Illustrated Physics
physics, drawn
Fluids

The silo that does not weigh what it holds

Pour water into a tall vessel and the pressure at the bottom is the depth times the density times g, whatever the shape above it. Pour grain in and the floor stops learning anything new after the first couple of metres, because the walls have quietly taken the rest — and the length over which they take it contains no property of the grain at all.
18 min read 8 figures What happens at the edgeOrder out of the random

Assumes: The pressure that only knows depth · The force that takes what it needs

A grain silo standing forty metres tall holds perhaps a thousand tonnes of wheat. The floor of it is designed for a pressure of about fifteen kilopascals — a sixth of an atmosphere, less than the pressure under a car tyre, and about what a metre and a half of water would deliver. The other thirty-eight and a half metres of grain are pressing on something, and it is not the floor.

The stress that stops growing with depth. Vertical stress against depth in a silo of radius 0.5 m holding grain of bulk density 1500 kg/m³, with a wall friction coefficient of 0.5 and Janssen's ratio K = 0.5. The straight line is what a liquid of the same density would do — ρgz, with no length in it anywhere. The curve is what grains do: wall friction, mobilised by the sideways stress the grains themselves exert, removes weight from the column at a rate proportional to the stress, so the stress saturates at ρgλ over a screening length λ = R/2μK = 1.00 m. Read off the drawn curve at the 8 m base, the stress is 14.7 kPa against the 117.7 kPa the liquid delivers — 88 per cent of the weight is standing on the walls. Another twenty metres of grain would move the floor's reading by less than a pascal.
Fig. 1 Vertical stress against depth in a silo half a metre in radius, for grain of bulk density 1,500 kg/m³, drawn against what a liquid of the same density would do. The straight line has no length in it and rises for ever. The curve has a length in it — Janssen’s λ = R/2μK, one metre here — and stops.

The reason is a single missing property, and the fact that it is missing rather than present is what makes the hydrostatic answer so persuasive and so wrong.

What hydrostatics is actually a consequence of

The pressure at depth hh in a liquid is ρgh\rho g h, and the shape of the vessel above that point makes no difference whatever — a conical flask, a cylinder and a flared beaker filled to the same depth read the same on the same gauge, even though the weights they contain differ by a factor of several.

Three vessels of very different contained weight, filled to one depth, have the same pressure at the base. What makes them equal is that a liquid transmits no shear: the walls can push perpendicular to themselves and in no other direction, so a sloping wall carries a vertical share of the load and a vertical wall carries none. Hydrostatics is not a law about liquids so much as a consequence of that one restriction — and grains, which can carry shear at their walls, are outside it from the first line.

That is not an axiom. It is a consequence of a liquid at rest having no shear strength: the only force a wall can exert on it is perpendicular to the wall. A vertical wall is therefore incapable of supporting any of the liquid’s weight, however tall it is, and every gram of what is above a point in the fluid ends up bearing on that point.

Grains break the assumption in the plainest possible way. Two grains in contact resist sliding past each other, and a grain against a steel wall resists sliding down it. There is a shear strength, it is frictional, and — as friction always is — it is a response with a ceiling rather than a force with a value, the ceiling being proportional to the normal force pressing the surfaces together.

What friction returns, against what it is asked for. The friction force on a block under a 50 N normal load, against the force applied to it. Below 27.5 N — the static limit μs·N — friction returns exactly what is asked for and nothing moves, so the curve is the 45° line and the coefficient never appears. At that point the surface gives way and the force drops to μk·N = 21.0 N, where it stays however hard the block is pushed. The gap above the flat line is the surplus that accelerates it: 24.0 N at the right-hand edge of the axis.
Fig. 2 Friction as a response rather than a force: it takes whatever value is needed to prevent sliding, up to a ceiling set by the normal force, and only then does anything move. The ceiling is the part that matters in a silo, because the normal force pressing grain against wall is itself produced by the weight the wall is being asked to carry.

Everything below follows from that one addition, and the arithmetic is a hundred and thirty years old.

Janssen’s slice

Take a horizontal slice of the column, of thickness dzdz, at depth zz. It is pushed down by the vertical stress σv\sigma_v on its upper face and by its own weight ρg dz\rho g\,dz per unit area, and pushed up by the stress on its lower face and by friction on the wall around its rim.

