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5 . 7 0 0 8 7 3 - F a K (015,, m < R ; 3 „ „ . G a , i . s s i a n contiibnlions, made oi lhogonal Ihrongl. a I f c n n i l e scnes. I'irsl-yicid and fiiliguc failure rales arc predlelcd f r o m llicse moMicnls, whicli arc oflcn simpler lo esllnialc ( f r o m cilhcr a lime iiislory (ir .•nialyliL-al iimdcl). Ilolh hardening nnd .soflcning nonlinear iiioilels are duveioped. ihese arc sliown Io be more Hcxiblc llian Ihe convenlional Charlier and Edgewoilh S C U M , w i l h Ihc abiliiy to rellecl wider ranges o f nonlinear bchi.vior. Aiialylical momcni-based cslimatcs ol spoclral den.silics. cicssing rales, probability dislributioh.s o f Ihc lespon.sc and lis cxlicnies, and faligiic damage rates arc formed. These me Imiiid lo compare well wilh cxaci rcsulis f o r varions nonlinear models, iiichiding imiilinear oscillalor responses and (luasi-slalic re-sponses lo Morison wave loads.

INTRODUCTION T o as,scs.s s l i n c l t i i n l l e l i i i b i l i t y against e x t r e m e l o a d s a n d f a l i g i i e , d y -n a m i c r e s p o -n s e s arc c o m m o -n l y m o d e l e d as G a u s s i a -n r a -n d o m processes i h i s n i o d c l IS g e n e r a l l y i n a p p r o p r i a t e , l i o w c v c r , w h e n s i r u c l u r a l b e h a v i o r IS n o n l i n e a r , l l i c c . x c i l a l i o n is n o n - G a u s s i a n ( e . g . , w i n d a n d w a v e l o a d s ) , o r l i o l l i . I n s u c h cases, G a u s s i a n m o d e l s m a y s i g n i r i c a n l l y m i s r e p r e s e n t the r i e q u c i i c y o f high response l e v e l s , w h i c h ' c o n l r i b u l e m o s t l o b o t h firsl-passngc aiKl l a l i g u c l a i l u r c s ( G r i g o r i u |98<(a, b ; L u l c ; e l a l . 1984) b c i m v a l c n l l i n e a r i z a l i o n ( C a i i g h e y 1963; A l a l i k and U i k u 1976; S p a n o s 1981) c a n a c c i n a l c l y e s l i m a l e gross response s t a t i s t i c s s u c h as r m s l e v e l s l i u l g i v e s no i n r o r m a l i o n o u these noii-Gau.ssian etTects. S i m p l e a n a l y t i c a l r c s t i l l s are d e v e l o p e d here l o p r c d i c i .Ihcsc .ejTecls, u s i n g H e r m i l e series niodcl.s based o n r e s p o n s e m o m c i i l s . V a r i o i i . s n o n l i n e a r m o d e l s have l i c e n f o r m u l a l c d t h r o u g h s c r i e s a p p r o x ^ i n i i i l i o i i s , o l l e n in l e r n i s o l l l c i m i l c p o l y n o m i a l s . F u l l p r o b a b i l i t y d i s t r i b u . n o n s h a v e been c s l i i n a l c d f r o m n o n l i n e a r response m o m c n i s w i t h G r a m -C h a i l i c r i i n d l i d g c w o r l h series ( e . g . , -C r a n d a l l 1980; O c h i 1986) T h e s e s e r i e s c a n b e h a v e c r r a l i c a l l y . h o w e v e r , y i e l d i n g n i u l l i m o d a l a n d e v e n n e g u h v e p r o b a b i l i l y d c n s i l i e s and c r o s s i n g rales f o r s i g n i f i c a n t n o n l i n e a r Hies. A l l c r i i a l i v c l y , a n o n l i i i c a r i l y o f k n o w n f i i n c l i o n a l f o r m ( e . g . . f o r c e t i e l l c c l i o n c u r v e ) can be c.xpand.cd i n t o a H e r m i l e s c r i e s t o r c n e c t n o n l i n e a r c o n l r i b u l l o n s l o Ihe response c o v a r i a n c e and s p c c l r a l d e n s i t y (Ivladsen et a l . 1986). T h i s m c l h o d i c c i i i i r c s an a n a l y t i c a l d e s c r i p t i o n o f t h e n o n l i n e a r i t y , a n d IS n o l g e n e r a l l y used l o r e f l c c l n o n G a u s s i a n r e s p o n s e c h a r a c l c r -i s l -i c s . A -i -i o l h e r v a l u a b l e I c c h n -i q u c f o r a n a l y t -i c a l n o n l -i n e a r m o d e l s -is M a r k o v d i l k i s i o n a n a l y s i s and s;lóclia5lic a v e r a g i n g ( C a u g l i c y 1971; L i n

' A c t i n g A s s l . I'rof., Dcpl. o f CÏv, Engrg., Stanford U n i v . , S t a n f o r d , • C A ' 9 4 3 O T ' N o l c . Oi.sciission open imlii March I , 1989, T o cxlend the closing dale one m o n l h , a written letiiicsl imisl be filed with tlic ASCE Manager o f Journals The manuscripl lor lliis paper was s a l w i t t c d for review and pos,siblc publication on i Occcmlier 29, 19X6. This paper is pari ofÜKjoiminl of li:„sineerine Mechanics, V o l I M , N o . 10, Oclober, 19KII. (?iASCE, ISSN 07.13-9.199/8R/OOin-l772/.'i;i 00 -t $ I S per page. I'apcr N o . 22X5.S.

