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Analysis of continuous arches on elastic piers. Wong, Dick Chao 1959

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ANALYSIS OP CONTINUOUS ARCHES ON ELASTIC PIERS by DICK CHAO WONG B . S c , N a t i o n a l Yunnan U n i v e r s i t y , C h i n a , 1 9 ^ 7 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF M.A.Sc. i n t h e Department o f 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 t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA September, 1 9 5 9 i i ABSTRACT This t h e s i s presents the i n v e s t i g a t i o n of the behaviour of continuous arches on e l a s t i c p i e r s about which l i t t l e i s c u r r e n t l y known. A s e r i e s of s t u d i e s were made to i n d i c a t e the e f f e c t s of p i e r dimensions on extreme f i b e r s t r e s s e s at a number of c r i t i c a l s e c t i o n s of arches. Such r e s u l t s are of p a r t i c u l a r i n t e r e s t t o the bridge designer. S i x numerical examples of symmetrical arch systems have been solved, using an i n t e r e s t i n g v a r i a t i o n of the for c e d i s t r i b u t i o n method of the l a t e P r o f . Hardy C r o s s . ^ The r e l a t i v e p r o p o r t i o n s of the system were based p r i m a r i l y upon a e s t h e t i c c o n s i d e r a t i o n s . I n the s i x s t r u c t u r e s which were i n v e s t i g a t e d two systems of arches and p i e r s , c a l l e d I and I I , were considered. Each system has f i v e spans. The v a r i a b l e span lengths are the same i n each system. The arches i n each (2) were s e l e c t e d from Whitney's paper and are -linear arches f o r dead load only. In System I the arch r i b s are l i g h t e r and f l a t t e r than i n System I I , and the p i e r s are more f l e x -i b l e . In each system the v a r i a t i o n of span lengths and arch r i s e s are such so th a t there i s no unbalanced dead load h o r i z o n t a l t h r u s t on the p i e r s . In both systems a l l p i e r s are of s i n g l e equal b a t t e r . Three d i f f e r e n t h e ights of p i e r 40', 60', and 80' were i n v e s t i g a t e d i n each system. F i g s . 1, Number i n s u p e r s c r i p t r e f e r s t o B i b l i o g r a p h y . i i i 2 and 3 c l a r i f y t h e f o r e g o i n g w h i l e T a b l e s 1 and 2 g i v e t h e p r o p e r t i e s o f t h e elements making up each system. I n System I twenty i n f l u e n c e l i n e s f o r u p p e r , l o w e r , r i g h t , and l e f t k e r n moments a t s p r i n g i n g s , crowns, and p i e r t o p s were c o n s t r u c t e d . I n System I I " p o r t i o n s " o f s i x t e e n i n f l u e n c e l i n e s f o r upper and lower k e r n moment a t s p r i n g i n g s and crowns s u f f i c i e n t t o e s t a b l i s h t h e t r e n d o f a l t e r a t i o n o f p r o p o r t i o n s were con-s t r u c t e d . The l a r g e number o f v a r i a b l e s i n v o l v e d i n t h e d e s i g n o f such i n d e t e r m i n a t e s t r u c t u r e s as c o n t i n u o u s a r c h systems makes i t i n a d v i s a b l e t o draw t o o d e f i n i t e c o n c l u s i o n s , but some r e s u l t s o f s t u d i e s o b v i o u s l y i n d i c a t e that.: ( l ) A l l con-t r o l l i n g L.L. f i b e r s t r e s s e s a r e g r e a t e r t h a n t h o s e i n f i x e d ended a r c h e s and i n c r e a s e as t h e h e i g h t o f p i e r s i n c r e a s e s , but t h e maximum D.L. + L.L. f i b e r s t r e s s e s do not e x h i b i t t h i s c h a r a c t e r i s t i c as might be e x p e c t e d but depend, o f c o u r s e , upon t h e r a t i o o f dead l o a d t o l i v e l o a d as w e l l as upon t h e p r o p o r t i o n s o f t h e s t r u c t u r e . (2) I t would appear t h a t t h e a n a l y s i s may be c o n f i n e d t o t h r e e spans f o r a r c h e s and two spans f o r p i e r s , save i n t h e case o f a l o n g c e n t r e span com-b i n e d w i t h v e r y f l e x i b l e p i e r s . I n such a case t h e complete s t r u c t u r e must be i n v o l v e d i n t h e a n a l y s i s . (3) The e f f e c t o f r o t a t i o n o f t h e p i e r t o p s on L.L. s t r e s s e s a t crowns i s s m a l l and almost independent o f p i e r h e i g h t , whereas t h e e f f e c t on L.L. s t r e s s e s a t s p r i n g i n g s i s somewhat g r e a t e r I v and i n c r e a s e s s l o w l y as t h e h e i g h t o f p i e r s i n c r e a s e s . The e f f e c t o f t r a n s l a t i o n o f t h e p i e r t o p s on L.L. s t r e s s e s a t s p r i n g i n g s and crowns i s u s u a l l y t h e g r e a t e s t and i n c r e a s e s r a p i d l y as t h e h e i g h t o f p i e r s i n c r e a s e s . The r e s u l t s p r e s e n t e d h e r e i n a r e , o f c o u r s e , t r u e o n l y f o r t h i s p a r t i c u l a r t y p e o f system, but t h e a u t h o r f e e l s t h a t t h e system chosen i s more r e p r e s e n t a t i v e o f a s t r u c t u r e w h i c h might be c o n s t r u c t e d t h a n a re t h e t y p i c a l t h r e e o r f o u r e q u a l span systems o f t h e t e x t book v a r i e t y w i t h w h i c h some w r i t e r s , more concerned w i t h s i m p l i c i t y t h a n r e a l i t y , have d e a l t . In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g of t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by t h e Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department of C/'h// L SIM/A/BG/?JA/& The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada. Date f^r. 5n 7 _- / P i " 9  V Contents Page Chapter I I n t r o d u c t i o n 1 Chapter I I Numerical Examples 5 Chapter I I I Conclusions 6 APPENDIX I Dead Load h o r i z o n t a l t h r u s t s on Arches f o r numerical Examples 65 I I Formulae f o r A n a l y s i s 71 BIBLIOGRAPHY 87 ACKNOWLEDGMENT The a u t h o r w i s h e s t o acknowledge h i s i n d e b t e d n e s s t o h i s s u p e r v i s o r , P r o f e s s o r A.H. F i n l a y , who spent many hours and t o o k much p a t i e n c e i n d i s c u s s i o n s on t h i s t h e s i s and gave v a l u a b l e s u g g e s t i o n s and u n f o r g e t t a b l e encourage-ment . September, 1959 Vancouver, B. C , Canada CHAPTER I I n t r o d u c t i o n More t h a n two thousand y e a r s ago a r c h e s were w i d e l y used f o r t h e f i r s t t i me by t h e Romans. I n t h e form o f masonry, c o n c r e t e , s t e e l and t i m b e r a r c h e s , t h e y have been i n wide use e v e r s i n c e . The t e r m i n a l f o r c e s e s s e n t i a l f o r a r c h a c t i o n t e n d t o c o u n t e r a c t s i m p l e beam moments and y i e l d economy w h i l e t h e u s u a l l y g r a c e f u l c u r v a t u r e o f t h e a r c h a x i s y i e l d s a p l e a s i n g appearance. These form two e s s e n t i a l advantages o f a r c h e s . I n t h e l a s t t h i r t y y e a r s , a s e r i e s o f a r c h e s o f r e i n f o r c e d c o n c r e t e r e s t i n g on s l e n d e r p i e r s have r e c e i v e d a g r e a t d e a l o f a t t e n t i o n and have o c c u -p i e d an i n c r e a s i n g l y i m p o r t a n t p o s i t i o n i n b r i d g e s i n t h i s c o n t i n e n t . The c o n t i n u o u s a r c h e s on e l a s t i c p i e r s have some o f t h e same advantages as does t h e s i n g l e a r c h . But t h i