Analytical Investigation for the Displacement of Beams with Flexible Supports

The closed form (analytical) solution for the displacement of a beam with semi-rigid supports under dynamic pulse loading has been developed. Essential (Dirichlet) boundary conditions are prescribed and the equation of motion and subsequent mixed (Robin) boundary conditions are derived using Hamilton’s principle (principle of least action). Using the exact assumed modes for various semi-rigid supports, the temporal displacements (generalised coordinates) are obtained. The displacement field is derived as a series solution with each term being the product of a generalised coordinate and an exact shape function. The derived exact shape functions, which depend upon a set of dimensionless parameters, are obtained through an eigenvalue analysis and define the associated eigenfunctions of the generalised coordinates. A table is presented to aid easy formulation of exact modes for varies beams using an intrinsic non-dimensional parameter, α. Using Galerkin’s weighted residual the equation of motion is transformed from a partial differential equation to an ordinary differential equation for easy calculations.