8/19/2019 Schofield1998_The Mohr-Coulomb Error Correction

1/3

PAPER

III

0

11

F08

OiII

)

errOr

COrreCI:iO11

by

Andrew

N

Schofield,

Cambridge University

Tenaghl

and

the Mohr Coulomb

error

In the

1st ISSMFE

Conference

(1936)

proceedings,

Terzaghi

writes that

newly

over-consolidated

clay

strength

fits

Mohr s

rupture

hypothesis

(he

used

the

word

hypothesis,

not criterion).

He

quotes

data for

drained

shear box

tests which his research

student Hvorslev fits

to

the

Mohr

Coulomb equation.

They

are

wrong. They

contradict

Coulomb s

paper

which, in

three separate

places,

states that

 I adherence

est

nulle

dans

les

terres

nouvellement

remuees

(newly

worked

soil

has no

cohesion),

and

they

have no data

for

soil

strength

on

the wet

side

of

critical states

(cs).

There is no

 true

cohesion on

the

dry

side of

the

critical

state. The

peak

strength of

dense

soil

paste

is

due

to

interlocking

and

friction

among

the

soil

particles.

When

soil

flows,

many

soil particles change

partners, and

there is no

time

to bond

particle to particle.

It

is

only

when soil is

left

to

age

and

creep

that

bonds

develop

at particle

contacts

and turn it into

soft

rock.

Renewed

strains

destroy this strength.

Broken

bonds

do not

resist

ground

failure

mechanisms. Bonds

are

not remade until

there

has been

time

for

ageing

and

creep,

long

after

a

failure

event.

Terzaghi s

Mohr

Coulomb error

isto

suppose

that

peak

strength

seen in

Hvorslev s

dense,

newly

remoulded shear

test samples

indicates

strong

cohesive

bonds,

when

it

really

indicates

dense

packing

of particles. Figure

1

shows

how

the

peak

strengths

are

caused

by

particle interlocking. Strength

in

Hvorslev s

tests

depends

on

packing geometry,

not on chemistry

of

bonds.

Taylof s

Interlocking;

peak,

and

ulmmate

strength

of

slff

clay

In

Figure

2,

dense

soil in

a

drained

box

is sheared

a

distance

x

by

shear

force T.

It

dilates and lifts the piston (and

the

normal force

o hat

acts on

it)

by

a

distance

y.

Taylor s

peak

strength

of

sand in drained

shear box

tests

involves

(i)

a

critical

state

friction

component and

(ii)

an

interlocking component,

which

increased with increase

in the

distance

of

the

state

of

the soil

at

failure

from the

ultimate states

in the

drained

test.

This

applies

both to

dense

sand and

stiff

clay.

In particular,

it

applies

to

any paste

made

by

mixing

fine

particles

and water;

for

example, a

mixture

of

cornflour

and

water

exhibits

dilatancy.

In Critical

State

Soil

Mechanics

(CSSM)

1968,

pp232/3,

Schofield

and

Wroth

describe

North London

retaining

walls failures, Figure

3,

and

the stiff

fissured

clay

behind them

that disintegrated

 into

a

rubble of

lubricated

blocks,

sliding

on

each

other

on

very

thin moist layers

of

soft,

lubricating

clay

paste .

How did

those

layers

become

moist and

 slick

?

That

stiff

over-consolidated

London

clay

was

a

paste

dilating

to

a

critical

state;

suction during shearing led

it

to

soften to a

critical

state after

peak

strength.

Earth pressure

in

that rubble of

clay

blocks

satisfies a

calculation

based on

critical

state friction

because the

effective

pressure

that acts

between

blocks,

and the strength with

which

they

adhere to

each

other,

increases

with

depth

in

the rubble;

(however,

note that

care

is

needed

to

detect old

slip

planes

where ground

in the

past

had

very large

displacements;

a

field

shear

box

Dais

Cohesion

Terzaghi

i/o eak

strength

(jj)

dy/dx

Ixl

Work

done

in

shearing

n

Y

Rgure

1:Peak srengths

are caused

hy

Darthde

Interloddng,

nol

chemlslryof

honda.

