The free body is a choice, and choosing it well is the whole skill

Every structural calculation begins with an act of violence. Something is cut away from everything it touches, and the forces that were doing the touching are drawn on the cut.

The cut is imaginary and the forces are not. That combination is what makes the free-body diagram strange, and it is worth being precise about, because the technique is usually presented as a step and is actually the entire subject.

What a cut creates #

Before the cut, the beam has a load and two reactions. After the cut, the left-hand piece has a load, one reaction, and two new quantities on its right-hand face: a shear force and a bending moment.

Those two did not exist as separate objects a moment earlier. They are what the material at that station was doing, made visible by removing everything on the other side of it and asking what has to replace it.

The logic runs backwards from the answer. The left-hand piece is not moving. Therefore the sums on it must vanish. Therefore whatever the right-hand piece was providing must have been exactly enough to make them vanish — and that is the definition of the internal forces, not a calculation of them.

Newton’s third law then guarantees consistency: the same actions appear on the right-hand piece with opposite signs. Either piece gives the same answer, so the choice between them is purely one of convenience. Cutting to the left of a complicated set of loads and working with the simple side is not a shortcut; it is the same calculation done with fewer terms.

That guarantee is also the source of the one piece of bookkeeping the subject cannot avoid. If the actions on the two faces are equal and opposite as vectors, then the same physical event — this station of the beam sagging — is described by a clockwise moment from one side and an anticlockwise moment from the other. A convention that took its sign from the vector direction would therefore give the same beam two different moment diagrams depending on which way the analyst happened to work, and the diagram would change sign in the middle if two people started from opposite ends.

Move the cut close to a support and both statements can be read off a drawing at once. The internal forces there are not small because the beam is doing little; they are what that particular free body needs, and it is a very short free body.

So the structural convention is not a vector convention at all. It is a statement about what the material is doing: a moment is positive when it puts the bottom of the section in tension, whichever face is being looked at, and a shear is positive when the left-hand piece is being pushed up relative to the right. Both are defined on the deformation rather than on an axis, which is exactly why they come out the same from either free body — and why the sign of a bending moment survives turning the drawing round, while the sign of a force component does not.

Where to cut #

The working rule is short: cut through what is wanted, and through as little else as possible.

A cut that passes through three unknown member forces in a plane frame leaves three unknowns and three equations, which is solvable but awkward. A cut through two leaves a choice of moment centre that kills one of them, and the remaining unknown falls out in a line.

Comparing the two cuts makes the point that the internal forces are not properties of a location in isolation. They are properties of a location given everything to one side of it, and moving the cut past a load changes what that everything contains.

The rule about cutting through as little as possible has a second half, which is that either piece will do and one of them is usually much shorter. Cut near the far support and the sensible free body is the half-metre stub on the right.

That is the entire economics of the technique. The two pieces give identical answers, so the choice between them costs nothing and buys the difference between one term and three — and on a real structure, between one term and a page.

That is why the shear diagram jumps at a point load and the moment diagram merely kinks. The free body gains a whole force at once, so the force sum steps; it gains that force at zero distance from the cut, so the moment sum is continuous.

The joint as a free body #

The same technique applied to a single joint of a truss gives the method of joints, and the free body is a point rather than a piece.

A joint is the smallest useful free body and the most restricted: with all forces passing through one point, the moment equation is satisfied automatically and only two equations survive. That limitation is the reason the method has an order — start at a support where only two members meet, solve it, and use the answers as knowns at the next joint. A joint further along the bottom chord may have four members meeting at it and no applied load, and the two equations still settle it, because two of the four forces arrived as answers from the joint before.

The whole-truss solver behind the truss figures does the same thing without the ordering, by writing all the joint equations at once and solving the system. The answers are identical; what disappears is the need to find a joint simple enough to start at — which is a convenience for a computer and a loss for a reader, because the ordering is where the structure’s own logic shows.

The boundary decides what is external #

A force is external or internal depending on where the boundary was drawn, and nothing else.

Take the whole beam as the free body and the shear at mid-span does not appear anywhere in the equations — it is internal, and internal forces cancel in pairs. Take half the beam and the same quantity is external, drawn on the cut face, and it is the unknown being solved for.

