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Strength model and design methods for bending and axial load interaction in timber members Buchanan, Andrew Hamilton 1984

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STRENGTH MODEL AND DESIGN METHODS FOR BENDING AND AXIAL LOAD INTERACTION IN TIMBER MEMBERS by ANDREW HAMILTON BUCHANAN B.E.(Hons), University of Canterbury, New Zealand, 1970 M.S., University of C a l i f o r n i a , Berkeley,'1972 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department Of C i v i l Engineering We accept th i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA January 1984 © Andrew Hamilton Buchanan, 1984 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r a n a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l m a k e i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s m a y b e g r a n t e d b y t h e h e a d o f m y d e p a r t m e n t o r b y h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t b e a l l o w e d w i t h o u t m y w r i t t e n p e r m i s s i o n . D e p a r t m e n t O f C i v i l Engineering  T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a 1 9 5 6 M a i n M a l l V a n c o u v e r , C a n a d a V 6 T 1 Y 3 1 MHr-r-h i i Abstract This thesis describes a model for predicting the strength of timber members in bending, and in combined bending and ax i a l loading, on the basis of a x i a l tension and compression behaviour of similar members. Both i n s t a b i l i t y and material strength f a i l u r e s are included. The model i s based on a s t r e s s - s t r a i n relationship which incorporates d u c t i l e non-linear behaviour in compression, and linear e l a s t i c behaviour associated with b r i t t l e fracture in tension. The model includes the ef f e c t s of v a r i a b i l i t y in timber strength, both within a member and between members. Size e f f e c t s which predict decreasing strength with increasing member size are quantified using separate parameters for member length and member depth. An extensive experimental program on a large number of timber members in str u c t u r a l sizes has been used to cali b r a t e and v e r i f y the model. Test procedures and results are described for members of d i f f e r e n t lengths tested to f a i l u r e in bending and in a x i a l loading. The a x i a l testing included both tension and compression loading using several end e c c e n t r i c i t i e s . Several alternative design methods based on the strength model are investigated and compared with existing methods. Recommendations are made for design methods for timber members subjected to combined bending and a x i a l loading. i i i Table of Contents Abstract i i L i s t of Tables ( v i i i L i s t of Figures ix Notation xv Acknowledgement x v i i i Chapter I INTRODUCTION . .. 1 1 . 1 BACKGROUND 1 1 .2 OBJECTIVES 2 1.3 DESIGN CODES 3 1.4 APPLICATIONS 4 1.5 THESIS ORGANIZATION 5 1.6 LIMITATIONS 6 Chapter II LITERATURE SURVEY . . - 7 2.1 INTRODUCTION 7 2.1.1 Background 7 2.1.2 Clear Wood and Commercial Timber 7 2.2 BENDING STRENGTH ...8 2.2.1 Bending Behaviour of Clear Wood 8 a. D i s t r i b u t i o n of Stresses 8. b. Size E f f e c t s 17 2.2.2 Bending Behaviour of Timber 19 a. Comparison With Clear Wood 19 b. In-grade Testing ....20 c. Derivation of Design Stresses 21 d. Size E f f e c t s 22 2.3 AXIAL TENSION STRENGTH 23 2.3.1 Axial Tension Strength of Clear Wood 23 2.3.2 Axial Tension Strength of Timber .^24 a. Effects of Defects °.24 b. Size E f f e c t s 26 2.4 AXIAL COMPRESSION STRENGTH 27 2.4.1 Axial Compression Strength of Clear Wood 27 2.4.2 Axial Compression Strength of Timber 28 2.4.3 Stress-Strain Relationship 29 2.4.4 Column Theory for Concentric Loading 33 2.4.5 Timber Columns 35 2.5 COMBINED BENDING AND AXIAL LOAD 38 2.5.1 Cross Section Behaviour 38 2.5.2 Members with Combined Bending and Compression ...42 2.5.3 In-Grade Testing 46 2.5.4 Members with Combined Bending and Tension 46 2.5.5 V a r i a b i l i t y in Wood Properties 48 2.6 SUMMARY 49 Chapter III SIZE EFFECTS 50 3.1 INTRODUCTION 50 3.2 CONVENTIONAL BRITTLE FRACTURE THEORY 51 iv 3.2.1 History 51 3.2.2 Applications of B r i t t l e Fracture Theory to Wood .52 3.2.3 Theory for Uniform Stress D i s t r i b u t i o n 54 3.2.4 Theory for Variable Stress D i s t r i b u t i o n 56 3.2.5 Coefficient of Variation 58 3.3 BRITTLE FRACTURE THEORY MODIFIED FOR TIMBER 59 3.4 DIFFERENT SIZE EFFECTS IN DIFFERENT DIRECTIONS 60 3.4.1 Size Effect Terminology 61 3.5 LENGTH EFFECT 62 3.5.1 Theory 62 3.5.2 Assumptions on Length Effect 63 3.6 DEPTH EFFECT 65 3.7 STRESS-DISTRIBUTION EFFECT 66 3.7.1 Tension Zone 66 3.7.2 Compression Zone 70 3.8 WIDTH EFFECT 71 3.9 EFFECT OF GRADING RULES 7 4 3.10 SUMMARY 75 Chapter IV EXPERIMENTAL PROCEDURES 76 4 .1 EXPERIMENTAL STAGES 76 4.2 TEST MATERIAL 79 4.2.1 Species 79 4.2.2 Grading 79 4.2.3 Moisture Content 80 4.2.4 Sample Selection 81 4.3 SAMPLE SIZES AND CONFIDENCE 82 4.3.1 Sample Sizes 82 4.3.2 Weibull D i s t r i b u t i o n 83 4.3.3 Confidence Calculation for Quantiles 83 4.4 TEST PROCEDURES 87 4.4.1 Bending 87 4.4.2 Axial Tension 89 a. Long' Boards 89 b. Short Boards 90 4.4.3 Axial Compression 91 a. Long Boards 91 b. Short Segments 92 4.4.4 Eccentric Compression 92 4.4.5 Combined Bending and Tension 94 a. Bending Followed by Tension 94 b. Eccentric Tension .....95 4.4.6 Data Aquisition 96 4.4.7 Modulus of E l a s t i c i t y 96 4.5 SUMMARY 98 Chapter V EXPERIMENTAL RESULTS 99 5.1 COMBINED BENDING AND AXIAL LOADING RESULTS 99 5.1.1 Presentation 99 5.1.2 Interaction Curves for Short Members 100 a. Test Results 1 00 b. Mode of Failure 106 V 5.1.3 Interaction Curves for Long Members 107 5.1.4 Axial•Load-slenderness Curves 107 5.2 SEPARATE BENDING AND AXIAL LOADING RESULTS 110 5.2.1 Test Results 110 5.2.2 Modes of Failure 112 5.3 LENGTH EFFECTS 114 5.3.1 Introduction 114 5.3.2 Compression Strength 115 5.3.3 Tension Strength 116 a. 38x89mm Boards 116 b. 38x140mm Boards 117 5.3.4 Bending Strength 118 a. 38x89mm Boards 119 b. 38x140mm Boards 120 5.3.5 Summary of Length Ef f e c t s 122 5.4 WEAK AXIS BENDING 123 5.5 SUMMARY 124 Chapter VI STRENGTH MODEL 126 6.1 INTRODUCTION 126 6.2 ASSUMPTIONS 127 6.3 CROSS SECTION BEHAVIOUR .127 6.3.1 Calculation Procedure 128 6.3.2 Neutral Axis Contours 131 6.3.3 Curvature Contours 132 6.3.4 Ultimate Interaction Diagram 134 6.3.5 Moment-Curvature Curves 134 6.4 COLUMN BEHAVIOUR 135 6.4.1 Failur e Modes 136 6.4.2 Calculation Procedure 139 6.5 TYPICAL OUTPUT 142 6.5.1 Axial Load-Moment Interaction Curves 142 6.5.2 Axial Load-Slenderness Curves 144 6.6 INPUT INFORMATION 145 6.6.1 Cross Section Dimensions 146 6.6.2 Tension and Compression Strengths 146 6.6.3 Modulus of E l a s t i c i t y 148 6.6.4 Shape of Stress-Strain Relationship in Compression 149 a. Shape of F a l l i n g Branch ;.151 b. Shape of Rising Branch 154 6.6.5 St r e s s - d i s t r i b u t i o n E f f e c t 156 6.6.6 Column Parameters 158 6.7 NON-DIMENSIONALI ZED PLOTS 158 6.8 SUMMARY 159 Chapter VII CALIBRATION AND VERIFICATION 160 7.1 INTRODUCTION 160 7.2 38x89mm BOARDS 161 7.2.1 Short Column Interaction Curves 161 7.2.2 Parameter Estimation 164 7.2.3 Long Column Interaction Curve for End Moments ..165 v i . 7.2.4 Long Column Interaction Curves for Mid-Span Moments 168 7.2.5 Axial Load - Slenderness Curves 170 7.3 38x140mm BOARDS 170 7.3.1 Short Column Interaction Curves 170 7.3.2 Long Column Interaction Curves 173 7.3.3 Axial Load-Slenderness Curves 173 7.4 REPRESENTATIVE STRENGTH PROPERTIES 173 7.5 APPLICABILITY OF STRENGTH MODEL ,.181 7.6 SUMMARY .181 Chapter VIII DESIGN METHODS FOR COLUMNS AND BEAM COLUMNS 183 8.1 INTRODUCTION ...183 8.1.1 Allowable Stress Design 183 8.1.2 Reliabi l i t y - B a s e d Design 183 8.1.3 Reliability-Based Design of Timber 185 8.1.4 Scope 186 8.2 EXISTING DESIGN METHDOS 186 8.2.1 Canadian Timber Code 186 a. Concentrically Loaded Columns 187 b. Combined Axial Load and Bending 188 c . Summary 190 8.2.2 NFPA Timber Code 191 a. Concentrically Loaded Columns 191 b. Combined Axial Load and Bending .191 c . Summary 193 8.2.3 Code Requirements for Steel 194 8.2.4 Canadian Concrete Code 198 8.2.5 Limit States Timber Codes 201 8.3 COLUMN CURVES FOR CONCENTRIC LOADING 205 8.4 PROPOSED DESIGN METHODS FOR ECCENTRICALLY LOADED COLUMNS 208 8.4.1 Type of Loading and Analysis 209 8.4.2 Input Strength Properties ...210 8.4.3 Design Approaches 210 8.4.4 Moment Magnification Factor 212 8.4.5 METHOD 1: B i l i n e a r Interaction Diagram 215 8.4.6 METHOD 2: Parabolic Interaction Diagram 218 8.4.7 METHOD 3: Ultimate Interaction Diagram 220 8.4.8 Comparison of Methods 1 to 3 221 8.4.9 METHOD 4: Published Design Curves 224 8.4.10 METHOD 5: Straight Line Approximation 225 8.4.11 METHOD 6: Axial Load-Slenderness Curves 227 8.5 COMBINED BENDING AND TENSION 231 8.6 SUMMARY 234 Chapter IX DESIGN RECOMMENDATIONS 235 9.1 STRUCTURAL ANALYSIS ...235 9.1.1 Strength Model 235 9.1.2 Second Order Structural Analysis 236 9.1.3 Simple Analysis 237 9.1.4 Code Format 237 v i i 9.2 APPROXIMATE DESIGN FORMULAE 238 9.2.1 Recommended Formulae 238 9.2.2 Example 240 9.2.3 Load Factors and Resistance Factors 240 9.2.4 Mimimum Moments .....241 9.3 DATA REQUIRED 241 9.3.1 In-grade Test Results 241 9.3.2 Size Effects 243 a. Length Ef f e c t s 244 b. Depth Effects 245 9.4 OTHER LOADING CASES . .. 245 9.4.1 Unequal End E c c e n t r i c i t i e s 246 9.4.2 Transverse Loads 246 9.4.3 Slenderness Ratio 248 9.4.4 Bi a x i a l Behaviour 249 a. Strength Under B i a x i a l Loading 250 b. S t a b i l i t y Under B i a x i a l Loading 251 9.5 LONG DURATION LOADING ...252 9.5.1 Strength Under Long Duration Loading 252 9.5.2 S t a b i l i t y Under Long Duration Loading ..253 9.6 MOISTURE CONTENT 254 9.6.1 Effect of Moisture on Compression Strength 254 9.6.2 Effect of Moisture on Strength Under Combined Loading 255 9.7 SUMMARY • 2 56 Chapter X SUMMARY 258 LITERATURE CITED 261 APPENDIX A - CALCULATIONS FOR SPECIAL CASES 272 A.1 ULTIMATE BENDING STRENGTH 272 A. 1.1 Background 272 A. 1.2 Bi l i n e a r -Stress-Strain Relationship with F a l l i n g Branch 273 a. Assumptions ...273 b. Calculations 274 c. Length E f f e c t s 278 d. Summary 279 e. Depth Effect in Compression 280 A.1.3 E l a s t o - P l a s t i c Stress-Strain Relationship 280 a. Assumptions 280 b. Calculations 281 A.2 ULTIMATE INTERACTION DIAGRAM 283 A.2.1 Background 283 A.2.2 Calculations 283 APPENDIX B - CALCULATION OF -INTERCEPT DEFINING INTERACTION DIAGRAM 288 APPENDIX C - TEST RESULTS 290 APPENDIX D - NOTATION 299 v i i i L i s t of Tables I. Summary of experimental stages 77 I I . Summary of lengths and sizes tested (stages 1 to 4) 78 II I . Material property ratios for representative groups 179 IV. Length effect parameter k, 244 V. Summary of equations for cal c u l a t i n g ultimate interaction diagram 286 ix L i s t of Figures 1. Actual and assumed stresses in a clear wood beam at f a i l u r e 9 2. Stress d i s t r i b u t i o n derived from a b i l i n e a r stress-s t r a i n relationship ..11 3. Stress d i s t r i b u t i o n assumed by Bechtel and Norris ....12 4. Stress d i s t r i b u t i o n s measured by Ramos 14 5. Stress d i s t r i b u t i o n proposed by Moe 15 6. Stress d i s t r i b u t i o n proposed by Bazan 17 7. Stress-strain relationships in a x i a l compression 30 8. Axial load-slenderness curve for concentrically loaded columns 36 9. Ultimate interaction diagrams for linear behaviour 39 10. Ultimate interaction diagrams for non-linear behaviour 40 11. Three-dimensional sketch of load vs. slenderness vs. moment 43 12. Typical log-log plot of f a i l u r e stress vs. volume ....56 13. Tension stress d i s t r i b u t i o n s 67 14. Ratio of a x i a l tension strength to maximum stress in extreme f i b r e 69 15. Compression stress d i s t r i b u t i o n s 71 16. Quarter-sawn board 73 17. P.d.f. of quantile estimators 86 18. Loading arrangement for bending test 88 19. Loading arrangement for ax i a l tests (a) tension (b) compression 90 20. Eccentric a x i a l loading (a) Compression (b) Tension 93 X 21. Combined bending and tension test (a) f i r s t stage (b) second stage 95 22. Test results for shortest length in eccentric compression 101 23. Percentile results for shortest length in eccentric compression 102 24. Test results for 38x89mm boards in eccentric compression and tension 104 25. Interaction diagram for mean test results of a l l lengths in eccentric compression 108 26. Axial load-slenderness curves. Mean test results for a l l lengths tested in eccentric compression 109 27. Comparison of tension, compression and bending test results 111 28. Ratio of tension strengths of 38x89mm boards 2.0m and 0.914m long 117 29. Ratio of tension strengths of 38x140mm boards 3.0 and 0.914m long. 118 30. Ratio of bending strengths of 38x89mm boards • 1.5 and 0.84m long .120 31. Ratio of bending strengths of 38x140mm boards, 3.0 and 1 . 5m long 121 32. Ratio of edgewise to flatwise bending strength 124 33. Cross section behaviour 128 34. Flow chart for ca l c u l a t i n g moment-curvature-axial load relationships for a cross section 130 35. Neutral axis contours for moment and axi a l load interaction 132 36. Curvature contours for moment and ax i a l load interaction 133 37. Ultimate interaction diagram for strength of cross section 133 38. Moment-curvature-axial load relationships 135 39. Column with a x i a l load and equal end e c c e n t r i c i t i e s .136 40. Typical interaction diagram for e c c e n t r i c a l l y loaded xi column 137 41. Column deflection curves 140 42. Interaction diagrams for slender columns 143 43. A x i a l load-slenderness diagram for several e c c e n t r i c i t i e s 145 44. Ultimate interaction diagrams for strength representative of several lengths 147 45. E f f e c t of modulus of e l a s t i c i t y on column behaviour .149 46. Stress-strain relationships in compression 150 47. Ultimate interaction diagrams for the b i l i n e a r stress-s t r a i n relationship, with varying slope of f a l l i n g branch 152 48. Ultimate interaction diagrams for the e l a s t o - p l a s t i c s t r e s s - s t r a i n relationship, with varying l i m i t i n g s t r a i n .152 49. Ultimate interaction diagram resulting from stress-s t r a i n relationship proposed by G1OS(1978) .. 153 50. Ultimate interaction diagrams for several stress-s t r a i n relationships 153 51. Stress-strain r e l a t i o n s h i p with f a l l i n g branch 155 52. Ultimate interaction diagrams with varying stress-d i s t r i b u t i o n parameter in tension 157 53. Ultimate interaction diagrams with varying stress-d i s t r i b u t i o n parameter in compression 157 54. Ultimate interaction diagram ca l i b r a t e d to test results for 38x89mm size 162 55. Predicted interaction diagram for end moments compared with test results for 38x89mm size 166 56. Predicted interaction diagram for mid-span moments compared with test results for 38x89mm size 169 57. Predicted a x i a l load - slenderness curves compared with test results for 38x89mm size 171 58. Ultimate interaction diagram calib r a t e d to test results for 38x140mm size 172 59. Predicted interaction diagram for end moments compared xi i with test results for 38x140mm size 174 60. Predicted interaction diagram for mid-span moments compared with test results for 38x140mm size 175 61. Predicted a x i a l load - slenderness curves compared with test results for 38x140mm size 176 62. Ultimate interaction diagrams for representative strength properties 178 63. Non-dimensionalized interaction diagrams for representative strength properties 180 64. Axial load-slenderness curves for representative strength properties (non-dimensionalized) .....180 65. Code column formula compared with model prediction (non-dimensionalized) 1 8 9 66. NFPA formula compared with model prediction (non-dimensionalized) 193 67. Axial load-moment interaction diagram for steel members 195 68. Axial load-moment interaction diagram from OHBDC ....202 69. OHBDC formula compared.with model prediction (non-dimensionalized) 204 70. Comparison of column curves for concentric loading (non-dimensionalized) 206 71. Design proposal for concentric loading compared with model prediction 209 72. Typical interaction diagram for a x i a l load and magnified moment 212 73. Interaction diagram showing t r a d i t i o n a l moment magnifier 214 74. Interaction diagram showing proposed moment magnifier 216 75. B i l i n e a r approximation to interaction diagram for magnified moments (non-dimensionalized) 217 76. Parabolic approximation to interaction diagram for magnified moments (non-dimensionalized) 219 77. Comparison of methods 1 to 3 with model prediction (non-dimensionalized) 222 xi i i 78. Straight l i n e approximations to interaction diagrams for unmagnified moment (non-dimensionalized) 226 79. Comparison of methods 1 and 5 with model prediction (non-dimensionalized) 228 80. Modified secant formula compared with model prediction 230 81. Parabolic approximation to interaction diagram 233 82. Bending moment diagrams for combined a x i a l and transverse loading 247 83. Bending about inc l i n e d neutral axis 250 84. Effect of moisture content on strength of timber in combined bending and a x i a l loading 256 85. B i l i n e a r s t r e s s - s t r a i n relationship with f a l l i n g branch 273 86. D i s t r i b u t i o n of stress and st r a i n in a rectangular beam assuming b i l i n e a r s t r e s s - s t r a i n relationship ...274 .87. Flow chart for cal c u l a t i n g ultimate bending moment for bi l i n e a r s t r e s s - s t r a i n relationship 279 88. E l a s t o - p l a s t i c s t r e s s s - s t r a i n relationship 281 89. D i s t r i b u t i o n of stress and s t r a i n in a rectangular beam assuming e l a s t o - p l a s t i c s t r e s s - s t r a i n relationship • 281 90. Flow chart for cal c u l a t i n g ultimate bending moment for el a s t o - p l a s t i c s t r e s s - s t r a i n relationship 284 9 1 . Stress d i s t r i b u t i o n s for nine di f f e r e n t combinations of a x i a l load and bending moment 285 92. Ultimate interaction diagram produced by hand calc u l a t i o n s , compared with computer calculated curve 287 93. Typical ultimate interaction diagram 288 94. Eccentric compression res u l t s , 38x89mm boards, 1.3m long 290 95. Eccentric compression results, 38x89mm boards, 1.8m long 290 96. Eccentric compression results, 38x89mm boards, 2.3m long 291 xiv 97. Eccentric compression results, 38x89mm boards, 3.2m long 291 98. Eccentric compression results, 38x1 40mm boards, 1.82m long ...292 99. Eccentric compression r e s u l t s , 38x140mm boards, 2.44m long 292 100. Eccentric compression res u l t s , 38x140mm boards, 3.35m long 293 101. Eccentric compression results, 38x1 40mm boards, 4.27m long 293 102. Axial tension r e s u l t s , 38x89mm boards, 2.0m long 294 103. Axial tension re s u l t s , 38x89mm boards, .914m long 294 104. Axial compression res u l t s , 38x89mm boards, 2.0m long 295 105. Bending test r e s u l t s , 38x89mm boards, 1.5m span 295 106. Bending test r e s u l t s , 38x89mm boards, ,84m span (edgewise) 296 107. Bending test r e s u l t s , 38x89mm boards, .84m span (flatwise) 296 108. Axial tension re s u l t s , 38x1 40mm boards, 3.0m long 297 109. Axial compression res u l t s , 38x140mm boards, 3.0m long 297 110. Bending test r e s u l t s , 38x140mm boards, 3.0m long 298 X V Notation a distance between loads on beam a r a t i o of depth A cross section area A r a t i o of magnification factors b r a t i o of cross section depth B axis intercept for interaction diagram c parameter in s t r e s s - s t r a i n equation c r a t i o of cross section depth C force in compression stress block Cm equivalent moment factor d depth of cross section subscript 1 reference depth e e c c e n t r i c i t y e st r a i n subscript stress subscripts 0 compression s t r a i n at e l a s t i c f a i l u r e 1 compression s t r a i n at peak stress t tension s t r a i n u upper l i m i t on compression str a i n y y i e l d s t r a i n in compression a allowable compression stress in long column b allowable bending stress c compression stress at f a i l u r e for short column ca allowable compression stress in short column cm compression stress at f a i l u r e modified for s t r e s s - d i s t r i b u t i o n effect e Euler buckling stress m f a i l u r e stress of extreme f i b r e in tension r modulus of rupture s asymptotic stress for large s t r a i n t f a i l u r e stress in a x i a l tension test ta allowable tension stress u compression stress at f a i l u r e for long column x extreme f i b r e stress in tension zone xvi F magnification factor G parameter in s t r e s s - s t r a i n equation I moment of i n e r t i a of cross section J r a t i o of slenderness ra t i o s k shape parameter for Weibull d i s t r i b u t i o n k size e f f e c t parameter subscripts 1 length e f f e c t parameter 2 depth ef f e c t parameter 3 s t r e s s - d i s t r i b u t i o n parameter K minimum slenderness r a t i o for long columns L length of member subscript e e f f e c t i v e length m ra t i o of slopes of s t r e s s - s t r a i n curves m scale parameter for Weibull d i s t r i b u t i o n M bending moment subscripts 1 larger end moment 2 smaller end moment u ultimate bending moment n parameter in s t r e s s - s t r a i n equation n sample size n r a t i o of extreme fibr e stresses in tension and compression p confidence P a x i a l compression force subscripts a compression capacity of short column e Euler buckling load u compression capacity of long column q quantile r " confidence i n t e r v a l r parameter in s t r e s s - s t r a i n equation r radius of gyration r r a t i o of stresses S section modulus xvi i T a x i a l t e n s i o n f o r c e s u b s c r i p t u t e n s i o n c a p a c i t y of member v d e f l e c t i o n v' s l o p e dy/dx v" d 2 y / d x 2 V volume of member s u b s c r i p t s 1 r e f e r e n c e volume e e q u i v a l e n t volume w w i d t h of c r o s s s e c t i o n x d i s t a n c e a l o n g member x s t r e s s s u b s c r i p t s o l o c a t i o n parameter i n w e i b u l l d i s t r i b u t i o n q v a l u e of x a t q u a n t i l e q y c o - o r d i n a t e z s t a n d a r d normal v a r i a b l e A d e f l e c t i o n of member <t> c a p a c i t y r e d u c t i o n f a c t o r 4> c u r v a t u r e xvi i i Many people have contributed to the success of t h i s thesis. My supervisor, Professor Borg Madsen, offered enthusiastic advice and encouragement throughout, and I am very proud to have been associated with him in t h i s work. Dr. Ken Johns, of the University of Sherbrooke, i n i t i a t e d part of the project and encouraged me to pursue i t . His co-operative research at Sherbrooke, with the assistance of Mr. Raymond Bleau, was a major contribution. Dr. Noel Nathan provided a computer program, and made many useful suggestions. Others at the University of B r i t i s h Columbia, including faculty members, technicians, and fellow students, helped in many ways. The Forintek Corp. made f a c i l i t i e s available, and Dr. Ricardo Foschi gave much useful advice. Fi n a n c i a l assistance was received from the Izaak Walton Killam Trust, the Programme of D i s t i n c t i o n in Timber Engineering at UBC, and the Natural Sciences and Engineering Research Council of Canada. The contributions of a l l the above, and others not mentioned, are g r a t e f u l l y acknowledged. My greatest thanks go to my wife Elsa, for her typing, and for her support and understanding without which this project would not have been possible. 1 I . INTRODUCTION 1 . 1 BACKGROUND Wood i s a complex organic material formed in trees. Wood is put to a large number of uses, an important one being the production of sawn timber for construction of buildings, bridges and other structures. Engineering design of timber structures requires knowledge of basic engineering properties of the material. In contrast to manufactured materials such as concrete and metals, timber has been the subject of very l i t t l e research. Timber research i s fundamentally d i f f e r e n t from that of manufactured materials because wood properties are much more variable and cannot be readily modified in a manufacturing process. Allowable stresses for structural design of timber members have t r a d i t i o n a l l y been derived from standard tests on small specimens of clear defect-free wood. Most timber used in s t r u c t u r a l engineering applications contains natural or man-made defects which may include knots, sloping grain, p i t h , checks or machining defects. These defects generally govern the strength of the material, and the f a i l u r e mode i s therefore d i f f e r e n t . Allowable stresses derived from small clear specimen tests are adjusted for the ef f e c t s of defects, but the results do not accurately r e f l e c t the strength of st r u c t u r a l timber. In recent years, design stresses for timber have been obtained ' more d i r e c t l y from f u l l size tests of commercially 2 available sizes, species and grades. This testing, c a l l e d "in-grade t e s t i n g " , has provided a great improvement in the understanding of timber as a structural engineering material, but this understanding i s by no means complete. Most in-grade testing has concentrated on simple span bending tests at a constant span-to-depth r a t i o , and some tension and compression tests at constant length. There have been no serious attempts to relate bending strength to tension and compression strengths. There , has been a conspicuous absence of information on strengths of d i f f e r e n t lengths and di f f e r e n t load configurations, and strength of members subjected to combined bending and a x i a l loading. This study attempts to improve the theoreti c a l basis of the in-grade testing concept by investigating these subjects. 1 . 2 OBJECTIVES The i n i t i a l objectives of this study were to investigate the behaviour of structural timber members subjected to combined bending and a x i a l loading, and to examine the hypothesis that the behaviour of timber members subjected to bending, or to combined bending and a x i a l loading, can be explained in terms of the a x i a l tension and compression strengths of similar members. This hypothesis has been long accepted for other materials, but i t s a p p l i c a b i l i t y to commercial qu a l i t y timber has not yet been c l e a r l y demonstrated. During the couse of this investigation i t became apparent that size e f f e c t s have a major influence on the . strength of 3 timber members. An additional objective, therefore, was to examine and quantify these size e f f e c t s . These objectives have been met by developing a strength model which can predict the strength of timber members in bending, and in combined bending and a x i a l loading. The model incorporates size e f f e c t s , and has been calib r a t e d and v e r i f i e d using the results of a large experimental program. The main input to the strength model i s a x i a l tension and compression strength from in-grade t e s t i n g . Input at any l e v e l in a d i s t r i b u t i o n of strength can be used to predict behaviour at that l e v e l . As such, the model i s not intended to predict strength of an individual board of timber. 1.3 DESIGN CODES Structural design has t r a d i t i o n a l l y been based on prescribed allowable stresses which are not to be exceeded when working loads are imposed on a structure. More recently, designers have become interested in the ultimate strength of structures and behaviour under maximum possible loads. Recent trends are toward r e l i a b i l i t y - b a s e d s t r u c t u r a l design which considers p r o b a b i l i t i e s of f a i l u r e under extreme loading conditions. This study contributes to the development of r e l i a b i l i t y -based design of timber structures by providing new information on the d i s t r i b u t i o n of. strength properties of timber members subjected to combined bending and a x i a l loading. The results of t h i s study can also be incorporated into conventional working stress design codes to provide improved design 4 methods. Current design codes consider the strength of timber to be independent of member length. This study shows that strength varies considerably with length, and a simple method of quantifying length effects i s proposed. 1.4 APPLICATIONS The results of t h i s study demonstrate that current design methods for combined bending and a x i a l loading are very conservative in some cases. This suggests that more e f f i c i e n t and economical designs may be possible for timber structures where combined loading occurs. The largest group of structural components in t h i s category are timber trusses, which, according to some estimates (Gromala and Moody 1983), are used in 95% of vnew domestic construction in the United States, and which are being used increasingly in larger commercial and i n d u s t r i a l structures. Other structural components with combined bending and a x i a l loading are wall studs and columns in many types of buildings. Recent research in timber truss design has been handicapped by a lack of understanding of the behaviour of timber members subjected to combined bending and a x i a l loading (FPRS 1979). 5 1.5 THESIS ORGANIZATION Chapter 2 describes previous attempts to understand the behaviour of both clear wood and timber members subjected to various types of loads. Chapter 3 discusses size e f f e c t s . One of the main obstacles encountered by previous researchers i s the phenomenon that the strength of timber i s s i g n i f i c a n t l y dependent on member si z e . The f i r s t part of Chapter 3 i s a survey of previous studies. In the second part, ex i s t i n g theories are modified and simple equations are derived for lat e r use. Chapters 4 and 5 describe experiments performed to investigate the strength of timber members subjected to bending and a x i a l loads, both separately and combined. The results of these experiments are used later to c a l i b r a t e and v e r i f y a strength model. The length e f f e c t theory introduced in Chapter 3 i s quantified in Chapter 5. Chapter 6 describes a conceptual model for predicting the strength of populations of str u c t u r a l timber members on the basis of tension and compression properties which can be obtained from in-grade t e s t i n g . The model includes the effects of v a r i a b i l i t y in timber strength, both within a member and between members. The model i s based on the i n i t i a l hypothesis that strength in bending, or in combined bending and a x i a l loading, can be explained in terms of ax i a l tension and compression properties. In Chapter 7 the strength model i s calibrated using 6 selected experimental r e s u l t s , then tested against many other • experimental r e s u l t s . The comparisons of predicted and measured results support the i n i t i a l hypothesis and demonstrate that t h i s strength model can be used to predict the behaviour of e c c e n t r i c a l l y loaded members of many lengths and e c c e n t r i c i t i e s . Chapters 8 and 9 are concerned with applications, describing existing design methods for various materials, and recommending how the results of t h i s study can be used as a basis for improved structural design of timber members. 1.6 LIMITATIONS The model developed in t h i s thesis i s a f l e x i b l e one, which i s believed to be applicable to timber from any source* However, i t has been ca l i b r a t e d with reference to tests of members of a single species group in two sizes of cross section. Application to other sizes, species and grades may require further c a l i b r a t i o n . The a n a l y t i c a l results of thi s study have only been v e r i f i e d with reference to to short duration tests on dry timber members, loaded with eccentric a x i a l loads applied with equal end e c c e n t r i c i t i e s about the major p r i n c i p a l axis and restrained against out-of-plane deformations. The significance of these l i m i t a t i o n s should be kept in mind before extrapolating the results to other s i t u a t i o n s . 7 I I . LITERATURE SURVEY 2.1 INTRODUCTION 2.1.1 Background This chapter provides a brief h i s t o r i c a l review of investigations into the strength of structural timber members subjected to bending, a x i a l loads and combined load, and describes the state-of-the-art at the commencement of thi s study. Many of the findings referred to in thi s chapter are used later in the thesis when developing a general model for the behaviour of str u c t u r a l timber members. The f i r s t part of the chapter chronicles the d i f f i c u l t i e s that have been encountered in trying to predict bending behaviour of wood and timber members from tension and' compression test r e s u l t s . This process i s much more d i f f i c u l t for wood than for other materials because of material v a r i a b i l i t y , non-linear s t r e s s - s t r a i n behaviour in compression, and the presence of very s i g n i f i c a n t size e f f e c t s . Investigations into .axial tension and a x i a l compression behaviour are described, and the discussion extended to include columns and beam-columns. 2.1.2 Clear Wood and Commercial Timber This thesis i s concerned with the strength of commercial quality sawn timber, containing natural or man-made defects. This material (which in certain sizes is c a l l e d "lumber" in North America) i s referred to as timber throughout this 8 t h e s i s . The term wood generally refers to clear defect-free wood, as often tested in small sizes in standard tests. Timber and clear wood behave quite d i f f e r e n t l y under most loading conditions. For example, the r a t i o of -tension to compression strength i s much greater in clear wood than in commercial timber, so clear wood tends to behave in a du c t i l e manner in a bending test, whereas commercial timber f a i l s with a sudden b r i t t l e fracture. For many years i t was believed that a complete understanding of clear wood behaviour would lead e a s i l y to an understanding of timber strength, but thi s has not developed. Emphasis has shifted to dire c t investigation of strength properties of timber, leading to the development of "in-grade" testing of commercial quality material in structural s i z e s . Investigations into the behaviour of both clear wood and timber are referenced in this chapter, and are discussed as two separate materials. 2.2 BENDING STRENGTH 2.2.1 Bending Behaviour of Clear Wood a. D i s t r i b u t i o n of Stresses In 1638 Gal i l e o recorded his attempts to describe . the d i s t r i b u t i o n of f l e x u r a l stresses in a bending member. His theory was in error because he assumed that the neutral axis was on the compression surface. Parent obtained the correct d i s t r i b u t i o n of stresses for an e l a s t i c beam in 1708, but i t was not u n t i l Coulomb confirmed these findings in 1773 that 9 they became generally accepted (Booth 1980). From that time u n t i l the present, most engineering design of timber beams has assumed linear e l a s t i c behaviour to f a i l u r e , with the neutral axis at mid-depth of the section. Early tests in bending indicated that wood does not always behave in t h i s manner. A bending test on a t y p i c a l defect-free wood member shows linear e l a s t i c behaviour up to a proportional l i m i t , beyond which compression y i e l d i n g occurs in the compression zone, the neutral axis s h i f t s towards the tension face, and with further loading the tension stresses continue to increase u n t i l a b r i t t l e tension f a i l u r e occurs. The assumed and actual stresses at f a i l u r e are shown in f igure 1. As early as 1841 Joseph Colthurst noted that for a bending test with f i r battens "extension and compression were equal up to three-quarters of the breaking load, but after t h i s compression yielded in a much higher r a t i o than extension". (Todhunter and Pearson 1886) Compression ^ Tension Figure 1 - Actual and assumed stresses in a clear wood beam at f a i l u r e 10 The t r a d i t i o n a l method of c a l c u l a t i n g t h e maximum s t r e s s i n a t i m b e r beam a t f a i l u r e has been t o measure t h e f a i l u r e l o a d , c a l c u l a t e t h e c o r r e s p o n d i n g b e n d i n g moment, and c a l c u l a t e a "modulus o f r u p t u r e " by d i v i d i n g t h e maximum b e n d i n g moment by t h e s e c t i o n m o d u l u s . F o r l i n e a r e l a s t i c b e h a v i o u r t h e modulus of r u p t u r e i s t h e maximum s t r e s s . I f any n o n - l i n e a r b e h a v i o u r o c c u r s t h e modulus o f r u p t u r e i s not t h e s t r e s s a t f a i l u r e , but m e r e l y a f i c t i c i o u s s t r e s s somehow r e l a t e d t o s t r e n g t h . T h i s r e l a t i o n s h i p between t h e modulus of r u p t u r e and s t r e n g t h i s n o t e a s i l y d e f i n e d , as i t i s a f u n c t i o n o f t h e non l i n e a r c o m p r e s s i o n b e h a v i o u r of t h e wood and t h e r a t i o between t e n s i o n and c o m p r e s s i o n s t r e n g t h s , w h i c h w i l l be e x p l a i n e d f u r t h e r i n t h i s t h e s i s O b s e r v a t i o n s of f l e x u r a l f a i l u r e s made i t c l e a r t h a t a s f a i l u r e l o a d s a r e a p p r o a c h e d , t h e l o n g i t u d i n a l f l e x u r a l s t r e s s e s a r e not l i n e a r l y d i s t r i b u t e d o v e r t h e c r o s s s e c t i o n . A s i m p l e m o d i f i c a t i o n t o e l a s t i c t h e o r y c an be made by a s s u m i n g t h a t wood has a b i l i n e a r e l a s t o - p l a s t i c s t r e s s - s t r a i n r e l a t i o n s h i p i n c o m p r e s s i o n , but r e m a i n s e l a s t i c i n t e n s i o n . T h i s m o d el, w h i c h p r o d u c e s a s t r e s s d i s t r i b u t i o n of t h e form shown i n f i g u r e 2 i s a r e a s o n a b l e a p p r o x i m a t i o n of a c t u a l b e h a v i o u r . T h i s d i s t r i b u t i o n o f s t r e s s e s i n a wood member a p p e a r s t o have been f i r s t p r o p o s e d by N e e l y ( ] 8 9 8 ) who^ , showed r e m a r k a b l e i n s i g h t i n t o t h e b e h a v i o u r of wood beams. He r e c o g n i z e d t h a t a b i l i n e a r e l a s t o - p l a s t i c s t r e s s - s t r a i n r e l a t i o n s h i p i n c o m p r e s s i o n c a n be u s e d t o c o n s t r u c t t h e c o m p r e s s i o n s t r e s s b l o c k shown i n f i g u r e 2. F o r c l e a r wood, 11 l/l in (a) stress - strain relationship (b) s t r e s s distribution Figure 2 - Stress d i s t r i b u t i o n derived from a b i l i n e a r s t r e s s - s t r a i n relationship which i s much stronger in tension than in compression, he suggested that f l e x u r a l capacity can be calculated from compression strength alone.' This study w i l l show that Neely's theory needs some modification for application to timber, but i t i s surprising how l i t t l e attention has been given to his work. Dietz (1942) tested f i v e beams of clear Douglas-fir to investigate the d i s t r i b u t i o n of stresses and str a i n s , and to compare the behaviour of the same material under dire c t a x i a l stresses in tension and compression. He was able to use the results of a x i a l tests to predict the bending strength of his test specimens su r p r i s i n g l y well compared with later investigators. Measured d i s t r i b u t i o n s of stra i n over the cross sections indicated some deviation from the usual assumption that plane sections remain plane. The measured st r a i n gradient was non-li n e a r , with larger strains near the tension face and smaller strains near the the compression face than would be predicted 12 for plane sections remaining plane. This phenomenon has not been reported by any later investigators. Dietz reported that the outermost compressive f i b r e in a beam possesses a much higher proportional l i m i t than does a similar f i b r e in a block subjected to a x i a l compression. This result has been reported elsewhere, as described below, but has not been found by a l l investigators. Bechtel and Norris (1952) car r i e d out a number of tests on small clear Sitka spruce beams using bending, tension, compression and shear specimens from each piece of wood. The compression tests produced a s t r e s s - s t r a i n curve of the shape shown by the s o l i d l i n e in figure 3(a). (a) s t r e s s - s t r a i n relationship (b) s t r e s s distribution Figure 3 - Stress d i s t r i b u t i o n assumed by Bechtel and Norris They used the simplifying assumption of perfect elasto-p l a s t i c behaviour shown by the dotted l i n e to obtain a d i s t r i b u t i o n of stresses as shown in figure 3(b). They then calculated the theoreti c a l f l e x u r a l strength of the beam using 13 th i s d i s t r i b u t i o n of stresses, material properties from the small clear tests, and a c r i t e r i o n of f a i l u r e under combined shear and normal stresses proposed by Norris(1955). As expected, shear strength governed the behaviour for short deep beams, fl e x u r a l strength for long shallow beams, and combined f l e x u r a l and shear stresses for intermediate beams. For the beams where shear f a i l u r e s were s i g n i f i c a n t (span-to-depth r a t i o less than about 10), the theory gave fle x u r a l strengths less than 5% over the measured strength. For beams governed by tension strength the theory overestimated the measured strength by 15% to 34%. The error was attributed to the s i m p l i f i c a t i o n made in figure 3(a), but the largest source of error may. well have been a b r i t t l e fracture size effect which was not recognized at that time. Size e f f e c t s result in a reduction of the f a i l u r e stress as the highly stressed volume of a member is increased. Comben(l957) carr i e d out a series of tests on 80mm deep beams of clear wood with s t r a i n gauges throughput the beam depth, and he also made tension and compression tests on wood from the same members. His findings were that plane sections remain e s s e n t i a l l y plane, wood in tension remains linear e l a s t i c u n t i l f a i l u r e , and compression stresses are the same at the proportional l i m i t for 1 both the compression specimens and for the extreme compression fibres in the bending specimen. These findings are not in agreement with those of Dietz. In a x i a l tension tests he found linear e l a s t i c behaviour 14 to f a i l u r e , as reported by most other investgators. Comben also tested beams of six d i f f e r e n t sizes and depths varying from 5mm to 80mm at constant span to depth r a t i o . He found a very s i g n i f i c a n t reduction in f a i l u r e stress with increasing s i z e . A study of three clear Douglas f i r beams was made by Ramos(l96l) with s t r a i n measurements over the depth. These experiments also confirmed that plane sections remain plane, and that the compression stress block in bending can be predicted from the a x i a l compression s t r e s s - s t r a i n relationship. He also noted that v i s u a l l y similar material can possess quite d i f f e r e n t s t r e s s - s t r a i n relationships, as shown in figure 4, which is a composite sketch of his measured stress d i s t r i b u t i o n s in three similar beams. Figure 4 - Stress d i s t r i b u t i o n s measured by Ramos Ramos proposed the use of the b i l i n e a r e l a s t o - p l a s t i c r e l a t i o n s h i p for p r a c t i c a l ' purposes, but he was unable to obtain s a t i s f a c t o r y v e r i f i c a t i o n of his theory because of machine malfunctions and the absence of direc t tension test r e s u l t s . 15 A s l i g h t l y d i f f e r e n t approach was taken by Moe(l96l) who looked more closely at the deformations on the compression side of glued laminated beams. He combined the physical phenomenon of l o c a l wrinkling of the compression fibres with a measured a x i a l s t r e s s - s t r a i n relationship far beyond the e l a s t i c range to propose the stress d i s t r i b u t i o n shown by the s o l i d l i n e in figure 5, which i s consistent with the model developed in this t h e s i s . His suggestion of the dotted l i n e in figure 5 as an approximation for ca l c u l a t i n g bending strength has not been pursued by others. 