The Analysis Of Orthotropic Skew Bridge Slabs by A. Coull

8/12/2019 The Analysis Of Orthotropic Skew Bridge Slabs by A. Coull

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Ap pl. Sci . Res. Vol . 16

THE ANALY SISO F O R T H O T R O P I S K E W BRI D G E S L A BS

by A COULLDept. of Civil Engineering The Unive rsity Southa mpto n Englan d

S umma r yA m e t h o d i s p r e s e n t e d f o r t i l e d i r e c t s t r e s s a n a l y s i s o f o r t h o t r o p i c s k e wbr idge s l abs .T h e m e t h o d o f a n a l y s i s e m p l o y s t h e P r i n c i p l e o f L e a s t W o r k i n c o n -

j u n c t i o n w i t h t h e a s s u m p t i o n t h a t t h e s t r e s s r e s u l t a n t s m a y b e e x p r e s s e d a sF ou r i e r s e ri e s i n t he c hordw ise co-ord ina t e t he coe f f i c i en t s be ing func t ionso f t h e s p a n w i s e p o s i t i o n o n l y . A s y s t e m o f o b li q u e c o - o r d i n a t e s is u s e d t os i m p l i f y t h e a n a l y s i s .NomenclatureO x, y)O u, v)o~,~lLCtr iMx, My, M~yMu, My, MuvSx, SypEz, Ev }Exy, GxyA,jF~, GtR~, S~C1, C~, C8DSu, Sv

s y s t e m o f o r t h o g o n a l c o - o r d i n a t e ss y s t e m o f o b li q u e c o - o r d i n a t e sa n g l e b e t w e e n a x e s Ou, Ovn o n - d i m e n s i o n M o b l i q u e c o - o r d i n a t e so b l i q u e s p a n o f sl a br i g h t s p a n o f s l a bt r a n s v e r s e s l a b w i d t hs l a b t h i c k n e s sc/ 2xil)b e n d i n g a n d t w i s t i n g m o m e n t s p e r u n i t l e n g t ho b l iq u e b e n d i n g a n d t w i s t in g m o m e n t ss h e a r f o r ce s p e r u n i t l e n g t hn o r m a l l o a d i n t e n s i t yo b l iq u e n o r m a l l o a d i n t e n s i t ye l a s t i c m o d u l i d e f i n i n g s t r e s s - s t r a i n r e l a t i o n s h i p s i n o r t h o -t r o p i c p l a t e ss t r a in ene rgy coe f f i c i en t sb e n d i n g m o m e n t f u n c t i o n st w i s t in g m o m e n t f u n c ti o n ss t a t ic a l b e n d i n g a n d t w i s t i n g m o m e n t c o n s t a n t sop era t o r d /d~/o b l i q u e s h e a r f o r c es p e r u n i t l e n g t h

1 7 8


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SKE W BRIDGE SL BS 179

1 . In t rodu ct ion . In spite of the fact t hat the majority of bridgesin this country are skewed in planform, few methods of analysisexist for such structures. Because of the complex boundary con-ditions involved, only approximate analytical or numerical solutionshave been produced.Previous solutions have aimed at the determination of the de-flection function, the corresponding moments and shear forces beingobtained by double and triple differentiat ion. Even if the deflectionsare reasonably accurate, substantial errors are liable to be intro-duced by the process of differentiation. In the present work, thisdifficul ty is avoided by a direct determinat ion of the stress distri-bution in orthotropic skew bridge slabs.

The partial differential equation of plate theory is reduced to aset of ordinary linear differential equations by the assumption thatthe load system and stress resultants in the plate may be expressedas Fourier series in the chordwise co-ordinate, the coefficients beingfunctions of the spanwise position only. The series are chosen tosatisfy both the free-edge boundary conditions and equilibriumequations for the plate, and the unknown coefficients are de-termined by minimisation of the strain energy. A system of ob-lique co-ordinates and oblique stress resultants is used to simplifythe analysis.

2 . A n a l y s i s . The structure considered is a thin skew ortho-tropic bridge slab of uniform chord and thickness, simply supportedalong two opposite edges O A and B C and free along the other twofig. 1).

