Civil Engineering Reference
In-Depth Information
λ
is a constant related to Young's modulus and Poisson ratio as
λ
1 1 2
.
In general, initial strains caused by shrinkage or temperature change
and/or initial stresses due to existing condition may exist at any point. Only
will the difference between actual and initial strains cause elastic stress
changes, and the total stresses should be the sum of elastic stresses and ini-
tial stresses. The elastic Equation 3.11 can be rewritten in a generic form as
[(
)(
)]
=
E
µ
+
µ
−
µ
0
(3.13)
σ
=
D
(
ε
−
ε
)
+
σ
0
where:
T
σ =
σ
σ
σ
τ
τ
τ
(3.14)
x
y
z
xy
yz
zx
are total stresses
T
=
0
0
0
0
0
0
σ
0
σ
σ
σ
τ
τ
τ
(3.15)
x
y
z
xy
yz
zx
are initial stresses
T
ε =
ε
ε
ε
γ
γ
γ
(3.16)
x
y
z
xy
yz
zx
are total strains
T
=
0
0
0
0
0
0
ε
0
ε
ε
ε
γ
γ
τ
(3.17)
x
y
z
xy
yz
zx
are initial strains
3.2.3 displacement functions of an element
To apply Equation 3.5 to obtain the total strain energy of an element,
displacements at any point within the element should be explicitly expressed
by nodal displacements of the element. This expression is called
element
displacement
or
shape functions
. Due to variations of geometry shape and
mechanical behavior of an element, there is no general theoretical defini-
tion on how the displacement at a point is related to all nodal displacements
of an element. Only certain types of element, for example, beam-bend-
ing element, have known theoretical displacement functions. As a generic
approach of FEM, these relationships have to be assumed according to dif-
ferent types of elements. The definition of the displacement function for a
certain type element plays a critical role in its behavior and convergence
of a solution. It is easy to understand that the error in the calculation of
strain energy due to an inaccurate or coarse displacement function can be
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