{"http:\/\/dx.doi.org\/10.14288\/1.0063508":{"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool":[{"value":"Applied Science, Faculty of","type":"literal","lang":"en"},{"value":"Civil Engineering, Department of","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider":[{"value":"DSpace","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeCampus":[{"value":"UBCV","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/creator":[{"value":"Moses, David Michael","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/issued":[{"value":"2009-07-27T19:21:10Z","type":"literal","lang":"en"},{"value":"2000","type":"literal","lang":"en"}],"http:\/\/vivoweb.org\/ontology\/core#relatedDegree":[{"value":"Doctor of Philosophy - PhD","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeGrantor":[{"value":"University of British Columbia","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/description":[{"value":"Models for the behaviour of structural composite lumber and solid wood are presented and\r\nverified for a variety of configurations. In particular, laminated strand lumber (LSL) is\r\ninvestigated for the influence of strand orientation and stacking sequence on its material\r\nproperties. LSL panels are modelled using laminate theory and the First Order Reliability\r\nMethod to determine the probability distributions of elastic properties and ultimate strengths.\r\nThe results are verified against properties determined from an extensive series of experiments on\r\ncustom-manufactured LSL panels. The laminate theory is capable of predicting panel properties\r\nand indicates that the properties are related to the degree of strand alignment. This modelling\r\ntechnique allows material designers to set target levels on panel properties and on variability by\r\nusing the properties of the constituent layers.\r\nThe behaviour of single-dowel mild steel connections in the five LSL panel types are\r\ndetermined experimentally for three end distances, three slenderness ratios and two edge\r\ndistances. (The connection specimens are cut parallel, perpendicular and at 45\u00b0 to the main\r\nstrand axis.) Three-dimensional finite element models of each tested connection are used to\r\npredict load-displacement behaviour, ultimate strength and mode of failure. The anisotropic\r\nplasticity constitutive model is used together with the Weibull weakest link strength criterion for\r\nbrittle materials to simulate the material behaviour of the LSL panels in the connection. The\r\nthree-dimensional models and the single-dowel connection experiments indicate that panels with\r\nsome strands oriented at \u00b1 45\u00b0 fail at higher loads than fully oriented panels. A parametric study\r\nof the model is used to indicate the critical variables in connection modelling and to isolate the\r\nmost significant material tests required for modelling.\r\nApplications of the three-dimensional model are illustrated for a number of cases,\r\nincluding: a) dowel embedment tests, b) shear block tests, and c) a simplified one-dimensional\r\nspring model for the single-dowel connection. The shear block model illustrates the effect of the\r\nnon-uniform shear stresses on the failure plane and the effects of the specimen notch. The one-dimensional\r\nspring model is shown to be useful for predicting the load-displacement behaviour\r\nand ultimate strength of multiple-bolt connections. These applications indicate the adaptability of\r\nthe model to changes in material properties and connection geometry, and its ability to predict\r\nload-displacement, ultimate strength and mode of failure.","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO":[{"value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/11284?expand=metadata","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/extent":[{"value":"10686211 bytes","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/elements\/1.1\/format":[{"value":"application\/pdf","type":"literal","lang":"en"}],"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note":[{"value":"CONSTITUTIVE AND ANALYTICAL MODELS FOR STRUCTURAL COMPOSITE LUMBER WITH APPLICATIONS TO BOLTED CONNECTIONS by DAVID MICHAEL MOSES B.Sc. (Hons.), Queen's University, 1993 M.Sc.(Eng.), Queen's University, 1995 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of Civil Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June 2000 \u00a9 David Michael Moses, 2000 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, 1 agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of C\\\\|\\L S^SMfeftlMS The University of British Columbia Vancouver, Canada Date fyjWL ^ T O f f l DE-6 (2\/88) A B S T R A C T Models for the behaviour of structural composite lumber and solid wood are presented and verified for a variety of configurations. In particular, laminated strand lumber (LSL) is investigated for the influence of strand orientation and stacking sequence on its material properties. LSL panels are modelled using laminate theory and the First Order Reliability Method to determine the probability distributions of elastic properties and ultimate strengths. The results are verified against properties determined from an extensive series of experiments on custom-manufactured LSL panels. The laminate theory is capable of predicting panel properties and indicates that the properties are related to the degree of strand alignment. This modelling technique allows material designers to set target levels on panel properties and on variability by using the properties of the constituent layers. The behaviour of single-dowel mild steel connections in the five LSL panel types are determined experimentally for three end distances, three slenderness ratios and two edge distances. (The connection specimens are cut parallel, perpendicular and at 45\u00b0 to the main strand axis.) Three-dimensional finite element models of each tested connection are used to predict load-displacement behaviour, ultimate strength and mode of failure. The anisotropic plasticity constitutive model is used together with the Weibull weakest link strength criterion for brittle materials to simulate the material behaviour of the LSL panels in the connection. The three-dimensional models and the single-dowel connection experiments indicate that panels with some strands oriented at \u00b1 45\u00b0 fail at higher loads than fully oriented panels. A parametric study of the model is used to indicate the critical variables in connection modelling and to isolate the most significant material tests required for modelling. Applications of the three-dimensional model are illustrated for a number of cases, including: a) dowel embedment tests, b) shear block tests, and c) a simplified one-dimensional spring model for the single-dowel connection. The shear block model illustrates the effect of the non-uniform shear stresses on the failure plane and the effects of the specimen notch. The one-dimensional spring model is shown to be useful for predicting the load-displacement behaviour and ultimate strength of multiple-bolt connections. These applications indicate the adaptability of the model to changes in material properties and connection geometry, and its ability to predict load-displacement, ultimate strength and mode of failure. ii TABLE OF CONTENTS ABSTRACT ii TABLE OF CONTENTS iii LIST OF TABLES vi LIST OF FIGURES viii LIST OF SYMBOLS xii ACKNOWLEDGEMENTS xv CHAPTER 1 Introduction \u2022\u2022 : 1 1.1 Objectives 2 1.2 S cope and Organization \u2022 \u2022 \u2022 \u2022 \u2022 \u2022 3 CHAPTER 2 Literature Review \u2022 4 2.1 Bolted Connections 4 2.2 Behaviour and Modes of Failure in Single- and Multiple-bolt Connections 5 2.3 European Yield Model 8 2.4 Design Code Requirements for Bolted Connections in Sawn Lumber 11 2.5 Existing Analytical Models 12 2.5.1. Fracture Models 12 2.5.2. One- and Two-dimensional Finite Element Models 14 2.5.3. Three-dimensional Finite Element Models 17 2.6 Failure Prediction 19 2.6.1. Failure Models 19 2.6.2. Size Effect 23 2.7 Material Properties 25 CHAPTER 3 Laminated Strand Lumber (LSL) 30 3.1 Background..; 30 3.2 Connection Behaviour of Non-wood Composites 32 3.3 LSL Panel Lay-ups .' \u2022\u2022\u2022 34 3.4 LSL Material Properties 35 3.4.1. LSL Properties - Experiments 35 3.4.2. LSL Properties - Test Results 37 3.5 Calculation of Panel Properties 39 3.5.1. Mechanics of Laminated Composite Materials 39 3.5.2. Reliability Method 41 3.5.3. Predictions of Panel Properties 42 3.6 Summary 44 iii CHAPTER 4 Single Dowel Connection Tests 59 4.1 Steel Dowels 59 4.2 Connection Test Set- up 59 4.2.1. Connection Test Set-up - Specimen Configurations 59 4.2.2. Connection Test Set-up - Procedure 61 4.3 Connection Test Results 62 4.3.1. Specimens loaded parallel to the main strand axis (Pa) 63 4.3.2. Specimens loaded perpendicular to the main strand axis (Pe) 68 4.3.3. Specimens loaded 45\u00b0 to the main strand axis (AN) 71 4.4 Summary \u2014 72 CHAPTER 5 Finite Element Model of Single Dowel Connection 95 5.1 Choice of Model 95 5.2 Model Geometry 95 5.3 Material Models and Constitutive Relations for Wood 98 5.3.1. Linear Elastic Constitutive Model for Wood 98 5.3.2. Non-linear Elastic Constitutive Modelling for Wood 99 5.3.3. Anisotropic Plasticity Modelling in Wood 100 5.3.4. Failure Prediction 101 5.4 Summary 102 CHAPTER 6 Verification of Finite Element Model 105 6.1 Calibration of Material Constants 105 6.2 Model Verification - Douglas fir 106 6.2.1. Tri-linear Elastic and Weibull Weakest Link 107 6.2.2. Anisotropic Plasticity and Weibull Weakest Link 108 6.3 Shear Block Model , 108 6.3.1. Shear Block Model Geometry 109 6.3.2. Shear Block Model Results 110 6.4 Bolted Connection Model in LSL 112 6.4.1. Load-displacement Results 113 6.4.2. Stress Distributions 113 6.4.3. Ultimate Strength and Displacement 115 6.4.4. Failure Modes \u2022\u2022 116 6.4.5. Hole Deformations 117 6.5 Sensitivity Study \u2022 117 6.5.1. Influence of Wood Properties 118 6.5.2. Influence of Contact Stiffness 120 6.5.3. Mesh Density 120 6.6 Summary 121 iv CHAPTER 7 Applications 144 7.1 Case 1: Single-dowel Connections - Geometry and Steel Effects 144 7.2 Case 2: Embedment Tests - Simulation and Verification 145 7.3 Case 3: Single-dowel Connections - Behaviour in OSB 146 7.4 Case 4: Multiple-dowel Connections - Spring Model 146 7.5 Summary 149 CHAPTER 8 Conclusions and Recommendations 156 8.1 Material Modelling 156 8.2 Experimental Findings 157 8.3 Analytical Model Findings 157 8.4 Recommendations 159 REFERENCES \u2022 1 6 0 APPENDIX A Load-displacement Curves for Single Dowel Connections: Experimental and Predicted !66 APPENDIX B Anisotropic Plasticity Theory 187 APPENDIX C Failure Prediction: Size Effect 191 v LIST OF TABLES Table 3-1. Laminate stacking sequences 45 Table 3-2. Test specimens for tension and compression tests ; 46 Table 3-3. Results from tension, compression and shear tests. 47 Table 3-4. Results from Poisson's ratio tests 48 Table 3-5. Shear properties from other studies 48 Table 4-1. Average steel dowel yield stress 74 Table 4-2. Connection specimen specifications 74 Table 4-3. Results summary for Pa specimens 75 Table 4-4. End distance and slenderness ratio effects on ultimate load (lbs.) for Pa specimens 76 Table 4-5. Statistical significance of test results for effects of end distance on ultimate load for Pa specimens (forp = 0.05) 77 Table 4-6. Statistical significance of test results for effects of slenderness ratio on ultimate load for Pa specimens (forp = 0.05) 77 Table 4-7. Edge distance effects on ultimate load (lbs.) for Pa specimens 78 Table 4-8. Statistical significance of test results for effects of edge distance on ultimate load for Pa specimens (for p = 0.05) 78 Table 4-9. End distance and slenderness ratio effects on ultimate displacement (inches) for Pa specimens 79 Table 4-10. Statistical significance of test results for effects of end distance on ultimate displacement for Pa specimens (for p = 0.05) 80 Table 4-11. Statistical significance of test results for effects of slenderness ratio on ultimate displacement for Pa specimens (for p = 0.05) 80 Table 4-12. Edge distance effects on ultimate displacement (inches) for Pa specimens 81 Table 4-13. Statistical significance of test results for effects of edge distance on ultimate displacement for Pa specimens (for p = 0.05) 81 Table 4-14. End distance and slenderness ratio effects on failure modes for Pa specimens 82 Table 4-15. Edge distance effects on failure modes for Pa specimens 83 Table 4-16. Results summary for Pe specimens 84 Table 4-17. End distance and slenderness ratio effects on ultimate load (lbs.) for Pe specimens 84 Table 4-18. Statistical significance of test results for effects of end distance on ultimate load for Pe specimens (for p = 0.05) 85 Table 4-19. Statistical significance of test results for effects of slenderness ratio on ultimate load for Pe specimens (forp = 0.05) 85 Table 4-20. End distance and slenderness ratio effects on ultimate displacement (inches) for Pe specimens. 86 Table 4-21. Statistical significance of test results for effects of end distance on ultimate displacement for Pe specimens (for p = 0.05) 86 Table 4-22. Statistical significance of test results for effects of slenderness ratio on ultimate displacement for Pe specimens (forp = 0.05) 87 Table 4-23. End distance and slenderness ratio effects on failure modes for Pe specimens 87 Table 4-24. Results summary for AN specimens 88 Table 4-25. End distance and slenderness ratio effects on ultimate load (lbs.) for type A loaded 45\u00b0 to main strand axis (AN) 88 Table 4-26. Statistical significance of test results for effects of end distance on ultimate load for AN specimens (for p = 0.05) 88 Table 4-27. Statistical significance of test results for effects of slenderness ratio on ultimate load for AN specimens (for p = 0.05) 88 Table 4-28. Hankinson's formula prediction for ultimate load (lbs.) for type A specimens loaded 45\u00b0 to main strand axis (AN) 89 Table 4-29. End distance and slenderness ratio effects on ultimate displacement (inches) for type A loaded 45\u00b0 to main strand axis (AN) 89 Table 4-30. Statistical significance of test results for effects of end distance on ultimate displacement for AN specimens (for p = 0.05) 89 Table 4-31. Statistical significance of test results for effects of slenderness ratio on ultimate displacement for AN specimens (forp = 0.05) 89 Table 4-32. End distance and slenderness ratio effects on failure modes for AN specimens 90 Table 6-1. Material properties for uniaxial behaviour of LSL with anisotropic plasticity model 122 Table 6-2. Material properties for behaviour of Douglas fir with tri-linear elastic model 123 Table 6-3 - Single dowel connection behaviour with Douglas fir 124 Table 6-4. Material properties for behaviour of Douglas fir with anisotropic plasticity model 125 Table 6-5. Material properties for shear behaviour of LSL with anisotropic plasticity model 126 Table 6-6. Predicted values of shear strength from shear blocks 127 Table 6-7. Stress concentrations factors for shear blocks at ultimate load. 128 Table 6-8. Predicted ultimate load in shear blocks due to shear or tension failure 129 Table 6-9. Observed and predicted failure modes for single-dowel connections in LSL .\" 130 Table 6-10. High-low values for sensitivity study of single dowel connection model 131 Table C-l . Result comparison for verification of volume integral of stresses 194 LIST OF FIGURES Figure 2-1. Test apparatus for single-member\/single-bolt study (after Patton-Mallory 1996) 26 Figure 2-2. Failure modes of single-bolt connections in tension in Douglas fir (after Patton-Mallory 1996)... 26 Figure 2-3. Sample European yield model failure modes and free-body diagrams 27 Figure 2-4. Half-hole bolt embedment test from ASTM D5764 28 Figure 2-5. Multiple-bolt connection spring model, (after Isyumov 1967) 28 Figure 2-6. Finite element model geometry (from Patton-Mallory 1996) 29 Figure 3-1. Elastic modulus in bending based on McNatt et al. (1992) 49 Figure 3-2. Laminated strand lumber orientation and co-ordinate system. 49 Figure 3-3. Shear block orientations and dimensions. Y-direction is primary fibre direction -50 Figure 3-4. Compression stress-strain curves and averages for type 'A' panels: a) perpendicular to main strand axis (X-direction), b) parallel to main strain axis (Y-direction), and c) perpendicular to surface (Z-direction) 51 Figure 3-5. Shear block load-displacement curves for type 'A' panels: a) in-plane (XY), b) interlaminar parallel to strands (YZ), and c) interlaminar perpendicular to strands (XZ) 52 Figure 3-6. Laminated composite panel: (a) layer co-ordinates relative to laminate co-ordinates, and (b) layer locations 53 Figure 3-7. Elastic property predictions for symmetric panels. Compression test data are plotted here and show mean values only. Shear properties at 0% and 100% orientation were assumed. Predicted data indicate \u00b1 1 standard deviation. Best fit lines for Ex and Ey are based on test data 54 Figure 3-8. Ultimate strength predictions in compression, a c uit., for symmetric panels. Test data show mean values only. Predicted data indicate \u00b1 1 standard deviation. Best fit line shown for a c U|t. is based on test data 55 Figure 3-9. Ultimate strength predictions in tension, o t u i t . , for non-symmetric panels. Test data show mean values only. Predicted data indicate \u00b1 1 standard deviation. 56 Figure 3-10. Ultimate strength predictions in tension, a tU| t., for symmetric panels. Test data show mean values only. Predicted data indicate \u00b1 1 standard deviation 57 Figure 3-11. Ultimate strength predictions in shear, a x y ui t., for symmetric panels. Test data show mean values only. Predicted data indicate \u00b1 1 standard deviation 58 Figure 4-1. Connection specimen layout. See Table 4-2 for details 91 Figure 4-2. Observed failure modes of connection specimens 92 Figure 4-3. Average ultimate loads of Pa specimens 93 Figure 4-4. Average ultimate loads of Pe specimens 94 Figure 5-1. Finite element model geometry 103 Figure 5-2. Tri-linear stress-strain curve for compression behaviour parallel-to-grain 103 Figure 5-3. Bi-linear stress-strain curve for normal stress in anisotropic plasticity model 104 Figure 5-4. Experimental stress-strain curve for cyclic loading of commercial grade LSL in compression parallel to the main strand axis 104 Figure 6-1. Stress-strain experimental curves for compression behavior in type 'A' LSL. Anisotropic plasticity curves shown in dark 132 Figure 6-2. Stress-strain experimental curves for compression behavior in type 'B' LSL. Anisotropic plasticity curves shown in dark 133 Figure 6-3. Stress concentrations in tension perpendicular-to-grain (i.e. perpendicular-to direction of loading) in Douglas fir. (a) anisotropic plasticity model, (b) tri-linear elastic model. lld = 5, e\/d= 4. 134 Figure 6-4. ASTM D143 shear block test set-up 135 Figure 6-5. Shear block finite element model geometry 135 Figure 6-6. Load-displacement curves for shear blocks in type 'A' LSL. Predicted curves shown in dark .136 Figure 6-7. Load-displacement curves for shear blocks in type 'B' LSL. Predicted curves shown in dark 137 Figure 6-8. Sample predicted shear stress in shear blocks along exterior (A-B) and interior (A'-B') at 100% loading 137 Figure 6-9. Stress contours of normal stress, a y, in type 'A' material at failure load, with eld = 4, edge distance \\.Sd for: (a) Pa4 specimen with 9.5 mm (3\/8 inch) dowel, lid = 4, and (b) Pal 1 specimen with 19 mm (3\/4 inch) dowel, lid = 2 138 Figure 6-10. Stress contours of normal stress, o x , in type 'A' material at failure load, with eld = 4, edge distance \\.5d for: (a) Pa4 specimen with 9.5 mm (3\/8 inch) dowel, lid = 4, and (b) Pal 1 specimen with 19 mm (3\/4 inch) dowel, lid = 2 139 Figure 6-11. Stress contours of normal stress, crz, in type 'A' material at failure load, with eld = 4, edge distance \\.5d for: (a) Pa4 specimen with 9.5 mm (3\/8 inch) dowel, lid = 4, and (b) Pal 1 specimen with 19 mm (3\/4 inch) dowel, lld= 2 140 Figure 6-12. Stress contours of shear stress, a x y , in type 'A' material at failure load, with eld = 4, edge distance l.5d for: (a) Pa4 specimen with 9.5 mm (3\/8 inch) dowel, lid = 4, and (b) Pal 1 specimen with 19 mm (3\/4 inch) dowel, l\/d=2 141 Figure 6-13. Type 'E' Pall specimen at failure load, with eld = 4, lid = 2, edge distance 1.5^ and 19 mm (3\/4 inch) dowel, stress contours (a) tension, crz and (b) shear, o x y 142 Figure 6-14. Ordered effects based on (a) ultimate loads, and (b) ultimate displacements. Outliers indicate significant variables 143 Figure 7-1. Influence of end distance, slenderness ratio and dowel yield strength on 13 mm (1\/2 inch) dowels in fully-oriented (type 'A') LSL 150 Figure 7-2. Embedment tests with 19 mm (3\/4 inch) dowels loaded: (a) parallel to main strand axis (fill hole test), and (b) perpendicular to main strand axis (half hole test) 151 Figure 7-3. Simulated Oriented Strand Board versus Laminated Strand Lumber for 9.5 mm (3\/8 inch) and 19 mm (3\/4 inch) dowels with lid = 4 and 2, respectively, and eld = 4 152 Figure 7-4. Load-displacement behaviour of 9.5 mm (3\/8 inch) dowels with end distances 2d, 3d and Ad. 152 Figure 7-5. Stress integral variation with load level for 9.5 mm (3\/8 inch) dowels with e\/d=4, lid = 4 and edge distance l.5d. Specimen Pa4a 153 Figure 7-6. Multiple-dowel connection geometry used for one-dimensional spring model study. Only the central LSL member is shown 154 Figure 7-7. Four-dowel connection load-displacement behaviour corresponding to cases in Figure 7-6 155 Figure A - l . Experimental and predicted load-displacement curves for type 'A' specimens loaded parallel to main strand axis, with lid - 4. Dark solid line is model prediction. Coeffient of variation in parentheses (%) 167 Figure A-2. Experimental and predicted load-displacement curves for type 'A' specimens loaded parallel to main strand axis, with lid = 3. Dark solid line is model prediction. Coeffient of variation in parentheses (%) 168 Figure A-3. Experimental and predicted load-displacement curves for type 'A' specimens loaded parallel to main strand axis, with lid = 2. Dark solid line is model prediction. Coeffient of variation in parentheses (%) 169 Figure A-4. Experimental and predicted load-displacement curves for type 'A' specimens loaded perpendicular to main strand axis. Dark solid line is model prediction. Coeffient of variation in parentheses (%) 170 Figure A-5. Experimental and predicted load-displacement curves for type 'A' specimens loaded 45\u00b0-to-main strand axis. Dark solid line is model prediction. Coeffient of variation in parentheses (%) 171 Figure A-6. Experimental and predicted load-displacement curves for type 'B ' specimens loaded parallel to main strand axis, with lid = 4. Dark solid line is model prediction. Coeffient of variation in parentheses (%) 172 Figure A-7. Experimental and predicted load-displacement curves for type 'B ' specimens loaded parallel to main strand axis, with lid = 3. Dark solid line is model prediction. Coeffient of variation in parentheses (%) 173 Figure A-8. Experimental and predicted load-displacement curves for type 'B' specimens loaded parallel to main strand axis, with lid = 2. Dark solid line is model prediction. Coeffient of variation in parentheses (%) -. 174 Figure A-9. Experimental and predicted load-displacement curves for type ' C specimens loaded parallel to main strand axis, with lid = 4. Dark solid line is model prediction. Coeffient of variation in parentheses (%) 175 Figure A-10. Experimental and predicted load-displacement curves for type ' C specimens loaded parallel to main strand axis, with lid = 3. Dark solid line is model prediction. Coeffient of variation in parentheses (%) 176. Figure A - l l . Experimental and predicted load-displacement curves for type ' C specimens loaded parallel to main strand axis, with lid = 2. Dark solid line is model prediction 177 Figure A-l2. Experimental and predicted load-displacement curves for type ' C specimens loaded perpendicular to main strand axis. Dark solid line is model prediction. Coeffient of variation in parentheses (%) 178 x Figure A-13. Experimental and predicted load-displacement curves for type 'D' specimens loaded parallel to main strand axis, with lid = 4. Dark solid line is model prediction. Coeffient of variation in parentheses (%) 179 Figure A-14. Experimental and predicted load-displacement curves for type 'D' specimens loaded parallel to main strand axis, with lid = 3. Dark solid line is model prediction. Coeffient of variation in parentheses (%) 180 Figure A-15. Experimental and predicted load-displacement curves for type 'D' specimens loaded parallel to main strand axis, with lid = 2. Dark solid line is model prediction. Coeffient of variation in parentheses (%) 181 Figure A-16. Experimental and predicted load-displacement curves for type 'D' specimens loaded perpendicular to main strand axis. Dark solid line is model prediction. Coeffient of variation in parentheses (%) 182 Figure A-17. Experimental and predicted load-displacement curves for type ' E ' specimens loaded parallel to main strand axis, with lid = 4. Dark solid line is model prediction. Coeffient of variation in parentheses (%) 183 Figure A-18. Experimental and predicted load-displacement curves for type ' E ' specimens loaded parallel to main strand axis, with lid = 3. Dark solid line is model prediction. Coeffient of variation in parentheses (%) 184 Figure A-19. Experimental and predicted load-displacement curves for type ' E ' specimens loaded parallel to main strand axis, with lid - 2. Dark solid line is model prediction. Coeffient of variation in parentheses (%) 185 Figure A-20. Experimental and predicted load-displacement curves for type ' E ' specimens loaded perpendicular to main strand axis. Dark solid line is model prediction. Coeffient of variation in parentheses (%) 186 Figure B-1. Yield surface for anisotropic plasticity accounting for differences in tension and compression yield stresses 190 Figure C - l . Simply-supported beam with point load at mid-span 195 Figure C-2. ANSYS finite element geometry 195 LIST OF SYMBOLS [A] extensional stiffness matrix (Pa-m, psi inch) a, b, d, L dimensions (mm, inch) * 2 2 [B] coupling stiffness matrix (Pa-m , psi-inch) [D] bending stiffness matrix (Pa-m , psi inch) d bolt or dowel diameter (mm, inch) Ej modulus of elasticity (MPa, psi) Eiang, ET tangent modulus (MPa, psi) e end distance (mm, inch) F connection strength (kN, lbs.) Fh Fy strength tensors of 2nd and 4 th rank (MPa'1, MPa\"2) Fv probability of failure \/ embedment strength (MPa, psi) \/ dimensionless constant for contact stiffness fy steel yield strength (MPa, psi) Gy shear modulus (MPa, psi) h, hi distance (mm, inch) \/, II integral function [J] Jacobian matrix K\\c, ^nc critical fracture toughness for mode I and II fracture (MN\/m 3 \/ 2 , ksiVin) K effective yield stress (MPa, psi) KN contact stiffness (kN\/mm, lbs.\/inch) k shape parameter for Weibull distribution kj spring stiffness Lj tensor used in anisotropic plasticity model \/, \/ length or thickness of wood member (mm, inch) Mi moment (kN-m, ft.-lb.) My, [M] anisotropic strength parameters in tensorial ' and matrix forms, respectively {M} vector of bending and twisting moments (kN-m\/m, ft.-lb.\/ft.) m scale parameter for Weibull distribution (MPa, psi) Ni element shape functions N connection strength at angle to grain (kN, lbs.) {N} vector of in-plane stress resultants P applied force, connection strength parallel-to-grain (kN, lbs.) Pmax, Pexp. ultimate connection strength from test (kN, lbs.) Ppred predicted ultimate strength (kN, lbs.) p probability of failure or p-value used in F-test Q connection strength perpendicular-to-grain (kN, lbs.) [Q] stiffness matrix [ Q ] transformed stiffness matrix [7] transformation matrix Vt specimen volume 3 3 V* reference volume (m , inch) Wj Gaussian weighting factors Wp plastic work per unit volume (N\/m, lbs.