In order to deal most conveniently with the free-edge boundary< C ~ ]

0

\ \Fig 1 Skew br idge s l ab


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S K E W B R I D G E S L A B S ] 8

O n t r a n s f o r m i n g i n t o t h e o b li q u e s y s te m , K i r c h h o f f s b o u n d a r yc o n d i t i o n s f o r t h e f r e e e d g e s o f t h e s l a b r e d u c e t o

1 ~ M u vM v = S , - - O. (5)l ~A n y s y s t e m o f n o r m a l l o a d i n g o f s k e w i n t e n s i t y p m a y b e e x -

p r e s s e d a s a F o u r i e r s e r ie s o f t h e f o r mP = p0 + X (Pl, co s i~ + p~, sin i~), (6)

w h e r e t h e c o e f f i c ie n t s o f t h e s e ri e s a r e f u n c t i o n s o 1 t h e s p a n w i s e c o -o r d i n a t e ~ o n l y , a n d n i s s o m e i n t e g e r , c h o s e n t o e n a b l e t h e s e r i e s(6 ) t o r e p r e s e n t a s c l o s e l y a s d e s i r e d t h e g i v e n l o a d f u n c t i o n .S t a t i c a l l y c o r r e c t s o l u t i o n s t o t h e e q u i l i b r i u m e q u a t i o n s (4 ), s a t is -f y i n g a ls o t h e f r e e - e d g e c o n d i t i o n s (5 ), m a y t h e n b e o b t a i n e d b ye x p r e s s i n g t h e s k e w s t r e s s - r e s u l t a n t s a s c o r r e s p o n d i n g se r ie sM u : M u o + Z ( M u l , c o s i ~ + Mu2 , sin i~),M y = Z{Mvx,(COS i~ -- 1) + My2, sin i~},M u v : M u , o + Z ( M ~ v l , s in i ~ + M urk , co s i~), (7)S u = S u + Z ( S u l , cos i~ + Su2, sin i~),

1 dM uvo 1 Z dM~v~,S v - - I d~ + l d~ + Z ( S v l s i n i ~ + S v 2 ( c s i ~ - - l ) }i n w h i c h , i n e v e r y c a se , s u m m a t i o n is to b e c a r r i e d o u t i n t h e r a n g ei = 1 t o n . T h e c o e f f i c i e n t s o f t h e s e r i e s a r e a g a i n a s s u m e d t o b ef u n c t i o n s o f t h e s p a n w i s e p o s i ti o n ~ o n l y .

S u b s t i t u t i o n o f e q u a t i o n s (6 ) a n d (7 ) i n t o (4) y i e l d s t h r e e e q u i -l i b r i u m c o n d i t i o n s w h i c h m u s t b e t r u e f o r a l l v a l u e s o f ~ . H e n c e ,o n e q u a t i n g c o r r e s p o n d i n g c o e f f ic i e n ts , a s e t o f ( 6n + 3 ) e q u a t i o n si n t e r m s o f t h e o r i g i n a l (1 0 n + 3 ) u n k n o w n f u n c t i o n a l c o e f f ic i e n ts o fe q u a t i o n s (7 ) is o b t a i n e d , e n a b l i n g t h e s t r e ss r e s u l t a n t s y s t e m t o b ee x p r e s s e d i n t e r m s o f a c h o s e n s e t o f 4 n a r b i t r a r y f u n c t i o n s .

I f t h e s e a r b i t r a r y f u n c t i o n s a r e d e f i n e d asM u l , = F i V ) , M u 2 , - ~ G i ~ ) ,M u v l , : R , ~ ] ), M u v 2 , = S l ~ ) ,

t h e b e n d i n g a n d t w i s t in g m o m e n t s , a n d s h e a r fo r ce s b e c o m e , o n


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S K E W B R I D G E S L A B S 83

g i v e n b y , i n th e s k e w s y s t e m ,2~

Mu = Mu d~,2nj~,u : ~ - Mu d~. (10)

