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Elastic stability of a pony truss Hrennikoff, Alexander 1933

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U.B.C. LIBRARY i cat wo. ^  to Ji^JLB^Mi k_4 ACC. no. & $ p fftg ELASTIC STABILITY OF A PONY THUbS. by Alexander Hrennikofi. A Thesis Submitted f o r 'une Degree of Master of Applied Science i n the Department of C i v i l Engineering.-The U n i v e r s i t y of B r i u i s h Columbia, February , 1923. TA8LS Of GOlfSMfS IS TR^DJCTlOri, A b r i e f sum up of the t h e s i s , ELASTIC STABILITY OF A fOEX TRUSS. 1. General presentation of the problem. a. Nature o f th® problem. b. Previous i n v e s t i g a t i o n s of the same problem. c . M>- i n asaustptloa© Involved. d. Q u a l i t a t i v e d i s c u s s i o n . 2* L a t e r a l Resistance of a Pony Truss f e r t l o a l w i t h no bending £a*n«nt a t the Top End. a. Basle d i f f e r e n t i a l equation. b. MxpressIons f o r the l a t e r a l r eaistaaee of vartlea& and the t o t a l l a t e r a l r e s i s t i n g f o r c e , e. C r i t i c a l value of the d i r e c t s t r e s s 3. l a t e r a l Ttml&%®mm of .* Pony Truss V e r t i c a l w i t h B#Mlng mmvat at th© top Sad, a. Basic d i f f e r e n t i a l e iUation. b. Expressions f o r the t o t a l l a t e r a l r e s i s t i n g force and the moment at the end. e* C r i t i c a l value of the d i r e c t s t r e s s d. Soste of the minor f a c t o r s disregarded, ©, Replacement o f the concentrated forces at the panel points w i t h d i s t r i b u t e d forces* 4. T o r s i o n a l information of the Buckled fop Gtoord. a . R e l a t i o n between the torque i n the 2op chord and the mo-mnta on the ends of v e r t i c a l s * b. R e l a t i o n bet wean oC &s& <T. The Eae.ro Method as Applied t o the Problem* a . Buckling of the structure as looked upon from the viewpoint of e l a s t i c energy* b* The energy equation. c . The shape of the b u c k l i n g curve. Work Done by E x t e r n a l Forces. a. Two possible methods of approach* b* Expression f o r d e f l e c t i o n of the bottom chord, c• Work done by uniform l o a d . d. Work done by concentrated load at the c e n t e r . E l a s t i c Energy of Deformed S t r u c t u r e . a . Energy of bending of the top chord. b. Energy of t w i s t and work of overcoming moiaents on the ends of v e r t i c a l s . c . Expression f o r ° ( t and work of overcoming the l a t e r a l r e s i s t i n g f o r c e . C r i t i c a l Yalue of the Uniform l o a d . a . iibcpresslon f o r the c r i t i c a l <\ b. iixp res s i on f o r y , f i r s t and second approximstions of i t , s i g n i f i c a n c e of d i f f e r e n t terms* c. Question of higher harmonics. C r i t i c a l Value of a Con-cent rated Load at the Center. I l l pages. 10, Reciprocal Influence L i n e of tha > C r i t i c a l Load. a* The ©lightness of dependence o f 33 the shape of b u c k l i n g curve on the p o s i t i o n of the l o a d . b. fhe r e c i p r o c a l Influence l i n e . 33 c. I t s a p p l i c a t i o n to movable l o a d s . J 4 11, Example* I l l u s t r a t i n g the A p p l i c a t i o n of formulas, a. Dimensions of the truss and s i z e s 35 of the members. b. C a l c u l a t i o n of t o r s i o 1 r i g i d i t y % of the too chord. c. Value of the c r i t i c a l uniform load; 40 f i r s t , d i s r e g a r d i n g decrease i n re s i s t a n c e of v e r t i c a l s , and sec-ondly, considering i t . d. Influence of the t o r s i o n a l 41 re s i s t a n c e of the top chord e. H e c i p r o c a l inf.Iuence l i n e , i t s 42 a p p l i c a t i o n to concentrated loads and i t s accuracy. 6 * CQ.-.'jGLiJo I OTv» 2o?ne P i n a l Remarks on the Method and the 43 Approximations and Assumptions Involved. fh© author o f t h i s paper endeavor© t o f i n d an expression f o r the Teliae of the load,, which causes c o l -lapse of c e r t a i n type of bridge without the top l a t e r a l bmeing* A f u a l i t a t i v e .study of the guest ion r e v e a l s , that the r e s i s t l a g a c t i o n of the web members i s g r e a t l y a f f e c t e d by the magnitudes of t h e i r a x e l s t r e s s e s ; and t h i s conclusion leads t o the n e c e s s i t y ©f determination of web r e s i s t a n c e s i n terras of loading* fhe energy method* c o n s i s t i n g i n comparison of the e l a s t i c energy of structure with the work done by the loading during b u c k l i n g , i s used as the method of attac k o-f the main problem* A f t e r e v a l u a t i o n of var i o u s terms i n the energy equation, expressions f o r the c r i t i c a l values of d i f f e r e n t types of loadings are found* An observation of the f a c t , that the shape of bu c k l i n g curve i s a f f e c t e d by the load i n g only very s l i g h t l y * leads to the idea of r e c i p r o c a l influence l i n e of c r i t i c a l l o a d i n g , a l l o w i n g an easy t re f i t -ment of the questions i n v o l v i n g movable load.-, fhe ap-p l i c a t i o n of the formulas developed i s demonstrated on an example,, and the paper i s concluded v i t h some f i n a l remarks, on the method and i t s assumptions- > 1 GMKRkZ PRES23TAT10B OF rmi PROBLEM. > The strength of engineering s t r u c t u r e s and parts of s t r u c t u r e i s sometloies governed not by the stresses i n the members, but by t h e i r e l a s t i c s t a b i l i t y . Thus, columns, s t r u t s and webs of compressed members can f a i l not on account of an excessive unit s t r e s s , but on account of t h e i r slender-ness, causing c o l l a p s e of the member, when the s t r e s s i s comparatively low. The problem of e l a s t i c s t a b i l i t y o f a h a l f -through t r u s s belongs t o the same c l a s s . I t has been tr e a t e d by s e v e r a l authors., and though not completely solved theore-t i c a l l y . , soue d e f i n i t e advance has been made toward i t s s o l u t i o n . Professor Timoshenlco i n Volume 94 of the "Transactions of the American Society of C i v i l Engineers" develops formulas and presents t a b l e s of c o e f f i c i e n t s to be used i n a c t u a l design- of a pony t r u s s with p a r a l l e l chords., s i m i l a r t o th© one shown on f i g * 1« The author of t h i s paper attempts to solve the same problem, but i n the course of h i s d i s c u s s i o n he takes i n t o c onsideration c e r t a i n f a c t o r s , l e f t out of question i n the paper Just mentioned, a olrouastanoe, due t o which the r e s u l t s of t h i s a n a l y s i s are generally d i f f e r e n t , from those of Professor TixGoshenJco*$* a»6 i t i s b e l i e v e d , acre eonforiaable to r e a l i t y . The c r i t i c a l load producing collapse of the 2 st r u c t u r e w i l l be thought o f , as applied a l l at the bottom chord* and. raay be e i t h e r i n the form of uniform load covering the whole'span, or i n the form of concentrated weights. M a t e r i a l w i l l be considered as p e r f e c t l y e l a s t i c i n a l l p a r t s of the structu r e under the c r i t i c a l l o a d , fhe str u c t u r e studied w i l l be l i k e the one on f i g . 