Diameter Of Hollow Shaft Subjected To Maximum Bending Moment Calculator

Formula Used:

\[ \text{Hollow Shaft Outer Diameter} = \left( \frac{\text{Maximum Bending Moment}}{\left( \frac{\pi}{32} \right) \times \text{Bending Stress} \times \left(1 - \text{Ratio of Inner to Outer Diameter of Hollow Shaft}^2\right)} \right)^{1/3} \] \[ d_o = \left( \frac{M_m}{\left( \frac{\pi}{32} \right) \times f_b \times \left(1 - k^2\right)} \right)^{1/3} \]

Hollow Shaft Outer Diameter (d_o):

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1. What is the Hollow Shaft Outer Diameter Formula?

The formula calculates the outer diameter of a hollow shaft subjected to maximum bending moment, considering bending stress and the ratio of inner to outer diameter. It ensures the shaft can withstand applied loads without failure.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ d_o = \left( \frac{M_m}{\left( \frac{\pi}{32} \right) \times f_b \times \left(1 - k^2\right)} \right)^{1/3} \]

Where:

\( d_o \) — Hollow shaft outer diameter (m)
\( M_m \) — Maximum bending moment (N·m)
\( f_b \) — Bending stress (Pa)
\( k \) — Ratio of inner to outer diameter
\( \pi \) — Archimedes' constant (≈3.1416)

Explanation: The formula derives from bending stress equations for hollow circular sections, accounting for the material's strength and geometric properties.

3. Importance of Hollow Shaft Design

Details: Proper shaft sizing is critical in mechanical engineering to ensure structural integrity, minimize weight, and optimize material usage while preventing failure under load.

4. Using the Calculator

Tips: Enter maximum bending moment in N·m, bending stress in Pa, and ratio k (0 ≤ k < 1). Ensure all values are positive and k is less than 1 to avoid division by zero.

5. Frequently Asked Questions (FAQ)

Q1: Why use a hollow shaft instead of a solid one?
A: Hollow shafts offer higher strength-to-weight ratio, reduced material cost, and sometimes better resistance to torsion and bending.

Q2: What is a typical value for k (diameter ratio)?
A: k typically ranges from 0.5 to 0.9, depending on application requirements and material constraints.

Q3: How does bending stress affect shaft design?
A: Bending stress must be below the material's yield strength to prevent permanent deformation or failure.

Q4: Can this formula be used for dynamic loads?
A: For dynamic or cyclic loading, additional factors like fatigue strength and stress concentrations should be considered.

Q5: What units should be used for input values?
A: Use consistent SI units: N·m for moment, Pa for stress, and dimensionless for k. Ensure unit compatibility to avoid errors.