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Rankine's earth pressure fallacy

Professor Andrew Schofield says engineers should abandon the concept of limiting stress vectors.

In 'A fundamental fallacy in earth pressure computations', Boston Society ofCivil Engineers, Terzaghi (1936), Karl Terzaghi comments that strain does not enter the computations. He concludes: 'The fundamental assumptions ofRankine's earth pressure theory are incompatible with the known relation between stress and strain in soils, including sand. Therefore the use of this theory should be discontinued. '

Terzaghi used the Mohr Coulomb equation +/- t = c' + s' tan f' for limiting stress vectors across planes. He never said what relation applied to stress and strain in sands. Rankine would not rely on cohesion in design; for him c' = 0.

Rankine 'theorised'that no vector ofpressure across any plane through a body ofearth makes an angle to the normal to that plane greater than the angle ofrepose f' d. A drained slope of loose earth at the angle of repose, f d, is shown in Figure 1. A slice of earth of unit width applies an inclined pressure gzcosf don a plane 'a'parallel to the slope at depth z. This pressure vector is shown touching a stress circle at a point that is also the pole ofplanes. Coulomb considered one family ofmany parallel planes 'a'on each ofwhich there was a limiting stress vector.

Rankine (1857) realised from the plane stress ellipse that ifthere is one family ofdirections 'a'in which there is anticlockwise limiting shear stress + t, there must be a second family of conjugate directions 'b'with clockwise limiting shear - t.

This insight led him to conceive of zones of active and passive stress. There are two real characteristic directions in these limiting stress zones. Terzaghi pointed out the problem of introduction of strain boundary conditions. We must escape from Rankine's 'earth pressure fallacy'by abandoning the whole concept of limiting stress vectors.

In the stress circle the principal stresses are s' 1and s' 2 = s' 3. The slope can be seen as made of many triaxial test specimens. The slope, once seen as slipping in simple shear on many parallel planes, is now seen as triaxial specimens all shortening and swelling at constant volume and constant effective pressure.

My PhD thesis (Schofield 1959) analysed triaxial test paths using isotropic continuum parameters (q, p, v). A unit volume ofsolids occupies a space v = (1+e) in an aggregate ofgrains.

Two basic stress parameters are derived from the principal stresses. The deviator stress is q = (s' 1- s' 2)and the spherical stress is p' = {(s' 1+ 2s' 2)/3} For any aggregate whose slope angle f ddoes not reduce as the slope height increases, an equation q = Mp' fits all states ofstress at increasing depths, where M =6sinf d/(3-sinf d). Coulomb's work predated the analysis ofan elastic continuum. The isotropic elastic continuum has two elastic constants. Change ofspherical effective stress p' divided by bulk modulus E equals the volumetric strain. Change ofdeviator stress divided by shear modulus G equals the deviator strain change.

In 1956, seeking a good alternative to envelopes to stress circles at failure of triaxial tests, I plotted progress of tests as paths in a space (q, p', v). They approached a critical state line (Roscoe, Schofield, and Wroth,1958) with two scalar equations q =Mp' and v + l in p' = G (Figure 2) In the three soil constants, l and G relate to classification tests of soil, and M (capital m) is an alternative description of internal friction which is more true and less restrictive than slip on planes or active and passive zones with characteristic directions.

Soil in critical states q = M p' exists both in slopes at rest at angles of repose, and in triaxial test specimens at the end of tests, and in the 'soil paste', left as gouge material on slip surfaces after progress of'Mohr Coulomb'rupture through stiffground.

A rational limit state design that takes account ofRankine's earth pressure fallacy is needed by Eurocode 7, for design with Coulomb's limiting stress vector across a surface in soil, or Rankine's active or passive zones. The original Cam-clay model established that soil paste is a perfectly plastic material. Schofield & Wroth (1968) 'Critical state soil mechanics'based their geotechnical teaching on plasticity theory. In design problems, disturbed soil properties may be used with undrained action on soil and c=c u, or with drained action on soil and f= f

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