Solving ordinary thin plate bending problem in engineering, only a few analytical solutions with simple boundary shapes have been proposed. When using numerical methods (e.g. the variational method) to solve the problem, the trial functions can be found only it exhibits a simple boundary shape. The R-functions can be applied to solve the problem with complex boundary shapes. In the paper, the R-function theory is combined with the variational method to study the thin plate bending problem with the complex boundary shape. The paper employs the R-function theory to express the complex area as the implicit function, so it is easily to build the trial function of the complex shape thin plate, which satisfies with the complex boundary conditions. The variational principle and the R-function theory are introduced, and the variational equation of thin plate bending problem is derived. The feasibility and correctness of this method are verified by five numerical examples of rectangular, I-shaped, T-shaped, U-shaped, and L-shaped thin plates, and the results of this method are compared with that of other literatures and ANSYS finite element method (FEM). The results of the method show a good agreement with the calculation results of literatures and FEM.
A mathematical analysis method is employed to solve the bending problem of slip clamped shallow spherical shell on elastic foundation. Using the slip clamped boundary conditions, the differential equations of the problem are simplified to a biharmonic equation. Using the [Formula: see text]-function, the fundamental solution and the boundary equation of the biharmonic equation, a function is established. This function satisfies the homogeneous boundary condition of the biharmonic equation. The biharmonic equation of the slip clamped shallow spherical shell bending problem on elastic foundation is transformed to Fredholm integral equation of the second kind by using Green’s formula. The vector expression of the integral equation kernel is derived. Choosing a suitable form of the normalized boundary equation, the singularity of the integral equation kernel is overcome. To obtain the numerical results, the discretization of the integral equation of the bending problem is conducted. The treatment of singular term in the discretization equation is to use the integration by parts. Numerical results of rectangular, trapezoidal, pentagonal, L-shaped and concave shape shallow spherical shells show high accuracy of the proposed method. The numerical results show fine agreement with the ANSYS finite element method (FEM) solution, which shows the proposed method is an effective mathematical analysis method.
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