Illustrated Physics
physics, drawn
Concept

Statically indeterminate — where it appears

A structure with more unknown reactions than equilibrium equations, so that statics has a family of exactly valid answers rather than one. A four-legged table is the simplest case; what selects a member of the family is how much each support gives, which is information the rigid-body idealisation threw away.

Named by 7 essays across 2 fields — each of them below, with the objects they name alongside it.

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.

The table statics cannot settle

A rigid top on three legs has one possible set of reactions and a rigid top on four has infinitely many, all of them balancing every force and every moment exactly. The extra leg does not make the problem harder; it makes it unanswerable, and the answer has to come from somewhere the model deliberately threw away.

mechanics · Free-body Where a heap stops flowing. The number of motions that cost nothing, against the mean number of contacts each grain has, for a patch of 37 grains. Every point is the rank of an actual rigidity matrix — one row per contact, one column per coordinate — subtracted from the number of coordinates, so it is a measurement of the network rather than a formula about it. The dashed line is Maxwell's count, two coordinates a grain minus half a contact each, which is what the counting alone predicts. The measured curve reaches its floor of three free motions — the rigid-body ones — at a coordination of 4.11, and Maxwell's line reaches zero at exactly four, which is twice the dimension of the plane. That is the isostatic point and the difference between the two numbers is the boundary of a finite patch, whose outer grains have fewer neighbours than they would in an infinite one. Above the threshold the two curves separate: the measured floor stays at three while the count keeps falling, and the gap is the redundancy — contacts whose forces no equilibrium equation can determine. Below it the pack has genuine mechanisms and will rearrange under any load at all. The transition is in the count and not in the material, which is why a fluid and a solid here are made of exactly the same grains.

The heap that becomes a solid

Sand poured into a jar flows; the same sand shaken down and pressed does not. Nothing about the grains has changed — not their size, their hardness, their friction or their density by more than a per cent — and the thing that changed is a count of contacts, which crosses a threshold set by the dimension of space and by nothing else.

fluids · Granular matter 5 balls whose contact force goes as the overlap to the three halves, touching. The velocity of every ball in a line of 5, against time in units of one binary contact, with the first arriving at unit speed. Nothing about collisions is assumed: neighbours push on each other with k times their overlap raised to the power 1.5, and the equations of motion are integrated. With the balls touching there is no such separation — several overlaps are non-zero at once and the disturbance crosses the line as a single compression wave. The far ball leaves at 0.989 of the striking speed and the others keep 0.011 between them, which is why a real cradle's balls do not quite come to rest. Momentum and energy are conserved to 4.4e-16 and 4.1e-8, so the difference between the two cases is the contact law and not the bookkeeping.

Five balls, and the law that does not choose

The usual account of a Newton's cradle says that momentum and energy conservation force one ball out at the striking speed. For three balls or more they do no such thing: the two laws leave a whole curve of possible outcomes, and what picks one is the shape of the force between two touching spheres.

mechanics · Momentum The stress a temperature change puts into a bar that cannot move. The stress in a member held between supports that will not let it change length, against how much its temperature changes, for four materials. Each line is EαΔT, computed here from a free expansion and the force needed to undo it, and checked by evaluating it for a bar half a metre long and one thirty-seven metres long: the two agree to every figure carried, because neither the length nor the cross-section appears in the answer. steel develops 2.40 MPa for every kelvin and reaches yield at 104 K; aluminium develops 1.59 MPa for every kelvin and reaches yield at 151 K; concrete develops 0.30 MPa for every kelvin and reaches cracking at 10 K; invar develops 0.17 MPa for every kelvin and reaches yield at 1655 K. Concrete reaches its cracking stress after ten kelvin, which is less than a sunny afternoon, and is why every slab has movement joints in it. Invar is in the comparison because it was made to have a small product: it is as stiff as steel and develops a fourteenth of the stress, which is a statement about the expansion coefficient and nothing else.

The load nobody applied

A redundant structure develops forces with nothing on it. Change its temperature and the same extra constraint that made statics unanswerable also refuses the expansion — and a restrained steel member reaches its yield stress after a hundred and four kelvin, a figure that contains no length, no area and no load.

mechanics · Free-body Four stress states in one beam, and only one of them has no tension. 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.

A state no load could reach

The free direction a redundant structure leaves open can be driven on purpose. Tighten a tendon through a concrete beam and its whole stress state moves into the half of the range the material is good at; tighten a bolt hard and the load it carries fluctuates by a fifth of what is applied to it. Both put the structure somewhere no arrangement of external loads could.

mechanics · Free-body Four tolerances, four elastic answers, one collapse load. The force in each of a table's four legs against the load on it, for four different manufacturing errors, with the legs made of a material that yields at 25 kilonewtons. While everything is elastic the short diagonal pair takes more than its share by a fixed amount that depends on the error and not at all on the load, so the load at which the first leg reaches its capacity runs from 25 to 100 kilonewtons — a spread of more than a factor of three. Past that point the yielded legs hold a constant force and the others take the rest, and the difference the tolerance made is erased. Every one of the four cases collapses at 100 kilonewtons, which is four times one leg's capacity, checked here to a part in a million across the four. The quantity nobody could compute and the quantity that decides whether the structure stands are not the same quantity.

The one number the tolerances cannot touch

A redundant structure's load sharing depends on stiffnesses and manufacturing errors that nobody knows. Its collapse load does not depend on either. Once members yield they hold a known force instead of a force proportional to a displacement, the compatibility equations that needed the unknowns drop out, and the load at which the structure becomes a mechanism follows from a work balance with no stiffness in it at all.

mechanics · Free-body A floor that has to hit a rod nobody dropped. The normal force on the lower end of a uniform rod at 63° with μ = 1.6, sliding outward at √(gl) and not yet turning, against time on a logarithmic axis, for a floor whose contact stiffness is 10⁴, 10⁵ and 10⁶ times mg/l. The rigid equations have no solution here, and the compliant floor shows why: the end digs in, the deeper it digs the harder friction twists it further down, and the force climbs until the sliding stops. The stiffer the floor, the higher and briefer the spike — 20, 56, 163 times the rod's weight, ending after 41, 18, 7 thousandths of √(l/g) — a collision with no approach.

The rod that friction cannot decide about

Newton's laws, a rigid rod and Coulomb's law of friction are the first things a mechanics course teaches, and in 1895 Paul Painlevé showed that together they can fail to give any answer at all. A rod sliding on a rough enough floor, at the right angle, has no motion consistent with the three — or has two. The way out is not a new law but a forgotten fact: real contacts give a little. With that, the rod jams, and the floor delivers a sudden blow to a rod that nothing struck — an impact without a collision, which is how a stick of chalk comes to hop.

mechanics · Friction

Named alongside it

The objects these essays reach for when they reach for this one.

Constraint countingStiffnessElasticityReaction forceStatic equilibriumPreloadDegrees of freedomImpulseResidual stressRigid bodyBucklingContact

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