ENG1011 Chap.4 Moments, Supports and Rigid-Body Equilibrium
Moments, Supports and Rigid-Body Equilibrium
Moments, Supports and Rigid-Body Equilibrium is Week 3's step from particles to bodies with size. A force that misses a point turns the body about it, and that turning effect is the moment: the force times its perpendicular distance from the point. The unit takes anticlockwise as positive and notes that any force whose line of action passes through the point has no moment about it.
Applied couples are pure turning effects with the same moment about every point, so they enter moment sums without a lever arm and never enter force sums. Supports are idealised by what they resist: a roller resists movement normal to its surface with one reaction, a pin resists movement in both directions with two, and a fixed support adds a reaction moment for three.
A planar rigid body in equilibrium satisfies two force equations and one moment equation, so three unknown reactions make a beam statically determinate.
The art is choosing the moment point where unknowns meet, and checking with a second moment equation about another support.
Rigid-body equilibrium is the heart of the Content Test, whose 2026 items included writing a beam's equilibrium equation, finding reactions in two different set-ups, and judging whether a supported beam is stable.
The practice test solutions repeatedly remind readers to answer the quantity asked and to use the sign convention stated in the question, which is where most lost marks in this week come from rather than the arithmetic itself.
What this chapter covers
- 01
The moment of a force and the perpendicular distance
- 02
Sign convention: anticlockwise positive
- 03
Couples and why they have no lever arm
- 04
Pin, roller and fixed supports and their reactions
- 05
Three equilibrium equations for a planar rigid body
- 06
Choosing the moment point to isolate one unknown
- 07
Reaction moments at built-in ends
- 08
Reading negative reactions as reversed directions
Worked example · free
Reactions of a simply supported beam with one load
- 1Moments about A with anticlockwise positive: 5 By minus 10 times 3 = 0, so By = 6 kN upward.
- 1Vertical balance: Ay + 6 minus 10 = 0, so Ay = 4 kN upward. No horizontal load acts, so Ax = 0.
- 1Check about B: 10 times 2 minus Ay times 5 = 20 minus 20 = 0.
Key terms
- Moment of a force
- The turning effect of a force about a point, equal to the force times its perpendicular distance.
- Couple
- A pair of equal, opposite, parallel forces whose only effect is a moment, the same about every point.
- Pin support
- A support that prevents movement in both directions but allows rotation, supplying two reaction components.
- Roller support
- A support that prevents movement normal to its surface only, supplying a single reaction.
- Fixed support
- A built-in support that prevents both movements and rotation, supplying two forces and a reaction moment.
Moments, Supports and Rigid-Body Equilibrium FAQ
What is the difference between a reaction moment and a resultant moment?
The resultant moment is the combined turning effect of the applied loads about a point. The reaction moment is what a fixed support supplies to cancel it. They are equal in size and opposite in sign, and practice questions ask specifically for the reaction.
Where should I take moments?
Choose a point where two unknown reactions act, usually a pin support, so both drop out of the moment equation. That leaves one unknown to solve directly, and the force equations then give the rest.
Why can a roller not carry a horizontal load?
A roller only pushes normal to the surface it rests on. Any horizontal component of the applied loads must therefore be carried by a pin elsewhere on the beam, which is why questions with an inclined load ask for the pin's horizontal reaction.
Does it matter where a couple is applied on a beam?
Not for reactions. A couple has the same moment about every point, so its position never changes a moment sum. Its position does matter later, when bending moment diagrams are drawn, because it causes a jump in moment at that point.
What does a negative reaction mean?
It means the reaction acts opposite to the direction you assumed. Keep the sign through the working and state the true direction in words, and enter the signed value exactly as the question's convention requires.
Exam move
Work every reactions question the same way: diagram, moments about the best point, force balances, then a check about a different point. Practise with overhanging beams, where one reaction often turns negative, and with cantilevers carrying couples, where the reaction moment is easy to confuse with the applied moment.
For stability questions, list the reactions, then check they stop sliding in x, sliding in y and rotation before answering.
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