\/in. ) w width (mm, inch) Xj t , Xie failure stress along principal material direction \/' in tension or compression (MPa, psi) X , Y, Z principal axes of orthotropy x, y, z material and global co-ordinate system a; tensor describing origin of yield surface Ai ,A2 dimension (mm, inch) A m a x ultimate connection displacement from tests (mm, inch) Apred ultimate predicted connection displacement (mm, inch) xiii Si strain tensor (mm\/mm, inch\/inch) sy strain at yield or strain in direction y (mm\/mm, inch\/inch) f-inner, f-outer, initial final inner, outer and initial hole diameters (mm, inch) {K} vector of curvatures (mm\/mm, inch\/inch) LI coefficient of friction Vjj Poisson's ratio 0 angle (\u00b0) a; stress tensor (MPa, psi) o+j, o\\j yield stress in tension or compression (MPa, psi) auit. vector of failure reference stress (MPa, psi) G u i t . ultimate stress (MPa, psi) a y stress at yield or stress in direction y (MPa, psi) CT* reference stress for Weibull distribution (MPa, psi) Tjj shear stress in material coordinate system (MPa, psi) \u00a3,,r\\,C, local (natural) isoparametric element co-ordinate system Abbreviations: A, B, C, D, E panel lay-up types AN specimens cut at 45\u00b0 to main strand axis LSL laminated strand lumber Pa specimens cut parallel to main strand axis Pe specimens cut perpendicular to main strand axis R randomly oriented layers SCL structural composite lumber xiv ACKNOWLEDGEMENTS I would first like to thank my fiancee Yasmin for all of her love and support during my years of work on this thesis. Thanks to her understanding and optimism, I have had the confidence to continue working, especially at the times when working on this project became onerous. Thank you Yasmin! There are many people who have also contributed to my well being over the years and these include my grandmother, parents and sister, and a number of close friends. I would next like to thank my supervisor, Dr. H. Prion, who has remained enthusiastic about this work and has shown his interest in curiosity-driven research starting from our forays into x-ray tomography to our more fruitful endeavours with the early tests of dowel connections in LSL. His support for this project and for the enrichment of his students (myself included) through conference attendance and professional outreach programmes will leave a lasting effect. I thank my colleagues and fellow graduate students for their assistance and for the discussions in which we exchanged ideas and friendship. I believe that on reading this thesis, the influence of each member of my advisory committee is apparent, therefore, I am convinced that the committee was true to its name and should be acknowledged for doing so. I would like to thank Dr. W. Boehner and Mr. B. Craig (also an advisory committee member), of Trus Joist MacMillan, the industry collaborators for this research programme. These individuals have been instrumental in supporting the objectives of this research through personal discussions. The assistance I received in the laboratories at the University cannot go unrecognized: Paul Symons, Doug Smith, John Wong and Bob Myronuk are the problem-solvers to whom I refer, in addition to the assistance of a number of undergraduate students. Finally, this project would not have evolved into its current form without the financial support provided by Trus Joist MacMillan, The Science Council of British Columbia, and Forest Renewal British Columbia. xv CHAPTER 1 INTRODUCTION Wood is one of the most common building materials available to engineers - it is also one of the most complex. As with any naturally occurring material, wood has variability in its structure and properties. Wood has distinctive cell structures, however, it is not homogeneous due to discontinuities caused by the presence of knots and other natural flaws -this leads to difficulties for engineers and builders. Nonetheless, wood structures have been built very successfully for centuries based on generations of experience. The development of modern building design codes, while reflecting the standard practices which developed as a result of years of experience, is substantiated by extensive laboratory tests and by extrapolations from these tests. This empirical approach to code development requires testing programmes with numerous replications of physical tests to address each of many design situations. To avoid this, and simplify the process of code development, researchers have attempted to establish rigorous models to predict the behaviour of wood in numerous applications; this includes the mechanical connections between wood members. Connection modelling is particularly complicated due to the many different parameters that govern the performance of the connections. Analytical models for connections are important tools that allow researchers to carry out parametric studies on connection geometry and material properties, particularly as new wood composites are developed. To do this, one must first develop a material model (independent of the connection model) of the behaviour of wood, accompanied by an appropriate failure or fracture criterion. Few constitutive models for linear and non-linear stress-strain behaviour exist for wood and wood composites. These models must account for material variability and failure prediction in addition to duplicating the true stress-strain behaviour of wood. A completed model of this sort has many possible applications including the prediction of: a) the performance of structural composite lumber (SCL) panels, and b) the behaviour of bolted connections in solid wood and in wood composites. In the first case, the final properties of a layered panel are required, whereas, in the second case, the response of 1 the a structural member cut from a finished panel is required. Although the modelling approach for each case may be different, the underlying material properties and the laws governing their behaviour are the same. Structural composite products play a particularly important role in the future of engineered wood design since they can be manufactured to specified stiffnesses and strengths. A model which predicts the effective properties of the panel (and their variability) from the properties of its individual layers will allow manufacturers to set target properties and performance levels for the panels. The use of SCL products in bolted connections requires a thorough analysis of the stress concentrations which occur in the region of a connection. The analysis must include the effects of connection geometry and the material properties of the wood composite and the connector material. Load-displacement behaviour, ultimate strength and failure mode should be predicted to provide designers with insight on performance and capacity of connections. 1.1 Objectives The primary objectives of this thesis are to develop and apply material models for wood composites, their mechanical properties and their behaviour in bolted connections. Based on this, the secondary objectives are to determine: Mechanical Properties of Panels a) the axial behaviour in tension and compression in three orthogonal material directions, including linear and non-linear stress-strain relationships and ultimate strengths; b) the shear behaviour in three orthogonal shear planes; c) the variability associated with each property; and d) the influence of changes in strand orientation, lay-up and stacking sequence. Single-dowel Connection Behaviour a) the load-displacement behaviour; b) the ultimate load, ultimate displacement and mode of failure; c) the influence of load orientation on connection behaviour; d) the influence of connection geometry; 2 e) the influence of panel lay-up; and f) the influence of dowel strength. The behaviour of multiple-dowel connections is a logical extension of these objectives for single-dowel connections. 1.2 Scope and Organization The approach followed in this thesis is to first determine the underlying material properties of individual layers in a wood composite, calculate the effective panel properties of the entire laminate, and, finally, determine the behaviour of a structural composite member in a bolted connection. To this end, a review of relevant literature is made in Chapter 2 with a focus on connection behaviour and material modelling. This is followed, in Chapter 3, by a brief review of laminated strand lumber, and its properties as a composite wood product. A description follows of the five custom-manufactured laminated strand lumber lay-ups for this thesis and the comprehensive series of experiments required to determine their mechanical properties. The development of a computer program to predict the panel properties and their variability is described in this chapter. Chapter 4 describes the extensive series of single-dowel connection tests carried out on the material described in Chapter 3. In Chapter 5, the proposed three-dimensional anisotropic plasticity material model, Weibull weakest link failure analysis and three-dimensional finite element model for wood composites, are described. The material model (including failure prediction) is calibrated and verified in Chapter 6 against a previous tri-linear elastic material model, a wood shear block model and against the results from the connection tests of Chapter 4. In Chapter 7, four applications of the material model are described, ending with a simplification of the three-dimensional finite element connection model to a one-dimensional spring model for multiple-dowel connections. The work concludes in Chapter 8 with a summary and a number of recommendations. Note on units: The test apparatus and previous experimental databases used for this thesis were all calibrated in Imperial units. The units have been converted to S.I. metric in the main text of the thesis. The following conversion factors apply: 1 lb. \u00ab 4.45 N 1000 psi = 1 ksi \u00ab 6.9 MPa 3 CHAPTER 2 LITERATURE REVIEW The topics of numerical modelling and material behaviour, as they apply to wood and wood composites, are discussed in this chapter. Particular emphasis is placed on the analysis and design of bolted connections along with a review of current design standards. The critical elements of bolted connection behaviour are then used to illustrate the requirements for the numerical analysis of wood and for bolted connections of wood members. 2.1 Bolted Connections The design of bolted connections in steel, wood and non-wood composites (such as polymer-based fibre-matrix materials) is not a new subject. Connection behaviour of these materials has been analyzed in numerous studies. Wood, however, differs from the other materials because of its fibre structure and variability. Wood and non-wood composites are both orthotropic but they exhibit different levels of ductility prior to failure. Large, permanent deformations of wood cells can occur in compression and result in ductile behaviour, whereas non-wood resin-based composites tend to have limited ductility and fail in a brittle manner. The orthotropic structure of these materials requires special attention during design compared to isotropic materials such as steel, since variables such as the direction of loading in wood and non-wood composites, can lead to a critical brittle failure in tension perpendicular to the longitudinal fibre axis. This behaviour is not present in steel design. Wood members also differ from non-wood composites by virtue of their size. Non-wood composites are typically very thin relative to their wood counterparts. This can lead to brittle failure of a bolted connection in non-wood composites whereas wood connections with steel bolts can become ductile when cells crush near the highly stressed surfaces while the steel bolt yields. This can only happen if the wood member is sufficiently thick to lead to a connection in which deformations are not uniform throughout the member thickness. In fact, the behaviour is three-dimensional in nature as the bolt moves in the direction of loading and deforms fibres by pushing them apart and crushing them. The result is a complex 4 behaviour of bolt and wood deformation that is distinctly different than the behaviour of connections in other materials. 2.2 Behaviour and Modes of Failure in Single- and Multiple-bolt Connections The performance of a bolted connection in wood can be measured by the relationship between applied load and connection displacement. In the laboratory, connection tests are performed to determine this load-displacement relationship up to connection failure. The applied load is compared against the slip in the connection where slip is defined as the displacement of the side members relative to the main member. The load is increased until the connection fails, thereby revealing the maximum strength of the connection. One can measure the linearity of the load-displacement relationship to decide i f the connection behaviour is brittle or ductile. Following the laboratory experiment, an examination of failed members gives an indication as to the cause of failure, such as low tension perpendicular-to-grain strength. These two pieces of empirical information (load-displacement and cause of failure) are the only outcomes of such laboratory tests. The load-displacement relationship and the final examination do not, however, provide enough information on the very complex behaviour of the bolts and the wood as the load is applied. As a result, many combinations of connection geometry have been tested over the years to elicit, empirically, the factors underlying the response of single- and multiple-bolt connections. To obtain a reasonable estimate of the fifth percentile of the strength distribution curve, a large number of replicates per case is required. A large number of single-bolt connections were tested by Trayer (1932) to determine design values for engineers. These values formed the basis of earlier design codes in Canada and the United States. In his investigation, and in those of many others who followed, the influence of species, bolt diameter and geometry on the ultimate strength of single-bolt connections was studied. Trayer was aware of the transition from ductile to brittle behaviour as a result of connection geometry and material properties of the bolt and wood. But he, like the others who followed (see Moss 1997), continued testing three-member connections. The drawback of such tests is the influence of other factors on the connection behaviour. These factors include the friction from side members, bolt end support conditions and bolt yield strength. To isolate the bolt and main member, Patton-Mallory (1996) tested a single member with a single bolt as shown 5 in Figure 2-1, thereby removing external effects. The failure modes of single-bolt connections were documented in detail for these tests in solid sawn Douglas-fir. The failure modes were found to be dependent on geometry of the wood member relative to the bolt size and loading direction, and on the material properties of the wood. Failure modes were grouped into two categories: brittle and ductile. Although brittle failure should be avoided in connection designs, the range of connection geometries were intentionally chosen to illustrate the transition from brittle to ductile failure. At least six modes of failure were documented for the single-bolt connections loaded in tension parallel-to-grain. The modes consisted of combinations of the three basic failure modes: tension perpendicular-to-grain, shear and crushing, as shown in Figure 2-2. Two types of brittle failure occurred: tension perpendicular-to-grain and shear plug. Tension failure perpendicular-to-grain occurred in specimens with small end distances. This mode occasionally produced a densified region of wood (due to cell buckling and crushing) under the bolt, which contributed to the tension failure by wedging apart the wood fibres. Shear plug failures occurred in connections with small end distances and where the wood member was thin (less than two bolt diameters in thickness). A third brittle mode not observed by Patton-Mallory is net section-failure. This can occur when the edge distance is too small, but it rarely governs in single-bolt wood connections since the tensile strength parallel-to-grain in wood is approximately fifty times greater than its perpendicular-to-grain strength. Combinations of these modes are common due to the heterogeneity of the material. Each of the ductile modes documented by Patton-Mallory (1996) involved crushing of cells under the bolt and resulted in elongation of the hole, as shown in Figure 2-2. Ductile modes sometimes resulted in small cracks near the bolt on the member surfaces without bolt yielding. These cracks did not lead to catastrophic failure. The hole became elongated evenly through the member. No bolt yielding occurred in connections with low member thickness. Bolt yielding did occur in thicker members. For the thickest members with large end distances, initial crushing of the wood or tensile stresses perpendicular-to-grain were thought to initiate small cracks on the member faces. These cracks did not extend through the member thickness. Thus, Patton-Mallory's work verified the transition between brittle and ductile failure modes based on member thickness, end distance and edge distance. Increasing member 6 thickness, end distance and edge distance led to ductile behaviour. The transition from brittle to ductile failure and the occurrence of certain failure modes in single-bolt connections each depend on the relative strengths in each of the principal directions of the member material. Building on the behaviour of a single bolt, we now turn to multiple-bolt connections where the additional effect of load sharing and interactions change the failure behaviour. Interaction between and within members was eliminated in Patton-Mallory (1996) by testing a solitary member with a single bolt. When testing multiple-bolt connections, it is difficult to avoid interaction due to friction between members and uneven load distribution since side members (steel or wood) are normally used to apply loads to the bolts. Friction between members is not considered to be as significant as the effect of load distribution between bolts in multiple-bolt connections. In Moss' (1997) review of research on multiple-bolt connections in wood, the uneven load distribution in laboratory specimens was found to lead to premature failure of connections. It is generally believed that the end bolts in a row of bolts tend to carry more load than the others. This is due to the relative stiffness of the members to each other and to the bolts, as well as oversizing of holes. The high load in the end bolt causes it to govern the capacity of the entire connection. End distance and member thickness can, therefore, be chosen to increase the resistance of the end bolt and, hence, the entire connection. This, however, adds additional geometric design constraints for engineers and architects. It was found that end distance and member thickness are not the only factors which affect multiple-bolt connection performance. Studies on the spacing between bolts and the spacing between rows of bolts have shown that increased spacing delays failure in multiple-bolt connections (for example, Masse et al. 1988). However, Yasumura et al. (1987) found the opposite effect. Not only has the spacing effect resulted in conservative design requirements, the contradiction in results has revealed that there are other effects which are not included in these empirically-driven research studies. Nonetheless, studies continue to be conducted on many versions of the same problem to determine the proper spacing requirements and preferred connection geometries to ensure predictable ductile behaviour. Before discussing ductility, it is important to examine the failure modes of multiple-bolt connections determined from these studies. 7 The failure modes of multiple-bolt connections are expressions of single-bolt failures. The tension perpendicular-to-grain failure for certain connection geometries has been found to initiate at the end of the member (Jorissen 1997) and results in a tension split along the line of bolts. A shear plug failure can also occur along the line of bolts. A group shear failure is also possible for the entire group of bolts where shear along two planes and tension parallel-to-grain govern the failure (Masse et al. 1988). These brittle failure modes can be delayed or even avoided by introducing ductility in a multiple-bolt connection. By using lower diameter bolts, ductility in connections can be developed through significant plastic deformation of the bolts and elongation of the holes as the wood crushes prior to total failure of the connection (Mischler 1998). A ductile connection, for single or multiple bolts, is more desirable from a design standpoint. One reason for this is the problem of load sharing in multiple-bolt connections. In a connection, some bolts will carry more load than others due to misalignment of bolt holes from fabrication tolerances, and due to the relative stiffness of the side members. In a brittle design, the governing bolt or bolts (likely to be at the end of a row of bolts) will fail and cause catastrophic failure of the entire connection shortly thereafter. In a ductile connection, the permanent deformation of wood under the heavily loaded bolts allows for a re-distribution of the load onto the bolts which were previously only lightly loaded. The connection may still fail suddenly, but only after significant deformation of the entire connection - an improvement over the brittle connection. The drawback of the ductile connection can be the reduction in connection stiffness and, possibly, reduced ultimate load. However, the benefits of this ductile design have been accepted by current design standards (see section 2.4) following the development of the semi-empirical European Yield Model for determining the allowable load on a bolt in a ductile connection. 2.3 European Yield Model The yield model was originally developed by Johansen (1949) to predict the strength ' of two- and three-member dowel connections in timber only. It is based on a free body diagram of a single bolt in a wood connection, as shown in Figure 2-3, and the corresponding equations of equilibrium. Steel-to-wood connections were later included in the model. The deformed shape of the connection is used to determine the appropriate forces on the free 8 body diagram. Each free body diagram is a representation of a mode of deformation or connection failure and includes the behaviour of the steel bolt, side members and can be used for two- and three-member connections. It is assumed that the steel bolt can yield and that non-recoverable deformations of the wood are possible. Thus, the yield strength of the bolt and the embedment or dowel bearing strength of the bolt in each member are included in these calculations. The embedment strength, \/ , is determined through an embedment test. One such test used in the United States (ASTM D5764 1996) applies a uniform load onto a bolt in a half-round hole as shown in Figure 2-4. A full hole test is used in Canada and Europe (Smith et al. 1998). The embedment test is a way of incorporating all ductile behaviour of wood into one load-displacement curve. As the dowel bears onto the wood block, a number of failure mechanisms occur, such as cell crushing and splitting along fibres. Hidden in the results are the effects of the three-dimensional stress distribution and all material elastic and strength properties. Nonetheless, the yield model. requires the embedment strength as a material constant to determine connection resistance values. The embedment strength (also referred to as the dowel bearing strength) is related to the density of wood and is determined from the load-displacement curve of the embedment test. A slope, parallel to the initial slope of the curve, is drawn with a five percent offset (0.05 times the dowel diameter) from the zero displacement position on the curve. The point at which the offset slope intersects the load-displacement curve is used as the yield strength value. This definition of strength is the one most commonly used for the yield model even though is does not necessarily correspond to the ultimate or maximum load achieved. A more accurate definition of embedment strength would include the entire load-displacement curve and not just a single value. For design purposes, however, such a simplification is useful. The yield model is semi-theoretical (or semi-empirical) since it does make some theoretical allowance of the deformation of the connector in the hole. It is empirical because it assumes the embedment strength to be a material property when in fact this \"strength\" is really the combination of many geometric and material factors. To account for some of these variables, many embedment tests are conducted for different bolt sizes and directions of loading (for example, parallel- or perpendicular-to-grain loading). Since the model cannot account for stress concentrations which lead to brittle failure, it is restricted to ductile 9 behaviour. To develop any of the failure modes prescribed by the yield model, minimum geometric requirements, such as member thickness and end distance, must be provided. As for ductile behaviour, only the maximum allowable load in a connection is determined with no indication of displacement to be used as a measure of connection ductility. Given the yield model prediction for the capacity of a single-bolt connection, the strength of a multiple-bolt connection is calculated as a multiple of the capacity of the single connector. Reduction factors must be applied to the calculated capacity of a single bolt to account for geometry and load distribution as described in section 2.2. Brittle modes of failure and load re-distribution which occur in multiple-bolt connections are not considered in the yield model. Brittle modes of failure occur when the member is relatively thin or when insufficient end distance, edge distance and\/or bolt-spacing are provided. An attempt to account for brittle fracture behaviour for this group of connection geometries was made by Jorissen (1998) for use with the European Yield model. He included shear and tension perpendicular-to-grain failures in the connection analysis by calculating stress distributions along potentially critical load paths within the wood member. The analysis assumed a beam on an elastic foundation to calculate tension perpendicular-to-grain and shear stresses. The change in stress distribution, as the applied bolt load was increased, was determined by assuming a certain load-displacement behaviour of the connection. The average stress (for each state, i.e. tension perpendicular and shear) was then compared to a fracture mechanics model to predict the ultimate strength. An additional peak stress was introduced at the point of contact between the bolt and wood; this was an assumption. This model was used only on relatively thin members where a uniform stress distribution could be assumed through the member thickness. Jorissen (1998) extended his model to a multiple-bolt connection by using a non-linear spring model to account for the non-uniform load distribution between bolts. For his series of tests, it was found that the model did predict brittle failure reasonably well. However, the assumed stress distributions and the lack of concern for the size effect (see section 2.6.2) do not make this model robust. Another limitation of this model is that the stress in the wood member is assumed to be uniform throughout the member thickness. This means that only situations where the wood member is relative thin can be studied. In this 10 case, the bolt is assumed to be rigid and the failure brittle. Jorissen suggests that the fracture model be used for the entire range of member thicknesses and then compared against the yield model results; the lesser of the two capacities would govern. However, there are many assumptions made about the stress distribution throughout the member thickness and the plane of the member which make this model inapplicable to cases where, for example, high tensile stresses which can lead to failure exist near the member surfaces. The original yield model has been adopted by the current design standards in Canada and the United States as a result of confirmatory studies by Larsen (1973), McLain and Thangjitham (1983) and Soltis et al. (1986). To extend the theory to multiple-bolt connections, a number of assumptions and restrictions have been included in the design code, as explained in the following section. 