RCC Numerical – Calculate ultimate moment of resistance of beam & maximum u.d.l

A  RCC beam of rectangular section 250 mm wide and 600 mm deep (effective) is reinforced on tension side by 4 bars of 20 mm diameter. The beam is subjected to mild exposure conditions. Use M20 and Fe415 grade materials. 

(i) Calculate ultimate moment of resistance of beam. 

(ii) Determine the maximum u.d.l., a simply supported beam can carry over a span of 6 m.

Given Data:

• Width of the beam (b) = 250 mm
• Effective depth of the beam (d) = 600 mm
• Number of tension bars = 4
• Diameter of tension bars (Ø) = 20 mm
• Grade of concrete = M20 (fck = 20 MPa)
• Grade of steel = Fe415 (fy = 415 MPa)
• Span of the beam (L) = 6 m
• Mild exposure conditions

Step 1: Calculate the area of tensile reinforcement (Ast)

The area of one bar is given by:

𝐴bar=𝜋4×𝜙2

Substituting 𝜙=20 mm:

𝐴bar=𝜋4×(20)2=314.16 mm2

Thus, the total area of steel (Ast):

𝐴st=4×314.16=1256.64 mm2

Step 2: Check if the section is under-reinforced or over-reinforced

For an under-reinforced section, the design moment of resistance is governed by the tension steel reaching its yield strength.

Calculate the neutral axis depth factor 𝑥𝑢/𝑑. The limiting value for Fe415 steel is:

𝑥𝑢,lim𝑑=0.48(for Fe415)

Hence, the limiting depth of the neutral axis is:

𝑥𝑢,lim=0.48×600=288 mm

Now, calculate the actual depth of the neutral axis using the equation:

𝐴st⋅𝑓𝑦0.36⋅𝑓𝑐𝑘⋅𝑏=𝑥𝑢

Substituting the values:

1256.64×4150.36×20×250=𝑥𝑢=241.54 mm

Since 𝑥𝑢<𝑥𝑢,lim, the section is under-reinforced.

Step 3: Calculate the ultimate moment of resistance (Mu)

The ultimate moment of resistance for an under-reinforced section is given by:

𝑀𝑢=0.87⋅𝑓𝑦⋅𝐴st⋅(𝑑−0.42⋅𝑥𝑢𝑑)

Substitute the values:

𝑀𝑢=0.87×415×1256.64×(600−0.42×241.54)×10−6𝑀𝑢=453.38 kNm

Step 4: Determine the maximum uniform distributed load (w)

For a simply supported beam with span 𝐿, the maximum moment due to a UDL is given by:

𝑀𝑢=𝑤⋅𝐿28

Rearrange to solve for 𝑤:

𝑤=8⋅𝑀𝑢𝐿2

Substituting the values:

𝑤=8×453.38×106(6000)2=100.75 kN/m

Final Answers:

1. Ultimate moment of resistance 𝑀𝑢 = 453.38 kNm
2. Maximum UDL the beam can carry = 100.75 kN/m