Civil Engineering Reference
In-Depth Information
The displacement relationship and its processing are an important part
of FEM. In addition to beam joints mentioned earlier, this process can
be used to simulate many other complicated mechanics situations, such as
spring or rigid body connections.
3.2.6 Nonlinearities
In the prior derivations of the global equilibrium equation, both the geome-
try relationship (Equation 3.5) and the material relationship (Equation 3.13)
are in linear forms. When displacements are small and strains are within
the linear range with stresses, as for most engineering problems, linear
solutions (Equation 3.13) are accurate and adequate. However, large dis-
placements and/or nonlinear constitutive material problems widely exist
in engineering practices. The geometric nonlinearity of long-span cable
bridges, discussed in Chapter 11, and the plastic behavior of middle- and
short-span bridges, discussed in Chapters 14, 15, and 17, are two typical
examples of these problems in bridge structural analyses. The approach to
the respective geometric nonlinear and material nonlinear problems is an
important part of FEM.
In general, when material nonlinearity is considered, the stresses and
strains relationship (Equation 3.13) would be
=
( )
(3.34)
σ σ ε
When geometric nonlinearity is considered, the strains will contain the sec-
ond order of displacement derivatives as
2
2
2
∂
∂
u
x
∂
∂
u
x
∂
∂
v
x
∂
∂
w
x
1
2
+
+
+
2
2
2
∂
∂
v
y
∂
∂
u
y
∂
∂
v
y
∂
∂
w
y
1
2
+
+
+
ε
ε
ε
γ
γ
γ
x
y
2
2
2
∂
∂
w
z
∂
∂
u
z
∂
∂
v
z
∂
w
z
1
2
+
+
+
z
(3.35)
ε=
=
∂
xy
∂
∂
u
y
+
∂
∂
v
x
+
∂
∂
u
x
∂
∂
u
y
+
∂
∂
v
x
∂
∂
v
y
+
∂
∂
w
x
∂
∂
w
y
yz
zx
∂
∂
v
+
∂
∂
w
y
+
∂
∂
u
z
∂
∂
u
y
+
∂
∂
v
z
∂
∂
v
y
+
∂
∂
w
z
∂
∂
w
y
z
∂
∂
w
x
+
∂
∂
u
z
+
∂
∂
u
x
∂
∂
u
z
+
∂
∂
v
x
∂
v
z
+
∂
∂
w
x
∂
∂
w
z
∂
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