11 iV 1* Figure 5 - Stress d i s t r i b u t i o n proposed by Moe Nwokoye(1975) tested a series of clear hardwood glued laminated beams. For those beams long enough to prevent a shear f a i l u r e , he found that a theory based on the' simple b i l i n e a r stress d i s t r i b u t i o n shown in figure 2 gave a very accurate prediciton of strength. The the o r e t i c a l estimate was based on the results of di r e c t tension and compression tests on specimens cut from the same beams. This accuracy i s in contrast to Bechtel and Norris who found that a similar theory 16 overestimated the strength of their beams. Nwokoye's results may indicate a d i f f e r e n t size effect in hardwoods compared with softwoods. Nwokoye confirmed that plane sections remained plane, and he produced evidence supporting his assumption that the extreme f i b r e stress in bending at the proportional l i m i t i s the same as the ultimate compressive strength p a r a l l e l to the grain. A test series using clear poplar beams was c a r r i e d out by Zakic(l973) who assumed a parabolic s t r e s s - s t r a i n curve in compression, producing a parabolic d i s t r i b u t i o n of stresses over the compression region at f a i l u r e . Zakic's analysis is not consistent with any stated s t r e s s - s t r a i n relationship, and strains, in excess of those producing maximum stress are not considered. Several unanswered questions have been raised by Nwokoye( 1974 ). Bazan(l980) proposed a refinement to e l a s t o - p l a s t i c theory. He approximated the s t r e s s - s t r a i n relationship in compression by l i n e a r e l a s t i c behaviour up to maximum stress, followed by a linear reduction in stress with increasing s t r a i n , as shown in figure 6(a), where the s o l i d l i n e shows actual behaviour and the dotted l i n e shows Bazan's approximation. This produces a compression stress block of the shape shown in figure 6(b). This approximation is suitable for small and intermediate levels of s t r a i n , but obviously cannot be extrapolated to very large s t r a i n s . Bazan attempted to predict bending strength of clear wood beams from the results of a x i a l tension and compression tests 17 in 0) (a) s t r e s s - s t r a i n relationship (b) stress distribution Figure 6 - Stress d i s t r i b u t i o n proposed by Bazan on specimens cut from the same board. He achieved reasonable success using the compression s t r e s s - s t r a i n r e l a t i o n s h i p shown above, and an empirical size e f f e c t factor in tension. Bazan's general procedure i s continued in this thesis but with application to commercial qua l i t y timber members with defects, loaded in both bending and combined loading. This study also uses a much more rigorous approach to b r i t t l e fracture size e f f e c t s in the tension zone. Timber with defects was included in Bazan's test programme, but was not subjected to mathematical analysis. b. Size Effects Observations over many years have found that the bending strength of large members tends to be less than that of smaller members under similar loading conditions. The f i r s t attempt to quantify a size e f f e c t theory in bending was made by Newlin and Trayer (1924). They defined the term "form factor" to relate the modulus of rupture of any cross section to that of a 51x51mm specimen and carried out 18 bending tests on beams of a large number of d i f f e r e n t cross sections including rectangular, circula'r, hollow and flanged. The material for a l l these tests was clear straight-grained Sitka spruce. On the basis of these tests they developed a theory for the observed size e f f e c t . This theory, which became known as the " f i b r e support theory", suggests that the maximum compressive stress which the extreme fibres in a beam can sustain i s related to the amount of l a t e r a l support which those f i b r e s receive from less highly stressed fibres nearer the neutral axis. As such, the theory predicts that the strength of a deep beam should be less than that of a shallow beam because in the deep beam the highly stressed compression fib r e s are further from lower stressed fibres which can offer l a t e r a l support. -The theory also predicts the lower strength of flanged beams where one or more narrow webs do not offer as much l a t e r a l support as a s o l i d rectangular web. In addition to explaining strength differences at ultimate load, the theory gives a possible explanation for the phenomenon observed by some "investigators that the stress in the extreme compression f i b r e at the proportional l i m i t in bending i s greater than the f i b r e stress at the proportional l i m i t in a compression p a r a l l e l to the grain t e s t . This phenomenon i s not explained by more recent b r i t t l e fracture theories. The f i b r e support theory was based on certain assumptions regarding the strength of wood in the du c t i l e compression 1 9 region of the beam. It was assumed throughout this development that the strength of wood in tension i s a material constant, and the theory does not predict a size effect in a x i a l tension. The more recent development of a b r i t t l e fracture explanation for size e f f e c t s in bending w i l l be introduced in chapter 3. 2 . 2 . 2 Bending Behaviour of Timber a. Comparison With Clear Wood The bending strength of timber members i s considerably less than that of clear wood specimens of the same size, and f a i l u r e s tend to be more b r i t t l e . The simple explanation for th i s difference in behaviour i s that the tension strength of timber' i s greatly reduced by various defects, whereas compression strength i s affected much less, r e s u l t i n g in a material which is often weaker in tension than in compression. In t h i s case bending strength i s governed by tension behaviour, and f a i l u r e s tend to be sudden and b r i t t l e , with a l l stresses in the e l a s t i c range. The s t r e s s - s t r a i n r elationship for timber tends to be of the same form as shown in figure 3 for clear wood, with compression strength s l i g h t l y reduced and tension strength greatly reduced. 20 b. In-grade Testing In recent years large scale in-grade testing programmes have provided a a wealth of new information on the behaviour of timber. In-grade testing refers to f u l l - s i z e testing of large samples of commercial timber in the sizes, grades and species groups in which i t i s produced and marketed. In Canada testing has been carried out on the p r i n c i p a l sizes, grades and species groups. The emphasis has been to provide lower 5th percentile strength values as a basis for specifying allowable design stresses (Madsen and Nielsen 1978a). In many cases representative samples of timber have been subjected to a proof load such that only about the 10% weakest boards have been broken, providing an accurate estimate of the lower 5th percentile strength. Unbroken boards have been returned to normal production. In other research-related studies, complete samples of boards have been tested to f a i l u r e under various loading and environmental conditions. Some important findings from in-grade testing are: 1. Commercial qua l i t y timber exhibits quite d i f f e r e n t f a i l u r e modes than clear wood in many loading cases. 2. V a r i a b i l i t y in strength,properties i s much greater for timber than for clear wood. 3. The difference in strength between various tree species i s much less for timber members than observed in clear wood, espe c i a l l y at the 5th percentile l e v e l . 4. Exist i n g grading rules do not separate timber into well defined strength classes. 5. Timber members exhibit a s i g n i f i c a n t size e f f e c t . 21 Large members f a i l at lower stresses than smaller members of the same species and grade. 6. The f a i l u r e stress in ax i a l tension members i s much less than the modulus of rupture obtained from bending t e s t s . 7. Moisture content has less effect on the strength of timber than on clear wood, at the low end of the strength d i s t r i b u t i o n . 8. Strength i s not strongly correlated with size or type of defect, or with density or modulus of e l a s t i c i t y . Several of the above results w i l l be discussed in more d e t a i l l a t e r . c. Derivation of Design•Stresses The conventional method of predicting the strength of wood with defects (timber) has been to apply modification factors to the strength values obtained from tests on small clear specimens. This procedure i s described by WiIson(1978). The standard method of determining modulus of rupture is to load to f a i l u r e small clear specimens 51x51mm, 760mm long, under centre-point loading over a simple span (ASTM 1981a). For wood with defects, modification factors or "strength r a t i o s " are used to modify the results of the small clear tests (ASTM 1981c), depending on slope of grain or knot size, assuming that knots have zero strength. In-grade testing has shown that the actual strength of timber i s very d i f f e r e n t from that predicted in thi s way. In-grade testing i s now being used as a direct measurement of strength properties. This method provides c h a r a c t e r i s t i c strength properties for a given population of 22 timber, but has never been intended to predict the strength of a single board. In-grade test results provide e s s e n t i a l input to any p r o b a b i l i s t i c design method for timber structures. One of the largest problems preventing e f f i c i e n t design of s t r u c t u r a l timber i s the absence of a non-destructive test method for predicting strength. Many attempts have been made to predict strength on the basis of non-destructive test parameters such as f l e x u r a l s t i f f n e s s , l o c a l slope of grain, density and knot size, but none have produced very useful r e s u l t s . Commercial "stress-grading machines" are used to separate wood into strength classes on. the basis of l o c a l flat-wise f l e x u r a l modulus of e l a s t i c i t y . This can provide a small improvement over t r a d i t i o n a l v i s u a l grading methods, but the c o r r e l a t i o n between strength and modulus of e l a s t i c i t y i s not very high. d. Size Effects For at least 70 years a size e f f e c t has been observed for timber members in bending. In an extensive report, Cline and Heim (1912) document that the bending strength of large members tends to be less than that of smaller members under similar loading conditions. Madsen and Nielsen (1976) have investigated this phenomenon more recently using in-grade testing methods. Size e f f e c t s w i l l be discussed in more d e t a i l l a t e r . 23 2.3 AXIAL TENSION STRENGTH 2.3.1 Axial Tension Strength of Clear Wood The f i r s t recorded test on wood in a x i a l tension p a r a l l e l to the grain was performed by Mariotte in 1680 (Booth 1980). Tension tests have not been as easy to perform as bending and compression tests because of the d i f f i c u l t y of making a connection stronger than the test specimen. Today there i s a standard test (ASTM 1980) using a 450mm long piece of clear wood necked down to 4.8x9.5mm over a 64mm gauge length. The evolution of t h i s test specimen has been described by Markwardt and Youngquist (1956). Since test methods for small clear specimens became standardized, tension strengths have been consistently higher than the modulus of rupture (Wood Handbook 1974). No serious attempt has been made to explain t h i s discrepancy which w i l l be discussed further under size e f f e c t s . As described by Galligan et al.(l974) t e n s i l e design stresses were taken as equal to the design bending stress u n t i l about 1965 in the b e l i e f that t h i s was a conservative interpretation of test r e s u l t s . Consequently, the results of small clear tension tests were not put to much use and tension testing received very l i t t l e attention. This was not a serious problem at that time because the lack the of suitable connection d e t a i l s prevented very high stresses from being developed in tension members of real structures. It was not u n t i l more e f f e c t i v e connections became available and commercial size material with defects was tested 24 that the possible magnitude of a size effect in tension was reali z e d , leading to renewed interest in tension strength of clear wood. Kunesh and Johnson (1974) car r i e d out tests of commercial sizes of clear Douglas-fir and Hem-fir and observed a s i g n i f i c a n t decrease in strength with increasing cross section dimension, the strengths being far less than in small clear specimens. This size e f f e c t in clear wood has been discussed by the author elsewhere (Buchanan 1983). 2.3.2 Axial Tension Strength of Timber a. Effects of Defects The a x i a l tension strength of timber with defects received l i t t l e attention u n t i l recently, in the beli e f that the modulus of rupture was a conservative estimate of tension strength. Renewed interest in a x i a l tension strength occurred after in-grade tension testing produced stresses at f a i l u r e much less than the modulus of rupture, as described above for clear wood. The effects of defects have been investigated by several investigators. Zehrt(1962) investigated the effect of sloping grain on small clear specimens and 38x89mm members and found that the strength depended not only on the r e l a t i v e tension strengths p a r a l l e l to and perpendicular to the grain, but also on the shear strength. Quite good agreement was obtained with an interaction formula developed by Norris(l955) for strength under combined normal and shear stresses. 25 Dawe(l964) investigated the eff e c t of knots on the tension strength of European redwood boards and found a strong c o r r e l a t i o n between strength and knot si z e . The f i r s t tension tests on f u l l size timber members were carried out in the mid 1960's. Nemeth(l965) reported some c o r r e l a t i o n between t e n s i l e strength and modulus of e l a s t i c i t y , but no s i g n i f i c a n t c o r r e l a t i o n between strength and density. Both Littleford(1967) and McGowan(1968) confirmed these findings and reported that knots, l o c a l l i z e d grain i r r e g u l a r i t i e s and general grain deviations were the predominant factors a f f e c t i n g t e n s i l e strength. In a later report, McGowan(1971) demonstrated that the ASTM strength ratios are not good indicators of tension strength. A general finding from these studies was that large knots reduced strength more than small knots, and edge knots more than centre knots, a result confirmed by Kunesh and Johnson (1972) and by Johnson and Kunesh (1975). Several attempts have been made to combine various c h a r a c t e r i s t i c s such as knot size, f l e x u r a l s t i f f n e s s and slope of grain to predict tension strength. Schneiwind and Lyon (1971),. Gerhards (1972) and Heimeshoff and Glos (1980) a l l report multiple correlation c o e f f i c i e n t s larger than 0.80, for southern pine, C a l i f o r n i a n redwood and German spruce respectively, but these' multiple c o r r e l a t i o n techniques have not yet been incorporated into grading rules. Heimeshoff and Glos compare their results with ten other studies including those mentioned here. 26 Orosz(l975) used matrix structural analysis to predict the effects of d i f f e r e n t sizes and locations of knots on tension strength, considering moments induced by e c c e n t r i c i t i e s within the test specimen. He showed that for large knots and short lengths, test machines with pinned ends and fixed ends w i l l produce d i f f e r e n t r e s u l t s . Although the factors a f f e c t i n g tension strength are s t i l l not well understood, allowable stresses in design codes have become more r e l i a b l e as they are now based on the results of in-grade a x i a l tension tests, at least in Canada. b. Size Effects Results of a very large in-grade tension testing programme are described by Madsen and Nielsen (1978b), who tested more than 25,000 boards in tension. These boards were of several sizes, grades and species groups, and were loaded with a proof load that broke approximately 10% of each sample, in order to produce an accurate estimate of the f i f t h percentile strength. A regression analysis of strength versus size at the f i f t h percentile l e v e l produced quite consistent results for a l l species and grades, with the f a i l u r e stress in a 38x89mm member being approximately 1.25 times that in a 38x286mm member. The current Canadian allowable stresses in tension (CSA 1980) are based on t h i s study. Similar results are reported by Johnson and Kunesh (1975) for Douglas-fir and Hem-fir in s t r u c t u r a l sizes without defects. For material with knots they found an even more pronounced size e f f e c t . None of these studies investigated 27 the e f f e c t of member length on strength. 2.4 AXIAL COMPRESSION STRENGTH 2.4.1 Axial Compression Strength of Clear Wood The load capacity of short wood columns depends on the strength of wood in compression p a r a l l e l to the grain. When wood i s loaded in compression p a r a l l e l to the grain i t exhibits linear s t r e s s - s t r a i n behaviour up to a proportional l i m i t at approximately two-thirds of the ultimate strength. Beyond the proportional l i m i t the s t i f f n e s s drops gradually, leading to a d u c t i l e crushing type of f a i l u r e at ultimate load. Characteristic compression wrinkles caused by l o c a l buckling of the wood fibres become v i s i b l e as ultimate load i s approached. The f i r s t s c i e n t i f i c tests to determine the strength of wood in compression p a r a l l e l to the grain were carried out on 25mm cubes. Tredgold(1853) reports the results of such tests on several species. The current standard test for compression p a r a l l e l to the grain (ASTM 1981a) uses a clear straight-grained specimen 51x51mm, 203mm long, and the results of extensive testing programmes have been published (Wood Handbook 1974, Jessome 1977). 28 2.4.2 Axial Compression Strength of Timber The compression strength of timber with defects is generally less than that of clear wood. The t r a d i t i o n a l method of assessing the effect of knots or sloping grain has been to apply "strength r a t i o s " to the strength of small clear specimens (ASTM 1981c). A more e f f e c t i v e method i s to carry out in-grade testing in s t r u c t u r a l sizes (ASTM 1981b) with l a t e r a l r e s t r a i n t s to prevent buckling. An alternative test method under development in the United States uses a short specimen containing the worst v i s i b l e defect in each board, but i t is often d i f f i c u l t to estimate which defect w i l l have the lowest strength. Several commercial sizes and species of Canadian timber have been subjected to in-grade testing in long lengths ( L i t t l e f o r d and Abbott 1978), (Madsen and Nielsen 1978b, I978d). Several interesting findings are reported. Failure is associated with a large number of d i f f e r e n t defects, but strength i s not strongly correlated with size or type of v i s i b l e defect. For 38mm thick material, there i s a trend towards decreasing compression strength as width i s increased, but this size effect i s less pronounced than found in bending or in a x i a l tension. Compression strength of commercial timber" varies s i g n i f i c a n t l y with changes in moisture content, in contrast to tension strength which i s e s s e n t i a l l y independent of moisture content (Madsen 1982). 29 2.4.3 Stress-Strain Relationship The shape of the s t r e s s - s t r a i n relationship in compression i s important input for the model developed in this t h e s i s . Not much test data i s available. Although the standard ASTM test for compression p a r a l l e l to the grain s p e c i f i e s the drawing of a load-deformation curve, t h i s has generally only been used to est a b l i s h the e l a s t i c modulus and the proportional l i m i t stress. Very few studies have been made regarding the mathematical form of the s t r e s s - s t r a i n curve beyond the e l a s t i c range. Section 2.2 showed how the complete curve i s required to understand bending behaviour, and referenced several studies. The simple e l a s t o - p l a s t i c b i l i n e a r approximation shown in figure 2 has been used quite often. More recently, Malhotra and Mazur (1970) used a s t r e s s - s t r a i n equation, previously proposed by Ylinen(1956), given by e = i [ c f - (1 - c ) f c l n [ l - f - ) ] (2 . T ) where e i s s t r a i n , f is stress, fc i s the maximum compression stress, E i s modulus of e l a s t i c i t y , and c i s a parameter depending on the shape of the curve. The curve described by t h i s equation, as shown in figure 7(a) i s tangent to the e l a s t i c modulus at the o r i g i n , and tangent to the ultimate compressive stress for large s t r a i n . Malhotra and Mazur describe the curve as a very good approximation to the results of 144 tests of clear eastern (a) (b) (c) Figure 7 - Stress-strain relationships in a x i a l compression spruce wood at various moisture contents. They give no consideration to the possible shape of the curve at strains beyond the ultimate load. 0'Halloran(1973) used the data of Goodman and Bodig (1971) to propose a mathematical equation for the stress-s t r a i n curve for clear dry wood in compression at various grain angles and grain orientations. The proposed equation i s f = Ee - Aen (2.2) where f i s stress, E i s modulus of e l a s t i c i t y , A and n are equation constants determined, by f i t t i n g the equation to a given set of experimental data. If the st r a i n at peak stress i s found to be a certain r a t i o , r, of the equivalent st r a i n under e l a s t i c conditions, the parameters A and n can be found from 31 n = r/(r-1) (2.3) and A = E n(rf c/E) n-1 (2.4) where f c i s the maximum stress. A t y p i c a l plot of a f i t t e d curve and experimental results is shown in figure 7(b). The equation cannot be used beyond maximum stress, because i t drops rapidly to negative stress l i m i t a t i o n , f a i l i n g to recognize that the shape of the f a l l i n g branch of the curve is needed to predict ultimate bending strength. It i s true that the s t r e s s - s t r a i n relationship beyond maximum load cannot be quantified e a s i l y in an a x i a l compression test, because i t i s largely a function of the test machine c h a r a c t e r i s t i c s and the rate of loading, but a description of s t r e s s - s t r a i n behaviour beyond ultimate load is essential to the development of an ultimate bending strength theory. A simple b i l i n e a r proposal by Bazan(!980) has already been i l l u s t r a t e d in figure 6. Bazan assumed without any supporting argument that the slope of the f a l l i n g branch i s a variable which can be a r b i t r a r i l y taken as that value which produces maximum bending moment for any neutral axis depth. A di f f e r e n t assumption used in t h i s study i s that the slope of the f a l l i n g branch i s a material property, whose value can be estimated as part of the c a l i b r a t i o n of the computer model to test r e s u l t s . values. O'Halloran claims that t h i s i s not a serious 32 A comprehensive study on the st r e s s - s t r a i n relationship of timber with defects, in compression p a r a l l e l to the grain, has been made by G1OS(1978). On the basis of extensive experimental testing Glos proposes a curve of the shape shown in figure 7(c), which i s similar to that reported for clear wood by Bechtel and Norris (1952), Moe (1961), and others. The curve is characterised by four parameters as shown in the figure. The equation of the curve i s given by e / e , + GAe/e.)7 f = ± ± ± a w h e r e G G 2 + G 3 ( e / e ; L ) + G 4 ( e / e ; L ) 7 (2.5) f 1 6E(l-f s/f c) G 2 - 1/E G- = 1/f - 7/6E 3 c G4 = G l / f s where f i s stress, e i s s t r a i n , E i s modulus of e l a s t i c i t y , fc is maximum compressive stress, fs i s the asymptotic compression stress for large s t r a i n and e, i s the stra i n at the maximum stress, as shown in figure 7(c). Glos has estimated the four parameters to define the shape of the curve from four measurable wood properties; density, moisture content, knot r a t i o and percentage compression wood. Multiple c u r v i l i n e a r regression techniques have been used to express expected values of each parameter in terms of the four properties, using lengthy regression equations. The general form of st r e s s - s t r a i n curve resulting from Glos's equation has been investigated in this study, 33 without attempting to v e r i f y the dependence of the equation on the wood properties mentioned above. 2.4.4 Column Theory for Concentric Loading The load capacity of short columns with concentric a x i a l loading i s d i r e c t l y related to the compression strength of the material. As the length of a concentrically loaded column increases, there i s a v t r a n s i t i o n from a crushing type of f a i l u r e based on compression strength to a buckling type of f a i l u r e based on l a t e r a l i n s t a b l i l i t y . For linear e l a s t i c materials, the load capacity of long columns i s proportional to the e l a s t i c modulus and inversely proportional to the square of the length, as described by the well known column formula published by Euler in 1744 (Timoshenko 1953) P = » 2 E I e L2 (2.6) where Pe i s the a x i a l load capacity (or Euler buckling load) for a column of length L pinned at both ends. E is the modulus of e l a s t i c i t y . For materials with non-linear s t r e s s - s t r a i n relationships, the combination of both material and geometric n o n - l i n e a r i t i e s makes a n a l y t i c a l treatment much more d i f f i c u l t . Closed form solutions for very simple cases are availa b l e , but numerical and other approximate methods are necessary in most real situations (Chen and Atsuta 1976a). For a l l materials, the t r a n s i t i o n between short column 34 behaviour and long column behaviour has been the subject of much debate, and there have been a great many formulae proposed to describe t h i s behaviour. Test results in this range tend to have far more scatter than for long or for short columns. The e a r l i e s t formulae were empirical expressions which were only useful in the limited range where they f i t t e d test r e s u l t s . Subsequent formulae were improved to produce results consistent with crushing strength for short columns and the Euler curve for long columns. Many formulae including the secant formula and the Rankine formula, which w i l l be discussed in Chapter 9, f a l l into t h i s category. The Rankine formula (or Rankine-Gordon formula), published by Rankine in 1898, was developed by Gordon using an assumed deflected shape and a l i m i t i n g f a i l u r e stress (Timoshenko 1953). A further development attributed to Engesser in 1889 (Bleich 1952) was the acceptance of the tangent modulus approach which uses Euler's formula with the tangent modulus of e l a s t i c i t y from a c u r v i l i n e a r s t r e s s - s t r a i n relationship. This method predicts a safe lower bound on strength regardless of loading path but requires detailed knowledge of the stress-s t r a i n r e l a t i o n s h i p . Chen and Atsuta (1976a) summarize the development of more accurate theories for p l a s t i c buckling of columns, which include allowance for e l a s t i c unloading of portions of the cross section that have been previously stressed beyond the proportional l i m i t . 35 2.4.5 Timber Columns According to Timoshenko(1953), the f i r s t recorded s c i e n t i f i c tests on long timber columns were by Musschenbrock in 1729, who found that the buckling strength was inversely proportional to the square of the length, a result proved t h e o r e t i c a l l y by Euler in 1744. Booth(l964) describes compression tests of many large members performed by Girard in 1798, where the ever-present problems of defects, load duration and end r e s t r a i n t did not allow satisfactory v e r i f i c a t i o n of Euler's theory. The f i r s t recorded wood column tests in America appear to be those of Bryson (1866) who tested 40 small specimens of varying length cut from one piece of dry white pine timber. Bryson obtained good agreement with two formulae attributed to Hodgkinson, noting the t r a n s i t i o n from a crushing type of f a i l u r e in short columns to a buckling type of f a i l u r e in long columns. Current formulae for timber columns appear to date from the work of Newlin and Trayer (1925) who c a r r i e d out detailed analysis of a large number of clear Sitka spruce columns. For a x i a l compression loading they related the buckling load to the s t r e s s - s t r a i n curve in compression for three d i f f e r e n t lengths of columns. For long columns, the maximum load was reached with a l l stresses in the e l a s t i c range as predicted accurately by the Euler formula. For short columns, f a i l u r e occurred by crushing of the wood in compression p a r a l l e l to the grain. 36 •d O O crushing strength 4 th power parabola short Euler curve intermediate long slenderness Figure 8 - Axial load-slenderness curve for concentrically loaded columns For intermediate length columns Newlin and Trayer suggested a t r a n s i t i o n between these two types of behaviour based on a tangent modulus approach. Using some approximations they obtained a parabolic t r a n s i t i o n curve tangent to the Euler curve at two-thirds of the crushing strength. They found the power of the parabola to depend on the r a t i o between proportional l i m i t stress and maximum compressive stress. The eighth power curve i n i t i a l l y proposed for Sitka spruce was changed to a fourth power curve to include other species. This fourth power curve was later v e r i f i e d by Newlin and Gahagan (1930) after tests on many large size columns of several grades and species, and has remained in many North American Codes ever since. The form of th i s relationship i s shown'in Figure 8. The formulae w i l l be given in Chapter 8. In B r i t a i n , according to Sunley(1955) and Burgess(1977), 37 the development of wood column formulae was l a r g e l y the work of Robertson(1925) who used an. 1886 a r t i c l e by Ayrton and Perry to e x p l a i n the r e s u l t s of h i s t e s t s on small c l e a r S i t k a spruce columns. The r e s u l t i n g "Perry-Robertson" formula assumes that the column remains e l a s t i c to f a i l u r e , but has an i n i t i a l d e v i a t i o n from s t r a i g h t n e s s . The f a i l u r e c r i t e r i o n i s a l i m i t i n g compression s t r e s s . The e f f e c t s of d e f e c t s and m a t e r i a l v a r i a b i l i t y can be c o n s i d e r e d by v a r y i n g the assumed i n i t i a l e c c e n t r i c t y . T h i s approach has become the b a s i s of the B r i t i s h code ( B S I 1980). Larsen(l973) compared column formulae from the codes of many c o u n t r i e s , and d e s c r i b e d the background to the formula i n an i n t e r n a t i o n a l code (CIB 1980). Burgess(1976) has shown that the CIB formula i s e s s e n t i a l l y the same as the Perry-Robertson formula. In Canada, Malhotra and Mazur (1970) have re-examined the tangent modulus formula or "Euler-Engesser" formula, and propose i t s use f o r c o n c e n t r i c a x i a l l o a d i n g . T h i s method has the advantage of being a continuous formula f o r a l l values of sl e n d e r n e s s , and i t has a sound t h e o r e t i c a l b a s i s i f the s t r e s s - s t r a i n curve i s always of the assumed form. Neubauer(1973) has reviewed the background to the Rankine formula, and m o d i f i e d i t to a simple cu b i c form which gi v e s good agreement to wood column t e s t r e s u l t s f o r a wide range of slenderness r a t i o s . S e v e r a l of these column formulae w i l l be compared i n more d e t a i l i n Chapter 8. 38 2.5 COMBINED BENDING AND AXIAL LOAD Combined bending and a x i a l load w i l l be discussed in two sections. The f i r s t i s the f a i l u r e c r i t e r i o n for a cross section subjected to combined bending and a x i a l loading. The second is the behaviour of members of any length subjected to a x i a l and bending loads, including the effects of member i deformations. 2.5.1 Cross Section Behaviour To calculate the ultimate strength of a member subjected to combined a x i a l and fl e x u r a l loading i t i s f i r s t necessary to know the nature of the interaction between a x i a l strength and f l e x u r a l strength at a cross section. This interaction can be presented graphically as an ultimate interaction  diagram which shows combinations of a x i a l load and bending moment that a cross section can r e s i s t . The t r a d i t i o n a l approach has been to assume a linear e l a s t i c material with a maximum normal stress f a i l u r e c r i t e r i o n . These assumptions produce a straight l i n e ultimate interaction diagram, as shown by the s o l i d l i n e s in figure 9(a) for material with weak tension strength, and figure 9(b) for strong tension strength, assuming constant compression strength. Most design formulae make the linear e l a s t i c assumption even more conservative by specifying design according to the dotted l i n e s in figure 9. If wood i s assumed to have a non-linear s t r e s s - s t r a i n 39 moment (a) weak in tension (b) strong in tension F i g u r e 9 - U l t i m a t e i n t e r a c t i o n d i a g r a m s f o r l i n e a r b e h a v i o u r r e l a t i o n s h i p i n c o m p r e s s i o n ( f o r example, a b i l i n e a r e l a s t o -p l a s t i c r e l a t i o n s h i p ) t h e u l t i m a t e i n t e r a c t i o n d i a g r a m becomes c u r v e d on t h e c o m p r e s s i o n s i d e , f o l l o w i n g t h e shape i l l u s t r a t e d i n f i g u r e 10. T h i s t y p e of i n t e r a c t i o n c u r v e has been d e s c r i b e d by G u r f i n k e l ( 1 9 7 3 ) and by L a r s e n and R i b e r h o l t ( 1 9 8 1 ) , but has r e c e i v e d v e r y l i t t l e e x p e r i m e n t a l v e r i f i c a t i o n . The shape of t h e c u r v e above t h e h o r i z o n t a l a x i s ( f o r 1 n e t a x i a l c o m p r e s s i o n ) depends on t h e r a t i o of t e n s i o n t o c o m p r e s s i o n s t r e n g t h s . F o r h i g h t e n s i o n s t r e n g t h ( o u t e r c u r v e ) , a s t r a i g h t l i n e i n t e r a c t i o n i n t h e c o m p r e s s i o n r e g i o n i s a r e a s o n a b l e a p p r o x i m a t i o n w h i c h has been u s e d i n many column 40 c o Figure 10 - Ultimate interaction diagrams for non-linear behaviour formulae, but thi s becomes less accurate as the r a t i o of tension to compression strength decreases. Newlin(l940) carried out testing which showed some deviation from a straight l i n e for small clear specimens. He suggested a parabolic interaction equation of the form r c where M i s the bending moment, S i s the section modulus, fr is the modulus of rupture, P is the a x i a l load, A i s the section area and fc i s the compresssive stress at f a i l u r e for a short column. As described by Wood(l950), Newlin conservatively dropped the exponent of 2 for short columns, but for slender 41 columns he used the parabolic re l a t i o n s h i p to j u s t i f y using bending strength as the l i m i t i n g stress, regardless of ax i a l load. The p o s s i b i l i t y of a tension f a i l u r e was not considered, so he may not have re a l i z e d how his results f i t t e d the more complete relationship shown in figure 10. The actual shape of the curves in the compression region also depends on the nature of the s t r e s s - s t r a i n relationship for wood in compression, e s p e c i a l l y the f a l l i n g branch, as w i l l be explored in Chapter 6. In the tension region, figures 9 and 10 both show a straight l i n e , based on the same extreme .fibre stress at f a i l u r e in bending or in tension. As discussed e a r l i e r , i t has quite recently been determined that the f a i l u r e stress in an a x i a l tension test i s considerably less than the maximum tension stress when a similar member f a i l s in bending. The tr a n s i t i o n between a x i a l tension behaviour and bending behaviour can be explained by b r i t t l e fracture theory (Johns and Buchanan 1982) which suggests a curve in the tension region, as shown by the dotted lines in figure 10. Very similar curves were obtained independently by Kersken-Bradley(1981) in an a n a l y t i c a l study of glued laminated wood members with d i f f e r e n t strength properties over the cross section. A more detailed analysis of the shape of the interaction diagram w i l l be described in Chapter 6. 42 2.5.2 Members with Combined Bending and Compression For concentrically loaded columns, i t has been shown that load carrying capacity depends on material strength for short columns, and on slenderness r a t i o for long columns. If the columns are also subjected to bending moment the problem becomes more complex because the load capacity also depends on the bending moment. A conceptual method of considering these e f f e c t s is shown in figure 11, which i s a three-dimensional sketch of load vs. slenderness vs. moment (Johns and Buchanan 1982). A member of given slenderness r a t i o can r e s i s t any combination of a x i a l load and moment inside the surface shown. The curve on the load vs. slenderness plane i s the relationship shown in figure 8. The curve on the- load vs. moment plane i s the a x i a l load vs. moment interaction for short columns shown in figure 10. The behaviour of timber members under combined bending and a x i a l loading (or the form of the surface shown in figure 11) has received very l i t t l e attention. Standard methods are available as long as the wood remains in the linear e l a s t i c range (Timoshenko and Gere 1961). For behaviour beyond the linear range, Chen and Atsuta (1976a) summarize extensive studies on other materials such as steel and concrete, none of which are d i r e c t l y applicable to timber. Differences with other materials are that wood behaves d i f f e r e n t l y in tension and compression, tension strength is s i g n i f i c a n t l y affected by the size of the stressed volume, and the material properties 43 Axial load BENDING STRENGTH Figure 11 - Three-dimensional sketch of load vs. slenderness vs. moment may vary considerably along the length of a member. The model developed in thi s thesis addresses these problems. A comprehensive experimental study of small clear wood members subjected to combined column and beam action by Newlin and Trayer (1925) has already been mentioned. Small defect-free members were subjected to eccentric a x i a l loads, and in 44 a n o t h e r t e s t , c o n c e n t r i c a x i a l l o a d a n d t r a n s v e r s e l a t e r a l l o a d t o g e t h e r i n a c o n s t a n t r a t i o . T h e i r t h e o r y " f o r c o n c e n t r i c a l l y l o a d e d c o l u m n s h a s b e e n b r i e f l y d e s c r i b e d . F o r c o m b i n e d b e n d i n g a n d a x i a l l o a d i n g t h e y p r o v i d e d a s e t o f d e s i g n c u r v e s w i t h o u t m u c h d e v e l o p m e n t o f t h e t h e o r y . S u b s e q u e n t l y N e w l i n ( l 9 4 0 ) d e v e l o p e d a f o r m u l a f o r c o m b i n e d b e n d i n g a n d a x i a l l o a d i n g , l a t e r a m p l i f i e d a n d e x p l a i n e d b y W o o d d 9 5 0 ) , a n d a d o p t e d f o r u s e i n t h e U . S . ( N F P A 1 9 8 2 ) . T h e N F P A f o r m u l a i s u s e f u l i n t h a t i t c o n s i d e r s s e p a r a t e l y t h e e f f e c t s o f a x i a l l o a d , m o m e n t d u e t o e c c e n t r i c a x i a l l o a d a n d m o m e n t d u e t o t r a n s v e r s e l o a d s . I t i s b a s e d o n a s s u m p t i o n s o f s i n u s o i d a l d e f l e c t e d s h a p e , l i n e a r e l a s t i c b e h a v i o u r t o f a i l u r e , a n d a l i m i t i n g c o m p r e s s i o n s t r e s s f a i l u r e c r i t e r i o n d e s c r i b e d a b o v e , a l l o f w h i c h n e e d t o b e r e -e v a l u a t e d . T h e f o r m u l a i s d i s c u s s e d f u r t h e r i n C h a p t e r 8 . I n E u r o p e , t h e P e r r y - R o b e r t s o n f o r m u l a a n d t h e C I B f o r m u l a , d e s c r i b e d a b o v e f o r c o n c e n t r i c l o a d i n g , b o t h i n c l u d e a t e r m f o r i n i t i a l o u t - o f - s t r a i g h t n e s s , w h i c h c a n a l s o b e u s e d f o r l a r g e e c c e n t r i c i t i e s t o p r o v i d e a d e s i g n f o r m u l a f o r c o m b i n e d b e n d i n g a n d a x i a l l o a d . J o h n s t o n ( 1 9 7 6 ) d e s c r i b e s o t h e r a p p l i c a t i o n s o f t h e s e c a n t f o r m u l a , a n d d i s c o u r a g e s t h e u s e o f e x p r e s s i o n s o f t h i s t y p e b e c a u s e t h e i n i t i a l y i e l d c r i t e r i o n c a n n o t b e a p p l i e d r a t i o n a l l y t o m a t e r i a l s w i t h n o n - l i n e a r s t r e s s - s t r a i n r e l a t i o n s h i p s . M a l h o t r a ( 1 9 8 2 ) f o u n d t h a t t h e P e r r y - R o b e r t s o n f o r m u l a u n d e r e s t i m a t e d t h e s t r e n g t h o f t i m b e r c o l u m n s t e s t e d i n 4 5 compression with small end e c c e n t r i c i t i e s . He obtained better agreement with two other formulae. One was the secant formula, modified to allow for higher f a i l u r e stresses in bending than in compression, and the other a formula derived by Jezek who considered the s t a b i l i t y of a column of ideal e l a s t o - p l a s t i c material. For wood columns with various e c c e n t r i c i t i e s and end r e s t r a i n t s , Hammond et al.(l970) obtained good agreement with test results using a theory based on the assumption of b i l i n e a r e l a s t o - p l a s t i c behaviour in compression, but they did not consider stressed volume effects in the tension region, nor did they propose a r e a l i s t i c design method. Larsen and Thielgaard (1979) have developed and v e r i f i e d a general theory for l a t e r a l l y loaded timber columns that includes b i a x i a l effects and l a t e r a l torsional buckling. The lim i t a t i o n s of this approach are the assumptions of linear e l a s t i c behaviour and a simple f a i l u r e c r i t e r i o n based on l i m i t i n g compression stresses. Bleau(l984) has used an energy approach to explain i n -plane behaviour, using the same data as t h i s study for c a l i b r a t i o n and v e r i f i c a t i o n . His theory, incorporating size e f f e c t s in a similar way to t h i s thesis, i s able to give a reasonable prediction of experimental re s u l t s . 46 2.5.3 In-Grade Testing The'only in-grade tests of commercial grades and sizes of timber under combined a x i a l compression and bending appear to be those by Zahn(l982) and Malhotra(1982). Zahn tested four groups of 1500mm long 38x140mm western hemlock with various i n i t i a l e c c e n t r i c i t i e s . He measured curvatures over a 450mm long gauge length, and used the resulting moment-curvature curves to predict the behaviour of longer members, using a Monte Carlo approach to combine 450mm long segments with representative properties. The prediction was checked with tests on 30 boards 2400 mm long. The computer simulation gave a good indication of average properties, but did not predict the f u l l d i s t r i b u t i o n accurately. Two disadvantages of Zahn's method are the d i f f i c u l t y of experimentally obtaining the moment-curvature relat i o n s h i p and the expense of carrying out Monte Carlo simula't ions . Malhotra (1982) reported a testing programme on 38x89mm and 64x89mm members of No. 1 grade eastern spruce, with e c c e n t r i c i t i e s up to 20mm. Reasonable agreement was obtained with formulae referred to in the previous section. 2.5.4 Members with Combined Bending and Tension The strength of timber members under combined a x i a l tension and bending has received very l i t t l e attention. Senft and Suddarth (1970) tested 38x89mm members of high grade commercial southern pine, and Senft(l973) carried out similar tests on a lower grade of western hemlock. For both 47 series of tests the members were loaded with a constant tension load as bending stresses were increased to f a i l u r e . The results indicated a deviation from the t r a d i t i o n a l straight l i n e interaction between bending and a x i a l load, the deviation being larger for the higher qu a l i t y material, as would be expected from the sketched curves in figure 10. Their proposed design equation i s very conservative compared with their test r e s u l t s . Suddarth, Woeste and Galligan (1978) have used the data obtained in the above tests as an example for c a l c u l a t i n g the r e l i a b i l i t y of wood members subjected to combined loading. Their contribution i s not a design method, but i s rather a step towards the development of r e l i a b i l i t y design of wood structures. Burgess(1980) used the assumptions of linear e l a s t i c behaviour and simple l i m i t i n g values of stress for f a i l u r e in tension and compression to calculate the strength of tension members with l a t e r a l loads. He found that equations previously derived for wood beam-columns with i n i t i a l curvature and given slenderness r a t i o can be modified s l i g h t l y and used for tension members, incorporating the fact that deflections due to l a t e r a l loads w i l l be reduced by a x i a l tension forces. His linear e l a s t i c assumption i s more accurate in tension than in compression because tension behaviour is generally linear to f a i l u r e whereas compression behaviour i s not. Burgess used simple f a i l u r e c r i t e r i a which do not include any size e f f e c t s , so the theory does not 48 explain the difference in f a i l u r e stresses between bending and tension t e s t s . 