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

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.

Saint-Venant problem, elastic foundation, symplectic, Hamilton principle, Legendre transformation
Received: 15 January 2009
Revised: 22 June 2009
Accepted: 9 July 2009
Published: 27 February 2010
C. F. Lü
Department of Civil Engineering
Zhejiang University
Hangzhou 310058
C. W. Lim
Department of Building and Construction
City University of Hong Kong
Tat Chee Avenue
Hong Kong
W. A. Yao
State Key Laboratory of Structural Analysis for Industrial Equipment and Department of Engineering Mechanics
Dalian University of Technology
Dalian 116024