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Modelling strength and stiffness of glued-laminated timber using machine stress rated lumber Xiong, Pingbo 1991

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MODELLING STRENGTH AND STIFFNESS OF GLUED-LAMINATED TIMBER USING MACHINE STRESS RATED LUMBER By PINGBO XIONG M. Eng., Anhui A g r i c u l t u r a l U n i v e r s i t y , 1985 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES Department of F o r e s t r y 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 September 1991 © Pingbo Xiong, 1991 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada D a t e 0 O-T- ^ . I ^ S \ DE-6 (2/88) ABSTRACT A l o c a l i z e d E - s i m u l a t i o n model and a st r e n g t h s i m u l a t i o n model, which are based on the t h e o r i e s of s t a t i o n a r y random process and the b i v a r i a t e standard normal d i s t r i b u t i o n , have been developed. A group of 2 x 6 2100f-1.8E SPF MSR lumber have been t e s t e d to o b t a i n the within-board compressive strengths. The t e s t E - p r o f i l e s and compressive str e n g t h data was used to provide the s t a t i s t i c a l i n f o r m a t i o n f o r E- s i m u l a t i o n model and cr-simulation model. The comparison between the within-board compressive s t r e n g t h t e s t data and the s i m u l a t i o n r e s u l t s shows th a t the E - s i m u l a t i o n approach and the cr-simulation approach can model the l o c a l i z e d s t i f f n e s s and stre n g t h behaviours s a t i s f a c t o r i l y . Using E - s i m u l a t i o n model and cr-simulation model three grades of MSR lumber have been generated with l o c a l i z e d MOE, t e n s i o n s t r e n g t h and compression s t r e n g t h p r o f i l e s on each board. With these generated MSR lumbers, d i f f e r e n t s i z e s and layups of glulam beams have been b u i l t and the e f f e c t of beam s i z e s and layups on the st r e n g t h of glulam beams has been simulated. The r e s u l t s obtained from glulam beam s i m u l a t i o n showed t h a t the beam s i z e s and layups d i d have s i g n i f i c a n t e f f e c t on the beam st r e n g t h p r o p e r t i e s . With one or two l a y e r s of higher grade lam i n a t i o n on the outer l a y e r of the beam and lower grade laminations i n the r e s t of inner l a y e r s of the beam, the glulam beam bending s t r e n g t h c o u l d be improved s i g n i f i c a n t l y . i i i TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS i i i LIST OF TABLES v i LIST OF FIGURES v i i i LIST OF NOTATIONS xv ACKNOWLEDGEMENTS x v i i i 1. INTRODUCTION AND OBJECTIVES 1 1.1 Objectives 3 1.2 Previous Study 4 2. COFI MSR LUMBER TEST PROGRAM 7 2.1 I n t r o d u c t i o n 7 2.2 M a t e r i a l s 7 2.3 Test Procedures 8 2.4 R e s u l t s and A n a l y s i s 8 3. VITHIN-BOARD COMPRESSION EXPERIMENT 10 3.1 I n t r o d u c t i o n 10 3.2 M a t e r i a l s 10 3.3 Experiment Design and Procedures 11 3.4 R e s u l t s 12 4. E - RANDOM PROCESS SIMULATION MODEL 14 4.1 I n t r o d u c t i o n 14 4.2 T h e o r e t i c a l Basis 14 i v 4.2.1 Random Processes 14 4.2.2 F o u r i e r Transform and Power S p e c t r a l Density.. 15 4.2.3 Mode l l i n g the L o c a l i z e d E-functions 19 4.3 E - f u n c t i o n S i m u l a t i o n Example 24 4.3.1 MSR E - p r o f i l e Treatment 24 4.3.2 S p e c t r a l A n a l y s i s of E - p r o f i l e 25 4.3.3 Reconstruction of E-functions 26 5. BIVARIATE STANDARD NORMAL SIMULATION MODEL 29 5.1 I n t r o d u c t i o n 29 5.2 B i v a r i a t e Normal D i s t r i b u t i o n 30 5.3 Model Development 32 5.4 cr-Profile S i m u l a t i o n Example 35 6. GENERATION OF E- AND (7-PROFILES FOR THREE GRADES 38 6.1 I n t r o d u c t i o n 38 6.2 Generation of E - p r o f i l e s f o r Three Grades 38 6.3 Generation of <r-profiles f o r Three Grades 39 7. GLULAM BEAM SIMULATIONS 44 7.1 I n t r o d u c t i o n 44 7.2 Glulam S i m u l a t i o n Model 45 7.3 Si m u l a t i o n Beam Layups 47 7.4 Si m u l a t i o n R e s u l t s 48 8. DISCUSSION OF GLULAM SIMULATION RESULTS 50 8.1 I n t r o d u c t i o n 50 8.2 Modulus of E l a s t i c i t y 50 8.3 Bending Strength 53 8.3.1 Grade E f f e c t 55 V 8.3.2 Depth E f f e c t 56 8.3.3 E f f e c t of Mixed Grade Layups 56 8.3.3.1 1650f-1.5E and 2100f-1.8E Combinations 57 8.3.3.2 1650f-1.5E and 2400f-2.0E Combinations 57 8.3.4 Comparison of bending strength f o r glulam 59 8.4 Ten s i l e Strength 60 8.4.1 Grade E f f e c t 60 8.4.2 Depth E f f e c t 61 8.4.3 E f f e c t of Mixed Grade Layups 62 8.5 Compressive Strength 63 8.5.1 Grade E f f e c t 63 8.5.2 Depth E f f e c t 64 8.5.3 E f f e c t of Mixed Grade Layups 64 9. SIZE EFFECTS ANALYSIS 66 9.1 Introduction 66 9.2 Size E f f e c t s i n Bending Strength 67 9.3 Size E f f e c t s i n Tensil e Strength 67 9.4 Size E f f e c t s i n Compressive Strength 68 10. CONCLUSIONS AND RECOMMENDATIONS 69 10.1 CONCLUSIONS 69 10.2 RECOMMENDATIONS FOR FURTHER VORK 70 11. REFERENCES 72 APPENDIX A : TABLE 1 - TABLE 21 75 APPENDIX B : FIGURE 1 - FIGURE 103 98 v i LIST OF TABLES Page 1. D e s c r i p t i o n of t e s t m a t e r i a l s and t e s t matrix from COFI .. 76 2. Summary s t a t i s t i c s f o r t e n s i o n MSR lumber from COFI t e s t . 77 3. Summary s t a t i s t i c s f o r compression MSR lumber from COFI t e s t 78 4. Summary of average beam p r o p e r t i e s 79 5. F i t t e d d i s t r i b u t i o n parameters f o r MOE and crc 81 6. D e s c r i p t i v e s t a t i s t i c s f o r t e s t E and GQ (2100f-1.8E) .... 82 7. D e s c r i p t i v e s t a t i s t i c s f o r generated ac (2100f-1.8E) 83 8. D e s c r i p t i v e s t a t i s t i c s f o r t e s t and generated MOE values 84 9. Lower-bound strengths f o r compression and t e n s i o n s i m u l a t i o n 85 10. D e s c r i p t i o n of s i m u l a t i o n beams 86 11. Parameters f o r 3-P Weibul l d i s t r i b u t i o n and t h e i r a s s o c i a t e d 5th and 50th p e r c e n t i l e values of simulated beam strength (Depth=9") 87 12. Parameters f o r 3-P Weibull d i s t r i b u t i o n and t h e i r a s s o c i a t e d 5th and 50th p e r c e n t i l e values of simulated beam strength (Depth=12") 88 13. Parameters f o r 3-P Weibull d i s t r i b u t i o n and t h e i r a s s o c i a t e d 5th and 50th p e r c e n t i l e values of simulated beam stre n g t h (Depth=18") 89 v i i 14. Simulated MOE, transformed MOE and t h e i r r a t i o s 90 15. Comparison of simulated and p r e d i c t e d bending s t r e n g t h ... 91 16. Test of mean values of bending s t r e n g t h f o r 24 beam layups 92 17. Test of mean values of t e n s i l e s t r e n g t h f o r 24 beam layups 93 18. Test of mean values of compressive s t r e n g t h f o r 24 beam layups 94 19. Bending s t r e n g t h s i z e parameter Fab 95 20. T e n s i l e s t r e n g t h s i z e parameter F s t 96 21. Compressive s t r e n g t h s i z e parameter F s c 97 v i i i LIST OF FIGURES Page 1. C u t t i n g p a t t e r n f o r t e n s i o n and compression specimen 99 2. CDF of t e n s i o n data from COFI t e s t (1650f-1.5E) 100 3. CDF of t e n s i o n data from COFI t e s t (2100f-1.8E) 101 4. CDF of t e n s i o n data from COFI t e s t (2400f-2.0E) 102 5. CDF of compression data from COFI t e s t (1650f-1.5E) 103 6. CDF of compression data from COFI t e s t (2100f-1.8E) 104 7. CDF of compression data from COFI t e s t (2400f-2.0E) 105 8. Short span, f l a t w i s e bending E - p r o f i l e 106 9. C u t t i n g and numbering p a t t e r n f o r compression specimen ... 107 10. Experimental setup f o r compressive str e n g t h specimen 13 11. MOE and compressive strength p r o f i l e s along the length ... 108 12. The cumulative d i s t r i b u t i o n f u n c t i o n (CDF) of t e s t and f i t t e d board mean of compressive strength 109 13. The cumulative d i s t r i b u t i o n f u n c t i o n (CDF) of t e s t and f i t t e d board standard d e v i a t i o n of compressive s t r e n g t h .. 110 14. Ensemble of MOE along the length of the board I l l 15. Ensemble of zero mean MOE along the length of the board .. 112 16. P r o b a b i l i t y d e n s i t y f u n c t i o n p(</>(wt)) 113 17. Ensemble average of t e s t E - p r o f i l e 114 18. Ensemble standard d e v i a t i o n of t e s t E - p r o f i l e 115 19. The cumulative d i s t r i b u t i o n f u n c t i o n (CDF) of t e s t and ix f i t t e d board mean of MOE 116 20. The cumulative d i s t r i b u t i o n f u n c t i o n (CDF) of t e s t and f i t t e d board standard d e v i a t i o n of MOE 117 21. Ensemble average of the amplitude spectrum 118 22. Generated E - p r o f i l e of one board 119 23. Ensemble average of t e s t and generated E - p r o f i l e s 120 24. Ensemble standard d e v i a t i o n of t e s t and generated E - p r o f i l e s 121 25. Test and simulated ensemble average of amplitude s p e c t r a . 122 26. The r e g r e s s i o n p l o t of board mean of MOE vs. compressive str e n g t h (2100f-1.8E) 123 27. The r e g r e s s i o n p l o t of board standard d e v i a t i o n of MOE vs. compressive strength (2100f-1.8E) 124 28. Gra p h i c a l demonstration of the transformation between the r e a l space and standard normalized space 125 29. The r e g r e s s i o n p l o t of t e s t modulus of e l a s t i c i t y (E) vs. within-board compressive strength (O-Q) 126 30. The r e g r e s s i o n of t e s t and simulated modulus of e l a s t i c i t y (E) vs. within-board compressive str e n g t h (<rc) .127 31. Cumulative d i s t r i b u t i o n of RC 128 32. Cumulative d i s t r i b u t i o n of RT 129 33. CDF of t e s t and generated minimum te n s i o n data (1650f-1.5E) 130 34. CDF of t e s t and generated minimum te n s i o n data (2100f-1.8E) 131 35. CDF of t e s t and generated minimum te n s i o n data X (2400f-2.0E) 132 36. CDF of t e s t and generated minimum compression data (1650f-1.5E) 133 37. CDF of t e s t and generated minimum compression data (2100f-1.8E) 134 38. CDF of t e s t and generated minimum compression data (2400f-2.0E) 135 39. Beam layups f o r three grade combinations (depth=9") 136 40. Beam layups f o r three grade combinations (depth=12") 137 41. Beam layups f o r three grade combinations (depth=18") 138 42. Beam with two s t i f f n e s s zones 139 43. Comparison of simulated and transformed MOE 140 44. CDF of bending strength (depth = 9") 141 45. CDF of bending strength (depth = 12") 142 46. CDF of bending strength (depth = 18") 143 47. The 5th p e r c e n t i l e values of bending strength v a r i a t e s with depth 144 48. The 50th p e r c e n t i l e values of bending str e n g t h v a r i a t e s w i t h depth 145 49. CDF of bending strength f o r 1650f-1.5E beams as a f u n c t i o n of beam depth 146 50. CDF of bending str e n g t h f o r 2100f-1.8E beams as a f u n c t i o n of beam depth 147 51. CDF of bending strength f o r 2400f-2.0E beams as a f u n c t i o n of beam depth 148 52. The 5th p e r c e n t i l e values of bending str e n g t h x i v a r i a t e s with grade 149 53. The 50th p e r c e n t i l e values of bending strength v a r i a t e s w i t h grade 150 54. CDF of bending str e n g t h with the combinations of 1650f-1.5E and 2100f-1.8E (depth = 9") 151 55. CDF of bending strength with the combinations of 1650f-1.5E and 2100f-1.8E (depth = 12") 152 56. CDF of bending strength with the combinations of 1650f-1.5E and 2100f-1.8E (depth = 18") 153 57. The 5th p e r c e n t i l e values of bending str e n g t h v a r i a t e s with the outer l a y e r percent of 2100fl-1.8E 154 58. The 50th p e r c e n t i l e values of bending s t r e n g t h v a r i a t e s with the outer l a y e r percent of 2100fl-1.8E 155 59. CDF of bending strength with the combinations of 1650f-1.5E and 2400f-2.0E (depth = 9") 156 60. CDF of bending str e n g t h with the combinations of 1650f-1.5E and 2400f-2.0E (depth = 12") .. 157 61. CDF of bending str e n g t h with the combinations of 1650f-1.5E and 2400f-2.0E (depth = 18") 158 62. The 5th p e r c e n t i l e values of bending str e n g t h v a r i a t e s with the outer l a y e r percent of 2400fl-2.0E 159 63. The 50th p e r c e n t i l e values of bending str e n g t h v a r i a t e s with the outer l a y e r percent of 2400fl-2.0E 160 64. CDF of t e n s i l e strength (depth = 9") 161 65. CDF of t e n s i l e s t r e n g t h (depth = 12") 162 66. CDF of t e n s i l e strength (depth = 18") 163 x i i 67. The 5th p e r c e n t i l e values of t e n s i l e s t r e n g t h v a r i a t e s with depth 164 68. The 50th p e r c e n t i l e values of t e n s i l e s t r e n g t h v a r i a t e s with depth 165 69. CDF of t e n s i l e s t r e n g t h (grade = 1650f-1.5E) 166 70. CDF of t e n s i l e s t r e n g t h (grade = 2100f-1.8E) 167 71. CDF of t e n s i l e strength (grade = 2400f-2.0E) 168 72. The 5th p e r c e n t i l e values of t e n s i l e s t r e n g t h v a r i a t e s with grade 169 73. The 50th p e r c e n t i l e values of t e n s i l e s t r e n g t h v a r i a t e s w i t h grade 170 74. CDF of t e n s i l e s t r e n g t h with the combinations of 1650f-1.5E and 2100f-1.8E (depth = 9") 171 75. CDF of t e n s i l e strength with the combinations of 1650f-1.5E and 2100f-1.8E (depth = 12") 172 76. CDF of t e n s i l e strength with the combinations of 1650f-1.5E and 2100f-1.8E (depth = 18") 173 77. CDF of t e n s i l e s t r e n g t h with the combinations of 1650f-1.5E and 2400f-2.0E (depth = 9") 174 78. CDF of t e n s i l e strength with the combinations of 1650f-1.5E and 2400f-2.0E (depth - 12") 175 79. CDF of t e n s i l e strength with the combinations of 1650f-1.5E and 2400f-2.0E (depth = 18") 176 80. The 5th p e r c e n t i l e values of t e n s i l e s t r e n g t h v a r i a t e s w i t h the outer l a y e r percent of 2100fl-1.8E 177 81. The 50th p e r c e n t i l e values of t e n s i l e s t r e n g t h v a r i a t e s x i i i w i t h the outer l a y e r percent of 2100fl-1.8E 178 82. The 5th p e r c e n t i l e values of t e n s i l e strength v a r i a t e s w i t h the outer l a y e r percent of 2400fl-2.0E 179 83. The 50th p e r c e n t i l e values of t e n s i l e s t r e n g t h v a r i a t e s w i t h the outer l a y e r percent of 2400fl-2.0E 180 84. CDF of compressive strength (depth = 9") 181 85. CDF of compressive strength (depth = 12") 182 86. CDF of compressive strength (depth = 18") 183 87. The 5th p e r c e n t i l e values of compressive str e n g t h v a r i a t e s with depth 184 88. The 50th p e r c e n t i l e values of compressive s t r e n g t h v a r i a t e s w i t h depth 185 89. CDF of compressive str e n g t h (grade = 1650f-1.5E) 186 90. CDF of compressive str e n g t h (grade = 2100f-1.8E) 187 91. CDF of compressive str e n g t h (grade = 2400f-2.0E) 188 92. The 5th p e r c e n t i l e values of compressive str e n g t h v a r i a t e s w i t h grade 189 93. The 50th p e r c e n t i l e values of compressive str e n g t h v a r i a t e s with grade 190 94. CDF of compressive strength with the combinations of 1650f-1.5E and 2100f-1.8E (depth = 9") 191 95. CDF of compressive strength with the combinations of 1650f-1.5E and 2100f-1.8E (depth = 12") 192 96. CDF of compressive strength with the combinations of 1650f-1.5E and 2100f-1.8E (depth = 18") 193 97. CDF of compressive strength with the combinations x i v of 1650f-1.5E and 2400f-2.0E (depth = 9") 194 98. CDF of compressive strength with the combinations of 1650f-1.5E and 2400f-2.0E (depth = 12") 195 99. CDF of compressive str e n g t h w i t h the combinations of 1650f-1.5E and 2400f-2.0E (depth = 18") 196 100. The 5th p e r c e n t i l e values of compressive s t r e n g t h v a r i a t e s w i t h the outer l a y e r percent of 2100fl-1.8E 197 101. The 50th p e r c e n t i l e values of compressive s t r e n g t h v a r i a t e s w i t h the outer l a y e r percent of 2100fl-1.8E 198 102. The 5th p e r c e n t i l e values of compressive s t r e n g t h v a r i a t e s with the outer l a y e r percent of 2400fl-2.0E 199 103. The 50th p e r c e n t i l e values of compressive s t r e n g t h v a r i a t e s with the outer l a y e r percent of 2400fl-2.0E 200 LIST OF NOTATIONS V a r i a b l e s : A( ) = Amplitude; b c - Width of core zone lamination i n glulam beam ; b^ = Width of face zone lamination i n glulam beam; E = Modulus of e l a s t i c i t y (MOE); E = Apparent modulus of e l a s t i c i t y ; E c = Core zone modulus of e l a s t i c i t y i n glulam beam ; E^ = Face zone modulus of e l a s t i c i t y i n glulam beam; E[ ] = Expected value; E(z,) = Mean MOE a t l o c a t i o n xi i n an E-process; {E(z)} = MOE process; F( ) = F o u r i e r transform; F*() = Complex conjugate of F( ); G( ) = One-sided power s p e c t r a l d e n s i t y ; I = Apparent moment of i n e r t i a ; I t = Transformed cross s e c t i o n moment of i n e r t i a ; k = Shape parameter i n Weibul l d i s t r i b u t i o n ; L = The length of the board; x v i m = Scale parameter i n Weibull d i s t r i b u t i o n ; p = Random number uniformly d i s t r i b u t e d on the i n t e r v a l ( 0 , 1 ) ; p( ) = P r o b a b i l i t y d e n s i t y f u n c t i o n ; R( ) = A u t o c o r r e l a t i o n f u n c t i o n ; S( ) = two-sided power s p e c t r a l d e n s i t y ; T = Transformed s e c t i o n f a c t o r i n S e c t i o n 4.8; t = Depth of core zone i n glulam beam ; t ^ = Depth of face zone i n glulam beam; X = Test l o c a l i z e d modulus of e l a s t i c i t y ( E ) ; X = Simulated l o c a l i z e d modulus of e l a s t i c i t y ( E ) ; X = Test board mean modulus of e l a s t i c i t y ( E ) ; X = Simulated board mean modulus of e l a s t i c i t y ( E ) ; Y = Test l o c a l i z e d s t r e n g t h (o -); Y = Simulated l o c a l i z e d s t r e n g t h (5-); Y = Test board mean strength (<f); Y = Simulated board mean stre n g t h (<r) ; Zj£ = Standard normalized X; Zy = Standard normalized Y; pxy = C o r r e l a t i o n c o e f f i c i e n t f o r vector (X, Y) i n the r e a l space; p-. = C o r r e l a t i o n c o e f f i c i e n t f o r v e c t o r (X, Y) i n the r e a l x v i i space; pN - C o r r e l a t i o n c o e f f i c i e n t f o r v e c t o r (X, Y) i n the standard normalized space. = Mean value; a - Standard d e v i a t i o n ; cr = Strength; cr = L o c a t i o n parameter i n W e i b u l l d i s t r i b u t i o n ; ac - Compressive str e n g t h ; a E ( x i ) ~ Standard d e v i a t i o n of MOE a t l o c a t i o n xi i n an E-process; aT = T e n s i l e s t r e n g t h ; < 7 ^ = Test board standard d e v i a t i o n f o r MOE; cr* = Simulated board standard d e v i a t i o n f o r MOE; cry - Test board standard d e v i a t i o n f o r s t r e n g t h ; cr^ = Simulated board standard d e v i a t i o n f o r s t r e n g t h ; (/>( ) = Phase angle; w = Angular frequency; oo = V a r i a b l e p e r t a i n i n g to i n f i n i t y ; ACKNOWLEDGEMENTS I would l i k e to express my s i n c e r e g r a t i t u d e to my s u p e r v i s o r , Dr. J.D. B a r r e t t f o r h i s i n v a l u a b l e advice and p a t i e n t guidance throughout the research work and i n pre p a r a t i o n of t h i s t h e s i s . I would l i k e to express my a p p r e c i a t i o n to Dr. R.O. Foschi f o r h i s advice and support. Thanks are a l s o due to Mr. Frank Lam and Mr. Yintang Wang f o r t h e i r h e l p f u l suggestions and a s s i s t a n c e during v a r i o u s stages of the work presented i n t h i s t h e s i s . The f i n a n c i a l support from the Department of Harvesting and Wood Science of the U n i v e r s i t y of B r i t i s h Columbia i s g r a t e f u l l y acknowledged. 