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Reliability of structures with load history-dependent strength and an application to wood members Yao, Zhao-Cheng 1987

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RELIABILITY OF STRUCTURES WITH LOAD HISTORY-DEPENDENT STRENGTH AND AN APPLICATION TO WOOD MEMBERS by ZHAO-CHENG YAO B.Sc.,Tong J i U n i v e r s i t y , 1 9 8 1 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES Department of C i v i l E n g i n e e r i n g We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1 9 8 7 © Zhao-cheng Yao, A p r i l , 1 9 8 7 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 of The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 DE-6(3/81) ABSTRACT Because of an i n t e r a c t i o n between the l o a d h i s t o r y and the m a t e r i a l ' s p r o p e r t i e s a r e d u c t i o n i n the s e r v i c e a b i l i t y of s a f e t y of a s t r u c t u r e may be seen as the r e s u l t of a random p r o c e s s of damage a c c u m u l a t i o n w i t h t i m e . A damage model i s proposed and v e r i f i e d t o d e a l w i t h t h i s t y p e of problem i n g e n e r a l , and s p e c i f i c a l l y w i t h d u r a t i o n - o f - l o a d e f f e c t s i n t i m b e r s t r u c t u r e s . Through c o m p a r i s o n s w i t h and d i s c u s s i o n of o t h e r damage a c c u m u l a t i o n models, the proposed model i s found t o be both s u i t a b l e t o r e p r e s e n t e x p e r i m e n t a l r e s u l t s and r e l a t i v e l y easy t o use. A r e l i a b i l i t y a n a l y s i s f o r a s i n g l e s t r u c t u r a l member w i t h a s e r v i c e l i f e of 30 y e a r s i s s t u d i e d f o r d i f f e r e n t snow l o a d i n g c o n d i t i o n s of t h r e e Canadian c i t i e s , u s i n g the damage model i n c o m b i n a t i o n w i t h R a c k w i t z - F i s s l e r ' s a l g o r i t h m . A s i m p l i f i e d a n a l y s i s method i s proposed and compared w i t h a l a r g e - s c a l e M o n t e - C a r l o s i m u l a t i o n . The r e s u l t s of the r e l i a b i l i t y a n a l y s i s a r e found t o be s a t i s f a c t o r y . F i n a l l y , t he i m p l e m e n t a t i o n of the r e l i a b i l i t y a n a l y s i s i n a s i m p l e d e s i g n g u i d e l i n e f o r l o a d d u r a t i o n e f f e c t s i s d i s c u s s e d . LIST OF TABLES TABLE(1) E x p e r i m e n t a l l o a d d a t a 14 TABLE(2) S h o r t term s t r e n g t h T s 15 TABLE(3) Parameters f o r the damage model 23 TABLE(4) Parameters f o r N i e l s e n ' s model 36 TABLE (5) Snow l o a d d a t a f o r 3 Canadian c i t i e s 51 TABLE(6) M o n t e - C a r l o vs R-F A l g o r i t h m 78 TABLE(7) R e s i s t a n c e f a c t o r 0 w i t h 0 = 3.0 89 TABLE(8) Performance and DOL adjustment f a c t o r s a t d i f f e r e n t t a r g e t r e l i a b i l i t i e s 91 i i i LIST OF FIGURES FIGURE(l) Experiment r e s u l t of Western hemlock lumber 4 FIGURE(2) E x p e r i m e n t a l l o a d h i s t o r y 18 FIGURE(3) C u m u l a t i v e d i s t r i b u t i o n u s i n g the damage model .24 FIGURE(4) C u m u l a t i v e d i s t r i b u t i o n u s i n g Gerhards' damage model (case 1 ) 28 FIGURE(5) C u m u l a t i v e d i s t r i b u t i o n u s i n g Gerhards' damage model (case 2) 29 FIGURE(6) P r o c e s s of c r a c k development 31 FIGURE(7) C u m u l a t i v e d i s t r i b u t i o n u s i n g N i e l s e n ' s f r a c t u r e mechanics model 37 FIGURE(8) Range and average v a l u e of t i m e - t o - f a i l u r e f o r d i f f e r e n t s t r e s s r a t i o 41 FIGURE(9) Extreme case f o r the damage model 42 FIGURE(10) Extreme case w i t h r s = 5 % of CDF f o r the damage model 44 FIGURE(11) Extreme case w i t h T s=95% of CDF f o r the damage model 45 FIGURE(12) Parameter adjustment and average v a l u e of t i m e - t o - f a i l u r e 47 FIGURE(13) T y p i c a l l o a d sequence f o r Quebec C i t y 54 FIGURE(14) T y p i c a l l o a d sequence f o r Vancouver ....55 FIGURE(15) P l a t e a u of the f a i l u r e s u r f a c e 63 FIGURE(16) F a i l u r e s u r f a c e and m u l t i p l e minima 65 FIGURE(17) Bounds f o r two f a i l u r e modes 68 FIGURE(18) M o n t e - C a r l o s i m u l a t i o n r e s u l t s 71 FIGURE(19) D i t l e v s e n bounds and M o n t e - C a r l o r e s u l t 72 FIGURE(20) Minimum J3 v a l u e f o r each number of y e a r s and M o n t e - C a r l o r e s u l t 75 FIGURE(21) R e s u l t s of R-F a l g o r i t h m and Mo n t e - C a r l o s i m u l a t i o n 77 FIGURE(22) 0 v a l u e s w i t h d i f f e r e n t y 79 i v FIGURE(23) R e l i a b i l i t y index 0 w i t h and w i t h o u t DOL (dead l o a d o n l y ) 83 FIGURE(24) R e l i a b i l i t y index 0 w i t h and w i t h o u t DOL (Quebec) , 85 FIGURE(25) R e l i a b i l i t y index 0 w i t h and w i t h o u t DOL (Winnipeg) 87 FIGURE(26) R e l i a b i l i t y index 0 w i t h and w i t h o u t DOL (Vancouver) 88 v ACKNOWLEDGMENT The author wishes to express his sincere appreciation and gratitude to his adviser, Dr.R.O.Foschi, for his invaluable advice, patient guidance and continuing encouragement during a l l stages of the work presented in th i s t h e s i s . v i DEDICATION TO MY DEAR PARENTS T a b l e of C o n t e n t s ABSTRACT i i LIST OF TABLES i i i LIST OF FIGURES i v ACKNOWLEDGMENT v i DEDICATION v i i 1. INTRODUCTION AND OBJECTIVES . 1 2. DAMAGE ACCUMULATION 3 2.1 DURATION OF LOAD EFFECT AND MATERIAL BEHAVIOR 3 2.2 A MODEL FOR DAMAGE ACCUMULATION 5 2.2.1 Ramp l o a d t e s t 8 2.2.2 Constant l o a d t e s t 9 2.2.3 A r b i t r a r y l o a d h i s t o r y t e s t 11 2.3 CALIBRATION OF THE DAMAGE MODEL TO EXPERIMENTAL DATA .13 2.3.1 S e l e c t i o n of p r o b a b i l i t y d i s t r i b u t i o n s f o r damage model parameters 14 2.3.2 C a l c u l a t i o n of t i m e - t o - f a i l u r e f o r the e x p e r i m e n t a l l o a d h i s t o r y 16 2.3.3 Model c a l i b r a t i o n p r o c e d u r e 20 2.4 COMPARISON WITH OTHER MODELS FOR STRENGTH DEGRADATION 25 2.4.1 Gerhards' c u m u l a t i v e damage model 25 2.4.2 N i e l s e n ' s f r a c t u r e mechanics model 30 2.4.3 F o s c h i - B a r r e t t damage model 38 2.4.4 The damage a c c u m u l a t i o n model: f u r t h e r d i s c u s s i o n 40 3. SINGLE MEMBER RELIABILITY ANALYSIS USING THE DAMAGE MODEL 48 3. 1 THE LOADING .49 3.1.1 Dead l o a d r e p r e s e n t a t i o n 49 3.1.2 L i v e l o a d (snow) r e p r e s e n t a t i o n 49 3.1.3 The l o a d c o m b i n a t i o n and l i m i t s t a t e d e s i g n e q u a t i o n 53 3.2 RELIABILITY ANALYSIS 57 3.2.1 R-F a l g o r i t h m and the p r o b a b i l i t y of f a i l u r e 57 3.2.2 Damage a c c u m u l a t i o n and the f a i l u r e f u n c t i o n 59 3.2.3 Some comments about u s i n g the R-F a l g o r i t h m w i t h t h i s t a s k .....61 3.2.3.1 The " p l a t e a u " of the f a i l u r e  f u n c t i o n G 61 3.2.3.2 M u l t i p l e ( l o c a l ) minima 62 3.2.4 The c a l c u l a t i o n of bounds f o r the p r o b a b i l i t y of f a i l u r e 66 3.3 NUMERICAL RESULTS 69 3.3.1 The M o n t e - C a r l o s i m u l a t i o n 70 3.3.2 The bounds of p r o b a b i l i t y of f a i l u r e 70 3.3.3 The r e s u l t from the R-F a l g o r i t h m 74 3.3.4 The e f f e c t of the l o a d r a t i o 7 76 3.3.5 The magnitude of the d u r a t i o n - o f - l o a d e f f e c t 81 3.3.5.1 Dead l o a d o n l y 81 3.3.5.2 Com b i n a t i o n of dead l o a d and snow  l o a d .82 3.4 DESIGN PROCEDURE AND CODE IMPLEMENTATION 86 4. CONCLUSION AND FURTHER RESEARCH 92 REFERENCES 94 i x 1. INTRODUCTION AND OBJECTIVES The s e r v i c e a b i l i t y or s a f e t y of a s t r u c t u r e may change over time as a r e s u l t of i n t e r a c t i o n s between the l o a d h i s t o r y and the m a t e r i a l ' s p r o p e r t i e s . T h i s i r r e v e r s i b l e phenomenon may be seen as the r e s u l t of an a c c u m u l a t i o n of damage over t i m e , and examples of such problems a r e : s t r e n g t h c h a n g i n g w i t h t i m e as a f u n c t i o n of the s t r e s s h i s t o r y ( f a t i g u e ) ; s t r e n g t h c h a n g i n g w i t h time because of v a r y i n g e n v i r o n m e n t a l c o n d i t i o n s ( i n f l u e n c i n g the r a t e of c o r r o s i o n , f o r example); s t i f f n e s s c h a n g i n g w i t h time as a f u n c t i o n of the l o a d h i s t o r y ( c r e e p and r e l a x a t i o n ) ; G e n e r a l l y s p e a k i n g , a c c u m u l a t i o n of damage i s a s t o c h a s t i c p r o c e s s , because the m a t e r i a l p r o p e r t i e s and the l o a d h i s t o r y a r e a l l random v a r i a b l e s . To p r e d i c t c u m u l a t i v e damage, a m a t h e m a t i c a l model i s r e q u i r e d t o i n v e s t i g a t e the r e l i a b i l i t y of the s t r u c t u r e f o r an i n t e n d e d s e r v i c e l i f e . S t r u c t u r e r e l i a b i l i t y r e s e a r c h has been i n t e n s i v e d u r i n g t h e p a s t decade. H a s o f e r and L i n d ( 1 9 7 4 ) i n t r o d u c e d the c o n c e p t of a r e l i a b i l i t y i ndex j3, i n v a r i a n t w i t h d e f i n i t i o n of the f a i l u r e f u n c t i o n . R a c k w i t z and F i e s s l e r ( 1 9 7 8 ) extended t h a t concept t o t h e c a s e s w i t h n o n - n o r m a l l y d i s t r i b u t e d v a r i a b l e s and proposed a n u m e r i c a l a l g o r i t h m f o r the c a l c u l a t i o n of 0. T h i s a l g o r i t h m i s a p o w e r f u l t o o l i n r e l i a b i l i t y a n a l y s i s , and g e n e r a l l y p e r m i t s 1 2 the c a l c u l a t i o n of a good a p p r o x i m a t i o n t o the p r o b a b i l i t y of f a i l u r e ( D i t l e v s e n , 1 9 7 8 ) . The main o b j e c t i v e of t h i s t h e s i s i s t o c o n s i d e r the s t r u c t u r a l r e l i a b i l i t y under c o n d i t i o n s of s t r e n g t h c h a n g i n g over time as a f u n c t i o n of the s t r e s s h i s t o r y . Thus, r e l i a b i l i t y i n s t r u c t u r e s under f a t i g u e c o n d i t i o n s f a l l w i t h i n the scope of the t h e s i s . More s p e c i f i c a l l y , t h e problem c o n s i d e r e d i s one of s t a t i c f a t i g u e as p r e s e n t i n tim b e r s t r u c t u r e s : t h e i r s t r e n g t h d e c r e a s e s as a f u n c t i o n of the i n t e n s i t y and d u r a t i o n of the a p p l i e d s t r e s s e s , even i f these s t r e s s e s a r e c o n s t a n t over time (Madsen et al.,1976,Foschi et al . ,1982). T h i s t h e s i s p r e s e n t s f i r s t a d i s c u s s i o n of damage a c c u m u l a t i o n models t o r e p r e s e n t s t r e n g t h d e g r a d a t i o n i n ti m b e r and then c o n s i d e r s a r e l i a b i l i t y a n a l y s i s f o r a s i n g l e s t r u c t u r a l member under dead l o a d o n l y or under a c o m b i n a t i o n of dead and snow l o a d s f o r a s e r v i c e l i f e of 30 y e a r s . F i n a l l y , t he i m p l e m e n t a t i o n of the r e l i a b i l i t y a n a l y s i s i n a s i m p l e d e s i g n g u i d e l i n e i s d i s c u s s e d . We s h a l l propose a new damage a c c u m u l a t i o n model and use i t i n the r e l i a b i l i t y a n a l y s i s v i a the R a c k w i t z - F i e s s l e r a l g o r i t h m f o r the index j3 a t the end of the s e r v i c e l i f e . 2. DAMAGE ACCUMULATION 2.1 DURATION OF LOAD EFFECT AND MATERIAL BEHAVIOR I t i s known t h a t wood i s s t r o n g e r under l o a d s of s h o r t - t e r m d u r a t i o n , and weaker i f the l o a d i s a p p l i e d f o r l o n g e r p e r i o d s . T h i s " d u r a t i o n - o f - l o a d e f f e c t " i n wood has been c l e a r l y shown by bending e x p e r i m e n t s (Madsen et al.,1976; F o s c h i et al.,1982) FIG(1) shows the t e s t d a t a from bending t e s t s on Western hemlock lumber r e p o r t e d by F o s c h i and B a r r e t t (1978). The d a t a , shown as c u m u l a t i v e p r o b a b i l i t y d i s t r i b u t i o n s f o r t h e t i m e - t o - f a i l u r e under a c o n s t a n t l o a d , c o r r e s p o n d t o two a p p l i e d s t r e s s l e v e l s . The h i g h e r l e v e l (4500 p s i or 31.0 MPa) c o r r e s p o n d e d t o the 2 0 _ t h p e r c e n t i l e of the s h o r t - t e r m s t r e n g t h d i s t r i b u t i o n . The lower l e v e l (3000 p s i or 20.7 MPa) c o r r e s p o n d e d t o the 5 - t' 1 p e r c e n t i l e of the s h o r t - t e r m s t r e n g t h d i s t r i b u t i o n . D e s i g n codes l i k e CAN3-086.1-M84 c o n t a i n " d u r a t i o n of l o a d " adjustment f a c t o r s f o r s t r e n g t h . These were d e r i v e d from t e s t s of s m a l l , c l e a r or d e f e c t - f r e e bending specimens of D o u g l a s - f i r under c o n s t a n t l o a d (Wood,1951). F u r t h e r r e s e a r c h showed t h a t the b e h a v i o r of f u l l - s i z e lumber d i f f e r s c o n s i d e r a b l y from t h a t o b s e r v e d i n the s m a l l specimen t e s t s . F u r t h e r m o r e , a t h e o r e t i c a l approach i s needed t o i n t e r p r e t the e x p e r i m e n t a l c o n s t a n t l o a d d a t a and t o p r e d i c t the t i m e - t o - f a i l u r e of wood i n s t r u c t u r a l s i z e s f o r d i f f e r e n t s e r v i c e l o a d h i s t o r i e s . 