Torsional Shear Stress When Shaft Is Subjected To Pure Torsional Moment Calculator

Torsional Shear Stress Formula:

\[ \tau_h = \frac{16 \times M_t}{\pi \times d_o^3 \times (1 - C^4)} \]

Torsional Moment in Hollow Shaft (M_t):

N·m

Outer Diameter of Hollow Shaft (d_o):

Ratio of Inner to Outer Diameter (C):

(dimensionless)

Torsional Shear Stress (τ_h):

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1. What is Torsional Shear Stress?

Torsional shear stress is the stress developed in a shaft when it is subjected to a twisting moment or torque. In hollow shafts, this stress varies across the cross-section and is maximum at the outer surface.

2. How Does the Calculator Work?

The calculator uses the torsional shear stress formula for hollow shafts:

\[ \tau_h = \frac{16 \times M_t}{\pi \times d_o^3 \times (1 - C^4)} \]

Where:

\( \tau_h \) — Torsional shear stress (Pa)
\( M_t \) — Torsional moment (N·m)
\( d_o \) — Outer diameter of hollow shaft (m)
\( C \) — Ratio of inner to outer diameter (dimensionless)
\( \pi \) — Mathematical constant pi (approximately 3.14159)

Explanation: This formula calculates the maximum shear stress in a hollow circular shaft subjected to pure torsion, accounting for the hollow geometry through the diameter ratio term.

3. Importance of Torsional Shear Stress Calculation

Details: Calculating torsional shear stress is crucial for designing shafts in mechanical systems to ensure they can withstand applied torques without failure. It helps determine the appropriate shaft dimensions and material selection for various applications.

4. Using the Calculator

Tips: Enter torsional moment in N·m, outer diameter in meters, and the diameter ratio (0 ≤ C < 1). All values must be positive, and the diameter ratio must be less than 1.

5. Frequently Asked Questions (FAQ)

Q1: What is the significance of the diameter ratio C?
A: The diameter ratio (C = d_i/d_o) accounts for the hollow nature of the shaft. As C increases, the shaft becomes thinner-walled, affecting the stress distribution.

Q2: How does hollow shaft design compare to solid shafts?
A: Hollow shafts can provide similar torsional strength with less material and weight compared to solid shafts, making them more efficient for many applications.

Q3: Where is this formula typically applied?
A: This formula is used in mechanical engineering for designing drive shafts, propeller shafts, and other rotating members in automotive, aerospace, and industrial applications.

Q4: What are the limitations of this formula?
A: This formula assumes pure torsion, homogeneous material, circular cross-section, and linear elastic behavior. It may not apply to shafts with stress concentrations or non-circular sections.

Q5: How does torsional stress relate to shaft failure?
A: Excessive torsional shear stress can cause yielding or fracture in shafts. Designers must ensure the calculated stress remains below the material's allowable shear stress with appropriate safety factors.