This is the deepest thing about the technique and the easiest to lose. The question “is this force internal or external?” has no answer until the boundary is stated. A column’s axial force is internal to the building and external to the column. The reaction at a foundation is external to the structure and internal to the structure-plus-ground.

Drawing the boundary badly usually shows up as a diagram with forces that cannot be determined. Drawing it well means the boundary crosses exactly the things that are wanted and nothing else.

What has to be drawn on it #

A free-body diagram is complete when four things are on it, and a diagram missing any of them will give an answer that is wrong without looking wrong.

Every applied load, including self-weight where it matters.

Every reaction the supports can supply, and only those. A roller drawn with a horizontal component invents a force; a pin drawn without one loses an equation.

Every internal action on every cut face — for a beam that is three quantities, since a cut can transmit axial force as well as shear and moment.

Nothing else. The commonest error is drawing a force that is already accounted for, usually by including both a load and the reaction it produces in a body that contains only one of them.

The diagram is then a complete statement of the problem, and the arithmetic that follows is mechanical. Nearly every error in statics is an error in the diagram rather than in the algebra after it, which is why the diagram is worth drawing even when the answer seems obvious.

Cutting a whole structure open #

The same technique applied to a frame rather than a beam is the method of sections, and it answers one question without solving the rest.

Choosing the cut is the skill. A cut through three members, with moments taken about the point where two of them meet, gives the third in one line. That is why the method of joints is used to solve a whole truss and the method of sections to interrogate one member of it.

Read that way, a shear diagram is not a new idea at all. It is the free-body calculation performed at every station and plotted, and its shape records what each successive free body contained. The five cuts drawn on this page are five ordinates of it: 6.2 at half a metre, 25.0 at two, 37.5 at the load, 22.5 at five and 3.8 at seven and a half. Join them and the moment diagram is what appears, without a single idea having been added to the one in the first figure.

One cut, three answers, three moment centres #

The claim that a good cut turns twenty lines into one is worth demonstrating on the truss above, with the numbers the solver returns.

Cut the six-panel truss vertically through the third panel. Three members are severed — the top chord, the bottom chord and the diagonal between them — and the left-hand piece carries the support reaction of $25$ and two panel loads of $10$ each, at one and two panels in.

For the top chord, take moments about the bottom joint where the other two cut members meet, at three panels from the support. The chord and the diagonal both pass through it and vanish. What remains is

$$F_{\text{top}} \times 0.85 = -\left(25\times3 - 10\times2 - 10\times1\right) = -45,$$

so $F_{\text{top}} = -52.94$: compression, and identical to the solver’s value.

For the bottom chord, move the moment centre to the top joint at two panels in, where the top chord and the diagonal meet. Now those two vanish instead:

$$F_{\text{bot}} \times 0.85 = 25\times2 - 10\times1 = 40,$$

giving $+47.06$: tension, again matching.

For the diagonal, no moment centre is needed at all — the two chords are horizontal, so vertical equilibrium of the whole left-hand piece contains only the diagonal. The shear passing the cut is $25 - 10 - 10 = 5$, and the diagonal’s vertical component is its force times $0.6477$, giving $7.72$. Matching once more.

Three unknowns, three equations, and each one solved on its own by choosing where to take moments. Nothing was iterated, no other member was touched, and the eighteen members outside the cut were never mentioned. The same three answers by the method of joints would require working inward from the support through five joints in the correct order.

That is the whole argument for choosing the free body deliberately. The equations available are always the same three; what a good cut does is arrange for two of the three unknowns to pass through the point moments are taken about, so that each equation contains one unknown rather than three. The skill is not in the algebra — it is in noticing where the lines of action cross.

The boundary can include the ground #

Every free body so far has been a piece of the structure. Nothing requires that, and enlarging the boundary is often the fastest route to an answer.

Consider whether a retaining wall or a tall crane base will overturn. The usual approach cuts at the underside of the base and draws the foundation reaction, which is an unknown distribution requiring an assumption about how the pressure varies. The alternative is to draw the boundary below the ground, taking the wall and a wedge of soil as one body. The bearing pressure is now internal and disappears entirely, and the equation is a comparison of two moments about the toe: the overturning moment from the retained material against the restoring moment from the weight of everything inside the boundary.

