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The validity of the simplified limit design method for the design of structures. Parkhill, Douglas Leonard 1958-12-31

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T H E  V A L I D I T Y  S I M P L I F I E D FOR  T H E  L I M I T D E S I G N  OF  T H E  D E S I G N OF  M E T H O D  S T R U C T U R E S  by DOUGLAS LEONARD PARKHILL Dip.CE.,  Melbourne T e c h n i c a l  College,  1944  A T h e s i s Submitted i n P a r t i a l F u l f i l m e n t the Requirements f o r  of  the Degree of  MASTER OF APPLIED SCIENCE i n the Department of CIVIL ENGINEERING  We accept t h i s t h e s i s as r e q u i r e d standard  conforming to  THE UNIVERSITY OF BRITISH COLUMBIA August,  1958  the  In the  presenting  this thesis  r e q u i r e m e n t s f o r an  of  British  it  freely available  agree that for  Columbia,  that  copying  gain  shall  Department  by or  not  of  advanced degree at  s h a l l make  for reference  and  study.  I  for extensive  M^cA  be  publication  '^'I  *o <£esr*-><j  ,  SfSg  by  the  Columbia,  Head o f  thesis my  It i s understood  of t h i s thesis  a l l o w e d w i t h o u t my  further  copying of t h i s  granted  representative.  The U n i v e r s i t y o f B r i t i s h Vancouver 3, Canada. pate  University  Library  his  be  the  of  the  p u r p o s e s may  Department o r  fulfilment  I agree that  permission  scholarly  in partial  for financial  written  permission.  i  ABSTRACT  P r a c t i c e i n the f i e l d of l i m i t design has tended to c e r t a i n r e s t r i c t i o n s on s t r u c t u r a l l o a d i n g patterns  place  i n order to  s i m p l i f y the c a l c u l a t i o n s i n v o l v e d i n the l i m i t design procedure. The loads  considered i n t h i s s i m p l i f i e d approach are assumed  e i t h e r remain constant  and f i x e d ,  or i f they vary then t h i s  to is  to be i n such a manner that t h e i r magnitudes stand i n a constant r e l a t i o n s h i p one to the o t h e r . Actual s t r u c t u r a l l o a d i n g s seldom s a t i s f y  these  restrictive  c o n d i t i o n s and the question n a t u r a l l y a r i s e s as to whether or not t h i s s i m p l i f i e d l i m i t design procedure i s v a l i d f o r general  use  i n p r a c t i c a l design problems i n which e x t e r n a l loads may be wholly independent i n t h e i r i n d i v i d u a l  actions.  T h i s question i s i n v e s t i g a t e d i n the present paper through the examination of s e v e r a l p r a c t i c a l forms of s t r u c t u r e  which  p o r t r a y the more adverse c o n d i t i o n s of independent and v a r i a b l e l o a d i n g to be met i n p r a c t i c e .  These s t r u c t u r e s  are,  respectively,  s i n g l e and double bay gable bents of l i g h t w e i g h t c o n s t r u c t i o n , and two forms of m u l t i s p a n bridge  girders.  The,study i n d i c a t e s that a l l of these s t r u c t u r e s  are  able  to support the u l t i m a t e loads p r e d i c t e d by the s i m p l i f i e d l i m i t  ii  design method; the a c t u a l u l t i m a t e l o a d s exceeding the p r e d i c t e d values by up to twenty percent. It  i s concluded 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  u l t i m a t e l o a d c a p a c i t y as p r e d i c t e d by the s i m p l i f i e d method.  iii  TABLE OF CONTENTS  SECTION 1. SECTION 2.  INTRODUCTION 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 P r e m a t u r e 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 G a b l e B e n t 3.02 A n a l y s i s o f D o u b l e Bay G a b l e B e n t 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  inclusive  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 r u c t u r a l  design techniques employing an Ultimate  Load or L i m i t Design concept, approached i n terms of i t s  i n which a problem s t r u c t u r e  t o t a l l o a d support c a p a c i t y at  is failure,  r a t h e r than i n terms of the i n t e r n a l d i s t r i b u t i o n of s t r e s s under working loads  as with the c l a s s i c a l  B r i e f l y stated,  elastic  theory.  these l i m i t design techniques have been  developed on the s u p p o s i t i o n 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 s t r u c t u r e occurs as the r e s u 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 c e r t a i n key l o c a t i o n s i n  members, which transform the s t r u c t u r e at i t s u l t i m a t e l o a d .  i n t o a c o l l a p s e mechanism  P l a s t i c behaviour of the s t r u c t u r e  is  fundamental to the theory and the a p p l i c a t i o n of l i m i t design confined to s t r u c t u r a l m a t e r i a l s istics  e x h i b i t a marked p l a s t i c  its  whose s t r e s s - s t r a i n  is  character-  range and f o r 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 h i n g e . to m a t e r i a l s  The method i s  inapplicable  devoid of p l a s t i c i t y or where the p l a s t i c  of l i m i t e d extent.  range  is  Some doubt a l s o e x i s t s under c e r t a i n circum-  2 stances as to the s u i t a b i l i t y of the method when a s t r a i n - h a r d e n i n g r e g i o n , w i t h i n the p l a s t i c range, The p r o p e r t i e s of s t r u c t u r a l steel,  fulfil  the d e s i r e d p l a s t i c  is  absent.  steels,  and p a r t i c u l a r l y m i l d  requirements almost i d e a l l y ,  as  evidenced by the t y p i c a l s t r e s s - s t r a i n r e l a t i o n s h i p f o r m i l d steel  shown i n F i g u r e 1(a)  Figure 1(b),  and the theory thus f i n d s i t s  i n the f i e l d of s t e e l M and 0 i n F i g u r e 1(b) i n the f i g u r e later  and the M-0 r e l a t i o n s h i p shown i n  structures. applies  greatest a p p l i c a t i o n  The exact r e l a t i o n s h i p between  to the m i l d s t e e l  beam i l l i s t r a t e d  (one a c t u a l l y taken from a gable bent example to be  studied i n d e t a i l )  and the closeness of t h i s exact r e l a t i o n -  s h i p , 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 consists,  essentially,  i n p r o p o r t i o n i n g the members of the  ure so that the c o l l a p s e l o a d of the whole s t r u c t u r e , critical  portion thereof,  is  or any  equal to the appropriate working  loads m u l t i p l i e d by a c e r t a i n l o a d f a c t o r ,  with t h i s l o a d f a c t o r  so s e l e c t e d as to provide a s u f f i c i e n t margin of s a f e t y f a i l u r e of the whole s t r u c t u r e or  are of constant  the l i m i t design procedure i s d i r e c t and simple as, of the indeterminacy of the s t r u c t u r e ,  loads  against  part.  In cases where the working loads  e n t i r e l y avoided.  struct-  elastic  magnitude  irrespective  analysis  can be  On the other hand, i n cases where the working  act and vary independently the l i m i t design procedure  3  becomes f a i r l y complex and a s o l u t i o n to the problem r e q u i r e s the e v a l u a t i o n of e l a s t i c moments i n the s t r u c t u r e as the step i n an e l a s t o - p l a s t i c  first  a n a l y t i c a l procedure.  In order to a v o i d 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 p r a c t i c e to ignore the true independent nature of most s t r u c t u r a l l o a d i n g s and to assume that a l l o f the loads a p p l i e d to a s t r u c t u r e behave i n the ' c o n s t a n t ' manner appropriate to the simpler design method. T h i s assumption o f constant l o a d i n g i s i n many cases a poor approximation to the a c t u a l l o a d s t a t e e x i s t i n g and the question n a t u r a l l y a r i s e s as to whether or not t h i s s i m p l i f i e d design method can be reasonably accepted as v a l i d f o r a l l cases of s t r u c t u r a l design l i k e l y to be met i n p r a c t i c e . I t i s known that v a r i a b l e l o a d i n g can be severer on a s t r u c t u r e than constant l o a d i n g , and, i n theory, a s t r u c t u r e designed i n accordance with the constant l o a d method may f a i l prematurely, if  it is  that i s  before i t s  designed l o a d c a p a c i t y i s  reached,  a c t u a l l y subjected to v a r i a b l e l o a d s of the same design  magnitude.  The question r a i s e d above concerning the v a l i d i t y of  the general use o f the s i m p l i f i e d method f o r p r a c t i c a l design purposes thus becomes one of whether i n p r a c t i c e a premature c o l l a p s e c o n d i t i o n can a c t u a l l y e x i s t ,  and i f so,  whether i t  reaches  s e r i o u s p r o p o r t i o n s or whether i t can always be expected  to f a l l  w i t h i n acceptable l i m i t s of the l o a d c a p a c i t y of the  s t r u c t u r e as i n d i c a t e d by the s i m p l i f i e d method.  4  T h i s question has been g i v e n very l i t t l e c r i t i c a l a t t e n t i o n to date and the purpose of t h i s present  study i s to attempt  o b t a i n an i n d i c a t i o n of the l i k e l i h o o d and p o s s i b l e premature c o l l a p s e  to  extent of the  c o n d i t i o n through the examination of  several  p r a c t i c a l forms of s t r u c t u r e which are p u r p o s e f u l l y s e l e c t e d  as  examples of the more adverse c o n d i t i o n s of l o a d i n g to be expected in  practice.  Co.o  failure <af straitn or about 0.1Z> i ^ / i ^ i  So.  Ao.o 9 XX 0.  vO  3o.  to.o  O.oZ  o  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  3ooo Z1o5  ^9  Irl.klpS.  ?ooo  cv:  Cy/r\y&  ox  \\c [«. O.ZS  o looo  lo  IS  (b)  Zo  25  M  /  \  /  \  So  M }  35  SECTION 2 GENERAL THEORETICAL CONSIDERATIONS  2*01  Conditions Governing  A. For Case of Constant  Structural Failure  Loading  The frame shown i n F i g u r e 2 i s acted upon by f i x e d working l o a d s of magnitude W, and under these p a r t i c u l a r loads i t w i l l be assumed that the s t r u c t u r e deforms e l a s t i c a l l y thus c r e a t i n g c e r t a i n e l a s t i c moments at a l l s e c t i o n s o f i t s members. >  ]__L  w,  I f the l o a d s are i n c r e a s e d  i n magnitude, with a constant  ratio  remaining between t h e i r values, the e l a s t i c moments w i l l be p r o p o r t i o n a t e l y i n c r e a s e d , and t h i s proportFigure 2  i o n a l i t y o f loads and moments w i l l continue, as the loads f u r t h e r  increase, u n t i l  at some l o a d value the e l a s t i c l i m i t s t r e s s o f  the frame m a t e r i a l 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 t h i s s e c t i o n and develops the s e c t i o n u n t i l  through  at some higher l o a d value a f u l l y m o b i l i z e d  p l a s t i c hinge i s created which o f f e r s a constant moment of r e s i s t ance to f u r t h e r deformation. At a s t i l l higher l o a d value the e l a s t i c l i m i t s t r e s s be reached  will  at some other key p o i n t o f the s t r u c t u r e and a second  6  hinge  created.  under  the  the  failure  i n i t i a l l y  additional  now  P.W  is  the  in  the  hinges  to  terms  plastic  in  moving  ments.  Thus:  through  total  work the  equation  simplified and of ing  the the  method,  done  geometry loads,  loads  possible the  one  the  i n c r e a s e d to  In  the  resulting  indicated  by  with  and w r i t i n g  Pigure  certain a  2 with elastic  similar  the  and  angles  failure  at  offer  no  theory.  form  of  their  of  the  basic the  can  hinge  rotation of  virtual  magnitude displace-  (1)  expression of  proportions  F by  the  which  the  of  a given  structure  the  value  temporarily frame  moments to  true  to  of  the  factor  at  for  an  of  will  be  P. condition  working the  work-  readily  a number  to  value  original  original  subjected  frame  failure  can be  load  the  above  the  c o l l a p s e mode  M existing (l)  of  true  value,  and  the  work  the  loads  failure  equation  to  virtual  the  the  transform  £ F . W . 5>  smallest  back  of done  2> o f  factor  exist  the  referring  the  given  analysis  in  by  =  the  to  load  in  work  c o l l a p s e mechanism,  thence  m e c h a n i s m s may  Mow,  W and  and  were  determined.  of  structure  Mp  distances  which,  exist  constant  equation  moments  constitutes in  an  the  internal  ZMp.©  This  points  until  collapse.  the  considered  the  external  of  developed  a mechanism able  mechanics of  relating  total  into  displacement  of  similarly  hinge  complete  simple  is  are  sufficient  structure  mechanism  written  points, to  rigid  a virtual  collapse be  load  resistance  This If  Additional  key  loads points  identical  7  virtual  displacement,  i n terms of working loads and moments:  Z M.6  =  (2)  £W. h  Again i n c r e a s i n g these loads to the f a i l u r e value F.W by m u l t i p l y i n g both s i d e s of equation (2)  ZF.M.e  =  2F.W.S ,  anid equating t h i s expression with (1), SMp.©  =  by F :  gives:  £ F.M.©  (3)  The p l a s t i c moments Mp o c c u r i n g i n these equations be considered as being p o s i t i v e .  should  E l a s t i c moments M at each hinge  are l i k e w i s e p o s i t i v e 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 as  treated  negative. Load f a c t o r F may be obtained from equation (3),  p r o p e r t i e s of the s t r u c t u r e and the d i s t r i b u t i o n of moments w i t h i n i t .  given the  elastic  T h i s equation thus provides an a l t e r n a t i v e  b a s i s to that of equation (l)  f o r o b t a i n i n g the f a i l u r e l o a d  f a c t o r f o r the constant l o a d i n g c o n d i t i o n . R e w r i t i n g equation (3), expressed  i n the f o l l o w i n g form:  F Now,  the r e q u i r e d quantity F may be  =  gM .e r  £M.