discussion_ buckling of one-story frames

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DISCUSSION

Buckling of One-Story FramesPaper by C.G. SCHILLING(2nd Quarter, 1983)

Discussion by Dan S. CorrentiAs pointed out by Schilling, practical methods are available for calculating a frame's magnification factor, elastic buckling load and thus, i ts secondary P-A momentsand other forces. Also presented was a method to determine a frame's critical elastic/inelastic buckling loadanalogous to the column strength curve adopted by AIS C.1Three steps for frame colum n design w ere presented whichinclude a check on the frame's buckling capacity, a checkon yielding instability at the frame joints including factored P-A moments and a check on individual columnbuckling and bending capacities.

REVIEW OF PRESENT COLUMN/STABILITYDESIGN APPROACHES AND C O M M E N T S

Of all three design steps outlined above, the latter twohave been suggested by the Structural Stability ResearchCouncil

2(SSRC) and other authors. The first design step,a check on the frame's buckling capacity, is an additionalrequirement as presented in the subject paper.An elaboration of the SSRC's 2 suggested procedurefor column design using factored P-A values follows:a. Check AISC stabili ty Form ula 1.6-la as for bracedframe using K = 1, since the P-A effect has already been included.

b. Check AISC strength Formula 1.6-lb using firstorder and P-A moments .Design Step A evaluates a column's resistance againstlateral buckling between end joints in conjunction withthe maximum moment at or near the buckling location.Design Step B evaluates a column's resistance againstyielding at its end joint.The additional design step (overall frame bucklingcheck) proposed by the subject paper evaluates the fram e's

Dan S. Correnti, S.E., is President of DC Consulting Engineers,Elm hurst, Illinois.

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resistance to sidesway buckling. This evaluation determines the overall frame strength for pure axial loadingconditions, similiar to an axial loaded post free to translate at one end but with some resistance to rotation. Thecolumns within the framework would not be subject tofirst order bending for this loading condition.As pointed out by SSRC, 2 actual column strength canbe closely determined using interaction equations of thetype specified in the AISC Code 1 when it is subjectedto both axial compression and bending. The basis for theinteraction equations is that the sum of the ratios of theapplied axial load and moment to the failure causingaxial load and mom ent, respec tively, if each were actingseparately, is less than unity. Since a column can failunder pure axial load by first order yielding or by ins tabi l i ty due to bi furcat ion of lateral d isplacement ,determination of column strength related to both typesof axial failure is required. The former type of failure(pure yielding) can occur at locations of the column wherebuckling is prohibited. The latter type of failure (instability) will always occur at loads less than yield strengthof the column, since real columns have initial geometricimperfections and residual stress. Therefore, actual columnstrength for instability type failure is related to both itselastic critical buckling load and yield load for stockiercolumns and to just the elastic critical buckling load forslender columns. Instability failures occur at locationswhere the column lateral displacements can bifurcate.

Since two types of column failure can occur, two typesof interaction equations are required for columns subjectto both axial and bending forces. One interaction equation is a strength equation (AISC Formula 1.6-lb) an dthe other is a stability equation (AISC Formula 1.6-la).The strength equation compares actual column forces toyield strength and should be utilized at locations of thecolumn where yielding could occur prior to instabilityfailure. This type of failure is discussed in a subsequentparagraph. Since the stability equation compares actualcolumn forces to its axial compressive strength (determined from considering instability due to bifurcation of

