Four supports, and a whole line of answers

At its defaults it draws four supports, and a whole line of answers. The same top on four supports at the corners of a square, with the load in the same place. Five sets of reactions are drawn and every one of them satisfies all three equilibrium equations exactly — worst residual 1.1e-16 across the whole family — so statics does not prefer any of them. They differ by a multiple of the pattern plus, minus, plus, minus around the square: pressing one diagonal pair harder and the other pair less adds no net force and no moment about either axis, which is exactly what it means for the problem to have a fourth unknown and only three equations. The rigid-body idealisation has not been applied carelessly here; it has been applied correctly, and the answer it gives is that there is no answer. Every member drawn keeps all four reactions positive, so the requirement that a leg can only push narrows the family without closing it. What decides is left out of the model entirely — how much each leg gives under load.

redundant-supports is one function in lib/figures/mechanics.js — motion, force, energy and rotation. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

The same top on four supports at the corners of a square, with the load in the same place. Five sets of reactions are drawn and every one of them satisfies all three equilibrium equations exactly — worst residual 1.1e-16 across the whole family — so statics does not prefer any of them. They differ by a multiple of the pattern plus, minus, plus, minus around the square: pressing one diagonal pair harder and the other pair less adds no net force and no moment about either axis, which is exactly what it means for the problem to have a fourth unknown and only three equations. The rigid-body idealisation has not been applied carelessly here; it has been applied correctly, and the answer it gives is that there is no answer. Every member drawn keeps all four reactions positive, so the requirement that a leg can only push narrows the family without closing it. What decides is left out of the model entirely — how much each leg gives under load.

Stress across the depth of a 300 by 600 millimetre concrete section at the middle of an eight-metre span, compression to the right, for four conditions. Under the load alone the bottom fibre is in tension at 11.1 MPa, which is four times what concrete can carry, so an ordinary reinforced beam cracks there and relies on steel to hold the crack together. With 1,500 kilonewtons of prestress 120 millimetres below the centroid and the full load applied, the section runs from 9.8 to 3.6 MPa and every fibre of it is in compression. The second case is the one that surprises: with the prestress applied and nothing whatever to oppose it, the top fibre is in tension at 1.7 MPa, because a force below the centroid bends the beam upwards. A prestressed beam is at its most vulnerable when nothing is on it, and what rescues it is its own weight: adding that alone brings the top back to 0.3 MPa of compression. Each stress block is checked by integrating it and recovering the force and the moment that produced it.

The range of positions a prestressing tendon may occupy, against distance from the support along half of an eight-metre span, for 1,500 kilonewtons of prestress in the section of the companion figures. The upper curve is what the beam will stand at transfer, when the prestress is at its largest and only the beam's own weight opposes it; the lower curve is what the full service load demands once the prestress has dropped by a fifth. The band between them narrows from the support to 75 millimetres at midspan, and it is the lower curve that rises — so the tendon is not merely permitted to dip towards midspan, it is required to, and its profile is forced to follow the bending moment. The band is checked to stay open along the whole span at this prestress and to close at 800 kilonewtons, where no position satisfies all four conditions at once.

The force in a preloaded bolt and the compression left in the parts it clamps, against the external load pulling them apart. The bolt is tightened to 30 kilonewtons before any load is applied, and the clamped members are four times as stiff as the bolt — so of every kilonewton applied, the bolt takes 0.20 and the members give up 0.80. A load swinging from nothing to twenty kilonewtons therefore fluctuates the bolt's force by 4 kN rather than 20, which is the whole of why a bolted joint is tightened hard. At 37.5 kN the clamping is used up, the joint separates, and from there the bolt carries everything: the two curves meet, and beyond that point the preload has bought nothing at all.

The four reactions again, this time with each leg treated as a spring rather than as a rigid point. The top is still rigid, so it settles into some plane, and a plane has three numbers in it — a height and two slopes. Three unknowns and three equations: the problem closes, and the answer is 0.186, 0.214, 0.386, 0.214 of the load. It lies on the family the rigid calculation produced, at the parameter 0.0262, to 3.5e-17 — so elasticity has not overruled statics, it has supplied the one piece of information statics was missing. The chart sweeps the load along a diagonal and shows the four reactions following it, with the near pair rising and the far pair falling. The stiffnesses used are 1, 2.2, 1, 0.45 relative to one another; making all four equal makes the answer the symmetric one, and the interesting cases are the unequal ones, where a stiff leg takes more than its geometric share simply for being stiff. The idealisation to unpack, and it is the one this whole figure is about, is that a real floor is not four independent springs either.

A rigid top on three supports, seen from above, with a load applied off centre. Three unknown reactions and three equations — the vertical forces balance, and the moments balance about two horizontal axes — so there is one answer and it is drawn: 0.194, 0.394, 0.411 of the load. Those three numbers are the barycentric coordinates of the load point in the triangle of support, which is why they sum to exactly one and why each is the area of the opposite sub-triangle divided by the whole. Nothing about the material enters: a top of steel and a top of cheese on legs of steel and legs of cheese give the same three numbers, provided both stay rigid enough to keep the geometry. Every reaction is positive, so the load lies inside the triangle of support and the top stands. Move it outside and one of these numbers goes negative, which is a leg pulling downwards — the tipping condition, and the only way this calculation can warn about it.

physicscheck asserts something about redundant-supports that could fail — it draws it and measures the result against a value reached some other way.

Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.