The Mechanics of Soil: An Introduction to Critical State Soil Mechanics [1 ed.] 0070841357

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CHAPTER

THIRTEEN BEHAVIOUR OF SOILS BEFORE.FAILURE

E

, 1 I I

,,1 :

:-j

l I

t

.t {

-I ,I 5-

J

I. r

,l

In the last three chapters

we have introduced the concepts of the critical state line and the state boundary surface for sands and clays. The stress ,atht followed by soils in drained and undrained tests have been identified, and methods of calculating volumetric strains to the ultimate condition on the eritical state line have been described. So far, however, we have not con$idered the rhagnitude of the shear strains and we have not considered the stress-strain behaviour of a sample early on in a test. In. order to consider deformations at an early stage of a test, it is ne@ssary, and indeed essential, to make a distinction between elastic and plastic strains and to devtlop a criterion which determines whether a particular loading path produces elastic or plastic strains. We will also discuss the application of the theories of elasticity and plasticity to the stress-strain behaviour of soil. We shall find that interpretation of soil deformations in terms of these theories gives extra insight into the critical state framework developed in the preceding chapters. It also allows quantitative estimates to be made of the shear and volumetric strains caused by loading. We go on to discuss the Cam-clay theory, which uses ideas of elasticity and plasticity expressed in quantitative mathematical terms. We have chosen to discuss the original simple Cam-clay theory, which is described by Schofield and Wroth (1968, pp. 134-.166) in more detail, because it is simple, it illustrates how the various concepts fit together, and it is the basis for much recent work on the stress-strain behaviour of soil. Nevertheless, readers should be aware that although predictions of the theory are broadly correct, they are unsatisfactory in some respects and the simple theory has been superseded by theories which, although conceptually similar to Cam-clay, are rnathematically more complex. The reader must be referred to recent research papers and conferences for an up-to-date account i

1}2 ELASTIC AND PLASTIC DEFORMATIONS: THE ELASTIC WALL

It

is first necessary to make a distinction between elastic (recoverable)-ffiI.' plastic (irecoverable) strains.'This distinction is commonly made in the

zia\nrMECHANrcs oF sorls discussion of the behaviour of rnetals. Thus, the behaviour of rnetal can often be idealized as shown in Fig. l3-1. For uniaxial applied stresses less

than o, the deformation is linearly elaslic and, if the metal is loaded and unloaded, the strains caused'on loading are fully recovered on unloading. However, if the metal is loaded beyond a stress'or; additional plastic strains occur and the state of the metal might be represented by point G. When the metal is unloaded it follows path GB and some (elastic) strain is recovered. However, at B, the metal has suffered a large irrecoverable plastic strain. If the metal is reloaded from B the deformation is linearly elastic for applied stresses less than o* which is greater than o". The stresses o, and o. at which the behaviour of the metal beoomes plasfic,are known as yield stresses and an effect of plastic straining from Y to-G is to iaiie the yield stress from o" to ori this effect is known as strain_@fufug;. If the metal is loaded "C biyond it *iU eventually fail at ltfi,E;ififfnss is o1. For soil, the distinction between recoverable and irrecoverable strain is best illustrated by behaviour during isotropic compression. The normal consolidation line for a clay is indicated by line ABC in Fig. 13-2. If the clay is unloaded from B, it moves along the swelling Iine BD. If it is reloaded from D, the soil retraces path DB to B, after which additional compressions occur as the sample moves down the normal consolidation line to. C" Similarly, if the sample is unloaded from C, it moves back along the swelling line to E. We should note that, at a fixed value of rnean nortrral effective stress, the szr.mple is at a lower specific volume at E than at D, i.e., some irrecoverable (plastic) strain has occurred on the path DBCE. We know that the strains are recoverable along the swelling lines DB and EC, and so the plastic strains must have occurred over path BC, that part of the path that Iies on the state boundary surface. There is a direct analogy with the Axial stress, oa O1

o-

.,?j,t'r., 0

B

Figure 13-1 Elastic-plastic behaviour oi rn"t"l

\

\

Axial strain, .

co

I !

l

BEI{AVIOUR OF SOIIS BEFORE TAILURE'265

.:

