Symplectic Eigenvalue Analysis Method for Bending of Beams Resting on Two-Parameter Elastic Foundations

Abstract

This paper develops a new symplectic eigenvalue analysis (SEA) method for deriving the symplectic analytical solutions for the bending of Timoshenko beams resting on two-parameter elastic foundations. The SEA method accommodates various beam–foundation models with arbitrary external loads and arbitrary boundary conditions. A generalized Hamiltonian functional is first derived by introducing the Lagrange multipliers into the total potential energy functional, and then the first-order dual equation and its corresponding boundary conditions are obtained. Once the variables are separated, the beam bending is transformed into an eigenvalue problem, and the corresponding eigenvalues and eigenvectors are used to derive the symplectic solutions. In addition, a new hybrid method, which incorporates the singular functions and the variable separation method, is proposed to obtain the particular solutions with respect to arbitrary external loads. The systematic SEA method provides a new general analytical framework for studying beam–foundation interaction problems, and the comprehensive analyses on the symplectic eigenvalues help to deeply understand the characteristics of the symplectic solutions. Three examples are presented to assess the applicability of the SEA method and to validate the accuracy of the symplectic solutions. The differences and similarities between the Pasternak elastic foundation and the generalized elastic foundation are quantitatively and qualitatively investigated by the SEA method.