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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), U n i v e r s i t y of Canterbury, New Zealand, 1970 M.S., U n i v e r s i t y 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 E n g i n e e r i n g  We accept t h i s t h e s i s as conforming to the r e q u i r e d  standard  THE UNIVERSITY OF BRITISH COLUMBIA January  ©  1984  Andrew Hamilton Buchanan, 1984  In  p r e s e n t i n g  r e q u i r e m e n t s o f  B r i t i s h  i t  f r e e l y  agree f o r  t h i s f o r  an  a v a i l a b l e  t h a t  I  u n d e r s t o o d  t h a t  f i n a n c i a l  by  h i s  o r  The  U n i v e r s i t y  1956 M a i n  of  Canada  1Y3  1  MHr-r-h  U n i v e r s i t y s h a l l  r e f e r e n c e  and  study.  I  e x t e n s i v e be  her o r  s h a l l  Civil  Mall  Vancouver, V6T  Of  the  the  L i b r a r y  by  the  of  p u b l i c a t i o n  n o t  be  a l l o w e d  Columbia  of  t h i s  I t  t h i s  w i t h o u t  make  f u r t h e r  head  r e p r e s e n t a t i v e s .  Engineering  B r i t i s h  c o p y i n g  g r a n t e d  p e r m i s s i o n .  Department  a t  o f  the  may  c o p y i n g  g a i n  degree  f u l f i l m e n t  t h a t  f o r  purposes  o r  p a r t i a l  agree  f o r  p e r m i s s i o n  s c h o l a r l y  i n  advanced  C o l u m b i a ,  department  f o r  t h e s i s  t h e s i s  o f  my  i s t h e s i s my  w r i t t e n  ii  Abstract T h i s t h e s i s d e s c r i b e s a model f o r p r e d i c t i n g the strength of timber members i n bending, axial  loading,  and  bending  and  Both i n s t a b i l i t y  and  material  f a i l u r e s are i n c l u d e d .  The  model i s based on a s t r e s s - s t r a i n r e l a t i o n s h i p which  incorporates d u c t i l e non-linear linear  combined  on the b a s i s of a x i a l t e n s i o n and compression  behaviour of s i m i l a r members. strength  in  elastic  behaviour i n compression,  behaviour a s s o c i a t e d with b r i t t l e  and  fracture in  tension. The  model i n c l u d e s the e f f e c t s of v a r i a b i l i t y  strength,  both  e f f e c t s which member  within predict  size  are  a  member  timber  and between members.  decreasing  quantified  in  strength  using  with  separate  Size  increasing  parameters f o r  member l e n g t h and member depth. An timber and  extensive  experimental  large  number  of  members i n s t r u c t u r a l s i z e s has been used to c a l i b r a t e  verify  described  the  model.  Test  procedures  f o r members of d i f f e r e n t  in bending and i n a x i a l both  program on a  tension  and  loading.  compression  and  lengths  The a x i a l loading  r e s u l t s are  t e s t e d to f a i l u r e testing  using  included  several  end  eccentricities. S e v e r a l a l t e r n a t i v e design model  are  investigated  methods based on the  and compared with e x i s t i n g methods.  Recommendations are made f o r design subjected  strength  methods f o r timber members  to combined bending and a x i a l  loading.  iii Table of Contents Abstract L i s t of T a b l e s L i s t of F i g u r e s Notation Acknowledgement Chapter I INTRODUCTION 1 . 1 BACKGROUND 1 .2 OBJECTIVES 1.3 DESIGN CODES 1.4 APPLICATIONS 1.5 THESIS ORGANIZATION 1.6 LIMITATIONS  (  . ..  i i viii ix xv xviii 1 1 2 3 4 5 6  Chapter II LITERATURE SURVEY . . 7 2.1 INTRODUCTION 7 2.1.1 Background 7 2.1.2 C l e a r Wood and Commercial Timber 7 2.2 BENDING STRENGTH ...8 2.2.1 Bending Behaviour of C l e a r Wood 8 a. D i s t r i b u t i o n of S t r e s s e s 8. b. S i z e E f f e c t s 17 2.2.2 Bending Behaviour of Timber 19 a. Comparison With C l e a r Wood 19 b. In-grade T e s t i n g ....20 c. D e r i v a t i o n of Design S t r e s s e s 21 d. S i z e E f f e c t s 22 2.3 AXIAL TENSION STRENGTH 23 2.3.1 A x i a l Tension Strength of C l e a r Wood 23 2.3.2 A x i a l Tension Strength of Timber ^.24 a. E f f e c t s of D e f e c t s °.24 b. S i z e E f f e c t s 26 2.4 AXIAL COMPRESSION STRENGTH 27 2.4.1 A x i a l Compression S t r e n g t h of C l e a r Wood 27 2.4.2 A x i a l Compression S t r e n g t h of Timber 28 2.4.3 S t r e s s - S t r a i n R e l a t i o n s h i p 29 2.4.4 Column Theory f o r C o n c e n t r i c Loading 33 2.4.5 Timber Columns 35 2.5 COMBINED BENDING AND AXIAL LOAD 38 2.5.1 Cross S e c t i o n Behaviour 38 2.5.2 Members with Combined Bending and Compression ...42 2.5.3 In-Grade T e s t i n g 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 i n Wood P r o p e r t i e s 48 2.6 SUMMARY 49 Chapter I I I SIZE EFFECTS 3.1 INTRODUCTION 3.2 CONVENTIONAL BRITTLE FRACTURE THEORY  50 50 51  iv  3.2.1 H i s t o r y 51 3.2.2 A p p l i c a t i o n s of B r i t t l e F r a c t u r e Theory t o Wood .52 3.2.3 Theory f o r Uniform S t r e s s D i s t r i b u t i o n 54 3.2.4 Theory f o r V a r i a b l e S t r e s s D i s t r i b u t i o n 56 3.2.5 C o e f f i c i e n t of V a r i a t i o n 58 3.3 BRITTLE FRACTURE THEORY MODIFIED FOR TIMBER 59 3.4 DIFFERENT SIZE EFFECTS IN DIFFERENT DIRECTIONS 60 3.4.1 S i z e E f f e c t Terminology 61 3.5 LENGTH EFFECT 62 3.5.1 Theory 62 3.5.2 Assumptions on Length E f f e c t 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 74 3.10 SUMMARY 75 Chapter IV EXPERIMENTAL PROCEDURES 4.1 EXPERIMENTAL STAGES 4.2 TEST MATERIAL 4.2.1 Species 4.2.2 Grading 4.2.3 Moisture Content 4.2.4 Sample S e l e c t i o n 4.3 SAMPLE SIZES AND CONFIDENCE 4.3.1 Sample S i z e s 4.3.2 W e i b u l l D i s t r i b u t i o n 4.3.3 Confidence C a l c u l a t i o n f o r Q u a n t i l e s 4.4 TEST PROCEDURES 4.4.1 Bending 4.4.2 A x i a l Tension a. Long' Boards b. Short Boards 4.4.3 A x i a l Compression a. Long Boards b. Short Segments 4.4.4 E c c e n t r i c Compression 4.4.5 Combined Bending and Tension a. Bending Followed by Tension b. E c c e n t r i c Tension 4.4.6 Data A q u i s i t i o n 4.4.7 Modulus of E l a s t i c i t y 4.5 SUMMARY Chapter V EXPERIMENTAL RESULTS 5.1 COMBINED BENDING AND AXIAL LOADING RESULTS 5.1.1 P r e s e n t a t i o n 5.1.2 I n t e r a c t i o n Curves f o r Short Members a. Test R e s u l t s b. Mode of F a i l u r e  76 76 79 79 79 80 81 82 82 83 83 87 87 89 89 90 91 91 92 92 94 94 .....95 96 96 98 99 99 99 100 1 00 106  V  5.1.3 I n t e r a c t i o n Curves f o r Long Members 5.1.4 A x i a l • L o a d - s l e n d e r n e s s Curves 5.2 SEPARATE BENDING AND AXIAL LOADING RESULTS 5.2.1 Test R e s u l t s 5.2.2 Modes of F a i l u r e 5.3 LENGTH EFFECTS 5.3.1 I n t r o d u c t i o n 5.3.2 Compression Strength 5.3.3 Tension Strength a. 38x89mm Boards b. 38x140mm Boards 5.3.4 Bending Strength a. 38x89mm Boards b. 38x140mm Boards 5.3.5 Summary of Length E f f e c t s 5.4 WEAK AXIS BENDING 5.5 SUMMARY Chapter VI STRENGTH MODEL 6.1 INTRODUCTION 6.2 ASSUMPTIONS 6.3 CROSS SECTION BEHAVIOUR 6.3.1 C a l c u l a t i o n Procedure 6.3.2 N e u t r a l A x i s Contours 6.3.3 Curvature Contours 6.3.4 U l t i m a t e I n t e r a c t i o n Diagram 6.3.5 Moment-Curvature Curves 6.4 COLUMN BEHAVIOUR 6.4.1 F a i l u r e Modes 6.4.2 C a l c u l a t i o n Procedure 6.5 TYPICAL OUTPUT 6.5.1 A x i a l Load-Moment I n t e r a c t i o n Curves 6.5.2 A x i a l Load-Slenderness Curves 6.6 INPUT INFORMATION 6.6.1 Cross S e c t i o n Dimensions 6.6.2 Tension and Compression Strengths 6.6.3 Modulus of E l a s t i c i t y 6.6.4 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 i n Compression a. Shape of F a l l i n g Branch b. Shape of R i s i n g Branch 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 6.6.6 Column Parameters 6.7 NON-DIMENSIONALI ZED PLOTS 6.8 SUMMARY  107 107 110 110 112 114 114 115 116 116 117 118 119 120 122 123 124 126 126 127 .127 128 131 132 134 134 135 136 139 142 142 144 145 146 146 148 149 ;.151 154 156 158 158 159  Chapter VII CALIBRATION AND VERIFICATION 160 7.1 INTRODUCTION 160 7.2 38x89mm BOARDS 161 7.2.1 Short Column I n t e r a c t i o n Curves 161 7.2.2 Parameter E s t i m a t i o n 164 7.2.3 Long Column I n t e r a c t i o n Curve f o r End Moments ..165  vi  .  7.2.4 Long Column I n t e r a c t i o n Curves for Mid-Span Moments 7.2.5 A x i a l Load - Slenderness Curves 7.3 38x140mm BOARDS 7.3.1 Short Column I n t e r a c t i o n Curves 7.3.2 Long Column I n t e r a c t i o n Curves 7.3.3 A x i a l Load-Slenderness Curves 7.4 REPRESENTATIVE STRENGTH PROPERTIES 7.5 APPLICABILITY OF STRENGTH MODEL 7.6 SUMMARY  168 170 170 170 173 173 173 ,.181 .181  Chapter VIII DESIGN METHODS FOR COLUMNS AND BEAM COLUMNS 183 8.1 INTRODUCTION ...183 8.1.1 A l l o w a b l e S t r e s s Design 183 8.1.2 R e l i a b i l i t y - B a s e d Design 183 8.1.3 R e l i a b i l i t y - B a s e d Design of Timber 185 8.1.4 Scope 186 8.2 EXISTING DESIGN METHDOS 186 8.2.1 Canadian Timber Code 186 a. C o n c e n t r i c a l l y Loaded Columns 187 b. Combined A x i a l Load and Bending 188 c . Summary 190 8.2.2 NFPA Timber Code 191 a. C o n c e n t r i c a l l y Loaded Columns 191 b. Combined A x i a l Load and Bending .191 c . Summary 193 8.2.3 Code Requirements f o r S t e e l 194 8.2.4 Canadian Concrete Code 198 8.2.5 L i m i t S t a t e s 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 A n a l y s i s 209 8.4.2 Input S t r e n g t h P r o p e r t i e s ...210 8.4.3 Design Approaches 210 8.4.4 Moment M a g n i f i c a t i o n F a c t o r 212 8.4.5 METHOD 1: B i l i n e a r I n t e r a c t i o n Diagram 215 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 218 8.4.7 METHOD 3: U l t i m a t e I n t e r a c t i o n Diagram 220 8.4.8 Comparison of Methods 1 to 3 221 8.4.9 METHOD 4: P u b l i s h e d Design Curves 224 8.4.10 METHOD 5: S t r a i g h t L i n e Approximation 225 8.4.11 METHOD 6: A x i a l Load-Slenderness Curves 227 8.5 COMBINED BENDING AND TENSION 231 8.6 SUMMARY 234 Chapter IX DESIGN RECOMMENDATIONS 9.1 STRUCTURAL ANALYSIS 9.1.1 Strength Model 9.1.2 Second Order S t r u c t u r a l A n a l y s i s 9.1.3 Simple A n a l y s i s 9.1.4 Code Format  235 ...235 235 236 237 237  vii  9.2 APPROXIMATE DESIGN FORMULAE 238 9.2.1 Recommended Formulae 238 9.2.2 Example 240 9.2.3 Load F a c t o r s and R e s i s t a n c e F a c t o r s 240 9.2.4 Mimimum Moments .....241 9.3 DATA REQUIRED 241 9.3.1 In-grade Test R e s u l t s 241 9.3.2 S i z e E f f e c t s 243 a. Length E f f e c t s 244 b. Depth E f f e c t s 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 R a t i o 248 9.4.4 B i 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 E f f e c t of M o i s t u r e on Compression S t r e n g t h 254 9.6.2 E f f e c t of M o i s t u r e 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 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 F a l l i n g Branch 273 a. Assumptions ...273 b. C a l c u l a t i o n s 274 c. Length E f f e c t s 278 d. Summary 279 e. Depth E f f e c t i n Compression 280 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 280 a. Assumptions 280 b. C a l c u l a t i o n s 281 A.2 ULTIMATE INTERACTION DIAGRAM 283 A.2.1 Background 283 A.2.2 C a l c u l a t i o n s 283 APPENDIX B - CALCULATION OF -INTERCEPT DEFINING INTERACTION DIAGRAM  288  APPENDIX C - TEST RESULTS  290  APPENDIX D - NOTATION  299  vi i i  L i s t of T a b l e s  I.  Summary of experimental stages  II.  Summary of l e n g t h s and s i z e s  77  tested  (stages 1 to 4)  78  III.  M a t e r i a l p r o p e r t y r a t i o s f o r r e p r e s e n t a t i v e groups 179  IV. V.  Length e f f e c t parameter k, Summary of equations f o r c a l c u l a t i n g u l t i m a t e i n t e r a c t i o n diagram  244 286  ix  L i s t of F i g u r e s  1. A c t u a l and assumed s t r e s s e s failure  i n a c l e a r wood beam at  2. S t r e s s d i s t r i b u t i o n d e r i v e d from a b i l i n e a r strain relationship  stress-  3. S t r e s s d i s t r i b u t i o n assumed by B e c h t e l and N o r r i s  9 ..11  ....12  4. S t r e s s d i s t r i b u t i o n s measured by Ramos  14  5. S t r e s s d i s t r i b u t i o n proposed by Moe  15  6. S t r e s s d i s t r i b u t i o n proposed by Bazan  17  7. S t r e s s - s t r a i n  30  relationships  i n a x i a l compression  8. A x i a l l o a d - s l e n d e r n e s s curve f o r c o n c e n t r i c a l l y columns  loaded  36  9. U l t i m a t e i n t e r a c t i o n diagrams for l i n e a r behaviour  39  10. U l t i m a t e i n t e r a c t i o n diagrams f o r n o n - l i n e a r behaviour  40  11.  Three-dimensional sketch of load moment  12.  Typical  13.  Tension s t r e s s d i s t r i b u t i o n s  67  14.  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 extreme f i b r e  69  15.  Compression s t r e s s d i s t r i b u t i o n s  71  log-log  v s . slenderness v s .  43  p l o t of f a i l u r e s t r e s s v s . volume ....56  16. Quarter-sawn board  73  17.  P.d.f. of q u a n t i l e  18.  Loading arrangement f o r bending t e s t  88  19.  Loading arrangement f o r a x i a l (a) t e n s i o n (b) compression  90  20.  Eccentric axial (a) Compression  estimators  loading (b) Tension  86  tests  93  X  21. Combined bending and t e n s i o n t e s t (a) f i r s t stage (b) second stage  95  22. Test r e s u l t s f o r s h o r t e s t length i n e c c e n t r i c compression  101  23. P e r c e n t i l e r e s u l t s f o r s h o r t e s t length i n e c c e n t r i c compression  102  24. Test r e s u l t s f o r 38x89mm boards i n e c c e n t r i c compression and t e n s i o n  104  25.  I n t e r a c t i o n diagram f o r mean t e s t r e s u l t s of a l l lengths i n e c c e n t r i c compression  108  26. A x i a l l o a d - s l e n d e r n e s s curves. Mean t e s t r e s u l t s f o r a l l lengths t e s t e d i n e c c e n t r i c compression 109 27. Comparison of t e n s i o n , compression and bending results  test  111  28. R a t i o of t e n s i o n s t r e n g t h s of 38x89mm boards 2.0m and 0.914m long  117  29. R a t i o of t e n s i o n strengths of 38x140mm boards 3.0 and 0.914m l o n g .  118  30. R a t i o of bending s t r e n g t h s of 38x89mm boards • 1.5 and 0.84m long  .120  31. R a t i o of bending s t r e n g t h s of 38x140mm boards, 3.0 and 1 . 5m long  121  32. R a t i o of edgewise to f l a t w i s e bending s t r e n g t h  124  33. Cross s e c t i o n behaviour 128 34. Flow c h a r t f o r c a l c u l a t i n g moment-curvature-axial load r e l a t i o n s h i p s for a cross section 130 35. N e u t r a l a x i s contours interaction 36. Curvature contours interaction 37. U l t i m a t e section  for moment and a x i a l  for moment and a x i a l  load  load  133  i n t e r a c t i o n diagram f o r s t r e n g t h of c r o s s  38. Moment-curvature-axial load r e l a t i o n s h i p s 39. Column with a x i a l  132  133 135  load and equal end e c c e n t r i c i t i e s .136  40. T y p i c a l i n t e r a c t i o n diagram f o r e c c e n t r i c a l l y  loaded  xi  column  137  41.  Column d e f l e c t i o n curves  140  42.  Interaction  143  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 eccentricities  145  44.  U l 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  147  45.  E f f e c t of modulus of e l a s t i c i t y  46.  Stress-strain  47.  U l t i m a t e i n t e r a c t i o n diagrams f o r the 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 v a r y i n g slope of f a l l i n g branch 152  48.  U l t i m a t e i n t e r a c t i o n diagrams f o r 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 r e l a t i o n s h i p , with v a r y i n g l i m i t i n g strain .152  49.  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 a i n r e l a t i o n s h i p proposed by G1OS(1978)  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 strain relationships  51.  Stress-strain  52.  U l t i m a t e i n t e r a c t i o n diagrams with v a r y i n g d i s t r i b u t i o n parameter i n t e n s i o n  stress-  53.  U l t i m a t e i n t e r a c t i o n diagrams with v a r y i n g d i s t r i b u t i o n parameter i n compression  stress-  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 to t e s t r e s u l t s f o r 38x89mm s i z 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 with t e s t r e s u l t s f o r 38x89mm s i z e 166  56.  P r e d i c t e d i n t e r a c t i o n diagram f o r mid-span moments compared with t e s t r e s u l t s f o r 38x89mm s i z e  169  57.  P r e d i c t e d a x i a l load - slenderness curves compared with t e s t r e s u l t s f o r 38x89mm s i z e  171  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 to t e s t r e s u l t s f o r 38x140mm s i z e  172  59.  Predicted  diagrams f o r slender columns  relationships  on column behaviour .149  i n compression  150  stress-  stress-  r e l a t i o n s h i p with f a l l i n g branch  .. 153 153 155 157 157 162  i n t e r a c t i o n diagram f o r end moments compared  xi i  with t e s t r e s u l t s f o r 38x140mm s i z e  174  60. P r e d i c t e d i n t e r a c t i o n diagram f o r mid-span moments compared with t e s t r e s u l t s f o r 38x140mm s i z e  175  61. P r e d i c t e d a x i a l load - s l e n d e r n e s s curves compared with t e s t r e s u l t s f o r 38x140mm s i z e  176  62. U l t i m a t e i n t e r a c t i o n diagrams f o r r e p r e s e n t a t i v e strength properties  178  63. Non-dimensionalized i n t e r a c t i o n diagrams f o r representative strength properties  180  64. A x i a l l o a d - s l e n d e r n e s s curves f o r r e p r e s e n t a t i v e s t r e n g t h p r o p e r t i e s (non-dimensionalized) .....180 65. Code column formula compared with model (non-dimensionalized)  prediction  189  66. NFPA formula compared with model p r e d i c t i o n (nondimensionalized)  193  67. A x i a l load-moment i n t e r a c t i o n diagram members  for steel  195  68. A x i a l load-moment i n t e r a c t i o n diagram  from OHBDC ....202  69. OHBDC formula compared.with model p r e d i c t i o n (nondimensionalized)  204  70. Comparison of column curves f o r c o n c e n t r i c (non-dimensionalized)  206  71. Design p r o p o s a l f o r c o n c e n t r i c model p r e d i c t i o n 72. T y p i c a l i n t e r a c t i o n diagram magnified moment 73. I n t e r a c t i o n magnifier  diagram  74. I n t e r a c t i o n diagram moment m a g n i f i e r  loading  loading  compared with  f o r a x i a l load and  209 212  showing t r a d i t i o n a l moment 214 showing  proposed 216  75. B i l i n e a r approximation t o i n t e r a c t i o n diagram f o r magnified moments (non-dimensionalized)  217  76. P a r a b o l i c approximation to i n t e r a c t i o n diagram f o r magnified moments (non-dimensionalized)  219  77. Comparison of methods 1 t o 3 with model (non-dimensionalized)  222  prediction  xi i i  78.  S t r a i g h t l i n e approximations t o i n t e r a c t i o n diagrams for unmagnified moment (non-dimensionalized) 226  79.  Comparison of methods 1 and 5 with model (non-dimensionalized)  80.  M o d i f i e d secant formula compared with model p r e d i c t i o n  230  81.  Parabolic  approximation to i n t e r a c t i o n diagram  233  82.  Bending moment diagrams f o r combined a x i a l and t r a n s v e r s e loading  247  83.  Bending about i n c l i n e d n e u t r a l  250  84.  E f f e c t of moisture content on s t r e n g t h of timber i n 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 r e l a t i o n s h i p with branch  273  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 ...274  .87.  Flow chart f o r c a l c u l a t i n g u l t i m a t e bending moment f o r bilinear stress-strain relationship 279  prediction  axis  falling  228  88.  Elasto-plastic stresss-strain relationship  281  89.  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 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 f o r c a l c u l a t i n g u l t i m a t e bending moment f o r elasto-plastic stress-strain relationship 284  91.  S t r e s s d i s t r i b u t i o n s f o r nine d i f f e r e n t of a x i a l load and bending moment  92.  combinations 285  U l t i m a t e i n t e r a c t i o n diagram produced by hand calculations, compared with computer c a l c u l a t e d curve  287  93.  Typical  288  94.  E c c e n t r i c compression r e s u l t s , 38x89mm boards, 1.3m long  290  E c c e n t r i c compression r e s u l t s , 38x89mm boards, 1.8m long  290  E c c e n t r i c compression r e s u l t s , 38x89mm boards, 2.3m long  291  95. 96.  u l t i m a t e i n t e r a c t i o n diagram  xiv  97. E c c e n t r i c compression r e s u l t s , 38x89mm boards, 3.2m long  291  98. E c c e n t r i c compression r e s u l t s , 38x1 40mm boards, 1.82m long  ...292  99. E c c e n t r i c compression r e s u l t s , 38x140mm boards, 2.44m long  292  100. E c c e n t r i c compression r e s u l t s , 38x140mm boards, 3.35m long  293  101. E c c e n t r i c compression r e s u l t s , 38x1 40mm boards, 4.27m long  293  102. A x i a l t e n s i o n r e s u l t s , 38x89mm boards, 2.0m long  294  103. A x i a l t e n s i o n r e s u l t s , 38x89mm boards, .914m long  294  104. A x i a l compression r e s u l t s , 38x89mm boards, 2.0m long  295  105. Bending t e s t r e s u l t s , 38x89mm boards, 1.5m span  295  106. Bending t e s t r e s u l t s , 38x89mm boards, ,84m span (edgewise)  296  107. Bending t e s t r e s u l t s , 38x89mm boards, .84m span  296  (flatwise)  108. A x i a l t e n s i o n r e s u l t s , 38x1 40mm boards, 3.0m long  297  109. A x i a l compression r e s u l t s , 38x140mm boards, 3.0m long  297  110. Bending t e s t r e s u l t s , 38x140mm boards, 3.0m long  298  XV  Notation a  distance  between loads on beam  a  r a t i o of depth  A  cross  A  r a t i o of m a g n i f i c a t i o n  factors  b  r a t i o of c r o s s  section  depth  B  axis intercept  f o r i n t e r a c t i o n diagram  c  parameter i n s t r e s s - s t r a i n equation  c  r a t i o of c r o s s  C  force  Cm  equivalent  d  depth of c r o s s  section  e  section  depth  i n compression s t r e s s block  subscript e  area  moment f a c t o r  1  eccentricity strain subscript 0 1 t u y stress subscripts  a b c ca cm e m r s t ta u x  section reference  depth  compression s t r a i n at e l a s t i c f a i l u r e compression s t r a i n at peak s t r e s s tension s t r a i n upper l i m i t on compression s t r a i n y i e l d s t r a i n i n compression allowable compression stress in long column a l l o w a b l e bending s t r e s s compression stress at f a i l u r e f o r short column allowable compression stress in short column compression s t r e s s at f a i l u r e m o d i f i e d for stress-distribution effect Euler buckling stress f a i l u r e s t r e s s of extreme f i b r e i n tension modulus of rupture asymptotic s t r e s s f o r l a r g e s t r a i n f a i l u r e stress in a x i a l tension test allowable tension stress compression stress at f a i l u r e f o r long column extreme f i b r e s t r e s s i n tension zone  xvi  F  magnification  factor  G  parameter  I  moment of i n e r t i a of c r o s s s e c t i o n  J  r a t i o of slenderness  k  shape parameter  k  size effect subscripts  K  minimum slenderness  L  length of member  in stress-strain  subscript  equation  ratios  for Weibull  distribution  parameter 1 length e f f e c t parameter 2 depth e f 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  e  r a t i o f o r long columns  effective  length  m  r a t i o of slopes of s t r e s s - s t r a i n  m M  s c a l e parameter bending moment subscripts 1 2 u  n  parameter  n  sample  n  r a t i o of extreme  p  confidence  P  a x i a l compression f o r c e subscripts a compression c a p a c i t y of short column e E u l e r b u c k l i n g load u compression c a p a c i t y of long column  q  quantile  r  " confidence  r  parameter  r  r a d i u s of g y r a t i o n  r  r a t i o of s t r e s s e s  S  s e c t i o n modulus  for Weibull  curves  distribution  l a r g e r end moment smaller end moment u l t i m a t e bending moment  in stress-strain  equation  size f i b r e s t r e s s e s i n t e n s i o n and compression  interval in stress-strain  equation  xvi i  T  axial  tension  subscript  force  u  tension  capacity  o f member  v  deflection  v'  slope  v" V  d y/dx v o l u m e o f member subscripts 1 r e f e r e n c e volume e e q u i v a l e n t volume  w  width of cross  x  distance  x  stress subscripts  y  2  dy/dx 2  section  a l o n g member o q  l o c a t i o n parameter i n w e i b u l l value of x a t q u a n t i l e q  co-ordinate  z  standard normal  A  d e f l e c t i o n o f member  <t>  capacity  4>  curvature  variable  reduction  factor  distribution  xvi i i  Many people have c o n t r i b u t e d to the success My  supervisor,  enthusiastic  Professor  advice  and  of t h i s t h e s i s .  Borg  encouragement  Madsen,  throughout, and  very proud to have been a s s o c i a t e d with him Dr. Ken  research  Raymond Bleau,  was  encouraged me  useful  to pursue  i t . His  suggestions.  The Ricardo  helped  a  computer  program,  Mr.  and  made  British fellow  i n many ways.  Forintek  Corp.  made  F o s c h i gave much u s e f u l  Trust,  Engineering  and  Others at the U n i v e r s i t y of  F i n a n c i a l a s s i s t a n c e was Killam  co-  a major c o n t r i b u t i o n .  Columbia, i n c l u d i n g f a c u l t y members, t e c h n i c i a n s , students,  initiated  at Sherbrooke, with the a s s i s t a n c e of  Dr. Noel Nathan provided many  I am  i n t h i s work.  Johns, of the U n i v e r s i t y of Sherbrooke,  part of the p r o j e c t and operative  offered  the  at UBC,  and  f a c i l i t i e s a v a i l a b l e , and  Dr.  advice.  r e c e i v e d from the  Programme  of  the N a t u r a l  Izaak  Distinction  Sciences  and  in  Walton Timber  Engineering  Research C o u n c i l of Canada. The  contributions  of  all  the  above,  and  others  not  mentioned, are g r a t e f u l l y acknowledged. My and  for  greatest her  thanks go to my  support  and  wife E l s a ,  understanding  p r o j e c t would not have been p o s s i b l e .  for  her  typing,  without which t h i s  1  I.  INTRODUCTION  1 . 1 BACKGROUND Wood i s a complex organic m a t e r i a l formed i n t r e e s . i s put t o a l a r g e number of uses, an important one production  of  sawn  b r i d g e s and other Engineering  timber  for construction  design  contrast  and metals, research.  of  timber  to  timber  has  Timber  more  been  research  variable  manufacturing  structures  requires  p r o p e r t i e s of the m a t e r i a l .  and  the is  subject  of  very  little  fundamentally d i f f e r e n t  because  cannot  be  stresses  for  from  wood  properties  are  readily  modified  in a  structural  members have t r a d i t i o n a l l y been d e r i v e d  in  of b u i l d i n g s ,  process.  Allowable  small  the  manufactured m a t e r i a l s such as concrete  that of manufactured m a t e r i a l s much  being  structures.  knowledge of b a s i c engineering In  Wood  specimens  design  from standard  of c l e a r d e f e c t - f r e e wood.  s t r u c t u r a l engineering  applications  of  timber t e s t s on  Most timber used  contains  natural  or  man-made d e f e c t s which may i n c l u d e knots, s l o p i n g g r a i n , p i t h , checks the  or  strength  therefore clear but  machining d e f e c t s . of  the  different.  These d e f e c t s g e n e r a l l y govern  material,  and  Allowable  stresses derived  specimen t e s t s are adjusted the  results  structural  do  not  failure  mode  from small  f o r the e f f e c t s of  accurately reflect  is  defects,  the s t r e n g t h of  timber.  In recent years, design obtained  the  ' more  directly  stresses  from f u l l  f o r timber  have  been  s i z e t e s t s of commercially  2  a v a i l a b l e s i z e s , s p e c i e s and "in-grade  testing",  understanding but  has  of timber  t h i s understanding Most  This  provided a great  testing,  called  improvement i n the  as a s t r u c t u r a l e n g i n e e r i n g  material,  i s by no means complete.  in-grade  testing  bending t e s t s at a  constant  tension  grades.  has  concentrated  span-to-depth  and compression t e s t s at constant  on simple  ratio, length.  span  and  some  There have  been no s e r i o u s attempts to r e l a t e bending s t r e n g t h to t e n s i o n and  compression  absence  of  strengths.  There , has  been  a  conspicuous  i n f o r m a t i o n on strengths of d i f f e r e n t  different  load  subjected  to  configurations,  and  strength  combined bending and a x i a l  attempts t o improve the  theoretical  t e s t i n g concept by i n v e s t i g a t i n g these  of  loading.  basis  lengths and  of  members  This  the  study  in-grade  subjects.  1 . 2 OBJECTIVES The the  initial  behaviour  combined  o b j e c t i v e s of t h i s study  of  bending  hypothesis  and  timber  axial  members  loading,  that the behaviour of timber  bending,  or  to  explained  in  terms  strengths accepted  structural  of  combined  the  axial  s i m i l a r members.  f o r other  commercial  of  bending  quality  materials, timber  were to i n v e s t i g a t e  and  subjected  to  members  and a x i a l tension  subjected  has  and  to  compression  has been long  i t s applicability not  the  l o a d i n g , can be  T h i s hypothesis but  examine  to  yet  been  to  clearly  demonstrated. During  the couse of t h i s  i n v e s t i g a t i o n i t became apparent  that s i z e e f f e c t s have a major i n f l u e n c e on  the . s t r e n g t h  of  3  timber  members.  An  a d d i t i o n a l o b j e c t i v e , t h e r e f o r e , was to  examine and q u a n t i f y these s i z e  effects.  These o b j e c t i v e s have been met by developing model  which  can  predict  the  a  s t r e n g t h of timber members i n  bending, and i n combined bending and a x i a l l o a d i n g . incorporates  size  effects,  and  has  been  and  program.  main input t o the s t r e n g t h model i s a x i a l t e n s i o n and  compression level  The model  calibrated  v e r i f i e d using the r e s u l t s of a l a r g e experimental The  strength  strength  from  in-grade  testing.  i n a d i s t r i b u t i o n of s t r e n g t h can  behaviour  at  that l e v e l .  be  Input at any  used  to  predict  As such, the model i s not intended  to p r e d i c t s t r e n g t h of an i n d i v i d u a l board of timber. 1.3 DESIGN CODES Structural prescribed  design  allowable  has  traditionally  stresses  which  been  More r e c e n t l y ,  have become i n t e r e s t e d i n the u l t i m a t e  strength  s t r u c t u r e s and behaviour under maximum p o s s i b l e loads. trends  are  considers  toward  of  failure  under  extreme  of  Recent  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  probabilities  on  are not to be exceeded  when working loads are imposed on a s t r u c t u r e . designers  based  which  loading  conditions. T h i s study c o n t r i b u t e s to the development of r e l i a b i l i t y based design on  the  of timber s t r u c t u r e s by p r o v i d i n g new  d i s t r i b u t i o n of. s t r e n g t h p r o p e r t i e s of timber members  subjected  t o combined bending and a x i a l l o a d i n g .  of  study  this  working  information  stress  can  also  design  be  codes  The  incorporated  into  to  improved  provide  results  conventional design  4  methods.  be  Current  design  independent  of  strength  codes c o n s i d e r member  the s t r e n g t h of timber to  length.  varies considerably  with  This  study  l e n g t h , and  shows  that  a simple  method  r e s u l t s of t h i s study demonstrate that c u r r e n t  design  of q u a n t i f y i n g length e f f e c t s i s proposed. 1.4  APPLICATIONS The  methods f o r  combined  conservative  i n some cases.  and where  economical  designs  combined  loading  structural  components  which, a c c o r d i n g are  bending  used  in  S t a t e s , and  which  Other  axial  be p o s s i b l e f o r timber  occurs. in  loading  this  The  largest  category  very  (Gromala and  v  being  used  structures group  of  are timber t r u s s e s ,  of new domestic c o n s t r u c t i o n are  are  T h i s suggests that more e f f i c i e n t  to some estimates  95%  commercial and  may  and  Moody  1983),  in the  United  increasingly  in  larger  industrial structures.  structural  components  a x i a l l o a d i n g are w a l l studs and  with  combined bending  columns  in  many  types  and of  buildings. Recent  research  in  timber  handicapped by a lack of understanding timber members subjected (FPRS 1979).  truss of  design the  to combined bending and  has  behaviour axial  been of  loading  5  1.5 THESIS ORGANIZATION Chapter  2  d e s c r i b e s p r e v i o u s attempts to understand the  behaviour of both c l e a r wood and timber  members  subjected  to  v a r i o u s types of l o a d s . Chapter obstacles  3  discusses  encountered  phenomenon  that  by  the  of  previous  of  of  the main  researchers is  is  the  significantly  part of Chapter  In  and simple  One  timber  The f i r s t  studies.  t h e o r i e s are m o d i f i e d  effects.  previous  strength  dependent on member s i z e . survey  size  3  is a  the second p a r t , e x i s t i n g  equations  are  derived  for  l a t e r use. Chapters investigate bending  4  and  the  and  5  describe  strength  axial  of  experiments  timber  members  to  subjected  to  l o a d s , both s e p a r a t e l y and combined.  r e s u l t s of these experiments are used l a t e r t o verify  performed  a s t r e n g t h model.  The l e n g t h e f f e c t  calibrate  theory  The and  introduced  in Chapter 3 i s q u a n t i f i e d i n Chapter 5. Chapter 6 d e s c r i b e s a conceptual s t r e n g t h of p o p u l a t i o n s basis  of  obtained effects  tension from  of  and  in-grade  variability  model f o r p r e d i c t i n g the  of s t r u c t u r a l timber compression testing. in  member and between members.  properties The  timber  on  the  which can be includes  the  both w i t h i n a  The model i s based on the i n i t i a l  that s t r e n g t h i n bending, or  and  l o a d i n g , can be e x p l a i n e d  and  model  strength,  hypothesis axial  members  in  combined  i n terms of a x i a l  bending tension  compression p r o p e r t i e s . In Chapter 7  the  strength  model  i s calibrated  using  6  selected  experimental  experimental measured  r e s u l t s , then t e s t e d a g a i n s t many other  results.  results  The  comparisons  support  the  of  initial  demonstrate that t h i s s t r e n g t h model can be the and  behaviour  predicted  and  hypothesis  and  used  to  predict  of e c c e n t r i c a l l y loaded members of many lengths  eccentricities. Chapters  describing  8  and  9  are  concerned  with  applications,  e x i s t i n g design methods f o r v a r i o u s m a t e r i a l s , and  recommending how the r e s u l t s of t h i s study can be  used  as  a  b a s i s f o r improved s t r u c t u r a l design of timber members. 1.6 LIMITATIONS The  model  developed  in  t h i s t h e s i s i s a f l e x i b l e one,  which i s b e l i e v e d t o be a p p l i c a b l e t o timber However, i t has been c a l i b r a t e d with members  of  section. require  a  single  Application further  axial major  tests  calibration.  group  principal  in  mind  situations.  in  to  verified  and  of  two s i z e s of c r o s s grades  with  reference  to  dry timber members, loaded with  axis  tests  may  The a n a l y t i c a l r e s u l t s of t h i s  loads a p p l i e d with equal end  deformations. kept  on  reference  to other s i z e s , s p e c i e s and  study have only been duration  species  from any source*  eccentricities  restrained  against  to  short  eccentric about  the  out-of-plane  The s i g n i f i c a n c e of these l i m i t a t i o n s should be before  extrapolating  the  results  to  other  •  7  II. 2.1  LITERATURE SURVEY  INTRODUCTION  2.1.1  Background This  chapter  investigations  provides  study. used  brief  historical  the  axial  loads  state-of-the-art  and  at  combined  of  load,  and  the commencement of t h i s  Many of the f i n d i n g s r e f e r r e d to i n t h i s later  review  i n t o the s t r e n g t h of s t r u c t u r a l timber members  subjected to bending, describes  a  chapter  are  i n the t h e s i s when d e v e l o p i n g a general model f o r  the behaviour of s t r u c t u r a l timber members. The  first  part of the chapter c h r o n i c l e s the d i f f i c u l t i e s  that have  been  behaviour  of  compression  test  for  wood  than  variability, compression,  encountered wood  and  to  members  predict  from  bending  tension  and'  T h i s process i s much more d i f f i c u l t  f o r other non-linear the  trying  timber  results.  and  in  materials  because  stress-strain  presence  of  very  of  material  behaviour significant  in size  effects. Investigations behaviour  are  into  .axial t e n s i o n and a x i a l  described,  and  the  discussion  compression extended  to  i n c l u d e columns and beam-columns. 2 . 1 . 2 C l e a r Wood and Commercial This  Timber  t h e s i s i s concerned with the s t r e n g t h of commercial  q u a l i t y sawn timber, c o n t a i n i n g n a t u r a l or man-made This  material  North America)  (which is  in certain sizes  referred  to  as  is called  timber  defects. "lumber" i n  throughout  this  8  thesis.  The  term wood g e n e r a l l y  wood, as o f t e n t e s t e d Timber  and  clear  wood  loading conditions. compression  in  r e f e r s to c l e a r d e f e c t - f r e e  small  sizes  behave  i s much  standard  tests.  q u i t e d i f f e r e n t l y under most  For example,  strength  in  the  ratio  greater  of -tension  to  i n c l e a r wood than i n  commercial timber, so c l e a r wood tends t o behave i n a  ductile  manner i n a bending t e s t , whereas commercial timber f a i l s  with  a sudden b r i t t l e f r a c t u r e . For  many  years  i t was  believed  that  a  complete  understanding of c l e a r wood behaviour would l e a d e a s i l y to understanding of timber s t r e n g t h , Emphasis  has  shifted  to  but t h i s has not developed.  direct  investigation  of s t r e n g t h  p r o p e r t i e s of timber, l e a d i n g to the development of t e s t i n g of commercial q u a l i t y m a t e r i a l Investigations  into  the  timber are referenced two  separate  an  behaviour  in of  i n t h i s chapter,  "in-grade"  structural both  and  are  sizes.  c l e a r wood and discussed  as  materials.  2.2 BENDING STRENGTH 2.2.1 Bending Behaviour of C l e a r Wood a. In  D i s t r i b u t i o n of S t r e s s e s  1638  Galileo  recorded  h i s attempts  d i s t r i b u t i o n of f l e x u r a l s t r e s s e s i n a theory  was  bending  d e s c r i b e . the member.  i n e r r o r because he assumed that the n e u t r a l  was on the compression s u r f a c e . distribution  to  of  Parent obtained  s t r e s s e s f o r an e l a s t i c  the  His axis  correct  beam i n 1708, but i t  was not u n t i l Coulomb confirmed these f i n d i n g s  in  1773  that  9  they became g e n e r a l l y accepted From that time u n t i l of  timber  beams  has  (Booth 1980).  the present,  assumed  most e n g i n e e r i n g  linear  elastic  behaviour to  f a i l u r e , with the n e u t r a l a x i s at mid-depth of the Early tests in always  behave  in  bending this  indicated  manner.  that  design  section.  wood  does  not  A bending t e s t on a t y p i c a l  d e f e c t - f r e e wood member shows l i n e a r e l a s t i c behaviour up to a proportional in  the  face, and  continue  with f u r t h e r l o a d i n g  to increase u n t i l a b r i t t l e  assumed and  f igure  yielding  occurs  compression zone, the n e u t r a l a x i s s h i f t s towards the  tension  The  l i m i t , beyond which compression  actual  stresses  at  the  tension  tension  stresses  failure  failure  are  occurs. shown  in  1.  Compression  ^ Tension  Figure  As  1 - A c t u a l and assumed s t r e s s e s in a c l e a r wood beam at f a i l u r e early  as  1841  Joseph  Colthurst  noted  that f o r a  bending t e s t with f i r battens "extension and compression were equal up to threeq u a r t e r s of the breaking l o a d , but a f t e r t h i s compression y i e l d e d in a much higher r a t i o than e x t e n s i o n " . (Todhunter and Pearson 1886)  10  The in  t r a d i t i o n a l method  a timber  load,  bending  a  the  "modulus by  stress  rupture"  the  section  behaviour  rupture  and  function  o f t h e non l i n e a r  and  strength  the ratio  will  between  be e x p l a i n e d Observations  of  compression  in this  flexural  that  relationship model,  shown  in figure  behaviour. appears  which  produces  2 i s  This  t o have  a  been  the  recognized  that  bilinear  compression  in stress  a  of  proposed  into  compression block  shown  stress  If  i snot somehow  t h e modulus of  behaviour  as of  i t is a the  strengths,  i tclear  wood which  stress-strain in  distribution approximation  of  in figure  2.  of t h e form of  who^,  actual member showed  wood beams.  elasto-plastic used  tension.  i n a wood  by N e e l y ( ] 8 9 8 )  be  section.  c a n b e made b y  elastic  stresses  as  flexural  the cross  theory  behaviour  can  that  longitudinal over  a stress  first  insight  of rupture  defined,  but remains  distribution  stress.  elasto-plastic  reasonable  remarkable  relationship  the  wood h a s a b i l i n e a r  This  elastic  between  to elastic  i n compression,  For linear  thesis  a r e not l i n e a r l y d i s t r i b u t e d modification  and  maximum  f a i l u r e s made  approached,  failure  the  and compression  stresses  assuming  moment,  maximum  easily  loads  simple  bending  ficticious  failure  A  are  not  tension  further  a  relationship  i s  the  t h e modulus  a t f a i l u r e , but merely  stress  measure  dividing  i s the  occurs  This  by  t h e maximum  modulus.  of rupture  to strength.  to  corresponding  of  t h e modulus  non-linear  related  the  moment  behaviour any  beam a t f a i l u r e h a s b e e n  calculate  calculate  of c a l c u l a t i n g  He  stress-strain to construct  For  clear  the  wood,  11 l/l in  (a)  s t r e s s - strain  Figure which  relationship  (b)  suggested  that  stronger  in  flexural  capacity  compression s t r e n g t h alone.'  it  distribution  2 - S t r e s s d i s t r i b u t i o n d e r i v e d from a b i l i n e a r stress-strain relationship  i s much  theory  stress  t e n s i o n than i n compression, he can  be  T h i s study w i l l  calculated  from  show that Neely's  needs some m o d i f i c a t i o n f o r a p p l i c a t i o n to timber, but  i s s u r p r i s i n g how l i t t l e  a t t e n t i o n has been  given  to h i s  work. Dietz  (1942)  tested  f i v e beams of c l e a r D o u g l a s - f i r to  i n v e s t i g a t e the d i s t r i b u t i o n of s t r e s s e s and s t r a i n s , compare  the behaviour of the same m a t e r i a l under d i r e c t  s t r e s s e s i n tension and compression. results test  and  He was able to  to  axial  use the  of a x i a l t e s t s to p r e d i c t the bending s t r e n g t h of h i s  specimens  surprisingly  well  compared  with  later  investigators. Measured  d i s t r i b u t i o n s of s t r a i n over the c r o s s  sections  i n d i c a t e d some d e v i a t i o n from the usual assumption that sections  remain plane.  l i n e a r , with strains  The measured s t r a i n g r a d i e n t  l a r g e r s t r a i n s near the t e n s i o n  face and  plane  was nonsmaller  near the the compression face than would be p r e d i c t e d  12  f o r plane  s e c t i o n s remaining  been r e p o r t e d by any l a t e r the  outermost  plane.  T h i s phenomenon  investigators.  compressive  fibre  D i e t z reported  t o a x i a l compression.  that  i n a beam possesses a much  higher p r o p o r t i o n a l l i m i t than does a s i m i l a r subjected  has not  f i b r e i n a block  T h i s r e s u l t has been  reported  elsewhere, as d e s c r i b e d below, but has not been found  by a l l  investigators. Bechtel  and  N o r r i s (1952) c a r r i e d out a number of t e s t s  on small c l e a r S i t k a compression  spruce  beams  using  bending,  and shear specimens from each p i e c e of wood.  compression t e s t s produced a s t r e s s - s t r a i n curve shown by the s o l i d l i n e  (a)  They  of the  The shape  in figure 3(a).  s t r e s s - s t r a i n relationship  Figure  tension,  (b) s t r e s s  distribution  3 - S t r e s s d i s t r i b u t i o n assumed by Bechtel and Norris used  p l a s t i c behaviour distribution  of  the s i m p l i f y i n g assumption of p e r f e c t e l a s t o shown  by  the  dotted  line  to  s t r e s s e s as shown i n f i g u r e 3(b).  c a l c u l a t e d the t h e o r e t i c a l  obtain  a  They then  f l e x u r a l s t r e n g t h of the beam using  13  this  distribution  of  s t r e s s e s , m a t e r i a l p r o p e r t i e s from the  small c l e a r t e s t s , and a c r i t e r i o n of f a i l u r e shear and normal As  s t r e s s e s proposed  expected,  shear  under  by N o r r i s ( 1 9 5 5 ) .  s t r e n g t h governed  the behaviour f o r  short deep beams, f l e x u r a l s t r e n g t h f o r long and  combined  beams.  flexural  and  shear  For the beams where shear  (span-to-depth  ratio  less  failures  than  beams  governed  by  shallow  about  tension  were  significant  10), the theory gave measured  strength  strength. the  overestimated the measured s t r e n g t h by 15% to 34%. was  fracture  source of e r r o r  size  The  the h i g h l y  may.  error  have  been  i n a r e d u c t i o n of the  carried  failure  gauges  time.  stress  throughput  and he a l s o made t e n s i o n and compression  from the same members. remain  essentially  elastic  until  as  plane,  wood  in  f a i l u r e , and compression  extreme  These  tension  s t r e s s e s are  1  fibres  in  beam  sections  remains  f o r both the compression  compression  the  t e s t s on wood  H i s f i n d i n g s were that plane  proportional limit  f o r the  specimen.  brittle  out a s e r i e s of t e s t s on 80mm deep  beams of c l e a r wood with s t r a i n  the  a  s t r e s s e d volume of a member i s i n c r e a s e d .  Comben(l957)  depth,  well  e f f e c t which was not r e c o g n i z e d at that  Size e f f e c t s result  and  theory  a t t r i b u t e d to the s i m p l i f i c a t i o n made i n f i g u r e 3 ( a ) , but  the l a r g e s t  at  beams,  s t r e s s e s f o r intermediate  f l e x u r a l s t r e n g t h s l e s s than 5% over the For  combined  the  linear  the  same  specimens bending  f i n d i n g s are not i n agreement with those of  Dietz. In a x i a l t e n s i o n t e s t s he found l i n e a r e l a s t i c  behaviour  14  to  failure,  also tested from  as  r e p o r t e d by most other i n v e s t g a t o r s .  beams of s i x d i f f e r e n t  sizes  and  Comben  depths  5mm t o 80mm at constant span to depth r a t i o .  very s i g n i f i c a n t r e d u c t i o n i n f a i l u r e s t r e s s  varying  He found a  with  increasing  size. A  study  of  three  clear  Douglas f i r beams was made by  Ramos(l96l) with s t r a i n measurements over experiments and  that  the  predicted  also  compression from  relationship. can  possess  the  He a l s o quite  shown i n f i g u r e stress  confirmed that  block  axial  compression  noted that  results.  in  visually  stress-strain  bending  can  be  stress-strain  similar  material  relationships,  as  i n three s i m i l a r beams.  4 - S t r e s s d i s t r i b u t i o n s measured by Ramos of  the  bilinear  f o r p r a c t i c a l ' purposes,  obtain s a t i s f a c t o r y machine  These  4, which i s a composite sketch of h i s measured  Ramos proposed the use relationship  depth.  plane s e c t i o n s remain plane,  stress  different  distributions  Figure  the  verification  malfunctions  and  of  but  elasto-plastic he was unable to  h i s theory  because  the absence of d i r e c t t e n s i o n  of test  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) looked  more  closely  side of glued  at  laminated  the deformations on the compression beams.  phenomenon of l o c a l w r i n k l i n g measured elastic solid  axial  He  combined  the  physical  of the compression f i b r e s with a f a r beyond  the  range to propose the s t r e s s d i s t r i b u t i o n shown by  the  line  stress-strain  in  figure  figure  5  as  an  relationship  5, which i s c o n s i s t e n t  developed i n t h i s t h e s i s . in  who  with the model  H i s suggestion of the  approximation  dotted  for calculating  line  bending  s t r e n g t h has not been pursued by o t h e r s .  11  iV  1*  Figure  5 - S t r e s s d i s t r i b u t i o n proposed by Moe  Nwokoye(1975) t e s t e d laminated  beams.  For  a series those  shear f a i l u r e , he found that bilinear  stress  clear  a  theory shown  of s t r e n g t h .  based  specimens  contrast  cut  from  on  the' simple  The t h e o r e t i c a l estimate was  the same beams.  t o B e c h t e l and N o r r i s  glued  i n f i g u r e 2 gave a very  based on the r e s u l t s of d i r e c t t e n s i o n and on  hardwood  beams long enough to prevent a  distribution  accurate p r e d i c i t o n  of  compression  tests  T h i s accuracy i s i n  who found that  a similar  theory  16  overestimated the  s t r e n g t h of t h e i r beams.  may  different  indicate  with  a  softwoods.  remained  assumption that proportional  and the  confirmed  he  produced  that  is  the  s t r e n g t h p a r a l l e l to the  plane  evidence  extreme f i b r e s t r e s s  limit  in  same as the  compression, over the not  bending  assumed a  parabolic a  the  c a r r i e d out curve  d i s t r i b u t i o n of  stresses  stress-strain relationship,  and  of those producing maximum s t r e s s are  not  S e v e r a l unanswered q u e s t i o n s have been r a i s e d  by  stated  Zakic's  in  is  with any  failure.  by  analysis  excess  considered.  at  u l t i m a t e compressive  stress-strain  parabolic  compression region at  in  his  grain.  producing  consistent  strains,  sections  supporting  A t e s t s e r i e s using c l e a r poplar beams was Z a k i c ( l 9 7 3 ) who  results  s i z e e f f e c t i n hardwoods compared  Nwokoye  plane,  Nwokoye's  Nwokoye( 1974 ). Bazan(l980) theory.  He  approximated  compression  by  f o l l o w e d by a strain, actual  proposed  as  linear shown  behaviour  shape  suitable  reduction  and  in  small  the produces figure and  to  stress-strain  in  in f i g u r e 6(a),  This  shown for  the  refinement  l i n e a r e l a s t i c behaviour up  approximation. the  a  relationship  with  where the  solid  line  l i n e shows  shows  6(b).  This  Bazan's  approximation  is  l e v e l s of s t r a i n ,  but  o b v i o u s l y cannot be e x t r a p o l a t e d to very l a r g e  beams from the  increasing  a compression s t r e s s block of  intermediate  Bazan attempted to p r e d i c t  in  to maximum s t r e s s ,  stress  dotted  elasto-plastic  strains.  bending s t r e n g t h of c l e a r wood  r e s u l t s of a x i a l t e n s i o n and  compression  tests  17  in 0)  (a)  s t r e s s - s t r a i n relationship  Figure on  (b) s t r e s s  6 - S t r e s s d i s t r i b u t i o n proposed by Bazan  specimens cut from the same board.  success using above, and  an  application  empirical  size  effect  procedure i s continued  factor  in  tension.  to commercial q u a l i t y timber members with d e f e c t s ,  i n both bending and combined l o a d i n g .  effects  reasonable  i n t h i s t h e s i s but with  uses a much more r i g o r o u s approach to  in  He achieved  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  Bazan's general  loaded  distribution  i n the t e n s i o n  Bazan's  test  zone.  T h i s study a l s o  brittle  fracture  Timber with d e f e c t s was  programme,  but  was  not  found  that  size  included  subjected  to  mathematical a n a l y s i s . b.  Size E f f e c t s  Observations over many strength  of  large  years  members  have tends  s m a l l e r members under s i m i l a r l o a d i n g The bending  first was  attempt to q u a n t i f y a made  to  be  the  l e s s than that of  conditions. size  by Newlin and Trayer  effect  (1924).  theory They  the term "form f a c t o r " to r e l a t e the modulus of rupture cross  bending  s e c t i o n to that of a 51x51mm specimen  and  in  defined of any  carried  out  18  bending  tests  sections  including rectangular,  The  material  Sitka  on  beams of a l a r g e number of d i f f e r e n t c r o s s  f o r a l l these t e s t s  was  clear  straight-grained  spruce. On  the  b a s i s of these t e s t s they developed a theory f o r  the observed s i z e e f f e c t . the  c i r c u l a ' r , hollow and f l a n g e d .  "fibre  support  T h i s theory,  theory",  suggests  compressive s t r e s s which the extreme sustain  is  related  to  the  which became known that  fibres  in  amount of l a t e r a l  those f i b r e s r e c e i v e from l e s s h i g h l y s t r e s s e d the  neutral  strength beam  axis.  As  such,  of a deep beam should  the  the  theory  a  support.  -The theory  beam  can  support which fibres  nearer  p r e d i c t s that the  because i n the deep beam the h i g h l y s t r e s s e d  lateral  maximum  be l e s s than that of a  f i b r e s are f u r t h e r from lower s t r e s s e d  as  shallow  compression  f i b r e s which can  a l s o p r e d i c t s the lower  offer  strength  of flanged beams where one or more narrow webs do not o f f e r as much l a t e r a l In ultimate  support as a s o l i d  addition  to  explaining  l o a d , the theory  phenomenon  observed  rectangular  limit  is  greater  i s not  differences  at  f o r the  by some " i n v e s t i g a t o r s that the s t r e s s i n the  proportional  limit  in  than the f i b r e s t r e s s at the p r o p o r t i o n a l  i n a compression  phenomenon  strength  gives a possible explanation  the extreme compression f i b r e at bending  web.  parallel  explained  to  the  grain  by more recent  test.  brittle  This  fracture  theories. The regarding  f i b r e support theory the strength  of  was based on c e r t a i n assumptions  wood  in  the  ductile  compression  19  region  of  the  beam.  It  was  assumed  throughout  this  development that the s t r e n g t h of wood i n t e n s i o n i s a m a t e r i a l constant, axial  and the theory  tension.  The  fracture explanation introduced  does not p r e d i c t  more  recent  for size  a  size  development  effects  in  effect  in  of a b r i t t l e  bending  will  be  i n chapter 3.  2 . 2 . 2 Bending Behaviour of Timber a. The  Comparison With C l e a r Wood  bending  strength  than that of c l e a r failures  of timber  wood  members i s c o n s i d e r a b l y  specimens  tend to be more b r i t t l e .  of  the  same  The simple  is  greatly  compression s t r e n g t h  reduced  by  various  i s a f f e c t e d much  size,  strength  defects,  less,  and  explanation for  t h i s d i f f e r e n c e i n behaviour i s that the t e n s i o n timber'  less  of  whereas  resulting  in a  m a t e r i a l which i s o f t e n weaker i n t e n s i o n than i n compression. In  this  case  behaviour, all  bending  stress-strain  the same form as shown compression greatly  i s governed  and f a i l u r e s tend t o be sudden  s t r e s s e s i n the e l a s t i c The  strength  strength  reduced.  and  tension  brittle,  with  range.  r e l a t i o n s h i p f o r timber in  by  figure  slightly  3  tends to be of  for clear  reduced  wood,  and t e n s i o n  with  strength  20  b.  In-grade T e s t i n g  In recent years  l a r g e s c a l e in-grade  provided  wealth  timber.  a  a  testing  programmes  have  of new i n f o r m a t i o n on the behaviour of  In-grade t e s t i n g r e f e r s to f u l l - s i z e t e s t i n g of l a r g e  samples of commercial timber  i n the s i z e s , grades and  species  groups i n which i t i s produced and marketed. In  Canada  t e s t i n g has been c a r r i e d out on the p r i n c i p a l  s i z e s , grades and species groups. provide  lower  specifying 1978a).  In  estimate boards  of have  design  many cases  the  have lower  been  been  stresses  (Madsen  and  to  Nielsen  r e p r e s e n t a t i v e samples of timber  to a proof  boards  has  p e r c e n t i l e s t r e n g t h values as a b a s i s f o r  allowable  been subjected weakest  5th  The emphasis  load such that only  been 5th  returned  broken,  providing  percentile to  about an  strength.  the  have 10%  accurate Unbroken  normal p r o d u c t i o n .  In other  r e s e a r c h - r e l a t e d s t u d i e s , complete samples of boards have been t e s t e d to f a i l u r e  under  various  loading  and  environmental  conditions. Some important  f i n d i n g s from in-grade  testing are:  1. Commercial q u a l i t y timber e x h i b i t s q u i t e d i f f e r e n t f a i l u r e modes than c l e a r wood i n many l o a d i n g cases. 2. V a r i a b i l i t y in strength,properties for timber than f o r c l e a r wood.  i s much  greater  3. The d i f f e r e n c e i n s t r e n g t h between v a r i o u s t r e e s p e c i e s i s much l e s s f o r timber members than observed i n c l e a r wood, e s p e c i a l l y at the 5th p e r c e n t i l e l e v e l . 4. E x i s t i n g grading r u l e s do not separate well defined strength c l a s s e s . 5.  Timber members e x h i b i t a  significant  timber  size  into  effect.  21  Large members f a i l at lower s t r e s s e s than smaller members of the same s p e c i e s and grade. 6. The f a i l u r e s t r e s s i n a x i a l t e n s i o n members i s much l e s s than the modulus of rupture obtained from bending tests. 7. Moisture content has l e s s e f f e c t on the s t r e n g t h of timber than on c l e a r wood, a t the low end of the s t r e n g t h distribution. 8. Strength i s not s t r o n g l y c o r r e l a t e d with s i z e or type of d e f e c t , or with d e n s i t y or modulus of e l a s t i c i t y . S e v e r a l of the above r e s u l t s w i l l  be d i s c u s s e d  i n more  detail  later. c. The  D e r i v a t i o n of  conventional  with d e f e c t s to  the  specimens.  of  p r e d i c t i n g the s t r e n g t h of wood  has been t o apply  values  obtained  modification  factors  from t e s t s on small c l e a r  T h i s procedure i s d e s c r i b e d by WiIson(1978).  The standard to  method  (timber)  strength  Design•Stresses  method of determining  modulus of rupture  is  load t o f a i l u r e small c l e a r specimens 51x51mm, 760mm long,  under c e n t r e - p o i n t l o a d i n g over a simple  span  For  factors  or " s t r e n g t h  of  small  wood  with  defects,  modification  r a t i o s " a r e used to modify the tests  results  the  1981a).  clear  (ASTM 1981c), depending on slope of g r a i n or knot  assuming that knots have zero s t r e n g t h . shown  (ASTM  that  the  actual  size,  In-grade t e s t i n g  s t r e n g t h of timber  i s very  has  different  from that p r e d i c t e d i n t h i s way. In-grade measurement  testing of  characteristic  i s now  strength  being  properties.  used This  s t r e n g t h p r o p e r t i e s f o r a given  as  a  direct  method  provides  population  of  22  timber,  but has never been intended  a s i n g l e board. to any  t o p r e d i c t the s t r e n g t h of  In-grade t e s t r e s u l t s provide e s s e n t i a l input  p r o b a b i l i s t i c design method f o r timber  One  of  the l a r g e s t problems p r e v e n t i n g  of s t r u c t u r a l timber method  to p r e d i c t s t r e n g t h  on  the  parameters  such  d e n s i t y and  knot s i z e , but  separate  design  basis  of  non-destructive  as f l e x u r a l s t i f f n e s s ,  Commercial  test  Many attempts have been made  none  have  "stress-grading  wood i n t o s t r e n g t h c l a s s e s  flat-wise  efficient  i s the absence of a n o n - d e s t r u c t i v e  for predicting strength.  results.  structures.  on.  test  l o c a l slope of g r a i n , produced  very  machines" the  f l e x u r a l modulus of e l a s t i c i t y .  useful  are used to  basis  of  local  T h i s 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 s t r e n g t h and modulus of e l a s t i c i t y not very  For  high.  d.  Size Effects  at  least  timber Heim  70  years  a s i z e e f f e c t has been observed f o r  members in bending. (1912)  document  In an e x t e n s i v e  that  the  bending  members tends to be l e s s than that of similar  loading  investigated this testing detail  is  methods. later.  conditions. phenomenon Size  report, Cline strength  smaller  more  effects  recently will  be  of l a r g e  members  Madsen and N i e l s e n  and  under  (1976) have  using  in-grade  discussed  in more  23  2.3 AXIAL TENSION STRENGTH 2.3.1  A x i a l Tension The  first  recorded  to the g r a i n was Tension  Strength  of C l e a r Wood  t e s t on wood in a x i a l t e n s i o n  performed by M a r i o t t e  i n 1680  parallel  (Booth  1980).  t e s t s have not been as easy to perform as bending  compression t e s t s connection  because  stronger  Today  there  of  the  than the t e s t  i s a standard  difficulty  of  and  making  a  specimen.  test  (ASTM 1980)  using a 450mm  long p i e c e of c l e a r wood necked down to 4.8x9.5mm over a  64mm  gauge  been  length.  The  e v o l u t i o n of t h i s t e s t  d e s c r i b e d by Markwardt  and  Youngquist  specimen has  (1956).  Since  methods f o r small c l e a r specimens became s t a n d a r d i z e d , strengths rupture made  have  been  c o n s i s t e n t l y higher  (Wood Handbook 1974).  to  explain  this  f u r t h e r under s i z e  were  u n t i l about  Galligan  taken  1965  interpretation  than the modulus of  serious  discrepancy  as  of t e s t  et  equal  i n the b e l i e f  attempt  which  has  w i l l be  serious  been  discussed  al.(l974)  tensile  to the design  that t h i s  results.  was  very  little  design  bending s t r e s s a  conservative  Consequently, the r e s u l t s of  small c l e a r t e n s i o n t e s t s were not put to much use and testing received  tension  effects.  As d e s c r i b e d by stresses  No  test  attention.  This  was  tension not  a  problem at that time because the lack the of s u i t a b l e  connection  d e t a i l s prevented  very  high  stresses  from  being  developed i n t e n s i o n members of r e a l s t r u c t u r e s . It  was  a v a i l a b l e and  not  until  more  effective  connections  commercial s i z e m a t e r i a l with d e f e c t s was  became tested  24  that the p o s s i b l e magnitude of a s i z e e f f e c t realized,  leading  c l e a r wood.  in  tension  was  t o renewed i n t e r e s t i n t e n s i o n s t r e n g t h of  Kunesh and Johnson (1974) c a r r i e d  out  tests  of  commercial s i z e s of c l e a r D o u g l a s - f i r and Hem-fir and observed a  significant  decrease  in  strength  s e c t i o n dimension, the strengths clear  being  with  increasing cross  f a r l e s s than i n  small  specimens. This  size effect  i n c l e a r wood has been d i s c u s s e d  by the  author elsewhere (Buchanan 1983). 2.3.2 A x i a l Tension a. The  Strength  of Timber  E f f e c t s of Defects  a x i a l t e n s i o n s t r e n g t h of  little  attention  until  modulus of rupture strength. after  was  timber  recently, a  with in  the  conservative  defects  received  belief  that the  estimate  of  Renewed i n t e r e s t i n a x i a l t e n s i o n s t r e n g t h  in-grade  tension  tension occurred  t e s t i n g produced s t r e s s e s at f a i l u r e  much l e s s than the modulus of rupture,  as d e s c r i b e d above f o r  c l e a r wood. The  e f f e c t s of d e f e c t s have been i n v e s t i g a t e d by s e v e r a l  investigators.  Zehrt(1962) i n v e s t i g a t e d the e f f e c t of s l o p i n g  g r a i n on small c l e a r specimens and 38x89mm members that  the  strength  depended not only on the r e l a t i v e  s t r e n g t h s p a r a l l e l t o and p e r p e n d i c u l a r on the shear s t r e n g t h . an  and  found tension  to the g r a i n , but a l s o  Quite good agreement was obtained  i n t e r a c t i o n formula developed by N o r r i s ( l 9 5 5 )  under combined normal and shear s t r e s s e s .  for  with  strength  25  Dawe(l964)  investigated  the  effect  of  knots  on  the  t e n s i o n s t r e n g t h of European redwood boards and found a strong c o r r e l a t i o n between s t r e n g t h and knot s i z e . t e s t s on f u l l 1960's.  The f i r s t  s i z e timber members were c a r r i e d out i n the mid  Nemeth(l965)  reported  some  correlation  t e n s i l e s t r e n g t h and modulus of e l a s t i c i t y , correlation  between  Littleford(1967) and  reported  general  and  strength  density.  deviations  demonstrated  that  the  i n d i c a t o r s of t e n s i o n A general  were  irregularities  the  predominant  In a l a t e r  report,  strength  ratios  ASTM  knots,  attempts  characteristics  McGowan(1971) are  not good  such  edge  knots  more  a r e s u l t confirmed by Kunesh and Johnson  have as  (1975).  been  knot  made  size,  slope of g r a i n to p r e d i c t t e n s i o n Lyon  factors  f i n d i n g from these s t u d i e s was that l a r g e knots  (1972) and by Johnson and Kunesh Several  and  strength.  reduced s t r e n g t h more than small knots, and centre  Both  confirmed these f i n d i n g s  that knots, l o c a l l i z e d g r a i n  grain  between  but no s i g n i f i c a n t  and  McGowan(1968)  a f f e c t i n g t e n s i l e strength.  than  tension  to  combine  various  f l e x u r a l s t i f f n e s s and  strength.  Schneiwind  (1971),. Gerhards (1972) and Heimeshoff and Glos  and  (1980) a l l  r e p o r t m u l t i p l e c o r r e l a t i o n c o e f f i c i e n t s l a r g e r than 0.80, f o r southern  pine,  Californian  redwood  and  r e s p e c t i v e l y , but these' m u l t i p l e c o r r e l a t i o n not  y e t been i n c o r p o r a t e d  i n t o grading  Glos compare t h e i r r e s u l t s with those mentioned here.  ten  rules.  other  German  techniques  spruce have  Heimeshoff and  studies  including  26  Orosz(l975)  used  matrix  the e f f e c t s of d i f f e r e n t tension  strength,  s t r u c t u r a l a n a l y s i s to p r e d i c t  sizes  and  knots and  in-grade b.  s t r e s s e s i n design  tension t e s t s , at l e a s t  described  l a r g e in-grade  by Madsen and N i e l s e n  25,000 boards i n t e n s i o n . and  produce an accurate regression  member. 1980)  grades,  10%  of  with  approximately  The  have  in Canada.  current  of  each  of the  For  and  consistent  the  failure times  sizes,  in  order  load to  p e r c e n t i l e strength. s i z e at the results  for  fifth all  s t r e s s in a 38x89mm that  Canadian a l l o w a b l e  in  a  38x286mm  s t r e s s e s in t e n s i o n  study.  Hem-fir  material  pronounced s i z e e f f e c t .  fifth  than  with a proof  sample,  s t r e n g t h versus  r e s u l t s are r e p o r t e d  Douglas-fir  t e s t e d more  were loaded  1.25  are based on t h i s  Similar  defects.  still  codes  These boards were of s e v e r a l  estimate  analysis  and  member being  for  for  based on the r e s u l t s of  (1978b), who  p e r c e n t i l e l e v e l produced q u i t e  (CSA  that  t e n s i o n t e s t i n g programme are  s p e c i e s groups, and  that broke approximately  species  by  Size E f f e c t s  R e s u l t s of a very  grades  showed  on  results.  r e l i a b l e as they are now  axial  induced  f a c t o r s a f f e c t i n g t e n s i o n s t r e n g t h are  not w e l l understood, a l l o w a b l e more  He  knots  t e s t machines with pinned ends  f i x e d ends w i l l produce d i f f e r e n t  become  of  moments  specimen.  short l e n g t h s ,  Although the  A  locations  considering  e c c e n t r i c i t i e s w i t h i n the t e s t large  and  with None of  in  by Johnson and structural  knots these  Kunesh (1975) sizes  without  they found an even more studies  investigated  27  the e f f e c t of member length on s t r e n g t h . 2.4 AXIAL COMPRESSION  STRENGTH  2.4.1 A x i a l Compression The strength wood  load  Strength  capacity  of Clear Wood  of short wood columns depends on the  of wood i n compression p a r a l l e l to the  is  loaded  in  compression  parallel  to  e x h i b i t s l i n e a r s t r e s s - s t r a i n behaviour up to limit  at  approximately two-thirds  Beyond the p r o p o r t i o n a l leading load.  to  a  limit  ductile  C h a r a c t e r i s t i c compression  buckling  the  a  When  grain i t  proportional  of the u l t i m a t e  the s t i f f n e s s  crushing  grain.  drops  strength. gradually,  type of f a i l u r e at u l t i m a t e wrinkles  caused  by  of the wood f i b r e s become v i s i b l e as u l t i m a t e  local load i s  approached. The  first  scientific  t e s t s to determine the s t r e n g t h of  wood i n compression p a r a l l e l to the g r a i n were c a r r i e d out 25mm  cubes.  on s e v e r a l  Tredgold(1853) r e p o r t s the r e s u l t s of such t e s t s  species.  The c u r r e n t grain  standard  (ASTM 1981a)  51x51mm,  203mm  uses  long,  t e s t f o r compression p a r a l l e l to the a  and  clear the  programmes have been p u b l i s h e d 1977).  on  straight-grained  specimen  r e s u l t s of e x t e n s i v e  testing  (Wood  Handbook  1974,  Jessome  28  2.4.2  A x i a l Compression Strength The  compression  g e n e r a l l y l e s s than method  of  strength that  assessing  been to apply  "strength  in-grade  lateral  testing  States  estimate  have been  The  commercial  and  Several  associated i s not  visible  defect.  with  For  has  to  carry  method short  under  specimen it  is  development  in  the  c o n t a i n i n g the worst often  difficult  to  strength.  s i z e s and  s p e c i e s of Canadian timber  in-grade  testing  1978),  (Madsen  in and  long  lengths  Nielsen  1978b,  reported.  Failure  correlated 38mm  with  size  or  t h i c k m a t e r i a l , there  type  but of  i s a trend  compression s t r e n g t h as width i s i n c r e a s e d , i s l e s s pronounced than found i n  bending  tension.  Compression  strength  of  commercial  s i g n i f i c a n t l y with changes i n moisture content, t e n s i o n s t r e n g t h which i s e s s e n t i a l l y content  traditional  a l a r g e number of d i f f e r e n t d e f e c t s ,  strongly  is  buckling.  i n t e r e s t i n g f i n d i n g s are  this size effect  or i n a x i a l  to  Abbott  towards d e c r e a s i n g  defects  s t r u c t u r a l s i z e s (ASTM 1981b) with  in each board, but  strength  but  in  a  subjected  (Littleford  is  wood.  which d e f e c t w i l l have the lowest  Several  I978d).  with  A more e f f e c t i v e method i s  test  uses  v i s i b l e defect  clear  timber  r a t i o s " to the s t r e n g t h of small c l e a r  r e s t r a i n t s to prevent  An a l t e r n a t i v e United  of  of  the e f f e c t of knots or s l o p i n g g r a i n  specimens (ASTM 1981c). out  of Timber  (Madsen 1982).  timber"  varies  in contrast  independent of  to  moisture  29  2.4.3  Stress-Strain The  shape  compression thesis.  of  the  much  ASTM  test  test  is  available.  compression  in  in this  Although  parallel  the  to the g r a i n  curve,  this  has  only been used to e s t a b l i s h the e l a s t i c modulus and  regarding  the  stress.  complete  curve  is  range.  studies of the  S e c t i o n 2.2  have  The  stress-strain  showed  simple  been  how  the  behaviour,  elasto-plastic  shown i n f i g u r e 2 has been used q u i t e  recently,  stress-strain  form  studies.  approximation More  few  r e q u i r e d to understand bending  and r e f e r e n c e d s e v e r a l bilinear  Very  mathematical  curve beyond the e l a s t i c  often.  relationship  of a load-deformation  the p r o p o r t i o n a l l i m i t made  data  for  s p e c i f i e s the drawing generally  stress-strain  i s important input f o r the model developed  Not  standard  Relationship  Malhotra  equation,  and  Mazur  (1970)  used  a  p r e v i o u s l y proposed by Y l i n e n ( 1 9 5 6 ) ,  given by  e  = i  where e i s s t r a i n , stress, depending this  E  is  [ c f  (1-c)  -  f  c  l n [ l  -  f i s s t r e s s , f c i s the maximum  modulus  of  elasticity,  on the shape of the curve.  equation,  as  shown  in  stress for large  Malhotra  and  Mazur  )  compression  and c i s a  The curve  figure  e l a s t i c modulus at the o r i g i n , and compressive  (2.T  f - ) ]  parameter  described  by  7(a) i s tangent to the  tangent  to  the  ultimate  strain.  describe  approximation to the r e s u l t s of 144  the tests  curve as a very good of  clear  eastern  (a) Figure  (b)  (c)  7 - Stress-strain relationships in axial compression  spruce  wood  at  various  moisture  contents.  c o n s i d e r a t i o n to the p o s s i b l e shape of the  They  give no  at  strains  curve  beyond the u l t i m a t e l o a d . 0'Halloran(1973) (1971) to propose a strain  curve  used  the  data  mathematical  for clear  dry  equation  =  Goodman for  Ee - Ae  The proposed  constants  determined,  given set of experimental data. is  found  to  stress-  at v a r i o u s equation i s (2.2)  n  where f i s s t r e s s , E i s modulus of e l a s t i c i t y , equation  and Bodig  the  wood i n compression  g r a i n angles and g r a i n o r i e n t a t i o n s . f  of  by  fitting  A  and  stress  r , of the e q u i v a l e n t s t r a i n  under e l a s t i c c o n d i t i o n s , the parameters A and n can be from  are  the equation to a  I f the s t r a i n at peak  be a c e r t a i n r a t i o ,  n  found  31  n = r/(r-1)  and  (2.3)  E  A =  n(rf /E)  n-1  c  where f  (2.4)  i s the maximum s t r e s s .  c  A t y p i c a l p l o t of a f i t t e d curve and experimental r e s u l t s is  shown  i n figure 7(b).  The equation cannot be used beyond  maximum s t r e s s , because i t drops r a p i d l y  to  values.  i s not  O'Halloran  limitation,  failing  claims  that  to recognize  this  negative  stress  a  serious  that the shape of the f a l l i n g  branch of the curve i s needed  to  strength.  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  It  i s true  that  predict  ultimate  beyond maximum load cannot be q u a n t i f i e d e a s i l y  bending  i n an  axial  compression t e s t , because i t i s l a r g e l y a f u n c t i o n of the t e s t machine  characteristics  and  the  rate  of  loading,  but a  d e s c r i p t i o n of s t r e s s - s t r a i n behaviour beyond u l t i m a t e  load i s  e s s e n t i a l to the development of an u l t i m a t e  bending  strength  theory. A  simple  been i l l u s t r a t e d supporting  bilinear in  proposal  figure  6.  by Bazan(!980) has already Bazan  assumed  without  any  argument that the slope of the f a l l i n g branch i s a  v a r i a b l e which can be a r b i t r a r i l y  taken as  that  value  which  produces maximum bending moment f o r any n e u t r a l a x i s depth. different the  assumption  used i n t h i s study i s that the slope of  f a l l i n g branch i s a m a t e r i a l property,  estimated  A  whose value  can be  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 .  32  A comprehensive of  study on the  timber with d e f e c t s ,  has been made experimental  i n compression  G1OS(1978).  by  stress-strain  On  p a r a l l e l to the g r a i n ,  the  basis  by  Bechtel  and N o r r i s  reported  extensive  for  clear  (1952), Moe (1961), and o t h e r s .  The curve i s c h a r a c t e r i s e d by four parameters figure.  of  t e s t i n g Glos proposes a curve of the shape shown  in f i g u r e 7 ( c ) , which i s s i m i l a r to that wood  relationship  as shown i n the  The equation of the curve i s given by f  e/e, +  =  7  ±  G  + G (e/  2  3  f G1  where  GAe/e.)  e ; L  s  e ; L  )  (2.5)  7  c  G3  =  1/f - 7/6E c  G  =  4  4  a  1/E  2  ±  + G (e/  6E(l-f /f ) -  G  ± )  G l  /f  s  where f i s s t r e s s , e i s s t r a i n , E i s modulus of e l a s t i c i t y , f c is  maximum  compression  compressive  stress,  fs  is  the  asymptotic  s t r e s s f o r l a r g e s t r a i n and e, i s  the  strain  at  the maximum s t r e s s , as shown i n f i g u r e 7 ( c ) . Glos  has  shape of the density,  estimated  curve  from  moisture  compression  wood.  the  four  four  content,  parameters  measurable knot  Multiple curvilinear  to d e f i n e the  wood  ratio  properties;  and  percentage  regression  techniques  have been used t o express expected values of each parameter i n terms  of  equations. from  Glos's  the  four  properties,  using  lengthy  The general form of s t r e s s - s t r a i n curve equation  has  been  investigated  regression resulting  in this  study,  33  without attempting  to v e r i f y  the dependence of the equation on  the wood p r o p e r t i e s mentioned above. 2.4.4 Column Theory f o r C o n c e n t r i c The loading  Loading  l o a d c a p a c i t y of short columns with c o n c e n t r i c is directly  material.  As  failure  based  r e l a t e d t o the compression s t r e n g t h of the  the  i n c r e a s e s , there  length  is a on  of a c o n c e n t r i c a l l y  transition  v  from  a  loaded  crushing  type  is  of  long  p r o p o r t i o n a l to the e l a s t i c modulus and i n v e r s e l y  p r o p o r t i o n a l to the square of the l e n g t h , as d e s c r i b e d well  of  instablility.  l i n e a r e l a s t i c m a t e r i a l s , the load c a p a c i t y  columns  column  compression s t r e n g t h to a b u c k l i n g type of  f a i l u r e based on l a t e r a l For  axial  known  column  formula  published  by  Euler  by  the  in  1744  (Timoshenko 1953) P  where Pe i s the a x i a l for  a  column  of  e  = »  2  L  E I  (2.6)  2  load c a p a c i t y  length  L  (or E u l e r  pinned  buckling  at both ends.  load)  E i s the  modulus of e l a s t i c i t y . For  materials  relationships, non-linearities difficult.  with  For  stress-strain  the combination of both m a t e r i a l and geometric makes  Closed  form  a v a i l a b l e , but numerical necessary  non-linear  analytical  s o l u t i o n s f o r very and  other  i n most r e a l s i t u a t i o n s a l l materials,  treatment  much  simple  approximate  (Chen and Atsuta  more  cases are  methods  are  1976a).  the t r a n s i t i o n between short column  34  behaviour and much  long column behaviour has been  debate,  and  there  have  been  a  proposed to d e s c r i b e t h i s behaviour.  the  great  Test  subject  of  many formulae  results  in  this  range tend to have f a r more s c a t t e r than for long or f o r short columns. The  earliest  formulae  were e m p i r i c a l expressions  were only u s e f u l in the l i m i t e d range where they results.  with  E u l e r curve  for long columns.  secant  formula 1898, and  crushing  formula  discussed  and  Rankine  fall  including  formula,  which  i n t o t h i s category.  The  1952)  elasticity  was  the  attributed acceptance  from a c u r v i l i n e a r  to  of  Engesser the  Rankine  with the  of l o a d i n g path but  tangent  of more accurate  the  modulus  regardless  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 -  columns, which i n c l u d e of  1889  stress-strain relationship.  r e l a t i o n s h i p . Chen and A t s u t a  development  in  tangent modulus  T h i s method p r e d i c t s a safe lower bound on s t r e n g t h  portions  be  f a i l u r e s t r e s s (Timoshenko 1953).  approach which uses E u l e r ' s formula  strain  will  the  developed by Gordon using an assumed d e f l e c t e d shape  A f u r t h e r development  of  formulae  the  Rankine-Gordon formula), p u b l i s h e d by Rankine in  a limiting  (Bleich  Many  test  results  s t r e n g t h f o r short columns and  the  i n Chapter 9, (or  was  fitted  Subsequent formulae were improved to produce  consistent  which  cross  (1976a)  summarize  t h e o r i e s for p l a s t i c  allowance section  s t r e s s e d beyond the p r o p o r t i o n a l  for  elastic  that  have  limit.  the  b u c k l i n g of  unloading  of  been p r e v i o u s l y  35  2.4.5  Timber Columns According  scientific in  1729,  to  Timoshenko(1953),  the  first  recorded  t e s t s on long timber columns were by Musschenbrock  who  found that the b u c k l i n g  proportional  to  theoretically  the  by  square  Euler  in  of  strength  the  was  inversely  length, a r e s u l t  1744.  Booth(l964)  proved  describes  compression t e s t s of many l a r g e members performed by G i r a r d in 1798,  where  the  ever-present  and  end  restraint  duration  v e r i f i c a t i o n of E u l e r ' s The be  first  those  varying  Bryson  l e n g t h cut  Bryson obtained Hodgkinson, failure  did  not  allow  defects,  load  satisfactory  wood column t e s t s i n America appear to (1866) who  from one  t e s t e d 40 small specimens of  p i e c e of dry white  good agreement with  noting  of  theory.  recorded  of  problems  the  two  transition  i n short columns to a b u c k l i n g  pine  timber.  formulae a t t r i b u t e d to  from  a crushing  type of  type of f a i l u r e  in long  columns. Current the  formulae f o r timber columns appear to  work of Newlin and  Trayer  (1925) who  date  from  c a r r i e d out d e t a i l e d  a n a l y s i s of a l a r g e number of c l e a r S i t k a spruce columns. a x i a l compression l o a d i n g they r e l a t e d the the  stress-strain  lengths reached  of columns. with  all  curve For  in  compression f o r three  long columns, the  s t r e s s e s in the e l a s t i c  a c c u r a t e l y by the E u l e r formula. occurred the  grain.  by  crushing  buckling  For  short  maximum  load  For to  different load  was  range as p r e d i c t e d columns,  failure  of the wood i n compression p a r a l l e l  to  36  crushing strength  •d O O  4 th power  parabola  short  Euler  intermediate  curve  long slenderness  Figure For  intermediate  suggested based  8 - A x i a l l o a d - s l e n d e r n e s s curve f o r c o n c e n t r i c a l l y loaded columns  a  length  transition  on  a  they  modulus  obtained  tangent t o the E u l e r  Newlin  and  curve  a  at  approach. parabolic  two-thirds  Using  of  the  between  proportional  compressive  stress.  The e i g h t h power curve i n i t i a l l y  for  spruce  found the power of the parabola to depend on  was  i n c l u d e other s p e c i e s . verified  by  Newlin  changed This  and  this  limit  to  fourth  Gahagan  l a r g e s i z e columns of s e v e r a l remained  curve  the c r u s h i n g  They  Sitka  some  transition  strength. ratio  Trayer  between these two types of behaviour  tangent  approximations  columns  i n many North American  a  stress  and  maximum proposed  f o u r t h power curve to  power  curve  was  later  (1930) a f t e r t e s t s on many  grades  and  species,  Codes ever s i n c e .  r e l a t i o n s h i p i s shown'in F i g u r e 8.  and has  The form of  The formulae w i l l be  given i n Chapter 8. In B r i t a i n , a c c o r d i n g t o Sunley(1955)  and  Burgess(1977),  37  the of  development  R o b e r t s o n ( 1 9 2 5 ) who  Perry  to explain  spruce  initial  material initial the  The  t h e column  deviation  limiting  variability eccentricty.  many c o u n t r i e s ,  the  Robertson In  The  clear  to f a i l u r e ,  The  failure  effects  column  (CIB  Sitka  formula  but has  an  criterion  is  of  defects  by v a r y i n g  and  t h e assumed  the  basis  of  f o r m u l a e from t h e c o d e s of  the background  1980).  to the formula i n  Burgess(1976)  is essentially  M a l h o t r a and Mazur formula  slenderness,  and  stress-strain  i t has  has  shown  t h e same a s t h e  Perry-  and  has  modified  good agreement  Several  (1970) have  re-examined  "Euler-Engesser" axial  loading.  formula,  the and  T h i s method  has  a c o n t i n u o u s f o r m u l a f o r a l l v a l u e s of a  sound  theoretical  c u r v e i s a l w a y s o f t h e assumed  Neubauer(1973)  slenderness  or  for concentric  a d v a n t a g e of b e i n g  detail  on s m a l l  and  formula.  i t s use  formula,  Ayrton  1980).  compared  code  modulus  propose  by  t h e work  "Perry-Robertson"  c a n be c o n s i d e r e d  formula  Canada,  tangent  the  CIB  article  remains e l a s t i c  and d e s c r i b e d  international  that  resulting  largely  T h i s a p p r o a c h has become  (BSI  code  1886  of h i s t e s t s  compression s t r e s s .  British  f o r m u l a e was  from s t r a i g h t n e s s .  Larsen(l973)  an  u s e d an.  the r e s u l t s  columns.  assumes t h a t  a  of wood column  basis  t o wood column  test  results  the  form.  reviewed the background i t to a simple cubic  if  t o the Rankine  form w h i c h  f o r a wide  gives  r a n g e of  ratios. of t h e s e column  i n Chapter  8.  formulae w i l l  be compared  i n more  38  2.5 COMBINED BENDING AND AXIAL LOAD Combined bending and a x i a l load w i l l be d i s c u s s e d sections.  The  s e c t i o n subjected second  first  i s the  to combined bending and a x i a l  loads,  Cross To  including  the  effects  The  of  c a l c u l a t e the u l t i m a t e s t r e n g t h of a member  to  know  the nature  and  f l e x u r a l s t r e n g t h at a c r o s s be  member  S e c t i o n Behaviour  combined a x i a l and f l e x u r a l l o a d i n g i t i s  can  loading.  i  deformations.  to  f o r a cross  i s the behaviour of members of any l e n g t h subjected to  a x i a l and bending  2.5.1  failure criterion  i n two  presented  subjected  first  necessary  of the i n t e r a c t i o n between a x i a l  graphically  section. as  diagram which shows combinations of  This  strength  interaction  an  ultimate  axial  load  interaction and  bending  moment that a c r o s s s e c t i o n can r e s i s t . The  traditional  elastic  material  criterion. interaction  approach  with  a  has  maximum  been normal  to assume a l i n e a r stress  These assumptions produce a s t r a i g h t diagram,  as  shown  failure  line  by the s o l i d l i n e s  ultimate in figure  9(a)  f o r m a t e r i a l with weak t e n s i o n s t r e n g t h , and f i g u r e  for  strong  tension  strength,  assuming constant  9(b)  compression  strength. Most design even  more  dotted If  formulae make the l i n e a r  elastic  assumption  c o n s e r v a t i v e by s p e c i f y i n g design a c c o r d i n g  to the  l i n e s i n f i g u r e 9. wood i s assumed to  have  a  non-linear  stress-strain  39  moment  (a)  weak  in  tension  Figure  relationship  9  in  (b) strong - Ultimate for linear  compression  plastic  r e l a t i o n s h i p ) the  curved  on  the  illustrated This  figure  type  of  Gurfinkel(1973) received the  very  curve  straight reasonable  line  ( f o r example,  ultimate  a  bilinear  elasto-  diagram  becomes  interaction  side,  interaction by  Larsen  following  curve  and  the  high  the  tension  interaction  approximation  in  which  been  shape  described  (1981),  verification.  horizontal on  has  Riberholt  experimental  the  depends For  diagrams  10.  little  above  compression) strengths.  and  tension  interaction behaviour  compression  in  in  axis  ratio  of  the has  (for  tension  strength  The 1  to  (outer  compression  been  used  in  but  by has  shape  net  of  axial  compression curve), region many  a is  a  column  40  c o  Figure formulae,  but  10 - U l t i m a t e i n t e r a c t i o n diagrams for n o n - l i n e a r behaviour this  becomes  less  t e n s i o n t o compression s t r e n g t h Newlin(l940) deviation  carried  from a s t r a i g h t  out line  accurate  as the r a t i o of  decreases. testing  which  showed  some  f o r small c l e a r specimens.  suggested a p a r a b o l i c i n t e r a c t i o n equation  He  of the form  c  r  where M i s the bending moment, S i s the s e c t i o n modulus, f r i s the modulus of rupture, P i s the a x i a l area  load, A i s the  section  and f c i s the compresssive s t r e s s at f a i l u r e f o r a short  column.  As d e s c r i b e d  dropped  the  by  Wood(l950),  Newlin  conservatively  exponent of 2 f o r short columns, but f o r slender  41  columns he used the p a r a b o l i c r e l a t i o n s h i p bending load.  strength The  as the l i m i t i n g  possibility  of  a  to  justify  using  s t r e s s , r e g a r d l e s s of a x i a l  tension  failure  was  c o n s i d e r e d , so he may not have r e a l i z e d how h i s r e s u l t s  a c t u a l shape of the curves i n the compression  a l s o depends on the nature of the for  fitted  r e l a t i o n s h i p shown i n f i g u r e 10.  the more complete The  wood  in  not  compression,  stress-strain  region  relationship  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 e x p l o r e d i n Chapter 6. In the t e n s i o n r e g i o n , f i g u r e s straight failure has  line,  based  on  the  9  same  i n bending or i n t e n s i o n .  and  10  both  tension  stress  As  discussed  transition behaviour  between can  i s considerably less  when  earlier,  as  similar  curves  Bradley(1981)  tension  shown  by  were in  suggests the  dotted  obtained  an a n a l y t i c a l  members with d i f f e r e n t  a  strength  the  maximum  and  lines  in  the  in figure  independently  by  The  bending  f r a c t u r e theory curve  it  stress in  i n bending.  behaviour  be e x p l a i n e d by b r i t t l e  and Buchanan 1982) which region,  than  a s i m i l a r member f a i l s axial  a  extreme .fibre s t r e s s at  q u i t e r e c e n t l y been determined that the f a i l u r e  an a x i a l t e n s i o n t e s t  show  (Johns tension  10.  Very  Kersken-  study of glued laminated wood properties  over  the  cross  section. A  more d e t a i l e d a n a l y s i s of the shape of the i n t e r a c t i o n  diagram w i l l be d e s c r i b e d i n Chapter  6.  42  2.5.2  Members with Combined Bending and For c o n c e n t r i c a l l y loaded  Compression  columns, i t has  been shown that  l o a d c a r r y i n g c a p a c i t y depends on m a t e r i a l s t r e n g t h columns,  and  on  slenderness  columns are a l s o  subjected  for  short  r a t i o f o r long columns.  to  bending  moment  the  If the problem  becomes more complex because the load c a p a c i t y a l s o depends on the bending moment. A conceptual  method of c o n s i d e r i n g these e f f e c t s i s shown  in f i g u r e 11, which i s a three-dimensional slenderness  vs.  member of given of  moment slenderness  and  r a t i o can  a x i a l load and moment i n s i d e the  on the load vs. in  (Johns  figure  8.  a x i a l load vs. figure  slenderness The  Buchanan 1982). resist  any  and  axial  11)  has  curve on the- load vs.  The  curve  relationship  shown  moment plane i s the  moment i n t e r a c t i o n for short columns shown  behaviour loading  of  (or the form of the s u r f a c e shown in  r e c e i v e d very  little  attention.  range  (Timoshenko  and  Gere 1961).  linear  range, Chen  and  Atsuta  s t u d i e s on other m a t e r i a l s are  directly  the  linear  compression, t e n s i o n s t r e n g t h size  elastic  For behaviour beyond  (1976a)  summarize  such as s t e e l and  applicable  figure  Standard methods are  to timber.  the  extensive  concrete,  none of  Differences  other m a t e r i a l s are that wood behaves d i f f e r e n t l y  the  in  timber members under combined bending  a v a i l a b l e as long as the wood remains in  and  combination  surface shown.  plane i s the  A  10.  The  which  sketch of load vs.  in  with  tension  i s s i g n i f i c a n t l y a f f e c t e d by  of the s t r e s s e d volume, and  the m a t e r i a l  properties  43  Axial  load  BENDING STRENGTH  Figure  11 - Three-dimensional slenderness v s .  sketch of l o a d v s . moment  may vary c o n s i d e r a b l y along the l e n g t h of a member. developed  i n t h i s t h e s i s addresses  A comprehensive experimental  The model  these problems. study of  small  clear  wood  members s u b j e c t e d to combined column and beam a c t i o n by Newlin and  Trayer  (1925) has already been mentioned.  Small d e f e c t -  f r e e members were subjected to e c c e n t r i c a x i a l l o a d s ,  and i n  44  another load  test,  concentric  together  in  concentrically combined design  bending  Subsequently  columns  and  without  axial  Woodd950),  and  the  effects  axial  and  moment  load  assumptions behaviour failure  of to  In  a  term  for  The  formula  i n i t i a l  large  combined  the  formula,  and  because rationally  is  to  a  the  load,  the  and  set  of  theory.  for  combined  explained  (NFPA that due  by  1982).  it  considers  to  eccentric  It  deflected  shape,  linear  elastic  compression  stress  limiting a l l  to  of  which  further  concentric  in  is  need  to  be  Chapter  8.  formula  and  loading,  both  which  provide  a  based  can  the  also  design  on  re-  CIB  include be  used  formula  for  load.  describes  i n i t i a l  a  For  loads.  discussed  axial  for  described.  of  moment  "  transverse  a  for  theory  formula  in  lateral  provided  U.S.  out-of-straightness,  discourages  the  they  Perry-Robertson  above  and  Johnston(1976)  in  above,  eccentricities  bending  briefly  useful  and  described  described  for  to  Their  amplified  axial  due  failure,  Europe,  formula,  is  of  transverse  development  use  sinusoidal  criterion  evaluated.  for  and  loading  later  formula  separately  been  developed  loading,  NFPA  has  much  adopted  load ratio.  axial  Newlin(l940)  and  The  constant  loaded  curves  bending  a  axial  the  other use  yield  materials  of  applications expressions  criterion  with  of of  secant  this  cannot  non-linear  the  be  type  applied  stress-strain  relat ionships. Malhotra(1982) underestimated  the  found  that  strength  of  the  Perry-Robertson  timber  columns  formula  tested  in  45  compression agreement  with small end e c c e n t r i c i t i e s . with  two  other  formula, m o d i f i e d to allow bending  than  formulae.  One  f o r higher  i n compression,  eccentricities  material. and  end  the  assumption  compression,  of  the  secant  stresses in  and the other a formula d e r i v e d  For  wood  restraints,  obtained good agreement with t e s t on  was  failure  by Jezek who c o n s i d e r e d the s t a b i l i t y of elasto-plastic  He obtained b e t t e r  a  column  columns  of  with  Hammond  et  ideal  various al.(l970)  r e s u l t s using a theory  bilinear  elasto-plastic  based  behaviour i n  but they d i d not c o n s i d e r s t r e s s e d volume  effects  in the t e n s i o n r e g i o n , nor d i d they propose a r e a l i s t i c  design  method. Larsen and T h i e l g a a r d (1979) have developed a  general  theory  for  includes b i a x i a l e f f e c t s limitations  of  e l a s t i c behaviour  this  and a  plane behaviour, calibration effects  simple  buckling.  The  are the assumptions of l i n e a r failure  criterion  based  on  stresses.  has using  used  an energy  the  and v e r i f i c a t i o n .  in a similar  verified  loaded timber columns that  and l a t e r a l t o r s i o n a l  approach  l i m i t i n g compression Bleau(l984)  laterally  and  same  approach to e x p l a i n i n -  data  as  this  study  for  H i s theory, i n c o r p o r a t i n g s i z e  way to t h i s t h e s i s ,  reasonable p r e d i c t i o n of experimental  i s able  results.  to  give  a  46  2.5.3  In-Grade  Testing  The'only in-grade t e s t s of commercial grades and s i z e s of timber  under combined  a x i a l compression and bending appear to  be those by Zahn(l982) and Malhotra(1982). Zahn t e s t e d four groups of 1500mm long hemlock  with  various  c u r v a t u r e s over a resulting  initial  450mm  long  moment-curvature  38x140mm  eccentricities. gauge  length,  curves t o p r e d i c t  western  He measured and  used  the behaviour of  longer members, using a Monte C a r l o approach to combine long  segments with r e p r e s e n t a t i v e p r o p e r t i e s .  was checked with computer  simulation  properties,  but  accurately. difficulty  tests  Two  on  30  gave  a  d i d not  and  good  predict  disadvantages  of e x p e r i m e n t a l l y  relationship  boards  the  of  of  long.  indication  obtaining  expense  mm  of  full  Zahn's  450mm  The p r e d i c t i o n  2400  the  the  The  average  distribution  method  are the  the  moment-curvature  carrying  out Monte C a r l o  simula't ions .  and  Malhotra  (1982) reported a t e s t i n g programme  64x89mm  members  of  No. 1  e c c e n t r i c i t i e s up to 20mm.  grade  on  eastern  Reasonable agreement  38x89mm  spruce, with was  obtained  with formulae r e f e r r e d to i n the p r e v i o u s s e c t i o n . 2.5.4  Members with Combined Bending and Tension The  strength  of  timber  members  under  t e n s i o n and bending has r e c e i v e d very l i t t l e  combined  attention.  Senft and Suddarth (1970) t e s t e d 38x89mm members of grade  commercial  southern  axial  high  p i n e , and S e n f t ( l 9 7 3 ) c a r r i e d out  s i m i l a r t e s t s on a lower grade of western hemlock.  For  both  47  s e r i e s of t e s t s tension The  load  members  as bending  results  straight  the  indicated  line  be  loaded  deviation  from  i n t e r a c t i o n between bending  expected  T h e i r proposed  f o r the  from  higher  the  Suddarth,  a  constant  the  traditional  and a x i a l  quality  l o a d , the  material,  sketched curves i n f i g u r e  design equation  with t h e i r t e s t  with  s t r e s s e s were i n c r e a s e d to f a i l u r e . a  d e v i a t i o n being l a r g e r would  were  i s very  conservative  as 10.  compared  results. Woeste  and G a l l i g a n  (1978) have used  the data  obtained i n the above t e s t s as an example f o r c a l c u l a t i n g reliability  of  wood  Their contribution step  towards  members s u b j e c t e d to combined l o a d i n g .  i s not a design method,  the  the  development  but  of r e l i a b i l i t y  is  rather  a  design of wood  structures. Burgess(1980) used behaviour  and  the  with  previously  of  linear  to c a l c u l a t e the s t r e n g t h  lateral  derived  loads.  for  wood  He  found  beam-columns  of  that  used  for  tension  d e f l e c t i o n s due  to l a t e r a l  tension  forces.  accurate  in  behaviour behaviour do  not  is  members,  His  tension  loads  linear than  in  with  include  Burgess used any  size  initial slightly  i n c o r p o r a t i n g the f a c t that will  be  elastic  reduced  by  assumption  compression  so  more  tension  compression  simple f a i l u r e c r i t e r i a  effects,  axial  is  because  g e n e r a l l y l i n e a r to f a i l u r e whereas  i s not.  tension equations  c u r v a t u r e and given slenderness r a t i o can be m o d i f i e d and  elastic  simple l i m i t i n g values of s t r e s s f o r f a i l u r e i n  t e n s i o n and compression members  assumptions  which  the theory does not  48  e x p l a i n the d i f f e r e n c e i n f a i l u r e s t r e s s e s between bending and tension  tests.  2.5.5 V a r i a b i l i t y  i n Wood P r o p e r t i e s  A l l of the column t h e o r i e s d e s c r i b e d assumed  that  the  relevant  deterministic qualities.  wood  to t h i s  point  have  properties  are  known  Newton and Ayaru (1972),  recognizing  that s t r e n g t h and s t i f f n e s s are very v a r i a b l e i n a  population  of  wood  members,  propose  using  s t r e n g t h and s t i f f n e s s and t h e i r the  strength  of  the known d i s t r i b u t i o n s of  known c o r r e l a t i o n to  wood columns.  predict  They use the Perry-Robertson  formula and assume that wood p r o p e r t i e s w i t h i n each column are constant. Suddarth and Woeste (1977) have attempted variability  i n s t i f f n e s s along  computer modelling long.  to  allow  for  the length of a long column by  a 4.4m column with  four segments each  1.1m  A slender column has been used so the l o a d c a p a c i t y i s  governed by e l a s t i c b u c k l i n g , not simulation  known  safety  distributions.  constant apparent  but  they  f o r each  Their p r i n c i p a l  i n c r e a s e s as v a r i a b i l i t y  decreases,  A  Monte  Carlo  was used to c a l c u l a t e the s t r e n g t h of one thousand  columns, the modulus of e l a s t i c i t y from  strength.  in  selected  f i n d i n g was that  stiffness  between  boards  d i d not compare v a r i a b l e s t i f f n e s s  average s t i f f n e s s along  each  whether the e x e r c i s e of using  board was a c t u a l l y  segment  board,  so  with  i t i s not  four segments f o r each  necessary.  No s t u d i e s appear t o s t r e n g t h v a r y i n g along  have  been  done  on  the l e n g t h of the board.  members  with  49  2.6  SUMMARY  This current It  chapter  briefly  reviewed  t h e development of  knowledge r e g a r d i n g t h e s t r e n g t h of  timber  members.  h a s been shown t h a t a l a r g e amount o f r e s e a r c h u s i n g s m a l l  clear  wood  specimens  understanding size  has  testing  useful  of  failed  structural  o f t i m b e r members  results.  continued  has  in  the  d i s c u s s i o n of s i z e  The  timber is  literature  first effects.  part  to  of  produce  behaviour,  starting survey the  to  a  good  but t h a t  full  produce  very  i n t h i s chapter i s next  chapter,  for  50  III .  SIZE EFFECTS  3 . 1 INTRODUCTION As  described b r i e f l y  i n the l i t e r a t u r e survey, m a t e r i a l s  such as wood e x h i b i t a s i z e e f f e c t which i s observed  i n the  f o l l o w i n g ways:  1.  Long  members  fail  at lower s t r e s s e s than  similarly  loaded short members.  2.  Bending members of a c e r t a i n depth tend  lower  stresses  than s i m i l a r l y  to  fail  at  loaded members of smaller  depth.  3.  In a x i a l t e n s i o n , members with l a r g e c r o s s  area  tend  to  fail  sectional  at lower s t r e s s e s than members with  smaller cross s e c t i o n a l area.  4.  For a  member  increases  as  decreases. bending in  These  axial  of  the  given  volume  size,  failure  modulus  of  rupture  in a  i s g e n e r a l l y g r e a t e r than the f a i l u r e  stress  tension.  s i z e e f f e c t s are widely accepted to be b r i t t l e  phenomena materials.  stress  subjected to tension stresses  For example, the  test  the  similar The  conventional b r i t t l e  to  those  first  observed  part  of  in this  fracture  other  brittle  chapter  reviews  f a i l u r e theory, and e x p l a i n s  why  timber  51  members r e q u i r e a somewhat modified  theory.  A modified  theory  i s developed and d i s c u s s e d . 3.2 CONVENTIONAL BRITTLE FRACTURE THEORY 3.2.1 H i s t o r y It  i s now beginning  wood i n t e n s i o n similar  to  is a  that  t o be recognized  statistical  observed  phenomenon  i n other  behaviour i s o f t e n e x p l a i n e d  by  sometimes c a l l e d  strength  A  theory  "statistical  that the s t r e n g t h of  "brittle  fracture  concrete.  yarns  m a t e r i a l s has been concept proposed by  and  Tucker(l927)  by Weibull(1939a,b) who v e r i f i e d h i s r e s u l t s  In brittle with will  simple solid  some fail  weakest  theory",  Major developments of the theory  many d i f f e r e n t b r i t t l e  terms, the weakest l i n k theory  i s made up of a l a r g e number of  who  were made  with  m a t e r i a l s , but a p p a r e n t l y  statistical  This  theory".  developed on the b a s i s of the weakest-link  studied  behaviour  b r i t t l e materials.  f o r the s t r e n g t h of b r i t t l e  P i e r c e ( l 9 2 6 ) who s t u d i e d cotton  with  tests  on  not wood. assumes that a  small  d i s t r i b u t i o n of s t r e n g t h .  elements The member  when the a p p l i e d s t r e s s exceeds the s t r e n g t h of the element  (as f o r example the weakest l i n k  with no s h a r i n g of load t o other subject to a size  effect  elements.  because  the  in a chain),  Such m a t e r i a l s are  larger  the  stressed  volume, the l a r g e r the p r o b a b i l i t y of that volume c o n t a i n i n g a weak  element.  m a t e r i a l s having  The  size  effect will  large v a r i a b i l i t y  of the c o n s t i t u e n t Alternatively  be more pronounced f o r  i n the s t r e n g t h  properties  elements. the  material  may  be  considered  to be a  52  homogeneous m a t e r i a l c o n t a i n i n g a l a r g e number of d e f e c t s with a statistical theory  d i s t r i b u t i o n of s i z e  (Jayatilaka,  i s the same in e i t h e r case.  1979).  The  Such m a t e r i a l i s sometimes  r e f e r r e d to as a " G r i f f i t h m a t e r i a l " . Weibull can  be  showed how  explained  exponential of  by  type,  the t e s t  the s t r e n g t h of a weakest l i n k a  cumulative  and how  distribution  system  of  the  the s t r e n g t h depends on the volume  specimen f o r uniform  or v a r y i n g  distributions  of  s t r e s s w i t h i n the specimen. Johnson(1953) Weibull's  theory,  distribution  improved  the  r e c o g n i z i n g that  proposed  theoretical the  by W e i b u l l  exponential  any  parent  parent  distribution,  then  the  distribution  exactly a Weibull d i s t r i b u t i o n ,  A p p l i c a t i o n s of B r i t t l e The  theory  first  to wood was  equations sizes.  study  to  for  of any  exact  samples  a  If the Weibull  member s t r e n g t h s i s number  Weibull's  of  elements.  brittle  that by Bohannan(1966), who the  by  of  F r a c t u r e Theory to Wood  applying  predict  in  f o r l a r g e sample s i z e .  p o p u l a t i o n of elements can be. d e s c r i b e d  distribution  3.2.2  value  for  type  i s a s y m p t o t i c a l l y the  extreme value d i s t r i b u t i o n of the s m a l l e s t from  basis  used  fracture Weibull's  s t r e n g t h of wood beams of v a r i o u s  He c a l i b r a t e d t h i s model with the average  results  of  t e s t s on three s i z e s of c l e a r dry s t r a i g h t - g r a i n e d D o u g l a s - f i r beams. Bohannan matched  by  found the  that  model  if  the  t e s t data was  strength  was  most a c c u r a t e l y  considered  to  be  53  dependent  on  length  independent of width. verified  these  geometrically depth  to  and  depth  of  the  His  main  c o n c l u s i o n was  s i m i l a r beams, the s t r e n g t h power 1/9.  