ultimate strenth of laterly laoded columns

8/13/2019 Ultimate Strenth of Laterly Laoded Columns

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lasi ic esignof Multi Story r mes

November 966

byLe Wu Hassan Kamalvand

ULTIM TE STRENGTH OFL TER LLY LO E O UMNS

R TZ N IN RINL SOR TORV LI Y

Engineering Laboratory Report: No 73 5


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_ _ _

Plast ic Design of Multi Story Frames

ULTIM TE STRENGTH OF L TER LLY LO DED COLUMNS

Le Wu Luand

Hassan Kamalvand

This work has been carried out as part of an investigationsponsored joint ly the Welding Research Council and the De-partment of the vy with funds furnished by the following:

American Iron and Steel ns t i tu teAmerican nst i tute of Steel Construction

Naval Ships Systems Command v l ac i l i t ies Engineering Command

Reproduction of th is report in whole or in part is permittedfor any purpose of the United States Government.

Fri tz Engineering LaboratoryDepartment of Civil EngineeringLehigh UniversityBethlehem Pennsylvania

November 1966Fri tz Engineer ing Laboratory Repor t No. 273.52


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SY OPS S

Analy t ical procedures are developed for computing them ximum carrying capacity of s t ee l columns subject to combinedax ia l th rus t and l a t e r a l load I t is assumed tha t the columnsare permitted to d ef le ct only in the plane of the app lie d lo adand th at f ailu re is always caused by excessive bending in thes me plane Numerical resul ts a re obtain ed for four types ofcolumns with different loading and support conditions and arepresented in the form of in teract ion curves re la t ing axia l thrus tl a t e ra l load and slenderness ra t io . he analyt ical ly obtainedresul ts are compared with the predictions based on an empiricalin teract ion formula and good agreement is observed

i


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T BLE OF CONTENTS

Page

SYNOPSIS i1 . INTRODUCTION 1

1 .1 Previous Work 2l .2 Scope of Investigation and Assumptions 2

2. GENER L PROCEDURE OF NUMERIC L INTEGR TION 52 .1 Integration Procedure 52.2 Moment Curvature Thrust Relationships 8

3. METHO OF SOLUTION N RESULTS FOR C SE a 104. METHO OF SOLUTION N RESULTS FOR C SE b 155. METHO OF SOLUTION N RESULTS FOR CASE c 176. METHOD OF SOLUTION N RESULTS FOR C SE Cd 197 COMP RISON OF RESULTS WITH INTER CTION FORMUL S 228 SUMM RY N CONCLUSIONS 9. CKNOWLEDGMENTS

10. NOT TION 2811. FIGURES 3012. REFERENCES 40

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INTRO U TION

Compression members subjected to l a t e ra l or tr an sver seloads occur frequently in building frames, bridge t russes andother important engineering st ructures. They are. usually proportioned to sat isfy some l imi t ing stress c r i t e r i a se t by speci f i cations or codes. The st resses developed a t any cross sect ion insuch a member consist of 1 the axial stress caused the com-pressive force, 2 the primary bending stress due to the l a te ra lload, and 3) the secondary bending stress produced by th e so -c alle dsecondary moment, whlch is the product of the resul t ing deflectionof the section and the compressive force. The l a s t stress in t ro-duces ins tabi l i ty ef fec t into the members and becomes part icular lyimportant fo r columns with high slenderness ra t ios and carryinglarge compressive forces. The procedures fo r computing the secondarymoments and st resses in elas t ic columns are described in books ons tab i l i ty theory. l 2 3

Although elas t ic analysis has been used extensively indesign computations, does not give accurate indicat ions of thet rue load-carrying capacity. Lateral ly loaded columns generallyf a i l by excessive bending af te r the st resses in some portions ofthe member exceed the e las t i c l imit . To determine the ultimatestrength of such a column, i s necessary to perform a s tabi l i tyanalysis tha t considers the elas t ic p las t ic behav io r o f the varioussec t ions. Unfor tu na te ly , th e required analysis i s often too complex


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TABLE OF CONTENTS

Page

SYNOPSIS i1 . INTRODUCTION

1 .1 Previous Work 1 .2 Scope of Investigation and Assumptions

2. GENERAL PROCEDURE OF NUMERICAL INTEGRATION 5

2 .1 Integration Procedure 52.2 Moment-Curvature-Thrust Relationships 8

3. METHO OF SOLUTION N RESULTS FOR CASE a 104. METHO OF SOLUTION N RESULTS FOR CASE b 155. METHO OF SOLUTION N RESULTS FOR CASE c 17 METHO OF SOLUTION N RESULTS FOR CASE d 197. COMPARISON OF RESULTS WITH INTERACTION FORMULAS 228. SUMM RY N CONCLUSIONS 59. ACKNOWLEDGMENTS 10. NOTATION 2811. FIGURES 3012. REFERENCES 40

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for prac ti ca l app li ca t ions, and recourse is sometimes made toempir ical formulas 3 4 formulas providing approximate estimatesof the column strength .