O n s u b s t i t u t i n g e q u a t i o n s (3) i n t o (1 0) , a n d c o m p a r i n g t h e r e -s u l t in g m o m e n t s a n d s h e a r f o rc e s w i t h (9 ), t h e c o n s t a n t s C 1 a n d C2a r e f o u n d t o b e

1

C1 = / | p o ( l - - ~ ) d ~ 7 - - R o , (11)d C

C2 = 0.T h e s o l u t i o n i s l e f t a t p r e s e n t i n t e r m s o f t h e c o n s t a n t C 3.F o r a n o r t h o t r o p i c p l a t e , t h e s t r e s s -s t r a in r e l a ti o n s f o r t h e c a se

o f p l a n e s t r e s s i n t h e 0 (x, y ) p l a n e m a y b e w r i t t e n~x = Exez + E x y e y ,f l y ~ Exyex + E y e y ,rxy ~ G x y y x y .

B y m a k i n g t h e u s u a l a s su m p t i o n t h a t s t ra i n s a re p r o p o r t i o n a l tot h e i r d i s t a n c e s f r o m t h e m i d d l e s u r fa c e o f t h e p l a t e , t h e s t r a ine n e r g y o f b e n d i n g m a y b e sh o w n t o b e

2 n

u - { A 1 M X A 2 M ~ A 3 M L 2 A 1 2 M ~ ,M v - -~ t 3o o -- 2A13M uMuv -- 2A23MvMuv} d~ d~, (12)

w h e r e t h e c o e f f i c i e n t s A a s s u m e t i le f o r m sA 1 = A{( E -- 2EzvGxv) co s 2 c~ si n 2 ~ + Gzy(Ex co s 4 ~ + E y s in 4 ~ )} ,A 2 = 4{E , ,G~ ,y} ,A 3 = A{E s in 2 ~ + 4ExGxv co s2 ~},A 12 = A{ Gxy( Ez co s 2 c~ - - E x v s in~ ~ )} ,A 13 = A{E cos c~s in 2 ~ + 2Gxv(Ex c o s 2 ~ - - Exv s i n 2 c~) c o s c~},A 23 = A{2ExGxv co s a} ,


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184 A. COULLi n w h i c h

o s ezl - - a n d E E x E y E x y .G x y ES u b s t i t u t i o n o f e q u a t i o n s ( 8) i n t o (1 2), f o l l o w e d b y i n t e g r a t i o no v e r t h e w i d t h o f t h e p l a te , y i e ld s a n e n e r g y i n t e g r a l o f t h e f o r m

13cl fU - - t3 / (F , , G, , R , , S , , 7) d~ .

T h i s i n te g r a l m u s t b e a m i n i m u m , b y v i r t u e o f t h e P r i n c i p l e ofL e a s t W o r k . M i n i m i s a t io n b y t h e c a l c u l u s o f v a r i a t i o n s l e a d s t o as e t o f 4 n h n e a r d i f f e r e n t i a l e q u a t i o n s w i t h c o n s t a n t c o e f f ic i e n ts ,t o g e t h e r w i t h a c o m p l e t e se t o f b o u n d a r y c o n d i ti o n s w h i c h a r is en a t u r a l l y i n t h e m i n i m i s in g p r o c e d u r e .

O n p e r f o r m i n g t h e p r oc e ss , it is f o u n d t h a t t h e c o n d i t i o n fo rm i n i m u m e n e r g y is t h a t t h e f u n c t i o n a l s F i , G ,, R , a n d S , o b e y t h ef o ll o w i n g s e t o f d i f f e r e n t i a l e q u a t i o n s(A~r~D 4 + 2 A 1 2r ~D 2 + A 1 ) F , - - 2(A2r~D~ + A12r ,D) R i +

