PracticalProblems in

Stability of SteelStructures

William F. Baker

Author

William F. Baker joinedSkidmore, Owings, & Merrill

LLP (SOM) in 1981 after receivinghis bachelor and master of sci-ence degrees in civil engineeringfrom the University of Missouriand the University of Illinois,respectively. Mr. Baker is thepartner in charge of structural andcivil engineering in the Chicagooffice of SOM. His experience isinternational and extends to avariety of structural types, includ-ing offices, hotels, churches,museums, educational facilities,exhibition/convention centers, air-ports, retail and mixed-use devel-opments. As structural/civil engi-neering partner, Mr. Baker leadsthe development of structural con-cepts and oversees the quality ofall work contributed by his team.

Mr. Baker is a registered pro-fessional engineer in Illinois,Missouri and New York. He is amember of numerous professionaland civic organizations including:American Concrete Institute,(ACI); American Institute of SteelConstruction (AISC); AmericanSociety of Civil Engineer (ASCE);Council on Tall Buildings andUrban Habitat (CTBUH) andStructural Stability ResearchCouncil (SSRC).

His recent publications and lec-tures include "Bare BonesBuilding," by W.F. Baker, S.H.lyengar, R.B. Johnson and R.C.Sinn, Civil Engineering,November 1996, pp. 43-45; "SteelBuildings and the Art of StructuralEngineering," R. Halvorson, W.F.Baker and R.C. Sinn, InProceedings, First ArgentineanConference on SteelConstruction, Buenos Aires,Argentina, October, 1995; "RussiaTower - 120-Story OfficeBuilding," May 1995, Council onTall Buildings and Urban Habitat,Amsterdam, The Netherlands.

Summary

The fundamental stability prob-lem that a structural steel

designer must solve is the deter-mination of the nominal compres-sion capacity (Pn) of a structuralmember or system. This paperwill discuss the calculation of Pn

for two systems: unbracedframes and lattice/built-up mem-bers. The traditional calculationof Pn for the columns of anunbraced frame using k-values isextremely cumbersome; in addi-tion it is difficult to properly correctthe k-values for deviations fromthe assumptions contained in theeffective length factor nomograph.This paper will demonstrate anapproach that allows the designerto directly address the stability ofan unbraced frame and calculatePn for the columns without usingk-values. The proper calculationof Pn for non-prismatic membersor lattice/built-up systems is notwell known. This paper will illus-trate how to use a computereigenvalue buckling analysis andthe tangent modulus factors,implied by the LRFD specifica-tions, to accurately determine thenominal compression capacity ofsuch systems.

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PRACTICAL PROBLEMS IN STABILITY OF STEEL STRUCTURES

INTRODUCTION

Engineers must address stability in the design of all steel structures. The stability concerns canbe the local buckling of flanges or webs, the lateral torsional buckling of flexural members, theflexural buckling of compression members, etc. A major stability design consideration is theevaluation of the stability of systems composed of steel members. This paper will focus ontwo important categories of system stability: unbraced frames and lattice/built-up systems.

There are several attributes that are desirable for a calculation methodology. The calculationsshould be easily performed using commonly available resources. In the current officeenvironment, this implies that calculations that require the use of hand-held calculators or PC-based finite element programs would be appropriate. The calculations should not be overlycomplex in order to minimize the risk of errors caused by misinterpretation or misuse of themethodology. The methodology should be based on the fundamentals of structural steelbehavior and the calculation process should assist the designer in understanding the behaviorof the structure being designed.

Often, calculation procedures seem to be a part of a cookbook recipe that hopefully will leadto a safe structure but in fact can lead to incorrect solutions if the "recipe" is not understoodor is misapplied. Sometimes, the procedure is so complex that the designer is at risk ofmaking errors. Unfortunately, the preceding statements can be made about commoncalculation methods used to determine the overall stability of a structure.

