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UBC Theses and Dissertations

The validity of the simplified limit design method for the design of structures. 1958

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T H E V A L I D I T Y O F T H E S I M P L I F I E D L I M I T D E S I G N M E T H O D F O R T H E D E S I G N O F S T R U C T U R E S by DOUGLAS LEONARD PARKHILL D i p . C E . , Melbourne Technical College, 1944 A Thesis Submitted i n P a r t i a l Fulf i lment of the Requirements for the Degree of MASTER OF APPLIED SCIENCE i n the Department of CIVIL ENGINEERING We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1958 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 the requirements f o r an advanced degree at the U n i v e r s i t y of 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 study. 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 copying o f 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 granted by the 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 . I t i s understood t h a t copying 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 allowed without my w r i t t e n p e r m i s s i o n . Department o f '^'I *o <£esr*-><j The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 3 , Canada. pate M^cA , SfSg i ABSTRACT Pract ice i n the f i e l d of l i m i t design has tended to place cer ta in r e s t r i c t i o n s on s t ructura l loading patterns i n order to s impl i fy the ca lcula t ions involved i n the l i m i t design procedure. The loads considered i n th i s s impl i f i ed approach are assumed to e i ther remain constant and f ixed , or i f they vary then t h i s i s to be i n such a manner that the i r magnitudes stand i n a constant re l a t ionsh ip one to the other. Actual s t ruc tura l loadings seldom sat i s fy these r e s t r i c t i v e conditions and the question natura l ly ar i ses as to whether or not t h i s s impl i f i ed l i m i t design procedure i s v a l i d for general use i n p r a c t i c a l design problems i n which external loads may be wholly independent i n the i r ind iv idua l act ions . This question i s invest igated i n the present paper through the examination of several p r a c t i c a l forms of structure which portray the more adverse conditions of independent and var iable loading to be met i n prac t i ce . These structures are, re spect ive ly , s ingle and double bay gable bents of l ightweight construction, and two forms of multispan bridge g i rders . The,study indicates that a l l of these structures are able to support the ultimate loads predicted by the s impl i f i ed l i m i t i i design method; the actual ultimate loads exceeding the predicted values by up to twenty percent. I t i s concluded that s t ructura l f a i l u r e i n pract ice can always be expected to occur within acceptable l i m i t s of the ult imate load capacity as predicted by the s imp l i f i ed method. i i i TABLE OF CONTENTS SECTION 1. INTRODUCTION SECTION 2. GENERAL THEORETICAL CONSIDERATIONS 2.01 C o n d i t i o n s G o v e r n i n g S t r u c t u r a l F a i l u r e 2.02 Premature C o l l a p s e SECTION 3. ANALYSIS OF SELECTED STRUCTURES 3.01 A n a l y s i s o f S i n g l e Bay Gable Bent 3.02 A n a l y s i s o f Double Bay Gable Bent 3.03 A n a l y s i s o f B r i d g e G i r d e r s SECTION 4. INTERPRETATIONS AND FURTHER CONSIDERATIONS 4.01 I n t e r p r e t a t i o n o f A n a l y t i c a l R e s u l t s 4.02 F u r t h e r C o n s i d e r a t i o n s R e g a r d i n g Premature C o l l a p s e SECTION 5. CONCLUSIONS SECTION 6. BIBLIOGRAPHY APPENDIX 1 TABLES 1 t o X i n c l u s i v e FIGURES 1 t o 24 i n c l u s i v e THE VALIDITY OF THE SIMPLIFIED LIMIT DESIGN METHOD FOR THE DESIGN OF STRUCTURES SECTION 1 INTRODUCTION Much thought has been devoted i n recent times toward the development of s t ructura l design techniques employing an Ultimate Load or Limit Design concept, i n which a problem structure i s approached i n terms of i t s to ta l load support capacity at f a i l u r e , rather than i n terms of the in terna l d i s t r i b u t i o n of stress under working loads as with the c l a s s i c a l e l a s t i c theory. B r i e f l y stated, these l i m i t design techniques have been developed on the supposition that the f a i l u r e of an i n i t i a l l y r i g i d structure occurs as the re su l t of the formation of l o c a l i z e d y i e l d points , or ' p l a s t i c hinges ' , at cer ta in key locat ions i n i t s members, which transform the structure into a collapse mechanism at i t s ultimate load . P l a s t i c behaviour of the structure i s fundamental to the theory and the appl ica t ion of l i m i t design i s confined to s t ructura l materials whose s t re s s - s t ra in character- i s t i c s exhibit a marked p l a s t i c range and for which a constant r e s i s t i n g moment at each hinge may be assumed over a wide range of angular displacement of the hinge. The method i s inappl icable to materials devoid of p l a s t i c i t y or where the p l a s t i c range i s of l i m i t e d extent. Some doubt also exis t s under cer ta in circum- 2 stances as to the s u i t a b i l i t y of the method when a strain-harden- ing region, within the p l a s t i c range, i s absent. The properties of s t ructura l s tee l s , and p a r t i c u l a r l y mild s tee l , f u l f i l the desired p l a s t i c requirements almost i d e a l l y , as evidenced by the t y p i c a l s t re s s - s t ra in re la t ionsh ip for mild s teel shown i n Figure 1(a) and the M-0 r e l a t ionsh ip shown i n Figure 1(b), and the theory thus f inds i t s greatest app l ica t ion i n the f i e l d of s tee l s tructures . The exact re l a t ionsh ip between M and 0 i n Figure 1(b) applies to the mild s teel beam i l l i s t r a t e d i n the f igure (one actua l ly taken from a gable bent example to be l a t e r studied i n de ta i l ) and the closeness of th i s exact r e l a t i o n - ship, over a wide range i n angle 0 , to the constant moment value assumed i n l i m i t design of 2705 f t . kips i s well demonstrated. The design procedure employed i n the l i m i t design concept consis ts , e s sent i a l ly , i n proportioning the members of the s truct- ure so that the col lapse load of the whole structure, or any c r i t i c a l port ion thereof, i s equal to the appropriate working loads m u l t i p l i e d by a cer ta in load factor , with th i s load factor so selected as to provide a su f f i c i en t margin of safety against f a i l u r e of the whole structure or part . In cases where the working loads are of constant magnitude the l i m i t design procedure i s d i rec t and simple as, i r re spec t ive of the indeterminacy of the structure, e l a s t i c analysis can be e n t i r e l y avoided. On the other hand, i n cases where the working loads act and vary independently the l i m i t design procedure 3 becomes f a i r l y complex and a so lut ion to the problem requires the evaluation of e l a s t i c moments i n the structure as the f i r s t step i n an e l a s to-p la s t i c a n a l y t i c a l procedure. In order to avoid the l a t t e r more complex ' v a r i a b l e 1 load design procedure i t i s becoming an accepted pract ice to ignore the true independent nature of most s t ruc tura l loadings and to assume that a l l of the loads appl ied to a structure behave i n the 'constant' manner appropriate to the simpler design method. This assumption of constant loading i s i n many cases a poor approximation to the actual load state ex i s t ing and the question na tura l ly ar ises as to whether or not t h i s s impl i f i ed design method can be reasonably accepted as v a l i d for a l l cases of s t ructura l design l i k e l y to be met i n prac t i ce . It i s known that var iable loading can be severer on a structure than constant loading, and, i n theory, a structure designed i n accordance with the constant load method may f a i l prematurely, that i s before i t s designed load capacity i s reached, i f i t i s actual ly subjected to var iable loads of the same design magnitude. The question ra i sed above concerning the v a l i d i t y of the general use of the s imp l i f i ed method for p r a c t i c a l design purposes thus becomes one of whether i n pract ice a premature col lapse condit ion can actual ly ex i s t , and i f so, whether i t reaches serious proportions or whether i t can always be expected to f a l l within acceptable l i m i t s of the load capacity of the structure as indicated by the s imp l i f i ed method. 4 This question has been given very l i t t l e c r i t i c a l a t tent ion to date and the purpose of th i s present study i s to attempt to obtain an i n d i c a t i o n of the l i k e l i h o o d and possible extent of the premature collapse condit ion through the examination of several p r a c t i c a l forms of structure which are purposefully selected as examples of the more adverse conditions of loading to be expected i n prac t i ce . Co.o So. Ao.o 9 XX 0. vO 3o. to.o o failure <af straitn or about 0.1Z> i ^ / i ^ i O.oZ a.o 4- o.od> o.o& oAo •3\rai»n - m i . p e r 1V1. ,-'<3) Typical slyess-strain curve for Mild Steel 9̂ cv: o 3ooo ?ooo looo Z1o5 I r l . k l p S . Cy/r\y& ox \\c [«. O . Z S M / / \ \ M } lo IS Zo 25 So (b) 3 5 SECTION 2 GENERAL THEORETICAL CONSIDERATIONS 2*01 Conditions Governing Structural Failure w, ]__L A. For Case of Constant Loading The frame shown i n Figure 2 i s acted upon by fix e d working loads of magnitude W, and under these pa r t i c u l a r loads i t w i l l be assumed that the structure deforms e l a s t i c a l l y thus creating certain e l a s t i c moments at a l l sections of i t s members. > If the loads are increased i n magnitude, with a constant r a t i o remaining between their values, the e l a s t i c moments w i l l be proportion- ately increased, and t h i s proport- i o n a l i t y of loads and moments w i l l continue, as the loads further increase, u n t i l at some load value the e l a s t i c l i m i t stress of the frame material i s reached at some c r i t i c a l section. Yielding commences i n the outer f i b r e s of this section and develops through the section u n t i l at some higher load value a f u l l y mobilized p l a s t i c hinge i s created which offers a constant moment of r e s i s t - ance to further deformation. At a s t i l l higher load value the e l a s t i c l i m i t stress w i l l be reached at some other key point of the structure and a second Figure 2 6 h i n g e c r e a t e d . A d d i t i o n a l h i n g e s a r e s i m i l a r l y d e v e l o p e d u n t i l u n d e r t h e f a i l u r e l o a d s u f f i c i e n t h i n g e p o i n t s e x i s t t o t r a n s f o r m t h e i n i t i a l l y r i g i d s t r u c t u r e i n t o a m e c h a n i s m a b l e t o o f f e r n o a d d i t i o n a l r e s i s t a n c e t o c o m p l e t e c o l l a p s e . T h i s i s t h e s i m p l e m e c h a n i c s o f t h e c o n s t a n t l o a d t h e o r y . I f n o w a v i r t u a l d i s p l a c e m e n t o f t h e s t r u c t u r e i n t h e f o r m o f t h e c o l l a p s e m e c h a n i s m i s c o n s i d e r e d a n e q u a t i o n o f v i r t u a l w o r k c a n b e w r i t t e n r e l a t i n g t h e t o t a l i n t e r n a l w o r k d o n e a t t h e h i n g e p o i n t s , i n t e r m s o f p l a s t i c m o m e n t s M p a n d a n g l e s o f r o t a t i o n t o t h e t o t a l e x t e r n a l w o r k d o n e b y t h e f a i l u r e l o a d s o f m a g n i t u d e P . W i n m o v i n g t h r o u g h t h e d i s t a n c e s 2> o f t h e i r v i r t u a l d i s p l a c e - m e n t s . T h u s : Z M p . © = £ F . W . 5> ( 1 ) T h i s e q u a t i o n c o n s t i t u t e s t h e b a s i c e x p r e s s i o n o f t h e s i m p l i f i e d m e t h o d , i n w h i c h , g i v e n t h e p r o p o r t i o n s o f t h e f r a m e a n d t h e g e o m e t r y o f t h e c o l l a p s e m e c h a n i s m , t h e t r u e f a i l u r e v a l u e o f t h e l o a d s , a n d t h e n c e t h e f a c t o r F b y w h i c h t h e o r i g i n a l w o r k - i n g l o a d s w e r e i n c r e a s e d t o t h e f a i l u r e v a l u e , c a n b e r e a d i l y d e t e r m i n e d . I n t h e a n a l y s i s o f a g i v e n s t r u c t u r e a n u m b e r o f p o s s i b l e m e c h a n i s m s m a y e x i s t a n d t h e t r u e c o l l a p s e m o d e w i l l b e t h e o n e r e s u l t i n g i n t h e s m a l l e s t v a l u e o f l o a d f a c t o r P . M o w , r e f e r r i n g b a c k t e m p o r a r i l y t o t h e o r i g i n a l c o n d i t i o n i n d i c a t e d b y P i g u r e 2 w i t h t h e f r a m e s u b j e c t e d t o w o r k i n g l o a d s W a n d w i t h c e r t a i n e l a s t i c m o m e n t s M e x i s t i n g a t t h e k e y p o i n t s a n d w r i t i n g a s i m i l a r e q u a t i o n t o ( l ) a b o v e f o r a n i d e n t i c a l 7 v i r t u a l displacement, i n terms of working loads and moments: Z M.6 = £W. h (2) Again increas ing these loads to the f a i l u r e value F.W by mul t ip ly ing both sides of equation (2) by F: ZF.M.e = 2F.W.S , anid equating th i s expression with (1), g ives : S M p . © = £ F.M.