One drawing solves the whole truss

Solving a truss one joint at a time works, and it has an odd property: every joint’s solution is discarded the moment the next joint is begun. Two equations are written, two member forces come out, and the pair of numbers is carried forward as a fact about the next joint while the working that produced them is thrown away.

Drawn rather than written, the working is not thrown away. It stays on the sheet, and the joints turn out to share it.

One joint, as a free body and as a polygon #

Take the left-hand support. Three forces act on it: the reaction, the bottom chord pulling to the right, and the end diagonal pushing back along its own line.

The algebraic route resolves those three into horizontal and vertical components and sets each sum to zero. The graphical route does something that is not a translation of it: it lays the three forces tip to tail in a chain, each one drawn to a scale of so many millimetres per kilonewton, each in its own true direction.

If the chain returns to its starting point, the forces sum to zero and the joint is in equilibrium. Closure is equilibrium. That single sentence replaces $\Sigma H = 0$ and $\Sigma V = 0$, and it replaces both at once rather than one after the other.

The construction runs the other way as well, which is what makes it a method and not an illustration. Draw the known force to scale. Draw a line in the direction of the first unknown member from the head of it, and a line in the direction of the second from the tail. Where they cross is the only point at which the chain can close, and the two segments it cuts off are the two forces. Nothing was solved; the intersection of two lines did the solving.

Two unknowns, and why the order is forced #

The construction above works because exactly two forces were unknown. Two unknown directions give two lines, two lines meet at one point, and one point gives one answer.

A joint with three unknowns gives three lines and no unique closure, and the drawing simply cannot be started there. That is the same restriction the algebraic method has — two equations cannot resolve three unknowns — arriving as a geometric fact rather than as a rank condition. Counting the unknowns before beginning tells which joints are available, and the sequence of solvable joints is identical whichever method is used.

So the drawing proceeds joint by joint in a forced order: the support, then the top node above it, then the next bottom node, and so on inward. Each step converts two unknowns into knowns and makes the next joint solvable.

Notice what the second polygon contains. One of its four sides is a side of the first, because the end diagonal is one member and its force is one number, appearing at both of the joints it connects. Drawn on separate sheets that shared edge is duplicated. Drawn on the same sheet it is drawn once.

The whole truss as one figure #

Extend that observation to every joint and the separate polygons collapse into a single drawing.

Bow’s notation is the bookkeeping that makes it possible. Letter the spaces rather than the members: the regions of the plane between adjacent external forces around the outside, and the triangular cells inside the truss. A member is then named by the two spaces it separates — the space above it and the space below — and it is named by the same pair from either end.

In the force diagram, each space becomes a point. A member’s force is the line between the two points naming it, its length is the magnitude to scale, and its direction is the member’s own direction. Because the member separates the same two spaces however it is approached, its line is drawn once and read by every joint that uses it.

The count is the point. This truss has twenty-one members and eleven joints. Six separate polygons of four or five sides each would be twenty-six drawn segments with most of them repeated; the reciprocal diagram has twenty-one lines and no repetition at all, because there are twenty-one members.

The two figures are called reciprocal because the relationship is symmetric: every line in one is perpendicular or parallel to a line in the other, every point in one corresponds to a polygon in the other, and either can be constructed from the other. Maxwell established that in 1864, Cremona gave the construction its modern form in 1872, and Bow supplied the lettering in 1873 — three people over nine years, for a result that a draughtsman could then use in an afternoon.

Which free body produced the number #

Every polygon on this page is a free body, and the free body is one joint: a small disc cut out of the structure with every member severed just outside it.

That cut is what makes the method work, and it is worth being precise about why. Cutting a joint out leaves only concurrent forces — every member force passes through the joint’s own point, because a pin transmits no moment and a two-force member’s line of action is its own axis. Concurrent forces have no moment equation to satisfy, so the three planar equilibrium equations reduce to two.

Two equations is exactly what a closed polygon asserts: the horizontal components sum to zero and the vertical components sum to zero, which are the two coordinates of the statement the chain returns to its start. The polygon is not a picture of the equations; it is the equations, drawn in the plane they live in, which is why three forces on a body must meet at a point and why that fact and this construction are the same fact.

What the joint free body cannot do is anything requiring a moment. It cannot find the force in a single member without solving everything before it, which is the gap the method of sections exists to fill and which no amount of drawing supplies.

The load line comes first, and everything hangs off it #

There is a step before the first joint that is easy to miss and is half the reason the whole diagram fits together.

Go round the outside of the truss and lay every external force — the five panel loads and the two reactions — tip to tail in the order they are met. Because they are all vertical here, the chain is a single straight segment: 12 down, five times, with 30 up at each end. It closes, because the truss as a whole is in equilibrium, and it is called the load line.

Every point on that line is a space around the outside of the structure, and every internal point of the finished diagram is located from it. The construction is therefore not a sequence of independent polygons that happen to share edges; it is one figure whose boundary is fixed before any member is considered, with the interior filled in joint by joint.

That has a practical consequence worth stating. The load line is drawn once and does not change when the truss’s interior does. A designer comparing a Pratt against a Howe against a Warren for the same loading redraws only the inside of the diagram, which is a fair description of what is actually being compared: the same external problem, three arrangements of the same total force.

It also explains the one thing about the construction that looks arbitrary. The order of the external forces around the outside must be the order they occur around the structure, not any convenient order, because that ordering is what makes each internal space adjacent to the right pair of boundary points. Get it wrong and the polygons still close individually while the figure as a whole is nonsense — the single failure mode of the method that does not announce itself.

What the shape of the diagram says about the truss #

Once the figure is complete it can be read as an object in its own right, and its geometry is a summary of the structure’s behaviour that no table of member forces provides.