i l ^ ' o r ^ £ o , V - n " . " ° ' " ' " " ' " " " ^ ^ l ' ' ' ^ p r o b a b i l i s t i c d y n a m ¬ I C S o r , . m c m o i y o f the r e s p o n s e . . , i ( s k ^ e w n e ^ s ' T u r i n r ' ^ T ' ^ ^ ' " r ' ' ' ^ ' " " ^ ^ Use r e s p o n s e m o m e n t s ( S k e w n e s s , kiirlosi.s e ( c . ) l o f o r m n o n - G a u s s i a n r e s p o n s e c o n l r i b u l l o n s m a d e o r t h o g o n a l i h r o u g h a H e r m i l e , series. T l i c m o d e l s p edic ull' p r o b a b i l i t y d i s t r i b u t i o n s and rales o f e x t r e m e s and f a t i g u e f o m I.ese m o m e n t s w i n c h are o f t e n s i m p l e r t o e s t i m a t e ( f r o m e i t h e r t i n e series o r j w i a l y l i c a m o d e l s ) . I f used w i l l , o b s e r v e d m o m e n t s b o m ' e s p o n s e ^ S f y o ' " n i r ' ' ^ ' ' ' ' - ' n e e d , 0 ' 5 s p e c i f y o r a n a l y z e a p r e c i s e n o n l i n e a r m o d e l . I n a n a l y t i c a l s t u d i e s these n o n l i n e a r m o d e l s can be c o m b i n e d w i l l , v a r i o u s L i n i e i l es. n i a b o n c c h n i c u e s ; f o r e x a m p l e , c l o s u r e s c h e m e s f o r m o m c n i s o f no n a r e s p o n s e s l o w h i l e noi.se ( C r a n d a l l I9KÜ; W u and L i n 1984), a n d genera m p i i l - o u l p u l r c l i i l i o n s f o r h i g h e r m o m c n i s o f l i n e a r i c s p o n i t on G a u s s i a n i n p u l ( L i n 1976; L u l c s and H u 1986) I h e s e H e r m i l e m o d e l s are s h o w n l o be m o r e H c x i b l c l h a n Ihe C h a d i e r b l n v ? o r - n , ' : ^ " ' " r

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w i d e r . . n g e o f n o n N n b c h a v i o i . r i i e y are aLso p a r l i c i i l a r l y I r a c l a b l e , r e d u c i n g n o n l i n e a r a n a l y s i s m m a n y ca.ses l o s i m p l e l i a n s f o r m a l i o n o f k n o w n r e s t i i i s f . ^ n i n'c r S ' T Ï ' - ' 1 1 3 f o r c r o s s i n g r a l e s , c x . r e m e and f.Uigue a i c f o u n d l o c o m p a r e w e j l w i t h Ihe e x a c t r e s u l t s f o r v a r i o u s n o n m e a r m o d e l s , i n c l u d i n g f u n c t i o n a l l y I r a n s l o r m e d G a u s s i a n processe w a ^ e I S i " ' ' " " ^ ^ ^ - ' • ^ ' ^ ^ - " " ' ^ - - ^ - ^ A p p r o x i m a l i i i g N o n i i o n u a l Kesptm.ses I ' o r b o l l , l i r s l - y i e l d a n d f a l i g u c a p p l i c h l i o n s i l i.s u.scrul l o e s l i m a l e v J x ) d e l e m i n e d b y he e n i n c p r o l i a b i l i l y d e n s i t y f u n c l i o n (l>DI^) of X(,) M t ) n l ^ n l T L T '''^- " ' " ^ '«^Giirding Ihe s i a l i c n a U . i c o f l l i e r e s p o n s e - l o

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I i n w h i c h / ; i . , d e n o t e s I h c r e s p o n s e m o d e ; a n d - mnx[„J.x); • ' ^ < , < + ' ^ ] KS a c o n v e n i e n t d e f i n i l i o n o f t h e m e a n r a l e o f r e s p o n s e " c y c l e s " T h e q u a n t i t y v ^ h ' o u p p r o x i m a l c s b o i h U i c f a i l u r e r a l e p e r c y c l e ( f o r e x -n ™ \ " . " u " '•'^•MH'i.se p e a k e x c e e d s l e v e l . r n n a r o w - b a n d v i b r a l i o u ( f o r f a l i g u c ) . E c , . la f o l l o w s i f I h c q n a n l i l i e s X O a n d XU w i n c h a r e i i n c o r r c l a l c d , a r c a s s u m e d i n d e p e n d e n t a s w e m

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n c x i b i e y c l t r a c t a b l e m o d e l s . •

W h e n o n l y c e n l r a l m o m e n i s o f .V(/) a r c . a v a i l a t i l e , E q . ' I f l ' i s ' a n a t u r a l g h o i c c m v i e w o f t h e w i d e l y used C h a r l i e r a n d E d g e v b i ^ h series f o r / S b sed o n these m o m e n t s , It is s h o w n in the n e x l ' s e c l i o n ; h r e v e , tha Ï sc l o f ' T ' ^ ' l ' " ° " l i n « n r i . i e s w h e t) sed o n a h m , c d n u m b e r o f m o m e n t s . I n Ihc f o l l o w i n g s e c t i o n an a (erna ,ve m o d e l ,.s d e n v e d b y a p p r o x i m a t i n g , in E q . lb, , ^ t l , e r ƒ M b e . l a , f r o m c c i , , a l m o m e n t s . H e r m i l e scries e s t i m a t e s o f r . and o f / ' ^ a i c s l i o w n ,0 p , o v , . l e u s e f u l m o d e l s o f s o f l e n i n g ( „ „ > 3) . n d har ing ( „ , - f . L M , '•"••'P'^'^".^^ y- ^ 1 ' " ' = H e r m i l e jTiodcls a,e as s i m p l e (o a p p l y : M - n ' ' ' ' ' ' ? ; ' " ^ ' r " ' ' " ' ^ ^ • ' " " " ' ^ c o c l t i c i e n l s m a y ari.se, a l b c i in a d d l e , c n l , o l c . I hese l l c n n i l e models a i e gcrie. ally m ö i e s t a b l e , h o w c v c . " v o . d m g he p o s s i b . h l y o f negative roF.sa„d c r o s s i n g l a l c s i i l h e f n n ^

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•moments . a „ Ihese H e r m i t c m o m c n i s are m o r e d i r e c t m e a s u r e s o f • n o n l i n e a r i t y . F o r l i n e a r (Gaussian) responses /i„ = 0 f o r ^ > 0

£ q . 2 I S t y p i c a l l y t r u n c a l e d at /V = 4 , because h i g h e r m o m e n t s ' m a y s h o w 'large s a m p l i n g v a r i a b i l i t y . C o m b i n i n g E q s . \a a n d 2 w i t l , N ^ 4 X