The same move settles a question that confuses people about buildings: what holds down a structure being pushed sideways by wind? Cut at ground level and the answer is a set of foundation reactions requiring analysis. Take the building and its foundations and the soil they sit in as the body, and the answer is the weight of the enlarged body, which is why an overturning check is essentially a weighing exercise and why lightweight buildings need holding-down piles that heavy ones do not.

The general principle is that enlarging the free body makes forces disappear, and shrinking it makes them appear. A quantity that is hard to determine can often be made internal by drawing the boundary further out — and a quantity that is wanted must be made external by drawing the boundary through it. Those two sentences are the whole strategy, and everything else is bookkeeping.

The limit on enlargement is that the enlarged body has to be genuinely in equilibrium, which means being honest about what is inside it. A soil wedge included in the body must be one that will actually move with the wall if it moves at all — the choice of wedge is a structural assumption about a failure mechanism, not a free choice, and choosing it optimistically is the standard way to prove that something unsafe is safe.

The free body that is not still #

Every equation on this site sets a sum to zero on the grounds that nothing is moving. A great deal of what structural engineers analyse is moving, and the technique survives with one addition.

For a body accelerating at $a$, Newton gives $\Sigma F = ma$ rather than zero. Moving the right-hand side across,

$$\Sigma F - ma = 0,$$

and the term $-ma$ can be drawn on the free-body diagram as though it were an applied force. That is d’Alembert’s principle, and its effect is to convert every dynamics problem into a statics problem with one extra arrow on the picture.

The arrow is not a real force — nothing is pushing — and it behaves exactly like one for the purpose of summing. A lift accelerating upward at $a$ has a cable tension of $m(g+a)$, which is the static answer with an inertia force added. A crane slewing a load has a horizontal inertia force on the hook. A vehicle braking on a bridge deck applies a longitudinal force that the bearings and abutments have to carry, and it appears in the analysis as a load case with no source.

The most consequential instance is seismic. The ground moves; the building’s mass resists being moved; and the resulting inertia forces are what shake the structure apart. The equivalent static force method — still the basis of most low-rise seismic design — is exactly this substitution: compute the accelerations the building will experience, multiply by the mass at each floor, apply the results as horizontal loads, and then do ordinary statics. The whole apparatus in this essay applies unchanged.

Which is a good note on which to end the technique’s claims. A free-body diagram is not a statement about a structure being still. It is a statement about a boundary and a complete accounting of everything crossing it, and if one of the things crossing it is an acceleration, the accounting still works.

Where the model stops #

Rigid bodies. The cut is made on the undeformed shape, and the pieces are assumed not to change shape as the forces are applied. That is a first-order assumption, and it fails when deflections are large enough to move the lines of action.

Statics only. The technique gives the internal forces of a determinate structure completely. For a structure with a redundant restraint it gives a family of answers, and picking among them needs the stiffness of the members — information the free body does not contain.

No local effects. The forces on a cut face are drawn as a shear, a moment and an axial force — three numbers standing in for a whole distribution of stress across the face. Saint-Venant’s principle says the substitution is safe a short distance away from where the load is applied, and unsafe close to it. Bearing failures, bolt tear-out and web crippling all live in the region where three numbers are not enough.

Two dimensions. A plane free body has three equations. A real one has six, and the three that were dropped are assumed to be looked after by something outside the picture.

The figures have a limit too, and it is one this site cannot get round. A cut face is drawn with a gap so the internal actions can be seen, and there is no gap — the material is continuous, and the shear and moment are distributions of stress rather than an arrow and a curved arrow. Every free-body diagram ever drawn commits this, and it is a good habit to remember that the arrow is a summary of something spread across an area.

The ladder from here #

Later rungs on this anchor: the method of sections, and answering one question without solving the whole truss. Free bodies with internal releases. Sign conventions, and why the sagging-positive convention exists at all. Saint-Venant’s principle. Three-dimensional free bodies. Superposition, which is a statement about free bodies added together. The free body in a moving frame, where an inertia force is added and statics is recovered. And the free body of a piece of material rather than a piece of structure, which is where stress analysis begins.

Newton drew free bodies without naming them. The phrase and the teaching convention are twentieth-century, and the practice of insisting on the diagram before the algebra is younger still than that.