Re-appraisal of Terzaghi’s soil mechanicsAndrew Schofield, Emeritus Professor, Cambridge University

• “Terzaghi and Peck” versus “Taylor” (Goodman p 213)

• Civil engineering plastic design

• Continuum of grains at repose (i) Coulomb’s and (ii) Rankine’s errors

• Yielding of a saturated soil paste

• Conclusion

D. W. Taylor (1900-55) Associate Professor, MIT

• K. H. Roscoe taught hıs students to respect D. W. Taylor

• The x- y =x in “Fundamentals of soil mechanics” led us to an understanding of the mechanics of soil as an elastıc-plastic continuum

“Terzaghi and Peck” versus “Taylor” (1948)

• Taylor's “interlocking” theory (1948)• Review of Taylor’s manuscript • John Wiley & Sons reply to Terzaghi• Critical state flow of grains without damage

Roscoe, Schofield, and Wroth (1958)

Taylor's “interlocking” theory (1948) i

• Workis x=x+y so strength is (friction) plus (interlocking) /=+y/x.

x sand in y shear box

y / y/x x x x

Increase of water content on slıck slıp planes shows that thıs applıes to“true” cohesıon of over-consolıdated clay

Peck’s review of Taylor’s manuscript

• “I am convınced that the theorıes of soıl mechanıcs and the results of laboratory tests serve only to guıde the engıneer toward a recognıtıon of the factors whıch may affect the desıgn and constructıon of a real project”

• from review sent to Wiley by R B Peck July 31 1944 quoted from page 213 “Karl Terzaghi; the Engineer as Artist” R E Goodman (1999)

John Wiley & Sons reply to Terzaghi

... (Taylor’s book) will be published by one of our competitors if we do not take it. Under the circumstances, we see nothing to do but publish it.

However, as I said in the first paragraph of this letter, we believe that each book will be judged on its own merits, and certainly we have no fears for the success of (Terzaghi & Peck).

E P Hamilton (President) December 17, 1946

Roscoe, Schofield, and Wroth (1958)

• Triaxial test paths approach steady flow in critical states with aggregates of grains at constant v specific volume

• As strain increases v and p are constant at a Critical-state (v p) v= { v+lnp} =

• q=p

wet

dry

Critical state flow of grains without damage

• Competent aggregate of selected sand grains flows in critical states v = v + ln p = with no dust or damage

• Soıl paste is unchanged in mixing or yielding on the “wet” side, v>

Plastic design in civil engineering

• Construction without plastic ductility• Plastic design of a steel frame, Baker (1948)• Plastic design of structures• Ductility and continuity in soil mechanics• Strains by the associated plastic flow rule

Construction without plastic ductility

Ductility can save life. The 1995 bomb at the Oklahoma Federal Centre, and similar damage in the 1999 Turkish earthquake, show the risk of brittle behaviour

Plastic design of a steel frame, Baker (1948)

• Cambridge text book example plastic design of shelter to resist floor load 20 lb/sq.ft falling 9 ft in bombed house; Mother and 3 children survived WW II 250kg bomb in Falmouth, UK

Plastic design of structures

• Small imperfections causes big local stress concentrations in elastic analysis of steel frames

• In practice plastic yield of steel relieves high stress

• Ductility of steel gives safety, rather than high yield strength

• Claddıng breaks up but framework survıves

Ductility and continuity in soil mechanics

• A paste of soil saturated with water is plastic, (from the Greek word plassein to mould, as in moulding pottery from clay).

• An aggregate of separate hard grains ın a crıtıcal state behaves as a ductile plastic continuum.

• Plastic design guıdes us to select construction materials and methods; soil is not plastic and ductile if over compacted to high peak strength

Strains by the associated plastic flow rule

For stability the product of any stress increment vector (i j) and the plastic strain rate flow vector may not be negative; i i

p + j jp

> 0.

In plastic flow, as a body yields under combined stresses i j with strain increments i

p jp, the

flow vector is normal to the yield locus at (i j).

ip j

p

i

i j

(i j)

j

Calladine’s associated plastic flow (1963)

• Yield loci for paste with v = (const) on wet side of Critical-states, satisfy the associated flow rule dpdv+dqd=0

• The Original Cam-clay locus was based on this plus Thurairajah’s dissipation function

A continuum of grains

• Some historical dates• Belidor and Navier• Coulomb’s error• Rankine Active slope at angle-of-repose• Drained angle-of-repose slope • Flow of grains with elastic energy dissipation• Elastic-plastic strains of aggregates of grains• Undrained and drained ultimate strength

Some historical dates

• Coulomb, at school in Mezieres, learned friction theory from a text book written by Belidor in 1737 (reprinted with notes by Navier in 1819) and a Dutch concept of (cohesion) = (adhesion). In his 1773 paper he reported new rock strength data

• Terzaghi (1936), in “A fundamental fallacy in earth pressure computation”, rejected Rankine’s theory of limiting statics of granular media, (Sokolovski), for lacking consideration of strains

Belidor’s friction hypothesis (1737)

• Belidor attributed sliding friction coefficients of 1/3 to the hemispherical geometry of roughness

• Navier (1819) called Belidor’s theory très-fautive but he offered no alternative to it.

