# Critical State Soil Mechanics

The preparation of the Cam Clay theoretical account in 1958 is possibly the most of import development in modern dirt mechanics. Mechanical belongingss of dirt have been unified elegantly and systematically into the theoretical model of the theoretical account: the critical province dirt mechanics. Since so, many theoretical accounts were developed within the theoretical model of Cam Clay theoretical account and they form theoretical accounts of the Cam Clay household.

Two most distinguished characteristics of theoretical accounts of the Cam Clay household are: ( 1 ) the hardening of dirts based on plastic volumetric distortion, and ( 2 ) the being of a critical province of distortion as the concluding failure province. A brief debut of the two characteristics is given in this session, and descriptions of the strengths of dirts based on critical province dirt mechanics is given in the undermentioned subdivision.

7.1.1: Hardening of dirt based on plastic volumetric distortion

As seen in the compaction theoretical account illustrated in Fig. 10, whenever the current emphasis goes beyond the historical upper limit emphasis, plastic volumetric distortion occurs and the elastic zone enlarges. The expansion of the elastic zone is seen as the hardening of dirt and is straight linked to the plastic volumetric distortion of the dirt.

Consequently, the undermentioned decisions are drawn from indurating mechanism of dirt.

The magnitude of plastic volumetric distortion is dependent on the alteration in size of the output surface, but independent of the stress way.

All emphasis provinces which have the same accretion of plastic volumetric strain constitute a individual output surface.

7.1.2: Prediction of the being of a critical province of distortion

Soil is a frictional stuff. When the emphasis ratio applied on a dirt component additions, dirt will finally make a point, where it has no opposition to further shear distortion. The dirt fails. This is a critical province of distortion. A Critical State of distortion is defined as

At a Critical State of distortion, a dirt has no opposition to shear distortion and the dirt can be distorted continuously with its emphasis province and nothingnesss ratio remain unchanged.

A critical province of distortion is a concluding failure province. The theoretical model, uniting systematically the mechanical belongingss of dirt into one simple and elegant system under the Cam Clay theoretical account, is referred to as the Critical State Soil Mechanics ( CSSM ) .

7.2: Strength of dirt described in the critical province dirt mechanics

The behavior of Fuji sand in triaxial trials is shown in Fig. 12 ( Tatsuoka, 1972 ) . The trials are drained trials. The axial emphasis additions with the restricting emphasis kept changeless. The denseness of the dirt for the three trials varies from really loose to really dense. It is seen that Fuji sand in a really heavy Fuji province has two strengths: a peak strength and a strength as really big shear strain. For the loose sand, the dirt has merely one strength, besides at really big shear strain. It besides appears that the strength as really big shear

0 5 10 15 20 25

Distortional strain ad ( % )

Volumetric strain Cv ( % )

-12

4

0

5 10 15 20 25

Dense =0.52

Loose eaˆz =0.78

Very loose =0.85

Distortional strain ad ( % )

Fig. 12 The shearing behavior of Fuji sand under triaxial

compaction trials ( Test informations after Tatsuoka, 1972 )

Shear emphasis ratio two

3.2

2.4

0.8

1.6

0

Dense =0.52

Loose eaˆz =0.78

Very loose =0.85

-4

-8

strain for the three samples are really near and appears to near a alone value as the distortional strain additions. The peak strength and the critical province strength of dirt interpreted by the critical province dirt mechanics are introduced in the followers.

7.2.1 Critical province strength

Under shearing ( increasing distortional strain ) , dirt reaches a concluding failure province, the critical province of distortion. At a critical province of distortion:

Therefore, a dirt can be distorted with no alteration in its emphasis, and no volumetric plastic distortion. At critical provinces of distortion, dirt has the undermentioned characteristics ( Fig. 13 ) .

The shear emphasis ratio c is changeless, denoted by I .

