-a and -b show the von-Mises stress of a typical parallel- and cross-layers ofCLT9C specimen, respectively. Both confined- and unconfined-parallel layers stresses areshown in the figure. It shows, confined-parallel layers have reached higher stress than theunconfined-parallel layers. On the other hand, in the cross-layers, stress on the surfacewhich is attached to the unconfined-parallel layer’s surface is higher than the stress on thesurface which is attached to the confined-parallel layer’s surface. It is due to the Poisson’seffect of the surrounding lamallae. Stresses in the lamellae are described in the next.731MNMXXYZCLT9 27.989629.455430.921132.386833.852535.318236.783938.249739.715441.1811E-LNODAL SOLUTIONSTEP=1SUB =18TIME=5.39756SEQV (AVG)DMX =15.2632SMN =27.9896SMX =41.18111MNMXXYZCLT9 1.175992.191363.206724.222085.237446.25287.268178.283539.2988910.3143E-LNODAL SOLUTIONSTEP=1SUB =18TIME=5.39756SEQV (AVG)DMX =15.1318SMN =.959142SMX =10.3143(a) (b)Figure 3.5: Von-Mises stress (√3J2) contour diagram of a typical CLT9C specimen withthe maximum and minimum value at the ultimate state: (a) an unconfined-parallel layer (1st sequence) and a confined-parallel layer (7st sequence), and(b) two cross-layers (2nd and 8th sequence)A stress prob is set up at the quarter and half distance along length-wise and width-wiseof a specimen. The location of probes is shown in Figure 3.6a and Figure 3.7a for a typicalCLT9C specimen. Figure 3.6 shows the von-Mises stress along the length at various widthsection of a confined- and unconfined-parallel layers of a typical CLT9C specimen. It showsthe confined-parallel layer’s stress fluctuates at the cross-layers’ gap position. In addition,confined-parallel layer’s stress shows constant across the width of the specimens whereas,unconfined-parallel layer’s stress varies along the width, with the highest values at the edge.In the similar fashion, Figure 3.7 shows the von-Mises stress along the width at variouslength section of a confined- and unconfined-parallel layers of a typical CLT9C specimen. Itshows, confined-parallel layers have higher stresses at zero length. This difference decreasestoward to mid length of the specimen because of Poisson’s effect. At mid length and midwidth, the unconfined-parallel layers have slightly higher stresses compared to the confined-parallel layers. At mid length and free edge, the confined-parallel layers have higher stressescompared to the unconfined-parallel layers.740 100 200 300 400 500 600 700 800 900 100026.52727.52828.52929.5301MNMXXYZCLT9 27.989629.455430.921132.386833.852535.318236.783938.249739.715441.1811E-LNODAL SOLUTIONSTEP=1SUB =18TIME=5.39756SEQV (AVG)DMX =15.2632SMN =27.9896SMX =41.1811************Unconfined parallel layerConfined parallel layer(a) (b)Figure 3.6: Von-Mises stress (√3J2) of a confined- and unconfined-parallel layers of CLT9Cspecimen at the ultimate limit state: (a) location of the probes along the lengthof the specimen at various width section, and (b) stress distribution27.5 28 28.5 29 29.5 30 30.5 31 31.5 32 32.50501001502002501MNMXXYZCLT9 27.989629.455430.921132.386833.852535.318236.783938.249739.715441.1811E-LNODAL SOLUTIONSTEP=1SUB =18TIME=5.39756SEQV (AVG)DMX =15.2632SMN =27.9896SMX =41.1811 **** ****Confined parallel layerUnconfined parallel layer(a) (b)Figure 3.7: Von-Mises stress (√3J2) of a confined- and unconfined-parallel layers of CLT9Cspecimen at the ultimate limit state: (a) location of the probes along the widthof the specimen at various length section, and (b) stress distribution753.4.2 Behaviour of lamellae-interfaceThe confining stress is measured at each contact surface. Figure 3.8a shows the contactstress of a typical CLT5C2 specimen’s contact elements. Figure 3.8b represents the octa-hedral normal stress (hydrostatic stress56) of the specimen. It shows, contact pressure ishigh close to the gap and low at the core. This is because, under high stress level, parallellayers relive stresses (in terms of strain produces from Poisson’s effect) through the gap.As a result, at the edge of cross-layers, stress is tensile whereas as at the core stress iscompressive in nature. This is due to expansion of parallel layers in the in-plane transversedirection.1MNMXXYZCLT5 -2.01246-.949394.1136691.176732.23983.302864.365925.428996.492057.55511NODAL SOLUTIONSTEP=8SUB =5TIME=7.00048CONTPRES (AVG)RSYS=0DMX =13.3846SMN =-2.01246SMX =7.555111MN MXXYZCLT5 -12.5261-10.7284-8.93075-7.13309-5.33544-3.53779-1.74013.0575181.855173.65282NODAL SOLUTIONSTEP=8SUB =5TIME=7.00048NLHPRE (AVG)RSYS=0DMX =13.3846SMN =-12.5261SMX =3.65282(a) (b)Figure 3.8: Nodal solution (average from the contact element’s integration points) at theultimate state of a CLT5C2 specimen: (a) contact stress, and (b) octahedralnormal stress (13I1)3.4.3 Failure modeTo identify the failure location, stress ratio (failure index, IF , Equation B.36) is calculated.Failure is predicted when IF ≥ 1. Failure locations of each specimen groups are presentedin Figure 3.9. Failure modes from experiments are also shown in the inset of each specimengroup. It shows, at failure, the maximum stress ratios of CLT3C, CLT5C, CLT7C, andCLT9C specimen groups are obtained as 1.01, 1.27, 1.05 and 2.04, respectively. In allcases, an unconfined-parallel layer reaches failure level first followed by failure occurs ina confined-parallel layer. As an example, for the 9-layer CLT specimen, when a confined-parallel layer reaches failure index greater than 1, an unconfined-parallel layer reachesalmost double at the same load level.56Hydrostatic stress is the average of the three normal stress components of any stress tensor.761MNMXXYZCLT3 .987863.990412.992961.99551.998061.000611.003161.005711.008261.01081E-LNODAL SOLUTIONSTEP=4SUB =999999TIME=4NLSRAT (AVG)RSYS=0DMX =2.47739SMN =.987842SMX =1.010911MNMXXYZCLT5 .573397.64075.708104.775458.842811.910165.9775191.044871.112231.17958E-LNODAL SOLUTIONSTEP=7SUB =7TIME=7NLSRAT (AVG)RSYS=0DMX =13.3828SMN =.573397SMX =1.179581MNMXXYZCLT7 .723306.759073.79484.830607.866374.902141.937908.9736741.009441.04521E-LNODAL SOLUTIONSTEP=21SUB =2TIME=20.2421NLSRAT (AVG)RSYS=0DMX =2.94544SMN =.719456SMX =1.045211MNMXXYZCLT9 .9836781.105731.227791.349841.47191.593951.716011.838061.960122.08217E-LNODAL SOLUTIONSTEP=1SUB =14TIME=2.80364NLSRAT (AVG)RSYS=0DMX =7.9174SMN =.983678SMX =2.08217(a) (b)(c) (d)Figure 3.9: Stress ratio contour diagram with the maximum and minimum value at theultimate state: (a) an unconfined-parallel layer (1st sequence) of a CLT3C1specimen, (b) an unconfined-parallel layer (1st sequence) and a confined-parallellayer (3rd sequence) of a CLT5C2 specimen, (c) an unconfined-parallel layer (1stsequence) and a confined-parallel layer (5th sequence) of a CLT7C specimen,and (d) an unconfined-parallel layer (1st sequence) and a confined-parallel layer(7th sequence) of a CLT9C specimen3.5 ConclusionIn this chapter, the numerical model development and simulation results of medium-scalespecimens are presented. In order to compute the ultimate strength of CLT composite,we developed a nonlinear material model, namely, Subroutine for Orthotropic Materials’Elasticity & Rate-independent Plasticity (SOME&RIP) and implemented into ANSYS asan UserMat library. We used SOLID186 element formulation to implement the SOME&RIPUserMat subroutine into ANSYS Mechanical APDL simulation platform. The subroutineis called at every material integration point of the elements during the solution phase. The77program passes in stresses, strains, and state variable values at the beginning of the timeincrement and strain increment at the current increment, then updates the stresses andstate variables to the appropriate values at the end of the time increment. In total 19input parameters effectively contribute to the SOME&RIP constitutive model under staticloading scenario. The input material properties of the confined and unconfined parallellayers are calibrated using the test results of the CLT3C1 and CLT5C2 groups.Numerical results are reported in terms of failure modes, displacements, stresses, strains,reaction forces and state variables of each specimen group. The study found that the aver-age strength of CLT3C, CLT5C, CLT7C and CLT9C specimen groups by FEM simulationsare 25.40 MPa, 21.30 MPa, 20.20 MPa, and 21.80 MPa, respectively, whereas, the MOE ofthe groups are 7285 MPa, 6500 MPa, 8696 MPa, and 7530 MPa, respectively. The maxi-mum deviation of FEM results from tests and analytical models that presented in Chapter2 is obtained as 7% and 13%, respectively. In general, FEM results are more conservativethan its test and analytical model counterparts.Stress analysis indicates that in a CLT system, confined-parallel layers reaches higher stressthan the unconfined parallel layer. On the other hand, in cross-layers, stress on the surfacewhich is attached to the unconfined-parallel layer’s surface is higher than the stress onthe surface which is attached to the confined-parallel layer’s surface. Furthermore, contactpressure is high close to the gap and low at the core due to reliving stress by parallel layersthrough the gap under high stress level (in terms of strain produces from Poisson’s effect).In all cases, an unconfined-parallel layer reaches failure level first followed by failure occursin a confined-parallel layer.78Chapter 4Stiffness of CLT: ExperimentalIn this chapter, we describe the experimental setup and stiffness57 results of three types ofmaterials- clear wood (CW), wood with defect (DW), and cross-laminated timber (CLT).The test setup and testing condition, failure modes, and mechanical properties of eachspecimen group58 are presented from section 4.2 to section 4.3. Then, testing data analysisand data diagnosis are described in section 4.4. In order to unify the testing results ofdifferent specimen groups, the adjustment of the experimental results to a specific condi-tion59 and statistical representative properties are laid out in section 4.5. The comparisonof nominal properties with its code’s counterpart is pointed out in section 4.6.4.1 OverviewOur primary objective is to investigate the effects of cross layers on the CLT’s stiffnessbehaviour. The hypothesis being tested is that the cross layers have some contributionstoward the elastic modulus of CLT composite. The modulus of elasticity (MOE) was con-sidered as the independent variable. In the experimental phase, we employed three types oftesting, namely, compression test, flexural test and transverse vibration test. In compres-sion test, the specimens were put under loading until its load carrying capacity reacheda peak and followed by a drop. On the other hand, in flexural testing, the specimenswere tested under a ‘proof’ loading condition so that no damage occur in the specimens.This was done because of the same specimens were used for the stability (buckling) testslater on. Both compression and flexural tests measure the static elastic modulus of CLT.Whereas, CLT’s dynamic elastic modulus were obtained by transverse vibration test.Table 4.1 states the information regarding specimens scale, species identity, testing stan-dards and loading protocol. The naming system and dimension of all specimen groups aredescribed in Chapter 2. The species’ grading information and geometric properties of thespecimens are also listed in Chapter 2.57Stiffness referred to the material’s intrinsic properties in terms of modulus of elasticity.58Clear wood parallel to loading; clear wood perpendicular to loading; wood contains defects; 3-, 5-, 7-,and 9-layer CLT elements.59Adjustment due to moisture content and sample size.79Table 4.1: Overview of the testKeywords DescriptionScaling Small-scale:Clear wood‖ (CW0)Clear wood⊥ (CW90)Wood contains defects (DW)Medium-scale:CLT0−90−0 (CLT3C)CLT0−90−0−90−0 (CLT5C)CLT0−90−0−90−0−90−0 (CLT7C)CLT0−90−0−90−0−90−0−90−0 (CLT9C)Full-scale:CLT0−90−0 (CLT3B)CLT0−90−0−90−0 (CLT5B)Species S-P-F #2, Visual gradingV2M1.1Standards ASTM: D4933-99, D2915-10ASTM: D143-14, D198-15, D4761-13, D4442-15ASTM: D2555-15, D245-06, D1990-14, EN 408-07Loading Displacement controlledMonotonic conservative loadingTransverse vibrationThird-point bending loading4.2 Small-scale and medium-scale specimensThe test setup, failure modes and mechanical properties of small- and medium-scale spec-imen groups are discussed in Chapter 2. The MOE test results of these specimens groupsare listed in Table 4.2. The average value of MOE obtained from the test is compared withthe code specified value. The CSA code tabulated value for the SPF#2 dimension lumberis 9500 MPa. It shows that the minimum and maximum values of MOE are obtained fromtest as of 4670 MPa and 20040 MPa, respectively, which are 49% lower and 208% higherthan the code specified value.Table 4.2: MOE test results of small- and medium-scale specimensEstimations SpecimensCLT3C1 CLT3C2 CLT5C1 CLT5C2 CLT7C CLT9C CW0 DW1 DW2Min [MPa] 5351 5666 4668 5035 6590 6120 6219 5909 7514Max [MPa] 9521 10143 14367 8451 9704 9453 20038 14866 17891Average [MPa] 6961 7875 6377 6244 8282 7533 11000 10513 11843SD [MPa] 930 1292 1801 876 1291 1235 3736 2028 2469COV [%] 13.36 16.40 28.24 14.04 15.59 16.39 33.97 19.29 20.855th P. [MPa] 5432 5750 3415 4803 6158 5502 4854 7177 7781CSA code specified MOE of a SPF#2 dimension lumber is 9500 MPa.804.3 Full-scale: 3- and 5-layer CLT (CLT3B and CLT5B)4.3.1 Test setupCommonly there are two test methods exists for the determination of the elastic modulus[Din00]|. The first method comprising static methods based on the application of a directstress and the measurement of the resultant strain, and the second method comprisingdynamic methods based on resonant vibration from flexural, torsional or ultrasonic pulseexcitation. We employed both methods to evaluate the MOE of CLT elements. Thedynamic test method was performed by [Hor13]|.4.3.1.1 Dynamic modulusThe determination of the dynamic elastic modulus, can be obtained by either longitudinalor flexural vibration. In our study, by means of transverse vibration testing, the dynamicMOE for the whole cross section of the full-scale specimen group was determined. Fig-ure 4.1 illustrates a typical free transverse vibration test setup. The specimen was simplysupported at both ends. To get the dynamic MOE, the beam has to vibrate under theaction of an impulse. Thus, the specimens were tapped in the middle in direction of the mi-nor axis, i.e. in the buckling direction of the CLT members. As part of this measurement,the weight of each specimen was recorded. Then the vibration response was measured asa function of the frequency, and the dynamic elastic modulus, MOED was calculated fromthe specimen dimensions and resonant frequency as follows:MOED =f3W L3Ck I(4.1)where f is vibration frequency (Hz), W is weight of the specimen (N), L is length (m), Iis moment of inertia (m4), and Ck is a calibration constant (often used as a value of 2.01).The results of this test is highlighted in Table 4.3.Figure 4.1: Transverse vibration test setupTable 4.3: Dynamic MOE test resultsEstimations SpecimensCLT3B2 CLT3B3 CLT3B4Min [MPa] 7320 8420 8120Max [MPa] 11670 10620 10970Average [MPa] 8995 9486 9290SD [MPa] 970 569 897COV [%] 10.78 6.00 9.665th percentile [MPa] 7400 8551 7814CSA specified MOE of a SPF#2 sawn lumber is 9500 MPa.814.3.1.2 Static bending modulusThe setup of the static bending MOE test is shown in Figure 4.2. The samples weresimply supported. Each specimen was tested in flatwise (Figure 4.2a, c) and edgewise(Figure 4.2b, d) orientation. The bending test was conducted according to ASTM D198via third-point loading. The third-point loading was adopted to achieve a constant bendingmoment between the two loading points which is more representative of a beam in service.Moreover, third-point loading provides a true modulus of bending60 devoid of shear betweenthe two loading points. We calculated both true and apparent MOE61 of the specimens. Incase of the third-point loading scenario, true MOE can be measured based on the relativedeflection of the loading points and the mid span where the internal shear force is zero.The apparent MOE in this case are based on the relative defection of the mid span andthe support. Here shear force and shear deflections are present in the outer two-thirds ofthe member.The test specimen had a length of 13 (edgewise) and 24 (flatwise) times the depth of thecross-section, and was symmetrically loaded at two points over a span of 1800 mm; thedistance between the two loading points was 600 mm (Figure 4.2). A yoke deflectometerwas used to measure deflection at the centre of the beam with respect to two points alongthe neutral axis. The gauge length of 600 mm and 1800 mm were used for calculating trueand apparent-MOE, respectively. The yoke consisted of a lightweight material suspendedbetween nails driven into the beam at its neural axis. Then, a deflection transducer attachedto the centre of the yoke with its stem attached to a nail driven into the beam at midspanat the neural axis (Figure 4.2).The threshold value of the applied load was set to 10 kN. The magnitude of the appliedload was <1% of the predicted ultimate capacity of the beam. The low threshold valuewas chosen such that load-deflection trend follows linear while no damage occurred in thespecimen. Keeping in mind MOE is sensitive to rate of loading, load was applied at aconstant loading-head movement so adjusted that maximum load is reached within (300 ±120)s. Load and deflection were continuously monitored until loading stop. Static bendingMOE was calculated for each specimen using their corresponding load–deflection diagramaccording to the following equation:MOEA =23P L3108 b h3 ∆L; MOET =P LL26004 b h3 ∆L600, (4.2)where MOEA (MPa) and MOET (MPa) are the apparent and true modulus of elasticity,respectively, L (mm) is span of beam, L600 (mm) is length of beam that is used to measure60The true MOE, a term used in the wood engineering community, also known as shear-free modulus, isthe MOE that contributes to the bending component of the total deflection.61The apparent MOE, is a hypothetical MOE used for simplicity and practicality due to having smallshear deflection under design loads, as compared to the bending deflection for prismatic bending membersand the vagueness to calculate shear rigidity of engineered wood products.82deflection between two load points, that is, shear-free deflection, P (N) is increment ofapplied load below proportional limit, b (mm) is width of beam, h (mm) is depth of beam,∆L (mm) is increment of deflection of beam’s neutral axis measured at midspan overdistance L and corresponding load P , and ∆L600 (mm) is increment of deflection of beam’sneutral axis measured at midspan over distance Lb and corresponding load P .(a) CLT5BET (b) CLT3B1FA(c) CLT5BEA(d) CLT5BFA600 mm 300 mm 300 mm 600 mm600 mm 300 mm 300 mm 600 mmFigure 4.2: A representative test setup of a specimen under third-point loading: (a) atypical CLT5BET specimen, (b) a typical CLT3B1FA specimen, (c) renderingof a CLT5BEA specimen, and (d) rendering of a CLT5BFA specimen834.3.2 Mechanical propertiesThe test result in terms of load-displacement curve is shown in Figure 4.3-a and -b forthe CLT3B1 and CLT5B specimen groups, respectively. It includes the results from theflat-wise and edge-wise specimens. Moreover, the true and apparent displacement areplotted. Under a load level, the edge-wise specimen has lower displacement than the flat-wise specimen due to higher flexural rigidity (EI). The descriptive statistics of this testresults are highlighted in Table 4.4. It shows the average true MOE values are higher thanthe apparent MOE values of the specimen groups.0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5012345678910(a) CLT3B1 (b) CLT5BAslopethroughaveragecurveA specimen curveff0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2012345678910A slope through average curveHHYA specimen curveffFigure 4.3: Test results of the specimen groups: (a) CLT3B1, and (b) CLT5BTable 4.4: Static MOE test results of full-scale specimensEstimations SpecimensCLT3B1EA CLT3B1ET CLT3B1FA CLT3B1FT CLT5BEA CLT5BET CLT5BFA CLT5BFTMin [MPa] 5637 5002 7819 8274 4860 4885 5502 6336Max [MPa] 8444 10124 11673 12368 8827 9767 7130 9561Average [MPa] 7111 7578 9858 10370 5849 6405 6257 7698SD [MPa] 932 1260 1129 1334 828 971 422 848COV [%] 13.11 16.63 11.46 12.86 14.16 15.17 6.75 11.025th P. [MPa] 5578 5505 8000 8175 4486 4807 5562 6303CSA code specified MOE of a SPF#2 dimension lumber is 9500 MPa.4.4 Testing data analysisThe descriptive statistics of the elastic modulus of the specimens are stated in Table 4.2,Table 4.3, and Table 4.4. The results of all specimens are also stated in A. The follow-ing subsection examines the test data quality whether it was sufficient to represent thecharacteristic properties of each specimen group.844.4.1 Sampling qualityIn order to compare between different groups, the test results from all groups are standard-ized according to procedure described in Chapter 2. Sphere with the 95% probability anddata points of the standardized strength, stiffness (MOE) and density parameters (univari-ate Gaussian distribution case) of all samples are also shown in Chapter 2. It shows, datafor all small-, medium-, and full-scale specimen groups are sufficient to measure the me-chanical properties. The variance of data and the intensity of correlation among mechanicalproperties are also presented in Chapter 2. For instance, for the CLT3C1 specimen group,it shows that strength is correlated with MOE by 0.78; whereas density is correlated withstrength and MOE by -0.31 and -0.24, respectively. The information regarding correlationis used in the reliability analysis later on.4.4.2 Central tendency and variabilityThe central tendency and data variability of MOE is measured by plotting five plots,namely, run sequence, lag, histogram, normal probability and box-and-whisker plot. Forexample, these plots are shown in Figure 4.4 for the CLT3B2D specimen group. It shows,the run sequence is random and good sparse of data on the lag plot. Moreover, thehistogram and normal probability plot ensure the data is normally distributed. The boxplot indicates the data sparsity and skewness of mean value.0 10 208000850090009500100001050011000115008000 9000 10000110008000850090009500100001050011000115008000 9000 100001100000.050.10.158000 9000 1000011000-2-1.5-1-0.500.511.52800085009000950010000105001100011500Figure 4.4: Representation of measuring the central tendency and data variability of thestiffness parameter of the CLT3B2D group854.4.3 Distribution functionThe probability distribution types and parameters of the specimen groups are then sought;Gaussian distribution seems most appropriate distribution based on the goodness-of-fitscores. Knowing the best fitted distribution, along with specimen data, Figure 4.5a rep-resent the smooth CDF and PDF curves of the CLT3B2D specimen group considering theGaussian probability distribution. All specimens’ distribution curves are drawn in Fig-ure 4.5b considering the Gaussian distribution. It includes the MOE in the parallel andperpendicular to grain direction as well. It shows the MOE of the CW0 and CLT5BFAspecimen groups have the highest and lowest variability, respectively. The adjustment ofthe test results to a specific condition is described in the subsequent section.7000 8000 9000 10000 11000 12000 13000 14000 1500000.10.20.30.40.50.60.70.80.91(a) (b)0 2000 4000 6000 8000 10000 12000 14000 16000 18000 2000000.10.20.30.40.50.60.70.80.91Figure 4.5: Probability density function and cumulative distribution function of the stiff-ness parameter: (a) CLT3B2D group, and (b) all specimen groups4.5 MOE results of the specimens4.5.1 Test measurementThe MOE results of each group are laid out in Figure 4.6 by box plot with whisker at± 1.5 of interquartile range and outliers. The jitter of specimen results of each groupare also shown by black dots in the figure respectively. Moreover, descriptive statisticsof the adjusted MOE of the specimen groups are presented in Table 4.5 and Table 4.6.It shows, CW and DW group have higher variability than their counterparts. The MOEresults shown here is not adjusted with a standard size and moisture content (MC) of thespecimens.86lllll llllllll lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllll lllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll lllllllll l l lllllllllllllllll llllllllllllllllllllllllllllllllllllll lll lllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllll ll l llll ll ll l lllll ll llllllllllllllllllllllllllll lllllllll05000100001500020000CLT3B2DCLT3B3DCLT3B4DCLT3B1EACLT3B1ETCLT3B1FACLT3B1FTCLT3C1CLT3C2CLT5BEACLT5BETCLT5BFACLT5BFTCLT5C1CLT5C2CLT7CCLT9CCW0CW90DW1DW2SpecimenMOE (MPa)s Outlier· Specimen resultFigure 4.6: Stiffness results of all specimen groups without adjustmentTable 4.5: Statistical data of MOE of small- & medium-scale specimens without adjustmentEstimations SpecimensCLT3C1 CLT3C2 CLT5C1 CLT5C2 CLT7C CLT9C CW0 DW1 DW2Min [MPa] 5351 5666 4668 5035 6590 6120 6219 5909 7514Max [MPa] 9521 10143 14367 8451 9704 9453 20038 14866 17891Average [MPa] 6961 7875 6377 6244 8282 7533 11000 10513 11843SD [MPa] 930 1292 1801 876 1291 1235 3736 2028 2469COV [%] 13.36 16.40 28.24 14.04 15.59 16.39 33.97 19.29 20.855th P. [MPa] 5432 5750 3415 4803 6158 5502 4854 7177 7781CSA code specified MOE of a SPF#2 dimension lumber is 9500 MPa.Table 4.6: Statistical data of MOE of full-scale specimens without adjustmentEstimations SpecimensCLT3B1EA CLT3B1ET CLT3B1FA CLT3B1FT CLT3B2D CLT3B3D CLT3B4D CLT5BEA CLT5BET CLT5BFA CLT5BFTMin [MPa] 5637 5002 7819 8274 7320 8420 8120 4860 4885 5502 6336Max [MPa] 8444 10124 11673 12368 11670 10620 10970 8827 9767 7130 9561Average [MPa] 7111 7578 9858 10370 8995 9486 9290 5849 6405 6257 7698SD [MPa] 932 1260 1129 1334 970 569 897 828 971 422 848COV [%] 13.11 16.63 11.46 12.86 10.78 6.00 9.66 14.16 15.17 6.75 11.025th P. [MPa] 5578 5505 8000 8175 7400 8551 7814 4486 4807 5562 6303CSA code specified MOE of a SPF#2 dimension lumber is 9500 MPa.874.5.2 Unifying test measurementIn order to compare the results of the different groups, MOE is calculated based on theeffective cross-sectional area. Therefore, at this level, CLT’s MOE is comparable to thethat of CW and DW groups. The effective MOE results of each group are plotted inFigure 4.7. Moreover, descriptive statistics of the adjusted MOE of the specimen groupsare presented in Table 4.7 and Table 4.8. It shows, CLT9C and CLT5BFA groups havethe highest and lowest MOE values, respectively. The variability of MOE values betweenthe groups is due to data have yet been adjusted to a standard sample size and moisturecontent.llllllllll llllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllll