(1963) Inelastic Buckling of Steel Frames Le-wu-lu 65p

7/29/2019 (1963) Inelastic Buckling of Steel Frames Le-wu-lu 65p

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.WELDED CONTINUOUS. FRAMES .. AND THEIR ..COMIJONENTS

INELASTIC BUCKLINGQF STEEL.FRAMESByLe-Wu Lu

This work has been.carried out as part of aninvestigation sponsored joint ly by the.WeldingResearch:Council and the Department of the Navyw i ~ h funds furnished by th e followin,g:

American Inst i tute of Steel ConstructionAmerican Iron and Steel Inst i tute'Office o f Naval Research (Contract Nonr.6l0(03Bureau of 'oShip's .Bureau of Yards and l)ocks

.Reoproduction of this report in whole or inpart is permitted fo r any purpose of the United. S t ~ t e s .Government.

Fritz Engineering Laboratory.Department of Civil EngineeringLehigh UniversityBethlehem, Pennsylvania

June 1963

.Fritz Engineering L ~ b o r a t o r y Report No. 276.7


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S Y N.O PSIS

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I t is well-known that a symmetrical frame carrying symmetrical loadbuckles into an asymmetrical configuration when the load ha s reached certainc r ~ t i c a l value. This phenomenon, oft en refer red to as "Frame. Jinstabili ty",.may occur at a load level. below the yield (elas.tic buckling), but morefrequently would take place when the applied load has c.used yielding in.some portion .of the ftame ( i n e l a s t i c b u c k l i ~ g ) . In this paper a .numerical

.1

method fo r th e determ ina tion .of the buckling strength of part ial ly yieldedframes is presented.

The proposed msthpd is an adaptation of the modified moment distr ibutionprocedure developed previously for analYZing the elas t ics ta ,b i l i ty of planeftames. In the present method the.st i ffness and c a r r y - o ~ e r factors of thevarious members aremodified for the.combined influence of axial force anf;lnonuniform yielding. The.effects of in i t ia l residual stresses and thesecondary bending moments resulting from deformations are included in theanalysis. Two examples are given to i l lustrate the application .of th e methodin constructing frame buckling curves.

E x p e r i m e n t ~ on.three sets of s teel frames ;abricated from a smallwide-flange shape were conducted. Satisfactory agreement between th e tes tresults and th e theoretical solution has been . o b s e r v e d ~

On .the hasis of the results of the theoretical and experimental studiespresented herein, th e va.lidity of a currently used column design rule isdiscussed and a .new method. is suggested.


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1.

.2 .

.3 .4.5.

T A.BoLE ,OF C.O.T.EN. T S

.SYNOPSIS. I : N T R O P U C ~ I Q N .

1.1 Previous R e s e a r ~ h .on Inelastic B u ~ k l i n g.1.2 Limiting Strengthof Frames1.3 . Scope of Investigation

DEVELO}OO;NT.OF ~ I { J ! : . T H E O R Y2.1 Analysis of PartiallYcYielded Frames2.2 Assumptions for Buckling Analysis2.3 Stiffness o f ~ e m b e r s After Yielding2.4 Meth.od of Solution

C O N S T R U C ~ I O N O F . A F ~ B U C K L I N G CURVEEXPERIMENTAL.STUDY.CpLUMN DESIGN ,IN.UNBRACED FRAMES

Pagei.1234.5

57711142023

.ii

5.1 Comparison of Resu lts with .. the.AISC .Desig;n ~ u l e 24

6.7.8.9.10.11.

5. 2 ~ e v e 1 o p m e n t of.a ,New Design ~ } ) p r o x i m a t i o nS ~ AND CONCLUSIONS.ACKNOWLEDGEMENTS.TABLES AND FIGURESAPPENDIX.NOTA'l;' IONREFERENCES

26303334515860


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1. liN T.RO D U C.T 1 0 N

.Whena.symmetrical frame, unrestrained from.sipesway, is .subjected .t o.symmetrically applied loads, i ts .deformation .configuration . w i l l a l s ~ besymmetricalas long as th e loads are below a certain cr i t ical v a l u e ~However, as the cr i t ical loading is re ached, th e frame may buckle suddenlyiQ,to an antisymmetrtcal c9nfiguration, and cOllsequently a large displacementdevelope in th e la teral direction. ..At this ins tant the frame has lostcompletely i t s res is tance. to any imposed la tera l force or deformation, andfailure by buckling h a thus te rminated. the load-carrying capacity.l,2

InFig . 1 schematic l o a d - d e f l e c t i o n . c u r v e s ~ of a portal frame . c o r r e ~sponding to v ar ious modefJ of failure are shown. I f the frame is preventedfromswaying.sidewise, th e .symmetrical.deflection form.shown in . inset (a)will be maintaiQ,ed at a l l s.tages of loading. The ultimate load of th e ,frame,.wu ' is reached when .thebendingmomeI1:t at ,the top of columns has a t ~ a i n e dth e limiting capacity. This load is indicated by point A on th e .load-deflection curve. The type of fai lu re .which is ' typical for braced fcameswill be referred to as "beam-column instabil i ty" in the subsequent discussions

3In a previous paper a method has been developed for determining the ultimatestrength for this type dd! failure.

I f no external bracing is provided for the f rame, s idesway buckling(a s shown in inset (b) of Fig. 1) may tak e p la ce a t poiI1:t B ( inelast icbuckling) or at point.C (elast ic buckling), depending on the slendernessrat io of the columns. In th e Case of .bucklingbelow the yield l im i t , a l lthe members are.elast ic , .and the cr i t ical load ca;n be readily determined pyth e methods developed by Masur, Chang and Donne1l4 or using the solution

5presented by th e aut ho r. For inealstic buckling, however, precise deter-


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mination of t ~ e . c r i t i c a l laaq becomes extremely laborious i f not impossible.This is ,mainly due to the nonuniform yi,elding preseQ,t in th e vaUous partsof t h e s t r u c ~ u r e . The problem is further complicated in, that the effect of

6residual stresses must a lso beconsided .

I t is the purpoSe of this paper to p ~ e s e n t anengineering.solution fortheinelas, t ic .sidesway buckling of single-story single-bay portal frames.~ combined influence of nonuniform ,yielding and in i t ia l stresses ~ r takeninto account through a nutnerical procedure previously developed for analyzing3braced ft:ames.

1 .1 PREVIOUS RESEARCH,ON .INELASTIC BUCKLINGSince inelastic buckling strength may be r egarded as th e l imiting

s,trength fo r unbraced frames, i t is important to consider this type offailure in proportioning the columns insuch.s t ruc tures , especially whenthe plast ic method is adopted in , the.design. In recent years several attemptshave been,made to ,develop methods for e st imat ing the illelastic buckling load

7of frames. Merchant in 1954 suggested that for practical calculations i tmight be reasonable to conside r the inelast ic buckling . s t rengthof an.:elastic-plast ic structure as some function.of the .elastic buckling Load and th e simpleplast ic load. These ,loads represent two extreme idealizations of the carrying

8 9 10capac ity o f a stee l .structure. Bolton, .Salem. and Low have tes,ted severalseries of model s teel frames to observe the magnitude of the frame buckling

. .effect . Their results have shown some degree of correlation with the empirica11approach proposed by Merchant . . Wood in his studies on pla:stic ins tabi l i ty

of mult i- story frames intorduced th e concept of ,"deter iorated cr i t ical load"as a theoretical tes t for the .stabi l i ty of par,tially yielded structures In


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. 12a recent survey prepared by Horne ;_the importance of co nsidering thedeformation ..effects in instabil i ty anaylsis was s tressed. The author in

13 ..an unpublished report has presented an anay1tica1s01ution to the .sideswaybuckling of portal frames in the plast ic range. This solution takes intoaccount the influence o"f residual stresses and inelast ic deformations. Thepresent p ~ p e r is a sunnnary and an extension .of th e work .contained in thatreport .

1.2 LIMITING .STRENGTH OF FRAMESIn s tudy ing the buckling problems associated with frames, i t is often

convenient to present the .solution in the form of a frame bucklingcurve.This curve giMes th e relationship between the height of the columns and thecr i t ical load of the s tructure . A typical plot of suh a curve is shown in

. .Fig.2. In this plot a l l th e frames are assumed to have a constant spanlength and acted upon by the same type of loading. The.comp1ete frame bucklincurve consists essentially of two parts : (1). Portion AB defines the elas t ic.puckling .strength, and (2 ) Portion .BC corresponds to bucklin.g in th e inelast icrange. PointB marks th e transit ion.between.e1astic and inelast ic bucklingand represents the column height for which buckling and yielding occursimultaneously.

Also shown i ~ . F i g . 2 are two curves representing the strength .o f theframes when they are proper1y.braced to prevent sidesway movement. LineEFgives the maximum load according to s i m p ~ e plast ic theory. This theory assumefai lure by symmetrical bending and ignores any reduction in the.moment c a p ~ c i t

of th e columns due to axial load and .due to th e .secondary moments in thecolumns resulting from their deformations. CurveDG represents th e ultimatestrength corresponding to fai lure by beam-column instabil i ty. This strength


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276.7 -4 can be determined by considering the reduc tion of plastic moment capacity a tthe top of th e columns due. to beam-column action .' I t . is interestipg. to notein Fig. 2 that curve: BC becomes coincident with curve DC after passing through

. .point C.' ~ h e r e f o r e , for frames with height less than that . indica ted by pointC., the reduction is strength due to sidesway buckling". would be negligibly sma

"L 3 SCOPE' OF' 'INVESTIGATION

.This paper. contains the results of an investigation of the followingphases:

I The.oretical development of a method for' the determination of the.. inelastic buckling. s tre ng th o f part ial ly yielded frames. 'The methodwUl beexpla ined with reference to the simple portal frame shown in Fig . 3. Theframe.carried simultaneously a uniformly.distributed load of intensi ty w onthe. beam ..and two' concentrated loads' P applied along the centerlines of the

columns . The load p, is re la te d to the uniform load by the parameter N. in the form p' = N ( ~ ) . To achieve proportional. loading" N will be held constant.2This condi tion o f loading. was originally' suggested for' investigation by

. I. Bleich . and is intended to simulate approximately the axial .loads and momentsoccurring in the lower stor ies of a m u l t i ~ s t o r y building

. 2. Experimental verif ication of the analytical solut ion,3. Comparison of thetheoretical. and experimental results with an

existing design.rule" and4 .. Development: of a new design approximation

. Throughout this invt::stigation, the frames are assumed to be suff i -ciently braced in the perpendicular direction. so that buckling can o c c u ~only in. the plane. of th e applied loads.


