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Prof. Younane N. Abousleiman Email: [email protected] Office: SEC, Suite P119
Torsion
Dr. A. Younane N. Abousleiman ‘s lecture notes
iPMI, www.pmi.ou.edu
The moment of a force about a point or an axis provides a measure of the tendency of the force to cause a body to rotate about the point or axis.
oz ⊥ plane xoy in which Fx lies; Fx causes the pipe to turn about the z-axis ⇒Fx causes a moment about the z-axis (Mo)z
The moment of a force about a point or an axis provides a measure of the tendency of the force to cause a body to rotate about the point or axis.
Fy passes through O ⇒ Fy does not cause the pipe to turn because the line of action of the force passes through O.
The moment of a force about a point or an axis provides a measure of the tendency of the force to cause a body to rotate about the point or axis.
ox ⊥ plane yoz in which Fz lies; Fz causes the pipe to turn about the x-axis ⇒Fz causes a moment about the x-axis (Mo)x
Drilling a well
Torsion resistance to balance TB
Torque transmission
Shallow drilling
T
F
F
Torque is a moment that causes twisting along the length of a bar.
T
Applications
T
Shear Stress Due To Torsion
The twist is the torsional deformation. For a circular shaft, the torque (or torsional moment) rotates each cross-section relative to the nearby ones.
The shear stress varies linearly along each radial line of the cross section
JTρτ =
Shear stress
J is the polar moment of inertia of the cross- sectional area, geometric property
T
JTc
=maxτ
ρ is the radial distance τ is the shear stress at a radial distance ρ τmax is the maximum shear stress in the shaft, which occurs at the outer surface T is the resultant internal torque acting at the cross section. Its value is determined from the method of sections and the equation of moment equilibrium applied about the shaft’s longitudinal axis J is the polar moment of inertia of the cross- sectional area c is the outer radius of the shaft
Shear stress
JTρτ =
JTc
=maxτ
T, reaction torque at fixed head
Φ L
Φ(x)
x
γ⋅∆= xBD dxdφργ =
Note: dΦ/dx is a function of distance x and is a constant for fixed x
In OBD:
O
φρ∆=BD
In ABD: γφρ ⋅∆=∆ x
(4)
τxy
z
x
y
The shear stress τxy (or σxy) acts on the surface normal to x (the first subscript) axis, pointing the y (the second subscript) direction
A A
xyτ
yxτ
yxτ
xyτ
At section A-A
Also a dimensionless quantity θπγ −=2
strainShear xy
γxy xyτ
y
x
xyτ
yxτ
yxτ
Pure shear: τxy = τyx
θ0 = π/2 θ = π/2- γxy
xyτxyτ
yxτ
yxτ
θ0 θ
max)( γρφργcdx
d==
o
(1), (2)
T
τ τ γ
Shear strain by definition θπγ ′−=2
θ'
τyx τ = τxy y
x
z
Element A A
C
o
(1), (2)
The shear strain within the cross section varies linearly along the radial line, from zero to maximum γmax.
γmax
γ
Derivation of shear stress τ = Tρ/J
maxγργc
=
T Element A
Element C
γ = 0 at center
dxdφργ =
dxdc φγ =max
Derivation of shear stress τ = Tρ/J
max)( γρφργcdx
d==
o
(1), (2)
T
τ τ γ
τyx
Hooke’s law:
maxmax )()( τργργτcc
GG ===Again the shear stress varies linearly along the radial line
τ = τxy y
x
z Element A
From torque equilibrium equation
∫
∫∫
=
==
A
AA
dAc
dAc
dAT
2max
max ])[()(
ρτ
τρρτρ
J = polar moment inertia
Jc
T maxτ=
JTc
=maxτ
Derivation of shear stress τ = Tρ/J
maxτρτc
=
JTρτ =
T
Shear stress
JTρτ =
JTc
=maxτ
ρ is the radial distance τ is the shear stress at a radial distance ρ τmax is the maximum shear stress in the shaft, which occurs at the outer surface T is the resultant internal torque acting at the cross section. Its value is determined from the method of sections and the equation of moment equilibrium applied about the shaft’s longitudinal axis J is the polar moment of inertia of the cross- sectional area c is the outer radius of the shaft
Polar moment of inertia J
4
0
3
0
22
22
)2(
cd
ddAJ
c
c
A
πρρπ
ρπρρρ
==
==
∫
∫∫
Solid shaft
ci
co
ρ
)(2
2
)2(
443
2
io
c
c
c
c
ccd
dJ
o
i
o
i
−==
=
∫
∫
πρρπ
ρπρρ
Tubular shaft
Shaft Design
τmax is maximum shear stress [N/m2 or psi] τallow is τfail/F.S. τallow is allowable shear stress [N/m2 or psi] τfail is failure shear stress [N/m2 or psi] F.S. is factor of safety (F.S. > 1) J is polar moment of inertia [m4 or in4] c is outer radius of the shaft [m or in]
allow
TcJ
τ≥
Shaft design:
Design criteria: the maximum shear stress τmax should be less than the allowable stress τallow
allowJTc ττ ≤=max
τmax τmax
τmax < τallow
τ
γ
τallow
τfail
τallow = τfail/F.S. τpl
T T
Usually, geometric parameters need to be designed!
