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Hint:
0 on the boundary. Let the origin of the
y, z
coordinate
system be in the center of the region. The fact that the solution should be symmetric
about the
z
and
y
coordinate axes should be reflected in the choice of the trial
functions. A possible choice is to use
Recall that
ψ
=
ψ
=
i
ψ
i
N
ui
,
with
j
π
z
cos
k
π
y
N
ui
=
cos
j
=
1
,
3
,
...
k
=
1
,
3
,
...
6
4
Answer:
(
M
t
)
exact
=
76
.
4
,
τ
max
=
2
.
96
7.25 Redo the torsion example of Problem 7.24. Use a trial solution that satisfies the
y
=+
2
2 boundary conditions. Choose, for example,
ψ
=
i
ψ
−
and
i
N
ui
,
with
y
2
y
2
x
2
,
y
2
y
2
N
u
1
=
(
4
−
)
,
=
(
4
−
)
=
(
4
−
)
u
2
u
3
y
2
x
2
y
2
,
y
2
x
4
N
u
4
=
(
4
−
)
=
(
4
−
)
...
u
5
7.26 The structural member shown in Fig. P7.26 consists of a thin linearly elastic rect-
angular sheet of unit thickness att
ac
hed to two edge bars. It is required to support
the longitudinal distributed force
p
on the bars. Assume the height of the sheet
does not change. Use the Ritz method to describe the stress distribution in the sheet.
Assume a displacement pattern of the form
(
x
)
u
1
sin
π
x
2
a
cosh
k
y
2
b
π
u
=
N
u
1
u
1
=
v
=
0
and
u
1
is the parameter to be determined. The bars have a constant cross-section
A
and the thickness of the sheet is
t
.
Hint:
Let
u
=
N
u
1
u
1
and
u
1
u
1
a
−
a
b
Edxdy
[
N
u
1
x
(
N
u
1
x
(
N
u
1
x
(
δ
W
=
δ
x, b
)
+
x,
−
b
)
]
EAdx
+
x, y
)
−
−
a
a
b
Bars
Plate
a
]
pdx
−
[
N
u
1
(
x, b
)
+
N
u
1
(
x,
−
b
)
=
0
−
a
Applied Loading
FIGURE P7.26
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