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2. D: I,V leL 0; po M'I N'T . 0' FT.H I T' H : 1:.0: R Y

2.1 ANALYSIS OF PARTIALLY YIELDED FRAMES

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In order to obta in. so1utions to the inelastic buckling problem describedabove, i t is f i r s t neces sa ry to develop a method by means of which thedistr ibution .o f bending moment and the . va ria tion o f y ie ld ing in various membecan be determined. Since the frame considered in this i ~ v e s t i g a t i o n is anindeterminate.structure, s tat ical conditions alone are not .sufficient toobtain.a11 t he r eact ion. components. An additional condition based on 'geomet-r ica1 .compatibil i ty has to be inc,orporated in the analysis. For th e frame.shown in Fig. 3 a commonly used compatibility condition . i s tha t at jo in tBor D the.s10pe of the beam should be equal to that of the column. ,With theaid of this condition the ,structure ,can then be analyzed by th e classical.slope-deflection method modified to take into account the effects of axialforces. 4 This method of solution, however, is applicable only when the frameis loaded w i t h i n , ~ h e , e 1 a s t i c range. If the applied load has exceeded .thee1as,tic l imit, some portions of ,the structure a re y ie lded ; and th e effect ofyielding ,will h ~ v e tobe considered in th e analysis. Obviously the analysisof a part ia l ly yielded structure is far more involved than"i that of an"elastic structure.

In a r e c e ~ t paper3 Oja1vo and the au thor h ~ v e presented a ,method foranalyzing symmetrical frames s tr es sed int o the inelast ic range. The method,as applied to the frame considered in this paper, may besu1lDIlarized asfq110ws: For given load w, construct the e n d ~ m O m e n t versus end-rotation c ~ r v e s for th e beams. and the, columns (MBD r ' 9 .and MBA - 9 BA curves),. .. / 14using the numerical integation procedure developed by von ,Karman which isbased on .the moment-curvature ,relationship of the', members. In computing these


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curves . i t : i s possible.to take into accountnot.only the. inelas t icact ion,,bu t .al so ~ h e . e f f e c t s of axial .force and the deformation due to bending.When the resul t ing .curves are plotted, . th e po in t of in tersect ion gives themoment and rotation ,a t jo in tB ,for the e q u i l i b r i ~ m c o ~ f i g u r a t i o n of the.structure. By knowing the.moment a t th:e to p .o f the column and the deformedshap,e of the .frame , : the :distribution of . m o m e ~ t can .then ,be determined froms ta t ica l c o ~ d i t i o n s .

In the development of t ~ method several assumptions were.made in orderto.reduce the amount o( numerical computat ions. The assumptions are:

(1 ) AIl.members a re p risma tic ,(2 ) The shear , fo rce presen t .a tany section of the fJ:',ames is small.and i ts effect on ,yielding may be negle.cted,

(3 ) ..Only deformations (elas t ic .or inelastic) due to bending are,con.sidered,

(4 ) 'rhe axial force in th,e.beamis small compared.with tl1e thrustin the columns and i ts effect .maybeignored,

(5 ) .No la teral (sidesway) displacements a t column ends are considered,(6 ) Np . transverse load is ap plied to the column except a t .the ends , and(7) .The method is non-historic. It is necessary to specify thatdllr ing

loading there is no.strain.reversal ,o f material stressed beyond the3. elast ic . l imit .

As wii l be .seen i n ~ h e l a te r .d i scussions, th is method of elast ic ..plast ic.anaiysis leads to convenient . w ~ y s of .determining the .s t i ffnesses of the ..beamand columns . By. knowing th.e .s t i ffnesses of the .various members a t a l l stagesof.loading, it is then possi,ble to determine the.buckling,strength ..of th e' . . :f r a m ~ b y any o ne.of the existing t e c h n i q ~ e s of b u c k l i n g , a n a ~ y s i s . ~ the


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present investigation ,. tb,emoment distr ibutionproced\lredue to ~ i n t e r , ..,Hsu,Koo and L o h l 5 ~ s adopted.because of i t s simplicity .

. 2.2 ASSUMPTIONS .FOR BUCKLING ANALYSIS. ~ i n c e the .metl:!.od of. buckling analysis developed in . this paper uti l izes

directly theres \ l l ts of elast ic-piast ic .analysis as described above, i is. a lsosubject to the assumptions stated in Sec t ion .2 . l . I n addition, thefollowing. two assumpti0rls are made:

(1) The.axial force i n t he .b eami s small . and i t s effect on tne b e n d i ~ gst i ffness may, be ignored i n t ~ e .buckling ,analysis . . The jus.tificat:ioof this assumption.has been discussed.ina previous paperS in~ o n r l e c t i o n .with .elastic.buckling problems.

(2)TIl.e f rame deforms in.a perfectly symmetrical form up to t ~ i l lstantof .buckling. Tb,is implies t ~ themethqd of inelastic.analysiscan be applied to determine .the yield configuration ~ t a r l Y loadlevel. below . that .which .causes sidesway.buckling.

2 ..3 'STIFFNESS .OF MEMBERS AFTER.YIELDINGAnalogous to the method commonly used in determining the inelast ic

buckling .strength .of central1y. 10a4ed colunms,. th e p r o c e d ~ r e heredevelopedalso . r e q ~ i ~ e s proper eva luat ion o f the reduction .of.bending.stiffness(buckling constant) of a l l the members due to.yielding. By using.thesereduced stif fnesses in the analys is , the problem of inelast ic buckling maybe treated in a manner similar to that .o f the .elastic'-case. Since the'. .-.modified moment distr ibution method developed.by Winter, et a1.;15 is adopted. in the analysis, i t . i s n e ~ e s s a r y t o obtain the fo11owing:buckling,constant:s:


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(1) For beams: .s t iffness f a c t o r . ~ ( ~ s s u m i n g far end fixed) andcarry-over factor Cb .(2 ) -For columns: ,;(;tiffness farr K ~ (assuming far end hinged)

,Stiffnessap.d'CarrY-Over Factors of the .Beam.For a given set of loads wand P, the -bending moment at B (or .DQ is

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fi,rst determined by the method of e l a s t i c _ - p l ~ t i c analysis. By knowing thetwo end moments_, the moment .diagram of the beam can by easily constructed_ bys.tatics. Figure .4a ,shows a typical _example of.s,uch -a diagram. t\ccording tothe elementary theory of strength_of materials, the flexural _behavior (inthe elast ic and inelas.tic range) of any fiectionof the beam is g o v e r n ~ dcompletely by the m o m e n ~ - c u r v a t u r e relationship .o f the member. In.theelastic range the_slope of the .moment-curvaturediagram is constan,t and equalto the flexural rigidity of thesect ipn EIb -When the applied moment exceedsthe elast ic l imit , t h e _ ~ l o p e (o r rigidity) starts to decrease_ a ~ approacheszero when the moment is near the plastic-moment Mp ' The effective flexuralrigidity (EIb) eff of the_sec tion _can thus be determined as the instantaneousslope on the M-0 diagram corresponding to t h applied moment.

In this paper, th e m o m e n t - c u r v a t u r e ( M ~ 0 ) c u r v e of a typical beamsection 27.WF 94 as shown, in Fig. Sa is adopted. The curve _wascollstructed

16according to the p r o c e d u r e - d ~ v e l o p e d by Ketter, Kaminsky and Beedle and is-,based on .an idealized e ~ a s t i c ' ; ' f u l l y plastic s t r e s s ~ s _ t r a i n relationship and al inearly varying .symmetrical residual stress, pattern with -a maximum compressivresidual stress at the flange t ips equal to 0.3 times the .yield _s.tress of thematerial. I t has been observed from _th,e resl.l1ts of extellsive c o m p u t a t i o n ~ ,that the M-0 curves, in . thei r nondimensional form, are approximately the same,for .most of theWF sections t h ~ t are commonly used as beams. Therefore the


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M - curve' constructed for ' th is particular section, after being properlynondlmensionalized, can be applied to other ' sections as well ...The' moment-curvature relationship given in FIg. Sa shows that the actual .yield moment i ~ only 70 percent 'o f thenominal.yieldmoment 1),. Thus,: yielding occurs at ' sections' where the moment. has exceeded 0.7. 1),. . ':rhis ,is ,indicated in Fig.4b for, the beam under consideration.

