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Then calculate the elements of the Jacobian matrix, using the shape functions of Eq.
(6.131),
4
∂
x
∂ξ
=
∂
N
i
∂ξ
1
4
[
x
i
=
−
x
1
(
1
−
η)
+
x
2
(
1
−
η)
+
x
3
(
1
+
η)
−
x
4
(
1
+
η)
]
i
=
1
4
∂
x
∂η
=
∂
N
i
∂η
1
4
[
x
i
=
−
x
1
(
1
−
ξ)
−
x
2
(
1
+
ξ)
+
x
3
(
1
+
ξ)
+
x
4
(
1
−
ξ)
]
i
=
1
(1)
4
∂
y
∂ξ
=
∂
N
i
∂ξ
1
4
[
y
i
=
−
y
1
(
1
−
η)
+
y
2
(
1
−
η)
+
y
3
(
1
+
η)
−
y
4
(
1
+
η)
]
i
=
1
4
∂
y
∂η
=
∂
N
i
∂η
1
4
[
y
i
=
−
y
1
(
1
−
ξ)
−
y
2
(
1
+
ξ)
+
y
3
(
1
+
ξ)
+
y
4
(
1
−
ξ)
]
i
=
1
For element
e
,x
1
=
x
4
=
2
,x
2
=
3
,x
3
=
5
,y
1
=
0
,y
2
=
2
,y
3
=
y
4
=
3
=
+
2
1
−
η
2
1
3
4
(
det
J
=
1
+
η
−
ξ)
(2)
1
+
ξ
2
−
2
1
The determinant
J
is not non-zero everywhere. Along
,
det
J
is zero, and the Jaco-
bian matrix is singular. This means that the coordinate transformation cannot be inverted
somewhere in the element, and implies that for this element an interior angle greater than
π
ξ
=
1
+
η
should be avoided.
A more general investigation of the effect of certain interior angles can be conducted
from the standpoint of det
J
. Because the evaluation of the stiffness matrix involves
dA
,
consider a small parallelogram area at a vertex of an element. This small area uses the two
boundaries of the element as its sides (see Fig. 6.44b). Let the length of these two sides be
dr
1
and
dr
2
. Then
dA
=
dr
1
·
dr
2
·
sin
θ
(3)
Since (Eq. 6.137)
dA
=
det
J
d
ξ
d
,
it follows that
det
J
η
=
dr
1
·
dr
2
·
sin
θ/(
d
ξ
d
η)
(4)
is small or close to 180
◦
,
det
J
will be very small. Also, if
It is apparent that if
is larger
than 180
◦
,
det
J
becomes negative. In general, an interior angle should not be too small or
too large. Thus, it is often recommended that interior angles less than 30
◦
θ
θ
or greater than
150
◦
be avoided.
6.7.3 Instabilities
Instabilities
or
spurious singular modes
in the elements occur due to deficiencies in the for-
mulation of the elements. These instabilities are numerical phenomena, not related to the
buckling treated in Chapter 11, which are discussed in numerous references on finite ele-
ments such as Cook, et al. (1989). Particular instability characteristics, which often entail
rank deficiencies
, are referred to by such names as
zero-energy modes, hourglass modes, kinematic
modes
, and
mechanisms
.
To gain some insight into this problem, consider a zero-energy mode, which by definition
corresponds to a displacement field that does not represent rigid body motion, yet produces
zero strain energy. A stiffness matrix constructed by numerical integration is based on values
obtained at the integration points of the quadrature rule. If a low order quadrature rule
(few integration points) is employed and the strains happen to be zero at the integration
points, a zero-energy mode occurs, leading to a stiffness matrix that is equal to zero.
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