© (3) The p l a s t i c moments Mp occuring i n these equations should be considered as being p o s i t i v e . E l a s t i c moments M at each hinge are l ikewise pos i t ive i f they occur i n the same sense as Mp, e l a s t i c moments of opposite sense to Mp must therefore be treated as negative. Load factor F may be obtained from equation (3), given the properties of the structure and the d i s t r i b u t i o n of e l a s t i c moments within i t . This equation thus provides an a l ternat ive basis to that of equation (l) for obtaining the f a i l u r e load factor for the constant loading condi t ion. Rewriting equation (3), the required quantity F may be expressed i n the fol lowing form: F = gMr.e (4) £M.© Now, of a l l possible mechanisms whereby the structure may 8 collapse the true f a i l u r e mode, that i s the one requ i r ing the smallest value of E, i s obviously obtained when the terms of the numerator are smallest i n comparison with those of the denominator. Two var iables are involved i n these terms, namely, moments and angles of r o t a t i o n . Variat ions i n the l a t t e r are continuous and usua l ly gradual and i t i s thus reasonable to expect that the hinge points of the true f a i l u r e mechanism w i l l be associated with sections of the frame where minimum values of Mp occur. M This associat ion of hinge points of the constant load mechanism with minimum values of the quantity Mp i s found to be M c lose ly followed i n numerous examples which have been considered. Such minima may exceed i n number the hinges required for the development of a mechanism but each i s a key point of the structure representing a possible hinge point of a f a i l u r e mechanism. It w i l l be noted that i n cases where Mp remains constant along a member of the structure the key points of th i s member w i l l be associated with those sections where maximum values of e l a s t i c moment occur. Equation (3) may also be written i n the form: £ M p . e - ej'.M.e = o or, £(Mp - E .M) .© = 0 (5) Now, i t w i l l be remembered that equation (3) represents the structure i n the f a i l u r e state under the appl icat ion of loads P.W. To obtain the above equation (5) the quanti t ies E.M.© have 9 been s u b t r a c t e d from both s ide s o f e q u a t i o n (3) and t h i s s u b t r a c t - i o n has , i n e f f e c t , removed the f a i l u r e l o a d s and l e f t the s t r u c t u r e i n a zero l o a d s t a t e . E q u a t i o n (5) thus r e f e r s to the comple te ly unloaded s t r u c t u r e and must be the v i r t u a l work e x p r e s s i o n f o r t h i s s t a t e . The q u a n t i t i e s (Mp - F.M) a re , then , r e s i d u a l moments r e m a i n i n g at the h inge p o i n t s o f the s t r u c t u r e under t h i s zero l o a d s t a t e . B * For Case o f V a r i a b l e L o a d i n g Cons ider now the s t r u c t u r e shown i n F i g u r e 2 s u b j e c t e d to the more genera l form of l o a d i n g i n which c e r t a i n o f the a p p l i e d l o a d s are assumed as f i x e d and c o n s t a n t , and the remainder are capable o f independent v a r i a t i o n . T h i s i s the v a r i a b l e l o a d i n g s i t u a t i o n which a r i s e s i n p r a c t i c a l s t r u c t u r e s . The assumed f i x e d l o a d s are ' d e a d ' l o a d s o f magnitude Wo and they c r e a t e e l a s t i c moments i n the s t r u c t u r e o f magnitude at any s e c t i o n . The rema in ing l o a d s are ' l i v e ' l o a d s o f magnitude WL. and the n u m e r i c a l - l y g r e a t e s t moment c r e a t e d at any s e c t i o n by t h e i r independent a c t i o n w i l l be r e f e r r e d to as M u . F a i l u r e o f the s t r u c t u r e occur s i n the form of a mechanism when dead and l i v e l o a d s are i n c r e a s e d i n magnitude by l o a d f a c t o r s F c and F 0 , r e s p e c t i v e l y . The zero l o a d s t a t e o f the frame at t h i s c o n d i t i o n o f f a i l u r e i s such t h a t : £ ( M p - F L . M C - F P .Mp ) .e = G, ( 6 ) t h i s b e i n g the work e x p r e s s i o n f o r a v i r t u a l d i sp lacement o f the 10 structure i n the form of the f a i l u r e mechanism. The convention for the signs of moments i s the same as that e a r l i e r establ ished, namely, a l l p l a s t i c moments Mp are considered as pos i t ive and the e l a s t i c moments ML. and M & are l ikewise pos i t ive i f they occur i n the same sense as Mp, i f of the opposite sense then they are negative. The l i v e load factor IV, i s now required and th i s may be read i ly obtained by rearranging the terms of equation (6), thus: K = g ( M p - Fo.Mc).Q (7) The true f a i l u r e mode for var iable loading therefore requires that terms involv ing (Mp - Pp.Mo).© are smallest with respect to those invo lv ing MU.C, and thus key points of the structure for var iable loading can be expected to be associated with sections where minimum values of the quantity (Mp - ffc.Mp) M u occur. Now, equation (6) defines the zero load state for the structure subjected to var iable loading and i t w i l l be evident from inspect ion of th i s equation that the res idual moments remain- ing at the hinge points i n th i s case are equal to (Mp - F^Mi,- Pi>Mp);; As l i v e loads F̂ .W are repeatedly and independently applied the structure w i l l eventually "shakedown1 to th i s pattern of res idual moments. This shakedown procedure may involve a number of appl icat ions of the loads, each of these e a r l i e r cycles of loading necess i ta t ing a readjustment of in te rna l stress from the key points 11 where overstress i n i t i a l l y occurs to adjacent sections of the s tructure . This transfer of stress takes place through increments of p l a s t i c y i e l d i n g at these key points , with each increment of y i e l d being associated with an increment of permanent de f lec t ion of the structure as a whole, and these def lect ions continue u n t i l the shakedown state i s reached. I f loads s l i g h t l y greater than the true f a i l u r e value are applied the structure i s unable to reach the shakedown state and the increments of permanent def lec t ion continue i n d e f i n i t e l y , with the structure deforming as a mechanism, u n t i l complete collapse eventually r e su l t s . This type of f a i l u r e under var iable loading i s therefore referred to as one of ' incremental c o l l a p s e ' . A second form of f a i l u r e under var iable loading must also be invest igated. It was previously stated, i n def ining the quant i t ies involved i n equation (6) , that M u represents the numerically greatest e l a s t i c moment which could occur at any section of the s tructure . Both pos i t ive and negative values of Uu w i l l generally exist at each key point and the quantity involved i n equation (6) w i l l be ei ther the pos i t ive or negative amount depending on the sense of the hinge rota t ion at the key po int . The t o t a l range i n l i v e load moment at any section w i l l be the algebraic summation of these positve and negative amounts. This t o t a l range i n moment M u at a section conceivably may exceed the e l a s t i c resistance of that sect ion, and i f th i s condit ion occurs the opposite extremes of l i v e load moment w i l l produce an 12 i n e l a s t i c y i e l d i n g a t t h e s e c t i o n f i r s t i n o n e d i r e c t i o n a n d t h e n i n t h e o t h e r . T h i s a l t e r n a t i n g p l a s t i c y i e l d i n t u r n r e s u l t s i n i n c r e m e n t s o f p e r m a n e n t s t r a i n w h i c h m u s t e v e n t u a l l y l e a d t o t h e f a i l u r e o f t h e s e c t i o n . T h i s c o n d i t i o n i s d e s c r i b e d a s o n e o f ' p l a s t i c f a t i g u e ' , o r , m o r e c o m m o n l y , ' a l t e r n a t i n g p l a s t i c i t y ' . I t o b v i o u s l y may o c c u r a t a n y s e c t i o n o f t h e s t r u c t u r e . Now, i f t h e moment a t w h i c h y i e l d s t r e s s i s f i r s t r e a c h e d i n t h e o u t e r f i b r e s o f t h e s e c t i o n i s r e f e r r e d t o a s My t h e n t h e t o t a l e l a s t i c r e s i s t a n c e o f t h e s e c t i o n , u n d e r r e v e r s a l s o f m o m e n t , i s t w i c e t h i s moment v a l u e , t h a t i s 2 . M y . T h e c o n d i t i o n o f a l t e r n a t i n g p l a s t i c i t y w i l l d e v e l o p when t h e f o l l o w i n g e q u a l i t y e x i s t s : FJ .ML = 2.My (8) f r o m w h i c h , FJ_ = 2 « M y (9) The c r i t i c a l s e c t i o n o f t h e s t r u c t u r e a t w h i c h t h i s c o n d i t i o n o f f a i l u r e i s f i r s t r e a c h e d e v i d e n t l y i s w h e r e t h e s m a l l e s t v a l u e o f My o c c u r s . T h i s s t a t e i s a t t a i n e d a t a l i v e l o a d f a c t o r e q u a l t o VI. C . Summary o f F a i l u r e C o n d i t i o n s I n b r i e f l y s u m m a r i z i n g t h e f i n d i n g s s o f a r r e a c h e d i n t h e s e t h e o r e t i c a l c o n s i d e r a t i o n s i t may be s a i d t h a t i n t h e c a s e o f v a r i a b l e l o a d i n g , t h a t i s i n t h e g e n e r a l c a s e t o be m e t i n p r a c t i c e i n w h i c h b o t h f i x e d d e a d l o a d s a n d i n d e p e n d e n t l y v a r y i n g l i v e 13 l o a d s a r e e n c o u n t e r e d , two s t a t e s o f f a i l u r e must be c o n s i d e r e d . F i r s t l y , i n c r e m e n t a l c o l l a p s e o f t h e s t r u c t u r e i n t h e f o r m o f a mechanism, t h e c o n d i t i o n o f t h e s t r u c t u r e a t t h e p o i n t o f i n c i p i e n t c o l l a p s e b e i n g d e f i n e d by e q u a t i o n (6), as f o l l o w s : £ ( M p - F L.M^ - F P.M D) .0 = 0, (6) t h e t e r m s o f t h i s e q u a t i o n e x t e n d i n g t o a l l h i n g e p o i n t s o f t h e f a i l u r e m echanism. S e c o n d l y , a l t e r n a t i n g p l a s t i c i t y , i n v o l v i n g o n l y one s e c t i o n o f t h e s t r u c t u r e , and f o r w h i c h i n c i p i e n t f a i l u r e i s d e f i n e d by t h e e q u a l i t y s t a t e d i n e q u a t i o n ( 8 ) , t h u s : F J . i L = 2.My (8) The t r u e f a i l u r e l o a d f o r t h e s t r u c t u r e u n d e r t h e c o n d i t i o n o f v a r i a b l e l o a d i n g w i l l c o r r e s p o n d t o t h e s m a l l e s t v a l u e o b t a i n - a b l e f o r e i t h e r F u o r F J . B o t h s t a t e s o f f a i l u r e r e q u i r e e l a s t i c a n a l y s i s o f t h e s t r u c t u r e f o r t h e d e t e r m i n a t i o n o f moments Mo, M u, and M,_. The s i m p l i f i e d c a s e o f c o n s t a n t l o a d i n g may be t h o u g h t o f as r e p l a c i n g t h e d e a d and l i v e l o a d s , t o w h i c h t h e s t r u c t u r e i s a c t u a l l y s u b j e c t , by an e q u i v a l e n t s y s t e m o f c o n s t a n t l o a d s . T h e s e c o n s t a n t l o a d s b e i n g r e s t r i c t e d t o s i m u l t a n e o u s a c t i o n and I f g e n e r a l symmetry o f s t r u c t u r e and l o a d i n g e x i s t s t h e n more t h a n one s e c t i o n o f t h e s t r u c t u r e may f a i l u n d e r a l t e r n a t i n g p l a s t i c i t y . * 14 i f they vary i n magnitude th i s must be i n such a fashion that a constant r a t io always exis t s between them. In th i s equivalent system the loads are thus of the same nature and the same load factor must be applied to each and every one. The condit ion of the structure at the point of i n c i p i e n t collapse i n th i s case i s expressed by equation (1), as fol lows: = 2-F.W. (1) or , a l t e rna t ive ly , i n the form given by equation (3), thus: The terms of these equations extending to each hinge point of the f a i l u r e mechanisms, araid to each load item i n the case of equation ( l ) . Equation (3) requires an e l a s t i c analys is of the structure for the determination of moments M, but th i s i s not necessary for equation ( l ) . The true f a i l u r e load for t h i s condit ion of constant loading w i l l correspond to the smallest value obtainable for load factor P. 15 2.02 Premature Collapse Having establ ished the conditions governing the f a i l u r e of structures subjected, on the one hand to actual var iab le loading, amd on the other hand to an assumed constant loading, i t i s now only necessary to make a comparison between Fu and F i n order to determine the extent of premature col lapse . I f r a t io s of Fu of F l e s s than 1.0 are obtained th i s w i l l indicate that premature collapse can take place, and the magnitude of th i s r a t io w i l l indicate the sever i ty of the premature collapse condi t ion . Such a comparison of f a i l u r e load values can be put on a p r a c t i c a l basis by se lec t ing p r a c t i c a l forms of structure for analys i s , but before th i s i s attempted here i t i s desirable to further extend the previous theore t i ca l evaluations. F i r s t l y , consider the c r i t e r i a establ ished for the l o c a t i o n of key points i n structures subject to mechanism f a i l u r e . For constant loading the c r i t e r i o n i s that these key points w i l l be at sections of the structure where minimum values of Mp occur. M For var iable loading the c r i t e r i o n i s minimum values of Mp - Fo.Mo Mu but inspect ion of t h i s quantity w i l l show that these minimum values occur when the combined effects of the e l a s t i c moment terms are largest with respect to Mp. As c r i t i c a l values for M, ML, and Mp, are commonly a maximum i n the same v i c i n i t y i n a structure these separate c r i t e r i a may amount to substant ia l ly the same thing and normally i t can be expected that the key points of the structure for both loading conditions w i l l also be i n the same 16 v i c i n i t y . In fact i n l i m i t design i t i s usual to assume i d e n t i c a l key points for both loadings . I f i t i s also assumed here temporarily that the same c r i t i c a l mechanism applies for both forms of loading then the r a t i o J F L for mechanism f a i l u r e may be writ ten as: F = ^M.e . Z ( M p - pP.Mp).e , F 2 M i . e 2 M p . G and making use of known re la t ionships t h i s expression may be read i ly resolved into the fo l lowing form : Fu_ = g M L c . e F £ M U . © i n which, 2MLC.e, and SW^.S, are the in te rna l and external work, respect ive ly , of l i v e loads W u applied i n the manner of constant loading, M t , as previously defined, are the numerically greatest moments produced by these same l i v e loads act ing independently as var iable loads, and 2 M L . © i s therefore the work done by these moments, and 2.Wj>.S i s the external work done by dead loads W^. Now, the interes t i n equation (10), and the reason for introducing i t , l i e s i n the l i g h t i t sheds i n i s o l a t i n g the main features or factors which influence the load factor r a t io JV. . F *- See Appendix 1. 