#### Vol. 4, No. 10, 2009

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A new analytic symplectic elasticity $\!$approach for beams resting on Pasternak elastic foundations

### C. F. Lü, C. W. Lim and W. A. Yao

Vol. 4 (2009), No. 10, 1741–1754
##### Abstract

Analytic solutions describing the stresses and displacements of beams on a Pasternak elastic foundation are presented using a symplectic method based on classical two-dimensional elasticity theory. Hamilton’s principle with a Legendre transformation is employed to derive the Hamiltonian dual equation, and separation of variables reduces the dual equation to an eigenequation that differs from the conventional eigenvalue problems involved in vibration and buckling analysis. Using adjoint symplectic orthonormality, a group of eigensolutions of zero eigenvalue, corresponding to the Saint-Venant problem, are derived. This approach differs from the traditional semi-inverse analysis, which requires stress or deformation trial functions in the Lagrangian system. The final solutions, which account for the effects of an elastic foundation and applied lateral loads, are approximated by an eigenfunction expansion. Comparisons with existing numerical solutions are conducted to validate the efficiency of this new approach.

##### Keywords
Saint-Venant problem, elastic foundation, symplectic, Hamilton principle, Legendre transformation