Exact transcendental stiffness matrices of general beam-columns embedded in elastic mediums

Abstract

Stiffness matrices of beams embedded in an elastic medium and subjected to axial forces are considered. Both the bending and the axial deformations have been incorporated. Two approaches for deriving the element stiffness matrix analytically have been proposed. The first approach is based on the direct force–displacement relationship, whereas the second approach exploits shape functions within the finite element framework. The displacement function within the beam is obtained from the solution of the governing differential equation with suitable boundary conditions. Both approaches result in identical expressions when the exact transcendental displacement functions are used. Exact closed-form expressions of the elements of the stiffness matrix have been derived for the bending and axial deformation. Depending on the nature of the axial force and stiffness of the elastic medium, seven different cases are proposed for the bending stiffness matrix. A unified approach to the non-dimensional representation of the stiffness matrix elements and system parameters that are consistent across all the cases has been developed. Through Taylor-series expansions of the stiffness matrix coefficients, it is shown that the classical stiffness matrices appear as an approximation when only the first few terms of the series are retained. Numerical results shown in the paper explicitly quantify the error in using the classical stiffness compared to the exact stiffness matrix derived in the paper. The expressions derived here gives the most comprehensive and consistent description of the stiffness coefficients, which can be directly used in the context of finite element analysis over a wide range of parameter values.