s t y p e o f s t r u c t u r e l i k e a l l a r c h e s i s s u i t a b l e o n l y i f th e r a t i o o f l i v e l o a d t o dead l o a d i s r a t h e r s m a l l , o t h e r -w i s e t h e l a r g e l i v e l o a d p r o d u c e s l a r g e bending moments i n a r c h e s and i n p i e r s r e s u l t i n g i n a l o s s o f economy. T h i s t h e s i s r e s t r i c t s i t s e l f t o t h e a n a l y s i s o f two c a r e f u l l y s e l e c t e d systems ( I and I I ) o f a r c h e s on e l a s t i c p i e r s . As has been s t a t e d i n t h e A b s t r a c t t h e s e systems were " d e s i g n e d " f o r dead l o a d , u s i n g r e s u l t s o f 1 2 (2) C.S. Whitney v ' a f t e r r e l a t i v e span lengths r i s e s and p i e r heights had been s e l e c t e d on the b a s i s of a e s t h e t i c s combined w i t h keeping the s t r u c t u r e f r e e of bending moments (save those due to r i b - s h o r t e n i n g ) under dead load. They present i n every way s t r u c t u r e s which could be b u i l t and i t i s hoped that the conclusions reached may a s s i s t designers. This t h e s i s w i l l not d e a l , save i n a general way, with the a n a l y t i c a l procedure, which has been proposed by , ( 1 , 3 , ^ , 5 , 6 , 7 , 8 , 9 , 1 0 ) „ _ . many people. For ana l y s i n g the problem a v a r i a t i o n of the "Moment D i s t r i b u t i o n Method" w i l l be used i n t h i s t h e s i s . The mentioned method, which might be termed "Force D i s t r i b u t i o n " , was developed by the l a t e P r o f . Hardy Cross. He d i s t r i b u t e d f o r c e f u n c t i o n s s u c c e s s i v e l y by pure t r a n s l a t i o n and pure r o t a t i o n of the n e u t r a l p o i n t of the j o i n t w i t h the n e u t r a l p o i n t s of the adjacent j o i n t s kept f i x e d t e m p o r a r i l y . The n e u t r a l p o i n t i s so chosen th a t i t t r a n s l a t e s only under the a c t i o n of a s u i t a b l y d i r e c t e d f o r c e a c t i n g upon i t and r o t a t e s only under the a c t i o n of a moment a c t i n g upon i t . The p i e r top i s "replaced" by the n e u t r a l p o i n t of the j o i n t i n order to di m i n i s h the d i s -advantage of slow convergence. The method has been popul-a r i z e d and improved i n a proper form of c a l c u l a t i o n s by P r o f . A.H. P i n l a y . ^ 1 1 ^ Besides t h a t , P r o f . F i n l a y has extended the c e n t r a l i d e a of the method to construct i n f l u e n c e l i n e s f o r arches on e l a s t i c p i e r s by applying a u n i t load on each span i n only one p o s i t i o n . Having found, f o r t h i s one p o s i t i o n , the s t r e s s f u n c t i o n d e s i r e d the 3 r e m a i n i n g o r d i n a t e s t o i t s i n f l u e n c e l i n e f o l l o w a t once from t h e f a c t t h a t t h e shape o f t h e d e s i r e d i n f l u e n c e l i n e i s s i m i l a r t o t h e more f a m i l i a r i n f l u e n c e l i n e i f t h e a r c h had been f i x e d - e n d e d . T h i s w i l l be c l e a r upon r e f e r r i n g t o P i g . 5« T h i s e x t e n s i o n has made t h e o r i g i n a l method more b r i l l i a n t and p o w e r f u l , which becomes t h e most i n g e n i o u s and p e r f e c t method i n c o n t i n u o u s a r c h a n a l y s i s . U s u a l l y t h e d e s i g n i s governed by t h e combined e f f e c t o f normal t h r u s t and a x i a l b ending moment, and s i n c e t h e f o r m e r i s a maximum under f u l l span l o a d i n g s , w h i l e t h e l a t t e r i s n o t , I t i s e v i d e n t t h a t t h e independent i n f l u e n c e l i n e s do n o t d i r e c t l y g i v e t h e l o a d i n g p r o d u c i n g t h e maximum combined s t r e s s a t any s e c t i o n . F o r d e s i g n i n g p u r p o s e s i t i s n e c e s s a r y t o c o n s t r u c t i n f l u e n c e l i n e s f o r maximum t o t a l f i b e r s t r e s s r a t h e r t h a n f o r maximum a x i a l moment and t h r u s t . At any s e c t i o n t h e extreme f i b e r s t r e s s may be o b t a i n e d by d i v i d i n g t h e k e r n moment by t h e a p p r o p r i a t e s e c t i o n modulus (see Appendix I I ) . Such use o f t h e concept o f k e r n moment y i e l d s t h e c o m b i n a t i o n o f normal t h r u s t and a x i a l b e n d i n g moment e f f e c t s and g i v e s d i r e c t l y t h e l o a d i n g c o n d i t i o n s f o r maximum t o t a l f i b e r s t r e s s , t h e r e f o r e t h e concept o f k e r n moment w i l l be used i n t h i s t h e s i s . I n c o n t i n u o u s a r c h system, i t s d i m e n s i o n s a r e based on t h e t o p o g r a p h i c c o n d i t i o n s and t h e t r a n s p o r t a t i o n r e q u i r e -ments. The a u t h o r has not c o l l e c t e d any d a t a o f such s t r u c t u r e s . But d e s i g n i s an e n g i n e e r i n g p r o b l e m and a l s o an a r c h i t e c t u r a l 4 problem t o o . Economy r e q u i r e s t h a t dead l o a d t h r u s t s s h a l l be b a l a n c e d a t t h e p i e r t o p s and a e s t h e t i c c o n s i d e r a t i o n s u s u a l l y I n d i c a t e s an u n e q u a l number o f a r c h e s w i t h span l e n g t h s d e c r e a s i n g towards t h e abutments. Prom t h e s e two b a s i c i d e a s t h e a u t h o r has r o u g h l y d e s i g n e d s i x d i f f e r e n t f i v e span c o n t i n u o u s s y m m e t r i c a l a r c h systems. They a r e made up o f r e a s o n a b l e d i m e n s i o n s and p l e a s i n g appearance and were used t o I n v e s t i g a t e t h e e f f e c t s o f p i e r d i m e n s i o n s upon extreme dead p l u s l i v e l o a d f i b e r s t r e s s e s a t a number o f c r i t i c a l s e c t i o n s o f a r c h e s . CHAPTER I I Numerical Examples In order t o make the i n v e s t i g a t i o n as sta t e d i n Chapter I , s i x examples w i l l be solved and the r e s u l t i n g i n f l u e n c e l i n e s compared w i t h the i n f l u e n c e l i n e s f o r f i x e d ended arches and the components of i n f l u e n c e l i n e s caused by p i e r top t r a n s l a t i o n and r o t a t i o n separately w i l l be shown (see A b s t r a c t ) . The c h a r a c t e r i s t i c s of the examples have been stat e d i n the A b s t r a c t . The bases of the p i e r s are f i x e d r i g i d l y , as are the outer ends of the r i b s at the abutments. The e f f e c t s of r i b - s h o r t e n i n g and p i e r - s h o r t e n i n g are not considered, as they are n e g l i g i b l e i n a l l cases. The p r o p e r t i e s of arch r i b s and p i e r s are l i s t e d i n Tables 1 and 2. The constants f o r a n a l y s i s are shown i n Table 3. The t h r u s t s produced on arches by applying a u n i t f o r c e ( h o r i z o n t a l d i r e c t e d f o r c e or couple) at the n e u t r a l p o i n t of J o i n t , B or C, are shown i n Tables 4 t o 7. The t h r u s t s on arches due t o a u n i t f o r c e at the n e u t r a l p o i n t of J o i n t , D or E, may be obtained by symmetry. The i n f l u e n c e l i n e s f o r kern moments at s p r i n g i n g s , crowns and p i e r tops of System I are shown i n F i g s 5 t o 24, i n c l u s i v e . 5 6 To study the r e l a t i v e importance of r o t a t i o n and t r a n s l a t i o n of the p i e r tops, the two e f f e c t s are p l o t t e d separately as shown i n P i g s . 