allow
TcJ
τ≥
Shaft design:
Design criteria: the maximum shear stress τmax should be less than the allowable stress τallow
allowJTc ττ ≤=max
τmax τmax
τmax < τallow
τ
γ
τallow
τfail
τallow = τfail/F.S. τpl
T T
T dt
TdP θ= fTTP πω 2==
Tdθ is work [N·m or lb·ft] ω = dθ/dt is the shaft’s angular velocity [rad/s] f (ω = 2πf) is frequency measuring the cycles the shaft rotates per second [Hz (1 Hz = 1 cycle/s)] P is power [W (1 Watts = 1 N·m/s) or hp (1 horsepower = 550 lb·ft/s)]
or
The power transmitted from a machine through shafts (tubes) is defined as the work performed per unit time:
Motor M provides an output power of P, developing a torque T in the shaft AB
Motor: output power P
f (ω = 2πf) is frequency [Hz (1 Hz = 1 cycle/s)] P is power [W (1 Watts = 1 N·m/s) or hp (1 horsepower = 550 lb·ft/s)] J is πc4/2 for solid shaft and π(co
4-ci4)/2 for
tubular shaft [m4 or in4]
allowfP
cJ
τπ2≥
Shaft design:
If the power P and rotation frequency f, instead of torque T, are given:
allowfJPc τπ
≤2
τmax τmax
τmax < τallow
ci
co c
Solid Tubular
)P/(2T fπ=
T T
Example 5.1
The motor delivers 500 hp to the steel shaft AB, which is tubular and has an outer diameter of 2 in. and an inner diameter of 1.84 in. Determine the smallest angular velocity at which it can rotate if the allowable shear stress for material is τallow = 25 ksi.
torque T delivered from motor
Motor: output power P
T
Reaction torque from gear G on AB, equal to T
T
Gear G
Example 5.1 Basic parameters: P = 500 hp co = 2 in/2 = 1 in ci = 1.84 in/2 = 0.92 in τallow = 25 ksi
T
T
Torque delivered from motor
Reaction torque
FBD of shaft AB
A
B
ci
co Need to calculate: the smallest angular velocity: ωmin
ffTTP
πωπω
22
===
f is frequency [Hz (1 Hz = 1 cycle/s)] P is power [W (1 Watts = 1 N·m/s) or hp (1 horsepower = 550 lb·ft/s)] J is πc4/2 for solid shaft and π(co
4-ci4)/2 for
tubular shaft [m4 or in4]
Shaft design:
If the power P and rotation frequency f instead of torque T are given:
allowfJPc τπ
≤2
τmax τmax
τmax < τallow
c
Solid Tubular
allowfP
cJ
τπ2≥
ci
co
T T
Solution 5.1
allowo fP
cJ
τπ2≥
The geometric parameters needed to satisfy the design criteria:
)psi100025(2in/ft12lb/s)/hpft(550hp500
in1in445.0 4
××⋅×
≥fπ
444444 in445.0in)92.01)(2/())(2/( =−=−= ππ io ccJci
co
Solution 5.1
rad/s2962 ≥fπ
rpm2830/radrevolution)2/1(s/min60rad/s296
rad/s296)2( minmin
=××=
==π
πω f
So the smallest angular velocity:
)psi100025(2in/ft12lb/s)/hpft(550hp500
in1in445.0 4
××⋅×
≥fπ
Angle of Twist
The angle of twist Φ(x) at location x is proportional to x, at x = L, Φ = TL/JG
T, reaction torque at fixed head
JGTL
=φ
Angle of Twist
Φ L
Φ(x)
x
T, reaction torque at fixed head
Φ L
Φ(x)
x
ρτ
ργφ dx
Gdxd ==
JTρτ =
