Civil Engineering Reference
In-Depth Information
Again the continuous variable,
w
in this case, is approximated in terms of discrete
nodal values, but we introduce the idea that not only
w
itself but also its derivatives
θ
can
be used in the approximation. In this case the continuous variable
w
is approximated by
w
in terms of nodal values as follows:
w
1
θ
1
w
2
θ
2
w
=
[
N
1
N
2
N
3
N
4
]
=
[
N
]
{
w
}
(2.21)
d
w/
d
x
at node 1, and so on. In this case, (2.21) can often be made exact by
choosing the cubic shape functions:
where
θ
1
=
1
L
3
2
3
N
1
=
3
(L
−
3
Lx
+
2
x
)
1
L
2
2
3
N
2
=
2
(L
x
−
2
Lx
+
x
)
1
L
2
3
N
3
=
3
(
3
Lx
−
2
x
)
(2.22)
1
L
3
2
N
4
=
2
(x
−
Lx
)
Note that the shape functions have the property that they, or their derivatives in this
case, equal one at a specific node and zero at all others.
Substitution in (2.20) and application of Galerkin's method leads to the four element
equations:
N
1
N
2
N
3
N
4
N
1
N
2
N
3
N
4
w
1
θ
1
w
2
θ
2
L
L
d
4
d
x
EI
4
[
N
1
N
2
N
3
N
4
] d
x
=
q
d
x
(2.23)
0
0
Again Green's theorem is used to avoid differentiating four times; for example
d
4
d
3
d
2
d
2
N
j
d
x
d
N
i
d
x
N
j
d
x
N
i
d
x
N
j
d
x
N
i
d
x
≈−
d
x
≈
d
x
+
neglected terms
(2.24)
4
3
2
2
Hence assuming
EI
and
q
are not functions of
x
, (2.23) becomes
N
1
N
2
N
3
N
4
w
1
θ
1
w
2
θ
2
d
2
EI
L
0
L
d
2
N
i
d
x
N
j
d
x
d
x
=
q
d
x
(2.25)
2
2
0
i,j
=
1
,
2
,
3
,
4