There is a alone relationship between the average effectual emphasis paˆ? and nothingnesss ratio vitamin E, irrespective of the initial state of affairs of the dirt or the stress way of the trial.

27

This relationship is additive in the vitamin E – lnpaˆ? infinite, and its gradient is the same as that of ICL in the compaction theoretical account, being e .

The features of critical provinces can be represented by the critical province line ( CSL ) in the paˆ? – q infinite and the vitamin E – lnpaˆ? infinite. They are described mathematically as.

I·

Q = I? ( 46 )

P

aˆ?

e=eCS a?’I»lnpaˆ? ( 47 )

Critical province shear emphasis ratio M

Shear emphasis Q

I? CSL

Mean effectual emphasis P ‘ ( kPa ) ( a ) CSL in the p’-q infinite

The critical province shear emphasis ratio is linked to the concluding failure clash angle ocs of dirt measured from conventional triaxial compress trial by the undermentioned equation

I? = 6 wickedness I†cs ( 48 )

3 a?’sin I†cs

The critical province clash angle for most san

vitamin D is 32A° A± 1A° , which gives about M = 1.28.

Determination of the concluding failure strength of dirts

Nothingnesss ratio vitamin E

vitamin E CS

eIC

( 1 ) Undrained concluding strength of dirt

ICL

I»

CSL

I»

p’=1 kPa

Mean effectual emphasis P ‘ ( kPa ) ( B ) CSL in the e-lnp ‘ infinite

Fig. 13 Characteristics of critical province of

distortion

P aˆ? = exp

aZ? aZ? aZY

Cesium I

a?’

Cs aZ? I» aZ

Three parametric quantities define a dirt province. They are the emphasis province on the dirt and its nothingnesss ratio, i.e. ,

( paˆ? , Q, vitamin E ) . Because any dirt province at a critical province of distortion must fulfill conditions expressed by equation ( 46 ) and ( 47 ) , the concluding failure province ofa dirt, the critical province of distortion, can

be determined if there is one more extra status. Some of common instances are discussed here.

During an undrained trial, volumetric distortion is non allowed. Therefore, the nothingnesss ratio ofthe dirt is kept the same and is equal to its initial value,

ei.

Based on equations ( 46 ) and ( 47 ) , the

undrained concluding shear strength

qcs,

the mean

effectual emphasis

paˆ?cs,

an

d the nothingnesss ratio European Union are

given by

( 49 )

28

aZ§ aZ›e vitamin E aZz

Cesium I

aZ› a?’ aZz

aZ? aZY

aZ? I» aZ

aZ?aZ? vitamin E vitamin E

Cs

aZ? Q = I? exp

vitamin E vitamin E

Cs I

=

aZ©

aZ?

aZ?

aZ?

( 2 ) Drained concluding strength for changeless paˆ? trials

The average effectual emphasis at the concluding critical failure province is known, paˆ?i, so

aZ§

aZ?aZ?

aZ?aZ©

personal computers = pi

qcs Ipaˆ?i ( 50 )

ei = eCS a?’ I» ln pi

The stress way for this type of trials is a perpendicular line in the paˆ? – Q infinite, as shown in Fig. 4. ( 3 ) Drained concluding strength for changeless Q trials

The shear strength at the concluding failure province is known, chi, so

I» ln

chi

aˆ? =

Cs

P

I?

qcs = chi

=

ei

European Union

a?’

aZ› Q aZz

( 51 )

I

aZ?aZ? I? aZ aZY

The stress way for this type of trials is a horizontal line in the paˆ? – Q infinite, as shown in Fig. 4. ( 4 ) Drained concluding strength for trials with additive emphasis waies

Suppose the gradient for the additive emphasis way is k, and the initial emphasis province of the dirt is ( paˆ?i, chi ) . Then the dirt province at the concluding failure province can be obtained from work outing the undermentioned equations

qcs a?’ chi = k personal computers a?’ pi

I?aˆ? P Cs

=eCSa?’I»lnp

qcs European Union

( 52 )

aZ§

aZ?

aZ?

aZ©

aˆ?

