Re-appraisal of Terzaghi

1
Re-appraisal of Terzaghis 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) Coulombs and
  (ii) Rankines errors
• Yielding of a saturated soil paste
• Conclusion

2
D. W. Taylor (1900-55) Associate Professor, MIT

• K. H. Roscoe taught his 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 elastic-plastic continuum

3
Terzaghi and Peck versus Taylor (1948)

• Taylor's interlocking theory (1948)
• Review of Taylors manuscript
• John Wiley Sons reply to Terzaghi
• Critical state flow of grains without damage
  Roscoe, Schofield, and Wroth (1958)

4
Taylor's interlocking theory (1948)
i

• Work?is ??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 slick slip planes
shows that this applies totrue cohesion of
over-consolidated clay
5
Pecks review of Taylors manuscript

• I am convinced that the theories of soil
  mechanics and the results of laboratory tests
  serve only to guide the engineer toward a
  recognition of the factors which may affect the
  design and construction 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)

6
John Wiley Sons reply to Terzaghi...
(Taylors 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
7
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
8
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
• Soil paste is unchanged in mixing or yielding on
  the wet side, v?gt ?

9
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

10
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
11
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

12
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
• Cladding breaks up but framework survives

13
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 in a
  critical state behaves as a ductile plastic
  continuum.
• Plastic design guides us to select construction
  materials and methods soil is not plastic and
  ductile if over compacted to high peak strength

14
Strains by the associated plastic flow rule
??ip ??jp ?i
??i ??j
(?i ?j) ?j
In plastic flow, as a body yields under combined
stresses ?i ?j with strain increments ??ip ??jp,
the flow vector is normal to the yield locus at
(?i ?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 gt 0.
15
Calladines associated plastic flow (1963)

• Yield loci for paste with v? (const) on wet
  side of Critical-states, satisfy the associated
  flow rule dp?dvdqd?0
• The Original Cam-clay locus was based on this
  plus Thurairajahs dissipation function

16
A continuum of grains

• Some historical dates
• Belidor and Navier
• Coulombs 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

17
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 Rankines
  theory of limiting statics of granular media,
  (Sokolovski), for lacking consideration of strains

18
Belidors friction hypothesis (1737)

• Belidor attributed sliding friction coefficients
  of 1/3 to the hemispherical geometry of roughness
• Navier (1819) called Belidors theory
  très-fautive but he offered no alternative to it.

19
Navier (1819) a footnote in his edition of
Belidor
20
Coulombs soil (1773) Friction

• Coulomb defined soil internal friction as the
  angle of repose ?d of drained slopes
• Grand rock face
• Canyon
• soil slope

21
Coulombs soil (1773) Cohesion

• In Coulombs 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
22
Terzaghi interprets Hvorslevs (1937) shear box
tests

• Terzaghi fitted true cohesion and friction to
  peak strengths found by Hvorslev in shear box
  tests, normalising them by equivalent pressure.

wet side of critical states
23
A point Terzaghi missed in interpreting test data

cs wet side
Hvorslevs 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? gt ?
24
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)

25
Rankine Active slope at angle-of-repose i
Stress on a sloping plane
?
z ??
?d
? z
cos ?d
26
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.
27
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 Belidors
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 Rankines
earth pressures fallacy. Let us replace
Rankines loose earth by an elastic-plastic
continuum.
28
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
29
Drained angle-of-repose slope ii

?
z ?d
?? ?d

??r ??a For q(??a
-??r) and p?(??a2??r)/3 in triaxial tests, and
q/p?3(??a??r)/(??a2??r)6sin?d/(3sin?d)?(cons
t), a continuum with (??a/??r)(1sin?d)/(1-sin?d)
(const), has constant slope angle ?d as q and p?
increase, without the assumption of slip in two
directions. Circle diameters increase with depth
z.
30
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

31
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
32
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
loci
Roscoe and Schofield (1963)
33
Plastic compression is explained by ? lines
v? ? line cs
? line (?-?)
lnp?

• 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?gt?
  lines

34
Undrained and drained ultimate strength

• q?p?

• Critical States
• Undrained strength ccu with vconst.,
  cu?/2exp(?-v)/ ?
• Drained strength in p?const. tests ?
  ?dsin-13/(16/?)
• See Schofield and Wroth (1968) CSSM

35
Fall cone tests of mixtures of clay and silt
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)

• Fall cone tests with 80 and 240gm cones give
  ?v?lnp??ln3
• If p?PL 100 p?LL then IP 1.71 ? (from CSSM)

36
Yielding of a saturated-soil paste

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

37
Taylor / Thurairajah (1961) dissipation function

• Taylors 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)

38
Original Cam-clay (1963)
q/?p?1-ln(p?/p?c)
q cs
(dv,d? ) v q ?p?
p?c
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 ln p? d?
-?dp?/p?. When integrated this gives
?/?1-ln(p?/p?c).
39
Original Cam-clay (figure from my 1980 Rankine
Lecture)
40
Original Cam-clay (1963) ?(?-?)

(?-?)
q/?p?1
v?

?
?

q/?p?0 ?
S
1
ln (p?/p?c)
41
Interim conclusions

• Coulombs zero cohesion Law is confirmed by
  data on the wet side of critical states
• Terzaghis Mohr-Coulomb error is clear
• Map of soil behaviour (Schofield 1980)
• Centrifuge work of TC2 up to 1998
• Choice between two liquefaction hypotheses

42
Coulombs zero cohesion Law is confirmed

• Cam-clay model fits test data on the wet side of
  critical, which confirms Coulombs law that
  newly disturbed soil paste has zero cohesion
• (CSSM figure paste data (kaolin-clay)(rock-flour
  ) (Lawrence1980))

43
Terzaghis 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.

44
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
45
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
46
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
47
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 Terzaghis Society.
  
• A concluding comment on the Report of TC2 to the
  Istanbul Conference, Schofield (1998) Lecture in
  Centrifuge 98 Vol 2 - IS Tokyo

48

• Terzaghis 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

49
Coulombs purpose in teaching soil mechanics

• jai tâché autant quil ma été possible de
  rendre les principes dont je me suis servi assez
  clairs pour quun Artiste un peu instruit pût les
  entendre sen servir
• Teton photo from US Dept of Interior Bureau of
  Reclamation