lllllll llllllllllllllllllll llllllllllllllllllllllllllllllllllllllll llllll llllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllllllll lllll lllllllllllllllllllllllllllllllll lllllllllllllllll lllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllllllll ll ll ll l ll l llllllllllllllllllllllllllllllllllllllllll0500010000150002000025000CLT3B2DCLT3B3DCLT3B4DCLT3B1EACLT3B1ETCLT3B1FACLT3B1FTCLT3C1CLT3C2CLT5BEACLT5BETCLT5BFACLT5BFTCLT5C1CLT5C2CLT7CCLT9CCW0CW90DW1DW2SpecimenMOEeff (MPa)s Outlier· Specimen resultFigure 4.7: Effective stiffness results of all specimen groups without adjustmentTable 4.7: Statistical data of the effective MOE of small- and medium-scale specimenswithout adjustmentEstimations SpecimensCLT3C1 CLT3C2 CLT5C1 CLT5C2 CLT7C CLT9C CW0 DW1 DW2Min [MPa] 8278 8499 7780 8392 12435 11015 6219 5909 7514Max [MPa] 14728 15214 23944 14085 17005 17015 20038 14866 17891Average [MPa] 10768 11813 10628 10462 14875 13559 11000 10513 11843SD [MPa] 1438 1938 3001 1510 1945 2223 3736 2028 2469COV [%] 13.36 16.40 28.24 14.44 13.07 16.39 33.97 19.29 20.855th P. [MPa] 8403 8626 5691 7978 11676 9903 4854 7177 7781CSA code specified MOE of a SPF#2 dimension lumber is 9500 MPa.88Table 4.8: Statistical data of the effective MOE of full-scale specimens without adjustmentEstimations SpecimensCLT3B1EA CLT3B1ET CLT3B1FA CLT3B1FT CLT3B2D CLT3B3D CLT3B4D CLT5BEA CLT5BET CLT5BFA CLT5BFTMin [MPa] 8456 7503 8094 8602 7647 8796 8482 8101 8141 6614 7684Max [MPa] 12666 15185 12283 13014 12191 11094 11459 14711 16278 8542 11479Average [MPa] 10667 11367 10294 10829 9396 9910 9705 9748 10674 7513 9244SD [MPa] 1398 1890 1177 1389 1013 594 937 1380 1619 489 1005COV [%] 13.11 16.63 11.44 12.83 10.78 6.00 9.66 14.16 15.17 6.51 10.875th P. [MPa] 8367 8258 8358 8544 7730 8933 8163 7477 8011 6709 7590CSA code specified MOE of a SPF#2 dimension lumber is 9500 MPa.4.5.3 Adjusting test measurementsThe test results is adjusted with sample size and moisture content. It is to be noted thatthe MOE values do not need to be adjusted with duration-of-load. During adjustment itis followed the specifications from ASTM standards and CSA codes.First, data quantity for CLT7C and CLT9C specimen groups were adjusted. These twogroups contain 5 members whereas others groups consist of more than 20 specimens. Be-cause of unknowing the distribution parameters, we simulated data for these two groupsby performing Bayesian analysis62 in R 3.5.2 environment [R C18]. In a nutshell, we hadto estimate a population proportion, and we had a prior distribution representing our be-liefs about the value of that proportion. We also calculated the likelihood function for theproportion given the data. Thus, based on the collected data, we updated the prior dis-tribution for the population proportion. That is, our goal was to calculate the conditionaldistribution of the proportion given the data and the prior following Bayes’ rule as follows:P (θ|D) = P (D|θ)P (θ)P (D) (4.3)where θ = parameters = {mean, SD} and D = data. For Bayesian inference we getP (θ|D) ∝ P (D|θ)P (θ)⇔ posterior ∝ likelihood× prior (4.4)We performed inference on θ using the joint posterior P (θ|D). One way to get informedabout joint posterior is if we had a large enough sample from P (θ|D), then, the distri-bution of the sample should have approximated P (θ|D). A computer algorithm, namely,Gibbs sampling was used to sample observations from the joint posterior. The algorithmrequired to know the conditional posterior distribution of each parameters in the model(e.g. P (mean | data, SD) and P (SD |Data, mean)). The input parameters of the algo-rithm were taken from the average test results of 3- and 5-layer CLT specimen groups. Thetraceplot of simulated MOE values and histograms of each run are plotted in Figure 4.8.62Bayesian analysis is a statistical paradigm that quantifies unknown parameters using probability state-ments.89(c) (d)(a) (b)Figure 4.8: Traceplot of the simulated MOE values by Bayesian analysis using Markovchain Monte Carlo sampling technique for the CLT7C group: (a) mean value,(b) standard deviation value, (c) probability density of mean, and (d) proba-bility density of standard deviationFollowing the above procedure, the stiffness for CLT7C and CLT9C groups are adjustedwith Bayesian update statistics and is presented in Figure 4.9. Moreover, descriptivestatistics of the adjusted MOE of the specimen groups are presented in Table 4.9. It isshown that after adjustment, the average stiffness of CLT7C and CLT9C are decreased.The reason is that the assumed prior distribution of MOE was lower than the likelihooddistribution of MOE of CLT7C and CLT9C specimen groups.llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllll lll lllllllllllllllllllllllllllllllllllll llllll lllllllllllllllll lllllllllllllllllllllllllllllllllllllllllll llllllll lllllllllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllll ll llll ll llllll l lll ll lllllllllllllllllllllllllllll lllllllll0500010000150002000025000CLT3B2DCLT3B3DCLT3B4DCLT3B1EACLT3B1ETCLT3B1FACLT3B1FTCLT3C1CLT3C2CLT5BEACLT5BETCLT5BFACLT5BFTCLT5C1CLT5C2CLT7CCLT9CCW0CW90DW1DW2SpecimenMOEeff (MPa)— Code specified MOEr Outlier· Specimen resultFigure 4.9: Effective stiffness results of all specimen groups adjusted with sample number90Table 4.9: Statistical data of the effective MOE of small- and medium-scale specimensadjusted with sample numberEstimations SpecimensCLT3C1 CLT3C2 CLT5C1 CLT5C2 CLT7C CLT9C CW0 DW1 DW2Min [MPa] 8278 8499 7780 8392 9593 6360 6219 5909 7514Max [MPa] 14728 15214 23944 14085 17271 15697 20038 14866 17891Average [MPa] 10768 11813 10628 10372 11763 10404 11000 10513 11843SD [MPa] 1438 1938 3001 1457 1561 1816 3736 2028 2469COV [%] 13.36 16.40 28.24 14.05 13.27 17.45 33.97 19.29 20.855th P. [MPa] 8403 8626 5691 7975 9195 7417 4854 7177 7781CSA code specified MOE of a SPF#2 dimension lumber is 9500 MPa.Then, the test data is adjusted with the moisture content (MC). The test data for MC ofeach group is presented in Chapter 2. The MC adjustment for MOE is calculated fromEquation 4.5 according to the ASTM standard. The adjustment equations are valid for MCvalues between 10 and 23%. We set our target MC as 15%. The graphical representationof this adjustment factor is depicted in Figure 4.10. It exhibits that the adjustment factorfor MOE does not vary with the MOE value.MOEnew = MOEold(1.857− (0.0237×MCnew))(1.857− (0.0237×MCold)) (4.5)where MOE in psi and MC in %.10152025303540455010 11 12 13 14 15 16 17 18 19 20 21 22 23MC (%)0.880.90.920.940.960.9811.021.041.061.08fMCFigure 4.10: Moisture content adjustment factor in the ASTM standard for the stiffnessparameter91With the factor derived earlier, the test data for MOE is adjusted with the moisture contentat 15% and presented in Figure 4.11. Moreover, descriptive statistics of the adjusted MOEof the specimen groups are presented in Table 4.10 and Table 4.11. For all group ofspecimens, the experimental mean MOE is approximately same as the code value.lllllllllll ll llllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllllllll lllllll ll llllllllllllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllll llllllllllllllllllllllllllllll llllll lll lllllllllllll llllllllllllll llllllllllllll lllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllll ll ll lll l ll lll lll lll ll llllllllllllllllllllllllllllllllll0500010000150002000025000CLT3B2DCLT3B3DCLT3B4DCLT3B1EACLT3B1ETCLT3B1FACLT3B1FTCLT3C1CLT3C2CLT5BEACLT5BETCLT5BFACLT5BFTCLT5C1CLT5C2CLT7CCLT9CCW0CW90DW1DW2SpecimenMOEeff adj (MPa)— Code specified MOEs Outlier· Specimen resultFigure 4.11: Effective stiffness results of all specimen groups adjusted with sample numberand moisture contentTable 4.10: Statistical data of the effective MOE of small- and medium-scale specimensadjusted with sample number and moisture contentEstimations SpecimensCLT3C1 CLT3C2 CLT5C1 CLT5C2 CLT7C CLT9C CW0 DW1 DW2Min [MPa] 7798 7778 7439 7805 9157 6003 6077 5314 6715Max [MPa] 13978 13914 23536 13015 16486 14820 19319 13402 16120Average [MPa] 10195 10791 10232 9591 11166 9839 10781 9457 10586SD [MPa] 1386 1793 3006 1332 1486 1728 3604 1822 2228COV [%] 13.59 16.61 29.38 13.89 13.30 17.56 33.43 19.27 21.055th P. [MPa] 7916 7842 5288 7400 8722 6997 4853 6460 6921CSA code specified MOE of a SPF#2 dimension lumber is 9500 MPa.92Table 4.11: Statistical data of the effective MOE of full-scale specimens adjusted withsample number and moisture contentEstimations SpecimensCLT3B1EA CLT3B1ET CLT3B1FA CLT3B1FT CLT3B2D CLT3B3D CLT3B4D CLT5BEA CLT5BET CLT5BFA CLT5BFTMin [MPa] 7687 6844 7402 7840 7246 8273 8136 7438 7528 6101 7125Max [MPa] 11631 13842 11212 11866 11484 10497 10925 13562 15036 7996 10677Average [MPa] 9746 10384 9408 9879 8918 9467 9361 9019 9879 6952 8569SD [MPa] 1294 1730 1089 1266 971 568 824 1273 1483 474 948COV [%] 13.28 16.66 11.58 12.81 10.89 6.00 8.80 14.12 15.01 6.82 11.065th P. [MPa] 7617 7539 7616 7797 7321 8533 8006 6925 7439 6172 7010CSA code specified MOE of a SPF#2 dimension lumber is 9500 MPa.4.6 CLT characteristic stiffness and the CSA code stiffnessThe adjusted MOE of all specimen groups and its code counterpart are shown in Fig-ure 4.12. The descriptive statistics of the adjusted MOE of the specimen groups arepresented in Table 4.10 and Table 4.11. Figure 4.13 depicts the normalized characteristicMOE value of all specimen groups with the code specified value. Moreover, Table 4.12lists the characteristic MOE results of all specimen groups. It shows, true MOE is likely tobe higher than the apparent MOE. For the 3-layer CLT specimens, the characteristic-trueMOE is 3.6 and 4.1 times higher than the characteristic-apparent MOE for the edge-wiseand flat wise configurations, respectively. Likewise, for the 5-layer CLT specimens, thecharacteristic true-MOE is 2.3 and 2.8 times higher than the characteristic-apparent MOEfor the edge-wise and flat wise configurations, respectively. For both 3- and 5-layer CLT, ithas found that the edge-wise MOE is higher than the flat-wise MOE. In addition, resultsshow that dynamic MOE is lower than the static MOE. The average characteristic-dynamicand -static MOE of 3-layer full-scale CLT specimens are obtained as of 9250 MPa and 9850MPa, respectively. The characteristic MOE of medium-scale specimens are higher thanthat of full-scale specimens. For 3-layer medium- and full-scale CLT specimens, the aver-age characteristic MOE are determined as of 10500 MPa, and 9600 MPa, respectively.Results record that the average characteristic MOE of small-, medium, and full-scale spec-imen groups are 10280 MPa, 10300 MPa, and 9300 MPa, respectively. In other words, theaverage characteristic MOE of small-, medium, and full-scale specimen groups are +8.2%,+8.4%, and -2.1% than the code specified value. For full-scale specimens, if we disregardcharacteristic-apparent MOE values due to account shear deformation, the average charac-teristic MOE is obtained as 9500 MPa which is the same value as the CSA code specifiedMOE value. Hence, it can be said that stiffness (MOE) of CLT specimens shows indifferentvalue, even if changing the types of test and size of the specimens.93lllll llllllll llllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllll lllllllllllllllllllllllll ll llllllllllllllllllllllllll lllllllllllll llllllllllllllll lllllllllllllllllllllllll lllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllllllll llll lllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllll llllllllllllll lllllllllllllllllllllllllllllllllll llllllllllllll lllllllllllllllllll ll ll lll llll ll l lll lllllllllllllllllllllllllllllllllllll0500010000150002000025000CLT3B2DCLT3B3DCLT3B4DCLT3B1EACLT3B1ETCLT3B1FACLT3B1FTCLT3C1CLT3C2CLT5BEACLT5BETCLT5BFACLT5BFTCLT5C1CLT5C2CLT7CCLT9CCW0CW90DW1DW2SpecimenMOEeff adj (MPa)— Code specified MOEs Outlier· Specimen resultFigure 4.12: Adjusted stiffness results of all specimen groupsCLT3C 1CLT3C 2CLT5C 1CLT5C 2CLT7CCLT9CCW0DW1DW2CLT3B 1EACLT3B 1ETCLT3B 1FACLT3B 1FTCLT3B 2CLT3B 3CLT3B 4CLT5B EACLT5B ETCLT5B FACLT5B FT-30-25-20-15-10-505101520Figure 4.13: Characteristic MOE compared with the code specified value94Table 4.12: MOE values compared to the CSA codeScale Specimen Characteristic MOE valueActual [MPa] Code [MPa] MOEactual−MOEcodeMOEcode × 100 [%]MediumCLT3C1 10195 9500 7.32CLT3C2 10791 9500 13.59CLT5C1 10232 9500 7.70CLT5C2 9591 9500 0.95CLT7C 11166 9500 17.53CLT9C 9839 9500 3.57SmallCW0 10781 9500 13.49DW1 9457 9500 -0.45DW2 10586 9500 11.43FullCLT3B1EA 9746 9500 2.59CLT3B1ET 10384 9500 9.31CLT3B1FA 9408 9500 -0.97CLT3B1FT 9879 9500 3.99CLT3B2 8918 9500 -6.12CLT3B3 9467 9500 -0.34CLT3B4 9361 9500 -1.47CLT5BEA 9019 9500 -5.06CLT5BET 9879 9500 3.99CLT5BFA 6952 9500 -26.82CLT5BFT 8569 9500 -9.804.7 ConclusionIn this chapter, the experimental setup and MOE results of small-, medium- and full-scalespecimens are presented. In the experimental phase, we employed three types of testing,namely, compression test, flexural test, and transverse vibration test. Compression testswere conducted on the small- and medium-scale specimens, whereas, flexural and trans-verse vibration tests were performed on the full-scale specimens.Test results are reported in terms of static and dynamic MOE of each specimen group. Dur-ing flexural test of the full-scale specimens, we calculated both true and apparent MOE.The investigation on the sampling quality and quantity, confidence interval, and correlationof MOE with other properties are elaborated in Chapter 2. The central tendency, datavariability, and probability distribution function of stiffness (MOE) parameter of specimensgroups are then determined. The MOE test results are adjusted with the moisture contentand sample size.95The study found that the average characteristic MOE of small-, medium, and full-scalespecimen groups are +8.2%, +8.4%, and -2.1% than the code specified value. The char-acteristic MOE of medium-scale specimens are higher than that of full-scale specimens.Moreover, the true MOE is likely to be higher than the apparent MOE. For both 3- and5-layer CLT, it was found that the edge-wise MOE was higher than the flat-wise MOE.In addition, results show that dynamic MOE is lower than the static MOE, for example,the ratio of average dynamic- and static-characteristic MOE of 3-layer full-scale CLT spec-imens is found to be 0.94. For full-scale specimens, if we disregard characteristic-apparentMOE values due to account shear deformation, the average characteristic MOE is obtainedas of 9500 MPa which is the same value as the CSA code specified MOE value.96Chapter 5Stiffness of CLT: NumericalIn this chapter, we describe the formulation of an archetype model of finite element method(FEM) to compute the modulus of elasticity (MOE) of cross-laminated timber (CLT)material. The development of the finite element (FE) model in the ANSYS MechanicalAPDL is described in section 5.2. Then, the calibration and verification of the FE model arelaid out in section 5.3. Finally, the MOE simulation results of the medium- and full-scaleCLT specimens are presented in section 5.4.5.1 OverviewThe stiffness of the material in terms of MOE is determined by the FEM analysis ofmedium- and full-scale CLT specimen groups. The experimental counterpart of thesespecimens groups are described in Chapters 2 and 4. The hypothesis being tested is that,similar to the experimental observations, whether the numerically computed MOE showsindifferent value, even if changing the testing type and scale of the specimens. In this con-text, the FE model to obtain the MOE of medium-scale specimens under axial compressionload is described in Chapter 3. A FE model of full-scale 3- and 5-layer CLT under third-point bending loading configuration is described in this Chapter. The same specimen isloaded in flat-wise and edge-wise orientation to examine the directional sensitivity of MOE.A group of 28 specimens in each group having different material properties are analyzed.The archetype model of FEM and results are presented in the following sections.5.2 Stiffness: archetype model of FEMMedium-scale 3-, 5-, 7-, and 9-layer CLT members are modelled in ANSYS Mechani-cal APDL platform [ANS17] incorporated with an UserMat material model, named afterSOME&RIP63, where material nonlinearity is explicitly considered (Chapter 3). For mod-elling the initial stiffness of the material under load where finding the MOE is of interest,material non-linearity is disregarded in the full-scale CLT’s FE model. The FE model de-velopment of medium- and full-scale specimens are described in the following subsection.5.2.1 Medium-scale: CLT3C, CLT5C, CLT7C and CLT9CDescribed in Chapter 3.63Subroutine for Orthotropic Materials Elasticity & Rate Independent Plasticity.975.2.2 Full-scale: CLT3B and CLT5BAn archetype model of FEM is developed in the ANSYS Mechanical APDL simulationenvironment for the static nonlinear analysis of CLT full-scale specimens. The FE modelsof a typical 3- and 5-layer CLT specimens are showing in Figure 5.1. It shows the elementmesh, boundary and loading conditions of each specimen. The mesh quality was examinedby whether the orthogonal quality and aspect ratio of an element cell within the recom-mended values by ANSYS guidelines without compromising the computational accuracy.Orthotropic elasticity has been considered as the material behaviour. Table 5.1 shows aset of material properties for the parallel- and cross-layers. It shows 9 parameters are usedto define the material properties; among them, 1 property, namely, elastic constant in thelongitudinal direction is considered as a random variable with the Gaussian distributionhaving mean and COV value of 10900 MPa and 28%, respectively. The other proper-ties used in the model are obtained from different published literature [CSA16; Hor13;LLL16; MP04; LO18]|. The distribution parameters of the property are calibrated withCLT3B1 and CLT5B specimen groups so that the displacements obtained from FEM anal-ysis matched with the recorded displacements from tests. Using the calibrated propertiesand the distribution parameters, 28 samples of each specimen group are simulated underapplied load.Table 5.1: Material properties of full-scale specimensLayerPropertiesEX? [MPa] EY EZ GXY GYZ GXZ PRXY PRYZ PRXZMean COV [MPa] [MPa] [MPa] [MPa] [MPa]Parallel (‖) 10900 0.28 EX/30 EX/30 EX/16 EX/160 EX/16 0.42 0.35 0.37Cross (⊥) EY‖ EX‖ EZ‖ GXY‖ GXZ‖ GYZ‖ 0.033 0.347 0.469? Gaussian distributionWe used SOLID186 element formulation to define the parallel- and cross-layers lamellae.A surface-to-surface contact definition is used for the glue line idealization; in particu-lar, the target surface (parallel layer’s surface) is modelled as TARGE170 and the contactsurface (cross layer’s surface) is modelled with CONTA174 element. Third point edgewiseand flatwise loading configuration of a simply supported beam is used as the loading andboundary conditions. Loading was applied as specified structural surface loads on the se-lected elements, whereas, roller support idealization is imposed on the selected elementsfor the boundary condition (Figure 5.1). We activated the large deformation feature in thesolution strategy. In addition, the load was subdivided into a series of load increments (e.g.100 N) which was applied over several load steps. The loading magnitude is selected sothat beam deflection remains linear elastic. Results in terms of displacements and reactionforces are stored as the output variables.981XYZCLT3B_E V-EU1XYZCLT3B_F V-EU1XYZCLT5B_E V-EU1XYZCLT5B_F V-EU(a) (b)(c) (d)Figure 5.1: A typical archetype model of FEM with the mesh, elements, loading &boundary condition of the CLT3B1EA, CLT3B1ET , CLT3B1FA, CLT3B1FT ,CLT5BEA, CLT5BET , CLT5BFA, CLT5BFT , CLT3B2, CLT3B3, and CLT3B4specimen groups: (a) CLT3B1EA, (b) CLT3B1FA, (c) CLT5BEA, and (d)CLT5BFA5.3 Model calibration and verificationThe calibration and verification of FE models of CLT specimens are described in this sec-tion. In this context, the medium-scale CLT specimens having 3-, 5-, 7-, and 9-layer underaxial compression loads are pointed out in Chapter 3, whereas, full-scale CLT specimensunder bending loading is described in the following subsection.5.3.1 Medium-scale: CLT3C, CLT5C, CLT7C and CLT9CDescribed in Chapter 3.995.3.2 Full-scale: CLT3B and CLT5BIn the FE models, the material properties are calibrated for the 3- and 5-layer CLT specimenso that displacements from the FEM simulations and test results are matched with eachother. The calibrated material properties data are shown in Table 5.1. The deformedshape of a typical 3- and 5-layer CLT specimens are showing in Figure 5.2. The deformedshape from tests are presented in the inset of the figures accordingly. Table 5.2 shows thedisplacements values subjected to 2 kN and 5 kN loading from test and FEM analysis. Itshows the maximum deviation between test and FEM results is 9%. Therefore, the modelcan predict the test observation reasonably. Based on this agreement, in order to verifythe model, FEM analysis was performed for the CLT3B2, CLT3B3 and CLT3B4 specimengroups. The results are discussed in the following section.1MNMXXYZCLT3B_E -4.69647-3.99651-3.29656-2.59661-1.89665-1.1967-.496747.203206.9031591.60311NODAL SOLUTIONSTEP=10SUB =1TIME=10UY (AVG)RSYS=0DMX =7.16419SMN =-4.69647SMX =1.603111MNMXXYZCLT3B_F -7.86708-6.93945-6.01183-5.08421-4.15658-3.22896-2.30133-1.37371-.446083.481542NODAL SOLUTIONSTEP=3SUB =1TIME=3UZ (AVG)RSYS=0DMX =7.87871SMN =-7.86708SMX =.4815421MNMXXYZCLT5B_E -2.93542-2.54745-2.15948-1.77151-1.38354-.995571-.6076-.21963.16834.55631NODAL SOLUTIONSTEP=3SUB =2TIME=3UY (AVG)RSYS=0DMX =4.6413SMN =-2.93542SMX =.556311MNMXXYZCLT5B_F -5.1429-4.54249-3.94209-3.34168-2.74128-2.14088-1.54047-.940068-.339664.260739NODAL SOLUTIONSTEP=5SUB =1TIME=5UZ (AVG)RSYS=0DMX =5.15432SMN =-5.1429SMX =.260739(a) (b)(c) (d)Figure 5.2: Simulated deformation contour results (magnifying scale: 15) of the full-scalespecimen groups with the maximum and minimum value at the final solutionstep: (a) CLT3B1EA, (b) CLT3B1FA, (c) CLT5BEA, and (d) CLT5BFA100Table 5.2: FEM calibration results of full-scale specimensSpecimen Displacement at 2 kN Displacement at 5 kNTest FEM TestFEMTest FEM TestFEM[mm] [mm] [mm] [mm]CLT3B1F 0.76 0.72 1.06 3.41 3.32 1.03CLT3B1E 0.25 0.23 1.09 1.52 1.41 1.08CLT5BF 0.21 0.22 0.95 0.55 0.59 0.93CLT5BE 0.13 0.13 0.97 0.39 0.42 0.935.4 Simulation results5.4.1 Medium-scale: CLT3C, CLT5C, CLT7C and CLT9CThe descriptive statistics of MOE obtained from tests and FEM analyses presented inChapter 3 is stated in Table 5.3. It shows the effective average MOE by FEM analysis ofeach specimen group reasonably agree with the test counterpart. Note that the statisticalestimations of FEM analysis are derived using the mean value from FEM results and otherdistribution parameters (e.g. distribution type and COV) from test data. Moreover, thedata presented in Table 5.3 is not adjusted with the moisture content and size factors. Forthe medium-scale specimens, the average MOE value of 10100 MPa is obtained by FEMsimulations using the adjustment factors derived in Chapter 3.Table 5.3: MOE results of medium-scale specimens from test and FEM analysisSpecimen Min [MPa] Max [MPa] Average [MPa] COV [%] 5thP. [MPa]Test FEM Test FEM Test FEM Test FEM Test FEMCLT3C1 5351 6208 9521 9521 6961 7240 13.36 10.82 5432 4780CLT3C2 5666 5609 10143 11664 7875 6930 16.40 15.58 5750 5865CLT5C1 4668 4575 14367 14510 6377 7142 28.24 25.70 3415 3312CLT5C2 5035 5387 8451 7099 6244 5183 14.04 11.79 4803 4947CLT7C 6590 6392 9704 8540 8282 6957 15.59 18.09 6158 5727CLT9C 6120 5508 9453 9736 7533 8587 16.39 18.03 5502 47315.4.2 Full-scale: CLT3B and CLT5BThe FEM predicted and test measured MOE of the full-scale CLT specimen groups areplotted in Figure 5.3. Moreover, descriptive statistics of MOE obtained from tests andFEM analyses is stated in Table 5.4. It shows the FEM estimations are well representedthe test observations. However, due to unable to account the shear modulus property101of CLT lamella in the FE model accurately, the predicted results shows relatively highervariability. Note that the data presented in Table 5.4 is not adjusted with the moisturecontent and size factors. Using the adjustment factors derived in Chapter 3, the average and5th percentile value of edge-wise-effective64-apparent MOE of full-scale specimens from theFEM analysis are obtained as 9800 MPa and 7300 MPa, respectively. Therefore, similar tothe experimental observations, FEM simulation results restate that for full-scale specimens,the average characteristic MOE value is the same as the CSA code specified MOE valueand it shows indifferent value, even if changing the types of test and size of the specimens.In general, the adjusted edge-wise-apparent MOE is comparatively higher than the flat-wise-apparent MOE. It is due to the shear contribution; deformation due to shear in theedge-wise loading specimens is higher than the flat-wise loading specimens. Results alsoshow, as expected true MOE is higher than the apparent MOE.4000 5000 6000 7000 8000 9000 10000 11000 12000 1300040005000600070008000900010000110001200013000Figure 5.3: Stiffness in the longitudinal direction of full-scale specimens from FEM simu-lations and test64Effective MOE is calculated according to Shear Analogy Method described in the CSA O86-14.102Table 5.4: MOE results of full-scale specimens from test and FEM analysisSpecimen Min [MPa] Max [MPa] Average [MPa] COV [%] 5thP. [MPa]Test FEM Test FEM Test FEM Test FEM Test FEMCLT3B1FA 7819 8053 11673 10506 9858 11040 11.46 10.65 8000 6480CLT3B1FT 8274 7943 12368 10637 10370 8918 12.86 14.28 8175 8748CLT3B1EA 5637 4679 8444 8022 7111 7538 13.11 11.80 5578 5857CLT3B1ET 5002 4702 10124 9111 7578 8487 16.63 15.30 5505 4569CLT3B2 7320 8491 11670 12954 8995 8185 10.78 9.59 7400 7104CLT3B3 8420 9767 10620 11682 9486 8822 6.00 5.82 8551 9064CLT3B4 8120 6658 10970 8995 9290 7897 9.66 9.76 7814 7971CLT5BFA 5502 4622 7130 6631 6257 5381 6.75 6.14 5562 5339CLT5BFT 6336 5386 9561 8509 7698 6851 11.02 11.90 6303 6114CLT5BEA 4860 4666 8827 9798 5849 5264 14.16 14.30 4486 3993CLT5BET 4885 4982 9767 8497 6405 6277 15.17 15.77 4807 52885.5 ConclusionIn this chapter, the numerical model development and simulation results of medium-scalespecimens under axial compression load and full-scale specimens under third-point bend-ing loading in the flat-wise and edge-wise directions are presented. Results are reportedin terms of MOE of the specimens. To obtain the MOE property, medium-scale 3-, 5-, 7-,and 9-layer CLT members are modelled in ANSYS Mechanical APDL platform with anUserMat material model, as elaborated in Chapter 3, where material nonlinearity is explic-itly considered. For the medium-scale CLT