薄板哈密顿求解体系及其变分原理

HAMILTONIAN SOLUTION SYSTEM FOR THIN PLATES AND ITS VARIATIONAL PRINCIPLE

将哈密顿求解体系推广应用于薄板弯曲问题。首先导出薄板哈密顿对偶微分方程,然后导出薄板哈密顿变分原理的泛函表示式HP。有两点值得指出第一,以挠度w、转角xy、弯矩xM和等效剪力xV取为对偶变量,与相关文献的取法不同。第二,对于薄板问题,由Hellinger-Reissner泛函HRP导出哈密顿泛函HP时既要消元,又要增元,与在厚板问题中只需要消元的推导方法不同。薄板哈密顿求解体系的理论成果将为研究薄板解析解和有限元解提供新的有效工具。

The Hamiltonian solution system was generalized to the thin plate problems. Firstly, the Hamiltonian dual differential equations for thin plates were derived. Then, the functional expression of Hamiltonian variational principle, ∏H, was obtained. Differing from those proposed in related reference, the deflection w, the rotation ψx, the moment Mx and the equivalent shear force Vx were taken as the dual variables; For thin plate problems, both elimination and addition of variables were needed when the Hamiltonian functional ∏H was derived using the Hellinger-Reissner functional ∏HR. The process is different from that used in the thick plate problems, where only elimination is utilized. The theoretical achievements of the Hamiltonian system for thin plates provide new effective tools for analytical and finite element solutions of thin plates.