A rectangular RCC beam section having a width of 200 mm and an overall depth of 300 mm is subjected to a factored shear force of 60 kN. Determine the nominal shear stress acting in the section if the effective cover is 50 mm, the grade of concrete is M 20, and the grade of steel is Fe 415.

🔬 Understanding the Core Concepts

To solve this problem, we need to understand a few key terms from Reinforced Concrete Cement (RCC) design.

What is Nominal Shear Stress (τ_v)?

Imagine a beam trying to resist being sliced vertically by a heavy load. The internal stress that develops to resist this slicing action is the shear stress. In RCC design, we calculate an average or "nominal" shear stress across the effective area of the beam. This value is crucial for determining if the beam needs special shear reinforcement (stirrups).

Key Parameters:

• Factored Shear Force (V_u): This isn't just the shear force from the loads; it's the shear force multiplied by a factor of safety (usually 1.5 as per IS 456). The problem already gives this value as 60 kN.
• Beam Width (b): The horizontal dimension of the beam's cross-section. Here, it's 200 mm.
• Overall Depth (D) vs. Effective Depth (d): This is the most critical concept here.
  • Overall Depth (D): The total height of the beam (300 mm).
  • Effective Depth (d): The distance from the top-most compression fiber to the centroid of the main tension reinforcement (the steel bars). This is the depth that is truly "effective" in resisting bending and shear.
• Effective Cover: The distance from the center of the main reinforcement to the nearest outer surface of the beam. It's used to calculate 'd' from 'D'.

🧮 Step-by-Step Calculation

Let's break down the solution into clear, logical steps.

Step 1: List the Given Data

• Width (b) = 200 mm
• Overall Depth (D) = 300 mm
• Factored Shear Force (V_u) = 60 kN = 60 × 1000 N = 60,000 N
• Effective Cover = 50 mm

Step 2: Calculate the Effective Depth (d)

The effective depth is what we need for our calculation, not the overall depth. The formula is:

d = 300 mm - 50 mm = 250 mm

Note: This is the most common place for errors. Always ensure you are using 'd' and not 'D' in the shear stress formula.

Step 3: Calculate the Nominal Shear Stress (τ_v)

The formula for nominal shear stress, as per IS 456:2000 (Clause 40.1), is:

Now, we substitute our values into this formula:

τ_v = 60,000 N / (200 mm × 250 mm)

τ_v = 60,000 / 50,000 N/mm²

τ_v = 1.2 N/mm²

🌐 Why is this calculation important?

Calculating τ_v = 1.2 N/mm² is just the first step in shear design. An engineer would then compare this value to two other critical values:

• Design Shear Strength of Concrete (τ_c): This is the amount of shear stress the concrete can safely resist on its own. It depends on the grade of concrete (M20) and the percentage of tension steel (which isn't given, but would be known in a full design problem). If τ_v > τ_c, shear reinforcement is mandatory.
• Maximum Shear Stress (τ_c,max): This is the absolute speed limit for shear stress in a beam, regardless of reinforcement. For M20 concrete, τ_c,max is 2.8 N/mm². Since our calculated τ_v (1.2 N/mm²) is less than 2.8 N/mm², the beam section is safe from diagonal compression failure and can be redesigned if needed.

💡 Study Tips & Common Mistakes

• D vs. d: Always double-check if you are using the effective depth (d) for shear calculations. This is the number one mistake students make.
• Unit Consistency: Ensure your force is in Newtons (N) and dimensions are in millimeters (mm) to get the stress in N/mm² (which is the same as a MegaPascal, MPa).
• Visualize It: Draw a quick sketch of the beam's cross-section. Mark the overall depth, the steel bars, and the effective depth. This helps solidify the concept.
• Context is Key: Remember that the M20 and Fe 415 information, while not used in *this specific calculation*, is critical for the subsequent steps of a full shear design (τ_c and τ_c,max).