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P 2
P 3
P 1
k w
2.0
6.0
10.0
3.0
21.0
-2.06
-0.48
Exact
solution
1.10
2.32
2.32
-2.06
-2.06
3.89
4
1 23
8
-0.35
-0.35
5
6
7
9
A
0.58
0.58
1.10
1.10
2.31
2.31
-2.06
-2.06
3.89
3.89
-0.65
-1.12
-0.65
-0.48
-0.48
12
1
2
3
5
10
11
8
B
0.58
-0.40
0.58
1.10
1.10
2.32
2.32
3.89 3.89
-2.06
-2.06
-0.48 -0.48
1 2 3 4 5
9 10 11 12
13 14 15 16 17 18 19 20 21 22 23
C
8
7
1.10
1.81 1.81
1.10
2.32
2.32
3.89
3.89
FIGURE 6.6
The exact moment distribution followed by the moment distributions for mesh refinements A, B, and C, using
nodal forces
p i 0 . For A, B, and C, the nodal values are arbitrarily connected with straight lines.
=
p i
k i
v i
Consider the case where the moment distributions are determined from the material law
w . Since the second derivative of the assumed
M
=−
EI
w(
x
)
leads to a linear expression,
and only the continuity of
are enforced at the nodes, a piecewise linear distribution
of the moment with possible jumps in value at the nodes can occur. See Fig. 6.7. Relative
to the case of nodal forces (Fig. 6.6), the approximation here, involving derivatives of the
deflection, is not as accurate. Remember that the equilibrium conditions for the element are
satisfied only in an average sense.
w
and
θ
Shear Force Distributions
Figure 6.8 displays the shear force distributions for the exact solu tio n as well as for the three
mesh refinements determined from the nodal forces
p i 0 . The nodal values of V
= k i
p i
v i
are arbitrarily linearly connected.
Shear forces distributions determined from the material law V
w are shown in
Fig. 6.9. The third derivative of the deflection assumption (cubic polynomial) is constant.
The use of this method to find the shear force leads to a piecewise constant distribution of
shear force between the nodes, whereas the exact solution is linear. Jumps can occur at the
element boundaries and may be used as an indication of error level. The use of derivatives
of the shape function results in a poorer approximation than that of Fig. 6.8.
=−
EI
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