1 ; i t s top chord i s of constant cross section,, the end v e r t i c a l s are abs o l u t e l y r i g i d , and a l l the intermediate v e r t i c a l s have the same cross se c t i o n constant along- t h e i r length* and are r i g i d l y f i x e d at the bottom ends and not influenced by the bending of f l o o r beams. SOKSQ a d d i t i o n a l assumptions w i l l be mad© l a t e r i n the course of the d i s c u s s i o n , 1st f i g . £ represent the plan view of a deformed pony t r u s s . Here ABB i s the s t r a i g h t bottorc chord* and i.CB— the buckled top chord.; the intermediate v e r t i c a l s appear on t h i s view as KG, FS e t c . , and the diagonals—as G-F* SD et c * fhe ordinatea of the buckled top chord such as $ ere considered i n f i n i t e s i a a l s of the f i r s t order* then* the l o n g i t u d i n a l displacements of the p o i n t s of the top chord become i n f i n i t e s i m a l s of the second order and a w not indicated on t h i s sketch. (Thus the p o i n t s A. and B represent the ex t r e m i t i e s of both the top and the bottom c h o r d s ) . She forces a c t i n g on the top chord at each panel point* such as point E* are two i n number* the force of the diagonal and the force of the v e r t i c a l ; the f i r s t can be resolved i n t o P and P, a c t i n g h o r i z o n t a l l y ^ a n d P e — 3 act l a g v e r t i c a l l y } the second resolves Into the h o r i z o n t a l force S {the bending resi s t a n c e of th© v e r t i c a l ) , , and the v e r t i c a l force ¥* ihleh- cancels P2 • t h i s leaves only I-, P, and S to act on the top chord at th© panel point E. P, i s th© force producing bu c k l i n g * and th® sum P + S « 1?—is the t o t a l r e s i s t i n g f o r c e , opposing the- b u c k l i n g * In aost cases, l a a d d i t i o n t o the f o r c e s discussed h e r e , there i s a aoaent fi0 » a p p l i e d t o the top chord at each panel point by the defl e c t e d v e r t i c a l , as is-evident from. f i g . 2; t h i s moment a f f e c t s bending -of the v e r t i c a l and produces t o r s i o n of the top chord* Angular c o n t i n u i t y between the v e r t i c a l and the top chord has great s t a b i l i z i n g e f f e c t * both on the structure as a 'whole, and on the i n d i v i d u a l v e r t i c a l as a s t r u t , and cannot be neglected without a s u b s t a n t i a l e r r o r . Only i n 'rare cases of chords with s i n g l e is-ebs and no p r o v i s i o n f o r c o n t i n u i t y ( f i g * 4 ) , the iaoa»nt at the top end of the v e r t i c a l i s absent*- I t i s needless to say* that the aoaent M„ a f f e c t s the magnitude of the r es i s t i n g force of the v e r t i c a l S* and that- both Mo and s are af f e c t e d by the, d i r e c t s t r e s s i n the v e r t i c a l . - In the process of b u d d i n g of the top chord' the for c e s here mentioned S, P, P, and the moment H 0 do work on' the top chord* which i s stored In the l a t t e r as p o t e n t i a l energy of bending (buckling) and t w i s t , fhe value of the c r i t i c a l load i s found f r o a the equation expressing i s a t h e a s t i -e a l l y t h i s statement* a f t e r assumption of a s u i t a b l e shape f o r the buckled chord* With t h i s q u a l i t a t i v e d i s c u s s i o n i n view, the 4 q u a n t i t a t i v e study of the subject w i l l be undertaken now. IATJJRAL RSSISEAJBTCE OF A POKY TRUSS ySRTICAL* DSFLBCTiilD A DISTANCE S » WITH RO BEHPIMfl MQMB38T AT THSI TOP| HD« Such a v e r t i c a l ( f i g * 5} s a t i s f i e s the c o n d i t i o n s of f i g * 4* where there i s no angular c o n t i n u i t y between the top chord and the v e r t i c a l . The distance <5~ i s date rained by buclciing of the chord| the d i r e c t s t r e s s i n the v e r t i c a l V i s supposed t o be 3aaown» and I t Is required t o f i n d the force S. fhe d i f f e r e n t i a l equation of the e l a s t i c curve i s : -V(?->l)-S(!,-x).-EI,$£ } <i> where S i s the Young's modulus and Iv i s the constant moment of i n e r t i a of the v e r t i c a l f o r bending out of the plane of the t r u s s . C a l l i n g by u the equation CD reduces t o whose general s o l u t i o n I s y = A Cos ux + 3 Sm ox + ^0}-*)  + & (S) The constants of i n t e g r a t i o n A ant B, as w e l l as the force S, are fount from the f o l l o w i n g three conditionss ? x*o ; y = o X'fi , y = $ These g i v e f o r S the f o l l o w i n g expression; s- -i—"4-—j- s 14} tan uh - uh T h i s equation shows p r o p o r t i o n a l i t y between S and & , hut the c o e f f i c i e n t of p r o p o r t i o n a l i t y depends on 1 and decreases g r e a t l y with increase of V« Mien uh- /ST/? increases t o , S be-cooes aero,, i . e . so force i s required t o d e f l e c t the s t r u t , and s t i l l f u r t h e r increase i n Tf makes S n e g a t i v e , which means that the s t r u t not only does not r e s i s t , d e f l e c t i o n , hut r e q u i r e s soeie support on the pert of the top chord and diagonal t o prevent i t s c o l l a p s e . C o l l a p s i n g load ¥ c r f o r l a t e r a l buckling corresponds to zero value of the denominator i n ecj. tan uh — oh - O ^ which g i v e s (uh)cr. = 4 - 4 9 . and Vcr - U tlv - (5) T h i s expression f o r c o l l a p s i n g d i r e c t s t r e s s i n 6 the v e r t i c a l of a pony t r a s s , when the top end of the v e r t i c a l i s permitted t o t u r n f r e e l y , must be considered as approximate only', because the col l a p s e i s accompanied by great angle changes, under which circumstances the b a s i c equation (1) ceases to apply* Returning t o f i g , E a , the t o t a l r e s i s t i n g force F = S-r* P, * e r f i P i s th© l a t e r a l h o r i z o n t a l component of the s t r e s s i n the d i a g o n a l . Since the v e r t i c a l component of the s t r e s s i n the diagonal i s V, and F- vs+ S = n Tan (ah) - uh tan (uh ) \/ g tan (uh) - uh h (7) f h l s equation show^s tha t F i s p r o p o r t i o n a l t o S and decreases with-Increase of V. For V*0 , F « 3 ft" 8. 7 / 3 ,/_ 9TTz £lv r- , r c Ely o-Por oh = ^7T or I/ - ~7g ^F" ; ^ = <-65 0 F o r uh » 7T o r V'-Z^h- (la) ? F= O. As ? increases ©DOT?© t i l l s value* F becomes negative, i . e . the diagonal i n combination with the v o r t i c a l cease t h e i r r e s i s t -ing a c t i o n on the chord and begin t o exert a d e f l e c t i n g or outward force on i t * I t it»st be mentioned at t h i s point,, that the a b i l i t y of the v e r t i c a l t o withstand a compressive s t r e s s up to the magnitude Tcr , given by the expression ( 5 ) , i s p r e d i -cated on the adequate support on the part of the top chord, otherwise, the collapse of the v e r t i c a l w i l l occur f o r the value of T aocaewhere between the expressions ( 7 a ) and ( 5 ) . Thus, should the end v e r t i c a l s be a b s o l u t e l y r i g i d , any of the intermediate ones can stand a compression up t o _ vWi£h ihe and Verticals not absolutely rigid j T c r = Z°' Z^z ; ©n the other hand / f a i l u r e w i l l i n e v i t a b l y r e s u l t , should a l l t h e v e r t i c a l s , i n c l u d i n g the end ones,be stressed t o the values a l i t t l e above t h e i r \/'_ respective V - j^. • T h i s brings up f o r c o n s i d e r a t i o n one of the possible ways,, i n which f a i l u r e of a pony t r u s s , due t o e l a s t i c i n s t a b i l i t y may occur. Ihen sotae of the v e r t i c a l s are stressed below t h e i r ? ' and others above i t , there i s a p o s s i b i l i t y of such f a i l u r e . 8 However, t h i s p o s s i b i l i t y i s purely t h e o r e t i c a l , 1.11 the p r a c t i c a l column formulas used i n design of t r u s s v e r t i c a l s are based on the value of c o l l a p s i n g s t r e s s not TTZ EI over 7-2— . a l l o w i n g a s u i t a b l e f a c t o r of h  7 s a f e t y , and since i t i s not the object of t h i s paper to d i s c u s s the adequacy of -compression formulas, the mode of e l a s t i c f a i l u r e due t o f a i l u r e of i n d i v i d u a l v e r t i c a l s , as o u t l i n e d above., i s dismissed as p r a c t i c a l l y i m p o s s i b l e . As i s evident from th© examples g i v e n , i t i s p o s s i b l e t o write g e n e r a l l y F = b, S y {8} where £>, depends s o l e l y on uh and can be calculated, from the eq« { ? ) * 2?h© diagram 1 of the f a c t o r b, i n terms of uh has been p l o t t e d t o f a c i l i t a t e the c a l c u l a t i o n s . fhough the type of the v e r t i c a l considered In t h i s chapter way be treated as a s p e c i a l ease of the v e r t i c a l with bending sioaent on the end, i t was thought a a v i s a b l e , to b r i n g t h i s esse up independently, i n order t o point out th© d i f f e r e n c e i n i n d i v i d u a l s t a b i l i t y of the two kinds of v e r t i c a l s and t o e x p l a i n on the more simple case the supporting influence of the top chord* In the f o l l o w i n g developments,, however., t h i s type w i l l be r e f e r r e d to only o c c a s i o n a l l y , aiid the s a i n d i s c u s s i o n w i l l be concerned e l t h the kind of the v e r t i c a l of the next chapter* LAT ;K/,L R.JJl.7i'i',JJL^ A PUXiY TVV5S V -iRTICAL, Piffll'kT ,3) A PLJTAUlS ? ^ , WITH B1IKBISQ EOffiiHIT AT TUB TOP SEP* The v e r t i c a l ; r e f e r r e d t o i n t h i s , s e c t i o n s a t -i s f i e s the co n d i t i o n s of f i g . 