---
title: "WSM beam safety check, 350x650"
subject: "Design of RCC Structure"
chapter: 2
key: "semester_7/design_of_rcc_structure:wsm-beam-safety-check-350x650"
source: https://www.bctengineer.com/q/semester_7/design_of_rcc_structure/wsm-beam-safety-check-350x650
site: BCT Engineering
---

# WSM beam safety check, 350x650

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

- Asked in: 2082 Bhadra, 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-beam-safety-check-350x650
- Whole chapter: https://www.bctengineer.com/md/semester_7/design_of_rcc_structure/chapter-2

## Question

A rectangular beam with 350 mm width and 650 mm effective depth is reinforced with five 25 mm diameter tension bars. Due to accidental reason the beam has to resist a service load moment of 150 kNm. Determine whether the beam is safe or not. Considering the safety of the beam, also determine the stress developed in concrete and steel at the state of failure as per WSM. Use M25 concrete and Fe415 steel.

## Short answer

The section is over-reinforced because actual neutral axis 248.62 mm exceeds critical 187.62 mm. Moment of resistance is 209.73 kNm, greater than 150 kNm, so the beam is safe. At failure state concrete stress is 8.5 N/mm2 and steel stress 150.68 N/mm2.

## Answer

**Given:** $b=350$ mm, $d=650$ mm, 5-25 mm dia, $A_{st}=5\times\frac{\pi}{4}(25)^2 = 2454.4$ mm$^2$, $M_{service}=150$ kN·m. M25: $\sigma_{cbc}=8.5$ N/mm$^2$; Fe415: $\sigma_{st}=230$ N/mm$^2$; $m=10.98$.

**Step 1: Constants** (M25: $\sigma_{cbc}=8.5$, Fe415: $\sigma_{st}=230$ N/mm$^2$, $m=280/(3\times8.5)=10.98$)

$$
\begin{aligned}
k &= \frac{m\sigma_{cbc}}{m\sigma_{cbc}+\sigma_{st}} = \frac{10.98\times 8.5}{10.98\times 8.5 + 230} = 0.289\\
x_c &= k d = 0.289\times 650 = 187.62\ \text{mm}
\end{aligned}
$$

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

$$
\frac{b x^2}{2} = m A_{st}(d-x)\ \Rightarrow\ \frac{350 x^2}{2} = 10.98\times 2454.4\,(650-x)
$$

$$
x = 248.62\ \text{mm}
$$

Since $x = 248.62 > x_c = 187.62$ mm, the section is **over-reinforced**: concrete reaches $\sigma_{cbc}=8.5$ N/mm$^2$ first, so concrete governs.

**Step 3: Moment of resistance**

$$
\begin{aligned}
M_r &= \frac12\sigma_{cbc}\,b\,x\left(d-\frac{x}{3}\right) = 0.5\times 8.5\times 350\times 248.62\times\left(650-\frac{248.62}{3}\right)\\
&= 209.73\ \text{kN·m}
\end{aligned}
$$

Stress in steel at this moment: $\sigma_{st} = \dfrac{m\,\sigma_{cbc}(d-x)}{x} = 150.68$ N/mm$^2$ $<$ 230 N/mm$^2$.

**Safety check:** $M_r = 209.73$ kN·m $>$ 150 kN·m. The beam is **safe**.

**Stresses at the service moment of 150 kN·m** (neutral axis $x=248.62$ mm):

$$
\sigma_c=\frac{M}{\frac12 b x(d-x/3)} = 6.08\ \text{N/mm}^2,\qquad \sigma_{s}=\frac{M}{A_{st}(d-x/3)}=107.76\ \text{N/mm}^2
$$

**Stresses at the limiting (failure) state as per WSM:** the section is over-reinforced, so concrete reaches its permissible stress first. At this state concrete stress $= 8.5$ N/mm$^2$ and steel stress $= 150.68$ N/mm$^2$ (less than 230 N/mm$^2$).

**Answer:** Beam is safe ($M_r=209.73$ kN·m $>150$ kN·m). At failure state: $\sigma_c = 8.5$ N/mm$^2$, $\sigma_s = 150.68$ N/mm$^2$. (At 150 kN·m: $\sigma_c=6.08$, $\sigma_s=107.76$ N/mm$^2$.)

## Check yourself

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

1. How is the critical (balanced) neutral axis x_c found in WSM?
   A. x_c = m sigma_cbc / (m sigma_cbc + sigma_st) x d
   B. x_c = sigma_st / sigma_cbc x d
   C. x_c = 0.48 d
   D. x_c = d / 3
   Answer: A. k = m sigma_cbc/(m sigma_cbc + sigma_st) and x_c = k d; here k = 0.289. (Key point: Critical neutral axis)

2. The actual x = 248.62 mm exceeds x_c = 187.62 mm. What does this indicate?
   A. Under-reinforced: steel governs
   B. Balanced section
   C. Over-reinforced: concrete reaches sigma_cbc first and governs
   D. Section has failed in shear
   Answer: C. x > x_c means over-reinforced, so concrete controls the moment of resistance. (Key point: Classification)

3. Which formula gives M_r for this section?
   A. sigma_st Ast (d - x/3)
   B. 0.87 fy Ast (d - 0.416 xu)
   C. 0.5 sigma_cbc b x (d - x/3)
   D. Q b d^2 with Q for the balanced section
   Answer: C. For an over-reinforced section the concrete stress limits sigma_cbc, so M_r = 0.5 sigma_cbc b x (d - x/3) = 209.73 kN.m. (Key point: Moment of resistance formula)

4. At the WSM failure state of this beam, what are concrete and steel stresses?
   A. Concrete 8.5 N/mm2, steel 230 N/mm2
   B. Concrete 8.5 N/mm2, steel 150.68 N/mm2
   C. Concrete 6.08 N/mm2, steel 107.76 N/mm2
   D. Concrete 6.08 N/mm2, steel 230 N/mm2
   Answer: B. Concrete reaches its permissible 8.5 while steel is only 150.68, below 230. The values 6.08 and 107.76 are at the 150 kN.m service moment. (Key point: Stresses at failure state)

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