2.5.5 V a r i a b i l i t y in Wood Properties A l l of the column theories described to thi s point have assumed that the relevant wood properties are known deterministic q u a l i t i e s . Newton and Ayaru (1972), recognizing that strength and s t i f f n e s s are very variable in a population of wood members, propose using the known d i s t r i b u t i o n s of strength and s t i f f n e s s and their known corr e l a t i o n to predict the strength of wood columns. They use the Perry-Robertson formula and assume that wood properties within each column are constant. Suddarth and Woeste (1977) have attempted to allow for v a r i a b i l i t y in s t i f f n e s s along the length of a long column by computer modelling a 4.4m column with four segments each 1.1m long. A slender column has been used so the load capacity i s governed by e l a s t i c buckling, not strength. A Monte Carlo simulation was used to calculate the strength of one thousand columns, the modulus of e l a s t i c i t y for each segment selected from known d i s t r i b u t i o n s . Their p r i n c i p a l finding was that safety increases as v a r i a b i l i t y in s t i f f n e s s between boards decreases, but they did not compare variable s t i f f n e s s with constant average s t i f f n e s s along each board, so i t i s not apparent whether the exercise of using four segments for each board was actually necessary. No studies appear to have been done on members with strength varying along the length of the board. 49 2.6 SUMMARY T h i s c h a p t e r has b r i e f l y r e v i e w e d the development of c u r r e n t knowledge r e g a r d i n g the s t r e n g t h of t i m b e r members. I t has been shown t h a t a l a r g e amount of r e s e a r c h u s i n g s m a l l c l e a r wood specimens has f a i l e d t o produce a good u n d e r s t a n d i n g of s t r u c t u r a l t i m b e r b e h a v i o u r , but t h a t f u l l s i z e t e s t i n g of t i m b e r members i s s t a r t i n g t o produce v e r y u s e f u l r e s u l t s . The l i t e r a t u r e s u r v ey i n t h i s c h a p t e r i s c o n t i n u e d i n the f i r s t p a r t of the next c h a p t e r , f o r d i s c u s s i o n of s i z e e f f e c t s . 50 III . SIZE EFFECTS 3 . 1 INTRODUCTION As described b r i e f l y in the l i t e r a t u r e survey, materials such as wood exhibit a size e f f e c t which i s observed in the following ways: 1 . Long members f a i l at lower stresses than s i m i l a r l y loaded short members. 2. Bending members of a certain depth tend to f a i l at lower stresses than s i m i l a r l y loaded members of smaller depth. 3. In a x i a l tension, members with large cross sectional area tend to f a i l at lower stresses than members with smaller cross sectional area. 4. For a member of given size, the f a i l u r e stress increases as the volume subjected to tension stresses decreases. For example, the modulus of rupture in a bending test i s generally greater than the f a i l u r e stress in a x i a l tension. These size effects are widely accepted to be b r i t t l e fracture phenomena similar to those observed in other b r i t t l e materials. The f i r s t part of t h i s chapter reviews conventional b r i t t l e f a i l u r e theory, and explains why timber 51 members require a somewhat modified theory. A modified theory i s developed and discussed. 3.2 CONVENTIONAL BRITTLE FRACTURE THEORY 3.2.1 History It i s now beginning to be recognized that the strength of wood in tension i s a s t a t i s t i c a l phenomenon with behaviour similar to that observed in other b r i t t l e materials. This behaviour i s often explained by " b r i t t l e fracture theory", sometimes c a l l e d " s t a t i s t i c a l strength theory". A theory for the strength of b r i t t l e materials has been developed on the basis of the weakest-link concept proposed by Pierce(l926) who studied cotton yarns and Tucker(l927) who studied concrete. Major developments of the theory were made by Weibull(1939a,b) who v e r i f i e d his results with tests on many di f f e r e n t b r i t t l e materials, but apparently not wood. In simple terms, the weakest link theory assumes that a b r i t t l e s o l i d i s made up of a large number of small elements with some s t a t i s t i c a l d i s t r i b u t i o n of strength. The member w i l l f a i l when the applied stress exceeds the strength of the weakest element (as for example the weakest link in a chain), with no sharing of load to other elements. Such materials are subject to a size e f f e c t because the larger the stressed volume, the larger the pr o b a b i l i t y of that volume containing a weak element. The size effect w i l l be more pronounced for materials having large v a r i a b i l i t y in the strength properties of the constituent elements. Al t e r n a t i v e l y the material may be considered to be a 52 homogeneous material containing a large number of defects with a s t a t i s t i c a l d i s t r i b u t i o n of size (Jayatilaka, 1979). The theory i s the same in either case. Such material is sometimes referred to as a " G r i f f i t h material". Weibull showed how the strength of a weakest link system can be explained by a cumulative d i s t r i b u t i o n of the exponential type, and how the strength depends on the volume of the test specimen for uniform or varying d i s t r i b u t i o n s of stress within the specimen. Johnson(1953) improved the theoretical basis for Weibull's theory, recognizing that the exponential type of d i s t r i b u t i o n proposed by Weibull i s asymptotically the exact extreme value d i s t r i b u t i o n of the smallest value in samples from any parent d i s t r i b u t i o n , for large sample s i z e . If the parent population of elements can be. described by a Weibull d i s t r i b u t i o n then the d i s t r i b u t i o n of member strengths i s exactly a Weibull d i s t r i b u t i o n , for any number of elements. 3.2.2 Applications of B r i t t l e Fracture Theory to Wood The f i r s t study applying Weibull's b r i t t l e fracture theory to wood was that by Bohannan(1966), who used Weibull's equations to predict the strength of wood beams of various sizes. He c a l i b r a t e d t h i s model with the average results of tests on three sizes of clear dry straight-grained Douglas-fir beams. Bohannan found that the test data was most accurately matched by the model i f strength was considered to be 53 dependent on length and depth of the test specimen, but independent of width. A small series of tests on other sizes v e r i f i e d these r e s u l t s . His main conclusion was that for geometrically similar beams, the strength i s proportional to depth to the power 1/9. Bohannan's theory predicts no size e f f e c t with varying cross sectional dimensions for an axial tension test . He did not report any attempt to compare his bending results with those of a x i a l tension t e s t s . Where depth effects for bending have been included in design codes they have either used in-grade test results d i r e c t l y , or have used the parameters derived by Bohannan. A comprehensive summary of the development of b r i t t l e fracture theory.has been made by Barrett(1974^, who used the theory to explain the e f f e c t of size on perpendicular-to-grain tension • strength of Douglas-fir. Using test results from d i f f e r e n t sizes of tension and bending specimens reported by a number of authors, Barrett obtained a straight l i n e log-log plot of f a i l u r e stress vs. stressed volume, as predicted by the theory. He used t h i s plot to calculate the parameters of the strength d i s t r i b u t i o n , but showed that i t i s not easy to define the parameters p r e c i s e l y . Barrett and others have produced further studies on b r i t t l e fracture e f f e c t s in wood, investigating perpendicular-to-grain tension strength (Barrett, Foschi and Fox 1975), shear strength (Foschi and Barrett 1975), and bending strength of glued laminated beams (Foschi and Barrett 1980). This author (Buchanan 1983) has shown that the accepted 54 theory does not accurately explain the relat i o n s h i p between bending strength and a x i a l tension strength for clear wood. Size e f f e c t s in wood are more complex than suggested by simple b r i t t l e fracture theory, but can be explained.if separate parameters are used to quantify length, depth, and width e f f e c t s , as described later in t h i s chapter. 3.2.3 Theory for Uniform Stress D i s t r i b u t i o n For a r e l a t i v e l y simple development of thi s theory, assume that the parent population of elements has a cumulative d i s t r i b u t i o n function (c.d.f.) of strength given by a Weibull d i s t r i b u t i o n r tx Xo >i (3.1) F ( x ) = 1 - e x p { - [ — J } where, x i s the strength, x 0 i s a lower l i m i t or minimum strength c a l l e d the "location parameter", m is a "scale parameter" with the same units as x, and k i s a dimensionless "shape parameter" which r e f l e c t s both the skewness and the spread of the d i s t r i b u t i o n . If samples of size n are drawn from th i s parent population, i t can be shown (Bury 1975) that the strength of the weakest element in each sample has a c.d.f. given by F ( x ) = 1 - exp{-n(^-l^)k} (3.2) -1 /k where m has been replaced by mn ' . Equation 3.2 can be re-arranged to give the strength at any quantile q in the d i s t r i b u t i o n 55 x = x +m i T 1 / k { l n [ ^ - ) } 1 / k (3.3) q o 1 q (For example q=0.5 would give the median or 50th percentile strength). Now consider two members of d i f f e r e n t sizes containing n, and n 2 elements. Equations 3.3 for each member can be combined to give the r a t i o of strengths of the two sizes at any quantile q x ( n . ) x + m n " 1 / k { l n ( * ) } 1 / k x q ( n 2 ) X Q + m n 2 { M ^ J } If the location parameter x 0 i s assumed to be zero, as is often done, the three-parameter model described above reduces to a two-parameter model and equation 3.4 i s greatly s i m p l i f i e d to x „ ( n i ) n 0 , -M4 = (-i-) 1 / k (3.5) x q ( n 2 ) ^ It can be seen in t h i s case that size e f f e c t s can be quantified by only the shape parameter k, and the r a t i o of sizes, regardless of the quantile, or the actual values of n, and n 2. A log-log plot of strength against volume becomes a straight l i n e of slope -1/k as shown in figure 12. In some cases i t may be a poor assumption to use the two-parameter form of the Weibull d i s t r i b u t i o n , p a r t i c u l a r l y when extrapolating to large volumes because this assumption implies strength reducing to zero as the volume becomes i n f i n i t e l y 56 log stress x(n 2 ) volume Figure 12 - Typical log-log plot of f a i l u r e stress vs. volume large. 3.2.4 Theory for Variable Stress D i s t r i b u t i o n The above development has assumed that the member i s subjected to a uniform d i s t r i b u t i o n of stresses. In the more general case where stresses vary within a member, equation 3.2 can be written as F(x) = l - e x p { - i - J (- dv} (3.6) v i v m where V i s the volume of the member, and V, i s a reference volume associated with the scale parameter m. For the two parameter case, the integral in t h i s equation can be evaluated for any non-uniform stress d i s t r i b u t i o n and the result can be expressed as 5 7 F ( x ) ( 3 . 7 ) where Ve i s an equivalent stressed volume. For a beam of span L with two symmetrically placed loads, distance a apart, and neutral axis at mid-depth, the integral can be evaluated over the tension region (assuming b r i t t l e which is seen to be a quite simple proportion of the t o t a l volume V, of the member. The r a t i o of strengths of beams of two sizes can now be predicted using equation 3 . 5 with n, and n 2 replaced by Ve, and Ve 2 respectively. For an a x i a l tension member the equivalent volume Ve in equation 3 . 7 • i s the t o t a l volume V, of the member, so the re l a t i v e strengths of bending and a x i a l tension members can also be compared using equation 3 . 5 . The equivalent highly stressed volume of a member loaded in bending i s much less than the t o t a l volume of the member stressed in tension, and this explains why tension stresses at f a i l u r e are greater in bending tests than in a x i a l tension tests. To make a log-log plot of strength vs. equivalent volume for bending members, a t r i a l and error approach i s necessary to find the value of k such that a l i n e of slope -1/k is obtained when the same value of k i s used in equation 3 . 8 to calculate the stressed volume. fracture in tension only) to give an equivalent volume of ( 3 . 8 ) 58 3 . 2 . 5 Coefficient of Variation The c o e f f i c i e n t of variation of the Weibull d i s t r i b u t i o n i s given by cv = [rq+2/iQ - r 2 g + i / k ) F 2 C V xo/m + r(l+l/k) ( 3 - 9 > where r i s the gamma function (Bury 1 9 7 5 ) . For the two-parameter model with x o = 0 , the c o e f f i c i e n t of vari a t i o n becomes a function only of the shape parameter k. A simple but accurate approximation i s then given (Leicester 1973) by cv - k-0.922 ( 3 > 1 0 ) Because the c o e f f i c i e n t of variation i s a function only of k, strength tests of members of di f f e r e n t sizes should a l l have the same c o e f f i c i e n t of va r i a t i o n , and the k value obtained from equation 3 .10 should be the same as that obtained in a log-log plot of strength against volume. The same should also apply for a comparison of bending tests and a x i a l tension te s t s . For t h i s ideal case of a perfectly b r i t t l e material following a two-parameter Weibull d i s t r i b u t i o n , i t would be possible, in theory, to carry out only one test series on members of only one size , and to predict the strengths of other sizes on the basis of a k value obtained from equation 3 . 1 0 . 59 Unfortunately neither clear wood nor timber with defects behave according to t h i s model, for tension stresses p a r a l l e l to the grain. Clear wood has been discussed elsewhere by the author (Buchanan 1983). A discussion of size effects in timber w i l l follow. 3.3 BRITTLE FRACTURE THEORY MODIFIED FOR TIMBER Up to t h i s point, t h i s chapter has been a review of previous work. Most of the remainder of the chapter describes a new approach. The conventional b r i t t l e fracture theory described in the previous section i s a very neat and useful formulation for materials that behave in a perfectly b r i t t l e manner. Unfortunately timber stressed in tension p a r a l l e l to the grain is not as well behaved as t h i s . Observations of size e f f e c t s in both clear wood and timber with defects show departure from the simple theory in the following ways: 1. For bending members, modulus of rupture decreases as length and depth of members are increased, but not as width i s increased. The ef f e c t s for length and depth are not always of the same order. 2. For a x i a l tension members the ef f e c t s of varying length and cross sectional area are d i f f e r e n t . 3. For a x i a l compression members, which f a i l in a 60 r e l a t i v e l y d u c t i l e manner, size e f f e cts are observed for varying length and cross section. These e f f e c t s are less in compression than in a x i a l tension or in bending, but are s t i l l s i g n i f i c a n t . 4. The k value which relates the strengths of several sizes (from equation 3.5 or figure 12) i s often not consistent with the value obtained from the c o e f f i c i e n t of v a r i a t i o n (equation 3.10). These departures from conventional b r i t t l e fracture theory are discussed below. 3.4 DIFFERENT SIZE EFFECTS IN DIFFERENT DIRECTIONS Size e f f e c t s result from v a r i a b i l i t y of strength properties within a, material, and are much more pronounced in b r i t t l e materials than in d u c t i l e materials. For a more homogeneous material such ' as concrete or metal, the v a r i a b i l i t y in material properties tends to be the same in a l l directions within a member, and between members. Timber i s very d i f f e r e n t in that v a r i a b i l i t y within a member depends on d i f f e r e n t b i o l o g i c a l and environmental factors in d i f f e r e n t orientations. V a r i a b i l i t y between members depends a d d i t i o n a l l y on the differences betweeen d i f f e r e n t trees and growing s i t e s . V a r i a b i l i t y along the length of a piece of timber is related to the frequency of knots and other defects up the height of a tree. V a r i a b i l i t y within a cross section depends on the type and location of defects in the width or depth of 61 any board, and on environmental and other factors that may have affected the properties of layers of wood produced from year to year. Because d i f f e r e n t size e f f e c t s result from dif f e r e n t sources, they w i l l be considered separately here, each being investigated for both tension and compression behaviour, and the results w i l l be used to discuss bending behaviour. 3.4.1 Size Effect Terminology Size effect terms are defined below. The term length e f f e c t refers to the phenomenon that when boards of d i f f e r e n t lengths are tested under similar loading conditions, long boards tend to be weaker than shorter boards. The term width effect i s used to describe the effect of cross section width (or breadth) on the strength of'a bending member. The term "depth" refers to the depth (or height) of a bending member. For a x i a l l y loaded members the d i s t i n c t i o n i s not clear, but in general the larger cross section dimension w i l l be considered as depth. The term depth effect describes the phenomenon that for several members of given length and width, loaded in geometrically similar ways, the stress at f a i l u r e tends to decrease as member depth i s increased. The term s t r e s s - d i s t r i b u t i o n effect describes the phenomenon that, for members of given size, the maximum stress at f a i l u r e tends to decrease as the highly stressed proportion of the depth is increased. 62 The depth effect and the s t r e s s - d i s t r i b u t i o n effect are very c l o s e l y related and could perhaps be quantified using the same parameters. However, for timber the parameters are sometimes d i f f e r e n t , so these two effects w i l l be considered separately. 3.5 LENGTH EFFECT 3.5.1 Theory The strength of a timber board varies along i t s length. One eff e c t of th i s v a r i a b i l i t y i s that the strength of long boards i s less than the strength of shorter boards, th i s phenomenon occurring not only for average values, but throughout a d i s t r i b u t i o n of strength values. Axial tension and compression are the most simple loading cases to consider, but apparently no results of in-grade a x i a l tension or compression tests for varying lengths have been previously reported. Almost a l l in-grade bending tests have been performed with a constant span-to-depth r a t i o of 17, and third-point loading. Results from these tests combine both length and depth e f f e c t s so that neither can be estimated independently. If a timber board under a x i a l loading i s assumed to be a chain-like material where the strength of any length i s the strength of i t s weakest l i n k , the theory already described can be used to quantify the length e f f e c t . Observations from i n -grade a x i a l tension and compression tests confirm that f a i l u r e i s usually at a single cross section. This allows the weakest 63 l i n k theory to be used for length e f f e c t s even though the f a i l u r e may not always be a b r i t t l e fracture. The theory described in equations 3.2 to 3.5 can be used to quantify length e f f e c t s and equation 3.5 can be si m p l i f i e d for design purposes to ^ - ( I 1 ) ( 3 ' 1 1 )  x2 L l where x, and x 2 are the strengths of members of length L, and L 2 respectively, at any point in the d i s t r i b u t i o n of strength and k, i s the length effect parameter. Note that k, may have di f f e r e n t values for tension and compression. 3.5.2 Assumptions for Length Eff e c t The assumption of segment strength being randomly dis t r i b u t e d along the length of a board may not always be v a l i d , depending on the nature of the wood. For example, fast grown trees of some species may have long clear sections of wood between large clusters of knots, and the strength of one segment may not then be independent of the strength of adjacent segments. A study by Riberholt and Madsen (1979) has investigated the d i s t r i b u t i o n of material strength along a board for several European species. They considered the distances between defects to be governed by a Poisson type of process, and used a second d i s t r i b u t i o n to describe the strength at those defects. The second d i s t r i b u t i o n was suggested to be a Weibull or log-normal d i s t r i b u t i o n . If a Weibull d i s t r i b u t i o n 64 i s used to describe the strength at defects, the predicted length e f f e c t can be quantified in exactly the same way as done in t h i s study, except that i t w i l l only be applicable for lengths long enough to contain a certain number of defects. Riberholt and Madsen's technique has not been pursued in t h i s study because visu a l inspections of the boards tested generally showed a large number of small •knots and other defects closely spaced unevenly along each board. This i s c h a r a c t e r i s t i c of the Canadian spruce-pine-fir (SPF) species group which does not contain regularly spaced large defects which are often found in faster grown trees. In view of other uncertainties, the assumption of wood strength varying in a random manner along each board i s considered to be a reasonable one. Another assumption in this development is that the lower bound on segment strength i s zero, leading to the two-parameter Weibull formulation. This assumption implies zero strength for i n f i n i t e length, which may be appropriate for low grade timber, but may be unreasonable for the highest grades. The assumption i s , i f anything, conservative and i s considered to be reasonable provided that the theory i s not used for extrapolation far beyond the range of data. 65 3.6 DEPTH EFFECT If depth effect i s assumed to be a b r i t t l e fracture phenomenon, the same theory used for length'effeet can be used to quantify depth e f f e c t , and equation 3.5 for depth becomes JL = (-1) 2 (3.12) where x, and x 2 are the f a i l u r e stresses of members of depths d, and d 2 respectively, and k 2 i s the depth ef f e c t parameter. Depth effects for commercial timber have been investigated in some d e t a i l for bending and for a x i a l tension, but only in a few tests for ax i a l compression. The trend of decreasing f a i l u r e stress with increasing depth has been described in Chapter 2. For a x i a l tension and compression a l l timber subjected to in-grade testing, has been of the same width (nominal 38mm), so the ef f e c t s of varying two cross sectional dimensions are not known. In a x i a l tension tests, f a i l u r e s are usually sudden b r i t t l e fractures, so b r i t t l e fracture theory is considered to be appropriate. In a x i a l compression tests, many f a i l u r e s exhibit some d u c t i l i t y , so b r i t t l e fracture theory may be a less suitable explanation for observed behaviour, but the same formula (equation 3.12) can be used to quantify the e f f e c t . Most bending tests have been c a r r i e d out at constant span-to-depth r a t i o s in which case i t i s not possible to separate length and depth e f f e c t s . However, the two effects can be separated i f bending tests are c a r r i e d out at di f f e r e n t span-66 to-depth r a t i o s as described in Chapter 5. 3.7 STRESS-DISTRIBUTION EFFECT 3.7.1 Tension Zone The s t r e s s - d i s t r i b u t i o n effect for tension stresses can be explained using b r i t t l e fracture theory even though some of the assumptions involved in that theory may not be s a t i s f i e d very p r e c i s e l y . This theory i s based on the assumption that the depth of any member i s made up of a large number of small elements, the strength of each being randomly selected from a d i s t r i b u t i o n of known parameters. The theory further assumes that the material i s perfectly b r i t t l e such that the f a i l u r e of any one of these tiny elements w i l l cause f a i l u r e of the member. This being the case, the two-parameter form of equation 3.6 can be re-written for depth, not considering any length e f f e c t s or width e f f e c t s . F(x) = 1 - e x P { - ^ - / (|)k3dy} (3.13) 1 d where x is the f a i l u r e stress, y i s the depth co-ordinate, d is the depth of the member and 6y i s the depth of a single element. k 3 and m are the shape and scale parameters, respectively, of the two-parameter Weibull d i s t r i b u t i o n of strength of the constituent elements of depth referred to above. k 3 w i l l be referred to as the s t r e s s - d i s t r i b u t i o n parameter. The integration only applies to that portion of the cross section stressed in tension. 67 Consider the case shown i n f i g u r e 13(a). For a x i a l F i g u r e 13 - Tension s t r e s s d i s t r i b u t i o n s t e n s i o n , where s t r e s s i s constant over the depth as shown in f i g u r e 13(b), x=ft, and equation 3.12 becomes F ( x ) = 1 - e x p { -(3.14) For the case shown i n f i g u r e 13(c) or (d), where s t r e s s e s vary l i n e a r l y over the depth, the s t r e s s x at any depth y from the n e u t r a l a x i s i s x = Z _ f cd m o r x = r f where r = ^—r m cd and equation 3.13 becomes 68 F ( r V - 1 - - , | - ? J C ^ 3 * } ( 3 - 1 6 > 1 r Under conditions of loading such as shown in figure 13(c) or (d) the extreme fibre stress at f a i l u r e , fm, can be calculated as a r a t i o of the a x i a l tension strength f t . For any prob a b i l i t y of f a i l u r e (for example, the median strength with F(x)=0.5) equations 3.14 and 3.16 can be equated and rearranged to give k " 1 / k 3 f m = 1/ r dr] f t ( 3. 1 7) r For the neutral axis within the member depth as shown in figure 13(c), t h i s becomes f = [ ' ] 3 f (3.18) m L k^+li t which for the special case of neutral axis at mid-depth becomes 1/k, f = [ 2(k,+l)l J f (3.19) m 3 t The l i n e in figure 14(a) marked c=0.5 shows thi s equation plotted for a range of values of k 3. This relationship can be used to estimate the value of k 3 from the ra t i o of strength of a x i a l tension tests to bending tests. If the test specimens are of d i f f e r e n t lengths, then length corrections should be made f i r s t . 69 J f f Q -J — i — i — i — i — i — i — i — i i i i i i i i i i i _ 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 !8JJ 50.3 S T R E S S D I S T R I B U T I O N P A R A M E T E R k, a o (a) i 1 1 1 1 1 1 1 1 1 1 1 r c=ao i r i -cc 0.0 2.3 4.0 €.0 8.0 10.0 12.0 14.0 S T R E S S D I S T R I B U T I O N P A R A M E T E R k 3 (b) 16.0 LB.O 20.0 Figure 14 - Ratio of a x i a l tension strength to maximum stress in extreme fi b r e Equation 3.19 assumes that the neutral axis i s at mid-depth. The two dotted l i n e s in figure 14(a) show that there i s not a l o t of difference i f the neutral axis i s at 40% or 45% of the depth from the tension surface. When the neutral axis is outside the beam depth as shown in figure 13(d), the integration y i e l d s 70 f = { c u _ l ^ " 1 ^ ( 3 . 2 0 ) m 3 J J t F i g u r e 14(b) shows the r a t i o of a x i a l t e n s i o n s t r e n g t h to maximum s t r e s s i n the extreme t e n s i o n f i b r e f o r a wide range of n e u t r a l a x i s depths, using equations 3.19 and 3.20. 3.7.2 Compression Zone The assumptions of b r i t t l e f r a c t u r e theory cannot be a p p l i e d to a d u c t i l e m a t e r i a l l i k e wood in compression. However i t i s p o s s i b l e that some other mechanism c o u l d r e s u l t in the maximum compression s t r e s s i n c r e a s i n g as the percentage of the depth s t r e s s e d i n compression decreases. For example, a wide face c e n t r e l i n e knot c o u l d reduce the a x i a l compression s t r e n g t h at the c r o s s s e c t i o n without having any s i g n i f i c a n t e f f e c t on bending s t r e n g t h . The " f i b r e support theory" d e s c r i b e d i n Chapter 2 i s a l s o c o n s i s t e n t with a within-member depth e f f e c t i n compression. As a mathematical t o o l to i n v e s t i g a t e t h i s phenomenon, the theory developed above fo r t e n s i o n w i l l be extended to i n c l u d e compression behaviour. Assume that wood in compression has an e l a s t o - p l a s t i c s t r e s s - s t r a i n r e l a t i o n s h i p . Consider the c r o s s s e c t i o n shown in f i g u r e 15(a). When s t r e s s e d i n a x i a l compression wood y i e l d s at a s t r e s s fc as shown in f i g u r e 15(b). In the cases shown in f i g u r e s 15(c) and (d) the y i e l d s t r e s s fern i s g r e a t e r than the s t r e s s fc depending on the percentage of the s e c t i o n depth that i s h i g h l y s t r e s s e d . The same procedure used i n t e n s i o n can be used here to 7 1 give (a) (b) (c) (d) Figure 15 - Compression stress d i s t r i b u t i o n s f - [a + 3 f cm L k ^ + l J c (3.21 ) - u + b^>k3+V1/k3 f cm k 3 + l (3.22) for the cases shown in figures 15(c) and (d) respectively, where the ratios a, b, and c are defined in the figure. 3.8 WIDTH EFFECT Conventional b r i t t l e fracture theory suggests that f a i l u r e stresses should decrease for any increase in member volume. For wood bending members a size effect has been, observed with length and depth of member, but not with width. For clear wood, Bohannan(1966) found that modulus of rupture was independent of width, and explained t h i s observation using 72 an i n t u i t i v e argument. He pointed out that bending stresses vary as the square of the depth of a member, but only l i n e a r l y with the width, so that some small i n i t i a l reduction in depth would be more l i k e l y to cause a "cascading" type of member f a i l u r e than a reduction in width. An extension to this argument can be made i f shear stresses are considered, and i t is assumed that a fl e x u r a l f a i l u r e i s i n i t i a t e d in some way by a l o c a l shear f a i l u r e . In a beam consisting of p a r a l l e l longitudinal f i b r e s , subjected to a shear force, shear i s transferred from fibres to adjacent ones above and below, but not to adjacent fibres side by side. For this reason an increase in beam depth may contribute to a size effect more than an increase in beam width. These arguments are not unreasonable, but f a i l to explain a cross section effect in ax i a l tension, which is of similar or greater magnitude than that observed in bending. Another possible explanation for the absence of a width effect in clear wood could be that the c e l l s in each new layer added to a tree's circumference are of very similar properties, but that v a r i a b i l i t y between layers and between annual rings causes depth e f f e c t s . This theory i s only s t r i c t l y applicable i f a l l members are "quarter-sawn" or "edge sawn" as shown in figure 16., There i s no suggestion that a l l members tested have been sawn in thi s way, but the theory may s t i l l contribute to a size effect which i s affected by only one cross sectional dimension, not two. For commercial timber, Madsen and Stinson (1982) have 73 Figure 16 - Quarter-sawn board found a s i g n i f i c a n t increase in modulus of rupture with increasing width, contrary to the predictions of b r i t t l e fracture theory. They attempted to explain these observations by looking at the projections of knots allowed by grading rules for various widths of timber, but were unable to separate grading effects from b r i t t l e fracture e f f e c t s . No results have been reported for in-grade bending tests of structural timber about the weak axis. Results for strong axis bending would suggest that the f a i l u r e stress for weak axis bending should be s i g n i f i c a n t l y greater, because of additive factors of decreasing depth and increasing width. Such an effect was found for some but not a l l sizes tested in th i s study, as described in Chapter 5. Width effects are pursued no further in this thesis, because a l l the test results for combined loading are for a single width of 38mm with bending about the strong axis. 74 3.9 EFFECT OF GRADING RULES When investigating size e f f e c t s in timber members with defects, i t is very d i f f i c u l t to separate the eff e c t of member sizes from the eff e c t of grading rules. Consider for example the effect of member depth. Some defects are limited to a certain size independent of member depth, others are permitted to be a certain percentage of the depth, so strength differences between di f f e r e n t sizes can be affected by the way in which grading rules control such defects. A similar problem exists with member length. When graded timber i s cut into short pieces to investigate length e f f e c t s , many of the short pieces become >a higher grade than- the o r i g i n a l pieces because there may be only one l i m i t i n g defect in a given long board. Conversely, i f a defect (such as wane) is limited to some percentage of the member length, then cutting into smaller lengths may down-grade a board. For these reasons, investigations into size effects using in-grade testing must be reported and interpreted c a r e f u l l y . In t h i s study i t i s simply stated that the results obtained are ch a r a c t e r i s t i c of material purchased as Number 2 and Better Grade in 4 .88m lengths, subsequently cut into shorter lengths for t e s t i n g . 75 3.10 SUMMARY This chapter has reviewed conventional size effect theory which i s based on a s t a t i s t i c a l strength theory for b r i t t l e materials. The theory can be modified and improved for application to timber, with size effects in d i f f e r e n t directions receiving separate consideration. The modified theory has been developed and discussed. 76 IV. EXPERIMENTAL PROCEDURES This chapter describes experimental procedures for a number of d i f f e r e n t tests c a r r i e d out to provide input to t h i s thesis. The experiments served a number of d i f f e r e n t purposes, and the results w i l l be referred to in several subsequent chapters. Most of the results relate to the strength model which w i l l be described in Chapter 6. Some results provide basic input to the model, some are used to c a l i b r a t e the model, and many more are used for v e r i f i c a t i o n . Readers who wish to follow the theoretical development of the thesis may proceed d i r e c t l y to Chapter 6. 4.1 EXPERIMENTAL STAGES The experimental work described in this thesis was carried out in fiv e stages under a co-operative testing programme. The experimental stages are summarized in Table I, and the sizes and lengths tested in stages 1 to 5 are summarized in Table I I . Stage 1 which was i n i t i a t e d and supervised by Dr. Ken Johns of the University of Sherbrooke, Quebec, consisted of e c c e n t r i c a l l y loaded compression tests on 38x140mm boards at the University of B r i t i s h Columbia (UBC) in early 1981. A detailed analysis of these results by the author in mid-1981 i d e n t i f i e d serious discrepancies between current theory and actual behaviour, stimulating the subsequent work described in t h i s thesis. Stage 2 which was supervised by the author, consisted of 7 7 Stage Material (Source) Size Test Location, Date Description 1 SPF (BC) 38x140 UBC (1981) Eccentric compression 2 SPF (BC) 38x140 UBC (1982) Bending A x i a l tension Axial compression Combined bending & tension 3 SPF (Quebec) 38x89 Sherbrooke (1982-83) Eccentric compression 4 SPF (Quebec) 38x89 UBC (1983) Bending A x i a l tension (short & long) Axial compression (short & long) Eccentric tension 5 SPF (BC) 38x140 UBC (1983) Bending ( d i f f e r e n t lengths & load configurations) Tension (short and long) Table I - Summary of experimental stages bending tests, compression tests, tension tests, and combined bending and tension tests, a l l using 38x140mm boards from the same sample as the Stage 1 te s t s . These tests were carried out at UBC in early 1982. Stages 1 and 2 have been summarized by Johns and Buchanan (1982). Stage 3 was carr i e d out on the same machine as Stage 1, relocated to the University of Sherbrooke, Quebec. This stage, which was supervised by Dr. Johns and Mr. Raymond Bleau, consisted of e c c e n t r i c a l l y loaded compression tests on 38x89mm boards, in late 1982 and early 1983. More d e t a i l s are 78 Material Approx. length (o) Eccentric Axial Loading Axial Compression Axial Tension Combined Tension & Bending Bending Eccentricity (nnn) 2 12 18 39 75 202 COMPRESSION 38x140 0.21 X SPF 0.91 X X X X X (BC) 1.82 X X 2.44 X X X X X X 3.00 X X X 3.35 X X 4.27 X X 0.135 X 0.45 X X X X X 0.91 X 0.84 X 1.30 X X X 38x89 1.50 X SPF 1.80 X X X X X (Quebec) 2.0 X X 2.3 X X X 3.2 X X X X X TENSION 0.45 X X X X Each x represents a test of approximately 100 specimens. Table II - Summary of lengths and sizes tested (stages 1 to 4) provided by Bleau(l984). Stage 4 was ca r r i e d out at UBC by the author in early 1983. This stage repeated the tests of Stage 2 for 38x89mm boards from the population of the Stage 3 t e s t s . It also included short compression tests, short tension tests, short bending tests, machine stress grading and e c c e n t r i c a l l y loaded tension tests, a l l described in more d e t a i l below. Stage 5 was a series of tests to investigate the effects of member length and load configuration on the bending strength of 38x140mm timber members. These tests were 79 designed and supervised by Professor Borg Madsen as part of a separate research program (Madsen 1983). Some preliminary results which are relevant to th i s study w i l l be described in th i s thesis. 4.2 TEST MATERIAL 4.2.1 Species The experiments described in th i s thesis were carried out on boards from three separate groups. The f i r s t group for stages 1 and 2 was 38x140mm (nominal 2x6 inch) spruce-pine-fir (SPF) timber from central i n t e r i o r B r i t i s h Columbia. The second group for stages 3 and 4 was 38x89mm (nominal 2x4 inch) SPF timber from Quebec. The t h i r d group of boards for stage 5 were SPF from northern i n t e r i o r B r i t i s h Columbia. No attempt was made to quantify the actual species in each group, but a l l three groups appeared to be predominantly spruce. The test results are believed to be c h a r a c t e r i s t i c of the commercial SPF species group in B r i t i s h Columbia and in Quebec respectively. 4.2.2 Grading The f i r s t two groups were purchased as "Number Two and Better" grade in 4.88 m lengths. After cutting to length, a l l the boards were regraded by a q u a l i f i e d grader. Almost two-thirds of the i n i t i a l lengths were of Select Structural Grade. After cutting to short lengths more than three-quarters of the sample became Select Structural Grade. For example, a long Number Two board with one defect could produce several Select Structural segments. 80 In the data analysis a preliminary attempt was made to separate by grade but there were too many Select Structural grade boards to produce sa t i s f a c t o r y results for the lower grades. A small number of boards down-graded for large knots or serious structural defects were noticeably weak, but many boards down-graded for wane or cosmetic defects were very strong. A l l the. results described are for a l l grades combined. These results are believed to be c h a r a c t e r i s t i c of "Number Two and Better" material as graded in 4.88m lengths. The boards tested in stage 5 were purchased in two separate grades, Select Structural and Number 2. The a x i a l tension tests and the flat-wise bending tests were carr i e d out on Number 2 grade only. The bending tests were carr i e d out separately on both grades. 4.2.3 Moisture Content A l l boards were purchased k i l n - d r i e d . No climate-controlled storage areas were available in either testing location. A l l the boards were kept indoors, and moisture content recorded at the time of testing using an e l e c t r i c a l resistance moisture meter. For the Stage 1 tests, the moisture content varied between 11% and 16% with an average value of 14.25%. At the time of the Stage 2 tests the average moisture content was 13.8%. The Stage 3 and 4 testing was ca r r i e d out over a period 81 of several months, and there were some minor moisture content changes during this period. The moisture content varied from 7% to 13%, with an average value of 10.4%. The test results are believed to be representative of material of thi s moisture content range. 4.2.4 Sample Selection Stages 1 and 2 (38x140mm) On a r r i v a l in the laboratory, s u f f i c i e n t timber for a l l the Stage 1 testing was designated, and samples for each length and e c c e n t r i c i t y were selected randomly from that material. The Stage 2 testing was carried out from the remainder of the shipment. Stages 3 and 4 (38x89mm) The whole of this group was randomly assigned to samples for combined bending and compression testing in Stage 3. When i t was decided to not test every length at every e c c e n t r i c i t y , approximately 300 boards 2.9m long and 200 boards 1.9m long became available for shipping to UBC for Stage 4 testing. For the Stage 4 testing, two samples of 90 boards 2.9m long were selected for the long tension and compression tests. Two samples of 90 boards 1.9m long were selected for bending t e s t s . One sample was tested f u l l length, the other was cut in two halves, one half length tested edgewise, the other flatwise. Ten boards 2.9m long were cut up for short compression tests. The remainder of the boards were cut into 0.97m lengths for combined bending and tension tests. Four samples of 80 boards (for four e c c e n t r i c i t i e s ) were selected 82 at random such that no sample had more than one segment from any board. Stage 5 (38x140mm) The samples for d i f f e r e n t tests were selected randomly on the basis of long-span f l e x u r a l modulus of e l a s t i c i t y as described by Madsen(1983). 4.3 SAMPLE SIZES AND CONFIDENCE 4.3.1 Sample Sizes The intention for most tests was to have a sample size of 100. In practice the useful sample sizes were usually s l i g h t l y less due to minor problems. The sample size for combined bending and tension in Stage 4 was reduced to 80, in order to obtain data at four e c c e n t r i c i t i e s from the limited number of boards available. The sample sizes used allow calculation of mean or median, values with considerable confidence, and upper and lower t a i l values with s i g n i f i c a n t l y less confidence. The 5th and 95th percentile values (.05 and .95 quantiles) have been used throughout t h i s study as indicators of behaviour at the t a i l s of the d i s t r i b u t i o n . Percentiles can be calculated d i r e c t l y from a cumulative plot of the ranked data, either using the two points either side of the percentile, or a weighted average of several points in the v i c i n i t y . An alternative approach, using a l l of the data, i s to f i t an appropriate d i s t r i b u t i o n a l model to the data, then to 83 calculate the percentile values from the f i t t e d d i s t r i b u t i o n . This method has been used throughout t h i s study, using the Weibull d i s t r i b u t i o n . 4.3.2 Weibull D i s t r i b u t i o n The Weibull d i s t r i b u t i o n i s a f l e x i b l e d i s t r i b u t i o n which has been widely used for studying the strength of wood and other materials. The Weibull d i s t r i b u t i o n i s most appropriate for describing material strength properties because for large sample sizes i t is the asymptotically exact d i s t r i b u t i o n for extreme values from any i n i t i a l d i s t r i b u t i o n that i s bounded in the d i r e c t i o n of the extreme value. Material strength f i t s t h i s description because i t tends to be governed by the strength of the weakest one of a large number of elements, p a r t i c u l a r l y when b r i t t l e f a i l u r e s occur.. None of the elements can have negative strength, so there i s a lower bound at zero (or hi-gher in some cases). In t h i s study, three-parameter Weibull . d i s t r i b u t i o n s have been f i t t e d to experimental data by estimating the Weibull parameters with maximum li k e l i h o o d equations. 4.3.3 Confidence Calculation for Quantiles Calculation of confidence intervals depends on the type of model f i t t e d to the data, and on the number of data points. An estimate of a quantile value can be assumed to be asymptotically normally d i s t r i b u t e d , for large sample size, regardless of the d i s t r i b u t i o n a l form of the parent population, with variance given by Bury (1975), as 84 v a r ( x ) = ( 4 . 1 ) q n [ f ( x q ) ] 2 where q i s the q u a n t i l e under c o n s i d e r a t i o n , n i s the sample s i z e , xq i s the q t h q u a n t i l e of the v a r i a b l e x, and f ( x q ) i s the p r o b a b i l i t y d e n s i t y f u n c t i o n ( p . d . f . ) . For the W e i b u l l d i s t r i b u t i o n x q = x o + m [ l n ( l / ( l - q ) ) ] 1 / k ( 4 . 2 ) and x - x , . x^ - x „ ; - 3 ^ ) k _ 1 e x p { - ( - ^ — m v m m c , . k r x q ~ x o ^ k - i r r x q %M ( 4 3 ) which combine t o g i v e f ( x q ) = i [ l n ( l / ( l - q ) ) ] ( 1 - 1 / k ) e x p { - l n ( l / ( l - q ) ) } ( 4 . 