A Practical Look at Frame Analysis, Stability, And Leaning Columns

The 2000 T.R.Higgins Lecture:A Practical Look

at FrameAnalysis,

Stability, andLeaning Columns

Louis F. Geschwindner

Author

Louis F. Geschwindner is pro-fessor of architectural engi-

neering at Pennsylvania StateUniversity and is a RegisteredProfessional Engineer. Hereceived his bachelor's degree inbuilding science from RensselaerPolytechnic Institute and both hismaster of science in architecturalengineering and his Ph.D. in civilengineering from PennsylvaniaState University. He has been afaculty member at PennsylvaniaState for over 30 years, teachingand conducting research in build-ing structures and he is also incharge of the undergraduate pro-gram in architectural engineering.

Professor Geschwindner ischairman of the Committee onDesign of Steel BuildingStructures of the AmericanSociety of Civil Engineers(ASCE). He is also a member ofthe Committee on Metals, theLRFD Committee and vice-chairof the Tension MembraneStructures Standards Committee,all of ASCE. Dr. Geschwindner isa member of the AISC Committeeon Specifications as well as itsTask Committee 10 - Stability.His memberships also include theMasonry Society and theAmerican Society for EngineeringEducation.

Teaching advanced level struc-tural courses in the five-yeararchitectural engineering programat Pennsylvania State University,has been the primary thrust of Dr.Geschwindner's academic career.He has participated as a lecturerin the American Institute of SteelConstruction's lecture series andhas developed a short course onLRFD for practicing professionals.He has received numerousawards for outstanding teaching,including the University-wideAMOCO Foundation OutstandingTeaching Award and the AT&TFoundation Award for Excellencefrom ASEE.

Summary

The analysis and design ofunbraced moment frames is a

fairly regular activity in structuralengineering practice yet it can bea complex structural engineeringproblem. Numerous analysismethodologies are available andthe many commercial softwarepackages used in practice providea variety of approaches to theproblem. Questions arise as towhether a first-order or second-order analysis is appropriate,should an elastic or inelasticanalysis be carried out, whatmoment magnifiers should beused when axial load and momentact together, and how should col-umn capacity be determined, witheffective length factors or someother approach. Stability of a col-umn, although often expressed asa function of the individualcolumn, is actually a function of allof the members in the story.Thus, column design is a storyproblem, not an individual columnproblem. When unbracedmoment frames support pin-ended columns, additional prob-lems arise. These pin-endedcolumns do not participate in thelateral resistance of the structure,but instead rely on the unbracedframe for their lateral stability.Thus, the frame must be designedto accommodate the loads thatare applied to it and provide suffi-cient stiffness to support any ofthese "leaning" columns.

Frame analysis may beapproached by a variety of pathsbut linear elastic analysis isperhaps the most common, aswell as the least complete. Asecond order inelastic analysis,while perhaps the most compre-hensive, is also the most com-plex. Whichever analysis methodis chosen, the design approachmust be compatible. Numerousapproaches have been presentedin the literature to address thedesign of frames both with andwithout leaning columns.Although a direct stability analysis

1-1© 2003 by American Institute of Steel Construction, Inc. All rights reserved.

This publication or any part thereof must not be reproduced in any form without permission of the publisher.

may be performed, the most com-mon approaches still appear to bethose that utilize some form ofsimplification. This paper willbriefly review first- and second-order elastic and inelastic analysisapproaches and a few of the com-monly used computer softwarepackages for unbraced momentframes. This will be followed witha discussion of the impact of lean-

ing columns on these analyses.Once analysis approaches havebeen identified, the designprocess, where the inclusion ofcolumn effective length comesinto play, will be addressed. Theuse of effective length to predictcolumn capacity will be discussedand its use to account for theinclusion of leaning columns willbe addressed. Effective length

calculations will be reviewed withparticular attention to theapproaches presented by Yura,Lim & McNamara, LeMessurier,and the equations found in theAISC LRFD Commentary. Theresults from these approaches willbe compared to those of a stabili-ty analysis for simple frames thathave been found in the literature.

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© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.

A Practical Look at Frame Analysis, Stability, and Leaning Columns

By Louis F. Geschwindner

Introduction

The analysis and design of unbraced moment frames is a fairly regular activity instructural engineering practice yet it can be a complex structural engineering problem.Numerous analysis methodologies are available and the many commercial softwarepackages used in practice provide a variety of approaches to the problem. Questions ariseas to whether a first-order or second-order analysis is appropriate, should an elastic orinelastic analysis be carried out, what moment magnifiers should be used when axial loadand moment act together, and how should column capacity be determined, with effectivelength factors or some other approach. Stability of a column, although often expressed asa function of the individual column, is actually a function of all of the members in thestory. Thus, column design is a story problem, not an individual column problem. Whenunbraced moment frames support pin-ended columns, additional problems arise. Thesepin-ended columns do not participate in the lateral resistance of the structure, but instead,rely on the unbraced frame for their lateral stability. Thus, the frame must be designed toaccommodate the loads that are applied to it despite the reduction in overall framestability that results from these "leaning" columns.

Frame analysis may be approached by a variety of paths. Linear elastic analysis isperhaps the most common, although the least complete. A second order inelasticanalysis, while perhaps the most comprehensive, is also the most complex. And there aremany approaches between these. Whichever analysis method is chosen, the designapproach must be compatible. Numerous approaches have been presented in the literatureto address the design of frames both with and without leaning columns. Although a directstability analysis may be performed, the most common approaches still appear to be thosethat utilize some form of simplification. This paper will briefly review first- and second-order elastic and inelastic analysis approaches and a few of the commonly used computersoftware packages for unbraced moment frames. This will be followed with a discussionof the impact of leaning columns on these analyses. Once analysis approaches have beenidentified, the design process, where the inclusion of column effective length comes intoplay, will be addressed. The use of effective length to predict column capacity will bediscussed and its use to account for the inclusion of leaning columns will be addressed.Effective length calculations will be reviewed with particular attention to the approachespresented by Yura, Lim & McNamara, LeMessurier, and the equations found in the AISCLRFD Commentary. The results from these approaches will be compared to those of astability analysis for simple frames that have been found in the literature.