is  test.  He  Bohannan's theory  d i d not  bending r e s u l t s with those depth  effects  r e p o r t any of  axial  test  that for  p r e d i c t s no for  an  to size  axial  attempt to compare h i s tension  tests.  f o r bending have been included  they have e i t h e r used in-grade  sizes  proportional  e f f e c t with v a r y i n g c r o s s s e c t i o n a l dimensions tension  specimen, but  A small s e r i e s of t e s t s on other  results.  the  test  Where  in design  codes  r e s u l t s d i r e c t l y , or  have  used the parameters d e r i v e d by Bohannan. A  comprehensive  summary  of  the development of  f r a c t u r e theory.has been made by Barrett(1974^, theory  to e x p l a i n the e f f e c t of s i z e on  tension • strength different  of  He  strength  used  test  r e s u l t s from  bending specimens reported a  the  perpendicular-to-grain  Using  B a r r e t t obtained  f a i l u r e s t r e s s vs.  the theory. the  Douglas-fir.  s i z e s of t e n s i o n and  number of authors, plot  of  who  brittle  straight  line  by a  log-log  s t r e s s e d volume, as p r e d i c t e d  used t h i s p l o t to c a l c u l a t e the parameters d i s t r i b u t i o n , but  by of  showed that i t i s not easy to  d e f i n e the parameters p r e c i s e l y . B a r r e t t and brittle to-grain  have  produced  further  studies  on  f r a c t u r e e f f e c t s in wood, i n v e s t i g a t i n g p e r p e n d i c u l a r tension  shear s t r e n g t h of glued  others  strength  (Foschi and  laminated  T h i s author  (Barrett, Barrett  1975), and  beams (Foschi and (Buchanan 1983)  Foschi  has  Barrett  and  Fox  bending  1975), strength  1980).  shown that  the  accepted  54  theory  does  not  a c c u r a t e l y e x p l a i n the r e l a t i o n s h i p between  bending s t r e n g t h and a x i a l t e n s i o n s t r e n g t h  for clear  wood.  S i z e e f f e c t s i n wood are more complex than suggested by brittle  fracture  theory,  parameters are used  to  e f f e c t s , as d e s c r i b e d  but  can  quantify  later  be e x p l a i n e d . i f  length,  in this  depth,  simple  separate  and  width  chapter.  3.2.3 Theory f o r Uniform S t r e s s D i s t r i b u t i o n For  a  relatively  assume that the parent distribution  simple  development  population  of  this  theory,  of elements has a cumulative  f u n c t i o n ( c . d . f . ) of s t r e n g t h given  by a  Weibull  distribution F ( x )  where, x  is  the  strength c a l l e d parameter"  =  1  -  r  strength,  the  t  X  x  e x p { - [ —  x  "location  0  o >i J  i s a lower l i m i t parameter",  m  with the same u n i t s as x, and k i s a  "shape parameter" which r e f l e c t s both spread  (3.1)  }  the  or minimum  is a  "scale  dimensionless  skewness  and  the  of the d i s t r i b u t i o n .  If  samples  population,  of  size  n  are  drawn  from  this  parent  i t can be shown (Bury 1975) that the s t r e n g t h of  the weakest element i n each sample has a c . d . f .  F(x)  given by  = 1 - exp{-n(^-l^) }  (3.2)  k  -1 /k where  m  has been r e p l a c e d by mn ' .  arranged t o give distribution  the  strength  at  Equation  any  3.2 can be r e -  quantile  q  i n the  55  x  q  =  x  +m i T  o  (For example q=0.5 would  {ln[^-)} 1 q  1 / k  (3.3)  1 / k  give the median  or  50th  percentile  strength).  and  Now  c o n s i d e r two members of d i f f e r e n t  n  elements.  2  combined  Equations  3.3  s i z e s c o n t a i n i n g n,  for  each  to g i v e the r a t i o of strengths of the  member two  can be  sizes  at  any q u a n t i l e q x  (n.)  x  x (n )  X  q  If  the  2  location  +m  n "  + m n  Q  1  /  {ln(  k  *  ) }  1  /  k  {M^J}  2  parameter  x  i s assumed to be zero, as i s  0  o f t e n done, the three-parameter model d e s c r i b e d above  reduces  to  greatly  a  two-parameter  model  and  equation  3.4  is  s i m p l i f i e d to  x  „ ( i)  n  n  x (n ) q  -M4 I t can  be  quantified sizes, and  seen by  2  in only  0  ^  ,  =  (-i-)  this  case  the  (3.5)  1/k  that  size  shape parameter  effects  can  k, and the r a t i o of  r e g a r d l e s s of the q u a n t i l e , or the a c t u a l values  n . 2  straight  l i n e of slope -1/k  n,  as shown i n f i g u r e 12.  be a poor assumption  form of the W e i b u l l  to use the  distribution, particularly  e x t r a p o l a t i n g to l a r g e volumes because t h i s assumption strength  of  A l o g - l o g p l o t of s t r e n g t h a g a i n s t volume becomes a  In some cases i t may parameter  be  reducing  to  zero  as the volume becomes  twowhen  implies  infinitely  56  log  stress  x(n ) 2  volume Figure  12 - T y p i c a l l o g - l o g p l o t of f a i l u r e s t r e s s v s . volume  large. 3.2.4 Theory f o r V a r i a b l e The  Stress D i s t r i b u t i o n  above development has  subjected  assumed  the  to a uniform d i s t r i b u t i o n of s t r e s s e s .  general case where s t r e s s e s vary w i t h i n can  that  member  is  In the more  a member, equation 3.2  be w r i t t e n as  F(x) = l - e x p { - i - J v  i  (v  where V i s the volume of the member, and volume  associated  with  the  dv}  (3.6)  m  V,  is a  s c a l e parameter m.  reference  For the two  parameter case, the i n t e g r a l i n t h i s equation can be evaluated f o r any non-uniform s t r e s s d i s t r i b u t i o n and the r e s u l t can be expressed as  57  (3.7)  F(x)  where Ve i s an e q u i v a l e n t  s t r e s s e d volume.  For a beam of span L with two symmetrically distance  loads,  a a p a r t , and n e u t r a l a x i s at mid-depth, the i n t e g r a l  can be e v a l u a t e d fracture  placed  over the  tension  region  (assuming  brittle  i n t e n s i o n only) t o g i v e an e q u i v a l e n t volume of  (3.8)  which  i s seen  to  be a q u i t e simple  volume V, of the member. two n  2  p r o p o r t i o n of the t o t a l  The r a t i o of s t r e n g t h s of  s i z e s can now be p r e d i c t e d using equation  3 . 5 with n, and  r e p l a c e d by Ve, and V e r e s p e c t i v e l y . 2  For an a x i a l  relative also  t e n s i o n member the e q u i v a l e n t volume  3 . 7 • i s the  equation  total  s t r e n g t h s of bending and a x i a l  be  compared  than  the  tension  members can  3 . 5 . The e q u i v a l e n t h i g h l y  using equation  in  bending  i s much  bending t e s t s than i n a x i a l  tension  bending  in  equivalent  volume  members, a t r i a l and e r r o r approach i s necessary  f i n d the value of k such that when  greater  tests.  To make a l o g - l o g p l o t of s t r e n g t h v s .  obtained  less  t o t a l volume of the member s t r e s s e d i n t e n s i o n , and  t h i s e x p l a i n s why t e n s i o n s t r e s s e s at f a i l u r e are  to  Ve i n  volume V, of the member, so the  s t r e s s e d volume of a member loaded  for  beams of  a  line  of  slope  -1/k i s  the same value of k i s used i n equation  c a l c u l a t e the s t r e s s e d volume.  3 . 8 to  58  3 . 2 . 5 C o e f f i c i e n t of V a r i a t i o n The c o e f f i c i e n t  of v a r i a t i o n of the W e i b u l l  distribution  i s given by  cv  =  C V  [rq+2/iQ - r g + i / k ) F x /m + r ( l + l / k ) 2  2  ( - > 3  o  where r  9  1975).  i s the gamma f u n c t i o n (Bury  For the two-parameter model with x = 0 , o  the c o e f f i c i e n t of  v a r i a t i o n becomes a f u n c t i o n only of the shape parameter k. simple 1973)  but  accurate  approximation  i s then given  A  (Leicester  by  cv  - -0.922 k  Because the c o e f f i c i e n t strength  tests  equation  of v a r i a t i o n  3  >  1  0  i s a f u n c t i o n only of  of members of d i f f e r e n t  the same c o e f f i c i e n t from  (  k  value  obtained  3 . 1 0 should be the same as that obtained  l o g - l o g p l o t of s t r e n g t h a g a i n s t volume. bending  k,  s i z e s should a l l have  of v a r i a t i o n , and the  apply f o r a comparison of  )  tests  in a  The same should a l s o and  axial  tension  tests. For  this  ideal  case  of  a  perfectly b r i t t l e material  f o l l o w i n g a two-parameter W e i b u l l d i s t r i b u t i o n , possible,  in  theory,  to  carry  members of only one s i z e , and other 3.10.  sizes  to  i t would  be  out only one t e s t s e r i e s on predict  the  strengths  of  on the b a s i s of a k value obtained from equation  59  Unfortunately behave to  according  the g r a i n .  author timber  n e i t h e r c l e a r wood nor timber  defects  t o t h i s model, f o r t e n s i o n s t r e s s e s p a r a l l e l  C l e a r wood has been d i s c u s s e d elsewhere by  (Buchanan will  with  1983).  A  discussion  of  the  size effects in  follow.  3.3 BRITTLE FRACTURE THEORY MODIFIED FOR TIMBER Up to t h i s p o i n t , t h i s p r e v i o u s work.  chapter  has  been  a  review  Most of the remainder of the chapter  of  describes  a new approach. The  conventional  previous  section  materials  that  Unfortunately is  brittle  is a behave  timber  f r a c t u r e theory  in  a  perfectly  1.  brittle  manner.  s t r e s s e d i n t e n s i o n p a r a l l e l t o the g r a i n  in both c l e a r wood and timber theory  i n the  very neat and u s e f u l f o r m u l a t i o n f o r  not as w e l l behaved as t h i s .  the simple  described  Observations  of s i z e  effects  with d e f e c t s show departure  i n the f o l l o w i n g ways:  For bending members, modulus of rupture decreases  length  and  from  depth  of  width i s i n c r e a s e d .  as  members are i n c r e a s e d , but not as  The e f f e c t s f o r l e n g t h and depth are  not always of the same o r d e r .  2.  For a x i a l t e n s i o n  members  the  effects  of  varying  l e n g t h and c r o s s s e c t i o n a l area are d i f f e r e n t .  3.  For  axial  compression  members,  which  fail  in a  60  r e l a t i v e l y d u c t i l e manner, s i z e e f f e c t s are observed varying in  l e n g t h and  compression  are s t i l l  4.  The  sizes  cross section.  but  significant.  k value which r e l a t e s the (from  equation  of v a r i a t i o n (equation These departures  3.4  These e f f e c t s are l e s s  than i n a x i a l t e n s i o n or i n bending,  3.5  c o n s i s t e n t with the value  discussed  for  or  strengths figure  obtained  12)  from  of  several  i s often  the  not  coefficient  3.10).  from c o n v e n t i o n a l  brittle  f r a c t u r e theory  are  below.  DIFFERENT SIZE EFFECTS IN DIFFERENT DIRECTIONS Size  effects  result  from  p r o p e r t i e s w i t h i n a, m a t e r i a l , and  variability  of  strength  are much more pronounced  in  b r i t t l e m a t e r i a l s than in d u c t i l e m a t e r i a l s . For  a  more  homogeneous  metal, the v a r i a b i l i t y same  such ' as concrete  in m a t e r i a l p r o p e r t i e s tends to be  i n a l l d i r e c t i o n s w i t h i n a member, and  Timber i s very d i f f e r e n t depends  on  additionally growing  on  height the  or the  between members. within  a  member  environmental f a c t o r s in  V a r i a b i l i t y between  members  the d i f f e r e n c e s betweeen d i f f e r e n t  depends trees  and  timber  is  other d e f e c t s up  the  sites.  V a r i a b i l i t y along related  i n that v a r i a b i l i t y  d i f f e r e n t b i o l o g i c a l and  different orientations.  on  material  to  the  of a t r e e . type and  the l e n g t h of  frequency  a  of knots and  piece  of  V a r i a b i l i t y within a cross section  depends  l o c a t i o n of d e f e c t s i n the width or depth of  61  any  board,  and  on  environmental and other f a c t o r s that may  have a f f e c t e d the p r o p e r t i e s of l a y e r s of wood  produced  from  year to y e a r . Because  different  sources, they w i l l investigated the  effects  result  from d i f f e r e n t  be c o n s i d e r e d s e p a r a t e l y here,  each  being  f o r both t e n s i o n and compression behaviour, and  results will  3.4.1  size  be used to d i s c u s s bending behaviour.  Size E f f e c t  Terminology  S i z e e f f e c t terms are d e f i n e d below. The term length e f f e c t boards of d i f f e r e n t  r e f e r s t o the phenomenon that when  l e n g t h s are t e s t e d under  similar  loading  c o n d i t i o n s , long boards tend t o be weaker than s h o r t e r boards. The  term  width e f f e c t  i s used to d e s c r i b e the e f f e c t of  c r o s s s e c t i o n width (or breadth) on the s t r e n g t h o f ' a  bending  member. The  term  "depth"  bending member. not  refers  For a x i a l l y  to the depth (or h e i g h t ) of a  loaded members the d i s t i n c t i o n i s  c l e a r , but i n g e n e r a l the l a r g e r c r o s s  will  section  dimension  be c o n s i d e r e d as depth. The  several  term  depth e f f e c t d e s c r i b e s the phenomenon that f o r  members  geometrically  of  similar  given ways,  length the  and  width,  loaded  in  s t r e s s at f a i l u r e tends to  decrease as member depth i s i n c r e a s e d . The  term  stress-distribution effect  describes  phenomenon t h a t , f o r members of given s i z e , the maximum at  f a i l u r e tends to decrease as the h i g h l y s t r e s s e d  of  the depth i s i n c r e a s e d .  the stress  proportion  62  The  depth e f f e c t 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 r e l a t e d and could perhaps be q u a n t i f i e d using the same  parameters.  However,  sometimes d i f f e r e n t ,  f o r timber  the  so these two e f f e c t s w i l l  parameters are be  considered  separately. 3.5 LENGTH EFFECT 3.5.1 Theory The  s t r e n g t h of a timber board v a r i e s along  One e f f e c t of t h i s v a r i a b i l i t y boards  is  less  than  phenomenon  occurring  throughout  a  and but  the not  i s that the  strength only  tests  reported. performed  Almost with  third-point length  no  and  a  loading. depth  results  of  Results effects  but  A x i a l tension  axial  bending  tension  tests  span-to-depth from these so  values,  this  or  lengths have been p r e v i o u s l y  a l l in-grade constant  long  l o a d i n g cases to c o n s i d e r ,  in-grade  f o r varying  of  of s h o r t e r boards,  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 .  apparently  compression  strength  f o r average  compression a r e the most simple  i t s length.  have  ratio  tests  been  of 17, and  combine  that n e i t h e r can be  both  estimated  independently. If a timber board under a x i a l chain-like  material  where  l o a d i n g i s assumed to be  the s t r e n g t h of any length  s t r e n g t h of i t s weakest l i n k , the theory be used to q u a n t i f y the l e n g t h e f f e c t .  i s the  already d e s c r i b e d can Observations from i n -  grade a x i a l t e n s i o n and compression t e s t s confirm i s u s u a l l y at a s i n g l e c r o s s s e c t i o n .  a  T h i s allows  that  failure  the weakest  63  l i n k theory  to be used f o r l e n g t h  effects  f a i l u r e may not always be a b r i t t l e The  theory  described  even  though  the  fracture.  i n equations  3.2 t o 3.5 can be used  to q u a n t i f y l e n g t h e f f e c t s and equation  3.5 can be  simplified  for design purposes t o  ^ x  where L  2  x, and x  -  2  (I ) 1  L  (  l  k, i s the length e f f e c t parameter.  different  values  assumption  distributed  along  of the  segment length  between l a r g e c l u s t e r s  adjacent A  not  study  several  strength of  then  be  being  of the wood.  of knots,  randomly  For example, clear  fast  sections  of  and the strength of one  independent  by R i b e r h o l t and Madsen of  European  used  a  defects.  Weibull  strength  a board may not always be  of  the  strength  of  material  species.  (1979) has i n v e s t i g a t e d  strength They  along  considered  between d e f e c t s to be governed by a Poisson  those  of  segments.  the d i s t r i b u t i o n  and  )  Note that k, may have  grown t r e e s of some s p e c i e s may have long  segment may  1  Effect  v a l i d , depending on the nature  wood  1  f o r t e n s i o n and compression.  3.5.2 Assumptions f o r Length The  '  are the s t r e n g t h s of members of length L, and  2  r e s p e c t i v e l y , at any p o i n t i n the d i s t r i b u t i o n  and  3  second  distribution  board  the  type  of  for  distances process,  to d e s c r i b e the s t r e n g t h at  The second d i s t r i b u t i o n  or log-normal d i s t r i b u t i o n .  a  was suggested to be  If a W e i b u l l  a  distribution  64  i s used to d e s c r i b e the s t r e n g t h length  effect  can  be  at  defects,  the  predicted  q u a n t i f i e d in e x a c t l y the same way  as  done i n t h i s study, except that i t w i l l only be a p p l i c a b l e f o r lengths  long enough to c o n t a i n a c e r t a i n number of d e f e c t s .  R i b e r h o l t and Madsen's technique has not been pursued this  study  because  visual  i n s p e c t i o n s of the boards t e s t e d  g e n e r a l l y showed a l a r g e  number  defects  unevenly  closely  spaced  of  small •knots  which  does  not c o n t a i n  and  along each board.  c h a r a c t e r i s t i c of the Canadian s p r u c e - p i n e - f i r group  in  other This i s  (SPF)  species  r e g u l a r l y spaced l a r g e d e f e c t s  which are o f t e n found in f a s t e r grown t r e e s . In view of other strength  varying  considered  u n c e r t a i n t i e s , the  in  a  to be a reasonable  Another assumption bound  on  segment  parameter W e i b u l l strength  random  is  zero,  l e n g t h , which may  leading  to  be a p p r o p r i a t e  to  be  provided  conservative  that  and  the theory  f a r beyond the range of  data.  lower  the  implies  be unreasonable for the h i g h e s t  assumption i s , i f anything,  extrapolation  each board i s  T h i s assumption  The  reasonable  wood  one.  formulation.  grade timber, but may  along  of  in t h i s development i s that the  strength  for i n f i n i t e  manner  assumption  for  twozero low  grades.  i s considered  i s not used f o r  65  3.6 DEPTH EFFECT If depth e f f e c t  i s assumed  to  be  a  brittle  fracture  phenomenon, the same theory used f o r l e n g t h ' e f f e e t can be used to q u a n t i f y depth e f f e c t , and equation 3.5 f o r depth becomes  JL  where  x, and x  d, and d  2  2  =  (-1)  are the f a i l u r e s t r e s s e s of members of depths  r e s p e c t i v e l y , and k  Depth e f f e c t s f o r commercial some  detail  stress  Chapter 2. subjected  i s the depth e f f e c t  2  For to  compression.  with axial  tension  in  t e n s i o n , but only i n a  The  increasing  in-grade  parameter.  timber have been i n v e s t i g a t e d  f o r bending and f o r a x i a l  few t e s t s f o r a x i a l failure  (3.12)  2  trend  of d e c r e a s i n g  depth has been d e s c r i b e d i n and  testing,  compression  has  been  a l l timber  of the same width  (nominal 38mm), so the e f f e c t s of v a r y i n g two c r o s s  sectional  dimensions are not known. In brittle  axial  tests,  f r a c t u r e s , so b r i t t l e  be a p p r o p r i a t e . exhibit  tension  some  In a x i a l ductility,  failures  are  usually  sudden  f r a c t u r e theory i s c o n s i d e r e d to  compression so b r i t t l e  tests,  many  failures  f r a c t u r e theory may be a  l e s s s u i t a b l e e x p l a n a t i o n f o r observed behaviour, but the same formula Most  (equation 3.12) can be used to q u a n t i f y  effect.  bending t e s t s have been c a r r i e d out at constant span-to-  depth r a t i o s i n which case i t i s length  the  and  depth  effects.  not  possible  to  separate  However, the two e f f e c t s can be  separated i f bending t e s t s are c a r r i e d out at d i f f e r e n t  span-  66  to-depth r a t i o s as d e s c r i b e d  i n 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 e f f e c t for tension  be e x p l a i n e d  using b r i t t l e  the assumptions i n v o l v e d  f r a c t u r e theory  i n that theory  s t r e s s e s can  even though some of  may not  be  satisfied  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  of each being  known  material  randomly  parameters.  The  i s perfectly brittle  s e l e c t e d from  theory  a distribution  f u r t h e r assumes that the  such that the f a i l u r e of any one  of these t i n y elements w i l l cause f a i l u r e of the member. T h i s being 3.6  can  be  the case, the two-parameter  re-written  form  of  f o r depth, not c o n s i d e r i n g  equation  any length  e f f e c t s or width e f f e c t s .  F(x)  =  1 - ex {-^-/ (|) dy} 1 d  (3.13)  k3  P  where x i s the f a i l u r e s t r e s s , y i s the depth is  the  depth  element.  k  of and  3  respectively,  of  the member and 6  y  m  are  the  of the c o n s t i t u e n t  above.  k  parameter.  will  be  shape  two-parameter  strength  3  the  elements  referred  co-ordinate,  d  i s the depth of a s i n g l e and  scale  Weibull of  parameters,  d i s t r i b u t i o n of  depth  referred  to  t o as the s t r e s s - d i s t r i b u t i o n  The i n t e g r a t i o n only a p p l i e s to  the c r o s s s e c t i o n s t r e s s e d i n t e n s i o n .  that  portion  of  67  Consider  the  Figure tension, figure  case  13 - T e n s i o n  where s t r e s s  13(b),  x=ft,  F(x)  For  the case  linearly neutral  shown  over  and e q u a t i o n  1  =  in  figure  stress  i s constant  the depth,  equation  x  3.13  =  rf  over 3.12  the depth  as  1 3 ( c ) o r ( d ) , where s t r e s s e s  where  in  becomes  x a t any d e p t h  = Z_f cd m  becomes  shown  (3.14)  the s t r e s s  m  For a x i a l  distributions  axis i s  or  13(a).  - exp{-  in figure  x  and  shown  r  =  ^—r  cd  vary  y from t h e  68  F ( r  -  V  1--,|-?J 1  C  ^  *  3  }  (  (d) the extreme f i b r e s t r e s s at f a i l u r e , a  ratio  of  the  p r o b a b i l i t y of f a i l u r e F(x)=0.5)  axial  tension  3.14  and  m  =  "  k  1/  13(c) or  fm, can be c a l c u l a t e d  strength  3.16  r  1  /  k  ft.  For any  can  be  with  equated  and  3  r dr]  For the n e u t r a l a x i s w i t h i n  f  the  ( .  t  3  member  depth  as  1 7  )  shown  in  13(c), t h i s becomes  f  which  >  t o give  f  figure  6  ( f o r example, the median s t r e n g t h  equations  rearranged  1  r  Under c o n d i t i o n s of l o a d i n g such as shown i n f i g u r e  as  -  3  f o r the  m  = [ ' ] k^+l L  special  f t  3  i  case  of  (3.18)  neutral  a x i s at mid-depth  becomes  f m  The  line  =  [ 2(k,+l)l 3  i n f i g u r e 14(a) marked  1/k, J  f t  c=0.5  p l o t t e d f o r a range of values of k . 3  used to estimate axial  tension  are of d i f f e r e n t made  first.  the value of k  3  (3.19)  shows  this  equation  T h i s r e l a t i o n s h i p can be  from the r a t i o of s t r e n g t h of  t e s t s to bending t e s t s . l e n g t h s , then l e n g t h  I f the t e s t corrections  specimens should  be  69  J f f  J — i — i — i — i — i — i — i — i 2.0  ao  4.0  6.0  i  8.0  STRESS  i  10.0  DISTRIBUTION  i  i  12.0  i  i  14.0  PARAMETER  i  1  1  1  1  1  1  1  1  1  i  16.0  i  i_  !8JJ  50.3  k,  (a) i  -  Q  1  r  1  i  r  c=ao  icc  0.0  2.3  4.0  €.0  8.0  STRESS  10.0  DISTRIBUTION  12.0  14.0  PARAMETER  k  LB.O  16.0  20.0  3  (b)  Figure  14 - Ratio of a x i a l tension  strength  to maximum  s t r e s s i n extreme f i b r e Equation depth. is  not  3.19  assumes  The two dotted a  that the n e u t r a l a x i s i s at mid-  l i n e s i n f i g u r e 14(a) show  that  there  l o t of d i f f e r e n c e i f the n e u t r a l a x i s i s at 40% or  45% of the depth from the tension  surface.  When the n e u t r a l a x i s i s outside  the beam depth as  i n f i g u r e 13(d), the i n t e g r a t i o n y i e l d s  shown  70  =  f  {  m  Figure  14(b)  neutral  3.7.2  the  axis  to a  However  of  ductile  the depth  effect  on  described  stressed  As  i n Chapter  the  section  2 i s also  result  percentage  For  the a x i a l  "fibre  for tension  behaviour.  example,  compression  support  with a  theory"  within-member  will  t h i s phenomenon, be  Assume  extended  that  to  wood  in  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 . section  stressed  in  axial  compression  shown i n f i g u r e  highly  decreases.  reduce  be  compression.  as t h e  t o o l to i n v e s t i g a t e  cross  the on  in  increasing  consistent  the  depending  3.20.  in compression.  compression  (d)  range  theory cannot  wood  The  Consider  and  to  w i t h o u t h a v i n g any s i g n i f i c a n t  strength.  mathematical  compression  wide  some o t h e r mechanism c o u l d  stress  t h e o r y d e v e l o p e d above  include  like  knot c o u l d  the c r o s s bending  a  strength  for a  fracture  i n compression  centreline  depth e f f e c t  fibre  e q u a t i o n s 3 . 1 9 and  brittle  that  of  at  (3.20) t  of a x i a l t e n s i o n  material  the maximum c o m p r e s s i o n  strength  ^  1  Zone  i t i s possible  face  ratio  depths, using  in  a wide  "  i n the extreme t e n s i o n  assumptions  applied  ^  3  Compression The  _ l  u  J J  shows  maximum s t r e s s of  c  15(b). yield  shown  in  figure  wood y i e l d s a t a s t r e s s  In t h e c a s e s shown stress  the percentage  fern i s g r e a t e r of  15(a).  the  in  figures  When f c as 15(c)  than the s t r e s s f c  section  depth  that  is  stressed.  The  same  procedure  used  in tension  c a n be  used here  to  71  (a) Figure  (b)  (d)  (c)  15 - Compression s t r e s s  distributions  give  f  cm  - [+ L  a  3  k^+l  J  f  (3.21 )  c  - u + ^> V b  k3+  1/k3  k +l  cm  3  for the cases shown i n f i g u r e s  15(c) and  f  (3.22)  (d)  respectively,  where the r a t i o s a, b, and c are d e f i n e d i n the f i g u r e . 3.8 WIDTH EFFECT Conventional  brittle  fracture  f a i l u r e s t r e s s e s should decrease volume. observed  For  wood  bending  theory  f o r any  members  a  suggests  increase  in  member  s i z e e f f e c t has been,  with l e n g t h and depth of member, but not with  For c l e a r wood, Bohannan(1966) found that modulus was independent  that  of  width. rupture  of width, and e x p l a i n e d t h i s o b s e r v a t i o n using  72  an  intuitive  argument.  He p o i n t e d out that bending s t r e s s e s  vary as the square of the depth of a member, but only with the width, so that some small i n i t i a l would  be  more  likely  argument  in  a l o c a l shear f a i l u r e . longitudinal transferred to  width.  An  failure  In  fibres,  a  is initiated  beam  subjected  fibres  increase  in  unreasonable, but f a i l axial  to  side  a  tension,  beam  this  and i t  i n some way by  consisting  to  from f i b r e s to adjacent  adjacent  an  type of member  extension  of  parallel  shear f o r c e , shear i s  ones above and below,  by  side.  width.  but  For t h i s reason an  increase i n beam depth may c o n t r i b u t e to a than  depth  can be made i f shear s t r e s s e s are c o n s i d e r e d ,  i s assumed that a f l e x u r a l  not  reduction i n  t o cause a "cascading"  f a i l u r e than a r e d u c t i o n  linearly  size  effect  more  These arguments are not  to e x p l a i n a c r o s s  section  effect  in  which i s of s i m i l a r or g r e a t e r magnitude than  that observed i n bending. Another p o s s i b l e e x p l a n a t i o n effect  f o r the absence of  i n c l e a r wood could be that the c e l l s  added  to  a  tree's  circumference  p r o p e r t i e s , but that v a r i a b i l i t y annual  rings  causes  depth  are  between  effects.  width  i n each new l a y e r of  very  layers This  a  similar  and  theory  between i s only  s t r i c t l y a p p l i c a b l e i f a l l members are "quarter-sawn" or "edge sawn" as shown i n f i g u r e 16., There i s no suggestion members still one  that a l l  t e s t e d have been sawn i n t h i s way, but the theory may  c o n t r i b u t e to a s i z e e f f e c t which i s a f f e c t e d  by  only  c r o s s s e c t i o n a l dimension, not two. For  commercial  timber,  Madsen  and S t i n s o n  (1982) have  73  Figure found  a  significant  i n c r e a s i n g width, f r a c t u r e theory. by  16 - Quarter-sawn  looking  at  increase  contrary  in  to  the  rules for various  of  predictions  of  brittle  timber,  but  fracture  were  unable  bending  a x i s bending additive  would  tests  Results for strong  suggest that the f a i l u r e s t r e s s f o r weak  should  factors  the weak a x i s .  to  effects.  r e s u l t s have been r e p o r t e d f o r in-grade bending  of s t r u c t u r a l timber about axis  rupture with  of knots allowed by grading  separate grading e f f e c t s from b r i t t l e No  of  to e x p l a i n these o b s e r v a t i o n s  projections  widths  modulus  the  They attempted  board  be  of  significantly  greater,  because  of  d e c r e a s i n g depth and i n c r e a s i n g width.  Such an e f f e c t was found f o r some but not a l l s i z e s t e s t e d  in  t h i s study, as d e s c r i b e d i n Chapter 5. Width  effects  because a l l the t e s t  are  pursued  no  further  in this  r e s u l t s f o r combined l o a d i n g  s i n g l e width of 38mm with bending about  thesis,  are  the s t r o n g a x i s .  for a  74  3.9 EFFECT OF GRADING RULES When i n v e s t i g a t i n g defects,  size effects  i t i s very d i f f i c u l t  f o r example  members  with  rules.  the e f f e c t of member depth.  d e f e c t s are l i m i t e d to a c e r t a i n depth,  timber  to separate the e f f e c t of member  s i z e s from the e f f e c t of grading Consider  in  size  independent  of  Some member  others are permitted to be a c e r t a i n percentage of the  depth, so s t r e n g t h d i f f e r e n c e s affected  by  the  way  in  between d i f f e r e n t s i z e s can be  which  grading  rules control  such  defects. A s i m i l a r problem e x i s t s with member l e n g t h . timber  i s cut i n t o short p i e c e s t o i n v e s t i g a t e  many of the short original  pieces  >a  higher  i s l i m i t e d to some into  percentage  smaller  of  lengths  may  the  i t i s simply  stated  c h a r a c t e r i s t i c of m a t e r i a l  for  than- the  member  (such as wane) length,  i n t o s i z e e f f e c t s using  that  the  carefully.  then For  in-grade In  this  r e s u l t s obtained are  purchased as Number  2  and  i n 4 . 8 8 m l e n g t h s , subsequently cut i n t o shorter  testing.  defect  down-grade a board.  t e s t i n g must be reported and i n t e r p r e t e d  Grade  grade  Conversely, i f a defect  these reasons, i n v e s t i g a t i o n s  study  length e f f e c t s ,  p i e c e s because there may be only one l i m i t i n g  in a given long board.  cutting  become  When graded  Better lengths  75  3.10  SUMMARY  T h i s chapter has  reviewed c o n v e n t i o n a l s i z e e f f e c t  which i s based on a s t a t i s t i c a l materials.  The  application  to  directions  theory timber,  receiving  can  be  with  and  theory  modified  size  separate  theory has been developed  strength  and  effects  consideration.  discussed.  for  theory brittle  improved f o r in  different The  modified  76 IV.  EXPERIMENTAL PROCEDURES  T h i s chapter d e s c r i b e s experimental procedures different The and  t e s t s c a r r i e d out to p r o v i d e input to t h i s  thesis.  experiments  purposes,  served  the r e s u l t s w i l l be  chapters.  Most  of  a number of d i f f e r e n t  referred  input  to  the  model,  the t h e s i s may 4.1  several  subsequent  Some are used  results  provide  to c a l i b r a t e  the  for v e r i f i c a t i o n .  wish to f o l l o w the t h e o r e t i c a l development of  proceed d i r e c t l y to Chapter  6.  EXPERIMENTAL STAGES The  carried  experimental out  programme. and  in  6.  some  model, and many more are used Readers who  to  the r e s u l t s r e l a t e to the s t r e n g t h model  which w i l l be d e s c r i b e d i n Chapter basic  f o r a number of  in  work  five  stages  The experimental  the s i z e s  and  described  lengths  under  in a  this  thesis  was  co-operative testing  stages are summarized i n Table I, tested  in  stages  1  to  5  and  s u p e r v i s e d by Dr.  are  summarized i n Table I I . Stage 1  which  was  initiated  Johns of the U n i v e r s i t y of Sherbrooke, eccentrically  loaded  compression  the U n i v e r s i t y of B r i t i s h Columbia detailed  analysis  this  consisted  of  t e s t s on 38x140mm boards at (UBC)  in  early  of these r e s u l t s by the author  i d e n t i f i e d s e r i o u s d i s c r e p a n c i e s between a c t u a l behaviour,  Quebec,  Ken  current  1981.  A  i n mid-1981 theory  and  s t i m u l a t i n g the subsequent work d e s c r i b e d i n  thesis. Stage 2  which was  s u p e r v i s e d by the author, c o n s i s t e d of  77  Material (Source)  Stage  1  SPF  Size  Test Location, Date  38x140  UBC  (BC)  2  Description  E c c e n t r i c compression  (1981)  SPF  UBC  38x140  (BC)  Bending  (1982)  Axial  tension  Axial  compression  Combined bending & t e n s i o n  SPF  3  38x89  (Quebec)  4  SPF  Sherbrooke  Eccentric  compression  (1982-83)  38x89  (Quebec)  UBC  Bending  (1983)  A x i a l t e n s i o n ( s h o r t & long) A x i a l compression ( s h o r t & long) Eccentric  SPF  5  38x140  (BC)  UBC  tension  Bending ( d i f f e r e n t l e n g t h s &  (1983)  load  configurations)  T e n s i o n ( s h o r t and l o n g )  Table I - Summary of experimental stages bending  t e s t s , compression t e s t s , t e n s i o n t e s t s , and combined  bending and t e n s i o n t e s t s , a l l u s i n g 38x140mm boards from same out  sample  as  the Stage  at UBC i n e a r l y  1982.  by Johns and Buchanan  1 tests.  These t e s t s were c a r r i e d  Stages 1 and 2 have been summarized  (1982).  Stage 3 was c a r r i e d out on the same machine as relocated  to  stage, which Bleau,  the was  the  University supervised  of by  Sherbrooke, Dr. Johns  Stage  Quebec.  and  1, This  Mr. Raymond  c o n s i s t e d of e c c e n t r i c a l l y loaded compression t e s t s on  38x89mm boards, i n l a t e 1982 and e a r l y  1983.  More d e t a i l s are  78  Approx. length (o)  Material  Eccentric  Axial Loading  Axial Compression  Axial Tension  Combined Tension & Bending  X  X  Eccentricity (nnn) 2  12  18  39  75  Bending  202  COMPRESSION 38x140 SPF (BC)  0.21 0.91 1.82 2.44 3.00 3.35 4.27  38x89 SPF (Quebec)  X X  X X  X  X  X  X  X  X X  X  X  X X  0.135 0.45 0.91 0.84 1.30 1.50 1.80 2.0 2.3 3.2  X  X  X  X  X X  X  X  X  X X X  X  X  X  X  X  X  X  X  X  X  X X  X  X  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 s i z e s (stages 1 to 4)  tested  p r o v i d e d by B l e a u ( l 9 8 4 ) . Stage 4 was c a r r i e d out at UBC by 1983.  This  stage  short  author  in  Stage  3  tests.  It  compression t e s t s , short t e n s i o n t e s t s ,  bending t e s t s , machine s t r e s s grading and e c c e n t r i c a l l y t e n s i o n t e s t s , a l l d e s c r i b e d i n more d e t a i l  member  strength  length of  and  38x140mm  load timber  configuration members.  also short  loaded  below.  Stage 5 was a s e r i e s of t e s t s to i n v e s t i g a t e of  early  repeated the t e s t s of Stage 2 f o r 38x89mm  boards from the p o p u l a t i o n of the included  the  on  These  the  effects  the  bending  tests  were  79  designed  and supervised by P r o f e s s o r Borg Madsen as part of a  separate r e s e a r c h program results this  (Madsen  1983).  Some  preliminary  which are r e l e v a n t to t h i s study w i l l be d e s c r i b e d i n  thesis.  4.2 TEST MATERIAL 4.2.1 S p e c i e s The experiments d e s c r i b e d i n t h i s t h e s i s were c a r r i e d out on boards from three separate groups.  The  first  group  for  stages 1 and 2 was 38x140mm (nominal 2x6 inch) s p r u c e - p i n e - f i r (SPF)  timber  from  central  interior  B r i t i s h Columbia.  The  second group f o r stages 3 and 4 was 38x89mm (nominal 2x4 inch) SPF timber from Quebec.  The t h i r d group of boards f o r 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 q u a n t i f y the a c t u a l s p e c i e s i n each group, but a l l three groups appeared to be predominantly spruce. The t e s t  r e s u l t s are b e l i e v e d to be c h a r a c t e r i s t i c of the  commercial SPF s p e c i e s group i n B r i t i s h Columbia and i n Quebec respectively. 4.2.2 Grading The  first  two  groups were purchased as "Number Two and  B e t t e r " grade i n 4.88 m l e n g t h s . the  A f t e r c u t t i n g to l e n g t h , a l l  boards were regraded by a q u a l i f i e d grader.  t h i r d s of the i n i t i a l  Almost  lengths were of S e l e c t S t r u c t u r a l  twoGrade.  A f t e r c u t t i n g to short lengths more than t h r e e - q u a r t e r s of the sample  became  Select  S t r u c t u r a l Grade.  For example, a long  Number Two board with one d e f e c t could produce s e v e r a l Structural  segments.  Select  80  In  the  data  a n a l y s i s a p r e l i m i n a r y attempt was made to  separate by grade but there were too grade  boards  grades. or  to  produce  many  Select  Structural  s a t i s f a c t o r y r e s u l t s f o r the lower  A small number of boards down-graded f o r l a r g e  serious  knots  s t r u c t u r a l d e f e c t s were n o t i c e a b l y weak, but many  boards down-graded f o r wane  or  cosmetic  described  defects are  were  very  strong.  A l l the. r e s u l t s  combined.  These r e s u l t s are b e l i e v e d to be c h a r a c t e r i s t i c  "Number Two and B e t t e r " m a t e r i a l as graded The  boards  tested  in  stage  5  f o r a l l grades of  i n 4.88m l e n g t h s .  were  purchased i n two  separate grades, S e l e c t S t r u c t u r a l and Number  2.  The  axial  t e n s i o n t e s t s and the f l a t - w i s e bending t e s t s were c a r r i e d out on  Number  2  grade o n l y .  The bending t e s t s were c a r r i e d out  s e p a r a t e l y on both grades.  4.2.3 Moisture Content All  boards  controlled location. content  were  storage  purchased  areas  A l l the boards recorded  at  were were  kiln-dried. available kept  No  climate-  in either  indoors,  and  testing moisture  the time of t e s t i n g using an e l e c t r i c a l  r e s i s t a n c e moisture meter. For the between  11%  Stage  1  tests,  content  varied  and 16% with an average value of 14.25%.  At the  time of the Stage 2 t e s t s the  the  moisture  average  moisture  content  was  13.8%. The  Stage  3 and 4 t e s t i n g was c a r r i e d out over a p e r i o d  81  of s e v e r a l months, and there were some minor moisture changes  during t h i s period.  content  The moisture content v a r i e d  7% t o 13%, with an average value of 10.4%.  The  test  from  results  are b e l i e v e d t o be r e p r e s e n t a t i v e of m a t e r i a l of t h i s moisture content  range.  4.2.4 Sample S e l e c t i o n Stages 1 and 2 (38x140mm) On  arrival  i n the l a b o r a t o r y , s u f f i c i e n t  the Stage 1 t e s t i n g length  and  material.  was  designated,  eccentricity  The Stage  2  were  and  selected  testing  was  timber f o r a l l  samples  f o r each  randomly  from that  carried  out  from  the  remainder of the shipment. Stages 3 and 4 (38x89mm) The  whole of t h i s group was randomly a s s i g n e d to samples  for combined  bending and compression t e s t i n g  i n Stage 3.  When  i t was decided to not t e s t every l e n g t h 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 a v a i l a b l e f o r s h i p p i n g t o UBC f o r Stage 4 t e s t i n g . For  the  Stage  long were s e l e c t e d  4 t e s t i n g , two samples of 90 boards 2.9m  f o r the long t e n s i o n and compression  Two samples of 90 boards 1.9m long were s e l e c t e d tests.  One  sample was t e s t e d f u l l  i n two h a l v e s , one h a l f  length  flatwise.  2.9m  Ten  boards  compression t e s t s . 0.97m  lengths  f o r bending  l e n g t h , the other was c u t  tested long  tests.  were  edgewise, cut  up  the  other  for  short  The remainder of the boards were cut  f o r combined  bending and t e n s i o n t e s t s .  samples of 80 boards ( f o r four e c c e n t r i c i t i e s )  were  into Four  selected  82  at  random  any  board.  such that no  sample had  more than one  segment from  Stage 5 (38x140mm) The the  samples for d i f f e r e n t t e s t s were s e l e c t e d  basis  of  described 4.3  long-span  Sample The  100.  of  elasticity  as  CONFIDENCE  Sizes  intention  In  slightly  modulus  on  by Madsen(1983).  SAMPLE SIZES AND  4.3.1  flexural  randomly  for most t e s t s was  practice l e s s due  combined  the  to minor  bending and  order to obtain  useful  to have a sample s i z e of  sample  problems.  tension  data at four  sizes The  were  sample  in Stage 4 was  usually size  for  reduced to 80,  e c c e n t r i c i t i e s from  the  in  limited  number of boards a v a i l a b l e . The values  sample s i z e s used allow c a l c u l a t i o n of mean or median, with c o n s i d e r a b l e c o n f i d e n c e , and  v a l u e s with s i g n i f i c a n t l y l e s s c o n f i d e n c e . percentile  values  (.05  and  throughout t h i s study as of  the  distribution.  points  either  average of s e v e r a l An  The  quantiles)  5th  lower  tail  and  95th  have been used  i n d i c a t o r s of behaviour at the Percentiles  from a cumulative p l o t of the two  .95  upper and  side points  of  can  be c a l c u l a t e d d i r e c t l y  ranked data, the  i n the  either  percentile,  using  the  or a weighted  vicinity.  a l t e r n a t i v e approach, using a l l of the  an a p p r o p r i a t e d i s t r i b u t i o n a l  tails  model  to  the  data, i s to f i t data,  then  to  83  c a l c u l a t e the p e r c e n t i l e values from the f i t t e d T h i s method has been used throughout Weibull 4.3.2  study,  using  the  distribution.  Weibull  Distribution  The W e i b u l l d i s t r i b u t i o n has  this  distribution.  been  widely  used  i s a f l e x i b l e d i s t r i b u t i o n which  f o r studying the s t r e n g t h of wood and  other m a t e r i a l s . The  Weibull  describing  distribution  material  strength  is  most  appropriate  properties  because  sample s i z e s i t i s the a s y m p t o t i c a l l y exact extreme  values  from any  initial  strength  of  particularly  the  it  weakest  for  M a t e r i a l strength  to  be  governed  by  fits the  of a l a r g e number of elements, the  elements can have negative s t r e n g t h , so there i s a lower  bound  zero  (or  parameter  hi-gher  Weibull  experimental  data  maximum l i k e l i h o o d 4.3.3  brittle  distribution  of  at  when  tends  one  for large  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 v a l u e . t h i s d e s c r i p t i o n because  for  Confidence  in  failures  some c a s e s ) .  .distributions by  model f i t t e d An  have  been  three-  fitted  to  for Q u a n t i l e s i n t e r v a l s depends on  the  type  to the data, and on the number of data p o i n t s . of  a  quantile  value can be assumed to be  a s y m p t o t i c a l l y normally d i s t r i b u t e d , f o r of  In t h i s study,  equations.  Calculation  estimate  regardless  None  e s t i m a t i n g the W e i b u l l parameters with  C a l c u l a t i o n of confidence of  occur..  the  distributional  large form  p o p u l a t i o n , with v a r i a n c e given by Bury (1975),  sample of as  the  size, parent  84  var(x  )  (4.1  = n[f(x )]  q  )  2  q  where  q  size, the  i s t h e 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 t h e sample  x q i s t h e q t h q u a n t i l e o f t h e v a r i a b l e x, a n d  probability density For  the Weibull  x  (p.d.f.).  distribution  =  q  function  f(xq) i s  x  o  + m [ l n ( l / ( l - q ) ) ]  1  /  (4.2)  k  and c  ,  .  k m  which combine  f(x )  x  -  q  x  ,  ~ o^k-i  ;-3  m  v  x  ^ )  k  .  _  r  x  x^ -  q  x„  %M  (4  exp{-(-^— m  1  r  i  [ l n ( l / ( l - q ) ) ]  (  1  -  1  /  k  )  exp{-ln(l/(l-q))}  (4.4)  where x , k, a n d m a r e t h e l o c a t i o n p a r a m e t e r , shape 0  and  s c a l e parameter Suppose  estimated 100  r %  standard  we  the  normal  is  z  that  s  t  ^  to  100 p %  be  within  population  an  distribution.  confident  interval  of  parameter  that  the  plus-or-minus  q u a n t i l e , then t r a n s f o r m i n g  to a  variable  2r  where p / 2 *  r e s p e c t i v e l y , of the W e i b u l l want  quantile of  3)  to give  =  q  x r  i e  v a  l  t h e a r e a under  x  u e  q  =  2 z  ,„ P/2  /var(x  ) q'  °f t h e s t a n d a r d the standard  (4.5)  normal  variable  normal curve i s p / 2 .  such  85  If  we, have  a  sample  of  s i z e n, we can c a l c u l a t e the  confidence p by r e a r r a n g i n g these equations  =  p/2  f (  Z  The  value of Zp/2 c  a  n  D  V/^q7  <'> 4  course  advance,  but  the  Weibull  values  m a t e r i a l can be  6  r e f e r r e d to standard normal t a b l e s to  e  o b t a i n the confidence p, f o r a two-sided Of  to give  test.  parameters  are  not  known  in  obtained from s i m i l a r t e s t s on the same  used  to  make  a  reasonable  estimation  of  confidence.  C a l c u l a t i o n s were made to estimate the confidence  with  quantiles,  which  calculated  from  the  fitted  d i s t r i b u t i o n , were i n the i n t e r v a l plus-or-minus population  values,  for a  sample  size  parameters were obtained from both bending of  SPF  timber  in  two s i z e s  S e l e c t S t r u c t u r a l and No.  timber  parameters r=0.l0, 0.97,  in  are  n=90  bending.  10.1,  and  1.70  the  figures  The  i s within  90.  the  Weibull  tension  tests  and  f o r No.  2  grade  l o c a t i o n , shape and s c a l e 34.3,  respectively.  With  4.6 gives a value p/2 °^ z  to an area of 0.335 under the standard  normal curve, or a confidence of 67% that percentile  of  (38x89mm and 38x135mm) f o r both  q=0.05, equation  which corresponds  and  10%  2 grades.  As an example c o n s i d e r 38x89mm  of  Weibull  the  estimated  5th  the i n t e r v a l between 0.9 times and 1.1  times the p o p u l a t i o n v a l u e .  line  T h i s procedure  is illustrated  in figure  17.  i s the p.d.f.  of the u n d e r l y i n g d i s t r i b u t i o n  The  solid  (the parent  86  00 CN  - 5%ile  .1  I'IS  50%ile  m ec m o or  p.d.f of quantile  CN  Area p f  estimator,  95%ile  I p.d.f  of  /'I/  underlying distribution  using:  '\y  order statistics normal Y approximation  1  I " — I 60.0  o.o  70.0  STRESS (flPfl)  Figure population).  17 - P.d.f. The  dotted  estimated q u a n t i l e s at the levels,  for  lines  are  5th,  50th,  the  p.d.f.'s  and  95th  The  the  percentile  two dotted l i n e s are very c l o s e ,  that the normal d i s t r i b u t i o n  even f o r t h i s case with a small  sample  parameter.  r,  The  terms  p  showing  i s a very good approximation  exact curve obtained using order s t a t i s t i c s  illustrated  of  a sample s i z e 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 .  more  of q u a n t i l e e s t i m a t o r s  and  size  referred  to a  (Bury  1975),  low  shape  and to  above,  are  f o r the 50th p e r c e n t i l e case o n l y .  Using a l l the samples r e f e r r e d to  above,  the  following  87  c o n c l u s i o n s can be drawn regarding the t e s t s i n t h i s  1.  5th  percentile  approximately  values  have  been  study:  estimated  with  70% confidence that they are w i t h i n 10%  of  population values.  2.  Median v a l u e s (50th p e r c e n t i l e s ) have been estimated  with approximately  90% confidence that  that  Confidences  they  are  w i t h i n plus-or-minus  These  5%.  confidence  95th p e r c e n t i l e s have confidence l i m i t s about  confidence  view  of  uncertainties testing  the  on  similar.  between the v a l u e s quoted  4.4.1  within  65%  3.  4.4  are  10% of p o p u l a t i o n v a l u e s , or approximately  mean v a l u e s are  in  they  f o r 5th and  50th  midway  percentiles.  i n t e r v a l s are c o n s i d e r e d q u i t e s a t i s f a c t o r y variable  regarding  nature  grade,  of  wood  and  various  s p e c i e s , source of supply  and  procedures.  TEST PROCEDURES Bending Bending t e s t s were c a r r i e d out on each group of timber  shown i n Table I I , to strength  model.  length e f f e c t s  obtain  Several  data  for  calibration  of  as the  lengths were t e s t e d to i n v e s t i g a t e  i n bending.  Bending t e s t s were performed  on an Olsen  900  kN universal  88  testing  machine.  controlled  was  applied  mechanically  i n about one minute.  Lateral  load p o i n t s prevented l a t e r a l b u c k l i n g .  recorded from the load  Figure All  i n d i c a t o r attached  with simple supports and  one-third  of 17, except f o r the short  ratio  boards was a l s o t e s t e d the  stage  configurations;  of  carried  of  9.5.  supports  at o n e - t h i r d  and a c e n t r a l point out  at  One  2  shown  sample  of  grade  in this thesis, and  points, load.  boards  Number was  three  load  two loads oneEach  two spans of 1.5m  f o r S e l e c t S t r u c t u r a l and Number  as  span t e s t of 38x89mm  5 bending t e s t s reported  two loads  span apart,  loading  i n f l a t - w i s e bending at t h i s span.  the boards were t e s t e d with simple  sample  was  A l l of these t e s t s had a span-to-  boards which was t e s t e d at a r a t i o  separately  Maximum load  point  depth  was  near  i n Table II were c a r r i e d out  i n f i g u r e 18.  tests  supports  to the machine.  schematically  quarter  a  18 - Loading arrangement f o r bending t e s t  the bending t e s t s l i s t e d  For  at  displacement rate of approximately 30 mm/min which  produced f a i l u r e the  Load  of  these  and 3.0 m, and  2  tested  grades.  One  in flat-wise  89  .bending at the s h o r t e r 4.4.2  Axial a.  span.  Tension  Long Boards  T e s t s of long boards were c a r r i e d out under s i m i l a r c o n d i t i o n s to previous 1978b).  in-grade  The  a x i a l t e n s i o n t e s t s (Madsen  Tension  tests  machine d e s c r i b e d schematically  which  out  by Madsen  and  Nielsen  figure  19(a).  grip  the  j a c k i n g system.  on  This  an  axial  (I978d)  and  machine has  The  when  forced  second  F a i l u r e load i s  system.  increased,  from a c a r e f u l l y c a l i b r a t e d h y d r a u l i c f l u i d The  friction  about any  axis.  machine's grips. 2.0 m,  grips The  standard  The  are  friction steel by  a  specimen i s s t r e s s e d in t e n s i o n is  jacking  shown  together  when the l e n g t h of the whole machine hydraulic  loading  with polyurethane-covered  board  to  6.  carried  in  were  i n Chapter  g r i p s 450mm long at each end,  hydraulic  Nielsen  r e s u l t s of these t e s t s are used as b a s i c input  the s t r e n g t h model d e s c r i b e d  plates  and  using  pressure  a  recorded gauge.  r i g i d l y mounted to prevent r o t a t i o n  38x140mm  boards  were  tested  l e n g t h with a f r e e length of 3.0m  38x89mm boards were t e s t e d over a f r e e  at  the  between  length  of  c o n t r o l l e d manually throughout  the  a l l other d e t a i l s remaining the same.  The  g r i p pressure  t e s t s , being  was  increased g r a d u a l l y  in-the g r i p s ,  with  perpendicular  to  care the  not  hydraulic  to  grain.  displacement rate c o n t r o l l e d j a c k i n g system.  i f the specimen began to cause  The by  the  excessive  l o a d i n g was electric  at a pump  F a i l u r e g e n e r a l l y occurred  slip  crushing uniform on  the  in about  90  1 9 - Loading arrangement f o r a x i a l (a) t e n s i o n (b) c o m p r e s s i o n  Figure 30 s e c o n d s  i f there  Although tests, due  there  to  was no s l i p p a g e  these  test  was p r o b a b l y  are  i n the g r i p s .  referred  some b e n d i n g  has been n e g l e c t e d ,  the cross b.  Short  Boards  boards  were  tested  on l e n g t h  effects.  described  faced  above was m o d i f i e d  The g r i p s a t one end  modified with  i n most  dividing  tension boards  each board.  Any  s t r e s s i n the  the a x i a l  force  s e c t i o n a l area.  Short  information  0.9m.  induced  and t h e t e n s i o n  b o a r d s has been c a l c u l a t e d by s i m p l y by  t o as a x i a l  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  such bending  tests  in  axial  The  tension  t o accept were  tension  testing  any l e n g t h  used  to  sandpaper  cut  from  unmodified.  a  heavy  machine  as s m a l l as  end, new g r i p s were made from a p a i r o f s t e e l coarse  obtain  duty  At  the  plates sanding  91  belt.  