1 > F - F o . £ W p . % F S W L . S . (10) 17 Two such factors predominate i n the equation and on these the ra t io i s d i r e c t l y dependent. These factors are: 1. . £Wp.S . This factor compares the external work done by loads We. and WL. The effect of t h i s factor i s such that the larger the dead loads are with respect to l i v e loads the more favourable w i l l be the re su l t on ra t io and the le s s the l i k e l i h o o d of premature col lapse . 2. gHuc .Q . Compares the in te rna l work done at hinge points by the l i v e loads act ing respect ive ly i n the manner of constant loads and as independent var iable loads. This factor evidently measures the r e l a t i v e extremes of l i v e load moment created i n the structure under these d i f ferent states of loading. The more var iable the actual l i v e loads are the smaller w i l l be the value of the in terna l work ra t io and the more adverse w i l l be the effect on % , , hence the greater the l i k e l i h o o d of premature col lapse . Conditions which can therefore be expected to tend to produce a lower l i m i t to the load factor r a t io w i l l be those i n which the l i v e loads are h ighly var iable and i n which dead loads are as small as poss ib le . Although these conclusions are based on an i d e n t i c a l form of mechanism f a i l u r e i t w i l l be evident that the same conclusions are relevant to the case of f a i l u r e through di f ferent mechanism forms and also to that of f a i l u r e by a l ternat ing p l a s t i c i t y . Now, these f indings present a l o g i c a l basis on which to 18 select the structures for analysis i n the present study. As the purpose of the study i s to es tabl i sh the l i k e l i h o o d and possible extent to which premature collapse may be expected to occur i n pract ice i t i s obviously l o g i c a l to choose for analysis forms of structure which sa t i s fy the loading requirements found necessary for small values of PL . F Two forms of structure which exhibi t such loading character- 1 i s t i c s are the gable bents shown i n Figures 3(a) and 3(b) . Dead weight i s minimized i n both cases by the use of l ightweight roof ing construction and tapered frame members, and l i v e loads are highly var iable - cons i s t ing of independent gravity and revers ib le sidesway forces . The s ingle bay and double bay bents were also s p e c i f i c a l l y chosen of s imi lar form to i l l i s t r a t e the effects of d i f ferent degrees of s t ructura l redundancy. These gable bents, then, can be expected to indicate the probable order of the lower l i m i t to the premature col lapse condi t ion . How, i t would also be of p r a c t i c a l interes t to examine the effect on F L of var ia t ions i n the dead to l i v e load r a t i o F and th i s i s best done by se lec t ing a second type of structure i n which the l i v e loads are highly var i ab le , as with the bents, but i n which the dead weight i s quite large i n magnitude i n comparison with the applied l i v e loads . A suitable example of such a loading condit ion i s the multispan plate g irder highway bridge shown i n Figure 3(c) , i n which the moving vehic le loads cause highly var iable l i v e moments but these loads are considerably smaller 19 i n magnitude than the combined dead weight of the s teel g irder system and concrete deck. Detai led analyses of the bents and bridge g irders w i l l be made i n the remaining sections of th i s paper. The several structures are actual designs, and dimensions and other de ta i l s used i n the study have been obtained from the f abr ica t ion and construction drawings. The o r i g i n a l designs were based on the e l a s t i c theory and although a comparison of p l a s t i c and e l a s t i c designs i s beyond the scope of t h i s paper the o r i g i n a l loading spec i f i ca t ions and design assumptions w i l l be also employed i n the p l a s t i c analyses i n order that such a comparison for these structures could be independently made. TTT TTT Tf F igu re 4-. 0.5-7& n <.r<ze[ bolts l-lofOĉ hoL-'f. Proper i ics of J SecY\or\ N o . o f Area of Section Pro pcv tie £=• Pfasfic Properties r M y k i p s ZP i n . 3 »m. k i p s A 5 . o o 6>. 3 3 oo . 5 n . s T Z o 1 & .55 6 .65 3 3 . 4 ?z . 6 7 4 5 Z 4 . 5 80S Z lo. 4 3 6 .85 t 4 S Z 7 . 5 3 l o 3 o . ] l o o s 3 I Z . I S 7 . 1 7 12)3 3 Z . 6 l o & o 3 5 . 6 1 1 £>o 4 ! 3 . 0 i 7 ,4ft 2 4 4 3 & . Z \zc*c> 4 1 . & I 3 o o & I 5>. 4 7 7 . 7 3 3 4 o 4 4 . o I 4 S O 4 & . 3 I S 3 o ^ . 1 7 . 1 3 © - t o 4 Z 5 4 5 . S I & 3 & S>4.£> 1 60S 7 ! 7 3 e .41 5 Z o SS . s \e>zs> 6 1 . Co Z o 3 o 2 0 . 45 8> . 7 Z 6 Z 7 6 1 . & Z o S S Z Z & S 2> Z Z . 1 I 3 . O S 7 5 1 (£»7. 3 Z Z 4 0 nc o Z 5 o S l o 2 3 . 7 7 3 . 44 & & Z 7 4 . Z Z 4 & o & 3 . 7 Z1Co &c*l. 2 4 . o o 3 . 3g> 3 o 4 7 5 . 3 Z 4 3 ^ & 4 . & Z&oo £ 4 . o o o .e>e> » 3 5 <£>3. 7 . 2 3 o o &z.<* Z1oS> Z 3 . l o 3 • & o 6 1 J <2>£>. £> ZZGo 7 7 . & Zh0?o \t Z ? . 4 7 0 . /z 7 o 6 <&Z. 3 Zolo 7 4 . 15 2 I • & S 7 . & S 5 6 J 5 4 . 7 1 &oo 6 , 4 . <£? Z ! S o 14 Z o . o I 7 . CS. S o & <i» ! £ , 7 o 5 7 . 3 1 8 3 5 I S i e>. ie> 7 . 4 3 4 3 5 4 & . 7 1 £ » 4 o S Z . 7 ! 7 4 o ! 6 1 7 . 5 4 7 . 1 5 3 7 0 , 4 2 . 3 1 4 f t 4 6 . Z 1 S 3 o J 7 S I £>. 7 3 S Z o 32>- Z / Z 5 5 4 3 . & 1 4 4 5 16 / & . o & C. 7 4 ZGS> 3 5 . . £> I I 7 o 1 3 o S > I'D 1 3 • 6 S 6 . S I Z Z Z 3 2 . 1 \oC>o ! 1 7 5 to 1 © J Z 6 . 3 4 5 5 ( . 4, 1 o 4 5 C 1 1. oo ] & I C O 2 7 . O 0 S o 2 5 . 7 3 £ > o 20 SECTION 3 ANALYSES OF SELECTED STRUCTURES 3.01 Analysis of Single Bay Gable Bent A. General Frame sections and deta i led dimensions of the bent are shown i n Figures 4 and 5. The column and roof beams are of b u i l t - up welded sect ion, as shown in Figure 4, and the properties of these are given i n Table 1. High strength bolted connections are employed throughout and deta i l s of these connections are shown i n Figure 5. Latera l s t i f feners for the ins ide flanges of columns, and beams are located i n Figure 4 and take the form of angle braces between flanges and side and roof pur l in s . The spacing of frames i s 20 , -0" on centre. The o r i g i n a l design of the bent was to AISC - 1952 spec i f i c - at ions . Loading conditions were f igured as follows: (1) Dead loading plus 30.0 pounds per square foot l i v e loading on the f u l l hor izonta l pro ject ion of the structure. (2) Dead loading plus 20.0 pounds per square foot wind loading act ing on the f u l l v e r t i c a l pro ject ion of the structure i n one or other hor izonta l d i r e c t i o n . (3) Dead loading plus 30.0 pounds per square foot on the h o r i - zontal pro ject ion plus 20.0 pounds per square foot i n e i ther d i r e c t i o n on the v e r t i c a l pro jec t ion . 21 Design loadings were computed on the basis of these conditions with dead loads assumed d i s t r ibuted uniformly over the roof span and estimated thus: Roof beams & 1.4 l b s . per sq. f t . Roof pur l ins @ 1.1 »' " " Roof sheeting @ 1.0 r s " " Total dead weight 3.5 l b s . per sq. f t . g iv ing for : Design condit ion (1), uniform load on f u l l hor izonta l pro ject ion = (30.0 + 3.5).20.0 = 670 l b s . per l i n . f t . Design condit ion (2), wind load on f u l l v e r t i c a l project ion = (20.0).20.0 = 400 l b s . per l i n . f t . plus dead load = (3.5).20.0 = 70 l b s . per l i n . f t . Design condit ion (3), wind load on v e r t i c a l pro jec t ion 400 l b s . per l i n . f t . uniform load on hor izonta l pro ject ion = 670 l b s . per l i n . f t . B. E l a s t i c Analysis A step by step der ivat ion of the e l a s t i c analysis i s unnecessary for the purposes of th i s study. The procedures used 22 are standard and well known. Moment d i s t r i b u t i o n methods were employed, with the f ixed end moments, s t i f fness coe f f i c i ent s , and carry-over factors required for the analysis determined by means of the column analogy method. Both supports were assumed as hinged. The properties o f . the frame elements used i n these ca lcula t ions are given for reference purposes i n Table 11, and i n the accompanying Figure 6. The symbols used carry t h e i r usual meanings. Result ing moments at sections throughout the frame are given i n Table 111 for the ind iv idua l gravity and wind loadings . Shears and thrusts are not considered as inves t iga t ion has shown that t h e i r effects are small and can be e n t i r e l y neglected i n the study. Moment diagrams for each loading condit ion, p lot ted from the values tabulated i n Table 111, are shown i n Figures 7, 8, and 9. For these diagrams the moments are, p lo t ted , i n the convention- a l manner, on the side of the frame at which they create compress- ive s tress . The severest f ibre stress created by the applied loadings i s worthy of note although th i s data i s not required for the present purpose. The severest stress condit ion i s found to occur i n the roof beams at the knee j o in t s where the combined stress f igures at 26.6 k s i . The value permitted by the AISC Code, for wind and a l l other loadings, equals 26.7 ksi> ( = 20.0 k s i plus T<* b le ! ! P v o p e r f f e s o f G <a b l e, Be i n f E U Scchon N o . Frame Ph x: f*. y ff. I ins?" I ft. 4 ' V e ^ h i c a l 3 L a H of <£ R i g lot ot <£. I loo I l o o I fh k ip. Ms.y.&s M s M s. y.As M s fh kip l o o 1 l o o I l o o r 1 1 .67 o . o 4 o . 3 4 6 . S 3 3 3 . 4 . o o 4 7 0 331 3 . 6 0 3 . 5 1 . o 3 . 7 2 3 65 o . 3 1.1 2 1.07 o . 1 1 z . e i 1 0 . 43 145 . o o 6 3 0 Z66 7 . 5 2 21 . ! 2 . 7 Z o . 3 6 3 5 10 o.o 6 . o 3 o . 10 4 . 6 & 1 2 . IS 1 3 3 .OO350 135 3 . (S 42.6 4 . 3 2 3 3 . 5 1 I 2 l o 2 5 1 . 4 1 2 . 8 4 o . Z 5 6 .55 1 3 . 0 1 Z 6 4 . o l 2 7 147 3 . 6 4 6 3 . 1 6 . o l 57.6 151 1453 1 . 3 1 0 . 3 b 1 .6*7 o.3Z 8 . 4 Z I 5 . 4 & 34 o . © 1 6 4 I 14 3 . 6 o 6o.6 7 . 6 7 - 7 3 . 6 1 8 7 1737 Z . 4 2 3 . o Co I . 07 I o . 23 1 7 . 1 3 4 Z 5 . o 2 o 5 31 . 3 3 . 4 o 3 6 . 6 9 . 3 3 0 7 . 6 ZZo 2 o 7 o 2 . 3 Z1.Z 7 1.07 O. 4 6 I Z . 16 10.73 S>Zo , o 2 S o 74.6 3 - Io 1 1 o . 6 I 1 . o o l o o 2 4 6 ZZGo 3 . 4 31 . o 0 1 . 0 7 O.S3 I4.O3 Z o . 4 5 6 2 7 . o 3 o Z 6 ! .3 0 . 7 o 1 2 2 . 1 1 Z . 6 6 ( l o Z 7 4 2 3 6 o 4 . o 3 4 . 8 3 l . 8 7 O.CoO i s . 3 o ZZ.W 7 5 ! . o 3 6 l 5 / .& 6 . 24 I 3 o . I 14 . 3 2 11 b 237 Z 4 5 o 4 . 5 3 7 . 1 IO 1 .61 O. (ob 17.77 tl.11 0 6 2 . o 4 2 4 44 . » 7 . & 3 1 3 3 . o I t . 3 8 125 3 1 4 Zt&o 5 . o 3t>.2 1 1 2.43 1. 66 1 3 . 12 2 3 . Io 6 7 6 . o 4 2 I 5 3 . 2 1 1 . 3 ! 2 1 6 4 3 . 7 o 4 3 5 3 1 3 3 6 ! o 14 .1 ! 6 o \Z £ . 4 3 4 . 2 Z 1 3 . 3 5 ZZ.47 7 7 6 . o 3 7 0 6 6 . 6 1 3 . o o 2 6 5 3 3 . Io I Z 4 o 3 o 6 4 l o o 31 . 6 4 2 o 13 2 . 4 3 6 . 5 7 Z o . 73 2 I . 8 5 6 6 5 . o 3 3 o 7 5 . 5 \b .1 3 2 6 1 3 7 Zt5o Z 3 4 4 6 2 o 4 3 . 2 7 7 3 14 2 . 4 3 6 .31 2 1 . 6 2 Z o . o t 5 3 3 . o 2 0 6 8 6 . 8 1 6 . 7 4 o 5 1 7 7 3 3 l o 2 6 3 B5oo 6 6 . 7 1 2 4 5 15 Z . 4 3 M . 2 5 2 2 . 4 5 5 Z o . o Z 5 1 3 3 . 3 7.2.5 Soo Z o 3 4 6 7 o 2 7 o CoOSO 8 4 . 4 I860 1(2, 2 . 4 3 1 3 . S 3 Z 3 . 2 3 [•7.54 4 4 6 . O 2 I 6 M £ 2 6 . 6 6 Z 4 2 3 6 6 3 3 o Z & 6 C&lo l o Z 2 7 4 o 1-7 Z . 4 3 I S . 5 3 2 4 . 12 I 6 . 3 I 361 . o l 0 3 136 32.6 7 3 Z Z 5 3 85 o o 241 7 3 o o 1 13 3 3 o o ie> Z . 4 3 I & . Z & 2 4 . 3 6 I 5 . o 0 321 . o l 5 5 1 6 ! 4 o . 1 1 ooo Z75 1 ! o 5 o Z 2 4 8 3 3 o 137 5 5 o o 13 2.43 Z o . 6 2 2 5 . 7 3 1 3 . 6 5 2 6 6 . o J Z 3 132 4 3 . 5 1277 2 6 6 14 I 7 o 2 o 3 l o 3 5 o 155 7 6 7 o Z o 2 . 4 3 22.36 Z 6 . 6 Z 1 2 . 6 3 2/7 • o l o S Z 3 6 6 2 . 3 1676 23 £ ! 