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(6) .in w h i c h .v„ - (.r - ,„^.)/tr^.. T h i s is c o n s i s l e n i w l l l l the c o n v e n t i o n a l o u , - m o n , e n l C l , a d , e r series f o r y , ( . v ) . T h e c o r r e s p o n d i n g Edge o , e n e s , w h i c h assumes t h a i r e s p o n s e c u m u l a i K s d e c a y in a s p e c i f i c fash?o c(ain,s a n a d d i t i o n a l i c r i n f o r a s y m m e l r i c respon.ses ( O c h i 1986). A d d 1 l i f i i a l ( c i m s m a y alst, be r c l a i i i c d in E q . 2 t o r e l l e c l m a r g i n a l response m o m c n i s o l s l i l l h i g h e r o r d e r s , n o n s l i i l i o n a r i l y , o r j o i n i ( H e r m i l e ) n , m c n l s o f A ' a , i d A ' ( W i n ( c r s l e i n 1987) t n c i m n c ; m o r . , l t ^ " ï ' p S - ' V ' " ; / YT'^'r'^^' ' i . p p r o . x i m a l l o n to Ihe rai o o f I D l - s , y v„(-v„)Mv„). I f f e w h i g h e r m o m c n i s arc r e t a i n e d y . v U ( , W - V ; , ) IS a p p i o x i i n a l c d b y a l o w - o r d e r p o l y n o m i a l a n d i b e i c s u l l i n R til! n 11 T ' ? ' i i ^ ' ^ ^ ' ^ ^ " ' ' ^ " 1 ^-vW in E q . I « - i s n o i m a r k e d l y d i l f e r e n t t h a n t h a i o f Ihe G a u s s i a n m o d e l . I l c c a u s c i l s tail b e h a v i o r is so i n H c x i b l e

hil 2 m a y p r o d u c e an i l l - b c h a v c d estimate o l / . v ( . v ) - m u l l i m o d a l and even'

n c g a t i v e - i n o r d e r t o r e p r o d u c e the m o n i c n i s o f a s i g n i h c a n l l y n o n l i n e a r r e s p o n s e I

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M n . T I I , ^ ' ' - . ' " ' ^ '^^'Gewoilh vai-iuni arc (1) P o s i l i v e and (2) Ihe w ' r b e h a v i o r is the s m a l l e r o f the t w o . ( T h e r e s p e c t i v e regions are b o u n d e d b y the c u r v e s s h o w n w h i c h r e p r e s e n m a x i m u m possible aj v a l u e s , and the a , = 0 a x i s . ) T h e ( a , a . ) v a l u e s o f v a r i o u s c o m m o n d i s t r i b u t i o n s are al.so s h o w n f o r r e f e r e n c e ' Ihc ^ g n o r m a l a n d c l i . - s q n a r e d i s l i i b u l i o n s c o r r e s p o n d l o lines o n Ihis d i a g r a m . W h i l e o t h e r d i s l r i b u l i o n s ( e . g . . R a y l e i g h , e x p o i i e n l i a l , c x l r c m c l y p e I e f l e c l o n l y s i n g l e p o i n t s . A l l h o u g h the C h a r l i e r series s h o w s g r e a l e s l a b i l i l y t h a n the E d g e w o i l h m o d e l , n e i l h e r p r o d u c e s a p o s i l i v e d e f m i l c f n l l , " ' ' " " ^ h a r d e n i n g rcspon.sc ( „ „ < 3, < (1). T h e e x i c n l o f s o f t e n i n g ( p o s i l vc v a l u e s o f /i„) a n d a s y m m e l r y ( n o n z e r o values o t / , , ) l h a l can be nmdclec a c c e p t a b l y is also l i . n i l c d . O f ||,e c o m m o n d i s l , i b i U i o n s s h o w n

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E v e n i f these series r e m a i n u n i m o d a l , l l i c y may nol a c c u r a l c l y m o d e l s i g n i n c a n l i . o n h n c a i i l i c . s . - f h i s is i l l u s l r a l c d b y E i g , 2, w h i c h s h o w s U i a i h c r r c s u l i s (Ec|. 6) l o r s y m m e l i i c respon.ses w h o s e f o u r l h m o m e n t s inciea.se t h r o u g h = 5 (/,, = | / | 2 ) . A l s o s h o w n arc exact r e s u l t s f o r v a r i o u s n o n l i n e a r I r a n s f o r m a l i o n s o f a G a u s s i a n p r o c e s s U(iy i h e n o n l i n -c a r p a r a m e t e r -c in F i g . 2 has been a d j u s t e d l o p r o d u -c e the d e s i r e d f o u r t h m o n i c n l . A s « i n c r e a s e s , the C h a r l i e r results r e t a i n I h c c l i a r a c l e r i s l i c O a i s s i a n - l i k c s h a p e , p a r a b o l i c o n t h e s e m i l o g scale s h o w n . T h i s p r o d u c e s sy.slcmntic a n d u n c o n . s e r v a l i v c e i r o i s i n c s l i m a l i n g t a i l b e h a v i o r . F v e n f o r m o d e r a t e r e s p o n s e l e v e l s , C h a r l i e r e s l i m a l c s o f v.lv) a n d f J x ) l , c c o m e mcrca.sing y ei r a l i c ( y e l u n i m o d a l . f a l l i n g w i . h j n the a c c e p l a b l e regions o f F l g . I f o r I h c cases s h o w n ) as g i o w s ,

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s c a ' J l c ' ï h i s t l c s t , ' , 7 ' ? " r ' ^'8- 2 s h o w . e h . l i v e l y l i u l e d e « n f . h V ! m o m e n t s are s u n i c i c n t t o c a p t u r e a g r e a t