©  (4)  of a l l p o s s i b l e mechanisms whereby the s t r u c t u r e may  8  collapse  the true f a i l u r e mode, that i s  the one r e q u i r i n g the  s m a l l e s t value of E , i s o b v i o u s l y obtained when the terms of the numerator are s m a l l e s t i n comparison with those of the denominator. Two v a r i a b l e s  are i n v o l v e d i n these terms,  angles of r o t a t i o n . u s u a l l y gradual  namely, moments and  V a r i a t i o n s i n the l a t t e r  and i t i s thus reasonable to expect  p o i n t s of the true f a i l u r e mechanism w i l l sections  are continuous and that the hinge  be a s s o c i a t e d with  of the frame where minimum values  This a s s o c i a t i o n  of Mp o c c u r . M of hinge p o i n t s of the constant l o a d  mechanism with minimum values of the q u a n t i t y Mp i s found to be M c l o s e l y followed i n numerous examples which have been c o n s i d e r e d . Such minima may exceed i n number the hinges r e q u i r e d f o r development of a mechanism but each i s representing a possible w i l l be noted that  a key p o i n t of the  structure  hinge p o i n t of a f a i l u r e mechanism.  i n cases where Mp remains constant  member of the s t r u c t u r e  the  along a  the key p o i n t s of t h i s member w i l l  a s s o c i a t e d with those s e c t i o n s  where maximum values of  It  be  elastic  moment o c c u r . Equation (3) may also be w r i t t e n i n the form:  £M .e p  or, Now,  it will  structure P.W.  -  ej'.M.e  £(Mp - E.M).©  be remembered that  i n the f a i l u r e  = =  equation  o 0 (3)  (5) r e p r e s e n t s the  s t a t e under the a p p l i c a t i o n of loads  To o b t a i n the above equation (5) the q u a n t i t i e s  E.M.© have  9 been s u b t r a c t e d i o n has, in  from b o t h s i d e s o f  in effect,  removed t h e f a i l u r e  a zero l o a d s t a t e .  unloaded s t r u c t u r e this  state.  load  B  *  E q u a t i o n (5)  (Mp -  the hinge p o i n t s of  (3)  loads  and t h i s  and l e f t  thus r e f e r s  and must be t h e v i r t u a l  The q u a n t i t i e s  r e m a i n i n g at  equation  to  structure  the completely  work e x p r e s s i o n  F.M) a r e ,  then,  the s t r u c t u r e  for  residual  moments  under t h i s  zero  F o r Case o f V a r i a b l e L o a d i n g  t h e more g e n e r a l loads  are  capable  structure  shown i n F i g u r e 2 s u b j e c t e d  form o f l o a d i n g i n w h i c h c e r t a i n o f  assumed  as  fixed  and c o n s t a n t ,  of independent v a r i a t i o n .  This  the  is  the v a r i a b l e  in practical  loads  o f m a g n i t u d e Wo and t h e y c r e a t e  are  'dead'  moments i n t h e  loads  structure  remaining loads  are  o f magnitude  'live'  g r e a t e s t moment c r e a t e d  action will  be r e f e r r e d t o  Failure  of  F c and F 0 ,  loads at  The assumed  fixed  elastic  any s e c t i o n .  The  as M u .  are  occurs  increased  i n t h e form o f  a mechanism  i n m a g n i t u d e by l o a d  factors  at  failure  respectively. this  condition of  that:  £(Mp - F L . M C - FP.Mp).e this  loading  o f m a g n i t u d e WL. and t h e n u m e r i c a l -  The z e r o l o a d s t a t e o f t h e frame such  at  are  any s e c t i o n by t h e i r i n d e p e n d e n t  the s t r u c t u r e  when dead and l i v e l o a d s  structures.  to  applied  and t h e r e m a i n d e r  s i t u a t i o n which a r i s e s  is  the  state.  C o n s i d e r now t h e  ly  subtract-  b e i n g t h e work e x p r e s s i o n  for  =  a virtual  G, displacement  (6)  of  the  10  s t r u c t u r e i n the form of the f a i l u r e mechanism. for  The convention  the signs of moments i s the same as that e a r l i e r  established,  namely, a l l p l a s t i c moments Mp are considered as p o s i t i v e and the e l a s t i c moments ML. and M  &  are l i k e w i s e p o s i t i v e 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 l o a d f a c t o r IV, i s now r e q u i r e d and t h i s may be r e a d i l y obtained by r e a r r a n g i n g the terms of equation (6),  K  =  thus:  (7)  g ( M - Fo.Mc).Q p  The true f a i l u r e mode f o r v a r i a b l e l o a d i n g t h e r e f o r e r e q u i r e s that terms i n v o l v i n g (Mp - Pp.Mo).© are s m a l l e s t with respect to those i n v o l v i n g M .C, U  and thus key p o i n t s of the  s t r u c t u r e f o r v a r i a b l e l o a d i n g can be expected to be a s s o c i a t e d with s e c t i o n s where minimum values of the q u a n t i t y  (Mp  -  M  ffc.Mp) u  occur. Now, structure  equation (6)  defines the zero l o a d s t a t e f o r  the  subjected to v a r i a b l e l o a d i n g and i t w i l l be evident  from i n s p e c t i o n of t h i s equation that the r e s i d u a l moments remaini n g at the hinge p o i n t s i n t h i s case are equal to (Mp - F^Mi,- Pi>Mp);; As l i v e loads F^.W  are repeatedly and independently a p p l i e d the  s t r u c t u r e w i l l e v e n t u a l l y "shakedown 1 to t h i s p a t t e r n of r e s i d u a l moments.  T h i s shakedown procedure may i n v o l v e a number of  a p p l i c a t i o n s of the l o a d s , necessitating  each of these e a r l i e r c y c l e s of l o a d i n g  a readjustment  of i n t e r n a l s t r e s s from the key p o i n t s  11  where o v e r s t r e s s i n i t i a l l y occurs to adjacent s e c t i o n s structure.  This transfer  of  the  of s t r e s s takes place through  increments of p l a s t i c y i e l d i n g at these key p o i n t s ,  with each  increment of y i e l d being a s s o c i a t e d with an increment of permanent d e f l e c t i o n of the s t r u c t u r e continue u n t i l  as a whole, and these  the shakedown s t a t e i s reached.  deflections  I f loads  slightly  g r e a t e r than the true f a i l u r e value are a p p l i e d the s t r u c t u r e  is  unable to reach the shakedown s t a t e and the increments of permanent d e f l e c t i o n continue i n d e f i n i t e l y , with the deforming as a mechanism, u n t i l results.  complete c o l l a p s e  structure  eventually  T h i s type of f a i l u r e under v a r i a b l e l o a d i n g i s  r e f e r r e d to as one of  'incremental  therefore  collapse'.  A second form of f a i l u r e under v a r i a b l e l o a d i n g must be i n v e s t i g a t e d . quantities  I t was p r e v i o u s l y s t a t e d ,  i n v o l v e d i n equation  (6),  also  i n d e f i n i n g the  that M u represents the  n u m e r i c a l l y g r e a t e s t e l a s t i c moment which c o u l d occur at any s e c t i o n of the s t r u c t u r e . U  u  will  Both p o s i t i v e  and negative values  of  g e n e r a l l y e x i s t at each key p o i n t and the q u a n t i t y i n v o l v e d  i n equation  (6) w i l l  be e i t h e r the p o s i t i v e or negative  amount  depending on the sense of the hinge r o t a t i o n at the key p o i n t . The t o t a l range i n l i v e l o a d moment at any s e c t i o n w i l l algebraic total  summation of these p o s i t v e  and negative  be the  amounts.  range i n moment M u at a s e c t i o n conceivably may exceed  e l a s t i c r e s i s t a n c e of that s e c t i o n , the opposite  This the  and i f t h i s c o n d i t i o n occurs  extremes of l i v e l o a d moment w i l l produce an  12  inelastic then in the  y i e l d i n g at  i n the  other.  increments failure  of  of  'plastic  It  obviously Now,  in  the  total  The the  the  the  or,  twice  this  condition of  following equality  This  any at  the  resistance  The  of  of  first  o f My o c c u r s .  This  C.  yield  is  eventually  the  results  lead  described  to  as  one  plasticity'.  structure.  stress  is  first  referred  to  a s My t h e n  section,  alternating  in turn  'alternating  of  and  that  under  reached  reversals  the  of  2.My.  is  plasticity  will  develop  when  exists:  2.My  (8)  =  2«My  (9)  of  reached state  =  the  structure  at  which t h i s  evidently  is  where  attained  at  a live  is  the  condition  smallest  value  load factor  equal  VI. Summary o f In  variable  Failure  briefly  theoretical  in  section  moment v a l u e ,  section  is  to  failure  commonly,  the  direction  condition is  section  FJ_  critical  plastic  which y i e l d  FJ.ML  from which,  i n one  s t r a i n w h i c h must  more  at  moment of  first  alternating  section.  fibres  elastic  section  permanent  may o c c u r  if  is  This  fatigue',  outer  moment,  of  the  Conditions  summarizing the  considerations  loading,  that  which both f i x e d  is  it  may  i n the  dead l o a d s  findings be  said  general  sofar that case  reached i n the  to  and i n d e p e n d e n t l y  in  case  be met varying  in  these of practice  live  13  loads  are  encountered,  Firstly, of  the  collapse  £(M  the  terms of  failure  being  -  F .M^  equation  of by  The  u  for  and The  as  the the  true  These  =  D  to  considered.  structure at  i n the  the  (6),  equation  extending  structure, equality  failure  either F of  as  form  point  of  follows:  0,  (6)  a l l hinge points  of  the  and  the  u  will or  f o r which  stated  load  structure  =  2.My  for  the  Both for  incipient  i n equation  correspond  FJ.  involving only  failure  is  thus:  (8) structure to  states  the  (8),  one  the of  under  the  smallest failure  determination  of  condition  value  obtain-  require  elastic  moments  Mo,  M,_.  simplified  replacing  actually  P  m u s t be  structure  alternating plasticity,  variable loading  analysis M,  by  the  F . M ) .0  -  L  of  the  defined  FJ.iL  able  of  failure  mechanism.  section  of  of  collapse  condition  p  this  Secondly,  defined  states  incremental  a mechanism,  incipient  two  the  subject,  constant  case of  dead and by  loads  an  constant  live  loads,  equivalent  loading to  system  being r e s t r i c t e d  to  may  which the of  be  thought  of  structure  is  constant  loads.  simultaneous  action  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 o n e 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 under a l t e r n a t i n g plasticity. *  14  if  they vary i n magnitude t h i s must be i n such a f a s h i o n that  constant r a t i o always e x i s t s between them. system the loads  In t h i s  equivalent  are thus of the same nature and the same l o a d  f a c t o r must be a p p l i e d to each and every one.  The c o n d i t i o n of  the s t r u c t u r e at the p o i n t of i n c i p i e n t c o l l a p s e i n t h i s case expressed by equation (1),  as  alternatively,  is  follows: =  or,  a  (1)  2-F.W.  i n the form g i v e n by equation (3),  thus:  The terms of these equations extending to each hinge p o i n t of the f a i l u r e mechanisms, (l).  Equation (3)  araid to each l o a d item i n the case of equation r e q u i r e s an e l a s t i c  analysis  of the  structure  f o r the d e t e r m i n a t i o n of moments M, but t h i s i s not necessary equation  for  (l).  The true f a i l u r e l o a d f o r t h i s c o n d i t i o n of constant l o a d i n g will  correspond to the smallest  value o b t a i n a b l e f o r l o a d f a c t o r P.  15  2.02 Premature C o l l a p s e Having e s t a b l i s h e d the c o n d i t i o n s governing the f a i l u r e of structures  subjected,  on the one hand to a c t u a l v a r i a b l e l o a d i n g ,  amd on the other hand to an assumed constant l o a d i n g , i t i s now only necessary to make a comparison between Fu and F i n order to determine the extent of premature c o l l a p s e . l e s s than 1.0  are obtained t h i s w i l l  c o l l a p s e can take p l a c e ,  If ratios  i n d i c a t e that  o f F u of F premature  and the magnitude of t h i s r a t i o  i n d i c a t e the s e v e r i t y of the premature c o l l a p s e Such a comparison of f a i l u r e l o a d values  will  condition.  can be put on a p r a c t i c a l  b a s i s by s e l e c t i n g p r a c t i c a l forms of s t r u c t u r e f o r a n a l y s i s , before t h i s i s  attempted here i t i s d e s i r a b l e to f u r t h e r extend  the previous t h e o r e t i c a l Firstly,  but  evaluations.  consider the c r i t e r i a e s t a b l i s h e d f o r the l o c a t i o n  of key p o i n t s i n s t r u c t u r e s  subject  to mechanism f a i l u r e .  For  constant l o a d i n g the c r i t e r i o n i s that these key p o i n t s w i l l  be  at s e c t i o n s of the s t r u c t u r e where minimum values of Mp o c c u r . M  For v a r i a b l e l o a d i n g the c r i t e r i o n i s minimum values of Mp - Fo.Mo Mu  but  i n s p e c t i o n of t h i s q u a n t i t y w i l l  show that these minimum  values occur when the combined e f f e c t s are l a r g e s t with respect  to Mp.  of the e l a s t i c moment terms  As c r i t i c a l values f o r M, M , L  and Mp, are commonly a maximum i n the same v i c i n i t y i n a s t r u c t u r e these separate c r i t e r i a may amount to s u b s t a n t i a l l y the same t h i n g and normally i t can be expected that the key p o i n t s of the s t r u c t u r e f o r both l o a d i n g c o n d i t i o n s w i l l  a l s o be i n the same  16  vicinity. key  In f a c t  i n l i m i t design i t  i s usual to assume i d e n t i c a l  p o i n t s f o r both l o a d i n g s . If  it is  a l s o assumed here t e m p o r a r i l y that the same  c r i t i c a l mechanism a p p l i e s  f o r both forms of l o a d i n g then the  r a t i o J F L f o r mechanism f a i l u r e may be w r i t t e n  as:  F  =  ^M.e . Z ( M p - p P .Mp).e ,  F  2Mi.e  2M .G p  and making use o f known r e l a t i o n s h i p s  t h i s expression may be  r e a d i l y r e s o l v e d i n t o the f o l l o w i n g form : Fu_  =  gMLc.e  F  £MU.©  1 > F-Fo. F  (10)  £Wp. %  SWL.S .  i n which,  2M .e, LC  and  respectively, constant Mt,  SW^.S,  are the i n t e r n a l and e x t e r n a l work,  of l i v e loads W u a p p l i e d i n the manner of  loading,  as p r e v i o u s l y d e f i n e d ,  are the n u m e r i c a l l y g r e a t e s t moments  produced by these same l i v e l o a d s variable loads,  and 2 M L . © i s  a c t i n g independently as  therefore  the work done by these  moments, and 2.Wj>.S i s Now,  the e x t e r n a l work done by dead loads W^. the i n t e r e s t  introducing i t ,  lies  f e a t u r e s or f a c t o r s  i n equation (10),  i n the l i g h t i t  and the reason  for  sheds i n i s o l a t i n g the main  which i n f l u e n c e the l o a d f a c t o r  r a t i o JV. . F  *See Appendix 1.  