6 3 6 o lt>Z I t l o o 172 l o 6 2 o Table III M orne*^i& f o r S i n g l e 6 a y Ben- f . F r o > T O « ,Seo4 i o n N o . \)ead load M o m e n t ir». k i p s Li^sa Load M o m c r i l i n kips. Mor r>0 to t 5 for s i S Cor\s.t#r->V Lo&<A \r-\ kipS. Mp iK-i kips Lo<»<s< i m ^ f ro^n lef t f r o r n r igHf Con <?lition I C o n d i t i o n •z Condit ion 3 3 U — 3 Z - 2 7 1 + 3 G I - ) 6 6 - 3 o 5 + 3 Z 3 + 5 6 1 1 s o - 5 7 - 4 6 3 + 5 67 - Z 3 3 5 4 f e + 5 3 o + 4-1 I & 3 o 1 , &Z - 7 o 7 + 7 4 3 - 4 3 Z - 7 &e> + 6 6 1 - 4 f c Z 0 3 o 3 u — J o o — ZZ? + 6 3 5 - 5 6 5 - 103 0 + 7 Z 7 - 1 3 5 Z S o S £>, — ! 77 - I0&5 +- 6 & 5 - 6 6 5 - ! Z I Z + 7 3 6 - 3 4 7 Z 7 o S — 106 S 3 o + 8 I 7 - 6 3 6 - I o 3 S +• 7 o 3 - Z7 I Z 5 6 o - 43 - 3 6 7 . & 4 Z - 5 3 I - 4 I 0 +• Z 3 7 Z 1 3 0 -f 3 + ZZ + 4 5 5 - 4 Z 5 + 2 5 + 4 5 8 4 © o 1 7 4 o + 35 +- 3 o / + 2 5 7 - 3 I 6 4 3 3 6 Z 3 2 + 6 5 3 1 4 4 5 4-7 + 4 0 1 + 3 6 - Z I I + 4 4 8 +• 6 3 + 4 8 4 1 1 7 5 C + 4? + - I 3 ( - ! 3 I + 3 3 8 - 8 3 4 Z 6 7 3 80 + 4 7 + 4 o J — z ; i + 3fe + 4 4 8 - t 6-4 + 1 3 7 1 1 7 5 +- 3 5 + 3 o 1 - 3 I 8 2 5 7 4 3 3 6 - - Z 3 7 + 1 8 1 4 4 5 + 3 +- ZZ - 47-5 + 4 5 5 + ZS> - 4 Z Z - 4 o o ! 7 4 o I S , - 4 3 - 3,0>1 - 5 3 / + 6 4 Z - 4 I 0 - £ 7 4 - 3 4 1 1 I 3 0 "R - 1 o & - 3 3 o - + 8 I 7 - 1 o 3 6 - 7 4 C * - •25 60 - 1 Z 7 - l o 6 5 - 6 6 5 + 8 6 5 — 1 Z / Z - 7 3 7 - 18 7 7 zn - 1 06 - 3>ZZ - 5 & S + 8 3 5 - I 0 J o - 6 7 3 - 1 5 3 5 Z £ o £ - 8 Z - 1 o 7 - 4 32 + 7 4 3 - 7 8 3 - E> 14 - I7Z-I Z o 3 o S< - 5 7 - 4 &e> - 2 33 + 5 8 7 - &4<o - 3 5 0 * - 8 4 5 IS S o 3 R — 3 2 — Z 7 I — I 6 £ » + 3 C I - 3 o 3 — f 2>8 — 4 6 3 1 1 & o Table I V M o m e n h [or S\r\gle £><auj S e m i . S e c l i ' o n N o. Dead load \Ao m<zr~its i r » . k i p s l i v e load tAom<zin\s f o r l/ic»r :<alo!e load A n a l y s i s i n . k i p s M p , t o . k i p s Yield M o m e > o l s M y , i o . k i p s Ne^ta+Ive VI t R<7ing£ i f - i - 3 2 3 6 ! - 4 3 7 7 3 6 1 1 6 o l o g o 5 U - 5 7 h&n - 7 6 6 1 3 7 5 1 5 S o 1 4 5 o - &z + 7 4 3 - 1 \ 31 1 8 7 5 7 o 3 o 1 8 2 5 \ - i°0 + 6 3 5 - 1 4 6 7 - 2 3 7 2 7 5 o i Z 2 4 © - \m 6 6 5 - 1 7 5 © 2 6 1 5 Z 7 o 3 2 3 o o I I , - 1 o 8 + 6 1 7 - 1 5 6 6 7 3 6 5 7 £ 6 o 2 ? 6 o I3L - 4 3 6 4 2 - 8 2 ) 6 1 5 4 o 7 1 3 o 1 &oo + 3 + 4 7 7 - 4 7 5 2>©2 ' 1 7 4 o 1 5 4 e > 1 7 , + 3 5 5 5 0 - 3 1 6 6 7 6 1 4 4 5 1 2 3 5 I S i , + 4-7 + 4 S 7 - 7 1 ! 6 4 8 I 1 7 5 1 060 c 4 Z + 3 5 6 - 1 3 1 4 6 7 3 6 o . 0 3 o ' O R + 4 7 + 4 3 7 - t i l 6 4 8 / I 7 5 1 0 6 0 -f 3 5 + 5 5 6 - 3 1 6 8 7 6 1 4 4 i 1 2 3 S + 3 + 4 7 7 - 4 7 5 3 o 2 1 7 4 o I S 4 o ! 3 K - 4 3 + 6 4 2 - 6 3 6 [ 5 4 o 2 l 3 o I 8 o o ' ' K - l o 6 oil - 1 £ 6 6 2 3 6 5 Z £ 6 o Z 7 6 o - 1 7 7 + e(oZ - 1 7 5 © 2 6 I S ZIoL 2 3 o o - !o& + 6 3 5 - 1 4 8 7 2 3 - Z 2 2 S o £ k Z 7 4 o 7 * - 81 +- 7 4 3 - 1 1 3 2 1 6 7 5 2 o 3 o 1 8 7 5 s K - 5 7 •f- 5 6 7 - 7 8 8 1 3 7 5 1 S 3 o 1 4 5 o 3 * - si 4- 3 6 1 - 4 3 7 7 3 6 1 1 S o 1 080  N O T E S : PESJcC't - O C O M D i r i c O s J 1 1. B t a s f i o rnarnet^\i> are *p\o\\e.<A o o (•he cowipt-essioo si<s>(e fhe f r^me, L o ^ ^ i o ^ : t 7 o las per ff- o o Viort-zo^ . projection Z- Scale o f r o o r o e n f s — [ i»n =• IE>o<3 ir-> k i p s . 3. Key poin^i of fr-wme S I O O K J O fh^5> — -£-z 4. Hxo^e p o i n t s o£ w i e c l ^ d ^ i s t v i s jhowO th<->S — «^ f i g u r e 1 + rsl o r e S : I. E l ^ s f i c b n o m e n l s are. p l o H e ^ o n f h e c o m p r e s s i o n s i < s ^ e o f f l o e Wtx<*~\Q. Z. S e ^ l e o f i ^ i o t r i e r i f s — I i n = l £ > o o i n kips. Figure S> P C S / G N J C O M D I T l O M Z Locttf^m^s : 7 0 lbs per f {. o n l o o r i j o n j ^ l pro j ect i o n . 4oo l b s p e r f f . o n v e r h c ^ l p r o j e c f i o n ( d c f i i o ^ f r o m l e f C o l l a p s e K4eclniaioisiv-) rsf o r L s t i n g c o m p r e s s i o n s i ' ^ e o f H o g f r v a r r - i e . Z . S e < 5 * ( e «?f m o m e r i l ^ — l i n c i E > 0 0 i r o k i p s 5. Ke<--j p<3iro)-s of fWirv->e sioowio tki^s — -^-^ 4. 'H'noge po in ts of yne.cV~t , a m s . » o s h o w n t h ^ & — • ^ DESJCN] C O N J P l f l O N .OCK^^^S : Colo Ibi p«< o n hor izoof<? pf"o je£ h o » n . 4-0<3 Iloi p« f f4 o n x / e H i c ^ t (_ * K O W I K - > ^ p o s i t i v e ^ " " " v l • n e ^ c a f i v / e o f NJOT E S. : f . Scale o f t ^ o o t - o e r - i f s — i i n = l ^ o o i n k i p s . £ . I C e y p o m f s c f f r a m e s h o w n i - W w ^ — Collapse iAec\n&oisws L O A P l K J G C O N D I T I O N L o a ^ i n ^ s : C 9 O O l b s p e r f f . o n h o r ' i z o n l f l l 4 o o lbs per f f . ovo v e r f i c ^ f pro je^ f 10n ( ^ < r f i r - i < ^ fro^-j e i l l o e r * i s d i ' g c f i o n ^ Figure IO 23 33 1/3 percent as per Clause 15(e) of the Code). Thus the actual stress f a l l s just within the allowable value. E l a s t i c values for shearing stress throughout the frame are well within acceptable l i m i t s . C. L imit Design Analysis by Constant Load Method From previous considerations i t i s known that two methods are avai lable for the determination of the load capacity of the bent under an assumed constant load condi t ion . These are, f i r s t l y , the v i r t u a l work equal i ty of equation (1) r e l a t i n g the in terna l and external work done under the f a i l u r e loads during a displace- ment of the structure i n the form of a collapse mechanism, and, secondly, by means of the zero load equation (5) def ining the in terna l work done by the res idua l moments at the hinge points during a mechanism displacement. Now, as the var ia t ions i n e l a s t i c moment throughout the bent have been determined already the l a t t e r procedure w i l l be adopted here as i t presents a far more d i rec t means of. s o l u t i o n - than does the former method. F i r s t consider the frame i n general terms, the number of hinge points necessary to develop a mechanism in general w i l l be two, and i f these hinge points are located at, say, frame sections (1) and (2), refer diagram below, equation (5) becomes: F = Mp, .G, + M m . Q x M, .©, + M 2 . © z 24 H where Mp,, M,, and are moments and a n g u l a r d e f o r m a t i o n s at s e c t i o n ( 1 ) , and Mp^, M t, and Qz, are l i k e q u a n t i t i e s a t s e c t i o n ( 2 ) . The r e s i d u a l moments a t these s e c t i o n s , as d e f i n e d by e q u a t i o n (5), are (Mp,-F.M,) a t s e c t i o n ( l ) , and (Mp l-£ i.M l) a t s e c t i o n ( 2 ) , w i t h moments s i g n s throughout conforming t o the p r e v i o u s l y e s t a b l i s h e d c o n v e n t i o n . The diagram shows the f o r c e s a c t i n g on the unloaded bent and i t i s a t once e v i d e n t from e q u i l i b r i u m r e q u i r e m e n t s t h a t r e a c t i o n s H must be equal and o p p o s i t e . Thus the r e s i d u a l moments must be d i r e c t f u n c t i o n s o f o r d i n a t e ' y 1 , and t h e s e moments can now be w r i t t e n as: M p, - P.M-, and, Mp z - F.M r and s o l v i n g f o r P g i v e s : y,.H , y T.H , a,.(i/y, ) + K.(l/yJ ( i i ) A n g ular d e f o r m a t i o n s Q, and Qz are thus i n v e r s e l y p r o p o r t - i o n a l t o o r d i n a t e s y, and yz, r e s p e c t i v e l y . Q u a n t i t i e s Mp, M, and y, are known f o r a l l frame s e c t i o n s and the v a l u e o f P c o r r e s p o n d i n g to any two s e c t i o n s can be r e a d i l y determined. I t i s now o n l y n e c e s s a r y to l o c a t e the s e c t i o n s r e s u l t i n g i n the 25 s m a l l e s t va lue o f F i n o rder to e s t a b l i s h the c o r r e c t f a i l u r e mechanism and the c o r r e c t f a i l u r e l o a d . The two h i n g e s w i l l not deve lop s i m u l t a n e o u s l y and the f i r s t one to form w i l l be at the s e c t i o n where y i e l d s t r e s s i s f i r s t reached i n the frame, tha t i s where the va lue o f Jd_ i s a Z e maximum. For d e s i g n c o n d i t i o n (3 ) , as r e p r e s e n t e d by the moment diagram i n F i g u r e 9, the maximum va lue of M_ o c c u r s at the knee 2e j o i n t s e c t i o n B* and t h i s t h e r e f o r e i s one hinge p o i n t . The c o r r e c t p o s i t i o n f o r the second h inge can now be r e a d i l y e s t a b l i s h - ed by t r i a l by means of e q u a t i o n (11) . Such a procedure f i x e s the second h inge at between s e c t i o n s 18L. and 19,_. R e l a t i v e v a l u e s o f r o t a t i o n Q f o r the two h inges are f i g u r e d to be 0.770 and 1 .000, f o r Qt and © z r e s p e c t i v e l y , and thus F , o b t a i n e d from e q u a t i o n (5) , i s as f o l l o w s : Up, .0 =1240(0 .770)= 950, M,.0 = 522(0.770)= 403, M p ^ . « = 2705(1.000)=' 2705, M z . © =1877(1.000)= 1877, Numerator = 3655, Denominator = 2280, and, F = 3655 = 1.605 2280 T h i s method has e s t a b l i s h e d the exact f a i l u r e mechanism and has produced the t rue minimum va lue f o r the f a i l u r e l o a d . I f , on the o t h e r hand, the approximate method suggested p r e v i o u s l y , o f l o c a t i n g the h i n g e s at s e c t i o n s of minimum Mp, i s M a p p l i e d i t i s found tha t the h i n g e s are l o c a t e d at knee j o i n t B^ 26 and at sect ion l & Y , g iv ing a load factor of 1.61. The difference between the exact and approximate methods i s thus of i n s i g n i f i c a n t consequence i n t h i s case. Other cases have also been considered and the difference between the two resu l t s has been found to be small i n every instance. On the basis of these f indings the approximate method for hinge point l oca t ion w i l l be adopted i n the remainder of th i s study. Now, i n addit ion to loading condit ion (3), considered above, i t i s also necessary to evaluate the effects of loading condit ion (1), as shown i n Figure 7. Sections of minimum M-n are also indicated i n the f igure . The general symmetry of the moment diagram i s at once evident and t h i s fact immediately suggests the p o s s i b i l i t y of a symmetrical f a i l u r e mode with hinges forming at a l l three key points , 1, 2, and 3 (see Figure 7). For t h i s mechanism r e l a t i v e values of © are found to be, ©, = © 3 = 1.000, and © l = 1.382, and so lv ing for F: 2705(1.000) - 2705 , 1212(1.000) = 1212 , 980(1.382) «= 1355 , 392(1.382) = 550 , 2705(1.000) = 2705 , 1212(1.000) = 1212 , Numerator = 6765 , Denominator = 2974 , and, F = 6765 = 2.27. 2974 That the sway mechanism with hinges at points 1 (or 3), and 2, produces an i d e n t i c a l value for F w i l l be apparent from the 27 relationship between Q's for the two mechanisms. Thus l i m i t i n g values for I' for loading conditions (1) and (3) are 2.27 and 1.61, respectively. Comparing these values i t would appear that condition (3) i s c r i t i c a l , however th i s condition involves a l l external loads including wind and a smaller load factor can be used for t h i s case than for condition (1) which excludes the wind loadings. Recommended design values for load factors conforming to AISC requirements are as follows: (a) 1.41, for a l l forces including wind, (b) 1.88, f o r a l l forces excluding wind. The actual additional margins of safety provided by the bent over and above these recommended values are 1.135 and 1.21 for conditions (3) and (1), respectively, and the former condition therefore d e f i n i t e l y governs. The f a i l u r e load factor for the bent under the loading assumptions of the constant load method i s therefore 1.61, corresponding to an additional margin of safety over s p e c i f i c a t i o n requirements of 1.135. The f a i l u r e mechanism i s as shown i n Figure 9. D. Limit Design Analysis for True Variable Loading The actual loading to which the bent i s subjected consists i n a constant dead load and a random pattern of applications of gravity l i v e load and wind, and the moments which must be taken into account i n the variable loading analysis w i l l be those 28 r e s u l t i n g from each o f t h e s e d i s t i n c t l y independent l o a d i n g s . As wind i s i n v o l v e d i n t h i s l o a d p a t t e r n the s p e c i f i c a t i o n r e quirement f o r l o a d f a c t o r w i l l be 1.41. I t w i l l be c l e a r t h a t the case o f g r a v i t y l o a d w i t h o u t wind need n ot be c o n s i d e r e d (and c e r t a i n l y not w i t h a h i g h e r s a f e t y r equirement) i n t h i s v a r i a b l e l o a d i n g a n a l y s i s as g r a v i t y a l o ne r e p r e s e n t s j u s t a p o r t i o n o f t h i s t o t a l a c t u a l l o a d i n g . Now, as p r e v i o u s l y d i s c u s s e d , two modes of f a i l u r e are p o s s i b l e under v a r i a b l e l o a d i n g . F i r s t l y , the s t r u c t u r e may f a i l as a mechanism through i n c r e m e n t a l c o l l a p s e w i t h t r u e f a i l u r e l o a d o b t a i n i n g when the f o l l o w i n g e x p r e s s i o n i s m i n i m i z e d : F u = £(Mp - P P.M P).0 (7) Secondly, f a i l u r e may occur through a l t e r n a t i n g p l a s t i c i t y a t a l o c a l i z e d s e c t i o n o f the s t r u c t u r e - w i t h f a i l u r e g i v e n when the f o l l o w i n g e x p r e s s i o n i s m i n i m i z e d : S o l u t i o n s f o r these e q u a t i o n s can be r e a d i l y o b t a i n e d once moments M u and Mi. are known. Both the wind and g r a v i t y l o a d s are i n v o l v e d i n the s e terms and the magnitude of the- c o n t r i b u t i o n due t o wind w i l l be composed o f the maximum l o a d a c t i n g on one s i d e o f the bent and an equal o r l e s s e r wind e f f e c t a c t i n g i n d e p e n d e n t l y as a r e v e r s e d l o a d i n g from the o p p o s i t e d i r e c t i o n . 