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H E R M I T E M O M E N T M O D E L S FOR NONLINEAR R E S P O N S E S Z n n t . '^'^^ ' " ^ a p p r o p r . a l e m o n o t o n e f u n c l i o n , A', l o a G a u s s i a n u t l . c r O) Is i n v e r s e , ' l , ; , s c d o n response m o m e n t s . I t is c o n v c n i e n l here d 1, i b u l i o n t-'Js c.g a „ > 3) a n d ' h a r d e n i n g - responses ( w i l h n a r r o w e r .nis e . g . , n , < 3) 1 licsc c h a r a c l c r i s l i c s m a y be due l o a c i u a l s o f l c n i n c / o n Z ' e ^ : . r r ' T " " " " " " " n o n - G a u s s i ^ n ex i u f M c r i n i t c M o m e n t IWodels o f S o K c i i i i i g Responses „ i n ' r c ' ; . ^ . ' ' " r " f " ^ " ' " ' ^ " ' ^ " ^ a s o f t e n i n g r e s p o n s e , (lie t r a n s f o r m a t i o n a t o a s t a n d a r d i z e d r e s p o n s e is l a k e n as a n A ' - l e n n H e r m i l e s e r i e . X - riix = K [ f y + / i 3 ( t y ' - l ) + / i , ( ( ; . i - 3 ( y ) H . . . . ] (7) ' r ! ? . ! ' ° ' ï f ' ' ' ' r ' ' • 3 l - ' ' - ' J = " - ^ l i z c d d i s t r i b u t i o n , w h i l e ic KS a s c a l i n g f a c t o r e n s u r i n g l h a l X„(i) has i m i l v a r i a n c e . T h e s e c o c L en s ; ; o 7 y n ' o m h r ' S 'EO " 7 " " 7 ' ? T " " " ' - ^ ' ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ _{o l y n o m i a l to E q . 7 a n d t a k i n g c x p e c l a l i o n s . I f i h e n o n l i n e a r i t y is} u f f i c i e n l l y m i l d so l h a l t e r m s o f o r d e r f , J „ c a n be n e g l e c t e d , ll,, is o u n d to be p . e c s c l y e q u a l l o Ihc H e r m i l e m o m e n t l,„ ( W i n t c r s l e i n 1985): K = I {8(1) = = " . • • • ) M o r e a c c u r a t e H e r m i l e m o d e l s m a y l,e f o r m e d b y i n c h i d i n g s e c o n d -o r d e r t e r m s ( -o f -o r d e r / , , „ / i „ ) i n i n a l c h i n g m -o m e n t s . T h i . ' g i v c s a -o u ' p M s ï l o f q u a d r a t i c e q u a t i o n s f o r the c o e l l i c i e i i l s h„ ( W i n l c r s l e i n 7987) -m m o m e n t s , Ihis s o l u l i o n is • / , H / y ^ - V ' + 3 6 / ; , - I y i + l - 5 ( a , : ^ - | •8 ~ T 8 (9ri) ' +6/54 . 4 - I - 2 V 1 + l . 5 ( a , - 3 )

:2> -1/2

•••J r- _{' " r i l l .} (9c)

a n d K a - / ; , ,,s H c r m K e m o m e n l s are m a t c h e d l o firsl o r d e r i n ' i h e n o n G a u s s m n c o n l n b u t i o n s . S i m i l a r l y . E q s . 7 and 9 « - c arc l o g c he said l o p r o d u c e ••.second-order" s o f t e n i n g H e r m i l e model.s.

. H g . I s h o w s l h a l Ihesc H e r m i l e m o d e l s b e h a v e s a c c e p l a b l v ( m o n o t o n - c a l l y , so l h a t dXIdU remains p o s i l i v e ) f o r a c o n s i d e r a b l y w der^ ^ ^ ^ ^ ^ ^ r^mihnear responses l h a n Ihe C h a r l i e r and E d g e w o r l h s e d e s . ÏÏe n s ' ( h E-lt. s h o w s (he s e c o n d - o r d e r s o f l e n i n g m o d e l l o be p a r l i c t d a r l y f l e x i b l e A l a,c r c s l n c t c d l o .soflening respon.ses , ( „ , > 3 ) ; I n i r d e n i n g c Ï Ï ^ ^ m o m e n l m o d e l s are s h o w n as f o l l o w s . . ^ ' " n b u c i m n c

C r o s s i n g K a l e s a n d P r o b a b i l i t y D i s l i i b u l i o n s

I f Ihe H e r m i l e series in E q . 7 is m o n o l o n c , ils c r o s s i n g s l a l i s l i c s -md r n s l - o r d e r p r o b a b d i l y d i s l r i b u l i o n s f o l l o w d l r e c U y f r o m G i u i s s i a n ' S u l i s

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F-.v(.v) = / ' [ A ' ( O s . r ] - < P K v ) ] , ( i l )

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" W - { v ^ ^ v ) ,!• c + i u ) r - i ^ / m T c ( i 2 „ ) In w h i c l i I

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(12/;) in i c r m s o f Ihe c o n s l a n l s a = h j i f , , , /, = | , / 3 A , , a n d <; = ^ , , 2 , , m o r e m o m e n t s are u s e d . E q . 7 m a y need lo be i n v e r t e d n u m e r i c a l l y (o find

F o i i r - m o m e n l H e r m i t e e s l i m a l c s .of v^{x) ( E q s . 10 a n d 1 2 « - / ; ) are shown

ic.sponscs. U n l i k e Iho C h a d i e r ,ind E d g e w o r t h r c s u l i s . the H e r m i l e m o d e l s c l e d i b c c o r r e c t Iransilion f r o m l o g - p a r a b o l i c lo r m . g h l y l o g - l i n e a r t '

e h a v i o r as the n o n l m c a r i t y increases. T h e . s e c o n d - o r d e r H c r m i n ode u e s h o w n 10 be p a r l l c u l a r l y a c c u r a t e , a v o i d i n g .he s y s t e m a t i c c o n e v t

S 9r U m c h i 1' " " f ''"f " " ' y ' " " ' " ^ y " ' ^ c o c i r l c i e n l s Zni t q . s . 9«--( l o i n c l u d e second-Older non-Gaussian e l f e c l s

J i v c n l o g n o r m a l processes, who.sc nonGaussian aspects incrcisc r m

-E d B c l ? ^ I ''.''='^'""^'"°''el (I^'e. 3 ) . I n c d n l r a s l , n e i l h e r Ihc C h a d i e r n o r

l t 7 , . TVT " ' ° ' o g n o r m a l ' s l o n g u p p e r l a i Thé

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FIG, 3. Crossing Rale ËsHmntfes (or Lognormal Responses

A p p l i c a t i o n (o N o n l i n e a r O s c i l l a t o r s A s a r u r l h c r e x a m p l e , c o n s i d e r the d i i i p l a c c m c n t r e s p o n s e , X{i), o f a n o s c i l l a t o r w i t l i nonliiPcar slilTness: . (13) vdV) + cX(i) + kiXu)] = /•(/) i f Ihe f o r c i n B f u n c t i o n / • ( / ) is G a u s s i a n w i l h z d r o mean a n d c d n s l a n i s p c c l r a l d e n s i t y S„, Ihc ( h e o r y ó f d i f f u s i o n processes gives Ihc exact r e s u l t s ( C a u g h c y 1971; L i n 1976):

U(x)

Mx) K(x) clx

(I4r7)

in t e r m s o f the s l r a i n e n e r g y U{x), Ihe i n i t i a l slifTncss k„ = K'id) a n d the m i l i a l v a r i a n c e al T r 5 , M , • I n the l i n e a r case. (7(.v) = A„ r / 2 a n d Ihc G a u s s i a n n i o d c l is o b l a i n c d .

