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) = 𝐷cover = 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=𝐴𝑠𝑓𝑦𝑑0.87𝑓𝑐𝑑𝑏

    where:

    • 𝑏 = width of the beam = 250 mm
    𝑥=1256166.670.8713.33250𝑥0.636 m636 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=0.48500=240 mm (as assumed, this value of 𝑥 is used).

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

    𝑧=𝑑𝑥2𝑧=4702402=470120=350 mm
  6. Calculate the ultimate moment of resistance (𝑀𝑢):

    𝑀𝑢=𝐴𝑠𝑓𝑦𝑑𝑧𝑀𝑢=1256166.67350𝑀𝑢73.29 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) = 𝐷cover = 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=𝐴𝑠𝑓𝑦𝑑0.87𝑓𝑐𝑑𝑏

    where:

    • 𝑏 = width of the beam = 250 mm
    𝑥=1256166.670.8713.33250𝑥0.636 m636 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=0.48500=240 mm (as assumed, this value of 𝑥 is used).

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

    𝑧=𝑑𝑥2𝑧=4702402=470120=350 mm
  6. Calculate the ultimate moment of resistance (𝑀𝑢):

    𝑀𝑢=𝐴𝑠𝑓𝑦𝑑𝑧𝑀𝑢=1256166.67350𝑀𝑢73.29 kNm

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



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