G. Venkateswara Rao

Concept of Coupled Displacement Field for Large Amplitude Free Vibrations of Shear Flexible Beams

Continuum solutions for solving the large amplitude free vibration problem of shear flexible beams using the energy method involves assuming suitable admissible functions for the lateral displacement and the total rotation. Use of even, single-term admissible functions leads to two coupled nonlinear temporal differential equations in terms of the lateral displacement and the total rotation, the solution of which is rather involved. This situation can be effectively tackled if one uses the concept of a coupled displacement field wherein the fields for lateral displacement and the total rotation are coupled through the static equilibrium equation of the shear flexible beam. This approach leads to only one undetermined coefficient, in the case of single-term admissible functions, which can easily be used in the principle of conservation of total energy, neglecting damping, to solve the problem. Finally, one gets a nonlinear ordinary differential equation of the Duffing type which can be solved using any available standard method. The effectiveness of the concept discussed above is brought out through the solution of the large amplitude free vibrations, in terms of the fundamental frequency, of uniform shear flexible beams, with axially immovable ends, using single-term admissible functions.

Simple Formula to Study the Large Amplitude Free Vibrations of Beams and Plates

A simple formula to study the large amplitude free vibration behavior of structural members, such as beams and plates, is developed. The nonlinearity considered is of von Karman type, and after eliminating the space variable(s), the corresponding temporal equation is a homogeneous Duffing equation. The simple formula uses the tension(s) developed in the structural members due to large deflections along with the corresponding buckling load obtained when the structural members are subjected to the end axial or edge compressive load(s) and are equal in magnitude of the tension(s). The ratios of the nonlinear to the linear radian frequencies for beams and the nonlinear to linear time periods for plates are obtained as a junction of the maximum amplitude ratio. The numerical results, for the first mode of free vibration obtained from the present simple formula compare very well to those available in the literature obtained by applying the standard analytical or numerical methods with relatively complex formulations.

Comparative study of thermal post-buckling analysis of columns using an equilibrium path method and an equivalent eigenvalue problem

Thermal post-buckling analysis of uniform slender and shear flexible columns with axially immovable ends is studied for various boundary conditions using a load-deflection curve (bifurcation diagram). Linear buckling load is computed as an eigenvalue problem and the location of initial perturbation in the non-linear analysis is chosen based on the linear eigenvector. Geometric non-linearity is considered using von-Karman strain displacement relations and load-displacement curves are established for the aforementioned boundary conditions. Elegant closed form solutions are obtained for the post-buckling load parameter to linear buckling load parameter as a function of amplitude ratio for the boundary conditions considered using least squares approximation. Numerical values obtained are in good agreement with available literature for the classical boundary conditions, and for the non-classical boundary conditions results are validated with the results obtained from the available FE formulation which is based on the equivalent eigenvalue problem. The procedure mentioned to evaluate the thermal post-buckling analysis is very general and it can be easily applied to any other structural configurations with anisotropic materials which exhibits a bifurcation buckling. The results obtained from the post-buckling analysis are used for evaluating the non-linear to linear frequency parameters of the classical boundary conditions and a brief discussion is presented on the numerical accuracy of the results obtained.