5 t o 20, i n c l u s i v e , f o r System I, and t h e i r r e s u l t s on f i b e r s t r e s s e s i n arches are shown i n P i g s . 43 and 44. Maximum t o t a l l i v e load lower kern moments at the sprin g i n g s and crowns are shown i n Tables 8 and 9 r e s -p e c t i v e l y . Maximum t o t a l l i v e load r i g h t kern moments at p i e r tops are shown i n Table 10. The r a t i o of dead load t o l i v e load i s an important f a c t o r i n the design, u s u a l l y , the dead load s t r e s s e s govern l a r g e l y . To f a c i l i t a t e the a n a l y s i s the r e l a t i v e l y small l i v e load was assumed un i f o r m l y d i s t r i b u t e d . A thousand pounds per l i n e a r foot per r i b of l i v e load i s assumed i n these examples. The maximum t o t a l kern moments at the s p r i n g -ings and crowns due to the combination of dead load and l i v e load are shown i n Tables 11 and 12, and the changes i n f i b e r s t r e s s e s i n arches from f i x e d ends are shown i n P i g s . 4 l and 42. The c o n t r o l l i n g compressive upper extreme f i b e r s t r e s s e s at the sp r i n g i n g s and crowns, and the compressive l e f t extreme f i b e r s t r e s s e s at the p i e r tops due t o the combination of dead load and l i v e load are shown i n Tables 13 and 14, r e s p e c t i v e l y . The s i g n convention i s the same as us u a l i n arch a n a l y s i s . Moments are considered as p o s i t i v e i f they produce 7 t e n s i o n on t h e i n s i d e f i b e r o f an a r c h o r on t h e r i g h t f i b e r o f p i e r , and normal f o r c e s as p o s i t i v e i f t h e y produce com-p r e s s i o n i n t h e member. Computations f o r t h e most p a r t were made w i t h t h e " P r i d e n " desk c a l c u l a t o r , and t h e a r e a s under I n f l u e n c e l i n e diagrams were c a r e f u l l y measured by means o f a p l a n i m e t e r . 8 9 Width 4 * - 0 " Depth v a r i e s iC • L . Diaphragm , 4 ' - 0 ' .Hangers l8'»xl8" @ 20' - 0 " c/fc 21' -0" S t r i n g e r s 9"xl4" n 3 @ 7 , -0"=21'-0" 3 '-6,'1. 23'-11" F l o o r beams I8"x36" @ 2 0 ' - 0 " c/c Diaphragm P i e r Width 5 ' - 0 " _ Depth v a r i e s -Columns $ I8"xl8*' -Rib Diaphragm Hal f S e c t i o n At Crown of Middle Span H a l f Section At Spring FIG. 3 CROSS SECTIONS OF SYSTEMS I & I I Table 1 P r o p e r t i e s of Arch Ribs System I I I Span AB & EF BC & DE CD AB & EF BC & DE CD L i n f t . 180 200 240 180 200 240 r i n f t . 28.6 36.175 60 35.8 45.55 80 S 1.455 1.543 1.756 1.5^3 1.756 2.24 m 0.325 0.300 0.275 0.375 0.275 0.200 y c i n f t - 6.89 8.47 13.45 8.87 10.21 16.31 W i n f t . 4 4 4 4 4 4 d c i n f t . 3 3.25 4 3.25 3.5 4.25 d s i n f t . 4.65 5.24 7.03 4.95 6.05 8.93 34°2' 37°17 ' 47052' 40°44' 45°20' 57°20' 11 Table 2 P r o p e r t i e s of P i e r s H p w dm A 2 System i n f t . i n f t . i n f t . i n f t . 40 5 6.5 8 1.2 I 60 5 6.5 8 1.3 80 5 6.5 8. 1.4 40 5 7.5 9 1.2 II 60 5 < 7.5 9 1.3 80 5 7.5 9 1.4 G H I ] THRUST LINE FOR TRANSLATION. Fig. 4 THRUST LINE FOR ROTATION s « >« Hp dB & & dAB & dEF d" AB Gn 1 d BA &, dEF dBC & dED dBC & d'ED d"BC & d ED dCB dDE d" CB G d"DE dcD dCD G t dDC d" CD G d DC hAB & hFE hCB & hDE mAB & mFE G mDE He hED hDC & hCD mBC & mED m DC G m C D in ft. in ft. in ft. in ft. in ft. in ft. in ft. in ft. in ft. in ft. in ft. in ft. in ft. in ft. % % % % % % % I 40 14. 91 16.61 21.72 17.09 29. 35 27.71 38. 15 21.58 37. 7S 21. 35 46. 55 65. 50 34. 85 4. 40 3. 28 9.93 19. 25 1.85 1.11 12.90 15.49 60 16. 11 21.04 21.72 17.26 29. 13 27.71 37.87 22. 16 36. 84 21. 53 46.55 64. 30 35.65 11. 34 8. 46 10.44 22. 27 5. 19 3. 10 15.68 19. 26 80 13. 66 22.45 21.72 16. 96 29. 63 27.71 38.40 22. 25 36. 50 21. 16 46. 55 63.90 37. 20 19.56 14. 60 9.72 21. 81 9.99 5.97 15.29 20. 82 40 16. 19 17.09 26.93 21.00 37.20 35. 34 48.75 27. 30 48.55 27. 14 63. 69 88. 30 47.70 2. 23 1.86 8. 35 17. 31 1.01 0.64 11.26 14.95 II 60 19. 69 22.50 26.93 21. 33 36.40 35. 34 47.90 28.00 47. 35 27.67 63.69 86. 86 48.70 6. 18 5.09 9. 49 21. 69 3. 15 1.83 14. 72 18.95 80 19.83 25.50 26.93 21. 33 36. 33 35. 34 47.90 28. 38 47.25 27.71 63. 69 86.00 49. 15 11.52 9.50 9.53 23. 11 6. 27 3. 63 15.95 20.95 Table 3 Constants far Analysis Only the horizontal components of the thrusts due to the distributed moments are shown in the figure, the vertical components may be obtained by proportion Table 4 Thrusts on Arches Produced by Applying a Unit Force to the Right at the Neutral Point of Joint,B.- (Minus sign means arch In tension) i Hp Arch AB Arch BC Arch CD Arch DE Arch EF ca CO PQ < X = < x < PQ X -CQ X o o = PQ X PQ o X PQ -o x PQ s O x Q o X « Q - O X X o X X w « w - Q w c Q X w X Q - w X Q = W X W X -w X £ X w - fe X o r>-<M -=* .=* o i CO ro rH o o 1 t— CVI •=t O 1 CO ro rH O o in ro CVI ro O VO rH rH o o ON -3" m o o i in oo CVI oo o ON -=»• m o o i vo rH rH O O OO oo o o o 00 vo o o ro co o o o i ro CO O O O oo CO o o o 1 CO VO -=* o o VO o o o o ON o o o o CVI o o o 1 vo o o o o o CVI o o o 1 o-ON o o o m o o o o ro CVI o o o in o o o o oo CVI o o o M o VO LPl rH VO rH rH 1 0-l>-m o o i in rH VO rH rH 1 t--t>-m o o i vo vo rH CO o o o in o o t-CVJ ON rH o 1 vo vo rH CO o t-CVI ON rH o 1 o o in o o in oo CVI o o CO ON m rH o rH O -=* o o 1 in ro CVJ o o rH o o o 1 CO ON in rH o t-m o o o ro co o o CO rH rH O o 1 t>-m o o o 00 rH rH o o 1 oo co -=* o o t>-vo o o o .00139 t-vo o o o ON oo rH o o o CO m m VO o CVI 1 t>-o CVI rH O 1 m m VO o CVI i o CVI rH o 1 -=t vo oo rH OO in o rH o CVI co CO o 1 •=* VO ro rH CVI 00 oo o 0O m o rH o vo t-o o m ro rH ro o CO ON ON O o 1 VO t-o o CO ON ON o o 1 in CO rH ro O t-CVi CVI o o co rH CVI rH O VO rH oo O O 1 t>-CVI CVJ o o vo rH OO o o 1 CO rH CVI rH o rH OO OO o o CVI VO oo o o r-t oo oo o o CVI VO ro o o o VO on CM CVI o 1 o in o o o « VO ro CVI CVJ o i o in o o o i in co rH o ON -=r o o o in vo CVI o o i in -=*-00 rH o m vo CVI o o i ON o o o rH rH O O o vo ro CVI o o CO oo o o o 1 rH rH o o o co ro o o o 1 VO ro cvi O O rH O o o o ro o o o l>-o o o o 1 rH o o o o t-o o o o 1 OO o o o rH O O o o t— o o o o rH o o o o o-o o o o M M b CO in CVI VO o t tr-m CVI o o i CO in CVI vo o m CVI o o -=t t-C\ -=? o CA CVI o o 00 o o rH o 1 .=*• c-o> .=* o CO o o rH O 1 ON -ST CVI o o •3-co o o o ON VO CO o o t-ON rH O o 1 -=t co o o o 1-ON rH o o 1 ON vo co o o ON rH O O o .00229 in m o o o ON rH O o o m in o o o 1 ON CVI CVI o o vo rH O o o VO in o o o VO rH o o o vo in o o o b CO OS in co rH rH 1 o VO VO o o 1 ON m CO rH rH 1 o vo vo o o i o\ co o OS O CVI vo o o VO in CVI CVI o 1 ON CO o ON o VO in CVI CVI o 1 CVI VO o o o o oo o o in rH ON rH O m •=* in o o » o o ro o o in in o o 1 m rH ON rH o CVI ON o o o VO .