Civil Engineering Reference
In-Depth Information
also be modeled beyond those boundaries with some confidence. If the
general equation is continuous at the boundaries, then this type of model
is appropriate. If not, then the model should end at the boundaries and for-
ward or central difference operators must be used. Figure 3.4 shows the
beam and the assumed deflections at 1/6th points of the beam. By symme-
try of the model, only four specific values of the deflection are unknown.
Z
4 k/ft
X
A
B
25′-0″
-Y
3
-Y
2
-Y
1
Y
0
Y
1
Y
2
Y
3
Y
2
Y
1
Y
0
-Y
1
-Y
2
-Y
3
Figure 3.4.
Example 3.10 Simple beam with difference operator.
The central-difference expressions with error of order
h
2
will be used
to solve for the values. Since the load is known, we will use the fourth
derivative relationship between load and deflection.
q
EI
y
−
4
y
+
6
y
−
4
y
+
y
y
′′ ′′=− =
i
+
2
i
+
1
i
i
−
1
i
−
2
i
4
h
EI
h
(
)
q
=−
y
−
4
y
+
6
y
−
4
y
+
y
i
+
2
i
+
1
i
i
−
1
i
−
2
4
Placing the central difference operator on
y
0
, the first equation can be writ-
ten from Figure 3.5:
-Y
3
-Y
2
-Y
1
Y
0
Y
1
Y
2
Y
3
Y
2
Y
1
Y
0
-Y
1
-Y
2
-Y
3
0
1
-4
6
-4
1
0
0
0
0
0
0
0
Figure 3.5.
Example 3.10 Simple beam with difference operator.
EI
h
EI
h
(
)
=−
(
(
)
−−
(
)
+−+
)
q
=−
−
y
4
y
6
y
4
y
y
6
y
0
2
1
0
1
2
0
4
4