2.4 Design Code Requirements for Bolted Connections in Sawn Lumber The criteria have been used in the Canadian wood design code (CSA 1984, CSA 1989) for the design of bolted connections in solid sawn lumber were originally based on empirical test results of connections (Trayer 1932) and, more recently, on the yield model (Johansen 1949, Larsen 1973, Whale et al. 1987, CSA 1994). The ductile effects of wood crushing and bolt yielding are incorporated into the yield model to establish allowable connections strengths. The model produces reasonable results for single-bolt connections with multiple members provided that brittle failures are prevented. Brittle failure modes, such as wood splitting, are not desirable from a design standpoint, consequently code stipulations for member sizing and connection geometry help to prevent brittle failure from occurring. The leap from single-bolt to multiple-bolt connections required more empirical evidence on the effects of bolt spacing, member sizing and connection geometry. It was determined that the strength of a bolted connection is not a linear function of the number of bolts in a connection due to unequal load-sharing in bolt groups (Masse et al. 1988). Bolts near the end of a row tend to carry more load. As a result of this and earlier studies, the Canadian design code adopted two group factors to discount the strength of multiple-bolt connections. These factors are based on: a) the number of bolts within a row, and b) the number of rows of bolts. The reduction for bolts within a row depends on the bolt slenderness ratio (member thickness to bolt diameter) and the bolt spacing within the row 11 (Yasumura et al. 1987). The intent of this group factor is to prevent very high loads in the end bolts because these loads tend to initiate brittle modes of failure. By penalising all bolt strengths in a connection to meet this condition, the Canadian design code requirements can result in inefficient connections. The other group factor is based on the number of rows but is overly conservative because it neglects the spacing between rows (Mischler and Gehri 1999). After the incorporation of the yield model equations into the recent edition of the Canadian wood design code (CSA 1994), connection design became more rational, however, the group factor may still result in overly conservative designs. In other countries, design equations similar to the Canadian standard are employed (Patton-Mallory 1989), however, questions remain regarding such topics as: 1) the amount of ductility in a connection; 2) the transition point between ductile and brittle failure modes as a result of material and geometric considerations; 3) the behaviour of multiple-bolt connections; behaviour in loading perpendicular-to-grain; 4) behaviour of connections with more than three members; and 5) connections involving wood composites. The analytical model developed in Chapter 5 will be used to address most of these points. 2.5 Existing Analytical Models There were a number of attempts to numerically model connection behaviour. The European yield model is an example of a simplified solution based on force and moment equilibrium in a ductile single-bolt connection. The group and spacing requirements in the Canadian wood design code (CSA 1994) resulted from analyses based on empirical evidence. No universally accepted model exists for the behaviour of wood or a bolted connection in wood although other analytical models for wood connections have been developed. These include the use of fracture mechanics for failure prediction, and two- and three-dimensional numerical models based on the finite element method, as will be discussed in this section, along with a number of numerical models for use with connections in non-wood composites. 2.5.1. Fracture Models Fracture mechanics in timber is typically used to calculate ultimate strength in tension perpendicular-to-grain and in shear. Crack growth leading to fracture results in brittle failure. The main difficulty in applying the theory of fracture mechanics to connections in wood is 12 the determination of the mixed mode fracture criterion for tensile and shear stresses. The problem is further complicated for cases where the stress distributions of tension perpendicular-to-grain and shear are not uniform throughout the member thickness. As a result, fracture analysis has been limited to connection members of small thickness where little or no bending of the bolt takes place. Smith and Hu (1994) used a two-dimensional finite element mesh with orthotropic material properties and linear elastic fracture mechanics to analyse a single-bolt connection loaded perpendicular-to-grain. The point on the circumference of the bolt hole with the highest perpendicular-to-grain stress was chosen as the point at which a crack would initiate. They found correspondence of this location with test results. The finite element mesh was then updated to include a crack at this location. Mixed mode crack growth was assumed using the criterion developed by Hunt and Croager (1982) which interrelates tension perpendicular-to-grain (Mode I) and shear (Mode II) stress intensity factors. Results were not conclusive and led to recommendations for further model development. Jorissen (1998) used fracture mechanics, with more success, to predict failure along the planes leading to shear plug failure of the type shown in Figure 2-2. This crack location was chosen based on experimental observations of single-bolt connections. Crack lengths of 0 and 20 mm were used to determine the effect on tension stresses perpendicular-to-grain and shear stresses along the failure planes. Using a mixed mode fracture criterion, Jorissen used two-dimensional beam theory and the theory of beams on elastic foundations to predict stresses along these planes. The foundation modulus was estimated from the stiffness of wood perpendicular-to-grain. The author included the effects of bending and shear deformation in his derivation of the beam model for the region around the hole. A deformation function must be chosen to fit prescribed boundary conditions. The author used functions from the literature which satisfied the boundary conditions. The estimates of peak tension perpendicular-to-grain stresses at the hole were lower than expected so the author added an assumed peak stress perpendicular-to-grain at the hole. This peak was necessary to allow crack initiation to be detected using the fracture theory. Jorissen (1998) expanded the analysis to multiple-bolt connections using a one-dimensional multiple-spring model, similar to Isyumov (1967) shown in Figure 2-5. The springs in this model represent main and side members, and the interaction of each bolt with 13 wood. The interaction stiffness can be non-linear, as can be the case for a single-bolt connection, to capture the ductile behaviour of a single-bolt connection until failure. Jorissen used the non-linear interaction stiffness and was able to model the redistribution of load on the bolts as loading proceeded. At every load step, the load in each bolt was calculated, the tension perpendicular-to-grain and shear stresses were determined for each bolt, and the stresses were checked against the fracture criterion to estimate the failure load. Jorissen verified this model with numerous connection tests in European spruce. The shortcomings of the Jorissen model include the limitations on connection geometry and the number of assumptions required to estimate the stress distribution. These assumptions are based on tests on one species of tree and thus this model requires more tests (some complicated to determine fracture mechanics properties) for the behaviour and material properties of other species. Though difficult to determine, the assumed interaction of fracture (mixed mode crack propagation) in tension and shear appeared to work; however, there was no indication of the sensitivity of the model to errors in this assumption. The assumption that stresses are uniform throughout the member thickness further limits the applications of this model. To determine the actual stress distribution around a bolt hole, and for cases with more complicated stress distributions throughout the member thickness, other methods such as the finite element method, are required. 2.5.2. One- and Two-dimensional Finite Element Models Foschi (1974) developed a finite element model of a'nail to determine the ductile load-displacement characteristic of the nail in wood. Elasto-plastic behaviour was incorporated into the beam bending equations for the nail elements. The nail was assumed to be resting on a non-linear elastic foundation representing wood crushing under the nail. This was an improvement over previous studies which assumed the foundation to be elastic. A cubic polynomial displacement function was used for the beam elements. Bearing tests of nails into wood loaded parallel-to-grain and perpendicular-to-grain were conducted to calibrate the foundation stiffness from load-displacement data. The results produced non-linear load-displacement responses of the head of the nail which corresponded to 14 experimental results. The model was used to determine the effect of depth of penetration on load-displacement. Foschi and Longworth (1975) used a similar beam bending model to analyse timber rivets. A semi-analytical finite element model was developed. The displacement field around each rivet was approximated by assuming that displacements in directions other than the direction of loading were zero. A displacement function was chosen to satisfy boundary conditions and to take advantage of symmetry in the model. The final displacements were used to calculate stresses around the rivet or group of rivets. Failure was predicted from these stresses, which will be discussed using a procedure described in section 2.6.2. There have been numerous two-dimensional studies using finite element analysis where a state of plane stress is assumed. For example, Backlund and Aronsson (1986) determined the stress distribution around a hole in an orthotropic plate in tension with no bolt. Wilkinson et al. (1981) addressed bolt-loaded holes. They incorporated linear elastic orthotropic properties with isoparametric quadrilateral elements for the wood, and assumed the bolt to be rigid. A contact interface between the wood and the rigid bolt was used to account for the effect of the changing area of contact with increasing load. This required a non-linear solution algorithm. The effect of friction over part of the contact surface was included. The authors compared the finite element analysis results for stresses around the hole to Moire fringe results from experiments and found good agreement. This indicates that the finite element method is effective for prediction of stresses in relatively thin plates where plane stress conditions exist. In subsequent studies by others, more detail was added to produce more realistic predictions of the stress distribution around the hole and in the plate. Some studies assumed cosinusoidal load distributions applied to the hole in place of uniform load application to represent the rigid bolt. Hyer et al. (1987), however, included elasticity of the bolt in the finite element model for a wood connection but found that bolt elasticity was not significant for the two-dimensional analysis. This was due to the very high modulus of elasticity of steel bolts compared to that of wood. It was found that the clearance around the bolt significantly affected the stress distribution by changing the location, magnitude and direction of the peak stress. This is an important consideration for the design of bolts in timber connections since the holes are typically drilled slightly oversize to allow for ease of construction. It was also 15 confirmed that friction between the bolt and the wood should be included in these models because friction was found to reduce the bearing stress at the hole edge and increase the peak circumferential tensile stress. The two-dimensional analysis of multiple-bolt connections paralleled. the developments in the analysis of single-bolt connections. Rowlands et al. (1982) included the features of a rigid pin, orthotropic wood properties, and a contact surface between the bolt and wood in a model with two bolts in one row. They analysed different combinations of load distribution between the bolts to determine the effect of load distribution on stress patterns. Rahman et al. (1991) used a similar model but incorporated material non-linearity due to crushing of wood fibres. Again, different load ratios between bolts were assumed between the two bolts. It was shown that the non-linear compression behaviour parallel-to-grain resulted in improved stress predictions along the region beginning under the bolt and extending to the end of the specimen. The results corresponded to measured Moire fringe patterns. There have been numerous, similar studies to those already mentioned (Patton-Mallory et al., 1997a). Topics such as oversized bolt holes, contact friction, end distance, and edge distance, have all been studied for their influence on stress distribution. The limitation of the two-dimensional models, as previously mentioned, is that they are valid for thin members only, thus restricting the results to very simple examples of connections. Improvements in the material model, such as non-linearity in compression, will not reflect the true distribution of stresses in three dimensions as long as two-dimensional models are used. Since the prediction of failure is dependent on accurate predictions of stress throughout the entire wood member, particularly in locations where stress concentrations exist, and in instances where the member aspect ratio causes deformations to vary throughout the member thickness, three-dimensional models of connection geometry and material behaviour are required (Patton-Mallory 1997). Patton-Mallory (1996) noted that in non-wood composite materials, the slenderness ratio lid, where \/ is the length of the bolt (or member thickness) and d is the bolt diameter, is usually small enough that stresses are close to uniform throughout the thickness. This justifies using a two-dimensional model. For wood, however, a three-dimensional model is more appropriate, especially for those cases where stresses are not uniform throughout the member thickness. The two-dimensional assumption is only valid 16 for cases where the failure mode is uniform throughout the thickness or where the presence of an edge does not significantly alter the through-thickness stress distribution. 2.5.3. Three-dimensional Finite Element Models The two-dimensional models have been used in the past to reduce the computational size of problems. Faster computers have allowed more complex models to be developed. Chen et al. (1995) used a three-dimensional model to study connections in composite materials where out-of-plane normal and shear stresses have been shown to be responsible for delamination leading to failure (Camanho and Matthews, 1997). Chen et al. (1995) predicted delamination in fibre-reinforced plastic composites due to interlaminar shear stresses. The material was assumed to be orthotropic and linearly elastic. The authors performed parametric studies on friction between the bolt and the composite laminate, hole size, bolt elasticity, and laminate stacking sequence. The improved stress distribution predictions allowed the authors to assess the point at which delamination (failure) initiates. Models of the failure mechanisms and the prediction of failure strength require accurate models for material behaviour. A three-dimensional finite element model for a new hollow dowel timber connector was developed by Guan and Rodd (1996), with linear elastic properties for the wood and elasto-plastic properties for the hollow metal connector. Moss (1997) recommended that the effect of the number of rows of bolts and slenderness ratios should only be studied once a detailed analysis of a single bolt connection is conducted. He also suggested that modes of failure be studied in a manner similar to that used for timber rivets by Foschi and Longworth (1975). The three-dimensional model is best suited for these types of studies on end distance, aspect ratio and bolt yield strength, for example, all of which influence connection behaviour (Patton-Mallory 1997). Analytical studies are also more efficient than testing hundreds of wood specimens in more empirical tests. Patton-Mallory (1996) developed a three-dimensional finite element model of a single-bolt connection, shown in Figure 2-6, in solid sawn Douglas fir. Solid brick elements were used for the steel bolt and the wood. A contact interface was placed between the bolt and wood and assumed to be frictionless. One-quarter of the geometry was modelled due to symmetry. The geometry of the model could be modified for changes in end distance, edge 17 distance, member thickness, bolt diameter, hole clearance, and support location. No side members were included in the model to isolate the behaviour of the connector. Non-linear elasticity was assumed for compression parallel-to-grain and shear. Linear elastic properties were assumed for all other stresses, including all tensile normal stresses. The bolt was modelled with elasto-plastic behaviour. The non-linear material model is crucial to the development of accurate load-displacement curves for connections in wood. Crushing behaviour in compression results in non-linear stress-strain behaviour in the three orthotropic material directions in wood. Patton-Mallory (1996) assumed non-linearity in the critical direction, parallel to loading and found excellent correspondence with experimental load-displacement curves for a range of end distances and member thicknesses. This model did not, however, account for the energy dissipation associated with the non-recoverable deformations due to crushing, thereby ignoring the conservation of energy assumptions made in the development of the elastic theory. As a result, limitations existed with the non-linear material model and required non-linearity in the shear stress-strain behaviour in two directions to prevent over-prediction of these stresses. It was also shown (Patton-Mallory 1996) that the elastic stiffness matrix would develop negative terms on the main diagonal if non-linear behaviour was applied to the two other orthotropic material directions in compression. This material model can also result in poor estimates of the state of stress and failure strength. Studies of the stress field in this three-dimensional model (Patton-Mallory et al. 1998a, 1998b) provided evidence of regions with stress concentrations. Contour plots of each normal and shear stress (six in total) indicated the locations of stress concentrations in the vicinity of the hole. Tensile stress perpendicular-to-grain and shear stresses in the wood member adjacent to the bolt were plotted. Predictions of failure strength of the connection, however, were, not performed. The maximum stress and Tsai-Wu failure criteria were used to provide qualitative information on the probable modes of failure for each connection geometry. This was the first numerical model to determine failure modes in three-dimensions for wood. Patton-Mallory (1996) recommended that improvements to the material model (non-linear constitutive behaviour and failure prediction) be made to improve stress calculations and lead to better predictions of ultimate connection load. 18 2.6 Failure Prediction The ultimate strength of bolted connections has already been discussed in general terms. Load-displacement behaviour was found to be either ductile or brittle (or a combination of both) for single- and multiple-bolt connections. Bolt yielding and wood crushing result in ductile load-displacement behaviour, while splitting in tension perpendicular-to-grain and along shear planes causes sudden brittle failure, even in cases with some ductility. Numerical models, however, should predict the ultimate strength of the connection at the real failure point in the load-displacement history. To do this, the stresses leading to each type of failure mode must be predicted. Ductile behaviour such as wood crushing and steel yielding may not lead directly to failure since the connection continues to carry load through large displacements. Brittle failure, however, results in a sudden drop or total loss of connection strength. Methods such as fracture mechanics and failure envelopes can be used to predict the point at which brittle failure occurs in wood and non-wood composites. In the case of wood, tension and shear are the brittle material properties. All failure models presented here are based on satisfying some criterion (usually based on stress) for particular modes of failure of the material. 2.6.1. Failure Models The three-dimensional displacement behaviour around a bolt-hole create a three-dimensional stress field in the bolt vicinity. Fracture mechanics, however, normally deals with the propagation of a single crack in a uniform stress field. Fracture criteria for mixed-stress fields in wood, termed mixed-mode fracture for tension-perpendicular-to-grain and shear acting at the same time, exist but there is some question over the accurate determination of the fracture toughness constants, Kic and Knc, for wood (Fonselius and Riipola 1992). Crack propagation in wood will always favour the direction along the grain (Ashby etal, 1985, Fonselius and Riipola, 1992, Patton-Mallory and Cramer, 1987). A crack can start in any direction but will always turn to the direction with the lowest fracture toughness. In wood, this is normally the direction along the grain because the fracture toughness, Ki, in other directions is approximately ten times greater. In addition to difficulties in determining the fracture toughness constants, there is currently, to the author's 19 knowledge, no criterion for three-dimensional mixed-mode stress cases. Assumptions must be made in the application of such criteria in mixed stress fields. Two-dimensional models which employ fracture mechanics, such as Smith and Hu (1994) were discussed in section 2.5.1. In addition, Petersson (1995) developed a mixed mode fracture model for timber beams with holes or notches for a combination of tension and shear stresses acting simultaneously. Jorissen (1998) applied this fracture model to his predictions of connection strength by assuming that the stress state was constant through the wood member thickness around the bolt hole. Jorissen assumed the location and length of the crack to make this model work for connections, however, it is not clear whether Jorissen's approach will gain acceptance (Mischler and Gehri 1999). The two-dimensional finite element models of bolted connections in wood of the past did not extend the analyses into the realm of failure prediction; they focussed primarily on the effect of geometry and material effects on stresses (Patton-Mallory et al, 1997b). Failure prediction in non-wood composites has been attempted. Arnold et al. (1990) used the concept of a characteristic distance for failure of bolt-loaded holes in non-wood composite materials. In simple terms, the characteristic distance is a length of material which must be subjected to critical stresses prior to total failure. Thus, once some part of the material has reached critical failure stresses, the analysis is continued until this critical stress is reached all along a pre-selected path. The path and its length are referred to as the characteristic distance. It is claimed that this method accounts for flaws in the material since failure will not occur until a flaw of sufficient size creates stresses that exceed the failure stresses. Researchers have used this approach for multiple-bolt connections in non-wood composites and have achieved good agreement with experimental results (Hyer and Chastain 1988, Cohen et al. 1995, Eriksson et al. 1995). The discussion to this point has focussed on failure criteria based on the initiation of a crack. The other group of failure criteria only consider the state of stress in a material (Rowlands 1985). This concept states that stresses or strains exceeding a critical value of stress (or strain) cause failure. These criteria can be isotropic or anisotropic and they can ignore or relate interactions between stresses. The most simple criteria are the Maximum Stress and Maximum Strain criteria. Failure is assumed to occur when one of the principal stresses (or strains) exceeds the limiting stress (or strain) for that direction. If the material is 20 isotropic, then the critical value is the same in all directions. These theories do not account for the influence of the other stresses or strains on the critical strength values - that is, they ignore the combined effects of stresses (or strains). However, these criteria are simple to apply. The critical strength values can be determined by testing the material under uniaxial stress in each of the principal material directions - this procedure is typically applied to wood to determine its strength. The Maximum Stress criterion is independent of material non-linearity because it uses specific values of stress. The Canadian wood design code (CSA 1994) uses a variant of the maximum stress criterion to determine connection strength for connections loaded at an angle to grain. The formula, developed by Hankinson (1921) for compression stresses in wood, relates the strength parallel-to-grain (0\u00b0) and strength perpendicular-to-grain (90\u00b0) to calculate strength at intermediate angles. This formula is based on empirical results and is only applicable to a limited set of conditions. However, it has been widely accepted in practice. Many more rigorous failure criteria exist, as described in Rowlands (1985) and Nahas (1986). These typically include the interactions between stresses (or strains) and are based on polynomials in stress (or strain). For orthotropic materials, the Tsai-Wu (Tsai and Wu 1971) criterion relates stresses in the three normal stress directions, three shear stresses, and their interactions, to critical values of each stress. The polynomial has the form: [2-1] f(cjk)= F , a , + F i j G j Gj = 1 where i,jand k = 1,2,3,...6 Failure occurs when^ct) is greater than 1. Ft and Fy are the strength tensors of'second and fourth rank, respectively. The polynomial represents a failure surface in stress space. Higher order tensors are typically removed from the polynomial in practice to make the criterion manageable. Differences in tension and compression strengths are accounted for in the parameters F, and Fu. These terms can be determined from the uniaxial tensile strength, Xih and compressive strength, Xic, in each direction, z, as follows: [2-2] Ff=- -%it % ic and it ' A ic 21 Fy terms are not clearly defined in this way although some researchers have attempted to find a theoretically-based equation for these terms (see, for example, Cowin, 1979, Liu 1984). For two-dimensional problems, the Mises-Hencky criterion, for example, incorporates the tension and compression strengths to give: [2-4] ^ 1 2 = - -2\\XuXXcXltX2c Each of the proposed interaction equations, such as equation [2-4], require experiments under various stress combinations and at magnitudes which vary over a range of angles to the axes of orthotropy (off-axis testing) followed by calibration of the equation through curve-fitting. Experiments have been proposed to determine these terms but they remain difficult to obtain. Tsai and Wu (1971), Wu (1972), and Narayanaswami and Adelman (1977) suggested a number of tests for determining the interaction coefficients, Fy, for non-wood composites. Testing for wood products was performed by Clouston (1995) for parallel strand lumber where it was shown that, for a two-dimensional model, the Tsai-Wu criterion with the interaction terms could be used to predict failure providing that size effects were incorporated in the prediction of failure strengths. Thus, it was shown that the failure theory produced adequate results with the interaction terms included. Some researchers have assumed the interaction terms to be zero in their analyses (Narayanaswami and Adelman 1977) because determining these terms requires a considerable amount of testing even for just one coefficient, as was shown by Clouston (1995). Theocaris and Philippidis (1990), however, indicated that this may be sufficient for two-dimensional analyses but, for three-dimensional effects, arbitrarily zeroing the interaction terms of the failure tensor should be avoided. For three-dimensional models of wood, no literature has been found to indicate that the interactions can be neglected (Patton-Mallory et al, 1997b). Rahman et al. (1991) compared three different polynomial type failure criteria and non-linear material properties for a two-dimensional finite element model of a bolted joint in wood. They found good agreement with experimental results and noted that failure initiated at about 20% of the ultimate load in the experiments - this was predicted by their model. It was noted that three-dimensional effects were not considered in this model and that this 22 could lead to an incorrect prediction of the mode of failure. They recommended that a three-dimensional model be developed. Patton-Mallory et al. (1998a, 1998b) used the maximum stress criterion and the Tsai-Wu criterion to provide information on regions with stress concentrations in a three-dimensional single-bolt finite element model. Non-uniform stress throughout the member thickness was predicted from the model, and the failure criteria further illustrated the regions most likely to fail. This model provided qualitative data on the likely mode of failure of the connection, but not the actual load at which failure would occur. Patton-Mallory et al. (1997b) reviewed research on failure prediction of bolted connections related to the effects of geometry on the strength and failure modes. It was suggested that considerable development of failure criteria is still required for bolted connections in wood. One analytical model for timber connections which has been used successfully to determine connection strength and mode of failure is the finite element model for timber rivets developed by Foschi and Longworth (1975). Their approach, as will be discussed shortly, considers the size effect in timber. 