, ~ no strain reversa l is assumed to take ,place: a t the moment of buck- l ing , the, st i ffness of ' the beam can be, determined by considering a beam ,o fvariable ' EI .. For' the, ' e last ic part 'the flexural rigidity of' ,the .beam . i sE lb ,while for th e p la 'sUc Mrt: the effective flexural rigidity. , is reduced as i f

the __ yielded por.tions' of the. beam were removed. The effect ive flexural rigidi (Elb>eif ' determined from the ~ - curve of Fig. Sa, 'i s plotted as a functionof-the applied moment in Fig. 5b. Figure.6a shows a symbolic representationof .a plast ic if ied .beam c o r r e s p ~ n d i n g to the ,yield configuration indicated-. - - . ~ :

"in Fig. 4b. - It is called the: "reduced beam" in ' th is paper . .The' st i ffnessof this beam can be evaluated by the method of column analogy which. is commonused in indeterminate analysis . The analogous column of the ~ e d u c e d b e a m isshowninYig.6b .The width of the column at: each section. is inversely pro-

portio na l to the' flexural r ig idi ty of th at s ec tio n . . To determine the, b e n d ~ n gmoment at B induced by a unit rotation at '..B, a unit: load of one radian isthen. applied to th e analogous column at the end B. The st i ffness of thebeamB is . equivalent, to the. stress on the,analogous column.at that 'point ,that is .' L L1 . 2' . '2

K... = 1 + .1..-. _~ = A I (1 )in which A is , the area of the analogous column. and 1. is the moment of inert ia


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about the . cen tro ida l ax is .. G-G . . SimUar ly the momen.t a t D is equa1.to:

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.mD1 L ,L1 .. 0'2'.2'= - -A I ( 2

The . c ~ r r y ~ o v e r factor is . simply the ra t io of the'moment a t D to tah,t a t .B, or

c;b (3)

If i t is known tha t b o t h e n d s ~ .of the beam would rota te through .the sameangle and in .the same .direct ion .a t the i n ~ t a n t of s idesway, the st i ffness,may be computeq,by usi1l$the analogous co1umn.shown in.Fig. 7b. Thecentrodia1 axis G-G is now a t the r igh t end of t ;he column .apd the area isassumed to be in f in i ty . The.st i ffness 1);11 of the beam' a t the l e f t end is

L ' T.l ' ~ . ~ .'2 . 2 (4).K" = '. I Ibin.which I is the moment of i n e r t i a a b o ~ t axis G-G. The carry-dver factori s not .needed .in th is case .

.Sti f fness of the. Columns.The .s t i f fness factor .K' of a c o l u m n w i t h . h i t ~ . g e d .ends can .be.determinedc

the slope of the moment.-rotation .curve o f a beam-column aa shown in .Fig. 8.. 1. Within the e ~ a s t i c range t h e . s t i f f n e ~ s is given by

in ,which

,A.2 h2"K' = - - ........ _ ~ ,c 1 ' ; O ~ h c o t X h

A =fp,EIc

EIc- h (5)


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As t h e a p p l i e d . m o m e n t i n ~ r e a s e s .beyondtqe elas t ic . l imit , tqe.s t if fness of.the ~ o l u m n decreases and.becomes zero when .themoment.reaches the.maximumvalue. At.this i n s t a n t t ~ e columnhas lost completely i ts resistance toany further increase of.bending m o m e ~ t . ,However"if a.moment of oppositesense is applied,. the column ,wil l .behaveelastica11y again and i ts st i ffnessis equal t o tqa t given by ~ q 5. 'J: 'his is indicated as lul1loading" in Fig. 8.

' rqevar iat ion o f th e s t i f f n e s ~ K with t ~ applied end moment fo r wideflange columns with.slenderness rat ios ranging from 40 to 120 and sl1bjected~ axla l forces of .0 .12,0 .2 , :0 .3 and 0.4 ~ is shown in Fig .. 9. The valueswere.obtained by measu.ring,slopes on . t h e m o m e n t ~ r o ~ a t i o n curves presented by

17Ojalvo and Fukumoto. . ~ ~ e r e s i d u a l stress p a t ~ e r n u s e d in. th e construction.o f these moment-rotation .curyes is the same .as that previously a d o p t ~ d f o ranalyzing ,the beam members. When the.axial force Pal1d th e end moment M ofa column are . speci fied , . i ts st if fness can be determined from Fig. 9 byinterpolation .

.2.4 METHODOF.SOLUTION'J;heproposed method of ~ o m p u t i n g t h e i n e l a s t i c b u ~ k l i n g s t r e n g t h o f . f r a m

.may be summarized as fo llows:(1) Perform.a complete elast ic-plast ic analysis of the.frame.by

assuming.that no sidesway i l1s tab i l i ty occurs a ta11 stages of. loadin(2) . S ~ e < ; t . a . s u i t a b l e . l o a d level wl . a ~ d .determine the moment a t the

coluI1in tops. I.~ e v a l u . e s of Kb , Cb and K c ~ a n then. be obtained.byt4e procedures described .above .

(3) I ~ t r o d u ~ e a n a r b i t r a r y la te ra l (sidesway) ,displacement. and performa moment distr ibution computation for the.frame, a c ~ o r d i n g . t o the

.al..,15procedu.resuggestedbyWinter, et lhe i n ~ r o d u c e d fixed-end


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moment can betaken to be proport iona l to the stif fness K of eachcolumn. . . Using the end moment values resulting from the distr ibu- .

, t ion process, the horizontal. shear:Q of each column. may be deter=mined .. The sum of these. shears, r=Q, should be positive i f the. selected .load w1 is below the cr i t ical value. ' This means that: a. lateral force. is required to produce a sidesway displacement. Inthe moment. dis tr ibution procedure, i t is required t;hat assumption,( 1). of Section. 2..2 be valid . Thus, the stabil i ty of the' frame maybe examined by considering the simplified, loading. system shown. in':,

5Fig . 10b As explained in a previous report , ' the buckling loadthus determined, will be' very close to the exact value. Althoughit is possible to o btain more ,precise results, with. the. same proce-

d u r e ~ b y taking ,the thrust in the horizontal, beam.into account, (Fig . 10c), the wOl:'k involved would be prohibitive.

(4) ."Repeat steps (2). and (3) for several values of w that are' in therange' between. the yield load and the ultimate. load. By plotting

,the' total . shear r Q against the load w for each case, a ,curve. such as that shown. in Fig .. 14 is obtained .. The intersection ofthis curve with. the.1oad, axis gives the cr i t i ca l , load of theframe which, will cause' i t to sway without the application of any1atera11oad.

, In, determining the. stif fness of the. members, tpe following' assumptionsa ~ adopted with r ~ g a r d to unloading of the .yie1ded portion:

(1) '. No strain reversal is assumed to take place for the plastic por-. t ion of the beam, a t the .ins tant of sidesway buckling.


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(2) .For the case . w h e ~ . ; t h e f i r s t plas t ic hinge forms a t the .ceQ-ter ofthe beam" no unloading .o f th,e columns is assumed . . This La the.s i tua t ion that usual ly occurs for t a l l frames or frames with , s lendecolumns.Both . assumptions (1 ) and (2) ,are inagreeme\l t with .th,e generally

18accep ted concept of ine1as. ti c buck l ing .due to Shanley.(3) When .no plas t ic hinge forms in the beam, one of the columns may be

assumed to unload. This ~ s s u m p t i o n ..wasadopted in . ea rl ie r i nv e st i-19,20gations and has..been checked with. experiments


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3 CONSTRUCTION OF A FRAME BUCKLING CURVE

The :procedure outlined above will be i l l u s t r a t e ~ by two completeexamples in this. section .in .connection.with.the.development of a frameb ~ c k l i n g curve. The dimensions and member ~ i z e o f the example frames are.shown in.Fig. 11. The.span length L .i s arbi t rar i ly chosen to be 80rx(88.2 f t) , - in which .rx is the radius o f gyra tion ,about the strong axis of the33 WF130 section. A value of 2.0 is assigned for the loading parameterN. The cross sectional properties and .the material .constants adopted in the

r . .= 13 . 23 in.x3Sx = 404.8iri.= 6699 in4xa- = 33 ksiy .

p = 1263 kips M = 1113 ft-kips M = 1282 ft-kipsy y' pThe ultimate load of thes turctures based on simple plast ic theory (corres-

computations a ~ as follows:2.A = 38 . 26 in

3E = 30 x 10 ksi

pondiQg to a beam mechanism). is Pp = 349 kips or, equivalently, wp = 2.64kips per f t .

In the elas t ic range, the buckling load of the frames can. ,be determined~ r o m th e solution presented by the author in .anear l ie r report. 5 The resultsare plotted .non-dimensionally as th e dot-da,shedline (curve .AF) in .Fig 18.This curve is valid only for f r ~ e s with.slenderness ratios. greater than .thatcorresponding to point Bshownon .the curve. At this point th e elas t icbuckling load is equal to the load which causes in i t ia l yielding at . the mosthighly stressed. section. For frames having .column. slendern.ess ra tio s lessthan that indicated .by point B, inelast ic buckling will govern their load-carrying capacity.


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19/65

2 7 6 . 7 ~ 1 6capacity i s found to be.wu = 2.38 kips per f t . Comparison of this ' load withthe maximum load based on' simple p l a s t i c load indicates a reduction of. 9 .8%of. the load-c;arrying capacity due to beam-column a c t i o n .

2) . S.e1ect a' t r i a l load w1 = 2.20 kips per f t and compute the. moment. v s .r o t a t i o n curve: for. j o i n t .B .This curve i n t e r s e c t s the moment-ro ta t ion curveof . the ' column a t point 01 , -The moment lfB a t the column top for. t h i s ' load i sequal to 0.830 My' .The a x i a l t h r u s t in. the columns i s P = 3 /2 x :2 ,. 20 x 88.2 =291 kips , and t h e r e f o r e L F ,291/1263 = 0 .. 230. . T h e s t i f f n e s s of the columns.p .ysubjected to t h i s combination of bending moment. and a x i a l force are foundfrom-Fig. 9 to be

.K' = 24.0MycK = 4 .. 9 l)r

(loading)(unloading)

. ' l'he' computations. involved i n the eva lu at ion o f the s t i f f n e s s . factor~ and c a r ~ y - o v e r C b of ;the.beam are contained i n the appendix.

. o f ..these' factors t hu s obt ai ned. are:Kb = 46.6 MyCb = 0.7125

The. values

Since there is no p l a s t i c hinge forming a t the c ent er s ec ti on of thebeam for. t h i s t r i a l . load (see. the c a ~ c u l a t i o n s c o n t a i n e d in. the appendix) ,t hen accordi ng to assumption (3) of Section 2.4 one of the columns maybe.assumed to unload i n the buckling a n a l y s i s . I f the frame i s assumed to

;sway. t o the r i g h t , the l e f t column w i l l be' the unloading c o 1 u m n ~

3) A f ixed end moment due to a l a t e r a l displacement a t the. column tops, of.:MyL' = 100. f t - k ip s i s a r b i t r a r i l y assigned to a c t on the unloading column.


