3.1 Show that a simply supported column is in a stable equilibrium at the bifurcation point of P _E = ? ² EI/ l ². Assuming the length of the column is inextensible, show that the column is in unstable equilibrium.

3.2 Derive the relationship between the lateral deflection ? and the axial force P for the pin-connected rigid bars supported by a spring as shown in Fig. 3.17.

Figure 3.17

3.3 A column is supported by two translation and two rotation springs as shown in Fig. 3.18. The boundary conditions for the column are

Figure 3.18

Derive the equation of equilibrium for the column from the condition that the potential energy is stationary;

3.4 Discuss about stability of the von Mises truss shown in Fig. 3.19. The two bars are pin-connected and have only tensile rigidity EA.

Figure 3.19

3.5 Show that the maximum deflection at mid-span of a simply supported beam-column with a concentrated load at mid-span is given by the second equation of (3.158).

[ Solution for Prob. 3.1]

Substitute w = a sin ? x/ l and P = P _E = ? ² EI/ I ² into Eq. (3.101) and get

(a)

Thus, the column is stable at the bifurcation point. If we assume the condition of inextensible column length, we have ds = dx or

rearranging,

The external work done...