Cs

29

Two common additive emphasis waies, discussed in subdivision 3.3, are ( 1 ) conventional triaxial compaction trials with the restricting emphasis kept changeless and in this instance k = 3, and ( 2 ) conventional triaxial extension trials with the axial emphasis kept changeless and in this instance k = – 1.5.

Example 5

The critical province clash angle for a sand is 32A° . For its CSL in the vitamin E – lnpaˆ? infinite, the gradient is 0.12 and European Union is 1.42. ( 1 ) Determine the values of M ; ( 2 ) Pull a study of the CSL in the paˆ? – q infinite and the vitamin E – lnpaˆ? infinite ; ( 3 ) Determine the concluding failure strength of a specimen of the sand with initial province as ( paˆ? = 75 kPa, q = 0, vitamin E = 0.85 ) . ( a ) under undrained state of affairs ; ( B ) under a changeless mean effectual emphasis trial ; and ( degree Celsius ) following a stress way with = 2

.

dq

dpaˆ?

A: Determine the values of M

400

800

600

200

0

CSL q=1.29p ‘

Shear emphasis Q kPa

30

100 200 300 400 500 600

Mean effectual emphasis P ‘ kPa

Nothingnesss ratioe

0.9

0.7

0.5

1.5

1.3

1.1

CSL e=1.42-0.12lnp ‘

1 10 100 1000

Mean effectual emphasis P ‘ kPa

Oxygen

Nothingnesss ratio e n Shear emphasis Q

CSL

C

I?

Bacillus

A

P ‘ ( kPa )

CSL

e=eCS -elnpaˆ?

Bacillus

C

Mean effectual emphasis P ‘ ( kPa ) Fig. 14 Variation of dirt strength

( 53 )

7.2.2 Peak strength

As seen in Fig. 12, under shearing dirt at heavy province may make a peak strength ( higher than the critical province strength ) . However, this strength of dirt lessenings with the addition of distortional strain, and becomes indistinguishable to the critical province strength finally. Two characteristics of the peak strength should be noticed.

The happening of a strength for dirt greater than the concluding critical province strength ( the extremum strength ) is possible merely if the dirt is under the CSL in the vitamin E – lnpaˆ? infinite ( Fig 14 ) . This is in the “ Dry ” side. As named by Schofield and Wroth ( 1968 ) , soil behavior with both peak and critical province strength is “ Dry ” behavior.

The peak strength is non stable. The minute a peak strength is reached, the strength of dirt will diminish with the farther distortional distortion.

An empirical equation proposed by Liu and Carter ( 2002 ) may be used to gauge the peak strength ratio cp of dirt based on its place in the vitamin E – lnpaˆ? infinite

=

Q P =

aˆ?

I·

P

P P

31

( 1 a?’ I¦ ) I? for A- & lt ; 0

I¦ , the province parametric quantity, defined by Been and Jefferies ( 1985 ) as

I¦ = vitamin E a?’ vitamin E CS + I» ln p aˆ? ( 54 )

Uniting the above two equations, we obtain q P

I· = = + a?’ a?’ aˆ? I? & lt ; a?’ I» aˆ?

( vitamin E e P ) vitamin E vitamin E P

1 ln for ln

P CS CS ( 55 )

P

P aˆ?

For A- & gt ; 0, there is merely one strength. The peak strength and the critical province strength may be considered as coincident.

7.2.3 Variation of dirt strength in the paˆ? – Q infinite

Mohr-Coulomb ‘s strength standard, written in the paˆ? – Q infinite, is given as

32

Q degree Fahrenheit = c + I? MC P aˆ? degree Fahrenheit ( 56 )

This standard is possibly the widely used standard to find the strength of dirts in geotechnical technology pattern. However, it is applicable for dirt conditionally. As shown in Fig. 14, Mohr-Coulomb ‘s strength standard is applicable to dirty in the scope of AB, where the dirt in the “ Dry ” side, i.e. , below the CSL.