In the vocabulary of the AISC-LRFD Specification (AISC, 1993), the calculation of thestability of a member or a system becomes the determination of the nominal axial strength, Pn.In this paper, calculation methodologies are presented that are intended to assist the structuralsteel designer with the calculation of Pn for unbraced frames and lattice/built-up systems.

UNBRACED FRAMES

It could be argued that the calculation procedure for Pn for unbraced frames is among the leastunderstood and most frequently misapplied procedures used in steel design. Traditionally,calculations have been based on the determination of effective length factors (k-values).These k-values are generally determined from a nomograph that is based on numerousassumptions. Unfortunately, these assumptions are generally not satisfied in commonlyoccurring structural systems. This leads to a complicated series of corrections that must bemade. The complexity of the corrections is such that the designer must expend great care toavoid errors.

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These assumptions and the corresponding corrections are discussed in the commentary toChapter C of the AISC-LRFD Specification. For an in-depth discussion of unbraced frames,the reader is referred to ASCE (1997), White and Hajjar (1997a) and White and Hajjar(1997b).

The stability of an unbraced frame is described by the term of the AISC-LRFDinteraction equations that are repeated here as Equations (1) and (2).

The following discussion will present a method that uses story drift to calculate

Drift Based Determination of Stability

The AISC formulae emphasize the calculation of Pn for an individual column when checkingthe stability of an unbraced frame. The traditional methods of calculation of the Pn of anindividual column, based on k-values, have two unfortunate side effects. First, the designerloses sight of the fact that stability is a system phenomenon (i.e. lateral collapse of an entirestory), not an individual column phenomenon. Second, greater stability can often be achievedby increasing the size of other members of the lateral system instead of the individual columnbeing checked. The following is a discussion of methods of calculating Pn based on the lateralstiffness of the unbraced frame.

The framing of a building is generally composed of two major groups: gravity-only framingand lateral framing. The gravity-only framing is designed to only resist the vertical loadsimposed by gravity with the assumption that the lateral stability is provided by other systems(shear walls, diagonal bracing, rigid moment resisting frames, etc.) The failure of a column inthe gravity-only framing occurs primarily through braced buckling. These columns arecommonly called leaning columns because they "lean" against the lateral system for sideswaystability. The lateral system resists the lateral loads (wind, seismic, etc.). It also providesstability to the gravity loads that are supported directly by the lateral system and to thegravity-only framing (leaning columns). The basic stability failure (in addition to localizedbuckling of a column in a braced mode) of the lateral system is through the simultaneoussidesway buckling of all the columns in a story.

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(1)

(2)

The lateral movement per unit height (drift) of a story under the action of a lateral load is afundamental indicator of lateral stability. This movement for a 50-year wind has traditionallybeen limited to values such as 1/400 or 1/500, based on an elastic first order analysis. Thefirst order drift for a given lateral load along with the total vertical load provides the basis fora stability analysis. In order to facilitate the discussion, the following terms are defined:

Lateral inter-story deflection of the story under considerationStory Height

First order drift index

Lateral shear in an individual columnLateral story shear causing D1

Total service vertical load in the story under considerationTotal factored vertical load in the story under consideration

The elastic buckling load of a story can be approximated by (Nair, 1981).This approximation is quite useful and is often given the term (LeMessurier, 1977).

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(4)

(5)

(6)

Although it is common to use a 50-year wind load in the calculation of the actualmagnitude of the load is not important because the lateral movement is proportional to thelateral load for a linear analysis. It is only important that the deflected shape approximate thebuckled shape. For this reason, it is often useful to use a lateral load that is proportional tothe gravity weight of the building. This is particularly true if the distribution of the weight ofthe building is not uniform.

This is the same estimate of that is used in the moment magnification term of theAISC-LRFD Specification. The formulation, as shown below, is commonly used as anapproximation of second order effects. A variation on the term is shown below as anamplification factor, This amplification factor can be used to estimate the second orderdrift, as is shown in Equation (6). This leads to the simple formulation of Equation (7).

