---
title: "MoR of beam 250x475 by WSM"
subject: "Design of RCC Structure"
chapter: 2
key: "semester_7/design_of_rcc_structure:mor-of-beam-250x475-by-wsm"
source: https://www.bctengineer.com/q/semester_7/design_of_rcc_structure/mor-of-beam-250x475-by-wsm
site: BCT Engineering
---

# MoR of beam 250x475 by WSM

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

- Asked in: 2079 Baisakh, 6 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/mor-of-beam-250x475-by-wsm
- Whole chapter: https://www.bctengineer.com/md/semester_7/design_of_rcc_structure/chapter-2

## Question

Determine the moment of resistance of a RC rectangular beam of overall dimension 250 mm x 475 mm reinforced with 3-16 mm dia. bars in tension side. Use M20 concrete and Fe415 steel in working stress method.

## Short answer

With d=442 mm and Ast=603.2 mm2, actual neutral axis is 139.49 mm, greater than balanced 127.56 mm, so over-reinforced. Concrete governs. Moment of resistance is 0.5×7×250×139.49×(442−139.49/3)=48.27 kN·m.

## Answer

**Assumption:** clear cover 25 mm, so $d=475-25-\dfrac{16}{2}=442$ mm. $b=250$ mm, $A_{st}=3\times\frac{\pi}{4}(16)^2=603.2$ mm$^2$.

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

$$
\begin{aligned}
k &= \frac{m\sigma_{cbc}}{m\sigma_{cbc}+\sigma_{st}} = \frac{13.33\times 7}{13.33\times 7 + 230} = 0.289\\
x_c &= k d = 0.289\times 442 = 127.56\ \text{mm}
\end{aligned}
$$

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

$$
\frac{b x^2}{2} = m A_{st}(d-x)\ \Rightarrow\ \frac{250 x^2}{2} = 13.33\times 603.2\,(442-x)
$$

$$
x = 139.49\ \text{mm}
$$

Since $x = 139.49 > x_c = 127.56$ mm, the section is **over-reinforced**: concrete reaches $\sigma_{cbc}=7$ N/mm$^2$ first, so concrete governs.

**Step 3: Moment of resistance**

$$
\begin{aligned}
M_r &= \frac12\sigma_{cbc}\,b\,x\left(d-\frac{x}{3}\right) = 0.5\times 7\times 250\times 139.49\times\left(442-\frac{139.49}{3}\right)\\
&= 48.27\ \text{kN·m}
\end{aligned}
$$

Stress in steel at this moment: $\sigma_{st} = \dfrac{m\,\sigma_{cbc}(d-x)}{x} = 202.35$ N/mm$^2$ $<$ 230 N/mm$^2$.

**Answer: $M_r=48.27$ kN·m** (over-reinforced, concrete governs).

## Check yourself

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

1. With 25 mm clear cover and 16 mm bars, what is d for D = 475 mm?
   A. 450 mm
   B. 442 mm
   C. 475 mm
   D. 434 mm
   Answer: B. d = 475 - 25 - 16/2 = 442 mm. (Key point: Effective depth)

2. x = 139.49 mm and x_c = 127.56 mm. The section is:
   A. Under-reinforced
   B. Balanced
   C. Over-reinforced, so concrete governs
   D. Doubly reinforced
   Answer: C. x > x_c, concrete reaches 7 N/mm2 first. (Key point: Classification)

3. Which moment formula applies?
   A. sigma_st Ast (d - x/3)
   B. 0.5 sigma_cbc b x (d - x/3)
   C. 0.87 fy Ast d
   D. Q b d^2 with steel stress
   Answer: B. Concrete governs: M_r = 48.27 kN.m. (Key point: Formula)

4. Why is the steel stress computed as 202.35 N/mm2?
   A. To verify it is below 230 N/mm2 permissible
   B. To find the modular ratio
   C. To replace M_r
   D. To find the cover
   Answer: A. If steel were above permissible the section would be invalid; 202.35 < 230. (Key point: Steel stress check)

## Related in Design of RCC Structure

- [WSM moment of resistance, 250x450 beam](https://www.bctengineer.com/question.md?key=semester_7%2Fdesign_of_rcc_structure%3Awsm-moment-of-resistance-250x450-beam) — Design of RCC Structure, chapter 2
- [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
- [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
- [Ultimate moment of L-beam, doubly reinforced](https://www.bctengineer.com/question.md?key=semester_7%2Fdesign_of_rcc_structure%3Aultimate-moment-of-l-beam-doubly) — Design of RCC Structure, chapter 4
- [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

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