I

Normal consolidation line

I

,-

It t

I I ! I I

i I

Figure l3-2 Elastic-plastic behaviour of clay in isotropic compression and swelling

occurrence of plastic ,t.uin, over the path YG for the metal specimen of Fig. I3-1. We can generalize this observation, and argue for soils that plastic (irrecoverable) strains only occur when the sample is tanersing the state boundary surface. Thus, for paths below the state boundary surface, the strains are purely elastic and recoverable. This hypothesis leads to some strong limitations on the paths that can be followed by specimens. For example, because irrecoverable (plastic) strain has occurred between points D and E of Fig. l3-2, it means that the test path followed by the specimen must have touched the state boundary surface between D and E. The path DBCE satisfies that requirementbecause section BC (the normal consolidation line) lies on the Roscoe surface. An alternative path'for the specimen to move from D to E is for it to be sheared at constant p'. Then, in order that the necessary irrecoverable strains occur, the test path must be such that q' increases so that the test path strikes the Roscoe state boundary surface at G (Fig. 13-3), above D, before the path traverses the state boundary surface to K, above E. As the value of q' reduces, the sample then deforms only elastically as it moves to E. The value of g'. at G fixes the value of q' Which must be applied to the sample at D (when p' is held constant) in order to

There is a range of otlrer paths by whigh the sample could move from D to E; all of them requile that the sample rioves across the state boundary surface. Convg.rsely, there is a rangerg{-paths which may be followed by a Sample at D without plastic deformition occurring. All paths that remain on.the curved vertical plane above the.surdling line BD, but below the state

266 tua MEcHANIqs oF sorls Critical state line

Normal consolidation

Swelling line

line

u

Figure 13-3 The test path frorn points D to E in q,

:

p,:

1,

space

boundary surface, will cause only elastic deformation of the soil; this curved surface, BJIH in Fig. 13-4 is called the elastic wail. of course there is an infinite number of elastic walls, each elastic wall being associated with a particular swelling line. If the state of a sampre is berow the state boundary surface, its behaviour is assumed to be elastic and stresses and strains may be related through the theory of elasticity. on the other hand, if the state of a sample ries on the state boundary surface, both plastic and elastic strains may occur and the plastic strains may be calculated from the theory of plasticity. The importance of the distinction between elastic and plastic stiaini is that elasfic strains are relatively small, while plastic strains are relatively large. Thus, if a loading on a soil stratum causes only elastic strains, *" -"*p..t that the soil deformations would be small. conversely, soil deformations and settlements will-be large if significant amounts of plastic straln occur in the soil stratum. For the 'theoretical calculation of soil deformations, the oirtir"ti.o, between elastic and plastic strains is importanf fo-r the.,two types of strain

areco1lpu[e{completeIydifferentIy.:..i.......,.---Jr

13.3 CALCT'LATION OF ELASTIC STRAINS The behaviour of an ideal isotropic elastic soil was discussed in sec..4-li. we began with the generalized form of Hooke's law in Eqs (a-66) and we

I

BEHAVIOUR, OF SOILS BEFORE FAILURE 267

Critica! state line

I I

I ..I I 1

,.

a

I!

I

.t

,

'i .tI

-;

't

t

t

Figure 13-4 The elastic wall

al

showed that increments of strain could be related to increments of effective stress more conveniently through the use of invariants by Eqs (4-76) and (4-77). These were

*

-t I

t

I

Etr: iEp'+0.69',

.I

6e"

--

'l

i" a; .i,

-

j :::r::

I

6..

6l' +1O;6Q',

(r3-2)

r(', the elastic bulk modulus, and G', the elastic shear moclulus, were constants over the appropriate increments of stress and strain. Equations (13-l) and (13-2) show that, for an ideal isotropic elastic soil, volumetric strains ane connected with p' and separated from q' and shear strains are connected with q' and separated from p'. , W. have now. poslulpted the- gxi.ste_npe 'of the elagtic yall BJIH in Fig. 134 and suggested that the path of an overconsolidated soil whose state lies below the state boundary surface must remain oo a particular elastic wall; consequently, the path followed by a sample of an overconsolidated soil, during loading oi unlohding will follow the line of iffisection of the elastic wall and the appropriate.drained or undrained loadifllgplanes discussed in Section i0.5. Thus, Fig. 13-5 shows the line of int€.rseCti&n DG of the elastic where

-.!

