Civil Engineering Reference
In-Depth Information
For the second equation, the central difference operator is placed on
y
1
and
is shown in Figure 3.6.
0
0
1
-4
6
-4
1
0
0
0
0
0
0
Figure 3.6.
Example 3.
10
Simple beam with difference operator.
EI
h
EI
h
(
)
=− −+−+
(
)
−+−+
(
)
q
=−
−
y
464
y
y
y
y
4 54
y
y yy
1
1
0
1
2
3
0
1
2
3
4
4
The third and fourth equations can be written by placing the central differ-
ence operator on
y
2
and
y
3
.
EI
h
EI
h
(
)
=− −+−
(
)
q
=− −+−+
y
464
y
y
y
y
y
4 74
y
y
y
2
0
1
2
3
2
0
1
2
3
4
4
EI
h
y yy
EI
(
)
=− −+
(
)
q
=−
y
−+−+
464
y
h
yyy
2 86
3
1
2
3
2
1
1
2
3
4
4
These four equations constitute a non-homogeneous linear algebraic set
and can be written in matrix form.
6000
45 41
1474
0286
y
y
y
y
q
q
q
q
0
0
−
−
EI
h
1
1
−
=
4
−
−
2
2
−
3
3
From the conditions of the beam, two simplifications can be made. First,
the load is uniform and all the values of
q
are the same. Second, the deflec-
tion at point 0 is known to be zero, so the first row and column can be
eliminated since they correspond to those values.
541
47 4
286
−
y
y
y
q
q
q
1
EI
h
−
−
−
=
2
4
−
3
This can be solved by many of the methods presented in Chapter 2. The
method of cofactors is used here, since the solution is small enough to
solve determinants directly.
