Structural Engineering & Statics
where the load goes
Concept

Friction angle — where it appears

The angle whose tangent is the ratio of shear strength to effective normal stress in a soil, written φ′. It sets how much strength a slip surface gains from the soil pressing on it, which is why a frictional soil fails shallow and a cohesive one deep.

Named by 4 essays across one field — 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 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°.

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.

equilibrium · Slope stability 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.

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.

equilibrium · Slope stability Every slip lies on a ray set by its own slope, not on the clay's line. The shear stress and assumed normal stress on the slip plane of each of the 655 slips among 6,000 clay cuttings of one site, its clay at a friction angle of 22° and a cohesion of 4 kPa, with a sample of the cuttings that stood. The clay's own line reaches 16.1 kPa at 30 kPa; the line through the slips has a friction angle of 24.1° and a cohesion of 0.7 kPa, 14.1 kPa there; the line between what fell and what stood, 18.8° and 6.3 kPa, 16.5 kPa there.

A slip measures the slope it fell from

A slipped cutting is a full-scale shear test, and a railway has hundreds of them, so a line through all those points ought to be the clay's strength, with the scatter averaged away. It is not. Each slip's point lies on a ray whose slope is its own cutting's angle, whatever the clay, so the line through them reads mostly the slopes that were built and the pore pressure the analyst assumed. More slips only make the same wrong answer more precise. The clay is in the cuttings that stood, and at the line between what fell and what did not.

equilibrium · Slope stability

Named alongside it

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

Back analysisCohesionPore pressureSlope stabilityBishop methodEffective stressFactor of safetyBearing capacityCollapse mechanismEarth pressureFailure surfaceFree body

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