3 TIME-to-FAILURE CUMULATIVE DISTRIBUTION from experiment for western hemlock 0.9-0 .8 -0 .7-LOG(Tf) in hours 5 2.2 A MODEL FOR DAMAGE ACCUMULATION The c u m u l a t i v e damage we are concerned w i t h stems from m a t e r i a l b e h a v i o u r a t the m i c r o s c o p i c l e v e l . Because knowledge of b e h a v i o u r a t t h i s l e v e l i s g e n e r a l l y i n c o m p l e t e , i t i s d i f f i c u l t t o p o s t u l a t e c o r r e s p o n d i n g models f o r damage a c c u m u l a t i o n , b a s e d on p h y s i c a l l a w s . As an a l t e r n a t i v e , c u m u l a t i v e damage models have been proposed which a r e p h e n o m e n o l o g i c a l i n n a t u r e , t h a t i s , t hey a r e based on our c o r r e c t u n d e r s t a n d i n g of the phenomena a t the mac r o s c o p i c l e v e l and upon e x p e r i m e n t a l d a t a . L e t us propose a g e n e r a l type of damage a c c u m u l a t i o n model p r e s c r i b i n g t h e r a t e of damage growth t o be: da — = F ( r , a ) (2-1) d t Here a i s the measure of damage, d e f i n e d such t h a t a=0 i d e n t i f i e s an undamaged i n i t i a l s t a t e and a=1 c o r r e s p o n d s t o f a i l u r e ; t i s t h e time and r ( t ) i s the a p p l i e d s t r e s s h i s t o r y . T h i s form of model p o s t u l a t e s t h a t damage a c c u m u l a t e s a t a r a t e which i s not o n l y a f u n c t i o n of the a p p l i e d s t r e s s TR but a l s o as a f u n c t i o n of a l r e a d y a c c u m u l a t e d damage a. EQ(2»1) can be expanded as a power s e r i e s i n a, da — = F 0 ( r ) + F , ( r ) a + F 2 ( r ) a 2 ••• dt (2-2) 6 M i n e r ' s r u l e can be d e r i v e d as a s p e c i a l case of EQ(2«2), d r o p i n g the dependence on p r e v i o u s damage, da — = F 0 ( T ) (2-3) dt Under a c o n s t a n t s t r e s s r , M i n e r ' s r u l e g i v e s a l i n e a r a c c u m u l a t i o n of damage over time,and i s commonly used t o c a l c u l a t e the a c c u m u l a t e d damage i n metals.We w i l l show t h a t t h i s r u l e i s not adequate f o r m a t e r i a l s l i k e wood. A p a r t i c u l a r form of E Q ( 2 • 2 ) , i n c l u d i n g the f i r s t damage-dependent term, i s : da > — = a { r ( t ) - a 0 r j b + c { r ( t ) - o 0 } n a (2-4) d t s s Here a,b , c , n , a 0 a r e model p a r a m e t e r s . They a r e a l l assumed c o n s t a n t f o r a g i v e n s t r u c t u r a l member,but v a r y between members. r ( t ) i s the s t r e s s h i s t o r y a p p l i e d t o the member. r g i s the " s t a n d a r d " s h o r t - t e r m s t r e n g t h of the member,measured i n a ramp t e s t of s h o r t d u r a t i o n . T h i s model i n c l u d e the parameter a0 or " t h r e s h o l d s t r e s s r a t i o " . The p r o d u c t O 0 T C d e f i n e s a t h r e s h o l d t h a t must be exceeded f o r damage t o grow. That i s , when r ( t ) - a 0 r s ^ 0 t h e r e w i l l be no damage a c c u m u l a t i o n . EQ(2«4) can be i n t e g r a t e d t o f i n d the damage a a t any time T. L e t : 7 f i = { r(t)-a0Ts } and ( 2 - 5 ) f 2 = { T(t)-a0Ts } n ( 2 - 6 ) So t h a t , da — = af , +cf 2 a dt M u l t i p l y i n g both s i d e by e x p ( - / c f 2 d t ) , da — e x p ( - / c f 2 d t ) dt = a f , e x p ( - / c f 2 d t ) + c f 2 e x p ( - / c f 2 d t ) a or da — e x p ( - / c f 2 d t ) - c f 2 e x p ( - J c f 2 d t ) a dt = a f , e x p ( - / c f 2 d t ) d — {a e x p ( - / c f 2 d t ) } = a f ! e x p ( - / c f 2 d t ) dt ( 2 - 8 ) I n t e g r a t i n g , {a- e x p ( - / c f 2 d t ) }I = / a f , e x p ( - / c f 2 d t ) dt T ( 2 - 9 ) we can thus c a l c u l a t e the damage a at a c e r t a i n time T. The i n t e g r a l must be c a r r i e d out onl y over the time i n t e r v a l s when r{t)-o0Ts > 0. 8 2.2.1 RAMP LOAD TEST L e t us c o n s i d e r the p a r t i c u l a r case of a ramp l o a d t e s t of s h o r t d u r a t i o n ,where the l o a d i n c r e a s e s w i t h time a t a c o n s t a n t r a t e K g. r ( t ) = K s t (2«10) f , = ( K Q t - a 0 r _ ) b (2-11) f 2 = ( K s t - a 0 r s ) n (2-12) Then, - / c f 2 d t = - ( K Q t - a 0 r c ) ( n + 1 } (2-13) K_(n+1) s s S i n c e damage w i l l grow o n l y a f t e r a time t 0 , f o r which K g t o - a o r s = 0 , EQ(2'9) becomes { a . e x p ( - — ( K s t - a 0 r _ ) ( n + 1 ) )}T K„(n+1) s s t 0 T = / a ( K s t - a 0 r s ) b exp{- * \ ( K Q t - a 0 r Q ) ( n + 1 } }dt K_(n+1) s s (2-14) to "s I f the s h o r t term s t r e n g t h r g was o b t a i n e d u s i n g t h i s ramp t e s t , EQ(2'14) can be used t o compute the parameter a f o r any c h o i c e of the r e m a i n i n g paremeters b , c , n , o 0 , T S . That i s , when t = T g, a = 1, K S T S = T S 9 when t = t 0 , a = 0, K Q t 0 = a 0 r , T h e r e f o r e , exp{- C ( r s - a 0 r s ) ( n + l ) } K g(n+1) s s T = f S a ( K q t - a 0 O b exp{- ( R q t - a 0 r J ( n + 1 } }dt t 0 3 5 K (n+1) s 5 (2-15) Because K g i s a l a r g e number i n r e l a t i o n t o c, EQ(2«15) can be w r i t t e n i n an approximate manner, T 1 * / s a ( K _ t - a 0 r c ) b d t (2-16) r and t h e r e f o r e 1 - ( r _ - a 0 r _ ) b + 1 (2-17) K e ( b + 1 ) , S S F i n a l l y K (b+1) a ^ ( r s - a 0 r s ) b + 1 (2-18) 2.2.2 CONSTANT LOAD TEST C o n s i d e r now the case of a l o a d w h i c h i s h e l d c o n s t a n t u n t i l f a i l u r e o c c u r s a t the time T c: r ( t ) = r c (2-19) 1 0 f, = ( r c - a 0 r s ) b (2-20) f2 = ( r c - a 0 r s ) n (2-21 ) - f c f 2 d t = - c ( r c - a 0 r s ) n t ( 2 - 2 2 ) I n t r o d u c i n g EQ(2-22) i n t o EQ(2»9) we h a v e : {a«exp(- c ( r c - a 0 r s ) n t ) } J T = / a ( r c - a 0 T s ) D e x p { - c ( T c - a 0 T S ) n t } d t ( 2 - 2 3 ) When f a i l u r e o c c u r s at-T=T , a=1. T h u s , T 1 = /° a ( r c - a 0 r s ) b e x p { c ( r c - a 0 r s ) n ( T c ~ t ) } d t ( 2 - 2 4 ) o r a ( T - OnTR)B 1 = —r2- \xT ( e x P { c ( r c - C T o T s ) T c } " l ) ( 2 * 2 5 ) F i n a l l y : 1 F C ( T - a 0 r ) n T = — Ln{ °- § + 1 j ( 2 - 2 6 c c ( r c - a 0 r s ) n a ( r c - a 0 r s ) b 11 2.2.3 ARBITRARY LOAD HISTORY TEST The i n t e g r a t i o n of EQ(2'9) i s d i f f i c u l t f o r an a r b i t r a r y l o a d h i s t o r y and i t g e n e r a l l y r e q u i r e s a n u m e r i c a l i n t e g r a t i o n p r o c e d u r e . Here, w i t h o u t l o s s of g e n e r a l i t y , w e w i l l c o n s i d e r t h a t an a r b i t r a r y l o a d h i s t o r y may be r e p r e s e n t e d by the F e r r y B o r g e s - C a s t a n h e t a l o a d model ( F e r r y Borges et al.,1971). For an a r b i t r a r y l o a d p r o c e s s , t h i s model assumes t h a t : 1. the l o a d h i s t o r y i s s u b d i v i d e d i n t o e l e m e n t a r y time i n t e r v a l s of d u r a t i o n A t . 2. the l o a d i s c o n s t a n t i n each e l e m e n t a r y i n t e r v a l . 3. the l o a d s i n the e l e m e n t a r y i n t e r v a l s a re i d e n t i c a l l y d i s t r i b u t e d and m u t u a l l y independent random v a r i a b l e s . L e t us f i n d the a c c u m u l a t i o n of damage i n the time At from t ^ _ 1 t o t ^ . L e t ' s assume t h a t the c o n s t a n t l o a d l e v e l d u r i n g the p e r i o d At i s 7 ^ . T h e n , from EQ(2«9), {a-exp(- c(T--a0rt,)n t)}H t : = / a ( r i - o r 0 r s ) D e x p { - c ( r i - a 0 r s ) n t } d t (2-27) fci-1 Of c o u r s e , EQ(2«27) i s o n l y v a l i d when TJ>O 0T . I n t e g r a t i n g , we have 12 u^expf- C(T^-a0rs)n t i } - a i _ 1 exp{- c ( r i - a 0 r s ) n t ^ } a( TI - f f o f j 1 5 1 ° s — exp{- c ( r i - a 0 r s ) n t} \i ' c ( r r a 0 r s ) n ^ - ' ^ o ^ ( 2 . 2 8 ) A f t e r some a l g e b r a i c o p e r a t i o n s , w e o b t a i n the r e l a t i o n s h i p : a i = a i - 1 e x p ( c ( r i - C T 0 r s ) n A t ) + | ( T i - C T o r s ) b _ n {exp( c ( r i - a 0 r s ) n A t ) - 1 } (2-29) L e t t i n g K 0 ( i ) = exp{ c ( T i - a 0 T s ) n At} (2-30) and K i ( i ) = | ( T r a 0 T S ) b _ n ( K 0 ( i ) - l ) (2-31) the damage can be w r i t t e n as a i = a i - 1 K ° ( 1 ) + K i ( 1 J i f ri > O 0 T S (2-32) or 1 1-1 i f rL < a 0 T S (2-33) For an a r b i t r a r y l o a d h i s t o r y w i t h a o=0, the c u m u l a t i v e damage a f t e r each i n t e r v a l of time i s : 1 3 a, = K,(1 ) a 2 = a,Ko(2)+K, (2) = K, ( 1 ) K 0 ( 2 ) + K , ( 2 ) a 3 = a 2 K 0 ( 3 ) + K 1 ( 3 ) = K , ( 1 ) K 0 ( 2 ) K 0 ( 3 ) + K 1 ( 2 ) K 0 ( 3 ) + K 1 ( 3 ) n n a = K,(1) n K 0 ( j ) + K,(2) n K 0 ( j ) + + j=2 j=3 +. K, (n-1 ) K 0 (n) + K, (n) or n-1 n a = 2 { K , ( i ) n K 0 ( j ) } + K,(n) (2-34) i=1 j=i+1 2.3 CALIBRATION OF THE DAMAGE MODEL TO EXPERIMENTAL DATA The e x p e r i m e n t a l d a t a used f o r the c a l i b r a t i o n of the model were t e s t r e s u l t s f o r Western hemlock lumber a t two c o n s t a n t s t r e s s l e v e l s ( F o s c h i and B a r r e t t , 1 9 8 2 ) . The t e s t m a t e r i a l was a sample of 2 x 6 lumber ( n o m i n a l s i z e ) , v i s u a l l y g r a ded, No.2 "and b e t t e r " g r a d e , t e s t e d i n bending w i t h a span of 138 i n (3.51 m) under t h i r d - p o i n t l o a d i n g . The two s t r e s s l e v e l s , t h e i r s u s t a i n i n g time and the sample s i z e s a r e l i s t e d i n TABLE(1). Data from the s h o r t term s t r e n g t h t e s t s a r e l i s t e d i n TABLE(2). 2.3.1 SELECTION OF PROBABILITY DISTRIBUTIONS FOR DAMAGE MODEL PARAMETERS 1 4 STRESS LEVEL SUSTAINING TIME SAMPLE SIZE psi (MP a) hour s (year(s)) pi eces 4500 (31. 0) 8760 (I. 0) 400 3000 (20. 7) 30660 (3. 5) 200 TABLE(1) E x p e r i m e n t a l l o a d d a t a SHORT-TERM STRENGTH r MEAN = 6936.46 psi (47.83 MPa) MEDIAN = 6421.41 psi (44.28 MPa) STD.DEV , = 2833.34 psi (19.54 MPa) C.of.V = 0.408 = 385285.0 psi/hour (2656.74 MPa/hour) Median time to f a i l u r e T = 1 minute TABLE(2) S h o r t term s t r e n g t h T S 1 6 P r o b a b i l i s t i c c h a r a c t e r i z a t i o n o f t h e random v a r i a b l e s i n t h e damage m o d e l c a n be d i v i d e d i n t o two p a r t s : 1) t h e c h o i c e o f s u i t a b l e p r o b a b i l i t y d i s t r i b u t i o n s a n d 2) t h e c h o i c e o f a p p r o p r i a t e v a l u e s f o r t h e p a r a m e t e r s o f t h o s e d i s t r i b u t i o n s . To c h o o s e an a p p r o p r i a t e d i s t r i b u t i o n f o r e a c h p a r a m e t e r i s n o t s i m p l e . T h e r e a r e n o t s u f f i c i e n t e x p e r i m e n t a l r e s u l t s u n d e r d i f f e r e n t l o a d s e q u e n c e s w h i c h w o u l d h e l p i n d i s c r i m i n a t i n g d i s t r i b u t i o n a l a s s u m p t i o n s . U s i n g r e s u l t s f o r a s i n g l e l o a d s e q u e n c e one f i n d s t h a t d i f f e r e n t d i s t r i b u t i o n s a p p e a r t o f i t t h e a v a i l a b l e d a t a e q u a l l y w e l l . L a c k i n g a more e x t e n s i v e d a t a b a s e , a s t r i c t p r o c e d u r e t o d e c i d e t h e d i s t r i b u t i o n t y p e f o r t h e m o d e l p a r a m e t e r s c a n n o t be e s t a b l i s h e d . N e v e r t h l e s s , t h e r e a r e some g u i d e l i n e s t o be f o l l o w e d i n t h e s e l e c t i o n : s i n c e a l l p a r a m e t e r s ( b , c , n , o - 0 ) must be p o s i t i v e , i t was assumed t h a t t h e y were l o g n o r m a l l y d i s t r i b u t e d a n d f u l l y c h a r a c t e r i z e d by t h e i r mean a n d s t a n d a r d d e v i a t i o n . The d i s t r i b u t i o n o f r g i s a s s u m e d known f r o m t h e a n a l y s i s o f t h e s h o r t - t e r m t e s t s . 