The principal factors governing the proportioning of members carrying compressive forces are overall column buckling, local buckling, and gross section area. The effect of overall column buckling depends on the slenderness ratio Kl/ r, where Kl is the effective length, in, of the column, l is the unbraced length, and r is the least radius of gyration, in, of the cross section. The effect of local buckling depends on the width-thickness ratios of the individual elements of the member cross section.

W shapes with depths of 8, 10, 12, and 14 in are most commonly used for building columns and other compression members. For unbraced compression members, the most efficient shape is one where the value of ry with respect to the minor axis approaches the value of rx with respect to the major axis.

When built-up sections are used as compression members, the element joining the principal load-carrying elements, such as lacing bars, should have a shear capacity of at least 2% of the axial load.

## Effective Column Length

Proper application of the column capacity formulas for ASD or LRFD depends on judicious selection of K. This term is defined as the ratio of effective column length to actual unbraced length.

For a pin-ended column with translation of the ends prevented, K 1. But in general, K may be greater or less than unity. For example, consider the columns in the frame in Fig. 7.26. They are dependent entirely on their own stiffness for stability against sidesway. If enough axial load is applied to them, their effective length will exceed their actual length. But if the frame were braced to prevent sidesway, the effective length would be less than the actual length because of the resistance to end rotation provided by the girder.

Theoretical values of K for six idealized conditions in which joint rotation and translation are either fully realized or nonexistent are given in Fig. 7.27.

Also noted are values recommended by the Column Research Council for use in design when these conditions are approximated. Since joint fixity is seldom fully achieved, slightly higher design values than theoretical are given for fixed-end columns.

Specifications do not provide criteria for sidesway resistance under vertical loading, because it is impossible to evaluate accurately the contribution to stiffness of the various components of a building. Instead, specifications cite the general conditions that have proven to be adequate.

Constructions that inhibit sidesway in building frames include substantial masonry walls, interior shear walls; braced towers and shafts; floors and roofs providing diaphragm action that is, stiff enough to brace the columns to shear walls or bracing systems; frames designed primarily to resist large side loadings or to limit horizontal deflection; and diagonal X bracing in the planes of the frames. Compression members in trusses are considered to be restrained against translation at connections. Generally, for all these constructions, K may be taken as unity, but a value less than one is permitted if proven by analysis.

When resistance to sidesway depends solely on the stiffness of the frames; for example, in tier buildings with light curtain walls or with wide column spacing, and with no diagonal bracing systems or shear walls, the designer may use any of several proposed rational methods for determining K. A quick estimate, however, can be made by using the alignment chart in an AISC Manual of Steel Construction.

The effective length Kl of compression members, in such cases, should not be less than the actual unbraced length.

## ASD of Compression Members

The allowable compressive stress on the gross section of axially loaded members is given by formulas determined by the effective slenderness ratios Kl/ r of the members. A critical value, designated Cc, occurs at the slenderness ratio corresponding to the maximum stress for elastic buckling failure (Table 7.12). This is illustrated in Fig. 7.28. An important fact to note: when Kl/ r exceeds Cc = 126.1, the allowable compressive stress is the same for A36 and all higher-strength steels.

unimportance of these members and the greater restraint likely at their end connections.

The full unbraced length should always be used for l.

Tables giving allowable stresses for the entire range of Kl/ r appear in the AISC ASD Manual of Steel Construction. Approximate values may be obtained from Fig. 7.28. Allowable stresses are based on certain minimum sizes of structural members and their elements that make possible full development of strength before premature buckling occurs. The higher the allowable stresses the more stringent must be the dimensional restrictions to preclude buckling or excessive deflections.

The AISC ASD specification for structural steel buildings limits the effective slenderness ratio Kl/ r to 200 for columns, struts, and truss members, where K is the ratio of effective length to actual unbraced length l, and r is the least radius of gyration.

A practical rule also establishes limiting slenderness ratios l / r for tension members:

For main members 240

For bracing and secondary members 300

But this does not apply to rods or other tension members that are drawn up tight (prestressed) during erection. The purpose of the rule is to avoid objectionable slapping or vibration in long, slender members.

The AISC ASD specification also specifies several restricting ratios for compression members. One set applies to projecting elements subjected to axial compression or compression due to bending. Another set applies to compression elements supported along two edges.

Figure 7.29 lists maximum width-thickness ratios, b/ t, for commonly used elements and grades of steel. Tests show that when b/ t of elements normal to the direction of compressive stress does not exceed these limits, the member may be stressed close to the yield stress without failure by local buckling. Because the allowable stress increases with Fy, the specified yield stress of the steel, widththickness ratios are less for higher-strength steels.

These b/ t ratios should not be confused with the width-thickness ratios described in Art. 7.20. There, more restrictive conditions are set in defining compact sections qualified for higher allowable stresses.

## LRFD of Compression Members