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.

   $$𝑑=\text{Total Thickness}-\text{Cover}-\frac{\text{Diameter of Reinforcement}}{2}$$$$𝑑=130\text{ mm}-20\text{ mm}-\frac{10\text{ mm}}{2}=130\text{ mm}-20\text{ mm}-5\text{ mm}=105\text{ mm}$$

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

   $${𝐴}_{𝑏}=\frac{𝜋\cdot (\text{Diameter}{)}^2}{4}=\frac{𝜋\cdot (10\text{ mm}{)}^2}{4}=78.54{\text{ mm}}^2$$

   The number of bars per meter (n):

   $$\text{Number of Bars per Meter}=\frac{1000\text{ mm}}{100\text{ mm}}=10$$

   Total area of reinforcement (A_s):

   $${𝐴}_{𝑠}=\text{Number of Bars}\times {𝐴}_{𝑏}=10\times 78.54{\text{ mm}}^2=785.4{\text{ mm}}^2$$

3. Moment of Resistance (Mu):

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

   $${𝑀}_{𝑢}=0.87\cdot {𝑓}_{𝑦}\cdot {𝐴}_{𝑠}\cdot (𝑑-\frac{𝑎}{2})$$

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

   $$𝑎=\frac{{𝐴}_{𝑠}\cdot {𝑓}_{𝑦}}{0.36\cdot {𝑓}_{𝑐}𝑘\cdot 𝑏}$$

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

   $$𝑎=\frac{785.4{\text{ mm}}^2\cdot 415\text{ MPa}}{0.36\cdot 20\text{ MPa}\cdot 1000\text{ mm}}=\frac{325,581}{7,200}\approx 45.4\text{ mm}$$

   Now, calculate $𝑑-\frac{𝑎}{2}$:

   $$𝑑-\frac{𝑎}{2}=105\text{ mm}-\frac{45.4\text{ mm}}{2}\approx 105\text{ mm}-22.7\text{ mm}\approx 82.3\text{ mm}$$

   Finally, the ultimate moment of resistance (Mu):

   $${𝑀}_{𝑢}=0.87\cdot 415\text{ MPa}\cdot 785.4{\text{ mm}}^2\cdot 82.3\text{ mm}$$$${𝑀}_{𝑢}\approx 0.87\cdot 415\cdot 785.4\cdot 82.3=23,541,222\text{ Nmm}$$

   Converting to kNm:

   $${𝑀}_{𝑢}\approx 23,541.2\text{ Nm}=23.54\text{ kNm}$$

Ultimate Moment of Resistance ${𝑀}_{𝑢}$ is approximately 23.54 kNm.