2.3.2 CALCULATION OF TIME-TO-FAILURE FOR THE EXPERIMENTAL  LOAD HISTORY I n o r d e r t o p e r f o r m t h e c a l i b r a t i o n , we n e e d t o know t h e t i m e - t o - f a i l u r e o f a t e s t s p e c i m e n , c o m p u t e d u s i n g t h e damage m o d e l , f o r t h e same l o a d h i s t o r y u s e d i n t h e 17 exper iment. In t h e t e s t , the l o a d , s t a r t i n g from z e r o , i n c r e a s e d a t then was kept c o n s t a n t u n t i l the end of the t e s t . T h e r e f o r e , the l o a d h i s t o r y was a c o m b i n a t i o n of ramp l o a d and c o n s t a n t l o a d (see F I G ( 2 ) ) . A c c o r d i n g l y , the f a i l u r e time p r e d i c t e d by t h e model must be c a l c u l a t e d i n two s t e p s . 1. For the ramp l o a d from t = t 0 t o t = t c , a " s t a n d a r d i z e d " r a t e K t o the p r e d e c i d e d l e v e l r , and (2-35) r ( t ) = K s t , r ( t c ) = K s t c = r c (2-36) a ( t c ) = a c (2-37) Then, from EQ(2'14) c n+1 {a'exp(-K s(n+1 ) ( K g t - O o T g ) (K st-o- 07" s) n+1 )dt (2-38) S i n c e c/K s << 1, and i n t r o d u c i n g EQ(2«l8),we get an a p p r o x i m a t e e x p r e s s i o n f o r the damage a'. Stress F I G ( 2 ) Experiment Load H i s t o r y 19 ( r c - a 0 T S ) b+1 ( r s - a 0 r s ) 2. For t h e c o n s t a n t l o a d from t=t t o the time of f a i l u r e r ( t ) = r c (2-40) T h e r e f o r e , by i n t e g r a t i o n of EQ(2'9), 1 = a c - e x p { c ( T C - O 0 T S ) n ( T f - t c ) } K s ( b + 1 ) b _ n r r : — „ ^ ^ b+i ( T C " ^ S ' c ( T s - a 0 r s ) •(exp{c ( r c - a 0 r s ) n ( T f - t c ) } - l ) (2-41) F i n a l l y 1 , 1+(a/c) ( r _ - a 0 r _ ) b-n T f = t„ + =• Ln{ ——:——- F^TTT } rc - o 0 T S ) a c+(a/c) ( r c - a 0 r s : However, 1 ) l f t he p i e c e i s v e r y weak,i.e. TQ > T S , i t f a i l s d u r i n g l o a d a p p l i c a t i o n and T F = ^ (2-43) K s 20 2 ) I f the p i e c e i s v e r y s t r o n g , i . e . r c < a 0 r s , t h e a p p l i e d s t r e s s l e v e l does not exceed the t h r e s h o l d and the p i e c e w i l l not accumulate damage.Thus T f = co ( no f a i l u r e ) (2-44) 2.3.3 MODEL CALIBRATION PROCEDURE A n o n - l i n e a r f u n c t i o n m i n i m i z a t i o n p r o c e d u r e was used t o d e t e r m i n e the c h a r a c t e r i s t i c d i s t r i b u t i o n a l parameters of each of t h e model v a r i a b l e s . The s t e p s i n t h i s p r o c e d u r e were as f o l l o w s : 1. For the random v a r i a b l e s i n the model, l e t : * b = b X, c = c X 2 * n = n X 3 c r 0 = o0 X 4 * * * * Where b ,c ,n , o0 a r e chosen s c a l i n g c o n s t a n t s and X 1 , X 2 , X 3 , X 4 a r e a l l l o g n o r m a l random v a r i a b l e s . For l o g n o r m a l d i s t r i b u t i o n s , t h e f o l l o w i n g r e l a t i o n s h i p h o l d s : Ln(X-) = Ln(M-)-0.5Ln(1+( — ) 2)+R n•/Ln(1+ ^ ) 2 (2-45) 1 1 M i n"V M i Where M^ i s the mean v a l u e of d i s t r i b u t i o n of X^, a^ i s the s t a n d a r d d e v i a t i o n of d i s t r i b u t i o n of and R i s a s t a n d a r d random normal v a r i a b l e , 0 < R < 1. 21 Choose i n i t i a l v a l u e s f o r each mean v a l u e and s t a n d a r d d e v i a t i o n of the model parameters.These means and s t a n d a r d d e v i a t i o n s a r e the c a l i b r a t i o n v a r i a b l e s . U s i n g the d i s t r i b u t i o n of s h o r t - t e r m s t r e n g t h r , g e n e r a t e , b y computer s i m u l a t i o n , a sample of NT t i m e - t o - f a i l u r e s T£,for each t e s t s t r e s s l e v e l , u s i n g EQ(2-42). Rank the f a i l u r e t i m e s and d e t e r m i n e the c u m u l a t i v e p r o b a b i l i t y d i s t r i b u t i o n c o r r e s p o n d i n g t o each s t r e s s l e v e l . Compare the s i m u l a t e d d i s t r i b u t i o n w i t h the e x p e r i m e n t a l d a t a and c a l c u l a t e a g l o b a l d e v i a t i o n measure The measure of d e v i a t i o n $ used was N T f • <i> = Z (1.0- ) 2 (2-46) i- 1 T d i i n w hich N i s the number of the p r o b a b i l i t y l e v e l s used f o r the c o m p a r i s o n , T ^ i s the s i m u l a t e d f a i l u r e time c o r r e s p o n d i n g t o the same p r o b a b i l i t y l e v e l as the t e s t r e s u l t T d i . Compute the g r a d i e n t s of EQ(2'46) w i t h r e s p e c t t o the c a l i b r a t i o n v a r i a b l e s . That i s , compute 9<i>/3M^ and d$/do^. T h i s was done n u m e r i c a l l y f o r each v a r i a b l e : a. Change v a r i a b l e X^ w i t h a p o s i t i v e increment AX^ and r e p e a t the c a l c u l a t i o n of $ u s i n g the same sequence of random numbers R n. L e t the r e s u l t be Repeat 22 f o r a n e g a t i v e v a r i a b l e change (~AX^) and o b t a i n V -b. C a l c u l a t e the p a r t i a l d e r i v a t i v e of = -= i - (2-47) 3X i 2AX i 7. The v a l u e of $ and t h e g r a d i e n t V$ f o r a c h o i c e of c a l i b r a t i o n v a r i a b l e s was used i n an a u t o m a t i c computer s e a r c h a l g o r i t h m u s i n g a quasi-Newton method,to l o c a t e the approximate g l o b a l minimum of $. In o r d e r t o a v o i d wide f l u c t u a t i o n s i n the c a l i b r a t i o n r e s u l t when t h e ' p r o c e d u r e i s s t a r t e d w i t h a d i f f e r e n t random seed, i t i s i m p o r t a n t t o a p p r o p r i a t e l y choose the sample s i z e NT, the s t e p AX^ f o r t h e g r a d i e n t c a l c u l a t i o n and the convergence c r i t e r i o n f o r the s e a r c h . In our c a s e , we chose a sample s i z e NT=1000, a s t e p AX^=0.001X^ and a convergence t a r g e t e=0.00l. The l a t t e r means t h a t the c a l c u l a t i o n s t e r m i n a t e d when changes i n X^ of t h e s i z e e-AX^ d i d not reduce the f u n c t i o n v a l u e f o r a l l i=1,2,•••,n. TABLE(3) shows the i n i t i a l a s s u m p t i o n s and the f i n a l r e s u l t of the c a l i b r a t i o n of t h e damage model t o the e x p e r i m e n t a l d a t a . For the i n i t i a l v a l u e s of the parameters,$=62052.89,for the f i n a l v a l u e s of the p a r a m e t e r s , <t> = 9.099. FIG(3) shows the f i t of the model t o the d a t a by comparing the c o r r e s p o n d i n g c u m u l a t i v e d i s t r i b u t i o n 2 3 INITIAL ASSUMPTION VARIABLE DISTRIBUTION MEAN STD.DEVIATION b Lognormal 50. 00 10. 00 c Lognormal 0. 10-io-5 0. 20-10-6 n Lognormal 1. 00 0. 20 Lognormal 0. 50 0. 10 FINAL RESULT VARIABLE DISTRIBUTION MEAN STD.DEVIATION b Lognormal 35. 204 6. 589 c Lognormal 0. 1559-IO'6 0. 9621 -IO'1 n Lognormal 1. 429 0. 139 o0 Lognormal 0. 578 0. 163 TABLE(3) Parameters f o r t h e damage model TIME-to-FAILURE CUMULATIVE DISTRIBUTION from experiment and from theory The Damage Model Legend EXPERIMENT THEORETICAL MODEL t II if n it .s-j^L ' ~~ t or 1 I I I 1 1 I I 1 -3 -2 -1 0 1 2 3 4 5 LOG(Tf) in hours 25 f u n c t i o n s . I t i s apparent t h a t the damage model i s c a p a b l e of f o l l o w i n g the e x p e r i m e n t a l t r e n d s v e r y w e l l . 2.4 COMPARISON WITH OTHER MODELS FOR STRENGTH DEGRADATION I t i s u s e f u l t o compare p r e d i c t i o n s from the damage model proposed here w i t h t h o s e from o t h e r models f o r s t r e n g t h d e g r a d a t i o n as a f u n c t i o n of s t r e s s h i s t o r y . 2.4.1 GERHARDS' CUMULATIVE DAMAGE MODEL A c u m u l a t i v e damage model has been p r o p o s e d ( G e r h a r d s , 1 9 8 6 ) , a c c o r d i n g t o which, the r a t e of damage growth i s g i v e n by: da r ( t ) — = exp(-a+b ) (2-48) where a i s the damage ( 0 ^ a < l ) , r ( t ) i s s t r e s s h i s t o r y , T s i s the s h o r t - t e r m s t r e n g t h measured i n a ramp t e s t and a, b a r e c o n s t a n t s f o r a g i v e n p o p u l a t i o n . T h i s i s then a p a r t i c u l a r c a s e of EQ(2'2) where the damage r a t e does not depend on the damage a l r e a d y accumulated. C o n s i d e r now the model b e h a v i o u r f o r two d i f f e r e n t s t r e s s h i s t o r i e s . 1)Ramp l o a d t e s t used t o de t e r m i n e r ^ : In t h i s c a s e , r ( t ) = K g t . T h e r e f o r e , 26 da K t — = exp(-a+b - 2 - ) (2-49) d t r„ And T_ K_t a ( t ) = {exp(-a+b )-exp(-a)} (2-50) S i n c e a t f a i l u r e , o=1, we can get an e x p r e s s i o n f o r the parameter a, 1 = e a ( e b - 1 ) (2-51) bK s or e = 7-FT T (2-52) r ( e b - 1 . 0 ) I t i s c l e a r t h a t , c o n t r a r y t o the model's d e f i n i t i o n , a and b cannot be b o t h c o n s t a n t s w h i l e r g i s a random v a r i a b l e . 2 )For the e x p e r i m e n t a l s t r e s s h i s t o r y of FIG(2) In t h i s c a s e , r ( t ) = T , t > T 0 da bK_ T b . _ exp(b -°- ) (2-53) d t r s ( e " - 1 . 0 ) r s And t h e r e f o r e , 27 B KC T a(t> = 7-TT exp(b -°- ) (t-T0)+a (2-54) r s ( e b - 1 . 0 ) . r g a c i s the accumulated damage a t time T 0 , and can be computed from EQ(2'50). F i n a l l y , the time t o f a i l u r e T f i s g i v e n f o r a= 1 , Tf. = T 0+ — ( e x p { b ( l - IS- )}-D (2-55) b K s A c a l i b r a t i o n of t h i s model t o the same Western hemlock bending d a t a i s shown i n F I G ( 4 ) , u s i n g t h e p r o c e d u r e p r e v i o u s l y d i s c u s s e d . I t i s apparent t h a t t h i s model i s t o o s t i f f t o r e p r e s e n t the d a t a t r e n d . I f one e n f o r c e s a b e t t e r f i t f o r the s h o r t t i m e s , as FIG(5) shows, i t i s even c l e a r e r t h a t the model i s not a b l e t o f o l l o w the da t a t r e n d a t l o n g e r t i m e s . T h i s f a c t r e v e a l s the e f f e c t of the a c c u m u l a t e d damage i n the p r o c e s s of damage a c c u m u l a t i o n : G e r h a r d s ' model cannot f i t the l o n g - t i m e e x p e r i m e n t a l r e s u l t s because i t l a c k s a damage-dependent f a c t o r . T h i s can be pr o v e d i f we attempt t o c a l i b r a t e our damage model w i t h the c o n s t r a i n t c = 0. More g e n e r a l l y , t h i s f a c t a l s o shows t h a t a s p e c i a l type of EQ(2'2) w i t h the f i r s t term o n l y , l i k e M i n e r ' s r u l e or Gerhards'model, can not p r o p e r l y r e p r e s e n t t h e r a t e of damage a c c u m u l a t i o n f o r m a t e r i a l s l i k e wood. TIME-to-FAILURE CUMULATIVE DISTRIBUTION from experiment and from theory Legend EXPERIMENT THEORETICAL MODEL Gerhards' Model - 3 LOG(Tf) in hours CD TIME-to-FAILURE CUMULATIVE DISTRIBUTION from experiment and from theory Legend EXPERIMENT THEORETICAL MODEL Gerhards' Model (case 2) -3 LOG(Tf) in hours V£> 30 From the e x p e r i m e n t a l d a t a , an upward t r e n d i s s i g n i f i c a n t a f t e r a c e r t a i n time i n the c u m u l a t i v e p r o b a b i l i t y d i s t r i b u t i o n . T h i s t r e n d can be p r e d i c t e d w i t h the i n t r o d u c t i o n of a damage-dependent f a c t o r . At f i r s t the f a i l u r e p r o c e s s i s c o n t r o l l e d by the s t r e s s - d e p e n d e n t term, when the c o n t r i b u t i o n from the damage-dependent f a c t o r i s t r i v i a l because a i s too s m a l l . A f t e r some t i m e , when a r e a c h e s a s u f f i c i e n t l y h i g h v a l u e , t h e damage-dependent f a c t o r becomes more and more i m p o r t a n t and the damage grows e x p o n e n t i a l l y , a c c e l e r a t i n g f a i l u r e . 