1 1. INTRODUCTION Glued-laminated timber, commonly r e f e r r e d to as glulam, i s a s t r u c t u r a l timber product made of elements glued together from smaller pieces of wood, e i t h e r i n s t r a i g h t or curved form, w i t h the g r a i n of a l l the laminations e s s e n t i a l l y p a r a l l e l to the length of the member. In Canada, glulam i s manufactured i n accordance w i t h the requirements of Canadian Standards A s s o c i a t i o n (CSA) Standard 0122-M, Structural Glued-Laminated Timber. Complete S p e c i f i c a t i o n data f o r glulam i s given i n CVC d a t a f i l e VS-2 Glued-Laminated Timber Specifications. Design c r i t e r i a f o r glulam i n l i m i t s t a t e design format are contained i n N a t i o n a l Standard of Canada CAN3-086.1-M84, Engineering Design in Wood. In North America and Europe, glulam i s used i n a wide v a r i e t y of a p p l i c a t i o n s , ranging from headers or support beams i n r e s i d e n t i a l framing to major s t r u c t u r a l elements i n roof framing of domed stadiums. Glulam may be produced i n any s i z e and any shape d e s i r e d , ranging from l a r g e long-span s t r a i g h t beams to complex curved-arch c o n f i g u r a t i o n s . Current production l e v e l s by the glulam i n d u s t r y are approximately 13 MMFBM (1 MMFBM = 10 6 Foot Board Measure) per year i n Canada (Ainsworth, 1989). For glulam beams, the most common design a p p l i c a t i o n i s as a 2 bending member with the primary design loads a p p l i e d p e r p e n d i c u l a r to the wide face of the laminations. To more e f f e c t i v e l y u t i l i z e the a v a i l a b l e lumber resources and to enhance the competitive p o s i t i o n of glulam i n the market p l a c e , such bending members are produced using engineering layups or combinations, i n c o r p o r a t i n g a range of species and s t r u c t u r a l grades of lumber. In these engineered layups, the highest q u a l i t y m a t e r i a l i s p o s i t i o n e d i n the member where the s e r v i c e l o a d i n g w i l l create the highest s t r e s s . Conversely, lower grade laminations are p o s i t i o n e d i n areas or zones where the s t r e s s w i l l be lower. The Canadian glulam i n d u s t r y uses l a m i n a t i n g stock based on the v i s u a l c r i t e r i a and supplementary E - r a t i n g . S p r u c e - P i n e - F i r (SPF) machine s t r e s s - r a t e d lumber i s not permitted i n the CSA Standard 0122-M, Structural Glued-Laminated Timber. Since the SPF species group has a very l a r g e volume i n standing timber and i s more r e a d i l y a v a i l a b l e from domestic s u p p l i e r s , i t i s o b v i o u s l y a reasonable choice to consider SPF machine s t r e s s - r a t e d lumber as an a l t e r n a t e source and type of m a t e r i a l f o r the l a m i n a t i n g stock, e s p e c i a l l y s i n c e the e x i s t i n g v i s u a l l y graded lami n a t i n g stock must be E-rated p r i o r to use. The work done i n t h i s r e p o r t i s aimed a t a s s e s s i n g the use of MSR lumber as the l a m i n a t i n g stock i n the glulam beam production. In order to evaluate the glulam beam st r e n g t h and s t i f f n e s s behaviour, a fundamental understanding i s needed about the c o r r e l a t i o n and the v a r i a t i o n of modulus of e l a s t i c i t y (MOE) and s t r e n g t h values 3 w i t h i n and between the laminations used to f a b r i c a t e the glulam beams. 1.1 OBJECTIVES This study i s aimed a t a c h i e v i n g f i v e main o b j e c t i v e s , namely: 1. To analyze t e n s i l e s t r e n g t h , compressive s t r e n g t h and MOE data from e v a l u a t i o n s of three grades (1650f-1.5E, 2100f-1.8E and 2400f-2.0E) of 2 x 6 SPF MSR lumber provided by the C o u n c i l of Forest I n d u s t r i e s of B r i t i s h Columbia (COFI) and to o b t a i n the necessary d i s t r i b u t i o n parameters r e q u i r e d f o r developing an E - s i m u l a t i o n model and s t r e n g t h s i m u l a t i o n model. 2. To t e s t a group of 2 x 4 2100f-1.8E s p r u c e - p i n e - f i r (SPF) machine-stress-rated (MSR) lumber and develop a data base f o r studying p a i r e d MOE and compressive str e n g t h p a r a l l e l to g r a i n v a r i a t i o n s w i t h i n and between laminations to be used i n glulam beams. 3. To develop an E - s i m u l a t i o n model, from which MOE data p o i n t s along the board length can be generated. The generated MOE data can be used as the input data i n a t e n s i l e and compressive s t r e n g t h s i m u l a t i o n model. 4. To develop a strength s i m u l a t i o n model, from which compressive s t r e n g t h p a r a l l e l to g r a i n (crc) and t e n s i l e s t r e n g t h p a r a l l e l to g r a i n (cy) data p o i n t s can be generated corresponding to the c o r r e l a t e d MOE data p o i n t s generated by the E - s i m u l a t i o n model. 5. To simulate the e f f e c t of beam layups on the s t r e n g t h of 4 glulam beams f a b r i c a t e d using r e g u l a r grades of 2 x 6 SPF MSR lumber. The a n a l y s i s i s performed using i n a computer program c a l l e d GLULAM. 1.2 PREVIOUS RESEARCH The l o c a l i z e d values of modulus of e l a s t i c i t y E and st r e n g t h a along the length of lumber i s r e q u i r e d to a c c u r a t e l y determine the s t i f f n e s s , s t r e n g t h and the f a i l u r e modes of timber s t r u c t u r e s . In recent years, some researchers have turned t h e i r a t t e n t i o n to the v a r i a t i o n of l o c a l i z e d MOE along the length of the board ( B e c h t e l , 1985; F o s c h i , 1987). Based on an a p p l i c a t i o n of the F o u r i e r transform, a method which can produce s a t i s f a c t o r y approximations of the d i s t r i b u t i o n of the l o c a l modulus of e l a s t i c i t y E(x) along the length of a board was presented (1987, F o s c h i ) . I t was concluded t h a t the use of the minimum modulus of e l a s t i c i t y along the length of a board could improve the p r e d i c t i o n of stre n g t h through a b e t t e r s t i f f n e s s and str e n g t h c o r r e l a t i o n . Based on the theory of s t a t i o n a r y random processes, Vang et a l . (1990) developed a procedure which may be used to generate r e a l i s t i c E(z) f u n c t i o n s r e p r e s e n t i n g the v a r i a t i o n i n modulus of e l a s t i c i t y along the length of a specimen. Mo d e l l i n g c o r r e l a t e d lumber p r o p e r t i e s has been an area of a c t i v e research f o r many years. These models are u s e f u l i n Monte Ca r l o s i m u l a t i o n s t h a t p r e d i c t the r e l i a b i l i t y of wood s t r u c t u r e s . Several authors have discussed the importance of accounting f o r the 5 concomitance between lumber str e n g t h p r o p e r t i e s (Suddarth et a l . , 1978; G a l l i g a n et a l . , 1979; R o j i a n i and T a r b e l l , 1984). One approach to the problem was presented by Voeste et a l . (1979). They presented a b i v a r i a t e a n a l y s i s t h a t simulated the c o r r e l a t e d p r o p e r t i e s of modulus of e l a s t i c i t y (MOE) and t e n s i l e s t r e n g t h ( c T ) . T h e i r technique began w i t h f i t t i n g a set of MOE data w i t h an a p p r o p r i a t e p r o b a b i l i t y d i s t r i b u t i o n u s i n g the method of maximum l i k e l i h o o d e s t i m a t i o n . Then a weighted l e a s t squares r e g r e s s i o n was conducted to r e l a t e to MOE. Random values of MOE were then generated from the f i t t e d d i s t r i b u t i o n and s u b s t i t u t e d i n t o the r e g r e s s i o n equation, adjusted by a randomly sampled r e s i d u a l , to generate a corresponding value of u^. This MOE -(Tj, p a i r was c o r r e l a t e d i n a manner s i m i l a r to the t e s t data. But a disadvantage w i t h t h i s approach i s th a t the dependent v a r i a b l e may need to be transformed to o b t a i n the d e s i r e d marginal p r o b a b i l i t y d i s t r i b u t i o n . The choice of transformations i s s u b j e c t i v e and o f f e r s only l i m i t e d f l e x i b i l i t y i n modelling marginal d i s t r i b u t i o n . T a y l o r and Bender (1988) presented an a l t e r n a t e method f o r s i m u l a t i n g c o r r e l a t e d lumber p r o p e r t i e s t h a t are not n e c e s s a r i l y normally d i s t r i b u t e d . This approach uses a tr a n s f o r m a t i o n of the m u l t i v a r i a t e normal d i s t r i b u t i o n to model the c o r r e l a t e d lumber p r o p e r t i e s , and had the advantages of e x a c t l y p r e s e r v i n g each marginal d i s t r i b u t i o n as w e l l as c l o s e l y approximating the c o r r e l a t i o n matrix of the v a r i a b l e s . Lam and Varoglu (1991a, 1991b) developed a model f o r the w i t h i n 6 member v a r i a t i o n of t e n s i l e s t r e n g t h p a r a l l e l to g r a i n i n nominal 38x89 mm No. 2 SPF lumber. They evaluated w i t h i n member t e n s i l e s t r e n g t h cumulative p r o b a b i l i t y d i s t r i b u t i o n s and the s p a t i a l c o r r e l a t i o n of the simulated data by window and semivariogram analyses. Foschi and B a r r e t t (1980) have developed a computer s i m u l a t i o n model of the stre n g t h and s t i f f n e s s of glued-laminated beams i n e i t h e r bending, compression or t e n s i o n . T h e i r approach was to use b a s i c data on the lami n a t i o n p r o p e r t i e s i n a f i n i t e element computer program to estimate v a r i a b i l i t y i n beam stre n g t h and s t i f f n e s s . The b a s i c data r e q u i r e d i n the model are MOE, t e n s i l e and compressive s t r e n g t h values and the d i s t r i b u t i o n of knots f o r va r i o u s l a m i n a t i n g grades i n a p a r t i c u l a r glulam beam layup. However, few experiments have been done to measure the v a r i a t i o n s of the l o c a l i z e d MOE and stre n g t h values along the board length f o r machine s t r e s s - r a t e d lumber. The model developed i n t h i s study incorporates some of the ideas by the previous r e s e a r c h e r s , and provides a general method to simulate the p a i r e d MOE and stre n g t h along the length of lumber. Using these generated MOE and stre n g t h values along the length of lumber, the str e n g t h and s t i f f n e s s p r o p e r t i e s of glulam beams w i l l be simulated to study the e f f e c t of beam layups on glulam beam strengths f a b r i c a t e d from MSR lumber. 7 2. COFI MSR LUMBER TEST PROGRAM 2.1 INTRODUCTION In the f a l l of 1987, the T e c h n i c a l S e r v i c e s Department of the C o u n c i l of F o r e s t I n d u s t r i e s of B r i t i s h Columbia (COFI) i n i t i a t e d a glulam beam research program to s p e c i f i c a l l y address an a l t e r n a t e source of raw m a t e r i a l supply f o r the glulam i n d u s t r y . The o b j e c t i v e of t h a t research program was to evaluate the s u i t a b i l i t y of machine s t r e s s - r a t e d (MSR) lumber f o r the manufacture of s t r u c t u r a l glulam products. This chapter o u t l i n e s some of the t e s t m a t e r i a l s , procedures and r e s u l t s obtained from COFI t e s t program (Ainsworth, 1989). 2.2 MATERIALS Three grades of 2x6 SPF MSR lumber, obtained from an MSR lumber producer l o c a t e d i n the i n t e r i o r of B r i t i s h Columbia, were s e l e c t e d by a Chief Grading Inspector. The grades s e l e c t e d were 1650f-1.5E, 2100f-1.8E and 2400f-2.0E. Two 6-foot t e s t specimens were s e l e c t e d from each of the 16-foot t e n s i o n and compression t e s t lumber groups. One specimen, c a l l e d Zone A, contained the minimum MOE zone lumber as determined by the Cook-B o l i n d e r s equipment. The other specimen, Zone B, was taken from the remaining p o r t i o n of the parent t e s t lumber (Fig u r e 1). 8 A d e s c r i p t i o n of the t e s t m a t e r i a l s and t e s t matrix from COFI i s provided i n Table 1. 2.3 TEST PROCEDURES The specimens from the t e n s i o n and compression t e s t groups were n o n - d e s t r u c t i v e l y t e s t e d i n the Cook-Bolinders machine to o b t a i n a f l a t w i s e bending E - p r o f i l e f o r each board. Jus t before t e s t i n g , an average moisture content of 16% was obtained from the randomly s e l e c t e d sample of ten p i e c e s . The t e n s i o n t e s t specimens from Table 1 and Figure 1 were t e s t e d i n a t e s t i n g machine with a gauge length of two f e e t (610 mm) between the g r i p s . The ramp load t e s t s were conducted at a r a t e of 4000 p s i per minute (27.8 MPa per minute). A l l specimens were t e s t e d to f a i l u r e . The compression t e s t specimens were t e s t e d i n a compression t e s t i n g machine with a gauge length of s i x f e e t (1830 mm). The ramp load t e s t s were conducted a t a displacement r a t e of 0.58 inches per minute (14.7 mm per minute). A l l specimens were t e s t e d to f a i l u r e . 2.4 RESULTS AND ANALYSIS A n a l y s i s of the t e n s i o n and compression specimens t e s t r e s u l t s c o n s i s t e d of a comparison between three MSR lumber grades (1650f-1.5E, 2100f-1.8E and 2400f-2.0E) using of cumulative d i s t r i b u t i o n f u n c t i o n s . 9 Summary s t a t i s t i c s f o r the tension and compression t e s t populations are provided i n Table 2 and Table 3. The cumulative d i s t r i b u t i o n functions f o r the three grades of tension and compression specimen t e s t r e s u l t s in Zone A and Zone B are provided from Figure 2 to 7. At f i r s t the COFI tension and compression t e s t data were planned to be used as the data base f o r the development of a strength simulation model. Later i t was recognized that i n s u f f i c i e n t strength data were a v a i l a b l e on each board in order to construct the simulation model f o r the within-board strength. Therefore, i t was decided that a more d e t a i l e d study of within-board compressive strength t e s t s should be completed in order to e s t a b l i s h a more complete l o c a l i z e d strength data base. The l o c a l i z e d strength experiment and the l o c a l i z e d strength and s t i f f n e s s simulation model developed from the d e t a i l e d compression studies w i l l be discussed i n the following chapters. 10 3. WITHIN-BOARD COMPRESSION EXPERIMENT 3.1 INTRODUCTION This chapter g i v e s a f u l l d e s c r i p t i o n of the procedures and r e s u l t s of the within-board compressive strength p a r a l l e l to g r a i n experiment. The purpose of t h i s experiment i s to develop a data base f o r p a i r e d within-board MOE and compressive str e n g t h p a r a l l e l to g r a i n ((Tgi) p r o p e r t i e s along the board length. The r e l a t i o n s h i p between MOE and compressive strength p a r a l l e l to g r a i n i^c) determined exp e r i m e n t a l l y w i l l provide a b a s i s f o r developing a s i m u l a t i o n method f o r generating c o r r e l a t e d property data f o r glulam beam stren g t h s t u d i e s . 3.2 MATERIALS The lumber f o r t h i s t e s t was s e l e c t e d from a MSR lumber producer l o c a t e d i n the i n t e r i o r of the province of B r i t i s h Columbia. Three grades of 2 x 4 inches (38mm x 89mm) SPF MSR lumber were s e l e c t e d . The s e l e c t e d grades were 1650f-1.5E, 2100f-1.8E, and 2400f-2.0E. Of these three grades, only grade 2100f-1.8E was t e s t e d f o r i t s within-board compressive s t r e n g t h p a r a l l e l to g r a i n . The other two grades were expected to be t e s t e d s e p a r a t e l y i n the f u t u r e . 