In-plane free vibration analysis of circular arches with varying cross-sections using differential quadrature method

Abstract

A differential quadrature (DQ) methodology recently developed by the authors is used to obtain a general and a computationally efficient and accurate DQ solution for free vibration of variable cross-section circular thin arches. As an improvement to the classical theory and in order to evaluate the higher order natural frequencies more accurately, the commonly used hypothesis of “the inextensibility of the central axis” is removed. This enables one to study the effects of slenderness ratio on the natural frequencies, especially at higher order modes. Rotary inertia is included in the formulation and its influence on natural frequencies is studied. Arches with different types of boundary conditions, including those with elastic constraint against rotation at their ends, are considered. For the cases where a change in the cross-sectional or material properties of the arch occurs, a numerical domain decomposition technique in conjunction with DQ methodology is developed and incorporated. To verify the accuracy of the methodology, the results are compared with those of exact solutions and/or other approaches such as finite elements, Rayleigh–Ritz, Galerkin, cell discretization methods, and other DQ methodologies. In particular, excellent solution agreements are achieved with those of exact solutions, the generalized differential quadrature rule and the optimized Rayleigh–Ritz method solutions.