Stability

Practical Problems in Stability of Steel Structures

William F. Baker

Author

Summary

illiam F. Baker joined Skidmore, Owings, & Merrill LLP (SOM) in 1981 after receiving his bachelor and master of science degrees in civil engineering from the University of Missouri and the University of Illinois, respectively. Mr. Baker is the partner in charge of structural and civil engineering in the Chicago office of SOM. His experience is international and extends to a variety of structural types, including offices, hotels, churches, museums, educational facilities, exhibition/convention centers, airports, retail and mixed-use developments. As structural/civil engineering partner, Mr. Baker leads the development of structural concepts and oversees the quality of all work contributed by his team. Mr. Baker is a registered professional engineer in Illinois, Missouri and New York. He is a member of numerous professional and civic organizations including: American Concrete Institute, (ACI); American Institute of Steel Construction (AISC); American Society of Civil Engineer (ASCE); Council on Tall Buildings and Urban Habitat (CTBUH) and Structural Stability Research Council (SSRC). His recent publications and lectures include "Bare Bones Building," by W.F. Baker, S.H. lyengar, R.B. Johnson and R.C. Sinn, Civil Engineering, November 1996, pp. 43-45; "Steel Buildings and the Art of Structural Engineering," R. Halvorson, W.F. Baker and R.C. Sinn, In Proceedings, First Argentinean Conference on Steel Construction, Buenos Aires, Argentina, October, 1995; "Russia Tower - 120-Story Office Building," May 1995, Council on Tall Buildings and Urban Habitat, Amsterdam, The Netherlands.

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he fundamental stability problem that a structural steel designer must solve is the determination of the nominal compression capacity (Pn) of a structural member or system. This paper will discuss the calculation of Pn for two systems: unbraced frames and lattice/built-up members. The traditional calculation of Pn for the columns of an unbraced frame using k-values is extremely cumbersome; in addition it is difficult to properly correct the k-values for deviations from the assumptions contained in the effective length factor nomograph. This paper will demonstrate an approach that allows the designer to directly address the stability of an unbraced frame and calculate Pn for the columns without using k-values. The proper calculation of Pn for non-prismatic members or lattice/built-up systems is not well known. This paper will illustrate how to use a computer eigenvalue buckling analysis and the tangent modulus factors, implied by the LRFD specifications, to accurately determine the nominal compression capacity of such systems.

PRACTICAL PROBLEMS IN STABILITY OF STEEL STRUCTURES INTRODUCTION

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

There are several attributes that are desirable for a calculation methodology. The calculations should be easily performed using commonly available resources. In the current office environment, this implies that calculations that require the use of hand-held calculators or PCbased finite element programs would be appropriate. The calculations should not be overly complex in order to minimize the risk of errors caused by misinterpretation or misuse of the

methodology. The methodology should be based on the fundamentals of structural steel behavior and the calculation process should assist the designer in understanding the behavior of the structure being designed. Often, calculation procedures seem to be a part of a cookbook recipe that hopefully will lead to a safe structure but in fact can lead to incorrect solutions if the "recipe" is not understood or is misapplied. Sometimes, the procedure is so complex that the designer is at risk of making errors. Unfortunately, the preceding statements can be made about common

calculation methods used to determine the overall stability of a structure. In the vocabulary of the AISC-LRFD Specification (AISC, 1993), the calculation of the stability 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 structural steel 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 least understood 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 numerous assumptions. Unfortunately, these assumptions are generally not satisfied in commonly occurring structural systems. This leads to a complicated series of corrections that must be made. The complexity of the corrections is such that the designer must expend great care to avoid errors.

