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 mm−cover−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 𝐴𝑏𝑎𝑟 = 𝜋𝑑24=𝜋×(20)24=314.16 mm2
   • Total Area of Steel 𝐴𝑠𝑡 = 3 × 314.16 mm² = 942.48 mm²

3. Find the Design Values:

   • For M20 concrete, 𝑓𝑐𝑘=20 MPa and 𝑓𝑦=415 MPa for Fe415 steel.
   • Design compressive strength of concrete 𝑓𝑐ck=0.67×𝑓𝑐𝑘=0.67×20=13.4 MPa
   • Design yield strength of steel 𝑓𝑦=415 MPa

4. Determine the Lever Arm (z):

   • Approximate lever arm 𝑧≈𝑑−𝑎2
   • 𝑎 (depth of the equivalent stress block) = 0.87𝑓𝑦𝐴𝑠𝑡0.36𝑓𝑐𝑘𝑏
   • Where 𝑏 is the breadth of the beam (230 mm).
   • So, 𝑎=0.87×415×942.480.36×20×230≈105.62 mm
   • Therefore, 𝑧=365−105.622=365−52.81=312.19 mm

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

   • 𝑀𝑢=0.87𝑓𝑦𝐴𝑠𝑡×𝑧
   • 𝑀𝑢=0.87×415×942.48×312.19×10−6 kNm
   • 𝑀𝑢≈105.64 kNm

So, the ultimate moment of resistance of the beam is approximately 105.64 kNm.