RCC Numerical - calculate ultimate moment of resistance of a beam using limit state of method

Using L.S.M., calculate ultimate moment of resistance of a beam 230 mm × 400 mm effective, reinforced on tension side with three bars of 20 mm diameter bars. Assume M20 concrete and Fe415 steel.

To calculate the ultimate moment of resistance (M_u) of a reinforced concrete beam using the Limit State Method (L.S.M.), follow these steps:

1. Determine the Effective Depth (d):

   • Given beam dimensions are 230 mm × 400 mm, with an effective depth $𝑑$ assumed to be $400\text{ mm}-\text{cover}-\text{bar diameter}$. Typically, the cover and bar diameter would be provided, but for simplicity, let’s assume the cover is 25 mm and the bar diameter is 20 mm.
   • Effective depth $𝑑$ = 400 mm - 25 mm - 10 mm (half of the bar diameter) = 365 mm

2. Calculate the Area of Steel (A_st):

   • The beam has three bars of 20 mm diameter.
   • Area of one bar ${𝐴}_{𝑏𝑎𝑟}$ = $\frac{𝜋{𝑑}^2}{4}=\frac{𝜋\times (20{)}^2}{4}=314.16{\text{ mm}}^2$
   • Total Area of Steel ${𝐴}_{𝑠𝑡}$ = 3 × 314.16 mm² = 942.48 mm²

3. Find the Design Values:

   • For M20 concrete, ${𝑓}_{𝑐𝑘}=20\text{ MPa}$ and ${𝑓}_{𝑦}=415\text{ MPa}$ for Fe415 steel.
   • Design compressive strength of concrete ${𝑓}_{𝑐\text{ck}}=0.67\times {𝑓}_{𝑐𝑘}=0.67\times 20=13.4\text{ MPa}$
   • Design yield strength of steel ${𝑓}_{𝑦}=415\text{ MPa}$

4. Determine the Lever Arm (z):

   • Approximate lever arm $𝑧\approx 𝑑-\frac{𝑎}{2}$
   • $𝑎$ (depth of the equivalent stress block) = $\frac{0.87{𝑓}_{𝑦}{𝐴}_{𝑠𝑡}}{0.36{𝑓}_{𝑐𝑘}𝑏}$
   • Where $𝑏$ is the breadth of the beam (230 mm).
   • So, $𝑎=\frac{0.87\times 415\times 942.48}{0.36\times 20\times 230}\approx 105.62\text{ mm}$
   • Therefore, $𝑧=365-\frac{105.62}{2}=365-52.81=312.19\text{ mm}$

5. Calculate the Ultimate Moment of Resistance (M_u):

   • ${𝑀}_{𝑢}=0.87{𝑓}_{𝑦}{𝐴}_{𝑠𝑡}\times 𝑧$
   • ${𝑀}_{𝑢}=0.87\times 415\times 942.48\times 312.19\times 10^{-6}\text{ kNm}$
   • ${𝑀}_{𝑢}\approx 105.64\text{ kNm}$

So, the ultimate moment of resistance of the beam is approximately $105.64\text{ kNm}$.