+ ~ A2ar~DSG~ -- (A2ar~D 2 + AI~) S , +t = l

+ 2 A 2 Z ( r ~ D 4 F j - 2 r ~ D S R j + 1 2 r~ D 2p l,) +1 = 1

+ 2A12poFr~ + (A2r~D~ + A12 )p i ,Fr~ ++ A~ .3 ~ l~r~Dpej ---- 0, (13a )

i = 1

2 2 (A r ~D 4 A 12r~D~ A 1) G ,A 2 3 ri D + A l a ) R , - - + + - -- - 2(A2r~D~ + A19.r ,D) S , +

b+ A23 Z ( r~D3 Fj - - 2r~DSRj + 12raDpl ,) +~=1

+ A s ~ (r~D2 GI + P2,12r~) --f = lAla (C l l r , - - f po l2r , d~) - -(Aur~D~ + A12) p~,12r~ = 0 (13b)


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SKEW BRDIGE SLABS 185

2(A2raD 3 + A12r , D) F , - - (4A2r~D 2 - - A3) R i - --- (A23r/2D2 4- A13) G~ - - 4A 2ar ,D S l 4 -+ 4 A 2 ~ ( r a D a F j - - 2r~D2Rj + 12raDpl j ) +

j = l4 , 2 A ~ 3 E (r~DUG~4 P2/2r~) - - 4A1 2(Cl l r i - - f po l2r i d2]) 4i=1 0+ 2A 212r aDp l , - - 2 3 P 2 f 2 r 2 = 0 (13c)

(A23r,~D 2 + A la) F , - - 4 A 2 3 r , D R i 4 2 ( A 2 raDa 4 A12r ,D) G, 44 (4A2r/2D~ -- A a) S , 4 A23P1,12r~ 44 2 A 212raOp2, = 0, ( i = 1, 2 . . . . n) . (13d)

T h e r e q u i re d b o u n d a r y c o n d i t io n s a r e d e r i v e d fr o m t h e i n t e g r a t e dt e r m s w h i c h a r is e in t h e m i n i m i s i n g p r o c e d u r e , in c o n j u n c t i o n w i t ht h e k n o w n p h y s i c a l ed g e re q u i r e m e n t s o f v a n i sh i n g n o r m a l m o m e n ta n d d e f l e c t i o n . I n t h i s c a s e , t h e a p p r o p r i a t e b o u n d a r y c o n d i t i o n sre d u c e to , a t ~ ----- 0, 1.F i = G~ = 0A ~ ( ( r ~ D 2 F I - - 2 r t D R t 4 Pl,/2r/2) 4

2 4D F; -- 2rjDRj +i=1A 2a{S~ -- 2(Ca 4 ~ (rD G , 4 f P2,12r,d~))} = 0, (14)i=1

A2(r~D 2G i 4 2r~DS~ 4- p2 ,12r~) - - A 23 R i = O.A t a n y s p a n w i s e s e c t i o n W, t h e o v e r a l l t w i s t i n g m o m e n t , a b o u t

t h e e d g e v = 0 , o f t h e i n t e r n a l s t r e s s r e s u l t a n t s i s g i v e n b yT ~ = f ( M z v 4 S x v ) d v ,

w h i c h b e c o m e s , o n u s i n g e q u a t i o n s (1) a n d (8 ),

T ~ = c { C s - - 2 ( r D G i 4 , 2 s , - - f p 2 1 2 r i d 0 -

- - ( C l l ~ - - f ( f P o ( ' )1 2 d ' r ) d ~ ] ) c o s ~ 4 2 ( C l - - f flo ld ~])} .


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186 A. COULLTh e to t a l tw i s t in g m o m e n t o n t h e b r id g e s la b, d u e t o t h e e x t e ln a l

load sys tem, is1 er = f f pv dv du,o o

which becomes , on subs t i tu t io n o f equa t ion 6),1T = f {pol2c - - Sp ~,l ri} d~7.o

The cond i t ion o f to r s iona l equ i l ib r ium fo r the s lab i s tha t T == T ~ = 0) - - T ~ = 1), in wh ich the un de t e rm ine d cons ta n t Cavan ishes . Thus , a s the in te rna l s t re ss re su l ta n ts a re chosen to be ins ta t ica l equ i l ib r ium wi th the app l ied loads , to r s iona l equ i l ib r ium i sa lways ma in ta ined i r re spec t ive o f the va lue o f the cons tan t Ca .

