An orthogonality relationship for thin plate theory and its variational principle

While Hamiltonian system was led to solution of elastictheory a new systematic methodology for theory of elasticity was establishedand a symplectic orthogonality relationship was presented (ZhongWanxie, 1995). For two-dimensional theory of elasticity a new dual vectorand a new dual differential matrix were presented by putting the old dualvector in a new order. It was discovered for isotropic materials that thesymplectic orthogonality relationship may be decomposed into 2 independentlyand symmetrically orthogonal sub-relationships (Luo Jianhui et al., 2002).The new orthogonality relationship includes the symplectic orthogonalityrelationship. The new orthogonal relationship was generalized intothree-dimensional elasticity problems in which a direction of coordinate isan orthogonal direction of materials (Luo Jianhui et al., 2003). Theresearch of a systematic methodology for bending theory of thin and thickplate has also been noticed. Some conclusion of the systematic methodologyfor bending theory of Reissner-Mindlin thick plate was obtained (Luo Jianhuiet al., 2004). Firstly, the Hamiltonian dual differential equations forthick plates were derived. Then, the functional expressions of Hamiltonianvariational principle were obtained by using the variable substitution andmultiplier method. At last, the new orthogonality relationship of thickplate theory was proposed. But the new orthogonality relationship of thickplate theory can not be degenerated into thin plate theory. Therefore it isnecessary to research the new orthogonality relationship of thin platetheory. Based on the analogy between plate bending problems and planeelasticity problems Hamiltonian system was applied to thin plate bendingproblems and its symplectic orthogonality relationship was presented (ZhongWanxie et al., 1999). For thin plate bending theory a new dual vector ispresented while the dual vectors based on the analogy are put in a neworder. A variational principle based on the new dual vector is proposed andalso demonstrated by a new method. The principal diagonal sub-matrixes ofthe dual differential matrix are zero matrixes. As a result of thepeculiarity of the dual differential matrix it is discovered that theorthogonality relationship of thin plate bending theory based on the analogymay be decomposed into 2 orthogonal sub-relationships. Based on the integralform (Luo Jianhui et al., 2002) of the systematic methodology forelasticity, the new orthogonal relationship is demonstrated. The neworthogonality relationship of theory of elasticity is generalized intoanisotropic thin plate bending theory. The theoretical achievements of theHamiltonian system for thin plates provide new effective tools for theresearch on analytical and finite element solutions of thin plates.