Moment Capacity Ratio Explained

This solution explains how to find the ratio between the moment-carrying capacity of a square cross-section beam and a circular cross-section beam, given they have the same characteristic dimension 'D'.

Square Cross-Section Properties

For a square cross-section with side length D:

• Moment of Inertia (I_square): The formula is \( I = \frac{BD^3}{12} \). Since the side length is D, we substitute B = D, giving \( I_{square} = \frac{D \cdot D^3}{12} = \frac{D^4}{12} \).
• Section Modulus (Z_square): This is calculated as \( Z = \frac{I}{y_{max}} \), where \( y_{max} \) is the distance from the neutral axis to the extreme fiber. For a square section, \( y_{max} = \frac{D}{2} \). Therefore, \( Z_{square} = \frac{D^4/12}{D/2} = \frac{D^4}{12} \times \frac{2}{D} = \frac{D^3}{6} \).

Circular Cross-Section Properties

For a circular cross-section with diameter D:

• Radius (R): \( R = \frac{D}{2} \).
• Moment of Inertia (I_circle): The formula is \( I = \frac{\pi R^4}{4} \). Substituting \( R = \frac{D}{2} \), we get \( I_{circle} = \frac{\pi (D/2)^4}{4} = \frac{\pi D^4}{64} \).
• Section Modulus (Z_circle): Calculated as \( Z = \frac{I}{y_{max}} \). For a circle, \( y_{max} \) is the radius, so \( y_{max} = R = \frac{D}{2} \). Thus, \( Z_{circle} = \frac{\pi D^4 / 64}{D/2} = \frac{\pi D^4}{64} \times \frac{2}{D} = \frac{\pi D^3}{32} \).

Calculating the Moment Capacity Ratio

The moment-carrying capacity (M) is directly proportional to the section modulus (Z), assuming the maximum allowable bending stress (\(\sigma_{max}\)) is the same for both sections. So, \( M = \sigma_{max} \times Z \).

The ratio of the moment capacities is:

Ratio = \( \frac{M_{square}}{M_{circle}} = \frac{\sigma_{max} \times Z_{square}}{\sigma_{max} \times Z_{circle}} = \frac{Z_{square}}{Z_{circle}} \)

Substituting the calculated section moduli:

Ratio = \( \frac{D^3 / 6}{\pi D^3 / 32} \)

Simplify the expression:

Ratio = \( \frac{D^3}{6} \times \frac{32}{\pi D^3} \)

Cancel out \( D^3 \):

Ratio = \( \frac{32}{6\pi} \)

Reduce the fraction:

Ratio = \( \frac{16}{3\pi} \)

Conclusion

The ratio of the moment carrying capacity of a square cross-section beam to a circular cross-section beam, both with dimension D, is \( \frac{16}{3\pi} \).