4 ) where x 0 , k, and m a r e the l o c a t i o n parameter, shape parameter and s c a l e parameter r e s p e c t i v e l y , of the W e i b u l l d i s t r i b u t i o n . Suppose we want t o be 100 p % c o n f i d e n t t h a t the e s t i m a t e d q u a n t i l e i s w i t h i n an i n t e r v a l of p l u s - o r - m i n u s 100 r % of the p o p u l a t i o n q u a n t i l e , then t r a n s f o r m i n g t o a s t a n d a r d normal v a r i a b l e 2 r x = 2 z ,„ / v a r ( x ) ( 4 . 5 ) q P /2 q ' where z p / 2 * s t^ i e v a l u e °f the s t a n d a r d normal v a r i a b l e such t h a t the a r e a under the s t a n d a r d normal c u r v e i s p / 2 . 85 If we, have a sample of size n, we can calculate the confidence p by rearranging these equations to give Zp/2 = f ( V / ^ q 7 <4'6> The value of Zp/2 c a n D e referred to standard normal tables to obtain the confidence p, for a two-sided test. Of course the Weibull parameters are not known in advance, but values obtained from similar tests on the same material can be used to make a reasonable estimation of confidence. Calculations were made to estimate the confidence with which quantiles, calculated from the f i t t e d Weibull d i s t r i b u t i o n , were in the int e r v a l plus-or-minus 10% of the population values, for a sample size of 90. Weibull parameters were obtained from both bending and tension tests of SPF timber in two sizes (38x89mm and 38x135mm) for both Select Structural and No. 2 grades. As an example consider the figures for No. 2 grade 38x89mm timber in bending. The location, shape and scale parameters are 10.1, 1.70 and 34.3, respectively. With r=0.l0, n=90 and q=0.05, equation 4.6 gives a value zp/2 °^ 0.97, which corresponds to an area of 0.335 under the standard normal curve, or a confidence of 67% that the estimated 5th percentile i s within the int e r v a l between 0.9 times and 1.1 times the population value. This procedure is i l l u s t r a t e d in figure 17. The s o l i d l i n e i s the p.d.f. of the underlying d i s t r i b u t i o n (the parent 86 00 CN - 5%ile m ec m o or CN o.o .1 'IIS 50%ile Area p f I p.d.f of underlying distribution 95%ile 'I \ p.d.f of quantile estimator, using: // 'y order statistics normal Y approximation I " — I 1 60.0 70.0 STRESS (flPfl) Figure 17 - P.d.f. of quantile estimators population). The dotted l i n e s are the p.d.f.'s of the estimated quantiles at the 5th, 50th, and 95th percentile l e v e l s , for a sample size of 59, using two d i f f e r e n t methods of c a l c u l a t i o n . The two dotted l i n e s are very close, showing that the normal d i s t r i b u t i o n is a very good approximation to a more exact curve obtained using order s t a t i s t i c s (Bury 1975), even for t h i s case with a small sample size and low shape parameter. The terms p and r, referred to above, are i l l u s t r a t e d for the 50th percentile case only. Using a l l the samples referred to above, the following 87 conclusions can be drawn regarding the tests in t h i s study: 1. 5th percentile values have been estimated with approximately 70% confidence that they are within 10% of population values. 2. Median values (50th percentiles) have been estimated with approximately 90% confidence that they are within 10% of population values, or approximately 65% confidence that they are within plus-or-minus 5%. Confidences on mean values are s i m i l a r . 3. 95th percentiles have confidence l i m i t s about midway between the values quoted for 5th and 50th percentiles. These confidence intervals are considered quite satisfactory in view of the variable nature of wood and various uncertainties regarding grade, species, source of supply and testing procedures. 4.4 TEST PROCEDURES 4.4.1 Bending Bending tests were carried out on each group of timber as shown in Table II, to obtain data for c a l i b r a t i o n of the strength model. Several lengths were tested to investigate length e f f e c t s in bending. Bending tests were performed on an Olsen 900 kN universal 88 testing machine. Load was applied mechanically at a controlled displacement rate of approximately 30 mm/min which produced f a i l u r e in about one minute. Lateral supports near the load points prevented l a t e r a l buckling. Maximum load was recorded from the load indicator attached to the machine. Figure 18 - Loading arrangement for bending test A l l the bending tests l i s t e d in Table II were car r i e d out with simple supports and one-third point loading as shown schematically in figure 18. A l l of these tests had a span-to-depth r a t i o of 17, except for the short span test of 38x89mm boards which was tested at a r a t i o of 9.5. One sample of boards was also tested in flat-wise bending at th i s span. For the stage 5 bending tests reported in th i s thesis, the boards were tested with simple supports and three load configurations; two loads at one-third points, two loads one-quarter span apart, and a central point load. Each of these tests was car r i e d out at two spans of 1.5m and 3.0 m, and separately for Select Structural and Number 2 grades. One sample of Number 2 grade boards was tested in flat-wise 8 9 .bending at the shorter span. 4.4 .2 Axial Tension a. Long Boards Tests of long boards were carried out under similar conditions to previous in-grade a x i a l tension tests (Madsen and Nielsen 1978b). The results of these tests are used as basic input to the strength model described in Chapter 6. Tension tests were car r i e d out on an a x i a l loading machine described by Madsen and Nielsen (I978d) and shown schematically in figure 19(a). This machine has f r i c t i o n grips 450mm long at each end, with polyurethane-covered steel plates which grip the board when forced together by a hydraulic jacking system. The specimen i s stressed in tension when the length of the whole machine is increased, using a second hydraulic jacking system. Failure load i s recorded from a c a r e f u l l y c a l i b r a t e d hydraulic f l u i d pressure gauge. The f r i c t i o n grips are r i g i d l y mounted to prevent rotation about any axis. The 38x140mm boards were tested at the machine's standard length with a free length of 3.0m between grips. The 38x89mm boards were tested over a free length of 2.0 m, a l l other d e t a i l s remaining the same. The grip pressure was controlled manually throughout the tests, being increased gradually i f the specimen began to s l i p in-the grips, with care not to cause excessive crushing perpendicular to the grain. The loading was at a uniform displacement rate controlled by the e l e c t r i c pump on the hydraulic jacking system. Failu r e generally occurred in about 90 Fi g u r e 1 9 - Loading arrangement for a x i a l t e s t s (a) t e n s i o n (b) compression 30 seconds i f there was no s l i p p a g e i n the g r i p s . Although these t e s t are r e f e r r e d to as a x i a l t e n s i o n t e s t s , there was probably some bending induced i n most boards due to v a r i a t i o n s i n wood p r o p e r t i e s w i t h i n each board. Any such bending has been neg l e c t e d , and the t e n s i o n s t r e s s in the boards has been c a l c u l a t e d by simply d i v i d i n g the a x i a l f o r c e by the c r o s s s e c t i o n a l area. b. Short Boards Short boards were t e s t e d i n a x i a l t e n s i o n to o b t a i n i n f o r m a t i o n on l e n g t h e f f e c t s . The ten s i o n t e s t i n g machine d e s c r i b e d above was mo d i f i e d to accept any l e n g t h as small as 0 . 9 m . The g r i p s at one end were used unmodified. At the mo d i f i e d end, new g r i p s were made from a p a i r of s t e e l p l a t e s faced with coarse sandpaper cut from a heavy duty sanding 91 b e l t . These plates were clamped together by the same hydraulic system used for the o r i g i n a l grips. The plates were connected to the rest of the machine through a tension member of variable length. The new connection was not as r i g i d as the o r i g i n a l one, as i t allowed rotation about the strong axis of the board and some torsional rotation, but previous tests indicate no s i g n i f i c a n t difference in results from th i s source (Madsen and Nielsen I978d). 4.4.3 Axial Compression a. Long Boards Axial compression tests of long boards were carr i e d out under similar conditions to previous in-grade compression tests. The results of these tests provide basic input information for the strength model. . / Compression tests of long boards were performed in the same machine as the tension tests, with the loading jacks reversed. A system of l a t e r a l supports faced with teflon pads prevented l a t e r a l buckling in either d i r e c t i o n , as described by Madsen and Nielsen (1978c) and i l l u s t r a t e d schematically in figure 19(b). The l a t e r a l supports were located with just enough clearance for the boards to be inserted e a s i l y without adjustment for each board. No attempt was made to force each board into a perfectly straight condition, so a very small amount of bending moment may have been present in addition to the applied a x i a l load. Any such bending has been neglected. 92 b. Short Segments A number of boards were cut into short segments to investigate the variation in compression strength within each board, as a means of quantifying length effects in compression. The segment length was 210mm for the 38x140mm boards, and 140mm for the 38x89mm boards. These segments were loaded in a x i a l compression p a r a l l e l to the grain. The testing machine was an Amsler universal testing machine of 1.0 MN capacity. Load was applied hy d r a u l i c a l l y through steel platens which were not free to rotate. Platen displacement and load were recorded continuously and plotted on an x-y recorder. Loading rate was approximately 4 mm per minute, which corresponds to a str a i n rate of approximately 0.02mm/mm per minute. 4.4.4 Eccentric Compression Eccentric compression tests were performed in a special purpose testing machine described by Johns and Buchanan (1982). Each end of the test specimen f i t t e d snugly into a steel "boot" through which a x i a l compression was applied at a pre-determined e c c e n t r i c i t y about the strong axis as shown in figure 20(a). A system of r o l l e r bearings and l a t e r a l supports prevented buckling about the weak axis. The machine can test lengths up to 5.0m at e c c e n t r i c i t i e s up to 202mm. Equal end e c c e n t r i c i t i e s were used for a l l t e s t s . The actual lengths and e c c e n t r i c i t i e s tested are shown in table I I . Axial load was provided by a hydraulic jack at one end, connected in series with a load c e l l . Lateral displacements 93 Eccentricity varies 2mm to 202mrn Test specimen 140 mm or 8 9 mm (a) (b) F i g u r e 20 - E c c e n t r i c a x i a l l o a d i n g (a) Compression (b) Tension were measured at three l o c a t i o n s along each specimen with l i n e a r v a r i a b l e displacement t r a n s d u c e r s (LVDT's). The load c e l l and LVDT's were scanned throughout the t e s t by a computer based data a c q u i s i t i o n system. Load was a p p l i e d at a constant displacement rate u n t i l maximum l o a d had c l e a r l y been reached. Time to f a i l u r e averaged about one minute. 94 4.4.5 Combined Bending and Tension Combined bending and tension tests were car r i e d out in two separate and quite d i f f e r e n t experiments. a. Bending Followed by Tension As part of the Stage 2 testing approximately 200 38x140mm boards 4.8m long were tested to f a i l u r e in combined bending and tension as described by Johns and Buchanan (1982). These tests were carried out in the machine already described for the long tension and compression tests. As shown schematically in figure 21(a) each board was i n i t i a l l y loaded in bending as a simply supported beam with a gravity load at the one-third points, before the ends were clamped with the hydraulic grips and a x i a l tension load . applied to produce f a i l u r e , as shown in figure 21(b). The tension load was recorded as in the a x i a l tension tests. The bending moment at f a i l u r e was calculated by superposition of three separate e f f e c t s ; moment due to gravity load bending, moment due to e c c e n t r i c i t y of a x i a l load, and secondary moment developed because the ends of the boards were p a r t i a l l y restrained against rotation during the te s t . These boards were tested in two groups with d i f f e r e n t levels of i n i t i a l ' gravity load. Several boards broke in Bending before any tension force was applied. For the group with high gravity loads about half broke in this manner. This testing procedure was not very s a t i s f a c t o r y because i t was not possible to increase bending moment and a x i a l load concurrently and because some sli p p i n g in the grips gave 95 ( b ) Figure 21 - Combined bending and tension test (a) f i r s t stage (b) second stage erroneous moment readings for the stronger boards in the sample. b. Eccentric Tension A much more sati s f a c t o r y testing procedure was used for the 38x89mm boards in Stage 4. Combined bending and tension tests were performed on four groups of 80 boards, each 970mm long. The boards were stressed in tension with equal end e c c e n t r i c i t i e s as shown schematically in figure 20(b), a di f f e r e n t e c c e n t r i c i t y for each group. This type of in-grade testing has not been reported elsewhere. A 275mm length at each end of each board was clamped between sandpaper-faced steel plates with the modified hydraulic jacking system used in the long tension tests. The bottom grip plates were connected to the floor with a saddle 96 which allowed rotation about the strong axis but no other movement. The top grip plates were connected to the loading system with a similar saddle which allowed rotation about the strong axis, and about the longitudinal axis. The load was applied through a 450 KN capacity MTS jack and load c e l l in seri e s . The system was operated under stroke control with a displacement rate of approximately 1.5mm per second, producing f a i l u r e in 15 to 30 seconds. Grip pressure was controlled manually as in the long tension tests. Lateral deflections were measured with three LVDT's. Load and deflection were scanned continuously with the computer-based data acqu i s i t i o n system. Maximum load and the corresponding deflections were recorded. 4.4.6 Data Aquisition For a l l the combined bending and a x i a l load testing c a r r i e d out at UBC, a computer data acqu i s i t i o n system was used. This system consists of a NEFF scanner on l i n e with a PDP-11 d i g i t a l computer. The scanner can scan up to 60 channels of input at high speed. A data a c q u i s i t i o n programme writes the data to disk f i l e at predetermined load or defl e c t i o n increments. A similar system was used for the Stage 3 testing at Sherbrooke. 4.4.7 Modulus of E l a s t i c i t y Modulus of e l a s t i c i t y is required for predicting the buckling strength of long columns, and for input to the strength model. Three methods were used to assess modulus of e l a s t i c i t y . 97 1 . A random sample of a l l the boards were subjected to a st a t i c bending test, and" the measured deflection used to calculate the modulus of e l a s t i c i t y . Approximately one f i f t h of the boards were tested. 2. For the boards tested in combined bending and compression at Sherbrooke, a Southwell plot was made. A detailed description of thi s i s given by Bleau(l984). For the long length boards with small e c c e n t r i c i t y , the modulus of e l a s t i c i t y determined in thi s manner was very close to that obtained in s t a t i c bending. 3. A l l of the 38x89mm boards subsequently tested in tension and compression were subjected to fle x u r a l s t i f f n e s s measurements. The boards were passed through a Cook-Bolinder stress-grading machine at the Western Laboratory of Forintek Canada Corp. This machine bends each board as a plank, passing i t between three r o l l e r s , the centre one offset to provide a midspan de f l e c t i o n over a 910mm single span. Each board is passed through twice, rotated 180 degrees between passes and the results averaged to eliminate the effect of crook. The load on the centre r o l l e r i s recorded every 100mm of board t r a v e l . An average modulus of e l a s t i c i t y over the 910mm span is calculated from the load, the deflec t i o n , and the board dimensions. The imposed def l e c t i o n of 5mm and the rate of travel of 40 m/min were both considerably lower values than used in a ty p i c a l commercial application in order to increase accuracy 98 and reduce the p o s s i b i l i t y of damage. 4.5 SUMMARY This chapter has described sample selection, testing machines, and experimental procedures used for testing more than 4000 test specimens in bending, tension, compression, and combined loading. Results are described in the next chapter. 99 V. EXPERIMENTAL RESULTS T h i s chapter c o n t a i n s a b r i e f d e s c r i p t i o n of experimental r e s u l t s , with some d i s c u s s i o n and e x p l a n a t i o n for observed behaviour. Combined bending and a x i a l l o a d i n g r e s u l t s are d i s c u s s e d i n some d e t a i l , as are those r e l a t i n g to l e n g t h e f f e c t s . Some of the other experimental r e s u l t s are analysed in more d e t a i l i n subsequent c h a p t e r s . 5.1 COMBINED BENDING AND AXIAL LOADING RESULTS 5.1.1 P r e s e n t a t i o n As d e s c r i b e d p r e v i o u s l y , f i v e d i f f e r e n t lengths of boards were t e s t e d at s e v e r a l d i f f e r e n t e c c e n t r i c i t i e s , f o r both 38x140 and 38x89mm s i z e s , with about 100 boards in each sample. Within each sample there i s a l a r g e amount of s c a t t e r i n the r e s u l t s . The s i m p l e s t method of comparing one sample with another i s to compare sample means, but t h i s does not n e c e s s a r i l y d e s c r i b e r e l a t i v e behaviour at the high or low ends of the d i s t r i b u t i o n . R e s u l t s can be compared at v a r i o u s q u a n t i l e s w i t h i n the d i s t r i b u t i o n . Throughout t h i s study, comparisons w i l l be made at the 5th p e r c e n t i l e , mean and 95th p e r c e n t i l e l e v e l s , the p e r c e n t i l e s being obtained from f i t t e d W e i b u l l d i s t r i b u t i o n s . There are s e v e r a l methods of p r e s e n t i n g r e s u l t s f o r combined bending and a x i a l load fo r given l e n g t h s . The two most u s e f u l methods are to p l o t a x i a l load-moment i n t e r a c t i o n 100 diagrams for each length, or to plot a x i a l load against length (or slenderness) for given e c c e n t r i c i t i e s . The two methods are related as shown in figure 11. 5.1.2 Interaction Curves for Short Members a. Test Results The shortest members tested had lengths between grips of 900mm and 450mm for the 38x140 and 38x89mm sizes, respectively. Buckling effects are very small at these lengths, so maximum recorded values of a x i a l load and moment represent the material strength of the weakest cross section in each board. Figures 22(a) and (b) show the test results for the shortest length of 38x89 and 38x140mm boards, respectively. Each dot represents the a x i a l load and mid-span moment at f a i l u r e for one board. The mid-span moment was calculated as the product of the ax i a l load and the maximum deflection from the l i n e of thrust. For each value of end e c c e n t r i c i t y , the points shown deviate from a r a d i a l l i n e because of di f f e r e n t amounts of deflection in each board. For each cloud of points representing one end e c c e n t r i c i t y , each point was defined in polar co-ordinates, and the centre of gravity of the cloud located. A l l points were assumed to be on a ra d i a l l i n e through the centre of gravity. A three parameter Weibull d i s t r i b u t i o n was f i t t e d to a l l the r a d i i , and 5th and 95th percentile values of radius were calculated from the d i s t r i b u t i o n parameters. This procedure resulted in each cloud of points being reduced to three c h a r a c t e r i s t i c strength values as shown in 101 CO • d O • J o . »—« CE 2 mm •• • Length 0.45m. 12mm eccentricity 39 mm 75 mm 202mm mm mm"" — — — -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1— 0.D 0.5 J.O 3.5 2.0 2.5 3.0 3.5 4.0 niD-SPRN HOriENT (KN.M) (a) 38 x 89 mm 2 mm Length 0.914 m. t • • • 18mm eccentricity _ V- • • . ••*. 39 mm » r. • • a _ a • • • V s . - j l ? f 75 mm • • • j * " • . * 202mm ^ • Co CXoc 9-n 1 1 T 1 1 1 1 1 1 i i i r 0.0 J.O 2.0 3.0 4.0 5.0 6.0 "?.0 B.O niD-SPflN HOriENT (KN.n) (b) 38x1 AO mm F i g u r e 22 - T e s t r e s u l t s f o r s h o r t e s t l e n g t h i n e c c e n t r i c c o m p r e s s i o n 102 5do CL o CT.1D X cc si-Length CU5m. i i i i i i i i i i i i i i 0.0 O.S 1.0 1.5 2.0 2.5 3.0 3.5 rflD-SPRN MOMENT (KN.M) 4.0 (a) 38 x 89 mm CEa dice 9-Length 0.9Hm. i i i i i i i i i ; i i i i i i 0.0 1.0 2.0 3.0 4.0 5.0 €.0 "7.0 B.D MID-SPRN MOMENT (KN.M) (b) 38x140mm Figure 23 - Percent i le resu l t s for shortest length eccentr ic compression 103 figure 23. The three l i n e s sketched through the points represent t y p i c a l behaviour at the 5th percentile, mean and 95th percentile levels of the d i s t r i b u t i o n . A l l the curves in figure 23 are convex away from the o r i g i n . Furthermore a s i g n i f i c a n t feature i s displayed at the f i f t h percentile l e v e l . As a x i a l compression i s increased from zero, moment capacity increases s i g n i f i c a n t l y before decreasing to zero at high levels of a x i a l compression. The simple explanation for th i s behaviour i s that when tested in bending, these weaker boards exhibit a tension f a i l u r e due to defects in the tension zone. When subjected to bending under the action of a moderate a x i a l compression force, the tension f a i l u r e i s suppressed and moment capacity is increased. The resemblance to reinforced concrete behaviour i s s t r i k i n g . The behaviour of these short members is c l a r i f i e d i f we add the results of combined bending and tension tests, as shown in figure 24 for 38x89mm material. This data i s not available for 38x140mm material. The top half of figure 24(a) is i d e n t i c a l to figure 22(a) and the top half of figure 24(b) is i d e n t i c a l to figure 23(a). Tension results have been added below the bending axis. A s t r i k i n g feature of thi s figure is the much larger strength v a r i a b i l i t y in tension than in compression. This difference produces changes in the ra t i o of tension to compression strength through the d i s t r i b u t i o n of strength values. The r a t i o of tension to compression strength has a pronounced ef f e c t on the shape of the interaction curve. At 104 o Z — g to UJ ct 0. o X < o o k <••; ' v.-o cn z X* -< 2mm Length 0.45m. 12mm eccentricity -.4*-. 39 mm tf • 75mm 202mm 202mm 1; • * 39 mm 12 mm 2 mm 0.0 1.0 2.0 3D A.0 niD-SPRN nOHENT (KN.M) 5.0 (a) data points Figure 24 - Test results for 38x89mm boards in eccentric compression and tension C ro K ) >-•• 3 —~ O CO o o 3 o r t o> 3 3 r t C »1 ro a O ' o i o 3 -3 Xi f D CO ro n-cn cn I-I ro o CO 3 c CU •n-3 to a r i - O ft) i-( 3 01 t o 0 0 O 3 oo vo 3 3 t r o PJ >-t co A X I A L TENSION (KN) AXIAL COMPRESSION (KN) 200 150 100 50 0.0 50 100 p • i m • i f l I I I i I I l i I i 1 i I i J 1 1 1 1 1 — i — i — i — i — i — i — i — i — i — i — i — i — L C D " 106 the 5th percentile l e v e l the low r a t i o of tension to compression strength results in the "nose" of the curve being high in the compression region. At the 95th percentile level the tension strength i s much greater than the compression strength, so the "nose" of the curve is well below the bending axis and the interaction curve in the compression region i s close to the straight l i n e assumed in design codes. Behaviour at the 95th percentile level i s of similar form to that expected at the 5th percentile l e v e l for clear wood specimens. Hence the shapes of the the 5th and 95th percentile curves on figure 24(b) are indicative of the differences between timber and clear wood behaviour referred to in Chapter 2. b. Mode of Fa i l u r e For the interaction curves shown in figure 24(b), the nose of the curve marks the t r a n s i t i o n between a compression dominated f a i l u r e above, to the tension dominated f a i l u r e below. In reinforced concrete terminology, a f a i l u r e at the nose of the curve would be c a l l e d a "balanced f a i l u r e . " The observed f a i l u r e modes generally followed this pattern, with f a i l u r e in the compression dominated f a i l u r e region being associated with d u c t i l e crushing of wood on the compression face, and f a i l u r e in the tension dominated f a i l u r e region being associated with b r i t t l e f a i l u r e in the tension part of the board. This i s purely a q u a l i t a t i v e assessment because i t i s extremely d i f f i c u l t to quantify the large number of f a i l u r e modes observed. 107 5.1.3 Interaction Curves for Long Members The test results i l l u s t r a t e d as clouds of points in figure 22 and as percentiles in figure 23 have also been produced for the more slender boards. These are a l l i l l u s t r a t e d in Appendix C, with the percentiles superimposed on the individual data points. To i l l u s t r a t e the general r e l a t i o n s h i p between strength and increasing slenderness, mean values for various lengths have been plotted on figure 25. In a l l cases the ax i a l load at f a i l u r e decreases with increasing slenderness, t h i s being caused by a t r a n s i t i o n from a material f a i l u r e for the shortest boards to an i n s t a b i l i t y f a i l u r e for the more slender boards. For low level s of a x i a l load the bending moment at f a i l u r e also tends to reduce with increasing length. This e f f e c t , which is much less pronounced than the- decrease in ax i a l load capacity, is largely due to the size effect with length introduced in Chapter 3 and quantified later in th i s chapter. 5.'1.4 Axial Load-Slenderness Curves The test results described above can be presented in a dif f e r e n t way by p l o t t i n g a x i a l load against slenderness r a t i o , defined as L/d where L i s the length of the member and d i s the relevant cross section dimension of a rectangular member. The resu l t i n g curves are shown in figures 26(a) and (b) for the 38x140mm and 38x89mm boards respectively, for mean values of a x i a l load at f a i l u r e . Behaviour at the 5th and 95th percentile levels follow a similar trend. 1 0 8 LENGTH O O 0 . 4 5 m H +. ! ,30 m <5 e> 1.80 m x- X 2 .30 m LB- B 3 .20 m i 1 1 1 1 1 1 1 1 1 i i i i 1 0 0 0.5 1.0 L5 2.0 2.5 3.0 3.5 MID-SPAN MOMENT (KN.M) (a) 38 x 89 mm LENGTH a B 0.31 m O- - - 0 1.82 m A — — A 2 . 4 4 m B H 3 . 3 5 m ^ —TO Q. (j) 4 4 2 7 m ** —i 1 1 1—~ —i 1 1 1 30 1.0 2.0 3.0 4.0 5.0 6.0 7.U H l D - S P f l N MOHENT (KN.M) (b) 38x140mm 2 5 - I n t e r a c t i o n d i a g r a m f o r m e a n t e s t r e s u l t s a l l l e n g t h s i n e c c e n t r i c c o m p r e s s i o n 1 0 9 a cc o 1-ex I—I cr *>-I. o C E g -M X <X • O.D ECCENTR IC ITY 0 O 2 m m H + 1 2 m m <$— . —0 39 m m \ \ X X 7 5 m m \ N \ • - CD 202 m m \ \ \ \ \ 1 \ \ v. . X - . — ^ CS ~ -x O.D B.O 16.0 24.0 32.D SLENDERNESS (L/D) 40.0 48.0 (a) 38x89 mm E C C E N T R I C I T Y CB ED 2 m m \ \ \ \ ts +- h 18mm o — - —e> 39 m m . X - X 7 5 m m \ B- CD 2 0 2 m m -H I B.O i i i r 32.D 16.0 24.0 SLENDERNESS [L/D) (b) 38x140mm 40.0 48.a Figure 26 results for - Axial load-slenderness curves. Mean test a l l lengths tested in eccentric compression 1 1 0 In each case the curve of most interest i s the top curve, representing 2mm nominal e c c e n t r i c i t y . This loading condition •is close to concentric loading. The f a i l u r e condition for small slenderness values, at the l e f t hand side of the figure, represents a material f a i l u r e under almost concentric axial compression. The f a i l u r e condition for large slenderness values represents a linear e l a s t i c buckling f a i l u r e . These curves are similar to the form shown previously in figure 8, and w i l l be compared with existing design code requirements in Chapter 9. 5.2 SEPARATE BENDING AND AXIAL LOADING RESULTS 5.2.1 Test Results As described previously, testing was c a r r i e d out in bending, in a x i a l tension, and in a x i a l compression (restrained to prevent buckling), for both the 38x89mm and the 38x140mm sizes. The results of these tests for the stage 2 and stage 4 testing are i l l u s t r a t e d in Appendix C, with a plot of the cumulative d i s t r i b u t i o n of the ranked data, overlayed with the f i t t e d Weibull d i s t r i b u t i o n and a number of relevant s t a t i s t i c s . The bending test results from stage 5 have not been included in Appendix C because they w i l l be reported elsewhere (Madsen 1983). It i s interesting to compare the modulus of rupture with the a x i a l tension and compression strengths for a given grade and size of material. Figures 27(a) and (b) show this comparison for the two sizes. There are several points of in t e r e s t . The strength in a x i a l tension i s much more variable 111 £2-m *©_ <X o m o in Q _ tfi o" LLi CC o =1 m O o-— TENSION • - COMPRESSION — BENDING 1 r~—r— 1 r -<0.0 50.0 6a 0 STRENGTH IrlPfl) (a) 38 x 89 mm i r 70.0 n i r 80.0 ao.o m o m 1 rx < CQ o cc Q_ i i UJ > cx , (_) TENSION ' / COHPRESSION / / / BENDING 1 / / l / / ; / /' < / / 7 / / 1 I 1 \ 7 / < /" / < / / ' 7 / • / 0.0 10.0 2ao 30J) 40.0 so.a eao STRENGTH (flPfl) ' 70J 30.0 100.0 (b) 38xU0mm Figure 27 - Comparison of tension, compression and bending test results 112 than in a x i a l compression, tension strength being greater than compression strength at the strong end, less at the weak end. The modulus of rupture i s much greater than the tension strength throughout the d i s t r i b u t i o n , and has a similar c o e f f i c i e n t of v a r i a t i o n . The relationship between these three strength properties w i l l be explored in d e t a i l l a t e r . In figure 24(b) i t was shown how the shape of the interaction diagram i s quite d i f f e r e n t at various levels in the d i s t r i b u t i o n of strength. The differences are not a result of the actual le v e l s in the d i s t r i b u t i o n , but are rather a r e f l e c t i o n of the changes in the r e l a t i v e values of tension and compression strengths, shown in figure 27. 5.2.2 Modes of Failure A very large number of d i f f e r e n t f a i l u r e modes were observed, many of them related to s p e c i f i c defects in the boards. In tension most f a i l u r e s were at a single cross section or in a length of board less than two or three times the largest cross section dimension, with l i t t l e d i s t r e s s observed elsewhere in the board. Some f a i l u r e s were preceded by v i s i b l e and audible crack growth, usually in tension perpendicular to the grain. A l l the f a i l u r e s were sudden b r i t t l e fractures associated with sudden release of stored energy. Many f a i l u r e s were associated with l o c a l or generalized sloping grain, often around knots.' For a x i a l compression restrained against buckling, 113 f a i l u r e s tended to f a l l into two categories. The f i r s t was a d u c t i l e crushing type of f a i l u r e with v i s i b l e compression wrinkles, as described for clear wood. The second was a more b r i t t l e f a i l u r e mode when two halves of the specimen separated and buckled sideways between l a t e r a l supports, with a longitudinal s p l i t t i n g f a i l u r e . This type of f a i l u r e was usually i n i t i a t e d at a knot near the centre of the board where the stress concentration induces tension stresses perpendicular to the grain. In both cases compression yi e l d i n g was often v i s i b l e at several cross sections as maximum load was approached. Failur e modes in bending tended to be consistent with those already described for tension and compression. Many of the weaker boards f a i l e d suddenly in the tension zone with no sign of compression y i e l d i n g . Some of these f a i l u r e s were without warning, others were preceded by v i s i b l e and audible cracking. The most common mode of f a i l u r e was compression yi e l d i n g near the top surface at one or more cross sections, eventually followed by a b r i t t l e f a i l u r e in the tension zone. Some boards snapped into two pieces while others hung together with a less sudden reduction in -load. Almost a l l boards f a i l e d in the central one-third of the span. No horizontal shear f a i l u r e s were observed, even for the shortest span. 114 5.3 LENGTH EFFECTS 5.3.1 Introduct ion A small but important part of the experimental procedure was designed to produce information on length e f f e c t s in a x i a l compression, a x i a l tension, and in bending. This section uses test results to estimate values of the parameter k, in equation 3.11. If cross section strength i s a .quantity that varies along the length of a t y p i c a l board, then the average strength of long boards can be expected to be lower than that of short boards. If cross section strength is assumed to be a random variable with length, knowledge of the amount of v a r i a b i l i t y within a t y p i c a l board can be used to quantify the length e f f e c t . The v a l i d i t y of this.assumption has been discussed in Chapter 3. There are apparently no previous reported studies on the effects of specimen length on the strength of timber in a x i a l tension or compression. Bending tests are usually carried out at a constant span-to-depth r a t i o , producing lower f a i l u r e stresses for longer and deeper members, but such tests do not allow length effects and depth effects to be quantified independently. Madsen and Nielsen (1976) report a preliminary study into the effect of length on bending strength, which produced results similar to those described below. 115 5.3.2 Compression .Strength Ten boards of 38x89mm size, 2.9m long were cut into 135mm long segments. Ten boards of 38x140mm size 4.0m long were cut into 210mm long segments. In both cases the number of segments varied from 15 to 19. The segments were consecutive along each board, although occasionally a small amount of clear wood was eliminated between segments in order to locate a s i g n i f i c a n t defect near the centre of a segment. Each segment was tested to f a i l u r e in a x i a l compression as described in Chapter 4. The following analysis was made for the segments within each board. The c o e f f i c i e n t of variation of segment strength within each board was calculated. For both sizes the c o e f f i c i e n t of variation of segment strengths within each board was in the range of 7% to 13% with an average value of 9%. A two parameter Weibull model was f i t t e d to the segment strengths within each board using a maximum li k e l i h o o d routine. For both sizes the shape parameter varied between 9 and 19, the average value being close to 13. This i s consistent with the c o e f f i c i e n t of var i a t i o n reported above using equation 3.9. The p o s s i b i l i t y of a three-parameter Weibull d i s t r i b u t i o n giving a di f f e r e n t result was investigated. In th i s case, however, the d i s t r i b u t i o n a l form of the data i s such that there is very l i t t l e difference between the two d i s t r i b u t i o n s and both predict similar length e f f e c t s , so the two parameter form has been used. Length effects can be calculated from equation 3.11. A 116 parameter of k,=13 produces a strength reduction factor of 0.948 for doubling length. Because of the assumption of a two-parameter Weibull d i s t r i b u t i o n , the predicted length ef f e c t is the same at a l l strength levels from the weakest to the strongest. This average figure, predicting a 5% reduction in strength for doubling length, could vary for individual boards, but i t w i l l be used as the length effect factor in compression for the rest of t h i s study. 5.3.3 Tension Strength It i s not possible to carry out tension tests of consecutive segments of a board as done in compression, because a considerable length of board i s required in grips at each end of the test specimen. Two approaches have been- used. A dire c t approach i s to carry out a x i a l tension tests of boards of d i f f e r e n t lengths. A second approach i s to use the results of bending tests of d i f f e r e n t lengths on the assumption that bending strength is governed by f a i l u r e s in the tension zone. a. 38x89mm Boards Axial tension tests were carr i e d out on two lengths of boards as described in Chapter 4. The strength d i s t r i b u t i o n s are shown with the other test results in Appendix C. The f i t t e d d i s t r i b u t i o n s have been used to calculate the r a t i o of the tension strengths of long boards to those of short boards, for the whole d i s t r i b u t i o n . The resulting plot is shown in figure 28. If the r a t i o i s taken to be 0.91 throughout the d i s t r i b u t i o n , then we can calculate the shape parameter for 1 17 O . D T 0 . 4 0 . 5 0 . 6 CUMULATIVE RANK 3.0 Figure 28 Ratio of tension strengths of 38x89mm boards 2.0m and 0.914m long within board segment strengths from equation 3.11, giving a value of k,=8.3, which corresponds to a strength reduction factor of 0.92 for doubling the length. b. 38x140mm Boards The 38x140mm material used in experimental stages 1 and 2 was not the subject of any length effect investigation in tension. At a later date however, comparative testing was carried out on two lengths of 38x140mm timber in Stage 5. Approximately 100 boards were tested at each length of 3.0m and 0.91 m. Figure 29 shows the r a t i o between the strengths of the two lengths, over the whole d i s t r i b u t i o n . The difference i s much larger than observed for the 38x89mm si z e . A possible explanation for this large difference i s that the material was ordered as No. 2 grade with as many strength reducing defects as possible. If the r a t i o of strength of long boards to short board.s is taken to 1 18 T i i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 r 0 . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 D.l D.B O . S 1 . 0 CUMULATIVE RANK Figure 29 - Ratio of tension strengths of 38x140mm boards 3.0 and 0.914m long. be 0.67, that corresponds to a shape parameter of k,=3.0, which implies a strength reduction factor of 0.80 for doubling the length, from equation 3.11. 5.3.4 Bending Strength A central hypothesis of thi s study i s that bending behaviour can be predicted from a x i a l tension and compression behaviour. Accordingly, length effects in bending should be able to be quantified from length effects in tension and compression. If fl e x u r a l strength i s governed solely by tension strength, length effects should be the same in bending and in tension. Commercial timber tends to be r e l a t i v e l y weak in tension, p a r t i c u l a r l y at the low end of the strength d i s t r i b u t i o n . In thi s case the length effect in bending is expected to be similar to that observed in a x i a l tension. As the r a t i o of tension to compression strength increases, the compression strength of the material has an 1 1 9 increasing influence on the bending strength, so for timber with high tension strength the length e f f e c t in bending is expected to approach a x i a l compression values. In order to investigate length effects in beams, i t becomes necessary to have an expression for the highly stressed length of the beam. E a r l i e r i t was shown how the integral in equation 3.6 can be evaluated over the volume of a beam to give the equivalent volume of equation 3.7. If depth ef f e c t s are excluded,, an equivalent stressed length Le can be calculated as 1 + r k i Le " T ^ r l L (5.1) where L i s the span- of the beam, a i s the distance between two symmetrical point loads and k, is the relevant length effect factor. a. 38x89mm Boards Bending tests of two d i f f e r e n t lengths of 38x89mm timber were carried out as described in Chapter 4. Figure 30 shows the r a t i o of strengths of the long to the short boards. As expected there i s a greater, length effect for the weak boards than for the strong boards in the d i s t r i b u t i o n , but the actual magnitude of the length effect i s larger than predicted from the tension and compression parameters, p a r t i c u l a r l y at the weak end. The reasons for this discrepancy are not c l e a r . One possible factor i s the short stressed length in the bending t e s t s . At the weak end, bending strength depends 1 20 "T 1 1 1 1 1 1 1 1 r 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 CUMULATIVE RRNK ] .D Figure 30 - Ratio of bending strengths of 38x89mm boards 1.5 and 0.84m long e n t i r e l y on tension strength with no contribution from compression strength. The tension strength length effect parameters were derived from test results at lengths of 0.9lm and 2.0m. In the short span bending test referred to here the span was 0.84m so the highly stressed length was less than 0.3m. It is possible that lengths as short as this may not contain many of the defects found in longer lengths, hence the unexpectedly high strength values. b. 38x140mm Boards Results from bending tests at two spans are available from the stage 5 tes t i n g . Figure 31(a) shows the ra t i o of strengths of the 3.0m span to the 1.5m span for three load configurations, for select s t r u c t u r a l grade. The three samples do not behave i d e n t i c a l l y , but there i s a consistent o v e r a l l trend for the strength of the long boards to be about 0.89 times that of the shorter boards, which corresponds to a shape parameter from 121 LU ' CH CO ^  2 a' o X CO o O cr 0.0 1 1 1 r 0.1 0.2 ~\ 1 1 1 1 1 1 r 0.3 i 1 1 r 0 . 4 0 . 5 0 . 6 CUMULATIVE RANK o.i 0 . 6 0 . 9 1 . 0 (a) Select Structural Grade cn D'1 1 1 1 1 r — i 1 1 1 1 1 1 1 1 r 0 . 0 0 . 1 ' 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 1 CUMULATIVE RANK i r 0 . 8 0 . 9 I.D (b) Number 2 Grade Figure 31 - Ratio of bending strengths of 38x140mm boards, 3.0 and 1.5m long equation 3.11 of k,=5.9. The range of values observed is from 0.80 to 1.02. This result i s very close to that obtained in the d i r e c t a x i a l tension tests, and a trend towards decreasing length effect at. the strong end of the d i s t r i b u t i o n can be seen as predicted. 122 Figure 31(b) shows the same plot for Number 2 grade boards, where a somewhat greater length e f f e c t can be seen. The three load configurations produce very similar results with an average r a t i o of about 0.84, which i s a l i t t l e more than the figure of 0.80 obtained from dire c t tension tests on the same material. The range of observed values i s from 0.79 to 0.90. A factor of 0.84 corresponds to a shape parameter of k,=4.0. 5.3.5 Summary of Length Effects The results presented in thi s section show considerable v a r i a b i l i t y b u t the following consistent trends can be seen: 1. For a x i a l compression the observed length effect produces a strength reduction factor of about 0.95 for doubling length. 2. For a x i a l tension and , for bending, the observed length effect appears to be grade dependent. For Select Structural grade the strength reduction factor for doubling length i s approximately 0.89. This figure w i l l be used for the remainder of thi s study. For Number 2 grade, the strength reduction factor for doubling length is approximately 0.84. 123 5.4 WEAK AXIS BENDING The size e f f e c t theory derived for timber members suggests that for boards of a given length, modulus of rupture should be less for edgewise bending than for flatwise bending. The reasons for thi s have been discussed in Chapter 3. For each of the two cross section sizes used in thi s study, a comparative test was made of bending strength about each p r i n c i p a l axis. The same span was used for each, loading orientation to eliminate any possible length e f f e c t . From the results of these tests, the rati o s of the strengths are shown in figure 32. The unexpected result shown in figure 32(a) is that the two orientations have almost exactly the same strength for the 38x89mm si z e . No obvious explanation i s available for this behaviour. This result would be predicted from b r i t t l e fracture theory applied to a perfectly b r i t t l e material, because the highly stressed volume is the same in each orientation, but i t i s inconsistent with other results observed in timber members. For the larger size, figure 32(b) shows a s i g n i f i c a n t difference between the two orientations, as expected, because there i s a large decrease in depth and increase in width when the orientation i s changed from edgewise to flatwise. The tentative conclusion from these few tests i s that bending strength properties obtained from strong axis bending tests can be conservatively used to predict weak axis bending strength, the degree of conservatism probably increasing as depth to width r a t i o i s increased. 124 O . D 0.1 ~i 1 1 1—n 1 1 1 1 1 1 1 1 1 r 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 1 0 . 8 0 9 CUMULATIVE RANK I.D (a) 38 x 89 mm I— CO o a 1 0 2 a'-LU CO l-H CT _J Ll_ LU CO Lu => I I I I I I I I I 1 1 1 1 1 1 1 1 1 1— 0 . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 D . 6 0 . 1 0 . 8 0 . 9 1 . 