The friction is μ\mu times the horizontal stress the grains exert outwards on the wall, and that horizontal stress is the one quantity in the problem that cannot be derived from statics. Janssen, in 1895, took it to be a fixed fraction KK of the vertical stress. That is a modelled assumption and it is the only one; everything else is a force balance:

dσvdz=ρg−2μKR σv.\frac{d\sigma_v}{dz} = \rho g - \frac{2\mu K}{R}\,\sigma_v .

The equation is a familiar one in disguise. A quantity is fed at a constant rate and drained at a rate proportional to itself, so it approaches a plateau exponentially — and the length over which it does so is

λ=R2μK,\lambda = \frac{R}{2\mu K},

which contains the radius of the silo, the friction coefficient, and Janssen’s constant. It contains no density, no grain size, and no fill height. The plateau it approaches is ρgλ\rho g \lambda.

The fraction the floor is allowed to feel. The share of the stored weight that reaches the floor of a silo, against how many screening lengths deep the fill is. A shallow fill behaves like a liquid and the floor carries all of it. Past about two screening lengths the curve has become the hyperbola λ/H — to 0.03 per cent at the right-hand edge — and that says something stronger than that the fraction is falling: a constant divided by a growing weight means the force on the floor has stopped changing, so every further tonne poured in is carried entirely by the walls. At 1λ the floor takes 63 per cent; At 2λ the floor takes 43 per cent; At 4λ the floor takes 25 per cent; At 8λ the floor takes 12 per cent.
Fig. 3 The share of the stored weight that reaches the floor, against how many screening lengths deep the fill is. Past about two lengths the curve is the hyperbola λ/H, and a constant divided by a growing weight means the force on the floor has stopped changing: every further tonne poured in is carried entirely by the walls.

The hyperbola is worth pausing on, because a falling fraction is less alarming than what it means. The force is constant. A silo half full and a silo overflowing press on their floors with the same total force, and the difference between them is entirely a difference in what the walls are carrying.

The gauge that measures the wrong quantity

Since the plateau is ρgR/2μK\rho g R/2\mu K, it is proportional to the radius and independent of everything about the filling.

What the plateau depends on, and what it does not. The saturation stress ρgR/2μK against silo radius, at three wall friction coefficients, for grain of bulk density 1500 kg/m³ and K = 0.5. Each is straight to within 0.01 pixels, measured as the largest departure of a drawn vertex from the chord through its own ends. Two things are worth reading off it. The plateau is proportional to the width of the silo, so a wide flat bin is nearly hydrostatic and a narrow tower is nothing like it. And the fill height appears nowhere on the figure, because it appears nowhere in the quantity: a silo floor cannot tell a full silo from a quarter-full one, provided the quarter is more than a few λ deep.
Fig. 4 The saturation stress against silo radius, at three wall friction coefficients. Straight lines through the origin, verified on the drawn vertices to a hundredth of a pixel. The fill height appears nowhere on the figure because it appears nowhere in the quantity.

A wide shallow bin is nearly hydrostatic — its screening length exceeds its fill, so the exponential never gets going. A narrow tower is nothing like hydrostatic and never was. The same reversal is available in the other direction: a hydraulic press multiplies a force because a liquid does transmit pressure undiminished, and a granular press would not, because most of the input would end up in the cylinder wall. The practical consequence is the one every structural engineer who works on silos knows and nobody else does: a load cell under the floor of a tall silo measures the width of the silo and the friction of the wall, and tells nobody how much grain is inside. The weighing is done on the wall.

It also runs the other way. The wall is carrying a shear that grows with depth toward ρgR/2\rho g R/2 per unit area of wall, and if the grain is discharged from the centre the flow pattern changes, KK changes with it, and the wall load can jump. Silos fail more often than tanks do, and they fail on the wall rather than the floor.

What the constant is an average over

KK is the modelled quantity, and it is worth asking what it is an average of. The picture that goes with a continuum stress is a smooth transmission of load from grain to grain. That is not what happens.

Put a grain at every site of a triangular lattice. Give each one unit weight, and let it pass its whole load to the two grains below it, splitting it in a ratio drawn at random for each contact. Nothing is created and nothing is lost: the total load in each row is exactly the number of grains above it.

