Question

What is the modular ratio to be used in the analysis of RC beams using working stress method if the grade of concrete is M20?

• A. 18.6
• B. 13.3
• C. 9.9
• D. 6.5

Hint:

For working stress method, always use the modified modular ratio based on permissible concrete stress.

Answer 1

The Correct Option is B

Step 1: Formula for modular ratio.
In working stress method, the modular ratio is given by:
\[ m = \frac{280}{3 \sigma_{cbc}} \]
Step 2: Permissible stress for M20 concrete.
For M20 concrete, permissible compressive stress in bending is:
\[ \sigma_{cbc} = 7 \text{ N/mm}^2 \]
Step 3: Substitute the value.
\[ m = \frac{280}{3 \times 7} = \frac{280}{21} = 13.33 \]
Step 4: Conclusion.
Thus, the modular ratio to be used is approximately 13.3.

Answer 2

Instead of just plugging M20 into the formula once, we can cross-check the result using the ratio behavior of the modular ratio across different concrete grades, since the formula \(m = \dfrac{280}{3\sigma_{cbc}}\) implies m is inversely proportional to the permissible compressive bending stress \(\sigma_{cbc}\).

For M15 concrete, the permissible bending compressive stress is well known to be \(\sigma_{cbc} = 5\ \text{N/mm}^2\), which gives \(m = \dfrac{280}{3(5)} = \dfrac{280}{15} \approx 18.7\), a widely quoted standard value for M15 in working stress method tables. Since M20 has a higher permissible stress (\(\sigma_{cbc} = 7\ \text{N/mm}^2\)) than M15, and m is inversely proportional to \(\sigma_{cbc}\), the modular ratio for M20 must be correspondingly smaller than 18.7, specifically by the ratio 5/7: \(m_{M20} = 18.7 \times \dfrac{5}{7} \approx 13.3\).

1. 18.6: This value is close to the modular ratio for the weaker M15 grade, not the stronger M20 grade, so assigning it to M20 is incorrect.
2. 13.3: This matches the value obtained above for M20 by scaling down from the known M15 result in proportion to the higher permissible stress, confirming it as correct for M20.
3. 9.9: This is even smaller than the M20 value calculated, and would instead correspond to a concrete grade stronger than M20 (higher \(\sigma_{cbc}\) than 7 N/mm2), so it does not apply here.
4. 6.5: This is smaller still, corresponding to an even higher-grade concrete than M20, so it does not match this question.

Scaling the well-known M15 modular ratio down in proportion to M20's higher permissible stress reproduces the same result as direct substitution.

Therefore, the correct answer is 13.3.