3.3 EXPERIMENT DESIGN AND PROCEDURES The experiment was designed to t e s t the within-board compressive s t r e n g t h p a r a l l e l to g r a i n (<rc) along the board length f o r SPF species group. The t e s t specimens were co n d i t i o n e d to a maximum moisture content (MC) of 12% i n the l a b o r a t o r y . The average of three MC readings i n each piece of lumber was obtained w i t h a r e s i s t a n c e - t y p e meter. They were then n o n - d e s t r u c t i v e l y t e s t e d i n the Cook-Bolinders s t r e s s grading machine i n F o r i n t e k Canada Corp. (Vancouver) to o b t a i n a short span, f l a t w i s e bending E - p r o f i l e f o r each board (Fi g u r e 8 ) . Data c o l l e c t i o n was performed by a computer a t a scan r a t e of 440 readings per second. This t r a n s l a t e d i n t o approximately 2100 readings per 16-foot specimen. Load readings were c o l l e c t e d a t a mid-span d e f l e c t i o n of 0.179 inch (4.55 mm) over a 2.99 f o o t (910 mm) span (simply supported and centre loaded). Each t e s t specimen was run through the Cook-Bolinders machine twice to e l i m i n a t e the e f f e c t s of specimen bow. The values f o r each pass were recorded, along w i t h the average values from the two passes. F o l l o w i n g the Cook-Bolinders e v a l u a t i o n , the 16-foot specimens were cut i n t o t h i r t y - t w o (32) 6-inch (152.4 mm) long compression specimens (Fig u r e 9 ) . The within-board compression specimens were t e s t e d a t room temperature with the MTS T e s t i n g Machine (MTS810) i n displacement c o n t r o l mode. A lo a d i n g r a t e of 0.018 in/min (0.457 mm/min) was used 12 which caused f a i l u r e i n about 2 minutes. The load c e l l c a p a c i t y used i n the t e s t was 250 KN. A f t e r the d e s t r u c t i v e t e s t , the average d e n s i t y of each within-board compressive str e n g t h t e s t piece was measured. The d e n s i t y was c a l c u l a t e d based on specimen weight and measured s i z e . The experimental setup i s shown i n Figure 10. 3.4 RESULTS The t e s t r e s u l t s f o r within-board compression specimens are given i n Table 4. In t o t a l 54 beams were t e s t e d . Each beam c o n s i s t e d of 32 w i t h i n -board compression specimen. Therefore, there were i n t o t a l 1728 specimen t e s t e d . F i gure 11 shows the MOE and compressive s t r e n g t h p r o f i l e obtained f o r specimen No. 1805 and demonstrates the general tendency f o r MOE and compressive str e n g t h to be c o r r e l a t e d . The cumulative d i s t r i b u t i o n f u n c t i o n s f o r the mean compressive s t r e n g t h and the standard d e v i a t i o n of compressive s t r e n g t h f o r the 54 members are shown i n Figures 12 and 13. In Figures 12 and 13, the data sets have been f i t t e d w i t h normal, lognormal, 2-parameter Weibull and 3-parameter W e i b u l l d i s t r i b u t i o n s . The normal d i s t r i b u t i o n was v i s u a l l y judged to best f i t the board average compressive str e n g t h and the board compressive s t r e n g t h standard d e v i a t i o n . 13 The parameters f i t t e d f o r normal, lognormal, 2-parameter V e i b u l l and 3-parameter V e i b u l l d i s t r i b u t i o n s f o r MOE and ac are given in Table 5. Figure 10 Experimental set up f o r compressive strength specimen 14 4. E - RANDOM PROCESS SIMULATION MODEL 4.1 INTRODUCTION The use of l o c a l i z e d values of modulus of e l a s t i c i t y (MOE) and streng t h (c) along the length of the lumber i s necessary t o a c c u r a t e l y determine s t i f f n e s s , s t r e n g t h and f a i l u r e mode of timber s t r u c t u r e s . These l o c a l i z e d MOE v a r i a t i o n s must be inc l u d e d , f o r example, i n s t r u c t u r a l a n a l y s i s of models of glulam beams where the s t i f f n e s s and streng t h of a beam are dependent on the l o c a l i z e d s t i f f n e s s and stre n g t h of each l a m i n a t i o n . This chapter w i l l d i s c u s s the t h e o r e t i c a l b a s i s of the random processes, F o u r i e r transform and power s p e c t r a l d e n s i t y and t h e i r a p p l i c a t i o n s i n l o c a l i z e d E - f u n c t i o n generation. 4.2 THEORETICAL BASIS 4.2.1 RANDOM PROCESSES A t i m e - s e r i e s or t i m e - h i s t o r y i s the c o l l e c t i o n of observations i n time. I f the observations can be p r e d i c t e d p r e c i s e l y , the process i s c a l l e d d e t e r m i n i s t i c . I f , however, the observations can only be def i n e d i n terms of p r o b a b i l i t y statements, the process i s r e f e r r e d to as s t o c h a s t i c (or random). Consider an ensemble of observed E-functions f o r a p a r t i c u l a r grade of lumber (2100f-1.8E) as shown i n Figure 14. Here, the 15 l o c a l i z e d MOE values along the length of a board represents a time s e r i e s . This ensemble i s r e f e r r e d to as a process. Due to the f a c t t h a t the observations can only be defined i n terms of p r o b a b i l i t y statements, the process i s r e f e r r e d to as random process. A random E-process can f o r m a l l y be defined as an i n f i n i t e s et or ensemble of " e q u a l l y - 1 i k e l y " sample E-functions w i t h some s t a t i s t i c a l or p r o b a b i l i s t i c i n f o r m a t i o n about the samples. A random E-process can be described by the mean E(z,) and the standard d e v i a t i o n <rE(rt) of the ensemble f o r any l o c a t i o n x-. I f the E(x,-) and "E^,-) a r e ^ e s a m e ^ o r a H l o c a t i o n s , the process {E(z)} i s s a i d to be s t a t i o n a r y . For the development of the model, i t i s assumed t h a t the E-process s a t i s f i e s t h i s s t a t i o n a r y c o n d i t i o n . Furthermore, i f the mean value of the ensemble i s equal to the mean value taken along the length of any sample E - f u n c t i o n , then the process {E(z)} i s s a i d to be ergodic. For the development of the model, i t i s assumed t h a t the E-process i s not an ergodic process. The E-process can be transformed i n t o a s t a t i o n a r y - e r g o d i c process w i t h zero mean by s u b t r a c t i n g the mean of each sample E-f u n c t i o n as shown Figure 15. 4.2.2 FOURIER TRANSFORM AND POWER SPECTRAL DENSITY Let E n(z) be a sample from zero mean continuous E-process {E(x)} (see Figure 15}. I f 16 oo I I E„(z) I dx < oo ( 4.1 ) -oo then the F o u r i e r transform F n(w) of En(a:) e x i s t s , a c o n d i t i o n which i s u s u a l l y s a t i s f i e d i n p r a c t i c e . The sample E n(x) uniquely corresponds to i t s F o u r i e r transform and conversely, the inverse F o u r i e r transform uniquely d e f i n e s the sample f u n c t i o n E n(x) through: F„( W j 0 = E„(r)e dx En(x) e,uX dx (4.2) OO L -oo 0 where £ i s the length of the board. The two f u n c t i o n s En(x) and Fn(w, Z.) are a F o u r i e r transform p a i r . Fn(w,Z.) i s i n general a complex f u n c t i o n of the frequency u. F n(w, 0 = Re(w) + Im(u) = | Fn(w, L) \ (4.4) where Re(w) and Jm(w) are the r e a l and imaginary p a r t of the F o u r i e r transform, r e s p e c t i v e l y . |F n(w,/.)| i s the amplitude or F o u r i e r spectrum of En(x) and <^ (w) i s the phase angle of the F o u r i e r transform, and given by: |Fn(w, 0 1 = ^Re2(u) + /m2(w) 17 (4.5) ^ = M £££ ) <4-6> For a sample E - f u n c t i o n (En(x)) over the f i n i t e length 0< x < L, the 2-sided power s p e c t r a l d e n s i t y f u n c t i o n of the sample i s de f i n e d as (Bendat and P i e r s o l , 1 9 8 6 ) : S E(n, W ) 0 = J F n > , 0 F „ ( W , 0 (4.7) where Fn*(w, L) i s the complex conjugate of F n(w, L). The two-sided power s p e c t r a l d e n s i t y f u n c t i o n of the un d e r l y i n g s t a t i o n a r y E-process i s given by (Bendat and P i e r s o l , 1 9 8 6 ) : S E(w)= lim E[S E(n, u, 0 ] (4.8) L—>oo where E[S E(n, u, /.)] i s the expected value of the power s p e c t r a l d e n s i t y over the ensemble of E- f u n c t i o n s . S u b s t i t u t i n g Eq. (4.7) i n t o Eq. (4.8) y i e l d s : SE(w) = Um I E [ | F > , O l 2 ] (4.9) L—*oo where | Fn(w, Z.) | i s the F o u r i e r amplitude spectrum. Since E n(z) i s r e a l , Sg(w) i s a r e a l , non-negative and even 18 f u n c t i o n : S E ( - W ) = SE(u>) ( 4.10 ) The one-sided power s p e c t r a l d e n s i t y f u n c t i o n GE(w) can be defi n e d as: G E( W) = 2 SE(u/) = \ lira E[ | F > , L) | 2 ] ( 4.11 ) where 0 < u < o o . A t r u e s t a t i o n a r y random process contains an i n f i n i t e number of sample f u n c t i o n s with i n f i n i t e l e n g t h , whereas r e a l E - f u n c t i o n s are few i n number and have f i n i t e length. Consequently, an estimate of the power s p e c t r a l d e n s i t y f u n c t i o n of the E-process GE(w) can be obtained by f i r s t computing the power s p e c t r a l d e n s i t y f u n c t i o n Gn(w, L) of each E - f u n c t i o n , and then averaging the ensemble of s p e c t r a l d e n s i t y components a t each frequency. The averaging i s intended to approximate the expected value i n Eq. (4.11), which can be replac e d by: G > , 0 = I | F > . O I 2 (4-12) G E ( « ) = i ! > > . o (4.i3) n = 1 where N i s the number of sample E-functions (sample s i z e ) . 19 4.2.3 MODELLING THE LOCALIZED E-FUNCTIONS A s t o c h a s t i c model f o r the s i m u l a t i o n of the E values along the length from MSR lumber data was set up. The model was developed from s p e c t r a l a n a l y s i s and random v i b r a t i o n theory, where the E - f u n c t i o n was t r e a t e d as s t a t i o n a r y , non-ergodic random process as discussed above. Using s t i f f n e s s t e s t data on lumber (MSR lumber d a t a ) , the power s p e c t r a l d e n s i t y of the u n d e r l y i n g E-process can be estimated. An ensemble of E-functions can then be r e c o n s t r u c t e d by combining the power s p e c t r a l d e n s i t y f u n c t i o n (or amplitude spectrum) of the process with the randomly s e l e c t e d phases. The r e c o n s t r u c t e d data have s i m i l a r s t a t i s t i c a l c h a r a c t e r i s t i c s and frequency content to the experimental data. The generated E-process can be modeled by the f o l l o w i n g cosine s e r i e s (Vang et a l . , 1990): N/2 ( 4.14 ) i = 1 where N i s the number of the d i s c r e t e frequencies i n c l u d e d i n S E(w);the amplitude A(u>j) and frequency uii are d e t e r m i n i s t i c , but the phase angle ei(wt) i s assumed to be a random v a r i a b l e . The p r o b a b i l i t y d e n s i t y f u n c t i o n f o r 4>^i) i s taken to be d i s t r i b u t e d u n i f o r m l y between 0 and 2ir as shown i n Figure 16. That i s , otherwise 20 ( 4.15 ) The ensemble mean of the Eq. (4.14) a t a s p e c i f i c l o c a t i o n x i s N/2 E l E ^ x ) ] = E [ £ A( WJcos(w , . x + # * , • ) ) ] i = 1 N/2 f 27T N $NA(W ,-) cos(W,x + #w,.)) K*(w,0) «W",0) .• = i l o J = 0 ( 4.16 ) The ensemble mean square of the Eq. (4.14) i s N/2 E[E3(x)] = E[( £A( W l.)cos( W l.x+^ W,.))) ] t = I N/2 ( 2? ) = E S A V i ) cos 2(W,x + 0K>) i»(0(w,.)) < ^ K » » = H 2 i = 1 ( 4.17 ) Since the E-process i s a s t a t i o n a r y process, the a u t o c o r r e l a t i o n f u n c t i o n f o r the random process E(x) can be de f i n e d as the average value of the product E(x)E(x + r) as the f o l l o w i n g R B ( r ) = E[E(i)E(x + r ) ] 21 ( 4.18 ) where RE(T) i s the a u t o c o r r e l a t i o n f u n c t i o n f o r E(z) From Eq. (4.17), (4.18) and l e t r - 0 N/2 .2/ x E[E5-(x)] = £ = R £(0) ( 4.19 ) i = i A l s o , the F o u r i e r transform of RE(T), and i t s i n v e r s e , are given by (Bendat and P i e r s o l , 1 9 8 6 ) : ' oo ( 4.20 ) M O = i ' SE(w) rfw ( 4.21 ) where SE(u>) i s c a l l e d the power s p e c t r a l d e n s i t y f u n c t i o n of the E processes and i s a f u n c t i o n of angular frequency of u. I f r = 0, then M ° ) = SE(w) dw ( 4.22 ) Equating Eq. (4.19) and Eq. (4.22), and knowing t h a t SE(w) i s a 22 r e a l , non-negative and even f u n c t i o n : j2 A ("••) t = i oo dw = 2 N/2 SE(u) du « 2 ^ S E ( W l ) 4 w (4.23) i = 1 from which the amplitude A(w f) can be c a l c u l a t e d to be: A(w,.) = 2^ SE(w,-) Au ( 4.24 ) For a random process with zero mean (E = 0 ) , the mean square value equals the standard d e v i a t i o n a ^  (x) of the process: 2 N/2 .2/ \ E[E*.(x)] = E[(E(X) - E) ] = 4 ( x ) = £ i = l ( 4.25 ) ^ (*) i - 1 A"( W.) ( 4.26 ) The E-process generated by Eq. (4.14) i s a s t a t i o n a r y , ergodic process wit h zero mean. To make the E-process to be non-ergodic with non-zero mean, Eq. (4.14) can be modified as f o l l o w s : N/2 E j O ) = E i + £ A(w,-)cos(w,.x + #w,.)) «' = 1 ( 4.27 ) 23 where i s a random value chosen from the d i s t r i b u t i o n of beam mean MOE values. I f , f o r example, the mean data can be approximated by a 3-parameter Weibu l l d i s t r i b u t i o n , E^- i s given by Ej = <r0 + m ( - l n ( l - p) ) ( 4.28 ) where <rQ, m and k denote, r e s p e c t i v e l y , the l o c a t i o n , s c a l e and shape parameters f o r the d i s t r i b u t i o n , and p i s a random number un i f o r m l y d i s t r i b u t e d between 0 and 1. The Nyquist number f o r the F o u r i e r transform i s 77 +1 ( i f N i s an even number), or N + 1 ( i f N i s an odd number), where N i s the number of f r e q u e n c i e s . Then: A( W l.) A K ) = j 0 MuN-i +2) * < 2 + 1 • > 2 + 1 ( 4.29 ) where N i s an even number; or A( W (.) A( W t.) = | 0 MUN -»' +1) i < i > N + 1 2 N + 1 2 N + 1 2 ( 4.30 ) 24 where N i s an odd number. Therefore the amplitude spectrum A(w,) i s symmetrical to the Nyquist frequency. This property i s used to reduce the number of harmonics i n Eq. (4.27) without considerable e f f e c t on the accuracy of the r e s u l t s . 4.3 E-FUNCTION SIMULATION EXAMPLE Tests of l o c a l i z e d E - functions can be obtained by bending a board over consecutive short spans, using a concentrated load a p p l i e d a t the middle of each span. This i s done by grading machines based on the so c a l l e d " d e f l e c t i o n method" where the d e f l e c t i o n i s constant and the c e n t r e - p o i n t load i s measured. From the d e f l e c t i o n measurements and simple equations from the beam theory, an estimate of the bending E values along the length of the board can be obtained. The l o c a l i z e d MOE values along the length of the boards of grade 2100f-1.8E, which were obtained from the Cook-Bolinders machine i n F o r i n t e k Canada Corp. (Vancouver), were analyzed and used as the input data i n the f o l l o w i n g E - f u n c t i o n s i m u l a t i o n program. In t o t a l , 54 boards were n o n - d e s t r u c t i v e l y t e s t e d i n the Cook-Bolinders machine. 4.3.1 MSR E-PROFILE TREATMENT In order to implement the E - f u n c t i o n s i m u l a t i o n program, the machine s t r e s s - r a t e d MOE data p r o f i l e s should be t r e a t e d f i r s t to determine the process s t a t i s t i c s . 25 A f t e r averaging the MOE values a t the same l o c a t i o n s along the length of the boards, an ensemble average of MOE along the length of the board was obtained, as shown i n Figure 17. Fi g u r e 18 i s the ensemble average of the standard d e v i a t i o n of MOE along the length. Eq. (4.14) generates an E - p r o f i l e f o r a zero mean process. In order to perform the F o u r i e r transform of each board r e c o r d and c a l c u l a t e i t s power s p e c t r a l d e n s i t y f u n c t i o n and the r e l a t e d amplitude spectrum, the zero mean MOE records are cons t r u c t e d by s u b t r a c t i n g from each one the corresponding mean value E. The cumulative d i s t r i b u t i o n f u n c t i o n ( c d f ) of board mean MOE values and the within-board standard d e v i a t i o n of MOE i s shown i n Figure s 19 and 20 r e s p e c t i v e l y . The r e s u l t s i n Figures 19 and 20 have been f i t t e d w i t h normal, lognormal, 2-parameter Weibull and 3-parameter Weibu l l d i s t r i b u t i o n s . The 3-parameter Weibull d i s t r i b u t i o n was v i s u a l l y judged to provide the best f i t f o r the board mean of MOE values and the board standard d e v i a t i o n of MOE. The parameters f o r the d i s t r i b u t i o n s are summarized i n Table 5. 4.3.2 SPECTRAL ANALYSIS OF E-PROFILE By performing Fast F o u r i e r Transform (FFT), the amplitude spectrum of the MOE rec o r d i s obtained. Then the ensemble average of the amplitude spectrum by averaging the amplitudes a t each frequency, shown i n Figure 21, i s constructed. 26 4.3.3 RECONSTRUCTION OF E-FUNCTIONS The p r i n c i p l e of generation of E-functions i s th a t the rec o n s t r u c t e d E - f u n c t i o n should have s i m i l a r s t a t i s t i c a l c h a r a c t e r i s t i c s and frequency content as the experimental E values. The t y p i c a l E - f u n c t i o n can be rec o n s t r u c t e d as i n the form of Eq. (4.27), from which the f i r s t p a r t E^ i s the random mean of the sample E - f u n c t i o n obtained from a f i t t e d cumulative d i s t r i b u t i o n f u n c t i o n ( F i g u r e 19) through the t e s t mean of the E - f u n c t i o n s ; the N/2 second p a r t , ^ A ( W - ) C O S ( W 1 J ; + ^ (w,-)) , i s the v a r i a t i o n of the E - f u n c t i o n » = 1 about the mean value along the length. From Eq. (4.26), the standard d e v i a t i o n crE(x) i n each generated board would be a constant value s i n c e the summation of the ensemble average of the amplitude spectrum i s a constant. Therefore we choose to normalize the ensemble average of amplitude spectrum by d i v i d i n g the standard d e v i a t i o n of the E-process o-g(x) . L a t e r , when r e c o n s t r u c t i n g E - f u n c t i o n s , the amplitude spectrum of each generated board i s c o r r e c t e d to achieve a random standard d e v i a t i o n cre(x). This random standard d e v i a t i o n <re(x) i s obtained from a cumulative d i s t r i b u t i o n f u n c t i o n f i t t e d to within-board standard d e v i a t i o n of the E-functions ( F i g u r e 20). With t h i s procedure the randomly generated E-fu n c t i o n s w i l l have a within-board standard d e v i a t i o n d i s t r i b u t i o n 27 which matches the t e s t data. Figure 22 i s a sample of a generated MOE p r o f i l e , F i g ure 23 and Figure 24 are the ensemble mean and ensemble standard d e v i a t i o n of the generated MOE p r o f i l e s f o r 2100f-1.8E MSR lumber. R e s u l t s as presented i n Figures 23 and 24 show th a t there i s a r e l a t i v e l y good agreement between the t e s t and generated data. As a way of v e r i f y i n g the agreement between t e s t and computer s i m u l a t i o n r e s u l t s , the two ensemble amplitude s p e c t r a had been compared i n Figure 25, which shows very good agreement. A summary of the steps to implement the E - f u n c t i o n s i m u l a t i o n program i s as f o l l o w s : 1. Obtain the E - p r o f i l e s along the length using the Cook-B o l i n d e r s grading machine; 2. C a l c u l a t e the mean and standard d e v i a t i o n of MOE f o r each board; 3. Obtain the f i t t e d cumulative d i s t r i b u t i o n f u n c t i o n of the mean MOE (E^) and standard d e v i a t i o n (<rE); 4. Construct zero mean E - p r o f i l e from steps 1 and 2; 5. Perform the Fast F o u r i e r Transform (FFT) to each zero mean E - p r o f i l e to o b t a i n i t s amplitude spectrum; 6. C a l c u l a t e the ensemble average of amplitude spectrum; 7. Normalize the ensemble average of amplitude spectrum by d i v i d i n g by the ensemble standard d e v i a t i o n <rE(x); 8. Generate random phase angles <#(wt) between 0 and 2ir; 9. Generate a random mean from a f i t t e d cdf obtained i n step 3; 10. Generate a random standard d e v i a t i o n <rE from a f i t t e d cdf obtained i n step 3 and ad j u s t amplitude spectrum; 11. Reconstruct the generated E - p r o f i l e according to the Eq. (4.27). 