17  Two such f a c t o r s ratio  is  predominate i n the equation and on these the  d i r e c t l y dependent.  1. . £ W p . S .  This factor  We. and W .  the dead l o a d s favourable  be the r e s u l t  l i k e l i h o o d of premature 2.  gHuc.Q .  is  such that the  to l i v e loads  on r a t i o  the  collapse.  a c t i n g r e s p e c t i v e l y i n the manner of  e v i d e n t l y measures the r e l a t i v e  extremes  This  factor  s t a t e s of  The more v a r i a b l e the a c t u a l l i v e l o a d s  smaller w i l l  are  be the value of the i n t e r n a l work r a t i o  more adverse w i l l  be the e f f e c t  l i k e l i h o o d of premature  which the l i v e loads  the and the  on % , , hence the g r e a t e r  be expected to tend to  to the l o a d f a c t o r  ratio will  be those i n  are h i g h l y v a r i a b l e and i n which dead  are as small as p o s s i b l e .  the  collapse.  C o n d i t i o n s which can t h e r e f o r e a lower l i m i t  constant  of l i v e l o a d moment  created i n the s t r u c t u r e under these d i f f e r e n t  produce  the more  and the l e s s  and as independent v a r i a b l e l o a d s .  loading.  larger  Compares the i n t e r n a l work done at hinge p o i n t s by  the l i v e loads loads  of t h i s f a c t o r  are with respect  will  are:  compares the e x t e r n a l work done by loads  The e f f e c t  L  These f a c t o r s  loads  Although these c o n c l u s i o n s are based  on an i d e n t i c a l form of mechanism f a i l u r e  it will  be evident  that  the same c o n c l u s i o n s are r e l e v a n t to the case of f a i l u r e through d i f f e r e n t mechanism forms and also to that of f a i l u r e by alternating Now,  plasticity. these f i n d i n g s present  a l o g i c a l b a s i s on which to  18  select  the s t r u c t u r e s  for analysis  i n the present study.  As the  purpose of the study i s to e s t a b l i s h the l i k e l i h o o d and p o s s i b l e extent to which premature c o l l a p s e may be expected to occur i n p r a c t i c e i t i s o b v i o u s l y l o g i c a l to choose f o r a n a l y s i s forms of s t r u c t u r e which s a t i s f y  the l o a d i n g requirements found necessary  f o r small values of PL . F  Two forms of s t r u c t u r e which e x h i b i t such l o a d i n g c h a r a c t e r 1 istics  are the gable bents shown i n F i g u r e s 3(a)  and 3(b)  .  Dead weight i s minimized i n both cases by the use of l i g h t w e i g h t r o o f i n g c o n s t r u c t i o n and tapered frame members, and l i v e l o a d s are h i g h l y v a r i a b l e - c o n s i s t i n g of independent g r a v i t y and r e v e r s i b l e sidesway f o r c e s .  The s i n g l e bay and double bay bents  were also s p e c i f i c a l l y chosen of s i m i l a r form to i l l i s t r a t e effects  the  o f d i f f e r e n t degrees of s t r u c t u r a l redundancy.  These gable bents,  then,  can be expected to i n d i c a t e the  probable order of the lower l i m i t to the premature  collapse  condition. How, i t would also be of p r a c t i c a l i n t e r e s t to examine the e f f e c t on F L o f v a r i a t i o n s i n the dead to l i v e l o a d r a t i o F  and t h i s i s best done by s e l e c t i n g a second type of s t r u c t u r e i n which the l i v e l o a d s are h i g h l y v a r i a b l e ,  as with the bents,  but  i n which the dead weight i s q u i t e l a r g e i n magnitude i n comparison with the a p p l i e d l i v e l o a d s .  A suitable  example of such a l o a d i n g  c o n d i t i o n i s the m u l t i s p a n p l a t e g i r d e r highway bridge shown i n Figure 3 ( c ) ,  i n which the moving v e h i c l e l o a d s cause h i g h l y  v a r i a b l e l i v e moments but these l o a d s are c o n s i d e r a b l y s m a l l e r  19  i n magnitude than the combined dead weight of the s t e e l  girder  system and concrete deck. D e t a i l e d analyses  of the bents and bridge g i r d e r s w i l l  made i n the remaining s e c t i o n s o f t h i s paper. structures  are a c t u a l designs,  be  The s e v e r a l  and dimensions and other  details  used i n the study have been obtained from the f a b r i c a t i o n and c o n s t r u c t i o n drawings.  The o r i g i n a l designs were based on the  elastic  theory and although a comparison of p 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 l o a d i n g  specifications  and design assumptions w i l l  the p l a s t i c analyses structures  and e l a s t i c  be also employed i n  i n order that such a comparison f o r these  could be independently made.  TTT  TTT  Tf  F i g u r e 4-.  0.5-7& n  <.r<ze[  bolts l-lofOc^hoL-'f.  J Proper  i ics  of  Area SecY\or\  Pro pcv tie £=•  of  of  Section  N o .  r  ZP  M y kips  A  . o o  6>. 3 3  oo . 5  & .55  6 .65  33  6 .85  t4S  ?z . 6 Z7 . 5  12)3  3Z  . 6  . Z  5  1  . 4  Z  lo.  3  IZ . I S  7 . 1 7  4  ! 3 . 0 i  7  ,4ft  244  3&  &  I 5>. 4 7  7 . 7 3  3 4 o  4 4 .  ^  17.13  © - to  4 Z 5  !  7 3  e .41  20.  45  8> .  .  7  4 3  7Z  Properties  Pfasfic  i n .  »m. k i p s  3  n.s  TZo  7 45  Z4.5  80S  3  3 o . ]  l o o s  35 . 6  1 1 £>o  l o  l o & o  \zc*c>  41 . &  I 3 o o  I4SO  4 &. 3  I S 3 o  4 5 . S  I&3&  S>4.£>  5 Z o  SS  \e>zs>  6 1 .  6 Z 7  61 . &  ZoSS  7 5 1  (£»7. 3  Z Z 4 0  o  . s  1  Co  60S  Z o 3 o  ZZ&S  nc  2>  ZZ . 1 I  3 . OS  lo  23. 77  3.  &&Z  74. Z  Z 4 & o  &3.7  Z1Co  &c*l.  24.oo  3 . 3g>  3 o 4  75 . 3  Z 4 3 ^  &4.&  Z&oo  £4. o o  o .e>e>  » 3 5  <£>3. 7 .  2 3 o o  &z.<*  Z1oS>  Z3. lo  3 • & o  61 J  <2>£>. £>  ZZGo  77. &  Zh0?o  <&Z. 3  Zolo  74.  1  44  \t  Z?.47  0.  /z  7o6  15  2 I • &S  7. & S  56 J  54. 7  14  Z o . o I  7 .  So&  <i»  IS  i  7.43  4 3 5  4& . 7  !6  1 7 . 54  7.15  370,  £>. 7 3  SZ o  e>. ie>  J7  SI  CS.  16  /&.o&  C.  I'D  1 3 • 6 S  6. SI  74  to C  1  1. oo  ] &  &oo  o  6 , 4 . <£?  Z  ! S o  835  !£,7o  5 7 . 3  1  1 £»4o  SZ. 7  ! 7 4 o  4 2 . 3  14  f t  4 6. Z  1 S 3 o  32>- Z  / Z 5 5  43. &  1445  ZGS>  3 5 . . £>  II 7 o  ZZ Z  32 . 1  \oC>o  1 © J  Z6.  3 4 5  5 ( . 4,  27. O  0 S o  25.7  ICO  Z5oS  1 3oS> ! 1 7 5 1 o 4 5 3£>o  20  SECTION 3 ANALYSES OF SELECTED STRUCTURES  3.01  A n a l y s i s of S i n g l e Bay Gable Bent  A. General Frame s e c t i o n s and d e t a i l e d dimensions of the bent shown i n Figures 4 and 5. up welded s e c t i o n ,  are  The column and roof beams are of b u i l t -  as shown i n Figure 4,  these are given i n Table 1.  and the p r o p e r t i e s  of  High strength b o l t e d connections  are  employed throughout and d e t a i l s of these connections are shown i n F i g u r e 5.  Lateral  stiffeners  f o r the i n s i d e flanges of columns,  and beams are l o c a t e d i n Figure 4 and take the form of angle braces between flanges and side and r o o f p u r l i n s . of frames i s  The spacing  2 0 , - 0 " on c e n t r e .  The o r i g i n a l design of the bent was to AISC - 1952 ations. (1)  Loading c o n d i t i o n s were f i g u r e d as  follows:  Dead l o a d i n g p l u s 30.0 pounds per square foot l i v e l o a d i n g on the f u l l  (2)  specific-  h o r i z o n t a l p r o j e c t i o n of the  structure.  Dead l o a d i n g plus 20.0 pounds per square foot wind l o a d i n g a c t i n g on the f u l l  v e r t i c a l p r o j e c t i o n of the s t r u c t u r e  in  one or other h o r i z o n t a l d i r e c t i o n . (3)  Dead l o a d i n g plus 30.0 pounds per square foot on the h o r i zontal p r o j e c t i o n plus 20.0 pounds per square foot e i t h e r d i r e c t i o n on the v e r t i c a l  projection.  in  21  Design l o a d i n g s  were computed on the b a s i s of these  c o n d i t i o n s with dead l o a d s assumed d i s t r i b u t e d u n i f o r m l y over the r o o f span and estimated  thus:  Roof beams  & 1.4  lbs.  Roof p u r l i n s  @ 1.1  »'  "  "  Roof sheeting  @ 1.0  rs  "  "  T o t a l dead weight giving  3.5 l b s .  per sq.  per sq.  ft.  ft.  for:  Design c o n d i t i o n  (1),  uniform l o a d on f u l l =  (30.0 + 3 . 5 ) . 2 0 . 0  Design c o n d i t i o n  projection  =  670 l b s .  per l i n .  ft.  (2),  wind l o a d on f u l l =  horizontal  vertical  (20.0).20.0  projection =  400 l b s .  per l i n .  ft.  =  70 l b s .  per l i n .  ft.  400 l b s .  per l i n .  ft.  per l i n .  ft.  p l u s dead l o a d =  (3.5).20.0  Design c o n d i t i o n  (3),  wind l o a d on v e r t i c a l  projection  uniform l o a d on h o r i z o n t a l  projection =  B. E l a s t i c  670 l b s .  Analysis  A step by step d e r i v a t i o n of the e l a s t i c a n a l y s i s unnecessary  f o r the purposes of t h i s  study.  is  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 i x e d end moments, s t i f f n e s s and c a r r y - o v e r f a c t o r s  coefficients,  r e q u i r e d f o r the a n a l y s i s determined by  means of the column analogy method.  Both supports were assumed  as h i n g e d . The p r o p e r t i e s o f . t h e frame elements used i n these c a l c u l a t i o n s are given f o r reference purposes i n the accompanying F i g u r e 6.  i n Table 11,  and  The symbols used c a r r y t h e i r usual  meanings. R e s u l t i n g moments at s e c t i o n s  throughout the frame  are  given i n Table 111 f o r the i n d i v i d u a l g r a v i t y and wind l o a d i n g s . Shears  and t h r u s t s  that t h e i r e f f e c t s  are not considered as i n v e s t i g a t i o n has shown are small and can be e n t i r e l y n e g l e c t e d i n the  study. Moment diagrams f o r each l o a d i n g c o n d i t i o n , the values t a b u l a t e d i n Table 111, 9.  p l o t t e d from  are shown i n F i g u r e s 7, 8,  For these diagrams the moments are, p l o t t e d , i n the convention-  a l manner, on the side o f the frame at which they create ive  and  compress-  stress. The severest f i b r e s t r e s s created by the a p p l i e d l o a d i n g s  i s worthy of note although t h i s data i s not r e q u i r e d f o r the present purpose.  The severest s t r e s s c o n d i t i o n i s found to occur  i n the roof beams at the knee j o i n t s where the combined s t r e s s figures  at 26.6 k s i .  The value permitted by the AISC Code, f o r  wind and a l l other l o a d i n g s ,  equals  26.7 ksi> ( = 20.0 k s i p l u s  T<* b  Frame  Scchon  le  !!  Pvoper ffes  I  y  1 .67 1.07  f*.  ff.  3  o.o4 o. 1 1 o . 10  4.6&  4  o.Z5  6.55  o.3Z O. 4 6  8.4Z I o . 23 I Z . 16  O.S3  I4.O3  b Co  1 .6*7 I . 07  7 0  1.07 1.07  o.34  z.ei  6.S3 10.43  Be  inf  EU  3 L a H of  1 2 . IS 13.01 I5.4& 17.13  135  34 o 4Z5  .©164 .o2o5  I 14 31 . 3  676 776  .o370  53.2 66.6  7.&3 11 . 3 !  .o33o .o206 .oZ51  I6.3I I5.o0 13.65  361 321 266  12.63  2/7  13.S3  1-7  Z.43  ie>  Z.43 2.43  IS.53 I&.Z& Zo.62  2.43  22.36  7.67-  6.24  24.36 25.73 Z6.6Z  2.43  6o.6  5/ .&  2 4 . 12  6.31 M .25  533 5Zo  1 .o 2.7  3.6o  44. »  .o42I  3.5 21 . ! 42.6  4.32 6.ol  .o36l .o424  ,o2So  Ms.y.&  63.1  75! 062  446  6.57  2.43 Z.43  loo I  3 . (S 3.64  .o3oZ  [•7.54  2.43  3.60 7.52  74.6 6 ! .3  22.45 Z3.23  13 14  ZZ.47  Z66  3.4o 3 - Io 0.7o  665  £.43  331  147  2 I .85 Zo.ot  2.43  \Z  .oo470 .oo630 .ol27  Z o . 73 21.62  1 1  I  '  .OO350  2 3 . Io  1 .61  O.CoO i s . 3 o O. (ob 17.77 1. 6 6 1 3 . 12 4.2Z 13.35  loo  s  Ms  36.6 9.33 1 o . 6I 1 .oo 122.1 1 Z.66  1  I3o. I 133. o 216  14 . 3 2 It.38 4 3 . 7o 3 3 . Io  3.7 Zo.3 33.5 57.6  s  23  I  65 5 10  M  s  fh kip  loo  r  1.1 6.o  lo25  o.3 o.o 1 .4  151  1453  1.3  10.3  1  1737 2o7o ZZGo  Z.4 2.3 3.4 4.o  23.o  63 1 I2  87  12.8  ZZo  loo ( lo  246 Z74  11 b  237 314  Z45o Zt&o  4.5  37. 1  5.o  313 3o6  36!o  14.1 31 . 6  3t>.2 !6o  Z34 263  462 o  125 435 I Z4o  265  75.5 86.8  \b .1 16.7  326  1 3 7 Zt5o  4o5  1 77 33 l o  .O2I6  33.3 M£  7.2.5 26.6  Soo 6Z4  Zo3 236  .ol03  136  32.6  73Z  .ol55 .oJZ3 • oloS  16!  4o. 1 43.5 62.3  1 ooo 1277 1676  loo  73.6 07.6  13.oo  132 Z36  R i g lot ot <£.  <£  M . y.As  fh k i p . l o o 1  Z64  ZZ.W tl.11  l. 87  ft.  4  33.4 145 133  S>Zo 627  3  13 Zo  ins?"  10.73 Zo.45  IO  15 1(2,  e,  I  No.  1 2  bl  Ve^hical  x:  I Ph  G <a  of  236o  4loo  633 o  27o Z&6  43.2 66.7 B5oo CoOSO 8 4 . 4 C&lo loZ  Z53  85 o o  241  73oo  Z75 266 23 £  1 !o5o 14 I 7 o  Z24  !636o  lt>Z  833o lo35o It loo  467o  2o3  1  13 137  155 172  Z1.Z 31 . o 34.8  42o 773 1245  I860 274o 33oo 55oo 767o lo62o  Table  >TO«  Fro  ,Seo4 i o n  load  —  U  3Z  1,  -  3 u  —  J  £>,  —  ! 77  57  &Z oo  — 106  -  43  -f  3  IS,  "  R  3  R  Mo m c r i l  Lo<»<s< i m ^  --  2  7 1  f ro^n +  left  for  i n kips.  3G I  Single  6ay  Morr>0 tot 5  I Z  +  7  -  347  Z7oS  o3S  +•  7o3  0  +•  636  - I  -  5 3 I  -  5 5  -  4 Z5  +  57  -  3 I 6  4  336  -  Z I I  +  4  ! 3 I  +  3  3fe  +  4  +  47  + 4o J  +-  35  +  3o 1  -  +  3  +-  ZZ  2  ZSoS  8 I 7  4?  3  135  &4Z  - I3( —z;i  —  7Z7  -!Z  +  57  +  -3  1  3,0>1  -  -  3o  lo65  3>ZZ 1o7  4 &e>  —Z 7  I  3Z  665  +  8Z  103 0  Z33  -  4 0  --  -  6&5  + 4  2  +  36  +  257  3 I 8  -  6  -  5&S  47-5  +  455  +  53 /  +  6 4Z  -  +  8  + +  65  - 4 32 - 2 33 — I 6£»  1 1s  565  +-  +  5 6  -  -  ZZ  +  53o  635  +  3Z3  i K - i kips  661  +  S3o  +  Mp  +  - 4  367.  3o5  3  54fe  43  -  -  •z C o n d i t i o n  7 &e>  67  7  +  Con <?lition I C o n d i t i o n  -  5  I0&5  for Cor\s.t#r->V Lo&<A \r-\ k i p S . s iS  +  +  -  Ben-f.  +  463  +  43  ) 66  - 7o7 — ZZ?  4-7  1 o& - 1 Z7  -  rigHf  -  35  --  frorn  +  3o/  - 1 06 S<  Load  +-  +  C  Li^sa  M orne*^i&  Moment ir». k i p s  No. 3  \)ead  III  4 I  25  +  36  -  458  Z7  I  653  1445  117 5  63  +  484  83  4  Z67  +  1  8  4 4 8  336-  -  Z37  +  -  4ZZ  -  4oo 341  ZS>  £ 7 4 74C*  -  865  -  737  -  -  673  4 I  0  835  -I0  +  7  -  -  E>  +  587  - &4<o  -  350*  +  3CI  -  —f  J o  783  3o3  14 2>8  1  1  +•  6-4  Z56o  4 © o  -  t  3o  Z  48 3  &3o  0  Z37  Z 32  - 1o36 —1 Z / Z  43  I Z  +  -  I 7  4-1 4fc  3  0  74o  3  80  37  1 1 75  18  1  445  !  7 4 o  1  I3 0  •25 18  o  77  - 1 535 - I7Z-I  - 845 —4 6 3  60  zn Z£  o£  Z o 3 o IS  S o  1 1&o  Table  IV  Dead Secli'on  o.  N  \Ao m<zr~its ir».  5  U  \  load  5  -  &z  -  i°0  -  \m  3  17,  +  ISi,  +  K  '' K  7*  s  K  3 *  tAom<zin\s  S\r\gle  £><auj  Semi.  f o r l/ic»r : < a l o ! e  A n a l y s i s  Yield  in. kips  Mome>ols Ne^ta+Ive VI 3 6 !  -  h&n  M p  t  7 3 6  4 3 7  ,  to.kips  My  , i o . kips  1 1 6 o  l o g o 1 4 5 o  - 7 6 6  1 3 7 5  + 743  - 1 \ 31  1 8 7 5  7 o 3 o  1 8 2 5  +  635  - 1467  2 3 7 2  7 5 o i  Z24©  665  - 175©  2615  Z 7 o 3  23 o o  -  1  5 S o  - 1 5 6 6  7365  7£6o  2 ? 6 o  6 4 2  - 82)6  1 5 4 o  7  3 o  1 &oo  477  - 4 7 5  2>©2'  1 7 4 o  1 54e>  35  5 5 0  -  6 7 6  1445  1235  4-7  + 4 S 7  - 7 1 !  6 4 8  I 1 75  1 060  4 6 7  3 6 o  + 6 17  43 +  3 1 6  1  4Z  + 3 5 6  -  +  47  +  4 3 7  -  t i l  6 4 8  / I 7 5  1 060  -f  35  +  5  -  3 16  8 7 6  1 4 4 i  1 2 3 S  +  3  +  477  - 4 7 5  3 o 2  1  -  + 6 4 2  - 6 3 6  [ 5 4 o  2 l 3 o  I 8 o o  lo6  oil  -  2 3 6 5  Z£6o  Z7 6 o  e(oZ  - 175©  2 6 I S  ZIoL  23oo  c  ! 3  [or  R<7ing£ i f - i  7  +  'OR  load  32  -  1 o 8  I3L  load  l i v e  kips  --  II,  M o m e n h  43  5 6  13 1  1 £ 6 6  .  0 3 o  7 4 o  IS4o  -  177  +  -  !o&  + 635  -  1 4 8 7  23-Z2  2 So£  81  +- 7 4 3  -  11 3 2  1 6 7 5  2 o 3 o  1 8 7 5  57  •f- 5 6 7  -  78 8  1 3 7 5  1 S 3 o  1 4 5 o  si  4- 3 6 1  - 4 3 7  7 3 6  11 S o  1 080  k  Z74o  NOTES 1. B t a s f i o  (•he  PESJcC't-O  : rnarnet^\i>  are  *p\o\\e.<A  cowipt-essioo si<s>(e  Z- Scale  of r o o r o e n f s  4. Hxo^e  points  o£  SIOOKJO  Lo^^io^ :  fh^5>  — -£-  th<->S  t 7 o las per ff- o o projection  z  wiecl^d^istvis  jhowO  1  o o  fhe f r ^ m e ,  . — [ i»n =• IE>o<3 ir-> k i p s .  3. K e y p o i n ^ i of fr-wme  COMDiricOsJ  — «^ f i g u r e  1  Viort-zo^  +  rsl o I.  r  e  P C S / G N J  :  E l ^ s f i c f h e  Z.  S  b n o m e n l s  c o m p r e s s i o n  S e ^ l e  of  i^iotrierifs  are.  p l o H e ^  si<s^e  —  o f  I  i n  floe  =  Locttf^m^s :  o n  Wtx<*~\Q.  l £ > o o  in  kips.  C  O  M  S>  I  T  l  O  M  Z  7 0 lbs per f {. o n p r o j ect i o n .  loorijonj^l  4oo  v e r h c ^ l  lbs  p e r  p r o j e c f i o n  Figure  D  ff.  o n  ( d c f i i o ^  f r o m  lef  Collapse  rsf o r L s  DESJCN] .OCK^^^S  ting  c o m p r e s s i o n  Z.  Se<5*(e  «?f  5.  Ke<--j p<3iro)-s of  4.  'H'noge  s i ' ^ e  m o m e r i l ^  points  —  fWirv->e of  of l i n  Hog c  iro  sioowio tki^s  4-0<3  k i p s  — -^-^  yne.cV~t , a m s . » o s h o w n  t h ^ &  —  Colo  CONJPlflON Ibi  p«<  on  horizoof<?  on  x/eHic^t  pf"oje£ h o » n .  frvarr-ie.  iE>00  :  K4eclniaioisiv-)  •  ^  Iloi p « f  f4  Collapse (_ * K O W I K - > ^  p o s i t i v e  ^"""vl  •ne^cafiv/e  o f  N J O T E S. : f.  £ .  Scale ICey  of  p o m f s  iAec\n&oisws  L O A P l K J G  t^oot-oer-ifs  c f  f r a m e  —  ii  n  s h o w n  = l ^ o oi i-Ww^  n  Loa^in^s :  k i p s .  C  on  O  N  D  I  T  I  C 9 O O  l b s p e r ff.  4oo  lbs per f f . ovo v e r f i c ^ f (^<rfir-i<^ eilloer*  Figure  IO  N  h o r ' i z o n l f l l  —  proje^f 10n  O  fro^-j  i s d i ' g c f i o n ^  23  33 1/3  percent as per Clause 15(e)  stress f a l l s  of the Code).  j u s t w i t h i n the allowable v a l u e .  Thus the a c t u a l  E l a s t i c values  shearing s t r e s s throughout the frame are w e l l w i t h i n  for  acceptable  limits. C. L i m i t Design A n a l y s i s by Constant Load Method From previous c o n s i d e r a t i o n s  i t i s known that two methods  are a v a i l a b l e f o r the determination of the l o a d c a p a c i t y of the bent under an assumed constant l o a d c o n d i t i o n . the v i r t u a l work e q u a l i t y of equation (1)  These are,  firstly,  r e l a t i n g the i n t e r n a l  and e x t e r n a l work done under the f a i l u r e loads d u r i n g a d i s p l a c e ment of the s t r u c t u r e  i n the form of a c o l l a p s e mechanism, and,  secondly, by means of the zero l o a d equation (5)  d e f i n i n g the  i n t e r n a l work done by the r e s i d u a l moments at the hinge p o i n t s d u r i n g a mechanism displacement. Now,  as the v a r i a t i o n s 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 f a r more d i r e c t means of. s o l u t i o n than does the former method. First  consider the frame i n general terms,  hinge p o i n t s necessary two, (1)  to develop a mechanism i n general w i l l  and i f these hinge p o i n t s are l o c a t e d a t , and (2),  r e f e r diagram below, equation (5)  F  the number of  =  M, p  .G, +  M  m  .Qx  M, .©, + M . © 2  z  say,  frame  becomes:  be  sections  24  where Mp,, M,, a n d moments a n d a n g u l a r at s e c t i o n  H  Q,  deformations  ( 1 ) , and Mp^, M , t  are l i k e  z  are  and  quantities at  section (2). The r e s i d u a l moments a t t h e s e s e c t i o n s , equation section  ( 5 ) , a r e (Mp,-F.M,) a t s e c t i o n ( 2 ) , w i t h moments s i g n s  previously  established  by  ( l ) , and (Mp -£ .M ) a t i  l  l  throughout conforming t o the  convention.  The d i a g r a m shows t h e f o r c e s and  as d e f i n e d  i t i s a t once e v i d e n t  a c t i n g on t h e u n l o a d e d b e n t  from e q u i l i b r i u m requirements  r e a c t i o n s H must be e q u a l and o p p o s i t e . moments must be d i r e c t f u n c t i o n s  that  Thus t h e r e s i d u a l  of ordinate  ' y , and t h e s e 1  moments c a n now be w r i t t e n a s :  M,  -  P.M-,  y,.H ,  Mp  -  F.M  y .H ,  p  and, and  solving f o rP  z  T  r  gives:  a,.(i/y, ) + A n g u l a r d e f o r m a t i o n s Q, a n d Q ional to ordinates and  y, and y , z  z  are thus i n v e r s e l y  respectively.  y, a r e known f o r a l l f r a m e s e c t i o n s  c o r r e s p o n d i n g t o any two s e c t i o n s i s now o n l y n e c e s s a r y t o l o c a t e  (ii)  K.(l/yJ  proport-  Q u a n t i t i e s Mp, M,  and t h e v a l u e o f P  c a n be r e a d i l y d e t e r m i n e d .  the sections  r e s u l t i n g i nthe  It  25  s m a l l e s t v a l u e o f F i n o r d e r to mechanism and t h e  correct failure  The two h i n g e s w i l l n o t first  one t o  first  reached i n  e s t a b l i s h the  form w i l l the  be at  frame,  correct  failure  load.  d e v e l o p s i m u l t a n e o u s l y and the  that  s e c t i o n where y i e l d i s where  the  stress  v a l u e o f Jd_  Z maximum.  For design c o n d i t i o n  diagram i n F i g u r e 9,  (3),  the  is  at  a  e  as r e p r e s e n t e d by t h e  t h e maximum v a l u e o f M_ o c c u r s  is  moment  the  knee  2e joint  s e c t i o n B* and t h i s  correct  position for  ed by t r i a l the  the  by means o f  second hinge at  therefore  i s one h i n g e p o i n t .  s e c o n d h i n g e c a n now be r e a d i l y equation  between  (11).  the  and 1 . 0 0 0 ,  respectively,  equation  and ©  t  (5),  is  z  two h i n g e s a r e  =1240(0.770)=  M ^.«  = 2 7 0 5 ( 1 . 0 0 0 ) = ' 2705,  Numerator  and,  F  fixes  Relative  f i g u r e d to  and t h u s F ,  establish-  be 0 . 7 7 0  obtained  from  as f o l l o w s :  Up,.0 p  Such a p r o c e d u r e  s e c t i o n s 18L. and 19,_.  values of r o t a t i o n Q f o r for Q  The  950,  M,.0  3655  403,  M . © =1877(1.000)=  1877,  Denominator  2280,  z  = 3655,  =  = 522(0.770)=  =  =  1.605  2280 T h i s method h a s e s t a b l i s h e d t h e has produced the If,  on t h e  previously, applied it  t r u e minimum v a l u e other hand,  the  for  the  failure  mechanism and  failure  load.  a p p r o x i m a t e method s u g g e s t e d  o f l o c a t i n g the h i n g e s at i s found that  exact  s e c t i o n s o f minimum Mp, M t h e h i n g e s a r e l o c a t e d at knee j o i n t  is B^  26 and at s e c t i o n l & Y , g i v i n g a l o a d f a c t o r of 1.61. between the exact  The d i f f e r e n c e  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 d i f f e r e n c e between the two r e s u l t s small i n every i n s t a n c e .  has been found to be  On the b a s i s of these f i n d i n g s the  approximate method f o r hinge p o i n t l o c a t i o n w i l l be adopted i n the remainder of t h i s Now, above,  study.  i n a d d i t i o n to l o a d i n g c o n d i t i o n  i t i s also necessary  c o n d i t i o n (1),  (3),  considered  to evaluate the e f f e c t s  as shown i n F i g u r e 7.  also i n d i c a t e d i n the f i g u r e .  of l o a d i n g  Sections of minimum M-n are  The general symmetry of the moment  diagram i s at once evident and t h i s f a c t  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 all  three key p o i n t s , 1,  2, and 3 (see F i g u r e 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 s o l v i n g f o r F : 2705(1.000) -  2705 ,  1212(1.000) = 1212 ,  980(1.382) «= 1355 ,  392(1.382) =  2705(1.000) = 2705 , Numerator and,  F  =  1212(1.000) = 1212 ,  = 6765 , 6765 2974  550 ,  Denominator =  = 2974 ,  2.27.  That the sway mechanism with hinges at p o i n t s 1 (or 3 ) , 2, produces an i d e n t i c a l value f o r F w i l l  be apparent from the  and  27 r e l a t i o n s h i p between Q's f o r the two  mechanisms.  Thus l i m i t i n g values f o r I' f o r l o a d i n g c o n d i t i o n s (3) are 2.27  and 1.61,  respectively.  (1)  and  Comparing these values i t  would appear that c o n d i t i o n (3) i s c r i t i c a l , however t h i s c o n d i t i o n i n v o l v e s a l l e x t e r n a l loads i n c l u d i n g wind and  a  smaller l o a d f a c t o r can be used f o r t h i s case than f o r c o n d i t i o n (1) which excludes the wind l o a d i n g s .  Recommended design  values  f o r l o a d f a c t o r s conforming to AISC requirements are as f o l l o w s : (a)  1.41,  f o r a l l f o r c e s i n c l u d i n g wind,  (b)  1.88,  f o r a l l forces excluding  wind.  The  a c t u a l a d d i t i o n a l margins of s a f e t y provided  and  above these recommended values  conditions  (3) and  are 1.135  (1), r e s p e c t i v e l y , and  by the bent over  and 1.21  for  the former c o n d i t i o n  t h e r e f o r e d e f i n i t e l y governs. The  f a i l u r e l o a d f a c t o r f o r the bent under the  assumptions of the constant  l o a d method i s t h e r e f o r e  corresponding to an a d d i t i o n a l margin of s a f e t y over requirements of 1.135. Figure  The  loading 1.61, specification  f a i l u r e mechanism i s as shown i n  9.  D. L i m i t Design A n a l y s i s f o r True V a r i a b l e Loading The  a c t u a l l o a d i n g to which the bent i s subjected  i n a constant  dead l o a d and  consists  a random p a t t e r n of a p p l i c a t i o n s of  g r a v i t y l i v e l o a d and wind, and  the moments which must be  i n t o account i n the v a r i a b l e l o a d i n g a n a l y s i s w i l l be  taken  those  28  r e s u l t i n g from each o f these d i s t i n c t l y  independent  loadings.  As w i n d 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 t h e s p e c i f i c a t i o n requirement f o r load f a c t o r w i l l It will  be 1.41.  be c l e a r t h a t t h e c a s e o f g r a v i t y l o a d  w i n d n e e d n o t be c o n s i d e r e d  (and c e r t a i n l y n o t w i t h a  without higher  s a f e t y r e q u i r e m e n t ) 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 alone represents  just a portion of this total  Now, a s p r e v i o u s l y d i s c u s s e d , p o s s i b l e under v a r i a b l e l o a d i n g .  t h e s t r u c t u r e may  collapse with true  l o a d o b t a i n i n g when t h e f o l l o w i n g e x p r e s s i o n  F  u  =  loading.  two modes o f f a i l u r e a r e  Firstly,  as a mechanism t h r o u g h i n c r e m e n t a l  actual  failure  i s minimized:  £(Mp - P . M ) . 0 P  fail  P  (7)  S e c o n d l y , f a i l u r e may o c c u r t h r o u g h a l t e r n a t i n g p l a s t i c i t y a t a localized  s e c t i o n o f t h e 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 t h e  following expression  i s minimized:  S o l u t i o n s f o r t h e s e e q u a t i o n s c a n be r e a d i l y o b t a i n e d moments M u and Mi. a r e known.  B o t h t h e w i n d and g r a v i t y  once  loads  are  i n v o l v e d i n t h e s e t e r m s and t h e m a g n i t u d e o f 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 t h e maximum l o a d a c t i n g on one  s i d e o f t h e b e n t and an e q u a l o r l e s s e r w i n d 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 opposite d i r e c t i o n .  29  The magnitude of t h i s r e v e r s e d wind e f f e c t s p e c i f i e d f o r the bent, nor i s  i s not a c t u a l l y  i t r e q u i r e d i n the e l a s t i c  analysis  where the l o a d i n g c r i t e r i o n need only be i n terms of the maximum wind a c t i n g i n one d i r e c t i o n .  The r e v e r s e d wind e f f e c t  is  also  of no i n t e r e s t f o r the previous constant l o a d a n a l y s i s f o r which the same c r i t e r i o n  applies.  The severest p o s s i b l e  c o n d i t i o n i n v o l v e s a reversed wind  equal i n magnitude to the maximum l o a d i n g of 20.0 l b s . Values o f  per sq.  ft.  and M u have been computed f o r t h i s r e v e r s e d l o a d i n g  and these values are given i n Table I V . The v a r i a t i o n i n throughout the bent, f o r both p o s i t i v e and negative amounts, shown i n F i g u r e 10,  together with the l o c a t i o n of s e c t i o n s where  minimum v a l u e s of the q u a n t i t y (Mp - F p . M P ) f a c t o r of 1.25  is  occur, a dead l o a d  being employed i n the values f o r the  latter  quantity. The symmetry o f t h i s f i g u r e again immediately suggests the p r o b a b i l i t y of a symmetrical f a i l u r e mechanism with hinges p o i n t s 1,  2, 3,  and 4, of the f i g u r e .  