SUPERCRITICAL BEHAVIOR OF BIMODAL OPTIMAL RODS

The Lagrange problem [ 1 ] for an elastic rod of fixed length and volume subject to a compressive axial force is to find the optimal shape of the rod such that the buckling force is maximal. This problem was solved in [2-10] with various boundary conditions. It turns out that in certain cases, for instance, for rigidly clamped ends, optimal rods have two linearly independent buckling modes. In such cases, the supercritical behavior of optimal rods is of particular interest. Some authors [11-14] believe that the existence of several buckling modes, which is characteristic of optimal structures, leads to the structural instability and substantial sensitivity to defects. The supercritical behavior of the pin-ended rods with a single buckling mode was investigated in [4,15]. In the present paper, the Lagrange problem is considered in the case of elastic clamping of the ends of the rod. Using the analytical relations obtained previously, we calculate the bimodal optimal solutions, i.e., the solutions with two linearly independent buckling modes, for various coefficients of clamping rigidity. The supercritical behavior of optimal rods is investigated using the perturbation technique. It is shown that this behavior is determined by four pairs of solutions which branch off from the trivial equilibrium solution for the force landa > landa(). The stability of these states of equilibrium is analyzed near the bifurcation point landa(). It is found that only two pairs of solutions are stable, which correspond to the symmetric and antisymmetric buckling modes. This conclusion is valid for any coefficient of clamping rigidity c > 0.