Biomedical Engineering Reference
In-Depth Information
where
∼
and
∼
contain the global nodal weighting factors and displacements,
respectively, and
K
is the global stiffness matrix. This equation should hold for
all weighting factors, and thus
K
∼
=
f
∼
.
(18.62)
18.5
Solution
As outlined in Chapter
14
the nodal displacements
∼
may be partitioned into two
groups. The first displacement group consists of components of
∼
that are pre-
scribed:
∼
p
. The remaining nodal displacements, which are initially unknown, are
gathered in
∼
u
. Hence:
∼
u
∼
p
.
∼
=
(18.63)
In a similar fashion the stiffness matrix
K
and the load vector
f
∼
are partitioned. As
a result, Eq. (
18.62
) can be written as:
K
uu
K
up
K
pu
K
pp
∼
u
∼
p
f
∼
.
u
=
(18.64)
f
∼
p
The force column
f
∼
is split into two parts:
f
∼
u
and
f
∼
p
. The column
f
∼
is known,
u
since it stores the external loads applied to the body. The column
f
∼
p
, on the other
hand, is not known, since no external load may be applied to points at which the
displacement is prescribed. In Eq. (
18.64
)
f
∼
is known and
f
∼
is unknown. The
u
p
following set of equations results:
K
uu
∼
u
=
f
∼
u
−
K
up
∼
p
,
(18.65)
f
∼
p
=
K
pu
∼
u
+
K
pp
∼
p
.
(18.66)
The first equation is used to calculate the unknown displacements
∼
u
. The result
is substituted into the second equation to calculate the unknown forces
f
∼
p
.
18.6
Example
Consider the bending of a beam subjected to a concentrated force (Fig.
18.2
). Let
the beam be clamped at
x
=
0 and the point load
F
be applied at
x
=
L
. It is inter-
esting to investigate the response of the bar for different kinds of elements using
a similar element distribution. Four different elements are tested: the linear and
the quadratic triangular element, and the bi-linear and bi-quadratic quadrilateral
