Civil Engineering Reference
In-Depth Information
5
Static Equilibrium of Linear
Elastic Solids
5.1
Introduction
This chapter describes six programs, which can be used to solve equilibrium problems
in small strain solid elasticity. The programs differ only slightly from each other and,
following the method adopted in Chapter 4, the first is described in some detail with changes
gradually introduced into the later programs. Program 5.1 deals with 2D plane strain or
axisymmetric analysis of rectangular regions using any of the 2D elements described in this
topic. Program 5.2 introduces 3D strain for the special case of non-axisymmetric strain of
axisymmetric solids. Program 5.3 introduces conventional 3D analysis of cuboidal meshes
offering a choice of hexahedral elements. Program 5.4 is a general program capable of
analysing geometrically more complex problems in 2 or 3D including the use of tetrahedral
elements. Program 5.5 repeats the analyses described by Program 5.3 using a mesh free
pcg technique in which global stiffness assembly is avoided entirely. This procedure lends
itself to vectorisation as shown in Program 5.6, which highlights some efficiency issues
which arise when programming for a vector computer.
The majority of examples in this chapter consider problems involving a regular (usu-
ally rectangular or cuboidal) geometry. This has been done to simplify the presentation
and minimise the volume of data required. The simple geometries enable the nodal coor-
dinates and numbers to be generated automatically once the user has provided the element
type and preferred numbering direction as data. This is done by geometry subroutines
(e.g.
geom rect
for rectangles and
hexahedron xy
for cuboids, see Appendix E for
geometry subroutine listings). For more complicated geometries, such as are possible using
Program 5.4, the geometry subroutines are replaced by
read
statements for the nodal coor-
dinates and numbering, and it is left to the user to find some other means of generating
this data. Once nodal coordinates, nodal numbering, and boundary conditions are known,
the next stage in all programs is to determine the element “steering vectors”
g
. These are
found from
num
and
nf
as in Chapter 4, using the library subroutine
num to g
.
