How wrong a drawing is

Every graphical construction met so far has ended by crossing two lines. Three forces meet where two known lines cross. Culmann’s line joins two crossings. A pole’s rays find a resultant where the first and last strings cross. A reciprocal figure places each point where two members’ lines cross. A three-hinged arch is two lines and a triangle, and a funicular through three points is two moves, each of which ends at a crossing.

None of those essays asked how precisely a crossing can be drawn. The pin essay came nearest. It found that friction turns each line into a band, and that bands crossing at small angles give answers too uncertain to use. But friction is a property of the structure. A pencil is a property of the drawing, and it turns every line into a band too. This essay measures that band’s effect. The answer is that a drawing’s accuracy is decided by its angles, that the angles are worst on the shallow arch, and that measured the right way round the drawing is as accurate as the arch.

A line is a band #

A line drawn with a sharp pencil is about a fifth of a millimetre wide. Wherever the true line is, it is somewhere inside that width. Two such lines cross, and the true crossing is somewhere inside the region where the two bands overlap.

The region is a parallelogram, and its shape is set entirely by the angle between the lines. At a right angle it is a square one width on a side, and the crossing is uncertain by at most the square’s diagonal. As the lines close together the parallelogram stretches along the bisector of the angle between them, until at 12° it is nearly ten widths long.

The stretching has a simple law. The long diagonal of the parallelogram is the width divided by the sine of half the angle between the lines. At 60° that is the width divided by a half, two widths. At 12°, the width divided by 0.105, which is 9.6. The law has no floor: two lines drawn a degree apart cross somewhere in a region more than a hundred widths long.

The angle, not the hand #

Plotted against the angle, the law makes a point that took the practitioners of graphic statics a long time to state plainly.

Nothing about the draughtsman is on this axis. A better hand draws a thinner line and places its ends more exactly, which shrinks the width, and the width scales the whole curve. It does not change its shape. The same hand, pencil and care are worth ten times less at 12° than at 90°. A construction’s accuracy is therefore a property of the geometry it has to draw, and a construction with small angles in it is inaccurate however it is drawn.

Direction is a band too. A line drawn through two points each placed to a tenth of a millimetre, 100 mm apart, has its direction uncertain by up to two thousandths of a radian — about a tenth of a degree. A line drawn parallel to another with set squares carries the same order of error across. A tenth of a degree is the unit the rest of this essay uses. It is not a claim about any draughtsman. It is the order of error a careful pencil and a steady rule leave in a line’s direction, and every result below scales with it.

The flat arch, drawn #

The method’s favourite structure is where the small angles live, and one figure shows why.

A load at the crown of a three-hinged arch is carried by two reactions pointing at the crown hinge. Their lines are the lines from the springings to the crown, and on an arch rising a twentieth of its span they are 5.7° off horizontal. The force triangle has two long sides meeting at an angle of 11.4° and a short side, the load, closing it. Its width is the thrust, 500 kN from a 100 kN load.

Now let each reaction line be drawn a tenth of a degree off. The triangle’s apex slides along the load line’s direction by an amount set by those two nearly parallel sides. The thrust, which is the apex’s distance from the load line, changes by the angular error divided by the product of the sine and cosine of the line’s slope. At 5.7° that product is a tenth, so the tenth of a degree becomes a two per cent error in the thrust. At a rise of a quarter of the span it is about half a per cent, and at a rise of a fiftieth of the span, four and a half.

The formula is short enough to carry. With the reaction lines at a slope α, the thrust is the load over twice the tangent of α. A small change dα in the slope changes the thrust by the load over twice the square of the sine of α, times dα. As a fraction of the thrust, that is dα divided by the product of the sine and the cosine of α. The product falls toward zero as the arch flattens, and the error grows as one over the rise, which is the same law as the parallelogram’s, arrived at through the triangle.

That is the same ill-conditioning the friction circle found and the hinged arch’s band repeated. A small sideways movement of either line moves the crossing a long way, because the lines are nearly parallel. What changes is only what moved the line: a pencil here, a pin’s friction there.

