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: Recall the standard modular ratio table used in working stress method (IS 456, Annex B) for different concrete grades: M15 gives about 18.7, M20 gives about 13.3, M25 gives about 11.3, M30 gives about 9.3, all decreasing as concrete grade or strength increases.
Step 2: Locate M20 in this standard table.
Step 3: Read off the tabulated value directly for M20 grade concrete.
\[ \boxed{m \approx 13.3} \]

Answer 2

A third way to approach this is to go back to the fundamental physical meaning of modular ratio, \(m = E_s/E_c\), and estimate it using a rough empirical elastic modulus for M20 concrete rather than the code's simplified working-stress formula.

The modulus of elasticity of steel is taken as \(E_s \approx 2 \times 10^5\ \text{N/mm}^2\). For concrete, a commonly used empirical estimate of the long-term, working-stress elastic modulus for M20 grade is of the order of \(E_c \approx 1.5 \times 10^4\ \text{N/mm}^2\) once long-term effects such as creep are accounted for (this is intentionally lower than the short-term modulus used for deflection checks, because the working stress method's modular ratio is meant to reflect sustained loading behavior).

Dividing these gives \(m = E_s/E_c \approx \dfrac{2\times10^5}{1.5\times10^4} \approx 13.3\), consistent with the value expected for M20 concrete.

1. 18.6: This would require a noticeably lower effective concrete modulus than the estimate used above, more typical of a weaker grade like M15, so it does not fit M20.
2. 13.3: This matches the ratio obtained from the physical \(E_s/E_c\) estimate for M20 concrete under sustained loading.
3. 9.9: This would require a noticeably higher effective concrete modulus than estimated for M20, more typical of a stronger grade of concrete.
4. 6.5: This would require an even higher concrete modulus still, corresponding to an even stronger grade of concrete than M20.

Estimating the ratio from the physical elastic moduli of steel and M20 concrete under sustained load reproduces essentially the same value as the code formula.

Therefore, the correct answer is 13.3.