lecture note for solid mechanics - kocw
TRANSCRIPT
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
Lecture Note for Solid Mechanics- Torsional Loading -
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
Torsional Loading
Typical example of torsional loading problems :
- circular shaft connecting an electric motor with othermachine.
- circular shaft transmits the torque from the armature tothe coupling
- determination of significant stresses and deformation of the circular shaft
areaarear dAdFTT
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
Torsional loading problem :
- Transverse cross section before twisting remains planeafter twisting
- Diameter of the section remains straight
Plane sections remain plane
Plane sections become warped
Torsional Loading
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
Torsional shear strain
Lc
ABBB
cc
'
tan
LEDDD
'
tan
If the small strain is assumed
L
cLc
Combining above equations
cc
Torsional shear strain becomes
surfaceaatstrainshearcedisaatstrainshear
twistofangle
c :tan:
:
Torsional Loading
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
If Hooke’s law is assumed (stress is proportional to strain)
cc
cc
dAdAc
TT cr 22
dAJwhere 2 : polar second moment of area
2
24
0
22 cddAJc
)( 4444
22
222
2
io
c
b
rrbc
ddAJ
For case (a) :
For case (b) :
A
dAT
Torsional Loading
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
dAdAc
TT cr 22
J
cJTT c
r
Unknown shear stress becomes
JT and
JcT
c
Shear stress is zero at the center of the shaft and increases linearly with respect to the distance from the axis of the shaft.
Shear strain is zero at the center of the shaft and increases linearly with respect to the distance from the axis of the shaft.
Therefore,
cc
JT and
Torsional Loading
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
Determination of angle of twist
L
JT
dLd or
orJcT
c
GJGLT
GLL
If T, G, or J is not constant along the length of the shaft
n
i ii
ii
JGLT
1
If T, G, or J is a function of distance of the length of the shaft
L
JGdxT
0
Torsional Loading
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
(a) Maximum shear stress in the shaft
GPaG 80
(b) Shear stress at the inside surface of the shaft
(c) Magnitude of angle of twist in 2m length
))(.())()(()(
max 6
33
10117181020010300
JcT
c
MPamN 934109234 26 .)(.
))(.())()(()(
6
33
10117181015010300
J
T
MPamN 226101926 26 .)(.
))(.)()(())(()(
69
3
10117181080210300
GJ
LT
radrad 0043700043650 ..
46
46
44
44
10117181011718
1502002
2
mmm
rrJ io
)(.)(.
)(
)(
Torsional Loading
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
(a) Maximum shear stress in the shaft
ksiJ
cTAB
ABABAB 2444
2312731215 .).(
))(()(
4444 13252
22incJ .)(
ksiJ
cTBC
BCBCBC 7754
13252125 .).(
))(()( (max)
(b) Rotation of end B w.r.t. end A
radJGLT
ABAB
ABABAB 012730
23127120001291215 .
).)(())()(()(
/
(c) Rotation of end C w.r.t. end B
radJGLT
BCBC
BCBCBC 011940
132512000125125 .
).)(())()(()(
/
(d) Rotation of end C w.r.t. end A
radBCABAC
0007950011940012730
...///
4446 231273
22incJ .)(
ksiG 00012,
Torsional Loading
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
Questions: (1) transverse plane is a plane of maximum shearing stress?(2) there are other significant stresses induced by torsion?
xyyx From moment equilibrium equation:
cossinsincos xyyxn 222 From stress transformation equations:
cossincossin xyxy 2200
)sin(coscossin)( 22 xyyxnt
)sin(cos)sin(cos 22220 xyxy
2sinxy
2cosxy
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
Stresses on an inclined plane are functions of angle of inclined plane.
2sinxyn
2cosxynt
A & C : maximum shear stressesB : maximum tensile stressD : maximum compressive stress
B
A
C D
Torsional Loading
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
(a) Steel rear axle of a truck Split longitudinally due to longitudinally running grains
(b) Thin-walled aluminum tube Buckled and teared along 45 deg planes
(c) Gray cast iron of brittle material Tensile failure
(d) Low carbon steel of ductile material Shear failure
Type of failures
Torsional Loading
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
(a) Maximum torque Tmax :
26108080 mNMPaJ
cT /)(maxmaxmax
(b) FS when failure occurs at T=12kN-m, if ultimate strengths are 205MPa in shear and 345MPa in tension.
MPa80maxmmt 6
46
46
44
10096141009614
69752
mmm
J
)(.)(.