29 The magnitude of t h i s reversed wind effect i s not actua l ly spec i f ied for the bent, nor i s i t required i n the e l a s t i c analysis where the loading c r i t e r i o n need only be i n terms of the maximum wind act ing i n one d i r e c t i o n . The reversed wind effect i s also of no interes t for the previous constant load analysis for which the same c r i t e r i o n appl ies . The severest possible condit ion involves a reversed wind equal i n magnitude to the maximum loading of 20.0 l b s . per sq. f t . Values of and M u have been computed for th i s reversed loading and these values are given i n Table I V . The var i a t ion i n throughout the bent, for both pos i t ive and negative amounts, i s shown i n Figure 10, together with the loca t ion of sections where minimum values of the quantity (Mp - F p . M P ) occur, a dead load factor of 1.25 being employed i n the values for the l a t t e r quantity. The symmetry of th i s f igure again immediately suggests the probab i l i ty of a symmetrical f a i l u r e mechanism with hinges at points 1, 2, 3, and 4, of the f i gure . Hinges 1 and 4 being at sections B L and B K , re spect ive ly , and hinges 2 and 3 at sections 19 L and 19 K , r e spect ive ly . Relat ive values of Q are f igured to This factor of 1.25 i s based on an evaluation of probable errors involved i n the quanti t ies entering into the design procedure when dead loads only are present. Deta i l s of th i s evaluation w i l l not be included here, but reference i s made to papers on the subject of Factors of Safety and par t i cu l a r - l y to the paper by A. Freudenthal, e n t i t l e d "Safety and Probab i l i ty of Structural F a i l u r e " , Proceedings, No. 468, ASCE, August 1954. 30 be, «, = 0 4 = 1 . 3 8 0 , and Qz - 0 3 = 1 . 0 0 0 . O n l y one h a l f o f t h e f r a m e n e e d be c o n s i d e r e d and F c , o b t a i n e d f r o m e q u a t i o n ( 7 ) , becomes: ( 2 7 0 5 - ( 1 . 2 5 ) 1 2 7 ) 1 . 3 8 0 = 3 5 1 0 , ( 1 7 5 0 ) 1 . 3 8 0 = 2413 , ( 1 1 7 5 - ( 1 . 2 5 ) 4 7 ) 1 . 0 0 0 = 1 1 1 5 , ( 4 3 7 ) 1 . 0 0 0 = 437 , N u m e r a t o r = 4 6 2 5 , D e n o m i n a t o r = 2850 , and, P L = 4625 - 1 . 6 2 . 2850 The c o r r e s p o n d i n g sway m e c h a n i s m w i t h h i n g e s a t p o i n t s 1 and 3 ( o r 2 and 4) a l s o p r o d u c e s a v a l u e f o r I\ o f 1 . 6 2 a s w i l l a g a i n be e v i d e n t f r o m t h e s i m i l a r i t y i n t h e r e l a t i v e v a l u e s o f 0. The l o a d f a c t o r f o r i n c r e m e n t a l f a i l u r e o f t h e b e n t i s t h u s 1 . 6 2 . C o n s i d e r i n g now t h e p o s s i b i l i t y o f a l t e r n a t i n g p l a s t i c i t y . I t i s e v i d e n t f r o m e q u a t i o n (9) t h a t t h e s m a l l e s t v a l u e f o r F J o c c u r s when My i s m i n i m i z e d , and r e f e r e n c e t o T a b l e I V shows t h a t Mu t h e s m a l l e s t v a l u e o f t h i s q u a n t i t y i s o b t a i n e d i n t h e r o o f beam a t t h e k n e e j o i n t s where My and ML a r e 2300 and 2615 i n . k i p s , r e s p e c t i v e l y . Thus t h e l o a d f a c t o r f o r f a i l u r e by a l t e r n a t i n g p l a s t i c i t y i s : F' = 2 . 2300 = 1 . 7 6 . 2615 L o a d f a c t o r P L i s s m a l l e r , a t 1 . 6 2 , and t h e r e f o r e i n c r e m e n t - a l f a i l u r e i s c r i t i c a l f o r t h e b e n t ; t h i s r e p r e s e n t i n g t h e t r u e l i v e l o a d f a c t o r t o p r o d u c e f a i l u r e o f t h e b e n t u n d e r a c t u a l 31 loading condit ions , as envisaged by the variable load method of analys i s . As previously noted a spec i f i ca t ion load factor of 1.41 i s appropriate to the l i v e loading involved and the addi t iona l safety margin over th i s spec i f i ca t ion requirement i s 1.62 , 1741 equals 1.15. The collapse mechanism may take the form of e i ther the symmetrical mode or the.sway mode, as shown i n Figure 10. 3.02 Analysis of Double Bay Gable Bent A. General The double bay bent to be considered i s a standardized design employing outside column and roof members of i d e n t i c a l . sect ion and dimensions with those used for the single bay bent. Reference i s made to Figures 4 and 5 for the dimensions and de ta i l s of these sect ions. A v e r t i c a l 12"x 6£ Mx36.0 l b . r o l l e d wide-flange section replaces the var iable sect ion f a b r i - cated columns for the central support, and deta i l s of the connect- ion between the wide-flange and roof members i s shown i n Figure 5. A l l other connections and s t i f f ener arrangements are as for the single bay bent. The spacing of frames i s 20'-0" on centre. Design requirements are also as described for the previous structure, and loading conditions are i d e n t i c a l and, b r i e f l y , as fo l lows: Design condit ion ( 1 ) , uniform load on hor izonta l pro ject ion of 670 l b s . per l i n . f t . Design condit ion ( 2 ) , wind load on v e r t i c a l pro ject ion of 400 l b s . per l i n . f t . plus dead load of 70 l b s . per l i n . f t . Design condit ion ( 3 ) , wind load on v e r t i c a l project ion of 400 l b s . per l i n . f t . plus load on hor izonta l pro ject ion of 670 l b s . per l i n . f t . 33 B. E l a s t i c Analysis Procedures employed i n the e l a s t i c analysis are as previous- l y described for the single bay bent. A l l supports are assumed hinged. Result ing moments at sections throughout the frame are given i n Table V for i n d i v i d u a l gravi ty and wind loadings. Shears and thrusts are not shown as the i r effects are again small and can be neglected i n the l i m i t design analys i s . Moment diagrams for each loading condit ion, p lot ted from the tabulated re su l t s are shown i n Figures 11, 12, and 13. For these diagrams moments are p lot ted on the side of the frame at whch they create compressive s tress . Fibre stresses are not required for t h i s study but i t i s in te re s t ing to note that the severest stress occurs i n the roof beams at the--central j o in t and i s equal to 24.6 k s i . This i s appreciably le s s than the allowable value but the condit ion i s to lerated i n the interes t s of standardization of components. ^ ' k lmit Design Analysis by Constant Load Method With e l a s t i c moments known a so lut ion for load factor F i s again most r ead i ly obtained i n terms of equation (5). As with the s ingle bay bent loading conditions (l) and (3) only need be considered. Invest igat ing loading condit ion ( l ) . The general symmetry of Figure 11 indicates that a symmetrical f a i l u r e mechanism i s to Tajoie V M o w e n h f o r D o u b l e bay S e . n i . 5ecf IOIO N o . L i t n . <ip& load A n a f o r C © r ~ > s t a r \ i ^ 9 i 4 , n n . k i p s H a s . 4 ic Momenf i n . k i p s loading W i n d Worn L . YJ\r\d f r o m K. Cor\d\\ibn Cond\hor\ z. Cond i t ion 3 . M P m . k<p£ - 3 5 - 3 o 4 t loo - 6 3 - 3 3 3 4 2 4 5 - 5 3 1 I 6 o - 6 1 - 5Z 7 + 4 7 3 - 1 6 1 - 5 6 8 4 3 6 6 - 1 5 3 1 5 3 © - s z - 7 7 3 4 5 2 3 - 2 3 4 - 8 7 | 4 4 3 7 - 3 4 2 2 © 3 © - Ito - ! o 3 Z 4 5 5 4 - 3 o 6 - 1 1 &z 4- 4 3 4 - 5 5 6 Z 5 o 5 & L - - 1 4 1 - 1 2 ) 4 + 5 4 7 - 3 6 © - 1 3 5 5 4 4 o I - 6 1 7 2 7 o 5 I I L L - l o 5 - 3 3 6 4 4 3 3 - 3 4 o - l o 4 5 4 3 5 o - 5 4 6 2 5 6 © ! 3 C L - 4 5 - 3 9 o 4 3 » l - 2 6 6 - 4 3 5 4 3 4 6 - 4 4 7 1 3 © - - + 2 5 4 - 2 3 7 - 4 " 4 ^ + 2 5 4 1 7 4 © 4- 2 6 + I l l 4 I 1 3 - 1 5 6 4- 2.4 8 4- ( 3 3 + 3 6 I I 4 4 S l t > L U + 3 5 + 7 5 5 - 4 © - 1 3 4 + 3 3 © - 5 + I D © 1 1 7 5 3 o 4- 2 5 f t 1 6 3 - 3 6 4- 2 8 8 - ( iS 4 1 7 5 3 6 o ' 3 L R 4- 2 3 + 24-7 - 1 6 4 - 4 4 2 7 6 - 1 3 5 4- 1 1 2 1 1 7 5 I f LIT + 1 3 4 1 6 ! - 1 6 5 4 1 3 5 4 1 bo - 1 4 6 4 1 5 1 4 4 5 - 1 1 - . 3 7 - 1 6 6 + Z 6 5 - t 0 8 - 1 7 7 - 2 7 4 l 7 4 o ! 3 L R • - 6 3 - 5 3 7 - 1 6 7 4- 3 3 6 - 6 o o - 2 3 o - 7 6 7 2 1 3 © M t R - 1 3 o - 1 1 ( Z - I 6 6 4 S 2 & - ! 2 4 2 - 2 3 8 - 1 4 1 o 2 5 6 o P L R - I 6 4 - 1 4 o 6 - 1 6 6 4 5 8 o - 1 & 7 © - 33 2 - 1 7 3 8 7 7 o 5 D c o l . - - - 7 4 6 +• 7 4 8 - - 7 4 8 - 7 4 8 1 6 8 2 D « u - 1 6 4 - 1 4 o 6 4 S g o - 1 6 8 - 1 5 7 o 4- 4 16 - 3 3 © 2 7 o 5 ! i e u - I 3 o - I I 11 +- & Z 8 - 1 6 8 - 1 2 4 2 4- 3 3 6 - 7 1 4 7 5 6 o I 3 R L - 6 3 - 5 3 7 4 3 3 6 - 1 6 7 - 6 o o 4 333 - 2 o 4 2 1 3 o I 5 «u - 1 1 - 3 7 4 2 6 5 - 1 6 6 - 1 o & 4 2 5 4 4 1-5 7 I 7 4 o 4 1 3 + 1 6 1 4 I 3 5 - 1 6 5 - 1 6 © 4- 1 6 4 +• 315 1 4 4 5 4 ZZ> 4- 2 4 7 — 4 — 1 6 4 4- 2 7 6 4- 2.5 + 2 7 7 / 1 7 5 C K 4- bo 4 2 S 6 - 3 6 - 1 6 3 4- - 2 8 8 - 6 6 4 1 3 2 3 & o 4 3 5 + 2 5 5 - 1 3 4 - 4 o 4- 3 3 o - 3 5 + 1 e>6 M 7 5 + 2 6 4 2 7 Z - 1 3 6 4- I 1 3 4- 2 4 8 - 1 7 o 4- 5 2 ( 4 4 5 1 5 K < - - - Z 3 7 4 7 5 4 - - 2 3 7 - 2 3 7 1 7 4 © ! S R R - 4 5 - 3 5 © 2 6 6 4 3 3 1 - 4 3 5 - 3 3 3 - 7 Z 3 2 1 3 © •'»„ - 1 o 3 - 3 3 6 - 3 4 o + 4 3 3 - 1 © 4 5 - 4 4 3 - 1 3 &5 2 5 6 o 6 R - 14-1 - l-Z ( 4- - 3 6 o + 5 4 2 - 1 3 5 5 - S o 1 - 1 7 15 7 7 o 5 3> RR - 1 ?o - l o 3 £ - 3 © 6 4- 5 5 4 - 1 1 5 2 - 4 2 6 - 1 4 3 8 2 5 o 5 1 « - 3 7 - 7 7 3 2 3 4 4 5 2 3 - 8 7 1 - 5 2 6 - 1 1 © 5 7 o 3 o ^ l?R - 6 I - 5 7 7 - 1 6 1 4 4 7 3 - 5 8 6 - 2 7 2 - 7 4 3 l 5 3 o 3> - se - 3 © 4 - • 6 3 4 2 8 o 3 3 3 — 1 2 4 - 4 7 6 1 1 8 © f e e b l e VJ M o m e n t s for D o u b l e E>e>nf. N o . De^«3< Load M o m e n t s , m. kips Live load Moments for \Jar\a\o\e. Load A n a l y s i s , in. k ips f l a s t IC Moments Mp, iVvlcipS Yield Moments My , 1V1 kips Greatest PosiJix/e M L Greatest Moment M L 35 + t 6 3 - 3 3 3 6 8 t 1 1 6 o 1 060 £ u - 61 4 Z 3 - 6 6 6 1 1 1 1 1 5 Oo 1 4 S o 7 L L - 3 7 + 5 2 3 - 1 o 1 3 I 5 4 7 ZoSo 16 2 5 - 1 I o + S £ 4 - I 3 3 o I 6 3 7 S o 5 2 2 4 o - 1 4 l +- 5 4 7 - 1 5 7 4 I I I 6 2 7 o 5 Z 3 o o 11 t t - 1 o 5 + 4 3 3 - 1 17 6 I 7 7 5 I S C c O 2 Z 6 o - 4 5 3 3 I - 6 7 & I o 6 3 Z.I S o I &oo - + 2 5 4 - 7 3 7 4^> J 1  4 o 1 5 4 o 7 6 + 3 3 5 - 1 3 6 5 3 I 4 4 5 1 2 3 5 + 3 5 + Z D S - 1 3 4 4 7 3 117 5 1 06 o c u S o + 2 5 6 - 1 6 3 4 t l 3 S>o 6 3 o + 7 3 + Z 4 7 - 1 6 4 4 I l 1 1 7 5 1 o G o 1 3 + Z 3 6 - 1 6 5 4 6 ! 1 4 4 5 1 2 3 5 - I I f 7 6 5 - Z 6 3 5 2 8 1 7 4 o l 6 4 o - 6 3 4- 3 3 6 - 1 o 4 - [loo 7 1 3 o 1 0OO - 1 3 o + 5 2 8 - \Z 6 o \do& 2 S 6 o Z 2 6 o - 1 6 4 5 8 o - I S 7 4 2 I 5 4 2 7 o S 2 3 o o - 4- 1 4 6 - 7 4 6 I 4 3 6 1 6 6 2 1 5 ) 5 - 1 6 4 + '5 & o - I 5 7 4 Z I 5 4 2 7 o 5 Z 3 o o - l 3 o 4- 5Z & - I t &o I 608 Z 5 6 o 2 2 6 o - 6 3 4- 3 3 6 - 1 o 4 Woo 2 I S o 1 6 o o is E L - I 1 + 2 6 5 - 7 6 3 5 2 6 1 7 4 o ! 5 4 o +• 1 3 + 2 3 6 - 1 6 5 4 6 I 1 4 4 5 1 2 3 5 +- 7 3 + 2 4 7 - 1 6 4 4 1 1 1 1 7 5 1 060 c * + 3 o 4 7 . 5 5 - 1 6 3 4 2 1 3 8 ° 8 3 o 3 5 f 2 3 5 - 1 3 4 4 - i 3 1 1 7 5 f 060 + 2 6 + 3 3 5 - 1 "O<0 5 31 I 4 4 5 1 7 3 5 - + 7.54 - Z 3 7 4 t> 1 1 7 4 o (5 4 o IS R« 4 5 4- 3 3 ( - 6 7 8 1 o6e> 7 1 S o 1 boo ) I ER - 1 o2> + 4 3 3 - 17 7 6 1 7 7 5 2 5 6 o 2 Z 6 o £> K - 1 4 J 4- 5 4 Z - 1 5 7 4 2 1 1 6 2 7 o S ZSoo ^ KB - 1 7 o 4- 5 5 4 - 13 3 6 1 © 3 7 Z 5 o 5 Z?4o 7 - 3 7 4- 5 X3> - l o 13 1 5 4 2 Z ° 3 o 1 6 2 5 - 6 1 4- - 6 8 6 11 17 I S 3 o l 4 5 o 3 RR - 3.5 4- •Z 6 3 - 32>3 6 6 2 1 ) e>o I o 6 o (he c o m p r e s s i o n Sxsle of the [rat^e . 1. Scale of m o v i e ^ f s — Im.* 1 l£>00 .in. lops . 3. Key points of frame shown ^<J5 - r 4. points of mCclr>ai^\iswi ( t ? ) Collapse M e c h ^ t n i s u n £ ! c y s t i c M o m e n j D : ̂ » <̂  r-^a t-r-> P E S I 6 N C O N P I T I O N I Lo<s.d<n^ ; 6 ^ 0 (t?s per ff. f i c ^ ^ r e 11 N O T E S 1. Scale o f f o o m e ^ J s - / m •= \*~,oo im. k i p s . P E 5 I 6 N C O N D I T l O f N 2 Lo<^oii>n <sj •' "7<0 lb* per ff. o n \~,of\-zc>r^o\\ projection 4-oo lbs per ff. o n v e r d c ^ l P » " 0 J « c f 'om ( a c t i n g f r o I T O leff J IZ I. £!^sfi<c m o r d e n t s 6\<rc <p\o\ke,ck. o n }(->e c o m p r e s s i o i o s i ^ e of f h e f r a m e . T. Si-ale, ol m o m e n t s — 1 i n = !E>00 i n k i p s . p o m f s o f ir-ame s k i o w n fkiuS — Ar. Hin^e poir-iks> o f M € c k < 9 n i s m Shown fhoS> —• a"2- (h) Collapse M e c h ^ r i i s i ^ i N O T E 5 : I. 5e<sWe ©f YTiome<\t<, — lio= l£>oomkips. poi'p-v/s cf frame showo fhcS — 3. \\\s~>ije points of m e t h a n i s i r i 5ho»~>n fl-wS — »̂ (b) C o l l a p s e KAeCr~i&r-\\S>\rr-i ( s h o w i n g p o S t \ i \ s e d*>n<A n z g a f i ^ e v a l u e s o f V A P . I A 6 . l E L O A P I N 6 C O N P I M C r S lo^c?( i'.o^ s : lbs per ff. o o l o o r i - i o - n W p r c > j e c f I ' O O 4oo lbs pe'' ff. or-i -JtrWca-l p e o j e c ^ i o n (jxc'tM^ia^ f r o r n ei l lner ^ i r e d lo - m j f i g u r e 14- 34 be expected. It has been determined that maximum e l a s t i c stress occurs i n the roof beams at the central j o in t and, as y i e l d i n g must f i r s t occur here, p l a s t i c hinges w i l l be created simultaneous- l y at these sect ions. Addit ional sections at which minimum values of Mp occur are located at j o in t s B and C and i t w i l l be f a i r l y M evident from inspect ion of Figure 11 that the c r i t i c a l mechanism w i l l conform to the mode indicated i n the same f igure . Only one ha l f of the structure on either side of centre l i n e need be considered i n evaluating F. The r e l a t i v e values of hinge rotat ion may be f igured from the geometrical concept of instantaneous centres, as fol lows: Relat ive frame dimensions and r a d i i of rotat ion of frame elements are shown in Figure 15, from which, Figure 15. Now, subst i tut ing these values for Q, with the appropriate values for Mp and M, into equation (5), gives: 2705(2.127) = 5750 , 1355(2.127) = 2880 , 980(2.254) = 2210 , 288(2.254) = 650 , 2705(1.127) = 3050 , 1570(1.127) = 1770 , 35 Numerator = 11010 , Denominator = 5300 , thus, F = 11010 = 2.08. 