Structural Concrete - Hassoun

12.2 Effective Column Length (Kl u ) 421
Figure 12.1 Rectangular and circular sections of columns, with radius of gyration r.
not be significant. The slenderness ratio is the ratio of the column height, l, to the radius of gyration,
r,wherer = √ I∕A, I being the moment of inertia of the section and A the sectional area.
For a rectangular section of width b and depth h (Fig. 12.1), I x = bh 3 /12 and A = bh. Therefore,
r x = √ I∕A= 0.288h (or, approximately, r x = 0.3h). Similarly, I y = hb 3 /12 and r y = 0.288b (or,
approximately, 0.3b). For a circular column with diameter D, I x = I y = πD 2 /64 and A = πD 2 /4; therefore,
r x = r y = 0.25D.
In general, columns may be considered as follows:
1. Long with a relatively high slenderness ratio, where lateral bracing or shear walls are required.
2. Long with a medium slenderness ratio that causes a reduction in the column strength. Lateral
bracing may not be required, but strength reduction must be considered.
3. Short where the slenderness ratio is relatively small, causing a slight reduction in strength.
This reduction may be neglected, as discussed in previous chapters.
12.2 EFFECTIVE COLUMN LENGTH (KL U )
The slenderness ratio l/r can be calculated accurately when the effective length of the column (Kl u )
is used. This effective length is a function of two main factors:
1. The unsupported length, l u , represents the unsupported height of the column between two
floors. It is measured as the clear distance between slabs, beams, or any structural member
providing lateral support to the column. In a flat slab system with column capitals, the unsupported
height of the column is measured from the top of the lower floor slab to the bottom
of the column capital. If the column is supported with a deeper beam in one direction than in
the other direction, l u should be calculated in both directions (about the x and y axes) of the
column section. The critical (greater) value must be considered in the design.
2. The effective length factor, K, represents the ratio of the distance between points of zero
moment in the column and the unsupported height of the column in one direction. For
example, if the unsupported length of a column hinged at both ends, on which sidesway is
prevented, is l u , the points of zero moment will be at the top and bottom of the column—that