2.6.2. Size Effect The material properties for wood are highly variable compared to manufactured non-wood composites. The ultimate strength of solid sawn wood in tension or shear, for example, normally has a coefficient of variation of approximately 20%. In particular, strength in the brittle modes (tension and shear) depends greatly on the size of the piece of wood being tested (Barrett 1974). Barrett found that the ultimate tension stress perpendicular-to-grain in Douglas fir was related to the stressed volume of the specimen. This was explained according to the weakest link theory (Weibull 1939) using a Weibull probability distribution of the strength data as a function of specimen volume. The Weibull theory postulates that for brittle materials, larger specimens are more likely to fail at lower stresses due to the increased probability of a flaw in that larger specimen volume. This concept was used in further developments of the Canadian wood design code (CSA 1994) for timber strength according to work by, for example, Madsen and Buchanan (1986). 23 The probability distribution of brittle failure can be represented as a function of the stress distribution over the volume of material with the following two-parameter Weibull distribution: [2-5] F , . \\ - . r ' K m i where Fy is the probability of failure, V* is a reference volume, k is the shape parameter, m is the scale parameter associated with the reference volume, and a is the stress (tension or shear, in any given direction). The failure stress for a given probability of failure,\/*, is: [2-6] o* =m[-\\xv{\\-p)]lk It can be shown that failure occurs when a uniform reference stress, or*, occurring over volume V*, satisfies the following inequality which relates a non-uniform state of stress over an arbitrary volume to the reference stress at the same probability of failure: [2-7] lakdV>akVt v This has a form similar to the maximum stress criterion where each stress is treated individually and stress interactions are assumed to have little effect. When the stress field is constant over a volume, the failure stresses for a given probability level, p, equation [2-7] can be simplified to: [2-8] \u00b0 1 CT2 Y2 v^iy where u\\ and 0*2 are the strengths of specimens of volumes, V\\ and Vi, respectively, and k is the shape factor (from above) determined from a Weibull fit of the cumulative probability distribution plot of experimental strength values. Foschi and Longworth (1975) used the weakest link strength model to predict the failure of timber rivets in Douglas fir. The finite element model developed in Foschi and Longworth (1975), as discussed in section 2.5.2, was used to analyse the stress field around a group of timber rivets. The stresses were analysed at each load step to determine i f any of them exceeded the critical stress determined from equation [2-7]. The tensile stresses, for example, in the model were integrated over the entire volume of the specimen to determine 24 the left-hand side of equation [2-7]. If the inequality was true for a given failure probability, then the finite element analysis was stopped. By checking each normal and shear stress at every load step, Foschi and Longworth (1975) were able to predict failure modes and strengths for tension perpendicular-to-grain and for shear, from the first of critical stresses to be exceeded. The three-dimensional finite element model, discussed earlier, is well suited for determining the volume integral throughout the wood member, because it includes the effects of uneven stress distribution through the member thickness. It is noted that the two-parameter Weibull distribution in equation [2-5] has been criticized for poor estimates of shear strength. This is an area in which future research is required. 2.7 Material Properties The prediction of failure is dependent on the underlying material model. From the review of literature, it is clear that there does not exist any widely accepted material model for wood which addresses the orthotropic nature of the material, its brittle characteristics in tension and shear, its ductile behaviour in compression and its variability. Each numerical model addresses some part of this, but no unified three-dimensional model has been developed for wood to the extent that is required for accurate predictions of connection strength. A new model will be proposed in Chapter 5 to address these points. 25 Displacement transducers High strength loading bolts Rigid steel frame Single wood member Single mild steel dowel FRONT VIEW SIDE VIEW Figure 2-1. Test apparatus for single-member\/single-bolt study (after Patton-Mallory 1996). No permanent bend in bolt Permanent bend in bolt \u2014 Tension perpendicular-to-grain \\ Q\u2014 ( c -Shear plug S Q\u2014r Crushing \/ hole elongation S ( o Figure 2-2. Failure modes of single-bolt connections in tension in Douglas fir (after Patton-Mallory 1996). i 26 F Wood crushing only Governing strength is lesser of: Side member crushing: F=dtJs Main member crushing: F = dtmfm where is bolt diameter, \/ is member thickness (main or side) \/ i s member embedment strength (main or side) F < 0 3 C 0 > ^2 2F Wood crushing and bolt yielding Equations of equilibrium: F~y\\ fsd (side member) F = yi fm d (main member) M,= X-y}fsd-My 1 2 M 2 = -y2fmd-My M, + M 2 = 0 From these, the governing strength is: F--fm+fs where \/\" d3 My = is the yield moment in the bolt, with yield strength fy. Figure 2-3. Sample European yield model failure modes and free-body diagrams. 27 Steel bolt Wood block with pre-drilled half-round 1.6 mm larger than bolt diameter. Figure 2-4. Half-hole bolt embedment test from ASTM D5764. F-4 F-F-AA\/V AAAr AA\/V M\/VV-i HAAq MAAq MAAr-j I \u00a3m I km I I -AAV AAAr AAAr -+-2F MAAT-J MAAT-I MAAg MAAg AAAr AAAr AAAr A:s = stiffness of the side members km = stiffness of the main member kh\u2022= stiffness of the bolt\/member interaction Figure 2-5. Multiple-bolt connection spring model (after Isyumov 1967). 3 symmetry mwM mm\u00ae \" * \u00bb n jm\u00bb * \u2022* '4 mm mm mtm 1 1 1 i i 1 411111 dii l l l it 1 syinmetry 2 displacement specif ied restrained i n 2 Figure 2-6. Finite element model geometry (from Patton-Mallory 1996). CHAPTER 3 LAMINATED STRAND LUMBER (LSL) Manufactured composite materials enable a fabricator to specify materials and stacking sequences to produce a final product which best suits a particular application. In the current research programme, this approach is applied to laminated strand lumber to improve the behaviour of bolted connections. In this chapter, a brief background of laminated strand lumber (LSL) is provided. This is followed by a description of connection improvements achieved in non-wood composites by modifying the alignment and stacking sequence of layers. The discussion proceeds to describe the fabrication of LSL with different arrangements of layers to achieve similar improvements in connection response in wood composites as were achieved for non-wood composites. A series of experiments to determine the mechanical properties of the finished LSL panels is described, along with the measured results. The chapter concludes with the numerical prediction of in-plane wood composite panel properties and their associated variability using lamination theory and reliability analysis. 3.1 Background Laminated strand lumber (LSL) is an oriented strand wood composite which can be produced with sufficient thickness to be used as structural building elements such as beams and columns. LSL and other composite wood products have lower variability than solid sawn lumber because the defects which are present in sawn lumber are distributed in LSL. Smaller diameter, faster growing species such as Aspen are typically used for this product. These species would not normally be used for structural applications, but once they have been made into LSL, properties can be achieved which are as good as, or better than, solid lumber. LSL is produced in large panels which can be cut into structural members. LSL is differentiated from flake board or oriented strand board (OSB) by its longer strands and greater panel thickness. The target size of each strand is 30.5 cm (12 inches) long, 2.5 cm (1 inch) wide, and less than 1 mm (0.037 inches) thick. The strands are produced mechanically by slicing strands from logs using knives. The strands are passed through dryers to reduce 30 their moisture content and then sorted by size to eliminate those strands which are considered to be short. The final steps of the process include spraying the strands with a pMDI isocyanate resin in a blender, orienting the strands as they are placed into a mat, and pressing the mat using steam, heat and pressure until the resin cures. The target density of the finished panel is controlled by the amount of strands in the panel and the pressing cycle. The material properties of LSL are dependent on the density of the panel, the strand species and the orientation of strands. Density and species are, generally, directly related to stiffness and strength in both solid wood and in wood composites. Strand orientation in the plane of the panel can be controlled to increase axial and bending stiffness and strength. Although most strands in LSL are oriented in one direction, limitations in the manufacturing process result in a significant portion of cross-aligned strands. This results in reduced stiffness and strength in the direction parallel to the strands and increased stiffness and strength in the orthogonal direction in the plane of the strands. By adjusting these variables, the manufacturing process can be modified to produce panels with material properties which suit a particular application (such as improving bolted connection behaviour). A number of studies illustrate the impact of strand alignment on axial and bending properties of members cut from panels. Hunt and Suddarth (1974) fabricated particleboard mats of average density 670 kg\/m3 (42 lbs.\/cu. ft.) using Aspen flakes arranged randomly throughout the mat thickness. They found the average in-plane tensile elastic modulus to be 5.7 GPa (830,000 psi). Hoover et al. (1992) fabricated panels using an electrostatic aligner to produce fully aligned panels from Aspen strands. With an average density of 690 kg\/m3 (44 lbs.\/cu. ft.), the elastic modulus in tension parallel to the strand alignment was reported to be 8.3 GPa (1,201,000 psi). Thus, the effect of orientation was to increase the in-plane axial tension modulus by 45%. Panels fabricated with three different layers of strands were studied by McNatt et al. (1992) to determine the effect of non-uniform strand alignment throughout the panel. They fabricated and tested eight rectangular three-layer panels with combinations of oriented and non-oriented layers of equal thickness. Average density was reported to be approximately 650 kg\/m3 (41 lbs.\/cu. ft.). The first set of panels had randomly oriented strands throughout the panel thickness. The elastic modulus of this panel was determined using a bending test and reported to be approximately 5.0 GPa (720,000 psi). The seven other lay-ups were 31 combinations of surface layers and core layers with strands in one of three orientations: a) fully aligned at 0\u00b0 (the long axis of the panel), b) fully aligned at 90\u00b0 (across the width of the panel), or c) randomly aligned. A disk orienter and a vane orienter were used to align strands at 0\u00b0 and 90\u00b0, respectively. Using data for these panels, the graph in Figure 3-1 was produced showing the influence of the degree of orientation on the elastic modulus in the strong direction of the panel. The degree of orientation is expressed as a percentage of the total panel thickness for each panel type. This graph indicates that the effective elastic modulus of a panel is dependent on the amount of strand alignment in the panel. 3.2 Connection Behaviour of Non-wood Composites Fibre-reinforced plastic composites are, generally, more brittle than wood composites but they both have orthotropic material properties which can be modified to suit particular applications. Although this is common practice for non-wood composites, the opportunity still exists for improvements in wood composites through changes in layer stacking sequence and orientation. Bolted connections in non-wood composites have been shown to benefit from changes in the layer structure resulting in changes in bearing strength and mode of failure of the connection. The effects of the layer lay-up on connection behaviour in composite laminates are primarily related to fibre orientation and stacking sequence. Fibre orientation was found to greatly influence the bolt bearing strength of laminates (Collings 1977, Quinn and Matthews 1977, and Hamada et al. 1995). While the bearing strength is largely dependent on the longitudinal compressive strength of the laminates in the 0\u00b0 direction, the plies at \u00b1 45\u00b0 help to \"diffuse the load path around the bolt hole\" (Collings 1977). It was shown that maximum bolt bearing strengths were achieved in quasi-isotropic laminates when a combination of 0\u00b0, 90\u00b0 and \u00b1, 45\u00b0 plies were used. Highly orthotropic laminates containing over 50% of 0\u00b0 plies, however, failed in shear at a lower maximum load. Hamada et al. (1995) found that the highest strength was obtained in a twelve layer quasi-isotropic laminate [0790\u00b0\/+457-45 o\/+45 o\/-45\u00b0] s confirming Collings' results1. This quasi-isotropic laminate also had very high net tension strength. 1 Subscript's' indicates symmetry around the last layer listed, i.e. layers are repeated in reverse order. 32 Quinn and Matthews (1977) showed that the stacking sequence significantly affected the strength of a single-bolt connection. The highest bearing strength was achieved by placing the 90\u00b0 plies near the outer surfaces of the laminate. The 90\u00b0 plies were believed to prevent delamination by reducing the inter-laminar stress and the through-thickness normal stress. In effect, the 90\u00b0 plies acted as a clamp on the material by imparting a lateral constraint on the inner plies. The mode of failure was also found to change with stacking sequence. Quinn and Matthews (1977) found that a net section tension failure occurred in laminates with 90\u00b0 layers at, or near, the surfaces. They attributed this to small edge distances and indicated that the benefit of increased connection strength was offset by the occurrence of this catastrophic failure mode. Laminates with 0\u00b0 layers on the surface primarily underwent bearing failure followed by shear failure. These laminates also had the lowest connection strengths. Other layer combinations resulted in mixtures of these failure modes. Based on these findings, and the results of preliminary tests on bolt behaviour in LSL, it is probable that shear failures and tension perpendicular-to-grain failures may be prevented in LSL connections through the use of appropriately arranged cross-aligned strands. Differences in practice, however, which exist between non-wood and wood composites must be considered since these may influence the modes of failure in connections. Washers are, generally, placed under the head and nut of bolts in each material - the washers for composite materials provide an external clamping force on the laminate which results in increased ultimate strength of the connection because delamination is prevented. Confinement resulting from the tightening of bolts in wood, however, cannot be relied upon for this type of constraint since shrinkage and creep of the wood can relieve the effect of the initial prestressing. Another difference is the hole clearance in the materials. Non-wood composites can be designed to exact specifications with regard to hole size. It has been shown that press-fit connections and those with oversize holes do behave differently. As indicated in Chapter 2, wood construction practice in North America normally requires holes to be drilled 1 mm (1\/16 inch) larger than the diameter of the bolt to allow for ease of construction. These differences also affect the approach used to produce improved material lay-ups for LSL. 33 In a similar fashion to tailoring of non-wood laminated composites for specific applications, the panel strand alignment and stacking sequence of wood composites can be changed to optimize member strength and, in particular, improve the behaviour of bolted connections. 3.3 LSL Panel Lay-ups To study the influence of strand orientation on connection behaviour, five lay-ups of LSL were fabricated. The panels were formed and pressed at the Alberta Research Council in Edmonton. The five panel lay-ups used for this study were: A) fully oriented; B) randomly oriented; C) surfaces oriented, core randomly oriented; D) surfaces randomly oriented, core fully oriented; and E) eight fully oriented layers aligned at angles 0\u00b0 and \u00b1 45\u00b0. The fully aligned (unidirectional) and randomly aligned (in-plane isotropic) lay-ups, 'A' and 'B', respectively, were chosen as two extreme cases in strand alignment. Type 'A' is a highly orthotropic material with high strength and stiffness in the direction parallel to the strand axis and low strength perpendicular to the strand axis. Type 'B' has the same properties in the plane of the panel, regardless of direction. Commercial grade TimberStrand\u00ae LSL is a combination of types 'A' and 'B' since not all strands are fully aligned. The three-layer panels, ' C and 'D', were formed to determine the effect of stacking sequence on in-plane properties and connection behaviour. Type 'E' lies mid-way between 'A' and 'B' in terms of percent strand alignment. Unlike commercial grade LSL, however, type 'E' panels have a known amount of cross-aligned strands. Letting R represent random layers, the laminate stacking sequence for each panel type is indicated in Table 3-1 along with the amount of aligned strands expressed as a percentage of total panel thickness. A summary of the panel fabrication process is provided here, while a detailed description can be found in Moses and Prion (1999). The Aspen strands used for these panels had average dimensions of 23 cm (9 inches) in length, 2 cm (0.9 inches) in width, and less than 1 mm (0.037 inches) in thickness. The length and width were found to be lower than the target values described in section 3.1 due to breakage during drying and blending. Finished panel dimensions were roughly 76 cm (30 inches) wide by 76 cm (30 inches) long, and 38 34 mm (1 V2 inch) thick. Layers were of equal thickness in panels consisting of three or eight layers. The density of all finished panels was approximately 690 kg\/m3 (43 Ibs.\/cu. ft.). A high degree of control in strand alignment was achieved by placing strands by hand for all panels including the layered panels ' C \\ 'D' and 'E' (a mechanical orienter is normally used in production). Strands were bundled by hand and sprinkled onto the mat in an oversized forming box. Fully aligned layers were formed by carefully placing strands in one direction. Random orientations were achieved by shaking strands by hand and dropping them from a greater distance onto the mats. The \u00b1 45\u00b0 layers were achieved in the same way as the fully oriented strands. A total of 45 panels were formed and pressed. Three type 'A' panels did not press well and suffered steam blowouts. Of the remaining 42 panels, some showed signs of poorly formed pockets, indicating regions of poor steam penetration. Since the panels were not surface planed, any imperfections were visible as unevenness in thickness. The final panel count is listed in Table 3-1. 3.4 LSL Material Properties The elastic, plastic and failure constants, characteristic of the five types of LSL panels were determined through a series of tension, compression, and shear tests. A summary of the tests methods and results is given here. For more details see Moses and Prion (1999). 3.4.1. LSL Properties - Experiments ASTM test methods were used where possible for these experiments. Test specimens were cut from the panels and then placed in a conditioning chamber with 65% relative humidity and 20 \u00b0C (68 \u00b0F) for two months prior to testing, resulting in 9% average moisture content. Axial properties were tested in three material directions: a) parallel to the longitudinal axis of the strands (the Y-direction of Figure 3-2), b) perpendicular to the longitudinal axis of the strands (the X-direction), and c) perpendicular to the panel surface (Z-direction). Shear properties were tested in three planes, corresponding to xxy, xyz, and Txz, as shown in Figure 3-2. Tension behaviour in the plane of the panel was determined using modified ASTM D1037 (ASTM 1991a) specimens with dimensions and number of replicates indicated in 35 Table 3-2. The specimen blanks were 25 mm (1 inch) wide and 12 mm (lA inch) thick but necked down in the centre to a 12 x 12 mm QA x lA inch) cross-section. 75 mm (3 inches) was left at each end of each specimen to allow for clamping by the wedge grips. Two effective lengths, as shown in Table 3-2,. were tested in each direction in the plane of the panel to determine the effect of size on ultimate strength. The length ratio was 4:1 for specimens loaded parallel to the main strand axis (Y-direction) and 2:1 for specimens loaded perpendicular to the main strand axis (X-direction). Strain was measured on these specimens using an extensometer to determine elastic modulus and ultimate stress. Internal bond tension specimens were tested according to ASTM D1037 to determine the tension properties perpendicular to the panel surface (Z-direction). Two sizes were tested as indicated in Table 3-2, to measure the size effect on ultimate tensile strength perpendicular to the panel surface. These specimens had a length ratio of approximately 2:1. No strain was measured. Compression in the plane of the panel was measured in the X- and Y-directions according to ASTM D143 (ASTM 1991b). Specimen dimensions listed in Table 3-2 were adjusted from the ASTM D143 specifications to allow for the 38 mm (1 lA inch) panel thickness. Load and displacement were measured and elastic modulus and ultimate stress were calculated according to the standard. Compression in the Z-direction was measured using a small block of dimensions listed in Table 3-2. Stress and strain were determined from load and platen displacement measurements. Shear blocks were cut according to ASTM D143 with dimensions adjusted to allow for the 38 mm (1 lA inch) panel thickness. Specimens were cut in three orientations, as shown in Figure 3-3, to measure interlaminar and in-plane shear strength. Two specimen sizes were cut in each orientation with a width ratio of 2:1 to determine the size effect in shear. The XY orientation was used to determine the edgewise shear of the panel (although shear failure is unlikely to occur in this orientation). The YZ and XZ orientations are equivalent to interlaminar shear parallel-to and perpendicular-to the main strand axis, respectively. Load and platen displacement were recorded during these tests. Failure modes were observed and documented, where possible, for all tests. 36 Poisson's ratios were measured for panel types 'A' and 'B' using three smaller compression specimens. The six Poisson's ratios were determined from these specimens. The two specimens loaded in the plane of the panel had dimensions 25 x 25 x 100 mm (1x1x4 inch) and the third, perpendicular to the plane of the panel, was 25 x 25 x 38 mm (1 x 1 x 1 Vi inch). Ten replicates were tested for each type of specimen. The Poisson's ratios were determined from data prior to the onset of non-linear stress-strain behaviour. This was a non-standard, non-destructive test method. Four steel pads were mounted to the sides of each specimen with a 25 mm (1 inch) gauge length to accommodate a bi-axial extensometer. The Poisson's ratio was determined from the ratio of transverse to axial strains for each specimen. 3.4.2. LSL Properties - Test Results The test results from these experiments are summarized in Tables 3-3 and 3-4. Detailed results are given in Moses and Prion (1999). Elastic moduli and ultimate strengths are listed in Table 3-3 for each type of panel. (It is noted that tension and in-plane shear specimens in the three- and eight-layer panels were non-symmetric in lay-up once they were cut to the specimen thickness from the original panel.) Poisson's ratios and elastic moduli (from axial measurements during Poisson's ratio tests) are listed in Table 3-4. The elastic moduli are notably different for each test method indicating some dependence on loading in tension versus compression and on large versus small (from Poisson's ratio tests) compression specimens. The ratio of elastic moduli for the principal material directions, from most flexible to most stiff, is roughly 1:7:130 for type 'A' panels; this indicates a high degree of orthotropy. Reported compression strengths are the average ultimate stresses at failure. The ratio of compression strengths is roughly 1:4 for type 'A' specimens. No failure was observed in the compression specimens loaded perpendicular to the panel surface, however, elastic moduli were determined in this direction. It was noted that the values of elastic modulus in this orientation for each panel type were similar, indicating little influence of panel lay-up on this property. Sample stress-strain curves for compression specimens in type 'A' panels are shown in Figure 3-4. The elastic modulus was determined assuming a bi-linear fit of the experimental data. A windowing procedure was used to determine the 'yield' point, initial slope and post-yield slope from the data points. The average values for each set of ten curves 37 are superimposed onto Figure 3-4. Compression specimens from panel types ' C , 'D' and 'E' were observed to delaminate. Ultimate strength data in tension and shear were fitted assuming a Weibull distribution for these brittle modes of failure, with shape and scale parameters k and m, respectively, shown in Table 3-3. These values each are averages of the measurements on two sizes of specimen and are based on a unit volume, V*. The ratio of tension strengths are approximately 1:4:40 for type 'A' panels. Crack arrest was noted to occur in tension specimens of the layered panels ('C, 'D' and 'E') where cracks were found to originate in weak layers but stopped growing or changed direction (beginning with growth perpendicular to the main strand axes and continuing with growth along the plane of the panel) until total brittle failure occurred across the specimen. It was noted that the ultimate tension strength perpendicular to the panel surface (from internal bond tests) did not appear to be influenced by the panel lay-up. There was a measurable size effect in the tension specimens unlike the shear specimens (Moses and Prion 1999), agreeing with results for solid sawn lumber. Bendtsen and Porter (1978) reported that no statistically significant difference in shear strength could be found between standard-sized Douglas-fir ASTM D143 shear blocks and ones which were 25% smaller. According to Foschi and Barrett (1976), one would have expected approximately a 5% difference in the strength for smaller specimens, however, at a 95% confidence level, this difference was not detectable in Bendtsen and Porter's experiments. The current study has a specimen size ratio of 1:2, which, for Douglas-fir, would have resulted in a 12% difference in strength. Shear blocks did not fail in shear in the XY plane. The failure of these specimens is best described as the separation of strands, crushing of the contact surfaces, and no signs of failure along the intended shear plane. Specimens in the interlaminar YZ and XZ orientations did fail along the intended shear plane. Sample load-displacement curves recorded for shear block specimens are shown in Figure 3-5. No estimate of shear modulus is easily made from this data, only ultimate strength. For wood products, ASTM D1037 (ASTM 1991a) and D3044 (ASTM 1991c) are commonly used to determine the shear modulus in each plane, however, in the current study, more material would have been required to perform these tests. Thus, shear moduli for type 38 'A' and 'B' specimens were assumed based on results for Aspen from Hunt and Suddarth (1974), Hunt et al. (1985), Hoover et al. (1992) and Bradtmueller et al. (1997), as indicated in Table 3-5. 