Moment-area theorem

Key takeaways

Moment-area theorem belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Moment-area theorem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Moment-area theorem from memory before moving on to harder problems.

Reference excerpt

The moment-area theorem is an engineering tool to derive the slope, rotation and deflection of beams and frames. This theorem was developed by Mohr and later stated namely by Charles Ezra Greene in 1873. This method is advantageous when we solve problems involving beams, especially for those subjected to a series of concentrated loadings or having segments with different moments of inertia.

Theorem 1 The change in slope between any two points on the elastic curve equals the area of the M/EI (moment) diagram between these two points.

θ B / A = ∫ A B ( M E I ) d x {\displaystyle \theta _{B/A}={\int _{A}}^{B}\left({\frac {M}{EI}}\right)dx}

where,

M {\displaystyle M} = moment

E I {\displaystyle EI} = flexural rigidity

θ A / B {\displaystyle \theta _{A/B}} = change in slope between points A and B

A , B {\displaystyle A,B} = points on the elastic curve

Theorem 2 The vertical deviation of a point A on an elastic curve with respect to the tangent which is extended from another point B equals the moment of the area under the M/EI diagram between those two points (A and B). This moment is computed about point A where the deviation from B to A is to be determined.

t B / A = ∫ A B M ( x ) E ( x ) I ( x ) x d x {\displaystyle t_{B/A}={\int _{A}}^{B}{\frac {M(x)}{E(x)I(x)}}x\;dx}

where,

M ( x ) {\displaystyle M(x)} = moment

E ( x ) I ( x ) {\displaystyle E(x)I(x)} = flexural rigidity (often constant, so written as: E I {\displaystyle EI} )

t A / B {\displaystyle t_{A/B}} = deviation of tangent at point A with respect to the tangent at point B

A , B {\displaystyle A,B} = points on the elastic curve

Rule of sign convention The deviation at any point on the elastic curve is positive if the point lies above the tangent, negative if the point is below the tangent; we measured it from left tangent, if θ is counterclockwise direction, the change in slope is positive, negative if θ is clockwise direction.

Procedure for analysis The following procedure provides a method that may be used to determine the displacement and slope at a point on the elastic curve of a beam using the moment-area theorem.

Determine the reaction forces of a structure and draw the M/EI diagram of the structure. If there are only concentrated loads on the structure, the problem will be easy to draw M/EI diagram which will results a series of triangular shapes. If there are mixed with distributed loads and concentrated, the moment diagram (M/EI) will results parabolic curves, cubic, etc. Then, assume and draw the deflection shape of the structure by looking at M/EI diagram. Find the rotations, change of slopes and deflections of the structure by using the geometric mathematics.

References

External links Area-Moment Method. (n.d.)

Worked examples

Example 1 — a first encounter with Moment-area theorem

Start with the simplest possible case. Write down what Moment-area theorem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Moment-area theorem before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Moment-area theorem ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Moment-area theorem

In research: Moment-area theorem appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Moment-area theorem in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Moment-area theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Moment (physics), Structural analysis, so understanding it makes those chapters shorter.
In everyday life: Look for Moment-area theorem outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Moment-area theorem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Moment-area theorem means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Moment-area theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Moment-area theorem in simple terms?

The moment-area theorem is an engineering tool to derive the slope, rotation and deflection of beams and frames. This theorem was developed by Mohr and later stated namely by Charles Ezra Greene in 1873.

Why does Moment-area theorem matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Moment-area theorem?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Moment-area theorem.