3 . The buckl i n g and t w i s t of the top chord determine the amounts of the angular and l i n e a r d e f l e c t i o n s at the top end c< and & ; ¥ i s Jcnown, and i t i s .required t o f i n d S and M0 f o r the given values of oC and & » The p o s i t i v e d i r e c t i o n of M, i n the f o l l o w i n g d i s c u s s i o n w i l l be taken as indicated on f i g . . 6. The d i f f e r e n t i a l equation of the e l a s t i c curve i s : which reduces t o where again The s o l u t i o n of (11) i s y = A, Cos ux + 3,<Sin ux + <f + + -y^~ x)-The constants of i n t e g r a t i o n Ay and B, , as.well as the untaiowns U0 and 3, arc found from the f o l l o w i n g four (11) (12) 10 c o n d i t i o n s : dx ~ i which determine the unknowns; uh 2tan _ ok Zian  u-h _ oh ' M tan 4 V S l i + Z M ' c H ) ^ °~ 2tan f - uh 2tanf- uh { w ) Adding to S the. l a t e r a l h o r i z o n t a l component i * of the s t r e s s i n the d i a g o n a l , the t o t a l r e s i s t i n g force F i s found as ' explained hef©res F = P+o - -T- o + -7TZ—UP, T  0 ~ T+—u7t 7 °C = /j / Tan - uh  2 tan - uh 2  v" 2 2 tan # 1/ r tan^V 2tanf -ah k 2tan u-h ~uh fhe lowest value of V making e i t h e r M0 or S i n f i n i t e . , corresponds t o = 7T t which gives 11 v ' = 4TT z EIV _ EIV ( I f ) , ©a expression almost twice greater than tii® expression (5) f o r a v e r t i c a l f r e e t o rotate at the t o p . I t m&j be repeated here* that although the I n d i v i d u a l v e r t i c a l s can stand a s t r e s s up t o l'Cr according t o eq. ( l ? ) , the ot h e r s , stressed considerably lower, must come t o t h e i r a i d . To a n t i c i p a t e danger t o s t a b i l i t y of the t r u s s from t h i s source i s again quit© unnecessary. H«turning t o equations l i b ) and ( l b ) , the f o l l o w i n g formulas are obtained a f t e r s u b s t i t u t i o n f o r ¥ of i t s expression i n terms of ( t/A ): F- 2 tan °2 f,L\2 Ely? tan (if EI„ 2 where t>2 ~ ^ ^ _ ^ a l s o .M0= +b2^d , U l ) . , uh L _ 1+ T where °3 - o ten Vh - u h (°h'-  U'' 12 The . c o e f f i c i e n t s bz ant b3 are p l o t t e d on th© diagram S* The equations (10) and ( g l ) solve the pr e l i m i n a r y problem of the f o r c e , r e s i s t i n g the buckling,, and of the aoaent producing t o r s i o n of the top chord.* I t may be n o t i c e d , that the force has been computed us caused by the eomblnea a c t i o s of bonding i n the v e r t i c a l anfi d i r e c t s t r e s s i n the diagonal} while the .moment—as caused only by bending i n the v e r t i c a l * A c t u a l l y , i t . i s not quite t r u e : the diagonal has some bending r e s i s t a n c e , and the fa«.t # that th© l a t e r a l h o r i z o n t a l component of i t s s t r e s s , P ( f i g * Ua)t Is applied w i t h i n the depth of the 'gusset p i s t e , that i s below the center of g r a v i t y .of the chord s e c t i o n , r e s u l t s i n some a d d i t i o n a l t w i s t i n g moment, augmenting that produced by the v e r t i c a l * these influences have been disregarded i n the a n a l y s i s as of minor Importance^ and the e f f e c t of ignoring them tends to aafce the r e s u l t s on the safe side* Since the f a c t o r s b, bz and b3 i n equations {B):$! 118) and (£1) depend on f ant through that on the loa d i n g of the b r i d g e , they are not known st the beginning, and the " t r i a l and e r r o r " metiiod must be resorted t o . Some expected value of ¥ must be assumed, that w i l l determine u and the b coefficients... Ihen the c r i t i c a l load I s found, th© stresses f are c a l c u l a t e d , ehe bleed against the assumed and, i f neeessary, the procedure should be repeated with new values of the c o e f f i c i e n t s . To f a c i l i t a t e the mathematical treetment of the 13 subject the concentrated a c t i o n of the v e r t i c a l s and diagon-als,, on the top chord at the panel points w i l l be replaced w i t h continuous a c t i o n a l l along the chord of th® i n t e n s i t y per- trait l e n g t h of th® chord equal to the force at the panel point d i v i d e d by the l e n g t h of the panel... The?*, f o r t r u s s w i t h v e r t i c a l s free .to r o t a t e at the top ends* ~d = °'JM  u - \ (23) and f o r truss with, v e r t i c a l s , having moments at th© top ends aad-In these formula.® d i s the panel l e n g t h o f the t r a s s , and f and a a r e , of course, the i n t e n s i t i e s of force and so-, s e n t , -acting on the chord per u n i t length o f I t . I t i s need-l e s s to- say,». that the .positive d i r e c t i o n of f and a should be regarded, as opposite to. what i s Indicated by arrows 14 f o r S and M<, on f i g u r e s 5 and 6» since i t i s the a c t i o n of web members on truss.,, that i s considered now* In f u r t h e r development the c o e f f i c i e n t s k, , kz ant. /c3 w i l l be considered as constant along the bridge; a c t u a l l y they vary f o r d i f f e r e n t v e r t i c a l s , but,,, o r d i n a r i l y , not g r e a t l y , so that t h e i r laean value should give s a t i s f a c -t o r y r e s u l t s * JOKSIOB&L 'PBFQBMAf 101 Of TUB mOSJim '10? CBOBD. fhe d e f i n i t e r e l a t i o n i n which the l i n e a r and angular deforaations o f the top chord & and d ( f i g . 6) stand t o each o t h e r , w i l l be found now. Figures 7 and fa represent a small length dx of the top chord with the corresponding continuous w a l l of v e r t i c a l s * d i r e c t i n g the a t t e n t i o n t o the t o r s i o n of t h i s elemental l e n g t h , the equation of e q u i l i b r i u m w i l l be T + m dx = T+dT niiere m i s the aoiaent at the end per u n i t length of the w a l l of v e r t i c a l s . , and 2 i s the t w i s t i n g torque at the point of the top chord with an abscissa along the length of the t r u s s x . This g i v e s the torque T produces on the length dx a change i n the angle of t w i s t dot . I f the constant t o r s i o n a l r i g i d i t y of the top chord i s c a l l e d C, then doi = ~^ ^ 15 or . T = C | f (so) S u b s t i t u t i n g Into (£9) f o r m s a t $ t h e i r expressions from (25) and ( 3 0 ) , the f o l l o w i n g d i f f e r e n t i a l equation r e s u l t s : - ^ S +*3 0C = C ^ (31) Shea a s u i t a b l e expression of S i n terms of x i s decided on, oC i s determined without d i f f i c u l t y , from t h i s equation* fhen f , m and T are e a s i l y found i n terms of x fro® (E4),t (£S) and ( 3 0 ) . l a t e r a l l y , when the v e r t i c a l s are connected t o the top chord according to f i g * 4 , there i s no moment at the top end of the v e r t i c a l s ana no t o r s i o n i n the top chord* Consequently, a l l that i s required to Jcnow In that case, i s the force f , which i s gives by the equation (23). With preliminary work thus completed the main problem of the e l a s t i c s t a b i l i t y of the pony t r u s s w i l l be a t t a c k e d . 