O n m in im i s in g t h e s t r a in e n e r g y b y d i f f e r e n t i a t i n g t h e i n t e g r a lwi th re spec t to Ca, and us ing the bou nd ar y cond it ions 14), thecons tan t C~ even tua l ly reduces to

1

C3 = -- f Z f P2~12r d~ @o o+Al:~- GZ-- ( f { f Poe~)12d~ Idv)dv}--

o o o 1

The p rob lem then reduces to the so lu t ion o f the se t o f l inea rs imul taneo us equa t ions 13), sub jec t to the bo un da ry cond i t ions14) and th e tors ional cond i t ion 15).

3. Exa mp le. As a pa r t icu la r example , in o rde r to compare thepresent resul ts with previous so lu t ions , the case is considered of askew s lab sub jec ted to a un i fo rmly d i s t r ibu ted load o f ob l ique in -ten si ty p0.~ Fo r a s im ple load sys tem of th is form, , th e ser ies wit hn = 1 in equa t ions 7) is assu me d adequ ate to descr ibe the s t ressdis t r ibut ion ,


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SKEW BRIDG E S LABS 87

T h e m o m e n t s a n d s h e a r fo rc es a re t h e n g i v e n b yM u = P o 12 * l - - ,1 2) + F cos ~ + G s in ~ .M y = r a D 2 F - - 2 r D R ) ( c o s ~ - - I ) + ( r2 D 2G + 2 r D S ) s i n ~ .M u v = C a q - r D G + R sin ~ -1- S cos ~.S u = Po/(1 - - 2.1 ) + -~r { ( rD F - - R) cos + r D G + S ) sin ~}.S v - ~ l -~ -{( rD R -- r2 D 2F ) sin ~e _~_ r 2 D 2 G @ r D S ) ( c o s ~ - - 1) - ~

+ (r2D2G q- 2 rD S)} .T h e g o v e r n i n g e q u a t i o n s (1 3) r e d u c e t o , r e s p e ct iv e l Y ,

( 3 A ~r 4 D 4 + 2 A l ~r 2 D 2 + A 1 ) F - - 2 ( 3 A 2 r a D a + A 1 2 r D ) R +q - A 2 a r a D a G - - ( A e a r 2 D 2 q - A l a ) S q - 2 A 1 2 P o 1 2 r 2 = O , (16a)

( 2 A 2 a r aD a ) F - - 2 ( A 2 a r 2 D 2 - - A l a ) R - -- - {2A2r4D 4 + (4A12 - - Aa) rg'D 2 + 2A1} G - -- - 4 ( A 2 r a D a q - A 1 2 / D ) S - - A l a P o l 2 r 1 - - 2,1) = 0. (16b )

2 3 A 2 r a D a + A I ~ r D ) F - - 1 2 A 2 r g D 2 - - A a ) R ++ A 2 a r 2 D 2 - - A l a ) G - - 4 A 2 ~ r D ) S - - 2 A 1 2 P o l ~ r 1 - - 2.1) = 0. (16c)(A23r2D 2 + A la ) F - - 4 A 2 a r D R q - 2 ( A 2 r a D a + A 1 2 r D ) G +

q - ( 4 A 2 r 2 D 2 - - A s ) S - - 0 , (16d)w h e r e r = c / 2 x l ) , a n d , f o r c o n v e n i e n c e , t h e s u f fi c es h a v e b e e no m i t t e d f r o m t h e u n k n o w n f u n c t i o n s .

T h e b o u n d a r y c o n d i t i o n s b e c o m e , a t .1 = 0 , 1,F = G = 0 ,

3A2 r2D2F - - 2 r D R ) - - A 2 3 ( S - - r D G - - 2Ca) = 0,A 2 r 2 D 2 G + 2 r D S ) - - A 2 a R = O ,

w h e r e1 A l a A 2 aC a - p o l 2 - - - - [ r2D2F - - 2 rDR]01 .1 2 A a A a

F o r t h e p a r t i c u l a r c a s e o f a n i s o t r o p i c p l a t e o f 4 5 s k e w . a n d w i t ht h e s a m e r i g h t s p a n a n d c h o r d , s o lu t io n s h a v e b e e n o b t a i n e d b y


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188 A. COULLJ e n s e n l ) , u s i n g f i n it e d i f fe r e n c e t e c h n i q u e s , b y R u s h t o n 2 ) , u s i n ga n e l e c tr i c al a n a l o g u e , b y D a y a ) , u s i n g t h e d y n a m i c r e l a x a t i o nm e t h o d , a n d b y t h e a u t h o r 4 ) , u s i n g a d i r e c t e x p e r i m e n t a l i n v e s t i -g a t i o n .

