Finite element solution methods for linear and nonlinear beam-on-foundation problems

Abstract

The objective of this research is to develop finite elements and iterative techniques for numerical solution of linear and nonlinear beam-on-foundation problems encountered in structural engineering. Although closed-form solutions are available for analyzing simple linear load cases of beam-on-foundation problems, complicated loading combinations or beam-on-nonlinear-foundation problems generally require sophisticated numerical solution methods. This research establishes new finite elements that yield exact solutions at the nodes for linear beam-on-foundation problems and new iterative techniques for rapid solution convergence of beam-on-nonlinear-foundation problems. The approach taken here in solving linear beam-on-foundation problems is unique in that the solution of the homogeneous portion of the fourth-order differential equation is used for the finite element shape functions. Furthermore, the complex form of the solutions is used rather than the real form. Elements that result from these shape functions are real and yield several major advantages for solving this class of problems, including (1) Achieving exact solutions at the finite element nodes for linear, self-adjoint problems; (2) Providing two, easily implemented, complex shape function elements that address the majority of linear beam-on-foundation problems. Nonlinear foundation problems are divided into two classes for solution here: those in which the foundation response and its first derivative are continuous functions of displacement (C$\sp1$-continuous functions), and those in which the foundation response can be modeled as a piecewise-linear function (C$\sp0$-continuous functions). An example of the first case is the lateral motion of a beam buried in soil where restraint forces vary continuously with deflection. This class of problems is addressed using numerical integration to compute element stiffnesses and quasi-Newton methods to perform iterative solution. An example of the second (piecewise-linear) class of foundations is that of a "compression only" foundation in which a beam may "liftoff" and lose contact with the foundation. A simple and straightforward nonlinear solution method is developed for this type of problem which exhibits rapid convergence properties.