Text book for the mechanics of materials
Shaft & Torsion
・Angle of Torsion & Specific Angle of Torsion
・Stress & Strain under Torque
・Design of Shaft Diameter
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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
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
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
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
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