Navier (1819); a footnote in his edition of Belidor

Coulomb’s soil (1773) Friction

• Coulomb defined soil internal friction as the angle of repose d of drained slopes

Grand rock faceCanyon

• soil slope

Coulomb’s soil (1773) Cohesion

• In Coulomb’s rock tests, cohesion in shear was slightly greater than adhesion in tension, so he considered it safe to design with tension data

• His wall design assumed that newly compacted soil has zero cohesion

error

Terzaghi interprets Hvorslev’s (1937) shear box tests

• Terzaghi fitted “true” cohesion and friction to peak strengths found by Hvorslev in shear box tests, normalısıng them by equıvalent pressure.

wet sıde of crıtıcal states

A point Terzaghi missed in interpreting test data cs wet side

Hvorslev’s data ended at a critical state point.

Terzaghi should have asked Hvorslev why he put equations in space where there were no peak strengths. Fillingthe space meant that he asked no questions about the wet side of critical states v= { v+lnp} >

Alternative strength components in soil paste

• For Belidor (and Navier) the 2 soil strength components were (cohesion) + (interlocking = friction)

• For Terzaghi (and Mohr) the 2 soil strength components were (true cohesion) + (true friction)

• Critical State Soil Mechanics has only 2 strength components (interlocking = cohesion) + (friction); it is a theory for dust with (true cohesion) = (zero)

Rankine Active slope at angle-of-repose i

Stress on a sloping plane z d

z cos d

Rankine Active slope at angle-of-repose ii

Stresses on sloping planes and on vertical planes are conjugate. Rankine hypothesised that d is a limiting angle z for both vectors of d

stress, and also thatboth these planes slip.

Rankine Active slope at angle-of-repose iii

Slip lines are lines of constant length.If vertical lines had constant length, all slope material z would move forward d

horizontally. If we accept Belidor’s error, (friction) = (dilation), no work is done or dissipated . Rankine (1851) should have deduced that slip planes are not planes of limiting stress.

Terzaghi called Rankine’s earth pressures “fallacy”. Let us replace Rankine’s “loose earth” by an elastic-plastic continuum.

Drained angle-of-repose slope i

Stresses on sloping planes and on vertical planes remain conjugate in a plastic continuum. Instead of z d two sets of slip planes d in a Rankine Active zone r a

let us have many ‘triaxial test’cylinders in constant volume shear, giving plastic flow at all depths z

Drained angle-of-repose slope ii

z d d r a

For q=(a -r) and p=(a+2r)/3 in triaxial tests, and q/p=3(ar)/(a+2r)=6sind/(3–sind)==(const), a continuum with (a/r)=(1+sind)/(1-sind)=(const), has constant slope angle d as q and p increase, without the assumption of slip in two directions. Cırcle dıameters ıncrease wıth depth z.

Flow of grains with elastic energy dissipation

• Elastic energy is lost on wood surfaces as fibre brushes spring free; Coulomb (1785)

• Frameworks of soil grains carry load (after Allersma). Elastic energy is stored and lost as frameworks buckle

Elastic-plastic strains of aggregates of grains

• Elastic compression and swelling states with specific volume v, spherical pressure p, fit v = {v + lnp}

• Plastic compression fits v = {v + lnp}

, are constantsElastic slope

Plastic slope

Taylor (1948) data

Elastic-plastic strains of aggregates of grains

• Elastic compression of aggregate fits v={v+ lnp}

• A yield locus defines how elastically compressed grains yield when sheared

line shift v=vp

gives plastic volume change (hardening)

v =vp plastic volume change

Roscoe and Schofield (1963)

loci

Plastic compression is explained by lines

• Elastic compression lines in plot of v=v+lnp against lnp go past the cs line v=v+(-)lnp= and yield at a line.

