Structural Engineering & Statics
where the load goes
Equilibrium

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.
9 min read 7 figures Drawing as calculationThe statics of things that do not move

A body acted on by exactly three forces, in equilibrium, has those three lines of action passing through a single point. Always. It is not an approximation and not a special case, and it means that knowing two of the three directions fixes the third by drawing rather than by algebra.

That theorem, and the closure of the force polygon that goes with it, made a drawing board into a calculating instrument. Bridges, roofs and cathedral vaults were analysed with a straightedge and a scale rule for most of the nineteenth century, and the drawings were not illustrations of the answers — they were the answers.

Three forces must meet at a pointA body held by two supports and one load. The reaction at the roller is vertical and the load's direction is given, so their lines of action fix a meeting point — and the pin reaction has to point at it.the loadroller: vertical onlypin: any directionall three lines meet here
Fig. 1 A body held by two supports and one load. The roller reaction must be vertical and the load’s direction is given, so those two lines of action fix a meeting point — and the pin reaction has to point at it.

Why concurrency is forced

The proof takes two lines and is worth having, because it explains why the theorem is about three and not four.

Two of the forces have lines of action that meet somewhere; call that point P. Taking moments about P, both of those forces contribute nothing, since a force through a point has no moment about it. The moment equation then says the third force’s moment about P must also vanish. A non-zero force has zero moment about a point only if its line of action passes through it.

So the third line goes through P too. The argument used nothing but the moment equation and the fact that there were exactly three forces — a fourth force would have somewhere else to put its moment, and the theorem evaporates.

The exception is the parallel case. If the first two lines are parallel they meet nowhere, and the third must be parallel to them as well, which is concurrency at infinity. A beam with two vertical reactions and one vertical load is exactly this case, and it is the reason the theorem looks useless for beams and is decisive for frames.

The polygon that has to close

The moment equation gives the direction of the third force. The force equation gives its magnitude, and drawn rather than written it becomes a closure condition.

A closed force polygonThe forces on a joint, laid tip to tail. Equilibrium is the statement that the polygon closes, and the gap when it does not is the out-of-balance force, to scale.load 60strut 84.9tie 60starts and ends here
Fig. 2 The forces on a joint laid tip to tail. Equilibrium is the statement that the polygon closes, and the closure is checkable with dividers.

Laying the forces head to tail, the resultant is the vector from the start of the first to the end of the last. Equilibrium means the resultant is zero, which means the last arrowhead lands exactly on the first tail.

A force polygon that does not closeThe forces on a joint, laid tip to tail. Equilibrium is the statement that the polygon closes, and the gap when it does not is the out-of-balance force, to scale.load 60strut 84.9tie 60out of balance: 19.2
Fig. 3 The same construction with one force too small. The gap that opens is the out-of-balance force, at the same scale as everything else in the drawing.

The gap is a measurement, not a symptom. Its length is the resultant and its direction is the direction the joint would accelerate. A drawing that failed to close told a nineteenth-century engineer both that something was wrong and exactly what was missing — which is more than a set of simultaneous equations does when a sign has been mistyped.

The two conditions together are complete. Concurrency handles rotation and closure handles translation, so a three-force body is fully solved by a drawing with no numbers in it beyond a scale.

What the drawing could do that algebra could not

Graphic statics was not a poor substitute for calculation. For several classes of problem it was strictly better, and the reasons are worth listing because they are not obvious now.

It scaled to many members. A Cremona diagram solves an entire truss by nesting all the joint polygons into one figure, with each member’s force appearing exactly once as a line in it. A fifty-member truss is a page of drawing; the same problem by hand algebra is a hundred simultaneous equations.

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. 4 One joint’s forces. In a Cremona diagram the polygon for each joint shares its sides with its neighbours, so the whole truss becomes a single tiling of triangles and every member length is read off once.

It made errors visible. An arithmetic slip produces a plausible wrong number. A drafting slip produces a polygon that does not close, and the size of the gap is the size of the mistake.

It solved for shape. The hardest structural questions are not “what force is in this member” but “what shape should this be”. The funicular polygon answers the second directly: hang the loads from a string, and the shape it takes is the shape that carries them in pure tension. Inverted, it is the arch that carries them in pure compression. Gaudí built the Colònia Güell chapel from a hanging model of strings and weights, photographed upside down.