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These assumptions and the corresponding corrections are discussed in the commentary to Chapter 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-LRFD interaction equations that are repeated here as Equations (1) and (2). (1)

(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 checking the stability of an unbraced frame. The traditional methods of calculation of the Pn of an individual column, based on k-values, have two unfortunate side effects. First, the designer loses sight of the fact that stability is a system phenomenon (i.e. lateral collapse of an entire story), not an individual column phenomenon. Second, greater stability can often be achieved by increasing the size of other members of the lateral system instead of the individual column being checked. The following is a discussion of methods of calculating Pn based on the lateral stiffness of the unbraced frame. The framing of a building is generally composed of two major groups: gravity-only framing and lateral framing. The gravity-only framing is designed to only resist the vertical loads imposed 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 in the gravity-only framing occurs primarily through braced buckling. These columns are commonly called leaning columns because they "lean" against the lateral system for sidesway stability. The lateral system resists the lateral loads (wind, seismic, etc.). It also provides stability to the gravity loads that are supported directly by the lateral system and to the gravity-only framing (leaning columns). The basic stability failure (in addition to localized buckling of a column in a braced mode) of the lateral system is through the simultaneous sidesway buckling of all the columns in a story.

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The lateral movement per unit height (drift) of a story under the action of a lateral load is a fundamental indicator of lateral stability. This movement for a 50-year wind has traditionally been limited to values such as 1/400 or 1/500, based on an elastic first order analysis. The first order drift for a given lateral load along with the total vertical load provides the basis for a stability analysis. In order to facilitate the discussion, the following terms are defined: Lateral inter-story deflection of the story under consideration Story Height First order drift index Lateral shear in an individual column Lateral story shear causing D 1 Total service vertical load in the story under consideration Total 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). (3)

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

This is the same estimate of that is used in the moment magnification term of the AISC-LRFD Specification. The formulation, as shown below, is commonly used as an approximation of second order effects. A variation on the term is shown below as an amplification factor, This amplification factor can be used to estimate the second order drift, as is shown in Equation (6). This leads to the simple formulation of Equation (7). (4)

(5)

(6)

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(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 of the total actual load on the story should be used. The lateral movement of a structure is always a primary concern of the structural steel designer. Even if the building has more than adequate lateral strength, the building's lateral movement must be controlled in order to avoid damage to cladding, partitions and other nonstructural elements, and to avoid disturbing the occupants of the building. Although the first order 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 should always calculate to see if it is excessive. Using the above approximation, the designer can quickly evaluate The following discussion will demonstrate how the lateral story shear drift index can be used to calculate

and the first order

The AISC-LRFD column formula for the calculation of the nominal strength of a column is expressed as: (8) (9) (10)

where: (11)

Equations (8) through (11) can be simplified to: (12) (13)

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The two values that are needed to calculate are the squash load, and the elastic buckling load, The calculation of is simply is the portion of the story lateral buckling strength that is attributed to an individual column and is denoted by for the column. 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

approximate but the result tends to over-estimate observed in the two frames shown below.

(a)

can be used to

for some frames. This can be

(b) Figure 1

Both frames have the same but have different The frame on the right has a that is 82% of tends to over-estimate because it does not account for the loss of flexural stiffness caused by the presence of axial loads effects) in the lateral force resisting columns. This phenomenon is extensively discussed by LeMessurier in his 1977 landmark paper (LeMessurier, 1977). The relationship between and is a function of the end conditions of the column in the unbraced frame and the amount of load supported by leaning columns. The AISC commentary suggests a refinement of that can be used as an estimate of for column i. (14)

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or (15) (16)

but

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 a frame 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. This must be compared with the in the out-of-plane direction and the lowest value used in the interaction equation for each column.

The use of has advantages over methods based on the determination of effective length factors (k-values) in that it is a much easier method to calculate and that it is based on information that is well known to the designer. In fact, the terms needed to calculate based on are approximately known prior to the beginning of the design. It is common for a target first order drift index to be established for a 50 year wind prior to the beginning of the frame design. Also, the weight of the building is generally known from past experience (i.e. an approximate building density of 10 pcf for service loads or 13 pcf for factored loads is common for steel framed office buildings) and the ratio of and can be estimated from the layout of the framing. Another advantage of the method for determining inter-story drift as a method of achieving a desirable

is that it emphasizes controlling

Because lateral stability is a story behavior phenomenon, the first term in the interaction equation, should be the same for every column in an unbraced frame unless out-ofplane 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 columns will 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 the method (drift method, corrected effective length method, eigenvalue buckling analysis method, etc.) used to calculate This variation is an inherent limitation of using elastic analysis to evaluate an inelastic behavior.

