Civil Engineering Reference
In-Depth Information
9
CHAPTER
Applications
in Solid Mechanics
9.1 INTRODUCTION
The bar and beam elements discussed in Chapters 2-4 are line elements, as only
a single coordinate axis is required to define the element reference frame, hence,
the stiffness matrices. As shown, these elements can be successfully used to
model truss and frame structures in two and three dimensions. For application
of the finite element method to more general solid structures, the line elements
are of little use, however. Instead, elements are needed that can be used to
model complex geometries subjected to various types of loading and constraint
conditions.
In this chapter, we develop the finite element equations for both two- and
three-dimensional elements for use in stress analysis of linearly elastic solids.
The principle of minimum potential energy is used for the developments, as
that principle is somewhat easier to apply to solid mechanics problems than
Galerkin's method. It must be emphasized, however, that Galerkin's method is
the more general procedure and applicable to a wider range of problems.
The constant strain triangle for plane stress is considered first, as the CST is
the simplest element to develop mathematically. The procedure is shown to be
common to other elements as well; a rectangular element formulated for plane
strain is used to illustrate this commonality. Plane quadrilateral, axisymmetric,
and general three-dimensional elements are also examined. An approach for
application of the finite element method to solving torsion problems of noncir-
cular sections is also presented.
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