It handled the indeterminate by approximation. A thrust line drawn inside the middle third of a masonry arch is a proof that a set of forces exists in equilibrium within the material — which, for a material with no tensile strength, is enough to say the arch stands. That is a lower-bound argument, and it is still the basis of masonry assessment.

Reading a real case

The figure at the top is the standard configuration: a body on a pin and a roller, with one inclined load.

The roller can only push perpendicular to its surface, so its line of action is known before anything is calculated. The load’s line of action is given. Extend both, mark the meeting point, and draw the line from the pin to it — that is the pin reaction’s direction, obtained without arithmetic.

Three forces must meet at a pointA body held by two supports and one load. The reaction at the roller is vertical and the load's direction is given, so their lines of action fix a meeting point — and the pin reaction has to point at it.the loadroller: vertical onlypin: any directionall three lines meet here
Fig. 5 The same body with the load leaning further over. The meeting point has moved and the pin reaction has swung with it, which is a statement about direction that no force equation on its own would give.

Then the polygon: draw the load to scale, draw a line through its tip parallel to the roller’s direction and a line through its tail parallel to the pin’s, and the triangle closes at their intersection. The two sides of the triangle are the two reactions, measured off with the same scale.

Two drawings, no equations, and the same answer the algebra gives.

The shape a drawing can find

The most useful thing graphic statics did was not solving frames but finding forms.

The cable and the arch are the same curveThe shape a cable takes under a uniform load is a parabola, and it carries that load in pure tension. Reflected, the identical curve carries the same load in pure compression, which is what an arch is.cable: pure tensionarch: pure compression
Fig. 6 A cable under a uniform load and its mirror image. The cable finds the shape that carries the load in pure tension, and inverted the same curve carries it in pure compression.

That inversion is Hooke’s principle, and it was used as a design method rather than an illustration — a hanging model of strings and weights, photographed and turned upside down, gives the elevation of an arch that has no bending in it anywhere.

A Warren truss of 6 panelsA Warren truss under equal panel-point loads. The joint equilibrium equations were assembled and solved; 10 members came out in tension, 11 in compression and 2 carrying nothing.tensioncompression2 carrying nothing
Fig. 7 A truss, which is the same idea discretised. Each panel is a triangle whose force polygon closes, and a nineteenth-century engineer solved the whole frame by nesting those polygons into one figure.

The connection between the two is worth stating: a truss is a funicular polygon with the members made rigid so that it can carry more than one load case, and the price of that generality is bending in the members whenever the load is not the one the shape was chosen for.

Where the model stops

Exactly three forces. With four, concurrency fails and the drawing loses its determining power. Frames are handled by cutting them into three-force pieces, which is the free-body technique again.

Rigid bodies and small deflections. The lines of action are drawn on the undeformed shape. Where a structure moves enough to change them, the geometry is being solved on a picture that no longer applies, and the load starts amplifying itself.

Determinate structures. A drawing is a statement of equilibrium and nothing else, so it can no more resolve a redundant frame than the equations can.

Drafting precision. The method’s accuracy is the accuracy of the drawing, which for a member sized by deflection was always ample and for a modern check is not — perhaps a part in two hundred on a large sheet. That was ample for a nineteenth-century truss and is not for a modern one, and it is the honest reason the technique was abandoned.

The figures have a specific limitation worth naming: the force polygon is drawn at a scale chosen to fit the canvas, and the scale is not marked. A real graphic-statics drawing carried an explicit force scale beside it, without which the polygon is a shape rather than a measurement. Every figure on this page shows the geometry and hides the calibration.

The ladder from here

Later rungs on this anchor: the Cremona or Maxwell diagram, and the reciprocal figure behind it. Bow’s notation. The funicular polygon and the pole. The thrust line in masonry, and the middle-third rule. Culmann’s graphical method for sections. Gaudí’s hanging models and their photographic inversion. Graphic statics for indeterminate problems, and where the approximation lies. And the modern revival of the reciprocal diagram in computational form design, which put the whole technique back to work with a computer holding the straightedge.

Culmann published Die graphische Statik in 1866 and the method was standard within fifteen years. It was taught in engineering schools until the 1960s and dropped almost everywhere within a decade of the pocket calculator.

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. cable: pure tensionarch: pure compression The shape that carries itself, and the arch that is its reflection Hang a chain and it takes the one shape that carries its load in pure tension. Turn the shape upside down and it carries the same load in pure compression. That is what an arch is. 2012.57.5shear 7.5moment 22.5the cut, at x = 5nothing was applied here — the internal forces are what the left-hand piece needs 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.