2.4.2 NIELSEN'S FRACTURE MECHANICS MODEL U n l i k e t h e damage models d i s c u s s e d u n t i l now, N i e l s e n ' s model approaches the problem u s i n g f r a c t u r e mechanics t h e o r y f o r v i s c o e l a s t i c b o d i e s (Nielsen,1978,1985) A c c o r d i n g t o N i e l s e n , w o o d may be c o n s i d e r e d as a c r a c k e d v i s c o e l a s t i c m a t e r i a l , a n d i t s m e c h a n i c a l b e h a v i o r can be d e s c r i b e d i n s u f f i c i e n t d e t a i l by c o u p l i n g the t h e o r i e s of v i s c o e l a s t i c i t y and f r a c t u r e mechanics. FIG(6) shows the p r o c e s s of c r a c k development w i t h time (Madsen et al .,1982) : 1. A c r a c k of l e n g t h 2c i n i t s i n i t i a l c o n d i t i o n , f r e e of s t r e s s e s ; 2. A s t e p l o a d a p p l i e d f a r away causes the c r a c k t o open. Here, R i s the w i d t h of the p l a s t i c i z e d zone,5 i s the c r a c k o p e n i n g d i s p l a c e m e n t . The a p p l i e d s t r e s s i s a, but i n the p l a s t i c i z e d zone, the s t r e s s e q u a l s a c o n s t a n t F I G ( 6 ) P r o c e s s of Crack Development 32 u l t i m a t e s t r e s s a-^; 3. Crack c o n t i n u e s t o open, the whole l e n g t h 2c and the w i d t h of the p l a s t i c zone R remains unchanged; 4. At the time t = t g , the c r a c k opening reaches a c r i t i c a l v a l u e 8 c r . The c r a c k l e n g t h 2c and the w i d t h of the p l a s t i c zone a r e s t i l l unchanged; 5. S t a r t i n g from t = t s , the c r a c k b e g i n s p r o p a g a t i o n . Crack o p e n i n g remain c o n s t a n t a t 8 c r , but the c r a c k l e n g t h and the w i d t h of the p l a s t i c zone i n c r e a s e ; 6. At the time t = t c r , the c r a c k grows a t h i g h speed and the body f a i l s . U s i n g t h i s c r a c k model and the e l a s t i c - v i s c o e l a s t i c c o r r e s p o n d e n c e p r i n c i p l e ( F l u g g e , 1 9 6 7 ) , N i e l s e n d e r i v e d the r e l a t i o n s h i p s f o r the c a l c u l a t i o n of the time t g r e q u i r e d t o s t a r t c r a c k p r o p a g a t i o n . These r e l a t i o n s h i p s depend on the l o a d h i s t o r y and a r e , r e s p e c t i v e l y : For ramp l o a d : ( ) 2 = 1.0+ ( ) b (2-56) K s t s (b+D(b+2) r For c o n s t a n t l o a d : ( ^ 1 ) J = 1 .0+( ^ ) b (2-57) a T Here a c r i s the s h o r t - t e r m s t r e n g t h measured i n a v e r y f a s t ramp t e s t ( i . e . K g = °°) , the parameters b and r c o r r e s p o n d t o the assumed c r e e p f u n c t i o n J ( t ) : J ( t ) = - {l.0+( - ) b} E T 3 3 (2-58) where r i s the d e f o r m a t i o n d o u b l i n g t i m e , and E i s the i n i t i a l modulus of e l a s t i c i t y . I f c ( t ) and c 0 a r e , r e s p e c t i v e l y , the c u r r e n t and o r i g i n a l c r a c k l e n g t h , N i e l s e n shows t h a t the n o n - d i m e n s i o n a l c r a c k l e n g t h r a t i o c ( t ) K i t ) = (2-59) c 0 obeys, a f t e r the time t g , the f o l l o w i n g r e l a t i o n s h i p : dic(t) _ Tr2 ( o c r U ( t ) ( a ( t ) / a c r ) 2 ) 1 + 1 / b dt 8qr a± ( 1 . 0 - K ( t ) ( a ( t ) / a Q r ) 2 ) 1 / b (2-60) The c o n s t a n t q i s d e f i n e d as (b+1)(b+2) . 7. q = ( ) 1 / b (2-61) F a i l u r e o c c u r s when the r a t e of c r a c k growth reaches an i n f i n i t e v a l u e , o r the denominator of EQ(2-60) v a n i s h e s : a ( T f ) 1 . 0 - K ( T f ) ( ) 2 = 0 (2-62) a c r I n t e g r a t i o n of the r e l a t i o n s h i p of EQ(2«60), w i t h the f a i l u r e c o n d i t i o n EQ(2«62), would p e r m i t the c a l c u l a t i o n of the t i m e - t o - f a i l u r e T^. But, g e n e r a l l y , t h i s i s v e r y d i f f i c u l t i f not i m p o s s i b l e , e x c e p t f o r a s p e c i a l c a s e . 34 To e x p l a i n t h i s , l e t us d e f i n e the q u a n t i t y : ( o - / a ( t ) ) 2 * = — ^ (2-63) Kit) which i n i t i a l l y ( a t t = t g ) t a k e s t he v a l u e *° = ( f f c r / f f o ) 2 f where a0 i s the s t r e s s a t t = t s . At f a i l u r e , f r o m EQ(2'62) * = 1. Thus, t h i s q u a n t i t y behaves as a damage parameter. The r a t e of growth of * can be d e r i v e d from EQ(2-60): d* ait) 7T2 o(t) 1 — +*{2 + ( ) 2 } = 0 (2-64) r l d t " a i t ) 8qr ' a, ' (¥-l)T7 E where a i t ) s t a n d s f o r the d e r i v a t i v e d a ( t ) / d t . I t i s c l e a r , t h a t o n l y when the a p p l i e d s t r e s s i s c o n s t a n t ( s t e p f u n c t i o n T Q ) , a = 0, the v a r i a b l e s t and * i n EQ(2'64), and t h e r e f o r e , the v a r i a b l e s t and K i t ) i n EQ(2'60) can be s e p a r a t e d and EQ(2'60) can be i n t e g r a t e d between the lower l i m i t ^,=1 t o the upper l i m i t K 2 = ( A C R / T C ) 2 g i v e n by EQ(2'62). The f a i l u r e - t i m e T^ can be c a l c u l a t e d a s : 8 q r cr r a, «2 ( 0 - l ) l / b T f = t + — ( )2 i -±- ) 2 / d9 *2 rc a c r 1 d (2-65) N i e l s e n ' s model was c a l i b r a t e d t o the same e x p e r i m e n t a l d a t a as the p r e v i o u s damage models. T h i s c a l i b r a t i o n was perfo r m e d a l l o w i n g f o r some a p p r o x i m a t i o n s : l ) B e c a u s e the s h o r t - t e r m s t r e n g t h under an i n f i n i t e r a t e of l o a d i n g i s i n f a c t i m p o s s i b l e t o get,we 35 .approximately chose the s h o r t - t e r m s t r e n g t h T s o b t a i n e d w i t h the r a t e K g. 2)Because of the d i f f i c u l t y i n the t i m e - t o - f a i l u r e c a l c u l a t i o n f o r a ramp load,we a p p r o x i m a t e l y chose a s t e p f u n c t i o n l o a d i n g i n s t e a d of the combined l o a d used i n the e x p e r i m e n t . To a v o i d p o s s i b l e b i a s due t o t h i s a s s u m p t i o n , the c a l i b r a t i o n was p e r f o r m e d u s i n g o n l y e x p e r i m e n t a l t i m e s - t o - f a i l u r e l o n g e r than 10 h o u r s . In N i e l s e n ' s model , t h e c a l i b r a t i o n parameters were the exponent b, the d o u b l i n g time r and the s t r e s s r a t i o r= (o-^/o ) . Each of them were assumed t o be independent,random v a r i a b l e s . S i n c e the d o u b l i n g time T and the s t r e s s r a t i o r c o u l d not be n e g a t i v e , t h e i r d i s t r i b u t i o n were assumed t o be l o g n o r m a l . The exponent b was assumed t o be n o r m a l , s i n c e the c a l i b r a t i o n showed i t t o have a low v a r i a b i l i t y . TABLE(4) c o n t a i n s t h e f i n a l r e s u l t s f o r the mean v a l u e and the s t a n d a r d d e v i a t i o n of each v a r i a b l e . F I G ( 7 ) shows the f i t of N i e l s e n ' s model t o the e x p e r i m e n t a l d a t a u s i n g the p a r a m e t e r s from T A B L E ( 4 ) . The r e s u l t i s s a t i s f a c t o r y . I t may be s a i d t h e r e f o r e , t h a t t h i s f r a c t u r e mechanics m o d e l , l i k e our damage model,can r e p r e s e n t s a t i s f a c t o r i l y the e x p e r i m e n t a l d a t a from Western hemlock lumber i n b e n d i n g . Some a d d i t i o n a l comments may be o f f e r e d about N i e l s e n ' s model. F i r s t , i t i s r a t h e r c o m p l i c a t e d t o use. In g e n e r a l , t h e r a t e of c r a c k growth i s c o n t r o l l e d by a 36 VARIABLE DISTRIBUTION MEAN STD. DEVIATION T Lognormal 1017.38 hours 536.11 hours Normal 0. 249 0. 0415 r Lognormal 2. 764 2. 883 (r=l. 0+r ) TABLE(4) Parameters f o r N i e l s e n ' s model 0.9-TIME-to-FAILURE CUMULATIVE DISTRIBUTION from experiment and from theory 0.8->. 0.7-m < 0.6-m O ce ^ 0.5H Legend EXPERIMENT THEORETICAL MODEL Nielsen's Model _i 0.4-2 <-> 0.3-J 0.2-0.1-i i •i -3 -2 LOG(Tf) in hours O J 38 f i r s t - o r d e r , n o n l i n e a r e q u a t i o n f o r / c ( t ) , w i t h c o e f f i c i e n t s which a r e a l s o f u n c t i o n s of time t . To f i n d a c l o s e d - f o r m s o l u t i o n t o t h i s e q u a t i o n i s g e n e r a l l y v e r y d i f f i c u l t and the o n l y r e c o u r s e i s t o u t i l i z e n u m e r i c a l i n t e g r a t i o n p r o c e d u r e s . T h i s f a c t c o m p l i c a t e s and makes cumbersome the u t i l i z a t i o n of t h i s model f o r s i m u l a t i o n s i n c a l c u l a t i o n s f o r r e l i a b i l i t y - b a s e d d e s i g n . Second,the parameters i n N i e l s e n ' s model r e f l e c t the v i s c o e l a s t i c b e h a v i o u r around the c r a c k t i p . The v a r i a b i l i t y i n these v a l u e s w i l l depend on v e r y l o c a l i z e d c o n d i t i o n s of g r a i n d e v i a t i o n s and d e n s i t y , and a r e l a t i v e l y wide f l u c t u a t i o n i s e x p e c t e d . F u r t h e r m o r e , some of th e s e parameters ( l i k e or o ) can not be s i m p l y d e r i v e d from t e s t s . So t h a t , a l t h o u g h more " p h y s i c a l " i n meaning than those f o r the damage models, t h e s e parameters can o n l y be o b t a i n e d by model c a l i b r a t i o n and not from independent, p h y s i c a l t e s t s . To i l l u s t r a t e t h i s f a c t , t he d o u b l i n g time r i s a p p r o x i m a t e l y 42 days from the model c a l i b r a t i o n t o the t e s t d a t a . T h i s i s t o o s h o r t a time f o r bending c r e e p of lumber. A r e c e n t r e p o r t by Hoyle et al . (1985) on bending c r e e p of Douglas f i r beams e s t i m a t e s d o u b l i n g t i m e s i n the o r d e r of 10 5 h o u r s . 2.4.3 FOSCHI-BARRETT DAMAGE MODEL In t h i s model, 39 da r ( t ) , — = a{ - a 0 } D + X a ( t ) (2-66) dt r s where a and X a r e independent random v a r i a b l e s l o g n o r m a l l y d i s t r i b u t e d , b and aQ a r e c o n s t a n t s and r g i s the s t a n d a r d s h o r t - t e r m s t r e n g t h . T h i s damage model was used by F o s c h i et al . t o f i t the Western hemlock e x p e r i m e n t a l d a t a and r e s u l t s were s a t i s f a c t o r y . But a f u r t h e r s t u d y showed t h a t , a l t h o u g h t h i s model can be s u c c e s s f u l l y used t o p r e d i c t the e x p e r i m e n t a l t r e n d s under ramp-and-constant l o a d c o n d i t i o n , i t had a s e r i o u s s h o r t c o m i n g which became apparent f o r o t h e r l o a d s i t u a t i o n s . The problem was i n the damage-dependent term X a ( t ) . Here,X was a random v a r i a b l e , b u t i t was a l s o a c o n s t a n t f o r a g i v e n member. Suppose the damage a has a l r e a d y accumulated t o a s i g n i f i c a n t amount and a l s o suppose t h a t d u r i n g the f o l l o w i n g t ime t h e s t r e s s r a t i o T ( t ) / r s was j u s t an i n f i n i t e s i m a l h i g h e r than the t h r e s h o l d . Then, a l t h o u g h the f i r s t term i n the model would be v e r y c l o s e t o z e r o , the damage w i l l s t i l l grow e x p o n e n t i a l l y because X i s a c o n s t a n t . One i s then f a c e d w i t h the u n r e a s o n a b l e p r e d i c t i o n t h a t a s i t u a t i o n of almost no s t r e s s w i l l l e a d t o f a i l u r e over t i m e i f the member has been p r e v i o u s l y s u f f i c i e n t l y damaged. The proposed new model f o r damage a c c u m u l a t i o n i s an e x t e n s i o n of t h i s e a r l i e r v e r s i o n by F o s c h i and B a r r e t t , i n t h a t the damage-dependent term i s ta k e n t o be a f u n c t i o n of 40 s t r e s s l e v e l a l s o , t h e r e b y p r e v e n t i n g the above mentioned u n r e a s o n a b l e s i t u a t i o n . 2.4.4 THE DAMAGE ACCUMULATION MODEL: FURTHER DISCUSSION Le t us now stu d y i n more d e t a i l the b e h a v i o u r of the damage model. FIG(8) shows the r e l a t i o n s h i p between a c o n s t a n t a p p l i e d s t r e s s r a t i o and f a i l u r e time,computed from the damage model. The s t r e s s r a t i o a i s d e f i n e d as the r a t i o between the c u r r e n t s t r e s s T and the s h o r t - t e r m s t r e s s r s : a = ( r / r s ) . For t he h i g h e r s t r e s s r a t i o s t he t i m e - t o - f a i l u r e has a r e l a t i v e l y low v a r i a b i l i t y . But f o r the lower s t r e s s r a t i o s , the t i m e s - t o - f a i l u r e a r e much more v a r i a b l e . T h i s may be due t o the assu m p t i o n of parameter independence.Thus, a specimen may f a i l i n a s h o r t time because e i t h e r t h e specimen i s too weak (T too l o w ) , o r the t h r e s h o l d s t r e s s r a t i o f o r t h a t specimen i s t o o low,or t h e damage parameter c i s t o o h i g h . S i n c e the t h r e s h o l d s t r e s s r a t i o i s a l s o a random v a r i a b l e , some f a i l u r e s o ccur a t s t r e s s r a t i o s as low as a=0.4,which i s lower than the mean v a l u e of the t h r e s h o l d . What i n f l u e n c e s do the i n d i v i d u a l p a r a m e t e r s have on the t i m e - t o - f a i l u r e ? The model parameters c and r s a r e the two more i m p o r t a n t . FIG(9) shows the model p r e d i c t i o n s f o r two c a s e s : 1) TQ and c e q u a l t o the 9 5 ~ t b p e r c e n t i l e of t h e i r r e s p e c t i v e c u m u l a t i v e d i s t r i b u t i o n f u c t i o n s ( C D F ) , and 2) T s and c e q u a l t o t h e i r 5 - t ^ p e r c e n t i l e s . D iQ 0) 0) D a > < O a> < cu h-1 c cn o r r i-h i-l n> »-3 « i j Ul H - H w 3 O ft) ^ XI I 00 CJ r t ^ r r O O 0) C o i-l r> r f 1.1 1-0.9 to 0.7-00 U l CC l/> 0.6-0.5-0.4-0.3 sample size: 300 o OO ODCCtID OO COGD O GDO dB OO [•n . i im»in»iiMB»i i inii i im (HDD CO O (3D O a n x s x D o o co o o aaiMK OCUJO ) CODOaDOOO co CO CO ODD i o < -3 -2 -1 —r 0 - r 3 SURVIVAL PROBABILITY 1.00% 2.67% 3.33% 5.67% 11.33% 21.67% 37.33% 69.67% 85.00% L0G(T) (in hours) (6)91.3 43 From FIG(9) i t i s c l e a r t h a t , f o