Formula Sheets: Torsion of Shafts | Strength of Materials (SOM) - Mechanical Engineering PDF Download

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F ormula Sheet: T orsion of Shafts
Introduction to T orsion
• Definition : T orsion is the twisting of a shaft due to an applied torque, caus-
ing shear stre sses and angular deformation.
• Assumptions : Linear elastic material, circular cross-section, small defor-
mations, plan e sections remain plane.
K ey F ormulas
• T orsion F ormula (Shear Stress) :
t =
Tr
J
where t = shear stress, T = applied torque, r = r adial distance from the
center ,J = polar moment of inertia.
• Maximum Shea r Stress :
t
max
=
TR
J
whereR = outer r adius o f the shaft.
• Angle of Tw ist :
? =
TL
GJ
where? = angle of twist (r adians),L = l ength of shaft,G = shear modulus.
• Polar Mom ent of Inertia (J ) :
– F or solid circular shaft:
J =
pR
4
2
– F or hollow circular shaft:
J =
p
2
(R
4
o
-R
4
i
)
whereR
o
= outer r adius,R
i
= inner r adius.
Power Tr ansmission
• Power Tr ansmitted b y Shaft :
P =T?
whereP = power (W atts),T = torque (N·m),? = angular velocity (r ad/s).
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Page 2


F ormula Sheet: T orsion of Shafts
Introduction to T orsion
• Definition : T orsion is the twisting of a shaft due to an applied torque, caus-
ing shear stre sses and angular deformation.
• Assumptions : Linear elastic material, circular cross-section, small defor-
mations, plan e sections remain plane.
K ey F ormulas
• T orsion F ormula (Shear Stress) :
t =
Tr
J
where t = shear stress, T = applied torque, r = r adial distance from the
center ,J = polar moment of inertia.
• Maximum Shea r Stress :
t
max
=
TR
J
whereR = outer r adius o f the shaft.
• Angle of Tw ist :
? =
TL
GJ
where? = angle of twist (r adians),L = l ength of shaft,G = shear modulus.
• Polar Mom ent of Inertia (J ) :
– F or solid circular shaft:
J =
pR
4
2
– F or hollow circular shaft:
J =
p
2
(R
4
o
-R
4
i
)
whereR
o
= outer r adius,R
i
= inner r adius.
Power Tr ansmission
• Power Tr ansmitted b y Shaft :
P =T?
whereP = power (W atts),T = torque (N·m),? = angular velocity (r ad/s).
1
• Relation w ith Rotational Speed :
P =
2pNT
60
whereN = rotational speed (RPM).
T orsional Rigidity and Stiffness
• T orsional Ri gidity :
GJ
whereGJ = torsional stiffness (N·m²/r ad).
• T orsional St iffness ( k ) :
k =
T
?
=
GJ
L
Combined Loading
• Equivalent T orque (for combined bending and torsion) :
T
e
=
v
T
2
+M
2
whereM = bending moment.
• Equivalent B ending Moment :
M
e
=
M +
v
M
2
+T
2
2
• Maximum Shea r Stress (von Mises or Tresca for combined loading) :
t
max
=
v
(
s
2
)
2
+t
2
wheres = normal stress from bending,t = shear stress from torsion.
Shafts in Series and Par allel
• Shafts in Se ries : T otal angle of twist is the sum of individual twists:
?
total
=
?
T
i
L
i
G
i
J
i
• Shafts in P ar allel : T otal torque is the sum of individual torques:
T
total
=
?
T
i
, ?
1
=?
2
=··· =?
n
2
Page 3


F ormula Sheet: T orsion of Shafts
Introduction to T orsion
• Definition : T orsion is the twisting of a shaft due to an applied torque, caus-
ing shear stre sses and angular deformation.
• Assumptions : Linear elastic material, circular cross-section, small defor-
mations, plan e sections remain plane.
K ey F ormulas
• T orsion F ormula (Shear Stress) :
t =
Tr
J
where t = shear stress, T = applied torque, r = r adial distance from the
center ,J = polar moment of inertia.
• Maximum Shea r Stress :
t
max
=
TR
J
whereR = outer r adius o f the shaft.
• Angle of Tw ist :
? =
TL
GJ
where? = angle of twist (r adians),L = l ength of shaft,G = shear modulus.
• Polar Mom ent of Inertia (J ) :
– F or solid circular shaft:
J =
pR
4
2
– F or hollow circular shaft:
J =
p
2
(R
4
o
-R
4
i
)
whereR
o
= outer r adius,R
i
= inner r adius.
Power Tr ansmission
• Power Tr ansmitted b y Shaft :
P =T?
whereP = power (W atts),T = torque (N·m),? = angular velocity (r ad/s).
1
• Relation w ith Rotational Speed :
P =
2pNT
60
whereN = rotational speed (RPM).
T orsional Rigidity and Stiffness
• T orsional Ri gidity :
GJ
whereGJ = torsional stiffness (N·m²/r ad).
• T orsional St iffness ( k ) :
k =
T
?
=
GJ
L
Combined Loading
• Equivalent T orque (for combined bending and torsion) :
T
e
=
v
T
2
+M
2
whereM = bending moment.
• Equivalent B ending Moment :
M
e
=
M +
v
M
2
+T
2
2
• Maximum Shea r Stress (von Mises or Tresca for combined loading) :
t
max
=
v
(
s
2
)
2
+t
2
wheres = normal stress from bending,t = shear stress from torsion.
Shafts in Series and Par allel
• Shafts in Se ries : T otal angle of twist is the sum of individual twists:
?
total
=
?
T
i
L
i
G
i
J
i
• Shafts in P ar allel : T otal torque is the sum of individual torques:
T
total
=
?
T
i
, ?
1
=?
2
=··· =?
n
2
T orsional Shear Stress Distribution
• Shear stress varies linearly from zero at the center to maximum at the outer
surface.
• F or non-circular sections (e.g., rectangular), use approximate methods or
finite element an alysis (not typically required for GA TE).
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