Torsion
A shaft is said to be under pure torsion when it is subjected to two equal and opposite couples in a plane perpendicular to the longitudinal axis of the shaft (that is, twisting couples) so that the magnitude of the twisting moment remains constant along the length of the shaft. The twisting moment (or torque) is the product of the force and the perpendicular distance between the pair of forces.
Torque T = p × d
Magnitude and representation of Torque Figure shows a bar or shaft of circular section subjected to torque T. Such a case is one of pure torsion.
Shaft is under pure torsion
Note: J/R is known as the torsional section modulus. GJ is known as the torsional rigidity of the bar or shaft. The relation for shear stress under pure torsion shows that the intensity of shear stress at any point in the cross-section is proportional to its radial distance from the centre; the variation with radial distance is linear.
Variation of Torsional Shear Stress Basic Relations for a Circular Shaft under Torsion
Shear stress distribution and internal torque
Consider a circular shaft of cross-sectional area A, subjected to an external torque T. The internal resisting couple is the result of shear stresses distributed over the cross-section. Equating internal resisting moment to the applied torque gives the fundamental relation:
τ(r) = T·r / J
where τ(r) is the shear stress at a radial distance r from the centre, and J is the polar moment of inertia (also called polar second moment, or polar moment) of the cross-section about the shaft axis. This relation implies that shear stress is zero at r = 0 and increases linearly to a maximum at the outer radius.
Angle of twist
The angle of twist θ (in radians) of a shaft of length L carrying a torque T is
θ = T L / G J
where G is the modulus of rigidity (shear modulus). This formula is valid for prismatic (constant cross-section) shafts made of linearly elastic, homogeneous material under small deformations.
Polar Moment of Inertia (J)
Polar moment J measures a section's resistance to torsion. For circular sections the standard expressions are:
Solid circular shaft
For a solid circular shaft of diameter d (or radius R = d/2),
J = π d^4 / 32 = π R^4 / 2
Hollow circular shaft
For a hollow circular shaft with outer diameter D and inner diameter d,
J = π (D^4 - d^4) / 32
Torsional Section Modulus and Maximum Shear Stress
The torsional section modulus Z is defined as
Z = J / c
where c is the outer radius (distance from neutral axis to extreme fibre). The maximum shear stress at the outer surface is
τ_max = T / Z = T·c / J
Shear Stress Distribution in Different Sections
The shear stress distribution for circular shafts is:
- Solid circular section - shear stress is zero at the centre and increases linearly with radius to a maximum at the outer surface.
- Hollow circular section - shear stress is minimum at the inner radius and increases linearly to a maximum at the outer radius; there is no shear stress in the hole region.
Power Transmitted by a Shaft
When a shaft rotates at N revolutions per minute (rpm) and transmits torque T (in N·m), the power P (in watts) transmitted is
P = 2π N T / 60
In SI units, P (W) = T (N·m) × angular speed ω (rad/s), where ω = 2πN/60.
Design of Shafts - Limiting Torque from Allowable Shear
In shaft design the resisting couple developed by internal shear stresses must equal the applied torque. Using τ_max = T·c / J, the maximum torque that can be transmitted safely (based on an allowable shear stress τ_allow) is
T_max = τ_allow · J / c
Maximum Torque Transmitted by a Circular Shaft
- Circular solid shaft
The maximum torque transmitted by a circular solid shaft is obtained from the maximum shear stress induced at the outer surface. Substituting J for a solid shaft gives a direct expression for T_max in terms of diameter and allowable shear.
- Hollow circular shaft
Torque transmitted by a hollow circular shaft is obtained in the same way as for a solid shaft by using the polar moment of inertia for a hollow section. Hollow shafts are commonly used when weight or material economy is important because they provide higher torsional rigidity per unit weight in many practical cases.
Composite Shafts
A composite shaft consists of two or more shaft segments of different materials, diameters or lengths joined together. Two important connection types are:
- Series connection - shafts are arranged end-to-end so that each segment carries the same torque.
- Parallel connection - shafts are arranged so that the angle of twist of each shaft is the same (they act in parallel to share torque).
Series connection
When shafts are in series, the torque in each segment is equal to the applied torque. The total angle of twist is the sum of the twists of each segment. For two shafts in series, with the same torque T, lengths L1 and L2, moduli G1 and G2, and polar moments J1 and J2:
θ_AC = θ_AB + θ_BC
and
θ_total = T L1 / (G1 J1) + T L2 / (G2 J2)
Parallel connection
For shafts loaded so that their angle of twist is the same, torque is shared between the parallel members. The total torque is the sum of individual torques and equality of angles of twist gives a relation between the individual torques:
T = T1 + T2
θ1 = θ2
From θ = T L / G J, we obtain
T1 L1 / (G1 J1) = T2 L2 / (G2 J2)
Strain Energy Stored in a Shaft under Torsion
The torsional strain energy stored in a shaft is equal to the work done in twisting it from zero torque to the final torque T. For a shaft of length L:
Work done = ∫0^T T' dθ = 1/2 T θ
Using θ = T L / (G J) leads to
U = 1/2 · T · θ = T^2 L / (2 G J) = (G J θ^2) / (2 L)
Torsional Stiffness
Torsional stiffness (denoted K) is the torque required to produce a unit radian twist. For a prismatic shaft of length L:
θ = T L / (G J)
so
K = G J / L
and
T = K θ
Practical Remarks and Applications
- Shafts in machinery (shafts for motors, gearboxes, axles) are almost always designed using the relations above to ensure the permissible shear stress and permissible angle of twist are not exceeded.
- Hollow shafts are used to reduce weight while retaining torsional rigidity. For the same material and outer diameter, an optimally chosen hollow shaft can transmit similar torque with less material than a solid shaft.
- When shafts are subjected to combined bending and torsion, interaction theories (like maximum distortion energy or von Mises) are used for safe design; the pure torsion formulas apply to the torsional component of stress only.
- Torsional vibration, stress concentrations near shoulders, keyways and fillets, and fatigue under cyclic torque must be considered in detailed design; the basic formulas provide the elastic behaviour and strength limits for static design.
Summary: For circular shafts under pure torsion the central relations are τ(r) = T·r / J, θ = T L / (G J), J expressions for the section (solid or hollow), torsional section modulus Z = J/c, strain energy U = T^2 L / (2 G J) and torsional stiffness K = G J / L. These formulae form the foundation for analysis and design of shafts in civil, mechanical and allied engineering applications.
FAQs on Torsion
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