=* VO o o rH o o 1 CVI ON o o o t-rH O o 1 vo •=*• vo o o oo ON o o o CVI ro •3-o o oo ON o o o cu oo •=r o o Table 5 Thrusts on Arches Produced by Applying a Unit Force to the ri g h t at the Neutral Point of J o i n t , C. (Minus sign means arch i n tension) H p Arch AB Arch BC Arch CD Arch DE Arch EF Syst PQ < X PQ = < X % X < - P Q X o PQ X o -PQ x o pq o X PQ -o x PQ = O x ft o w Q - o X Q = o ac 8 X - 8 X X w X - B X w = « X ft w X « - w X ft * w X to w X - 6 X w X w X o -=j-t-00 o o o 1 vo 00 o o 1 t-oo o o o 1 VO 00 •=f o o 1 ON ON rH O 1 •=r CM -=* O o CM rH rH O O 1 ON ON rH O 0.00112 CM ^ • o o •=* o rH rH O vo ON o o o CO CO CM O O 1 o rH rH O CO CO CM o o 1 vo ON o o o r-i O o o CO CM o o VO in o o o i • -=fr rH o o o VO m o o o i CO CM O O CO rH O o o vo o o o CO r-i o o o •=t vo o o o M o vo o t*-vo o o 1 m m r-i O 1 o r-vo o o 1 in m rH o 1 t-CO CO -=r o i vo co CO rH o cr* vo m o o i t~ CO 00 •=* o 1 cr\ vo in o o i VO CO CO rH O VO O CO o CM t-•=r o o VO 00 o o 1 t--VO O CO o VO r~ oo o o 1 CM o o in ON O O O £— in o rH o c-in CM o o i in ON o o o t-in CM o o i m o rH o CO rH O O t-ON CM o o CO -=»-rH o o t-ON CM o o o 00 vo 00 CM o 1 CO CO 00 CVI o 1 vo t-co CVI o 1 CO CO 00 CVI o o> oo 00 CO o 1 CVI CVI o vo 00 -=f r-i o 1 o\ 00 co CO o i VO CO -=r rH O 1 CM •=r CM O VO in CO in o CO rH CM rH o CO vo ON rH O 1 VO in oo in o CO VO ON rH o CO rH CM rH O O ON CO o o in ON CO CM O ON o VO o o 1 o ON CO o o ON O VO O o 1 m ON CO CM O 00 CO vo o o 00 ON VO o o CO CO VO o o 00 ON vo o o o -=r CVI o o o 1 vo rH CVI O o 1 CVI o o o 1 vo rH CM o o 1 CO ON ON o o i CM rH CM O O ON -=r o o o i CO ON o o o 1 ON o o o 1 CM rH CM O O 00 CO vo o o -3 --=r o o o -.00128 00 CO vo o o CO CM rH O O 1 •=t •=r o o o .=1-o o o o rH O O VO CM O o o 1 •=1-o o o o VO CM O o o 1 rH O o CO o o o o VO CM o o o co o o o o VO CM o o o r-i M o vo vo r-i CM o o 1 o -5* oo o o 1 vo rH CVI o o o 00 o o 1 rH CVI O CO o 1 ON r-i 00 O o CM O o 1 rH CM O CO O 1 CM O O 1 ON rH CO o o t -rH CO rH O ON CO CM o o ON •=r o o t -rH 00 rH O -=t ON O O 1 ON CO CM o o ON CO o o o CO ^~ m o o in CO rH O O 1 ON CO o o o in CO rH o o i CO m o o rH l-i o o o ON CO rH o o rH r-i o o o ON CO rH o o o 00 ON CO 00 o o 1 CVI r-i r-i O ON CO CO o o CVI rH rH O 1 rH 00 in o i •=* CM t-rH O CM 00 t— o o 1 -=}" rH CO in o t CM CO r-o o i CM t-rH O C -CO VO CO o in vo VO o o in vo rH rH O 1 t-co vo CO o m vo rH rH o 1 m VO VO o o CM in rH o o .01370 CM VO co o o 1 CM in rH o o CM VO CO O O 1 o r~ co rH O CO ON rH o o CM t-co o o CO ON rH o o CM t-co o o Table 6 Thrusts on Arches Produced by Applying a Unit Couple In the clockwise d i r e c t i o n at the Neutral Point of Joint,B. (Minus sign means arch i n tension ) System Arch AB Arch BC Arch CD Arch DE Arch EF System Hp < X pq < pq -SS X a X o - PQ X o = pq X PQ 0 X pq - 0 X PQ = 0 X 0 X - 0 X p s O X 8 X - B X = B X w Q X w -p X X P w X -a X P s W X w X fe - W X E X w c f e X o -=r -4-o o o o 1 ON o\ m o o i -.00004 -.00599 -.00007 r o CVI in o o -.00073 -.00007 -.00073 00 CVI m 0 0 .00005 .00062 -.00011 .00005 -.00011 .00062 T0000" .00013 -.00003 T0000* -.00003 .00013 T0000* .00003 .00001 .00003 M o vo -3-r-i o o o t t -ro vo o o i -.00014 c— ro vo O O t -.00019 rH in in o o -.00104 ON rH O O O 1 0 rH O O 1 r-i in in 0 0 .00015 .00087 -.00022 .00015 -.00022 .00087 .00003 .00027 -.00007 ..00003 t— 0 0 0 0 1 .00027 .00003 CO 0 0 0 0 .00003 .00008 b CO ro 00 o o o I o r-i VO O o -.00033 -.00610 in oo o o o i CVI ro in o o -.00113 -.00035 -.00113 CVJ 00 in 0 0 .00027 .00093 0 r o 0 0 0 1 .00027 0 r o O O 0 1 .00093 .00007 vo CO 0 0 0 I-. 00009 .00007 -.00009 vo 00 O O O .00010 1 .00010 .00010 .00010 b -.00001 VO o o o 1 -.00001 VO O -3-O O 1 -.00003 .00399 t -o o o 1 00 O O O O 1 t>--=T O O O 1 .00399 .00003 CVI -3-0 0 0 -.00006 .00003 -.00006 CVI •=* 0 0 0 0 .00007 -.00001 0 -.00001 .00007 0 .00001 0 T0000* M M o VO -.00005 -=t vo -=J-o o 1 in o o o o • j-. 00464 -.00010 rH in -=* o o -=J-t>-o o o 1 -.00010 -3-r-O O 0 1 rH m 0 0 loooo* .00064 -.00014 .00007 rH O O O 1 .00064 .00001 .00017 -.00004 T0000* 0 0 0 0 .00017 .00001 .00004 .00001 .00004 o OO -.00014 ON c-^J-o o • r-i O o o 1 •=r o o i -.00019 r-i VO -3" O O -.00089 -.00019 ON 00 0 0 0 .00461 .00014 VO c— 0 0 0 -.00021 .00014 -.00021 VO 0 0 0 .00004 .00025 -.00007 .00004 -.00007 .00025 .00004 i.0000* .00004 i.0000* Table 7 Thrusts on Arches Produced by Applying a Unit Couple i n the clockwise d i r e c t i o n at the Neutral Point of Jo i n t , C. (Minus sign means arch i n tension) 6 Arch AB Areh BC Arch CD Arch DE 1 \rch EP <t> +> co Ep PQ pq <£ < o o o PQ PQ pq o o o p w w w Q w w >» < =< PQ -PQ PQ -pq =PQ o - p = o o - p - o o - Q = Q - Q = o w - w = w w -w -CO X X x X x x X X X X X X X X X X X X X X X X X X CM o CM o O o IfN o in o CM ON rH CM rH ON o CO -=r CO o CO '.=* CO o rH co rH CO rH r- rH t- o rH m o m rH o VO rH o rH vo o rH o rH O o O o O O OO o oo o o oo o o o oo o o o o o o o o o o O o O o o O O o o o o o o o o o o o o o o o o o o o o o o o o O O o o o o o o o o o o o o o o o o o o o 1 1 1 1 1 i 1 1 1 1 1 1 _ CO CO CO CO CO oo m CO in oo VO 00 ON vo ON 00 CM m CM in CO CO CO 00 o lT\ o in o CM ON in CM in ON o t- t - o t- t - rH ON CM rH CM ON rH CM rH CM H O r-l o rH O o •=r O o o co o o o oo o o o o o o o o o o o o o o O o o O O o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o 1 1 1 1 i I 1 1 1 1 1 1 vo CVI vo CVI .=* ON VO -=1- vo ON o CM rH o rH CM CM rH ON CM ON rH o CO o CO o ON o ON o CO vo vo co rH 00 ON rH ON 00 CM rH CM CM CM rH CO oo CO oo CO o r-l o rH o o -=»• o -=r o o oo o o o oo o H o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o 1 1 1 1 I 1 1 1 1 1 1 1 VO vo VO VO -=r VO CO -=t co VO rH oo vo rH vo co rH rH t- rH r- rH rH t- rH t-o o -3- o -3- o VO O vo o oo ro o co CO o o o o -=}- o o o o o o o o o o CM o CM o o CM o o o CM o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o 1 1 1 1 1 1 1 1 1 1 1 1 ^. rH CVI iH CVI CM o t- CM o CM rH ON CM ON rH in 00 vo m vo oo m t- in t-M o cvi t- CVI iH t- oo rH oo r— o ON in o in ON o VO rH o rH vo o rH o rH M vo o o o o O o oo O oo o o CM o o o CM o o o o o o o o o o o o o o o o o O o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o 1 1 1 1 1 1 1 i 1 1 1 1 m in in CM m in in CM vo 00 CO vo CO 00 CM vo oo CM CO vo CM CM -=* o .=1- co •=* CO CM CO t— CM t— 00 o rH t- o c- rH rH 00 CM rH CM oo rH CM rH CM CO o o o o O o oo o oo o o oo o o o CO o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o 1 » 1 1 1 i 1 1 1 1 1 1 ON I T T a b l e 8 Max. L . L . Kerns Moments a t S p r i n g i n g s Max M K L a t Hp In f t V a l u e o f when on L i v e Load Span A B i n f t B C i n f t C D i n f t D E i n f t E P i n f t C o n s i d e r i n g 5-Spana T o t a l % L P i n f t C o r -r e s . F . E . V a l -u e s T o t a l <0b1 C o n s i d e r i n g 3-Spans o n l y P i n f t max <fi E r r o r A o f A r c h A B 80 + 3 1 7 - 0 0 -1015.00 + 666.00 - 25.50 + 222.00 - 2 1 . 2 0 + 42.30 - 10 .60 + 2 1 . 2 0 +1268.50 -1073.30 158.00 178.00 +1205.00 - 1 0 6 2 . 7 0 - 5.00 - 0.99 60 + jf54.50 9.00 + 549*50 + 116.20 + 2 1 . 2 0 0 - 31.80 - 2 1 . 2 0 - 7.40 +1141.40 - 9 4 9 . 4 0 1 4 2 . 2 0 +1120.20 157.20 - 9 4 2 . 0 0 - I.85 - 0.78 40 + 602.50 ^ 7 o 2 . 0 0 + 3 ^ 9 . 4 0 30 6 3 . 4 0 « 0 - 2 1 . 2 0 0 +1025.30 0 - 825.50 127.80 136.70 +1025.30 - 825.50 F i x e d Ends + 8 0 3 . 0 0 - 6 0 3 . 0 0 B o f A r c h A B 80 74.00 - 952.00 + 7 4 0 . 0 0 - 63.50 + 208.00 60 - 7 9 3 . 0 0 + 6 6 6 . 0 0 - 7 4 . 0 0 - "34.90 + 179.80 + 6 3 . 4 0 + 2 1 . 2 0 +1106.60 - 1 0 5 0 . 4 0 137^ 174T00 80 +1022.00 - 1 0 5 0 . 4 0 #0 4 2 . 3 0 0.20 - 4 . 2 0 +1046.60 - 911.40 130.10 151 .00 +1004.30 - 4 . 0 4 - 907.20 40 +-3|7,50 -666.00 + 5 i i L o o - 8 4 . 6 o + 105.80 - T 2 . 3 0 + 10.60 ~eT.20 + 961.90 -799.10 119.80 132.50 + 951.30 - 792.90 - 0.46 1.10 0.78 F i x e d Ends + 8 0 3 . 0 0 - 603.00 B o f A rch B C 80 + 856.00 - 2 1 . 2 0 + 158.70 - 1 1 7 4 . 0 0 + 317.40 31.70 84.60 + 31.80 - 2.00 +1448.50 -1228". 90 150.40 +1332.10 173.20 -1226.90 - 8 . 