Determine the equivalent nominal shear stress in a rectangular RCC beam of 200 mm width and 400 mm effective depth if the shear force is 8 kN and torsional moment is 2 kN-m at factored loads.
Equivalent Nominal Shear Stress in RCC Beams
When a rectangular Reinforced Concrete (RCC) beam is subjected to both shear force and torsional moment, it is essential to determine the equivalent nominal shear stress. This stress helps in understanding the combined effect of these forces on the beam and is crucial for proper reinforcement design. The calculation involves first finding an equivalent shear force, which accounts for the torsional moment, and then using this equivalent shear force to compute the nominal shear stress.
Beam Parameters and Given Data
To calculate the equivalent nominal shear stress, we first need to identify the given parameters of the rectangular RCC beam:
- Width of the beam (\(b\)): 200 mm
- Effective depth of the beam (\(d\)): 400 mm
- Factored shear force (\(V_u\)): 8 kN = \(8 \times 10^3\) N
- Factored torsional moment (\(T_u\)): 2 kN-m = \(2 \times 10^6\) N-mm
Equivalent Shear Force Calculation
According to IS 456:2000, Clause 41.3, the equivalent shear force (\(V_e\)) due to the combined effect of factored shear force (\(V_u\)) and factored torsional moment (\(T_u\)) is given by the formula:
\[ V_e = V_u + 1.6 \frac{T_u}{b} \]
Let's substitute the given values into this formula to find the equivalent shear force:
\[ V_e = 8000 \, \text{N} + 1.6 \times \frac{2 \times 10^6 \, \text{N-mm}}{200 \, \text{mm}} \]
First, calculate the term related to the torsional moment:
\[ 1.6 \times \frac{2 \times 10^6}{200} = 1.6 \times 10000 = 16000 \, \text{N} \]
Now, add this to the factored shear force:
\[ V_e = 8000 \, \text{N} + 16000 \, \text{N} \]
Therefore, the equivalent shear force is:
\[ V_e = 24000 \, \text{N} \]
Nominal Shear Stress Determination
Once the equivalent shear force (\(V_e\)) is calculated, the equivalent nominal shear stress (\(\tau_v\)) can be determined. The nominal shear stress is calculated by dividing the equivalent shear force by the cross-sectional area of the beam, which is the product of its width and effective depth. The formula for nominal shear stress is:
\[ \tau_v = \frac{V_e}{b \times d} \]
Now, substitute the calculated equivalent shear force and the beam dimensions into the formula:
\[ \tau_v = \frac{24000 \, \text{N}}{200 \, \text{mm} \times 400 \, \text{mm}} \]
Calculate the cross-sectional area:
\[ 200 \, \text{mm} \times 400 \, \text{mm} = 80000 \, \text{mm}^2 \]
Finally, calculate the equivalent nominal shear stress:
\[ \tau_v = \frac{24000 \, \text{N}}{80000 \, \text{mm}^2} \]
\[ \tau_v = 0.3 \, \text{N/mm}^2 \]
Summary of Calculation
The steps involved in determining the equivalent nominal shear stress are summarized below:
| Step | Description | Formula/Calculation | Result |
|---|---|---|---|
| 1 | Convert given units to consistent units (N and mm) | \(V_u = 8 \text{ kN} = 8000 \text{ N}\) \(T_u = 2 \text{ kN-m} = 2 \times 10^6 \text{ N-mm}\) |
8000 N \(2 \times 10^6\) N-mm |
| 2 | Calculate Equivalent Shear Force (\(V_e\)) | \(V_e = V_u + 1.6 \frac{T_u}{b}\) \(V_e = 8000 + 1.6 \times \frac{2 \times 10^6}{200}\) \(V_e = 8000 + 16000\) |
24000 N |
| 3 | Calculate Equivalent Nominal Shear Stress (\(\tau_v\)) | \(\tau_v = \frac{V_e}{b \times d}\) \(\tau_v = \frac{24000}{200 \times 400}\) \(\tau_v = \frac{24000}{80000}\) |
0.3 N/mm2 |
Based on the calculations, the equivalent nominal shear stress in the rectangular RCC beam is \(0.3 \, \text{N/mm}^2\).
Important Questions from Torsion
Additional longitudinal reinforcement is provided at faces if depth of member subjected to torsion exceeds _______ mm.
As per IS:456-2000, the side face reinforcement should be provided along the two faces of a beam, when the depth of the web in a beam exceeds
In an RCC beam, side face reinforcement is provided if its depth exceeds:
