Page 172 - Aircraft Stuctures for Engineering Student

156 Structural instability
I
A ~ P=Pc, (bifurcation point)

0
Lateral deflection at mid-height
Fig. 6.4 Behaviour of a perfect pin-ended column.

theoretically possible for the column to take one of three deflection paths. Thus, if the
column remains undisturbed the deflection at mid-height would continue to be zero
but unstable (Le. the trivial solution of Eq. (6.3), u = 0) or, if disturbed, the
column would buckle in either of two lateral directions; the point at which this
possible branching occurs is called a bifurcation point; further bifurcation points
occur at the higher values of PcR(4~2EI/12, 9.ir2EI/12,. . .).

We have shown that the critical stress, Eq. (6.8), depends only on the elastic modulus
of the material of the column and the slenderness ratio l/r. For a given material the
critical stress increases as the slenderness ratio decreases; i.e. as the column becomes
shorter and thicker. A point is then reached when the critical stress is greater than the
yield stress of the material so that Eq. (6.8) is no longer applicable. For mild steel
this point occurs at a slenderness ratio of approximately 100, as shown in Fig. 6.5.

t
900 -

c
N
E
600-
z
v
a
b"
300 -
Yield stress -- --- -

I
0' I I I *
100 200 300
(l/d
Fig. 6.5 Critical stress-slenderness ratio for a column.