Torsion of Shafts
Torsion of solid and hollow shafts
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Shear Stress in the Shaft
When a shaft is subjected to a torque or twisting, a shearing stress is produced in the shaft. The shear stress varies from zero in the axis to a maximum at the outside surface of the shaft.
The shear stress in a solid circular shaft in a given position can be expressed as:
σ = T r / Ip (1)
where
σ = shear stress (MPa, psi)
T = twisting moment (Nmm, in lb)
r = distance from center to stressed surface in the given position (mm, in)
Ip = "polar moment of inertia" of cross section (mm4, in4)
The "polar moment of inertia" is a measure of an object's ability to resist torsion.
Circular Shaft and Maximum Moment
Maximum moment in a circular shaft can be expressed as:
Tmax = σmax Ip / R (2)
where
Tmax = maximum twisting moment (Nmm, in lb)
σmax = maximum shear stress (MPa, psi)
R = radius of shaft (mm, in)
Combining (2) and (3) for a solid shaft
Tmax = (π/16) σmax D3 (2b)
Combining (2) and (3b) for a hollow shaft
Tmax = (π/16) σmax (D4 - d4) / D (2c)
Circular Shaft and Polar Moment of Inertia
Polar moment of inertia of a circular solid shaft can be expressed as
Ip = π R4/2 = π D4/32 (3)
where
D = shaft outside diameter (mm, in)
Polar moment of inertia of a circular hollow shaft can be expressed as
Ip = π (D4 - d4) /32 (3b)
where
d = shaft inside diameter (mm, in)
Diameter of a Solid Shaft
Diameter of a solid shaft can calculated by the formula
D = 1.72 (Tmax/σmax)1/3 (4)
Torsional Deflection of Shaft
The angular deflection of a torsion solid shaft can be expressed as
θ = 584 L T / (G D4) (5)
where
θ = angular shaft deflection (degrees)
L = length of shaft (mm, in)
G = modulus of rigidity (Mpa, psi)
The angular deflection of a torsion hollow shaft can be expressed as
θ = 584 L T / (G (D4- d4)) (5b)
Torsion Resisting Moments of Shafts of Various Cross Sections
| Shaft Cross Section Area | Maximum Torsional Resisting Moment - Tmax - (Nm, in lb) |
Nomenclature |
| Solid Cylinder Shaft | ( π/16) σmax D3 | |
| Hollow Cylinder Shaft | ( π/16) σmax (D4 - d4) / D | |
| Ellipse Shaft | ( π/16) σmax b2 h | h = "height" of shaft b = "width" of shaft h > b |
| Rectangle Shaft | (2/9) σmax b2 h | h > b |
| Square Shaft | (2/9) σmax b3 | |
| Triangle Shaft | (1/20) σmax b3 | b = length of triangle side |
| Hexagon Shaft | (1/1.09) σmax b3 | b = length of hexagon side |
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