Page 214 - Aircraft Stuctures for Engineering Student

198 Structural instability

P.6.2 A pin-ended column of length 1 and constant flexural stiffness EZ is
reinforced to give a flexural stiffness 4EZ over its central half (see Fig. P.6.2).
Considering symmetric modes of buckling only, obtain the equation whose roots
yield the flexural buckling loads and solve for the lowest buckling load.

Ans. tanp1/8 = l/d, P = 24.2EZ/12.

€I 4EI EI
P - 3 P

Fig. P.6.2

P.6.3 A uniform column of length 1 and bending stiffness EZ is built-in at one end
and free at the other and has been designed so that its lowest flexural buckling load
is P (see Fig. P.6.3).
Subsequently it has to carry an increased load, and for this it is provided with a
lateral spring at the free end. Determine the necessary spring stiffness k so that the
buckling load becomes 4P.

Am. k = 4Pp/(pl- tan pl).

Fig. P.6.3

P.6.4 A uniform, pin-ended column of length I and bending stiffness EZ has an
initial curvature such that the lateral displacement at any point between the
column and the straight line joining its ends is given by

42
vo=a-(l-z) (see Fig. P.6.4)
P
Show that the maximum bending moment due to a compressive end load P is
given by

Mmax = -- 8aP (swy - 1)
(W2
where

X2 = PIE1