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FIGURE P7.21
7.21 A simply supported beam carries the load shown in Fig. P7.21. Use a two-term trigono-
metric series to calculate the approximate midspan deflection. Use the Ritz and the
Galerkin methods and compare to the exact solution.
7.22 Use the Trefftz method to determine the displacement and internal forces along the
beam of Fig. P7.22.
Answer:
For the trial solution,
p
0
L
4
360
EI
ξ
6
2
3
w
=
N
p
+
N
u
w
=
+
ξ w
+
ξ
w
+
ξ
w
1
2
3
the Trefftz method gives
p
0
L
4
360
EI
(
2
3
6
2
3
6
w
=
3
ξ
−
4
ξ
+
ξ
)
=
0
.
0576
ξ
−
0
.
0768
ξ
+
0
.
0192
ξ
which is also the exact beam theory solution.
FIGURE P7.22
7.23 Compare the Trefftz method with that of Ritz. How do these approaches differ from
Galerkin's method? If you wish, use the beam and results of the previous problem to
assist in preparing your answer.
7.24 From Example 7.8 the force form of the torsion problem involves the governing equa-
tion
2
2
∂
z
2
+
∂
ψ
ψ
y
2
=−
2
∂
∂
Find the twisting moment
M
t
corresponding to the above equation being satisfied
in a 6
rectangular region. Also calculate the peak shear stress. Use a
trigonometric trial solution.
×
4
(
z
by
y
)
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