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Torsion
Torsion is defined as the twisting of a shaft due to torque applied on the shaft. The torque applied is along the longitudinal axis.
e.g. applying two opposite moments along the same shaft
Image 1: a shaft with two opposite moments applied
Torsion in Circular Shafts
The cross section of the circular shaft retains its shape on the same plane.
Image 2: a cylindrical shaft under torsion
Torsion in Square Shafts
The cross section of the square shaft is distorted when torque is applied.
Image 3: a square shaft under torsion
Torsional Stress and Strain
When torsion is applied on a cylindrical shaft, the cross-sectional area of the shaft is still planar (still circular) with no distortion present compared to a cube shaped shaft. When a torque is applied to the shaft, the shaft will twist to an angle of θ. The bar length, l, and stiffness, G, which is the shear modulus, will vary with the twist angle.
Different from stress due to axial loads, shearing stresses due to torsional loads cannot be assumed to be uniform. Maximum shear stress is furthest away from the centre and decreases proportionally as distance decreases.
Equation (1) shows the relationship between shear stress, radius of the bar, torsion applied, second moment of area, length, twist angle and shear modulus.
Image 4: shear stress on a shaft
(1)
Where,
τ Shear stress
r Radius
T Torque applied on shaft
J Polar second moment of area
G Shear modulus
θ Twist angle
L Length
Polar Second Moment of Area
Polar second moment of area refers to how resistant a part is to torsion and it is represented by the symbol J (Science Dictionary, n.d.).
For a solid cylinder, the polar second moment of area is calculated using equation (2).
(2)
where rmax is the radius of the cylinder.
Then, for a hollow cylinder, equation (3) is used to calculate its polar second moment of area.
(3)
Similarly, rmax is the external radius of the hollow cylinder, and rint is the internal radius of the hollow cylinder.
The maximum amount of shear stress applied on a shaft can be calculated using equation (4)
(4)
To calculate principal shear stress, equation (5) can be used.
(5)
References
References
Science Dictionary. (n.d.). What is polar second moment of area? Retrieved April 30, 2019, from Science Dictionary: http://thesciencedictionary.org/polar-second-moment-of-area/
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