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FIGURE 8.7
The fictitious boundary conditions in Example 8.2.
From Chapter 5, the exact values are
w
1
w
2
w
3
w
0
.
000864
(
44%
)
=
p
0
L
4
EI
0
.
002048
(
25%
)
(10)
0
.
002352
(
19%
)
0
.
001536
(
16%
)
4
where the percent error of the approximate value is given in parentheses.
The errors in the displacement calculation are immense. Better accuracy would be ob-
tained with a finer grid or by using an improved or multiple position difference formula.
The bending movement and shear force can be obtained from the equations
w
w
M
=−
EI
and
V
=−
EI
(11)
From Table 8.1, simple central derivatives for
M
and
V
at node
i
are
h
2
M
i
=−
EI
(
1
/
)(w
i
−
1
−
2
w
i
+
w
i
+
1
)
(12)
2
h
2
V
i
=−
EI
(
1
/
)(
−
w
i
−
2
+
2
w
i
−
1
−
2
w
i
+
1
+
w
i
+
2
)
These lead to the following results
Bending Moment
Shear Force
M
exact
p
0
L
2
M
fnt diff
p
0
L
2
V
exact
p
0
L
V
fnt diff
p
0
L
ξ
% diff
% diff
0
−
0
.
0667
−
0
.
06215
6.8
0.4
0.40238
0.6
0.4
0.024
0.02675
11.5
0.08
0.082
2.5
0.6
0.0293
0.3113
6.1
−
0
.
02
−
0
.
0179
10.3
1.0
0
0
—
−
0
.
10
−
0
.
0979
2.1
The above example dea
lt
with a member with distributed loadin
g.
In solving the differ-
ential equation
EI
i
v
p
for a beam with a concentrated force
P
at point
i,
it
is
oft
en
convenient to model the concentrated force as a distributed loading. Usually,
p
i
=
w
=
P
/
h
will suffice.
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