Understanding the Calculation of Torsional Torque in Shafts

Torsional torque calculation determines the twisting force acting on rotating shafts. It is essential for mechanical design and safety.

This article explores formulas, variables, tables, and real-world examples for precise torsional torque analysis in shafts.

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Calculate torsional torque for a steel shaft with 50 mm diameter under 500 Nm load.
Determine maximum shear stress in a hollow shaft transmitting 1000 Nm torque.
Find the angle of twist for an aluminum shaft 2 meters long under 300 Nm torque.
Evaluate torsional rigidity for a composite shaft with given dimensions and torque.

Comprehensive Tables of Common Values for Torsional Torque Calculations

Material	Shear Modulus (G) [GPa]	Typical Shaft Diameter (d) [mm]	Allowable Shear Stress (τ_allow) [MPa]	Common Torque Range (T) [Nm]	Length Range (L) [m]
Steel (AISI 1045)	79.3	20 – 100	50 – 150	10 – 5000	0.5 – 5
Aluminum (6061-T6)	26.9	15 – 80	30 – 90	5 – 1500	0.3 – 4
Cast Iron	40.0	25 – 120	20 – 70	15 – 3000	0.5 – 6
Stainless Steel (304)	77.2	20 – 90	40 – 120	20 – 4000	0.5 – 5
Composite (Carbon Fiber)	30 – 60 (varies)	10 – 50	50 – 200 (varies)	5 – 2000	0.2 – 3

Fundamental Formulas for Calculating Torsional Torque in Shafts

Accurate torsional torque calculation relies on understanding the relationship between torque, shear stress, shaft geometry, and material properties. Below are the essential formulas with detailed explanations.

1. Torsional Shear Stress Formula

The shear stress induced by torque in a circular shaft is given by:

τ = T × r / J

τ = Shear stress at radius r (Pa or N/m²)
T = Applied torque (Nm)
r = Radial distance from shaft center to point of interest (m)
J = Polar moment of inertia of the shaft cross-section (m⁴)

For solid circular shafts, the maximum shear stress occurs at the outer surface (r = c, where c is the outer radius):

τ_max = T × c / J

2. Polar Moment of Inertia (J)

The polar moment of inertia quantifies the shaft’s resistance to torsion. It depends on the shaft’s cross-sectional geometry.

Solid circular shaft:

J = π × d4 / 32

d = shaft diameter (m)

Hollow circular shaft:

J = π × (d_o4 – d_i4) / 32

d_o = outer diameter (m)
d_i = inner diameter (m)

3. Angle of Twist (θ)

The angle of twist over the length of the shaft is calculated by:

θ = T × L / (G × J)

θ = Angle of twist (radians)
L = Length of the shaft (m)
G = Shear modulus of the material (Pa)

4. Torsional Rigidity (K)

Torsional rigidity defines the shaft’s resistance to twisting deformation:

K = G × J / L

K = Torsional rigidity (Nm/rad)

5. Power Transmission and Torque Relationship

In rotating shafts, torque relates to power and angular velocity:

T = P / ω

T = Torque (Nm)
P = Power transmitted (Watts)
ω = Angular velocity (rad/s)

Angular velocity can be calculated from rotational speed (N in rpm):

ω = 2 × π × N / 60

Detailed Explanation of Variables and Typical Values

Torque (T): The twisting moment applied to the shaft, typically measured in Newton-meters (Nm). Common industrial shafts experience torques from a few Nm to several thousand Nm depending on application.
Shear Stress (τ): The internal stress caused by torque, measured in Pascals (Pa) or Megapascals (MPa). Materials have allowable shear stress limits to prevent failure.
Radius (r or c): Distance from the shaft center to the point where stress is calculated. Maximum stress occurs at the outer radius (c = d/2).
Polar Moment of Inertia (J): A geometric property representing resistance to torsion. Larger J means less twist for the same torque.
Length (L): Length of the shaft segment under consideration, affecting angle of twist.
Shear Modulus (G): Material property indicating rigidity against shear deformation. Steel typically has G ≈ 79 GPa, aluminum ≈ 27 GPa.
Angle of Twist (θ): The angular displacement caused by torque, important for mechanical alignment and performance.
Power (P) and Angular Velocity (ω): Used to relate mechanical power transmission to torque.