:

(13-r)

\

zf,g rHr MEcHANrcs oF sorI.s

Critical state line

Undrained plane

Normal consolidation !ine

u

Figure 1&5 Intersection of an elastic wall and an undrained plane

wall and an undrained plane QRST for a constant volume loading or unIoading test. The path DG rises vertically from D to G, which is on the state boundary surFace. If the sample is loaded beyond G it will suffer plastic strains as its state traverses the state boundary surface along its intersection with the undrained plane towards its ultimate state at F at the intersection of the critical state line and the uudrained plane. For undrained loading of saturated soil, when 6ro:0, Eq. (13-l) has

the consequence that

6p':0.

(r 3-3)

This confirms that the stress path DG in Fig. l3-5 rises vertically up the intersection of the undrained plane and the elastic wall. It was for this reason that we sketched undrained effectiye stress paths on overconsolidated samples, in Fig. 11-15 and elsewhere, as being vertical until they reached the state boundary surface. Figure 13-6 shows the intersection DG of an elasfic wall and a drained plane QRST, and this is the path followed by a sample of isotropic elastic soil during loading or unloading in a drained triaxiaf compression test; the line DG is. not straight because the elastic *ail is curved in plan and there is a reduction in volume associated with an increase of p,. It was for this

\

::i.t

I

i,

1 I

i

I I ll

! -t

BEIIAVIOUR OF SOrI S BEFORE FAILURE ?69

reason that we showed in Fig. ll-22 atd elsewhere, that there was com-samples pression of overconsolidated in draiued triaxiat tests before the sample reached the.state boundary surface. If the sample is loaded beyond G (Fig. l3-Q, it will suffer plastic strains as its state trave'rses the state boundary surface along its inters*tion with the drained plane towards its ultimate state at F, where $e critical state line and the drained plaae intersect. We now have.sufrcient informatioh to calculate the shear and volumetric strains in a lample of ideal isotropic elastic soil as it is loaded or unloaded in drained triaxial compression along DG. The elastic wall is vertically above a swelling line BDF introduced in chapter 7 and is therefore given by

- o*-xltp'

(r34)

= -x(6p'lp').

(r3-s)

(+54),6e": -6olo, we have 6e": (rclop)Ep'.

(r 3-6)

o 6u

Hencq from Eq.

I J

Thus, the bulk modulus

K' t

for the soil is given by

Kt :

up,

l*,

(13-7)

1

J

tI

p,

l

Figure 13-6 Intersection otan elastic wall and d drained plane

\

"l

270

r.laE.

MEcHANrcs oF sorls

and using the result for elastic materials that

(i 3-8)

(l 3-e) So, from Eq. (13-2), (13-10)

Equations (13-6) and (13-10) define the-.stress-strain behaviour of a soil undergoing ideal isotropic elastic deformations during a drained triaxiat compression test along a path such as DG in Fig. t3-6. They are also valid for any loading path in which the state of the soil remains below the state boundary surface. rn particular, Eqs. (r3-O and (13-J0) are valid for any undrained loading path such as DF in Fig. l3-5, in which case Eeo : Ep' : 0 and increments of shear strain are related to increments of by Eq. (ig-ro). 4, Equation (13-7), together-with the definition of bulk modulus K' in terms of .E' and z', gives

K'

:

op'|rc

and, hence,

:

E' l1(l

-2r')

E' :3up'(1-2v)lx.

(13-1

r)

(13-t2)

values for E' ar.d v' may be obtained directly from the results of a drained triaxial compression test in which Ao'r: Ao,":0 and axial and radial strains are rneasured. From the generarized Hooke's taw in Eqs (+67), putting Doj: 6oi:9

E' :6o'J6eo, y'

=

-Ee./6eo.

