V.B. Glavardanov

Optimal shape of a twisted and compressed rod

Abstract We formulate and solve the problem of determining the shape of an elastic rod stable against buckling and having minimal volume. The rod is loaded by a concentrated force and a couple at its ends. The equilibrium equations are reduced to a single nonlinear second-order equation. The eigenvalues of the linearized version of this equation determine the stability boundary. By using Pontryagins maximum principle we determine the optimal shape of the rod.

Buckling of a twisted and compressed rod

We consider the problem of determining the stability boundary for an elastic rod under thrust and torsion. The constitutive equations of the rod are such that both shear of the cross-section and compressibility of the rod axis are considered. The stability boundary is determined from the bifurcation points of a single nonlinear second order differential equation that is obtained by using the first integrals of the equilibrium equations. The type of bifurcation is determined for parameter values. It is shown that the bifurcating branch is the branch with minimal energy. Finally, by using the first integral, the solution for one specific dependent variable is expressed in terms of elliptic integrals. The solution pertaining to the complete set of equilibrium equations is obtained by numerical integration.

Twisted axially loaded rod with shear and compressibility

SummaryStability of a twisted and axially compressed elastic rod is analysed using the Euler method of adjacent equilibrium configuration. The constitutive equations of the rod are assumed in the form that takes into account both shear of the cross-sections and compressibility of the rod axis. It is shown that bifurcation points of the non-linear system of equations describing equilibrium are determined from the linearized system of equations. The influence of shear and compressibility on the critical values of twisting couple and compressive force is obtained. Moreover, the bifurcation pattern is examined by using the Liapunov-Schmidt method.

VIBRATION OF CIRCULAR PLATE WITH AN INTERNAL ELASTIC RING SUPPORT UNDER EXTERIOR EDGE PRESSURE

In this paper, we analyze the transverse vibration of a circular plate loaded by uniform pressure along its edge. The plate is supported by an elastic ring support being coaxial with the plate. At its edge the plate is clamped but the radial displacement is allowed. Apart from this problem, the heated plate clamped at its edge, but without the possibility of radial displacement, is also analyzed. The analytical solution of governing equation is obtained in the form of Bessels functions. Using the analytical solution, the frequencies of transverse vibrations depending on loads, elastic ring stiffness and the location of ring are obtained. The results show that the lowest frequencies vibrations can be either symmetric or asymmetric having one or two nodal diameters. It is also shown that multiple vibration frequencies can occur for special values of load and ring stiffness.

Stability of a pipe through which a string is pulled

Abstract Stability of an elastic pipe through which a string is pulled with the constant velocity is studied by the Liapunov– Schmidt method. It is assumed that imperfections in shape (small initial deformation) and loading (distributed load along the axis of the pipe) are present. Stability boundary is obtained from the eigenvalues of the linearized equations. It is shown that the bifurcation is super-critical. The conditions guaranteeing that imperfections introduced here form a universal unfolding are stated. The post-critical shape of perfect pipe is determined by numerical integration of the corresponding system of equations. For this system of equations we also found two new first integrals. For the case of Bernoulli–Euler model of the pipe the post-critical shape is expressed in terms of elliptic integrals.