Outer Diameter of Shaft Given Torsional Shear Stress Calculator

Formula Used:

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

Torsional Moment (M_t):

N·m

Torsional Shear Stress (τ_h):

Ratio of Inner to Outer Diameter (C):

(dimensionless)

Outer Diameter of Hollow Shaft (d_o):

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1. What is the Outer Diameter of Hollow Shaft Formula?

The formula calculates the outer diameter of a hollow shaft subjected to torsional loading, considering the torsional moment, allowable shear stress, and the ratio of inner to outer diameter. It ensures the shaft can safely transmit torque without exceeding the material's shear stress limit.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

\( d_o \) — Outer diameter of hollow shaft (m)
\( M_t \) — Torsional moment (N·m)
\( \tau_h \) — Torsional shear stress (Pa)
\( C \) — Ratio of inner to outer diameter (dimensionless)
\( \pi \) — Mathematical constant pi (≈3.1416)

Explanation: The formula derives from torsion theory for hollow circular shafts, accounting for the reduced cross-sectional area due to the hollow center.

3. Importance of Outer Diameter Calculation

Details: Proper shaft sizing is critical for mechanical design to ensure structural integrity, prevent failure under torsional loads, and optimize material usage while maintaining safety factors.

4. Using the Calculator

Tips: Enter torsional moment in N·m, torsional shear stress in Pa, and diameter ratio (0 ≤ C < 1). All values must be positive, with diameter ratio strictly less than 1.

5. Frequently Asked Questions (FAQ)

Q1: Why is the diameter ratio important in this calculation?
A: The diameter ratio affects the polar moment of inertia of the hollow shaft, which directly influences its torsional stiffness and stress distribution.

Q2: What are typical values for torsional shear stress?
A: Allowable torsional shear stress varies by material, typically ranging from 40-200 MPa for steel shafts, depending on the application and safety factors.

Q3: How does hollow shaft compare to solid shaft performance?
A: Hollow shafts provide better weight-to-strength ratio and can transmit more torque per unit weight compared to solid shafts of the same outer diameter.

Q4: What are the limitations of this formula?
A: This formula assumes homogeneous, isotropic material, pure torsion loading, and circular cross-section. It may not account for stress concentrations or combined loading conditions.

Q5: How should safety factors be applied?
A: The calculated diameter should be multiplied by an appropriate safety factor (typically 1.5-3.0) depending on the application, loading conditions, and material properties.