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lateral displacements) and its moment strength, the equation should be utilized at locations of the column wherelateral displacements can bifurcate.An unbraced frame can fail in one of two differentmodes of buckling. One mode of buckling results frombifurcation of lateral displacements of an individual column member between its ends, say mid-height. The other-mo de of buckling results from bifurcation of sides waydisplacements of the upper ends of the frame columnsrelative to their lower ends. Since bifurcation of displacements (and thus, buckling) can occur at both thesecolumn locations (mid-height and end), the stability interaction equation should be utilized at both these points.Present recommended design procedure after performing a P-A analysis requires use of the stability interactionequation only for mid-height buckling for individual columns and not for sidesway buckling of frame columnswhere displacement bifurcation occurs at the frame joints.Present procedure requires use of the strength interactionequation at the joints. The thinking is that factored P-Amoments are already included in the bending portion ofthe interaction equation. But, as a result, the actual axialload is compared to the column's yield strength and notits actual axial strength. The column's actual axial strengthis determined from both its yield strength and its elasticsidesway buckling capacity when acting in conjunctionwith the other frame members. The frame axial strengthand thus, related individual column axial strength, canbe determined using Eq. 7 or 8 of the subject paper.Equation 7 is based on elastic buckling at loads less thanP y /2, whereas Eq. 8 is based on buckling of weakenedframe columns at loads greater than P y 12 due to theeffects of residual stresses and geometric imperfections.Frame axial strength, therefore, would always be lessthan its yield strength value, P y.

RECOMMENDED COLUMN/STABILITY DESIGN APPROACH

As mentioned earlier, it is necessary to utilize the stability interaction equation at points on the column wherelateral displacements can bifurcate, since its actual axialstrength will be less than its yield strength. Thus, for anunbraced frame where two different buckling modes arepossible, two uses of the stability interaction equationare required for each column. One application would bea check on the Euler-type buckling between column endsin conjunction with the actual moment occurring at thislocation. The other application would be a check on thesidesway type buckling in conjunction with the actualmom ent occurring at the appropriate end of the columns.The former application of the stability interactionequation is presently recommended by SSRC 2 and is applied as for a braced frame using K = 1. The latterapplication of the stability interaction equation can beperformed for each column, using the following AISC

stability interaction formula as a guide.AISC Formula 1.6-la all terms are defined in AISCSpecification Sect. 1.6J a , ^ mx Jbx , *-' my J by ^ jFa (1 " fJF'ex) Fbx (1 - faIF 'ey) Fby ^

1. The first term (fa/F a) is replaced by the ratio of thetotal gravity load P acting on the story to the sideswaycritical buckling value P cr of the story as determinedfrom the appropriate Eq. 7 or 8 of the subject paper.a. If the frame is braced in one orthogon al direction ,then P cr is calculated for the unbraced direction.b. If the frame is unbraced in both orthogonal directions, then P cr is calculated for both directions,and the lowest value is used.

2. The second and third terms of the above equation arereplaced by the ratio, fbIF b. All other values areomitted.a. fb is the maximum bending stress that occurs atthe column ends and results from the first order

moment plus the P-A moment due to sideswaycaused by factored loads. Per SSRC,2 P-A moments are calculated using ultimate displacementsequal to the working load displacements multiplied by a displacement factor F. The approachfor calculation of the P-A moments is also described in the subject paper under "Yielding Ins tabi l i ty" except that in i t ial s idesway due toconstruction misalignment need not be includedsince this effect is included in the safety factorused in the determination of P cr of the first term.This can be considered rational since the P cr valuewas developed as the allowable gravity load actingon an imperfect frame in the same manner that Fawas developed, the allowable axial stress on animperfect column.b. Fb, allowable bending stress, is determined in thenormal fashion, assuming bending only exists.In the above approach for symmetrical frames subjected to symmetrical gravity loads only, the P-A moments would be zero and only the first order columnmoments need be considered for the second and thirdterms of the interaction equation. The requirement toassume some initial misalignment is, thus, avoided forgravity loading only cases. A displacement factor F foruse in determining the factored P-A values is not present ly recommended by SSRC. 2 To avoid the use of dis

placement factors a different approach in the applicationof the above interaction equation may be used, which isconsistent with present AISC requirements.1. The first term, fa/Fa, is replaced by P /P cr , the sameas outlined above.2. For the second and third terms, the following modi

fications are required:a. Deter mine /^ by adding first order column end mo-

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ments that are no t associated with sidesway movement to those column end moments (magnified)that are associated with sidesway movement. Themagnified column end moments, associated withsidesway, are determined by multiplying the associated first order moments by the followingmagnification factor m.m = 1/(1-P/P cr)where , P = total gravity load on storyP cr = Eq. (7) from subject paper= (12/23) x (shl 1.2)