JGTLdx
JGTdx
JGTd
L
=== ∫0
φφ
dxdφργ =
Derivation of Φ
T, reaction torqueat fixed head
ΦL
JGTL
=φ
The angle of twist for constant J, G, and T:
Φ(x)
The angle of twist Φ(x) at location x is proportional to x, at x = L, Φ = TL/JG
Angle of Twist
Φ is the angle of twist between two ends [rad] T is the applied torque [N·m or lb·ft] J is the shaft’s polar moment of inertia [m4 or in4] G is the shear modulus [Pa (1 Pa = 1 N/m2) or psi (1 psi = 6895 Pa]
x
∑= JGTLφ
If J, G, and T are constants for each segment: Angle of Twist
T(x) is the internal torque at arbitrary position x [N·m or lb·ft] J(x) is the shaft’s polar moment of inertia at x [m4 or in4]
Constant J, G, T for each segment
J(x), T(x) varying with x
In general case both J and T are functions of x:
∫=L
dxGxJ
xT
0 )()(φ
x
Φ
Φ
x
dx
γ
dΦ
ρτ
ργφ dx
Gdxd ==
)()(xJ
xT ρτ =∫==L
dxGxJ
xTdxGxJ
xTd0 )(
)()()( φφ
ρdxγdφdxγρdφ =⇒⋅=⋅
Derivation of Φ
∫=L
dxGxJ
xT
0 )()(φ
Φ is the angle of twist between two ends T(x) is the internal torque at the arbitrary position x, found from the method of sections and the equation of moment equilibrium applied about the shaft’s axis J(x) is the shaft’s polar moment of inertia, a function of position x G is the shear modulus
The angle of twist
Similarity between rotation Φ under torsion and elongation δ under axial load:
EAFL
=δ
∫=L
dxGxJ
xT
0 )()(φ ∫=
L
dxExA
xF
0 )()(δ
JGTL
=φ
T → F G → E J → A
(Torsion) (Axial load)
OR: ∑= JGTLφ ∑= EA
FLδ
The A-36 steel bar (Est = 200 Gpa, cross-sectional area 300 mm2) is subjected to loads as shown in the figure. Determine the displacement of end D relative to end A.
0.25 m 0.25 m 0.5 m
10 kN 20 kN
50 kN
FBD: FA
FA+10-20-50=0
FA=60 kN
Calculation of FA:
0=→+ ∑ xF
10 kN 20 kN 50 kN
60 kN PAB
PAB= 60 kN
Calculation of PAB:
60-PAB= 0
FBD:
FBD:
60+10-PBC=0
PBC=70 kN
60 kN PBC
Calculation of PBC:
10 kN
0=→+ ∑ xF
0=→+ ∑ xF
PCD-50=0
PCD
Calculation of PCD:
PCD=50 kN
D
50 kN
0=→+ ∑ xF
P (kN)
x A B
-60
-70
C
-50
D
60 kN
Tension: positive; Compression: negative.
10 kN 20 kN 50 kN
The displacement of end D relative to end A.
]m )0GPa][300(1 [200m) kN)(0.5 50
]m )0GPa][300(1 [200m) kN)(0.25 70
]m )0GPa][300(1 [200m) kN)(0.25 60
262626 −−−−
+−
+−
==∑ (((/ EA
FLADδ
mm 9.0/ −=ADδ
Negative sign means that end D moves towards end A.
P (kN)
x A B
-60
-70
C
-50
D
Right-hand rule: The Torque and angle will be positive, if the thumb is directed outward from the shaft when the fingers curl to give the tendency for rotation.
FBDs:
Torque Diagram:
Torque Diagram:
The angle of twist of the end A with respect to the end D:
Example 5.2
The 20-mm-diameter A-36 steel shaft is subjected to the torque shown. Determine the angle of twist of the end B.