The chains a uniform load breaks into. A 12-row triangular packing in which every grain weighs the same and passes its whole load to the two grains below it, in a ratio drawn at random for each contact. Nothing is lost: the total load in each row is exactly the number of grains above it, checked here to 1.9e-16. The line widths are the contact forces, and they are uniform nowhere — the load travels in chains with quiet regions between them. In the bottom row the heaviest grain carries 1.91 times the mean and the lightest 0.15 times it, out of a rule containing no heterogeneity at all. This is what Janssen's constant is an average over, and why a silo wall is designed for a pressure it will never see uniformly.
Fig. 5 A twelve-row packing under that rule, with line widths proportional to the contact forces. The load travels in chains, with quiet regions between them, and the heaviest grain in the bottom row carries nearly twice the mean while the lightest carries a seventh of it. The rule contains no heterogeneity at all; the heterogeneity is what iterating a random split produces.

The chains are visible in real packings — photoelastic discs under load light up in exactly this pattern — and they are the reason a silo wall is designed against a pressure it will never experience uniformly. A gauge the size of a grain and a gauge the size of a hand report different things, and only the second reports what Janssen’s KK describes. That is the same distinction that separates the pressure of a gas from the impacts that make it: a gas’s pressure is a rate of arrival averaged over an area and a time, and shrinking either until individual events resolve destroys the quantity rather than measuring it more finely.

The closed form, and the depth at which it becomes true

The distribution of loads in this model has a closed form. With two supports and a uniform split, the steady state is P(f)=4f e−2fP(f) = 4f\,e^{-2f} in units of the mean, a gamma distribution of variance exactly one half. It is quoted everywhere.

The closed form, and how deep a pile has to be to obey it. The load on the grains of the bottom row, normalised to the mean, against the distribution 4f·e^(−2f) the q-model is quoted as having. The bars are 900 piles 12 grains deep — 13500 grains in all — with mean exactly 1, as conservation requires, and variance 0.237 against the closed form's exactly 0.5: an integrated absolute difference of 0.335. That variance is checked against the lattice's own second-moment recursion, which gives 0.236 without sampling anything. The dashed outline is the same model run 96 grains deep, at variance 0.411 and a difference of 0.088. Nothing about the rule has changed between them — the closed form is an asymptote in depth and a shallow pile has not reached it. What both already have is the shape that matters: zero probability of a grain carrying nothing, and an exponential rather than Gaussian tail, so a grain at five times the mean is rare and not vanishingly rare.
Fig. 6 The bottom-row loads of nine hundred twelve-row packings, against that closed form, with the same model run ninety-six rows deep as the dashed outline. The sampled mean is exactly one, as conservation requires, and the variance is 0.237 against the closed form’s 0.5. The rule has not changed between the two histograms. Only the depth has.

A twelve-grain pile has less than half the steady-state variance. The reason it is not a defect in the sampling is worth establishing separately, because the second moments of the lattice satisfy a closed recursion and can be computed exactly at any depth without sampling anything.

How far down a pile the closed form starts to be true. The distance still to run — ½ minus the variance of the normalised load — against the depth of the pile, both logarithmically, from 10 rows to 10000. Every point is exact rather than sampled: the second moments of the lattice satisfy a closed recursion, which is what is iterated here. The dashed line through all 7 has slope −0.471; through the deepest three, −0.494, against the −0.5 of an inverse square root. So the q-model does reach the distribution it is quoted as having, and it reaches it at the rate a random walk closes anything — to halve what is left, dig four times as deep. A pile 10 grains deep has 43 per cent of the steady-state variance and one 100 deep has 79 per cent, which is worth knowing before a closed form is compared against a photograph of a laboratory packing twenty beads high.
Fig. 7 The distance still to run — a half minus the variance — against depth, both logarithmically, computed exactly. The line through the deepest three points has slope −0.494 against the −0.5 of an inverse square root. So the closed form is right, and it is an asymptote: to halve what is left, dig four times as deep.

The square root is not a coincidence either. The correlation between two neighbouring grains’ loads is inherited from their shared ancestry a few rows up, and the number of rows over which two sites at a fixed separation become independent grows as the square of that separation — which is the same diffusive spreading a random walk has, running sideways through the packing while the load runs down it.

