The strongest layout leans on one bolt

Every earlier essay on the bolt group was a check. A layout was given — three rows of two at 75 mm, or a ring, or a rectangle with its long side across the load — and the question was what it carried, in which direction it was weakest, how the instantaneous centre changed the answer, and finally, in the essay on the bolt that was never fitted, what it carried with one bolt gone. That last question ended on a design question it could not answer by checking: if a bolt may be missing, where should the six go?

It also ended on a prediction. The layout that carries most with all its bolts concentrates its resistance in the corners, because the polar moment rewards distance; concentrating resistance in the corners is what makes losing a corner expensive; so a layout chosen for the worst single omission ought to spread its bolts more evenly and carry less when complete. That is the ordinary shape of a robustness argument, and it is the shape this essay tests rather than repeats.

The prediction is half wrong, and the half that is wrong is the interesting one. The most robust layout does carry less when complete — by one per cent. It does not spread its bolts evenly; the evenly spread layout is worse on both counts. And the layout that carries most turns out to be strong for a reason that has almost nothing to do with the corners, and fragile for a reason that has everything to do with one bolt.

Six bolts, one plate, two questions #

The problem is set up to be as plain as possible. A bracket plate has room for a field of bolt centres 150 mm square. Six bolts go anywhere inside it, no two closer than 60 mm — three diameters of an M20, a spacing any detailer would accept. The load acts through a point 200 mm to the right of the field’s centre, which is where the web of a supported beam would put it, and it may point in any direction in the plane of the plate. Each bolt is good for 100 kN, and the group’s capacity is the elastic method’s: the load at which the worst bolt, at the worst direction, reaches 100.

Two searches are then run over the same plate. The first looks for the layout whose complete capacity is largest — the ordinary optimum, what a designer would draw if strength were the only question. The second looks for the layout whose worst omission is largest: of the six groups of five that remain when each bolt in turn is missing, it maximises the weakest. That is a minimax, and it is exactly the question the last essay left open.

The two familiar layouts are there for scale. Two columns of three at the edges of the field is what most brackets look like, and it carries 193.5 kN. A ring of six on the largest circle the field holds is the layout that spreads its bolts most evenly — every bolt the same distance from the centre, every bolt contributing the same $r^2$ to the polar moment — and it carries 167.2.

The two searched layouts are stronger than either, and by a lot: 234.3 and 231.6 kN, a fifth more than the two-column layout. Nothing about that is subtle and most of this essay is not about it. What matters is the third bar group in the figure, where the strongest layout loses 36 per cent of its capacity to one omission, against the fourth, where the most robust loses 25 per cent to any of six.

The two answers differ by one bolt #

Look first at what a plate without room does, because it sets the scale for what a plate with room can do. In a field half as wide, 75 mm by 150, every layout that fits is some rearrangement of two columns, and the three answers sit within four per cent of one another. The strongest layout found beats the plain two columns by two per cent and is less robust than they are; the most robust beats them by two per cent on the omission. When the plate decides the layout, the layout has nothing left to decide, and neither search finds anything a detailer would not have drawn anyway.

Back in the square field there is room, and the searches use it. The strongest layout found puts two bolts on the right-hand edge at about 30 mm either side of the load’s line, two more along the top and bottom edges at 35 mm right of centre, one in the top left corner — and the last one not in the bottom left corner but 50 mm in from it, at $(-25, -75)$.

The most robust layout found is the same layout with that one bolt moved into the corner.

That is not an approximation or a summary; it is what the two searches return, from sixty random starts each, to within a fraction of a millimetre on the five bolts they share. The ordinary optimum and the minimax differ by one bolt position, and the difference between the two positions is fifty millimetres along the bottom edge of the plate. The strongest layout is the robust one with its symmetry broken, and the symmetry it broke was carrying the robustness.

What the broken symmetry buys, and what it costs #

With the sixth bolt in the corner the layout is symmetric about the load’s line. Its centroid sits 11.7 mm to the right of the field’s centre, so the load’s lever arm about it is 188.3 mm, and its polar moment is 48,443 mm². Pull the sixth bolt in by fifty millimetres and three things happen. The centroid moves 8.3 mm further right and a little down, toward the load, and the lever arm shortens to 180.0 mm — 4.4 per cent. The polar moment falls to 41,848 mm², 13.6 per cent, because the bolt that moved was one of the two furthest from the centroid and $J$ counts that distance squared. And the group is no longer symmetric, so which bolt governs, and at which direction, changes.

