Structural Engineering & Statics
where the load goes
Generator

A bearing capacity is a mechanism, and here it is

Rendered here at the parameters it defaults to, with every essay that calls it — which is the same list as the blast radius of changing it.
A bearing capacity is a mechanism, and here it is. Prandtl's collapse mechanism under a 2.5 m footing in a soil of 30° friction. A rigid wedge is driven down with the footing at 60° 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 30° 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 12.0 m from the centre, which is 9.6 times the footing's half width. That is why two footings closer together than about four widths do not have separate bearing capacities.

A bearing capacity is a mechanism, and here it is. Prandtl's collapse mechanism under a 2.5 m footing in a soil of 30° friction. A rigid wedge is driven down with the footing at 60° 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 30° 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 12.0 m from the centre, which is 9.6 times the footing's half width. That is why two footings closer together than about four widths do not have separate bearing capacities.

5 essays call bearing-wedge. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Changing this generator changes every one of these figures.

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. Equilibrium

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.

 The group is not weaker; it is very much softer. A 3 × 3 pile cap on the left, with each pile's share of 9.0 MN and 4.5 MNm in meganewtons — N/n plus M·y/Σy², the same three terms in the same order as a bolt group under an eccentric load and a section under biaxial bending. The corner piles take 1.25 times the average and a pile added at the centroid would change that by nothing at all, because it adds to neither second moment. On the right is the effect a bolt group cannot have: the piles share ground, so the stress bulbs overlap and the group settles 3.9 times as much as a single pile at the same load per pile, rising to 14.2 for 144 of them. The capacity check everyone makes — block failure against the sum of the piles — comes out at 4.54 here and does not govern at all. The check nobody tabulates is the one that does. Structural form

Nine piles, and four times the settlement

A pile cap divides its load between its piles by the same three terms a bolt group uses and a section under biaxial bending uses. What a bolt group does not have is neighbours it shares ground with — and the group effect that matters is not the strength check everybody makes, but a stiffness effect nobody tabulates.

 The circle is searched for, and the first guess is 39 per cent optimistic. The same slope with 81 trial circles evaluated, each one through the toe and each one giving its own factor of safety. There is no equation whose solution is the answer: the slip surface is a shape the ground chooses, so the calculation is a search over shapes and the answer is the smallest number found — 1.191 against 1.650 for the circle a first guess puts through the toe from above the middle of the slope, which is 39 per cent optimistic. A slope analysis that reports one circle has reported nothing. Equilibrium

The surface that has to be searched for

Every other check in this collection is made at a section somebody drew. A slope has no section — the failure surface is a shape the ground chooses, so the calculation is a search over shapes, and the answer is the smallest number found rather than the solution of anything.

 Every one of these soils failed that slope. For a 10 m slope at 35° in soil of 19 kN/m³ with a pore pressure ratio of 0.25, over a firm stratum 5 m below its toe, which has slipped: the pairs of effective cohesion and friction angle that put its critical circle at a factor of safety of exactly one, by Bishop's method. At no friction the cohesion is 32.1 kPa; at 20° it is 12.4; at 40°, 2.0; and the line reaches no cohesion at 48.1°, where a skin of soil slides parallel to the face. Labelled, the depth of each pair's slip below the original ground: 13.6 m at 0°, 8.8 m at 5°, 7.4 m at 10°, 5.2 m at 20°, 3.6 m at 30°, 2.3 m at 40°. Equilibrium

A failed slope gives a line, not a soil

A slope that has slipped is the one full-scale strength test geotechnics ever gets, and the standard way to read it is to set its factor of safety to one and solve for the soil. One equation cannot give two strengths. What it gives is a line of cohesions and friction angles, every one of which fails the slope exactly, and the repair designed on one point of that line can be worth a factor of 1.07 or of 1.51. The slip itself says which point: each soil on the line fails along a different surface, and the depth of the scar is the second equation.

 Each slip is a shear test at one normal stress. The soil's strength, τ = 8 + σ′·tan 24° (line), and the mean effective normal stress and shear stress along the critical slip of each of five slopes that fail in it: 55° and 3.8 m high, 8 kPa; 45° and 5.2 m high, 13 kPa; 35° and 8.2 m high, 23 kPa; 28° and 14.1 m high, 42 kPa; 22° and 41.5 m high, 109 kPa. Every slip is one point on the line. Two slips fix the line as two shear-box tests do, and fix it well only if their normal stresses are far apart: the steep, short slopes test the soil where cohesion is most of its strength, the flat, tall ones where friction is. Equilibrium

Two slips are a shear test

One failed slope gives a line of soils, every one of which fits the failure. Two failed slopes give two lines, and where they cross is a soil that fits both. Whether the crossing means anything depends on how differently the two slopes loaded the ground: two slips at one angle cross only at zero cohesion, two at nearby angles cross anywhere in a twenty-degree band, and a flat, tall slope beside a steep, short one pins the soil to within two degrees — because each slip is a shear test at one normal stress, and a strength line needs two tests far apart.

The library, page 1 of 7 — where bearing-wedge sits