The moment of resistance for a balanced section using the limit state design method

The moment of resistance for a balanced section in reinforced concrete design is given by the equation Mu.lim = Kbd^2. Here, Mu.lim represents the moment of resistance, K is a constant, b is the width of the beam, and d is the effective depth of the beam.

To find the value of K, we need to consider the material properties of the concrete and steel being used.

Material properties:
- Concrete grade: M20
- Steel grade: Fe 415 (HYSD)

Step 1: Determine the characteristic strength of the concrete (fck):
The characteristic strength of the concrete is the value below which the probability of failure is very low. For M20 grade concrete, the characteristic strength (fck) is 20 MPa.

Step 2: Determine the characteristic strength of the steel (fy):
The characteristic strength of the steel is the value below which the probability of failure is very low. For Fe 415 steel, the characteristic strength (fy) is 415 MPa.

Step 3: Calculate the modular ratio (m):
The modular ratio (m) is the ratio of the elastic moduli of steel and concrete. It is given by the equation m = Es/Ec, where Es is the modulus of elasticity of steel and Ec is the modulus of elasticity of concrete.

For Fe 415 steel, the modulus of elasticity (Es) is approximately 200 GPa.
For M20 grade concrete, the modulus of elasticity (Ec) can be taken as 5000√fck (in MPa). Therefore, Ec = 5000√20 = 10,000 MPa.

Thus, the modular ratio (m) = Es/Ec = 200/10,000 = 0.02

Step 4: Calculate the lever arm factor (k):
The lever arm factor (k) is the distance from the centroid of the tension reinforcement to the outermost compression fiber. It depends on the shape of the stress block and can be determined from the relevant design codes or guidelines. For a rectangular stress block, the lever arm factor (k) is given by the equation k = 0.156 + 0.87m.

Substituting the value of m = 0.02, we get k = 0.156 + 0.87*0.02 = 0.156 + 0.0174 = 0.1734

Step 5: Calculate the value of K:
The value of K can be determined by rearranging the moment of resistance equation: K = Mu.lim / (bd^2)

Since Mu.lim = Kbd^2, we can substitute the value of K from the previous step:
K = (Mu.lim) / (bd^2) = (Mu.lim) / (0.1734bd^2)

The value of K is specific to the particular design condition and loading scenario. Therefore, it cannot be determined without additional information or specific design criteria.

Hence, the given options (a) 2.98, (b) 2.76, (c) 1.19, and (d) 0.89 are incorrect,