The purpose of this paper i s to develop eff ic ient methods,part icular ly adaptable to computer prograrruning for performinge l as t ic p la s ti c s t ab i li ty analysis for a variety of columns and topresent numerical results in a form suitable for des ign u se .

1.1 PREVIOUS W RIn c on tr ast to the extensive work done on columns sub-

jected to combined axial force and end moments 2 5 6 7 8 only a fewattempts have been made to study the strength of la teral ly loadedcolumns. Wright developed an approximate formula for the case ofa column loaded by a concentrated load at the midspan [Case a int s paper].9 The same case was also studied by Ketter w developed f 6 1 hdnteractlon curves or a varlety 0 co umns. n emplrlca met 0for estimating the ultimate strength of lateral ly loaded columnswas proposed by Horne and Merchant using the modified Rankine For-mula. The val idi ty of the method has not been verified by comparing t he e st imat ed strength with the strength determined eitherfrom exact solutions or from laboratory experiments.

1.2 S OPE OF INVESTIG TION ND SSUMPTIONSThe loading and support conditions of the four cases of


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l a tera l ly loaded columns in ve st ig ated i n the paper are shown inFig. 1. The columns are assumed to be prismatic and made of tf asro l led wide-flange. shapes. In a l l the cases , the l a te ra l load ,R or w causes bending moment about the major axis of the crosssection.

For each case, a method of solut ion is f i r s t developedand numerical resul ts are then given for columns with s lendernessra t ios ranging from 20 to 100. The resul ts a re p res en te d as interaction curves. All computations were performed on A 6 yield s t ress =6 ksi) s tee l columns, but the resul ts can also be used for othercolumns with different yield s t ress levels by the proper adjust ingof the slenderness ra t io see Summary and Conclusions). The Offiputed ultimate strength is compared with the ult imate strength. pre-

~ t e by th e emp iri ca l interact ion formula contained in the AISCSpecification.

The following assumptions are made in the solut ions:1. The s t ress s t ra in properties of the column material

are elas t ic and perfect ly plas t ic and the effect of stra in hardeningis neglected.

2. For a given combination of axial force and bending

f ina l valuesof the axial force and bending moment and tha t the actualhis tory of loading does not affect the resul t ing curvature.

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3. he effect of shear i s small and ca n be neglected.4. Weak axis buckling and la te ra l to rs iona l buckling

are. effectively p re ve nte d so that fa i lure i s always caused excessive bending in the plane of the applied l a t e ra l load.

In performing numerical calculations i t i s furtherassumed tha t the axial force is applied f i r s t and maintained ata constant value as the l a t e ra l load in cre as es o r d ec re ase s.


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2 . GENERAL PROCEDURE OF NUMERICAL INTEGRATION

The methods of solution to be developed subsequentlyfor the individual cases ut i l ize a common numerical integrationprocedure. I t was f i r s t used von ~ r m n in h is studies oneccentrically loaded columnslO and was l a te r modified by Chwalla,Ojalvo, and others for use in th e ana ly sis of restrained columns. l2I t is fur ther modified in th i s paper to take into account the effectof l a t e ra l load and var ia t ion in end conditions.

2.1 I ~ T G R T I O N PROCEDUREFigure 2 shows a portion of a la tera l ly loaded column

whose deformed. eonf igura t ion in the e las t ic and elas t ic p las t icrange is to be determined. The applied forces consis t of the axialforce, P, l a t e ra l load, q, end moment, MO and support reaction,VO They are considered as the known quanti t ies in this discussion.I t is required to determine the deflect ion curve of the member whenthe i n i t i a l slope, eO a t the l e f t end is assigned a specif ic value.This canbe accomplished by applying a segment-by-segment integrationprocess, start ing from the l e f t For the f i r s t segment, whoselength is chosen to be Pl , the deflection a t i t s mid-point shownas a dot in Fig. 2) is approximately equal to

PI8a l

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and the corresponding bending moment is

M = P V PI - M - I Moment ue to q oj 2a l a l 0 0 _Applied Over Pl/2This moment is assumed to be the average moment of the entiresegment. The average curvature, al ,can be determined from themoment-curvature-thrust M--P relat ionship which includes both

13 14the elas t ic and ine las t ic range of cross sect ional response. The propert ies of the M--P relat ionships used in th is paper wil lbe brief ly described l a te r . When 0a l is known, the slope andd efle ctio n a t the end of the f i r s t segment can be computed fromthe expressions

8I = 80 - 0alP I

and the corresponding bending moment i s given by

M1 = V - M _ [Moment ue to q]I I 0 Applied Over PI

3 ,

5

The values of 81 and 61 determined from Eqs. 3 and 4 wil lbe used as the i n i t i a l values to s t a r t the in tegrat ion for thesecond segment. Again, the deflect ion and bending moment a t themid-point of the segment are f i r s t computed

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6 )

an d

P 2 [Moment Due to q ]M a2 + Vo 1 + ) - MO -a2 A ppl i ed Over p 1 + P 2 / 2 7 )

