Civil Engineering Reference
In-Depth Information
In Equation 9.114, the strain-displacement matrix is given by
∂
∂
00
x
∂
∂
0
0
y
00
∂
∂
[
N
]
[0]
[0]
z
[
B
]
=
[
L
][
N
3
]
=
(9.115)
[0]
[
N
]
[0]
∂
∂
∂
∂
0
[0]
[0]
[
N
]
y
x
∂
∂
∂
∂
0
z
z
∂
∂
∂
∂
0
z
y
and is observed to be a
6
×
3
M
matrix composed of the first partial derivatives
of the interpolation functions.
Application of the principle of minimum potential energy to Equation 9.114
yields, in analogy with Equation 9.22,
[
B
]
T
[
D
][
B
]d
V
{}={
f
}
(9.116)
V
as the system of nodal equilibrium equation for a general three-dimensional stress
element. From Equation 9.116, we identify the element stiffness matrix as
[
B
]
T
[
D
][
B
]d
V
[
k
]
=
(9.117)
V
and the element stiffness matrix so defined is a
3
M
×
3
M
symmetric matrix,
as expected for a linear elastic element. The integrations indicated in Equa-
tion 9.117 depend on the specific element type in question. For a four-node,
linear tetrahedral element (Section 6.7), all the partial derivatives of the volume
coordinates are constants, so the strains are constant—this is the 3-D analogy to
a constant strain triangle in two dimensions. In the linear tetrahedral element, the
terms of the
[
B
]
matrix are constant and the integrations reduce to a constant
multiple of element volume.
If the element to be developed is an eight-node brick element, the interpolation
functions, Equation 6.69, are such that strains vary linearly and the integrands
in Equation 9.117 are not constant. The integrands are polynomials in the spatial
variables, however, and therefore amenable to exact integration by Gaussian quad-
rature in three dimensions. Similarly, for higher-order elements, the integrations
required to formulate the stiffness matrix are performed numerically.
The eight-node brick element can be transformed into a generally shaped
parallelopiped element using the isoparametric procedure discussed in Sec-
tion 6.8. If the eight-node element is used as the parent element, the resulting