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l-FIG. 4, SInllsllcs o l Nonlinear Osclllalpra

Fig. '1 con.sidens I w o l o r n i s o f n o n l i n c n r .slinncs.s: u l i i l i n e a r - c l a s l i c c . s e and

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c o n l n u K . u . s l y s o f l e n i n g m o d e l . These m o d e l s

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he n n - 1 v l i c ,1 values d i . s p l . i y c d . ( T o n i a x n m z e nonlinear e i ï e c i s in Ihc b i l i n e a r c i s e Ihn F f f % " I o s ; : " ^ r '''''' ' ' ° ' " ' " ' ' k u r l o s L ' l ; " : b e c ^ e ï J e. ^ al.so s h o w s l o u r - m o m e n l e s l i m a l c s o f Vy{x) and r U) Frnm i l „ C h a r l i e r and H e r m i l e series. T h e s c eslimales o f . . ^ ^ ^

ol i n 1 , ,, " ' ^ f " ' e ' l y n a m i c a s s u m p t i o n in F q . ! « ) and are o b l m n e d l o r Ihc H e r m d e m o d e l b y d i i ï e r e n l i a t i n g E q . 11 ( W i n S s l c . ' n

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or n ^ " . ' " o d c i s w i l l , the g i v e n m o m e n l s -.inc g c a t e r n e x , b , l , t y l o m o d e l m a r k e d l y n o n - G a u s s i a n tail i X v i o ' o n S s' i ie^^^^^ N n ? ^ I • ' ' ^ " • ' " 1 " ' ^ a s s i i m p l i o n . i n E q . 1/; is no l o n g c i s a l i s f i c d . N o l c , h o w e v e r , l h a l Ihe Irufc d y n a m i c b e h a v i o r c a n n o t be I 1:7.00

i d c n l i f i e d b y c e n l r a l m o m e n l s , a n d l h a l Ihe H e r m i t e m o d e l s d e f i n e d here g e n e r a l l y l e a d l o m o r e c o n s c r v a l i v c e s t i m a t e s o f Vx(x) l h a n the C h a r l i e r a n d E d g e w o r l h m o d e l s , O f c o u r s e , these H e r m i l e m o d e l s can be a l t e r e d to t a k e a d v a n t a g e o f a d d i l i o i i a l d y n a m i c i n f o r m a t i o n . F o r e x a m p l e , v^lx) can be c s t i m a l e d b y c o m b i n i n g the f o i c g o i n g H e r m i l e e s l i m a l e o f / . v W w i l h E q . I « , i f this d y n a m i c m o d e l is k n o w n l o be m o r e a c c t n a l c l h a n E q . \b, N o n l i n c n r V i b r a l i o n F r a t l i k s

I J c c a u s c lie]. 7 is a s s i M i i e d l o h o l d a l all instants i n l i m e , i l a l s o I'clales •'U b i l r a r y f r a c l i l e s o f l i n e a r and n o n l i n e a r r e s p o n s e s . C o n s i d e r , f o r e x a m -p l e , Ihe u -p -p e r / ; - l ' r a c l i l e s - o f Ihe n o n l i n e a r r e s -p o n s e and i l s m a x i m a , v , and /n,, ( i . e . , w i t h p r o b a b i l i t y i h e r e s p o n s e e x c e e d s l e v e l .v,, al an arbi'l'rary I m i e , a n d l e v e l r/i,, a l an a r b i l r a r i l y s e l e c t e d m a x i m u m ) . E q . 7 I h c n rehiles .V,, a n d / / I , , l o l l i c i r l i n e a r analogs: - "',v 4" Ktr,-^ ' ( / ; ) ! / ( „ / / c „ , [ -

_'(/')]

_{. (15;,)} III,) = / ; l y i i « r , v ' - 2 _{+ I / ; „ 7 / c „ - | [ V - 2 l n (/;)] (15/;)} T h e s e r c s u l i s m a y be u.sefnl f o r d e s i g n aghirisi v i b r a t i o n - i n d u c e d f a i l u r e . I n I h e l i n e a r case, li„ = 0 f o r a l l / i > 3 and K = 1 so lhat Eqs. I 5 Ü - / ; y i e l d f r a c l i l e s o f G a u s s i a n a n d R a y l e i g h d i s l r i b u l i o n s , c o n s i s l e n i w i t h l i n e a r t h e o r y f o i ' l i g h t l y d a m p e d r e s p o n s e s . N o t e l h a l i f o n l y nr.sl-oidcr m o d e l s are u s e d ( E q s , 8 r t / ; ) , E q . \5a b e c o m e s s i m i l a r l o C o r n l s h F i s h c r e x p a n -s i o n -s f o r r a n d o m v a r i a b l e f r a c l i l e -s , a l l h o u g h the-se l y p i c a l l y i n c l u d e a d d i t i o n a l n o n - H c i i n i l e p o l y n o m i a l s based o n Ihe E d g e w o r l h a s s i i m p l l o n r e g a r d i n g c u m u l a n l S r H e r m i t e M o m e n t M o d e l s o f H a r d e n i n g Kcspon.sös T o m o d e l (hc n a r r o w e r d i s l r i b u l i o n l a i l s o f hai d e n i n g (a., < 3) r e s p o n s e s , a H e r m i l e s e r i e s can be a p p l i e d l o m o d e l Ihc f i i n c l i o n a l I r a n s f o r m a l i o n l o a G a u s s i a n p r o c e s s . T h i s e q u i v a l e n t G a u s s i a n f i a c t i l e , d e n o t e d ;((v) in E q . 12(1, c a n be l a k e n as ( W i n l c r s l e i n 1987): 1 / ( . V ) = ^ « ' C V ) - . V „ ^ 2 ll„/lL'„.dx,) . . . ₍₁₆₎

Hi w h i c h .r,| = (x - iiix)Ury , F i g . 1 s h o w s that Ihis m o d e l r e m a i n s m o n o t o n e

