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Plastic design of columns and its inherent uncertainties Christoffersen, Per Trond 1962

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PLASTIC DESIGN OF COLUMNS AND ITS INHERENT UNCERTAINTIES by  PER TROND CHRISTOFFERSEN Dipl.Ing., SWISS FEDERAL INSTITUTE OF TECHNOLOGY, 1 9 4 9 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE In the Department of  /  CIVIL ENGINEERING  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA August, 1 9 6 2  In presenting  ;:  this thesis i n p a r t i a l fulfilment of  the r e q u i r e m e n t s f o r an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . for extensive  I f u r t h e r agree t h a t p e r m i s s i o n  c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be  g r a n t e d b y t h e Head o f my Department o r by h i s  representatives.  I t is- u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r . f i n a n c i a l • gain' s h a l l riot be a l l o w e d w i t h o u t my w r i t t e n  Department o f  civil  Bnginpgring  The U n i v e r s i t y o f B r i t i s h Vancouver 8, Canada. D  a  t  e  Auguot  27th, 1962  Columbia,  permission.  ii ABSTRACT The two well-known methods for the plastic design of steel columns subjected to combined bending and normal force were reviewed and compared. The validity of some of the basic design assumptions was examined, and the influence of unknown variables affecting practical applications was investigated. It was assumed that the existing column c r i t e r i a provide a true prediction of failure within the possible range of yield stress values of mild structural steel. It was found that bending moments i n columns vary greatly depending on the assumed conditions of load application, the method of analysis, and the actual values of the yield stresses i n the different structural members. Some of the recommended methods of analysis seem inadequate and in many cases unsafe.  Elastic analysis appears to  represent a necessary part of plastic design of rigid frames• A procedure based on an elastic analysis with subsequent redistribution of bending moments after the formation of plastic hinges was used to illustrate the possible variations i n column end moments. This procedure takes  iii  into account the effect of the individual structural members having different yield stresses. It i s shown that only through careful analysis and c r i t i c a l evaluation of unknowns can the main object of plastic design be achieved:  The design of structures  with consistent factors of safety.  iv  TABLE OF CONTENTS Page I II  THE DETERMINATION OF END MOMENTS IN COLUMNS (A) The E f f e c t o f V a r i a t i o n  III  IV  V VI VII  1  INTRODUCTION  o f the Y i e l d S t r e s s  5  (B) The E f f e c t o f O v e r s i z e d Members  17  (G) The C o n d i t i o n s o f Loading  19  (D) The Moment D i s t r i b u t i o n  25  THE COLUMN DESIGN METHODS (A) G e n e r a l  30  (B) The American Method  36  (C) The Cambridge Method  50  (D) Comparison o f the Two Design Methods  63  EXAMPLES Example 1  6$  Example 2  32  Example 3  #4  CONCLUSIONS  91  TABLES AND ILLUSTRATIONS  93  BIBLIOGRAPHY  ToJ&les I- \2  131  - 1 -  I  INTRODUCTION  In most basic literature on plastic design i t i s emphasized that conventional elastic design i s highly inconsistent with regard to the factor of safety of different types of structures• It i s shown that statically indeterminate structures have a great reserve of strength beyond the stage when the yield stress f i r s t appears i n a structural member.  The  elastic design does not make f u l l use of this reserve, and consequently produces statically indeterminate structures  possessing greater factors of safety than  those of a statically determinate structure, such as the simply supported beam. Since the simply supported beam has been proven safe by past experience, the obvious conclusion i s that the rigid frames i n the past have been unnecessarily overdesigned.  The main object of the  plastic design method i s to provide a tool, by which a l l types of structures can be designed with a uniform and consistent factor of safety. The factor of safety commonly used in plastic design i s the one which exists in a simply supported beam designed by the conventional elastic method. Thus the simply supported beam has become a yardstick for adequate safety of statically indeterminate structures.  -  2  -  When discussing factors of safety, i t i s important to differentiate between the calculated or apparent factor of safety and the true factor of safety. The apparent factor of safety i s determined theoretically based on numerous assumptions with regard to materials and workmanship, usually before the structure i s b u i l t . It i s therefore only as accurate as the assumptions upon which i t i s based.  The true factor of safety i s  the ratio of the failure load to the working load of a structure after i t has been built.  Since few structures  are loaded to failure, their true factors of safety are seldom known. There are many factors affecting the true safety of a structure, and some of these w i l l not affect a l l types of structures to the same extent.  The introduction of  a constant apparent factor of safety for a l l types of structures w i l l therefore not necessarily provide a uniform true factor of safety.  It i s obvious that  inferior workmanship w i l l have a much greater effect on an all-welded, rigid frame than on a simply supported beam consisting of a single rolled section. Similarly inaccurate design assumptions and design methods w i l l be applied more frequently in the design of rigid frames than in the design of simply supported beams. Many of the rigid frames designed and erected i n the past may  - 3have excessive apparent factors of safety, but are their true factors of safety excessive? Since true factors of safety of actual structures i n existence are not known, i t i s impossible to design a rigid frame to have exactly the same true factor of safety as a simply supported beam. It i s , however, possible to design a rigid frame having a true factor of safety equal to or greater than that of a simply supported beam. Such a design must make allowance for a l l factors which w i l l affect the rigid frame to a greater extent than they w i l l affect the simple beam, either by considering them i n the analysis and in the construction, or by the use of a higher apparent factor of safety. In the following an attempt w i l l be made to show the effect of some such factors, and how they can be considered in the design, with particular reference to the design of columns i n r i g i d frames. In the past much work has been carried out to establish relatively accurate criteria for predicting the failure conditions of columns. Much of the progress to date can be attributed to the efforts of the research groups at Cambridge University i n Great Britain and at Lehigh University i n the U.S.A. The findings and recommendations of the Cambridge group are compiled i n Ref.1, and those of the Lehigh group i n Ref.2.  - 4-  Ref. 3 i s a paper which outlines a method for the plastic design of columns as developed by the Cambridge group. The paper was discussed i n Ref. 4, where valid objections were raised to many of the assumptions made i n the method. Among the items questioned was the method for determining end moments i n columns, which w i l l be discussed further in the following.  II  THE DETERMINATION OF END MOMENTS IN COLUMNS  (A) The e f f e c t of V a r i a t i o n of the Y i e l d Stress The p l a s t i c design method assumes a certain f i x e d value f o r the y i e l d stress of s t r u c t u r a l s t e e l .  This value  i s usually the s p e c i f i e d minimum guaranteed y i e l d stress, which i n North America f o r A7 s t e e l i s 33 k s i . The corresponding value i n Great B r i t a i n i s 1 5 . 2 5 tons per sq i n . or 3 4 * 2 k s i .  It has been found that y i e l d stress values vary considerably.  This i s discussed  on page 37 of R e f . 2  where i t i s stated that s t u b column" tests best reveal n  the basic y i e l d stress l e v e l of the material, and that t h i s basic y i e l d stress f o r A7 s t e e l may vary from 25 k s i to 4# k s i .  This statement i s based on the r e s u l t s of  extensive t e s t programs carried out at Lehigh University, Pennsylvania.  Ref.5 i s a summary report on tests c a r r i e d  out p r i o r to 1 9 5 7 , which confirms that the above statement refers to values of the lower y i e l d stress as determined by compression tests on f u l l s t e e l sections. These values w i l l i n the following be assumed to represent the lower and higher l i m i t s of the y i e l d stress, although i t i s possible that by very slow or very rapid a p p l i c a t i o n of loads the range of v a r i a t i o n can be widened even f u r t h e r . Creep under loads of long duration may have the e f f e c t of lowering the lower l i m i t of the y i e l d s t r e s s .  Although i t  perhaps i s possible that the y i e l d stress may vary from one  end of a rolled member to the other, i t i s in the following assumed that the yield value i s constant throughout the length of each individual structural member. Consider the simple portal frame shown in F i g . l , loaded by the uniformly distributed  specified failure load W . p  As the load i s applied gradually, the frame w i l l at f i r s t be elastic and the bending moments at a l l points w i l l be directly proportional to the load.  When the load approaches the speci-  fied failure load Wj,, plastic hinges may develop at points A and B, either in the beam or in the columns.  If a plastic  hinge forms i n the beam, the bending moment at this i s equal to the plastic moment Mp^p^y*  w h e r e  z  i s t p  location h  e  plastic  section modulus of the beam, and cr^. i s the yield stress i n the beam. Ignoring strain hardening the maximum moment which the beam can transfer to the columns is directly proportional to the yield stress. It i s usual in plastic design to assume the plastic moment which corresponds to a yield stress of 3 3 k s i .  With the  assumed variation of the yield stress, however, the maximum moment which the beam can transfer to the columns w i l l be from 7 6 percent to 1 4 5 percent of the usually assumed value. If plastic hinges develop in the columns at points A and B, the maximum bending moment at these locations i s equal to the reduced plastic moment M = Z' c*. , where Z\ i s the P p y ' P 1  - 7 reduced p l a s t i c section modulus of the column i n the presence of the normal force N.  In t h i s case the p l a s t i c  hinge moment i s not d i r e c t l y proportional to the y i e l d stress, because the reduced p l a s t i c section modulus Z » i s also a function of the y i e l d s t r e s s .  p  I f the y i e l d  stress increases, the portion of the column section required to carry a c e r t a i n normal force N w i l l decrease, and the reduced p l a s t i c section  modulus Z  will  T  increase.  P  The moment M'p w i l l therefore increase at a f a s t e r rate than the y i e l d stress f o r a given constant normal f o r c e . I f , on the other hand, the y i e l d stress decreases, the section modulus Z* w i l l also decrease, with the r e s u l t p  that the moment M p decreases at a f a s t e r rate than the f  yield stress.  The column moment i s therefore, when a  p l a s t i c hinge forms i n the column, subject to a greater range of v a r i a t i o n than the y i e l d s t r e s s . To obtain an impression of the possible v a r i a t i o n of the p l a s t i c hinge moment M  F P  , consider the column  section shown i n F i g . 2 . The section i s a close approximation of an American standard 8WF31, and the expressions used i n the following are taken from Ref. 6 .  - £-  The plastic section modulus i s Z  - (Ap + 1/4 A )h - ( 3 . 4 6 + 0 . 5 5 ) ( 7 . 6 ) - 3 0 . 4 S i n . , 3  p  w  and the plastic moment i s Mp = 30.48" Sy . The section i s acted upon by the bending moment  and  the normal force N: and i f the normal force can be developed by the portion /3h of the web as shown i n Fig.2, i . e . N i A ^ C y , the reduced plastic moment can be expressed as follows: i L — /a  ii  n a  With a l l units i n kips and inches, and with the dimensions shown i n Fig.2, the expressions are as follows: A  '  _  __N  [^  N  =  (T  6  +  2 . 2 (T.  y  y  ¥  i  - <?%> !]  1  2  (7.6)  s  = 30.48" (T - 0 . 8 6 !L  y  <r  y  When the normal force cannot be developed by the web alone, i . e . -N > A^ (Ty, i t w i l l also occupy a portion A  F  of each flange.  Then N = (A^ + 2 * A ) <T, or p  and Mp  - A^TTf  *-  y  - Z£ (Ty - (1 -*) A h (T p  y#  With the numerical values from Fig. 2 :  H!_~,- -> 6.92 22  y  =  O* ^| 1  y  - 0.313  9  N VJ  =  ( 1-0.144 cr  34.6  <r  y  -  T  + 0.313) ( 3 . 4 6 ) ( 7 . 6 ) cr  3.79N  The expressions above have been used t o compute the values shown i n Table 1.  The f i r s t column i n d i c a t e s  the normal f o r c e s N i n k i p s , the second the y i e l d  stresses  <Ty i n k s i , and the t h i r d the c r o s s - s e c t i o n a l areas required to c a r r y the normal f o r c e s i n square inches. The f o u r t h column shows the reduced p l a s t i c moments M£  i n k i p - i n c h e s , and the f i f t h gives the r a t i o of  M£ to Mp^,  t n e  l a t t e r being the reduced p l a s t i c moments  f o r a y i e l d s t r e s s of 33 k s i . The l a s t column shows the r a t i o s of the y i e l d s t r e s s e s to 33 k s i . A comparison of the f i f t h and the l a s t columns shows the d i f f e r e n c e i n the r a t e of v a r i a t i o n of the column moment M' and the y i e l d s t r e s s o"*„. P y The r e s u l t s are shown g r a p h i c a l l y i n F i g . 3 and F i g . 3 shows the r a t i o  Fig.4.  i n f u n c t i o n of the y i e l d P33  stress  0^.  f o r d i f f e r e n t values of the normal f o r c e  The functions are l i n e a r when  N.  N>A (r ., i . e . when the w  y  normal f o r c e cannot be c a r r i e d by the web  alone. This  i s because i n the expressions used i t was  assumed that  the flange areas were concentrated i n two points at the distance h/2  from the c e n t r o i d of the s e c t i o n . The  made i n t h i s approximation i s s m a l l .  The functions  error are  - 10 not s t r i c t l y linear for N<A^Gy, but the deviations from linear functions are very small and cannot be determined by s l i d e rule computations. The functions therefore appear as straight lines i n Fig.3 also i n this case. The line for N=0 represents the ratio directly proportionally to the yield  m  M p M 33 p  which varies stress (Ty.  Fig.4 shows the ratio, - w f — i n function of the normal ^>33 force N for different yield values. force i s equal to the yield load  When the normal  N = A <r , the column  cannot resist bending and the moment becomes zero. For yield stresses <T < 33 ksi, the ratio also becomes y M» zero when the yield load N i s reached. ^33 This occurs for <T = 25 ksi when N = 223 kips, and for <T = 29 ksi when N= 264 kips. For C * 33 ksi the yield load i s N - 300 kips. For this T, M* load Ml = 0 . For <r > 33 ksi the ratio p approaches "P33 i n f i n i t y when the column load approaches 300 kips, because 3 3  the denominator  a  PP  r o a c n e s  zero.  From Table 1 and Figs. 3 and 4 i t i s seen that there i s very l i t t l e difference between the variation of the moment M£  and the variation of the yield stress  for small  normal forces, say N 1 0 . 1 5 N . In this range the moment  - 11  -  w i l l be practically proportional to the yield stress, such as i s the case when the plastic hinge forms in the beam. For greater normal forces, however, the deviation is considerable.  In the case of N= 0 . 6 N  =  k i  P  s  i t would usually be expected that a plastic hinge w i l l form for the moment  =  ^  2  kip-inches. With the  assumed possible variation in yield stress from 25 ksi to 43 ksi the plastic hinge may form for bending moments anywhere from 185.5  kip-inches to 931 kip-inches. In  this case the yield stress increases 92 percent from 25 ksi to 43 ksi, while the moment increases 429 percent from 185.5  kip-inches to 931  kip-inches.  The findings above are s t r i c t l y speaking limited to the cross-section shown in Fig.2, but there i s no reason to suspect that the relationships w i l l be much different for other wide flange sections. Returning to the portal frame shown in F i g . l , the third and f i n a l possibility to be considered i s that a l l members remain elastic under the specified failure load. This i s theoretically possible provided the actual yield stresses are higher than assumed in the design. In discussing this further, the portal frame of F i g . l w i l l be called Frame 1 .  Two other frames, Frames 2 and  3 , shown in Figs. 5 and 6 respectively, w i l l be  considered  - 12 at the same time. Frame 2 i s identical with Frame 1 except that the columns of Frame 2 are hinged at their lower ends.  Frame 3 has only one column which i s fixed  at i t s lower end, and the beam i s hinged at point B. It i s assumed that sidesway i s prevented. In Ref.3 i t was assumed that the end moments of the beams were equal to the so-called "required" plastic beam moments, which were determined for optimum beam economy by setting the end moments of the beams equal to their midspan moments. Such "required" plastic beam moments for the frames under consideration would be as follows: Frames 1 and 2  Mp - 0.0625 W L F  M„ = 0.0853 W„L P * Designating the elastic flexural stiffness of the beams Frame 3:  as K , and that of the columns as K , and assuming that b  Q  the yield stress i s nowhere exceeded, the bending moment at the joint A can be expressed as follows:  Frame 1:  ^he  moment  coefficients  for  the three frames a r e  shown a s f u n c t i o n s o f t h e s t i f f n e s s  K. b  ratio  i n F i g . 7. c f o r the "required" p l a s t i c K  A l s o shown a r e t h e c o e f f i c i e n t s beam moments. the  "required" plastic  ratio value for  I t i s s e e n t h a t t h e e l a s t i c moments  i s less K, is  Frame 3  °  than  exceed  moments o n l y when t h e s t i f f n e s s  a certain  value.  