The builder’s error is the same size #

At this point the natural conclusion is that the drawing fails on flat arches and a calculation is needed instead. The next figure is the reason that conclusion is wrong.

The second line is not about drawing at all. It is how much the thrust of the real arch changes if the crown hinge ends up a thousandth of the span higher or lower than drawn — 20 mm on a 20 m span, which is a plausible tolerance for setting out a steel arch’s crown. The thrust is the crown moment over the rise, so a rise one per cent wrong makes a thrust one per cent wrong, and on a shallow arch 20 mm is a large fraction of the rise.

The two curves run side by side across the whole range, because they are the same curve twice. A reaction line drawn a tenth of a degree off is the reaction line of an arch whose crown is about 18 mm higher or lower, since tilting a line from the springing through the crown is the same as moving the crown. So the drawing’s answer is exactly right for an arch that differs from the intended one by less than the arch will be built to. The drawing is not approximately right about this arch. It is exactly right about a neighbouring arch that is equally likely to be the one that gets built.

That way of reading an error is the surprising connection this essay ends on, and it comes from the other end of the twentieth century. Numerical analysts working on the first electronic computers — Wilkinson is the name attached to it — stopped asking how far a computed answer was from the true one. They asked instead for which nearby problem the computed answer is exactly true. An answer is acceptable if the nearby problem is closer to the intended one than the intended one was known to begin with. That is backward error, and graphic statics passes it on shallow arches for the same reason a computer does. The sensitivity is in the problem, not in the method, and no method can be more accurate than the problem’s own data.

The pole was always the draughtsman’s tool #

Seen with the law in hand, one freedom that earlier essays treated as incidental turns out to be the method’s main defence against it. The pole construction left the pole free and showed that every pole finds the same resultant, the same reactions and the same bending moment. It is also true that different poles find them with different accuracy.

A pole far from the load line draws a shallow polygon whose strings are all nearly horizontal. The resultant is found where the first and last strings cross, and two nearly horizontal strings cross at a small angle, so its position is found poorly. A pole close to the load line draws a deep polygon with steep strings, and at the extreme the strings are nearly vertical and nearly parallel to the loads, and the crossing is poor again. Between the two there is a pole whose first and last strings cross at something near a right angle, and it finds the resultant about as precisely as a pencil can. The choice of pole changes nothing in the answer. It changes how well the answer is drawn, and the freedom that made the pole’s position seem arbitrary is what lets a draughtsman keep every crossing open.

The same reasoning runs through every construction graphic statics has. Culmann’s line can pair four forces in three ways, and the pairing whose crossings are least oblique is the one to draw. A bracket whose three supports nearly meet at a point has no good pairing at all, and that is a fact about the bracket. A flat arch has no pole or pairing to choose. Its crossing is fixed by its hinges, which is why its accuracy is a fact about the arch.

A truss drawn joint by joint #

A truss diagram has many crossings rather than one, and the errors meet each other on the way through.

The method of joints drawn as a diagram proceeds joint by joint, each time solving for the two members that are still unknown. The forces already found enter each new joint with the errors they were found with, and the two new lines add errors of their own. By the end every force in the truss has been found once, and two or three joints remain whose forces are all known. Those joints are the check. Their polygons should close, and the amount by which they do not is the closure error the method was always checked by.

In this truss, a fifth of its span deep, the worst member is out by 0.32 per cent of the largest force and the last joint closes to within 0.12 per cent. Both are small, and they are of a similar size. The check has warned of an error about as large as the one there is. This is the case the method’s reputation was built on.

The shallow truss #

Make the truss shallow and the same construction behaves differently.

At a fortieth of its span deep, every diagonal crosses the chords at about 8°, and every joint where a diagonal and a chord are the two unknowns is a crossing of nearly parallel lines. The worst member is now out by 1.84 per cent of the largest force — and the largest force is itself eight times what it was in the deep truss. The closure at the last joint has hardly changed: 0.08 per cent. The check is twenty-four times too optimistic, and nothing about the drawing announces it.