0 CUMULATIVE RANK (b) 38xU0mm Figure 32 - Ratio of edgewise to flatwise bending strength 5.5 SUMMARY This chapter has described the trends observed in a large experimental programme, and has referred to Appendix C where many more results are summarized. The results of most interest are those for eccentric a x i a l loading tests which 125 have not been c a r r i e d out on t h i s s c a l e p r e v i o u s l y . E x p e r i m e n t a l r e s u l t s have been used t o q u a n t i f y l e n g t h e f f e c t s , which a l s o have not been p r e v i o u s l y i n v e s t i g a t e d t o t h i s e x t e n t . 126 VI. STRENGTH MODEL 6.1 INTRODUCTION This chapter describes a theoret i c a l model for predicting the strength of timber members. The model uses tension and compression strength information from in-grade testing to predict strength in bending or in eccentric a x i a l loading for members of any length. The theory and assumptions in the model w i l l be described, along with the basis of the computer program used to make the necessary numerical c a l c u l a t i o n s . The program w i l l be described in two separate parts. The f i r s t part of the computer program calculates the behaviour of a cross section of given geometry and material properties. This part has been written s p e c i f i c a l l y for timber. The second part of the program uses this information to calculate the load capacity of an e c c e n t r i c a l l y loaded column of any length, and can be used for members of any material. The organization of the program i s based on a similar program developed by Nathan(1983a) for reinforced and prestressed concrete, and v e r i f i e d for those materials by Alcock and Nathan (1977). The f i r s t part has, however, been completely re-written because the f a i l u r e mechanisms in wood are quite d i f f e r e n t from those in concrete. The second part i s almost exactly as described by Nathan. 127 6.2 ASSUMPTIONS The following assumptions are made: 1. Plane sections remain plane. 2. Timber stressed in tension behaves in a linear e l a s t i c manner u n t i l b r i t t l e fracture occurs at a l i m i t i n g tension stress. 3. Timber stressed in compression behaves . in a non-line a r ductile manner. Any shape of s t r e s s - s t r a i n curve can be used. A l i m i t i n g compression str a i n may be sp e c i f i e d . 4. Stress-strain relationships are independent of rate of loading. 5. Axial tension and compression strengths decrease as member length i s increased. 6. In both tension and compression, the maximum attainable stress at a cross section i s a function of the proportion of the section subjected to that stress. 7. If moment varies along a member, f a i l u r e occurs at the cross section subjected to maximum moment. 8. Modulus of e l a s t i c i t y i s constant along the length of each board. 9. No torsional or out-of-plane deformations are considered. Duration of load e f f e c t s are not considered. Shear f a i l u r e s are not considered. Some of these l i m i t a t i o n s are discussed further in Chapter 9. 6.3 CROSS SECTION BEHAVIOUR The ultimate interaction diagram and the moment-curvature-axial load relationships for a cross section are derived using a simple step-by-step procedure to obtain a x i a l load and moment capacities for a range of neutral axis depths and curvatures. Recall that the ultimate interaction diagram shows l i m i t i n g combinations of a x i a l load and bending moment 128 that a cross section can r e s i s t . 6.3.1 Calculation Procedure For a cross section such as that shown in figure 33(a), a t y p i c a l c a l c u l a t i o n begins by selecting a neutral axis depth and applying an i n i t i a l curvature, <p, to the section, to produce the strains shown in the t y p i c a l case of figure 33(b). The following procedure i s used to determine what combination of a x i a l load and moment would be necessary to produce t h i s condition. Figure 33 - Cross section behaviour The depth of the section is divided into a number of segments. For each segment the mid-height strain i s calculated, and used to calculate the corresponding stress, using an input s t r e s s - s t r a i n relationship as shown in figure 33(c). The resulting stress d i s t r i b u t i o n i s shown in figure 33(d). The force in each segment i s calculated, producing the d i s t r i b u t i o n shown in figure 33(e). A l l of the tension and compression segment forces are combined into a single force for each stress block as shown in figure 33(f), and these are 129 combined to give the net a x i a l force and bending moment about the centroidal axis, as shown in figure 33(g). At t h i s stage four items of information are stored before repeating the ca l c u l a t i o n with increased curvature. The stored information i s 1. Neutral axis depth (input) 2. Section curvature (input) 3. Net a x i a l load (output) 4. Bending moment (output) For the chosen neutral axis location, the cal c u l a t i o n i s repeated with increasing curvature u n t i l either a l i m i t i n g compression s t r a i n i s exceeded, the extreme f i b r e stress in tension exceeds a l i m i t i n g value (tension f a i l u r e occurs), or the moment drops to zero. The procedure described . to this point i s then repeated for a number of other neutral axis locations. For each neutral axis location the limiting" tension stress is calculated from the input a x i a l tension strength using equation 3.18 or 3.20 with k 3 as the s t r e s s - d i s t r i b u t i o n parameter for tension. To include a s t r e s s - d i s t r i b u t i o n effect for compression in the ca l c u l a t i o n , the compression strength is, modified at each curvature increment using equation 3.21 or 3.22 with k 3 as the compression s t r e s s - d i s t r i b u t i o n parameter. This procedure i s i l l u s t r a t e d schematically in the flow chart of figure 34. Output from a t y p i c a l run for a 38x140mm member w i l l be used in the next sections to i l l u s t r a t e several 130 a s p e c t s of the program. Increment curvature Select f i r s t n.a. depth f modify tensic for t h i s n jn strength a. depth apply i n i t i a l curvature 41 YES YES modify comp. strength for t h i s n.a. and this increment n.a. depth TfT -calculate internal tension and compression forces from s t r e s s - s t r a i n relations calculate net ax i a l load and moment YES F i g u r e 34 - Flow c h a r t f o r c a l c u l a t i n g moment-curvature-a x i a l l o a d r e l a t i o n s h i p s f o r a c r o s s s e c t i o n 131 6.3.2 Neutral Axis Contours Figure 35 i l l u s t r a t e s the relat i o n s h i p between a x i a l load and moment for forty neutral axis contours. Compression i s po s i t i v e . This p a r t i c u l a r figure has been obtained using a b i l i n e a r s t r e s s - s t r a i n relationship with a f a l l i n g branch as shown in figure 6. Each r a d i a l l i n e represents a single neutral axis location. For any one neutral axis location, each successive point on the line represents the combination of a x i a l load and moment required to produce a pa r t i c u l a r curvature. Curvature has been increased step-by-step u n t i l a tension f a i l u r e occurred, or u n t i l the moment dropped to zero (as seen in the top left-hand corner). The curves have been constructed from straight segments between points of increasing curvature. The neutral axis locations have been selected with five outside the section on the tension side, eighteen outside the section on the compression side, and the remainder within the section, to provide a suitable range of cal c u l a t i o n points. This figure of neutral axis contours is used to construct an envelope, or ultimate interaction diagram, showing possible combinations of a x i a l load and bending moment for thi s cross section. The actual l i n e s shown in figure 35 are not used for any other purpose in t h i s study. 1 32 o.o n 1 1 1 r — r 2.0 3.0 4.0 MOMENT (KN.M) 6.0 Figure 35 - Neutral axis contours for moment and axi a l load interaction 6.3.3 Curvature Contours Figure 36 shows the same points as figure 35, plotted in another way to show the relationship between a x i a l load and moment for t h i r t y curvature contours. Each l i n e represents a single value of section curvature. Different points on a li n e of constant curvature represent the combinations of a x i a l load and moment required to produce that curvature for various 2 0 0 . 0 134 neutral axis locations. In both figures 35 and 36, the contours are straight in the range of linear e l a s t i c behaviour. The contours become curved when the wood i s stressed in compression beyond the proportional l i m i t . If figures 35 and 36 were to be superimposed, the intersection points would represent the number of combinations of neutral axis locations and curvatures for which internal forces were calculated. 6.3.4 Ultimate Interaction Diagram Figure 37 i s the ultimate interaction diagram for the cross section, being the envelope of a l l the points shown in figures 35 and 36. This curve i s constructed from.the neutral axis contours shown -in figure 35, and consists of straight l i n e segments between a f a i l u r e point at each neutral axis location. Any point inside the curve represents a combination of a x i a l load and moment that the cross section can r e s i s t . 6.3.5 Moment-Curvature Curves Figure 38 shows the moment-curvature relationships for several levels of a x i a l load, P. Pa is the ax i a l compression strength of the material. A l l curves start at the o r i g i n , but each has been sh i f t e d s l i g h t l y f o r l c l a r i t y of presentation. The program computes these curves from the data already presented. Each l e v e l of a x i a l load can be represented by a horizontal l i n e such as the l i n e A-B on figure 36. Each intersection of the horizontal l i n e with a curvature contour provides a value of moment and curvature which is plotted on figure 38. The f i n a l point on each moment curvature curve 135 0 . 0 0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 0 . 1 0 . 1 2 0 . 1 4 0 . 1 6 0 . 1 8 0 . 2 CURVATURE d/m) Figure 38 - Moment-curvature-axial load relationships represents the point on the ultimate interaction diagram for the respective a x i a l load. For high a x i a l loads the moment-curvature curves have a f a l l i n g branch beyond maximum moment, but that has not been plotted because the information i s never used in the subsequent c a l c u l a t i o n s . A l l the information required for ca l c u l a t i o n of long column s t a b i l i t y behaviour i s contained in the moment-curvature-axial load curves. 6.4 COLUMN BEHAVIOUR The second section of the computer program investigates the behaviour of columns of any length under the action of eccentric a x i a l loads with equal end e c c e n t r i c i t i e s and no l a t e r a l load as shown in figure 39. 136 Figure 39 - Column with a x i a l load and equal end eccentric i t i e s 6.4.1 Failure Modes The behaviour -of an e c c e n t r i c a l l y loaded compression member w i l l be reviewed in t h i s section, in order define some terms used in the explanation of the computer model. Consider the possible behaviour of a member with equal end e c c e n t r i c i t i e s as shown in figure 39, as the axi a l load P is increased to f a i l u r e . If the end e c c e n t r i c i t y i s a distance e, the moment at the ends of the member w i l l always be P times e. The moment at mid-span w i l l be P(e+A) where A is the mid-span deflection as shown. Figure 40(a) i s an interaction diagram of ax i a l load vs. moment. The outer curved l i n e i s the ultimate interaction diagram representing material f a i l u r e (figure 37). As an example of t y p i c a l loading, l i n e O-A shows the load path for ax i a l load and end moment as the a x i a l load P is increased. The corresponding load path for mid-span moment is shown by 137 O O 5 P 2 ^ M a t e r i a l fa i lu re c _ D \ \ E / / / / // A ) B // 1/ M o m e n t ( a ) a o a x < Mater ia l fa i lure End moments at failure Figure 40 - Typical interaction diagram for e c c e n t r i c a l l y loaded column the curved l i n e O-B. The horizontal distance between lines O-A and O-B represents the amount by which the i n i t i a l moment, P times e, has been magnified to P(e+A). In thi s case the member f a i l s at an a x i a l load P, when the mid-span load path O-B intersects the material strength interaction diagram at 138 point B. This i s described as a material f a i l u r e . If the same member i s loaded with a smaller e c c e n t r i c i t y , the load path for end moments could be shown by li n e 0-C, and the load path for mid-span moments by l i n e 0-D. In thi s case an i n s t a b i l i t y f a i l u r e occurs when the a x i a l load reaches a maximum value P 2. The mid-span moment at f a i l u r e i s shown by point D, which is well inside the material strength curve. If the member- were loaded with a system under load control (for example, gravity loads) to load P 2, deformations would' increase rapidly and a material f a i l u r e would follow immediately. If the member were loaded under conditions of controlled displacement the load path shown by the extension c of the l i n e 0-D could be followed to eventual material f a i l u r e at point E. • If thi s process i s repeated many times for the same member using a f u l l range of e c c e n t r i c i t i e s , figure 40(b) can be produced. The s o l i d l i n e i s the same ultimate interaction diagram. The dotted l i n e Pu-D-B-Mu is the locus of points such as B and D in figure 40(a), representing combinations of ax i a l load and mid-span moment (magnified moment) just causing f a i l u r e . For low ax i a l loads this l i n e co-incides with the ultimate interaction diagram, indicating material f a i l u r e s . For higher a x i a l loads i n s t a b i l i t y f a i l u r e s occur. The chain-dotted l i n e Pu-C-A-Mu i s the locus of points such as A and C in figure 40(a), representing combinations of a x i a l load and end moment (unmagnified moment) just causing f a i l u r e . For any ax i a l load the horizontal distance between the two dotted 139 li n e s represents the ' moment magnification due to member deformations at f a i l u r e . 6.4.2 Calculation Procedure A computer program for ca l c u l a t i n g points on the curves shown in figure 40(b) w i l l be described below. The program can consider a column of any length made up of a number of segments of "equal length. A method described by Galambos(1968) is used to develop column def l e c t i o n curves for a given a x i a l load, to determine the maximum end ec c e n t r i c i t y , e, at which that load can be applied to the column. As the column is symmetrically loaded, the slope is always zero at mid-span. For the a x i a l load under consideration, the moment at mid-span is i n i t i a l l y set to the material f a i l u r e moment for that load (a point on the ultimate interaction diagram). The corresponding mid-span de f l e c t i o n , e+A, (from the l i n e of a x i a l load) is the f a i l u r e moment divided by the ax i a i load. To find the actual values of e and A i t i s necessary to calculate the deflected shape of the member. A column deflection curve is obtained by proceeding along the column, segment-by-segment from mid-span, ca l c u l a t i n g the deflection at each node. Consider calculations for a t y p i c a l segment of length Ax, such as that shown in figure 41(a). If the deflection v 0 and slope v' 0 are known at the st a r t i n g node x 0, then the moment, M 1 r at the mid-point of the segment (point x,) is approximately 140 Figure 41 - Column deflection curves (6.1 ) The curvature, <£1f at point x, can be obtained from the moment-curva.ture-axial load relationship (figure 38). The curvature i s assumed to be constant along the segment. Deflections are assumed to be small such that <£=v". The displacement, v 2, and slope, v' 2, at the next node, x 2, are calculated from = v • + v * ( A x ) o o v ' (6.2) = v t _ + n ( A x ) (6.3) 141 The moment M2 at node x 2 i s the product of P and v 2 . This calculation i s repeated u n t i l the end of the column is reached. The deflection of the end of the column from the li n e of a x i a l load represents the e c c e n t r i c i t y , e 0 , at which th i s a x i a l load would produce this deflected shape. Once the deflected shape has been obtained in thi s way, the ca l c u l a t i o n i s repeated with the starting mid-span moment M0 reduced by a small amount to M,, to generate a second column deflection curve, and a corresponding end e c c e n t r i c i t y e,. If e, i s less than e 0, as shown in figure 41(b), the second calculation i s ignored, and the result of the f i r s t c a l c u l a t i o n represents a material f a i l u r e for thi s a x i a l load, with maximum end ec c e n t r i c i t y e 0. If e, is greater than e 0, as shown in figure 41(c), then both of these calculations represent unstable configurations. The maximum end e c c e n t r i c i t y i s found by continuing to decrease the starting mid-span moment in small steps and repeating the calcu l a t i o n u n t i l the end e c c e n t r i c i t y passes a maximum and begins to decrease. This maximum end e c c e n t r i c i t y is the required value. Failure of the member in thi s case i s an i n s t a b i l i t y f a i l u r e . The ca l c u l a t i o n described i s for a single a x i a l load. It can be repeated for other a x i a l loads as necessary. 142 6.5 TYPICAL OUTPUT 6.5.1 Axial Load-Moment Interaction Curves Axial load - moment interaction curves can be used to demonstrate the results of these calculations for several column lengths. The calculations described in the previous section have been carried out for columns of several lengths, each at forty d i f f e r e n t load le v e l s between zero and maximum load. Figure 42(a) i s an interaction, diagram showing the combinations of a x i a l load P and mid-span moment P(e+A) just producing f a i l u r e for t y p i c a l 38x140mm columns. The outer l i n e i s the ultimate interaction diagram representing material strength. The inner curves correspond to the curve Pu-D-B-Mu in figure 40(b). For low a x i a l load the inner curves coincide with the ultimate interaction diagram indicating that behaviour under these loads i s governed by material f a i l u r e s . For high' a x i a l load the curves move inside the ultimate interaction diagram, indicating that behaviour under these loads i s governed by i n s t a b i l i t y f a i l u r e s . These curves have been plotted d i r e c t l y from the computer output. Some of the curves are not very smooth, because of rather large steps used in part of the numerical analysis. The curves can be smoothed by reducing the mid-span moment in smaller steps during the analysis of column behaviour, but this involves a corresponding increase in computing costs, and the curves shown are considered to, be s u f f i c i e n t l y accurate for the purposes of thi s study. Figure 42(b) i s the corresponding interaction diagram for 143 o o 6 . 0 END riOnENT (KN.H) (b) End moments F i g u r e 42 - I n t e r a c t i o n diagrams f o r s l ende r columns 144 a x i a l load P and end moment, P times e. Again the outer l i n e i s the ultimate interaction diagram. In thi s case the inner curves correspond to the curve Pu-C-A-Mu in figure 40. For any l e v e l of ax i a l load, the horizontal distance between a point on any curve of figure 42(b) and" the point on the corresponding curve of figure 42(a) represents the moment magnification PA due to member deformations. 6.5.2 Axial Load-Slenderness Curves Another method of presenting these results i s to plot a x i a l load at f a i l u r e against slenderness (or length) for given values of end e c c e n t r i c i t y . A r a d i a l line'such as that shown by l i n e O-R on figure 42(b) represents behaviour for a certain end e c c e n t r i c i t y . The intersection points of the rad i a l l i n e with the interaction curves for each of the lengths shown can be used to make a plot of a x i a l load against slenderness for that end e c c e n t r i c i t y . In these plots slenderness is defined as the non-dimensional r a t i o L/d where L i s the length of the member and d is the cross sectional dimension in the dire c t i o n under consideration. These d e f i n i t i o n s w i l l be used throughout t h i s thesis. A family of such curves i s shown in figure 43. The top curve, for zero e c c e n t r i c i t y , represents the intersections of the interaction diagrams with the v e r t i c a l axis of figure 42(b). The disadvantages of i l l u s t r a t i n g the results in this way are that there i s no indication as to whether a material or i n s t a b i l i t y f a i l u r e occurs, and there i s no way of seeing how much the moments are magnified before f a i l u r e : 145 i r D.O 8.0 16.0 24.0 32.0 SLENDERNESS (L/d) i r 40.0 48.0 F i g u r e 43 - A x i a l l o a d - s l e n d e r n e s s diagram f o r s e v e r a l e c c e n t r i c i t i e s 6.6 INPUT INFORMATION The f o l l o w i n g i n f o r m a t i o n i s r e q u i r e d as input to t h i s computer model. Each of these items w i l l be d i s c u s s e d more f u l l y below. A. Cross s e c t i o n behaviour: 1. Cross s e c t i o n dimensions 2. Tension and compression s t r e n g t h s 3. Modulus of e l a s t i c i t y 4. Shape of s t r e s s - s t r a i n r e l a t i o n s h i p 5. Depth e f f e c t parameters B. Column behaviour 1. Column le n g t h 146 2. Segment l e n g t h f o r column d e f l e c t i o n curves 3. Reduction r a t i o f o r mid-span moment 6.6.1 Cross S e c t i o n Dimensions The program can accomodate any shape of c r o s s s e c t i o n with up to twenty c o r n e r s . However, i t has only been c a l i b r a t e d and v e r i f i e d f o r r e c t a n g u l a r s e c t i o n s , so use with other shapes would r e q u i r e f u r t h e r study and v e r i f i c a t i o n . In p a r t i c u l a r , depth e f f e c t s r e q u i r e much more i n v e s t i g a t i o n f o r non-rectangular s e c t i o n s . 6.6.2 Tension and Compression Strengths The a x i a l s t r e n g t h of the m a t e r i a l i n both t e n s i o n and compression i s e s s e n t i a l input i n f o r m a t i o n . Because of the i n f l u e n c e of member s i z e upon s t r e n g t h , the,input s t r e n g t h s should be f o r the same s i z e of member as that under c o n s i d e r a t i o n . In t h i s study the input t e n s i o n and compression s t r e n g t h s have been obtained from in-grade a x i a l s t r e n g t h t e s t i n g of m a t e r i a l of the same grade, s p e c i e s and c r o s s s e c t i o n a l dimensions as the member under c o n s i d e r a t i o n . To p r o v i d e accurate r e s u l t s , these s t r e n g t h s from i n -grade t e s t r e s u l t s must be c o r r e c t e d f o r l e n g t h before input, using the simple theory proposed e a r l i e r . As an example of the i n f l u e n c e of the l e n g t h e f f e c t , f i g u r e 44 shows the u l t i m a t e i n t e r a c t i o n diagram for four d i f f e r e n t lengths of a t y p i c a l column. The d i f f e r e n c e s between the l i n e s are s i g n i f i c a n t , p a r t i c u l a r l y i n the t e n s i o n r e g i o n . The v a r i a b i l i t y i n wood s t r e n g t h p r o p e r t i e s between boards i s accounted for by i n p u t t i n g s t r e n g t h at s p e c i f i c 147 — i 1 1 1 1 1 1 1 1 1 1 1 0.0 1.0 2.0 3.0 4.D 5.0 6.0 n o n E N T (KN.ru F i g u r e 44 - Ul t i m a t e i n t e r a c t i o n diagrams f o r s t r e n g t h r e p r e s e n t a t i v e of s e v e r a l lengths l o c a t i o n s w i t h i n the s t r e n g t h d i s t r i b u t i o n . For example, mean a x i a l s t r e n g t h s are used to p r e d i c t mean s t r e n g t h in bending and i n combined l o a d i n g , and 5 t h p e r c e n t i l e a x i a l s t r e n g t h s to p r e d i c t 5 t h p e r c e n t i l e s t r e n g t h s i n bending, and combined l o a d i n g , and so on. A f u l l d i s t r i b u t i o n of input s t r e n g t h p r o p e r t i e s i s not e s s e n t i a l because the model can be used to p r e d i c t behaviour at any l e v e l w i t h i n a d i s t r i b u t i o n , from a x i a l t e n s i o n and compression s t r e n g t h s at the same l e v e l . 148 6 . 6 . 3 Modulus of E l a s t i c i t y Timber is assumed to have a linear s t r e s s - s t r a i n r e l a t i o n s h i p to f a i l u r e in tension, and a non-linear r e l a t i o n s h i p in compression. In the linear e l a s t i c range the modulus of e l a s t i c i t y i s assumed to be the same in tension as in compression, and hence in bending. Modulus of e l a s t i c i t y varies along the length of timber boards, but because deformations in a l l segments of the board influence i n s t a b i l i t y behaviour, i t has been assumed that the average value of modulus of e l a s t i c i t y within a board can be used for i n s t a b i l i t y c a l c u l a t i o n s . This average value is independent of board length, so unlike strength properties, modulus of e l a s t i c i t y i s not subject to a length e f f e c t . Figure 45 shows the effect of modulus of e l a s t i c i t y on column behaviour, with no change in tension or compression strength. Figure 45(a) is a t y p i c a l interaction diagram of a x i a l load vs. mid-span moments, and figure 45(b) is the corresponding diagram for end moments. In both cases the s o l i d l i n e s have been constructed for material with a modulus of e l a s t i c i t y of 10,000 MPa, and the dotted lines have been obtained using a modulus of e l a s t i c i t y of 7,500 MPa. It can be seen that the modulus of e l a s t i c i t y has no effect on the ultimate interaction diagram for cross section strength (outer curve), and only a very small e f f e c t on squat columns with a slenderness r a t i o of L/d=6.5 (which was the shortest length tested). For longer members, however, whose strength is 1 49 o 0 . 0 1 . 0 2 . 0 3 . 0 4 . 0 5 . 0 6 . 0 3 . 0 1 . 0 2 . 0 3 . 0 4 . 0 5 . 0 E . O niD-SPRN nODENT (KN.M) END HOHENT (KN.n) (a) Mid-span moments (b) End moments Figure 45 - Effect of modulus of e l a s t i c i t y on column behaviour governed by s t a b i l i t y considerations, the a x i a l load capacity tends to be in direct proportion to the modulus of e l a s t i c i t y . The modulus of e l a s t i c i t y has been measured using several d i f f e r e n t methods, as described in Chapter 4. 5th percentile values of modulus of e l a s t i c i t y have been input with 5th percentile strength values, and so on. Although the corr e l a t i o n between strength and s t i f f n e s s of timber boards is not very good, th i s procedure has produced good r e s u l t s . 6.6.4 Shape of Stress-Strain Relationship in Compression Uncertainties about the precise form of the c u r v i l i n e a r stress s t r a i n r e l a t i o n s h i p require some discussion. Consider the four s t r e s s - s t r a i n relationships shown in figure 46, a l l of which have been introduced in Chapter 2. Note that only the compression region has been shown, as linear e l a s t i c behaviour in tension i s assumed throughout this thesis. 150 Figure 46 - Stress-strain relationships in compression The form of curve shown in figure 46(a), proposed by Glos(l978) on the basis of a large study on commercial quality timber with defects, appears to be the best available representation of actual behaviour. Figure 46(b) shows a similar r i s i n g branch, followed by a linear f a l l i n g branch to approximate behaviour at large s t r a i n s . Figure 46(c) shows a b i l i n e a r r e l a t i o n s h i p with a linear f a l l i n g branch, and figure 46(d) shows the familiar e l a s t o - p l a s t i c relationship, with an upper l i m i t on compression s t r a i n . The sequence described in figures 46(a) to (d)- i s considered to represent a progression of decreasing accuracy in representing real timber properties but increasing s i m p l i c i t y for strength c a l c u l a t i o n s . To compare these curves, the falling- 1 and r i s i n g branches must be investigated separately. . 151 a. Shape of F a l l i n g Branch The shape of the f a l l i n g branch in the compression region of the s t r e s s - s t r a i n curve can have a s i g n i f i c a n t e f f e c t on the ultimate interaction diagram. For example, figure 47 shows an interaction diagram produced by using the b i l i n e a r stress-s t r a i n curve of figure 46(c). The dotted l i n e s show the eff e c t of varying the slope of the f a l l i n g branch. Figure 48 shows that very similar behaviour can be produced by using an ela s t o - p l a s t i c s t r e s s - s t r a i n relationship with varying values for the l i m i t i n g s t r a i n . If the s t r e s s - s t r a i n relationship of figure 46(a) i s used, varying the asymptotic stress for large st r a i n has a similar e f f e c t on the ultimate interaction diagram, which also develops a kink as shown- in figure 49. This s t r e s s - s t r a i n curve has not been used further in this study because the kink creates d i f f i c u l t i e s in calcu l a t i o n of the ultimate interaction diagram from the neutral axis contours, and is not expected to be seen in tests results representative of a large number of boards. Figure 50 demonstrates that almost i d e n t i c a l ultimate interaction diagrams are obtained using the s t r e s s - s t r a i n curves of figures 46(b) and (c) using the same slope of f a l l i n g branch, 0.02 times the modulus of e l a s t i c i t y . Almost exactly the same curve has also been obtained by using the el a s t o - p l a s t i c s t r e s s - s t r a i n relationship, with an upper l i m i t on s t r a i n of 0.009mm/mm. For the remainder of thi s study the f a l l i n g branch of the st r e s s - s t r a i n relationship has been T i 1 1 1 r — i 1 1 1 1 1 1 o i.a 2.a i.a 4.0 v.) o.a MOMENT (KN.M) Figure 47 - Ultimate interaction diagrams for the bi l i n e a r stress-strain relationship, with varying slope of f a l l i n g branch Figure 48 - Ultimate interaction diagrams for the e l a s t o - p l a s t i c s t r e s s - s t r a i n relationship, with varying l i m i t i n g s t r a i n F i g u r e 4 9 - U l t i m a t e i n t e r a c t i o n diagram r e s u l t i n g from s t r e s s -s t r a i n r e l a t i o n s h i p proposed by G 1 O S ( 1 9 7 8 ) F i g u r e 50 - U l t i m a t e i n t e r a c t i o n diagrams f o r s e v e r a l s t r e s s -s t r a i n r e l a t i o n s h i p s 154 -assumed to be linear as shown in figures 46 (b) and (c). The actual slope of the f a l l i n g branch i s discussed with reference to c a l i b r a t i o n of the strength model in Chapter 7. b. Shape of Rising Branch The shape of the r i s i n g branch of the s t r e s s - s t r a i n curve in compression affects only the predicted behaviour of columns loaded concentrically or with very small e c c e n t r i c i t i e s . The curves proposed by Glos (1978), Malhotra and Mazur (1970) and O'Halloran (1973), a l l introduced in Chapter 2, are not s i g n i f i c a n t l y d i f f e r e n t in the r i s i n g branch. O'Halloran's curve has been chosen for the remainder of t h i s study simply because i t has the simplest computational form. One variable which needs to be quantified i s the s t r a i n corresponding with peak stress, indicated by e, in figure 51(a). This can be defined as a r a t i o of the corresponding linear e l a s t i c s t r a i n e 0. For t h i s study the s t r a i n at peak stress e, has been taken as 1.35 times e 0, t h i s being the average result of many experiments reported by Glos on German spruce timber with defects. The data reported by 0'Halloran(1973) gives a figure of 1.25 for tests on small clear wood specimens. To i l l u s t r a t e the effect of this r a t i o , figure 51(b) shows a graph of a x i a l load against slenderness for concentrically loaded columns, as predicted by the model using three values for the r a t i o e,/e 0. A r a t i o of 1.0 i s the simple b i l i n e a r s t r e s s - s t r a i n relationship. The differences are seen to be small but s i g n i f i c a n t in t h i s case of concentric column loading. Similar curves plotted for 1 55 VI in 1 1 1 1 1 1 1 1 I I I ! £LC 8.D WLO 24J) 32.D U1.0 49.3 SLENDERNESS (L/d) (b) F i g u r e 51 - S t r e s s - s t r a i n r e l a t i o n s h i p w i t h f a l l i n g b r a nch e c c e n t r i c l o a d i n g show t h a t the d i f f e r e n c e s d i s a p p e a r , even f o r s m a l l e c c e n t r i c i t i e s . 156 6.6.5 S t r e s s - d i s t r i b u t i o n E f f e c t The o r i g i n and s i g n i f i c a n c e of the s t r e s s - d i s t r i b u t i o n e f f e c t has been d i s c u s s e d i n Chapter 3. A rough estimate of the parameter can be obtained by comparing in-grade t e s t r e s u l t s f o r members with d i f f e r e n t depths but a more accurate estimate w i l l be made i n the c a l i b r a t i o n phase. The importance of the s t r e s s - d i s t r i b u t i o n e f f e c t can be seen i n f i g u r e s 52 and 53. F i g u r e 52 shows the shape of a t y p i c a l u l t i m a t e i n t e r a c t i o n diagram f o r f i v e d i f f e r e n t values of k 3 f o r t e n s i o n . The s o l i d l i n e , f o r no s i z e e f f e c t , r e s u l t s i n a constant f a i l u r e s t r e s s i n the extreme t e n s i o n f i b r e , r e g a r d l e s s of n e u t r a l a x i s depth. Decreasing values of the parameter r e s u l t i n i n c r e a s i n g moment c a p a c i t y f o r a given a x i a l t e n s i o n s t r e n g t h . F i g u r e 53 shows u l t i m a t e i n t e r a c t i o n , diagrams f o r the same m a t e r i a l , t h i s time i n t r o d u c i n g a s t r e s s - d i s t r i b u t i o n e f f e c t i n compression. A comparison with f i g u r e 47 shows that the s t r e s s - d i s t r i b u t i o n e f f e c t i n compression has an i n f l u e n c e s i m i l a r to that of the f a l l i n g branch of the s t r e s s - s t r a i n c u r ve. These two f a c t o r s may not be independent and cannot be q u a n t i f i e d s e p a r a t e l y . Although a s t r e s s - d i s t r i b u t i o n e f f e c t f o r compression has been i n c l u d e d i n the model and demonstrated here, i t has not been used i n c a l i b r a t i o n and v e r i f i c a t i o n c a l c u l a t i o n s , because such a s i m i l a r r e s u l t can be obtained by v a r y i n g the slope of the f a l l i n g branch of the s t r e s s - s t r a i n curve. Figure 52 - Ultimate interaction diagrams with varying st r e s s - d i s t r i b u t i o n parameter in tension • Figure 53 - Ultimate interaction diagrams with varying s t r e s s - d i s t r i b u t i o n parameter in compression 158 6.6.6 Column Parameters The computer program calculates column defl e c t i o n curves for any length s p e c i f i e d . For the step-by-step construction of column deflection curves a column segment length must be input. Any length of segment may be used, short segment lengths providing increased accuracy at greater computing costs. A s e n s i t i v i t y study showed that calculated results were r e l a t i v e l y i nsensitive to segment length, for segment lengths up to four times the section depth. Chen and Atsuta (1976b) report a study which found that a segment length of four times the radius of gyration gives s u f f i c i e n t l y accurate results, t h i s corresponds to 1.16 times the section depth. A segment length of 1.5 times the section depth has been.-used throughout this study. A f i n a l input parameter for construction of column def l e c t i o n curves i s the rate at which mid-span moment is to be reduced in the step-by-step procedure. A value of 0.04 times the maximum moment has been used. 6.7 NON-DIMENSIONALIZED PLOTS The a x i a l load - moment interaction diagrams and the a x i a l load - slenderness diagrams i l l u s t r a t e d in t h i s chapter have been shown with real units representative of the material used in the experimental programme. Later in t h i s thesis i t w i l l become desireable to non-dimensionalize these plots for more general discussion purposes. In both cases the a x i a l load axis can be non-159 dimensionalized by dividing a l l a x i a l loads by the maximum a x i a l load for the material. The same can be done for moment values on the horizontal axis of the interaction diagrams. 6.8 SUMMARY This chapter has described a strength model which can be used to predict the load capacity of timber members of any length under eccentric a x i a l loading. Input to the model i s a x i a l tension and compression strength, the st r e s s - s t r a i n relationship, and certain size effect parameters. 160 VII. CALIBRATION AND VERIFICATION 7.1 INTRODUCTION This chapter relates the test data from Chapter 5 with the strength model for eccentric a x i a l loading described in Chapter 6. Test data are available for two sizes: 38x89mm and 38x140mm. The smaller size was subjected to a more extensive test programme so these results are considered f i r s t . The strength model i s f i r s t c a l i b r a t e d using only the average test results in tension and compression for the shortest length, then i t i s v e r i f i e d using test results for the longer lengths in compression, and 5th and 95th percentiles. Test data i s compared with the model in three d i f f e r e n t ways. 1. The f i r s t comparison uses an interaction diagram of a x i a l load vs. moment to compare the measured moments at the ends of the boards (unmagnified moments) with those predicted by the model. 2. The second comparison i s similar except that mid-span moments (magnified moments) are compared. For the shortest length tested there was very l i t t l e mid-span def l e c t i o n at f a i l u r e , so end moments and mid-span moments were s i m i l a r . A l l f a i l u r e s of the shortest length boards were material f a i l u r e s , so for thi s length comparisons have only been made for mid-span moments. 161 3. The t h i r d comparison uses a plot of a x i a l load vs. slenderness to compare the maximum measured load with the model prediction for several e c c e n t r i c i t i e s . A l l comparisons are made at 5th percentile, mean and 95th percentile levels in the d i s t r i b u t i o n . 7.2 38x89mm BOARDS 7.2.1 Short Column Interaction Curves Figure 54 shows a comparison of strength predicted by the model, compared with test results for the shortest length (0.45m) of 38x89mm boards, for 5th percentile, mean and 95th percentile l e v e l s . The lines are the ultimate interaction diagrams calculated by the model. The points for combined loading are the test results for 0.45m long boards, the same points as shown in figure 24(b). The points on the v e r t i c a l axis represent the results of a x i a l loading tests, corrected to a 0.45m length using equation 3.11. Only these two points were used as input to the strength model when cal c u l a t i n g each curve. The points on the horizontal axis represent the results of bending tests, also corrected for length using equation 3.11. There are two points each for the 5th percentile, mean, and 95th percentile values, because the bending tests at two lengths gave s l i g h t l y d i f f e r e n t r e s u l t s . Once the strength model had been set up as described in Chapter 6, and input data obtained from a x i a l tension and compression test r e s u l t s , values were required for two 1 6 2 Length 0.45m. — I 1 1 1 1 1 1 r— 0.0 1.0 2.0 3.0 4.0 MOMENT (KN.M) 5.0 F i g u r e 54 - U l t i m a t e i n t e r a c t i o n diagram c a l i b r a t e d t o t e s t r e s u l t s f o r 38x89mm s i z e p arameters t o c a l i b r a t e the model. These a r e the depth e f f e c t parameter i n t e n s i o n and a compression parameter, both d i s c u s s e d i n the next s e c t i o n . I t would have been d e s i r e a b l e t o o b t a i n the parameter e s t i m a t e s from independent s o u r c e s , 163 but t h i s was not possible. The two parameters were varied on a t r i a l and error basis to give the good v i s u a l f i t shown in figure 54 for the mean test r e s u l t s . Estimation of the parameters i s described in the next section. The strength model was cal i b r a t e d to the test data for the mean test results only. The same parameters were then used to compare the model with test results at the t a i l s of the strength d i s t r i b u t i o n . The resulting f i t at the 5th and 95th percentile levels i s seen to be quite good, which demonstrates the power of t h i s model, and shows that the various parameters used in the model do not vary very much within a very wide d i s t r i b u t i o n of strength values. A few data points do not appear to f i t the model prediction very well. The points representing eccentric tension tests with the smallest e c c e n t r i c i t y have a weaker strength than predicted, which can be explained by a problem encountered with the testing equipment for that p a r t i c u l a r test. A mis-alignment caused these specimens to be stressed with some bending about the weak axis in addition to the intentional bending about the strong axis, which may account for the observed strength values being lower than the general trend. The points representing eccentric compression tests with the smallest e c c e n t r i c i t y also have a weaker strength than predicted by the model. This may be explained by a length e f f e c t , because although these boards had a clear length of 0.45m between the loading boots, the t o t a l length was 1.05m, 164 and most load was probably introduced by end bearing, for thi s case with very small e c c e n t r i c i t y . This difference in lengths could contribute to the observed r e s u l t s . 7 . 2 . 2 Parameter Estimation The estimation of the two parameters to ca l i b r a t e the strength model are described below. St r e s s - d i s t r i b u t i o n parameter The s t r e s s - d i s t r i b u t i o n parameter k 3 was i n i t i a l l y estimated by comparing mean values of tension and bending strengths as described in Chapter 3. Minor adjustments were then made to provide a better f i t to a l l the data. The f i n a l value of k 3 = 7 . 0 corresponds to a x i a l tension strength being approximately 0 . 6 7 times bending strength, for the same length of test specimen. A comparison can also be made with the depth effect parameter k 2 , which i s expected to be somewhat si m i l a r . The value of k 3 = 7 . 0 used here i s similar to values of k 2 observed in bending tests, but greater than values of k 2 observed in a x i a l tension t e s t s . This difference i s probably related to grading rules. Compression parameters There are two clos e l y linked parameters which aff e c t the shape of the interaction diagram in the compression region. These are the s t r e s s - d i s t r i b u t i o n parameter in compression and the slope of the f a l l i n g branch of the st r e s s - s t r a i n curve. As both have similar effects on predicted behaviour, and neither can be quantified independently, only one has been 165 used in thi s c a l i b r a t i o n . The p o s s i b i l i t y of a stress-d i s t r i b u t i o n effect has been ignored, and the slope of the f a l l i n g branch of the s t r e s s - s t r a i n curve has been selected to give a good f i t to the data. The value used i s 0.02 times the r i s i n g slope in the e l a s t i c region, which represents a very gradual drop in stress with increasing s t r a i n beyond peak stress. Limited test data such as that of G1OS(1978) suggest a steeper f a l l i n g branch which would not give such a good f i t to the data in the compression region. It i s possible that the effects of a steeper f a l l i n g branch are being offset somewhat by a s t r e s s - d i s t r i b u t i o n effect in compression, but this cannot be quantified. 7.2.3 Long Column Interaction Curve for End Moments Figure 55 shows the predicted long column interaction diagram for end moments compared with the test results for four d i f f e r e n t lengths, at three levels within the d i s t r i b u t i o n . The model prediction curves have been obtained using the input data described in Chapter 6, and the two parameters obtained by c a l i b r a t i o n to mean values of the shortest length as described above. Each point on a curve represents a combination of end moment and a x i a l load corresponding to the load capacity of the column for a given e c c e n t r i c i t y and column length. Each plotted point represents 5th percentile, mean, or 95th percentile of maximum recorded a x i a l load and the corresponding end moment for a sample of columns of given length. Each point was obtained from a large number of test 1 6 6 3.2m. i j C 1 Lengl i" / " <i j i i 1 d •o / / / < ,'BJ S I com 0 '08 0 '09 0'Ofr (NX) fJticn IblXti z o C O B C O S 0'0t> 0"DS (N>t) atton n a i x t i O'O i r O'OB C O S O'Ot" O ' K (N>i) aacn luixu F i g u r e 55 - P r e d i c t e d i n t e r a c t i o n diagram f o r end moments compared w i t h t e s t r e s u l t s f o r 38x89mm s i z e 167 r e s u l t s as d e s c r i b e d i n Chapter 5. The f i t of the p r e d i c t e d curves to the data i s reasonable. In general the model tends to underestimate the f a i l u r e moment, but the di s c r e p a n c y i s not l a r g e . No d e f i n i t e e x p l a n a t i o n i s a v a i l a b l e f o r t h i s underestimate, but some c o n t r i b u t i n g f a c t o r s w i l l be d i s c u s s e d . One p o s s i b i l i t y i s that the computer model assumes that s t r e n g t h and s t i f f n e s s are uniform along each board, and that f a i l u r e w i l l always occur at mid-span. In p r a c t i c e , the f a i l u r e s were o f t e n at a de f e c t away from mid-span, r e q u i r i n g a l a r g e r load and moment than would have been necessary had the de f e c t been at mid-span. A r e l a t e d e x p l a n a t i o n i s that the input v a l u e s of a x i a l t e n s i o n and compression s t r e n g t h f o r the model have been m o d i f i e d to the len g t h of the column, assuming that the f u l l l e n g t h of the column i s sub j e c t e d to uniform s t r e s s . T h i s i s a c o n s e r v a t i v e assumption f o r columns which d e f l e c t c o n s i d e r a b l y before f a i l u r e , and may c o n t r i b u t e to the underestimate of the model. The accuracy of the f i t i s con s i d e r e d to be q u i t e a c c e p t a b l e , and i f the p r e d i c t e d curves are to be used as a design t o o l they g e n e r a l l y represent a s l i g h t l y c o n s e r v a t i v e p r e d i c t i o n of s t r e n g t h . 