Civil Engineering Reference
In-Depth Information
Displacement functions at some nodes of a less-used 12-node element are
1
32
(
)
−
(
)
(
)
2
2
N
=
1
+
ξ
1
+
η
9
ξ
+
η
10
1
1
32
(
)
−
(
)
(
)
1
1
9
2
2
1
0
N
=
−
ξ
+
η
ξ
+
η
(3.21)
2
9
32
9
32
(
)
(
)
(
)
(
)
(
)
(
)
2
2
N
=
1
+
η
1
−
ξ
1 3
−
ξ
,
N
=
1
−
ξ
1
−
η
1 3
+
η
6
7
To investigate the characteristics of displacement function, three-dimensional
(3D) views of some of the earlier functions are shown in Figures 3.3 through 3.5.
The functions in Equations 3.19 through 3.21 are linear, square, and cubic,
respectively. From the earlier definitions and 3D views shown in Figures 3.3
through 3.5, it can be seen that the conditions in 1, 3, and 4 are met. To check
the continuous condition, the element edge 1-5-2 of the eight-node element
can be used as an example. Displacement functions of all nodes other than
1, 5, and 2 are 0, which means the interpolation of any displacement along the
edge merely depends on nodal displacements at nodes 1, 5, and 2. Therefore,
any displacement at any point along the edge will obtain the same value by
interpolation from either of the adjacent elements.
3.2.4 Strain energy and principles of minimum
potential energy and virtual works
When applying external forces, strains and stresses will be present over
the entire elastic body. The total strain energy accumulated by increasing
external loads from zero to a given load will be used to measure the internal
1
1
1
4
2
2
4
3
3
(a) Displacement function
N
1
(b) Displacement function
N
2
Figure 3.3
(a, b) 3D views of displacement functions of a four-node rectangle element
(linear function).
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