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276.7 -17

Th h f i d d f h 1 d 1 i 1 -_ 24.0 100en t e xe en .moment 0 . t e oa ing,co umn . s equa .to ~ 48.9 X =49.0 ft-kips. These moments are distributed and.ba1anced as shown in Fig. 13 .

Theresu1ting.shear force may be computed from express ions der ived byWinter;15 For the l e f t column

Q= _1_L ' hP(M. ,1 - -'- x M x h)- .K I FL'.-L

1=. --h (53.10 - 48.9x1113 x 100 x 66.2) (6)and for the . rightco1umn

Q = _1_ (M I -R h R p- . -. x l).R x h)

1= - (43.25 .h~ h total shear .force is

291. . . . . . . ------ x '49 x 662)24.0xll137.92= - h (7)

. ~ : .This shows that theframe. is la te ra l ly stable at . this t r ia l load.

(8)

4) Se1ectw2 =2.28 kips per f t as the second t r i a l load and repeatSteps 2 and 3. The to ta l . result ing,shear force for this t r ia l load isL Q:= O. 61/h. This indicates that the selected load is very close to thetrue .buck1ing load. In Fig. 14 the to ta l shear LQ is p1otted against th eload w for these two t r i a l s , ~ h e c r i t i c a l .load is determdned as the inter-s ec tio n o f this curve withthe.w-axis, that is wci = 2.283 kips per f t . rhetotal load . c ~ r r e s p ~ n d i n g to this value of .w is ~ c =,3/2 x 2.283 x .88.2 =.302 kips,. therefore tqera t io P:cr/pp= 302/340 ";0.865. :rhis furnisqes onepointon. the inelast ic.bucklingcurve.


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,Case 2 - Frame with h = 80rx

'-18

1) }Carry out a complete elast . ie-plast ic analysis in a manner similar' to that ' d e s ~ r i b e d in Case, 1. ,The resul t ing moment-rotation relationshipof, the, column is shown in Fig. 15.

2) ,A sa f i r s t t ry , a .load of wi = 2,.20 kips per' f t is selected. The'point of , in tersect ion of the moment-rotation curve-of the. beam for thisvalue of w with that of the column is marked as 01 in the, figure. The corresponding moment a t joint', B is 0 , . 1 7 0 ~ My . The axial force, in the columnis P = 291 kips and the rat ioY/P = 0.230. For this combination. of axia l, " , y , .force and end moment, the'column st i ffness is

K' ..= 21.0 M (loading)"c , y " , ". I t may be ' seen from Fig. ,15 that the bending'moment a t the center of

th e beam is equal to MP for ' th is t r i a l load. ,Then according to assumption(2) of S ectio n 2 .4 , n eith er of the columns should be assumed to unload.

,Therefore, the beam will be, bent in an antisymmetrical form a t the, instant'ofbuckling. "The st i ffness factor of the beam may be. determined by the

'I

simplified procedure shown in Fig. 7. Numerical computations involved are"similar to those of th e f i r s t case. They are. also included in the appen-dix. The' s t ~ f f n e s s of the beam t hus dete rmined is ~ = 70.9 My.

3), .' Introduce a fixed end moment due to lateral , displacement of'MF' =100 f t- kip s for each column.These moments can be distr ibuted and,.balanced in one cycle' as indicated on Fig. ,16 .. The resul t ing s hea r fo rc e, is : -L Q =


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276.7 =19

_This indicates that the t r i a l .load is higher' than- the., cr i t i ca l load andthat_a smal le r value of w should be assumed for the next t ry .

- 4) ,Use.w = 2.06 kips p e r f t a s the second t r i a l load and repeat Steps- 2 and- 3. ,The resulting shear, f o r ~ e - at the column tops is LQ-- = -4 .90/h.indicating that the selected w is s t i l l h ighe r- than the actual cr i t i ca l

__ load. - By us ing t he se resul t s . the inelastic buckling load of the frame. -- -. c ~ be d e ~ e r m i n e d graphically as shown in Fig. -17. ,_The, value _of wcr - isequal to 2 ~ 0 5 kips

- "oc r 271_- - - - = - - - 0.776.P --349>pshown in Fig .-18 .

-er' f t and the- to taLload P=271 kips, thus th e ratio. .-.This gives ano ther point on the inelastic buckling curve

. Similar analyses may be performed for f r a m ~ s w i ~ h different: values of:h / r . -These analyses will resul t in_ a series of points in Fig. 18, each of

.- -

_which gives the buckling load of a particular frame. - By passing a curvefrom point: B through th ese point s an inelastic: buckl ing curve is obtained.At point C this curve becomes tangent. to curveDG which defines the- strengthof theframes i f they are braced to prevent buckling. For any frame-with

,a _slenderness-ratio less than that corresponding to point C, i t s load;'carry ing capaci ty wil l not be affected by _lateral i n s t a b i l i ~ y . Therefore-within this region the problem of . frame stabil i ty may be safely ignoredand th e des ign can be based on the plastic- strength.


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276.7'. : .

. 4. S' T.-U D'Y

-20

In the course o f this invest igat ion .experiments on model s t ee l frames,were conducted to' check the val id i ty of the proposed theory. .The t e s t pro-

gram included three sets of welded rectangu1ar ' frames f a b r i c a ~ e d from a. .. .... . '.5small wide-flange' shape (2 '8 ' WF 3.}25). Figure .19 shows' the' dimensions of

the . te s t frames and the s ect ion p rope rt ie s of the WF. shape. .The span .. 1ength, ;, .

.Lwas .kept constant for a l l the' frames alld was equal to 80 times the radiusof gyrat.ion r x of' the sect ion; and th e h eig hts hof. the three frames were- . " ", - ' ... '. J t-

so chosen> that the corresponding slenderness ra tio s o f the columns were equalto 40., 60 and 80. The three se ts of tes t frames are. designated asW-I , W ~ 2 ,~ n d W - 3 . in order of the i r ' column height in 'F ig . 19.

Since the members of the frames were s ub je cte d to .bending moments about thei r major axis . i t was necessary to. brace th e frames in the. di rec-t ion perpendicular ' to the plane of. loading. Past experience in conductingframe' buckling , tests had . indicated tha t th e b ra cin g should be at tached . insuch ,a ..manner. tha t no sidesway res t ra in t . i s offered to the s tructures a t.the in i t i a t ion of buckling .. For th i s reason i t was decided to use. a' two-frame system' in a l l the t e s t s . For each. t e s t . two iden t ica l frames werefabricated and purl ins and cross braces were attached between them to ac tas . th e b ra cin g members . .. The pur.1ins were spaced a t a distance equal to 45r ,.'ywhere r i s th e rad iu s of . gyrat ion about the minor axis of the' WF sec t ion .y

.. 21This spacing was. based on the recommendation made' by Lee and Galambos.

The uniformbeam.load w assumed in the theore t ica l development was re -placed by t h r e e ~ ~ n c e n t ~ a t e d l o a d s .P1appl ied as shown in Fig. 19. I t wasobs.erved that the dis t r ibu t ion of bending moment around the frames due to


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276.7 -21

these concentrated .10ads is approximately the same as that produced by theuni fo rm load .. All the loads' were applied to the frames by dead weights magni-fied by five lever systems .. Figure 20 shows a general view of the' test setupand the fixtures. used for transmitting th e loads to their points of app1ica-t ion . :A l l the: levers and .10ading fixtures were so.arranged that th ey couldsway freely with the frames a t any stage of tes t . '.The. loads were, applied insuccessive' increments, and th e deflections of ' the beam and columns weremeasured 'after ' each load application. .The increment. of .10ad was gradually.reduced ,a s the applied load neared the p redicted load .. Figure 21 shows the

. deformed 'shape of the test ' frame W-3 af ter a l l the loads were ,removed. Typi-cal sidesway buckling may be. seen. Details of the tes t procedure and the

22experimental techniques emp1yed can be found in a separate report .

Information pertaining to these model experiments, including the framedimensions , loading parameter'N, theoretical predictions and the test re-su1ts, ..is. summarized in Table .1. The. "test load" reported in the table isnot the ..buckling load, but. the maximum .10ad observed from each test . Dueto the unavoidable imperfection of the ' test specimens, i t was diff icul t to

" .

detect precisely when the' tes t frames star ted to buckle. However ,in generall i t t l e increase. in .10ad can be expected af ter the ini t ia t ion of sideswaymau.ement, so th e ultimate.1oad observed from the tes ts should .be very closeto:.the actual buckling .load.

, I t may be seen from the comparisons given in the las t column of Table, l . tha t satisfactory correlat ion between the theory and the experiments ' has' been obtained.For' frames,W-1 and,W-3, the observed loads a rea few percent,higher't.han the predictions, while the experimental and predicted loads are


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-:.22

ap p r o x i m at el y e q u a l f o r frame W-2. ..The av er ag e d is c r e p a n c y o f the t h r e et es t s is -a bo ut 3. 6% . .This. shows t h a t t h e procedure developed in this p ap eri s cap ab l e o f p r e d i c t i n g t h e ine las t ic b u c k lin g s t r e n g t h o f frames w i t h ar e a s o n a b le de gr e e of: a c c u r a ~ y .


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2 7 6 ~ 7

5. ' COLUMN: DESIGN IN UNBRACED FRAMES

~ 2

. .l\s pointed out earlier" in ' the Introduction, i t . is Important to consider the possibil i ty of overall buckling in proportioning'co1umns in building frames which ' a renot braced to prevent sidesway. - .This has' been recognized

23by the 1961 ! ) p e c ~ f i c . a t i o n of ~ ~ American Inst i tute ' of Steel Construction ..In a llowab1e-s t ress design, the'Specification .. requires that the compression

!.

members in.unbraced frames be. designed for' their "effective.lengths" corres-ponding' to the sidesway buckling mode. (See' Section. 1.8 of the: SpecificatIon)

. Various types of design cha rt s, ta ble s and approximate formulas are available24for estimating. the:effective..1ength. of the columns in a variety of frames.

