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Application of limit design to high-strength aluminum alloy beams Katramadakis, Tony 1962

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APPLICATION OF LIMIT DESIGN TO HIGH - STRENGTH ALUMINUM ALLOY BEAMS  by TONY KATRAMADAKIS B.Sc. Robert's College I s t a n b u l , 1957  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR 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 conforming t o the required standard  THE UNIVERSITY OF BRITISH COLUMBIA May, 1962 ^  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study.  I further agree that permission  for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his  representatives.  It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  Department of  C7'"77~na  /y7&y7 ff &  The University of British Columbia, Vancouver 3, Canada. Date  /&/  / 9 61-  ii  ABSTRACT  The t h e o r y o f l i m i t  d e s i g n o r i g i n a l l y was  for structural steel construction.  Tests carried  developed  o u t on m i l d  s t e e l beams and f r a m e s a r e i n agreement w i t h t h e t h e o r y .  Un-  f o r t u n a t e l y a l i m i t e d number o f t e s t s have been c a r r i e d out on  other d u c t i l e m a t e r i a l s such as l i g h t  alloys.  Therefore  more t e s t s a r e r e q u i r e d i n o r d e r t o i n v e s t i g a t e w h e t h e r t h e theory  of l i m i t  design i s also applicable, with  m o d i f i c a t i o n t o aluminum a l l o y s . predicted  i n limit  The f a i l u r e  or without  mechanism  d e s i g n m a t e r i a l i z e s i n s t e e l frames n o t  o n l y because s t e e l  i s very ductile  b u t a l s o because s t e e l has  s t r a i n hardening.  Aluminum a l l o y s  exhibit very l i t t l e  hardening. objects. of l i m i t  I n the r e s e a r c h d e s c r i b e d here The f i r s t  t h e r e were two  o b j e c t was t o i n v e s t i g a t e t h e a p p l i c a b i l i t y  d e s i g n t o aluminum a l l o y s .  The s e c o n d o b j e c t was t o  check e x p e r i m e n t a l l y t h e t h e o r y o f i n e l a s t i c bending. load  strain  t e s t s were c a r r i e d  on c o n t i n u o u s  Three  beams made o f aluminum  a l l o y t o s e e i f t h e mechanism c o n d i t i o n was a t t a i n e d b e f o r e failure  o f t h e beam.  Moments and d e f l e c t i o n s p r e d i c t e d b y  the theory o f i n e l a s t i c  bending  were compared a g a i n s t measure-  ment o f beam moments and d e f l e c t i o n s . bending  considers the effect  Tables  of unit  of s t r a i n  The t h e o r y  of inelastic  hardening.  f u n c t i o n d e r i v e d from t h e s t r e s s -  s t r a i n d i a g r a m o f aluminum a l l o y  (65S-T6) a r e p r e s e n t e d  so  iii that they may he used when the theory is  of i n e l a s t i c bending  applied. The  f i r s t t e s t f a i l e d prematurely due t o c r i p p l i n g  of the compression f l a n g e s .  In the second and the t h i r d t e s t  the mechanism c o n d i t i o n o f l i m i t design was reached s h o r t l y before  f a i l u r e o f the t e n s i o n s i d e o f the beam under the load  p o i n t by f r a c t u r e .  Thus the type o f f a i l u r e i n d i c a t e s that  not a l l s t r u c t u r e s w i l l achieve the mechanism c o n d i t i o n . The  f a i l u r e l o a d and the r a t i o ofmomentsat f a i l u r e , as  p r e d i c t e d by the theory 15.53  o f i n e l a s t i c bending was equal t o  Kips and 1.13 r e s p e c t i v e l y .  Test r e s u l t s i n d i c a t e d a  f a i l u r e l o a d o f 16 Kips and a r a t i o ofmomeriUat f a i l u r e equal t o 1.1.  The l o a d - d e f l e c t i o n curves were the same as the curves  from the theory. was 5.57 inches of 5.46 i n c h e s .  At f a i l u r e the d e f l e c t i o n under the load compared t o computed t h e o r e t i c a l d e f l e c t i o n  iv  ACKNOWLEDGEMENT  The w r i t e r t a k e s his is  indebtedness dedicated.,  criticism. British  this  opportunity to  to Dr. A . Hrennikoff to  for his untiring interest,  He a l s o  Columbia f o r  made h i s  studies  Research  Counsil for  wishes the  to  whom t h i s guidance  thank the U n i v e r s i t y  original assistantship  i n Canada p o s s i b l e , the  grant  express  of  and the  a special  work and of  which  National scholarship.  TABLE OP CONTENTS  Page 1  INTRODUCTION PART I  General T h e o r e t i c a l Considerations . . . .  101  An i n t r o d u c t i o n to l i m i t design  102  An i n t r o d u c t i o n to the theory of i n e l a s t i c tending  PART I I  Unit Functions and Test Arrangements . . .  10  23  201  Tension and compression t e s t s to determine the s t r e s s - s t r a i n curve of high strength aluminum a l l o y  202  Evaluation of i n e l a s t i c theory u n i t f u n c t i o n f o r I beams made of aluminum a l l o y  203  Test arrangement  PART I I I Beam Test Results 301  Results of beam t e s t No.l  302  Results of beam t e s t No.2  303  Results of beam t e s t No.3  PART IV 401  P r e d i c t i o n of the Theory of I n e l a s t i c Bending 117 T h e o r e t i c a l p r e d i c t i o n s on the behavior of the t e s t beams  CONCLUSIONS AND RECOMMENDATIONS References 43 Figures 14 Tables  58  142  INTRODUCTION  F o r many y e a r s and d e s i g n  of  structures  on u n i t  stress.  develop  his  load  limit  of  are  "Limit yield  the  is  claimed  analysis  specifically  T . A . Van den Broek  Design",  stress  their  l a w o r more  remained f o r  i n which the regarded  i n favour  as of  to  ultimate  the the  design theory  following:  Rationality. -  1.  have based  on Hooke's  The a d v a n t a g e s  design  structure  has  than the  criterion. of  It  theory  rather  engineers  can be d e t e r m i n e d  T h e maximum s t r e n g t h f a r more a c c u r a t e l y  of  by  the  limit  design.  2. beyond because  the  elastic  its  necessary  Economy.  of  The r e s e r v e  limit  3.  Simplicity. -  for  the  (b)  supports,  disregarded.  elastic Also,  points and  (d)  of  strength  of  a  structure  c a n be u t i l i z e d w i t h a s s u r e d  maximum s t r e n g t h  beams and f r a m e s . stresses,  -  c a n be a c c u r a t e l y Most of  the  solution is  stress  settlement  analysis  eliminated for  concentration of  determined.  complicated  such imperfections  supports,  as  safety  (a)  (c)  redundant  residual spreading  usually  can  be  2 T e s t s c a r r i e d out by J.F. Baker on m i l d s t e e l beams and f u l l - s c a l e p o r t a l frames support the c l a i m f o r l i m i t design.  U n f o r t u n a t e l y v e r y few t e s t s have been c a r r i e d  out on other d u c t i l e m a t e r i a l s such as l i g h t a l l o y s .  There-  f o r e , i t i s the purpose of t h i s t h e s i s t o i n v e s t i g a t e whether the theory o f l i m i t design, o r i g i n a l l y developed  for structural  s t e e l , i s a l s o a p p l i c a b l e , with or without m o d i f i c a t i o n , t o h i g h s t r e n g t h aluminum a l l o y s . According to e l a s t i c methods, a s t r u c t u r e i s designed so that the s t r e s s e s due t o the working loads w i l l never exceed a c e r t a i n p e r m i s s i b l e working s t r e s s . represent the maximum loads expected of the s t r u c t u r e .  The working loads d u r i n g the l i f e  time  The working s t r e s s , which i s d i r e c t l y  r e l a t e d t o the y i e l d s t r e s s , ensures a margin of s a f e t y to account  f o r u n p r e d i c t a b l e overloads, d e f e c t i v e workman-  s h i p , d e f e c t i v e m a t e r i a l and so on.  Since e l a s t i c  design  f a i l s t o a l l o w f o r the great reserve o f s t r e n g t h and d u c t i l i t y of s t e e l beyond the e l a s t i c l i m i t , the l o a d and deformations of the s t r u c t u r e a s s o c i a t e d with f a i l u r e cannot be To take i n t o account  determined.  as f u l l y as p o s s i b l e the  d u c t i l i t y o f m a t e r i a l s such as s t e e l , the theory of l i m i t d e s i g n assumes that beyond the e l a s t i c range, the m a t e r i a l can undergo an i n f i n i t e s t r a i n under constant s t r e s s without f a i l u r e .  F o r example i n a beam or r i g i d frame loaded  3  beyond i t s elastic capacity the section of maximum bending moment w i l l induce resistance from more and more inner fibers, each in turn-reaching the yield stress, until yield stress ultimately spreads to the neutral axis.  Then the beam sec-  tion attains a condition of deformability without limit. According to the theory of limit design, such a section of a redundant beam w i l l act as a hinge except instead of transmitting a zero bending moment, i t w i l l transmit a constant bending moment.  The hinge is termed a plastic hinge and the  bending moment (developed at the plastic hinge) is called plastic moment.  The formation of a plastic hinge reduces  the redundancy of the structure by one degree.  If the  structure is determinate,it w i l l deform without limit.  Such  a condition i s palled a mechanism. If the structure is indeterminate, the load w i l l increase u n t i l a sufficient number of plastic hinges are formed to transform the structure (or part of i t ) into a mechanism. This limiting condition for the entire structure (assuming uniform cross sections) is reached when as many sections as can "equalize" their bending moments are developing simultaneously  the same resisting moment.  Thus before the formation of a mechanism, redistribution or equalisation of moments must take place. This usually necessitates large angle change which in turn requires large curvature and therefore large strains in the  4 cross section. I t i s t a c i t l y ; assumed i n l i m i t design t h a t the s t r u c t u r e cannot f a i l by a c t u a l p h y s i c a l breaking a t some o f the e a r l i e r formed hinges before i t begins t o a c t as a mechanism. Apparently the b a s i s o f t h i s assumtion shape o f the standard  l i e s i n the  (^-$) curve f o r s t r u c t u r a l s t e e l  assumed i n l i m i t design, f i g u r e 1, where the angle-change i n c r e a s e s i n d e f i n i t e l y under the constant p l a s t i c moment. However, the a b s o l u t e v a l u e o f angle-change t h a t a beam can maintain without breaking depends on the d i s t a n c e over which the p l a s t i c moment i s a c t i n g . F o r example a 12" I beam can e a s i l y s u s t a i n an angle-change o f 5° over 12" o f l e n g t h but i t i s very questionable i f the same angle-change can m a t e r i a l i z e over 1 of l o n g i t u d i n a l l e n g t h o f the same beam without breaking i t . Figure 1  To i n v e s t i g a t e the' x- p o s s i b i l i t y ) of such a f a i l u r e at the p l a s t i c hinge and the c o n d i t i o n s t h a t would prevent i t , a r e f e r e n c e to the (M-<J>) curve beyond the e l a s t i c  range  i s necessary.  l/> /}_  H  Ptastic  l/i  p  l/>  p  3  1< . 1  / /x  R  E>//  o  / / D  B  xc  F  (Zcc/p  Figure 2  <z.nc/.  ..Figure 3  F i r s t assume an i d e a l i z e d case o f  (M-4?) diagram ,  f i g u r e 2 , c o n s i s t i n g o f two s t r a i g h t l i n e s , the e l a s t i c part OA and the p l a s t i c p a r t AB, and c o n s i d e r an example of bending of a f i x e d ended beam loaded with two g r a d u a l l y i n c r e a s i n g 1  7  concentrated loads each o f magnitude P a t a d i s t a n c e of ^ 6 from each end.  See f i g u r e 3 .  During the e l a s t i c deformation the r a t i o of maximum negative to maximum p o s i t i v e moment i s 2.  As the i n t e n s i t y  of the loads i n c r e a s e s t o some s p e c i f i c value of P (say^? )  6  such as t o cause.- p l a s t i c moments a t the f i x e d ends, the moment diagram takes the shape CDKLEF and the f i x e d end 2  p l a s t i c moments become equal t o •g'?/ . Up t o t h i s p o i n t a l l the angle  changes a r e e l a s t i c .  I f P exceeds P« then p l a s t i c hinges form a t the ends w i t h t h e value o f the end moment unchanged, whereas the p o s i t i v e moment a t K and L s t a r t s i n c r e a s i n g beyond and  up t o  the new moment diagram approaches the l i m i t moment  diagram CD'K'L'E'F. . Having i n mind t h a t the angle and  change between A  0 i s zero a l l the time, we observe that when P approaches  the l i m i t l o a d  the moment between the loads  increases  twice i n magnitude which o b v i o u s l y y i e l d s p o s i t i v e angle change twice as great as the one c r e a t e d by/? , and the l e n g t h o f p o s i t i v e moment r e g i o n DO i n c r e a s e s t o D'O. Yet the r e g i o n of negative moment s h r i n k s t o the l e n g t h D'A and the negative bending moment gets s m a l l e r everywhere except at one p o i n t A where the moment remains the same.  This  means t h a t a l l the i n c r e a s e i n the p o s i t i v e angle  change  p l u s the l o s s o f some n e g a t i v e angle  change must be balanced  i n one s e c t i o n A over an i n f i n i t e s i m a l l e n g t h which w i l l c r e a t e i n f i n i t e s t r a i n s and break the beam a t the ends bef o r e the l i m i t c o n d i t i o n v i s u a l i z e d i n l i m i t d e s i g n i s materialized.  Now assume a more g e n e r a l case o f  diagram  f i g u r e 4 and l e t <fr represent an angle change o c c u r i n g on a u n i t l e n g t h o f the beam corresponding t o the bending moment M. I f dtf r e p r e s e n t s an increment o f moment o c c u r i n g on the l e n g t h of the beam determined  di  by the shear-  i n g f o r c e , V , then from equations of e q u i l i b r i u m we get:  dev -  ^  (i)  and the angle change on l e n g t h d£ i s equal t o dM  d4<f>  Figure 4  V  cj>  (2)  G r a p h i c a l l y , the expression <f>d'M represents the shaded area o f the h o r i z o n t a l s t r i p ABDC f i g u r e 4, and the t o t a l a v a i l a b l e angle change between any two p o i n t s on the curve such as K and N (assuming a constant s h e a r i n g f o r c e ) i s equal t o the summation o f a l l these s t r i p s between the two mentioned p o i n t s . N  P r e s e n t i n g i t i n mathematical  form, we have that the  t o t a l angle change i s equal to  =  Ud£  Consequently  -  v  v  the t o t a l angle change that a  c e r t a i n beam can undergo depends on the magnitude o f the area bound between the {M-<j> ) curve and the  (3)  8 M axis.  Figure 4. This area -is limited^ i f the material lacks strain  hardening even though the curve may extend horizontally without limit, whereas with strain hardening as indicated by the dotted line i n figure 4, the area increases appreciably thus providing the necessary angle change required at the hinge section. So without strain hardening failure loads predicted by limit design w i l l not materialize.  Most ductile materials  such as mild steel have the necessary characteristics but some light alloys have very l i t t l e strain hardening and are probably unfit for limit design. To date, l i t t l e c r i t i c a l attention has been given to these questions. One of the purposes of this present investigation i s to attempt to obtain an indication of the likelihood of a premature failure of a structure, made of light alloys with l i t t l e strain hardening, with the test of several statically indetermined beams. Also the extend of equalization of moments predicted by limit design before the mechanism condition i s reached, w i l l be investigated. For the accurate interpretation of the test results i t was necessary to introduce the theory of inelastic bending presented by Dr. A. Hrennikoff in 1948.  The inelastic bend-  ing theory has certain simplifying assumtions.  Therefore,  9  •the second purpose of the research i s to test the inelastic bending theory experimentally, by comparing actual loads, moments and deformations with the ones predicted by the theory.  10  PART I GENERAL THEORETICAL CONSIDERATIONS 101  An introduction to limit design.  Definition of Factor of Safety and Explanation of the Term "Failure"* For many decades engineers have based their analysis and design of structures on Hooke's Law or more specifically on unit stress. Thus the stress in a structure under the most severe combination of loads should not exceed the working stress at the most highly stressed point. During the last 40 years and especially the past 15 years, proposals have been made to switch from the unit stress criterion to failure condition as the basis of design, in order to provide an adequate and uniform factor of safety on the basis of load rather than unit stress. This factor of safety may be defined as the ratio of failure load to working load i.e. Wf By working load is meant the heaviest or'the most severe load to be supported by the structure, while the failure load is of the same kind as the working load but of greater intensity so that i t can produce failure.  11 The  term " f a i l u r e " i n g e n e r a l i s v e r y wide, and  i s used t o s i g n i f y d i f f e r e n t c o n d i t i o n s such as rupture due to t e n s i o n o r reverse s t r e s s e s , c o l l a p s e due t o l o c a l or g e n e r a l i n s t a b i l i t y or l a r g e deformation.  