r a s t r e s s r a t i o a=1,these two ca s e s j u s t r e p r e s e n t the extremes of the 90% range of s h o r t - t e r m t i m e s - t o - f a i l u r e i n F I G ( 8 ) . There i s a c r o s s i n g p o i n t a t a s t r e s s r a t i o a=0.9. Above t h a t point,weak p i e c e ( r s = 5 % of CDF) f a i l s f i r s t and s t r o n g p i e c e ( r s = 9 5 % of CDF) f a i l s l a t e r . T h i s i s r e a s o n a b l e and u n d e r s t a n d a b l e . But below the c r o s s - o v e r p o i n t , t h e s i t u a t i o n i s i n r e v e r s e . T h i s s u g g e s t s t h a t the parameter c p l a y s an i m p o r t a n t r o l e f o r l o n g e r t i m e s t o f a i l u r e . In f a c t , t h e parameter c p l a y s e x a c t l y the same r o l e of the c r e e p d o u b l i n g time i n N i e l s e n ' s model. When s t r e s s r a t i o a > 0.9, the specimen i s under a ramp or an a l m o s t ramp t e s t . The t i m e - t o - f a i l u r e i s q u i t e s h o r t and, under t h e s e c i r c u m s t a n c e s , the s h o r t - t e r m s t r e n g t h r g c o n t r o l s the s i t u a t i o n no ma t t e r what c i s . But when the s t r e s s r a t i o i s l o w e r , the t i m e - t o - f a i l u r e of the specimen i s i n f l u e n c e d not o n l y by T s , but a l s o by c. I f a p i e c e has a h i g h s h o r t - t e r m s t r e n g t h r , but a low c (low c r e e p ) , i t can s u s t a i n a l o a d f o r a l o n g e r p e r i o d than a p i e c e w i t h the same r s and h i g h e r c ( h i g h e r c r e e p ) . FIG(10) and FIG(11) s h o w , r e s p e c t i v e l y , t h e r e s u l t s f o r 4- "U i- "U T s e q u a l t o the 5 p e r c e n t i l e and T s e q u a l t o the 95 p e r c e n t i l e of the CDF, w h i l e c t a k e s v a l u e s of 5 and 9 5 ~ t h p e r c e n t i l e of i t s CDF. Both p l o t s show once a g a i n the e f f e c t of r s and c. For a h i g h e r T s , the damage a c c u m u l a t i o n becomes slowe r and t i m e - t o - f a i l u r e i s i n c r e a s e d when c i s d e c r e a s e d (see oiiva ss3dis FIG(IO) Extreme Case w i t h r =5% of CDF f o r the Damage Model F I G ( l l ) treme Case w i t h r s=95% of CDF f o r the Damage Model 46 F I G ( 1 0 ) ) . For a lower r , parameter c s h o u l d be i n c r e a s e d t o s h o r t e n the t i m e - t o - f a i l u r e (see F I G ( 1 1 ) ) . FIG(12) shows t h a t i t i s p o s s i b l e t o e l i m i n a t e the e f f e c t of r s ( s t r e n g t h e f f e c t ) by an a p p r o p r i a t e c o m b i n a t i o n of c r e e p c h a r a c t e r i s t i c s (as r e p r e s e n t e d by the parameter c ) . 47 F I G(12) Parameter Adjustment and Average V a l u e of T i m e - t o - F a i l u r e 3. SINGLE MEMBER RELIABILITY ANALYSIS USING THE DAMAGE MODEL The damage model w i l l now be used f o r the r e l i a b i l i t y a n a l y s i s of a s i n g l e member under l o a d f o r a p r e s c r i b e d s e r v i c e l i f e . In p a r t i c u l a r , we w i l l d e a l w i t h the st u d y of r e l i a b i l i t y under d u r a t i o n - o f - l o a d e f f e c t s by u s i n g the R a c k w i t z - F i e s s l e r a l g o r i t h m as a more e f f i c i e n t a l t e r n a t i v e t o M o nte-Carlo computer s i m u l a t i o n . The purpose of r e l i a b i l i t y a n a l y s i s i s t o c o n s i d e r the u n c e r t a i n t i e s i n the d e s i g n v a r i a b l e s , i n c l u d i n g the ones a f f e c t i n g the l o a d i n g and the ones a f f e c t i n g the s t r e n g t h or r e s i s t a n c e of s t r u c t u r e s , and t h e r e b y d e v e l o p a d e s i g n p r o c e d u r e f o r p r a c t i c a l a p p l i c a t i o n s w i t h a p r e s c r i b e d margin of s a f e t y . 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 must ta k e i n t o account the b e h a v i o u r of the s t r u c t u r a l system r a t h e r than o n l y t h a t of a s i n g l e member. In f a c t , the u s u a l l y h i g h v a r i a b i l i t y i n s i n g l e member p r o p e r t i e s i s compensated by the a c t i o n of a redundant system, and c o n s i d e r a t i o n of such a c t i o n i s the o n l y way i n which a r e a l i s t i c r e l i a b i l i t y assessment can be made f o r s t r u c t u r a l a p p l i c a t i o n of wood.An example of r e l i a b i l i t y e s t i m a t i o n s f o r t i m b e r systems has been p r e s e n t e d by F o s c h i ( 1 9 8 4 ) . N e v e r t h l e s s , many codes a r e based on " s i n g l e member" d e s i g n , w i t h " l o a d s h a r i n g " f a c t o r s a p p l i e d t o ta k e i n t o a c c o u n t the added r e l i a b i l i t y of a system. In t h i s c o n t e x t , r e l i a b i l i t y of s i n g l e members i s a f i r s t s t e p i n the c o n s i d e r a t i o n of the more g e n e r a l problem. 48 49 3.1 THE LOADING In t h i s work, we have c o n s i d e r e d the s u p e r p o s i t i o n of a permanent dead l o a d and a l i v e l o a d ( s n o w ) . 3.1.1 DEAD LOAD REPRESENTATION A l t h o u g h the dead l o a d i s u s u a l l y c o n s i d e r e d t o be an a c t i o n w hich does not v a r y w i t h time or i n p o s i t i o n , i t must be c l a s s e d as an u n c e r t a i n q u a n t i t y f o r the pur p o s e s of r e l i a b i l i t y a n a l y s i s , s i n c e i n g e n e r a l i t s magnitude w i l l not be known w i t h p r e c i s i o n . The dead l o a d D may be assumed as a n o r m a l l y d i s t r i b u t e d random v a r i a b l e w i t h c o e f f i c i e n t of v a r i a t i o n C D and a mean M Q . I f the d e s i g n dead l o a d D n i s taken e q u a l t o the mean v a l u e M D , the r a t i o L D of the dead l o a d t o the d e s i g n dead l o a d w i l l be:, D L D = — = 1.0+C Dp n (3-1 ) D n where p n i s a s t a n d a r d normal random v a r i a b l e . 3.1.2 LIVE LOAD (SNOW) REPRESENTATION As a l i v e l o a d , snow i s an i n t e r m i t t e n t l o a d of a c y c l i c random c h a r a c t e r . The d i s t r i b u t i o n f o r the a n n u a l maximum snow l o a d S a t a g i v e n l o c a t i o n can be d e t e r m i n e d from weather d a t a and can be r e p r e s e n t e d by an Extreme Type 1 d i s t r i b u t i o n (Gumbel), { - L n ( - L n ( p ) ) } S = U+ c 50 (3-2) or p = e -e -C(S-U) (3-3) where p i s a random number u n i f o r m l y d i s t r i b u t e d between 0 and 1, C and U a r e d i s t r i b u t i o n p a r a m e t e r s . The p a r a m e t e r s C and U have been o b t a i n e d from wheather d a t a . TABLE(5) l i s t s t h e s e parameters f o r t h r e e Canadian c i t i e s . A d e s i g n snow l o a d S n i s chosen f o r a 30-year r e t u r n p e r i o d , c o r r e s p o n d i n g t o a p r o b a b i l i t y p = 29/30 ( i . e . , S n may be exceeded w i t h p r o b a b i l i t y 1/30 ). Thus, the r a t i o of S/S n of a n n u a l maximum t o d e s i g n l o a d i s a l s o G u m b e l - d i s t r i b u t e d a c c o r d i n g t o : S * {-Ln(-Ln(p))} — = U + * (3-4) S n c CU w i t h U*= ; C*= CU+3. 3843 CU+3.3843 I f we d e f i n e p 0 as the p r o b a b i l i t y c o r r e s p o n d i n g t o no snow i n one y e a r , we g e t , f r o m EQ(3'3) and the c o n d i t i o n S=0, t h a t - D c u p 0 = e e (3-5) 51 CITY C U a b / f t 2 ) - ' (kN/m2)-1 I b/ft 2 kN/m2 Vancouver Winnipeg Que bee 0. 181 0. 160 0. 140 3. 779 3. 340 2. 923 2. 69 13. 20 39. 30 0. 129 0. 632 1. 882 Note: Loads adjusted from ground a fact or 0. 8 (NBC,1985). to roof I oads wi t h TABLE(5) Snow l o a d d a t a f o r 3 Canadian c i t i e s 52 For e x a m p l e , i n Vancouver p o=0.l96 or the p r o b a b i l i t y of no snow i n a y e a r i s a p p r o x i m a t e l y once e v e r y 5 y e a r s . But i n Quebec p 0 — 0 , meaning t h a t the s i t u a t i o n of no snow d u r i n g a year i s a l m o s t i m p o s s i b l e t h e r e . In g e n e r a l , the shape of the snow l o a d c y c l e i s not known,but the d u r a t i o n of the w i n t e r may be assumed t o be 5 months l o n g (November 1 - A p r i l 1 ) . Here we w i l l use the Bo r g e s - C a s t a n h e t a model and assume t h a t the a n n u a l l o a d comes i n NS segments of e q u a l d u r a t i o n , t h a t w i t h i n each segment the l o a d remains c o n s t a n t , and t h a t segment l o a d s a r e independent of each o t h e r and e q u a l l y d i s t r i b u t e d . For c o n s i s t e n c y , the d i s t r i b u t i o n f o r the segment l o a d s must be such t h a t the d i s t r i b u t i o n f o r the maximum of the NS segments be e q u a l t o t h a t f o r the a n n u a l maximum l o a d , F m = v , (S) . Thus i f F(S) i s the segment l o a d d i s t r i b u t i o n ( c o n d i t i o n a l or g i v e n t h a t snow p r e s e n t ) , a n d p e i s the p r o b a b i l i t y of snow i n a segment,we must have: F m a x ( S ) = O . 0 - p e + p e F ( S ) ) N S (3-6) On t h e o t h e r hand,p e must s a t i s f y ( l . 0 - p e ) N S = po (3-7) I n t r o d u c i n g EQ(3-3) i n t o EQ(3«6), the d i s t r i b u t i o n of the segment l o a d S i s found t o be: 1 S = U+ - {-Ln(-NS.Ln(1.0-p e+p ep))} 53 (3-8) where p i s an u n i f o r m random number, 0 < p ^ 1. T h i s i s a m o d i f i e d Extreme Type 1 (Gumbel) d i s t r i b u t i o n , a n d the d i s t r i b u t i o n of the r a t i o Lg of the l o a d S t o the d e s i g n l o a d S n becomes: S * 1 L s = — = U + —5p {-Ln(-NS- Ln( 1 .0-p e+p ep) )} (3-9) S n C where U and C are as d e f i n e d i n EQ(3«4). 3.1.3 THE LOAD COMBINATION AND LIMIT STATE DESIGN EQUATION The l o a d c o m b i n a t i o n we c o n s i d e r i s the s u p e r p o s i t i o n of the dead l o a d and the snow l o a d . A l o a d sequence f o r NY y e a r s , w i t h NS l o a d segments per w i n t e r , w i l l c o n t a i n a t o t a l of (NY-NS+1) independent random v a r i a b l e s : (NY-NS) l i v e l o a d v a r i a b l e Lg p l u s the dead l o a d v a r i a b l e L D . FIG(13) and FIG(14) shows t y p i c a l s i m u l a t e d l o a d sequences f o r NY=5 and NS=10 f o r Quebec C i t y and Vancouver. The l o a d a t any segment d u r i n g t h e w i n t e r w i l l be the sum S+D,and i t can be w r i t t e n i n terms of the l o a d r a t i o s Lg and L D as f o l l o w i n g : S+D = S n ( L D 7 + L s ) (3-10) where 7 i s the r a t i o of d e s i g n dead l o a d t o d e s i g n l i v e l o a d : 54 H H , ' — i • - p 3 r in o d O U V H a v o i F I G ( 1 3 ) T y p i c a l Load Sequence f o r Quebec C i t y a 0) w O U V H a v o i T y p i c a l Load FIG(14) Sequence f o r Vancouver 56 7 = -0- (3-11) F u r t h e r m o r e , the L i m i t S t a t e D e s i gn e q u a t i o n i n the Canadian code CAN3-086.1-M84 has the f o l l o w i n g form: a d D n + a s S n = ^R<o.o5) (3-12) where and a q a r e the l o a d f a c t o r s c o r r e s p o n d i n g , r e s p e c t i v e l y t o dead and l i v e l o a d . They a r e common t o a l l m a t e r i a l s , and a r e f i x e d a t a^=1.25 and a s=1.50. 0 i s the performance f a c t o r a p p l i e d t o the c h a r a c t e r i s t i c s t r e n g t h R ( 0 0 5 ) , w h i c h i s taken t o be the 5 p e r c e n t i l e of the s h o r t - t e r m s t r e n g t h d i s t r i b u t i o n , 7 S ( O . O 5 ) • Assuming t h a t EQ(3«12) s a t i s f i e d , the d e s i g n l i v e l o a d S n can be w r i t t e n ( o . O 5 ) . S = (3-13) n (1.257+1.50) Combining EQ(3'13) and EQ(3«10), the segment l o a d w i l l be e x p r e s s e d a s : ^ ( 0 . 0 5 1 S+D = ( L n 7 + L q ) (3-14) (1.257+1.50) U ^ D u r i n g the summer months, of c o u r s e , L s=0 and o n l y dead l o a d i s p r e s e n t . 57 3.2 RELIABILITY ANALYSIS The damage model was used t o d e f i n e the f a i l u r e p r o c e s s and t o d e r i v e the f a i l u r e f u n c t i o n used i n the r e l i a b i l i t y a n a l y s i s . As a q u i c k e r and a c c u r a t e method,the R-F a l g o r i t h m was used i n the c a l c u l a t i o n of the r e l i a b i l i t y index 0 a t the end of the s e r v i c e l i f e , a n d t h e s e r e s u l t s were compared w i t h those from a l a r g e - s c a l e M o n t e - C a r l o s i m u l a t i o n . 