HRBF Meshless Method For Buckling Analysis Of Thin Plates

The stability theory of plates and shells is a branch of solid mechanics. With the development of modern industrial technology, it's of great significance to analyze the buckling behavior. At present, numerical methods for buckling and post-buckling behavior analysis are mainly the finite element method and perturbation method. Finite element method is difficult to construct unit with C1 continuity which is applied to Kirchhoff theory for thin plates. In addition, finite element method is prone to shear locking phenomenon resulting in great error when it's used in the Mindlin plate theory. Perturbation method does not apply to industrial computing. Meshless method is easy to construct unit with C1 continuity and calculated with the conditions of industrialization, which is a new numerical method for the structural analysis. Meshless method applied to buckling analysis is still under development.As buckling problem of plate is a higher order boundary problem, so displacements and rotation angles on the boundary can be as essential conditions of the compulsory. In order to ensure the accuracy of solution, a new Hermite Radial Basis Functions (HRBF) interpolation is introduced in this paper. The HRBF interpolation makes the shape functions and their derivatives have the properties of Kronecker delta. The computing time for Collocation method is short , but the stability is not good, especially in the case that shape functions need to calculate higher derivatives. In buckling problem of plate, the strong form has to calculate the fourth-order partial derivatives, but with weak form, it is reduced to second-order partial derivatives, which makes the algorithm more stable, so Galerkin method is selected to discrete the energy equation. Based on the meshless method. The HRBF interpolation is constructed in this paper. After that, energy form of buckling problem of a plate is given and then discreted. According to the discreted equation, the buckling load of the rectangular plate, which is in a variety of boundary conditions and loading cases, is calculated by numerical method. Comparing with the previous results and analytical solutions shows the validity and accuracy of the method in this paper. Further, exploratory study with buckling conditions of a laminated plate with piezoelectric materials is carried on, and then obtains boundary line of stability under the action of in-plane load and electric field. Finally, HRBF meshless method is applied to the initial post-buckling analysis of thin plate, the result shows that the method is suitable to solve the plate buckling problem with high precision.