Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
Lecture Note for Solid Mechanics- Pressure Vessels and Axial Loading Applications -
Prof. Sung Ho YoonDepartment of Mechanical EngineeringKumoh National Institute of Technology
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
Text book : Mechanics of Materials, 6th ed.,
W.F. Riley, L.D. Sturges, and D.H. Morris, 2007.
Prerequisite : Knowledge of Statics, Basic Physics, Mathematics, etc.
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
Theory of elasticity
Determining internal forces and deformations at all points subjected to external forces
To obtain general solutions for complex loading and geometry
Pressure Vessels and Axial Loading Applications
Mechanics of materials
Real structural elements are analyzed as idealized models subjected to simplified loading and restraints
To obtain approximate solutions Practical to most design problems
Deformations of axial loaded members
Uniform member
Multiple Loads/sizesEAPLand
ELL
n
i ii
iin
ii AE
LP11
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
Nonuniform deformation (Varying axial force or geometry)
- Axial strain :
- Incremental deformation :
- Hooke’s law :
dxEAPd
x
x
dxd
dLd
dxd
x
x
EAP
E
L
x
xLdx
EAPd
00
Pressure Vessels and Axial Loading Applications
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
(Example) Rigid yokes B and C are fastened to 2-in square steel (E=30,000 ksi) bar AD. Determine (a) maximum normal stress in the bar (b) change in length of the segment AB (c) change in length of segment BC (d) change in length of complete bar.
(Sol)
)(5.204
82maxmax Tksi
AP
(a)
inAELP
ABAB
ABABAB 0656.0
)4(000,30)12)(8(82
(b)
inAELP
BCBC
BCBCBC 006.0
)4(000,30)12)(5(12
(c)
in
inAELP
CDBCABAD
CDCD
CDCDCD
0776.0018.0006.00656.0
018.0)4(000,30)12)(4(45
(d)
Pressure Vessels and Axial Loading Applications
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
(Example) Determine (a) elongation of the bar due to its own weight W in terms of W, L, A and E (b) elongation of the bar if the bar is also subjected to an axial tensile force P at its lower end
(Sol)
Axial force is a function of x, the distance from the free end of the bar.The weight of the segment is
AxVW xx
ALWAE
WLE
LALW
EL
EL
Exxdx
E
xdxE
AxdxEA
dxEAP
LL
LLL
x
x
222
22
1
22
2
0
2
0
000
Pressure Vessels and Axial Loading Applications
(a)
(b) Elongation would be found using the method of superposition.
EAPLPW
EAL
EAPL
EAWL
22
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
Deformations in a system of axially loaded bars
Determining axial deformations and strains in pin-connected deformable bars
Axial deformations of the bars in the system through a study of geometry of the deformed system
BABBBBABAB
BBifAB
vuvLvLLL
LuvLLL
22222
22
22
In a similar manner
Axial deformation in the bar AB
Axial deformation in the bar BC
222
22222
22
sin2sin
cos2cos2
sincos
BB
BBBCBC
BBBC
vRvR
uRuRRR
RvRuR
cossincossin
BEAB
BBBC uv
Pressure Vessels and Axial Loading Applications
BBEBBBBEBE
BBifBE
uvuLuLLL
LvuLLL
22222
22
22
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
Statically indeterminate axially loaded members
Statically determinate system
- reactions at supports and the forces in the individual members can be found by solving equilibrium equations
Statically indeterminate system
- equilibrium equations are not sufficient for the determinations of reactions at supports and the forces in the individual members
- additional equations involving the geometry of deformations are needed
RWaT
WaTRM B
0
(1) Assume that cable and lever are rigid
Pressure Vessels and Axial Loading Applications
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
cos
cos
2 WaLEARR
RWa
LEA
EATL
(2) Assume that cable is deformable
cos
0cos
RWaT
WaTRM B
(Example) The cable is rigid.If W=100 lb, a=30 in, R=15 in, therefore T=200 lb
(Example) The cable is a 3/32 in diameter steel (E=29,000 psi) wire.If W=100 lb, a=30 in, R=15 in, L=45 in,
Then =0.002997 rad=0.1717 deg, therefore T=199.999 lb
%0005.0)100(999.199
999.199200%
D
Pressure Vessels and Axial Loading Applications
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
(Example) The cable is a 3/32 in diameter aluminum (E=10,600 psi) wireIf W=100 lb, a=30 in, R=15 in, L=45 in,
Then =0.00820 rad=0.4698 deg, therefore T=199.993 lb.
%0035.0)100(993.199
993.199200%
D
Tension in the wire changes very little when the stiffness of the wire is changed by a factor of almost 3 and both values are essentially the same as that obtained using rigid wire assumption.
Pressure Vessels and Axial Loading Applications
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
(Example) A ½-in diameter alloy-steel bolt (E=30,000 ksi) passes through a cold rolled brass sleeve (E=15,000 ksi). The cross sectional area of the sleeve is 0.375 in2. Determine the normal stresses produced in the bolt and sleeve by tightening the nut ¼ turn (0.020 in).
(Sol)
BS
BS
BSx FFF
5236.021
4375.0
0:02
The FBD contains two unknowns : statically indeterminateAs the nut is turned, 0.020 in, if the sleeve is not present.If sleeve is present, the movement is resisted.