F o r Frame 1 t h i s K f o r Frame 2 i t i s ^ = 0.50, and h  = 0.6?, K  b  ~ 0.61.  Stiffness  ratios  °  i n this  range  c are  quite usual,  even f o r s i n g l e  In m u l t i - s t o r e y and m u l t i - s p a n at  storey, single  frames t h e e l a s t i c  t h e beam ends a r e e v e n more l i k e l y  "required" plastic  span  moments b e c a u s e o f t h e g r e a t e r  a t t h e beam ends b y s e v e r a l o t h e r members  attached  to the joints. of concentrated  l o a d s t h e above  b u t t h e same p r i n c i p l e s  different,  The  above e x p r e s s i o n s  the  s t r e s s e s i n t h e members r e m a i n b e l o w t h e y i e l d  a r e o f course  t h e extreme f i b r e s .  t h e maximum v a l u e p l a s t i c moment M  being  apply.  only v a l i d  as l o n g as stress.  means t h a t t h e b e n d i n g moment must n o t  e x c e e d t h e y i e l d moment M at  restraint  relationships  are  F o r a beam t h i s  moments  t o exceed t h e  provided  In t h e case  frames.  a t which y i e l d f i r s t  I t may be o f i n t e r e s t  o f t h e y i e l d moment  P33  corresponding  appears  t o determine  a s compared t o t h e  to a yield  stress  o f 33 k s i .  - 14 Assuming a y i e l d 1.12,  cx=  flange to  = 43 k s i and  (T  s e c t i o n s , the  ratio  of the  factor  v a l u e f o r wide  actual plastic  moment =  «43  " P 3 3  "  moment i s  1.45  Substituting =  I  #  M^  "P33  =  = oc My 1  -  3 0  M  t h e e x p r e s s i o n becomes  P33  T h i s means t h a t t h e beam t h e o r e t i c a l l y up  t o a bending  assumed p l a s t i c In the and  case  <T  modulus and  2,  Fig.  (43  used  ~g  = T~- +  y M'  2  usually  bending  just  produce  t h e maximum  . where S i s t h e  A  A - 9.12  ^  of the  o f a c o l u m n s u b j e c t e d t o combined  fibre  elastic  section  S  the reduced  in.  t h e maximum e l a s t i c  2  yield  moment i n t h e  W i t h t h e column s e c t i o n and  27.4  S=  i n . , and 3  c o l u m n moment i s My  )(27.4)  = 1312  -  3.ON.  t o compute t h e v a l u e s o f My  =  were i n p a r t t a k e n  puted  directly  from  units i n Table  shown i n  w i t h Gy=43 k s i , N)S  This expression shown i n T a b l e  f r o m T a b l e 1,  2 a r e i n k i p s and  was  2.  h i n g e moment  and  t h e e x p r e s s i o n s used  c o l u m n shows t h e r a t i o s  presence  ( <Ty -  v a l u e s o f t h e u s u a l l y assumed p l a s t i c  M^2  All  i n excess  elastic  moment.  t h e n o r m a l f o r e e N.  The  percent  at the highest s t r e s s e d f i b r e ,  stress i s  of  moment 30  can r e m a i n  normal f o r c e o f s u c h magnitude as w i l l  yield  -  a shape  t h e l a t t e r b e i n g a not unusual  t h e u s u a l l y assumed p l a s t i c  =  *V  stress  -  i n p a r t com-  f o r Table  inches.  The  1.  last  o f t h e maximum p o s s i b l e e l a s t i c  - 15 -  column moments My to the usually assumed plastic hinge moments  T h  e results as shown graphically i n  Fig. a. Table 2 and Fig. 8 show that columns can remain elastic for bending moments considerably higher than the plastic hinge moments M^-j usually assumed i n plastic design. This effect i s more pronounced i n columns than i n beams, except for normal forces less than one third of the yield load N ^^, where N ^^ = 33A i s the yield load y  y  corresponding to a yield stress of 33 k s i . For a normal force equal to 0.6 Ny^^ - 1B0 kips, which i s the maximum load at present allowed by the North-American design specifications,  the column can theoretically remain  elastic up to a bending moment of 1.67 ^^3* It was shown i n Fig. 7 for the three simple single span frames that the elastic end moments of the beams w i l l not exceed the "required" plastic moments provided the beams are sufficiently s t i f f as compared to the columns.  In this case the column moments cannot be  greater than the values usually assumed i n plastic design under the specified failure loads, even i f the actual yield values are much higher than anticipated.  This i s  generally so when the maximum elastic beam moment occurs near midspan rather than at the beam ends.  If, however,  in Frames 1 and 2 the yield stress i s low i n column B and high i n the other two members, a plastic hinge w i l l  - 16 form i n column B for a smaller moment than anticipated, resulting i n an increase of the moment at A beyond the expected value.  Thus even i n this case can an unfavour-  able variation of the yield stress cause an unexpected increase of the column moments. It has i n the foregoing been shown that the assumed possible variation of the yield stress has a great influence on the magnitude of end moments i n columns. This influence i s sufficiently great to make i f practically impossible to determine the column moments with some degree of accuracy.  It w i l l seldom be known  whether the members of a structure w i l l become plastic or remain elastic under the specified failure load, and the column moments can become much greater than usually assumed i n plastic design.  - 17 -  (B) The Effect of Oversized  Members  When, i n practical design, the required plastic section modulus Zp has been computed for a team loaded by a specified failure load, the beam section w i l l be selected. In most cases the selected section w i l l have a plastic section modulus slightly i n excess of the computed required value, and often architectural or deflection requirements w i l l dictate the use of a stronger beam than required to resist stresses.  Stronger sections  are also at times substituted because the specified sections are not immediately available. The effects of such oversized beams on the end moments in columns i s much the same as the effect of beams with high yield values discussed previously.  The plastic  hinges may form at much higher moment values than anticipated, or they may not form at a l l under the specified failure loads.  Combinations of oversized  beams and high yield stresses w i l l emphasize this further. As an example determine how much a fixed-ended beam supporting a uniformly destributed specified failure  load  must be oversize i n order to remain elastic under this load.  Assuming a shape factor oc=1.12 and a high yield  stress of (T « 43 k s i , the end moments for which yield w i l l f i r s t appear at the outer fibres i s M, = 1.3 M^o as determined previously.  The bending moment at midspan  f •  - 18 -  is then 0 . 5 My = 0 . 6 5 ^ 3 3 * andthe sum of the end moment and the moment at midspan i s 1 . 5 My = 1 . 9 5 ^ 3 3 * a yield stress C  With  = 33 ksi, the sum of the end moment  and the moment at midspan under the specified failure load would have been 2Mp^^, which i s 2 . 5 percent higher than the 1 . 9 5 ^ 3 3 above. This means that the selected beam section, i n order to remain elastic, must have an elastic section modulus 2 . 5 percent greater than the required plastic section modulus, or a plastic modulus 15 percent greater than the required.  The plastic section  moduli of the beams of the four storey extension to Cambridge University Engineering  Laboratory, described  in Ref. 3 , exceed the "required" by from 4 . 5 to 21 percent.  Here the columns were designed to withstand the  "required* plastic moments of the beams and not those of the sections actually selected.  The assumption was made  that "the provision of a beam stronger than the required minimum w i l l not weaken the resistance of the structure". In the design of a simply supported beam this assumption would undoubtedly be correct, but i n the design of columns subjected to instability i t s validity i s questionable. It seems to the writer that a f i n a l analysis after a l l sections have been selected is essential, thereby eliminating the effect of oversized beams on the column moments.  - 19 -  (C) The Conditions of Loading In the conventional elastic design of rigid frames i t is customary to consider different arrangements of l i v e loads to produce the most unfavourable condition for each structural member. In the plastic design of beams this becomes less important.  The exact degree of restraint  at the beam ends i s immaterial as long as the restraint is sufficient to produce plastic hinges.  Consequently  i t i s usually assumed that the loads remain i n a constant ratio to each other.  However, i n the design of  columns i t becomes important to consider different arrangements of live loads, and the question arises how the loads should be assumed applied to produce a failure condition which w i l l result i n a satisfactory margin of safety and s t i l l remain within reasonable limits of probability.  Since very l i t t l e i s laid down i n the  North-American specifications for plastic design i n this regard, the writer proposes to discuss this problem further. In plastic design the specified failure load i s assumed to be equal to the working load multiplied by the load factor k-^ . North-American specifications use k^ = while k^ = 1 . 7 5 i s used by Ref. 1 .  1.6*5,  The load factors are  applied both to dead and live loads. Eef. 3 recommends a method for determining column end  -  20 -  moments, which w i l l be d i s c u s s e d F i g . 9 represents  two bays o f a m u l t i - s t o r e y ,  span r i g i d frame, loaded loads.  W-, and W  ?  i n the f o l l o w i n g . multi-  by u n i f o r m l y d i s t r i b u t e d  are the t o t a l s p e c i f i e d f a i l u r e  loads  on the two beams, and g^ and gg a r e t h e r e s p e c t i v e r a t i o s of dead loads t o t o t a l I t would i n g e n e r a l  loads.  be necessary t o determine whether  the most severe l o a d i n g c o n d i t i o n f o r the columns a t j o i n t E would be a s s o c i a t e d w i t h a clockwise counter-clockwise  r o t a t i o n of t h i s j o i n t .  assumed t h a t a clockwise joints and  or a  I t i s here  r o t a t i o n w i l l govern.  The  a r e i n i t a i l l y assumed l o c k e d a g a i n s t r o t a t i o n ,  the t o t a l unbalanced j o i n t moment M-^ a t E i s d e t e r -  mined as the d i f f e r e n c e between the end moments o f t h e beams a t E .  The maximum clockwise  unbalanced  joint  moment w i l l occur when the g r e a t e s t p o s s i b l e l o a d i s a p p l i e d t o beam EF and t h e s m a l l e s t p o s s i b l e l o a d t o beam DE. In Ref. 3 t h e l o a d i n g shown i n F i g . 9 i s recommended. Here beam EF i s loaded  by i t s f a c t o r e d t o t a l l o a d T/iL,  w h i l e beam DE supports the f a c t o r e d dead l o a d gjW^. I t i s assumed t h a t p l a s t i c  hinges w i l l form a t the  ends o f beam EF, w h i l e beam DE w i l l remain e l a s t i c . The t o t a l unbalanced j o i n t moment i s then  - 21 -  The function in Fig. 1 0 .  ^ ^—  =  16  " 12 W L —  *  s s n o w n  a s  li  n  e  1  It should be noted that the unbalanced joint  moment becomes zero for ^1^1^1 WL 2  in the case of W^L-^ = ^2^2  = 0.75•  This means that  2  *  an<  ~ &2 ~ ^ . 7 5 the columns  w i l l be designed for normal forces only at joint E. This does not seem to be a reasonable assumption. The loading condition as recommended i n Ref. 3 and shown in Fig. 9 assumes that the smallest possible load on beam DE i s represented by the factored dead load, i . e . the working dead load multiplied by the load factor k^. There does not seem to be any reason why this load cannot be smaller.  Theoretically the dead load on beam  DE can be applied with a load factor equal to unity, which of course w i l l produce a much greater unbalanced joint moment. This condition i s physically entirely possible, but may represent an unnecessarily conservative assumption.  design  A more reasonable approach i s the one  recommended by Ref. 7 for ultimate load design of reinforced concrete structures, where, in certain cases, the design w i l l be governed by the use of two different load factors for dead and live loads.  Where the meet  severe condition for a structural member i s created by applying the smallest possible load to a span, e.g. span DE in Fig. 9 , Ref. 7 recommends a load factor of k  2  = 1.2 for dead loads.  - 22 If a minimum load factor of k  2  = 0 . 7 5 k^ i s used for dead  loads on beam DE i n Fig. 9 , the unbalanced joint moment at E becomes  The function  = 2 2 16 W  MUg v / f f —  WL 2  =  2  -  L  -T-T—  16^  °'  7 5  Sl l l 12 W  L  0.75 giW-iLi  1  2  ^  i s shown as  the dotted line 1' i n Fig. 10. With a load factor for total loads of K-^ = 1 . 8 5 , the reduced load factor for dead loads becomes k  2  = 0.75k-^ = 1 . 3 8 5 , which is not  unreasonable. So far i t has been assumed that plastic hinges w i l l form in beam EF and that beam DE remains elastic.  Due to the  possibility of oversized beams and variations of yield stresses discussed previously several other conditions are theoretically possible. First there i s the possibility of both beams remaining ^TE elastic. The function ...ft— for this case i s shown 2 2 ¥  L  in Fig. 10 as line 2 for f u l l y factored dead loads, and as the dotted line 2» for dead loads with the reduced load factor k = 0 . 7 5 ^ . 2  The third possibility is that both beams become plastic. The lines 3 and 3 ' in Fig. 10 indicate the function ^ ^ in this case, for f u l l y and partially factored dead loads respectively.  1  - 23 -  Lines 4 and 4  1  represent the last possibility, with  beam DE plastic and beam EF elastic.  This condition  requires a low yield stress i n beam DE, while beam EF must be oversize or have a high yield stress.  Line 4  is for f u l l y factored dead loads on beam DE, while line 4 ' assumes the same dead loads partially factored. It may appear unreasonable to assume that beam EF can ¥ L 2  2  remain elastic with an end moment of — — •  It must  be remembered, however, that only beam moments with "locked" joints are being considered, and that the end moment of beam DE w i l l decrease when the joint i s released and the moments are distributed. As mentioned previously i t i s practically impossible to predict whether the structural members of a r i g i d frame w i l l remain elastic or become plastic under the specified failure loads because of the great variation in actual yield stresses. It i s therefore equally impossible to select one of the lines of Fig. 10 as representing the correct value of the unbalanced joint moment. The writer believes i t i s basically wrong to attempt to decide which members are elastic and which are plastic i n a structural frame with "locked" joints. Fig. 10 shows the confusion which confronts the designer in this case.  The decision should not be made before  the moments have been distributed, as w i l l be discussed in the next chapter.  - 24 Much work needs to be carried out by the writers of codes and design specifications regarding the loads to be considered i n plastic design.  Serious consider  ation should be given to the use of a reduced load factor for dead loads where minimum loading governs the design.  - 2 5 -  (D) The Moment D i s t r i b u t i o n When the unbalanced j o i n t moments have been determined as discussed i n the preceding chapter, they must be d i s t r i b u t e d to s a t i s f y the c o n d i t i o n o f s t a t i c s . A simple approximate procedure f o r the d i s t r i b u t i o n of an unbalanced j o i n t moment to the members connected t o the j o i n t i s o u t l i n e d i n Ref. 3 .  The method ignores moments  c a r r i e d over from adjacent j o i n t s , and attempts t o compensate f o r t h i s by overestimating the s t i f f n e s s of the columns.  This i s done by assuming the beams hinged  at t h e i r f a r ends, w h i l e the columns are assumed hinged at t h e i r f a r ends o r a t midheight, depending on whether they are bent i n s i n g l e o r double curvature r e s p e c t i v e l y . The w r i t e r has found the method t o y i e l d good r e s u l t s i n several cases, but i t should not be considered u n i v e r s a l . In many cases i t may not give adequate r e s u l t s unless i t i s modified t o s u i t p a r t i c u l a r c o n d i t i o n s . In the f o l l o w i n g a step by step procedure, used by the w r i t e r t o determine end moments i n columns i n the Examples, w i l l be explained.  The procedure attempts  to eliminate the e f f e c t of some of the unknown f a c t o r s described p r e v i o u s l y , and takes the form of an a n a l y s i s , i . e . the s i z e s o f a l l members are assumed known. As mentioned i n the preceding chapter, the w r i t e r r disagrees w i t h the p r i n c i p l e of assuming d e f i n i t e p l a s t i c  r  - 26 hinge locations before a moment distribution has been carried out.  The f i r s t step ofthe procedure i s there-  fore a moment distribution assuming a l l members to remain elastic.  In this case only lines 2 and 2' of  Fig. 10 need to be considered, depending on whether the dead load i s f u l l y or partially factored. The second step consists of assigning yield stress values to a l l members of the frame, except the column under consideration.  The yield stress values w i l l be  within a predetermined possible range, e.g. 25 ksi < cr ^ 4# ksi, and w i l l be chosen such that the most unfavourable condition of bending is produced for the column under consideration.  For the center columns  in Fig. 9 this would be achieved by assigning the maximum yield stress to beam EF and the minimum yield stress to beam DE, with the intent of keeping beam EF elastic and making beam DE plastic. The yield moments M  and the plastic moments M  of the  beams are computed from the assigned yield stress values and compared with the elastic moments found by the moment distribution of the f i r s t step. Wherever the elastic moments are equal to or smaller than the yield moments M , the beams remain elastic. At y' locations where the elastic moments are equal to or greater than the plastic moments i t i s assumed that plastic hinges  - 27 -  form.  The differences between the elastic moments and  the plastic moments are redistributed at these locations. In certain cases i t may be found that the elastic moment i s greater than the yield moment My, but smaller than the plastic moment Mp.  