An analytic model for the free in-plane vibration of beams of variable curvature and depth

In this study a flexible and powerful model is presented for the in-plane vibration of a beam whose curvature and depth are arbitrary, constrained only to such smoothness as can be accommodated by a quartic polynomial in the centerline arc length. The model includes the effects of shear deformation, rotary inertia, and centerline extensibility. The equations of motion are solved by an extension of the classic Galerkin method wherein the effect of boundary residuals is included. Timoshenko models are compared to other theoretical models and to experimental results; the comparisons are done for a variety of support schemes. A detailed study of a cantilever beam of variable curvature and depth is done and, finally, the utility and flexibility of the model is demonstrated for an arch whose shape is given only by a scale drawing, devoid of any other information except material properties.