specimens, the average adjusted-characteristicMOE value of 10100 MPa is obtained by FEM simulations.On the other hand, for modelling the initial stiffness of the material under a load wherefinding the MOE is of interest, material non-linearity is disregarded in the full-scale CLT’sFE model, as described in this chapter. The FE model is developed in the ANSYS Me-chanical APDL simulation environment for the static nonlinear analysis of CLT full-scalespecimens using 9 parameters orthotropic elastic material model. The material propertiesare calibrated so that the displacements obtained from FEM analysis matched with therecorded displacements from tests. The maximum deviation between test results and FEMresults is obtained as 9%. Therefore, the model can predict the test observation reason-ably. Using the adjustment factors derived in Chapter 3, the average and 5th percentilevalue of edge-wise-effective-apparent MOE of full-scale specimens from the FEM analysisare obtained as 9800 and 7300 MPa, respectively. It was also observed from the FEMsimulations that true MOE is higher than the apparent MOE. Therefore, similar to theexperimental observations, FEM simulation results restate that for full-scale specimens,the average characteristic MOE value is the same as the CSA code specified MOE valueand it shows indifferent value, even if changing the types of test and size of the specimens.103Chapter 6Stability of CLT: ExperimentalIn this chapter, we describe the experimental setup and results of full-scale cross-laminatedtimber (CLT) under axial compression load. The test setup, failure modes, and mechanicalproperties of full-scale specimen groups are presented in section 6.2. The testing dataanalysis is described in the section 6.3. In order to unify the test results of differentspecimen groups, the adjustment of the experimental results to a specific condition65 andstatistical representative properties are laid out in section 6.4. The nominal capacities ofthe specimens groups are pointed out in section 6.5.6.1 OverviewOur primary objective is to investigate the effects of cross layers on the CLT’s load carryingcapacity behaviour. The hypothesis being tested is that the cross layers have some con-tributions toward the ultimate capacity of CLT element across various lengths. The loadcarrying capacity is considered as the independent variable. In the experimental phase, weemployed compression testing on 3-layer CLT elements having effective slenderness ratioof 12.60, 13.16, 16.50 and 19.76. Moreover, 5-layer CLT is tested under the same testingprotocol. In the test, the specimens were loaded until peak load was reached followed by adrop. Table 6.1 states the information regarding specimens scale, species identity, testingstandards and loading protocol. The naming system and dimension of all specimen groupsare described in Chapter 2. The species’ grading information and geometric properties ofthe specimens are also listed in Chapter 2.Table 6.1: Overview of the testKeywords DescriptionScaling Full-scale: CLT0−90−0 (CLT3B)CLT0−90−0−90−0 (CLT5B)Species V2M1.1Standards ASTM: D4933-99, D2915-10ASTM: D198-15, D4761-13, D4442-15ASTM: D2555-15, D245-06, D1990-14Loading Displacement controlled monotonic conservative loading65Adjustment due to size, moisture content, and duration-of-load.1046.2 Full-scale: 3- and 5-layer CLT (CLT3B and CLT5B)Compression test on the full-scale 3- and 5-layer of CLT is conducted as per the descriptionbelow. The testing of CLT3B2 CLT3B3 and CLT3B4 specimens groups was done by [Hor13]|.6.2.1 Test setupThe test setup we adopted was originally developed by [LO18]| for the investigation ofthe performance of Glulam columns at UBC. Fixed end conditions for the column testset-up was chosen because of the high axial forces in full size member testing it is verydifficult to achieve frictionless pinned end boundary conditions or to estimate the amountof rotational restraint caused by friction. Figure 6.1 shows a schematic drawing for thefull-scale compression test. The needed rigid supports were designed and manufacturedafter plans of [LO18]|. At one end, the support consists of a steel shoe and a set of steelroller guides that allow axial movement while rotation about the vertical axis is prevented.At the other end, the support is designed as a fixedly rigid steel shoe in order to restrainrotation. The fixation at both ends is achieved by tightening the bolts. To get horizontallevelling, a piece of wood was put underneath the column in the fixed-support end.The specimens were placed on edge (thickness side) into the supports. In order to preventbuckling in vertical direction, i.e. around the strong axis, two lateral supports were installedas shown in Figure 6.2. These supports consisted of roller plates and metal plates in orderto reduce the impact of friction on the deflection of the specimens. With this arrangement,it was ensured that buckling would only occur in horizontal direction, i.e. around weakaxis. At one end, the hydraulic actuator pushed the steel shoe in the fixed support trans-ferring the axial force to the CLT column. At the other end, the custom-made supportwas rigidly attached to a strong steel column (Figure 6.2). Prior to load application, thecurvature of every specimen was checked with a string. Based on this measurement, thecolumns were placed into the test set-up in that way that initial curvature was in directionof the intended buckling direction. The initial curvature was not measured separately, butit could be noted that this value was very small for all specimens.Prior to every test, the friction between the specimen and support ends was measured.The column was placed in the test set-up. The bolts for the support at the actuator sitewere tightened by hand in order to obtain lower values for friction, whereas the end ofthe column in the rigid shoe on the other side was not clamped. Furthermore, a gap wasleft between the vertical supporting area and the wood. Hence, while the actuator pushedthe CLT member until the gap was closed, the actuator load was recorded. For achievingrigid fixation at the end support the bolts were tightened by a torque wrench.Axial forcewas applied at the displacement controlled speed of 3.0 mm/min for this testing. The testcontinued until displacement occurred with no load increase. The peak load was reached inmore than 24 min, the minimum required in the standard ASTM 198-09, and the durationof the tests did not take longer than 40 minutes. Axial load and test machine stroke105were recorded. Lateral displacements resulting from column buckling were measured inthe middle of each cross sectional dimension by linear voltage displacement transducers(LVDT) with bars at quarterpoint and endpoint as well as string pots at the midpoint(Figure 6.2a). The string pots were attached to a screw placed in the middle of the crosssection in the cross layer as well as the brackets for the LVDTs were attached to the columnin the middle layer.(b)(c)(d)(e) (f) (g)(a)Figure 6.1: Test setup of full-scale CLT specimen groups: (a) a CLT3B1 specimen underaxial compression loading, (b) a CLT5B specimen with the lateral support,(c) side-view of a CLT5B specimen, (d) a CLT3B1 specimen with the lateralsupport that restrains buckling in the vertical direction, (e) rolling supportcondition at the loading end of a specimen, (f) fixed-in-vertical but free-in-lateral support condition at mid-span, and (g) fixed support condition at thefar end of a specimen106400 mm200 mm200 mm200 mm200 mm400 mm400 mm200 mm300 mm(a) Top view(b) Side viewFigure 6.2: Rendering of a representative test setup of full-scale CLT specimens: (a) topview a CLT3B1 specimen under axial compression loading with the end supportconditions and location of LVDTs, and (b) side view of a typical specimen withthe lateral supports that restrain buckling in the vertical direction6.2.2 Failure modesFailure modes of the CLT3B and CLT5B specimens are shown in Figure 6.3 and Figure 6.4,respectively. It shows wood crushing failure of the parallel layers. In the post-ultimatephase, with increasing the displacement-controlled loading, lamella splitting occurs.Compression failure occurred for the specimens of CLT3B1, CLT3B2 and CLT5B specimengroups. Here, compressive wrinkles formed before the peak load. However, for the spec-imens of CLT3B4, no compressive wrinkles occurred before the peak load. In this case,although a clearly visible bending deformation was observed, however, no rolling shearfailure was registered (Figure 6.3).In general, all tested specimens failed in compression in the area of the highest over-alldeformation, i.e. in-between the quarter points along the specimens’ length. Only brittlefailure was observed near the end points. None of the first-failure locations marked at thefinger joint location. Note that it has already observed for the behaviour of small-scale CLTspecimens (Chapter 2), the failure of finger joints depends on an interaction with knots.This means, finger joints do not represent a particular weakness in the CLT lamellae, butcontribute to the overall failure behaviour.107(a)(b)Figure 6.3: Representative failure modes of full-scale 3-layer CLT specimens: (a) typicallateral deformation of a CLT3B4 specimen, and (b) buckling followed by woodcrushing failure at the mid-span of a CLT3B3 specimen(a)(b)(c)(d) (e)Figure 6.4: Representative failure modes of the CLT5B group: (a, b, c) buckling followedby splitting at the wood-adhesive interface, (d) buckling followed by woodcrushing and splitting of a lamella, and (e) buckling deformation of a specimen1086.2.3 Mechanical properties6.2.3.1 Full-scale: 3-layer CLT (CLT3B)The mechanical properties of a typical CLT3B1 specimen group are depicted in Figure 6.5.All specimens’ load-displacement trajectories which were recorded by 7 LVDTs are drawn.The displacement pattern along the length of the specimens at 60% of the ultimate loadlevel is plotted in Figure 6.5b. The average volumetric mass density, and ultimate compres-sive capacity of 20 specimens are obtained as 485 kg/m3, and 340 kN, respectively. More-over, descriptive statistics of the ultimate compression capacity of the specimen groups arepresented in Table A.2. It shows the compressive capacity of the specimen groups decreaseswith increasing the slenderness ratio.(a)(b)A specimen curveAAAUAverage curveAAAAUA specimen curveAAAKFigure 6.5: Test results of the CLT3B1 specimens: (a) load-displacement curve, and (b)lateral displacement along the length of the specimens at 60% of the ultimateload carrying capacity level109Table 6.2: Loading capacity of CLT3B groupsEstimations SpecimensCLT3B1 CLT3B2 CLT3B3 CLT3B4Min [kN] 287 272 218 176Max [kN] 466 404 314 287Average [kN] 356 332 273 220SD [kN] 42 31 28 32COV [%] 11.78 9.24 10.19 14.595th percentile [kN] 287 281 227 167Table 6.3: Loading capacity of CLT5BEstimations SpecimenCLT5BMin [kN] 392Max [kN] 581Average [kN] 527SD [kN] 45COV [%] 8.595th percentile [kN] 4526.2.3.2 Full-scale: 5-layer CLT (CLT5B)The mechanical properties of a typical CLT5B specimen group is depicted in Figure 6.6. Allspecimens’ load-displacement trajectories which were recorded by 7 LVDTs are drawn. Thedisplacement pattern along the length of the specimens at 60% of the ultimate load levelis plotted in Figure 6.6b. The average volumetric mass density, and ultimate compressivecapacity of 20 specimens are obtained as 435 kg/m3, and 527 kN, respectively. Moreover,descriptive statistics of the ultimate compression capacity of the specimen group are pre-sented in Table 6.3. It shows the compressive capacity of the CLT5B group is higher thanthe CLT3B counterpart.(a)(b)A specimen curveAAAUAverage curveAAAAUA specimen curveAAAKFigure 6.6: Test results of the CLT5B group: (a) load-displacement curve, and (b) lateraldisplacement along the length of the specimens at 60% of the ultimate loadcarrying capacity level1106.3 Testing data analysisThe spectrum and average line of all specimen groups’ lateral displacement at 60% of theultimate load carrying capacity level along the length of the elements are portrayed in Fig-ure 6.7-a, -b, respectively. It shows that both CLT3B1 and CLT5B specimen groups havingthe same length, CLT3B1 specimen group has more deflection than CLT5B counterpart.On the other hand, for the CLT3B2, CLT3B3 and CLT3B4 specimen groups, deflection iscontrolled by the length of the group— higher the length higher the mid-span deflection.The results of all specimens are tabulated in A. The following subsection diagnosis the testdata quality and quantity whether it was sufficient to obtain the characteristic capacity ofeach specimen group.0 500 1000 1500 2000 2500 3000 3500 4000 4500024680 500 1000 1500 2000 2500 3000 3500 4000 450002468(a)(b)Figure 6.7: Test results of the CLT3B1, CLT3B2, CLT3B3,CLT3B4, and CLT5B specimengroups: (a) spectrum of the load-displacement curves, and (b) average line ofthe lateral displacement along the length of each specimen group at 60% oftheir ultimate load carrying capacity level6.3.1 Sampling qualityIn order to compare capacities between different groups, the test results from all groups arestandardized according to procedure described in Chapter 2. Sphere with the 95% proba-bility and data points of the standardized capacity, stiffness (MOE) & density parameters111(univariate Gaussian distribution case) of all samples is also shown in Chapter 2. It showsdata for all full-scale specimen groups are sufficient to measure the mechanical properties.The variance of data and the strength of correlation among mechanical properties are alsopresented in Chapter 2. For example, for the CLT3B2 specimen group, the correlation coef-ficient between capacity and MOE is calculated as 0.86; whereas the correlation cofficientsbetween density and capacity, and between density and stiffness are obtained as -0.31, and-0.24, respectively. The correlation values are used in the reliability analysis later on.6.3.2 Central tendency and variabilityThe central tendency and data variability of compression loading capacity of CLT spec-imens is diagnosed by plotting five plots, namely, run sequence, lag, histogram, normalprobability and box-and-whisker plot. For example, these plots are shown in Figure 6.8for the CLT3B2 specimen group. It shows, the run sequence is random and good sparse ofdata on the lag plot. Moreover, the histogram and normal probability plot show the datacan be considered lognormally distributed.0 5 10 15 20300350400450300 350 400 450300350400450300 350 400 45000.050.10.150.20.250.30.35300 350 400 450-2-101234280300320340360380400420440460Figure 6.8: Representation of measuring the central tendency and data variability of thestability capacity parameter of the CLT3B2 group1126.3.3 Distribution functionThe probability distribution parameters are then sought out based on the goodness-of-fitparameters of the distributions. For example, the probability density function (PDF) andcumulative density function (CDF) of CLT3B2 specimen group is computed and plotted inFigure 6.9a considering the lognormal probability distribution. Moreover, the sorted dis-tribution of specimen results are indicated by a line on the PDF curve. A continuous CDFcurve is computed from distribution parameters (mean and standard deviation of lognor-mal distribution) and plotted by solid red line. Thus, the probability density function andcumulative density function of each specimen group is computed and plotted in Figure 6.9bconsidering the lognormal probability distribution. It shows the load carrying capacity ofthe CLT3B4 specimen group has the highest variability, whereas, CLT5B specimen grouphas the lowest variability.0 100 200 300 400 500 600 700 80000.10.20.30.40.50.60.70.80.91(a) (b)0 100 200 300 400 500 600 700 80000.10.20.30.40.50.60.70.80.91Figure 6.9: Probability density function and cumulative distribution function of the stabil-ity capacity parameter of the CLT3B1, CLT3B2, CLT3B3,CLT3B4, and CLT5Bspecimen groups: (a) CLT3B2 specimen, and (b) full-scale specimen groups6.4 Stability capacityThe axial compression testing method were employed to measure the ultimate capacity ofCLT structures having intermediate range slenderness ratio. The test measurements arepresented in the following subsection.6.4.1 Test measurementThe ultimate capacity of each group are laid out in Figure 6.10 by box plot with whiskerat ± 1.5 of interquartile range and identifying the outliers which lie beyond the whiskerfence. The jitter of specimen results of each group are also shown by black dots in thefigure respectively. Moreover, descriptive statistics of the ultimate compression capacity113of the specimen groups are presented in Table A.2 and Table 6.3. It shows CLT3B4 grouphave higher variability than their counterparts. The capacity results shown here is notadjusted with moisture content, size of specimens and duration-of-loads. In the followingsubsection results are adjusted.llllllllllllllllllll lllllllllllllllllll lllllllllll lllllllllllllllllllllllllllllllll llllllllllllllllll lllllllllllllllllllllll0200400600CLT3B1CLT3B2CLT3B3CLT3B4CLT5BSpecimenUltimate capacity (kN)r Outlier· Specimen resultFigure 6.10: Load carrying capacity of full-scale specimen groups without adjustment6.4.2 Unifying test measurementIn order to compare the results of the different groups, the compression capacity of thespecimen groups is calculated based on the same width of 184 mm. Therefore, at this level,CLT’s capacity is comparable to each other. The adjusted compression capacity is plottedin Figure 6.11. Moreover, descriptive statistics of the adjusted capacity of the specimengroups are presented in Table 6.4. It shows CLT5B group have higher capacity values thantheir CLT3B counterpart. The next section describes the adjustment of the data.Table 6.4: Statistical data of the effective load carrying capacity without adjustmentEstimations SpecimensCLT3B1 CLT3B2 CLT3B3 CLT3B4 CLT5BMin [kN] 313 278 222 180 399Max [kN] 505 413 320 293 534Average [kN] 392 339 279 225 489SD [kN] 44 31 28 33 36COV [%] 11.15 9.24 10.19 14.59 7.365th percentile [kN] 320 288 232 171 430114llllllll lllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll0100200300400500CLT3B1CLT3B2CLT3B3CLT3B4CLT5BSpecimenUltimate Capacity adj (kN)r Outlier· Specimen resultFigure 6.11: Effective (with the same width of 184 mm) load carrying capacity of full-scalespecimen groups without adjustment6.4.3 Adjusting test measurementThe test results are adjusted with the moisture content, duration-of-load, and size of thespecimens. During adjustment, it is followed the specifications from ASTM standards andCSA codes. The adjustment for moisture content (MC) is calculated according to theASTM standard as described in Chapter 2. We set our target MC as 15%. The duration-of-load adjustment factor for the capacity parameter is also depicted in Chapter 2. Asdiscussed in Chapter 2, duration-of-load factor is taken as 1.25. Then, the adjusted testdata for load carrying capacity of full-scale 3- and 5-layer CLT specimens are presentedin Figure 6.12. The descriptive statistics of the adjusted test results is highlighted inTable 6.5. In general, 3-layer CLT specimens having different slenderness ratio shows thesimilar variability in the test results.Table 6.5: Statistical data of adjusted capacity of full-scale specimensEstimations SpecimensCLT3B1 CLT3B2 CLT3B3 CLT3B4 CLT5BMin [kN] 276 261 216 179 417Max [kN] 411 375 306 269 492Average [kN] 332 314 264 218 457SD [kN] 31 26 25 27 23COV [%] 9.41 8.43 9.51 12.39 5.145th percentile [kN] 281 270 223 174 418115lllllllllllllllllllll lllllllllllll lllllllllllll lll lllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllll lllll0100200300400500CLT3B1CLT3B2CLT3B3CLT3B4CLT5BSpecimenUltimate Capacity adj (kN)r Outlier· Specimen resultFigure 6.12: Effective load carrying capacity of all specimen groups adjusted with the mois-ture content and duration-of-loading6.5 Characteristic capacityThe adjusted experimental capacities of all specimen groups are shown in Figure 6.13. Thedescriptive statistics of the adjusted test results is stated in Table 6.5. It can be shownthat for all specimen groups, capacity decreases with increasing slenderness ratio.llllllllllllllllllllllllllllllllll lllllllllllll lll lllllllllllllllllllllllllllllllll lllllllll llllllll lllllllllllllllllllllll0100200300400500CLT3B1CLT3B2CLT3B3CLT3B4CLT5BSpecimenUltimate Capacity adj (kN)r Outlier· Specimen resultFigure 6.13: Characteristic capacity of full-scale specimen groups1166.6 ConclusionIn this chapter, the experimental setup and load carrying capacities of full-scale specimensare presented. In the experimental phase, we employed axial compression testing on 3-layerCLT elements having effective-slenderness ratio of 12.60, 13.16, 16.50 and 19.76. Moreover,5 layers CLT having slenderness ratio of 8.75 is tested under the same testing protocol.Test results are reported in terms of failure modes, ultimate load carrying capacity anddisplacement behaviour of each specimen group. The investigation on the sampling qualityand quantity, confidence interval, and correlation of MOE with other properties are elab-orated in Chapter 2. The central tendency, data variability, and probability distributionfunction of ultimate load carrying capacity parameter of all specimen groups are then de-termined. The test results are adjusted with the moisture content, duration-of-load, andsize factors.The study found that the characteristic capacities of full-scale 3-layer CLT specimenshaving effective slenderness ratio of 12.60, 13.16, 16.50, and 19.76 are 281 kN, 270 kN, 223kN, and 174 kN, respectively. The characteristic capacity of 5-layer CLT having slendernessratio of 8.75 is measured as 418 kN. For all specimen groups, capacity decreases withincreasing slenderness ratio. In general, all tested specimens failed in compression. In thepost-ultimate phase, with increasing the displacement-controlled loading, lamella splittingoccurs.117Chapter 7Stability of CLT: NumericalIn this chapter, we describe the development of a numerical model to compute the ulti-mate load carrying capacity of cross-laminated timber (CLT) structures idealized as beam-column elements. The development of the model is described from section 7.2 to section 7.6.The calibration and verification of the model are laid out from section 7.7 to section 7.8.The beam-column capacity curve of CLT structures obtained by simulations and the CSAcapacity curve is discussed in section 7.9.7.1 OverviewA finite element method (FEM) simulation program, named after, Analysis of UniversalBeam-Columns (AnUBC), is developed in MATLAB computing environment [MAT17] forthe stability analysis of CLT structures. The capability of AnUBC program is comparedwith other FEM simulation environments available commercially and academically, for ex-ample, ANSYS66, SeismoStruct67 and SATA68 in Table 7.1. It shows the AnUBC programis outperformed others in terms of having multi-layered fibre element, in which, nonlinearmaterial model can be assigned each layer separately. This resembles the condition of aCLT cross-section as practical as possible. Moreover, developing a program would give usfreedom to include more features in the long run and save computation effort for solving aspecific problem compared to using a general-purpose software.7.2 AnUBC development: multi-layer fibre elementIn order to examine structural stability of beam-column elements including material andgeometric nonlinearity, the development of a finite element program, namely AnUBC, isdescribed in this section. The nonlinear procedures for a prismatic beam column ele-ment require three fundamental equations, namely, equilibrium, compatibility, and force-deformation, that describe both the member and section behaviour [Red93; Bat82]|. With-out derivation, these equations are stated in Table 7.2 and the variables are shown inFigure 7.1. The connection between these equations can be provided by either displace-ment or force formulation algorithms using multi-layer fibre element. The formulations areimplemented into the AnUBC program based on [Fer09; NZL13; PPC99; Bit+16; KC17]|.66ANSYS is a commercially available FEM analysis software developed by Ansys Inc., USA.67SeismoStruct is a commercially available FEM package developed by Seismosoft, Italy.68SATA (Structural Analysis of Truss Assemblies) is a FEM program developed by Dr. Song and Prof.Lam, UBC.118Table 7.1: Features in the finite element analysis programsFeatures ProgramsANSYS SeismoStruct SATA AnUBCSolutionstrategyArc control X X X 7Load control X X X XDisplacement control X X X XLoads Cyclic, ground motions X X 7 7Monotonic X X X XDeformation Flexural X X X XShear X 7 7 XElement Multi-layered fibre 7? 7 7 XNonlinearityMaterial (CLT) 7? 