Varignon Theorem: The Principle of Moments With Examples

The Varignon theorem, often called the principle of moments, states that the moment of a force about any point equals the sum of the moments of its components about the same point. It sounds almost too obvious to need a name, yet it is one of the most used tools in structural engineering. Every time an engineer finds where a distributed load acts, resolves an inclined force into easier pieces, or checks whether a footing is loaded within its middle third, Varignon’s principle is doing the work.

This article states the theorem, sketches why it is true, works through three practical examples with units, and lists the mistakes that commonly creep into hand calculations. It also clears up a source of confusion: there is a second, unrelated Varignon theorem in geometry.

The Varignon theorem stated

For a system of concurrent forces F₁, F₂, …, F_n acting at a point A, with resultant R = F₁ + F₂ + … + F_n, the moment of R about any point O equals the sum of the moments of the individual forces about O:

M_O(R) = M_O(F₁) + M_O(F₂) + … + M_O(F_n)

In two dimensions, the moment of a force about a point is the force multiplied by the perpendicular distance from the point to its line of action, with a sign for direction. The most common everyday use is the special case of a single force split into its x and y components. With the force applied at coordinates (x, y) relative to O and components F_x and F_y, the moment about O is:

M_O = xF_y − yF_x

taking anticlockwise moments as positive. This is often far easier than finding the perpendicular distance to an inclined line of action.

Where it comes from

The theorem is named after Pierre Varignon (1654–1722), a French mathematician who published it in 1687 in his Projet d’une nouvelle mécanique, a work that treated mechanics through the composition of forces. Varignon was one of the early adopters of the new calculus and corresponded with Leibniz and Johann Bernoulli.

In modern vector notation the proof is a single line. The moment of a force F applied at position r is the cross product r × F. Because the cross product distributes over addition:

r × (F₁ + F₂) = r × F₁ + r × F₂

which is the theorem. The forces must act at the same point (or, more generally, the principle applies to any force system once its resultant and line of action are properly defined), and all moments must be taken about the same point.

A note on the other Varignon theorem

In geometry, Varignon’s theorem states that joining the midpoints of the sides of any quadrilateral produces a parallelogram. It is named after the same mathematician, whose proof was published posthumously in 1731, but it has nothing to do with forces. When engineers say “Varignon’s theorem” they mean the principle of moments.

Worked example 1: moment of an inclined force

A force of 10 kN acts at 30° above the horizontal, pointing up and to the right, at a point 2.0 m to the right of and 1.5 m above a point O. Find its moment about O.

F_x = 10 cos 30° = 8.66 kN
F_y = 10 sin 30° = 5.00 kN
M_O = xF_y − yF_x = 2.0 × 5.00 − 1.5 × 8.66 = 10.00 − 12.99 = −2.99 kNm

The negative sign means the moment is clockwise. As a check, the perpendicular distance from O to the line of action is |M| / F = 2.99 / 10 = 0.299 m, which you could also find geometrically, with more effort. The component route avoids trigonometry on awkward angles and is much less prone to error.

Worked example 2: where a distributed load acts

A distributed load can be replaced by a single resultant force for the purpose of finding reactions. Varignon’s principle locates that resultant: its moment about any point must equal the sum, or integral, of the moments of the distributed load.

For a uniformly distributed load w over a length L, the resultant is wL acting at the midpoint. For a triangular load rising from zero to w over length L, the resultant is wL/2. Taking moments about the zero end:

∫ (wx/L) · x dx from 0 to L = wL²/3

So the resultant acts at x̄ = (wL²/3) / (wL/2) = 2L/3 from the zero end, that is, L/3 from the heavy end. For a triangular load of peak 12 kN/m over 3.0 m, such as soil or water pressure on a wall, the resultant is 12 × 3.0 / 2 = 18 kN, acting 2.0 m from the zero end, or 1.0 m from the base if the load peaks at the bottom. This is the familiar “one third of the height” used for lateral earth pressure on retaining walls.

Worked example 3: eccentric load on a footing

A square pad footing 1.5 m wide carries a vertical column load of 600 kN at its centre and a horizontal load of 40 kN applied 1.0 m above the underside of the footing. Ignore the footing’s self-weight for simplicity. Where does the resultant act on the base?

Taking moments about the centre of the base, the vertical load has zero moment and the horizontal load has a moment of 40 × 1.0 = 40 kNm.
By Varignon’s principle, the resultant of the two forces must produce the same 40 kNm about that point. Its vertical component is 600 kN, so it crosses the base at an eccentricity e = 40 / 600 ≈ 0.067 m from the centre.
The middle-third limit is B/6 = 1.5 / 6 = 0.25 m. Since e is less than 0.25 m, the whole base remains in compression.

The base pressures then follow from N/A ± 6M/(B² · B) for a square footing: 600 / 2.25 ± 6 × 40 / 3.375 = 266.7 ± 71.1, giving about 338 kN/m² at one edge and 196 kN/m² at the other. Our article on the pad foundation explains how these pressures are compared with the ground’s capacity.

Where engineers use the principle

Support reactions. Taking moments about one support, with each load replaced by its resultant, gives the other reaction directly.
Centroids and centres of gravity. The centroid of a composite area or the centre of gravity of a structure is found by setting the moment of the total equal to the sum of the moments of the parts, the same step used to locate the neutral axis before applying the parallel axis theorem.
Overturning checks. Retaining walls, signs, canopies and tall structures are checked by comparing the moments of overturning forces with those of restoring forces about the toe.
Bolt and weld groups. An eccentric load on a connection is replaced by a force through the centroid of the group plus a moment, a direct application of the same idea.
Bending moment calculations. Each value on a bending moment diagram is a sum of moments of the forces to one side of the section.

Common mistakes

Mixing reference points. All moments must be taken about the same point. Switching points halfway through a calculation is the most common error.
Inconsistent sign convention. Decide whether clockwise or anticlockwise is positive and keep to it, especially with the xF_y − yF_x form.
Wrong coordinates. x and y are the coordinates of a point on the force’s line of action, measured from the moment centre, not from some other origin.
Locating distributed loads at the wrong point. A triangular load does not act at its midpoint. Use the centroid of the load diagram.
Forgetting moments, not just forces, are transferred. Moving a force to a new point to simplify a problem requires adding a couple equal to the force times the distance moved.

Frequently asked questions

What does the Varignon theorem state?

It states that the moment of a resultant force about any point equals the algebraic sum of the moments of its component forces about the same point, which lets engineers replace a force by its components, or a set of forces by their resultant, when taking moments.

Why is it useful in structural engineering?

It simplifies moment calculations for inclined forces, locates the resultant of distributed loads, finds centroids and eccentricities, and underpins reaction, overturning and connection calculations that are carried out daily in design.

Who was Varignon?

Pierre Varignon was a French mathematician who lived from 1654 to 1722 and published the principle of moments in 1687 in his work on mechanics, Projet d’une nouvelle mécanique.

Is Varignon’s theorem the same as the principle of moments?

In mechanics the two names are generally used for the same idea, while the separate geometric theorem bearing Varignon’s name concerns the midpoints of a quadrilateral and is unrelated to forces.

Does the theorem apply in three dimensions?

Yes, in three dimensions moments are vectors given by the cross product of position and force, and because the cross product distributes over addition, the moment of the resultant still equals the sum of the moments of the components.

Need loads, reactions or footing pressures checked by a practising engineer? Talk to our consultancy or get in touch.