Question

The maximum bending stress induced in a beam of rectangular section is 160 N/mm$^2$. If the depth of beam is increased by two times, keeping all the parameters same, the maximum bending stress induced in the beam is

• A. 20 N/mm$^2$
• B. 40 N/mm$^2$
• C. 80 N/mm$^2$
• D. 160 N/mm$^2$

Hint:

For rectangular beams, the section modulus $Z$ is proportional to $d^2$.
Since bending stress $\sigma = M/Z$, the stress is inversely proportional to $d^2$.
This means doubling the depth makes the beam four times stronger in bending.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks how the maximum bending stress in a rectangular beam changes when its depth is doubled, assuming the applied load and other dimensions remain the same.

Step 2: Key Formula or Approach:
The maximum bending stress ($\sigma_{max}$) is given by the flexure formula:
\[ \sigma_{max} = \frac{M}{Z} \] where $M$ is the maximum bending moment and $Z$ is the section modulus.
For a rectangular cross-section of width $b$ and depth $d$:
The moment of inertia is $I = \frac{bd^3}{12}$.
The distance to the extreme fiber is $y_{max} = d/2$.
The section modulus is $Z = \frac{I}{y_{max}} = \frac{bd^3/12}{d/2} = \frac{bd^2}{6}$.
So, the stress formula becomes:
\[ \sigma_{max} = \frac{M}{bd^2/6} = \frac{6M}{bd^2} \]

Step 3: Detailed Explanation:
From the formula, we can see the relationship between stress and depth:
\[ \sigma_{max} \propto \frac{1}{d^2} \] Let $\sigma_1$ and $d_1$ be the initial stress and depth, and $\sigma_2$ and $d_2$ be the final stress and depth.
We are given:

• $\sigma_1 = 160$ N/mm$^2$.

• The depth is "increased by two times", which can be interpreted as the new depth is twice the old one, $d_2 = 2d_1$.


Using the proportionality, we can write a ratio:
\[ \frac{\sigma_2}{\sigma_1} = \left(\frac{d_1}{d_2}\right)^2 \] \[ \frac{\sigma_2}{160} = \left(\frac{d_1}{2d_1}\right)^2 = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \] \[ \sigma_2 = \frac{160}{4} = 40 \text{ N/mm}^2 \]

Step 4: Final Answer:
The new maximum bending stress induced in the beam is 40 N/mm$^2$.