This is a pattern worth naming, because it recurs whenever a stochastic model is compared with an experiment. The distribution a model settles into and the distribution a finite realisation of it shows are different objects, and the gap closes as a power of the size. A laboratory packing twenty beads deep is at about half the asymptotic variance. Comparing a photograph of one against 4fe−2f4fe^{-2f} and reporting a discrepancy would be reporting the depth of the box.

Where the grains stop being a continuum

The angle a heap of grain settles at is the second place the friction shows itself, and it is the cleanest measurement of a granular material there is: a pile poured onto a flat surface builds up until its surface reaches an angle at which a grain on the surface is on the point of sliding, and then grows sideways instead.

Sliding on a slope begins when the along-slope component of the weight exceeds the friction ceiling, which happens at tan⁡θ=μ\tan\theta = \mu and at no other angle. So the angle of repose of a heap is a direct reading of the friction coefficient, taken with a protractor — which is why a pile of dry sand and a pile of gravel have visibly different profiles, and why the same material poured wet gives a different number again.

That single number is doing a great deal of work in the arithmetic above, and it is not a constant of the material. It is closer in spirit to a mean free path than to a density: a statistical property of a disordered arrangement, well defined as an average and not attached to any grain. It depends on humidity, on how the grains were poured, on whether the pile is being built up or drained, and on how much the whole thing has been shaken. A silo filled slowly and a silo filled fast have different packing fractions and different KK.

And there is a dynamic failure of the continuum picture that a silo makes audible.

Apply the same steady pull through two holders of different stiffness and one contact slides smoothly while the other builds, gives way, jumps and repeats. Discharging grain does the second thing at the wall. A silo can emit a low periodic honk audible a hundred metres away, produced by the whole column stick-slipping down its own steel, and the frequency depends on the discharge rate rather than on any resonance of the structure — which is how it is told apart from a vibration and why slowing the flow sometimes stops it.

The number that decides everything, and how badly it is known

Everything above rests on two coefficients, and it is worth stating plainly how well either is known.

The wall friction coefficient μ can be measured directly, by shearing a sample of the grain across a plate of the wall material under a known normal load, and for wheat on smooth steel it comes out at about 0.3 and for wheat on corrugated steel at about 0.5. That is a factor of nearly two in λ, and therefore a factor of nearly two in the plateau, from a choice of wall finish that has nothing to do with the grain at all.

The area that is not the area. The real area of contact against the load, for a surface of hardness 1000 MPa. Contact happens only at asperities, which flatten until they can carry the load, so the real area is the load divided by the hardness: 0.0500 mm² under 50 N, whatever the block looks like. Two faces differing 4-fold in apparent area — 400 mm² against 100 mm² — touch over 0.013% and 0.050% of themselves, and over the same absolute area. That is the whole of why the coefficient of friction carries no area in it.
Fig. 8 Why a friction coefficient is a number at all: the true area of contact between two rough surfaces is set by the load and the hardness of the softer material, not by the apparent area, so the friction force is proportional to the load and independent of how much surface is nominally touching. A silo wall and a grain touch over a small fraction of what looks like contact, and the fraction rises with depth exactly as fast as the load does.

Janssen’s K is worse. It is not measurable in the same direct way, because it is a ratio between two components of an internal stress rather than a property of a boundary; it is inferred from the very pressures the model is being used to predict. Values between 0.3 and 0.6 are all defensible, and the two ends differ by a factor of two in λ. Multiplying the two uncertainties gives a plateau known to within a factor of about four before any grain has been poured — which is why silo codes are written around measured pressures with generous factors rather than around this equation.

None of that damages the argument. The structure of the answer — a saturation, over a length proportional to the radius, independent of the fill — survives any value of either coefficient, and it is the structure that overturns the hydrostatic intuition. What the uncertainty damages is the number, and the honest form of the claim is that a silo floor reads a pressure set by the silo’s width to within a factor of a few, rather than a pressure set by its contents to within anything at all.

The same exponential, wrapped round a post

Janssen’s equation has the form of a load drained at a rate proportional to itself, and that structure is not peculiar to grain. The cleanest other instance is a rope round a bollard.

Take a rope over a cylindrical post and pull one end. The tension in the rope falls as it goes round, because friction against the post takes a share proportional to the normal force pressing it there — and the normal force at each point is proportional to the tension. Same rule, same solution: the tension falls exponentially with the angle wrapped, T=T0e−μθT = T_0 e^{-\mu\theta}.