29 5. BIVARIATE STANDARD NORMAL DISTRIBUTION SIMULATION MODEL 5.1 INTRODUCTION Mode l l i n g c o r r e l a t e d lumber p r o p e r t i e s has been an area of a c t i v e research f o r many years. In t h i s s e c t i o n , we w i l l d i s c u s s a b i v a r i a t e approach of s i m u l a t i n g c o r r e l a t e d lumber p r o p e r t i e s data. I t i s w e l l known t h a t f o r lumber, compressive ( G C ) and t e n s i l e strengths p a r a l l e l to g r a i n ( G T ) are p o s i t i v e l y c o r r e l a t e d w i t h the modulus of e l a s t i c i t y (MOE). A knowledge of these c o r r e l a t i o n s leads to the development of the mechanical lumber grading machine. For s t r u c t u r a l r e l i a b i l i t y a n a l y s i s of wood s t r u c t u r e s , the lumber p r o p e r t i e s ac, aT, and MOE are needed as a compatible s e t , f o r each piece of m a t e r i a l . For a beam strength a n a l y s i s , as i n the case of GLULAM beam st r e n g t h s i m u l a t i o n s , each element must be assigned a value of MOE, <7j. and GQ , r e s p e c t i v e l y . In the previous s e c t i o n s , the generation of l o c a l i z e d MOE values along the length of the boards has been discussed. With the generated MOE values along the length of the boards, c o r r e l a t e d s t r e n g t h values can a l s o be generated by a B i v a r i a t e Standard Normal D i s t r i b u t i o n S i m u l a t i o n Program (BNSIM) as described i n the f o l l o w i n g paragraphs. A computer s i m u l a t i o n model was presented f o r s i m u l a t i n g c o r r e l a t e d random v a r i a b l e s t h a t preserves the marginal d i s t r i b u t i o n of each v a r i a b l e as w e l l as the c o r r e l a t i o n between the v a r i a b l e s . The 30 method i s i l l u s t r a t e d using modulus of e l a s t i c i t y (MOE) and compressive s t r e n g t h (<rc) data f o r 2 x 6 inches (38x140 mm) 2100f-1.8E Spr u c e - P i n e - F i r (SPF). 5.2 BIVARIATE NORMAL DISTRIBUTION The random vec t o r ( X, Y) has a b i v a r i a t e normal d i s t r i b u t i o n i f the j o i n t p r o b a b i l i t y d e n s i t y f u n c t i o n (p.d.f.) of ( X, Y ) i s given by (5.1) In (5.1), X and Y have marginal N(/j ; r, (rx2) and N(//y, cr y 2) d i s t r i b u t i o n s w i t h the means u„, u„ and the standard d e v i a t i o n s a , a , r e s p e c t i v e l y . ( X, Y) has a variance-covariance matrix the c o r r e l a t i o n c o e f f i c i e n t of X and Yis p. According to the c o n d i t i o n a l p r o b a b i l i t y formula 31 (5.2) Then, the c o n d i t i o n a l d i s t r i b u t i o n of Y, given X = x, has the c o n d i t i o n a l p.d.f. /(y | x ), where " " • J - ^ r - { - ^ ( ( ^ - ^ X ! ? ) + ( ^ ) 2 - < w > ( ^ ) 2 } S i m p l i f y i n g the above equation by = «y 1 1 - P2 ( 5.4 ) V* = A*y + P ^  ( * - Hx) ( 5.5 ) where a; i s a random v a r i a b l e from x — \ix + RN • ax (0 < RN < 1 ) and RN i s a random normal number with mean 0 and standard d e v i a t i o n of 1. 32 Then / ( H O = 0-N2T 1 exp (5.6) = %(0, 1) (5.7) y = a z + n (5.8) where cr* i s a constant; z* i s a random number ( 0 < z* < 1 ). Therefore, the c o n d i t i o n a l d i s t r i b u t i o n of Y, has the s i m p l i f i e d c o n d i t i o n a l p.d.f. / ( y \ x), where 5.3 MODEL DEVELOPMENT A tra n s f o r m a t i o n of the b i v a r i a t e standard normal d i s t r i b u t i o n i s chosen to model the two c o r r e l a t e d lumber stre n g t h and s t i f f n e s s p r o p e r t i e s . The approach has the advantage of e x a c t l y p r e s e r v i n g each marginal d i s t r i b u t i o n as w e l l as the c o r r e l a t i o n between MOE and str e n g t h p r o p e r t i e s obtained from t e s t data. This s e c t i o n w i l l d e scribe the procedures which are used to generate a c o r r e l a t e d random p a i r (X, Y) or given X, to generate the random value Y, where X could be chosen as a MOE and Y the compressive (5.9) 33 s t r e n g t h a t a p a r t i c u l a r l o c a t i o n w i t h i n a board. The t h e o r e t i c a l c o n s i d e r a t i o n has already been discussed i n the above s e c t i o n . The study of within-board MOE and compressive s t r e n g t h property v a r i a t i o n showed t h a t both means and standard d e v i a t i o n s of MOE and compressive st r e n g t h vary from board to board. In the b i v a r i a t e standard normal d i s t r i b u t i o n s i m u l a t i o n model, we assume t h a t : a) . The c o r r e l a t i o n c o e f f i c i e n t p-~ f o r X mean and Y mean i s the / r xy same as p f o r X and Y, as shown i n Figure 26. b) . The c o r r e l a t i o n c o e f f i c i e n t pxy f o r X standard d e v i a t i o n and Y standard d e v i a t i o n i s zero, as shown i n Figure 27. The procedures f o r modelling the c o r r e l a t e d v a r i a b l e s are as f o l l o w s : 1. Obtain E - p r o f i l e s from the Cook-Bolinders grading machine and the s t r e n g t h p r o f i l e s from the within-board compressive s t r e n g t h t e s t s . 2. Estimate the parameters f o r the best f i t t i n g cumulative d i s t r i b u t i o n f u n c t i o n s of means and standard d e v i a t i o n s f o r both v a r i a b l e s X and Y. The mean and standard d e v i a t i o n cumulative d i s t r i b u t i o n f u n c t i o n s F(r) and F(j^) could be f i t t e d by normal, lognormal, 2-parameter Weibull and 3-parameter V e i b u l l d i s t r i b u t i o n s i n the model developed, as i n Figure 19, Figure 20, and Figure 12, Fig u r e 13. 34 3. Estimate the c o r r e l a t i o n c o e f f i c i e n t pxy i n the r e a l space by: E (** -1) (y* - y) i = 1  ( E ( y , - y ) 2 £ (y , -y) 2 ) ^ - ; ~ < 5- 1 0) 2 « = 1 «' = 1 where xi and are the observations of X and Y; x and j/ are the means of the X and Y; n i s the number of the observations of J and Y. 4. Transform X from r e a l space i n t o standard normalized space X:-X w i t h i t s mean equals 0 and standard d e v i a t i o n equals 1, i.e. —a • 5. Transform Y from r e a l space i n t o standard normalized space Y.-Y with i t s mean equals 0 and standard d e v i a t i o n equals 1. i.e. Zy "Y 6. Estimate the c o r r e l a t i o n c o e f f i c i e n t pN f o r Zx and Zy i n the standard normalized space. 7. Generate random board mean X and Y from the best f i t t i n g cumulative d i s t r i b u t i o n f u n c t i o n s of mean f o r X and Y assuming t h a t they have the c o r r e l a t i o n c o e f f i c i e n t pxy (as shown i n Fi g u r e 26). 8. Generate random board standard d e v i a t i o n cr- and <7* from the yi. Y best f i t t i n g cumulative d i s t r i b u t i o n f u n c t i o n s of standard d e v i a t i o n f o r cr- and cr- assuming t h a t t h e i r c o r r e l a t i o n c o e f f i c i e n t i s zero (as JC Y shown i n Figure 27). 9. Generate random Z-% and the c o r r e l a t e d random Zy by B i v a r i a t e Standard Normal D i s t r i b u t i o n S i m u l a t i o n Program (BNSIM). Or 35 10. Use Zx as input, generating the c o r r e l a t e d random ZY by B i v a r i a t e Standard Normal D i s t r i b u t i o n S i m u l a t i o n Program (BNSIM). 11. Transform Z% from standard normalized space back to r e a l space with i t s mean equals X and standard d e v i a t i o n equals a - , i . e . JL Xi = X + <r~- Zy i f step 9 has been adopted. 12. Transform Zy from standard normalized space back to r e a l space with i t s mean equals Y and standard d e v i a t i o n equals <T^_ , i . e . Y,. = Y + * y • ZY . The t r a n s f o r m a t i o n between the r e a l space and standard normalized space f o r the d i f f e r e n t d i s t r i b u t i o n s i s shown s c h e m a t i c a l l y i n Figure 28. 5.4 ^--PROFILE SIMULATION EXAMPLE Fo l l o w i n g the above procedures i t i s known th a t given X-, Yj- can be generated from b i v a r i a t e standard normal d i s t r i b u t i o n model with the c o r r e l a t i o n c o e f f i c i e n t pT„. This s e c t i o n w i l l g i v e a s i m u l a t i o n example f o r generating compressive s t r e n g t h < T c - p r o f i l e s i n accordance w i t h c o r r e l a t e d E-p r o f i l e s . In order to implement the b i v a r i a t e standard normal d i s t r i b u t i o n model, the d i s t r i b u t i o n parameters f o r the cumulative d i s t r i b u t i o n f u n c t i o n s of means and standard d e v i a t i o n s f o r MOE and ac are obtained from an experiment. Table 6 shows the d e s c r i p t i v e s t a t i s t i c s f o r the 36 t e s t MOE and <rc f o r a sample of 2x4 2100f-1.8E machine s t r e s s - r a t e d lumber. The cumulative d i s t r i b u t i o n f u n c t i o n s of mean and standard d e v i a t i o n of MOE are shown i n Figures 19 and 20. The cumulative d i s t r i b u t i o n f u n c t i o n s of mean and standard d e v i a t i o n of <JQ are shown i n Figures 12 and 13. From Figures 19, 20, and Figures 12, 13, the random board mean X and Y, and the random board standard d e v i a t i o n cr~ and cr- are X Y generated. A 3-parameter V e i b u l l d i s t r i b u t i o n w i l l be used to f i t the MOE mean and standard d e v i a t i o n d i s t r i b u t i o n s . Normal d i s t r i b u t i o n s are used to f i t compressive s t r e n g t h (CTQ) mean and standard d e v i a t i o n d i s t r i b u t i o n s . F i g u r e 29 shows the experimental r e g r e s s i o n between modulus of e l a s t i c i t y (E) and within-board compressive s t r e n g t h (crc) of the 2100-1.8E grade 2 x 4 MSR lumber. The length of short compression specimen i s 6 inch (152.4 mm). The r e g r e s s i o n equation of the form <TQ - a x E + b obtained with c o e f f i c i e n t s a = 0.0025 and b = 10.32. The c o r r e l a t i o n c o e f f i c i e n t p between t e s t E and ac i s 0.654. The i n d i v i d u a l Xi and Y,- values from Figure 29 were normalized board by board. The c o r r e l a t i o n c o e f f i c i e n t pN f o r Zx and Zy i n the standard normalized space was found to be 0.40. Then, using Zx as in p u t , the c o r r e l a t e d random Zy w i l l be generated by B i v a r i a t e Standard Normal D i s t r i b u t i o n S i m u l a t i o n Program 37 BNSIM) assuming t h a t c o r r e l a t i o n c o e f f i c i e n t pN = 0.40. F i n a l l y , transforming Zy from standard normalized space back to normal space with i t s mean equals Y and standard d e v i a t i o n equals , i . e . Yt = T + <ry • ZY . Table 7 shows the d e s c r i p t i v e s t a t i s t i c s f o r the generated <JC. F i g u r e 30 shows the experimental and generated MOE vs. compressive s t r e n g t h (O-Q) of grade 2100-1.8E. The c o r r e l a t i o n c o e f f i c i e n t pxy between t e s t E and <rc i s 0.654. The generated c o r r e l a t i o n c o e f f i c i e n t Pzt, = 0.649. By v i s u a l check i t can be seen t h a t the t e s t and simulated x y MOE and ac data p a i r s are i n agreement and the c o r r e l a t i o n between MOE and the compressive s t r e n g t h i s preserved. 38 6. GENERATION OF E- AND a- PROFILES FOR GLULAM BEAM ANALYSIS 6.1 INTRODUCTION In Chapter 7, a s i m u l a t i o n program c a l l e d GLULAM w i l l be used to simulate the st r e n g t h of glulam beams. The glulam beams w i l l be f a b r i c a t e d using combinations of three MSR grades (1650fl.5E, 2100f-1.8E, and 2400f-2.0E). The methods described i n Chapter 4 and Chapter 5 to generate E - p r o f i l e s and <T-profiles are used to generate the c o r r e l a t e d MOE vs. compressive str e n g t h and MOE vs. t e n s i l e s t r e n g t h p r o f i l e s . The simulated p r o p e r t i e s f o r t h i s study were developed u s i n g data from a study of 2 x 6 (38 mm x 140 mm) MSR lumber t e n s i l e and compressive s t r e n g t h behaviour provided by COFI ( C o u n c i l of Forest I n d u s t r i e s of B r i t i s h Columbia), which i s summarized i n chapter 1 (Ainsworth, 1989). 6.2 GENERATION OF E-PROFILES FOR THREE GRADES In order to generate l o c a l i z e d E - p r o f i l e s u s i n g the E - f u n c t i o n s i m u l a t i o n model developed i n Chapter 4, t e s t l o c a l i z e d E - p r o f i l e s from grading machine are needed. In t h i s study, E - p r o f i l e s f o r the three grades of MSR lumber were generated u s i n g the E - f u n c t i o n s i m u l a t i o n procedures. Test E - p r o f i l e s f o r three MSR grades were provided from COFI. These data were used to generate E - p r o f i l e s u s i n g 39 techniques d e s c r i b e d i n s e c t i o n s 4.2 and 4.3. Three lengths (16, 21, and 32 f e e t r e s p e c t i v e l y ) of the generated board MOE p r o f i l e s were produced f o r each grade. A t o t a l of 1000 board were generated f o r each length/grade combination. The t o t a l number of generated l o c a l i z e d E-p r o f i l e data p o i n t s on each board are 32, 42, and 64 r e s p e c t i v e l y i n accordance with the three board spans, i. e. the data p o i n t s are spaced a t 6 inches (152.4 mm). A d e s c r i p t i o n of the s t a t i s t i c s f o r the t e s t and generated E-values f o r three grades i s given i n Table 8. For each grade, 1000 boards were generated f o r a n a l y s i s of glulam beam behaviour. 6.3 GENERATION OF <T -PR0FILES FOR COFI DATA The GLULAM beam s i m u l a t i o n program, to be discussed i n the next chapter, r e q u i r e s data f o r l o c a l i z e d t e n s i l e s t r e n g t h p a r a l l e l to g r a i n (crT) , l o c a l i z e d compressive str e n g t h p a r a l l e l to g r a i n (crc) and the c o r r e l a t e d l o c a l i z e d modulus of e l a s t i c i t y (MOE). Using the E-f u n c t i o n s i m u l a t i o n model, l o c a l i z e d E - p r o f i l e s f o r three grades of 2 x 6 SPF have already been generated. In t h i s s e c t i o n , the l o c a l i z e d < r T - p r o f i l e s and a c - p r o f i l e s c o r r e l a t e d with l o c a l i z e d E - p r o f i l e s w i l l be generated using the B i v a r i a t e Standard Normal D i s t r i b u t i o n S i m u l a t i o n Program (BNSIM). The modelling technique developed from short span (within-board) compressive s t r e n g t h t e s t r e s u l t s would be used i n order to generate the l o c a l i z e d <7y-profiles and cr^-prof i l e s c o r r e l a t e d w i t h the 40 l o c a l i z e d E - p r o f i l e s . Together with the three grades of E - p r o f i l e , compressive st r e n g t h ac and t e n s i l e strength (TT t e s t r e s u l t s c o l l e c t e d from COFI, the strength <7-profiles s i m u l a t i o n can be expanded from one 2x4's grade (2100f-1.8E) to the other three 2x6's grades (1650-1.5E, 2100f-1.8E and 2400f-2.0E). The s i m u l a t i o n procedure r e q u i r e s knowledge of the mean and standard d e v i a t i o n of stre n g t h along the length of the t e n s i o n and compression members. The UBC t e s t (see Chapter 2 ) , usin g 152.4 mm (6 inch) long specimens, provides estimates of the mean and standard d e v i a t i o n of the compression s t r e n g t h . For the COFI data there were only two t e s t r e s u l t s a v a i l a b l e (Zone A and Zone B) and these strengths w i l l represent the minimum strengths f o r the s p e c i f i c t e s t zones evaluated. This data would tend to underestimate the a c t u a l mean stre n g t h of the m a t e r i a l . To compensate f o r the underestimate of s t r e n g t h , a st r e n g t h adjustment f a c t o r R was c a l c u l a t e d using the UBC data. The UBC compressive st r e n g t h data were used to simulate the COFI experiment. Zones A and Zone B were s e l e c t e d f o l l o w i n g the procedures of the COFI study. Each 72 inches zone contains 12 continuous segments. The minimum stre n g t h i n the t e s t zone f o r both Zone A and Zone B was determined from the t e s t data. The r a t i o RQ was c a l c u l a t e d f o r each specimen i n the UBC t e s t a ccording to 41 * ° = C , + % ) / 2 < " > where <? = average compressive s t r e n g t h ; <rA = minimum compressive str e n g t h i n Zone A; <7g = minimum compressive str e n g t h i n Zone B. The median value of RQ, c a l c u l a t e d from UBC compression t e s t r e s u l t s , i s approximately 1.15 (see Figure 31). The value R - 1.15 w i l l be used as r a t i o RC and a p p l i e d i n the l a t e r compressive s t r e n g t h c r ^ - p r o f i l e s i m u l a t i o n s . In order to o b t a i n the RT value, the data r e s u l t s from Lam and Varoglu (1991a) within-board t e n s i l e s t r e n g t h t e s t have been analyzed. The median value of RT, c a l c u l a t e d from Lam and Varoglu's t e n s i l e s t r e n g t h t e s t r e s u l t s , was about 1.20 (see Figure 32). The value RT = 1.20 w i l l be used as r a t i o RT and a p p l i e d i n the t e n s i l e s t r e n g t h <rT-p r o f i l e s i m u l a t i o n s . Before s t a r t i n g to simulate the l o c a l i z e d c ^ - p r o f i l e s and c r ^ - p r o f i l e s , the f o l l o w i n g assumptions are made i n order to f a c i l i t a t e the implementations of the program BNSIM: a ) . Assume th a t the d e r i v e d d i s t r i b u t i o n parameters f o r the standard d e v i a t i o n s of grade 2100f-1.8E compressive s t r e n g t h from UBC t e s t r e s u l t s were s u i t a b l e f o r the d i s t r i b u t i o n s of the standard 42 d e v i a t i o n s of the compressive and t e n s i l e s t r e n g t h f o r a l l the MSR grades; b) . Assume t h a t the standard normalized c o r r e l a t i o n c o e f f i c i e n t PN (PN = 0-4) d e r i v e d from UBC t e s t r e s u l t s i s s u i t a b l e f o r compressive str e n g t h and t e n s i l e strength f o r a l l the MSR grades; c) . Assume th a t the R a t i o Factors Rc = 1.15 and RT = 1.20 are s u i t a b l e f o r compressive strength and t e n s i l e s t r e n g t h s i m u l a t i o n model f o r a l l the MSR grades; d) . Assume th a t the w i t h i n board c o r r e l a t i o n c o e f f i c i e n t between the standard d e v i a t i o n s of MOE and compressive str e n g t h crc (or t e n s i l e s t r e n g t h cr^) i s zero and i t i s grade independent. With the above assumptions and the procedures developed i n the s e c t i o n (5.3), l o c a l i z e d t e n s i l e s t r e n g t h p a r a l l e l to g r a i n (o"j.) and l o c a l i z e d compressive strength p a r a l l e l to g r a i n (cc) have been generated i n accordance with the c o r r e l a t e d l o c a l i z e d modulus of e l a s t i c i t y (MOE) . While s i m u l a t i n g the l o c a l i z e d compressive s t r e n g t h O~Q and l o c a l i z e d t e n s i l e strength (T T, a lower bound on board s t r e n g t h was adopted i n order to e l i m i n a t e some values which were out of the range of the minimum compressive strength and t e n s i l e s t r e n g t h obtained from COFI t e s t r e s u l t s (see Table 9). In order to compare the generated l o c a l i z e d t e n s i l e s t r e n g t h and compressive str e n g t h with COFI t e s t r e s u l t s , two minimum t e n s i l e 43 str e n g t h and compressive strength values were s e l e c t e d from Zone A and Zone B i n each generated board. Zone A and Zone B are chosen as desc r i b e d i n Chapter 2. Cumulative d i s t r i b u t i o n f u n c t i o n s of the t e s t and generated t e n s i l e s t r e n g t h and compressive s t r e n g t h of Zone A and Zone B from three grades are provided from Figures 33 to 38. The length of the generated board i n t h i s a n a l y s i s i s 16 f e e t . The comparisons of the CDF of t e s t t e n s i l e s t r e n g t h and compressive str e n g t h w i t h generated t e n s i l e s t r e n g t h and compressive s t r e n g t h show us th a t the strength <7-simulation model can simulate t e n s i l e s t r e n g t h and compressive strength reasonably w e l l . 