(3)

(7)

The value of used in the calculation of is dependent on the purpose of the calculation.For strength, should be used for For serviceability, or the designer's estimate ofthe total actual load on the story should be used.

The lateral movement of a structure is always a primary concern of the structural steeldesigner. Even if the building has more than adequate lateral strength, the building's lateralmovement must be controlled in order to avoid damage to cladding, partitions and other non-structural elements, and to avoid disturbing the occupants of the building. Although the firstorder drift is useful for the calculation of (as it shall be demonstrated later in this paper),it is the second order drift that the building actually experiences. The designer shouldalways calculate to see if it is excessive. Using the above approximation, the designer canquickly evaluate

The following discussion will demonstrate how the lateral story shear and the first orderdrift index can be used to calculate

The AISC-LRFD column formula for the calculation of the nominal strength of a column isexpressed as:

(8)

(9)

(10)

(11)

(12)

(13)

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where:

Equations (8) through (11) can be simplified to:

The two values that are needed to calculate are the squash load, and the elastic bucklingload, The calculation of is simply is the portion of the story lateral bucklingstrength that is attributed to an individual column and is denoted by for thecolumn. If the applied factored load, were increased by a factor until the story buckles,then the buckling load for the story is and is equal to Therefore the for each

column is The task now is to calculate can be used to

approximate but the result tends to over-estimate for some frames. This can beobserved in the two frames shown below.

(a) (b)

Figure 1

(14)

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Both frames have the same but have different The frame on the right has a thatis 82% of tends to over-estimate because it does not account for the loss offlexural stiffness caused by the presence of axial loads effects) in the lateral forceresisting columns. This phenomenon is extensively discussed by LeMessurier in his 1977landmark paper (LeMessurier, 1977). The relationship between and is a function ofthe end conditions of the column in the unbraced frame and the amount of load supported byleaning columns. The AISC commentary suggests a refinement of that can be used as anestimate of for column i.

is the fraction of the total story vertical load supported on leaning columns.

The last inequality (Equation (16)) is used in order to detect braced buckling of a column in aframe and to avoid cases where the correction in Equation (15) is inadequate.

can now be calculated for all of the columns that participate in the unbraced frame. Thismust be compared with the in the out-of-plane direction and the lowest value used in

the interaction equation for each column.

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or

but

(15)

(16)

The use of has advantages over methods based on the determination ofeffective length factors (k-values) in that it is a much easier method to calculate and that itis based on information that is well known to the designer. In fact, the terms needed tocalculate based on are approximately known prior to the beginning of the design. Itis common for a target first order drift index to be established for a 50 year wind prior to thebeginning of the frame design. Also, the weight of the building is generally known from pastexperience (i.e. an approximate building density of 10 pcf for service loads or 13 pcf forfactored loads is common for steel framed office buildings) and the ratio of and canbe estimated from the layout of the framing.

Another advantage of the method for determining is that it emphasizes controllinginter-story drift as a method of achieving a desirable

Because lateral stability is a story behavior phenomenon, the first term in the interactionequation, should be the same for every column in an unbraced frame unless out-of-plane buckling or braced in-plane buckling controls the strength of an individual column.However, it can be seen from the preceding equations that the for different columnswill be equal only if the ratio of are equal for all columns of an unbraced frame(or if the on all of the columns are so small that they are all controlled by Equation (13)).This variation of values for different columns in the same story is independent of themethod (drift method, corrected effective length method, eigenvalue buckling analysismethod, etc.) used to calculate This variation is an inherent limitation of using elasticanalysis to evaluate an inelastic behavior.