The worst member is in the top chord, and that is no accident either. In a shallow truss the chords carry the bending moment as a couple over a small depth, so they are the members sized by force and the ones a designer reads most carefully. Every chord force here is found at a joint where a diagonal crosses the chord at 8°. The members whose forces matter most are the ones found at the worst crossings, which is what makes a shallow truss the hard case for the method rather than an unlucky one.

The check that does not see it #

One truss and one set of errors could be luck, so the next figure averages the same comparison over twenty-four sets of errors at every depth.

The worst member’s error grows sevenfold as the truss flattens. The closure gap does not move at all. The reason is a property of the construction that is easy to miss. A joint with two unknown members is closed by the construction itself. Two lines are drawn in the unknown members’ directions to close the polygon, and they always close it, whatever angle they cross at. A shallow crossing puts a large error into the two forces it produces and leaves no gap anywhere to show for it. The gap at the end comes only from the few joints where a single unknown had to satisfy two directions at once, and those come at the far end of the construction, beside the last support, where the truss’s geometry is much the same whatever its depth.

That is worth setting beside the claim made for the reciprocal figure, that each unused crossing is an equilibrium equation checked without being written. It is true, and it is a check of consistency: the drawing agrees with itself. It is not a check of accuracy, because a drawing made with consistently wrong angles agrees with itself perfectly. Closure proves the diagram is a diagram of some frame in equilibrium. It does not prove it is a diagram of this one.

What the numbers rest on #

Every figure here makes the same modelling choice, and it deserves stating because the results depend on it.

The errors are random and independent. Each line is off by its own amount, a seeded draw of up to a tenth of a degree either way. Real drawing errors are partly systematic: a set square a little out of true, a scale slightly stretched. A systematic error in direction, applied equally to every line, rotates the whole diagram and leaves every force exactly right, because the diagram is a drawing of directions relative to each other. Only the differences between lines’ errors matter, and independent errors are the pessimistic case.

The draughtsman is not clever. A practitioner who knew where the small angles were drew those joints at a larger scale, or constructed the shallow crossings from a known point by calculation, or re-ordered the construction so that the long lines were drawn first. None of that is modelled, and all of it helps. The figures measure the method used plainly, which is the case the reputation came from.

Only direction errors are modelled. Lengths are scaled off the drawing too, and a length read off to a fifth of a millimetre on a force drawn 50 mm long is 0.4 per cent. That error adds to everything here without changing its shape.

The assumption that makes the error the arch’s #

The whole argument for the shallow arch rests on one assumption: that the arch’s rise is not known more precisely than about a thousandth of its span. For a steel arch set out on site, that is the right order. For a precast concrete arch cast in a stiff mould it might be several times better, and then the drawing’s tenth of a degree would be the larger error, and a calculation would be the right tool.

The comparison is the useful thing, not either number. An answer is good enough when its error is smaller than the uncertainty in what it is an answer about. The flat arch’s thrust is uncertain by a couple of per cent before anyone draws or computes anything, because its rise is uncertain and it shortens under its own thrust. A method that adds another couple of per cent has not failed, and a method that claims four significant figures has not succeeded. It has only reported the answer for an arch that will not be built.

Still open: the pencil a computer uses #

The analysis that replaced the drawing board does the same arithmetic in a stiffness matrix, and it has a pencil width of its own. Every number carries about sixteen significant figures, and every operation rounds the last of them. A computer’s line is a band a ten-thousand-billionth wide instead of a fifth of a millimetre.

The small angles do not go away. A shallow arch or a shallow truss produces a stiffness matrix whose rows are nearly parallel — the same geometry, written as numbers — and the amplification that turned a tenth of a degree into two per cent turns sixteen figures into fewer. On any structure a draughtsman could draw that loss is harmless. On a structure close to a mechanism it is not. The matrix’s conditioning is the angle between two lines, measured in a form nobody draws, and it is where the question of how wrong a calculation can be goes next.