A.. convenient alignment chart , reconnnended for use.by the Column' Research25 26: Council, is included in t h e ~ Connnentary on the Specification. The chart

also provides a rapid means of computing the approximate elast ic: buckling'load of structural frames .

. When the plast ic method is used in the design of. an unbraced fram!!,th e usual approach is t o p ropo rtion the members f i rs t on the basis of theirplastic ' strength ( th i s yields the member sizes for the t r i a l frame), and

; .. then'modify the columns to take account of the possible reduction in strength.due to ins tabi l i ty . The second s tep requ ir es a close estimate of the ine1as-t ic buckling strength of the t r i a l f r a m ~ . As seen' in the previous discussion,the procedure' for determining the inelast ic buckling strength. is usually verytedious. - I t would be.impractica1 to per form such an analysis in an actualdes.ign . For this reason, the AISC Specification places l imitations on theslenderness rat io and the intensity of axial thrust in columns to safeguard

.,

agains t poss ib le fai lure due to buckling. I t: is understood that if the


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276.7 -24

columns in an unbraced frame are designed to meet the specified l imits, thereduction in load-carrying capacity due to sic;lesway buckling would be verysmall and can be ignored for pract ical purposes. In the following the valid-ity . of the design' l imitations will be discussed in the l ight of th e resul ts

. obtained from this investigation.

5.1 COMPARISON OF' RESULTS WITH THE AISC. DESIGN RULEIn deSigning an unbraced frame by the plastic method, the Specification

st ipulates that. i ts columns be proportioned to ' satisfy th e following rule,: -. ~ + h . -10.'-P' 70 - .. y . r x . ' (10) .

in which'P . is .the axi al f or ce .i n the column when the frame carries i ts . c o m ~putedul t imateload; i t is equal to.Pu for the symmetrical frame considered.. ; . 'i n this study. This rule. is applicable to columns in continuous frames,where sidesway is not prevented .1) by diagonal bracing, . 2) by attachmentto an adjacent structure ' having. ample la te ra l s tabi l i ty or . 3) by floor slabsor roof decks secured horizontally .by walls or bracing systems paral lel ' to

. th e plane of the frames.

< The above. rule was derived from an approximate solution of the. inelas.. ~ 20t ic ' frame. buckling problem. : The analyt ical procedure presented herein has. . been used to check th e validity of' th is ru le. Two groups of portal frameswere selected for this check. In a l l the frames, the beam and the columnswere assumed' to be of. the same' size . The frames in. the' f i r s t group have, aconstant span.length of L,=70 f t and variable. heights ranging from 0.2L to1.2l-. ,Frame buckling curves, similar to the one shown in Fig. 18, were con-

, structed for these. frames. '.The structural shapes used in the computations


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. 276.7 -25

were: 33 :WF:130, 27 WF. 102, 21- WF 73 and .18 I 54 . 7. For. the frames in the'second group a'span length of. 90 f twas chosen, and computations were carriedout for' two shapes: . 36 WF 260 and 33 WF130.

A loading parameter. of N = 2.0 was used for a l l the frames. "(See Fig.3) :This choice of: N. would produce approximately the loading condition that

"". '. Imay occurinthe.bot tom story of a three-story building. When applied to a, two-story frame, which is permitted to be designed plastically by th e Spec i-

.' . ' " . .f ication, the value N = 2.0 provides for additional sources of column loadssuch.as cranes, pipe' supports and. miscellaneous' loading.

"

. Figure ,22, shows a comparison between the t h ~ o r e t i c a 1 1 y computed side-'. J

sway buckling loads and the design rule . Also shown are the resul ts obtained\

from the model frame experiments previously d ~ s c r i b e d . All the buckling" .:: j ',.,

. loads are' ,expressed as percentages of the ultimate load P. which was deter-, u ,mtned. by assuming, that the frames: were' restrained from sidesway and that, fai lure was due to the i ns tab il it y o f the column members .. ,The. l ine. definedby the rule therefore should represent' the l i m ~ t within which the. Sideswaybuckling load should. be equal to 100% of the' computed u1timate.10ad. I t maybe seen from Fig . 22 that , within the range of:P:/Py and h /rx that have beencovered by.the resul ts presented in this stu4y, the rule' is somewhat con-

i ..,;

servative. ,The' buckling strengths' of some frames having combinations of'Pip' -.and h /rx "considerably outside of the safe region def ined .by the. ,rule. y " . ,. , .are more .than 95% of the ultimate strength." This: indicates that somel iberalization of the l imitat ions on slenderness rat io and axial force may

.13.be possible.


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276.7 -26

5.2 'DEVELOPMENTOF. A, NEW' DESIGN'APPROXIMATION: .The above discussion has shown that.theAISC rule is adequate.foruse. . . -,

in proportioning columns to avoid possible reduction in strength due' tosideswaybuckling . I t ' i s fe l t , however, that a' different t y p ~ of . design

". ".' lapproximation, developed by considering' the. overall behavior of frames, may

, ,prove more useful in future applications to multi-story' bu ild in gs . F ur th er -. ,.\ I .more, as will be: seen in the subsequent discussion, the proposed design, 'approximation leads to a' convenient way of estimating' the inelastic ' buckling

I ~

strength; and thus makes possible the use: of this strength as a basis ofd ~ s i g n i n g .unbraced frames.

", .

, .I n studying the s tabi l i ty of a, centrally-.10aded column, . it. is sometimes~ ,...

. useful to consider inelastic ' buckling as a type' of f ai lu re r esul ti ng fromTherefore the inelastic buck-

,. '!'the combined:effects o f y ie ld ing and buckling.,, I". :r"Ungstrength.may be expressed as a function of the axia1yie1d ..10ad of: the. . . ~

'7column section and the Euler load .. Following th e same reasoning"Merchant-', . ~ '. ' ", ..suggested that for pract ica1.purposes the . ine1asticbuckling load of'a struc- turemay also be empirically expressed as a function of the ultimate .load. , .according to the plast ic theory ,and .the elas t ic . buckling load.. . , . This func t ion'i:"depends 'on the type of struc.ture and on the .10ading condition under consid-. , " , ,_ . " ' eration. A great deal of experimental work has' been conducted on various

; ~ "

types of model frames in. an attempt to establish some simple relationships- . . .. 8 , 9 , l a ~ 1 1 that are u ~ e f u 1 in pract ical design.. , Un fortunately the resultsobtained so far are rather inconclusive. "A similar attempt is made here, .using the data given in Fig. 22

A nondimensiona1p10t of the inelastic buckling loads of',the frames. ', - . ",' . .. ..

described .in the previous section is shown in.F.ig . 23 . Two' independent


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276. T -27

parameters are used to nondimensionalize the inelas t ic . buckling loads (Pcr) i ;namely the plastic failure load P and the elast ic buckling load (P ) .u . cr e

. I t should be pointed ou t that the load Pu ' used here'. is the ult imate: :load. cor-. responding to failure by beam-column instabi l i ty , and is not the simpleplastic. load P as was originally, suggested by Merchant. A wider' scat terpof the points:had beenabserved. when Pp was used in th e plot.

-'The following may be.observed from Fig. 23:{Per) i.1. For frames with P . less than .about 0.4, the . inelastic buck. . cr)e

. ling load may be expected to be close to the ultimate load P .,".', . . . ; .., u. I t is therefore not necessary' to consider side sway. buckling inthe design of such f rames.

2.

3.

(Pcr)iFrames with ---=::::-.=-.::- 0.8 are .lik ely to buckle. in. the elast icpu. range; and hence their design should be based on.a l imitingallowable st ress .A straight l ine p a s s ~ n g through the points Gl ~ 0 . 4 , . 1 . 0 ) and G2(1.0, 0.8) can be used to approximate the lim itin g stre ng th o fthe f rames which may fa i l by inelastic buckling. ,This straight' .line is . given. by

= 3..4 ..( 11 ):{Pcr) iwhich is applicable.in.the region.where . 0.4.and.


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276.7 ~ 2

. The above observations were made on the basis of the theoretical andexperimental results obtained for the simple frame considered in this studyand for a constant . loading parameter N = 2.0. Obviously, for other 'struc-tures and loading conditions, design approximations different from thatgiven, by, Eq . (11), may have to' be developed. I t is believed, however, that

. th e proposed approximation shown in Fig. 23 can be applied to the various

. types of frames which are permitted to be designed plast ical ly by t he p res en t:: Specification.

The. procedure of using the new design approximation is .as follows:1. Perform a pre limina ry des ign using simple plastic. strength as

' the. design basis .2. Revise the column. sizes to allow for the reduction in strength

due to beam-column action. This can be done conv.enient1y byusing the beam-column tables contained in the AISC,Specifica-

-ion . The ~ l t i m a t e .10adPu of the t r i a l frame is thus equalto the design ultimate .10ad (design load multiplied by a loadfactor)

. 3. Compute th e elast ic . buckling load of the t r i a l frame' ..( P ~ r ) e 'For this purpose the a ~ i g n m e n t . charts contained in theCommen-

from t he expre ss ion.. tary on the AISC, Specification ..may be used .

. ( P c r ) ~Determine the ratio --,~ P c r ) e(P . ) 3.4_ ~ c & r ....Q1_ = ._.__-----{ p ~ r ) e . 1 +'3 (:c;r)e

Pu(12)


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. 276.7 ~ 2

I f th is ratio is less than 0.4, then. the member.' sizes chosenfor th e t r i a l frame are satisfactory and .no further revisionof . the design will be needed.

5. I f the rat io is greater than 0.4, the c a r ~ y i n g capacity of the. t r ia l f rame would be affected by overall buckling and willgenerally be somewhat less than the design ultimate load. Thisindicates that a sl ight increase in the size of the columns. (or

. the beams) of the t r i a l frame is necessary

. 6. Repeat Steps 3 and 4 for the second t r i a l frame. The selectedmember sizes will be satisfactory i f the inelastic: bucklingstrength ( P ' c ~ ) i of the frame is equal to or greater than thedesign ultimate load.

', i

The. procedure outlined above has been used in the design. of severalunbraced building frames. " I t was found that in most. cases no more thantwo t r ia ls are needed in each. design.