Analysis

The state of the art of structural analysis encompasses a wide range of possibleapproaches to the determination of member response to structural loading. Each new

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approach adds or subtracts some aspect of frame or member behavior in an attempt toproperly model the true behavior of the structure. Before looking at the requirements ofthe current LRFD specification, it will be helpful to categorize these analysis approachesand discuss their characteristics. Figure 1 shows a comparison between the load-deflection curves of a series of analysis approaches.

First-order elastic analysis (1): The first and most common approach to structuralanalysis is the linear elastic analysis. In this case, deformations are assumed to be smallso that the equations of equilibrium may be written about the undeformed configuration.In this case, superposition is valid, any inelastic behavior of the material is ignored andthe load-deflection curve is linear. This is the approach used in the development of thecommon analysis tools of the profession, such as slope-deflection, moment distributionand the stiffness method found in most commercial computer software. In a first-orderanalysis, the deformations are determined and then used in turn to determine forces. Oncethe forces are determined, the analysis is complete.

Figure 1. Load-deflection history

Linear buckling analysis (2): An analysis carried out for the determination of the elasticbuckling load will result in the determination of a single critical buckling load for aframe. The critical buckling load may be determined through an eigenvalue solution orthrough a number of iterative schemes based on equilibrium equations written about thedeformed configuration. This is the type of analysis that yields the critical buckling loadof the single column and is the basis upon which the effective length factor is founded. Itcan be seen in Figure 1 that the results of this analysis do not provide a load-displacementcurve but rather the single value of load at which the structure fails.

Second-order elastic analysis (2): When the equations of equilibrium are written aboutthe deformed configuration of the structure and the deflections corresponding to a given

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set of loads are determined, the resulting analysis is a second-order analysis. This is theanalysis generally referred to as the P-delta analysis. Two components of these second-order effects are included in the analysis. When the influence of axial force and membercurvature is included, it is said that the effects are included and when the sideswayeffects are included it is said that the effects are included. It is seen in Figure 1 thatthe load-deflection history obtained through this analysis approach the value obtainedfrom the eigenvalue solution. This analysis usually requires an iterative solution so it is abit more complex than the first-order analysis. Because of the problems inherent withiterative solutions, many researchers have proposed one-step approximations to thesecond-order analysis.

First-order plastic-mechanism analysis (3): As load is increased on a structure, it isassumed that defined locations within the structure will reach their plastic capacity. Whenthat happens, the particular location continues to resist that plastic moment but undergoesunrestrained deformation. These sections are called plastic hinges. Once a sufficientnumber of plastic hinges have formed so that the structure will collapse, it is said that amechanism has formed. No additional load can be placed on the structure. This limit canbe seen in Figure 1.

First-order elastic-plastic analysis (4): If the approach to determination of the collapsemechanism tracks the development of individual hinges, more information is obtainedfrom this analysis than from the mechanism analysis. It is clear however, that if the sameassumptions as to hinge length and geometry are maintained, the limit of the elastic-plastic analysis will be the mechanism analysis as seen in Figure 1.

Second-order inelastic analysis (5): This approach to analysis combines the sameprinciples of second-order analysis discussed previously with the plastic hinge analysisjust discussed. It is obvious that this approach to analysis is much more complex. It does,however, yield a more complete and accurate picture of the behavior of the structure,depending on the completeness of the model used. This type of analysis is what has cometo be known as "advanced analysis." The load-deformation curve for a second-orderinelastic analysis is shown in Figure 1.

In summary, it can be seen that as more complex behavior is taken into account in theanalysis, the predicted load level is reduced. Thus, designers must exercise extremecaution when using an analysis approach with which they are unfamiliar.

Design

The approach taken for member design must be consistent with the approach chosen foranalysis. There are currently three approaches to the design of steel structures acceptableunder US building codes as they incorporate AISC Specifications (6,7). Currently, thebest tool for steel design is the Load and Resistance Factor Design Specification (LRFD).However, the Plastic Design (PD) approach is also permitted and the Allowable StressDesign Specification (ASD) is still used.

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The LRFD specification stipulates, in Section C1, that "Second order effects shall beconsidered in the design of frames." The comparable statement in the ASD specificationstates, in Section A5.3, that "Selection of the method of analysis is the prerogative of theresponsible engineer." And in Section C1 that "frames...shall be designed to provide theneeded deformation capacity and to assure overall frame stability." The normal analysismethod is one of the linear elastic approaches. The satisfaction of the deformationcapacity and the assurance of stability are left to the engineer.

In order to use the Plastic Design Specification for structural design, a first-ordermechanism analysis must be carried out. Additionally, other restrictions found within thespecification must be satisfied.

Thus, regardless of the specification used, the engineer is required to address the secondorder effects. This may be done using a first-order analysis and a code providedcorrection for second order effects or it may be addressed through direct use of a second-order analysis.

Impact of Second-order Effects

Two different second-order effects will impact on the design of a single column. The first,illustrated in Fig. 2a for a column in which the ends are prevented from displacing, is theresult of the deflection along the length of the column. It can be seen that the momentsalong the column will be increased due to the column deflection, by an amount Thisincrease in moment due to member deflection is referred to as the member effect.

