2. Torsional Shear Stress

For a circular shaft subjected to torque T, the shear stress at a radius r is:

τ = \frac{T r}{J}

Where:

• τ: Shear stress (Pa)
• T: Applied torque (N·m)
• r: Radial distance from center (m)
• J: Polar moment of inertia (m⁴)

This formula calculates the shear stress (τ) in a circular shaft when it is subjected to a torque (T). The shear stress is dependent on two main factors: the distance from the center of the shaft (r) and the shaft's geometry, captured in the polar moment of inertia (J).

• τ (shear stress) is a measure of how much force is being distributed over a particular area of the shaft.
• T represents the torque applied, which is the twisting force.
• r is the distance from the center of the shaft to the point where we want to find the shear stress. The further from the center, the higher the shear stress.
• J is a geometric property of the shaft that indicates how it resists twisting; it varies depending on whether the shaft is solid or hollow.

Imagine you're trying to twist a thick rubber band. The closer you grip it to the center, the easier it is to twist. However, if you grip it towards the end, it becomes harder to twist because more force is required due to the distance from the center. This example relates to how shear stress is higher at points further from the center of a shaft.

For a solid circular shaft: J = \frac{\pi d^4}{32}

The polar moment of inertia (J) is a measure of an object's ability to resist twisting when subjected to torque. For a solid circular shaft, the formula J = (π d^4) / 32 shows how J is computed based on the diameter (d) of the shaft.

• This formula illustrates that as the diameter increases, the polar moment of inertia increases significantly, enhancing the shaft's ability to withstand torsion.

Consider a thick wooden dowel versus a thin one: if you try to twist both, the thick dowel (solid shaft with a larger diameter) is much harder to twist than the thin dowel because it has a higher polar moment of inertia, allowing it to resist the twisting force more effectively.

For a hollow circular shaft: J = \frac{\pi (d_o^4 - d_i^4)}{32}

This formula calculates the polar moment of inertia for hollow circular shafts, which have an outer diameter (d_o) and an inner diameter (d_i). The value of J shows how effective a hollow shaft is at resisting torsion compared to a solid shaft.

• Similar to the solid shaft, a larger difference between outer and inner diameters results in a greater polar moment of inertia. This is beneficial for reducing weight while still providing structural integrity.

Think of bicycle tires: the outer part is thick and sturdy, while the inside is hollow. This design allows the tire to be lightweight yet strong enough to support the weight of the bike and rider, much like how a hollow shaft can resist twisting forces compared to a solid one.

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