(r 3-r 3)

(13-14)

rn this treatment of

elastic behaviour in soils, the value of young,s modulus E' as given by Eq. (13-12) depends on the current values of u and p', on the value of r which defines the srope of a swelling line as well as on v'. Even though we assume' that v' is constant, the value of .8, will not be constant and the soil behaviour, even if it is isotropic and qlastic, will not be linear; Eqs (13-6) ind (13-10) ar6, therefore, valid only ior incrernents of loading sufficiently small so that the value of E, may ue assumed to be constant. However, in mahy cises tho change of specific volume u during a Ioading path which .uur", only elastic strains, and which therefore, remains on a particular elastic wall, is relatively small and so the value of E'lp' will remain approximately constant. Hence, we may write E' lP'

t

I

\ 1

\

_i:

:

3u(I -2v')ltoj. From geometry of the Mohr's rhe arcle o,o, and o,oare related by

iii;:?ir(.);;[:,i"rticurar ,;;,rr;-or11

i6;;d;;iig.

, ll +sind'\ o": (I;E-C?

T

I" I I I

)o'r:

Ko'6 (say).

(t+2e)

Equation (r+zg) does not contain the third principar stress oj and so the value of oi is irrerevant for trris raiiuie critqoo. Td; ur.lir"rn"rive bilities for faiture of th9,di;;;;;;;;;e possiis greater than oi g'e', o1:11op;;; the soir."t;;iuirirg so thar oi so, may u" ruiri"g such that the extreme stresses are o[

uid

i

o;; "[ ^;i;i. tr,J; ,i'rl]*o,,ities must be

I l

:

'

.

ub

Figtrre I 4-t 7 The

o',

(a)-.-.". rvr

oh1--Couioiil i"if or"

cri

terion

,'

ROUTINE SOIL TESTS AND CRITICAL STATE MODEL

i

:

,:: i

.t i =l t

incorporated into the general failure criterion. One way of representing the general failure criterion is as follows: : g. (1+30) @L- K")(oL- K")(sL'KC.X"L- Ko)(o',"- Ko)(o!.- Kq The first bracket of Eq. (1430) is exactly equivalent to Eq. (L4-29)', the six possible failure conditions apply when each of the brackets in turn are equal to zero. The best way of examining the form of the surface corresponding to Eq. (1a-30) is to study thelnlers,ection of the surface with a plane perpendicular to the space diagonal, i.e., a plane on which (o'r+ o!o+ o) is constant and -equal to oi, say. Becausg of the form of Eq. (l+30), we expect the interrc"iion to- have six separatq segments, each segment corresponding to conditions when one of the six brackets is equal to zero. The expression (1+31) (o'"-Ko'):Q which passes through the origin of the axes. The intersection of this plane and the plane normal to the space diagonal must be a'straight line, so we need only to define two points on the line to fix its position iq principal stress space. The line of intersection AB of the two planes is shown in Fig. 14-18, which is a view down the space diagonal. We may locate the point A, for which o'": o'6, and the point B' for which oL: oL. In the first case, the M6hr-Coulomb failure criterion is oL K"L KoL and (t4-32) oL+ oL*o," o,rrr+ (zrK)r. o,o dpfines a plane

!

I .T

j T E

$

t I

3I5

:

in o'o:

o'u: o'" space

:

:

:

Now the radius OA is J(3)"1,*

: (,l2llZ)(o'"-"L); hence,

oA:ffi";

(1+33)

I

I I

I

In the second is

case,

for which o'":

o'o,

o'o: Ys;: o'"and

oL

:

oL+ oL* o'"

the Mohr-Coulomb failure criterion

:

and the radius OB is given as before by

o'"[2 + (l I K))

{l)

r!"r;

:ffi"'a

(14-34)

hence,

(14-3s)

and the intersection of the Mohr-Coulomb failure criterion and the plane normal to the space diagonal is the line AB in Fig. 1418,By symmetry, or by-repeating this argume4t for the six brackets jn 1ur1 of Eq. (t+-:Oy, ihe complete Mohr-Coulomb failure locus can be obtained in this as shtwn in fig- 14-lg; it takes the form of,-an irregular bexagon view. Stress siates A, C, E correspond to triaxial compression 4nd states B, D, F to trihxial extension. The extended von Mises criterion is circular,

r

ii

--!

t.1'l

t# 3f

trre MEcHANrcs oF sorLs

i:: :!:

o:a

'.:

A lr tt T'

lt

n tl lt l. I

I

-l / I

I

f,:l

lf

I I

i

i-i I

t

.,-! I

\

i I

::i

..) '1

I

\..*.

