Civil Engineering Reference
In-Depth Information
where
are the displacement vectors of the nodes of the triangular element
and
etc. for (i, j, m)
9.4
Rectangular plane stress elements
Another type of finite element commonly used for the analysis of deep
beams is the rectangular element. However, right-angled rectangular
elements are not suitable for beams with inclined or curved edges, and it is
more convenient to use the quadrilateral element, which must be formulated
through the use of natural co-ordinates (ξ, η) and co-ordinate transformation
techniques.
9.4.1
Isoparametric quadrilaterals
Based on some mathematical manipulation, regular shaped elements can be
distorted into desired irregular shapes with either straight or curved edges
which can then be made to coincide nearly with the curved boundary of a
structure. In general a one-to-one correspondence must exist between points
on the original element and those on the distorted one. The co-ordinates of a
point on the parent element and on the distorted one are related by means of
interpolation functions or shape functions. If the same shape functions are
used to represent the relationships of the displacements as well as the
geometric co-ordinates system, the procedure is known as the isoparametric
finite element formulation (Zienkiewicz, 1971).
Figure 9.3
Natural co-ordinates of a parent square element and element co-ordinates of a
quadrilateral.
