Torsion of Shafts

Torsion of solid and hollow shafts

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 / I_p (1)

where

σ = shear stress (MPa, psi)

T = twisting moment (Nm, in lb)

r = distance from center to stressed surface in the given position (mm, in)

I_p = "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:

T_max = σ_max I_p / R (2)

where

T_max = maximum twisting moment (Nm, in lb)

σ_max = maximum shear stress (MPa, psi)

R = radius of shaft (mm, in)

Combining (2) and (3) for a solid shaft

T_max = (π/16) σ_max D³ (2b)

Combining (2) and (3b) for a hollow shaft

T_max = (π/16) σ_max (D⁴ - d⁴) / D (2c)

Circular Shaft and Polar Moment of Inertia

Polar moment of inertia of a circular solid shaft can be expressed as

I_p = 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

I_p = 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 (T_max/σ_max)^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

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