3.5 Calculation of Panel Properties The in-plane axial and out-of-plane bending properties of layered panels can be predicted using the mechanics of laminated composite materials. Panel types ' C , 'D' and 'E' are combinations of the fully oriented and randomly oriented panels 'A' and 'B', respectively, thus their properties can be predicted based on the properties of the individual layers within them. The variability in properties can also be estimated by using the First Order Reliability Method (FORM). 3.5.1. Mechanics of Laminated Composite Materials Multiple-layered composite materials can be represented as a stack of thin plates which have continuity at the lamination interfaces. In-plane elastic properties form the stiffness matrix in each layer using Hooke's law for two-dimensional unidirectional materials (plates): where 1 and 2 are material directions analogous to X and Y in Figure 3-2, and 6 is the in-plane shear (XY). Since unidirectional layers can be stacked at any angle, 9, with respect to the global longitudinal axis of the entire laminate, as shown in Figure 3-6, the stiffness matrix [Q] must be transformed to give global stresses: where [T] is the transformation matrix, and is a function of 9. Global stresses and global strains are related through a transformed stiffness matrix [Q] which incorporates the effect of the transformation for each layer such that: [3-1] 12 \\ = \\Qhoca,} [3-2] [3-3] 39 The transformed stiffnesses of each layer can be assembled into a global stiffness matrix for the laminate by relating applied loads and moments to strains and curvatures, respectively, while accounting for the position and thickness of each layer. Let the midplane of a laminate be the plane lying at exactly half the total thickness, h, of the laminate as shown in Figure 3-6. The distance from the midplane to the interface between each of r layers, is hr, and is required to determine the total force carried by each layer due to the linear strain distribution across the laminate thickness. Normal and shear forces, {N*, Ny, iV x y} are determined by integrating stresses over the thickness of the laminate. Bending of the laminate can occur under axial loads if the stacking pattern is non-symmetric. Thus, moments, {M*, My, M x y}T, and curvatures, {KX, KY, KXY}T, are introduced into the stiffness matrix for the entire laminate. Using a partitioned stiffness matrix for the global behaviour of the laminate, the summation of the transformed stiffnesses of individual layers and the distances hr, is: _B D\\{K} where the extensional stiffness [A] is derived from: P-5] 4, =![((?,,)]('<,-'',-,). ;=l,2,3;;=l,2,3 [3\"4! IM r=l [B] and [D] are the coupling stiffness and bending stiffness, respectively. Effective elastic axial constants of symmetric laminates can be determined from [A] alone, as follows: 1 \u201e 1 ^ 1 [3-6] E = \u2014 E y = \u2014 Gxy hAn hA22 hA66 \u2014 ^12 \u2014 ^12 Vxy~~~A~ Vyx~ A A \\ 1 ^22 For details on bending properties, the general theory of laminates, and the determination of effective properties of non-symmetric laminates, see Kaw (1997). The global in-plane elastic properties from equations [3-4] to [3-6] were used to predict the properties of ' C \\ 'D' and 'E' (each of which is based on layers of 'A' and 'B') with the following assumptions: a) The fully oriented layers (type 'A' panels) were assumed to be uni-directional. Strands were placed by hand in a controlled manner resulting in a negligible amount of cross-orientation. 40 b) The randomly oriented layers (type 'B' panels) were assumed to be in-plane isotropic. Strands were shaken by hand above the mat and dropped into place resulting in a random strand pattern. c) Properties within each layer were assumed to be homogeneous throughout that layer. d) Perfect bonding was achieved between layers, and e) All properties remained linear and elastic. This approach and these assumptions remove the influence of off-axis strands from the analysis by using aggregate properties of 'A' and 'B' panels. The ultimate strength of the laminate is assumed to be governed by the first layer to reach its failure stress. This is a conservative assumption since other layers will continue to carry load. The laminate is checked for failure in compression and tension. A failure criterion is used to account for combinations of stresses in each laminate which occurs, for example, when axial loads are applied at an angle to a layer's primary strength axis. The Tsai-Wu criterion, described in the Chapter 2, can be used. It incorporates axial and shear stresses in the prediction of failure while allowing for differences between tensile and compressive strengths (Tsai and Wu, 1971). In this two-dimensional situation, only one interaction term, F\\2, is required, and is given by equation [2-4]. To perform the ultimate strength analysis on in-plane axial loads only, each of the three forces {N} are applied individually, as unit loads. Axial forces, Nx> and Ny are applied first in tension and then in compression. The stiffness terms in [3-4] and ultimate strengths (first ply failure) can then be determine through the composite laminate analysis just described. This procedure was programmed into a FORTRAN subroutine. 3.5.2. Reliability Method The procedure for panel analysis can be used exactly as described above to determine the mean values of laminate strengths if the underlying material properties (elastic and ultimate strength) are normally distributed. Wood properties are commonly prescribed with probability distributions to account for variability, thus a technique is required which includes variability in layer properties in the prediction of the variability of the complete laminate properties. The technique must also be able to accommodate other probability 41 distributions (such as Weibull) since brittle strength in wood is commonly described using the Weibull distribution and not a normal distribution. The first order reliability method (FORM) can meet these needs when used in association with the subroutine described in section 3.5.1. The following procedure was used to determine the properties of panel types ' C , 'D' and 'E' based on the distributions in properties of the 'A' and 'B' panels. The reliability method is used to determine the probability of exceedence of a given value. For example, the probability, P, of the panel elastic modulus, Ex, exceeding 10 MPa: [3-7] P = Prob(\u00a3 x> 10 MPa) can be determined using reliability analysis for a given lay-up based on the probability distributions of the properties of each layer. The analysis can be repeated many times for different exceedence values to reveal the cumulative distribution function of Ex. From this function, the mean and standard deviation are determined. A similar procedure is applied to each of the ultimate strengths (these are fitted with appropriate distributions). For details on the reliability method, see Foschi et al. (1997). A software package using FORM (Foschi et al., 1997) was linked with the composite laminate analysis subroutine from section 3.5.1. As indicated earlier, the mean values of normally distributed data can be determined without the reliability method, however, the reliability method provides standard deviations on the values - this information can be used in the production process to assist in setting target levels for properties within an assigned confidence interval. 3.5.3. Predictions of Panel Properties The properties for panel types 'A' and 'B' given in Tables 3-3 to 3-5 were entered into the composite materials subroutine with the reliability analysis program (Foschi et al., 1997) to predict the behaviour of the symmetric panels ' C , 'D' and 'E'. Ex, Ey, Gxy, and ultimate strengths in tension, compression and shear were calculated. The predictions were compared to experimental results and to the quantity of aligned strands in each panel indicated in Table 3-3. Elastic properties determined from compression tests are shown in Figure 3-7. The primary elastic modulus, Ey, was found to increase with increasing strand alignment in the model and experiments. Conversely, Ex was found to decrease with increasing strand 42 alignment. Increasing variability was noted in model predictions in this orientation, however, the test values appeared to be lower than predicted for type 'D' and 'E' panels. The model predictions may have been off due to assumed input values of shear modulus, G x y , distributions for 0% (type 'B') and 100% (type 'A') strand orientation. The predictions of shear modulus for the other panel types are shown in Figure 3-7. Test panels 'E' had layers aligned at \u00b1 45\u00b0 resulting in the highest shear modulus prediction compared to other percentage alignments. This indicates that the strand orientation measure on the abscissa does not entirely account for the panel lay-up. Ultimate strength predictions in compression are shown in Figure 3-8. Compressive strength parallel to the direction of strand alignment, Y-direction, was found to increase with increasing strand alignment. Strength decreased in the transverse direction, X-direction, with increasing strand alignment. Test data fit within reasonable bounds of the measured variability. Ultimate strength predictions in tension were made assuming two layers for panels ' C and 'D' and three layers for 'E' panels. This corresponded to the test specimens which were cut to a reduced panel thickness of 16 mm (0.64 inches) from 38 mm (1 V2 inches). Hence, the two and three layer specimens were no longer symmetric in lay-up, thus, the reported failure stress is not comparable to the actual stress required to fail the entire symmetric panel thickness in tension. The failure stress predicted by the model incorporated the size effect and should be conservative since the failure procedure was terminated upon reaching the stress required to fail the weakest layer. The results are shown in Figure 3-9. Note that the percentage of aligned strands has changed to 78% for type ' C panels, 10% for 'D' and 33% for 'E', as a result of the reduced cross-section. The predictions were very close for this data set indicating that the model can accommodate a variety of lay-up configurations. Strength parallel to the strand alignment was noted to increase with increasing strand alignment although the relationship does not appear to be linear. Perpendicular to strand strength was noted to decrease with increasing alignment. Based on the convincing agreement between experimental and model predictions, ultimate tensile strength for the full thickness symmetric panels was predicted and results are shown in Figure 3-10. A linear relationship appears to link ultimate tensile strength to strand alignment parallel to (Y-direction) and perpendicular to (X-direction) the main strand axis. 43 Ultimate shear stress in the plane of the panels was predicted and compared to experimental results as shown in Figure 3-11. Predictions matched well with the experimental results. 3.6 Summary The prediction of properties of LSL composite panels was made using a combination of mechanics of composite materials and reliability theory. The influence of strand alignment on the in-plane properties of the panels was included in the model and the variability in results matched well with experimental results. The material properties and behaviour discussed in this chapter form the basis of the properties which are required for constitutive modelling of this material. Such a model is required for advanced numerical modelling of bolted connections using LSL, as will be discussed in Chapter 5. 44 Table 3-1. Laminate stacking sequences. Panel Type Sequence1'2 % aligned strands3 # panels produced A [0\u00b0] 100 20 B [R] 0 7 C [0\u00b0\/R\/0\u00b0] 66 5 D [R\/0\u00b0\/R] 33 5 E [0\u00b0\/+457-45\u00b0\/0] s 50 5 Notes: 1. R indicates random orientation. 2. Subscript's' indicates plane of symmetry in stacking (i.e. an 8-layer laminate). 3. Expressed as percentage of total panel thickness. 45 Dimensions Number of Replicates (for each panel type) Tension X-direction \u2022c^ x\u2014 A <4 \/ \u2022 w = 0.5 inches t = 0.5 inches 1 = 3 inches w = 0.5 inches t = 0.5 inches \/ = 6 inches 30 30 Y-direction w = 0.5 inches \/ = 0.5 inches \/ = 3 inches w = 0.5 inches \/ = 0.5 inches \/ = 12 inches 30 30 Z-direction w = 2 inches \/ = 2 inches \/= 1.5 and 0.7 inches 30 in each of two sizes Compression X- and Y-directions w = 1.5 inches \/ = 1.5 inches \/ = 8 inches 10 in each direction Z-direction w = 1.5 inches r = 1.5 inches \/ = 1.5 inches 46 Table 3-3. Results from tension, compression and shear tests. B C 2 D 2 % oriented strands7 100 0 66 33 50 Tension Tension Parallel (Y-direction) Tension Perpendicular (X-direction) \u00a3 y ( x 103 psi) Suit. (m* in psi, k) V* (cu. in.) \u00a3 x (x 103 psi) Cfult. (m* in psi, k) V* (cu. in.) 1814(28) 8150, 5.7 1.0 123 (38) 747, 4.5 1.0 779 (35) 3827, 3.7 1.0 779 (35) 3827,3.7 1.0 1380 (28) 7527,5.1 1.0 302 (58) 1298,2.6 1.0 > 720 (30) 3585,6.1 1.0 582 (52) 2805, 3.8 1.0 1011 (42) 4655, 5.5 1.0 246 (45) 1323,3.6 1.0 Internal Bond6 (Z-direction) Cult. (m* in psi, k) V* (cu. in.) 207, 6.4 1.0 195,6.6 1.0 177,5.9 1.0 183,6.3 1.0 229,4.7 1.0 ComDression Parallel (Y-direction) Perpendicular (X-direction) Perpendicular to thickness3 (Z-direction) \u00a3 y ( x 103 psi) o-\u201eit. (psi) \u00a3 x ( x 103 psi) cuu. (psi) \u00a3 z ( x 103 psi) 1698(11) 5346 (11) 95 (22) 1238 (14) 13(14) 826 (19) 2891 (28) 826(19) 2891 (28) 15 (10) 1255 (12) 3932 (15) 384 (23) 1644(16) \u202216(13)\\ 1137(14) 3838(12) 363 (12) 1782 (10) 17(18) 1397 (30) 3763 (12) 209(18) 1581 (10) 14(14) Shear In-plane (XY) Interlaminar Parallel (YZ) Interlaminar Perpendicular (XZ) . x^y ult. ( T W * in psi, k) V* (sq. in.) t xz ult. (m* in psi, k) V* (sq. in.) tyz ult. (m* in psi, k) V* (sq. in.) 2003, 10.6 1.0 881,8.1 1.0 477, 6.9 1.0 2664, 8.0 1.0 713,7.6 1.0 713,7.6 1.0 2525, 9.0 1.0 735,7.6 1.0 718,7.3 1.0 2604, 6.3 1.0 1075,5.9 1.0 498,9.1 1.0 2585, 8.9 1.0 1097,6.4 1.0 491, 12.4 1.0 Notes: 1. V* is volume associated with m*. 2. Tension and in-plane shear values for ' C \\ 'D' and 'E' are based on non-symmetric laminates. 3. Estimated from machine stroke. 4. Values in parentheses are coefficients of variation (%). 5. Shear strengths calculated according to ASTM D143. 6. Ez not measured on Internal Bond specimens. 7. Expressed as a percentage of total panel thickness. 47 Table 3-4. Results from Poisson's ratio tests. Type'A' Type'B' % oriented strands 100 0 Vxy Vyx v y z v K Ey parallel \u00a3 x - perpendicular Ez - perpendicular to surface 0.048 (0.02) 0.741 (0.17) 1.00 (0.01) 0.021 (0.01) 0.351 (0.12) 0.083 (0.05) 1851 (500) 130(51) 24 (5) 0.298 (0.11) 0.298(0.11) 0.601 (0.38) 0.029(0.02) 0.601 (0.38) 0.029 (0.02) 592(189) 592 (189) 25 (6) Note: Standard deviation indicated in parentheses. Table 3-5. Shear properties from other studies. Hoover et al. (1992) Hunt et al. (1985) Hunt and Suddarth (1974) Bradtmueller etal. (1997) Current Study Type'A' Current Study Type'B' Alignment (%) Density (pcf) 100 44 66 43.4 0 42 Unstated 36 100 43 0 43 Interlaminar Shear G y z parallel (psi) G x z perpendicular (psi) 61,440 25,570 32,000 27,000 - 17,000 19,000 61,000' 26,000' 50,000' 50,000' Edgewise Shear Gxv(psi) 275,200 214,000 308,000 _ 275,000' 300,000' Note: 1. Assumed values. 48 2000 '35 W 0 A r\u2014 1 i 1 0 25 50 75 100 % Oriented Strands Figure 3-1. Elastic modulus in bending based on McNatt et al. (1992). Figure 3-2. Laminated strand lumber orientation and co-ordinate system. z A Figure 3-3. Shear block orientations and dimensions. Y-direction is primary fibre direction. 50 X-direction COMPRESSION 2,000 Strain (inches\/inch) Y-direction COMPRESSION 8,000 0 0.0025 0.005 Strain (inches\/inch) Z-direction COMPRESSION 1,500 0 0.075 0.15 Strain (inches\/inch) Figure 3-4. Compression stress-strain curves and averages for type 'A' panels: a) perpendicular to main strand axis (X-direction), b) parallel to main strain axis (Y-direction), and c) perpendicular to surface (Z-direction) 51 XY-plane Large specimen 8,000 -I 0 0.1 0.2 Displacement (inches) YZ - plane Large specimen 4,000 ~i 0 0.05 0.1 Displacement (inches) XZ - plane Large specimen 2,500 ~i 0 0.075 0.15 Displacement (inches) Figure 3-5. Shear block load-displacement curves for type 'A' panels: a) in-plane (XY), b) interlaminar parallel to strands (YZ), and c) interlaminar perpendicular to strands (XZ). 52 (a) Layer 1 Layer 2 Layer 3 h 0 = h\/2 Midplane hr., Layer r (b) Figure 3-6. Laminated composite panel: (a) layer co-ordinates relative to laminate co-ordinates, and (b) layer locations. 53 \u2022o o \u2022a a tu 2000 1000 Ey 25 50 75 100 % Oriented Strands \u2022a o s J3 2000 1000 i Ex 25 50 75 % Oriented Strands 100 -a o s S3 500 250 Gxy 25 50 75 % Oriented Strands 100 O Test Data \u2022 Predictions Figure 3-7. Elastic property predictions for symmetric panels. Compression test data are plotted here and show mean values only. Shear properties at 0% and 100% orientation were assumed. Predicted data indicate \u00b1 1 standard deviation. Best fit lines for \u00a3 x and Ey are based on test data. 54 on Compression Y-direction Ultimate Stress 6000 3000 25 50 75 % Oriented Strands 100 Compression X-direction Ultimate Stress 6000 3000 0.11 3d 0.08 >0.33 >0.26 Ad 0.11 >0.28 >0.39 TypeB End distance Slenderness 2d 3d Ad 2d 0.24 0.17 0.25 3d 0.14 >0.32 >0.27 Ad 0.24 >0.37 >0.33 Type C End distance Slenderness 2d 3d Ad 2d 0.08 0.21 >0.33 3d 0.20 - >0.45 Ad 0.21 >0.30 >0.39 Type D End distance Slenderness 2d 3d . Ad 2d 0.08 0.22 0.28 3d 0.16 - >0.46 Ad 0.23 >0.39 >0.47 Type E End distance Slenderness 2d 3d Ad 2d 0.08 0.27 >0.31 3d 0.14 - >0.44 Ad 0.13 >0.43 >0.42 79 Table 4-10. Statistical significance of test results for effects of end distance on ultimate displacement for Pa specimens (forp = 0.05). Type A Type B TypeC Type D Type E End End End End End End End End End End blender dist. dist. dist. dist. dist. dist. dist. dist. dist. dist. -ness 2d-3d 3d-4d 2d-3d 3d-4d 2d-3d 3d-4d 2d-3d 3d-4d 2d-3d 3d-4d 2d No (a) No No Yes (a) Yes No Yes (a) 3d (a) (a) (a) (a) \u2022 - - - - -4d (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) Note: (a) Not included due to ductile behaviour of one or both of the test groups being compared. Table 4-11. Statistical significance of test results for effects of slenderness ratio on ultimate displacement for Pa specimens (forp = 0.05). Type A Type B Type C Type D Type E FnH Slender Slender Slender Slender Slender Slender Slender Slender Slender Slender -ness -ness -ness -ness -ness -ness -ness -ness -ness -ness dist. 2d-3d 3d-4d 2d-3d 3d-4d 2d-3d 3d-4d 2d-3d 3d-4d 2d-3d 3d-4d 2d No No No No Yes No Yes No No No 3d (a) (a) (a) (a) - - - - - -4d (a) (a) (a) (a) (a) (a) (a) (a) (a) (a) Note: (a) Not included due to ductile behaviour of one or both of the test groups being compared. 80 Table 4-12. Edge distance effects on ultimate displacement (inches) for Pa specimens. Type A Edge distance Slenderness Id l.Sd 2d >0.06 >0.11 Ad >0.23 >0.39 TypeB Edge distance Slenderness Id l.5d 2d 0.05 0.25 Ad 0.04 >0.33 TypeC Edge distance Slenderness Id l.Sd 2d 0.09 >0.33 Ad 0.26 >0.39 Type D Edge distance Slenderness Id l.5d 2d 0.08- 0.28 Ad 0.14 >0.47 Type E Edge distance Slenderness Id l.Sd 2d 0.07 >0.31 Ad 0.20 >0.42 Table 4-13. Statistical significance of test results for effects of edge distance on ultimate displacement for Pa specimens (forp = 0.05). Type A Type B TypeC Type D Type E Slenderness Id-l.Sd ld-l.5d Id-l.Sd ld-l.Sd ld-l.Sd 2d (a) Yes (a) Yes (a) Ad \"(a) (a) (a) (a) (a) Note: (a) Not included due to ductile behaviour of one or both of the test groups being compared. Table 4-14. End distance and slenderness ratio effects on failure modes for Pa specimens. Type A End distance Slenderness A, 2d A2 A, 3d A2 A, 4d A2 2d 0.04 0.03 B 0.07 0.06 B\/D >0.08 >0.02 .(4) 3d 0.05 0.03 B >0.15 >0.14 D >0.13 >0.13 D 4d 0.10 0.08 B 0.11 0.12 D >0.21 >0.19 D Type B End distance Slenderness A, 2d A2 A, 3d A2 A, 4d A2 2d 0.11 0.03 B >0.53 >0.05 B 0.47 0.00 B 3d 0.16 0.05 B 0.18 0.14 B\/D 0.37 0.33 B\/D 4d 0.13 0.11 B\/D 0.19 0.19 D 0.19 0.16 B\/D TypeC End distance Slenderness A, 2d A2 A, 3d A2 A, 4d A2 2d 0.15 0.03 B 0.21 0.00 B 0.31 0.01 B\/D 0.17 0.07 B - - - 0.23 0.17 B\/D 4d 0.20 0.17 B\/D 0.20 0.19 D >0.19 >0.18 D Type D End distance Slenderness A, 2d A2 A, 3d A2 A, 4d A2 2d 0.13 0.01 B 0.31 0.00 B\/D 0.35 0.01 B 3d 0.16 0.04 B - - - 0.26 0.21 B\/D 4d 0.18 0.11 B\/D >0.17 >0.16 D >0.22 >0.22 D Type E End distance Slenderness A, 2d A2 A, 3d A2 A, 4d A2 2d 0.09 0.03 B 0.35 0.01 B >0.29 >0.00 D 3d 0.11 0.04 B - - - >0.21 >0.17 D 4d 0.11 0.08 B 0.25 0.24 B\/D >0.26 >0.26 D Notes: 1) A| = final outer diameter of hole minus initial outer diameter of hole (inches) 2) A2 = final outer diameter of hole minus final inner diameter of hole (inches) 3) B = brittle failure, D = ductile failure 4) No failure achieved due to grip limitations. 82 Table 4-15. Edge distance effects on failure modes for Pa specimens. Type A Edge Distance Slenderness Id A , A 2 \\.5d A , A 2 2d 4d >0.02 >0.02 - ( 5 ) 0.18 0.17 B\/D >0.08 >0.02 - ( 5 ) >0.21 >0.19 D Type B Edge Distance Slenderness \\d A , A 2 \\.5d A , A 2 2d 4d 0.13 0.02 B 0.04 0.01 B 0.47 0.00 B 0.19 0.16 B\/D TypeC Edge Distance Slenderness Id A , A 2 l.Sd A , A 2 2d 4d N\/A N\/A B 0.17 0.17 B\/D 0.31 0.01 B\/D >0.19 >0.18 D Type D Edge Distance Slenderness Id A , A 2 \\.5d A i A 2 2d 4d 0.22 0.03 B 0.21 0.10 B\/D 0.35 0.01 B >0.22 >0.22 D Type E Edge Distance Slenderness Id A , A 2 l.Sd A i A 2 2d 4d 0.16 0.02 B 0.22 0.16 B\/D >0.29 >0.00 D >0.26 >0.26 D Notes: 1) A] = final outer diameter of hole minus initial outer diameter of hole (inches) 2) A 2 = final outer diameter of hole minus final inner diameter of hole (inches) 3) B = brittle failure, D = ductile failure 4) N\/A = not available. 5) No failure achieved due to grip limitations. Table 4-16. Results summary for Pe specimens. End distance, e Bolt size, d Lay-up Ultimate load p 1 max Little or no effect. Pmaxtrft \/'max 1\" % aligned strands t (more or less) Ultimate displacement Amax Little or no effect. Little or no effect. A m a x l with 100% alignment Failure mode Tendency for mode to go from brittle to ductile as e t Brittle Net section failure in all panel types. Note: t = increased, 4 = decreased Table 4-17. End distance and slenderness ratio effects on ultimate load (lbs.) for Pe specimens. Type A End distance Slenderness 2d Ad 2d 3489 3409 Ad 1422 1571 TypeB End distance Slenderness 2d Ad 2d 5483 6931 Ad 2212 2255 TypeC End distance Slenderness 2d Ad Ad 1972 2470 Type D End distance Slenderness 2d Ad Ad 2346 2474 Type E rind distance Slenderness 2d Ad Ad 2251 2451 84 Table 4-18. Statistical significance of test results for effects of end distance on ultimate load for Pe specimens (for p - 0.05). Type A TypeB TypeC Type D Type E End End End End End Slenderness dist. dist. dist. dist. dist. 2d-Ad 2d-Ad 2d-Ad 2d-Ad 2d-Ad 2d No No ( a ) Yes No No Ad No No . N\/A N\/A N\/A Notes: (a) Significant forp = 0.09. (b) N\/A = not available due to specimen breakage at ultimate load. Table 4-19. Statistical significance of test results for effects of slenderness ratio on ultimate load for Pe specimens (for p = 0.05). Type A TypeB End Slenderness Slenderness distance 2d-Ad 2d-Ad 2d Yes Yes Ad Yes Yes 85 Table 4-20. End distance and slenderness ratio effects on ultimate displacement (inches) for Pe specimens. Type A End distance Slenderness 2d Ad 2d 0.09 0.08 Ad 0.08 0.13 Type B End distance Slenderness 2d Ad 2d 0.24 0.25 Ad 0.24 >0.33 TypeC End distance Slenderness 2d Ad Ad 0.20 >0.52 Type D End distance Slenderness 2d Ad Ad >0.36 >0.54 Type E End distance Slenderness 2d Ad Ad >0.35 >0.42 Table 4-21. Statistical significance of test results for effects of end distance on ultimate displacement for Pe specimens (for p = 0.05). Type A Type B TypeC Type D Type E End End End End End Slenderness dist. dist. dist. dist. dist. 2d-Ad 2d-Ad 2d-Ad 2d-Ad 2d-Ad 2d No No (a) (a) (a) Ad No (a) N\/A N\/A N\/A Notes: (a) Not included due to ductile behaviour of one or both of the test groups being compared, (b) N\/A = not available due to specimen breakage at ultimate load. 86 Table 4-22. Statistical significance of test results for effects of slenderness ratio on ultimate displacement for Pe specimens (forp = 0.05). Type A TypeB End Slenderness Slenderness distance 2d-Ad 2d-Ad 2d No No Ad No (a) Notes: a) Not included due to ductile behaviour of one or both of the test groups being compared. Table 4-23. End distance and slenderness ratio effects on failure modes for Pe specimens. . Type A End distance Slenderness A, 2d A 2 A, Ad A2 2d \u2022N\/A N \/ A B N \/ A N \/ A B Ad N \/ A N \/ A B N \/ A N \/ A B TypeB End distance Slenderness A, 2d A , A, Ad A 2. 2d 0.11 0.03 B\/D 0.47 0.00 B\/D Ad 0.13 0.11 B 0.19 0.16 B Type C End distance Slenderness A, 2d A2 A, Ad A2 Ad 0.16 0.05 B 0.18 0.16 B\/D Type D End distance Slenderness A, 2d AT A, Ad A2 Ad 0.25 0.19 B\/D >0.28 >0.26 D Type E End distance Slenderness A, 2d A2 A, Ad A2 Ad 0.22 0.15 B\/D 0.24 0.16 B\/D Notes: 1) A, = final outer diameter of hole minus initial outer diameter of hole (inches) 2) A2 = final outer diameter of hole minus final inner diameter of hole (inches) 3) B = brittle failure, D = ductile failure 4) N\/A = not available due to specimen breakage at ultimate load. 87 Table 4-24. Results summary for A N specimens. End distance, e Bolt size, d Ultimate load P f ef P t dt p 1 max 1 e 1 'max ' \" 1 * max Ultimate displacement A m a x f e\\ Little or no effect. _ .. , Change from brittle Change from brittle Failure mode . . . * , ., , i to ductile as e I to ductile as d 4-Note: t = increased, 4 = decreased Table 4-25. End distance and slenderness ratio effects on ultimate load (lbs.) for type A loaded 45\u00b0 to main strand axis (AN). Type A End distance Slenderness 2d Ad 2d 4067 6145 Ad 1925 2402 Table 4-26. Statistical significance of test results for effects of end distance on ultimate load for A N specimens (forp = 0.05). . Type A 0 1 , End distance Slenderness \u201e , . . 2d-Ad 2d Yes Ad Yes Table 4-27. Statistical significance of test results for effects of slenderness ratio on ultimate load for A N specimens (for p = 0.05). Type A End Slenderness distance 2d-Ad 2d Yes Ad Yes 88 Table 4-28. Hankinson's formula prediction for ultimate load (lbs.) for type A specimens loaded 45\u00b0 to main strand axis (AN). End distance Slenderness Ultimate load parallel P Ultimate load perpendicular Q Hankinson's formula (equation [3-2]) ' N Actual load at 45\u00b0 N Actual 2d 2d 4447 3489 3910 4067 0.96 4d 2d 7272 3409 4642 6145 0.76 2d 4d 1793 1422 1586 1925 0.82 4d 4d 2329 1571 1876 2402 0.78 Table 4-29. End distance and slenderness ratio effects on ultimate displacement (inches) for type A loaded 45\u00b0 to main strand axis (AN). Type A End distance Slenderness 2d 4d 2d 0.07 0.24 4d 0.13 >0.39 Table 4-30. Statistical significance of test results for effects of end distance on ultimate displacement for AN specimens (for p = 0.05). Type A \u201e. . End distance Slenderness \u201e , . , 2d-4d 2d Yes Ad (a) Note: a) Not included due to ductile behaviour of one or both of the test groups being compared. Table 4-31. Statistical significance of test results for effects of slenderness ratio on ultimate displacement for AN specimens (forp - 0.05). T y P e A End Slenderness distance 2d-4d 2d No 4d (a) Note: (a) Not included due to ductile behaviour of one or both of the test groups being compared. 89 Table 4-32. End distance and slenderness ratio effects on failure modes for AN specimens. Type A End distance Slenderness A, 2d A 2 \u2022 A, Ad A 2 2d 0.05 0.01 B 0.20 0.07 B\/D Ad 0.06 0.03 B 0.16 0.15 B\/D Notes: 1) A! = final outer diameter of hole minus initial outer diameter of hole (inches) 2) A 2 = final outer diameter of hole minus final inner diameter of hole (inches) 3) B = brittle failure, D = ductile failure 90 Pal Pa2 Pa3 Pa4 Pa5 Pa6 Pa7 Split perpendicular to direction of loading (uniform through thickness) Split perpendicular to direction of loading Gagged through thickness, type 'E' only) Net section failure (tension parallel to direction of loading - on one or both sides of hole) Shear plug failure (not typically. through entire thickness of specimen) Hole elongation (uniform through thickness or greater elongation at surfaces) Delamination (between layers of fully oriented and randomly oriented strands) Hole elongation and initiation of splits in tension perpendicular to direction of loading \/-\u00ab A2 Ai ^initial Hole elongation uniform through thickness Hole elongation greater at surfaces (Dotted lines indicate initial and final hole limits) B = brittle failure D = ductile failure Figure 4-2. Observed failure modes of connection specimens. 