Every space a point, every joint a polygon

Four essays have now been spent on one direction of travel. Three forces must meet, four pair off along Culmann’s line, any number are gathered by a pole and a string, and a real pin turns each line into a band. Every one of those constructions starts from a structure and some loads and ends with a drawing of forces.

The drawing it ends with has a structure of its own, and nothing so far has looked at it as one. The truss drawn in one figure used Cremona’s diagram as a solver — twenty-one members, twenty-one lines, no repetition — and moved on. This essay stays with the diagram. Read as an object, it is a second drawing of the same frame in which two kinds of thing have traded places. Maxwell saw that the trade runs in both directions. That fact turns an analysis method into a design method, and there is a theorem under it about which frames can take part at all.

Letters on the spaces, not on the members #

The trick that makes the diagram a single figure is Robert Bow’s, from 1873, and it is a piece of bookkeeping. Do not name the members. Name the spaces between them.

A space outside the truss is the region between two neighbouring external forces as one goes round it. There are five here: the region under the truss between the two reactions, and four more between the reactions and the three loads. A space inside is a triangular cell, and there are six. Every member separates exactly two spaces, and every external force separates two outer ones.

Now draw each space as a point. A member is then the segment joining the points of the two spaces it separates, drawn parallel to the member and as long as the force in it. The external forces go the same way: each separates two outer spaces, so the five outer points lie on one straight line — the load line — spaced by the loads and reactions.

The result is the figure on the right, and its accounting is exact. The frame’s 13 members and 5 external forces are 18 lines in the diagram. The frame’s eleven spaces are eleven points, although only eight of them are distinct, which comes up below. And each of the frame’s eight joints is a closed polygon, the lines of every member meeting at that joint, laid tip to tail.

A joint is a polygon because it is in equilibrium #

That last statement is not a convention. It is the method of joints, and it is where the whole figure gets its right to exist.

Cut the ringed joint free. The forces on it are the five members it holds, and since a pin carries no moment they all pass through one point. Concurrent forces in equilibrium close into a polygon when laid tip to tail. The same five forces appear in the diagram as the five segments that separate the five spaces meeting at that joint, taken in order round it. So the joint’s force polygon is sitting in the diagram already, and it closes because the joint balances.

Every other joint is the same. The diagram is therefore eight force polygons that share their edges: each member’s line is a side of the polygon at both of its ends, walked once in each direction. A member drawn once and read twice is the whole economy of the method.

The economy has a second face, and it is the one that makes a diagram trustworthy. Placing the points needs only one crossing per space: start from any space, cross a member, and the next space’s point is fixed by that member’s force. Eleven spaces need ten crossings. There are eighteen lines, so eight crossings are never used to place anything, and each of them either closes or does not. In the drawing above all eight close, to within the last digit the arithmetic carries. Each unused crossing is an equilibrium equation checked without being written.

That is where the count of eight distinct points comes from, too. The two outer verticals carry nothing, so the two cells either side of each one sit at the same point. Cells 2 and 5 then land there as well, because the bottom chord carries the same 20 kN in both of its middle bays along the same line. A member that carries nothing is a member whose two spaces are one point, and a diagram with fewer points than the frame has spaces is a diagram reporting idle members before anybody has looked for them.

Tension and compression, read off the order of the letters #

Bow’s lettering does one more piece of work, and it is the reason draughtsmen kept it after the diagram itself had been superseded. It tells the sense of every force without a sign convention.

Go round a joint clockwise and name each member by the two spaces it separates, in the order they are met. The ringed joint’s diagonal on the left, going clockwise from cell 2 into cell 3, is member 2–3. In the force diagram, the arrow from point 2 to point 3 is the force that member exerts on that joint. If the arrow points toward the joint in the frame, the member is pushing on it, which is compression. If it points away, the member is pulling, which is tension.

Go round the joint at the other end of the same member, and the clockwise order meets the two cells the other way round: 3–2. The arrow reverses, and it points toward that joint too, because a compression member pushes on both of its ends. A single line in the diagram, read in opposite directions from its two ends, gives the equal and opposite forces the member exerts on the two joints it joins. The sense comes out of the geometry of the lettering. No sign was ever assigned; it is where the letters are.

One more exchange balances the count. The external forces behave like members that all meet at a single joint far away — the point from which the loads come and to which the reactions go — and that joint’s polygon is the load line. Count the frame with it included, nine joints, eighteen members and eleven spaces, and count the diagram as eleven points, eighteen lines and nine polygons. Both satisfy Euler’s rule for a map drawn on a sheet, that joints minus lines plus faces is two. Joints and spaces have simply swapped roles.

The same exchange run backwards #

Maxwell named the relation reciprocal in 1864 because it is symmetric. The clearest way to see that is on a frame with no loads, whose diagram therefore has no load line to confuse matters.