44 7. GLULAM BEAM SIMULATIONS 7.1 INTRODUCTION The behaviour of any s t r u c t u r e subjected to loads i s dependent on the p r o p e r t i e s of i t s c o n s t i t u e n t s . In the case of glulam beams, the engineering p r o p e r t i e s of the laminations were h i g h l y v a r i a b l e and d i f f i c u l t to p r e d i c t . For t h i s reason, a great deal of research has been ta r g e t e d a t f i n d i n g new ways to determine str e n g t h c h a r a c t e r i s t i c s of glulam beam members. One method of e s t i m a t i n g the s t a t i s t i c a l d i s t r i b u t i o n of glulam beam stre n g t h would be to i n i t i a t e a l a r g e - s c a l e d e s t r u c t i v e t e s t i n g program. An appr o p r i a t e s t a t i s t i c a l d i s t r i b u t i o n could then be f i t t e d to the data from which design s t r e s s l e v e l s could be e s t a b l i s h e d . Assuming t h a t the sample s i z e was s u f f i c i e n t l y l a r ge r e p r e s e n t a t i v e of the p o p u l a t i o n , t h i s approach would y i e l d accurate r e s u l t s . However, the cost a s s o c i a t e d w i t h a t e s t i n g program of t h i s magnitude would be p r o h i b i t i v e . A computer model could e l i m i n a t e p a r t of the need f o r l a r g e -s c a l e d e s t r u c t i v e t e s t i n g . This s e c t i o n discusses a computer s i m u l a t i o n model to p r e d i c t the performance of s t r u c t u r a l g l u e -laminated beams, f a b r i c a t e d from r e g u l a r grades of MSR lumber, under un i f o r m l y d i s t r i b u t e d loads. 45 7.2 GLULAM SIMULATION MODEL This computer model, GLULAM, was o r i g i n a l l y developed by Drs. Foschi and B a r r e t t (1980), who used a f i n i t e element approach to estimate the p r o b a b i l i t y d i s t r i b u t i o n of stre n g t h and s t i f f n e s s of glulam beams. The GLULAM model uses a Monte Carlo s i m u l a t i o n technique to compute the bending, t e n s i l e and compressive strengths of the glulam beams. In t h i s manner, any beam s i z e or layup can be e a s i l y analyzed. A b r i e f d e s c r i p t i o n of beam s i m u l a t i o n i n t h i s model f o l l o w s : Beam layups are chosen f o r the GLULAM model i n the same manner th a t a c t u a l beams are assembled i n a lamina t i n g p l a n t . Laminations are s e l e c t e d according to grade requirements s p e c i f i e d by the beam layup. One beam length lamination i s made up of s e v e r a l pieces of l a m i n a t i n g lumber connected by e n d - j o i n t s . The length of each piece of lumber i n each lam i n a t i o n i s randomly generated. The program i n d i c a t e s the number and l o c a t i o n of e n d - j o i n t s chosen f o r each l a m i n a t i o n . Each beam length l a m i n a t i o n c o n s i s t s of a s e r i e s of 152.4 mm (6 inch) long elements. Each element i n a p a r t i c u l a r piece of lumber i s assigned a modulus of e l a s t i c i t y (MOE), a t e n s i l e s t r e n g t h (o"y) , a n <^ a compressive str e n g t h ( c c ) . The model i s based on the observation t h a t glulam beams behave e l a s t i c a l l y to f i r s t f a i l u r e and th a t most bending f a i l u r e s of glulam beams i n i t i a t e i n the te n s i o n zone. The f a i l u r e s t r e n g t h of the beams 46 was c a l c u l a t e d u s i n g the weakest l i n k theory. F a i l u r e i s assumed to occur when the t e n s i l e s t r e n g t h p a r a l l e l to the g r a i n of any element i n a la m i n a t i o n i s exceeded. This i s r e f e r r e d to as " f i r s t f a i l u r e " . The element s t r e s s e s are c a l c u l a t e d a t the centre of the l a m i n a t i o n . The subsequent p r o g r e s s i v e f a i l u r e s could a l s o be expected f o l l o w i n g the f i r s t f a i l u r e i n GLULAM program. The higher u l t i m a t e loads a s s o c i a t e d w i t h p r o g r e s s i v e f a i l u r e were ignored i n t h i s study. A b r i e f o u t l i n e of the procedures used f o r the GLULAM s i m u l a t i o n program f o l l o w s : 1) . The elements i n each lam i n a t i o n of the beam are assigned i n d i v i d u a l l y a MOE, a compressive str e n g t h (<r c), and a t e n s i l e s t r e n g t h ((TT) values a p p r o p r i a t e f o r the lami n a t i o n grade; 2) . The length of each piece of lumber i n each l a m i n a t i o n i s randomly generated and the en d - j o i n t l o c a t i o n s are determined and the en d - j o i n t strengths are assigned throughout the beam; 3) . F i n i t e element a n a l y s i s i s used to p r e d i c t f a i l u r e i n the te n s i o n zone. 4) . The u l t i m a t e bending s t r e s s i s recorded and the above steps are repeated u n t i l s u f f i c i e n t data have been generated. 5) . The s t a t i s t i c a l d i s t r i b u t i o n of bending str e n g t h i s then estimated u s i n g data from step 4. 47 7.3 SIMULATION BEAM LAYUPS Strength and s t i f f n e s s values of glulam are dependent on the grade layup of glulam beams, i.e. the grade and l a m i n a t i o n p r o p e r t i e s of the outer and inner laminations. For bending a p p l i c a t i o n s the outer laminations are of a higher grade than the inner l a m i n a t i o n s . The outer laminations g e n e r a l l y c o n t r o l the bending s t r e n g t h of the glulam beam. T e n s i l e and compressive strengths p a r a l l e l to g r a i n of glulam beams may be c o n t r o l l e d by e i t h e r the inner or the outer l a m i n a t i o n p r o p e r t i e s . Three depths of glulam beams have been evaluated. These depths are 9, 12, and 18 i n c h , r e s p e c t i v e l y . The r a t i o between span and depth i s 21. Therefore the three spans a s s o c i a t e d w i t h the three depths are 16, 21, and 32 f e e t , r e s p e c t i v e l y . To d e f i n e the depth of each beam (which c o n t r o l s the number of laminations i n each design combination), a uniform t h i c k n e s s of 1.5 inch (38 mm) was assumed f o r each l a m i n a t i o n . For 9, 12, and 18 inch depth beam, the r e q u i r e d number of laminations are 6, 8, 12, r e s p e c t i v e l y . Glulam beams with pure and mixed grade combinations were s t u d i e d f o r each of the three beam depths are shown s c h e m a t i c a l l y i n Figures 39, 40, and 41. In Figure 39, A, D and G are the pure grade combinations of 2400f-2.0E, 1650f-1.8E and 2100f-1.5E, r e s p e c t i v e l y ; B and C are the mixed combinations with grade 2400f-2.0E on the outer l a y e r s and 1650f-1.5E i n the inner l a y e r s ; E and F are the mixed combinations with grade 2100f-2.0E on the outer l a y e r s and 1650f-1.5E i n the inner 48 l a y e r s . In Fi g u r e 40, A, E and H are the pure grade combinations of 2400f-2.0E, 1650f-1.8E and 2100f-1.5E, r e s p e c t i v e l y ; B, C and D are the mixed combinations w i t h grade 2400f-2.0E on the outer l a y e r s and 1650f-1.5E i n the inner l a y e r s ; F and G are the mixed combinations w i t h grade 2100f-2.0E on the outer l a y e r s and 1650f-1.5E i n the inner l a y e r s . In Figure 41, A, F and I are the pure grade combinations of 2400f-2.0E, 1650f-1.8E and 2100f-1.5E, r e s p e c t i v e l y ; B, C, D and E are the mixed combinations with grade 2400f-2.0E on the outer l a y e r s and 1650f-1.5E i n the inner l a y e r s ; G and H are the mixed combinations with grade 2100f-2.0E on the outer l a y e r s and 1650f-1.5E i n the inner l a y e r s . In s h o r t , three beam grades and three beam s i z e s were v a r i o u s l y combined i n the s i m u l a t i o n design of t h i s study, r e s u l t i n g i n 24 d i f f e r e n t beam groups. 7.4 SIMULATION RESULTS The grade combinations f o r the three depths are 7, 8, and 9, r e s p e c t i v e l y (see Figures 39, 40, and 41). One hundred beam s i m u l a t i o n r e p l i c a t e s were chosen f o r each one of the 24 beam groups, p r o v i d i n g a t o t a l of 2400 simulated beams (see Table 10). The o b j e c t i v e of t h i s study was to assess the performance of MSR lumber, thus, the stre n g t h of e n d - j o i n t s i n each beam has been assigned s u f f i c i e n t l y high i n order to e l i m i n a t e the p o s s i b i l i t y of e n d - j o i n t f a i l u r e s . 49 The r e s u l t s of beam s i m u l a t i o n s as portrayed by the three-parameter Wei b u l l d i s t r i b u t i o n parameters and t h e i r a s s o c i a t e d 5th and 50th p e r c e n t i l e values f o r the three beam depths are shown i n Tables 11, 12 and 13, r e s p e c t i v e l y . In Table 11, B9A-B represents bending s t r e n g t h of combination A i n 9" depth, B9A-C represents compressive s t r e n g t h of combination A i n 9" depth, B9A-T represents t e n s i l e s t r e n g t h of combination A i n 9" depth, and so on. S i m i l a r r e p r e s e n t a t i o n s are shown i n Tables 12 and 13. 50 8. DISCUSSION OF SIMULATION RESULTS 8.1 INTRODUCTION This chapter w i l l d i s c u s s the r e s u l t s obtained from GLULAM beam st r e n g t h s i m u l a t i o n s . The e f f e c t s of grades, depths and the combinations of beam layups on the beam p r o p e r t i e s , such as the MOE, bending s t r e n g t h , compressive str e n g t h p a r a l l e l to g r a i n and t e n s i l e s t r e n g t h p a r a l l e l to g r a i n , w i l l be analyzed. Since the r a t i o of the length and the depth of a l l the simulated glulam beams has the same value, 21, i n t h i s study, the depth e f f e c t might be b e t t e r c a l l e d s i z e e f f e c t . Here we use the term depth e f f e c t j u s t f o r the purpose of convenience. The simulated modulus of e l a s t i c i t y and bending s t r e n g t h values w i l l be compared to r e s u l t s of the transformed cross s e c t i o n a n a l y s i s . Figure 42 i s an example diagram of one of 24 beam combinations which shows the transformed s e c t i o n and s t r e s s d i s t r i b u t i o n . The other 23 combinations w i l l have the s i m i l a r transformed cross s e c t i o n s and s t r e s s d i s t r i b u t i o n s . 8.2 MODULUS OF ELASTICITY The average simulated MOE values f o r each beam layup can be compared with the average transformed MOE val u e s , c a l c u l a t e d by usi n g 51 a transformed cross s e c t i o n a n a l y s i s (see Figure 42) and the nominal MOE f o r each grade of lumber i n the beam combination (Table 14). The nominal MOE values f o r grade 1650f-1.5E, 2100f-1.8E, and 2400f-2.0E used i n the transformed s e c t i o n a n a l y s i s are 1 . 5 x l 0 6 p s i , 1 . 8 x l 0 6 p s i , and 2 . 0 x l 0 6 p s i , r e s p e c t i v e l y . The equations used f o r c a l c u l a t i n g transformed bending MOE values are presented i n the f o l l o w i n g s e c t i o n s . The beam s t i f f n e s s can be determined as (see Figure 42): EI = E y I t ( 8.1 ) where E = apparent MOE; I = apparent moment of i n e r t i a , b^(2t^ + t c ) 3 / 12; E^ = face (or outer) l a y e r lamination MOE; It = transformed cross s e c t i o n moment of i n e r t i a . Since the face l a y e r l a m i n a t i o n Ey i s s e l e c t e d as the r e f e r e n c e , the width of the core l a y e r l a m i n a t i o n b c w i t h i t s E c can be r e l a t e d w i t h the width of the face l a y e r lamination b^ with i t s E^ usi n g the f o l l o w i n g equation: The moment of i n e r t i a of the transformed cross s e c t i o n I ( i s c a l c u l a t e d from the f o l l o w i n g equation: 52 hf(2tf+tcf t c 3 ( b / - b c )  Lt - 12 12 <• 8 > d ' The r a t i o of the transformed moment of i n e r t i a to the apparent moment of i n e r t i a can be determined: L E / ( 2 y t e ) 3 - tc3(E/-Ec) I - E / ( 2 V t c ) 3 { 8 ' 4 } This r a t i o w i l l be c a l l e d the transformed s e c t i o n f a c t o r and denoted by T (Moody, 1974). Then, from Eq. (8.1) E can be c a l c u l a t e d from the f o l l o w i n g equation: EfV E / ( 2 t / + t c ) 3 - t c 3 ( E f - E c ) E = Jf- = — i — Z L vc_v f cJ ( 8.5 ) I 2 t / + t c ) 3 As Table 14 shows, the average simulated MOE values f o r the 24 beam layup groups were a l l w i t h i n 2% of t h e i r transformed MOE values (w i t h the average value of 0.4% h i g h e r ) , which v e r i f i e d the GLULAM program works w e l l to simulate the s t i f f n e s s p r o p e r t i e s of the glulam beams. Figure 43 compares the average simulated MOE values w i t h average transformed MOE values c a l c u l a t e d by using a transformed s e c t i o n a n a l y s i s f o r the twenty-four (24) beam layup groups. The average 53 simulated MOE values f o r the twenty-four (24) groups ranged from 1 . 4 7 8 x l 0 6 p s i to 2.054 x 10 6 p s i , while the transformed MOE values ranged, from 1 . 5 x l 0 6 p s i to 2.0 x l O 6 p s i (nominal v a l u e s ) . A r e g r e s s i o n a n a l y s i s suggested a l i n e of best f i t as: Y = 0.2276 + 0.8707X ( 8.6 ) where Y i s the simulated MOE and X i s the transformed MOE, both i n terms of m i l l i o n l b / i n 2 ( p s i ) . The c o e f f i c i e n t of determination (-R2) was 0.997. 8.3 BENDING STRENGTH P r e d i c t a b i l i t y of bending str e n g t h can be measured by comparing the p r e d i c t e d bending str e n g t h with the simulated bending s t r e n g t h (Table 15). The procedures of c a l c u l a t i n g the p r e d i c t e d bending s t r e n g t h w i l l be described i n the f o l l o w i n g s . For a two-zone beam (Figure 42), the transformed s e c t i o n f a c t o r , T, can be expressed as i n Eq. (8.4): E f ( 2 t / + t c ) 3 - t c 3 ( E f - E c ) T = -Z±—f—^- % V / c ( 8.7 ) E / ( 2 t / + t c ) 3 where: E^ and E c = moduli of e l a s t i c i t y f o r the face zone and core zone shown i n Figure 42; 54 t ^ and t c = depths shown i n Figure 42. The 5th p e r c e n t i l e MOR values f{ f o r grades of 1650f-1.5E, 2100f-1.8E and 2400f-2.0E are 3465, 4410 and 5040 p s i r e s p e c t i v e l y (obtained from "SPS 2 NLGA S p e c i a l Products Standard f o r Machine S t r e s s Rated Lumber"). In order to avoi d inner l a m i n a t i o n over s t r e s s e s i n the depths t c (ASTM, 1990): * * ( HThr-) ( i j ) 4 < 8 - 8 > I f f2 i s l e s s than the q u a n t i t y c a l c u l a t e d f o r the r i g h t s i d e of the equation (8.9), /x i s l i m i t e d to a value t h a t w i l l s a t i s f y an e q u a l i t y . For use of with p r o p e r t i e s of the simulated p h y s i c a l s e c t i o n , f-y can be m u l t i p l i e d by T to y i e l d a value of a l l o w a b l e combination bending s t r e s s , or the p r e d i c t e d bending s t r e n g t h , / : f= AT ( 8.9 ) In order to account f o r the e f f e c t of depth a s i z e e f f e c t (12/ef) 1/ 9 was m u l t i p l i e d to the al l o w a b l e combination bending s t r e s s / where d i s the beam depth (ASTM, 1990). I t i s shown t h a t (see Table 15) most of the simulated bending s t r e n g t h from the 24 beam layup groups were w i t h i n 30% higher than the p r e d i c t e d bending s t r e n g t h except f o r the pure grade combinations by grade 1650f-1.5E. The simulated bending strengths i n those combinations a t t a i n e d only 85, 76 and 71 percent of the p r e d i c t e d 55 values a t the depths of 9, 12 and 18 i n c h , r e s p e c t i v e l y . Table 16 shows the hypothesis t e s t s of e q u a l i t y of means f o r bending s t r e n g t h of twenty-four (24) beam layup groups. I t i n d i c a t e d t h a t beam lay-up has a s i g n i f i c a n t e f f e c t on bending s t r e n g t h s . Within the beam layups of 9 inch depth, i t showed t h a t the d i f f e r e n c e s between beam9-A and beam9-B, and beam9-F and beam9-G are not s i g n i f i c a n t a t a = 0.05 l e v e l . This was expected because the outer l a y e r laminations (grades 2100f-1.8E and 2400f-2.0E) c o n t r i b u t e most to the beam bending s t r e n g t h , whereas the inner l a y e r laminations (grade 1650f-1.5E) do not have much i n f l u e n c e on the beam bending s t r e n g t h . S i m i l a r phenomena can a l s o be found i n the beam layups of 12 inch depth and 18 inch depth. The beam layup r e p r e s e n t a t i o n s i n Table 16 are the same as i n Tables 11, 12, 13 and Figures 39, 40, and 41. 