{((iildx > 0) f o r m o s t h a r d e n i n g r e s p o n s e s , i n c l u d i n g a l l s y m n i e l r i c ( a , = 0) cases. I n c o m p a r i s o n w i l h I h c C h a i l i e i / E d g c w o r ( h m o d e l s , (hc c o m b i n a -t i o n o f lhe.se H e r m i l e h a r d e n i n g m o d e l s ( E q . 16) a n d Ihe s e c o n d - o r d e r H e r m i t e s o f l e n i n g m o d e l s ( E q s . 7 a n d 12rt) can r e p r e s e n i a f a r g r e a t e r range o f ( u j , H ^ ) v a l u e s . M u l l i v a r i a l e e x l c n s l o n s o f b o t h s o f l e n i n g a n d h a r d e n i n g s c r i e s can a l s o be e s t a b l i s h e d ( W i n l c r s l e i n and ü j e r a g c r 1987). I f E q . 16 is m o n o l o n c , l l can b e c o m b i n e d w i ( h E q s . 10 a n d 11 l o e s l i m a l e cro.ssing r a l e s a n d f u l l p r o b a b i l i l y d i s l r i b u l i o n s . F l g . 5 s h o w s Ihese e s l i m a l e s o f v^{x) to be o f s i m i l a r a c c u r a c y l o the s o f l e n i n g n i o d c l r e s u l t s in F j g s . 2 - ^ . I n c o n l i a s l . C h a d i e r a n d E d g e w o r l h e s l i m a l e s b e c o m e

1781

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FIG. 5. c r o s s i n g Rates for Hardening Transformations of Normal Processes

- ? : ! i Ï s c a l e ' ' : F S I ^ ' ""^^ 3 . 5 0 , ( n o , s h o w n o n Ihe

N O N L I N E A R S P E C T R A , E X T R E M E S , AND F A T I G U E , HcspoosL' C o v a r i a n c e a n d S p c c l r a l D u n s i f y

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I. .1 = 3 (18)

i n w l u c l i [.y(>(<^)]„ s i g n i f i e s (lie / i l h - f o i d c o n v o l u l i o n o r S.XM) ( E q <)c ensures t l i a t / ? ; ^ { O j = t r i ) . S i g n i f i c a n t l y , w h i l e d i i ï e r e n c c s h c l w e e n m a r g i n a l m o m e n t s o f X{0 a n d UU) are o f first o r d e r i n fi„ , t h e i r s p e c t r a d i f i er b y o n l y s e c o n d - o r d e r I c r m s . T h e f u n c t i o n a l t r a n s f o r m a t i o n can t h u s s u b s l a n l i a l l y n i t e r he s t a t i c d i s t r i b u t i o n o f a n o r m a l p r o c e s s w i t h o u t g r e a t l y c h a n E i n u its c o r r e l a t i o n s t r u c t u r e . C o n s i d e r f o r e x a m p l e , the f o l l o w i n g u s e f u l s p c c l r a l m o d e l f o r Ihe u n d e r l y i n g G a u s s i a n p r o c e s s : .V|;((o; (i)n, 8) = -8(0,, Ü) -I- ül(| (19)

in w h i c h co„ a n d 8 i c f i e c l Ihc c e n l r a l f r e q u e n c y a n d b a n d w i d i h o f lAl) T h e c o r r e s p o n d i n g n o n l i n e a r s p c c l r u n i in E q . I« b e c o m e s

S,,im) =. (

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\ / 2 S ) K<fA-)' .V|/(w; o,n, R) 1 2f,l

oiD, \ / 3 5 ) + ^6',/(iu; 3iü,|, Y ^ 5 ) + - - - J (20)

Thi.s s h o w s t h e m a g n i t u d e s o f .sub- a n d s u p e r h a r m o n i c s i n d u c e d b y the n o n l i n e a r i t y . A g a i n , i n c o m p a r i s o n to c h a n g e s i n n i i i i g i n a l d i s t r i b m i o n s a n d m o m e n t s , I h i s r e d i s l i i b u l i o n o f p o w e r is a s e c o n d - o r d e r c n c c l . E x i r c i n c s n n d F i r s t - r n s s n g c K a i l u r c s

F o r first-passage f a i l u r e s ( e . g . , first y i e l d ) , i l is ii.scliil l o o b t a i n s l a l i s l i c s o f t h e e x t r e m e v a l u e

(21) C = m a x { [ A " ' ( r ) ] " " , O s / s T } ( / , = 1 , 2 )

T h u s , E d e n o t e s Ihe m n x i m i i i n v a l u e o f A ' ( 0 i f / i = 1 o r o f iA'(OI i f ' i = 2 a n d is o f interest in first pa.ssagc b e y o n d o n e - a n d t w o - s i d e d b a r r i e r s I f Ihe H e r m i l e m o m e n t s c r i e s is m o n o t o n i c ( F i g . I ) , the c o n v e n t i o n a l F o i s s o i i m o d e l f o r u p c r o s s i n g s g i v e s

P[E < A-] = ex p • - ,1 Vn 7' e x p ₍₂₂₎

m w i n c h the G a u s s i a n f r a c l i l c i,(.x) is g i v e n b y E q . 10 f o r s o f l e n i n g H e r m i t e m o d e l s , a n d b y E q . 16 f o r h a r d e n i n g H e r m i l e .scries. I he e x t r e m e f r a c l i l c

e„ ( s u c h l h a l P[E < c ] = p) is al.so e x p l i c i t l y c s t i m a l e d , ;is i n E q s . \'^<i-b

f o r the s o f t e n i n g m o d e l : 'V = >>tx + i<<r,v ",, -1- X ' ' - . / / < • „ - i ( ' g _(2.1) in w h i c h « = the c o r r e s p o n d i n g f r u c l i l c o f G a u s s i a n e x t r e m e v a l u e s F o r e x a m p l e , uf, = 2 In [/M.|,7/ln ( / > " ' ) ] f o r the P o i s s o n m o d e l . E q . 22 c a n also be r e f i n e d l o a c c o u n t f o r r e s p o n s e b a n d w i d t h a n d c l u s t e r i n g efiTccts ( W i n l c r s l e i n 19H4): / ' ( f = S x J = e x p _{P ^ - d i M T - c x p - ^ ^ ^ | - 2 i | . - ^ „ ( x ) j (24a)}