3.2.1 R-F ALGORITHM AND THE PROBABILITY OF FAILURE Le t X=(X,,X 2,•• •, X n ) denote t h e v e c t o r of d e s i g n v a r i a b l e s which a r e , i n general,random v a r i a b l e s . W e w i l l assume t h a t a l l X^ a r e u n c o r r e l a t e d w i t h known d i s t r i b u t i o n s . L e t G(X") be a f u n c t i o n of v e c t o r X which d e s c r i b e s the b e h a v i o u r of the s t r u c t u r a l system w i t h r e s p e c t t o a g i v e n l i m i t state.G(X") w i l l be c a l l e d the f a i l u r e f u n c t i o n . I f G(X") t a k e s on n e g a t i v e v a l u e s when the system f a i l s t o meet the performance r e q u i r e m e n t ( i . e . the event of f a i l u r e w i l l be: G(R) < 0 ) , s u r v i v a l w i l l c o r r e s p o n d t o G(X) > 0, and G=0 w i l l c o r r e s p o n d t o the " l i m i t s t a t e " between f a i l u r e and s u r v i v a l . G e n e r a l l y , the p r o b a b i l i t y of f a i l u r e P£=Prob(G<0) can be e v a l u a t e d from the m u l t i p l e i n t e g r a l : P f = / f x ( S ) d $ (3-15) 58 Where f x ( $ ) i s 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 of the random v e c t o r X and i s the f a i l u r e domain i n t h e space of d e s i g n v a r i a b l e s . The i n t e g r a l of EQ(3'15) i s v e r y d i f f i c u l t t o com p u t e , i f the number of the random v a r i a b l e s exceeds two. The R-F a l g o r i t h m i s proposed as an a l t e r n a t i v e t o compute a r e l i a b i l i t y i n d e x 0 from which P j <J>(-0),where $ i s s t a n d a r d normal d i s t r i b u t i o n f u n c t i o n . The p r o c e d u r e of the R-F a l g o r i t h m i s as f o l l o w s : 1. The o r i g i n a l random v a r i a b l e s X^ are t r a n s f o r m e d i n t o the " n o r m a l i z e d " v a r i a b l e s u^: (3-16) where and a r e the mean v a l u e and the s t a n d a r d d e v i a t i o n of X^ r e s p e c t i v e l y . The random v a r i a b l e u^ has t h e r e f o r e , a mean of z e r o and s t a n d a r d d e v i a t i o n of 1. 2. The R-F a l g o r i t h m d e t e r m i n e s the minimum d i s t a n c e between the o r i g i n (mean p o i n t ) and the f a i l u r e s u r f a c e G=0 i n the space of n o r m a l i z e d v a r i a b l e s . The p o i n t on the s u r f a c e c o r r e s p o n d i n g t o t h i s minimum d i s t a n c e i s c a l l e d the " d e s i g n " or "most l i k e l y f a i l u r e " p o i n t . 3. The R-F a l g o r i t h m computes " e q u i v a l e n t normal" v a l u e s of mean and s t a n d a r d d e v i a t i o n f o r those random v a r i a b l e s X^ which are not n o r m a l l y d i s t r i b u t e d , by impos i n g the c o n d i t i o n t h a t the " e q u i v a l e n t normal" v a r i a b l e h a v e , l o c a l l y , the same c u m u l a t i v e p r o b a b i l i t y and the 59 same p r o b a b i l i t y d e n s i t y as t h a t c o r r e s p o n d i n g t o the a c t u a l d i s t r i b u t i o n f o r X^ . 4. The R-F a l g o r i t h m i s an i t e r a t i v e p r o c e d u r e t o f i n d the " d e s i g n p o i n t " and the minimum d i s t a n c e between the o r i g i n and t h a t p o i n t . T h i s d i s t a n c e i s the r e l i a b i l i t y i n dex |3. 5. A f t e r o b t a i n i n g the index 0,we can c a l c u l a t e a p p r o x i m a t e l y the p r o b a b i l i t y of f a i l u r e t h r o u g h the f o r m u l a : P f - (3-17) where $ i s the s t a n d a r d normal d i s t r i b u t i o n f u n c t i o n . T h i s p r o c e d u r e and EQ(3'17) w i l l r e s u l t on the e x a c t p r o b a b i l i t y P^ , o n l y when the f u n c t i o n G i s a l i n e a r c o m b i n a t i o n of tho s e v a r i a b l e s . 3.2.2 DAMAGE ACCUMULATION AND THE FAILURE FUNCTION L e t a i t ) be the damage accumulated by the l o a d sequence up t o time t . S i n c e a(t)=1 w i l l c o r r e s p o n d t o the f a i l u r e a t time t , the f a i l u r e f u n c t i o n G may be w r i t t e n a s : G = 1-a(t) (3-18) Thus,G>0 means s u r v i v a l a t t , G<0 means f a i l u r e and G=0 r e p r e s e n t s the l i m i t s t a t e f o r the s e r v i c e l i f e t . 60 We use the damage model t o c a l c u l a t e the a c c u m u l a t e d damage. Le t the l o a d c o n d i t i o n be as showed i n FIG(13) or FI G ( 1 4 ) . From EQ(2.32) and EQ(2«33) we can get f o l l o w i n g r e c u r r e n c e r e l a t i o n s h i p s : a i = a - . ^ o d l + K , ( i ) s i f TI=(D+S-)>a0r^ (3-19) 1 i - 1 i f T i = ( D + S i ) < a 0 T S (3-20) i n which K 0 ( i ) = exp( c ( r i - a 0 T S ) N A-) (3-21) K , ( i ) = - ( r i - a 0 r s ) b " n ( K 0 ( i ) - 1 ) (3-22) For each c y c l e the s t r e s s T^=D+S^ i s g i v e n by EQ(3»14). The R-F a l g o r i t h m a l s o r e q u i r e s the g r a d i e n t of the f a i l u r e f u n c t i o n G ( b , c , n , a 0 , r g , L D , L s 1 ' L S 2 ' * * * L S ^' I N g e n e r a l , the p a r t i a l d e r i v a t i v e of G w i t h r e s p e c t t o a v a r i a b l e z can a l s o be computed by a r e c u r s i v e scheme as f o l l o w s : 3G da — = - — (3-23) dz dz 61 and 9aj 9a i-1 9z 9z K 0 ( i ) + o i _ 1 d K 0 ( i ) dK,(i) 9z 9z i f r i > a 0 r s (3-24) or 9a^ 9a^ _ ^  9z 9z i f r i < a 0 r s (3-25) 3.2.3 SOME COMMENTS ABOUT USING THE R-F ALGORITHM WITH THIS  TASK The R-F a l g o r i t h m i s a " f a s t p r o b a b i l i t y i n t e g r a t i o n " method f o r g e n e r a l r e l i a b i l i t y a n a l y s i s . When used i n the i n v e s t i g a t i o n of d u r a t i o n - o f - l o a d e f f e c t s , t h i s a l g o r i t h m poses some s p e c i a l problems which w i l l be d i s c u s s e d now. 3.2.3.1 The " p l a t e a u " of the f a i l u r e f u n c t i o n G The R-F a l g o r i t h m a p p r o x i m a t e s the f a i l u r e s u r f a c e G(X) w i t h a ta n g e n t a t the p o i n t R , i . e . , a l i n e a r h y p e r p l a n e of random v a r i a b l e v e c t o r X.The i n t e r c e p t of t h i s h y p e r p l a n e w i t h the p l a n e G=O,0 o,is used t o r e p r e s e n t a p p r o x i m a t e l y the i n t e r c e p t of r e a l f a i l u r e s u r f a c e , 0. The d i s t a n c e from the o r i g i n t o 0 O i s regar d e d as an a p p r o x i m a t i o n of the 0 i n d e x . The 62 e v a l u a t i o n i s c o n t i n u e d u n t i l the d i f f e r e n c e between the d i s t a n c e from o r i g i n t o 0 and the d i s t a n c e from o r i g i n t o 0 O i s l e s s than a s m a l l v a l u e e. A l t h o u g h we c o u l d not know the e x a c t shape of the f a i l u r e s u r f a c e i n the m u l t i - d i m e n s i o n a l space, our work shows t h a t the damage i s v e r y s m a l l e x c e p t f o r those c o m b i n a t i o n s of v a r i a b l e s near the c o n d i t i o n G=0. a c c o r d i n g l y , t h e f u n c t i o n G has a n e a r l y f l a t p l a t e a u (G =* 1 ) around the mean p o i n t and a q u i t e sudden d r o p t o G=0 as the l i m i t s t a t e i s a p p r o a c h e d . S c h e m a t i c a l l y t h i s i s r e p r e s e n t e d i n F I G ( 1 5 ) . I f the f i r s t c h o i c e of X" g i v e s no damage,G^1, the c o r r e s p o n d i n g h y p e r p l a n e i may have d i f f i c u l t y i n f i n d i n g an i n t e r c e p t w i t h the p l a n e G=0, i n t r o d u c i n g n u m e r i c a l problems. In o r d e r t o s o l v e t h i s d i f f i c u l t y , t h e domain of the v e c t o r X" was kept w i t h i n a narrow band around G=0. I f , f o r a g i v e n X , the a b s o l u t e v a l u e of the f u n c t i o n G was |G| > G , the l o a d components of X" were p r o p o r t i o n a l l y i n c r e a s e d (or d e c r e a s e d ) u n t i l the c o r r e s p o n d i n g G s a t i s f i e d |G| < G . The a l g o r i t h m was then r e - s t a r t e d from the a d j u s t e d p o i n t X . In t h i s work, * the v a l u e G = 0.85 was used. 3.2.3.2 M u l t i p l e ( l o c a l ) minima I f the s u r f a c e G=0 i s f a r from f l a t i n the t r a n s f o r m e d space, i t c o u l d e x h i b i t s e v e r a l l o c a l minima.(see F I G ( 1 6 ) ) . Each minima w i l l c o r r e s p o n d t o a d i f f e r e n t d e s i g n p o i n t , w h i c h , i n t u r n c o r r e s p o n d s t o a 0.5 Z g i— o z U. P l a t e a u of FIG(15) the F a i l u r e S u r f a c e 64 d i f f e r e n t f a i l u r e mode. In our c a s e , t h e d i f f e r e n t f a i l u r e modes were shown t o be d i f f e r e n t t y p e s of l o a d sequences over the NY y e a r s , o r more p r e c i s e l y , the d i f f e r e n t f a i l u r e modes c o r r e s p o n d e d t o the d i f f e r e n t p o s i t i o n of a h i g h l o a d segment i n the l o a d s e q u e nce,with a l l r e m a i n i n g segments t a k i n g on c o n s t a n t l o a d v a l u e s . T h e h i g h l o a d segment c o u l d come a t the b e g i n n i n g , a t any time in-between or a t the end of the s e r v i c e p e r i o d c o n s i d e r e d . C o n s e q u e n t l y , d i f f e r e n t |3 v a l u e s can be computed a c c o r d i n g t o these l o c a l minima,when the i t e r a t i o n i s s t a r t e d from d i f f e r e n t i n i t i a l p o i n t s . The p r o b l e m , t h e r e f o r e , i s c h a r a c t e r i z e d by (NY•NS) d i f f e r e n t f a i l u r e modes ( t h e t o t a l number of l i v e l o a d segments i n the NY y e a r s ) . I t was found t h a t s t a r t i n g the R-F a l g o r i t h m w i t h segment l o a d s a l l e q u a l t o each o t h e r c o n v e rged t o a s i t u a t i o n where the h i g h l o a d segment o c c u r s a t the b e g i n n i n g of t h e s e r v i c e p e r i o d and c o r r e s p o n d s t o the l o w e s t j3 i n the s e t . The p r o b a b i l i t y of f a i l u r e Pf a f t e r NY y e a r s i s the p r o b a b i l i t y c o n t e n t i n the f a i l u r e zone G<0 of F I G ( 1 6 ) . I f the s u r f a c e G=0 were r e l a t i v e l y f l a t around p o i n t 1, as shown by the l i n e 1-1 i n the f i g u r e , P^ c o u l d be d e t e r m i n e d v e r y a c c u r a t e l y from E Q ( 3 • 1 7 ) , u s i n g the r e l i a b i l i t y index j31 .Since the s u r f a c e p r e s e n t s s e v e r a l minima, i t i s apparent t h a t t h e t r u e p r o b a b i l i t y of f a i l u r e must be g r e a t e r than what would be p r e d i c t e d by 6 5 FIG(16) F a i l u r e S u r f a c e and M u l t i p l e Minima 66 EQ(3«17) and the minimum 0,. For t h i s reason,we have t o c o n s i d e r the r e l a t i o n s h i p between the g l o b a l r e l i a b i l i t y index j3 and the l o c a l i n d e c e s j 3 ^ , ( i = 1,N). 3.2.4 THE CALCULATION OF BOUNDS FOR THE PROBABILITY OF  FAILURE How t o d e r i v e the g l o b a l p r o b a b i l i t y of f a i l u r e P f of the whole p r o c e s s , i f t h e p r o b a b i l i t i e s of the s i n g l e f a i l u r e modes a r e known? T h i s i s a v e r y d i f f i c u l t p r o b l e m , e s p e c i a l l y when 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 s of the f a i l u r e modes a r e unknown. In o r d e r t o overcome t h e s e d i f f i c u l t i e s and t o get a r e a s o n a b l e e s t i m a t i o n of the g l o b a l P^,we w i l l go on w i t h f i r s t - o r d e r system r e l i a b i l i t y a n a l y s i s and c a l c u l a t e the bounds on t h e p r o b a b i l i t y of f a i l u r e . L e t P^ be the p r o b a b i l i t y of f a i l u r e f o r the i - t ^ f a i l u r e mode. L e t Pf be t h e p r o b a b i l i t y of f a i l u r e of the system. Then we can have the f o l l o w i n g rough bounds ( C o r n e l l , 1 9 6 7 ) n max P i < P f < Z Pi (3-26) i = 1 C l o s e r bounds on P^ can be g i v e n i n terms of the f a i l u r e p r o b a b i l i t i e s i n any mode and the j o i n t f a i l u r e p r o b a b i l i t i e s between any two modes. 67 These c l o s e r bounds a r e g i v e n by ( D i t l e v s e n , 1 9 7 9 ) n i - 1 P,+ 2 max{P.- L P i H , 0 } < P f i=2 j=1 J n n < I P i - I max. P- • (3-27) i = 1 i = 2 3<i 1 3 where P^j i s the j o i n t f a i l u r e p r o b a b i l i t y between modes i and j . The P^ w i l l be c a l c u l a t e d t h r o u g h R-F a l g o r i t h m . i . e . PL = * ( - j 3 i ) (3-28) But how about p i j ? FIG(17) shows the p r o j e c t i o n of the f a i l u r e s u r f a c e f o r the two f a i l u r e modes on the p l a n e spanned by the o r i g i n and the two d e s i g n p o i n t s . P ^ j i s the p r o b a b i l i t y c o n t e n t i n the a n g l e AED. P^j i s l a r g e r than each of the p r o b a b i l i t y c o n t e n t s i n the a n g l e s AEC or BED,but l e s s than t h e i r sum. T h i s g i v e s : max{$(-0-)*(-/3_- *(-/3 H)*(-/3, H ) } < P ^ <*(-/?. )$(-/?, • )+ *(-0.)*(-/3. .) 