The average curvature, a 2 o f th e segment i s then found from theM P r e l a t i o n s h i p . The slope, d e f l e c t i o n , an d bending moment a tth e en d o f th e segment can then be determined from the equations

8 )

9 )

an d

M2 = po + V + p M - [Moment Due t o q ] )2 a p1 2 a Applied Over p 1 + P2Repeated cal cu l at i o n s can be c a r r i e d ou t fo r as many

segments as necessary. The cal cu l at i o n s y be terminated whenc e r t a i n s p eci fi ed c o n d i t i o n s are sat isf ied For i n st a n c e , inanalyzing columns with a given length, L, th e i n t e g r a t i o n y beterminated when th e to ta l accumulated l e n g t h i s equal to L/2. t

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i s also possible to terminate t he cal cu la tions when the slope ofthe def lect ion curve becomes zero or negative. f the integrat ionprocess is terminated f ter completing n segments Fig. 2), thel s t se t of the numerical resul ts gives the values of the slope,deflection, and bending moment of the point which is located ata distance PI P p from the l e f t end.2

The procedure described above can be effectively pro-grammed on a dig i t l computer, and extensive computations can bemade for many columns with different loading and support cond it ions.The in i t i l cond it ions requ ired to s t r t the in tegrat ion and thec r i t e r i used to terminate are, of course, somewhat di f ferentfor the different cases. They wil l be discussed in some det i lwhen the method of solut ion for each case is presented.

2.2 MOMENT CURV lURE THRUST REL TIONSHIPSThe M P relat ionships used in the computations were de-

termined for the 8W31 section by a separate computer program. nth is program a moment vs. curvature curve was developed for a con-sta nt a xial thrust by d iv id ing th e cross section into a large numberof f in i te elements. The strains of the elements are related to thecurvature of the sec t ion and the s tresse s to the applied bendingmoment. The relationship between the applied moment and the resul t ing curvature can therefore be found through equilibrium andcompatibil i ty conditions of these elements. The detai ls of the

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method and the computer program are described elsewhere. 14

The basic program can be easily modified to take intoaccount the ef fec t of residual stresses. In th is study, onlyresidual s tr es se s r esu lt ing from dif feren t i l cooling rate areconsidered. The dis t r ibut ion and magnitude of the res idual stressesadopted in the calculations are the same as those used in the previous studies on beam-columns. 5 ,6,13 When the resul t ing M Prelat ionships are used in the numerical in tegrat ion process, thef in l resul ts deformations and ultimate strength wil l automaticallyinclude the inf luence of res idual stresses.

Although the ~ P curves and other cross sectionalpropert ies used in th e an aly sis were based on the W l sect ion,the numerical resul ts obtained, f ter proper nondimensionalization;are valid for other column s ec tions a ls o. t has been found inanother study th t the M P curves of the 8W sect ion are close

h M t 15to t e average -p-P curves most co umn sec lons.

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3. METHO OF SOLUTION N RESULTS FOR SE aThe problem to be solved for this case is as follows:

for a given column with known length and cross sect ional propert ies subjected to a specified axial force, determine the maximuml a t e ra l load, t h a t can be s af el y c ar ri ed by the member. Themaxf i r s t step in the solut ion is to find a systematic approach fo rdetermining the response of the column to the varying l a te ra l load.The desired response is usually represented by a l oad versus centerdeflect ion R 00 curve or a load versus end slope R 80 curve.Once the complete response curve i s o t ~ i n e d the maximum load canbe easily determined from the peak of the curve.

Referring to Fig. 2, the specia l cond it ions appl icableRto Case a are: MO = 0, q 0, and Va =2. With t he se condi ti on s,

the in tegrat ion process may proceed according to the scheme de-scribed above af te r Rand 80 are assigned specif ic values . Incarrying out the actual numerical computat ions , i s convenientto f i r s t specify a value of R and to perform repeated computationsfor a number of selected eO values. The procedure i s then repeatedfor other R values . In a l l the computations, the in tegrat ion isterminated a t a point where the slope of the deflect ion curve be-comes zero. This i s done because the actual deflect ion curve ofthe column has a zero slope a t the midspan. The distance from thel e f t end of the column to the point of zero slope obviously wil l

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vary with the assumed in i t i a l slope eO.

Since the purpose of the computations is to obtain nu-merical resul ts fo r se veral columns with specif ied slendernessra t ios , is more convenient to se lec t the segment lengths asmultip les o r fractions of the radiu s of gyration of the cross sec-t ion . Both the slenderness ra t io and the radius of gyration usedin the subsequent d is cu ss ions a re computed for the major axis.In order to improve the accuracy of the f in a l r es ul ts , the lengthse le cte d f or a given segment is varied according to the bendingmoment found a t the end of the previous segment. The followingcr i te r ia are used in the selection of segment length

p = 2r i fp = r i f

o sIMI 8 Mpc0.8 M M 0.9 Mpc pc

l l a

l I b

p = O.II i f 0.9 M M M pc - pc l I e

in which M i s the reduced plast ic moment corresponding to thepcspecif ied axial force. an example i f a t the end of the tenthsegment the bending moment MlO is found to be 0.85 Mpc then thelength for the eleventh segment should be PI1 = I Thus s tar t ingfrom the l e f t end the segment length decreases from 2r to r andf inal ly to O.lr i f the bending moment equals or exceeds 0.9 MpcFor certain cases in which the bending moment is always less than

