Text book for the mechanics of materials Shaft & Torsion ・Angle of Torsion & Specific Angle of Torsion ・Stress & Strain under Torque ・Design of Shaft Diameter Note: Your feedback is welcome!

1. Shaft & Torsion
1. Angle of Torsion & Specific Angle of Torsion
2. Stress & Strain under Torque
3. Design of Shaft Diameter
Goal
1/18

2. Shaft & Torque & Torsion
Shaft
Torque
Torsion：Twisting due to applied torque
Torque：Moment around axis line
[N・m]T
Torsional moment
2/18
Axis line

3. Machine Elements
Subjected to Torque
Ref. JAXA
3/18
Ref. waterstarmotors.clearmechanic.com

4. Angle of Torsion
Specific Angle of Torsion
TA B
B’
φ(x)
θ
ℓ
x
θ
dφ(x)
dx
=
：Specific angle of torsionθ
[ rad/m ]
Angle of torsion per unit length
r
：Angle of torsion
[ rad ]
θ
Angle between two ends 4/18
Fixed

5. Round Bar
Uniform cross-section
=ℓ θθ
θ=
θ
ℓ
5/18
is constant in every cross-sectionθ

6. Shear Deformation by Torsion
a
b c
d a’
b’
c’
d’
γ
a
b c
d a’
b’ c’
d’
AfterBefore
6/18Shear deformation on shaft surface
No axial force

7. Shear Strain
A B
B’ θ
T
γ
ℓ
r
BB’ = γℓ = rθ
γ= r
ℓ
θ = rθ 7/18
Fixed

8. Shear Strain & Stress
γ = rθ
ℓ
θ= r
τ = G γ
= Grθ
ℓ
θ= Gr
r
γ
r
τ
・angle of torsion
・radius
Strain distribution
Stress distribution
Max. on surface
8/18
G：Shear modulus [Pa]
・length
Proportional to
Inverse
proportional to

9. Relationship
between Torque and Shear Stress
τ (ρ)
ρ
dρ
r
dA = 2πρdρ
T T =
A
ρ・τ (ρ) dA
τ (ρ)= τ
r
ρ
= ρ dρ32π τr
r
0
=
2
πr3
τ
τ
τ =
πr3
2
T
9/18

10. Polar moment of inertia of Area
=Ip
A
ρ dA2
=
r
0
・2πρdρρ2
2
π
r4
=
ρ
dρ
dA = 2πρdρ 10/18

11. Torsional Rigidity
θ= T
πr4
2
G
=
Ip
T
G
1
IpG : Torsional rigidity
Torque to produce twist of 1[rad]
Ip
2
π
r4
=
2
π
r4
1
T=
11/18
∵
G

12. Relationship with Torque
32
πd3
（ ）d: diameter
τ =
πr3
2
T
γ = T=
G
τ
πr3
2
G
θ = r
γ
= T
πr4
2
G
T
T
πr3
32
G
T
πr4
64
G
=
r
Ip
T
=
r
T
IpG
=
Ip
T
G
1
12/18

13. [Exercise] Stepped Round Bar
θAB
πr4
2
G
ℓ1T
1
= θBC
πr4
2
G
ℓ2T
2
=
θAB θBC＋θ =
π
2
G
T ℓ1
r4
1
＋
ℓ2
r4
2
（ ）=
T
ℓ1 ℓ2
G
A B C
θ
Left end is fixed
T is applied at C
Q. Angle of twist
at right end ？
: Torque on cross-sectionT
r1 r2
13/18
Fixed

14. [Exercise]
Allowable Stress & Shaft Diameter
Allowable shear stress：τa
Torque：T
Q. Shaft diameter？
Max. shear stress
Max. stress Allowable−
<
=
πr3
2
Tτmax
τ τa−
<
max
r 3
πτa
2T−>
14/18

15. Hollow Shaft
If hollow
r
τ
Stress is small in center
＝Bearing torque is small
Not much decrease in capacity of bearing torque
Decrease in mass
15/18
Stress distribution

16. Shear Stress of Hollow Shaft
Torque
r
rin
T
n=
rin
outr
out
=
πr3
2 T
τ
（ ）n−1
4
out
τ
outr
rin
T =
A
ρ・τ (ρ) dA
outr
= ρ dρ32π τ
rinoutr
τ (ρ)= τ
ρ
outr
=
2 outr
π τ rin
44
−（ ）rout
16/18

17. [Exercise]
Solid vs. Hollow Round Shaft
Allowable shear stress ：τa Torque：T
Radius Mass (Vol.)
Solid
Hollow
Hollow
Solid 1.022 0.78
2.2 % increase 22 % decrease
r 3= πτa
2T πr2
ℓV=
V=
n =
2
1
π ℓrin
22
−（ ）routoutr = 3
πτa
2T
（ ）n−1 4
17/18

18. Summary
1. Angle of Torsion, Specific Angle of Torsion
2. Stress & Strain under Torque
3. Design of Shaft Diameter
：Specific angle of Torsionθ [ rad/m ]
Angle of torsion per unit length
Allowable shear stress ：τa
18/18
：Angle of Torsion [ rad ]θ
Angle between two ends