In such cases some inelastic deform-  ation w i l l take place resulting i n a reduction of the bending moment below the elastic value. To determine the bending moment accurately would be very d i f f i c u l t since the deformation i n this range could be s i g n i f i cantly affected by the difference between the upper and the lower yield points.  It i s sufficient to chose either  the elastic moment or the yield moment My, depending on which i s the safer with regard to bending i n the column. In most cases i t i s found that plastic hinges i n beams form f i r s t at the beam ends.  The elastic analysis may,  however, indicate that plastic hinges w i l l form f i r s t within the beam spans.  This can occur when the s t i f f -  ness of a beam i s great relative to the stiffnesses of other members connected to i t s ends, as indicated by Fig. 7. In such cases i t i s necessary to redistribute the moments within the particular span, thereby reducing the positive moment from the elastic value to the plastic moment Mp, and increasing the end moments. This redistribution can be performed very easily when the two end moments found by the elastic analysis are of equal  -  23 -  magnitude, i . e . when the plastic hinge forms exactly at midspan.  If, after the elastic analysis, the midspan  moment i s M > Mp andthe end moments are Mg, the difference c  M  Q  - Mp i s simply redistributed at midspan by reducing  the midspan moment to Mp and increasing the end moments to M  E  + M  c  - Mp.  Generally, however, the plastic hinge w i l l not be located exactly at midspan, and the redistribution becomes very difficult.  Deformations must be computed, and, since the  two end moments w i l l not increase by the same amount, the location of the maximum span moment w i l l change. The writer has not pursued this problem further as i t does not occur i n the Examples. The columns are considered i n a similar way. Here the reduced plastic moments M^ and the reduced yield moments M^  are used. It may become necessary to estimate the  normal forces since their values are affected by the end moments in the beams. This i s however not usually d i f ficult.  At times i t w i l l be more convenient to compute  the elastic fibre stresses due to the normal forces and the bending moments from the f i r s t elastic moment distribution, and compare these with the assigned yield values to determine whether the member i s elastic or not. During the moment distribution and redistribution, the column under consideration i s assumed to remain elastic. The f i n a l column moments are then the maximum possible.  -  29  -  If i t i s found that plastic hinges may form i n the column, i t s adequacy w i l l also be checked for this condition, using the reduced plastic moment  as the  maximum end moment. The procedure can be summarized briefly as follows: First step.: The most unfavourable loading condition for the column i n question i s assumed, and the distribution of bending moments determined assuming the frame to remain elastic. Second step: Yield stress values are assigned to a l l members except the column under consideration, and bending moments are redistributed until they do not at any location exceed the plastic hinge moments of the respective members. The procedure w i l l be illustrated further in the Examples.  -  III (A)  30 -  THE COLUMN DESIGN METHODS General  In the preceding chapter i t was shown that end moments i n columns can deviate greatly from those computed with the usual assumptions of the plastic design method. In the following the effects of such deviations on the design and safety of the column w i l l be discussed. Consider f i r s t a simply supported beam designed for a yield stress of 33 ksi and a load factor k^= 1 . 8 5 .  If the actual  yield stress of the beam i s greater than 33 ksi or the beam is oversized, i t s factor of safety w i l l be greater than 1 . 8 5 . If the yield stress i s only 2 5 k s i , the beam w i l l s t i l l have a remaining factor of safety of 1 . 4 0 against overloading and other effects, i . e . i t w i l l s t i l l be capable of supporting a load equal to 7 5 . 9 percent of the specified failure load before failure occurs.  The beam i s unaffected by the  sequence of load application, as the loads may be applied to parts of the span i n an arbitrary manner. In the case of columns forming parts of r i g i d frames the conditions may not be the same. Columns are acted upon at their ends by normal forces and bending moments which vary i n magnitude during the application of loads to the structure. As long as the loads are small and increase i n constant ratios to each  - 31 other, there i s virtually a proportionality between the end moments and the normal force i n a column, i . e . the eccentricities at the column ends remain constant.  As  the loads increase further the normal force w i l l affect the bending moments i n the column because the column axis no longer i s straight.  This has the effect of  reducing the flexural stiffness of the column and w i l l normally cause the column end moments to increase at a slower rate than the normal force. The eccentricities at the column ends no longer remain constant, and the principle of superposition i s no longer valid.  This  phenomenon i s usually ignored i n the design of building structures, and w i l l also be ignored i n the following. When the loads reach such magnitudes that plastic deformations take place at certain locations in the structure, the rate of increase of the column end moments may be greatly affected. Considering the simple portal frame shown i n Fig. 1, the formation of plastic hinges at the beam ends w i l l make the moments at the upper column ends remain constant while the normal force continues to increase with further loading of the beam. If the plastic hinges form in the columns, the end moments w i l l decrease with further increase of the loads because the plastic section modulus Z£ of the columns decreases.  - 32  The sequence of  load application can have similar effect  on the eccentricities at the column ends.  Fig. 11 shows  a two storey r i g i d frame, designed to support certain live and dead loads on both beams. It i s assumed that the most severe condition for the lower columns i s represented by f u l l loading of both beams. This condition can be arrived at i n several ways. One i s to assume, that a l l loads are applied gradually at the same time.  The normal forces and the end moments in  the lower columns w i l l then increase proportionally (ignoring the reduction of the column stiffness discussed previously) u n t i l the yield stress i s reached somewhere i n the frame. Another way i s to assume a l l dead loads applied f i r s t , and that live loads are then applied to the lower beam only.  The end moments i n the lower columns w i l l at this  stage have their maximum values, and they w i l l decrease when live loads are subsequently applied to the upper beam, while their normal forces w i l l continue to increase. The variation of the eccentricities at the column ends during the load application w i l l be quite different i n the two cases, although the conditions may become identical when a l l loads f i n a l l y have been applied. It w i l l be shown later that the factor of safety of columns under certain conditions depend, not only on  - 33 the f i n a l end moments and normal forces under f u l l loads, but also on the variations of the eccentricities at the column ends before the loads have been applied fully.  In such cases columns are quite different from  the simply supported beam, which i s unaffected by the way the loads are applied. In order to discuss the factors of safety of columns under different conditions i t becomes necessary to assume failure c r i t e r i a for columns subjected to combined bending and normal force.  The c r i t e r i a used are  the two methods referred to earlier, developed by research groups at Lehigh University and Cambridge University respectively, Ref. 8 i s a progress report on the design of compression members by a joint committee of the Welding Research Council and the American Society of C i v i l Engineers. The recommendations in the report are the same as the ones of the Lehigh University research group.  This  report forms the basis for the plastic design of columns contained i n the specifications of the American Institute of Steel Construction and the Canadian Standards Association, and thereby governs most practical applications in North America.  The recommended design method w i l l  therefore in the following be referred to as the "American Method". The method developed by the Cambridge University research  -  34 -  group as described i n Ref. 1 and Ref. 3 , w i l l i n the following be referred to as the "Cambridge Method". The  two methods are completely different i n scope, based  on different assumptions, and with different design procedures.  Both assume some imperfection i n the column, the  American method a certain pattern of residual stresses, based on experimental evidence, and the Cambridge method an assumed i n i t i a l curvature.  The American method only con-  siders columns bent about the major axis, and assumes the presence of adequate bracing to prevent lateral-torsional buckling and force the columns to f a i l i n the plane of bending.  The Cambridge method i s quite general, but has no  solution for some of the many possible combinations of end moments. Failure i s assumed to occur through l a t e r a l torsional buckling. There are many restrictions on the use of the American method at present.  The structural frame may not exceed two  storeys i n height, the column loads may not exceed 60 percent of the yield load N , and the slenderness ratios of the columns must be within certain specified limits.  Obviously  the originators have f e l t that there s t i l l i s insufficient knowledge about the general behavior of columns at failure, and have chosen a careful approach. The Cambridge method on the other hand contains very few restrictions on i t s use, and gives the impression of being quite generally applicable.  ~35It should be pointed out that the American method has been written i n the form of a building code or design specification, intended to have a certain legal status, whereas the Cambridge  i method, as i t i s known to the writer, i s i n textbook form and as such only has academic status.  It i s quite possible that  the Cambridge method, when incorporated i n building codes, also w i l l have many restrictions on i t s use. Both the above mentioned design methods w i l l for a given normal force indicate the maximum end moments which can be resisted by a column.  It i s assumed that the normal force and the end  moments are increased gradually until the failure condition i s reached.  The methods do not attempt to consider the interaction  between a column and the other members of a structural frame at the approach of failure, when end moments often decrease and even change direction, thereby enabling the column to resist a greater load before i t f i n a l l y f a i l s .  The column i s considered  isolated and loaded by gradually increasing normal force and end moments which are maintained until the column collapses. This assumption i s undoubtedly conservative in many cases when columns form parts of r i g i d frames. However, since more accurate failure criteria are not available, the writer w i l l i n the f o l lowing assume that the two design methods indicate true conditions of failure. This assumption i s necessary i n order to make i t possible to discuss the factors of safety of columns under different conditions. When therefore, i n the following, "failure" is referred to, this means "failure" as indicated by the above design methods, and not necessarily the true collapse condition.  - 36 -  (B)  The American Method  It was already mentioned that the American method i s very limited in i t s scope and contains many restrictive requirements.  When the structure can be made to comply  with a l l the requirements, however, the method i s simple and easy to apply, and lends i t s e l f to practical design. The design recommendations give expressions for the allowable end moments of columns under a certain axial force.  Depending on the ratio fb of the end moments  three different cases are considered. Case 1; The columns are bent in double curvature by end moments of numerically equal magnitude, fb = -1. The maximum allowable end moment M  Q  M  expressions  i s given by the  P  ^0 = 1.18 ( 1 -  M  y  ) and ^0 < 1.0  where P i s the axial load and Py i s the yield load which would produce yield over the whole cross-section of the column. Case 2: The columns are bent in double curvature by unequal end moments, -l</3< 0. end moment M  Q  M  o = B - G—£  The maximum allowable  i s given by the expressions and o M  3080  < 1.0  185000  and y  G = 1.11 +  r 190" L /  ( 4) L  -  2  9000  720000  i s the slenderness ratio of the column i n the plane  of bending. Case 3 :  The columns are bent i n single curvature by  unequal end moments, 0<(b<  + 1.0.  The maximum allow-  The slenderness ratio L/ may not exceed 100 for Case r  1 columns, and 120 for columns of Cases 2 and 3* The reason for limiting the slenderness ratio of Case 1 columns to L/ = 100 i s because very slender columns r  have a tendency to "unwind" from double to the more severe single curvature bending. The design expressions above are approximations of interaction curves for an 3WF31 section, calculated by an iterative procedure.  In the calculations the column  was considered isolated, subjected to a constant normal force and end moments of increasing magnitude. There i s only one set of expressions independent of whether the column ends are elastic or plastic, and the designer does not have to choose between two d i f ferent methods for the two different conditions.  - 3d -  The interaction curves computed from the above design expressions are shown i n Fig. 12 for Case 1 and Case 2 columns, and i n Fig. 13 for Case 3 columns. The dotted curve shown i n both figures represents the values of the plastic hinge moment  of the column section i n  Fig. 2. The design expression for Case 1 columns i s an approximate expression for the plastic hinge moment  of the  column. It i s seen i n Fig. 12, that the solid line representing Case 1 bending i s f a i r l y close to the dotted curve representing the plastic hinge moment M^, particularly for heavier column loads. The error introduced by this approximation i s not on the side of safety.  It i s the intention to allow plastic hinges  to form at the ends of Case 1 columns for a l l slenderness ratios and normal forces within the stated limits. Because of the uncertainty i n determining end moments i n columns of rigid frames, the writer considers Case 1 bending unrealistic and not suited for practical design. There does not seem to be any reason why the end moments should remain equal although the usual simple analysis may indicate that they do.  It w i l l not require much  variation i n the physical properties of the structural members to transfer a column from Case 1 bending into the more severe condition of Case 2 bending.  For slender  columns the difference between the conditions i s considerable, especially for higher normal forces, as can be observed  from Fig. 12. Since, however, the design expression for Case 2 columns i s supposed to be valid for a wide range of moment ratios /3 , i t must necessarily be somewhat conservative at the lower limit of the range, i . e . when (h i s close  to -1.  The unexpected transfer of  a column from Case 1 bending to Case 2 bending w i l l therefore i n many cases not be as serious as indicated by Fig. 12. The writer believes there i s merit i n omitting the present Case I bending from the design specifications, and instead divide the present Case 2 into two ranges,  -16fi<  -0.5 and -0.5£/3< 0, with  two different design expressions.  The present design  expression for Case 2 would be used for the range -0.5£/3< 0, and a new, less conservative expression for the range -1 i/3<-0.5. A comparison of the interaction lines for Case 2 bending in Fig. 12, with the dotted curve representing the plastic hinge moment  shows, that according to the American  method most Case 2 columns w i l l f a i l before plastic hinges can develop at their ends.  Only for relatively  light column loads can such plastic hinges develop. A similar comparison of the curves i n Fig. 13 shows that plastic hinges cannot form at the ends of Case 3 columns according to this method.  - 4P -  T h i s means t h a t i f t h e column end moments become g r e a t e r than a n t i c i p a t e d i n the d e s i g n , p l a s t i c hinges w i l l i n most cases not form a t t h e column ends and r e l i e v e the column from b e i n g bent e x c e s s i v e l y . Unless such columns happen t o be o v e r s i z e d f o r some reason, even s l i g h t unexpected  i n c r e a s e s o f t h e end moments beyond t h e v a l u e s  assumed i n t h e d e s i g n must t h e o r e t i c a l l y  cause premature  failure• The d i s t r i b u t i o n o f bending moments over the l e n g t h o f a Case 2 column i n the l i m i t i n g case o f fb = 0 i s i d e n t i c a l w i t h t h e d i s t r i b u t i o n o f bending moments over the upper and lower h a l v e s o f a Case 1 column. be expected t h a t the i n t e r a c t i o n  I t would t h e r e f o r e  curve f o r a Case 2  column w i t h the s l e n d e r n e s s r a t i o L /  r  •» 5 0 should be  i d e n t i c a l w i t h t h a t f o r Case 1 columns, s i n c e t h e l a t t e r i s v a l i d f o r a l l s l e n d e r n e s s r a t i o s up t o L/ = 1 0 0 . r  F i g . 1 2 i n d i c a t e s t h a t t h e d i f f e r e n c e between t h e f a i l u r e l o a d s o f these two cases i s about 1 0 p e r c e n t . ference i s introduced expressions. with L/  r  This d i f -  by s i m p l i f i c a t i o n o f the d e s i g n  The i n t e r a c t i o n  curve f o r a Case 2 column  - 5 0 i s s l i g h t l y below the i n t e r a c t i o n  curve  f o r t h e p l a s t i c hinge moment M * w h i l e t h e i n t e r a c t i o n curve f o r Case 1 columns i s s l i g h t l y above the curve for As an example c o n s i d e r the frame shown i n F i g . 