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SJTU
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海
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大
学
Introduction to Solid MechanicsVm211
Chapter 14 Combined Loadings
Combined
Loadings
SJTU
上
海
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大
学
Introduction to Solid MechanicsVm211
CHAPTER OUTLINE
1.
ThinWalled Pressure Vessels
2.
State of Stress Caused by Combined Loadings
SJTU
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海
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大
学
Introduction to Solid MechanicsVm211
THINWALLED PRESSURE VESSELS
±
Cylindrical or spherical vessels are commonly used in
industry to serve as boilers or tanks
±
When under pressure, the material which they are made
of, is subjected to a loading from all directions
±
However, we can simplify the analysis provided it has
a thin wall
SJTU
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大
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Introduction to Solid MechanicsVm211
THINWALLED PRESSURE VESSELS
±
“Thin wall”
refers to a vessel having an innerradius
thickness ratio of 10 or more (
r/t
≥
10
)
±
When
r
/
t
= 10, results of a thinwall analysis will
predict a stress approximately 4% less than the actual
maximum stress in the vessel
SJTU
上
海
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大
学
Introduction to Solid MechanicsVm211
THINWALLED PRESSURE VESSELS
±
Assumption taken before analysis is that
the thickness of
the pressure vessel is uniform
or constant throughout
±
The pressure in the vessel is understood to be the
gauge
pressure
, since it measures the pressure
above
atmospheric pressure which is assumed to exist both
inside and outside the vessel’s wall
SJTU
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大
学
Introduction to Solid MechanicsVm211
THINWALLED PRESSURE VESSELS
Cylindrical vessels
±
A gauge pressure
p
is developed within the vessel by a
contained gas or fluid, and assumed to have
negligible
weight
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SJTU
上
海
交
通
大
学
Introduction to Solid MechanicsVm211
THINWALLED PRESSURE VESSELS
Cylindrical vessels
±
Due to uniformity of loading, an element of the vessel
is subjected to normal stresses
σ
1
in the
circumferential
or hoop direction
and
2
in the
longitudinal or axial
direction
SJTU
上
海
交
通
大
学
Introduction to Solid MechanicsVm211
THINWALLED PRESSURE VESSELS
Cylindrical vessels
±
We use the method of sections and apply the equations
of force equilibrium to get the magnitudes of the stress
components
±
For equilibrium in the
x
direction, we require
Σ
F
x
=0;
2
[
σ
1
(
tdy
)]
−
p
(
2rdy
)
= 0
pr
t
1
=
SJTU
上
海
交
通
大
学
Introduction to Solid MechanicsVm211
THINWALLED PRESSURE VESSELS
Cylindrical vessels
±
As shown,
σ
2
acts uniformly throughout the wall, and
p
acts on the section of gas or fluid. Thus for equilibrium
in the
y
direction, we require
Σ
F
y
=0;
σ
2
(2
π
rt
)
p
(
r
2
)
= 0
pr
2
t
2
=
SJTU
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大
学
Introduction to Solid MechanicsVm211
THINWALLED PRESSURE VESSELS
Cylindrical vessels
±
For Eqns ,
1
,
2
= normal stresses in the hoop and axial directions,
respectively. Each is assumed to be constant
throughout the wall of the cylinder, and each subjects
the material to tension
pr
t
1
=
pr
2
t
2
=
SJTU
上
海
交
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大
学
Introduction to Solid MechanicsVm211
THINWALLED PRESSURE VESSELS
Cylindrical vessels
±
p
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This note was uploaded on 08/09/2011 for the course EE 211 taught by Professor Liuxila during the Summer '09 term at Shanghai Jiao Tong University.
 Summer '09
 LiuXila

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