Calculating the Percentage Area of Steel in a Singly Reinforced Beam

Given:
- Permissible compressive stress = 20 MPa
- Permissible tensile stress = 250 MPa
- Neutral axis depth limit = 0.3 times effective depth
- Economic section required

To Find:
- Percentage area of steel required

Solution:

1. Determine the depth of the beam

Let the effective depth of the beam be d. Then, the neutral axis depth limit is 0.3d. Therefore, the depth of the beam is:

Depth = d + 0.3d = 1.3d

2. Calculate the moment of resistance

The moment of resistance of the beam can be given by:

M = (0.87 * fy * Ast * (d - 0.4 * x)) + (0.36 * fck * b * x * (d - (0.42 * x)))

Where:
- fy = Yield strength of steel
- Ast = Area of steel
- x = Distance of the neutral axis from the compression face
- fck = Characteristic compressive strength of concrete
- b = Width of the beam

3. Calculate the area of steel required

The area of steel required can be calculated using the moment of resistance and the permissible tensile stress. Let Ast_req be the required area of steel. Then:

Ast_req = (M / (0.87 * fy * (d - 0.4 * x))) * (1 / 1000000)

4. Check for the permissible compressive stress

Once the area of steel is calculated, it is important to check if the compression stresses are within the permissible limit. This can be done using the following formula:

P = 0.4 * fck * b * (d - 0.4 * x)

Where P is the compressive force acting on the beam and should be less than 0.4 times the characteristic compressive strength of concrete.

5. Determine the percentage area of steel

Once Ast_req is known, the percentage area of steel can be calculated using the following formula:

Percentage area of steel = (Ast_req / (b * d)) * 100

6. Check if the section is economical

The section is said to be economical if the percentage area of steel falls between 0.8% to 4% of the total cross-sectional area of concrete.

Conclusion:

In conclusion, the percentage area of steel required for an economic section can be calculated by finding the moment of resistance, the area of steel required, checking for the permissible compressive stress and finally determining the percentage area of steel. It is important to note that the section is economical only if the percentage area of steel is between 0.8% to 4% of the total cross-sectional area of concrete.