b. Determine the allowable bending stress Fb as outlined above.c. The sidesway instability interaction equation becomesPIPcr + fbx IFbx + fby IF by ^ 1

where the terms are defined above.As is consistent with present AISC procedure, the 12/23 factor used in the determination of the magnificationfactor m above is equivalent with the use of the undefined factor F suggested by SSRC 2 in the approach outlined earlier. That is, F = the ratio of the factored loadlateral deflection to the design lateral load deflection.Application of F accounts for the effect of ultimatelydisplaced frame while design gravity loads are acting onit. The 12/23 factor, which had accounted for this effect,can still be used, as demonstrated next.As presently accepted, final working momentsincluding effects of secondary analysis can be determined by multiplying the first order moments bya magnification factor.For example,

where, M fM tPP.

M f = M t 1(1 - PIP e)= final working load momentincluding secondary effects= initial working load firstorder moment= working gravity load= elastic critical buckling load= sh/1.2

To find the ultimate final moment including secondary effects, assuming elastic frame behavior,multiply all working load values in the above equation by the load factor B,

BM f = BM (/(1 - BPIP e)where BM f = ultimate final moment which includesthe effects of the ultimate gravity load BP acting onthe ultimately displaced frame caused by the ultimate moment BM t.

To determine the final design moment includingsecondary effects, divide both sides of the aboveequation by the load factor B.M d = Milil - BPIP e)

where M d = design moment.M d determined here is equivalent to the design moment that would_be obtained from a conventionalP-A analysis if F were set equal to B.Thus , M d determined by the above equation is asummation of the first order moment and the secondary moment of working gravity loads acting onan ultimately displaced frame. Use of B equal to23/12 is consistent with present AISC requireme nts.For wind loading analysis divide B by 1.33.For an unbraced frame laterally loaded at floor levels,the stability interaction equation for sidesway bucklingwill almost always control except for special conditions.For example, a slender column being part of a framehaving other heavier stocky columns may be controlledeither by the strength interaction equation or the stabilityinteraction equation for Euler-type buckling betwee n column ends. Or a column subjected to substantially moreaxial load than the others may also be controlled by theabove two types of interaction equations.

The analysis/design procedure described herein can beapplied to multi-story frames that approach the behaviorof shear cantilevers (as opposed to flexural cantilevers)as proposed by Nair.3 Determination of the sideswaybuckling strength and magnification factor for each storyis required. These values are calculated from the relativelateral displacement of the story being considered, theactual total shear load acting in the story, and the actualtotal gravity load acting on the story, all determined froma conventional first order analysis procedure. Once thebuckling strength and magnification factor for each storyare obtained, the design approach for each story is thesame as outlined herein.

SUMMARYWith the development of sidesway buckling strengthequations by the subject paper which takes into accountthe effects of residual stresses and initial geometric imperfections, frame sidesway stability interaction withcolumn end moments can be utilized for determinationof overall frame strength via interaction checks of individual columns. Although stability interaction checks forbuckling between column ends and strength interactionchecks are required for individual columns, the sideswaystability interaction check proposed here will usuallycontrol the column design.

A rational design procedure was proposed for gravityloaded only frames where first order sidesway does notoccur. This avoids the need for assumptions regardinginitial geometric imperfections which are difficult to assess accurately. A lso, a rational approach for con sidering

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the effect of working gravity loads acting on a ultimatelydisplaced frame was proposed in lieu of using an undefined displacement factor (F) as required for the conventional P-A analysis procedure by SSRC. 2Also, as indicated, the proposed design approach canbe extended for use in design of multistory frames thatapproach the behavior of shear cantilevers.REFERENCES

1. American I nstitute of Steel Construction, Inc. Manual ofSteel Construction 8th Ed., Ch icago, III. , 1980.2. Structural Stability Research Council Guide to StabilityDesign Criteria for Metal Structures 3rd Ed., John Wiley& Sons, New York, N.Y., 1976.3. Nair, R.S. Lecture Notes , Lateral Load Resisting Systemsfor Multistory Buildings Presented at the 1980 ACI Chicago Area Chapter Seminar, ' 'Selected Topics in ConcreteDesign and Construction," M arch 20, 1980.

