Module 2 · Lesson 2.2
Moments, couples and equilibrium
Turning effects and the three equations that keep a body still.
Why this matters
Forces on their own do not explain why a beam stays put. A structure can have balanced forces and still spin. The moment equation is what stops that, and it is the equation that does most of the work when you find reactions.
By the end of this lesson you should be able to
- Calculate a moment using the perpendicular distance
- Explain why a couple has the same moment about every point
- Use ΣH = 0, ΣV = 0 and ΣM = 0 together
A moment is the turning effect of a force about a point. It is the force multiplied by the perpendicular distance from the point to the force's line of action. The word perpendicular is doing a lot of work in that sentence.
What it calculates: The turning effect of a force about a chosen point.
- M
- Moment about the point (kN·m)
- F
- Size of the force (kN)
- d⊥
- Perpendicular distance from the point to the line of action (m)
This assumes
- The force and the point lie in the same plane
In plain terms: Doubling the force doubles the turning effect; so does doubling the lever arm. A force whose line of action passes straight through the point has no turning effect about it at all — that is the fact you exploit to eliminate unknowns.
Predict first
A force acts at the end of a spanner. Its line of action passes exactly through the centre of the nut. What is its moment about the nut?
A couple is two equal and opposite forces on different lines of action. There is no net force at all, but there is a turning effect, equal to one force times the distance between the lines. Because the net force is zero, the couple's moment comes out the same about every point you choose. That is why an applied moment on a beam can be drawn anywhere along it without changing the sums.
For a body to stay still in a plane, three things must all be true at once. The forces must not push it sideways, they must not push it up or down, and they must not spin it.
What it calculates: The three conditions that must hold for a plane structure to remain at rest.
- ΣH
- Sum of all horizontal force components (kN)
- ΣV
- Sum of all vertical force components (kN)
- ΣM
- Sum of moments about any chosen point (kN·m)
This assumes
- The structure is in one plane and is not accelerating
- All forces and reactions have been included, including self-weight if it matters
In plain terms: Three equations means you can find at most three unknowns from equilibrium alone. The moment equation works about any point you like, so choose the point that kills off the unknowns you care least about.
Worked example
Will this bracket rotate?
Given
- A horizontal arm is pinned at A
- A downward force of 12 kN acts 2.5 m to the right of A
- An upward force of 12 kN acts 2.5 m to the left of A
Find
Whether the arm is in equilibrium.
Assumptions
- The arm is rigid and weightless
- Both forces are vertical
Practice
A force of 18 kN acts vertically downwards, 3.5 m to the right of a support. What is its moment about the support? Give the magnitude.
Practice
Two equal and opposite forces of 15 kN act on a rigid bracket, on parallel lines 1.2 m apart. What is the magnitude of the couple they form?
Practice
A force of 40 kN acts on a bracket. The perpendicular distance from the bolt group to the force's line of action is 1.75 m. What moment does it apply about the bolts?
A set of parallel forces can always be replaced by a single resultant, and finding it is one of the most useful small skills in statics. Two conditions fix it completely:
- Its magnitude is the algebraic sum of the forces, R = ΣF, counting one direction as positive.
- Its position is wherever it produces the same moment as the original set about any point you choose: R·x̄ = ΣF·x.
The second condition is the reason the first is not enough. Two systems can have the same total force and behave completely differently, because they turn the structure by different amounts.
If ΣF happens to be zero the forces do not reduce to a single resultant at all — they reduce to a couple, of magnitude ΣF·x, which has the same moment about every point and no resultant force whatever.
Worked example
Replacing a set of parallel loads with one resultant
Given
- Four downward loads on a horizontal beam, measured from its left-hand end:
- 20.0 kN at 1.00 m, 40.0 kN at 3.00 m, 30.0 kN at 5.00 m, 10.0 kN at 8.00 m
Find
The magnitude and position of the single resultant.
Practice
Four downward loads act on a beam, measured from its left end: 20.0 kN at 1.00 m, 40.0 kN at 3.00 m, 30.0 kN at 5.00 m and 10.0 kN at 8.00 m. What is the magnitude of their resultant, in kN?
Practice
For that same set of loads, where does the resultant act, in metres from the left end?
Summary
- Moment = force × perpendicular distance
- A force through the point has no moment about it — use this to eliminate unknowns
- A couple has no resultant force but the same moment about every point
- Plane equilibrium needs all three of ΣH = 0, ΣV = 0 and ΣM = 0
This is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions. Module overview and checkpoint