An alternative formulation of the boundary value problem for the Timoshenko beam and Mindlin plate

The traditional formulations of the boundary value problems for the Timoshenko beam and Mindlin plate theories do not allow the bending and shearing deflections to be determined uniquely. An alternative formulation is proposed in which the bending deflection is regarded as a fundamental variable in place of the angle of rotation due to bending. Using the total deflection as an accompanying variable, the governing equations and boundary conditions can be derived on the basis of Hamiltonʹs principle. This formulation is shown to afford unique results for the bending and shearing deflections, with natural frequencies equal to or higher than those determined by traditional methods for certain boundary conditions. The proposed formulation represents a deductive approach to determining the total deflection, providing consistency for both dynamic and static analyses.