1 4 , where the beams a r e simply supported a t t h e i r o u t e r ends and  - 41 -  rigidly connected to the column at their inner ends. The column has a slenderness ratio of L/r«»100, and i s bent i n single curvature when the beams are loaded. uniformly distributed failure load i s W  FQ  The specified  on each beam.  It i s assumed that the collapse of the structure w i l l occur through column failure. Suppose the frame designed i n such a way that failure of the column i s expected to occur for the combination of normal force and end moments corresponding to point A i n Fig.  13.  The behavior of the frame before failure depends  on the relative flexural stiffnesses and the physical properties of the members. I f the eccentricities at the column ends would remain constant while the loads are applied, the end moment-normal force relationship would be as indicated by the straight line OA i n Fig. 13. This would produce economical beam sections i f the relative flexural s t i f f nesses allow the elastic moments at the beam ends to equal those i n the spans. If the elastic end moments are considerably greater than those in the span i t w i l l be more economical to l e t plastic hinges form at the beam ends before the f u l l design loads have been applied. In this case the moment-normal force relationship at the column ends may be approximately as indicated by the broken line OBA i n Fig. 13. Here the end moments increase at a faster rate to begin with u n t i l plastic  -  42  hinges form at the beam ends (point B i n Fig. 1 3 ) , and then remain constant while the loads are increased further until the column f a i l s (point A).  I f the yield stresses  of the beams are considerably higher than anticipated i n the design the plastic hinges may not form at the beam ends, and the end moments may continue to increase, resulting i n premature column failure at point C i n Fig. 1 3 . In this ease the increased strength of the beams reduces the failure load of the column. It can be questioned whether this reduction of the failure load of the column also reduces the failure load of the structure as a whole.  It can easily be shown that this  is the case for the structure shown i n Fig. 1 4 . From simple statics the expected failure load P of the column Q  can be expressed P  c  - -rgSl + ^  , from which W  F  Q  = 2(P -  M  Q  Q  i s the  numerical value of the expected column end moments at failure, P  Q  and M  Q  correspond to point. A i n Fig. 1 3 .  Similarly the load on each beam at the actual failure represented by point G i n Fig. 13 i s W  F  - 2 ( P - ^F ), l p  L  where Pp is the actual failure load of the column and Mp the corresponding numerical value of the column end moments. Since Pp < P  Q  and Mp > M  0  i t can easily be seen from the  expressions above that Wp < Wp , i . e . that the structure Q  as a whole w i l l suffer premature failure.  - 43 -  If the relative flexural stiffnesses of the members are such that the elastic beam moments are greater in the spans than at the inner beam ends, plastic hinges may form in the beam spans before the f u l l design loads are applied. Such a case i s illustrated by the broken line ODA i n Fig. 13*  Here the end moments increase at a slower rate to  begin with (line OD), and then at a faster rate after the plastic hinges have developed i n the beam spans (line DA). If i n this case the yield stresses of the beams are higher than anticipated i n the design, and the plastic hinges do not form, the end moments may continue to increase at the same rate (line DE) u n t i l column failure occurs at point E. The column i s then capable of supporting a greater load at failure because the end moments are lower than anticipated.  In this case Pp > P and Mp < M , and Q  Q  by the same reasoning as used before the actual load on each beam at failure i s W  p  > Wp , i . e . the structure i s Q  strengthened by the increased beam strengths. Thus i t i s shown that the variations of the eccentricities at the ends of columns during loading can affect the safety of structures when the yield stresses of certain members are greater than assumed in the design.  As already men-  tioned the sequence of load application can produce similar variations of the end eccentricities. The conclusions above refer to the rigid frame shown i n  -  44  -  Fig. 1 4 , and are not necessarily valid for other types of structures. Also the relationships of column loads to column end moments during loading as indicated by the dotted lines i n Fig. 13 are somewhat arbitrary.  Since,  however, the interaction curves of the American method as shown in Fig. 12 and Fig. 13 are determined for an isolated column of a specific cross section, they do not necessarily represent true failure conditions of columns forming parts of r i g i d frames.  A more accurate  numerical evaluation of the above example therefore has no purpose, and the writer has consequently only attempted to illustrate the possible behavior of the frame in principle. It i s seen from Fig. 1 3 , that the reduction of the load factor of the column at failure i s produced by conditions causing the column end moments to increase at a high i n i t i a l rate, e.g. as represented by the dotted line OB.  To be safe one should therefore assume that the l i v e  loads which tend to increase the column end moments greatly are applied f i r s t , and the other live loads later. It was assumed above, that the structure in the example is designed with care, and that the anticipated failure condition as represented by point A i n Fig. 13 i s the true failure condition provided the actual yield stresses of the beams are not different from the values assumed in  -  the' design.  45 -  I f a more a r b i t r a r y design procedure i s  employed such as assuming the end moments equal to the "required" p l a s t i c beam moments without considering the beam sections a c t u a l l y used, the f i n a l end moments can be greatly underestimated.  This means that the actual  end moments w i l l increase at a f a s t e r rate during loading than anticipated i n the design, e.g. as indicated by l i n e OF i n F i g . 13. In such cases the provision of stronger beams than required can cause premature f a i l u r e .  In  more complex structures inaccurate moment d i s t r i b u t i o n methods can have s i m i l a r e f f e c t s , although t h i s i s hardly so i n the considered  example because of the s i m p l i c i t y  of the frame. It was assumed i n the discussion above, that the y i e l d stress of the column i s 33 k s i , since the i n t e r a c t i o n curves i n F i g . 12 and F i g . 13 are only v a l i d i n t h i s case.  In order to discuss the behavior of columns with  y i e l d stresses d i f f e r e n t from 33 k s i , i t becomes necessary to assume f a i l u r e c r i t e r i a f o r such columns.  It i s in-  dicated i n Ref. 8, that the i n t e r a c t i o n curves i n F i g . 12 and F i g . 13 can be used as approximations f o r columns having y i e l d stresses the slenderness slenderness  (T d i f f e r e n t from 33 k s i , provided  r a t i o used i n the design i s the actual  r a t i o of the column m u l t i p l i e d by the f a c t o r  — Z • Although t h i s i s only an approximation, i t w i l l  - 46 -  be assumed to represent a f a i r l y accurate failure criterion for the purpose of this discussion. The assumed failure criterion indicates that the resistance of a column against failure does not increase linearly with an increase of the yield stress. slenderness ratio  A column having a  = 1G0 and a y i e l d stress o" = 4 3 ksi  w i l l be designed for a modified  y  slenderness ratio column with a yield  stress  C_ ™ 25 ksi w i l l be designed for a modified slender-  Fig. 15 shows three interaction curves for the above column subjected to Case 2 bending, corresponding to the yield stresses 25 k s i , 33 k s i , and 4 3 k s i respectively.  Also  shown are three dotted curves representing the relationships between the column loads and the bending moments for which plastic hinges form at the column end. The dotted curves are determined for the column section shown i n Fig. 2 . The column loads P and the bending moments M  Q  been related to the yield load  an<  *  Mp33 corresponding to the yield stress  t h e  Pl  and a s t i c  have moment  <5y * 33 k s i to f  make a direct comparison possible. The behavior of columns with yield stresses different from the value assumed i n the design w i l l now be discussed, based on the interaction curves i n Fig. 15 for a Case 2 column with slenderness ratio L/r = 1 0 0 .  - 47 -  Suppose that the column is part of a r i g i d frame which has been analyzed carefully assuming the yield stress equal to 33 k s i i n a l l members, and l e t the computed failure condition be as indicated by point A i n Fig. 1 5 . If the actual yield stress of the column i s only 25 k s i , the column w i l l f a i l before the anticipated failure condition (point A) can be reached, and the failure load w i l l depend on the variation of the eccentricities at the column ends during loading as discussed previously. If the structure i s such that the end eccentricities remain constant during loading (line OA), Fig. 15 indicates that failure w i l l occur at point B. However, the structure can also be such that the column end moments increase more rapidly to begin with and reach their maximum values before the f u l l design load i s applied.  Such a  case i s illustrated i n Fig. 15 by the broken line OCA for a column having a yield stress of 33 k s i . I f the yield stress of the column i n this case i s only 25 ksi, failure w i l l occur at point D. Since the column load at point D is less than at point B, the two assumed structures do not have the same factor of safety. It was stated previously that a simply supported beam, designed for a yield stress of 33 k s i , i s capable of supporting approximately 7 6 percent of i t s design load i f i t s actual yield stress happens to be only 25 k s i . In Fig. 15 the  - 43 -  column load at any point B on the interaction curve for <5y = 25 ksi i s always greater than 7 6 percent of the column load at the corresponding point A on the interaction curve for  (T =» 33 k s i . This means that provided the eccent-  r i c i t i e s at the column ends remain constant during loading, the column w i l l have a greater safety against unexpected low yield stress values than the simply supported beam. This i s not necessarily so when the eccentricities at the column ends vary during loading.  E.g. the column  load at point D i n Fig. 15 can possibly be less than 76 percent of the load at point A. In such cases the column can have less safety against unexpected low yield stresses than the simple beam. This i s even more so i f the column end moments are underestimated by the use of arbitrary design methods as discussed previously. The interaction curves i n Fig. 15 indicate that below certain values of the column load P plastic hinges w i l l form at the ends of the columns before failure occurs. These values are approximately 0 . 1 1 5  ^y^y  0 * 1 3 7 Py-j^>  and O . I 6 5 y 3 3 > corresponding to the yield stresses P  2$ k s i , 33 k s i and 4 3 k s i respectively.  Fig. 15 indicates  that the safety of columns designed for loads P < 0 . 1 3 7 P _ _ V  w i l l not be affected by the variation of the eccentricities at the column ends during loading.  I f the yield stress i n  the column i s only 25 k s i , i t s failure load w i l l always be at least P=0.115Py^^ according to Fig. 1 5 , provided the  -  49 -  column has sufficient rotation capacity. A rapid i n i t i a l rate of increase of the column end moments w i l l cause the plastic hinge to form for a smaller normal force, e.g. at point E i n Fig. 1 5 , but for further increase of the column load the column end moment w i l l decrease along the dotted curve of the plastic hinge moment occurs for P = 0 . 1 1 5 P of 0 . 1 3 7 ^ 3 3 ' **  he c  °l  until failure  .. Since this load i s 84 percent  u m n w  i H apparently have a greater  safety against unexpected low yield stress values than the simply supported beam. It must be realized, however, that the interaction curves in Fig. 15 are crude approximations of failure conditions, particularly for very light column loads.  The numerical  values of column loads determined from Fig. 15 and the resulting conclusions regarding the behavior of lightly loaded columns are therefore undoubtedly highly inaccurate. They have been included here to illustrate the implications of the American method, but should not be considered as an indication of the true behavior of columns.  -  (C)  50 -  The Cambridge Method  Of a l l the different combinations of end moments possible, only the case of columns subjected to bending about the major axis w i l l be considered i n the following.  The reader  i s referred to Ref. 1 for a complete explanation of the method, and for the necessary tables and graphs without which the method would become very laborious.  Only what  i s required for the purpose of this discussion w i l l be explained here.  The notations used w i l l in general  be those used i n Ref. 1 , except when these contradict those normally used i n plastic design i n North America. When a column i s checked for i t s a b i l i t y to resist a certain combination of normal forces and end moments by the Cambridge method, the elastic fibre stresses at the column ends are f i r s t computed.  If these stresses do  not exceed the lower yield stress f  L  , usually assumed  to equal 1 5 * 2 5 tons/sq i n . (approximately 3 4 ksi), the column i s said to have elastic ends.  I f the yield stress  is exceeded at one or both ends, the column i s said to have one or both ends plastic respectively.  The column  is assumed to f a i l i n lateral torsional buckling when the maximum fibre stress at midheight of the column f i r s t reaches the lower yield stress f . The condition to be L  satisfied to prevent failure can be written: p + N f x  x  + G <f . L  - 51 -  The combined stress p+N f + C i s the maximum compressive stress at midheight of the column and occurs only at one p t i p of one flange of the section.  p==j i s the mean axial  stress due to the normal force P, and f  i s the fibre  stress at midheight caused by the external end moments and  , which bend the column about i t s major axis.  This bending deforms the column, and produces eccentricities of the load P throughout i t s length, resulting i n additional bending moments and increased deformations. The increased fibre stress at midheight due to this additional bending i s taken into account by the magnification the product N f  factor N . Thus  i s the fibre stress at midheight caused  by the end moments  and M £ combined with the bending  moment at midheight produced by the eccentric column load P, a l l bending the column about i t s major axis. The stress C i s the bending stress at the flange tips due to bending about the minor axis.  Since i t i s assumed  here that the column has no external end moments about i t s minor axis, the bending about this axis i s produced by the eccentricities of the column load only.  It is  assumed i n Ref. 1, that the column has a certain i n i t i a l curvature i n the plane of the major axis, and that this curvature increases when the load i s applied.  The increase  depends on whether the column ends are elastic or p l a s t i c . With elastic ends the eccentricity at midheight and the  - 52 -  stress C are smaller than when the column has plastic ends.  Gf the three terms of the expression for the  maximum fibre stress at midheight of the column, only the stress C depends on whether the column ends are elastic or plastic.  The other two terms are not affected  by this condition. The method assumes that the unsupported length 1  of the  column with regard to bending about the major axis i s equal to the unsupported length L with regard to bending about the. minor axis, i.e. L ^ L ^L. form  not be applied to cases where L  It can i n i t s present i s different from  L , unless either L =0 or L = 0, which would mean that y x y the column i s prevented from deflecting i n one of the two principal directions.  With L =0 the stress at column  midheight due to bending about the major axis becomes N f = G, and with L = 0 the stress due to bending about the minor axis becomes 0 = 0 . The failure equation w i l l thus i n both cases be simplified. In Ref. 1 expressions are derived for N , f , and C. The expression for N  x  i s the f i r s t term of a Fourier  series taking into account the relationship between the normal force and the deflection of the column i n the plane of the web. The following approximate expression i s proposed for British standard rolled sections:  - 53 -  N  8=1 Y  1  +  ( L }2 p(300 r ) , where L/r  J*.  i s the slenderness ratio  X  of the column about i t s major axis. In order to compute the magnitude of the bending fibre stress f at midheight due to the end moments M* and X  X  M" , i t i s assumed i n Ref, 1, that these moments are equivalent to a uniform moment M^. applied  over the f u l l  length of the column. This equivalent uniform moment i s expressed through the larger end moment M  x ~  Yp-  ^x . The coefficient  as follows:  , which only  depends on the ratio /3 of the end moments, i s i n Ref, 1 given both graphically and in table form.  Fig. 16, which  is reproduced from Ref. 1, shows the relationship between ^jp and (b . The bending stress f i s computed from x  where S represents the section modulus of the column about i t s major axis. The fibre stress C caused by bending about the minor column axis i s i n Ref, 1 also expressed as the f i r s t term of a Fourier series.  