7 7 XStructural (Lagrangian) X X X 7Structural (Corotational) X X 7 XRestraint Separate in both ends X X X XUser interface X X X 7?Available with writing subroutineTable 7.2: Fundamental equationsMember SectionEquilibrium P = aT q s =∫A aTs σdACompatibility v = au = aseForce-deformation q = Kv + q0v = fq + v0s = Ksee = fss123123456P ,uq,vX2, U2X1, U1MNV s =MNVe =φaγFigure 7.1: Definition of variables for the FEM formulation1197.2.1 Displacement formulation algorithmThe displacement formulation assumes the strong form of compatibility i.e. the deforma-tions along the member are interpolated from the member end displacements [Bat82]|. Thealgorithm for displacement formulation is given in Figure 7.2.StartuAssumev = aue = BvStrain at the jth fibre:j = asjeSectionAnalysisσj , ETjs = ∑jaTsjσjAjKs =∑jaTsjETjAjq = L2∑i[BT (ζi)s(ζi) − NT (ζi)w(ζi)]wiK = L2∑i[BT (ζi)Ks(ζi)B(ζi)]wiP = aT qKT = aTKa∆u = K−1T (Pext − P )u = u + ∆u‖∆u‖ < toleranceu , PStopyesnoFigure 7.2: Displacement formulation algorithm for the AnUBC program1207.2.2 Force formulation algorithmThe force formulation assumes the strong form of equilibrium i.e. the section forces alongthe member are interpolated from the member end forces [Red93; Bat82]|. The algorithmfor force formulation is given in Figure 7.3.StartuAssumev = auMemberAnalysisqk (assume)sk = Bqkf = L2∑iBT (ζi)fs(ζi)B(ζi)wivk = L2∑iBT (ζi)e(ζi)wi∆q = f−1(v − vk)qk+1 = qk + ∆q‖v − vk‖ <tolerancefq = qk+1SectionAnalysise (assume)Strain at the jth fibre:j = asjeσj , ETjs = ∑jaTsjσjAjKs =∑jaTsjETjasjAj∆e = K−1s (sk − s)ei+1 = ei + ∆e‖∆e‖ <tolerancef is = k−1seiP = aT qKT = aT f−1a∆u = K−1T (Pext − P )u = u + ∆u‖∆u‖ <toleranceu , PStopyesnoyesnoyesnoFigure 7.3: Force formulation algorithm for the AnUBC program1217.2.3 Implementation resultThe displacement and force formulation algorithms described above have been implementedinto the AnUBC program for a simply supported beam subjected to monotonic end mo-ments. A base case example was established to look at the effect of various parameters onthe performance of the two formulations. For this purpose a 6 m simply supported beamwith a 300 × 250 mm rectangular cross section was selected. The beam is composed of ahomogeneous bilinear material with a elastic modulus of 200 GPa and a yield strength of250 MPa. The properties of the material and structure are chosen such that the AnUBCprogram outputs can be verified with the hand calculations or results obtained by an-other FEM programs. The moment-curvature-thrust curve of the considered cross-sectionis computed for different axial to elastic yield load ratios, and plotted in Figure 7.4. Itshows the moment-curvature relationships become nonlinear as yielding starts to occur inthe member. Moreover, moment capacity deceases with increasing the level of axial load.0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.205001000150020002500Figure 7.4: Moment-curvature-thrust results of a beam-column structural memberThe implementation results of displacement formulation algorithm is shown in Figure 7.5.It shows the number of elements significantly affects the accuracy of the displacement for-mulation. The results approach the true solution as the number of elements increases.Whereas, the number of integration points has little effect on the accuracy of the displace-ment formulation. Results also shows the displacement formulation is more computation-ally efficient, but at the cost of lower accuracy (Figure 7.5c).1220 0.02 0.04 0.06 0.08 0.1 0.1202000400060008000Displ 2 elementsForce 2 elementsDispl 10 elementsForce 10 elementsDispl 20 elementsForce 20 elements0 1 2 3 4 5 6-0.500.52 4 6 8 10 12 14 16 18 2000.050.10.150.2(a)(b)(c)Figure 7.5: Simulation results by varying the number of elements parameter of a beam-column structural member: (a) moment-rotation curves, (b) curvature alongthe length of the member, and (c) trend of the computational time requiredOn the other hand, the implementation results of force formulation algorithm is shown inFigure 7.6. The force formulation does not directly depend on the number of elements,but on the location of the integration points. The accuracy of the force formulation issignificantly improved by increasing the number of integration points. Increasing the num-ber of integration points allows the piece-wise linear curvature diagram to more closelyapproximate the true curvature diagram (Figure 7.6b). Therefore, for larger structuresit is evident that the reduction in degrees of freedom is possible by the force formulationleading to more computationally efficient analysis than the displacement formulation.1230 0.02 0.04 0.06 0.08 0.1 0.12 0.1402000400060008000Displ 2 Int PtsForce 2 Int PtsDispl 4 Int PtsForce 4 Int PtsDispl 10 Int PtsForce 10 Int Pts0 1 2 3 4 5 6-0.500.52 3 4 5 6 7 8 9 1000.050.10.150.2(a)(b)(c)Figure 7.6: Simulation results by varying the number of integration points parameter ofa beam-column structural member: (a) moment-rotation curves, (b) curva-ture along the length of the member, and (c) trend of the computational timerequired7.3 AnUBC development: solution controlTwo solution control algorithms, namely, load control and displacement control algorithm,are implemented into the AnUBC program based on [VK05; KB00; IA80; SG04; Cri97;Pos07; Cha02]|. The description of these algorithms with the implementation results isdescribed in the following subsection.1247.3.1 Load control algorithmIn load control technique, the structure would not be analyzed by the entire applied externalforce Pext, but to a fraction of Pext or load level λ. In this approach, for each load level, adisplacement is calculated. The standard load control approach is based on an equilibriumequation defined as the difference between the internal force Pint and a prescribed externalload λPext, where λ is the load level, such thatr(di) = Pint(di)− λPext = 0 (7.1)The vector r is known as residuum or out of balance vector, which has to vanish at equi-librium state. For brevity, r(di) will be written most of the time as ri. As a general rule,a Newton-Raphson (N-R) iterative scheme is adopted to solve the nonlinear Equation 7.1with respect to the displacement. Following the procedure of the standard N-R method,the linearizion of the residuum equation yields,ri+1 = ri + ∂r∂dδdi + Higher order term = 0 (7.2)Neglecting the higher order terms and defining∂r∂d = Kt (7.3)where Kt is again the tangent stiffness matrix, Equation 7.2 can be re-expressed simply asδdi = −K−1t ri (7.4)Then, an iterative scenario is implemented in which, after solving Equation 7.4 for thenodal displacement increments δdi, the overall displacements are updated throughdi+1 = di + δdi (7.5)until convergence is achieved, i.e. ri+1 ≤ tolerance. Next, the iterations stop and proceedwith a new load level λ. Concerning the computational cost, there are several ways tocode the algorithm. We adopted the full Newton-Raphson scheme which is based on thecomputation of the tangent stiffness matrix K−1t at each iteration. In this case, the com-putational cost per iteration is more expensive but results in quadratic convergence.The use of the N-R method has one big advantage: it keeps the symmetry and bandedproperties of the tangent stiffness matrix. However, the calculation of the inverse of Ktposses the major part of the computational cost of the standard load control method andfurthermore, the inversion of Kt cannot be performed if the tangent stiffness matrix issingular. Hence, the idea of prescribing the load increment is very simple, however, themethod is unstable after a limit point, being unable to follow the equilibrium path as shownin Figure 7.7a.1257.3.2 Displacement control algorithmIn contrast to the load control, which prescribes the external load level λPext, the naturalcounterpart is to use the nodal displacement component d as an independent variable.First, let us assume that we have found equilibrium state of a given structure configuration.This stage correspond to an iteration i = 0, i.eλ = λ0 and d = d0 (7.6)Following after, instead of varying the load parameter λ, the qth component of d is incre-mented by, δd(q). Consequently, the updated displacement vector reads withd0(q) = d0(q) + δd(q) (7.7)Notice that δd(q) is not a vector but the size (scalar) of the increment for the constrainedqth component of the displacement vector d. An iterative scheme to compute the non-constrained displacement components of vector d and the load parameter λ is thereforeneeded. The residuum Equation 7.1, corresponding to the load level λ, can be expandedby mens of a truncated Taylor series, so thatr(λi + δλi) = r(λi) + ∂r(λi)∂λδλi (7.8)which finally yieldsr(λi + δλi) = r(λi)− δλi Pext (7.9)Substituting Equation 7.9 into Equation 7.4, the iterative nodal displacement δdi can beexpressed through the new unknown load level, λi+1 = λi + δλi, givingδdi = −K−1t r(di, λi + δλi) = −K−1t r(di, λi) + δλi K−1t Pext (7.10)where Kt is the tangent stiffness matrix corresponding, in a full N-R scenario, to δdi.Now, two incremental displacement vectors δd¯ and δdt can be defined in such a way thatEquation 7.10 becomesδdi = δd¯i + δλidit (7.11)withδd¯i = −K−1t ri and δdit = −K−1t Pext (7.12)The vector δd¯ stems from the usual residuum vector r and δdt, also known as the tangentialsolution, from the external load vector Pext which is multiplied by λ to obtain the actualload. It is to be noted that another way to find Equation 7.11 is the arc-length method.126Here, the key of the displacement control method is that the displacement of the qthcomponent of d is constrained, such that δd(q) = 0. Theretofore the following condition,coming from Equation 7.11 has to be satisfiedδd¯i(q) + δλi δdit(q) = δdi(q) = 0 (7.13)Therefore, solving for δλi from Equation 7.13 givesδλi = −δd¯i(q)δdit(q)(7.14)Finally, the displacements & the load level are updated until convergence is achieved as:di+1 = di + δdi and λi+1 = λi + δλi (7.15)It is to be noted that the displacement control method becomes unstable after passingturning points, and therefore, cannot follow the equilibrium path as shown in Figure 7.7b.λ 1λ 2λ 3d1d2d3-r3-r2-r1...Kt1Kt3Kt2d1d2d3Loadlevel (λ)Displacementcomponent (d)LimitpointCannot befollowedLoad level (λ)Displacement component (d)TurningpointCannot befollowedMightconverge againLimitpointdှဿdှဿdှဿλi1λi2di1di2λi-1λi1λi2λin(a) (b)Figure 7.7: Solution strategy for getting the convergence points: (a) load control algorithm,and (b) displacement control algorithm7.3.3 Implementation resultsThe implementation results of solution control strategies are shown in Figure 7.8. It shows,solution by load control algorithm stops once it reaches the limit point. Whereas, displace-ment control algorithm coverages the solution even after passing the limit point. However,displacement control algorithm takes more computational effort than the load control.1270 1 2 3 4 5 6012 1040 1 2 3 4 5 6-6-4-200 1 2 3 4 5 6-2-10 10-39 9.2 9.4 9.6 9.8 10 10.2 10.4 10.6 10.8 110510(a)(b)(c)(d)Figure 7.8: Simulation results by varying the solution control algorithm of a beam-columnstructural member: (a) load-displacement curves, (b) displacement along thelength of the member, (c) curvature along the length of the member, and (d)computational time required7.4 AnUBC development: shear deformationIn general shear and bending rotations are considered uncoupled, requiring deformationprediction of shear deformable beams to be obtained from a superposition of bendingand shearing components. In contrast, the approach adopted here based on [Bor+12;128Cri97; MOS12; SR12; PPC99; KC17; Fer+14; WRL00; Red97; Ede06; PFP10]|, is to usean analytical bending and shear rotation interdependent shape functions. These shapefunctions are derived based on the beam’s geometric and material parameters to expressthe total beam cross-sectional rotation in terms of bending and shear rotation as follows:Θ = Θbϕb + Θsϕs (7.16)where, θ represents the total beam cross-sectional rotation, θb the bending rotation andθs the average shear rotation in line based on first order shear deformation theory. Thesymbols ϕb and ϕs are the analytical interdependent shape functions, which account forthe distribution of bending and shear rotation, respectively. The asymmetrical bendingmode can be decomposed to realize the bending and shear rotation present as shown inFigure 7.9. The end rotation at node-1 due to applied nodal moments m1 and m2 is:Θ1,1b =m1L3EI Θ1,2b = −m2L6EI Θ1,1s =m1LGA¯Θ1,2s =m2LGA¯(7.17)where E is the Young’s modulus; G is the shear modulus; I is the moment of inertia; L isthe span of the beam and A¯ is the shear area. The superscripts i, j refers to the rotationat node i due to moment applied at node j.LQQm2QQm1Θm1m2m1m22Θ1Θb1,1Θb2,2ΘsΘsQQFigure 7.9: Rendering of the bending and shear kinematics of a typical shear deformablebeam under asymmetrical bending and shear loadingThe distribution of bending rotation (Θ1b = Θ1,1b + Θ1,2b ) and shear rotation (Θ1s = Θ1,1s +Θ1,2s ) in the presence of an average shear and bending rotation, in a shear deformable beam129element can then be written as:ϕb =Θ1bΘ1b + Θ1s= 11 + Ω (7.18)ϕs =Θ1sΘ1s + Θ1s= Ω1 + Ω (7.19)where, the bending shear stiffness factor Ω isΩ = 12EIL2GA¯(7.20)Then, the contribution from the bending rotation filed dwdx and shear rotation field Θs(x)to the total cross-sectional beam rotation field Θ(x) of the shear deformable beam modelcan be written as:Θ(x) = dwdxϕb + Θs(x)ϕs (7.21)where w(x) is the beam transverse displacement field and dwdx corresponds to the slope ofthe beam neutral axis and cross-sectional bending rotation. Now, the relationship betweena generalized bending moment field M(x) and beam cross-sectional rotation filed Θ(x) canbe written as:Θ(x) =∫ L0M(x)EIdx (7.22)(dwdxϕb + Θs(x)ϕs)=∫ L0M(x)EIdx (7.23)ddx(dwdxϕb + Θs(x)ϕs)=M(x)EI(7.24)On expanding the Equation 7.24, two distinct moment-curvature equations can be written:EI1d2wdx2= M1(x) and EI2dθsdx2= M2(x) (7.25)where, I1 and I2 are moment of inertial of a beam, having a modified width b1 = bϕb andb2 = bϕs respectively, b being the width of the beam. Then the bending curvature69 andshear curvature70 in Equation 7.25 can be defined as:kb =d2wdxand ks =dθsdx(7.26)This way, the curvature of shear deformable beam is given by two curvature terms— asecond order and a first order term in the spatial coordinate x; hence, the formationcircumvents the shear locking phenomenon.69Beam curvature from bending deformation filed.70Beam curvature from shear deformation filed.1307.4.1 Finite element formulationBased on the total potential energy Π, the weak form of the shear deformable beam elementunder a distributed loading q(x) can be written as follows:dΠ =12∫ L0[δkbEI1 kb] dx+12∫ L0[δksEI2 ks] dx−∫ L0q(x)δwdx = 0 (7.27)For an element e, in the domain [x1 x2], a non-dimensional coordinate ξ can be written as:0 ≤ ξ = x¯L≤ 1 (7.28)where x¯ is a local beam coordinate, with origin x1 away from a global origin. Then,the choice of interpolation function for w(x) and Θs(x) is dictated by the variation indexin the energy functional Π. Consequently, the minimum requirements are C1 continuityensured by Hermite cubic interpolation functions for nodal freedoms arising from w andC0 continuity satisfied by linear Lagrangian interpolation for Θs.7.4.1.1 Hermite Interpolation Function for w(x)Using a cubic polynomial approximation for w(x) can be written:w(x¯) =a1 + a2x¯+ a3x¯2 + a4x¯3 = HTa (7.29)where H is the column vector [1 x¯ x¯2 x¯3]T , and a stands for the column vector of ai for1 ≤ i ≤ 4. The nodal displacements can be labelled as:w¯1 = w(0) w¯2 = Θ(0) = Θ¯1,w¯3 = w(l) w¯4 = Θ(L) = Θ¯2 (7.30)The Equation 7.30 can be expressed in terms of a and w¯ = [w¯1 Θ¯1 w¯2 Θ¯2]T as follows:w¯ = Ca (7.31)whereC =1 0 0 00 1 0 01 L L2 L30 1 2L 3L3 ∴ C−1 =1 0 0 00 1 0 0−3/L2 −2/L 3/L2 −1/L2/L3 1/L2 −2/L3 1/L2 (7.32)Then w(x) can be written in terms of Hermite interpolation functions Nw as follows:w(x) = Nww¯ = HTC−1 w¯ =[Nw1 Nw2 Nw3 Nw4]w¯ (7.33)The entries of Nw are:Nw1 =(1− 3ξ2 + 2ξ3) Nw2 = −Lξ(1− ξ)2Nw3 =ξ2(3− 2ξ) Nw4 = Lξ2(1− ξ) (7.34)1317.4.1.2 Linear interpolation functions for θs(x)Approximating Θs(x) using a linear polynomial can be written as follows:Θs(x¯) =c1 + c2x¯ = DT c (7.35)where D is the column vector [1 x¯]T , and c stands for the column vector of ci for 1 ≤ i ≤ 2.The nodal displacements can be labelled as:Θ¯s1 = Θ¯1 = Θs(0) Θ¯s2 = Θ¯2 = Θs(L) (7.36)The Equation 7.36 can be expressed in terms of c and Θ¯s = [Θ¯1 Θ¯2]T as follows:Θ¯s = Sc (7.37)whereS =[1 01 L]∴ S−1 =[1 0−1/L 1/L](7.38)Then Θs(x) can be written in terms of Hermite interpolation functions NΘs as follows:Θs(x) = NΘsΘ¯s = DTS−1 Θ¯s =[NΘs1 NΘs2]Θ¯s (7.39)The entries of NΘs are:NΘs1 =(1− ξ) NΘs2 = ξ (7.40)7.4.1.3 Governing equationsUsing Equation 7.33 and Equation 7.39, the variations in the field variables— w(x) andΘs(x), and the curvatures— kb and ks can be written as:δw =Nwδw¯ δkb = δ[(Nw)′′ w¯]= δ [Bww¯] = Bwδw¯ (7.41)δΘs =NΘsδΘ¯s δks = δ[(NΘs)′Θ¯s]= δ[BΘsΘ¯s]= BΘsδΘ¯s (7.42)Then, the weak form of the Equation 7.27 can be written as:∫ 10[(Bwδw¯)EI1Bww¯]Ldξ +∫ 10[(BΘsδΘ¯s)EI2BΘsΘ¯s]Ldξ−∫ 10q(x) (Nw)T δw¯Ldξ = 0 (7.43)132After algebraic operations, the load vector f e, and stiffness matrices— Kekb from linearvarying curvature, and Keks from constant curvature, can be written as follows:Kekb =∫ 10[(Bw)T EI1Bw]Ldξ Keks =∫ 10[(BΘs)TEI2BΘs]Ldξ (7.44)f e =∫ 10q(x) (Nw)T Ldξ (7.45)Then, the assembled beam element stiffness matrix Ke = can be written as:Ke = Kekb + Keks (7.46)Following after, the governing finite element equation to be solved becomes:Ked¯e − f e = 0 (7.47)where d¯e = [w¯1 Θ1 w¯2 Θ2]T is the vector of nodal displacement for the assembled beamelement. As an example, for a constant value of EI1 and EI2, the beam element stiffnessmatrix Ke can be obtained as:Ke =EIϕbL312 6L −12 6L6L 4L2 + ϕsϕbL2 6L 2L2 − ϕsϕbL2−12 −6L 12 −6L6L 2L2 − ϕsϕbL2 −6L 4L2 + ϕsϕbL2 (7.48)= EIL3(1 + Ω)12 6L −12 6L6L 4L2 + ΩL2 6L 2L2 − ΩL2−12 −6L 12 −6L6L 2L2 − ΩL2 −6L 4L2 + ΩL2 (7.49)7.4.2 Implementation resultsThe implementation results of shear deformation algorithm are shown in Figure 7.10. Itshows, keeping the same cross-section with increasing length of the structure, shear con-tribution decreases. The decrease rate gets asymptotic after a certain length, for example,for the considered case: structure over 8 m length.1330 1 2 3 4 5 6 7 8 9 10-80-60-40-2000 1 2 3 4 5 6 7 8 9 10-0.02-0.0100.010.024 5 6 7 8 9 1000.511.52(a)(b)(c)Figure 7.10: Simulation results considering with- and without-shear deformation of a beam-column structural members having different lengths: (a) displacements alonglength of the members, (b) curvatures along length of the members, and (c)trend of shear deformation contribution of the members7.5 AnUBC development: out-of-straightnessA corotational formulation seeks to separate rigid body motions from strain producingdeformations at the local element level. This is adopted into the AnUBC program byattaching a local element reference frame, which rotates and translates with the beam el-ement. The formulation is based on [Bor+12; Cri97; LBH11; UR05; LBH11]|. To accountlarge displacements and rotations of a structure at the global level, corotational formula-tion required three ingredients, namely, the angle of rotation of a co-rotating frame, therelations between global and local variables, and a variationally consistent tangent stiffnessmatrix. A beam element in its initial and current configurations is shown in Figure 7.11.It shows the beam has translated, rotated and has local flexural deformations. The axialdeformation, rotation and flexural deformation are then derived accordingly.1347.5.1 Angle of rotation of the co-rotating frameFor the beam in its initial configuration, the global nodal coordinates are defined as (X1, Y1)for node-1 and (X2, Y2) for node-2. Then, for the beam element in its current configuration,the global nodal coordinates are calculated as (X1 + u1, Y1 + w1) for node-1 and (X2 +u2, Y2 + w2) for node-2, where for example, u1 is the global nodal displacement of node-1in the X direction and w1 is the global nodal displacement of node-1 in the Y direction.Thus, the original (L0) and current (L′) length of the beam can be written as:L0 =√(X2 −X1)2 + (Y2 − Y1)2 (7.50)L′ =√((X2 + u2)− (X1 + u1))2 + ((Y2 + w2)− (Y1 + w1))2 (7.51)The axial force in the beam (N) having axial stiffness of EA due to local axial displacement(ul =L′2−L20L′+L0 ) can be written as:N = EAulL0(7.52)The current angle of the co-rotating frame with respect to the global coordinate system isdenoted as β′ and can be expressed as:cosβ′ = (X2 + u2)− (X1 + u1)L′sin β′ = (Y2 + w2)− (Y1 + w1)L′(7.53)For flexural deformed beam as shown in Figure 7.11b, using the initial (β0) and final (β′)angles of the beam, the local nodal rotations can be derived as71:θ1l = arctan(cosβ′ sin β1 − sin β′ cosβ1cosβ′ cosβ1 + sin β′ sin β1)θ2l = arctan(cosβ′ sin β2 − sin β′ cosβ2cosβ′ cosβ2 + sin β′ sin β2)where β2 = θ2 + β0 (7.54)Then the local end moments (M¯1, M¯2) and shear forces (V1, V2) of the element can beobtained as follows:{M¯1M¯2}= 2EIL0[2 11 2]{θ1lθ2l}V1 =M¯1 + M¯2L′V2 = −V1 (7.55)71In MATLAB the arctan2 function is employed to compute the angles.135Initialβ0X , uCurrent1Y l β'L'L0Initialβ0Currentαθ1θ1lθ2θ2lInitialβ0CurrentL'∂ ule 2 ∂d21e1∂αMovement fromcurrent stateY , wX , u X , uY lY lY l Y l111 11X l X l X lX l2 2222β'β'2X lY , wY , w(a) (b) (c)Figure 7.11: Rendering of the initial and current configuration for a typical beam element inthe corotating frame: (a) ignoring flexural deformation, (b) including flexuraldeformation, and (c) having a small movement from the current configuration7.5.2 Relation between global and local variablesAs shown in Figure 7.11c, the relationship between local and global nodal displacementscan be found considering a small movement δd21 from the current configuration as follows:δul = eT1 δd21 ={cosβ′sin β′}Tδd21 ={cosβ′sin β′}T {δu2 − δu1δw2 − δw1}(7.56)where, e1 is the unit vector lying along the line drawn between the beam nodes in thecurrent configuration. In subsequent writing c = cosβ′ and s = sin β′ is used. Then,Equation 7.56 can be rewritten as follows:δul =[−c −s 0 c s 0]δp = rT δp (7.57)where δp is the variation of the global displacement vector pT = [u1 w1 θ1 u2 w2 θ2]. Thus,Equation 7.57 gives the relationship between the infinitesimal local axial deformation andthe infinitesimal global nodal displacements. Then, to find the relationship between localinfinitesimal nodal rotations and the global nodal displacements, first, the unit vector e2is found as eT2 = [−s c]. Next, a small rigid rotation from the current configurations, asshown in Figure 7.11c, gives, for an infinitesimal angle change, an arc length change ofL′δα = eT2 δd21, so thatδα = 1L′{−sc}T {δu2 − δu1δw2 − δw1}= 1L′[−c −s 0 c s 0]δp = 1L′zT δp (7.58)To find δθl, the variation of local nodal rotations (θ1l = θ1 + β0 − β′, θ1l = θ2 + β0 − β′)136gives:δθl =δ{θ1 + β0 − β′θ2 + β0 − β′}={δθ1 + δβ0 − δβ′δθ2 + δβ0 − δβ′}={δθ1 − δαδθ2 − δα}(7.59)=[[0 0 1 0 0 00 0 0 0 0 1]− 1L′[zTzT]]δp = AT δp (7.60)Finally, to get a relationship between local variables and global variables the completevector of infinitesimal local strain producing deformations can be written as:δpl =δulδθ1lδθ2l =[rTAT]δp = Bδp (7.61)where B = −c −s 0 c s 0−s/L′ c/L′ 1 s/L′ −c/L′ 0−s/L′ c/L′ 0 s/L′ −c/L′ 1 (7.62)From the equilibrium of virtual work, we can write:δpTv qi = Nδulv + M¯1δθ1lv + M¯2δθ2lv = δpTlvqli = (Bδpv)Tqli = δpTv BTqli (7.63)where qi is the vector of global internal forces for element i, qTli = [N M¯1 M¯2] and asubscript v implies a virtual quantity. Realizing that the δpv are arbitrary virtual dis-placements, then global internal forces can be derived as:qi = BTqli (7.64)7.5.3 Consistent tangent stiffness matrixTaking the variation of Equation 7.64, leads to the variationally consistent tangent stiffnessmatrix as follows:δqi =BT δqli + δBTqli = BT δqli +NδB1 + M¯1δB2 + M¯2δB3 = Kt1δp + Ktσδp (7.65)where B2, for example, is the second column in the matrix BT , Kt1 is the transformedmaterial stiffness at the global level and Ktσ is the geometric stiffness. Now, taking thevariation of Equation 7.52 and Equation 7.55 and organizing the results in matrix formleads to:δqli = δNδM¯1δM¯2 = EAL01 0 00 4r2 2r20 2r2 4r2 δpl = Clδpl (7.66)137where r =√I/A is the radius of gyrations. Then, using Equation 7.66 in the first part ofEquation 7.65 yields the standard transformed global tangent stiffness matrix as:BT δqli =BTClδpl = BTClBδp = Kt1δp (7.67)Therefore, Kt1 = BTClB (7.68)Again, taking the variation of the first column of BT yields:δB1 = δr = δ[−c −s 0 c s 0]T=[s −c 0 −s c 0]Tδβ = zδβ′ (7.69)Figure 7.11c shows that δβ′ = δα, hence, it can written as:δB1 =1L′zzT δp (7.70)Now, taking the variation of the second column of BT yields:δB2 = δ(− 1L′z)= δ(− 1L′)z +(− 1L′)δz (7.71)where, δz = δs−c0−sc0=cs0−c−s0δα = −zδα = −1L′rzT δp (7.72)Finding δ(− 1L′ ) and using L′ = L0 + ul, Equation 7.71 can be written as:δB2 =1L′2zrT δp = 1L′2(rzT + zrT)δp (7.73)By inspection of BT it is evident that δB3 = δB2. Therefore, using Equation 7.70 andEquation 7.73 in Equation 7.65 yields:Ktσ =NL′zzT + M¯1 + M¯2L′2(rzT + zrT)(7.74)Finally, combining above derivations the variationally consistent tangent stiffness matrixcan be written as:Kt = BTClB +NL′zzT + M¯1 + M¯2L′2(rzT + zrT)(7.75)1387.5.4 Implementation resultsThe implementation results of corotation algorithm is shown in Figure 7.12. It shows,corotation formulation can predict curvature of a structure more accurately. It is due toits capability to trace the displacement field in a more discretized way, hence ensuring abetter convergence solution. The simulation results of a structure either applying initiallateral load at mid-span or defining an initial out-of-straightness as a displacement profileare shown in Figure 7.13. It shows corotation formulation can converge solution even athigh out-of-straightness value. Moreover, corotation algorithm provides computationallymore economic solution.0 0.05 0.1 0.15 0.2 0.25 0.3012 1040 1 2 3 4 5 6-3-2-10 10-10 1 2 3 4 5 6-101 10-40 2 4 6 8 10 12 14 16 18 201.081.11.12(a)(b)(c)(d)Figure 7.12: Simulation results considering with- and without-corotational algorithm of abeam-column structural member: (a) load-displacement curves, (b) displace-ment along length of the member, (c) curvature along length of the member,and (d) computational time required1390 1 2 3 4 5 600.511.52 1040 1 2 3 4 5 6-2-1.5-1-0.50 10-39 9.2 9.4 9.6 9.8 10 10.2 10.4 10.6 10.8 114.555.566.5(a)(b)(c)Figure 7.13: Simulation results by varying the out-of-straightness parameter of a beam-column structural member: (a) load-displacement curves, (b) curvature alonglength of the member, and (c) computational time required7.6 AnUBC development: support restraintsSupport conditions are implemented into the AnUBC program as defining spring elementbased on [Wri08; Son09; KB00; VK05; Fer09]|. Spring elements are concentrated stiffnessvalues along selected degree-of-freedoms (DOFs). The element formulation was conductedin the local coordinate system of the spring element, which defined by two additionalreference nodes, each with three DOFs. The nodal displacement vector, a, of a springelement can be expressed as:aT = [ui vi θi uj vj θj ] (7.76)140where i and j are the node numbers; u, v are the translational and θ is the rotationaldisplacement. The element displacement vector, u, is calculated as the difference betweenthe nodal displacements of the two spring nodes in all three DOFs. The kinematic matrix,B, for the calculation of u can be expressed as:u =uvθ = B a =1 0 0 −1 0 00 1 0 0 −1 00 0 1 0 0 −1 a (7.77)The load and displacement relationship in an individual DOF of the spring element wascharacterized by an exponential model based on [FFY89]|:Fs = (m0 +m1|∆|)[1− e−k|∆|m0](7.78)where ∆ is the spring displacement, Fs is the spring force, and m0, m1 and k are theparameters representing the intercept and slope of the asymptote and the initial stiffness ofthe force and displacement curve. The secant and tangent moduli of the force displacementcurve are calculated from the quotient and the first derivative of the spring force, Fs, withrespect to the displacement, ∆, as:Espringtan =m1 +(k + km1 ∆m0−m1)e−k|∆|m0 (7.79)Espringsec =(m0 +m1|∆|)[1− e−k|∆|m0]|∆| (7.80)Thus, the spring force vector, fs, is calculated based on the element displacement vector,u, as:fs =Fs,uFs,vFs,θ = Dspringsec u =Espringsec,uEspringsec,vEspringsec,θuvθ (7.81)where Dspringsec is the secant modulus matrix, and Espringsec,u , Espringsec,v and Espringsec,θ are the secantmoduli of the force and displacement curves in