The first effect is worth more than the second by just enough to leave a net gain of 2.7 kN, which is 1.2 per cent of the group. The optimum lives on that margin. It is a flat optimum, the kind every optimisation arrives at when two competing effects are nearly balanced, and the search finds its top by trading most of one quantity for slightly more of another.

The cost is in the other bar chart. With the sixth bolt pulled in, the top left corner is the only bolt on the far side of the group at full distance, and the group’s resistance to twisting leans on it. Take it away and the capacity falls from 234.3 kN to 149.7. The same omission in the symmetric layout costs almost nothing extra, because the bottom left corner is there to share what the top left corner was doing. A symmetric layout has every far bolt backed up by its mirror image, and the optimum spent the backup to buy one per cent.

That is the shape the best design is the most sensitive one found in a tube whose wall thickness was optimised: the optimum sat exactly where two buckling modes arrived together, which is the one proportion imperfections hurt most. Here the imperfection is a missing part rather than an out-of-straightness, and the same thing happens for the same reason. An optimiser pushes a design toward the point where everything is working at once, and a design where everything is working at once has nothing in reserve for the failure the optimisation did not include — a determinate truss is the limiting case.

The minimax makes every omission cost the same #

Look again at the most robust layout’s six omission bars. They read 176, 175, 175, 175, 175 and 175 kN. That is not a coincidence of this particular plate; it is the signature of a minimax, and it is worth understanding before trusting the number.

A search that maximises the smallest of six quantities keeps pushing on whichever one is smallest. Moving a bolt to help the weakest omission usually hurts some other omission, and the search continues until the other omissions have been dragged down to meet it — until there is no move that raises the smallest without lowering another below it. At that point several of the six are equal, because if one were strictly larger the search could have spent some of its surplus on the weakest. Six equal omissions is what a minimax looks like when it has converged. It is the same property that makes a fully stressed truss have every member at its limit, and the same one that makes the best polynomial approximation of a curve touch its error bound, alternately above and below, at as many points as it has freedoms to spend.

The direction sweep shows what the equality does and does not mean. The group of five without bolt 6 has its trough at 78 degrees and the complete group has its trough at 277, a little under half a turn away. At its trough the five-bolt group is 25 per cent below the complete one. But over 46 per cent of the directions, the group of five still carries more than the complete group does at its weakest direction. A minimax over omissions is paid for at one direction and enjoyed at all the others, which is the same observation the essay on a load that could come from anywhere made about the complete group: a capacity quoted at its weakest direction is a statement about one direction, and a bracket whose load has a known direction can be worth much more.

Evenly spread is not robust #

The ring is the test of the prediction, because it is the layout an argument about evenness would draw. Every bolt is 75 mm from the centre, every bolt contributes the same to the polar moment, and no bolt is special. If robustness were a matter of not concentrating resistance, the ring would be the most robust layout in the plate.

It is the least robust of the four in the only unit that matters on site: with its worst bolt missing it keeps 117.8 kN, less than any of the others. It carries 167.2 kN complete, so that is a loss of 30 per cent — more than the two-column layout’s 28 and the robust layout’s 25, and beaten only by the strongest layout’s 36.

The reason is the one the last essay found when it priced each omission. Losing a bolt costs three things at once: the equal share rises because there are fewer bolts, the polar moment falls because a contributor is gone, and the centroid moves away from the gap, which changes the load’s lever arm. On a ring the third effect is as large as it can be. The centroid of six bolts on a circle is the circle’s centre; remove one and the centroid of the other five moves a fifth of the radius directly away from it — 15 mm on this ring — and when the missing bolt was on the side nearer the load, that is 15 mm added to the lever arm on every one of the remaining five. Evenness does nothing about that. It makes every bolt equally important, which is a different thing from making none of them important.