-:.!

,+

I -I

I

I I

I

-?'"/

-,

Figure 14-18 The Mohr-Coulomb criterion plotted on the ptane perpendicula.r to the space diagonal

aud, if the two criteria are fitted at point A, coresponding to triaxial compression, a direit compailson may be made between the two criteria. It can be seen from Fig. 1+19 that there is a layge difference between the two criteria, especiatly at points B, D, F, which correspond to triaxial extension. The shape of the.complete Mohr-coulomb failure surface in principal stress space is shown in Fig. l+20.

i

r{oulomb

/ I

\ \

t

I von Mises

I

I

\

I

\

/c

Figure 14-19 The Mohr-Coulohb criterion anil the extended von Mises eriierion

t

i-

\tr

ROUTINE SOIL,TESTS AND CRITICAL STATE MODEL -.

3I7

i'

::: e-:

ti

ll l-:i f'"! I'

ll FI ll

I,J

I1

ii

1t l:r.i 4. ..

l.i t'l

1t IJ

n t.l

ft Lt " Li

n. it

tI =I I

,l ri

:

Figure 1420 The Mohr-Coulomb criterion in principal stress space

The two failure criteria have been discussed at length because it is found that each criterion applies to different stages of soil deformation. Thus, the effective stress patls for undrained tests in a true triaxial apparatus on isotropically normally consolidated samples are of the form sketched in fie. i4-2t, where tni initiat isotropic state is represented by point I. The efective stress paths define a smooth axisymmetric surface directly analogous to the Roscoe surface observed in standard triaxial compression tests. The observation that the surface is axisymmetric suggests that the pre-failure behaviour is governed by a function of tbe von Mises type. Nevertheless, at failure, it is found that'$'is approximately the same for all tests, that is, failure is governed by the Mohr-Coulomb 'criterion. The geometry of the intersection of the axisymmetrical Roscoe surface with the itregular hexagonal cone defining the Mohr-Coulomb criterion has not been established experimentally with any certainty. Nevertheliss, the ,Roscoe surface itself will have th9 approximate geometry.shg$ i1 Iiiig l+2Z,with the line of intersectionsketched as ABCD . . . Points A,.C co-rrespond to triaxial compression and points B, D to triaxial extenqion. '. ,,The Roscoe surface shovm in Fig. l4-22is that for one fixed r-ay.9f specific volume. There will bc a succession of.ych surfaces, all gebmetrically similar, but of different sizes, fss diftrqnt dpecific volumes, as shown in

Fie.l*23:

6

l:::i li:J

ii 31\ rur

MEcHANrcs oF sorls

n ta TT

FI Ill

II la tt

lt I

.ll

il

-n U

tr ft fi

Figure 14'21 Etrective stress paths isotropically consolidated samples

in principal

stress space

for undrained tests on

For overconsolidated samples of clay, we expect that there will be some state boundary surface in principal stress space analogous to (and containing) the Hvorslev surface observed for triaxial compression tests. It was shown in Chapter 1l that the Hvorslev surface for triaxial compression tests was given by the equation q'

:

gPl.+hP',

(t4-36)

where g and h are 'soil constants and pi is the equivalent pressure. Unfortunateln ther6 is little or no experimental evidence available concerning the behaviour of overconsolidated clays in stress states other than those which can be imposed in the triaxial apparatus, and so the form of the generalized Hvorslev surface in principal stress space is uncertain. However, P{ry (1956) performed an extensive series of triaxial tests in which samples of weald cJay were failed in both compression and extension and so we can at Ieast sxamine one compiete'section of the generaiized Hvorsley.surface. The failure state.$ oisiicimens tested in comprission and extension,''drained and undrained and with a wide va.riety of applied stress paths, are plotted in tt|lpl: p'lpl spacq in Fig. t4-24. The convention adopted is that values of Q' aie plotted upwards for compression tests and downwards for extension tests; Fig. 14.24 is, therefore, a normalized section through the gen"eralized

