---
title: "WSM moment of resistance, 250x450 beam"
subject: "Design of RCC Structure"
chapter: 2
key: "semester_7/design_of_rcc_structure:wsm-moment-of-resistance-250x450-beam"
source: https://www.bctengineer.com/q/semester_7/design_of_rcc_structure/wsm-moment-of-resistance-250x450-beam
site: BCT Engineering
---

# WSM moment of resistance, 250x450 beam

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

- Asked in: 2075 Chaitra, 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-moment-of-resistance-250x450-beam
- Whole chapter: https://www.bctengineer.com/md/semester_7/design_of_rcc_structure/chapter-2

## Question

A rectangular RC beam of overall dimensions 250 mm x 450 mm is reinforced with 4-16 mm dia. bars in tension at an effective cover of 40 mm. Calculate the moment of resistance of the beam using working stress method. Adopt M20 concrete and Fe415 grade steel.

## Short answer

Moment of resistance is 47.11 kN·m. Section is over-reinforced, concrete governs; using Mr=0.5*sigma_cbc*b*x*(d-x/3).

## Answer

**Given:** $b=250$ mm, $D=450$ mm, effective cover 40 mm so $d=410$ mm, 4-16 mm dia, $A_{st}=4\times\frac{\pi}{4}(16)^2=804.2$ mm$^2$. M20: $\sigma_{cbc}=7$, $m=13.33$; Fe415: $\sigma_{st}=230$ N/mm$^2$.

**Step 1: Constants** (M20: $\sigma_{cbc}=7$, Fe415: $\sigma_{st}=230$ N/mm$^2$, $m=13.33$)

$$
\begin{aligned}
k &= \frac{m\sigma_{cbc}}{m\sigma_{cbc}+\sigma_{st}} = \frac{13.33\times 7}{13.33\times 7 + 230} = 0.289\\
x_c &= k d = 0.289\times 410 = 118.33\ \text{mm}
\end{aligned}
$$

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

$$
\frac{b x^2}{2} = m A_{st}(d-x)\ \Rightarrow\ \frac{250 x^2}{2} = 13.33\times 804.2\,(410-x)
$$

$$
x = 149.48\ \text{mm}
$$

Since $x = 149.48 > x_c = 118.33$ mm, the section is **over-reinforced**: concrete reaches $\sigma_{cbc}=7$ N/mm$^2$ first, so concrete governs.

**Step 3: Moment of resistance**

$$
\begin{aligned}
M_r &= \frac12\sigma_{cbc}\,b\,x\left(d-\frac{x}{3}\right) = 0.5\times 7\times 250\times 149.48\times\left(410-\frac{149.48}{3}\right)\\
&= 47.11\ \text{kN·m}
\end{aligned}
$$

Stress in steel at this moment: $\sigma_{st} = \dfrac{m\,\sigma_{cbc}(d-x)}{x} = 162.63$ N/mm$^2$ $<$ 230 N/mm$^2$.

**Answer: $M_r=47.11$ kN·m.**

## Check yourself

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

1. D = 450 mm and effective cover = 40 mm. What is d?
   A. 410 mm
   B. 450 mm
   C. 400 mm
   D. 434 mm
   Answer: A. d = 450 - 40 = 410 mm. (Key point: Effective depth)

2. x = 149.48 mm exceeds x_c = 118.33 mm. Meaning:
   A. Under-reinforced
   B. Over-reinforced, concrete governs
   C. Balanced
   D. Safe at 230 N/mm2
   Answer: B. Concrete reaches 7 N/mm2 first. (Key point: Neutral axis)

3. Which formula gives M_r = 47.11 kN.m?
   A. 0.5 x 7 x b x x x (d - x/3)
   B. 230 x Ast x (d - x/3)
   C. 7 x Ast x d
   D. 0.87 fy Ast d
   Answer: A. Concrete governs the over-reinforced section. (Key point: Moment of resistance)

4. The steel stress at M_r is 162.63 N/mm2. What does this show?
   A. Steel has yielded
   B. Steel stays below permissible 230 N/mm2
   C. The beam is unsafe
   D. The modular ratio is wrong
   Answer: B. Since 162.63 < 230, concrete is the limiting material. (Key point: Steel stress)

## Related in Design of RCC Structure

- [WSM moment of resistance, 4.2 m beam](https://www.bctengineer.com/question.md?key=semester_7%2Fdesign_of_rcc_structure%3Awsm-moment-of-resistance-4-2) — Design of RCC Structure, chapter 2
- [WSM moment of resistance, 230x350 beam](https://www.bctengineer.com/question.md?key=semester_7%2Fdesign_of_rcc_structure%3Awsm-moment-of-resistance-230x350-beam) — Design of RCC Structure, chapter 2
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- [WSM moment of resistance, 250x500 beam](https://www.bctengineer.com/question.md?key=semester_7%2Fdesign_of_rcc_structure%3Awsm-moment-of-resistance-250x500-beam) — Design of RCC Structure, chapter 2
- [WSM moment of resistance, 300x550 beam](https://www.bctengineer.com/question.md?key=semester_7%2Fdesign_of_rcc_structure%3Awsm-moment-of-resistance-300x550-beam) — Design of RCC Structure, chapter 2

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