These  hydraulic  plates  were  clamped  together  by  system used f o r the o r i g i n a l g r i p s .  the same  The p l a t e s were  connected to the r e s t of the machine through a t e n s i o n of  variable  length.  The new connection  the o r i g i n a l one, as i t allowed  member  was not as r i g i d as  r o t a t i o n about the strong a x i s  of the board and some t o r s i o n a l r o t a t i o n , but  previous  tests  i n d i c a t e no s i g n i f i c a n t d i f f e r e n c e i n r e s u l t s from t h i s  source  (Madsen and N i e l s e n 4.4.3  I978d).  A x i a l Compression a.  Axial  Long Boards compression t e s t s of long boards were c a r r i e d out under  s i m i l a r c o n d i t i o n s to p r e v i o u s The  r e s u l t s of these  in-grade  t e s t s provide  compression  b a s i c input i n f o r m a t i o n f o r  the s t r e n g t h model. Compression  .  tests  of  same machine as the t e n s i o n  tests,  with  A system of l a t e r a l supports  prevented  lateral  enough  into  a  jacks  supports  were  schematically in  located  with  f o r the boards to be i n s e r t e d e a s i l y  adjustment f o r each board. board  loading  faced with t e f l o n pads  (1978c) and i l l u s t r a t e d  The l a t e r a l  clearance  the  b u c k l i n g i n e i t h e r d i r e c t i o n , as d e s c r i b e d  by Madsen and N i e l s e n 19(b).  /  long boards were performed i n the  reversed.  figure  tests.  perfectly  without  No attempt was made to f o r c e  each  s t r a i g h t c o n d i t i o n , so a very  small  amount of bending moment may have been present the a p p l i e d a x i a l l o a d .  just  in addition  Any such bending has been  to  neglected.  92  b.  Short Segments  A number of boards were cut i n t o short segments t o i n v e s t i g a t e the  variation  means of  i n compression s t r e n g t h w i t h i n each board, as a  quantifying  segment  length  was  length  effects  i n compression.  The  210mm f o r the 38x140mm boards, and 140mm  for the 38x89mm boards. These to  segments were loaded i n a x i a l compression  the g r a i n .  testing  The  machine  hydraulically rotate.  of  through  Platen  t e s t i n g machine was an Amsler 1.0  MN  capacity.  steel  platens  displacement  and  Load  4  universal  was  applied  which were not free t o load  c o n t i n u o u s l y and p l o t t e d on an x-y r e c o r d e r . approximately  parallel  were  recorded  Loading r a t e was  mm per minute, which corresponds to a s t r a i n  r a t e of approximately 0.02mm/mm per minute. 4.4.4  Eccentric  Compression  E c c e n t r i c compression t e s t s were performed purpose  testing  (1982).  Each end of the t e s t  steel  machine  described specimen  by  in a  Johns  fitted  and  special Buchanan  snugly  into  a  "boot" through which a x i a l compression was a p p l i e d at a  pre-determined e c c e n t r i c i t y about the strong a x i s as shown i n figure  bearings  and  lateral  supports prevented b u c k l i n g about the weak a x i s .  The  machine  can  20(a).  test  A  lengths  system  of  roller  up t o 5.0m a t e c c e n t r i c i t i e s up t o 202mm.  Equal end e c c e n t r i c i t i e s were used f o r a l l t e s t s .  The  actual  l e n g t h s and e c c e n t r i c i t i e s t e s t e d a r e shown i n t a b l e I I . Axial  load  was p r o v i d e d by a h y d r a u l i c  connected i n s e r i e s with a load c e l l .  jack at one end,  Lateral  displacements  93  Eccentricity varies 2mm to 202mrn  Test  specimen  140 mm or 8 9 m m  (b)  (a)  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) C o m p r e s s i o n (b) T e n s i o n were  measured  linear cell  variable  at  three  locations  displacement transducers  a n d LVDT's were s c a n n e d  based data a c q u i s i t i o n displacement  rate  Time t o f a i l u r e  along  until  each specimen  (LVDT's).  throughout the t e s t  system. maximum  averaged about  by a  The  with load  computer  L o a d was a p p l i e d  at a constant  load  been  had c l e a r l y  one m i n u t e .  reached.  94  4.4.5  Combined Bending and Tension Combined  two  bending  boards  4.8m  tension  tests  long  2  testing  were  carried  long  200  38x140mm  i n combined bending  by Johns and Buchanan (1982).  out i n the machine already  tension  and  compression  These  described for As  shown  i n f i g u r e 21(a) each board was i n i t i a l l y  loaded  in bending as a simply one-third  approximately  were t e s t e d to f a i l u r e  as d e s c r i b e d  schematically  the  t e s t s were c a r r i e d out i n  Bending Followed by Tension  As part of the Stage  the  tension  separate and q u i t e d i f f e r e n t experiments. a.  and  and  tests.  supported beam with a g r a v i t y  points,  before  h y d r a u l i c g r i p s and a x i a l  load  at  the ends were clamped with the  tension  load . a p p l i e d  to  produce  f a i l u r e , as shown i n f i g u r e 21(b). The tests.  tension The  bending  superposition load  load  bending,  was  moment  of three moment  recorded at  as i n the a x i a l  failure  was  tension  calculated  by  separate e f f e c t s ; moment due to g r a v i t y 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 partially  restrained against  r o t a t i o n during  These boards were t e s t e d i n levels  of  initial'  Bending before  gravity  any t e n s i o n  two  load.  the t e s t .  groups Several  f o r c e was a p p l i e d .  with  different  boards broke i n For  the  group  with high g r a v i t y loads about h a l f broke i n t h i s manner. This  t e s t i n g procedure was not very  s a t i s f a c t o r y because  i t was not p o s s i b l e t o i n c r e a s e bending moment and a x i a l concurrently  and  because  some  slipping  in  load  the g r i p s gave  95  (b)  F i g u r e 21 - Combined bending and t e n s i o n (a) f i r s t stage (b) second stage erroneous moment readings  f o r the stronger  test  boards  i n the  sample. b. A  E c c e n t r i c Tension  much  more  s a t i s f a c t o r y t e s t i n g procedure was used f o r the  38x89mm boards i n Stage 4.  Combined bending and t e n s i o n  were performed on four groups of 80 boards, each 970mm The  boards  were  e c c e n t r i c i t i e s as different  stressed shown  eccentricity  i n tension  schematically f o r each group.  in  with  tests long.  equal end  figure  20(b),  a  T h i s type of in-grade  t e s t i n g has not been r e p o r t e d elsewhere. A 275mm l e n g t h a t each end  each  with  was  clamped  sandpaper-faced  hydraulic  j a c k i n g system used i n the long t e n s i o n t e s t s .  grip  plates  board  between  bottom  steel  of  the m o d i f i e d The  p l a t e s were connected t o the f l o o r with a saddle  96  which allowed r o t a t i o n about movement.  The  axis,  strong  and  The  displacement failure  about  other  jack and  The load  the  load  was  cell  in  operated under stroke c o n t r o l with a  rate of approximately  i n 15 to 30  no  to the l o a d i n g  the l o n g i t u d i n a l a x i s .  system was  1.5mm  per second,  producing  seconds.  G r i p pressure was  controlled  tension  tests.  LVDT's.  Load and d e f l e c t i o n were  the  but  saddle which allowed r o t a t i o n about  a p p l i e d through a 450 KN c a p a c i t y MTS series.  axis  top g r i p p l a t e s were connected  system with a s i m i l a r strong  the  manually  as  in  the  long  L a t e r a l d e f l e c t i o n s were measured with three scanned  computer-based data a c q u i s i t i o n  continuously  system.  with  Maximum load  and  the corresponding d e f l e c t i o n s were recorded. 4.4.6  Data  Aquisition  For a l l the carried used.  out  combined  at  UBC,  T h i s system  PDP-11  digital  bending  the  deflection  c o n s i s t s of a NEFF scanner  computer.  data  to  The  disk  increments.  Stage  3 t e s t i n g at  4.4.7  Modulus of  A  scanner  elasticity.  system  on l i n e  can  testing  scan  with  was a  up to 60  A data a c q u i s i t i o n programme  file  at  similar  predetermined system  was  used  load  or  f o r the  Elasticity  strength  s t r e n g t h model.  load  Sherbrooke.  Modulus of e l a s t i c i t y buckling  axial  a computer data a c q u i s i t i o n  channels of input at high speed. writes  and  of  long  is  required  columns,  for  and  predicting for  the  input to the  Three methods were used to assess modulus  of  97  1. static  A random sample of a l l the boards were s u b j e c t e d t o a bending  test,  and" the measured  c a l c u l a t e the modulus of e l a s t i c i t y .  d e f l e c t i o n used to  Approximately one  fifth  of the boards were t e s t e d . 2.  For the boards  compression  tested  at Sherbrooke,  a  i n combined  Southwell  plot  bending was  made.  d e t a i l e d d e s c r i p t i o n of t h i s i s given by B l e a u ( l 9 8 4 ) . long  length  boards  e l a s t i c i t y determined obtained i n s t a t i c 3.  A l l of  with  the 38x89mm  were s u b j e c t e d to  flexural  i t between three r o l l e r s ,  This  machine  bends  the r e s u l t s  rotated  and  crook.  The load on the c e n t r e r o l l e r  The  board  the c e n t r e  passes  the board  each  one  travel.  180  Forintek  as a plank, offset  degrees  to Each  between  averaged t o e l i m i n a t e the e f f e c t of i s recorded every  100mm  An average modulus of e l a s t i c i t y over the  i s calculated  from the l o a d ,  the d e f l e c t i o n ,  and  dimensions. imposed  d e f l e c t i o n of 5mm and the rate of t r a v e l of  40 m/min were both c o n s i d e r a b l y lower v a l u e s than typical  of  midspan d e f l e c t i o n over a 910mm s i n g l e span.  board i s passed through twice,  910mm span  stiffness  The boards were passed through a Cook-Bolinder  passing  board  that  boards subsequently t e s t e d i n  Corp.  of  to  bending.  Canada  a  For the  close  s t r e s s - g r a d i n g machine a t the Western Laboratory  provide  A  small e c c e n t r i c i t y , the modulus of  i n t h i s manner was very  t e n s i o n and compression measurements.  and  commercial  application  used  in a  i n order to increase accuracy  98  and reduce the p o s s i b i l i t y  of damage.  4.5 SUMMARY T h i s chapter machines,  and  has  described  experimental  than 4000 t e s t specimens combined l o a d i n g .  sample  selection,  testing  procedures used f o r t e s t i n g more  i n bending, t e n s i o n , compression, and  R e s u l t s are d e s c r i b e d i n the next chapter.  99  V.  This  chapter  results,  discussed  Some of in  the  description  and  bending  and  axial those  experimental for  loading  results  observed  results  relating  experimental  subsequent  of  explanation  are  other  COMBINED BENDING AND  5.1.1  brief  i n some d e t a i l , as  more d e t a i l  5.1  a  some d i s c u s s i o n Combined  effects. in  contains  with  behaviour.  EXPERIMENTAL RESULTS  to are  are  length analysed  chapters.  AXIAL LOADING RESULTS  Presentation As  described  were t e s t e d 38x140  at  and  previously,  several  38x89mm  five different  different  sizes,  lengths  of  eccentricities,  with  about  100  boards  for  boards  both  in  each  sample. Within the  each  results.  another  is  ends  of  percentile Weibull  most  sample  relative  the  will  be  levels,  large  comparing  means, at  R e s u l t s can  be  the  the  percentiles  the  being  with  does  not  or  low  high  compared a t  percentile,  various  this  study,  mean and  obtained  in  sample  this  Throughout  5th  scatter  one  but  behaviour  distribution. made a t  amount of  from  95th fitted  distributions.  There combined  compare  is a  method of  distribution.  within  comparisons  there  simplest  describe  the  quantiles  The to  necessarily  sample  are  several  b e n d i n g and  useful  methods  axial  methods a r e  to  load plot  of  for  presenting  given  axial  lengths.  load-moment  results The  for two  interaction  100  diagrams f o r each l e n g t h , or t o p l o t a x i a l (or  slenderness)  load a g a i n s t  f o r given e c c e n t r i c i t i e s .  length  The two methods  are r e l a t e d as shown i n f i g u r e 11. 5.1.2 I n t e r a c t i o n Curves f o r Short Members a.  Test  Results  The  s h o r t e s t members t e s t e d had lengths between g r i p s of 900mm  and  450mm f o r the 38x140  Buckling  effects  recorded  values  and  a r e very of  axial  38x89mm  sizes,  small at these load  respectively.  l e n g t h s , so maximum  and moment  represent  the  m a t e r i a l s t r e n g t h of the weakest c r o s s s e c t i o n i n each board. Figures  22(a) and  (b) show  the t e s t  s h o r t e s t l e n g t h of 38x89 and 38x140mm Each  dot  failure the  represents  the a x i a l  f o r one board.  product  points  shown  boards,  load and mid-span moment a t  eccentricity,  For each cloud of p o i n t s  r e p r e s e n t i n g one end e c c e n t r i c i t y , each p o i n t was co-ordinates,  located. through  and  of  to  gravity.  be  values  of  distribution  parameters.  radius  were  T h i s procedure r e s u l t e d i n each reduced  to  three  characteristic  on  a  radial  A three parameter  d i s t r i b u t i o n was f i t t e d t o a l l the r a d i i , percentile  defined  in  the c e n t r e of g r a v i t y of the c l o u d  A l l p o i n t s were assumed the centre  the  from a r a d i a l l i n e because of d i f f e r e n t  amounts of d e f l e c t i o n i n each board.  polar  as  load and the maximum d e f l e c t i o n from  For each value of end  deviate  respectively.  The mid-span moment was c a l c u l a t e d  of the a x i a l  the l i n e of t h r u s t .  r e s u l t s f o r the  and  cloud  of  Weibull  5th and  calculated  line  95th  from the  points  being  s t r e n g t h values as shown i n  101  Length  2 mm 12mm  0.45m.  eccentricity  •• •  • d  O  CO  39  •  mm  Jo. »—« CE  75  mm  202mm — 1  -1  0.D  1  1  1  J.O  0.5  1  mm  —— 1  3.5  1  1  2.0  1  mm""  1  2.5  1  1  1  3.0  1—  3.5  4.0  niD-SPRN HOriENT (KN.M) (a)  38 x 89 m m L e n g t h 0.914 m.  2 mm 18mm  t• • • _ V-  •  •  . ••*.  Co  •  •V .-  jl?  s  _  a  •• •  75 m m  f  j * " •  •• .  9-  *  202mm •  ^ n  0.0  Figure  39 m m  » r. • a  CXoc  eccentricity  J.O  22 - T e s t  1  1  2.0  T  1  1  1  1  1  1  3.0 4.0 5.0 niD-SPflN HOriENT (KN.n) (b) 38x1 AO mm  results f o r shortest compression  i  i  6.0  length  i  "?.0  r  B.O  in eccentric  102  Length CU5m.  5do CL  o CT.1D  X cc  si-  i  i  0.0  i  O.S  i  i  1.0  i  i  1.5  i 2.0  i  i  i  2.5  i  i  3.0  i  4.0  3.5  rflD-SPRN MOMENT (KN.M)  (a)  38 x 89 mm Length 0.9Hm.  CEa  dice  9-  i  0.0  i  1.0  i  i  2.0  i  i  i  i  i ; i  i  i  3.0 4.0 5.0 €.0 MID-SPRN MOMENT (KN.M)  i  i  "7.0  i B.D  (b) 38x140mm  F i g u r e 23 - P e r c e n t i l e r e s u l t s for s h o r t e s t e c c e n t r i c compression  length  103  f i g u r e 23.  The  represent  three  typical  lines  sketched  behaviour  through  significant level.  from  When  moderate  subjected  axial  to  bending  compression  to r e i n f o r c e d concrete The  behaviour  zero,  decreasing  figure  under  force,  moment  to zero at  explanation  the  the  i n the t e n s i o n action  tension  i s increased.  The  24  bending  and  tension  f o r 38x89mm m a t e r i a l .  T h i s data  i d e n t i c a l to f i g u r e 23(a).  compression.  i s not  24(b)  r e s u l t s have been added  feature of t h i s f i g u r e i s  variability  in  tension  than  in  T h i s d i f f e r e n c e produces changes i n the r a t i o of  tension  to  strength  values.  The  Tension  A striking  strength  as  The top h a l f of f i g u r e 24(a)  is  the bending a x i s .  i f we  tests,  i d e n t i c a l to f i g u r e 22(a) and the top h a l f of f i g u r e  larger  a  resemblance  is  the much  of  failure is  of these short members i s c l a r i f i e d  a v a i l a b l e f o r 38x140mm m a t e r i a l .  below  for  behaviour i s s t r i k i n g .  the r e s u l t s of combined in  from  The simple  a  f i f t h percentile  f a i l u r e due to d e f e c t s  suppressed and moment c a p a c i t y  shown  the  Furthermore  i s that when t e s t e d i n bending, these weaker  boards e x h i b i t a t e n s i o n zone.  at  i n c r e a s e s s i g n i f i c a n t l y before  behaviour  A l l the curves i n  origin.  i s increased  l e v e l s of a x i a l compression.  this  add  the  i s displayed  As a x i a l compression  capacity high  feature  points  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 of the d i s t r i b u t i o n . f i g u r e 23 a r e convex away  the  compression  ratio  of  s t r e n g t h through the d i s t r i b u t i o n of  tension  to  compression  strength  pronounced e f f e c t on the shape of the i n t e r a c t i o n curve.  has At  a  104  Length  2mm  Z  v.-  —  g  12mm eccentricity  k <••; '  o  -.4*-.  to UJ  39 mm  ct  0.  tf  o  0.45m.  •  75mm  X  <  202mm  o o 202mm  1;  •*  39 mm  o cn z  X*  -  < 12 mm  2 mm  0.0  1.0  2.0  3D  A.0  5.0  niD-SPRN nOHENT (KN.M)  (a) data  points  F i g u r e 24 - Test r e s u l t s f o r 38x89mm boards i n e c c e n t r i c compression and t e n s i o n  C  ro K)  p >-••  3  —~ O  CO  o o o> 3 rt  »1  O  o  3 rt 3  C  ro a  '  o i o 3 -3 Xi f D CO  ro ncn cn I-I ro o CO 3 c CU 3  •nto  ri-  O i-(  a  ft) 3 01 O 3  to 00  oo vo 3 3 tr o PJ  >-t co  CD"  •  i m  •  i  200 f l  I  I  I  AXIAL TENSION (KN) 150 100 i I I l i I i 1  i  I  50 i J  1  1  1  A X I A L COMPRESSION (KN) 0.0 50 100 1 1—i—i—i—i—i—i—i—i—i—i—i—i—L  106  the  5th  percentile  compression high  level  the  low  ratio  of  tension  s t r e n g t h r e s u l t s i n the "nose" of the curve  i n the compression  region.  the t e n s i o n s t r e n g t h i s much  being  At the 95th p e r c e n t i l e  greater  than  the  to  level  compression  s t r e n g t h , so the "nose" of the curve i s w e l l below the bending axis  and  the  i n t e r a c t i o n curve i n the compression  region i s  c l o s e t o the s t r a i g h t l i n e assumed i n design codes. Behaviour at the 95th p e r c e n t i l e l e v e l  i s of s i m i l a r  to that expected at the 5th p e r c e n t i l e l e v e l specimens.  Hence  p e r c e n t i l e curves differences  the on  between  shapes  for clear  wood  of  the  the  24(b)  are  indicative  of the  timber and c l e a r wood behaviour  referred  figure  5th  form  and  95th  to i n Chapter 2. b.  Mode of F a i l u r e  For the i n t e r a c t i o n curves shown i n f i g u r e 24(b), the nose the curve marks the t r a n s i t i o n between a compression failure  above,  to  the  t e n s i o n dominated  of  dominated  f a i l u r e below.  In  r e i n f o r c e d 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 The  observed  p a t t e r n , with f a i l u r e region  being  compression  failure in  the  failure."  modes  compression  of  face, and f a i l u r e  because  the  followed  dominated  this  failure  a s s o c i a t e d with d u c t i l e c r u s h i n g of wood on the i n the t e n s i o n dominated  region being a s s o c i a t e d with b r i t t l e part  generally  board.  failure  in  the  This i s purely a q u a l i t a t i v e  i t i s extremely d i f f i c u l t  of f a i l u r e modes observed.  failure tension  assessment  to q u a n t i f y the l a r g e number  107  5.1.3  I n t e r a c t i o n Curves f o r Long Members The  test  results  illustrated  f i g u r e 22 and as p e r c e n t i l e s produced  for  illustrated  the  more  and  illustrate  slender  been  the general  a  transition  mean v a l u e s  from  a  material  failure  also  to  failure  capacity,  strength lengths  this  failure  being f o r the  f o r the more slender bending  moment  reduce with i n c r e a s i n g l e n g t h .  e f f e c t , which i s much l e s s pronounced  introduced  superimposed  slenderness,  For low l e v e l s of a x i a l load the  length  a l l  In a l l cases the a x i a l load  boards.  load  been  are  for various  boards to an i n s t a b i l i t y  axial  These  percentiles  shortest  tends  also  r e l a t i o n s h i p between  p l o t t e d on f i g u r e 25.  by  have  boards.  at f a i l u r e decreases with i n c r e a s i n g caused  23  of p o i n t s i n  points.  i n c r e a s i n g slenderness,  have  clouds  figure  i n Appendix C, with the  on the i n d i v i d u a l data To  in  as  than  the- decrease  at This in  i s l a r g e l y due to the s i z e e f f e c t with  i n Chapter 3 and q u a n t i f i e d  later  in  this  chapter. 5.'1.4 A x i a l Load-Slenderness Curves The  test  d i f f e r e n t way ratio,  defined  results by  The  plotting  axial  above can be presented i n a load  against  as L/d where L i s the length  d i s the r e l e v a n t member.  described  cross  section  dimension  slenderness  of the member and of  a  rectangular  r e s u l t i n g curves a r e shown i n f i g u r e s 26(a) and  (b) f o r the 38x140mm and 38x89mm boards r e s p e c t i v e l y , f o r mean v a l u e s of a x i a l load at f a i l u r e .  Behaviour  95th p e r c e n t i l e l e v e l s follow a s i m i l a r  at  trend.  the  5th and  108  LENGTH  i 00  1  1 0.5  1  1 1.0  1  1 L5  1  1 2.0  1  O  O  H  +. ! , 3 0 m  <5  e> 1.80 m  x-  X  2.30 m  LB-  B  3.20 m  i 2.5  i  i 3.0  MID-SPAN MOMENT (KN.M) (a)  0.45m  i  1  3.5  38 x 89 mm LENGTH a  B 0.31 m  O- - - 0  A — ^  —TO  1.82 m  — A 2.44 m  B  H 3.35 m  Q.  (j) 4 2 7 m 4  **  —i 30  1.0  2.0  3.0  1  1  —i  1—~  4.0  5.0  H l D - S P f l N MOHENT (KN.M)  1  1  6.0  1  7.U  (b) 38x140mm 25  - I n t e r a c t i o n d i a g r a m f o r mean t e s t results all lengths i n eccentric compression  109  ECCENTRICITY  0  O  2mm  H  +  12mm  <$— . —0 \  \ \  75mm  X  X  •-  CD 2 0 2 m m  N  \  a cc o  39 m m  \  1-  \ \ \  \ 1  ex  \ \  I—I  cr *>-  v.. X-.  — ^  CS  O.D  ~ -x  16.0  B.O  24.0  32.D  40.0  48.0  SLENDERNESS (L/D)  38x89 mm  (a)  I.  ECCENTRICITY CB +-  \  \  \  \  ts  h 18mm  o — - —e> . \  o  ED 2 m m  39 m m  X-  X  75mm  B-  CD  202mm  CEg-  M X <X •  -H  O.D  I  B.O  i  i 16.0  i  r 24.0  32.D  40.0  48.a  SLENDERNESS [L/D)  (b)  38x140mm  F i g u r e 26 - A x i a l l o a d - s l e n d e r n e s s curves. Mean t e s t r e s u l t s f o r a l l lengths t e s t e d i n e c c e n t r i c compression  110  In each case the curve of most i n t e r e s t i s the top curve, representing  2mm  nominal e c c e n t r i c i t y .  •is c l o s e t o c o n c e n t r i c small  slenderness  represents  a  The  v a l u e s , at the l e f t  material  compression. values  loading.  The  a  failure  condition  for  hand side of the f i g u r e ,  condition  for  large  l i n e a r e l a s t i c buckling  w i l l be compared with e x i s t i n g design  axial  slenderness  failure.  curves are s i m i l a r t o the form shown p r e v i o u s l y and  condition  f a i l u r e under almost c o n c e n t r i c  failure  represents  This loading  in  These  figure  8,  code requirements i n  Chapter 9. 5.2 SEPARATE BENDING AND AXIAL LOADING RESULTS 5.2.1 Test As  Results  described  bending,  in  previously,  axial  tension,  testing and  ( r e s t r a i n e d to prevent b u c k l i n g ) , 38x140mm and  sizes.  The  in  was  axial  been  the f i t t e d Weibull  i n Appendix C, with a p l o t  elsewhere It the and  data,  in  overlayed  d i s t r i b u t i o n and a number of relevant  The bending t e s t r e s u l t s from stage  included  compression  r e s u l t s of these t e s t s f o r the stage 2  stage 4 t e s t i n g are i l l u s t r a t e d  statistics.  out i n  f o r both the 38x89mm and the  of the cumulative d i s t r i b u t i o n of the ranked with  carried  Appendix  C  5  have  because they w i l l be reported  (Madsen 1983).  i s i n t e r e s t i n g to compare the modulus of rupture  a x i a l t e n s i o n and compression strengths size  comparison interest.  not  of  material. for  the  Figures  two  The s t r e n g t h  sizes.  27(a)  and  f o r a given (b)  show  with grade this  There are s e v e r a l p o i n t s of  in a x i a l tension  i s much more v a r i a b l e  111  —  TENSION  •-  COMPRESSION  —  BENDING  £2m *©_  <X o m o in Q _ tfi  o"  LLi CC  o  m O o-  =1  1  r~—r—  r  1  i  -  <0.0  70.0  6a 0  50.0  STRENGTH IrlPfl)  r  n  80.0  i  r ao.o  mo  30.0  100.0  (a) 38 x 89 mm TENSION  '  /  COHPRESSION /  1  BENDING  l  ; / < /  /  /  /  /  /' /  7  /  m rx < 1  I  CQ o cc Q_ i i UJ > cx ,  \  (_)  / /  /  1  1  7  / < /" / < / / ' 7  / 0.0  10.0  •  2ao  /  30J)  so.a eao STRENGTH (flPfl) '  40.0  70J  (b) 3 8 x U 0 m m F i g u r e 27 - Comparison of t e n s i o n , compression and bending t e s t r e s u l t s  112  than i n a x i a l compression, t e n s i o n s t r e n g t h being  greater  than  compression s t r e n g t h at the strong end, l e s s at the weak end. The  modulus of  strength  rupture  throughout  is  the  much  greater  distribution,  than and  the  tension  a  similar  has  c o e f f i c i e n t of v a r i a t i o n . The  r e l a t i o n s h i p between these three s t r e n g t h  w i l l be explored In  in d e t a i l  figure  24(b)  later.  i t was  shown  how  i n t e r a c t i o n diagram i s q u i t e d i f f e r e n t at the  distribution  of  strength.  r e s u l t of the a c t u a l rather  a  properties  levels  The  in  the  shape of the  various  levels  differences  the  in  are not a  distribution,  but  are  r e f l e c t i o n of the changes i n the r e l a t i v e values of  t e n s i o n and compression s t r e n g t h s ,  shown i n f i g u r e 27.  5.2.2 Modes of F a i l u r e A very observed,  large many  number of  of  them  different  related  failure  modes  to s p e c i f i c d e f e c t s  were i n the  boards. In t e n s i o n most f a i l u r e s were at a s i n g l e or  in  a  length  of  board  l e s s than two or three  l a r g e s t c r o s s s e c t i o n dimension, with elsewhere i n visible  and  perpendicular brittle energy.  the  board.  audible  Some crack  t o the g r a i n .  fractures Many  associated  failures  cross  were  little  failures growth,  A l l the with  section times the  d i s t r e s s observed were  preceded  usually failures  by  in  tension  were  sudden  sudden r e l e a s e of stored  associated  with  local  or  g e n e r a l i z e d s l o p i n g g r a i n , o f t e n around knots.' For  axial  compression  restrained  against  buckling,  113  failures  tended to f a l l  ductile  crushing  type  w r i n k l e s , as d e s c r i b e d brittle and  i n t o two of  failure  sideways  the  The  between  This  of  a  concentration  perpendicular  to  yielding  often  was  maximum load was  the  induces  grain. visible  In  with  failure  at  tension  both  cases  several  to  a l r e a d y d e s c r i b e d f o r t e n s i o n and  of  compression  compression  cross  s e c t i o n s as  be  consistent  compression.  yielding.  Some of these  The  most  common  mode of f a i l u r e was  y i e l d i n g near the top s u r f a c e at one e v e n t u a l l y followed by a b r i t t l e Some boards snapped i n t o two  failed  a  less  sudden  or more  failure  reduction  in  no were  audible  compression  cross  sections,  in the t e n s i o n zone.  p i e c e s while others hung  i n the c e n t r a l o n e - t h i r d of the  shear f a i l u r e s  and  with  Many of  failures  warning, others were preceded by v i s i b l e  cracking.  with  was  stresses  the weaker boards f a i l e d suddenly i n the t e n s i o n zone with  without  a  approached.  F a i l u r e modes i n bending tended  sign  more  separated  supports,  type  a  compression  second was  lateral  was  i n i t i a t e d at a knot near the centre of the board where stress  those  first  halves of the specimen  longitudinal splitting failure. usually  The  with v i s i b l e  f o r c l e a r wood.  f a i l u r e mode when two  buckled  categories.  -load. span.  together  Almost a l l boards No  horizontal  were observed, even f o r the s h o r t e s t span.  114  5.3  LENGTH EFFECTS  5.3.1  Introduct ion A  was  small but  designed  important  part of the experimental  to produce 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  compression, a x i a l t e n s i o n , and test results equation  procedure  to  estimate  i n bending.  values  of  in a x i a l  T h i s s e c t i o n uses  the  parameter  k,  in  3.11.  If c r o s s s e c t i o n s t r e n g t h i s a .quantity that v a r i e s along the  length  of  a t y p i c a l board, then the average s t r e n g t h of  long boards can be expected to be lower boards.  If  a  effect. Chapter  l e n g t h , knowledge of the amount  typical The  board  can  of  of  short  variability  be used to q u a n t i f y the  v a l i d i t y of this.assumption  has  length  been d i s c u s s e d i n  3.  There are apparently effects  no previous  reported  s t u d i e s on  of specimen length on the s t r e n g t h of timber  t e n s i o n or compression. at a constant stresses  ratio,  producing  f o r longer and deeper members, but effects  independently. into  the  and  Madsen and effect  depth Nielsen  effects  lower  be  out  failure  such t e s t s do to  the  in a x i a l  Bending t e s t s are u s u a l l y c a r r i e d  span-to-depth  allow l e n g t h  study  that  c r o s s s e c t i o n s t r e n g t h i s assumed to be a random  v a r i a b l e with within  than  not  quantified  (1976) r e p o r t a p r e l i m i n a r y  of l e n g t h on bending s t r e n g t h , which  produced r e s u l t s s i m i l a r to those  d e s c r i b e d below.  115  5.3.2  Compression .Strength Ten boards of 38x89mm s i z e , 2.9m long were c u t i n t o 135mm  long segments. i n t o 210mm segments  Ten boards of 38x140mm s i z e 4.0m long were c u t  long  segments.  In  cases  the number  of  v a r i e d from 15 t o 19. The segments were c o n s e c u t i v e  along each board, although clear  both  occasionally  a  small  amount  of  wood was e l i m i n a t e d between segments i n order to l o c a t e  a s i g n i f i c a n t d e f e c t near the c e n t r e of a segment. Each segment was t e s t e d to f a i l u r e as  described  i n Chapter 4.  for the segments variation  of  calculated. segment  within segment  in axial  compression  The f o l l o w i n g a n a l y s i s was made  each  board.  strength  The  within  coefficient each  For both s i z e s the c o e f f i c i e n t  of  board  of was  variation  of  s t r e n g t h s w i t h i n each board was i n the range of 7% t o  13% with an average value of 9%. A two parameter W e i b u l l model was f i t t e d strengths  within  each  board  using  a  to  the segment  maximum  likelihood  routine.  For both s i z e s the shape parameter v a r i e d between  and  the average  19,  value  being  close  of  variation  c o n s i s t e n t with the c o e f f i c i e n t using  equation  Weibull  3.9.  The  distribution  investigated.  possibility  giving  a  to  13.  This i s  reported  above  of a three-parameter  different  result  was  In t h i s case, however, the d i s t r i b u t i o n a l  of the data i s such  that  there  9  i s very  little  difference  between  the two d i s t r i b u t i o n s and both p r e d i c t s i m i l a r  effects,  so the two parameter  form has been  form  length  used.  Length e f f e c t s can be c a l c u l a t e d from equation  3.11.  A  116  parameter  of  k,=13  0.948 f o r d o u b l i n g two-parameter effect  length.  Weibull  a s t r e n g t h r e d u c t i o n f a c t o r of  Because of  the  assumption  distribution,  the  predicted  of  strength  T h i s average f i g u r e , p r e d i c t i n g for  doubling  boards, but i t w i l l  length  to  a 5% r e d u c t i o n  l e n g t h , c o u l d vary  be used as the  a  length  i s the same at a l l s t r e n g t h l e v e l s from the weakest  the s t r o n g e s t . in  produces  for individual  effect  factor  in  tests  of  compression f o r the r e s t of t h i s study. 5.3.3  Tension Strength It  is  consecutive  not  possible  segments  of  because a c o n s i d e r a b l e  to a  carry  board  direct  approach  boards of d i f f e r e n t results  of  is  to  Two  carry  lengths.  bending  tension  done  in  approaches have been- used. out a x i a l t e n s i o n t e s t s of  A second approach i s to use  tests  of  different  lengths  assumption that bending s t r e n g t h i s governed the t e n s i o n a. Axial  compression,  length of board i s r e q u i r e d i n g r i p s at  each end of the t e s t specimen. A  as  out  by  on  failures  the the in  zone.  38x89mm Boards t e n s i o n t e s t s were c a r r i e d out on two lengths of boards  as d e s c r i b e d  i n Chapter 4.  shown  the other  with  distributions  The  strength  test results  distributions  i n Appendix C.  have been used to c a l c u l a t e  the  The  ratio  are fitted  of  the  t e n s i o n s t r e n g t h s of long boards to those of short boards, f o r the  whole  figure  28.  distribution.  The  resulting  I f the r a t i o i s taken to be  distribution,  then  we  can c a l c u l a t e  plot  0.91  is  shown i n  throughout  the  the shape parameter f o r  1 17  T  O.D  0.4  0.5  CUMULATV IE RANK  R a t i o of t e n s i o n strengths of 38x89mm boards 2.0m and 0.914m long  F i g u r e 28  w i t h i n board segment strengths value  of  k,=8.3,  which  from equation  giving  a  reduction  the l e n g t h .  38x140mm Boards  The  38x140mm m a t e r i a l used i n experimental  not  the subject of any l e n g t h e f f e c t  At  a  stages  1 and 2  was  investigation in tension.  l a t e r date however, comparative t e s t i n g was c a r r i e d out  on two lengths of 38x140mm timber 100  3.11,  corresponds to a s t r e n g t h  f a c t o r of 0.92 f o r doubling b.  3.0  0.6  boards  were  tested  i n Stage  5.  Approximately  a t each length of 3.0m and 0.91 m.  F i g u r e 29 shows the r a t i o between the  strengths  of  the two  l e n g t h s , over the whole d i s t r i b u t i o n . The 38x89mm  difference size.  difference  A  i s that  i s much possible  larger  than  explanation  of  for  the m a t e r i a l was ordered  with as many s t r e n g t h reducing d e f e c t s as ratio  observed f o r the this  as No.  possible.  large 2 grade If the  s t r e n g t h of long boards t o short board.s i s taken to  1 18  0.0  i  T  1  i  0.1  1  0.2  1  1  0.3  1  1  0.4  1  1  0.5  1  1  1  0.6  1  D.l  CUMULATIVE RANK  1  1  1  r  O.S  D.B  1.0  F i g u r e 29 - R a t i o of t e n s i o n s t r e n g t h s of 38x140mm boards 3.0 and 0.914m long. be 0.67, that corresponds to  a  shape  parameter  which i m p l i e s a s t r e n g t h r e d u c t i o n f a c t o r  of  k,=3.0,  of 0.80 f o r doubling  the l e n g t h , from equation 3.11. 5.3.4  Bending A  Strength  central  hypothesis  of  this  study  behaviour can be p r e d i c t e d from a x i a l behaviour.  Accordingly,  able to be q u a n t i f i e d compression.  If  flexural  length  i n tension.  in  i s governed  at  distribution.  case the length e f f e c t  expected t o be s i m i l a r As  the  increases,  ratio  the  the  low  end  of  to that observed i n a x i a l  of  tension  compression  and  s o l e l y by  tends to be r e l a t i v e l y weak  in t e n s i o n , p a r t i c u l a r l y this  tension  should be the same i n bending  Commercial timber  In  compression  i n bending should be  effects  strength  tension strength, length e f f e c t s and  t e n s i o n and  length e f f e c t s  from  i s that bending  to  strength  the  strength  i n bending i s tension.  compression  strength  of the m a t e r i a l has an  119  i n c r e a s i n g i n f l u e n c e on the bending s t r e n g t h , with  high  tension  strength  the length e f f e c t  expected to approach a x i a l compression In order becomes  to  investigate  necessary  to  length  have  an  s t r e s s e d l e n g t h of the beam. integral beam  i n equation  3.6  so  effects  it  can be evaluated  calculated  i n bending i s  in for  was  beams, the  shown  it  highly how  the  over the volume of a  to give the e q u i v a l e n t volume of equation  e f f e c t s are excluded,, an e q u i v a l e n t  timber  values.  expression  Earlier  for  3.7.  If depth  s t r e s s e d l e n g t h Le can  be  as 1 L  e  "  +  r i k  T^rl  L  (5.1)  where L i s the span- of the beam, a i s the d i s t a n c e between symmetrical  point  loads and  k,  i s the r e l e v a n t length  two  effect  factor. a.  38x89mm Boards  Bending t e s t s of two carried  out  as  different  described  r a t i o of s t r e n g t h s of expected  there  the  lengths of 38x89mm timber  were  in Chapter 4.  F i g u r e 30 shows the  long  short  to  the  i s a greater, l e n g t h e f f e c t  boards.  As  f o r the weak boards  than f o r the strong boards in the d i s t r i b u t i o n , but the a c t u a l magnitude of the l e n g t h e f f e c t the  tension  weak end. One  The  possible  bending t e s t s .  and  i s l a r g e r than  predicted  compression parameters, p a r t i c u l a r l y at  reasons for t h i s discrepancy factor At the  is  the  weak  short end,  are  stressed  bending  not  from the  clear.  length in the  strength  depends  1 20  "T  1  0.2  1  1  0.3  1  1  0.4  1  1  r  1  0.5  ] .D  0.6  CUMULATIVE RRNK  F i g u r e 30 - R a t i o of bending strengths of 38x89mm boards 1.5 and 0.84m long entirely  on  tension  compression s t r e n g t h . parameters and 2.0m.  strength The  with  tension  were d e r i v e d from t e s t  strength  length  from effect  In the short span bending t e s t r e f e r r e d to here the  It  length  was  less  than  i s p o s s i b l e that lengths as short as t h i s may not  c o n t a i n many of the d e f e c t s found i n longer unexpectedly high s t r e n g t h b.  contribution  r e s u l t s at lengths of 0.9lm  span was 0.84m so the h i g h l y s t r e s s e d 0.3m.  no  l e n g t h s , hence the  values.  38x140mm Boards  R e s u l t s from bending t e s t s at two spans are a v a i l a b l e from the stage 5 t e s t i n g .  F i g u r e 31(a) shows the r a t i o of strengths of  the 3.0m span t o the 1.5m span f o r three load for  select  s t r u c t u r a l grade.  i d e n t i c a l l y , but there  configurations,  The three samples do not behave  i s a c o n s i s t e n t o v e r a l l trend  f o r the  s t r e n g t h of the long boards t o be about 0.89 times that of the shorter  boards,  which  corresponds t o a shape parameter  from  121  LU '  CH  CO ^ 2 a'  o  X CO  o  O  cr  0.0 1  1 0.2 r  1 0.1  ~\ 0.3  1  1  0.4  1  0.0  1  1  0.1  1 ' 0 . 2  r—i  1  0.5  1  1  0.6  CUMULATIVE RANK  (a) Select  cn D ' 1  1  1  0.3  1  o.i  i  1  0.6  1  r  0.9  1.0  r  I.D  Structural Grade  1  0.4  r  1  1  0.5  1  1  0.6  CUMULATIVE RANK (b) Number  2  1 0.1  r  i 0.8  0.9  Grade  F i g u r e 31 - R a t i o of bending strengths of 38x140mm boards, 3.0 and 1.5m long equation 3.11 of k,=5.9. 0.80 t o 1.02. the d i r e c t length  The range of values  This result  observed i s from  i s very c l o s e to that  obtained  a x i a l t e n s i o n t e s t s , and a trend towards  effect  at. the  seen as p r e d i c t e d .  in  decreasing  strong end of the d i s t r i b u t i o n  can be  122  F i g u r e 31(b) shows boards, The with  where  the  same  average  produce  same m a t e r i a l .  to 0.90.  grade  very  similar  results  r a t i o of about 0.84, which i s a l i t t l e more  than the f i g u r e of 0.80 obtained the  f o r Number 2  a somewhat g r e a t e r l e n g t h e f f e c t can be seen.  three load c o n f i g u r a t i o n s an  plot  from d i r e c t t e n s i o n t e s t s  The range of observed v a l u e s  on  i s from 0.79  A f a c t o r of 0.84 corresponds to a shape parameter of  k,=4.0. 5.3.5 Summary of Length E f f e c t s The  r e s u l t s presented  variabilitybut  1.  For  in this section  show  the f o l l o w i n g c o n s i s t e n t trends can be seen:  axial  compression  the  observed length  produces a s t r e n g t h r e d u c t i o n f a c t o r of doubling  2.  considerable  about  effect  0.95 f o r  length.  For  axial  tension  and , for  bending, the observed  l e n g t h e f f e c t appears to be grade dependent.  For  Structural  factor for  doubling be  used  grade  the  strength  l e n g t h i s approximately  reduction 0.89.  This figure  f o r the remainder of t h i s study.  0.84.  will  For Number 2  grade, the s t r e n g t h r e d u c t i o n f a c t o r f o r doubling i s approximately  Select  length  123  5.4  WEAK A X I S  The  BENDING  size  effect  theory  derived  f o r timber  members  suggests that f o r boards of a given l e n g t h , modulus of rupture should be l e s s f o r edgewise  bending than f o r f l a t w i s e  bending.  The reasons f o r t h i s have been d i s c u s s e d i n Chapter 3. For  each  of  the  two  c r o s s s e c t i o n s i z e s used i n t h i s  study, a comparative t e s t was made of bending each  principal axis.  strength  about  The same span was used f o r each, loading  o r i e n t a t i o n to e l i m i n a t e any p o s s i b l e length e f f e c t .  From the  r e s u l t s of these t e s t s , the r a t i o s of the s t r e n g t h s a r e in  f i g u r e 32.  that  the  strength  two  The unexpected r e s u l t shown i n f i g u r e 32(a) orientations  f o r the  38x89mm  have  brittle  fracture  almost  size.  a v a i l a b l e f o r t h i s behaviour. from  exactly  orientation,  but  the  is  same  No obvious e x p l a n a t i o n i s  T h i s r e s u l t would be  predicted  theory a p p l i e d to a p e r f e c t l y  m a t e r i a l , because the h i g h l y s t r e s s e d volume i s each  shown  the  brittle same  in  i t i s i n c o n s i s t e n t with other r e s u l t s  observed i n timber members. For  the l a r g e r s i z e , f i g u r e  difference  32(b)  shows  a  significant  between the two o r i e n t a t i o n s , as expected,  because  there i s a l a r g e decrease i n depth and i n c r e a s e i n width the  orientation The  i s changed from edgewise  tentative  conclusion  from  to f l a t w i s e .  these few t e s t s i s that  bending s t r e n g t h p r o p e r t i e s o b t a i n e d from strong a x i s tests  when  bending  can be c o n s e r v a t i v e l y used t o p r e d i c t weak a x i s bending  s t r e n g t h , the degree of c o n s e r v a t i s m depth t o width r a t i o i s i n c r e a s e d .  probably  increasing  as  124  1  ~i  0.1  O.D  0.2  1—n  1  0.3  0.4  1  1  0.5  1  1  0.6  1  1  0.1  CUMULATIVE RANK  1  1  0.8  1 0  9  r  I.D  (a) 38 x 89 mm  — I CO  o  a  10  2  a'-  LU CO l-H CT _J Ll_ LU CO  Lu  =>  I 0.0  I 0.1  I  I  I  0.2  I  I  I  I  0.4  0.3  1  1  0.5  1 D.6  1  1  1  0.1  1  1  0.8  1  1—  0.9  1.0  CUMULATIVE RANK  (b) 38xU0mm Figure  32 - R a t i o  of edgewise t o f l a t w i s e bending strength  5.5 SUMMARY T h i s chapter has d e s c r i b e d experimental many  more  interest  programme, results  are those  are  the trends observed i n a l a r g e  and has r e f e r r e d t o Appendix C where summarized.  The  results  f o r eccentric a x i a l loading  of  most  t e s t s which  125  h a v e n o t been c a r r i e d o u t on t h i s  scale  E x p e r i m e n t a l r e s u l t s have been u s e d effects, this  which  extent.  previously. to  a l s o have n o t been p r e v i o u s l y  quantify  length  investigated to  126  VI.  STRENGTH MODEL  6.1 INTRODUCTION T h i s chapter  d e s c r i b e s a t h e o r e t i c a l model f o r p r e d i c t i n g  the s t r e n g t h of timber members. compression  strength  predict strength  The model  information  from  uses  tension  in-grade  and  t e s t i n g to  i n bending or i n e c c e n t r i c a x i a l  loading  for  members of any l e n g t h . The  theory  described, to  make  along the  computer  assumptions  in  the  model  with the b a s i s of the computer  necessary  w i l l be d e s c r i b e d the  and  numerical  i n two separate  program  calculates  program  calculations.  parts. the  The  been w r i t t e n s p e c i f i c a l l y  first  behaviour  the program capacity can  uses  this  information  of an e c c e n t r i c a l l y  loaded  to  used  part  of  of a c r o s s This  part  The second part of calculate  the  load  column of any length, and  be used f o r members of any m a t e r i a l . The  o r g a n i z a t i o n of the program i s based  program  developed  p r e s t r e s s e d concrete, Alcock  and  completely are  f o r timber.  be  The program  s e c t i o n of given geometry and m a t e r i a l p r o p e r t i e s . has  will  quite  by  Nathan(1983a)  and  verified  Nathan (1977).  and by  part has, however, been  from those i n c o n c r e t e .  i s almost e x a c t l y as d e s c r i b e d  similar  materials  r e - w r i t t e n because the f a i l u r e mechanisms different  a  for reinforced  f o r those  The f i r s t  on  by Nathan.  i n wood  The second part  127  6.2 ASSUMPTIONS The  f o l l o w i n g assumptions are made:  1.  Plane s e c t i o n s remain  plane.  2. Timber stressed i n tension behaves elastic manner u n t i l brittle fracture l i m i t i n g tension s t r e s s .  in a occurs  linear at a  3. Timber s t r e s s e d i n compression behaves . i n a nonlinear d u c t i l e manner. Any shape of s t r e s s - s t r a i n curve can be used. A limiting compression strain may be specified. 4. Stress-strain of l o a d i n g .  r e l a t i o n s h i p s are independent of r a t e  5. A x i a l t e n s i o n and compression s t r e n g t h s member length i s i n c r e a s e d .  decrease  as  6. In both t e n s i o n and compression, the maximum a t t a i n a b l e s t r e s s at a c r o s s s e c t i o n i s a f u n c t i o n of the p r o p o r t i o n of the s e c t i o n s u b j e c t e d t o that s t r e s s . 7. I f moment v a r i e s along a member, f a i l u r e occurs the c r o s s s e c t i o n subject e d t o maximum moment. 8. Modulus of e l a s t i c i t y each board.  i s constant  at  along the l e n g t h of  9. No torsional or out-of-plane deformations are considered. Duration of load e f f e c t s are not c o n s i d e r e d . Shear failures are not c o n s i d e r e d . Some of these l i m i t a t i o n s are d i s c u s s e d f u r t h e r i n Chapter 9.  6.3 CROSS SECTION BEHAVIOUR The  ultimate  interaction  diagram  c u r v a t u r e - a x i a l load r e l a t i o n s h i p s f o r a derived  using a simple  step-by-step  and cross  the  section  procedure to o b t a i n  load and moment c a p a c i t i e s f o r a range of n e u t r a l a x i s and  curvatures.  momentare axial depths  R e c a l l that the u l t i m a t e i n t e r a c t i o n diagram  shows l i m i t i n g combinations of a x i a l  l o a d and  bending  moment  128  that a c r o s s  s e c t i o n can r e s i s t .  6.3.1 C a l c u l a t i o n For  a cross  typical  The  s e c t i o n such as that  calculation  and a p p l y i n g produce  Procedure shown i n f i g u r e 33(a), a  begins by s e l e c t i n g a n e u t r a l a x i s  an i n i t i a l  curvature,  <p, to  the  depth  section,  to  the s t r a i n s shown i n the t y p i c a l case of f i g u r e 33(b).  following  procedure i s used to determine what combination  of a x i a l load and moment would be necessary  to  produce  this  condition.  Figure The  depth  segments.  using 33(c). 33(d).  of  For  calculated,  and  an input The  33 - Cross s e c t i o n the  each used  section  segment  behaviour  i s divided  the  i n t o a number of  mid-height  strain  is  to c a l c u l a t e the corresponding s t r e s s ,  s t r e s s - s t r a i n r e l a t i o n s h i p as shown  in  figure  r e s u l t i n g s t r e s s d i s t r i b u t i o n i s shown i n f i g u r e  The f o r c e  i n each segment i s c a l c u l a t e d , producing the  distribution  shown i n f i g u r e 33(e).  compression  segment  forces  A l l of  the  tension  are combined i n t o a s i n g l e  for each s t r e s s block as shown i n f i g u r e 3 3 ( f ) , and these  and force are  129  combined  to give the net a x i a l f o r c e and bending moment about  the c e n t r o i d a l a x i s , as shown i n f i g u r e 33(g). At t h i s stage repeating  the  four items of information a r e s t o r e d  calculation  with  increased  before  curvature.  The  stored information i s  For  1.  N e u t r a l a x i s depth  2.  S e c t i o n curvature  3.  Net a x i a l load  (output)  4.  Bending moment  (output)  the  repeated  chosen with  compression  neutral  (input) (input)  axis  location,  i n c r e a s i n g curvature strain  until  the c a l c u l a t i o n i s either  a  limiting  i s exceeded, the extreme f i b r e s t r e s s i n  t e n s i o n 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  for a number neutral  of  axis  calculated equation  other  neutral  location  from 3.18  d e s c r i b e d . t o t h i s point i s then  the or  the  input  3.20  with  axis  locations.  limiting"  axial k  3  tension  tension  repeated For  each  stress  strength  is  using  as the s t r e s s - d i s t r i b u t i o n  parameter f o r t e n s i o n . To in  include 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  the  calculation,  each curvature  f o r compression  the compression s t r e n g t h  increment using equation  is, modified at  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  c h a r t of f i g u r e 34. member w i l l  i s illustrated  s c h e m a t i c a l l y i n the flow  Output from a t y p i c a l run f o r a  be used i n the next s e c t i o n s to i l l u s t r a t e  38x140mm several  130  a s p e c t s o f t h e program.  S e l e c t f i r s t n.a. depth f  increment n.a. depth  modify tensicjn s t r e n g t h f o r t h i s n a. depth  TfT  -  apply i n i t i a l c u r v a t u r e 41 Increment curvature YES  YES  modify comp. s t r e n g t h f o r t h i s n.a. and t h i s  calculate internal tension and compression f o r c e s from s t r e s s - s t r a i n relations  c a l c u l a t e net a x i a l l o a d and moment YES  F i g u r e 34 - F l o w c h a r t f o r c a l c u l a t i n g m o m e n t - c u r v a t u r e a x i a l load relationships f o r a cross section  131  6.3.2 N e u t r a l Figure and  moment  positive. bilinear  Axis  Contours  35 i l l u s t r a t e s the r e l a t i o n s h i p between a x i a l load f o r f o r t y neutral axis contours. T h i s p a r t i c u l a r f i g u r e has been  stress-strain  Compression i s  obtained  using  a  r e l a t i o n s h i p with a f a l l i n g branch as  shown i n f i g u r e 6. Each location. point  radial  on the l i n e represents  been  occurred, top  represents  single  increased  axis  successive  the combination of a x i a l load and  step-by-step  until  a  Curvature  tension  failure  or u n t i l the moment dropped to zero (as seen i n the The curves have been c o n s t r u c t e d  from  segments between p o i n t s of i n c r e a s i n g c u r v a t u r e .  The  n e u t r a l a x i s l o c a t i o n s have been s e l e c t e d the  neutral  to produce a p a r t i c u l a r c u r v a t u r e .  left-hand corner).  straight  a  For any one n e u t r a l a x i s l o c a t i o n , each  moment r e q u i r e d has  line  s e c t i o n on the t e n s i o n  with  five  s i d e , eighteen o u t s i d e  on the compression s i d e , and the remainder w i t h i n  outside  the s e c t i o n the s e c t i o n ,  to p r o v i d e a s u i t a b l e range of c a l c u l a t i o n p o i n t s . This  f i g u r e of n e u t r a l a x i s contours i s used to c o n s t r u c t  an envelope, or u l t i m a t e  i n t e r a c t i o n diagram, showing p o s s i b l e  combinations of a x i a l load and bending moment f o r t h i s section. any  cross  The a c t u a l l i n e s shown i n f i g u r e 35 are not used f o r  other purpose i n t h i s study.  1 32  n  o.o  1  2.0  MOMENT  1  3.0  r — r  1  (KN.M)  F i g u r e 35 - N e u t r a l a x i s contours load 6.3.3 Curvature Figure  f o r moment and a x i a l  interaction  Contours  36 shows the same p o i n t s as f i g u r e 35, p l o t t e d i n  another way t o show the r e l a t i o n s h i p between moment  f o r t h i r t y curvature  contours.  s i n g l e value of s e c t i o n c u r v a t u r e . of constant and  6.0  4.0  curvature  represent  moment r e q u i r e d to  produce  axial  Each l i n e  load  and  represents a  D i f f e r e n t p o i n t s on a l i n e  the combinations of a x i a l that  curvature  load  f o r various  200.0  134  neutral axis locations. In the  both  f i g u r e s 35 and 36, the contours are s t r a i g h t i n  range of l i n e a r e l a s t i c behaviour.  curved  when  the  proportional  limit.  superimposed, number  of  curvatures  the  i s stressed If  figures  intersection  combinations  of  contours  Figure  i n compression beyond the 35  and  points  neutral  36  were  would axis  to  be  represent  the  locations  and  I n t e r a c t i o n Diagram  37  is  the  ultimate  i n t e r a c t i o n diagram f o r the  s e c t i o n , being the envelope of a l l the p o i n t s  f i g u r e s 35 and 36. axis  become  f o r which i n t e r n a l f o r c e s were c a l c u l a t e d .  6.3.4 Ultimate  cross  wood  The  contours  T h i s curve i s c o n s t r u c t e d  from.the  in  neutral  shown - i n f i g u r e 35, and c o n s i s t s of s t r a i g h t  l i n e segments between a f a i l u r e p o i n t location.  