---
title: "WSM moment of resistance, 250x500 beam"
subject: "Design of RCC Structure"
chapter: 2
key: "semester_7/design_of_rcc_structure:wsm-moment-of-resistance-250x500-beam"
source: https://www.bctengineer.com/q/semester_7/design_of_rcc_structure/wsm-moment-of-resistance-250x500-beam
site: BCT Engineering
---

# WSM moment of resistance, 250x500 beam

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

- Asked in: 2073 Shrawan, 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/wsm-moment-of-resistance-250x500-beam
- Whole chapter: https://www.bctengineer.com/md/semester_7/design_of_rcc_structure/chapter-2

## Question

Find the moment of resistance of a RCC beam 250 mm wide and 500 mm effective depth if it is reinforced with 3-16 mm dia bars. The permissible stresses for concrete and steel are given as 7 MPa and 230 MPa. The value of modular ratio is taken as 13.33.

## Short answer

Moment of resistance is 59.08 kN·m. Over-reinforced, concrete governs.

## Answer

**Given:** $b=250$ mm, $d=500$ mm, 3-16 mm dia, $A_{st}=3\times\frac{\pi}{4}(16)^2=603.2$ mm$^2$, $\sigma_{cbc}=7$, $\sigma_{st}=230$ N/mm$^2$, $m=13.33$.

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

$$
\begin{aligned}
k &= \frac{m\sigma_{cbc}}{m\sigma_{cbc}+\sigma_{st}} = \frac{13.33\times 7}{13.33\times 7 + 230} = 0.289\\
x_c &= k d = 0.289\times 500 = 144.30\ \text{mm}
\end{aligned}
$$

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

$$
\frac{b x^2}{2} = m A_{st}(d-x)\ \Rightarrow\ \frac{250 x^2}{2} = 13.33\times 603.2\,(500-x)
$$

$$
x = 150.04\ \text{mm}
$$

Since $x = 150.04 > x_c = 144.30$ mm, the section is **over-reinforced**: concrete reaches $\sigma_{cbc}=7$ N/mm$^2$ first, so concrete governs.

**Step 3: Moment of resistance**

$$
\begin{aligned}
M_r &= \frac12\sigma_{cbc}\,b\,x\left(d-\frac{x}{3}\right) = 0.5\times 7\times 250\times 150.04\times\left(500-\frac{150.04}{3}\right)\\
&= 59.08\ \text{kN·m}
\end{aligned}
$$

Stress in steel at this moment: $\sigma_{st} = \dfrac{m\,\sigma_{cbc}(d-x)}{x} = 217.65$ N/mm$^2$ $<$ 230 N/mm$^2$.

**Answer: $M_r=59.08$ kN·m.**

## Check yourself

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

1. Which values are used for M_r here?
   A. d = 500, m = 13.33, sigma_cbc = 7, sigma_st = 230
   B. d = 550, m = 18.67
   C. d = 500, sigma_st = 140
   D. d = 450, m = 10
   Answer: A. These are stated in the question. (Key point: Given data)

2. x = 150.04 mm vs x_c = 144.30 mm. The section is:
   A. Under-reinforced
   B. Over-reinforced, slightly
   C. Balanced exactly
   D. Doubly reinforced
   Answer: B. x > x_c, concrete governs. (Key point: Classification)

3. Which formula and value are correct?
   A. 0.5 sigma_cbc b x (d - x/3) = 59.08 kN.m
   B. sigma_st Ast (d-x/3) = 59.08 kN.m
   C. 0.5 sigma_cbc b x d = 59.08
   D. Q b d^2 = 50
   Answer: A. Concrete-governed moment of resistance is 59.08 kN.m. (Key point: Result)

## Related in Design of RCC Structure

- [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, 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
- [Bent-up bars, 600 mm effective depth](https://www.bctengineer.com/question.md?key=semester_7%2Fdesign_of_rcc_structure%3Abent-up-bars-600-mm-effective) — Design of RCC Structure, chapter 5, asked 2×
- [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
- [Footing 350x350, 8-16 mm bars, reversible moment](https://www.bctengineer.com/question.md?key=semester_7%2Fdesign_of_rcc_structure%3Afooting-350x350-8-16-mm-bars) — Design of RCC Structure, chapter 11
- [Bent-up bars shear, 550 mm effective depth](https://www.bctengineer.com/question.md?key=semester_7%2Fdesign_of_rcc_structure%3Abent-up-bars-shear-550-mm) — Design of RCC Structure, chapter 5, asked 3×

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