Analytical Singular Finite Elements For Analysis Of Thin Plate Bending Problems

Thin plate, as an important structural element, has been widely used in structrural engineering. Local stress singularities often occur in a plate resulting from a crack or a V-shaped hole in engineering practices. Finite element method (FEM) is one of very efficient numerical methods. However, the application of conventional FEM for numerical analyses of Kirchhoff plate bending problems with local stress singularities from a crack or a V-shaped notch often causes stiffness problem, so dense meshes are required near the singular points in order to ensure the accuracy of solutions. This leads to low computational efficiency and the unsatisfactory precision of solutions. Therefore, it is worthwhile to investigate how to improve the efficiency and precision in analysis of local stress singularities in a thin plate.In this doctoral dissertation, analysis of the thin plate bending problems with stress singularities and related numerical methods was systematically studied based on symplectic duality system.(1) Based on the general solution of bending problem of an annular sector thin plate, symplectic eigensolutions of the bending in homogeneous and bi-material annular sector thin plate under various boundary conditions were obtained, using boundary conditions described in dual variables. To begin with, symplectic eigensolutions of thin plates were obtained after the analysis of the bending problem of a homogeneous annular sector using boundary conditions of both straight sides clamped, and one straight side free and the other straight side clamped. Stress singularities around a V-shaped notch under correlative boundary conditions were discussed. Furthermore, methodology of the symplectic duality system was introduced to the solution to bi-material annular sector thin plate bending. In the symplectic geometry space composed of original variables and their dual ones, symplectic dual equations for the related problem and boundary conditions on both straight sides and compatibility conditions along the interface described in dual variables were given. Three particular solutions related to the inhomogeneous boundary conditions were solved for the first time. For a bi-material wedge plate, they are the solutions to a unit bending moment, a unit torsion moment and a unit concentrated vertical force acting at the vertex. Symplectic eigensolutions for the homogeneous boundary conditions were also obtained. For these solutions, the system of forces at the end face is under self-equilibrium. In summary, some new symplectic eigensolutions to the annular sector thin plate bending problems were provided in this dissertation. These results extend the application fields of the symplectic duality system, and lay a foundation for construction of analytical singular finite elements for the related problems.(2) Using the symplectic eigensolutions of bending problems of the homogeneous and the bi-material annular sector thin plate as the displacement models, three analytical singular finite elements were constructed respectively. Local-global method was applied with these three elements to analyze the thin plate bending problems with V-shaped notches, bi-material interface cracks and bi-material interface V-shaped notches. Due to the use of the analytical eigensolutions within the singular elements, the displacement models can exactly reflect the stress singularities near the singular points. Because of the application of the analytical singular finite elements, dense meshes near the singular points are not required any more. The use of one such singular element instead of tens of or more conventional ones avoids the stiffness problem caused by conventional finite elements in solution to the stress singularity problems. Solution precision and efficiency are improved. Stress intensity factors which reflect the local stress singularities can be determined easily and directly without other numerical methods, such as extrapolation. To verify the validity of these new methods, many numerical examples were presented in this dissertation, comparing with some benchmark solutions. Numerical results show that application of these analytical singular finite elements can improve the analysis precision in problems involving stress singularities and has good numerical stability. The present analytical singular finite element is an effective technique for the analysis of thin plate bending problems with stress singularities.