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276.7

6. S.U MMA R'YA N D co NC LU S. 1 0 N:S

-30

. This paper presents a numerical method for the determination of thesidesway buckling strength of par t ia l ly . yielded steel frames. The method.hasbeen i 1 1 u s t r a t ~ d with reference to the simple rectangular frame. shownin Fig. 3. Extension of . the procedure. to othertypes of frames is possible:so long as .the applied loads_and the frame geometry are symmetrical. and i ffai lure-is characterized by sidesway buckling' in the plane of the frames.It i s.be li eved that the proposed method may also' be adapted to the studyof the dynamic response of elast ic -plast ic structures, when. the.- effect of. axial force. is of considerable importance.

The inelastic. buckling s tre ng th o f a given frame subjected to a'. s.pecified system of loads can be . determined in the following manner:

(1) _A complete elast ic -plast ic ana ly sis o f the frame is made- using the. graphical method developed in a previouspaper. 3 Inusing this method, i t is possible to take. into account.' such effectsas axial force,. residual s t ress , and bending moment result ingfrom elast ic as well as inelastic -deformations. This analysisassumed that no sidesway buckling occurs a t any stage of.10ad-.ing and, therefore, the symmetricaL deformation configuration is maintained.

(2) ,A t r i a l load which. is higher than the load causing in i t ia ly ie ld ing in the frame. i s selected .. Corresponding to th isload, the st i ffness and carry-over factors' of the. beam can bedetermined by applying Eqs. 1, 3 or 4, and the.st i ffness for the column is obtained from.charts given in Fig .. 9.


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276.7 -31

(3) ,A s tabi l i ty check i s made: for ' the frame carrying. the' t r i a l load.This : .i s convenien tly done by applying the modified moment dis t r i -

. 15bution procedure. developed by, Winter J e t a1. A specif iedamount of l a t e ra l displacement i s ..introduced to the frame and, )the result ing moment of the column. top. i s determined by a momentdis t r ibu t ion computation.using. the s t i f fness factors obtained:\ ".',previously. Dividing th is moment by the column height givesthe hor izontal shear of each column .. I f the sum of the shears.

r

is posi t ive then the t r i a l load is less than the buckling load.(4) . A. h ~ g h e r .load i s chosen for ' the next t r i a l and the above steps

are repeated. The c r i t i c a l condit ion i s reached:when the sumof the result ing shears becomes zero. ,.The load at 'which th isoccurs determines the inelas t ic 'buckling s tr en gt h o f the frame.

The proposed method has' been applied to determine the b u c k l ~ ~ g-: ".

s tre ng th o f the frame'shown in Fig . . l l and the resu l ts are presented inthe form of a f rame buck ling curve in Fig., lB... .. ,Experiments were conducted on three sets of model s t ee l frames fabri=

. 'I

cated from .!i small wide"'\f1ange shape. The dimensions of the frames are given. . in,. Fig. 19 and. the t e s t setup i s shown in Figs., 20 and 21. I t : has been noted. from the comparison given in Table. I. tha t the. average. discrepancy betweenthe theory and t he expe r. iment si s about: 3.6%.

..:;"

Extensive 'calculat ions of t he bucklin g st rength of . s ix se rie s o f por=. ta l .frames were made.by using the procedure .presentedherein. Figure: 22. shows. a comparison of the results with the column design rule (Formula (20


35/65

276.7 -32

.

contained in th e 1961 AISC.Specification.. I t ~ h a s been observed that, . within. 'P' , ,

the range of the two variables( r and....h-. ) that, were covered .by the ca1cu-.. y r x.1ations , t he rule is somewhat cOllservative .

A new design approximation taking into account the. overall behavior. of unbracedf rames ' has been. developed. It is based on the interactionrelation shows' in Fig. 23 and expressed mathematically by Eq. ,11. ,Thismethod. would lead to a more rational design of the columns, especiallywhen the s t ruc ture i s ' designed to support 'heavy gravity loads.


36/65

276.7

7 . A C' K. N O WJLED GEM EN' T S

. -33

This paper is based on part of a thesis submitted to the Graduate-Faculty of Lehigh University in par t ia l fulfi l lment of the requirements. for the Degree of Doctor of Philosophy in C iv il Engineering. > The thesis. was written under the supervision of Professor George C. Driscol l , J r .

The study leading to this paper is part of th e g ene ra l investiga-tions "w:&LDEDCONTINUOUS FRAMEl) AND' T H ~ I R C O M P O ~ N T ~ " being carried out atthe Fritz Engineering Laboratory, Lehigh University, under th e generaldirection of Professor .Lynn S .. Beedle. The investigation is sponsoredjointly by the Welding Research Council and the Department of the Navy,with funds furnished bythe.American Inst i tute of Steel Construction, Ameri-can Iron and Steel Inst i tu te , Office of Naval Research, Bureau of Ships, andBureau of Yards and Docks .. Technical guidance for the project is providedby the Lehigh Project Subcommittee of the St ruc tu ral S t ee l Committee of the

Welding Research Council. ' Dr. T. R.,Higgins is Chairman of the Lehigh Pro-jec t Subcommittee.

The author wishes to express his appreciation to Dr. Morris Ojalvofor f rui t ful discussions and to Mr. Yu-Chin Yen for his effort in conductingthe e x p e r i m e n t s ~

:' .


37/65

'.,.1

.8. ,.T.A.B'L'E.S ,A:ND ,F I.GU R.E,S. . .,' \, ' .._'. - ....

-34.:: "


38/65

TABLE I COMPARISON OF TEST RESULTS WITH THEORETICAL PREDICTIONSSIMPLE BEAM-COWMN PREDICTED TESTFRAME COLUMN SLENDERNESS LOADING PLASTIC INSTABILITY Pu BUCKLING Per LOAD PexpNO. HEIGHT RATIO PARAMETER LOAD L C ~ ~ D LOAD Pp Pexp(in) h/rx N Pp Pp Per Per(kips) (kips) (kips) (kips)

W-I 43.8 40 2.0 12.43 II . 53 0.928 10.65 0.857 11.17 1.049W-2 65.7 60 2.0 12.43 11.47 0.923 10.18 0.819 10.14 0.996W-3 8"7.6 80 1.8 11.43 10.44 0.913 8.61 0.753 9.16 1.064

Ave. =1.036

IUI


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276.7 -36

Horizontal Deflection

Beam-ColumnInstabilityWJ J ..U

A

----"lasticBuckling

-Ultimate---.-----------_-0--_WuLoad

wy _Yielcl __

FIG. I ILLUSTRATION OF FRAME BUCKLING

E Simple Plastic Load '. F,wp ------ ------------- --- ~ - _ . _ -DC' GBeam-ColumnInstabilityInelasticBuckling

Load WElastic} Buckling A

LI- -IColumn Height

FIG.2 LOAD CARRYING CAPACITY OF FRAMES


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276.7 -37

H

pW

B T I 1 I \ I \ .\ \ \ \ I \ I DIb C

Ie Ie h

tL A ...rT " "- L

P=N ( W2L) t is =(I +N) w2=-FIG.3 FRAME DIMENSIONS AND LOADING

II (0) 1I II II II I /YieldedI I ZoneC ::

(b)

FIG. 4 MOMENT DIAGRAM AND VARIATION OFYIELDING OF THE BEAM


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276.7 -38

.M..=0.7My .27YF94

-I~ p - - - - - - - - - - -~

l!L .6My

0 .5 1.0 1.5 20 2.5 0 .2 .6 .8 1.0.! (EIb)eff.+y EIb(a) (b)

FIG. 5

(e) Loads on theAnalogousColumn

B ~ ~ ~ " " ( ~ ~ ~ " i ' ~ ~ D(b) Analogous-L I ..L ColumnElbJElb)effo E1b

J I Rod.

FIG.6 DETERMINATION OF Kb AND Cb BY COLUMNANALOGY .


42/65

-39

Analogous ColumnA= coIGG is finite

Load on Column

(a) I 1=--4;cJ. 8 --- 1?(b) ~ / 4 W ~

IIGI I Rad.(e) t

276.7

FIG. 7 DETERMINATION OF KbFOR MEMBER WITHEQUAL END ROTATION

MomentM

Loading

Unloadingis

M

Rotation 8

FIG.8 STIFFNESS OF A BEAM-COLUMN WITH CONSTANTAXIAL FORCE


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276.7

1.2

.2

120

: =0.12'1P'1 =Axial YieldLoad

.!l=40rc

1.2

.2

120

ISI f=0.2'1

-40

o

.2

o

20

20

40K'cMy

40K'-.My

60

pif= 0.3'1

60

80

80

o

1.0

o

20

20

40K'My

40

60

p'P,'"= 0.4'1

60

80

80

FIG. 9 VARIATIONS OF COLUMN STIFFNESS FACTOR K WITH APPLIED MOMENT(STRONG AXIS BENDING)


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276.7

H H

- 41

is' is PH . ~ . . . . . - ~

P = ( I + N ) ~(a)

Loading Condition forDetermining BucklingConstants

(b)Loading Conditionfor Analyzing BucklingLoad

(c)

FIG. 10 SIMPLIFICATION OF LOADING CONDITION

p p

h (variable)

H A L =eOr H

FIG. II ILLUSTRATIVE EXAMPLE


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276.7 -42

1.2,---------------.. . . . ,

.04

h=60r:

".'.,........ :tI I : 1.91 kips/ft............-.-.- .

.02 .038e(Radians)

"\ '.\"

\\

\ \

-\ P / ~ = O . l 2\ ~ ~ - - - - -'. / , ', / 0.20, / -----0.2-4--' . . / ,. , ..., ' . I" _ __ - - - - ~". 't.""_ w=2.28 kips/ft'f. / 0 ' - .-._.-._._._.