1783

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~m (co) <lu) (240) l i o n s hasc.1 on s n e e l r n l n o n e^. 'n 9 ^ ^^^^ c o m p a r i s o n t o c a l c n l a -l . i t ^ -l . - n e . , i i e i , e y

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(25/;)

peak a l l e v e l V m I ' r " ' T r o w - b a n d p r o c e s s U(0 has a

. V ( 0 ( W i . n e r s S I9H7). P ' " " " ' ^0) a n d B e c a u s e .7 has R a y l e i g h d i s t r i b u t i o n w i t h N T ^ I - 9 P n ->< a p p l i e d t o ( i n d the m e a n damage c v M ' } r ó M v " , , - r ^ ' '.' 7 = • t [ 0 ( r ) J ' • • ' n / ( 2 V 2 , r , ) ' ^ = I + / ; ( / ; - I)/i4

(26)

1704

D u e O lts first-Older n a t u r e , h o w e v e r , e r i o r s i n E q . 26 m a y b e c o m e m o r e s i g n i f i c a n t f o r , larger values o f b

(or

g r e a t e r l

o n l i S l l s )

A

second-order

r e s u l t

has

been o b t a i n e d i n ' W i n t e i s t e i n (

S

in

which an

7 ••

I

2V^J (27n)

w i d i V„ determined from (he .sccoiul-ordcr .soflening H e r m i l e m o d e l :

V„ = ; ; ; r = - ( l - l - / . l / i . ) - l

F o r l i n e a r r e s p o n s e s , K = 1 a n d /,., = = 0 so l h a l Ee|. 27/; g i v e s V., = 0.523. coii.sislcnl w i ( l , (he R a y l e i g h distribution. ( E q . 27 / s l i g h l l y o v e cs-S u l l m n d ^ n r ^ T " 1 '"'-^ ' ° t h e ^ , p r o x i m a l c r 1 0 (he W c i b u l l m o d e l . ) C o m p a r e d (o (he firs(-ordcr es(ima(e ( E q . 2 6 ) , the second-o r d e r c s i m a t e second-o y in E q . 2 7 . is s h second-o w n in t h e f second-o l l second-o w i n g (second-o m second-o e a c c u i ^ . e l y P i e d i c ( f a t i g u e d u e (o w a v e loads f o r large values o f b a n d a , .

n n n d w i d l h EITcces on Kaligne I ) i i m i , p c

E q s . 26 o r 27ri m a y be c o m b i n e d w i l h a s e c o n d c o r r e c t i o n f a c t o r (o r e n e c l b a n d w i d d , e l l e c ( s o n mean d a m a g e o b l a i n c d b y r a i n f i o w c o u n ino e^l5, W m ( e r s . c i n 1984; W i i s c h i n g a n d L i g h t 1980). 1, i.s n o . i T c . ho c er l l at this s e c o n d f a c t o r g c n c r n l l y decreases w i ( h b a n d w i d ( h f o Gaussian f o n h d r ' m o t ; ' " " ^ ' ^ ' ' ' ^ ^ ' ^ ' " ^ '^'e'^' n n d hence 0 to Fas 26 nd 9 7 " '^"".'^ " ' - ' " a t i o n s T h u s (hc n a r r o w - b a n d m o d e l l e a d i n g to E q s . 26 a n d 2 7 « , w h i c h a v o , d s l l i e need f o r b a n d w i d d , i n f o r m a d o n is u s u a l l y c o n s e r v a t i v e w i . h r e s p e c l (o the r a i n l l o w - c o n n d n g a s s u m p i i o n . ' H E R M I T E M O D E L S O F M O R I S O N W A V E F O R C E S n u S " d ^ n r i [ ! ; n e I ^ T ' ' " ? ' ' ' ^ " " ' ^ ^ ' - ' ^ ^ ' " ' « e s f r o m n o n l i n e a r Uld d a g l o r c e s T h e s e f^orces are l y p i c a l l y c o m b i n e d w i l h i n e r ( i a l e[rcc(s t h l o u g h M o r i s o n ' s f o r m u l a , w h i c h gives (he f o r c e per u n i ( l c n g ( h on a fixed c y l i n d e r ( e . g . , Sai p k a y a a n d Isaacson 1981): ^

n o = k,„V{l) + k,,V(n\VU)\ (28)

ill t e r m s o f the w a v e p a n i c l e v e l o c K y . V{i), a n d (he d r a g and mass c o e m c c n t s . k , a n d k,„. R e s u l l s a.ssume VU) lo be Gaii.ssian w i l h zero m e a n . T h e s e a s s u m p h o n s p e r m i t c o m p a r i s o n w i t h v a r i o u s r e s e a r c h re-sul s _ p a r l i c u l a i l y f o r c x l r e m e . s - i n Ihis case (see. f o r e x a m p l e S a r n k t v a and I s a a c s o n 1981; M a d s e n el a l . 1986). T h e assunVplion.s r E o t ' n e c c V s a r y 10 a p p l y (he H e r m d c modcLs. h o w e v e r , a n d m a y be g e n e r a h z c d I r o u ï m o m c n l - b a s e d H e i m i t e m o d e l o f VU) based o n o b s e r v e d w a v e s l a l i s l i c s

It I S c o n v e n i e n t to c o n s i d e r Ihe n o r m a l i z e d f o r c e (and quasi-s(a(ic r e s p o n s e ) . XU) = m / y , . E x a c ( c a l c u l a l l o n o f „ , ( . ) r c ^ e . s é u m e H é a l

lUegrsition, as w e l l as k n o w l e d g e of llie v a r i a n c e s o f VU) and VU) A s the b a n d w i d t h o f F ( , ) d e c r e a s e s , h o w e v e r , a s i m p l e a.sy n p l o > i c resu t s a v a i l a b l e ( e g . , Mad.sen c l a l . 1986): MOOC r c s m i is

5.C0 G a u s s i a n : . " t T E x a c t T r u c r m s • R i f s l - o ^ ' d e r " ' j E g u w J m ^ o r ^ - - - ^ ^ S e c o n d - o r d e r H e r m i l e : 5.Ö0

D.or)