The cloud says the same thing in bulk. Across two hundred and forty random layouts the complete capacity and the worst-omission capacity are strongly correlated, at 0.84: a random layout that is strong when complete is, far more often than not, the one that survives losing a bolt. Most of the time strength and robustness are the same property, because both are bought by the same geometry. The trade-off the prediction imagined exists only at the very top of the cloud, among layouts that are already stronger than everything a detailer would draw by habit — and there it is not a trade-off between strong and robust layouts but a choice between two nearly identical strong ones.

The spread in the lost share is the other number to carry away. A missing bolt costs between 19 and 36 per cent of the group depending on the layout alone, a factor of 1.9 with the same six bolts in the same plate. The last essay found the loss depending on which bolt was missing by a factor of ten on one fixed layout; this one finds it depending on the layout by a factor of two for the worst bolt. Together they say that “a sixth of the bolts” is never the right mental model for what an omission costs, at either scale.

The lever arm both searches are pulling on #

Why is the ordinary optimum so much stronger than the two-column layout, which has a larger polar moment — 56,250 mm² against 41,848? The answer is the lever arm, and it is the variable both searches are really working on.

The load point is fixed on the plate. The group’s centroid is not — it is wherever the average of the bolts is — and every millimetre the centroid moves toward the load is a millimetre off the torque the bolts have to resist. The two symmetric familiar layouts put their centroid in the middle of the field and accept a 200 mm lever arm. The searched layouts crowd bolts toward the load’s side — four of their six are in the right-hand half of the field — and cut it to 180 or 188.

The torsional share of the worst bolt is $V d\, r_{\max}/J$, and a layout can improve it by raising $J$ or by lowering $d$. The two-column layout has maximised $J$ inside the field, which is the obvious move and what every textbook layout does. The optimum gives up a quarter of $J$ to take a tenth off $d$ and a little off $r_{\max}$, and comes out 21 per cent ahead. The textbook instinct is to push bolts apart; the arithmetic says to push them toward the load, until the loss of polar moment catches up with the gain in lever arm.

That is also what makes the optimum fragile. A layout crowded toward the load has its far side thinly populated, and the far side is where the resistance to twisting lives. The strongest layout has one bolt at full distance in the far top corner and one pulled in at the bottom; the robust layout has both far corners occupied. The difference in lever arm is 8 mm, and the difference in what one omission costs is eleven points.

The whole exchange, priced #

The price of robustness can be measured directly by asking the minimax search a constrained question: what is the most robust layout whose complete capacity is at least some fraction of the best? At 100 per cent only the strongest layout qualifies, and its worst omission is 149.7 kN. At 99 per cent the answer jumps to 174.8, and it stays there all the way down to 80 per cent.

That is a step, not a curve. All of the robustness there is to buy costs one per cent of strength, and no further sacrifice buys any more. A designer who accepted a group ten or twenty per cent weaker in the hope of making it tolerant of a missing bolt would get exactly what the one-per-cent layout gives and nothing extra — and would very likely get less, because a weaker layout chosen by hand is not the one the search found.

The practical form of it is simple. The robust layout is not a compromise between strength and robustness, and it is not found by weakening the strong one. It is found by searching among the strong layouts for the one that does not lean on a single part, and in a plate with room there nearly always is one, because the strong layouts form a flat plateau and the plateau has many points on it.

Where the design rule comes from #

Checked against the four layouts, a rule suggests itself that is worth stating because it is not in any code and is not obvious.

Put the bolts toward the load, and never leave a far corner alone. The first half is where the strength comes from: the centroid is a design variable, and the lever arm is the most powerful lever the designer has, more powerful than the polar moment it is usually traded against. The second half is where the robustness comes from: a far bolt without a partner at the same distance is a bolt the group’s torsional resistance depends on, and its absence costs the most.

Neither half requires a search. A detailer drawing a bracket by hand can put four of six bolts on the load’s side of the field and keep both far corners, and will land within a few per cent of the robust layout found here. What the search adds is the measurement that the rule costs one per cent rather than ten, and the finding that the layout an optimiser would produce, left to itself, breaks the second half to buy the last per cent of the first.

