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 ($𝑑$) = $𝐷-\text{cover}$ = 500 mm - 30 mm = 470 mm

2. Calculate the design strengths of the materials:

   • For M20 concrete: ${𝑓}_{𝑐𝑘}=20\text{ MPa}$
     • Design strength of concrete, ${𝑓}_{𝑐𝑑}=\frac{{𝑓}_{𝑐𝑘}}{1.5}=\frac{20}{1.5}=13.33\text{ MPa}$
   • For Fe250 steel: ${𝑓}_{𝑦}=250\text{ MPa}$
     • Design strength of steel, ${𝑓}_{𝑦𝑑}=\frac{{𝑓}_{𝑦}}{1.5}=\frac{250}{1.5}=166.67\text{ MPa}$

3. Calculate the effective area of steel reinforcement (${𝐴}_{𝑠}$):

   • Given ${𝐴}_{𝑠}=1256{\text{ mm}}^2$

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:

   $$𝑥=\frac{{𝐴}_{𝑠}\cdot {𝑓}_{𝑦𝑑}}{0.87\cdot {𝑓}_{𝑐𝑑}\cdot 𝑏}$$

   where:

   • $𝑏$ = width of the beam = 250 mm
   $$𝑥=\frac{1256\cdot 166.67}{0.87\cdot 13.33\cdot 250}$$$$𝑥\approx 0.636\text{ m}\approx 636\text{ mm}$$

   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, $𝑥=0.48\cdot 500=240\text{ mm}$ (as assumed, this value of $𝑥$ is used).

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

   $$𝑧=𝑑-\frac{𝑥}{2}$$$$𝑧=470-\frac{240}{2}=470-120=350\text{ mm}$$

6. Calculate the ultimate moment of resistance (${𝑀}_{𝑢}$):

   $${𝑀}_{𝑢}={𝐴}_{𝑠}\cdot {𝑓}_{𝑦𝑑}\cdot 𝑧$$$${𝑀}_{𝑢}=1256\cdot 166.67\cdot 350$$$${𝑀}_{𝑢}\approx 73.29\text{ kNm}$$

   Thus, the ultimate moment of resistance for the given R.C. beam is approximately $73.29\text{ kNm}$.



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 ($𝑑$) = $𝐷-\text{cover}$ = 500 mm - 30 mm = 470 mm

2. Calculate the design strengths of the materials:

   • For M20 concrete: ${𝑓}_{𝑐𝑘}=20\text{ MPa}$
     • Design strength of concrete, ${𝑓}_{𝑐𝑑}=\frac{{𝑓}_{𝑐𝑘}}{1.5}=\frac{20}{1.5}=13.33\text{ MPa}$
   • For Fe250 steel: ${𝑓}_{𝑦}=250\text{ MPa}$
     • Design strength of steel, ${𝑓}_{𝑦𝑑}=\frac{{𝑓}_{𝑦}}{1.5}=\frac{250}{1.5}=166.67\text{ MPa}$

3. Calculate the effective area of steel reinforcement (${𝐴}_{𝑠}$):

   • Given ${𝐴}_{𝑠}=1256{\text{ mm}}^2$

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:

   $$𝑥=\frac{{𝐴}_{𝑠}\cdot {𝑓}_{𝑦𝑑}}{0.87\cdot {𝑓}_{𝑐𝑑}\cdot 𝑏}$$

   where:

   • $𝑏$ = width of the beam = 250 mm
   $$𝑥=\frac{1256\cdot 166.67}{0.87\cdot 13.33\cdot 250}$$$$𝑥\approx 0.636\text{ m}\approx 636\text{ mm}$$

   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, $𝑥=0.48\cdot 500=240\text{ mm}$ (as assumed, this value of $𝑥$ is used).

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

   $$𝑧=𝑑-\frac{𝑥}{2}$$$$𝑧=470-\frac{240}{2}=470-120=350\text{ mm}$$

6. Calculate the ultimate moment of resistance (${𝑀}_{𝑢}$):

   $${𝑀}_{𝑢}={𝐴}_{𝑠}\cdot {𝑓}_{𝑦𝑑}\cdot 𝑧$$$${𝑀}_{𝑢}=1256\cdot 166.67\cdot 350$$$${𝑀}_{𝑢}\approx 73.29\text{ kNm}$$

   Thus, the ultimate moment of resistance for the given R.C. beam is approximately $73.29\text{ kNm}$.