Civil Engineering Reference
In-Depth Information
Now, the mathematical complications arise in computing the strain components
as given by Equation 9.55 and rewritten here as
∂
∂
∂
u
0
=
x
∂
x
u
v
ε
x
ε
y
xy
∂
∂
∂
v
0
{
ε
}=
=
(9.70)
y
∂
y
∂
u
y
+
∂
v
∂
∂
∂
∂
∂
∂
x
y
x
Using Equation 6.83 with
=
u
,
we have
∂
u
∂
u
x
∂
x
r
+
∂
u
y
∂
y
r
=
∂
∂
∂
∂
∂
r
(9.71)
∂
∂
x
∂
s
+
∂
y
∂
u
u
x
u
y
s
=
∂
∂
∂
∂
∂
s
with similar expressions for the partial derivative of the
v
displacement. Writing
Equation 9.71 in matrix form
∂
u
∂
u
∂
u
∂
x
∂
y
∂
x
∂
x
∂
r
∂
r
∂
r
=
=
[
J
]
(9.72)
∂
u
∂
u
∂
u
∂
x
∂
y
∂
y
∂
y
∂
s
∂
s
∂
s
and the Jacobian matrix is identified as
∂
x
∂
y
J
11
J
12
∂
r
∂
r
[
J
]
=
=
(9.73)
J
21
J
22
∂
x
∂
y
∂
∂
s
s
as in Equation 6.83. Note that, per the geometric mapping of Equation 9.68, the
components of
[
J
]
are known as functions of the partial derivatives of the inter-
polation functions and the nodal coordinates in the
xy
plane. For example,
4
∂
x
∂
N
i
∂
1
4
[(
s
s
)
x
4
]
(9.74)
a first-order polynomial in the natural (mapping) coordinate
s
. The other terms
are similarly first-order polynomials.
Formally, Equation 9.72 can be solved for the partial derivatives of dis-
placement component
u
with respect to
x
and
y
by multiplying by the inverse of
the Jacobian matrix. As noted in Chapter 6, finding the inverse of the Jacobian
matrix in algebraic form is not an enviable task. Instead, numerical methods are
used, again based on Gaussian quadrature, and the remainder of the derivation
here is toward that end. Rather than invert the Jacobian matrix, Equation 9.72
J
11
=
r
=
x
i
=
−
1)
x
1
+
(1
−
s
)
x
2
+
(1
+
s
)
x
3
−
(1
+
∂
r
i
=
1
