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where
P
2
A
sin
σ
=
=
stress in each member
θ
Pb
2
E A
sin 2
δ
=
=
displacement at the load
P
θ
sin
θ
10
9
Pa.
15.
SolveProb. 14 if the allowable displacement ischanged to 2
and
E
=
200
×
.
5 mm.
16.
r
1
r
2
L
= 1.0 m
L
= 1.0 m
P
= 10 kN
The cantileverbeam of circular cross sectionistohave the smallest volume pos-
sible subject to constraints
σ
1
≤
180MPa
σ
2
≤
180MPa
δ
≤
25 mm
where
8
PL
π
σ
1
=
r
1
=
maximum stress in left half
4
PL
π
σ
2
=
r
2
=
maximum stress in right half
7
r
1
+
4
PL
3
3
1
r
2
δ
=
=
displacement atfree end
π
E
and
E
=
200 GPa. Determine
r
1
and
r
2
.
17.
Find the minimum of the function
2
x
2
3
y
2
z
2
,
,
=
+
+
+
+
−
F
(
x
y
z
)
xy
xz
2
y
and confirm the result analytically.
18.
r
h
b
The cylindricalcontainer has a conical bottom and an open top. If the volume
V
of the containeristobe 1.0 m
3
, find the dimensions
r
,
h
and
b
that minimize the
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