all three DOFs of the spring element. Then,the internal force vector and stiffness matrices are derived based on the principle of virtualwork. Considering the virtual work done by the spring force, fs, at a virtual displacement,δu can be written as:δWint = −fsT δu = −fsT B δa = −Pint δa (7.82)from which the internal force vector, Pint, can be calculated by:Pint = fsTB =(Dspringsec u)TB = aT BT Dspringsec B = aT K (7.83)141where K is the stiffness matrix and can be calculated by K = BT Dspringsec B. Then, thetangent stiffness matrix, Ktan, is calculated as the first derivative of the internal forcevector, Pint, with respect to the nodal displacement, a, as:Ktan =∂Pint∂a =∂(fsTB)∂a =∂(fsTB)∂u∂u∂a = BT Dspringtan B (7.84)where Dspringtan is the tangent modulus matrix and can be calculated similarly to Equa-tion 7.81 with the secant moduli replaced by the tangent moduli.7.6.1 Implementation resultsThe FEM implementation results of structures considering different spring constants areshown in Figure 7.14. It shows, with the low and high spring constant values, the structurebehaves like having hinged and fixed supports, respectively. With low spring constant value,structure deforms more than its counterpart of structure having high spring constant value.0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 200.511.52 1040 1 2 3 4 5 6-2-1.5-1-0.500 1 2 3 4 5 6-6-4-202 10-4(a)(b)(c)Figure 7.14: Simulation results by varying the end support conditions of a beam-columnstructural member: (a) load-displacement curves, (b) displacement alonglength of the member, and (c) curvature along length of the member1427.7 Stability of CLT: FE model calibrationFollowing after the development of the AnUBC program successfully, finite element (FE)analyses of CLT structures are performed with the consideration of the non-linear materialmodel, shear deformation, and initial imperfection. To obtain faster simulation withoutcompromising accuracy, a parametric simulation was performed with changing the elementdiscretization numbers. Finally, the study was ended up for CLT structures having thenumber of elements as 22 as an optimal number, and followed by, carried out all analyseswith this number of elements by displacement control solving scheme. In order to computethe CLT’s load carrying capacities accurately, the initial imperfection is quantified. We ex-plored three methods to find the initial imperfection, namely, Southwell plot, probabilisticmodel and FEM analysis. Then, a spring constant is calibrated for the specimen groups.The material model, initial imperfection of specimen groups and the calibration of springconstant are described in the following subsection.7.7.1 Material modelA polynomial model as described in [LO18]|is assumed to represent the non-linear parallel-to-wood-grain stress-strain relationship as follows:σˆ =E0 ˆ ,FtE0> ˆ > 0(rˆ − 2)Fc(ˆp)3+ (3− 2rˆ)Fc(ˆp)2+ E0ˆ , 0 > ˆ > pEd(ˆ− p) + Fc , p > ˆ > u(7.85)where σˆ and ˆ are the stress and corresponding strain; Ft and Fc are the tensile and com-pressive strength; rˆ is the nonlinear quantity which can be obtained as 1.25 FcE0 ; E0 and Edare the initial modulus of elasticity and the slope of the falling branch of the stress straincurve, respectively; p is the strain corresponding to the compression strength, Fc; and uand FtE0 are the maximum compressive and tensile strains, respectively.The material model is shown schematically in Figure 7.15. In the FE model calibration,the average compressive strength and MOE values of CLT3C specimen group from testing(Chapter 2 and 4) are used as the values of Fc and E0, respectively, for the parallel layers ofCLT structures. Because of the other properties were not sensitive to FE model calibration(i.e. spring constant calibration), the CSA O86 specified design values and values publishedin the literature [Hor13; LLL16; MP04; LO18]|are used for the other input properties. Thevalues of rˆ, Ed and p are taken as 1.4, 1000 MPa and 0.003435 respectively, which wereobtained from the CLT3C specimens’ tests. Furthermore, the cross layers’ strength andMOE properties are considered as 15 and120 of the corresponding parallel layers’ properties.Due to no load drops was observed during the testing of CW90 specimen group (Chapter2), no strength degradation was considered for the cross layers’ material model i.e. Edvalue is considered as zero after reaching the limit point. The value of shear modulus isconsidered as 116 of E0 value. The material properties of both layers of CLT structures arepresented in Table 7.3.143Ft/E0FtFcEdE0Fc/E0σˆu p ˆFigure 7.15: Schematic material modelTable 7.3: Material properties dataLayer PropertiesFc Ft E0 Ed p u rˆ[MPa] [MPa] [MPa] [MPa]Parallel -30.4 15.3 9640 -1000 -0.0039 -0.0037 1.4Cross -3.49 2.47 305 -30 -0.0031 -0.0043 3.27.7.1.1 Size and stress distribution effectsSince CLT is a composite system having wide variety of knots within the lumbers andcross-section, size and stress distribution effects are considered in the present study. Inthis context, for the case of uniform stress volume and at an identical probability of failure,strength adjusted was calculated based on [Wei39]|as follows:Fc1Fc2=(V2V1) 1k′(7.86)where, Fc1 is the average compression strength from test at volume 1 (V1) of a specimen;Fc2 is the adjusted average compression strength for volume 2 (V2); and k′ = (COV )−1.085where COV = coefficient of variation. The adjusted materials properties are shown inTable 7.6 as the mean value of the distribution.7.7.2 Quantifying initial imperfectionInitial imperfection of CLT specimen groups are quantified by three approaches, namely,Southwell plot method, statistical approach and FEM analysis.7.7.2.1 Southwell approachThe Southwell plot method enables to determine the buckling capacity of a compressionmember in a non-destructive manner. The method can be derived based on [Sou32; BL10;SH06; CA77]|by considering the governing differential equation of an end-restrained com-pression member under an axial force, P, with an initial geometric imperfection, w0(x) asfollows:EI(w′′ − w′′0) + Pw + Vx −MA = 0 (7.87)where ′′ signifies the curvature quantity of initial and current displacement functions, Vxis the shear force and MA is the end moment induced at joint A. Considering the initial144imperfection in Fourier series as, w0(x) =∞∑n=1an sinnpixKeL, the differential equation becomes:EIw′′ + Pw + Vx −MA = EI∞∑n=1(npiKeL)2an sinnpixKeL(7.88)Applying the appropriate boundary conditions the solution yields:w(x) =∞∑n=1PnPn − P an sinnpixKeLand w − w0 =∞∑n=1PPn − P an sinnpixKeL(7.89)where, Pn =n2pi2EIKeL2(7.90)[Sou32]| reported that, as long as the imperfection is such that a1 exists, then a criticalcondition arises as P → pi2EIKeL2. Furthermore, since in a test, P and the net deflection ofthe midpoint [ w(L/2)−w0(L/2)= δ] are the measurable parameters, then constructing aplot from which P1 = Pcritical can be determined through using of the experimental data.Consequently, the equation of the plot holds as follows:P1(δP)− a1 = δ (7.91)Thus, plotting a graph with δP along the y-axis and δ along the x-axis will attain a slopeof 1/Pcritical and an intercept at y = 0 of a1/Pcritical. From there, the value of the loadcarrying capacity, Pcritical and the imperfection, a1 can be determined.Exercising the above method, the load carrying capacity and initial imperfection of aspecimen (specifically, 6th observation) from CLT3B3 specimen group is presented in Fig-ure 7.16a. It shows the slope and intercept of the equation is 0.002005 and 0.01846,respectively. Therefore, the ultimate capacity and the initial imperfection can be obtainedas 498 kN and 9.2 mm, respectively. The Southwell plot for all specimens of CLT3B3specimen group is shown in Figure 7.16b. The initial imperfection obtained by Southwellplot method for all specimen groups are stated in Table 7.4. Compared with other groups,it shows, Southwell approach cannot predict the imperfection very well. It is due to theincapability of the method to capture the inelastic behaviour of the structures.1450 1 2 3 4 5 600.010.020.030.040.050.060 1 2 3 4 5 600.010.020.030.040.050.06(a) (b)Figure 7.16: Southwell plot: (a) a typical specimen, (b) all specimens of CLT3B4 groupTable 7.4: Quantifying the initial imperfection by different methods and spring constantSpecimen Estimation Initial imperfection Spring constant[kN-m/rad]Southwell plot Probabilistic FEMCLT3B1Average [mm] 1.90 (L/950)? 1.41 (L/1280) 0.90 (L/2000) 0.8×102COV [%] 41.5 27.8 30CLT3B2Average [mm] 1.16 (L/2070) 1.88 (L/1280) 1.20 (L/2000) 2.8×102COV [%] 25.2 27.8 30CLT3B3Average [mm] 1.07 (L/2940) 2.46 (L/1280) 1.58 (L/2000) 2.8×102COV [%] 15.6 27.8 30CLT3B4Average [mm] 1.68 (L/2320) 3.05 (L/1280) 1.95 (L/2000) 2.8×102COV [%] 18.7 27.8 30CLT5B Average [mm] 1.60 (L/1130) 1.41 (L/1280) 0.90 (L/2000) 0.8×102COV [%] 45.8 27.8 30?Values of the initial imperfection in the parenthesis are shown as a fraction of the unsupported length ofthe corresponding specimen groups1467.7.2.2 Probabilistic approachInitial imperfection i.e. out-of-straightness of CLT structures also determined by a prob-abilistic approach based on [CA77; Baz17; Bjo72; Sim90]|. Most specifications for thedelivery of CLT elements contain straightness requirements for the members. For example,ANSI/APA PRG 320-2018 [ANS12]|specifies the limit of out-of-straightness as less than 1.6mm. The measurements during testing, and manufacturing tolerance of CLT’s crooked-ness indicates the magnitude of the out-of-straightness will follow a negatively skeweddistribution. Therefore, the Type I distribution for the smallest values is utilized here, inorder to arrive at a similarity with the distribution of the CLT strength72. Therefore, thedistribution function and the probability density function can be written as:Fé(é) =1− exp[−eµ(é−q)] for −∞ ≤ é ≤ ∞ (7.92)fé(é) =µ exp[µ(é− q)− eµ(é−q)] (7.93)where é has been used to denote the initial out-of-straightness, the factor q represents themode of the distribution, and µ is a measure of the dispersion. The mean value (é¯) andstandard deviation (σé) of the out-of-straightness (é) can be written as:é¯ = ζµ≈ q − 0.577µσé =pi√6µ≈ 1.282µ(7.94)The distribution function and the probability density function with mode q = 0 and µ = 1of a transformed out-of-straightness (ét) can be written as:Fét(ét) = 1− exp[−eét ] (7.95)fét(ét) = exp[ét − eét ] (7.96)Now, the maximum allowable out-of-straightness is assumed as L/1000, and it is furtherassumed that values larger than this may occur with a probability of 2.5%. This corre-sponds to a value of é of 1.3 (antisymmetry with the transformed, Type I extreme valuedistribution). On the other hand, the smallest possible out-of-straightness is considered aszero, corresponding to a perfectly straight column. It is arbitrarily assumed that this valueoccurs with a probability of 1%, and the value of é thus becomes -4.6. Therefore:émax = q +éµ= q + 1.3µ= L1000 émin = q +éµ= q − 4.6µ= 0 (7.97)Solving the µ and q from the Equation 7.97 and using Equation 7.94, it can be derived as:é¯ ≈q − 0.577µ= 0.78( L1000)−0.5775.9 .(L1000) ≈ 0.68L1000 =L1470 (7.98)σé =piµ√6≈ 1.2825.9 (L1000) = 0.22(L1000) =L4600 (7.99)72In fact, CLT’s strength is found to be lognormally distributed (Chapter 2).147Figure 7.17 shows the probability density functions for the initial out-of-straightness andits transformed counterpart, ét. It shows that the most frequently occurring value, averagevalue and COV of the out-of-straightness is L/1280, L/1470 and 32%, respectively. Theinitial imperfection value of all CLT specimen groups obtained by probabilistic approachis listed in Table 7.4.0 1 2 3 400.10.20.30.40.5-5 -4 -3 -2 -1 0 1 20.10.20.30.4(a) (b)Figure 7.17: Probabilistic characteristics of the out-of-straightness variable: (a) actualvalue, (b) standardized value7.7.2.3 FEM approachIn the FE model, it was assumed that the initial deflection is of sinusoidal shape with themaximum value at the mid-span. For the CLT3B2, CLT3B3 and CLT3B4 specimen groups,during experiments an initial preload was applied through the lateral supports that wasequivalent to Length1500 (Chapter 6).Then, the FE model of each specimen groupwas run with the Length1500 value as the ini-tial mid-span value of a sinusoidal deflectioncurve. Then, the sensitivity of the initial de-flection was explored. The sensitivity is eval-uated by calculating the slope of the moment(=load × mid-span displacement) versus an-gle (=mid-span displacementmid-span length ) curve in the elasticzone. Table 7.5 lists the FEM simulation re-sults. It shows the maximum difference of thevariables from the case of Length1500 is 2.5%, thus,small enough to disregard the variation.Table 7.5: Sensitivity of initial deflectionInitialdeflectionSlopeMomentAngleDifference L1500[kN-m/rad] [%]L?500 196.9 2.5L?1000 200.9 0.5L?1500 201.7 0.0L?2000 202.2 0.1L?4000 202.4 0.2?Unsupported length of each specimen group148The initial imperfection values of all CLT specimen groups obtained by FEM analysiscompared with Southwell plot and probabilistic approach, and are listed in Table 7.4.It shows, values obtained by the FEM simulations are close to that of the probabilisticapproach.7.7.3 Spring constant calibrationThe spring constant is calibrated by the least square method where the spring constantyields the minimum value of the sum-of-square-error (Se), written as:Se =n∑i=1(mTesti −mFEMi)2where, m = ∂(moment)∂(angle) (7.100)In Equation 7.100, mi is the slope of the moment-angle curves in the linear elastic zone ofthe ith observation from the respective test and FEM analysis. Here, ‘moment’ is definedas (load × mid-span displacement), and ‘angle’ as mid-span displacementmid-span length . The moment-anglecurves of a typical specimen group is depicted in (Figure 7.18a). It shows, the moment-anglecurves of FEM analysis change with the spring constant (Ks). The value of spring con-stant that minimizes the sum-of-square-error is determined by the 1st derivative of ∂Se∂Ks = 0.Following the above procedure, the spring constant is calibrated with the help of therecorded lateral deformation during the test and analyzing the FE models. In the FEmodels, the initial lateral deflection is assumed to be sinusoidal shape with the value ofLength1500 at the mid-point of CLT structures. Moreover, average compressive strength andMOE values of CLT3C specimen group from test are used as the input material properties(Table 7.3). The slope (mi) is obtained from the test data and FEM analysis consider-ing a value of spring constant varying from 1.0×102 to 6.0×102 kN-m/rad. Then, thesum of square error is quantified. The sum of square error and the 1st derivative of thesum of square error of a typical specimen group are presented in Figure 7.18b. It showsthe best-fit spring constant of CLT3B4 group is calibrated as a value of 2.8×102 kN-m/rad.In this study, the specimens of CLT3B1 and CLT3B4 groups are used to calibrate the springconstants. In the experiment, the same supporting condition was used for the CLT3B1 andCLT5B specimen groups, hence, the same spring constant value is assumed for these twogroups. Then, since the supporting condition for the CLT3B2 and CLT3B3 groups was thesame as the CLT3B4 group, the calibrated spring constant value is assumed to be the samefor these three groups. The calibrated spring constant values of each specimen group arestated in Table 7.4.1490 0.005 0.01 0.015 0.02 0.025 0.030123456789101112130 1 2 3 4 5 6105-150-100-50050100150Moment21.2 2.4 3.6 4.8 6.0 7.25.04.83.3222(b)(a)Figure 7.18: Spring constant calibration result of CLT3B4 specimen group: (a) simulationresults at various spring constants with the test observation, and (b) errormeasurement at each run resulting the optimal value7.8 Stability of CLT: FE model verificationThe FE model prediction is verified with the test result of each specimen group consideringstochastic material data. The calibrated spring constant of each specimen group is usedas the boundary condition accordingly. Material properties and verification results aredescribed in the following subsection.7.8.1 Stochastic material dataTo generate stochastic material data, we considered both compressive strength and MOEproperties as one-dimensional stationary stochastic fields. Thus, the realization of theproperties are computed using the distribution parameters shown in Table 7.6. The prob-ability distribution of the material properties was sought from the CLT3C experimentaldatabase considering the size and stress distribution effects. Then, in the FE models, thediscretized elements were assigned with the generated material properties data earlier andfollowed by performed the static nonlinear analysis.Table 7.6: Stochastic parameters of material propertiesProperties Distribution type Distribution parametersMean COV [%]Fc [MPa] Lognormal 30.4 14MOE [MPa] Gaussian 9640 281507.8.2 FE model verification resultsThe developed FE models were run until failure which was attributed to either the ma-terial’s limit state behaviour or large deformation of structures caused convergence issuesof the solution. To improve the convergence of numerical solution, very small load incre-ments and displacement control iterative solution strategy were used. The FEM resultsare depicted in Figure 7.19. It shows the cumulative density functions (CDFs) of columncapacities are in good agreement between the FEM and experimental results for all groups.During model verification it was observed that with increasing Fc value shifted the capacityCDF curves to the right; and with increasing COV of Fc, the CDF curves shifted to theleft. Moreover, unlike CLT3B3 and CLT3B4 specimen groups, the CDF curves of CLT3B1,CLT3B2 and CLT5B groups are more sensitive to Fc change (MOE has very little effecton the CDF curve shifting and/or changing its COV). A descriptive statistics of modelverification result is stated in Table 7.7. It shows the difference between FEM predictedand experimental observed average peak load is below 2%. Moreover, the difference of thestandard deviation between FEM predicted and test results is found as high as 13%.0 100 200 300 400 500 600 700 80000.10.20.30.40.50.60.70.80.91Figure 7.19: The simulated ultimate load carrying capacity results of the full-scale CLTspecimen groups with the test observations, and the CDF curvesTable 7.7: FEM and test results of the load carrying capacity of full-scale CLT specimensEstimations SpecimensCLT3B1 CLT3B2 CLT3B3 CLT3B4 CLT5BTest FEM Test FEM Test FEM Test FEM Test FEMMin [kN] 287 269 272 270 218 197 176 139 392 442Max [kN] 466 466 404 390 314 338 287 271 581 587Average [kN] 356 349 332 335 273 278 220 221 527 524SD [kN] 42 45 31 35 28 30 32 27 45 38COV [%] 11.78 12.88 9.24 10.31 10.19 10.74 14.59 12.29 8.59 7.275th percentile [kN] 287 275 281 278 227 229 167 176 452 4611517.9 Stability of CLT: P-M interaction capacityThe interaction strength at the beam-column level (P-M) is more complex, depending oncross-section stiffness and strength as well as member length, end restraint, and initialgeometric imperfections. In the design of composite members, depending on the strategyused by the provisions to account for beam-column stability, the beam-column interactionstrength is often taken as the cross-section interaction strength or an interaction surfacethat reduces the cross-section strength to account for member stability [CA77; Ona13;CL87]|. The P-M interaction of CLT’s cross-sectional capacity is determined through FEMsimulations. The elastic limit envelope for multi-layer rectangular cross-section of a typical5-layer CLT by FEM analysis is shown in Figure 7.20. We utilized second-order elasticmethod to determine points on the P-M interaction surface by increasing moment at aconstant axial load. It shows the P-M interaction is represented by a straight line, resemblesthe design equation as:PPu+ MQMu= 1 (7.101)where Pu is the ultimate axial capacity if the load P acted alone and Mu is the ultimatebending moment capacity under the lateral load Q when P = 0. In addition, the loadcarnying capacity curve in relation to slenderness ratio is calculated by following the CSAO86 code design equation as follows:Pr = φc FcAeff KZc,eff Kc,eff = φc FcAeff KZc,eff[1.0 +FcKZc,eff C3c,effN E05KSEKT]−1(7.102)where, Pr is factored capacity, φc is performance factor for compression, Aeff is effec-tive cross-section area, KZc,eff is effective size factor, Kc,eff is effective slenderness factor,Cc,eff is effective slenderness ratio, Fc is adjusted compression strength, N is fitting param-eter, KSE is service condition factor and KT is treatment factor, and E05 = 5th percentileMOE. Figure 7.20 depicts the surfaces of elastic P-M interaction with the CSA code ca-pacity. It shows the loading capacity decreases with increasing slenderness ratio.050010015022000250104 206 308 4010 50050100150200250Axial load (kN)Figure 7.20: 2nd order elastic load carrying capacity in relation to slenderness ratio andloading eccentricity of a typical CLT5B specimen152The cross-sectional strength of a typical 5-layer CLT up to plastic limit is depicted inFigure 7.21. It shows, similar to the elastic envelop surface, the load carrying capacities ofthe plastic envelop surface decrease with increasing slenderness ratio. Moreover, the P-Minteraction can be represented by a parabola. Thus, for the parabolic interaction limitstate design equation, it can be written as follows:(PPu)2+ MQMu= 1 (7.103)050010015020002503005 10203010 4050050100150200250300Axial load (kN)Figure 7.21: 2nd order plastic load carrying capacity in relation to slenderness ratio andloading eccentricity of a typical CLT5B specimenThe load carrying capacities including the elastic and plastic envelops of a typical CLT5Bspecimen are drawn in Figure 7.22. It shows the load carrying capacity drops with increas-ing the applied moment and slenderness ratio. The gap between the elastic and plasticenvelops signifies that, considering P-M linear interaction, will reserve the full utilization ofthe ultimate capacity of CLT structure. The safety indices of the structures considering P-M linear- and parabolic-interactions are determined in the reliability analysis as describedin Chapter 9.Axial load (kN)Figure 7.22: Elastic and plastic range load carrying capacity of a typical CLT5B specimen,and the contour of P-M interaction curves from the plastic analysis result1537.10 ConclusionIn this chapter, the numerical model development and numerical results of full-scale spec-imens are presented. In order to compute the ultimate load carrying capacity of full-scale CLT specimens, a finite element program, named after, Analysis of Universal Beam-Columns (AnUBC), is developed in the MATLAB computing environment. For examiningthe computation efficiency, both displacement and force formulations for multi-layer fi-bre element are implemented into the AnUBC program. Result shows that given the sameelement number of a beam-column structure, the displacement formulation is more compu-tationally efficient than the force formulation, but at the cost of lower accuracy. Whereas,for larger structures it is evident that the reduction in degrees of freedom (number ofelements) is possible by the force formulation leading to more computationally efficientanalysis than the displacement formulation. Two solution control algorithms, namely, loadcontrol and displacement control algorithms, are implemented into the AnUBC program.In addition, shear deformation included in the model, hence deflection field has been cor-rected accordingly. Hermitian cubic polynomial interpolation function has been consideredfor bending formulation and linear interpolation function has been taken for shear defor-mation. To account large displacements and rotations of a structure at the global level,corotational formulation is acquired in the model. Result shows, corotational formulationcan converge even at high out-of-straightness value with a reasonable computational time.Following after the development of the AnUBC program successfully, FEM analyses offull-scale CLT structures are performed with the consideration of non-linear parallel-to-wood-grain stress-strain relationship, size effect of material properties, shear deformation,initial imperfection, and different support conditions. The initial imperfection is quanti-fied by three approaches, namely, Southwell plot, probabilistic model and FEM analysis.The study calculated the initial imperfection of full-scale specimens ranges from L/950to L/2940. The boundary condition in terms of spring constant is determined with thehelp of the recorded lateral deformation during the test and analyzing the FE models.Result found that the calibrated spring constants for the CLT3B1 and CLT5B specimengroups as 0.8×102 kN-m/rad and for the rest of the specimen groups as 2.8×102 kN-m/rad.The FE model prediction is then verified with the test result of each full-scale specimengroup considering stochastic material data. The FEM study found that the average peakloads of CLT3B1, CLT3B2, CLT3B3, CLT3B4, and CLT5B specimen groups are 349 kN,335 kN, 278 kN, 221 kN and 524 kN, respectively. The difference between FEM predictedand experimental observed average peak load is below 2%, whereas, the difference of thestandard deviation parameter is found as high as 13%. Moreover, the P-M interaction ofCLT’s cross-sectional capacity is determined through FEM simulations. We utilized secondorder elastic method to determine points on the P-M interaction surface by increasingmoment at a constant axial load. Result shows that the P-M interaction can be representedby a straight line and parabola at the elastic and plastic limit, respectively.154Chapter 8Structural capacity of CLTIn this chapter, we describe the compressive capacity of cross-laminated timber (CLT)elements across various slenderness ratios. Different methods73 to obtain load carryingcapacity of CLT structures are described from section 8.2 to section 8.5. Then, comparisonbetween different methods are presented in section 8.6. The characteristic capacity of CLTspecimens with its code’s counterpart is pointed out in section 8.7. In order to better fitthe code-prescribed design equation with the experimental results, sensitivity of the designequation is also examined.8.1 OverviewOur primary objective is to investigate the effects of cross-layers on the CLT’s compressioncapacity across various slenderness ratios. The hypothesis being tested is that the cross-layers have some contributions toward the compressive resistance of CLT structures. Theload carrying capacity was considered as the independent (indicator) variable. To examinethe hypothesis, we employed different methods to calculate the CLT’s compression capacity.In