Civil Engineering Reference
In-Depth Information
1
1
1
5
8
5
8
2
4
2
4
7
6
6
7
3
3
(a) Displacement function
N
1
(b) Displacement function
N
2
1
1
1
1
8
5
4
2
2
4
6
7
6
7
3
3
(c) Displacement function
N
5
(d) Displacement function
N
8
Figure 3.4
(a-d) 3D views of displacement functions of a eight-node rectangle element
(square function).
works that the external loads are done. The multip
li
cation of stress and
strain at any point gives the strain energy density
U
. Taking the spatial
strain and stress problem, as shown in Figure 3.1, as the example to illus-
trate a generic approach, the accumulated strain energy density starting
from the beginning to any equilibrium point is the shaded area as shown in
Figure 3.6. It can be expressed as follows:
ε
γ
ε
ε
y
xy
x
z
∫
∫
∫
∫
U
=
σ ε
d
+
σ ε
d
+
σ ε
d
+
τ
d
γ
x
x
y
y
z
z
xy
xy
0
0
0
0
(3.22)
γ
γ
µ
yz
zx
∫
∫
∫
T
+
τ
d
γ
+
τ
d
γ
=
σ ε
d
yz
yz
zx
zx
0
0
0
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