Structural Engineering & Statics
where the load goes
Equilibrium

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

Cutting a structure open is not a step in the method. It is the method — and where the cut is made decides whether the answer takes one line or twenty.
10 min read 7 figures The statics of things that do not moveThe load must go somewhere

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.

The same beam, cut at x = 5A beam separated at one station. On the exposed face a shear force and a bending moment appear, equal and opposite on the two pieces, with values obtained by summing the forces on whichever piece is easier.2012.57.5shear 7.5moment 22.5the cut, at x = 5nothing was applied here — the internal forces are what the left-hand piece needs
Fig. 1 A beam separated at one station. On the exposed face a shear force and a bending moment appear — nothing was applied there, and both are equal and opposite on the two pieces.

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.

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.

The same beam, cut at x = 2A beam separated at one station. On the exposed face a shear force and a bending moment appear, equal and opposite on the two pieces, with values obtained by summing the forces on whichever piece is easier.2012.57.5shear 12.5moment 25.0the cut, at x = 2nothing was applied here — the internal forces are what the left-hand piece needs
Fig. 2 The same beam cut on the other side of the load. The shear has changed sign and the moment is smaller, because the free body to the left now contains different things — and the structure has not changed at all.

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.

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.

Joint 0 of the truss, cut outOne joint of the truss with every force acting on it. Two equations — the horizontal and vertical sums — are enough for a joint with no more than two unknown member forces, which is the whole method.HV29.4-38.6reaction 0.0reaction 25.0ΣH = 0 and ΣV = 0, and nothing else is needed
Fig. 3 One joint of a truss, cut out, with every force acting on it. Two equations are available, so a joint with no more than two unknown member forces can be solved immediately.

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.

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.

Joint 1 of the truss, cut outOne joint of the truss with every force acting on it. Two equations — the horizontal and vertical sums — are enough for a joint with no more than two unknown member forces, which is the whole method.HV29.429.40.0ΣH = 0 and ΣV = 0, and nothing else is needed
Fig. 4 A joint in the bottom chord, away from the support. Four members meet here and there is no applied load, and the two equations available still settle it because two of the four forces are already known from the joint before.

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.

A beam, its loads and its reactionsA free body: the beam cut away from everything it touches, with the forces that were doing the touching drawn on it. The reactions are computed from the loads, so their arrows are to scale relative to each other.1289.510.5ΣM about one support gives the other reaction; ΣF then gives the first
Fig. 5 The whole beam as a free body. Every internal force has vanished from the picture, because internal forces occur in equal and opposite pairs and cancel within any body they are internal to.

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.

A Pratt truss of 6 panelsA Pratt truss under equal panel-point loads. The joint equilibrium equations were assembled and solved; 10 members came out in tension, 9 in compression and 2 carrying nothing.tensioncompression2 carrying nothing
Fig. 6 A loaded truss. Cutting it through three members exposes three unknowns, and three equations settle them — without touching any of the other eighteen.

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.

Load, shear and moment — a simple spanThe applied load, the shear force it produces and the bending moment that follows, drawn one above another to the same horizontal scale. Shear is the integral of the load and moment is the integral of shear.4 per unit lengthshear16.0moment32.0 at x = 4.00the moment peaks exactly where the shear passes through zero
Fig. 7 The shear and moment along a beam. Every ordinate on these curves is a separate free body, cut at that station and summed — the diagram is the technique applied continuously.

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.

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 distribution of stress: three numbers standing in for a distribution of stress across the whole 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.

Where this goes next

1289.510.5ΣM about one support gives the other reaction; ΣF then gives the first Everything adds to nothing, and that is the whole of statics A structure that stays put obeys two statements — the forces on it sum to zero, and so do the moments. Every number in the subject comes out of those two sentences. 2012.57.5shear 7.5moment 22.5the cut, at x = 5nothing was applied here — the internal forces are what the left-hand piece needs What a cut reveals, and why it was there all along Cut a beam anywhere and two quantities appear on the face — a shear force and a bending moment. Nothing was applied there. They are what the material was already doing. the loadroller: vertical onlypin: any directionall three lines meet here Three forces must meet at a point, and a drawing can find it A body held by exactly three forces has their lines of action concurrent. That is a theorem, it is enough to solve for direction and magnitude, and for a century it was done with a straightedge.