CIVIL ENGINEERING FORMULAS

COLUMN FORMULAS 95

whereEmodulus of elasticity of the column material, psi (Mpa)
Acolumn cross-sectional area, in^2 (mm^2 )
rradius of gyration of the column, in (mm)

Figure 3.8 shows some ideal end conditions for slender columns and corre-
sponding critical buckling loads. Elastic critical buckling loads may be obtained
for all cases by substituting an effective length KLfor the length Lof the
pinned column, giving

(3.26)

In some cases of columns with open sections, such as a cruciform section,
the controlling buckling mode may be one of twisting instead of lateral
deformation. If the warping rigidity of the section is negligible, torsional buck-
lingin a pin-ended column occurs at an axial load of

P (3.27)

GJA

Ip

P

2 EA

(KL/r)^2

Type of column Effective length Critical buckling load

L L

π^2 EI

L^2

4π^2 EI

L^2

2π^2 EI

L^2

π^2 EI

4 L^2

L

2

~0.7L

2 L

L/4

L/2

L/4

0.7

L

0.3

L

L

~ ~~

FIGURE 3.8 Buckling formulas for columns.