In l i m i t design f a i l u r e i s  i n v a r i a b l y used i n the l a s t sense.  T h i s type o f f a i l u r e i s  always a s s o c i a t e d with y i e l d s t r e s s . deformations,  In view of extensive  l i m i t d e s i g n doesm'ot r e q u i r e any s e t o f l i m i t s  f o r deformations  o r s t r e s s e s , c l a i m i n g that once y i e l d i n g  has occurred a t s p e c i a l s e c t i o n s , the deformation w i l l be so great that the s t r u c t u r e may be considered as f a i l e d . S t r e s s D i s t r i b u t i o n Under F l e x u r e The a b i l i t y o f s t e e l t o deform p l a s t i c a l l y i s i l l u s t r a t e d g r a p h i c a l l y i n f i g u r e 5a which i n l i m i t i s approximated as shown i n f i g u r e 5b,  design  i . e . by i g n o r i n g the  peak corresponding t o the upper y i e l d p o i n t .  When the beam  i s s t r e s s e d w i t h i n the e l a s t i c l i m i t , s t r a i n s and s t r e s s e s are l i n e a r ( f i g u r e 5d and 5 e ) .  E l a s t i c behavior of the beam  i s p o s s i b l e only w i t h i n the e l a s t i c range, the upper l i m i t of which i s d e f i n e d by the coordinates o f p o i n t A.  I f the  curvature o f the beam i n c r e a s e s f u r t h e r so that the s t r a i n i n . the extreme f i b e r becomes three times as great as the y i e l d s t r a i n , then assuming l i n e a r s t r a i n d i s t r i b u t i o n the corresponding s t r e s s d i s t r i b u t i o n over.the s e c t i o n w i l l be as i s shown i n f i g u r e 5 g , where there i s a constant i n the outer  stress  o f the depth o f the beam and a l i n e a r l y v a r y i n g  12 s t r e s s d i s t r i b u t i o n i n the middle, curvature w i l l that  extend t h e y i e l d s t r e s s  figure  5k.  distribution will  Beyond t h a t ,  the actual  of  o v e r t h e s e c t i o n so  when t h e o u t s i d e f i b e r s a r e s t r e s s e d  5b t h e s t r e s s  and  Jj*urther i n c r e a s e  to point  1  figure  t a k e t h e shape shown i n  s t r a i n hardening stresses  will  appear  s t r e s s d i s t r i b u t i o n w i l l be a s i n f i g u r e  5j,  which i s s i m p l i f i e d i n l i m i t  d e s i g n by assuming c o n s t a n t y i e l d  stress  figure  a l l over the section,  The  r e s i s t i n g p l a s t i c moment M p c o r r e s p o n d i n g t o s t r e s s  distribution in limit  5^.  of figure  5 t i s t h e one a s s o c i a t e d  with f a i l u r e  design,  where  ^ r  ana  =A  y  () 6  where y i s t h e d i s t a n c e the  o f t h e center, o f g r a v i t y  area from t h e c e n t r o i d a l Moment C u r v a t u r e  The  axis.  Relation.  r e l a t i o n between c u r v a t u r e and s t r e s s  o u t e r f i b e r i s g i v e n by t h e simple  -i-  IZ  -  i n the  expression  —  (7)  h  derived  from simple geometrical, .considerations.  elastic  r a n g e t h e above e x p r e s s i o n s  so  o f one h a l f  In the  c a n he r e l a t e d  t o moment  that  / Plotting  M equation  (7) beyond t h e e l a s t i c l i m i t ,  a s s u m i n g no s t r a i n h a r d e n i n g and a p p l y i n g  e a c h t i m e moments  13  of opposite sense, we get the graph shown i n figure 6 , consisting of a straight line GA and a sharplybent part AB, which approaches ordinate M p asymptotically. For standard Vf jr£ Z  being small, Mp-My  is also small. A7omenT  v. s.  Figure 6  beams,  This  proposition justifies the replacement of GAB  by OFD so that the moment at a section i s either elastic or plastic. The very fact that Mp is associated with indefinite increase i n curvature explains the consideration of Mp as criterion for failure i n the sense of large deflections. Application of Limit Design to Statically Indeterminate Frames Under Simple Proportional Loading. During the elastic loading condition where a l l stresses developed i n the frame are elastic, there are special locations where the bending moments are maximum. As the loads are increased in magnitude, with a constant ratio remaining between their valued the greatest elastic moment becomes plastic and the beam section forms a plastic hinge, transmitting constant moment of resistance for farther deformation.  14 At  a still  higher load, y i e l d  o t h e r key p o i n t  and a s e c o n d h i n g e w i l l d e v e l o p .  further increase one g r e a t e r rigid  s t r e s s w i l l be r e a c h e d a t some  o f t h e l o a d t h e number o f h i n g e s becomes  t h a n t h e number o f r e d u n d a n t s ,  structure w i l l  as, a w h o l e .  the  initially  t r a n s f o r m i n t o a mechanism u n a b l e t o  o f f e r any r e s i s t a n c e t o c o m p l e t e  This  I f by  statement  collapse.  holds true  i f the s t r u c t u r e  I t i s a l s o p o s s i b l e , however, t o h a v e a  mechanism f o r a p a r t  fails partial  o f t h e s t r u c t u r e w i t h l e s s number o f  hinges. I f the approximate known, t h e f a i l u r e l o a d the  l o c a t i o n s of the hinges are  c a n be d e t e r m i n e d  by s t a t i c s  o r by  p r i n c i p l e o f v i r t u a l work s i n c e t h e s t r u c t u r e has  come s t a t i c a l l y  be-  determined.  T h i s p r o p o s i t i o n l e a d s t o two  basic  theorems o f  analysis. 1.  S t a t i c P r i n c i p l e ( l o w e r bound) w h i c h  t h a t any a s s u m p t i o n s  o f b e n d i n g moments a t t h e  locations of p l a s t i c hinges consistant  states  possible  w i t h s t a t i c s and  not  e x c e e d i n g t h e p l a s t i c v a l u e s o f moments w i l l r e s u l t i n c o r r e s p o n d i n g v a l u e s o f l o a d s w h i c h w i l l be l e s s o r e q u a l to the f a i l u r e  2.  load.  K i n e m a t i c P r i n c i p l e ( u p p e r bound) w h i c h  t h a t any a s s u m p t i o n  o f mechanism due  to p l a s t i c  hinges  states will  15 result in a value of the load, found hy statics or principle of virtual work, which w i l l be greater or equal to the failure load. The exact solution of course satisfies both theorems. As explained in the introduction, the theory of limit design i s based on the assumption that the structural material can undergo extensive local strains, without causing an immature failure due to physical breaking of an early hinge.  This assumption i s not favored by material lacking  strain hardening. To investigate the realistic behavior of aluminum alloys with very l i t t l e strain hardening, a more exact theory of bending i s necessary.  Dr. A.P. Hrennikoff in  1948 has developed such a theory which is introduced briefly in the next section.  102.  An Introduction to the Theory of Inelastic Bending  Assumptions of the Exact Theory. The assumption of limit design, that the moments developed at the locations of the plastic hinges equalize during the process of formation of a mechanism, w i l l be examined with reference to exact theory of inelastic bending.  The theory i s more exact than limit design but is also  16 based on certain assumptions introduced for the sake of simplicity. 1.  They are as follows: Distribution of strains over the cross-section is linear.  2.  The stress-strain relation in bending is the same as i n simple tension or compression and the stress-strain curves in tension and compression are identical.  3.  The flexural members are symmetrical about their neutral axes.  4.  The bending moment diagram of the loaded structure i s bound by straight lines.  A curved  diagram is approximated by a polygonal shape. 5.  Normal forces are ignored.  6. Deformations due to shearing forces are neglected. 7. Instability is not considered to be a factor. 8. Deformations of the structure are assumed small. Unit Strain - Bending Moment Relation in a Rectangular Beam. -.. "  The functional relationship between G  (unit stress) and e (unit strain) may be expressed as an equation G = f ( O , a graph, or a table.  17 Then using the simplifying assumptions of the theory, i t is possible to derive expressions for the moment, curvature, angle change, deflection and shear stresses i n terms of the stress-strain curve.  These relations have been  derived by Dr. Hrennikoff and w i l l only be briefly introduced here. To find a relation between bending moment and corresponding unit strains, e , in the outer fiber of the beam we must introduce a variable, m,, which is defined as the statical moment about G axis of an area under the.stressstrain curve taken to a variable point A on the curve provided that the variable base e i s reduced to unity. (Figure 7e, )  e,  Unit Strain  6?  .  -b  C  d _  Q  Figure 7 Presenting i t mathematically, we get the following expression:  18  The value of m,, can be easily computed i f 6 and G are related mathematically.  Otherwise i t must be determined by summation,  Now i f we equate the external"'moment to resisting moment and notice that the internal moment i s proportional to the breadth and the square of the depth h we have:  M-  2m  Hit - ^ JJ2  d°>  Unit Stain-Bending Moment Relation in an I-Beam. In the case of I-beams or channels i t i s also possible by introducing a new variable m to relate the bending moment to the unit strain in the outer fiber. To simplify the analysis, the web depth i s assumed to be equal to the distance between the center of flanges and the flange area i s assumed to be concentrated at the extremities of the web. Figure 8a. Thus for the particular section shown in figure 8b the moment about the neutral axis i s equal to sr? =  fe-ede  +• JTG.  (12)  where e and e are the strain and stress of the flange respectively, and K i s the ratio of the flange area to £ of the web area.  19  >/ K 2  C  b  d  Figure 8  Now equating external moment to resisting moment and observing that in an I beam the internal moment i s proportional to the product A„h  , we have the following  expression: (  1  3  )  Unit Strain - Shear Stress Relation in an I Beam. The analysis of shear stress requires the introduction of a new variable Cj defined as the sum of the normal stresses developed by the section shown in figure 8. When K = 0 then CJ is referred as cj, . Thus:  (14) o (15)  20  Unit Strains - Angle Change Relation i n an I Beam. The relationship between the angle ehange and the flange strain i s given mathematically in the following manner: ' 6 dm  ^ V  =  (16).  J*/k  *  where cj> i s the angle change between a point of contraflexure and a point of 6o flange strain, V the shear force, and n  0  the unit function for the angle change. See figure 9. rr> moment € flange stra'm ni ;., rio u/7/t funcHon ^for <j> Uo unit function for o 0  a  For a member under constant moment <4> = J .  n  xo-6o-  (17)  where 5T„ is the length of the beam.  Figure 9 Unit Strain-Deflection Relation in an I Beam., The mathematical expression relating the unit strains and the deflection is as follows:  21 where  i s the d e f l e c t i o n a t a point  a t a n g e n t drown t o a p o i n t  £o f l a n g e s t r a i n ,  of  the u n i t f u n c t i o n f o r d e f l e c t i o n . member u n d e r c o n s t a n t  of contraflexure  S e e f i g u r e 9.  from  and Uo i g For a  moment  (19) Formulas f o r d e f l e c t i o n s o f n o n t r i a n g u l a r ( t r a p e z o i d a l ) moment d i a g r a m s c a n be d e r i v e d geometrical  relations.  Maximum S h e a r i n g  The  They a r e g i v e n  Stress  greatest  i n refer.  simple (I).  i n a n I Beam.  shearing  a x i s and c a n be d e r i v e d w i t h  from  stress  reference  T©  i s a t the n e u t r a l  t o f i g u r e 10.  oLx.  Figure  From s t a t i c s J^f*. then  t°  -To  =0  t>c/z =  -  A ^  10  i A«/j ^  ~? J ~  t  _f£_ dxr  =  _v Aw  ^  - L A„c/j  (  2  0  ( 2 1 )  dm  )  The function fix. can he computed point hy point hy taking increments of both OJ and m corresponding to the same increments in e .  23  PART I I  UNIT FUNCTIONS AND TEST ARRANGEMENTS  201.  T e n s i o n and C o m p r e s s i o n  S t r e s s - S t r a i n Curve  o f H i g h S t r e n g t h Aluminum A l l o y .  As was m e n t i o n e d of unit  T e s t s t o Determine t h e  i n s e c t i o n I . f o r the computation  f u n c t i o n s o f h i g h s t r e n g t h aluminum a l l o y  (65S-T6),  t h e f u n c t i o n a l r e l a t i o n s h i p between s t r e s s and s t r a i n must be known.  S i n c e no s u c h r e l a t i o n was a v a i l a b l e ,  three  t e n s i o n t e s t s and one c o m p r e s s i o n t e s t were c o n d u c t e d on round  samples  c u t f r o m b o t h a 6 i n c h I beam ( d e s i g n a t e d  28008 A l c a n ) and a 4 i n c h H beam ( d e s i g n a t e d 29001 A l c a n ) .  T e n s i o n c y l i n d i c a l specimens web and f l a n g e web i n t e r s e c t i o n , 0.24 was  were c u t f r o m  w i t h d i a m e t e r s o f 0.3 i n s . ,  i n s . , and 0.5 i n s . r e s p e c t i v e l y .  The l o a d i n g machine  a 60,000 l b s B a l d w i n Soutwark h y d r a u l i c t e s t i n g  S t r a i n measurements f o r t h e  flange,  machine.  s p e c i m e n were r e c o r d e d b o t h  by Cambridge e x t e n s o m e t e r  (gauge l e n g t h 4") and (SR-4 t y p e  A7)  s t r a i n gauges  e l e c t r i c a l resistance  F o r specimens  (gauge l e n g t h  ^-").  h a v i n g a d i a m e t e r l e s s t h a n 0.5 i n s .  o n l y t h e Cambridge e x t e n s o m e t e r was u s e d .  E x t e n s i v e deforma-  1" t i o n n e a r f a i l u r e were measured t o -J-QQ a c c u r a c y w i t h on a 4" gauge l e n g t h .  Compression  cylindrical  calipers  specimens c u t  24 f r o m t h e web f l a n g e  intersection,  1.5 i n s . l o n g , were a l s o t e s t i n g machine. by  two  to  avoid possible  tions  0.5 i n s . i n d i a m t e r and  compressed  i n t h e 60,000  hydraulic  S m a l l s t r a i n measurements were c o n d u c t e d  SR-4 s t r a i n gauges p l a c e d d i a m e t r i c a l l y moment e f f e c t .  a d i a l gauge l o c a t e d  opposite  To measure l a r g e  deforma-  u n d e r t h e head o f t h e t e s t i n g  machine was u s e d . For  s m a l l d e f o r m a t i o n t h e F e d e r a l d i a l gauges  showed e r r o n e o u s r e a d i n g s due t o t h e c r u s h i n g e f f e c t specimen's  on t h e  surfaces. S i n c e creep i s a s s o c i a t e d w i t h p l a s t i c deforma-  tion,  the test  hours.  d u r a t i o n was v a r i e d  The s t r e s s - s t r a i n  curves a r e presented  f o r t h e t e n s i o n t e s t and f i g u r e  A each t e s t  f r o m 30 m i n u t e s t o 2\ i n figure  12 f o r t h e c o m p r e s s i o n  11  test.  summary o f t h e main m e c h a n i c a l p r o p e r t i e s o f  i s given i n table  1.  O b s e r v i n g t h e s t r e s s - s t r a i n diagram o f each men, we s e e t h a t  h i g h s t r e n g t h aluminum a l l o y s  have  speci-  certain  mechanical p r o p e r t i e s which a r e s i m i l a r t o those o f s t e e l . L i k e s t e e l , t h e y were c a p a b l e o f a l m o s t p e r f e c t l y havior. of  3 3  stressed and  The t e n s i o n s p e c i m e n s e x t e n d e d  elastic  e l a s t i c a l l y to a  and a s t r a i n o f a p p r o x i m a t e l y  .35 p e r c e n t .  p a s t t h e e l a s t i c l i m i t , t h e r e l a t i o n between  s t r a i n was e n t i r e l y d i f f e r e n t  from that  bestress When  stress  i n structural  steel.  25  Specifically, the excessive deformation did not begin at any sharply distinguishable point on the stress strain diagram. After a rapid increase of inelastic deformation at approximately 40 '? /i £ stress, large inelastic deformaK  s  n  tion took place (without almost any strain hardening) until the ultimate stress of 43 ^'fVins  2  was reached. At that stress,  "necking" took place and the specimen fractured at a lower stress after considerable "necking". In the compression tests, the elastic part and some of the early plastic parts of the curve were similar to the tension tests.  After some yielding, a,higher compres-  sion stress was observed apparently due to the increase in the cross section area . After further deformation the test was stopped due to buckling of the specimen which, in turn, caused some decrease in the slope of the stress-strain curve. Comparing the stress-strain diagram of high strength aluminum alloys to steel, two basic differences in their mechanical properties are readily recognized. First, the light alloys have very l i t t l e stain hardening and second, they lack a well defined yield point. The lack of strain hardening, observed in the stress-strain diagram of the specimens as was mentioned in the introduction, i s believed to be of primary importance in  26 limit design since the threat of premature failure i s associated with i t . The absence of a well defined yielding, however, is of secondary importance in limit design because while the presence of a well defined yield condition may be desir.eable for the sake of having a clear cut limiting state on which to base working values, (loads and so on)',, i t s absence constitutes an advantage i n that relatively:" higher loads can be carried by the redundant structure due to the continuing a b i l i t y of the structural member to develop increasing resistances. Deformations in members made of such metal become excessive only when they exceed a value that will have to be agreed upon in advance.  Therefore, the limit loads or limit  moments likewise have to be arbitrarily defined.  Thus for  limit design calculations, the stress at 0.2 percent offset from the i n i t i a l elastic line arbitrarily will be defined as the yield stress. This w i l l hereafter be referred to as yield stress Gy which corresponds to the point where the metal starts deforming inelastically with very l i t t l e increase in stress. Referring to the test results presented i n table I we observe a considerable variation i n the physical properties of the material such as modulus of elasticity, strain at ultimate stress, strain at failure and stress at failure.  27 The  reason f o r t h i s unexpected v a r i a t i o n of the t e s t r e s u l t s  i s the s i z e o f the specimen ( r a t i o o f c r o s s - s e c t i o n a l area t o gauge l e n g t h ) and d i f f e r e n c e o f m a t e r i a l .  V a r i a t i o n s due  to time e f f e c t a r e not a p p r e c i a b l e w i t h i n the l i m i t s of 20 minutes t o 2|- hours. Modulus o f E l a s t i c i t y The between Wo.l.  9300  modulus o f e l a s t i c i t y v a r i e d approximately ^p/inS "  to  1  .  11,300  In t e n s i o n t e s t  s t r a i n s measured by the Cambridge extensometer gave a  modulus o f e l a s t i c i t y equal t o  9300  *' /jm'' p  , whereas s t r a i n  measured by SR-4 gauges gave a modulus o f e l a s t i c i t y of 9550 ip//ns' . lc  x  I n view o f these d i s c r e p a n c i e s another compres-  s i o n t e s t was conducted.  The specimen was a 6-1 beam with  2 A=3»311 i n s and L=12 i n c h e s . the e l a s t i c l i m i t .  The deformation d i d not exceed  E l e c t r i c a l s t r a i n gauges with a  gauge l e n g t h were placed symmetrically of the f l a n g e s and the web.  opposite a t the center  M i r r o r extensometers with 6"  gauge l e n g t h were a l s o placed symmetrically c e n t e r of the f l a n g e s . averaged.  inch  Symmetrically  opposite at the  opposite s t r a i n s were  The r e s u l t s i n f i g u r e 13 show a c l o s e agreement  between the m i r r o r extensometer and f l a n g e s t r a i n gauges g i v i n g a modulus of e l a s t i c i t y of 10,400 ^'pjmf'.  The web  s t r a i n s gave a modulus o f e l a s t i c i t y o f 10,620 ^'p/ms ' . t  1  In the beam t e s t s , d e f l e c t i o n s were measured at different locations.  D e f l e c t i o n s measuredjduring the e l a s t i c  28  behavior of the beam, were also used i n determining the modulus of e l a s t i c i t y . t i o n Ity.  These computations are given i n sec-  The modulus of e l a s t i c i t y computed from beam test  No.l. and No.3. was 9,660 Kfp/ s and from beam test No.2. 2  n  was 10,400 Kif>J,fts  .  z  In order to compute moments during the  test, using the flange s t r a i n measurements, a modulus of e l a s t i c i t y of 10,400 K r p / s / n  z  was used.  For the computation of unit functions a modulus of e l a s t i c i t y of 9,550 Ki'p/ms ' was used. 2  The variations observed  i n the modulus of e l a s t i c i t y w i l l not affect appreciately the unit functions. Stress-Strain Diagram f o r the Theory of Inelastic Bending As stated i n the assumptions of the theory the s t r e s s - s t r a i n curve i s the same i n tension and compression. Between the two curves, preference should be given to a tension s t r e s s - s t r a i n curve since the beam w i l l f a i l at the tension side.  From the s i x tension tests, the s t r e s s - s t r a i n  diagram of test number No.l. with the largest area (0.5 inches diameter) was chosen f o r the computation of unit functions. The cross-section  area of the test beam, r e s i s t i n g the f l e x u r a l  stresses, being large i n size, would more l i k e l y develop a stress s t r a i n relationship similar to the large size specimen, rather than the small size. j  The unit functions m, n, u, q, and ZJ. depend only dm  29  on the s t r e s s - s t r a i n curve of the material, and the geometry of the. cross section of the beam which i s reflected i n the value of K.  Thus i n order to find the moments, the shears,  and deformations i n a beam of a p a r t i c u l a r cross section i t i s necessary to have tabulated a l l the unit functions versus each value of the flange s t r a i n  .  The theory of i n e l a s t i c bending w i l l be compared with the beam tests i n order to get an indication of i t s accuracy i n predicting moments, deflections and f a i l u r e conditions beyond the e l a s t i c l i m i t of the materials. 202.  Evaluation of Inelastic Theory Unit Functionsfor  I Beams- Made of Aluminum A l l o y (65 S - T 6) Up to the e l a s t i c l i m i t the evaluation of unit functions m, n, q, ^-L , already presented, i s a very simple dm  proposition since the functional relationship between stress and s t r a i n i s l i n e a r .  Beyond the e l a s t i c l i m i t since no  simple mathematical r e l a t i o n between the two variables i s available, a step by step summation of small increments must be made.  For each shape of I beam as given by the  parameter, K, (ratio of the flange area to one h a l f the web area) a set of unit function was computed.  The values of K  ranging from 0 to 3 included the rectangular section (K = 0) and the common shapes of H or I beams l i s t e d i n the manufacturer's  catalogues.  Table £ and figure 14 present  the unit functions m,n,q and  above the e l a s t i c l i m i t f o r  TABLE  Test  Ho.  Specimen Location  Specimen Diameter in.  /  M E C H A N I C A L P R O P E R T I E S P R O M T E N S I O N AND C O M P R E S S I O N T E S T S  Duration of test  Modulus of Elasticity Kip/in .  Proportional Limit Kip/in .  Yield Stress a t 2fo o f f s e t Kip/in .  Ultimate Stress Kip/in .  Ultimate Strain Per Cent  Failure Stress Kip/in .  Elongation Per Cent  Tension 1.  F i l l e t of 0.4996 I beam  1/2  hr.  9540 9300  (SR-4) (Cambridge)  36.3  39.9  43.4  9.0  30.4  14.8  2.  F i l l e t of 0.4997 H beam  2/3  hr.  9550  (Cambridge)  34.5  39.9  42.7  6.8  31.0  11.3  3.  Flange of 0.3002 I beam  2 1/4  h r . 9600  (Cambridge)  31.0  39.5  43.0  7.3  27.0  10.5  4.  F l a n g e of 0.2998 H beam  1/2  hr.  9300  (Cambridge)  30.0  38.9  42.0  6.0  25.8  9.5  5.  Web o f I beam  0.2394  1/3  hr.  9300  (Cambridge)  33.5  39.5  42.7  6.8  28.7  9.0  6.  Web o f H beam  0.2386  2 2/3  h r . 9300  (Cambridge)  31.5  40.0  42.4  .5.0  26.0  7.5  1.  Fillet^6f H beam  0.500  (SR-4)  32.0 "  39.2  -  -  -  2.  F i l l e t of 0.500 I beam  33.0  40.5  -  -  -  Compression 9960 1/2  hr.  11000  (SR-4)  AO  J/  MILLIMETERS- '/J C E N T I M E T E R  36  o  2  3  4-  S  4>  STrcr/s?  7,  %  8  9  10  39 800  750  45  m 700  6O0  ' m  SOO\  HI  iili i  • 3 SOl  250  IOC  5  e,  7  8  /o  42 TABLE 2 UNIT FUNCTIONS FOR A l 65S-T6 BEAMS (a) K - 0  ,-3  x/o  3.8 3.9 4.0 4.2 4.4 4.6 4.8 5.0 5,3 5.6 6.0 7.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0 22.0 24.0 26.0 28.0 30.0 32.0 36.0 40.0 44.0 48.0 52.0 56.0 60.0 64.0 68.0 72.0 76.0 80.0 84.0 88.0 90.0 92.0 94.0 96.0 100.0 104.0  m /in 12.10 12.42 12.73 13.31 13.84 14.32 14.75 15.13 15.64 16.08 16.57 17.47 18.06 18.78 19.19 19.46 19.64 19.77 19.87 19.95 20.02 20.08 20.15 20.20 20.26 20.37 20.46 20.55 20.63 20.71 20.79 20.86 20.94 21.00 21.06 21.11 21.16 21.20 21.24 21.26 21.28 21.30 21.31 21.33 21.35  3  22.99 24.21 25.43 27.83 30.11 32.26 34.27 36.15 38.77 41.16 44.02 49.84 54.30 60.79 65.30 68.78 71.56 73.72 75.54 77.20 78.81 80.41 82.16 83.85 85.61 89.22 92.79 96.36 100.22 104.27 108.49 112.84 117.30 121.52 125.44 129.20 133.19 136.88 140.49 142.19^ 143.82 145.31 146.64 148.80 150.02  (b) K = 0.5  ^fox/O 185. 200. 216. 247. 278. 308 337. 366. 406. 444. 490. 590. 669. 788. 874. 941. 995. 1038. 1074. 1107. 1139. 1171. 1207. 1241. 1276. 1350. 1423. 1496. 1575. 1659. 1746. 1837. 1930. 2019. 2101. 2182. 2265. 2343. 2420. 2456. 2490. 2522. 2550. 2596. 2623.  m  5  dm  1.50 1.51 1.53 1.57 1.62 1.67 1.73 1.80 1.87 1.98 2.17 2.37 2.80 3.20 3.72 4.13 4.62 5.11 5.34 5.30 5.10 4.69 4.41 4.25 3.93 3.77 3.61 3.37 3.H 2.94 2.81 2.71 2.68 2.75 2.80 2.76 2.75 2.79 2.82 2.89 2.97 3.20 3.61 4.47 14.89  3.8 3.9 4.0 4.2 4.4 4.6 4.8 5.0 5.3 5.6 6.0 7.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0 22.0 24.0 26.0 28.0 30.0 32.0 36.0 40.0 44.0 48.0 52.0 56.0 60.0 64.0 68.0 72.0 76.0 80.0 84.0 88.0 90.0 92.0 94.0  30.25 30.97 31.53 32.48 33.22 33.84 34.35 34.83 35.44 35.95 36.50 37.44 38.09 38.88 39.34 39.68 39.89 40.02 40.17 40.30 40.44 40.58 40.72 40.85 40.96 41.19 41.36 41.55 41.76 41.94 42.11 42.28 42.42 42.50 42.60 42.70 42.79 42.86 42.92 42.94 42.95 42.95  J  57.48 60.23 62.45 66.38 69.52 72.34 74.70 77.08 80.21 83.01 86.16 92.31 97.14 104.31 109.36 113.82 116.97 119.13 121.91 124.62 127.95 131.43 135.21 139.06 142.38 150.23 156.66 164.43 174.04 183.09 192.70 202.56 211.06 216.27 223.34 230.52 237.54 243.69 249.02 250.27 250.99 251.55  1.20 1159 1243. 1.25 1313. 1.31 1438. 1.37 1542. 1.46 1636. 1.54 1717. 1.60 1799. 1.66 1909. 1.74 2009. 1.86 2123. 2.09 2350. 2.29 2533. 2.64 2809. 2.98 3006. 3.30 3183. 3.57 3308. 4.35 3394. 4.36 3505. 3.76 3614. 3.34 3749. 2.93 3890. 2.71 4044. 2.54 4201. 2.56 4337. 2.41 4659. 2.38 4924. 2.32 5246. 2.02 5647. 1.89 6025. 1.86 6429. 1.80 6845. 1.82 7205. 2.06 7427. 2.17 7727. 1.98 8034. 1.96 8334. 2.01 8597. 2.14 8826. 2.46 8879. 4.17 8910. 5.79 8934. 14.20  43 TABLE 2 UNIT FUNCTIONS FOR A l 65S-T6 BEAMS  (e)  K =  a  m x/o  1.0  (d) -3  x/O  /n  3*8 4 8 . 4 0 3 . 9 49.52 4 . 0 50.33 4 . 2 51.66 4.4 52.59 4.6 53.37 4 . 8 53.95 5;o 54.53 5 . 3 55.24 55.83 5.6 6 . 0 56.42 7 . 0 57.42 8 ; 0 58.11 1 0 . 0 58.98 1 2 . 0 59.49 14.0 59.91 1 6 . 0 60.14 1 8 . 0 60.27 2 0 . 0 60.47 2 2 . 0 60.65 2 4 . 0 60.87 2 6 . 0 61.08 2 8 . 0 61.39 3 0 . 0 61.50 3 2 . 0 61.66 3 6 . 0 62.02 4 0 . 0 62.26 4 4 . 0 62.55 4 8 . 0 62.88 5 2 . 0 63.16 5 6 . 0 63.44 6 0 . 0 63.70 64.0 63.91 6 8 . 0 64.00 7 2 . 0 64.15 7 6 . 0 64.29 8 0 . 0 64.42 8 4 . 0 64.52 8 8 . 0 64.60 90.0 64.61  91.96 96.26 99.46 104.93 108.93 112.43 115.14 118.01 121.65 124.86 128.30 134.78 139.98 147.82 153.43 158.87 162.39 164.55 168.27 172.03 177.09 182.44 188.25 194.28 199;15 211;25 220.52 232.49 247.86 261.91 276.92 292.29 304.81 311.02 321.24 331.74 341.88 350.50 357.54 358.35  2967 1.13 3178. 1.17 3338. 1.22 3616. 1.28 3825. 1.36 4010. 1.45 4156. 1 . 5 1 1.55 4311. 4 5 1 1 . 1.64 4690. 1.77 4883. 2.02 5251. 2 . 2 1 .51 5552. 22.80 6011. 6343. 2;99 6668. 3.17 6879. 4 i l 2 7009. 3.84 7234. 3.02 7462. 2 . 6 1 7769. 2.26 8095. 2 . 1 1 8450. 1:99 8 8 2 1 . 2.02 9121. i ; 9 3 9869. i ; 9 2 10445. 1.88 11192. 1.65 12156. 1.57 13041. 1.55 13991. 1 . 5 1 14969. 1.54 15768. 1.77 16164. 1.88 16819. i ; 6 7 17494. i : 6 6 18146. 1 . 7 1 18702. 1.83 19157. 2.22 19209. 1 0 ; 97  -3  3.8 3.9 4.0 4.2 4.4 4.6 4.8 5.0 5.3 5.6 6.0 7.0 8.0 10.0 12.0 14.0 16.0 18;0 20.0 22.0 24.0 26;o 28.0 30.0 32.0  36;o 40;o 44:0 48.0 52.0 56.0 60.0 64.0 68.0 72.0 76.0 80.0 84.0 88.0  K = 1.5  m  n ,  2  in  66*55 68.07 69.13 70.83 71.97 72.89 73.55 74.23 75.04 75.70 76.35 77.39 78.14 79.08 79.64 80.14 80.39 80.52 80.77 81.00 81.29 81.58 81.87 82.15 82.36 82.84 83.16 83.55 84.01 84.39 84.76 85.12 85.39 85.50 85.69 85.88 86.05 86.18 86.28  -3  ty/tfX/O  126.45 132.28 136.47 143.48 148.34 152.51 155.58 158.94 163.10 166.72 170.44 177.24 182.82 191.34 197.50 203.91 207.81 209.97 214.64 219.45 226. 3 233.46 241.29 249.50 255.91 272.27 284.39 300.56 321.67 340.72 361.13 382.01 398.57 405.76 419.13 432.97 446.23 457.30 466.07  5610 6003. 6290. 6780. 7128. 7430. 7654. 7902. 8213. 8486. 8769. 9292. 9726. 10395. 10884. 11396. 11709. 11883. 12260. 12649. 13199. 13788. 14427. 15101. 15628. 16979. 17985. 19333. 21102. 22706. 24432. 26206. 27617. 28232. 29376. 30563. 31703. 32657. 33413.  1.09  1.12 1.17 1.22  1.30  1.39 1.44 1.48 1.56 1.70 1.96 2.14 2.40 2.65 2.75 2.89 3.91 3.46 2.60 2.22 1.94 1.82 1.73 1.76 1.69 1.69 1.66 1.48 1.42 1.41 1.38 1.40 1.61 1.70 1.51 1.50 1.54 1.66 2.05  44 TABLE £ UNIT FUNCTIONS FOE. A l 65S-T6 BEAMS (e) K = X/O'  K//»  3.8 3.9 4.0 4.2 4.4 4.6 4.8 5.0 5.3 5.6 6.0 7.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0 22.0 24.0 26.0 28.0 30.0 32.0 36.0 40.0 44.0 48.0 52.0 56.0 60.0 64.0 68.0 72.0 76.0 80.0 84.0 88.0  84.70 86.62 87.93 90.01 91.34 92.42 93.15 93.93 94.84 95.58 96.27 97.37 98.16 99.18 99.79 100.36 100.64 100.77 101.07 101.35 101.72 102.08 102.45 102.80 103.06 103.67 104.06 104.55 105.13 105.61 106.09 106.54 106.88 107.00 107.24 107.47 107.68 107.84 107.96  z  2.0  U n K/,n x./o~ z  160.93 168.31 173.48 182.03 187.75 192.60 196.02 199.86 204.54 208.57 212.59 219.71 225.66 234.85 241.56 248.95 253.22 255.38 261.01 266.86 275.38 284.47 293.33 304.71 312.68 333.28 348.25 368.62 395.49 419.54 445.35 471.74 492.33 500.51 517.03 534.20 550.58 564.11 574.60  9087 9719 10171 10931 11450 11895 12212 12572 13014  13397  13783 14472 15055 15961 16629 17368 17797 18015 18582 19175 20039 20967 21974 23040 23860 25990 27545 29669 32486 35020 37752 40558 42754 43630 45399 47242 49004 50462 51594  ( f ) K = 2.5  U -3  c/tr> x/o 1.07 1.10 1.14 1.18 1.26 1.34 1.39 1.42 1.50 1.64 1.91 2.80 2.30 2.52 2.56 2.66 3.74 3.17 2.32 1.99 1.75 1.65 1.58 1.61 1.55 1.55 1.53 1.38 1.33 1.-32 1.30 1.32 1.50 1.58 1.41 1.40 1.44 1.54 1.-92  3.8 3.9  K/in<  102.85 105.17 106.73 4vO 109.18 4.2 110.72 4.4 4.6 111.94 112.75 4.8 113.63 5.0 114.64 5.3 5 . 6 115.45 116i?20 6.0 117:34 7.0 118.19 8.0 119.28 10.0 12.0 119.94 14.0 120.59 16*0 120.89 18.0 121.02 20.0 121.37 22.0 121.70 24.0 122.14 26.0 122.58 28 sO 123.02 30.0 123.45 32.0 123.76 36.0 124.49 40.0 124.96 44.0 125.44 48.0 126.26 52.0 126.84 56*0 127.41 60.0 127.96 64.0 128.36 68.0 128.50 72.0 128.78 76.0 129.06 80.0 129.31 84.0 88.0 129.50 129.64  K/rfx/c* 195.42 204.33 210.50 220.58 227.16 232.68 236.45 240.79 245.98 250.42 254.73 262.18 268.51 278.37 285.63 293.99 298.64 300.80 307.37 314.28 324.52 335.49 347.37 .359.93 369.44 394.30 412.12 436.69 469.31 498.36 529.57 561.47 586.08 595.26 614.92 635.42 654.92 670.92 683.12  13398 14326. 14980 16068 16792 17407 17830 18321 18913 19424 19923 20793 21538 22709 23578 23583 25114 25406 26202 27042 28290 29633 31091 32639 33815 36900 39123 42201 46307 49983 53951 58024 61179 62357 64887 67530 70049 72119 73700  1.06 1.08 1.12 1.-16 1.23 1.30 l.\35 1.38 1.45 1.-59 1.86  2.03  2.22 2.41 2.42 2.49 3.58 2.94 2.12 1.83 1.62 1.54 1.48 1.-51 1*46 1.46 1.46 1.31 1.27 1.27 1.25 1.27 1.41 1.50 1.37 1.37 1.37 1.40 1.84  45 TABLE 2 UNIT FUNCTIONS FOR A l . 65S-T6 BEAMS (g) K =  3.0  (h) K = 2.752 x/o-  3  3.8 3.9 4.0 4.2 4.4 4.6 4.8 5.0 5.3 5.6 6.0 7.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0 22.0 24.0 26.0 28.0 30.0 32.0 36.0 40.0 44.0 48.0 52.0 56.0 60.0 64.0 68.0 72.0 76.0 80.0 84.0 88.0  121.00 123.72 125.53 128.36 130.09 131.47 132.35 133.33 134.44 135.33 136.12 137.32 138.21 139.38 140.09 140.81 141.14 141.27 141.67 142.05 142.57 143.08 143.69 144.10 144.46 145.32 145.86 146.55 147.38 148.06 148.74 149.38 149.85 150.00 150.33 150.65 150.94 150.94 151.32  229.90 240.36 247.51 259.13 266.57 272.77 276.89 281.72 287.42 292.27 296.87 304.64 311.35 321.89 329.70 339.03 344.05 346.21 353.74 361.70 373.66 386.51 400.41 415.14 426.21 455.31 475.99 504.76 543.12 577.17 613.78 651.19 679.84 690.00 712.82 736.65 759.27 777.72 791.65  18545 19825 20716 22191 23153 23963 24507 25148 25912 26566 27190 28253 29177 30639 31731 33042 33750 34055 30519 36248 37950 39786 41779 43898 45495 49712 52721 56928 62566 67596 73029 78605 82891 84415 87841 91427 94838 97625 99731  1.05 1.07 1.10 1.14 1.20 1.27 1.32 1.34 1.41 1.54 1.82 1.98 2.14 2.32 2.29 2.35 3.44 2.75 1.98 1.71 1.53 1.46 1.41 1.43 1.39 1.40 1.38 1.26 1.23 1.23 1.21 1.23 1.37 1.44 1.30 1.29 1.32 1.40 1.73  t'PW  3.90 114.47 4.00 116.43 4.20 118.95 4.40 120.58 4.60 121.61 4.80 122.59 5.00 123.53 5 . 3 0 124.59 5.60 125.58 6.00 126.36 7.00 127.55 8.00 128.14 10.00 129.27 12.00 130.23 14.00 130.78 16.00 131.10 18.00 131.22 20.00 131.59 22.00 132.09 24.00 132.43 26.00 132.91 28.00 133.38 30.00 133.86 32.00 134.34 36.00 135.00 40.00 135.50 44.00 136.14 48.00 136.91 52.00 137.54 56.00 138.17 60.00 138.65 64.00 139.14 68.00 139.34 72.00 139.67 76.00 140.00 80.00 140.18 84.00 140.37 88.00 140.54 90.00_140.56 92.00 140.58 94.00 140.46 96.00 140.34 100.00 139.95 104.00 139.13 108.00 138.03 112.00 136.36 118.00 132.99  *>M  222.43 230.19 240.50 247.53 252.17 256.79 261.38 266.84 272.25 276.73 284.47 288.92 299.11 309.69 316.79 321.57 323.68 330.73 341.12 349.04 360.86 373.73 387.50 402.28 424.84 443.92 470.57 505.94 537.41 571.23 599.39 629.31 642.47 665.59 689.80 704.41 719.24 734.47 736.16 737.71 726.39 714.75 676.31 . -  16975 1.05 17871 1.11 19084 1.15 19926 1 . 2 3 20488 I . 3 6 21051 1.31 21616 I . 3 6 22294 1.41 22971 1.52 23535 1.83 24517 2.15 25086. 2.38 26399 2.19 27771 2.23 28698 2.65 29323 3.57 29600 2.86 30527 1.88 31896 1.76 32944 1.67 34512 1.50 36226 1.44 38066 1.39 40048 1.41 43085 1.42 45666 1.47 49285 1.26 54115 1.28 58434 1.25 63097 1.22. 