---
title: "Sections with stress distribution, moment expressions"
subject: "Design of RCC Structure"
chapter: 4
key: "semester_7/design_of_rcc_structure:sections-with-stress-distribution-moment-expressions"
source: https://www.bctengineer.com/q/semester_7/design_of_rcc_structure/sections-with-stress-distribution-moment-expressions
site: BCT Engineering
---

# Sections with stress distribution, moment expressions

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

- Asked in: 2080 Baisakh, 6 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/sections-with-stress-distribution-moment-expressions
- Whole chapter: https://www.bctengineer.com/md/semester_7/design_of_rcc_structure/chapter-4

## Question

What are balanced, under-reinforced and over-reinforced sections? Explain with neat sketches of the stress distributions and expressions for moment resistance of each section.

## Short answer

Balanced: x_u = x_u,max, M_u,lim = 0.36 fck b x_u,max (d - 0.42 x_u,max) = 0.138 fck b d² (Fe415). Under-reinforced: x_u < x_u,max, steel yields, M_u = 0.87 fy A_st (d - 0.42 x_u) = 0.87 fy A_st d (1 - (A_st fy)/(b d fck)). Over-reinforced: x_u > x_u,max, concrete crushes first, moment limited to M_u,lim.

## Answer

Let $x_u$ be the depth of the neutral axis and $x_{u,max}$ its limiting value ($0.48d$ for Fe415). In the stress block, the compressive force is $C=0.36f_{ck}b\,x_u$ at lever arm $z=d-0.42x_u$.

```text
  Strain             Stress block
   0.0035          0.446fck  C=0.36fck.b.xu
   |\               |=====|
   | \ xu           |     | <- 0.42xu from top
 --+--- NA        --+-----+
   |  \             |
   |   \ es         T=0.87fy.Ast
```

#### 1. Balanced section ($x_u=x_{u,max}$)
Concrete reaches 0.0035 and steel reaches yield strain $\varepsilon_y=0.87f_y/E_s+0.002$ together.

$$
M_{u,lim}=0.36f_{ck}b\,x_{u,max}(d-0.42x_{u,max})=0.138f_{ck}bd^2\ (\text{Fe415})
$$

#### 2. Under-reinforced section ($x_u<x_{u,max}$)
Steel yields first; $x_u=\dfrac{0.87f_yA_{st}}{0.36f_{ck}b}$.

$$
M_u=0.87f_yA_{st}(d-0.42x_u)=0.87f_yA_{st}d\left(1-\frac{A_{st}f_y}{bdf_{ck}}\right)
$$

Failure is ductile.

#### 3. Over-reinforced section ($x_u>x_{u,max}$)
Concrete crushes first; steel stress is below $0.87f_y$. As per IS 456 the moment is limited to

$$
M_u=M_{u,lim}=0.36f_{ck}b\,x_{u,max}(d-0.42x_{u,max})
$$

Failure is brittle and sudden, so over-reinforced sections are not allowed.

## Check yourself

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

1. What is the compressive force in the equivalent stress block?
   A. 0.446 fck b xu
   B. 0.36 fck b xu acting at 0.42 xu from the top
   C. 0.67 fck b xu at xu/2
   D. 0.87 fy b xu
   Answer: B. The average stress is 0.36 fck over depth xu, with lever arm d - 0.42 xu. (Key point: Compression force)

2. How is xu found in an under-reinforced section?
   A. xu = 0.87 fy Ast / (0.36 fck b)
   B. xu = 0.48d
   C. xu = Ast / b
   D. xu = 0.36 fck b / (0.87 fy Ast)
   Answer: A. Equating C = T gives 0.36 fck b xu = 0.87 fy Ast. (Key point: Under-reinforced xu)

3. What is Mu,lim for Fe415?
   A. 0.138 fck b d^2
   B. 0.148 fck b d^2
   C. 0.36 fck b d^2
   D. 0.87 fy Ast d
   Answer: A. Substituting xu,max = 0.48d into the moment expression gives 0.138 fck b d^2. (Key point: Balanced moment)

4. What moment is used for an over-reinforced section per IS 456?
   A. The actual moment from the steel provided
   B. Limited to Mu,lim
   C. Zero
   D. 1.5 Mu,lim
   Answer: B. Steel stress is below 0.87 fy, so capacity is limited to Mu,lim and such sections are avoided. (Key point: Over-reinforced moment)

## Related in Design of RCC Structure

- [Define balanced, under, over-reinforced sections](https://www.bctengineer.com/question.md?key=semester_7%2Fdesign_of_rcc_structure%3Adefine-balanced-under-over-reinforced-sections) — Design of RCC Structure, chapter 4
- [Under, over and balanced sections](https://www.bctengineer.com/question.md?key=semester_7%2Fdesign_of_rcc_structure%3Aunder-over-and-balanced-sections) — Design of RCC Structure, chapter 4, asked 6×
- [Significance of under and over-reinforced sections](https://www.bctengineer.com/question.md?key=semester_7%2Fdesign_of_rcc_structure%3Asignificance-of-under-and-over-reinforced) — Design of RCC Structure, chapter 4
- [Balanced, under, over sections with strain diagrams](https://www.bctengineer.com/question.md?key=semester_7%2Fdesign_of_rcc_structure%3Abalanced-under-over-sections-with-strain) — 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

## Related in other subjects

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- [Shear stress distribution in an I-section](https://www.bctengineer.com/question.md?key=semester_4%2Fstrength_of_materials_mechanical%3Ashear-stress-i-section-distribution) — Strength of Materials, chapter 5