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

# State and explain moment area theorems

Theory of Structures I · Chapter 4: Deflection of Beams

- Asked in: 2066 Kartik, 6 marks; 2075 Bhadra, 6 marks; 2068 Bhadra, 4 marks (asked 3 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-explain
- Whole chapter: https://www.bctengineer.com/md/semester_4/theory_of_structures_i/chapter-4

## Question

State and explain the theorems of the moment area method with a simple example.

## Short answer

The moment area method uses two theorems: Theorem I states the change in slope between two points equals the area of the M/EI diagram. Theorem II states the vertical deviation of a point from the tangent at another point equals the moment of the M/EI area about the point. For a cantilever with end load P, slope at free end is PL^2/(2EI) and deflection is PL^3/(3EI).

## Answer

The moment-area method finds slopes and deflections of beams from the area and the moment of the $M/EI$ diagram.

#### Theorem I
**The change of slope between two points A and B of the elastic curve equals the area of the $M/EI$ diagram between those points:**

$$
\theta_B - \theta_A = \int_A^B \frac{M}{EI}\,dx
$$

#### Theorem II
**The vertical deviation of a point A from the tangent drawn at another point B equals the moment of the $M/EI$ area between A and B about the vertical through A:**

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

Here $x$ is measured from A, and $\bar x_A$ is the horizontal distance from A to the centroid of the area.

```text
 fixed A *----------------------------* B1   tangent at A (horizontal)
         |'''--..__                   |
         |         ''--..__           | deviation of B from the
         |    elastic curve ''--..    | tangent at A = deflection at B
         |                        ''--* B (free end)
```

#### Simple example
A cantilever of length $L$ with a point load $P$ at the free end B, fixed at A. The tangent at A is horizontal.

$M = -Px$ ($x$ from B), so the $M/EI$ diagram is a triangle with base $L$ and peak $\dfrac{PL}{EI}$ at A.

$$
\text{Area} = \frac12\times L\times\frac{PL}{EI} = \frac{PL^2}{2EI}
$$

**Slope at B** (Theorem I, since $\theta_A = 0$):

$$
\theta_B = \frac{PL^2}{2EI}
$$

**Deflection at B** (Theorem II, deviation of B from the tangent at A): the centroid of the triangle is at $\dfrac{2L}{3}$ from B:

$$
\Delta_B = \frac{PL^2}{2EI}\times\frac{2L}{3} = \frac{PL^3}{3EI}
$$

These are the standard cantilever results.

## Check yourself

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

1. What does Theorem I give?
   A. Change of slope = area of the M/EI diagram between the points
   B. Deflection = area of M/EI
   C. Slope = moment of area
   D. Deflection = M/EI
   Answer: A. Theorem I relates slope change to the M/EI area. (Key point: Theorem I)

2. In Theorem II the area of M/EI is multiplied by what?
   A. Distance from B to the centroid
   B. The beam length
   C. EI
   D. Distance from the point A to the centroid of the area
   Answer: D. The deviation of A from the tangent at B equals area times the distance of its centroid from A. (Key point: Theorem II)

3. For a cantilever with end load P, why is the slope at B equal to PL^2/(2EI)?
   A. Theorem II applies
   B. M/EI is constant
   C. The load is at B
   D. The slope at fixed A is zero, so the change of slope equals the triangle area
   Answer: D. Since theta_A = 0, theta_B equals the triangle area 1/2 x L x PL/EI. (Key point: Cantilever example)

## Related in Theory of Structures I

- [State and prove moment area theorems](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Amoment-area-theorems-prove) — Theory of Structures I, chapter 4, asked 4×
- [Moment area: slope and deflection at E and at 4 m](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Amoment-area-e-4m-a) — 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: 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
- [Moment area: B, C, maximum deflection](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Amoment-area-200-300kn) — 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
- [Height and area of simple surge shaft](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Asurge-shaft-height-area) — Hydro Power, chapter 5
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- [BMD of a simple beam and a frame](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Abmd-simple-beam-and-frame-10knm) — Strength of Materials, chapter 1