66993 1.26 71149 1.39 72982 1.53 76207 1.29 79592 1.49 81640 1.26 83720 1.43 85858 2.87 86096 2.97 86314 .35 84724 .73 83089 .74 77703 .89  46 K = 0, 0 . 5 , 1 , 1 . 5 , 2, 2 . 5 , 2 . 7 5 2 , corresponds  t o the t e s t  The v a l u e o f K=2.752  a detailed  a n a l y s i s o f t h e beam  beam.  Before going into tests,  3-  i t i s necessary to establish  the flange s t r a i n In  beforehand  e a t w h i c h a beam w i l l  a simply supported  fail  a t f a i l u r e corresponds  This value of flange s t r a i n  Thus t h e f l a n g e  t o t h e maximum v a l u e o f m.  isa little  corresponding to the ultimate stress, of  i n bending.  beam t h e f a i l u r e l o a d i s  g o v e r n e d by t h e v a l u e o f t h e maximum moment. strain  the value of  g r e a t e r than the s t r a i n  d e p e n d i n g on t h e v a l u e  K. In  beam w i l l  establishing  fail  i n bending  the f l a n g e s t r a i n 6 a t which the t e s t a s t u d y o f t h e moment d i a g r a m o f t h e  beam i s n e c e s s a r y . I  <  J B  A  (CL)  m ^  U  Figure 15  42."  „  (  *  _  A7§"  M  cMc  r-  P  *(b) l  47 C o n s i d e r a moment d i a g r a m as corresponding to point that  the  value  l o a d and  of flange  extended over the v a l u e  This under the unit  6c u n d e r t h e  load point  and  w i l l decrease the  i n order  the  C.  Also  increase  d e f l e c t i o n under the  increase  continuously  contribute  point.  c  or  to t h i s  load  s t r a i n s and Thus t h e  point.  the  i n t e r n a l support  break the  load. will  o f d e f l e c t i o n no  failure  strain  c o r r e s p o n d t o t h e maximum .value  In the  case of the  condition w i l l  beam by €  c  has  point.  increase  This  m.  moment d e c r e a s e  r u n n i n g "-resting machine load  of  strains  moment H  to the  has  i n p o s i t i v e a n g l e change i s a v a i l a b l e e x c e p t a t  s e c t i o n under the infinite  point  to m a i n t a i n e q u i l i b r i u m w i t h the  the  To  due  moment o v e r t h e  Furthermore the  load  assume  condition w i l l resultinhigher flange  load point  increase  s u p p o r t , and  c o r r e s p o n d i n g t o maximum v a l u e  f u n c t i o n KM « a t p o i n t  under the to  strain  point  15<JL •  shown i n F i g u r e  the  create  f r a c t u r e under the  under the  load  load  point  o f m.  will r  t e s t beam t h e  l o a d was -not  a  3 point  load b u t • i t  was  a c t i n g over a width of ^  c o r r e s p o n d i n g moment d i a g r a m i s assumed t o be shown i n F i g u r e  15b.  A c t u a l l y the  inner  inches. as  the  s u p p o r t was  The one  also  3» wide but  s i n c e the  s t r a i n s were r a t h e r low  w i t h o u t a p p r i c i a b l e e r r o r t o be  a point  a c t u a l d i s t r i b u t i o n of the  over the  known.  Thus t h e  load  i t can  support.  be  Also  assumed the  i n c h width i s not  shown moment d i a g r a m i s a c l o s e  approximation.  48  Considering  t h e moment d i a g r a m o f F i g u r e 15b a n d a s s u m i n g t h a t  the f l a n g e s t r a i n 6c under t h e l o a d p o i n t has extended over i t s value  corresponding  t o maximum v a l u e  o f m, we g e t h i g h e r  s t r a i n s a n d a d e c r e a s e d moment a t p o i n t C. d i t i o n s i n order t o maintain crease  running  take place.  t e s t i n g machine w i l l  Furthermore, the  increase the deflec-  t i o n under t h e l o a d , which i n t u r n w i l l n e c e s s i t a t e a n g l e changes f o r c o n s i s t a n c y  con-  e q u i l i b r i u m w i t h t h e l o a d an i n -  i n moment a t p o i n t B w i l l  continuously  Under these  flange  of deformations.  positive  F r o m t h e moment  d i a g r a m shown i n F i g u r e 15b i t i s a p p a r e n t t h a t a l l t h e a n g l e •5 change w i l l value  take place a t point C over the ^ inch length.  o f t h e a n g l e change a v a i l a b l e u n d e r c o n s t a n t  The  moment i s  g i v e n by t h e e x p r e s s i o n  which i s a f i n i t e  quantity.  Thus when t h e f l a n g e s t r a i n €c u n d e r t h e l o a d extends over i t s value  corresponding  point  t o t h e maximum v a l u e o f  m t h e beam w i l l n o t b r e a k , u n t i l t h e f l a n g e s t r a i n €c r e a c h e s its  f a i l u r e s t r a i n corresponding For  to failure  stress.  c o m p a r i s o n i n s e c t i o n - I V two a n a l y s i s o f beam  t e s t s were made.  The f i r s t a n a l y s i s assumed a moment d i a g r a m  a s shown i n F i g u r e 15a and  t h e s e c o n d a n a l y s i s assumed a moment  diagram s i m i l a r t o F i g u r e 15b. 203.  T e s t Arrangement The  l i g h t metal a l l o y s a r e capable o f almost  p e r f e c t l y e l a s t i c behavior  b u t when s t r e s s e d p a s t  the e l a s t i c  49  limit  they  mentioned  exhibit very i n the  b e a m made o f by the  equalization of  at.  of  limit  tests  introduction, a statically  limit  design,  order to  to  test  not  ical  properties  not  were made.  are  load since load given  at  many s e c t i o n s  and a t to  the  case  center  plastic  given  the  (,.;:.shape .: ..."  as  fixed  test  o f beam  of of  will  inelastic limit  the  a fixed  develop  arrangement  same  r e d i s t r i b u t i o n o f moments  in  the  reached moments  plastic concentrated is  required  simultaneously under  can take  special  care  a considerable place.  order  limit-  t h e i r bending  end beam w i t h a  that  mechan-  bending.  design  resisting  Therefore, so  little  deflections  i n d e t e r m i n a t e beam i s  can "equalize"  ends.  Its  were used  n o r e d i s t r i b u t i o n o f moments  hinges  the  of  very  i n t a b l e .3.  measurements  theory  three  on f o u r by f o u r H beams,  aluminum a l l o y  theory  simultaneously  In the the  out  measurements  the  the  theory,  a structural material.  for a statically  developing  moments.  as  These  experimentally  conditions  when as  the  materialize.  carried  test,  According to ing  visualized  p r e d i c t e d by  experimentally the  i n tension are  During the  check  behave as  failure,  was  indeterminate  This d u c t i l e material which e x h i b i t s  hardening i s used  and s t r a i n s  As  More s p e c i f i c a l l y  4--\tL/fc.made o f h i g h s t r e n g t h  strain  of  will  f a i l u r e were  29001 Alcan).  to  design.  o f moments b e f o r e  In load  s t r a i n hardening.  such a m a t e r i a l w i l l  theory  theory  little  Also  it  the  must  be  amount is  50 d e s i r a b l e that the shear no  need f o r s t i f f e n e r s ,  inelastic  analysis.  f o r c e i s s m a l l , and t h a t t h e r e i s which otherwise  To f u l f i l l  t i n u o u s beam i t was n e c e s s a r y  The  these  will  complicate the  c o n d i t i o n s i n a con-  t o ilow'erv one o f t h e s u p p o r t s .  dimensiongof the t e s t  arrangements a r e g i v e n i n f i g u r e 16 and t h e  '  P  .  3  4-2"  geometric  &  46'- . | -I"  .  of the s e c t i o n are  42*  g i v e n i n T a b l e 3-  1  732' 7es/  the negative reached  properties  The  beam was  originally  0r-/y./?ga/not{  s e t on s u p p o r t s B and  F i g u r e 16  D, and a t t h a t  s u p p o r t A was n o t i n a c t i o n .  a c e r t a i n value  of approximately  time  When t h e l o a d P 8 K i p s , t h e end A  d e f l e c t e d upward and came i n c o n t a c t w i t h t h e s u p p o r t A. t h a t time  At  t h e p o i n t B on t h e beam was \ \ i n c h b e l o w t h e  straight  l i n e AD.  The beam span., o f 132 i n c h e s between  supports  was c h o s e n t o f i t t h e t e s t i n g m a c h i n e .  d i m e n s i o n ) b e t w e e n l o a d and s u p p o r t s  were p u r p o s l y  such  i n order t o decrease  thus  a v o i d t h e u s e o f s t i f f e n e r s w h i c h would have  outer  The g i v e n chosen as  t h e r e a c t i o n s a t t h e s u p p o r t s and  t h e a n a l y s i s and i n t e r p r e t a t i o n o f t h e beam t e s t . s u p p o r t B 14/' below t h e s t r a i g h t  complicated Having  l i n e AD i n c r e a s e d t h e r a t i o  Mc of  -—  i n the e l a s t i c  r a n g e , and c r e a t e d a c o n s i d e r a b l e  amount o f moment r e d i s t r i b u t i o n b e f o r e was  reached.  the f a i l u r e c o n d i t i o n  The beam section was so chosen, that i t s strength would not exceed the capacity of the testing machine. Limit Design Failure Load. Under the assumption that the beam is unaltered by the settlement and spread of supports, the value of the limit design failure load can be found quite easily.  Since failure  condition is reached only after equalization of moment then plastic moment M p under the load and over the support should give the limiting condition.  Consequently the value of the  failure load can be found from equilibrium.  From virtual  work equation:  Mi  Figure 18 With a plastic section modulus 2Tp of 5 . 6 ins^ and a yield stress Cy of 3 9 . 9 Kif>s/ £ m  beam  , the plastic moment of the test  which amounts to 224 Kip-ins • With L = 90 ins  the failure load P is from equation (22), 14.65 K i p s . The Mp outer beam reaction$at A and D are or 5.33 K|p5 . The  52 t o t a l r e a c t i o n a t the i n t e r n a l support D e s c r i p t i o n o f Support  The  At  and L o a d i n g  beams were s u p p o r t e d  01sen mechanical  i s 14.65 K i p s .  and l o a d e d  b l o c k 3/4" wide  on t h e t o p f l a n g e o f t h e beam was u s e d .  and  i n a Tinius  t e s t i n g machine a s shown i n F i g u r e 17.  the load p o i n t a simple  calculations,  Point  the f a i l u r e  l o a d expected  l o a d p o i n t i s 14.65 K i p s .  From l i m i t at the inner  stress  i n compression  In this (39.9  formula  ^ ?/ms i  -{; i s t h e t h i c k n e s s o f t h e web bearing  design support  I n o r d e r t o c h e c k t h e web  c a p a c i t y a g a i n s t c r i p p l i n g t h e f o r m u l a £p t (ex.. + 2k) t h e A . I . S . C . was u s e d .  resting  L  g i v e n -in  i s the f a i l u r e  f o r aluminum  alloy,),  (.255 i n s ) , a., t h e l e n g t h o f  (3/4") and k t h e d i s t a n c e i r o m t h e o u t e r f a c e o f t h e  f l a n g e t o t h e web o f t h e t o e o f t h e f i l l e t  (.625 i n . ) .  S u b s t i t u t i n g these values  we g e t t h a t  :(a, + 2k) = 20 .' K  against  P  i n t o the formula  Thus a 3/4"  crippling.  wide b l o c k p r o v i d e s  The i n n e r s u p p o r t  enough s a f e t y  was p r o v i d e d w i t h p i n ,  supported  b y r o c k e r a r r a n g e m e n t s on t h e b a s e o f t h e t e s t i n g  machine.  On t o p o f t h e p l a t e ,  r e s t i n g on t h e p i n , a 3/4"  wide b l o c k was u s e d .  T h i s arrangement p r o v i d e d v e r t i c a l p i n  support  h o r i z o n t a l forces being transmitted i n  and p r e v e n t e d  t o t h e beam.  xhe  r e a c t i o n due t o o u t e r s u p p o r t  a t D was  spread  o v e r a 4 x 3{ p l a t e r e s t i n g on a p i n and i s u p p o r t e d b y r o c k e r  53 a r r a n g e m e n t w h i c h i n t u r n was r e s t i n g t e s t i n g machine.  ( F i g u r e 1 7 ) . The o u t e r s u p p o r t  t o t h e b a s e o f t h e t e s t i n g machine made f o r t h e t e s t ,  at A attached  ( F i g u r e 1 7 ) , was  providing vertical  h o r i z o n t a l f o r c e s from b e i n g c a r r i e d  Beam  on t h e b a s e o f t h e  support  specially  and p r e v e n t i n g  i n t o t h e beam.  Collars. •io p r e v e n t  l a t e r a l buckling of the  f l a n g e s , between, {the o u t e r s u p p o r t  compression  and t h e l o a d p o i n t , two  c o l l a r s were a t t a c h e d t o t h e b a s e o f t h e t e s t i n g which prevented movement.  lateral  machine  i n s t a b i l i t y and a l l o w e d v e r t i c a l beam  The c o l l a r s were made o f a b o l t e d frame work o f  a n g l e s and p i p e s  a s shown i n f i g u r e 1 7 .  Measurement o f S t r a i n s  of Deflection.  E l e c t r i c a l r e s i s t a n c e type s t r a i n gauges, gauges and s u r v e y o r ' s l e v e l were u s e d deflections  of the test  symmetrically  beam.  t o measure s t r a i n s and  S t r a i n gauges were l o c a t e d  on t o p and b o t t o m o f t h e c e n t e r l i n e  f l a n g e i n o r d e r t o measure t h e b e n d i n g  strain.  shows t h e l o c a t i o n o f a l l s t r a i n g a u g e s . that  the t e s t  beam was s t a t i c a l l y  i n t h e beam c o u l d n o t be f o u n d a d d i t i o n a l measurements. at the  directly  of the  F i g u r e 18  Due t o t h e f a c t  indeterminate from  t h e moments  statics  without  T h e r e f o r e , s t r a i n gauges were p l a c e d  s e c t i o n s where o n l y e l a s t i c b e n d i n g elastic  dial  theory o f bending,  would o c c u r .  t h e moments a t e a c h  Using section  54  could be determined since the flange strains were known. Thus, the extend of equalization of moments could be determined. To measure strain due to shear, strain gauges were placed at 45 degrees to the beam axis on opposite side of the web and right on the neutral axis. As was mentioned i n the introduction one of the purposes of the beam test was to test experimentally the accuracy of the deformations predicted by the inelastic bending theory.  Federal dial gauges  were used to measure beam deflections under the load. Extensive deformations were measured by using the surveyors level and sighting on a steel rod. to  The steel rod, reading  of an inch accuracy, was placed on the top of the  head of the testing machine.  55 Table 3a.  Mechanical Properties of Aluminum 65S-T6 i n Tension  (from test shown i n F i g .  )  Modulus of Elasticity-  9,540 kips i n f  Proportional Limit  36.3  "  Yield Stress (0.2 per cent o f f set)  39.9  "  Ultimate Stress  43.4  "  Strain at Proportional Limit  0.0038  Strain at Ultimate Stress  0.09";  Table 3 b Section Properties of 4 'in. H beam (29001 Alcan) From measurements of the beam section d = 4.00 i n .  Y  b = 4.00 i n . t  w  = 0.255 i n .  t f = 0.313 i n . f i l l e t radius  T  = 7/16 i n .  A = 3.51 i n f X  -x  The following properties are needed for calculation: h = d-tf = A = t h w l w  3.688 i n .  =  w  0.94 i n f  Af = 1/2 (A-A ) =  1.294 i n f  w  =  2.752  x-x  =  9.74 i n f  Z = | I d x-x  =  4.87 in?  =  5.602 in?  K = Af/k J  w  r  K  =2/ y - A d  Sk—s  V  56  58  • ^ i  •^i fPl  j!|N  I,  i  it©  VO  0.  :.'*0-  CO  5  Til  3.  >:  I N V 3  N  1  0  <0  59  PART I I I BEAM TEST. RESULTS In order t o have an i n d i c a t i o n of the extent o f r e d i s t r i b u t i o n o f moments, v i s u a l i z e d by the theory o f l i m i t design, three h i g h s t r e n g t h aluminum a l l o y beams (designated 29001 Alcan) were loaded t o f a i l u r e .  The observed  failure  c o n d i t i o n s , d e f l e c t i o n s and moments o f the t e s t beams were compared a g a i n s t the p r e d i c t i o n s o f Dr. H r e n n i k o f f ' s theory of i n e l a s t i c 501.  bending.  R e s u l t s of Beam Test N o . l . S t r a i n and d e f l e c t i o n readings observed  t e s t N o . l . a r e given i n t a b l e s (4) and ( 5 ) .  during  S t r a i n gauge  readings, l o c a t e d symmetrically, on top and bottom o f f l a n g e s , were w i t h i n &fo agreement.  I n computing the t e s t r e s u l t s the  average top and bottom value of the s t r a i n s was used. The dimensions o f beam t e s t N o . l . a r e g i v e n i n f i g u r e 19. The  L o c a t i o n s o f s t r a i n gauges a r e g i v e n i n f i g u r e 18.  overhanging  portion  m a t e r i a l the p o r t i o n  was  .  In order to economize  was reused i n t e s t No.2.  The beam was o r i g i n a l l y s e t on supports B and D, and a t that time the negative support A was not i n a c t i o n .  At the  60'  load of approximately F  c  D  3-6"  4-'-o" -f-  8 Ki'ps the beam came in  contact with  support A, becoming statically  indeterminate  with elastic deformaFigure 19 imately 9 K i p s  tion up to approx-  * After an increase i n the load above 9 K i p s ,  inelastic strains were observed i n the flanges under the load and the beam began to deflect more rapidly with an increase of load. As the load increased beyond 9  K'ips  the flange  strains over the load increased rapidly while the remainder of the beam stayed elastic.  At a load of 13.5 K7ps, the  strains of the flange above the inner support were close to yielding but s t i l l elastic.  At a Moad, of 13.7 K)ps the  compression flange of the beam under the load crippled. Figure 17.  Apparently this failure condition was caused by  excessive compression stresses present in the flange and the unfavorable orientation of the flanges under the load. Figure 21. The large angle change under the load point created enough eccentricity to cause buckling of the flanges which were completely unsupported. The vertical component of Figure 21  I  60>  the stresses deflected the flanges i n a manner shown i n figure 2 1 / b . cross-section. The premature failure of the test beam caused by crippling of the compression flange was not the type of failure expected and the bending failure condition was not reached during the test. The strain gauges for measuring shear strain were located at the neutral axis, under the load and over the inner support, making an angle of 4 5 degrees with the center line of the web. These strains throughout the test remained velastic.  \  ^  This decreased the section modulus of the  '  Moments of Beam Test No.l. As already pointed out, the theory of limit design  predicts that before the mechanism condition is reached the moments under the load and over the inner support equalize. In order to see whether the predicted redistributing of moments did take place, the beam moments at the approach of failure had to be found. The measured strains for determining the beam moments at each stage of the loading are shown i n table Afor the 1 2 gauges. Gauges ( 9 ) and ( 4 ) , located under the load and over the inner support, were not reliable for computing the beam moments due to the effect of the concentrated  point load and support.  Comparing the strain readings of  symmetrically located strain gauges such as ( 2 ) and (8) we: observe that there i s as much as 8 percent variation.  There-  fore for determining the moment at each location of the gauges, the top and bottom flange strain readings were averaged. Using the conventional elastic theory, the moments at the locations of the gauges can be calculated as follows:  (23)  € EZL  where & is the measured average flange strain, £  i s the  modulus of elasticity and Z i s the section modulus. Neglecting the spread of load and supports and assuming a moment diagram of straight lines, the moments at any point on the beam can be evaluated from the moments at the locations of the gauges. Before going into any'further analysis i t i s advisable to investigate the accuracy of the moment$,at the location of the strain gauges calculated by equation ( 2 3 ) . The section modulus i s a geometric property of the cross section area of the test beam. Its value was found to be 4.Ql>ni  . The possible error is negligible.  The modulus  of elasticity E, i s a mechanical property of the material. As was observed in section II., i t varies with location. The modulus of 10,400  K  ips/ $ In  z  i s a value determined from  63 t h e beam d e f l e c t i o n s .  Not a g r e a t d i s c r e p a n c y from  modulus w o u l d be e x p e c t e d  this  a t a n y beam c r o s s s e c t i o n .  The a v e r a g e f l a n g e s t r a i n , 6 , was measured by e l e c t r i c a l r e s i s t a n c e s t r a i n gauges.  One s o u r c e  i s t h e d i s c r e p a n c y between t h e l i s t e d  gauge f a c t o r  m a n u f a c t u r e r ) and t h e a c t u a l one. a + 1.5  percent  to  (by t h e  The m a n u f a c t u r e r  e r r o r i n t h e gauge f a c t o r .  p o s s i b l e e r r o r may a r i s e  of error  listed  The o t h e r  f r o m i m p e r f e c t g l u i n g ; , o f t h e gauge  t h e beam.  Consequently  able',  the only source  o f error of consider-  m a g n i t u d e i s t h e e r r o r due t o f l a n g e gauge In order t o eliminate t h i s  readings.  e r r o r , t h e beam moments  s h o u l d be c a l c u l a t e d b y t h e f o l l o w i n g manner. 1.  Assume t h e a v e r a g e f l a n g e s t r a i n £ i s i n c o r r e c t  by a f a c t o r o f  which i s constant  throughout  the t e s t .  T h e n t h e t r u e moment w i l l be g i v e n b y t h e e q u a t i o n :  M=^ez  (24)  where pie a r e t h e a c t u a l s t r a i n s . 2. the product constant,!^,  S i m p l i f y t h e above e x p r e s s i o n b y s u b s t i t u t i n g o f the t h r e e constant  single  so t h a t t h e new e x p r e s s i o n f o r moment w i l l b e  as f o l l o w s  »A  is  A7where  q u a n t i t i e s by a  i s i n KIP-ins  f  £  Ke i s i n percent  (25) c..{ /SJ?A?J^-  and(fis i n  64 Kip-ins.  The  d u r i n g the of K, the  constant  stage  which i s  moment a t  of  s i m p l e beam a c t i o n .  assumed any  K c a n he d e t e r m i n e d  to  be  further  constant  from the  After  the  throughout  l o a d i n g s t a g e up t o  moments evaluation  the  failure  test, can  be  determined. Calculation The equation were at in  the  evaluation  3  tively,  z  0.163  and £  Using the  for  to  equation  ^  2  M  locations  of  versus  at  the  K"„ Kz  and K 3 , of  (1) (2) (3J w h i c h D were  determined  of  a n d b o t t o m s t r a i n s 6,, (2), and  (1),  load.  7 Kips  from the  Figure  the  graph,  same  locations  (lj  (2) a n d  \C -  07^— ,183  ~  0.2215  -  78.4  0303  :  22.  load  respectively.  M  •  respec- .  strains  w e r e 0.183, 0.2215 a n d  the  Kip-ins,  (3)  corresponding  At  =  *3 -  top  locations  a load  a n d 78.4  Yi  constants  from support  respectively.  moments  that  to  average  , measured  3  percent  97, 112,  we g e t  the  were p l o t t e d  determined to  the  No.l.  manner.  corresponding  Thus , £  of  30 a n d 66 i n c h e s  26,  following  £*  of Test  (25) c o r r e s p o n d i n g  First and  o f Moments  6  - 530  Kip-ins  507  Kip-ins  481  Kip-ins  the  statically  (3) w e r e  equal  65  After determining K'I , \< , and « , between supports B and D, .. z  3  the evaluation of the remaining constant coefficients K^and K5 (corresponding to locations (6) and (5) at 26 and 30 inches from support A, respectively) was determined using the following relations: If  Ji'Mt.  and  g  Jii  = M - H Mc+-- • •  (26)  5  =  cds -H d - h -  • •  6  (  2  7  )  where Ms-, M.& andMn are the moments of sections located between E> and £ and at a distance of  » cU ando/ respectively from, n  outer support D.. -  then  ^Mi  ^/7^  j£AJ  =- R.D£JLI c  = £ 5 ^ 5 + £*/(6 +-• •  (28) fen^n  (29)  where r?D i s the reaction at the outer supportD. Equating (28) and ( 2 9 ) we get that Rp£dl  Since Ks »  = £sK5 +- e&K<,+  6nKVi  and Kn are approximately equal to each other,  then they can be replaced by an average value of K such as K whereK can be the average value of the coefficients of such locations as (l) and (2) between the load and outer support D. then t  h  u  s  (3D  K =* KoJtdi-  6 R+ S  e*K+-...-  eo*<  (  3  2  )  66 or  £<J-l  &D  where  <^€i  •Then  f?o K  but  o  =  "  -  €  *  (JJ)  K-^£L  +  £ & 4  (34)  € n  K  (35)  ~HrV  (36)  Thus equating 35 to 36 and s u b s t i t u t i n g Mn by fCne^we get the equation ^ . d.»i -^e^L Solving f o r K v ) w e get t h a t :  Kn =  K  k  (37)  4lL &  (38) '  Thus formula (38) can be used to determine any constant c o e f f i c i e n t between A and B such as K G and Ks , provided the value of K has been predetermined by averaging the c o e f f i c i e n t s of such l o c a t i o n s as (1) and (2), between C and  D.  For t e s t N o . l .  ,  » and  was found t o be equal  to 530 K i p - i n s 507 K i p - i n s and 481 K i p - i n s T  h  e  n  j ^  =  J<i±i^ 3  =  530 + 507 + 481  m  3 Q 6  K i p  respectively. _.  n s  3.  To e v a l u a t e , the average s t r a i n s , € a n d 5  correspond-  ing to l o c a t i o n s (5) and ( 6 ) r e s p e c t i v e l y , were p l o t t e d against the load.  Figure 23. Thus f o r a load of 10 Kips the  corresponding average s t r a i n s , £ 5 and £ 4 measured from the graph were equal to 0.023and 0.033" percent r e s p e c t i v e l y .  Then  =  Ut-^s  =  Jfdi = d td(,  and  Thus u s i n g  K=  equation  After  x  0  ,  °|  g  :  , Hi  presented  i n the  (2),  (3),  spread  of  straight  following  = 5fO  Kip-ins  constant  I'd  , Ka , f £  throughout  |<T^ a n d  3  the  test,  » l^r a n d M g d u r i n g a l l l o a d i n g were  computed.  These  the  i n the next  computed  moments  corresponding to  '  stages  computations  a table  (5) a n d  were f i r s t possible  for  page  are  (see  the  dn  distance  support,  location).  effect  a b e n d i n g moment  e a c h l o a d i n g s t a g e was  straight  sketched  as  the  locations of  diagram  plotted  in  manner*  through the  coordinates  (6) a n d n e g l e c t i n g  l o a d and s u p p o r t s ,  which determined,the  outer  56 i n c h e s .  form of  lines  The  was  =  percent  Kip-ins..  502  3  .0621*  6). Using  the  be  ,  failure,  of  get  =  coefficients,  to  and up t o  (1),  we  |°  Q  the  assumed  m o m e n t s , M;  table  26  (38)  x  l  evaluating  which are  30 +  x -§|j=  (5Q6)  Ks = 506  the  =  s  .033-  .0E9 +  l i n e s A E , ED, and E P , f i g u r e  shape so  of  that  the  they  plotted  the  as  the  diagram,  close  determined by  such  for  the  dn  from e i t h e r -  c o r r e s p o n d i n g moment a t  an a d d i t i o n a l a i d  as  previously defined  s t r a i n gauge measured  a n d Mn w a s  Then as  c o u l d pass as  points,  , a n d Mn ( w h e r e to  b e n d i n g moment  24,  that  constraction  68 o f t h e moment d i a g r a m Considering load  t h e f o l l o w i n g r e l a t i o n was  t h e f r e e body d i a g r a m  o f beam segment u n d e r t h e  s u c h a s t h e one shown i n f i g u r e  Figure  24b  24  and w r i t i n g t h e e q u a t i o n s o f e q u i l i b r i u m , following  used;  we o b t a i n t h e  expression:  VL + V * where VL the  (39)  and V ^ a r e t h e s h e a r f o r c e s  r i g h t o f the l o a d  t o the l e f t  and t o  P.  R e p l a c i n g the shear f o r c e s  i n terms  of the  moments we g e t :  (40) w h i c h can:' be r e a d i l y e x p r e s s e d i n more c o n v e n i e n t as  or  D  m  m  tv\  -4-2.  -4g  form  (41)  = 2 2 A P  where m  i s the distance  diagram  o f f i g u r e 24.  (42) FG  shown i n t h e b e n d i n g moment  T h i s a d d i t i o n a l c o n d i t i o n was  quite  "Vv  v>  -V  •S  s N.  vj  i  ^ Nl  N:  0 Nt Vl  V; >  <X vl  •  N  *Q » x>  \  o 0  NN X  V*  Ift; 1  Ci  1 X  •ti  E 0  §  o  0  X\ X  ro o  VO  X  Ni  •«  Ni  8  0  N:  §  N  X X  "i  Ci  1  N»  «i  <i  «i  NV  V \  VO  "X  « x*  vv  V:  x  X l«i  S  VP  0  0  M  N.  I  N  XI 3  N  rs Oi  N Ml  «i  4i  N  In  •3  •?  X X  N \  I  N  ^N  ?  Ci  VO  N  •A  X  V  N.  N  •«n  N  Ni Ni  O O  x  \  <a  Q  Ni  0  ft  X  \  I 1 N\ N  VI «i  NV V  A  N \  U  Ci  0 \  O  N!  o  Ni  Ni X> X,  8i  2.  ^.  X?  Vi Q 0 N.N N  <*>  § 0,  55' vo Q)  •  *s >•«.  Q <0 O >»  % N N.  «Q "i <N  to  < 4 \  N.  §  1 —  M «VJ <M N  s  i  I  N  0 0 vi  vv,  *>  «i <©  X~  x  X •V  V  vi  v»  ; NV  %  §•  •N  Q tN  <Ji  "i  «i  rN  vo  "i  •Vi  VO  V  K  m  O  Q x'  X  N  • VTi \  *i  S  N Vo N Ov  *  1  o o  X X N  *  1  0  ft  XV .\  1  0  !  1 N  1  io Oi  <s «i <vr  ? X\  0 X  O) VO  s <h VO  X  *• a) Ni  Ni  «vi  s  vi  a  I  $ Vfl  X V  N  *\  N vo Cv  N  Vo  &  1  2  Q (i  0\  Oi  s \  N  X  X  R 15  X X  X X  Ni  O Q  o>  §  s  vi <0  3  ^  I  <^  *»N  8 !?  o  N.\ \  Oi  vo  vi K  X  NV  «>  N?  •  5  %  o\  x  m  X  Ni  §  N X  vt»  X  fN.  N X  vo  to «i  O ' «\ VTi .  $ vo  s VO  1  x^ «v  X  v\>  ft  N  X  N  X  <vj  X viv X  ! Oi  l\  X <i  rx  N  ^X X  ro  Oi  *i  l»v.  a X  x  *vj  *vj  N  "\  Ci  vO  X  0 «i  5? Ci  N.  Ci N N  "I N X  vfl  M  X  !  <^>  Ni  5?  Ni  ff)  vo  vp  X  X  X (Yi  s  vCi >>  \  Ci «\  •5  Ci  «\  I vt  X  N N  H  IN ?  Oi  <i  2 isX <5  •»i  1  IN  X*  s  X  •Vl  N  flv,  «>.  X  X  vs  ^  ,  N  X  X  Ci  'X xV  ?  vo >  «0 Ci  •vj  !  $ x  X X  Ci  X  N  <N  VO  Ni  vo  ?!  X X  <n  X  x VX)  cs 5!  Ci  I  °v| «i  0  X <i VO  ov. NCi  X N  X  Q  N  "i N Ci  3  10  Qo  V  x^  X . X  v»  X  VI X X  Vi  fli  N  X  Ci  X" X  >Vi  I  "0 N  N  VO Pi N  ( * > *)  t  X v5  s  ix«i Ni \  N  fx  X)  11  v« In  1X ! vo  !  X fi  1  *)  Ci i»i  l*i  ? ?  s  1)  X  N  Oi  S ?X ?  N «0  X VJ °\ vo  (»» N  si  Ci  to fv]  I  X  5  S  N  V  N  $  x-  v&  K  Vo  vi  *1 ft  1X X  vo  jo  vo  Nv  Ci  Ci Ci  it Ni N  K  «i  IS N <&  0 li  N  N  V  <V1  X  X  «i  0  VO  Ci  0  X  N  Ci  k  &  N  Ci V*  0\  X  "S  <s\ \ N  W  <VJ  «i  Di  a)  x  x  VO  X  Ot  N  X  ft  v> 0  o  IX  N ft  Oi  vo  D  8 " •*>  JN X* N «i K X  «\.  Ci  X N %  o» OQ Ol  v»  X X X  cb  g  X UV  x  (VJ  5)'  «i  •3!  ?i  ov  v»  X  X  to X  Ni  ro  «i "i  V  <o  0)  x-  ?  0  o  N  Oi «i  ?  o  vfi  «>  x  M Ix  1  N5  N5  S N  2? «v.  w  X  N  3  X  81  > 0  Ci ti  ?'  i*J  N.  n b  Hi  Q  VO  Ci  3  X Ci  Ci  o>  X X  !  X  1 k  1  Oi  K  X  x^  rx  X XN  Ci  | ! X 1^  88.  x  X  ix  X  Ci  !  X  X  Ni  (0 VO  N  rN «i  o  vO \0  ! 1 !X 1X  r*  o\  ?N  «  3  Ni  X  X  tvl  X  *i  !  8 v)  vo  v»  X X  0\ vt VO \ N  rs  ?v vo  6  •ti  N.  <*.  X  vo  V)  "V  N  $  X  0 N  in  <vi  N  Oi  Si  VO VO  vs  X  X  *»»N  vi  0i  Nv  vi  <\  Oi  >  *>  \  3  *i  «i N <i  Ni  X X  N  <»i  Q  \  «\ <i  Oi  Ni  "I  X*  NN  X  s  3  N  SI  <n  v>  X v*  '§  VB  N  a  O  % Q  CO  •  Ni  X \  I  VO  $ N «5  wv  N.  N  3 «v  N  «i  SI  vo  X  I  •v  "v,  N 10 V*  X  vi  X N  N Cl  i  x  X  1  Ni  Ni  •V < V 4  X X  m CV  VI  X  •»o ")  3 X  X  & <•>  VO  av.  \  VO  Oi  0  N  vi •0  «i  «vj O Ci  rx  X X  o  s x'  X  T2  /  o cuoL  C/bs)  a  6>  ^  2  2>SOO  9  3OOO  .' 6  3~O->0  JS  isn a/er X o  Sec? 4-  Tfemari: s  oaC  50  BOO  0  /£> OO  .3SO  &76>  .3 2 4-.  .736  3 OO  . VOO  .7^5&  7/.S  O  • 700  7000  s s>  vsoo  2  73 30  -407.3-  2. 6  83SO  3JS-3-  723*2.  7 2D 2  33  3350  2 03. s~  /.-^-<z 3  7.3 3 /  -3~6  AS £>/  2J50  S>ao  7S-9/  / c? 0  2JZO  2.Z £ /  .370  6  36.  /0300  47  ^6  OiDO  ,*4  £03.ST  Majk/rra  7/4 5  7.4-SO J>7ar/s 0  7-4-30  77-eset  7 94D 7 S> &sr  -50 S2  4 76  24 O 7  4 76  -36 37 GO  /reset  7  2 ^  2 7 72  22/s  O  ^3S-  TA  B LE  5  79  £> / 0/  /. & 74 ^  2>2  JST 3-5-  a.  J, 1  MOHElslT CQEPUTATIOITS OP TEST U O . l  Load  K,  M,  M  2  pa rcar> t  5  K-;r>s.  K. ,v*r.  percent.  0.2265  530  120.5  0.2745  507  140  0.1842  481  88.5  10.450  0.2403,  530  127.0  0.2920  507  148  0.1850  481  12.050  0.2542  530  135  0.3097  507  157  0.1480  12.5  0.2525  530  133.8  0.3084  507.  156  13.0  0.2490  530  132  0.3048  507  X3 • 5  0.2414  530  128  0.2980  507  K.  9.550  €  3  ,  €  5 ,  *r- int. K-;»i  M-Jnt.  percent  • 0.0245  50.6  12.4 0.0215  89  0.0427  506  2;i; .6 0.0374. . 5Q2 18.8  481  71.2  0.1149  506  58.4  0.1005  502 50.. 3  0.1229  481  59  0.1504  506  76.5  0.1314  50.S 66.0  154.5  0.0895  481  43  0.1934  50.6  2 98.0 0.1693  mz  8a 3  152  0.0457  481  22  0.2477  506  126.0  50Z  109..  f<l-/nS,  T/?3LE  tot.  6.  t-tns.  0.2166  .50.2 1©.8  78  89  81  valuable i n improving the accuracy of moment diagrams, which i n turn provided an i n d i c a t i o n of the magnitude of the moments under the load and over the inner support during each loading stage up t o f a i l u r e . Moment diagrams of Test No.l. f o r each loading procedure are shown i n f i g u r e 25. Also the extent of e q u a l i z a t i o n of moments under the load and over the inner support, (derived from the p l o t t e d bending diagrams) i s presented  i n the form of a graph,  f i g u r e 26. 302.  Results of Beam Test No.2. Up t o a load of 13 Kips, the beam of t e s t No.2.  behaved s i m i l a r l y to the beam t e s t No.l. 9 Kips the beam deformed e l a s t i c a l l y .  Up to approximately  As the load increased  beyond 9 K i p s , the flange s t r a i n s under the load increased rapidly.  At a load of 13.5 Kips the s t r a i n s of the flange  above the inner support were close to y i e l d i n g but s t i l l e l a s tic.  At t h i s stage, signs of c r i p p l i n g of the compression  flange were 'apparent. s i m i l a r to t e s t  Thus to prevent an e a r l y f a i l u r e  No.l. two 4"" b o l t s were inserted between  the top and bottom flange on each side of the web where the signs of e a r l y c r i p p l i n g were observed. At a load of 15,450 l b s .  the beam f a i l e d  by f r a c t u r e due to bending under the load. f o r t u n a t e l y the l a s t >readings were taken  Un-  82 when t h e l o a d v a l u e was e q u a l t o 14, 500 l b s . a complete set  o f measurements  testing a third  and c r e a t e d  This  prevented  the n e c e s s i t y  of  beam.  S t r a i n and d e f l e c t i o n r e a d i n g s  observed  t e s t N o . 2 : a r e g i v e n i n t a b l e 7 , and 8  during  The l o c a t i o n of  s t r a i n g a u g e s a r e g i v e n i n f i g u r e /;&<. C a l c u l a t i o n of-Moments of Test  Beam N o . 2 .  The method u s e d - i n d e t e r m i n i n g t h e moments T e s t Beam N o . 2 . was e x a c t l y  the  of  same a s i n T e s t N o . l .  The e v a l u a t i o n o f t h e c o n s t a n t s K\ , K z , and K 3 of equation  (25) c o r r e s p o n d i n g t o l o c a t i o n s  ( 1 ) , (2) and  (3) w h i c h w e r e a t 26, 30, and 66 i n c h e s f r o m s u p p o r t D were d e t e r m i n e d i n the  f o l l o w i n g manner.  F i r s t t h e a v e r a g e s t r a i n s , e, €zt and 6-i t  ing to locations against  ( 1 ) , (2) and (3) r e s p e c t i v e l y ,  the l o a d , see f i g u r e  correspond-  were p l o t t e d  27.  Thus f o r a l o a d o f 7 K i p s t h e  corresponding  s t r a i n s 6, , 62. and £ 3 m e a s u r e d f r o m t h e g r a p h w e r e 0.208, 0,242 and 0.167 p e r c e n t  respectively.  U n d e r t h e same l o a d  t h e s t a t i c a l l y d e t e r m i n e d moments a t l o c a t i o n s and (3) w e r e e q u a l t o 97, 112, and 78.4 K i p - i n s Thus u s i n g t h e  equation  ( 1 ) , (2) respectively.  