Re-appraisal of Terzaghi’s soil mechanics
Andrew 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)
• Work is x=x+y so
strength is (friction) plus
(interlocking) /=+y/x.
y
i
x
y
/

sand in
shear box
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
ip jp
In plastic flow, as a body
yields under combined
i
stresses i j with strain
i j
increments ip jp, the
flow vector is normal to
(i j)
the yield locus at (i j).
j
For stability the product
of any stress increment vector (i j) and the
plastic strain rate flow vector may not be negative;
i ip + j jp > 0.
Calladine’s associated plastic flow (1963)
• Yield loci for paste with
v = (const) on wet side
of Critical-states, satisfy
the associated flow rule
dpdv+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 face
Canyon
•
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. Filling
the 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

Stress on a sloping plane
d
z
 z cos d
i

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 that
both 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 elasticplastic 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
d
ii

z d
r

a
For q=(a -r) and p=(a+2r)/3 in triaxial tests, and
q/p=3(ar)/(a+2r)=6sind/(3–sind)==(const),
a continuum with (a/r)=(1+sind)/(1-sind)=(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 constants
Plastic slope
Elastic 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)
loci
v
=vp
plastic
volume
change
Roscoe and Schofield (1963)
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
80gm
v
240gm
v
Plasticity index IP is loss of
water content for strength
increase by factor of 100
(triaxial test data; Lawrence MPhil 1980)
ln(penetration)
• Fall cone tests with 80
and 240gm cones give
v=lnp=ln3
• If pPL = 100 pLL then
IP= 1.71  (from CSSM)
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
pdvp + qd = pd
(p,d are orthogonal)
Original Cam-clay (1963)
q
cs
(dv,d )
pc
p
q/p=1-ln(p/pc)
v
q = p
dpdv + dqd = 0
associated flow
pdv + 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
d= -dp/p. When integrated this gives /=1-ln(p/pc).
ln p
Original Cam-clay (figure from my 1980 Rankine Lecture)
+(-)
Original Cam-clay (1963)
(-)
v
q/p=1
S


q/p=0

1
ln (p/pc)
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 cracking
3
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
A Casagrande Boston
There is a unique
critical void ratio and a
risk of liquefaction in
any embankment built
with higher void ratio
CVR
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