Real-World Application Examples of Torsional Torque Calculation

Example 1: Solid Steel Shaft Transmitting Power in an Industrial Motor

An industrial motor transmits 15 kW of power at 1500 rpm through a solid steel shaft (AISI 1045). Determine the torque, maximum shear stress, and angle of twist for a shaft diameter of 50 mm and length 2 meters.

Step 1: Calculate Angular Velocity (ω)

ω = 2 × π × N / 60 = 2 × 3.1416 × 1500 / 60 = 157.08 rad/s

Step 2: Calculate Torque (T)

T = P / ω = 15000 W / 157.08 rad/s = 95.5 Nm

Step 3: Calculate Polar Moment of Inertia (J)

d = 50 mm = 0.05 m

J = π × d4 / 32 = 3.1416 × (0.05)4 / 32 = 3.07 × 10-7 m4

Step 4: Calculate Maximum Shear Stress (τ_max)

c = d/2 = 0.025 m

τ_max = T × c / J = 95.5 × 0.025 / 3.07 × 10-7 = 7.77 × 106 Pa = 7.77 MPa

Step 5: Calculate Angle of Twist (θ)

G = 79.3 GPa = 79.3 × 109 Pa

L = 2 m

θ = T × L / (G × J) = 95.5 × 2 / (79.3 × 109 × 3.07 × 10-7) = 0.0079 radians ≈ 0.45°

Interpretation: The shaft experiences a maximum shear stress of 7.77 MPa, well below typical allowable limits for AISI 1045 steel (50-150 MPa), and a small angle of twist, indicating safe operation.

Example 2: Hollow Aluminum Shaft in Aerospace Application

A hollow aluminum shaft (6061-T6) with outer diameter 80 mm and inner diameter 60 mm transmits 1200 Nm torque. Calculate the maximum shear stress and angle of twist for a shaft length of 1.5 meters.

Step 1: Calculate Polar Moment of Inertia (J)

d_o = 0.08 m, d_i = 0.06 m

J = π × (d_o4 – d_i4) / 32

= 3.1416 × (0.084 – 0.064) / 32

= 3.1416 × (4.096 × 10-5 – 1.296 × 10-5) / 32

= 2.72 × 10-6 m4

Step 2: Calculate Maximum Shear Stress (τ_max)

c = d_o / 2 = 0.04 m

τ_max = T × c / J = 1200 × 0.04 / 2.72 × 10-6 = 17.65 × 106 Pa = 17.65 MPa

Step 3: Calculate Angle of Twist (θ)

G = 26.9 GPa = 26.9 × 109 Pa

L = 1.5 m

θ = T × L / (G × J) = 1200 × 1.5 / (26.9 × 109 × 2.72 × 10-6) = 0.0247 radians ≈ 1.42°

Interpretation: The maximum shear stress of 17.65 MPa is within the allowable range for 6061-T6 aluminum (30-90 MPa). The angle of twist is moderate, suitable for aerospace shaft design where weight and deformation are critical.

Additional Considerations and Advanced Topics

Stress Concentrations: Real shafts often have keyways, shoulders, or fillets causing localized stress increases. Correction factors must be applied to τ_max.
Fatigue Analysis: Repeated torsional loading requires fatigue strength evaluation, especially in rotating machinery.
Non-Circular Shafts: For shafts with non-circular cross-sections, torsional analysis becomes complex and may require finite element methods.
Temperature Effects: Material properties like G and allowable shear stress vary with temperature, impacting torque capacity.
Composite Shafts: Anisotropic materials require specialized formulas considering directional stiffness and strength.

Authoritative Resources for Further Study

ASME Boiler and Pressure Vessel Code – Standards for mechanical design including torsion.
ASTM International – Material property standards and testing methods.
NPTEL Mechanical Engineering Lectures – Detailed courses on strength of materials and torsion.
Engineering Toolbox – Practical calculators and tables for shaft torsion.

Mastering the calculation of torsional torque in shafts is critical for engineers designing reliable rotating machinery. By applying the formulas, understanding material properties, and considering real-world factors, one can ensure optimal shaft performance and safety.