AMME1802 Chap.4 Free-Body Diagrams and Equilibrium
Free-Body Diagrams and Equilibrium
Equilibrium calculations inherit every decision made in the free-body diagram. This chapter models pins, rollers, cables, smooth contacts and fixed supports, then selects moment centres that eliminate unknowns. It also distinguishes external forces on an entire structure from internal forces revealed only after a joint, member or section is isolated.
A support model comes from the motions it prevents: a planar pin supplies two force components, a smooth roller one normal reaction and a fixed support forces plus a couple. The number of independent equations limits the unknown reactions that statics can determine. Moment centres are chosen strategically to eliminate lines of action, while a second moment balance provides an independent check.
Internal forces disappear on a whole-system diagram and reappear only when the boundary cuts their connection. Naming the body before every equation set prevents forces from migrating between diagrams.
What this chapter covers
- 01
Body and system boundaries
- 02
Support reaction models
- 03
Planar equilibrium equations
- 04
Moment-centre selection
- 05
Three-dimensional equilibrium
- 06
Determinate, indeterminate and unstable support arrangements
- 07
Physical admissibility of negative support reaction results
- 08
A final balance check about a second independent moment centre
Worked example · free
Beam support reactions
- +1Draw pin components, the roller reaction and both applied loads.
- +1Sum moments about the pin to find the roller reaction.
- +1Use vertical force balance for the pin vertical component.
- +1Use horizontal force balance for the pin horizontal component.
- +1Check with a moment balance about the roller.
Key terms
- Free-body diagram
- A boundary model showing every external load and reaction on an isolated body.
- Reaction force
- A force supplied by a support or contact that prevents an allowed motion.
- Static equilibrium
- The condition of zero resultant force and zero resultant moment.
- System boundary
- The imagined enclosure separating included bodies from external interactions.
- Fixed support
- A support that prevents planar translation and rotation and therefore supplies forces and a couple.
- Smooth contact
- A frictionless contact that supplies only a reaction normal to its surface.
- Static determinacy
- The condition in which equilibrium equations are sufficient to find the reaction unknowns.
- Unstable structure
- A support arrangement that leaves at least one rigid-body motion insufficiently restrained.
Free-Body Diagrams and Equilibrium FAQ
What belongs on a free-body diagram?
Include every external force and couple crossing the chosen boundary, along with necessary geometry and coordinate directions. Do not draw a third-law partner that acts on another body, and do not retain internal connection forces when both connected parts remain inside the system.
How many equations are available in two dimensions?
A planar rigid body provides three independent scalar equilibrium equations: two force components and one moment equation. If the support model creates more unknown reaction components, additional bodies, constitutive information or a different model may be needed.
What does a negative reaction mean?
Unknown reaction directions may be assumed. A negative solution means the real force acts opposite the assumed arrow. Interpret that result within the physical support model, because a cable cannot push and a unilateral contact cannot pull.
How can I tell whether a structure is determinate?
Count independent reaction unknowns against available rigid-body equilibrium equations, then inspect the geometry for redundant or missing constraints. Equation counting is necessary but not sufficient because poorly arranged supports can still permit motion or create dependent equations.
Assessment move
Spend more time on the isolated-body sketch than on the first equation. Practise support models from words alone, then solve beams by changing the moment centre and checking the same reactions. Label every equation set with the body it belongs to. Build a support flash set with a sketch on one side and permitted reaction components on the other.
Include inclined rollers, cables, pins and fixed supports rather than only horizontal beams. For each beam problem, count unknowns before solving and identify the moment centre that removes the largest useful group. After finding reactions, repeat one moment balance about a different point and make sure the total external force closes.
Add boundary drills in which the same assembly is drawn once as a whole system and once as a separated member; circle which pin forces vanish internally and which become external. When a solved reaction is negative, rewrite the final arrow and explain whether the support can physically supply it.
Finish with one indeterminate and one unstable support arrangement so equation counting becomes a modelling test rather than a mechanical prelude. Redraw several physical supports without loads and annotate what translation or rotation each prevents. Then add loads and predict the reaction directions before choosing unknown components.
Practise overhang beams because they expose incorrect assumptions about where a support or endpoint lies. Include a problem with an unknown load position so the signed distance must be interpreted on the beam. For connected bodies, compare a whole-system free-body diagram with two separated diagrams and verify that the internal pin pair cancels only in the whole system.
Write one line after every solution explaining whether each reaction is physically admissible for the support. This turns a negative result into model information instead of a sign to discard.