9000 Thickness: 2d i 6000 3000 E D D I A C A Edge distance: 1.5d 1.5d Id 2d 3d 4d End distance 6000 5000 4000 3000-2000-1000 0 Thickness: 3d A B Edge distance: 1.5d 1.5d 2d 3d End distance 3000 2500 2000 1500 1000 500 0 Thickness: 4d Edge distance: 1.5d 1.5d A g D B Id 2d 3d 4d End distance 1.5d 4d 1.5d 4d 1.5d 4d Note: Letters correspond to panel types: A is fully oriented, B is randomly oriented, C is [0\u00b0\/R\/0\u00b0], D is [R\/0\u00b0\/R], E is [0\u00b0\/+45\u00b0\/-45 o\/0\u00b0] s Figure 4-3. Average ultimate loads of Pa specimens. 93 3000 2500 2000 1500 1000 500 0 Thickness: 4d C A 2d 4d End distance Note: Letters correspond to panel types: A is fully oriented, B is randomly oriented, C is [0\u00b0\/R\/0\u00b0], D is [R\/0\u00b0\/R], E is [0\u00b0\/+45%45o\/0\u00b0]s Figure 4-4. Average ultimate loads of Pe specimens. 94 CHAPTER 5 FINITE ELEMENT MODEL OF SINGLE DOWEL CONNECTION The primary objective of this thesis was to develop a model to predict the load-displacement behaviour and ultimate strength of single dowel connections in any wood composite. To this end, the proposed model is described in this chapter. 5.1 Choice of Model A three-dimensional finite element model was selected as the best choice for this problem because of its ability to predict the effects of the following: a) end distance, edge distance, dowel slenderness ratio and other geometric parameters; b) through-thickness variation in deformation and stresses; c) non-linear, orthotropic properties of the wood composite; d) elastoplastic behaviour of the dowel; e) orientation of dowel loading; f) modifying material properties in layers throughout the member thickness; and g) size-effect and governing mode of failure. The geometry of the finite element model and the material laws will now be discussed. 5.2 Model Geometry The model geometry was chosen to simulate the real test specimens as closely as possible. A single member with no side members was modelled with a single dowel passing through the member. As was the case of the connection experiments described in chapter 4, this removed the uncertainty caused by load-distribution between multiple fasteners, the effect of friction between adjacent members, and the effects of a tightened bolt. Some of the model parameters were chosen based on work by Patton-Mallory (1996) described in chapter 2. Three-dimensional eight-noded, quadrilateral isoparametric brick elements were used to represent the wood and steel in the connection. The model was developed using the commercial software package ANSYS v.5.3 (1996). The model geometry and boundary 95 conditions are shown in Figure 5-1. Using symmetry, only one quarter of the actual wood member and dowel were modelled. The wood member was defined to have an oversized hole as used in the experiments, with specified end distance, edge distance and thickness. The highly stressed region in the wood member was expected to extend from the point where the dowel contacts the wood member to the free end of the member and through the member thickness. A higher mesh density of quadrilateral bricks was used in this highly stressed region of the wood. Roughly five elements per centimetre thickness (nine elements per inch) were deemed to be sufficient to predict load-displacement behaviour (Patton-Mallory 1996). To determine the effect of changes in mesh density around the hole, strain energy was checked in the post-processor for a variety of mesh densities. The model results were found to be insensitive to small changes to the mesh density shown in Figure 5-1 (i.e. little or no change in strain energy). Patton-Mallory (1996) found that stresses in the wood member were more or less uniform at the fixed end of the wood member. This point was located 12.7 cm (5 inches) from the centre of the bolt (in the negative Y-direction in Figure 5-1) away from the bolted end allowing the model to be cut off at that point. This location corresponds to the baseline for displacement measurements in the experiments, thus displacements relative to this location in the model can be compared directly to experimental results. The steel dowel elements extended out from the face of the wood member by 1.27 cm (0.5 inches) to simulate the points where the load was applied during experiments. Surface-to-point contact elements were used for the interface between the wood and the bolt on the surface shown in Figure 5-1. Only one-quarter of the hole circumference and bolt circumference were covered with contact elements since these regions were expected to come into contact with each other. The contact elements extended through the thickness of the specimen. If the hole were the same diameter as the bolt, then the entire hole circumference and bolt circumference would have required contact elements. The contact stiffness, KN, required to ensure compatibility between the steel and wood was found to affect the initial stiffness of the load-displacement curves for each specimen. For the ANSYS contact elements, this stiffness can be estimated from the following formula: [5-1] KN - fEh 96 where KN is the contact stiffness in force per unit length, E is the elastic modulus in the primary direction of loading, h is the square root of the smallest element contact surface, and \/ is a dimensionless 'compatibility' constant which is typically between 0.01 and 100 (ANSYS, 1992). For the current set of analyses, \/ was found to be dependent on the configuration (i.e. loading orientation, slenderness ratio and lay-up). Trial analyses with existing experimental data for load-displacement behaviour of connections were conducted from which \/ was found to be 0.041, on average, with a lowest value of 0.005 and highest value of 0.15. This corresponded with contact stiffnesses, KN, of between 1750 N\/cm (1000 lbs.\/inch) and 17 500 N\/cm (10,000 lbs.\/inch). In addition to the stiffness developed by the surfaces coming into contact, friction between the steel and wood was permitted assuming a coefficient of friction for sliding as [i=0.7 (Smith 1983). The unbolted end of the member was restricted from displacements in the Y-direction by applying a zero-displacement boundary along the plane normal to the Y-direction. The reaction on this plane is equal to one-quarter of the entire applied load. The two planes of symmetry; in the wood and dowel were modelled with zero-displacement boundary conditions in the planes normal to the X- and Z-directions. Displacements were applied in the positive Y-direction to the end of the dowel to simulate the experimental displacement-controlled conditions. The solution was non-linear due to material non-linearity and the discontinuous load-displacement characteristics of the contact elements. Displacements to the end of the dowel were applied in small equal increments to assure proper convergence. A force convergence criterion was used. On average, the displacement increment was 0.05 mm (0.002 inches) per step up to a total of 2.8 mm (0.11 inches). If convergence could not be achieved, the number of load steps was increased or the solution was terminated at a smaller maximum displacement. The latter condition typically corresponded with the true physical behaviour of connections, which were found to fracture at that point. One model was used for each connection geometry and set of material properties. Load-displacement and stress data were saved at each step. 97 5.3 Material Models and Constitutive Relations for Wood In the past, linear elastic models have been assumed to be adequate for wood under most bad applications. However, due to the highly stressed zones of tension, compression and shear associated with bolted connections, non-linear behaviour plays an important role and a more sophisticated model is required. Patton-Mallory (1996) used a tri-linear elastic model to predict connection behaviour using a three-dimensional finite element model. The anisotropic plasticity model was chosen for wood in this thesis for a number of reasons, as will be explained shortly. 5.3.1. Linear Elastic Constitutive Model for Wood From the theory of elasticity, the three-dimensional stress-strain relationship for orthotropic materials is: [5-2] 1 v 21 V 31 0 0 0 E2 El \u00a3 1 v 12 Ei 1 T2 V 32 E> 0 0 0 v 13 v 2 3 1 0 0 0 a 2 s2 E2 \u00a33 0-3 2s12 2s13 0 0 0 1 0 1 0 0,2 \u00b0\"l3 2 e 2 3 , 0 0 0 0 0 1 G23. \u00b0 2 3 . 0 0 0 0 0 where Eh and vy are the elastic moduli, shear moduli and Poisson's ratios, respectively, defined according to the right-handed Cartesian co-ordinate system 1,2,3. These twelve constants can be determined from tension, compression and shear tests; however, only nine are required since the Poisson's ratios and elastic moduli are related as follows: [ 5 - 3 ] Equation [5-2] is valid for linear elastic behaviour only. It is known, however, that wood in compression behaves in a non-linear manner and permanently deforms under loading. 98 5.3.2. Non-linear Elastic Constitutive Modelling for Wood Rahman et al. (1991) performed a two-dimensional finite element analysis of a pin in a hole in an orthotropic wood plate assuming material non-linearity due to crushing of wood fibres, and determined the stress distribution around the hole. They showed that a non-linear compression model parallel-to-grain (with pin loading parallel-to-grain) resulted in improved stress predictions under the pin. These results corresponded to measured Moire fringe patterns. Patton-Mallory et al. (1997b) used a tri-linear elastic analysis in three-dimensions for a model of a single bolt in a piece of Douglas fir, as mentioned previously. The tri-linear elastic stress-strain behaviour assumed for this model is shown in Figure 5-2 for normal stress parallel-to-grain. The permanent crushing of fibres parallel-to-grain was found to be the greatest influence on displacements of the bolt relative to the piece of wood (including elasto-plastic behaviour for the bolt). This tri-linear assumption resulted in good load-displacement predictions compared to experimental results. However, as indicated by Patton-Mallory (1996), improvements are still required in the material model for wood. Two drawbacks were associated with the tri-linear elastic model. Firstly, at high strains, negative terms were found to develop on the main diagonal of the stiffness matrix if similar tri-linear elastic behaviour was used in the two other orthotropic material directions in compression. As a result, non-linearity in the shear stress-strain behaviour was introduced to improve the numerical stability of the model. Secondly, the prediction of stresses in the perpendicular-to-grain directions were inaccurate, as will be shown in chapter 6. This means that the ultimate strength of the connection cannot be well predicted using stress-based failure criteria. An elastic model does not account for energy dissipation associated with the non-recoverable deformations. due to wood crushing. Plasticity models prevent this form of instability from occurring. For the current study, it was proposed that the behaviour in compression will be modelled with the anisotropic plasticity model, while the tension and shear behaviour will remain linear elastic, until reaching ultimate failure. 99 5.3.3. Anisotropic Plasticity Modelling in Wood The anisotropic plasticity material model has been shown to be effective when predicting failure of non-wood composites in two-dimensional finite element models (Vaziri et al, 1991). In the current application, the anisotropic plasticity model was applied in three-dimensions using the TB,ANISO option in ANSYS v.5.3. Unlike the non-linear elastic models, this material model does not require modification of the elastic stiffness matrix. Instead, it accounts for permanent deformation and energy dissipation in three orthogonal planes. This model is based upon the yield criterion by Hill (1947) for orthotropic materials. The model was modified by Shih and Lee (1978) to accommodate differences in compression and tension yield stresses in each direction. The work hardening model described in Valliappan et al (1976) is assumed. An associative flow rule is assumed. Details on this model are presented in Appendix B. A sample bi-linear stress-strain curve for the anisotropic plasticity model is shown in Figure 5-3 indicating the yield stress and tangent modulus for one material direction. The two other normal stress-strain curves and three shear stress-strain curves are also required for this model. This results in eighteen additional constants beyond the nine required for orthotropic elastic materials. Two criteria must be satisfied when using this model. To satisfy the requirement for plastic incompressibility, the yield stresses CT+, and o\\, in tension and compression, respectively, in direction i are inter-related: [5-4] ^ - a - x + ^ - ^ + ^ - a - , = 0 0\" + xa_ t 0\\rO\"_r <7+zCT_2 To maintain a closed yield surface: [5-5] M, 2 , + M22 +M22 -2(MUM22 +M22Mii + M n M 3 3 ) < 0 where: [5-6] Mu = \u2014 \u2014 \u2014 , (\/' = 1,2, 3) and [5-7] * = a + xo_ x. Equation [5-5] must be satisfied at all levels of straining since the yield stresses will change as work hardening proceeds. These restrictions on allowable yield stresses and 100 tangent moduli make fitting of experimental results difficult. This is particularly true for highly orthotropic materials. Making the yield stresses and tangent moduli more equal increases the chances of satisfying these equations. Equation [5-4] is better satisfied if the yield stresses in tension and compression are set equal to each other for each direction. To assist in determining these values, a spreadsheet was developed using an optimization algorithm to satisfy these conditions up to 20% plastic strain. For wood, the ductile behaviour in compression is of concern i.e. non-linear stress-strain behaviour. It is important to model this correctly since stresses in a connection quickly exceed the proportional limit of the wood in the region under the steel dowel. Under cyclic loading in compression, commercial LSL can be seen to behave plastically, as shown in a typical experimental curve in Figure 5-4, where permanent deformations occur on unloading once the yield point is exceeded. The unloading curve follows the initial elastic modulus. This behaviour occurs in each of the three orthogonal directions with a different yield stress in each direction, as shown for the current study in Figure 3-4. Thus, the proposed model does reflect the behaviour of wood. 5.3.4. Failure Prediction Ductile behaviour in compression is simulated using the anisotropic plasticity model as described above; however, brittle modes of fracture in tension and shear can result in total failure of a connection. To predict ultimate strength due to tension or shear failure, the Weibull weakest link theory is used. The Weibull weakest link theory, described in section 2.6.2, is used to predict ultimate strength while accounting for size effect. Failure of wood in tension or shear occurs when the stresses exceed a critical level. The maximum stress criterion is assumed here, meaning that at each load increment, the three tension stresses and three shear stresses are independently compared against critical values. It is assumed that little or no interaction between these stresses exists. The magnitude of the critical stress values is predicted using the Weibull weakest link theory. Given the probability distribution of failure stresses af in Table 3-3, and their dependence on material volume, critical limit values for a unit volume of material with a 101 given probability of failure are set. During post-processing, each of the six stresses cr, are integrated according to: for comparison against the critical stresses cr\/ for each load increment to determine if failure has occurred. The first stress to exceed its critical level indicates the load at failure. The mode of failure is associated with this exceeded stress. The evaluation of the volume integral on stresses from the finite element nodal stresses was carried out with a user-programmable subroutine in ANSYS v.5.3. The implementation of this theory using the finite element method is shown in Appendix C. The formulation is valid for three-dimensional brick elements with eight or twenty Gauss integration points. As indicated in Appendix C, the finite element solution matched well with the closed-form verification problem. 5.4 Summary The current model employs the theory of anisotropic plasticity to model the non-linear behaviour of wood. By accounting for differences in the three orthotropic material directions, this plasticity model prevents the development of the instability which occurs in the global stiffness matrix using only elastic models. This results in improved stress predictions, on which the ultimate loads are based. The spreadsheet discussed in section 5.3.3 was used to modify experimentally-determined material properties to satisfy the model constraint equations. Using this spreadsheet, one can determine the sensitivity of the material properties to changes such as setting the tension and compression yield stresses in each direction equal to one another. This will be discussed in more detail in Chapter 6. [2-7] V 102 ,End region 0 displacement in Z-direction for all surface nodes contact interface steel bolt 0 displacement in X-direction for all surface nodes 0 displacement in Y-direction for all surface nodes Close-up of end region wood member Figure 5-1. Finite element model geometry. One-quarter of real specimen geometry due to symmetry. 0\"-y2 CT.ylj 5E Tyl E-yl \u00a3.y2 . 0 1 E Ty2 Figure 5-2. Tri-linear stress-strain curve for compression behaviour parallel-to-grain. 103 yield point in tension S compression Figure 5-3. Bi-linear stress-strain curve for normal stress in anisotropic plasticity model. 6000 T \u00a3 3000 -0 0 0.0025 0.005 0.0075 0.01 Strain (inches\/inch) Figure 5-4. Experimental stress-strain curve for cyclic loading of commercial grade LSL in compression parallel to the main strand axis. 104 CHAPTER 6 VERIFICATION OF FINITE ELEMENT MODEL In this chapter, the three-dimensional finite element model discussed in Chapter 5 is calibrated, verified and checked for sensitivity to a variety of parameters. In the first section of this chapter, the eighteen anisotropic plasticity constants (as discussed in section 5.3.3) for LSL are determined. Then, the tri-linear elastic model is used to verify the performance of the finite element model geometry against the previously calibrated Patton-Mallory (1996) model for Douglas fir. This model is used to predict connection failure using the Weibull weakest link theory (as discussed in section 5.3.4) and then reanalysed for failure using the anisotropic plasticity model with properties for Douglas fir. To determine the shear failure stresses of LSL, the anisotropic plasticity model is used in a finite element model of the ASTM D143 shear blocks (as discussed in section 3.4.1). The single dowel connection model is then revisited to predict the behaviour of the five lay-ups of LSL for the geometries tested in the experiments of Chapter 4. Finally, a sensitivity study on material properties and the finite element model is conducted. 6.1 Calibration of Material Constants The non-linear compressive behaviour of LSL was determined in the experiments described in Chapter 3. Figures 6-1 and 6-2 show the uniaxial stress-strain behaviour for the fully oriented type 'A' and the randomly oriented type 'B' lay-ups, respectively. For each series of curves, the average elastic moduli, yield stresses, a+\u201e and tangent moduli, Ejj, were determined by averaging the curves and using a windowing technique to find the yield stresses and the tangent moduli. A sample set of averages is indicated by the dark lines in Figure 3-4 for the type 'A' lay-up. The values determined in this manner for the type 'A' and the type 'B' lay-ups were not found to satisfy the constraint equations [5-4] and [5-5] for the anisotropic plasticity model and, therefore, required adjustment. Using a spreadsheet and optimization routine, values were determined which satisfied equations [5-4] and [5-5] for 105 plastic strains of up to 20%. The values determined in this way are listed in Table 6-1 for types 'A' and 'B' LSL. Figures 6-1 and 6-2 show the model compression curves as dark lines. The values shown in Table 6-1 were found to be the most feasible solution. It was found that the equations were more easily satisfied by making yield stresses and tangent moduli for each direction more equal (as indicated in section 5.3.3 for highly orthotropic materials). For this solution, the yield stresses were made equal in tension and compression. The introduction of a yield point in tension is fictitious since no ductility was found to occur in the tension tests described in chapter 3. However, in the interest of using the anisotropic plasticity model, the presence of a yield point was assumed in tension. In reality, the failure stress, CT*\u201e in tension was found to be less than this assumed yield stress, o+\u201e in two of the three principal directions and brittle failure will be predicted to govern in these directions prior to the onset of the fictitious tensile yielding. For the type 'A' lay-up, the failure stresses in the X- and Z-directions in tension are lower than the yield stresses in these directions, as indicated in Table 6-1. Type 'B' had very low tensile strength in the Z-direction compared to the yield stress. It was assumed that yielding in tension in the other directions has no affect on the solution for the single-dowel connection model since brittle failure in these material directions would not tend to govern. Shear properties are required but will be discussed in section 6.3 with the analysis of the shear block results. 6.2 Model Verification - Douglas fir Patton-Mallory (1996) conducted a rigorous series of verification analyses on two-dimensional and three-dimensional finite element models to check displacements and stresses around a pin-loaded hole. Patton-Mallory also developed and verified the three-dimensional single dowel connection model using the commercial software package ABAQUS v.5.4 (ABAQUS 1994). This model was compared against experimental results for Douglas fir and good agreement was found for load-displacement behaviour in a number of connection geometries. Because the current model in this thesis was based on the Patton-Mallory model, it was run using the same user-programmable tri-linear elastic model as used by Patton-106 Mallory (see Figure 5-2) to verify its behaviour. The load-displacement results were used to assess the model. 6.2.1. Tri-linear Elastic and Weibull Weakest Link The assumed material properties for this model in Douglas fir are listed in Table 6-2. Each of the nine geometries tested and modelled by Patton-Mallory were reconstructed using the current model geometry in ANSYS v.5.3, with tri-linear elastic material properties for wood (not anisotropic plasticity), bilinear kinematic hardening for steel, and the contact interface. All analyses were allowed to proceed up to 1 mm (0.04 inches) displacement. Load-displacement curves were found to be very similar, indicating that the current model performed equally well. Failure prediction of these models was conducted using the algorithm for the Weibull weakest link theory from section 5.3.4. In this case, failure strength in Douglas fir was assumed to be controlled by tensile stresses perpendicular-to-grain and shear along the grain. Strength parameters for Douglas fir in tension and in shear were determined by Barrett et al. (1975) and Foschi and Barrett (1976) and are listed in Table 6-2 based on a unit volume and 50% probability of failure. The resulting failure strengths of the connections are listed in Table 6-3. The connection strengths based on the tri-linear model were found to be lower than the experimental values in all cases. In addition, failures were found to occur at very low displacements. Brittle failure was predicted in all cases due to the development of very high tension stresses perpendicular-to-grain by the tri-linear model. Shear failure in the critical longitudinal direction (XY-plane) was never detected as would be expected given that the assumed shear yield stress was much lower than the reference shear stress at failure. This prevented shear stresses from growing large enough to cause failure. The experimental results on Douglas fir indicated that shear failure did occur in specimens with small lid and eld; however, this was accompanied by a tension-perpendicular-to-grain split. It was postulated that the anisotropic plasticity model would prevent the development of high perpendicular-to-grain tension stresses in the experimental connections which did not fail, thereby improving the failure prediction for Douglas fir. 107 6.2.2. Anisotropic Plasticity and Weibull Weakest Link Based on the known behaviour of Douglas fir from the literature, the values listed in Table 6-4 were chosen for the anisotropic plasticity model. The connections described above were then re-analysed. Load-displacement behaviour was found to be very similar to the behaviour of the experiments and the tri-linear model. This is explained by the use of a non-linear stress-strain model in compression in the direction of loading and the elastoplastic behaviour of the steel dowel. However, ultimate strength and displacement predictions were found to be much improved by using anisotropic plasticity instead of the tri-linear elastic model. Table 6-3 shows that by using the anisotropic plasticity model, the failure loads were closer to, and more consistent with, those observed in experiments. In particular, brittle failures were found to occur with small slenderness ratios, lid, whereas ductile behaviour occurred in all other cases. The improved behaviour is explained by the prediction of stresses in the other material directions. The connection, shown in Figure 6-3 with the steel dowel removed, shows that in regions under the steel dowel at ultimate load, the perpendicular-to-grain stresses are over-predicted using the tri-linear elastic model. A zone of high tension stress is developed at the point of contact between the dowel and wood (noting that the hole is larger than the dowel diameter) with peak stress in the tri-linear model predicted to be more than four times greater than those predicted using the anisotropic plasticity model. The tri-linear elastic model also predicted stresses in the compression zone to the side of the tension region to be more than one and a half times greater than the stresses predicted using the plasticity model. In addition, the tension zone at the end of the specimen, where cracks are known to develop (Jorissen 1997 and discussed in section 2.2), is much higher, in relative terms, using the plasticity model. These deficiencies in the stress predictions of the tri-linear model led to the poor prediction of ultimate strength, thus, the anisotropic plasticity model is an improvement. 6.3 Shear Block Model The ASTM D143 (ASTM 1991b) shear block has been used for many decades to characterize the parallel-to-grain shear strength of solid wood. In this thesis, shear blocks were also used to determine the ultimate shear strength of LSL according to the ASTM 108 protocol. The block, shown in Figure 6-4, is placed into a steel test frame which applies a load onto the notched surface of the wood shear block, effectively shearing the notched portion of the block relative to the main block body until the specimen fails. The load at failure is used as an estimate of ultimate shear strength in this plane where the nominal shear strength, Tn0minai, is assumed to be equal to the ultimate load, Pmaximum, divided by the shear area, \/ x w, shown in Figure 6-4. This test specimen has been criticized, however, for failing prematurely due to: a) the non-uniform stress distribution over the failure plane, b) the stress concentration caused by the presence of the notch, and c) the presence of bending stresses along the failure plane. As a result, shear strengths determined from this test have been shown to be lower than the actual shear strength of wood. To determine the actual probability distribution of ultimate shear strengths in LSL, a finite element model of the shear block specimen was developed using the anisotropic plasticity model and the Weibull weakest link theory for failure prediction. Assumed shear properties for the model for lay-ups 'A' and 'B' are listed in Table 6-5. Values of shear yield stresses were chosen to prevent yielding from occurring in shear since yielding in shear has not yet been documented for LSL. In addition to finding the shear strength constants for the Weibull weakest link theory, this finite element model was used to determine: a) the effectiveness of the anisotropic plasticity model, b) the stress distribution along the shear plane, and c) the effect of the stress concentration. 