---
title: "State and prove moment area theorems"
subject: "Theory of Structures I"
chapter: 4
key: "semester_4/theory_of_structures_i:moment-area-theorems-prove"
source: https://www.bctengineer.com/q/semester_4/theory_of_structures_i/moment-area-theorems-prove
site: BCT Engineering
---

# State and prove moment area theorems

Theory of Structures I · Chapter 4: Deflection of Beams

- Asked in: 2079 Chaitra, 6 marks; 2075 Baisakh, 4 marks; 2073 Bhadra, 4 marks; 2070 Bhadra, 6 marks (asked 4 times)
- Video: [Moment Area Theorems Explained - Structural Analysis](https://www.youtube.com/watch?v=4W47rgTE09M) — structurefree
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/theory_of_structures_i/moment-area-theorems-prove
- Whole chapter: https://www.bctengineer.com/md/semester_4/theory_of_structures_i/chapter-4

## Question

State and prove the theorems of the moment area method for determining deflections and slopes of a beam.

## Short answer

Moment-area theorems: I) The change in slope between two points equals the area of the M/EI diagram between them. II) The deviation of one point from the tangent at another equals the moment of that M/EI area about the first point. Proof: integrating dθ = (M/EI) dx and x dθ.

## Answer

The moment-area method uses the area and the moment of the $M/EI$ diagram to find slopes and deflections of beams.

#### Basis
For small deflections the curvature is $\dfrac{d^2y}{dx^2} = \dfrac{M}{EI}$. Consider two points A and B on the deflected beam and an elemental length $dx$ at a distance $x$ from A.

```text
 A *
   |'.
   |  '.       elastic curve
 t |    '-._
   |        ''-.._____|_|________* B
 A1*-------------------------------- tangent at B (extended)
   |<-------- x ------>|dx|
```

The tangents at the two ends of $dx$ meet at an angle $d\theta$:

$$
d\theta = \frac{dx}{\rho} = \frac{M}{EI}\,dx
$$

#### Theorem I (slope)
**The change in slope between the tangents at two points A and B on the elastic curve equals the area of the $M/EI$ diagram between those points.**

Proof: integrate $d\theta$ from A to B:

$$
\theta_{AB} = \theta_B - \theta_A = \int_A^B \frac{M}{EI}\,dx = \text{Area}_{AB}\ \text{of the } \frac{M}{EI}\text{ diagram}
$$

#### Theorem II (deviation)
**The vertical deviation of point A from the tangent drawn at B equals the moment, about A, of the area of the $M/EI$ diagram between A and B.**

Proof: the tangents at the ends of the element $dx$ intercept a small vertical distance on the vertical through A:

$$
d\Delta = x\,d\theta = x\,\frac{M}{EI}\,dx
$$

where $x$ is the distance from A to the element. Adding up these intercepts from A to B gives the total deviation $t = AA_1$:

$$
\Delta_{A/B} = \int_A^B x\,\frac{M}{EI}\,dx = \text{Area}_{AB}\times\bar{x}_A
$$

where $\bar x_A$ is the distance from A to the centroid of the $M/EI$ area between A and B.

#### Notes
- A positive $M/EI$ area means a counter-clockwise (positive) change of slope from A to B, and A lies above the tangent drawn at B.
- The theorems are valid for linear elastic beams with small slopes. The deviation is measured perpendicular to the original beam axis.

## Check yourself

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

1. For an element dx, the tangents at its ends meet at an angle d(theta). What is it?
   A. EI dx / M
   B. M EI dx
   C. M dx / EI
   D. dx / (M EI)
   Answer: C. With curvature 1/rho = M/EI, d(theta) = dx/rho = (M/EI) dx. (Key point: Basis)

2. What does Theorem I state?
   A. Change in slope between two points equals the area of the M/EI diagram between them
   B. Slope equals M/EI at a point
   C. Deviation equals the area of M/EI
   D. Change in slope equals the moment of the M/EI area
   Answer: A. Integrating d(theta) between A and B gives the area of the M/EI diagram. (Key point: Theorem I)

3. What does Theorem II state?
   A. Deviation of A from the tangent at B equals the moment of the M/EI area between them about A
   B. Deviation equals the area times 2
   C. Deviation equals the area alone
   D. Deviation equals the slope change
   Answer: A. d(Delta) = x d(theta), integrated gives area times the distance of its centroid from A. (Key point: Theorem II)

## Related in Theory of Structures I

- [Conjugate beam: slopes and deflections under loads](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Aconjugate-ss-beam-two-loads-3-6-5-4-5) — Theory of Structures I, chapter 4, asked 2×
- [Moment area: slopes at A, B, deflection at D](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Amoment-area-beam-60kn-12m) — Theory of Structures I, chapter 4
- [State and explain moment area theorems](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Amoment-area-theorems-explain) — Theory of Structures I, chapter 4, asked 3×
- [Moment area: slopes at A, B, deflection at D](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Amoment-area-slopes-a-b-defl-d-2065) — Theory of Structures I, chapter 4
- [Moment area vs conjugate beam method](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Amoment-area-vs-conjugate) — Theory of Structures I, chapter 4
- [Moment area: overhang beam free-end deflection](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Amoment-area-overhang-10kn-m-2078) — Theory of Structures I, chapter 4

## Related in other subjects

- [Moment-area theorems with proof](https://www.bctengineer.com/question.md?key=semester_4%2Fstrength_of_materials_mechanical%3Amoment-area-theorems-statement-proof) — Strength of Materials, chapter 8
- [Beam with two symmetrical loads by moment-area](https://www.bctengineer.com/question.md?key=semester_4%2Fstrength_of_materials_mechanical%3Asymmetrical-beam-two-loads-moment-area) — Strength of Materials, chapter 8
- [State and prove De Morgan's theorems](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Ade-morgan-state-prove) — Basic Electronics Engineering, chapter 6, asked 4×
- [Prove De Morgan's theorems; XOR using NAND](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Ademorgan-prove-xor-nand) — Digital Logic, chapter 2, asked 3×