Civil Engineering Reference
In-Depth Information
isoparametric element has planar faces and is analogous to the two-dimensional
quadrilateral element. If the parent element is of higher-order interpolation func-
tions, an element with general (curved) surfaces results.
Regardless of the specific element type or types used in a three-dimensional
finite element analysis, the procedure for assembling the global equilibrium equa-
tions is the same as discussed several times, so we do not belabor the point here.
As in previous developments, the assembled global equations are of the form
[
K
]
{
}={
F
}
(9.118)
with [
K
] representing the assembled global stiffness matrix,
{
}
representing the
column matrix of global displacements, and
{
F
}
representing the column matrix
of applied nodal forces. The nodal forces may include directly applied external
forces at nodes, the work-equivalent nodal forces corresponding to body forces
and forces arising from applied pressure on element faces.
9.7 STRAIN AND STRESS COMPUTATION
Using the stiffness method espoused in this text, the solution phase of a finite
element analysis results in the computation of unknown nodal displacements as
well as reaction forces at constrained nodes. Computation of strain components,
then stress components, is a secondary (postprocessing) phase of the analysis.
Once the displacements are known, the strain components (at each node in the
model) are readily computed using Equation 9.104, which, given the discretiza-
tion in the finite element context, becomes
[
L
]
u
{
ε
}=
v
w
=
[
L
][
N
3
]
{}=
[
B
]
{}
(9.119)
It must be emphasized that Equation 9.119 represents the calculation of strain
components for an
individual element
and must be carried out for
every
element
in the finite element model. However, the computation is straightforward, since
the [
B
] matrix has been computed for each element to determine the element
stiffness matrix, hence the element contributions to the global stiffness matrix.
Similarly, element stress components are computed as
{}=
[
D
][
B
]
{}
(9.120)
and the material property matrix [
D
] depends on the state of stress, as previously
discussed. Equations 9.119 and 9.120 are general in the sense that the equations
are valid for any state of stress if the strain-displacement matrix [
B
] and the
material property matrix [
D
] are properly defined for a particular state of stress. (In
this context, recall that we consider only linearly elastic deformation in this text.)
The element strain and stress components, as computed, are expressed in
the element coordinate system. In general, for the elements commonly used in