1 J r 1 J 1 i J i f P i j > 0 (3-29) 0 < P- • < min{*(-/3. • • ) , • • ) } i f P i j < 0 (3-30) FIG(17) Bounds f o r Two F a i l u r e Modes 69 The d i s t a n c e (i- • and /3- • a r e found by g e o m e t r i c a l c o n s i d e r a t i o n s . They can be e x p r e s s e d a s : 0- • i s the c o n d i t i o n a l r e l i a b i l i t y i n d e x , t h a t i s , the r e l i a b i l i t y index f o r f a i l u r e i n mode i g i v e n t h a t the s a f e t y margin of f a i l u r e i n j mode e q u a l s z e r o . From F I G ( 1 7 ) , the c o r r e l a t i o n c o e f f i c i e n t = cost ' — . I t can be c a l c u l a t e d by the c r o s s p r o d u c t of the v e c t o r s of d i r e c t i o n c o s i n e s i n the two f a i l u r e modes. 3.3 NUMERICAL RESULTS As an example,we have i n v e s t i g a t e d the case of 30 y e a r s of snow w i t h 7=0.4 and 0=0.8 i n Quebec C i t y . In a d d i t i o n , t h r e e number of l o a d segments per year were c o n s i d e r e d : NS=1,5,and 10.Thus the snow l o a d was assumed t o have a c o n s t a n t v a l u e l a s t i n g f o r 5 months,1 month or 2 w e e k s . A l l damage model ' p a r a m e t e r s , as w e l l as the s h o r t - t e r m s t r e n g t h , were assumed l o g n o r m a l l y d i s t r i b u t e d as d e s c r i b e d e a r l i e r i n t h i s t h e s i s . U s i n g the same parameter s e t and a l a r g e sample s i z e of 100,000 r e p l i c a t i o n s , M o n t e - C a r l o s i m u l a t i o n produced the p r o b a b i l i t y of f a i l u r e i n p e r i o d s from 1 year t o 30 y e a r s of s e r v i c e . The r e s u l t s were t a k e n as benchmark f o r the a n a l y s i s u s i n g the R-F a l g o r i t h m . / ( i - p i : j 2 ) (3-31) (3-32) 70 A l s o , the r e l i a b i l i t y i n d e x /? w i t h a f i x e d p e r i o d of 30 y e a r s and d i f f e r e n t d e a d - l i v e l o a d r a t i o s 7 was c a l c u l a t e d f o r 3 t y p i c a l Canadian c i t i e s . The r e s u l t s were used i n comparison w i t h t h o s e c a l c u l a t e d w i t h o u t c o n s i d e r a t i o n of d u r a t i o n - o f - l o a d e f f e c t . 3.3.1 THE MONTE-CARLO SIMULATION FIG(18) shows the r e s u l t s of Mo n t e - C a r l o s u m u l a t i o n w i t h NS=1, NS=5 and NS=10 f o r Quebec C i t y . I t i s c l e a r , t h a t the case of NS=1 i s the most severe,and t h a t the p r o b a b i l i t y of f a i l u r e d e c r e a s e d when the l o a d segments were of s h o r t e r d u r a t i o n . However,since the d i f f e r e n c e s between NS=5 and NS=10 a r e s m a l l e r than between NS=1 and NS=5, i t i s e x p e c t e d t h a t a h i g h e r NS w i l l g i v e r e s u l t s not t o o d i f f e r e n t from NS=10. T h i s p e r m i t s t o ta k e the r e s u l t s f o r NS=10 as a not o v e r l y c o n s e r v a t i v e a p p r o x i m a t i o n . I t s h o u l d a l s o be mentioned t h a t the c a l c u l a t i o n s a r e based on the independence of the d i f f e r e n t l o a d v a r i a b l e s , an a s s u m p t i o n which w i l l be compromised f o r h i g h e r NS. 3.3.2 THE BOUNDS OF PROBABILITY OF FAILURE FIG(19) shows t h e D i t l e v s e n bounds on the p r o b a b i l i t y of f a i l u r e f o r t h e case of NS=5. Bounds c a l c u l a t i o n i s l e n g t h y and ex p e n s i v e , b e c a u s e the computer has t o c o n s i d e r a l l of the f a i l u r e modes ( i n our c a s e , the t o t a l number of f a i l u r e modes i s 150 ). o:l i — z =s or UJ a. 6 in Q c < CD o _J CO o <D a: INTE UJ z $ a: UJ CL DP PL d Ul 6 (A LJ LOAD LOAD m /I i: i • If / / / 7 / 3mVA sf CM FIG(18) Monte-Carlo S i m u l a t i o n R e s u l t s f o r Quebec C i t y (with 0=0.8, 7=0.4) FIG(19) D i t l e v s e n Bounds and Mon t e - C a r l o R e s u l t ( w i t h 0=0.8, 7=0.4) 73 From F I G ( 1 9 ) , t h e upper bound of r e l i a b i l i t y i ndex 0 (or e q u a l l y , t h e lower bound of the p r o b a b i l i t y of f a i l u r e ) i s r e l a t i v e l y independent of the number of y e a r s c o n s i d e r e d , but the lower bound of r e l i a b i l i t y i ndex 0 (or e q u a l l y , the upper bound of the p r o b a b i l i t y of f a i l u r e ) shows a s t r o n g e r dependence. No ma t t e r the NS v a l u e , a l l r e s u l t s of the M o n t e - C a r l o s i m u l a t i o n f a l l between the bounds, but t h e s e bounds a r e not narrow, p a r t i c u l a r l y a t l o n g s e r v i c e l i v e s . One reason f o r this , m a y be the h i g h l y n o n - l i n e a r c h a r a c t e r i s t i c of our f a i l u r e f u n c t i o n . EQ(3'27) i s a g e n e r a l f o r m u l a , i t can g i v e the bounds of a m u l t i - v a r i a b l e p r o b a b i l i t y of f a i l u r e u s i n g o n l y the i n d i v i d u a l p r o b a b i l i t y of f a i l u r e and j o i n t p r o b a b i l i t y of f a i l u r e of ev e r y two f a i l u r e modes P — . B u t i n our c a s e , the j o i n t p r o b a b i l i t y of f a i l u r e was c a l c u l a t e d by^ the f i r s t - o r d e r r e l i a b i l i t y a n a l y s i s , w h e r e the j o i n t f a i l u r e s e t i s a p p r o x i m a t e d by the s e t bounded by the tangent h y p e r p l a n e s a t the d e s i g n p o i n t s f o r two f a i l u r e modes. The u n s a t i s f a c t o r y r e s u l t s u g g e s t s , t h a t a more a c c u r a t e a p p r o x i m a t i o n of the f a i l u r e s u r f a c e must be chosen.For example,a second-order system r e l i a b i l i t y a n a l y s i s may be used,where the f a i l u r e s u r f a c e w i l l be a p p r o x i m a t e d by a q u a d r a t i c s u r f a c e w i t h the same c u r v a t u r e s a t the d e s i g n p o i n t . 74 3 . 3 . 3 THE RESULT FROM THE R-F ALGORITHM Because of the e x i s t a n c e of m u l t i p l e minima ( d i f f e r e n t f a i l u r e modes), the p r o b a b i l i t y of f a i l u r e f o r the minimum /3 u n d e r e s t i m a t e s the a c t u a l p r o b a b i l i t y of f a i l u r e P j . FIG(20) shows t h e comparison of the minimum /3 f o r each number of y e a r s t o the M o n t e - C a r l o s i m u l a t i o n f o r the case NS=5. Thus, i f 0! i s the minimum 0 f o r the case of NS=1, and Pf{NS=l} i s the p r o b a b i l i t y of f a i l u r e f o r NS=1,we w i l l have: * ( - 0 i ) < P f{NS= U ( 3 - 3 3 ) On the o t h e r hand, the M o n t e - C a r l o s i m u l a t i o n shows t h a t : P f{NS>l} < P f{NS= U ( 3 - 3 4 ) T h e r e f o r e , i t i s p l a u s i b l e t h a t a good a p p r o x i m a t i o n t o P£{NS>1} (which i s t h e case of i n t e r e s t ) c o u l d be found by u s i n g : P f {NS>1 } * ( - / J , ) ( 3 * 3 5 ) FIG ( 2 1 ) shows the r e l i a b i l i t y index j 3 c o r r e s p o n d i n g t o the minimum /3 f o r NS=1 . I t i s apparent t h a t t h i s index can be used v e r y a c c u r a t e l y t o approximate the t r u e r e l i a b i l i t y f o r NS=10, and w i t h somewhat l e s s a c c u r a c y f o r the case of 7 5 3mVA FIG(20) Minimum j3 f o r Each Number of Y e a r s ,and M o n t e - C a r l o R e s u l t ( w i t h 0=0.8, 7=6.4) 76 NS=5. We can say, t h e r e f o r e , t h a t f o r NS>1, the R-F a l g o r i t h m g i v e s us d i f f e r e n t p r o b a b i l i t i e s of f a i l u r e f o r t h e d i f f e r e n t f a i l u r e modes, but when NS=1 i s adopted, the minimum 0 from the R-F a l g o r i t h m g i v e s us an a p p r o x i m a t i o n t o the p r o b a b i l i t y of f a i l u r e f o r the c o m b i n a t i o n of a l l f a i l u r e modes. TABLE(6) shows the comparison of the p r o b a b i l i t y of f a i l u r e , u s i n g the M o n t e - C a r l o s i m u l a t i o n or the R a c k w i t z - F i e s s l e r a l g o r i t h m . The c a l c u l a t i o n of |3, w i t h the R a c k w i t z - F i e s s l e r a l g o r i t h m i s v e r y f a s t , e v e n f o r 30 y e a r s . The convergence t o 0, i s o b t a i n e d when a l l segment l o a d s a r e i n i t i a l l y assumed e q u a l t o each o t h e r . The r e s u l t i n g mode of f a i l u r e c o r r e s p o n d s t o a l o a d sequence w i t h t h e h i g h segment l o a d o c c u r i n g a t the f i r s t w i n t e r , f o l l o w e d by a l l segments l o a d s of e q u a l but lower magnitude. 3.3.4 THE EFFECT OF THE LOAD RATIO y FIG(22) shows the v a l u e s of 0, a t 30 y e a r s f o r two c o m b i n a t i o n of r a t i o y and the performance f a c t o r <p. A s m a l l e r mean dead l o a d i n c r e a s e s the r e l i a b i l i t y (7=0.2), w h i l e a l a r g e r r a t i o 7=0.4 d e c r e a s e s i t . A c o n s e r v a t i v e a p p r o x i m a t i o n f o r code i m p l e m e n t a t i o n would be t o use a r e a s o n a b l y h i g h r a t i o 7.(7=0.4 i s the r a t i o used i n the s o f t c a l i b r a t i o n of the p r e s e n t Canadian L i m i t S t a t e s D e s i g n v e r s i o n of the code) 3H1VA 0 FIG(21) R e s u l t s of R-F A l g o r i t h m and M o n t e - C a r l o S i m u l a t i o n ( w i t h 0=0.8, 7=0.4) 78 PROBABILITY OF FAILURE YEAR(S) M- C (NS=1) M-•C (NS=10) R-F (NS=1) I 0. 2079'10- 2 0. 0889-10-2 0. 07 92-10- 2 5 0. 7 487•10- 2 0. 3519-10-2 0. 3498-10- 2 10 1. 2225'10- 2 0. 5638-10-2 0. 5606-10- 2 15 1. 5738-10- 2 0. 7 263-IO'2 0. 7 243-10- 2 20 1. 8671-10- 2 0. 87 27-10-2 0. 8493-10- 2 25 2. 1229-10- 2 0. 9903-10-2 0. 9539-10- 2 30 2. 357 2-10' 2 1. 0982-IO'2 1. 0444-10- 2 M-C: Monte-Carlo simulation R-F: Rac kwi t z-Fi e s s I e r algorithm, minimum (3 For the case: 7 = 0 . 4 , 0 = 0 . 8 T A B L E ( 6 ) M o n t e - C a r l o vs R-F A l g o r i t h m SAFETY INDEX (8 AND RESISTANCE COEFFICIENT 0 WITH LOAD DURATION EFFECT 80 D i f f e r e n t 7 v a l u e s can a l s o be used t o study the e f f e c t of d i f f e r e n t t y p e s of l o a d i n g d i s t r i b u t i o n . EQ(3'12) can be w r i t t e n , a d G D D n + a s G S S n = ^ R <o.o 5 > (3-36) i n w h i c h , the parameters G Q and Gg a r e g e o m e t r i c f a c t o r s t o t r a n s f o r m t o t a l a p p l i e d l o a d s i n t o s t r e s s e s . For a r e c t a n g u l a r beam of span L, w i d t h B and depth H, t h e s e f a c t o r s a r e : 1) f o r u n i f o r m l y d i s t r i b u t e d l o a d , ( 0 . 7 5 L / B H 2 ) ; 2) f o r c o n c e n t r a t e d l o a d a t midspan, (1.5L/BH 2). Now,.the s t r e s s a t any l o a d segment can be w r i t t e n as D+S = G s S n { 7 ( ^ ) L D + L S } (3-37) EQ(3«13) becomes ( 0 . 0 5 ) G~S_ = (3-38) b n {1.257(G D/G S)+1.50} Combining EQ(3'37) and EQ(3'38), the segment s t r e s s w i l l now be e x p r e s s e d as ( 0 0 5 ) GT-> (7( — ) ' L D + L S } ( 3 ' 3 9 ) 1 { 1 .257(G D/G S) + 1 .50} G s I f b o th dead and l i v e l o a d a r e u n i f o r m l y d i s t r i b u t e d , G n = Gg the e x p r e s s i o n of segment s t r e s s i s as g i v e n by EQ(3-14). 81 Now,consider the s i t u a t i o n t h a t the dead l o a d i s u n i f o r m l y d i s t r i b u t e d but the l i v e snow l o a d i s c o n c e n t r a t e d . The r a t i o of two geometric f a c t o r s i s G n 0.75L BH 2 1 = . = - (3-40) G s BH 2 1.5L 2 I f 7* = 7 ( G D / G S ) , EQ(3-39) i s i d e n t i c a l t o EQ(3-14). Thus, the c u r v e f o r 7=0.2 i n FIG(22) c o r r e s p o n d s e i t h e r t o 1) 7=0.2 and u n i f o r m l y d i s t r i b u t e d l i v e l o a d or t o 2) 7=0.4 and c o n c e n t r a t e d l i v e l o a d . 3.3.5 THE MAGNITUDE OF THE DURATION-QF-LOAD EFFECT To i n v e s t i g a t e t he magnitude of the d u r a t i o n - o f - l o a d e f f e c t ( D O L ) , we have c o n s i d e r e d two c a s e s i n t h r e e g e o g r a p h i c a l p l a c e s . 3.3.5.1 Dead l o a d o n l y Suppose the s t r u c t u r e i s under dead l o a d o n l y . What w i l l be the d i f f e r e n c e i n t h e p r o b a b i l i t y of f a i l u r e w i t h and w i t h o u t c o n s i d e r a t i o n of DOL e f f e c t ? For t h i s c a s e , EQ(3'11) w i l l become D 0 7 = — = » (3-41 ) s n Computer r e s u l t s were o b t a i n e d u s i n g a l a r g e y (7=1000). FIG(23) shows the r e l a t i o n s h i p between r e l i a b i l i t y i ndex 0 and d i f f e r e n t v a l u e s of <p w i t h 7=1000 and a s e r v i c e 8 2 p e r i o d of T=30 y e a r s . T h i s f i g u r e a l s o shows the r e l i a b i l i t y of the s t r u c t u r e i f d u r a t i o n - o f - l o a d e f f e c t s a r e not taken i n t o a c c o u n t . In t h i s c a s e , the s h o r t - t e r m s t r e n g t h d i s t r i b u t i o n of the member i s compared w i t h the d i s t r i b u t i o n f o r the h i g h e s t l o a d t h a t the member c o u l d r e c e i v e i n a 30 year p e r i o d w h i ch, i n t h i s c a s e , i s j u s t the d i s t r i b u t i o n f o r t h e dead l o a d . I t i s c l e a r t h a t the d u r a t i o n - o f - l o a d e f f e c t can not be i g n o r e d . FIG(23) shows t h a t the p r o b a b i l i t y of f a i l u r e w i l l be much more severe w i t h the DOL e f f e c t than w i t h o u t i t . F o r example, i f a t a r g e t r e l i a b i l i t y 0=3.0 i s chosen f o r the d e s i g n , the performance f a c t o r 0 would be 0=0.71 i f do not c o n s i d e r the d u r a t i o n - o f - l o a d e f f e c t . I n c l u d i n g i t , the performance f a c t o r would be lowered t o 0=0.37. 