Figure 12: Beam in flexure with no
shear ( 
). Analytical and predicted deflected shape. The
deflections have been magnified by a factor of 100. The deflected
beam appears to be ``discontinuous'' due to the small number of panels.
Figure: 2-D beam in flexure with no shear.
It can be shown (by direct substitution of the displacements into the equilibrium equations) that the boundary conditions shown in Figure 11 produce the following displacement and stress distribution throughout the beam:
and
with A being an arbitrary coefficient.
We address this case as ``the beam in flexure with no shear''. The
``analytic'' deflected shape of a beam with a 
height to length ratio
is shown in Figure 12 together with the predicted via the
quadratic saw-tooth method and for 16 total number of panels (5 on the top
or bottom, and 3 on each of the other sides). The accuracy of the quadratic
saw-tooth appears to be remarquable. Notice that at least three numbers per
side are needed for the finite difference scheme to work. In addition, the
panels on the vertical sides are spaced equally. On the horizontal
sides the panels are spaced in such a way so that the panels closest to the corners
have the same size as the panels at the adjacent vertical sides,
while a ``blending''
scheme is used to determine the panel size in between [9].
This so called blended spacing avoids panel size discontinuities and
has been found to improve the convergence of the method.