---
title: "Steel requirement and doubly reinforced moment"
subject: "Design of RCC Structure"
chapter: 2
key: "semester_7/design_of_rcc_structure:steel-requirement-and-doubly-reinforced-moment"
source: https://www.bctengineer.com/q/semester_7/design_of_rcc_structure/steel-requirement-and-doubly-reinforced-moment
site: BCT Engineering
---

# Steel requirement and doubly reinforced moment

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

- Asked in: 2076 Chaitra, 2+2+4 marks
- Video: [Case 1: Finding the Ultimate Moment Capacity of Doubly Reinforced Beam](https://www.youtube.com/watch?v=D25hlfVzPLQ) — Engr. Ailene De Vela - Bendicio
- 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/steel-requirement-and-doubly-reinforced-moment
- Whole chapter: https://www.bctengineer.com/md/semester_7/design_of_rcc_structure/chapter-2

## Question

Describe the requirement of steel as reinforcement in RCC structure. Explain about moment of resistance of doubly reinforced section. Derive the formula.

## Short answer

Steel carries tension, resists shear and torsion, controls cracking, and provides ductility and bond. Doubly reinforced section is needed when M exceeds Qbd^2; Mr=M1+M2. M1=Qbd^2 and M2=(1.5m-1)Asc σcbc[(x-d')/x](d-d'), with neutral axis equation bx^2/2+(1.5m-1)Asc(x-d')=mAst(d-x).

## Answer

#### Requirement of steel as reinforcement

Concrete is strong in compression but weak in tension (tensile strength is about 10% of compressive strength), so it cracks at low loads. Steel is therefore provided as reinforcement for the following reasons.

- **Tension:** steel carries the tensile force in beams, slabs, footings and ties, with concrete protecting it from corrosion and fire.
- **Shear and torsion:** stirrups and bent-up bars resist diagonal tension.
- **Compression:** longitudinal bars share the load in columns and raise capacity of beams (doubly reinforced section).
- **Control of cracks and shrinkage:** distribution and temperature steel limit crack width.
- **Bond:** steel has a similar coefficient of thermal expansion to concrete ($\approx 1.2\times10^{-5}$ per $^\circ$C), and ribs/deformations give good bond.
- **Ductility:** steel gives warning before failure.

#### Moment of resistance of doubly reinforced section

A beam is doubly reinforced when the applied moment exceeds the balanced moment $Q\,b\,d^2$ of the singly reinforced section, or when depth is restricted, or for reversal of moment. Compression steel $A_{sc}$ is added at effective cover $d'$ from the compression face. The moment of resistance is

$$
M_r = M_1 + M_2
$$

where $M_1 = Q\,b\,d^2$ is the balanced-section moment and $M_2$ the additional moment taken by the compression steel and the additional tension steel.

#### Derivation

**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. Why is steel provided in concrete?
   A. Concrete is weak in tension and cracks at low loads
   B. Concrete is weak in compression
   C. Steel makes concrete lighter
   D. Steel removes the need for cover
   Answer: A. Concrete tensile strength is about 10 percent of its compressive strength, so steel carries tension. (Key point: Why steel is needed)

2. Why do steel and concrete work well together?
   A. Similar thermal expansion coefficient and good bond from ribs
   B. Identical elastic modulus
   C. Steel is non-corroding
   D. Concrete has high tensile strength
   Answer: A. Both expand about 1.2x10^-5 per deg C and deformed bars bond well. (Key point: Bond compatibility)

3. When is a beam made doubly reinforced?
   A. When the applied moment is less than the balanced moment
   B. When the moment exceeds the balanced moment Q b d^2, depth is restricted, or moment reverses
   C. When span is short
   D. When no shear steel is needed
   Answer: B. Compression steel adds capacity beyond the singly reinforced balanced moment. (Key point: When doubly reinforced)

4. In M_r = M1 + M2, what is M1?
   A. Moment from compression steel
   B. Balanced-section moment Q b d^2
   C. Dead load moment
   D. Shear moment
   Answer: B. M1 is the balanced singly reinforced moment; M2 is the extra moment taken by compression steel and extra tension steel. (Key point: Components of M_r)

## Related in Design of RCC Structure

- [Derive moment capacity of doubly reinforced WSM](https://www.bctengineer.com/question.md?key=semester_7%2Fdesign_of_rcc_structure%3Aderive-moment-capacity-of-doubly-reinforced) — Design of RCC Structure, chapter 2
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