V

Taylor

cs

cs

There

is

no

 true

cohesion on the

dry

side

of the critical

state. The

peak

strength

of dense

soil

paste

is

due

to

interlocking

and

friction

among

the

soil particles.

test

on such a

surface

may

find

the

strength

there to

have

fallen

below

the

critical

state value).

Movement

may

stop

for

long

periods,

and

ageing

may

then

begin

to

create

bonds,

but

renewed

deformation of a

rubble

of

fissured

clay

soon strains

a

clay

paste layer

and

returns it to

the

critical

state. As

intact

blocks

come under new stress,

their

strength

to

resist

rupture

will

always

exceed the critical

state

strength

of

the paste

on

their

boundaries;

CSSM analysed

the

pressure

of stiff

fissured

London

clay

on

the

dry

side

of

critical state

in

terms

of a material

with an

angle

of

friction based

on

the

critical

state

parameter

M

(denoting

capital

mu).

Tdx

=

Iitf dx+

a dy

Tl/r5

ll

+

dyldx

dy/

Strength

=

friction + interlocking

/

X

Figure

2:

Dense soll

In

a

drained

sheaihox.

In as figures

the spectrum

of

colours

from

red

to

violet

Indicates

strengths at Increasingly

high

slress

ratios.

 Apparent

cohesion

of soil

During

critical

state

flow

of

soil,

the

undrained

cohesion

results

from effective

stress and

critical

state friction,

not chemical

bonds

between soil

particles. All

that

small

clay

mineral

particles

and

chemicals do

during steady

plastic

deformation

of

soft

soil

is

30

GRQUND ENGINEERING

AUGUST 1998

8/19/2019 Schofield1998_The Mohr-Coulomb Error Correction

2/3

p~

ER

0,20

0.30

Water

content

Tension

cracks

Future

corregtl0

on

of

the

Mohr

Coulomb

error

will

invigorate

geotechnical

teaching,

research,

and

laboratory

testing.

100

Bank

before

Feet

-40

50

-30

Distance

from

soft

zone

(mm)

—

lip

—

one

t

-20

Retaining

wall

Slip

zone

—

ncovered

slip planes

I

Soil samples

f ilure of

retain

n

wall

n example

of

g

on

term aiu

-0

Prager

or

a

sim milar

e Drucker

is

obvious

r

eth

t

oh

ive

d

te

e

n

interlocking

is

neg

with

unsta ble

failures.

here

At

large

strain, so

state line w

 cal

state strength.

P +P

c~l

le

a

simple

cs

the wet side;

th

tr

gth

is called

calculatio

gi

 ves

e

mpress to

ki

g

and emit

o

le

w o

acking

ater co

o from

its

cs

p

spe

im

tate

spec

2.7

the

critical

s a e

he

left

o

 

where

X=0.16,

x=

line,

w

M=0.89, hence

p,

-,

=

p

c~t

s to

-

=140.78kN

m,

ggregates

to

causes

ag

d

an

c„—in water an

ak

tr

gthround slips

at

pe

s

cs

ed

Pl

Wet side

p crit

In

p 1

ater suction,ore wa

be

measure .

50

s

ction x

(friction .

i i.

it

remains

astic ma e

 

100

at

constant

vo

of water

content

M

ff

Observations o

wa

taconstan e

one

e

auseitisa

of

a

slip

zon

CSSM

analyses

ss

and

critical

sta

e

r

 efafnfng

walls fagnra

f

offl

o

11

3:

Neth

Londonrefafn

ng

eo

 alysis

in an

office

usin

g p

yo

so

v

ritical stat

 r

i

model,

ar

rial

 

S

ill

am

clay

asserts

that the

plastic

vo

articles.

