RCC Numerical - Calculate maximum UDL (including self weight) it can carry over entire span.

 A singly reinforced beam 230 mm × 500 mm effective cross-section is simply supported and has 6 m. effective span. Beam is reinforced with 4 Nos. of 16 mm φ bars in tension. Calculate maximum UDL (including self weight) it can carry over entire span. Use concrete of grade M20 and steel Fe415 grade.



To calculate the maximum uniformly distributed load (UDL) a singly reinforced concrete beam can carry, we need to:

1. Determine the beam's self-weight.
2. Calculate the ultimate moment of resistance (Mu) based on the concrete and steel grades.
3. Use the span and the moment of resistance to find the maximum UDL that the beam can carry.

Step-by-Step Calculation:

1. Self-weight of the beam:

• Dimensions of the beam = 230 mm × 500 mm (0.23 m × 0.5 m)
• Unit weight of concrete (γ) ≈ 25 kN/m³

$$\text{Self-weight per meter}=\text{cross-sectional area}\times \text{unit weight of concrete}$$$$=0.23\times 0.5\times 25=2.875\text{kN/m}$$

2. Moment of resistance (Mu):

For a singly reinforced beam, the ultimate moment of resistance is given by:

$${𝑀}_{𝑢}=0.138{𝑓}_{𝑐𝑘}𝑏{𝑑}^2$$

Where:

• ${𝑓}_{𝑐𝑘}=20{\text{N/mm}}^2$ (grade of concrete M20)
• $𝑏=230\text{mm}$ (width of the beam)
• $𝑑=500\text{mm}$ (effective depth of the beam)

Substitute values:

$${𝑀}_{𝑢}=0.138\times 20\times 230\times (500{)}^2\text{Nmm}$$

Convert ${𝑀}_{𝑢}$ into kNm:

$${𝑀}_{𝑢}=\frac{0.138\times 20\times 230\times 500^2}{10^6}=158.7\text{kNm}$$

3. Maximum UDL calculation:

For a simply supported beam, the relationship between the UDL ($𝑤$) and the ultimate moment of resistance (${𝑀}_{𝑢}$) is:

$${𝑀}_{𝑢}=\frac{𝑤{𝐿}^2}{8}$$

Where:

• $𝐿=6\text{m}$ (span of the beam)
• ${𝑀}_{𝑢}=158.7\text{kNm}$

Rearranging the formula to solve for $𝑤$:

$$𝑤=\frac{8{𝑀}_{𝑢}}{{𝐿}^2}$$$$𝑤=\frac{8\times 158.7}{6^2}=35.27\text{kN/m}$$

This is the total UDL the beam can carry, including its self-weight.

4. Net UDL carried by the beam:

Now, subtract the self-weight:

$$\text{Net UDL}=35.27\text{kN/m}-2.875\text{kN/m}=32.395\text{kN/m}$$

Final Answer:

The maximum uniformly distributed load (including self-weight) that the beam can carry is approximately 35.27 kN/m, and the net UDL the beam can carry is 32.4 kN/m.