168 7.2.4 Long Column Interaction Curves for Mid-Span Moments Figure 56 shows a comparison of the model prediction and the data, using interaction curves of ax i a l load and mid-span moment. A l l of the curves and data points show much larger moments than in figure 55 because the moments have been amplified due to deformations within the members. The f i t of the data to the model i s not as good in this case, as the model sometimes underestimates the moment at f a i l u r e by a considerable amount. The factors discussed in the previous section s t i l l apply, but another point must be included. Each data point i s calculated from the measured mid-span deflection at maximum ax i a l load. In most cases the f a i l u r e i s an i n s t a b i l i t y f a i l u r e associated with rapid increase in l a t e r a l deflections at very small changes in load. The load was not applied very slowly, so the measured deflections could be considerably larger than that which would just cause f a i l u r e under steady state conditions. A similar problem has been reported in tests on concrete members (Nathan 1983b). In view of these problems, more emphasis should be given to end moments than mid-span moments when checking the v a l i d i t y of the computer model. 169 F i g u r e 56 - P r e d i c t e d i n t e r a c t i o n d i a g r a m f o r m i d - s p a n moments c o m p a r e d w i t h t e s t r e s u l t s f o r 38x89mm s i z e 170 7.2.5 Axial Load - Slenderness Curves Figure 57 shows the axi a l load vs. slenderness curves for the 38x89mm boards at the 5th percentile, mean and 95th percentile l e v e l s . The top (dotted) l i n e i s the predicted curve for concentric loading (not tested), and the next five l i n e s are for the test e c c e n t r i c i t i e s , defined in the key to the data points. The data points are the same points from the test results presented previously. The f i t of the model to the data i s very good. In some cases the model underestimates the a x i a l load recorded in the tests. 7.3 38x14 0mm BOARDS 7.3.1 Short Column Interaction Curves Figure 58 shows the ultimate interaction diagrams from the model, calibrated to mean values of test results for the shortest length (0.914m) of 38x140mm,boards. A l l the parameters used for thi s prediction are as for the 34x89mm material with, one exception. The stress-d i s t r i b u t i o n parameter in tension, which relates the tension strength to the bending strength, i s d i f f e r e n t . The value of k3=7.0 used for the smaller size has been changed to k3=9.0 to provide a better f i t to the test data°for the larger cross section. The model predictions for the 5th percentiles, mean, and 95th percentiles have been made using the same parameters. General observations about the goodness of f i t in this figure are similar to those described for the smaller size material. Note that there are only two points in the tension region, one each at the 5th percentile and mean l e v e l s . As (a) 5 th % i l e (b) Mean (c) 95th % i l e F i g u r e 57 - P r e d i c t e d a x i a l l o a d - s l e n d e r n e s s c u r v e s compared w i t h t e s t r e s u l t s f o r 38x89mm s i z e 172 Length 0.91 Am. \ Mean / / / / / 9 5 t h %ile I 1 1 1 1 1 1 1 1 1 1 1 1 T -0.0 ID 2.D 3.0 4.0 S.0 6.0 7.0 noriENT cKN.ro 8.0 F i g u r e 58 - U l t i m a t e i n t e r a c t i o n diagram c a l i b r a t e d t o t e s t r e s u l t s f o r 38x140mm s i z e d e s c r i b e d i n Chapter 4, the t e s t method f o r d e r i v i n g these p o i n t s was not v e r y s a t i s f a c t o r y , so they s h o u l d be viewed w i t h c a u t i o n , a l t h o u g h they do tend t o support the p r e d i c t e d c u r v e s . 173 7.3.2 Long Column Interaction Curves Figure 59 shows the predicted long column interaction curves for end moments compared with the test r e s u l t s . Figure 60 shows the predicted long column interaction curves for mid-span moments compared with the test r e s u l t s . In both cases the comparison of the model and test r e s u l t s i s of similar form to that previously shown for the 38x89mm size, but the f i t i s not as good and the model i s seen to be very conservative in many cases. No s p e c i f i c explanation i s available other than the points raised in discussion of the 38x89mm re s u l t s . 7.3.3 Axial Load-Slenderness Curves Figure 61 shows the predicted curves of a x i a l load vs. slenderness compared with the test results for the 38x140mm boards. The l i n e s are as described for the smaller size, and the same comments apply. 7.4 REPRESENTATIVE STRENGTH PROPERTIES When investigating possible design methods in the next chapter i t w i l l be convenient to refer to material with certain representative strength properties, rather than s p e c i f i c grades, species, and sizes of timber. The load capacity of timber members is influenced by one or more of the following, depending on the nature of the loads. tension strength compression- strength bending strength modulus of e l a s t i c i t y Figure 59 - Predicted interaction diagram for end moments compared with test results for 38x140mm size 175 (NX) ouoi "ibixti (NX) ami it i ixti Figure 60 - Predicted interaction diagram for mid-span moments compared with test results for 38x140mm size cn (a) 5th % i l e (b) Mean (c) 95th % i l e F i g u r e 61 - P r e d i c t e d a x i a l l o a d - s l e n d e r n e s s c u r v e s compared w i t h t e s t r e s u l t s f o r 38x140mm s i z e 1 7 7 The r a t i o between tension and compression strengths has a major influence on the shape of the ultimate interaction diagram for cross section strength. The r a t i o of tension to bending strength also a f f e c t s the shape of the ultimate interaction diagram because th i s r a t i o i s an estimator for the st r e s s - d i s t r i b u t i o n parameter. For long columns the modulus of e l a s t i c i t y has a major influence on the load capacity. A small survey was carr i e d out on the results of the tests reported in Chapter 5 and on the results of several other in-grade tests carried out at the University of B r i t i s h Columbia. For the r a t i o of in-grade tension strength to i n -grade compression strength, most results were in the range of 0.55 to 0.85. For the r a t i o of tension strength to bending strength, corrected for length e f f e c t s , most results were in the range of 0.6 to 0.8. The modulus of e l a s t i c i t y was between 300 and 350 times the a x i a l compression strength in most cases. These figures are representative of 5th percentile results for SPF and Hem-fir material in sizes 39x89mm to 38x235mm, grades Number 2 to Select Structural, and moisture content approximately 15%. On the basis of this small survey, three groups of representative properties have been chosen, one average group and two groups representing the l i m i t s of observed results, these having properties that produce the largest and the smallest "nose" in the compression region of the a x i a l load moment interaction diagram. Figure 62 shows ultimate interaction diagrams for material of these three representative groups, non-178 AVERAGE STRONG W E A K MOMENT Figure 62 - Ultimate interaction diagrams for representative strength properties dimensionalized to a compression strength of 1.0. Material that i s r e l a t i v e l y weak in tension and in bending produces a large "nose" in the compression region. This is representative of low grade material, and w i l l be referred to as weak material in the next chapter. Material that is r e l a t i v e l y strong in tension and in bending produces a smaller "nose" in the compression region. This i s representative of high grade material, and w i l l be referred to as strong material. The curves representing weak and strong material ( should be considered upper and lower bounds for the material described. Material represented by the central l i n e on figure 179 62 w i l l be referred to as average material. The ratios and r e l a t i v e strengths are summarized in Table I I I . In a l l cases Weak Average Strong Ratios Ft/Fc .55 .70 .85 Ft/Fb .80 .70 .60 Relat ive Fc 1.0 1 .0 1 .0 Strengths Ft .55 .70 .85 Fb .69 1 .0 1 .42 Table III - Material property ratios for representative groups the r a t i o of modulus of e l a s t i c i t y to compression strength has been taken as 300. Figure 63 shows interaction diagrams of end moments, for columns of three slenderness r a t i o s , and material of the three representative strength groups. The curves have been non-dimensionalized so that the shapes of the curves can be compared. Figure 64 shows a x i a l load - slenderness curves for the three representative strength groups, for columns with several end e c c e n t r i c i t i e s . Again these plots have been non-dimensionalized so that r e l a t i v e shapes can be compared. 180 Figure 64 - Axial representative strength load-slenderness curves for properties (non-dimensionalized) 181 7.5 APPLICABILITY OF STRENGTH MODEL The s t r e n g t h model d e s c r i b e d i n t h i s t h e s i s i s a d e t e r m i n i s t i c model w h i c h does no s t a t i s t i c a l c a l c u l a t i o n s . To use t h e model t o c a l c u l a t e s t r e n g t h a t any p e r c e n t i l e i n t h e d i s t r i b u t i o n , i t i s n e c e s s a r y t o use i n p u t d a t a f o r t h a t p e r c e n t i l e , and t o c a r r y o u t a d e t e r m i n i s t i c c a l c u l a t i o n . The t e s t r e s u l t s d e s c r i b e d i n C h a p t e r 5 showed v e r y l a r g e v a r i a b i l i t y i n s t r e n g t h . In t h i s c h a p t e r t h e s t r e n g t h model has been c a l i b r a t e d u s i n g mean t e s t r e s u l t s , and has g i v e n a good p r e d i c t i o n of s t r e n g t h t h r o u g h o u t t h e d i s t r i b u t i o n . T h i s d e m o n s t r a t e s t h e v e r s a t i l i t y o f t h e model, and shows t h a t the p r e d i c t e d b e h a v i o u r of t i m b e r members under combined b e n d i n g and a x i a l l o a d depends o n l y on the'-values of t h e i n p u t d a t a , and n o t t h e l o c a t i o n i n t h e d i s t r i b u t i o n of s t r e n g t h . T h e s e f i n d i n g s a r e s i g n i f i c a n t b e c a u s e t h e y i n d i c a t e t h a t t h e s t r e n g t h model can p o t e n t i a l l y be used f o r t i m b e r from any s o u r c e , a t any l e v e l i n a d i s t r i b u t i o n of s t r e n g t h , p r o v i d e d t h a t i n p u t d a t a i s a v a i l a b l e and t h a t s u i t a b l e c a l i b r a t i o n i s c a r r i e d o u t . 7.6 SUMMARY T h i s c h a p t e r has b r o u g h t t o g e t h e r t h e t e s t r e s u l t s d e s c r i b e d i n C h a p t e r 5 and t h e s t r e n g t h model d e s c r i b e d i n C h a p t e r 6. The model has been c a l i b r a t e d u s i n g o n l y mean t e s t r e s u l t s f o r t h e s h o r t e s t b o a r d s t e s t e d . The s t r e n g t h model c a l i b r a t e d i n t h i s way has been u s e d t o p r o v i d e a r e a s o n a b l y a c c u r a t e p r e d i c t i o n of member s t r e n g t h , f o r a wide range of 182 V member l e n g t h s , throughout the d i s t r i b u t i o n of s t r e n g t h v a l u e s . T h i s i s c o n s i d e r e d t o be q u i t e a s e v e r e t e s t f o r the model. D i s c r e p a n c i e s between the p r e d i c t e d c u r v e s and the t e s t r e s u l t s a re g e n e r a l l y not l a r g e , and where they occur they t e n d t o be c o n s e r v a t i v e . On the b a s i s of th e s e r e s u l t s the model w i l l be used i n the next c h a p t e r t o i n v e s t i g a t e a number of p o s s i b l e d e s i g n methods f o r t i m b e r members s u b j e c t e d t o combined bending and a x i a l l o a d i n g . 183 VIII. DESIGN METHODS FOR COLUMNS AND BEAM COLUMNS 8.1 INTRODUCTION Structural designers require a simple yet e f f e c t i v e method for s i z i n g a member to safely r e s i s t prescribed a x i a l loads and bending moments. One of the objectives of this study is to propose design methods which can be used to check whether a member of selected dimensions has s u f f i c i e n t load carrying capacity. This chapter consists of a very brief review of design philosophy and current code requirements for combined bending and a x i a l loading in timber, steel and concrete, followed by several proposals for new design methods for timber members. 8.1.1 Allowable Stress Design For many years structural design was based on the behaviour of structures under working load conditions and e l a s t i c behaviour. Design codes specified allowable stresses that were not to be exceeded when the structure was loaded with the maximum anticipated loads, without consideration of behaviour at ultimate loads. The allowable stresses were derived from strength tests on materials using large factors of safety. 8.1.2 R e l i a b i l i t y - B a s e d Design A s i g n i f i c a n t advance in structural design was made when design codes began to place more emphasis on the behaviour of structures loaded to conditions near f a i l u r e . "Ultimate strength" design of reinforced concrete and " p l a s t i c " design of steel structures involved checking that a structure would 1 8 4 not actually collapse when loaded with maximum anticipated loads increased by a substantial "load factor". "Limit states" design codes consider structural f a i l u r e as an ultimate l i m i t state to be investigated along with a number of s e r v i c e a b i l i t y l i m i t states. In recent years there has been a lot of attention given to quantifying the amount of structural safety provided by various design methods. Design codes for steel and reinforced concrete have been improved in an attempt to provide a certain minimal pr o b a b i l i t y of f a i l u r e over the l i f e of a structure. Design codes based on t h i s approach may be c a l l e d "probability-based" or " r e l i a b i l i t y -based" l i m i t states design codes. Accurate cal c u l a t i o n of prob a b i l i t y of f a i l u r e requires detailed knowledge of the d i s t r i b u t i o n s of both load and resistance in the overlapping region. This information is not readily available, so approximate methods have beeen developed to calculate a " r e l i a b i l i t y index" from just the mean and standard deviation of load and resistance. There have been many developments in t h i s area since early work by Cornell(1969) and others. Recent developments towards consistent design codes for di f f e r e n t materials in North America are summarized by Galambos et al.(1982). Despite the best attempts to quantify structural safety, observations of structural f a i l u r e s have demonstrated that even when best available estimates of load and resistance d i s t r i b u t i o n s are used, some allowance must, be made for other factors more d i f f i c u l t to quantify such as poor workmanship, 185 design mistakes and unforseen circumstances (Blockley 1980). 8.1.3 Reliability-Based Design of Timber Rel i a b i l i t y - b a s e d design methods for timber structures are less advanced than for other materials. The ca l c u l a t i o n of loads i s e s s e n t i a l l y the same for structures of any material, with a large amount of uncertainty in most cases. The c a l c u l a t i o n of resistance depends on detailed knowledge of the strength properties and behaviour of the structural material. The strength of timber under various loading conditions i s not well defined and i s much more variable than for other materials. Sexsmith and Fox (1978) were among the f i r s t to demonstrate how r e l i a b i l i t y - b a s e d l i m i t states design methods could be applied to glued laminated beams. Foschi(1979) discussed some potential problems with application of the r e l i a b i l i t y index concept to timber structures. Goodman et a l . ( l 9 8 l ) have compared the l e v e l of safety implied by code sp e c i f i c a t i o n s in several international codes. Another investigation of r e l i a b i l i t y - b a s e d design of timber members i s described by Ellingwood(1981). Goodman et al.(l983) summarize a large investigation into r e l i a b i l i t y - b a s e d design of wood transmission l i n e structures. Malhotra(1983) has investigated the r e l i a b i l i t y index implied by several alternative design methods for timber compression members. 186 8.1.4 Scope The results of t h i s study w i l l be used in this chapter to propose design methods for checking that a selected member has the capacity to r e s i s t specified loads. In this context i t is not important whether the loads are specified by a design code in a working stress format or in a l i m i t states format. The actual load and resistance factors in each case must be determined by others. It i s beyond the scope of this study to develop r e l i a b i l i t y - b a s e d design methods for timber members. A p r i n c i p a l contribution of t h i s study has been to produce information on the strength d i s t r i b u t i o n of timber members under, various combinations of a x i a l and f l e x u r a l loadi-ng, which w i l l be useful input to the eventual development of r e l i a b i l i t y - b a s e d design methods. 8.2 EXISTING DESIGN METHDOS 8.2.1 Canadian Timber Code The current Canadian Code (CSA 1980) i s based on allowable design stresses providing a certain factor of safety against f a i l u r e under specified working loads (Wilson 1978). The Timber Design Manual (1980) provides a working guide to the code with many design aids. The background to these requirements have been described e a r l i e r in t h i s thesis. 187 a. Concentrically Loaded Columns For concentrically loaded columns design i s based on three d i f f e r e n t slenderness classes, as described in Chapter 2 and shown in figure 8. Slenderness r a t i o i s defined as the r a t i o of e f f e c t i v e length, L, to the appropriate cross sectional dimension, d. The allowable stress, fa, in a column i s given by L/d < 10 f a = fca ( 8- 1> 10 < L/d < K f a = f c J l - l / 3 ( ^ f ] (8.2) L/d > K f - , 2 3 3 E (8.3) 3 (L/d) 2 The t r a n s i t i o n between intermediate length and long columns i s at a slenderness r a t i o K, given by K = 0.591 | — (8.4) ca where E i s the modulus of e l a s t i c i t y and fca i s the allowable stress p a r a l l e l to the grain in a short column. These formulae include safety factors. The notation has been changed from that in the code for consistency. Although quite simple in concept, the design formulae are unwieldy and awkward to use because there i s a d i f f e r e n t formula for each 1 8 8 of three ranges of slenderness. A number of continuous column formulae which could be used for a l l slenderness ratios w i l l be described later in t h i s chapter. The experimental phase of thi s study did not include concentrically loaded columns, so direc t v e r i f i c a t i o n of the code column formula i s not possible. However, the strength model has been calibrated to results of tests for a large number of e c c e n t r i c i t i e s , and can with s l i g h t extrapolation, be used to predict behaviour at zero e c c e n t r i c i t y . Figure 65 compares the code formulae with behaviour predicted by the model for concentrically loaded columns. The plot has been non-dimensionalized to maximum load using a modulus of e l a s t i c i t y 300 times the material strength. There is a moderate discrepancy between the two curves for intermediate column lengths, the code formula overestimating column strength in t h i s range. b. Combined Axial Load and Bending For members subjected to combined a x i a l compression and bending the code sp e c i f i e s a linear interaction between the a x i a l load capacity of a concentrically loaded column and the moment capacity in pure bending. The formula i s UA + *l± < i (8.5) fa fb where P i s the a x i a l load, A i s the area of cross section, fa is the allowable a x i a l stress under concentric loading for the pa r t i c u l a r slenderness r a t i o , M i s the bending moment 189 nODEL PREDICTION = ~l i i i I 1 1 1 1 1 1 1 1 0-D 8.0 16.0 24.0 32.0 40.0 48.0 SLENDERNESS (L/d) Figure 65 - Code column formula compared with model prediction (non-dimensionalized) including that due to a x i a l loads, S is the section modulus, and fb is the allowable bending stress. The code provides no guidance for ca l c u l a t i n g the bending moment due to a x i a l load and column deflections, but the Timber Design Manual (1980) suggests equations for simple cases and a t r i a l and error method for others. For short columns equation 8 .5 i s consistent with assumptions of linear e l a s t i c behaviour, a l i m i t i n g value of compression stress, with no consideration of a f a i l u r e in the tension zone. E a r l i e r chapters have shown that these assumptions are not consistent with observed behaviour. As an example of the inadequacies of the code provisions for short 190 columns, refer to figure 23. The form of equation 8.5 i s represented by the straight dotted l i n e which i s seen to be very conservative compared with the inner s o l i d l i n e sketched through the test r e s u l t s . For longer columns refer to the four parts of figure 56. The inner dotted curves are the model predictions of 5th percentile strength. Equation 8.5 would be represented by straight l i n e s with the same axis intercepts, which would again be very conservative. A similar linear interaction formula i s spec i f i e d for combined bending and a x i a l tension. c. Summary To summarize, present Canadian code requirements are unsatisfactory for the following reasons: 1. For concentrically loaded columns of intermediate lengths, the code formula overestimates strength. The awkward formulae for intermediate and long columns could be replaced by a single continuous formula for a l l slenderness r a t i o s . 2. For combined bending and a x i a l compression the code interaction equation i s a poor representation of actual behaviour, grossly underestimating strength in many cases. 3. Moment magnification due to slenderness e f f e c t s is not adequately provided for. 191 8.2.2 NFPA Timber Code A timber design code widely used in the United States i s that produced by the National Forest Products Association (NFPA 1982). This code i s also based on allowable stresses. a. Concentrically Loaded Columns Design of concentrically loaded columns i s similar to the Canadian code requirements of equations 8.1 to 8.4. The t r a n s i t i o n between short and intermediate columns is at a slenderness r a t i o of 11 rather than 10, and the safety factors are s l i g h t l y d i f f e r e n t . b. . Combined Axial Load and Bending For combined a x i a l compression and bending, a general formula is used which includes both eccentric a x i a l loads and transverse loads. ' The derivation by Newlin(l940) and Wood(l950), described b r i e f l y in Chapter 2, used assumptions more applicable to clear wood than to sawn timber. The formula i s P/A M M/S + P/A(6 + 1.5 J) ( e / d ) { 8 > 6 ) T~ + f - P/A * 1 W H E R E J = M^S* . 0< J < 1 (8-7) and M/S i s the bending stress resulting from l a t e r a l loads, fb is the allowable stress in bending, e is the e c c e n t r i c i t y of the a x i a l load. A l l other terms are as defined in the previous section. 192 The code formula w i l l be compared with behaviour predicted by the strength model for e c c e n t r i c a l l y loaded columns, assuming that the allowable stresses in the code are representative of actual behaviour, and ignoring safety factors. For the case of no transverse loads (M/S=0) the moment allowed by the code formula can be obtained from equation 8.6 as f. - JP/A a where M i s now the end moment P times e. This has been plotted for comparison with the model prediction in figure 66. For low slenderness r a t i o (L/d=lO) the code formula i s seen to be conservative at low a x i a l loads and unsafe at high a x i a l loads. For columns with L/d=20 the code formula is again very conservative for low a x i a l loads, mainly because i t predicts only 80% of actual bending strength when a x i a l load is zero (This can be seen by puttting P=0 and J=1 in equation 8.8). The small unsafe region near the v e r t i c a l axis is due to the discrepancy between the code formula and the model prediction for t h i s slenderness r a t i o , as shown in figure 65. For slender columns (L/d=40) the code i s again conservative for low a x i a l loads. The comparison i l l u s t r a t e d in figure 66 has been made for material with "average" strength properties as defined in Chapter 7. For "strong", or "weak" material the model predicts very d i f f e r e n t interaction curves as shown in figure 1 9 3 riODEL PREDICTION NFPA FORriULA U N H f l G N I F I E D FlOHENT Figure 66 - NFPA formula compared with model prediction (non-dimensionalized) 63, but equation 8.8 produces almost i d e n t i c a l curves. This comparison has not been plotted, but a comparison of figures 66 and 63 shows that the NFPA formula w i l l result in large unsafe areas for strong material and large conservative areas for weak material. c. Summary 1 Compared with the Canadian code, the NFPA code i s an improvement in that moment magnification i s incorporated into a r e l a t i v e l y simple formula that handles both eccentric a x i a l loads and transverse loads. However, by comparing the NFPA formula with the prediction of the strength model, i t has been shown that the formula does not accurately predict the 194 s t r u c t u r a l behaviour of timber compression members at ultimate loads. 8.2.3 Code Requirements for Steel The current Canadian code for steel design (CSA 1978) i s in a l i m i t states format. Structural members are designed such that their load capacity, reduced by a performance factor, i s not exceeded by spec i f i e d factored loads. The Handbook of Steel Construction (1980) provides a commentary and working guide to the code with many design aids. The Canadian s t e e l code i s based on similar p r i n c i p l e s to many other codes for steel design. For concentrically loaded columns, design follows the same general procedure as for timber, with a parabolic t r a n s i t i o n from material f a i l u r e in short columns to Euler buckling in long columns. For combined bending and a x i a l loading the code spe c i f i e s that strength and s t a b i l i t y be checked separately. When designing a member to r e s i s t some combination of a x i a l load P and bending moment M, the s t a b i l i t y requirement i s s a t i s f i e d in i t s simplest form i f P , 1 M . , / n — + T=W7f- M~ < 1 (8.9) u e u where Pu i s the a x i a l strength of the column under concentric loading (including possible e f f e c t s of buckling), Mu i s the moment capacity and Pe i s the Euler buckling load from equation 2.6. This formulation assumes that the interaction 195 diagram for ax i a l load and mid-span moment in slender beam columns i s the straight l i n e Pu-Mu on Figure" 67, which is an empirical relationship v e r i f i e d by experimental studies MOMENT Figure 67 - Axial load-moment interaction diagram for steel members (Galambos 1968). To s a t i s f y t h i s requirement, a design combination of P and M must l i e to the l e f t of the curved dotted l i n e Pu-Mu, the horizontal distance between the dotted l i n e and the s o l i d l i n e representing moment magnification caused by l a t e r a l deflections in the member. The moment magnification factor i s given by F = 1 - P/P, (8.10) which i s a close approximation to the exact expression for 196 linear e l a s t i c behaviour (Timoshenko and Gere 1961). For short columns in which the a x i a l strength i s not reduced by s t a b i l i t y e f f e c t s the interaction diagram becomes the straight l i n e Pa-Mu on Figure 67, where Pa i s the axi a l compression strength of the column material. If the beam-column i s loaded with d i f f e r e n t moments at each end the design equation becomes r + I=P7P- M- < 1 { 8 - 1 1 ) u e u where Cm i s a factor to account for the d i s t r i b u t i o n of moments along the member which can be approximated by C m = 0.6 + 0.4 M2/M1 > 0.4 (8.12) where M, and M2 are the larger and smaller end moments respectively. Galambos(1968) and Johnston(1976) show that th i s i s a reasonable approximation for a wide range of loading cases. The product of Cm and M is an equivalent uniform moment that should lead to the same long column strength as the actual moment diagram. For some unsymmetrical loading conditions the load capacity of a beam-column w i l l be governed by the moment capacity of a p l a s t i c hinge at one end. In thi s case the load capacity w i l l be governed by strength rather than by s t a b i l i t y considerations, and the design must be checked against an interaction diagram for cross section strength. The interaction formula for strong-axis bending i s 1 97 M < 1.18 (1 - !-) , M < M (8.13) u u which i s shown by the l i n e Pa-B-Mu in figure 67. The b i l i n e a r shape appears because the web of an I-section can carry some ax i a l load without reducing the p l a s t i c moment capacity of the flanges. The shaded l i n e on figure 67 shows the resulting design envelope for a possible situation with unsymmetrical end moments. The possible e f f e c t s of l a t e r a l - t o r s i o n a l buckling in bending can be accommodated by using a bending strength value Mu that represents the strength at which buckling occurs. B i a x i a l behaviour i s included by extending the interaction formula for s t a b i l i t y to C M C M , ' x L_ + rc* x + m y _ Z _ < i (8.14) P 1 - P / P M 1 - P / P M u e x u x e y u y where the subscripts x and y refer to actions about the x and y axes., A similar formula is specified for material strength under b i a x i a l loading. These formulae assume linear interaction between x-axis and y-axis behaviour. This assumption i s very conservative (Johnston 1976), but an appendix to the code provides more detailed formulae for more accurate design in certain cases. It can be seen that the steel requirements have some s i m i l a r i t i e s to those for timber, but are s i g n i f i c a n t l y better because 1. Strength and s t a b i l i t y effects are considered separately. 198 2. The specified interaction between a x i a l load and moment i s more representative of actual behaviour at ultimate loads. 3. Moment magnification due to second-order effects i s included. 4. The effect of unequal end moments i s included. 5. B i a x i a l e f f e c ts are accounted for. 6. The possible e f f e c t s of l a t e r a l t o rsional buckling can be included. 8.2.4 Canadian Concrete Code The current Canadian code (CSA 1977) for reinforced concrete design is written in a l i m i t states format, often referred to as "ultimate strength design". The Canadian code is similar to many other reinforced concrete codes. As with s t e e l , s t r u c t u r a l members are designed such that their ultimate load capacity, reduced by a capacity reduction factor, i s not exceeded by sp e c i f i e d factored loads. A detailed background to the column provisions is given by MacGregor et al. ( l 9 7 0 ) , and is used as the basic reference for this section. The design method for reinforced concrete compression members i s s i g n i f i c a n t l y d i f f e r e n t from the methods described above for steel and timber. The code provisions are much less extensive than for s t e e l , partly because long slender members occur much less frequently in concrete than in s t e e l , and because members with hinged end conditions are rare. No s p e c i f i c provision is made for concentrically loaded compression members. A l l members must be designed to r e s i s t a minimum nominal bending moment due to an e c c e n t r i c i t y of 10% 199 of the member's dimension about either axis. . Although an accurate second-order s t r u c t u r a l analysis i s recommended, slenderness effects are generally accounted for by magnifying the bending moment from a l l sources by a magnification factor, F, given by C F = m 1-P/4>P (8.15) where P i s the factored a x i a l load, Pe is the Euler buckling load, 0 is a capacity reduction factor, and Cm i s the c o e f f i c i e n t for di f f e r e n t end moments exactly as used in the steel code. The member i s then sized and reinforced such that the factored a x i a l load combined with the magnified factored bending moment does not cause a material f a i l u r e at any cross section. The strength of the cross section is determined from f i r s t p r i n c i p l e s by standard methods or from published design charts (ACI 1970). The shape of the ultimate interaction diagram for cross section strength is very similar to that obtained for timber members in t h i s study. This method of design assumes that a l l f a i l u r e s are material f a i l u r e s , not i n s t a b i l i t y f a i l u r e s . The design method considers i n s t a b i l i t y only i n d i r e c t l y , in that equation 8.15 can only be used sensibly for a x i a l loads less than the Euler buckling load. Equation 8.15 i s an approximation based on linear e l a s t i c theory, but real behaviour of reinforced concrete i s non-linear. To overcome th i s problem the code 2 0 0 includes an empirical expression for the s t i f f n e s s EI which leads to approximately correct results for members which have material f a i l u r e s . These approximations are apparently not serious d e f i c i e n c i e s in the design method because very few reinforced concrete columns are very slender. For members f a i l i n g in an i n s t a b i l i t y mode (slender prestressed members for example) the apparent s t i f f n e s s EI would have to be further reduced empirically, or a more rati o n a l method method used (Nathan 1983a). The expression for EI also includes a factor to allow for the p o s s i b i l i t y of creep under long duration loading. The p o s s i b i l i t y of a similar provision for timber members is discussed in Chapter 9. The p r i n c i p a l advantages of the reinforced concrete method are that 1. A single design procedure can be used, for a l l types of compression members. 2. No concentric loading column formula is required. 3. The short column interaction curve used for a l l design can be derived from f i r s t p r i n c i p l e s , or simply obtained from published graphs or tables. The primary disadvantage i s that the design method for slender columns is not an accurate representation of real behaviour. The approximations introduced are sati s f a c t o r y for reinforced concrete but may not be for more slender columns made of timber. ( 201 8.2.5 Limit States Timber Codes A proposal for a l i m i t states design format for timber structures was made by SexsmitM1979). Two new codes contain provisions similar to some of those suggested by Sexsmith. The Canadian code proposal (CSA 1983) i s almost i d e n t i c a l to the exis t i n g working stress code previously described, with new load factors and resistance factors which change the format of the design equations without any s i g n i f i c a n t conceptual changes. A second code in l i m i t s states format i s the Ontario Highway Bridge Design Code (OHBDC 1982) referred to as OHBDC in t h i s chapter. The design provisions of OHBDC for compression members represent a major change from existing requirements. There are several improvements but the requirements are s t i l l lacking in some respects. The main change i s the deletion of a design method for concentric loading, recognizing that i t i s impossible to load a timber member with zero end e c c e n t r i c i t y . For a x i a l loading a minimum end e c c e n t r i c i t y of 0.05 times the cross section dimension i s speci f i e d , together with an i n i t i a l bow (or crook) at mid-length of 1/500 times the e f f e c t i v e length. The design equation is a e u where a l l the terms are the same as defined previously for steel and concrete. This equation assumes that a l l f a i l u r e s 2 0 2 are material f a i l u r e s ; i n s t a b i l i t y f a i l u r e s are only considered i n d i r e c t l y in that the equation can only be used for a x i a l loads less than the Euler buckling load. To investigate the possible consequences of t h i s s l i g h t l y incorrect approach, a comparison with the more correct formulation of the steel code w i l l be made. Consider figure 68 which i s an interaction diagram of a x i a l load vs. moment. MOMENT Figure 68 - Axial load-moment interaction diagram from OHBDC Pa i s the concentric a x i a l load capacity of a short column, Pu is the concentric a x i a l load capacity of the long column under consideration, and Mu is the bending capacity. Figure 68 has been constructed, to scale, to i l l u s t r a t e the behaviour of a slender column whose concentric load capacity Pu i s half of 203 the short column load capacity Pa. The curve Pu-C-Mu has been obtained from the OHBDC formula (equation 8.16). The curve Pu-B-Mu has been constructed from the more correct steel formula (equation 8.11). The steel code suggests that t h i s member, subjected to a x i a l load A, can just carry an end moment represented by point B, which corresponds to a magnified mid-span moment represented by point D. OHBDC suggests that this member can carry a larger end moment represented by point C which corresponds to a magnified mid-span moment causing material f a i l u r e at point E. The OHBDC approach i s seen to be conceptually incorrect and s l i g h t l y unsafe. The OHBDC formula i s compared with the prediction of the strength model in figure 69. To make a v a l i d comparison i t has been necessary to work backwards from the linear interaction diagram for magnified moments, to compare the unmagnified moments that the designer begins with, shown as dotted l i n e s . The dotted lines are a l l curved toward the ori g i n because the moment value of each point on the linear interaction diagram has been reduced by the code magnification factor to give a design moment. For short columns (L/d=l0) the OHBDC formula i s seen to be very conservative, except at high a x i a l loads, even more so than the NFPA formula. For longer columns the formula becomes more accurate. Figure 69 has been plotted for "average" strength material, but the OHBDC curves w i l l not change for "strong" or "weak" material, so a comparison with figure 63 shows that the formula w i l l be 204 o cr o DODEL PREDICTION - OHBDC FORrlllLH \ \\ \ \ CO _ C3 \ CO _ \ \ , \ \ \ . V \ x \ A A U 1 0 \ _ \ ^ \ ^ J V d = 20 \ X. \ \ Ol _ \ \ \ L/d = 40 \ •= = ' * — / a " - ^ ^ \ / <=>' I I I I 1 I I 1 I I I I I 0.0 0.2 0.4 0.6 0.8 I.D 1.2 UNHflGNIFIED HOHENT Figure 69 - OHBDC formula compared with model prediction (non-dimensionalized) very conservative for weak material. The OHBDC formula does not accurately represent behaviour at ultimate loads. The linear interaction diagram is conservative and the moment magnification factor w i l l be shown to be unconservative, but these two effects cancel out somewhat to produce the curves shown in figure 69. For b i a x i a l loading the OHBDC speci f i e s a linear interaction formula almost i d e n t i c a l to that in the steel code. The s p e c i f i c a t i o n of minimum end e c c e n t r i c i t y and i n i t i a l bow suggests that b i a x i a l loading should be considered in every case but the code i s not clear on this matter. In summary, the OHBDC represents a major development in 205 tha t i t s p e c i f i e s minimum moments and i n c l u d e s a des ign equa t ion fo r a m p l i f i c a t i o n of a l l moments, but some of i t s assumpt ions are not c o n s i s t e n t wi th the r e s u l t s of t h i s s tudy . 8.3 COLUMN CURVES FOR CONCENTRIC LOADING Some of the des ign methods to be proposed fo r beam-columns w i l l r e q u i r e a method fo r c a l c u l a t i n g the s t r eng th of the column under c o n c e n t r i c l o a d i n g . A b r i e f i n t r o d u c t i o n in Chapter 2 d e s c r i b e d some of the d i f f i c u l t i e s in c a l c u l a t i n g the s t r e n g t h of columns of m a t e r i a l s w i th non- l i nea r p r o p e r t i e s . Any accu ra te approach r e q u i r e s d e t a i l e d knowledge of the s t r e s s - s t r a i n r e l a t i o n s h i p s , and such i n fo rma t i on i s not a v a i l a b l e fo r t imber members, e s p e c i a l l y when d e a l i n g wi th t y p i c a l p r o p e r t i e s of p o p u l a t i o n s of members r a the r than w i th i n d i v i d u a l spec imens. For s l ende r columns the Eu l e r equa t ion can be used , but fo r s h o r t e r l eng ths some form of e m p i r i c a l formula must be used . There are s e v e r a l e m p i r i c a l formulae a v a i l a b l e that can be used fo r any s l ende rness r a t i o . The best formula i s l i k e l y to be the one wi th the s i m p l e s t computa t iona l form produc ing a good f i t to expe r imen ta l r e s u l t s . F i g u r e 70 shows a compar ison of s e v e r a l c u r v e s . The s o l i d l i n e i s the column behav iour p r e d i c t e d by the computer model . The do t t ed l i n e s are i d e n t i f i e d in the key, and w i l l be d i s c u s s e d b r i e f l y h e r e . The l i n e marked CODE i s the c u r r e n t code formula (CSA 1980, NFPA 1982), which has been d i s c u s s e d e a r l i e r wi th r e f e r ence to equa t ions 8.1 to 8 .3 . 206 16.0 24.0 32.0 SLENDERNESS (L/d) 48.0 Figure 70 — Comparison of column curves for concentric loading (non-dimensionalized) The curve marked MALHOTRA i s almost a perfect f i t to the model prediction. This curve has been proposed by Malhotra and Mazur (1970) by using tangent modulus theory based on the str e s s - s t r a i n relationship of equation 2.1. The resulting column formula is TT2E + f c ( L / r ) 2 2c(L/r) 2 [TT 2E + f c ( L / r ) 2 ] 2 - 4TT2E f c(L/r) 2 4c 2(L/r) 4 (8.17) where fu i s the maximum a x i a l stress that the column can support, fc i s the failure, stress for a short column of the same material, and r i s the radius of gyration. The term c, defining the shape of the s t r e s s - s t r a i n relationship in 207 equation 2.1, i s taken as 0.90 both here and by Malhotra. The formula has a sound theoretical background but i s cumbersome to use. The l i n e marked CUBIC RANKINE i s a cubic modification to the t r a d i t i o n a l Rankine formula, proposed by Neubauer(1973). The formula i s f f = £ U i + f c ( W d ) 3 ( 8 . 1 8 ) E 4 0 where d i s the cross sectional dimension in the d i r e c t i o n that gives the largest slenderness r a t i o L/d, and a l l other terms are as used above. The number 40 has been chosen to provide the best f i t to the model prediction, compared with a value of 50 used by Neubauer to give a good f i t to tests on small clear Douglas-fir columns. Neubauer describes the theore t i c a l development and the subsequent simplification, for use with timber, but the formula must be considered empirical because one parameter has been obtained from a c u r v e - f i t t i n g exercise. The l i n e marked PERRY-ROBERTSON i s the curve obtained from the Perry-Robertson formula which i s the basis of B r i t i s h and European codes. The formula i s _ f e + ( ^ i ) , . _ r*e + c n + i ) . • ( 8 . 1 9 ) u 2 / v 2 ' c e , ca where n = — Terms not previously defined are fe, the stress in the column 208 due to application of the Euler buckling load; c, the assumed i n i t i a l deviation from straightness and a, the distance" of the extreme f i b r e on the concave side from the neutral axis. Robertson(1925) and Sunley(l955) report that p r a c t i c a l l y a l l experimental values l i e between the two curves obtained by taking n=0.00l L/r and n=0.003 L/r. The figure of 0.001 has been used to give the best f i t in t h i s case. Allen and Bulson (1980) describe how the Perry-Robertson formula must be regarded as an empirical formula despite the logic by which i t was derived, because of the inadequacy of several assumptions. If one of these curves is to be selected for design purposes, there i s l i t t l e between them as they a l l give a very good f i t to the strength predicted by the computer model, and several have previously been v e r i f i e d experimentally for timber columns. The one with the simplest computational form therefore becomes favoured, and this is the cubic Rankine formula. Figure 71 shows the curve resulting from the proposed formula, compared with the model prediction. The curve i s not an exact f i t , but compared with ' the present formula i t i s no less accurate, and is a l o t simpler to use. 8.4 PROPOSED DESIGN METHODS FOR ECCENTRICALLY LOADED COLUMNS This section w i l l describe several alternative design methods for e c c e n t r i c a l l y loaded timber columns, based on the experimental and a n a l y t i c a l results of e a r l i e r chapters. 