3.3.5.2 Combination of dead l o a d and snow l o a d The second case t o be c o n s i d e r e d i s the s t r u c t u r e under a c o m b i n a t i o n of dead and snow l o a d . FIG(24) shows a comparison of the r e l i a b i l i t y of the s i n g l e member a f t e r 30 y e a r s w i t h and w i t h o u t d u r a t i o n - o f - l o a d e f f e c t f o r Quebec C i t y . The d e s i g n d e a d - l i v e l o a d r a t i o was taken 7=0.4. For a t a r g e t r e l i a b i l i t y index 0=3.0, the performance f a c t o r must be 0=0.60 i f the d u r a t i o n - o f - l o a d e f f e c t i s c o n s i d e r e d . Without i t , t h e performance f a c t o r must be 0=0.78. 83 3mVA tf FIG(23) R e l i a b i l i t y Index 0 w i t h and w i t h o u t DOL (Dead Load o n l y ) 84 Because the weather f a c t o r s a r e d i f f e r e n t f o r d i f f e r e n t c l i m a t e zones, the r e l i a b i l i t i e s of the s t r u c t u r e under snow-and-dead l o a d were a l s o s t u d i e d f o r two o t h e r Canadian c i t i e s . I t was assumed t h a t the same s t r u c t u r e (same c r o s s - s e c t i o n ) was s u b j e c t e d t o the d i f f e r e n t snow regimes. The 7=0.4 f o r Quebec C i t y was chosen as a s t a n d a r d , the " e q u i v a l e n t " 7 v a l u e s f o r another two c i t i e s were c a l c u l a t e d w i t h a assumption t h a t the d e s i g n dead l o a d was the same everywhere. T h e r e f o r e , For Quebec: 7„ = — (3-42) q S nq For W i nnipeg: 7w = v (3*43) bnw T a k i n g each d e s i g n snow l o a d e q u a l t o the c o r r e s p o n d i n g 30 y e a r - r e t u r n snow l o a d , the f o l l o w i n g " e q u i v a l e n t " 7 v a l u e s were c a l c u l a t e d : For W i n n i p e g : 7 W=0.74; For Vancouver: 7 =1.19. to m KJ <M 3mVA si FIG(24) R e l i a b i l i t y Index 0 w i t h and w i t h o u t DOL (Quebec) 86 FIG(25) and FIG(26) show the r e l i a b i l i t y index 0 w i t h and w i t h o u t t h e DOL e f f e c t f o r Winnipeg and Vancouver r e s p e c t i v e l y . Comparing F I G ( 2 4 ) , FIG(25) and F I G ( 2 6 ) , i t i s c l e a r t h a t the b i g g e s t d i f f e r e n c e between the r e l i a b i l i t y i n d i c e s /3 w i t h and w i t h o u t the DOL e f f e c t appears i n Quebec,where the snow l o a d i s v e r y heavy and l a s t s l o n g e r . TABLE(7) l i s t s the r e s i s t a n c e f a c t o r <f> w i t h a t a r g e t r e l i a b i l i t y i n d e x 0=3.0. I t once a g a i n d e m o n s t r a t e s t h a t the DOL e f f e c t must be ta k e n i n t o c o n s i d e r a t i o n t o m a i n t a i n a t a r g e t r e l i a b i l i t y a t the end of the i n t e n d e d s e r v i c e l i f e . 3.4 DESIGN PROCEDURE AND CODE IMPLEMENTATION C o n s i d e r i n g the p r e v i o u s r e s u l t s , t h a t the d u r a t i o n - o f - l o a d e f f e c t w i l l degrade the s t r e n g t h of the s t r u c t u r e and t h e r e f o r e i n f l u e n c e i t s s a f e t y , the d e s i g n e q u a t i o n EQ(3«13) may be w r i t t e n a s : 1.25D n+1.50S n = 0 R ( o . o 5 ) K d (3-44) where <$> i s the performance f a c t o r w i t h o u t the c o n s i d e r a t i o n of d u r a t i o n - o f - l o a d e f f e c t and i s a d u r a t i o n - o f - l o a d adjustment f a c t o r f o r d i f f e r e n t l o a d i n g c o n d i t i o n s . For t h e s p e c i a l case of snow l o a d p l u s dead l o a d over a s e r v i c e l i f e of 30 y e a r s , TABLE(7) g i v e s performance and d u r a t i o n - o f - l o a d adjustment f a c t o r s a t d i f f e r e n t t a r g e t 87 r~- <£> m KJ <N 3H1VA Si FIG(25) R e l i a b i l i t y Index 0 w i t h and w i t h o u t DOL (Winnipeg) SAFETY INDEX p AND LOAD DURATION EFFECT for Vancouver 89 SINGLE MEMBER, SNOW + DEAD LOAD CITY WITH DOL WITHOUT DOL DIFFERENCE Vaneouver 0. 62 0. 71 + 14.5% Winnipeg 0. 64 0. 7 4 + 15. 6% Que be c 0. 60 0. 78 + 30. 0% TABLE(7) R e s i s t a n c e f a c t o r <p w i t h 0 = 3.0 90 r e l i a b i l i t i e s . I t i s ap p a r e n t t h a t K^=0.77 c o u l d be used r e g a r d l e s s of the t a r g e t s e l e c t e d . TABLE(8) i s c a l c u l a t e d f o r Quebec C i t y w i t h 7=0.4. For o t h e r p l a c e s , where the snow l o a d i s l e s s s e v e r e , t h e s e adjustment f a c t o r s may be c o n s e r v a t i v e . T h i s a p p r o x i m a t e p r o c e d u r e f o r the d e t e r m i n a t i o n of r e l i a b i l i t y a t the end of the i n t e n d e d s e r v i c e l i f e i s q u i c k , a c c u r a t e and w i l l p e r m i t code c a l i b r a t i o n w i t h o u t the demands of M o n t e - C a r l o s i m u l a t i o n s . S i n c e the s t r e s s e s produced by dead and snow l o a d s a r e p r o p o r t i o n a l t o the square of the span, i t i s seen t h a t the f a c t o r K^=0.77 i m p l i e s t h a t a r e d u c t i o n f a c t o r 0.88 must be a p p l i e d t o the a l l o w a b l e span computed w i t h o u t d u r a t i o n - o f - l o a d e f f e c t s . T h i s 12% r e d u c t i o n i n span a g r e e s w i t h p r e v i o u s r e s u l t s from s i m u l a t i o n s of complete s y s t e m s ( F o s c h i , 1 9 8 4 ) . 91 TARGET 0 0 WITHOUT DOL <j> WITH DOL K QUEBEC CITY 4- 0 0. 52 0. 40 0.77 3.0 0.78 0.60 0.77 2.5 0.95 0.7 3 0.77 Note: y=0.40 (Design snow I oad= 2. 5 • De s i g n dead load) WINNIPEG 4- 0 0.48 ' 0. 42 0. 88 3- 0 0. 74 0. 64 0. 86 2-5 0.90 0.79 0.88 Note: 7=0. 7 4 VANCOUVER 4- 0 0. 45 0. 40 0. 89 3. 0 0. 70 0. 62 0. 89 2.5 0. 86 0. 76 0. 88 Note: 7 = 7 . 19 TABLE(8) Performance and DOL adjustment f a c t o r s at d i f f e r e n t t a r g e t r e l i a b i l i t i e s 4 . CONCLUSION AND FURTHER RESEARCH T h i s t h e s i s p roposes a new damage model,which not o n l y c o n s i d e r s the i n f l u e n c e of the l o a d h i s t o r y , b u t a l s o the i n f l u e n c e of e x i s t i n g damage on the r a t e of damage a c c u m u l a t i o n . The model i n v o l v e s f o u r p a r a m e t e r s . They are a l l random v a r i a b l e s and t h e i r d i s t r i b u t i o n c h a r a c t e r i s t i c s were o b t a i n e d t h r o u g h c a l i b r a t i o n of the model t o e x p e r i m e n t a l d a t a . T h i s damage model was found s u i t a b l e t o r e p r e s e n t a c c u r a t e l y the e x p e r i m e n t a l t r e n d s of Western hemlock lumber i n b e n d i n g . Compared t o o t h e r c u m u l a t i v e damage approach, the model p r o v e d the importance of i n c l u d i n g the damage-dependent term i n the damage a c c u m u l a t i o n f o r m u l a ; compared t o the f r a c t u r e mechanics approach, the model was shown t o be e a s i e r t o use i n r e l i a b i l i t y c a l c u l a t i o n s . The model was used w i t h the R-F a l g o r i t h m t o i n v e s t i g a t e the d u r a t i o n - o f - l o a d e f f e c t i n a s i n g l e lumber member i n b e n d i n g , f o r t h r e e Canadian c i t i e s under two l o a d regimes: dead l o a d o n l y or dead and l i v e snow l o a d . The p r e d i c t i o n s of the R-F a l g o r i t h m have been v e r i f i e d u s i n g a l a r g e - s c a l e M o n t e - C a r l o s i m u l a t i o n . They show t h a t the d u r a t i o n - o f - l o a d e f f e c t cannot be i g n o r e d i n d e s i g n and the code s h o u l d c o n t a i n a " d u r a t i o n of l o a d a d j u s t m e n t f a c t o r " t o t a k e the e f f e c t i n t o a c c o u n t . The work i n t h i s t h e s i s i s o n l y a b e g i n n i n g . F u r t h e r r e s e a r c h s h o u l d c o n s i d e r the f o l l o w i n g t o p i c s . F i r s t , the methods d e s c r i b e d i n t h i s t h e s i s s h o u l d be f u r t h e r m o d i f i e d 92 93 t o c o n s i d e r p o s s i b l e c o r r e l a t i o n s between the model p a r a m e t e r s ; second, the methods s h o u l d be i n v e s t i g a t e d f o r o t h e r l o a d c a s e s , i n c l u d i n g s h o r t - d u r a t i o n , p u l s e - l i k e l o a d s ( e x t r a o r d i n a r y l i v e l o a d , wind l o a d , e t c ) or the c o m b i n a t i o n of more than two l o a d s ; t h i r d , the methods s h o u l d be used t o i n v e s t i g a t e the d u r a t i o n - o f - l o a d e f f e c t i n s t r u c t u r a l members o t h e r than bending e l e m e n t s . F u r t h e r m o r e , the e x t e n s i o n of t h e s e methods t o s t r u c t u r a l system i s ve r y i m p o r t a n t f o r wood s t r u c t u r e s , and s h o u l d r e c e i v e s p e c i a l a t t e n t i o n . REFERENCES B a r r e t t , J.D., and F o s c h i , R.O., " D u r a t i o n of Load and P r o b a b i l i t y of F a i l u r e i n Wood,Part 1: M o d e l l i n g Creep R u p t u r e , P a r t 2: Constant,Ramp and C y c l i c L o a d i n g s " , Canadian Journal of C i v i l Engi ne e r i ng, V o l . 5 ,No. 4 , 1 978 Borges, J . F . , and C a s t a n h e t a , M., "Structural Safety",2nd ed., N a t i o n a l C i v i l E n g i n e e r i n g L a b o r a t o r y , L i s b o n , P o r t u g a l , 1 9 7 1 D i t l e v s e n , 0., "Narrow R e l i a b i l i t y Bounds f o r S t r u c t u r a l Systems", Journal of Structural Mechanics, ASCE,Vol.7,No.4,1979 F l u g g e , W., Vi s c o e l a s t i ci ty B l a i s d e l l P u b l i s h i n g Company, London,1967 F o s c h i , R.O., " R e l i a b i l i t y of Wood S t r u c t u r a l Systems", Journal of Structural Engineering D i v i s i o n ASCE,Vol.110,No.12, 1 984 F o s c h i , R.O., and B a r r e t t , J.D., "Load D u r a t i o n E f f e c t s i n Western Hemlock Lumber", Journal of Structural Engineeri ng D i v i s i o n ASCE,Vol.108,No.7,1982 G e r h a r d s , C , and L i n k , L., " E f f e c t of L o a d i n g Rate on Bending S t r e n g t h of D o u g l a s - F i r 2 by 4's", Forest Products Journal, Vol.36,No.2,1986 H a s o f e r , A.M., and L i n d , N.C., "Exact and I n v a r i a n t Second-Moment Code Format", Journal of the Engineering Mechanics Di vi si on ASCE,Vol.100,No.EM1,1974 H o y l e , R., G r i f f i t h s , M. and I t a n i , R., " P r i m a r y Creep i n D o u g l a s - F i r Beams of Commercial S i z e and Q u a l i t y " , Wood and Fiber Scince, 17(3) Madsen, B., and B a r r e t t J.D., " T i m e - S t r e n g t h R e l a t i o n s h i p Lumber", Dept. of Civel Engi neer i ng, Univ. of B.C. , St ructura I Research Series, Re port No.13,Vancouver,B.C.,Canada,1976 Madsen, B., and Kenneth, J . , " D u r a t i o n of Load E f f e c t s i n Lumber, P a r t 1: A F r a c t u r e Mechanics Approach, P a r t 2: E x p e r i m e n t a l Data", Canadi an Journal of C i v i l Engi neeri ng,Vol.9,No.3, 1 982 Madsen, H.O., Krenk, S. and L i n d , N.C., Methods of Structural Safety, P r e n t i c e - H a l l Inc.,N.J.,1986 M i n e r , M.A., "Cumulative damage i n f a t i g u e " Journal of Applied Mechanics ASME,Vol.12,A159-A164,1945 94 95 N i e l s e n , L.F., "Crack F a i l u r e of Dead-,Ramp- and Combined-Loaded V i s c o e l a s t i c M a t e r i a l s " Proceedings of F i r s t International Conf e r e nee on Wood Fracture, B a n f f , A l b e r t a , C a n a d a , A u g u s t , 1 9 7 8 N i e l s e n , L.F., "Wood as a Cracked V i s c o e l a s t i c M a t e r i a l , P a r t 1: Theory and A p p l i c a t i o n s , P a r t 2: S e n s i t i v i t y and J u s t i f i c a t i o n of a Theory", Proceedings of International Workshop on Duration of Load in Lumber and Wood Products,Richmond,B.C.,Canada,September,1985 R a c k w i t z , R., and F i e s s l e r , B., " S t r u c t u r a l R e l i a b i l i t y under Combined Random Load Sequences", Journal of Computers and Structures ,Vol.9,1978 T h o f t - C h r i s t e n s e n , P., and Baker, M., Structural R e l i a b i l i t y Theory and It's Applications, S p r i n g e r - V e r l a g , New York,N.Y.,1982 Wood, L., " R e l a t i o n of S t r e n g t h of Wood t o D u r a t i o n of Load", U.S. Department of A g r i c u l t u r e , Report No. 1916, F o r e s t P r o d u c t s L a b o r a t o r y , Madison,Wis.,1951 

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