Question 5.1.6: Determine the critical compression force required to make a ......

The magnitude of the sought after critical force, T_{cr}, depends on the boundary conditions at the ends of the rod. Consider first a rod with hinged ends, as shown in Figure 5.5. In this case both ends of the rod are free to turn; hence, there is no bending moment there and, therefore, the radius of curvature is equal to zero. The unperturbed straight form of the rod becomes unstable when another equilibrium, with a small bending x(z), also exists.

Thus, if the rod is under a given compressioin force T , the equilibrium requirement for its portion between the upper end, z = 0, and z is as follows: the moment of the external force, which is equal to −Tx(z), must be balance by the bending moment there, i.e.,

−T x = \frac{EI}{R(z)} ≈ EI \frac{d^{2}x}{d^{2}z}

A non-trivial, x(z) ≠ 0, solution of this equation with the boundary conditions x(0) = x(L) = 0 does exist when (T/EI)^{1/2}L = π, which yields the critical compression force T_{cr} = π^{2}EI/L^{2}. If the cross section of the rod is not circular, its moment of inertia I depends on the plane of bending. Therefore, in this case the instability threshold is determined by bending in the plane with the minimum moment of inertia.

Consider now another case, when both ends of the rod are clamped. Then the boundary conditions require that x(z) and its derivative, dx/dz, are equal to zero at z = 0,L. However, the curvature of the rod differs from zero at the ends, and, therefore, the respective bending moments should be accounted for in deriving the equilibrium equation. They are not known beforehand and are established at such a value that the necessary boundary conditions are satisfied. Thus, by denoting the bending moment at z = 0 by M_{0}, one now gets the following equilibrium equation:

−T x +M_{0} = EI \frac{d^{2}x}{d^{2}z}

Its solution with x(0) = x^{′}(0) = 0 reads

x(z) = M_{0}T^{−1}\left\{1 − \cos[(T/EI)^{1/2}z]\right\},

and the boundary conditions at the other end, z = L, which are x(L) = x^{′}(L) = 0, are satisfied when (T/EI)^{1/2}L = 2π. Therefore, the critical force in this case is equal to T_{cr} = 4π^{2}EI/L^{2}, and, as before, the instability threshold is determined by bending in the plane with the minimum moment of inertia I.