Ratio of Diameters given Bending Stress of Hollow Shaft Calculator

Formula Used:

\[ C = \left(1 - \frac{32 \times M_{b h}}{\pi \times d_o^3 \times \sigma_{b h}}\right)^{1/4} \]

Bending Moment in Hollow Shaft (M_{b h}):

N·m

Outer Diameter of Hollow Shaft (d_o):

Bending Stress in Hollow Shaft (σ_{b h}):

Ratio of Inner to Outer Diameter (C):

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1. What is the Ratio of Diameters Formula?

The ratio of inner to outer diameter of a hollow shaft is calculated using the bending moment, outer diameter, and bending stress. This ratio helps in determining the optimal cross-sectional geometry of hollow shafts for given loading conditions.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ C = \left(1 - \frac{32 \times M_{b h}}{\pi \times d_o^3 \times \sigma_{b h}}\right)^{1/4} \]

Where:

\( C \) — Ratio of inner to outer diameter
\( M_{b h} \) — Bending moment in hollow shaft (N·m)
\( d_o \) — Outer diameter of hollow shaft (m)
\( \sigma_{b h} \) — Bending stress in hollow shaft (Pa)
\( \pi \) — Mathematical constant (approximately 3.14159)

Explanation: The formula calculates the diameter ratio that satisfies the given bending stress constraint for a specific bending moment and outer diameter.

3. Importance of Diameter Ratio Calculation

Details: Calculating the optimal diameter ratio is crucial for designing hollow shafts that can withstand specified bending moments while maintaining acceptable stress levels, ensuring structural integrity and material efficiency.

4. Using the Calculator

Tips: Enter bending moment in N·m, outer diameter in meters, and bending stress in Pascals. All values must be positive and valid for meaningful results.

5. Frequently Asked Questions (FAQ)

Q1: What is the typical range for diameter ratio in hollow shafts?
A: The diameter ratio typically ranges from 0.5 to 0.9, depending on the application and material properties.

Q2: Why use hollow shafts instead of solid ones?
A: Hollow shafts provide better strength-to-weight ratio, reduced material usage, and can accommodate internal components or fluids.

Q3: What happens if the calculated ratio is negative or imaginary?
A: This indicates that the input parameters are not physically feasible - either the bending stress is too low or the outer diameter is too small for the given bending moment.

Q4: Can this formula be used for dynamic loading conditions?
A: This formula is primarily for static bending analysis. For dynamic loading, additional factors like fatigue and impact loads should be considered.

Q5: How does temperature affect the diameter ratio calculation?
A: Temperature changes can affect material properties (like modulus of elasticity and yield strength), which may require adjustment of the bending stress value used in the calculation.