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FIGURE 6.19
Principal stresses at the centroid of the elements of the
model of Fig. 6.14b.
FIGURE 6.20
A beam model of the structure of Fig. 6.14a.
A comparison can be made between these midspan values and those obtained with the
finite element solution. For the structure of Fig. 6.14a, the appropriate displacement is u y of
global node 3, which from (8) is
10 4 m. This is significantly smaller than the exact
value; consequently, it can be observed that the behavior of the approximate structural
model is too stiff. For a comparison of the moment, first compute the
0
.
88
×
σ x stresses along
the vertical line between nodes 3 and 9 using (50):
σ x 9 =−
692
.
53 kPa,
σ x 6 =−
57
.
40 kPa,
and
33 kPa. The resultant moment about node 6 will be calculated. For the sake
of this calculation,
σ
=
807
.
x 3
x 6 will be set to zero since 57.40 is small in comparison to 692.53 and
807.33. The total force in the stress triangle between nodes 6 and 9 is 692
σ
.
53
×
1
.
0
×
0
.
2
/
2
=
69
.
25 kN, where the area on which the stress acts is 1
.
0
×
0
.
2. The total force between nodes
6 and 3 would be 807
73 kN. These forces are assumed to act through
the centroid of the triangles as shown in Fig. 6.21. The moment at node 6 of the beam then
would be
.
33
×
1
.
0
×
0
.
2
/
2
=
80
.
=
.
×
/
+
.
×
/
=
.
M
69
25
2
3
80
73
2
3
99
99 kNm
(13)
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