A Hybrid Membrane Element Based on the Hamilton Variational Principle

The traditional hybrid element is usually formulated based on the complementary energy principle or the Hellinger-Reissner variational principle. This paper presents a new strategy for developing hybrid elements by the Hamilton variational principle with dual mixed variables. A new rectangular hybrid element for plane elasticity is constructed. Firstly, the functional ∏ H of Hamilton variational principle is derived from the functional ∏ HR of Hellinger-Reissner variational principle. The variables in ∏HR are displacements (u, v) and stresses (σ x , σ y , τ xy ), while the variables in ∏ H are dual mixed variables (u, v) and (σ x , τ xy ) (stress σ y is eliminated). Secondly, the displacement fields are defined with the nodal displacement vector: $$ q = [u_1 v_1 u_2 v_2 u_3 v_3 u_4 v_4 ]^T , $$ and stress fields are defined by the stress parameter vector: $$ \beta = [\beta _1 \beta _2 \beta _3 ]^T . $$ Finally, the stiffness matrix K e of the hybrid element is obtained from the functional ∏ H . Results of numerical examples show that the new hybrid element HHR4 is more accurate than the isoparametric element Q4, and is more efficient than the traditional hybrid element models.