16 $m. BSERGY mssmm AS. APPLUSD £0 THE QUEST I Oil QF s m s f i c STABILITY OF tee POHY TRUSS. In the process of b u c k l i n g , the ends of the top chord ar® not permitted by^bhev^Bdvert i c a l s to have e i t h e r l i n e a r or angular deformation l a t e r a l l y , b u t , of course, the l o n g i t u d i o n a l displacements are t a k i n g p l a c e . The key t o the s o l u t i o n i s the "Energy Method"*, e x t e n s i v e l y used by Professor Tiiaoshenko* and i t i s f e l t , that i t s b r i e f explanation i n r e l a t i o n t o the present #uestio» w i l l be not out of place h e r e . Suppose,, that the load i s placed on the b r i d g e , and the t r u s s ©sabers become stressed with the primary d i r e c t s t r e s s e s and undergo c e r t a i n deformations* as a r e s u l t of •v«hleh a d e f i n i t e amount of e l a s t i c energy i s stored i n the deformed s t r u c t u r e * Considering no® the p o s s i b i l i t y of col l a p s e due t o e l a s t i c i n s t a b i l i t y * i m g i n e th© top chord buckled by a small amount, i t s a x i s assuming c e r t a i n appropriate curved shape .in h o r i z o n t a l plane*. T h i s b u c k l i n g aut©matleslly b r i n g s i n t o play the re s i s t a n c e s f and m on the part of the v e r t i c a l s and d i a g o n a l s , and the moments m cause some t o r s i on. of the chord* the primary stresses i n the truss, members* and the corresponding o r i g i n a l amount of the e l a s t i c energy of the structure before b u c k l i n g , are not aff e c t e d by t h i s process, and thus the agency, producing the 1? on-ekllng, i s c a l l e d upon to supply a l l the a d d i t i o n a l e l a s t i c energy W brought about by buckling., which conies under the f o l l o w i n g f o u r it©as: W5 —'Energy required f o r the bu c k l i n g of the top chord proper. Wr —Energy required f o r the t o r s i o n of the top chord. Wf and Vm-—Energy required t o overcome ^ and m, the web r e s i s t a n c e s t o bu c k l i n g o f the top chord. Thus. W = W6 + WT + Wf + Wtn • (S2) As the b u c k l i n g deformation takes p l a c e , the two ends of the top chord com© c l o s e r together., and the h o r i z o n t a l l o n g i t u d i n a l components of the stresses i n the diagonals,, c a l l e d on f i g . Za t do some p o s i t i v e work Vf.9 as aay be seen from fi g . - 8.. fhe e l a s t i c energy IV depends on the amount of buc k l i n g and on the siz.es and shapes of the cross sections o f the t r u s s members; a a t o the load, on the truss,, i t a f f e c t s W only as far., &s the c o e f f i c i e n t s , fc* and x3 i n the equations {.24} and {2-5} are a f f e c t e d ; and these c o e f f i c i e n t s , as w e l l as the corresponding amount of energy, decrease with an increase i n l o s d . The work V, on the other hand,, i s d i r e c t l y p r o p o r t i o n a l to the load on the bridge,, and a l s o depends on the amount of b u c k l i n g , but not on the s i z e s of the t r u s s members. I t i s ev i d e n t , that as long as W > U, the assumed buckling cannot be brought about by the load alone }without the a i d of soua outside agency. As the load on the bridge i n c r e a s e s , U increases and f o r c e r t a i n 18 value of the load W = U •> or WB + WT + Wf +Wm - U . {S3) The losd of t h i s laagnltude i s quite s u f f i c i e n t alone to eying about the buck! l u g , - t h i s load i s the required c r i t i c a l l o a d* The curve of b u c k l i n g i s a sine l i k e curve, whose number of waves i s determined by the r e l a t i v e s t i f f n e s s e s of the chord and the v e r t i c a l s , when s t i f f n e s s of the v e r t i c a l s i s email compared t o that of the chord, the curve i s one wave curve* but as the s t i f f n e s s of the v e r t i c a l s i n c reases, the number of -waves w i l l increase to two, three or even more (see f i g . 9 ) . The curve of any shape can be represented as a harmonic s e r i e s of sine curves, i t i s s u f f i c i e n t l y accurate to t h i n k of the curve of b u c k l i n g as the sum of two sine curves, the primary one, roughly o u t l l n g l n g the shape of the curve, and the secondary one, whose a d d i t i o n modifies ts e shape of th© primary* b r i n g i n g i t i n t o a greater conformity with the a c t u a l buckling curve* itien the load on the bridge i s symmetrical, both curves crust be e i t h e r with odd or even number of ?iaves» Thus, f i g . 10a represents the sua of one save primary and three wave secondary sine curves, and f i g * 10 k -the sum of two » v e primary ana four wove secondsry* A combination of even and odd sine curves, l i k e the one on f i g * 10 c .# would be considered impossible f o r symmetrical loading of the t r u s s , as amking the two halves of 19 the r e s u l t a n t curve d i s s i m i l a r * l a tiie f o l l o w i n g d i s c u s s i o n the equation of the bu c k l i n g curve ( f i g . 10a aad 10 i? } w i l l be assumed* S = apS,n > anSw ^Lx f (34) where the number of waves i n the primary p end In the second-ary n are two consecutive odd or even numbers* With the shape of the buck l i n g curve decided on., the various tonus i n the equations (2£) and (o3) w i l l be determined* SIPaSSSIQJf FOR HisJ a'QRK POffft BY ilgTSKKAL FORCES To f i n d the work U done by the buckling forces ( f i g . 8 ) , the d i r e c t method would be to express these forces i n terms of the load and to m u l t i p l y them by the decrease of respective distances SS, , GQ( and AA, ,. caused by buc k l i n g ; however, there i s another method, believed t o ce more I n s t r u c t i v e , which w i l l be followed here* Thinking of the top chord as separate frora the rest of the s t r u c t u r e , i t i s the forces P, , that do the work U, but considering the whole, truss,- the forces P, , now interns 1 f o r c e s between the chord and the diagonals, do no work, and a l l the work n a t u r a l l y comes from the load on the bridge* As the top chord buckles by an i n f i n i t e s i m a l amount i n h o r i z o n t a l plane, the bottom chord d e f l e c t s i n the v e r t i c a l p lane, the ordinate s' Of. d e f l e c t i o n being i n f i n l -20 tesimals of the second order,, and the load applied to the Bottom chord does on i t s lowering the same amount of work U. I t say he necessary t o mention, that the d e f l e c t i o n j u s t r e f e r r e d t o , i s the one due e x c l u s i v e l y t o the b u c k l i n g of the top chord, and has nothing t o do with the d e f l e c t i o n caused by the d i r e c t stresses i n the t r u s s , nor with the a d d i t i o n a l d e f l e c t i o n caused by bending of the v e r t i c a l s . By the way.of explanation of the nature of the a d d i t i o n a l d e f l e c t i o n r e f e r r e d to i n the previous paragraph, i t may be s a i d , that as the v e r t i c a l bends,, i t s top end lowers p u l l i n g down the diagonal member connected t o the top of i t . Since the diagonal i s assurued to remain s t r a i g h t , i t s bottom end a l s o goes down, lowering with i t the next v e r t i c a l , f h i s causes an a d d i t i o n a l d e f l e c t i o n , over and above that produced by b u c k l i n g of the top chord proper, and, consequently, an a d d i t i o n a l work done by the load on the b r i d g e . However, i n the method of approach adopted i n t h i s paper, the influence of t h i s f a c t o r i s to be taken i n t o account by reduction i n the resistance of v e r t i c a l s , caused by d i r e c t s t r e s s e s ; hence, the a d d i t i o n a l d e f l e c t i o n and i t s work should not be considered h e r e . Considering the influence of buckling of the top Chord alone, as t h i s takes p l a c e , the plane of the t r u s s deforms i n t o curved surface of such geometrical nature, that the magnitude of any angle i n the plane of the t r u s s i s preserved, and a l l the v e r t i c a l s 37e.uain perpendicular to