shown  Any p o i n t  at  each  neutral  i n s i d e the curve represents  of a x i a l load and moment that the c r o s s  axis  a combination  s e c t i o n can r e s i s t .  6.3.5 Moment-Curvature Curves Figure  38  shows  the moment-curvature r e l a t i o n s h i p s f o r  s e v e r a l l e v e l s of a x i a l load, P. strength each The  of the m a t e r i a l .  has  Pa i s the a x i a l  A l l curves s t a r t at the o r i g i n , but  been s h i f t e d s l i g h t l y  program  presented.  computes Each  these  of  for clarity l  curves  from  line  A-B  38.  the  presentation. data  on  figure  already  36.  the h o r i z o n t a l l i n e with a curvature  p r o v i d e s a value of moment and c u r v a t u r e figure  of  l e v e l of a x i a l load can be represented by a  h o r i z o n t a l l i n e such as the intersection  compression  The f i n a l point  which i s  Each contour  plotted  on each moment curvature  on  curve  135  0.0  0.02  0.04  0.06  0.1  0.08  0.12  0.14  CURVATURE d/m)  0.16  0.2  0.18  F i g u r e 38 - Moment-curvature-axial load r e l a t i o n s h i p s represents the  respective  curvature but  the p o i n t on the u l t i m a t e i n t e r a c t i o n a x i a l load.  curves  diagram f o r  For high a x i a l loads the moment-  have a f a l l i n g branch beyond maximum  moment,  that has not been p l o t t e d because the i n f o r m a t i o n  i s never  used i n the subsequent c a l c u l a t i o n s . All column  the  information  stability  required  behaviour  c u r v a t u r e - a x i a l load  f o r c a l c u l a t i o n of long  i s contained  in  the  moment-  curves.  6.4 COLUMN BEHAVIOUR The  second  s e c t i o n of the computer program i n v e s t i g a t e s  the behaviour of columns of any l e n g t h eccentric lateral  axial  loads  with  equal  l o a d as shown i n f i g u r e 39.  under  the  action  of  end e c c e n t r i c i t i e s and no  136  Figure  39 - Column with a x i a l load and equal end eccentric i t i e s  6.4.1 F a i l u r e Modes The behaviour member  -of an  eccentrically  compression  w i l l be reviewed i n t h i s s e c t i o n , i n order  terms used i n the e x p l a n a t i o n the  loaded  possible  behaviour  eccentricities increased distance  to e,  as  be P times e.  of the computer model. a  member  with  Consider  equal  end  shown i n f i g u r e 39, as the a x i a l load P i s  failure.  the  of  d e f i n e some  If  the  end  eccentricity  is  a  moment at the ends of the member w i l l always  The moment at mid-span w i l l be P(e+A)  where  A  i s the mid-span d e f l e c t i o n as shown. Figure moment.  40(a) i s an i n t e r a c t i o n diagram of a x i a l load v s .  The  outer  curved  diagram r e p r e s e n t i n g  material  example  loading,  axial The  of  typical  line  i s the u l t i m a t e i n t e r a c t i o n  failure  load  37).  As  an  l i n e O-A shows the load path f o r  load and end moment as the a x i a l corresponding  (figure  load P  is  increased.  path f o r mid-span moment i s shown by  137  ^ Material  failure  O O  5  P  c  2  \  _D  \E  / / / / // //  A  )B  1/  Moment (a)  Material  failure  a o a x < End  moments at failure  F i g u r e 40 - T y p i c a l  i n t e r a c t i o n diagram f o r e c c e n t r i c a l l y loaded column  the  curved l i n e O-B.  The h o r i z o n t a l  d i s t a n c e between l i n e s 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 member  fails  O-B i n t e r s e c t s  case  the  at an a x i a l load P, when the mid-span load  path  the m a t e r i a l  to  P(e+A).  strength  In  this  interaction  diagram  at  138  p o i n t B.  T h i s i s d e s c r i b e d as a m a t e r i a l  failure.  If the same member i s loaded with a s m a l l e r the  load path  the l o a d path an  eccentricity,  f o r end moments c o u l d be shown by l i n e 0-C, and f o r mid-span moments by l i n e 0-D.  instability failure  maximum value P .  occurs  when the a x i a l  In t h i s  case  load reaches a  The mid-span moment at f a i l u r e  i s shown  by  p o i n t D, which i s w e l l i n s i d e the m a t e r i a l s t r e n g t h curve.  If  2  the  member- were loaded with a system under l o a d c o n t r o l ( f o r  example,  gravity  increase  loads)  rapidly  immediately. controlled  and  to a  load  P , 2  material  failure  I f the member were loaded displacement  deformations  the l o a d path  under  would  would' follow  conditions  of  shown by the extension  c  of the l i n e 0-D could be followed to eventual at  material  failure  p o i n t E. • I f t h i s process  member  using a f u l l  be produced. diagram.  i s repeated  dotted  line  ultimate  For  low  interaction  For higher  axial  line  interaction  Pu-D-B-Mu i s the locus of p o i n t s r e p r e s e n t i n g combinations  axial  causing  l i n e c o - i n c i d e s with the  diagram, i n d i c a t i n g  material  loads i n s t a b i l i t y f a i l u r e s  occur.  failures. The c h a i n -  r e p r e s e n t i n g combinations of a x i a l  end moment (unmagnified load  loads t h i s  of  Pu-C-A-Mu i s the locus of p o i n t s such as A and C  in f i g u r e 40(a),  axial  f i g u r e 40(b) can  load and mid-span moment (magnified moment) j u s t  failure.  dotted  f o r the same  i s the same u l t i m a t e  such as B and D i n f i g u r e 40(a), axial  times  range of e c c e n t r i c i t i e s ,  The s o l i d l i n e  The  many  the  moment) j u s t  horizontal  causing  distance  failure.  load  and  For any  between the two dotted  139  lines  represents  the ' moment  magnification  due  to  member  deformations at f a i l u r e . 6.4.2 C a l c u l a t i o n Procedure A  computer  program f o r c a l c u l a t i n g p o i n t s on the curves  shown i n f i g u r e 40(b) w i l l be d e s c r i b e d below. The program can c o n s i d e r a column of any l e n g t h of  a  number of segments of "equal  by Galambos(1968) for  a  given  length.  made  A method  described  i s used t o develop column d e f l e c t i o n  axial  load,  to  determine  e c c e n t r i c i t y , e, at which that load  can  the  be  up  curves  maximum  applied  end  to the  column. As always  the  column  zero  at  consideration,  i s symmetrically  mid-span.  For  the  i n t e r a c t i o n diagram). (from  the  A i t i s necessary  along  A  line  of  axial  load  under  set to the  deflection,  i s the f a i l u r e moment  To f i n d the a c t u a l values of e and  to c a l c u l a t e  column,  mid-span  load)  the  column d e f l e c t i o n curve  the  axial  (a p o i n t on the u l t i m a t e  The corresponding  d i v i d e d by the a x i a i l o a d .  member.  the slope i s  the moment at mid-span i s i n i t i a l l y  m a t e r i a l f a i l u r e moment f o r that l o a d  e+A,  loaded,  deflected  shape  i s obtained  segment-by-segment  of the  by proceeding  from  mid-span,  c a l c u l a t i n g the d e f l e c t i o n at each node. Consider such and  as  that shown i n f i g u r e 41(a).  slope v '  moment,  calculations for a typical  M  1r  approximately  0  are known at the at  the  mid-point  starting  segment  of length Ax,  If the d e f l e c t i o n v node  of the segment  x , 0  then  0  the  (point x,) i s  140  F i g u r e 41 - Column d e f l e c t i o n  curves (6.1 )  The  curvature,  <£  at  1f  moment-curva.ture-axial curvature  is  point  load  assumed  to  x,  can  be obtained from the  relationship be  constant  (figure along  38).  The  the segment.  D e f l e c t i o n s are assumed to be small such that <£=v". The displacement, v , and s l o p e , v ' , at the 2  x , are c a l c u l a t e d  2  next  node,  from  2  =  v  o  • +  =  (6.2)  v * ( A x ) o ' v  vt  _  +  n  ( A x )  (6.3)  141  The moment M This  calculation  i s reached. line  of  at node x  2  2  i s the product of P and v . 2  i s repeated u n t i l  The d e f l e c t i o n  the end of the column  of the end of the column from  a x i a l load r e p r e s e n t s the e c c e n t r i c i t y , e , at which 0  t h i s a x i a l load would produce t h i s d e f l e c t e d Once the d e f l e c t e d the M  0  calculation  e,.  deflection  to  way,  M,,  to  generate  a  second  curve, and a corresponding end e c c e n t r i c i t y shown  0  calculation  calculation  this  i s repeated with the s t a r t i n g mid-span moment  I f e, i s l e s s than e , as  second  shape.  shape has been obtained i n  reduced by a small amount  column  the  is  ignored,  in  figure  41(b),  the  and the r e s u l t of the f i r s t  represents a m a t e r i a l f a i l u r e  for t h i s a x i a l load,  with maximum end e c c e n t r i c i t y e . 0  If e, i s greater than e , as shown i n f i g u r e 0  both of these c a l c u l a t i o n s The  maximum  end  repeating  mid-span  the c a l c u l a t i o n  until  maximum and begins to decrease. i s the r e q u i r e d v a l u e . an i n s t a b i l i t y  then  represent unstable c o n f i g u r a t i o n s .  eccentricity  decrease the s t a r t i n g  41(c),  Failure  is  found  moment  in  by  continuing  small  steps  to and  the end e c c e n t r i c i t y passes a T h i s maximum end e c c e n t r i c i t y of the member i n t h i s case  is  failure.  The c a l c u l a t i o n  described i s for a single a x i a l load. I t  can be repeated f o r other a x i a l loads as necessary.  142  6.5 TYPICAL OUTPUT 6.5.1  A x i a l Load-Moment I n t e r a c t i o n Axial  load  demonstrate column  -  moment  the r e s u l t s  lengths.  The  Curves  i n t e r a c t i o n curves can be used to  of  these  calculations  for  c a l c u l a t i o n s d e s c r i b e d i n the p r e v i o u s  s e c t i o n have been c a r r i e d out f o r columns of s e v e r a l each  at  load.  forty different  F i g u r e 42(a)  combinations  of  producing f a i l u r e line  ultimate  38x140mm  columns.  interaction  The  outer  representing material  diagram  under these loads i s governed load  i n t e r a c t i o n diagram, loads i s governed  the  curves  indicating  by i n s t a b i l i t y  Some  of  that  by m a t e r i a l f a i l u r e s .  move  that  indicating  i n s i d e the u l t i m a t e  behaviour  under  these  failures.  These curves have been p l o t t e d d i r e c t l y  from the computer  the curves are not very smooth, because of  r a t h e r l a r g e steps used The  the  For low a x i a l load the inner curves c o i n c i d e  high' a x i a l  output.  showing  The inner curves correspond to the curve Pu-D-B-Mu  the  behaviour For  diagram  load P and mid-span moment P(e+A) j u s t  for typical  in f i g u r e 40(b). with  interaction,  i s the u l t i m a t e i n t e r a c t i o n diagram  strength.  lengths,  load l e v e l s between zero and maximum  i s an  axial  several  i n part of  the  numerical  analysis.  curves can be smoothed by reducing the mid-span moment i n  smaller steps d u r i n g the a n a l y s i s  of  column  behaviour,  but  this  i n v o l v e s a corresponding i n c r e a s e i n computing c o s t s , and  the  curves  shown  for the purposes  are c o n s i d e r e d to, be s u f f i c i e n t l y a c c u r a t e  of t h i s  study.  F i g u r e 42(b) i s the corresponding i n t e r a c t i o n diagram f o r  143  o o  END riOnENT (b)  Figure  6.0  (KN.H)  End moments  42 - I n t e r a c t i o n d i a g r a m s  for  slender  columns  144  a x i a l load P and end moment, P times e. is  the  u l t i m a t e i n t e r a c t i o n diagram.  Again the outer  In t h i s case the inner  curves correspond to the curve Pu-C-A-Mu i n any  level  point  of  axial  on any curve of  corresponding  curve  load,  the h o r i z o n t a l  figure of  figure  40.  For  d i s t a n c e between a  42(b) and" the  figure  line  point  on the  42(a) r e p r e s e n t s the moment  m a g n i f i c a t i o n PA due t o member deformations. 6.5.2 A x i a l Load-Slenderness Curves Another method of p r e s e n t i n g these axial  load  at  failure  against  slenderness  given values of end e c c e n t r i c i t y . shown  by  i s to  plot  (or length) f o r  A radial line'such  as  that  l i n e O-R on f i g u r e 42(b) represents behaviour f o r a  c e r t a i n end e c c e n t r i c i t y . radial  results  line  with  the  The  intersection  interaction  curves  points  of the  f o r each of the  lengths shown can be used to make a p l o t of a x i a l load end  eccentricity.  slenderness  f o r that  slenderness  i s d e f i n e d as the non-dimensional r a t i o L/d where  L i s the length of the member and d dimension  in  the  direction  i s the  under  d e f i n i t i o n s w i l l be used throughout t h i s A family curve, the  plots  sectional  consideration.  These  thesis. 43.  The top  f o r zero e c c e n t r i c i t y , r e p r e s e n t s the i n t e r s e c t i o n s of the  vertical  The disadvantages of i l l u s t r a t i n g  way a r e that or  these  cross  of such curves i s shown i n f i g u r e  i n t e r a c t i o n diagrams with  42(b).  In  against  axis  the r e s u l t s  there i s no i n d i c a t i o n as to whether  instability  of  a  figure in this material  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  D.O  Figure  6.6  INPUT The  43 - A x i a l  16.0  24.0  i  r  32.0  (L/d)  SLENDERNESS  i  r  40.0  48.0  l o a d - s l e n d e r n e s s diagram eccentricities  for several  information  as input t o t h i s  INFORMATION following  computer m o d e l . fully  8.0  Each  of these  i s required  items w i l l  be  discussed  below. A.  B.  Cross  section  behaviour:  1.  Cross  2.  T e n s i o n and c o m p r e s s i o n  3.  Modulus of e l a s t i c i t y  4.  Shape o f s t r e s s - s t r a i n  5.  Depth e f f e c t  Column 1.  section  behaviour  Column l e n g t h  dimensions  parameters  strengths  relationship  more  146  6.6.1  2.  Segment  3.  Reduction  Cross The  with  Section  program  up  to  calibrated other  particular,  influence should  further  grade  and  Compression strength  member  for  the  using  the  this  ultimate  of  column.  significant, The boards  theory of  material  is  the  use  with  the  length  in for  i n the wood by  input from  effect,  strength  and axial  species  and  consideration. from  before  an  different  tension  under  in-grade  figure  between  the  tension  strengths  As  and  strengths  that  for length  earlier.  differences  particularly  as  same g r a d e ,  these  of  the,input  member under  for four  tension  Because  member the  corrected  proposed  The  accounted  of  results,  the  variability  of  study  i n t e r a c t i o n diagram  typical  so  i n both  information.  have been o b t a i n e d  accurate  simple  material  size  r e s u l t s must be  influence  and  upon s t r e n g t h ,  same  strengths  provide  the  sections,  been  Strengths  of  size  In  testing  test  study  only  sections.  s e c t i o n a l d i m e n s i o n s as To  the  require  has  section  for  of  compression  cross  for rectangular  it  cross  d e p t h 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  consideration.  strength  However,  i s e s s e n t i a l input  be  shape of  In  axial  compression  corners.  any  verification.  Tension The  f o r m i d - s p a n moment  accomodate  verified  non-rectangular 6.6.2  ratio  can  s h a p e s would  f o r column d e f l e c t i o n c u r v e s  Dimensions  twenty and  length  input,  example 44  shows  lengths the  in-  of  lines  of the a are  region. properties  inputting strength  at  between specific  147  —i  0.0  1  1  1.0  1  1  2.0  1  nonENT  Figure  and  1  1  4.D  1  1 5.0  1 6.0  (KN.ru  44 - 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 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 l e n g t h s  locations axial  1  3.0  within  the s t r e n g t h  distribution.  F o r example, mean  strengths  a r e used  t o p r e d i c t mean s t r e n g t h  i n combined  loading,  and 5 t h p e r c e n t i l e a x i a l  predict  5th  loading,  and so on.  properties predict axial  percentile  is  strengths  A full  not e s s e n t i a l because  and c o m p r e s s i o n  bending,  distribution  b e h a v i o u r a t any l e v e l  tension  in  of  in  bending  strengths and  input  to  combined strength  t h e model c a n be u s e d t o  within strengths  a  distribution,  a t t h e same  level.  from  148  6 . 6 . 3 Modulus of  Elasticity  Timber  assumed  is  failure  to  relationship  to  relationship  in compression.  modulus of e l a s t i c i t y in compression, and Modulus boards, but influence  of  linear  tension,  stress-strain  and  a  non-linear  In the l i n e a r e l a s t i c  range the  i s assumed to be the same i n t e n s i o n  as  hence in bending.  elasticity  timber  because deformations i n a l l segments of the  board  instability  for  a  length of  average value used  in  have  v a r i e s along  behaviour, i t has  been assumed that  of modulus of e l a s t i c i t y w i t h i n a board  instability  calculations.  independent of board l e n g t h , so modulus of e l a s t i c i t y Figure  the  45  i s not  shows  unlike  subject  the e f f e c t  column behaviour, with  This  no change  can  be  average value i s  strength  to a l e n g t h  properties, effect.  of modulus of e l a s t i c i t y in  the  tension  or  on  compression  strength. Figure load  vs.  45(a) mid-span  corresponding solid of  a typical  moments,  diagram  for  elasticity  seen  ultimate  end  10,000 MPa,  figure  that  45(b)  moments.  and  the dotted  slenderness For  ratio  of  longer  on  of 7,500 MPa. no e f f e c t  squat columns  L/d=6.5 (which was members,  however,  modulus  l i n e s have been  i n t e r a c t i o n diagram f o r c r o s s s e c t i o n s t r e n g t h small e f f e c t  the  In both cases the  the modulus of e l a s t i c i t y has  only a very  axial  is  f o r m a t e r i a l with a  using a modulus of e l a s t i c i t y  c u r v e ) , and  tested).  of  i n t e r a c t i o n diagram of  and  l i n e s have been c o n s t r u c t e d  obtained be  is  the s h o r t e s t whose  It on  can the  (outer with  a  length  strength  is  1 49 o  0.0  1.0  2.0  3.0  4.0  5.0  6.0  3.0  1.0  Figure  nODENT  3.0  4.0  5.0  E.O  END HOHENT (KN.n)  (KN.M)  (a) Mid-span moments niD-SPRN  2.0  (b) End moments  45 - E f f e c t 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 c o n s i d e r a t i o n s ,  tends to be i n d i r e c t p r o p o r t i o n  the a x i a l load  capacity  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 d i f f e r e n t methods, as d e s c r i b e d values  of  percentile  modulus strength  of  elasticity  values,  c o r r e l a t i o n between s t r e n g t h not very 6.6.4  so  four  percentile  been input with 5th on.  Although  the  and s t i f f n e s s of timber boards i s  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  stress strain  about  in Compression  the p r e c i s e form of the c u r v i l i n e a r  r e l a t i o n s h i p r e q u i r e some d i s c u s s i o n .  Consider  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 shown i n f i g u r e 46, a l l  of which have been introduced the  and  have  5th  good, t h i s procedure has produced good r e s u l t s .  Uncertainties  the  i n Chapter 4.  several  compression  region  behaviour i n t e n s i o n  has  i n Chapter 2. been  shown,  Note  that  as l i n e a r  i s assumed throughout t h i s t h e s i s .  only  elastic  150  F i g u r e 46 - 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 The  form of curve shown  G l o s ( l 9 7 8 ) on the b a s i s timber  with  defects,  representation similar  in  of a l a r g e appears  of a c t u a l  46(a),  proposed  by  study on commercial q u a l i t y to  be  behaviour.  the  Figure  best  available  46(b)  shows  a  r i s i n g branch, followed by a l i n e a r f a l l i n g branch to  approximate behaviour at l a r g e bilinear 46(d)  figure  i n compression  strains.  F i g u r e 46(c) shows  r e l a t i o n s h i p with a l i n e a r f a l l i n g  a  branch, and f i g u r e  shows the f a m i l i 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 , with an  upper l i m i t on compression s t r a i n . The  sequence  considered in  to  in  figures  for  real  timber  strength  properties  calculations.  curves, the f a l l i n g - and r i s i n g branches must 1  separately.  .  46(a)  to  (d)- i s  represent a p r o g r e s s i o n of d e c r e a s i n g accuracy  representing  simplicity  described  but To  increasing  compare be  these  investigated  151  a. The the  Shape of F a l l i n g Branch  shape  of the  stress-strain  f a l l i n g branch i n the curve can  ultimate interaction interaction  If the used, similar  effect  creates  show the  f a l l i n g branch.  F i g u r e 48  be produced by using  relationship  the asymptotic on the  an  with v a r y i n g values  of  stress  figure  been used f u r t h e r in  diagram from the  expected to be seen i n t e s t s  figure  46(a)  diagram, which  49.  This  also  stress-strain  i n t h i s study because the  calculation  of  kink  the u l t i m a t e  n e u t r a l a x i s contours, and i s results  is  for l a r g e s t r a i n has a  ultimate interaction  difficulties  interaction  stress-  lines  relationship  develops a kink as shown- in curve has not  47 shows an  strain.  stress-strain  varying  on the  bilinear  The dotted  behaviour can  stress-strain  limiting  using the  slope of the  shows that very s i m i l a r  for the  example, f i g u r e  by  46(c).  of v a r y i n g the  elasto-plastic  For  produced  s t r a i n curve of f i g u r e effect  have a s i g n i f i c a n t e f f e c t  diagram.  diagram  compression region of  not  r e p r e s e n t a t i v e of a l a r g e  number of boards. F i g u r e 50 demonstrates interaction  diagrams  curves of f i g u r e s falling  and  stress-strain  s t r a i n of 0.009mm/mm.  f a l l i n g branch of  the  ultimate  stress-strain  the same  slope of  modulus of e l a s t i c i t y . been  obtained  relationship,  For  identical  using  (c) using  same curve has a l s o  elasto-plastic  almost  a r e obtained  branch, 0.02 times the  e x a c t l y the  on  46(b)  that  the  by  Almost  using the  with an upper  limit  remainder of t h i s study  the s t r e s s - s t r a i n  relationship  the  has been  T  o  i  1  i.a  1  1  r — i  2.a  i.a  1  1 4.0  MOMENT (KN.M)  1  1  v.)  1  1  o.a  Figure 47 - Ultimate i n t e r a c t i o n diagrams for the 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 v a r y i n g slope of f a l l i n g branch  F i g u r e 48 - U l t i m a t e i n t e r a c t i o n diagrams f o r the e l a s t o - p l a s t i c stress-strain relationship, with v a r y i n g l i m i t i n g s t r a i n  Figure 49 - Ultimate 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 p r o p o s e d by G1OS(1978)  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 strain relationships  154  -assumed  t o be l i n e a r as shown i n f i g u r e s 46  a c t u a l slope of the f a l l i n g branch  (b) and  The  Shape of R i s i n g  7.  Branch  shape of the r i s i n g branch of the s t r e s s - s t r a i n  compression  affects  The  i s d i s c u s s e d with r e f e r e n c e  to c a l i b r a t i o n of the s t r e n g t h model i n Chapter b.  (c).  only  curve  in  the p r e d i c t e d behaviour of columns  loaded c o n c e n t r i c a l l y or with very small e c c e n t r i c i t i e s . curves  proposed  O'Halloran  by Glos  (1973),  significantly  because  (1978), Malhotra and Mazur (1970) and  all  introduced  different  curve has been chosen  in  in  Chapter  indicated  by  are  study  form.  One  simply variable  i s the s t r a i n corresponding e, i n f i g u r e 51(a).  For  0  this  taken as 1.35 experiments defects. of  1.25  study  the  times e , reported  by  Glos  on  graph  of  axial  of  many  German spruce timber  f o r t e s t s on small c l e a r wood  a  strain  r e s u l t of  The data r e p o r t e d by 0'Halloran(1973)  To i l l u s t r a t e the e f f e c t shows  be  s t r a i n at peak s t r e s s e, has been  t h i s being the average  0  with  T h i s can  d e f i n e d as a r a t i o of the corresponding l i n e a r e l a s t i c e .  not  O'Halloran's  of t h i s  i t has the s i m p l e s t computational  stress,  2,  the r i s i n g branch.  f o r the remainder  which needs to be q u a n t i f i e d peak  The  gives a figure  specimens.  this  load  with  ratio,  against  figure  51(b)  slenderness  for  c o n c e n t r i c a l l y loaded columns, as p r e d i c t e d by the model using three v a l u e s f o r the r a t i o e , / e . 0  simple are  bilinear  seen  concentric  to  stress-strain  be column  small  but  loading.  A  ratio  of  relationship.  1.0  The  significant  in  Similar  curves  is  the  differences  this  case  plotted  of for  1 55 VI  in  £LC  1  1 8.D  1  1 WLO  1  1 24J)  SLENDERNESS  1  1 32.D  I  I  I  ! 49.3  U1.0  (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  falling  branch eccentric  loading  for  eccentricities.  small  show t h a t  the differences  disappear,  even  156  6.6.5  Stress-distribution The  effect the  origin  results  can  will  The  of  for  the  tension  this  that  These  quantified  of  here,  has  test  accurate  effect  the  in  f o r no s i z e the  c a n be  shape  for five different  depth.  extreme  of  a  values effect, tension  D e c r e a s i n g v a l u e s of  moment c a p a c i t y  for a given  i n t e r a c t i o n , diagrams  introducing  effect  the f a l l i n g may  with f i g u r e  47 shows  has an  included  because  the slope  that  influence  branch of the s t r e s s - s t r a i n  n o t be i n d e p e n d e n t  been  f o r the  stress-distribution  and c a n n o t  Although a s t r e s s - d i s t r i b u t i o n  calculations,  curve.  a  i n compression  i t has not been  o b t a i n e d by v a r y i n g  stress-strain  of  phase.  line,  A comparison  two f a c t o r s  compression  verification  axis  in-grade  b u t a more  52 shows  solid  ultimate  time  separately.  demonstrated  be  shows  stress-distribution  curve.  comparing  estimate  strength. 53  to  A rough  depths  diagram  in increasing  i n compression.  similar  for  The  of n e u t r a l  result  same m a t e r i a l ,  the  Figure  interaction  regardless  effect  by  in a constant f a i l u r e stress  Figure  3.  of the s t r e s s - d i s t r i b u t i o n  tension.  parameter  axial  obtained  52 and 53.  ultimate  fibre,  i n Chapter  with d i f f e r e n t  importance  typical  results  of the s t r e s s - d i s t r i b u t i o n  be made i n t h e c a l i b r a t i o n  in figures  3  be  f o r members  estimate  k  significance  has been d i s c u s s e d  parameter  seen  and  Effect  used  in in  the  be  effect  model  and  calibration  and  such a s i m i l a r  of the f a l l i n g  r e s u l t can  b r a n c h of  the  Figure 52 - Ultimate i n t e r a c t i o n diagrams with varying s t r e s s - d i s t r i b u t i o n parameter in t e n s i o n •  F i g u r e 53 - U l t i m a t e i n t e r a c t i o n diagrams with v a r y i n g 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  for any of  computer program c a l c u l a t e s column d e f l e c t i o n curves length s p e c i f i e d .  column  input.  deflection  Any  lengths  length of  providing  For the  step-by-step  construction  curves a column segment l e n g t h must be segment  may  increased  be  accuracy  used, at  short  greater  segment computing  costs. A s e n s i t i v i t y study showed that c a l c u l a t e d relatively  results  i n s e n s i t i v e to segment l e n g t h , f o r segment  up to four times the s e c t i o n depth.  Chen and  Atsuta  were  lengths (1976b)  report a study which found that a segment length of four times the  radius  of  g y r a t i o n gives s u f f i c i e n t l y accurate  t h i s corresponds to 1.16 l e n g t h of this  1.5  times the s e c t i o n depth.  A  results, segment  times the s e c t i o n depth has been.-used throughout  study. A  final  input  parameter  for  construction  of  column  d e f l e c t i o n curves i s the r a t e at which mid-span moment be  reduced  in  the  step-by-step  times the maximum moment has 6.7  procedure.  is  A value of  to 0.04  been used.  NON-DIMENSIONALIZED PLOTS The  axial  a x i a l load -  moment  l o a d - slenderness  have been shown with  purposes.  diagrams i l l u s t r a t e d  and  in t h i s  these  the  chapter material  programme.  in t h i s t h e s i s i t w i l l become  dimensionalize  diagrams  r e a l u n i t s r e p r e s e n t a t i v e of the  used i n the experimental Later  interaction  plots  In both cases the  desireable  for  more  general  axial  load  axis  can  to  non-  discussion be  non-  159  dimensionalized axial  by  dividing  load f o r the m a t e r i a l .  a l l a x i a l loads by the maximum The same can be done f o r moment  v a l u e s on the h o r i z o n t a l a x i s of the i n t e r a c t i o n diagrams. 6.8  SUMMARY  This  chapter has d e s c r i b e d  a s t r e n g t h model which can be  used to p r e d i c t the load c a p a c i t y of length  under  a x i a l tension  timber  eccentric axial loading. and  compression  members  of any  Input t o the model i s  strength,  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 , and c e r t a i n s i z e e f f e c t parameters.  160  VII.  CALIBRATION AND  VERIFICATION  7.1 INTRODUCTION This  chapter  relates  the t e s t data  the s t r e n g t h model f o r e c c e n t r i c a x i a l  from Chapter 5 with  loading  described  in  Chapter 6. Test  data  38x140mm. test  are  available  so  these  s t r e n g t h model i s f i r s t  then  sizes:  38x89mm and  The smaller s i z e was s u b j e c t e d to a more  programme  results  f o r two  r e s u l t s are c o n s i d e r e d  extensive first.  The  c a l i b r a t e d using only the average t e s t  i n t e n s i o n and compression f o r the  i t i s v e r i f i e d using t e s t  shortest  length,  r e s u l t s f o r the longer  lengths  in compression, and 5th and 95th p e r c e n t i l e s . Test data  i s compared with the model i n  1.  first  three  different  ways.  The  axial  comparison uses an i n t e r a c t i o n diagram of  load v s . moment to compare the measured moments at  the ends of the boards (unmagnified  moments)  with  those  p r e d i c t e d by the model.  2.  The second comparison  moments  (magnified  i s s i m i l a r except  moments)  are  compared.  s h o r t e s t l e n g t h t e s t e d there  was  very  deflection  end  moments  moments were length  at  failure,  similar.  so  that mid-span  A l l failures  boards were m a t e r i a l f a i l u r e s ,  little  of  For the mid-span  and  mid-span  the  shortest  so f o r t h i s  length  comparisons have only been made f o r mid-span moments.  161  3.  The t h i r d comparison uses a p l o t of a x i a l  slenderness  load  vs.  to compare the maximum measured load with the  model p r e d i c t i o n f o r s e v e r a l e c c e n t r i c i t i e s .  All  comparisons  are made  at  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 i n the d i s t r i b u t i o n . 7.2 38x89mm BOARDS 7.2.1  Short  Column I n t e r a c t i o n Curves  F i g u r e 54 shows a comparison of s t r e n g t h p r e d i c t e d by the model, compared with t e s t (0.45m)  of  f o r the  shortest  The l i n e s  calculated  by  are  the  the model.  ultimate  interaction  The p o i n t s f o r combined  l o a d i n g are the t e s t r e s u l t s f o r 0.45m long boards, points  as  shown i n f i g u r e 24(b).  a x i s represent to  length  38x89mm boards, f o r 5th p e r c e n t i l e , mean and 95th  percentile levels. diagrams  results  The p o i n t s on the v e r t i c a l  the r e s u l t s of a x i a l l o a d i n g  a 0.45m length using equation  the same  tests,  corrected  3.11. Only these  two p o i n t s  were used as input to the s t r e n g t h model when c a l c u l a t i n g  each  curve.  the  results equation  The of  points bending  3.11.  percentile,  on  tests,  There  mean,  the  and  are 95th  horizontal also two  axis  represent  c o r r e c t e d f o r length points  percentile  each  f o r the 5th  v a l u e s , because the  bending t e s t s at two lengths gave s l i g h t l y d i f f e r e n t Once the s t r e n g t h model had been set up as Chapter  6,  and  compression  test  input  data  results,  obtained values  using  results.  described  in  from a x i a l t e n s i o n and  were  required  for  two  162  Length 0.45m.  1  —I  0.0  1  1.0  1  1  2.0  1 3.0  1  r—  4.0  5.0  MOMENT (KN.M)  Figure  54 - 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 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  parameters  to calibrate  the model.  parameter  in  and  discussed  i n the next s e c t i o n .  to obtain  tension  the parameter  a  These a r e t h e d e p t h  compression  estimates  parameter,  I t w o u l d h a v e been from  independent  effect both  desireable sources,  163  but  t h i s was not p o s s i b l e .  The two parameters were v a r i e d on  a trial  and e r r o r b a s i s t o give the good v i s u a l  figure  54  f o r the  mean  parameters i s d e s c r i b e d The the  test  results.  test  strength  results  only.  distribution.  95th p e r c e n t i l e demonstrates  levels  the  r e s u l t s at  seen  of  to  this  be  prediction tension  data very  tests  points  do  well. with  The  the  model,  test.  with  the  some  bending  not  not  appear  points  vary  of  which  very  much  values. to  f i t the model  representing  eccentric  s m a l l e s t e c c e n t r i c i t y have a weaker  testing  by  about  the  the observed s t r e n g t h values  a  problem  equipment f o r that p a r t i c u l a r  weak  be  stressed  a x i s i n a d d i t i o n to the  i n t e n t i o n a l bending about the strong a x i s , which for  tails  good,  A mis-alignment caused these specimens to  with  for  and shows that the  s t r e n g t h than p r e d i c t e d , which can be e x p l a i n e d encountered  the  quite  w i t h i n a very wide d i s t r i b u t i o n of s t r e n g t h few  data  The r e s u l t i n g f i t at the 5th and  is  power  test  The same parameters were then  v a r i o u s parameters used i n the model do  A  of the  i n the next s e c t i o n .  used to compare the model with t e s t the  shown i n  Estimation  s t r e n g t h model was c a l i b r a t e d to the  mean  fit  being  may  account  lower than the general  trend. The the  points representing  smallest  eccentricity  p r e d i c t e d by the model. effect,  because  e c c e n t r i c compression t e s t s  with  also  than  have a weaker s t r e n g t h  T h i s may be  although  explained  by  a  length  these boards had a c l e a r length of  0.45m between the l o a d i n g boots, the t o t a l l e n g t h  was  1.05m,  164  and most load was probably  introduced by end b e a r i n g ,  case with very small e c c e n t r i c i t y . c o u l d c o n t r i b u t e to the observed 7.2.2  Parameter The  for this  T h i s d i f f e r e n c e i n lengths  results.  Estimation  estimation  of  the  two parameters t o c a l i b r a t e the  s t r e n g t h model are d e s c r i b e d below. Stress-distribution The  parameter  stress-distribution  estimated  by  comparing  then  made to provide  mean  values  i n Chapter 3 .  strengths as d e s c r i b e d  3  initially  of t e n s i o n and bending Minor  adjustments  tension  were  The f i n a l  strength  being  0 . 6 7 times bending s t r e n g t h , f o r the same l e n g t h  approximately  specimen.  A comparison can a l s o k ,  parameter  2  be  made  3  bending  tests,  the  depth  to values of k  but greater than values of k  a x i a l tension t e s t s . grading  with  effect  which i s expected t o be somewhat s i m i l a r .  value of k = 7 . 0 used here i s s i m i l a r in  was  3  a b e t t e r f i t to a l l the data.  value of k = 7 . 0 corresponds to a x i a l  of t e s t  k  parameter  T h i s d i f f e r e n c e i s probably  2  2  The  observed  observed i n related  to  rules.  Compression parameters There are two c l o s e l y shape  of  the i n t e r a c t i o n  linked  parameters which a f f e c t  diagram i n the compression  These a r e the s t r e s s - d i s t r i b u t i o n  branch of the  As  effects  have  similar  n e i t h e r can be q u a n t i f i e d  region.  parameter i n compression and  the slope of the f a l l i n g both  the  on  stress-strain  curve.  p r e d i c t e d behaviour, and  independently,  only  one  has  been  165  used  in  this  calibration.  The  possibility  d i s t r i b u t i o n e f f e c t has been ignored, and falling  branch  slope  in  of the  with  r e g i o n , which r e p r e s e n t s a very increasing  strain  beyond  peak  L i m i t e d t e s t data such as that of G1OS(1978) suggest  stress.  a steeper f a l l i n g the  slope  The value used i s 0.02 times the  the e l a s t i c  gradual drop i n s t r e s s  to  the  a stress-  of the s t r e s s - s t r a i n curve has been s e l e c t e d to  g i v e a good f i t to the d a t a . rising  of  data  branch  i n the compression  the e f f e c t s of a somewhat  by  t h i s cannot  which would not give such a good f i t  steeper  region.  falling  I t i s p o s s i b l e that  branch  are  a stress-distribution effect  being  offset  i n compression, but  be q u a n t i f i e d .  7.2.3 Long Column I n t e r a c t i o n Curve f o r End Moments F i g u r e 55 shows the diagram four  f o r end  different  distribution.  predicted  moments  shortest  lengths,  at  three  levels  obtained  by  in  Chapter  calibration  to  6, mean  results for within  the  and  the  two  values of the  l e n g t h as d e s c r i b e d above.  and  axial  represents  l o a d corresponding  a  combination  plotted  point  percentile  of  corresponding  represents maximum end  moment  5th  percentile,  recorded for a  of  end  to the load c a p a c i t y of  the column f o r a given e c c e n t r i c i t y and column  length.  interaction  The model p r e d i c t i o n curves have been obtained  Each p o i n t on a curve moment  column  compared with the t e s t  using the input data d e s c r i b e d parameters  long  axial  length. mean, load  Each  or  95th  and  the  sample of columns of given  Each p o i n t was obtained from a l a r g e number  of  test  3.2m.  166  i  i"1 z / " <i j i o  Lengl  jC  i 1  d  •o / / / < ,'BJ COB  COS  (N>t) atton  com  0'08  0'09  0'Ofr  (NX) fJticn IblXti  i  0'0t>  0"DS  SI O'O  naixti  r O'OB  COS  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 d i a g r a m f o r e n d 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  167  results  as d e s c r i b e d  The  f i t  reasonable. failure  of  i n Chapter the  In g e n e r a l  contributing  factors will  the  available  computer  uniform  along  defect  away from m i d - s p a n ,  span.  have  this  e a c h b o a r d , and  necessary  b u t some  One  possibility  strength  of t h e column, assuming  a  conservative  i s subjected  assumption  considerably  before  underestimate  of the model.  The  accuracy  acceptable, design  and  they  generally  of s t r e n g t h .  model  may  f i t is  represent  of a x i a l  have  that  stress.  columns  contribute  a r e t o be a slightly  to  been  the f u l l This  which  considered  and i f the p r e d i c t e d c u r v e s  tool  prediction  the  the  to uniform  for  failure,  of  for  moment  been a t m i d -  values  modified  o f t h e column  always  were o f t e n a t a  strength  length  will  had t h e d e f e c t the input  is  and s t i f f n e s s  failure  t e n s i o n and c o m p r e s s i o n the length  the  underestimate,  that  i s that  is  definite  r e q u i r i n g a l a r g e r l o a d and  A related explanation  to  data  underestimate  In p r a c t i c e , the f a i l u r e s  been  the  No  model assumes t h a t  a t mid-span.  would  for  to  i s not l a r g e .  be d i s c u s s e d .  occur  than  curves  moment, b u t t h e d i s c r e p a n c y is  are  predicted  t h e model t e n d s t o  explanation  that  5.  is  deflect to  the  be q u i t e  used  as  a  conservative  168  7.2.4 Long Column I n t e r a c t i o n Curves f o r Mid-Span Moments F i g u r e 56 shows a comparison of the model p r e d i c t i o n the  data, using i n t e r a c t i o n curves of a x i a l  moment.  A l l of the curves and data p o i n t s  moments  than  in  figure  55  a m p l i f i e d due t o deformations The case,  as  because  the  model  sometimes  mid-span failure  section  still  underestimates The  factors  apply, but another  Each data p o i n t i s c a l c u l a t e d d e f l e c t i o n at maximum a x i a l i s an  instability  larger  moments have been  f i t of the data to the model i s not as good  previous  included.  much  w i t h i n the members.  f a i l u r e by a c o n s i d e r a b l e amount. the  l o a d and mid-span show  the  and  failure  load.  from  in  this  the moment at discussed  in  p o i n t must be the  measured  In most cases the  associated  with  rapid  i n c r e a s e i n l a t e r a l d e f l e c t i o n s at very small changes i n l o a d . The  load  was  not  applied  very  slowly,  so  the  measured  d e f l e c t i o n s c o u l d be c o n s i d e r a b l y l a r g e r than that which would j u s t cause f a i l u r e under steady s t a t e c o n d i t i o n s .  A  similar  problem has been r e p o r t e d i n t e s t s on concrete members (Nathan 1983b). In to  end  validity  view of these problems, more emphasis should be given moments  than  mid-span  of the computer model.  moments  when  checking  the  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 m o m e n t s 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  A x i a l Load - Slenderness Figure  57  shows  Curves  the a x i a l l o a d v s .  slenderness  f o r the 38x89mm boards at the 5th p e r c e n t i l e , percentile curve  levels.  The  top  f o r concentric loading  lines  are  l i n e i s the p r e d i c t e d  (not t e s t e d ) , and the  previously.  i s very good.  7.3 38x14 0mm Short  The f i t of  In some cases  the a x i a l load recorded  next  the  model  to  the model underestimates  i n the t e s t s .  Column I n t e r a c t i o n Curves 58  shows  the u l t i m a t e  i n t e r a c t i o n diagrams from  the model, c a l i b r a t e d t o mean values of t e s t r e s u l t s s h o r t e s t length  the  five  BOARDS  Figure  All  95th  The data p o i n t s are the same p o i n t s from the  t e s t r e s u l t s presented  7.3.1  and  f o r the t e s t e c c e n t r i c i t i e s , d e f i n e d i n the key to  the data p o i n t s .  the data  (dotted)  mean  curves  the  34x89mm  (0.914m) of 38x140mm,boards. parameters  material  distribution  f o r the  used f o r t h i s p r e d i c t i o n are as f o r  w i t h , one  parameter  exception.  The  stress-  i n t e n s i o n , which r e l a t e s the t e n s i o n  s t r e n g t h to the bending s t r e n g t h , i s d i f f e r e n t .  The value  of  k =7.0 used f o r the s m a l l e r s i z e has been changed to k =9.0 to 3  3  provide  a  section. and  better  f i t to the t e s t data°for the l a r g e r c r o s s  The model p r e d i c t i o n s f o r the 5th p e r c e n t i l e s , mean,  95th p e r c e n t i l e s have been made using the same parameters. General  figure  are  material. region,  observations  about the goodness of  similar  those d e s c r i b e d  to  f i t in  this  f o r the smaller  size  Note that there are only two p o i n t s i n the one  each  tension  at the 5th p e r c e n t i l e and mean l e v e l s .  As  (a) 5 t h % 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 A m .  \  /  Mean  /  / /95th  I  0.0  1  1  ID  1 2.D  1  1 3.0  1  1  4.0  1  1 S.0  1  /  1 6.0  %ile  1  T  -  8.0  7.0  noriENT cKN.ro  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 d i a g r a m 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 described points  i n Chapter  was  not  4, t h e t e s t  very  method  satisfactory,  for deriving  so t h e y s h o u l d be  w i t h c a u t i o n , a l t h o u g h t h e y do t e n d t o s u p p o r t curves.  these viewed  the p r e d i c t e d  173  7.3.2  Long Column I n t e r a c t i o n Curves Figure  curves  59  shows  the  p r e d i c t e d long column  interaction  for end moments compared with the t e s t r e s u l t s .  60 shows the p r e d i c t e d long column i n t e r a c t i o n curves span moments compared with the t e s t In both cases results  is  of  be  very  explanation  mid-  of  the  model  and  test  form to that p r e v i o u s l y shown for the  the f i t i s not as good and  conservative is  for  results.  comparison  similar  38x89mm s i z e , but to  the  Figure  in  available  many  other  the model i s seen  cases.  than  the  No  specific  points raised in  d i s c u s s i o n of the 38x89mm r e s u l t s . 7.3.3  A x i a l Load-Slenderness Curves F i g u r e 61  shows the p r e d i c t e d curves  slenderness  compared  boards.  l i n e s are as d e s c r i b e d  The  with  the t e s t  of a x i a l  load  vs.  r e s u l t s f o r the 38x140mm  f o r the smaller  size,  and  the same comments apply. 7.4  REPRESENTATIVE STRENGTH PROPERTIES When  chapter certain  investigating  i t will  be  convenient  representative  s p e c i f i c grades, s p e c i e s , c a p a c i t y of timber  possible to  and  strength  strength  modulus of  to  material  with  rather  than  The  load  properties,  sizes  of  timber.  members i s i n f l u e n c e d by one  compression- s t r e n g t h bending  refer  strength  f o l l o w i n g , depending on the nature tension  design methods in the next  elasticity  of the  loads.  or more of  the  Figure  59 - 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 with t e s t r e s u l t s f o r 38x140mm s i z e  175  (NX) ouoi "ibixti  (NX) ami  itiixti  F i g u r e 60 - P r e d i c t e d i n t e r a c t i o n diagram f o r mid-span moments compared with t e s t r e s u l t s f o r 38x140mm s i z e  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  177  The r a t i o between t e n s i o n  and  major  shape  influence  on  the  compression of  strength  also  affects  has a  the u l t i m a t e i n t e r a c t i o n  diagram f o r c r o s s s e c t i o n s t r e n g t h . bending  strengths  The r a t i o of  the  shape  of  tension  to  the u l t i m a t e  i n t e r a c t i o n diagram because t h i s r a t i o i s an estimator  f o r the  s t r e s s - d i s t r i b u t i o n parameter.  modulus  of e l a s t i c i t y has a major A  small  t e s t s reported other  grade  i n f l u e n c e on the load c a p a c i t y .  was  carried  i n Chapter 5 and  on  out on the r e s u l t s of the the  results  of  several  For the r a t i o of in-grade t e n s i o n s t r e n g t h  compression t o 0.85.  strength,  between  corrected  300  cases.  percentile  to i n -  s t r e n g t h , most r e s u l t s were i n the range of  For the r a t i o of t e n s i o n  the range of 0.6  most  the  in-grade t e s t s c a r r i e d out at the U n i v e r s i t y of B r i t i s h  Columbia.  0.55  survey  For long columns  to  bending  f o r length e f f e c t s , most r e s u l t s were i n  to  and  strength  0.8.  The  modulus  of  elasticity  was  350 times the a x i a l compression s t r e n g t h i n  These results  figures  are  f o r SPF  and  representative Hem-fir  material  of  5th  in sizes  39x89mm to 38x235mm, grades Number 2 to S e l e c t S t r u c t u r a l , and moisture content approximately 15%. small  survey,  On  the  basis  of  this  three groups of r e p r e s e n t a t i v e p r o p e r t i e s have  been chosen, one average group and two groups r e p r e s e n t i n g the l i m i t s of  observed  results,  these  produce the l a r g e s t and the s m a l l e s t region of the a x i a l Figure material  of  62  having  properties  that  "nose" i n the compression  load moment i n t e r a c t i o n diagram. shows  these  ultimate  three  interaction  representative  diagrams groups,  for non-  178  AVERAGE STRONG WEAK  MOMENT  F i g u r e 62 - Ultimate i n t e r a c t i o n diagrams f o r representative strength properties dimensionalized that  to  i s relatively  large  "nose"  a  compression s t r e n g t h of 1.0. M a t e r i a l  weak i n t e n s i o n and i n bending in  the  compression  region.  r e p r e s e n t a t i v e of low grade m a t e r i a l , and w i l l as  weak  material  relatively  the  next  chapter.  This i s  high  be  grade  should  be  described.  This  be r e f e r r e d  a is to  M a t e r i a l that i s  i n t e n s i o n and i n bending produces a smaller  "nose" i n the compression r e g i o n .  material. (  strong  in  produces  material,  and  will  The curves r e p r e s e n t i n g considered  weak  representative  referred and  of  to  as strong  strong  material  upper and lower bounds f o r the m a t e r i a l  M a t e r i a l represented by the c e n t r a l l i n e  on f i g u r e  179  62 w i l l be r e f e r r e d to as average m a t e r i a l . relative  strengths  are summarized i n Table  Weak Ratios  Ft/Fc Ft/Fb Fc Ft Fb  Relat ive Strengths  Table  III.  ratios  Strong  .55 .80  .70 .70  .85 .60  1.0 .55 .69  1 .0 .70 1 .0  1 .0 .85 1 .42  the r a t i o of modulus of e l a s t i c i t y  for  and  In a l l cases  Average  III - M a t e r i a l property r a t i o s groups  been taken as  The  representative  to compression s t r e n g t h  has  300.  F i g u r e 63 shows i n t e r a c t i o n columns of three slenderness representative  strength  dimensionalized  so that  diagrams of end moments,  r a t i o s , and  groups. the  The  shapes  m a t e r i a l of the  for three  curves have been  of  the  curves  can  nonbe  compared. Figure three end  64  shows a x i a l  representative eccentricities.  dimensionalized  load - slenderness  curves f o r the  s t r e n g t h groups, f o r columns with Again  these  so that r e l a t i v e  plots  shapes can  have  several  been  be compared.  non-  180  F i g u r e 64 - A x i a l load-slenderness curves f o r r e p r e s e n t a t i v e s t r e n g t h p r o p e r t i e s (non-dimensionalized)  181  7.5 A P P L I C A B I L I T Y The  OF  STRENGTH  strength  model  MODEL  described  deterministic  model  which  To  use  model  to calculate strength  the  distribution,  the  percentile, The  variability has good  been  results in  and and  axial  the v e r s a t i l i t y  load  of timber  depends  results,  only  under  on t h e ' - v a l u e s  findings are significant  strength  model  can p o t e n t i a l l y  a t any l e v e l  input  data  for  that  calculation. very  large  the strength  model  and has  given  the d i s t r i b u t i o n .  of t h e model, members  data  5 showed  chapter  throughout  of  because be u s e d  in a distribution  i s a v a i l a b l e and that  a  a t any p e r c e n t i l e i n  i n Chapter  test  i s  calculations.  t o use input  In t h i s mean  thesis  statistical  not the l o c a t i o n i n the d i s t r i b u t i o n  source, that  strength.  behaviour  These the  described  p r e d i c t i o n of strength  predicted  this  out a d e t e r m i n i s t i c  c a l i b r a t e d using  demonstrates  no  i t i s necessary  and t o c a r r y  test  does  in  This  a n d shows  that the  combined  bending  of the input  data,  strength. they  indicate  f o r timber  of  a  strength,  that  from  any  provided  suitable calibration  is  carried out.  7.6  SUMMARY This  chapter  described  in  has  Chapter  5  Chapter  6.  results  f o r the shortest  calibrated accurate  The model  in  this  brought  the  and t h e s t r e n g t h  has been  way  together  tested.  has been  p r e d i c t i o n o f member  model  c a l i b r a t e d using  boards  used  strength,  test  The  described in  only  mean  strength  to provide for a  results  a  wide  test model  reasonably range  of  182  V  member  lengths,  values.  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  model.  Discrepancies  test they  throughout  the  between  distribution  the  r e s u l t s are g e n e r a l l y not l a r g e , tend  t o be c o n s e r v a t i v e .  t h e model w i l l  t o c o m b i n e d b e n d i n g and a x i a l  strength for  the  p r e d i c t e d c u r v e s and t h e and  where  they  On t h e b a s i s o f t h e s e  be u s e d i n t h e n e x t c h a p t e r  number o f p o s s i b l e d e s i g n methods  of  to  occur results  investigate  a  f o r t i m b e r members s u b j e c t e d  loading.  183  VIII. 8.1  DESIGN METHODS FOR  COLUMNS AND  BEAM COLUMNS  INTRODUCTION Structural  designers  require  a  simple  method f o r s i z i n g a member to s a f e l y r e s i s t loads  and  bending  moments.  study i s to propose design whether  a  member  carrying  capacity.  One  and  and  a x i a l loading  s e v e r a l proposals 8.1.1  axial  methods which can  be used to  brief  this check  sufficient  review  of  load  design  code requirements for combined bending  i n timber, s t e e l and f o r new  design  concrete,  followed  by  methods for timber members.  Allowable S t r e s s Design For  many  years  structural  behaviour of s t r u c t u r e s elastic  behaviour.  