Question

In limit state method of design, if $x_u$ is the depth of the neutral axis, then the distance of centroid of the compressive force from the extreme compression fibre is

• A. $0.57 x_u$
• B. $0.45 x_u$
• C. $0.42 x_u$
• D. $0.36 x_u$

Hint:

For Limit State Design as per IS 456, memorize these two key values for the concrete stress block:
- Total Compressive Force, $C = \textbf{0.36} f_{ck} b x_u$.
- Depth of Compressive Force, $\bar{y} = \textbf{0.42} x_u$.
These are fundamental to all flexural calculations.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The question asks for the location of the resultant compressive force in the concrete stress block, as defined by the Limit State Method in IS 456:2000.

Step 2: Key Formula or Approach:
In the Limit State Method, the stress distribution in the compression zone of a concrete beam at failure is assumed to be non-linear. The code (IS 456:2000) specifies an idealized rectangular-parabolic stress block.
- For a depth from the neutral axis of 0 to $4/7 x_u$, the stress is parabolic.
- For a depth from $4/7 x_u$ to $x_u$, the stress is constant at $0.446 f_{ck}$.
The total compressive force ($C$) is the area of this stress block. The location of this force is at the centroid of the stress block.
For this specific rectangular-parabolic shape, the code specifies that the centroid is located at a distance of

$0.42 x_u$ from the extreme compression fiber (the top surface of the beam).

Step 3: Detailed Explanation:
This is a standard value from the Indian concrete design code (IS 456:2000) that needs to be memorized.
- The total compressive force is given by $C = 0.36 f_{ck} b x_u$.
- The lever arm ($z$) between the tensile force in the steel and the compressive force in the concrete is given by $z = d - 0.42 x_u$, where $d$ is the effective depth of the beam.
This confirms that the centroid of the compressive force is located at a distance of $0.42 x_u$ from the top.

Step 4: Final Answer:
The distance of the centroid of the compressive force from the extreme compression fibre is $0.42 x_u$.