L

I

ncuTi:iE sc'i;. tEsts AliD cRITTcAL srATE lrooer 319

L L

bL=oL

L Roscoe

surface

L

/,,, ont /,rfrl

,.t,

/,tI

' -,/ri'

L

" '// l/rr,/z

,,i y', tr,;ti./l rctqr, lri* /,, qa f, /'1"4u1 (y',nsrt" /qls {ur

L

A, B, (tD 7"rry ,/ t"$,/ttty-,

I

-r, 4la, I f/( fu/lt/

L

ila t/Lhr- 4,lu,o

J

l-

,L

L

Eyglrl.asurface which includes the soace di

1

(_

pointsfromtriaxialffi

l

:.

{q._ e

.l

(o',,

: o'^:

nts A and D,

ol and the re,s

ively,

q' :0.72(p'+0.107p'.)

(t+37)

M 0.58(p' + 0.107p'.).

(14-38)

and folgSjeng[orl

'l

{-

q'

1_ I

1=

I

I

/ Y** 's

from Fie. I4:i9I The data points of Fig. 14-24 clearly define two straight lines, one for compressibn and one for extension, but itre lines are ptace"J the p'lp'u axis. Both lines intersect the p,lO,o axis "r;;.,;i;;i at p,lp,":_0.1;i, 1b:lt but for compresston

,L

J{L

11 )q-

Figure 14-22 The Roscoe surface in principal stress space

:)*

l,V/.'= 2!" d- rv1

:

The slopes of the two lines from the,?k.-:ji-gii p-,1:0.lo7p,rare such that they correspond exactly with lines : lgo i7, 'fcx the compression (Eq. d, (14-21)) and extension (Eq. (r4z{!3)) cases, respectiverv. w. *ay tt e..[ie interpret the lines by a Mohr-Eouio*b failurl criterion with tgJ47l and..c'

: 0.0363p1.,

n. u4' -..

,,-

_:

/,:

l. .i t.,::

. ,{,: t-'.r'11-2-r '

,I-

:..

.

. .1..'.:1., , '

'" '

.+.,'

^_ , .1.

;.r

. l,'ttt -j ria 1 irlrl

320 rgs MEcHAMcs oF sorls

= o,b = or,

N,(

ML

u

,r.

Figure 14-23 Roscoe surfaces corresponding to different specific volumes

If

i A,o)

the extension and compression poinfs of_the generalized Hvorslev

_$&e qa4 bq lt

-

able to suppose that the complete state boundary surlace for one constant ygl11glg1yglll4 haye thg 14-25. The generalized Hvorslev sut&cq i.s qs-stqe"4_tg_!q g! lqggulqt hqxry.i,ql-cqng whose (virtua lies the gtgs. q normal effective pressure. The IT' e-? 9-l oon $skl -? tssslivtivee mean l ""llglP-?r " Loscoe surface surlace is circular in cross-section near the space snace diagonal, diasonal- but either e "ti'it'" o"'age and l5-5) Fis' stress (point B' therefore, be less t#"^t'* ;;[ -oi'F"t dwiator stress (point c)' It is' ihe critical siate woutd probably b" ;i;;.;; everypeak deviator stress being mobilized therefore, unsafe a;;r, ;;,h" where on the sliP surface state strength parameters for an It would b" #;;; use the critical on the slip we can- argu: that the deviator stress analysis of the cutting, for state critical the vafue corresponding to surface will not ,";;; b"low would be togi"ut' but conservative' and it conditious. Thi' ";;;;;;-i' of safety in the design than would commonly reasonable to use r"'rJ", .*gins to peak conditions were chosen' be used if paramete"

'o*t'ponding

I

l 1

I i

eg' strain in a sloPc Figure 15-6 Contours crf ihear

E, I

j

I

,:

