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a
b
The final element stiffness matrix is the sum of
k
B
and
k
V
in the form
b
a
,
α
=
β
=
k
i
=
K
[
k
B
+
ζ
k
V
]
(13.109)
When the element matrices are assembled and boundary conditions imposed, the global
stiffness equation appears as
(
K
B
+
ζ
K
V
)
V
=
P
/
K
(13.110)
t
2
.If
k
s
is constant, it can be ob-
From Eq. (13.104),
ζ
can be defined as
ζ
=
6
k
s
(
1
−
ν)/
served that for very thin plates
ζ
can assume a very large value. Also, as
t
→
0
,
ζ
→∞
.
Co
n
sequently, the unrealistic result of
V
=
0
can be obtained, regardless of the magnitude
of
P
This phenomenon is called
locking,
which is intended to imply that the displacement is
“locked”. Locking also occurs for thin beam elements when traditional shear deformation
effects are taken into account and the displacement and slope are represented by indepen-
dent shape functions. As discussed in Section 12.2.2, Chapter 12, for beams and plates it
may be help to employ a
k
s
that varies with the thickness.
A classical method of alleviating the locking problem is to make the
K
V
matrix singular
so that
.
K
V
can be finite. One way to accomplish this is to reduce the rank of the matrix
k
V
and use a low order numerical integration scheme to integrate
k
V
.
ζ
This is called
reduced
integration
. It is known that if the number of strains at the integration points is less than
the degrees of freedom available, then the singularity exists for the global stiffness matrix
[Zienkiewicz, 1977]. Let
h
be the total number of integration points,
k
the number of strains
used in the formation of the stiffness matrix
k
V
,
and
j
the total number of degrees of
freedom in
V
(with suitable restraints against rigid body motion). Then, if
j
0
,
the stiffness matrix
k
V
will be singular. Usually, a single point Gauss integration scheme is
used for the formation of
k
V
−
hk
>
After the element stiffness matrices
k
V
are assembled to form
the global stiffness matrix
K
V
and the boundary conditions are applied, the matrix
K
V
is
still singular and, thus, locking will be avoided. Another technique to eliminate the locking
problem is to impose the Kirchhoff assumption at discrete points. This leads to an element
called the
discrete Kirchhoff theory
(
DKT
) element, which will be treated later.
Variations of this 12 DOF rectangular element of this section can improve the performance
of the element. For example, the addition of the twist
.
2
y
as a degree of freedom tends
to improve the accuracy of an element based on Kirchhoff plate theory, but decreases the
efficiency as it would now be a 16 DOF element. Rectangular elements limit the modeling
options, especially near irregular boundaries. Triangular elements which are discussed in
the following section, provide much greater flexibility.
∂
w/∂
x
∂
13.5.2
Triangular Plate Elements
Plates with irregular boundaries require the use of nonrectangular elements, e.g., triangular
elements. For example, the meshes for stress concentration regions are often modeled with
triangular elements. Such elements are also used to form quadrilateral elements, as rect-
angular elements are not easily generalized into quadrilateral shapes. The transformation
of coordinates of the type described in Chapter 6, Section 6.7 may be performed, but un-
fortunately the constant strain criterion is then violated [Zienhiewicz, 1977]. Typically, the
quadrilateral element is treated as the composition of four three-node triangular elements.
Usually, triangular elements are defined by the diagonals of the quadrilateral elements.
As in the case of rectangular elements, stretching and compression effects, obtained, for
example, from a plane stress analysis, can be superimposed on the plate bending triangular
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