---
title: "WSM resistance and stresses, 230x400"
subject: "Design of RCC Structure"
chapter: 2
key: "semester_7/design_of_rcc_structure:wsm-resistance-and-stresses-230x400"
source: https://www.bctengineer.com/q/semester_7/design_of_rcc_structure/wsm-resistance-and-stresses-230x400
site: BCT Engineering
---

# WSM resistance and stresses, 230x400

Design of RCC Structure · Chapter 2: Working Stress Method of Design

- Asked in: 2080 Baisakh, 8 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/design_of_rcc_structure/wsm-resistance-and-stresses-230x400
- Whole chapter: https://www.bctengineer.com/md/semester_7/design_of_rcc_structure/chapter-2

## Question

An RC beam 230 mm wide and 400 mm deep (effective depth of 380 mm with area of steel 580 mm$^2$) has permissible stress in concrete and steel 5 N/mm$^2$ and 140 N/mm$^2$ respectively. Find the moment of resistance of the section and actual stresses in concrete and steel.

## Short answer

Actual neutral axis is 147.85 mm, below balanced depth 152.02 mm, so the section is under-reinforced. Moment of resistance is 26.85 kN·m. Actual stresses are 4.78 N/mm2 in concrete and 140.00 N/mm2 in steel.

## Answer

**Given:** $b=230$ mm, $d=380$ mm, $A_{st}=580$ mm$^2$, $\sigma_{cbc}=5$, $\sigma_{st}=140$ N/mm$^2$ (M15 concrete, mild steel), $m=18.67$.

**Step 1: Constants** ($\sigma_{cbc}=5$, $\sigma_{st}=140$ N/mm$^2$, $m=\dfrac{280}{3\times5}=18.67$)

$$
\begin{aligned}
k &= \frac{m\sigma_{cbc}}{m\sigma_{cbc}+\sigma_{st}} = \frac{18.67\times 5}{18.67\times 5 + 140} = 0.400\\
x_c &= k d = 0.400\times 380 = 152.02\ \text{mm}
\end{aligned}
$$

**Step 2: Actual neutral axis** (taking moments of areas about the neutral axis, transformed section)

$$
\frac{b x^2}{2} = m A_{st}(d-x)\ \Rightarrow\ \frac{230 x^2}{2} = 18.67\times 580.0\,(380-x)
$$

$$
x = 147.85\ \text{mm}
$$

Since $x = 147.85 < x_c = 152.02$ mm, the section is **under-reinforced**: steel reaches $\sigma_{st}=140$ N/mm$^2$ first, so steel governs.

**Step 3: Moment of resistance**

$$
\begin{aligned}
M_r &= \sigma_{st}A_{st}\left(d-\frac{x}{3}\right) = 140\times 580.0\times\left(380-\frac{147.85}{3}\right)\\
&= 26.85\ \text{kN·m}
\end{aligned}
$$

Stress in concrete at this moment: $\sigma_{c} = \dfrac{\sigma_{st}\,x}{m(d-x)} = 4.78$ N/mm$^2$ $<$ 5 N/mm$^2$ (safe).

**Actual stresses at $M_r$:** concrete $\sigma_c=4.78$ N/mm$^2$, steel $\sigma_s=140.00$ N/mm$^2$.

**Answer:** $M_r=26.85$ kN·m; actual stresses are $\sigma_c=4.78$ N/mm$^2$ and $\sigma_s=140.00$ N/mm$^2$.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. x = 147.85 mm vs x_c = 152.02 mm. Which is true?
   A. Over-reinforced, concrete governs
   B. Under-reinforced, steel reaches 140 N/mm2 first
   C. Balanced exactly
   D. Not safe
   Answer: B. x < x_c means steel governs. (Key point: Classification by comparison)

2. Which gives the moment of resistance here?
   A. sigma_st Ast (d - x/3)
   B. 0.5 sigma_cbc b x (d - x/3)
   C. sigma_cbc Ast d
   D. Q b d^2 only
   Answer: A. 140 x 580 x (380 - 147.85/3) = 26.85 kN.m. (Key point: Moment formula)

3. How is the actual concrete stress found at M_r?
   A. sigma_c = sigma_st x / (m (d - x))
   B. sigma_c = m sigma_st
   C. sigma_c = sigma_st (d - x)/ x
   D. sigma_c = sigma_st / m
   Answer: A. From strain compatibility, giving 4.78 N/mm2, below 5. (Key point: Concrete stress)

4. What are the actual stresses at M_r?
   A. Concrete 5.00, steel 125
   B. Concrete 4.78, steel 140
   C. Concrete 4.78, steel 230
   D. Concrete 7, steel 140
   Answer: B. Steel is at permissible 140 and concrete at 4.78 N/mm2. (Key point: Actual stresses)

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