6.3.1. Shear Block Model Geometry The same type of brick elements used for the single dowel connection model were used for the shear block model. The geometry and boundary conditions for a single shear block are shown in Figure 6-5. A higher mesh density was used in regions of stress concentrations. One shear block model was used for each of the shear block sizes and orientations shown in Figure 3-3 resulting in six separate models. The analysis was carried out for panel types 'A' and 'B'. The steel test frame was simulated using rigid brick elements for the base and loading platen. The base was restricted from moving in each of the three degrees of freedom, whereas the loading platen was only allowed to move in the direction of loading, as shown in Figure 6-5. Loading was controlled by the displacement of the platen over the same ranges found in the experiments for each block type. Elements along the 109 vertical side of the notch were on rollers, as shown in the figure, to simulate the presence of the platen. Rollers were placed on the back side of the specimen to simulate the presence of the supporting back plate, as indicated in Figures 6-4 and 6-5. The entire model is only half of the actual shear block (using appropriate boundary conditions along the plane of symmetry of the specimen), thereby reducing the model size. Surface-to-point contact elements were used for the interfaces between the steel frame and the wood specimen at the two locations shown in Figure 6-5. The contact elements allowed for gaps to form between the steel and the wood block as the block rotated under load. Previous models by Foschi and Barrett (1976) and Cramer et al. (1984) simplified the analysis by overlooking this effect. However, it has been postulated that normal stresses, perpendicular to the shear plane, may influence the failure behaviour (Liu, 1984) at the loading platen interface, thus contact elements were used to introduce the effect of friction and to develop these normal stresses. The coefficient of friction, u, between the steel and wood was set to 0.7. Since the shear strengths in LSL were unknown, the model was loaded up to the experimental ultimate load (or displacement), at which point the integral of equation [2-7] could be evaluated to back-calculate the reference shear stress for a unit volume. The model was used to determine the shear strengths in each of the three shear planes. 6.3.2. Shear Block Model Results The experimental load-displacement curves are shown in Figures 6-6 and 6-7 for type ' A ' and ' B ' lay-ups, respectively. The corresponding curves produced by the model, adjusted to account for initial seating of the specimens, are also shown in the figures as dark lines. The predicted load-displacement behaviour was close to that observed in the experiments. The displacements shown in the figures were due to a combination of compression and shear deformations. Compression properties were found to dominate the load-displacement behaviour. Failure in the models was assumed to occur at the final displacement level measured in the experiments, as mentioned previously. Since this failure was assumed to have occurred in shear, the reference shear stress, a*, shown in Table 6-6, was determined at that final load step using equation [2-7]. The average values of a* for each plane were taken as the actual 110 reference strength for this material. Thus, the averages of the shear strengths for the large and small size type 'A' specimens in planes X Y , Y Z and X Z are 20.4 MPa (2958 psi), 8.3 MPa (1204 psi) and 5.4 MPa (781 psi), respectively. For type 'B', shear strengths in planes X Y and X Z are 23.4 MPa (3393 psi) and 8.1 MPa (1169 psi), respectively. The strength values for each specimen type were compared against the average nominal ultimate shear stresses as shown in Table 6-6. If the ratio of these values, P, is greater than 1, then the specimen did fail prematurely and the nominal strength value is conservative, p was found to be 1.7, on average. Foschi and Barrett (1976) found the value of P to be approximately 2 for the shear plane parallel-to-grain in Douglas fir and that P was dependent on the value of the shape parameter, k. The shear stress along two sections was analysed to determine the distribution of stress. Section A-B, running from the notch to the bottom of the specimen along the outside of the specimen, is shown in Figure 6-5; section A'-B' runs through the middle of the specimen. A preliminary analysis using this three-dimensional model, assuming isotropic properties for the wood, indicated that the peak stresses occurred at the notch. The stress concentration factor (peak stress divided by nominal stress) was found to be roughly two for this shear block using isotropic properties - this agrees with findings by Coker and Coleman (1935). Figure 6-8 shows the typical stress distribution along the outside section A-B and at the centre of the specimen, A'-B', at ultimate load. The distribution is not uniform over the sections. The peak values at the notch ('0' inches in Figure 6-8) are used to determine the stress concentration factor. From the two curves shown in Figure 6-8, there does not appear to be much difference between shear stresses in the centre of the specimen and those along the outside. This indicates that a two-dimensional geometric model is probably adequate for predicting shear stresses (but not necessarily for ultimate strength). Using the anisotropic plasticity material model, the stress concentration factor was at least two, on average, as indicated in Table 6-7. The difference in values between the three planes may, in part, be attributed to assumed values of the shear moduli, given in Table 6-5. Cramer et al. (1984) indicated that including the effects of crushing through a non-linear compression stress model would result in stress concentration factors lower than their calculated value of 2.36 due to the release of strain energy through ductile behaviour. This is not always the case as shown in Table 6-7. I l l It was previously mentioned that a two-dimensional model would be sufficient to determine the stress distribution. The current three-dimensional model can be used, however, to determine if brittle failure occurred in tension in the X-, Y- or Z-directions. Using the reference stress values determined from the tension tests (shown in Table 6-6) at p - 0.5, it was found that delamination failure (tension in the Z-direction) occurred at roughly 40% of the ultimate displacement as shown in Table 6-8 for the type 'A' lay-up, and approximately 44% for the type 'B' lay-up. The specimen could still carry load, as was observed in experiments, but the analysis indicated that the tensile stress concentration in the region of the notch would produce delamination. 6.4 Bolted Connection Model in LSL The single bolt connection model (described in chapter 5) was used to predict the behaviour of the LSL connection specimens discussed in chapter 4. The material constants for oriented and randomly oriented layers are listed in Tables 6-1, 6-5, and 6-6 for uniaxial and shear properties. One model was developed for each of the connection geometries and lay-ups listed in Table 4-2. r Multiple-layer panels (types ' C , 'D', and 'E') were modelled by changing the material properties through the thickness of the wood composite. For example, lay-ups ' C and 'D' were modelled by assigning type 'A' properties to fully oriented layers and type 'B' properties to randomly oriented layers. The type ' E ' lay-up with layers at \u00b1 45\u00b0 were modelled using only type 'A' properties but with the principal material axes oriented at the appropriate angles. Failure prediction was calculated for each layer by applying the Weibull weakest link theory to the local stresses within that layer. The strength values corresponding to that layer were used in the criterion (i.e. type 'A' or 'B'). It is noted that the type 'A' members tested at 45\u00b0 (AN specimens) and all type ' E ' members require a half-model rather than the quarter-model described in Chapter 5. This is a result of the non-zero displacement condition along the plane where x is equal to zero (see Figure 5-1). This displacement occurs because of the cross-orientation of layers. Two sample half-models were created for these cases to determine if the displacements caused significant changes in load-displacement and ultimate load predictions, or if a quarter-model would suffice. For the A N specimens with lid = 4, load-displacement was found to be affected by 112 about 4% when using only a quarter-model. Failure predictions were unaffected, however, in-plane shear stresses, although never found to govern, were found to be roughly 17% higher in the half-model. Type 'E' members with lid = 4 were found to have no change in load-displacement behaviour, likely as a result of the cross-plies of \u00b145\u00b0 layers. Failure prediction was unchanged. The quarter-model was used for the remaining analyses. It is cautioned, however, that other lay-ups may be significantly affected by this assumption and should, therefore, be modelled using one-half of the specimen geometry. For each connection model, load-displacement curves, stress concentrations, ultimate strengths, failure modes and hole deformations were predicted. These will now be discussed. 6.4.1. Load-displacement Results Each model was analysed for displacements of up to 2.5 mm (0.1 inch) unless it failed to converge prior to this, regardless of whether or not the failure analysis predicted failure earlier than this. Failure analysis was performed in the post-processor and thus had no effect on the solution phase. However, the curves reported in this section are truncated at the predicted failure loads. In cases where failure was not predicted to occur, a greater-than symbol is shown preceding the listed ultimate loads and displacements. Load-displacement curves are, thus, plotted up to the lesser of: a) the predicted failure displacement, or b) the displacement at the end of the analysis. Predicted load-displacement curves are shown superimposed on the experimental curves in Appendix A. Generally, the shape of each predicted curve was reasonably well-matched with the experimental curves. The predicted initial stiffness was found to be very close to the actual stiffness, regardless of dowel size, lay-up, loading orientation and connection geometry. In cases that exhibited connection ductility, such as most connections with 9.5 mm (3\/8 inch) and 13 mm (1\/2 inch) dowels, the flattened portion of the curve was seen to develop. In cases where no connection ductility was observed, the model predicted a linear load-displacement curve corresponding to the experimental results. 6.4.2. Stress Distributions Normal and shear stress distributions throughout each connection are useful indicators of locations of stress concentrations and expected modes of failure. Comparing 113 contour plots of stresses for different connection geometries leads to a number of observations. Consider the connections in type 'A' fully oriented material, shown in Figure 6-9. Both connections have eld = 4 and edge distance of 1.5d, however 9.5 mm (3\/8 inch) and 19 mm (3\/4 inch) dowels were used in Figures 6-9(a) and (b), respectively. This resulted in different specimen dimensions, as indicated in the figure, and slenderness ratios, lid, of 4 and 2. The uniformity of stress distribution throughout the specimen thickness is an indicator of the expected modes of failure for each connection geometry. For example, Figure 6-9 shows that peak normal stresses in the Y-direction, a y, (parallel to the loading direction) are non-uniform throughout the specimen thickness for large lid, and uniform throughout the thickness when lid was small. (Note that the plot is only one-quarter of the entire connection due to two planes of symmetry.) Peak normal stresses in the X-direction are shown in Figure 6-10. These plots indicate that these stresses are non-uniform throughout the thickness with large lid, and uniform with small lid. It is clear that a tension field develops in the loaded region of wood, a compression field develops on the unloaded side of the hole, and another compression field develops adjacent to the tension field on the loaded side. The plasticity model will allow for stress release in these compression zones while the tension field may lead to brittle failure. The distribution of Z-direction tension stresses at the edge of the hole is dependent on lid, as shown in Figure 6-11. The shear stress distribution for XY shear stresses is also dependent on lid, as shown in Figure 6-12. This shear stress is seen to be at it highest on the face of the wood member, offset from the member centreline. A different pattern was observed to occur in multiple-layer material. For example, Figure 6-13(b) shows non-uniform shear stress distribution in the type 'E' connection with lid - 2. Compared to the similar plot in Figure 6-12(b) for the same specimen but with type 'A' material, two peak shear stresses appeared along the dowel\/wood contact interface at the \u00b1 45\u00b0 layers. The opposite signs of the type 'E' peaks are representative of the layer orientations. The magnitude of shear stress on one of the peaks was higher than the peak stress found in the type 'A' specimens, however, the failure analysis indicated that this specimen failed due to tension in the Z-direction. Thus, stress distribution plots on their own are not indicators of the governing modes of failure. 114 Although stress distribution plots provide an indication of the location and magnitude of peak stresses, only the strength criterion can be used to determine whether the entire stress state will lead to brittle failure. 6.4.3. Ultimate Strength and Displacement The ultimate load, Ppredicted, and ultimate displacement, Ap r e (iicted5 were determined for a 50% probability of failure for each connection. The results are listed in Appendix A with each specimen's load-displacement curve. In cases where no failure was predicted to occur, the greater-than symbol is shown preceding the predicted load and displacement values and indicates that no brittle fracture was detected up to the listed maximum displacement. It was found that analyses that reached the maximum 2.5 mm (0.1 inch) displacement without failure corresponded to ductile experimental behaviour with no brittle fracture. This occurred, for example, in specimens with lid = 4 and eld = 4. When brittle fracture was found to occur, the maximum ultimate load and displacement were always found to be less than the experimental average. The ratios of predicted to experimental loads and displacements for brittle fractures are listed in Appendix A. Connections loaded parallel to the main strand axis were predicted to fail, on average, at approximately 71% of the average experimental ultimate load, e^xperiment- Connections loaded perpendicular to the main strand axis were predicted to fail, on average, at 64% of e^xperiment-Adjustments to the failure probability level could result in improved predictions. Experimental load data had roughly a 20% coefficient of variation based on only five specimens, thus the predicted loads are considered reasonable. These results are averaged across all dowel sizes, lay-ups and connection geometries for each loading orientation. Connections with type 'A' lay-up loaded at 45\u00b0 to the main strand axis were predicted to fail at 60% of the ultimate experimental load. It was expected that the model would not produce these conservative results for this loading orientation as a result of the stress interaction between shear and tension which occurs with off-axis loading. A review of the shear stress data for these specimens indicated that the shear strength was notably higher than the shear stresses which developed and that tension perpendicular to the main strand axis was the cause of failure, thus, there would be little effect of interaction. 115 Connections in multiple-layer panels were assumed to fail once the stresses in any one layer were found to exceed capacity. This is a conservative estimate since load-sharing among layers adjacent to the failed layer would occur in reality. Consider Figures 6-13(a) and (b) for type 'E' specimens. Shear stress, a x y , was found to govern the strength in the + 45\u00b0 layers, not tension stress, a z, as was the case for all other lay-ups with this specimen geometry (see Table 6-9). Figure 6-13(a) shows a disruption in the o z tension field on the edge of the hole. Figure 6-11(b) shows the same specimen in type 'A' material which has a continuous a z tension field. Ultimate displacements were always under-predicted in cases where fracture occurred, as shown in Appendix A; however, the coefficient of variation of the experimental ultimate displacements for specimens that experienced fracture was quite high, roughly 37% on average (compared to 17% for ultimate load in the same specimens). Thus, ultimate displacements can be used as indicators of ductility but variability must be considered with respect to the actual values. The shape of the load-displacement curve is a more notable indicator of connection ductility prior to fracture. 6.4.4. Failure Modes Table 6-9 summarizes the predicted and observed modes of failure for each specimen. In the experiments, it was found that brittle fracture was prevented with increasing end distance in type 'A' material. The analysis predicted this transition from brittle failure to no fracture for most specimens when the end distance was increased. The model results generally corresponded with the observed failure modes. Brittle fracture was mainly found to be predicted in X- and Z-direction tension (very few specimens failed with shear governing) for specimens loaded parallel to the main strand axis (Y-direction loading). Specimens loaded perpendicular to the main strand axis re-direction) were always predicted to have brittle fracture. Type 'A' specimens loaded in this fashion were predicted to fail in tension in the X-direction, as was observed in the experiments. The complicated failure patterns observed in multiple-layer specimens (described in section 4.3.1.3), however, were predicted in simplified form using the model since only the first layer to fail was assumed to govern ultimate strength. 116 It is noted that only the failure criterion was able to determine when failure occurred - not plots of stress concentrations. The relative strengths in each direction resulted in one mode governing over another, not the presence of localized stress concentrations. 6.4.5. Hole Deformations One final measure of the effectiveness of the model is the total deformation which occurs at the hole. In Chapter 4, Ai was defined as the permanent elongation of the hole at the specimen surface in the direction of loading. Table 4-13, for example, lists measured values of Ai between 1 and 13 mm (0.04 and 0.5 inches). Although this large deformation is difficult to model, it is an indicator of significant plastic deformation in the wood. The largest deformations occurred in specimens that never experienced brittle fracture. Since the end of the dowel was only displaced up to 2.5 mm (0.1 inches) in the model, results cannot be compared to these final experimental deformations. However, they were checked to determine if the model behaviour was within reasonable bounds. Typical Y-direction compression plastic strains were roughly 12% indicating a high degree of permanent deformation. For the specimens shown in Figures 6-9 to 6-13, the predicted permanent elongation of holes at the end of the analyses were found to be between 0.5 and 1 mm (0.02 and 0.04 inches) at 2.5 mm (0.1 inches) displacement. Values from Table 4-13 indicate the observed elongation in experiments to be between 2 and 7 mm (0.08 and 0.29 inches) with average final displacement of roughly 8 mm (0.3 inches). Thus, the predicted values are reasonable. 6.5 Sensitivity Study Since there are many material and modelling variables involved in the single-bolt connection model, it was decided that a parametric study should be performed to determine the effect of each parameter and to decide which variables warrant more attention in future studies. In this section, the material properties for the wood composite will be studied, followed by a brief discussion on the effects of contact stiffness, KN, and mesh density. 117 6.5.1. Influence of Wood Properties Assumptions were made in section 6.1 to estimate the parameters used in the anisotropic plasticity model. First, the yield stresses and tangent moduli were estimated from the experimental stress-strain curves. Then, these values were used as seed values for the optimisation spreadsheet to determine admissible quantities to be used in the anisotropic plasticity model according to equations [5-4] and [5-5]. It is possible, therefore, that some of these variables may not be close to their \"true\" values. This sensitivity study was performed to determine: a) the effects of approximating material constants and, b) the most significant material properties. Determining the variables which lead to the greatest change in model output will help to pinpoint the essential experiments for determining material properties. This will minimize experimental effort in the future as new materials are developed. In the following discussion, the single-bolt finite element model for connection Pa5 in type 'A' material (see Table 4-2) with l\/d=3 and eld =2 was modified to perform a multiple variable analysis of variance. Only material properties were modified. These included the nine elastic constants and eighteen plasticity constants. Because of the large number of factors (i.e. the parameters manipulated during the experiment), an initial screening analysis was performed on each variable to determine its effect on two model outputs: the ultimate strength and the ultimate displacement (more sophisticated output parameters such as ductility could also be developed). From this preliminary analysis, nine factors that most influenced the output (out of the initial 27 factors) for this model geometry were selected. This was due, in part, to the direction of loading, which limited the effect of, for example, two of the shear moduli. The factors chosen for this study are listed in Table 6-10. The yield stresses in tension and compression were treated as equal, thus a change in the value in tension corresponded to an equal change in compression. The upper and lower values for each factor are listed in the table. Bounds on these quantities exist physically and due to the constraints imposed by equations [5-4] and [5-5]. To test all combinations of these factors, a total of 29 = 512 computer runs would be required (including the two-level, high and low, values). To make this more manageable, and to account for possible interactions between factors, a reduced number of runs were performed using a fractional factorial experimental design (Montgomery 1991). Designing this study using fractional factorial design meant that a more 118 reasonable analysis with only 64 runs (29-4) was allowed. The optimal choice of combinations for this design was chosen based on standard catalogues (JMP 1996), the, development of which is beyond the scope of this thesis. This \"experiment\" was not repeatable because only one result is possible for each computer run (as opposed to a laboratory experiment in which one would expect variability in results). Since there was only one run associated with each combination of variables and since 64 effects (63 factor effects and the mean) were being estimated, no estimate of the error in the sum of squares could be made from the analysis of variance, thus no calculation of the F statistic (a measure of the significance of each factor) was possible. It is widely accepted, however, that the estimates of effects (the change in response due to a change in one or more factors) with the largest absolute values indicate the factors that are most significant (Montgomery 1991). Ranking these estimates and plotting them on normal probability paper results in all negligible effects lying close to a straight line whereas significant factors will lie away from a straight line. This will be illustrated shortly. Once this initial analysis is performed, the effects that are deemed to be negligible can be lumped into the error term of the sum of squares to allow for calculation of the significance of each remaining effect through the analysis of variance. The analysis was run once for each model output (i.e. once for maximum load and once for maximum displacements). The plot of effects versus probability for maximum load are shown in Figure 6-14. From this plot, four factors were isolated from the effects because they did not lie on the straight line. These corresponded to the independent factors Ex, Ez, o \u00b1 x and a+z, and their interactions. Six interactions with very low estimates of effects (those which lie on the line) were chosen to be lumped into the error term. The data was then re-analysed to determine the factors with significant effects. Assuming a sufficiently small p-value (0.01 for the F-test), the same factors, Ex, Ez, a \u00b1 x and a+z, were found to be significant. The same trend was found by performing a similar analysis on maximum displacement data. Thus, for loading in the Y-direction, the elastic moduli in the X - and Z-directions significantly affected the model output for maximum load and displacement. In addition, the yield points in the X- and Z- directions influenced the model outcome. This explains the poor ultimate strength predictions by the tri-linear elastic model described in section 6.2 since no stiffness degradation was considered in these directions. 119 Thus, it appears that compression tests in the X- and Z-directions must be performed to accurately determine the elastic moduli and yield stresses in these directions. The magnitudes of the tangent moduli were not found to be significant for the range of values tested, thus, simplifying the analysis procedure required for satisfying equations [5-4] and [5-5] up to 20% plastic strain can be simplified in the future. This does not eliminate others tests, however, since ultimate strengths in tension and shear were not included in this study, and must be determined a priori. In addition, the shape of the load-displacement curve was not considered as an output parameter in this study. However, as previously mentioned, the non-linear material behaviour in the direction of loading (Y-direction in this case) does affect its shape, therefore, these properties are also significant. 6.5.2. Influence of Contact Stiffness It is common for contact stiffness to affect the results of finite element models that include contact elements on an interface. To assess the effect of KN (from equation [5-1]), a number of analyses were performed over a range of KN values. Extreme values of AJV produce divergence (if KN is too high) or zero stiffness (if KN is too low) resulting in the dowel passing through the wood member (i.e. no compatibility enforced). Working within these bounds, KN was found to affect behaviour. It was found that: 1) the shape of the load-displacement curve became stiffer with increasing KN, 2) low KN delayed failure until higher displacements, and 3) failure occurred at the same load level regardless of KN. The results indicate that KN did not influence the governing failure stresses associated with a particular load level; however, the associated displacement was affected. 6.5.3. Mesh Density As indicated in section 5.2, the effect of mesh density around the hole was checked using strain energy as a measure. The results of the failure analysis were compared for one specimen geometry with a number of mesh densities. Little change was noted in the failure prediction for the meshes described in section 5.2. 