0 4 - 0.16 60 ±__I7Jb_59_ + 232.50 c o f A rch B C - 31.70 - 1 0 1 5 . 0 0 + 2 6 4 . 4 0 + 63,50 <0 - 39.10;- 5-00 +1131.90 -1090.80 138.20 153780 +1268.40 -1085.80 ^j4,76_ - 0 . 4 6 40 + 5 7 1 . 0 0 - 4 2 . 3 0 + 4 0 2 . 0 0 - 867.50 + 190.40 ;+ 2 1 . 2 0 42.30 - 9 - 0 0 ;o F i x e d Ends + 962.50 - 7 0 9 . 0 0 80 + 5 0 8 . 0 0 3 - 2 0 + 169.20 - 1 1 8 5 . 0 0 + 603.00 *0 +1184.60 - "961.10 123.20 j+1163.40 !- 1.79 135.60!- 952.10 - 0.94 +201.00 - 10.60" + 74.00 0 i . 2 0 +i ; -1283.4b" 161.60 181.00 +1280.20 - 1 2 7 2 . W -17.68 - O.83 60 + 4 2 3 . 0 0 7.^40 + 296.20 + 539.50 -1079.00 - ^ 7 . 8 0 +137.50 - 9.50 + 4 2 . 3 0 +1438.50 0 •1183.70 149.30 167.00 +1258.70 -12.50 - 1 1 7 4 . 2 0 - 0 . 8 0 40 + 3 2 7 . 9 0 - 1 0 . 6 0 +' 4 8 1 . 5 0 + 4 0 2 . 0 0 - 9 5 2 . 0 0 - 9 0 . 0 0 + 63-50 + 1 0 . 6 0 - 7 . 4 0 +1285.50 - 1 0 6 0 . 0 0 133.60 +1211.40 149.50 -1052V60 - 5 . 7 6 - 0 . 7 0 F i x e d Ends + 9 6 2 . 5 0 - 7 0 9 . 0 0 C o f A r c h C D 80 + 4 0 2 . 0 0 8 . 5 0 + 7 4 0 . 5 0 95.20 + 4 1 2 . 5 0 - 1 4 4 0 . 0 0 ±370 .^50 +190.40 - 2 1 . 2 0 - 4 . 2 0 +2115.90 -I569V1O 144.90 142.60 +1523.50 -28.00 -1556.40 - 0 . 8 1 60 + 285.50 + 666.00 - 12 .70 - 105780 + 560.50 -1312.00 +317.50 - 2 4 . 4 0 +137.50 +1967.00 - 1 4 - 5.30 6o;20 134.80 132.80 +1544.00 -1442.20 - 2 1 . 5 0 - 1.23 40 + 137.50 - 2 1 . 2 0 + 497.00 - 99.50 + 835.50 - 1 0 9 0 . 0 0 +222.20 - 31^80" + 63.50 i .50 +1755.70 120.20 -1251.00 113.90 +1554.70 - 1 1 . 4 4 -1221.30 - 2.37 F i x e d Ends +1460.00 - 1 1 0 0 . 0 0 18 Table 9 Max. L.L. Lower Kern Moments at Crowns Wax M K L at Crown of Arch H P i n f t Value of ^ L when Li v e Load on Spa P n Considering 5 spans Considering 1-Span f o r + MK L & 3-Span f o r -M K L T o t a l % L P In f t # of Cor-r e s . F.E. V a l -ues T o t a l % L P i n f t E r r o r A B i n f t B C i n f t C D i n f t D E i n f t E F i n f t A B 80 +370.50 + 7.40 + 3.70 ZO * 0 +381.60 212.00 +370.50 - 2.91 - 31.70 -179.90 - 63.50 ^0 zO -275.10 289.80 -275.10 ^ 0 60 +317.50 + 8.50 + 4.20 zb zo +330.20 183.40 +317.50 - 3.85 - 47.50 -158.80 - 52.90 ^0 <y0 -259.20 273.00 -259.20 zo 40 +254.00 + 10.60 + 5.30 &0 +269.00 150.00 +254.00 - 5.90 - 74.00 -105.80 - 21.20 zo «0 -201.00 211.60 -201.00 ^ 0 Fixed Ends +179-90 - 95-00 B C 80 + 3.70 +487.00 + 7.40 zo ZO +498.10 209.20 +487.00 - 2.23 -201.00 - 63.50 -127.00 - 52.90 <z0 -444.00 300.00 -391.10 -11.90 60 + 4.20 +412.50 + 8.50 zo ZO +425.20 178.60 +412.50 - 2.99 -169.20 - 84.60 -105.80 - 31.70 0 -391.30 264.00 -359.60 - 8.10 40 + 5.30 +338.50 + 10.60 ZO z 0 +364.40 152.90 +338.50 - 7 . 1 0 -IO5.8O -105.80 - 74.00 - 10.00 •xO -295.60 199.50 -294.60 - 3-40 Fixed Ends +238.00 -148.00. C D 80 + 5-00 + 10.00 +508.00 + 10.00 + 5-00 +538.00 181.70 +508.00 - 5.68 - 52.90 -121.70 - 84.60 -121.70 -52.90 -433.80 256.00 -328.00 -24.40 60 + 5.30 + 10.50 +455.00 + 10.50 + 5.30 +486.60 164.00 +455.00 - 6.50 - 42.30 -105.80 -105.80 -105.00 -42.30 -402.00 238.00 -317.40 -21.00 40 + 5.50 + 10.60 +391.50 + 10.60 + 5.50 +423.70 143.00 +391.50 - 7.60 - 15.90 - 79.30 -148.00 - 79.30 -15.90 -338.40 202.00 -306.60 - 9.70 Fixed Ends +296.30 -169.00 Table 10 Max. L.L. Right Kern Moments at P i e r Tops Value of Mj—^prp when P L i v e Load Considering 5 Spans Considering 2-Span only P i e r Hp on Span T o t a l M a x MKRPT P T o t a l » M Max KRPT P % E r r o r A B B C C D D E E F i n f t i n f t i n f t i n f t i n f t i n f t i n f t i n f t 80 +1883.00 + 0 + 1 1 6 . 3 0 & 0 &o + 1 9 9 9 . 3 0 +1883.00 - 6 .18 BG 0 -1713.00 8.50 0 &o -1721.50 -1713.00 - 0.50 40 +1079.00 + 52.90 + 74.10 0 ^0 +1206.00 +1131.90 - 6.14 - 10 . 6 0 - 973-00 - 3 1 .70 ^ 0 -1015 .30 - 983.6O - 3-12 80 + 0 +1958.00 + 84 . 6 0 +201.00 +95.20 +2338.80 +2042.60 -12.65 CH - 84 . 6 0 0 -1588.00 - 10.60 - 3-20 -1686.40 -1588.00 - 5.83 40 + 21.20 +1196.00 + 476.00 +158.80 +42.30 +1894.30 +1672.00 -11.73 - 158.80 - 84 . 6 0 -1080.00 - 21.20 - 1 0 . 6 0 -1355.20 -1164.60 -14.05 Table 11 Max. D.L. + L.L. Lower Kern Moments at Springings % L at H p i n f t Due to Dead Load of Arch Due to Live Load k - f t T o t a l DL+LL k - f t % of Corresponding F.E. Values A B k - f t B C k - f t C D k - f t D E k - f t E F k - f t T o t a l k - f t A Of Arch A B 80 60 40 F.E. -4830 -3006 -1103 1383 4635 3750 2296 1760 833 370 295 100 0 147 0 0 2007 1677 1563 1269 1141 1025 3276 2818 2588 149.75 128.75 118.25 B Of Arch A B 80 60 40 F.E. -6075 -4385 -2340 1383 4900 4285 3140 1518 1219 557 460 276 32 147 0 0 950 1395 1389 1383 1107 1047 962 803 2057 2442 2351 2186 94.00 111.50 107.30 B Of Arch B C 80 60 40 F.E. 5775 5150 3660 -7355 -5670 -3373 I838 2540 1975 1299 599 424 88 220 0 0 1779 1879 1674 1838 1449 1332 I I 8 5 963 3228 3211 2 8 5 9 2801 115.10 114.60 112.20 C Of Arch B C 80 60 40 F.E. 3490 2 8 7 5 2195 -7360 - 5 6 7 2 -3407 1838 4545 3952 2 7 3 5 1 3 7 8 928 406 512 293 73 2565 2376 2002 1838 1555 1439 1286 963 4120 3 8 1 5 3288 2801 147.00 136.00 117.50 C Of Arch C D 80 60 40 F.E. 2722 1888 805 4575 4060 2875 -9015 - 6 5 8 5 -2232 3144 2530 2124 1 3 7 8 1288 915 381 2100 2402 3207 3155 2116 1967 1 7 5 6 1460 4216 4369 4963 4615 91.30 94.75 107.50 Table 12 Max. D.L.+ L.L. Lower Kern Moments at Crowns Span Hp In f t Due to D.L. of Arch Due t o LL k - f t T o t a l DL+ LL k - f t % of Correspond F.E. Value A B k - f t B C k - f t C D k - f t D E E F k - f t k - f t T o t a l k - f t A B 80 60 40 F.E. 2343 1867 1244 587 -1250 -1087 - 689 -525 -427 -139 ^0 .^ 0 ^0 ZO #0 ^0 567 353 416 587 382 330 270 180 949 683 686 767« 123.90 89.10 89.35 B C 80 60 40 F.E. -1364 -1141 - 695 3065 2375 1687 652 -1049 - 853 - 556 -383 -230 - 73 ^0 269 151 363 652 498 425 364 238 767 576 727 890 86.25 64.85 81.70 C D 80 60 40 F.E. - 332 - 256 - 72 - 809 - 691 - 498 3715 3060 2135 1117 -809 -691 -498 -332 -256 - 72 1433 1166 995 1117 538 487 424 296 1971 1653 1419 1413 139.50 116.90 100.20 ro Table 13 Max. D.L. + L . L . Compressive Upper Extreme Fiber Stresses In ; Irenes Span Location In f t Total Max MKL k - f t f = MRLCT* cu • j -p s i # of allowable f i b e r stress 1,350 p s i A B & E F Springings A & F 80 60 40 3276 2818 2588 1580 1360 1250 117.00 100.70 92.60 Crowns 80 60 40 949 683 686 1100 790 794 81.50 58.50 58.80 B C & D E Springings C & D 80 60 40 4120 3815 3288 1564 1446 1248 116.00 107.00 92.40 Crowns 80 60 40 767 576 727 760 572 722 56.30 42.40 53.50 C D Springing C & D 80 60 40 4216 4369 4963 890 922 1048 66.00 68.30 77.60 Crowns 80 60 40 1971 1653 1419 1282 1076 922 95-00 79.70 68.30 * I based on t o t a l depth of r i b s t e e l Ignored. Table 14 Max. D.L. + L.L. Compressive Left Extreme Fiber Stresses at Pier Tops D.L. of Arch • MKRPT Total f _MC Pie r Hp Mj^ prp Due to due Max. c 1 A B B C C D D E E F Total to L.L. MKRPT i n f t k- f t k - f t k - f t k - f t k - f t k - f t k - f t k - f t p . s . l . BG 80 13,020 -12,410 945 0 0 1555 1999 3554 702 & EJ 40 7,385 - 6,670 372 0 0 IO87 1206 2293 453 CH 80 - 585 14,190 -13,180 1380 637 2442 2339 4781 599 &DI 40 - 952 8,055 - 5,300 998 220 3021 1894 4915 640 ro 15 10 - 5 - 1 0 - I S - 5 IS 10 - 5 . 1 0 . 1 5 - 5 - 1 -1 I 1 1 i 1 ! i i i 1 i | 1 i I I - t , 1 I i 1 I | I i i 1 1 l i ! __LJ _. i 1 i • T " 1 1 i i 1 4 - 1 - - l - L f - L 1 i -! 4 . 1 ! 1 | 1 I 1 1 i 1 1 0 : i S . . . 4 4 . . INF R *-I -Al :R©\ m I ii L0HI i C 5YS1 i 1 ~! l Fn t.r 11: i i l r •— J i i i 1 1 _ i i I i i i i i I _ ! 