Stiffness-Matrix Formulation of Beams with Shear Effect on Elastic Foundation

Abstract

A stiffness matrix for a beam element with shear effect on an elastic foundation is developed using the differential-equation approach for plane-frame analysis. Small-displacement theory and linear-elastic material are assumed. The contribution of the shear stresses to the deflection of the beam in addition to the bending stresses is considered. Shear effect is represented by a second-order term of the derivative of beam deflection in the differential equation. This fact is also taken into account in the boundary conditions of the beam element. The complex variables are used in the solution of the differential equation. Stiffness and loading matrices for a beam element resting on Winkler foundation are derived by means of a computer program coded in FORTRAN language. Using this element, the effects of the shear forces on deflections and internal forces in the beams on an elastic foundation are investigated. The results of this finite-element analysis are exact for the Winkler assumption. It is shown that the deflections due to shear effect are influenced particularly with the modulus of the foundation and for the beams that have small span to height ratio. Also, numerical examples are given to demonstrate this fact.