)(
mkNmNc
JT
04.15)10(04.15)10)(75(
)10)(096.14)(10)(80(
3
3
66max
max
MPa
JcT
o
xyn
29556021009614
10751012
22
6
33
.)(sin))(.(
))()((
sinsin
MPaJcT
xynt 923122 .coscos
2462955
345 ..
n
ultFS
4269231
205 ..
nt
ultFS
and
overall FS
Torsional Loading
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
Power is the time rate of doing work
TdtdT
dtdWPower k
Work done by a constant torque : TWk
(where is angular displacement in radian)
(where is angular velocity in rad/min)
(Example 6-8) Determine minimum permissible diameters of two shafts if allowable shearing stress is 20ksi and angle of twist in 10-ft of propeller shaft is not exceed 4-deg.
hpandrpmwithengine 80020014 :propellertogearboxratio
TPower ))((),( 220000033800 1T
shaftcrankforlbftT 010,211
Assume no power loss, shaftpropellerforlbftT 252,52
))(())(,()(
31
41
102012010212
c
cTcJ
For crank shaft,
inc 00221 .
ind 00441 .
1HP=550ft-lb/s
Torsional Loading
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
))(())(,()(
32
42
10201225252
c
cTcJ
For propeller shaft,
inc 26112 .
ind 52222 .
2612154831210121210122525
1804
2
42
6
..))()((
))(())(,(
ccGJ
LT
Therefore, the propeller shaft should be diameter of 3.10-in.
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
Circular bar subjected to axial load P and torque T
AP
due to axial load P
JT due to torque T
Shearing stresses exist on both transverse and longitudinal planes
yxxy
Once stresses on the planes in (d) are known, stresses on any other plane through the point, as well as principal stresses and maximum shearing stress at the point can be found.
As long as the strains are small, these stresses can be computed separately and superimposed on the element
Torsional Loading
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
(Example 6-12) Determine (a) x- and y-components of stresses, (b) principal stresses and maximum shearing stress at the point for point A on outside surface.
)(.).()( CMPa
APA
x 4625104
102002
3
(Sol)
(a)
MPaJcTA
xy 791520502
05010304
3
.).(
).)((
(b) ).,,.(),,( 7915204625 xyyx
321537312
791522
046252
04625
22
22
22
21
..
).(..
,
xy
yxyxpp
Principal stresses and maximum shearing stress are
0
05166321537312
59140321537312
3
2
1
zp
p
p
CMPaTMPa
)(...)(...
MPa321532
05166591402
.).(.
minmaxmax
Torsional Loading
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
Circular bar subjected to torque T
JTc
c max
Stress concentrations occur in the vicinity of abrupt changes in diameter.
JTcKmax
Stress concentration factor K as
),(dr
dDKKwhere
Kt : based on net section
(Example 6-14) For stepped shaft of D=4-in and d=2-in, determine min. fillet radius if max. shearing stress is 8ksi under torque of 6280 in-lb.
psiJ
Tc 39981216280
4 )()(
max
00239988000 .tK
indrdr
120060060 ...
Torsional Loading
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
Shearing stresses in circular shaft are proportional to the distance from its axis.
Shearing stresses for rectangular cross sections are not proportional to the distance from its axis.
Violate free boundary condition
Shearing stresses at the corners of the rectangular bar must be zero.
Torsional Loading
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
Every cross sections, except for circular cross sections, will warp (not remain plane) when the bar is twisted.
GbaTL
baT
3
2
max
Torsional Loading
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
(Example 6-17) Determine maximum shearing stress and the angle of twist for 12-in length.
b/a=3/(1/2)=6Aluminum alloyG=4000-ksiT=2500 in-lb
30.
radGba
TLpsi
baT
067010400032130
12250011011
321302500
333
22
.))()(()(.
)(,
)()(.max
Torsional Loading
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
Elementary torsion theory : circular sections, thin-walled sections
lengthunitforceshearinternal(q) flowshear
tq
where is average shearing stress across the thickness, t
BA
BBAA
BA
qqtt
ttqq
dxqdxqVV
31
31
3131
Shear flow on the cross section is constant even though the wall thickness varies.
Torsional Loading
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
dsqqdsdFTr )()(
tAAqTr 22
AtT
2
(Example 6-18) Determine maximum torque that can be applied to the section if the maximum shearing stress must be limited to 95 MPa.
mNtq /)())()(( 336 101901021095
mN
AqT
1750101901035021002
236 ))()()()((
where is the area enclosed by the median line.
mNtq /)10(285)10)(3)(10(95 336
mN
AqT
2625)10)(285)(10)(350)(2100(2
236
Torsional Loading
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
Homework
(6-14), (6-47), (6-64), (6-76), (6-92), (6-126)