On the Finite Deflection Dynamics of Thin Elastic Beams

A dynamic theory of thin beams undergoing large deflection but small strain is derived. Geometric nonlinearities are preserved but the material is assumed to behave linearly. Contributions due to rotatory inertia, shear deformation, and axial stress resultants are included. The resulting equations are analyzed by characteristic techniques. Wave-propagation speeds, jump properties, and their physical significances are discussed. A simplifying assumption generates a modified Timoshenko beam equation which is valid for large deformation.

Method of Initial Functions in Two-Dimensional Elastodynamic Problems

The method of initial functions is formulated for the two-dimensional elastodynamic problems both in plane stress and plane strain. Knowing the stresses and displacements on a given layer of an elastic body in a plane state, the stresses and displacements at any other layer or point can be obtained by the method developed in this paper. The method is illustrated in the case of a deep beam subjected to time-dependent transverse loads. A particular solution for a straight-crested wave traveling along an infinite beam is obtained. Exact value of the dynamic shear coefficient for the well-known Timoshenko beam equation is derived. This method is also applied to the dynamic analysis of layered beams.