Three forces must meet at a point, and a drawing can find it

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.

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.

Run the same argument with two forces instead of three and it gives a result that is used far more often than the theorem itself. A body acted on by exactly two forces has them equal, opposite, and collinear — take moments about any point on one line of action, and the other force’s moment must vanish, so its line passes through that point too; do it again at a second point and the two lines coincide. There is no freedom left.

That is the whole justification for the pin-jointed member. A truss member with loads applied only at its two ends is a two-force body, so whatever it carries must act along the line joining the pins — which is why a member’s direction is known before its force is, why every member force in a truss is axial rather than a force with a bending component, and why a straight member is the efficient shape for one. It is also the reason the three-force construction is drawable at all: in the figure above, the strut’s line of action is available for extending only because it is a two-force member and therefore has a direction fixed by its own geometry.

The theorem says nothing about the member’s shape, and that is worth drawing rather than asserting. The force runs along the chord between the pins whether the member is straight, bowed or bent double, because the two end forces are the only forces there are and equilibrium leaves them no choice. What changes with the shape is not the force but where it acts relative to the material.

That is a large moment obtained from no load case at all, and it is the reason a strut is drawn straight and then checked for how straight it actually is. The cost of the bow can be stated as a pure ratio, because both stresses come from the same force: the bending stress is $Py/Z$ and the axial is $P/A$, so the force cancels and what is left is the offset measured against the section’s own depth.

The steepness at the left-hand end is the practical content. A bow of a tenth of the section depth is already six per cent, which is why fabrication tolerances on struts are written as a fraction of the length and why a member delivered with a visible kink is rejected rather than recalculated.

The corollary is the one that catches people out. A member loaded anywhere between its ends is not a two-force body, so its force is no longer along its axis and it is bending for a second reason on top of the first. Hanging a service off the middle of a truss diagonal, or loading a chord between panel points, breaks the assumption that made the whole drawing possible — and which body was cut in the first place is what decides whether the assumption was ever available.

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.

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.

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. The polygon for each joint shares its sides with its neighbours, so the whole frame becomes a single tiling of triangles and every member force is measured once rather than recomputed at both of its ends.

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.

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.

Hang a cable and load it, and it takes the one shape that carries that load without any bending in it at all. Invert the curve and it carries the same load 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.

Every segment of that hanging cable is a two-force member, which is why the construction of the previous section is the whole apparatus: a chain of pins with load only at the pins, each link carrying force along its own length. The funicular polygon is the two-force theorem applied as many times as there are links.

The connection to the frame 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. Each panel is a triangle whose own force polygon closes, and the rigidity is what lets the shape stay put when the load stops matching it.

The free body that makes it useful: a wall #

The theorem earns its keep on bodies that are not beams, and the clearest of them is a retaining wall or a masonry dam, where concurrency locates something no equation is asked for directly.

Draw the free body: the wall, cut from the soil behind it and from the ground beneath it. Exactly three forces act. Its own weight, acting vertically through its centre of gravity. The horizontal thrust of the retained material, acting through the centroid of a triangular pressure distribution, which for a wall of height $h$ is a third of the way up. And the reaction of the foundation, which is whatever the ground supplies.

Two of those three are fully known — magnitude, direction and line of action. The theorem then says the third passes through the point where the first two cross, and following that line down to the base gives something a designer very much wants: not the size of the foundation reaction but where on the base it acts.

Laying those three tip to tail gives a triangle whose closing side is the foundation reaction, so its inclination is read off the drawing rather than assumed: a horizontal thrust of 90 against a weight of 100 closes with a reaction of 134.5 leaning 42 degrees from the vertical. That number is the one the sliding check wants, and it arrives from the same construction that located the line.

That position is the whole design problem. A rectangular base can push but cannot pull, so if the reaction lands outside the middle third of the base width, part of the base would have to be in tension to keep the resultant in equilibrium and instead the wall simply lifts off along that edge. The middle-third rule — the eccentricity must not exceed a sixth of the base width — is therefore not a rule about stress at all. It is the concurrency theorem plus the observation that soil and masonry do not pull.

The same free body answers two more questions with no extra work. Whether the wall slides is whether the drawn reaction leans further from the vertical than friction allows. Whether it overturns is whether the reaction’s line of action has walked off the toe entirely. Three forces, one meeting point, three failure modes located on one drawing — which is why the technique survived in retaining-wall and masonry work for decades after it was abandoned for frames.