s e c t i o n s B L and B K , r e s p e c t i v e l y , 1 9 L and 1 9 K ,  respectively.  at  Hinges 1 and 4 being at  and hinges 2 and 3 at  sections  R e l a t i v e values of Q are f i g u r e d to  T h i s f a c t o r of 1.25 i s based on an e v a l u a t i o n of probable e r r o r s i n v o l v e d i n the q u a n t i t i e s e n t e r i n g i n t o the design procedure when dead loads only are p r e s e n t . D e t a i l s of t h i s e v a l u a t i o n w i l l not be i n c l u d e d here, but reference i s made to papers on the subject of F a c t o r s of Safety and p a r t i c u l a r l y to the paper by A. F r e u d e n t h a l , e n t i t l e d "Safety and P r o b a b i l i t y of S t r u c t u r a l F a i l u r e " , Proceedings, No. 468, ASCE, August 1954.  30  be,  «, = 0  = 1.380,  4  and Q  - 0  z  frame n e e d be c o n s i d e r e d  3  = 1.000.  Only one h a l f  and F , o b t a i n e d  from  c  of the (7),  equation  becomes:  (2705 -  (1.25)127)1.380  = 3510  ,  (1750)1.380  =  2413  ,  (1.25)47)1.000  = 1115  ,  (437)1.000  =  437  ,  = 4625  ,  = 2850  ,  (1175 -  Numerator  PL  and,  =  4625  -  Denominator  1.62.  2850 The 3  and  corresponding  ( o r 2 a n d 4)  a g a i n be e v i d e n t The  load factor  sway m e c h a n i s m w i t h h i n g e s  the similarity  f o r incremental  i nthe relative  failure  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 It  i sevident  occurs  the at  from  (9)  equation  w h e n My i s m i n i m i z e d , Mu  smallest value  of this  as w i l l o f 0.  o f t h e bent i s thus  1.62.  of alternating  and reference  t o Table  quantity i sobtained L  1  values  plasticity.  that the smallest value  w h e r e My a n d M  t h e knee j o i n t s  respectively.  f o r I\ o f 1 . 6 2  also produces a value  from  at points  f o r FJ  I V shows  i n the roof  that  beam  a r e 2300 a n d 2615 i n . k i p s ,  Thus t h e l o a d f a c t o r  forfailure  by a l t e r n a t i n g  plasticity i s : F'  Load f a c t o r al  failure  live  =  2300 2615  =  PL i s smaller, at 1.62,  i scritical  load factor  2.  f o r t h e bent;  t o produce f a i l u r e  this  1.76.  and t h e r e f o r e  increment-  representing the true  o f t h e bent under  actual  31  loading conditions, analysis. is  as envisaged by the v a r i a b l e l o a d method of  As p r e v i o u s l y noted a s p e c i f i c a t i o n l o a d f a c t o r  appropriate  of  1.41  to the l i v e l o a d i n g i n v o l v e d and the a d d i t i o n a l  s a f e t y margin over t h i s  specification  requirement i s 1.62  ,  1741 equals 1.15.  The c o l l a p s e mechanism may take the form of  either  the symmetrical mode or the.sway mode, as shown i n F i g u r e 10.  3.02  A n a l y s i s of Double Bay Gable Bent  A. General The double bay bent to be considered i s  a standardized  design employing o u t s i d e column and r o o f members o f i d e n t i c a l . s e c t i o n and dimensions with those used f o r the s i n g l e bay bent. Reference i s made to F i g u r e s 4 and 5 f o r the dimensions and details  of these s e c t i o n s .  A vertical 12"x 6£ x36.0 l b . M  r o l l e d wide-flange s e c t i o n r e p l a c e s the v a r i a b l e s e c t i o n f a b r i cated columns f o r the c e n t r a l support,  and d e t a i l s of the connect-  i o n between the wide-flange and r o o f members i s shown i n F i g u r e 5.  A l l other connections and s t i f f e n e r  the s i n g l e bay bent.  arrangements are as  The spacing of frames i s  for  20'-0" on c e n t r e .  Design requirements are a l s o as d e s c r i b e d f o r the previous structure,  and l o a d i n g c o n d i t i o n s are i d e n t i c a l and, b r i e f l y ,  as  follows: Design c o n d i t i o n  (1),  uniform l o a d on h o r i z o n t a l p r o j e c t i o n of 670 l b s . Design c o n d i t i o n  per  lin.ft.  (2),  wind l o a d on v e r t i c a l p r o j e c t i o n of 400 l b s .  per l i n .  ft.  70 l b s .  per l i n .  ft.  wind l o a d on v e r t i c a l p r o j e c t i o n of 400 l b s .  per l i n .  ft.  p l u s l o a d on h o r i z o n t a l p r o j e c t i o n of 670 l b s .  per l i n .  ft.  plus dead l o a d o f Design c o n d i t i o n  (3),  33  B. E l a s t i c A n a l y s i s Procedures employed i n the e l a s t i c l y d e s c r i b e d f o r the s i n g l e bay bent.  analysis  are as  A l l supports  are  previousassumed  hinged. R e s u l t i n g moments at s e c t i o n s  throughout the frame are given  i n Table V f o r i n d i v i d u a l g r a v i t y and wind l o a d i n g s . thrusts  are not shown as t h e i r e f f e c t s  n e g l e c t e d i n the l i m i t design  Shears and  are again small and can be  analysis.  Moment diagrams f o r each l o a d i n g c o n d i t i o n , p l o t t e d from the t a b u l a t e d r e s u l t s  are shown i n F i g u r e s 11, 12,  and 13.  For  these diagrams moments are p l o t t e d on the s i d e of the frame whch they create compressive  at  stress.  F i b r e s t r e s s e s are not r e q u i r e d f o r t h i s study but i t  is  i n t e r e s t i n g to note that the severest s t r e s s occurs i n the r o o f beams at the--central j o i n t and i s equal to 24.6 k s i .  This  is  a p p r e c i a b l y l e s s than the allowable value but the c o n d i t i o n i s t o l e r a t e d i n the i n t e r e s t s  of s t a n d a r d i z a t i o n of components.  ^ ' k l m i t Design A n a l y s i s by Constant Load Method With e l a s t i c  moments known a s o l u t i o n f o r l o a d f a c t o r F i s  again most r e a d i l y obtained i n terms of equation (5). the s i n g l e bay bent l o a d i n g c o n d i t i o n s (l)  and (3)  As with  only need be  considered. Investigating loading condition ( l ) .  The general symmetry  of F i g u r e 11 i n d i c a t e s that a symmetrical f a i l u r e mechanism i s  to  Tajoie V  M o w e n h  for  bay  Double  L  5ecf  IOIO in.kips  -  4 7 3  -  161  -  4  5 2 3  -  234  -  - !o3Z  4  554  -  3 o 6  -12)4  +  547  -  36©  l o 5  -  3 3 6  4  4 3 3  -  3 4 o  45  -  39o  4  3 » l  -  +  254  2 6  +  Ill  35  +  755  3 o  4-  25ft  2 3  +  24-7  1 3  4  11  -  35  -  3o4  t +  61  -  5Z  -  773  &L-  - Ito -14 1  IILL  -  !3CL  -  4-  +  '3LR  4-  I f LIT  +  -  6 3  R  -  1 3 o  PLR  -  !3  L  M  t  R  •  I 6 4  -  Dcol. D«u  -  !ieu  -  I 3  I 3  R  L  I5«u  C  K  -  -  1 6 4  o  -  6 3 11  -  ^ 9 i 4  ,  nn. k i p s  14 o 6  -  4  37  ^ l?R  -  6 I  -  -  se  -  16 4  -  4  4  276  -  135  4-  1 12  1 175  1  146  15  1 4 4 5  1 7 7  S 2 &  -  16 6  4  5 8 o  -  746  +• 7 4  37  4  265  - l-Z  ?o  175  3 3 6  14-1 1  4  4  -  ( 4-  -lo3£ 773 577 3 © 4  I 35  -  168 167 166 1 65  08  6  oo  -  23o  -  ! 2 4 2  -  23 8  -  1 &7©  -  33  -  -  8  168  t  -  bo  -  -  -  -  2  748  16  4  274  l 7 4 o  7 6 7  2 13©  1 4 1 o  2 5 6 o  1738  77o5  7 4 8  1 6 8 2  -  33©  27o5  336  -  7 1 4  7 5 6 o  4  333  -  2o4  2 1 3 o  1 o&  4  2 54  4  1-5 7  I 7 4 o  1 6©  4-  164  +•  315  1 4 4 5  2.5  +  277  / 1 75  1 5 7o  4-  4  1242  4-  6 oo  —  16 4  4-  276  4-  3 6  -  16 3  4-  -288  -  6 6  4  1 32  1 34  -  4o  4-  33o  -  35  +  1 e>6  2 4 8  -  1 7 o  4-  52  (445  237  -  237  174©  333  -  7 Z 3  213©  1 3 &5  256o  4  3&o M  75  136  4-  I 1 3  Z37  4  7 5 4  266  4  331  -  435  -  34o  +  4 3 3  -  1 ©45  -  36o  +  5 4 2  -  1355  -  So 1  77o5  3©6  4-  1152  -  14 3 8  25 o5  234  4  5 2 3  871  -  5 26  -  1 1 ©5  7 o 3 o  - 161 -• 63  4  4 7 3  -  4 26  --  17 15  554  5 8 6  -  272  -  7 4 3  l 5 3 o  4  1 24  -  4 7 6  118©  -  35© 3 3 6  3 6 o  5  ( iS  537  1o 3  1 1 7 5  -  Sgo  -  I 44S  ID©  1 7 4 ©  -  &Z8  45  36 I  7 13©  2 8 8  +-  -  (33  +  44 2 54  33©  14 o 6  -  +  4-  4  27 Z  26  " 4 ^  256©  3 6  I 6 6  -  4  2 7 o 5  134  -  255  3 4 6  -  61 7 5 4 6  -  -  +  35  435  4  Z5o5  4©  3 3 6  4  lo45  I  5 5 6  163  4-  -  4 o  3 5 o  &z  153© 2 ©3©  +  167  2S6  4 4  11  153 342  4-  -  4  1 3 5 5  -  2.4 8  4  4- bo  4 3 4  1I 6 o  4-  -  —  4 3 7  4-  m . k<p£  +  Z65  247  3 6 6  4  P  156  II  1 6 1  4  8 7 |  M  -  1 35  4  568  5 3  Momenf  -  4  11  245  H a s .4 ic  I 1 3  +  -  -  4  3 3 3  237  165  4-  1 «  3>  Z  2 6 6  166  +  -  R  .37  --  63  4  1 3  3> RR  6  1 6 !  loo  -  ZZ>  -  •'»„  -  4  K  R  4  4  +  R  7  - 537 -11 (  -  15 < ! S  Ana  load  YJ\r\d Cor\d\\ibnCond\hor\ Condition loading Worn L . f r o m K . 3. z.  - sz  L U  <ip&  Wind  N o .  lt>  C©r~>star\i  for itn.  Se.ni.  -  2 8 o  4-  -  3 3 3  —  4 4 3  feeble  VJ  De^«3< Load Moments, No.  m. kips  Moments  -  7  L  L  11  tt  Greatest PosiJix/e M +  61  t  37  + +  S £4  - 14 l  +-  547  1 o5  + u  + -  -  E L  c*  +  ZDS  So  +  25  ) I  ER  £> K ^ KB 7 3  RR  6  73  +  Z47  1 3  +  Z36  II  f  765  1 64  4-  336  + 5 28 5 8o  4-  146  +  '5 & o  Yield  L  Mp,  Moments  iVvlcipS 11 6o  1 060  1 5  16 2 5  I 637  Oo ZoSo ZSo5  224  1574  III 6  27o5  Z3 oo  1 17  I 775  I S C c O  2Z6o  333  68t  -  666  1111  - 1o 1 3  I 547  -  I 33 o  6  67 &  I o63  1235 1  1 36  5 3I  1445 4 o  -  134  473  117 5  1 63  4tl  -  -  3  o  1 5 4o  06 o  63o  S>o  164  4 I l  117 5  1 oGo  1 65  46!  1445  1 235  Z [loo \Z o \do& 52 8  1o4- 6 3  I S 74 6 7 4 6  4-  5Z &  4-  336  - I 574 I 1 o 4&o  -  1  o  I &oo  Z.I S  -  -  14 So  4^> J  737  -  My , 1V1 kips  t  174o  l64o  7 13 o  1 0OO  2 S6o  Z26o  2 I5 4  27 oS  23oo  I 4 36  1662  15)5  Z I 54  2 7o5  Z3oo  Z56o  226o  Woo  2 ISo  1 6oo  I  608  -  I1  +  265  -  5 26  174o  !54o  +•  1 3  +  236  -  1 65  4 6 I  1445  1235  +-  73  +  2 4 7  -  1 64  4 1 1  1175  1 060  +  3o  4  7.55  -  163  4 21  38°  35  f  2 3 5  1 3 4  4-i3  1175  f  5 31  I 44 5  1735  4 t> 1  1 7 4o  (5 4 o  1 o6e>  7 1 So  1775  256o 27oS  +  I S R«  2 5 4  35  - l3o - 63 is  +  33 5  -  3  433  +  63  Moment M  -  f l a s t IC Moments  L  3 3 I  76  - 13o - 164  -  +  45  -  c  Z  - 1I o  -  E>e>nf.  Greatest  63  45 2 3  -  Double  Live load Moments for \Jar\a\o\e. Load A n a l y s i s , in. k i p s  35  £ u  for  -  -  -  26  +  335  -  +  7.54  45  4-  33 (  1 o2>  +  4 3 3  1 4J  4-  Z  5 5 45 4  17o  4-  37  4-  61  4-  3.5  4- •Z 6 3  5 X3>  -  763  1  "O<0  83o  060  -  Z  -  17 7 6  - 1574  2 116  -  13 3 6  1 ©37  Z5o5  l o 13  1542  Z°3o  Z?4o 1625  686  11 17  IS3o  l45o  -  -  -  6 7 83 7  32>3  66 2  1)  e>o  boo ZSoo  21Z 6 o  Io  6o  (he c o m p r e s s i o n  Sxsle  1.  Scale of m o v i e ^ f s  3.  Key p o i n t s of frame points  4.  of the [rat^e .  — Im.*  1  l£>00  .in. lops .  shown ^<J5 - r  of mCclr>ai^\iswi  £! c y s t i c  P  E  S  I  6  N  Collapse  (t?)  CONPITION  I  M  o  m e n j  Lo<s.d<n^ ;  f i c ^ ^ r e  Mech^tnisun  D : ^» <^ r-^a t-r->  6^0  11  (t?s  per  ff.  N O T E S  1.  P E 5 I 6 N  Scale  C O N DITlOfN  of  2  foome^Js  -  / m  Lo<^oii>n <sj •'  •=  \*~,oo  im. k i p s .  "7<0 lb* per ff. o n  4-oo lbs per ff. o n (acting IZ  \~,of\-zc>r^o\\  verdc^l froITO  projection P»" J«cf 0  leff  J  'om  I.  £!^sfi<c  mordents  }(->e c o m p r e s s i o i o  T. Si-ale,  ol  pomfs  Ar. Hin^e  6\<rc <p\o\ke,ck.  moments of  poir-iks>  o n  s i ^ e of f h e f r a m e . —  ir-ame  of  1  =  in  !E>00  s k i o w n fkiuS  M€ck<9nism  in kips. —  Shown fhoS> —• a"2-  (h)  Collapse  Mech^riisi^i  NOT E5 :  ©f YTiome<\t<, — lio= l £ > o o m k i p s . poi'p-v/s cf frame showo fhcS — points of 5ho»~>n fl-wS — »^  I.  5e<sWe  3.  \\\s~>ije  methanisiri  (b)  ( s h o w i n g  VAP.IA6.lE  LOAPIN6  poSt\i\se  CONPIMCrS  d*>n<A lo^c?(  KAeCr~i&r-\\S>\rr-i  C o l l a p s e  n z g a f i ^ e  v a l u e s  of  l b s per ff. o o l o o r i - i o - n W  i'.o^s:  4oo  lbs pe'' ff. or-i  -JtrWca-l  prc>jecf I'OO peojec^ion  (jxc'tM^ia^ f r o r n eillner ^ i r e d l o - m j  figure  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 c e n t r a l j o i n t and, as y i e l d i n g must f i r s t  occur here,  plastic  l y at these s e c t i o n s .  hinges w i l l be created  Additional sections  simultaneous-  at which minimum values  of Mp occur are l o c a t e d at j o i n t s B and C and i t w i l l be f a i r l y M evident from i n s p e c t i o n of F i g u r e 11 that the c r i t i c a l mechanism will  conform to the mode i n d i c a t e d i n the same f i g u r e .  h a l f of the s t r u c t u r e  on e i t h e r side of centre l i n e need be  considered i n e v a l u a t i n g F.  The r e l a t i v e values of hinge r o t a t i o n  may be f i g u r e d from the geometrical centres,  as  Only one  concept of  instantaneous  follows:  R e l a t i v e frame dimensions and r a d i i of r o t a t i o n of elements  are shown i n Figure 15,  frame  from which,  F i g u r e 15. Now, s u b s t i t u t i n g  these values  values f o r Mp and M, i n t o equation  f o r Q, with the  (5),  appropriate  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 thus,  F  =  11010 ,  =  Denominator  11010  =  =  5300 ,  2.08.  5300 Check a n a l y s i s  of the a l t e r n a t i v e mechanisms confirms  as the true value of l o a d f a c t o r will  f o r l o a d i n g c o n d i t i o n (1).  be noted that on account of the symmetrical mode of  four r e q u i r e d f o r a general c o l l a p s e  Sections of the  at which minimum values of Mp occur are l o c a t e d i n M  F i g u r e 13.  It  should f i r s t  be observed that hinges can develop  at only two of the three key p o i n t s at j o i n t D, as w i l l evident i f the s t a t i c a l  e q u i l i b r i u m of t h i s j o i n t i s  be  considered;  one of these hinges must be at the key p o i n t i d e n t i f i e d  as 3 as y i e l d s t r e s s i s f i r s t section.  than the  of the frame.  C o n s i d e r i n g now l o a d i n g c o n d i t i o n (3).  