1 r i i F R,F If F If 1 1 1 1 1 I i -. i 1 4 • ! — — H --t- -- . — - -• - — -• . - — - . . . -• V n RTHp =;6 n i F T . i t : 1 O i "i P F p a SO t. i r i 1 // \ 1 i J_ •tj V i 1 t i 'li i\ i 1 • 1 1 11 >\ \ \\ ! i 1 i ; ! " t J I 'II 7 1 | 1 i _ .11 1 Ii V \ 1 i 1. 1 1 i i i 1 1 i ! 1 ! 1 \ t - 4 _ ! 1 J, '/ > 1 T ; A / [\ 1 I [ ! 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B - -- E . . . - - -j - - I - -- - I .... - - 1 1 • - J -- -I : I i i I " i ^1 A\ r US E a: r. • _< I s El si A DF r 31 ; •i L * - -- < - - - -> E - -i . . . - - ------ - - — - - -' i i- i i I ' I J l ! J_LL FIG! EOliINI LUfilCEaiNESIB >; ! I I 4-__L! i I Lj.i_L • i i -3»r fOPhJ IXSD ENDS i I I i mix 15 10 -5 -10 -15 - V ? C»C MPONENT OP Mi SiWSEP ;B¥ 1 RAN^LATl€iN-OF PiBRl -l^yPS M i l l J- l-J_!_L 4-LL 42 r J d ^ - ^ S a t t o H J W ^ i w 5 s J C S P A N . A B : bmxl F O R . J AT A , S Y S T E M : I T . id:rr;.:r„jr 10 T T j _ u J _ j _ j j _ i i i i T i i s r -i—)—: r TT •TT TrrrtT" - • i - i - j - " — i — ' F O R tfp-60 4 - , _ , . j _ , r t r . Y ^ F r.. f, ™ F O R H P * 8 0 ;. j -j -j- -j -T t ~ ~ r - ' - t :.; ;:R)R Hp^o f t . f t . f t . r r 4-j. I. 1.-1.;.iTj r a n LIT ! " i .... ,..! 4 +-P-LuL -|-T- i-p-- / ~BTi.rt.n.:i y;xri:| [•;:.; :r [;j.|::::[Tr .L!:TI..;'I '1:3: T ;• x L L i n i 1 - M ' -5 -10 •15 .::^-r!:.t±i:r -j_;_t . 1-4 t rC'T-iXj:;];; •TTV^ . : : . B .1 .J • rt. 1 I L L . ; 1 : . i ! : 1 ' i : • . . . . . 4 .... ILL"" {:±.U:LL;|. t.iirj ri-.1 .1.1 i l l !: L . CHANGES-, IN Jfo HUE TO T R A N S L A T I O N O F P I E R T O P S : < C H A N G E S ; I N M K L '.DUE: TO ROTAT-ION j O F P I E R T O P S '< j . j . ; . .1 43 44 45 46 47 48 49 52 1* 10 IFIG, 7.3$:iteiUENCEXIWAS ..(SPANJJBC;ONLY) .FOR1 l\i t i r r i - T Jj_lJ-4J.-lw . J-; • i ! i : i : : ; :rt j : i : i j :r l : ! n i * i r ' r •—L.J .i..R-4-._;.-. . . .J .1 : . L . L B..T:jl;.iT-:2i ..•_rj. |_L,.1J. FOR F i X m J i M ^ ItlfrfTJ FO^R: Hpriio; f t . FOR;flp«60;n. FORLHp-Solft. .4. . . . . ! . I • : T r r~j T' T t t i . ] : . L i - l -U I- • •H < -.L. i t i t : . i_r 11 ri'Lhli i J ntfi: : : : : . / : . ; . 4_J v i -4-i-4..L J .1 ATB OF: ARCH BC, SYSTEI-f -f :;:'i:!:[. r r r i ; - 4 - - , - :-: 1 -) I I " T !" . 4 - L -5 -10 -15 -20 w AN •i-h 1/ / r ! t . | . ! j . . i i , ^  ... , . j » . . , :.. ;. !--- U--;- 4-J • j-. . . . . j . . - ! -.L.j j .;....!..(. Lj. , . :..! } ;..|..:.!..! 4. -"-i-4fw|j'-T"; 1!'.:!" L;".n_ . L ...... -; 1- • I ! J_Li_ 0 . J . - T B : TO TRANSLAT ON :0F: PIER' •OPS: -10 TI:;.;:-!.:::: T . - r t CHANGES :IN;%j DUE iTO ROTATION OF- PIER _i 1 J. • ,— L— s—, 1—, , 1—1—;—~ i—i  LT-'..;4 I i : • i- , .1 _:.J_i < I , 4.;.:.... •!:;••;•; Mil-53 54 55 56 5 7 58 L-lBJ ^USJ-J_.u;« : -*-f-tOJ-:--'"-r--!-•+ to- - ; - : - ; • j- i g> -' <D 0 -; ; 0,1-4 -4- :-iQ,rtS-i • J i . B . ^ i - j - ' a -co n -a> «) U X • - • : S to - - -O • • - •- • io : • - • C- •- vr • t~ O r r f ft • -+ CQ. -!•--! • <D r • , -r • . •= - -- ^ U • k-_j—«-o—-Q--« o 1 X in '• - • •• -d O • £ v - • : : .'bo • , • - • > < - ~ • £1 O _:-k2a: 4 i-FIG. R a t i o o f P i e r 1 Height^ to- Arch! R i s e 4 j-i I't-... r• 41 CHANGES IN;MAX. D.L. i+ L.L. COMPRESSIVE UPPER FIBER STRESSES AT SPRINGINGS FROM FIXED ENDS -i . . , .| -+ t T i l l -r t I 4 5 9 60 c o -co., • r -r i i n • IV .'• V EFFECT OF TRANSLATION EFFECT OF ROTATION. •i'T" E F i AT :A dP ARCH AB & F OF ARCH AT. .1 OF. .ARCH. .AB....1&. E._;O.P .ARCH EF_ AT Al A l B OF ARCH BC & E OP, ARCH DE | c :OP :ARCH BC & D :OF :c. OR D :OP ARCH CD ARCH DE FIG R a t i o of p i e r Height to Arch R i s e 43- CHANGES IN MAX. L.L. COMPRESSIVE UPPER FIpERi I : STRESSES AT. SPRINGINGS FROM FIXED ENDS.DUE TO TRANSLATION: AND* ROTATION: ; ] .. ! .j. . . . . . ' 1 - ! ~ 4 • 61 UPPER-FiBER.:.:. -FIG . . .M:. CHANGES..IN.-MAX..:-L..L. 1-COMPRESSH STRESSES AT CROWNS FROM FIXED ENDS DUE TO - r - - 1 TRANSLATION AND • ROTATION • ... i. •I- i r-' i-^.i-i i ! ! i r T ._Li__ CHAPTER I I I Conclusions By i n s p e c t i n g the areas of the i n f l u e n c e l i n e s , the lower kern moments i n arches, and the r i g h t kern moments i n p i e r s w i l l govern the design i n the arch systems analyzed, and t h a t i s probably t r u e i n gener a l . System I has been almost completely analyzed f o r dead load and l i v e load by means of q u a n t i t a t i v e i n f l u e n c e l i n e s . In System I I q u a n t i t a t i v e i n f l u e n c e l i n e s were constructed f o r the same s t r e s s f u n c t i o n s but or d i n a t e s computed f o r but the span i n v o l v e d . These showed (vide P i g s . 5 to 20 and P i g s . 25 t o 40) such s i m i l a r i t y i n shape as t o j u s t i f y i n the author's o p i n i o n the deductions pre-v i o u s l y r e f e r r e d t o although q u a n t i t a t i v e values f o r dead load and l i v e load kern moments at s p r i n g i n g and crown of the arches i n System I I were not computed. Tables 8 to 10, i n c l u s i v e , i n d i c a t e t h a t f o r Max. D.L. + L.L. s t r e s s e s , (save f o r very f l e x i b l e p i e r s ) s t u d i e s may be confined t o three spans f o r arches and two spans f o r p i e r s . E r r o r s i n v o l v e d are not s i g n i f i c a n t i n comparison w i t h the v a r i a t i o n s of the e l a s t i c c h a r a c t e r -(12 13) i s t i c s of the s t r u c t u r e s . ' Tables 8 and 9 a l s o i n d i c a t e the e f f e c t of p i e r dimensions on changes from f i x e d end c o n d i t i o n s . A l l 62 63 maximum p o s i t i v e L.L. lower kern moments and maximum nega-t i v e L.L. lower kern moments at the s p r i n g i n g s and crowns are g r e a t e r than those f o r f i x e d ends, and the d i f f e r e n c e i n c r e a s e s as the height of p i e r s i n c r e a s e s from 40' to 80', but the maximum lower kern moments due t o the combination of dead load and l i v e load do not of course e x h i b i t t h i s c h a r a c t e r i s t i c as i s shown i n Tables 11 and 12, and P i g s . 41 and 42 where kern moments are replaced by f i b e r s t r e s s e s . P i g s . 43 and 44 i n d i c a t e the r e l a t i v e e f f e c t of r o t a -t i o n and t r a n s l a t i o n of the p i e r tops on extreme f i b e r s t r e s s e s at sp r i n g i n g s and crowns. The e f f e c t of r o t a t i o n on crowns i s q u i t e small and almost independent of p i e r h e i g h t , and that on s p r i n g i n g s i s somewhat gre a t e r and increases slowly as the height of p i e r s i n c r e a s e s . The e f f e c t of t r a n s l a t i o n on s p r i n g i n g s and crowns I s u s u a l l y the greatest and Increases r a p i d l y as the height of p i e r s i n c r e a s e s . Tables 13 and 14 i n d i c a t e the D.L. + Max. L.L. f i b e r s t r e s s e s i n arches and i n p i e r s . The moment of I n e r t i a used i n the computations i s based on the gross area of con-c r e t e s e c t i o n only. The data chosen f o r the numerical examples should be s a t i s f a c t o r y as the u s u a l amount of s t e e l used i n arches i s s m a l l . The r e s u l t s presented h e r e i n are only t r u e f o r t h i s p a r t i c u l a r type of continuous arch system. A c t u a l l y the problem i s very complicated and the large number of v a r i a b l e s i n v o l v e d i n the design of continuous a r c h systems makes i t i n a d v i s a b l e t o draw t o o d e f i n i t e c o n c l u s i o n s , save t h a t t h e e f f e c t s o f t r a n s l a t i o n o v e r -shadow t h o s e o f r o t a t i o n i n almost a l l c a s e s . I n t h i s r e s -p e c t a t l e a s t t h e a u t h o r i s i n agreement w i t h some p r e v i o u s w o r k ers i n t h i s l i t t l e e x p l o r e d f i e l d . 