A triangle with a joint inside it is the smallest frame with a stress and no load. Six members on four joints give six unknowns against eight equations. Three of those equations are used up in holding the frame still, so six members against five independent equations leaves one member’s worth of freedom — one redundant, in the count’s own terms. That freedom is a set of forces in equilibrium with nothing applied: tension in the spokes, compression in the sides, in fixed proportions and at any scale.

Its reciprocal is drawn the same way as the truss’s. Four spaces — three cells and the outside — become four points. Six members become six lines. Four joints become four closed polygons, and in the picture three of them are the reciprocal’s inner cells while the fourth, belonging to the bracing joint, is its outer boundary.

The reciprocal of a braced triangle is a braced triangle, and the point inside it is the region outside the frame. That is the sort of result that looks like a coincidence until one asks what the reciprocal’s reciprocal would be.

It is the original frame. Take the force diagram, pretend its lines are bars and its points pins, find the stress it can carry, and draw that stress’s force diagram. Every line comes back parallel to a member of the triangle it started from, and the joints come back where they were, scaled.

Neither drawing is the original and neither is the diagram. Either can be read as a frame and the other as its forces. That symmetry is what Maxwell meant by reciprocal, and it is what makes the rest of this essay possible.

A stress is a polyhedron #

Maxwell’s paper went one step further, and the step is the theorem the previous essay pointed at. It says which plane frames have a reciprocal figure at all.

The first condition is plain from the construction. The diagram needs spaces to letter, and a frame drawn with two members crossing where there is no joint does not divide the sheet into spaces. It has no reciprocal until a joint is put at the crossing. The second condition is subtler and more beautiful. The frame must carry a stress — loads included as members meeting far away — and a plane frame carries a stress exactly when it is a polyhedron seen from directly above.

The recipe is short. Every space of the frame becomes a flat face. The reciprocal point of that space, turned through a right angle, becomes the face’s slope. Across a member, the two faces’ slopes differ by that member’s force turned through a right angle, which is perpendicular to the member itself. So the two faces meet along a line parallel to the member, and with one face pinned down the other is fixed to meet it exactly on the member. Going round a joint, the slopes come back where they started exactly when the joint’s force polygon closes. Equilibrium at every joint is the same statement as the faces fitting together into one surface.

The pyramid on the braced triangle has faces whose slopes are 0.59, 0.56 and 0.56. Those are the forces in the three sides of the triangle, because each face meets the flat outside along one side. Each face’s steepness is the force in the member at its foot.

The counting follows from the picture. A bare triangle, with no joint inside it, has one space inside and the outside. Lift it and the inner face must meet the flat outside along all three sides, which leaves it only the flat plane itself. So the lift is flat and the stress is zero, which is the determinate frame’s statement in another language. Put a joint inside and the three inner faces can rise together into a pyramid of any height. There is one family of lifts, so there is one family of stresses. Brace a square with both diagonals and a joint where they cross, and the lifts are a four-sided pyramid of any height, again one family. The same square drawn with the diagonals crossing and no joint has no faces to lift, which is the planar condition arriving from the other side.

This is the surprising connection the essay was written for, and it runs well beyond the braced triangle. The statement that a redundant plane frame can carry a stress with no load is a fact about equilibrium. It is the same fact as the statement that the frame is the shadow of a polyhedron with flat faces, which is a fact about solid geometry. A frame drawn so that it could not be the view of any such solid carries no stress. It is determinate, or it is a mechanism. A frame that could be such a view carries exactly as many independent stresses as there are independent ways of lifting it.

Drawing the forces first #

The symmetry has a practical consequence, and it is the one that brought graphic statics back into use a century after the calculator made its analysis work unnecessary. If either figure can be drawn from the other, a designer can draw the force diagram first. That chooses the forces the structure is to carry, and the frame they belong to follows.

The bowstring is a frame designed that way round. Ask for a truss whose web does nothing — whose diagonals separate cells that share a point in the diagram — and whose bottom chord carries one force from end to end. That diagram is already known: it is the funicular’s force diagram, a load line with a pole, and its rays fix the directions of the top-chord segments. The frame belonging to it has a top chord that is the funicular of the hanger loads, which for five equal loads is a parabola through the panel points. The diagonals can then be put in for stiffness under other loadings, knowing that under this one they will be idle.

Nothing about this required a force to be computed. The choice was made in the force diagram, where it is a choice about positions of points, and the geometry of the frame was read off it.

