Understanding Torsional Stress

Torsional stress, also known as shear stress due to torque, occurs when an object is subjected to a twisting force around its longitudinal axis. This type of stress is commonly encountered in mechanical systems and structural components, such as shafts, beams, and rods, where torque transmission is essential. Understanding torsional stress is critical in engineering to ensure components can withstand the applied forces without failure.

What is Torsional Stress?

Torsional stress arises when a torque or twisting moment is applied to a material. The stress is distributed over the cross-section of the material and varies with the radius of the cross-section. Torsional stress is mathematically represented as:

τ = (T * r) / J

Where:

• τ: Shear stress at a given radius (r) (Pa or N/m²)
• T: Applied torque (Nm)
• r: Radial distance from the axis of the shaft (m)
• J: Polar moment of inertia of the cross-section (m⁴)

The polar moment of inertia (J) depends on the geometry of the cross-section. For a circular shaft:

J = (π * d⁴) / 32

Where d is the diameter of the shaft.

Angle of Twist

In addition to stress, the angle of twist (θ) is another important parameter in torsion. It represents the angular displacement of one end of the shaft relative to the other and is given by:

θ = (T * L) / (G * J)

Where:

• θ: Angle of twist (radians)
• L: Length of the shaft (m)
• G: Modulus of rigidity of the material (Shear Modulus)(Pa)

Key Concepts in Torsional Stress

• Elastic Range: Within the elastic range, materials follow Hooke's law, and shear stress is proportional to shear strain.
• Plastic Range: Beyond the yield point, the material undergoes plastic deformation.
• Failure: Failure occurs when the material exceeds its torsional strength. This may result in brittle fracture or ductile yielding, depending on the material.

Worked Examples

Worked Example 1: Solid Circular Shaft

Problem:

A solid circular steel shaft of diameter d = 50 mm and length L = 1.5 m is subjected to a torque T = 200 Nm. The modulus of rigidity of steel is G = 80 GPa. Determine:

1. The maximum shear stress in the shaft.
2. The angle of twist in radians.

Solution:

Step 1: Calculate the polar moment of inertia (J)

J = (π * d⁴) / 32

J = (π * (0.05)⁴) / 32 = 3.067 × 10⁻⁸ m⁴

Step 2: Calculate maximum shear stress (τ_max)

τ_max = (T * r) / J

τ_max = (200 * 0.025) / (3.067 × 10⁻⁸) = 162.95 MPa

Step 3: Calculate the angle of twist (θ)

θ = (T * L) / (G * J)

θ = (200 * 1.5) / (80 × 10⁹ * 3.067 × 10⁻⁸) = 0.122 radians

Final Results:

• Maximum shear stress: 162.95 MPa
• Angle of twist: 0.122 radians

Worked Example 2: Hollow Circular Shaft

Problem:

A hollow circular shaft has an outer diameter of d_o = 100 mm and an inner diameter of d_i = 60 mm. It is subjected to a torque T = 500 Nm. Calculate the maximum shear stress.

Solution:

Step 1: Calculate the polar moment of inertia (J)

J = (π * (d_o⁴ - d_i⁴)) / 32

J = (π * (0.1⁴ - 0.06⁴)) / 32 = 4.578 × 10⁻⁷ m⁴

Step 2: Calculate maximum shear stress (τ_max)

τ_max = (T * r) / J

τ_max = (500 * 0.05) / (4.578 × 10⁻⁷) = 54.6 MPa

Final Result:

• Maximum shear stress: 54.6 MPa

Applications of Torsional Stress

• Drive Shafts: Automotive drive shafts experience torsional stress while transmitting power from the engine to the wheels.
• Rotating Machinery: Turbines, propellers, and drills are subjected to torsional forces during operation.
• Structural Elements: Bridges and towers may encounter torsional stress due to dynamic loading, such as wind or seismic activity.

Conclusion

Torsional stress analysis is vital in ensuring the structural integrity and functionality of components subjected to torque. Understanding the relationship between torque, shear stress, and material properties allows engineers to design safe and efficient systems.

Explore other engineering stress topic:

• Bending Stress: Exploring how materials deform under bending loads, including the calculation of bending stress and its significance in structural engineering.

• Shear Stress: Discussing the internal forces that cause layers of material to slide past each other, relevant in contexts like beam design and material failure analysis.

• Axial Stress: Examining stress caused by axial loads, including tension and compression, and their effects on structural elements.

• Combined Stress Analysis: Analyzing situations where multiple types of stresses occur simultaneously, such as in complex loading scenarios, and how to evaluate their combined effects.

References

1. Hibbeler, R. C. (2020). Mechanics of Materials. Pearson Education.
2. Gere, J. M., & Goodno, B. J. (2018). Mechanics of Materials. Cengage Learning.
3. Shigley, J. E., Mischke, C. R., & Budynas, R. G. (2011). Mechanical Engineering Design. McGraw-Hill Education.

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