The expression takes into  account the relationship between the normal force and the deflection of the column perpendicular to the plane of the  web. Fig. 17 and Fig. 18 show the stress C i n function  of the slenderness ratio L / r about the minor axis, and y  the  parameter p . Both sets of curves are taken from Ref.l, f  -  54 -  but have been rearranged for easier explanation.  The  corresponding charts i n Ref. 1 contain a greater number of curves than those shown i n Fig. 17 and Fig. 13. The f parameter p = p + x , where T i s a constant of the T 2  1  column section taking into account i t s torsional r i g i d i t y . Values of T for different British rolled sections are tabulated i n Ref. 1.  The curves i n Fig. 17 are for columns  with elastic ends, and those i n Fig. 13 are for columns with one or both ends plastic.  The letter C i s not used  in Ref. 1 as the symbol for the fibre stress due to bending about the minor axis, but i s used here for convenience. A comparison of the graphs of Fig. 17 and Fig. 13 shows that columns with elastic ends have considerably greater resistance against lateral torsional buckling than columns with one or both ends plastic.  Thus i f a column i s  designed to remain elastic under the specified failure load, but actually becomes plastic, i t s resistance w i l l be reduced and premature failure may occur.  On the  other hand a column which i s designed to become plastic w i l l , i f i t remains elastic under the specified failure load, have a greater factor of safety than anticipated in the design.  In this respect the Cambridge method i s  different from the American method, since the latter makes use of the same design expressions independent of  -  55  -  whether the column ends are elastic or plastic. Another observation which can be made from Fig. 17 and Fig. IS i s that the curves become very steep for values of C i n excess of 4 tons/sq i n . In this range the strength of the column appears to be extremely sensitive to varif . Since f i s a function of the ations of p' = p + _x JJ( x 2  end moments i t seems that these must be determined very carefully to provide an adequate factor of safety, Ref, 1 contains a number of charts for direct design of columns having plastic hinges at one or both ends.  One  such chart i s reproduced i n Fig. 19 for a column section with T » 1 0 0 , e.g. British Standard Section 1 0 inx8 i n at 55 lbs. The charts have been arrived at by making certain simplifying assumptions to the mathematical design expressions, and permit the allowable mean axial stress p to be determined directly provided the minor axis slenderness L/r , the ratio fh of the end moments, and the constant T of the section are known. These charts allov; a number of observations to be made. Contrary to the American method, the Cambridge method allows plastic hinges to form at the ends of columns bent in single curvature.  This occurs only i n the range 0^/2><+ 0 . 6 ,  i.e. when the smaller end moment does not exceed 6 0 percent of the larger end moment, and only for light column  -  56 -  Considering the British Standard Section 1 0 i n x 8 i n  loads.  at 55 lbs., the allowable column loads for fb = 40.5 are as follows: L/r  y  =  0  L/r  x  =  0  -  0.16P 0.14P O.llP  P  20  50  8.7  21.8  j  150  100  J  43.5  65.5  0.06P  J  0.02P  J  J  For a narrower section, British Standard 22 i n x 7 i n at 75 lbs, the corresponding values are: L/r  y  L/r  x  P  0  50  20  70  =  0  3.1  7.8  =  0.16P J  O.llP  0.05P  10.9  0  J  Jr  As expected, the narrow section w i l l f a i l at lower column loads than the wide section. The allowable loads increase as the smaller end moment decreases.  When i t becomes  zero, i . e . fb » 0, the allowable loads for the above sections are as follows: British Standard Section 1 0 i n x 8 i n at 55 lbs L/r  -  0  L/r  -  0  P  -  P  y  50  20 8.7 y  y  0.48P  y  British Standard Section L/r  -  0  L/r  «  0  P  =  P  3.1 y  87.0  65.5  0.24P  0.12P  y  y  130  100  7.8  0.76P  200  20.3  15.6  0.48P  y  0.13P  y  0  250 109  0.05P  22 i n x 7 i n at 75 lbs  50  20  y  43.5  21.8  0.76P  150  100  y  0.01P  y  - 57 For small values of the slenderness ratio L/r , the allowable loads are identical for both sections, but for higher slenderness ratios the allowable loads decrease more rapidly for the narrow section than for the wide section. It may appear that the American method i s more conservative than the Cambridge method by not allowing plastic hinges to form at the ends of columns subjected to single curvature bending, i . e . for 0 < (3 <  +1.  This i s because  the American method uses the same design expression for the whole range from fi = 0 to ft = +1.  The expression  must be based on the most severe case within this range, namely (3= +1, for which the Cambridge method does not allow plastic hinges to form at the column ends either. The Cambridge method i s thus more refined than the American method inasmuch as the columns can be designed for exactly the ratios of end moments to which they are subjected, provided the accurate ratios are known. Fig. 19 allows the effect of variations of the ratio (b on the allowable mean axial stress!to be determined directly for the British Standard Section 10 i n x S i n at 55 l b s . Assuming a variation of the ratio (b of 0.1, e.g. from fb = +0.4 to fi = +0.5,  i t i s seen that the per-  centage reduction of the allowable column loads i s greater  - 58 -  for l i g h t l y loaded columns than for heavily loaded columns, and greater for columns bent i n single curvature than for columns bent i n double curvature. For light column loads i n the order of 0 , 1 5 P the assumed v  variation of the ratio (i can cause a reduction of 50 percent of the allowable load for single curvature columns, whereas the same variation only causes a reduction of 10 to 15 percent i n the case of double curvature columns. Intermediate column loads i n the order of 0 . 4 0 P have y  corresponding reductions of approximately 4 0 percent and 10 to 15 percent, and for heavy column loads of say 0 . 8 0 P  the reductions are 2 0 percent and 5 to 10 percent for  single and double curvature bending respectively.  Thus  the importance of arriving at reliable values of the end moments i n columns i s again emphasized. In order to discuss the behavior of columns with yield stresses different from 1 5 . 2 5 tons/sq i n . i t again becomes necessary to establish failure c r i t e r i a for such cases.  It i s assumed that the maximum possible range  of variation of the yield stresses extends from 1 1 . 2 tons/ sq i n . (25 ksi) to 2 1 , 5 tons/ sq i n , (48 ksi), and that the Cambridge method can be extended to include such yield stresses without losing i t s validity. Because of the complexity of the Cambridge method, i t  -  59 -  cannot be discussed as easily as the American method. A specific example w i l l therefore be used to illustrate the effect of low yield stresses i n the column material. It has already been shown by the American method that the variation of the eccentricities at the column ends during the application of loads to a structure can greatly affect i t s factor of safety.  This w i l l now be illustrated  by use of the Cambridge method. Consider a column of British Standard Section 13 in.x 6 in.at 55 lbs, 13 f t 6 i n . long, forming part of a r i g i d frame. Assume that the analysis of the frame for the specified failure load shows the column bent i n double curvature by the end moments  = 1 0 0 tons f t and M£ = 10  tons f t , and subjected to the normal force P = 25 tons. Assuming the usual yield stress of f ^ = 1 5 . 2 5 tons/sq i n . the column ends are elastic under this condition, and the computed maximum fibre stress at midheight of the column is p + N X  + C - 1 . 5 5 + 6 . 9 5 + 6 . 7 5 - 1 5 . 2 5 tons/sq i n .  = f , where the stress C i s determined from Fig. 1 7 . L  Since the maximum combined stress i s exactly equal to the yield stress, the column i s just adequate without being overdesigned. Now assume that the actual yield stress of the column i s only f ^ = 1 1 . 2 tons/sq i n . Obviously the column must f a i l before the design condition i s reached, and the magnitude  -  60  -  o f t h e column l o a d a t f a i l u r e depends on t h e magnitude o f t h e column end moments, i . e . on t h e v a r i a t i o n o f t h e eccentricities question  The  i s whether the column can support a l o a d equal  to 7 3 . 5 percent a simply  a t the column ends d u r i n g l o a d i n g .  o f the s p e c i f i e d  f a i l u r e l o a d such as  supported beam can do w i t h the same reduced  yield stress. I t i s f i r s t assumed, t h a t t h e r i g i d frame behaves i n such a way under l o a d i n g t h a t t h e column l o a d and t h e end moments i n c r e a s e p r o p o r t i o n a l l y , i . e . the eccentr i c i t i e s a t the column ends remain c o n s t a n t .  The c o n d i t i o n  t o be checked i s then as f o l l o w s : M» = 7 3 . 5 t o n s f t , M" » 7 . 3 5 tons f t , and P - 1 8 . 3 5 tons, where t h e moments and the normal f o r c e a r e 7 3 . 5 percent of the design values.  Computations show t h a t the column  ends a r e f u l l y e l a s t i c f o r t h i s c o n d i t i o n , and the computed f i b r e s t r e s s a t column midheight i s p+N f x  x  + C - 1.14 +5.10+1.15 -  7.39  tons/sq  in.<f  L  I t seems t h a t t h e column i s f u r t h e r from f a i l u r e under t h i s c o n d i t i o n than under the o r i g i n a l d e s i g n  condition.  T h i s i s because o f t h e great r e d u c t i o n o f the s t r e s s C due t o t h e l i g h t e r column l o a d . ponents p and N f  The other s t r e s s com-  decrease p r a c t i c a l l y a t the same r a t e as  the y i e l d s t r e s s . Now suppose t h a t the r i g i d frame behaves such d u r i n g  loading,  - 61 -  that the column end moments i n i t i a l l y increase at a much faster rate than the column load. has been explained previously.  How this can occur  When i n this case the  column load reaches P = 18.35 tons, the end moments w i l l be greater than  = 7 3 . 5 tons f t and M £ = 7 . 3 5 tons f t ,  e.g. M£ = 85 tons f t and M £ = 8 . 5 tons f t , a not unreasonable increase of 1 5 . 5 percent.  Computations show that  for P= 18.35 tons the yield stress f ^ «= 1 1 . 2 tons/sq i n . w i l l be reached at one end of the column for an end moment of 7 8 . 5 tons f t , i . e . this end must according to the Cambridge method be considered plastic, although a plastic hinge has not been f u l l y developed.  The stress C can  therefore no longer be determined from Fig. 1 7 , but must be determined from Fig. 18. For M £ » 85 tons f t , M£. 8 . 5 tons f t , and P = 18.35 tons the computed maximum fibre stress at column midheight becomes P+N f + C = x  x  1 . 1 4 + 5.38 + 1 0 . 5 0 - 1 7 . 5 2 tons/sq i n . Since this computed stress i s much greater than the yield stress f  L  = 1 1 . 2 tons/sq i n . the column must f a i l before the  assumed condition i s reached.  In this case the column  is not able to support 7 3 . 5 percent of the specified failure load. The above example shows that the factor of safety of a column which has a lower yield stress than anticipated in the design can be greater or smaller than that of a  - 62 -  simply supported beam, depending on the ratio of the column end moments to the normal force during the application of loads to the structure. From this and other similar examples the writer has found that according to the Cambridge method columns do not possess less safety against unexpected low yield values than a simply supported beam, unless the ratio of the end moments to the normal force i s greater during loading than at the anticipated failure condition, or the lower yield stress causes the column ends to become plastic when they are expected to remain elastic according to the design analysis. In spite of the difference between the two- approaches, the Cambridge method i n general confirms the conclusions found by the American method. The importance of determining the column end moments accurately i s emphasized even more by the Cambridge method because the exact value of the moment ratio (3 i s used i n the design.  - 63 -  (D)  Comparison of the Two Design Methods  Since the American and the Cambridge methods are based on completely different assumptions a direct comparison i s not possible i n a general form. The failure conditions considered by the American method can be looked upon as special cases of the more general Cambridge method, and the most convenient way of obtaining a direct comparison is therefore to reduce the Cambridge method to the same special cases. It must f i r s t be assumed that lateral-torsional buckling is prevented by adequate bracing, and that the columns therefore w i l l f a i l by bending about their major axes. The stress C due to bending about the minor axes becomes zero, and the failure criterion can be written P  +  Vx  -  f  L  *  As i n the general case failure i s assumed to occur when the maximum fibre stress at midheight of the column reaches the yield stress f . Since the stress C i s no L  longer included in the expression i t i s of no consequence whether the column ends are elastic or plastic. The design expressions of the American method were derived for columns of 8WF31 section and a yield stress of 33 k s i . The same section and yield stress w i l l be used i n the comparison.  The properties of the considered column section  - 64 are as f o l l o w s : A - 9.12 i n . , S = 27.4 i n . , r = 3.47 i n . , Z = 30.48 i n . 2  3  x  x  3  p  With f = 33 k s i the p l a s t i c moment i s Mp= 1007 k i p i n . , L  and the y i e l d load P = 301 k i p s .  The modulus of e l a s t i c i t y  y  i s assumed as E = 29000 k s i . In order t o e s t a b l i s h i n t e r a c t i o n curves f o r the Cambridge method s i m i l a r t o those of the American method, the b a s i c failure c r i t e r i o n i s written  f  =  L  p  .  Values o f the normal f o r c e P are assumed, and the mean P  a x i a l s t r e s s p*  i s computed.  N  x  i s determined from t h e  f o l l o w i n g expressions which are taken from Ref. 1: N  x  l* J, where  = 1  When f  r  ^ =  TC  kjH P 2  2 E  i s determined, the maximum allowable equivalent  uniform moment M  i s obtained from the expression M = f S .  J*.  J v .  The maximum allowable column end moment i s then found from the expression M*. = M  x  "VF*-,  r a t i o £ o f the end moments. between |3 and  yjjr  where  VF"  depends only on the  F i g . 16 shows the r e l a t i o n s h i p  • The values o f H». found t h i s way  represent the values o f the l a r g e r column end moments which, combined w i t h the assumed column loads, w i l l j u s t produce y i e l d at the extreme f i b r e o f the column a t midheight. The cases o f bending considered are the f o l l o w i n g : Case 1:  M  x  - -M£ , [2> = -1  - 65 -  Case 2 : M" = 0 ,  /J - 0  Case. 3 : M£ -  /3 - +1  ,  Case 1 considers the column bent i n double curvature, while Cases 2 and 3 represent single curvature bending. The computed values are listed i n Tables 3 , 4 , and 5 for the three different cases of bending.  A l l units are  in kips and inches. The corresponding interaction curves are shown by solid lines in Figs. 2 0 , 2 1 and 2 2 . The dotted lines represent the interaction curves of the American method, as well as the plastic hinge moment of the column as noted.  It i s seen that some of the  interaction curves l i e above the dotted curve representing It i s of course impossible that the end moments can exceed the plastic hinge moment M£. The interaction curves for the Cambridge method merely indicate the end moments for which failure w i l l occur at midheight of the column, without considering the conditions at the column ends.  Wherever the interaction  i t means that plastic hinges w i l l form at the column ends before failure occurs at column midheight.  The maximum  possible end moment Fig. 2 0 shows the interaction curves for Case 1 bending. It i s seen that the two design methods give practically identical results for slenderness ratios L/r < 80 as long as  - 66 P £ 0.60P , which i s the heaviest column load allowed by the American method.  Both methods allow plastic hinges  to form at the column ends i n this range. For more slender columns, e.g. L / r = 100, the Cambridge method becomes more x  conservative for column loads between 0 . 4 P  V  and 0 . 6 P  and does not allow plastic hinges to form at the column ends.  The American method allows this for a l l slenderness  ratios up to L/r = 1 0 0 .  The Gambridge method, however,  allows much greater column loads than the American method for columns with small end moments, e.g. for for L/r = 30, and H» < 0 . 2 5 r  X  M_ P  X  < 0.45Mp,  for L/r - 1 0 0 . X  Fig. 21 shows the interaction curves for Case 2 bending. Here the two methods are i n good agreement for short columns with loads P < 0 . 6 P . y  The Cambridge method i s  slightly less conservative for slender columns with loads P > 0 . 4 P , and much more l i b e r a l for columns with small V  end moments, particularly for columns of short and intermediate lengths. Interaction curves for Case 3 bending are shown i n F i g . 22. It i s seen that the Cambridge method i n this case i s more conservative than the American method for column loads P < 0 . 6 P y , particularly for short columns. For columns of short and intermediate lengths with small end moments the Cambridge method again i s much more l i b e r a l by allowing column loads i n excess of 0 . 6 P .  -  67  -  The writer expected that there would be more consistent differences between the results of the two methods, with the Cambridge method being the more conservative since i t assumes failure to occur when the maximum fibre stress at midheight f i r s t reaches the yield stress.  This conservative  assumption i s partly offset by the inclusion of residual stresses in the American method, since such stresses are not considered by the Cambridge method.  