5300 Check analysis of the a l ternat ive mechanisms confirms th i s as the true value of load factor for loading condit ion (1). It w i l l be noted that on account of the symmetrical mode of f a i l u r e the number of hinges involved in the mechanism i s greater than the four required for a general collapse of the frame. Considering now loading condit ion (3). Sections of the structure at which minimum values of Mp occur are located i n M Figure 13. It should f i r s t be observed that hinges can develop at only two of the three key points at j o i n t D, as w i l l be evident i f the s t a t i c a l equi l ibrium of th i s j o in t i s considered; further , one of these hinges must be at the key point i d e n t i f i e d as 3 as y i e l d stress i s f i r s t reached i n the frame at t h i s sect ion. A second hinge must be at key point 7 as stress here i s equal to that at key point 3 and these two hinges w i l l therefore develop simultaneously. With these facts recognized the simplest procedure i s now to regard the t o t a l mechanism for the frame as the compounding of two elementary, mechanisms with one of these i n each bay. A br i e f review of the previous analysis of the s ingle bay bent w i l l reveal that the c r i t i c a l t o t a l mechanism must consist of hinges at key points 2, 3, 6, and 7, as shown i n Figure 16. Again using the concept of instantaneous centres, refer 36 Figure 16. Figure 17. Figure 17, the angles of rota t ion at hinge pqints are f igured as fol lows: B"0 H N" = 1 8 . 7 thus, DEN' DO'K ' BAH Q. 0: 41.7 44.5 (0.449) 33.0 18.7 (0.605) 47.3 50.8 (0.239) 31.0 0.391 + 0.239 0.605 + 0.239 0.605 + 0.449 0.449 0.605 0. 239 0.391 0.630 0.844 1.054 37 1.000 + 0.449 1.449 and s u b s t i t u t i n g these values of Q, and the appropriate values of Mp and M, i n t o equation (5), gives: 1310(0.630) = 825 , 2705(0.844) = 2285 , 1175(1.054) = 1240 , 2705(1.449) = 3920 , 320(0.630) = 205 , 1738(0.844) = 1465 , 272(1.054) = 287 , 1715(1.449) = 2483 , thus, Numerator P = 8270 , 8270 Denominator 1.86. = 4440 , 4440 F a i l u r e l o a d f a c t o r s f o r l o a d i n g conditions (1) and (3) are therefore 2.08 and 1.86, r e s p e c t i v e l y . However allowance must be made f o r the d i f f e r e n t forms of l o a d i n g i n v o l v e d i n these co n d i t i o n s , as p r e v i o u s l y discussed i n the case of the s i n g l e bent. The a d d i t i o n a l margins of safety involved i n the above f a c t o r s , over and above the appropriate recommended-values, f i g u r e at 2.08 - 1.11, f o r l o a d i n g c o n d i t i o n ( l j , and 1.86 - 1.88 1.41 1.32, f o r l o a d i n g c o n d i t i o n (3). The governing l o a d i n g c o n d i t i o n f o r the frame under the assumptions of the constant load method i s thus a c t u a l l y c o n d i t i o n (1). The c r i t i c a l l o a d f a c t o r i s therefore 2.08, and the corres- ponding f a i l u r e mechanism i s as shown i n Figure 11. 38 D. l i m i t Design Analysis for True Variable Loading The actual pattern of loading to be considered i s as previously discussed for the s ingle bay bent, and as with the e a r l i e r bent a reversed wind loading of 20.0 l b s . per sq. f t . i s assumed. The var ia t ions i n maximum pos i t ive and negative moments corresponding to th i s condit ion are tabulated i n Table VI and p lot ted i n Figure 14. The general symmetry of th i s moment diagram i s at once evident and th i s fact again suggests the probab i l i ty that the c r i t i c a l mechanism w i l l also be symmetrical i n form. Such a f a i l u r e mode involves a t o t a l of s ix hinges at key points 1, 2, 3, 5, 6, and 7. Only one ha l f of the frame need be considered and, from previously , the re l a t ive values of hinge ro ta t ion 6 at points 1, 2, and 3, f igure at « , = 2.127, Qz = 2.254, and © 3 = 1.127, and the equation for F u becomes: 2.127(2705 - (1.25)141) = 5380 , (1574) 2.127 = 3350 , 2.254(980 - (1.25)30) = 2120 , (258)2.254 = 580 , 1.127(2705 - (1.25)162) = 2820 , (1574)1.127 = 1780 , Numerator = 10320 , Denominator = 5710 , from which, F u = 10320 = 1.80. 5710 Further consideration of the a l ternat ive modes of f a i l u r e invo lv ing other possible combinations of key points , proves th i s to be the true mechanism for incremental col lapse . 39 Por f a i l u r e through a l ternat ing p l a s t i c i t y on the other hand reference to Table VI shows that the smallest value of My M . i s associated with the central column jo in t D where values of My and M L . are 1515 and 1496 i n . k ips , re spect ive ly . The l i m i t i n g value for F^ from equation (9) i s thus: ?i = 1515 (2) = 2.03 149? The c r i t i c a l f a i l u r e load for the condit ion of var iable loading i s thus produced through incremental collapse and corres- ponds to a f a i l u r e load factor of 1.80. The addit ional margin of safety provided by the bent, over and above spec i f i c a t ion requirements i s 1.80 , equals 1.28. The mode of f a i l u r e i s as 1.41 shown i n Figure 14. l£'-(J 14-o 74- a" 46 - 6' 11 "x l V ZoW 4' J 8 - o W e b 6 - S x 5 / & " f n 4̂  l^m^e 2 6 - o" Sf i f fences <a G ' - o " I3Z ' - o ' P e r i l s o f C<ar- iTi l e v e r - G i r d e r D c s i ^ Z? - o " 3 4 - o " 4 6 - o " Pl 2 o " x 5 / & " ! Z7."> v 6 - Zo' x 5 / « - Web o'-S" * &̂ \Y\ro\ja\Y~)0\jk - A I f 3 - o ' I St- o" Pef<?if!s o f C o o i i n u o a s O i r ^ l e r Desi^ r ~ > F i g u r e 1 6 F\a>^>r-e I'D 40 3.03 Analysis of Bridge Girders General The complete three span, two lane, highway bridge, from which two examples to be l a t e r considered are taken, i s shown i n general arrangement i n Figure 3(b). The two examples chosen for study involve : 1. The actual bridge, cons i s t ing of two cant i lever g irders which overhang the central p iers and support simple g irder spans at each end of the bridge v i a bearings at the outer ends of the cant i l ever s . 2. A hypothetical continuous g irder var iant , which consists of twin girders continuous throughout the ent ire length of the bridge. General s t ructura l de ta i l s of the plate girders and deck of the actual bridge are shown i n Figures 18 and 19. The main girders are. of bu i l t -up welded sect ion throughout, cons i s t ing of ASTM A-7 s t ructura l s tee l webs and flanges. The deck i s composed of cast- in-place concrete. No long i tud ina l shear connectors are provided between deck and girders and composite action i s therefore not allowed for i n the designs. In the case of design 1, invo lv ing the cant i lever g i rders , interes t w i l l be confined to the portions of the cant i levers between the centra l supports i n which reversals i n moment take place due to the movement of the l i v e loads . The simple end beams are of no 41 specia l in teres t from the point of view of the study. The cant i lever g irders are s t a t i c a l l y determinate and are therefore not subject to incremental col lapse , however as a re su l t of the reversals i n moment they are subject to a l ternat ing p l a s t i c i t y and must be examined for th i s mode of f a i l u r e . The continuous girders of design 2 are s t a t i c a l l y indeterminate and may be subject to f a i l u r e through e i ther incremental collapse or a l ternat ing p l a s t i c i t y . Both of these p o s s i b i l i t i e s must therefore be considered for th i s case. Both the actual and hypothetical structures are designed to withstand the standard AASHO H20-S16 loading i n each lane, to the fol lowing spec i f i ca t ions : (a) GSA Speci f icat ions for Steel Highway Bridges, 1952, (b) AASHO Standard Speci f icat ions for Highway Bridges, (c) ACI Bu i ld ing Code Requirements for Reinforced Concrete. In a l l instances where a l ternat ive requirements oceured i n these several spec i f i ca t ions the severest condit ion was accepted as governing. T&lole> VI ! M o v n e n l s Por 6 a r i f i l e v / e r G i r d e r - 6 t r d e r - N o . i o f f . k i p s f o : M ^ x i m u w - > T o \ c\ | M o m a n t s # f h k i p s . M o vn £r>f S j f h k i p s M £\* l»V) c>vn L ' v e L o < a d M o n o e n h s P h k i p s Lo<ac/i I t O ^ L o & ck i i o <j i n S p a n 0 l o # < ^ m ^ m S p a n © L i v / e * L o < a ^ i i o ^ M c n e n f s (cols. 1+ 4 ) (cck 7 + S + 5 ) Po 5 i +1 v / « M o m e n t (ool. 4 ) M o n o e n f s Cools. 5 + 5 ) Col. J . C o l . t. C e l . S . C o l . 4 - . C o l . £>. C o / . 7 . C o l . & . C o l . 2>. C o l . l o . - 3 5 6 S - S o l o - - - - 6 5 7 5 - - S o l o 3 o I o 4 f 6- - Z 3 3 o + 2 6 o £ > - 6 - 1 5 + Z I 3 o - 3 3 6 o + ? 6 o S - 2 3 4 5 5 S £ o 6 . + . 7 8 5 - I 7 6 o + 3£> 1 o - 1 1 l o +- 4 3 3 £ - Z 1 o5 + 3 6 1 o - - Z & S o 6 5 o o e>, + t>oo - 1 5 o £ 4 3 T o o - I 3 6 o +• 4 6 o o - Z o S o + 3 7 o o - - Z & 0 5 6 5 6 5 + 1&£> - I « o 4- 3 6 1 O - / CrOZ + 4 3 3 5 - Z l o i + 3 6 1 o - Z 8 t ) o 6 5 o o - 4 1 5 6 S o + £ 6 o £ ~ Z \ oo + 7 j e>o - 3 3 6 0 + Z 6 o 5 - 7 3 4 5 5 5 5 o - 3 5 6 5 - - - Z1\ & - - 6 5 7 5 - - 3 o l o S o l o A s <a*:le l o t f ^ m ^ j s • s h o w n i n figure 7.0 (o) are r e v e r s i b l e n n <̂  / i n n r o n r \ c ?ir~ > e o + s s h o w n in c o l u m n 3 b e f r c a n s p o s e d , w>Wr> i n v e r s i o n , f o c o l i / r m o S . C o l u m n J> n o o m e o l ' s i7ir<z S i m i l a r l y (• n s f e r < a b l C , w i l l - , i n v e r s i o n , F o C o l u i n r i 2> , Ma^imuh-i m o m g n l j s h o w o io> c o l c " ^ n S 6 , 7 , ^> ^ em<si IO, a r e f o r f l o e s e v ' e r e s f o f l l n ^ s e ^>^ le ! o « < s T m < g COr~><5( i f | o n S . * M o f e : C o n c e a l r « * ) e d i F c m e l L o a d i n g s , m k i p s ^—3.$S »-1-2 6io &.lo 0.12. 9.40 '3 .7O 9.4O e.3o 6.S0 5.4o 9 . 7 o 5.46 8-IX &-3o 6.3o 6.12 3.55 I5.S" Z4.o' 24.0' 24-. o' I5.S' Zfc.o' 3o.o' 24.0' 2*.o' 24.0' 3o.o' Z 6 . 0 ' 15.5' 24.o' Z4.o' 24'. 0' IS.S'( 'If Z, 3, 4, 5, 6, 7, 0, 3 , . lo, II, 12, IS, 14, 15, 1 7 1 !o3.o 26 . o' "Pis<ribui«a l o a d of 1.72-7 yn. 132.©' Z6.0' loS.o1 (,̂0 Dead Loadings, of Cantilever- Girder Design C Cor\cesiircxted fa\nel lo&dingz m k/p£ 4.fe6 5.93 6.3(7 8-5o Z>Ao /o.4o 2>.4o 0.3o 6.Z0 3.4-0 to.4o 3.4o S.30 &.30 0 .B3 4.G6 27 .0' 24 . 0 ' 24.0* J 24.0' 3o.o' 3 0.0 24 .o' 24.o' 24.0 j 3o. 0' 3o.o' 24.o' 24.0' 24.0 11.0 . ' l 3Z •S i 7 i I2z I3t 14,. I 6 t I7-, I2t>. © "Distributed t o * d of I .7I7 K/f|-. 132. o' I25 .o ' (b) Dead-loadings' of Continuous. Girder Destan J i_ Sfe.4fc 56.4-" I 4 I ' 1 l4.o 14.0 I I 1 I !7<o. o 15-2. o ' 1 2 3 . o S p a n <J) i p ^ c i (z) (c) Li^se Loadings t o r C>\r<de<<' D e s i ^ r i s F i g u r e 2 o S p a n (3) •= tooo ff.kr'ps 3o fh 42 3.031 Analysis of Canti lever Girder Design A. E l a s t i c Analysis Dead weights of the s teel system and concrete deck for the cant i lever design are shown i n Figure 20(a). The i n t e n s i t i e s and form of l i v e load considered i s shown i n Figure 20(c), t h i s conforming to the AASHO H20-S16 pattern of loading and inc luding the spec i f ied impact allowance. The combinations of these basic loadings which were assumed i n the e l a s t i c design for maximum pos i t ive and negative moments over the centre span port ion of the cant i levers are as fol lows: (1) For maximum pos i t ive moments, the entire dead load system of the cant i levers and end beams (Figure 20(a)), together with the l i v e load (Figure 20(c)) applied between the centre span supports only . (2) For maximum negative moments, the entire dead load system of the cant i levers and end beams (Figure 20(a)), together with the act ion of l i v e loads (Figure 20(c)) applied i n both outer spans simultaneously. Maximum e l a s t i c moments derived from these combinations of load are f igured to be: (1) Maximum pos i t ive moment, at midspan = +4600 f t k ips , (2) Maximum negative moments, at supports = — 6575 f t k ip s . The extreme f l exura l stress permitted by spec i f i c a t ion for 43 A-7 s teel i s 20.0 k s i . Actual stress r e a l i z e d at the midspan section i s f igured at 20.1 k s i , and for the support sections at 19.9 k s i . B. L imi t Design Analysis by Constant Load Method As the centre span i s s t a t i c a l l y determinate only one hinge i s required for the development of a f a i l u r e mechanism and l i m i t design analysis for constant loading simply consists i n the determination of the c r i t i c a l p l a s t i c moment of resistance for the span, which w i l l be e i ther at the midspan sect ion or at the support sections, and comparing t h i s with the appropriate applied moment. Considering f i r s t l y the maximum pos i t ive moment condit ion at midspan. This moment i s 4600 f t . kips and thus th i s value can be substituted for M i n equation (5). The p l a s t i c moment of resistance Mp i s also involved i n th i s equation and i t s value may be ca lculated as fol lows: P l a s t i c moment of web = (99.0) .0.625(33.0)._1 = 4200 f t . k ips . 