Civil Engineering Reference
In-Depth Information
can be solved via Cramer's rule. Application of Cramer's rule results in
∂
u
J
12
∂
r
∂
u
∂
u
J
22
∂
u
∂
1
s
∂
r
x
=
=
[
J
22
−
J
12
]
∂
|
J
|
|
J
|
∂
u
∂
s
(9.75)
∂
u
J
11
∂
r
∂
u
∂
u
J
21
∂
u
∂
1
s
∂
r
y
=
=
[
−
J
21
+
J
11
]
∂
|
J
|
|
J
|
∂
u
∂
s
or, in a more compact form,
∂
u
∂
u
J
22
∂
x
1
−
J
12
∂
r
=
(9.76)
∂
u
−
J
21
J
11
∂
u
|
J
|
∂
y
∂
s
The determinant of the Jacobian matrix
|
J
|
is commonly called simply the
Jacobian.
Since the interpolation functions are the same for both displacement compo-
nents, an identical procedure results in
∂
v
∂
v
J
22
∂
x
1
−
J
12
∂
r
=
(9.77)
∂
v
−
J
21
J
11
∂
v
|
J
|
∂
y
∂
s
for the partial derivatives of the
v
displacement component with respect to global
coordinates.
Let us return to the problem of computing the strain components per
Equation 9.70. Utilizing Equations 9.76 and 9.77, the strain components are
expressed as
∂
u
∂
u
∂
∂
u
∂
r
r
∂
x
∂
∂
u
u
∂
ε
x
ε
y
xy
J
22
−
J
12
0
0
∂
v
1
∂
s
s
{
ε
}
=
(9.78)
=
=
=
[
G
]
0
0
−
J
21
J
11
∂
y
|
|
∂
v
∂
v
J
−
J
21
J
11
J
22
−
J
12
∂
u
y
+
∂
v
∂
r
∂
r
∂
∂
x
∂
v
∂
v
∂
s
∂
s