In an elastic analysis, the stability demand is distributed amongst the columns based on theirrelative elastic stability capacities. Unfortunately, columns with lower values of have alower inelastic stability capacity relative to other columns in the system than is implied byelastic analysis because of capacity reductions from inelastic effects (i.e. local yielding) in themore highly stressed columns. When column capacity Equations (8) through (11) (orEquations (12) and (13)) are applied to individual columns, it is done with the impliedassumption that all other columns have the same adjustments (tangent modulus corrections)for inelastic effects. To the extent that the inelastic effects are not equal, the story capacitybased on an elastic analysis and the column capacity equations may overestimate orunderestimate the actual total story stability {story capacity

It has been the author's experience that the story stability based on an elasticanalysis and the column capacity equations may be underestimated or overestimated by morethan 10%, however, the maximum potential unconservatism has not been rigorouslyestablished. It is for this reason that the term for all of the columns in an unbracedframe should be based on the column with the highest value of unless a moresophisticated analysis is performed.

The uniform, actual value of can be determined only if additional calculations aredone that accommodate an inelastic redistribution of stability demand (from the columns withhigh values of to columns with low values of

ColumnMark

12345

All Others

Individual Column LoadDistribution

3.0%1.0%5.5%0.5%2.5%

Column Load Distributionfor Both Rigid Frames

6.0%2.0%11.0%1.0%5.0%

75.0%100.0%

TABLE 1

All steel is ASTM A572-Grade 50.

Note:

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Figure 3 - Example 1

The following example (Figures 2 and 3 and Table 1) will demonstrate the calculation ofbased on The frame was chosen because it violates many of the assumptions of the k-value nomograph (extremes in load variations, unequal bay sizes, etc.) and would requireextensive corrections if that method were used. However, the procedure is relatively straightforward if the method is used. In order to equate to the AISC commentary for the k-value nomograph relative to columns with pinned bases, a rotational spring stiffness of6EI/GL with G equal to 10 was used at the base of each column to approximate the stiffnessthat is expected from actual column base details. The lateral analysis calculates a first orderelastic drift of 0.362 inches. Based on this information, the designer can now calculate thefollowing:

The second order drift index at service loads is approximately:

(This may be excessive for some applications).

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The story elastic buckling load is estimated by:

24 kip x 497 = 11,900 kips.

correcting based on Equation (15) leads to:

= 11,900 x (0.85 + 0.15 x 0.75) = 11,500 kips.

The value of 11,500 kips compares closely with a value of 11,700 kips from a computerbased elastic eigenvalue analysis.

is then prorated to the columns of the lateral load resisting frame by the ratio ofand is calculated for the individual columns using Equation (12) or (13). The results aresummarized in Table 2. An inspection of the results shows that the column with the highestvalue of has the highest value of This is because they all have the same ratio of

(for this problem the ratio is 0.435) but the columns that are closest to yield strengthhave the greatest strength reductions because of inelastic effects. The highest ratio ofwill be a conservative estimate of the actual ratio of and the lowest ratio will be anunconservative estimate. Unless an inelastic redistribution of the stability demand is made, thedesigner should use the highest value of for all of the columns of the unbraced frame.If greater precision is required, then an inelastic stability demand redistribution analysis shouldbe performed.

The following section will describe one method suggested by White and Hajjar (White andHajjar, 1997b) for redistributing the stability demand based on inelasticity.

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TABLE 2ColumnMark

12345

(kips)

15050

27525

125

(kips)

780660850660660

0.1920.0760.3240.0380.189

(kips)

345115632

57287

1.7(kips)

19432007247520431675

(kips)

302101484

50252

0.5840.5840.6680.5840.584

In-planefor

Equations(1)&(2)

0.6680.6680.6680.6680.668

Corrections for Inelastic Redistribution

The AISC-LRFD column equations can be formulated in terms of a tangent modulus/out-of-straightness correction to the elastic buckling value. After some manipulation, Equations (12)and (13) can be rewritten as:

A value 0.877 is included in Equations (13) and (14) to account for out-of-straightness.

