Nonlinear analysis of wave progagation using transform methods and plates and shells using integral equations
Abstract
Two diverse topics of relevance in modern computational mechanics are treated. The first involves the modeling of linear and nonlinear wave propagation in flexible, lattice structures. The technique used combines the Laplace Transform with the Finite Element Method (FEM). The procedure is to transform the governing differential equations and boundary conditions into the transform domain where the FEM formulation is carried out. For linear problems, the transformed differential equations can be solved exactly, hence the method is exact. As a result, each member of the lattice structure is modeled using only one element. In the nonlinear problem, the method is no longer exact. The approximation introduced is a spatial discretization of the transformed nonlinear terms. The nonlinear terms are represented in the transform domain by making use of the complex convolution theorem. A weak formulation of the resulting transformed nonlinear equations yields a set of element level matrix equations. The trial and test functions used in the weak formulation correspond to the exact solution of the linear part of the transformed governing differential equation. Numerical results are presented for both linear and nonlinear systems. The linear systems modeled are longitudinal and torsional rods and BernoulliEuler and Timoshenko beams. For nonlinear systems, a viscoelastic rod and Von Karman type beam are modeled. The second topic is the analysis of plates and shallow shells undergoing finite deflections by the Field/Boundary Element Method. Numerical results are presented for two plate problems. The first is the bifurcation problem associated with a square plate having free boundaries which is loaded by four, self equilibrating corner forces. The results are compared to two existing numerical solutions of the problem which differ substantially. <The second problem involves a simply supported rhombic plate. &The bending moments in the nonlinear model are compared to those in the linear model. The intent being to study the effect the nonlinearity and inplane boundary conditions have on the singular bending moments known to occur in the linear problem.
 Publication:

Ph.D. Thesis
 Pub Date:
 1992
 Bibcode:
 1992PhDT........26P
 Keywords:

 Finite Element Method;
 Laplace Transformation;
 Linear Systems;
 Nonlinear Systems;
 Plates (Structural Members);
 Shallow Shells;
 Wave Propagation;
 Boundary Conditions;
 Deflection;
 Differential Equations;
 Rods;
 Timoshenko Beams;
 Structural Mechanics