the bottom end top chords, as they were before the buckling E l ( d e f l e c t i o n due t o the d i r e c t stresses need not he considered h e r e ) . The d i f f e r e n t i a l equation of the d e f l e c t e d bottom > chord i s . h d'x . (S5} , as may be seen fro© f i g * 11a and 116 , representing d i a g r a m a t l e a l l y the plan and the e l e v a t i o n o f a deformed t r u s s * Her© A i s th© l o n g i t u d i n a l displacement toward the center of the t r u s s of any point A of the top chord* due to bu c k l i n g * A = / Cos if) 7  d x = 2 (36} from (24): = f(paFGsJfx+.nanCosaf. T h i s i s s u b s t i t u t e d i n t o (S€>)* and i n the i n t e g r a t i o n * the f o l l o w i n g formulas are made use of* remembering that ( p+n ) and ( p ~n ) are both even* ri Cos* J*Exdx = I - f 4P1T t X and J I c 7T P+n p+n n n-p. n-p Thsm T . pap + na»{4 Z 4nTT -  s,n -rv — T — [ r T T + - T p - y (37) 22 f f a i s expression f o r A i s su b s t i t u t e d i n t o (&5), and on i n t e g r a t i n g of the re s u l t a n t equation end s u b s t i t u t i n g the i n i t i a l e onditions %= o, when x= © and x = l , the f o l l o w i n g expression f o r the downward d e f l e c t i o n of the bottom chord, due to the buc k l i n g of the top chord, re s u i t s i 2 X + -tan O/f! —f~-x — Sm z 2F * , Sin 2 2<g/D""x (P,n)2 t (n-p)* (38) The work H don© by the load applied at the bottom chord can now be e a s i l y found* Hh.es uniform load <^  covers the whole span Jo When the load i s a s e r i e s of concentrated weights % >the work Uc = Z ( < ? j J . (38b) where the ordinate* z are taken under the weights Let Q = /V Qt ; (38c) where H i s a number d e f i n i n g the unknown c r i t i c a l i n t e n s i t y of a set of weights, having a d e f i n i t e r a t i o among themselves, l i k e , f o r example, a Cooper's loading; and — are the known values of -weigrxts corresponding to some a r b i t r a r y u n i t i n t e n s i t y of the s e t , say, Cooper's E* 10. Then the equation (28b) takes the form: f o l l o w i n g re s u i t s l&ea t l i * i n t e g r a t i o n in. (S8a) i s performed, the expression f o r the work tone by the uniform load (39) Iteteraining the ordinate J at the center of the bridge from t&@), the expressions f o r the work U, , done by a s i n g l e concentrated load Q, at the center,, a r e obtained from fihen both' p and /? are odd : IT 2'Z Iben both p and /7 are even: 8 + Z (39b) Witn. C known,, the terms on the l e f t hand side of the e f n a t i o n w i l l now be determined. g£aggIC BKffRSYOF PSPOHMED STRUCTlIRg* C a l l i n g the constant moment of i n e r t i a of the top chert tvt bending i n the h o r i z o n t a l plane l c *• the e l a s t i c energy ©f bending (buckling) of the top chord WB i s found from a 2 j I 2 2 ' d^\ clx . * (40) 2 , 2 1 " 2 2 2 2 a d P 7T c PIT n IT o - r?7T f i l l s i s s u b s t i t u t e d i n t o ( 4 0 ) . sines r l z . \ Sin ^-j-x - Y and o a f t e r s i m p l i f i c a t i o n ™B = + °"  n J • l a expressing the e l a s t i c energy of t w i s t of the top chord WT „ combine i t with Wm m th.© energy required to overcome the resistance of bending moments on the ends of the i d e a l i s e d v e r t i c a l s * rf 2 e WT + Wfn = f j^dx + j I mctdx . (42) £5 From (29) and (m) and t h i s gives m = dT dx to 2 oC cLxLt (4S) As the end v e r t i c a l s are a b s o l u t e l y r i g i d , cX = 0 f o r both - x = l and x = o, and co n s e q u e n t l y , the expression i n . the square brackets and the sum fWT + Wm ) are both equal t o %»t&# l a understanding t h i s Important r e s u l t one must r e a l i s e , that as the buckling of the top chord progresses, and the angle ck increases i t s p o s i t i v e v a l u e , the moment m also increases n u m e r i c a l l y , remaining negative,, i . e . i n the d i r e c t i o n opposite t o <X , and, consequently, the work of over-coming m i s negative* I t i s i n t e r e s t i n g . t o note, that i t i s f o r the whole top chord and not f o r any small p o r t i o n of i t , t hat the work of t o r s i o n and the work of overcoming the iBoment m balance completely, f h u s , f o r the eleaexit dx at the center,., f = 0,a»d the elementary energy of tw i s t d(WT ) = 0, while the negative elementary work of bending d (Wm) i s the greatest here; on the other hand,the p o s i t i v e d (W T ) at the end i s the greatest due to the greatest f , but d ( l m ) 26 I s here zero* ; Though th© sua (wT+wm) i s z e r e , i t would fee wrong t o t h i n k , t h a t the t o r s i o n a l resistance of th© top chord i s immaterial t o the s t a b i l i t y of the s t r u c t u r e * The f a c t i s , that the magnitudes f of the r e s i s t i n g f o r c e s on th© ends of th© v e r t i c a l s are considerably affected by the t o r s i o n a l r i g i d i t y of the top chord* The f o l l o w i n g i s th© expression f o r Wf r the work spent i n overcoming the r e s i s t i n g f o r c e s f : J f$ d* • (44) The equation (24) gives f i n terms of <T and oC and. c< must be determined from the d i f f e r e n t i a l equation 101)* S u b s t i t u t i o n i n t o ( g l ) of the expression f o r *b from (34) b r i n g s t h i s equation i n t o forms The general s o l u t i o n of t h i s equation i s (X = AZ Cosh (N/f x) +BZ Smh ( v ^ x j + (46) c o n d i t i o n s at- the ends ©.re. sueto., that when x= © or x = 1, o<, = ©I- t h i s makes the constants o f i n t e g r a t i o n Introduce new symbol yvi , so tha t _ 02 tZ Ely 4TTZC 4TT z hd C then . pz CTT2 62 • ( 4 8 ) , as f o l l o w s from t E f ) and (ES)* S u b s t i t u t i n g these Into {46}» the f o l l o w i n g expression f o r oC i s obtained .than from (£4) f-**a> ('-p^)^* + ^-(l-^szY"^- (50) and Wf = ^ ifSdx = ^ ' V P ^ | W 4 I n\4^)- tm The corresponding expression f o r v/ ^  when no t o r s i o n of the top chord i s possible,, i s found fro® {ZZl* 4 ^  p J " (51a) 28 I n the f o l l o w i n g d i s c u s s i o n , however, only the expression (51) w i l l be used, because (51a) i s a s p e c i a l case of i t corres-> ponding to the value of t o r s i o n a l r i g i d i t y of the top chord 0 = 0 and ju. = oo CRITICAL. .VJOtW OF Ti:|S OKI FORM LOAD QOTBRXlia. THE IgOLl SPAM. Both'sides of the equation (33) being known, the en-ergy equation f o r the uniform load on the bridge 1st 7T4fJ li h ! 4 + aid- * (52) L e t °-n  =-)) af (53), where y i s an unknown. S u b s t i t u t i n g t h i s expression f o r y i n t o (52), c a n c e l l i n g dp and making the necessary transformations, the f o l l o w i n g expres-s i o n result® f o r the compressive s t r e s s a t the center of-the top chord corresponding to the c r i t i c a l u n l f o r a load C£ en the- bridge? 8h This can be represented i n a for a s i m i l a r to the Euler s (53) formula, 8h " 0 f z (54) 29 © a i l i n g - 4 r r z £ I c - 4^z -fte~T: C55):, %$m f o l l o w i n g Is the expression f o r ^ from (5.3} If /-' t (Tr z* ,)u z Q , Pn(p^n z) (56) th® f i r s t term of the numerator i n t h i s expression «.©aes from th© handing r e s i s t a n c e of the top chord proper; the second*-from the web re s i s t a n c e to bending and th® t o r -s i o n a l r e s i s t a n c e of th® chord, and i t contains the c o e f f i c -i e n t s IJ and , and the r a t i o » As, may be seen from (53) and {¥{}., tj i s a f a c t o r i n v o l v i n g the r a t i o of tim f l e j t b u r a l r i g i d i t i e s of the v e r t i c a l s and the top chord, and yU. —a f a c t o r depending on the r a t i o of f l e x t u r a l r i g -i d i t y o f v e r t i c a l s to the t o r s i o n a l r i g i d i t y of th® chord'; both* s.s f a r as they depend on 1SZ » deporsd on the loading of the v e r t i c a l s * the unknown y (the r a t i o of the amplitude® of the secondary and the p r i m a r y sine waves of the budded top chord) i s found so as to «afee y a ulniwum... This involves s o l u t -ion o f a quadratic equation, whose general form i s complicated, out each case taken i n d i v i d u a l l y presents no d i f f i c u l t y . The f i r s t approximation of ^ t s u f f i c i e n t l y accurate f o r th® pr e l i i a l n a r y estimation of the c r i t i c a l l o a d , corresponds to y = o, l» e , when the secondary sine wave Is neglected. 