Figure 2. Influence of Second-Order EffectsThe column in Fig. 2b is part of a structure that is permitted to sway laterally an amountAs a result, the moment required in the end of the column to maintain equilibrium in thedisplaced configuration is given as This member moment is referred to as the structureeffect, since the lateral displacement of the column ends is a function of the properties of allof the members of the structure participating in sway resistance.

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The deflections, and shown in Fig. 2 are second-order deflections, resulting from theapplied loads plus the second-order forces. Although it requires a second-order analysis todetermine the magnitude of the forces, both of these effects appear to be straightforward forthe individual column of Fig. 2. However, when columns are combined to form frames, theinteraction of all of the members of the frame significantly increases the complexity of theproblem. The addition of columns which do not participate in lateral resistance but which docarry gravity load brings further complexity to the problem.

Computer Approaches

Although the influence of second order effects was easily observed in the single columnof Figure 2, the analysis of an unbraced frame of any realistic proportions could beexpected to require the use of some computer software. The particular software beingused will dictate the methods of analysis available to the engineer. Based on theirshowing in a recent AISC survey (8), five commercial software packages were reviewedto determine their approach to a second order analysis. They are STAAD-III,GTSTRUDL, RISA-2D, ETABS, and RAM FRAME. All of these programs indicate, intheir literature that they include some form of second-order analysis.

STAAD-III (9) adopts a simplified method to perform the P-delta analysis. The usersmanual provides the following steps as a guide for using its P-delta analysis: 1. Perform afirst order analysis to obtain deflections and member forces based on external loads, 2.Create an additional load vector by combining the previously calculated member forcesand deflections with the external loads, 3. Use this new load vector and the originalstiffness matrix to calculate a revised set of deflections, 4. Determine member forces andreactions from these revised deflections. Although this is not an exact approach, theprogram authors indicate their belief that their approach has been justified by theliterature.

GTSTRUDL (10) uses a direct stiffness formulation about the displaced geometry of thestructure. A secant stiffness approach is used where the structure stiffness matrix ismodified in each step of the analysis until the resulting change in displacements or forcesfrom one step to the next is within the bounds specified or the maximum number ofspecified cycles is reached.

In RISA-2D (11), the P-delta effects are "accurately approximated." Their approach is tosolve the first-order problem and determine the member forces. From these memberforces and displacements, a member shear is determined as These shears areadded to the original loads and the first-order problem is resolved. The newdisplacements are compared to the previously calculated displacements. If the differenceis small enough, the analysis is complete. If a more accurate solution is needed,additional cycles are performed until the required level of accuracy is reached.

ETABS uses a geometric stiffness correction in which the effects are "exactlyrepresented"(12). The approach is to use the load P determined from the story mass to

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modify the stiffness matrix, which is then used throughout the analysis, both static anddynamic. This method has been presented in the literature by the program authors (13).

RAM FRAME (14) has also adopted the geometric stiffness method. However, theyindicate that certain modifications have been made in the implementation of the methodwithin the specific formulation of their program.

It appears that commercial software for second-order analysis of steel frames is readilyavailable to the design engineer. Thus, it can be expected that the use of a second-orderanalysis will become a more realistic approach for normal design.

Predicting the Critical Buckling Load

When an analysis tool is available to determine the critical buckling load of a frame, thereis no need to worry about "predicting" that load through some other means. Thus, itmight be said that if all structural analysis were carried out as a second order bucklinganalysis, there would be no need to spend time discussing the correct approach todetermining a K-factor to use in design. It seems that ever since the K-factor wasintroduced into the 1961 AISC Specification, it has generated extensive discussion andmisunderstanding (15). To understand the debate over the K-factor, one must understandwhat the K-factor is intended to accomplish. It will be helpful to consider the criticalbuckling load of a particular column, determined by one of the second-order analysisprograms, as It will also be helpful to remember that the critical buckling load of theperfect column, as derived by Euler, is given as

(1)

Since the column in a steel frame is not likely to be a perfect column, but rather a realcolumn with multiple imperfections, its critical buckling capacity can be said to besomewhat less than the Euler column, thus

(2)

If that reduction factor is defined as it is seen that

(3)

Thus, the K-factor is simply a mathematical adjustment to the perfect column equation totry to predict the capacity of an actual column. Every method or equation that is proposedis simply trying to predict accurately the actual column capacity as a function of theperfect column.

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Perhaps the most commonly used approach to the determination of K-factors is thenomograph found in the commentary to the LRFD and ASD Specifications (6,7). Theequation upon which the nomograph is based is given here as Eq. 4 (3).

with

(4)

and the A and B subscripts referring to the ends of the column under consideration.

The assumptions used in the development of the nomograph are detailed in the Commentaryto the Specification. One of these important assumptions is "all columns in a story bucklesimultaneously." Although this assumption was essential in the derivation of this usefulequation, it is also one that is regularly violated in practical structures. This assumption iscritical since it eliminates the possibility that any column in an unbraced frame mightcontribute to the lateral sway resistance of any other column. A reasoned analysis of thebehavior of columns in actual structures would seem to indicate that strong columns, oneswhose load is not near their capacity, should be able to help restrain weaker columns. Thus,other approaches to determining the K-factor should be considered.

Figure 3. Symmetric Portal Frame

Second-Order Analysis and Leaning Columns

Without leaning columns: Two simple frames will be discussed so that the impact of second-order analysis and leaning columns may be understood. The symmetric frame shown in Fig.3a is subjected to a symmetrically placed gravity load. A first order analysis yields the

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forces shown. Note that there are no column moments and thus, there will be no axial forceand moment interaction. When a lateral load is added as shown in Fig 3b, forces andmoments as shown result from a first-order analysis. In this case, all moment is due to thelateral load and these moments must be amplified to account for the second-order effects.For both cases, axial capacity may be determined using the K-factor from the nomograph.