The most con-  sistent differences occur for Case 3 columns with loads P < 0.6P  as shown i n Fig. 2 2 . Here the differences  decrease with increasing column loads and slenderness ratios, indicating that the effects of the residual stresses on Case 3 columns possibly are more pronounced for heavy column loads and slender columns. The greatest differences between the results of the two methods occur for columns with small end moments. The reason for this i s that the American method does not allow column loads i n excess of P = 0 . 6 P , even for very small end moments. The originators of the two methods appear to have different views on the adequacy of the present knowledge about the failure of columns. While the many restrictions of the American method reflect some doubt and hesitation, no such doubts seem to be present among the originators of the Cambridge method.  - 68 -  IV  EXAMPLES  Example 1 The purpose of this example i s to demonstrate the possible variations of the column moments depending on the assumptions made and the methods used i n the analysis.  The bend-  ing moments w i l l be determined for different ratios of dead loads to total loads, and by different approaches. A comparison of the results w i l l be made at the end of the example.  The example w i l l also serve to illustrate  the procedure for determining bending moments described previously. Fig. 23 shows a one storey rigid frame consisting of two equal spans, each required to support a specified uniformly distributed failure load of W = 60 kips.  A l l members are  assumed to be 14WF30, which i s the most economical American beam section i n this case. A l l columns are fixed at their lower ends, and i t i s assumed that the frame i s prevented from sidesway. A. Ratio of Dead Loads to Total Loads g = 0.75 1. Analysis i n Accordance with Ref. 3« The loading condition i s as shown i n Fig. 2 4 . It i s assumed that plastic hinges w i l l form at the ends of beam EF, and that beam DE w i l l remain elastic.  - 69  -  With the joints locked against rotation, the total unbalanced moment at joint E i s  The design moment  for column EB i s thus Mgg = 0 .  The  design moment for the exterior columns i s taken as the WT  required plastic moment of the beams, i . e . Mp^ 112.5  ~  =  kip f t .  2 . Analysis with Reduced Load Factor for  Dead Loads  The loading condition i s as shown on Fig. 2 5 .  The  minimum load on beam DE has a load factor equal to 75 percent of the load factor for total loads. (a) Elastic Analysis To obtain an elastic moment distribution the frame has been analyzed by the slope-deflection method for a unit load on one span.  The loading condition and the resulting  bending moments are shown in Fig. 26.  The bending moments  are indicated in the form of coefficients c, such that 2  at any point M = cwL  = cWL.  Because of symmetry the  bending moments at a l l joints and at midspans for any uniformly distributed load on the beams can be determined from Fig. 26 by superposition.  The moments due to the  loading shown in Fig. 25 are thus as follows:  Mj} = (0.0696)(G.75)gWL - (0.0072)WL  =  57.4  kip f t .  M Q ( 0 . 0 5 2 3 ) ( 0 . 7 5 ) g W L - (0.0054)WL  =  43.2  kip f t .  - 70 -  (0.0757)(0.75)gWL + (0.018)WL -  108.3 kip f t  (0.0757) WL + (0.018)(0.75)gWL = 1 5 4 . 4  kip f t  (0.0577)(WL-0.75gWL)  45.6  kip f t  (0.0523)WL -(0.0054)(0.75)gWL =  88.7 kip ..ft  (0.0696)WL -(0.0072)(0.75)gWL = 117.9 kip f t No particular sign convention i s used for the bending moments. Only numerical values are used, and the bending moments are drawn on the tension side of the members. The elastic moment distribution i s shown i n Fig. 2 7 . The column loads corresponding to the above bending moments are as follows: Column DA:  P = 15.17 kips  Column EB: P - 49.81 kips Column FC: P = 28.78 kips. With a section modulus of S = 4 1 . 8 i n . and a cross2 3  sectional area of A = 8 . 8 1 i n . , the elastic fibre stresses are as follows: Beam DE, at D at G at E Beam EF, at E at H at F Column DA, at D: Column EB, at E:  G" - 1 6 . 4 7 k s i 6" = 1 2 . 3 9 ksi S" = 3 1 . 2 0 k s i er m 4 4 . 3 0 ksi 6" « 2 5 . 4 5 k s i S" = 3 3 . 8 0 k s i  o- = 1 8 . 1 9 k s i 6" = 1 8 . 7 3  ksi  -  Column FC, at F:.,  ;  71 -  (T = 3 7 . 0 7 k s i  With the previously assumed possible v a r i a t i o n of y i e l d stresses from 2 5 k s i to 4 8 k s i , i t i s possible that the whole frame w i l l remain e l a s t i c under the considered loading.  In t h i s case the column moments w i l l be as  shown i n F i g . 2 7 . (b) Formation of P l a s t i c Hinges and Redistribution of Bending Moments. Y i e l d stress , = 33 k s i For the purpose of r e d i s t r i b u t i n g the bending moments i t i s convenient to determine the d i s t r i b u t i o n of a unit moment at j o i n t E as shown i n F i g . 28. The bending moments i n F i g . 28 have been derived very simply from F i g . 26. Assuming the y i e l d stress equal to 33 k s i , and that the loads on both spans are increased proportionally to those shown i n F i g . 2 5 , the f i r s t p l a s t i c hinge w i l l form i n beam EF at j o i n t E when the bending moment reaches the p l a s t i c moment M^.= 1 2 9 . 5 k i p f t of the beam. The moments shown i n F i g . 27 must therefore be modified by d i s t r i b u t i n g the difference ^ use of F i g . 28. MJ = H  G  =  ~ \  =2 / f  *^  k  i  p  f  t  b  y  t  n  e  The modified moments are as follows:  Mp + ( 0 . 0 9 5 ) ( 2 4 . 9 ) +  F  (°-°73)(24.9).  =  59.8 k i p f t 4 5 . 0 kip f t  - 72 -  M  ED  M|  B  M» M  H  M  F  ^ED " ( ° » ^ ° ) ( ^ » 9 )  =  = Mgg - ( 0 . 7 6 0 ) ( 2 4 . 9 )  -  = Mp  =  =  F  2  =  +  = M  2  ^°«30)(24.9) + (0.40)(24.9)  p  -  1 0 2 . 8 kip f t 2 6 . 7 kip f t 1 2 9 . 5 kip f t 9 6 . 2 kip f t  =  1 2 7 . 9 kip f t  It i s seen that the moment Mp i s very close to the plastic moment Mp.  It i s therefore necessary to determine whether  a plastic hinge has formed i n column FC. only a small difference between the and M  F  Since there i s  end moments M'g  F  of beam EF, i t i s sufficiently accurate to assume  the load on column FC equal to P - 0 . 5 W = 3 0 . 0 kips. For this load the plastic hinge moments of the column i s determined as M£ = 1 2 7 . 4 kip f t . The f i n a l value of the bending moment at joint F i s Mg - M£ - 1 2 7 . 4 kip f t . The bending moment at H increases slightly to Mg = M^ + (0.5)(M  F  - M») - 9 6 . 5 kip f t .  The f i n a l distribution of the bending moments i s shown in Fig. 2 9 , as well as the location of the plastic hinges.  It i s seen that the yield moment My = S <T  y  = 115 kip f t i s exceeded only in the vicinity of the plastic hinges. (c) Most Unfavourable Combination of Yield Values 25 ksi < (Ty < 48 ksi  The maximum possible bending moments in column EB and FC  - 73 due to the loading shown in Fig. 25 w i l l be determined assuming that  the yield values ofthe individual structural  members vary between 25 ksi and 48 k s i .  The yield stress  is assumed to be constant within any one member. Column EB w i l l be considered f i r s t .  It i s seen from the  elastic moment distribution of Fig. 27 that the column moment Mgg w i l l increase i f a plastic hinge forms i n beam DE at joint E. A yield stress of O" = 2 5 ksi i s therefore assumed for beam DE, with a corresponding plastic moment of  = 98.0 kip f t .  The difference  M^-M^  - 10.8 kip f t i s distributed i n accordance with Fig. 28. Because i n this case the plastic hinge i s located on the l e f t side of joint E, i t i s necessary to imagine the frame turned around so that column DA takes the place of column FC when the coefficients are picked from Fig. 28. The modified bending moments are as follows: 61.7 kip f t 46.4 kip f t 98.0 kip f t M» = Mgg + (0.76)(10.8)  53.8 kip f t  k F " (°r24)(10.8)  151.8 kip f t  M = M + (0.073)(10.8)  89.5 kip f t  B  M ,  EF  = l y  H  Mp -  H  + (0.095)(10.8)  118.9 kip f t  -  74  To make the condition even more severe for column EB, i t i s assumed that the yield stress of column FC i s 25 k s i . A plastic hinge w i l l then form i n the column at joint F, causing an increase of the moment i n beam EF at joint E. In order to determine the plastic hinge moment M* of P25 &  the column at joint F, the column load must be estimated. = 9 5 . 0 kip f t , the difference M -  First estimating  p  M^ J_ = 2 3 . 9 kip f t would, when distributed, increase M|  F  2  by the amount ( 0 . 4 0 ) ( 2 3 . 9 ) - 9 . 6 kip f t . The coefficient 0 . 4 0 is here taken from Fig. 28 at joint F. This can be done because a plastic hinge has already developed in beam DE at joint E, so that this beam w i l l not resist a further clockwise rotation of joint E. This makes joint E equivalent to the simple knee joint F with respect to the distribution of moments. Based on the estimated beam end moments M^ + 9 . 6 kip f t and M^,. = 9 5 . 0 kip f t , F  the column load i s determined as P = 27.8 kips, for which the plastic hinge moment of the column i s calculated to  MJ- -  9 5 . 6 kip f t .  The difference M - M« p  and  - 2 3 . 3 kip f t i s distributed,  the adjusted moments are as follows:  Mg - MJ Mg = MJ  = =  6 1 . 7 kip f t 4 6 . 4 kip f t  75 -  ^D  = 9 8 . 0 kip f t  =  = 6 3 . 1 kip f t  B  - M'  Mg  F  = Mg  +  (0.40)(23.3)  = 161.1 kip f t  - M  +  (0.30)(23.3)  -  9 6 . 5 kip f t  =  9 5 . 6 kip f t  Mg  EB  F  R  +  (0.40)(23.3)  Mg  The column load calculated with these modified bending moments i s P= 2 7 . 8 kips, i.e. the same as the estimated value.  The elastic maximum fibre stress i n beam EF at  joint E i s cT= 4 6 . 3 k s i .  For the beam to remain elastic  in this case i t s yield stress must at least equal this elastic fibre stress, i . e . 6" > 4 6 . 3 ksi in beam EF. Fig. 30 shows the f i n a l bending moments arid the location of the plastic hinges. These bending moments represent the most severe for column EB under the assumed loading. N w column FC w i l l be considered. Going back to the Q  elastic moment distribution shown i n Fig. 2 7 , i t i s again f i r s t assumed that the yield stress of beam DE i s 25 k s i , resulting i n the formation of a plastic hinge i n this beam at joint E, and the modified moments shown on page 73.  The next step i s to determine whether a plastic  hinge can form i n column EB at joint E, as this would increase the bending moment at joint F. With the assumed loading and the modified bending moments i t i s found that the maximum fibre stress i n column EB i s (T = 2 1 . 0 ksi, and that therefore no plastic hinge can form i n the column.  - 76 -  T  he only remaining possibility for increasing the bending  moment at joint F i s to assume the formation of a plastic hinge in beam EF at joint E. As the moment decreases at joint E i t w i l l increase at joint F u n t i l the moments are of the same magnitude at both ends of beam EF.  Since the  yield stress i s assumed to be constant throughout the length of each member, the plastic moment must be the same at both beam ends.  of beam EF  The problem i s there-  fore to determine the maximum possible value the plastic moment M  p  can have i n this case.  Since i n this case the formation of the hinge in beam DE no longer has any significance,  i t i s convenient  to begin by again considering the elastic moment distribution of Fig. 2 7 . As the end moment Mg decreases F  by the amount AMg to M , the moment Mp increases by F  the amount AMp to Mp.  p  From Fig. 2 8 i t i s seen that  AMp = 0 . 4 AMgp, and the f i n a l end moments are therefore M - Mgp - AMgp = Mp + AMp = Mp + 0 . 4 AMgp, from which p  AMgp =J^J- (Mgp - Mp) = 2 6 . 1 kip f t .  The optimum value  of the plastic moment i s M » Mg - AMg = 1 2 8 . 3 kip f t . p  The difference Mgp -  F  F  - 2 6 . 1 kip f t i s redistributed at  joint E by the application of the coefficients and the f i n a l moments are as follows: Mg• - Mp + ( 0 . 0 9 5 ) ( 2 6 . 1 )  -  5 9 . 9 kip f t  of Fig. 2 8 ,  77 Mg =  + (0.073)(26.1)  -  45.1 kip f t  - M^ - (0.240) (26.1) = 102.5 kip f t » 25.8 kip f t Mg - Mgg - (0.760)(26.1) = 128.3 kip f t Mg  D  B  MJ = M  H  + (0.30)(26.1)  Mg - My + (0.40)(26.1)  -  96.5 kip f t  - 128.3 kip f t  The plastic moment M^ = 128.3 kip f t of beam EF corresponds to a yield stress of C  y  = 32.8 k s i . The  f i n a l moment distribution i s shown i n Fig.31, with the location of the plastic hinges. This condition produces the highest bending moment i n column FC under the assumed loading. (d) Most Unfavourable Combination of Yield Values 33 ksi < CT £ 48 ksi y  The purpose of considering this range of the yield stresses is to determine the influence on the column moments of the yield stress being higher than anticipated i n the design. The plastic beam moment for  = 33 ksi i s  =  ^ 9.5 2  kip f t . Referring to the elastic moment distribution of Fig. 27, i t i s seen that no plastic hinges can form to increase the bending moments i n column EB.  The only  plastic hinges which can develop w i l l reduce these bending moments. F i g . 27 therefore represents the most severe condition for column EB in this case, with Mgg =45.6 kip f t .  - 78 -  For column FC the most severe condition i s created by assuming (T = 33 ksi i n beam EF,  A plastic hinge w i l l  •7  then form i n beam EF at joint E, and the elastic moments shown i n Fig, 27 w i l l be modified by the distribution of the difference M  -  EF  = 24.9 kip f t . The modified  moments were already determined for this condition on page 71-72, and they are shown i n F i g . 3 2 . The bending moment Mp = 127,9 kip f t w i l l cause some yielding i n the beam EF although a plastic hinge i s not f u l l y developed. Some inelastic deformation w i l l take place at the joint and reduce the moment somewhat. The column moment Mp = 127,9 kip f t i s therefore somewhat conservative. 3 . Analysis with Full Load Factor for Dead Loads The loading condition i s as shown i n Fig. 3 3 , The minimum load on beam DE has the same load factor as the total load on beam EF.  The maximum column moments were deter-  mined for the four different conditions of elasticity and placticity considered previously, namely a) Complete elasticity, b) (T - 33 k s i , y  c) 25 ksi £  C  d) 33 ksi 6. or  48 k s i ,  £  £  48 k s i .  The f i n a l column moments and the corresponding column loads are compiled i n Table 6.  - 79 -  B. Ratio of Dead Loads to Total Loads g = 0 . 5 0 The preceding computations were repeated for this ratio of dead loads to total loads.  The maximum column moments were  determined i n accordance with the recommendations of Ref. 3 , as well as more accurately with reduced and f u l l load factor for dead loads. C. Ratio of Dead Loads to Total Loads g - 0 . 2 5 The maximum column moments were also determined for this ratio of dead loads to total loads.  The same assumptions  were made and the same conditions considered as previously. D. Discussion of the Results 1. Interior Column EB (a) Comparison of Methods 1 and 3(b) These methods are both based on the same loading conditions and a uniform yield stress of 33 k s i . The differences in the results are caused by the methods of moment distribution only, and are relatively small. difference occurs for g = 0 . 5 0 .  The greatest  Here the bending moment and  the column load as determined through method 3(b) exceed those of method 1 by 7 . 5 and 3 . 0 percent respectively. The approximate moment distribution method of Ref. 3 i s therefore relatively accurate i n this case.  In the case  of g=0.75 both methods produced the same answer, Mgg = 0 , although the condition of the structure was different for the two methods. Method 1 assumed beam EF plastic and beam  - 80 -  DE e l a s t i c  i n an unbalanced  joint  moment  equal to  Method 3 ( b ) i n d i c a t e d t h a t b o t h beams become p l a s t i c  zero. and  resulting  t h a t t h e c o l u m n moment  i s zero because t h e p l a s t i c  moments o f b o t h beams a r e e q u a l . (b) C o m p a r i s o n o f M e t h o d s 2 a n d 3 This  comparison i l l u s t r a t e s  the effect  l o a d f a c t o r s f o r dead l o a d s used paring  i n t h e example.  differences are r e l a t i v e l y become v e r y g r e a t .  small, i n other cases  I t i s apparent  t h a t such  l o a d i n g c o n d i t i o n s deserves  (c) Comparison o f Methods  they  differences  close attention.  (b) a n d ( d )  c o m p a r i s o n o f 2 ( b ) w i t h 2 ( d ) a n d 3 ( b ) w i t h 3 ( d ) shows  the i n f l u e n c e o f high y i e l d the  I n some c a s e s t h e  be i g n o r e d , a n d t h a t t h e p r o b l e m o f s e l e c t i n g t h e  critical  The  By com-  2(a) w i t h 3 ( a ) , 2(b) w i t h 3 ( b ) , e t c . , i t i s seen  that the differences vary greatly.  cannot  o f t h e two d i f f e r e n t  c o l u m n moment.  s t r e s s e s on.the magnitude o f  I t i s seen  that the differences are  t o o g r e a t t o be i g n o r e d .  The  bending  moments d e t e r m i n e d  corresponding normal f o r c e s , with those  determined  ( d ) , and t h e  a r e i n most c a s e s  by t h e e l a s t i c  (d) C o m p a r i s o n o f M e t h o d s  The  by assumption  analysis,  identical method ( a ) .  ( c ) and (d)  c o m p a r i s o n o f 2 ( c ) w i t h 2 ( d ) and 3 ( c ) w i t h 3 ( d ) shows  - 31 -  how the column moment can vary when the yield stress i s less than 33 ksi i n certain structural members. It i s seen that the column moment can increase by over 100 percent . The corresponding variations of the normal force are  small and of l i t t l e importance compared to the variations  of the bending moment. 2. Exterior Column FC It i s seen that there i s l i t t l e variation i n the column moments determined by methods (b), (c) and (d). These moments are a l l approximately 14 percent higher than those determined by method 1, which i n accordance with Ref. 3 were assumed equal to the "required" plastic beam moment j£ . The difference i s mainly due to the fact that the most economical beam section happens to possess a plastic modulus approximately 15 percent greater than required. The variation of the normal force i s small. The ratio g of dead loads to total loads has no influence on the maximum column moment except in the case of the whole frame remaining elastic, and the influence of variations i n the yield stresses of the beams i s small. The high degree of f i x i t y of the beam at the interior support also limits the variation of the bending moments in the exterior column.  - 82 -  Example 2  The  purpose  Example 1 .  of this  example i s t h e same a s t h a t o f  In this  c a s e , however, t h e b e n d i n g  i n the exterior  column w i l l  extent by a h i g h degree  n o t be a f f e c t e d t o t h e same  of f i x i t y  a t t h e f a r end o f t h e  The s t r u c t u r e t o be c o n s i d e r e d i s shown i n F i g . 3 4 ,  beam.  and r e p r e s e n t s h a l f o f t h e f r a m e o f Example 1 . dimensions,  cross-sections,  are  t h e same.  The  elastic  elastic yield that  For  analysis  under  stress  loading,  shows t h a t  the specified  and ether  The assumptions  t h e frame can remain  failure  load provided the  i n t h e beam i s n o t l e s s t h a n 3 6 . 9 k s i , a n d  i n t h e columns n o t l e s s t h a n 4 0 . 3 k s i .  a yield  hinges w i l l  For  moments  yield  stress  o f 33 k s i i n a l l members,  form a t t h e upper  ends o f b o t h  columns.  s t r e s s e s b e t w e e n 25 k s i a n d 4 8 k s i t h e most  u n f a v o u r a b l e c o n d i t i o n i s o b t a i n e d by assuming s t r e s s o f 25 k s i i n c o l u m n DB w h i l e t h e beam elastic.  A plastic  moment  decreases from the e l a s t i c  moment M  c  plastic  hinge t h e n forms  increases correspondingly.  a  yield  remains  i n c o l u m n DB a n d t h e value while t h e T h i s determines the  maximum v a l u e o f M^.  The  same c o n d i t i o n d e t e r m i n e s t h e maximum column moment  -  83 -  when t h e y i e l d s t r e s s e s a r e assumed t o v a r y between 33 k s i and 4 8 k s i .  I n t h i s case t h e p l a s t i c hinge i n  column DB forms a t a y i e l d s t r e s s o f 33 k s i . The r e s u l t i n g column moments a r e shown i n Table 7» I t i s seen t h a t t h e r e i s l i t t l e d i f f e r e n c e between the column moments determined by methods 2 ( a ) , 2(b) and 2 ( d ) , whereas method 2(c) r e s u l t s i n a moment a p p r o x i mately 10 percent g r e a t e r .  Again t h e lower moment d e t e r -  mined by method 1 i s mainly caused by the beam being s l i g h t l y overdesigned. The moments under 2(b) and 2(d) a r e v e r y c l o s e t o the c o r r e s p o n d i n g moments o f Example 1, but those under 2(a) and 2(c) show a g r e a t e r d i f f e r e n c e because fixity  o f the g r e a t e r  a t one end o f the beam i n Example 1.  The example shows t h a t t h e r e i s a g r e a t e r p o s s i b l e v a r i a t i o n i n t h e column moments i n t h e s i n g l e - s p a n frame than i n the e x t e r i o r columns o f a m u l t i - s p a n frame, but t h a t  this  v a r i a t i o n i s much s m a l l e r than the p o s s i b l e v a r i a t i o n o f the moments i n t h e i n t e r i o r columns o f a m u l t i - s p a n frame* The normal f o r c e i s the same f o r the cases 1, 2(a) and 2(b), but i n c r e a s e s s l i g h t l y w i t h the bending moment f o r cases 2(c) and 2 ( d ) .  -  84 -  Example 3 An extension to the Cambridge University Engineering Laboratory at Cambridge, England, has been chosen for this example.  The design of this structure, which was  erected i n 1 9 5 6 , was described i n Ref. 3 . Fig. 35 shows a typical, four-storey, r i g i d frame from this building. The sizes of beams and columns are noted. Table 8 gives further data of the individual structural members.  Only two column sizes were used, and the two  different sections were joined by welding four feet above the f i r s t floor.  The  one-storey portion of the structure,  which i s located on one side of the four-storey wing, i s ignored i n this example.  It i s assumed that sidesway i s  prevented by the end walls. With a load factor of 1 . 7 5 the specified failure loads are as shown i n Table 9 which also shows the ratios g of dead loads to total loads. The column loads include an allowance of 3*5 tons per floor for the weight of the exterior wall.  The live loads  on columns are reduced for columns supporting two or more floors i n accordance with British Standard 4 4 9 ( 1 9 4 8 ) . Table 10 shows the maximum factored column loads. The writer has made an elastic moment analysis of the frame with the beams loaded one at the time, by a uniformly  - 85 -  distributed load of 1.0 ton per lineal foot. The analysis was made by the slope-deflection method, and the stiffness of columns 1-2 and l - 2 ' was determined by the momentf  area principle to take into account the variation of the moments of inertia of these columns. The results are shown i n Table 11, where the moments are considered positive when they tend to turn the joint counter-clockwise. Four values are given for each bending moment according to which of the four beams i s loaded.  The tabulated  values allow the bending moments to be determined at the ends of a l l members due to any combination of uniformly distributed beam loads by superposition. 1. Analysis i n Accordance with Ref. 3 This analysis i s described i n Ref. 3 and w i l l not be repeated here.  Only a few explanatory comments are  included for the sake of clarity. Column 3-4 The required plastic moment of the roof beam i s M = j£  = 910 tons i n . This moment would in accordance with  the recommendations of Ref. 3 normally be assumed as the maximum bending moment at the top of the column. The designer chose, however, not to use this value, and assumed instead that a plastic hinge would form at the top of the column. With the selected column section and the given normal force, the plastic hinge i s M  f  p  = 1090 tons i n . ,  which therefore represents the moment for which the column  - £6 -  was designed.  The moment at the lower end of the column  was determined assuming only dead load on beam 3-3'. Column 2-3 This column was not analyzed i n Ref. 3 because of the assumption that column 1-2 would govern. Column 1-2 Although this column has a variable cross-section, i t was analyzed as i f i t consisted of the smaller cross-section throughout.  Because the heavier column section was extended  four feet above the f i r s t floor the column ends were assumed to remain elastic under the specified failure load. Column 0-1 This column was also assumed elastic i n the design. 2. Analysis with  Reduced load Factor for Dead Loads  The frame was analyzed for the two conditions of loading shown i n Fig. 36. (a) Elastic Analysis This was carried out by the use of the moment values of Table 11. The elastic fibre stresses at the beam ends were computed and found to be within the assumed maximum possible yield stress of 21.5 tons/sq i n .  -  87 -  (b) Formation of P l a s t i c Hinges i n Beams It was assumed that the y i e l d stress i n a l l beams i s equal to 15*25 tons/sq i n . , and the p l a s t i c beam moments were determined as follows: Beam 4 - 4 » : Beam 3 - 3 » :  = 1105 tons i n . "P  = 1 3 4 8 tons i n .  Beam 2 - 2 » :  Mp = 1352 tons i n .  Beam 1 - 1 » :  Mp = 2 1 5 6 tons i n .  For Loading 1 i t was found that the e l a s t i c end moments of beam 2 - 2 ' exceed the p l a s t i c moment of the beam and that p l a s t i c hinges therefore w i l l form.  The difference  between the e l a s t i c and the p l a s t i c moments was r e d i s t ributed at j o i n t s 2 and 2» and the moments adjusted accordingly throughout the frame.  A l l columns were assumed  to remain e l a s t i c . S i m i l a r l y beam 3 - 3 ' w i l l have p l a s t i c ends f o r Loading 2 , with a corresponding r e d i s t r i b u t i o n The end moments of beam 4 - 4 1 - 1 '  1  of the bending moments.  f o r Loading 1 , and beam  f o r Loading 2 are s l i g h t l y i n excess of the y i e l d  moments, My of these beams, but p l a s t i c hinges w i l l not develop f u l l y . 3 . Analysis with F u l l Load Factor f o r Dead Loads The two conditions of loading considered are shown i n Fig. 3 7 .  The frame was analyzed e l a s t i c a l l y with subsequent  -  redistribution  aa -  of moments due to the formation of plastic  hinges as previously.  The locations of the plastic hinges  for Loadings 3 and 4 were the same as for Loadings 1 and 2 respectively. 4 . Analysis with Full Loads on A l l Beams This loading condition consisted of f u l l y factored dead and live loads on a l l beams simultaneously. (a) Elastic Analysis The elastic analysis showed that the elastic fibre stresses at the beam ends were within the assumed maximum yield stress of 2 1 . 5 tons/sq i n . (b) Formation of Plastic Hinges i n Beams With a yield stress of 15.25 tons/sq i n . plastic hinges were found to form at the ends of beams 2 - 2 * and 3 3.'.  5 . Discussion of Results The end moments determined by the various methods and assumptions are compiled i n Table 1 2 . With the assumed maximum possible yield stress of 2 1 . 5 tons/sq i n . a l l the moments shown are theoretically  possible.  They do not  necessarily represent the most severe conditions for the columns since only symmetrical conditions were considered in the analysis, and the most unfavourable combinations  - 89 -  of yield stresses were not assumed to the same extent as i n the previous examples. Comparing the column moments from Ref. 3 with the others r  in Table 1 2 , i t i s seen that column 3 - 4 i s very slightly overdesigned.  Had the designer used the required plastic  moment of the roof beam 4 - 4 * instead of assuming a plastic hinge at the top of the column the result could have been different. The writer has checked column 2 - 3 for different conditions and found i t satisfactory i n a l l cases. The greatest difference between the column end moments determined i n Ref. 3 and those determined by the other methods i s found for columns 0 - 1 and 1 - 2 . The approximate moment distribution method of Ref. 3 has not considered the relative stiffnesses of these columns properly, resulting i n too small end moments i n column 1 - 2 and an excessive end moment i n column 0 - 1 . With a l l beams f u l l y loaded a plastic hinge can develop at the upper end of column 1 - 2 , "whereas the lower end w i l l remain elastic because of the heavier column section.  Because the column consists  of two different sections the unusual condition arises whereby the moment at the plastic end i s smaller than that at the elastic end. The writer has applied the ordinary Cambridge failure criterion to this condition, and has tried to make some allowance for the s t i f f e r lower part of  - 90 the column.  I t seems t h a t t h i s column must f a i l as soon  as i t beeomes p l a s t i c a t i t s  upper end, which i s p o s s i b l e  b e f o r e t h e f u l l s p e c i f i e d f a i l u r e l o a d has been reached. The w r i t e r c o n s i d e r s t h i s column somewhat underdesigned, but the e x t e r i o r w a l l may p r o v i d e s u f f i c i e n t b r a c i n g t o prevent  lateral torsional  buckling.  Because o f t h e i n a c c u r a t e moment d i s t r i b u t i o n the column 0-1 i s somewhat  overdesigned.  - 91 -  V  CONCLUSIONS  In the foregoing the writer has attempted to show how the safety of columns i n rigid frames depends on a number of variables which are not normally considered i n plastic design.  These variables include variable yield stress  values, loading assumptions, variation of eccentricities during loading, and inaccurate methods of analysis. It has been shown that it i s practically impossible to predict accurately the magnitude of the end moments of columns, and that i t therefore i s equally impossible to predict the true factors of safety of rigid frames. At the same time i t has been found that the failure loads of columns, as indicated by the two presently used design methods, greatly depend on the magnitude of the column end moments and their ratio. It i s possible to design r i g i d frames of equal or greater factors of safety than that of a simply supported beam provided conservative values are assumed for unknown variables and careful analysis i s employed.  This involves,  however, procedures which are normally not used i n plastic design.  The methods of analysis employed by the writer  in the examples would overcome some of the uncertainties and provide safe designs in many cases.  Such analysis i s ,  however, extremely laborious, especially when repeated t r i a l s may be necessary, and would be prohibitive for  - 9 2practical design.  With the presently used simplified  methods the writer cannot see the justification for using the same load factor for columns as for simply supported beams, since the true failure load of a column cannot be predicted by far as accurately as that of a simply supported beami  -  93 -  Table 1  N  A l l units i n kips and inches. N . M« . P. , M«  P33  2  5  25 29  33 37 41 45 48 50  25 29  33 37 41 45 48 100  150  25  29 33 37 41 45 45.5 48 25 29  33 37 41 45 48  I  1.00 0.36 0.76 0.68 0.61 0.56 0.52  2.00  1.72  1.51 1.35 1.22 1.11 1.04  740 865 989 1112  0.750 0.875 1.000 1.123  5 33L 0.759 0.879 1.000 1.120 1.240 1.360  1360 1450  1.250 1.376 1.468  675 809 940 1070 1195 1320 1420  0.719 0.860 1.000 1.140 1.270 1.406 1.510  0.759 0.879 1.000 1.120 1.240 1.360  488 626  0.636 0.818 1.000 1.180 1.359 1.540 1.560 1.680  0.759 0.879 1.000 1.120 1.240 1.360 1.379 1.452  0.517 0.760 1.000 1.242 1.480  0.759 0.879 1.000 1.120 1.240 1.360  1237  4.00 3.45 3.03 2.70 2.43 2.22 2.20 2.08  905 1040 1180 1195 1285  6.00 5.17 4.55 4.05 3.65 3.33 3.13  437 -575 715 852 990 1092  766  297  1.720 1.900  1.452  1.452  1.452  - 94 -  Table 1  C 25 29 33 37  41  200  185.5 324 462 601 740  3.75  981  25 29 33 37  8.00 6.90 6.06 5.40 4.88  25 29 33 37  41 45 48  29 33 37  41 45  48  M  7.20 6.20 5.45 4.87 4.39  4.00  45 48  264  y  45 48  41  228  „•  N  N  180  (Continued)  4.45 4.17 9.12 7.86  6.90 6.16 5.56 5.06 4.75 9.12  8.00  7.14 6.45 5.87 5.50  377  107  M3  0.402  0.701  33 0.759 0.879  1.000  1.000  1.600  1.240  1.300  1.120  1.900 2.125  1.360 1.452  0.280  246  0.645  0.759 0.879  522 660 798  1.367 1.725  1.120 1.240  902  2.360  1.360 1.452  0 140  0 0.508 1.000  0.759 0.879  332  276 416 554 692 796  0  140  280  420 560 665  1.000 2.082  1.000  1.000  1.504  1.120  2.510  1.360 1.452  2.008  2.880 0 1.000 2.000 3.000 4.000  4.750  1.240  0.879  1.000  1.120 1.240 1.360 1.452  - 95 -  Table 2  N  0 25  50 72.6  100 150  My  1312 1237 1162 1094 1012 862  180  772  228 264  628  200  300  712 520  412  M» „ P33  14 •  1007 989  1.30 1.25 1.24  940  865 766 575 462 382 276 140 0  P33  1.26  1.32 1.50 1.67 1.86  2.28 3.71 oo  -  96  -  Table 3 Case 1 b e n d i n g . -  ^ 0.0 0.2 0.4 0.6 0.8 1.0  0.0 60.2 120.4 180.6 240.8 301.0  0.00 6.60 13.20 19.80 26.40 33.00  - 80,  JT - 0 . 0 2 2 3 5 P  0.0 60.2 120.4 180.6 240.8 301.0  0.00 6.60 13.20 19.80 26.40 33.00  100,  0.0 60.2 120.4 180.6 240.8 301.0  -  0.391  0.00140 p  P  0.0 0.2 0.4 0.6 0.8 1.0  -  20, P  P/Py 0.0 0.2 0.4 0.6 0.8 1.0  /3>= - 1 . 0  1 ~7F  *x 0.00000 0.00925 0.01850 0.02775 0,03700  N  x  f  x  M  M /M  x  x  p  1.000 1.012 1.024 1.036 1.049  33.00 26.10 19.30 12.75 6.30 0.00  904 715 529 349 172 0  2310 1828 1350 890 440 0  2.30 1.82 1.34 0.88 0.44 0.00  1,000 1.220 1.533 2.097 2.830  33.00 21.60 12.90 6.30 2.33 0.00  904 592 353 172 64 0  2310 1512 902 440 164 0  2.30 1.50 0.90 0.44 0.16 0.00  33.000 19.100 9.480 3.420 0.415 0.000  904.00 523.00 259.00 93.50 11.36 0.00  2310 1340 662 239 29 0  2.30 1.33 0.66 0.24 0.03 0.00  X  *x0.00 6.60 13.20 19.80 26.40 33.00  0.0000 0.1475 0.2950 0.4625 0.5900  0.0350 p 0.0000 0.2305 0.4610 0.6915 0.9220  1.000 1.382 2.090 3.860 15.880  - 97 -  Table Case  2  bending,  = 20,  /J -  0.  1  =  ""^F  4  0.56$  tf" = 0.00140 p x  P/Py  P  P  0.0 0.2 0.4 0.6 0.6* 1.0  0.0 60.2 120.4 160.6 240.6* 301.0  0.00 6.60 13.20 19.80 26.40 33.00  - 80,  jr - 0.02235 p  0.0 0.2 0.4 0.6 o.a 1.0  0.0 60.2 120.4 180.6 240.8 301.0  0.00 6.60 13.20 19.80 26.40 33.00  V*x  = 120, JT = 0.0504 p  0.0 0.2 0.4 0.6 0.6* 1.0  0.0 60.2 120.4 180.6 240.8 301.0  N  0.00000 0.00925 0.01850 0.02775 0.03700  x  f  x  M  M'/M x' p  x  1.000 1.012 1.024 1.036 1.049  33.00 26.10 19.30 12.75 6.30 0.00  904 715 529 349 172 0  1600 1264 936 -618 305 0  1.59 1.26 0.93 0.61 0.30 0.00  1.000 1.220 1.533 2.097 2.830  33.00 21.60 12.90 6.30 2.33 0.00  904 592 353 172 64 0  1600 1048 625 305 113 0  1.59 1.04 0.62 0.30 0.11 0.00  33.00 16.12 5.61 0.026 0.00  904 441 154 0.71 0  1600 780 272 1.26 0  1.59 0.77 0.27 0.00J3 0  x  0.0000 0.1475 0.2950 0.4625 0.5900  x  0.00 6.60 13.20 19.80 26.40 33.00  0.0000 0.3325 0.6650 0.9975 1.3300  1.000 1.635 3.525 510.000 00  - 98 -  Table 5 Case 3 b e n d i n g ,  /3 = +1.0,  - 20, p  0.0 0.2 0.4 0.6 0.8 1.0  0.0 0.00 60.2 6.60 120.4 13.20  Vr*  -  0.0 0.2 0.4 0.6 0.8 1*0 ^  0.0 0.2 0.4 0.6 0.8 1.0  180.6  240.8 301.0  19.80  26.40  33.00  *x  0.00000 0.00925 0.01850 0.02775 0.03700  80,  8" - 0.02235 p  0.0  0.00 6.60 13.20  60.2  120.4  180.6  240.8 301.0  - 120,  1.00  0.00140 p  P  P/Py  -  N  x  1.000 1.012  f  X  33.00  26.10  1.036 1.049  1.024  19.30 12.75 6.30 0.00  1.000 1.220 1.533 2.097 2.830  33.00 21.60 12.90 6.30 2.33 0.00  1.000 1.635 3.525 510.000  33.00  M  M  x  x  904 715 529 349 172 0  904 715 529 349 172 0  904  904  M«/M  p  0.90 0.71  0.53 0.35 0.17 0.00  x  19.80  26.40  33.00  0.0000 0.1475  0.2950 0.4625 0.5900  592 353 172 64 0  592 353  0.900 0.590 0.350  64 0  0.064 0.000  172  0.170  a" = 0.0504 p X  0.0 0.00 60.2 6.60 120.4 13.20 180.6  240.8 301.0  19.80  26.40  33.00  0.0000 0.3325 0.6650 0.9975 1.3300  00  16.12  5.61  0.026  0.00  904  441  154  0.71  0  904  441 154 0.71  0  0.90  0.44 0.15 0.0007 0  - 99  -  Table 6 Bending moments in kip f t . Normal forces in kips Interior. Column Ratio g- 0 . 7 5  P  In accordance with Ref. 3 Reduced Load Factor for Dead Loads (a) Elastic Analysis (b) C - 33 ksi •(c) 25 ksi < Cy 1 48 ksi (d) 33 ksi < 6y £ 43 ksi 3 . Full Load Factor for Dead Loads (a) Elastic Analysis (b) = 33 ksi (c) 25 ksi £ cjy < 48 ksi (d) 33 ksi < Gy < 48 ksi 1. 2.  y  EB  M  Exterior Column P F FC r  0  52.50  112.5  30.00  45.6  49.80 48.38  117.9 127.4  28.78  50.27  128.3  29.93 30.00  49.80  127.9  29.95  26.0 0  55.79 54.12  115.5  28.50  63.0  26.7 63.1 45.6  54.76  127.4 128.3  29.93 30.00  29.8  55.50  127.9  29.95  32.2  45.18  112.5  30.00  65.0 50.7  43.33 42.76  120.3 127.4  29.07 29.93  y  72.9  45.12  30.00  y  65.0  43.83  128.3 127.9  52.0 34.6  47.82 46.51  118.7  Ratio g • 0 . 