65 Appendix I Dead Load Horizontal Thrusts on Arches f o r Numerical Examples Type of Structure - Open spandrel Data - See Figs. 1, 2, and 3 and Table 1. Slope of roadway surface: Deck Load per Linear foot per Rib. - Assuming that the panel loads are uniformly d i s t r i b u t e d over the spacing of the f l o o r beam, 20»-o" C/C. 1 100 Weight of handrail (50$ of area removed by openings) 4" sidewalk slab Curb beam Floor slab 6" curtain wall Stringers (9"xl4" ® 7'-0" C/C) l8"x36" f l o o r beam and bracket » 125 lbs = 305 lbs = 85 lbs =1640 lbs = 187 lbs - 460 lbs Total Deck Load Weight of hanger (or column) l8"xl8" @ 20'-0*' C/C - 935 lbs =3737 l b s / l i n . f t . / r i b =338 l b s / foot of height 66 System I Span A B: L = 180• r = 28.6 «c - 3' d s = 4.65' g •= 1.455 (assumed) tan 0 S = 2 r k ^ g 2 - 1 = 0.675 3 L(g - 1) 03 = 34°2' d^ m - 2 = 0.325 d| Cospf8 wc - 3,737 + 5'8{H&) +3(4)150 = 5,635 lbs w8 = 3,737 + 3 2 - 5 ( ? 3 8 ) 2 + 4.65(4)150(1.207) - 8,202 lbs check g = Hi = 8,202 = 1.445 o.k. wc 5,635 The dead load horizontal thrust l a : H d - wC (h)2{—l " 5 ,635(90) 2 ^ ; 4 ^ 5 - 1) c 855,000 lbs 2 r k 2 2 o.6 ( 0 . 9 2 l ) 2 Span B C: L = 200* ' r » 36.175' d c = 3.25' d s = 5.24-67 g = 1.543 (assumed) tan 0 S - L (4.328) - O.761 L 0s = 37°17 ' m = 0.30 wc = 3,737 + 3.25 (4) 150 = 5,687 lbs ws = 3,737 + ' + 5 .24 (4) 150 (1.257) - 8,802 lbs check g = W B m 1.545 o.k. *c* The dead load horizontal thrust i s : H d - 0.1358 w c L « 855,000 lbs Span C D: L = 240» r = 6 0 ' d c = 4-w c w s 1 d 8 = 7.03 1 g = 1.756 (assumed) tan 0 S = 1.105 0 8 = 47°52' m = 0.275 3,737 + 1 3 > 7 5 ( 3 3 8 ) + 4(4)150 = 6,370 lbs. 20 3,737 + 33.5(338)2 + 7.03(4)150(1.491) - 11,169 lbs check g = *S = 1.745 o.k. 68 The dead load horizontal thrust i s : H d = 0.1397 W C L 2 = 855,000 lbs r System I I Span A B: L « 180' r = 35.8' d c - 3.25' d s = 4.95' S = 1.5^3 (assumed) tan # 8 = 0.861 0B m 40°44' m = 0.375 wc = 3,737 + 7 , 9 T 3 ( 3 3 8 ) + 3.25(4)150 = 5,822 lbs 20 ws = 3,737 + 39-5(338)2 + 4.95(4)150(1.3197) = 8,990 lbs 20 check g « 2J§ = 1.543 o.k. w c The dead load horizontal thrust i s : *d H, » 0.138 w c L = 715,000 lbs r 69 Span B C: L = 200' r - 45.55' d c - 3.5" d s - 6.05' g = 1.756 (assumed) tan 0 3 = 1.012 0 B = 45°20* m = 0.275 wc = 3,737 + 3.5(4)150 = 5,837 lbs ws = 3,737 + 39.5(338)2 + 6.05(4)150(1.4217) = 10,240 lbs 20 check g = ^ £ = 1.756 o.k. w c The dead load horizontal thrust i s : H d = 0.1397 W c I j 2 = 715,000 lbs r Span C D: L » 240' r = 80' d c = 4.25' d a « 8.93' g = 2.24 (assumed) tan 0 S = I.56 0 S « 57°20' m =0.20 70 w = 3,737 + 2 4 - 1 2 5 ( 3 3 8 ) + 4 > 2 5 x 4 x 1 5 0 = 6,694 lbs 20 ws - 3,737 + 3 9 ' 5 ( 3 3 8 ) 2 + 8.93(4)150(1.853) » 15,013 lbs check g = = 2.24 o.k. The dead load horizontal thrust i s : 2 H d = 0.1483 w c L = 715,000 lbs 71 Appendix II Formulae for Analysis The derivation of formulae Is not a special feature of this thesis and w i l l not be included In detail. Rotation and Translation Factors of a Symmetrical Arch rib . - A single symmetrical span of a continuous arch is taken into consideration, one end i s kept fixed r i g i d l y and the other end i s free to move. Figure 45 In Fig. 45 the left end of aroh, L R, i s kept fixed and the right end i s given a unit clockwise rotation without any translation. From the neutral point method, the resultant expressions for the rotation factorB are found: 72 and, HR = -VJ M: ds EI rds T y ^ s ( J EI J EI J fv2 ds r E I L _ 2 2 ds • ET f y 2 f x2 ds J E l J ET L2 2 ds r x 2 ds E l (1) (2) (3) (4) In Fig. 46 the l e f t end of arch, L R, s t i l l remains fixed and the right end i s given a unit length of horizontal Figure 46 displacement to the right without any rotation. The resultant expressions for the translation factors are: The rotation and translation factors in Eqns. 1 to 6, inclusive, are in the same form and with the opposite sign for the l e f t end of the arch moves and the right end i s kept fixed. The axes of the co-ordinates, X and Y, to which values of x and y are referred in the foregoing expressions, pass through the neutral point, o, and are directed horizon-t a l l y and vertically. (2) According to Whitney's paper, the equation of arch axis i s , and the distance determining the location of the neutral point Is, y - y c - —L* (cosh zk - 1) g - 1 (g-l)(l+m) 2r where, g m I s Cos0, a z 2x L and, k Cosh'-'-g 74 Rotation and Translation Factors for a P i e r with Fixed Base. - F i g . 47 represents an e l a s t i c p i e r , fixed at the base. When the top i s given a unit clockwise rotation without any t r a n s l a t i o n the resultant expressions for the rotation factors Figure 47 are: H: PT and, MpT ( y | Z . f y2 ^ (|v_ 7/ Ely) E l y EL, - V2 & (& E l y j E l y (7) (8) In Pig. 48 when the pier top is given a unit length of horizontal displacement to the right without any . . 4=,1 Figure 48 rotation the resultant expressions for the translation factors are: The y co-ordinates of the pier axis are measured from the pier top. In order to evaluate the foregoing expressions the values of integrals have been determined. They are as follows: For Pier. -fdy_ EI„ f Tdy_ EX,, r2dy_ = H/l+q) 2 E I T q< 2EI - q2 H? Ely EI T(q-l ) 3 In q + - - -l—0 q 2 q 2 . 3 2 Distribution and Carry-Over Factors for Continuous Arch System. - They may be found by considering a two span system. The outer ends of the three members (by member i s meant each individual arch span or pier) are rigi d l y fixed and 77 the inner ends, or say Joints, are allowed to displace. For convenience of reference to the d i s t r i b u t i o n pro-i n Figs. 4 9 and 50, i n c l u s i v e , are those which the members exert on t h e i r terminal points. P o s i t i v e directions are to ri g h t for thrust and clockwise f o r moment. In F i g . 4 9 Joint B moves h o r i z o n t a l l y a unit length to the right without any ro t a t i o n . From Eqns. 5 and 6 the forces at the ends of the arches as shown i n F i g . 49(b) for accompanying the movement are: cedure described by Prof. A . H . Finlay, ( 1 1 ) the forces shown H A B = " H : 1 ( 1 1 ) BA ( 1 2 ) 1 ( 1 3 ) ( 1 4 ) and, From Eqns. 9 and 1 0 the forces at the p i e r top as shown in Fi g . 49(b) are: Figure 49(a) B ^ r BD H BD D Mi CB Figure k9(b) Thrust line for translation H" HcBL NPB where distribution i s being made Figure U9(c) 79 |y & and, M*_ = - J ^ EI ( l 6) *3 E ^ | ^ " ( r 3 E I y The co-ordinates x^ and y-^  are referred to the arch A B, %2 a n d 3*2 t o t n e a r c n B c» a n d y3 t 0 t n e P i e r B D, the or i g i n of those axes i s specified as before. The force system shown i n F i g . 49 (b) i s equivalent to that shown i n F i g . 49 (c) where the thrusts are acting along the thrust l i n e s for t r a n s l a t i o n . The distances from the thrust l i n e s to the springing l i n e are determined as below: d x =!!AB . !!BA . B ( 1 7 ) HAB "BA d 2 = !|c = ^gB = b 2 ( l 8) HBC HCB and, d 3 = - ^BD = ( y 3 |f (19) "BD fdy E I y In F i g . 