Discussion by Alfred ZweigSchilling's paper regarding an approximate frame buckling analysis is a valuable contribution to the extensiveliterature of frame buckling. He tries to validate his approximation by comparing its results with the answersobtained by a more rigid investigation based on a finiteelement analysis.The writer was interested to compare Schilling's approximation with the results of a frame buckling analysiswhich is based on the deformation method. This methodwas used by the writer for analysis of one-story framesin a paper published in September 1968 in the StructuralJournal of the American Society of Civil Engineers underthe ti t le, "Buckling Analysis of One Story Frames."Applied to the frames used in Schilling's comparison itleads to the following trigonometric equations:

(a) For the frames with pinned bases:

D {b xmi + 4 ) 2 b\rrii- 2 -(b3m 3 + 4) b3m 3

M i b3m 3 (dimi -\-d3m3)= 0 ( 1 )

(b) For the frames with fixed bases:D =

(miani + 4 ) 2 m xs\2 -(m 3an3 + 4) m 3s3 = 0m\S\ m 3s3 - ( C 1 / W 1 + C 3 / W 3 )

= 0(2)

Alfred Zweig is a Consulting Engineer in Laguna Hills, California.

Where e sin e - e2 cos e3 - 2h; u3 si rc-> 3 2d-x 3 sinc., " 3 2 -

ibiC\dxS\

mxm2

ehence with Lan d E

P er

- 6 sin 6 - 2 cos 00 2 sin 0

1 0 - 0 cos 00

3sin 0- 0 sin 0 - 2 cos 0

0 3 cos 00 - 0 cos 00 2 - 0 2 cos 0

- 0 sin 0 - 2 cos 0= 4= 3= 6= 3= 12= c y f l i= C2IB X= L (p cr/Eiy>2= 100 in.= 29,000 ksi= 290 0 2 C2

ForB lt C x and C 2 see Table D l .Equations 1 and 2 were applied to all portal framesanalyzed by Schilling in Table D l , i.e ., all frames w iththe exception for those frames in his table which haveeither a second bay with members B2 and C3, or a diagonal strut D l or a ratio A/I = 0 .1 . The results of thiscomparison are shown in Table 1.Table 1 substantiates S chilling 's claim to the accuracyof his approximation which to the writer's opinion seemssufficiently close for practical purposes, maybe with theexception of Case 5 for the frame with pinned bases,where the deformation method gives about a 15% smallervalue for P cr than Schilling's approximation. It is onlya question of whether the accuracy could not be improved for all frames with pinned bases by revising thecorrection factor of 1.2 for these frames.

Schi l l ing 's approximat ion wil l be especial ly welcomed by those who have used the more accurate methods as the writer has done. The advantage of his approachbecomes immediately apparent when comparing Eqs. 1and 2 (which, without the use of a computer would bevery difficult to solve) with the rather simple and elegantEq. B6, which can be solved without recourse to anycomputer.

Schilling's method, however, would be of l i t t le practical value if it could not be extended to the case wherenot only a single column in a given frame is loaded buteach of the columns is subject to an axial compression