30 fben V / (57) .' Th© number of waves p In th® primary curve i s un-known at the s t a r t , and i s determined by t r i a l s fron (57) so that the corresponding ^ i s the smallest; a f t e r t h a t , the second approximation of (f I s found from (56) f o r the same p. To f a c i l i t a t e design , diagrams 3 and 4 g i v i n g the f i r s t approximations of )f i n terns of yiA and ^ have been p l o t t e d f o r = 1.333 (the maximum pos s i b l e value) and As may be seen f r o a the diagrams, the ft -curve f o r every yn. c o n s i s t s of a number of s t r a i g h t p a r t s . The f i r s t from the l e f t part corresponds to a s i n g l e wave of b u t t l i n g , the second—to the doable wave e t c . Of course, the boundary between two types of the buckling curve, as deter-mined by a corner point on }f ~1. curve, i s only approx-imate, i n s o f a r as the curves of the diagraa give only the f i r s t approximation of ^ , hence, when c a l c u l a t i n g the second approximation f o r a combination of ^ . and JU. near a corner point on the diagram, the p o s s i b i l i t y of e i t h e r type of the buck l i n g curve must b© i n v e s t i g a t e d . It may be mentioned here, that i n a d d i t i o n to the primary and the secondary a t e r t i a r y wave could have been introduced i n t o th© curve of buckling by some simple m o d i f i -c a t i o n of the equation ( 5 6 ) , r e s u l t i n g i n a seemingly more 31 accurate expression f o r ft „ However, the accuracy would nave been Imaginary. As the number of waves Increases, the replacement of the a c t u a l truss v e r t i c a l s w i t h an equivalent continuous w a l l becomes generally i n c o r r e c t . In an extreme case, when the number of waves i n the primary, secondary or te r t i a r y , - as the case say be, equals the number of panels Cfig.,12), the wort: of oTorcoalag the corresponding r e s i s t a n c e of the v e r t i c a l s Wf i s a c t u a l l y zero, since the v e r t i c a l s c o i n c i d e with the nodal points,- As a r e s u l t , the part of the second term i n the numerator i n (5-5) and (57)* c o r r e s -ponding to t h i s wave, must be wiped out—-a circumstance d i s -couraging the refinement of i n t r o d u c t i o n of higher harmonies as too t h e o r e t i c a l . On the other hand, as lon g as the number of panels i s an exact m u l t i p l e (but not equal) o f the number of waves, the replacement referred t o , holds ex a c t l y t r u e . Judging frora few examples solved, i t i s expected, that the t h e o r e t i c a l e r r o r Introduced fey negle c t i n g the t e r t i a r y la never greater than 2%. COHCBMTRATSD LOAD AT fliS CSM'BR t u r n i n g to the question of a s i n g l e concentrated load at the c e n t e r , the energy equation (33) i s again the b a s i s of the analysis., but while the l e f t hand side of i t , the e l a s t i c energy of the deformed s t r u c t u r e , has exactly the sasae expression as f o r the unifona l o a d , the r i g h t hand si d e , the wort done by the l o a d , must be represented by the equations (39a) and (39b). Instead of '(39)* Hetr&eing the steps taken i n connection w i t h determination of ^  for the u n i -form load* th© r e s u l t a n t equations f o r a s i n g l e concentrated load Q at the center, corresponding to the equations (54), (56) and (5?) are: Qt 7TZ E Ic 8h For p and n odd (54a) 1-For p and n oven (56a) 4 The f i r s t approximations of £ correspond as .follows}-y (56b) ,ng. to j aero' are for p oddt TT Z _ 4 I-f o r p even; (5-V-/) 7 7T (57») (57b) have been These f i r s t approximations o f the c o e f f i c i e n t Jf, p l o t t e d on the diagram 5 tor d i f f e r e n t yU and , and f o r the value of = 1,333 ' The previous d i s c u s s i o n < o f the s i g n i f i c a n c e of d i f -f e r e n t terms i n the expression f o r ^ and of the shape of the b u c k l i n g curve holds equally true i n t h i s case. 33 RECIPROCAL iagLiJgKCS LlSE OF tm CRITICAL LOAD. If the p o s i t i o n of a singles concentrated load on the span varies., the work U done i n lowering of the load on account of b u c k l i n g of the top chord., a l s o v a r i e s In propor-t i o n to the ordinate % of the d e f l e c t i o n curve of the bo:.torn e&ord. A& to the shape of the b u c k l i n g curve and the e l a s -t i c energy of the deformed etruetare W? they change only s l i g h t l y , as w i l l be seen from the numerical example, and, as an approximation, aay b® considered constant... This c i r -cumstance i s very important, and i t euggests an i n t e r e s t i n g use. for the curve (385*. as a curve, whose ordinate© are ap-proximately i n v e r s e l y proportional to the load coneentrations producing c o l l a p s i n g e f f e c t on the bridge, i f placed at the points o f t b # l r a b s c i s s a e . The p r i n c i p l e , i f developed a l i t t l e f u r t h e r leads to the idea of a curve, which say be a p p r o p r i a t e l y termed wth© r e c i p r o c a l influence l i n e of the C o l l a p s i n g load:,** Oeing the n o t a t i o n of the e a r l i e r part of the paper, •let the top chord buckle t o c e r t a i n s u i t a b l e shape, f i r s t , un-der the a c t i o n o f uniform l o a d of c r i t i c a l i n t e n s i t y cj over the whole span, and secondly,, under a set of concentra-ted load© of c r i t i c a l - i n t e n s i t y H,. i n some d e f i n i t e p o s i t i o n on the bridge. ?hen the energy equation (33), combined with tne expressions for the work done by the loads (38d) and (39), gives the f o l l o w i n g r e l a t i o n s * 34 W Si. 8h and N -(58) Dividing one by the other whore the v a r i a b l e £ = S i and substituting z from (38) (59) (60) i -y (61) The £ ourve i s quite easy to v i s u a l i s e due to I t s p h y s i c a l meanings Since I t s main part,; th© f i r s t term i n the numerator, i s parabola,, the curve i s so-ewhat parabolic in form, w i t h maximum ordinate u s u a l l y a t the center* The r a t i o of th® amplitudes of the secondary and the primary sine waves, designated by l e t t e r y, mast be considered as constant f o r the whole span and having the value corresponding to the u n i -form, . c r i t i c a l load on the bridge* fa® shape of £ curve and the magnitudes of I t s ordinate® are thus quit® determined by two f a c t o r s y and p^ of which the. second i s of major importance (n i s test BOX* a f t e r p s w s e c u t l v e number of the same ki n d , I.e. odd I f p i s odd# and even when p is'even)* With £ curve constructed, and th© c r i t i c a l value of 35 t h * t r a l f o m l o a d {c}f } p r e v i o u s l y deter®iaed, th© © r l t i e a l in-tensity I of 'any group of concentrated weighta can be e a s i l y found froa the equation {60) . The c r i t i c a l value of a s i n g l e concentrated l o a d q i s . found from s Q - — (62) which i s a s p e c i a l ease of {60},. The equation (60) can a l s o be eas-i l y extended t o a combination of the uniform and the concen-t r a t e d load®.. Being an Influence l i n e , the curve £ provides also the means of f i n d i n g th© most unfavourable p o s i t i o n of the moving load,, f o r which M becomes a alnio.ua. However, i t must be remembered,, that these simple r e l a t i o n s based on the p r i n c i p l e of s u p e r p o s i t i o n , hold only approximately t r u e , s i n c e the bending r e s i s t a n c e of the v e r t i c a l s (coefficients b 2 and b 5 ) and the r a t i o y