1 ~ ~ 1 . 0;" .. ! ! : ? - = . ~ Q ~ ! P ~ / ~ ~ . '

.01

.4

.2

.8

1.0

FIG. 12 ELASTIC-PLASTIC ANALYSIS OF FRAMEWITH h =60rx


46/65

276.7=..48.80-23.04m ' ~ g.:!:"+1.84t2.68-1.31-0.61to.30

Cb= 0.7125M = +100.00 0.488FL -51.20+11. 80 an-8.37 0

+1.-I.+0.31ML = +53.10

FIG.13

-32.34-34.77+22.95+8.0r-5.29-5.69+3.76+ 1.31:8:1+0.61

+43.2!5 MR

-43

o 2.2w (kips per tt)

4030

IQ xh2010

FIG.14 DETERMINATION OF CRITICAL LOAD (h -60rx)


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276.7

1 . 0 , . - - - - - - - - - - - - - - - -

-44

.8

.6

.4

.2

o

p

II,,,,I. L=80rx _I

.01 .02 .03'88 ( Radians)

h=80rx

04

FIG. 15 ELASTIC-PLASTIC ANALYSIS OF FRAMEWITH h=80rx


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276.7 -45- 7 Z I ~ - n l ~0.77151

MFL=+100.00 an- 22.85 CDML = +77.85

I--

10.7715:g +100.00 MFRN -22.8;

+77.85MR--o

FIG. 16

Wcr=2.05

2.2.1o-10-20-30-40-50-60-70

IQ x h

FIG. 17 DETERMINATION OF CRITICAL LOAD (h-aOrx)


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..

.InelasticBuckling

.8

E . Simple Plastic Load . FID - - - - - - - - - ~ - - - - - - - - - ~ - - - - - - - - - ~ \ -

~ D r - - - - - - - . . . . . : - ( C r - = = = : : : : : : : : : : : : : - - ; ; - - - - - ~ ' \ ~ GBeam-Column\ . Instability,

' \. , Elastic,Buckling

. ~

0 20 40 .60 80 100 120Slenderness Ratio of Column x

FIG. 18 ILLUSTRATION OF FRAME BUCKLING CURVE


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276.7 -47

.

P PL fl L Fr L Pw ~ ~ "3 3" 6

P= N ( t ~ ) h15= (I+N)--

".

=40rJlJ-I)60rx(w-2)80rx(W-3)

. L_=_8_0_r,c...x=_8_7._.6_'IDimensions of the Test Frames

A =1.043 i n Ix=1.251 in.4 :Sx=0.953 in.3rx=1.095 in.Zx= 1.067in.3rx =0.421 in.Cross-Sectional Properties

1.813" - I iT 207 ".156"__-x=I t) x-__.

'"

F.IG.19


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276.7

FIG. 20 TESTING OF FRAME W-3

FIG. 21 FRAME W-3 AFTER TESTING

-48


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276.7 -49

0.7, . . . . -------------------- ,

1000 ' 90

L=90' . 36YF260 33YFI30I Model Frame Tests

97 92 86 ':170100ct'0 0 94 0 89 0 800. .. '.alOO 961P96 87a .87 a819!100 AI88 ., ~ 0 9 9 98 9 4 93A I82 ,860L 70'o 33YFI30a 27YFI02A21YF7318 I 54.7

10 20 30 40 50 60 70h/fx

o

FIG. 22 THEORETICAL AND EXPERIMENTAL RESULTSCOMPARED WITH THE AlSO RULE


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276.7 - 5 0

1.0 G1(0.4,1.0)I

0.8 ~ 2(1.0.0.8)0.6(Per) iP

L= 70' L =90'.4 o33VFI30 . -36YF260a27YFI02 33YF130

0.2 A21 YF73018 I 54.7I Model Frame Tests

0 0.2 0.4 0.6 0.8 1.0-(Pcr)i( ~ r ) e

FIG. 23 PROPOSED DESIGN APPROXIMATION


54/65

276.7,-519 .. A' P PE: N'D l 'X

DETERMINATION: OF' STIFFNESS'OF-PARTIALLY:PLASTIC BEAMS;'BY COLUMN'ANALOGY

This appendix con ta in s th e numerical computations' involved in deter-mining the . st i ffnessesandcarry-over factors' for the beams of the two. frames considered in Art. 3. The dimensions of these frames and the. ~ p plied loads (selected t r i a l loads), are as follows:

, Member' size'Radius of. gyration, r xSpan. length, L. Column height, hDistributed load, w

Case 1' .33.WF130

',13023 in ... 80 r'x60 r x2.20 kips/f t .

Case. 233:WF.. 130

..13023 , in.80 rx80 r x

. 2.20 kips/f t .

. For each case the moments a t the ends of. the beam are f i r s t deter-.mined by the- graphical method of elastic-plastic , analysis. ,The moment. at. a l l the sections of the beam can then. be' computed by s ta t ics . '. From. the computed moment values the effective, f lexural r igidi ty of a l l the'. sections can be determined from the flexural rigidity-moment relat ion-

ship of. the 33.'.,WF. 130. section (similar to that shown in Fig. 5) . Thus,in effect , a' beam of variable El is obtained. In the buckling analysis,it is r equi re d t o evaluate the st-iffness , factor and carry-over factor

,of. this beam. -.This can be done conveniently by the method of columnana logy . .Theprecedure . of applying t ~ i s method was discussed in Section2'03. Detailed computat ions for the beams of the frames considered here

,wil l .be explained below.


55/65

.27&.7 -52

!'. Case. 1 - Frame with h = 60 r x ..The moment. at: the to p o f the column corresponding to w = 2.20,. kips / f t

h found from' Fig. 12 to be' 0..83'!)r =924. f t -k ips .. By' s t a t i cs the momenta t the center i swL L'M C= l t4 - ~ =2136 - 924. = .1212 f t -k ips

ori= 1089.My

This. indicates ' tha t the center ' s ec tio n o f the beam is not ful ly plas t i f iedat the t r i a l load ' . since equals 1.15 My' . Figure Al shows the dis t r ibut ion of bending moment of the beam and the corresponding yield configura-

t iona The s t i f fness of t hi s p ar tia ll y yielded .beam.will be computed ~ the method. of column analogy. Numerical computations are shown in ~ a b l eAI. I t is conveni en t ( and also accurate enough) to divide. t he y ie ld edpor tio n in to segments, each having a length of one foot . ,Within each seg-

ment the f lexural r ig id i ty may. be assumed to be constant . .In column (.l)of Table Al are . l isted the: lIs ta t ion" numbers o r d is ta nc es from the origin

'. G-G of the beginning and end of each. segment. Each segment is one footlong' except the 22.7 f tunyie lded segment la be lle d.1 9.9 -4 2.6 . In column(2) the distance from the o r i g i n G ~ to the ce nter se ct io n o f each.seg-ment i s l i s ted

. The moment a t the center section of each .. segment .is computed f romthe known values of and and i s l i s ted in column.(3) of Table ~ L i n

(EI ) dimensionless form. '.The effective. f lexural . r igidity . b e ff o f theseEI'' b- sect ions. is then determined from the f lexural r ig id i ty -moment curveconstructed for. th e 3 3.W 130 , s imila r: to that shown in -F ig . 5 . Thisgiyes the .va lues shown in column (li:).


56/65

276.7 -53

. ',The area and the moment of . i n e r t i a of the analogous column can then.be calculated numerically as tabulated i n T a b 1 ~ . A,-1:. columns ( 5 ) , ( ~ ) , and

Since the: width of the analogous column a t any sect ion i s inverselypropor t ional to the f lexural r i g i d i t y of t h a t s ec ti o n , the reciprocal of

. the: values o f column (4) gives d i r e c t l y the, w i d t h . li s t ed i n column ( 5 ) of.'. the analogous column a t the center sect ions . For example, , the width of, t h e f ir s t segment i s 1 ,'E'11. .= "E7 I46 , where Elb i s the f l e x u r a l0.134 ... b , b.

r i g i d i t y of the' section. i n the e l a s t i c range. The area of . each segmento f the analogous column can.be computed by mult iplying i t s width, i n column,(5) by i t s le ng th . T hearf!as are , l i s t e d i n column ( 6 ) ~ o The moment of

. i n e r t i a of each segment with respect to ax is G-Gmay be computed by using. the p a r a l l e l - a x i s theorem. .The values obtained for a l l the segments are

' l i s t e d incolumU.,< 7) .. By sunnning columns' (6 ) and (7). v e r t i c a l l y andmultiplying by two,

187.04obtained are, EI .b

the t o t a l area. and the t o t a l moment of i n e r t i a t h u s61,067.08and respect ively ..El

b'

The s t i f f n e s s of the beam o r the moment a t end Binduced by a n i m -, posed u n i t r o t a t i o n a t B i s given by Eq.. 1:

= ....1=-_.-+,187.04. .'E! '.. . b1 ( 44. 1),. .< 44. 1)

61.067.08. E l l,

= 0.003719.Elbwhere, the u n i t r o t a t i o n applied a t end B i s represented by a uni t loadapplied t o the analogous' column a t point B. The s t i f f n e s s , , factor may beexpressed, i n terms of My. by s u b s t i t u t i n g ,My/fJy for Elb , t h a t is

0.-003719: K., = '" ' . M = 46.63 M~ D , ( J y , Y y


57/65


58/65

276.7 -55

TABLE A1. DETERMINATION ,OF BEAM'STIFFNESS BY COLUMN ANALOGY(h.= 60 r x)

(1) (2) (3) (4) (5) (6) (7)Dlbstance ( l ' Width Moment ofSegment 'from I ~ Eb) eff 1.0 Area Inert ia AboutAxis q-G E ~ (4) Axis --G";G' '," '

0 1 0.5 1.088 0.134 7.46 7.46 2.491 2 1.5 1.087 0.134 7.46 ,7.46 17.412 - 3 2.5 1.083 0.140 7.14 .7.14 45.2i3 - 4 3.5 1.077 0.160 6.25 .6.25 .77.084 - 5 4.5 1.069 0.172 5.81 5.81 118.135 - 6 5.5 1.059 0.190 5.26 5.26 159.566 - 7 6.5 1.047 :0.216 4.63 4.63 196.017 - 8 7.5 .1.033 0 . 2 ~ 0 4.00 4.00 225.338 - 9 8.5 1.0.18 0.290 3.45 3.45 249.559 - 10 .9.5 1.000 0.338 2.96 2.96 .267.39