Solution 5.2
Basic parameters: c = 20 mm/2 = 10 mm G = 75 Gpa LBC = 800 mm LCD = 600 mm LDA = 200 mm
In order to determine the angle of twist of end B, ΦB/A, we first need to determine the internal torque along the shaft BA
80 N·m
B
TBC = -80 N·m
80 N·m
B
TCD = -60 N·m
C
20 N·m
80 N·m
B
TDA = -90 N·m
C
20 N·m
D
30 N·m
C D
A
-80
T (Nm)
-60 -90
B C D A
Torque diagram
The angle of twist of end B:
48
44
4
m1057.1m)01.0)(2/(
)2/(
−×=
=
=
π
π cJc
{ }
°−=−=××
⋅−+⋅−+⋅−=
++=++=
−
74.5rad1.0)N/m1075)(m1057.1(
m)2.0m)(N90(m)6.0m)(N60(m)8.0m)(N80(2948
//// JGLT
JGLT
JGLT DADACDCDBCBC
ADDCCBAB φφφφ
-80
T (N·m)
-60 -90
B C D A
Torque diagram
Solution 5.2 x
Solution 5.2
{ }
°−=−=××
⋅−+⋅−+⋅−=
++=++=
−
74.5rad1.0)N/m1075)(m1057.1(
m)2.0m)(N90(m)6.0m)(N60(m)8.0m)(N80(2948
//// JGLT
JGLT
JGLT DADACDCDBCBC
ADDCCBAB φφφφ
300 Nm 400 Nm
500 Nm
600 Nm
400 mm
600 mm
300 mm
300 Nm 400 Nm
500 Nm
400 mm
600 mm
300 mm E600 Nm
Figure (a) Figure (b)
The solid 20-mm-diameter A-36 steel (G = 75 GPa) shaft is used to transmit the torques applied to the gears, as shown in FIGURE a.
(a) Draw the free body diagrams and determine the three internal torques in sections between A-C, C-D, and D-B;
(b) Draw the torque diagram along the shaft AB;
(c) Calculate the absolute maximum shear stress on the shaft;
(d) Calculate the twist angle of end A relative to C;
(e) Calculate the twist angle of end A to B;
(f) If the external torque of 600 Nm originally applied at the end B is moved to point E (in the middle of BD, i.e., DE = EB = 300 mm, see FIGURE b), redraw the torque diagram along the shaft AB and determine the twist angle of end B relative to D.
600 Nm 300 Nm
500 N
TAC = -300 Nm
600 Nm 300 Nm
TB = ?
500 mm
300 mm
TCD = -300+400 = 100 Nm
TCD
600 Nm 300 Nm
TB = ?
500 mm
600 mm
300 mm TDB
TDB = -300+400+500 = 600 Nm
500 Nm
-300
T (Nm)
100
600
A C D B
Torque diagram
300 Nm 300 Nm 400 Nm
300 Nm 400 Nm
(a)
(b)
Maximum shear stress on the shaft:
48
44
4
m1057.1m)01.0)(2/(
)2/(
−×=
=
=
π
π cJc
MPaJ
cTJ
T
382)m1057.1(m01.0mN600
48maxmaxmax
max
=×
×⋅=== −
ρτ
-300
T (Nm)
100
600
A C D B
Torque diagram
(c)
c
°−=−=××
⋅−== − 4.4rad0764.0
)N/m1075)(m1057.1(m)3.0m)(N300(
2948/ JGLT ACAC
CAφ
48
44
4
m1057.1m)01.0)(2/(
)2/(
−×=
=
=
π
π cJ
-300
T (Nm)
100
600
A C D B
Torque diagram
(d)
{ }°==
××++−
=
++=++=
− 1.15rad263.0)1075)(1057.1(
)6.0)(600()4.0)(100()3.0)(300(98
//// JGLT
JGLT
JGLT DBDBCDCDACAC
BDDCCABA φφφφ(e)
600 Nm 300 Nm
TB = ?
500 mm
600 mm
300 mm TDE
TDE = -300+400+500 = 600 Nm
500 Nm
-300
T (Nm)
100
600
A C D B
Torque diagram
300 Nm 400 Nm
(f)
300 Nm 400 Nm
500 Nm
400 mm
600 mm
300 mm E600 Nm
E 0
TEB
TEB = -300+400+500-600 = 0
c
{ }°==
××⋅+
=
+=+=
− 75.8rad152.0)N/m1075)(m1057.1(
m)3.0m)(N600(02948
/// JGLT
JGLT EDEDBEBE
DEEBDB φφφ
48
44
4
m1057.1m)01.0)(2/(
)2/(
−×=
=
=
π
π cJ
-300
T (Nm)
100
600
A C D B
Torque diagram
E 0
(f)