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0.9 M segment l ~ t h equal to O.lr i s also used -for the_ pcl a s t few increments before terminating the computations. Thisadjustment permits a more precise determination of the location a twhich the slope of the deflect ion curve becomes zero. When theintegrat ion process carried out f9r a given case is completed, thet o t a l distance included in the computations is usually given as amUltiple of r say Ar.

To provide a systematic, way of specifying the value ofthe l a t e r a l load, a reference load is used. This load, denotedby Rpc20 i s defined as the plast ic load of the shortest columnL/r = 20 considered in the computations and is given by

Rpc20 M M= _ p_c _ pcL s 12

In a l l the numerical computations, the l a t e r a l loads are alwaysspecified as fractions of the reference load. In a similar manner,the axial force, P is specified as a fract ion of the axial yieldload, P .y

resul ts of. computations made for the case withP = 0.4 P are shown in Fig. 3. Each curve gives the relat ionshipybetween the i n i t i a l slope, 80 and the resul t ing zero-slope dis-tance Ar or A for a specified value of R. I n i t i a l l y an increasein eo resu lts in a corresponding increase in A The rate o f i n -crease in A i s gradually reduced as eO increases; and, corresponding

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to some assumed 8 a maximum A is eventually reached. At th ispoint a fur ther increase in causes a decrease in A Althoughthe relationship between A and eO tends to reverse for larger eOvalues, the bending moment MA t the point of zero slo pe is foundto in cre as e continuously with 90 . For some specif ied i n i t i l slope,say eop the bending moment in some segment may reach the reducedpl s t ic moment M o the section before the slope of the de-pcf lect ion curve becomes zero. When th is occurs, the integrat ionprocess i s discontinued and no further computations wil l be per-formed for l rger assumed i n i t i l slopes. Each 80A curve thereforeterminates t a maximum i n i t i l slope equal to eOp . A t o t l oftwenty-three eO A curves was prepared sixteen are shown in Fig. 3from which the l t e r l load versus end slope relat ionships given inFig. 4 were obtained.

The procedure for determining the l oad-end s lope u ~ v si s i l lus t r t ed in Fig. 3 for a column with L/r 60. A vert ic ll ine is f i r s t drawn from the point where A equals 30 . The pointsof in tersect ion of th is l ine with the various 80A curves give theend slopes of the column when it is loaded the specif ied l te r lloads. The resul ts obta in ed a re plotted as small circles in Fig. 4,and, through these ci rc les the desired load-end s o ~ curve label led 60 can be constructed. The peak of the curve gives theultimate load of the column. The l te r l load i s nondimensionalizedwith respect to the individual pl s t ic load, Rpc not with respect

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to t he r ef er en ce load Rpc20 as was done in Fig. 3 This permitsa cl o s er examination o f the e f f e c t o f ins t i l i ty on the s t ren g t han d e h v i o ~ o f the f i ve cO lumns.

The procedures described above for, o bt ai ni ng t he 80ch art s and th e RR - eo curves have been r ep e at ed f or th e followingpcs pe ci fi ed a x ia l forces: p = 0.2 P 0 .6 P 0.8 P and 0 .9 P .Y Y Y YThe maximum l ter l l oa ds o bt ain ed f o r t h e s e cases t o g e t h e r withthose determined from Fig. 4 are summarized in the form of ultimates t ren g t h i n t e r a c t i o n curves in Fig. 5 Each curve is fo r a par-t i cu l r column an d gives th e combinations of a x i a l force an d l ter lload th t can be safely supported th e column. The l ter l loadi s now nondimensionalized with r e s p e c t to th e simple p l a s t i c load,R o f th e i n d i v i d u al columns assuming t h a t no a x i a l force is applied to th e columns). The inte r a c tion curves can be d i rect l yused in analysis an d design computations an d a l s o in checking th ev al i d i t y o f th e ex i s t i n g design approximations. 3 ,4,9


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4. M THO OF SOLUTION AND RESULTS FOR CASE b)

The numerical in tegra t ion procedure i l lus t ra ted inFig. 2 is again used to develop the load-deformation curves ofCase b) columns by ut i l iz ing the appropriate boundary and load-ing condit ions. An elementary analysis wil l show that the bendingmoment is general ly higher a t the midspan than at the two ends.The center portion of the column always yields f i r s t Althoughsome yielding would also occur near the ends, no plas t ic hingeshave been found to form a t these locat ions when the column isloaded by the maximum l a t e ra l load. Hence, t he app ropr ia te con-di t ions to be used in the integration process are eo = 0, q = 0,

Rand Vo =2. The t rying variable that must be assumed before s ta r ting , the in tegrat ion i s the end moment MO

For a specif ied value of R a number of MO values tf i r s t assumed and computations are carried out for each assumedMO to determine the ze ro- slope d istance Ar. When the integrationprocess reaches the point of zero slope, the aeflect ion a t tha tpoint, 0A is also determined. chart , s imilar to that given inFig. 3, can then be prepared to give the relat ionships between 0Aand A fo r a ser ies of specified R values. From th i s chart thel a t e ra l load versus center deflection R - 8 curves of the columnscan be determined. The maximum l a te ra l loads tha t can be resis ted the columns are aga in g iven the peaks of the curves.