Using the previous design example, the designer can re-run the lateral analysis after reducingthe moment of inertias for each column by the value for that column. It is suggested that theinitial estimates of (denoted as ) that are used to calculate be based on where

The "b" subscript refers to the column in the unbraced frame with the highestfrom the preceding iteration. The new lateral analysis results in a new inter-story drift index

that can be used to calculate an inelastic for the story. It is now necessary to calculatethe elastic for each of the columns so that a new estimate of can be made. This isaccomplished by dividing Equation (15) by the value used in the analysis for each columnand calculating a new and thus a new The modified Equation (15') is given below:

(17)

(18)

(19)

This procedure is repeated until the designer is satisfied with the range of values. Thesteps are outlined as follows:

1. Perform an elastic lateral analysis and calculate2. Calculate for each column of the lateral load resisting frame using Equation (15'). Use

for the first pass. Check for braced buckling using Equation (16).3. Calculate for each column of the lateral load resisting frame using Equations (12) and

(13).4. Calculate for each column of the lateral load resisting frame and use the highest

value for all columns of the frame. The designer can stop at this point. Ifdesired, inelastic analysis to more accurately estimate may be used as follows.

5. Estimate (denoted as for each column as where6. Calculate for each column using in Equations (18) and (19). Revise the moment of

inertia of each column of the unbraced frame to be7. Perform a standard lateral analysis and calculate8. Repeat steps 2 through 4.

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(15')

ColumnMark

12345

Calculated In-Plane1st-RunElastic

0.5840.5840.6680.5840.584

2nd-RunStiff. Red

0.5870.5870.6240.5870.587

3rd-RunStiff. Red

0.5940.5940.6100.5940.594

4th-RunStiff. Red

0.5970.5970.6040.5970.597

5th-RunStiff. Red

0.5980.5980.6010.5980.598

TABLE 3

An interesting variation on the first example is to fix the bases of the columns in the rigidframe. This is shown as example 2 in Figure 4. The first order deflection is reduced to 0.133inches which corresponds to a of 1357. The resulting calculated values of areshown in the "1st-Run Elastic" column of Table 4. For example 2, column No. 3 has athat is over twice that of some of the other columns. If an inelastic analysis iteration such asthat described above is made, it becomes clear on the next iteration from Equation (16) thatcolumn No. 3 buckles in a braced buckling mode before the story buckles. Column No. 3 isthen treated as a pinned leaning column for subsequent inelastic analysis iterations. Theconvergence of is shown in Table 4. It can be derived from Tables 1 and 4 that thetotal story capacity increases by 10% from the from the 1st to the 5th run for this example.

Figure 4 - Example 2

ColumnMark

12345

Calculated In-Plane1st-RunElastic

0.3200.2150.4680.2140.317

2nd-RunStiff. Red

0.3260.236

Buckled0.2360.324

3rd-RunStiff. Red

0.3210.311

Pinned0.3110.321

4th-RunStiff. Red

0.3180.314

Pinned0.3140.318

5th-RunStiff. Red

0.3170.315

Pinned0.3150.317

ColumnMark

12345

0.3570.282

Pinned0.2820.355

TABLE 4 TABLE 5

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For example (1), the convergence of is shown in Table 3. It can be derived fromTables 1 and 3 that the total story capacity increases by 3% from the from the 1st to the 5thrun for this example.

In a rigid frame, when one column has a significantly larger than the other columns,and the designer does not want to perform an iterative inelastic analysis, he/she may considerthe column as pinned (Figure 5) and calculate the for the remaining columns on thatbasis. If this were done on the preceding problem (Figure 5), the results from the elasticanalysis would be as shown in Table 5. A value of 0.360 for for all of the columns isclose to the value from the more involved iterative process as shown in Table 4. The value of

for the pinned column would be based on k 1.0.

Figure 5 - Example 3

A final unbraced frame example is shown in Figure 6. For brevity, only the results are shownin Table 6. The total story capacity decreases by 11% from the elastic analysis to theconverged inelastic analysis. For accuracy, these results are based on an eigenvalue bucklinganalysis. This example illustrates why the value of must be based on the column withthe highest value of unless an inelastic analysis is made. For the example in Figure 6, avalue of 0.799 should be used for for both columns if only an elastic analysis is made.