that were not with  the  derived  under  design  working  was  based  load  on  the  conditions  and  Design codes s p e c i f i e d a l l o w a b l e  to be exceeded when  the  structure  stresses  was  loaded  maximum a n t i c i p a t e d loads, without c o n s i d e r a t i o n  behaviour at u l t i m a t e  of  prescribed  of s e l e c t e d dimensions has  current  effective  of the o b j e c t i v e s of  T h i s chapter c o n s i s t s of a very philosophy  yet  from  strength  loads.  The  allowable  t e s t s on m a t e r i a l s  stresses  using  of  were  large factors  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 s t r u c t u r a l design  design  was  made  when  codes began to place more emphasis on the behaviour of  structures strength"  loaded design  to  conditions  near  of r e i n f o r c e d concrete  failure. and  of s t e e l s t r u c t u r e s i n v o l v e d checking that a  "Ultimate  "plastic"  design  structure  would  184  not  actually  collapse  loads i n c r e a s e d "Limit as  an  when  by a s u b s t a n t i a l "load  ultimate  limit  structural  structural  s t a t e to be i n v e s t i g a t e d  been a l o t of a t t e n t i o n  states.  In recent  given t o q u a n t i f y i n g  s a f e t y provided by various  anticipated  factor".  s t a t e s " design codes consider  number of s e r v i c e a b i l i t y l i m i t has  loaded with maximum  failure  along with a years  there  the amount of  design methods.  Design  codes f o r s t e e l and r e i n f o r c e d concrete have been improved  in  an attempt to p r o v i d e a c e r t a i n minimal p r o b a b i l i t y of f a i l u r e over  the  life  of  a  structure.  approach may be c a l l e d based" l i m i t  "probability-based"  or  "reliability-  s t a t e s design codes.  Accurate  calculation  d e t a i l e d knowledge of resistance  Design codes based on t h i s  of p r o b a b i l i t y of f a i l u r e  the  distributions  i n the overlapping  region.  of  This  requires  both  load  information  and  i s not  r e a d i l y a v a i l a b l e , so approximate methods have beeen developed to  calculate  a  "reliability  standard d e v i a t i o n many  consistent  and  design  from j u s t the mean and  of load and r e s i s t a n c e .  developments  Cornell(1969)  index"  in  this  others. codes  area Recent  There  since  have  early  work  developments  for different  been by  towards  materials  i n North  America are summarized by Galambos et a l . ( 1 9 8 2 ) . D e s p i t e the best attempts to q u a n t i f y observations  of  structural  even when best a v a i l a b l e distributions  failures  estimates  of  structural  safety,  have demonstrated load  and  that  resistance  a r e used, some allowance must, be made f o r other  f a c t o r s more d i f f i c u l t  to q u a n t i f y  such as  poor  workmanship,  185  design mistakes and unforseen circumstances ( B l o c k l e y  1980).  8.1.3 R e l i a b i l i t y - B a s e d Design of Timber Reliability-based  design  methods  f o r timber  are  l e s s advanced than f o r other m a t e r i a l s .  of  loads  is essentially  the  same  The  in  most  of any cases.  c a l c u l a t i o n of r e s i s t a n c e depends on d e t a i l e d knowledge of  the  strength  properties  and  The  conditions  i s not w e l l d e f i n e d  other  demonstrate be  and  timber  the  under  structural  various  loading  and i s much more v a r i a b l e than  Fox  (1978)  were  among  how r e l i a b i l i t y - b a s e d l i m i t  applied  discussed  some  reliability al.(l98l)  of  of  materials.  Sexsmith  could  strength  behaviour  material.  for  calculation  for structures  m a t e r i a l , with a l a r g e amount of u n c e r t a i n t y The  structures  to  glued  potential  laminated problems  index concept to timber have  specifications  compared in  the  to  s t a t e s design methods beams.  with  Foschi(1979)  a p p l i c a t i o n of the  structures.  the l e v e l of s a f e t y  several  first  international  Goodman  implied  codes.  et  by code Another  i n v e s t i g a t i o n 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).  a  investigation  large  transmission the  summarize  i n t o r e l i a b i l i t y - b a s e d design of wood  line structures.  reliability  Goodman et a l . ( l 9 8 3 )  index implied  Malhotra(1983) has i n v e s t i g a t e d by  several  methods f o r timber compression members.  alternative  design  186  8.1.4  Scope The  propose  r e s u l t s of t h i s study w i l l be used  i n t h i s chapter to  design methods f o r checking that a s e l e c t e d member has  the c a p a c i t y to r e s i s t  s p e c i f i e d loads.  In t h i s context i t i s  not important whether the loads are s p e c i f i e d by a design code in  a  working  s t r e s s format or i n a l i m i t  a c t u a l load and determined It  resistance  factors  in  s t a t e s format. each  case  The  must  be  by o t h e r s . is  beyond  reliability-based principal  the  scope  design  contribution  methods of  i n f o r m a t i o n on the s t r e n g t h under, v a r i o u s  of  for  this  of  study  to  axial the  study  timber has  distribution  combinations  which w i l l be u s e f u l input  this  develop  members. been to  of and  to  timber  A  produce members  f l e x u r a l loadi-ng,  eventual  development  of  r e l i a b i l i t y - b a s e d design methods. 8.2 8.2.1  EXISTING DESIGN METHDOS Canadian The  Timber Code  current  Canadian  Code  (CSA  1980)  a l l o w a b l e design s t r e s s e s p r o v i d i n g a c e r t a i n against The  f a i l u r e under s p e c i f i e d working  Timber  Design  the code with many requirements  is  aids.  The  have been d e s c r i b e d e a r l i e r  on  f a c t o r of s a f e t y  loads (Wilson  Manual (1980) p r o v i d e s a working design  based  background in t h i s  1978). guide to to  thesis.  these  187  a. For  C o n c e n t r i c a l l y Loaded Columns  concentrically  different shown  loaded  columns  design  i s based on  slenderness c l a s s e s , as d e s c r i b e d i n Chapter  i n f i g u r e 8.  Slenderness  of e f f e c t i v e l e n g t h , L, to dimension,  d.  The  three 2  r a t i o i s d e f i n e d as the  the  appropriate  allowable stress,  cross  and ratio  sectional  f a , i n a column i s given  by  L/d  <  10  f  10 < L/d < K  L/d  >  f  K  a  a  =  f  -  f  r a t i o K, given  K  =  0.591  8  , 2 3 3  (8.3)  E  (L/d)  2  length and  long columns i s  by  (8.4)  |— ca  where E i s the modulus of e l a s t i c i t y and  f c a i s the  stress  short  parallel  formulae  include  to  the  safety  grain  in  factors.  a The  in  concept,  the  design  formulae  awkward to use because there i s a d i f f e r e n t  are  allowable  column.  notation  changed from that i n the code f o r c o n s i s t e n c y . simple  1  (8.2)  c  t r a n s i t i o n between intermediate  at a slenderness  (->  = f J l - l/3(^f]  3  The  ca  These  has  been  Although  quite  unwieldy  formula  for  and each  188  of three ranges of s l e n d e r n e s s . formulae  which  in this  experimental  concentrically  has  been  this  study  i s not p o s s i b l e .  calibrated  be used t o p r e d i c t behaviour 65  d i d not  include  columns, so d i r e c t v e r i f i c a t i o n of the  to  However,  compares  at zero  the  the  strength  r e s u l t s of t e s t s f o r a l a r g e  number of e c c e n t r i c i t i e s , and can with  Figure  slight  extrapolation,  eccentricity.  code  formulae  with  behaviour  p r e d i c t e d by the model f o r c o n c e n t r i c a l l y loaded columns. p l o t has been non-dimensionalized modulus is  a  of e l a s t i c i t y moderate  intermediate  For  load  between  the  two  The  using  curves  a  There for  overestimating  range.  Combined A x i a l Load and Bending  members  bending  maximum  l e n g t h s , the code formula  column s t r e n g t h i n t h i s b.  to  300 times the m a t e r i a l s t r e n g t h .  discrepancy  column  will  chapter.  phase of  loaded  code column formula model  column  c o u l d be used f o r a l l slenderness r a t i o s  be d e s c r i b e d l a t e r The  A number of continuous  subjected  the  code  to  combined  specifies a linear  axial  compression  and  i n t e r a c t i o n between the  a x i a l l o a d c a p a c i t y of a c o n c e n t r i c a l l y loaded column and the moment c a p a c i t y i n pure bending.  UA + *l± f  where  a  f  The formula i s  < i  (8.5)  b  P i s the a x i a l l o a d , A i s the area of c r o s s s e c t i o n , f a  i s the a l l o w a b l e a x i a l s t r e s s under c o n c e n t r i c l o a d i n g f o r the particular  slenderness  ratio,  M  i s the  bending  moment  189  nODEL  = ~l  i  0-D  i  8.0  i  I  16.0  1  1  1  24.0  PREDICTION  1  1  32.0  1  1  40.0  1 48.0  SLENDERNESS (L/d)  F i g u r e 65 - Code column formula compared with model p r e d i c t i o n (non-dimensionalized) including and  that  due to a x i a l l o a d s , S i s the s e c t i o n modulus,  fb i s the a l l o w a b l e bending  guidance  stress.  f o r c a l c u l a t i n g the bending moment due to a x i a l  and column d e f l e c t i o n s , but the Timber suggests  The code p r o v i d e s  equations  f o r simple  Design  cases  Manual  and a t r i a l  no load  (1980)  and e r r o r  method f o r o t h e r s . For  short  assumptions compression tension  of  columns  equation  8.5  i s consistent  l i n e a r e l a s t i c behaviour, a l i m i t i n g value of  s t r e s s , with no c o n s i d e r a t i o n of a f a i l u r e  zone.  with  Earlier  chapters  have  shown  i n the  that  these  assumptions are not c o n s i s t e n t with observed behaviour.  As an  example of the inadequacies of the code p r o v i s i o n s  f o r short  190  columns,  refer  to  figure  23.  The form of equation  represented by the s t r a i g h t d o t t e d l i n e which i s very  c o n s e r v a t i v e compared with the inner s o l i d  through four  the t e s t  parts  results.  of  figure  For longer 56.  be  represented  by  seen  to  be  line  sketched  refer  t o the  The inner d o t t e d curves a r e the  model p r e d i c t i o n s of 5th p e r c e n t i l e would  columns  8.5 i s  strength.  straight  Equation  8.5  l i n e s with the same a x i s  i n t e r c e p t s , which would again be very c o n s e r v a t i v e . A similar  linear  combined bending c. To  unsatisfactory  present  for  Canadian  code  requirements  are  f o r the f o l l o w i n g reasons:  the  code  awkward formulae replaced  slenderness  2.  i s specified  and a x i a l t e n s i o n .  For c o n c e n t r i c a l l y  lengths,  be  formula  Summary  summarize,  1.  interaction  by  loaded  formula  behaviour,  of  overestimates  intermediate  strength.  f o r i n t e r m e d i a t e and long columns a  single  continuous  formula  The could  for a l l  ratios.  For combined bending  interaction  columns  equation grossly  and a x i a l compression  the  code  i s a poor r e p r e s e n t a t i o n of a c t u a l underestimating  strength  in  many  cases.  3.  Moment  magnification  not adequately  provided f o r .  due to slenderness e f f e c t s i s  191  8.2.2  NFPA Timber Code A timber design code widely used i n the U n i t e d States  that  produced  (NFPA 1982). a.  by  the  National  T h i s code  F o r e s t Products A s s o c i a t i o n  i s a l s o based on a l l o w a b l e s t r e s s e s .  C o n c e n t r i c a l l y Loaded Columns  Design of c o n c e n t r i c a l l y loaded Canadian  code  requirements  t r a n s i t i o n between short and  columns  of  is  equations  intermediate  similar 8.1  to  are s l i g h t l y  columns  is  combined used  transverse  more  at  a  factors  Bending  a x i a l compression and bending, a general formula  which  Wood(l950),  The  different.  b. . Combined A x i a l Load and  is  the  to 8.4.  slenderness r a t i o of 11 rather than 10, and the s a f e t y  For  is  includes  loads. '  both  The  derivation  described b r i e f l y  applicable  to  eccentric  clear  axial  by  in Chapter 2, wood  than  to  loads  and  Newlin(l940) used  and  assumptions  sawn timber.  The  formula i s  P/A  T~  W  H  E  R  M  M/S + P/A(6 + 1.5 J ) ( e / d )  +  J  E  and M/S  =  f  - P/A  M^S*  .  {  *  0<  J<1  i s the bending s t r e s s r e s u l t i n g from l a t e r a l  axial  previous  load.  section.  All  other  terms  >  6  )  (8-7)  i s the a l l o w a b l e s t r e s s i n bending, e i s the the  8  1  are  loads, fb  eccentricity as  of  d e f i n e d i n the  192  The  code  predicted  formula  by  the  will  be  strength  compared  model  factors. moment  of  actual  For the case of allowed  equation  by  the  and  transverse  code  loaded  s t r e s s e s i n the code  behaviour, no  behaviour  for eccentrically  columns, assuming that the a l l o w a b l e representative  with  formula  ignoring  loads can  are  safety  (M/S=0) the  be obtained  from  8.6 as f. - JP/A  a  where M i s now the end plotted  conservative  loads.  times  e.  This  ratio  at  (L/d=lO) the code formula  low a x i a l  80%  of  small  this  axial The  material Chapter predicts  very  i t predicts  a c t u a l bending s t r e n g t h when a x i a l load i s zero  between the code formula  equation  slenderness  ratio,  as  and the model shown  in  slender columns (L/d=40) the code i s again low  axial  8.8).  unsafe region near the v e r t i c a l a x i s i s due to the  discrepancy for  been  i s seen to  i s again  f o r low a x i a l loads, mainly because  (This can be seen by p u t t t i n g P=0 and J=1 i n The  has  loads and unsafe at high  For columns with L/d=20 the code formula  conservative only  P  f o r comparison with the model p r e d i c t i o n i n f i g u r e 66.  For low slenderness be  moment  prediction  f i g u r e 65. For  conservative  for  loads. comparison with 7.  illustrated  "average" For  strength  "strong",  very d i f f e r e n t  i n f i g u r e 66 has been made f o r  or  properties  "weak"  as d e f i n e d i n  material  i n t e r a c t i o n curves  the  model  as shown i n f i g u r e  193  riODEL PREDICTION NFPA FORriULA  U N H f l G N I F I E D FlOHENT Figure  66 - NFPA formula compared with model p r e d i c t i o n (non-dimensionalized)  63,  but equation 8.8 produces almost i d e n t i c a l  comparison  has  This  not been p l o t t e d , but a comparison of f i g u r e s  66 and 63 shows that the NFPA formula unsafe  curves.  will  result  in  large  areas f o r strong m a t e r i a l and l a r g e c o n s e r v a t i v e  areas  for weak m a t e r i a l . c.  Summary  Compared  with  1  the  Canadian  code,  the  NFPA  code  improvement  i n that moment m a g n i f i c a t i o n  a relatively  simple formula that handles both e c c e n t r i c  loads  and  transverse  loads.  is  i s incorporated  an into  axial  However, by comparing the NFPA  formula with the p r e d i c t i o n of the s t r e n g t h 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 u l t i m a t e loads. 8.2.3  Code Requirements f o r S t e e l The  c u r r e n t Canadian code f o r s t e e l design  in a l i m i t such  s t a t e s format.  that  factor,  load  capacity,  i s not exceeded  Handbook and  their  of  Steel  Structural  by  steel  Construction  code  transition buckling  factored  many  designed  a performance  (1980) p r o v i d e s  with  loads.  The  a commentary  design  aids.  The  design.  For c o n c e n t r i c a l l y loaded general  by  are  is  i s based on s i m i l a r p r i n c i p l e s t o many  other codes f o r s t e e l  same  reduced  specified  working guide t o the code  Canadian  members  (CSA 1978)  procedure  as  columns,  design  f o r timber,  from m a t e r i a l f a i l u r e  in  with  short  follows a  columns  the  parabolic to  Euler  i n long columns.  For combined bending and a x i a l l o a d i n g the code s p e c i f i e s that  strength  designing and  and  stability  a member to r e s i s t  bending  be  checked  requirement  P  +  ,  1  M  .  ,1  T=W7fM~ < e u  is satisfied  (8.9) /  where Pu i s the a x i a l s t r e n g t h of the column under (including  possible Pe  n  concentric  e f f e c t s of b u c k l i n g ) , Mu i s the  moment c a p a c i t y  and  equation  This formulation  2.6.  P  form i f  —u  loading  When  some combination of a x i a l load  moment M, the s t a b i l i t y  in i t s simplest  separately.  i s the  Euler  buckling  load  from  assumes that the i n t e r a c t i o n  195  diagram f o r a x i a l columns  load and mid-span  moment  in  slender  beam  i s the s t r a i g h t l i n e Pu-Mu on Figure" 67, which i s an  empirical  relationship  verified  by  experimental  studies  MOMENT F i g u r e 67 - A x i a l load-moment  i n t e r a c t i o n diagram f o r  s t e e l members (Galambos  1968).  To  satisfy  combination of P and M must dotted  this  l i e to  requirement,  the  left  of  a  design  the  curved  l i n e Pu-Mu, the h o r i z o n t a l d i s t a n c e between the dotted  l i n e and the  solid  line  representing  moment  magnification  caused by l a t e r a l d e f l e c t i o n s i n the member. The moment m a g n i f i c a t i o n f a c t o r i s given by  F  which  is a  close  =  1  - P/P,  approximation to the exact  (8.10)  expression f o r  196  linear elastic  behaviour  (Timoshenko and Gere 1 9 6 1 ) .  For short columns i n which reduced the  by  stability  the  axial  strength  i s not  e f f e c t s the i n t e r a c t i o n diagram becomes  s t r a i g h t l i n e Pa-Mu on F i g u r e 6 7 , where Pa  i s the  axial  compression s t r e n g t h of the column m a t e r i a l . If  the  beam-column  each end the design  i s loaded  equation  r  +  where Cm i s a moments along  M,  and  M  The  actual  { 8  -  account  f o r the  distribution  2  the  larger  Galambos(1968)  product  >  1  and  0.4  and  Johnston(1976)  show  of  Cm  diagram.  case the load c a p a c i t y w i l l stability  checked a g a i n s t strength.  an  that  approximation f o r a wide range of l o a d i n g and  M i s an e q u i v a l e n t  lead to the same long  moment  s m a l l e r end moments  column  be  For some unsymmetrical  governed  considerations, interaction  The i n t e r a c t i o n formula  and  diagram  by  as  loading  be governed  strength  the  uniform  strength  by the moment c a p a c i t y of a p l a s t i c hinge at one end.  by  of  (8.12)  c o n d i t i o n s the load c a p a c i t y of a beam-column w i l l  than  1 1 )  u  0.6 + 0.4 M /M  are  2  moment that should the  =  m  i s a reasonable  cases.  to  1  the member which can be approximated by  respectively. this  M- < e  factor  C  where  becomes  I=P7P-  u  with d i f f e r e n t moments at  In t h i s rather  design must be  f o r cross  section  f o r s t r o n g - a x i s bending i s  1 97  M  <  1.18 (1 - !-) ,  u  M < M  (8.13)  u  which i s shown by the l i n e Pa-B-Mu i n f i g u r e 67. shape  appears  axial  because the web  l o a d without  flanges. design  The  reducing  for  bilinear  of an I - s e c t i o n can c a r r y some  the p l a s t i c moment c a p a c i t y of  shaded l i n e on f i g u r e  envelope  The  67  shows  the  a p o s s i b l e s i t u a t i o n with  the  resulting  unsymmetrical  end moments. The  possible effects  bending Mu  can  the s t r e n g t h  behaviour  formula  is  for s t a b i l i t y  L_ P  C  +  1  u  -  rc*  axes.,  under  interaction assumption appendix accurate  buckling  M  +  x  M  P/P  ex  ux  loading. between  is  which  buckling  by extending  the  in  value  occurs.  interaction  to  very  Cm  1-P/P  y  M  _Z_ M  ey  ,  < i  '  uy  is specified These  x-axis  and  conservative  x  (8.14)  y r e f e r to a c t i o n s about the x  A s i m i l a r formula  biaxial  at  included  where the s u b s c r i p t s x and y  lateral-torsional  be accommodated by using a bending s t r e n g t h  that represents  Biaxial  of  for material  formulae  assume  y-axis  behaviour.  (Johnston  1976),  and  strength linear This but  an  to the code p r o v i d e s more d e t a i l e d formulae f o r more design  It can  in c e r t a i n cases.  be seen that  s i m i l a r i t i e s to those  the  steel  f o r timber,  requirements  have  but are s i g n i f i c a n t l y  some better  because 1. Strength separately.  and  stability  effects  are  considered  198  2. The s p e c i f i e d moment i s more ultimate loads.  i n t e r a c t i o n between a x i a l load and representative of a c t u a l behaviour at  3. Moment m a g n i f i c a t i o n included.  due to second-order  effects i s  4.  The e f f e c t of unequal end moments i s i n c l u d e d .  5.  B i a x i a l e f f e c t s are accounted f o r .  6. The p o s s i b l e can be i n c l u d e d .  e f f e c t s of l a t e r a l t o r s i o n a l b u c k l i n g  8.2.4 Canadian Concrete Code The  current  concrete  design  Canadian  code  i s written  (CSA  1977)  in a limit  s t a t e s format, o f t e n  r e f e r r e d to as " u l t i m a t e s t r e n g t h d e s i g n " . is  similar  steel,  t o many other  structural  ultimate  load  factor,  i s not  detailed  capacity,  to  are  designed  reduced  exceeded  background  The Canadian  r e i n f o r c e d concrete  members  by  by  the  column  codes.  such  a  specified  for reinforced  their  reduction  factored  provisions  As with  that  capacity  code  loads.  A  i s given by  MacGregor et a l . ( l 9 7 0 ) , and i s used as the basic reference f o r this section. The members  design  method  for reinforced  i s significantly different  above f o r s t e e l and timber. extensive occur  concrete  compression  from the methods d e s c r i b e d  The code p r o v i s i o n s are much l e s s  than f o r s t e e l , p a r t l y because long slender  much  less  frequently  in  concrete  members  than i n s t e e l , and  because members with hinged end c o n d i t i o n s are r a r e . No s p e c i f i c p r o v i s i o n i s made f o r c o n c e n t r i c a l l y compression members. minimum  nominal  loaded  A l l members must be designed t o r e s i s t a  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 accurate  second-order  slenderness  about  either  structural  a x i s . . Although  analysis  is  an  recommended,  e f f e c t s are g e n e r a l l y accounted f o r by  magnifying  the bending moment from a l l sources by a m a g n i f i c a t i o n  factor,  F, given by C F  where  P  load, 0  =  m  1-P/4>P  i s the f a c t o r e d a x i a l is a  coefficient  capacity  (8.15)  l o a d , Pe i s the E u l e r  reduction  factor,  and  buckling  Cm  i s the  f o r d i f f e r e n t end moments e x a c t l y as used i n the  s t e e l code. The factored  member i s then s i z e d and axial  load  combined  reinforced  with  the  bending moment does not cause a m a t e r i a l section.  p r i n c i p l e s by standard  charts  (ACI 1970).  magnified  The  shape  the  obtained  f o r timber members i n t h i s  T h i s method of failures,  method c o n s i d e r s  section  of  for cross  f a i l u r e at any  design not  strength  buckling  load.  linear elastic  concrete  cross  i s very  design  interaction  s i m i l a r to that  study.  assumes  instability  i n s t a b i l i t y only  ultimate  that  a l l failures  failures.  indirectly,  8.15 can only be used s e n s i b l y f o r a x i a l loads  on  the  factored  methods or from p u b l i s h e d  diagram  Euler  that  The s t r e n g t h of the c r o s s s e c t i o n i s determined from  first  material  such  The  i n that less  are  design equation  than  the  Equation 8.15 i s an approximation based  theory,  i s non-linear.  but To  real  behaviour  of  reinforced  overcome t h i s problem the code  200  i n c l u d e s an e m p i r i c a l e x p r e s s i o n f o r the leads  to approximately  material  failures.  serious  deficiencies  in  an  reduced  These approximations in  the  apparent  to  allow  duration loading. timber  apparently  not  design method because very few very  slender.  stiffness  EI  would  The e x p r e s s i o n f o r EI  f o r the  possibility  The p o s s i b i l i t y  For  members  have  to  be  principal  advantages  of  also  includes  a  creep under long  of a s i m i l a r p r o v i s i o n f o r  members i s d i s c u s s e d i n Chapter  The  are  e m p i r i c a l l y , or a more r a t i o n a l method method  used (Nathan 1983a). factor  which  i n s t a b i l i t y mode (slender p r e s t r e s s e d members  for example) the further  EI  c o r r e c t r e s u l t s f o r members which have  r e i n f o r c e d concrete columns are failing  stiffness  of  9.  the  reinforced  concrete  method are that 1. A s i n g l e design procedure of compression members. 2.  can be used, f o r  No c o n c e n t r i c l o a d i n g column formula  a l l types  i s required.  3. The short column interaction curve used f o r a l l design can be d e r i v e d from f i r s t principles, or simply obtained from p u b l i s h e d graphs or t a b l e s . The  primary  disadvantage  slender columns i s not  an  accurate  behaviour.  The approximations  reinforced  concrete  but  made of timber. (  i s that the design method f o r representation  of  real  introduced are s a t i s f a c t o r y f o r  may not be f o r more slender columns  201  8.2.5 L i m i t States Timber Codes A proposal structures  for a limit  s t a t e s design  was made by SexsmitM1979).  format  for  timber  Two new codes c o n t a i n  p r o v i s i o n s s i m i l a r to some of those suggested by Sexsmith. The  Canadian code proposal  (CSA 1983) i s almost  identical  to the e x i s t i n g working s t r e s s code p r e v i o u s l y d e s c r i b e d ,  with  new load f a c t o r s  and  the  format  design  of  conceptual  the  resistance equations  in  Bridge  this  requirements  The  members  requirements.  There are  end  zero  design  are  still  recognizing  member with  states  format  provisions  represent  change i s the d e l e t i o n  minimum  without  change  any  significant  is  the  Ontario  Design Code (OHBDC 1982) r e f e r r e d to as OHBDC  chapter.  compression  loading,  which  changes.  A second code i n l i m i t s Highway  factors  that  improvements  design  method  i t i s impossible  of  but  i n some r e s p e c t s .  eccentricity.  eccentricity  dimension i s s p e c i f i e d ,  a  OHBDC  0.05  For times an  The main concentric  to load a timber axial  loading  the c r o s s initial  a  section (or  crook) at mid-length of 1/500 times the e f f e c t i v e l e n g t h .  The  equation i s  a  where  with  for  the  bow  design  together  for  major change from e x i s t i n g  several  lacking of  end  a  of  a l l the  terms  s t e e l and c o n c r e t e .  e  u  are the same as d e f i n e d p r e v i o u s l y f o r  T h i s equation assumes that  a l l failures  202  are  material  failures;  considered i n d i r e c t l y for a x i a l To  instability  failures  i n that the equation can  loads l e s s than the E u l e r b u c k l i n g  are  only  be  used  only  load.  i n v e s t i g a t e the p o s s i b l e consequences of t h i s  incorrect  approach,  formulation  of the s t e e l code w i l l be made.  68 which i s an  a  comparison  with  the  slightly  more  correct  Consider  i n t e r a c t i o n diagram of a x i a l l o a d vs.  figure moment.  MOMENT  Figure Pa  68 - A x i a l load-moment i n t e r a c t i o n diagram from OHBDC  i s the c o n c e n t r i c  i s the c o n c e n t r i c consideration,  column  a x i a l l o a d c a p a c i t y of the long column under  and Mu  been c o n s t r u c t e d , slender  a x i a l l o a d c a p a c i t y of a short column, Pu  i s the bending c a p a c i t y .  Figure  to s c a l e , to i l l u s t r a t e the behaviour whose  concentric  load c a p a c i t y Pu  68 of  has a  i s h a l f of  203  the short column load c a p a c i t y Pa. obtained  from  the OHBDC formula  Pu-B-Mu has been formula  constructed  (equation  8.11).  represented  magnified mid-span suggests  that  by  from  the more  load  A,  can  point  B,  which  moment  this  (equation 8.16).  represented  member  can  moment  causing  correct  just  that t h i s  carry  point  conceptually  an end  a  to a  D.  OHBDC  l a r g e r end moment  to a  magnified  m a t e r i a l f a i l u r e at p o i n t E.  approach i s seen t o be  steel  corresponds  by  carry  represented by p o i n t C which corresponds span  The curve  The s t e e l code suggests  member, s u b j e c t e d t o a x i a l moment  The curve Pu-C-Mu has been  incorrect  and  mid-  The OHBDC slightly  unsafe. The  OHBDC formula  i s compared with the p r e d i c t i o n of the  s t r e n g t h model i n f i g u r e 69. has  been  necessary  to  To make a  work  backwards  i n t e r a c t i o n diagram f o r magnified unmagnified  moments  dotted l i n e s . origin  because  that  high a x i a l longer  The d o t t e d l i n e s  compare  the  a r e a l l curved  toward  the  For  by the code m a g n i f i c a t i o n short  columns  (L/d=l0)  i s seen t o be very c o n s e r v a t i v e , except at  l o a d s , even more so than the formula  the NFPA  formula.  becomes more a c c u r a t e .  has been p l o t t e d f o r "average" OHBDC  linear  the moment value of each p o i n t on the l i n e a r  formula  columns  to  the  the designer begins with, shown as  f a c t o r t o g i v e a design moment. OHBDC  comparison i t  from  moments,  i n t e r a c t i o n diagram has been reduced  the  valid  strength  material,  For  F i g u r e 69 but the  curves w i l l not change f o r " s t r o n g " or "weak" m a t e r i a l ,  so a comparison with f i g u r e 63 shows that the formula  will  be  204  DODEL PREDICTION OHBDC FORrlllLH  \ \\ \\ CO _ C3  o cr o  \  \  CO _ \  \  , \  V \  \ . \AAU10  x  \  _  \  ^JVd Ol _  ^  \  = 20  \  \  \  X. \ \  \  L/d = 40 •=  \  = '  a  <=>' I 0.0  I  I 0.2  I  1  *  —  I  0.4  /  1  I 0.6  "  I 0.8  - ^ ^ \ /  I  I  I.D  I 1.2  I  UNHflGNIFIED HOHENT  F i g u r e 69 - OHBDC formula  compared with model p r e d i c t i o n  (non-dimensionalized) very c o n s e r v a t i v e The at  f o r weak m a t e r i a l .  OHBDC formula  ultimate  loads.  does not a c c u r a t e l y represent The  linear  interaction  behaviour  diagram  is  c o n s e r v a t i v e and the moment m a g n i f i c a t i o n f a c t o r w i l l be shown to  be  unconservative,  but  these  somewhat t o produce the curves For  biaxial  interaction  formula  loading  the OHBDC  code.  The  initial  bow suggests that b i a x i a l  in every  effects  cancel  out  shown i n f i g u r e 69.  almost i d e n t i c a l  specification  two  of  to  minimum  specifies that end  a  i n the  linear steel  e c c e n t r i c i t y and  l o a d i n g should be considered  case but the code i s not c l e a r on t h i s matter.  In summary, the OHBDC represents a major  development  in  205  that  it  specifies  equation  for  amplification  assumptions are 8.3  moments  of  all  not c o n s i s t e n t  of  columns w i l l the  the  design  require  Chapter the  strength  properties. the  stress-strain  typical  for  individual For  slender  be u s e d f o r  good f i t  any  several  reference  its  study.  of  materials  and  beam-  strength  of  introduction  difficulties  in  in  calculating  with  non-linear  detailed  such  knowledge  information when d e a l i n g  empirical  is with  than  with  used,  but  is  line  The b e s t  f o r m u l a must  formula  computational  that is  be can  likely  form p r o d u c i n g a  results. comparison  t h e column b e h a v i o u r are  of  several  curves.  p r e d i c t e d by t h e  identified  in  the  key,  The  computer and  will  here.  marked  1982),  empirical  formulae a v a i l a b l e  ratio.  simplest  70 shows a  e q u a t i o n c a n be  form of  slenderness  be d i s c u s s e d b r i e f l y  1980, NFPA  this  p o p u l a t i o n s of members r a t h e r  The d o t t e d l i n e s  The  the  A brief  members, e s p e c i a l l y  to experimental  line  model.  of  of  be p r o p o s e d f o r  approach r e q u i r e s  some  one w i t h t h e  Figure solid  the  columns the E u l e r  lengths  There are  t o be t h e  some  specimens.  shorter  used.  of  results  calculating  relationships,  timber  properties  to  loading.  columns  Any a c c u r a t e  not a v a i l a b l e  for  of  but  a design  LOADING  a method f o r  2 d e s c r i b e d some of  includes  moments,  methods  column under c o n c e n t r i c  and  w i t h the  COLUMN CURVES FOR CONCENTRIC Some  of  minimum  CODE  which  to equations  8.1  has to  is  the c u r r e n t been  8.3.  code  discussed  formula earlier  (CSA with  206  16.0  24.0  48.0  32.0  SLENDERNESS (L/d) Figure  The  70 — Comparison of column curves f o r c o n c e n t r i c l o a d i n g (non-dimensionalized) curve marked MALHOTRA i s almost a p e r f e c t  model p r e d i c t i o n . and  T h i s curve has been  Mazur (1970) by using  stress-strain  proposed  f i t to the  by  Malhotra  tangent modulus theory based on the  r e l a t i o n s h i p of  equation  2.1.  The  resulting  column formula i s TT E + 2  f (L/r)  2c(L/r)  where  fu  [TT E + f ( L / r ) ] 2  2  2  2  c  c  - 4TT E f 2  4c (L/r)  2  2  i s the maximum  axial  material,  defining  the  and r i s the radius shape  of  the  2  (8.17)  4  s t r e s s that the column can  support, f c i s the f a i l u r e , s t r e s s f o r a short same  c(L/r)  column  of g y r a t i o n .  stress-strain  of the  The term c,  relationship  in  207  equation 2.1,  i s taken as 0.90  formula  a sound t h e o r e t i c a l background but i s cumbersome  to  has  The  l i n e marked CUBIC RANKINE i s a c u b i c m o d i f i c a t i o n  t r a d i t i o n a l Rankine formula, proposed  to  by Neubauer(1973).  formula i s f  =  f  i  U  +  f  c  E  £  (Wd) 40  (8.18)  3  where d i s the c r o s s s e c t i o n a l dimension g i v e s the l a r g e s t are  The  use. The  the  both here and by M a l h o t r a .  as  i n the d i r e c t i o n  s l e n d e r n e s s r a t i o L/d, and a l l  used above.  The  other  number 40 has been chosen  that terms  to provide  the best f i t to the model p r e d i c t i o n , compared with a value of 50 used by Neubauer to give a good f i t to t e s t s on small c l e a r Douglas-fir  columns.  development  and  Neubauer  the  describes  subsequent  the  theoretical  s i m p l i f i c a t i o n , f o r use with  timber, but the formula must be c o n s i d e r e d  empirical  one parameter has been obtained from a c u r v e - f i t t i n g The  line  marked  PERRY-ROBERTSON  because exercise.  i s the curve obtained  from the Perry-Robertson  formula which i s the b a s i s of  and European codes.  formula i s  _ u  f  e  + (^i),.  The  _  2  r* /  , where  e  + cn+i). 2  v  n  =  • '  British  (  8  .  1  9  )  c e  ca —  Terms not p r e v i o u s l y d e f i n e d are f e , the s t r e s s i n the  column  208  due  to a p p l i c a t i o n of the E u l e r b u c k l i n g l o a d ; c, the assumed  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 and a, the distance" of the extreme f i b r e on the concave Robertson(1925)  and  side  from  n=0.00l  two  L/r and n=0.003 L / r .  curves  The  describe  how  the  regarded as an e m p i r i c a l was  Perry-Robertson  one  purposes,  of  these  obtained  A l l e n and formula  by has  Bulson  must  be  formula d e s p i t e the l o g i c by which i t  d e r i v e d , because of the inadequacy If  axis.  f i g u r e of 0.001  been used to give the best f i t i n t h i s case. (1980)  neutral  Sunley(l955) r e p o r t that p r a c t i c a l l y a l l  experimental values l i e between the taking  the  curves  there i s l i t t l e  is  of s e v e r a l  assumptions.  to be s e l e c t e d f o r design  between them as they a l l give a very  good f i t to the s t r e n g t h p r e d i c t e d by the computer model,  and  several  for  have  previously  timber columns.  becomes  formula.  Figure  proposed  formula,  favoured, 71  shows  compared  curve i s not an exact  8.4  verified  and  this  the  curve  with  f i t , but  PROPOSED DESIGN METHODS FOR section  will  form  i s the c u b i c Rankine from  the  the model p r e d i c t i o n .  The  compared  i t i s no l e s s a c c u r a t e , and  This  experimentally  The one with the simplest computational  therefore  formula  been  resulting  with ' the  present  i s a l o t simpler to use.  ECCENTRICALLY LOADED COLUMNS  describe  s e v e r a l a l t e r n a t i v e design  methods f o r e c c e n t r i c a l l y loaded timber columns, based on experimental and a n a l y t i c a l r e s u l t s of e a r l i e r c h a p t e r s .  the  209  MODEL P R E D I C T I O N PROPOSED FORriULR  ex o _J  _J  o'  cr i—i x cr  i  i B.O  O.D  r  i 16.0  i  i 24.0  i  i 32.0  40.0  48.0  SLENDERNESS (L/d) Figure 8.4.1  71  - Design proposal f o r c o n c e n t r i c l o a d i n g compared w i t h model p r e d i c t i o n  Type of L o a d i n g This  eccentric member.  axial  is  this  stage,  concerned  will  no  loads w i l l  with  be  at  each  columns  subjected  transverse loads a p p l i e d to be  d i s c u s s e d i n Chapter  assume t h a t t h e  eccentricities  eccentricities  Analysis  loads, with  Transverse  At equal  chapter  and  applied  end.  The  introduced at'a  axial  case  later  the  9.  load  of u n e q u a l stage  to  in  has end this  chapter. The single known. of  design  members,  methods  f o r which the a x i a l  T h e s e methods a r e  a second  presented  order  intended  structural  in  this  l o a d s and  chapter end  for design  analysis.  Chapter  are  moments  without 9 will  the  for are use  discuss  210  how the r e s u l t s of t h i s t h e s i s can be used when a second  order  analysis i s available. 8.4.2 Input Strength  Properties  T h i s s e c t i o n assumes that c e r t a i n available  input  information  is  f o r the t r i a l member which i s to be designed.  The  information i s 1.  The a x i a l  t e n s i o n c a p a c i t y Tu, of the  member,  which  i s the product of the c r o s s s e c t i o n area A, and the a x i a l tension 2.  failure stress f t .  The  axial  compression  c a p a c i t y Pa, of the member,  which i s the product of the c r o s s s e c t i o n area A, and the a x i a l compression f a i l u r e s t r e s s f c , f o r a short  column.  3.  The bending moment c a p a c i t y Mu, of the member,  which  is  the product of the s e c t i o n 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  t e n s i o n , compression and bending s t r e n g t h values  to  above are a l l assumed to depend on the length and depth of  the member, as Chapter 9. of member 8.4.3  described  in  Chapter  3  and  referred  summarized  in  Modulus of e l a s t i c i t y i s assumed to be independent size.  Design Approaches This  section  b r i e f l y c a t e g o r i z e s the s i x design  that w i l l be d e s c r i b e d  i n the remainder of t h i s  methods  chapter.  Suppose that a column i s to be designed to r e s i s t a given axial member  load P with equal i s selected,  end a  eccentricities  design  method  e.  If  a  trial  i s r e q u i r e d to check  211  whether  t h e member  applied  load.  The figure  has  deflected  39.  bending  shape  The a x i a l  moment  "unmagnified"  M  load at  moment  "magnified"  moment  magnification  sufficient  at  3 will  and  interaction and  midspan  Methods  used In  the  method  solid  accurate  concrete  with envelope)  (and t h e  times  e.  line.-  where F i s a A.  calculating  1, 2 the  moment FM a t m i d moment  with  In both  correct  approximations  interaction  to  is-the  an  methods 1  The d i f f e r e n c e  this  The  between  the  same  same a p p r o a c h  code.  interaction  f o r timber  the  The  member  d e s i g n methods of  the  i s shown i n i s ' P.  FM=P(e+A),  magnified  Conceptually  moment w i l l diagram  and  be  compared  representing  in figure  approach,  design,  which is  72.  material  Conceptually  is  similar  with  is  used  for  to  t h e OHBDC  members.  Methods 4 and 5 w i l l  failure  that  3 the magnified  less  P  72, p r o p o s e d  shown by t h e d o t t e d c u r v e  reinforced  moments,  is  be d i f f e r e n t  i n the s t e e l  isa  the is  be made w i t h  the  diagram.  ultimate  proposal  of  ( o r f a i l u r e envelope)..  1 and 2 w i l l  failure, this  by  member  F, hence t h e m a g n i f i e d  compare  diagram  shown  interaction as  factor  will  resist  due t o t h e member d e f l e c t i o n  2, c o m p a r i s o n s w i l l  diagram  ends  to  t h e member  p r o v i d e an a p p r o x i m a t e method  magnification span,  typical  mid-span)  With r e f e r e n c e to f i g u r e and  a  throughout  the  at  factor  of  strength  the  compare end moments, o r  corresponding  f o r end moments.  interaction  unmagnified diagram  These two methods  do  (or not  212  o o  Material  Za u  Mid - span  p  I  moments at failure  failure  v.  p  M  F M M Moment  u  F i g u r e 72 - T y p i c a l i n t e r a c t i o n diagram f o r a x i a l and magnified involve  calculation  effects  of  construction  moment of  of  a  moment  m a g n i f i c a t i o n f a c t o r , because the  magnification  the  interaction  have  using  approximate  of  loaded  considered Method  from  the  4  in will  strength  p u b l i s h e d graphs, while method 5 w i l l propose an formula.  Method 6 w i l l use  been  diagram.  propose o b t a i n i n g the i n t e r a c t i o n diagram model  load  take a d i f f e r e n t approach by proposing  a x i a l load v s . slenderness  the  formulae f o r e c c e n t r i c a l l y  columns.  8.4.4 Moment M a g n i f i c a t i o n Factor Methods 1, 2 and 3 w i l l a  moment  magnification  r e q u i r e a method f o r  factor.  Most  design codes use the  e l a s t i c moment m a g n i f i c a t i o n f a c t o r given by For  the timber  calculating  equation  columns i n v e s t i g a t e d in- t h i s study,  a m p l i f i c a t i o n of mid-span moment at  failure  is  8.10.  the a c t u a l  considerably  213  more  than  predicted  m a t e r i a l behaviour figures  73(a)  dimensionalized for  members  by t h i s equation,  in (b)  compression. and  (c),  because of n o n - l i n e a r  For which  are  i n t e r a c t i o n diagrams of a x i a l  with  respectively.  slenderness  In  each  case  ratios the  of  outer  u l t i m a t e i n t e r a c t i o n diagram f o r a c r o s s curve  example,  consider  typical  load vs. 10,  20  nonmoment  and  s o l i d curve section.  40,  i s the  The  next  (chain dotted) represents combinations of a x i a l load and  mid-span  moment  The  inner curve  and  end  moment  between  these  causing  failure  (small  causing two  failure.  curves  dots)  has  The  represents  horizontal the  member.  been obtained  The  actual  intermediate  from the curve  It i s is  c o n s i d e r a b l y more than p r e d i c t e d by t h i s equation,  so that  if  parameters  in  amplification  8.10.  of end  failure  other  the  distance  at  the  that  load  m a g n i f i c a t i o n of  moments by a m p l i f y i n g the moments using equation apparent  length.  ( s o l i d l i n e ) i s the combinations of a x i a l  moment due to deformations i n the curve  f o r that p a r t i c u l a r  t h i s method c o u l d be q u a n t i f i e d i t  would be unsafe to  use  equation  moment  and  a  magnifier,  more  8.10  for calculating  accurate  expression  the  would be  necessary. Suppose that the m a g n i f i c a t i o n f a c t o r i s t o be by  increased  an e m p i r i c a l f a c t o r A at maximum load, with no increase at  zero a x i a l l o a d , and l i n e a r extremes.  The equation  F  interpolation  between  these  two  becomes  1 1-P/P  [l + (A-l) |-] e  e  (8.20)  214  F i g u r e 73 - I n t e r a c t i o n diagram showing moment m a g n i f i e r  traditional  215  T r i a l c a l c u l a t i o n s show that a s u i t a b l e value  for A  i s 2.0,  producing "a m a g n i f i c a t i o n f a c t o r of 1 + P/P  (8.21)  e  1 - P/P  e  The  previous  equation  interaction  diagrams  are shown i n f i g u r e 74.  to  be  this  conservative a  reasonable  approach f o r design purposes which w i l l be used f o r  the r e s t of t h i s  study.  8.4.5 METHOD 1: B i l i n e a r F i g u r e 75(a) shows axial  using  Although rather  for slender columns, t h i s i s c o n s i d e r e d empirical  replotted  load v s .  l i n e s passing  I n t e r a c t i o n Diagram a  magnified  interaction  diagram f o r  moments, superimposed with  straight  from c o n c e n t r i c a x i a l column s t r e n g t h to a p o i n t  on the h o r i z o n t a l a x i s at capacity.  typical  These  a  ratio  of  B  times  l i n e s can be used as a design  the moment approximation  for the i n t e r a c t i o n diagram, t e r m i n a t i n g at the v e r t i c a l  line  through  This  the moment c a p a c i t y , which cuts o f f the "nose".  approximation i s c o n s e r v a t i v e as shown by the shading, quite  but i s  simple. The  resulting  design  equation,  providing  a stability  check f o r any l e n g t h of column, i s  C  FM  —  < B ( l - !-) , M  v  u  P ' u  C FM <M m  u  where F i s the m a g n i f i c a t i o n f a c t o r from equation the  factor  (8.22)  8.21, Cm  f o r unequal end e c c e n t r i c i t i e s from equation  is 8.12  216  Figure  74  I n t e r a c t i o n d i a g r a m showing moment m a g n i f i e r  proposed  217 O  HRGNIFIED HOHENT  (b) W e a k  25  ^  30 40  *"*"  *~  ______  1  *~  •  "  .  - — — i  50  -11  o.o '  1 1  1 1  0.2  1  — 1 r t 1  1 i  1 i  1 i  1 i  iI  y ~~ ———_~— —  r—  a." HRGNIFIED o.c 0.1)HOH1 .0T EN  (c) S t r o n g  HRGNIFIED HOHENT F i g u r e 75 - B i l i n e a r approximation to i n t e r a c t i o n diagram f o r magnified moments (non-dimensionalized)  218  and  B i s the  shape that  of a  horizontal axis  the  ultimate  i n t e r c e p t , which depends  interaction  suitable semi-empirical  diagram.  I t can  f o r B can  be  value  on  the  be  given  shown by  1.35 =  where f t / f c  i s the  compression  strength  expression this seen  the  for  one  made, as  specified outer  weak  of  i n the  in figure  TT  <  ~h  x  8.4.6  i s the  METHOD 2: Another  curves figure  is  to  1,  Figures  75(b)  line  maximum end Parabolic  This  (c) show  and  s t r o n g m a t e r i a l , and  i t is  two  end by  moments the  are  strength  member, a s t r e n g t h code, u s i n g  different  of  the  cross  c h e c k must  the  be  approximation  75  M  i  <  M  u  (  8  '  2  4  )  moment.  I n t e r a c t i o n Diagram  code  a  type  The  particularly  and  this  a  assume  Newlin(1940) .  axial  d e r i v a t i o n of  e m p i r i c a l method of a p p r o x i m a t i n g  76.  straight  >  The  to  cases.  steel  curve  u where M,  the  strength  B.  i s governed  end  <  tension  material.  i n most  load capacity  the  axial  s i t u a t i o n s where the  at  (8.23)  in appendix  reasonable  section  for  of  f o r the  approximation  For and  ratio  i s given  t o be  F7f~ t C  family of  curves formula,  as  they  of  parabolae  expression are but  less  the  was  interaction as  suggested  conservative  more c o n s e r v a t i v e  show d e c r e a s i n g  shown i n  than  than  by the  Method  moment c a p a c i t y  even  219  i 0.4  r  1  0.6  J .0  0.B  MAGNIFIED MOMENT  F i g u r e 76 - P a r a b o l i c approximation to i n t e r a c t i o n diagram f o r magnified moments (non-dimensionalized) for small a x i a l  loads.  A design  equation  for stability  based  on t h i s approach would be C F M m M  u  and  <  1  the s t r e n g t h check would become  f!_12 *-P ' a  where  +  M  < i  a little  u  A refinement  to  l e s s c o n s e r v a t i v e would be to use a  v a r i a b l e exponent rather than the f i g u r e and  (8.26)  1  the terms are a l l as used i n Method 1.  make these curves  8.25  (8.25)  of  2  in  equations  8 . 2 6 . For example, the exponent c o u l d be increased  to 4 or more f o r slender columns, but t h i s p o s s i b i l i t y has not  220  )  been c o n s i d e r e d worth p u r s u i n g . T h i s design method strong  or  weak  will  material.  that t h i s method w i l l  be  produce  even  more  conservative  Diagram  r e i n f o r c e d concrete design method can be m o d i f i e d t o  produce a method of s i m i l a r difference  being  that  form t o Method 1, the  the member  approximation  The  bilinear  t o the u l t i m a t e i n t e r a c t i o n diagram shown by the  straight  section.  significant  i s checked f o r m a t e r i a l  f a i l u r e only, r e g a r d l e s s of slenderness r a t i o .  outer  f o r weak  f o r strong m a t e r i a l .  METHOD 3; U l t i m a t e I n t e r a c t i o n The  curves f o r  A comparison with f i g u r e 63 shows  m a t e r i a l , but may become unsafe 8.4.7  identical  lines  in figure  75  will  be  used i n t h i s  The r e s u l t i n g design equation i s C  _E  F M M  p  < f !_£_), B  C FM  P ' a  1  u  m  < M  (8.27)  u  which i s the same as equation 8.22 with Pu r e p l a c e d by Pa. strength  check  necessary  f o r unequal end e c c e n t r i c i t i e s .  Method  3  conservatively between  points  identical  is  similar  assumes Pa  to  a  equation  to straight  and Mu,  and  the line uses  8.24  OHBDC  would  A  a l s o be  code  which  interaction  diagram  the  unconservative  expression of equation 8.3 as the m a g n i f i c a t i o n f a c t o r .  221  8.4.8 Comparison of Methods 1 to 3 Figures  77(a),  (b)  and  (c)  show  proposed design methods 1 to 3, f o r the groups  of  material  properties.  made by working backwards from magnified  moments,  as  method with those figure  makes  marked.  three,  representative  These comparisons have been  the  described  code, to compare unmagnified  a comparison of the  interaction  diagram  for  i n d i s c u s s i o n of the OHBDC  moments (design moments) f o r each  predicted  by  the  strength  model.  Each  the comparison f o r three slenderness r a t i o s , as  In each case the s o l i d l i n e  i s behaviour  the s t r e n g t h model, and a d i f f e r e n t dotted l i n e  p r e d i c t e d by is  used  for  each proposed method. Consider slenderness  the upper group-of  curves  i n f i g u r e 77(a), f o r a  r a t i o of L/d=10 and average m a t e r i a l p r o p e r t i e s .  Method 1 i s seen to be the best f i t to significant  discrepancy  being  at  the  model,  low a x i a l  the  only  loads where the  "nose" was cut o f f the i n t e r a c t i o n diagram i n f i g u r e 75. The more  p a r a b o l i c approximation  conservative,  although  p r e d i c t i o n and the Method Method  3  i s considerably  i t does c o - i n c i d e with the model curve  on  the  vertical  axis.  produces the same r e s u l t s as Method 1 f o r low a x i a l  loads, but f o r high a x i a l region  1  of Method 2  and  meets  the  loads  i t moves  into  v e r t i c a l a x i s at a l o a d  the  unsafe  corresponding  with a x i a l c r u s h i n g f a i l u r e of the m a t e r i a l . For more slender column behaviour, group  of  curves  consider  i n f i g u r e 77(a) f o r L/d=20.  the  central  The curves f o r  Methods 1 and 2 have moved a l i t t l e c l o s e r together, and  both  222  UNnflGNIFIED nOHENT  UNnflGNIFIED HOHENT  0.0  0.2  0.4  0.6  '  0.8  1.0  1.2  UNnflGNIFIED HOriENT Figure  77  - Comparison of methods 1 t o 3 w i t h prediction (non-dimensionalized)  model  223  are c o n s e r v a t i v e Method  3  is  throughout, Method 2 being  again  the same as method 1 f o r low a x i a l  moving i n t o the unsafe method  3  curve  region  for  the  vertical  meets  c o r r e s p o n d i n g t o the curves  Euler  buckling  the  more  axial  axis  loads.  The  at an a x i a l  load  accurate  c a p a c i t y Pu, f o r columns of t h i s For much more slender L/d=40,  Pe,  load  whereas  concentric  intermediate  the  column  length.  columns with a slenderness  r a t i o of  the bottom group of curves on f i g u r e 77(a) shows that  i s not much d i f f e r e n c e between the  being  slightly  concentric Pe,  high  loads,  f o r Methods 1 and 2 meet the v e r t i c a l a x i s at an a x i a l  load corresponding to  there  somewhat more so.  conservative.  three  methods, a l l  For t h i s slenderness  r a t i o the  column c a p a c i t y Pu i s almost e x a c t l y the E u l e r  so a l l the curves meet at t h i s point on the v e r t i c a l For  materials  properties, described  with  different  relative  load axis.  strength  f i g u r e s 77(b) and (c) show s i m i l a r p l o t s to those  f o r f i g u r e 77(a).  Method 1 i s again  the  best f i t ,  although i t becomes s l i g h t l y unsafe f o r strong m a t e r i a l at low slenderness  ratio.  The formula f o r Method 2 does not i n c l u d e  the  r e l a t i v e s t r e n g t h p r o p e r t i e s , so t h i s curve  on  figures  77(a),  strong m a t e r i a l .  of  group. results  Method  and  producing the 1,  which  the  poor  most  consistent  areas.  