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The maximum loads determined fo r th e f iv e s ele ct edcolumns subjected to various s p eci fi ed a x i a l loads are pre-sented in Fig. 6. The l t e r l load is again nondimensionalizedw ith respec,t to the simple pl s t ic load o f th e i n d i v i d u a l member

1 3 )

A comparison of th e i n t e r a c t i o n curves o f Fig. 6 w ith tho se ofFig. 5 i n d i cat es th t th e e f f e c t o f ins t bi l i ty is l es s pronouncedin fixed-end columns than in simply-supported columns. The fixed-end columns are usually s t i f fe r an d have l e s s d e f l e c t i o n s t th emaximum l o a d . Consequently, the secondary moment which causesth e ins t bi l i ty effect is also l e ss

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5. METHO OF SOLUTION N RESULTS FOR SE c

Because of the d if fe re nc e i n loading conditio n, th emethod developed previously for Case a columns is not direct lyapplicable to th is case, although the basic numerical integrationprocedure can s t i l l be used. In the present case, the reactionVa a t the l e f t end depends not only on the l a t e r a l load, q, butalso on the span length, L. t is not possible to fully specifyVa and to s ta r t the in tegra t ion without actually knowing thezero-slope distance, Ar which should be equal to L/2 i f theassumed i n i t i a l slope eo is correct . For this reason, eachcolumn has to be analyzed i nd iv idual ly for a number of selectedl a t e r a l loads in order to determine i t s complete load-deformationre la t ionship.

Figure 7 shows the procedure used to obtain the endslopes of a given column when it is loaded y specified axial andl a t e r a l loads. The slenderness ra t io of the column is equal to60, and the specified loads are P 0.4 aR d w = 0.66 w iny pcwhich w is the plast ic load of the member and i s given yp

w ppc 7 14To determine the end slopes, a number of t r i a l eO values are f i r s t

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selected and the numerical in tegra t ion process is carr ied outfor each value. The appropriate conditions fo r integrat ion are:MO 0, w and a =wL/2. In each c a ~ e when the accumulatedlength covered by the computations reaches L/2 or 30r, the in -tegrat ion i s terminated and the slope 830 r e c o r d e ~ Unless the80 happens to be correct , 830 i s usually di f fe ren t from zero.When the assumed 80TS are plotted against the recorded 8 30 s thecurve given in Fig. 7 i s obtained. The correct end slopes for thecolumn are found to be 0.0144 rad. and 0.0224 rad. These resu l t sdetermine two points on the load versus end slope curve Fig. 8 .Additional points on the curve can be obtained y r ep ea tin g th eprocedure for other specif ied l a te ra l loads . When a l l the f ivecolumns are analyzed by th is method, the 80 curves shown inwpFig. 8 a re obta in ed , from which the maximum l a t e r a l loads can bedetermined.

Figure 9 summarizes the resul ts of ult imate strengtncomputations made for the five columns. The maximum l a te ra l loadi s nondimensionalized with respect to the simple plas t ic load, Wwhich is determined from Eq. 14 by subst i tu t ing M for M AP pclose examination of the in teract ion curves given in Figs. 5 and 9ind icates tha t on a nondimensional basis the reduction in load-car ry ing capac it y due to i ns ta b il ity i s greater in Case c columnsthan in Case a columns. This can again be a t t r ibu ted to thedifference in the resul t ing deflect ion produced by the twotypesof loading.

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6. METHOD OF SOLUTION AND R SULTS FOR CASE Cd

The determination of th e u ltim ate strength of Case Cdcolumns i s complicated the fac t tha t the columns may fa i l intwo different modes. For columns in which the bending moment iscaused primarily. the l a t e ra l load, plast ic hinges tend to forma t the ends prior to the attainment of the maximum load. This. s i tuat ion i s shown as Case I in Fig. 10. The available conditionsto be used in the integration process are: MO =Mpc q = w and

Vo =:2 and the unknown quantity that must be determined by trialis the end slope eO. The problem is thus seen to.be the same asthat for Case c) columns and the method of solution describedpreviously fo r that case can be direct ly applied.

