Civil Engineering Reference
In-Depth Information
relative size of the elements in the vicinity of the hole is maintained relative to
elements far removed from the discontinuity.
The automeshing capabilities of finite software as briefly described here are
extremely important in reducing the burden of defining a finite element model of
any geometric situation and should be used to the maximum extent. However,
recall that the results of a finite element analysis must be judged by human
knowledge of engineering principles. Automated model definition is a nicety of
modern finite element software; automated analysis of results is not.
Analysis of results is the postprocessing phase of finite element analysis.
Practical models contain hundreds, if not thousands, of elements, and the com-
puted displacements, strains, stresses, and so forth are available for every ele-
ment. Poring through the data can be a seemingly endless task. Fortunately, finite
element software has, as part of the postprocessing phase, routines for sorting
the results data in many ways. Of particular importance in stress analysis, the
data can be sorted in ascending or descending order of essentially any stress
component chosen by the user. Hence, one can readily determine the maximum
equivalent stress, for example, and determine the location of that stress by the
associated element location. In addition, with modern computer technology, it is
possible to produce color-coded stress contour plots of an entire model, to visu-
ally observe the stress distribution, the deformed shape, the strain energy distri-
bution, and many other criteria.
9.9 TORSION
Torsion (twisting) of structural members having circular cross sections is a com-
mon problem studied in elementary mechanics of materials. (Recall that earlier
we developed a finite element for such cases.) A major assumption (and the
assumption is quite valid for elastic deformation) in torsion of circular members
is that plane sections remain plane after twisting. In the case of torsion of a non-
circular cross section, this assumption is not valid and the problem is hence more
complicated. A general structural member subjected to torsion is shown in Fig-
ure 9.13a. The member is subjected to torque
T
acting about the
x
axis, and it is
assumed that the cross section is uniform along the length. An arbitrary point
located on a cross section at position
x
is shown in Figure 9.13b. If the cross sec-
tion twists through angle
, the point moves through arc d
s
and the displacement
components in the
y
and
z
directions are
v
=−
z
(9.132)
w
=
y
respectively. Since the angle of twist varies along the length of the member, we
conclude that the displacement components of Equation 9.132 are described by
v
=
v
(
x
,
z
)
w
=
w
(
x
,
y
)
(9.133)
