Structural Engineering & Statics
where the load goes
Concept

Earth pressure — where it appears

The horizontal stress soil applies to a structure retaining it, expressed as a coefficient times the vertical stress. The coefficient is not a soil property but a movement one: it falls to its active value when the wall moves away and rises toward its passive value when the wall pushes in.

Named by 8 essays across 2 fields — each of them below, with the objects they name alongside it.

A bearing capacity is a mechanism, and here it is. Prandtl's collapse mechanism under a 3.0 m footing in a soil of 32° friction. A rigid wedge is driven down with the footing at 61° to the horizontal; a fan of radial shear turns the stress through exactly ninety degrees on a logarithmic spiral whose growth rate is tanφ; and a passive wedge at 29° has to be pushed up and out of the way. Nothing here is empirical — every angle is a function of φ alone — and the mechanism reaches 15.9 m from the centre, which is 10.6 times the footing's half width. That is why two footings closer together than about four widths do not have separate bearing capacities.

The ground is a mechanism

Bearing capacity is met as a formula with three terms and a table of coefficients, and that presentation hides what it is. Underneath is a plastic collapse mechanism — a rigid wedge, a fan of radial shear on a logarithmic spiral, and a passive wedge that has to be pushed up and out of the way — and every coefficient in the table is a property of that one drawing.

equilibrium · Bearing capacity The prop load is decided by the digging, not by the hole. Prop forces in a 12 m excavation propped at 3 levels, with the force each prop reaches at any stage of the sequence drawn thick and the force the finished arrangement gives it drawn thin. The pale dots are the individual stages. The middle prop reaches 214 kN while the dig is at 9.5 m and finishes at 88 — a factor of 2.44 between the two, and the larger one is not in the final analysis anywhere. Terzaghi and Peck's apparent pressure diagram, a rectangle of 49.4 kN/m², reproduces the total of the staged maxima to 2% — which is what it is: an envelope of measured prop loads, back-figured into a pressure, and a shape nothing on a wall is ever loaded with.

Every prop has its own worst day

A braced excavation has no finished state worth analysing. It is dug in stages, a level of props goes in at each stage, and a prop's force is largely fixed the moment it is installed — so the force to design it for is the largest it sees during a sequence that appears on no calculation sheet, and which for the middle prop here is nearly two and a half times what the finished arrangement gives.

structures · Propped excavation The lining that carries less for being weaker. Bending moment and hoop thrust in a circular lining, against the lining's own bending stiffness, both as fractions of the free-ring values. The ground arrives already stressed — 500 kPa vertically and 300 horizontally at K₀ = 0.6 — and the difference between them tries to squash the hole into an ellipse. A lining stiff enough to refuse absolutely collects the whole distortion pressure, p₂R²/3 = 300 kNm/m; one flexible enough to go with the ground collects nothing, because there is no curvature change left to resist. The thrust is the flat line: it comes from the mean stress rather than the difference, so it does not move at all. Putting 8 joints in this ring drops the moment to 34% of the solid one and leaves the thrust exactly where it was, which is why a segmental lining is jointed and why the intuition carried over from a beam is inverted here.

The lining that is stronger for being weaker

A tunnel lining is not loaded. The ground arrives already stressed and the hole wants to squash into an ellipse; the lining's only job is to refuse, and how much moment it collects depends entirely on how hard it refuses. Make it stiffer and it takes more. Make it flexible — put joints in it, make it thin — and it takes almost none, while the hoop thrust it carries does not move at all.

structures · Tunnel ring Two identical pipes, and one carries three times the other. Load per metre on a buried conduit against the depth of cover, in trench widths, with the weight of the prism of soil directly above it drawn between them. A conduit laid in a narrow trench is stiffer than nothing and softer than the sides: the backfill settles relative to the undisturbed ground, the friction on the trench walls acts upward, and the conduit gets 64% of the prism. Lay the same conduit on the ground and build an embankment over it and it is now stiffer than the fill beside it, the interior prism settles less, the friction acts downward, and it gets 172% — a factor of 2.71 between two pipes with nothing different but which way the ground moved. The equation is Janssen's, the same one a silo wall obeys, with a trench for a silo; both curves start on the prism line, because with no depth there is no shear to redistribute anything. This is why a flexible pipe is buried rather than a rigid one: making the conduit weaker moves it down the page.

