RCC Numerical - Ultimate Moment of resistance of the beam

A beam having dimension 230 × 450 mm effective is reinforced with 4 bars of 16 mm diameter on tension side. Calculate the ultimate moment of resistance of the beam if M20 grade concrete and Fe415 steel is used. 

To calculate the ultimate moment of resistance of a singly reinforced rectangular beam, we can use the formula:

$$M_u = 0.87 f_y A_s d \left(1 - \frac{A_s f_y}{f_{ck} b d}\right)$$

Where:

• $M_u$ = Ultimate moment of resistance
• $f_y$ = Yield strength of steel (for Fe415, $f_y = 415 \, \text{N/mm}^2$)
• $A_s$ = Area of steel reinforcement
• $d$ = Effective depth of the beam (450 mm in this case)
• $f_{ck}$ = Characteristic compressive strength of concrete (for M20, $f_{ck} = 20 \, \text{N/mm}^2$)
• $b$ = Width of the beam (230 mm in this case)

Step 1: Calculate the area of steel reinforcement ($A_s$)

The area of steel reinforcement is calculated as:

$$A_s = \text{Number of bars} \times \frac{\pi}{4} \times \text{Diameter}^2$$

For 4 bars of 16 mm diameter:

$$A_s = 4 \times \frac{\pi}{4} \times 16^2 = 4 \times 201.06 = 804.25 \, \text{mm}^2$$

Step 2: Use the simplified formula for under-reinforced beams

If the beam is under-reinforced, we can use the approximate simplified formula for ultimate moment of resistance:

$$M_u = 0.87 f_y A_s d \times \left(1 - \frac{A_s f_y}{f_{ck} b d}\right)$$

Given values:

• $f_y = 415 \, \text{N/mm}^2$
• $A_s = 804.25 \, \text{mm}^2$
• $d = 450 \, \text{mm}$
• $f_{ck} = 20 \, \text{N/mm}^2$
• $b = 230 \, \text{mm}$

Let me calculate it for you.

The ultimate moment of resistance ($M_u$) of the beam is approximately $109.60 \, \text{kN-m}$.