In columns hav ing h igh slenderness ra t io and subjected\to heavy axial load, the bending moment a t the center may be ampli-

f ied signif icant ly by the secondary moment. As a resul t the centermoment may become larger than the moment at the two ends, and i n i t i a lyielding i s l ikely to occur near the midspan. Although in somecases, l im i ted y ie ld ing can also take place a t the ends, no plast ichinges are found to develop a t these locat ions. Th e ends of thecolumn can therefore be assumed to remain fixed for the purpose ofanalysis. This case is shown as Case I I in Fig. 10, and the knownconditions eO 0, wL I t be easilyre: q = w, and V = cana 2recognized that apart from the difference in l oading cond it ion the


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problem to be solved is essent ia l ly the same as th t involved inthe analysis of Case b columns. For a given column subjectedto specified axial and l te r l loads, the quantity to be determinedby the t r ia l-and-error procedure is the end moment MO A ser ies oft r i l MO values is f i r s t selected and numerical in tegrat ion iscarr ied out for each selected value to determine the slope e atthe midspan. A plot, similar to the 8 versus 8c plot given inFig. 7, can then be obtained between the t r i l MO value and theresul t ing slope 8c From th is plot the correct MO value which produces zero slope at the midspan can be determined.

For cer ta in combinations of slenderness r t io and axialforce, may be di ff i u l t to predict which case is the governingcase. In such s i tua t ions , t r i l calculat ions must be performedfor both cases.

Figure 11 summarizes the numerical resul ts obtained forthe five selected columns. The l te r l lead is again nondimension-alized with respe ct to the simple plast ic load, which is givenpby

wp 6M

p1 2 15

The interact ion curves presented in Fig. 11 can be compared withthose given in Fig. 6 to assess the effect of vari at io n i n loading

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condition The same curves can also be compared with the in ter-action curves in Fig to evalu ate th e in flu ence of end restr in t

21


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7. COMP RISON OF RESULTS WITH INTER CTION ORMUL S

In designing members carrying combined axial compressionand bending s t resses , use is frequently made of the so-calledinteract ion formulas. Three such formulas are given in Part 1 ofthe ISC Spec if ica ti on for use in al lowable-stress design. A1though these formulas were originally developed for pred ic ti ng thestre sse s in the elas t ic range, studies have shown tha t they also

16 17provided good e st imate s o f the ultimate s t r e ng t h . Thesestudies were made primarily for columns which are loaded by combined axial force and end moments. In the following, a similarstudy wil l be made for the four cases of la tera l ly loaded columnsanalyzed in the paper.

The formula which wil l be used in the comparison isFormula 7a) given in the Specification. I t can be writ ten interms of ultimate loads as

1 .0 16a)

for Case a) and Case b) columns, or as

P ew m = 1.0 p wCl 1 p pe

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16b


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for Case c) and Case d columns. There are three new terms,P P and C in these equations; they are defined as follows:e r e m

P = cr i t ica l load of the member_loaded coneI centr ical ly by axial force only. Fer thecolumns under consideration, i t is thebuckli ng l oad about th e strong axis.

P = e las t ic cr i t ica l load for buckling in theeplane of the l a t e r a l load.

C a coeff ic ient depending on loading andmsupport conditions. I t assumes the followingvalues for the cases of columns consideredherein

Case a)

ase b)

Case c

Case d)

C 1 0.2 P p-m eC 1 P 0.6 pm eC = 1 .0mC 1 P= 0 .4 m Pe

17a)

17b)

17c)

17d)

The maximum loads determined by the analyt ical proceduresfor the four types of columns are compared in Fig. 12 with the in ter-action formula in a manner suggested by Mason Fisher and Winter. 17 .

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In g en eral, the interact ion formulas are seen to give good pred ic tio ns fo r l l the invest igated. For Cases a and c ,the formula tends to undere-stimate the carrying capaci ty in thelow axial force range, but the difference is usually small. Thesame trend can also be observed for Case b and Case d columns,except th t in these cases the difference on conservative sideseems to be somewhat larger fo r low slenderness r t io columns. may be concluded from this study th t the IS interact ion:formula is valid not only in p redict ing the el st i range st ressesbut also in estimating the ultimate strength.

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SU RY P ON LUSIONS

general method for determining the load deformationrelationships of la tera l ly loaded columns in the elas t ic andelas t ic p las t ic range has been developed. I t employs a segment-by segment integration process using th e a va il ab le moment curvature-thrust data as the basic impact. By properly ut i l iz ing the boundaryand support conditions method can be effectively used to ob-ta in s olu tio ns to a variety of column problems. Specific appli-cations have been made to the four cases of columns shown inFig. 1. For each case the general method was applied to obtainu lt im a te s tr en gth solut ions for five selected columns with slender-ness ra t ios ranging from to 100. he resul ts are given inFigs. 5 6 9 and 11 in the form of interaction curves relat ingthe axia l thrus t maximum l a tera l load and slenderness rat io .