10. - 11 .10.5 ' .0.980 0.400 2.50 ,2.50 275.84.1L - 12 .11.5 0.958 0.470 2.13 2.13 281.8712 - 13 12.5 0.935 ,0.550 1.82 .1.82 .284.53P - 14 ,13.5 0.909 0.640 1. 56 1. 56 284.4414 - 15 14.5 0.882 0.720 1.39 1.39 292 3715 - 16 15.5 0.852 0.800 1.25 1.25 300 ..4116 - 17 16.5 0.820 .0.872 .1.15 1.15 313.1817 - 18 17.5 0.787 0.930 1.08 1.08 330.8418 .., 19 18.5 0.751 0.-972 .1.03 1.03 352.6119 - 19.9 . 19.45 0.716 .0.996 1.00 0.. 90 340.5319.9 - 42.6 .31.25 < 0.700 1.000 1.00 22.70 23,142.6942.6 - 43.1 42.75 0.723 0.990 1.01 0.51 992. 9343.1 - 44.1 . 43.6 0.787 0.930 1.08 ,1.08 2053.13

[= 93.52 30,533.54


59/65


60/65

276.7 -5 7

B C ~ . 8 3 0 M ,1.089My

44.1 44.1GI

FIG. AI DETERMINATION OF BEAM STIFFNESS FORTHE CASE h=60rx

1.15My Mp

G[ . O . ~ ~ 21.9' 4- 21.4: I.,

GFIG. A2 DETERMINATION OF. BEAM STIFFNESS FORTHE CASE h=80rx


61/65

276.7.. 10. : N'.O"T A, T' IO N

A = cross sectional area

. ...58

cEH

.hIK

= c a r ~ y - o v e r ' factor= Young's modulus= horizontal reaction at ' base

.. = height of frame= 'moment 'of inert ia= .st iffness of member

K" =sUf fne s s 'of. member' With ..:far :end .h i ~ g e d. K ' ~ = stif fness of member' having equal. end rotationsL = s.pan:,lengthH = bending momentH' = column moment resulting from moment distribution computations

= fixed-end moment= fullplastic,moment= .nomina1 yield ,moment

'm

N

p

-..' p

=,=====,=

moment at a point on an analogous columnloading parameter relating the concentrated 10ad,P to the. 'uniformly distributed load,w.concentrated load applied at column topaxial. yield loadc o n c e n t r a t e d , b e a m l o ~ d

to ta l axial force in.a column = (1 + N}wL/2t o t a l . ~ x i a l force in a column when the applied load is equal tothe simple plastic: ' load

- ' to ta l. ax ia l force. in a' column .when the applied load is equal to,the, computed ultimate load takingfnto account the effect. of. beam-column . instabil i ty


62/65

276.7

Per cr i t ical .value of' P

-59

{pct)"e .. = Per c o r r e s p o : ~ ~ i i n g to elas t ic buckling

:Pexp =

_.= Per. corresponding to inelasticbuckling

P .observed from e x p e r ~ m e n t. cr.Q = shear' force at column top

r .=. radius of gyrations = s e ~ t i o n modulus

. w = . intensity of uniformly dist r ibuted .load

wcr

=

=

,=

ultimate value of w .based on simple plastic theoryultimate value of w computed by, considering the effect of beamcolumn, ins tab i l i tycr i t ical value of w

z = .plastic modulusb = horizontal deflection of columne = end r ota ti on o f memberA = IP/EI co-y yield s tress of material." = curvature"y = c u r v a t u r e ' c o r r e ~ p o n d i n g to , ini t ia l .yielding


63/65

276.7 . ~ 6

11 . ' RE '. FER E N C: E S: , , , ' : : .

.1.

2.

3.

4.

5.

6.

Bleich, F 'BUCKLING STRENQTH OF METAL STRUCTURES , C H A ~ E R VI, McGraw-Hill, Book Co., Inc. ,New.York, . 1952. Ghwapa, E.DIE,STABILITAET LOTRECHT BELASTETER RECRTECKRAHMEN, Der

B a u i n g ~ n i e ~ r , " V o l . :19, )938, p.69Ojalvo, M. and Lu ,L.W.,, ANALYS.IS 'OF FRAMES LOADED INTO: THE PLASTIC RANGE, J ourna 1 ofthe Engineering ~ M ~ c h a r i i c s DiVisioh, ASCE, Vol., 87 ,No . EM4,

Proc . ' Paper' 2884, A\igUst, .1961, p .. 35 .Masur, ~ L F . , 'Chang, I.C. and Donnell, L.R.STABILITY'OF :FRAMES: IN; THE PRESENCE,OF PRIMARY BENDING .MOl4ENTS,Jo urn al o f the Engineering Mechanics' DiVIsion, ASCE', Vol. 81',No. EM,4, Proc Paper' 2882 '. August , i 9 6 l , p .19'

. , , ' ;

..Lu, L.W.'STABilr-ITY OF'FRAMES UNDER PRIMARY BENDING MOMENTS, ' Journalof . the;Structura l Divis ion , ASCE, Vol. 89, No. ST3, .Proc.Paper: 3';>47, June, . 1963', p. 35

,'Galambos, T.V. and Ketter , R.L.C O L ~ S : UNDER COMBINED BENDING AND THRUST, Transact ions, ASCE,Vol. ~ 2 6 ; Par t : I , '1961, ' p ' ~ l '

.. 7 Merchent., W.THE'.F,I\ILURE,LOAD'OF RIGID JOINTED F R A ~ O R K S ~ INFLUENCEDBY STABILITY,' The'Structura ' t Engineer , .Vol. 32, .No. 7, July1954, 'p .185 .

8 . Bolton, A.STRT,JCTURAl- F ~ O J U { , P h . D . Dissertat ion, .Manchester College. of Technology , 1957

9. Salem, A.R. , F ~ : INSTABILITY IN THE PLASTIC RANGE, Ph.D. Dissertat ion. ,Manchester 'College: of Technology, ,195'8

. 10.

.11.

Low, M . ~ ., SOME .. ~ D E L .TESTS.,ON: MULTI-STORY RIGID .STEELFRAMI!:S, Proceedi n g ~ , ICE " Vol. 13 ,Paper ,No. 6347, Ju ly '1959, p 287

Wood, R.B ..T H : & ' ~ ~ B I . L I T Y " O E TALL, BVILDINGS, Proceedings, I C ~ , Vol . l l ,Paper' No'. 6280, S ~ p t e m b e r ' 1 9 5 8 , p.69.


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12.

. -61

Wood; R.H . THE STABILITY . o F E L A S T I C ~ P l A S T I C S T R U C T U R E S . in. "PROGRESS' IN,SOLID'MECHANICS" edited by ' LN. Sneddon and R. Hil l , NorthHoT1and'PublishingCo., Amsterdam,' .1961

.13. . Lu , L.W.STABILITY OF ELASTIC AND PARTIALLY PlASTIC FRAMES, Ph.D.Disser ta t fon, .Lehigh U n i v e r s i t y ' ~ i960 ~ Universi ty Microfi lms,Inc . , .Ann Arbor, Michigan

. 14. von .Karman, T.UNTERSUCHUNGEN UBER KNICKFESTIGKEIT, .Mittei1ungen uber'Forschilngsarbelten, herausgegeben vom Verein DeutscherIngeh1.eure,: No. 81, 1910: Also in "Collected .Works ofTheodore von Karman", 'Butterworths 'Scientif ic ' Publ icat ions ,London, Vol. 1, '1956, p.90 .

.. 15. .Winter, G., Hsu, P.T . , Koo, B. and.Loh,.M,.H.B U C K L I N G . O F T ~ U S S E S ' A . N D RIGiD FRAMES, Cornell Universi ty Eng.

< Expt. Sta Bun . 36,1948. 16.

17.

Ketter , R.L., Kaminsky, E.L. , and Beedle, L"S,PLASTIC DEFOlUiATION: OF WIDE-FLANGE B E A M ~ C O L U M N S , Trans. ASCE,Vol. "120,1955, p.1028Oja1vo, ~ and Fukumoto ,. Y.

NOMOGRAPHS F()R THE. SOLUTION'OF B E A M ~ C O L U M N PROBLEMS, .We1dingResearch' Council Bulle t in No. 78', June J962

18. Shanley, F .R .. . I ~ T I C COWMN THEORY, J . of .Aeronaut;.ica1 ' Sc iences , Vo l..15, No .5 , .1947, p. 261

.19. Gurney,: T.R.FRAME. INSTABILITY 'OF' PARTIALLY PLASTICSTRUcTIJRES, Brit ishWelding Research Association R e p ~ r t FE 1/56/68, 1957

:. ".'

20. . W R G ~ A S C E . Connni t. teeCOMMENTARY ON ,PLASTIC DES IGN 'IN STEEL, ASCE 'Manua1 No .. 41,. .196'i', p.100 . .... "

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Lee " G.C. and Galambos, T.V., POST-BUCKLING. STRENGTH'. OF W I D E ~ F L A N G E B E ~ , Journa l, o f the.Engineering Mechanics Division, ASCE, Vol. .88 , .No. EM1 ~ P r o c .Paper' 3059, February .1962, p .59 . .

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.Yen, ~ Y . G . , L u , L . W . , and Drisco l l , G.C., J r .TEST8..0NTHE; STABILITYOF WELDEDSTEELFRAMJi:S, Welding ;Research. Council Bullet in No. 81, .September. 1962 .


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S T R U C r o ~ L . STEEL-FOR BUILDINGS, AISC, New York, .1961. Lu , :L.W.- A SURVEY 'OF LITERATURE:ONTIlE STABILITY :.OF FRAMES, Welding

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GVIDE TO DESIGN CRITERIA FOR METAL COMP:RESSION: MEMBJRS, CRC,1960 " . . - .

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(1963) Inelastic Buckling of Steel Frames Le-wu-lu 65p

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