The three-force body a rigger draws #

The theorem is still in daily use in one place, and it is not a drawing office. A load lifted on two slings from a single hook is a body held by exactly three forces — the weight down, and one tension along each leg — and they are concurrent because they all pass through the hook.

Two of the three directions are given by the rigging geometry and the third by gravity, so the triangle is fully determined and closes at one point. What it says is not intuitive, and the reason is the same one that made the polygon worth drawing: the closing condition is about directions, and the vertical component of a leg is the only part of it doing the lifting.

At sixty degrees the arrangement looks generous: each leg carries a little over half the load and the compression through the object is modest. The curve on the right is the whole of what happens as the legs are let out, and it is not linear anywhere near the working range.

Thirty degrees is the number worth carrying, because it is where a sling carries the whole of the load it is sharing. Below it the growth is violent: at fifteen degrees each leg is at 1.93 times the weight. The rule about minimum sling angles is therefore not a convention but a reading off a force triangle, and the spreader beam that appears on every awkward lift exists to take the horizontal component as a designed strut rather than as compression through something that was never checked for it.

It is the same construction as the retaining wall and the same construction as the pin-and-roller frame at the top of the page. Three forces, one meeting point, and a triangle that closes — solved on a drawing, in a yard, by somebody who has never heard of Culmann.

Bow’s notation, and why the polygons nest #

The step that made graphic statics a production technique rather than a demonstration is a labelling scheme, which is a strange thing for a structural method to hinge on.

Bow’s notation names the spaces between the forces rather than the forces themselves. Going round a joint clockwise, the regions between successive members are lettered A, B, C, and a member lying between region A and region B is then called member AB. In the force diagram, each region becomes a single point, and the force in member AB is the line from point a to point b — measured once, read by every joint that touches it.

The consequence is that adjacent joints share sides automatically. A polygon drawn for one joint has already drawn most of the polygon for its neighbour, so the whole truss collapses into a single tiling in which every member appears exactly once as a segment. The saving is not modest: a fifty-member truss requires about fifty measured lines rather than a hundred simultaneous equations, and the tiling has to fit together, so an error anywhere shows up as a figure that will not close.

There is a deeper fact underneath, which Maxwell established in 1864 and which is the reason the whole thing works. The form diagram and the force diagram are reciprocal figures: each has one line perpendicular to each line of the other, each vertex of one corresponds to a face of the other, and either can be recovered from the other. Maxwell showed that a pair of reciprocal plane figures exists exactly when both are projections of a single polyhedron in three dimensions — so the possibility of solving a truss with one drawing is a statement about polyhedra, and a frame that admits a reciprocal diagram is one that can carry a self-stress.

That result was a curiosity for a century. It is now the basis of computational form-finding, where the designer manipulates the force diagram and the program returns the geometry that goes with it — designing by choosing the forces first and letting the shape follow, which is the exact inverse of the usual direction and is only possible because the two figures determine each other.

What the method cost #

The technique was abandoned, and it is worth being clear about what for, because the usual explanation — that computers arrived — is only half of it.

One load case per drawing. A graphic solution is a picture of one set of forces. Change the load and the entire figure is redrawn. A modern building is checked against dozens of combinations of dead, imposed, wind, snow and accidental load, with partial factors differing between them, and the drawing office labour of doing that graphically would be enormous. Superposition helps and does not rescue it.

No stiffness. Every drawing here is a statement about equilibrium, so the whole apparatus stops precisely where statics stops. The nineteenth century met this by designing determinate structures on purpose — the pin-jointed truss with its statically determinate count was popular partly because it was the thing the available method could solve.

Precision that does not compound well. A part in two hundred is fine once. Through a chain of twenty nested polygons the errors accumulate, and the closing gap at the end has to be judged: too small to matter, or a mistake somewhere in the chain.

What was lost with it is worth naming too, since it was not nothing. A force diagram displays the whole state of a structure at once, at scale, in a form where the relative size of every member force is visible without reading a single number — and where a change to the geometry moves the diagram in a way that shows immediately what it did. Very little in a modern analysis output does that.

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.

Coplanar as well as concurrent. In three dimensions the theorem is stronger than it first appears and correspondingly less useful: three forces in equilibrium must not only meet at a point but lie in a common plane, since any component out of the plane of the other two has nothing to balance it. A body held by three forces in space is therefore a planar problem in disguise, and a body held by three forces that genuinely are not coplanar is not in equilibrium at all — which is a useful thing to check before drawing anything.

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.