Coulomb s

n

p

ive

a

better

insig

lay

theory

g oh

Co

1o

worked

so

behaviour thannt ee

eakstrengths

Av

ates soil

at

s

res

effective

these

high

pea

s

 al

and

radial

e

t

is

The axi

o

k.

d

rupture

o

d

unstable.

giv

an

oun

effective

ca

d

at

represen

ator

tates inevi

 

triaxial

deviattered

strength . s

specimens

a

sion

and

ic

compress

have an in

E

as

imens in

strength

e

dry

si

de

of

cr

t

in the

Mohr

swe

ral follows lines

by

gener

on the wet

side.

tudy

of

yielding

on

V

FE

m

Itlng

stress and

define eac

ag

n of

articles.

lane

ar

Each

aggr,

en

Two

equa

i

ect

to

shear

s o

 brium

 lar

critical

„dx+dz/dy=0

o „/

h

it

t

whic

Diy

side

wi

hr

pressure

p

t on

if

combined

wi

e.

Frictional

flow

o

Cou

om

b

equation

 ti

al state

oil

in

a cri  c

Vx

ol

tio b

e

A

five

which have a

so

Drain

h

racteristics or

g

ess

conditions

on

thodo

c a

five

test

p

hi criticises re

o

b

solutions for

all .

cri

g

p

tli

trains,

but

has equation

ies in

h

Successive s

tress state

ohesion;

the

in

the

equations

 oints

(q,p,

ve

strain

d

undrained

triaxi

s

0

calculations exist

FlgnrnA: Crnkal

slafos

,

a

laboratory

tes

tes

d

AUGUST

1998

ROUND

ENG

INEERING A

8/19/2019 Schofield1998_The Mohr-Coulomb Error Correction

3/3

PAPER

p dv„+qdk=MP

dk

bene

Original

cam

clay

=

Mp (M

=

0.888)

An

equation

is

calculated

below for

a

yield

locus,

Figure

B,

that

defines

the

resistance

of soil

to

some

applied

stress

(q,p )

when

q/p 0,

gives

(dq/dp +dv„/dk)

=0.

The equation

for

frictional

wor

ing

by

the

yield

stress

during

u

uctile plastic

strain

cu

80

C

60

(dvk

dk)

40

20

p c

p L

VK

Undrained test

path

Wet

cs

side

RteHT-

Rgtnu

D:

Ilntttutntnt

test tutttt.

s(de

In

p

(q/p +dv„/dk)

=M

.

Eliminating

dv„/dk

from

these

equations

and

 l

ntegrating gives

the

origin

cam

clay

equation

(q/Mp )

=1-ln(p /p, ).

Confirmation of cam

clay

theory

came

from an

drained test of a

triaxial

specimenonthewe

si

sideof

cs,

repr

uc

d ed

here

from

CSSM

edigures

7.12.

Any

undrain

test

has

v=(constant)

=v„tdnp hic

is

an

inclined

line in

Figure

D.

At points along

this

test

path

there

is an

 ersectionwitheac

yi

eld

n

er

locus

as

the

specimen

yi

elds

and

hardens.

ts fit

he fact

that

data points

the prediction

for

the

undrained

test

path

Figure

C

(

 ven

in

detail

in

CSSM)

given

in

th

re was no cohesion

state,

n the

wet

side

of

critical

s

hence no cohesion on

the

dry

side

either.

0

0

20 40 60 80

p

(Ib/ink)

Correc5on

5

of

the

Mohr Coulomb

enw

M

hr

Coulomb

error

w,

dl

bo

 on

of

the

Mo

r

geo

e

g

small

roug

ric

 1

1S

ro

erties that can

e

e

ith

critical

state

prop

2.2m

more

wet)

than

the critical

til t

bl

d

ib

(ib

d

)th

th

iti

1

t

t

it

n

which

gouge

with

rupture

planes

on

w

ic

n unstable

manner,

wi

r channels

that sudden

y

h

Cul

 

bl

G

cts

do

not fit

t e

o

1

roblems

iq

nd solves

stmp

 alcontrolof

cons

ruc

 

heory

fits the

facts

a

t

tion,

numerical

mo

 t

t

t

th d

sing

pr

d

bl o

tall

m

h

At

nt

re

4)

to

demonstrate failure

mec

 

n

cen r

1 d 1 o

that

hemical,therma,

an

ill

be

prepared

wit c

e m r e e

model soft

rock

bodies.

Rgure

4:A

drum

centrltuga

32

GROUND

ENGINEERING AUGUST

1998