---
title: "Moment of resistance of doubly reinforced beam"
subject: "Design of RCC Structure"
chapter: 4
key: "semester_7/design_of_rcc_structure:moment-of-resistance-of-doubly-reinforced"
source: https://www.bctengineer.com/q/semester_7/design_of_rcc_structure/moment-of-resistance-of-doubly-reinforced
site: BCT Engineering
---

# Moment of resistance of doubly reinforced beam

Design of RCC Structure · Chapter 4: Design of beams: Behavior in Flexure

- Asked in: 2071 Chaitra, 12 marks
- Video: [Doubly Reinforced Beam - Moment of Resistance Problem](https://www.youtube.com/watch?v=I1fAK_pd7zk) — Stan Academy
- 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/moment-of-resistance-of-doubly-reinforced
- Whole chapter: https://www.bctengineer.com/md/semester_7/design_of_rcc_structure/chapter-4

## Question

A RC beam 300 mm x 500 mm is reinforced with 5-25 mm bars in tension and 5-12 mm bars in compression each at a clear cover of 25 mm. If effective span of the beam is 4.30 m, find the moment of resistance of the beam at ultimate state. Use M25 concrete and Fe 415 grade steel.

## Short answer

Neutral axis depth 256.28 mm exceeds x_u,max = 222 mm; limit to 222 mm. Moment of resistance is 304.99 kN·m (over-reinforced section).

## Answer

**Given:** $b=300$ mm, $D=500$ mm, 5-25 mm tension bars ($A_{st}=2454.4$ mm$^2$), 5-12 mm compression bars ($A_{sc}=565.5$ mm$^2$), clear cover 25 mm, M25, Fe415.

**Effective depths:** $d=500-25-\dfrac{25}{2}=462.5$ mm; $d'=25+\dfrac{12}{2}=31$ mm. (The 4.30 m span is not needed for the moment of resistance.)

**Step 1: Neutral axis.** Assume the tension steel yields. Equilibrium of forces:

$$
0.36f_{ck}b\,x_u+A_{sc}(f_{sc}-f_{cc})=0.87f_yA_{st},\qquad f_{cc}=0.446f_{ck}=11.15\ \text{N/mm}^2
$$

Total tension $=0.87\times415\times2454.4=886.2$ kN. Solving by trial (with $f_{sc}$ from the Fe415 design curve at $\varepsilon_{sc}=0.0035(1-d'/x_u)$) gives $x_u=256.28$ mm.

**Step 2: Check section type.**

$$
x_{u,max}=0.48d=0.48\times462.5=222.0\ \text{mm}
$$

Since $x_u=256.28$ mm $>x_{u,max}$, the tension steel is more than needed (over-reinforced). The tension steel will not reach $0.87f_y$ and IS 456 limits the neutral axis depth to $x_{u,max}$. So take $x_u=x_{u,max}=222.0$ mm.

**Step 3: Compression steel stress**

$$
\varepsilon_{sc}=0.0035\left(1-\frac{31}{222.0}\right)=0.00301\ \Rightarrow\ f_{sc}=354.0\ \text{N/mm}^2
$$

**Step 4: Ultimate moment**

$$
\begin{aligned}
M_u&=0.36f_{ck}b\,x_{u,max}(d-0.42x_{u,max})+A_{sc}(f_{sc}-f_{cc})(d-d')\\
&=0.36\times25\times300\times222.0(462.5-0.42\times222.0)+565.5(354.0-11.15)(462.5-31)\\
&=221.33+83.66=304.99\ \text{kN·m}
\end{aligned}
$$

**Answer:** $M_u=304.99$ kN·m (limited by concrete, since the section is over-reinforced in tension).

## Check yourself

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

1. How are d and d' found from clear cover 25 mm?
   A. d = D - 25; d' = 25
   B. d = D; d' = 12
   C. d = D - 12; d' = 25/2
   D. d = D - cover - 25/2 (tension bar); d' = cover + 12/2 (compression bar)
   Answer: D. Distance is measured to the bar centre, so half the bar diameter is added. (Key point: Effective depth)

2. Trial gave xu = 256.28 mm > xu,max = 222 mm. What does this mean?
   A. The section is over-reinforced and xu is limited to xu,max
   B. The section is under-reinforced
   C. The steel reaches 0.87 fy
   D. The span is too short
   Answer: A. Beyond xu,max tension steel does not yield; IS 456 limits the NA depth to xu,max in computing Mu. (Key point: Over-reinforced result)

3. What value of xu is used for Mu?
   A. xu = 256.28 mm
   B. xu = xu,max = 0.48d = 222 mm
   C. xu = d
   D. xu = d'
   Answer: B. For over-reinforced sections the moment is limited by concrete, using xu,max. (Key point: Moment with limited xu)

4. Is the 4.30 m span needed for the moment of resistance?
   A. Yes, Mu = wL^2/8
   B. Yes, to find d
   C. No, Mu depends only on section and materials
   D. Yes, to find fsc
   Answer: C. Moment of resistance is a section property; span matters only to find the applied moment. (Key point: Role of span)

## Related in Design of RCC Structure

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