RCC Numerical – Calculate ultimate moment of resistance of R.C. Slab

Calculate ultimate moment of resistance of R.C. Slab 130 mm thick, reinforced with 10 mm φ bars @ 100 mm c/c. Cover to center of bars is 20 mm. M20 concrete and Fe415 steel.

To calculate the ultimate moment of resistance (Mu) of a reinforced concrete slab, follow these steps:

Given Data:

• Slab Thickness (d): 130 mm
• Reinforcement Bars: 10 mm diameter @ 100 mm c/c
• Cover to Center of Bars: 20 mm
• Concrete Grade: M20 (f_ck = 20 MPa)
• Steel Grade: Fe415 (f_y = 415 MPa)

Step-by-Step Calculation:

1. Effective Depth (d): Effective depth (d) is the distance from the top fiber of the slab to the centroid of the tensile reinforcement.

   𝑑=Total Thickness−Cover−Diameter of Reinforcement2𝑑=130 mm−20 mm−10 mm2=130 mm−20 mm−5 mm=105 mm

2. Area of Reinforcement (A_s): The area of one bar (A_b) is:

   𝐴𝑏=𝜋⋅(Diameter)24=𝜋⋅(10 mm)24=78.54 mm2

   The number of bars per meter (n):

   Number of Bars per Meter=1000 mm100 mm=10

   Total area of reinforcement (A_s):

   𝐴𝑠=Number of Bars×𝐴𝑏=10×78.54 mm2=785.4 mm2

3. Moment of Resistance (Mu):

   For a singly reinforced rectangular section, the formula for the ultimate moment of resistance is:

   𝑀𝑢=0.87⋅𝑓𝑦⋅𝐴𝑠⋅(𝑑−𝑎2)

   Where 𝑎 is the depth of the neutral axis, which can be approximated as:

   𝑎=𝐴𝑠⋅𝑓𝑦0.36⋅𝑓𝑐𝑘⋅𝑏

   Assuming the width 𝑏 is 1000 mm (for a unit width slab), compute 𝑎:

   𝑎=785.4 mm2⋅415 MPa0.36⋅20 MPa⋅1000 mm=325,5817,200≈45.4 mm

   Now, calculate 𝑑−𝑎2:

   𝑑−𝑎2=105 mm−45.4 mm2≈105 mm−22.7 mm≈82.3 mm

   Finally, the ultimate moment of resistance (Mu):

   𝑀𝑢=0.87⋅415 MPa⋅785.4 mm2⋅82.3 mm𝑀𝑢≈0.87⋅415⋅785.4⋅82.3=23,541,222 Nmm

   Converting to kNm:

   𝑀𝑢≈23,541.2 Nm=23.54 kNm

Ultimate Moment of Resistance 𝑀𝑢 is approximately 23.54 kNm.