further,  It  failure  the number of hinges i n v o l v e d i n the mechanism i s greater  structure  this  reached i n the frame at  this  A second hinge must be at key p o i n t 7 as s t r e s s here  equal to that at key p o i n t 3 and these two hinges w i l l  is  therefore  develop s i m u l t a n e o u s l y . With these f a c t s recognized the simplest procedure i s now to regard the t o t a l mechanism f o r the frame as the compounding of two  elementary, mechanisms with one of these i n each bay.  review of the previous a n a l y s i s  A brief  of the s i n g l e bay bent w i l l  reveal  that the c r i t i c a l t o t a l mechanism must c o n s i s t of hinges at key p o i n t s 2, 3, 6, and 7, as shown i n F i g u r e 16. Again u s i n g the concept of instantaneous  centres,  refer  36  Figure 16.  Figure 17. Figure 17, the angles o f r o t a t i o n at hinge p q i n t s are f i g u r e d follows: B"0HN"  thus,  = 1 8 . 7 41.7  0.449  DEN'  44.5 33.0  (0.449)  0.605  DO'K'  18.7 47.3  (0.605)  0. 239  BAH  50.8 31.0  (0.239)  0.391  Q.  0.391 + 0.239  0.630  0:  0.605 + 0.239  0.844  0.605 + 0.449  1.054  as  37 1.000 + 0.449  1.449  and s u b s t i t u t i n g these v a l u e s o f Q, and the a p p r o p r i a t e o f Mp and M, i n t o e q u a t i o n  (5), gives:  1310(0.630)  =  825 ,  320(0.630)  =  205 ,  2705(0.844)  =  2285 ,  1738(0.844)  =  1465 ,  1175(1.054)  =  1240 ,  272(1.054)  =  287 ,  2705(1.449)  =  3920 ,  1715(1.449)  =  2483 ,  =  8270 ,  =  4440 ,  Numerator thus,  values  P  Denominator 1.86.  8270 4440  Failure load factors f o r loading conditions are t h e r e f o r e 2.08 and 1.86, r e s p e c t i v e l y .  (1) and (3)  However a l l o w a n c e  must be made f o r the d i f f e r e n t forms o f l o a d i n g i n v o l v e d i n these c o n d i t i o n s , as p r e v i o u s l y d i s c u s s e d i n t h e case o f t h e s i n g l e bent.  The a d d i t i o n a l margins o f s a f e t y i n v o l v e d i n the above  f a c t o r s , over and above the a p p r o p r i a t e  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 , 1.88 1.32, f o r l o a d i n g c o n d i t i o n ( 3 ) .  and 1.86 1.41  -  The g o v e r n i n g l o a d i n g c o n d i t i o n f o r the frame under t h e assumptions o f the constant (1).  l o a d method i s thus a c t u a l l y c o n d i t i o n  The c r i t i c a l l o a d f a c t o r i s t h e r e f o r e 2.08, and the c o r r e s -  ponding f a i l u r e mechanism i s as shown i n F i g u r e 11.  38 D. l i m i t Design A n a l y s i s f o r True V a r i a b l e Loading The a c t u a l  p a t t e r n of l o a d i n g to be considered i s  p r e v i o u s l y d i s c u s s e d f o r the s i n g l e earlier  bay bent,  and as with the  bent a reversed wind l o a d i n g of 20.0 l b s .  assumed.  The v a r i a t i o n s  i n maximum p o s i t i v e  per sq.  i n Table VI and  symmetry of t h i s moment diagram i s  at  once  evident and t h i s f a c t again suggests the p r o b a b i l i t y that c r i t i c a l mechanism w i l l  also be symmetrical i n form.  f a i l u r e mode i n v o l v e s a t o t a l 5,  6,  and 7.  =  2, and 3,  2.254(980 1.127(2705 -  figure  2,  considered  at « , =  2.127, Q  z  =  2.254,  and © 3  becomes:  (1.25)141)  =  5380 ,  (1574) 2.127  =  3350 ,  (1.25)30)  =  2120 ,  (258)2.254  =  580 ,  (1.25)162)  =  2820 ,  (1574)1.127  =  1780  =  5710 ,  Numerator from which,  Such a  the r e l a t i v e values of hinge r o t a t i o n 6 at  1.127, and the equation f o r F u 2.127(2705 -  the  of s i x hinges at key p o i n t s 1,  Only one h a l f of the frame need be  and, from p r e v i o u s l y , p o i n t s 1,  is  14.  The general  3,  ft.  and negative moments  corresponding to t h i s c o n d i t i o n are t a b u l a t e d p l o t t e d i n Figure  as  Fu  = 10320 , =  Denominator  10320  =  ,  1.80.  5710 Further c o n s i d e r a t i o n of the a l t e r n a t i v e  modes of  i n v o l v i n g other p o s s i b l e combinations of key p o i n t s , to be the true mechanism f o r incremental  collapse.  failure  proves  this  39  Por f a i l u r e through a l t e r n a t i n g 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. is  a s s o c i a t e d with the c e n t r a l column j o i n t D where values of My  and M L . are 1515 and 1496 i n . k i p s , value f o r F^ from equation (9)  ?i  =  1515 149?  is  respectively.  The l i m i t i n g  thus:  (2)  =  2.03  The c r i t i c a l f a i l u r e l o a d f o r the c o n d i t i o n of loading i s  thus produced through incremental c o l l a p s e  ponds to a f a i l u r e l o a d f a c t o r of 1.80.  , equals 1.28.  and c o r r e s -  The a d d i t i o n a l margin  of safety provided by the bent, over and above requirements i s 1.80 1.41 shown i n F i g u r e 14.  variable  specification  The mode of f a i l u r e i s  as  l£'-(J  14-o  74- a"  46 - 6' 11 "x  Web  6 - S x / " 5  &  J8-o ZoW '  l V  4  S f i f f e n c e s <a G ' - o "  fn 4^ l^m^e  I 3 Z ' - o'  2 6 - o"  Perils  of  C<ar-iTi l e v e r -  Z? - o " 2o"x5/&"  Web  o'-S" * ^&  G i r d e r  34 !  - o"  Z7.">  v  6  -  Dcsi^  46-o" Zo' x 5 / « -  \Y\ro\ja\Y~)0\jk -A If3  - o'  Pef<?if!s  I Stof  C o o i i n u o a s Figure  16  Oir^ler  Desi^r~>  o"  Pl  F\a>^>r-e I'D  40  3.03  A n a l y s i s of Bridge G i r d e r s  General The complete three span, which two examples to be l a t e r  two l a n e , highway bridge, from considered are taken,  general arrangement i n F i g u r e 3 ( b ) .  is  shown i n  The two examples chosen f o r  study i n v o l v e : 1.  The a c t u a l b r i d g e ,  c o n s i s t i n g of two c a n t i l e v e r  girders  which overhang the c e n t r a l p i e r s and support simple g i r d e r spans at each end of the bridge v i a bearings ends of the 2.  at the outer  cantilevers.  A h y p o t h e t i c a l continuous g i r d e r v a r i a n t , of twin g i r d e r s  which c o n s i s t s  continuous throughout the e n t i r e l e n g t h of  the b r i d g e . General s t r u c t u r a l  details  of the p l a t e g i r d e r s  the a c t u a l bridge are shown i n F i g u r e s 18 and 19. girders  and deck of  The main  are. of b u i l t - u p welded s e c t i o n throughout, c o n s i s t i n g of  ASTM A-7 s t r u c t u r a l of c a s t - i n - p l a c e  steel  concrete.  webs and f l a n g e s .  composed  No l o n g i t u d i n a l shear connectors  p r o v i d e d between deck and g i r d e r s not allowed f o r i n the  The deck i s  and composite a c t i o n i s  therefore  designs.  In the case of design 1,  i n v o l v i n g the c a n t i l e v e r  girders,  i n t e r e s t w i l l be confined to the p o r t i o n s of the c a n t i l e v e r s the c e n t r a l supports  are  i n which r e v e r s a l s  to the movement of the l i v e l o a d s .  between  i n moment take place due  The simple end beams are of no  41  s p e c i a l i n t e r e s t from the point of view of the study.  The  c a n t i l e v e r g i r d e r s are s t a t i c a l l y determinate and are  therefore  not  subject to incremental c o l l a p s e ,  however as a r e s u l t of the  r e v e r s a l s i n moment they are subject to a l t e r n a t i n g p l a s t i c i t y and must be examined f o r t h i s mode o f  failure.  The continuous g i r d e r s of design 2 are  statically  indeterminate and may be subject to f a i l u r e through e i t h e r incremental 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 .  Both of these  p o s s i b i l i t i e s must t h e r e f o r e be considered f o r t h i s Both the a c t u a l and h y p o t h e t i c a l s t r u c t u r e s  case.  are  designed  to withstand the standard AASHO H20-S16 l o a d i n g i n each l a n e , to the f o l l o w i n g  specifications:  (a)  GSA S p e c i f i c a t i o n s f o r S t e e l Highway B r i d g e s ,  1952,  (b)  AASHO Standard S p e c i f i c a t i o n s f o r Highway B r i d g e s ,  (c)  ACI B u i l d i n g Code Requirements f o r R e i n f o r c e d Concrete. In a l l i n s t a n c e s where a l t e r n a t i v e requirements oceured i n  these s e v e r a l as governing.  specifications  the severest c o n d i t i o n was accepted  T&lole>  6  Movnenls  VI!  M^ximuw->  trderio  ff.  kips  fo  M o m a n t s  : L iv/e  N o .  L o & ck i i o <j  Lo<ac/i I t O ^  in Col.  J.  Col.  -  *  t.  Span  Cel.  lo#<^ m m Span  0  S .  Col.  -  -  S o l o  4 f 6-  -  Z 3 3 o  +  26o£>  -  1+4)  M o m e n t (cck  7+S+5)  Co/.  £>.  -  3 5 6 S  M £\* l»V) c>vn M o vn £r>f S j  fh  kips  Po 5 i +1 v / «  (cols. Col.  kips.  Mcnenfs  © 4-.  T o \ c\ | fh  #  *  Lo<a^iio^  ^  Girder-  6arifilev/er  Por  -  6-15  7.  Cools. 5 + 5 )  4)  Col.  &.  Col.  2>.  Lo<ad  Monoenhs Ph Col.  kips l o .  -  -  S o l o  +  ?6oS  -  2 3 4 5  5 S £ o  o5  +  3 6 1 o  -  -Z&So  6 5 o o  -  Z o S o  +  3 7 o o  -  -Z&05  6 5 6 5  -  6 5 7 5  Z I 3 o  -  3 3 6 o  +- 4 3 3 £  -  +•  4 6 o o  +  (ool.  L've Monoenfs  3 o I o  6.  + .  78 5  -  I7 6 o  +  3£> 1 o  -  11  e>,  +  t>oo  -  1 5 o £  4  3 T o o  -  I  +  1&£>  -  I  o  4-  3 6 1 O  -  / CrOZ  +  43 3 5  -  Z l o i  +  3 6 1 o  -  Z8t)o  6 5 o o  -  4 15  S o  +  £ 6 o £  ~  Z \ oo  +  7 j e>o  -  3360  +  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  Mofe  :  A s  6  -  <a*:le  c o l i / r m o  s h o w n  S .  i n v e r s i o n 7 ,  -  lotf^m^js  nr\c?ir~>eo+s  6 ,  «  ,  • s h o w n  in  C o l u m n Fo C o l u i n r i  ^> ^ e m < s i  COr~><5( i f | o n S .  IO,  J> 2>  36o  3  n o o m e o l ' s  ,  a r e f o r  -  &  figure  i n  c o l u m n  l o  7.0 (o) are r b e  i7ir<z  frcans p o s e d S i m i l a r l y  Ma^imuh-i floe  Z 1  sev'eresf  (•  m o m g n l j o f  lln^se  ,  -  e v e r s i b l e w>Wr>  n s fer<a b l C , s h o w o ^>^le  n n <^ / i n n  i n v e r s i o n ,  r o f o  will-,  io> c o l c " ^ n S !o«<sTm<g  Concealr«*)edi  Fcmel  Loadings,  m  kips  ^—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 24.0' Z4.o' 24-. o' I5.S' 24.0' I5.S" Zfc.o' 3o.o' 24.0' 3o.o' Z4.o' 2*.o' 15.5' 24.o' 24'. 0' Z 6 . 0 ' IS.S'( 3,  Z,  'If  4,  5,  7,  6,  0,  26.  C  Cor\cesiircxted 4.fe6 27.0'  fa\nel lo&dingz  o'  Dead  (,^0  .  lo,  II,  load 132.©'  Loadings,  of  3Z  IS,  14,  loS.o  Z6.0'  Cantilever-  Girder  7i  0'  Design  to*d  of  I.7I7  K  I3t  Dead-loadings'  14,.  I 6  /f|-.  132. o'  I2t>. ©  I25.o'  of Continuous.  Girder  Destan  Sfe.4fc 56.4-" I4I' l4.o 14.0  I  !7<o. o  15-2. o '  S p a n <J)  ip^ci  Li^se  Loadings  I  1I  1  i_  (c)  1  to.4o 3.4o S.30 &.30 0.B3 4.G6 3o.o' 24.o' 24.0' 24.0 11.0 I2z  "Distributed  J  17  m k/p£  •Si  (b)  15,  of 1.72-7 yn.  5.93 6.3(7 8-5o Z>Ao /o.4o 2>.4o 0.3o 6.Z0 3.4-0 3o.o' 24.0' 24.0* J 24.0' 3 0.0 24.o' 24.0 j 3o. 24.o'  .'l  12,  1  "Pis<ribui«a  !o3.o  3,  123.o S p a n (3)  (z)  t o r C>\r<de<<' D e s i ^ r i s  Figure  2o  t  I7-,  •= tooo  3o  fh  ff.kr'ps  42  3.031  A n a l y s i s of C a n t i l e v e r Girder Design  A. E l a s t i c A n a l y s i s Dead weights of the s t e e l  system and concrete deck f o r  c a n t i l e v e r design are shown i n F i g u r e 20(a). and form of l i v e l o a d considered i s  the  The i n t e n s i t i e s  shown i n F i g u r e 2 0 ( c ) ,  this  conforming to the AASHO H20-S16 p a t t e r n of l o a d i n g and i n c l u d i n g the s p e c i f i e d impact allowance. The combinations of these b a s i c l o a d i n g s which were assumed i n the e l a s t i c  design f o r maximum p o s i t i v e and negative  moments over the centre span p o r t i o n of the c a n t i l e v e r s are  as  follows: (1)  For maximum p o s i t i v e moments, the e n t i r e dead l o a d system of the c a n t i l e v e r s and end beams (Figure together with the l i v e l o a d (Figure 20(c)) between the centre span supports  (2)  20(a)),  applied  only.  For maximum negative moments, the e n t i r e dead l o a d system of the c a n t i l e v e r s and end beams (Figure together with the a c t i o n of l i v e loads  20(a)),  (Figure 20(c))  a p p l i e d i n both outer spans simultaneously. Maximum e l a s t i c moments d e r i v e d from these combinations of l o a d are f i g u r e d to be: (1)  Maximum p o s i t i v e moment, at midspan  = +4600 f t  kips,  (2)  Maximum negative moments, at supports  = — 6575 f t  kips.  The extreme f l e x u r a l  s t r e s s permitted by s p e c i f i c a t i o n  for  43  A-7 s t e e l  is  section i s 19.9  20.0 k s i .  f i g u r e d at  Actual s t r e s s r e a l i z e d at the midspan 20.1 k s i ,  and f o r the support s e c t i o n s  at  ksi.  B. L i m i t Design A n a l y s i s by Constant Load Method As the centre span i s  statically  determinate only one  hinge i s r e q u i r e d f o r the development of a f a i l u r e mechanism and l i m i t design a n a l y s i s  f o r constant l o a d i n g simply c o n s i s t s  i n the determination of the c r i t i c a l p l a s t i c moment of  resistance  f o r the span, which w i l l be e i t h e r at the midspan s e c t i o n or the support s e c t i o n s ,  and comparing t h i s with the  at  appropriate  a p p l i e d moment. C o n s i d e r i n g f i r s t l y the maximum p o s i t i v e moment c o n d i t i o n at midspan.  T h i s moment i s 4600 f t .  can be s u b s t i t u t e d  f o r M i n equation  k i p s and thus t h i s (5).  value  The p l a s t i c moment of  r e s i s t a n c e Mp i s a l s o i n v o l v e d i n t h i s equation and i t s may be c a l c u l a t e d as  follows:  P l a s t i c moment of web "  • flanges*  M  thus,  value  = (99.0) .0.625(33.0)._1 . 12 (99.88) 20.0(0.875) 3 3 . O J . 12  t o t a l p l a s t i c moment of s e c t i o n  = 4200 f t . = 4800  kips  "  =9000 f t .  " kips  As only one hinge i s i n v o l v e d the value f o r Q i n both numerator and denoninator of the equation f o r F i s , the same and thus the equation F  =  9000 46*00  becomes: = 1 . 9 6  of  course,  44  S i m i l a r l y f o r the maximum negative moment c o n d i t i o n at the supports.  The value f o r M i n t h i s case i s 6575 f t .  and f o r Mp i s 12560 f t . F  =  kips,  12560 6575  and thus: =  1.91  The value of F at the support s e c t i o n s t h i s i s therefore  kips  i s the smaller and  the l i m i t i n g v a l u e .  