A similar frame, with two different columns, is shown in Fig. 4a, loaded with anonsymmetric gravity load. For an LRFD-based design and a first-order analysis, thesway and non-sway moments must be separated. To do this, the first step is to restrainsidesway and perform a first-order analysis. A fictitious restraining force, AJR, results asshown in Fig. 4b. Next, the fictitious force is removed by applying CJF = -AJR,removing the restraint, and carrying out another first-order analysis. These results areshown in Fig. 4c. The capacity of the columns must then be checked according to thecode specified interaction equations. Regardless of method, LRFD or ASD, the columnbuckling capacity for the first term of the interaction equation can be determined from thenomograph and the column moments will be amplified to account for second-ordereffects using the in-plane K-factor in the calculation of the amplification factors.Normally, the sway effects of the non-symmetric load will be quite small. If a lateral loadis added, as shown in Fig. 4d, a similar approach can be used; however, the swaymoments will now be significant.

Figure 4. Nonsymmetric Portal Frame

When the frame of Fig. 3a is subjected to a second-order analysis, there will be no changefrom the results for a first-order analysis. Thus, the design approach should be the same;K-factors from the nomograph and no moments to be amplified. If a second-orderanalysis is performed for the frame of Fig. 3b, the results will be different from the first-

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order results. The buckling capacity may still be determined through the nomograph K-factor; however, the moments will not need to be amplified to account for sway since thatwas already accomplished through the second-order analysis.

When the frames of Fig. 4 are subjected to a second-order analysis, different momentsresult, which account for sway. The buckling capacity may be determined from the K-factors of the nomograph and moments may be used directly, with no need foramplification.

With Leaning Columns: When an unbraced frame is called upon to provide lateralrestraint for leaning columns, the first- and second-order analysis may or may not besufficient, depending on the simplifications used. In addition, the nomograph approach(Eq. 4) to determining buckling load may not be a good indicator of column capacity.When a leaning column is added to the frame of Fig. 3a, the resulting structure is asshown in Fig. 5. A first-order analysis will yield the same member forces for theunbraced frame as had been determined for the structure in Fig. 3a. Thus, it appears thatthe leaning column has no impact on the original structure. If the structure is subjected toa second-order analysis, again, no change will be noted.

Figure 5. Symmetric Frame withLeaning Column

Figure 6. Nonsymmetric Framewith Leaning Column

If a leaning column is added to the frame of Fig. 4, as shown in Fig. 6, and both thegravity and lateral loads shown are applied, a first-order analysis will again repeat theresults from the frame of Fig. 4. If a second-order analysis is performed, the results willbe different from those previously determined. They will account for the amplification ofmoment due to sidesway of the structure and both loads P and Q; but, as for the frame ofFig. 5, no account will be taken of the reduced buckling capacity due to the presence ofthe leaning column.

If the buckling load for a frame member is to be determined through an approach other thana complete buckling analysis, a model that will reasonably predict the capacity of a frameincluding these leaning columns is needed. Numerous approaches intended to account forthe effect of leaning columns and the sharing of lateral resistance have been presented in the

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literature. These approaches offer a wide range of mathematical complexity and practicalusefulness. Four approaches that have been presented in the literature for including theleaning column in the determination of column capacity will be discussed along with somesimplified equations that are included in the Commentary of the LRFD Specification. Asalways, the designer is called upon to decide on the appropriate approach to use in aparticular design situation.

Effective Length

Modified Nomograph Equation (16): The derivation of Eq. 4 is available in numerousreferences, including (3). Following the same procedures and assumptions, with the additionof the leaning column, as shown in Figure 7, a new equation may be developed.

Figure 7. Restraining and Leaning Columns

Viewing the structure in its displaced equilibrium configuration, the leaning column and therestraining column are separated as shown in Fig. 7b and c. The load Q on the leaningcolumn CD must be balanced by the horizontal force, at D, for equilibrium of theleaning column. This force must then be applied as a load at B on the restraining columnAB.

Equations of equilibrium at the joints of column AB and the sway equilibriumequation can be written for the structure in the displaced configuration. Member endmoment equations are then written using the slope deflection method, incorporating thestability functions (17) necessary to account for the influence of axial load on column AB.Combining these equations and setting the determinate of the coefficients equal to zero willyield the following buckling condition equation.

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If the leaning column load is zero, Eq. 5 reduces to Eq. 4. Since neither of these equationscan be solved explicitly, an iterative approach may be used or, in the case of the framewithout leaning columns, the nomograph already discussed may be used.

The Yura Approach (18): This is perhaps the easiest approach to develop since it relies on astraightforward interpretation of the physical problem. For the unbraced frame shown in Fig.8, equilibrium will be established for the structure in the undeflected configuration and againin the deflected configuration. The first-order, undeflected equilibrium configuration forcesare shown in Fig. 8a. If the frame is permitted to displace an amount A, equilibrium in thisdisplaced configuration will be as shown in Fig. 8b. In order for column EF to be inequilibrium, a lateral force, as shown at F is required. This force must be equilibratedby an equal and opposite force shown at B. Thus, when column AB buckles, it buckles witha moment of at its base. It is observed that this is the same moment that wouldresult if the individual column AB were to buckle under the axial load of (P + Q). Theassumption that the buckling load is (P + Q) is only slightly conservative for the individualcolumn AB, since the deflected shape due to an axial load and a lateral load differ onlyslightly. In order to insure sufficient lateral resistance to buckling for column EF, columnAB must be designed to carry a fictitious load (P + Q).

