or b

where M1 and M2 have the same definitions as noted earlier in the formula for Cb.

If the unbraced compression flange length ly exceeds the above limits or the section is noncompact,

Figure 7.3. (Continued )

For pipe sections the allowable bending stress in all directions is taken as

provided the section is compact, otherwise Allowable Shear Stresses

The allowable shear stress Fu is taken as 0.40Fy (AISC F4-1). For very slender webs, where h/tw > 380/^JFy, a reduction in the allowable shear stress applies and must be seperately investigated (AISC F-4).

7.1.3. Members Subject to Compression Buckling of Columns

In structural design, a column is considered slender if its cross-sectional dimensions are small compared to its length. The degree of slenderness is measured in terms of the ratio l/r, where l is the unsupported length of the column and r is the radius of gyration. Whereas a stocky column fails by crushing or yielding, a slender column does so by buckling.

Before the AISC design equations are examined, a review of column behavior is useful for understanding the design parameters. Since the derivations of the column buckling formulas may be found in strength-of-material textbooks, the emphasis here is only on the column behavior as related to design.

Euler enunciated more than 200 years ago that a straight concentrically loaded pin-ended slender column fails by buckling at a critical load p 2 EI

where E, I, and l are the familiar notations for Young's modulus, moment of inertia, and the unsupported length of the column. Dividing Pc by the cross-sectional area A of the column, the expression for the critical load may be written in terms of the critical average stress fc on the gross section of the column

Substituting I _ Ar2, where r is the radius of gyration, gives the critical stress equation _ P 2 E

A plot of the critical stress versus the slenderness ratio, called a column curve, is shown in Fig. 7.4, illustrating the reduction in column strength as the slenderness increases. Stocky columns do not fail by buckling but do so by yielding or crushing of the material. There is a limiting slenderness ratio below which failure occurs by crushing, while for larger values, the mode of failure is by buckling.

Figure 7.4. Euler stress. PJA versus Kl/r.

The expression for buckling load Pc is for an idealized column supported by fric-tionless supports, a condition that exists rarely in practice. Building columns are connected to beams which restrain column rotation, thereby inducing end moments. Aditionally, columns experience lateral deflections. Therefore, to determine the critical loads for practical cases, the idea of an effective length of column is used in design. The effective length is expressed as a product of actual length times a factor K, called the effective length factor. The critical load for practical cases is given by the relation p 2 EI Column Curves

To understand the performance of compression members, consider again the curves in Fig. 7.4, which show the failure stress versus the slenderness ratio Kl/r for three grades of steel. Three things are clear from the figure: the yield strength of steel is very significant for short columns, of decreasing significance through the intermediate range, and of no consequence in the performance of long columns. The most efficient use of the strength of steel is made by selecting columns in the intermediate range. To achieve large values of I and r for a given area A, a section that has the area distributed as far from its centroid as possible offers the best choice, other things being equal. The most efficient sections are those with rx/ry = 1. Of the available wide-flange sections, those with b/d ~ 1 are most efficient for columns.

Compression members are divided into two classes by their values of Kl/r, with the value of Cc = 2E/Fy dividing the two classes. Short columns are defined by very low values of Kl/r. In this range, the Euler curve for critical load is approaching infinity. However, when the axial load becomes sufficient to cause yield stress, failure occurs by compression yielding although collapse is unlikely.

Failure for intermediate length columns is initiated by the tendency for buckling instability. The failure curve shows a smooth transition between the yield and the buckling conditions. The two curves become tangent at a value of Kl/r = Cc, somewhat arbitrarily chosen in the AISC specifications as Cc = 2p2 E/Fy.

The allowable axial compressive stress value Fa for compact or noncompact sections is evaluated as follows:

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