209 ex o _J _J o ' cr i—i x cr MODEL P R E D I C T I O N PROPOSED FORriULR i i r i i i i i O.D B.O 16.0 24.0 32.0 SLENDERNESS (L/d) 40.0 48.0 F i g u r e 71 - Design p r o p o s a l f o r c o n c e n t r i c l o a d i n g compared with model p r e d i c t i o n 8.4.1 Type of Loading and A n a l y s i s T h i s chapter i s concerned with columns su b j e c t e d to e c c e n t r i c a x i a l loads, with no t r a n s v e r s e loads a p p l i e d to the member. Transverse loads w i l l be d i s c u s s e d i n Chapter 9. At t h i s stage, assume that the a p p l i e d a x i a l l o a d has equal e c c e n t r i c i t i e s at each end. The case of unequal end e c c e n t r i c i t i e s w i l l be i n t r o d u c e d a t ' a l a t e r stage i n t h i s c h a p t e r . The design methods presented i n t h i s chapter are f o r s i n g l e members, f o r which the a x i a l loads and end moments are known. These methods are intended f o r design without the use of a second order s t r u c t u r a l a n a l y s i s . Chapter 9 w i l l d i s c u s s 210 how the results of thi s thesis can be used when a second order analysis is avai l a b l e . 8.4.2 Input Strength Properties This section assumes that certain input information i s available for the t r i a l member which is to be designed. The information is 1. The a x i a l tension capacity Tu, of the member, which i s the product of the cross section area A, and the a x i a l tension f a i l u r e stress f t . 2. The a x i a l compression capacity Pa, of the member, which i s the product of the cross section area A, and the ax i a l compression f a i l u r e stress fc, for a short column. 3. The bending moment capacity Mu, of the member, which is the product of the section modulus S, and the modulus of rupture f r . 4. The modulus of e l a s t i c i t y , E. The tension, compression and bending strength values referred to above are a l l assumed to depend on the length and depth of the member, as described in Chapter 3 and summarized in Chapter 9. Modulus of e l a s t i c i t y i s assumed to be independent of member siz e . 8.4.3 Design Approaches This section b r i e f l y categorizes the six design methods that w i l l be described in the remainder of this chapter. Suppose that a column i s to be designed to r e s i s t a given a x i a l load P with equal end e c c e n t r i c i t i e s e. If a t r i a l member i s selected, a design method i s required to check 211 whether the member has s u f f i c i e n t s t r e n g t h to r e s i s t the a p p l i e d l o a d . The d e f l e c t e d shape of a t y p i c a l member i s shown i n f i g u r e 39. The a x i a l l o a d throughout the member i s ' P. The bending moment M at the ends of the member (and the "unmagnified" moment at mid-span) i s P times e. The "magnified" moment at midspan i s FM=P(e+A), where F i s a m a g n i f i c a t i o n f a c t o r due to the member d e f l e c t i o n A. With r e f e r e n c e to f i g u r e 72, proposed design methods 1, 2 and 3 w i l l provide an approximate method of c a l c u l a t i n g the m a g n i f i c a t i o n f a c t o r F, hence the magnified moment FM at mid-span, and w i l l compare that magnified moment with an i n t e r a c t i o n diagram (or f a i l u r e envelope).. In both methods 1 and 2, comparisons w i l l be made with the c o r r e c t i n t e r a c t i o n diagram shown by the s o l i d l i n e . - The d i f f e r e n c e between Methods 1 and 2 w i l l be d i f f e r e n t approximations to the same i n t e r a c t i o n diagram. C o n c e p t u a l l y t h i s i s - t h e same approach as used i n the s t e e l code. In method 3 the magnified moment w i l l be compared with the u l t i m a t e i n t e r a c t i o n diagram r e p r e s e n t i n g m a t e r i a l f a i l u r e , shown by the dotted curve i n f i g u r e 72. Conc e p t u a l l y t h i s i s a l e s s a c c u r a t e approach, which i s i s used f o r r e i n f o r c e d concrete design, and i s s i m i l a r to the OHBDC pro p o s a l f o r timber members. Methods 4 and 5 w i l l compare end moments, or unmagnified moments, with the corresponding i n t e r a c t i o n diagram (or f a i l u r e envelope) f o r end moments. These two methods do not 212 o o Z p u a I p M F M M u M o m e n t Figure 72 - Typical interaction diagram for a x i a l load and magnified moment involve c a l c u l a t i o n of a magnification factor, because the effe c t s of moment magnification have been considered in construction of the interaction diagram. Method 4 w i l l propose obtaining the interaction diagram from the strength model using published graphs, while method 5 w i l l propose an approximate formula. Method 6 w i l l take a d i f f e r e n t approach by proposing the use of a x i a l load vs. slenderness formulae for e c c e n t r i c a l l y loaded columns. 8.4.4 Moment Magnification Factor Methods 1, 2 and 3 w i l l require a method for cal c u l a t i n g a moment magnification factor. Most design codes use the e l a s t i c moment magnification factor given by equation 8.10. For the timber columns investigated in- this study, the actual amplification of mid-span moment at f a i l u r e i s considerably Material failure Mid - span moments at failure v. 213 more than predicted by this equation, because of non-linear figures 73(a) (b) and (c), which are ty p i c a l non-dimensionalized interaction diagrams of a x i a l load vs. moment for members with slenderness ra t i o s of 10, 20 and 40, respectively. In each case the outer s o l i d curve i s the ultimate interaction diagram for a cross section. The next curve (chain dotted) represents combinations of a x i a l load and mid-span moment causing f a i l u r e for that p a r t i c u l a r length. The inner curve ( s o l i d line) i s the combinations of a x i a l load and end moment causing f a i l u r e . The horizontal distance between these two curves represents the magnification of moment due to deformations in the member. The intermediate curve (small dots) has been obtained from the curve of end moments by amplifying the moments using equation 8.10. It i s apparent that the actual amplification at f a i l u r e i s considerably more than predicted by this equation, so that i f the other parameters in t h i s method could be quantified i t would be unsafe to use equation 8.10 for cal c u l a t i n g the moment magnifier, and a more accurate expression would be necessary. Suppose that the magnification factor i s to be increased by an empirical factor A at maximum load, with no increase at zero a x i a l load, and linear interpolation between these two extremes. The equation becomes material behaviour in compression. For example, consider F 1-P/P 1 [l + (A-l) |-] (8.20) e e 214 Figure 73 - Interact ion diagram showing t r a d i t i o n a l moment magnifier 215 T r i a l calculations show that a suitable value for A i s 2.0, producing "a magnification factor of 1 + P/P 1 - P/P e e (8.21) The previous interaction diagrams replotted using this equation are shown in figure 74. Although rather conservative for slender columns, t h i s i s considered to be a reasonable empirical approach for design purposes which w i l l be used for the rest of thi s study. 8.4.5 METHOD 1: Bi l i n e a r Interaction Diagram Figure 75(a) shows a t y p i c a l interaction diagram for ax i a l load vs. magnified moments, superimposed with straight l i n e s passing from concentric a x i a l column strength to a point on the horizontal axis at a r a t i o of B times the moment capacity. These li n e s can be used as a design approximation for the interaction diagram, terminating at the v e r t i c a l l i n e through the moment capacity, which cuts off the "nose". This approximation i s conservative as shown by the shading, but is quite simple. The resulting design equation, providing a s t a b i l i t y check for any length of column, i s C F M — < B(l - !-) , C F M < M (8.22) M v P ' m u u u where F i s the magnification factor from equation 8.21, Cm is the factor for unequal end e c c e n t r i c i t i e s from equation 8.12 216 F i g u r e 74 I n t e r a c t i o n diagram showing proposed moment mag n i f i e r 217 O HRGNIFIED HOHENT (b) W e a k 25 ^ 30 *"*" 1 ______ 40 *~ " - — — i *~ • . y ~~ ———_~— — 50 — 1 1 1 1 r t i i i i i -1 1 1 1 1 1 1 I r— o.o ' 0.2 a." o.c 0.1) 1.0 HRGNIFIED HOHENT (c) S t r o n g HRGNIFIED HOHENT Figure 75 for - B i l i n e a r approximation to interaction diagram magnified moments (non-dimensionalized) 218 and B i s the h o r i z o n t a l a x i s i n t e r c e p t , which depends on the shape of the u l t i m a t e i n t e r a c t i o n diagram. I t can be shown that a s u i t a b l e s e m i - e m p i r i c a l value f o r B can be given by 1.35 = F7f~ (8.23) t C where f t / f c i s the r a t i o of a x i a l t e n s i o n s t r e n g t h to a x i a l compression s t r e n g t h f o r the m a t e r i a l . The d e r i v a t i o n of t h i s e x p r e s s i o n i s given i n appendix B. F i g u r e s 75(b) and (c) show t h i s approximation f o r weak and strong m a t e r i a l , and i t i s seen to be reasonable i n most cases. For s i t u a t i o n s where the two end moments are d i f f e r e n t and the l o a d c a p a c i t y i s governed by the s t r e n g t h of the c r o s s s e c t i o n at one end of the member, a s t r e n g t h check must be made, as s p e c i f i e d i n the s t e e l code, using the approximation f o r the outer curve i n f i g u r e 75 TT < <x ~h > M i < M u ( 8 ' 2 4 ) u a where M, i s the maximum end moment. 8.4.6 METHOD 2: P a r a b o l i c I n t e r a c t i o n Diagram Another e m p i r i c a l method of approximating the i n t e r a c t i o n curves i s to assume a f a m i l y of parabolae as shown in f i g u r e 76. T h i s type of e x p r e s s i o n was suggested by Newlin(1940) . The curves are l e s s c o n s e r v a t i v e than the s t r a i g h t l i n e code formula, but more c o n s e r v a t i v e than Method 1, p a r t i c u l a r l y as they show de c r e a s i n g moment c a p a c i t y even 219 i 1 r 0.4 0.6 0.B J .0 MAGNIFIED MOMENT Figure 76 - Parabolic approximation to interaction diagram for magnified moments (non-dimensionalized) for small a x i a l loads. A design equation for s t a b i l i t y based on t h i s approach would be u C F M m M < 1 (8.25) and the strength check would become f!_12 + < i 1 (8.26) *-P ' M a u where the terms are a l l as used in Method 1. A refinement to make these curves a l i t t l e less conservative would be to use a variable exponent rather than the figure of 2 in equations 8 .25 and 8 . 2 6 . For example, the exponent could be increased to 4 or more for slender columns, but thi s p o s s i b i l i t y has not 220 ) been considered worth pursuing. This design method w i l l produce i d e n t i c a l curves for strong or weak material. A comparison with figure 63 shows that this method w i l l be even more conservative for weak material, but may become unsafe for strong material. 8.4.7 METHOD 3; Ultimate Interaction Diagram The reinforced concrete design method can be modified to produce a method of similar form to Method 1, the s i g n i f i c a n t difference being that the member is checked for material f a i l u r e only, regardless of slenderness r a t i o . The b i l i n e a r approximation to the ultimate interaction diagram shown by the outer straight l i n e s in figure 75 w i l l be used in thi s section. The resulting design equation is C F M p _E < B f ! _ £ _ ) , C FM < M (8.27) M 1 P ' m u u a which is the same as equation 8.22 with Pu replaced by Pa. A strength check i d e n t i c a l to equation 8.24 would also be necessary for unequal end e c c e n t r i c i t i e s . Method 3 i s similar to the OHBDC code which conservatively assumes a straight l i n e interaction diagram between points Pa and Mu, and uses the unconservative expression of equation 8.3 as the magnification factor. 221 8.4.8 Comparison of Methods 1 to 3 Figures 77(a), (b) and (c) show a comparison of the proposed design methods 1 to 3, for the three, representative groups of material properties. These comparisons have been made by working backwards from the interaction diagram for magnified moments, as described in discussion of the OHBDC code, to compare unmagnified moments (design moments) for each method with those predicted by the strength model. Each figure makes the comparison for three slenderness r a t i o s , as marked. In each case the s o l i d l i n e i s behaviour predicted by the strength model, and a di f f e r e n t dotted l i n e i s used for each proposed method. Consider the upper group-of curves in figure 77(a), for a slenderness r a t i o of L/d=10 and average material properties. Method 1 i s seen to be the best f i t to the model, the only s i g n i f i c a n t discrepancy being at low a x i a l loads where the "nose" was cut off the interaction diagram in figure 75. The parabolic approximation of Method 2 is considerably more conservative, although i t does co-incide with the model prediction and the Method 1 curve on the v e r t i c a l axis. Method 3 produces the same results as Method 1 for low a x i a l loads, but for high a x i a l loads i t moves into the unsafe region and meets the v e r t i c a l axis at a load corresponding with a x i a l crushing f a i l u r e of the material. For more slender column behaviour, consider the central group of curves in figure 77(a) for L/d=20. The curves for Methods 1 and 2 have moved a l i t t l e closer together, and both 222 UNnflGNIFIED nOHENT UNnflGNIFIED HOHENT 0.0 0 .2 0.4 0 .6 ' 0.8 1.0 1.2 UNnflGNIFIED HOriENT F i g u r e 77 - C o m p a r i s o n o f m e t h o d s 1 t o 3 w i t h m o d e l p r e d i c t i o n ( n o n - d i m e n s i o n a l i z e d ) 223 are conservative throughout, Method 2 being somewhat more so. Method 3 i s again the same as method 1 for low a x i a l loads, moving into the unsafe region for high a x i a l loads. The method 3 curve meets the v e r t i c a l axis at an a x i a l load corresponding to the Euler buckling load Pe, whereas the curves for Methods 1 and 2 meet the v e r t i c a l axis at an a x i a l load corresponding to the more accurate concentric column capacity Pu, for columns of th i s intermediate length. For much more slender columns with a slenderness r a t i o of L/d=40, the bottom group of curves on figure 77(a) shows that there is not much difference between the three methods, a l l being s l i g h t l y conservative. For this slenderness r a t i o the concentric column capacity Pu is almost exactly the Euler load Pe, so a l l the curves meet at th i s point on the v e r t i c a l axis. For materials with d i f f e r e n t r e l a t i v e strength properties, figures 77(b) and (c) show similar plots to those described for figure 77(a). Method 1 is again the best f i t , although i t becomes s l i g h t l y unsafe for strong material at low slenderness r a t i o . The formula for Method 2 does not include the r e l a t i v e strength properties, so this curve i s i d e n t i c a l on figures 77(a), (b) and (c), and i t becomes unsafe for strong material. Method 3 remains unsafe in several areas. The proposal producing the most consistent results is that of Method 1, which i s the preferred method from this group. The conservatism in t h i s method, where i t exi s t s , results from the poor approximation of the interaction diagrams for magnified moments shown in figure 75, and the 224 conservative nature of the proposed moment magnification factor for low a x i a l loads, as shown in figure 74. A general observation i s that there i s very l i t t l e difference between the alternatives for slenderness ratios of L/d2:20. Method 1 becomes much better than the other methods for slenderness ra t i o s of L/d<20, p a r t i c u l a r l y for weak or low grade material. Better accuracy would result from a more precise but much more complicated formula for the shape of the interaction diagram, or from published interaction diagrams for a range of material properties and slenderness r a t i o s . If the publishing of families of curves is to be seriously considered i t would be better to_ publish diagrams for unmagnified moments, as w i l l be suggested next. 8.4.9 METHOD 4; Published Design Curves Figure 42(b) showed a family of non-dimensionalized interaction curves of ax i a l load vs. unmagnified moment, at f a i l u r e for a range of slenderness r a t i o s . If curves of this nature were made available in published form, a designer could simply check that a given combination of a x i a l load P and moment M f a l l s inside the curve for the part i c u l a r slenderness r a t i o . Published design curves are used extensively for reinforced concrete design (ACI 1970). The designer should be aware of the derivation of these curves. A point on any curve represents a combination of a x i a l load and end moment that would just cause a member to f a i l . The mid-span moment at f a i l u r e would be s i g n i f i c a n t l y more than t h i s end moment. 225 Both m a t e r i a l f a i l u r e s and i n s t a b i l i t y f a i l u r e s have been c o n s i d e r e d i n producing the cu r v e s . The shape of these curves depends on s e v e r a l f a c t o r s , the most important being the r a t i o of t e n s i o n to compression s t r e n g t h , and the modulus of e l a s t i c i t y . A number of d i f f e r e n t f a m i l i e s of curves would be necessary to cover a s u i t a b l e range of p r a c t i c a l v a l u e s . They would be non-d i m e n s i o n a l i z e d so that one graph c o u l d be used f o r any s i z e of c r o s s s e c t i o n . Unequal end e c c e n t r i c i t i e s c o u l d be i n c l u d e d by m u l t i p l y i n g the design moment M by the f a c t o r Cm from equation 8.12 before comparing with the p u b l i s h e d curves. In t h i s case i t would a l s o be necessary to check c r o s s s e c t i o n s t r e n g t h , comparing P and M with the corresponding curve (the outer curve on f i g u r e 42(b), f o r L/d=0) 8.4.10 METHOD 5: S t r a i g h t Line Approximation Some of the curves i n f i g u r e 42(b) are almost s t r a i g h t , and the p o s s i b i l i t y of a s t r a i g h t l i n e i n t e r a c t i o n i s suggested. Because the curves tend to turn upwards near the v e r t i c a l a x i s , a s t r a i g h t l i n e between Mu on the h o r i z o n t a l a x i s and, say, 0.9 Pu on the v e r t i c a l a x i s w i l l be c o n s i d e r e d . The r e s u l t i n g l i n e s are shown on f i g u r e 78. They are very c o n s e r v a t i v e f o r short columns, and only have a few small unsafe areas f o r longer columns. These l i n e s produce a very simple design equation P C M + m < 1 ( 8 . 2 8 ) 0.9 P M u u 2 2 6 Figure 78 diagrams - Straight l i n e for unmagnified approximations to interaction moment (non-dimensionalized) 227 The terms are a l l as defined previously. Once again a strength check is necessary for unsymmetrical loading, and the b i l i n e a r approximation of equation 8.24 could be used for that purpose. Method 5 can be compared with Method 1 by superimposing the straight l i n e s of figure 78 on figure 77, Such a comparison i s shown in figure 79(a), for average strength material. The s o l i d l i n e s are the model prediction values which represent Method 4 i f produced as published curves. Method 5 i s very conservative for short columns (for example, the outside curves where L/d=lO), but not much di f f e r e n t for longer columns. The corresponding comparisons for strong and weak material are shown in figures 79(b) and (c). This approach i s purely empirical, and just happens to work in some cases because the shape of the mid-span moment interaction .diagram, reduced by the moment magnification factor, produces an interaction diagram for end moments consisting of nearly straight l i n e s . A disadvantage of t h i s method i s that designers are not alerted to the real behaviour of slender compression members with moment magnification. Consequently this method does not provide much guidance for handling unusual design situations. 8.4.11 METHOD 6; Axial Load-Slenderness Curves Another quite d i f f e r e n t approach to the design of e c c e n t r i c a l l y loaded columns can be made by using a x i a l load-slenderness curves for various e c c e n t r i c i t i e s . With reference to the three-dimensional sketch shown in figure 11, the 228 nODEL PREDICTION HETHOD 1 HETHOD 5 A v e r a g e UNnflGNIFIED HOHENT nODEL PREDICTION HETHOD 1 HETHOD 5 W e a k t.2 UNHHGNIFIED HOHENT nODEL PREDICTION HETHOD 1 HETHOD 5 1.2 UNHRGNIFIED HOHENT F i g u r e 7 9 - C o m p a r i s o n o f methods 1 and 5 w i t h model p r e d i c t i o n ( n o n - d i m e n s i o n a l i z e d ) 229 interaction diagrams discussed in the previous sections represented cuts through the surface p a r a l l e l to the load moment plane. This section describes behaviour represented by ra d i a l cuts perpendicular to the load-moment plane, a l l passing through the slenderness axis. The derivation of ax i a l load-slenderness relationships from the strength model has been described in Chapter 6 and a t y p i c a l plot was shown in figure 43. There have been many proposals for formulae to construct a x i a l load-slenderness curves for e c c e n t r i c i a l l y loaded columns. Several were introduced in Chapter 2, where their assumptions and lim i t a t i o n s were discussed b r i e f l y . The most common approach has been to use as assumed deflected shape and a simple f a i l u r e c r i t e r i o n based on a l i m i t i n g compression stress. The secant formula and the Perry-Robertson formula are both in t h i s category. Both can be modified to allow for dif f e r e n t f a i l u r e stresses in bending and in compression, and approach also used in the derivation of the NFPA formula, so these three formulae are a l l based on very similar assumptions. A di f f e r e n t approach was taken by Jezek who considered s t a b i l i t y behaviour of an e l a s t o - p l a s t i c material. It i s not appropriate to compare a l l these formulae here, however an example w i l l be used to compare a x i a l load-slenderness curves from one formula with results predicted by the strength model. Consider the modified secant formula, given by Malhotra(1980) as 230 f = u + f - / [f ( i + + f ]2 - 4f f ( l - 1.04e e c ) (8.29) 2(1 -in which e i s the e c c e n t r i c i t y of load, and d is the member thickness in the di r e c t i o n of berfding. Malhotra found that the modified secant formula gave a reasonably good prediction of test results for timber columns loaded with small e c c e n t r i c i t i e s . Figure 80 compares the modified secant formula with results predicted by the strength model for a range of e c c e n t r i c i t i e s . The model prediction in thi s case MODEL PREDICTION MODIFIED SECANT — i 1 1 1 1 1 1 1 1 1— O ao 16.0 24.0 32.0 40.0 SLENDERNESS (L/dJ 48.0 Figure 80 - Modified secant formula compared with model prediction has been made using the average representative strength ratios 231 previously described and the f i t is reasonably good. This may not be the same in other cases, however. Model predictions for d i f f e r e n t strength properties were shown in figure 64. The other formulae have not been plotted here, but i t can be shown that the Jezek formula and the NFPA formula give quite similar f i t s , and the Perry-Robertson formula tends to underestimate column strength. This method has several disadvantages compared with the a x i a l load-moment interaction diagrams discussed e a r l i e r . In most cases the formulae have been based on s i m p l i s t i c assumptions for clear wood, which may not be v a l i d for commercial quality timber. The formulae cannot be e a s i l y used for cases of unequal end e c c e n t r i c i t i e s , and they do not provide the designer with as much insight into member behaviour as methods based on moment magnification. Design methods based on a x i a l load-slenderness curves have not been pursued further in this thesis. 8.5 COMBINED BENDING AND TENSION Existing code requirements for combined bending and tension are a l l based on a l i m i t i n g stress f a i l u r e c r i t e r i o n which produces a linear interaction between bending capacity and a x i a l tension capacity f - + W * 1 (8.30) u u where T i s the design tension force, Tu i s the a x i a l tension strength, M i s the design moment and Mu i s the moment 232 capacity. It can be seen from a t y p i c a l case such as figure 54 that the straight l i n e interaction w i l l be a conservative approach in a l l cases. It w i l l be most accurate for material with a low r a t i o of tension to compression strength and a r e l a t i v e l y i n s i g n i f i c a n t s t r e s s - d i s t r i b u t i o n e f f e c t . This li n e a r interaction formula becomes very conservative for high grade or clear material where the tension strength i s high compared with the compression strength. Some machine stress graded material may also come into t h i s category. A useful approximation for a l l cases is to assume that the ultimate interaction diagram for material strength can be approximated by a parabola as shown in figure 81. (As an aside, t h i s parabola was considered for describing compression too,, but was found to be much less accurate than the methods described. The s o l i d l i n es in figure 81 are the same as the lines in figure 54, being the model prediction for the ultimate interaction diagram for the 38x89 mm material tested. Each dotted l i n e i s a parabola which has been forced to pass through the a x i a l tension and compression strengths on the v e r t i c a l axis, and the bending strength on the horizontal axis. Each parabola i s seen to be a reasonable f i t to the model prediction, p a r t i c u l a r l y in the tension region. A design combination of a x i a l tension T and bending moment M is inside such a parabola i f [ -L T 2 r 1+r 1-r 12 + M 1+r J M [ 1 " ( < 1 (8.31 ) u u 233 MOriENT (KN.ri) Figure 81 - Parabolic approximation to interaction diagram where r i s the r a t i o of a x i a l tension strength to a x i a l compression strength, f t / f c . In the symmetrical case when r=1.0, equation 8.31 s i m p l i f i e s to LT ' M u u which i s conservative for r greater than 1.0. A conservative design method i s to use equation 8.30 where tension strength i s less than compression strength, and 234 equation 8.32 in other cases. The more accurate, but unwieldy, equation 8.31 might be incorporated into a computer-based design procedure. 8.6 SUMMARY This chapter has reviewed existing design methods for members with combined bending and a x i a l loading, for s t e e l , concrete and timber. Several possible new design methods for timber members have been investigated. Of the six alternative methods described, only Methods 1 and 4 w i l l be recommended for design purposes. Method 1 provides approximate formulae for hand ca l c u l a t i o n , suitable for inclusion in a design code. This method provides a reasonable prediction of the actual behaviour of an e c c e n t r i c a l l y loaded column. Method 4 i s a more accurate method in which published design curves would be used by the designer to check whether a design combination of a x i a l load and unmagnified moment i s safe. Applications of both these methods w i l l be discussed in a broader context in the next chapter, together with several related topics. 235 IX. DESIGN RECOMMENDATIONS This chapter recommends design methods for timber members subjected to combined bending and a x i a l loading, on the basis of proposals described in Chapter 8. This chapter also includes discussion of several related topics. 9 . 1 STRUCTURAL ANALYSIS 9 . 1 . 1 Strength Model This study has demonstrated that the ultimate strength of a single timber member subjected to eccentric a x i a l load with equal end e c c e n t r i c i t i e s can be modelled by a ra t i o n a l analysis, using a computer-based numerical procedure. This rat i o n a l analysis, c a l l e d the "strength model", includes the e f f e c t s of both geometric and material non-linearit.ies. The most accurate available method of analysing and designing such a member would be to use the strength model. Such a procedure i s not r e a l i s t i c for most designers, but may be considered by designers of large volume structural components such as mass-produced roof trusses. This approach would be more a t t r a c t i v e i f the model were able to handle unequal end e c c e n t r i c i t i e s and transverse loads. A less costly and more widely applicable approach would be to produce a series of design charts based on the rational analysis, as suggested in Method 4 of the previous chapter. These design charts could be produced for any desired combinations of strength values, for d i f f e r e n t sizes, grades, 236 moisture contents and so on. 9.1.2 Second Order Structural Analysis Many structural designers now have access to computer programs for second order s t r u c t u r a l analysis, which formulate equilibrium on the deformed shape of the structure. This type of analysis calculates "magnified" moments throughout the structure. Most programs of thi s type are based on linear e l a s t i c material behaviour, so calculated member deflections w i l l not be as accurate as those calculated using the strength model. This should be taken into account by designers using second order analysis programs for timber structures. Programs are available which include the ef f e c t s of non-linear material properties, but none are s p e c i f i c a l l y appliccable to timber, which exhibits much less d u c t i l i t y than s t e e l , for example. The major advantages of second order programs over the strength model are that they can handle r i g i d frames of many members, as well as arches and any other st r u c t u r a l form, with any pattern of loads. The use of a second order program can be combined with the results of thi s study to produce a design method which is a major improvement over current methods. The second order analysis w i l l provide information on i n s t a b i l i t y f a i l u r e s , but design c r i t e r i a for material f a i l u r e s can come from t h i s thesis. The designer requires the ultimate interaction diagram for the member under consideration, and thi s can be provided in the form of a computer output curve, or as a 237 simple approximation using equations 8.23 and 8.24. Appendix A includes a method for calcu l a t i n g the ultimate interaction diagram which i s more direct than the strength model, more accurate than the approximate equations, but rather tedious. 9.1.3 Simple Analysis Structural designers often w'ish to carry out an approximate design of a structural member without the use of sophisticated computer programs as described above. If a x i a l loads and end moments in a member are determined by hand analysis, or using a f i r s t order s t r u c t u r a l analysis computer program, the methods described in the previous chapter are suitable for c a l c u l a t i n g whether a chosen size has s u f f i c i e n t strength. .The most accurate of these i s to use published design charts from the strength model, as proposed in Method 4. A less accurate method, more suitable for specifying as code formulae, is to use the approach described as Method 1. This approach is summarized in the next section. 9.1.4 Code Format If a timber design code i s to have the f l e x i b i l i t y to allow these approaches, i t should be a two-level code which recommends rati o n a l analysis, but also provides the formulae for an optional approximate method. Many codes for other materials are based on t h i s philosophy. 238 9.2 APPROXIMATE DESIGN FORMULAE 9.2.1 Recommended Formulae If a designer i s not in a position to use the results of a r a t i o n a l analysis, the following approximate procedure from Chapter 8 i s recommended for checking that a selected compression member can r e s i s t a design a x i a l load P and a bending moment M about one p r i n c i p a l axis. 1 . Obtain the a x i a l compression capacity for a short column Pa, the a x i a l tension capacity Tu, the bending moment capacity Mu, and the modulus of e l a s t i c i t y E, for the member. Derivation of thi s input data i s described in the next section. A l l of these values, except E, depend on the size of the member, also as described in the next section. 2. Calculate the load capacity Pu, of the member under concentric a x i a l compression loading. P a P u * „ ( 9 . 1 ) i+_c (L/d) ; E 4 0 where fc/E is the r a t i o of the f a i l u r e stress of a short column to modulus of e l a s t i c i t y , and L/d i s the largest slenderness r a t i o about either p r i n c i p a l axis. 3. Check that the design combination of a x i a l load P and mid-span moment M s a t i s f y the s t a b i l i t y formula 239 _E < B n - £_} C FM < M (9.2) u u where F i s a moment magnification factor given by 1 + P/P F = 1 - P/P e ± (9.3) where Pe i s the Euler buckling load. Cm i s a factor to allow for unequal end moments Cffl = 0.6 + 0.4 M2/M1 > 0.4 (9.4) where M, and M2 are the larger and smaller end moments, respectively. B i s a factor used to approximate the shape of the interaction diagram, given by B = (9.5) t c where f t / f c i s the r a t i o of a x i a l tension to a x i a l compression strengths defined in step 1. 4. If end moments are not equal, check that the maximum end moment M, s a t i s f i e s the strength formula ^ < B(l - £-), MX < Mu (9.6) u a 240 9.2.2 Example The formulae described above w i l l be si m p l i f i e d when material properties are included for a pa r t i c u l a r species and grade of timber. Consider the SPF timber of No. 2 and Better grade tested in t h i s study. The r a t i o of fc/E i s 1/300, and the r a t i o of f t / f c i s 0.7. Using these figures, the formula for concentric loading becomes P p = 2 . , (L/d) 3 (9.7) 12000 and the s t a b i l i t y check for combined bending and compression becomes C F M < M (9.8) m u and the strength check, for the end of a member becomes (9.9) M, < M 1 u 9.2.3 Load Factors and Resistance Factors The design formulae presented here do not include load or resistance factors. These can be added to the formulae using established methods once the o v e r a l l design philosophy i s formulated. Some codes use larger safety factors for slender columns than for squat columns because of the greater danger of accidental e c c e n t r i c i t y or overload causing f a i l u r e , and the greater consequences of such f a i l u r e . This can be e a s i l y C F M m  M < 1.71 (1 h u £ « 1-71 ( i - f ) . u a 241 incorporated. 9.2.4 Mimimum Moments The proposal of 0HBDC(1983) for minimum end ec c e n t r i c i t y and i n i t i a l lack-of-straightness appears to be a sensible one that should be incorporated into code requirements. To avoid the need to design every member for b i a x i a l bending i t is suggested that these e f f e c t s be considered in only the most slender d i r e c t i o n when no other s i g n i f i c a n t moment exists, and in both directions i f s i g n i f i c a n t applied moment ex i s t s . The actual values suggested by OHBDC appear reasonable, but have not been investigated in t h i s study. 9.3 DATA REQUIRED 9.3.1 In-grade Test Results To use the design methods proposed in thi s thesis for any timber members of given size, species, grade and moisture content, the following information is required for the member under consideration. 1. The a x i a l tension capacity Tu, of the member, which is the product of the cross section area A, and the axi a l tension f a i l u r e stress f t . 2. The a x i a l compression capacity Pa, of the member, which i s the product of the cross section area A, and the a x i a l compression f a i l u r e stress fc, for a short column. 3. The bending moment capacity Mu, of the member, which i s the product of the section modulus S, and the modulus of rupture f r . 242 4. The modulus of e l a s t i c i t y , E. Each of these properties should be derived from in-grade testing on the actual material in question. As long as codes are based on lower 5th percentile values, the values are a l l required at that quantile. Values obtained d i r e c t l y from test results must be modified with safety factors, load duration factors and so on. The strength model can be used to derive Mu from Tu and Pa, provided that certain c a l i b r a t i o n factors are available. However the bending strength Mu i s the most often measured strength property, so design values can be obtained d i r e c t l y from test r e s u l t s . The f i r s t three items above are subject to size e f f e c t s as described in the next section. In Canada the current design code (CSA 1980) s p e c i f i e s allowable stresses in a x i a l tension based on the results of in-grade testing, so code figures are a r e l i a b l e estimate of strength. A large amount of in-grade testing has been carried out on bending members, and steps are being taken to change the code bending stresses to r e f l e c t in-grade test r e s u l t s . In-grade testing of bending members has provided information on modulus of e l a s t i c i t y , so data is available to specify modulus of e l a s t i c i t y at the 5th percentile l e v e l . The l a t e s t draft code (CSA 1983) includes a proposal to use a 5th percentile modulus of e l a s t i c i t y for column design simply calculated as 0.74 times the average value (for sawn timber). 243 In-grade Compression Strength The information not yet available i s a x i a l compression strength. A few small studies of in-grade a x i a l compression strength have been referred to in t h i s thesis, but there are many grades, species and sizes which have not been tested. Some results in t h i s area are expected to be available shortly. The allowable stresses in the code are very much di f f e r e n t from values suggested by in-grade test results for some species. It has been shown how the interaction diagram for timber members i s sensitive to the r a t i o of tension and compression strengths. If th i s r a t i o i s calculated from allowable stresses in the current code there w i l l be some extremely misleading r e s u l t s . • ' No s i g n i f i c a n t improvements can be made to design methods u n t i l comprehensive in-grade a x i a l compression test results are a v a i l a b l e . 9.3.2 Size Effects Axial tension and compression strengths, and bending strength, are affected by member siz e . If strength values based on in-grade testing are not available for any size, they can be estimated using the following size effect formulae which were discussed in Chapter 3. 244 a. Length Effects The basic formula for length effects i s equation 3.11. x, L 0 1/k, . _J. = (-1) 1 (9.10) *2 L l where x, and x 2 are strengths of lengths L, and L 2 respectively, and k, i s the length effect parameter. The tentative values of k, found in t h i s study are given in Table IV, with d i f f e r e n t values for two grades. The figure in brackets is the corresponding strength reduction factor for doubling the.length of a member. Number 2 Select Structural Axial Compression Axial Tension Bending 13 (.95) 4 (.84) 4 (.84) 13 (.95) 6 (.89) 6 (.89) Table IV - Length effect parameter k. Most in-grade testing i s carried out at standard lengths of 3.0m for a x i a l tests and 17 times the depth for bending tests. For lengths shorter than these test lengths i t i s conservative to assume no length e f f e c t , or to use a large number for the length e f f e c t factor. For longer lengths i t is conservative to use a low number for the length effect factor. The length effect factors described here have been based on experimental data from only two lengths in most cases, and extrapolation beyond t h i s range may produce d i f f e r e n t r e s u l t s . These considerations should be investigated further before 2 4 5 incorporating length effect provisions into design codes. b. Depth E f f e c t s If test results are available for one depth of cross section, the strengths of other depths can be predicted from x , d ~ l / k 9 JL - 2 ( 9 . 1 1 ) *2 1 where x, and x 2 are strengths of members of depths d, and d 2 respectively, and k 2 i s the depth effect parameter. This formula i s l i k e l y to be used less often than equation 9 . 1 0 because there are only a few standard depths of cross section and most have been tested. Values of k 2 can be d i f f i c u l t to obtain because i t is d i f f i c u l t to separate depth e f f e c t s from grading e f f e c t s , and for most bending tests length and depth have both been changed in a constant r a t i o . Limited test data on SPF material suggests a value of k 2 = 4 in tension, and k 2 in the range 8 to 15 for bending. The bending value i s very sensitive to the length effect factor k, used to correct in-grade bending test results for length. A larger value of k 2 for tension than for bending has also been observed in clear wood (Buchanan 1 9 8 3 ) . 9 . 4 OTHER LOADING CASES Almost a l l of the experimental and a n a l y t i c a l investigations described in thi s thesis have been for a x i a l l y loaded members with equal e c c e n t r i c i t i e s at each end. Structural designers are often confronted with, situations where end e c c e n t r i c i t i e s are unequal or where a combination of 246 a x i a l loa.ds and transverse loads occur. Examples are wind loads on roof-supporting columns, p u r l i n loads on top chords of trusses, and many others. 9.4.1 Unequal End E c c e n t r i c i t i e s For a l l of the design method proposals incorporating a x i a l load-moment interaction diagrams, a simple method of including unequal end e c c e n t r i c i t i e s has been introduced with the Cm factor. The accuracy of t h i s factor for timber members has not been investigated in thi s study, but investigations of i t s a p p l i c a b i l i t y to steel (Johnston 1976) suggest that this i s a suitably conservative method for considering unequal end eccentr ic i t i e s . 9.4.2 Transverse Loads In real design situations an i n f i n i t e number of combinations of transverse loads and eccentric a x i a l loads are possible. No simple code formula can provide an accurate design method for a l l cases. A second order structural analysis can calculate deflections throughout a structure for any loading. If such an analysis is not carried out an approximate method of handling transverse loads becomes necessary. This discussion assumes that the structure i s braced against sidesway. Special consideration i s necessary for columns of unbraced structures. If transverse loads tend to increase the mid-span moment due to axi a l loads as shown in figure 82(a), the combined t o t a l mid-span moment should be used d i r e c t l y in the s t a b i l i t y design formula. If the 247 transverse loads tend to eliminate the mid-span moment as shown in figure 82(b), load capacity is l i k e l y to be based on cross section strength at the ends, so the strength design formula can be used. Many other combinations are possible, and in a l l situations a number of load combinations must be considered and a large measure of engineering judgement may be necessary, aided by an understanding of the potential modes of f a i l u r e . In a l l cases the design method recommended for a single member i s useful because i t provides the designer with an indication of the load capacity of the material, an interaction diagram for slender columns, and the amount of moment magnification to be expected as a result of second order deflections. 1 t Load Moment (a) t Load Moment (b) Figure 82 - Bending moment diagrams for combined a x i a l and transverse loading 248 A recent extension to one of the source programs used in t h i s study can calculate behaviour of beam columns under any combination of a x i a l loads, transverse loads, or applied moments (Nathan 1983b), and could be used as the basis of a more detailed study in t h i s area in the future. 9.4.3 Slenderness Ratio For any of these design methods not u t i l i z i n g a second order analysis, i t i s necessary to determine the "slenderness r a t i o " of the member. Throughout th i s thesis the term slenderness r a t i o has been used as L/d where L i s the length of the member and d i s the cross sectional dimension in the d i r e c t i o n under consideration, for a rectangular cross section. For a non-rectangular section d must be replaced by r\/12 where r is the radius of gyration, but i t should be noted that t h i s study has not considered non-rectangular sections because within-member size e f f e c t s have not been investigated for such cases. The length L should be the "effective length", which is the actual length for a column pinned at both ends. For a member in a structure, i t s e f f e c t i v e length i s the length of a pinned-pinned column which would buckle at a load equal to the load in the member when the structure buckles. Standard methods are available for estimating the e f f e c t i v e length for a variety of end conditions, but for some structures such as arches and frames, estimation is very d i f f i c u l t without a second order analysis. 