Figure 8. Equilibrium forces for Yura derivation

In order to compare this approach to others presented in the literature, it is helpful to convertit to an effective length approach. If column AB is to be designed to carry the load P buthave the capacity (P + Q), a modified effective length factor will be required. is definedas the effective length factor that would be determined from the nomograph or Eq. 4, whichdoes not account for the leaning column. In this case is defined as the effectivelength factor that will account for the leaning column. Thus, based on the buckling loadbeing (P + Q)

(6)

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If the column is to be designed to carry the actual applied load, P, with the leaning columnaccounted for through then

(7)

Solving equations 6 and 7 for their corresponding K's and taking the ratio yields

(8)

which may be solved for as

(9)

If column AB from Fig. 8a were designed to carry the load P using the effective lengthfactor it would provide sufficient lateral restraint to permit column EF to be designed tocarry the load Q using K = 1.0.

For frames with more than one leaning column and more than one restraining column,and will replace P and Q. It should also be noted that this approach maintains theassumption that all restraining columns in a story buckle in a sidesway modesimultaneously.

Lim & McNamara Approach (19): Another approach that will account for the leaningcolumn was proposed by Lim and McNamara for columns of unbraced tube buildings. Theirdevelopment is also based on the assumption that all columns in the restraining framebuckle in a sidesway mode simultaneously; however, they developed the sway bucklingequation through the use of stability functions and an eigenvalue solution.

The resulting effective length factor, accounting for leaning columns is given in their paperas

(10)

where and are as defined earlier, and is the eigenvalue solution for aframe without leaning columns and is the eigenvalue solution for a frame with leaningcolumns. The authors suggest that for normal column end conditions, shouldprovide a K-factor on the conservative side by at most 2%. Substituting for n and using

the Lim & McNamara approach gives the same K-factor as the modified Yuraapproach where

(11)

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Thus, for the story buckling approach, a single multiplier for each story will be sufficient tomodify the individual nomograph K-factors to account for leaning columns.

LeMessurier Approach (20): In his landmark paper, LeMessurier presented a more complex,yet still very practical approach for frames with and without leaning columns. The basicequations were developed for a single cantilever column and then extended to the generalframe. Where the previous approach determined a constant value for a story by which thenomograph value of was modified, this approach determines a constant value for a storywhich then multiplies the individual column moment of inertia divided by the column load,

, for each column, i. Thus, the contribution of each column to the lateral resistance isaccounted for individually. The effective length factor for each column that participates inresisting sidesway buckling, Eq. 46c from the original paper, expressed in the notation ofthis paper, is given by

(12)

where

(13)

(14)

= effective length of column i, accounting for leaning columns.

= 0 for leaning columns.

= load on restraining column, i.

= moment of inertia for column, i,

= load on the restraining columns in a story.

= load on the leaning columns in a story

= sum of for each column in the story.

= sum of for each column participating in lateral sway resistance

Commentary Equations (6): Although use of Eq. 12 is not particularly complex, the secondedition of the Commentary to the LRFD Specification presented two modified LeMessurierequations that were thought to be of value to the practicing engineer. Although the third

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edition of the commentary will be significantly simplified in the relevant section, it isanticipated that there will still be two simplified LeMessurier equations presented. One isbased on the story buckling model while the other is based on a story stiffness model.

For the story buckling model, it is assumed that there is no reduction in column stiffness dueto the presence of axial load. This is accomplished by taking for all columns, whichleads to Substitution of these values into Eq. 12 yields:

(15)

Which reduces to

(16)

Equation 16 can be recast into the form of the commentary equation, which is currentlyproposed to be

(17)

For a structure in which only one column can be considered to provide lateral stability, thesummation in Eq. 16 are unnecessary and the equation reduces to

(18)

which is the same as the equation that resulted from the modified Yura and Lim &McNamara approaches, Eqs. 9 and 11 respectively.

For the story stiffness model, stiffness reduction due to axial load is included as though allcolumns were cantilevers with a buckled shape in the form of a half sine curve as shown inFig. 2b, thus Since the leaning columns have no lateral stability of their own,= 0.0 for all leaning columns. The equation given in this paper as Eq. 12 is just one form ofthe effective length factor equations given by LeMessurier. Another form that uses, as ameasure of lateral stiffness, the ratio of lateral displacement of a story to the lateral load, isalso available through the same derivation (19). Equation 46d from the original paper, in thenotation of this paper, is given as

(19)

where the total lateral load supported by the level under consideration, is thecorresponding lateral displacement of the level and is the total load on the

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given story. In order to account for on the leaning columns, the load on these leaningcolumns must be subtracted from the total load on the story so that

Making this substitution and factoring out yields

(20)

This equation is somewhat simplified in the second edition commentary as

(21)

If the stiffness reduction due to axial load is applied to all columns, that is the leaningcolumns are not excluded, then would be applied to the total load on the storyand the separation taken to arrive at Eq. 20 would not be necessary. Thus, Eq. 19 wouldbecome

(22)

This equation, recast in the form of the third edition commentary equation is

(23)

These simplifications may not really be necessary since, in the original form, the equationspresented by LeMessurier are not much more complex.

Examples

The following examples will show how these approaches may be used to evaluatecolumns in unbraced frames.

Example 1: The unbraced frame with leaning columns as shown in Fig. 9 is to be checkedfor strength and stability. This is the frame introduced by Higgins (21) andmodified with a load factor of 1.43 to permit a check by LRFD. The frame is braced out ofthe plane.

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Yura Approach: Since sway will likely control the design of column AB, a W10x39 will beinvestigated. Due to symmetry, one half of the load on the leaning columns will be assignedto a single column, AB.

Figure 9. Symmetric Frame for Example 1

Note that to account for the pin ended beam, the length is modified by the factor 2.

For strength in the y-axis, thus, use LRFD E2-2.kips > 57 kips thus, the strength is adequate.

For stability about the x-axis, therefore from LRFD E2-2,= 228.0 kips > 207 kips, thus the column will be sufficient to provide stability for theremaining leaning columns.