The consequence is the reason mooring works. Three turns round a bollard is nearly nineteen radians, and with a friction coefficient of 0.3 that is a factor of nearly three hundred: a person holding the free end with a hundred newtons restrains thirty kilonewtons at the other. Nothing is gripping; the exponential is doing all of it, and adding a fourth turn multiplies the holding force by another six.

The parallel with the silo is exact. In both, friction removes load at a rate proportional to the load present, so the natural variable is a ratio rather than a difference, and the answer is set by a dimensionless quantity — angle times friction for the rope, depth over radius times friction for the silo. And in both, the striking behaviour comes from a mechanism the simpler description leaves out: a frictionless post is a pulley and a frictionless silo is a tank of liquid.

The dip under the middle of a heap

The strongest evidence that a granular column is not a continuum with a simple constitutive law is a measurement anybody would predict wrongly.

Pour sand into a conical heap on a plate and measure the vertical stress along the plate. The obvious expectation is a maximum directly beneath the apex, where the pile is deepest. What is measured, for a heap poured from a point, is a local minimum there — a dip of twenty or thirty per cent below the surrounding ring of maximum stress.

The chains are why. Grains added at the top run down the flanks of the growing heap and their load is carried outward along the arches that formed as the pile was built, so the interior of the cone is partly shielded by an arch of its own material, in the same way the middle of a silo is shielded by its walls.

The decisive part is what happens when the heap is made differently. Build the same cone by raining grain uniformly over the whole area, rather than pouring it from one point, and the dip does not appear: the stress profile has its maximum at the centre after all.

Two piles of the same material, the same shape and the same weight, resting on the same plate, with different stress distributions underneath them. No description that assigns a stress to a shape can produce that, because the shape is identical. What distinguishes them is the history of how the contacts were laid down — which is exactly the quantity Janssen’s KK is quietly averaging over, and the reason it is different for a silo being filled and one being emptied.

Where the model stops

KK is not derived, and cannot be. The balance is exact; the closure is not. A vertical stress does not determine a horizontal one without a constitutive assumption, and Janssen’s constant KK is that assumption in its simplest possible form. Measured values run from about 0.3 to about 0.6 and differ between filling and discharge, which moves λ\lambda by a factor of two and the plateau with it.

The model is one-dimensional and the failures are not. Averaging the stress across a horizontal slice discards exactly the structure the force chains have, and a real silo’s wall load varies round its circumference by more than the safety factor. Nothing in the equation knows that.

Discharge is a different problem entirely. Everything here is static. A silo being emptied develops a flow channel, the stress state switches from the filling state to the discharge state through a travelling front, and the wall pressures overshoot the static values substantially. The switch is the reason silos are designed for two load cases rather than one, and it is the one an essay about a static column cannot reach.

Nor is any of this thermodynamics. A packing has an enormous number of arrangements and might be expected to visit them, and entropy counts arrangements — but grains are far too heavy for thermal motion to move them, so a packing does not explore its own configurations and has no temperature in the ordinary sense. Every argument above is mechanical, and the statistics in the force-chain model come from the arbitrariness of the packing rather than from any agitation of it.

And a silo as a structure to be designed is not this subject. What is computed here is a stress profile from a friction coefficient and a radius. Sizing a wall against it, choosing a factor of safety, and deciding where the load cells go are structural questions and belong to the collection that owns them.

What the pictures cannot show

The Janssen curve is drawn as a single smooth line, and there is no such thing in a silo. The stress at a given depth is a distribution, whose width is the force-chain figure’s whole subject, and drawing the mean as a curve is the same simplification as drawing a material’s relaxation time as a point.

Nor can the force-chain figure show what happens when the load changes. The chains are not permanent structures; they rearrange, sometimes suddenly, and a grain that was carrying five times the mean can be carrying a tenth of it after a tap on the side of the vessel. The drawing is a snapshot of something whose interesting property is that it is not one.

Where this ladder goes next

What has been established is that a bag of grains is a material of its own kind: it has a shear strength like a solid, it flows like a liquid, and the two behaviours are separated by a stress rather than by a temperature. The screening length is the first of several places where a granular system introduces a length that neither of its neighbouring descriptions contains.