120 6.6 Summary In this chapter, a number of analyses were used to verify the proposed material model and to establish confidence in its behaviour. For example, it was shown that the anisotropic plasticity model produced more reasonable ultimate strength results than the tri-linear elastic model for connection specimens. It was also shown that the material model, together with the three-dimensional single dowel connection model, predict load-displacement, ultimate strength, ultimate displacements and failure modes reasonably well. In the next chapter, the model will be used to predict behaviour in four new applications. 121 Table 6-1. Material properties for uniaxial behaviour of LSL with anisotropic plasticity model. Type A Type B Fully oriented panels Randomly oriented panels (psi) (psi) 95,000 800,000 Ey 1,700,000 800,000 Ez 13,000 15,000 4452 8200 50,000 8200 3617 3300 O-x 960 2400 o-y 3500 2400 810 1300 Tension values (yield and ultimate) 0\"+\/ \u2022 (K=l,p=0.5) a*\/ (V=\\,p=0.5) X 960 689 2400 3299 Y 3500 7642 2400 3299 Z 810 195 1300 169 122 Table 6-2. Material properties for behaviour of Douglas fir with tri-linear elastic model. Uniaxial constants (psi) Shear constants (psi) 120,000 140,000 \u2022ETXI 120,000 G\u00b1Txyl 70,000 \u2022ETX2 120,000 G\u00b1Txy2 1,400 Ey 2,000,000 G y Z 140,000 \u00a3 - T y l 1,000,000 G \u00b1 T y z l . 70,000 \u00a3 - T y 2 20,000 G+Tyz2 1400 G-yl 5075 G\u00b1xyl = \u00b0 \u00b1 y z l 840 G.y! 6598 V\u00b1cy2 = <3\u00b1>:2 945 Ez 120,000 Gxz 14,000 120,000 G \u00b1 T x z l 14,000 120,000 G\u00b1 T x z 2 14,000 Tension X a +x (V=\\,p=0.5) 469 Shear XY \u00b0 \u00b1xy (V=\\,p=0.5) 2526 123 Table 6-3 - Single dowel connection behaviour with Douglas fir. Experimental Results Tri-linear Elastic Model Anisotropic Plasticity Model (Patton-Mallory, 1996) Geometry Average p-0 .5 lid eld edge\/rf Load @ 0.04 inches (lbs.) Ultimate Load (lbs.) Ultimate Disp. (inches) Ultimate Load (lbs.) Ultimate Disp. (inches) Ultimate Load (lbs.) Ultimate Disp. (inches) 2 4 1.5 3057 3057 0.05 2248 0.02 2563 0.04 2 7 1.5 3057 3372 0.07 2271 0.02 >2630 >0.04 2 10 1.5 3057 3260 0.07 2316 0.02 >2675 >0.04 5 4 1.5 3192 >3957 >0.13 2585 0.02 >2765 >0.04 5 7 1-5 3507 >4294 >0.13 2608 0.02 >2765 >0.04 5 10 1.5 3507 >3957 >0.13 2608 0.02 >2788 >0.04 7 4 1.5 3305 >3822 >0.13 2653 0.03 >2720 >0.04 7 7 1.5 3552 >3912 >0.13 2698 0.03 >2743 >0.04 7 10 1.5 3552 >3844 >0.13 2698 0.03 >2743 >0.04 Note: All tests used 12 mm (1\/2 inch) diameter mild steel dowels. 124 Table 6-4. Material properties for behaviour of Douglas fir with anisotropic plasticity model. Uniaxial constants (psi) Shear constants (psi) 120,000 40,000 Ey 2,000,000 G y 2 40,000 Ez 120,000 Gxz 4000 574 GjXy 661 \u00a3 T y 20,300 G T y z 661 574 7 1100 2497 (8) >0.28 (24) >2399(6) >0.23 (13) >2330 (6) >0.39 (23) 1445 >1643 >1632 >1647 0.05 >0.1 >0.1 >0.1 0.81 0.45 Figure A - l . Experimental and predicted load-displacement curves for type 'A' specimens loaded parallel to main strand axis, with lid = 4. Dark solid line is model prediction. Coefficient of variation in parentheses (%). Ultimate load prediction is for p = 0.50. 167 \u2022o \u00ab o mi 10000 7500 5000 2500 Pa5A All brittle 0.1 0.2 0.3 0.4 0.5 Displacement (inches) Pa6A o -1 10000 7500 5000 2500 1 brittle 4 ductile 0.1 0.2 0.3 0.4 Displacement (inches) 0.5 o 10000 7500 5000 2500 4 Pa7A AD ductile 0.1 0.2 0.3 0.4 Displacement (inches) 0.5 lid eld edgeld Experimental Maximum P (lbs) A (inches) Predicted Maximum P (lbs) A (inches) Ratios P A 'pred. pred. P A 1 exp. \"exp. Pa5a 3 2 1.5 Pa6a 3 3 1.5 Pa7a 3 4 1.5 2707 (22) 0.08 (58) >4280(10) >0.33 (39) >4364(12) >0.26(59) 2740 0.04 >2982 >0.07 >2986 >0.07 1.0 0.5 Figure A-2. Experimental and predicted load-displacement curves for type 'A' specimens loaded parallel to main strand axis, with l\/d = 3. Dark solid line is model prediction. Coefficient of variation in parentheses (%). Ultimate load prediction is for p = 0.50. 168 M o 10000 7500 5000 2500 10000 O Pa8A AD brittle 0.1 0.2 0.3 0.4 Displacement (inches) 0.5 Pal OA No failures or ductilit achieved (limited b displacement in wedge grip) 0.1 0.2 0.3 0.4 Displacement (inches) 0.5 a o 10000 7500 5000 J 2500 \u2022a \u00ab o -J 10000 7500 5000 2500 Pa9A 3 brittle 2 ductile 0.1 0.2 0.3 0.4 Displacement (inches) 0.5 P a l l A A l l ductile (limited b displacement in wedge grip) 0.1 0.2 0.3 0.4 Displacement (inches) 0.5 lid eld edge Id Experimental Maximum P (lbs) A (inches) Predicted Maximum P (lbs) A (inches) P 1 pred. P ' exp. Ratios A pred. AeXp. Pa8a 2 2 1.5 4447 (26) 0.05 (27) 3723 0.04 0.84 0.8 Pa9a 2 3 1.5 7428(15) 0.09 (52) 5300 0.06 0.71 0.7 Pa 10a 2 4 1.0 >6188(13) >0.06 (29) 3854 0.04 - -Pal la 2 4 1.5 >7272(18) >0.11 (40) 5884 0.06 - -Figure A-3. Experimental and predicted load-displacement curves for type 'A' specimens loaded parallel to main strand axis, with lid = 2. Dark solid line is model prediction. Coefficient of variation in parentheses (%). Ultimate load prediction is for p = 0.50. 169 \u2022o et o mi 10000 7500 5000 2500 J TJ M O mi 10000 7500 5000 J 2500 PelA All brittle 0.1 0.2 0.3 0.4 Displacement (inches) Pe3A All brittle 0.1 0.2 0.3 0.4 Displacement (inches) 0.5 0.5 \u2022a a o mi 10000 7500 -) 5000 2500 -I 0 \u2022o a o 10000 7500 5000 2500 0 Pe2A All brittle 0 0.1 0.2 0.3 0.4 Displacement (inches) Pe4A 0.1 0.2 0.3 0.4 Displacement (inches) 0.5 AU brittle r 0.5 lid eld edgeld Experimental Maximum P (lbs) A (inches) Predicted Maximum P (lbs) A (inches) p 1 pred. P 1 exp. Ratios A pred. Aexp. Pela 4 2 3 1422 (23) 0.08 (69) 1125 0.05 0.79 0.63 Pe2a 4 4 3 1571 (35) 0.13(56) 1140 0.05 0.73 0.38 Pe3a 2 2 3 3489 (9) 0.09(11) 1685 0.04 0.48 0.44 Pe4a 2 4 3 3409(18) 0.08 (31) 1729 0.04 0.51 0.50 Figure A-4. Experimental and predicted load-displacement curves for type 'A' specimens loaded perpendicular to main strand axis. Dark solid line is model prediction. Coefficient of variation in parentheses (%). Ultimate load prediction is for p = 0.50. 170 AN1A \u2022o a o 10000 7500 5000 2500 0 All brittle 0.1 0.2 0.3 0.4 Displacement (inches) 0.5 \u2022o (9 O 10000 7500 5000 2500 AN2A 1 brittle after long plateau 4 ductile 0.1 0.2 0.3 0.4 Displacement (inches) 0.5 AN3A 10000 \u2022o o o 7500 -I 5000 2500 0 -I All brittle A 0 0.1 0.2 0.3 0.4 Displacement (inches) 0.5 TS M O 10000 -, 7500 5000 2500 AN4A 3 brittle 1 ductile 1 unusable specimen 0.1 0.2 0.3 0.4 Displacement (inches) 0.5 lid eld edgeld Experimental Maximum P (lbs) A (inches) Predicted Maximum P (lbs) A (inches) Ratios pred. ' pred. \u2022*exp. ANla 4 2 AN2a 4 4 AN3a 2 2 AN4a 2 4 1925 (13) 0.13 (62) >2402(3) >0.39(10) 4067(14) 0.07(12) 6144(10) 0.24(29) 1436 >1465 2055 3333 0.07 >0.07 0.03 0.04 0.75 0.51 0.54 0.54 0.43 0.17 Figure A-5. Experimental and predicted load-displacement curves for type 'A' specimens loaded 45\u00b0-to-main strand axis. Dark solid line is model prediction. Coefficient of variation in parentheses (%). Ultimate load prediction is for p = 0.50. 171 \u2022a a o J 10000 7500 5000 2500 0 4 PalB 2 brittle 3 ductile 0 0.1 0.2 0.3 0.4 Displacement (inches) 0.5 e 10000 7500 5000 2500 Pa2B AU ductile 0.1 0.2 0.3 0.4 Displacement (inches) 0.5 Xi \u2022a a o -J 10000 7500 5000 2500 Pa3B All brittle 0.1 0.2 0.3 0.4 Displacement (inches) 0:5 \u2022a a o 10000 7500 5000 2500 Pa4B 2 brittle 3 ductile 0.1 0.2 0.3 0.4 Displacement (inches) 0.5 lid eld edgeld Experimental Maximum P (lbs) A (inches) Predicted Maximum P (lbs) A (inches) p 1 pred. P 1 exp. Ratios A pred. Aexp. Palb 4 2 1.5 2212(19) 0.24 (57) 1370 0.06 0.62 0.25 Pa2b 4 3 1.5 >2276(10) >0.37 (41) >1386 >0.06 - -Pa3b 4 4 1.0 1296 (37) 0.04 (54) 1194 0.04 0.92 1.0 Pa4b 4 4 1.5 >2255 (9) >0.33 (49) 1402 . 0.07 - -Figure A-6. Experimental and predicted load-displacement curves for type 'B ' specimens loaded parallel to main strand axis, with lid = 4. Dark solid line is model prediction. Coefficient of variation in parentheses (%). Ultimate load prediction is for p = 0.50. 172 a o 10000 7500 J 5000 2500 \u2022a \u00ab o \u2014I 10000 7500 5000 2500 Pa5B All brittle 0.1 0.2 0.3 0.4 Displacement (inches) Pa7B 4 brittle 1 ductile 0.1 0.2 0.3 0.4 Displacement (inches) 0.5 0.5 Pa6B \u2022o cs a 10000 7500 5000 J 2500 3 brittle 2 ductile 0.1 . 0.2 0.3 0.4 Displacement (inches) 0.5 lid eld edgeld Experimental Maximum P(lbs) A (inches) Predicted Maximum P (lbs) A (inches) Ratios P A 1 pred. \" pred. P A ' exp. \"exp. Pa5b 3 2 1.5 Pa6b 3 3 1.5 Pa7b 3 4 1.5 3224(12) 0.14(40) >4013 (28) >0.32(59) >3818 (24) >0.27(77) 2485 0.04 2697 0.05 2698 0.05 0.77 , 0.29 Figure A-7. Experimental and predicted load-displacement curves for type 'B' specimens loaded parallel to main strand axis, with lid =3. Dark solid line is model prediction. Coefficient of variation in parentheses (%). Ultimate load prediction is for p = 0.50. 173 \u2022o o \u20141 10000 7500 5000 2500 w O _1 10000 7500 5000 2500 Pa8B All brittle 0.1 0.2 0.3 0.4 Displacement (inches) PalOB All brittle 0.1 0.2 0.3 0.4 Displacement (inches) 0.5 0.5 Pa9B \u2022o n o \u20221 10000 7500 5000 2500 \u2014 AU brittle i f \\( I i 1 M O 0.1 0.2 0.3 0.4 Displacement (inches) Pal IB AD brittle 0.1 0.2 0.3 0.4 Displacement (inches) 0.5 0.5 Experimental Maximum Predicted Maximum Ratios lid eld edgeld P(lbs) A (inches) P (lbs) A (inches) p 1 pred. P A pred. Pa8b 2 2 1.5 5483 (24) 0.24 (39) 3655 0.04 0.67 0.17 Pa9b 2 3 1.5 7070(14) 0.17(47) 3916 0.05 0.55 0.29 Pa 10b 2 4 1.0 3339 (25) 0.05(11) 2322 0.03 0.70 0.60 Pal lb 2 4 1.5 6931 (14) 0.25 (63) 3918 0.05 0.57 0.20 Figure A-8. Experimental and predicted load-displacement curves for type 'B ' specimens loaded parallel to main strand axis, with lid = 2. Dark solid line is model prediction. Coefficient of variation in parentheses (%). Ultimate load prediction is for p = 0.50. 174 R O -1 10000 7500 J 5000 \u2022o \u00ab o 10000 7500 5000 2500 PalC 1 brittle 4 ductile T 0.1 0.2 0.3 0.4 Displacement (inches) 0.5 Pa2C \u00ab o -J 10000 7500 J 5000 2500 All ductile 0.1 0.2 0.3 0.4 Displacement (inches) 0.5 Pa3C 4 brittle 1 ductile 0.1 0.2 0.3 0.4 Displacement (inches) 0.5 \u2022a M o 10000 7500 5000 2500 Pa4C AD ductile 0.1 0.2 0.3 0.4 Displacement (inches) 0.5 lid eld edgeld Experimental Maximum P (lbs) A (inches) Predicted Maximum P (lbs) A (inches) Ratios 'pred. exp. * pred. *exp. Pale Pa2c Pa3c Pa4c 4 2 4 3 4 4 4 4 1.5 1.5 1.0 1.5 1945 (8) 0.21 (56) >2285 (8) >0.3 (13) 2169(14) ,0.26(51) >2472(5) >0.39(14) 1620 >1579 >1595 >1577 0.11 >0.1 >0.1 >0.1 0.83 0.52 Figure A-9. Experimental and predicted load-displacement curves for type ' C specimens loaded parallel to main strand axis, with lid - 4. Dark solid line is model prediction. Coefficient of variation in parentheses (%). Ultimate load prediction is for p = 0.50. 175 o - J 10000 7500 5000 2500 \u2022o a o J 10000 7500 5000 2500 Pa5C All brittle 0.1 0.2 0.3 0.4 Displacement (inches) Pa7C 1 brittle 4 ductile 0.5 0 0.1 0.2 0.3 0.4 0.5 Displacement (inches) lid eld edge Id Experimental Maximum P (lbs) A (inches) Predicted Maximum P (lbs) A (inches) Ratios P A 1 pred. pred. P A 1 exp. \"exp. Pa5c 3 2 1.5 Pa7c 3 4 1.5 3515(14) 0.20(44) >4589(9) >0.45(21) 2746 0.04 >3059 >0.07 ' 0.78 0.20 Figure A-10. Experimental and predicted load-displacement curves for type ' C specimens loaded parallel to main strand axis, with l\/d=3. Dark solid line is model prediction. Coefficient of variation in parentheses (%). Ultimate load prediction is for p = 0.50. 176 R O -J 10000 7500 A 5000 2500 10000 R O J 7500 J 5000 2500 Pa8C AD brittle 0.1 0.2 0.3 0.4 Displacement (inches) PalOC AD brittle 0.1 0.2 0.3 0.4 Displacement (inches) 0.5 0.5 \u2022O \u00a9 10000 7500 J 5000 J 2500 JO \u2022o R O mi Pa9C AD brittle 0.1 0.2 0.3 0.4 Displacement (inches) 0.5 PallC 1 brittle 4 ductile 0.1 0.2 0.3 0.4 Dis place ment (inches) 0.5 Experimental Maximum Predicted Maximum Ratios lid eld edgeld P (lbs) A (inches) P (lbs) A (inches) p 1 pred. P ' exp. A pred. Aexp. Pa8c 2 2 1.5 5089 (25) 0.08 (20) 4073 0.04 0.80 0.50 Pa9c 2 3 1.5 7375 (18) 0.21 (9) 5166 0.04 0.70 0.19 Pa 10c 2 4 1.0 6748 (7) 0.09(13) 3295 0.03 0.49 0.33 Pallc 2 4 1.5 >8043 (12) >0.33 (44) 5494 0.05 - -Figure A- l 1. Experimental and predicted load-displacement curves for type ' C specimens loaded parallel to main strand axis, with lid = 2. Dark solid line is model prediction. Coefficient of variation in parentheses (%). Ultimate load prediction is for p = 0.50. 177 10000 7500 5000 o PelC AD brittle 0.1 0.2 0.3 0,4 Displacement (inches) 0.5 o -J 10000 7500 5000 2500 -Pe2C 1 brittle 4 ductile 0.1 0.2 0.3 0.4 Displacement (inches) 0.5 lid eld edgeld Experimental Maximum P (lbs) A (inches) Predicted Maximum P (lbs) A (inches) Ratios \u2022 P A ' pred. \"pred. P A ' exp. \"exp. Pelc 4 2 3 Pe2c 4 4 3 1972(18) 0.20(29) >2469 (8) >0.52 (9) 1348 0.06 1381 0.07 0.68 0.30 Figure A-12. Experimental and predicted load-displacement curves for type ' C specimens loaded perpendicular to main strand axis. Dark solid line is model prediction. Coefficient of variation in parentheses (%). Ultimate load prediction is for p = 0.50. 178 o mi 10000 7500 J 5000 2500 J 10000 B O \u2022 J 7500 J 5000 2500 PalD 3 brittle 2 ductile 0.1 0.2 0.3 0.4 Displacement (inches) Pa3D 1 brittle 4 ductile 0.1 0.2 0.3 0.4 Displacement (inches) 0.5 0.5 o 10000 7500 5000 2500 o mi 10000 7500 5000 2500 Pa2D All ductile 0.1 0.2 0.3 0.4 Displacement (inches) Pa4D AH ductile 0.5 0.1 0.2 0.3. 0.4 0.5 Displacement (inches) lid eld edgeld Experimental Maximum P (lbs) A (inches) Predicted Maximum P (lbs) A (inches) Ratios pred. exp. pred. *exp. Paid Pa2d Pa3d Pa4d 4 2 4 3 4 4 4 4 1.5 1.5 1.0 1.5 2119(12) 0.23(46) >2409 (7) >0.39 (20) 1839(18) 0.14(52) >2408 (2) >0.47 (19) 1366 >1242 1238 >1415 0.05 >0.04 0.04 >0.07 0.64 0.67 0.22 0.29 Figure A-13. Experimental and predicted load-displacement curves for type 'D' specimens loaded parallel to main strand axis, with lid = 4. Dark solid line is model prediction. Coefficient of variation in parentheses (%). Ultimate load prediction is for p = 0.50. 179 TJ a o -) 10000 7500 5000 2500 Pa5D AD brittle 0.1 0.2 0.3 0.4 Displacement (inches) 0.5 R O - J 10000 7500 5000 2500 J Pa7D 1 brittle 4 ductile 0.1 0.2 0.3 0.4 Displacement (inches) 0.5 lid eld edgeld Experimental Maximum P (lbs) A (inches) Predicted Maximum P (lbs) A (inches) Ratios P A pred. \" pred. P A ' exp. \"exp. Pa5d 3 2 1.5 Pa7d 3 4 1.5 3096(32) 0.16(35) >4624 (6) >0.46 (25) 2648 0.04 2829 0.05 0.86 0.25 Figure A-14. Experimental and predicted load-displacement curves for type 'D' specimens loaded parallel to main strand axis, with lid = 3. Dark solid line is model prediction. Coefficient of variation in parentheses (%). Ultimate load prediction is for p = 0.50. 180 ( TJ \u00ab O -J 10000 7500 5000 2500 T> \u00ab o -J 10000 7500 5000 2500 Pa8D All brittle PalOD AD brittle 0.1 0.2 0.3 0.4 0.5 Displacement (inches) 0.1 0.2 0.3 0.4 0.5 Displacement (inches) TJ \u00ab O -J 10000 7500 5000 2500 J \u2022o O 10000 7500 5000 2500 Pa9D 4 brittle 1 ductile 0.1 0.2 0.3 0.4 Displacement (inches) PallD AD brittle 0.1 0.2 0.3 0.4 Displacement (inches) 0.5 0.5 Experimental Maximum Predicted Maximum Ratios lid eld edgeld P (lbs) A (inches) \/'(lbs) A (inches) p 1 pred. P 1 exp. Aprerf. A e x p . Pa8d 2 2 1.5 6065 (12) 0.08 (22) 4000 0.04 0.66 0.50 Pa9d 2 3 1.5 7856(21) 0.22 (24) 4640 0.04 0.59 0.18 PalOd 2 4 1.0 5550(14) 0.08 (37) 2802 0.03 0.50 0.38 Palld 2 4 1.5 7327 (5) 0.28 (25) 4651 0.04 0.63 0.14 Figure A-l5. Experimental and predicted load-displacement curves for type 'D' specimens loaded parallel to main strand axis, with lid - 2. Dark solid line is model prediction. Coefficient of variation in parentheses (%). Ultimate load prediction is for p - 0.50. 181 \u2022o a o 10000 7500 5000 PelD AH ductile with brittle splitting 0.1 0.2 0.3 0.4 Displacement (inches) 0.5 \u2022o a o -J 10000 7500 5000 J 2500 Pe2D AD ductfle 0.1 0.2 0.3 0.4 Displacement (inches) 0.5 lid eld edgeld Experimental Maximum P (lbs) A (inches) Predicted Maximum P (lbs) A (inches) Ratios P A 'pred. ^pred. P A 1 exp. \"exp. Peld 4 2 3 Pe2d 4 4 3 >2346 (4) >0.36 (23) >2474 (9) >0.54 (3) >1353 >0.05 >1394 >0.05 -Figure A-16. Experimental and predicted load-displacement curves for type 'D' specimens loaded perpendicular to main strand axis. Dark solid line is model prediction. Coefficient of variation in parentheses (%). Ultimate load prediction is for p = 0.50. 182 \"S o 10000 7500 5000 2500 J PalE All Med with brittle splitting. 0.1 0.2 0.3 0.4 Displacement (inches) 0.5 \u2022a a o mi 10000 7500 5000 2500 Pa2E 1 brittle 4 ductile 0.1 0.2 0.3 0.4 Displacement (inches) 0.5 \u2022o \u00ab o -J 10000 7500 5000 2500 J Pa3E 4 brittle 1 ductile .0.1 0.2 0.3 0.4 Displacement (inches) 0.5 \u2022o \u00ab o mi 10000 7500 5000 2500 Pa4E All ductile 0.1 0.2 0.3 0.4 Displacement (inches) 0.5 lid eld edgeld Experimental Maximum P (lbs) A (inches) Predicted Maximum P(lbs) A (inches) Ratios P A pred. pred. P A ' exp. \"exp. Pale 4 2 1.5 Pa2e 4 3 1.5 Pa3e 4 4 1.0 Pa4e 4 4 1.5 1937(11) 0.13 (64) >2334(9) >0.43(13) 2176(13) 0.20(57) >2535 (4) >0.42 (13) 1470 0.06 >1639 >0.10 1468 0.06 >1644 >0.10 0.76 0.46 0.67 0.30 Figure A-17. Experimental and predicted load-displacement curves for type 'E' specimens loaded parallel to main strand axis, with lid = 4. Dark solid line is model prediction. Coefficient of variation in parentheses (%). Ultimate load prediction is for p = 0.50. 183 \u2022o \u00ab o J 10000 7500 5000 2500 \u2022o \u00ab o 10000 7500 5000 2500 Pa5E All brittle 0.1 0.2 0.3 0.4 Displacement (inches) 0.5 Pa7E All ductile 0.1 0.2 0.3 0.4 0.5 Displacement (inches) lid eld edgeld Experimental Maximum P (lbs) A (inches) Predicted Maximum P (lbs) A (inches) Ratios P A ' pred. ^pred. P A ' exp. \"exp. Pa5e 3 2 1.5 Pa7e 3 4 1.5 3064(21) 0.14(46) >4833 (6) >0.44 (24) 2917 0.07 >2869 >0.06 0.95 0.5 Figure A-l8. Experimental and predicted load-displacement curves for type 'E' specimens loaded parallel to main strand axis, with l\/d = 3. Dark solid line is model prediction. Coefficient of variation in parentheses (%). Ultimate load prediction is for p = 0.50. 184 o mi 10000 7500 5000 J 2500 Pa8E All brittle 0.1 0.2 0.3 0.4 Displacement (inches) PalOE \u00a3 \u2022O O 10000 7500 5000 J 2500 An brittle 0.1 0.2 0.3 0.4 Displacement (inches) 0.5 0.5 Pa9E \u2022o M O mi 10000 7500 J 5000 J 2500 AH brittle 0.1 0.2 0.3 0.4 Displacement (inches) 0.5 PallE 10000 \u2022o M O mi AU ductile 0.1 0.2 0.3 0.4 Displacement (inches) 0.5 Experimental Maximum Predicted Maximum Ratios lid eld edgeld P(lbs) A (inches) P (lbs) A (inches) p pred. P exp. A pred. Aexp. Pa8e 2 2 1.5 5490(12) 0.08(16) 4457 0.05 0.81 0.60 Pa9e 2 3 1.5 8501(3) 0.27 (35) 5193 0.06 0.61 0.21 PalOe 2 4 1.0 5515(16) 0.07(18) 3370 0.04 0.61 0.57 Palle 2 4 1.5 >8341 (9) >0.31 (53) 4978 0.05 - -Figure A-19. Experimental and predicted load-displacement curves for type 'E' specimens loaded parallel to main strand axis, with lid = 2. Dark solid line is model prediction. Coefficient of variation in parentheses (%). Ultimate load prediction is for p = 0.50. 185 \u2022o O 10000 7500 J 5000 2500 J PelE 2 brittle 3 ductile 0.1 0.2 0.3 0.4 Displacement (inches) 0.5 Pe2E \u2022O \u00ab o -1 10000 7500 5000 2500 3 brittle 2 ductile ^ 0.1 0.2 0.3 0.4 Displacement (inches) 0.5 lid eld edgeld Experimental Maximum \/'(lbs) .A (inches) Predicted Maximum P (lbs) A (inches) Ratios P A ' pred. pred. P A ' exp. \"exp. Pele 4 2 3 Pe2e 4 4 3 >2251 (14) >0.35(42) >2451 (10) >0.42(30) 1147 0.04 >1266 >0.07 -Figure A-20. Experimental and predicted load-displacement curves for type 'E' specimens loaded perpendicular to main strand axis. Dark solid line is model prediction. Coefficient of variation in parentheses (%). Ultimate load prediction is for p = 0.50. 186 APPENDIX B ANISOTROPIC PLASTICITY THEORY The anisotropic plasticity model is based on a modification of the von Mises theory for isotropic materials. Hill (1948) extended the model for orthotropic materials. Valliappan et al. (1976) showed the requirements for implementing this theory into analysis for work hardening materials. An associative flow rule is assumed. Shih and Lee (1978) modified the theory to account for the distortion of the yield surface due to differences in tension and compression yield stresses. This results in a yield surface with the origin offset, as shown in Figure B - l . Hardening of the material following yielding results in growth of the yield surface. For the anisotropic plasticity model, the growth is non-proportional to the change in yield stresses and results in non-uniform expansion of the yield surface, i.e. hardening is not isotropic (Vaziri etal, 1991). Stress-strain behaviour is assumed to follow the bi-linear curves shown in Figure 5-3 in each of the principal material directions and in shear. The theoretical development of the model is as follows. The yield criterion: [B-l] F({a},Wp,{a}) = 0 defines the stress at which yielding occurs, where {a} is the vector of the current stress state, W is the amount of plastic work over the entire stress history, and {a} is a vector which accounts for differences in tension and compression yield stresses. When F is greater than zero, yielding occurs. Otherwise, there is no further yielding, {a} causes the origin of the yield surface to be shifted. The plastic work is determined over the entire history of loading. Increments of plastic work are used to calculate the change in the yield surface throughout loading. The shape of the yield surface changes according to increments in plastic work throughout the loading history. Equation [B-l] can be re-written to account for the different yield stresses in each material direction in addition to differences between compressive and tensile yield stresses: [B-2] F = {cs}T[M]{o}-{~~ - q - , + q + , - q - , = 0 a+xO_x V+yV-y \u00b0 ~ + z a - z Equation [5-4] must be satisfied throughout all levels of plastic straining. Two other factors limit the magnitudes of the yield stresses that can be used in the anisotropic plasticity model thrpughout plastic straining. Firstly, the yield surface must be a closed surface to prevent numerical instability, and secondly, [M] must be positive definite. Thus, the following condition must be satisfied for three-dimensional problems: [5-5] M\\x + M222 + M\\2 - 2(MuM22+M22Mx + M , , M 3 3 ) < 0 Equations [5-4] and [5-5] place severe restrictions on the values which can be used for yield stresses and tangent moduli since they must continue to be satisfied once the yield surface starts to harden, as previously mentioned, and must be checked for validity at higher levels of plastic strain. There must, therefore, be a compromise between using the exact uniaxial test values for these quantities and using values which satisfy these equations. More details on the implementation of this theory can be found in any of the references cited in this appendix. 189 190 APPENDIX C F A I L U R E PREDICTION: SIZE E F F E C T The Weibull weakest link theory postulates that for brittle materials, larger specimens are more likely to fail at lower stresses due to the increased probability of a flaw in that larger specimen volume. Barrett (1974) found this to be true for the ultimate tension stress perpendicular-to-grain in Douglas fir. This concept was used to develop lumber strength values for the Canadian wood design code CSA (1994). It has also been used to determine the strength of timber rivet connections listed in the standard (Foschi and Longworth, 1975). The probability of brittle failure based on a two-parameter Weibull distribution is a function of the stress distribution over the volume of material: [2-5] Fv=\\-e where Fy is the probability of failure, V* is a reference volume, k is the shape parameter, m is the scale parameter associated with the reference volume, and cr is the stress (tension or shear, in any given direction or plane, respectively). The reference stress is based on experimental results shown in Table 3-3. [2-6] a' =m[-\\n{\\-p)]lk is the failure stress for a given probability of failure, p. It can be shown that failure occurs when the reference stress, a*, occurring over volume V*, satisfies the following inequality which relates a variable state of stress over an arbitrary volume to the reference stress for a uniform state of stress for the same probability of failure in both volumes: [2-7] \\rjkdV>a'kV' V This form is similar to the Maximum Stress criterion where each stress is treated individually and stress interactions are assumed to have little or no effect. When the stress field is constant over a volume, equation [2-7] can be simplified to give the failure stresses for a given probability level, p: fir V [2-8] a 2 Mk 191 where <3\\ and o\"2 are the strengths of specimens of volumes, V\\ and Vj, respectively. This theory was implemented in ANSYS with a user-programmable subroutine to calculate stresses in wood in the post-processor. The following algorithm for calculating the volume integral in equation [2-7] was carried out at each load-step in the post-processor to determine when or if failure occurred. At each load step, this algorithm was applied to each of the three normal tensile stresses and each of the three shear stresses. In this way, the governing mode of failure could be determined. The volume integral, \/: [C-1] I=l dx dy dz dn 9n an dx dy dz di\\dr)d(;= Jdt\\dr\\d\u00a3> where [J] is the Jacobian. Combining equations [C-1] and [C-3]: 1 1 1 [C-4] \/= j jja*|j|^r|J<; -1-1-1 in local co-ordinates. The determinant of [J] is equal to the volume of the element in global co-ordinates and is calculated from the nodal co-ordinates and the derivatives of shape functions [N]: [C-5] N,.=i(l\u00b14Xl\u00b1TlXl\u00b1Ofori=l,2,...,8 o 192 For Gaussian integration, [C-4] is re-written in terms of stresses evaluated at each Gauss point, \/, multiplied by the appropriate Gauss weighting values, Wt. From [C-4] we get the approximate Gauss evaluation of the integral: [C-6] \/ = III\u00b0A|J|^^ ' j >\u2022 A 2x2x2 (8-point) integration scheme is used for the SOLID45 elements. The stresses at one Gauss point are determined: [C-7] o = $ > , a , 1 Combining [C-6] and [C-7]: [C-8] JWiWJWr Equation [C-8] is determined for each element and then summed for all elements. The total is equal to the volume integral over the entire model geometry in global co-ordinates.'The operation is repeated for each stress component and k value. Accuracy is dependent on the number of integration points, N, and is ensured if the order of the integral is less than 2N-1. The 2x2x2 point scheme can be used for larger values of k provided that the number of elements is increased in regions of high stress gradients. This algorithm was verified with the following closed form solution for a simply-supported beam with isotropic material properties. Consider the beam shown in Figure C-l. It can be shown that the volume integral with respect to bending stresses, a x , for the beam in Figure C-l is: 1 [C-9] 2bdL (PLd^k (* + l) 2 21 k+\\ A similar integral for beam shear can be found using Gauss integration: f a V \\L - a. [C-10] \\\\\\rkxydV = {TmJ{ba)\\ + II where: [C-lOa] for k= 5.0, and: 77 = 1.3545^ 193 3 PIL-a\\ [C-lOb] x = - \u2014 \u2014 Using equation [2-8], equations [C-9] and [C-10] can be raised to the \\lk power to give the stress related to a unit volume. This value can be compared against the reference stress. A three-dimensional finite element model using isoparametric brick elements was constructed in ANSYS to test the volume integral calculation. The model geometry is shown in Figure C-2. For the loading and dimensions given in Figure C-l , it was found that the finite element solution matched well with the closed form solution from equations [C-9] and [C-10], as shown in Table C-1. Table C-l. Result comparison for verification of volume integral of stresses. Theory ANSYS Ratio Bending Stress equation [C-9] Shear Stress equation [C-10]1 13.1 MPa 13.1 MPa (1900 psi) (1900 psi) 1.19 MPa 1.24 MPa (173 psi) (180 psi) 194 a A b P = 100 kN (22.5 kips) b = 0.1 m (4 inches) d= 0.3 m (12 inches) 1 = 3 m (10 feet) a= 1.5 m(5 feet) I=bd^l\\2 Figure C - l . Simply-supported beam with point load at mid-span. Figure C-2. A N S Y S finite element geometry. 195 ","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/hasType":[{"value":"Thesis\/Dissertation","type":"literal","lang":"en"}],"http:\/\/vivoweb.org\/ontology\/core#dateIssued":[{"value":"2000-11","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/isShownAt":[{"value":"10.14288\/1.0063508","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/language":[{"value":"eng","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeDiscipline":[{"value":"Civil Engineering","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/provider":[{"value":"Vancouver : University of British Columbia Library","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/rights":[{"value":"For non-commercial purposes only, such as research, private study and education. 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