Lim & McNamara Approach: Again, a W10x39 will be considered for column AB. Usingthe ratio of As before, so

As already shown, the column will be adequate for strength.Now, checking for stability, andThus, applied load which shows that the column is also adequatefor stability.

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LeMessurier Approach: Since the W10x39 column was shown to be adequate, it will againbe checked. Using the values obtained above, for column AB,

Thus, with from above, the W10x39 will be sufficient to provide lateralrestraint.

Modified Nomograph Equation: An iterative solution of Eq. 5 with Q and P asgiven above yields As can be seen, this value compares quite well with the valuesalready obtained and the column will be adequate.

Commentary Equations: For the first simplified equation, Eq. 16, as would beexpected from the derivation shown above, since there is only one restraining column.

The use of the second simplified equation, which is based on Eq. 19, requires an analysis ofthe structure to determine the ratio of lateral displacement to load. Since the analysis carriedout for the previous approaches assumed that the pin connection at the base was not a truepin but one which resulted in this must be included in the analysis. With thisprovision accounted for, an arbitrary lateral load of 5 kips results in a deflectionin. Using Eq. 20 yields

Using the equation from the second edition commentary, Eq 21, results in

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The proposed equation for the third edition, in the form of Eq. 22, yields

Using any of these K values, the column will prove to be adequate to provide the requiredlateral resistance. The results obtained from Eqs. 20 and 21 appear to be quite similar tothose determined from the other approaches presented. The result from Eq. 22 appears to besomewhat distant from the other results, although the column will still prove to be adequate.

Example 2: The frame shown in Fig. 10 was used by Cheong-Siat-Moy (22) to show thatYura's approach would not work and by de Buen (23) to present his new approach. The trussis assumed to provide sufficient rotational restraint at the top to permit that end of columnAB to be treated as a fixed end while the bottom of the column is pinned, thus from the

nomograph, Recognizing that Yura's approach will yield a larger column thanrequired and that LeMessurier's approach will give the correct results, as shown by the twopreviously cited references, the Lim & McNamara approach will be compared to theLeMessurier approach. For this example, the column will be taken as a W12x136 and50 ksi.

Figure 10. Frame for Example 2

Lim & McNamara: With one half of the leaning column load resisted by the singlerestraining column, n = 990/330 = 3.0. Thus,

Using thus f rom LRFD E2-3,= 8.48 ksi, thus Since this approach is essentially the same as theYura approach, this result is not entirely unexpected.

LeMessurier: For a fixed-pinned column, and so that Eq. 12 yields:

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as shown in Ref. 22 and 23.

Using thus from LRFD E2-3,thus

Modified Nomograph Equation: Using G values consistent with a true fixed end and a truepinned end, along with an iterative solution of Eq. 5 results in Thisresult is similar to that previously obtained from the LeMessurier approach.

Commentary Equations: As was shown for example 1, where only one column provideslateral support, the results from Lim & McNamara and the simplified equation, Eq. 16, areidentical, The LeMessurier analysis, Eq. 12, assumed that the columns were fixedat the upper end and had true pins at the lower end. This results in which is theassumption used to develop the second simplified equation, Eq. 20. With the lateraldisplacement calculated for a cantilever beam with a 5 kip load, in. This yields

from Eq. 20 and from Eq. 22. The results from Eq. 20 are identical tothe previously calculated value using the complete LeMessurier approach and the resultsfrom Eq. 22 are the same as that from the Lim & McNamara approach, as would beexpected from the derivations of the simplified equations.

Example 3: An interesting problem originally proposed by Zweig (24) is shown in Fig. 11.A portion of a large, unbraced one story industrial building with deep roof trusses is shown.The trusses, which frame in each direction, exhibit an infinitely large stiffness whencompared to the columns. In order to equalize sway restraint in each direction, alternatecolumns have their strong axes turned 90°. The nomograph approach with and= 10 yields Without considering leaning columns and using theW12x65 proposed by Zweig would be slightly undersized for the required Bythis approach, each column in the building would be the same, designed to carry 234 kipsabout its weak axis. However, if the strong axis column is used to brace the weak axiscolumn, there should be some savings available. Each of the approaches previouslydiscussed will be used to check a W12x53 column to determine whether this smaller columnwould be adequate. Throughout this example, the columns will be taken as pairs, one strongaxis and one weak axis column.

Yura Approach: Using the capacity of the W12x53 for sway buckling about thex-axis, column 1, and the y-axis, column 2, will be added. The combined capacity must beequal to or greater than the total load 2(234) = 468 kips. For

The

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Figure 11. Frame for Example 3

combined capacity is 432 + 127 = 559 kips which is greater than the 468 kips combinedload. Thus, the W12x53 will be acceptable for all of the columns.

Lim & McNamara Approach: If column 1 is taken as the restraining column and column 2as the leaning column, n = 234/234 = 1.0. Thus, for column 2, K = 1.0 and KL = 1.0(20) =20 ft. For the W12x53 buckling about the y-axis, For column 1,

and For buckling about the x-axis,280 kips > 234 kips. Thus, the W12x53 is adequate for both columns in each direction.

LeMessurier Approach: With and for both columns 1 and 2, and

Thus, from Eq. 12

which yields and Thus, for column 2,thus and For column 1,

which is the same as for column 2 so Thisshows that the W12x53 column is adequate for both carrying the load and providing lateralrestraint.

Modified Nomograph Equation: In this case, with end stiffness and load ratio as for theother approaches, Eq. 5 yields for the restraining column. The leaning columnwould use K=1.0 as with the Lim & McNamara approach. Again, the W12x53 will beacceptable.

