Free vibration of a circularly curved Timoshenko beam normal to its initial plane using finite curved beam elements

Abstract

This paper presented a simple technique to derive the stiffness and mass matrices of a horizontally circular curved beam element so that the possibility of practical applications of the presented property matrices should be greater than that of the existing ones. Where the element stiffness matrix was derived from the force–displacement equations and the element consistent mass matrix was derived from the kinetic energy equations. The effect of rotary inertias due to both bending and torsional vibrations and the effect of shear deformation were considered in the formulation. Since all element property matrices of the curved beam element were derived based on the local polar coordinate system (rather than the local Cartesian one), their coefficients were invariant for any curved beam element with constant radius of curvature and subtended angle. For this reason, one did not need to transform the property matrices of each curved beam element from the local coordinate system to the global one to achieve the overall property matrices of the entire curved beam before they were assembled. However, the simple transformation matrices between the local polar coordinate system and the global Cartesian coordinate system were also presented so that the element property matrices of the curved beam elements can be assembled with those of the conventional straight beam elements (or the other curved beam elements with different radii of curvatures and/or subtended angles) to achieve the overall property matrices of the entire “hybrid” structural system. The good accuracy of the presented approach has been verified by both the existing analytical (exact) solutions for the continuum curved beam and the numerical (approximate) solutions for the discretized curved beam composed of the conventional straight beam elements using either the consistent mass model or the lumped mass model.