T h i s p a r t i c u l a r p l a t e p r o b l e m i s u s e f u l a s a t e s t o f t h e a n a l y s i s,s i n ce i t r e p r e s e n t s a h ig h d e g r e e of s k e w , a n d t h e s p a n : c h o r d r a t i oi s s u f f i c i e n t l y l o w t o g i v e w i d e v a r i a t i o n s o f s t r e ss i n t h e c h o r d w i s ed i r e c t i o n .

[ v ~

/

t~6A ) ? o.sF i g 2 a

Fig. 2. Com parison betw een stress resu ltan t distributio ns at spa,nwisepositions ~ = 0.5 and 0.35, for skew angle of 45.O v e r a l l s t a t i c a l c o n d i t i o n s a r e a l w a y s s a t i s f i e d , b u t , w i t h a u n i -

f o r m l o ad , c h e c k s o n th e n u m e r i ca l w o r k a r e a f f o r d e d b y t h e f a c tt h a t t h e f u n c t io n s F a n d S m u s t b e s y m m e t r ic a l , a n d G a n d Rm u s t b e a n t i - s y m m e t r i c a l , a b o u t t h e c e n t r e li n e o f t h e s l ab .

T h e r e su l ts f ro m t h e p r e s e n t t h e o r y a re c o m p a r e d w i t h t h e p r e v i-o u s s o l u t i o n s , f o r t w o t y p i c a l s p a n w i s e p o s i t i o n s , i n f i g s . 2 ( a ) a n d


12/13

SKEW BRIDGE SLABS 89

2 (b ), t h e v a l u e s b e i n g e x p r e s s e d a s ra t i o s o f t h e m i d - s p a n b e n d i n gm o m e n t g iv e n b y o r d i n a r y b e a m t h e o r y . A v a l u e o f P o i s so n s r a ti oo t 0 . 3 w a s u s e d i n t h e c a l c u l a t i o n s .

R e a s o n a b le a g r e e m e n t is o b t a i n e d b e t w e e n c h o r d w i se b e n d i n ga n d t w i s t in g m o m e n t s , a n d b e t w e e n s pa n w i se b e n d i n g m o m e n t s a tm i d - s p a n . G r e a t e r d i s c r e p a n c i e s o c c u r b e t w e e n s p a n w i s e b e n d i n gm o m e n t s a t t h e t y p i c a l o f f - c e n t r a l a s y m m e t r i c a l p o s i t i o n i n v e s t i -g a t e d , s i n c e t h e f e w t e r m s u s e d i n t h e s e r i e s a p p r o x i m a t i o n s a r ei n c a p a b l e o f r e p r o d u c i n g a c c u r a t e l y t h e l a rg e v a r i a t io n s o f m o m e n t

i

. / ,o U S

ACUTE EDGE ~/ EDG E ~ . ~4,9

s ~ i > /~o ~ . i i . ~ . .

LEGEND (5 -I P R E S E N T T HE OR Y

0 JEN SEN S RESULTSX R U S H T O N S R E S U L T S

----- DAY'S RESULTS.. . . EXPERIMENTAL RESULTS

s ) ? = 0 . 3 5F i g. 2 b

F ig . 2. C o m p a r i s o n b e t w e e n s t r e s s r es ul ta nt d is tr ib ut io ns a t s p a n w i s ep o si t io n s ~ = 0 . 5 a n d 0 . 3 5 f o r s k e w ~ n g l e o f 4 5 .