Tensile stress in the bolt and compressive stressin the sleeve
1002020.0000,15
)6(000,30
)6(
SBSB
S
SS
B
BBBs E
LE
L
)(6.25)(8.48 CksiTksi SB
Pressure Vessels and Axial Loading Applications
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
Thermal effects
When temperature change takes place while a member is restrained, thermal stresses are induced in the member
LE
LT
LLTTtotal
0
If the temperature of the bar increases (T positive), the induced stress must be negative and the wall must push on the ends of the rod.
If the temperature of the bar decreases (T negative), the induced stress must be positive and the wall must pull on the ends of the rod.
Pressure Vessels and Axial Loading Applications
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
(Example) The 10-m section of steel rail (E=200GPa, =11.9x10-6/oC) has a cross section of 7500mm2. Both ends of the rail are tight against adjacent rigid rail. For increase in temperature of 50oC, determine (a) the normal stress in the rail (b) the internal force on the cross section of the rail.
(Sol) Since temperature increases, the deformation shownin figures are reversed.
)(0.119)10(0.119)10200)(50)(109.11(
0
26
96
CMPamNET
LE
LT
LLTTtotal
(a)
)(893)10(5.892)10)(7500)(10(0.119
3
66
CkNNAF
(b)
Pressure Vessels and Axial Loading Applications
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
Stress concentrations
If there exists a discontinuity in the structural or machine element, the stress at the discontinuity may be considerably greater than the nominal stress on the section
Stress concentration factor (K) : Ratio of the maximum stress to the nominal stress
Maximum normal stress
A : gross area or net area depending on the value used for K
APK
Pressure Vessels and Axial Loading Applications
gg A
PKt
t APKor
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
Pressure Vessels and Axial Loading Applications
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
Stress distributions based on theory of elasticity
: uniform tensile stress in the plane in the regions far from the hole
23212
2312
12
23412
12
4
4
2
2
4
4
2
2
4
4
2
2
2
2
sin
cos
cos
ra
ra
ra
ra
ra
ra
ra
r
r
On the boundary of the hole at r=a
0221
0
r
r
)cos(
At =0, 3
4
4
2
2 322 r
ara
The localized nature of a stress concentration can be evaluated by the consideration of tangential stress along the x-axis
Pressure Vessels and Axial Loading Applications
arat 3074.1
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
(Example) Machined part is 20-mm thick. Determine the maximum safe load P if a safety factor of 2.5 with respect to failure by yielding is specified.
(Sol)
25.060155.16090
drdD
The yield strength is 360 MPa.The allowable stress based on safety factor of 2.5 is 360/2.5=144MPa
kNN
KAP tta
7.106)10(7.106
62.1/)10)(20)(60)(10(144/3
66
62.1tK
3.09027 wd
At the hole,
30.2tK
kNN
KAP tta
9.78)10(9.78
30.2/)10)(20)(2790)(10(144/3
66
At the fillet,
kNP 9.78max
Pressure Vessels and Axial Loading Applications
Therefore,
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
Thin-walled pressure vessels
Thin-walled when the ratio of the wall thickness to the radius of the vessel is so small that the distribution of the normal stress on a plane perpendicular to the surface of the vessel is uniform throughout the thickness of the vessel
If the ratio of the wall thickness to the inner radius of the vessel is less than 0.1, the maximum normal stress is less than 5% greater than the average
(1) Spherical pressure vessels
nyx
Based on symmetry of loading and geometry
tpr
rptrPR
a
a
2
20
2
Axial (or meridional) stress in a sphere
Pressure Vessels and Axial Loading Applications
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
(2) Cylindrical pressure vessels
tprtpr
rLpLtPQ
a
h
h
x
2
2202
From a summation of forces in the x-direction
(3) Thin shells of revolution
tp
rr t
t
m
m
Pressure Vessels and Axial Loading Applications
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
(Example) Cylindrical pressure vessel with inside diameter of 1.5m is constructed by wrapping a 15mm thick steel plate. The butt welded seams form an angle of 30deg with a transverse plane through the cylinder. Determine the normal stress and shear stress when the internal pressure in the vessel is 1500kPa.
00.755.37,
5.37)10(5.37)015.0(2
)75.0)(10)(1500(2
0.75)10(0.75015.0
)75.0)(10)(1500(
263
263
xyhyax
a
h
MPaMPaTherefore
MPamNt
pr
MPamNtpr
)(24.16238.160)30cos()30sin()0.755.37(
)sin(coscossin)()(9.46875.46
0)30(sin)0.75()30(cos)5.37(
cossin2sincos
22
22
22
CMPaMPa
TMPaMPa
xyyxnt
xyyxn
Pressure Vessels and Axial Loading Applications
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
(Example) Cylindrical pressure vessel with inside diameter of 1.5m is constructed by wrapping a 15mm thick steel plate. The butt welded seams form an angle of 30deg with a transverse plane through the cylinder. Determine the normal stress and shear stress when the internal pressure in the vessel is 1500kPa.
)(412041230)174)(41)(8(2)8(250
0cos2
0cos
2
2
Tpsipsi
xtxp
dFdP
m
m
m
AAp
psipsirrt
pinr
indxyddxdyr
t
t
t
t
m
m
m
m
80008003246.819.140
412341
250246.84178
19.14021
411 5.12
22
5.12
Pressure Vessels and Axial Loading Applications
Determining m
Determining t
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
(Homework)
(5.4), (5.19), (5.49), (5.59), (5.79), (5.100)
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