8.3.1 GRADE EFFECT Figures 44 to 46 are the cumulative d i s t r i b u t i o n f u n c t i o n s ( c d f ) of bending str e n g t h of three grades (1650f-1.5E, 2100f-1.8E and 2400f-2.0E) under three depths (9, 12, and 18 in c h e s ) . Figures 47 to 48 show the v a r i a t i o n i n 5th p e r c e n t i l e and 50th p e r c e n t i l e values of bending strength of three grades (1650f-1.5E, 2100f-1.8E and 2400f-2.0E) with depths. C l e a r l y i n d i c a t e d from Table 16 and Figur e s 44 to 48, the 56 bending strengths of the pure grade combinations, made up of 1650f-1.5E, 2100f-1.8E, and 2400f-2.0E lumber r e s p e c t i v e l y , are d i f f e r e n t s i g n i f i c a n t l y a t the 0.05 l e v e l of s i g n i f i c a n c e . 8.3.2 DEPTH EFFECT Figures 49 to 51 are the cumulative d i s t r i b u t i o n f u n c t i o n s ( c d f ) of bending s t r e n g t h of three depths (9, 12, and 18 inches) f o r the three grades (1650f-1.5E, 2100f-1.8E and 2400f-2.0E). Figures 52 to 53 show the v a r i a t i o n of the 5th p e r c e n t i l e and 50th p e r c e n t i l e values of bending strength of three depths (9", 12", and 18") as a f u n c t i o n of grade. From Table 16 and Figures 49 to 53, the bending strengths of pure grade combinations, made up of 1650f-1.5E, 2100f-1.8E, and 2400f-2.0E r e s p e c t i v e l y , are s i g n i f i c a n t l y d i f f e r e n t ( a t a = 0.05 l e v e l ) a t the three beam depths (9, 12 and 18 inc h e s ) . 8.3.3 EFFECT OF MIXED GRADE LAYUPS Twenty-four d i f f e r e n t beam combinations, which can be d i v i d e d i n t o two groups, were assigned. One group was the combinations among grades 1650f-1.5E and 2100f-1.8E, i n which grade 2100f-1.8E i s used f o r the outer laminations and grade 1650f-1.5E f o r the inner laminations. The other group was the combinations among grades 1650f-1.5E and 2400f-2.0E, i n which grade 2400f-2.0E i s used f o r the outer laminations and grade 1650f-1.5E f o r the inner laminations. The e f f e c t 57 of mixed grade layups on the bending strength f o r the two groups w i l l be discussed s e p a r a t e l y i n the f o l l o w i n g s e c t i o n s . 8.3.3.1 1650f-1.5E AND 2100f-1.8E COMBINATIONS Figures 54 to 56 are the cumulative d i s t r i b u t i o n f u n c t i o n s ( c d f ) of bending str e n g t h w i t h the combinations of grades 1650f-1.5E and 2100f-1.8E under three depths (9, 12, and 18 in c h e s ) . Figures 57 to 58 are the 5th p e r c e n t i l e and 50th p e r c e n t i l e values of bending str e n g t h as a f u n c t i o n of the outer l a y e r percent of 2100fl-1.8E. G e n e r a l l y speaking, the 5th and 50th p e r c e n t i l e values of bending str e n g t h increased with i n c r e a s i n g outer l a y e r percent of grade 2100f-1.8E (see Figure 57 and Figure 58). From Table 16, Figure 39 to Figure 41, i t shows th a t the f o l l o w i n g group t e s t s of bending strength mean values are not s i g n i f i c a n t l y d i f f e r e n t a t the 0.05 l e v e l of s i g n i f i c a n c e : BEAM9-F vs. BEAM9-G (depth - 9 inches) and BEAM18-H vs. BEAM18-I (depth = 18 inches).That means as the outer l a y e r percent of grade 2100f-1.8E increased, the bending strength would not be improved too much a t those two beam depth groups (see Figures 54 to 56). 8.3.3.2 1650f-1.5E AND 2400f-2.0E COMBINATIONS Figures 59 to 61 are the cumulative d i s t r i b u t i o n f u n c t i o n s ( c d f ) 58 of bending s t r e n g t h w i t h the combinations of grades 1650f-1.5E and 2400f-2.0E under three depths (9, 12, and 18 i n c h e s ) . F i g u r e s 62 to 63 are the 5th p e r c e n t i l e and 50th p e r c e n t i l e values of bending str e n g t h as a f u n c t i o n of the outer l a y e r percent of 2400fl-2.0E. S i m i l a r as i n the above s e c t i o n , the 5th and 50th p e r c e n t i l e values of bending str e n g t h increased w i t h i n c r e a s i n g outer l a y e r percent of grade 2400f-2.0E (see Figures 62 and 63). From Table 16, Figures 49 to 51 and Figures 59 to 61, i t shows t h a t the f o l l o w i n g t e s t of bending str e n g t h mean values are not s i g n i f i c a n t l y d i f f e r e n t a t the 0.05 l e v e l of s i g n i f i c a n c e : BEAM9-A vs. BEAM9-B (depth = 9 i n c h e s ) , BEAM12-A vs. BEAM12-B, BEAM12-B vs. BEAM12-C (depth = 12 i n c h e s ) , BEAM18-A vs. BEAM18-B, BEAM18-B vs. BEAM18-C, and BEAM18-A vs. BEAM18-C (depth = 18 in c h e s ) . That was because t h a t the outer l a y e r laminations (grades 2400f-2.0E) c o n t r i b u t e most to the beam bending s t r e n g t h , whereas the inner l a y e r laminations (grade 1650f-1.5E) do not have much i n f l u e n c e on the beam bending s t r e n g t h . As discussed i n the l a s t two s e c t i o n s , we found t h a t as long as the inner laminations (grade 1650f-1.5E) was w i t h i n 50% of the t o t a l depth the beam bending str e n g t h would not be i n f l u e n c e d adversely. I t i s i n t e r e s t i n g to see t h a t , w i t h one l a y e r of grade 2100f-1.8E or 2400f-2.0E on the outer l a y e r of the beam and the r e s t inner 59 l a y e r of grade 1650f-1.5E, the beam bending s t r e n g t h can be increased 197. or 59% (from 5287 x 10 6 p s i to 6293 x l O 6 p s i or 8406 x 10 6 p s i ) r e s p e c t i v e l y when the beam depth i s 9 inch. The beam bending s t r e n g t h can be increased 35% or 61% (from 4384 x 10 6 p s i to 5902 x 10 6 p s i or 7 0 6 4 x l O 6 p s i ) r e s p e c t i v e l y when the beam depth i s 12 in c h . With two l a y e r of grade 2100f-1.8E or 2400f-2.0E on the outer l a y e r of the beam and the r e s t inner l a y e r of grade 1650f-1.5E, the beam bending s t r e n g t h can be increased 46% or 76% (from 3771 x 10 6 p s i to 5492 x l O 6 p s i or 6655 x l O 6 p s i ) r e s p e c t i v e l y when the beam depth i s 18 in c h . The above r e s u l t s i n d i c a t e d t h a t w i t h one or two l a y e r s of higher grade lam i n a t i o n on the outer l a y e r of the beam and lower grade laminations i n the r e s t of inner l a y e r s of the beam, the glulam beam bending s t r e n g t h could be improved s i g n i f i c a n t l y . 8.3.4. COMPARISON OF BENDING STRENGTH FOR GLULAM From c u r r e n t glulam standard (CAN3-086-M84), lodgepole pine-spruce and D. f i r - l a r c h glulam beams with v i s u a l l y graded l a m i n a t i o n , grade 20f-EX and 24f-EX, have the assigned maximum bending s t r e s s f 6 = 2000 psi and f 6 = 2400 psi. The corresponding 5th p e r c e n t i l e values are 2.1x2000 = 4200 psi and 2.1x2400 = 5040 psi, r e s p e c t i v e l y (see Figures 57 and 62, h o r i z o n t a l l i n e ) . In F igure 57, i t i s shown th a t the combinations BEAM9-E, BEAM12-G and BEAM18-H (see Figures 39, 40 and 41), which are made up of 33%, 50% and 66% grade 2100f-1.8E with corresponding depths of 9, 12 and 18 60 inches r e s p e c t i v e l y , w i l l produce higher bending s t r e n g t h values than grade 20f-EX f b . I f the depth i s 9 inches, the combination BEAM9-F, which i s made up of 66% grade 2100f-1.8E, can even produce higher bending s t r e n g t h values than grade 24f-EX f f c. S i m i l a r l y , In Figure 62, i t i s shown t h a t the combinations BEAM9-C, BEAM12-C and BEAM18-D (see Figures 39, 40 and 41), which are made up of 33%, 50% and 50% grade 2400f-2.0E with corresponding depths of 9, 12 and 18 inches r e s p e c t i v e l y , w i l l produce higher bending s t r e n g t h values than grade 24f-EX ff,. The s i m u l a t i o n r e s u l t s show th a t the Spruce - P i n e - F i r MSR lumber can be used to produce layups which achieve the f 6 = 2000 psi and f 6 = 2400 psi design s t r e s s l e v e l . 8.4 TENSILE STRENGTH Table 17 shows the hypothesis t e s t s of e q u a l i t y of means f o r t e n s i l e s t r e n g t h p a r a l l e l to g r a i n of twenty-four (24) beam layup groups. I t i n d i c a t e d t h a t beam layup has a s i g n i f i c a n t e f f e c t on mean t e n s i l e strengths . 8.4.1 GRADE EFFECT Figures 64 to 66 are the cumulative d i s t r i b u t i o n f u n c t i o n s ( c d f ) of t e n s i l e s t r e n g t h of three grades (1650f-1.5E, 2100f-1.8E and 2400f-2.0E) under three depths (9, 12, and 18 in c h e s ) . 61 Figures 67 to 68 show the v a r i a t i o n of the 5th p e r c e n t i l e and 50th p e r c e n t i l e values of t e n s i l e s t r e n g t h of three grades (1650f-1.5E, 2100f-1.8E and 2400f-2.0E) as a f u n c t i o n of depths. I t i n d i c a t e d from Table 17 and Figures 64 to 68 th a t a l l of the t e n s i l e strengths of the pure grade combinations, made up of 1650f-1.5E, 2100f-1.8E, and 2400f-2.0E lumber r e s p e c t i v e l y , are d i f f e r e n t s i g n i f i c a n t l y a t the 0.05 l e v e l of s i g n i f i c a n c e . 8.4.2 DEPTH EFFECT Figure s 69 to 71 are the cumulative d i s t r i b u t i o n f u n c t i o n s ( c d f ) of t e n s i l e s t r e n g t h of three depths (9, 12, and 18 inches) under three grades (1650f-1.5E, 2100f-1.8E and 2400f-2.0E). Figures 72 to 73 are the 5th p e r c e n t i l e and 50th p e r c e n t i l e values of t e n s i l e s t r e n g t h of three depths (9, 12, and 18 inches) as a f u n c t i o n of grades. From Table 17 and Figures 69 to 73, i t shows t h a t the t e n s i l e s t r e n g t h mean values of the pure grade combinations, made up of 1650f-1.5E, 2100f-1.8E, and 2400f-2.0E r e s p e c t i v e l y , are s i g n i f i c a n t l y d i f f e r e n t ( a t or = 0.05 l e v e l ) with three beam depths (9, 12 and 18 in c h e s ) . Figure 69 and f i g u r e 72 show t h a t the 5th p e r c e n t i l e values of grade 1650f-1.5E are very c l o s e among the three depths. 62 8.4.3 EFFECT OF MIXED GRADE LAYUPS The mixed grade beam layups, shown i n Figure 49 to Figure 51, have two kinds of grade combinations among the three grades. That i s grade 1650f-1.5E combined with grade 2100f-1.8E or with grade 2400f-2.0E. I t i s discussed i n the f o l l o w i n g s e c t i o n . F i g u r e s 74 to 76 are the cumulative d i s t r i b u t i o n f u n c t i o n s ( c d f ) of t e n s i l e s t r e n g t h w i t h the combinations of grades 1650f-1.5E and 2100f-1.8E under three depths (9, 12, and 18 i n c h e s ) . Figures 77 to 79 are the cumulative d i s t r i b u t i o n f u n c t i o n s ( c d f ) of t e n s i l e s t r e n g t h with the combinations of grades 1650f-1.5E and 2400f-2.0E under three depths (9, 12, and 18 i n c h e s ) . Figures 80 to 81 are the 5th p e r c e n t i l e and 50th p e r c e n t i l e values of t e n s i l e s t r e n g t h as a f u n c t i o n of the outer l a y e r percent of grade 2100fl-1.8E. F i g u r e s 82 to 83 are the 5th p e r c e n t i l e and 50th p e r c e n t i l e values of t e n s i l e s t r e n g t h as a f u n c t i o n of the outer l a y e r percent of grade 2400fl-2.0E. Although the 5th and 50th p e r c e n t i l e values of t e n s i l e s t r e n g t h increased with i n c r e a s i n g outer l a y e r percent of grade 2100f-1.8E and 2400f-2.0E (see Figures 74 to 83), the amount of increase suddenly went up when the outer l a y e r percent of grade 2100f-1.8E and 2400f-2.0E tend towards 100 percent. This i s because t h a t t e n s i o n p a r a l l e l to g r a i n high s t r e s s e s are d i s t r i b u t e d across the t o t a l c r o s s - s e c t i o n . 63 The t e n s i l e s t r e n g t h i s dependent on the stre n g t h p r o p e r t i e s of a l l l a m i n a t i o n s , and to a c e r t a i n degree the weakest laminations c o n t r o l the s t r e n g t h . Therefore, any lower grade laminations i n the centre of the beam could reduce the beam t e n s i l e strength d r a m a t i c a l l y . 8.5 COMPRESSIVE STRENGTH Table 18 shows the hypothesis t e s t s of e q u a l i t y of means f o r compressive s t r e n g t h p a r a l l e l to g r a i n of twenty-four (24) beam layup groups. I t i n d i c a t e d t h a t the beam layup has a s i g n i f i c a n t e f f e c t on the mean compressive strengths. 8.5.1 GRADE EFFECT Figure s 84 to 86 are the cumulative d i s t r i b u t i o n f u n c t i o n s ( c d f ) of compressive str e n g t h of three grades (1650f-1.5E, 2100f-1.8E and 2400f-2.0E) under three depths (9, 12, and 18 in c h e s ) . F i g u r e s 87 to 88 are the 5th p e r c e n t i l e and 50th p e r c e n t i l e values of compressive strength of three grades (1650f-1.5E, 2100f-1.8E and 2400f-2.0E) as a f u n c t i o n of depths. I t i n d i c a t e d from Table 18 and Figures 84 to 88 t h a t a l l of the compressive strengths of the pure grade combinations, made up of 1650f-1.5E, 2100f-1.8E, and 2400f-2.0E r e s p e c t i v e l y , are d i f f e r e n t s i g n i f i c a n t l y a t the 0.05 l e v e l of s i g n i f i c a n c e . 64 8.5.2 DEPTH EFFECT Figure s 89 to 91 are the cumulative d i s t r i b u t i o n f u n c t i o n s ( c d f ) of compressive str e n g t h of three depths (9, 12, and 18 inches) under three grades (1650f-1.5E, 2100f-1.8E and 2400f-2.0E). Figures 92 to 93 are the 5th p e r c e n t i l e and 50th p e r c e n t i l e values of compressive str e n g t h of three depths (9, 12, and 18 inches) as a f u n c t i o n of grades. From Table 18 and Figures 89 to 93, i t shows t h a t the compressive str e n g t h mean values of pure grade combinations, made by 1650f-1.5E, 2100f-1.8E, and 2400f-2.0E r e s p e c t i v e l y , are s i g n i f i c a n t l y d i f f e r e n t ( a t a = 0.05 l e v e l ) with three beam depths (9, 12 and 18 inc h e s ) . 8.5.3 EFFECT OF MIXED GRADE LAYUPS The mixed grade beam layups, shown i n Figure 49 to Figure 51, have two kinds of grade combinations among the three grades. That i s grade 1650f-1.5E combined with grade 2100f-1.8E or with grade 2400f-2.0E. I t i s discussed i n the f o l l o w i n g s e c t i o n . Figures 94 to 96 are the cumulative d i s t r i b u t i o n f u n c t i o n s ( c d f ) of compressive str e n g t h with the combinations of grades 1650f-1.5E and 2100f-1.8E under three depths (9, 12, and 18 in c h e s ) . F i g u r e s 97 to 99 are the cumulative d i s t r i b u t i o n f u n c t i o n s ( c d f ) of compressive str e n g t h with the combinations of grades 1650f-1.5E and 65 2400f-2.0E under three depths (9, 12, and 18 i n c h e s ) . F i g u r e s 100 to 101 are the 5th p e r c e n t i l e and 50th p e r c e n t i l e values of compressive s t r e n g t h as a f u n c t i o n of the outer l a y e r percent of 2100fl-1.8E. Figures 102 to 103 are the 5th p e r c e n t i l e and 50th p e r c e n t i l e values of compressive s t r e n g t h as a f u n c t i o n of the outer l a y e r percent of 2400fl-2.0E. The 5th and 50th p e r c e n t i l e values of compressive s t r e n g t h increased w i t h i n c r e a s i n g outer l a y e r percent of grade 2100f-1.8E and 2400f-2.0E (see Figures 71 to 80). The amount of increase was not suddenly changed as i n the case of t e n s i l e s t r e n g t h r a t h e r a very smooth increase occurred when the outer l a y e r percent of grade 2100f-2.8E and 2400f-2.0E tend towards 100 percent. 66 9. SIZE EFFECTS ANALYSIS 9.1 INTRODUCTION In t h i s chapter, the s i z e e f f e c t s i n bending, t e n s i l e and compressive strengths have been considered. S i z e e f f e c t i n lumber have been observed by Madsen from bending t e s t performed a t d i f f e r e n t lengths, f o r constant depths and the same load i n g p a t t e r n . Madsen concluded t h a t the observed s i z e dependence could be e x p l a i n e d by the changes i n length, and proposed an a d j u s t i n g f a c t o r based on the a p p l i c a t i o n of Weibull weakest l i n k theory. Thus, lengths and i 2 - The parameter k was found to be approximately 4.2 by c a l i b r a t i o n of Eq. (9.1) to Madsen's experimental data. In recent years, the use of volume-effect f a c t o r i s g e t t i n g more and more popular. This volume-effect f a c t o r accounts f o r a l l three parameters of volume: width, depth and length. For glulam beams of equal width subjected to the same loa d i n g c o n f i g u r a t i o n , the strengths and <T2 may take the form of the f o l l o w i n g equation a t a ( 9-1 ) where o-y and <r2 are the bending strengths corresponding to the 67 p a r t i c u l a r p r o b a b i l i t y l e v e l : zi (Yi\l/k <?2 ~ \ V1 ) ( 9.2 ) where V-^ and V2 are the beam volumes, D1, D2, Wlf W2, L\ and L2 are the beam depths, widths and lengths. 9.2 SIZE EFFECTS IN BENDING STRENGTH Glulam beam stre n g t h s i m u l a t i o n undertaken us i n g a constant r a t i o of span to depth, L/D = 21. Q The s i z e parameter F^ = ^ i n bending str e n g t h i s c a l c u l a t e d f o r three grades (1650f-1.5E, 2100f-1.8E and 2400f-2.0E) and two p r o b a b i l i t y l e v e l s (5th p e r c e n t i l e and 50th p e r c e n t i l e ) (Table 19). 9.3 SIZE EFFECTS IN TENSILE STRENGTH The s i z e parameter F t i n t e n s i l e strength i s c a l c u l a t e d f o r three grades (1650f-1.5E, 2100f-1.8E and 2400f-2.0E) and two p r o b a b i l i t y l e v e l s (5th p e r c e n t i l e and 50th p e r c e n t i l e ) (Table 20). Since V,- = Z?t- Wi Li and the simulated beams have constant length to depth r a t i o s {i.e. L^- C C i s a co n s t a n t ) , then ( 9.3 ) 68 9.4 SIZE EFFECTS IN COMPRESSIVE STRENGTH The s i z e parameter F g c i n compressive s t r e n g t h i s c a l c u l a t e d f o r three grades (1650f-1.5E, 2100f-1.8E and 2400f-2.0E) and two p r o b a b i l i t y l e v e l s (5th p e r c e n t i l e and 50th p e r c e n t i l e ) (Table 21). 