depend somewhat on the- p o s i t i o n and amount of the l o a d i n g . The r e l a -t i o n s are more nearly t r u e , when the concentrated weights, causing c o l l a p s e o f the t r u s s # a r e present i n pairs symmetri-cal, about the center o f the bridge, otherwise, the assumption, that the two sin® waves composing the b u c k l i n g curvt are both e i t h e r odd or even, w i l l be i n c o r r e c t , and the e r r o r In using th© r e c i p r o c a l Influence l i n e of the c r i t i c a l l o a d , w i l l be g r e a t e r . I n order to i l l u s t r a t e how- the developed- formulas are used i n Checking design o f a pony t r u s s f o r e l a s t i c s t a -b i l i t y , an example w i l l be given her®. Let f i g u r e 13 represent the pony t r u s s required to be checked.. The end v e r t i c a l ® are supposed to be a b s o l u t e l y 36 rigid* and the constant ©actions of the top chord and i n t e r -mediate v e r t i c a l s are given las f i g , 14 and 15. Sections o f the bottom chord and the diagonals are immaterial. Such m t r u s s i s good f o r the t o t a l equivalent uniform load of about 1 K ,f / f t of one t r u s s , which corresponds to. an ordinary highway bridge with wooden deck and on® lane of t r a f f i c , fhe t o r s i o n a l r i g i d i t y of the top chord C has nothing t o do with the polar' moment o f i n e r t i a o f the s e c t i o n , and Since the sethod by which. I t i s determined I s not a matter of common knowledge, i t w i l l be taken here i n f u l l d e t a i l . I t i s based on the s o - c a l l e d hfdTOdynamie analogy, according to which the problem- of t o r s i o n of a sh a f t i s reduced to the a a t n t m a t i c a l l y equivalent problem- of steady c i r c u l a t i n g motion of f r i c t l o n l e s a - f l u i d In a ve s s e l In the shape of the s h a f t , fki* T e l o c i t y of f l u i d at any point i s proportional and i n the. same d i r e c t i o n &§ the t o r s i o n a l s t r e s s , t h i s motion of f l u i d I n side the abaft i s easy to v i s u a l i s e * $«-f«rring t o f i g . 16, •representing * somewhat simplified s e c t i o n of the member i n question,- under the a c t i o n of a torque f ; t h © v e l o c i t i e s (or streesea) In d i f f e r e n t parts o f the s e c t i o n w i l l be p a r a l l e l t o the, walls as shown by arrows and inversely p r o p o r t i o n a l to the respective thicknesses t . Thus* f l u i d i n the web of the cnannel on reaching Vm flange w i l l swerve p a r a l l e l to i t and proceed to the gauge l i n e of r i v e t s , ' through, which i t w i l l enter the cover p l a t e and aove along i t i n the d i r e c t i o n opposite to that i n the f l a n g e . The stresses T } T, and T2 w i l l bear the- r e l a t i o n : . zt = <c,% = <L2t2 m> 37 The por t i o n s of the flange and ©over p l a t e outside of the gauge l i n e s of r i v e t s , shown dotted, do not co n t r i b u t e t o the t o r s i o n a l r e s i s t a n c e o f the section.* the l a t t i c i n g at the bottom flange introduces no s p e c i a l complications. I t s duty i s s i m i l a r to that of the cover plat©, namely, to bind the flanges of the two channels together, and i n performing i t eaeti l a t t i c e bar develops a force L, whose l a t e r a l component Is equal to the t o t a l shearing force i n th© cover p l a t e ; L Sin 60" = T21 ez (69) fbe same a l s o f o l l o w s from the e q u i l i b r i u m of forces In h o r i -z o n t a l d i r e c t i o n . R e f e r r i n g to f i g , 1? the equation of moments is? 7" =. T* 0.303* 7-53*7.053 + T, x 0-4Jx/.224x2*7. 53 + -h r 2x o.3/2 x 9-5 x 8-3/2 . Expressing T, and Tz In terms o f T from (53), the f o l l o w i n g expressions result* T \ r -45-62 T 70-8 T 47-1 > (71) and from (5$) L = 3.346 T = T J3-8 J then a torque T a c t s on the member, the- angle of twist per u n i t length of th© member Is - J - , and the'work 38 dene by til® torque, when-It gradually Increases In value froa rjO. aero Is —— . The I n t e r n a l work of• deformation of a u n i t 2C , voluwe o f - e l a s t i c laaterial i n shear i s , where <q i s the modulus of shear, and the i n t e r n a l work of a u n i t length of / i l a t t i c e bar l a d i r e c t s t r e s s L i s . Heaeaberlng, that Z EA the l e n g t h of the l a t t i c e bar, corresponding to the u n i t length o f the aesber i s • ' „ = ? * ^® energy equation w i l l Los 60 take the form: + J | 9.5*0.5,2 + YTTih^rZ • t?2) Expressing the stresses In terms of T f r o a (71), c a n c e l l i n g T and using the r e l a t i o n —z~ = 2-5 •» the t o r s i o n a l r i g i d i t y o f th© s e c t i o n comes out C = 100-4 <J . (73) I t may be mentioned In passing, that the greatest p o r t i o n of th© e l a s t i c energy of t o r s i o n of the aestber Is st o r e d i n i t s weakest p a r t , the l a t t i c e , and should the l a t t e r he replaced w i t h a cover p l a t e , the amount of i n t e r n a l energy w i l l decrease g r e a t l y , and, consequently, C w i l l r i s e i n proportion*. The t o r s i o n a l r i g i d i t y C being known, the eoeffle-' •tMMJi-maA *2 %M the equations (5*5) and (57) can now be de-termined frow (47) and (55) , but before that the c o e f f i c i e n t s b 2 and b 3 , c h a r a c t e r i z i n g the resis t a n c e of the v e r t i c a l s 39 must be found. As f i r s t approximation their maximum values w i l l b© taken (see diagram 2 ) . = 4- and 4-£ . then •> 4F2- Aoi C 39-5- 14* iz>5 100.4 1 " 7= J * - -C - i = /25* 4/ _ . . . t 4rr z I C 39-5 /43x/2-5 172.7 ~ Find f i r s t approximation of ^ from (57)., making the number of waves p = l » 2, 3 and 4 • f o r P = 1 = 12335 P = 2 P= 3 = 28,33 and P = 4 ^ = 33.96. fnese f i g u r e s s be a t r i p l e wave, and the approximate ^- 28*23, which value can a l s o be read o f f the diagram of ^ « For the second approximation s u b s t i t u t e i n t o (55) p= 3 and n = 5. The resulting expression i s v-_ 1823 y z + 404 "~ 40.6 y 2 - I5.93y t 14-S (74) The value of y that makes t h i s expression .a minimum, is y = — 0,342 (75), and the second approximation of ^ from (74) is ^ = 24.9 (75),about 12% less than the f i r s t one. Knowing ^ , the value of the compression s t r e s s In the cen-tral panel of the top chord, corresponding to the c o l l a p s i n g load on the bridge Is found from (54) : %0 i £ . = 24.9 J ^ ^ J Z ^ 8k 0 [ z (125* IZ) 2 Kip. in) and the corresponding i n t e n s i t y of the c r i t i c a l uniform load This f i g u r e has been obtained on the basis o f the ffiaxiauas values f o r the c o e f f i c i e n t s of re s i s t a n c e of v e r t i c a l s b 2 and b 3 ; a c t u a l l y , as the stress i n the v e r t i c a l s Increases, t h e i r r e s i s t i n g capacity decreases, and i t i s we l l to in q u i r e at t h i s p o i n t , how much th© s t a b i l i t y of the bridge w i l l be a f f e c t e d by the d i r e c t stress®© i n the v e r t i c a l s , corresponding to the uniform load on the truss q = 3 .92 KipM. The Intermediate v e r t i c a l s are numbered from th® outside of the t r u s s , and th© r e s u l t s of c a l c u l a t i o n s Involv-in g formulas (12) ;(19) and (22), or the diagram 2 yare ta b u l a -ted below. Intermediate Verticals Stress Kips inch uh " b2 1 • 171.5 .0120 2*013 11.18 3*42 2 122,5 ..01015 1.707 11.41 3.60 3 73*5 ,00785 1,321 11.64 3.78 h 24.5 • 00*54 0,762 11.90 3*95 5 0 0 0 12. 