RCC Numerical – Find the ultimate moment of resistance the beam

A singly reinforced beam 230 mm × 480 mm, effective to the centre of 4-20 mm diameter Fe415 bars is cast using M20 concrete. Find the ultimate moment of resistance the beam can offer.

To find the ultimate moment of resistance of the singly reinforced beam, we need to perform the following steps:

Given Data:

• Beam dimensions: Width 𝑏=230 mm, Depth 𝐷=480 mm
• Effective depth 𝑑=effective depth to the center of reinforcement
• Number and diameter of reinforcement bars: 4 bars of 20 mm diameter
• Concrete grade: M20
• Steel grade: Fe415

Steps to Calculate the Ultimate Moment of Resistance:

1. Calculate the Effective Depth (d):

   • Assume the cover is 25 mm.
   • Effective depth 𝑑=𝐷−cover−diameter of bars2
   • 𝑑=480 mm−25 mm−20 mm2
   • 𝑑=480 mm−25 mm−10 mm
   • 𝑑=445 mm

2. Calculate the Area of Steel Reinforcement (Ast):

   • Area of one bar 𝐴𝑠𝑡=𝜋𝑑24
   • For 20 mm diameter bars: 𝐴𝑠𝑡=𝜋×(20)24
   • 𝐴𝑠𝑡=3.14×4004
   • 𝐴𝑠𝑡=314 mm2
   • Total area for 4 bars: 𝐴𝑠𝑡(𝑡𝑜𝑡𝑎𝑙)=4×314
   • 𝐴𝑠𝑡(𝑡𝑜𝑡𝑎𝑙)=1256 mm2

3. Determine the Design Strengths:

   • For M20 concrete, 𝑓𝑐𝑘=20 MPa and 𝑓𝑦=415 MPa for Fe415 steel.
   • Design strength of concrete 𝑓𝑐𝑘,𝑑𝑒𝑠𝑖𝑔𝑛=𝑓𝑐𝑘1.5
   • 𝑓𝑐𝑘,𝑑𝑒𝑠𝑖𝑔𝑛=201.5≈13.33 MPa
   • Design strength of steel 𝑓𝑦,𝑑𝑒𝑠𝑖𝑔𝑛=𝑓𝑦1.15
   • 𝑓𝑦,𝑑𝑒𝑠𝑖𝑔𝑛=4151.15≈361 MPa

4. Calculate the Moment of Resistance:

   • For singly reinforced sections, the ultimate moment of resistance 𝑀𝑢 is given by:

     𝑀𝑢=0.87⋅𝑓𝑦⋅𝐴𝑠𝑡⋅(𝑑−𝑎2)

     where 𝑎=𝐴𝑠𝑡⋅𝑓𝑦0.36⋅𝑓𝑐𝑘⋅𝑏

   • First, calculate 𝑎:

     𝑎=1256⋅3610.36⋅13.33⋅230𝑎=45381611498.8≈39.5 mm

   • Then, calculate 𝑀𝑢:

     𝑀𝑢=0.87⋅361⋅1256⋅(445−39.52)𝑀𝑢=0.87⋅361⋅1256⋅(445−19.75)𝑀𝑢=0.87⋅361⋅1256⋅425.25𝑀𝑢≈0.87⋅361⋅1256⋅425.25𝑀𝑢≈62.26×425.25≈26,536 kN-mm𝑀𝑢≈26.54 kNm

Ultimate Moment of Resistance

The ultimate moment of resistance of the beam is approximately 26.54 kNm.