69 10. CONCLUSIONS AND RECOMMENDATIONS 10.1 CONCLUSIONS A group of 2 x 6 2100f-1.8E SPF MSR lumber has been t e s t e d to o b t a i n the within-board compressive strengths. The t e s t E - p r o f i l e s and compressive s t r e n g t h data were used to provide the s t a t i s t i c a l i n f o r m a t i o n f o r E - s i m u l a t i o n model and <r-simulation model. An E - s i m u l a t i o n model and a u-si m u l a t i o n model are demonstrated to simulate l o c a l i z e d s t i f f n e s s and stre n g t h along the length of board members. The E - s i m u l a t i o n model, which i s based on the theory of s t a t i o n a r y random processes, estimated the ensemble of amplitude spectrum by performing the f a s t F o u r i e r transform (FFT) to zero mean t e s t E - p r o f i l e s . This amplitude spectrum then combined w i t h random phases to r e c o n s t r u c t an E - p r o f i l e having the same s t a t i s t i c a l p r o p e r t i e s and frequency content as the t e s t data. The cr-simulation model, using the re c o n s t r u c t e d E - p r o f i l e s as the input data, i n v o l v e s s i m u l a t i n g from a b i v a r i a t e standard normal d i s t r i b u t i o n , and then transforming the generated standard normalized s t r e n g t h v a r i a b l e back to i t s r e a l space to o b t a i n the exact marginal d i s t r i b u t i o n w h i l e maintaining the c o r r e l a t i o n c o e f f i c i e n t between the s t i f f n e s s and st r e n g t h . 70 The s i m u l a t i o n r e s u l t s from E - s i m u l a t i o n model and c - s i m u l a t i o n model showed th a t the s t a t i s t i c a l c h a r a c t e r i s t i c s of MOE and strengths are preserved and t h a t the c o r r e l a t i o n s between MOE and strengths are preserved. GLULAM beam s i m u l a t i o n model has been used to simulate the glulam beam stre n g t h with twenty-four beam grade combinations. The GLULAM computer program r e q u i r e s the MOE, t e n s i l e s t r e n g t h and compressive s t r e n g t h p r o f i l e s generated from the E - s i m u l a t i o n model and c - s i m u l a t i o n model as i t s input data. D i s t r i b u t i o n s w i t h d i f f e r e n t beam combinations of simulated bending, t e n s i l e and compressive strengths of glulam beams were obtained from the glulam beam s i m u l a t i o n . The r e s u l t s obtained from glulam beam s i m u l a t i o n showed t h a t SPF MSR lumber can be used as the la m i n a t i n g stock, and the beam layups d i d have a s i g n i f i c a n t e f f e c t on the beam stren g t h p r o p e r t i e s . With one or two l a y e r s of higher grade lam i n a t i o n on the outer l a y e r of the beam and lower grade laminations i n the r e s t of inner l a y e r s of the beam, the glulam beam bending str e n g t h could be improved s i g n i f i c a n t l y . The s i m u l a t i o n r e s u l t s show th a t SPF MSR lumber can provide 5th p e r c e n t i l e s t r e n g t h p r o p e r t i e s e q u i v a l e n t to D o u g l a s - f i r beams c u r r e n t l y produced. 10.2 RECOMMENDATIONS FOR FURTHER WORK I t i s recommended t h a t more grade of within-board s t r e n g t h t e s t s 71 (both t e n s i o n and compression) be c a r r i e d out i n order to e n r i c h the l o c a l i z e d s t r e n g t h data base. The f i n g e r - j o i n t s t r e n g t h needs to be inc o r p o r a t e d i n t o the s i m u l a t i o n model i n order to enhance the accuracy of the s i m u l a t i o n r e s u l t . F u l l - s c a l e glulam beam stre n g t h needs to be t e s t e d i n order to v e r i f y the v a l i d i t y of the E- s i m u l a t i o n and cr-simulation model i n p r a c t i c e . To f i n d the optimum beam layups, the beam st r e n g t h a n a l y s i s w i t h more d i f f e r e n t beam grade and s i z e combinations should be achieved. Further a n a l y s i s should a l s o be included to determine the cost e f f e c t i v e n e s s of SPF MSR m a t e r i a l as an a l t e r n a t i v e source of lam i n a t i n g stock. Strength modelling should be extended to consider trends i n the MOE and st r e n g t h data. With the help of using l a g - s t a t i s t i c s , the a u t o c o r r e l a t i o n s among the s t i f f n e s s and strengths of the adjacent elements i n the generated boards can be revealed so th a t the accuracy of the s i m u l a t i o n model could be improved. 11. REFERENCES Ainsworth, D.M. 1989. Opp o r t u n i t i e s f o r standard lumber products i n the glulam i n d u s t r y . B. Sc. Thesis. Dept. of H a r v e s t i n g and Wood Science, F a c u l t y of F o r e s t r y , U n i v e r s i t y of B r i t i s h Columbia. ASTM. 1990. ASTM D3737-89a Standard t e s t method f o r e s t a b l i s h i n g s t r e s s e s f o r s t r u c t u r a l glued laminated timber (glulam). Annual Book of ASTM Standards: 455-470. B e c h t e l , F.K. 1985. Beam s t i f f n e s s as a f u n c t i o n of pointwise E with a p p l i c a t i o n to machine s t r e s s r a t i n g . Proceedings, I n t e r n a t i o n a l Symposium on For e s t Products Research, CSIR, P r e t o r i a , South A f r i c a . Bendat, J.S. and A.G. P i e r s o l . 1986. Random data: a n a l y s i s and measurement procedure. Wiley I n t e r s c i e n c e , New York. CWC. 1988. Canadian Wood C o n s t r u c t i o n , Glued-laminated timber design. CWC D a t a f i l e WD-3. Canadian Wood C o u n c i l , Ottawa, Canada. CSA. 1984. Engineering Design i n Wood (Working S t r e s s N a t i o n a l Standard of Canada). CAN3-086-M84. Canadian Standards A s s o c i a t i o n , Ontario, Canada. F o s c h i , R.O. and J.D. B a r r e t t . 1980. Glued-laminated Beam str e n g t h : a model. ASCE, Journal of S t r u c t u r a l D i v i s i o n , V o l . 106 (No. ST8): 1735-1754. 73 [8] F o s c h i , R.O. 1987. A procedure f o r the determination of l o c a l i z e d modulus of e l a s t i c i t y . Holz a l s Roh-und Werkstoff, 45: 157-1260. [9] G a l l i g a n , V.L., R.A. Johnson, and J.R. Ta y l o r . 1979. Examination of the concomitant p r o p e r t i e s of lumber. Proceedings of the Metal P l a t e Wood Truss Conference. Forest Products Research S o c i e t y , Madison, VI. Page 65-70. [10] Lam, F. and E. Varoglu. 1991. V a r i a t i o n of t e n s i l e s t r e n g t h along the length of lumber. P a r t 1: Experiment. Vood Science and Technology ( i n p r e s s ) . [11] Lam, F. and E. Varoglu. 1991. V a r i a t i o n of t e n s i l e s t r e n g t h along the length of lumber. P a r t 2: Model development and v e r i f i c a t i o n s . Vood Science and Technology ( i n p r e s s ) . [12] Marx, CM. and R.C. Moody. 1981. Strength and s t i f f n e s s of small glued-laminated beams with d i f f e r e n t q u a l i t i e s of te n s i o n laminations. USDA Forest S e r v i c e Research Paper FPL 381. Forest Products Laboratory, Madison,VI. [13] Moody, R.C. 1974. Design c r i t e r i a f o r l a r g e s t r u c t u r a l glued-laminated beams usin g mixed species f o r v i s u a l l y graded lumber. USDA Forest S e r v i c e Research Paper, FPL 236. Forest Products Laboratory, Madison,VI. [14] Newland, D.E. 1975. Random v i b r a t i o n s and s p e c t r a l a n a l y s i s . Longman Group L i m i t e d , London England. [15] R o j i a n i , K.B. and K.A. T a r b e l l . 1984. R e l i a b i l i t y of wood members under combined s t r e s s . Fourth ASCE conference on 74 P r o b a b i l i s t i c Mechanics and S t r u c t u r a l R e l i a b i l i t y , American S o c i e t y of C i v i l Engineers, NY. Page 86-89. [16] Suddarth, S.K., F.E. Voeste, and V.L. G a l l i g a n . 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 a p p l i e d to wood members i n bending/tension. USDA For e s t S e r v i c e Research Paper FPL 302. Forest Products Laboratory, Madison, VI. [17] T a y l o r , S.E. and D.A. Bender. 1988. S i m u l a t i n g c o r r e l a t e d lumber p r o p e r t i e s using a modified m u l t i v a r i a t e normal approach. Transaction of the American S o c i e t y of A g r i c u l t u r a l Engineers, 31(1): 182-186. [18] T a y l o r , S.E. and D.A. Bender. 1989. A method f o r s i m u l a t i n g m u l t i p l e c o r r e l a t e d lumber p r o p e r t i e s . Forest Products J o u r n a l , 39(7/8): 71-74. [19] Vang, Y., R.O. F o s c h i , and A. F i l i a t r a u l t . 1990. Random modelling of m a t e r i a l p r o p e r t i e s i n r e l i a b i l i t y s t u d i e s of laminated beams. I n t e r n a t i o n a l Timber Engineering Conference, Tokyo, Japan. [20] Voeste, F.E., S.K. Suddarth and V.L. G a l l i g a n . 1979. Si m u l a t i o n of c o r r e l a t e d lumber p r o p e r t i e s data - A r e g r e s s i o n approach. Vood Science, 12(2): 73-79. [21] Yang, C.Y. 1986. Random v i b r a t i o n of s t r u c t u r e s . V i l e y I n t e r s c i e n c e , New York. APPENDIX A TABLE 1 TO TABLE 21 76 Table 1 D e s c r i p t i o n of t e s t m a t e r i a l s and t e s t matrix from COFI Test Mode Grade Nominal Length Sample Dimension ( f e e t ) s i z e 1650f-1.5E 2 x 6 16 63 Tension 2100f-1.8E 2 x 6 16 63 2400f-2.0E 2 x 6 16 63 1650f-1.5E 2 x 6 16 63 Compression 2100f-1.8E 2 x 6 16 63 2400f-2.0E 2 x 6 16 63 Table 2 Summary s t a t i s t i c s f o r t e n s i o n MSR lumber from COFI t e s t Grade 1650f -1.5E 2100f -1.8E 2400f -2.0E Zone A Zone B Zone A Zone B Zone A Zone B s i z e (n) 60 60 57 57 63 63 mean (psi) 3713 .1 4166 8 5091 .3 6285 3 7078.3 7678. 7 S.D. (psi) 1368 .7 1234 9 1554 .6 1832 8 1612.0 1676. 0 5th pet (psi) 1459 .9 2135 4 2534 .0 3270 3 4426.6 4921. 7 minimum (psi) 1361 .2 1793 7 1438 .9 1639 9 4162.1 4255. 9 maximum (psi) 7785 .4 7400 4 8648 .2 9297 5 10560.6 11340.4 median (psi) 3557 .4 4143 1 5095 .7 6205 9 6884.9 7427. 8 78 Table 3 Summary s t a t i s t i c s f o r compression MSR lumber from COFI t e s t Grade 1650f-1.5E 2100f-1.8E 2400f-2.0E Zone A Zone B Zone A Zone B Zone A Zone B s i z e (») 63 63 mean (psi) 3689.9 3874 6 S.D. (psi) 430.0 529.3 5th pet. (psi) 2982.5 3004 0 minimum (psi) 2795.0 3008 9 maximum (psi) 4891.1 5692 9 median (psi) 3685.8 3775 1 58 58 60 60 4392. 1 4520 .9 5006. 7 5047 .4 460.6 535. 1 574.3 571. 7 3634. 4 3640 .7 4062. 0 4107 .0 3560. 1 3365 .5 3812. 6 3879 .2 5795. 2 5931 .0 6448. 6 6819 .4 4365. 6 4457 .1 5016. 8 4995 .9 79 Table 4 Summary of average beam p r o p e r t i e s MOE W ac & Specimen Mean S.D. D e n s i t y ( 2 ) ( 3 ) M.C.(4) No. (MPa) (MPa) Mean S.D. Mean S.D. (%) (MPa) (MPa) (g/cm 3) (g/cm 3) 1802 15038. ,39 436. ,78 52. ,37 2. 80 0, .557 0. 018 11, .0 1804 13506. .30 1108. ,35 44. ,28 4. 83 0, .494 0. 014 10, .0 1805 13953. ,06 564. ,51 46. .83 3. 05 0, .499 0. 006 10 .0 1807 13417. .98 381. ,97 44. .02 4. 15 0, .507 0. O i l 10 .0 1808 13682. ,52 794. ,83 44. ,47 5. 80 0, .476 0. 012 10, .5 1809 13334. ,35 740. ,05 42. ,96 4. 09 0, .544 0. 038 10, .0 1810 15575. ,77 1057. .10 49. ,27 6. 16 0, .502 0. 010 10, .0 1811 14761. ,65 405. .98 48. ,47 4. 69 0 .506 0. 008 9 .5 1812 14723. .22 874. .50 49. ,21 4. 50 0 .549 0. 005 9 .8 1813 16003. .64 1433. .76 52. ,77 6. 25 0 .485 0. 009 10 .0 1814 13353. .18 763. .06 46. ,36 4. 60 0 .488 0. 012 10 .0 1815 15009. .40 601. ,82 50. ,28 4. 03 0 .427 0. 010 9 .3 1816 12856. .37 375. ,41 39. ,96 3. 77 0 .501 0. 015 9 .3 1817 13710. .02 505. ,32 46. ,04 4. 15 0 .483 0. 008 9 .5 1818 13090. .96 452. ,30 45. ,64 2. 38 0 .505 0. 014 9 .9 1819 13468. .14 263. ,14 48. ,45 2. 67 0 .484 0. 010 10 .0 1820 13574. .48 533. ,76 46. ,34 3. 29 0 .517 0. 017 9 .3 1821 14379, .09 772. .55 47. .44 3. 11 0 .573 0. 011 9 .8 1822 17421, .42 1016. .90 54. .90 3. 95 0 .503 0. 014 10 .3 1823 14285, .38 982. .19 46. .24 4. 45 0 .565 0. 016 10 .2 1825 15839, .68 829. .25 51. .70 4. 08 0 .519 0. 017 10 .0 1826 14931, .23 716. .36 48. .79 3. 11 0 .478 0. 005 10 .0 1827 12802, .26 435. ,95 44. .44 3. 34 0 .548 0. 018 9 .5 1828 15461, .29 560. ,04 48. .53 4. 29 0 .477 0. 006 11 .0 1829 13683. .84 509. .57 44. .94 1. 59 0 .535 0. 010 10 .0 1830 14483, .48 564. .19 49, .45 4. 95 0 .548 0. 011 11 .0 1831 14684, .60 483. .99 46, .50 4. 68 0 .494 0. 012 10 .0 1832 14431, .55 618. .86 47. .26 3. 63 0 .451 0. 014 10 .2 1833 13551, .16 385. .56 39. .93 2. 79 0 .540 0. 005 9 .2 1834 18172, .17 663. .53 54. .98 4. 18 0 .550 0. 014 9 .5 1836 15463, .11 571, .83 52. .93 4. 14 0 .432 0. 011 10 .2 1837 12713 .66 593. .95 36. .73 3. 66 0 .495 0. 007 9 .5 1838 13664, .38 527. .90 42. .48 3. 43 0 .528 0. O i l 10 .0 1839 13164, .26 448. .97 46. .28 3. 32 0 .517 0. 010 10 .0 1840 14619 .36 534, .07 46, .72 3. ,51 0 .517 0. 014 10 .2 80 1841 14144 .22 544 .20 47 .63 3. 39 0, .490 0. 015 9 .8 1842 13934 .72 571 .51 46 .40 3. 60 0, .501 0. O i l 9 .2 1843 13832 .98 429 .24 44 .82 3. 11 0. .529 0. 010 9 .0 1844 16950 .28 954 .76 48 .74 3. 59 0, .542 0. 024 9 .5 1845 15095 .44 548 .90 47 .96 4. 18 0. .531 0. 009 10 .0 1846 15566 .12 548 .91 48 .63 3. 16 0. .530 0. 014 10 .5 1847 16017 .49 469 .17 46 .35 4. 35 0. .513 0. 008 10 .0 1848 13537 .45 562 .67 42 .28 3. 41 0. .549 0. 031 10 .5 1849 15550 .91 980 .45 49 .45 5. 71 0. .467 0. O i l 9 .7 1850 12730 .68 467 .14 42 .04 3. 33 0. .468 0. O i l 10 .2 1851 13845 .46 463 .27 39 .20 3. 93 0. .519 0. O i l 10 .0 1853 14296 .12 471 .19 44 .53 3. 89 0, .474 0. 007 10 .0 1854 13203 .80 368 .95 42 .95 3. 05 0. .504 0. 009 10 .0 1855 13961 .10 402 .41 44 .58 3. 01 0. .478 0. 007 9 .5 1856 13252 .70 614 .09 43 .61 2. 80 0. .625 0. 032 10 .0 1857 15507 .83 1600 .55 49 .29 7. 76 0, .499 0. 012 8 .5 1858 13635 .93 243 .28 46 .96 2. 18 0. .499 0. 012 10 .0 1859 13120 .80 490 .04 45 .71 3. 17 0. .486 0. 007 10 .0 1860 16559 .82 514 .49 51 .66 4. 18 0. .536 0. 009 10 .5 Average 14399 625 .07 46 .68 3. 87 0. .510 0. 012 9 .9 (1) Averages are f o r 25 s e c t i o n t e s t s . (2) Averages are f o r 32 s e c t i o n t e s t s . (3) Based on weight and volume at time of t e s t . (4) Determined f o l l o w i n g t e s t using r e s i s t a n c e - t y p e meter. 81 Table 5 F i t t e d d i s t r i b u t i o n parameters f o r MOE and cr*  Board Mean Board S.D. Compressive Compressive D i s t r i b u t i o n MOE Strength MOE Strength (MPa) (MPa) (MPa) (MPa) = 14399 — 46.68 = 625 - 3 .87 Normal = 1232 a — 3.77 a = 267 cr - 1 .07 A* 14399 V- - 46.69 t* = 623 - 3 .88 Lognormal c - 1195 a 3.82 cr = 243 a - 1 .07 m 14990 m - 48.39 m = 706 m - 3 .62 2-P We i b u l l Jfc = 10.81 k = 13.33 k = 2.45 k - 4 .27 °"o 12630 - 25.73 297 1 .51 3-P We i b u l l m = 1962 m 22.68 m = 363 m — 2 .42 k 1.42 k = 5.98 k = 1.32 k - 2 .61 *): (j, Mean; <T Standard d e v i a t i o n ; <T0 L o c a t i o n parameter; m Scale parameter; k Shape parameter. 82 Table 6 D e s c r i p t i v e s t a t i s t i c s f o r t e s t E and O~Q (2100f-1.8E) Board Mean Board S. D. Compressive Compressive MOE Strength MOE Strength Dimension Length S i z e (MPa) (MPa) (MPa) (MPa) (i n c h ) ( i n c h ) (n) Mean 14399 46.68 625 3.87 S.D. 1220 3.73 264 1.06 2 x 4 6 1375 C.O.V. 8.47 7.99 42.24 27.39 83 Table 7 D e s c r i p t i v e s t a t i s t i c s f o r generated ac (2100f-1.8E) Board Mean Board S. D. Compressive Compressive MOE Strength MOE Strength Dimension Length S i z e (MPa) (MPa) (MPa) (MPa) (i n c h ) ( i n c h ) (n) Mean 14399 46.00 625 3.26 S.D. 1220 3.26 264 0.80 2 x 6 6 2500 C.O.V. 8.47 7.09 42.24 24.54 84 Table 8 D e s c r i p t i v e s t a t i s t i c s f o r t e s t and generated MOE values Test MOE Generated MOE ( f o r three grades) ( f o r three grades) 1650f 2100f 2400f 1650f 2100f 2400f Dimension ( i n c h ) 2 x 6 2 x 6 2 x 6 2 x 6 2 x 6 2 x 6 Length ( f o o t ) 16 16 16 16 16 16 S i z e (n) 123 115 121 3000 3000 3000 Mean (MPa) 10353 12686 14494 10366 12618 14489 S. D. (MPa) 1167 1224 1047 1156 1149 858 85 Table 9 Lower-bound strengths f o r compression and t e n s i o n s i m u l a t i o n Grade Compressive Strength T e n s i l e Strength (MPa) (MPa) 1650f-1.5E 2100f-1.8E 2400f-2.0E 19 22 26 10 17 27 Table 10 D e s c r i p t i o n of s i m u l a t i o n beams Depth of Beam (inch) Span of Beam ( f o o t ) Number of Laminations* Number of Beams• 9 16 6 700 12 21 8 800 18 32 12 900 * Lamination t h i c k n e s s i s 1.5 inches. 87 Table 11 Parameters f o r 3-P V e i b u l l d i s t r i b u t i o n and t h e i r a s s o c i a t e d 5th and 50th p e r c e n t i l e values of simulated beam s t r e n g t h (Depth=9") Layups m k 5 % i l e 507.il. B9A-B* 5099 5235 2.704 6844 9670 B9A-C** 4732 1943 2.592 5350 6419 B9A-T*** 2907 3184 3.808 4367 5799 B9B-B 4335 6010 3.181 6697 9691 B9B-C 4473 1965 3.731 5359 6254 B9B-T 891 3314 3.296 2236 3856 B9C-B 4076 4843 3.061 5911 8372 B9C-C 4482 1515 2.616 4969 5799 B9C-T 1355 1882 2.311 1876 2961 B9D-B 1330 4406 3.147 3045 5252 B9D-C 3454 1986 3.139 4225 5221 B9D-T 1324 1367 1.923 1616 2454 B9E-B 2280 5626 3.518 4698 7349 B9E-C 2518 3148 6.349 4490 5489 B9E-T 1331 1742 2.355 1825 2822 B9F-B 4299 2928 2.462 5175 6822 B9F-C 4160 1846 3.182 4886 5805 B9F-T 1017 2693 3.129 2059 3412 B9G-B 4322 3662 2.451 5412 7475 B9G-C 4078 1935 2.547 4681 5754 B9G-T 2454 2061 2.971 3212 4276 * Bending; ** Compression; *** Tension; Ug L o c a t i o n parameter; m Scale parameter; k Shape parameter. 88 Table 12 Parameters f o r 3-P Weibu l l d i s t r i b u t i o n and t h e i r a s s o c i a t e d 5th and 50th p e r c e n t i l e values of simulated beam strength(Depth=12") Layups m k 5 % i l e 5 0 % i l e B12A-B 5272 4212 2.190 6357 8835 B12A-C 4740 1240 2.422 5104 5806 B12A-T 3630 1815 2.578 4203 5204 B12B-B 4975 4344 2.255 6139 8667 B12B-C 4693 1275 2.302 5044 5780 B12B-T 1107 2834 3.125 2203 3627 B12C-B 4657 4309 2.686 6083 8416 B12C-C 4745 965 2.119 4983 5557 B12C-T 1488 1718 2.424 1993 2965 B12D-B 0 7648 5.461 4440 7152 B12D-C 3991 1340 2.730 4442 5163 B12D-T 1540 1187 1.728 1753 2500 B12E-B 1020 3778 2.792 2324 4333 B12E-C 2947 2020 4.169 3938 4797 B12E-T 1372 1004 1.846 1573 2195 B12F-B 2088 4224 3.644 3958 5908 B12F-C 1804 3332 8.751 4177 4999 B12F-T 1530 1077 1.746 1727 2403 B12G-B 1035 5876 5.577 4485 6537 B12G-C 4292 1090 2.334 4597 5224 B12G-T 1463 1519 2.360 1894 2764 B12H-B 3196 4194 2.998 4748 6906 B12H-C 3840 1678 3.138 4491 5333 B12H-T 2050 2140 3.468 2959 3975 B, C, T, <r0, m, k Same as i n Table 11. 