0.1 INERTIA DOMINATED

FIG, 6, lExlremcs of Morison Wave Loads 10 PMC. DOMINATED - ~ = c x p >'.v(.r) '2(1 CNP I 5 ^ {29(0 (29/;) l i a s e d ö n Die l i i i i l l c s s d r a g p n i i u n c l e r r / = i . , r 5 / i ! - - - r i r e l a l e d d i r e c t l y . 0 the |lburiC L i n S' c^ f A ( | ' ' "^^ ° ' ' « 4 - 3

_ Jtl (Pi

24 • - 3.25 _{'I + " 3 ^} (30;

Co.30 =

KyiTl^P

[ „ „ ,ö +

fuiiiLa -

3'(o.5o)] • , . (31)

!'i ' / / " " ^ w a ' ' ' l ? ' ^ ' ! ! " "

^^'^«='"6

Tor a s l a n d a r d G a u s s i a n ; e.g., /,„ ,„ = [ 2 i n

u n d y (Eqs_ 8 « - ^ ) . w i n l e Ihe .seeond-order e s l i m a l e bases a n d K o n as n E q , 9. T w o G a u s s i a n i c s u l l s are also f o r m e d b y r e p l a c i n g K l V l w i l h a

n e a r l e r m o f Ihe f o r m p c r . K . T h e c u r v e s s h o w n use Ihe values p = \ ^ = 1./3 w h i c h p r e s e r v e s Ihe v a r i a n c e o f .V(/), a n d p = V s h = I 60 Ihe

^SrSi

h ^ ; M S 3 ) S " ^ ^ " ' ^ ' " ' ^

-• m Ï Ï l ï ^ T T t ' " r l ' " ' ' " " " ' ^ ^ " n n d c e s l i m a l e s Ihe res ^ '

•incc. D e u u l s e Ihe I r a c l i l e c,,,„ is n o r m a l i z e d i n F i g , 6 w i l l , ,e, peel lo Ihe c o n c c i r e s p o n s e v a r i a n c e ,,1. = I + 3./^, Ihe G a u s s i a n m o d e l w i c ,

I n S t ^ H e r m i l e

I . u g e i than Ihosc of the a c i u a l r e s p o n s e T h e s e c o n d - o r d e r H e r m i l e r c s u l l

E x n c L I l e r n i i l e ; D e c o u p l e d

S e c o n d - o r d e r

0.1 0.5 , 5 , 0

INEIiTlA DOMINATED d ^^^^^ DOMINATED

^ ^ ^ ^

v a l u c s ( M a d s e n e l al I98(,) A s t . J r '* ' ^ ^ " ^''''''''^e

77>r(\/2<l)

(32)

SUMMARY AND CONCLUSIONS!

exlrcines

' ' «J'^lribulioi, a i n i j t i o n s , a n d E q s . 22-24 f o r

h a S n ^ e " / l ? S ^ ; ^

- R e n i n g

and

p.ed;";'?,;.'',S;;;'';',,;;°

"t''

. „ o d d s . « o k • „

" C S ,

,1;',™'"'*-,

' ' " " ' ' < ' ' ' " = l ' ' " - « < '

m o , n c „ , - K

1788

model IS either uncertain or analytically i n l r a c l a b l e . In such c a s e s il rnav prove u s e l u l to require knowledge of only a limiled number of response moments. T l i e results s h o w n here (e.g., F i g s . 2 - 7 ) al.so suggest thai a simp e our-momcnt description may be sufficient to e s l i m a l e Ihc dislribu-lion (ails that govern r c l i a b i l i l y , against both extremes and fatigue, r a l h c f a c c u r a t e l y .

A C K N O W L E D G M E N T S 'i T h e research work p r e s e n l e d herein is supported'by ilic O n i c e of N a v a l

R e s e a r c h under C o n l r a c i No. N()()()|.|-«7.I<-()<l7.'i and by a joint granl from A . i>. V e r i l a s R e s e a r c h , .Saga I ' e l i o l c u m AAS, and S l a l o l l ( N o r w a y ) This

supporl IS .gralefully a c k n o w l e d g e d . ^

APFSENDIX I , R E F E R E N C E S

Alalik, T S . , and Uiku, S. (1976). ••Slochaslic lineimtóliün uf

nH,lli-dcga-e-^7it7.: M . l o S i ' - "'^""y -i'^'»lions." A</,.„„-... Af,ni:

" " r ; ; ; ; ^^S^höSl^'ll^S' ^0.0....;.. l,.ans,a,ion p r c c c s . . . ^ ^ J.. E.„r,.

^''mMni

•'• "'-'I': A S C I I ,

wm).

'

i-uwislwu^s::!

1 i u : S : ï S ' V Y^''^'' ^- ^^"^^'^^

^'Tj.b T

!• "Non-normi,l slochaslic icspolise of line,-,.

systems. 7. t/i^-r;.. M e t / , , , A.SCl', 112(2), 127-141.

Madsen, H Krenk. S.. and Lind. N . C . (1986). Mf,lu„h of sln,aural .safvly^

INenllcc-Hall. I n c linglewooil Cliirs, N J •

' ' t ^ ^ , : ! ^ r i ! ' ^ n ' ^ w ! t ^ ' ' ' nindonr processes m ocean cngincc, in«,'^

n r . ! i i ; i ! . " S f " " " ^ <l'^«f>)- "Sloch.slicaveraeine; an approximate melhöd of solvmg landoin vihralion problems." Inl. ./. Nonliiwor Med,., 21(2) 1 | I-I34 Saipkaya. F, and saacson, M . (I9RI). Medumic. of nu,yc force'on oklore

•unicliirc't. Van Noslrand Reinhold, New York. N . Y . "JJ^tor, ^^TcT'j4[\)^]'J^ '^'"'^'"'•""^ lincarizaliDn in siruclural dynamics, ' Ap,,L ,\l„d,.

" ^ i i n r n i ' r ! , ! ? - " ^ ^ ^ ^ ^ ' ^ ^ = 1 ^ :

I S s r ^ ^ h l ^ r S ^ ^ ; - ë r " "^^^

^ ) Z o ! f N f - ' V ' ^ f ) - "^'""'«"'-'^^'•^"l Hernnle models ofrandon, vibralion."

^^sn-^?rn!''''; ''""',!^''^'";c^.^' "I'^'lit^'-c .under wide hand fanUom

slic.sscs ./. i> r„a. Dn:, ASCII. 1116(7), I59.1-IC-07. ^ • ""f "Cunn.lanl.neglecl closure for non-linear