HOMEWORK PROBLEMS
Week 10: Due Friday 18 November
1.
A cylindrical pressure vessel of internal diameter 80 inches and wall thickness 1 inch has
two flat ends of thickness 1.5 in, each of which is secured to the cylindrical part by 20 bolts,
each of 1 inch diameter. The crosssection of the vessel is shown in Figure 1. The pressure of
gas inside the vessel is 300 psi. Find (i) the axial and circumferential membrane stresses in the
cylinder wall and (ii) the tensile stress in the bolts.
180
80
1.5
1
20 bolts, 1 in. diameter
each end
all dimensions in inches
Figure 1
Solution
The tensile stresses in the vessel wall can be obtained using the formulae
σ
z
=
pR
2
t
;
σ
θ
=
pR
t
.
Here, we obtain
σ
z
=
300
×
40
2
×
1
= 6000
psi
;
σ
θ
=
300
×
40
1
= 12
,
000
psi
.
1
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If we
cut through the bolts between the ends of the vessel and the cylindrical portion, we expose the
freebody diagram of Figure 1.1. The right figure shows an end view, showing how all 20 bolts
are sawn through and the left figure shows a side view, in which the forces acting on the plate
due to the pressure
p
and the tensile stresses in the bolts can be seen.
p
σ
bolts
σ
bolts
sawnthrough bolts
Figure 1.1
The stress
σ
bolts
acts on a total area equal to 20 times that of a single bolt — i.e.
A
bolts
= 20
×
π
×
1
2
4
= 15
.
71
in
2
and the pressure
p
acts on a circular area of diameter 80 inches and hence
A
p
=
π
×
80
2
4
= 5026
in
2
.
For axial equilibrium, we have
σ
bolts
A
bolts

pA
p
= 0
or
σ
bolts
=
300
×
5026
15
.
71
= 95
,
987
psi
.
Notice that equilibrium statements relate to forces, not stresses, and to write an equilibrium
equation involving unknown stresses, we first need to convert them to forces by multiplying by
the area on which they act.
2
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 Fall '07
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 Statics, Force, in2, vessel, maximum tensile stress, Tensile stress

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