Matrix Methods of Analysis - Structural and Stress Analysis - page 568

F IGURE 17.12 Finite element idealization of a flat
plate with a central hole
skins, shells, etc., do not possess such natural subdivisions and must therefore be arti-
ficially idealized into a number of elements before matrix methods can be used. These
finite elements, as they are known, may be two- or three-dimensional but the most com-
monly used are two-dimensional triangular and quadrilateral shaped elements. The
idealization may be carried out in any number of different ways depending on such
factors as the type of problem, the accuracy of the solution required and the time and
money available. For example, a coarse idealization involving a small number of large
elements would provide a comparatively rapid but very approximate solution while
a fine idealization of small elements would produce more accurate results but would
take longer and consequently cost more. Frequently, graded meshes are used in which
small elements are placed in regions where high stress concentrations are expected,
e.g. around cut-outs and loading points. The principle is illustrated in Fig. 17.12 where
a graded system of triangular elements is used to examine the stress concentration
around a circular hole in a flat plate.
Although the elements are connected at an infinite number of points around their
boundaries it is assumed that they are only interconnected at their corners or nodes.
Thus, compatibility of displacement is only ensured at the nodal points. However, in
the finite elementmethod a displacement pattern is chosen for each element whichmay
satisfy some, if not all, of the compatibility requirements along the sides of adjacent
elements.
Since we are employing matrix methods of solution we are concerned initially with
the determination of nodal forces and displacements. Thus, the system of loads on
the structure must be replaced by an equivalent system of nodal forces. Where these
loads are concentrated the elements are chosen such that a node occurs at the point of
application of the load. In the case of distributed loads, equivalent nodal concentrated
loads must be calculated.
The solution procedure is identical in outline to that described in the previous sections
for skeletal structures; the differences lie in the idealization of the structure into
finite elements and the calculation of the stiffness matrix for each element. The latter