Weak Form Quadrature Element Analysis Of Vibration Of Beams And Thin Plates On Nonlinear Foundation

Structures in civil engineering normally do not exist independently but combine with soil foundation. Also, some components in mechanical engineering cling to bonds. Therfore, it is important to study the mechanical behavior of structures interacting with foundation or bonds. Due to the complexity and nonlinearity of the foundation, linear computation brings out inevitable errors. It has become necessary to introduce the nonlinear system to meet the demands of engineering designs nowadays. This thesis aims at vibration problems of beam or thin plate structures on nonlinear foundation and analyzes them with the weak form Quadrature Element Method(QEM).The QEM is a new versatile numerical method based on variational principles. The essential idea of the QEM is that the variational description of a problem is formulated first. Numerical integrations and the Differential Quadrature Method(DQM) are then introduced to characterize the problem in terms of a weighted sum of field variables at all quadrature nodes. As a result, a set of algebraic equations are obtained after employing the variational principle. The QEM inherits the high computational efficiency of the DQM and enjoys flexibility and wide applicability of weak form descriptions.The present work focuses on studying vibrations of structures on nonlinear elastic foundations, analyzing the influences of nonlinearity to vibration frequencies and modes. Winkler foundation and Pasternak foundation with nonlinearity are considered, respectively, and influence of shear layer to vibration behavior is studied. In addition, the difference of vibration between Bernoulli-Euler beams and Timoshenko beams on nonlinear foundation is discussed. Results are tabulated for engineering design reference.The other aim of the work is to extend the application of the QEM to vibration problems of structures on nonlinear foundations. The analysis ends up with solution of nonlinear generalized eigenvalue problems. An iterative scheme is employed which is able to solve all nonlinear generalized eigenvalue problems. Results demonstrate the efficiency of the iterative scheme and the advantages of the QEM over the finite element method.