Commentary Equations: For the first simplified equation, Eq. 16, for bothcolumns. Thus,

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As was the case in example 1, the application of second simplified equation requires theanalysis of the structure with a connection stiffness at the base that will result inWith a 5 kip lateral load, the resulting lateral deflection, Thus, from Eq. 20,

while Eq. 21 yields

and Eq. 22 yields

As can be seen from these K values, the W12x53 will work satisfactorily for all columns inthe structure when checked through these simplified equations. The significance of thisexample, as it was when originally published, is to show that the sharing of lateral stiffnesscan be used to the advantage of the structure.

Example 4: The frame shown in Fig. 12, introduced by Geschwindner (25), will be usedto compare the simplified methods for determination of effective length factors with atrue buckling analysis. The frame is supported in such a way that in plane behavior willbe critical. The columns AB and CD as well as the beam BC are W12x136. The othermembers are of such a size that their individual characteristics will not control. Theresults of a GTSTRUDL buckling analysis for equal loads on columns AB and CD yields

Using Eq. 3 this is equivalent to K=2.232. The nomograph equationyields K=2.166 or

Figure 12. Frame for Example 4 with Leaning Columns

When equal loads are also applied to columns EF, GH, and JK, GTSTRUDL yields528.7 kips or K=3.414. The loading on the structure shows two equal loads on therestraining columns and three equal loads on the leaning columns. This gives P = 2 and Q= 3. From the modified nomograph equation, K=3.289 and Eq. 11 yieldsK = 3.425 and The LeMessurier equation, using G = 100,000 to represent thepin end, yields K = 3.295 and

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No matter what approach is taken to account for the leaning columns, it is clear thatthey have a significant impact on the stability of the structure. It is also evident that asecond-order elastic analysis will yield the same forces for members AB, CD, and BC,whether there are loads on the leaning columns or not. Thus, more than a second-orderelastic analysis is needed for the complete design of the structure.

Example 5: Factored loads are now applied to the frame of example 4, as shown in Fig.13. First- and second-order elastic analyses are performed and, along with the resultsfrom example 4, a check on column CD, with is carried out.

Using the results from the first order analysis, and andfrom the LeMessurier analysis including the leaning columns, K=3.295 andFrom this, so LRFD Eq. H1-1a is used. Since thecolumn moment is from a first order analysis, it must be amplified. This will beaccomplished using the second suggested equation for Since there are no moments inthe non-sway analysis, The results of the first-order analysis give alateral deflection due to the 20 kip load of 1.738 in. Using these values, so that

The interaction equation becomes

Since this is less than 1.0, the column will be adequate.

If the results of the second-order analysis are used, andft. Again, the effect of the leaning columns will be included from the LeMessurieranalysis so that, and LRFD Eq. H1-1a is usedagain. Since the column moment results from a second-order analysis, there is no need toamplify it prior to using the interaction equation, thus

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Again, the column is seen to be adequate. It is interesting to note that there is an increasein the column axial load due to the second order effects that is not included in thesimplified code approach to second-order analysis and that the second-order momentsobtained from the two approaches are quite similar.

Example 6: An interesting structure was presented by Baker (26) to demonstrate theproblems associated with effective length determination when the assumptions of thenomograph are violated. The frame shown in Figure 14 represents one of two frames

Figure 14. Example 6 Frame with lateral service load

participating in providing lateral resistance for the building. This frame carries a gravityload of and provides lateral stability to columns carrying an additional= 1875 kips, both being one half of the total load for the building. The lateral deflectionof the frame due to is with The total gravity load is

and

The results presented by Baker were in the form of column capacities, However, areview of his equations shows that his solution actually uses Eq. 21. Table 1 shows the K-factors for each of the five columns in this frame as determined through nomograph, thefour simplified equations, and an elastic buckling analysis. It can be seen from Table 1that the use of the nomograph, Eq. 4, does not predict effective length values that wouldsubsequently produce accurate elastic buckling values for the columns of this frame. Thisis due to the fact that this frame significantly violates the assumptions used to develop thenomograph. If this structure had been designed with those values, the columns wouldhave had an expected capacity significantly larger than their true capacity.

The goal in determining an appropriate K-factor is to predict the results of a linearbuckling analysis. The results of a linear buckling analysis from GTSTRUDL, convertedto effective length, are also given in Table 1. The values determined from Eqs. 16, 20, and21 appear to be fairly consistent although slightly below the elastic buckling values. The

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results of Eq. 22 are significantly higher than the elastic buckling values, although theywill yield a conservative solution.

Table 1. Summary of Effective Length Calculations for Example 6

Conclusions

Structural systems which combine unbraced frames with simple gravity columns provide thestructural engineer with a system requiring special attention. In addition, unbraced framesthat do not meet the restrictive assumptions permitting use of the nomograph also poseinteresting problems. It is clear that leaning columns result from the structural framingarrangement, not from the use of a particular design philosophy. It is also clear that aspecific accounting must be made for the leaning column, even when a second-orderanalysis has been carried out. In addition, the design philosophy and the analysis approachmust be compatible.

This paper presented a brief discussion of the full range of approaches that might be used tocarry out a structural analysis. It is well understood that the critical buckling load of a framecan be determined through an eigenvalue analysis. It is this load that the K-factor isattempting to predict. Although the K-factor has been a controversial topic from its initialintroduction, it remains a useful tool to measure column capacity. Perhaps the mosttroubling aspect associated with the use of the K-factor has been the assumptions included inthe most common predictor equations.

Four approaches from the literature for determination of the K-factor were presented, alongwith several simplified equations derived from those procedures. It was shown that throughan iterative solution of Eq. 5, a more accurate value of could be obtained than that fromthe nomograph, Eq. 4, when leaning columns are present. In this case, the leaning columnloads are accounted for; however, the other limitations of the nomograph solution are stillpresent.

