Slenderness Effects on Concrete Columns

The ACI 318 Building Code requires that primary column moments be magnified to provide safety against buckling failure. Detailed procedures, formulas, and design aids are provided in the Code and Commentary.

For most unbraced frames, an investigation will be required to determine the magnification factor to allow for the effects of sidesway and end rotation. The procedure for determination of the required increase in primary moments, after the determination that slenderness effects cannot be neglected, is complex. For direct solution, the requirements of Sec. 10.10, ACI 318-99 can be met by a P- analysis.
(See, for example, J. G. MacGregor and S. E. Hage, ‘‘Stability Analysis and Design of Concrete,’’ Journal of the Structural Division, ASCE, Vol. 103, No. ST10, October 1977.)
The direct P- method of MacGregor and Hage is based upon an equation for a geometric series that was derived for the final second-order deflection as a function of the first-order elastic deflection. This direct P- analysis provides a very simple method for computing the moment magnifier S when the stability index Q is greater than 0.04 but equal to or less than 0.22.

The approximate method of ACI 318-99 may also be used to determine the moment magnifier. This approximate method is a column-by-column correction based upon the stiffness of the column and beams, applied primary design column end moments, and consideration of whether the entire structure is laterally braced against sidesway by definition. (See ACI 318-99, Sec. 10.11).
The ACI 318 Building Code permits slenderness effects to be neglected only for very short, braced columns, with the following limitations for columns with square or rectangular cross-sections:

end of a column. As these ratios become less, the limiting heights can be increased.
When the total stiffnesses of the columns and the floor systems are equal at each end of the column (a common assumption in routine frame analysis), the two ratios  1.00, and the limiting heights increase about 30%. With this increase, the slenderness effects can be neglected for most columns in frames braced against sidesway.
A frame is considered braced when other structural elements, such as walls, provide stiffness resisting sidesway at least 6 times the sum of the column stiffnesses resisting sidesway in the same direction in the story being considered.