A linear model for the static and dynamic analysis of non-homogeneous curved beams

A simple one-dimensional mechanical model is presented to analyse the static and dynamic feature of non-homogeneous curved beams and closed rings. Each cross-section is assumed to be symmetrical and the “resultant loads” are acted in the plane of symmetry. The internal forces in a cross-section are replaced by an equivalent force-couple system at the origin of the cylindrical coordinate system used. The equations of motion and the boundary conditions are expressed in terms of two kinematical variables. The first kinematical variable is the radial displacement of cross-sections and the second one is the rotation of the cross-sections. Each of them depends on the time and the polar angle. Assumed form of the displacement field assures the fulfillment of the classical Bernoulli-Euler beam theory. Rotary inertia is included in the equations of motion. Natural frequencies for simply supported laminated composite curved beams and non-homogeneous circular rings are obtained by exact solutions. The application of the model presented is illustrated by examples.