All the resul ts obtained in this study are for wide-flange columns subjected to l a te ra l loads producing bending momentabout the major axis of the cross sect ion. I t has been assumedtha t fai lure is always due to excessive bending in the plane ofthe applied load and t ha t l at er al t o rs io n a l buckling or local buck-l ing does not occur throughout the loading history. In a l l thecalculat ions the influence of cooling residual stresses were takeninto account through the use of the special moment curvature thrustrelationships t ha t in clu de th e effec t of these s t resses . Althoughthe in terac t ion curves were prepared for A36 s tee l with a nominal

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yield s t ress of ksi , they can be applied to s teels of otheryield s t ress levels by subst i tu t ing an equivalent slendernessrat io

18

This subst i tu t ion wil l yield exact resul ts i f the res idual s t resshas the same dis t r ibut ion pattern over the cross sect ion and thesame proportion of the yield stress fo r the different s tee ls

The maximum l a t e ra l loads determined the analyt icalprocedures have been compared in Fig. 12 with the ISC interact ionformula and good agreement has been observed.

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9. KNOWLEDGMENTS

The work described in this paper was f i r s t carried outby the junior author when he was a Byllesby Research Fellow inthe ritz Engineering Laboratory Department of Civi l EngineeringLehigh University. The research was performed under the supervision of the senior author who formulated the analyt ical proceduresand prepared portions of the computer program.

The senior author has been closely associated with ageneral investigation on Plast ic Design of Multi-Story Framesfrom which the subject matter studied in the paper was developed.The investigation is sponsored the Welding Research Counc il andthe U S. Navy Department. Funds a re s upplied by the American Ironand Steel Ins t i tu te American Ins t i tu te of Steel ConstructionNaval Ship Systems ommand and Naval aci l i t ies Engineering ommandTechnical guidance for the project is provided by the Lehigh ProjectSubcommittee of the S truc tu ra l S te el Committee of the Welding Research Council. Dr. T R Higgins is Chairman of the Lehigh ProjectSubcorrunittee.

Dr. P. Parikh was responsible for the development ofthe moment-curvature relationships used in the computations. Hisassistance is grateful ly acknowledged.

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CmL

MMappcrpepyqR

10. NOT T ON

a coeff ic ient contained in the AISC in teract ion formulalength of columnbending momentaverage bending moment of segmentax ia l forcec r i t i c l load of column i f axial force alone existedel s t ic c r i t i c l load about s tro ng a xisa xia l y ie ld loadgeneral distri uted l te r l loadconcentrated l t e r l load

R maximum value of R according to plast ic theory. Thepcreduction in moment-carrying capacity due to ax ia l forceis i nc luded but t he effect of column in st il i ty is not.

R maximum value of R according to simple pl st ic theorypr radius o f gyra tio n about major axisv sh ear fo rceVa shear force t beginning of f i r s t segment support reactionw uniformly distri uted l te r l loadw maximum value of w according to pl st ic theory See Rpc w maximum value of w according to simple pl st ic theoryp deflect iono average deflection of segmenta

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e slope t j ~ t n t l end slope

A = a parameter used to define zero slope distancep length of segment = yield s t ress of materialy curvature

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R Rp p e - . ~ t ~ ~

j L 0 bw wp p p p L I 1 L Ic d

Fig Cases of Lateral ly Loaded ColurrmsStudied in the Paper

p ~ - - - - - - - ~ - - - - - - . . . . a . - - . . L - - - I ~ - - L . . .Mo t ~ I on I I I I I PI I I :I I I I VI I I : I I n~Fig. 2 Numerical ntegra t ion Scheme

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I p = 0 4 IyILNI JI

0 040

0 035

0 03080 R a d i ~ n s

0 025

2

0 015

0 010

0 005

o 5 5 25 3 40 A 45 5 55

Fig 3 A Sample 80 A Chart


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R4 0 P ~ ~ p =0.8

0.6JLRpc 0.4 lao

0.2 = 0.4Y

0.005 1 15 0.020 0.025

Fig. 4 Lateral Load Vs End Slope Curves for Five Columnsubjected to Concentrated Load

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1

ajor Axis ~ e n d i n g

1

R:.. uoo ---- ----o

5p

o

5

Fig Ultimate Strength nter ct ion Curves for SimplySupported olumns Subjected to Concentrated Load

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1

Major Axisen ing :

5RRp

r

5

Fig Ultimate Strength Interact ion Curves for Fixed End olumns Subjected_to Concentrated Load

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0.028

=0.4Y = =60r =0.66

0.0160.012 0.01440.008

~

0.004 -0.003 -0.002 -0.001 o 0.001 0.002830 RadiansFig. 7 Determination of End Slopes of Simply SupportedColumn Subjected to Uniformly Distr ibuted Load

l 0

8060

w

=0 4Y

~a40 0

0 008 0.012 0016 0.020 0 024 0 028 0.03280 Radians

0.004

6

8 .