00 4.00 Average of 9 v e r t i c a l s . • 11,59 3.72 41 Using b 2 = 1.1.59 4 >^ = 1.233 the values obtained f o r yu. and q are y u = 26*72 and = 496 With these f i g u r e s f o r yu and n ABO) f o l l o w i n g the @&ae V(79) procedure., the r a t i o y=-0«23 (31) ) the second approximation of ft comes to jf = 24*2 (82) and cj = 3.SI ^ - ( S 3 ). fh«*« valuea o f -and cj say be considered as f i n a l , and they are only about 3$ l e s s than those, c a l c u l a t e d on the baste of maximum r e s i s t a n c e of v e r t i c a l s . I t Is believed,, that such email d i f f e r e n c e .Justifying the use of (76) t o save a l l the »oric of c a l c u l a t i n g the .stresses and res i s t a n c e s of v e r t i c a l s - , i s general In aoet cases, A more s u b s t a n t i a l e r r o r In t h i s connection seems more l i i s s l y to be expected only, i f the v e r t i c a l s are very slender compared to top chordj and t h e i r r i g i d i t y f o r bending out of the plane of the t r u s s i s not much greater than f o r bending l a the plane of the t r u s s . r e s i s t a n c e of the top chord i s disregarded, as h&e been don© by f r o feasor f i n o s h e n l o , the e r r o r i s eoneiderabl®. I t Is interesting to notice,, that I f the torsional 42 which i s 17* l e s s than (76)* t h i s shews, that neglect of th© t o r s i o n o f th® top chord r e s u l t s In considerable inaceur-aoy. Coming now to th® studies of s t a b i l i t y of the span under the- concentrated l o a d s , and the accuracy of the approx-imate Influence l i n o method, the curve £ (diagram 6) Is constructed f o r the'span under c o n s i d e r a t i o n , a l l o w i n g y = — 0*2'3> as was determined f o r the uniform load on the bridge. The f o l l o w i n g three cases of concentrated loads are studied* I . Single concentrated load 4 at the c e n t e r . 2* ' Two concentrated loads -y- each, at the•quarter p o i n t s . 3* TWQ concentrated loads -y each, at one eighth p o i n t s . The c r i t i c a l values % are found in two ways« by the in f l u e n c e l i n e method, f a r a u l a (62), and by the d i r e c t method, us i n g formulas ^54a) and (56a) f o r the c e n t r a l l o a d , and mod-i f y i n g the denominator i n (56a) f o r the other two eases, with the value of y in each case corresponding to ami-mura. The various constants used In c a l c u l a t i o n s are as.have been found above; /I = 26.72} ^ =• 496; cj = 3*81 , = 476*2 fcip. . The r e -s u l t s are- presented In the t a b l e below. Influence him Method D i r e c t Method Z . V I»oad Q. at € enter 1.556 306,3 -»265 15,50 305.2 a loads f at i Points 4 1.094 435.5 -.205 22,10 435.0 2 loads -| at -g poirits .581 820.0 -.178 41.62 319.0 43 The t a b l e shows,, that although the shape of the b u c k l i n g curve v a r i e s somewhat$ as the p o s i t i o n of the concen-t r a t e d load changes, which i s manifested by.the v a r i a t i o n i n the value of y , the e r r o r of using the influence l i n e me-thod, based on constant y , Is so s m a l l , as to be b a r e l y detectable i n slid® r u l e work* fhe accuracy for s e v e r a l con-centrated loads w i l l e v i d e n t l y be even .greater, fhe other e r r o r Inherent i n the Influence l i n e method, when i t f a l l s to take i n t o account the change i n r e s i s t a n c e of v e r t i c a l s , i s also a matter ofVon® or two percent, as can be e a s i l y shown by c a l c u l -a t i o n . With t h i s d i s c u s s i o n i n view, the v a l i d i t y and s u f f i c -i e n t accuracy of the i n f l u e n c e l i n e method i n d e a l i n g w i t h the movable load nay be considered as duly substantiated. SOME FINAL REMARKS OB TEE M TiiPD AMD THE APPROXI ! AT IONS AND ASSUMPTIONS INVOLVED Design, of a truss- for s t a b i l i t y can be g r e a t l y sim-p l i f i e d by p l o t t i n g two sets of curves? f i r s t , jf curves {eeeend approximations) f o r d l f f e r e n t y U , and 4 ^ 7 s i m i l a r to the f i r s t approximations! plotted on the diagrams 3 and 4, and secondly, £ curves- f o r d i f f e r e n t p and y. Determination of the c r i t i c a l uniform load w i l l be then reduced to a simple c a l c u l a t i o n of tj and and reading the corresponding y o f f the diagram. The c r i t i c a l value of the -movable load w i l l be found by proper placing of the load on the £ curve w i t h s u i t a b l e p and y. 44 I t should not be l o s t out of s i g h t , that the various s t a b i l i t y formulas brought up In t h i s paper are a p p l i c a b l e only as long as the m a t e r i a l In any part of the s t r u c t u r e , loaded with the c r i t i c a l l o a d , remains below the e l a s t i c l i m -i t , , and as soon as t h i s point i s exceeded, the c r i t i c a l value® obtained become too h i g h , This l i m i t s the. a p p l i c a b i l i t y of the above formulas only t o very slender structures.- Thus, in the t r u s s of f i g , 13, Just considered, th® c r i t i c a l uniform-l o a d <\ =3«S1 -^f eauB-es the u n i t compression atres3.--.0j i n the f i r s t intermediate v e r t l e a l ~ ~ 2 S , 2 ,: and i n the c e n t r a l panels of the top chord—45 J^ch.*- , For the com-mon s t r u c t u r a l s t e e l , the l a t t e r • f i g u r e I s above the e l a s t i c l i m i t , and, consequently, the a c t u a l value of CJ w i l l be below 3 ,SI - -^p- .. However, as Professor f imoshenko r i g h t l y p oints out, w i t h the present tendency to introduce construc-t i o n materials of the higher strength and th© higher ©la?tic l i m i t , the l a t e r a l dimensions of members decrease and with' them, the f i e l d of a p p l i c a t i o n of t h e o r e t i c a l formulas, based on perfect e l a s t i c i t y of the m a t e r i a l , increases g r e a t l y . I t must be pointed o u t ; t h a t although the develop-ment of the s t a b i l i t y formulas was purely mathematical.,a due cognizance was taken of th© most o f the p h y s i c a l f a c t o r s of Importance. I t i s t r u e , that the d i f f i c u l t y of th© problem, required v a r i o u s i d e a l i z a t i o n s of c o n d i t i o n s , such as s u b s t i t -u t i o n of a continuous w a l l f o r the actual v e r t i c a l s , and the assumption of constancy o f the sections o f Intermediate ver-t i c a l s and th© top chord, .but i n those assumptions th® 45 r e a l i t y was not I d e a l i z e d out of existence; furthermore, a l -though the a c t u a l t r u s s say not have a l l the v e r t i c a l s the same, and the top chord of constant s e c t i o n , the formulas, nevertheless, can be used j u d i c i o u s l y , as Is done i n ©one other engineering problamaj an example—calculation o f mom-ents i n continuous t r u s s e s on the b a s i s of constant section.. Of l e s s e r f a c t o r s , l e f t out of c o n s i d e r a t i o n , may be mentioned? On th© safe s i d e— a. - The bending r e s i s t a n c e o f diagonals and t h e i r s t a b i l i z i n g e f f e c t on the top chord i n i t s tendency to t w i s t and b u c k l e . b. « The strengthening w i t h brackets of the bottom end connec-t i o n s of v e r t i c a l s t o the f l o o r s beams, r e s u l t i n g i n i n -creased r e s i s t a n c e of v e r t i c a l s t o bending, On the other side—the undesirable e f f e c t of d e f l e c t i o n of f l o o r beams on the l a t e r a l t i l t i n g of v e r t i c a l s and buckling of th© top Chord* however, - I f the ends of a l l f l o o r beams slope by the same angle, t h i s undesirable e f f e c t i s not f e l t . The methods used In t h i s paper- can be applied to the trusses w i t h the end posts of th© same r i g i d i t y as that of the Intermediate v e r t i c a l s , and also t o the types of t r u s -ses d i f f e r e n t from the one on. f i g . 1. Inpmm if Geffiamts.V par Uniform Load Covering U Whole Span \ \ Plotted on Base of n for Different M \ f£ axis. 50.1 fhr Uniform Load: Covering 'Vh/ko/e• Sfiah . S ! Plotted an "Bake' of H for Different M. \ I - : ,. : | -: 4k*/A \ - ; - i : i -. y — „ _ y . . . J _ i .._ r . , ! i . . . . i _ J - ,, i i - . - - J . T . -i ........'.. //y>l - j i~ •• - - - - j • .... . . ~ v - . . ... i <c/vif - ! 61 ; : ' '. 2 0 0 1 ••• 400] v:v. — 77 

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