Question

Two beams of equal cross sectional areas are subjected to equal bending moment. If one beam has square cross section and the other has circular section, then

• A. Both beams will be equally stronger
• B. Circular section beam will be stronger
• C. Square section beam will be stronger
• D. Cannot comment

Hint:

For a given cross-sectional area, material distributed further from the neutral axis increases the moment of inertia and section modulus.
A square section distributes its corners further away from the neutral axis than a circular section, making it more efficient in bending.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
This question compares the structural bending strength of two beams having different cross-sectional shapes (square vs. circular) but equal cross-sectional areas and subjected to the same bending moment.

Step 2: Key Formula or Approach:
The bending strength of a beam is determined by its section modulus ($Z$):
\[ \sigma_{\text{max}} = \frac{M}{Z} \]
For a given bending moment $M$, the beam with the larger section modulus ($Z$) will experience lower maximum bending stress ($\sigma_{\text{max}}$) and is therefore structurally stronger.

Step 3: Detailed Explanation:

• Let the cross-sectional area of both beams be $A$.

• For the square beam of side $a$:
\[ A = a^2 \implies a = \sqrt{A} \]
The section modulus of a square is:
\[ Z_{\text{square}} = \frac{I}{y} = \frac{a^4/12}{a/2} = \frac{a^3}{6} = \frac{A^{1.5}}{6} \approx 0.167 A^{1.5} \]

• For the circular beam of diameter $d$:
\[ A = \frac{\pi}{4} d^2 \implies d = \sqrt{\frac{4A}{\pi}} \]
The section modulus of a circle is:
\[ Z_{\text{circular}} = \frac{\pi d^3}{32} = \frac{\pi}{32} \left(\frac{4A}{\pi}\right)^{1.5} = \frac{A^{1.5}}{4\sqrt{\pi}} \approx 0.141 A^{1.5} \]

• Comparing the section moduli:
\[ Z_{\text{square}} > Z_{\text{circular}} \]

• Since the square section has a larger section modulus, it experiences lower maximum bending stress and is stronger.

Step 4: Final Answer:
The square section beam will be stronger.