---
title: "Derive moment capacity of doubly reinforced WSM"
subject: "Design of RCC Structure"
chapter: 2
key: "semester_7/design_of_rcc_structure:derive-moment-capacity-of-doubly-reinforced"
source: https://www.bctengineer.com/q/semester_7/design_of_rcc_structure/derive-moment-capacity-of-doubly-reinforced
site: BCT Engineering
---

# Derive moment capacity of doubly reinforced WSM

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

- Asked in: 2079 Bhadra, 6 marks
- Video: [Moment of resistance of Doubly R/f beam using WSM ( Steps)](https://www.youtube.com/watch?v=u84N9Hthbq8) — Govt. Polytechnic College Alirajpur
- 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/derive-moment-capacity-of-doubly-reinforced
- Whole chapter: https://www.bctengineer.com/md/semester_7/design_of_rcc_structure/chapter-2

## Question

Derive equations for bending moment carrying capacity of a double reinforced rectangular section using assumptions for working stress method given by IS 456:2000.

## Short answer

Using 1.5m for compression steel, neutral axis is bx^2/2+(1.5m-1)Asc(x-d')=mAst(d-x). Moment of resistance is Mr=0.5σcbc bx(d-x/3)+(1.5m-1)Asc σcbc[(x-d')/x](d-d'). Assumptions include plane sections, no tension in concrete, and perfect bond.

## Answer

**Assumptions (IS 456:2000, Annex B-1.2)**
- Plane sections remain plane after bending.
- Concrete is elastic in compression and carries no tension.
- Perfect bond between steel and concrete; stress in steel $=$ modular ratio $\times$ stress in surrounding concrete.
- For compression steel, effective modular ratio is $1.5m$ (to allow for creep).

**Notation:** $b$ = width, $d$ = effective depth, $d'$ = effective cover to compression steel, $x$ = depth of neutral axis, $A_{st}$ = tension steel, $A_{sc}$ = compression steel, $m$ = modular ratio.

```text
   Section    Strain      Stresses/forces
   +------+   |\         Cc  (concrete)
 d'| Asc  |   | \ x      Cs  (comp. steel)
   |      |  -+--- NA
   |      |   | /
   | Ast  |   |/         T   (tension steel)
   +------+
```

**Stress in concrete at the level of compression steel**

By similar triangles of the strain diagram:

$$
\sigma_c' = \sigma_{cbc}\,\frac{x-d'}{x}
$$

Stress in compression steel $\sigma_{sc} = 1.5\,m\,\sigma_c'$. The concrete displaced by the steel is allowed for by using $(1.5m-1)A_{sc}$ as the equivalent concrete area.

**Neutral axis** (first moment of the equivalent area about the neutral axis is zero)

$$
\frac{b x^2}{2} + (1.5m-1)A_{sc}(x-d') = m A_{st}(d-x)
$$

**Moment of resistance**

Taking moments about the tension steel, with concrete compression $C_c = \tfrac12\sigma_{cbc}\,b\,x$ acting at $x/3$ from the top:

$$
M_r = \frac12\sigma_{cbc}\,b\,x\left(d-\frac{x}{3}\right) + (1.5m-1)A_{sc}\,\sigma_{c}'\,(d-d')
$$

Substituting $\sigma_c'$:

$$
M_r = \frac12\sigma_{cbc}\,b\,x\left(d-\frac{x}{3}\right) + (1.5m-1)A_{sc}\,\sigma_{cbc}\,\frac{x-d'}{x}\,(d-d')
$$

The first term is the moment of the singly reinforced (balanced) part $M_1 = Q\,b\,d^2$ when $x = x_c$, and the second is the additional moment $M_2$ from the compression steel. The tension steel then follows from $A_{st}=\dfrac{M_1}{\sigma_{st}jd}+\dfrac{M_2}{\sigma_{st}(d-d')}$.

**Checks:** $\sigma_{c}\le\sigma_{cbc}$, $\sigma_{st}\le\sigma_{st,perm}$ and $\sigma_{sc}=1.5m\sigma_c'\le\sigma_{sc,perm}$ (190 N/mm$^2$ for Fe415).

## Check yourself

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

1. What effective modular ratio is used for compression steel in WSM?
   A. m
   B. 1.5 m
   C. 0.5 m
   D. 2 m
   Answer: B. 1.5m allows for creep of concrete. (Key point: Compression steel modular ratio)

2. By similar triangles, the concrete stress at the level of compression steel is:
   A. sigma_cbc (x - d')/x
   B. sigma_cbc d'/x
   C. sigma_cbc (d - x)/x
   D. sigma_cbc x/(x - d')
   Answer: A. Strain varies linearly from the neutral axis, so stress at depth d' is sigma_cbc (x - d')/x. (Key point: Stress in concrete at compression steel)

3. Which equation fixes x for a doubly reinforced section?
   A. b x^2/2 = m Ast (d - x)
   B. b x^2/2 + (1.5m - 1) Asc (x - d') = m Ast (d - x)
   C. b x^2/2 + Asc (x - d') = Ast (d - x)
   D. b x = (1.5m) Asc
   Answer: B. The equivalent area (1.5m - 1)Asc allows for concrete displaced by compression steel. (Key point: Neutral axis equation)

4. What are the two terms in M_r?
   A. Moment of concrete block, and moment of compression steel force about tension steel
   B. Moment of tension steel and shear
   C. Dead load and live load moments
   D. Moment of the balanced section and the span moment
   Answer: A. M_r = 0.5 sigma_cbc b x (d - x/3) + (1.5m - 1) Asc sigma_c' (d - d'). (Key point: Moment of resistance terms)

## Related in Design of RCC Structure

- [Steel requirement and doubly reinforced moment](https://www.bctengineer.com/question.md?key=semester_7%2Fdesign_of_rcc_structure%3Asteel-requirement-and-doubly-reinforced-moment) — Design of RCC Structure, chapter 2
- [Design steps of doubly reinforced section](https://www.bctengineer.com/question.md?key=semester_7%2Fdesign_of_rcc_structure%3Adesign-steps-of-doubly-reinforced-section) — Design of RCC Structure, chapter 4
- [Rectangular column with working-load moment](https://www.bctengineer.com/question.md?key=semester_7%2Fdesign_of_rcc_structure%3Arectangular-column-with-working-load-moment) — Design of RCC Structure, chapter 10
- [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
- [Moment of resistance of doubly reinforced beam](https://www.bctengineer.com/question.md?key=semester_7%2Fdesign_of_rcc_structure%3Amoment-of-resistance-of-doubly-reinforced) — Design of RCC Structure, chapter 4
- [WSM moment of resistance of T-section](https://www.bctengineer.com/question.md?key=semester_7%2Fdesign_of_rcc_structure%3Awsm-moment-of-resistance-of-t) — Design of RCC Structure, chapter 2

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