\

354 rse

MEcHANrcs oF sorls

15.4 WET AND DRY STATES : l'--t

tl n

ti t-i

a ['i

LI

n LJ l:',

t

l::l

II 11

'

lve will now discuss the factors which deterrnine whether the immediate (undrained) or the Iong term (drained) ,orrg,rr'rr is the.rower. The behaviour of a ,o.*atty consoridatsd ""rrro,. ,oiii, air*silo n.st. rmagine that a normafly consoridated ,oil sraa"nrv subjected to an increasing deviator stress, but that "r"rn.nii, the totar ,run normar stress p is held constant. stress p"mr rouowed are indicatJJhe on the q,,: p, and a: p' ptots of Fig. t5-7. Tiie erement at point A on the normal consoridation rine, and, when "r,r,-ir';ild; the in*easirrg JJl stress is appried, the so, deforms at constant vorume and moves-towaJs", the criticar state Iine at B. The corresponding deviator stress failure will be qi]. rf the soil erement is ailowed to drain asat the J";il;, stress is appried slowly, the erement wiil move up the conslant p, pathAC and arrive at the critical state point c, with thl correspondin'g ;;;;; stress 96. If the element is loaded in.increments, witt dr;inage Inowea-u"r*."r, incremenrs, the path foflowed wiil be trre patr, srrown dotted on Fig. r5_7; provided the

II

:ffixr3,;firffiinr*gr,

FI

We note that gi.is greater thln q,r,and, thus, the drained strength of the specimen is greaterthan the undraint strength. Thus, if the soir has time to compress

il n

I

,t

;I

ir,. ,p".iren will,,,ri

rril"se

ro the point

c

^ffir. during deformation, it wi, hardei ;J',h; point on the critical state rine wifl move rr". n t" c. crearry, depending on the rate and pattern of loading, there are an infinite number of intermediate points on the critical state tine at which sampres may fail. Howeu.r, in generar, the Ionger the loading process takes, the stronger will be the sample at failure. The same argument wi, appri to aiisamptes whose initiar state is between the critical state line and the normat consoridation we shalt define a, such r"d;;-;;iJing .*.t,, line ;;;. ,: p, diagram. for they exist at states with higher specific volumes .wetrer) ,rrti; 6;;il water -of than those sarnpres on the criticai ".ri"r,*'l. rr"i. iir? at the same varue p,.rwe may, therefore, state, generally, that wet samples gain strength with time, and thus, the immediate stabirity of soil structures constructed from wet soirs is likely to be criticar. r, ir *"r,i *ti"r- that wer samfte, *iil suffer rarge plastic deformations as they -or. orr. the Roscoe ,u.iau.. on paths which are analogous to consolidaiioo proorrrr. we will now consider the u"r,uuiou. of a heav,y specimen. Iniriauy, the specim"r;;;;,ri'",-o"'#i,,irloverconsoridated

*"

q,:

p,

an;d

t we must be carefur not to misunderstand this definition of the term .wet,. The soil is always assumed r" u" with warer fring the pore spaces. A 'wet' soil has a water content ""n.,orl"ir'l"iur",.a larger than the water content al its ;ritical same value of p'and its state in r-:p, rpr"}., to the rigiioiii"""rrt;.ur state at the state rinej theconverse is true i*,"L'i s o i p,space.ries ro the left of the Itt ;,XrJ'soit, criticat "ra

l: li t:::i l1:;

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'ri

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a

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sotr. PARAMETERS non oesrcN 355

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it fi

tI

n ti n II i-l

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Figure

1$7 The

p, spac6 stress paths in q, i p, arld o i

for normauy consolidated soil

u:p,plotsofFig.l5.S.WenowsuPposethatthesampleissubjected-toa rapidlyincreasingdeviatorstressbutthatthetotalmeannormalstressp. path,will follow path

i&: o

J.

if oo;;;;g"

cari occur, the stress B' The corre*;pf" *iff fuif "J"r the critical statetine at point en is ";a-rfr. if sufficient time "ll".Y:1-1:: sponding deviator stress is oi,' Conyjrsely' thg..-P',,= constairt stress :' the sample to drain, the sziriple -will move a"lq+g firotir6n iiitf"te bt'C; i11d"'r' path ACD, passing if'oueh apeak p.oiii oo iU. D with the deviator stress eventually reaching the critical starc line ai-p"i" ';"'"''

is held'coirstart-

rl ,l

'l !