249 9.4.4 B i a x i a l Behaviour This thesis has investigated the in-plane behavior of timber members subjected to bending about one of the p r i n c i p a l axes of the cross section. There are many design situations where b i a x i a l behaviour must be considered, and for lack of any better information the methods of the steel cbde are suggested. The steel code formula for s t a b i l i t y under b i a x i a l loading is shown as equation 8.14. The same formula is sp e c i f i e d in the OHBDC for timber design. In equation 8.14 the term Pu is the lowest a x i a l load capacity of the column considering the worst case for buckling about either p r i n c i p a l axis. Moments about the x and y axes are considered separately and are magnified separately using the corresponding Euler load as Pe in each case. This equation assumes a linear interaction between x-axis and y-axis e f f e c t s , which is a very conservative assumption for steel members and probably also for timber members. More accurate empirical expressions are available for steel members (Johnston 1976) but not for timber members. The steel code formulae modified for timber members using design proposal Method 1 become C F M C F M„ P mx x x my y y , RCI - — ) M M P ux uy u (9.12) C F M C F M mx x x + my y y < 1 M M ux uy for s t a b i l i t y , where Fx and Fy are calculated from equation 2 5 0 9.3 using the corresponding Euler load in each case. The formula for strength becomes M M T , M M _JL_ + < B ( l - — ) , -2- + -X_ < i 9.13 M M ' P J' M M ux uy a ux uy Some suggestions for further research in thi s area are made below. a. Strength Under B i a x i a l Loading The strength model developed in th i s . t h e s i s could be used to investigate cross section strength under b i a x i a l bending, because the general form of the computer program allows members of any cross section to be input. In theory i t would be possible to change the angle of the neutral axis as shown in figure. - 83 "for many d i f f e r e n t angles, but a serious d i f f i c u l t y would be encountered with depth e f f e c t s . For Figure 83 - Bending about inclined neutral axis rectangular sections subjected to bending about one p r i n c i p a l axis i t has been easy to include a depth effect and no width effect on the basis of experimental r e s u l t s . For the section of figure 83 i t i s not easy to separate depth effects and 251 width e f f e c t s . If both depth and width e f f e c t s were assumed to be the same (as in a perfectly b r i t t l e material) then stresses could be numerically integrated over the section at each step. Stresses at f a i l u r e may well be higher for b i a x i a l behaviour than for in-plane behaviour because in general there w i l l be smaller volumes of highly stressed material further from the neutral axis. One interesting study on the possible effect of knots on b i a x i a l strength has been described by Riberholt and Nielsen (1976). b. S t a b i l i t y Under B i a x i a l Loading The s t a b i l i t y behavior of timber members under b i a x i a l loading has received very l i t t l e attention. The most useful study i s by Larsen and Thielgaard (1979), who have developed some theory and design equations based on linear e l a s t i c behavior and a simple f a i l u r e c r i t e r i o n of l i m i t i n g compression stress. They v e r i f i e d their theory with a series of tests using combined a x i a l loads and strong-axis bending. The methods developed in this thesis could in theory be extended to b i a x i a l behaviour, but that would be a major undertaking. Cross section strength can be investigated quite simply as shown in the previous section, but for longer columns the computer program would have to be expanded to keep track of deflections about both p r i n c i p a l axes. An accurate analysis should also include the possible effects of l a t e r a l t o rsional buckling. The complexities are such that any a n a l y t i c a l study into b i a x i a l behaviour would, require an extensive experimental study, to provide c a l i b r a t i o n and 252 v e r i f i c a t i o n . Work in thi s area on other materials i s , described by Chen and Atsuta (1976b). 9.5 LONG DURATION LOADING The behaviour of timber members under long duration loading is of increasing interest and concern. As existing conservative design methods are improved, leading to more e f f i c i e n t and economical designs, the possible effects of long duration loading must be c a r e f u l l y considered. Long duration loads af f e c t the load capacity of timber members from both strength and s t a b i l i t y considerations. 9.5.1 Strength Under Long Duration Loading •It i s now well recognized that the strength of timber members decreases under the effects of long duration loads. Recent theoret i c a l developments in this area include those by Barrett and Foschi (1978) who proposed a damage accumulation model which has subsequently been incorporated into r e l i a b i l i t y studies (Foschi 1982), and Johns and Madsen (1982) who investigated .crack growth perpendicular to the grain in bending members. These and other theories show quite good agreement with experimental studies on bending members. Several experimental duration of load studies on a x i a l tension members are known to be in progress, but apparently a x i a l compression strength under long duration loads has received no attention. There i s no conceptual problem in incorporating the effe c t of duration of load on strength into the strength model 253 or proposed design methods. Duration of load factors simply have to be applied to the a x i a l tension and compression strengths, appropriate to the loading under consideration. 9.5.2 S t a b i l i t y Under Long Duration Loading The load capacity of slender compression members is governed by s t a b i l i t y rather than strength considerations, and th i s i s a function of modulus of e l a s t i c i t y and creep rather than material strength. U n t i l t h i s point, one of the assumptions of t h i s thesis has been that s t r e s s - s t r a i n relationships are independent of time. For many materials including wood thi s assumption i s not v a l i d , and specimens loaded with constant stress exhibit increasing st r a i n with time. This phenomenon, known as "creep", creates problems with increasing deflections in bending members and tension members, but is p o t e n t i a l l y far more serious in compression members where s t a b i l i t y f a i l u r e s can occur. The term "creep buckling" refers to a s t a b i l i t y f a i l u r e of a compression member under constant loads, as l a t e r a l deflections slowly increase over time u n t i l the member buckles. A s i g n i f i c a n t investigation of t h i s problem has been made by Kallsner and Noren (1978) who used both a numerical approach d i v i d i n g the column into a number of laminae, and an approximate method based on a f i c t i t i o u s modulus of e l a s t i c i t y assuming linear e l a s t i c behaviour, to produce a number of strength reduction curves for long term loading. Creep buckling has not been investigated in t h i s thesis. 254 The most d i r e c t way of studying creep buckling would be to carry out a time-step analysis, incorporating the effects of long duration loading on the s t r e s s - s t r a i n relationship. Such an analysis would not be easy with the strength model, because of the structure of the program. However,.some indication of long term behaviour could be obtained by carrying out a number of computer runs with the s t r e s s - s t r a i n relationship modified to simulate e f f e c t s of creep. It would be necessary to know the effects of long duration loading on compression behaviour (and, perhaps, tension behaviour), and this information could be obtained from duration of load tests on a x i a l compression members. Creep data for bending members could be useful for a preliminary investigation. In the meantime designers can only be made aware of the possible consequences of creep buckling, with some extra- conservatism in the design equation for members which might f a i l in an i n s t a b i l i t y mode. The p o s s i b i l i t y of creep buckling being influenced by c y c l i c changes in moisture content has been investigated by Humphries and Schniewind (1982). 9.6 MOISTURE CONTENT 9.6.1 Effect of Moisture on Compression Strength The a x i a l compression strength of wood i s sensitive to moisture content, whereas the a x i a l tension strength i s r e l a t i v e l y unaffected by changes in moisture content. The only s i g n i f i c a n t study of the e f f e c t s of moisture content on a x i a l tension and compression strengths i s reported by 255 Madsen(1982). 9.6.2 Effect of Moisture on Strength Under Combined Loading The strength model described in this thesis can be used to predict the effect of moisture content for any combination of a x i a l and bending loads. The data of Madsen(1982) has been used to produce figure 84 which shows ultimate interaction diagrams for two d i f f e r e n t moisture contents. The material in thi s case i s No. 2 and Better SPF, 38x140 mm size. Results have only been shown for 11% and 25% moisture content, but intermediate values follow a uniform trend between these two l e v e l s . Axial tension strength has been taken to be independent of moisture content as shown by the v e r t i c a l axis intercepts in the tension region. Axial compression strength on the other hand i s very sensitive to moisture content. Figure 84 shows predictions for a l l combinations of a x i a l load and bending. One prediction of considerable interest i s the bending strength on the horizontal axis. It is c l e a r l y demonstrated how bending strength at the 5th percentile l e v e l i s independent of moisture content whereas at the 95th percentile l e v e l (representative of strong material, such as small clear test specimens) moisture content has a major influence on bending strength. This graphical representation strongly supports the explanation of Madsen(l982) for the effect of moisture content on bending strength, which was previously not well understood. Future studies into the effect of moisture content on 256 CO CM . I DRY(11%m.c.) WET(25°/om.c.) 95th %ile i i 1 1 1 r~ O.D 2.0 4.0 B.O nOPIENT (KN.fl) 8.0 Figure 84 - Eff e c t of moisture content on strength of timber in combined bending and ax i a l loading bending strength should concentrate on behaviour in a x i a l tension and compression. 9 . 7 SUMMARY In t h i s chapter, the strength model has been recommended as the basis of an ultimate strength design method for timber members. One of the alternative design methods investigated in Chapter 8 has been recommended as an improved approximate method for designing timber members subjected to combined 257 bending and a x i a l l o a d s . Many f a c t o r s a f f e c t i n g the be h a v i o u r of s t r u c t u r a l t i m b e r members have been i n t r o d u c e d and d i s c u s s e d b r i e f l y , and recommendations have been made f o r f u t u r e r e s e a r c h . 258 X. SUMMARY The major achievements of thi s study are summarized below. 1. This study has demonstrated that the behaviour of timber members in bending, and in combined bending and a x i a l loading, can be predicted from observed behaviour in a x i a l tension and compression tests. 2. The interaction between a x i a l and f l e x u r a l strength of ec c e n t r i c a l l y loaded timber members has been investigated experimentally for a wide range of e c c e n t r i c i t i e s and lengths. 3. The effect of member length on strength has been investigated experimentally for a x i a l tension, a x i a l compression and bending. 4. A computer-based strength model has been developed to predict the cross section strength of timber members under any combination of a x i a l and fl e x u r a l loads. 5. This study has demonstrated how the strength model can be used to predict the load capacity of timber members of any length subjected to eccentric a x i a l loading, considering both i n s t a b i l i t y and material strength f a i l u r e s . 6. The strength model improves the theoreti c a l basis of the in-grade testing concept by re l a t i n g in-grade bending test 2 5 9 results to those in. a x i a l tension and compression, and by quantifying length e f f e c t s . 7 . The strength model can be used as the basis of a rational design method for timber members subjected to eccentric a x i a l load. 8. New approximate formulae have been proposed for designing timber members subjected to combined bending and a x i a l loading. The main features of the recommended approximate design method include: a. A new formula for the moment-axial load interaction at a timber cross section subjected to combined loading where M i s the^applied moment, Mu is the moment capacity of the section, P is the applied load, Pa i s the a x i a l compression strength of the section, and B is a factor r a t i o of the a x i a l tension strength f t to the a x i a l compression strength fc of the material, given by 0 fr - -U-f-J u a ( 1 0 . 1 ) which relates the shape of the interaction diagram to the B 1.35 "  T7 r t c ( 1 0 . 2 ) 260 The a x i a l strengths f t and fc should be obtained from i n -grade test r e s u l t s . b. A new formula for the moment-axial load interaction of e c c e n t r i c a l l y loaded timber compression members of any length F_M u u - B ( l - i - ) M ' P ' (10.3) where Pu is the load 1 capacity of the member under concentric a x i a l loading, including any effects of slenderness, and F i s a moment magnification factor given by 1 + P/P F = 1 1 - P/P 6 (10.4) The term Pe i s the Euler buckling load for the column. This thesis has i d e n t i f i e d many areas in which accurate and e f f i c i e n t design of timber members has not been possible because of lack of information on material properties and behaviour. The results of the thesis have provided a large amount of new information which can be used to improve design methods, but much remains to be done. Areas related to thi s thesis topic which s t i l l require further investigation include size e f f e c t s , creep buckling, b i a x i a l behaviour and other load configurations. 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In t h i s appendix the same assumptions ( l i s t e d in section 6.2) are used to obtain a closed form solution, both for strength in pure bending and for the ultimate interaction diagram of short columns. A.1 ULTIMATE BENDING STRENGTH A.1.1 Background The bending strength of timber beams has been a subject of interest since the time of G a l i l e o . Bending is a loading condition encountered far more frequently than combined bending and ax i a l loading. The l i t e r a t u r e survey describes many unsuccessful attempts to predict the bending strength of both clear wood beams and timber beams from the results of a x i a l tension and compression tests. A major new contribution of the present study i s a r e l a t i v e l y simple model to predict the bending strength of timber members from the a x i a l tension and a x i a l compression strengths. Recall that throughout th i s study timber member strength generally refers to strengths of defined populations of timber, at certain levels in the d i s t r i b u t i o n . Figure 27 compares d i s t r i b u t i o n s of bending, tension, and compression strengths obtained from in-grade testing of the timber studied in t h i s thesis. The strength model can be used to calculate the bending strength from the a x i a l tension and compression strengths, at any le v e l in the d i s t r i b u t i o n , or for any member whose tension and compression strengths are known. To calculate bending strength from the results of a x i a l tension and compression tests, a strength model has to incorporate size e f f e c t s with length and depth, and non-linear s t r e s s - s t r a i n behaviour in compression. These factors have been described for the general case in Chapter 6 and w i l l be described here for the special case of bending with no a x i a l load. For material that i s weaker in tension than in compression, f a i l u r e i s governed by tension strength alone, so non-linear compression behaviour need not be considered. Calculations for bending strength are developed for two separate cases. The f i r s t assumes a b i l i n e a r s t r e s s - s t r a i n r e l a t i o n s h i p with a f a l l i n g branch. The second assumes an el a s t o - p l a s t i c s t r e s s - s t r a i n relationship with an upper l i m i t 273 on compression s t r a i n . A.1.2 B i l i n e a r Stress-Strain Relationship with F a l l i n g Branch a. Assumptions The theory in t h i s section is an extension of theory described by Bazan(l980), which attempts to predict the bending strength of large clear wood beams from the a x i a l tension and compression strengths of small clear specimens. The major differences in the present study are that in-grade a x i a l tension and compression test results are used as input, and size e f f e c t s are included more e x p l i c i t l y . This study uses Bazan's assumptions of linear e l a s t i c behaviour to f a i l u r e in tension and a b i l i n e a r s t r e s s - s t r a i n r e l a t i o n s h i p in compression with a f a l l i n g branch after maximum stress, as shown in figure 85. stress brittle . fracture Figure 85 - B i l i n e a r s t r e s s - s t r a i n relationship with f a l l i n g branch The slope of the f a l l i n g branch is assumed to be a constant r a t i o , m, of the modulus of e l a s t i c i t y as shown. The general computer program for combined a x i a l and bending loads can use any shape of s t r e s s - s t r a i n curve in compression. However, comparative computer runs have shown that for c a l c u l a t i n g the strength at a cross section, the b i l i n e a r 274 r e l a t i o n s h i p with a f a l l i n g branch can be used to give almost e x a c t l y the same r e s u l t s as more ac c u r a t e c u r v e s . b. F i g u r e C a l c u l a t i o n s 86 shows the d i s t r i b u t i o n of s t r e s s and s t r a i n i n a r e c t a n g u l a r beam i n the i n e l a s t i c range. The depth of the s e c t i o n i s d, and a,b and c represent r a t i o s of d as shown i n the f i g u r e . v x X v \ \ x . w . section (a) strain (b) F i g u r e 86 - D i s t r i b u t i o n of s t r e s s and s t r a i n i n a r e c t a n g u l a r beam assuming b i l i n e a r s t r e s s - s t r a i n r e l a t i o n s h i p The extreme f i b r e s t r e s s i n t e n s i o n fx, i s taken to be a f a c t o r n times the y i e l d s t r e s s of the m a t e r i a l i n compression f c . The f a l l i n g branch of the s t r e s s - s t r a i n r e l a t i o n s h i p r e s u l t s i n the extreme compression f i b r e having a s t r e s s which i s a f a c t o r r of the y i e l d s t r e s s i n compression. Using the n o t a t i o n shown on the f i g u r e , the i n t e r n a l t e n s i o n and compression f o r c e s , T and C, are T = ^ nf wed ( A . 1 ) 2 c C = i f wd[b + a(l+r)] 2 c a ^ 1 T i ; j (A.2) I n t e r n a l a x i a l f o r c e e q u i l i b r i u m , T=C, y i e l d s b + a(l+r) = nc (A.3) 275 From the geometry of the s t r e s s diagram a = 1 - b - c ( A.4) and b = c / n (A.5) From equations A.3 to A.5 n(l+r) r \ c _ (n+r)(n+l) ( A « 6 ) from which and a = C " - 1 ) (A.7) , _ 1+r b " Cn+rUn+l-* (A.8) ( )(n+l) The bending moment can be c a l c u l a t e d by t a k i n g moments of the i n t e r n a l f o r c e s about the n e u t r a l a x i s to give 2 M - f ^g- [2nc 2 + .2b2 + (l+2r)a 2 + 3(l+r)ab] (A. 9) S u b s t i t u t i n g f o r a, b and c give s M = £ e ^ i [ n + (2n-l)r } ( ^ ] Q ) The development to t h i s p o i n t i s i d e n t i c a l to that d e s c r i b e d by Bazan. I f the slope of the f a l l i n g branch of the s t r e s s - s t r a i n r e l a t i o n s h i p i s a given r a t i o , m, of the modulus of e l a s t i c i t y , as shown i n f i g u r e 85, then from the s t r e s s and s t r a i n diagrams of f i g u r e 86, the r e d u c t i o n i n s t r e s s at the extreme compression f i b r e can be w r i t t e n as (l-r) f = mE(e - e ) v 1 c v c y (A.11) 276 From the s t r a i n diagram e c a+b „ T ~ e y (A.12) and the s t r e s s - s t r a i n r elationship gives e y - f c/E (A. 13) Substituting and rearranging equations A.11 to A.13 gives an expression for r in terms of m, a and b = 1 - ma/b (A. 14) Substituting the values of a and b from equations A.7 and A.8 give an expression for r in terms of m and n = /l-m(n2-l) (A.15) Equations A.10 and A.15 can be combined to give a new expression for the bending moment, M. M = wd< n + ( 2 n - l ) A - m(n2-l) n + / l - m(n2-l) (A.16) The only unknown in thi s equation i s n, the r a t i o of the extreme f i b r e tension stress at ultimate moment, to the maximum compression stress. The extreme f i b r e tension stress fx cannot exceed the f a i l u r e stress in tension fm, which i s related to the a x i a l tension strength f t by m -k3+l C ] 3 *, (A.17) where k 3 i s the s t r e s s - d i s t r i b u t i o n parameter described in Chapter 3, and c i s the neutral axis depth r a t i o as shown in figure 86. To calculate the value of n, and hence the moment capacity, four possible ratios of tension to compression strength must be considered. Each of the four cases produce a di f f e r e n t internal stress d i s t r i b u t i o n at failure.-277 Case 1 i s f o r timber weak i n t e n s i o n , where the f a i l u r e s t r e s s i n bending fm i s l e s s than the compression s t r e n g t h f c . F a i l u r e i n t h i s case i s a s s o c i a t e d with a t e n s i o n f a i l u r e and no compression y i e l d i n g . Simple e l a s t i c theory can be used to c a l c u l a t e the moment c a p a c i t y , the modulus of rupture being fm from equation A.17 using c=0.5 f o r n e u t r a l a x i s at mid-depth. Case 2. i s f o r timber with an intermediate r a t i o of t e n s i o n to compression s t r e n g t h such that maximum moment i s s t i l l a s s o c i a t e d with a t e n s i o n f a i l u r e , but a f t e r some compression y i e l d i n g has oc c u r r e d . In t h i s case the extreme f i b r e t e n s i o n s t r e s s fx, at f a i l u r e , w i l l again be equal to fm given by equation A.17. The extreme f i b r e s t r e s s fx has been d e f i n e d as n f c , so equation A.17 becomes times 1/k, (A.18) In t h i s case the n e u t r a l a x i s i s no longer at mid-depth. Equations A.6 and A.15 can be combined to give an ex p r e s s i o n f o r c, which s u b s t i t u t e d i n t o equation A.18 g i v e s an exp r e s s i o n r e l a t i n g n to the known a x i a l s t r e n g t h s fc and f t . n(l + A - m(n2-l) ) - (n + A - m(n2-l) ) (n+l)(k+l) 1/k, (A.19) T h i s equation can be s o l v e d i t e r a t i v e l y f o r n, which used i n equation A.16 to c a l c u l a t e the maximum moment. can be Case 3 i s f o r timber with a higher r a t i o of t e n s i o n to compression s t r e n g t h , such that a t e n s i o n f a i l u r e occurs a f t e r the moment has passed a maximum value , accompanied by c o n s i d e r a b l e compression y i e l d i n g . In t h i s case the maximum moment i s independent of te n s i o n s t r e n g t h , and i t i s necessary to f i n d the value of n that maximizes the moment i n equation A.16. D i f f e r e n t i a t i n g with r e s p e c t to n and equating to zero produces 1 + m - mn3 + ( l - m(n 2-l)) 3 / 2 = 0 (A.20) which can be s o l v e d i t e r a t i v e l y f o r n, given a value of the m a t e r i a l p r o p e r t y m. T h i s value of n can be used i n equation A.16 to c a l c u l a t e maximum moment. In p r a c t i c e i t i s necessary to c a r r y out c a l c u l a t i o n s f o r both Cases 2 and 3 and use the lower of the two values of n i n 278 equation A.16. Case 4 i s the extreme case for timber which i s much stronger in tension than in compression, where maximum moment i s associated with d u c t i l e compression yieldin g and no tension f a i l u r e occurs. This type of f a i l u r e w i l l be familiar to the reader who has t r i e d unsuccessfully to snap a green branch on a l i v i n g tree, finding that a p l a s t i c hinge forms, but the branch does not break. Equation A.20 must be used again in this case to find n, and the moment from equation A.16. c. Length Effects This derivation has not included length e f f e c t s . They can be included using equation 5.3 which gives an equivalent stressed length L 2 for a beam of span L, loaded with two symmetrical loads a distance aL, apart. 1 + ak 1/L 1 k x+l (A.21) and equation 3.11 which relates the strength of two similar members of di f f e r e n t lengths. X *2 (A.22) The procedure for modifying in-grade tension and compression test results for length, as input for ca l c u l a t i n g bending strength, i s as follows: 1. Calculate the equivalent stressed length of the beam, L 2, from equation A.21 using the tension length effect parameter a s k , . 2. Adjust the in-grade tension strength result obtained from testing of a length L , , to a value compatible with the equivalent stressed length of the beam, L 2, using equation A.22 and the tension length e f f e c t parameter as k i . The result i s f t . 3. Repeat steps 1 and 2 using in-grade compression test results and the compression length effect parameter as k 1 to calculate the compression strength f c . 279 Modify axial tension and compression strength to values consistent with stressed length of beam ^ ; modify tension strength for depth assuming neutral axis mid depth modified ^tension strength less thanj compression ^strength?^ YES calculate max moment assuming no tension failure calculate moment at tension f a i l u r e take lowest one ^_ calculate f a i l u r e moment from e l a s t i c theory V V ULTIMATE MOMENT ULTIMATE MOMENT Figure 87 - Flow chart for ca l c u l a t i n g ultimate bending moment for b i l i n e a r s t r e s s - s t r a i n relationship d. Summary A summary of the procedure for cal c u l a t i n g ultimate bending moment i s shown in the flow chart of figure 87, and described b r i e f l y as follows. The length effect calculations have just been described. The depth effect c a l c u l a t i o n can i n i t i a l l y be made using equation A.18 with c=0.5. The moment associated with a tension f a i l u r e i s found by cal c u l a t i n g a value of n to sa t i s f y equation A.19 and by using that value in equation A.16. Maximum moment assuming no tension f a i l u r e i s found by substituting a value of n from equation A.20 into equation A.16. It i s not d i f f i c u l t to write a computer program to carry out these c a l c u l a t i o n s . 280 e. Depth E f f e c t i n Compression The d e r i v a t i o n to t h i s p o i n t has been based on the assumption t h a t the compression s t r e n g t h at a c e r t a i n c r o s s s e c t i o n i s a m a t e r i a l constant independent of the s t r e s s e d depth. As i n the more general case a compression depth e f f e c t can be in t r o d u c e d - using equation 3.18 which r e l a t e s the maximum compression s t r e s s to the s i z e of the compression s t r e s s b l o c k , g i v i n g a mo d i f i e d compression s t r e n g t h fern . - l / k _ f = [a + j-lU 3 f (A.23) cm L k 3+l J c T h i s value of fem can be used i n place of the term fc i n equation A.18 to give a more general form of equation A.19, i n c o r p o r a t i n g a compression depth e f f e c t . n [ a + r ^ ] 3 C 3 t - / k3c 1 3t 1 r c where the s t r e s s - d i s t r i b u t i o n parameter k 3 has s u b s c r i p t s t and c f o r t e n s i o n and compression r e s p e c t i v e l y . T h i s equation can be sol v e d by i t e r a t i o n f o r n, as before, using equations A.6 to A.8 and A.15-for a, b, and c. An a l t e r n a t i v e value of n can be found from equation A.20, as b e f o r e . Once the c o r r e c t value of n i s found i t can be used i n equation A.16 to giv e the u l t i m a t e bending moment, again using fem i n place of f c . A.1.3 E l a s t o - P l a s t i c S t r e s s - S t r a i n R e l a t i o n s h i p a. Assumpt ions I t was shown i n Chapter 6 that f o r c a l c u l a t i n g the s t r e n g t h of a c r o s s s e c t i o n , a model based on an e l a s t o - p l a s t i c s t r e s s -s t r a i n r e l a t i o n s h i p can produce very s i m i l a r r e s u l t s to one based on a more accurate curve, p r o v i d e d that an upper l i m i t on compression s t r a i n i s s p e c i f i e d . Such a model i s a l e s s a c c u r a t e r e p r e s e n t a t i o n of a c t u a l behaviour, but i t produces a s i m i l a r s t r e s s d i s t r i b u t i o n w i t h i n the c r o s s s e c t i o n . T h i s s e c t i o n uses an e l a s t o - p l a s t i c s t r e s s - s t r a i n r e l a t i o n s h i p with l i m i t i n g s t r a i n to produce equations f o r u l t i m a t e bending s t r e n g t h which are s l i g h t l y more simple than those d e r i v e d i n the p r e v i o u s s e c t i o n . F i g u r e 88 shows the s t r e s s - s t r a i n r e l a t i o n s h i p . Once again the s t r e n g t h i s c a l c u l a t e d from the r e s u l t s of in-grade t e s t s i n a x i a l t e n s i o n and compression. A l l the other assumptions made p r e v i o u s l y are used again here. 281 Figure 88 - E l a s t o - p l a s t i c s t r e s s s - s t r a i n relationship b. Calculat ions Figure 89 shows the d i s t r i b u t i o n of stress and st r a i n in a rectangular beam in the i n e l a s t i c range. s e c t i o n s t r a i n ( a ) (b) Figure 89 - Dis t r i b u t i o n of stress and s t r a i n in a rectangular beam assuming e l a s t o - p l a s t i c s t r e s s - s t r a i n relat ionship From this figure the internal tension force i s the same as equation A.1. T = i nf wed 2 c (A.25) and the internal compression force becomes C = J f wd(2a + b) / c (A.26) 282 The same procedure as before produces the location of the neutral axis and the size of the p l a s t i c zone a = Z£ (A.27) n+1 b = (A.28) (n+1)2 2n c = (n+1)2 < A' 2 9 ) and an expression for the bending moment Assume now that the l i m i t i n g s t r a i n has just been reached and that no tension f a i l u r e has occurred. From the geometry of the s t r a i n diagram e u a + b „ , — = — b — (A.31) y Substituting equations A.26 to A.29 into equation A.31, a value for n can be obtained e n = 2 — - 1 (A.32) e y This value of n substituted into equation A.30 gives the bending moment when the l i m i t i n g compression s t r a i n i s reached, and the extreme f i b r e tension stress i s obtained d i r e c t l y from (A.33) This extreme f i b r e tension stress must be compared with the tension strength for this neutral axis depth, which is calculated from equations A.17 and A.29. If a tension f a i l u r e has not occurred, the previously calculated moment i s the desired value. If a tension f a i l u r e has occurred i t becomes 283 necessary to calculate a lower moment associated with that event, using equation A.19 with the slope m of the f a l l i n g branch being zero, giving n [ 2 n j " 1 7 " 3 = f t ( A.34) (n+l) 2(k 3+l) f c This equation must be solved i t e r a t i v e l y for n as before, and the subsequent value used in equation A.30 to calculate the ultimate moment. A flow chart for thi s procedure is shown in figure 90. The e l a s t o - p l a s t i c model contains just as many steps as the b i l i n e a r model, but the equations are somewhat simpler. The b i l i n e a r model w i l l also be used in the next section for cal c u l a t i n g the shape of the ultimate interaction diagram. A.2 ULTIMATE INTERACTION DIAGRAM A.2.1 Background Chapter 6 described a computer program for producing ultimate interaction diagrams .for a wide range of input parameters. This section describes a s i m p l i f i e d c a l c u l a t i o n procedure based on an e l a s t o - p l a s t i c s t r e s s - s t r a i n relationship in compression as shown in figure 88. The calculations in thi s section are only for cross section behaviour. Interaction diagrams for long columns cannot be produced by such simple procedures, and a numerical computer program i s necessary to produce accurate results for cases where i n s t a b i l i t y f a i l u r e s occur. This section f i r s t describes a number of di f f e r e n t regions in which the calculations must be performed, then provides a summary of the equations derived for each region. The actual derivations, which proceed e a s i l y from f i r s t p r i n c i p l e s , have not been included. A.2.2 Calculations Figure 91 shows nine d i f f e r e n t cases to be considered to produce a complete interaction diagram. Each case represents a combination of a x i a l load and bending moment just causing f a i l u r e . 284 Modify axial tension and compression strength to values consistent with stressed length of beam : y. modify tension strength for depth assuming neutral axis mid depth modified ^tension strength less than compression jtrengthj^ NO YES calculate moment and extreme fibre tension stress when limiting compression strain reached NO \ / calculate moment at tension failure \ / YES use moment for limiting compression strain calculate failure moment from elastic theory ULTIMATE MOMENT ULTIMATE MOMENT ULTIMATE MOMENT Figure 90 - Flow chart for c a l c u l a t i n g ultimate bending moment for e l a s t o - p l a s t i c s t r e s s - s t r a i n relationship Case 1 i s pure a x i a l tension behaviour. The f a i l u r e stress f t is the tension strength obtained from an in-grade a x i a l tension t e s t . Case 2 represents combined bending and tension with the whole of the section subjected to tension stresses. Failure occurs when the extreme fibre tension stress reaches a f a i l u r e stress fm given by equation 3.17. Case 3 i s the par t i c u l a r s i t u a t i o n where the whole section i s in tension, but with zero stress at the top edge. Case 4 i s similar to the previous two cases, but now there are compression stresses near the top edge, and in case 5 the 285 c o m p r e s s i o n s t r a i n s e x c e e d t h e y i e l d s t r a i n . In c a s e s 3 t o 5 f a i l u r e o c c u r s when t h e extreme f i b r e t e n s i o n s t r e s s r e a c h e s a f a i l u r e s t r e s s g i v e n by e q u a t i o n A.17. Case 6 l o o k s v e r y s i m i l a r t o c a s e 5, but now t h e c o m p r e s s i o n s t r a i n a t t h e t o p f i b r e r e a c h e s i t s l i m i t i n g v a l u e b e f o r e a t e n s i o n f a i l u r e o c c u r s . Case 7 i s a p a r t i c u l a r s i t u a t i o n where t h e b o t t o m f i b r e s t r e s s i s z e r o , w i t h t h e r e s t of t h e s e c t i o n i n c o m p r e s s i o n . In c a s e 8 t h e whole o f t h e s e c t i o n i s i n c o m p r e s s i o n w i t h y i e l d i n g o c c u r r i n g o v e r most of t h e d e p t h . The f i n a l d i a g r a m shows c a s e 9 w h i c h i s p u r e a x i a l c o m p r e s s i o n . . The maximum c o m p r e s s i o n s t r e s s i n c a s e s 5 t o 9 i s assumed t o be t h e f a i l u r e s t r e s s from an i n - g r a d e a x i a l c o m p r e s s i o n t e s t . In a l l of t h e c a s e s 1 t o 9 t h e n e t a x i a l l o a d c a n be c a l c u l a t e d by summing a l l o f t h e i n t e r n a l t e n s i o n and c o m p r e s s i o n s t r e s s e s i n t h e s t r e s s b l o c k s shown. The b e n d i n g moment a t t h e s e c t i o n c a n be c a l c u l a t e d by t a k i n g moments o f t h e i n t e r n a l f o r c e s a b o u t t h e c e n t r o i d a l a x i s of t h e s e c t i o n . The a c t u a l c a l c u l a t i o n p r o c e d u r e i s summarized i n T a b l e V w h i c h shows t h e s e q u e n c e of c a l c u l a t i o n s a f t e r s e l e c t i n g a n e u t r a l a x i s d e p t h . The numbers i n t h e l e f t - h a n d column a r e t h e c a s e numbers r e f e r r e d t o a b o v e . F i g u r e 92 shows t h e r e s u l t s o f t h e s e hand c a l c u l a t i o n s p e r f o r m e d a t e i g h t n e u t r a l a x i s d e p t h s , w h i c h a r e s u f f i c i e n t t o p r o d u c e a c l o s e a p p r o x i m a t i o n t o t h e c u r v e p r o d u c e d by t h e more g e n e r a l computer model, p r e v i o u s l y v e r i f i e d by t e s t r e s u l t s . # Select n.a. Depth Tension Failure Stress Calculate Strains Check Top Strain Calculate b = Calc a = Calculate Axial Load Calculate Moment 1 c =oo f t f wd 0 2 c > 1 f m ( _ c _ r l / k f wd (— - 1) 2c f wd2 1 m 6 2c 3 c = 1 f wd/2 m f wd2 1 m 6 2 A 0 < c < 1 f m e = — t E 1-c p — p i e > e y c f wd (— - 1) 2c f wd2 1 m 6 2c 5 e < e < e y c u e y c 6 t 1-b-c b c 2 f wd (a + - - — ) c 2 2b * w c i 1 ' w i - a t + v » r — - + ^—(— - c V I 6 c c t e > e c u i (1-c) u c 6 2 b 2 7 c •= 0 e _y e u f wd (a + -) c 2 f — [3a(l-a) + b(- - 3a-b)j c 6 2 8 c < 0 e (l-c)+c e u 1-b f wd (a + b + b C j c 2 2(b-c) _ wd2 i a N . b c f ,b 1, f [- (1-a) + [a + J c 6 2 b-c 2 2 + I b U ) la + ^ + i j ] 2 b-c 3 2 9 c = - oo f wd c , 0 287 F i g u r e 92 - U l t i m a t e i n t e r a c t i o n diagram produced by hand-c a l c u l a t i o n s , compared with computer c a l c u l a t e d curve 288 APPENDIX B - CALCULATION OF INTERCEPT DEFINING INTERACTION DIAGRAM T h i s appendix d e v e l o p s ' a s e m i - e m p i r i c a l e x p r e s s i o n fo r the h o r i z o n t a l a x i s i n t e r c e p t tha t d e f i n e s the shape of the b i l i n e a r approx imat ion to the u l t i m a t e i n t e r a c t i o n d iagram. F i g u r e 93 shows a t y p i c a l u l t i m a t e i n t e r a c t i o n d iagram. M c 1.0 B M o m e n t F i g u r e 93 - T y p i c a l u l t i m a t e i n t e r a c t i o n diagram Po in t B i s the h o r i z o n t a l a x i s i n t e r c e p t of a p r o j e c t i o n of the upper s t r a i g h t l i n e p o r t i o n of the d iagram. If a po in t on the s t r a i g h t l i n e p o r t i o n of the diagram can be l o c a t e d , as shown by p o i n t C in f i g u r e 93, the va lue of B can be ob ta ined from the geometry of the d iagram. The c o - o r d i n a t e s of po in t C can be ob t a i ned from the approximate hand c a l c u l a t i o n method d e s c r i b e d in Appendix A. T y p i c a l v a lues of E/fc=300 and a maximum s t r a i n of 0.01 can be used w i th a n e u t r a l a x i s depth of c=0.3 to make the c a l c u l a t i o n s shown on l i n e 6 of Tab l e V. The c o o r d i n a t e s of. C are found to be p = 0.592 f A (B.1) c c M = 0 . 8 4 4 f wd^ (B.2) 289 The moment value Mc must be related to the bending strength Mu (which becomes 1.0 when non-dimensionalized). If the neutral axis i s assumed to be at mid-depth, the moment capacity is M = f (B.3) u m o where fm i s the f a i l u r e stress in bending, which i s related to the a x i a l tension stress by equation 3.16. -l/k_ fm^= k&F>] ^ (B*4) where k i s the within-member depth effect parameter. Substitution of equations B.3 and B.4 into equation B.2 gives A reasonably conservative value of k=6.0 (corresponding to f a i l u r e stress in tension being 65% of that in bending), and the non-dimensionalized value of Mu=1.0 give f M = 0.544 ^  (B.6) c t From the geometry of figure 93 M B = 1 - P c (B.7) Substituting from equations B.1 and B.6 (with the term fcA non-dimensionalized as 1.0) equation B.7 becomes B = YHT-  ( B - 8 ) t c This expression has been incorporated into the recommended design formulae. 290 APPENDIX C - TEST RESULTS £•/ eccentricity y O 5th percentile a Mean • . . A • A . 95th percentile m ft • • -, • 0.0 0.5 3.0 3.5 2.0 2.5 3.0 3.5 H-ID-SPRN MOMENT (KN.M) 4.0 Figure 94 - Eccentric compression r e s u l t s , 38x89mm boards, 1.3m long 4.5 — m . . • J i • * . .. .0'— 202_rni!L — I — 0.5 "i r r 1 r 4 . 0 1 1 r 2.0 2.5 3.0 3.5 n i D - S P R N MOMENT (KN.M) Figure 95 - Eccentric compression r e s u l t s , 38x89mm boards, 1.8m long o.o ^ i i i r 3 . 0 3 . 5 4 . 5 291 o o o _ o 5th percentile • Mean w'o _ A 95th percentile .AO^ [RL LORE 60.0 X cr o 9-o R-o -----€) ••" * o 0 i i i i i i i i i i i i i i i i i i i .0 0.5 3.0 3.5 2.0 2.5 3.0 3.5 4.0 4.5 5 MID-SPRN noriENT (KN.ru F i g u r e 96 - E c c e n t r i c c ompression r e s u l t s , 38x89mm boards.* 2.3m l o n g o O -o _ oo cn CL CD -n i D - S P R N MOMENT (KN.M) F i g u r e 97 - E c c e n t r i c c ompression r e s u l t s , 38x89mm boards, 3.2m l o n g 292 8-Q CCD QfN. CLCD o 9' eccentricity 7 O 5th percentile / • Mean A 95th percentile A-• • .. a -0.0 1.0 2.0 3.0 4.0 5.0 €.0 7.0 n i D - s P R N n o n E N T (KN.n) F i g u r e 98 - E c c e n t r i c compression r e s u l t s , 38x140mm boards, 1.82m l o n g £.0 Q a a a ^ 1 0 . Ctoo 9-• • -fi \ mm 1——[ 1 1—"I 1 1 1 1 1 1 1 1 1 1— 0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 fi.O n i D - s P f l N n o n E N T (KN.n) F i g u r e 99 - E c c e n t r i c compression r e s u l t s , 38x140mm boa r d s , 2.44m l o n g 293 a o 8-CTo 0 r \ i . CCco 9" o 5th percentile • Mean A 95th percentile 'V . . ..... 4 —I 1 1 I 1 1 1 1 1 1 1 1 I I 1 1 D.C J.O 2.0 3.0 4.0 5.0 6.0 7.0 fl.fl MID-SPAN n o n E N T (KN.n) Figure 100 - Eccentric compression res u l t s , 38x140mm boards, 3.35m long 1 1 1 1 1 1 1 1 1 r 2.0 3.0 4.0 5.0 6.0 niD - s PRN n o n E N T (KN.n) Figure 101 - Eccentric compression res u l t s , 38x140mm boards, 4.27m long 294 Figure 102 - Axial tension results, 38x89mm boards, 2.0m long AXIAL TENSION STRENGTH SHORT BOARDS INTERVAL CHI-SQUARE FIT: 13.72 3-PAR WEIBULL (DASHED)= SHAPE = 2.6974 SCALE = 33.093 LOCA. = 4.311 o.o 10 0 2 0 . 0 3 0 . 0 4 0 . 0 TENSION 1 5 0 . 0 STRESS 6 0 . D (MPA) "I 7 0 . 0 —1 6 0 . 0 9 0 . 0 1 0 0 . 0 Figure 103 - Axial tension results, 38x89mm boards, .914m long 295 COMPRESSION STRENGTH WERKEST POINT IN BOARD DATA (SOL ID L I N E ) : N=88 5 7 I L E : 5 07 I LE - . MERN: 9 5 * 1 L E : C OF D I S P : DRTR 2 3 . 0 7 31 . 5 9 31 . 8 9 3 9 . 5 4 0 . 1 5 3 S T . D V . 3 . 9 6 0 . 6 3 4 . 8 8 0 . 7 2 WEIBULL 2 3 . 1 9 3 2 . 3 0 31 . 6 4 3 8 . 9 1 0 . 1 5 0 S T . D V . 1 .40 0 . 6 3 4 . 8 0 0 . 7 6 10 INTERVAL CH I - SQUARE F I T : 1 5 . 8 6 3 -PAR WEIBULL (DASHED) : SHAPE = 7 . 8 5 5 9 SCALE = 3 3 . 8 4 5 LOCA. = 0 . 0 0 0 l 1 50. D 60.0 STRENGTH (MPA) o.o 10.0 20.0 30.0 40.0 COMPRESSION 70.0 80.0 I 90.0 100.0 Figure 104 - Axial compression results, 38x89mm boards, 2.0m long EDGEWISE BENDING LONG SPAN DRTR (SOLID L I N E ) : N=88 5 / I L E • 5 0 / I L E MERN 9 5 / J L E C OF D I S P DRTR 2 2 . 7 0 5 3 . 4 7 51 . 4 4 7 6 . 6 0 0 . 3 0 7 S T . D V . 3 . 5 5 2 . 3 9 1 5 . 7 9 3 . 2 9 WEJBULL 2 5 . 8 6 . 51 . 7 3 51 .51 7 6 . 3 9 0 . 2 9 7 10 INTERVAL CHI -SQUARE F I T : 6 . 5 4 3 -PAR WEIBULL (DASHED) : SHAPE = 3 . 7 5 5 7 SCALE = 5 7 . 0 3 8 LOCA. = 0 . 0 0 0 S T . D V . 3 . 2 8 2 . 1 1 1 5 . 2 9 3 . 1 5 o.o 20.0 40.0 60.0 80.0 100.0 BENDING STRENGTH. 120.0 (MPAJ — i — 140. 160.0 1B0.0 200.0 Figure 105 - Bending test results, 38x89mm boards, 1.5m span 296 EDGEWISE BENDING SHORT SPAN ODi£> go" O 0_ U J 57ILE: 50JILE: MEAN: 95;iLE: OF DISP: DATA (SOLID LINE): N=88 DATA 36.70 60.72 60.33 82.26 0.229 ST.DV. 1 .63 5.77 13.82 6.59 WEIBULL 36.14 61 .02 60.29 81 .91 0.230 INTERVAL CHI-SQUARE FIT: 12.00 3-PAR WEIBULL (DASHED): SHAPE = 4.9718 SCALE = 65.690 LOCA. = 0.000 i 180.0 ST.DV. 3.46 1 .88 13.88 2.55 o.o —r 20.0 —r—. 1 1 1 1 40.0 60.0 60.0 100.0 120.0 BENDING STRENGTH (MPfl) 140.0 200.0 F i g u r e 106 - Bending t e s t r e s u l t s , 38x89mm boards, .84m span (edgewise) FLATWISE BENDING SHORT SPAN cr o CE ti-er _J ZD 5JILE 50JILE MEAN 95/ILE C OF DISP: 0.242 DATA (SOLID LINE): N=87 DATA ST.DV. 33.31 5.49 57.92 2.07 58.44 14.19 80.13 3.56 11 INTERVAL CHI-SQUARE FIT: 3-PAR WEIBULL (DASHED) SHAPE = 3.7061 SCALE = 51.751 LOCA. = 11 .742 WEIBULL ST.DV 34.96 3.00 58.62 1 .95 58.45 14.03 81 .32 2.92 0.240 4.92 20.0 1 1 I 80.0 100.0 120.0 BENDING STRENGTH (MPA) 140.0 —\ 160.0 1B0.O 200.0 F i g u r e 107 - Bending t e s t r e s u l t s , 38x89mm boards, .84m span ( f l a t w i s e ) 2 9 7 TENSION TEST 38 X 140 S-P-F DRTR (SOLID LINE): N=102 DRTR ST.DV. WEIBULL ST.DV 5/ILE: 13.23 1 .79 12.61 1.07 50/ILE: 26.64 4.15 26.20 1 .37 MERN: 27.60 10.82 27.59 10.74 95/ILE: 46.20 4.56 47.37 2.99 OF DISP: 0.392 0.389 10 INTERVAL CHI -SQUARE FIT: 21.92 3-PAR WEIBULL (DASHED): SHAPE SCALE LOCA. = 1.8971 = 22.083 = 7.996 i — 80.0 0.0 10.0 20.0 30.0 40.0 50.0 TENSION STRESS (MPA) 60.0 70.0 90.0 100.0 Figure 108 - Axial tension results 38x140mm boards, 3.0m long COMPRESSION TEST 38 X 140 S-P-F DATA (SOLID LINE): N=97 5*1LE: 50/1LE: MEAN: 95/ILE: C OF DISP: DATA 18.89 27.27 26.83 32.78 0. 144 ST.DV. 1 .95 0.44 3.87 0.82 WEIBULL ST.DV 20.00 1 .07 27.21 0.47 26.83 3.78 32.36 0.56 0 . 140 12 INTERVAL CHI-SQURRE FIT: 7.53 3-PAR WEIBULL (DASHED): SHAPE = 8.4536 SCALE = 28.423 LOCA. - 0.000 o.o 20.0 30.0 40.o COMPRESSION 50.0 STRESS 60.0 (MPA) 70.0 - I -80. 90.0 100.0 Figure 109 - Axial compression res u l t s , 38x140mm boards, 3.0m long 298 F i g u r e 110 - Bending t e s t r e s u l t s , 38x140mm b o a r d s , 3.0m l o n g 

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