The chords of this truss are parallel, so every top chord force is horizontal and every bottom chord force is horizontal, and in the force diagram all of those lines are horizontal too. The chord points therefore lie on two horizontal rows, and the vertical distance between the rows is the same for every panel — which is the graphical statement that the couple carried by the two chords is the moment at that section divided by the depth, a result that ordinarily takes a paragraph of algebra.

The diagonals slope, and their lines run between the two rows. Their lengths grow toward the supports and shrink toward mid-span, which is the shear diagram appearing in the force figure without anyone having drawn one. The reciprocal diagram of a parallel-chord truss is a picture of its shear and moment diagrams superimposed, and reading it that way is how a nineteenth-century designer chose a depth.

A truss with sloping chords produces a diagram in which the chord points are not in rows, and the departure from horizontal is exactly the amount of shear the chords themselves carry — the effect that makes a pitched truss’s web members lighter than a parallel one’s and which is otherwise a separate calculation.

Tension and compression, read off rather than deduced #

An algebraic solution returns a sign, and the sign has to be interpreted against a convention adopted at the start. The drawing returns a direction, and the direction is read directly.

Go round the joint clockwise in the space diagram, naming the spaces in the order they are met. Each member’s force is then read in the force diagram in the corresponding order — from the first-named point toward the second. If that direction points away from the joint, the member is pulling and is in tension; if it points toward the joint, the member is pushing and is in compression.

The same rule read from the joint at the other end of a member gives the opposite travel direction and therefore the opposite-looking arrow, which is correct: a member in tension pulls both its joints toward itself. That is a place where the drawing is clearer than the algebra, in which the same member force appears in two sets of equations with signs that depend on which node the equations were written at.

The closure is a check that algebra does not have #

The property that made the method survive its own obsolescence is what happens when something is wrong.

An error in a hand calculation produces a wrong number that looks exactly like a right one. An error in this construction produces a visible gap, whose size is the residual and whose direction says what is missing. The check is not an extra step; it is the same step.

That property compounds. Because the diagram is one figure rather than six, an error made early does not stay local — the whole construction fails to close on itself when the last joint is drawn, and the accumulated closure of the complete figure is a check on every member force in it at once. A truss is a structure whose analysis has an internal consistency condition, and this is the only method that displays it.

What it cannot do #

Three limitations follow from the construction rather than from the drawing.

It needs a joint with two unknowns to begin. A complex truss — one that is statically determinate by the count but has no joint anywhere with fewer than three unknown members — cannot be started. The classical answer is Henneberg’s method: substitute a member, solve, and superpose a correcting self-stress state, which is algebra rather than drawing.

It is planar. The whole construction rests on two lines meeting at a point, and in space three planes meet at a point instead. A space frame has three equations at every joint and the graphical equivalent needs two projections drawn together, which is possible and is not simple.

It gives forces and not deflections. Nothing in the reciprocal diagram knows about member areas or the modulus, so the deflection of a truss needs a separate construction entirely — the Williot-Mohr diagram, which is another reciprocal figure with displacements in place of forces and which is the reason the two subjects were taught together for a century.

Why it stopped being used, and where it did not #

The method’s decline had nothing to do with correctness. A drawing’s precision is the precision of the paper, roughly three significant figures at a sensible scale, and that was ample when the material properties were known to two. What ended it was that a computer does not get tired at the tenth joint, and that the same matrix that solves a determinate truss solves an indeterminate one with no change of method, which this construction cannot do at all.

Where it survived is where the answer wanted is a shape rather than a number. The funicular polygon that finds a cable’s form is the same reciprocal relationship with the truss replaced by a chain, and it is still the fastest way to see what a hanging form does when a load moves. The line of thrust in a masonry arch is a funicular polygon that has to stay inside a boundary, which is a question about a curve’s position and is barely a numerical question at all.

The common feature is that the unknown is a geometry. A method whose output is a picture is well matched to a question whose answer is a picture, and badly matched to one whose answer is a table — which is a reasonable summary of why graphic statics is currently taught to architects and not to engineers.

Where the model stops #

Every joint is a frictionless pin. The construction has no way to represent a moment at a joint, so it computes the idealised forces and not the ones a welded frame actually carries — the secondary bending a rigid joint adds is invisible to it by construction.

Every load arrives at a panel point. A load applied between joints puts bending into the chord, which is not a member force and has no line in the diagram.

The geometry is the undeformed one. Forces are read off the drawn shape, and the drawn shape is the one before loading. That is the standard first-order assumption and it fails for a slender compression chord at exactly the moment it matters.

And the scale is a decision with consequences. A force diagram drawn at a scale that fits the largest force on the sheet reads the smallest ones badly, and the two zero-force members in this truss are read as points rather than as lines — which is correct, and is also how a member carrying a small but real force gets recorded as carrying none.

The ladder from here #

Later rungs on this anchor: subdividing a panel, and what the shorter compression member is worth against the joints it costs. The chord that runs through several panels, which is a continuous beam whatever its joints are. Counters and tension-only diagonals, where a member is absent under one load case and present under the other. The counting rule in three dimensions, where m + r = 2j becomes 3j and a mechanism can hide inside a satisfied count. And the Williot-Mohr diagram, which is this construction’s other half and answers the question the force diagram cannot.

The reciprocal diagram has had an unusual afterlife. Maxwell’s 1864 paper was about the geometry of reciprocal figures and mentions frameworks almost in passing; the construction became a design tool in the hands of Cremona and Culmann, was the standard method for forty years, and was then displaced so completely that a generation of engineers met it only as a historical note. It returned in the 2000s in computational form, because the reciprocal relationship turns out to be exactly the right structure for form-finding — a problem in which the forces are chosen and the geometry is the unknown, which is the direction Maxwell’s theorem was always symmetric in.