I t i s i n t e r e s t i n g to here note that as there i s no essential  d i f f e r e n c e between t h i s l i m i t design a n a l y s i s  g i r d e r and the procedure of e l a s t i c design f o r determinate s t r u c t u r e s  of the  statically  i t i s not s u r p r i s i n g that these values of  l o a d f a c t o r f o r the g i r d e r are not much d i f f e r e n t  from the  l i m i t i n g l o a d f a c t o r requirement of the AISC Code, of  1.88.  C. L i m i t Design A n a l y s i s f o r True V a r i a b l e Loading As p r e v i o u s l y mentioned the f a i l u r e of the g i r d e r under v a r i a b l e l o a d i n g can only be a s s o c i a t e d with the s t a t e of alternating p l a s t i c i t y . i s given by equation  For t h i s s t a t e the f a i l u r e l o a d  factor  (9).  The l a r g e s t range i n l i v e l o a d moment occurs at the midspan s e c t i o n and i s equal to 6565 f t .  kips,  r e f e r Table V l l .  The c o r r e c t value f o r My must be computed on the b a s i s of a l i n e a r s t r e s s d i s t r i b u t i o n across the g i r d e r s e c t i o n v a r y i n g from zero at the n e u t r a l a x i s to the y i e l d s t r e s s value of ksi  at the extreme f i b r e s ,  and f o r the midspan s e c t i o n t h i s  33.0 is  45  equal  to  7530  ft.  P'  The are  kips  In  and  =  girder  reduced  failure.  kips.  the  from  (2)  at  midspan:  7530  =  section  18  ft.  2 0 % | "  to  20"x f",  this  case  yield  Pi  Thus  the  moment  =  (2)  from  range is  midspan, should  in  equal  2.30  moments to  6880  6880  =  where  also i s  be  flange  examined  equal  ft.  areas  kips,  to  6365  for ft.  thus:  2.14  6365 The is  limiting  therefore  alternating reduced  2.14  load with  plasticity,  flange  area  factor failure at  for  occuring,  points  sections  of  failure  18  the  ft.  in  under the  from  girder.  variable state  midspan  loading  of in  the  O M « » n l i  C- i r - d e r -  X)e,ad load  live  Load  M o m e n t s  Pa ne J  a x i m \jm  M No.  fh  kfps  I**  Kan^e  Total  Moime^fs,  ff k i ' p s .  in  Plastic  W e f«rf  Momenls,  K/lovn emt.Sj  M ,  My,  ft. k i p s  fh k i p s  Po&  if i ^ C  Z435  4  4 3 5 o  36IS  +  £ 6 6 c?  35  4  4365  4  Z3o£>  -  615  -  -  6455  1 Z 5 6 o  4  i ie>o  -  157c?  7 6 1o  3 6 6  -7 6 1 o  6l  Moments  P  -  +  +  -ZT&5  4  3o3S  -  5 t o  +  11 o o  4  -Z - 7 8 5  -  11 o  4-  4 4 o  +  1 g>65  -  1o 1 6  2 68c?  342>&  4  4Zc?  -  Zt>6c?  3 3 6c?  33o  4-  1t Z o  -  I Z 4 o  +  87 &  4  Z3S5  -  .1 Z 4 o  3635  4  3Z3o  -  4-  3 85  4  Z4Z5  -  I 2 4-c?  3665  4  5 4 / c?  -  -ZS5  7 6 r o  6 12>o.  4  675  +  Z3S6  -  1 2 4 o  3 6 35  4-  3-Z3o  -  365  -7 6 I o  61  t>o  2 7 6 o  4  | | S o  -  1 S7o  6!  S o  -  -  6 4 55  -  5 75  33o  Z  kips  Girder".  t l 3 o  '  ZZZc  Negofive  ft.  Corifinuous  4  -  !3  Positive  ,  for  -  -ZT6  4  I 6 L D  -  1  Z4 o  65  -  1 1 l o o  356c?  1 11o o  36&  ( M o o  3L&o  7 6 1 o  7 6 1  o  o  6 I 3>o lo32 o  1 5 o o  I o 3 t o  -  3 4 3 5  4-  4 Z o  -  Zt>6o  3 3 6o  4  4 4 o  4-  I 866  -  1 e? 1 £  2 8 8 o  +  23o6  4  2 1 6 o  4- Z -7ft5  -  77o  3656  4  4t>65  -  ! M o o  35&o  4  -Z-753  4  3ot>5>  -  i t o  36  IS  4-  6 8 £ o  -  ( M o o  3  4  "Z 1 3 o  •+.. -z ? Z o  -  Z -7 5  •Z455  4  4 3 & o  -  1 11 o o  3&6o  \ZbQ> o  n  6 I o  61 2>o  5.&C?  C(o.o  1 Span  PIe*s f c  Failure Mode  I (b)  Failure. Mode  i>> g e Ff. i  H  Pf.  II l o o  l7_5C*o  1  o  lt;Co  M , 3  6 7  o  So  Btoo  1  3 3 o o  1 ( » i o  1n  Ph  ^  £  Pf.  Z  o  35 4 o  L i V e Load M o i n e n l s M , ff. k ' p s  Cor^ii^^ous.  for  6-Z.oo  1 6 4 o  25 6  6too  3 o 3 S  Z% o  61oo  27o  «  Ph 3  87  oo  Pf.  Pf.  3  o  £of«fiOin  H ir-igg Pf. 1  Pf.  £  /. o o  o. I  3S  1. o o  o.33S  I. o o  o.&o  Loo  7.  (  Pf.  9  3  ?.36o  4-z. 5  7t)6o  dxr-cier-.  SUM  oo  1. o o  l-l Tn g « Wi n<3<j Pf. 1 Ph Pf. 7  PenovrnrWor  Te^vns.  fcr-tv-v S j  M . <9  M . e p  o  4 o 3 4 o -Suiv*  4T  7 . I3>  66.5 o  7 - 6 G  1o o of  Lo<5^d  f<acf o r 3  o. 3 t > 5  1 o & S o  o. 806  I  I. o o  M . 9  18  u  o  oo  7.oo  f  T S 3 o  So>rn  (Mp-hpMp)e 6 4> 4  oo  rac\c-r  7.  Wumeraf o r D^nommrtfor-  o. I3t>  1.  Load  3 - ? 3 o  1 4  of  ). o o  o o  Suvn o f  of  fJurri&raboi'  11 7 1  D  i  I. 7  d>\ro\er~.  l<o o S o  H m ^ e Rof«fior% ©  1.  o  (D  C o ^ f i n o o u s  u  1  6 3 8 o  3  o  4 1 5  Hm Pf. -z  Hin < ^ Pf. 7 .  Factors  3 5 3 S  341  lt56,o  M o m e n t s (Mp ~ f ff. kips Hm^e  Pf.  3 3 o o  l 7 5 6 o  loc\<d  ff. k i p s H  o  55 5  Z  iV-> c j e P>. "2  oi  fo\al El a s h e M o m e n t s  H  -ro? i ©  Pf. I  1 (c^  Momer-i f s ff. b p s  "7  Z  Derivator*  for  Mo^es  failure  IX  Span  ©  7 o  Z I  v  oo  K 3. ic&  4-z.c*&  t-M  45 S o  7.6I  l o 1 1 o  2 .7 I  46  3.032  A n a l y s i s of Continuous G i r d e r Design  A. E l a s t i c A n a l y s i s Structural details  of the continuous g i r d e r design  shown i n Figures 18 and 19.  Dead weights of the s t e e l  and concrete deck are shown i n F i g u r e 20(b). v e h i c l e l o a d i n g used i n the design i s ,  are  system  The H20-S16  of course,  i d e n t i c a l with  that considered f o r the previous case, r e f e r Figure 2 0 ( c ) . The procedure used i n the e l a s t i c continuous g i r d e r s for  a n a l y s i s of the  i n v o l v e s the p r e p a r a t i o n of i n f l u e n c e l i n e s  c r i t i c a l sections  of the g i r d e r ,  and the c o n s t r u c t i o n of an  envelope of maximum p o s i t i v e and negative moments across each span.  D e t a i l s of the c a l c u l a t i o n s necessary  for this  analysis  need not be given here but the f i n a l moment envelopes, l i v e l o a d i n g and combined l i v e and dead l o a d i n g s , Figure 22.  f o r both  are shown i n  Maximum moments at panel p o i n t s of the g i r d e r s  are  also tabulated i n Table V i l l . Extreme f i b r e s t r e s s e s at c r i t i c a l s e c t i o n s noted here f o r sake of i n t e r e s t ,  of the  girders,  are f i g u r e d to be:  1.  Near m i d - s e c t i o n of outer spans  -  20.1 k s i ,  2.  At m i d - s e c t i o n of centre span  -  18.2 k s i ,  3.  At centre supports  -  19.5 k s i .  B. L i m i t Design A n a l y s i s by Constant Load Method The key p o i n t s of the g i r d e r s are a s s o c i a t e d with  sections  where minimum values of _Mp occur, and a reference to F i g u r e 19 M  47  will  i n d i c a t e that a number of such s e c t i o n s must e x i s t  the g i r d e r s .  The number of hinge p o i n t s r e q u i r e d f o r  across  the  development of a f a i l u r e mechanism on the other hand may be any i n t e g e r between two and f i v e , partial  depending on whether f a i l u r e  is  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  c o l l a p s e of the g i r d e r .  The l i m i t design a n a l y s i s  on f i r s t  s i g h t would thus  to be a formidable task i n v o l v i n g a l l p o s s i b l e key p o i n t s i n e i t h e r p a r t i a l or t o t a l  collapse.  appear  combinations of The problem  however i s c o n s i d e r a b l y s i m p l i f i e d 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 i n v e s t i g a t i n g each elementary mode i n t u r n and superimposing these modes f o r a l l l i k e l y combinations of failure. The elementary f a i l u r e modes, individually, F i g u r e 23.  i n v o l v i n g each span  are r e a d i l y i d e n t i f i e d and these are shown i n  There are s e v e r a l key p o i n t s i n each outer span and  there are thus s e v e r a l these spans.  a l t e r n a t i v e elementary modes f o r each of  Now, i t w i l l be seen from the form of the  ary mechanisms that any p o s s i b l e  elements  complex -mechanism, i n v o l v i n g  some combination of the elementary modes, must e x h i b i t a f a i l u r e l o a d f a c t o r with a value which stands somewhere between the l i m i t i n g load factors which are combined. therefore  appropriate to each of the elementary modes The l o a d f a c t o r of the complex mechanism  cannot p o s s i b l y be l e s s than the smallest  load  factor  48  obtained f o r the separate elementary modes.  I t , i s thus evident  that the c r i t i c a l mechanism f o r the g i r d e r s must be an elementary mode and only these forms need t h e r e f o r e Thus f o r elementary mode l b i n span (3))  be c o n s i d e r e d . (1)(or mode 3b i n span  the p o i n t of maximum t o t a l moment near the midpoint of the  girder,  which i s  a s e c t i o n of minimum M , p  i s l o c a t e d 51 f t .  from  U  the end of the g i r d e r where the moment i s  equal to +5850 f t .  kips.  The corresponding moment at support B, with l i v e l o a d so p l a c e d to give the above maximum p o s i t i v e moment, f i g u r e s kips. as  at -4260  Values of hinge r o t a t i o n at the two points are  ft.  calculated  follows: 5i.o  ^  o  PAN  =  1.0 51.0  0.0196  PBN  =  1.0 78.0  0.0128  Figure 24 0.0196 - 0.0128  =  0.0324  PBN thus r e l a t i v e values of Q  z  factor  and Q  3  are lk.000  F determined by means of equation  (5)  and 0.395,  and l o a d  becomes:  ,  5850(1.000)  =  5850 ,  =  4950 ,  4260(0.395)  =  1680 ,  =  16050 ,  =  7530 ,  11100(1.000)  =  12560(0.395)  Numerator  0.0128  11100  Denominator  49  and thus,  -  F  16Q50  =  2.13  7530  S i m i l a r c a l c u l a t i o n s r e q u i r e d f o r a l l of the other elementary mechanisms are given i n Table I X . f o r the f a i l u r e l o a d f a c t o r  The value of 2 . 1 3 obtained  i n mechanism l b i s l i m i t i n g and t h i s  thus represents  the c r i t i c a l l o a d f a c t o r f o r the g i r d e r s under  the assumptions  of the constant l o a d method.  G. L i m i t Design A n a l y s i s f o r True V a r i a b l e Loading C o n s i d e r i n g f i r s t l y the s t a t e of incremental  collapse.  The envelopes of maximum p o s i t i v e and negative l i v e l o a d moments shown i n Figure 22 represent  the v a r i a t i o n s  negative values of M L across the g i r d e r s .  i n p o s i t i v e and This f i g u r e  shows the v a r i a t i o n i n dead l o a d moment IL . 0  also  Now, i t w i l l  be  evident from i n s p e c t i o n of t h i s f i g u r e that the key p o i n t s of the g i r d e r s 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 , s e c t i o n s Mp - F p . M p ,  of minimum  are at the same l o c a t i o n s as those e a r l i e r  established  Mi_  f o r constant l o a d i n g and t h e r e f o r e the elementary modes of ure f o r the present  form of l o a d i n g w i l l  those considered p r e v i o u s l y .  Also,  be e x a c t l y the same as  the e a r l i e r comments on the  l o a d f a c t o r of combined mechanisms w i l l incremental c o l l a p s e  fail-  and thus a n a l y s i s  also apply e q u a l l y to can again be l i m i t e d to  these same elementary.modes. Thus f o r elementary mechanism l b , e l a s t i c moments at midspan hinge, as obtained from F i g u r e 1 9 , kips, 1 ^  = 3095 f t .  kips,  are M D = 2 7 5 5  the  ft.  and f o r 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 ,  and,  F  u  =  10890 "4 265  4 265 ,  =  2.55  The a l t e r n a t i v e e l e m e n t a r y mechanisms a r e a n a l y s e d i n T a b l e I X and r e f e r e n c e value of F  u  to t h i s table w i l l  for failure  Considering  now  i s that  the state  The l a r g e s t v a l u e f o r M  u  of a l t e r n a t i n g  %l  =  The s m a l l e s t  with the smallest  condition.  respectively,  (2) 6190 3665  load  t h i s i s therefore  loading  plasticity. centre  v a l u e f o r My f o r one f o r a l t e r n a t i n g  The v a l u e s o f t h e s e moments a t t h i s s e c t i o n a r e  3665 and 6190 f t . k i p s ,  and  above f o r mode l b .  g i r d e r t h i s s e c t i o n must be t h e c r i t i c a l  plasticity.  the l i m i t i n g  occurs at the midpoint o f the  span and as t h i s c o i n c i d e s the  obtained  show t h a t  and t h u s :  =  3.38  f a c t o r obtained f o r the g i r d e r s  the true  i s 2.55  l o a d f a c t o r f o r the v a r i a b l e  Table  SKjrs~\m c\r  X  op  ^, e S u 1 f S .  Load factor Values (or fbe following 1f e  1  z  3  Govemno^ failure Lo^d factor' T ' for" structure writer Constant l o a d i n g 11 w i i f ( forfsdfi'o LiVvi  of  i'tiV->g  $fructures:  tn  di  Load facror~ e S i <sj i o ^ value  F  Desi^tn  \o \/<alue  failure- Load V&.c\or F ' f o r s\/uc\<jre ur-ider \Jar~\able Loadi^^  C a n t i lev'er" C o n t i n u o u s S r i d ^ Girder &ndgg 6irder  Simple. S>cvy Cable Bent  Double S a y Gable S e n t .  1.6 1  Z . 0 6  1. 4-1  1• & &  1. 14-  1.1/  !.<?Z  1.13  1. Q>Z  1. So  Z . 14-  Z . 55  1.4-1  1. 4l  1. &&  ! .  LIS  1. 2 &  1-14-  1 . hQ?  1. o |  1. 1 &  1. 1 Z  \ .Zo  1 . t>1  .  1-66  -z. 1 2>  1.  66  1  4  S  u  Livvoiti'-i^ (  ^afio  for  des  of  Li*Y-ufi>--->cJ)  7  Lo#^  f^-e»o<5if u r e  ' f- ' L  Factor"  i<3rV)  v^lue  Design Collapse  ^ (correc-te^ F '  =  fo  Via/oe  ^afi'o  i t e ^ 6> it€v^ .3  8&  51  SECTION 4 INTERPRETATIONS AND FURTHER CONSIDERATIONS 4.01  I n t e r p r e t a t i o n of A n a l y t i c a l ^ R e s u l t s A summary of the r e s u l t s  obtained i n the previous analyses  i s to be found i n Table X, but before proceeding with a d i s c u s s i o n of these r e s u l t s  i t i s desirable  to c l a r i f y s e v e r a l  previous  conceptions. The f i r s t further smallest  p o i n t to be mentioned i s  the n e c e s s i t y  for  e l a b o r a t i o n i n the d e f i n i t i o n of l o a d f a c t o r F .  The  numerical value obtained f o r t h i s l o a d f a c t o r  i s not  n e c e s s a r i l y the governing value f o r f a i l u r e l o a d , i n the case of the double bay bent.  as was found  Combinations of working  loads  c o n t a i n i n g wind f o r c e s have a d i f f e r e n t  (1.41  f o r the AISC Code) from combinations which exclude wind  forces  (1.88),  and f u l l  allowance must be made f o r t h i s i n  e s t a b l i s h i n g the governing value of F . bent example the smallest forces  2.08.  