r

\ 356 rHr MEcHANIcs oF sorls 4'Q

,

Critical state line

Qs

qD

p' Figure I5-8 The stress paths ia q' i p and

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r

2

p'spaces for an overconsolidated soil

being equal to q'r. The value of 4i is certainly less than Qb, as will be the peak deviator stress qA. Th" sample, therefore, has a high strength when it is tested rapidly, for it remains undrained, and a much lower strength when it is allowed time to soften and swell. If some drainage of the sample were allowed, the sampre would reach the critical state line at a position intermediate between B and D. However, in general, the more a sample is allowed to drain and soften, the weaker it

will be. The sarne argument will apply for all samples whbse initial qtateq ary below and to the left of the critical state line. Such sdmples will be termed tdry';:for they are at lower specific volumes (i.e., with lower'water contents or'drier') than samples at the same value ofp'on'thecritical stateline.For ciry-,sariiples then, the Iong term strength will bE tess than the immediate str'ength, and so long term stability will be expected to be critical. For simplicity, the discussion so far has been based on stress paths in

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PARAIYIETERS

1

FoR DESIGN 357

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which the total mean noffnal stress has been held constaut. This assqmption

I

is of course unrealistic f,or most practical situations, and so we will now consider whether the same argument applles for other total stress paths. A range of possible total stress paths, l-6, may be applied to the normally consolidated sample A (Fig. l5-9), the effective stress path of which in any undrained compression test is represented by path AB. For paths 14, the failure value of the deviator stress would be higher than gi, if drainage were allowed. For path 5, the sample would fail with q' : qL whether or not drainage was allowed. Thus, for all total stress paths to thc right of path 5, the long term strength of the soil would be higher than the immediate

,

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.

-

strength.

tt i ll

Qr

Critical state

4'

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.,: Figure

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'Range of total stress paths applied,to a normally csnsolidated sample

t

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358 rns

MECHANTcs oF

sorls

for paih 6, the reCuction ofp' ri'itir increase of q'is so markeC that, in a fully drained test, the sample reaches the critical state line at a point where the deviator stress is less than qu. For path 6, therefore, the long term strength is less than the immediate strength of the sample. Con'r'ersci;-,

Critical state line

Norrnal consolidation

(-ri ti ca

state line

Figure 15-10 Range of total stresi paths applied to an overconsolidated sample

3 }"7 SOIL PARAMETERS FOR DESIGN

L ata

t +**

on t

ali san:ples I sirniiar nrgun:ent coi:lci .be fr:1:o,;.lc through.fcr stare 9f the samples, as well as *.;;id;;iin. ".i,i."I statp line;.the initialw(ether the drainid or the un-

the applied stress paths, would aetermine drained strengths lYere higher' . embankments' In practice, ,r;;; putii rorigwed. by soil near5 footings' and so the l5-9 path on Fig. or retaining rvalls *oura be to the righi of However, for cuttings, when the samples would ,t,.ngtt..n with tim"e. reduced during excavation' it is lateral, and perhaps vertical, stresses are of a normally consolidated clay is possible that the tJng t.r'n sirength even iess than the immediate strength' paths may be applied to samples The same argument concerning stress

1," I

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onthedrysideofthecriticalstateline.ThesampleA(Fig'15-10)when state line at point B' For total stress rested undrained ;.;;i.;-the critical p does not' change much as q increases' the drained

L

paths

l-3,

where

thin the undrained deviator strength q'"' strengths of samplis will be less to the undrained strength' path 4 is such that the drained strength is identical

Forapathsuchas5,wherepincreaseS.rapidlyasqincreases,thedrained wilt te larger than the undrained strength of the overconsolidated specimen strength qir. .t - ^-tli^.

.L

Anyspecimenwhichisinitiallyclosetothecriticalstatelineinu:p, Spacewillshowanincreirseofstrengthinthelong]termforanyloading Conversely, an extremely heavily paith in lvhich p increases" significanily. path in rvhich p increases

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a loa