The equations proposed by LeMessurier are generally recognized as the most accurate ofthose presented. There are two approaches to the use of the LeMessurier equations. Onerequires the determination of which may be accomplished through the nomograph, as is

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normally done, or by an iterative solution of equation, Eq. 4. The other approach uses thelateral stiffness of the frame, as measured by its lateral deflection due to a lateral load. Eitherof these approaches will provide a practical solution to determining column capacity. Thus,it is not unrealistic to use the LeMessurier equations for effective length factors in normalengineering practice.

The commentary to the LRFD Specification provides simplified equations, based on theLeMessurier equations, which are felt to be useful to the designer. The examples presentedhere allow for a comparison of results between several of these equations. It appears thatthere is some wide variation in results, depending on the choice of approximation. Theassumptions used to develop these simplified equations are presented so the engineer will bein a better position to decide which should be used in a particular situation.

It was also shown that the use of a second-order analysis does not automatically account forthe presence of leaning columns, but does meet the need for determination of moments foruse in the interaction equations. Thus, both the second-order effects and buckling capacitymust be determined for the analysis and design to be complete. It is simply a matter ofdeciding which approach is to be used.

Although the LeMessurier approach is not overly complicated to use, designers wishing touse an even simpler approach may find that the Lim & McNamara equation for providesa sufficiently accurate way to account for leaning columns, particularly in preliminary stagesof design. In addition, the LeMessurier equation, which is based on the lateral deflection ofthe frame, provides a straightforward approach to the actual calculations, as suggested byBaker. Although simplified equations are presented in the Commentary to the 1993 LRFDSpecification, and will be included in the 1999 version, there is really no need to use them.Once the elastic-buckling load of the frame has been determined and the appropriate amountattributed to the individual columns, design by any approved method may proceed.

References

1. West, Harry H., Analysis of Structures. John Wiley and Sons, New York, 1989.

2. Galambos, T. V., Structural Members and Frames. Prentice-Hall, Inc., EnglewoodCliffs, NJ, 1968.

3. Disque, R. O., Applied Plastic Design in Steel. Van Nostrand Reinhold Co. New York,1971.

4. Chen, W. F., Goto, Y., and Liew, J. Y. R., Stability Design of Semi-Rigid Frames. JohnWiley and Sons, New York, 1996.

5. Chen, W. F. and Toma, S., Advanced Analysis of Steel Frames, CRC Press, BocaRaton, 1994.

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6. Load and Resistance Factor Design Specification for Steel Buildings. 2nd edition, AISC,Chicago, IL, 1994.

7. Specification for Structural Steel Buildings - Allowable Stress Design and PlasticDesign. AISC, Chicago, Ill., 1989.

8. AISC, "1998 Structural Engineering Software Survey," Modern Steel Construction.AISC, Jan. 1998, pp 62-63.

9. STAAD-III Structural Analysis and Design Program User's Manual. Revision16, Research Engineers, Inc., Marlton, N.J., 1992

10. GTSTRUDL User's Manual. Georgia Tech Research Corp., Atlanta, GA,

11. RISA-2D User's Guide, RISA Technologies, Lake Forest, CA.

12. ETABS User's Manual, Computers and Structures Inc., Berkeley, CA. 1988.

13. Wilson, E.L. and Habibullah, A., "Static and Dynamic Analysis of Multi-storyBuildings including P-Delta Effects," Earthquake Spectra Journal, EERI, Vol. 3, No.2,1987, pp.289-298.

14. The RAM Structural System, RAM Frame, Version 6, RAM International, Carlsbad,CA, 1998.

15. Higgins, T. R., "Effective Column Length - Tier Buildings", Engineering Journal,AISC Vol. 1, No.4, 1964, pp. 12-15.

16. Geschwindner, Louis F., "A Practical Approach to the Leaning Column", EngineeringJournal, AISC Vol. 31, No. 4, 1994, pp. 141-149.

17. Chen, W.F. and Lui, E.M.. Stability Design of Steel Frames, CRC Press, Boca Raton,FL., 1991.

18. Yura, J. A., "The Effective Length of Columns in Unbraced Frames," EngineeringJournal, AISC, Vol. 8, No. 2, 1971, pp. 37-42.

19. Lim, L. C. and McNamara, R. J., "Stability of Novel Building System," StructuralDesign of Tall Steel Buildings. Vol. II-16, Proceedings, ASCE-IABSE InternationalConference on the Planning and Design of Tall Buildings, Bethlehem, Pa. 1972, pp.499-524.

20. LeMessurier, Wm. J., "A Practical Method of Second Order Analysis," EngineeringJournal, AISC, Vol. 14, No. 2, 1977, pp. 49-67.

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21. Higgins, T. R., "Column Stability under Elastic Support", Engineering Journal, AISC,Vol. 2, No. 2, 1965, pp. 46-49.

22. Cheong-Siat-Moy, F., "Column Design in Gravity-Loaded Frame," Journal of StructuralEngineering, ASCE, Vol. 117, No. 5, May 1991, pp.1448-1461.

23. de Buen, Oscar, "Column Design in Steel Frames Under Gravity Loads," Journal ofStructural Engineering, ASCE, Vol. 118, No. 10, October, 1992, pp. 2928-2935.

24. Zweig, A., Discussion of "Column Stability Under Elastic Support," by T. R. Higgins,Engineering Journal, AISC, Vol. 2, No. 3, 1965, pp. 105-106.

25. Geschwindner, L.F., "Practical Design of Unbraced Moment Frames with LeaningColumns," Proceedings of Structures Congress 13, Boston, Mass., April 2-5, 1995,ASCE, pp. 527-542.

26. Baker, W. F., "Practical Problems in Stability of Steel Structures," Proceedings of theNational Steel Construction Conference, Chicago, IL, AISC 1997, pp. 2.1-2.24.

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Documents

A Practical Look at Frame Analysis, Stability, And Leaning Columns