results  is  i s the p r e f e r r e d method from t h i s  The conservatism i n from  identical  ( c ) , and i t becomes unsafe f o r  Method 3 remains unsafe i n s e v e r a l  The proposal that  (b)  is  this  method,  approximation  diagrams f o r magnified moments shown i n  of  where  i t exists,  the  interaction  figure  75,  and  the  224  conservative  nature  of  f a c t o r f o r low a x i a l A general difference L/d2:20. for  the  proposed  moment  magnification  loads, as shown i n f i g u r e 74.  observation  i s that  there  i s very  little  between the a l t e r n a t i v e s f o r slenderness r a t i o s of  Method 1 becomes much b e t t e r than the  slenderness r a t i o s of L/d<20, p a r t i c u l a r l y  other  methods  f o r weak or low  grade m a t e r i a l . B e t t e r accuracy would r e s u l t more  complicated  formula  from a more p r e c i s e but much  f o r the  shape of the i n t e r a c t i o n  diagram, or from p u b l i s h e d i n t e r a c t i o n diagrams f o r a range of m a t e r i a l p r o p e r t i e s and slenderness r a t i o s . of  f a m i l i e s of curves  If the p u b l i s h i n g  i s to be s e r i o u s l y c o n s i d e r e d  be b e t t e r to_ p u b l i s h diagrams f o r unmagnified be suggested 8.4.9  42(b) showed  i n t e r a c t i o n curves of a x i a l failure  moments, as w i l l  next.  METHOD 4; P u b l i s h e d Design Figure  i t would  a  Curves  family  of  load vs.  non-dimensionalized  unmagnified  f o r a range of slenderness r a t i o s .  moment, at  I f curves of t h i s  nature were made a v a i l a b l e i n p u b l i s h e d form, a designer c o u l d simply check that a given combination moment M f a l l s ratio.  of  Published  design  curves  are  (ACI 1970).  used  r e p r e s e n t s a combination cause  a  of a x i a l l o a d  member to f a i l .  f a i l u r e would be s i g n i f i c a n t l y more  P and  slenderness  extensively for  The designer should be  aware of the d e r i v a t i o n of these c u r v e s .  just  load  i n s i d e the curve f o r the p a r t i c u l a r  r e i n f o r c e d concrete design  would  axial  A p o i n t on any curve and  end  moment  that  The mid-span moment at  than  this  end  moment.  225  Both  material  considered The most  failures  i n producing  being  strength,  and  different  families  suitable  range  dimensionalized cross  included from  case  of  so t h a t  vertical axis The  conservative unsafe  areas  simple  design  lines for  They  end  number to  for  be n e c e s s a r y  P and M w i t h 42(b),  be  factor  Cm  the p u b l i s h e d c u r v e s . t o check c r o s s  section  Approximation  a  42(b) a r e  straight tend  line  line  straight,  interaction  t o t u r n upwards  axis  figure  columns,  f o r longer columns.  almost  near  is the  between Mu on t h e h o r i z o n t a l  a r e shown on short  could  the corresponding curve (the  in figure  straight  size  f o r L/d=0)  Line  of  a  be nonany  eccentricities  with  of  cover  would  c o u l d be u s e d  and, s a y , 0.9 Pu on t h e v e r t i c a l resulting  A  necessary  values.  Because the c u r v e s a  been  compression  t h e d e s i g n moment M by t h e  i t would a l s o  axis,  have  f a c t o r s , the  to  elasticity.  8.12 b e f o r e c o m p a r i n g  possibility  tension  would be  Unequal  on f i g u r e  suggested.  of  one g r a p h  Some o f t h e c u r v e s the  of  practical  8.4.10 METHOD 5: S t r a i g h t  and  ratio  of c u r v e s  comparing  curve  depends on s e v e r a l  modulus  by m u l t i p l y i n g  strength, outer  the  failures  the curves.  the  section.  equation  In t h i s  instability  shape o f t h e s e c u r v e s  important  of  and  and  will  78.  be c o n s i d e r e d . They  are  very  o n l y have a few s m a l l  These l i n e s  produce  a  very  equation  P 0.9 P  C  + u  M  m  M  < 1 u  (8.28)  226  F i g u r e 78 - S t r a i g h t l i n e approximations to i n t e r a c t i o n diagrams for unmagnified moment (non-dimensionalized)  227  The  terms  are  all  as  defined  s t r e n g t h check i s necessary  previously.  Once  again a  f o r unsymmetrical l o a d i n g , and  b i l i n e a r approximation of equation  8.24  the  c o u l d be used f o r that  purpose. Method 5 can be compared with Method 1 the  straight  comparison  lines  of  i s shown i n  material.  The  figure figure  solid  lines  which represent Method 4 i f Method  78  on  79(a),  by  superimposing  figure for  average  produced  as  The  corresponding  some  interaction factor,  cases  for and  and ( c ) . just  happens  to  because the shape of the mid-span moment  .diagram,  produces  different  comparisons f o r strong  T h i s approach i s p u r e l y e m p i r i c a l , and in  curves.  f o r short columns ( f o r example,  weak m a t e r i a l are shown in f i g u r e s 79(b)  work  a  strength  published  the o u t s i d e curves where L/d=lO), but not much columns.  Such  are the model p r e d i c t i o n values  5 i s very c o n s e r v a t i v e  longer  77,  reduced  an  by  the  interaction  moment  diagram  c o n s i s t i n g of n e a r l y s t r a i g h t l i n e s .  magnification  for  end  A disadvantage  moments of  this  method i s that d e s i g n e r s are not a l e r t e d to the r e a l behaviour of  slender  compression  members  with moment m a g n i f i c a t i o n .  Consequently t h i s method does not provide handling  unusual design  much  guidance  for  situations.  8.4.11 METHOD 6; A x i a l Load-Slenderness Curves Another  quite  different  approach  to  the  design  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 slenderness to  the  curves  for various e c c e n t r i c i t i e s .  three-dimensional  sketch  shown  in  With figure  of  load-  reference 11,  the  228  nODEL PREDICTION HETHOD 1 HETHOD 5  Average  UNnflGNIFIED HOHENT nODEL PREDICTION HETHOD 1 HETHOD 5  Weak  UNHHGNIFIED HOHENT  t.2  nODEL PREDICTION HETHOD 1 HETHOD 5  UNHRGNIFIED HOHENT Figure  79  1.2  - Comparison of methods 1 and 5 w i t h prediction (non-dimensionalized)  model  229  interaction  diagrams  represented  cuts  moment plane. radial  cuts  discussed  through  the  in  the  surface  previous  p a r a l l e l t o the load  T h i s s e c t i o n d e s c r i b e s behaviour perpendicular  passing through  to  the  the slenderness a x i s .  load-slenderness  relationships  been d e s c r i b e d i n Chapter  from  sections  represented by  load-moment  plane, a l l  The d e r i v a t i o n of a x i a l the  s t r e n g t h model has  6 and a t y p i c a l p l o t  was  shown  in  f i g u r e 43. There axial  have been many p r o p o s a l s f o r formulae  load-slenderness  columns.  Several  curves  were  for  to construct  eccentricially  i n t r o d u c e d i n Chapter  loaded  2, where t h e i r  assumptions and l i m i t a t i o n s were d i s c u s s e d b r i e f l y .  The  most  common approach has been to use as assumed d e f l e c t e d shape and a  simple  stress. are  failure  criterion  The secant  both  based on a l i m i t i n g  formula and  i n t h i s category.  the  f a i l u r e s t r e s s e s i n bending  approach  also  three  assumptions.  formula  and i n compression,  and  used i n the d e r i v a t i o n of the NFPA formula, so formulae A  are  different  all  based  approach  c o n s i d e r e d s t a b i l i t y behaviour It  Perry-Robertson  Both can be m o d i f i e d to allow f o r  different  these  compression  on  was  very  similar  taken by Jezek  of an e l a s t o - p l a s t i c m a t e r i a l .  i s not a p p r o p r i a t e to compare a l l these formulae  however an  example  slenderness  curves  the s t r e n g t h model.  will  be  who  used  to  compare  axial  here, load-  from one formula with r e s u l t s p r e d i c t e d by Consider  given by Malhotra(1980) as  the  modified  secant  formula,  230 + f f  u  + f ]2 - 4f f ( l - 1.04e ) e c  - / [f ( i +  =  in  2(1 -  which  e  i s the e c c e n t r i c i t y of l o a d , and d i s the member  t h i c k n e s s i n the d i r e c t i o n of berfding. the of  (8.29)  Malhotra  found  that  m o d i f i e d secant formula gave a reasonably good p r e d i c t i o n test  results  f o r timber  eccentricities.  Figure  columns  80  compares  formula with r e s u l t s p r e d i c t e d by the range  of  eccentricities.  loaded the  with  small  modified  secant  strength  The model p r e d i c t i o n  model  for a  in this  case  MODEL PREDICTION MODIFIED SECANT  O  —i  1  ao  1  1  1  16.0  1  24.0  1  1  32.0  SLENDERNESS (L/dJ  1  1—  40.0  48.0  F i g u r e 80 - M o d i f i e d secant formula compared with model p r e d i c t i o n has been made using the average  representative strength ratios  231  p r e v i o u s l y d e s c r i b e d and the f i t i s reasonably good. not be the same i n other cases, for d i f f e r e n t  however.  Model  This  may  predictions  s t r e n g t h p r o p e r t i e s were shown i n f i g u r e  64.  The other formulae have not been p l o t t e d here, but i t can be  shown  that  quite similar  the  fits,  Jezek  formula and the NFPA formula give  and the Perry-Robertson  formula  tends  to  underestimate column s t r e n g t h . This  method  has s e v e r a l disadvantages compared with the  a x i a l load-moment i n t e r a c t i o n diagrams most  cases  assumptions commercial for  for  of  the  behaviour  formulae clear  have  wood,  q u a l i t y timber.  cases  provide  the  unequal  end  designer  as  on  simplistic  not  be  valid  for  formulae cannot be e a s i l y  used  may  as  much  and they do not  insight  into  8.5  further  in t h i s  code  produces  not  been  TENSION  requirements  t e n s i o n are a l l based on a l i m i t i n g which  Design  thesis.  COMBINED BENDING AND Existing  member  based on moment m a g n i f i c a t i o n .  methods based on a x i a l l o a d - s l e n d e r n e s s curves have pursued  In  based  eccentricities,  with  methods  been  which  The  discussed e a r l i e r .  a linear  for  combined  stress  bending  failure  and  criterion  i n t e r a c t i o n between bending  capacity  and a x i a l t e n s i o n c a p a c i t y  fu  +  W  u  *  (8.30)  1  where T i s the design t e n s i o n f o r c e , Tu i s the  axial  strength,  is  M  is  the  design  moment  and  Mu  tension  the moment  232  capacity. 54  I t can be seen from a t y p i c a l case such  that  the s t r a i g h t  line  approach in a l l cases. with  a  low  relatively linear  ratio  stress-distribution  i n t e r a c t i o n formula  tension  a l s o come i n t o t h i s  This  for high is  high  Some machine s t r e s s  category.  A u s e f u l approximation for a l l cases the  effect.  strength  the compression s t r e n g t h .  graded m a t e r i a l may  material  becomes very c o n s e r v a t i v e  grade or c l e a r m a t e r i a l where the with  for  t e n s i o n to compression s t r e n g t h and a  insignificant  compared  figure  i n t e r a c t i o n w i l l be a c o n s e r v a t i v e  I t w i l l be most accurate  of  as  is  to  assume  that  u l t i m a t e i n t e r a c t i o n diagram f o r m a t e r i a l s t r e n g t h can  approximated by a parabola a s i d e , t h i s parabola too,,  but  was  was  as shown  in  considered  figure  81.  be  (As  an  f o r d e s c r i b i n g compression  found to be much l e s s accurate  than the methods  described. The figure  solid 54,  being  interaction dotted through  l i n e s i n f i g u r e 81 are the same as the l i n e s in  diagram  line  is a  the  Each  for  the  prediction  the 38x89 mm which  tension  is  has  and  bending  parabola  model p r e d i c t i o n , design  model  parabola  axial  v e r t i c a l a x i s , and axis.  the  for  the  material tested.  Each  been  pass  forced  to  compression strengths on strength  on  the  seen to be a reasonable  particularly  ultimate  in  the  tension  combination of a x i a l t e n s i o n T and  the  horizontal f i t to the region.  bending moment M i s  i n s i d e such a parabola i f  [L  T  u  2r 1+r  1-r 1+r  12 J  +  M M  [1  u  "  (  A  <  1  (8.31 )  233  MOriENT (KN.ri)  F i g u r e 81 - P a r a b o l i c approximation to  interaction  diagram where r i s the compression  ratio  strength,  r=1.0, equation  of  axial  ft/fc.  In  strength  the symmetrical  to  axial  case when  8.31 s i m p l i f i e s to  L  T ' u  which i s c o n s e r v a t i v e  M  u  f o r r g r e a t e r than 1.0.  A c o n s e r v a t i v e design method where  tension  is  to  use  equation  8.30  t e n s i o n s t r e n g t h i s l e s s than compression s t r e n g t h , and  234  equation  8.32  unwieldy,  equation 8.31  based design 8.6  in  other  cases.  The  more  accurate,  but  might be i n c o r p o r a t e d i n t o a computer-  procedure.  SUMMARY This  chapter  has  reviewed  members with combined bending concrete and  and a x i a l  possible  been  investigated.  new  Of  the  Method  1  provides  six  provides  behaviour  steel,  a  reasonable  alternative  methods  be recommended f o r design  approximate formulae  c a l c u l a t i o n , suitable for inclusion method  for  design methods f o r timber members  d e s c r i b e d , only Methods 1 and 4 w i l l purposes.  loading,  timber.  Several have  e x i s t i n g design methods f o r  in a  design  prediction  of an e c c e n t r i c a l l y loaded column.  of  f o r hand  code. the  Method  This actual  4  is  a  more a c c u r a t e method i n which p u b l i s h e d design curves would be used  by the designer to check whether a design combination  a x i a l l o a d and unmagnified  moment i s s a f e .  A p p l i c a t i o n s of both these methods w i l l broader  context  related  topics.  in the next  of  chapter,  be d i s c u s s e d in a  together  with  several  235  IX.  This  chapter  subjected of  DESIGN RECOMMENDATIONS  recommends  design  methods  to combined bending and  proposals  described  in  f o r timber  members  a x i a l l o a d i n g , on the  Chapter  8.  This  basis  chapter  also  i n c l u d e s d i s c u s s i o n of s e v e r a l r e l a t e d t o p i c s . 9 . 1 STRUCTURAL ANALYSIS 9.1.1  Strength Model T h i s study has demonstrated that the u l t i m a t e s t r e n g t h of  a s i n g l e timber equal  end  member subjec t e d to e c c e n t r i c a x i a l load  eccentricities  can  be  a n a l y s i s , using a computer-based rational  analysis,  modelled  numerical  by  a  with  rational  procedure.  This  c a l l e d the " s t r e n g t h model", i n c l u d e s  the  e f f e c t s of both geometric and m a t e r i a l n o n - l i n e a r i t . i e s . The  most  designing  such  accurate  considered  method  a member would be to use  Such a procedure i s not be  available  by  realistic  designers  of  analysing  the s t r e n g t h model.  f o r most d e s i g n e r s , but of  large  volume  components such as mass-produced roof t r u s s e s . would  be  unequal end  more  attractive  if  e c c e n t r i c i t i e s and  structural  This  transverse  applicable  a n a l y s i s , as suggested i n Method 4 of the design  charts  could  approach  loads. approach  to produce a s e r i e s of design c h a r t s based on the  These  may  the model were able to handle  A l e s s c o s t l y and more widely be  and  be  combinations of s t r e n g t h v a l u e s ,  produced  previous for  for d i f f e r e n t  would  rational chapter.  any  desired  sizes,  grades,  236  moisture contents and so on. 9.1.2 Second Order Many  Structural Analysis  structural  programs f o r second  designers  now  have access to computer  order s t r u c t u r a l a n a l y s i s , which  e q u i l i b r i u m on the deformed shape of the s t r u c t u r e . of a n a l y s i s  calculates  "magnified"  moments  formulate T h i s type  throughout  the  structure. Most  programs  of  m a t e r i a l behaviour, be  t h i s type are based  on l i n e a r  elastic  so c a l c u l a t e d member d e f l e c t i o n s w i l l  not  as a c c u r a t e as those c a l c u l a t e d using the s t r e n g t h model.  T h i s should be taken order  analysis  i n t o account  programs f o r timber  a v a i l a b l e which i n c l u d e the properties,  by  but  effects  The major advantages of second model  structures. of  using  second  Programs are  non-linear  material  none are s p e c i f i c a l l y a p p l i c c a b l e t o timber,  which e x h i b i t s much l e s s d u c t i l i t y  strength  designers  than s t e e l , f o r example. order  are that they can handle  programs rigid  over  the  frames of many  members, as w e l l as arches and any other s t r u c t u r a l form, with any p a t t e r n of l o a d s . The the  use of a second  order program can  be  combined  with  r e s u l t s of t h i s study to produce a design method which i s  a major improvement over c u r r e n t methods.  The  a n a l y s i s w i l l provide i n f o r m a t i o n on i n s t a b i l i t y design thesis. diagram  criteria The  for material  designer  f o r the  failures  requires  the  can  second  order  f a i l u r e s , but come from  ultimate  this  interaction  member under c o n s i d e r a t i o n , and t h i s can be  p r o v i d e d i n the form of a  computer  output  curve,  or  as  a  237  simple  approximation  using equations  8.23  A i n c l u d e s a method f o r c a l c u l a t i n g the diagram  which  is  more  direct  and  8.24.  ultimate  Appendix interaction  than the s t r e n g t h model, more  a c c u r a t e than the approximate equations, but r a t h e r t e d i o u s . 9.1.3  Simple  Analysis  Structural approximate  designers  design  often  w'ish  to  carry  out  of a s t r u c t u r a l member without  an  the use of  s o p h i s t i c a t e d computer programs as d e s c r i b e d above. If a x i a l loads and end moments i n a member are by hand a n a l y s i s , or using a f i r s t order computer  program,  the  methods  determined  structural  described  in  the  analysis previous  chapter are s u i t a b l e f o r c a l c u l a t i n g whether a chosen s i z e  has  sufficient  s t r e n g t h . .The  use  published  design  in Method 4. specifying as Method 1. 9.1.4  most accurate of  these  to  c h a r t s from the s t r e n g t h model, as proposed  A  less  accurate  as code formulae,  method,  more  suitable  for  i s to use the approach d e s c r i b e d  T h i s approach i s summarized i n the next s e c t i o n .  Code Format If a timber  allow  these  design code i s to  approaches,  have  the  an  optional  approximate  m a t e r i a l s are based on t h i s  flexibility  to  i t should be a t w o - l e v e l code which  recommends r a t i o n a l a n a l y s i s , but a l s o p r o v i d e s for  is  method.  philosophy.  the  formulae  Many codes f o r other  238  9.2 APPROXIMATE DESIGN FORMULAE 9.2.1 Recommended Formulae If a designer a  i s not i n a p o s i t i o n to use the r e s u l t s  of  r a t i o n a l a n a l y s i s , the f o l l o w i n g approximate procedure from  Chapter  8  compression  i s recommended member  can  f o r checking  resist  a  that  design  a  selected  a x i a l load P and a  bending moment M about one p r i n c i p a l a x i s .  1.  Obtain the a x i a l compression  column  Pa,  the  axial  capacity  for a  short  t e n s i o n c a p a c i t y Tu, the bending  moment c a p a c i t y Mu, and the modulus of e l a s t i c i t y E, f o r the  member.  Derivation  in the next s e c t i o n . depend  on  the  of t h i s input data i s d e s c r i b e d  A l l of  these  values,  except  E,  s i z e of the member, a l s o as d e s c r i b e d i n  the next s e c t i o n .  2.  C a l c u l a t e the load c a p a c i t y Pu, of the  concentric  a x i a l compression  member  under  loading.  P Pu i+  _c  *  (L/d) E  where  a  „  (9.1)  ;  40  f c / E i s the r a t i o of the f a i l u r e s t r e s s of a short  column t o modulus of e l a s t i c i t y , and L/d i s the slenderness  3.  largest  r a t i o about e i t h e r p r i n c i p a l a x i s .  Check that the design  combination of a x i a l  mid-span moment M s a t i s f y the s t a b i l i t y  formula  load P and  239  _E  <  B  n - £_}  u  C  FM  <  (9.2)  M  u  where F i s a moment m a g n i f i c a t i o n  f a c t o r given by  1 + P/P F  =  ±  1 - P/P  (9.3)  e  where Pe i s the E u l e r b u c k l i n g Cm i s a f a c t o r to allow  C  ffl  where  M,  =  and M  respectively.  2  load.  f o r unequal end moments  0.6 + 0.4 M /M 2  1  >  0.4  (9.4)  a r e the l a r g e r and smaller end moments,  B isa  factor  used  to  approximate  the  shape of the i n t e r a c t i o n diagram, given by  B  where  ft/fc  i s the  compression strengths  4.  =  (9.5)  t c ratio  defined  of  axial  i n step 1.  I f end moments are not equal,  check that the  end moment M, s a t i s f i e s the s t r e n g t h  ^  < B(l - £ - ) , u  a  t e n s i o n to a x i a l  M  X  <  maximum  formula  M  u  (9.6)  240  9.2.2 Example The  formulae  described  above  will  be s i m p l i f i e d when  m a t e r i a l p r o p e r t i e s are i n c l u d e d f o r a p a r t i c u l a r  species  grade of timber.  2 and B e t t e r  grade  tested  Consider  the SPF timber  i n t h i s study.  the r a t i o of f t / f c  i s 0.7.  of No.  and  The r a t i o of f c / E i s 1/300, and  Using  these  figures,  the  formula  for c o n c e n t r i c l o a d i n g becomes  P  p  and  the  2 . , (L/d) 12000  =  (9.7)  3  s t a b i l i t y check f o r combined bending and compression  becomes C  m M  FM  <  1.71 (1  h  C  u  and  <  (9.8)  M u  the s t r e n g t h check, f o r the end of a member becomes  £  «  1-71  u  9.2.3  F M m  ( i - f ) . a  M, 1  < M  Load F a c t o r s and Resistance The  design  u  Factors  formulae presented  resistance factors.  (9.9)  here do not i n c l u d e load or  These can be added t o the formulae  established  methods  formulated.  Some codes use l a r g e r s a f e t y f a c t o r s f o r  columns  than  once  f o r squat  the  overall  design philosophy i s slender  columns because of the greater danger  o f a c c i d e n t a l e c c e n t r i c i t y or o v e r l o a d the  using  causing  g r e a t e r consequences of such f a i l u r e .  failure,  and  T h i s can be e a s i l y  241  incorporated. 9.2.4 Mimimum Moments The and  proposal  initial  of 0HBDC(1983) for minimum end  l a c k - o f - s t r a i g h t n e s s appears t o be a s e n s i b l e one  that should be i n c o r p o r a t e d the  need  to  design  i n t o code requirements.  every  slender d i r e c t i o n when no other  avoid  i n only  the most  s i g n i f i c a n t moment e x i s t s , and  both d i r e c t i o n s i f s i g n i f i c a n t a p p l i e d moment e x i s t s .  a c t u a l values not  To  member f o r b i a x i a l bending i t i s  suggested that these e f f e c t s be considered  in  eccentricity  suggested by OHBDC appear reasonable,  been i n v e s t i g a t e d i n t h i s  but  The have  study.  9.3 DATA REQUIRED 9.3.1  In-grade Test  Results  To use the design methods proposed i n t h i s t h e s i s f o r any timber content,  members  of  given  size,  the f o l l o w i n g i n f o r m a t i o n  s p e c i e s , grade and moisture i s r e q u i r e d f o r the member  under c o n s i d e r a t i o n . 1.  The  a x i a l t e n s i o n c a p a c i t y Tu, of the member, which  i s the product  of the c r o s s s e c t i o n 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 c a p a c i t y  which i s the product a x i a l compression 3.  Pa,  of  the member,  of the c r o s s s e c t i o n area A, and the  f a i l u r e s t r e s s f c , f o r a short column.  The bending moment c a p a c i t y Mu, of the member, which  i s the product of rupture f r .  of the s e c t i o n modulus S, and the  modulus  242  4. Each  The modulus of e l a s t i c i t y ,  of  these  properties  E.  should  be  d e r i v e d from  t e s t i n g on the a c t u a l m a t e r i a l i n q u e s t i o n . are  based  on lower  Values  obtained d i r e c t l y  r e s u l t s must be m o d i f i e d with s a f e t y  The  so  factors,  strength  the  from t e s t  load  model can be used to d e r i v e Mu  bending  strength  Mu  f a c t o r s are  duration  results.  The  first  from Tu  and  available.  i s the most o f t e n measured  s t r e n g t h p r o p e r t y , so design values can be from t e s t  codes  on.  Pa, p r o v i d e d that c e r t a i n c a l i b r a t i o n However  As long as  5th p e r c e n t i l e v a l u e s , the values are a l l  r e q u i r e d at that q u a n t i l e .  f a c t o r s and  in-grade  obtained  directly  three items above are s u b j e c t to  s i z e e f f e c t s as d e s c r i b e d i n the next s e c t i o n . In  Canada  the  c u r r e n t design code (CSA  a l l o w a b l e s t r e s s e s i n a x i a l t e n s i o n based on in-grade  testing,  strength.  code  modulus  members, and  bending  of  steps are being  s t r e s s e s to r e f l e c t  elasticity,  data  d r a f t code (CSA  1983)  modulus  c a l c u l a t e d as 0.74  of  includes  a  elasticity  to  in-grade t e s t  members has so  taken  provided  of  change  results.  information  i s a v a i l a b l e to s p e c i f y  modulus of e l a s t i c i t y at the 5th p e r c e n t i l e l e v e l .  percentile  results  so code f i g u r e s are a r e l i a b l e estimate of  In-grade t e s t i n g of bending on  the  specifies  A l a r g e amount of in-grade t e s t i n g has been c a r r i e d  out on bending the  1980)  proposal for  times the average value  to  The use  column design  latest a  5th  simply  ( f o r sawn t i m b e r ) .  243  In-grade Compression Strength The  i n f o r m a t i o n not yet a v a i l a b l e  strength.  A  is  axial  compression  few small s t u d i e s of in-grade a x i a l  compression  s t r e n g t h have been r e f e r r e d to i n t h i s t h e s i s , but many  grades,  Some r e s u l t s shortly.  The  different  species in  there  are  and s i z e s which have not been t e s t e d .  this  area  allowable  are  expected  stresses  to  be  available  i n the code are very much  from v a l u e s suggested by in-grade t e s t  results  for  some s p e c i e s . It  has been shown how  the i n t e r a c t i o n diagram f o r timber  members i s s e n s i t i v e to the r a t i o of t e n s i o n strengths.  If  this  ratio  s t r e s s e s i n the c u r r e n t code  is  and  calculated  there  will  be  compression  from  allowable  some  extremely  misleading r e s u l t s . No s i g n i f i c a n t until are 9.3.2  •  '  improvements can be made to design methods  comprehensive  in-grade  a x i a l compression t e s t  results  available. Size  Effects  A x i a l tension strength,  are  and  affected  compression by  strengths,  member s i z e .  and  bending  If s t r e n g t h values  based on in-grade t e s t i n g are not a v a i l a b l e f o r any s i z e , they can be estimated u s i n g  the  following  which were d i s c u s s e d i n Chapter  3.  size  effect  formulae  244  a. The  Length E f f e c t s  b a s i c formula  f o r l e n g t h e f f e c t s i s equation x, . _J. *2  where  x,  and  x  r e s p e c t i v e l y , and The  2  L = (-1) l 0  1/k,  (9.10)  1  L  are  strengths  of  lengths  k, i s the l e n g t h e f f e c t  t e n t a t i v e v a l u e s of k, found  i s the corresponding  L,  and  L  2  parameter.  i n t h i s study are  in Table IV, with d i f f e r e n t values f o r two in brackets  3.11.  grades.  The  given figure  strength reduction factor for  d o u b l i n g t h e . l e n g t h of a member.  Number 2 A x i a l Compression A x i a l Tension Bending  13 4 4  Select  (.95) (.84) (.84)  13 6 6  Table IV - Length e f f e c t parameter  Most 3.0m  in-grade  testing  Structural (.95) (.89) (.89) k.  i s c a r r i e d out at standard  f o r a x i a l t e s t s and  lengths of  17 times the depth f o r bending  For l e n g t h s s h o r t e r than these t e s t  lengths i t i s c o n s e r v a t i v e  to assume no l e n g t h e f f e c t , or to use a l a r g e number length  effect  to use a low The on  factor.  experimental  for  the  For longer lengths i t i s c o n s e r v a t i v e  number f o r the l e n g t h e f f e c t  length e f f e c t  tests.  factor.  f a c t o r s d e s c r i b e d here have been  data from only two  e x t r a p o l a t i o n beyond t h i s range may These c o n s i d e r a t i o n s should  be  based  lengths i n most cases, produce d i f f e r e n t  investigated  further  and  results. before  245  i n c o r p o r a t i n g l e n g t h e f f e c t p r o v i s i o n s i n t o design b. If  codes.  Depth E f f e c t s  test  r e s u l t s are a v a i l a b l e f o r one depth of c r o s s  section,  the s t r e n g t h s of other depths can be p r e d i c t e d from  x ,  d~  JL *2  where  x,  and x  2  l / k  -  9  (9.11)  2  1  are s t r e n g t h s of members of depths d, and d  r e s p e c t i v e l y , and k T h i s formula  i s the depth e f f e c t  2  is  likely  to  be  2  parameter.  used  less  often  than  9 . 1 0 because there are only a few standard depths of  equation  c r o s s s e c t i o n and most have been t e s t e d . Values of k difficult for  in a constant  9.4  obtain  Limited  test  data  a value of k = 4 i n t e n s i o n , and k 2  effect  The bending factor  results for length. bending  to  because  i t is  t e s t s l e n g t h and depth have both been changed  ratio.  15 f o r bending. length  can be 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  most bending  suggests  2  value i s very  k, used to c o r r e c t A l a r g e r value of k  has a l s o been observed  on 2  SPF  material  i n the range 8 to sensitive  to the  in-grade bending 2  test  f o r t e n s i o n than f o r  i n c l e a r wood (Buchanan  1983).  OTHER LOADING CASES Almost  investigations  all  of  the  experimental  and  analytical  d e s c r i b e d i n t h i s t h e s i s have been f o r 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. S t r u c t u r a l d e s i g n e r s are o f t e n c o n f r o n t e d with, s i t u a t i o n s where end e c c e n t r i c i t i e s are unequal  or where a combination of  246  axial  loa.ds  and  transverse  loads occur.  Examples are wind  loads on r o o f - s u p p o r t i n g columns, p u r l i n loads on  top  chords  of t r u s s e s , and many o t h e r s . 9.4.1 Unequal End E c c e n t r i c i t i e s For axial  a l l of  the  design  method proposals i n c o r p o r a t i n g  load-moment i n t e r a c t i o n diagrams,  including  a  simple  method  of  unequal end e c c e n t r i c i t i e s has been introduced with  the Cm f a c t o r .  The accuracy  of t h i s f a c t o r f o r timber members  has not been i n v e s t i g a t e d i n t h i s study, but i n v e s t i g a t i o n s of i t s a p p l i c a b i l i t y to steel is  (Johnston  1976) suggest  that  this  a s u i t a b l y c o n s e r v a t i v e method f o r c o n s i d e r i n g unequal end  eccentr ic i t i e s . 9.4.2 Transverse In  real  combinations possible.  design  design method  any  situations  an  infinite  number  of  of t r a n s v e r s e loads and e c c e n t r i c a x i a l loads are No  analysis  Loads  simple  code  formula  f o r a l l cases.  A  can provide an accurate second  order  can c a l c u l a t e d e f l e c t i o n s throughout  loading.  approximate  I f such method  an of  analysis handling  structural  a structure for  i s not transverse  carried  out an  loads  becomes  necessary. T h i s d i s c u s s i o n assumes against  sidesway.  Special  that  the  mid-span  directly  in  the  is  braced  i s necessary f o r  I f t r a n s v e r s e loads  tend  to  moment due to a x i a l loads as shown i n  f i g u r e 82(a), the combined t o t a l used  structure  consideration  columns of unbraced s t r u c t u r e s . increase  the  stability  mid-span design  moment formula.  should  be  I f the  247  t r a n s v e r s e loads tend t o shown  i n f i g u r e 82(b),  eliminate  and  can  be  mid-span  used.  so  the  i n a l l s i t u a t i o n s a number of load  necessary,  aided by an understanding  failure.  In  a l l cases  the  magnification  must  judgement may be  load  capacity  to  be  expected  the designer  of  the  and  the  amount  t  Moment (a)  Load  of  as a r e s u l t of second  1  Load  with  m a t e r i a l , an  order d e f l e c t i o n s .  t  be  the design method recommended f o r a  i n t e r a c t i o n diagram f o r slender columns, moment  design  of the p o t e n t i a l modes of  s i n g l e member i s u s e f u l because i t provides of  strength  combinations  and a l a r g e measure of e n g i n e e r i n g  indication  as  Many other combinations are p o s s i b l e ,  considered  an  moment  load c a p a c i t y i s l i k e l y t o be based on  c r o s s s e c t i o n s t r e n g t h at the ends, formula  the  Moment (b)  F i g u r e 82 - Bending moment diagrams f o r combined and t r a n s v e r s e l o a d i n g  axial  248  A  recent extension  t h i s study  to one  of the source  programs used in  can c a l c u l a t e behaviour of beam columns  combination  of  axial  loads,  transverse  loads,  moments (Nathan 1983b), and c o u l d be used as the more d e t a i l e d study 9.4.3  Slenderness For  any  i n t h i s area  of  of  the  slenderness of  the  r a t i o has  to determine the  For a non-rectangular  this  consideration,  for  a  thesis  the the  term length  rectangular  has  not  cross  s e c t i o n d must be r e p l a c e d  r\/12 where r i s the r a d i u s of g y r a t i o n , but considered  cases.  length",  which  both ends.  i t should be noted  non-rectangular  sections  The is  length  the  L  should  be  the  "effective  a c t u a l l e n g t h for a column pinned  For a member in a s t r u c t u r e , i t s e f f e c t i v e  a load equal  to the load in  Standard  methods  the  member  without  arches  the  and  c o n d i t i o n s , but  frames,  a second order a n a l y s i s .  estimation  at  structure  are a v a i l a b l e f o r e s t i m a t i n g  e f f e c t i v e l e n g t h f o r a v a r i e t y of end s t r u c t u r e s such as  when  at  length  the l e n g t h of a pinned-pinned column which would buckle  difficult  by  within-member s i z e e f f e c t s have not been i n v e s t i g a t e d  for such  buckles.  a  and d i s the c r o s s s e c t i o n a l dimension in the  section.  is  of  "slenderness  been used as L/d where L i s  under  because  basis  i n the f u t u r e .  Throughout  direction  that t h i s study  or a p p l i e d  these design methods not u t i l i z i n g a second  member.  member  any  Ratio  order a n a l y s i s , i t i s necessary ratio"  under  the  for some is  very  249  9.4.4  Biaxial  Behaviour  T h i s t h e s i s has timber  investigated  the  in-plane  members s u b j e c t e d to bending about one  behavior  of  of the  principal  are many design s i t u a t i o n s where b i a x i a l  behaviour  axes of the c r o s s s e c t i o n . There  must be c o n s i d e r e d , and methods of the s t e e l formula  for  equation  8.14.  cbde  stability The  timber  design.  axial  load  for  f o r lack of any are  biaxial  same formula 8.14  The  loading  is specified the term  y  are  considered  using  the  case.  T h i s equation  axis  and  y-axis  assumes a l i n e a r effects,  which  More  accurate  is  the  lowest  and  and  are  magnified  E u l e r load as Pe interaction is  case  a  very  probably  i n each  between  x-  conservative  also  for  timber  e m p i r i c a l expressions are a v a i l a b l e  s t e e l members (Johnston The  shown as  Moments about the x  separately  corresponding  assumption f o r s t e e l members members.  is  code  c a p a c i t y of the column c o n s i d e r i n g the worst  axes  separately  steel  the  i n the OHBDC for  Pu  b u c k l i n g about e i t h e r p r i n c i p a l a x i s .  and  for  suggested.  under  In equation  b e t t e r information  1976)  but not  s t e e l code formulae modified  f o r timber  f o r timber  members.  members using  design proposal Method 1 become C  F  mx M  x  M  C  x  ux  my  F  M  y  M„  y  ,  RCI -  uy  P  P  — )  u  (9.12) C  F  mx M  for  stability,  x  ux  M  x  C +  F  my  y  M  M  y  <  1  uy  where Fx and Fy are  calculated  from  equation  250  9.3  using  the  corresponding  Euler  l o a d i n each case.  The  formula f o r s t r e n g t h becomes  M  M  M  M  T,  _JL_ +  M uy  ux  <  B  ( l - — ) J,  P ' a  '  M  - 2 - + -X_ < i ux  M  Some suggestions f o r f u r t h e r research  M  9.13  uy  in this  area  a r e made  below. a. The  Strength  strength  Under B i a x i a l  model developed i n t h i s . t h e s i s c o u l d be used to  investigate cross because  the  Loading  section  general  form  strength of  under  the  computer  members of any c r o s s s e c t i o n to be input. be  possible  in  f i g u r e . 83 " f o r would  Figure rectangular axis  program  In theory  many be  different  encountered  angles,  with  but  depth  83 - Bending about i n c l i n e d n e u t r a l  s e c t i o n s subjected  allows  i t would  a  serious  e f f e c t s . For  axis  to bending about one p r i n c i p a l  i t has been easy to i n c l u d e a depth e f f e c t and no width  e f f e c t on the b a s i s of experimental r e s u l t s . of  bending,  to change the angle of the n e u t r a l a x i s as shown  -  difficulty  biaxial  figure  83  For the  section  i t i s not easy t o separate depth e f f e c t s and  251  width e f f e c t s . to  be  the  If both depth and  same  (as  in  were  assumed  a p e r f e c t l y b r i t t l e m a t e r i a l ) then  s t r e s s e s c o u l d be n u m e r i c a l l y each s t e p .  width e f f e c t s  i n t e g r a t e d over the  S t r e s s e s at f a i l u r e may  section  w e l l be higher  for b i a x i a l  behaviour than for in-plane behaviour because i n general will  be  smaller  of  knots  on  One  interesting  biaxial  R i b e r h o l t and N i e l s e n b.  possible  been d e s c r i b e d  by  S t a b i l i t y Under B i a x i a l Loading  has  r e c e i v e d very l i t t l e a t t e n t i o n .  by  Larsen  theory and  behavior  and  Thielgaard  members under b i a x i a l The  (1979),  their  combined a x i a l  most u s e f u l study  who  to  undertaking.  is  have developed some  based on l i n e a r  elastic  theory  loads and  with  a  strong-axis  series  behavior  of t e s t s using  bending.  methods developed in t h i s t h e s i s c o u l d in  extended  loading  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 s t r e s s .  verified  The  of timber  design equations  a simple  simply  on the  (1976).  stability  They  study  s t r e n g t h has  The  and  there  volumes of h i g h l y s t r e s s e d m a t e r i a l f u r t h e r  from the n e u t r a l a x i s . effect  at  biaxial Cross  as shown  in  behaviour,  but  that  s e c t i o n s t r e n g t h can be the  previous  theory  be  would be a major investigated quite  section,  but  for  longer  columns the computer program would have to be expanded to keep track  of  d e f l e c t i o n s about both p r i n c i p a l axes.  An  accurate  a n a l y s i s should a l s o i n c l u d e the p o s s i b l e e f f e c t s  of  lateral  torsional  buckling.  analytical  study  extensive  The  into  experimental  complexities  biaxial study,  are  behaviour to  provide  such  would,  that  any  require  an  calibration  and  252  verification.  Work  in  this  d e s c r i b e d by Chen and Atsuta  area  on  other  materials  is ,  members  under  long d u r a t i o n  (1976b).  9.5 LONG DURATION LOADING The loading  behaviour  of  timber  i s of i n c r e a s i n g i n t e r e s t and  conservative efficient  design  methods  are  concern.  improved,  and economical designs, the p o s s i b l e e f f e c t s of long  affect  the  load  s t r e n g t h and s t a b i l i t y  c a p a c i t y of timber  Loading  •It i s now w e l l recognized  the  decreases  that  model  and  reliability who  has  been  loads.  i n c l u d e those  (1978) who proposed a damage subsequently  timber  by  accumulation  incorporated  into  s t u d i e s (Foschi 1982), and Johns and Madsen (1982)  investigated  bending members. agreement  of  under the e f f e c t s of long d u r a t i o n  Foschi  which  duration  members from both  strength  Recent t h e o r e t i c a l developments i n t h i s area Barrett  Long  considerations.  9.5.1 Strength Under Long Duration  members  existing  l e a d i n g t o more  d u r a t i o n l o a d i n g must be c a r e f u l l y c o n s i d e r e d . loads  As  with  .crack  growth p e r p e n d i c u l a r  These and other experimental  S e v e r a l experimental  theories  studies  on  to the g r a i n i n  show bending  quite  members.  d u r a t i o n of load s t u d i e s on a x i a l  members are known to be  in  progress,  but  good  apparently  tension axial  compression s t r e n g t h under long d u r a t i o n loads has r e c e i v e d no attention. There  is  no  conceptual  problem  in  i n c o r p o r a t i n g the  e f f e c t of d u r a t i o n of load on s t r e n g t h i n t o the s t r e n g t h model  253  or proposed design have  to  be  methods.  applied  to  strengths, appropriate 9.5.2  the  load  this  is  of  been  not  is  rather than s t r e n g t h c o n s i d e r a t i o n s ,  and  "creep",  and  strain  with  creates  serious  creep rather  this  i n c l u d i n g wood  time.  problems  independent of  this  assumption  with constant This  with  t e n s i o n members, but  is  is  stress exhibit  phenomenon,  increasing  thesis  known  as  deflections  in  potentially  i n compression members where s t a b i l i t y  far  failures  occur. The  of  compression  of the assumptions of  specimens loaded  bending members and more  slender  strength.  For many m a t e r i a l s  increasing  compression  Loading  that 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 are  valid,  simply  members  U n t i l t h i s p o i n t , one  time.  t e n s i o n and  a f u n c t i o n of modulus of e l a s t i c i t y and  than m a t e r i a l  has  factors  to the l o a d i n g under c o n s i d e r a t i o n .  capacity  governed by s t a b i l i t y  of load  axial  S t a b i l i t y Under Long Duration The  can  Duration  a  term "creep b u c k l i n g " r e f e r s to a  compression  deflections  member  slowly  under  increase  constant  over  time  stability l o a d s , as until  the  failure lateral member  buckles.  A significant  i n v e s t i g a t i o n of t h i s problem has  made  Kallsner  Noren (1978) who  by  and  used both a  numerical  approach d i v i d i n g the column i n t o a number of laminae, and approximate method based on a f i c t i t i o u s modulus of assuming  linear  elastic  behaviour,  to  been  an  elasticity  produce a number of  s t r e n g t h r e d u c t i o n curves f o r long term l o a d i n g . Creep b u c k l i n g has  not been i n v e s t i g a t e d i n t h i s t h e s i s .  254  The  most d i r e c t way  carry  out  of studying creep  a time-step  buckling  would  be  to  a n a l y s i s , i n c o r p o r a t i n g the e f f e c t s of  long d u r a t i o n l o a d i n g on the s t r e s s - s t r a i n  relationship.  Such  an a n a l y s i s would not be easy with the s t r e n g t h model, because of the s t r u c t u r e of the program.  However,.some i n d i c a t i o n  long term behaviour c o u l d be obtained of  by c a r r y i n g out a number  computer runs with the s t r e s s - s t r a i n  to simulate the  e f f e c t s of creep.  r e l a t i o n s h i p modified  I t would be necessary  to  know  e f f e c t s of long d u r a t i o n l o a d i n g on compression behaviour  (and, perhaps, t e n s i o n behaviour), be  of  obtained  members.  t h i s information  from d u r a t i o n of load t e s t s on a x i a l  Creep data  preliminary  and  compression  for bending members could be u s e f u l for a  investigation.  In the meantime designers  be made aware of the p o s s i b l e consequences of creep with  some  members  extra- conservatism  which  possibility  might  of  fail  creep  Schniewind  9.6  MOISTURE CONTENT  9.6.1  moisture  has  axial  the  an  instability  being  design  buckling,  equation  for  mode.  The  influenced  by  cyclic  Strength  a x i a l compression s t r e n g t h of wood content,  significant tension  only  been i n v e s t i g a t e d by Humphries  on Compression  whereas  the  r e l a t i v e l y u n a f f e c t e d by changes only  in  can  (1982).  E f f e c t of Moisture The  in  buckling  changes in moisture content and  could  and  study  axial in  is  tension  moisture  sensitive  to  strength  is  content.  of the e f f e c t s of moisture content  compression  strengths  is  reported  The on by  255  Madsen(1982). 9.6.2  E f f e c t of Moisture on Strength Under Combined Loading The  strength  model d e s c r i b e d i n t h i s t h e s i s can be used  to p r e d i c t the e f f e c t of moisture content f o r any of a x i a l and bending l o a d s . used  to  produce  figure  combination  The data of Madsen(1982) has been 84 which shows u l t i m a t e  interaction  diagrams f o r two d i f f e r e n t moisture c o n t e n t s .  The m a t e r i a l i n  t h i s case i s No.  2 and B e t t e r SPF,  size.  have  shown  only  been  38x140 mm  Results  f o r 11% and 25% moisture content, but  intermediate values f o l l o w a uniform trend between  these  two  levels. Axial  tension  s t r e n g t h has been taken to be  of moisture content as shown by the v e r t i c a l in  the  tension  region.  Axial  independent  axis  compression  intercepts  s t r e n g t h on the  other hand i s very s e n s i t i v e to moisture c o n t e n t . shows  predictions  bending. bending  One  is  how  independent  percentile level small  clear  a l l combinations  prediction  strength  demonstrated  for  on  of  the  considerable  horizontal  of  a x i a l load  and  interest  axis.  It  is  moisture  content  whereas  test  specimens)  the  at  studies  on  content  This graphical  explanation  e f f e c t of moisture content p r e v i o u s l y not well  moisture  of  bending  the  is clearly level  the  ( r e p r e s e n t a t i v e of strong m a t e r i a l ,  supports  Future  84  bending s t r e n g t h at the 5th p e r c e n t i l e  i n f l u e n c e on bending s t r e n g t h . strongly  of  Figure  such  95th as  has a major  representation  Madsen(l982) f o r the strength,  which  was  understood. into  the  e f f e c t of moisture content on  256  DRY(11%m.c.) WET(25°/om.c.)  95th %ile  CO CM . I  O.D  i  i  1  2.0  1  1  4.0  r~  B.O  nOPIENT (KN.fl)  8.0  F i g u r e 84 - E f f e c t of moisture content on s t r e n g t h of timber i n combined bending and a x i a l l o a d i n g bending s t r e n g t h should  concentrate  on  behaviour  in  axial  t e n s i o n and compression. 9.7  SUMMARY In  t h i s chapter,  the s t r e n g t h model has been  recommended  as the b a s i s of an u l t i m a t e s t r e n g t h design method f o r members.  One  of the a l t e r n a t i v e design methods i n v e s t i g a t e d  i n Chapter 8 has been recommended method  for  timber  designing  timber  as an  members  improved  approximate  subjected  to combined  257  bending and a x i a l l o a d s . of  structural  discussed future  timber  briefly,  research.  Many f a c t o r s a f f e c t i n g members  and  have  been  recommendations  the  behaviour  introduced have  and  been made f o r  258  X.  SUMMARY  The  major achievements of t h i s study are summarized below.  1.  T h i s study has demonstrated that the behaviour  of  members i n bending, and i n combined bending and a x i a l can  be p r e d i c t e d from observed behaviour i n a x i a l  compression  2.  The i n t e r a c t i o n  between a x i a l  flexural  f o r a wide range of e c c e n t r i c i t i e s  investigated  t e n s i o n and  of  members  member  length  experimentally  for  has  strength  experimentally  effect  timber  and  loaded  The  loading,  tests.  eccentrically  3.  timber  on  been i n v e s t i g a t e d and l e n g t h s .  strength  axial  of  has  tension,  been axial  compression and bending.  4.  A computer-based s t r e n g t h  model  has  been  developed  to  p r e d i c t the c r o s s s e c t i o n s t r e n g t h of timber members under any combination of a x i a l and f l e x u r a l  5.  loads.  T h i s study has demonstrated how the s t r e n g t h model can be  used t o p r e d i c t the l o a d c a p a c i t y of length  subjected  to e c c e n t r i c a x i a l  i n s t a b i l i t y and m a t e r i a l s t r e n g t h  6.  timber  members  loading, considering  testing  concept  by  any both  failures.  The s t r e n g t h model improves the t h e o r e t i c a l  in-grade  of  relating  basis  in-grade  of the  bending  test  259  r e s u l t s to those in. a x i a l q u a n t i f y i n g length  7.  tension  and  compression,  and  by  effects.  The s t r e n g t h model can be used as the b a s i s of a r a t i o n a l  design method f o r timber members subjected  to eccentric  axial  load.  8.  New approximate formulae have been proposed f o r d e s i g n i n g  timber  members  subjected  to  combined  bending  and  axial  loading. The  main  f e a t u r e s of the recommended approximate  design  method i n c l u d e :  a.  A new formula f o r the moment-axial  at a timber c r o s s s e c t i o n subjected  interaction  to combined  loading  fr - - U - f - J  0  where  load  u  (10.1)  a  M i s t h e ^ a p p l i e d moment, Mu i s the moment  of the s e c t i o n , P i s the a p p l i e d load, Pa compression  strength  of  is  capacity  the  axial  the s e c t i o n , and B i s a f a c t o r  which r e l a t e s the shape of the i n t e r a c t i o n diagram t o the r a t i o of the a x i a l  tension  strength  ft  to  the  axial  compression s t r e n g t h f c of the m a t e r i a l , given by  B  "  1.35  7  T  t  r  c  (10.2)  260  The  axial  strengths  f t and f c should  be obtained  from i n -  grade t e s t r e s u l t s .  b.  A  new formula f o r the moment-axial l o a d i n t e r a c t i o n  of e c c e n t r i c a l l y loaded  timber compression members of any  length F_M M  where Pu  B(l-i-) ' ' u  -  i s the  (10.3)  P  u  load  capacity  1  loading,  of  the  including  member  concentric  axial  any  slenderness,  and F i s a moment m a g n i f i c a t i o n  under  effects factor  of  given  by F  1  + P/P  1  - P/P  1  =  (10.4)  6  The  term Pe i s the E u l e r b u c k l i n g  load f o r the column.  T h i s t h e s i s has i d e n t i f i e d many areas i n efficient  design  of  because of lack of behaviour.  The  timber  information  results  amount of new information methods,  but  configurations.  has  material  not been p o s s i b l e properties  and  which can be used to improve  design  Areas r e l a t e d to t h i s  r e q u i r e f u r t h e r i n v e s t i g a t i o n include  concepts  laminated timber members.  and  a large  b i a x i a l behaviour and other  T h i s t h e s i s has concentrated  the  accurate  of the t h e s i s have provided  s i z e e f f e c t s , creep b u c k l i n g ,  but  on  much remains to be done.  t h e s i s t o p i c which s t i l l  timber,  members  which  could  also  be  solely  on  load sawn  a p p l i e d to glued  261  LITERATURE  ACI,  CITED  1970. U l t i m a t e S t r e n g t h D e s i g n Handbook, V o l 2. 17A. American Concrete I n s t i t u t e .  ACI SP-  Alcock,W.J. and N.D.Nathan, 1977. Moment M a g n i f i c a t i o n of P r e s t r e s s e d C o n c r e t e C o l u m n s . J o u r n a l o f PCI 22(4):50-61.  Tests  Allen,H.G. and P . S . B u l s o n , 1980. B a c k g r o u n d t o B u c k l i n g . M c G r a w - H i l l Book Company (UK) L t d . , L o n d o n . 582p. ASTM,  1981a. ASTM D143-52 S t a n d a r d Method o f T e s t i n g S m a l l C l e a r Spcimens of Timber. A n n u a l Book o f ASTM S t a n d a r d s 22:59-116.  ASTM,  1981b. ASTM D 1 9 8 - 7 6 S t a n d a r d Method o f S t a t i c T e s t s o f Timbers i n S t r u c t u r a l S i z e s . A n n u a l Book of ASTM S t a n d a r d s 22:121-146.  ASTM,  1981c. ASTM D245-74 S t a n d a r d Methods f o r E s t a b l i s h i n g S t r u c t u r a l G r a d e s and R e l a t e d A l l o w a b l e P r o p e r t i e s f o r V i s u a l l y G r a d e d Lumber. A n n u a l Book o f ASTM S t a n d a r d s 22:147-170.  B a r r e t t , J . D . , 1974. E f f e c t o f S i z e on T e n s i o n Perpendiculart o - G r a i n Strength of D o u g l a s - f i r . Wood a n d F i b e r 6(2):126-143. B a r r e t t , J . D . , R . O . F o s c h i , and S.P.Fox, t o - G r a i n S t r e n g t h of D o u g l a s - f i r . 2(0:50-57.  1975. PerpendicularCan.Jo.Civ.Eng.  Barrett,J.D. and R . O . F o s c h i , 1978. D u r a t i o n o f L o a d and P r o b a b i l i t y o f F a i l u r e i n Wood. P a r t s I and I I . Can.Jo.Civ.Eng.5:505-532. Bazan,I.M.M., 1980. U l t i m a t e B e n d i n g S t r e n g t h o f Timber Beams. Ph.D. D i s s e r t a t i o n . Nova S c o t i a T e c h . College. H a l i f a x , N.S.269 p. Bechtel,S.C. and C . B . N o r r i s , 1952. S t r e n g t h o f Wood Beams and R e c t a n g u l a r C r o s s S e c t i o n a s A f f e c t e d by Span-Depth Ratio. USDA F o r e s t S e r v i c e . For.Prod.Lab.Rep. M.R1910. 42 p . B l e a u , R., 1984. Comportment d e s P o u t r e - C o l o n n e s en B o i s . M.Sc.A T h e s i s . Department of C i v i l Engineering, U n i v e r s i t y o f S h e r b r o o k e , Quebec. B l e i c h , F., 1952. B u c k l i n g S t r e n g t h o f M e t a l S t r u c t u r e s . M c G r a w - H i l l Book Company. New Y o r k . 