The same rule has a familiar form in pile groups, where a cap’s piles share a load by the same three terms a bolt group uses. A pile group is usually symmetric for reasons that have nothing to do with robustness — the column is central and the load reverses — and the corner pile has its mirror image across the cap. A pile group under a permanent eccentric load, shifted toward the load to shorten the eccentricity, is the case where this essay’s warning would bite, and it is common: an edge column on a combined cap.

The free body, and what the search is allowed to move #

Every capacity on this page is the elastic method applied to a free body of the bracket plate with its bolts cut, exactly as in the earlier essays, and the bolt forces are checked to sum to the load in both directions and about the centroid. The one thing that changes is that the bolt positions are unknowns.

The search is a pattern search: from each of sixty random starting layouts, each bolt is moved in turn by a step in eight directions, a move is kept if it improves the objective and keeps every bolt inside the field and every pair at least 60 mm apart, and the step halves when no move helps. The minimax search is also started from the strength optimum, which guarantees it never reports a layout less robust than the one the strength search found — and from random starts alone, on three different seeds, it finds the same symmetric layout to the millimetre, so the one-bolt difference is not an artefact of where it began. The capacity reported for each layout is not the search’s own estimate but a closed-form calculation: the weakest-direction capacity of a bolt group is $R_b / \max_i \|M_i\|$, where $M_i$ is the 2 × 2 matrix mapping the load’s direction to bolt $i$’s force, and its norm is available exactly. The swept capacity in the direction figure agrees with it to half a per cent.

Two things about this should be said plainly. The search is a search, not a proof: a local optimum is reported as the optimum when sixty starts found nothing better, and in a narrower field the searches were seen to disagree with one another by several per cent from different seeds. In the square field they agree to a fraction of a per cent across three seeds, which is why the square field is the one the argument is built on. And every number the argument turns on — the 234.3, the 231.6, the 149.7, the 174.7 — is exact for the layout reported, whether or not a better layout exists.

What an elastic search on a rigid plate leaves out #

The instantaneous centre method. Every capacity here is elastic. The last essay found that the instantaneous centre method loses a larger share to a missing bolt than the elastic method does, because it has already spent the reserve the elastic method leaves idle; a search under that method would very likely find the same one-bolt difference with a larger price attached, but it has not been run.

A load point that moves. The load acts through one fixed point. A crane girder bracket or a beam seat whose bearing can shift sees its load move along a line, and a layout crowded toward one point is a layout that is wrong for the others. The symmetric robust layout is less sensitive to that than the strongest one, which is a second argument for it and not one the figures price.

The plate. Bolt holes crowded toward the load are holes crowded into the part of the plate that also carries the load into the beam, and the metal between the holes is not in this calculation. Edge distances are represented only by the field’s boundary.

The fabricator. Both searched layouts crowd their bolts toward the load, and a plate drilled from the wrong face is their mirror image — a layout crowded away from the load, with the lever arm lengthened instead of shortened. The two-column layout is the only one of the four that cannot be drilled wrong that way, which is a reason it is drawn so often that no capacity figure records.

A missing bolt is chosen by nature, and it is not #

That the bolt that goes missing is chosen by nature rather than by the load. A minimax over single omissions treats every bolt as equally likely to be absent and asks about the worst. Real omissions are not like that: a bolt is left out because it clashes with something, and the clash is usually at a corner, near a stiffener or a flange. A designer who knows which positions are at risk should protect those rather than all six, which is a different and easier problem. And the analysis stops at one: two missing bolts from six is a different geometry again, and nothing here says the one-bolt minimax is the two-bolt one.

Still open: the bracket that is also pulled off the column #

Every load on this page lies in the plane of the plate. A real bracket’s load rarely does: a beam seated on it is some distance out from the column face, so the same load that twists the group in its plane also tries to peel the plate off the column, putting the upper bolts in tension while they carry their share of shear.

A bolt carrying both is checked on an interaction, and the codes do not agree on its shape — one draws an ellipse, another a pair of straight lines. That disagreement is a known quantity for one bolt. For a group it is not, because the bolt that governs in shear is chosen by the plan geometry this essay has been searching over, and the bolt that governs in tension by how far it sits above the compression edge. When those are different bolts the group has no single worst bolt at all, and its capacity is a surface over the two loads rather than a number — with the layouts found here sitting somewhere on it that nobody has yet measured.