the calculations, 3- and 5-layer CLT specimens were considered whereas, 7- and 9-layerCLT specimens were discarded due to having limited test data point of these groups onthe capacity curve. Table 8.1 states the information regarding specimens scale, speciesidentity, testing standards and loading protocol. The naming system and dimension ofall specimen groups are described in Chapter 2. The species’ grading information andgeometric properties of the specimens are also listed in Chapter 2.Table 8.1: Design case overviewKeywords DescriptionScaling Medium and full-scale:CLT0−90−0 (CLT3)CLT0−90−0−90−0 (CLT5)Species V2M1.1Standards CSA O86-2014Loading Axial compression73Experimental, theoretical, numerical and empirical methods.1558.2 Experimental methodsCLT compression capacity can be determined by conducting experiments under axial com-pression load. Specimens including both stubby and slender CLT elements were tested toinvestigate their axial compressive behaviour (Chapters 2, 4 and 6). Changing in mem-ber’s lengths, sectional sizes and material strengths were used to quantify the influence ofmember geometry and constituent material properties on the structural behaviour of CLTelements. The capacity of CLT structures were determined by loading up to the limit state.The results are described in the following subsection.8.2.1 Capacity by experimentsThe experimental setup, failure modes and mechanical properties of CLT structures aredescribed in Chapter 2 and Chapter 6. Displacement controlled loading was applied untilthe specimen drops its load carrying capacity. The adjusted experimental load carryingcapacity of all specimen groups are shown in Figure 8.1. It includes the data frequency,box-and-whisker statistics summary, and each specimen result with the outliers. Moreover,the descriptive statistics of the test results are presented in Table 8.2. It can be shownthat for all group of specimens, capacity decreases with increasing slenderness ratio. Loadcarrying capacity of 5-layer CLT is higher than 3-layer CLT. It is due to the effectivecross-sectional area of 5-layer CLT was higher than the CLT3 specimen group.llllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllll llllllllllllllllllllllllllllllllllllllllll llllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllll0200400600CLT3C1CLT3C2CLT3B1CLT3B2CLT3B3CLT3B4CLT5C1CLT5C2CLT5BSpecimenUltimate Capacity adj (kN)r Outlier· Specimen resultFigure 8.1: Characteristic ultimate load carrying capacity of 3- and 5-layer CLT specimens156Table 8.2: Statistical data of characteristic capacity of 3- and 5-layer CLT specimensEstimations SpecimensCLT3C1 CLT3C2 CLT3B1 CLT3B2 CLT3B3 CLT3B4 CLT5C1 CLT5C2 CLT5BMin [kN] 324 331 276 261 216 179 429 456 417Max [kN] 457 537 411 375 306 269 565 647 492Average [kN] 366 434 332 314 264 218 510 552 457SD [kN] 32 49 31 26 25 27 34 48 23COV [%] 8.67 11.22 9.41 8.43 9.51 12.39 6.74 8.69 5.145th P. [kN] 314 354 281 270 223 174 454 473 4188.2.2 Capacity by Soutwell plotAs described in Chapter 7, the Southwell plot is a graphical method of determining a struc-ture’s critical load74 experimentally. The underlying assumption of this method is if theload-deflection curve approaches asymptotically the value of load given by the inverse slopeof the straight line. Following the procedure presented in Chapter 7, the capacity calculatedby Southwell plot approach, Euler equation and experimental results of 28 samples fromCLT3B4 specimens group are presented in Figure 8.2. In general, capacity calculated bySouthwell approach is lower than the Euler capacity and experimental results. The reasonSouthwell plot is lower than the Euler counterpart is that it considers initial imperfectionwhereas the other does not. On the other hand, due to materials non-linearity and supportfixity, Southwell approach yields lower value than the test results.1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28050100150200250300Figure 8.2: Capacities of CLT3B4 group by Southwell plot, experiments and Euler equation74Critical load is the load level at which, the compression member comes to be in a state of unstableequilibrium.157The capacities calculated by Southwell plot approach and experimental results of CLT3B1,CLT3B2, CLT3B3, CLT3B4, and CLT5B specimen groups are presented in Figure 8.3. Inaddition, the comparative results between experiments, Euler and Southerwell plot aregiven in Table 8.3. It shows, in general, capacity calculated by Southwell plot shows highervariability than the experimental results. The reason is that in the experimental case,materials and structural non-linearities and support fixity of CLT structures contribute tothe overall capacity. Southwell plot determine the capacity based on the initial load recordsfrom the experiment which was sometimes contained error data points. In addition, loadeccentricity and non-homogeneity nature of the CLT material could play a role to showthe difference.0 5 10 15 20 25 3002004006008001000KeLFigure 8.3: Capacities of full-scale specimen groups by Southwell plot with the test resultsTable 8.3: Capacity of full-scale specimens by test, Southwell plot and Euler equationSpecimengroupTest result Southwell Euler SouthwellTest resultSouthwellEulerMean COV Mean COV Mean COV[kN] [%] [kN] [%] [kN] [%] Mean COV Mean COVCLT3B1 350 12.9 482 41.5 581 1.38 3.22 0.83CLT3B2 332 9.2 359 25.2 521 1.08 2.73 0.69CLT3B3 273 10.2 202 15.6 293 0.74 1.53 0.69CLT3B4 220 14.6 171 18.7 231 0.78 1.28 0.74CLT5B 527 8.6 773 45.8 1924 1.47 5.35 0.408.3 Theoretical methodsThe capacity of CLT compression members is calculated considering elastic and inelasticbuckling theory. These two approaches are presented in the following subsection.1588.3.1 Capacity by elastic buckling theoryUsing second-order differential equations75, the elastic buckling loads for a compressionmember with a boundary condition can be expressed as:Px =pi2EIx(KxL)2, Py =pi2EIy(KyL)2, Pφ =[pi2EIwKzL2+GJ]1x20 + y20 +Ix+IyA(8.1)where Px is the flexural buckling load about the x-axis, Py is the flexural buckling loadabout the y-axis, Pφ is the torsional buckling load about z-axis, (x0, y0) is the shear centreof the cross-section, Kx,Ky, and, Kz are the effective length factor, E is the elastic mod-ulus, G is the shear modulus, L is length, Ix is the moment of inertia about x-axis, Iyis the moment of inertia about y-axis, Iw is the warping moment of inertia, and J is thepolar moment of inertia. All geometric properties of CLT specimens are calculated basedon effective cross-section.Following the procedure described above, the buckling capacity of a typical CLT3B1 spec-imen group is plotted in Figure 8.4. Note that in calculation, geometric properties ofa CLT specimen are kept constant except varying the length to get the capacity curve.Figure 8.4 shows, at lower slenderness ratio (less than 11), torsional buckling about theperpendicular-to-cross-section-plane axis governs the capacity, whereas, for higher slender-ness ratios, flexural buckling about the minor axis76 governs. Therefore, we can disregardflexural buckling about major axis for the entire range of the slenderness ratios. In ad-dition, for lower slenderness ratio, CLT’s failure is controlled by material’s failure ratherthan failure in buckling as described in Chapter 2. Therefore, we can also eliminate thetorsional buckling phenomenon for CLT elements.0 5 10 15 20 25 30 35 40 45 5001002003004005006007008009001000KeLFigure 8.4: Capacity curves of CLT3B1 group considering flexural and torsional buckling75Equations of equilibrium are formulated in deformed state.76The major and minor axes are defined as the highest and lowest moments of inertia, respectively.1598.3.2 Capacity by inelastic Engesser’s theoryThe elastic buckling theory derived earlier is assumed the load is always applied throughthe centroid of the element’s cross-sectional area and the member is perfectly straight. Thisis quite unpractical, since manufactured compression member are never perfectly straight,nor the application-of-the-load is known with great accuracy. Then, the critical loads areobtained by making certain assumptions regarding the strain and stress distributions in thecross-section of the CLT structures. The effect of this inelasticity has taken into account bymodifying the elastic modulus according to the two inelastic member theories: the tangentmodulus and the double modulus theories based on [CL87; Sha47]|. The equation to derivethe inelastic capacity is described as follows:Ptangent =pi2EtangentI(KeL)2, Pdouble modulus =pi2 4EEtangent(√E+√Etangent)2I(KeL)2(8.2)The above equations required the constitutive material model. Following the experimentalresults described in Chapter 2, the calibrated constitutive material model in terms of stress-strain relationship and its associated tangent modulus, Et curve of CLT3C1 specimen groupare presented in Figure 8.5. The tangent modulus shows plateau in the beginning up toa strain of 0.0018. With increasing displacement, it drops sharply until a strain of 0.06.This is followed an asymptotically horizontal shape. Inelastic member capacity derivedby tangent modulus and double modulus of typical CLT3B1 specimen group is plottedin Figure 8.6. The CSA code design capacity curve is also shown. It shows, inelasticmember capacity is higher than code design capacity for shorter elements and equal toEuler capacity for longer members.0 0.002 0.004 0.006 0.008 0.01051015202530010002000300040005000600070008000Figure 8.5: Tangent modulus curve of CLT3C1 group with its stress-strain curve1600 5 10 15 20 25 30 35 40 45 500100200300400500600700KeLFigure 8.6: Capacity curves of CLT3B1 group considering tangent and double modulus8.4 Numerical methodsThe CLT compression capacity obtained numerically by considering two approaches, namely,finite element method (FEM) simulations and a numerical method considering residualstress distribution. These two approaches are presented in the following subsection.8.4.1 Capacity by FEMThe compression capacity of medium- and full-scale CLT specimens by FEM simulationsare described in Chapters 3 & 7. The FEM-predicted capacities are drawn in Figure 8.7.The descriptive statistics of FEM & test results is shown in Table 8.4. It shows the FEMand test results are matched with each other very well.0 5 10 15 20 250100200300400500600700Figure 8.7: Load carrying capacity of 3- and 5-layer CLT specimens by test and FEM161Table 8.4: Statistical data of capacity of 3- and 5-layer CLT specimens by test and FEMEstimations SpecimensCLT3C1 CLT3C2 CLT3B1 CLT3B2 CLT3B3 CLT3B4 CLT5C1 CLT5C2 CLT5BTest FEM Test FEM Test FEM Test FEM Test FEM Test FEM Test FEM Test FEM Test FEMMin [kN] 324 319 331 329 287 269 272 270 218 197 176 139 429 401 456 452 392 442Max [kN] 457 537 466 466 404 390 314 338 287 271 565 647 581 587Average [kN] 366 434 356 349 332 335 273 278 220 221 510 552 527 524SD [kN] 32 49 42 45 31 35 28 30 32 27 34 48 45 38COV [%] 8.67 11.22 11.78 12.88 9.24 10.31 10.19 10.74 14.59 12.29 6.74 8.69 8.59 7.275th P. [kN] 314 354 287 275 281 278 227 229 167 176 454 473 452 4618.4.2 Capacity by residual stress effectsAnother numerical method considering residual stress distribution, followed by adjustingthe material properties, is developed to determine the CLT’s capacities. The residual stressmay found in timber either due to drying process of the lumber77 or dimensional changes oftimber. In fact, timber in an unstressed state may undergo dimensional changes in termsof shrinkage78 and movement79 due to having variations in its moisture content and/ortemperature [Din00; SLG03; Hoa00; GAH10]|. Moreover, the degree of dimensional changesof timber is different on the three principal axes, for example, shrinkage in the longitudinal,tangential, and radial directions is <0.1%, 4.8% and 2.7%, respectively [Din00]|.77The drying sequence of a lumber can be divided into three stages. During stage I, the piece is free ofstress and defects (Figure 8.8a). Eventually, the moisture content (MC) of wood near the surface (shell)drops below the fibre saturation point (FSP), leads to shrinking the shell. But the shell cannot shrink asmuch as it wants because the fully swollen core holds it in an oversized position. In stage II, the shelltherefore develops tension perpendicular-to-grain around the outside of the board (Figure 8.8b). As dryingcontinues further, the shell surface begins to level out at a low MC. Subsequently, the core continues drying,eventually drops below FSP, and attempts to shrink, making a transition to stage III (Figure 8.8c).(a) (b) (c)@@RGrowth ring RayAbove FSP Surface checks follow ray’s planeShell below FSP (tension)Core above FSP (compression) Slight cup Honeycomb checkCore below FSP (tension)Shell below FSP (compression)Figure 8.8: Residual stress due to drying: (a) above fibre saturation point (FSP) through-out, (b) shell below FSP and core above FSP; core moisture migrates outwardto the shell, (c) below FSP throughout; eventually reaches uniformly low equi-librium moisture content78Shrinkage is the changes in dimension when green timber is dried to very low MC (e.g. 12%).79Movement is the changes in dimension that arise in timber of low MC due to seasonal or daily changesin the relative humidity of the surrounding atmosphere.162Due to above reason, in our study, we considered residual stress distributions for the CLTspecimens. However, we were unable to measure the residual stress level in the experiments.Therefore, we calibrated the residual stress distribution by a numerical method. In thenumerical method, it is assumed, as an axial force is applied to the sections, the stressdistribution over the cross section will change. As the stress in any fibre equals or exceedsthe yield stress, that particular fibre will yield, and any additional load will be carried bythe fibres that are still elastic. This gradual yielding process has taken into account andcorresponding stress-strain curve has attained numerically. Then, compression capacity iscalculated using inelastic tangent modulus theory. The algorithm of finding compressioncapacity considering residual stress distributions is shown in Figure 8.9.BeginDiscretize the cross-section into fibers alongthe length and/or width.For each fibre, save the areaof fibre (Afib), the distancesfrom the centroid yfib and xfib,Ix−fib and Iy−fib, and the fibrenumber, Nfib in the matrix.Discretize residualstress distribution.Calculate residual stress(σri−fib) each fibre.Nfib∑i=1σri−fib= 0?Calculate effective resid-ual strain (ri) for each fi-bre ri = σri−fib/Efib.Assume centroidal strain, . Calculate total strain foreach fiber, tot = + ri .Assume a material stress-strain curve for each fibre.Calculate stress in each fibre, σfib.Calculate axial force, P =Nfib∑fib=1σfibAfib.Calculate average stress, σ = P/Nfib∑fib=1Afib.Calculate the tangent (EI)Txand (EI)Ty for the σ,(EI)Tx =Nfib∑fib=1(ET−fib{y2fibAfib + Ix−fib}).(EI)Ty =Nfib∑fib=1(ET−fib{x2fibAfib + Iy−fib}).Calculate the critical (KL)xand (KL)y for the σ,(KL)x−cr = pi√(EI)Tx/P .(KL)y−cr = pi√(EI)Ty/P .Achieved upto desiredstrain level?EndyesyesnonoFigure 8.9: Finding compression capacity considering residual stress algorithm163After implementing the Figure 8.9 algorithm in MATLAB computing environment success-fully, a parametric study is performed with varying the stress levels and distribution profiles.The schematic profiles of residual stress on the cross-section of 3-layer CLT are presentedin Figure 8.10d. Three types of profiles are considered. Profile1 is referred as linear distri-bution along the width and entire thickness of the cross-section, profile2 is defined as lineardistribution along the width and linear along each layer separately, and profile3 is referred asthe parabolic distribution. Based on the geometric properties of CLT3B1 specimen group,the parabolic profile is calculated along the width direction as (− 1810×width2+ 29×width−5)and along the thickness direction as (− 102601 × thickness2 + 2051 × thickness− 5). Aiming tomatch the numerical results with the test capacities, followed by performing the parametricstudy, the calibrated stress distributions of a typical 3-layer CLT specimen are obtained.The calibrated stress distributions of three profiles are presented in Figure 8.10-a,b,c. Themaximum stress level is found as high as 4.3 MPa which is about 25% of the characteristicstrength of a 3-layer CLT specimen.Residual stress (MPa)Residual stress (MPa)Residual stress (MPa)420-2-4420-2-4420-2-4(a)(b)(c)LinearParabolicLinearlayer wiseCCCCTTTC CTC CTCCCCTTTC = Compressive stressT = Tensile stressFigure 8.10: Residual stress of a typical CLT3B1 specimen: (a) linear distribution, (b)linearlayer wise distribution, and (c) parabolic distribution, (d) schematic profileThe capacity curves considering three calibrated residual stress profiles of CLT3B1 groupare plotted in Figure 8.11. The code design curve & capacity curves by elastic & inelastictheories are also visualized. It shows, linear1 profile matches with the code design curvevery well. Moreover, test results of different groups are also nicely fitted with the capacitycurve using profile1 stress distribution. Similar trends are observed for CLT5B group.1640 5 10 15 20 25 30 35 40 45 500100200300400500600700KeLFigure 8.11: Capacity curves of CLT3B1 groups considering residual stress distribution8.5 Empirical methodThe compression capacity of CLT is calculated following a design formula. The designformulae specified in the current codes of practice in Canada, namely, CSA O86 are in theform of empirical equations. The method is described in the following subsection.8.5.1 Capacity by CSA O86 codeIn the CSA code, the factored compressive resistance parallel-to-grain is calculated accord-ing to Equation 7.102 in Chapter 7. This equation depends on effective cross-section area,short member strength, size factor, and slenderness factor (Kc) of the member. The factorsused in the code design equations are explained in the following subsection.8.5.2 Slenderness factor, KcIn the code, the slenderness factor, Kc, as stated in Equation 2.3 in Chapter 2, is used torelate slenderness ratio to the loading capacity. The factor depends on dimension of thespecimen in the buckling direction, effective length and material properties. To determineeffective length, a factor introduced in the code, namely, effective length factor (Ke). TheKe factor is discussed in the following subsection.8.5.3 Effective length factor, KeThe Ke-factor in the code is simply a mathematical adjustment to the perfect columnequation to predict the capacity of an actual column. Mathematically, the effective lengthfactor or the elastic Ke-factor is defined as:Ke =√PEulerP=√pi2EIL2 P(8.3)165where PEuler is the Euler load, the elastic buckling load of a pin-ended column; P is theelastic buckling load of an end-restrained column; E is the modulus of elasticity; I isthe moment of inertia in the flexural buckling plane; and L is the unsupported lengthof column. The equation for obtaining effective length factor for end restraints columnis adopted based on [CL87; Hel07]|. Thereby, evaluating the relative stiffness factors GAof end A and GB of end B, the effective length factor Ke, can be obtained solving thefollowing equation:GAGB4(piKe)2+(GA +GB2)(1− pi/Ketan (pi/Ke))+ 2 tan (pi/2Ke)pi/Ke− 1 = 0 (8.4)Following the above equation, the effective length factor curve is computed and plotted inFigure 8.12. Figure 8.12a is computed at a specific spring constant and Figure 8.12b isobtained for a specific geometric property. The calculated effective length factor of CLTspecimen groups is indicated on the figures. Moreover, the code recommended value isdrawn. It shows, code has uniform value regardless the variation of the member’s sizeand intensity of support restraint condition, whereas, the computed factor decreases withincreasing slenderness ratio or support restraints’ conditions.0 10 20 30 40 500.50.550.60.650.70.750.80.850.90.9510 2 4 6 8 101080.50.550.60.650.70.750.80.850.90.951(a) (b)Figure 8.12: Effective length factor: (a) varying with slenderness ratio, and (b) varyingwith spring constantThe calculated effective length factor varying with length and spring constant is presentedin Figure 8.13. Moreover, effective length factors of CLT3B and CLT5B specimen groups166are shown in the figure. It shows that for short structures (low slenderness ratio), springconstant has negligible effect on the effective length factor. It is due to failure of thesestructure is controlled by material’s intrinsic behaviour. Due to discrepancy found betweenthe code and calculated Ke values, we employed different methods to calculate the factorand compared the values as described in the next subsection.00 50.24100.40.63201080.823011400500.50.550.60.650.70.750.80.850.90.951Figure 8.13: Effective length factor in relation with spring constant and slenderness ratio8.5.3.1 Ke by different methodsThe methodology to obtain Ke corresponding to capacity curves of different methods isdescribed in Figure 8.14. In tangent modulus method, first Ktangente is calibrated usingexperimental load carrying capacity. Corresponding tangent modulus, strain and % ofelastic fibre (equivalent to EtE ) are then obtained. The calibrated Ktangente is then justifiedwith the experimental observation of strain level and elastic fibre quantity. Finally,Ktangenteis founded. In the test method, Kteste is obtained by comparing the characteristic capacitywith the code capacity. In the FEM, first, support restraint parameters (spring constant,SFEMc ) are calibrated based on the test capacity. Load carrying capacity is computedusing pin-pin support restraint condition and calibrated SFEMc . KFEMe then calculatedusing Equation 8.3. In the residual-stress-effect method, first, stress distribution profileis calibrated using the FEM capacity. Kresiduale is then obtained using the computedcapacity. In the solving-transcendental-equation method, Kequatione is calibrated using theFEM capacity. Then, corresponding support restraint parameter is justified with the FEMcalibrated spring constant. After justification, Kequatione is established. The values obtainedby different methods are listed in Table 8.5. The code specified value is also shown. Itshows Ke calculated by numerical methods are more accurate.167TangentGet testcapacity,stress-strainrelation, Eof materialDeterminetangentmodulus, EtAssume a KeCalculatecapacity,Ptangent =pi2 Et I(Ke L)2||Ptest −Ptangent||≤ toler-ance?Get EtFind %of elasticfibre, f ′t andstrain, ′tIs f ′t , ′t≈ testobserva-tion?DiscardNoteKtangenteGet stress-strainrelation, Eof materialGet char-acteristiccapacity fromtest, PtestAssume a KeCalculatecapacityaccording tocode designequation,Pcode||Ptest −Pcode|| ≤toler-ance?NoteKtesteGet testcapacity,stress-strainrelation, Eof materialAssumea springconstant, ScDetect initialimperfectionCalculatecapacity,PFEM||P∆test−P∆FEM ||≤ toler-ance?Verify ScNote PFEMCalculateKFEMe =√P pin−pinEulerPFEMGet FEMcapacity,stress-strainrelation, Eof materialAssumeresidualstress distri-bution, SDCalculatecapacity,PresidualFindKresiduale||KFEMe −Kresiduale ||≤ toler-ance?Verify SDNoteKresidualeGet stress-strainrelation, Eof materialAssumea springconstant, ScSolve tran-scendentalequationNoteKequationeCalculatecapacity,Pcode||PFEM−Pcode|| ≤toler-ance?NoteSequationc||SFEMc −Sequationc ||≤ toler-ance?DiscardNoteKequationeTest FEM Residual Equationyesyesyesyesyesyesyesnono nononono noFigure 8.14: Finding effective length factor for the compression member168Table 8.5: Effective length factor obtained by different methodsSpecimen group KeKteste Ktangente KFEMe Kresiduale Kequatione KcodeeCLT3C1 1 0.95 1 1 1 0.65CLT3C2 0.99 0.99 0.99 1 1 0.65CLT3B1 0.88 0.91 0.86 0.87 0.86 0.65CLT3B2 0.65 0.89 0.66 0.67 0.67 0.65CLT3B3 0.63 0.65 0.65 0.65 0.64 0.65CLT3B4 0.62 0.63 0.60 0.61 0.62 0.65CLT5C1 1 1 1 1 1 0.65CLT5C2 1 1 1 1 1 0.65CLT5B 0.92 0.98 0.94 0.94 0.95 0.658.6 Capacity comparisonAt this stage, the capacity obtained by different methods are compared and presented inTable 8.6. It shows, the capacity predicted by Southwell plot and theoretical methods arenot accurate; maximum discrepancy between the capacities predicted and test results isfound to be as much as 79%. On the other hand, CSA code design equation can predict thecapacity reasonably well but depends on the short member compression strength. Note thatthe code capacity in Table 8.6 is calculated using the tabulated material properties from[CSA16]|and performance factor of 1.0. Therefore, it can be said that design procedure tocalculate CLT’s capacity by CSA O86 code is acceptable. The capacities of CLT structureusing the CSA code design equations are described in the next section.Table 8.6: Comparison of load carrying capacity obtained by different methodsSpecimengroupCapacity by different methods in kNExperimental Theoretical Numerical EmpiricalTest∗ Southwell∗ Elastic TangentmodulusDoublemodulusResidualstress1Residualstress2Residualstress3FEM∗ CSA code†CLT3C1 314 - - 354 390 320 323 342 318 215CLT3C2 354 - - 399 440 361 364 386 338 217CLT3B1 287 153 581 236 252 280 283 299 275 202CLT3B2 281 210 521 212 225 283 285 299 278 183CLT3B3 227 150 293 197 208 230 231 240 229 152CLT3B4 167 118 165 166 168 169 169 171 176 119CLT5C1 454 - - 510 546 460 463 481 433 324CLT5C2 473 - - 519 551 477 478 496 470 343CLT5B 452 191 1924 423 446 455 458 476 461 318∗ 5% lower tolerance limit value, †Using the material properties from [CSA16]|and performance factor of 1.01698.7 CSA code capacityThe ultimate capacities of 3- and 5-layer CLT compression members with the capacitycurves computed using the CSA O86 code design equation are shown in Figure 8.15 andFigure 8.16, respectively. The test capacities of the CLT specimen groups are presentedby box-and-whisker plot. It can be seen that the calculated capacity with the specifiedproperties underestimates the CLT’s test capacity by 42% when slenderness ratio is lessthan 6. With increasing slenderness ratio (SR), the gap between calculated capacity curvesusing code specified strength and characteristic strength decreases gradually. At about SRof 50, capacity curves using code specified strength and characteristic strength meet witheach other. The reason is that the capacity of member with high SR is primarily controlledby MOE and the characteristic MOE value of CLT specimens does not differ from the codespecified MOE value.Table 8.7 enumerates the difference between test results and capacity calculated using codespecified compression strength and different performance factor. Results show calculatedcapacity using the performance factor prescribed in the CSA code is underestimated testresults by a significant amount. For