0.4

0 2

Fig. 8 te r l Load Vs. End Slope Curves fo r Simply Supported.Columns Subjected to Uniformly Distr ibuted Load

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1

0 5

o

w pP rrrrr rrr rrTl

T MO jOr x s Ben ding

0 5w

1

Fig 9 Ultimate Strength Interaction Curves for Simply SupportedColumns Subjected to Uniformly Distributed Load

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I

Fig 10 Deformation Modes of Case d Columns

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1

_T_ Major Axis Bending

0 5

P L I ~ . . . . . . . I . . . I . o I . . . a . . . . . . . t ~ ~ p

= = r I60

801

o

1.0

0 5

Fig 11 Ultimate Strength Interaction Curves for Fixed Endolumns Subjected to Uniformly Distr ibuted oad

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l.R RP--e I o p p_P L p 0 I L er

L ..LT . rD0.5 o 20 A 0.5 o 20 40 0 406 60 A 60A 80 D A 80o , o 00

Cm: -O.2i Cm 1-0.60 0.5 CmR l.0 0 0.5 - Cm R top P

0) 1- Pe Rp b 1- PelRp

1.0p I 11 I 116-P p__ ~ pP L p L IPcr

0 LT 0 0.5 o 20 0.5 o 20 40 o 40

6 60 :} 60A 80 6- 80o ~ o o

em=1.0 em=1-0.40 Cmw 1.0 0 Cmw

I-J: )wp P1- Pelwpc Pe d) .

Fig 12 Comparison of Analytical Results With AISC Formula

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. . ~

R F R N S

1. Timoshenko 8. P. and Gere J. M.THEORY OF ELASTIC STABILITY Chapter 1 2nd EditionMcGraw Hill Book Co. Inc New York 1961

2. Column Research Counci lGUIDE TO DESIGN CRITERIA FOR METAL COMPRESSION MEMBERSChapter 6 2nd Edition edited by B. G. JohnstonJohn Wiley Sons Inc. New York 19663. Horne M. R. and Merchant W

THE STABILITY OF FRAMES Pergamon Press Inc New York1965

4. AISCCOMMENT RY ON THE SPECIFICATION FOR THE DESIGNFABRICATION ND ERECTION OF STRUCTURAL STEEL FORBUILDINGS AISC New York N. Y. 1963

5. Galambos T. V. and Ketter R. L.COLUMNS UNDER COMBINED BENDING ND THRUST Trans.ASeE Vol. 126 Part I Proc. Paper 1990 1961 p. I

6. Ketter R. L.FURTHER STUDIES ON THE STRENGTH OF BEAM COLUMNSJournal of the Structural Division ASCE Vol. 87No. ST6 Proc. Paper 2910 August 1961 p. 1357. Ojalvo M. and Fukumoto Y.

NOMOGRAPHS FOR THE SOLUTION OF BEAM COLUMN PROBLEMSBullet in No. 78 Welding Research Counci l New York19628. VanKuren T. C. and Galambos T. V.

BEAM COLUMN EXPERIMENTS Journal of the StructuralDivision ASCE Vol. 90 No. ST2 Proc. Paper 3876April 1964 p. 22 39. Wright D. T.

THE DESIGN OF COMPRESSED BEAMS The Engineering JournalCanada Vol. 39 February 1956 p. 127

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10. von Karman T. ..UNTERSUCHUNGEN DBER KNICKFESTIGKEIT Mittei lungen uber .Fors chungsarbeiten herausgegeben vom Verein DeutscherIngenieure, No. 81, Berlin, 191011. Chwalla, E.

AUSSERMITTIG GEDRUCKTE BAUSTAHLSTABE MIT ELASTISCHEINGESPANNTEN ENDEN liND VERSCHIEDEN GROSSENANGRIFFSHEBELN Der Stahlbau, Vol. 15, Nos. 7 and 8,- 1937, pp. 49 ~ n d 5712. Oja1vo, M.RESTRAINED COLUMNS Thesis presented to Lehigh Universitya t Bethlehem, Pa. in 1960 in part ial fu l f i l lment of therequirements for the degree of Doctor of Philosophy University Microfilms, Inc. , Ann Arbor, Mich.)13. Ketter , R. L., Kaminsky, E. L., and Beedle, L. S.

P ~ S T I DEFORMATION OF WIDE-FLANGE BEAM-COLUMNSTransaction, ASCE Vol. 120, 1955, p. 1028

14. Parikh, B. P.ELASTIC-PLASTIC ANALYSIS N DESIGN OF UNBRACED MULTISTORY STEEL FRAMES Thesis presented to Lehigh Universitya t Bethlehem, Pa. , in 1966, in par t ia l fulf i l lment of therequirements for the degree of Doctor of Philosophy Universi ty Microfilms, Inc- , Ann Arbor, M ich.)15. Lu, L. W.COLUMNS Lecture N o 4 in Lecture Notes on PLASTICDESIGN OF MULTI-STORY FRAMES Lehigh University, 196516. Galambos, T. V.COMBINED BENDING N COMPRESSION Chapter 11, inffSTRUCTURAL STEEL DESIGN Ronald Press, New York196417. Mason R. E., Fisher, G. P., and Winter, G.ECCENTRICALLY LOADED HINGED STEEL COLUMNS Journalof the Engineering Mechanics Division, ASCE Vol. 84,No. EM4 Prac. Paper 1792, October, 1958, pp. 1792-1

ultimate strenth of laterly laoded columns

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