Modular Ratio in RC Beams Analysis using Working Stress Method:

The modular ratio is an important factor used in the analysis and design of reinforced concrete (RC) beams using the working stress method. It helps to determine the stress distribution in both concrete and steel components of the beam.

Modular Ratio:
The modular ratio, denoted as n, is the ratio of the modulus of elasticity of steel (Es) to the modulus of elasticity of concrete (Ec). It is given by the equation:

n = Es / Ec

Where,
Es = Modulus of elasticity of steel
Ec = Modulus of elasticity of concrete

Modulus of Elasticity of Steel (Es):
The modulus of elasticity of steel is a material property that represents the stiffness of steel. It is a measure of how much a material can deform under an applied load. The modulus of elasticity for mild steel is generally taken as 200 GPa (GigaPascals) or 200,000 MPa (MegaPascals).

Es = 200,000 MPa

Modulus of Elasticity of Concrete (Ec):
The modulus of elasticity of concrete is a measure of the stiffness of concrete. It depends on the grade of concrete and is determined experimentally. The modulus of elasticity for different grades of concrete can be estimated using empirical formulas.

For grade M20 concrete, the modulus of elasticity (Ec) can be estimated using the formula:

Ec = 5000 * √(fck)

Where,
fck = Characteristic compressive strength of concrete at 28 days

For M20 grade concrete, the characteristic compressive strength is 20 MPa.

Ec = 5000 * √(20)
Ec = 5000 * 4.47
Ec = 22,350 MPa

Calculating Modular Ratio:
Now, substituting the values of Es and Ec into the formula for the modular ratio:

n = Es / Ec
n = 200,000 / 22,350
n ≈ 8.96

Answer:
Therefore, the modular ratio to be used in the analysis of RC beams using the working stress method for M20 grade concrete is approximately 8.96.

Note:
The correct option from the given choices is B, which states that the modular ratio is 13.3. However, this is not the correct value for M20 grade concrete. The correct value is approximately 8.96.