RCC numerical – Calculate the ultimate moment of resistance of R.C. beam

Calculate the ultimate moment of resistance of R.C. beam 250 mm × 500 mm. Reinforcement of 1256 mm2 is placed at a distance of 30 mm from the bottom face. Use M20 concrete and Fe250 steel.

To calculate the ultimate moment of resistance for a reinforced concrete (R.C.) beam, we follow these steps:

1. Determine the effective depth (𝑑):

   • Overall depth of the beam (𝐷) = 500 mm
   • Cover to the reinforcement = 30 mm (from the bottom face)
   • Effective depth (𝑑) = 𝐷−cover = 500 mm – 30 mm = 470 mm

2. Calculate the design strengths of the materials:

   • For M20 concrete: 𝑓𝑐𝑘=20 MPa
     • Design strength of concrete, 𝑓𝑐𝑑=𝑓𝑐𝑘1.5=201.5=13.33 MPa
   • For Fe250 steel: 𝑓𝑦=250 MPa
     • Design strength of steel, 𝑓𝑦𝑑=𝑓𝑦1.5=2501.5=166.67 MPa

3. Calculate the effective area of steel reinforcement (𝐴𝑠):

   • Given 𝐴𝑠=1256 mm2

4. Calculate the depth of the neutral axis (𝑥): To simplify, we assume that the beam is under-reinforced. For under-reinforced sections, the neutral axis depth can be approximated using:

   𝑥=𝐴𝑠⋅𝑓𝑦𝑑0.87⋅𝑓𝑐𝑑⋅𝑏

   where:

   • 𝑏 = width of the beam = 250 mm

   𝑥=1256⋅166.670.87⋅13.33⋅250𝑥≈0.636 m≈636 mm

   Since the calculated 𝑥 exceeds the effective depth (𝑑), this indicates that the beam is in a condition of over-reinforcement. For an over-reinforced section, the neutral axis is taken at the maximum depth of 𝑥=0.48𝐷.

   Hence, 𝑥=0.48⋅500=240 mm (as assumed, this value of 𝑥 is used).

5. Calculate the lever arm (𝑧): The lever arm is approximately given by:

   𝑧=𝑑−𝑥2𝑧=470−2402=470−120=350 mm

6. Calculate the ultimate moment of resistance (𝑀𝑢):

   𝑀𝑢=𝐴𝑠⋅𝑓𝑦𝑑⋅𝑧𝑀𝑢=1256⋅166.67⋅350𝑀𝑢≈73.29 kNm

   Thus, the ultimate moment of resistance for the given R.C. beam is approximately 73.29 kNm.



To calculate the ultimate moment of resistance for a reinforced concrete (R.C.) beam, we follow these steps:

1. Determine the effective depth (𝑑):

   • Overall depth of the beam (𝐷) = 500 mm
   • Cover to the reinforcement = 30 mm (from the bottom face)
   • Effective depth (𝑑) = 𝐷−cover = 500 mm – 30 mm = 470 mm

2. Calculate the design strengths of the materials:

   • For M20 concrete: 𝑓𝑐𝑘=20 MPa
     • Design strength of concrete, 𝑓𝑐𝑑=𝑓𝑐𝑘1.5=201.5=13.33 MPa
   • For Fe250 steel: 𝑓𝑦=250 MPa
     • Design strength of steel, 𝑓𝑦𝑑=𝑓𝑦1.5=2501.5=166.67 MPa

3. Calculate the effective area of steel reinforcement (𝐴𝑠):

   • Given 𝐴𝑠=1256 mm2

4. Calculate the depth of the neutral axis (𝑥): To simplify, we assume that the beam is under-reinforced. For under-reinforced sections, the neutral axis depth can be approximated using:

   𝑥=𝐴𝑠⋅𝑓𝑦𝑑0.87⋅𝑓𝑐𝑑⋅𝑏

   where:

   • 𝑏 = width of the beam = 250 mm

   𝑥=1256⋅166.670.87⋅13.33⋅250𝑥≈0.636 m≈636 mm

   Since the calculated 𝑥 exceeds the effective depth (𝑑), this indicates that the beam is in a condition of over-reinforcement. For an over-reinforced section, the neutral axis is taken at the maximum depth of 𝑥=0.48𝐷.

   Hence, 𝑥=0.48⋅500=240 mm (as assumed, this value of 𝑥 is used).

5. Calculate the lever arm (𝑧): The lever arm is approximately given by:

   𝑧=𝑑−𝑥2𝑧=470−2402=470−120=350 mm

6. Calculate the ultimate moment of resistance (𝑀𝑢):

   𝑀𝑢=𝐴𝑠⋅𝑓𝑦𝑑⋅𝑧𝑀𝑢=1256⋅166.67⋅350𝑀𝑢≈73.29 kNm

   Thus, the ultimate moment of resistance for the given R.C. beam is approximately 73.29 kNm.