89 Table 13 Parameters f o r 3-P V e i b u l l d i s t r i b u t i o n and t h e i r a s s o c i a t e d 5th and 50th p e r c e n t i l e values of simulated beam strength(Depth=18") Layups <xQ m k 5 % i l e 507.ile B18A-B 3398 5343 3. .118 5459 8148 B18A-C 4337 1101 3. .784 4839 5336 B18A-T 3509 1398 2. .919 4014 4742 B18B-B 5107 3175 1. .922 5784 7884 B18B-C 4236 1217 3. .561 4765 5334 B18B-T 1573 2011 2. .297 2125 3287 B18C-B 4895 3518 2. .250 5835 7731 B18C-C 4186 1207 3. .453 4697 5271 B18C-T 1743 1118 1. .661 1930 2640 B18D-B 4129 3725 2. .520 5275 7350 B18D-C 3692 1516 4. .355 4458 5086 B18D-T 1526 1042 2. .053 1771 2398 B18E-B 1640 5487 4. .567 4503 6704 B18E-C 3821 1176 3. .780 4357 4888 B18E-T 1506 818 1. .749 1656 2169 B18F-B 1522 2530 2. .707 2367 3732 B18F-C 2658 1897 4. .171 3589 4395 B18F-T 1431 509 1. .806 1529 1847 B18G-B 1734 4119 4. .383 3826 5523 B18G-T 802 4028 1. .195 3600 3766 B18G-T 1414 817 1. .935 1590 2090 B18H-B 3858 2575 2. .889 4326 6126 B18H-C 2840 2188 5. .818 4153 4894 B18H-T 1551 1059 1. .934 1779 2427 B18I-B 3059 3255 3. .148 4336 5956 B18I-C 3191 1822 4. .629 4150 4874 B18I-T 2464 1242 2. .455 2834 3534 B, C, T, aQ, m, k Same as i n Tabl e 11. Table 14 Simulated MOE, transformed MOE and t h e i r r a t i o s Simulated MOE Transformed Layups MOE Simulated MOE .„„ Transformed MOE Mean COV (1 0 6 p s i ) (7.) ( 1 0 6 p s i ) BEAM9-A 2. .054 4.6 2 .000 1.027 BEAM9-B 2. .006 3.7 1 .981 1.012 BEAM9-C 1. .858 3.9 1 .852 1.003 BEAM9-D 1. .479 4.5 1 .500 0.986 BEAM9-E 1. .701 4.5 1 .711 0.994 BEAM9-F 1. .786 4.3 1 .789 0.998 BEAM9-G 1. .808 4.6 1 .800 1.004 BEAM12-A 2. .054 3.8 2 .000 1.027 BEAM12-B 2. .028 3.0 1 .992 1.018 BEAM12-C 1. .958 3.5 1 .938 1.010 BEAM12-D 1. .795 3.6 1 .789 1.003 BEAM12-E 1. .447 4.2 1 .500 0.965 BEAM12-F 1. .666 3.9 1 .673 0.996 BEAM12-G 1. .754 3.6 1 .763 0.995 BEAM12-H 1. .805 3.9 1 .800 1.003 BEAM18-A 2, .049 2.7 2 .000 1.025 BEAM18-B 2, .033 2.2 1 .998 1.018 BEAM18-C 2, .005 2.2 1 .981 1.012 BEAM18-D 1, .951 2.3 1 .938 1.007 BEAM18-E 1, .863 2.2 1 .852 1.006 BEAM18-F 1. .478 2.4 1 .500 0.986 BEAM18-G 1, .701 2.5 1 .711 0.994 BEAM18-H 1, .783 2.7 1 .789 0.997 BEAM18-I 1, .805 2.6 1 .800 1.003 Mean 1.831 3.459 S. D. 0.189 0.846 COV (7.) 10.33 24.46 1.821 0.165 9.06 1.004 0.014 1.44 91 Table 15 Comparison of simulated and p r e d i c t e d bending s t r e n g t h Transformed Allowable Simulated Simulated Simulated Simulated Section Bending l Bending Layups Bending Tension Compression Moment of Stress Strength Strength Strength Inertia f \ u / Allowable (5%ile psi) (5%ile psi) (5%ile psi) T.- (psi) (psi) Bending BEAM9-A 6844 4367 5350 1.000 5040 5204 1.32 BEAM9-B 6697 2236 5359 0.991 4995 5157 1.30 BEAM9-C 5911 1876 4969 0.926 4667 4819 1.23 BEAM9-D 3045 1616 4225 1.000 3465 3578 0.85 BEAM9-E 4698 1825 4490 0.951 4194 4330 1.08 BEAM9-F 5175 2059 4686 0.994 4384 4526 1.14 BEAM9-G 5412 3212 4881 1.000 4410 4553 1.19 BEAM12-A 6357 4203 5104 1.000 5040 5040 1.26 BEAM12-B 6139 2203 5044 0.996 5024 5024 1.22 BEAM12-C 6083 1993 4983 0.969 4884 4884 1.25 BEAM12-D 4440 1753 4442 0.895 4511 4511 0.98 BEAM12-E 2324 1573 3938 1.000 3465 3465 0.67 BEAM12-F 3958 1727 4177 0.930 4101 4101 0.97 BEAM12-G 4485 1894 4497 0.979 4317 4317 1.04 BEAM12-H 4748 2959 4591 1.000 4410 4410 1.08 B E A M 18-A 5459 4014 4839 1.000 5040 4818 1.08 BEAM18-B 5784 2125 4765 0.999 5035 4813 1.20 BEAM18-C 5835 1930 4697 0.991 4995 4775 1.22 BEAM18-D 5275 1771 4458 0.969 4884 4669 1.13 BEAM18-E 4503 1656 4357 0.926 4667 4461 1.01 BEAM18-F 2367 1529 3589 1.000 3465 3312 0.71 BEAM18-G 3826 1590 3600 0.951 4194 4009 0.95 BEAM18-H 4326 1779 4150 0.994 4384 4191 1.03 BEAM18-I 4336 2834 4153 1.000 4410 4216 1.03 Mean 4918 2280 4556 0.978 4499 4466 1.08 S. D. 1213 841 471 0.030 497 500 0.17 COV (%) 24.67 36.88 10.33 3.11 11.04 11.20 15.47 92 Table 16 Test of mean value for 24 beam layups (Bending strength) LAYUPS BEAM9- BEAM12- BEAM18- SIZE AVERAGE S. D. A BCD EFG ABCDE FGH ABCDEFGH I (n) 106 psi 106psi BEAM9-A x * * • * • • • • 100 9770 1911 B • * • • • *•'» 100 9721 1849 C . . . . • x x x • x 100 8406 1521 D ... • • • • • • • • X • • 100 5287 1400 E • •' • • • X • • • X . . . . x ... . 100 6893 1109 F X . . . X • • • X • • • x 100 7348 1578 G * x • x 100 7574 1432 BEAM12-A x . . . . . . • • • 100 9003 1788 B x • * • 100 8819 1782 C • • • • • • x 100 8491 1517 D * * * X • * • X * * • * • 100 7064 1457 E . . . • * • 100 4384 1274 F • • XX 100 5902 1153 G • . . . . x ... . 100 6470 1115 H . . . . x ... . 100 6942 1341 BEAM18 - A xx 100 8183 1634 B x 100 7917 1494 C 100 8010 1447 D 100 7432 1384 E . . . . 100 6655 1229 F • • • 100 3771 887 G • • 100 5492 951 H X 100 6153 848 I 100 5974 995 — Significant at or = 0.05 level — Not significant at a = 0.05 level 93 Table 17 Test of mean value for 24 beam layups (Tensile strength) LAYUPS BEAM9- BEAM12- BEAM18- SIZE AVERAGE S. D. A BCD EFG A B C D E F G H A B C D E F G H I (n) 10" psi 10" psi BEAM9-A 100 5787 829 B • x X 100 3870 958 C • x • • • * x • 100 3023 743 D . . . • V • X • X • • . . . x • • • X • 100 2534 640 E • • • • X • • * X X • • x 100 2878 679 F • x • x X 100 3431 821 G 100 4295 666 BEAM12 - A 100 5242 661 B X 100 3645 862 C 100 3011 657 D • X * • • * X x • • • X • 100 2596 621 E . . . . . . . x . x . . 100 2262 491 F * * * * • X * * * x • 100 2488 560 G • • • x 100 2809 596 H 100 3975 603 BEAM18 - A • • 100 4757 465 B • • 100 3359 800 C 100 2741 603 D . . . x • 100 2447 463 E • X • • 100 2233 422 F • * * 100 1883 255 G * • 100 2138 385 H • 100 2489 494 I 100 3565 471 Significant at a = 0.05 level Not significant at a = 0.05 level 94 Table 18 Test of mean value for 24 beam layups (Compressive strength) BEAM9- BEAM12- BEAM18- SIZE AVEF1AGE S. D. LAYUPS ABCD EFG A B C D E F G H ABCDE FGH I (n) 10" psi 10 "psi BEAM9 - A 100 6456 705 B 100 6248 523 C • * X X 100 5827 546 D 100 5231 611 E 100 5449 534 F X 100 5814 565 G 100 5805 785 BEAM12 - A 100 5838 478 B 100 5823 519 C 100 5599 420 D * • x * • * X x * * * * - 100 5183 468 E X 100 4783 489 F • • • * * x x • • X X 100 4958 431 G X 100 5259 446 H 100 5344 530 BEAM18 - A 100 5332 291 B 100 5333 341 C 100 5271 346 D 100 5074 354 E * * X X 100 4884 312 F • • • 100 4383 461 G • • 100 4664 381 H X 100 4868 399 I 100 4858 402 * Significant ator = 0.05 level x Not significant at a = 0.05 level Table 19 Bending str e n g t h s i z e parameter Property Grade Fsb Level 1650f-1.5E 0.356 5 % i l e 2100f-1.8E 0.317 2400f-2.0E 0.329 1650f-1.5E 0.485 5 0 % i l e 2100f-1.8E 0.330 2400f-2.0E 0.244 Table 20 T e n s i l e strength s i z e parameter F Property Level Grade 1650f-1.5E 0.079 5 % i l e 2100f-1.8E 0.176 2400f-2.0E 0.121 1650f-1.5E 0.411 507„ile 2100f-1.8E 0.276 2400f-2.0E 0.286 Table 21 Compressive str e n g t h s i z e parameter F, Property Grade F s c Level 1650f-1.5E 0.235 5 X i l e 2100f-1.8E 0.175 2400f-2.0E 0.144 1650f-1.5E 0.246 507.ile 2100f-1.8E 0.238 2400f-2.0E 0.263 APPENDIX B FIGURE 1 TO FIGURE 103 4 1 6 f e e t ^ Minimum E • Piece with Minimum E Zone A Other piece Zone B Figure 1 Cutting pattern for tension and compression specimen Tensile strength in Zone A and Zone B (MPa) Figure 2 CDF of tension data from COFI test (1650f-1.5E) »—» 8 Tensile strength in Zone A and Zone B (MPa) Figure 3 CDF of tension data from COFI test (2100f-1.8E) o 1 0.9 0.8 -0.7 -0.6 -0.5 0.4 0.3 0.2 H 0.1 0 ,++ t a Zone A + Zone B 20 40 60 Tensile strength in Zone A and Zone B (MPa) Figure 4 CDF of tension data from COFI test (2400f-2.0E) 80 o to Compressive strength in Zone A and Zone B (MPa) Figure 5 CDF of compression data from COFI test (1650f-1.5E) o 0 H i 1 1 1 1 1 1 1 1 1 r 20 24 28 32 36 40 44 Compressive strength in Zone A and Zone B (MPa) Figure 6 CDF of compression data from COFI test (2100f-1.8E) Compressive strength in Zone A and Zone B (MPa) Figure 7 CDF of compression data from COFI test (2400f-2.0E) 16 Q_ 16 14 O 13 O O UJ 12 o 11 10 Minimum E point Length (1000 mm) Figure 8 Short span, flatwise bending E-profile o 0\ 16 foot | | | | | 32 1.5 inch 6 inch Figure 9 Cutting and numbering pattern for compression specimen o 18 55 16 14 12 10 MOE DATA -B -COMPRESSION DATA 45 35 25 0 1 2 3 4 5 Length (1000 mm) Figure 11 MOE and compressive strength profiles along the length CD a . c a> L— +-CO > "co co a> k_ Q. E o O o oo 3 0 40 60 Test data Normal fit Lognormal fit 2P Weibull fit 3P Weibull fit _ i 6 0 70 Compressive strength (MPa) Figure 12 CDF of test and fitted board mean of compressive strength 0 2 4 6 8 10 Standard deviation (MPa) Figure 13 CDF of test and fitted board S.D. of compressive strength M x ) t Length of the board Figure 14 Ensemble of MOE along the length of the board E„(x) 0.0 mi Ai 1 /V n WOT • E (x) 0.0 J. k. Length of the board Figure 15 Ensemble of zero mean MOE along the length zn 113 1 2 3 4 Length (1000 mm) Figure 17 Ensemble average of test E-profile 2 0 0 0 Modulus of Elasticity (1000 MPa) Figure 19 CDF of test and fitted board mean of MOE I—» ON Figure 20 CDF of test and fitted board standard deviation of MOE r—» r—' -0 6 0 0 5 0 0 Figure 21 Ensemble average of the amplitude spectrum 00 Figure 22 Generated E-profile of one board 16 Test data Generated data 0 1 2 3 4 5 Length (1000 mm) Figure 23 Ensemble average of test and generated E-profiles 500 0 1 2 3 4 5 Length (1000 mm) Figure 24 Ensemble standard deviation of test and generated E-profiles to 7 0 0 6 0 0 5 0 0 4 0 0 3 0 0 2 0 0 1 0 0 Test data Simulated data 10 2 5 3 0 15 2 0 F r e q u e n c y (co) F i g u r e 2 5 T e s t a n d s i m u l a t e d e n s e m b l e a v e r a g e o f a m p l i t u d e s p e c t r a 3 5 65 10 0 500 1000 1500 2000 Board standard deviation of MOE (MPa) Figure 27 Regression plot of board S.D. MOE vs. compressive strength Z X | in standard Xi in real distribution normal distribution ( X , , Y , ) 1 ""' 1 - r " i • -Y, in real distribution Z Y l in standard normal distribution Figure 28 Graphical demonstration of the transformation between the real space and standard normalized space 65 CO Q_ c > to CD Q. E o O 60 55 50 45 40 35 30 • • • Fitted regression line C = 8.0343 + 0.0027 x E P = 0.654 25 10 12 14 16 18 Modulus of elasticity (10 MPa) 20 Figure 29 Regression plot of test MOE vs. within-board compressive strength ON CO D) C 2 C O > 00 00 CD CL E o O 65 60 55 5 0 45 40 3 5 3 0 25 • 10 Jo"*1 • qb • • • A Ig A (DD • • AP QA • HA A A A • • • Test data (p = 0.654 ) A Simulated data (p = 0.649 ) 12 14 16 18 2 0 Modulus of elasticity (103 MPa) Figure 30 Regression plot of test and simulated MOE vs. compressive strength to 1 0.8 "8 0.6 O I 0.4 • E o 0.2 R c at 50 percentile - 1.15 1.2 Ratio R c Figure 31 Cumulative distribution function of ratio R c 1.4 S 3 CXI / R T at 50 percentile - 1.20 1.4 Ratio R T Figure 32 Cumulative distribution function R T 1.9 Tensile strength (MPa) Figure 33 CDF of test and generated minimum tension data (1650f-1.5E) h-» O 0 20 40 60 Tensile strength (MPa) Figure 34 CDF of test and generated minimum tension data (2100f-1.8E) Tensile strength (MPa) Figure 35 CDF of test and generated minimum tension data (2400f-2.0) Compressive strength (MPa) Figure 36 CDF of test and generated minimum compression data (1650f-1.5E) Compressive strength (MPa) Figure 37 CDF of test and generated minimum compression data (21 OOf-1.8E) 4^  J ' '/ Compressive strength (MPa) Figure 38 CDF of test and generated minimum compression data (2400f-2.0E) Figure 39 Beam layups for three grade combinations - 9 in. beam Figure 40 Beam layups for three grade combinations - 12 in. beam Figure 41 Beam layups for three grade combinations - 18 in. beam Figure 42 Beam with two stiffness zones 2.2 1.4 1.6 1.8 2 2.2 Trnasformed MOE (10* psi) Figure 43 Comparison of simulated MOE with transformed MOE Bending strength (1000 psi) Figure 44 CDF of bending strength for three grades (depth = 9 inch) 1 3 5 7 9 11 13 15 17 19 Bending strength (1000 psi) Figure 45 CDF of bending strength for three grades (depth = 12 inch) Bending strength (1000 psi) Figure 46 CDF of bending strength for three grades (depth = 18 inch) 10 9 8 7 6 5 4 3 2 1 sa 2400f-2.0E + 2100M.8E o 1650f-1.5E J L J I I L 11 13 Depth (inch) 15 17 Figure 47 The 5th percentile value of bending strength vs. depth 4^ m 2400f-2.0E + 2100f-1.8E o 1650M.5E J I I I L _ I I I I L 9 11 13 15 17 Depth (inch) Figure 48 The 50th percentile value of bending strength vs. depth Bending strength (1000 psi) Figure 49 CDF of bending strength for 1650f-1.5E beams as a function of beam depth ON Bending strength (1000 psi) Figure 50 CDF of bending strength for 2100f-1.8E beams as a function of beam depth 1 3 5 7 9 11 13 15 17 19 Bending strength (1000 psi) Figure 51 CDF of bending strength for 2400f-2.0E beams as a function of beam depth CC ii DEPTH=9" + DEPTH=12° o DEPTH 18" 2400f-2.0E 2100f-1.8E Grade 1650M.5E Figure 52 The 5th percentile value of bending strength vs. grade 10 I _ J _ : I I I 2400f-2.0E 2100M.8E 1650M.5E Grade Figure 53 The 50th percentile value of bending strength vs. grade Figure 54 CDF of bending strength with the combination of 1650f-1.5E and 2100f-1.8E (depth = 9 inch) Bending strength (1000 psi) Figure 56 CDF of bending strength with the combinations of 1650M.5E and 2100f-1.8E (depth = 18 inch) Figure 57 The 5th percentile value of bending strength vs. the outer layer percent of 2100f-1.8E U l ii DEPTH=9" + DEPTH=12" o DEPTH=18" J i i i i i i i i i i 0 20 40 60 80 100 Outer layer percent of 2100f-1.8E (%) Figure 58 The 50th percentile value of bending strength vs. the outer layer percent of 2100f-1.8E Bending strength (1000 psi) Figure 59 CDF of bending strength with the combination of 1650M.5E and 2400f-2.0E (depth = 9 inch) U l ON Bending strength (1000 psi) Figure 60 CDF of bending strength with the combination of 1650f-1.5E and 2400f-2.0E (depth = 12 inch) Bending strength (1000 psi) Figure 61 CDF of bending strength with the combination of 1650f-1.5E and 2400f-2.0E (depth = 18 inch) Figure 62 The 5th percentile value of bending strength vs. the outer layer percent of 2400f-2.0E vo 01 DEPTH=9" + DEPTH=12" o DEPTH=18" J I I I I I I I I L _ 0 20 40 60 80 100 Outer layer percent of 2400f-2.0E (%) Figure 63 The 50th percentile value of bending strength vs. the outer layer percent of 2400f-2.0E Tensile strength (1000 psi) Figure 64 CDF of tensile strength for three grades (depth = 9 inch) Tensile strength (1000 psi) Figure 65 CDF of tensile strength for three grades (depth = 12 inch) 2400f-2.0E CT18-A) 2100f-1.8E (T18-I) 1650M.5E rri8-F) 3-P Weibull fitting 1 3 5 7 9 Tensile strength (1000 psi) Figure 66 CDF of tensile strength for three grades (depth = 18 inch) SB 2400f-2.0E + 2100M.8E o 1650M.5E Depth (inch) Figure 67 The 5th percentile value of tensile strength vs. depth 2400f-2.0E + 2100f-1.8E o 1650f-1.5E 11 13 15 Depth (inch) 17 Figure 68 The 50th percentile value of tensile strength vs. depth U l Tensile strength (1000 psi) Figure 69 CDF of tensile strength under three depth (1650f-1.5E) Tensile strength (1000 psi) Figure 70 CDF of tensile strength under three depth (2100f-1.8E) Figure 72 The 5th percentile value of tensile strength vs. grade Figure 73 The 50th percentile value of tensile strength vs. grade o Tensile strength (1000 psi) Figure 77 CDF of tensile strength with the combination of 1650f-1.5E and 2400f-2.0E (depth = 9 inch) ^ Tensile strength (1000 psi) Figure 78 CDF of tensile strength with the combination of 1650M.5E and 2400f-2.0E (depth = 12 inch) Tensile strength (1000 psi) Figure 79 CDF of tensile strength with the combination of 1650M.5E and 240W-2.0E (depth = 18 inch) Figure 80 The 5th percentile value of tensile strength vs. the outer layer percent of 2100f-1.8E Figure 81 The 50th percentile value of tensile strength vs. the outer layer percent of 2100f-1.8E i—> oo 20 40 60 80 Outer layer percent of 2400f-2.0E (%) 100 Figure 82 The 5th percentile value of tensile strength vs. the outer layer percent of 2400f-2.0E • DEPTH=9" + DEPTH=12" o DEPTH=18" Figure 83 The 50th percentile value of tensile strength vs. the outer layer percent of 2400f-2.0E Compressive strength (1000 psi) Figure 85 CDF of compressive strength for three grades (depth = 12 inch) Figure 87 The 5th percentile value of compressive strength vs. depth oo + 2100f-1.8E o 1650f-1.5E — „ J i i « 1 7 Depth (inch) Figure 88 The 50th percentile value of compressive strength vs. depth Compressive strength (1000 psi) Figure 89 CDF of compressive strength under three depth (1650f-1.5E) 00 ON Compressive strength (1000 psi) Figure 90 CDF of compressive strength under three depth (2100f-1.8E) Compressive strength (1000 psi) Figure 91 CDF of compressive strength under three depth (2400f-2.0E) 0 0 oo 7 6 5 4 3 2 h • DEPTH=9" + DEPTH = 12" o DEPTH 18" 2400f-2.0E 2100M.8E Grade 1650f-1.5E Figure 92 The 5th percentile value of compressive strength vs. grade 10 8 7 6 5 4 3 2 h 1 • DEPTH=9" + DEPTH= 12" o DEPTH 18" 24001-2.0E 2100M.8E Grade 1650f-1.5E Figure 93 The 50th percentile value of compressive strength vs. grade VO o Compressive strength (1000 psi) Figure 94 CDF of compressive strength with the combination of 1650f-1.5E and 2100M.8E (depth = 9 inch) Compressive strength (1000 psi) Figure 95 CDF of compressive strength with the combination of 1650M.5E and 2100M.8E (depth = 12 inch) Compressive strength (1000 psi) Figure 96 CDF of compressive strength with the combination of 1650M.5E and 2100M.8E (depth = 18 inch) vo Compressive strength (1000 psi) Figure 97 CDF of compressive strength with the combination of 1650f-1.5E and 2400f-2.0E (depth = 9 inch) vo 4*. Compressive strength (\0Q0 psO Figure 98 CDF of compressive strength with the combination of 1650M.5E and 2400f-2.0E (depth = 12 inch) Compressive strength (1000 psi) Figure 99 CDF of compressive strength with the combination of 1650f-1.5E and 2400f-2.0E (depth = 18 inch) ON Figure 100 The 5th percentile value of compressive strength vs. the outer layer percent of 2100f-1.8E Figure 101 The 50th percentile value of compressive strength vs. the outer layer percent of 2100f-1.8E B DEPTH=9' + DEPTH=12" o DEPTH 20 40 60 80 Outer layer percent of 2400f-2.0E (%) Figure 102 The 5th percentile value of compressive strength vs. the outer layer percent of 2400f-2.0E Figure 103 The 50th percentile value of compressive strength vs. the outer layer percent of 2400f-2.0E 

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