83  f<  =  07208" "  =  '4 0.163  4 6 6  K  T  inS  4 8 I K'p-'  78  =  After determining Kt  ,ns  , >\2 , and K3 between support? B and D,  the evaluation .of the remaining constant coefficients K g , , K& and Kg" corresponding to locations ( 8 ) , (7), (6) and ( 5 ) which were at 10.8, 14.8, 26, and 30 inches respectively from support A, was determined exactly the same way as in test No.l. Using equation  i\n =•  any constant coefficient between A and B such as Kg , KV » f<  fc  and ks  can be found,provided the value of K has been  predetermined by averaging the coefficients of such locations as (l) (2) between G and D. For test No.2. K, » \<z and  were found to be  equal to 466, 463.5 and 470 KYp-ins respectively then  =  K,-rKz *fc-s_  466 + 463.5 + 470 , _ .  3 To evaluate and  & 6  5  k  -  ^ the average, strains £  , € , 6& 7  1=5- corresponding to locations ( 8 ) , (7), (6) and ( 5 )  respectively were plotted versus the load, figure 28. Thus for a load of 9 K/P  s  ^  e  corresponding  m  average s t r a i n s , graph  , (  , £  7  were e q u a l t o 0 . 0 0 8 6 ,  and  6  65  0.0116,  measured f r o m t h e  0.021,  0.024  percent  respectively. £ + £ +€c+&s = 0 . 0 0 8 6 + 0 . 0 1 1 6 + 0 . 0 2 1 + 0 . 0 2 4 = 0 . 0 6 5 2 percent  Then  -  also^L,.  = dfl+d +d*+cls- = 1 0 . 8 + 1 4 . 8 + 2 6 + 3 0 = 8 1 . 6 i n c h e s .  8  7  7  Thus u s i n g e q u a t i o n \s \\5  - Afifi * 0 * 0 6 5 2 = 466.5 Q^g  v  — A&fi, ^ 0 « l 0 6 5 2  U" KV  — A66 R 0 . 0 6 5 2 = 466.5 81.6  K  _ Afifi s Q.i0652 = 466.5 gjffl  e  (38)  we g e t t h a t  =  x Q - 1?_ 024 Y  26  v  x  _  _ x  458  14.8 0.0115 10.8 0  t  0  o  466 Kip - i n s  a  6  _ =  466  484  A f t e r e v a l u a t i n g t h e c o e f f i c i e n t s , K, ^  2  » fC » for , K& , 3  ,' and kg w h i c h a r e assumed t o he c o n s t a n t  the t e s t , all  ,K  t h e moments, M  , M* » Nl, ,Ms t M ^ M r a r i d M f t d u r i n g  l o a d i n g s t a g e s and up t o f a i l u r e ,  computations table 9  are presented  throughout  were computed.  i n the form  o f a t a b l e (see  ).  U s i n g t h e v a l u e s o f t h e computed moments ing  to locations, ( 1 ) ,  bending ing  These  (2),  (3),  (5),  moment d i a g r a m w i t h s t r a i g h t  (6), lines  (7)  and  correspond(8)  a  f o r each l o a d -  s t a g e was p l o t t e d u s i n g e x a c t l y t h e same method a s i n  test No.l.  86  ft  00  *J  Ml xj rt,  Ci Ci  1  xi X  I oi  «0  «0 X  o «» Ci  vo «v»  Vfl  I  X  ? 1  N X  s> Ci  N  XV  t> CJ x  o XV t-  cv Ci  0 tx <o v*  «i  >  v» Vo i «i  cr. Oi «i  N O  a  X  VO Ni N  Cv Ci Ci  Q IX  r> VU  X  X  X  Ci Ci r0. x  •  VJ  <a Ci  X  I  3  C O vo  X  IVi N N  vo  va  X X so X  $  VO Ni  VO  «i  «\  ti  X  N N «vl  X. X  XO V  X  Xi v«  TV  Vo  "0  X  X  CT\ «i  Ci Ci  3V  ti X  vi O  N  x!  «i  N  VO  VO  vS ON  X  I 1  X  I  Ci  I  3  ^  IM  S Vt  N  O) . «i v,  X -o  X  VO  v*  "o  Ci ti  X  I  «i >0  X  R  X X \  *  . 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X  K  *  Vb  <0  x  0  Q CJ  c<?  x  r>  Vjj  s<xl>  1  5!  Ci  MI  lx  IK  ci Cx  vD  «i X  « ft  {X  xV  > Oa 01  NX X*  *x  rx  X "* <ft  !  5  ">  « "1 5}  Xx,  •o <h < N X  ?  X  x «i rx  N COUi Ci « oo a> 0 cv 0 i ft ft tv vO C Ci ti N N x N -\  N  Ci  I  VO  ti  X  00,  vu  IN  ti  «o  1> v<oo 1 3 5 00 <vxt vo I«v Ci N"0 <Vl 0i Ci Ci CO o v «v •o X, a > Ci 0 01 i Ci ti Ci tv N «v Ci \ «i  N.  v* <»i «i  ti  <VI  X »i  X  Q <)  Ct  N  «t .  ^«  C^  fl N  tx  1  § X  IX  «»  X  ti X  ti  -0 x?  !  ^8  vo  x  vX X X N X  fx  Y  ti  X  X X X X  Is  —  X  s  X VO N X  X  VO 3i  xi OJ 2)  X  !3  vs  vp  .N  ">  X  I  JN  X X  vi  $ 1 ti i  X  N  X  l 1  Ci  $ Oi  io * ft •0 1 Oi fx  «\  «M . X -N  X  X  N O Oi  Ml  N  OV  '8 N N  tx  O-U  Ci  Oi  ti -  NO  X  X  xV OV N  *, *o X i  %  v»  <V|  VX)  k  VO  v»  x.  «V1  X  X  X  ") •0  X  v> o  -i rx  ti .  VO  vo o»  Vo  X  X  <VJ  vJx  Ci  $  X  txv  Vvi  X X  iJ  5!  v* 0)  <JV  X «V| ty, <  1  X  X  X  M  X X  v^  r^  X  *>  X X <V4  X  oo  X  X Wi  X  X  fx  Co  ?!  Q  XV  00  NV  xi  Ci X  X  Ni  I  -  C» Vi  v\» e Nl «v vO Ci Ci Ci a ti vu V0 5-) <sOi \ cx N CXO. V0 Cv 8 5 5 1 ! I <o Ix Ns R o •f X v 5 <*) 1 1 CN *O 1 1 00 % 3 ! vo vo ^0 •0 VXJ O Ci i vt est •0 1 8| av v o ) % cx C i f•00 v*> ! > 1 IN1 a i *i C l <V| o« •o o » CHi < 0 vu 10 8 i 8 S i *V| ti Ci C i <vj ft Ci, 1 !«0 1 > «l ? m 0 cn « Vo 'o Vo <x «VI «0 > 0 oi CO "v> «0 »0 vO *vj X 8? < v <  1  Ci  K  a  I  ov »> m <vj OV  8  ^?  Ci  oo  Oi  •N  N  1 X (N  s  X  X X  sXT x N«v Ci ft m 0\  wXt xjf 1  v» x» VO' Ci 1  «s  IX.  t>  S  «V|  V  K "0»  tx  X  X  K  ^ Ci  X  ">  <X  I  «0 VO X  Ci X  X  X  X  X  i*) «>  VO  t-  X  ^*  tv  N  X  X  X •  CQ  88  O  load  ujia/er  2S-  SCO  2  2  ^  4  2  407.S"  74  2S> +  /s  SO  3 S4-  / 7  £>. 0 o  20  3. 3S~  i  Ws  2.085-  722  / / 3S~ 7  2S3T  2^. 49  7  2.S/  7-323-  77&/77£?/'/rS  .  40.  360  jp a ao, e 2 ^ms e t • 6 £2.  243  22  /•  ,33  72  OS  2.  /  Sf  /ooo /S>£ •4-/  /.3  3 3  7.  4&8  7  2.  043  3-3S*  >t "  /SoO  3.4-OS  */ ff // ff  /s //  gouge  4-43  /~eset  2**reset  /. 3  7.6 00  S^O  ^2.&S6  B2o  ^2.9  <57o  3.02^  2. /^J>2/65  7 7 3.S  743  S  3  <SS.S  7.042  jge>"//g <r / or>o/ 2  /ooo 3S>  2.449  3  2-277  s 4. 2 2.  32  2>. 2 73~  2.  72  22  J.  62  ^  3-7>T  4.  S/ rs  3~  2. 0 3S-  /' SS  5.  34 3£  -4.  o43  //3~  1, „  ~7~^A 23 2 £  2.6  77  / ooO  S S c><y  srso  3./2  e  7' oo'^ it  y  373£ 4s<s:sS>  ~3~. 3 3  S70  2&/S  3.  /*£-  -^23  39  7 5  6  6 3> 8  .£43-  SS S.'  62  A  P  gexo_ge7  4JS2  /3.S  ^6,  /'from  &  .3S>S  /. 6-40  S  //•  /3. O  !  .392  /  o  7.245  0  c  Sao  J7  \44  *  by /. eve/  goo  /O  i I  £>/a7  /OOO 70S  &  32  Aj,  S  3~0  25  P  3./9o  e»«>_g e 2  O CP J.20Z  <4U5  3 a/  7es/  /Vo-  £.  reset  89  MOMENT COMPUTATIONS OP TEST HO. 2 .  Load £/'p  pf  ami  K  percent  kT/p - 'Oi.  * , kr.'/* - "is. parctnt 6  M *3 kT-ins. AT- ir>i. Perczo t 3  8  0.2340 466 109.0  0.2715  463.5  125.7  0.1810  470 , 85  9  0.2543 466 118.5  0.2947  463.5  136.5  0.1863  470  10  0.2754 466 128.2  0.3193  463.5  148  0.1881.:  470 , 88.3  11.5  0.2964 466 138.  0.3445  463.5  159.5  0.1649.. . 470  13  0.3058 466 142.5  0.3560,:  463.5  165.  13.5  0.3087 466 143.8  0.3602  463.5  14.5  0.3065 466 142.8  0.3580  463.5  Ar- ;*s.  M AT- 'r>S.  e  Ar-;»s.  M e. A~ir>S. J-~k.rco.ni  0.0068  4 5 8 . 3.115 0.0037  466  1.75 0.00265  484 1.285  466  5.54 0.0083  484 4.03  5  /~krce.n £  kr- '"t.  7  y  0.0078  466.  0.0240  466.  11.4  0.0213  4 5 8 . 9.75  0.04395  466.  20.5  0.0387.:  4 5 8 . 17.74 0.0213.; : 466  77.5  0.1014  466.  47.2  0.0896. : 4 5 8 . 4 1 . 1  0.0498  0.1118.  470 , 52.5  0.1885  466.  87.8  0.1672.*  4 5 8 . 76.7  166.8  0.0978  470, , 46  0.2127..  466.  99.0  0.1885  165.8  0.0451..  470, , 21.2  0.2823.;  466.  3131.5  0.2483.  87.6  TABLE  9  3.63  0.0119  Af-'tos.  0.0153.'  484 7.45  466 23.2  0.0361  484 17.5  0.0980  466 45.6  0.0710  484 34.4  4 5 8 . 86.4  0.1083  466 50,5  0.0775  484 37.6  4 5 8 . IL3.8  0.1420  466 66.2  0.1025  484 49.75  9.9  SU'  St//  ft//  95  96 T h e s e moment  diagrams  p r o v i d e d an i n d i c a t i o n of  t h e m a g n i t u d e o f t h e moments u n d e r t h e l o a d and o v e r t h e i n n e r support d u r i n g each l o a d i n g  Moment d i a g r a m s  s t a g e up t o f a i l u r e .  o f t e s t No.2. f o r e a c h  loading  a r e shown i n f i g u r e 30. The  e x t e n t o f e q u a l i z a t i o n o f moments u n d e r the  l o a d and over the i n n e r support  i s presented as a graph  (see f i g u r e 3 1 ) . 303.  R e s u l t s o f Beam T e s t No.3. Up t o a l o a d  o f a p p r o x i m a t e l y 13 KTips  t h e beam  o f t e s t No.3. behaved s i m i l a r l y t o t h e beam o f t e s t and No.2.  At a load  No.l.  o f a p p r o x i m a t e l y 13.5 K<ps s i g n s o f  c r i p p l i n g o f the compression order t o prevent an e a r l y  f l a n g e were o b s e r v e d .  failure  similar to test  In No.l.,  two s p r e a d e r s were i n s e r t e d between t h e t o p and b o t t o m f l a n g e on e a c h s i d e o f t h e web where c r i p p l i n g was  first  observed.  i n bend-  At a load  o f 16,060 l b s t h e beam f a i l e d  i n g under the l o a d .  The  last  r e a d i n g s were t a k e n when t h e l o a d  was  e q u a l t o 16,000 l b s . S t r a i n and d e f l e c t i o n r e a d i n g s observed d u r i n g t e s t No.3. a r e g i v e n i n t a b l e 10 and // .  91 97  Observing the flange strains €, , d , £ , G , 3  A  corresponding to location (1), ( 2 ) , ( 3 ) and ( 4 ) which were at 16, 18, 2 4 , and 30 inches respectively from support D, we see that up to a load of 15 Kips, they increase continuously with the.load. Beyond 15 Kips and up to failure there is some noticeable fluctuation in their values, although at 15.5 Kips some increase i s observed.  On the  other hand strain readings between support A and B show a continuous increase in their values up to failure load.  Figure 32 When the value of the flange strain C under the c  load corresponds to the maximum value of m then the moment. at section-C and C| attains i t s maximum value, which corresponds to point A on the (m-e) diagram. Adjacent sections correspond to any point between K and M on the (m-£) diagram.  The theory  98 of  i n e l a s t i c bending  p r e d i c t s moments, d e f o r m a t i o n s , and  f a i l u r e loads corresponding t o t h i s  condition.  As was m e n t i o n e d i n p a r t I I , due t o t h e f a c t t h a t the l o a d i s a p p l i e d over a ^ i n c h w i d t h f u r t h e r i n c r e a s e o f t h e l o a d w i l l n o t n e c e s s a r i l y b r e a k : t h e beam. w i l l be c a p a b l e for  o f p r o v i d i n g a c e r t a i n amount o f a n g l e change  c o n s i s t a n c y o f d e f o r m a t i o n a l o n g t h e beam l e n g t h C.  without breaking.  The beam w i l l  f a i l by f r a c t u r e on t h e  t e n s i o n s i d e when t h e f l a n g e s t r a i n its  The beam  6  C  under the l o a d  f a i l u r e value corresponding to f a i l u r e  reaches  stress.  C a l c u l a t i o n o f Moments o f Beam T e s t No.3. The method o f e v a l u a t i n g t h e moment o f t e s t beam No.3. was e x a c t l y s i m i l a r t o t h e p r e v i o u s two t e s t s . The e v a l u a t i o n o f t h e c o n s t a n t s K« » K2., lc% , K V , KQ and k> o f e q u a t i o n (25) c o r r e s p o n d i n g t o l o c a t i o n ( 1 ) , (2),  ( 3 ) , ( 4 ) , (6) and (7) w h i c h were a t 16, 18, 24, 30,  63 and 69 i n c h e s r e s p e c t i v e l y f r o m s u p p o r t D i s a s f o l l o w s : F i r s t t h e a v e r a g e s t r a i n s £, , £ Q  7  corresponding to locations  t  t  6  3  t C* 4  £  6  and  (1), ( 2 ) , (3)>..(4), (6) and  (7) r e s p e c t i v e l y were p l o t t e d v e r s u s t h e l o a d f i g u r e 33. The s t r a i n r e a d i n g €  s  b e i n g t o o c l o s e t o t h e l o a d was n o t  jr;eliable, Thus f o r a n a r b i t r a r y l o a d o f 10 K i p s t h e c o r r e sponding  strains  €. , € , £, , € . z  a n d £7 m e a s u r e d f r o m  99 t h e g r a p h h y e x t e n d i n g t h e s t r a i g h t l i n e b e y o n d 8 K i p s were 0.1735, 0.1983, 0.265, 0.326, 0.258 and 0.2015 respectively.  U n d e r t h e same l o a d t h e c o r r e s p o n d i n g  s t a t i c a l l y determined (4),  percent  moments a t l o c a t i o n s  (1), (2),  (3),  ( 6 ) and ( 7 ) were e q u a l t o 8 5 . 3 5 , 96.0, 1 2 8 . 0 , 160.0  126.0 and 98.0 K i p - i n c h e s r e s p e c t i v e l y . Using 1 quation Ki  =  85.35 0.1735  =  492  Kip-ins  K  =  96 0.1983  =  480  Kip-ins  126 0.265  =  483  Kip-ins  160 0.326  =  491  Kip-ins  126 0.258  =  488  Kip-ins  98 0.2015  =  483  Kip-ins  K  3  Ky  =  =  A f t e r determining  Ki ,  , K  , Y4.»  3  b e t w e e n s u p p o r t s D and B t h e e v a l u a t i o n constant c o e f f i c i e n t s , locations  K & and  Ky  of the remaining  , ^s> and K i , c o r r e s p o n d i n g t o 0  ( 8 ) , ( 9 ) and ( 1 0 ) w h i c h w e r e 24, 1 8 , a n d 16 i n c h e s  r e s p e c t i v e l y f r o m s u p p o r t A was d e t e r m i n e d  by u s i n g  t h e same method, a s i n t h e p r e v i o u s two t e s t s . Using the equation : -2dU  e  «  exactly  I  100 any  c o e f f i c i e n t between A and B s u c h a s  constant  Kio c a n be e v a l u a t e d p r o v i d e d t h e v a l u e o f K h a s b e e n  and  predetermined as  ,  by a v e r a g i n g t h e c o e f f i c i e n t s o f such  locations  ( 1 ) , ( 2 ) , ( 3 ) , ( 4 ) , (6) and (7) between C and D.  For test  No.3»»  » K  »  3  and ^ 7  , K4.,  was  found  t o be e q u a l t o  491.5, 483.5, 482.5, 490, 488 and 483 K i p - i n c h e s  K = 492 +  Then  To e v a l u a t e J?e  6  corresponding  480 + 48? + 491 + 488 +  the average  strains  to locations  were p l o t t e d v e r s u s  ( 8 ) , (9) and  the load,  (figure  =  4 Q 6 < 5  £g , £<> and  Figure  (10) r e s p e c t i v e l y ,  o f 10 K i p s t h e c o r r e s p o n d  , £9 and £;o measured f r o m  strains  34) were r e s p e c t i v e l y  £10  34.  Thus f o r a n a r b i t r a r y l o a d i n g average  483  respectively.  the  e q u a l t o 0.022, 0.0166,  graph 0.0152  percent. Then ^  and  =  jfcLi =  Using  €g + ^ 9 + e  0.022+0.0166+0.0152 = 0.0538 p e r c e n t  =  cLe + d$ <-d  equation  = 24+18+16 =. 58  l0  inches  (38) we g e t  ^  = 486.5 x ° - j ? - p  ^9  = 486.5 x ° ' ° | ?  K  = 486.5 x - ° - ^ 8  n  ( 0  8  8  x ^ l l g  x  0  t  Q i  6  6  ^_16^  =  490  =  =  491.5  4  7  ?  Kip-ins  Kip-ins  g . ^ ^  101 After  the  K7 »  evaluation of  > ks>  a  n  the  coefficient  ^10 w h i c h a r e  d  t o he  the  test,  and  kf  d u r i n g a l l l o a d i n g s t a g e s and up t o  computed. table  These  Mi , M  computations are  to  (10)  locations  (1),  the  previous  over  failure.  presented  the  (3),  two t e s t s .  the  6  > M  7  ,  k^, -  through-  , Hg ,  failure as  £  My  were  a table  (see  the  (4),  (6),  straight  and over 36.  These  magnitude  Moment d i a g r a m s  The  Figure  c o m p u t e d moments  of  (7),  (8),  lines  for  the  same  and  (9) each  method  moment d i a g r a m s the  correspond-  as  provided  moment u n d e r t h e  load  i n n e r support d u r i n g each l o a d i n g stage up  shown i n F i g u r e  load  , H4 » M  s t a g e was p l o t t e d u s i n g e x a c t l y  an i n d i c a t i o n of and  (2),  of  a b e n d i n g moment d i a g r a m o f  loading in  ?  3  12). Using the values  ing  , M  z  fC ,  constant  out  ) p  t h e moments  assumed  & , KV,  test  No.3« f o r  each l o a d i n g  are  35.  extant the  of  to  of  e q u a l i z a t i o n o f moment u n d e r  inner support  is  presented  as  the  a graph  in  105  ,  .  .  ,  i  o 8 »  ' SS  «  ^  fi  ?-  »  i  5' t  <fc  0)  vo  H  S  3  §  51  S •»  '  *  • >  n  0  s  A  r-i  *  I ?-  y  $.  » » 5 . 1 ?•  i s i i n - i ! A  N  §  O  O  * * » *  * §,  -S.  §,  ^  t  S  H I ! ! 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BJ-£  /  7 or//&/'£' 7 OOO/. r  /  /fe&c//h?s  J  fO  0/ 7^^  //Vo.  3.  106  MOMENT COMPUTATIONS OP TEST NO. 3  Load  K  M,  K- /ns.  «z  it-'OS.  M  z  fcZ-'OS.  12.5  .1875 4-92.0 92  13.5  .1908 492., 0 93.6 .2175 480.0 105  14.5 14.8  .2135 480.6 103  M4  ^ 3 kT-ios.  P<Zrc<Zn r * f f€-ios. k?-/r>S-  .2842 483.0 137  .3556 49© 174  ^5  pesc<i/ti  kr~ins.<  .2092 488 102  6  «7  7  e,  My A^-,'oS.  1  pzscxnt:  .1275 483 61.6 .0926 490  jpa.r<g»r  M  K,  9  k?-;»s.  K-/nS.  D  percent  f- 'OS.  45.5 .0696 491.5 34.2 .0615 477 29.3  .2892 483.0 139.5 .3624 49<D 177  .1793 488  87.5 .0862 483 41.6 .1387 490  68  .1052 491.5 51.6 .0922 477 44  .1939 4 9 a . 0 95  .2208 480.0 106.5 .2940 483.6 144.0 .3690 49d) 181  .1482 488  72.2 .0436 483 21.1 .1862 490  91  .1411 491.5 69.2 .1234 477 58.6  .1960 492.. 0 96  .2232 480.0 107.5 .2977 483.0 144  .3740 490) 183  .1446 488  70.3 .0374 483 18.1 .1955 490  95.7 .1481 491.5 73  .2989 483.0 145  .3757 49(D 184  .1319 488  64.5 .0243 483 11.7 .2090 490 103  .1589 491.5 78  15  .1955 492..0 95.6 .2228 480.0 107.4 .2968 483.0 143  .3734 49(D 182  .1321 488  64.5 .0214 483 10.3 .2109 490 103  .1602 491.5 78.6 .1397 477 66.8  15.3  .1962 492.0 96  .2233 480.6 107.5 .2974 483.0 143.5 .3741 49(1) 183  .1219 488  59.5 .0085 483  4.1 .2246 490 110  .1707 491.5 83.6 .1486 477 70.61  15.5  .1979 49a. 0 97  .2252 480.0 108.5 .2998 483.© 145  .3769 49 (J>  .1182 488  57.5 .0026 483  1.2 .2326 490 114  .1767 491.5 86.5 .1539 477 73.5  15.7  .1978 4Q&.0 97  .2253 480.0 108.5 .3001 483.6 145  .3771 49 (!)185  .1161 488  56.6 ;0066 483 -3.2 .2421 490 118.5 .1841 491.5 90.5 .1603 477 76.5  15.04 .1971 492 .6 96.5 .2243 480.6 108  185  .1295 477 62 .1385 477 66  15.88 .1971 492. .0 96.5 .2243 480.6 108  .2984 483.0 144  .3745 49 (J) 183  .0974 488  47.5 :0231 483 -11.2 .2580 490 126  .1961 491.5 96.2 .1708 477 81.5  16  .2979 483.6 144  .3749 49(D 183.5 .0928 488  45.3 :0296 483 -14.3 .2639 490 129  .2004 491.5 98.3 .1746 477 83  .1968 492. .0 96.6 .2238 480.6 108  TABLE  /Ei  107  ft  0 0  0  g  *  s  t  o  10©  Sis/ - c//jf  ^t/sctsa^  110.  