Civil Engineering Reference
In-Depth Information
Equations 4.18-4.21 are solved simultaneously to obtain the coefficients in terms
of the nodal variables as
a
0
=
v
1
(4.22)
a
1
=
1
(4.23)
3
L
2
(
v
2
−
1
L
(2
a
2
=
v
1
)
−
1
+
2
)
(4.24)
2
L
3
(
v
1
−
1
L
2
(
a
3
=
v
2
)
+
1
+
2
)
(4.25)
Substituting Equations 4.22-4.25 into Equation 4.17 and collecting the coeffi-
cients of the nodal variables results in the expression
1
v
1
+
x
3
x
2
L
2
2
x
3
L
3
2
x
2
L
x
3
L
2
v
(
x
)
=
−
+
−
+
1
3
x
2
L
2
v
2
+
x
3
L
2
2
x
3
L
3
x
2
L
+
−
−
2
(4.26)
which is of the form
v
(
x
)
=
N
1
(
x
)
v
1
+
N
2
(
x
)
1
+
N
3
(
x
)
v
2
+
N
4
(
x
)
2
(4.27a)
or, in matrix notation,
v
1
1
v
2
2
v
(
x
)
=
[
N
1
N
2
N
3
N
4
]
=
[
N
]
{}
(4.27b)
where
N
1
,
N
2
,
N
3
, and
N
4
are the interpolation functions that describe the dis-
tribution of displacement in terms of nodal values in the nodal displacement
vector
{}
.
For the flexure element, it is convenient to introduce the dimensionless
length coordinate
x
L
=
(4.28)
so that Equation 4.26 becomes
v
(
x
)
2
3
)
v
1
+
2
3
)
2
3
)
v
2
=
(1
−
3
+
2
L
(
−
2
+
1
+
(3
−
2
2
(
+
L
−
1)
2
(4.29)
where
0
≤ ≤
1
.
This form proves more amenable to the integrations required
to complete development of the element equations in the next section.
As discussed in Chapter 3, displacements are important, but the engineer is
most often interested in examining the stresses associated with given loading
conditions. Using Equation 4.11 in conjunction with Equation 4.27b, the normal
