Method of Sections
One cut, and one moment centre chosen so the two members you did not want vanish from the equation.
Skip to the animationCut the truss through the member you want, take one half as a free body, and choose the moment centre where the two members you do not want intersect — so they vanish from the equation and one line gives the answer.
The cut
Cut through the member of interest and take either side as a free body. That piece is a rigid body, so it obeys ΣH = 0, ΣV = 0 and ΣM = 0 — three equations. The cut may therefore expose at most three unknown members.
Either half gives the same answer, so take the side with fewer loads and reactions on it. The arithmetic is shorter for no loss of rigour.
The move that makes it worth doing
Any moment centre gives a valid equation. A careless one leaves several unknowns in it, forcing all three equations to be solved together — which works, and discards the entire advantage of sectioning.
Instead, take moments about the point where the two members you do not want intersect. A force through the moment centre has zero moment arm, so both vanish, leaving one equation in one unknown.
Choosing the moment centre well is the whole skill of this method. Everything else is ordinary statics.
Which equation for which member
| Wanted member | Use | Because |
|---|---|---|
| Top or bottom chord | ΣM about where the other chord meets the diagonal | Both unwanted members pass through that point |
| Diagonal, parallel chords | ΣV = 0 on the cut | Horizontal chords have no vertical component |
| Diagonal, non-parallel chords | ΣM about where the two chords meet | They intersect somewhere; that point is the centre |
| Vertical member | ΣV = 0, or ΣM about a chord intersection | Whichever isolates it |
Combining the two methods
Sections gives one member quickly; joints gives all of them. The practical combination is to section once to obtain three member forces in the middle of a long truss, then continue with joints outward from there — avoiding the propagation from a support entirely.
This is also how a spot check is done. If a computer analysis reports a member force, one cut and one moment equation will confirm it in a minute, which is a habit worth having.
The numbers you will be asked for
- Free body equations
ΣH = 0 · ΣV = 0 · ΣM = 0
Three, so the cut may expose at most three unknowns.
- Chord force
F = M / d
Moment about the opposite joint, over the truss depth.
- Diagonal, parallel chords
F = V / sin θ
From ΣV on the cut; θ is the diagonal's inclination.
- Cut rule
at most 3 unknown members
Otherwise the piece has more unknowns than equations.
Advantages and disadvantages
Advantages
- Reaches any member directly, without solving the truss up to it.
- One well-chosen moment equation gives the answer in a line.
- Ideal for spot-checking a computer result.
- Can seed a joints analysis in the middle of a long truss.
Disadvantages
- Only practical when the cut exposes three or fewer unknowns.
- Requires judgement in choosing the cut and the moment centre.
- Gives one or three member forces, not all of them.
- Awkward where the geometry offers no convenient intersection.
Watch it work
Check yourself
question 1 / 4
One question at a time. Pick an answer to see why it is right or wrong, then move on — there is no score to keep and nothing is saved.