49 (c) the point, o, represents the neutral point of the Joint, B, and the distance, d B, determining i t s location, i s found: -MgD -MgA -MgC dg = "BD + H B A + H B C The unbalanced thrust at the neutral point of the Joint , B, i s : Ho - -HBA " H B C " H^ D (20) ^ -0 Figure 50(a) Thrust lin e for rotation Figure 50(b) #1 NPB where distribution i s being made Figure 50(c)' #2 #1 The ve r t i c a l thrusts on arches and pier are not shown #2 Oniy the horizontal components of the thrusts on arches are shown, and the ver t i c a l components may be obtained by proportion. 81 In Pig. 50 while the neutral point of Joint, B, has rotated through a unit angle in the clockwise direction, the Joint, B, moves a horizontal distance, d B, to the right and rotates the same angle with the neutral point, the forces set up in the arches and pier as shown in Fig. 50 (b) are: H A B - - Z%1*L ( 2 D J y l EI M A+oC AB C 4 + o C NBA = 4+oC H B C * b 1(d B+ b1 ) J 1 EI b 1 ( d B + b J y l EI d B + b 2 H C B " , 2 ds EI rdsi J EI 1 fds _1 EI 4 c2 ds 1 EI (22) (23) (24) 4 + * WBC S b 2 ( d B + b 2) rv2 ds 211 M CB fyU* 2 EI _ + fdS 2 E l " fx2 ds 2 EI _2 fi5s f J EI J x2 ds 2 i l (25) (26) « B D -- fr3 i f (27) D B > 3 ft - Iy3 j and, M^"— E I J'* EI ( 28) In replacing the force system shown in Fig, 50(b) the thrusts are acting along the thrust lines for rotation as shown in Fig. 50(c). The distances determining the loca-tion of the thrust lines are: dx = _JB • (29) AB 1 r (30) H*+« ( 3 i> i BC d 2 - !!CB (32) HCB and d = ~__BD (33) 3 A+cC BD The unbalanced moment at the neutral point of the joint, B, i s : From Fig. 49(c) the distribution factors, h B A and h B C , and the carryover factors, h^g and hgg, at the neutral point of the Joint, B, for thrust are: 8 3 h B C - ^BC ( 3 6 ) K h A T , - H A B A B Z— ( 3 7 ) and, h C B - 5S2 ( 3 8 ) H o From F i g . 50(c) t h e d i s t r i b u t i o n f a c t o r s , m ^ and irigg, and t h e c a r r y o v e r f a c t o r s , m^g and m^g, f o r moment a r e : mBA = ( d x + d B } ( 3 9 > m B C = 1^ 1. (dg + d B ) (40) o and, M C B - ^CB (dg + d c) (42) o The d i s t r i b u t i o n and c a r r y o v e r f a c t o r s f o r t h e p i e r a r e u n n e c e s s a r y , but t h e y may be found by s t a t i c s . Extreme F i b e r S t r e s s . - L e t mn i n F i g . 51 r e p r e s e n t any s e c t i o n o f t h e member. R i s t h e t o t a l r e s u l t a n t f o r c e a t t h e s e c t i o n and may be r e s o l v e d as shown i n t o t h e sh e a r V 84 • N Upper kern p o i n t N e u t r a l a x i s < Lower kern p o i n t Figure 51 and normal t h r u s t N. Then the compressive upper extreme f i b e r s t r e s s i s : / O T • j[ + gT- • N f k 2 + 6 ° T ) , V(9 + f_ ) CT , M L ° T C T / I KL j - . . . . ( ^ 3 ) and the compressive lower extreme f i b e r s t r e s s i s : M C, j r , - N - U L = N CL £ k 2 - e For a r e c t a n g u l a r s e c t i o n the distance from the kern p o i n t to the n e u t r a l axiB I s : k2 Cr h 5 (45) where, k 2 - I A Kern Moment i n Arch. - Let AC i n F i g . 52 represent a h a l f span of the arch r i b . The kern moments i n the arch a r e : At t h e crown, M K L - M C + H 6 and, = M c - ^ H At t h e s p r i n g i n g l i n e , \ L - M g + (HCos^ s + V S l n # s ) V F i g u r e 52 86 K e r n Moment a t P i e r Top. - L e t BD i n P i g . 5 3 r e p r e s e n t an e l a s t i c p i e r , s u p p o r t i n g t h e a r c h e s A B and B C. The r i g h t k e r n moment a t t h e p i e r t o p i s : F i g u r e 53 and t h e l e f t k e r n moment I s : % L P T = M K ^ T ) V ( 5 1 ) where, V « V± + V"2 and, M » M 0 - M, 87 B i b l i o g r a p h y 1. Cross, H. " A n a l y s i s of Continuous Frames by D i s t r i b u t i n g Fixed-End Moments," Tra n s a c t i o n s , ASCE, V o l . 96, 1932, p. 1. 2. Whitney, C.S., "Design of Symmetrical Concrete Arches," Tr a n s a c t i o n s , ASCE, V o l . 88, 1925, p. 931. 3. Whitney, C.S., " A n a l y s i s of Continuous Concrete'Arch System," Transactions, ASCE, V o l . 90, 1927, p. 1094. 4. Cross, H., and Morgan, N.D., "Continuous Frames of Reinforced Concrete," John Wiley & Sons, Inc., New York, 1932, p. 316. 5. H j o r t , A.P., "Design of Continuous Arches on E l a s t i c P i e r s , " Proceedings, ACI, V o l . 29, 1932/33, P. 143. 6. H r e n n i k o f f , A., " A n a l y s i s of M u l t i p l e Arches," T r a n s a c t i o n s , ASCE, V o l . 101, 1936, p. 388. 7. Maught, L.C., " S t a t i c a l l y Indeterminate S t r u c t u r e s , " John Wiley & Sons, Inc., New York, 1946, p. 230. 8. Morgan, V.A., "An Exact Method of A n a l y s i s of Continuing P a r a b o l i c Arches," Concrete and C o n s t r u c t i o n a l Engineer-i n g , London, V o l . 47, No. 11, Nov. 1952, p. 343-9. Michalos, J . , and G l r t o n , D.D., "Continuous Arches on E l a s t i c P i e r s , " Proceedings, ASCE, V o l . 8 l , Nov. 1955, Paper No. 827, p. 827-1. 10. R i l e y , W.E., " A n a l y s i s of Continuous Arches on F l e x i b l e P i e r s , " Proceedings, ACI, V o l . 28, 1956/57, P. 999 11. F i n l a y , A.H., D i s c u s s i o n on " A n a l y s i s of M u l t i p l e Arches," T r a n s a c t i o n s , ASCE, V o l . 101, 1936, p. 410, and l e c t u r e s given to h i s graduate c l a s s i n the U n i v e r s i t y of B r i t i s h Columbia, Canada, 1958. 12. Cross, H., "Dependability of the Theory of Concrete Arches," B u l l e t i n No. 203, Univ. of I l l i n o i s Eng. Experiment S t a t i o n , 1930. 13. Wilson, W.M., Kluge, R.W., and Coombe, J.V., "An I n v e s t i -g a t i o n of R i g i d Frame Bridges," B u l l e t i n No. 308, Univ. of I l l i n o i s Eng. Experiment S t a t i o n , 1938, P t . 2 . 88 N o t a t i o n s L = t h e o r e t i c a l span o f n e u t r a l a x i s o f r i b . r = r i s e o f n e u t r a l a x i B o f r i b . W « w i d t h o f r i b d_ = depth o f r i b a t s p r i n g i n g l i n e . d c » depth o f r i b a t crown. y c = t h e d i s t a n c e from the r i b a x i s a t the crown t o t h e n e u t r a l p o i n t o f t h e r i b . b = t h e v e r t i c a l d i s t a n c e from t h e s p r i n g i n g l i n e t o th e n e u t r a l p o i n t o f t h e r i b . 0 S = t h e a n g l e between t h e h o r i z o n t a l and t h e ta n g e n t t o t h e n e u t r a l a x i s a t the s p r i n g i n g l i n e . I = the moment o f I n e r t i a o f t h e r i b a t any s e c t i o n . I 8 *» t h e moment o f i n e r t i a o f t h e r i b a t t h e s p r i n g i n g l i n e . I c = t h e moment o f i n e r t i a o f t h e r i b a t t h e crown. m «? _ i s I 8 Cos0s E = modulus o f e l a s t i c i t y . P = u n i f o r m l i v e l o a d p e r l i n e a r f o o t o f span. w s » dead l o a d p e r l i n e a r f o o t a t s p r i n g i n g l i n e . w„ 8 3 dead l o a d p e r l i n e a r f o o t a t crown, c g to W S Hp = h e i g h t o f p i e r . w p « w i d t h o f p i e r . dip = depth o f p i e r a t t h e t o p . cL^ = the depth of the exterior piers at the top d r p g » the depth of the Interior piers at the top d B « depth of pier at the bottom q = dB dT Iy « the moment of inertia of the pier at any section. I,p = the moment of inertia of the pier at the top. A = the area of the cross-section of the rib or pier. = the lower kern moment for the continuous arch on elastic piers. MjCU = the upper kern moment for the continuous arch on ^KRPT ™ ^ e r^S^t kern moment at the pier top. MKLPT ~ t h e l e f' t kern moment at the pier top. 

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