508 p .  262  B l o c k l e y , D . I . , 1980. The Nature of S t r u c t u r a l Design and Safety. Halstead P r e s s . U.K. Bohannan,B., 1966. E f f e c t of S i z e on Bending S t r e n g t h of Wood Members. USDA For.Serv.Res.Paper FPL 56. 30 p. Booth,L.G., 1964. The S t r e n g t h T e s t i n g of Timber During the 17th and 18th C e n t u r i e s . Jo.Inst.Wood S c i e n c e . No.13(530). Bryson,W., 1866. S t r e n g t h of Cast Iron and Timber P i l l a r s : A S e r i e s of Tables Showig the Breaking Weight of Cast Iron, D a n t z i c Oak and Red Deal P i l l a r s . J.Franklin Inst. 81:312~322. BSI, 1971. CP112:Part 2:1971. The S t r u c t u r a l Use of Timber, Metric Units. B r i t i s h Standards I n s t i t u t i o n , London. 124p. Buchanan,A.H., 1983. E f f e c t of Member Size on Bending and Tension S t r e n g t h of Wood. Proc. Wood E n g i n e e r i n g Meeting. IUFRO S5.02. Madison, Wisconsin, USA. Burgess,H.J., 1977. Column Design Theory. CIB-W18 T e c h n i c a l Paper. C o n s e i l I n t e r n a t i o n a l du Batiment. Burgess,H.J., 1976. Comparison of Larsen and Perry Formulas for S o l i d Timber Columns. CIB Paper W18 6-2-3. C o n s e i l I n t e r n a t i o n a l du Batiment. Burgess,H.J., 1980. T i e s with L a t e r a l Load. Proc. Wood E n g i n e e r i n g Group S5:02. Oxford, U.K. Bury,K.V., 1975. S t a t i s t i c a l Models i n A p p l i e d John Wiley and Sons. New York. 624 p.  IUFRO  S c i e n c e.  Chen,W.F. and T . A t s u t a , 1976a. Theory of Beam Columns. Vol.1. In-Plane Behaviour and Design. McGraw-Hill Book Company. New York. 513p. Chen,W.F. and T.Atsuta, 1976b. Theory of Beam Columns. Vol.11. Space Behaviour and Design. McGraw-Hill Book Company. New York. 732p. CIB, 1980. CIB S t r u c t u r a l Timber Design Code. Working Wl8~Timber S t r u c t u r e s . C o n s e i l I n t e r n a t i o n a l du Batiment. Cline,M. and A.L.Heim, 1912. T e s t s of S t r u c t u r a l USDA F o r e s t S e r v i c e B u l l e t i n 108. 123p.  Group  Timber.  Comben,A.J., 1957. The E f f e c t of Depth on the S t r e n g t h P r o p e r t i e s of Timber Beams. DSIR Forest Products Research S p e c i a l Report No.12. HMSO, London. 32p.  263  C o r n e l l , A . C . , 1969. A P r o b a b i l i t y Based S t r u c t u r a l Code. J o u r n a l 66:974-985.  ACI  CSA,  1977. Code f o r the Design of Concrete S t r u c t u r e s f o r B u i l d i n g s . N a t i o n a l Standard of Canada. CAN3-A23.3-M77. Canadian Standards Assoc. 131p.  CSA,  1978. Steel Structures for Buildings - Limit States Design. N a t i o n a l Standard of Canada. CAN3-S16.1-M78. Canadian Standards Assoc. I03p.  CSA,  1980. Code f o r E n g i n e e r i n g Design i n Wood. N a t i o n a l Standards of Canada. CAN3-O86-M80. Canadian Standards Assoc. I39p.  CSA,  1983. Code f o r E n g i n e e r i n g Design i n Wood - L i m i t States Design. D r a f t f o r Comment. Proposed N a t i o n a l Standard of Canada. CAN3-086.1-M. Candadian Standards Assoc.  Dawe,P.S., 1964. The E f f e c t of Knot S i z e on the T e n s i l e S t r e n g t h of European Redwood. Wood 29(11):49-51. Dietz,A.G.H., 1942. S t r e s s - S t r a i n R e l a t i o n s i n Timber Beams of D o u g l a s - f i r . ASTM B u l l , No.118. ppl9-27. Ellingwood,B., 1981. Reliability Proc. ASCE 107(ST1):73-87.  of Wood S t r u c t u r a l Elements.  Foschi,R.O., 1979. A D i s c u s s i o n on the A p p l i c a t i o n of the S a f e t y Index Concept to Wood S t r u c t u r e s . Can.Jo.Civ.Eng. 6(1):51-58. Foschi,R.O., 1982. S t r u c t u r a l A n a l y s i s of Wood F l o o r Proc. ASCE 108(ST7):1557-1574.  Systems.  Foschi,R.O. and J . D . B a r r e t t , 1975. L o n g i t u d i n a l Shear Strength of D o u g l a s - F i r . Can.Jo.Civ.Eng. 3(2):198-208. Foschi,R.O. and J . D . B a r r e t t , 1980. Glued-Laminated Beam S t r e n g t h : A Model. Proc. ASCE 106(ST8):1735-1754. FPRS, 1979. Metal P l a t e Wood T r u s s Conference Proceedings. Proc. P-79-28. F o r e s t Products Research S o c i e t y . Madison, Wisconsin, USA. 259p. Galambos,T.V., 1968. S t r u c t u r a l Members and Frames. P r e n t i c e H a l l Inc. Englewood C l i f f s , N.J. 373p. Galambos,T.V., B.Ellingwood, J.G.McGregor and C . A . C o r n e l l , 1982. P r o b a b i l i t y Based Load C r i t e r i a . Proc. ASCE 108(ST5):959-997. Galligan,W.L., C.C.Gerhards and R.L.Ethington, 1979. E v o l u t i o n of T e n s i l e Design S t r e s s e s f o r Lumber.  USDA  264  For.Serv.  FPL Gen.Tech.Rep.  No.28.  9p.  Gerhards,C.C., 1972. R e l a t i o n s h i p of T e n s i l e S t r e n g t h of Southern Pine Dimension Lumber t o Inherent Characteristics. USDA For.Serv.Res. Paper FPL 174. 3lp. Glos,P., 1978. B e r i c h t e zur Z u v e r l a s s i g k e i t s t h e o r i e der Bauwerke: Zur Bestimmung des F e s t i g k e i t s v e r h a l t e n s von B r e t t s c h i c h t h o l z b e i Druckbeanspruchung aus Werkstoff-und Einwirkungskenngrossen. ( R e l i a b i l i t y Theory f o r Timber S t r u c t u r e s : Determination of Compression S t r e n g t h Behaviour of Glulam Components from I n t e r a c t i o n of M a t e r i a l P r o p e r t i e s ) . Heft 34/1978. Laboratorium f u r den K o n s t r u c k t i v e n Ingenieurbau. Technische U n i v e r s i t a t Munchen. 335p. Goodman,J.R., M.D.Vanderbilt, and M . E . C r i s w e l l , 1983. R e l i a b i l i t y - B a s e d Design of Wood Transmission L i n e S t r u c t u r e s . Jo.Struc.Eng. 109(3):690-704. Goodman,J.R., Z.Kovacs and J.Bodig, 1981. Code Comparisons of F a c t o r Design f o r Wood. Proc. ASCE 107(ST8):1511 -1527. Goodman,J.R. and J.Bodig, 1970. O r t h o t r o p i c E l a s t i c P r o p e r t i e s of Wood. Proc. ASCE 96(ST11):2301-2319. Gromala,D.S. and R.C.Moody, 1983. Research Needs i n S t r u c t u r a l A n a l y s i s of Light-Frame Wood S t r u c t u r e s . Proc. Wood E n g i n e e r i n g Meeting. IUFRO S5.02. Madison, Wisconsin, USA. G u r f i n k e l , G . , 1973. Wood E n g i n e e r i n g . Southern F o r e s t Products A s s o c i a t i o n . New O r l e a n s . 540p. Hammond,W.C., J . O . C u r t i s , O.M.Sidebottom and B.A.Jones, 1970. C o l l a p s e Loads of Wooden Columns with V a r i o u s 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 . Trans.Am.Soc.Agr.Eng. 13(6):737-742. Handbook of S t e e l C o n s t r u c t i o n , 1980. Canadian S t e e l C o n s t r u c t i o n . Wilowdale, O n t a r i o .  I n s t i t u t e of  Heimeshoff,B. and P.Glos, 1980. Z u g f e s t i g k e i t und Beige-EModul von F i c h t e n - B r e t t l a m e l l e n . ( T e n s i l e Strength and MOE i n Bending of Spruce Laminates). Holz a l s Roh- und Werkstoff 38:51-59. Humphries,M. and A.P.Schniewind, 1982. Behaviour of Wood Columns under C y c l i c R e l a t i v e Humidity. Wood S c i . 15(1):44-48. Jayatilaka,A.S., Materials.  1979. F r a c t u r e of E n g i n e e r i n g B r i t t l e A p p l i e d Science P u b l i s h e r s , L t d . London.  265  Jessome,A.P., 1977. Strength and Related P r o p e r t i e s of Wood Grown i n Canada. F o r e s t r y T e c h n i c a l Report 21. M i n i s t r y of F i s h e r i e s and Environment. Ottawa. Johns,K.C. and A.H.Buchanan, 1982. Strength of Timber Members i n Combined Bending and A x i a l Loading. Proc. IUFRO Wood Engineering Group S5.02. Boras, Sweden. 343368. Johns,K.C. and B.Madsen, 1982. Duration of Load E f f e c t s i n Lumber, P a r t s I, II and I I I . Can.Jo.Civ.Eng. 9(3):502536. Johnson,A.I., 1953. Strength, Safety and Economical Dimensions of S t r u c t u r e s . Swedish State Committee f o r B u i l d i n g Research. B u l l e t i n No.22. 159p. Johnson,J.W. and R.H.Kunesh, 1975. T e n s i l e Strength of S p e c i a l D o u g l a s - F i r and Hem-Fir 2-Inch Dimension Lumber. Wood and F i b e r 6(4):305-318. Johnson,B.G., 1976. Guide to S t a b i l i t y Design C r i t e r i a f o r Metal S t r u c t u r e s . S t r u c t u r a l S t a b i l i t y Research C o u n c i l . . John Wiley and Sons. New York. 616p. Kallsner,B. and B.Noren, 1978. Creep Buckling of Wood Columns. Proc. IUFRO Wood E n g i n e e r i n g Group S5.02. Vancouver. 381-406. Kersken-Bradley,M., 1981. B e r i c h t e zur Z u v e r l a s s i g k e i t s t h e o r i e der Bauwerke: Beanspruchbarkeit von B a u t e i l q u e r s c h n i t t e n b e i streuenden Kenngrossen des K r a f t v e r f o r m u n g s v e r h a l t e n s i n n e r h a l b des Q u e r s c h n i t t s . ( R e l i a b i l i t y Theory f o r Timber S t r u c t u r e s : Load Capacity of Member Cross S e c t i o n s with Load Displacement Constants V a r y i n g Within the Cross S e c t i o n ) . Heft 56/1981. Laboratotium f u r den K o n s t r u k t i v e n Ingenieurbau. Technische U n i v e r s i t a t Munchen. 246p. Kunesh,R.H. and J.W.Johnson, 1972. E f f e c t of S i n g l e Knots on T e n s i l e Strength of 2x8-Inch D o u g l a s - f i r Dimension Lumber. For.Prod.Jo. 22(1):32-36. Kunesh,R.H. and J.W.Johnson, 1974. E f f e c t of S i z e on T e n s i l e S t r e n g t h of C l e a r D o u g l a s - f i r and Hem-Fir Dimension Lumber. For.Prod.Jo. 24(8):32-36. Larsen,H.J., 1973. The Design of S o l i d Timber Columns. CIB Paper W18 2-2-1. CIB Working Commission Wl8-Timber Structures. Larsen,H.J. and E . T h i e l g a a r d , 1979. L a t e r a l l y Loaded Columns. Proc. ASCE 105(ST7):1347-1363.  Timber  266  Larsen,H.J. and H . R i b e r h o l t , 1981. Beregning af Traekonstruktioner. 2. Udgave. S e r i e F. No.76. Department of S t r u c t u r a l E n g i n e e r i n g . Technical U n i v e r s i t y of Denmark. L e i c e s t e r , R . H . , 1973. E f f e c t of S i z e on the Strength of Structures. CSIRO For.Prod.Lab. D i y . B u i l d . R e s . Tech.Paper No.71. M e l b o u r n e , A u s t r a l i a . 13p. Littleford,T.W. and R.A.Abbott, 1978. P a r a l l e l - t o - G r a i n Compressive P r o p e r t i e s of Dimension Lumber from Western Canada. Information Report VP-X-180. Western F o r e s t Products Lab. Vancouver. 25p. L i t t l e f o r d , T . W . , 1967. T e n s i l e Strength and Modulus of E l a s t i c i t y of Machine Graded 2x6 D o u g l a s - f i r . Information Report VP-X-12. Western F o r e s t Products Lab. Vancouver. l 3 p . Madsen,B., 1982. Recommended M o i s t u r e Adjustment F a c t o r s f o r Lumber S t r e s s e s . Can.Jo.Civ.Eng. 9(4):602-610. Madsen,B., 1983. E f f e c t of Load C o n f i g u r a t i o n on Strength of 2x6 Inch Lumber. In P r e p a r a t i o n . Madsen,B. and P.C.Nielsen, 1976. In-Grade T e s t i n g : S i z e I n v e s t i g a t i o n s on Lumber Subjected to Bending. S t r u c t u r a l Research S e r i e s Rep. No.15. Department of C i v i l E n g i n e e r i n g . U n i v e r s i t y of B r i t i s h Columbia. Vancouver. Madsen,B. and P.C.Nielsen, 1978a. In-Grade T e s t i n g : Bending T e s t s i n Canada. June 1977-May 1978. S t r u c t u r a l Research S e r i e s Rep.No.25. Department of C i v i l E n g i n e e r i n g . U n i v e r s i t y of B r i t i s h Columbia. Vancouver. Madsen,B. and P.C.Nielsen, 1978b. In-Grade T e s t i n g : Tension T e s t s i n Canada. Prepared f o r N a t i o n a l Lumber Grades A u t h o r i t y . Department of C i v i l E n g i n e e r i n g . University of B r i t i s h Columbia. Vancouver. Madsen,B. and P.C.Nielsen., 1978c. In-Grade T e s t i n g : I n v e s t i g a t i o n of Test Parameters i n P a r a l l e l - t o - G r a i n Compression. S t r u c t u r a l Research S e r i e s Rep. No.26. Department of C i v i l E n g i n e e r i n g . U n i v e r s i t y of B r i t i s h Columbia. Vancouver. Madsen,B. and P.C.Nielsen, I978d. In-Grade T e s t i n g : I n v e s t i g a t i o n of Test Parameters i n P a r a l l e l - t o - G r a i n T e n s i o n . S t r u c t u r a l Research S e r i e s Rep.No.24. Department of C i v i l E n g i n e e r i n g . U n i v e r i s t y of B r i t i s h Columbia. Vancouver. 32p. Madsen,B.  and T . S t i n s o n ,  1982.  In-Grade T e s t i n g of Timber  267  Four Inch or More i n T h i c k n e s s . Unpublished Report. Department of C i v i l E n g i n e e r i n g . U n i v e r s i t y of B r i t i s h Columbia. Vancouver. Malhotra,S.K., 1982. A n a l y s i s and Design of Timber Columns Subjected t o E c c e n t r i c Loads. Proc. Canadian Soc. C i v i l Eng. 1982 Annual Conference. 117-132. Malhotra,S.K. and S.J.Mazur, 1970. B u c k l i n g S t r e n g t h of S o l i d Timber Columns. Trans.Eng.Inst. of Canada. P u b l i s h e d i n the Eng.Jo. 13(A-4):I-VII. Malhotra,S.K. and Bazan,I.M.M., 1980. U l t i m a t e Bending S t r e n g t h Theory f o r Timber Beams. Wood S c i . 13(1):5063. Malhotra,S.K. and R.E.MacDonnell, 1983. L i m i t States Design of Timber Compression Members - Comparative Study of V a r i o u s P r e d i c t i o n Models. Proc. Canadian Soc. Civil Eng. 1983 Annual Conference. Markwardt,L.J. and W.G.Youngquist, 1956. Tension Test Methods f o r Wood, Wood-Base M a t e r i a l s , and Sandwich C o n s t r u c t i o n s . USDA F o r . S e r v . FPL Rep.2055. 16p. McGowan,W.M., 1968. P a r a l l e l t o Grain T e n s i l e P r o p e r t i e s of V i s u a l l y Graded 2x6 Inch D o u g l a s - f i r . Information Report VP-X-46. Western Forest Products Lab. Vancouver. 48p. McGowan,W.M., 1971. P a r a l l e l - t o - G r a i n T e n s i l e P r o p e r t i e s of Coast and Interior-Grown 2x6 Inch D o u g l a s - f i r . Information Report VP-X-87. Western Forest Products Lab. Vancouver. '46p. McGregor,J.G., J.E.Breen and E.O.Pfrang, 1970. Design of Slender Concrete Columns. J o u r n a l ACI 67(1):6~28. Mindess,S., 1977. The F r a c t u r e of Wood i n Tension P a r a l l e l t o G r a i n . Can.Jo.Civ.Eng. 4(4):412-416. Moe,J., 1961. The Mechanism of F a i l u r e of Wood i n Bending. Publication Int'l. Assoc. f o r Bridge and S t r u c t u r a l Engineering. 21:163-178. Nathan,N.D., 1983a. Slenderness of P r e s t r e s s e d Concrete Columns. J o u r n a l of PCI 28(2):50-77. Nathan,N.D., 1983b. R a t i o n a l A n a l y s i s of Slender P r e s t r e s s e d Beam Columns and W a l l s . Submitted t o J o u r n a l of PCI. Neely,S.T., 1898. R e l a t i o n of Compression-Endwise to Breaking Load of Beam. Progress i n Timber P h y s i c s . USDA Div. of For. C i r c u l a r No. 1 8:1 3-1.7.  268  Nemeth,L.J., 1965. C o r r e l a t i o n between Tens,ile Strength and Modulus of E l a s t i c i t y f o r Dimension Lumber. Proc. 2nd Symp. on Non-Destructive T e s t i n g of Wood. Spokane, Washington, USA. 391-416. Neubauer,L,W., 1973. A R e a l i s t i c and Continuous Formula. For.Prod.Jo. 23(3):38~44.  Wood Column  Newlin,J.A., 1940. Formulas f o r Columns with Side Loads and Eccentricity. B u i l d i n g Standards Monthly. December 1 940. Newlin,J.A. and J.M.Gahagan, 1930. T e s t s of Large Timber Columns and P r e s e n t a t i o n of the FPL Column Formula. U.S. Dept.Agr. Tech B u l l . No.167. 43p. Newlin,J.A. and G.W.Trayer, 1924. Form F a c t o r s of Beams Subjected to Transverse Loading Only. Nat.Adv.Comm.Aero. Rep. No.181. Reprinted as USDA For.Serv. FPL Rep. No.1310. 1941. I9p. Newlin,J.A. and G.W. T r a y e r , 1925. S t r e s s i n Wood Members Subjected to Combined Column and Beam A c t i o n . Nat.Adv.Comm.Aero. Rep. No.188. Reprinted as USDA For.Serv. FPL Rep. No.1311. 1941. 13p. Newton,D.A. and R.. Ayaru, 1972. S t a t i s t i c a l Methods i n the Design of S t r u c t u r a l Timber Columns. S t r u c t u r a l Engineer 50(5):191-195. NFPA, 1982. N a t i o n a l Design S p e c i f i c a t i o n f o r Wood C o n s t r u c t i o n . N a t i o n a l F o r e s t Products Assoc. Washington D.C. 8 l p . N o r r i s , C . B . , 1955. Strength of O r t h o t r o p i c M a t e r i a l s Subjected to Combined S t r e s s e s . USDA For.Serv. FPL Rep. No.1816. 34p. Nwokoye,D.N., 1974. I n e l a s t i c Bending of Wood Beams. D i s c u s s i o n on a r t i c l e by B.D.Zakic. Proc. ASCE 100(ST9):1963-1966. Nwokoye,D.N., 1975. Strength V a r i a b i l i t y of S t r u c t u r a l Timber. The Struc.Eng.; The J o . of the I n s t . S t r . Engrs. 53(3):139-145. London. 0'Halloran,M.R., 1973. C u r v 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 for Wood i n Compression. Ph.D. T h e s i s . Colorado State Univ. 129p. OHBDC, 1982. O n t a r i o Highway Bridge Design Code. Wood S t r u c t u r e s , i n c l u d i n g Commentary. Min.Transp.Commun. Downsview, O n t a r i o .  S e c t i o n 13.  269  Orosz,I., 1975. D e r i v a t i o n of T e n s i l e Strength R a t i o by Second Order Theory. Wood S c i . and Tech. 9:25-44. P i e r c e , F . T . , 1926. T e n s i l e T e s t s f o r Cotton Jo.Text.Inst. 17:T355-T368.  Yarns.  Ramos,A.N., 1961. S t r e s s - S t r a i n D i s t r i b u t i o n i n D o u g l a s - F i r Beams w i t h i n the P l a s t i c Range. USDA For.Serv. FPL Research Rep.2231. 30p. R i b e r h o l t , H . and P.C.Nielsen, 1976. Timber under Combined Compression and Bending S t r e s s . Rapport Nr.R.70. S t r u c t u r a l Research L a b o r a t o r y . T e c h n i c a l U n i v e r s i t y of Denmark. 24p. Riberholt,H. and P.H.Madsen, 1979. Strength D i s t r i b u t i o n of Timber S t r u c t u r e s . Measured V a r i a t i o n of the Cross S e c t i o n a l Strength of S t r u c t u r a l Lumber. Report R.114. S t r u c t u r a l Research L a b o r a t o r y . T e c h n i c a l U n i v e r s i t y of Denmark. 70p. Robertson,A., 1925. The Strength of S t r u t s . I n s t n . C i v . E n g r s . No.28.  Sel.Eng.Pap.  Schiewind,A.P. and D.E.Lyon, 1971. T e n s i l e Strength of Redwood Dimension Lumber I . R e l a t i o n to Grade and Working S t r e s s . For.Prod.Jo. 21(7):18-27. S e n f t , J . F . , 1973. F u r t h e r S t u d i e s i n Combined Bending and Tension Strength of S t r u c t u r a l 2 by 4 Lumber. For.Prod.Jo. 23(10):36-41. Senft,J.F. and S.K.Suddarth, 1970. Strength of S t r u c t u r a l Lumber under Combined Bending and Tension Loading. For.Prod.Jo. 20(7):17-21 . Sexsmith,R.G., 1979. Proposed L i m i t States Design Format f o r Wood S t r u c t u r e s . F o r i n t e k Corp. Western Forest Products Lab. T e c h n i c a l Rep. No.7. 21p. Sexsmith,R.G. and S.P.Fox, 1978. L i m i t S t a t e s Design Concepts f o r Timber E n g i n e e r i n g . For.Prod.Jo. 28(5):4954. Smith,F.W. and D.T.Penney, 1980. F r a c t u r e Mechanics A n a l y s i s of Butt J o i n t s i n Laminaed Wood Beams. Wood S c i . 12(4):227-235. Suddarth,S.K. and F.E.Woeste, 1977. I n f l u e n c e s of V a r i a b i l i t y i n Loads and Modulus of E l a s t i c i t y on Wood Column S t r e n g t h . Wood S c i . l0(2):62-67. Suddarth,S.K., F.E.Woeste and W.L.Galligan, 1978. D i f f e r e n t i a l R e l i a b i l i t y : P r o b a b i l i s t i c Engineering  270  A p p l i e d to Wood Members i n Bending/Tension. For.Serv. Res.Paper. FPL 302. 16p.  USDA  S u n l e y , J . F . 1955. The Strength of Timber S t r u t s . DSIR For.Prod. Research S p e c i a l Report. No.9. HMSO London. 28p. f  Timber Design Mannual, 1980. Metric E d i t i o n . Timber I n s t i t u t e of Canada. 512p.  Laminated  Timoshenkb,S.P., 1953. H i s t o r y of Strength of M a t e r i a l s . McGraw-Hill Book Co. New York. 452p. Timoshenko,S.P. Stability.  and J.M.Gere, 1961. Theory of E l a s t i c McGraw-Hill BOok Company. New York.  541p.  Todhunter,I. and K.Pearson, 1886 and 1893. A H i s t o r y of the Theory of E l a s t i c i t y and the Strength of M a t e r i a l s , Vols. 1 and 2. Cambridge, England. Tredgold,T., 1853. Elementary P r i n c i p l e s of C a r p e n t r y . London. 334p. Tucker,J., 1927. A Study of the Compression Strength D i s p e r s i o n of M a t e r i a l s with A p p l i c a t i o n s . J.Franklin Inst. 204:751-781. Weibull,W., 1939a. A S t a t i s t i c a l Theory of the Strength of Materials. Proc.Royal Swed.Inst.Eng.Res. No.151. Stockholm. 45p. Weibull,W., 1939b. The Phenomenon of Rupture i n S o l i d s . Proc.Royal Swed.Inst.Eng.Res. No.153. Stockholm. 55p. Wilson,C.R., 1978. Commentary on S e c t i o n 4 "Sawn Lumber" of CSA 086-1976 Code f o r the E n g i n e e r i n g Design of Wood." Cna.Jo.Civ.Eng.5(1):11-17. Wood Handbook, 1974. Washington D.C.  USDA A g r i c u l t u r a l Handbook 72.  Wood, L.W., 1950. Formulas f o r Columns with Side Loads and Eccentricity. USDA F o r . S e r v . FPL Rep.No.R1782. 23p. Y l i n e n , A., 1956. A MEthod of Determining the B u c k l i n g S t r e s s and the Required C r o s s - S e c t i o n a l Area f o r C e n t r a l l y Loaded S t r a i g h t Columns i n E l a s t i c and I n e l a s t i c Range. IABSA P u b l i c a t i o n s . 16:529-549. Zahn, J . J . , 1982. Strength of Lumber under Combined Bending and Compression. USDA For.Serv.For.Prod.Lab.Res.Paper. FPL 391. 13p. Zakic, B.D.,  1973.  I n e l a s t i c Bending of Wood Beams.  Proc.  271  ASCE 99(ST10):2079-2095. Zehrt, W. H., 1962. P r e l i m i n a r y Study of the F a c t o r s A f f e c t i n g T e n s i l e Strength of S t r u c t u r a l Timber. USDA For.Serv.FPL Research Rep.2251. 49p.  272  APPENDIX A - CALCULATIONS FOR SPECIAL CASES  Chapter 6 d e s c r i b e d a g e n e r a l computer model which can p r e d i c t the load capacity of members of any l e n g t h under many combinations of a x i a l and f l e x u r a l l o a d s . In t h i s appendix the same assumptions ( l i s t e d i n s e c t i o n 6.2) a r e used to o b t a i n a closed form solution, both f o r strength i n pure bending and f o r the u l t i m a t e i n t e r a c t i o n diagram of short columns. A.1  ULTIMATE BENDING STRENGTH  A.1.1  Background  The bending s t r e n g t h of timber beams has been a subject of interest s i n c e the time of G a l i l e o . Bending i s a l o a d i n g condition encountered f a r more f r e q u e n t l y than combined bending and a x i a l loading. The l i t e r a t u r e survey d e s c r i b e s many u n s u c c e s s f u l attempts to p r e d i c t the bending s t r e n g t h of both c l e a r wood beams and timber beams from the r e s u l t s of a x i a l t e n s i o n and compression t e s t s . A major new c o n t r i b u t i o n of the present study is a relatively simple model to p r e d i c t the bending s t r e n g t h of timber members from the a x i a l t e n s i o n and a x i a l compression strengths. Recall that throughout t h i s study timber member s t r e n g t h g e n e r a l l y r e f e r s to s t r e n g t h s of d e f i n e d p o p u l a t i o n s of timber, at c e r t a i n l e v e l s i n the d i s t r i b u t i o n . F i g u r e 27 compares d i s t r i b u t i o n s of bending, t e n s i o n , and compression s t r e n g t h s obtained from in-grade t e s t i n g of the timber s t u d i e d i n t h i s t h e s i s . The s t r e n g t h model can be used to c a l c u l a t e the bending s t r e n g t h from the a x i a l t e n s i o n and compression s t r e n g t h s , at any l e v e l i n the d i s t r i b u t i o n , or for any member whose t e n s i o n and compression s t r e n g t h s are known. To c a l c u l a t e bending strength from the r e s u l t s of a x i a l t e n s i o n and compression t e s t s , a s t r e n g t h model has to i n c o r p o r a t e s i z e e f f e c t s with length and depth, and n o n - l i n e a r stress-strain behaviour i n compression. These f a c t o r s have been d e s c r i b e d f o r the general case i n Chapter 6 and w i l l be described here f o r the s p e c i a l case of bending with no a x i a l load. For m a t e r i a l that i s weaker in tension than i n compression, f a i l u r e i s governed by t e n s i o n s t r e n g t h alone, so n o n - l i n e a r compression behaviour need not be c o n s i d e r e d . Calculations f o r bending s t r e n g t h are developed f o r two separate c a s e s . The f i r s t assumes a b i l i n e a r stress-strain relationship with a f a l l i n g branch. The second assumes 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 an upper limit  273  on compression A.1.2  strain.  Bilinear Stress-Strain  R e l a t i o n s h i p with F a l l i n g Branch  a. Assumptions The theory i n t h i s s e c t i o n i s an e x t e n s i o n of theory d e s c r i b e d by Bazan(l980), which attempts to p r e d i c t the bending strength of large clear wood beams from the a x i a l tension and compression s t r e n g t h s of small clear specimens. The major differences i n the present study are that in-grade a x i a l t e n s i o n and compression t e s t r e s u l t s are used as input, and s i z e e f f e c t s are i n c l u d e d more e x p l i c i t l y . This study uses Bazan's assumptions of l i n e a r e l a s t i c behaviour to f a i l u r e i n t e n s i o n and a b i l i n e a r stress-strain relationship i n compression with a f a l l i n g branch after maximum s t r e s s , as shown i n f i g u r e 85.  stress  brittle . fracture  F i g u r e 85 - B i l i n e a r s t r e s s - s t r a i n f a l l i n g branch  relationship  with  The slope of the falling branch i s assumed to be constant r a t i o , m, of the modulus of e l a s t i c i t y as shown.  a  The g e n e r a l computer program f o r 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 i n compression. However, comparative computer runs have shown that for calculating the s t r e n g t h at a c r o s s s e c t i o n , the b i l i n e a r  274  r e l a t i o n s h i p w i t h a f a l l i n g b r a n c h c a n be used t o g i v e e x a c t l y t h e same r e s u l t s as more a c c u r a t e c u r v e s .  almost  Calculations b. Figure 86 shows t h e d i s t r i b u t i o n o f 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 t h e i n e l a s t i c r a n g e . The depth of the section i s d, and a,b and c r e p r e s e n t r a t i o s o f d a s shown i n the f i g u r e .  v x X  v  \  \ x  . w .  section (a) Figure rectangular  strain (b)  86 - D i s t r i b u t i o n o f s t r e s s and s t r a i n i n a beam a s s u m i n g 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 f x , i s t a k e n t o 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 fc. The falling 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 t h e extreme c o m p r e s s i o n f i b r e h a v i n g a s t r e s s w h i c h 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 c o m p r e s s i o n . Using the n o t a t i o n t e n s i o n and c o m p r e s s i o n  T  C  Internal  axial  force  =  shown on the f i g u r e , f o r c e s , T and C, a r e  =  ^ nf wed 2 c  equilibrium,  b + a(l+r)  =  internal  (A.1)  i f wd[b + a(l+r)] 2 c ^ a  the  (A.2)  1 T i ; j  T=C,  nc  yields  (A.3)  275  From t h e g e o m e t r y o f t h e s t r e s s a  =  diagram  1 - b - c  ( .4) A  and  b  From e q u a t i o n s  c  /  (A.5)  n  A.3 t o A.5  c  from  =  n(l+r) (n+r)(n+l)  _  (  A  «  r\ 6  )  which  a  = C"- )  (A.7)  1  and ,  _ "  b  1+r (n+r)(n+l) Cn+rUn+l-*  (A.8)  The b e n d i n g moment c a n be c a l c u l a t e d by t a k i n g i n t e r n a l f o r c e s about t h e n e u t r a l a x i s t o g i v e  moments o f t h e  2  M  -  f ^ g - [2nc  Substituting  2  + .2b  2  + (l+2r)a  f o r a, b a n d c M  The d e v e l o p m e n t by B a z a n .  =  £  e  ^ i  to t h i s  [  + 3(l+r)ab]  2  (A. 9)  gives  n  (2n-l)r  +  point  }  i s identical  (  to  that  ^  ]  Q  )  described  If t h e s l o p e of t h e f a l l i n g b r a n c h o f t h e s t r e s s - s t r a i n relationship is a given ratio, m, of the modulus of elasticity, a s shown i n f i g u r e 85, t h e n f r o m t h e s t r e s s and s t r a i n d i a g r a m s o f f i g u r e 86, t h e r e d u c t i o n i n s t r e s s a t the e x t r e m e c o m p r e s s i o n f i b r e c a n be w r i t t e n a s  (l-r) f c v  1  =  mE(e c v  - e ) y  (A.11)  276  From the s t r a i n diagram a+b „ T~ y  ec  and  (A.12)  e  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 gives e  -  y  f /E  (A. 13)  c  Substituting and r e a r r a n g i n g equations e x p r e s s i o n f o r r i n terms of m, a and b =  A.11 to A.13 gives an  1 - ma/b  (A. 14)  S u b s t i t u t i n g the values of a and b from equations give an expression f o r r i n terms of m and n = Equations expression  /l-m(n -l)  M =  A.8  (A.15)  2  A.10 and A.15 can be combined f o r the bending moment, M. wd<  A.7 and  n + (2n-l) A  - m(n -l)  to  give  a new  2  (A.16)  n + / l - m(n -l) 2  The only unknown i n t h i s equation i s n, the r a t i o of the extreme fibre tension s t r e s s at u l t i m a t e moment, to the maximum compression s t r e s s . The extreme f i b r e t e n s i o n s t r e s s fx cannot exceed the failure s t r e s s i n t e n s i o n fm, which i s r e l a t e d to the a x i a l t e n s i o n s t r e n g t h f t by m  -k +l ] C  3  *,  3  (A.17)  where k i s the s t r e s s - d i s t r i b u t i o n parameter described i n Chapter 3, and c i s the n e u t r a l a x i s depth r a t i o as shown i n f i g u r e 86. 3  To c a l c u l a t e the value of n, capacity, four p o s s i b l e r a t i o s of s t r e n g t h must be c o n s i d e r e d . Each of different internal stress distribution  and hence the moment tension to compression the four cases produce a at failure.-  277  C a s e 1 i s f o r t i m b e r weak i n t e n s i o n , where t h e f a i l u r e s t r e s s i n b e n d i n g fm i s l e s s than the compression strength f c . Failure 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 c o m p r e s s i o n y i e l d i n g . S i m p l e e l a s t i c t h e o r y c a n be u s e d t o c a l c u l a t e t h e moment c a p a c i t y , t h e modulus o f r u p t u r e b e i n g fm from e q u a t i o n A.17 u s i n g c=0.5 f o r n e u t r a l a x i s a t m i d - d e p t h . C a s e 2. i s f o r t i m b e r w i t h an i n t e r m e d i a t e r a t i o of t e n s i o n to compression strength such that maximum moment i s s t i l l a s s o c i a t e d w i t h a t e n s i o n f a i l u r e , b u t a f t e r some compression y i e l d i n g has o c c u r r e d . In t h i s case t h e extreme f i b r e t e n s i o n stress fx, at failure, will a g a i n be e q u a l t o fm g i v e n by e q u a t i o n A.17.  fc,  The e x t r e m e f i b r e s t r e s s s o e q u a t i o n A.17 becomes  f x h a s been d e f i n e d a s  n  times  1/k, (A.18) In this case t h e n e u t r a l a x i s i s no l o n g e r a t m i d - d e p t h . E q u a t i o n s A.6 and A.15 c a n be c o m b i n e d t o g i v e an expression for c, which substituted into equation A.18 g i v e s an e x p r e s s i o n r e l a t i n g n t o t h e known a x i a l s t r e n g t h s f c and f t .  1/k,  n(l + A - m(n -l) ) 2  (A.19)  - (n + A - m(n -l) ) (n+l)(k+l) 2  This used  e q u a t i o n c a n be s o l v e d i t e r a t i v e l y f o r n, which can i n e q u a t i o n A.16 t o c a l c u l a t e t h e maximum moment.  be  Case 3 i s f o r timber with a higher ratio of t e n s i o n t o compression s t r e n g t h , such t h a t a t e n s i o n f a i l u r e o c c u r s a f t e r the moment has passed a maximum value, accompanied by considerable compression yielding. In t h i s c a s e t h e maximum moment i s i n d e p e n d e n t o f t e n s i o n s t r e n g t h , and i t i s n e c e s s a r y t o f i n d t h e v a l u e o f n t h a t m a x i m i z e s t h e moment i n equation A.16. D i f f e r e n t i a t i n g w i t h r e s p e c t t o n and e q u a t i n g t o z e r o produces  1 + m - mn + ( l - m ( n - l ) ) 3  2  3/2  = 0  (A.20)  w h i c h c a n be s o l v e d i t e r a t i v e l y f o r n, g i v e n a v a l u e of the material p r o p e r t y m. T h i s v a l u e o f n c a n be u s e d i n e q u a t i o n A.16 t o c a l c u l a t e maximum moment.  both  In p r a c t i c e i t i s n e c e s s a r y t o c a r r y o u t c a l c u l a t i o n s f o r C a s e s 2 and 3 a n d u s e t h e lower o f t h e two v a l u e s o f n i n  278  equation A.16. Case 4 i s the extreme case f o r timber which i s much stronger in tension than i n compression, where maximum moment i s a s s o c i a t e d with d u c t i l e compression y i e l d i n g and no t e n s i o n failure occurs. T h i s type of f a i l u r e w i l l be f a m i l i a r to the reader who has t r i e d u n s u c c e s s f u l l y to snap a green branch on a living 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 i n t h i s case t o f i n d n, and the moment from equation A.16.  c. Length E f f e c t s This d e r i v a t i o n has not i n c l u d e d length e f f e c t s . They can be i n c l u d e d using equation 5.3 which g i v e s an e q u i v a l e n t s t r e s s e d l e n g t h L f o r a beam of span L, loaded with two symmetrical loads a d i s t a n c e aL, a p a r t . 2  1 + ak /L 1  k +l  1  (A.21)  x  and equation 3.11 which r e l a t e s the s t r e n g t h of two s i m i l a r members of d i f f e r e n t l e n g t h s . X  *2  (A.22)  The procedure f o r modifying in-grade tension and compression test results f o r length, as input f o r c a l c u l a t i n g bending s t r e n g t h , i s as f o l l o w s : 1. C a l c u l a t e the e q u i v a l e n t s t r e s s e d l e n g t h of the beam, L , from equation A.21 using the t e n s i o n length effect parameter a s k , . 2  2. Adjust the in-grade t e n s i o n s t r e n g t h r e s u l t obtained from t e s t i n g of a length L , , t o a value compatible with the equivalent stressed l e n g t h of the beam, L , using equation A.22 and the t e n s i o n length e f f e c t parameter as ki. The r e s u l t i s f t . 2  3. Repeat steps 1 and 2 using in-grade compression t e s t r e s u l t s and the compression l e n g t h e f f e c t parameter as k to c a l c u l a t e the compression s t r e n g t h f c . 1  279  Modify a x i a l 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?^  calculate max moment assuming no tension failure  YES  c a l c u l a t e moment at t e n s i o n f a i l u r e  ^_  take lowest one  V ULTIMATE MOMENT  calculate failure moment from e l a s t i c theory  V ULTIMATE MOMENT  F i g u r e 87 - Flow c h a r t f o r c a l c u l a t i n g u l t i m a t e bending moment f o r 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 d. Summary A summary of the procedure f o r c a l c u l a t i n g ultimate bending moment i s shown i n the flow chart of f i g u r e 87, and d e s c r i b e d b r i e f l y as f o l l o w s . The l e n g t h e f f e c t c a l c u l a t i o n s have j u s t been d e s c r i b e d . The depth e f f e c t calculation can i n i t i a l l y be made using equation A.18 with c=0.5. The moment 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 i s found by c a l c u l a t i n g a value of n to satisfy equation A . 1 9 and by using that value i n equation A.16. Maximum moment assuming no t e n s i o n f a i l u r e i s found by substituting a value of n from equation A.20 i n t o equation A.16. I t i s not d i f f i c u l t t o w r i t e a computer program t o c a r r y 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 t o t h i s p o i n t h a s been b a s e d on t h e a s s u m p t i o n that the compression s t r e n g t h a t a c e r t a i n c r o s s s e c t i o n i s a material constant independent of the s t r e s s e d depth. As i n t h e more g e n e r a l c a s e a compression depth effect c a n be introduced- using equation 3.18 w h i c h relates t h e maximum compression s t r e s s t o the s i z e of the compression stress b l o c k , g i v i n g a m o d i f i e d c o m p r e s s i o n s t r e n g t h fern  f  .  =  cm  L  [ + j-lU k +l a  -l/k_ 3  J  3  f c  (A.23)  This value o f fem c a n be u s e d i n p l a c e o f t h e t e r m f c i n e q u a t i o n A.18 t o g i v e a more g e n e r a l form o f e q u a t i o n 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  + k  r  3c  ^ ] 1  3  C  3 t  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 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 c o m p r e s s i o n r e s p e c t i v e l y . This equation c a n be s o l v e d by i t e r a t i o n f o r n, a s b e f o r e , u s i n g equations A.6 t o A.8 and A . 1 5 - f o r a, b, a n d c . An a l t e r n a t i v e v a l u e o f n c a n be f o u n d from equation A.20, a s b e f o r e . Once t h e c o r r e c t v a l u e o f n i s f o u n d i t c a n be u s e d i n e q u a t i o n A.16 t o give t h e u l t i m a t e b e n d i n g moment, a g a i n u s i n g fem i n p l a c e o f fc. 3  A.1.3  Elasto-Plastic a.  Stress-Strain  Relationship  Assumpt i o n s  I t was shown i n C h a p t e r 6 t h a t f o r c a l c u l a t i n g t h e s t r e n g t h o f a c r o s s s e c t i o n , a model b a s e d on an e l a s t o - p l a s t i c stressstrain relationship c a n p r o d u c e v e r y s i m i l a r r e s u l t s t o one b a s e d on a more a c c u r a t e c u r v e , p r o v i d e d t h a t an upper limit on compression strain i s specified. 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 b e h a v i o u r , 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 within the cross s e c t i o n . This section uses an elasto-plastic stress-strain relationship with limiting strain t o produce e q u a t i o n s f o r u l t i m a t e b e n d i n g s t r e n g t h w h i c h a r e s l i g h t l y more s i m p l e than those derived i n the previous s e c t i o n . F i g u r e 88 shows t h e stress-strain relationship. Once a g a i n t h e s t r e n g t h i s c a l c u l a t e d from t h e r e s u l t s o f in-grade t e s t s i n a x i a l tension and c o m p r e s s i o n . A l l the o t h e r a s s u m p t i o n s made p r e v i o u s l y a r e u s e d a g a i n h e r e .  281  F i g u r e 88 - E l a s t o - p l a s t i c  stresss-strain  relationship  b. C a l c u l a t ions Figure 89 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.  section (a)  strain (b)  F i g u r e 89 - 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 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 t h i s f i g u r e as equation A.1.  the i n t e r n a l t e n s i o n force  T  = i nf wed 2 c  is  the  same  (A.25)  and the i n t e r n a l compression f o r c e becomes C  = J f wd(2a + b) / c  (A.26)  282  The same procedure as before produces the l o c a t i o n of the n e u t r a l a x i s and the s i z e of the p l a s t i c zone a  b  = Z£ n+1  (A.27)  =  (A.28) (n+1)  2  2n  c  and  an e x p r e s s i o n  =  (n+1)  < '  2  A  2 9 )  f o r the bending moment  Assume now that the l i m i t i n g s t r a i n has j u s t been reached and that no t e n s i o n f a i l u r e has o c c u r r e d . From the geometry of the s t r a i n diagram e  —  u y  a + b = — b —  S u b s t i t u t i n g equations A.26 t o value f o r n can be obtained  A.29  „ , (A.31)  into  equation  A.31, a  e  n  =  2— e  - 1  (A.32)  y  This value of n s u b s t i t u t e d i n t o equation A.30 g i v e s the bending moment when the l i m i t i n g compression strain is reached, and the extreme fibre t e n s i o n s t r e s s i s obtained d i r e c t l y from (A.33) T h i s extreme f i b r e t e n s i o n s t r e s s must be compared with the tension strength for this n e u t r a l a x i s depth, which i s c a l c u l a t e d from equations A.17 and A.29. I f a t e n s i o n f a i l u r e has not o c c u r r e d , the p r e v i o u s l y c a l c u l a t e d moment i s the desired value. I f a t e n s i o n f a i l u r e has o c c u r r e d i t becomes  283  necessary to c a l c u l a t e a lower moment a s s o c i a t e d with that event, using equation A.19 with the slope m of the f a l l i n g branch being zero, g i v i n g  n  [  2  j"  n  1 7  "  (n+l) (k +l) 2  3  3  = f t f  ( .34) A  c  T h i s equation must be s o l v e d i t e r a t i v e l y f o r n as before, and the subsequent value used i n equation A.30 t o c a l c u l a t e the u l t i m a t e moment. A flow c h a r t f o r t h i s procedure i s shown i n  figure  90.  The elasto-plastic model c o n t a i n s j u s t as many steps as the b i l i n e a r model, but the equations are somewhat simpler. The bilinear model w i l l a l s o be used i n the next s e c t i o n f o r c a l c u l a t i n g the shape of the u l t i m a t e i n t e r a c t i o n diagram. A.2  ULTIMATE INTERACTION DIAGRAM  A.2.1  Background  Chapter 6 d e s c r i b e d a computer program for ultimate interaction diagrams .for a wide range parameters.  producing of input  T h i s s e c t i o n d e s c r i b e s 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 stress-strain relationship in compression as shown i n f i g u r e 88. The c a l c u l a t i o n s i n t h i s s e c t i o n are only f o r c r o s s s e c t i o n behaviour. Interaction diagrams f o r long columns cannot be produced by such simple procedures, and a numerical computer program i s necessary to produce accurate r e s u l t s f o r 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 d e s c r i b e s a number of d i f f e r e n t regions i n which the c a l c u l a t i o n s must be performed, then p r o v i d e s a summary of the equations d e r i v e d f o r each region. The a c t u a l d e r i v a t i o n s , which proceed easily from first p r i n c i p l e s , have not been i n c l u d e d . A.2.2  Calculations  F i g u r e 91 shows nine d i f f e r e n t cases to be c o n s i d e r e d to produce a complete i n t e r a c t i o n diagram. Each case represents a combination of a x i a l l o a d and bending moment just causing failure.  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^  YES  NO  calculate moment and extreme fibre tension stress when limiting compression strain reached  NO  YES  \/  calculate moment at tension failure  use moment for limiting compression strain  calculate failure moment from elastic theory  \/ ULTIMATE  ULTIMATE  ULTIMATE  MOMENT  MOMENT  MOMENT  F i g u r e 90 - Flow chart f o r c a l c u l a t i n g u l t i m a t e bending moment f o 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 Case 1 i s pure a x i a l t e n s i o n behaviour. The f a i l u r e s t r e s s f t is the t e n s i o n strength obtained from an in-grade a x i a l tension t e s t . Case 2 r e p r e s e n t s combined bending and t e n s i o n with the whole of the s e c t i o n subjected t o t e n s i o n s t r e s s e s . F a i l u r e occurs when the extreme f i b r e t e n s i o n s t r e s s reaches a f a i l u r e s t r e s s fm given by equation 3.17. Case 3 i s the p a r t i c u l a r s i t u a t i o n where the whole s e c t i o n i s in t e n s i o n , but with zero s t r e s s a t the t o p edge. Case 4 i s s i m i l a r to the previous two cases, but now there are compression stresses near the t o p edge, and i n case 5 the  285  compression s t r a i n s exceed the y i e l d s t r a i n . I n 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 e x t r e m e f i b r e tension 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 looks v e r y s i m i l a r t o c a s e 5, b u t now the compression s t r a i n at the top f i b r e reaches i t s l i m i t i n g value before a tension failure occurs. Case 7 i s a p a r t i c u l a r i s z e r o , with the r e s t In case yielding  8 the whole occurring over  s i t u a t i o n where t h e bottom f i b r e of the s e c t i o n i n c o m p r e s s i o n . of the most of  stress  section i s in compression the depth.  The final diagram shows case 9 which is pure 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 i s a s s u m e d t o be t h e f a i l u r e s t r e s s from an in-grade compression test.  with  axial 5 to 9 axial  In a l l of the cases 1 to 9 the net a x i a l l o a d c a n be calculated by summing a l l of the internal tension and compression 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 bending 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 of the i n t e r n a l f o r c e s about the c e n t r o i d a l a x i s of the section. 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 s u m m a r i z e d i n T a b l e V which shows the sequence 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 neutral axis depth. The numbers i n t h e l e f t - h a n d column are the case numbers referred to above. F i g u r e 92 shows t h e r e s u l t s of 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 neutral axis depths, which are sufficient to produce a close approximation to the curve produced by the more general c o m p u t e r m o d e l , 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 .  #  1  c  2  c > 1  3  Tension Failure Stress  Select n.a. Depth  f  =oo  Check Top Strain  Calculate Strains  Calculate  Calc  b =  a =  t  c = 1  f  wd  0  f  wd (— - 1) 2c  f  f f m  A (  e > e y c  _c_ l/k r  e 5  c  —  1-c c  y  < e  c •= 0  8  c  < e  y  u  6  2  6  m  wd  f  2 2  6  m  1  1 2c  c b c wd (a + - - — ) 2  f  t > e c u  i  (1-c) u  e u  c  2  1-b-c  e _y e u  f c  1-b  e c < 0  2c  t  pi  e  6  wd  1  e e  0 < c < 1  p  7  t  =  2  6  f  wd ( — - 1) 2c  f  wd m  wd/2  m  f  m — E  Calculate Moment  Calculate Axial Load  +  b  f  2  c  (l-c)+c  j 2(b-c)  _ f c  c = - oo  wd  0  wd c  ,  -  +  ^—(— - c V I  b  [3a(l-a) + b ( - - 3a-b)j 2  6  N . b c ,b [- (1-a) + [a + 2 b-c 2  2  i a  6  +Ib f  v » r —  2  2 9  +  6  — c  b C  1'wi-at  w c i  c  2b  wd (a + -) 2  wd (a +  f  *  f  U  )  b-c  1, J  2  la + ^ + i j ] 3 2  2  287  Figure  92 - 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 p r o d u c e d calculations, compared with computer c a l c u l a t e d curve  by  hand-  288  APPENDIX B - CALCULATION OF INTERCEPT DEFINING DIAGRAM  INTERACTION  This appendix develops' a s e m i - e m p i r i c a l e x p r e s s i o n f o r the horizontal axis intercept that defines the shape of the b i l i n e a r a p p r o x i m a t i o n to the 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 . Figure  93 shows a t y p i c a l  ultimate  M  interaction  B  1.0  c  diagram.  Moment Figure  93 - T y p i c a l  ultimate  interaction  diagram  Point B is t h e 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 t h e upper s t r a i g h t l i n e p o r t i o n o f t h e d i a g r a m . I f a p o i n t on t h e s t r a i g h t l i n e p o r t i o n of t h e d i a g r a m can be located, as shown by p o i n t C i n f i g u r e 9 3 , t h e v a l u e of B c a n be o b t a i n e d from t h e geometry of t h e d i a g r a m . The c o - o r d i n a t e s of p o i n t C can be o b t a i n e d from t h e a p p r o x i m a t e hand calculation method d e s c r i b e d in Appendix A. Typical values of E/fc=300 and a maximum s t r a i n of 0.01 c a n be u s e d w i t h a n e u t r a l a x i s d e p t h 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 T a b l e V . The c o o r d i n a t e s of. C a r e f o u n d t o be p  = 0.592 f A  c  M  c  =  0.844  f  wd^  (B.1)  (B.2)  289  The moment value Mc must be r e l a t e d to the bending s t r e n g t h Mu (which becomes 1.0 when non-dimensionalized). I f the n e u t r a l a x i s i s assumed to be at mid-depth, the moment c a p a c i t y i s M u  =  (B.3)  f mo  where fm i s the f a i l u r e s t r e s s i n bending, which i s r e l a t e d to the a x i a l t e n s i o n s t r e s s by equation 3.16. -l/k_ f  ^= k&F>  ^  ]  m  where  k  is  the  within-member  depth  *  (B  effect  4)  parameter.  S u b s t i t u t i o n of equations B.3 and B.4 i n t o equation B.2 gives  A reasonably c o n s e r v a t i v e value of k=6.0 (corresponding to f a i l u r e s t r e s s i n t e n s i o n being 65% of that i n bending), and the non-dimensionalized value of Mu=1.0 give M  =  f 0.544 ^  (B.6)  t  c  From the geometry of f i g u r e 93 B  =  (B.7)  M 1-P c  Substituting from non-dimensionalized  equations B.1 and B.6 (with the term fcA as 1.0) equation B.7 becomes B  T h i s e x p r e s s i o n has been design formulae.  =  YHTt c  incorporated  (  into  the  B  -  8  )  recommended  290  APPENDIX C - TEST RESULTS £•/  eccentricity  O 5th percentile  y • . .  a Mean  A  A . 95th percentile  •  ft  m  •  0.0  •  0.5  3.0  3.5  -,  •  2.0  2.5  3.0  H-ID-SPRN MOMENT (KN.M)  3.5  4.0  4.5  F i g u r e 94 - E c c e n t r i c compression r e s u l t s , 38x89mm boards, 1.3m long  —  m  .  .  • Ji • *  202_rni!L  . .. .0'— o.o  — I — 0.5  ^ 3.0  3  i  .  i  5  i  r  2.0  1  2.5  1  r  3.0  "i  n i D - S P R N MOMENT (KN.M)  r  3.5  r  1  r  4.0  F i g u r e 95 - E c c e n t r i c compression r e s u l t s , 38x89mm boards, 1.8m long  4.5  291  o o o_  'o _  •  Mean  A  9 5 t h percentile  .AO^  [RL LORE 60.0  w  o 5th percentile  X  cr  o 9-  -----  o Ro o  €) ••" 0 .0  i  i 0.5  i  i 3.0  i  i 3.5  * i  i 2.0  i  MID-SPRN  Figure  o O  i 2.5  i  i 3.0  i  noriENT (KN.ru  i 3.5  i  i 4.0  96 - E c c e n t r i c c o m p r e s s i o n r e s u l t s , 38x89mm boards.* 2.3m l o n g  -  o _  cnoo CL  CD  -  n i D - S P R N MOMENT (KN.M) Figure  97 - E c c e n t r i c c o m p r e s s i o n r e s u l t s , 38x89mm b o a r d s , 3.2m l o n g  i  i i 4.5  5  292  eccentricity  /7  8-  O 5th percentile •  Mean  A 95th percentile  Q  A-  CCD QfN.  CLCD  •  •  .. a -  o 9'  0.0  2.0  1.0  Figure  3.0  4.0  5.0  n i D - s P R N n o n E N T (KN.n)  €.0  7.0  £.0  98 - E c c e n t r i c c o m p r e s s i o n r e s u l t s , 38x140mm b o a r d s , 1.82m l o n g  Q a a a ^10  .  Ctoo  • • -fi  9-  \ mm  1——[  0  1.0  1  1—"I  2.0  1  3.0  1  1  4.0  1  1  1  5.0  1  6.0  1  1  7.0  n i D - s P f l N n o n E N T (KN.n) Figure  99 - E c c e n t r i c c o m p r e s s i o n r e s u l t s , 38x140mm b o a r d s , 2.44m l o n g  1—  fi.O  293  a  o  8-  o  5th percentile  •  Mean  A 95th percentile  CTo 0  r\i.  CCco  'V  9"  —I  D.C  1  J.O  .  1  I  1  .  1  2.0  1  .....  1  1  4.0  3.0  4  1  1  5.0  1  I  7.0  6.0  MID-SPAN n o n E N T (KN.n) Figure  100 - E c c e n t r i c compression r e s u l t s , 38x140mm boards, 3.35m long  1  2.0  Figure  1  1  3.0  1  1  4.0  1  1  5.0  1  1  6.0  I  r  N nonENT (KN.n) 101 - E c c e nntirDi- cs P Rcompression results, 38x140mm boards, 4.27m long  1  1  fl.fl  294  F i g u r e 102 - A x i a l t e n s i o n r e s u l t s , 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  20.0  30.0  40.0  1 50.0  60.D  TENSION STRESS (MPA)  "I 70.0  —1 60.0  F i g u r e 103 - A x i a l t e n s i o n r e s u l t s , 38x89mm boards, .914m long  90.0  100.0  295  COMPRESSION  STRENGTH  WERKEST P O I N T  IN BOARD DATA  (SOLID  LINE):  ST.DV.  DRTR  WEIBULL  23.07  3.96  23.19  1 .40  507ILE-.  31 . 5 9  0.63  32.30  0.63  MERN:  31 . 8 9  4.88  31 . 6 4  4.80  95*1LE:  39.54  0.72  38.91  0.76  10  0.150  0.153  INTERVAL  CHI-SQUARE  3-PAR  20.0  Figure  EDGEWISE LONG  30.0  1  l  40.0  COMPRESSION  50. D  7.8559  =  33.845  LOCA.  =  0.000  I  80.0  S T R E N G T H (MPA)  100.0  90.0  104 - A x i a l compression r e s u l t s , 38x89mm boards, 2.0m long  BENDING  SPAN DRTR  (SOLID L I N E ) :  DRTR •  22.70  3.55  25.86 .  53.47  2.39  51 . 7 3  2.11  51 . 4 4  15.79  51 . 5 1  15.29  76.60  3.29  76.39  3.15  10  80.0  BENDING  100.0  INTERVAL  120.0  STRENGTH. (MPAJ  0.297  0.307  CHI-SQUARE  3-PAR  60.0  ST.DV.  5/ILE  C OF D I S P  40.0  N=88 WEJBULL  50/ILE 95/JLE  20.0  ST.DV.  3.28  MERN  o.o  15.86  (DASHED):  SCALE  70.0  60.0  FIT:  WEIBULL  SHAPE =  10.0  ST.DV.  57ILE:  C OF D I S P :  o.o  N=88  WEIBULL  FIT:  SHAPE =  3.7557  SCALE  =  57.038  LOCA.  = 0.000  — i — 140.  6.54  (DASHED):  160.0  F i g u r e 105 - Bending t e s t r e s u l t s , 38x89mm boards, 1.5m span  1B0.0  200.0  296  EDGEWISE BENDING SHORT SPAN DATA (SOLID LINE): N=88 WEIBULL DATA ST.DV. 36.14 57ILE: 36.70 1 .63 61 .02 50JILE: 60.72 5.77 60.29 MEAN: 60.33 13.82 81 .91 95;iLE: 82.26 6.59 0.230 OF DISP: 0.229  ODi£>  go" O 0_ UJ  o.o  —r  20.0  1 60.0  —r—.  40.0  1 60.0  ST.DV. 3.46 1 .88 13.88 2.55  INTERVAL CHI-SQUARE FIT: 12.00 3-PAR WEIBULL (DASHED): SHAPE = 4.9718 SCALE = 65.690 LOCA. = 0.000  1  1  100.0  120.0  BENDING STRENGTH (MPfl)  i  140.0  180.0  200.0  F i g u r e 106 - B e n d i n g t e s t r e s u l t s , 38x89mm b o a r d s , .84m span ( e d g e w i s e )  FLATWISE BENDING SHORT SPAN DATA (SOLID LINE): N=87 DATA ST.DV. WEIBULL 33.31 5.49 34.96 5JILE 58.62 50JILE 57.92 2.07 58.45 MEAN 58.44 14.19 81 .32 95/ILE 80.13 3.56 0.240 C OF DISP: 0.242  cr o CE ti-  er _J ZD  20.0  1  80.0  1  ST.DV 3.00 1 .95 14.03 2.92  11 INTERVAL CHI-SQUARE FIT: 4.92 3-PAR WEIBULL (DASHED) SHAPE = 3.7061 SCALE = 51.751 LOCA. = 11 .742  100.0  I 120.0  BENDING STRENGTH (MPA)  140.0  —\ 160.0  F i g u r e 107 - B e n d i n g t e s t r e s u l t s , 38x89mm b o a r d s , .84m span ( f l a t w i s e )  1B0.O  200.0  297  TENSION TEST 38 X 140 S-P-F  5/ILE: 50/ILE: MERN: 95/ILE: OF DISP:  DRTR (SOLID LINE): N=102 DRTR ST.DV. WEIBULL 13.23 1 .79 12.61 26.64 4.15 26.20 27.60 10.82 27.59 46.20 4.56 47.37 0.392 0.389  ST.DV 1.07 1 .37 10.74 2.99  10 INTERVAL CHI-SQUARE FIT: 21.92 3-PAR WEIBULL (DASHED): SHAPE = 1.8971 SCALE = 22.083 LOCA. = 7.996 0.0  10.0  20.0  30.0  40.0  50.0  TENSION STRESS (MPA)  60.0  70.0  i —  80.0  90.0  100.0  Figure 108 - A x i a l t e n s i o n r e s u l t s 38x140mm boards, 3.0m long COMPRESSION TEST 38 X 140 S-P-F  5*1LE: 50/1LE: MEAN: 95/ILE: C OF DISP:  DATA (SOLID LINE): N=97 DATA ST.DV. WEIBULL 18.89 1 .95 20.00 27.27 0.44 27.21 26.83 3.87 26.83 32.78 0.82 32.36 0. 144 0 . 140  ST.DV 1 .07 0.47 3.78 0.56  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  Figure  40.o  50.0  60.0  COMPRESSION STRESS (MPA)  70.0  -I  -  80.  109 - A x i a l compression r e s u l t s , 38x140mm boards, 3.0m long  90.0  100.0  298  F i g u r e 110 - B e n d i n g t e s t 38x140mm b o a r d s , 3.0m  results, long  

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