Ill ]j-,)-r^l4-  112  115  116'  PART IV PREDICTIONS OF THE THEORY OF INELASTIC BENDING  401.  Theoretical Predictions on the Behavior of the  Test Beams. As was mentioned i n the introduction, one of the purposes of the beam tests was to check the accuracy of the theory of inelastic bending, by comparing predicted moments and deformations against those attained during the test. The theory and the unit functions are presented in section I. During elastic deformations the elastic theory is the same as the theory of inelastic bending.  The limit of  elastic deformation is reached when the extreme fiber stress under the load point reaches  3 6 . 3 K'ip-s/ns  2  • Observing the  recorded strain of Test No.l. and No.3. we see that the stress of 3 6 . 3 ^'py^s*-  i s  reached just before the negative support  A comes into action. point is M=  36.3Z  The corresponding moment under the load (177 Kip-ins ) The moment over inner  support B is zero since the beam i s s t i l l behaving as a simple beam. The corresponding load at the elastic limit from statics, i s 7.9  Kips  which checks very closely with the  117=  actual load. After support A comes into action at approximately 8 Kips the extreme fiber stress under the load point exceeds the proportional limit of 3 6 . 3  K\ps/  readings between supportsA and B,up show perfect linearity.  » although the strain  2 ins  to a load of .'9 k'ips ,  F-i8Ures23 »28 ,3 4-,.  The explanation of this phenomenon lies in the very nature of the stress-strain diagram of the material. First, as seen i n table / , the proportional limit seem to increase with the size of the cross-section area of the sample. The exact variation is not known, but since the tension area of the beam is about seven times that of the speciman ' an increase i n the proportional limit of the beam above 3 6 . 3  Ki? /ins ' s  2  i s  n o  * improbable.  Second, referring to  the stress-strain diagram of the material we see that the  °  Strou'n  (percent)  Figure 3 7 transition abrupt.  from the proportional limit to, yielding is not  Point B which corresponds to a stress of 3 9 . 5  almost lies on the straight line OAE.  Figure 37.  118 Thus w i t h a p r o p o r t i o n a l l i m i t of 39.5 Ki'ps/jns 2  and u s i n g e l a s t i c methods, p r e d i c t e d deformations q u i t e a c c u r a t e , up t o  should he  3 Kips.  E l a s t i c S o l u t i o n o f the Test Beam. (I) Simple beam a c t i o n . The e l a s t i c s o l u t i o n o f the t e s t beam d u r i n g the stage o f simple beam a c t i o n can be used t o determine the modulus o f e l a s t i c i t y o f the beam. deformations  A term t a k i n g account o f  due t o shear f o r c e w i l l be i n c l u d e d .  -a  tc  J£ L  3 8  Figure f o r o/ x.<Ca. the d e f l e c t i o n u !  b  due t o bending i s equal t o  1 -  for  <«>  o<fc<£x. the d e f l e c t i o n  J s  due t o shear only i s equal (44)  Aw GL  where G is the shear modulus and A  w  i s the area of the web.  119 foro<=c <<x the t o t a l d e f l e c t i o n due t o shear and t e n d i n g i s equal t o  0  QB11L  {  In equation ratio.  AwG)  U  (^-5)  = £ (\4-w-,) where y- i s Poissons  The value o f y- g i v e n i n the A l c o a s t r u c t u r a l hand-  book i s 0.33*  The value o f I , which i s the moment o f i n e r t i a  about the s t r o n g a x i s , i s 9.74 ms " 4  /\  w  At  , i s equal t o .94 m s x  2  .  The area o f the web  .  = 48" by equation (-4-5)  ,  =1610_P  b  where y pis At z  E  L  i s t h e d e f l e c t i o n under t h e l o a d g i v e n - i n inches and  the l o a d g i v e n i n Kips. E  w i l l have the u n i t s o f \C\ysJ £- . Jn  = 36" by equation  =  where y  N  1495  (47)  P  i s the d e f l e c t i o n a t 36" from support B g i v e n i n inches  and P i s the l o a d g i v e n inK'ips . ET s  w i l l have the u n i t s o f  2.  F o r Test N o . l . , No.2., and No.3. the d e f l e c t i o n s versus the l o a d were p l o t t e d . when P = 6 KT/ps y^  F o r beam Test N o . l . and No.3.  i s 1.0 inches and y^ i s 0.926 inches  g i v i n g E v a l u e s o f 9,660 ^ T ^ n s  2  .(Fi<j-4oJ  F o r beam Test No.2., when P = 6 K«'ps , y inches and  y.  N  L  i s 0.925  i s 0.86 inches g i v i n g E v a l u e s o f 10.400  ^'f^f-  12Q (£) Indeterminate Beam A c t i o n . The e l a s t i c s o l u t i o n o f the indeterminate beam was accomplished by s a t i s f y i n g a c o n d i t i o n of c o m p a t a b i l i t y . In order t o s i m p l i f y the c a l c u l a t i o n s the d e f l e c t i o n due to shear was n e g l e c t e d .  Figure 3 9 Thus r e f e r r i n g t o f i g u r e 39 found by equating &  c  t o c^> .  P a r t 1. Where  n,  _ +-— - tc/a  =  A  A  =  0 (48)  G  =  ' d  c  42  , the s o l u t i o n i s  121  _ (48)Q 42x48* Q ~ " 4fl ~ 2EI  / B  • f • • c  _ "  42(3 x 48  3EI  +  frgfa HaE!  ^  . 5 48 . 5 (48)Q 4 42 4 6EI Y  +  A  X  +  +  A  +  Q42 x (48) - (48) R 2£I 6EI  3  Part 2. <?„ = tc/ +  (0  c  //  =  / c  9c  - 9 ) 42  9Q (42Jd?  (42)Q  2EI  GET  08c = - 42 x 48Q - 48 Q 2  El2  EI  f*  _  90Q42  2  °  " 2E1  (42)Q " 6EI  48* x42 Q 2EI  (42)R ( 4 2 ) V G£I ' &EI  48R(42)* 2£I +  ?  48R " 2  Fl2  48x42* R 2'EI  42*R P42* 42 x48 Q 6EI " 6FI " f x 2  '48 x42R Q42* 5 2EI " 3 E I " 4" 2  Part 3 . Equating  §  c  to <$ and simplifying we get  544.8x10 Q E/I  0  121.7xl0 R EI 3  Substituting R by /13_2 <0 + >. 90  12.35x10* P _ EI "  2  110 8  42 p ) and 90 y  simplifying we get -Q  + 0.266 P =  2.35 x 10* EI  In the, .'las.ttest,.-(No.3.) when P = 8.6 at A was barely touching.  (48) , the negative support  This makes (?= 0.  122 Then s o l v i n g equation r-  0.266 x 8.6  E The  (48) f o r Ez  =  10,000 Kips/.  2.35x9.74x10 =  /  n  ^  s  d e f l e c t i o n under the l o a d i s given by  &  =%|20P  +  1.1  = ^|^  N  According S=  P  .  (50)  S u p p o r t B isc^iven  + 1.27  to equation  1.27".  N  inner  4 9 )  equation: t  The- def lecTibn at 36 i n c h e s f r o m by the equation! £  i  2  (51) (50) and (51) a t P=0  c5c = 1.1" and  T h i s i s due t o the assumption that although P=0  the end A i s brought i n contact with the support A. I n e l a s t i c S o l u t i o n o f the Test Beam. Analysis  No.l. The  The  load and supports a r e assumed to a c t on a p o i n t .  i n e l a s t i c s o l u t i o n o f the t e s t beam involved the same C  pros§es&as the e l a s t i c s o l u t i o n except that i n s t e a d o f .a s i n g l e equation  a t r i a l and e r r o r procedure was r e q u i r e d .  For d i f f -  erent bending s t r a i n s ( 6 c ) under the l o a d p o i n t , i n c l u d i n g the s t r a i n corresponding presented.  t o maximum value  of m, a s o l u t i o n w i l l be  T h e s o l u t i o n s was f o u n d by a s s u m i n g a f l a n g e  strain  6 o v e r the i n n e r support and then s a t i s f y i n g a c o n d i t i o n of B  compatability.  F i g u r e 41  123 R e f e r r i n g t o f i g u r e 41 , the i n e l a s t i c s o l u t i o n i s found by equating fr referenced).  t o So  c  For g  . F o r the u n i t f u n c t i o n s see  v a l u e o f 1.844ins was used.  a  used i n the equations are shown i n figure-4-1. d e r i v a t i o n the dimensions  A l l symbols  Throughout the  o f inches were used.  Part 1. c  -  where  / a  I  2  + * _ (42)_Ue_ 1_ . _5  tA/&  4£  ~ <  1  ' 44 42 8  f22.8)M?  168 "  +  m'  5 +  e  168  2  •Phor.il"  5  -u 22.8x48 u  &  , 5x48  1250  /  r  H '  Part 2. (To =  Xc D i% - (e-rGsc)*.  where 4^=  ^1  l u T7  U c  = -*2?_ ^ J.844 u  c  _= &5 5 >8 u^ 5 ce  t  6  then o" = 0  simplifying ct= i |  &  =  22.8 u e  958  958-%  +  958  26(ns-nJ  B  ife  +  1093-^f-  + ^252  5 _ j Q3 j  4  4-Z. (52)  r\e>-tlc  rn (3  (53)  124 The the  s o l u t i o n f o rd i f f e r e n t bending s t r a i n s under  load c o n s i s t s o f assuming a c e r t a i n flange  over t h e i n n e r support so t h a t The  arithmetic  table  strain  £s  §c w i l l be e q u a l t o S  •  D  o f t h e c o m p u t a t i o n s i s shown i n f o r m o f a  (Bee t a b l e 13.)  T h e o r e t i c a l d e f l e c t i o n s a n d moments  b a s e d on t h i s a n a l y s i s t o g e t h e r w i t h t h e t e s t d e f l e c t i o n s and  moments a r e g i v e n  i n t a b l e 14 and f i g u r e 42 and 43  respectively. I t s h o u l d be m e n t i o n e d t h a t t h e beam d e f l e c t i o n s during  t h e t e s t s were m e a s u r e d r e l a t i v e t o t h e o r i g i n a l  t i o n o f t h e beam when P was e q u a l t o z e r o .  posi-  I n f i g u r e 42 b o t h  a c t u a l and t h e o r e t i c a l d e f l e c t i o n s a r e p l o t t e d w i t h  reference  t o h o r i z o n t a l l i n e AD. A n a l y s i s No.2. I n t h e c a s e o f t h e t e s t beam t h e l o a d was n o t a p o i n t l o a d b u t i t was a c t i n g o v e r a w i d t h o f j i n c h e s .  The . c o r r e -  s p o n d i n g moment d i a g r a m i s assumed t o be a s t h e one shown i n f i g u r e 4 1 . a . A c t u a l l y t h e i n n e r s u p p o r t was a l s o ^ but  wide  s i n c e t h e s t r a i n s w e r e r a t h e r l o w i t c a n b e assumed  out a p p r e c i a b l e  e r r o r t o be a p o i n t  support.  Also  with-  the actual  3"  d i s t r i b u t i o n of the load the  o v e r t h e j w i d t h i s n o t known t h u s  shown moment d i a g r a m i s a c l o s e a p p r o x i m a t i o n .  The i n -  e l a s t i c s o l u t i o n o f t h e t e s t beam was a c c o m p l i s h e d b y s a t i s f y ing a condition of compatability,  as i n a n a l y s i s No.l.  The  o n l y d i f f e r e n c e b e t w e e n t h e two a n a l y s i s i s i n t h e moment diagrams.  /?jTS///77ecC\  Afo^e/?/- Z?/&yrosr7 ^ A / y 7 < f / ^ r ^ W ^ ^ 4 ^ ^ -  Pigure  For  different  41.a.  bending strains  (£ ) c  under the  i n c l u d i n g the  s t r a i n c o r r e s p o n d i n g t o maximum v a l u e  solution will  be p r e s e n t e d .  ing a flange satisfying figure  strain  the  41.a •  ^  value  of  1.844  over  o  r  inelastic  of  the  inches  were  inner support  solution is see  i n s was u s e d . 41.a.  s o l u t i o n was  compatability.  uni^ functions  shown i n f i g u r e  dimensions  B  a condition of  t o SD  are  £  The  reference  (l).  A l l symbols used  Throughout used.  the  o f m,  found by and  a  assum-  then  Referring  found by  load  to  equating  Sc  F o r -g- a i n the  derivation  equations the  126 Part  (1)  Sc = | +  0(*Bc,) + ^V  +  fi  {&+0 c, ) B  (f) -  where  O  5  U 1 . _55 _ _ ™ _ ^ .U& _ £a/* <^/& ++ £| _ (A2) (42) _1 " — ^ " 1.844 42 168 ~ ' ^ B  +  1  =  +  2 2 2 2  8 8  ^  .  . , _5_ ^168  n  +  ^  ^gg,  n e g l e c t i n g t h e terms -  (  ^  3/4 -  - !  +  Part  2 2  '  8  x  ^.625 ^  +  ^ 7 ^ 5  +  (  ^  ^  ^  (  ^  (2)  where  Jo  =  940-i^- + 1 6 . 9 ^ -  950-2  3  - - 1.238 - 1073  ^^-^)  J  )  127 The  s o l u t i o n f o r d i f f e r e n t bending  s t r a i n s under  the l o a d c o n s i s t s o f assuming a c e r t a i n f l a n g e s t r a i n the i n n e r s u p p o r t s so t h a t ^ a r i t h m e t i c o f computations (see t a b l e 1 3 a ) .  w i l l be e q u a l t o do  .  -  £ over 8  The  i s shown i n f o r m o f a t a b l e ,  Up t o a f l a n g e s t r a i n  (£ ) under t h e l o a d , c  c o r r e s p o n d i n g t o maximum v a l u e o f m, t h e t h e o r y o f i n e l a s t i c b e n d i n g was s u c c e s s f u l l y a p p l i e d .  When t h e f l a n g e s t r a i n  e  c  u n d e r t h e l o a d ^ e x c e e d e d i t s v a l u e c o r r e s p o n d i n g t o maximum v a l u e o f m t h e n i n o r d e r t o compute moment.and d e f o r m a t i o n s c e r t a i n reasonable assumption  and m o d i f i c a t i o n h a d t o be made.  A n a l y s i s o f t h e T e s t Beam when t h e F l a n g e  Strain  Under t h e l o a d Exceeds i t s Value Corresponding t o Maximum V a l u e o f m.  ,  ,  4-2"  1  I  -u r  * * s/  .  1  F i g u r e 4-1 b I n o r d e r t o c a l c u l a t e t h e moments a n d d e f o r m a t i o n s o f t h e t e s t beam f o r f l a n g e s t r a i n s g r e a t e r t h a n t h o s e c o r r e sponding  t o maximum v a l u e o f m, t h e beam was a n a l y s e d i n t h e  f o l l o w i n g manner. R e f e r r i n g t o f i g u r e 4-1 b we o b s e r v e  t h a t when t h e  f l a n g e s t r a i n &c u n d e r t h e l o a d e x c e e d s i t s v a l u e  corresponding  128 to maximum value of m then the moment and strain attained under the load are given by L on the m-e diagram. Point 1 gives a slightly less moment than point A. Assuming that the testing machine is running providing a constant or increased load at point C then the moment over the inner support w i l l increase i n order to maintain equilibrium.  Knowing the moments at point B and C, the  adjacent moments to point C can be easily calculated since they w i l l be located along the line B' 0* C , and C D of the moment diagram. The strains and moments of the adjCjgbent sections to point C are given by such points as M' or R on 1  the m-e diagram. M* and R' are located on the lines MN and KR respectively which are parallel to line OP.  Unfortunately  the corresponding strains of points M' and R' are not known. This indicates that the corresponding unit functions  V) a n d V  of the adjucent sections to C are unknown. Also, although the unit function nn at section C can be found from the m-e c  diagram, the rest unit functions LU and He are not known. Thus an exact analysis of the test beam is not possible. The author tried a reasonable solution making the following assumptions. , For the test beam under consideration the maximum value of m is equal to 140.58 Kip/ins  corresponding to a  strain of 9.2$.. For a strain of 11.8$ the corresponding unit function m is 133 Kip/ins . 2  Comparing the two values of m  we. see a difference of only 5$, which is quite small.  This  129 i n d i c a t e s t h a t t h e e l a s t i c r e b o u n d due t o d e c r e a s e at  of strains  t h e a d j a c e n t s e c t i o n s t o p o i n t 0 i s s m a l l a n d c a n be  neglected.  A l s o i t i s a reasonable assumption r)  the u n i t f u n c t i o n s  c  a n d >J  t o consider  a t p o i n t C, a n d C e q u a l t o  t  t h o s e c o r r e s p o n d i n g t o maximum v a l u e o f m,- \{(\€.c)> 9-2/0 In ing Q= c  t a b l e 13a t h r e e s o l u t i o n s a r e p r e s e n t e d  first to a ^  = 11.8$ and £  B  = 1.2$, s e c o n d  10.4$ a n d £& = .7$ a n d f i n a l l y a £  c  . correspond-  to a  = 10$ a n d £  B  = .6$.  Computed moments and d e f o r m a t i o n s a r e i h r e a s o n a b l e agreement with the test. Deflections during Inealstic The  Deformation.  d e f l e c t i o n under the l o a d  i s g i v e n by t h e  equation (52), where & = 2 . 6 * The  +  1085*:  +  1  2  g  g  ^/^^^-n,))  d e f l e c t i o n lVj a t 36 i n c h e s f r o m s u p p o r t B i s g i v e n b y t h e  equation.  A . = '2.6.7 + 1065±!±  + ^  :  , (uB +  »~+^fie-n»T ^ 5  These d e f l e c t i o n s a r e c a l c u l a t e d i n t a b l e 13a f o r different strains (6  G under the l o a d i n c l u d i n g t h e s t r a i n c  =9.2$) which corresponds  w e l l as a s t r a i n o f 11.8$.  t o t h e maximum v a l u e o f m,as  Deflections are l i s t e d  i n t a b l e 15fl4  and p l o t t e d w i t h t h e t e s t d e f l e c t i o n s i n f i g u r e 4 2 . deflection;, curves from t h e i n e l a s t i c bending  The l o a d  t h e o r y shown  i n f i g u r e 4£ f o l l o w t h e same s h a p e a s t h e d e f l e c t i o n  curves  130  from the t e s t measurements.  The d e f l e c t i o n s from the theory  up t o f a i l u r e are i n close agreement w i t h the t e s t d e f l e c tions.  At f a i l u r e the d e f l e c t i o n under the load was 5 . 5 7  inches compared t o computed t h e o r e t i c a l d e f l e c t i o n of 5.46 inches. Moments during I n e l a s t i c Deformation. Values of the t h e o r e t i c a l loads and moments when f a i l u r e occurs are presented i n t a b l e 14 and 15 with the t e s t values.  The maximum r e s i s t i n g moment predicted by the theory  corresponding t o the maximum m ( 1 4 0 . 5 8 K i p / i n s ) i s equal to  M  = * A^(m) = i x ( . 9 4 ) x 3 . 6 8 7 .(140.58) = 244 K i p - i n s .  This value checks very c l o s e l y w i t h the maximum t e s t moment. Up t o the t h e o r e t i c a l f a i l u r e load of 1 5 . 5 3 Kips the t h e o r e t i c a l and t e s t moments agree q u i t e c l o s e l y . (Figure 43) The theory of i n e l a s t i c bending can predict f a i l u r e moments which correspond to the maximum value of m.  I f m extends  beyond that value then the theory of i n e l a s t i c bending i s not a p p l i c a b l e , without c e r t a i n modifications, which e f f e c t the accuracy of the theory. Loads during I n e l a s t i c Deformation. The^lo.ad P can be 'expressed i n terms of the moments fJ[ and He, as shown c  ~  41.625'*  47.625  131 or  P  =  >> Aw  I mc  mc + Ma 4i-7. 6J2 5  The f a i l u r e l o a d p r e d i c t e d b y t h e t h e o r y t o 15.53 K i p s  was e q u a l  compared t o t h e t e s t l o a d o f 16.06 K i p s .  