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Table 1

Columnwithfixedbase

Columnwithpinnedbase

Case 1Case 2Case 3Case 4Case 5Case 6Case 1Case 2Case 3Case 4Case 5

Pcr in kipsaccording toFiniteElementMethod

Schilling1,4503,5304,3194,9445,8003,900

8291,0651,2341,450

890

DeformationMethod

Zweig1,4093,4514,2304,8205,8333,673

8191,0911,1961,408

75 3

ApproximateMethodSchilling

1,4503,3144,0604,7125,8003,780

725967

1,1601,450

865

e2.2043.4503.8194.0774.4075.0331.6801.8952.032

- 2.2042.278

load. The writer has solved this problem in a generalway in the above mentioned paper, which was used asa basis for a computer program by Albert Kahn Associates Inc. of Detroit, Mich. It served as a useful columndesign aid for factories with large unobstructed spaceswhere the overall column stability depends only on therigid interaction between column and roof trusses (novertical bracing is permitted) and all columns are subjectto the same vertical load.By alternately turning adjacent I-shaped columns 90,it is possible to utilize the inherent strength of this column arrangement and achieve a substantial saving in theresulting steel tonnage . W ithout a rational buckling analysis, these and similar savings would not have been attainable. It would be useful and of practical value ifSchilling's approximation could be extended to this andsimilar problems.To make practical use in all cases of Schilling's approximation, he proposes the factor of safety in his Eqs.5 and 6. The designer should be made aware, however,of the fact that this proposal per se could violate theAISC Code, which stipulates the K factor should not beless than 1. In using the expression in Eq. B6 for P crthe corresponding K value is unknown and may be lessthan 1. If the Specification requirement of K ^ 1 is notjustified and should be rescinded, the designer should bemade aware that by following Schilling's proposal in thisrespect, he might be violating the Specification unlesshe checks the guiding K value.Finally, a few random observations about Schilling's

paper: the term "allowable crit ical frame buckling load"used by Schilling seems to be a contradiction in itself.If a load is crit ical, according to common usage, theframe will buckle under this load and this quantity, therefore, cannot be allowable. If, on the other hand, you adda factor of safety to this load, it turns into the allowableload and, therefore, it is not any longer the critical loadin the generally accepted terminology. It may lead toless confusion if the attribute "allowable" would beomitted when referring to a critical load.The values given for P cr in Table D l are obviouslythe critical, not allowable loads in kips (not stated so inthis table). A value of 29,000 ksi was used in Eqs. B5and B6, although not stated in the paper.The writer wishes to express thanks to Albeit KahnAssociates Inc. of Detroit , Mich, for permitting the useof the Digital VAX-11/780 Computer to solve Eqs. 1and 2, and to Charles T. Robinson, Assistant ChiefStructural and Civil Engineer of this organization forproviding the figures of the 6 values in Table 1.

Discussion by John SpringfieldThe author has identified correctly the prinicipal shortcomings of current practical approaches to frame stability. These are the inability of the effective length methodto cope well with either complex frames or to take account of lateral forces (except through the beam-columninteraction equations) and, w ith the P-A m ethod, the lessthan satisfactory approach to the gravity load only loadcase.Within the limits of a uniform single story array ofrigid frames, the author has proposed a method whichpromises to overcome these problems. This method proposes to determine the critical elastic buckling load, defined as that load under which a laterally disturbed framewill sway uncontrolled to collapse.I wish to comment on three aspects, namely: The validity of the basic approach The difficulties in applying the method to irregularframeworks The approach to inelastic bucklingSchilling has improved the commonly used P-A methodby taking account of the curvature of the beam-columnbetween end points. This writer's concern was thatSchilling used the P-A method to predict the elasticbuckling load of a frame. Rather than arguing on theoretical grounds, this writer has compared Schilling'smethod, applied to a flag-pole under vertical and horizontal loading, with the results provided by Dr. KirbyJohn Springfield is Vice President of Carruthers & Wallace Ltd.,Consulting Structural Engineers, Rexdale, O ntario, Canada.

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of Sheffield University for this problem, derived usingboth the stability functions and the "Euler amplificationfactor" methods. The problem is illustrated in Fig. 1.These results are tabulated in Table 1, together with thenormal P-A results which do not include Schilling's 1.2factor.

r i; />s,

Figure I

The author's value would be:H h3A =

A -

3EI-l.2Ph 2

H h33 - 1 . 2 T T 2 P / ~W

T a b l e 1. Values of Coeff icient K0. 0.06 0.10 0.16 0.20 0.22 0.24

1. Stability 0.333 0.437 0.552 0.915 1.645 2.714 8.242function

2. Amplification 0.333 0.439 0.556 0.926 1.667 2.778 8.333factor3. Linear 0.333 0.415 0.497 0.704 0.975 1.207 1.5844. Schilling 0.333 0.437 0.551 0.905 1.584 2.535 6.347P - A5. Ratio 1:3 1.0 1.052 1.111 1.300 1.688 2.24 9 5.2036. Ratio 1:4 1.0 1.001 1.002 1.011 1.039 1.071 1.299