example, with the current practice value of perfor-mance factor of 0.8 and code specified properties, the calculated capacity is obtained asalmost half of its experimental counterpart for medium-scale specimens. The calibrationof performance factor is done in the reliability analysis later on.lllll01002003004005006000 10 20 30 40 50Effective slenderness ratioUltimate Capacity (kN)CLT3B1CLT3B2CLT3B3CLT3B4CLT3C1CLT3C2t Outlier+ Specimen result— Euler capacity— Testdesign capacity with φc = 1.0- - Testdesign capacity with φc = 0.8— Codedesign capacity with φc = 1.0- - Codedesign capacity with φc = 0.8Figure 8.15: Load carrying capacity curves of the 3-layer CLT with the test observations17001002003004005006007000 10 20 30 40 50Effective slenderness ratioUltimate Capacity (kN)CLT5BCLT5C1CLT5C2t Outlier+ Specimen result— Euler capacity— Testdesign capacity with φc = 1.0- - Testdesign capacity with φc = 0.8— Codedesign capacity with φc = 1.0- - Codedesign capacity with φc = 0.8Figure 8.16: Load carrying capacity curves of the 5-layer CLT with the test observationsTable 8.7: Capacity comparison with different values of the performance factor parameterSpecimen group Test∗ Code†design capacity [kN] Test∗Codeφc=0.8Test∗Codeφc=1.0[kN] φc = 0.8 φc = 1.0CLT3C1 314 172 215 1.83 1.46CLT3C2 354 174 217 2.04 1.63CLT3B1 287 162 202 1.78 1.42CLT3B2 281 146 183 1.92 1.54CLT3B3 227 122 152 1.87 1.49CLT3B4 167 95 119 1.75 1.40CLT5C1 454 259 324 1.75 1.40CLT5C2 473 274 343 1.72 1.38CLT5B 452 254 318 1.78 1.42∗5% lower tolerance limit value †Using the material properties from [CSA16]|8.7.1 Sensibility of the fitting parameterFor economic and safe design, the capacity curve should be placed as near as possible tothe Euler curve for higher slenderness ratios. On the other hand the calculated values171should not be higher than the Euler curve in order to prevent buckling failure. In the CSAcode design equation, mentioned in Equation 7.102 in Chapter 7, the factor, N controlsthe shape of the polynomial capacity curve. The suggested value of the empirical curveshape parameter N in the CSA O86 code is 35. The sensitivity of N parameter is exploredto seek the optimal value of this parameter. To check the best fit of the characteristiccapacity curve and the test results, the least square method was applied.The ultimate capacities of 3- and 5-layer CLT compression members with the calculatedcapacity curves considering different N values are shown in Figure 8.17 and Figure 8.18,respectively. Moreover, Table 8.8 states the difference between test capacity and capacitycalculated using characteristic strength from test. The maximum difference between testcapacity and calculated capacity using characteristics strength and N of 35 is found to be3%. Therefore, it can be said that the calculated capacity with the characteristic propertiesand performance factor of 1.0 fits with the test result of CLT specimens very well with Nvalue of 35.lllll01002003004005006000 10 20 30 40 50Effective slenderness ratioUltimate Capacity (kN)CLT3B1CLT3B2CLT3B3CLT3B4CLT3C1CLT3C2t Outlier+ Specimen result— Euler capacity— Testdesign capacity with φc = 1.0 and N=35- - Testdesign capacity with φc = 1.0 and N=30. . Testdesign capacity with φc = 1.0 and N=40— Codedesign capacity with φc = 1.0 and N=35Figure 8.17: Load carrying capacity curves of the 3-layer CLT with the test observationsand sensitivity of the curve fitting parameter17201002003004005006007000 10 20 30 40 50Effective slenderness ratioUltimate Capacity (kN)CLT5BCLT5C1CLT5C2t Outlier+ Specimen result— Euler capacity— Testdesign capacity with φc = 1.0 and N=35- - Testdesign capacity with φc = 1.0 and N=30. . Testdesign capacity with φc = 1.0 and N=40— Codedesign capacity with φc = 1.0 and N=35Figure 8.18: Load carrying capacity curves of the 5-layer CLT with the test observationsand sensitivity of the curve fitting parameterTable 8.8: Capacity comparison with different values of the curve fitting parameterSpecimen group Test∗ Test†design capacity [kN] Test∗Test†N30Test∗Test†N35Test∗Test†N40[kN] N30 N35 N40CLT3C1 314 320 320 321 0.98 0.98 0.98CLT3C2 354 343 343 344 1.03 1.03 1.03CLT3B1 287 278 284 290 1.03 1.01 0.99CLT3B2 281 268 279 287 1.05 1.01 0.98CLT3B3 227 207 220 233 1.10 1.03 0.97CLT3B4 167 157 166 179 1.06 1.01 0.93CLT5C1 454 450 450 451 1.01 1.01 1.01CLT5C2 473 472 472 473 1.00 1.00 1.00CLT5B 452 395 448 453 1.15 1.01 1.00∗5% lower tolerance limit value †Using the characteristic material properties from test1738.8 ConclusionIn (specimens’ length ≤ 250 mm)ter, the load carrying capacities of 3- and 5-layer CLTspecimens having different slenderness ratios under axial compression load are presented.Four methods, namely, experimental, theoretical, numerical and empirical methods areemployed to calculate the load carrying capacity.In the experimental method, CLT compression capacity was determined by conducting ex-periments and using Southwell plot. In general, capacity calculated by Southwell approachis lower than the experimental results. Due to materials non-linearity and support fixity,Southwell approach yields lower value than the test results.In the theoretical method, CLT compression capacity is calculated considering elastic (Eu-ler), inelastic tangent modulus and inelastic double modulus theory. The capacity predictedby Southwell and theoretical methods are not accurate; maximum discrepancy between thecapacities predicted and test results is found to be as much as 79%.In the numerical method, CLT compression capacity was obtained by considering residualstress profiles coupling with tangent modulus theory, and performing FEM simulations.Three types of residual stress profiles are considered; profile1 is referred as linear distribu-tion along the width and entire thickness of the cross-section, profile2 is defined as lineardistribution along the width and linear along each layer separately, and profile3 is referredas the parabolic distribution. Result shows, test capacities of different CLT groups arenicely fitted with the capacity curve obtained by considering profile1 stress distribution.Another numerical approach, the finite element models are described in Chapters 3 and 7.Result shows the FEM capacity has reasonably matched with the test capacity.In the empirical method, CLT compression capacity was calculated by CSA O86 code. Theeffective length factor (Ke) in the code design equation is quantified by test, theoreticaland numerical approaches. Result shows that Ke calculated by numerical methods aremore accurate. Moreover, the design procedure by CSA code can predict the capacityreasonably well but depends on the short member’s compression strength.The study found that the calculated capacity with the specified properties underestimatesthe CLT’s test capacity by 42% when slenderness ratio is less than 6. With increasingslenderness ratio (SR), the gap between calculated capacity curves using code specifiedstrength and characteristic strength decreases gradually. At about SR of 50, capacitycurves using code specified strength and characteristic strength meet with each other. Thereason is that the capacity of member with high SR is primarily controlled by MOE and thecharacteristic MOE value of CLT specimens does not differ from the code specified MOEvalue. The sensitivity of curve fitting parameter (N) is then explored to seek the optimalvalue of this parameter. Result shows the capacity curve having characteristic propertiesand performance factor of 1.0 fits with the characteristic capacity of a CLT specimen verywell with N value of 35.174Chapter 9Structural reliability of CLTMany sources of uncertainty are inherent in structural design. Code requirements haveevolved to include design criteria that take into account some of the sources of uncertaintyin design [WDM10]|. Such criteria are often referred to as reliability80-based design criteria[NC00]|. In this chapter, we describe the procedures to calculate the reliability index ofcross-laminated timber (CLT) elements subjected to axial compression load. The value ofperformance factor recommended by CSA code is also examined. The reliability analysisprocedure is described from section 9.1 to section 9.4. Then, reliability analysis results arepresented in section 9.5. Finally, the reliability-based design capacity of CLT structures ispointed out in section 9.6.9.1 Reliability analysis approachEvery real-life system has a capacity (or resistance) for doing something and is subjectedto some sort of demand (or load). The parameters of the demand (loading) and the load-carrying capacities of structural members are not deterministic quantities [AT07]|. Bothcapacity and demand may change depending on various factors such as, physical variationfactor, statistical variation factor, and model variation factor; and those factors can beviewed as random variables81. When demand exceeds the capacity of the system, a sys-tem failure is reached, given that the system cannot offer the service that it was designedto provide. Therefore, structures must be designed to serve their function with a finiteprobability of failure. The probability of failure (Pf ) is related to the overlapping areaof the load and resistance distributions. In addition to the Pf , another commonly usedprobabilistic measure of safety is the reliability index (β).To find the reliability index, β, the theory of reliability is briefed in this section based on[NC00; AT07; Bod92; Alb07; DM05]|. As previously mentioned, the resistance (R) anddemand (S) of a system both depend on random variables. Consequently, they each havea probability distribution (fs(S) and fR(R)), which in turn combine to generate a jointprobability density function, fRS(R,S). If the joint probability density function is known,80Reliability is often referred as equal to the probability that a structure will not fail to perform itsintended function.81A random variable is defined as a function that maps events onto intervals on the axis of real numbers.175the probability of failure (i.e. falling in the failure region) can be calculated directly:Pf =∫ ∞−∞∫ r≤s−∞fRS(R,S) dr ds (9.1)However, most of the times, the probability distributions of R and S are too complexbecause of the amount of factors affecting them, so they cannot be determined in a mathe-matical way and would require some sort of simulation-based analysis to be run beforehand.For simplicity, a reliability analysis entails the calculation of the probability of failure, de-noted by Pf , which is defined using:Pf ≡ P[g(X) > 0]=∫g(X)<0fX(X)dX (9.2)where X ={X1, X2, ..., XN}T is an N-dimensional random vector, g(X) is the limit statefunction such that g(X) < 0 is defined as system failure, and fX(X) is a joint probabilitydensity function of the random variable X.In most engineering applications, the exact evaluation of the equation shown above isvery difficult or often impossible to obtain since fX(X) is generally non-Gaussian andmay be correlated normal, and g(X) is highly non-linear. In our study, to handle the non-Gaussian fX(X), a transformation form the original X-space into the independent standardnormal U-space is sought out. Besides, mean of correlated normal random variables (µX)is transformed to uncorrelated normal variables (µY) using:{µX} = [T ]{µY} (9.3)where [T] is a transformation matrix, made up of the orthonormal eigenvectors corre-sponding to the eigenvalues of the covariance matrix of correlated normal variables X. Inaddition, First-order Reliability Method (FORM) is employed which approximates g(X)using a First Order Taylor Series Expansion to attenuate its non-linearity. In effort tofind a more accurate and robust estimation of Pf , importance sampling technique is alsoexplored. The FORM and importance sampling method are discussed in the followingsubsection.9.1.1 FORMTo calculate the probability of failure of the system with limit state function g(X) usingFORM, it is necessary to find the most probable point (MPP), which is defined as thepoint x? on the limit state function (g(u) = 0) closest to the origin in the standard normalU-space. The limit state function in U-space is defined as g(u) = g(X(u))= g(X) usingthe Rosenblatt transformation. After finding the MPP, the distance from the MPP tothe origin is commonly called the Hasofer-Lind reliability index and denoted by β. Usingthe reliability index, β, FORM can approximate the probability of failure using linearapproximation of the limit state function from the standard normal cumulative densityfunction (CDF), Φ as:PFORMf∼= Φ(−β) (9.4)1769.1.2 Importance samplingIn general, sampling methods evaluate the limit-state function at many realizations of therandom variables and yield an approximate value of the failure probability.As an example, failure probabilityby Monte Carlo simulation is calcu-lated as follows:PMonteCarlof ≈1NN∑i=1I[g(Xˆi)]= NFN(9.5)where NF is the number of systemfailures, N is the number of simu-lations, Xˆi is the value of the ran-dom variables in the i-th simulation,and g(Xˆ) is the limit state function.The Monte Carlo analysis algorithmis shown in Figure 9.1a.In order to get an accurate and ro-bust estimate of the failure prob-ability with far fewer samples, weinvoked importance sampling (IS)method instead of crude MonteCarlo technique, as the algorithmshown in Figure 9.1b. The failureprobability can be statistically ap-proximated by:P ISf ≈1NN∑i=1{I[Xˆi]fX(Xˆi)h(Xˆi) }(9.6)where h(Xˆ) is the sampling den-sity function and fX(Xˆ)) is the jointprobability density function (PDF)of the variables involved.Begini = 0Generatea randomvalue forevery variableinvolvedEvaluate theequation byreplacing eachvariable withits generatedrandom valuei++MonteCarlo datai < N?EndyesstorenoBeginDefine the limitstate function, g(X)Define each random vari-able’s, Xi, probabilitydistribution, as well as theresulting joint PDF, fX(Xi)Find the design point,x? using FORMGenerate a vector ofnormally distributedrandom numbers,Xˆi, with x? as meanCalculate h(Xˆi)as ϕ(x1, x?1, σx1)ϕ(x2, x?2, σx2)...ϕ(xN , x?N , σxN), whereϕ(xj , x?j , σxj)is the PDFof j-th component of XˆiCalculate the failure prob-ability of the iteration:Pfi =I[g(Xˆi) ≤ 0] fX(Xˆi)h(Xˆi)i < N?The final failuteprobability is:Pf =1NN∑i=1Pfi(b)(a)noyesFigure 9.1: Sampling algorithm: (a) Monte Carlo,(b) importance samplingTherefore, for importance sampling method, once P ISf is calculated, β can be obtainedfrom the inverse of standard normal cumulative density function, Φ−1 as β ∼= Φ−1(−P ISf ).9.1.3 MethodologyTo implement the concept stated above, the methodology that is followed in this studyis shown in Figure 9.2 based on [AT07; Bod92; FFY89; DM05]|. In the beginning stage,random variables are identified. The load carrying capacity of CLT is affected primarily by177a list of random factors, such as, compressive strength and MOE of material, end fixation,initial imperfection of structure, and load eccentricity [CLT11; FFY89]|. The probabilitydistributions of these variables are sought out based on the testing data. An appropriatelimit state function is then assumed to estimate the structural performance criteria. Thelimit state function is evaluated based on considering a design equation prescribed in acode. Then the safety indicies of CLT structures are calculated at different performancefactors used in the design equation considering the mechanical properties of CLT materialderived earlier in Chapters 2 and 4, and its code specified properties using first orderreliability method (FORM) and importance sampling method.StatisticalprocedureDefine design variables,statistical parameters,and limit state functionFORM SamplingEvaluate limitstate functionStop criteria ofthe relibability?Calculate relia-bility index (β)ExperimentalprocedureTest structuresunder loadingNumericalprocedureFinite elementmodel of structuresCalculate structuralresponse: loadcarrying capacitynoyesFigure 9.2: Methodology of reliability study9.2 Reliability analysis: limit state functionTo formulate the problem, typical 3-layer CLT system, under an eccentric axial load, P withan eccentricity, e and a lateral load, Q is illustrated in Figure 9.3. The system’s reliabilitycan be modelled by what is known as its limit state function82 (g) as the difference between82The limit state function is what separates the safe region from the failure region where g < 0 denotesfailure; g > 0 denotes safe; and g = 0 denotes the limit-state surface.178capacity, R, and demand, S. We adopted the limit state function of CLT beam-columnelements as follows:g =R− S (9.7)g =√[P 2max +Q2maxL2]−√[(Pd + Pl)2 + (Qd +Ql)2L2](9.8)g =Pmax ×√(1 + f2 L2)− Pln ×√[(γpdp + lp)2 + f2 (γqdq + lq)2 L2](9.9)where Pmax and Qmax satisfy the interaction equation for axial load and moment, respec-tively; Pd and Pl refer to the dead and live load components of the total axial load P ,respectively; Qd and Ql correspond to the dead and live load components of the lateralload Q, respectively; L is the member length; f is the ratio of Lateral loadAxial load ; γp is the ratio ofPdnPln; γq is the ratio of QdnQln ; dp is the ratio ofPdPdn; dq is the ratio of QdQdn ; lp is the ratio ofPlPln;lq is the ratio of QlQln ; and (Pdn, Pln) and (Qdn, Qln) are the dead and live load componentsof the nominal (design) loads of Pn and Qn, respectively. In this study, Pmax is obtainedby the finite element method (FEM) analysis of CLT structures considering material andgeometric nonlinearities as described in Chapter 7. To minimize computational effort, re-sponse surface method (RSM) is exercised. With the help of RSM, reliability analysis isperformed using RELAN simulation environment83.LP PQLeFigure 9.3: Typical beam-column layout under loading9.2.1 Design equationsTo incorporate capacity and demand variables in the limit state function stated above, adesign equation according to CSA O86 code is evoked. In the study, three scenarios ofinteraction between axial load (P) and moment (M), namely, P-M no-interaction, P-Mlinear-interaction and P-M parabolic-interaction, are considered in the design equations toquantify the reliability indicies of CLT structures. The design equations of beam-columnmembers considering different P-M interaction levels are described in the following subsec-tion.83Foschi, R. 0., Li, H., Folz, B., Yao, F., and Zhang, J. RELAN- Reliability Analysis Software, V9.0.University of British Columbia, Vancouver, Canada. 2010.1799.2.1.1 P-M no-interactionThe design equation for CLT structures to get the ultimate capacity under short-termconcentric loading (i.e. in absence of lateral loads that create moment), can be written as:PPu=1 (9.10)αdζd Pdn + αl ζl Pln =φc FcAeff KZc,eff Kc,eff (9.11)where Pu is the ultimate capacity if the load P acted alone as described in Equation 7.102in Chapter 7; αdand αlare the load factors for dead and live loads, respectively; ζd and ζlare the dead and live load geometric factors which convert the applied loads to compressioncapacity. Thus, using Equation 9.11, the nominal live load, Pln can be represented as:Pln =φc FcAeff KZc,eff Kc,effαdξ γ + αl(9.12)where γ is the ratio of Dead loadnominalLive loadnominal ; and ξ is the ratio ofζdζl. In the study, the variablesof ξ, αd, and αlare taken as 1.0, 1.25, and 1.50, respectively. In addition, three values ofγ are considerer as 0.25, 1.0 and 4.0.9.2.1.2 P-M linear-interactionThe design equation for CLT structures to get the ultimate capacity under axial and lateral(centre-point) loading considering the P-M linear-interaction, can be written as:PPu+ MQMu=1(9.13)αdζd Pdn + αl ζl Plnφc FcAeff KZc,eff Kc,eff+ αd ζd (−Pdn en +Qdn L/4) + αl ζl (−Pln en +Qln L/4)φb Fb Seff=1(9.14)where Mu is the ultimate bending moment capacity under the lateral load Q when P = 0;en is the nominal loading eccentricity, obtained by averaging the corresponding values ateach end of the beam-column elements; φb is the performance factor for bending; Fb is theadjusted bending strength; and Seff is effective section modulus. Thus, when the lateralload is applied as a centre point load, using Equation 9.14, the nominal live load, Pln canbe calculated as:Pln =(αdξ γp + αlφc FcAeff KZc,eff Kc,eff+ αd ξ (−enγp + fγqL/4) + αl (−en + fL/4)φb Fb Seff)−1(9.15)where, γp and γq is the ratio of Dead loadnominalLive loadnominal for the axial load P and lateral load Q,respectively. In the study, both of γp and γq are taken as a three-value as 0.25, 1.0 and 4.0.1809.2.1.3 P-M parabolic-interactionThe design equation for CLT structures to get the ultimate capacity under axial and lateral(centre-point) loading considering the P-M parabolic-interaction, can be written as:(PPu)2+ MQMu=1(9.16)(αdζd Pdn + αl ζl Plnφc FcAeff KZc,eff Kc,eff)2+ αd ζd (−Pdn en +Qdn L/4) + αl ζl (−Pln en +Qln L/4)φb Fb Seff=1(9.17)Thus, when the lateral load is applied as a centre point load, using Equation 9.17, thedesired value of the nominal live load, Pln can be obtained as the smaller positive root ofthe second degree equation as follows:(αdξ γp + αlφc FcAeff KZc,eff Kc,eff)2P 2ln +(αdξ (−enγp + fγqL/4) + αl (−en + fL/4)φb Fb Seff)Pln − 1 = 0(9.18)Solving Equation 9.18 we get,Pln =−(αdξ (−enγp + fγqL/4) + αl (−en + fL/4)φb Fb Seff)±√√√√√√√√√√(αdξ (−enγp + fγqL/4) + αl (−en + fL/4)φb Fb Seff)2+4(αdξ γp + αlφc FcAeff KZc,eff Kc,eff)22(αdξ γp + αlφc FcAeff KZc,eff Kc,eff)2(9.19)9.3 Reliability analysis: resistance variablesThe capacity (resistance) of a beam-column element is influenced by the randomness inthe material properties, axial load eccentricities, initial out-of-straightness, boundary con-ditions and random error. The resistance variables involved in the limit sate function arepresented in this section.1819.3.1 Probability distributionThe probability distributions of resistance variables are sought from the experimental stud-ies described in Chapters 2, 4, 6 and 7. The characteristic compression strength of CLTstructure, Fc is found to be lognormally distributed, as stated in Chapter 2. The mod-ulus of elasticity (MOE) has taken to be normally distributed, as reported in Chapter 4.The correlation values between strengths and stiffness (MOE) of CLT is shown in Chapter2. The initial out-of-straightness, Di is assumed to be normally distributed, as stated inChapter 7. The end restraint in terms of spring constant, Cs is also taken to be Gaussiandistributed, as reported in Chapter 7. The distribution parameters of the variables areshown in Table 9.1. Considering these random variables, a parametric study is then per-formed by FEM analyses in the AnUBC program as decribed in Chapter 7 to generate theresponses of CLT specimen groups. Two variables, namely, spring constant and initial out-of-straightness, having 10 data sets and 28 replicas at each set were selected for simulatingthe responses of each specimen group. The parametric results of all specimen groups areplotted in Figure 9.4. The cumulative density functions (CDFs) with the capacity resultsfrom the stability test presented in Chapter 6, are also drawn in the figure. Using theparametric results, a response surface is generated in the following subsection.Table 9.1: Random variables’ distribution parametersProperties Distribution type Distribution parametersMean COV [%]Fc [MPa] Lognormal 50.8 14MOE [MPa] Gaussian 9500 28Di [mm] Gaussian L/1000 30Cs [N-mm/rad] Gaussian 2.8 ×108 100 100 200 300 400 500 600 700 80000.10.20.30.40.50.60.70.80.91SetSetSetSetSetSetSetSetSetSetFigure 9.4: Parametric results from the FEM simulations with the test observations1829.3.2 Response surfaceIn order to reduce computational demand to evaluate failure probability in reliability anal-ysis, we used response surface method (RSM) to estimate structural capacity of CLTstructures based on [Dav03; Flo97; Li09]|. RSM replaces the actual structural responsewith explicit functions of the random variables of interest. The response surface model forthe mean (µ) and standard deviation (σ) of each group’s load carrying capacity can bewritten in terms of dependent (capacity) and independent (regressors) variables as follows:µ¯k= θ1µf1µ(xˆ) + θ2µf2µ(xˆ) + ...+ θnµfnµ(xˆ) (9.20)σ¯k= θ1σf1σ(xˆ) + θ2σf2σ(xˆ) + ...+ θnσfnσ(xˆ) (9.21)where {µ¯k, σ¯k} = capacity variable of the group k, θn = model parameters, xˆ = regressorsvector, fn(xˆ) = explanatory functions, and Ξ = model error. The regressors vector isconsisted of two variables, spring constant (Cs) and initial out-of-straightness (Di).When constructing the response surface model, data quality and model assumptions areneeded to be examined. Unusual data can be problematic in linear models because theycan unduly influence the results of the analysis, and because their presence may be asignal that the model fails to capture important characteristics of the data. The qualityof model is examined through calculating outliers, residual, leverage and influence. Forthe independent variables, the distance of an observation from the other observations isexamined by leverage plot as shown in Figure 9.5. It shows, capacity is more influenced byout-of-straightness variable (stepper slope) than the spring constant variable. High-leverageobservations (e.g. 2nd and 10th observations) tend to have small residuals, because theseobservations can coerce the regression surface to be close to them. Also these data pointsexert more influence on the model than a point that is close to the centre.−6 −4 −2 0 2 4−6−4−20246Out_of_straightness | othersCapacity | othersllllllllll−2 −1 0 1−4−20246Spring_constant | othersCapacity | othersllllllllll(b)(a)Figure 9.5: Plot of leverage points for checking the fitness of data for developing the ca-pacity regression model of a typical CLT3B2 group: (a) relation with out-of-straightness, and (b) relation with spring constant183The outliers are detected by constructing quantile-comparison plot for the studentizedresiduals as shown in Figure 9.6a. The two dotted lines are drawn at ± 2 standard errorto examine the departure of the model from linearity due to sampling variation. It showsthe largest studentized residual belongs to the 10th observation. In addition, the data isheavy-tailed at bottom rather than at top. Then, the influence of each observation onthe regression coefficients is measured by ‘Cook’s distance’ plot, as shown in Figure 9.6b.It shows ‘Cook’s distance’ of the 1st set of data points is 1.1 which indicates it is highlyinfluential data point and requires attention. After careful investigation, it turns out thathigh value of initial out-of-straightness (Di) and low value of MOE was attributed to thisdata point. Since MOE and Di are random variables having possibility to possess highvariability the warnings is ignored. Next, the leverage of regression is assessed by ‘Hatvalues’ as shown in Figure 9.6c. The lines on the plot are drawn at (mean + 2 standarderror) on the vertical axis, and at (2 × average hat-values) On the horizontal axis. Itshows three observations exceed the cutoffs which have substantial impact on the regressionmodel. 