Thus f o r the double bay  value of F i s 1.86  a c t i n g , but the governing value i s  applies  safety requirement  with wind and other  a c t u a l l y that which  to the c o n d i t i o n of g r a v i t y loads a c t i n g alone,  that  is,  The governing value of F must t h e r e f o r e be thought of i n  terms of the margin of safety over l i m i t i n g code requirements r a t h e r than as the n u m e r i c a l l y smallest  value of l o a d  factor  obtainable. Then sagaih,  the same c o n s i d e r a t i o n s w i l l  apply i n deter-  52  mining the premature c o l l a p s e r a t i o J \ , , and the n e c e s s i t y f o r F s u b s t i t u t i n g a p p r o p r i a t e margins of s a f e t y f o r the F u and F values,  when d i f f e r e n t  safety requirements are i n v o l v e d ,  will  be  apparent. The importance of r e c o g n i z i n g these f a c t s i s w e l l i n the r e s u l t s  obtained f o r the double bay bent.  numerical values of l o a d f a c t o r and 1.86,  respectively,  collapse  i s 0.97.  a c t u a l l y 1.15  condition i s i n fact  I f the  smallest  are considered F u and F are  and t h e . r a t i o  r a t i o on the other hand i s  illistrated  1.80  The c o r r e c t e d  and an apparent  premature  the reverse when proper account  is  made of the s a f e t y requirement a s p e c t . Returning now to Table X. ratios  are shown i n item 7 of the t a b u l a t i o n .  evident that these values premature c o l l a p s e examples for  The c o r r e c t premature  are a l l greater  i s not a f a c t o r  considered.  The smallest  It w i l l  than 1.0  i n any of the  structural  value obtained i s  that of for  1.01  the  girders.  The l i g h t w e i g h t s t r u c t u r e s which, on the--average, weight s t r u c t u r e s , (10),  be at once  and thus  the s i n g l e bay bent and the l a r g e s t value i s 1.20  continuous  collapse  may be s a i d to e x h i b i t  ratios  are lower than those, f o r the h i g h dead-  as i s of course to be expected from equation  and i t would seem reasonable  to conclude that where  structures  are subjected to h i g h l y v a r i a b l e l o a d i n g s the a c t u a l c o l l a p s e w i l l be somewhere between zero to twenty percent greater  load  than the  f a i l u r e l o a d p r e d i c t e d by the constant l o a d method, depending on  53  the p r o p o r t i o n of l i v e to dead loads present with the smaller dead loads t e n d i n g to produce the smaller d i f f e r e n c e s  in failure  load values. I t w i l l be noted that the more h i g h l y redundant  structures  of both types p o r t r a y r a t i o s which are d e f i n i t e l y l a r g e r than those f o r the corresponding simpler forms,  although t h i s  could  be c o i n c i d e n t a l as there appears to be no t h e o r e t i c a l b a s i s f o r a tendency one way or the o t h e r . Finally,  i t should be recognized that the values obtained  f o r F L , and thus also f o r the premature c o l l a p s e r a t i o ,  are,  in  three out of the four cases c o n s i d e r e d , dependent on the value chosen f o r the dead l o a d f a c t o r F  D  .  Dead l o a d i n g i s not i n v o l v e d  i n the f a i l u r e of the c a n t i l e v e r g i r d e r s and as f a r  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 e f f e c t s ;  however t h i s  not so i n the case of the continuous g i r d e r s . f o r dead l o a d s ,  A smaller  over those used f o r l i v e l o a d s ,  acknowledged as j u s t i f i a b l e  i n view of the f a c t  that  and t h i s value i s  the  certainty  and also doubtful behaviour, of l i v e  The value used i n t h i s study f o r F p , of 1.25, fact  factor  must be  magnitude of these loads i s known with f a r greater than i s the magnitude,  is  recognizes  loads.  this  e n t i r e l y c o n s i s t e n t with the somewhat  higher values recommended by code f o r l i v e l o a d i n g s .  However,  apart from t h i s l o g i c a l b a s i s f o r the use of a f a c t o r of 1.25 i s i n t e r e s t i n g to reason the e f f e c t  it  of the choice of a h i g h e r F ^  54  value.  I f the f i g u r e o f 1.41  is  selected  a l i v e l o a d f a c t o r and i t i s d i f f i c u l t reasonably be g r e a t e r  than t h i s )  ( t h i s value i s used as  to see how F p could  then the l i v e l o a d f a i l u r e  i 1 ^ f o r the continuous g i r d e r s works out to be 2.39, 2.55.  factor  i n s t e a d of  Now, the constant l o a d f a i l u r e f a c t o r i s 2.13 and thus the  premature c o l l a p s e r a t i o i s reduced from 1.20 increase i n F ; D  to 1.12 with the  however t h i s reduced value has not a l t e r e d the  sense of the r e s u l t f o r the continuous g i r d e r s as f a r as the present study i s concerned f o r the r a t i o i s 1.0,  still  g r e a t e r than  i n d i c a t i n g that premature c o l l a p s e does not take p l a c e . Arguments r e g a r d i n g the value to be assigned to F & are  thus l a r g e l y academic as f a r as the f i n a l is  purpose of t h i s paper  concerned f o r no s i g n i f i c a n t change can be grought about i n  the premature c o l l a p s e r a t i o s f o r any of the four s t r u c t u r a l examples c o n s i d e r e d .  55  4.02  F u r t h e r C o n s i d e r a t i o n s Regarding Premature C o l l a p s e Premature c o l l a p s e i s t h e r e f o r e not a f a c t o r i n the  s t r u c t u r a l examples considered, and as these s t r u c t u r e s s p e c i f i c a l l y chosen to represent would perhaps seem reasonable  were  severe cases of l o a d i n g ,  to e x t r a p o l a t e  it  t h i s f i n d i n g to the  extent o f c o v e r i n g a l l problems of s t r u c t u r a l design l i k e l y be met i n p r a c t i c e .  I t must however be agreed that even severer,  and not at a l l e x c e p t i o n a l , cases are very l i k e l y to f a l l the scope of d e s i g n .  The smallest  premature c o l l a p s e  within  ratio  recorded f o r the four cases considered i s that of 1.01, s i n g l e bay bent,  to  f o r the  and i t would not t h e r e f o r e take very much to  create the severer c o n d i t i o n s l e a d i n g to a r a t i o of l e s s than 1.0.  Thus a small degree of premature c o l l a p s e ,  t h e o r e t i c a l e v a l u a t i o n of l o a d f a c t o r ,  based on the  must be accepted as  highly  probable w i t h i n the range of p r a c t i c a l design problems. Insofar  as these t h e o r e t i c a l e v a l u a t i o n s are concerned i t  must be remembered that c e r t a i n b a s i c assumptions i n the d e r i v a t i o n of l o a d f a c t o r  equations.  are  involved  One such assumption  i s that r e g a r d i n g the magnitude of the r e s i s t i n g moment at hinge p o i n t s which was e a r l i e r r e f e r r e d to as the p l a s t i c moment and f o r which a constant value was assumed throughout the e n t i r e range of angle change at the h i n g e . moment i s  i d e a l i z e d , as w i l l  i s made to F i g u r e 2, which r e s u l t s  T h i s constant value f o r the  be r e a d i l y appreciated i f  and e n t i r e l y n e g l e c t s  i f strains  reference  the increase i n moment  at the hinge s e c t i o n extend i n t o the  56  strain-hardening region.  How, i t has been t h e o r e t i c a l l y demon-  s t r a t e d by exact methods of a n a l y s i s p o i n t s formed i n a s t r u c t u r e w i l l  that the e a r l i e s t  hinge  almost i n v a r i a b l y extend i n t o  t h i s s t r a i n - h a r d e n i n g r e g i o n and these hinges t h e r e f o r e must o f f e r moments of r e s i s t a n c e value Mp.  which are greater  than the  'plastic1  I t f o l l o w s that the true l o a d c a p a c i t y of a s t r u c t u r e  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 m o b i l i z e d  and the mechanism s t a t e m a t e r i a l i z e s , than the computed l i m i t design v a l u e .  must be somewhat  greater  This f a c t has been demon-  s t r a t e d i n t e s t cases i n v o l v i n g incremental c o l l a p s e i n which experimental continuous beams were subjected to v a r i a b l e concentrated l o a d s ,  and the r e p o r t e d r e s u l t s  of up to nineteen percent .  It  show l o a d  increases  i s doubtful that i n c r e a s e s  of  t h i s order can be expected i n a l l instances of incremental c o l l a p s e and no i n c r e a s e i s  allowable under c o n d i t i o n s i n v o l v i n g a l t e r n a t i n  p l a s t i c i t y where f a i l u r e i s a s s o c i a t e d  with the e l a s t i c  s t a t e at  only one s e c t i o n of the s t r u c t u r e ,  nevertheless,  i n general,  some  increase i n a c t u a l f a i l u r e l o a d i s  i n d i c a t e d over and above the  computed value f o r i\. A second assumption embodied i n the t h e o r e t i c a l i s that the working loads constant  a p p l i e d to the s t r u c t u r e f o r both  and v a r i a b l e l o a d i n g c o n d i t i o n s must be of i d e n t i c a l  magnitude i f e x a c t l y comparable f a i l u r e l o a d f a c t o r s obtained.  evaluations  Now, t h i s assumption i s a l s o ,  are to be  s t r i c t l y , not q u i t e  valid  as with the constant l o a d i n g c o n d i t i o n only one a p p l i c a t i o n of the  57  failure  load is  necessary  i n o r d e r to  state r e s u l t i n g i n f a i l u r e variable  loading failure  repeated  cycles  whether f a i l u r e  of  is,  then,  the  occurs  structure,  takes place  the f a i l u r e  nating plasticity. there  of  p r o d u c e t h e mechanism  loads  only are  justification  of  for  somewhat l e s s  constant  loading,  it  for  using a safety  the former  a number o f  applied,  does n o t m a t t e r  the l a t t e r .  the e f f e c t  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  the  course,  ratio  of  a structure  theoretical of  occurence  requirement)  for  for  slightly  less  smaller  would a g a i n of  is  the  have  structure  computed v a l u e .  As a r e s u l t load of  so  alter-  used  Such  (or  above  safety  is  which approach  requirement which i s  c o n d i t i o n than f o r smaller  or  than those  working loads of  and t h i s  e i t h e r u s i n g working loads  l o a d i n g which are (for  under  the p r o b a b i l i t y o f l o a d  variable  considered)  after  through incremental c o l l a p s e  I n terms  or,  whereas  t h e s e two i n f l u e n c e s will  value of F L ,  be g r e a t e r and t h i s  the  than that  actual  collapse  i n d i c a t e d by  increased load capacity  also  proportionately increase  this  r e s u l t i n g increase  the premature  the must,  collapse  Fu.  F  Now,  tend to o f f s e t  the e f f e c t  of  a severer  r e d u c i n g the r a t i o )  and i t  that  i n problems o f  in actual  unlikely  that  fact true  ratios  i n the l o a d f a c t o r  will  loading condition (in  would seem n o t u n r e a s o n a b l e  of l e s s  ratio  practical t h a n 1.0  design  will  it  occur.  to  conclude  is  highly  58  SECTION 5 CONCLUSIONS  The f o l l o w i n g g e n e r a l  conclusions  a r e i n d i c a t e d by t h e  p r e s e n t study: 1.  The t h e o r e t i c a l f i n d i n g s suggest t h a t 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 t o o c c u r w i t h i n acceptable  l i m i t s o f the V a l u e f o r u l t i m a t e l o a d computed  by means o f t h e s i m p l i f i e d l i m i t design method, and t h e method would t h e r e f o r e appear t o be e n t i r e l y v a l i d f o r d e s i g n purposes. 2.  For p r a c t i c a l structures the actual ultimate load  will  g e n e r a l l y range between a v a l u e equal t o , t o perhaps more than twenty p e r c e n t g r e a t e r than, t h e v a l u e p r e d i c t e d by t h e s i m p l i f i e d method;  t h e 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 p r o p o r t i o n o f l i v e t o dead l o a d p r e s e n t and t h e v a r i a b l e q u a l i t y o f t h e l i v e 3.  loads.  In computing t h e t h e o r e t i c a l c o l l a p s e l o a d s o f a s t r u c t u r e subjected t o the constant  and v a r i a b l e forms o f  loading 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  s a f e t y r e q u i r e m e n t s e x i s t f o r d i f f e r e n t combinations o f l i v e loading.  As a r e s u l t o f these d i f f e r i n g s a f e t y r e q u i r e m e n t s  i t appears t h a t t h e s m a l l e s t n u m e r i c a l frequently w i l l not represent  values f o r load f a c t o r  the governing conditions f o r  59 design. A smaller safety requirement f o r dead, l o a d s , against l i v e l o a d s ,  as  would a l s o appear to be j u s t i f i a b l e .  60  SECTION 6 BIBLIOGRAPHY  The s i n g l e bay and double bay bents are of B u t l e r B u i l d i n g s Inc.  standard designs  Lecture notes i n course on " L i m i t Design and I n e l a s t i c Bending", 1957, by D r . A . P . H r e n n i k o f f , U n i v e r s i t y of British-Columbia. Proceedings, AISC N a t i o n a l Conferences, 1955 and 1956. Containing r e p o r t s on i n v e s t i g a t i o n s at Lehigh U n i v e r s i t y .  61  APPENDIX 1  In for  terms o f mechanism f a i l u r e  constant  equations  the l o a d f a c t o r s  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 ,  (4)  and ( 7 ) ,  as  are  F and F u , g i v e n by  follows:  F  (4)  F,  and  thus r a t i o  £(Mp -  FP.Mo).©  IV. becomes: F  ^M.© 2ML.0  F  .  S(Mp-  E x p a n d i n g t h e r i g h t hand s i d e replacing £ M p . ©  of  F^.Mc) . ©  this  expression  and  by F.2M.0 :  SM.© ..0  F_u  F  now,  (7)  £M.©  ^M  F  L c  . G - f Sffip.G  .6  .  F 2 . M . © - F o 2Mo.O F S.M.©  ,  and t h u s  :  £M,_C.© + £ M D . ©  - _Fo2y .© F  £ M L C .0  1 +  F - F p a IP . © F *2M  .0 «£M, . ©  1 +  F-F  .©  F  P  £WD.S  £W .S, L  b  1  

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