The results give the values of the coefficient K whichwould be used to predict the lateral deflection of the tipof the flagpole:H h3K EI

With no axial load, K = lA.Kirby gives the following expressions:

1 H h 3A - [2s/m-s(l+c)](l+c) EI

1 1 H h 33 1-P/.25 P e EI

The stability functions have their usual values.Pe = TT2 EI/h 2, the 0.25 is derived from an effectivelength of 2h .The usual P A method gives

H h3~ 3EI-Ph 2

Setting p = TT2 EI/h 21

3 - T T 2 P

P h 2where P = ^ -

EI

Comparison of the coefficients confirms what was wellknown already, that the linear P-A method is a poorpredictor of second order effects when the slendernessratio approaches the upper limits of design practice. (Thislimit has been suggested by P.F . Adams as p = 0.145 :the values in the table confirm this.) Much more importantly, the values demonstrate the remarkable improvement effected by Schil l ing's s imple suggest ion ofincreasing the P A effects by 20% . Finally, the valuesshow how good is the approximation using the amplification factor, (l-P/P e). This is used by both Galambosand Kirby. The snag with this seemingly simple amplification factor is that it is necessary to know the elasticeffective length, which is what we are trying to avoid.Regardless of the theoretical rigor of Schilling's method,it appears to provide good results.The difficulties in applying the method to irregularframeworks and the author's approach to inelastic buck-

l j P | Pm vfi

Figure 2

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ling are really the same problem. There is no difficultyin determining the elastic critical load, for however irregular the framework, the frame stiffness s = H /A ca nbe determined from first order elastic analysis. H owever,as soon as the author tries to use the Johnson parabolato predict the inelastic buckling load, his method breaksdown w ith frames of the type shown in Figs. 1 and 2.These frames are diagrammatic representations of typesof framework used in practical structures. Figure 1 represents framework in which there are simply connectedcolumns supporting gravity load, stabilized by adjacentrigidly framed columns. Figure 2 is representative ofrigid frames in which there is wide variation in columnstiffness in a given plane. The author's magnificationfactor is satisfactory since it contains the summation ofall gravity loads on the structure and the frame stiffness.However, the logic for using P y/2 , where P y representsthe yield load of all the columns in the frame, as thepoint of intersection of the elastic buckling curve withthe inelastic transition curve is incorrect. The squash loadof the pinned colum n in Fig. 1 has no bearing on thetransition between elastic and inelastic frame buckling.The frame in Fig. 2 suffers from a similar problem butis more difficult to resolve. In this writer's view, thereis no validity to distributing the squash load in proportionto the gravity load on the columns since the cross-sectionof the pinned columns could be selected for some reasonindependent of column design. For the method to be ofwide practical use in the design of one-story frames, itis essential that the author develop another means ofdetermining the on-set of inelastic buckling.In this writer's opinion, the method proposed by theauthor is of sufficient merit to warrant further development. In the customary second order analysis approach,

the designer has to decide the degree of second ordereffect which is acceptable. The writer has not seen recommendations on this matter published but has adoptedthe following limits:Ratio of total (first and second order effects) to firstorder effectsLow-rise buildings: 1.3High-rise buildings: 1.2.The author's proposal permits the designer to arriveat a single solution under gravity load. To this degree itshares the design simplicity of the effective length methodwhen applied to simple frames.The method proposed by the author demonstrates a

requirement which in this writer's opinion is of extremeimportance in the assessment of frame stability. This isthe need to determine in any design the stiffness of theframework to lateral displacement when carrying its gravityload. A fundamental fault with the effective length methodof treating stability was that it was possible for an engineer to design a framework and satisfy the design specification with regard to stability, when he had no ideawhatsoever of the structure's lateral sway characteristics,

either under lateral load or under gravity load only. Admittedly, determination of lateral deflection was difficultprior to computer structural analysis programs. However, there is no longer any justification for avoiding theneed to determine this important parameter.

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discussion_ buckling of one-story frames

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