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In an elastic analysis, the stability demand is distributed amongst the columns based on their relative elastic stability capacities. Unfortunately, columns with lower values of have a lower inelastic stability capacity relative to other columns in the system than is implied by elastic analysis because of capacity reductions from inelastic effects (i.e. local yielding) in the more highly stressed columns. When column capacity Equations (8) through (11) (or Equations (12) and (13)) are applied to individual columns, it is done with the implied assumption 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 capacity based on an elastic analysis and the column capacity equations may overestimate or underestimate the actual total story stability {story capacity It has been the author's experience that the story stability based on an elastic analysis and the column capacity equations may be underestimated or overestimated by more than 10%, however, the maximum potential unconservatism has not been rigorously established. It is for this reason that the term for all of the columns in an unbraced frame should be based on the column with the highest value of unless a more sophisticated analysis is performed. The uniform, actual value of can be determined only if additional calculations are done that accommodate an inelastic redistribution of stability demand (from the columns with high values of to columns with low values of TABLE 1

Column Mark 1 2 3 4 5 All Others

Individual Column Load Distribution 3.0% 1.0% 5.5% 0.5% 2.5%

Column Load Distribution for Both Rigid Frames 6.0% 2.0% 11.0% 1.0% 5.0% 75.0% 100.0%

Note: All steel is ASTM A572-Grade 50.

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Figure 3 - Example 1 The following example (Figures 2 and 3 and Table 1) will demonstrate the calculation of based on The frame was chosen because it violates many of the assumptions of the kvalue nomograph (extremes in load variations, unequal bay sizes, etc.) and would require extensive corrections if that method were used. However, the procedure is relatively straight forward if the method is used. In order to equate to the AISC commentary for the kvalue nomograph relative to columns with pinned bases, a rotational spring stiffness of 6EI/GL with G equal to 10 was used at the base of each column to approximate the stiffness that is expected from actual column base details. The lateral analysis calculates a first order elastic drift of 0.362 inches. Based on this information, the designer can now calculate the following:

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 computer based elastic eigenvalue analysis.

is then prorated to the columns of the lateral load resisting frame by the ratio of and is calculated for the individual columns using Equation (12) or (13). The results are summarized in Table 2. An inspection of the results shows that the column with the highest value 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 strength have the greatest strength reductions because of inelastic effects. The highest ratio of will be a conservative estimate of the actual ratio of and the lowest ratio will be an unconservative estimate. Unless an inelastic redistribution of the stability demand is made, the designer 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 should be performed. TABLE 2

Column Mark

1 2 3 4 5

(kips)

150 50 275 25 125

(kips)

780 660 850 660 660

(kips)

0.192 0.076 0.324 0.038 0.189

345 115 632 57 287

1.7 (kips)

1943 2007 2475 2043 1675

(kips)

302 101 484 50 252

0.584 0.584 0.668 0.584 0.584

In-plane for Equations (1)&(2) 0.668 0.668

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

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0.668 0.668 0.668

Corrections for Inelastic Redistribution

The AISC-LRFD column equations can be formulated in terms of a tangent modulus/out-ofstraightness correction to the elastic buckling value. After some manipulation, Equations (12) and (13) can be rewritten as: (17) (18) (19)

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 reducing the moment of inertias for each column by the value for that column. It is suggested that the initial 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 highest from 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 calculate the elastic for each of the columns so that a new estimate of can be made. This is accomplished by dividing Equation (15) by the value used in the analysis for each column and calculating a new and thus a new The modified Equation (15') is given below:

(15') This procedure is repeated until the designer is satisfied with the range of steps are outlined as follows:

values. The

1. Perform an elastic lateral analysis and calculate 2. 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. If desired, inelastic analysis to more accurately estimate may be used as follows. 5. Estimate (denoted as for each column as where 6. Calculate for each column using in Equations (18) and (19). Revise the moment of inertia of each column of the unbraced frame to be 7. Perform a standard lateral analysis and calculate 8. Repeat steps 2 through 4.

