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FIGURE 13.18
Triangular element and the area coordinate system.
stiffness matrix. These plate bending elements with in-plane deformation effects can be
used in buckling analyses and in the modeling of curved surfaces such as occur with shells.
Derivations of three representative kinds of triangular elements are provided in this
section.
Three Node, Nonconforming, Nine DOF Element
Element Variables and Trial Functions
Consider the triangular element in Fig. 13.18a and use Kirchhoff plate theory in the devel-
opment of a stiffness matrix, so that no shear deformation effects are included. This element
has a node at each of its three vertices, with three DOF at each node, i.e., the deflection
w
,
θ y =− y .
As the triangular element has nine DOF, include only nine terms in a polynomial shape
function. Here, an immediate difficulty arises as the complete cubic polynomial expansion
contains ten terms (Chapter 6, Section 6.5.5), and any omission has to be made rather
arbitrarily to retain symmetry. All ten terms could be retained and two coefficients made
equal to limit the number of unknowns to nine. This leads to a serious problem, in that
the matrix, corresponding to N u in the form of
θ x =− x , and
the slopes
w =
N u
w , becomes singular for certain
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