The diagram also shows what the remaining freedom is. The pole A can slide along the horizontal through the middle of the load line. Moving it away from the load line lengthens every ray’s horizontal component, so the bottom chord carries more force and the top chord flattens. Moving it toward the load line does the opposite. For these loads the pole’s distance is the bottom chord’s force, and that force times the truss’s depth is fixed at the 90 kN·m the loads put at mid-span. So choosing the chord force is choosing the depth: 37.5 kN at 2.4 m, 22.5 kN at 4 m. A designer who draws the pole has decided both at once, from a drawing in which neither appears as a number.

The web that carries nothing is not the economy #

It would be natural to assume that a truss with an idle web is the efficient one. The diagram says otherwise, and a second theorem of Maxwell’s, from his 1870 paper on reciprocal figures, is what makes that visible.

Multiply each member’s force by its length and add them up, tension and compression separately. For the bowstring the two totals are 543 kN·m each. For the parallel-chord truss they are 511 kN·m each. The two are equal in both, and that equality is no coincidence of these frames.

Maxwell’s load-path theorem says that tension times length summed over the tension members, less compression times length summed over the compression members, depends only on the external forces and the points at which they act. It does not depend on the frame between them at all. For loads and reactions all applied on one straight line, as these are, the difference is zero, so every truss that could carry this loading has exactly as much tension-length as compression-length. For the roof truss, whose loads are applied up to 3 m above its supports, the difference is fixed at −60 kN·m, and the totals come to 190 and 250.

What the theorem leaves free is the sum, and the sum is the material. The bowstring’s is 1,087 kN·m and the parallel-chord truss’s 1,021. The truss with an idle web needs six per cent more material, because its chords are longer and carry the full 37.5 kN thrust from end to end, where the parallel-chord truss’s chords carry that force only in the middle bays. The web was not the waste. It is how the parallel-chord truss moves force out of its chords near the supports.

The same accounting shows where the bowstring’s economy does lie, and it is in its depth. Drawn 4 m deep instead of 2.4, with the pole moved to suit, its chords carry 22.5 kN and its total falls to 851 kN·m, a sixth less than the parallel-chord truss at the shallower depth. The comparison that the idle web seemed to settle was never a comparison between two webs. It was between two force diagrams, and the smaller diagram is the lighter truss, whatever its web is doing.

That is the argument Michell took up in 1904, and it starts here. With the difference fixed, minimising total material is the same as minimising the tension-length alone. And since the force diagram is where force and length appear together, the question of the lightest frame is a question about which reciprocal figure is smallest.

What the figures are and are not doing #

Each figure in this essay is a statement of equilibrium and nothing else, and that bounds what it can say.

Every force here is statics’ alone. The roof truss and both 12 m trusses are statically determinate, so their forces are fixed by equilibrium and each has one force diagram. The braced triangle is redundant, and its stress is defined only up to a scale. A redundant truss under load has a family of possible diagrams, one for each amount of self-stress added, and which one the structure takes depends on its members’ stiffness. That is a question of compatibility, and the reciprocal figure is silent on it.

The joints are pins. A reciprocal figure needs every member to be a line of force, which is true of a member loaded only at its ends through frictionless pins. The same joints drawn with friction turn each line into a band, and bands do not close into polygons. Welded or bolted joints carry moments that no diagram of lines can hold. The chord of a real truss is a continuous beam, and its bending lives outside the figure.

The lift is a picture of the stress, not of any structure. The pyramid does not exist. It is the geometry the stress would have if forces were slopes, and its height depends on a scale chosen to turn a force into a slope, so only its proportions mean anything. What the lift proves is the counting — how many independent stresses a frame can carry. What it does not prove is that the braced triangle is safe. Its stress is at any scale, so the lift says nothing about how large a stress the members will actually carry.

The assumption the diagram rests on #

The one assumption that governs everything above is that the frame is drawn in a plane, without crossings, with every member a straight line between two joints. Bow’s letters need spaces, and spaces need a map.

That assumption is broken more often than it looks. A cross-braced panel drawn with its two diagonals passing each other without a joint has no spaces, and no reciprocal until a joint is put where they cross. That joint changes nothing structurally, because each diagonal is straight through it, and in the diagram it becomes a parallelogram. A space truss, which needs three equations at every joint, has no reciprocal figure in the plane at all. Its analogue is a reciprocal in three dimensions, in which joints become polyhedra and members become faces, and the drawing board has nothing to say about it.

Still open: a drawing asked to pass through fixed points #

Every construction in these essays has so far been free to put its polygon wherever it liked. The pole was a free choice, and so was the point where the string began. A three-hinged arch takes that freedom away: its thrust must pass through three hinges fixed by the structure, and its reactions must pass through two of them.

That makes it the first structure met here in which the drawing’s freedoms are all spoken for. The arch’s own geometry — three points, and nothing about the rib between them — is enough to decide its reactions, and a load moving across it moves the construction along two straight lines. What those lines are, and why the shape of the rib never enters, is the next question.