12 " M • f langes* (99.88) 20.0(0.875) 33.OJ. = 4800 " " 12 thus, t o t a l p l a s t i c moment of sect ion =9000 f t . kips As only one hinge i s involved the value for Q i n both numerator and denoninator of the equation for F i s , of course, the same and thus the equation becomes: F = 9000 = 1 . 9 6 46*00 44 S imi l a r ly for the maximum negative moment condit ion at the supports. The value for M i n th i s case i s 6575 f t . kips and for Mp i s 12560 f t . k ips , and thus: F = 12560 = 1.91 6575 The value of F at the support sections i s the smaller and t h i s i s therefore the l i m i t i n g value. It i s in te re s t ing to here note that as there i s no essent ia l difference between t h i s l i m i t design analysis of the g irder and the procedure of e l a s t i c design for s t a t i c a l l y determinate structures i t i s not surpr i s ing that these values of load factor for the girder are not much di f ferent from the l i m i t i n g load factor requirement of the AISC Code, of 1.88. C. L imi t Design Analysis for True Variable Loading As previously mentioned the f a i l u r e of the g irder under var iable loading can only be associated with the state of a l ternat ing p l a s t i c i t y . For t h i s state the f a i l u r e load factor i s given by equation (9). The largest range i n l i v e load moment occurs at the mid- span section and i s equal to 6565 f t . k ips , refer Table V l l . The correct value for My must be computed on the basis of a l i n e a r stress d i s t r i b u t i o n across the girder section varying from zero at the neutral axis to the y i e l d stress value of 33.0 k s i at the extreme f ib re s , and for the midspan sect ion th i s i s 4 5 e q u a l t o 7 5 3 0 f t . k i p s . T h u s a t m i d s p a n : P ' = ( 2 ) 7 5 3 0 = 2 . 3 0 T h e g i r d e r s e c t i o n 1 8 f t . f r o m m i d s p a n , w h e r e f l a n g e a r e a s a r e r e d u c e d f r o m 2 0 % | " t o 2 0 " x f " , s h o u l d a l s o b e e x a m i n e d f o r f a i l u r e . I n t h i s c a s e t h e r a n g e i n m o m e n t s i s e q u a l t o 6 3 6 5 f t . k i p s a n d t h e y i e l d m o m e n t i s e q u a l t o 6 8 8 0 f t . k i p s , t h u s : P i = ( 2 ) 6 8 8 0 = 2 . 1 4 6 3 6 5 T h e l i m i t i n g l o a d f a c t o r f o r f a i l u r e u n d e r v a r i a b l e l o a d i n g i s t h e r e f o r e 2 . 1 4 w i t h f a i l u r e o c c u r i n g , i n t h e s t a t e o f a l t e r n a t i n g p l a s t i c i t y , a t p o i n t s 1 8 f t . f r o m m i d s p a n i n t h e r e d u c e d f l a n g e a r e a s e c t i o n s o f t h e g i r d e r . O M « » n l i f o r C o r i f i n u o u s G i r d e r " . C- i r - d e r - P a n e J N o . X)e,ad load fh k f p s l i v e Load M o m e n t s , ft. k i p s Total M o i m e ^ f s , ff k i 'ps. P l a s t i c M o m e n l s , M P , ft. k i ps W e f«rf K / lovn emt .S j M y , fh k i p s P o s i t i v e M a x i m \jm N e g o f i v e K a n ^ e i n M o m e n t s P o & if i ^ C 4 t l 3 o + ZZZc - - Z T 6 Z 4 3 5 4 4 3 5 o - 1 1 l o o 3 5 6 c ? + - Z T & 5 4 3 o 3 S - 5 t o 3 6 I S + £ 6 6 c? - 1 1 1 o o 3 6 & o + 1 1 o o 4 -Z -785 - 1 1 o 3 5 6 5 4 4 3 6 5 - ( M o o 3 L & o 4- 4 4 o ' + 1 g>65 - 1 o 1 6 2 6 8 c ? 4 Z 3 o £ > - 6 1 5 7 6 1 o 6 I 3>o - 342>& 4 4 Z c ? - Zt>6c? 3 3 6c? - - 6 4 5 5 1 Z 5 6 o l o 3 2 o 3 3 o 4- 1 t Z o - I Z 4 o 4 i ie>o - 1 5 7 c ? 7 6 1 o 1 5 o + 8 7 & 4 Z 3 S 5 - .1 Z 4 o 3 6 3 5 4 3 Z 3 o - 3 6 6 -7 6 1 o 6 l o 4- 3 8 5 4 Z 4 Z 5 - I 2 4-c? 3 6 6 5 4 5 4 / c? - -ZS5 7 6 r o 6 12>o. 4 6 7 5 + Z 3 S 6 - 1 2 4 o 3 6 3 5 4- 3 - Z 3 o - 3 6 5 -7 6 I o 61 t>o 3 3 o 4 I 6 L D - 1 Z 4 o 2 7 6 o 4 | | S o - 1 S 7 o 7 6 1 o 6 ! S o - 3 4 3 5 4- 4 Z o - Z t > 6 o 3 3 6 o - - 6 4 55 \ZbQ> o I o 3 t o ! 3 Z 4 4 4 o 4- I 8 6 6 - 1 e? 1 £ 2 8 8 o + 2 3 o 6 - 5 7 5 n 6 I o 61 2>o I * * 4 2 1 6 o 4- Z -7ft5 - 7 7 o 3 6 5 6 4 4 t > 6 5 - ! M o o 3 5 & o 4 -Z-753 4 3ot>5> - i t o 3 6 IS 4- 6 8 £ o - ( M o o 3 5 . & C ? 4 "Z 1 3 o •+.. -z ? Z o - Z -7 5 • Z 4 5 5 4 4 3 & o - 1 1 1 o o 3 & 6 o C(o.o 1 S p a n © Span (D f a i l u r e M o ^ e s for Cor^ii^^ous. d>\ro\er~. IX D e r i v a t o r * oi loc\<d Factors for C o ^ f i n o o u s dxr-cier-. Failure PIe*s f c Momer-i f s ff. b p s fo\al El a s h e M o m e n t s M , ff. k i p s £ o f « f i O i n ( 9 S U M of fJurri&raboi' Suvn o f PenovrnrWor Load rac\c-r Mode H i>> g e Ff. i H iV-> c j e P>. "2 Pf. 3 Pf. 1 H 1 n ^ £ Ph Z Pf. 3 H ir-igg Pf. 1 Pf. £ Pf. 3 Te^vns . M p . e f c r - t v - v S j M . <9 f -ro? i © o 3 3 o o 3 5 3 S /. o o o. I 3 S 3 - ? 3 o 7 . 4 T I (b) II l o o l 7 _ 5 C * o 1. o o o . 3 3 S l<o oS o T S 3 o 7 . I3> "7 1 o l 7 5 6 o 3 3 o o 4 1 5 o I. o o o.&oD 1 1 7 1 o 6 6 . 5 o 7 - 6 G Z l t ; C o 1 ( » i o l t 5 6 , o 3 4 1 o 3 5 4 o L o o 7 . oo 1. o o 4 o 3 4 o 1 4 1 o o Failure. Moments (Mp ~ f ff. k ips L i V e Load M o i n e n l s M u , ff. k ' p s H m^e Rof«f ior% i © - S u i v * of Wumeraf o r (Mp-hpMp)e S o > r n of D^nommrtfor- M u . 9 Lo<5^d f<acf o r K Mode Hm^e Pf. I H i n < ^ « Pf. 7. Ph 3 Pf. 1 H m Pf. -z Pf. 3 l-l Tn g « Pf. 1 Wi n<3<j Pf. 7 Ph 3 5 5 5 o 6-Z.oo 1 6 4 o 2 5 6 o ). oo o. I3t> 6 4> 4 o Z I o o 3. ic& 6 t o o 3 o 3 S Z % o 1. o o o. 3t>5 1 o & S o 4-z.c*& t-M 1 (c^ 6 7 So 6 1 o o 2 7 o o I. o o o. 806 I 1 8 7 o v 4 5 S o 7 . 6 I Z B t o o 6 3 8 o 8 7 oo ? . 3 6 o 7 4-z. 5 7 t ) 6 o 1. oo 7 . o o I. o o l o 1 1 o 2 . 7 I  46 3.032 Analysis of Continuous Girder Design A. E l a s t i c Analysis Structural de ta i l s of the continuous girder design are shown i n Figures 18 and 19. Dead weights of the s teel system and concrete deck are shown i n Figure 20(b). The H20-S16 vehic le loading used i n the design i s , of course, i d e n t i c a l with that considered for the previous case, refer Figure 20(c). The procedure used i n the e l a s t i c analysis of the continuous girders involves the preparation of influence l i n e s for c r i t i c a l sections of the g irder , and the construction of an envelope of maximum pos i t ive and negative moments across each span. Deta i l s of the ca lcula t ions necessary for th i s analysis need not be given here but the f i n a l moment envelopes, for both l i v e loading and combined l i v e and dead loadings, are shown i n Figure 22. Maximum moments at panel points of the girders are also tabulated i n Table V i l l . Extreme f ib re stresses at c r i t i c a l sections of the g irders , noted here for sake of interes t , are f igured to be: 1. Near mid-section of outer spans - 20.1 k s i , 2. At mid-section of centre span - 18.2 k s i , 3. At centre supports - 19.5 k s i . B. Limit Design Analysis by Constant Load Method The key points of the girders are associated with sections where minimum values of _Mp occur, and a reference to Figure 19 M 47 w i l l indicate that a number of such sections must exist across the g i rders . The number of hinge points required for the development of a f a i l u r e mechanism on the other hand may be any integer between two and f i v e , depending on whether f a i l u r e i s p a r t i a l and l i m i t e d to only one span or whether a l l spans are involved i n a t o t a l collapse of the g i rder . The l i m i t design analysis on f i r s t s ight would thus appear to be a formidable task invo lv ing a l l possible combinations of key points i n e i ther p a r t i a l or to t a l col lapse . The problem however i s considerably s impl i f i ed once the elementary modes of f a i l u r e of the system are i d e n t i f i e d and the procedure of analysis then becomes one of inves t iga t ing each elementary mode i n turn and superimposing these modes for a l l l i k e l y combinations of f a i l u r e . The elementary f a i l u r e modes, invo lv ing each span i n d i v i d u a l l y , are read i ly i d e n t i f i e d and these are shown i n Figure 23. There are several key points i n each outer span and there are thus several a l ternat ive elementary modes for each of these spans. Now, i t w i l l be seen from the form of the elements ary mechanisms that any possible complex -mechanism, invo lv ing some combination of the elementary modes, must exhibit a f a i l u r e load factor with a value which stands somewhere between the l i m i t i n g load factors appropriate to each of the elementary modes which are combined. The load factor of the complex mechanism therefore cannot poss ibly be l e s s than the smallest load factor 48 obtained for the separate elementary modes. I t , i s thus evident that the c r i t i c a l mechanism for the girders must be an elementary mode and only these forms need therefore be considered. Thus for elementary mode l b i n span (1)(or mode 3b i n span (3)) the point of maximum to ta l moment near the midpoint of the g i rder , which i s a section of minimum Mp, i s located 51 f t . from U the end of the g irder where the moment i s equal to +5850 f t . k ip s . The corresponding moment at support B, with l i v e load so placed to give the above maximum pos i t ive moment, f igures at -4260 f t . k ip s . Values of hinge rota t ion at the two points are ca lcula ted as fol lows: 5i.o ^ o Figure 24 PAN = PBN = 1.0 51.0 1.0 78.0 PBN 0.0196 0.0128 0.0196 - 0.0128 = 0.0324 0.0128 thus r e l a t i v e values of Qz and Q3 are lk.000 and 0.395, and load factor F determined by means of equation (5) becomes: 11100(1.000) = 11100 , 12560(0.395) = 4950 , 5850(1.000) = 5850 , 4260(0.395) = 1680 , Numerator = 16050 , Denominator = 7530 , 49 and thus, F - 16Q50 = 2 . 1 3 7530 Similar ca lcula t ions required for a l l of the other element- ary mechanisms are given i n Table I X . The value of 2 . 1 3 obtained for the f a i l u r e load factor i n mechanism l b i s l i m i t i n g and th i s thus represents the c r i t i c a l load factor for the girders under the assumptions of the constant load method. G. L imi t Design Analysis for True Variable Loading Considering f i r s t l y the state of incremental co l lapse . The envelopes of maximum pos i t ive and negative l i v e load moments shown i n Figure 22 represent the var ia t ions i n pos i t ive and negative values of M L across the g i rders . This f igure also shows the va r i a t i on i n dead load moment IL0. Now, i t w i l l be evident from inspect ion of th i s f igure that the key points of the girders for the var iable loading condit ion, sections of minimum Mp - F p . M p , are at the same locat ions as those e a r l i e r establ ished Mi_ for constant loading and therefore the elementary modes of f a i l - ure for the present form of loading w i l l be exactly the same as those considered previously . Also, the e a r l i e r comments on the load factor of combined mechanisms w i l l also apply equally to incremental col lapse and thus analysis can again be l i m i t e d to these same elementary.modes. Thus for elementary mechanism l b , e l a s t i c moments at the midspan hinge, as obtained from Figure 1 9 , are M D = 2755 f t . k ips , 1 ^ = 3095 f t . k ips , and for the hinge at support B , M p = 50 3495 f t . k i p s , M L « 2960 f t . k i p s . Thus l o a d f a c t o r F u becomes: (11100 - (1.25)2755)1.000 = 7650 , (3095)1.000 = 3095 , (12560 - (1.25)3495)0.395 = 3240 , (2960)0.395 = 1170 , 10890 , 4 265 , and, F u = 10890 = 2.55 "4 265 The a l t e r n a t i v e elementary mechanisms are analysed i n Table IX and r e f e r e n c e to t h i s t a b l e w i l l show that the l i m i t i n g value of F u f o r f a i l u r e i s that obtained above f o r mode l b . Cons i d e r i n g now the s t a t e of a l t e r n a t i n g p l a s t i c i t y . The l a r g e s t value f o r M u occurs at the midpoint of the centre span and as t h i s c o i n c i d e s with the sma l l e s t value f o r My f o r the g i r d e r t h i s s e c t i o n must be the c r i t i c a l one f o r a l t e r n a t i n g p l a s t i c i t y . The values of these moments at t h i s s e c t i o n are 3665 and 6190 f t . k i p s , r e s p e c t i v e l y , and thus: %l = (2) 6190 = 3.38 3665 The sm a l l e s t l o a d f a c t o r obtained f o r the g i r d e r s i s 2.55 and t h i s i s t h e r e f o r e the true l o a d f a c t o r f o r the v a r i a b l e l o a d i n g c o n d i t i o n . Table X SKjrs~\m c\r op ^, e S u 1 f S . 1 f e t n Load factor Values (or fbe following $ f r u c t u r e s : Simple. S>cvy Cable Bent Double Say Gable Sent . Cant i lev'er" S r i d ^ Girder Con t inuous &ndgg 6irder 1 Govemno^ failure Lo^d factor' T ' for" structure writer Constant l o a d i n g - 1 . 6 1 Z . 0 6 1 . t>1 -z. 1 2> z 11 w i i f Load facror~ ( for- di e S i <sj i o ^ 1. 4-1 1 • & & . 1 - 6 6 1. 6 6 3 f sdf i 'o o f F value \o L i V v i i'tiV->g D e s i ^ t n \/<alue 1. 14- 1 . 1 / ! . < ? Z 1 . 1 3 4 failure- Load V&.c\or 1 F u ' f o r s\/uc\<jre ur-ider \Jar~\able Loadi^^ 1. Q>Z 1 . S o Z . 14- Z . 5 5 S L i v v o i t i ' - i ^ Lo#^ Factor" ( for d e s i<3rV) 1.4-1 1 . 4 l 1 . & & ! . 8 & ^ a f i o o f ' f-L' v^ lue fo Li*Y-ufi>--->cJ) D e s i g n V i a / o e L I S 1 . 2 & 1 - 1 4 - 1 . hQ? 7 f^-e»o<5if u r e C o l l a p s e ^ a f i ' o ^ ( co r rec - te^ = i t e ^ 6> F ' it€v^ .3 1 . o | 1 . 1 & 1 . 