Structural Instability - Structural and Stress Analysis - page 694

21.3 F AILURE OF C OLUMNS OF ANY L ENGTH

Empirical or semi-empirical methods are generally used to predict the failure of a
column of any length: these then form the basis for safe load or safe stress tables given
in Codes of Practice. One such method which gives good agreement with experiment
is that due to Rankine.

RANKINE THEORY

Suppose that P is the failure load of a column of a given material and of any length.
Suppose also that P S is the failure load in compression of a short column of the same
material and that P CR is the buckling load of a long slender column, again of the same
material. The Rankine theory proposes that

1
P =

1
P S +

1
P CR

(21.26)

Equation (21.26) is valid for a very short column since 1/ P CR → P S ;

the equation is also valid for a long slender column since 1/ P S is small compared with
1/ P CR ; thus P

0 and P then →

→ P CR . Therefore, Eq. (21.26) is seen to hold for extremes in column

length.

Now let σ S be the yield stress in compression of the material of the column and A its
cross-sectional area. Then

P S =

σ S A

Also from Eq. (21.23)

π 2 EI
L e
Substituting for P S and P CR in Eq. (21.26) we have

P CR =

1
P =

1
σ S A +

1
π 2 EI / L e

Thus

π 2 EI / L e + 1

P =

σ S A
σ S A π 2 EI / L e

so that

σ S A π 2 EI / L e
π 2 EI / L e + σ S A
Dividing top and bottom of the right-hand side of this equation by π 2 EI / L e we have

P =

σ S A

P =

1 + σ S AL e /π 2 EI