Understanding the behavior of a singly reinforced section at the limiting state of collapse is crucial in structural design. According to IS 456:2000, the stress distribution in the compression zone of concrete at the ultimate limit state is represented by a specific stress block diagram. This stress block consists of two distinct parts: a parabolic portion and a rectangular portion.

Concrete Stress Block Diagram (IS 456)

The Indian Standard Code of Practice for Plain and Reinforced Concrete (IS 456:2000) specifies the stress-strain relationship for concrete in compression at the ultimate limit state. Key aspects of this relationship are:

• The maximum compressive strain in concrete at the extreme fiber in bending is taken as \(0.0035\).
• The stress-strain curve for concrete is considered parabolic up to a strain of \(0.002\).
• Beyond a strain of \(0.002\) and up to the ultimate strain of \(0.0035\), the stress is assumed to be constant at a value of \(0.446 f_{ck}\) (where \(f_{ck}\) is the characteristic compressive strength of concrete).

This idealized stress-strain curve translates into a specific stress block diagram for the analysis of the compressive force in concrete. Let \(x_u\) be the depth of the neutral axis from the extreme compression fiber.

Depths of Parabolic and Rectangular Stress Block Portions

To determine the individual depths of the parabolic portion and the rectangular portion within the stress block, we analyze the strain distribution across the concrete section. The strain distribution is assumed to be linear from the extreme compression fiber to the neutral axis.

Let:

• \(\epsilon_{max} = 0.0035\) be the maximum strain at the extreme compression fiber.
• \(\epsilon_{transition} = 0.002\) be the strain at which the parabolic part transitions to the rectangular part in the stress-strain curve.
• \(x_u\) be the depth of the neutral axis from the extreme compression fiber.

Consider a linear strain distribution diagram. Using principles of similar triangles, we can find the depth from the extreme compression fiber where the strain is \(0.002\). Let this depth be \(d_R\).

From the strain diagram, the strain at any depth \(y\) from the extreme compression fiber is given by:

$$\epsilon_y = \frac{\epsilon_{max}}{x_u} (x_u - y)$$

We are interested in the depth \(d_R\) where \(\epsilon_y = 0.002\):

$$0.002 = \frac{0.0035}{x_u} (x_u - d_R)$$

Rearranging the terms to solve for \(d_R\):

$$\frac{0.002}{0.0035} = \frac{x_u - d_R}{x_u}$$

$$\frac{4}{7} = 1 - \frac{d_R}{x_u}$$

$$\frac{d_R}{x_u} = 1 - \frac{4}{7}$$

$$\frac{d_R}{x_u} = \frac{3}{7}$$

Therefore, the depth of the portion where the strain varies from \(0.0035\) to \(0.002\) is \(d_R = \frac{3}{7} x_u\). This is the depth of the rectangular portion of the stress block, as the stress is considered constant over this region.

The remaining depth up to the neutral axis will correspond to the parabolic portion of the stress block. Let this depth be \(d_P\).

$$d_P = x_u - d_R$$

$$d_P = x_u - \frac{3}{7} x_u$$

$$d_P = \frac{4}{7} x_u$$

So, the depth of the parabolic portion of the stress block is \(d_P = \frac{4}{7} x_u\).

Ratio of Depths Calculation

The question asks for the ratio of the depth of the parabolic portion to the depth of the rectangular portion of the stress block.

$$\text{Ratio} = \frac{\text{Depth of Parabolic Portion}}{\text{Depth of Rectangular Portion}}$$

$$\text{Ratio} = \frac{d_P}{d_R}$$

$$\text{Ratio} = \frac{\frac{4}{7} x_u}{\frac{3}{7} x_u}$$

$$\text{Ratio} = \frac{4}{3}$$

Thus, the ratio of the depth of the parabolic portion block to the rectangular portion block at the limiting state of collapse of a singly reinforced section is \(4:3\).