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

Calculated In-Plane Column Mark 1 2 3 4 5

1st-Run 2nd-Run 3rd-Run 4th-Run 5th-Run Elastic Stiff. Red Stiff. Red Stiff. Red Stiff. Red 0.594 0.597 0.584 0.587 0.598 0.594 0.597 0.584 0.598 0.587 0.604 0.668 0.624 0.610 0.601 0.584 0.584

0.587 0.587

0.594 0.594

0.597 0.597

0.598 0.598

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

Figure 4 - Example 2 TABLE 4

Column Mark 1 2 3 4 5

TABLE 5

1st-Run Elastic 0.320 0.215 0.468 0.214 0.317

Calculated In-Plane 5th-Run 2nd-Run 3rd-Run 4th-Run Stiff. Red Stiff. Red Stiff. Red Stiff. Red 0.321 0.318 0.317 0.326 0.236 0.311 0.314 0.315 Pinned Pinned Pinned Buckled 0.311 0.314 0.315 0.236 0.321 0.318 0.317 0.324

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Column Mark 1 2 3 4 5

0.357 0.282 Pinned 0.282 0.355

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 consider the column as pinned (Figure 5) and calculate the for the remaining columns on that basis. If this were done on the preceding problem (Figure 5), the results from the elastic analysis would be as shown in Table 5. A value of 0.360 for for all of the columns is close 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 shown in Table 6. The total story capacity decreases by 11% from the elastic analysis to the converged inelastic analysis. For accuracy, these results are based on an eigenvalue buckling analysis. This example illustrates why the value of must be based on the column with the highest value of unless an inelastic analysis is made. For the example in Figure 6, a value 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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TABLE 6

Column Mark A B Total

Elastic Analysis (kip) 589 944 1533

0.799 0.499

Inelastic Analysis (kip) 0.689 683 683 1366

0.684

LATTICE AND BUILT-UP SYSTEMS

The determination of for non-prismatic compression members or systems is difficult because the LRFD column formulae (Equations (8), (9) and (10)) assume a constant cross section. The systems shown in Figure 7 cannot be evaluated in the same manner as a simple rolled wide-flange column. However, Equations (17), (18) and (19) can be used with eigenvalue buckling or other numerical analysis procedures to calculate for a member or system.

Figure 7

Consider the tapered member shown in Figure 8. The full elastic moment of inertia is shown in 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 analysis program 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 for the member or system can be used with Equations (17), (18) and (19). A more accurate value for 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 value for the system by analysis. When the estimated value for and the resulting buckling values are equal, they represent a value of that can be used in design. Figure 9c shows the reduced 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 fundamental behavior of a system being designed clear to the structural steel designer. These methods permit the determination of the compressive strength, of an unbraced frame or lattice/builtup system using standard analysis tools. The calculation of for an unbraced frame focuses on story stability. The stability of a story is a function of the lateral stiffness of the story in the presence of destabilizing forces from leaning columns and out-of-straightness. The lateral stability of a story is also a function of the softening of the flexural stiffness of the columns that participate in the braced frame from and tangent modulus effects. The calculation methodology suggested here addresses these issues and is easy to use.

The evaluation of lattice/built-up systems also directly addresses the issues of tangent modulus and 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 the building code, the more likely it is that a safe and efficient design will be produced. The methods presented in this paper are intended to help the designer understand the stability of unbraced frames and lattice/built-up systems and produce designs that are consistent with the AISC-LRFD specification. REFERENCES 1.

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

2.

ASCE (1997) Structural Engineering Institute Technical Committee on Load and Resistance Factor Design, "Effective Length and Notional Load Approaches for Assessing Frame Stability: Implications for American Steel Design." Committee Report, 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 Multistory Buildings, Structural Stability Research Council, Proceedings 1981.

6.

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

7.

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

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Fig. 1

Fig. 6

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Fig. 3 - EXAMPLE

Fig.

4

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EXAMPLE

2

Fig. 5

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EXAMPLE

3

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1

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