13/13

190 SKEW BRIDGE SLABS

a c ro ss th e c h o r d o f th e p la te . H o w e v e r , t h e m a x i m u m v a lu e s,w h i c h a r e o f g r e a t e s t i n t e r e s t f o r d e s i g n c a l c u l a t i o n s , a r e g i v e nf a i r l y a c c u r a t e l y .

T h e u s e o f f u l l - r a n g e F o u r i e r s e r i e s c o m p e l s t h e s t r e s s r e s u l t a n t st o h a v e t h e s a m e m a g n i t u d e a t t h e f r e e e d g es , a t e a c h s p a n w i s ep o s i ti o n . T h i s r e s t r i c t i o n i s m o s t s e r io u s in t h e c a s e o f t h e s p a n w i s em o m e n t s , w h i c h h a v e t h e g r e a t e s t c h o r d w i s e r a n g e o f v a lu e s . F o rt h e g i v e n n u m b e r o f t e r m s i n t h e a s s u m e d s e r i e s , g r e a t e r a c c u r a c ym u s t b e e x p e c t e d f o r s m a l l e r s k e w a n g l e s , w h e r e c h o r d w i s e s t r e s sv a r i a t i o n s a r e n o t s o g r e a t.

4 C o n c l u s i o n s A n a p p r o x i m a t e m e t h o d h a s b e e n p r e s e n t e d fo rt h e d i r e c t d e t e r m i n a t i o n o f t h e s t r e s s d i s t r i b u t i o n i n o r t h o t r o p i cs k e w b r id g e s la b s. A n y d e g r e e o f a p p r o x i m a t i o n m a y b e m a d e , d e -p e n d i n g o n t h e n u m b e r o f t e r m s u s e d in t h e a s s u m e d s e ri es fo r th es t re s s r e s u l t a n t s i n t h e p l a t e , t h e p r o b l e m f i n a l l y r e d u c i n g to a s e to f l i n e a r d i f f e r e n t i a l e q u a t i o n s , w i t h c o n s t a n t c o e f f ic i e n ts a s a r e s u l to f u s i n g a n o b l iq u e s y s t e m o f b o t h c o - o r d i n a t e s a n d s t r e s s re s u l t a n t s .A l t h o u g h l a b o r i o u s, t h e s o l u t io n o f th e r e s u l t i n g s e t o f e q u a t i o n s i sf a i r ly s t r a i g h t f o r w a r d w i t h t h e a i d o f m o d e r n c o m p u t i n g f a ci li ti es .

T h e f r e e - e d g e b o u n d a r y c o n d i t io n s , w h i c h a r e u s u a l l y d i f f ic u l t t od e a l w i t h a l o n g s k e w e d g e s , a r e s a t i s f i e d r e a d i l y b y t h e u s e o f a no b l iq u e s y s t e m o f s tr e s s - r e s u lt a n t s .

T h e m e t h o d p r e s e n t e d i s a n e x t e n s i o n o f a n a n a l y s i s d e v e l o p e do r i g i n a l l y f o r t h e d i r e c t d e t e r m i n a t i o n o f t h e s t r e s s d i s t r i b u t i o n i nr i g h t b r i d g e s la b sS . W i t h s k e w s l ab s , i t i s n o l o n g e r p o s s i b le t os i m p l if y t h e p r o b l e m b y s p l i tt in g a n y l o a d s y s t e m i n t o s y m m e t r i c a la n d a n t i s y m m e t r i c a l c o m p o n e n t s w i t h r e s p e c t t o t h e s p a nw i s ec e n t r e l i n e .Received 7th October, 1965.

REFERENCES1) JENSEN, V. P., Analyses of Skew Slabs, Engineering Experiment Stat ion, Universi ty

of Illinois, Bulletin 332, 1941.2) RUSHTON, K. R., Electrical Analogue Solutions for the Deformat ion of Skew Plates,Report, Department of Civil Engineering, Univers ity of Birmingham, July, 1963.3) DAY, A. S., The Engineer ~19 1965) 218.4) COULL,A., The Struc tura l Engineer 42 1964) 235.5) CO,ILL, A., Quart . J . Mech. Appl. Math. 1~ 1964) 437.

The Analysis Of Orthotropic Skew Bridge Slabs by A. Coull

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