5 0 In accordance with Ref. 3 Reduced Load Factor for Dead Loads (a) Elastic Analysis (b) Cy - 33 ksi (c) 25 ksi < C < 48 ksi (d) 33 ksi < C ^ 48 ksi 3 . Full Load Factor for Dead Loads (a) Elastic Analysis (b) Cy - 33 ksi (c) 25 ksi ^ Cy < 48 ksi (d) 33 ksi < Cy < 43 ksi 1. 2.  63.1 52.0  48.76 47.82  127.4  29.95  28.88  128.3  29.93 30.00  127.9  29.95  -  1 0 0  -  Table 6 (Continued) Bending moments in kip f t . Normal forces in kips. Interior Column Ratio g - 0 . 2 5 1 .  2.  In accordance with Ref. Reduced Load Factor for Dead Loads (a) Elastic Analysis (b) C 3 3 ksi (c) 25 ksi £ G y £ (d) 3 3 ksi. £ Gy 6: 4a Full Load Factor for Dead Loads (a) Elastic Analysis (b) cjy - 3 3 ksi (c) 25 ksi 6. Gy £ 4a (d) 3 3 ksi < Gy < 4a  ^B 3  y  3 .  ksi ksi  ksi ksi  P  EB  Exterior Column M  F  6 4 . 3  3 7 . 8 6  112.5  P  FC  30.00  8 4 . 3  3 7 . 8 6  1 2 2 . 7  2 9 . 3 5  7 4 . 6  3 7 . 1 4  1 2 7 . 4  29.93  93.0  3 9 . 2 7  1 2 8 . 3  30.00  8 4 . 3  3 7 . 8 6  •127.9:  29.95  7 8 . 0  3 9 . 8 5  122.0  2 9 . 2 6  6 6 . 8  3.9.01  1 2 7 . 4  2 9 . 9 3  8 6 . 4  41.22  128.3  30.00  7 8 . 0  3 9 . 8 5  127.9  29.95  >  - 101 -  i-  i  Table 7 Bending moments i n kip f t . Normal forces i n kips.  C Column Moment  1. In accordance with Ref. 3 2. More Accurate Analysis (a) Elastic Analysis (b) Cy - 33 ksi (c) 2$ ksi ± CT £ 48 ksi (d) 33 ksi £ Cy £ 48 ksi y  j  CA Load  M  P  112.5  30.00  128.6 127.4  30.00 30.00  141.8  31.54 30.06  129.1  Table 8 Section  S in  Z P in  X 3  3  A A 2 in  x in  r  T y in tons in^  r  Upper columns 12 x6" at 54 lbs 62.63 72.72 15.89 4.86 1.33 85.3 Lower columns 12"xl2" at 83 lbs109.21 24.41 5.18 3.06 48.2 n  Beam Beam Beam Beam  4-4' 3-3 2-2' 1-1' 1  14 x6" at 46 lbs 63.22 14 x6" at 57 lbs 76.19 I6"x6" at 50 lbs 77.26 20"x6£" at 65 lbs122.62 n  M  72.53 88.43 88.79 141.8  Table 9 Beam  _ _ D.L. " T.L.  D.L. tons  L.L. tons  T.L. tons  4-4'  28.3  8.5  W =36.8  g =0.77  3-3»  38.2  W =52.3  2-2'  14.1  g =0.73  38.2  14.1  W =52.3  1-1'  33.9  42.3  W =76.2  g =0.73  4  3  2  x  s  4  3  2  g-j-0.44  - 102 -  Table 10 Column  D.L. Tons  3-4  17.7  0-1  83.1  2-3 1-2  40.3 62.9  L.L. Tons  T.L. Tons  4.2  21.9 51.5  11.2 16.8 32.6  79.7  115.7  Table 11 Elastic bending moments i n tons-inches produced by uniformly distributed loads of 1.0 ton per lineal foot on each beam in turn. Beam loaded:  3-3»  2-2'  +921.4  -40.4 +40.4  +10.85 -10.85  +269.8 -49.1 -220.7  +462.0 -974.0 +512.0  -122.2 -30.1 +152.3  +139.0  +538.6  -34.9  -952.3  "2,1  -60.0 +15.0 +45.0  -104.1  +413.7  -113.5 -29.5 +143.0  Joint 1: l,2 l,l' l,0  +14.75 -6.35 -8.40  -34.3 +14.8  +136.5  +522.0  -58.9  +19.5  -77.6  -841.0 +319.0  Joint 4: M  4.V  M  4,3  Joint 3: M  3,4  M  3,3'  M  3,2  Joint 2: 2,3 "2,2M  M  M  M  4-4' -921.4  1-1' -2.27 +2.27 +25.80  +6.35 -32.15  - 103 -  Table 12 Bending Moments in Tons-Inches  ".,3 1. Ref. 3 2 . Reduced load factor D.L. (a) Elastic Analysis Loading 1 Loading 2 (b) f =15.25 tons/sq i n . inbeams Loading 1 Loading 2 3 . F u l l load factor D.L. (a) Elastic Analysis Loading 3 Loading 4 (b) f^=15.25 tons/sq i n . inbeams j' Loading 3 Loading 4 4. Full loads a l l beams (a) Elastic Analysis (b) f =15.25 tons/sq i n . i nbeams  M  3,4  M  3,3» 3,2  +1090 +575 - 1 2 9 5  M  M  2,  M 3  2,2»  -1295  M  2,l  M  l 2  +467 +917  l,l«  M  M  l,0  -1885 +968  +1048 +527 -941 +414 +818 -1543 +654 +857 -1533 +726 +386 - 9 3 8  - 7 3 6 +131 +725 +605 +552 +1279 -1973 +694  +1050 +551 -935 +384 +710 -1352 +644 +746 -1348 +602 +353 -930  +642 +573 -724 +146 +577 +1287 -1976 -639  +1060 +669 - 1 2 2 3 +554 +829 -1561 +847 +879 -1602 +723 +530 -1211  +732  +1062 +696 -1217 +521 +711 -1352 +836 +759 - 1 3 4 8 +589 +494 -1202  +641 +698 -933 +235 +708 +1331 -1996 +665  +1080 +897 -1628 +731 +743 -1613 +1071 +800 - 1 3 4 8 +548 +562 -1352  +870 +1334 -2018 +634 +790 +1358 -2006 +648  L  L  +728 -946 +218 +681 +1322 -1992 +o70  -104 -  WF  FRAME  I  FIG . I  AF = 3 . 4 6 i n  2  D  <o y  .-  1/2(1-/3) h *Aw =2.20in  2  & h t = 0.29in l/2(l-/3)h  3  Z 3  AF = 3 . 4 6 i n  2  FIG.  2  FIG.  3  -106 -  NORMAL  FORCE  F I G.  4  N Kips  - 107 -  -  WF  •> ;v. '.IT:-, •  FRAME  2  6 I  F I G.  5  .• .< ' • "' '4', . . " ' •  oB  FRAME  FIG.  6  3  -108-  0.15  1/8  p" R E Q U I R E D ' P L A S T I C MOMENT, FRAME 3  BfE A M  0.0858 1/12 FRAME c  / ^ " R E Q U I R E D " P L A S T I C BE AM MOMENT, FRAMES 1 A ND 2  .  FRAME 1  FRAME  2  •  0.00  0.25  0.50 0.610.67 0.75  STIFFNESS  RATIO  1.00  ^ Kc  FIG.  7  1.25  -109-  NORMAL  F I G  FORCE  .  8  Nkips  -no-  D  E  '  L  -  FIG.  2  9  777777  7777777  FIG.  II  0.10  FIG.  10  -112 -  FIG.  12  -113-  F I G .  13'  -114 -  w FO  4  FIG.  14  -115 -  -116-  00  1 1 1  ro o o SLENDERNESS COLUMNS  WITH  RATIO PLASTIC  i=ENDS  ro ro o  i  !  ro o  1 1 ro  CD  o  l  ! ro 00 O  1 OJ  O O  -120-  FIG.  20  - 121 -  F I G .  21  -122 -  F I G .  22  -12 3 -  W =,60  kips  W = 60  FIG.  45  kips  23  W .= 6 0 kips  ki  A 777T  C  77777  FIG.  24  -12  4 -  - 125 -  Fl G  FIG.  30  -127-  Fl G .  299  31  13.4  FIG.  32  64.0  - 128  $W  =45  kips  -  W = 60  FIG-  W = 60  kips  33  kips  II  x L =  FIG.  30-0"  34  -129 -  14" x  6"  4 6 lbs  at  o  _I  I4"x 6"  at  57  lbs.  O I  3'  (0 J3  in  I6"x 6"  at  50  lbs.  o  _x  o 4—  2 0 x 6 1/2  at  65 l b s .  F  rO 00 i  °I "cvi  o * 3 3 ' - 0"  FIG.  35  -130 -  3'  2'  0*  0*  AO"  Looding I  Loading 2  FIG.  -  AO  36  „  w  g  2  v  Loading 3  2  w  4  4'  2  2'  w,  g iWi  OA  W  94  4'  AO'  F IG  OA  37  Loading  4  AO'  -  VII  131  -  BIBLIOGRAPHY  Ref. 1 :  Baker, J.F., Home, M.R., Heyman, J:"The Steel Skeleton, Vol. II", Cambridge University Press, 1956.  Ref. 2 :  Beedle, L.S. "Plastic Design of Steel Frames", John Wiley & Sons, Inc., 1 9 5 8 .  Ref. 3 :  Baker, J.F. "The Plastic Method of Designing Steel Structures", Journal of the Structural Division, Proceedings of the American Society of C i v i l Engineers, Paper  2005,  Vol.  85,  No.  ST4,  April 1 9 5 9 . Ref. 4 :  Hrennikoff, A. Discussion of Ref. 3 , Journal of the Structural Division, Proceedings of the American Society of C i v i l Engineers, Vol. 8 5 , No. S T 7 , Part 1 , September 1 9 5 9 .  Ref. 5 : Beedle, L.S. and Huber, A.W. "Residual Stress and the Compressive Properties of Steel, a Summary Report". Report No.  Fritz Engineering Laboratory  220A.2?.  Pennsylvania.  Lehigh University, Bethlehem,  July 1 9 5 7 . Revised November 1 9 5 7 .  -  Ref. 6:  132  -  Hrennikoff, A. "Theory of Limit or Plastic Design", Lectures at the University of British Columbia.  Ref. 7 :  "Building Code Requirements for Reinforced Concrete", A.C.I. 3 1 8 - 5 6 .  American Concrete  Institute, Detroit, Michigan, 1 9 5 6 . Ref. 8 :  "Commentary on Plastic Design i n Steel, Compression Members", Progress Report No. 5 of the Joint  WRC-ASCE Committee on Plasticity Related  to Design.  Journal of the Engineering Mechanics  Division, Proceedings of the American Society of C i v i l Engineers. Paper 2 3 4 2 . Vol. 8 6 , No. EMI, Part 1 , January I 9 6 0 .  - 93 Table 1  N 25  A l l units i n kips and inches. N  25 29  33 37 41 45 48 50  25 29  33 37 41 45 48 100  25 29  33 37 41  45 45.5 48 150  %  Gy  25 29  33 37 41  45 48  *m  1.00  740  0.76  989 1112  5L  33  865  0.750 0.875 1.000  0.61 0.56  1237  0.52  1360 1450  1.250 1.376 1.468  1.360 1.452  2.00  675  0.719  0.759 0.879 1.000 1.120 1.240  0.86 0.68  1.72  1.123  0.759 0.879 1.000 1.120 1.240  1.51 1.35. 1.22 1.11 1.04  809 940 1070 1195 1320 1420  4.00 3.45 3.03 2.70 2.43 2.22 2.20 2.08  488 626 766 905 1040 1180 1195 1285  1.560  1.630  1.379 1.452  6.00 5.17 4.55 4.05 3.65 3.33 3.13  437 575 715 852 990 1092  297  0.517  0.759 0.379 1.000 1.120 1.240  0.860  1.000 1.140 1.270 1.510  1.360 1.452  0.636 0.818 1.000 1.180 1.359 1.540  0.759 0.879 1.000 1.120 1.240  1.406  0.760  1.000 1.242 1.480 1.720  1.900  1.360  1.360  1.452  -  Table 1  94 -  (Continued)  N  N  M  180  25 29 33 37  41  45  48 200  25 29 33 37  41  228  264  4.17  25 29 33 37 41 45 48  9.12 7.86 6.90 6.16 5.56 5.06 4.75  29 33 37  9.12 8.00 7.14 6.45 5.87 5.50  45 48  107 246 382 522 660 798 902  0.280 0.645 1.000 1.367 1.725  0.759 0.879 1.000 1.120 1.240 1.360 1.452  0 140 276  0 0.508 1.000 1.504  8.00 6.90  45 43  41  0.759 0.879 1.000 1.120 1.240 1.360 1.452  185.5 324 462 601 740 377 981  5.40 4.88 4.45  33  0.402  7.20 6.20 5.45 4.37 4.39 4.00 3.75  6.06  P33  416  554 692 796 0 140  280  420 560 665  0.701 1.000 1.300 1.600 1.900 2.125  2.082 2.360  2.008 2.510  0.759 0.879 1.000 1.120 1.240  2.880  1.360 1.452  0 1.000 2.000 3.000 4.000 4.750  ©.879 1.000 1.120 1.240 1.360 1.452  W WWMH P H O O M O Oavn o ->o vn ro 04^-03-OOOOroOvnO •  •p- vn O-O - O t f t O O H M U ) r- rO?Or- -<lCM- vO O V O M ,  J  ,  roooo-rorororo^-ro^oro  •-3  M rovo-p-vnO OS-vOvO O  O-P--O co- o-^j o o-p- oa o O O ro ro v n ovn O vO - O  rt  VO  vo  VOW -' -< Jf-ir- r- r-'r-' r  r  r  ,  J  f\ -o ro ooovnvo ro ro rovo t- CO-O-O o ro O-p-vn o w  1  vo vo  S» cr a> ro  vn  -96  Case 1 bending. (b= -1.0 L  -  Table 3 1 ~vT - 0.391  /'x - 20, *r  - 0.00140 p P *x  P/Py  P  0.0 0.2 0.4 0.6 0,8 1.0  0.0 60.2 120.4 180.6 240,8 301.0  0.00 6.60 13.20 19.80 26.40 33.00  = 80,  6" - 0.02235.P  0.0 60.2 120.4 180.6 240.8 301.0  0.00 6.60 13.20 19.80 26.40 33.00  ^  0.0 0.2 0.4 0.6 0.8 1.0  = 100, 0.0 0.2 0.4 0.6 0.8 1.0  0.0 60.2 120.4 180.6 240.8 301.0  0.00000  0.00925 0.01850 0.02775 0,03700  0.0000  0.1475 0.2950 0.4625 0.5900  N  x  1.000 1.012 i1.024 1.036  .1.049  1.000 1.22©', 1.533 2.097 2.830  f  x  M  33.00 26.10 19.30 12.75 6.30 0.00  904 715 529 349 172 0  33.00  904  12.90 6.30 2.33 ©.00  353 172  21.60  t  M/Mp  x  x  2310 1828 1350 ^ 890 440 0  592  64  0  2.30 1.82 1.34 0.88 0.44 0.00  2310 2.30 1512 1.50 902 0.90 440 0.44 164 0.16 0 0.00  0.0350 p 3* x = v  0.00 6.60 13.20 19.80 26.40 33.00  0.0000 0.2305 0.4610 0.6915 0.9220  1.000 33.000 904.00 1.382 19.100 523.00 2.090 9.480 259.00 3.860 93.50 3.420 15.880 11.36 0.415 0.00 0.000  2310 1340 662  2.30 1.33 0.66  29 0  0.03 0.00  239  0.24  - 97 -  Table 4 1 (b - 0 , ~ vT  Case 2 bending, L  / x = 20, r  £ . « 0.00140. D  P  P  *x  0.0 60.2 120.4 180.6 240.8 301.0  0.00 6.60 13.20 19.30 26.40 33.00  0.00000 0.00925 0.01850 0.02775 0.03700  - 80,  y  0.0 0.2 0.4 0.6 0.8 1.0  0.0 60.2 120.4 180.6 240.8 301.0  0.00 6.60 13.20 19.80 . 26.40 33.00  ^'x  °  2T - 0.0504 p  0.0 0.2 0.4 0.6 0.8 1.0  0.0 60.2 120.4 180.6 240.8 301.0  P/Py 0.0 0.2 0.4 0.6 0.8 1.0 ^ x  1 2 0  >  - O.565  =  N  x  f  x  M  x  M  t/M  p  1.000 1.012 1.024 1.036 1.049  33.00 26.10 19.30 12.75 6.30 0.00  904 715 529 349 172 0  1600 1264 936 618 305 0  1.59 1.26 0.93 0.61 0.30 0.00  1.000 1.220 1.533 2.097 2.830  33.00 21.60 12.90 6.30 2.33 0.00  904 592 353 172 64 0  1600 1048 625 305 113 0  1.59 1.04 0.62 0.30 0.11 0.00  33.00 16.12 5.61 0.026 0.00  904 441 154 0.71 0  1600 780 272 1.26 0  1.59 0.77 0.27 0.0033 0  - 0.02235„p 0.0000 0.1475 0.2950 0.4625 0.5900  X  0.00 6.60 13.20 19.80 26.40 33.00  0.0000 0.3325 0.6650 0.9975 1.3300  1.000 1.635 3.525 510.000 0 0  - 9a Table 5 Case 3 bending, /3 = + 1 . 0 , L  r  0.0 0.2 0.4 0.6  o.a  1.0 L  p  P  0.0 60.2 120.4 180.6 240.8 301.0  0.00 6.60 13.20 19.80 26.40 33.00  / x - 80, r  0.0 0.2 0.4 0.6  o.a 1.0 L  0.00140,p  / x - 20,  P/Py  / x r  0.0 0.2 0.4 0.6  o.a 1.0  0.0 60.2 120.4 180.6 240.8 301.0 - 120, 0.0 60.2 120.4 180.6 240.8 301.0  - 1.00  0.00000 0.00925 0.01850 0.02775 0.03700  *x - ° ' 0.00 6.60 13.20 19.80 26.40 33.00  0 2 2  x  M  x  "x  1.000 1.012 1.024 1.036 1.049  33.00 26.10 19.30 12.75 6.30 0.00  904 715 529 349 172 0  904 715 529 349 172 0  0.90 0.71 0.53 0.35 0.17 0.00  1.000 1.220 1.533 2.097 2.830  33.00 21.60 12.90 6.30 2.33 0.00  904 592 353 172 64 0  904 592 353 172 64 0  0.900 0.590 0.350 0.170 0.064 0.000  33.00 16.12 5.61 0.026 0.00  904 441 154 0.71 0  904 441 154 0.71 0  0.90 0.44 0.15 0.0007 0  N  f  x  3V  0.0000 0.1475 0.2950 0.4625 0.5900  a" = 0.O5O4J? 3 X 0.00 6.60 13.20 19.80 26.40 33.00  0.0000 0.3325 0.6650 0.9975 1.3300  1.000 1.635 3.525 510.000 00  - 99 -  Table 6 Bending moments i n kip f t . Normal forces i n kips  0  52.50  Exterior Column P •F FC 112.5 30.00  45.6 26.7 63.1 45.6  49.80 48.38 50.27 49.30  117.9 127.4 128.3 127.9  28.78 29.93 30.00 29.95  26.0 0 63.0 29.8  55.79 54.12 54.76 55.50  115.5 127.4 128.3 127.9  28.50 29.93 30.00  3  32.2  45.18  112.5  30.00  ksi ksi  65.0 50.7 72.9 65.0  43.83 42.76 45.12 43.83  120.3 127.4 128.3 127.9  29.07 29.93 30.00  ksi ksi  52.0 47.82 34.6 46.51 63.1 48.76 52.0 47.82  118.7 127.4 128.3 127.9  28.88 29.93 30.00 29.95  Interior Column Ratio g- 0.75  M  1. In accordance with Ref. 3 2. Reduced Load Factor for Dead Loads (a) Elastic Analysis (b) Gy - 33 k s i (c) 25 ksi < Gy ^ 48 k s i (d) 33 ksi < 6y £ 48 ksi 3. Full Load Factor for Dead Loads (a) Elastic Analysis (b) fjy - 33 k s i (c) 25 ksi 6 Gy ^ 48 ksi (d) 33 ksi < Gy £ 48 k s i  EB  P  EB  M  r  29.95  Ratio g = 0.50 1. In accordance with Ref. 2. Reduced Load Factor for Dead Loads (a) Elastic Analysis (b) Gy - 33 k s i (c) 25 k s i < CT < 48 (d) 33 k s i < G" ^ 48 3. Full Load Factor for Dead Loads (a) Elastic Analysis •(b) C = 33 k s i (c) 25 k s i £ CT < 48 (d) 33 k s i < C < 48 y  y  y  y  y  29.95  -  100  -  Table 6 (Continued) Bending moments i n kip f t .  Normal forces in kips. Interior Column  Ratio g = 0 . 2 5  M  1 . In accordance with Ref. 3 2 . Reduced Load Factor for  Dead Loads (a) Elastic Analysis (b) C - 33 ksi (c) 25 ksi £ 6y £ (d) 33 ksi £ <r £ 3 . Full Load Factor for Dead Loads (a) Elastic Analysis (b) <T - 33 ksi (c) 25 ksi £ Cy £ (d) 33 ksi < (Ty ^ y  EB  P  EB  Exterior Column M  F  P  FC  64.3  37.86  112.5  30.00  84.3 74.6  37.86 37.14  122.7  29.35 29.93  48 ksi  93.0  48 ksi  39.27  127.4 128.3  30.00  84.3  37.86  127.9  29.95  78.0 66.8  39.85 39.01  122.0 127.4  29.26  48 ksi  86.4  41.22  128.3  30.00  48 ksi  78.0  39.85  127.9  29.95  y  29.93  - 101 -  Table 7 Bending moments i n kip f t . Normal forces i n kips.  CA Load P  Column Moment  1. In accordance with Ref. 3 2. More Accurate Analysis (a) Elastic Analysis (b) Cy - 33 k s i (c) 25 ksi < Cy ^ 43 k s i (d) 33 k s i £ cT £ 48 k s i y  112.5  30.00  128.6 127.4 141.8 129.1  30.00 30.00 31.54 30.06 •  Table 8 S  Section  in  P in Z  X 3  3  A . 2 in  T y in tons in^  x in r  r  Upper columns 12"x6" at 54 lbs 62.63 72.72 15.89 4.86 1.33 85.8 Lower columns 12"xl2" at 83 lbs109.21 24.41 5.18 3.06 48.2 Beam Beam Beam Beam  4-4' 3-3' 2-2' 1-1'  14 x6" at 46 lbs 63.22 14"x6 at 57 lbs 76.19 I6"x6" at 50 lbs 77.26 20"x6£" at 65 lbs122.62 n  n  72.53 88.43 88.79 141.8  Table 9 Beam  D.L. tons  L.L. tons  T.L. tons  _ _ D.L. g  4-V 3-3' 2-2' 1-1'  23.3 38.2 38.2 33.9  8.5 14.1 14.1 42.3  W =36.8 W=52.3 W=52.3 V/^76.2 4  3  2  =  t  :  l  :  g=0.77 4  g =0.73 3  g=0.73 g^O.44 2  - 102 -  Table 1 0 Column 3-4 2Q3 1-2 0-1  D.L. Tons 17.7 40.3 62.9 83.1  L.L. Tons 4.2 11.2 16.8 32.6  T.L. Tons 21.9 51.5 79.7 115.7  Table 11 E l a s t i c bending moments i n tons-inches produced by uniformly d i s t r i b u t e d loads of 1 . 0 ton per l i n e a l foot on each beam in turn. Beam loaded:  3-3»  2-2'  -921.4  -40.4  +10.85  -2.27  +921.4  +40.4  -10.85  +2.27  +269.8  +462.0  -122.2  +25.80  "3,3-  -49.1  -974.0  -30.1  3,2  -220.7  +512.0  +152.3  +6.35 -32.15  Joint 4: M  4,4-  M  4,3  Joint 3: "3,4 M  Joint 2:  4-4»  2,3 "2,2-  -60.0  +139.0  +538.6  +15.0  -34.9  -952.3  "2,1  +45.0  -104.1  +413.7  +14.75 -6.35 -8.40  -34.3 +14.8  +136.5 -58.9 -77.6  M  1-1'  -113.5 -29.5 +143.0  Joint 1: M  l,2  M  l,l»  M  l,0  +19.5  +522.0 -841.0 +319.0  - 103 -  Table 12 Bending Moments i n Tons-Inches M  Ref, 3 2 . Reduced load factor D.L. (a) Elastic Analysis Loading 1 Loading 2  1.  (b) f = 1 5 . 2 5 tons/sq i n . i nbeams  4,3  "3,4 *1,V  +1090 +575  l >2  l,0  2,2'  "2,1  -1295  +467  +818 +386  -1543  -736 + 1 3 1 +725 +605 +552 +1279 -1973 +694  -1352 -930  +642  +829 -1561 +723 +530 ~ 1*2*1*1  +732 +681  +728 -946 +218 +1322 -1992 + 6 7 0  -1352 -1202  +641 +708  +698 -933 +1331 - 1 9 9 6  M  3,2  "2,3  -1295  +527 - 9 4 1 + 4 1 4 +654 +857 -1583 +726  +1048  M  -938  M  +917  M  -1385  +968  L  Loading 1 Loading 2 3 . F u l l load factor D.L. (a) Elastic Analysis Loading 3 Loading if (b) f = 1 5 . 2 $ tons/sq i n . i nbeams Loading 3 Loading 4 4 . Full loads a l l beams (a) Elastic Analysis (b) f = 1 5 . 2 5 tons/sq i n . i nbeams  +1050 + 5 5 1 +644 +746  -1348  + 3 8 4 +710 +602 +353  +1060  -1223  +554  +669  +847  +879  +1062  +696  -935  -1602  +577  -724 +146 +1287 -1976 - 6 8 9 +578  L  L  +336 +1080  +1071  -1217 +521 +759 -1348 +589  +897  +800  +711 +494  - 1 6 2 8 + 7 3 1 +743 -1613 -1348 +548 +562 -1352  +235 +665  +870 +1384 -2018 +634 +790 +1358 -2006 +648  -104 -  WF  FRAME  I  FIG  4  AF = 3 . 4 6 i n  z  <*y 1/2(1-/3) h  *Aw 2.20in 8  2  c  N  CD  A  h  II  t = 0.29in  l/2(l-/3)h  AF = 3 . 4 6 i n  2  FIG.  2  -105-  -106  25  50  100  NORMAL  150  FORCE  FIG/4  180  200  228  N Kips  250  264  300  - 107 -  WF  FRAME  F I G .  5  WF  FRAME  F I G .  6  2  -108-  -109-  5.0  4.0  o f< cc  3.0  2.0  -  _  g y =48 us i  1 25  72.6  1  1  1 50  100  150  NORMAL  FIG .  180 200  FORCE  8  N kips  228  1 250 264  — 300  -112 -  F I G . 12  -113-  F I G .  1 3  -115 -  -116 -  FIG.  16  -120-  Fl G .  20  -121 -  FIG.  21  -122 -  FIG.  22  -123 -  W • 6 0 kips  A  W = 60 kips  B  C  •in II  7777?  77777-  L = 30'-0"  L = 30'-0"  23  FIG.  W = 60 kips  gW • 45 k i p s  A  C  B  777T  11  FIG.  24  nt  - 1 2 4  -  W = 6 0 kips  0.75 g W = 33.75 kips  77777  FIG. 25  FIG.  26  77777  -126 -  -127-  - 128 -  W i 60 kips  gW = 45 kips  7777  FIG.  33  W • 60 kips  FIG.  34  ,_ I  20!-0"  _ II  —I  12-6  _  9-0  II  ^  I  _  9-0  4r0. 4*  -oro  •J>.  o_  x~  0)  I CD OJ  Ol O  O)  cr  m  ro 12x12  at  83 lbs.  12 x 6  at  fj)  ro CO I  Ol  cr  oi  x  54 lbs.  cr o>  - 1 3 0  w, 0.75  -  4"  g W 3  3  g  0.75  g  W  4  2'  W  2  0' Loading 2  Loading  F I G . 36  94  4' q  w  3  w  9l  W  4  W 3  2  3' g  2'  2  w  2  w,  w.  0' Loading 3  F I G . 37  9h &?\  2  W,  g, W,  0*  OA  4  3'  W-  0.75  0.75  Oi  Loading  4  

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