The habit worth carrying away is the one the whole essay turns on. When a familiar law contains no length, ask what would have to be added to give it one. Hydrostatics has no length because a liquid has no shear strength; adding friction adds a length, and the length immediately becomes the only thing that matters. The same move explains why a gas has a viscosity that does not depend on its density, why a wave in a periodic medium has a forbidden band, and why a slipping contact chatters at some stiffnesses and not others: in each case a length or a time has been introduced by a mechanism that the simpler description omits, and the new quantity organises everything.

What is left on this ladder is the dynamic half: what a granular material does when it is made to move, where the same friction that produced the screening length produces a flow with a yield stress, a segregation of large grains from small, and a jamming transition that has no counterpart in either a solid or a liquid.

Part 1 of 6

This essay is one argument about Granular matter. The others:

The slope a heap of grains settles at. The steepest slope a cohesionless heap can hold, against the friction between its grains. A slab of thickness h on a slope is pushed down it by the weight's along-slope component and held by friction acting on the weight's across-slope component, and both are proportional to the same ρgh — so the thickness cancels and the criterion is tan θ = μ. Checked here at thicknesses of 2 mm, 50 mm, 2000 mm, the ratio of driving to holding stress differs by 2.2e-16, which is zero. That is why the quantity is an angle: nothing about the size of the pile, the size of the grains, the density or the strength of gravity survives into it, and a heap of sand on the Moon stands at the same slope as one on Earth. The marked materials are glass beads at 24°, dry sand at 33°, crushed gravel at 40°. The band between the two curves is the hysteresis: a slope steeper than 31.0° will keep flowing once started, and one shallower than 35.0° will not start — so a pile has a range of stable angles rather than one, and which it is found at depends on how it was built. The angle that does not know the size of the heap Part 2 — Pour sand and it makes a cone with a definite slope, and pouring more makes a larger cone with the same slope. The reason the answer is an angle rather than a length is a cancellation — the force pulling a surface layer downhill and the friction holding it back are both proportional to its weight, so everything about the size of the pile divides out — and everything that puts a length back in is a story about cohesion.

Where this goes next

Three vessels, one pressure. Three vessels filled to the same depth of 3 m. The pressure on each base is 29.4 kPa — identical, because pressure is set by depth — while the weight of water each holds differs by a factor of 4.7. The base of the flaring vessel carries more force than the water standing over it weighs. The pressure that only knows depth A litre of water and a swimming pool press equally hard on a floor at the same depth. Pressure in a still fluid is a scalar with no direction of its own, it depends on how far down and on nothing else, and the shape of the container falls out of the arithmetic entirely. What friction returns, against what it is asked for. The friction force on a block under a 50 N normal load, against the force applied to it. Below 30.0 N — the static limit μs·N — friction returns exactly what is asked for and nothing moves, so the curve is the 45° line and the coefficient never appears. At that point the surface gives way and the force drops to μk·N = 22.5 N, where it stays however hard the block is pushed. The gap above the flat line is the surplus that accelerates it: 22.5 N at the right-hand edge of the axis. The force that takes what it needs Static friction has no value of its own. It supplies exactly what equilibrium demands and not a newton more, right up to the moment it cannot — which is the only instant in the whole business at which a coefficient of friction means anything at all. The same steady pull, twice. Spring force divided by the normal load, against time, with the far end drawn away at 100 µm/s in both traces and every property of the contact the same. The soft holder at 0.4 k_c produces events every 3.75 s, each of which reaches 0.94 m/s — 9.4e+3 times the speed it is being pulled at. The stiff one at 1.6 k_c settles to a straight line and stays on it. The chatter a stiffer holder removes A brake squeals, a bow sounds a string, a fault slips in jerks. The usual explanation is that static friction exceeds kinetic friction — and that explanation, taken seriously, predicts the jerking would happen no matter how the thing were held. It does not, and what decides is a length nobody mentions.

What links here

Essays that reach for this one mid-argument — the half of a link its own author cannot write down.

What this makes readable

Essays that declare this one a prerequisite.

The objects named here

The third axis, after the field and the reading path: the things themselves, and every essay that touches each one.

Angle of reposeConservationForce-chainFrictionGranular matterHydrostatic pressureScreening lengthShear strengthStick-slipStress