The pipe decides what the soil weighs

A buried conduit is not loaded by the soil above it. It is loaded by whatever share of that soil the relative movement leaves it — and which way the shear on the sides of the prism acts depends on whether the conduit settles more or less than the ground beside it. Two identical pipes under identical fill, one carrying two thirds of the prism and one carrying nearly twice it.

equilibrium · Soil arching The summer that is worse than the one before it. The earth pressure behind an integral abutment, summer by summer, as a multiple of the at-rest value it started at. A 60 m deck expands by 10.8 mm at each end and pushes the abutment into the backfill. Granular soil under cyclic strain densifies, so the same movement next year needs a higher pressure to achieve, and K climbs from 0.38 toward 0.96 — a factor of 2.49 on the force, reached after about a century. The design load on an integral abutment describes the bridge's whole life rather than a load case, and it is the only load in this collection that gets larger because time has passed rather than because something was added.

The summer that is worse than the last

An expansion joint is a hole in a deck that leaks salt water onto the bearings underneath it. Remove it and the thermal movement does not go away — it goes into the soil behind the abutment, twice a day for a hundred and twenty years, and granular soil under cyclic strain gets denser.

structures · Integral bridge The roller's pressure, not the soil's. Horizontal pressure down the back of a wall, for backfill of 20 kN/m³ and φ′ = 30° behind a wall 3.0 m high, compacted by a roller of 60 kN per metre of its width. Dashed, the soil's own active pressure, Kₐ·γz (Kₐ = 0.33), and at-rest pressure, K₀·γz (K₀ = 0.50). Solid, Ingold's compaction envelope: rising as the passive limit γz/Kₐ to 27.6 kPa at 0.46 m, then constant at √(2pγ/π) = 27.6 kPa all the way down, since the active pressure would not reach it until 4.15 m. Its thrust is 76.6 kN per metre of wall against the active 30.0, and its moment about the base 106.3 kN·m against 30.0.

The pressure the roller leaves behind

A retaining wall is designed for the pressure of the soil it holds, a triangle growing with depth. The soil behind most walls was not tipped there. It was placed in layers and rolled, and each pass of the roller pushed the soil sideways against the wall harder than the soil's own weight ever could, and left most of that push locked in when it moved on. Behind a 3 m wall compacted by an ordinary vibrating roller the locked-in pressure is a constant 28 kPa from half a metre down, the thrust two and a half times the active triangle's and the moment at the base three and a half.

equilibrium · Lateral pressure Where a bending stem sheds its compaction pressure. The share of the rigid build-up's pressure shed at each depth by a cantilever stem 3 m high, bending stiffness 67,500 kN·m² a metre (a concrete stem about 300 mm thick), backfilled in 250 mm lifts of soil at 20 kN/m³ and 30°, each compacted by a roller of 60 kN a metre: nothing at the base, which does not move; little in the top lift, which nothing is placed after; most, 20 per cent, at 0.9 m down, where the pressure was locked early and the stem went on moving away from it.

The stem that bends away from the roller

A roller compacting backfill against a wall leaves a lateral pressure behind it that the soil's weight alone would never make, and for a wall that cannot move that residual is Ingold's envelope. A cantilever stem moves as the fill goes up, and a residual locked in against it relaxes as the stem bends away. But not everywhere: the base never moves, so it keeps its pressure, and the last lifts have nothing placed after them, so they keep theirs. A 300 mm stem sheds an eighth of the rigid wall's base moment and the softest stem a third — never more, because the two ends of the stem are where the relief cannot reach.

equilibrium · Lateral pressure Propped before it is filled, the stem keeps the roller's pressure. For a 3 m stem of stiffness 67,500 kN·m² a metre, fixed at its base, propped at its top by a slab cast first, backfilled in 250 mm lifts with a 60 kN/m roller: the lateral pressure on the stem after the last lift, in kPa, against depth — propped at its top before the fill (solid), the same stem as a free cantilever (dashed), a stem too stiff to move (dotted) and the active pressure. The propped stem moves at most 0.15 mm, in its middle, where the free cantilever's top moved 2.61 mm; it relaxes almost nothing, and its pressure lies on the stiff stem's along the whole height — the roller's residual, 27 kPa from half a metre down, four times the active pressure at mid-height.

The prop that holds what the roller left

A basement wall whose ground-floor slab is cast before the backfill spans between its base and the slab, and as the fill is compacted against it the wall bows by a sixth of a millimetre. A free cantilever stem leaned sixteen times as far and shed an eighth of the roller's pressure; the propped wall sheds one per cent. Its base moment falls to a third, which looks like a gain, but the pressure has not gone: the slab now carries 22.6 kN a metre against the 6 an active-pressure design gives it and the 9 of soil at rest.

equilibrium · Lateral pressure

Named alongside it

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

Active pressureRetaining wallStiffnessCompactionFree bodyImposed deformationLower-bound theoremPassive pressureAt rest pressureConstruction sequenceLateral pressureLoad path

All concepts