1st observation has a relatively large residual but at a low-leverage point, while 9thobservation has high leverage but a small studentized residual.−1.5 −0.5 0.5 1.5−1012t QuantilesStudentized Residuals(fit1)l lllll l l ll2 4 6 8 100.00.20.40.60.81.01.2Obs. numberCook's distanceCook's distance1950.1 0.3 0.5 0.7−1012Influence PlotCircle size is proportial to Cook's DistanceHat−ValuesStudentized Residualsllllll ll(c)(b)(a)Figure 9.6: Plots for checking the quality of capacity regression model of a typical CLT3B2specimen group: (a) regression model result with the ± 2 standard error, (b)Cook’s distance, and (c) influence point plotAfter the diagnosis of data & model quality as stated above, taking into account theresponse surface fitting errors, the mean & standard deviation of the peak response can bewritten as:µk=[θ1µf1µ(xˆ) + θ2µf2µ(xˆ) + ...+ θnµfnµ(xˆ)].[1− Ξµ](9.22)σk=[θ1σf1σ(xˆ) + θ2σf2σ(xˆ) + ...+ θnσfnσ(xˆ)].[1− Ξσ](9.23)184where Ξµ and Ξσ are the random variables represnting the response surface fitting errors,which are assumed to follow the normal distribution. Using FEM results of each group,the fitting errors of the ith combination of the random variables can be calculated as:Ξiµ =µ¯ik− µikFEMµ¯kΞiσ =σ¯ik− σikFEMσ¯k(9.24)For all specimen groups, the mean and standard deviation of the fitting errors was obtained.The average values of the fitting errors are shown in Table 9.2. The merit of each candidateexplanatory function is judged by the COV of the corresponding model parameters aslisted in Table 9.2. The model quality is measured by R factor and obtained Ξ values. Theregression predicted results and FEM observations are plotted in Figure 9.7. It shows themodels can suitability predict the responses of all CLT specimen groups.0 100 200 300 400 500 600 70001002003004005006007000 10 20 30 40 50 600102030405060(b)(a)Figure 9.7: Regression model results of 3- and 5-layer CLT specimen groups: (a) loadcarrying capacity, (b) standard deviationThen, it is assumed that the peak response follows a lognormal distribution. Therefore, itcan be written as:Pmax =µk√1 + ν2kexp(RN√ln(1 + ν2k))(9.25)where νkis the coefficient of variation which can be calculated as µkσk; and RN is thestandard normal variate, RN (0, 1). The response surface with the input realizations arepresented in Figure 9.8. It shows the sparsity of data points across the considered randomvariables. Using the obtained response surface, the reliability indices are calculated usingFORM and importance sampling method.185Table 9.2: Response surface modelModel specification ModelexplanatoryfunctionsModelparametersSecond moment informationabout model parametersModel qualityinspectorMean COV θ1 θ2 θ3 R factor ΞmeanCLT3B1µCLT3B1f1=1 θ1 18682 0.42 1 -0.19 -0.980.99 166f2= D2iCsθ2 -5345 3.39 - 1 0.024f3=Cs θ3 1390 1.80 - - 1σCLT3B1f1=1 θ1 9823 0.85 1 -0.9 -0.910.99 29f2= DiC2s θ2 42 7.33 - 1 0.67f3=CsDiθ3 652 3.13 - - 1CLT3B2µCLT3B2f1=1 θ1 39134 0.81 1 -0.02 -0.980.99 681f2= Di θ2 -2123 3.19 - 1 -0.16f3=Cs θ3 1265 8.23 - - 1σCLT3B2f1=1 θ1 19749 5.14 1 -0.01 -0.980.98 215f2= Di θ2 -2604 8.25 - 1 -0.16f3=Cs θ3 2453 13.40 - - 1CLT3B3µCLT3B3f1=1 θ1 26432 0.43 1 -0.08 -0.960.99 243f2= Di θ2 -1209 2.01 - 1 -0.16f3=Cs θ3 1915 1.88 - - 1σCLT3B3f1=1 θ1 13645 1.04 1 -0.08 -0.960.99 30f2= D2i θ2 -854 3.53 - 1 -0.16f3=Cs θ3 1115 4.15 - - 1CLT3B4µCLT3B4f1=1 θ1 24573 2.78 1 -0.96 0.860.98 312f2= Di θ2 -1040 6.30 - 1 -0.96f3=DiCs θ3 -2570 5.60 - - 1σCLT3B4f1=1 θ1 34243 11.64 1 -0.9 -0.910.99 29f2= C2sDiθ2 -818 5.80 - 1 0.67f3=D2i θ3 1745 3.13 - - 1CLT5BµCLT5Bf1=1 θ1 22041 0.56 1 -0.31 -0.520.98 427f2= Di θ2 -1218 3.72 - 1 -0.31f3=Cs θ3 10960 4.38 - - 1σCLT5Bf1=1 θ1 19365 4.67 1 -0.05 -0.970.99 86f2= DiC2s θ2 -1058 6.23 - 1 -0.15f3=D2i θ3 1491 5.95 - - 1− Symmetric value10810860400.50.5 1 1.512 22.5 31.510522.530.20.40.60.811.21.41.61.822.2105Figure 9.8: Response surface of CLT3B2 specimen group with the input variables1869.4 Reliability analysis: demand variablesIn practice, the CLT structures are subjected to dead and live loads. The randomness ofload (demand) variables is described in this section.9.4.1 Probability distributionThe probability distribution of load variables are sought from published literatures basedon [FFY89; NC00]|. The normalized dead load, dp stated in Equation 9.9 has taken to benormally distributed, with a mean value of 1.0 and a coefficient of variation of 10%. Thenormalized live load, lp parameter has been described as below:lp = Rr ×Rg (9.26)where the ratio, Rr represents the variability between roof load and predicted roof loadusing the adjusting coefficient stated in NBCC 2015 [NBC15]|, to convert ground to roofloads; and the ratio, Rg is a random variable expressing the variability in total ground load.The variable Rr has taken to be lognormal distributed with the statistical parameters givenin in Table 9.3. The variability in the ground load ratio, Rg is calculated based on a highprobability (value of 29/30) of non-exceedance of the maximum load return in 30 years for6 different locations across Canada. The statistical distribution of Rg is shown in Table 9.3.Table 9.3: Loading distribution parametersLocation R∗r,mean R∗r,cov R†g,mean [kPa] R†g,covVancouver 0.8 1.153309 0.286945Halifax 0.6 0.45 1.110357 0.214543Arvida 0.6 0.45 1.083876 0.167046Ottawa 0.6 0.45 1.105234 0.205532Saskatoon 0.6 0.45 1.098912 0.194295Quebec City 0.6 0.45 1.066857 0.135274∗ Lognormal distribution, † Gumbel distribution9.5 Reliability analysis resultsTo get the reliability index, the limit state function stated in Equation 9.9 is evaluatedusing the capacity and demand variables described earlier. Treating the non-Gaussian &correlated normal random variables, as stated in section 9.1, the PDFs of the resistanceand demand random variables are shown in Figure 9.9a. The histogram of joint PDF ofthese variables is plotted in Figure 9.9b. Finally, random variables were transformed intostandard normal space and reliability index is obtained. The results of reliability analysesof all specimen groups are stated in C.18700.02-100.040.0600.080.1100.120.1420 40302030 10040-100150.050.1150.150.210100.255 5(b)(a)Figure 9.9: Graphical representation of the random variables: (a) PDFs of response anddemand variables, and (b) histogram of their joint probability densityThe reliability index obtained using response surface method with FORM and importancesampling of a typical CLT3B2 specimen is shown in Figure 9.10. It shows, higher valueof the safety index implies lower risk for failure, but also a more expensive structure.To ensure safety and economy, therefore, the calibration of reliability-based code designequations requires the definition of target reliability index. To set a target reliability index,we relied upon published literatures. [FFY89]|reported that in the development of targetreliability index for wood structures in the reliability-based CSA O86 design code, twoobjectives were kept in mind: a) compatibility with reliability levels of other materials forthe same structural applications, and b) compatibility with previous experience. Moreover,the compatibility between wood and light-weight steel would be achieved if reliability indexfor wood would vary from around 2.4 to 2.8, with an average of about 2.6 [FFY89]|.In this study, we put an effort to find a value for the performance factor (φc) used in theCSA O86 design equation for compression capacity of CLT structures which would satisfyabove mentioned both criteria— achieving uniformity in safety across different materials,and without unduly penalizing previous design practise. Since at low f i.e. at low ratioof Lateral loadAxial load , the reliability index, β is most sensitive to the compression strength, Fc andbecomes most critical while, at higher f, it becomes more sensitive to the tensile strength,Ft, we use f = 0 in the subsequent reliability analysis. Moreover, reliability analysesare performed for three values of Dead loadLive load as 0.25, 1.0 and 4.0, however, reliability indicesconsidering Dead loadLive load of 0.25 yield the lowest. Therefore, the subsequent results are reportedin the following subsections based on this condition.188030.51231.51222.50 130-1-1-2 -2-3 -300.511.522.5.020.40 60.81.0Proportion0.000.330.671.00 Importance samplingFigure 9.10: Graphical representation of the reliability index (β) with the joint probabilitydistribution and contour diagram in the standard normal space, and threecriteria for the limit state function evaluation (g > 0 denotes safe, g=0 denotesneutral, and g < 0 denotes failure)9.5.1 P-M no-interactionReliability analyses of 3- and 5-layer CLT structures were conducted considering charac-teristic strength obtained from test (designated by F testc ) and the CSA O86 code specifiedstrength value (designated by F codec ). By doing so, we meet twofold objectives. First, howmuch conservative the code specified value is in terms of safety index. Higher than intendedvalue of safety index (βtarget) would penalize cost of the structure (material) at the end.Second, what would be the safety index of structures using characteristic strength value.Thus, the characteristic strength value is judged by comparing whether it can achieve equalsafety index as the code specified design strength value.Reliability analysis results of 3- and 5-layer CLT specimens at six different locations inCanada using design Equation 9.10 are shown in Figure 9.11. Two juxtapose figures fromleft to right are representing the reliability analysis results using F codec and F testc , respec-tively. Range of βtarget and minimum β corresponding to the current code specified per-formance factor, φc of 0.8 are shown in the graphs. Moreover, the minimum and averageresults of each specimen group are drawn in the figure and listed in Table 9.4. It showsβ decreases gradually with increasing φc. Most of the cases, minimum β obtained for theVancouver location. This is because of the intensity of demand variable of Vancouver ishigher than other considered locations. In addition, higher slender structures (e.g. CLT3B3and CLT3B4) yield minimum β. This is due to utilizing resistance variables stated in Ta-ble 9.1; comparatively wider spectrum of capacity distribution is obtained for the higherslender structures.189The results show that the current design practise in the CSA O86 code having φc of 0.8and code specified strength value, F codec is excessively conservative. With these properties,the average safety index of structures at six locations in Canada is computed as 3.8 whichis 46% higher than the average βtarget of 2.6. On the other hand, the minimum β with thecharacteristic strength value, F testc and φc of 0.8 is obtained as 2.7, whereas the average βat this level is 3.1. In another scenario, setting the average βtarget of 2.6 as the objectivevalue and looking at the average trend-line of β, a performance factor, φc of 0.9 is obtained.0.4 0.5 0.6 0.7 0.8 0.9 11234567890.4 0.5 0.6 0.7 0.8 0.9 1123456789(b)(a) Figure 9.11: Reliability analysis results of 3- and 5-layer CLT specimens at six differentlocations in Canada considering P-M no-interaction in the design equation: (a)using code specified material properties, and (b) using characteristic materialproperties190Table 9.4: Reliability indices considering Dead loadLive load of 0.25 and P-M no-interactionSpecimen EstimationPerformance factor, φc0.4 0.6 0.8 0.9 1.0F testc Fcodec Ftestc Fcodec Ftestc Fcodec Ftestc Fcodec Ftestc FcodecCLT3B1Min 4.41 4.14 3.57 3.58 2.95 3.40 2.70 3.25 2.46 3.08(4.41)? (4.18) (3.57) (3.62) (2.98) (3.43) (2.73) (3.28) (2.49) (3.08)Average 5.31 4.91 4.25 4.09 3.51 3.75 3.18 3.43 2.86 2.85(5.31) (4.95) (4.29) (4.09) (3.54) (3.79) (3.18) (3.47) (2.89) (2.88)CLT3B2Min 4.49 4.22 3.64 3.74 3.03 3.53 2.78 3.34 2.54 2.99(4.53) (4.26) (3.64) (3.74) (3.06) (3.56) (2.78) (3.34) (2.54) (3.02)Average 4.92 4.30 3.84 3.81 3.18 3.60 2.90 3.41 2.64 3.05(4.92) (4.34) (3.84) (3.85) (3.21) (3.60) (2.93) (3.44) (2.64) (3.05)CLT3B3Min 4.32 4.06 3.49 3.42 2.88 3.27 2.62 3.16 2.38 3.16(4.37) (4.06) (3.52) (3.42) (2.90) (3.30) (2.62) (3.19) (2.41) (3.16)Average 4.63 4.78 3.69 4.19 3.00 3.89 2.72 3.68 2.46 3.35(4.63) (4.78) (3.72) (4.24) (3.03) (3.89) (2.74) (3.68) (2.46) (3.38)CLT3B4Min 4.16 3.84 3.33 3.26 2.73 3.02 2.47 2.80 2.16 2.37(4.20) (3.84) (3.36) (3.29) (2.73) (3.02) (2.49) (2.83) (2.18) (2.39)Average 4.41 4.02 3.48 3.39 2.83 3.13 2.54 2.89 2.28 2.47(4.45) (4.02) (3.52) (3.39) (2.85) (3.13) (2.57) (2.92) (2.28) (2.47)CLT5BMin 4.32 4.04 3.49 3.47 2.88 3.27 2.62 3.10 2.39 2.84(4.37) (4.08) (3.52) (3.47) (2.88) (3.27) (2.65) (3.13) (2.39) (2.87)Average 4.65 4.37 3.67 3.80 3.00 3.54 2.72 3.32 2.46 2.96(4.7) (4.37) (3.67) (3.8) (3.03) (3.54) (2.72) (3.36) (2.48) (2.99)?Results in the parenthesis are obtained by the importance sampling method9.5.2 P-M linear-interactionReliability analysis results of 3- and 5-layer CLT specimens at six different locations inCanada using design Equation 9.13 are shown in Figure 9.12. Two juxtapose figures fromleft to right are representing the reliability analysis results using F codec and F testc , respec-tively. Moreover, the minimum and average results of each specimen group are drawn inthe figure and listed in Table 9.5.The trends observed here are the same as the P-M no-interaction case. The results showthat the current design practise in the CSA O86 code having φc of 0.8 and code specifiedstrength value, F codec is excessively conservative. With these properties, the average safetyindex of structures at six locations in Canada is computed as 3.8 which is 46% higherthan the average βtarget of 2.6. On the other hand, the minimum β with the characteristicstrength value, F testc and φc of 0.8 is obtained as 2.7, whereas the average β at this levelis 3.1. In another scenario, setting the average βtarget of 2.6 as the objective value andlooking at the average trend-line of β, a performance factor, φc of 0.9 is obtained.1910.4 0.5 0.6 0.7 0.8 0.9 11234567890.4 0.5 0.6 0.7 0.8 0.9 1123456789(b)(a) Figure 9.12: Reliability analysis results of 3- and 5-layer CLT specimens at six differentlocations in Canada considering P-M linear-interaction in the design equa-tion: (a) using code specified material properties, and (b) using characteristicmaterial properties192Table 9.5: Reliability indices considering Dead loadLive load of 0.25 and P-M linear-interactionSpecimen EstimationPerformance factor, φc0.4 0.6 0.8 0.9 1.0F testc Fcodec Ftestc Fcodec Ftestc Fcodec Ftestc Fcodec Ftestc FcodecCLT3B1Min 4.68 5.48 3.80 4.70 3.11 4.12 2.81 3.86 2.56 3.64(4.68)? (5.53) (3.84) (4.70) (3.11) (4.16) (2.84) (3.9) (2.59) (3.64)Average 5.31 6.01 4.25 4.91 3.51 4.09 3.18 3.75 2.86 3.43(5.36) (6.07) (4.29) (4.91) (3.54) (4.09) (3.18) (3.75) (2.89) (3.47)CLT3B2Min 5.19 5.60 4.19 4.85 3.48 4.27 3.15 3.99 2.88 3.77(5.19) (5.60) (4.19) (4.85) (3.51) (4.27) (3.15) (4.03) (2.91) (3.77)Average 6.33 6.35 4.84 5.41 3.86 4.59 3.40 4.21 3.08 3.98(6.40) (6.35) (4.88) (5.46) (3.86) (4.64) (3.40) (4.21) (3.12) (3.98)CLT3B3Min 4.15 5.36 3.36 4.55 2.76 3.97 2.51 3.73 2.28 3.51(4.15) (5.36) (3.40) (4.55) (2.76) (4.01) (2.54) (3.73) (2.28) (3.51)Average 4.70 5.91 3.83 4.89 3.09 4.22 2.78 3.95 2.45 3.70(4.70) (5.97) (3.83) (4.89) (3.09) (4.27) (2.78) (3.99) (2.47) (3.74)CLT3B4Min 4.57 5.04 3.71 4.24 3.07 3.66 2.81 3.42 2.57 3.20(4.57) (5.09) (3.71) (4.28) (3.07) (3.70) (2.83) (3.42) (2.60) (3.23)Average 5.06 5.38 3.93 4.50 3.22 3.85 2.93 3.58 2.67 3.34(5.11) (5.38) (3.97) (4.50) (3.22) (3.85) (2.93) (3.58) (2.67) (3.38)CLT5BMin 4.64 5.33 3.78 4.55 3.11 3.97 2.80 3.71 2.56 3.49(4.69) (5.33) (3.82) (4.55) (3.14) (4.01) (2.80) (3.75) (2.58) (3.53)Average 5.30 5.88 4.17 4.93 3.37 4.22 3.01 3.92 2.71 3.68(5.35) (5.88) (4.17) (4.98) (3.40) (4.22) (3.04) (3.96) (2.74) (3.68)?Results in the parenthesis are obtained by the importance sampling method9.5.3 P-M parabolic-interactionReliability analysis results of 3- and 5-layer CLT specimens for six different locations inCanada using design Equation 9.16 are shown in Figure 9.13. Two juxtapose figures fromleft to right are representing the reliability analysis results using F codec and F testc , respec-tively. Moreover, the minimum and average results of each specimen group are drawn inthe figure and listed in Table 9.6.The trends observed here are the same as the P-M no-interaction case. The results showthat the current design practise in the CSA O86 code having φc of 0.8 and code specifiedstrength value, F codec is excessively conservative. With these properties, the average safetyindex of structures at six locations in Canada is computed as 3.7 which is 42% higherthan the average βtarget of 2.6. On the other hand, the minimum β with the characteristicstrength value, F testc and φc of 0.8 is obtained as 2.6, whereas the average β at this levelis 2.9. In another scenario, setting the average βtarget of 2.6 as the objective value andlooking at the average trend-line of β, a performance factor, φc of 0.88 is obtained.1930.4 0.5 0.6 0.7 0.8 0.9 11234567890.4 0.5 0.6 0.7 0.8 0.9 1123456789(b)(a)@RFigure 9.13: Reliability analysis results of 3- and 5-layer CLT specimens at six differentlocations in Canada considering P-M parabolic-interaction in the design equa-tion: (a) using code specified material properties, and (b) using characteristicmaterial properties194Table 9.6: Reliability indices considering Dead loadLive load of 0.25 and P-M parabolic-interactionSpecimen EstimationPerformance factor, φc0.4 0.6 0.8 0.9 1.0F testc Fcodec Ftestc Fcodec Ftestc Fcodec Ftestc Fcodec Ftestc FcodecCLT3B1Min 4.55 5.29 3.61 4.47 2.94 3.89 2.64 3.64 2.40 3.42(4.59)? (5.34) (3.61) (4.52) (2.97) (3.89) (2.67) (3.64) (2.40) (3.45)Average 5.16 5.87 4.10 4.75 3.36 3.94 3.03 3.58 2.70 3.26(5.16) (5.92) (4.10) (4.75) (3.39) (3.97) (3.06) (3.58) (2.70) (3.29)CLT3B2Min 5.00 5.37 3.92 4.55 3.23 3.97 2.91 3.70 2.65 3.49(5.05) (5.37) (3.96) (4.55) (3.23) (3.97) (2.94) (3.74) (2.65) (3.52)Average 5.90 6.00 4.42 4.99 3.53 4.32 3.13 3.90 2.82 3.66(5.90) (6.06) (4.46) (5.04) (3.56) (4.32) (3.16) (3.94) (2.82) (3.66)CLT3B3Min 4.01 5.20 3.24 4.40 2.65 3.81 2.40 3.57 2.17 3.35(4.01) (5.25) (3.27) (4.44) (2.68) (3.85) (2.43) (3.57) (2.19) (3.35)Average 4.45 5.52 3.66 4.69 2.94 4.04 2.64 3.77 2.32 3.52(4.49) (5.58) (3.66) (4.69) (2.97) (4.08) (2.67) (3.77) (2.34) (3.52)CLT3B4Min 4.05 4.54 3.17 3.69 2.56 3.07 2.29 2.81 1.99 2.58(4.09) (4.54) (3.17) (3.73) (2.58) (3.10) (2.29) (2.84) (2.01) (2.58)Average 4.55 4.93 3.38 3.95 2.68 3.24 2.38 2.95 2.09 2.69(4.59) (4.93) (3.38) (3.99) (2.68) (3.24) (2.40) (2.98) (2.09) (2.71)CLT5BMin 4.41 5.04 3.48 4.21 2.82 3.62 2.53 3.36 2.28 3.14(4.46) (5.09) (3.51) (4.21) (2.82) (3.62) (2.55) (3.36) (2.30) (3.14)Average 4.91 5.49 3.80 4.54 3.04 3.87 2.71 3.54 2.40 3.29(4.91) (5.54) (3.84) (4.54) (3.04) (3.91) (2.71) (3.54) (2.40) (3.29)?Results in the parenthesis are obtained by the importance sampling method9.5.4 ComparisonReliability analysis results of 3- and 5-layer CLT structures considering P-M no-, linear-and parabolic-interaction are shown in Figure 9.14. Two juxtapose figures from left toright are representing the reliability analysis results using F codec and F testc , respectively.The minimum and average β considering the three design equations are listed in Table 9.7.The results show that for the given performance factors for compression and bending inthe CSA O86 code, namely, φc = 0.80 and φb = 0.90, respectively, the performance of CLTstructures maintenance the adequate reliability level. This conclusion is valid regardless ofthe format used for the design equations. However, β considering P-M linear interaction,yields the most conservative results, leading to higher reliability levels. Then, in terms ofconsidering P-M interaction, the simpler linear interaction relationship is more conservativethan the parabolic interaction. Hence, the minimum β, leading to the most economicdesign, obtained for the design equation considering P-M parabolic-interaction. However,this advantage is obtained at the cost of increased computational complexity.1950.4 0.5 0.6 0.7 0.8 0.9 11234567890.4 0.5 0.6 0.7 0.8 0.9 1123456789(b)(a)RFigure 9.14: The minimum reliability analysis results of 3- and 5-layer CLT specimensconsidering P-M no-interaction, P-M linear-interaction and P-M parabolic-interaction in the design equation: (a) using code specified material properties,and (b) using characteristic material propertiesTable 9.7: Reliability indices considering different design equationsEstimation EquationPerformance factor, φc0.4 0.6 0.8 0.9 1.0F testc Fcodec Ftestc Fcodec Ftestc Fcodec Ftestc Fcodec Ftestc FcodecMinP 4.16 3.84 3.33 3.26 2.73 3.02 2.47 2.80 2.16 2.37(4.20)? (3.84) (3.36) (3.29) (2.73) (3.02) (2.49) (2.83) (2.18) (2.39)P+M 4.15 5.04 3.36 4.24 2.76 3.66 2.51 3.42 2.28 3.20(4.15) (5.09) (3.40) (4.28) (2.76) (3.70) (2.54) (3.42) (2.28) (3.23)P2+M 4.01 4.54 3.17 3.69 2.56 3.07 2.29 2.81 1.99 2.58(4.01) (4.54) (3.17) (3.73) (2.58) (3.10) (2.29) (2.84) (2.01) (2.58)AverageP 4.41 4.02 3.48 3.39 2.83 3.13 2.54 2.89 2.28 2.47(4.45) (4.02) (3.52) (3.39) (2.85) (3.13) (2.57) (2.92) (2.28) (2.47)P+M 4.70 5.38 3.83 4.50 3.09 3.85 2.78 3.58 2.45 3.34(4.70) (5.38) (3.83) (4.5) (3.09) (3.85) (2.78) (3.58) (2.47) (3.38)P2+M 4.45 4.93 3.38 3.95 2.68 3.24 2.38 2.95 2.09 2.69(4.49) (4.93) (3.38) (3.99) (2.68) (3.24) (2.40) (2.98) (2.09) (2.71)?Results in the parenthesis are obtained by the importance sampling method1969.6 Reliability-based design capacityIn this section, the compression capacity of 3- and 5-layer CLT structures are determinedaccording to CSA O86 code design equation using the reliability results found earlier. Thebound of reliability analysis results using Equation 9.10 is shown in Figure 9.15. The re-sults show that the current design practice having φc of 0.8 and code specified strengthvalue, F codec is excessively conservative. The discrepancy can be quantified in terms ofload carrying capacity across different slenderness ratios of CLT structures. With the codespecified value and characteristic value from test, the capacity curves of 3- and 5-layer CLTspecimens using code design equation are calculated and presented in Figure 9.16 and Fig-ure 9.17, respectively. The capacity using characteristic strength and performance factor,φc of 0.8 and 0.9 are drawn on the figures. Results show, both capacity curves lie belowthe test observations. The gap between calculated capacity curves and test results signifiesthe magnitude of the safety level— higher the level safer the structures. The safety levelsare quantified earlier in terms of the reliability indicies.Therefore, it is evident that the safety index of a structural capacity calculated usingthe characteristic mechanical properties of CLT that were established in Chapters 2, 3, 4and 5 are compatible with the capacity calculated using CSA O86 code specified materialproperties. At this safety level, due to meeting the target reliability index, CLT’s designcapacity is also compatible with others building materials such as, steel and concrete. Forutilizing CLT capacity efficiently and economically, using the characteristic properties anda performance factor of 0.9 instead the current prac
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Strength, stiffness, and stability of solid continua: gravity loading scenario on cross-laminated timber Moniruzzaman, P.K.M. 2019
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Title | Strength, stiffness, and stability of solid continua: gravity loading scenario on cross-laminated timber |
Creator |
Moniruzzaman, P.K.M. |
Publisher | University of British Columbia |
Date Issued | 2019 |
Description | Given the variety of wood species available, understanding of cross-laminated timber (CLT) materials is by no means complete. This dissertation serves to advance the state-of-the-art in understanding the material and structural response of the CLT system and developing engineering tools for modelling and predicting such responses. The investigation consisted of an experimental study, numerical study and reliability analysis. The hypothesis being tested is that the cross-layers have some contributions towards the CLT’s behaviour under the axial compression load. In this context, to evaluate the physical and mechanical properties of CLT-lamella(sawn lumber), testing was done on the small-scale (specimens’ length ≤ 250 mm) clear wood and wood contains defects specimens. Then, a medium-scale (495 mm ≤ specimens’ length ≤ 1000 mm) 3-, 5-, 7- and 9-layer CLT columns and a full-scale (specimens’ length ≥ 2400 mm) 3- and 5-layer CLT elements have been tested. In addition, to characterize the stiffness (modulus of elasticity) of CLT materials, we employed three types of testing, namely, compression test, flexural test, and transverse vibration test. A numerical study is then employed. In order to compute the strength and stiffness of medium-scale CLT composite, we developed a nonlinear material model, namely, Subroutine for Orthotropic Materials’ Elasticity & Rate-independent Plasticity (SOME&RIP), and implemented into ANSYS as an UserMat library. In addition, a finite element tool, namely, Analysis of Universal Beam-Columns (AnUBC), considering the material and structural nonlinearities for the stability analysis of full-scale CLT structures is developed in MATLAB. Finally, reliability analysis is carried out considering the sources of uncertainties that can be resulted from production, construction, material and loading conditions. Results show that characteristic strengths of the medium-scale 3-, 5-, 7-, and 9-layer CLT specimen groups are 42%, 21%, 64% and 65% higher than the code specified strength, respectively. Moreover, characteristic stiffness is approximately the same as its code’s counterpart. Following the reliability analysis, we conclude that for utilizing CLT capacity efficiently and economically, using the characteristic properties and a performance factor of 0.9 instead the current practice value of 0.8 is recommended in the CSA O86 code design equation. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2019-11-21 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0385812 |
URI | http://hdl.handle.net/2429/72348 |
Degree |
Doctor of Philosophy - PhD |
Program |
Forestry |
Affiliation |
Forestry, Faculty of |
Degree Grantor | University of British Columbia |
GraduationDate | 2020-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
AggregatedSourceRepository | DSpace |
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