NEW SOLUTION SYSTEM FOR PLATE BENDING AND ITS APPLICatION
板弯曲求解新体系及其应用

The governing equation for plate bending is biharmonic equation, the traditionalsolution methodology is the semi-inverse one that causes limitations. The Airy stress function isusually applied traditionally for plane elasticity problems, it also satisfies biharmonic equation.The analogy between plate bending and plane elasticity problems had been noticed long before,but their solution systems are different each other, and the analogy relationship had not been usedsystematically. In this paper, the analogy is set up and perfect further.The deflection w for plate bending correspond to the Airy stress function in plane elasticity,conversely, the displacements in plane elasticity correspond to two bending moment functionsop. I gb. in plate bending. Based on the analogy between plate bending and plane elasticity problems,Hamiltonian system can also be applied to plate bending problem, that is, the problem can be solvedin symplectic space that consists of curvatures and bending moment functions. So correspondingto the principle of minimum potential energy and the Hellinger-Reissuer variational principle forplane elasticity, the new variational principles of minimum complementary energy and the Pro-H-R in terms of bending moment for plate bending can be proposed respectively. Carrying outthe variations, a set of new governing equations and solution for plate bending clajssical theory ispresented.The new methodology presents the analytical solutions in plate strip via the methods ofseparation of variables and eigenfunction-vector expansion, it breaks through the limit of traditionalsemi-inverse solution. The new one for plate simply support on both sides is equivalent to the Levysingle trigonometrical series expansion method, but it is not the same as the classical semi-inverseone, is derived rationally and analytical. Therefore it can easily be applied to other lateral boundaryconditions, which is very difficult for semi-inverse method, such as the both sides free plate givenin this paper. The results show that the new methodology will have vast application vista.