1 Z \ .Zo 51 SECTION 4 INTERPRETATIONS AND FURTHER CONSIDERATIONS 4.01 Interpretat ion of Analytical^ Results A summary of the resul t s obtained i n the previous analyses i s to be found i n Table X, but before proceeding with a discuss- ion of these resu l t s i t i s desirable to c l a r i f y several previous conceptions. The f i r s t point to be mentioned i s the necessity for further elaboration i n the d e f i n i t i o n of load factor F. The smallest numerical value obtained for th i s load factor i s not necessar i ly the governing value for f a i l u r e load, as was found i n the case of the double bay bent. Combinations of working loads containing wind forces have a d i f ferent safety requirement (1.41 for the AISC Code) from combinations which exclude wind forces (1.88), and f u l l allowance must be made for th i s i n e s tab l i sh ing the governing value of F. Thus for the double bay bent example the smallest value of F i s 1.86 with wind and other forces act ing, but the governing value i s actual ly that which applies to the condit ion of gravi ty loads act ing alone, that i s , 2.08. The governing value of F must therefore be thought of i n terms of the margin of safety over l i m i t i n g code requirements rather than as the numerically smallest value of load factor obtainable. Then sagaih, the same considerations w i l l apply i n deter- 52 mining the premature collapse r a t io J \ , , and the necessity for F subst i tut ing appropriate margins of safety for the F u and F values, when di f ferent safety requirements are involved, w i l l be apparent. The importance of recognizing these facts i s well i l l i s t r a t e d i n the resu l t s obtained for the double bay bent. I f the smallest numerical values of load factor are considered F u and F are 1.80 and 1.86, respect ive ly , and the . ra t io i s 0.97. The corrected r a t i o on the other hand i s actual ly 1.15 and an apparent premature collapse condit ion i s i n fact the reverse when proper account i s made of the safety requirement aspect. Returning now to Table X. The correct premature col lapse r a t io s are shown i n item 7 of the tabulat ion. It w i l l be at once evident that these values are a l l greater than 1.0 and thus premature collapse i s not a factor i n any of the s t ructura l examples considered. The smallest value obtained i s that of 1.01 for the s ingle bay bent and the largest value i s 1.20 for the continuous g i rders . The l ightweight structures may be said to exhibi t r a t io s which, on the--average, are lower than those, for the high dead- weight structures , as i s of course to be expected from equation (10), and i t would seem reasonable to conclude that where structures are subjected to highly var iable loadings the actual col lapse load w i l l be somewhere between zero to twenty percent greater than the f a i l u r e load predicted by the constant load method, depending on 53 the proportion of l i v e to dead loads present with the smaller dead loads tending to produce the smaller differences i n f a i l u r e load values. It w i l l be noted that the more highly redundant structures of both types portray ra t ios which are d e f i n i t e l y larger than those for the corresponding simpler forms, although th i s could be coincidental as there appears to be no theoret ica l basis for a tendency one way or the other. F i n a l l y , i t should be recognized that the values obtained for F L , and thus also for the premature collapse r a t i o , are, i n three out of the four cases considered, dependent on the value chosen for the dead load factor F D . Dead loading i s not involved i n the f a i l u r e of the cant i lever girders and as far as the gable bents are concerned, i n which dead weight i s low, any change i n the value of F o would have immaterial ef fects ; however th i s i s not so i n the case of the continuous g i rders . A smaller factor for dead loads, over those used for l i v e loads, must be acknowledged as j u s t i f i a b l e i n view of the fact that the magnitude of these loads i s known with far greater cer ta inty than i s the magnitude, and also doubtful behaviour, of l i v e loads. The value used i n th i s study for F p , of 1.25, recognizes th i s fact and th i s value i s en t i re ly consistent with the somewhat higher values recommended by code for l i v e loadings. However, apart from th i s l o g i c a l basis for the use of a factor of 1.25 i t i s in tere s t ing to reason the effect of the choice of a higher F ^ 54 value. I f the f igure of 1.41 i s selected ( this value i s used as a l i v e load factor and i t i s d i f f i c u l t to see how F p could reasonably be greater than this) then the l i v e load f a i l u r e factor i 1 ^ for the continuous girders works out to be 2.39, instead of 2.55. Now, the constant load f a i l u r e factor i s 2.13 and thus the premature collapse ra t io i s reduced from 1.20 to 1.12 with the increase i n FD; however th i s reduced value has not a l tered the sense of the resu l t for the continuous girders as far as the present study i s concerned for the ra t io i s s t i l l greater than 1.0, ind ica t ing that premature collapse does not take place. Arguments regarding the value to be assigned to F & are thus l a r g e l y academic as far as the f i n a l purpose of t h i s paper i s concerned for no s ign i f i cant change can be grought about i n the premature collapse ra t ios for any of the four s t ruc tura l examples considered. 55 4.02 Further Considerations Regarding Premature Collapse Premature collapse i s therefore not a factor i n the s t ructura l examples considered, and as these structures were s p e c i f i c a l l y chosen to represent severe cases of loading, i t would perhaps seem reasonable to extrapolate th i s f ind ing to the extent of covering a l l problems of s t ruc tura l design l i k e l y to be met i n prac t i ce . It must however be agreed that even severer, and not at a l l exceptional , cases are very l i k e l y to f a l l within the scope of design. The smallest premature collapse r a t i o recorded for the four cases considered i s that of 1.01, for the s ingle bay bent, and i t would not therefore take very much to create the severer conditions leading to a r a t io of les s than 1.0. Thus a small degree of premature col lapse , based on the theore t i ca l evaluation of load factor , must be accepted as highly probable within the range of p r a c t i c a l design problems. Insofar as these theore t i ca l evaluations are concerned i t must be remembered that certa in basic assumptions are involved i n the der ivat ion of load factor equations. One such assumption i s that regarding the magnitude of the r e s i s t i n g moment at hinge points which was e a r l i e r referred to as the p l a s t i c moment and for which a constant value was assumed throughout the ent i re range of angle change at the hinge. This constant value for the moment i s idea l i zed , as w i l l be read i ly appreciated i f reference i s made to Figure 2, and e n t i r e l y neglects the increase i n moment which resu l t s i f s t ra ins at the hinge section extend into the 56 strain-hardening region. How, i t has been t h e o r e t i c a l l y demon- strated by exact methods of analysis that the ea r l i e s t hinge points formed i n a structure w i l l almost invar i ab ly extend into t h i s strain-hardening region and these hinges therefore must offer moments of resistance which are greater than the ' p l a s t i c 1 value Mp. It follows that the true load capacity of a structure at f a i l u r e , that i s when the f i n a l hinge becomes f u l l y mobil ized and the mechanism state mater ia l izes , must be somewhat greater than the computed l i m i t design value. This fact has been demon- strated i n test cases involv ing incremental collapse i n which experimental continuous beams were subjected to var iable concentrated loads, and the reported resu l t s show load increases of up to nineteen percent . It i s doubtful that increases of t h i s order can be expected i n a l l instances of incremental collapse and no increase i s allowable under conditions involv ing a l t e rna t in p l a s t i c i t y where f a i l u r e i s associated with the e l a s t i c state at only one section of the structure, nevertheless , i n general, some increase i n actual f a i l u r e load i s indicated over and above the computed value for i\. A second assumption embodied i n the theoret ica l evaluations i s that the working loads applied to the structure for both constant and var iable loading conditions must be of i d e n t i c a l magnitude i f exactly comparable f a i l u r e load factors are to be obtained. Now, th i s assumption i s a lso, s t r i c t l y , not quite v a l i d as with the constant loading condit ion only one appl ica t ion of the 57 f a i l u r e l o a d i s neces sary i n o rder to produce the mechanism s t a te r e s u l t i n g i n f a i l u r e o f the s t r u c t u r e , whereas under v a r i a b l e l o a d i n g f a i l u r e takes p l a c e o n l y a f t e r a number o f r e p e a t e d c y c l e s o f the f a i l u r e l o a d s are a p p l i e d , and t h i s i s so whether f a i l u r e occur s through i n c r e m e n t a l c o l l a p s e or a l t e r - n a t i n g p l a s t i c i t y . In terms o f the p r o b a b i l i t y o f l o a d occurence t h e r e i s , t h e n , j u s t i f i c a t i o n f o r e i t h e r u s i n g working l o a d s f o r v a r i a b l e l o a d i n g which are somewhat l e s s than those used f o r cons tant l o a d i n g , o r , ( f o r i t does not matter which approach i s cons idered) u s i n g a s a f e t y requirement which i s s l i g h t l y l e s s f o r the former c o n d i t i o n than f o r the l a t t e r . Such s m a l l e r work ing l o a d s (or s m a l l e r s a f e ty requirement) would aga in have the e f f e c t o f i n c r e a s i n g the t r u e l o a d c a p a c i t y o f the s t r u c t u r e above the computed v a l u e . As a r e s u l t o f these two i n f l u e n c e s the a c t u a l c o l l a p s e l o a d o f a s t r u c t u r e w i l l be g r e a t e r than t h a t i n d i c a t e d by the t h e o r e t i c a l va lue o f F L , and t h i s i n c r e a s e d l o a d c a p a c i t y must, o f course , a l s o p r o p o r t i o n a t e l y i n c r e a s e the premature c o l l a p s e r a t i o Fu. F Now, t h i s r e s u l t i n g i n c r e a s e i n the l o a d f a c t o r r a t i o w i l l t end to o f f s e t the e f f e c t o f a severer l o a d i n g c o n d i t i o n ( i n r e d u c i n g the r a t i o ) and i t would seem not unreasonable to conclude t h a t i n a c t u a l f a c t i n problems o f p r a c t i c a l d e s i g n i t i s h i g h l y u n l i k e l y tha t t rue r a t i o s o f l e s s than 1.0 w i l l o c c u r . 58 SECTION 5 CONCLUSIONS The f o l l o w i n g general conclusions are i n d i c a t e d by the present study: 1. The t h e o r e t i c a l f i n d i n g s suggest that s t r u c t u r a l f a i l u r e i n p r a c t i c e can always be expected to occur w i t h i n acceptable l i m i t s of the Value f o r u l t i m a t e l o a d computed by means of the s i m p l i f i e d l i m i t design method, and the method would therefore appear to be e n t i r e l y v a l i d f o r design purposes. 2. For p r a c t i c a l s t r u c t u r e s the ac t u a l u l t i m a t e l o a d w i l l g e n e r a l l y range between a value equal t o , to perhaps more than twenty percent greater than, the value p r e d i c t e d by the s i m p l i f i e d method; the exact r e l a t i o n s h i p f o r a p a r t i c u l a r s t r u c t u r e depending on the proportion of l i v e to dead l o a d present and the v a r i a b l e q u a l i t y of the l i v e loads. 3. In computing the t h e o r e t i c a l c o l l a p s e loads of a str u c t u r e subjected to the constant and v a r i a b l e forms of lo a d i n g i t i s e s s e n t i a l to appreciate the f a c t that d i f f e r e n t safety requirements e x i s t f o r d i f f e r e n t combinations of l i v e l o a d i n g . As a r e s u l t of these d i f f e r i n g safety requirements i t appears that the smallest numerical values f o r load f a c t o r f r e q u e n t l y w i l l not represent the governing co n d i t i o n s f o r 59 design. A smaller safety requirement for dead, loads, as against l i v e loads, would also appear to be j u s t i f i a b l e . 60 SECTION 6 BIBLIOGRAPHY The s ingle bay and double bay bents are standard designs of Butler Bui ldings Inc. Lecture notes i n course on "Limit Design and Ine la s t i c Bending", 1957, by Dr. A . P . Hrennikoff , Univers i ty of Br i t i sh-Columbia . Proceedings, AISC National Conferences, 1955 and 1956. Containing reports on invest igat ions at Lehigh Univer s i ty . 61 APPENDIX 1 In terms o f mechanism f a i l u r e the l o a d f a c t o r s F and F u , f o r cons tant and v a r i a b l e l o a d i n g r e s p e c t i v e l y , are g i v e n by equat ions (4) and (7 ) , as f o l l o w s : F (4) F, £ ( M p - F P . M o ) . © (7) and thus r a t i o IV. becomes: F F ^ M . © . S ( M p - F ^ . M c ) . © 2 M L . 0 Expanding the r i g h t hand s ide o f t h i s e x p r e s s i o n and r e p l a c i n g £ M p . © by F.2M.0 : F_u F S M . © . F2 . M . © - Fo 2Mo.O . .0 F S . M . © now, £ M . © ^ M L c . G - f Sffip .G , and thus : F . 6 £ M L C . 0 .© . 0 £ M , _ C . © + £ M D . © - _Fo2y b.© F 1 + F - F p a I F * 2 M P . © 1 «£M, . © 1 + F - F P £ W D . S F £W L.S,

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