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Chapter Objectives Determine stresses developed in thin-walled pressure
vessels
Determine stresses developed in a members cross
section when axial load, torsion, bending and shear
occur simultaneously.
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1. Thin-Walled Pressure Vessels
2. Review of Stress Analyses
In-class Activities
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THIN-WALLED PRESSURE VESSELS
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Assumptions:
1. Inner-radius-to-wall-thickness ratio 102. Stress distribution in thin wall is uniform or constant
Cylindrical vessels:
2
:stressalLongitudin
:directionHoop
2
1
t
pr
t
pr
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THIN-WALLED PRESSURE VESSELS (cont)
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Spherical vessels:
22
t
pr
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EXAMPLE 1
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A cylindrical pressure vessel has an inner diameter of 1.2 m
and a thickness of 12 mm. Determine the maximum internalpressure it can sustain so that neither its circumferential nor
its longitudinal stress component exceeds 140 MPa. Under
the same conditions, what is the maximum internal pressure
that a similar-size spherical vessel can sustain?
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EXAMPLE 1 (cont)
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The stress in the longitudinal direction will be
The maximum stress in the radial direction occurs on the material at
the inner wall of the vessel and is
The maximum stress occurs in the circumferential direction.
Solutions
(Ans)MPa28N/mm8.212
600140
2
1
p
p
t
pr
MPa701402
12
MPa8.2(max)3 p
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EXAMPLE 1 (cont)
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The maximum stress occurs in any two perpendicular directions on an
element of the vessel is
Solutions
(Ans)MPa5.6N/mm6.5
122
600
140
2
2
2
p
p
t
pr
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REVIEW OF STRESS ANALYSES
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Normal forceP leads to:
Shear force V leads to:
Bending moment M leads to:
A
Pstressnormaluniform ,
It
VQondistributistressshear ,
beam)curved(foror
beam)straight(for,
yRAe
MyI
Myondistributistressallongitudin
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REVIEW OF STRESS ANALYSES (cont)
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Torsional moment T leads to:
Resultant stresses by superposition:
Once the normal and shear stress components for each
loading have been calculated, use the principal of
superposition to determine the resultant normal and shearstress components.
tube)walled-thinclosed(for
2
shaft)circular(for,
tA
T
JTondistributistressshear
m
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EXAMPLE 2
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A force of 15 kN is applied to the edge of the member shown in
Fig. 83a. Neglect the weight of the member and determine the
state of stress at points B and C.
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EXAMPLE 2 (cont)
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For equilibrium at the section there must be an axial force of 15 000 N
acting through the centroid and a bending moment of 750 000 Nmm
about the centroidal or principal axis.
The maximum stress is
Solutions
MPa75.3
40100
15000
A
P
MPa25.1110040
12
1
5075000
3max I
Mc
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EXAMPLE 2 (cont)
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The location of the line of zero stress can be determined by proportional
triangles
Elements of material at B and Care subjected only to normal oruniaxialstress.
Solutions
mm3.33
100
1575
x
xx
(Ans)on)(compressiMPa15
(Ans)(tension)MPa5.7
C
B
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EXAMPLE 3
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The tank in Fig. 84a has an inner radius of 600 mm and athickness of 12 mm. It is filled to the top with water having a
specific weight ofw = 10 kNm3. If it is made of steel having aspecific weight ofst = 78 kNm
3, determine the state of stress atpointA. The tank is open at the top.
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EXAMPLE 3 (cont)
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The weight of the tank is
The pressure on the tank at levelA is
For circumferential and longitudinal stress, we have
Solutions
kN56.311000
600
1000
61278
22
ststst VW
kPa10110 zp w
(Ans)kPa9.77
56.3
(Ans)kPa50010
2
10006002
1000612
2
100012
1000600
1
st
st
A
W
t
pr
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EXAMPLE 4
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The member shown in Fig. 85a has a rectangular cross
section. Determine the state of stress that the loading produces
at point C.
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EXAMPLE 4 (cont)
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The resultant internal loadings at the section consist of a normal force, a
shear force, and a bending moment.
Solving,
Solutions
kN89.32kN,21.93kN,45.16 MVN
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EXAMPLE 4 (cont)
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The uniform normal-stress distribution acting over the cross section is
produced by the normal force.
At Point C,
In Fig. 85e, the shear stress is zero.
Solutions
MPa32.125.005.0
1045.16 3
A
Pc
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EXAMPLE 4 (cont)
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Point Cis located at y = c = 0.125m from the neutral axis, so the normal
stress at C, Fig. 85f, is
Solutions
MPa16.6325.005.0
125.01089.323
21
3
I
Mcc
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EXAMPLE 4 (cont)
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The shear stress is zero.
Adding the normal stresses determined above gives a compressive
stress at Chaving a value of
Solutions
MPa5.6416.6332.1 I
Mcc
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EXAMPLE 5
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The rectangular block of negligible weight in Fig. 86a is
subjected to a vertical force of 40 kN, which is applied to its
corner. Determine the largest normal stress acting on a section
throughABCD.
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EXAMPLE 5 (cont)
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For uniform normal-stress distribution the stress is
For 8 kN, the maximum stress is
For 16 kN, the maximum stress is
Solutions
kPa125
4.08.0
40
A
P
kPa3754.08.0
2.083
121max
x
xx
I
cM
kPa3758.04.0
4.0163
121max
y
xy
I
cM
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EXAMPLE 5 (cont)
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By inspection the normal stress at point Cis the largest since each
loading creates a compressive stress there
Solutions
(Ans)kPa875375375125 c
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EXAMPLE 6
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EXAMPLE 5 (cont)
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EXAMPLE 5 (cont)
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EXAMPLE 5 (cont)
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EXAMPLE 5 (cont)
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EXAMPLE 5 (cont)