Biomedical Engineering Reference
In-Depth Information
Table 18.1 Comparison of the relative accuracy of
different element types for the beam bending case.
Element type
u
c
/u
L
Linear triangle
0.231
Bi-linear quadrilateral
0.697
Quadratic triangle
0.999
Bi-quadratic quadrilateral
1.003
F
h
L
x
=
0
x
=
L
Figure 18.2
A beam that is clamped on one side and loaded with a vertical concentrated force at the other
side.
element. The meshes for the linear triangular and the bi-linear quadrilateral
element are shown in Fig.
18.3
. The displacement at
x
=
L
can be computed
using standard beam theory, giving
FL
3
3
EI
,
u
L
=
(18.67)
with
bh
3
12
,
I
=
(18.68)
where
b
is the width of the beam and
h
the height of the beam and
E
the Young's
modulus of the material. The ratio
h
/
L
of height over length equal to 0.1 is chosen.
Using the meshes depicted in Fig.
18.3
, the results of Table
18.1
are obtained,
which presents the ratio of
u
L
and the computed displacement
u
c
at
x
=
L
.
It is clear that the displacement of the beam, using a mesh of linear trian-
gles is much too small. The poor performance of the linear triangle can easily
be understood; because of the linear interpolation of the displacement field
u
, the
associated strains computed from
