Stability of bars with variable rigidity compressed by a distributed force

A variable cross-section bar is considered. The bar is not uniform in length. The bar is compressed by a variable longitudinal force distributed along its axis. The stability loss in the straightline shape of the bar’s equilibrium is discussed when a curved shape is also possible. The critical combination between rigidity and the longitudinal force is a result of using an integral representation for the solution to the original stability equation with variable coefficients with the aid of the solution to a similar equation with constant coefficients. The integral representation contains the Green function of the original equation. This function is determined by the method of perturbations. The numerical results obtained by the derived formulas are compared with the known exact solutions to the stability equations for various particular cases.     

Stability of a straight bar of variable rigidity

We consider the longitudinal compression of a straight bar whose rigidity is a periodic integrable function of the longitudinal coordinate. For a hinged bar with one clamped end, we obtain approximate analytic formulas that permit obtaining the critical compressing loads under which an adjacent, curved form of equilibrium is possible. In the case of a bar of stepwise varying rigidity that consists of a single period (the limit case), we compare the results obtained by our formulas with the already known exact solutions of the stability equation. A good agreement between the approximate and exact results is shown.     

On the torsional rigidity for bars with L- and +-cross-section

This paper is a continuation of the senior author’s previous papers^[1-3]. using the harmonic continuation technique, the torsional rigidity for bars with L- and +-cross-section can be easily found.Numerical results are shown in Tables 1-3 respectively.     

Bounds for the torsional rigidity of inhomogeneous cylindrical bars

The object of this paper is the uniform torsion of inhomogeneous, isotropic, linearly elastic cylindrical bar. The aim is to give lower and upper bounds for the torsional rigidity of the bar with doubly connected cross section. The outer and inner boundary curves of cross section are similar curves. The level lines of the function which gives the change of the shear rigidity on the cross section are also similar curves to the boundary curves. The application of derived bounding formulae is illustrated by examples. An approximated formula to determine the shear stresses is also presented.     

含有无穷大弯曲刚度段变截面刚架的求解

采用扩大平衡隔离体范围的方法, 直接由平衡方程求得位移法典型方程系数, 解决了难以写出转角位移方程等困难, 方便地求解了含有无穷大弯曲刚度段的变截面刚架.     

Stability of bars made of linear elastic materials with voids

Summary Dynamic stability of an elastic bar with voids is considered. Using the Lyapunov approach some new sufficient stability conditions are obtained and explicit expressions for the critical load are derived.     

Series solution for large deflections of a cantilever beam with variable flexural rigidity

In this paper large deflection and rotation of a nonlinear Bernoulli-Euler beam with variable flexural rigidity and subjected to a static co-planar follower loading is studied. It is assumed that the angle of inclination of the force with respect to the deformed axis of the beam remains unchanged during deformation. The governing equation of this problem is solved analytically for the first time using a new kind of analytical technique for nonlinear problems, namely the Homotopy Analysis Method (HAM). The present solution can be used for the analysis of a wide range of loads, material/cross section properties and lengths for beams undergoing large deformations. The results obtained from HAM are compared with results reported in previous works. Finally, the load–displacement characteristics of a uniform cantilever beam with different material properties under a follower force applied normal to the deformed beam axis are presented.     

Stability of motion of a particle with variable constraints

A mechanical system represented by a particle moving over an arbitrary surface that rotates about a vertical axis with constant angular speed is considered. Sufficient conditions for the parametric asymptotic stability of this system are established