Biomedical Engineering Reference
In-Depth Information
It is customary, and convenient, to gather all the nodal displacements
u
xei
and
u
yei
in one column, indicated by
∼
e
, according to
node 1
⎡
⎤
u
xe
1
u
ye
1
⎣
⎦
node 2
u
xe
2
u
ye
2
∼
e
=
.
(18.48)
.
node
n
u
xen
u
yen
Using this definition, the strain column for an element
e
can be rewritten as
∼
=
B
∼
e
,
(18.49)
with
B
the so-called
strain displacement matrix
:
⎤
⎡
⎣
∂
N
1
∂
x
∂
N
2
∂
x
∂
N
n
∂
x
0
0
···
0
⎦
∂
N
1
∂
∂
N
2
∂
∂
N
n
∂
0
0
···
0
B
=
.
(18.50)
y
y
y
∂
N
1
∂
y
∂
N
1
∂
x
∂
N
2
∂
y
∂
N
2
∂
x
∂
N
n
∂
y
∂
N
n
∂
x
···
w
. So, patching everything together, the
Clearly, a similar expression holds for
∼
w
:
σ
may be written as:
double inner product
ε
w
)
T
w
:
σ
=
(
∼
ε
∼
w
)
T
H
∼
=
(
∼
T
e
B
T
H B
∼
e
,
=
∼
(18.51)
where
∼
e
w
structured in the
same way as
∼
e
. This result can be exploited to elaborate the left-hand side of
Eq. (
18.24
):
stores the components of the weighting vector
e
(
∇
w
)
T
:
σ
d
=
∼
B
T
H B
d
∼
e
.
(18.52)
e
e
The element coefficient matrix, or stiffness matrix
K
e
is defined as
B
T
H B d
.
K
e
=
(18.53)
e
