---
title: "WSM total udl on 300x500 beam"
subject: "Design of RCC Structure"
chapter: 2
key: "semester_7/design_of_rcc_structure:wsm-total-udl-on-300x500-beam"
source: https://www.bctengineer.com/q/semester_7/design_of_rcc_structure/wsm-total-udl-on-300x500-beam
site: BCT Engineering
---

# WSM total udl on 300x500 beam

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

- Asked in: 2076 Asoj, 8 marks
- Video: [Design of Singly Reinforced Beam | Working Stress Method | #WSM | Moment of Resistance | Rcc](https://www.youtube.com/watch?v=DFwdx2_tC8E) — GATE HUB
- 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-total-udl-on-300x500-beam
- Whole chapter: https://www.bctengineer.com/md/semester_7/design_of_rcc_structure/chapter-2

## Question

A rectangular RC beam of overall dimensions 300 mm x 500 mm is reinforced with 4-25 mm dia. bars in tension at an effective cover of 50 mm. The beam is simply supported over an effective span of 5 m. Calculate the total uniformly distributed load the beam can carry inclusive of its self weight. Use working stress method of design. Adopt M20 concrete and Fe415 grade (TOR) steel.

## Short answer

Total UDL including self weight is 26.42 kN/m. The section is over-reinforced; moment of resistance from concrete is 82.56 kN·m, giving w=8M/L^2.

## Answer

**Given:** $b=300$ mm, $D=500$ mm, effective cover 50 mm so $d=450$ mm, 4-25 mm dia, $A_{st}=4\times\frac{\pi}{4}(25)^2=1963.5$ mm$^2$, $L=5$ m. M20: $\sigma_{cbc}=7$; Fe415: $\sigma_{st}=230$ N/mm$^2$; $m=13.33$.

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

$$
\begin{aligned}
k &= \frac{m\sigma_{cbc}}{m\sigma_{cbc}+\sigma_{st}} = \frac{13.33\times 7}{13.33\times 7 + 230} = 0.289\\
x_c &= k d = 0.289\times 450 = 129.87\ \text{mm}
\end{aligned}
$$

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

$$
\frac{b x^2}{2} = m A_{st}(d-x)\ \Rightarrow\ \frac{300 x^2}{2} = 13.33\times 1963.5\,(450-x)
$$

$$
x = 206.24\ \text{mm}
$$

Since $x = 206.24 > x_c = 129.87$ mm, the section is **over-reinforced**: concrete reaches $\sigma_{cbc}=7$ N/mm$^2$ first, so concrete governs.

**Step 3: Moment of resistance**

$$
\begin{aligned}
M_r &= \frac12\sigma_{cbc}\,b\,x\left(d-\frac{x}{3}\right) = 0.5\times 7\times 300\times 206.24\times\left(450-\frac{206.24}{3}\right)\\
&= 82.56\ \text{kN·m}
\end{aligned}
$$

Stress in steel at this moment: $\sigma_{st} = \dfrac{m\,\sigma_{cbc}(d-x)}{x} = 110.29$ N/mm$^2$ $<$ 230 N/mm$^2$.

**Step 4: Total UDL**

$$
w=\frac{8M_r}{L^2}=\frac{8\times82.56}{5^2}=26.42\ \text{kN/m}
$$

Self weight $=0.30\times0.50\times25=3.75$ kN/m, so the external (superimposed) load is $22.67$ kN/m.

**Answer:** total UDL including self weight $=26.42$ kN/m.

## Check yourself

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

1. Overall depth 500 mm and effective cover 50 mm give d equal to:
   A. 500 mm
   B. 475 mm
   C. 450 mm
   D. 425 mm
   Answer: C. d = 500 - 50 = 450 mm. (Key point: Effective depth)

2. x = 206.24 mm vs x_c = 129.87 mm. What governs?
   A. Steel yielding at 230 N/mm2
   B. Concrete reaching 7 N/mm2 (over-reinforced)
   C. Neither
   D. Shear only
   Answer: B. x > x_c, so M_r = 0.5 sigma_cbc b x (d - x/3) = 82.56 kN.m. (Key point: Governing criterion)

3. Total UDL on a simply supported beam is:
   A. 8 M_r / L^2
   B. M_r / L^2
   C. 2 M_r / L
   D. M_r L / 8
   Answer: A. w = 8 x 82.56 / 25 = 26.42 kN/m. (Key point: UDL formula)

4. The requested UDL is inclusive of self weight. What is the self weight?
   A. 0.30 x 0.50 x 25 = 3.75 kN/m
   B. 0.30 x 0.45 x 25
   C. 1.5 kN/m
   D. 25 kN/m
   Answer: A. b x D x 25 gives 3.75 kN/m, so external load is 22.67 kN/m. (Key point: Self weight inclusion)

## Related in Design of RCC Structure

- [WSM udl capacity of 25x60 cm beam](https://www.bctengineer.com/question.md?key=semester_7%2Fdesign_of_rcc_structure%3Awsm-udl-capacity-of-25x60-cm) — 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
- [Design simply supported beam, 6 m span](https://www.bctengineer.com/question.md?key=semester_7%2Fdesign_of_rcc_structure%3Adesign-simply-supported-beam-6-m) — Design of RCC Structure, chapter 4
- [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
- [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×
- [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×

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