Biomedical Engineering Reference
In-Depth Information
Clearly a similar expression holds for the weighting function:
e
=
T
dx
∼
e
.
dw
h
dx
d
∼
(14.34)
Substitution of this result into the left-hand side of Eq. (
14.25
) and considering
one element only yields:
T
T
dw
h
dx
c
du
h
dx
d
∼
dx
∼
e
c
d
∼
dx
=
∼
e
dx
dx
e
e
T
d
∼
dx
c
d
∼
e
=
∼
∼
e
dx
dx
e
e
e
T
d
∼
dx
c
d
∼
=
∼
dx
∼
e
.
(14.35)
dx
In the last step use has been made of the fact that
∼
e
and
∼
e
are both independent
of the coordinate
x
. Likewise, the integral expression on the right-hand side of Eq.
(
14.25
) yields for a single element:
e
T
∼
e
fdx
=
∼
T
w
h
fdx
=
∼
∼
fdx
.
(14.36)
e
e
e
Notice that the integral appearing on the right-hand side of Eq. (
14.35
) is in fact a
matrix, called the element coefficient or (in mechanical terms) stiffness matrix:
T
d
∼
dx
c
d
∼
K
e
=
dx
.
(14.37)
dx
e
Similarly, the integral on the right-hand side of Eq. (
14.36
) is the element array
corresponding to the internal source or distributed load:
f
∼
e
=
∼
fdx
.
(14.38)
e
Substitution of the expression for the element coefficient matrix and element
column in Eq. (
14.25
) yields
n
el
n
el
e
K
e
∼
e
=
e
f
∼
∼
∼
e
+
B
.
(14.39)
e
=
1
e
=
1
Step 3. Assembling the global set of equations
The individual element contri-
butions
T
∼
e
K
e
∼
e
,
(14.40)
using the local unknowns and weighting values (
∼
e
and
∼
e
, respectively) only, may
also be rewritten in terms of the global unknowns
∼
and the associated weighting
