Exact two-dimensional solutions for in-plane natural frequencies of laminated circular arches

Exact analysis on the in-plane free vibration of simply supported laminated circular arches is carried out based on the two-dimensional theory of elasticity. The method of separation of variables is employed to expand all the variables into Fourier series about the longitudinal coordinate, so that the system of partial differential equations is reduced to the ordinary one about the radial coordinate. The state space method is used to derive a series of simultaneous first-order differential equations. Due to the variable coefficients posed by the radial coordinate, analytical solutions are rather unpractical, and hence the approximate laminate model is adopted to translate the state equation into the one with constant coefficients. The relation between the state vector at the inner and outer surface of the arch is finally obtained according to the continuity conditions at interfaces and the traction free conditions at the two lateral surfaces. The formulation is validated by comparing the present results to those available in literature. Effects of geometric parameters and stacking sequences on the natural frequencies of circular arches are investigated and discussed. Numerical results presented in this paper provide benchmarks for future analysis of circular arches.