Civil Engineering Reference
In-Depth Information
juncture of the two flexure elements; that is, at node 2. This is
not
the usual situation in
finite element analysis. The formulation requires displacement and slope continuity but,
in general, no continuity of higher-order derivatives. Since the flexure element developed
here is based on a cubic displacement function, the element does not often exhibit mo-
ment (hence, stress) continuity. The convergence of derivative functions is paramount to
examining the accuracy of a finite element solution to a given problem. We must exam-
ine the numerical behavior of the derived variables as the finite element “mesh” is refined.
4.7 FLEXURE ELEMENT WITH AXIAL LOADING
The major shortcoming of the flexure element developed so far is that force load-
ing must be transverse to the axis of the element. Effectively, this means that the
element can be used only in end-to-end modeling of linear beam structures.
If the element is formulated to also support axial loading, the applicability is
greatly extended. Such an element is depicted in Figure 4.11, which shows, in ad-
dition to the nodal transverse deflections and rotations, axial displacements at the
nodes. Thus, the element allows axial as well as transverse loading. It must be
pointed out that there are many ramifications to this seemingly simple extension.
If the axial load is compressive, the element could buckle. If the axial load is ten-
sile and significantly large, a phenomenon known as
stress stiffening
can occur.
The phenomenon of stress stiffening can be likened to tightening of a guitar
string. As the tension is increased, the string becomes more resistant to motion
perpendicular to the axis of the string.
The same effect occurs in structural members in tension. As shown in Fig-
ure 4.12, in a beam subjected to both transverse and axial loading, the effect of
the axial load on bending is directly related to deflection, since the deflection at
a specific point becomes the moment arm for the axial load. In cases of small
elastic deflection, the additional bending moment attributable to the axial loading
is negligible. However, in most finite element software packages, buckling and
stress stiffening analyses are available as
options
when such an element is used
in an analysis. (The reader should be aware that buckling and stress stiffening ef-
fects are checked
only if the software user so specifies.
) For the present purpose,
we assume the axial loads are such that these secondary effects are not of concern
and the axial loading is independent of bending effects.
v
i
v
j
i
u
i
j
u
j
i
j
Figure 4.11
Nodal displacements
of a beam element with axial
stiffness.