The.  r e a s o n o f t h i s s m a l l d e s c r e p a n c y p r o b a b l y i s due t o t h e f a c t t h a t t h e s t r e s s s t r a i n d i a g r a m o f t h e t e s t beam may be somewhat d i f f e r e n t t h a n t h e s t r e s s s t r a i n d i a g r a m o f s p e c i m e n N o . l w h i c h was u s e d i n o r d e r  t o compute t h e u n i t  functions.  I n beam t e s t No.2 t h e f a i l u r e l o a d was 1 5 4 5 K i p s compared t o 16.06 K i p s  o f t e s t beam No.3.  The r e a s o n o f t h i s  difference  i s t h e r e d u c e d s e c t i o n m o d u l u s o f t e s t beam No. 2 u n d e r t h e l o a d due t o t h e c o n c e n t r a t e d I n t e s t No.3 b e f o r e  angle change.  See f i g u r e 2 ) .  s t a r t i n g t h e t e s t two s p r e a d e r s on e a c h  s i d e o f t h e web were u s e d .  This prevented a decrease of the  s e c t i o n m o d u l u s o f t h e beam. Moments a n d D e f l e c t i o n P r e d i c t e d b y t h e T h e o r y o f Limit  Design. The p l a s t i c c o l l a p s e l o a d  i s g i v e n by p l a s t i c  o c c u r i n g a t t h e l o a d p o i n t and o v e r t h e i n n e r s u p p o r t .  hinges The  p l a s t i c moment o f t h e beam s e c t i o n i s 224 K i p - i n s .  D ^  Figure Prom v i r t u a l w o r k e q u a t i o n  3  43  P&tfd  - M 9P  £• 0 M  P  = O  132  P = Mp(0.655)  =  14.65 K i p s  The moment under the load and over the inner support i s i^p  (224 KVp-ins ). Therefore the limit design  theory predicts perfect equalization of moments when failure takes place. The deflection before failure occurs when under the load there i s a plastic hinge and above the inner support a plastic moment has just developed. Referring to figure43, the deflection under the &c is equal +o  load  \  where B  5 42 Z  / e  °  Fl  3  ET  3  ,  4x42  t</ -  ' _L H P i 4 § l -  &  =  2.68  =  5.06 inches  Vs  £ EI +  2  2 (48) _ _Mp U s } 3  EI  %  The significant difference between the test moment and the limit design moment is that the theory of limit design  133 assumes no s t r a i n h a r d e n i n g , t h u s u n d e r e s t i m a t e s carrying  capacity  Configuration  The of f r a c t u r e is  o f T e s t Beam a f t e r F a i l u r e .  i n the tension  shown i n f i g u r e 17.  crippling in  of the section.  beam f a i l e d u n d e r t h e l o a d  t h e web p r e v e n t e d  flange.  The h o l t s  point  i n t h e form  The s e c t i o n a f t e r inserted  failure  on e a c h s i d e o f  e a r l y f a i l u r e o f t h e beam, due t o t h e  o f the compression  f i g u r e 17.  t h e moment  flanges,  Y i e l d i n g over the inner  which i s c l e a r l y s u p p o r t was  b u t n o t a s much a s t h e y i e l d i n g u n d e r t h e l o a d .  seen  apparent  134 7 Mnjt Function  2JL 26  . @ C  U  6 = 9 . Z7» e = . 3 2 % m=140.58 m-103 n=.7377 n=.165 u=86.31 u=11.3  2.68  M r  /O  1  -5  -  ins.  0  1.163  6  =9.2 % m=140.58 na.7377 u=86.31  £=.35% m-112 n*.196 us14.7  2.68  € =9.2 % m=140.58 n-.7377 u=86.31  6=.34% m=106 n=.18 u=12.7  12 7 2.68 7 ^ 6 ^ 1 . 1 3 1.24  6=7.4^ m=139.8 n«.6776 u=77.88  6- = • 28 /o m=90 n« . 126 u-7.57  2.68  6-7.4$ m=139.8 n«.6776 u=77.88  6=.29% m=93 n=.135 u-8.36  2.68  = 7.4$ ma139.8 n=.6776 u=77.88  € =.3% m=96.4 n=.144 u=9.3  2.68  € =5$ m=137.2 n«.5214 u=56.23  €-.22$ m=70.6 n=.078 u=3.67  2.68  £*5$ m=137.2 n«.5214 u=56.23  £=.21$ m=67.5 ne.071 u=3.19  2.68  £-2.6$ m « 132.9 n=.3608 u=34.51  er.lO$ m=32.1 i 2.68 n=. 016 u=.344  .344 _ (32.ir»33  £=.5$ m=123.5 n=.2614 u-21.61  6-.03$ m=9.63 2.68 ns.00144 u=.0092  .0092 _ -i (9.63f -  6 * .39;!: m=114.4 n-.2224 u=16.97  €= . 01$ m=3.21  e  n=.00016 u=.00034  n i i ) ^  1  *  1  1.28  7  / ' / ? S  -j  1  /3  358*®  mi  no' X  /  6»-  0  /2-  4  / K  r5  X / 6  2  /TO. 4  BB1,69 9 7 . 6 + 1 4 0 . 5 8 ( - . 5 7 2 7 ) »  (243.58f  ens.  zns.  F' 1  57  101. 01+140. 58(-. 5417),  x. /Q'  ens.  inS.  1  6 5  99. 0+140.58(-. 558)-  1.89  85.45+139.8(-.552)i  86. 24+139. 8(.-.5426)i  1.05  +.5417 252 = 2.15  2.34 4.2  Co, over e sit wetted.  •1.08  -1.25  + .558 246.6" »2.28  2.48 4.35  3.78  -.892  -1.25  .552 229.8 -2.39  2.61  4.25  3.94  3.78  -.925  -1.25  .542 232.8 = 2.34  2.56  4.16  3.94  3.78  -.958  -1.25  2.86  -.204  -1.25  2.99  2*86  -.67  -1.25  34 51 "[132.9)* = 1.95  1.87  -.32  -1.25  21.61 _ (123.5f = 1.42  1.36  -.1  -1.25  1.24  -.033  -1.25  .49  4.45  4.38  4.2  -1.12  .434 4.35  4.38  4.2  ,195 3.9  77.8 (139.8r = 3.94  .235 3.935  =10.2 9.3 (96.4r  1  ,  G  1J09  (236. 2 ) ^  1  ,  87.18+139.8 (-.534)-  8  .274  4.04  =12.18  3.67 „, (70.6)* * t  (207.87? * 2  o  59.9+137.(-.4434)=  3 2  •0.058 3.42  = -2.0 3.03  „  T 6 7 7 B T -  7  .763  (204.7)^ 2  3 8  59.42+137.(-.45)=  -.107  3.34  -.495  2  -.74  2.05  56.23 ( 1 3 7 . 2f = 2.99  = -3.6  2.68  34.85+132.9(-.343)-  ,364  ,  5  4  =-10.75  1  .00054 (3.2)* "  i > Q 3 3 !  ,11  0.036  rissfisT 5  (117.6)*  7.2  64  21.62+123.5(-.26)=-10.5  16.97+114.4(-.2214 )« -.74 B  ~8.25  (^a^/?^farsons '/^te/osAc 3  1.976  cnS.  -1.25  -1.015 -1.25  -8.25  iC9/x®  £. underesknnaked.  4.2  = 20.8  (229.8)^  -3  2.57 4.5  86.31 (140.58)? =4.38  -25  (246. 6 ) * '  177B. m.  /"7  +.5727 243.5" -2.35  .363 4.2  = 17.1  (252^6  ins.  /6  /4-  16.97 (114.47 rl.3  f  ge^cS/soy Theory C^/PG^s/s /'J  s  |  .534 236.2* = 2.26  2.47 4.04  .4434. 2.34 3.3 207.8" = 2.14 .45 204.7 = 2.2  s  ,343 165 :2.08  2i4  117."6  = 1.87  3.34  2.27 2.57  .26 , 2.13 133.1 = 1.96 .2214  6s upJeres^/naitPcL  2.14  - 2.04 2.0 0  €B overes'firna't'ecl-  TABLE 1 4 a MOMENTS AND DEFLECTIONS CALCULATED PROM INELASTIC BENDING THEORY .(A/V/UyS/6 /)  K-  /ns.  198 214  8.97 9.91 11.46 13.08  230  238 243 244  14.32  14.74  K-  ,ns.  5.5 17. o 56.0 117. 0 167.0 184. 0  'ns.  2.0 2.14 2.57 3.34 4.04 4.35  TABLE 1 4 b TEST MOMENT  T e s t No.  I  II"-'  s<;/=s.  c  9.55 10.95 12.05 12.50 13.00 13.50  200.0 207.5 220.0 220.0 220.0 212.5  9.0 10.0 11.5 13.5  192.5 207.5  12.5 13.5  240.0 246.0 253.0 253.0  14.5  III  M  14.8 15.3 15.5  15.7 15.88 16.0  225.0 232.5 230.0  256.0  " 254.0 253.0 • 253.0  K-,nS.  20.0 32.0  85.0  110.0 140.0  180.0 15.0  27.5  65.0 137.5  185.0 80.0  120.0  167.0 194.0 200.0  207.0  213.0  227.0 I'  <&€ C, e>rC  & a .36$ m = 115 n a .207 u a 15.8  <c = 9,2% m - 140.58 n « .737 tt a 86.31  in n  a a a  ii  <r _ .32$ HI a 103 n a .165 u a 11.3  <r = 7.6% m * liiO n = ..690 u = 79.6  7.6%  €  m « lUo n = .,690 u = 79.6  m n u  = 8.1$' m * 140.4  €  a a a a a  n » .719  in n  U  u a  83.7  -  6.4$ 139.1 n « .629 u = 71.15  &  6.4$  m » 139.1 n * .629 u * 71.15  U =  56.23  5$  m = 137.2 n * .521 U a  56.23  ^ »  2.6%  in «• 132.9 n » .3608 U  «  34.5  <f- .5$ m = 123.5 n = .2611* u = 21.61  n  a  ,  I  .39$  114.5 .222 17.0 .46$ 121.6 .252 20.5 ,29 m .135 8i36  u  a  e  a  .32  8  103  m n  n = .521  a  a a  €  137..2.  a  M  m n u  £ - 5$ Hi »  .5356 124.6 .267 22.3  88-  .165  a  11.3  =  .28  a  90  -»  .126 7.57  6T a m sn a u =  .25 80 .1 5.34  e a' .139 a ia. 6 a .027 a .75  m n u  6 _ .01$ m a 12.8 n a .00256 u a .0218  7 V8  & s  e = 9.2$ in = 140.58 n = .737 u = 86.31  «  3~  <4-  /  2.67  2.67  2.67  2.67  2.67  15.8  /Jtz  1.3  1.2  (I35T  /hs  1.16  17 =1.29 (lira  1.4  20.5  .1.39 u  >  2.67  nn  — i  cm) • 7.57  2.67  7  (241) =1.69  1.51 ( 5 5 1 3 )  1.05  (9*3y *  11.3  =1.42  (2BTT3)  y  B  2.67  1.58  ml  11.3 =_ 1.07 7Jo3T  (T2T75)  a l.^k  (225)  'te  (12575)  1.16  .98  a.835  1  ,  5  L  I  -1.47  (2427Q  (2177?)  ( 1 7 4 3 )  //?  102.1 + l4o.6(-.53) - 27.6  .521  108.6 + l40.6(-,47) = 43  .75  9 0 . 9 + li*0(-.5255 * 17.4  .36  4.19  96.6 + 11*0 (-.1*68) - 31.1  .59  4.66  i*.49  5.0  4.88  104.2- + i4o.4(-.467) 38.7  82.45 + 139.K-.464) = 18  .378  63.8 + 1 3 7 . 2 ( - . 3 9 5 ) = 9.8  .247  =2.12  61.57 * 137.2(-.1*21) * 3.87  .10  3.75  =3.3  3 5 . 2 + 132.9(-.33l*)  -.379  2.758  71  .0218 _ 1 no (l2.a) " *  -9.25  21.62. + 123.5(-.26) =.-10.5  w  86.31  , ,  (1403) ~ ^  3.96  ft  3 8  4.38  4.05  83.7  =4.27  a  .238  *l.  -SSOg)  71.15 B'a (XB7T)  AA  -/238  4.13  1.52  -1.14  -1.238  4.13  1.52  -1.38  -1.238  1.28  -1.02  -1.238  3.82.  1.28  -1.22  -1.238  4.01  1.42  -1.32  -1.238  -.918  -1.238  1.08  3.66  3.44  1.08  2# -  .47 =i _ 77 (255T2)  -1.0  -1.238  U  (  (  =2.14  .468 =1.84 (2553)  -.687  2.127  56.23 =3 (13T2)  34.5 (B2T9)  =  1  qc;  2.82  .845  -1.238  2.82  ,845  -*79  -1.238  1.84  .44  -.41  -1.238  .085  -.127  -1.238  1.36  1.9  4.93  2.3  5.34  £ Underestimated  1^97  4.61  D-K.  a. 8  1.92  4.8  .494  •2.13  2.28  4.65  2.03  4.29  1.87  3.4  2.08  3.68  2.07  3.702  2.05  2.177  .464 •1.9  .395  (22772)  i , y i J  21^61 . ! (1233)  -.89  5.5  .467  242a 3.93  2.23  261.6  252TT  3 , 6 6  4.21  707^X®  f/3/#7c //O -3  -  3.82  3.44  /7  /<4  ,-3  s  79.51 + 139.1(-.494) = 10.5  (2127T) «1.85  .1*67  2.67  2.67  "  1.01  a Q-DC;  /3  / x/O  2.67  J2_  -1*74  4gr =1.93  .334  1W3  .26  al.Q?:  =1.-91  13X3"  ^ m n u  » 10.4 = 139.13 » ^737 = 86.31  & = 11.8 m = 133 n - .737 u = 86.31 ^ = 11.8^ IB = 133 n a .737 * 86.31 11  m = 126.4  n » ,2767 U a 23.53 V -  .7$ ffl. a 127.5 n = .284 U a 24.51 ^= ,$6% m = 125.6 n » .272 u « 22.97 e *\.z% m = 130.2 n » .309 u • 27.7  2.67  2.67  23.53  4 1 , 7 1.6  24.5 _-, ,  1.64  K  (1273) ~  1  ,  5  1  (2553)  109.84 + 140(-.46) = 45.3  .78  110.82 + 1 3 9 . K - . 4 5 3 ) = 47.8  .822  5.05  5.12  86.31  86.31  i,  m  1,  „), 1,^  4.13  1.69  -1.4  -1.238  .46 25574  4.18  1.76  -1.43  -1.238  1.85  5.03  .453 . i 7 2553  1.82  5.09  .465 25X5  1.93  5.85  1.74  5.49  73  ' °  1  2.67  2.67  6  O.K.  B  O-'K.  £ 6 Underestimated  6 B Over estimated  * if A7V  2 ^ 7^<s  6 = ioj6 m = 140 n « .737 tt = 86.31  £ Underestimated  22.97  1.64  B l  -  U 6  1.58  (25F3)  1.78 26372  =1.44  109.28 + l33(-.465) = 47.4  .874  114 + 133(.428) » 57  1.01  5.124  86.31 T133T  5.46  «4.85  4.55  2.0  -1.39  -1.238  4.55  2.0  -1.56  -1.238  ai  •  8n  J  QrK.  TABLE 15 MOMENTS AND DEFLECTIONS CALCULATED FROM INELASTIC BENDING THEORY (ANALYSIS 2)  Load Kips  M  Sc  K ins. r  ins.  &  '  K-ins.  10.16  214  22.2  2.15  11.865  230  72  2.73  13.62  238  139  3.72  14.60  241  179  4.25  15.095  243  198  4.64  15.375  243  211  4.84  15.49  244  216  4.97  15.53  243  219  5.04  15.5  241.5  221  5.1  15.14  231  226  5.46  138  1 3 9  140  145'  146  147  148  1 4 9  150  151  mm  152  153  CONCLUSIONS AND RECOMMENDATIONS Theory o f L i m i t Design As  was m e n t i o n e d i n t h e p r e v i o u s  theory, o f l i m i t design  chapter the  assumes t h a t when a r e d u n d a n t  structure  i s l o a d e d b e y o n d t h e e l a s t i c l i m i t r e d i s t r i b u t i o n o f moments t a k e s p l a c e u n t i l a mechanism i s formed. d e f l e c t s e x c e s s i v e l y under constant  load.  Then t h e s t r u c t u r e When t h e t e s t beam  was s t r e s s e d up t o t h e e l a s t i c l i m i t , a t a l o a d - o f 9.5 t h e r a t i o o f maximum p o s i t i v e moment t o maximum moment was e q u a l t o 1 0 .  Kips,  negative  L a t e r on a t t h e f a i l u r e l o a d o f 16  K i p s t h e r a t i o o f 1.1 was r e a c h e d .  Thus a n  appreciable  r e d i s t r i b u t i o n o f moments t o o k p l a c e u n t i l a mechanism was' formed - b u t t h e s t r u c t u r e f a i l e d by f r a c t u r e o f t h e t e n s i o n flange under t h e load p o i n t . theory  of l i m i t design  Consequently, although the  predicted the f a i l u r e load of the test  beam, t h e t y p e of. f a i l u r e was d i f f e r e n t f r o m t h e one v i s u a l i z e d by l i m i t  design. 3" The  favorable  a p p l i c a t i o n o f the load over a ^  e f f e c t i n t h e f i n a l r e d i s t r i b u t i o n o f moments w h i c h  i n turn increased other  w i d t h had a  t h e c a r r y i n g c a p a c i t y o f t h e beam.  Thus,  s t r u c t u r a l c o n f o r m a t i o n s made o f a l u m i n u m a l l o y s may •  f a i l before conclusion  t h e mechanism c o n d i t i o n i s d e v e l o p e d . f r o m t h e t e s t beams w i t h r e g a r d  l i m i t design  Another  t o the theory of  i s the danger o f premature f a i l u r e o f a s t r u c t u r e  154 due  to crippling  of the compression f l a n g e .  I f the depth of  t h e beam i s s m a l l t h e n t h e a n g l e change u n d e r t h e l o a d be  large.  This  favorable  condition as explained  f o rcrippling  Theory of I n e l a s t i c  will  i n s e c t i o n ! ! bvary  o f the compression  flanges.  Bending  I n a r e d u n d a n t s t r u c t u r e t h e f a i l u r e p r e d i c t e d by the theory  o f i n e l a s t i c bending occurs a t the tension  of the cross  section.  side  The r e s i s t i n g moment a t t h e f a i l u r e  s e c t i o n r e a c h e s a maximum v a l u e d e t e r m i n e d b y t h e maximum value  of unit function The  m.  flange s t r a i n e a t f a i l u r e under t h e assumption  o f a p o i n t l o a d a l s o c o r r e s p o n d s t o t h e maximum v a l u e  of m  s i n c e u n d e r n o r m a l c o n d i t i o n s t h e beam c a n n o t p r o v i d e t h e required out  concentrated  a n g l e change u n d e r t h e l o a d p o i n t  breaking. The  was v e r y  a p p l i c a t i o n o f t h e load over a ^ i n c h w i d t h  favourable  without breaking  i n p r o v i d i n g t h e r e q u i r e d a n g l e change  t h e beam.  This provided which i n t u r n increased  further redistribution the value  f a i l u r e l o a d and t h e r a t i o by t h e t h e o r y and  with-  o f 1~  o f t h e moments  of the f a i l u r e load.  a t f a i l u r e , as p r e d i c t e d  o f i n e l a s t i c b e n d i n g was e q u a l t o 15.53  1.13 r e s p e c t i v e l y .  o f 16. K i p s and a r a t i o  The  Kips  Test r e s u l t s indicated a f a i l u r e o f Mi  a t f a i l u r e e q u a l t o 1.1.  load  155 One o f t h e p u r p o s e s o f t h e r e s e a r c h was t o check t h e moments and d e f l e c t i o n s p r e d i c t e d by t h e t h e o r y o f i n e l a s t i c bending experimentally. (9.5  K i p s ) up t o 15.53  Prom t h e e l a s t i c  d e f l e c t i o n curves  e r r o r . Pig.43  f r o m t h e t h e o r y up t o 15.53  K i p s f o l l o w e d t h e same shape as t h e t e s t d e f l e c t i o n Both curves  limit  K i p s t h e moments p r e d i c t e d by t h e i n -  e l a s t i c b e n d i n g t h e o r y were w i t h i n a 9 p e r c e n t The  load  were a l m o s t  i n p e r f e c t agreement. P i g u r e  When t h e f l a n g e s t r a i n the' v a l u e c o r r e s p o n d i n g  curves.  6  C  under the l o a d  42.  exceeded  t o maximum v a l u e o f m t h e t h e o r y o f  i n e l a s t i c b e n d i n g was n o t a p p l i c a b l e .  In order t o analyse  t h e beam f u r t h e r , c e r t a i n m o d i f i c a t i o n s had t o be made. P r e d i c t i o n s and t e s t were i n c l o s e agreement. Shortcomings o f the T e s t s When t h e t e s t  arrangement was d e c i d e d upon, t h e  danger o f premature f a i l u r e  o f t h e beam due t o c r i p p l i n g o f  t h e c o m p r e s s i o n f l a n g e was n o t r e a l i z e d . third  test  t o prevent'an  early failure,  t h e l o a d p o i n t were s u p p o r t e d spreaders  I n t h e s e c o n d and the f l a n g e s under  a g a i n s t c r i p p l i n g by two  i n s e r t e d on e a c h s i d e o f t h e web.  Recommendations  The  s i m p l e s t way o f c h e c k i n g  the theory  b e n d i n g e x p e r i m e n t a l l y i s by t e s t i n g a s i m p l y under a p o i n t l o a d .  of inelast  supported  beam  I f t h e t e s t moments and d e f o r m a t i o n s  up  156 t o and i n c l u d i n g f a i l u r e by  a r e i n agreement w i t h t h o s e  the theory of i n e l a s t i c  predict provided  bending then the theory  t h e moments and d e f o r m a t i o n s that,  the flange  its  i s a c t i n g on a p o i n t  should  o f a redundant  structure  t r w i l l not exceed i t s v a l u e  strain  0  c o r r e s p o n d i n g t o t h e maximum v a l u e o f m. load  predicted  the flange  T h e o r e t i c a l l y i f the  strain  £ c a n n o t exceed  v a l u e c o r r e s p o n d i n g t o t h e maximum v a l u e o f m.  Applying  the load  o v e r a -| i n c h w i d t h , p r o v i d e d  a  d e f i n i t e a n g l e change u n d e r t h e l o a d  which i n t u r n r e d i s t r i b u t -  ed  the carrying  f u r t h e r t h e moments and i n c r e a s e d  the  beam.  The No.l  c r i p p l i n g o f the compression flanges  i n test  c a u s e d a p r e m a t u r e f a i l u r e - b e f o r e t h e mechanism  tion visualized i n limit  To  avoid  deeper s e c t i o n that  capacity of  condi-  d e s i g n was r e a c h e d .  c r i p p l i n g of the compression flanges  i s recommended.  t h e s h e a r and a x i a l  Also  f o r c e are  i t i s further  insignificant.  a  recommended  157  REFERENCES  Hrennikoff, A. Inelastic Bending with Reference to Limit D esign. Transactions A.S.C.E. Vol.113. 1948. ' x  Baker, Home and Heyman, Steel Skeleton, Volume I I . Cambridge University Press, 1956. Timoshenko, S. Strength, of Materials: Part I: Elementary Theory and Problems. D. Van Nostrand, 1940.  

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