Figure 6 - Example 4

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ColumnMark

ABTotal

Elastic Analysis(kip)

589944

1533

0.7990.499

Inelastic Analysis(kip)

683683

1366

0.6890.684

TABLE 6

LATTICE AND BUILT-UP SYSTEMS

The determination of for non-prismatic compression members or systems is difficultbecause the LRFD column formulae (Equations (8), (9) and (10)) assume a constant crosssection. The systems shown in Figure 7 cannot be evaluated in the same manner as a simplerolled wide-flange column. However, Equations (17), (18) and (19) can be used witheigenvalue buckling or other numerical analysis procedures to calculate for a member orsystem.

Figure 7

Consider the tapered member shown in Figure 8. The full elastic moment of inertia is shownin Figure 9a and the squash load, is shown in Figure 9b. The full elastic buckling load,can be determined using a commercially available computer eigenvalue buckling analysisprogram or by hand calculated numerical methods that are available in the literature (Bleich,1952).

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Figure 8

For a conservative estimate of the squash load, on the smallest cross section and forthe member or system can be used with Equations (17), (18) and (19). A more accurate valuefor can be calculated on an iterative basis using estimates for in Equations (18) and (19)to reduce the moment of inertia at each cross section and calculating the buckling valuefor the system by analysis. When the estimated value for and the resulting buckling valuesare equal, they represent a value of that can be used in design. Figure 9c shows thereduced moments of inertia and the buckling strength of the system.

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Figure 9

CONCLUSION

The calculation methods described in this paper are intended to make the fundamentalbehavior of a system being designed clear to the structural steel designer. These methodspermit the determination of the compressive strength, of an unbraced frame or lattice/built-up system using standard analysis tools.

The calculation of for an unbraced frame focuses on story stability. The stability of a storyis a function of the lateral stiffness of the story in the presence of destabilizing forces fromleaning columns and out-of-straightness. The lateral stability of a story is also a function ofthe softening of the flexural stiffness of the columns that participate in the braced frame from

and tangent modulus effects. The calculation methodology suggested here addressesthese issues and is easy to use.

The evaluation of lattice/built-up systems also directly addresses the issues of tangent modulusand out-of-straightness and can be used on a wide range of design problems.

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The better understanding the designer has on the limit state that is being considered by thebuilding code, the more likely it is that a safe and efficient design will be produced. Themethods presented in this paper are intended to help the designer understand the stability ofunbraced frames and lattice/built-up systems and produce designs that are consistent with theAISC-LRFD specification.

REFERENCES

1. AISC (1993), Load and Resistance Factor Design Specification for Structural SteelBuildings and Commentary, AISC, Chicago, IL, 1993.

2. ASCE (1997) Structural Engineering Institute Technical Committee on Load andResistance Factor Design, "Effective Length and Notional Load Approaches forAssessing Frame Stability: Implications for American Steel Design." CommitteeReport, ASCE, New York, (to appear in 1997).

3. Bleich, F. (1952), Buckling Strength of Metal Structures, McGraw-Hill Book Co.,New York, NY, 1952.

4. LeMessurier, W. J. (1977), A Practical Method of Second Order Analysis, Part 2 -Rigid Frames, AISC Engineering Journal, 2nd Qtr., 1977.

5. Nair, R. S. (1981), A Simple Method of Overall Stability Analysis for MultistoryBuildings, Structural Stability Research Council, Proceedings 1981.

6. White, D. W. and Hajjar, J. F. (1997a), Buckling Models and Stability Design of SteelFrames: A Unified Approach, Journal of Constructional Steel Research, (to appear in1997).

7. White, D. W. and Hajjar, J. F. (1997b), Accuracy and Simplicity of Calcualtions forStability Design of Steel Frames: A Unified Approach, Journal of Constructional SteelResearch, (to appear in 1997).

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Fig. 1

Fig. 6

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Fig

. 2

Fig. 3 - EXAMPLE 1

Fig. 4 - EXAMPLE 2

Fig. 5 - EXAMPLE 3

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