• Plastic compression in tests is observed to fit predicted stable yielding in (-) gap of v> lines

v line cs

line (-)

lnp

Undrained and drained ultimate strength

• Crıtıcal States• Undrained strength

c=cu with v=const., cu=/2exp{(-v)/ }

• Drained strength in p=const. tests = =d=sin-13/(1+6/)

• See Schofield and Wroth (1968) CSSM

• q=p

Fall cone tests of mixtures of clay and silt

• Fall cone tests with 80 and 240gm cones give v=lnp=ln3

• If pPL = 100 pLL then IP= 1.71 (from CSSM)

Plasticity index IP is loss of water content for strength increase by factor of 100(triaxial test data; Lawrence MPhil 1980)

80gm 240gm

v v

ln(penetration)

Yielding of a saturated-soil paste

• Taylor / Thurairajah (1961) dissipation function• Paste mechanics Original Cam-clay (1963)

Taylor / Thurairajah (1961) dissipation function

• Taylor’s dissipation x- y =x (note ,x are orthogonal)

• Undrained and drained triaxial test data, including data of change of elastic energy, fit a function pdvp + qd = pd (p,d are orthogonal)

Original Cam-clay (1963) q/p=1-ln(p/pc)

pc p

dpdv + dqd = 0 associated flow

pdv + qd = p d dissipation function cs

dv/d = -(dq/dp)= -(q/p). Introduce =q/p so d/dp=1/p(dq/dp-q/p)= -/p. Hence ln p d= -dp/p. When integrated this gives /=1-ln(p/pc).

q cs (dv,d ) v

q = p

Original Cam-clay (figure from my 1980 Rankine Lecture)

Original Cam-clay (1963) +(-)

1

ln (p/pc)

S

(-)

q/p=1 v

q/p=0

Interım conclusions

• Coulomb’s zero cohesion “Law” is confirmed by data on the wet sıde of crıtıcal states

• Terzaghı’s Mohr-Coulomb error ıs clear• Map of soil behaviour (Schofield 1980) • Centrifuge work of TC2 up to 1998• Choice between two liquefaction hypotheses

Coulomb’s zero cohesion “Law” is confirmed

• Cam-clay model fits test data on the wet side of critical, which confirms Coulomb’s “law” that newly disturbed soil paste has zero cohesion

• (CSSM figure; paste data (kaolin-clay)+(rock-flour) (Lawrence1980))

Terzaghı’s Mohr-Coulomb error

• Terzaghi and Hvorslev wrongly claimed that true cohesion and true friction in the Mohr-Coulomb model fits disturbed soil behaviour. Geotechnical practice using Mohr-Coulomb to fit undisturbed test data has no basis in applied mechanics.

• Critical State Soil Mechanics offers geotechnical engineers a basis on which to continue working.

• The original Cam-clay model requires modification to fit effects of anisotropy and cyclic loading. Good centrifuge tests of soil-paste models achieves this.

Map of soil behaviour (Schofield 1980)

Regimes of soil behaviour 1

1 ductile plastic 2

2 dilatant rupture 3

3 cracking 3 2 1

(fracture with high hydraulic gradient causes clastic liquefaction)

A centrifuge test of a model made of soil paste will display integrated effects in behaviour mechanisms

Choice between two liquefaction hypotheses A

CVR

A Casagrande Boston

There is a unique critical void ratio and a risk of liquefaction in any embankment built with higher void ratio

Choice between two liquefaction hypotheses B

B Casagrande Buenos Aires

Even a dense sand if heavily loaded can liquefy.

Reject both A and B. Sand yields, it is stable, on the wet side of critical states

Figure from Schofield and Togrol 1966

Centrifuge work of TC2 up to 1998

• We should claim a fundamental significance for centrifuge tests of models made of reconstituted soil, and explain how our tests can correct some errors that were made in Harvard. If it led to serious discussions in Istanbul, it would be good for Terzaghi’s Society.

• A concluding comment on the Report of TC2 to the Istanbul Conference, Schofield (1998) Lecture in “Centrifuge 98 Vol 2 ”- IS Tokyo

• Terzaghi’s low expectation for applied mechanics was in error when he said at Harvard (1936)...(the) possibilities for successful mathematical treatment of problems involving soils are very low

• When I asked Bjerrum “What should Universities teach in soil mechanics?” he replied “Universities should not teach soil mechanics; they should teach mechanics” ( teaching in the spirit of K. H. Roscoe)

• ISSMGE should correct error. We all should teach Plasticity and Critical State Soil Mechanics and promote centrifuge model tests with soil paste

Coulomb’s purpose in teaching soil mechanics

• j’ai tâché autant qu’il m’a été possible de rendre les principes dont je me suis servi assez clairs pour qu’un Artiste un peu instruit pût les entendre & s’en servir

• Teton photo from US Dept of Interior Bureau of Reclamation