Structural Engineering & Statics
where the load goes
Concept

Janssen — where it appears

Janssen's solution for the pressure in a tall bin of granular material, in which wall friction carries part of each slice's weight. The pressure rises with depth towards a limit set by the bin's size and the wall's friction, not by its height.

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

A tank grows without limit; a silo stops. Vertical pressure against depth in a 8 m silo of a solid weighing 9 kN/m³, beside the straight line a liquid of the same weight would have produced. The free body is a slice: its own weight in, the wall friction out, and the friction is μK times the vertical pressure that generates it. The result saturates at γR/μK = 89 kN/m² and reaches 63% of it at one characteristic depth, R/μK = 9.9 m. At the base the pressure is 85 kN/m² against a liquid's 270 — 69% less — and the wall has taken 69% of the stored weight down with it. The exponent is the capstan's, and for the same reason.

The pressure that stops growing

A tank of liquid presses harder the deeper it gets, without limit. A silo of grain does not. Wall friction carries part of the weight, the pressure that generates the friction is proportional to the pressure being carried, and the equation that follows is the one that describes a rope round a bollard.

equilibrium · Silo pressure The pressure on the wall jumps where the bin meets the hopper. The normal pressure on the wall of a bin 8 m across holding 30 m of a solid weighing 9 kN/m³, wall friction 0.45 and K = 0.45, over a conical hopper of half-angle 20° with wall friction 0.40, the solid's internal friction 35°, down the bin and then down the hopper: in the bin, K times Janssen's vertical stress (solid) and that times the 1.35 discharge factor (dotted); in the hopper, filling (light) and emptying (dark). At the foot of the bin the wall carries 38.1 kPa, 51.4 with the discharge factor. On the hopper's top edge it carries 75.8 kPa filling and 95.7 emptying — 1.99 and 2.51 times the bin wall's. Half way down the hopper, 46.3 and 36.7 kPa.

The pressure that switches at the hopper

Below a silo's bin a conical hopper funnels the grain to its outlet, and the wall pressure there does not keep growing with depth as a liquid's would. It peaks at the hopper's top edge. Filling, the edge carries twice the bin wall's pressure just above it; emptying, the grain converges on the outlet, its largest stress turns from vertical towards horizontal, and the edge carries two and a half times. Half a metre above the apex the stress is 2.6 kPa whether the bin holds 5 m of grain or 60 — which is why a silo empties at the same rate full or nearly empty.

equilibrium · Silo pressure

Named alongside it

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

FrictionSiloArchingCapstanCharacteristic depthDischargeFree bodyHoop tensionHydraulic radiusHydrostatic pressureLateral pressure ratioPrincipal stress

All concepts