The general strength standard for dirt may be divided into three scopes to analyze.

Strength of dirts on the “ Wet ” side

If a dirt province in the vitamin E – lnpaˆ? infinite is above the CSL, i.e. , A- & gt ; 0, the dirt is on the “ Wet ” side. Dirts on the “ Wet ” side have merely one strength, the concluding critical province strength. The critical province strength of dirts is represented by line BC, and their belongingss are introduced in subdivision 7.2.1.

No tensile emphasis line

Farinaceous stuffs such as littorals or clays in reconstituted provinces have no echt coherence, and can non prolong a tensile emphasis. The boundary oaˆ?min & gt ; 0 in the paˆ? – q infinite is = 3

Q

,

paˆ?

represented by line OA. On the left side of line OA, tensile emphasis occurs.

Strength of dirts on the “ Dry ” side

Merely when dirt on the “ Dry ” side, the dirt has a coherence c. The strength of the dirt can depict by equation ( 56 ) , the Mohr-Coulomb ‘s strength standard. Cohesion degree Celsius may be treated as changeless. Clash angle oMC and parametric quantity MMC is related by the undermentioned equation

= 6sinoMC ( 57 )

Megahertz

I?MC 3 a?’ sino

The whole strength envelope OABC is shown in Fig. 14.

Two errors are normally made in using the Mohr-Coulomb ‘s strength standard for finding the strength of dirts.

The extension of the Mohr-Coulomb ‘s strength standard to the left side of AB and therefore implies dirt has a tensile strength.

The extension of the Mohr-Coulomb ‘s strength standard to the right side of AB ( beyond the critical province strength ) . This implies the ultimate clash angle of dirt will go on lessening after the critical province clash angle unlimited with the average effectual emphasis.

Terzaghi, the laminitis of modern dirt mechanics, made both errors in widening the pertinence of Don Taylor ‘s experimental information. And I hope you will non do the same error in your technology designs or safety cheque.

33

Example 6

The critical province clash angle for a Leighton Buzzard sand in situ is 31A° measured from conventional triaxial compaction trials. The strength envelope detected for the sand can be described by Mohr-Coulomb ‘s strength standard with c= 40 kPa and oMC = 23A° . ( 1 ) Determine the values of dirt parametric quantities M for critical province stength and MMC for Mohr-Coulomb ‘s strength ; ( 2 ) Pull the strength envelope of this Leighton Buzzard sand and depict the features of the strength of the sand ; ( 3 ) Discourse the strength of the sand under a drained conventional triaxial compaction trial with the initial emphasis province of the dirt being ( paˆ? = 30 kPa, q = 0, ) .

A: Determine dirt parametric quantities M and MMC ;

Shear emphasis Q kPa

250

200

150

100

50

0

A

3

( 30, 0 )

vitamin D

3

0.9

Bacillus

1.24

C

0 50 100 150 200

Mean effectual emphasis P ‘ kPa

34

7.2.4 Residual province strength of clayey dirts

After probes on landslides in the late fiftiess, it was found that the shearing opposition of dirt in a figure of instances was much smaller than the “ concluding ” critical province strength measured in the research lab. The construct of residuary strength is formed ( Skempton, 1964 )

Residual strength is defined as the shear strength of a dirt that can be mobilised on a polished sliding surface, after it has been formed through the dirt due to the alliance of its Platypoecilus maculatus atoms. For any given dirt it is the minimal strength come-at-able. There are four major facets of the residuary strength, viz.

Dirt must hold adequate plate-like atoms so that a smooth slickensided surface can be formed.

The skiding surface of well-aligned dirt atoms must be for the residuary strength to be mobilised.

The skiding surface of well-aligned Platypoecilus maculatus atoms can ease residuary failure merely along that surface.

4. The residuary sliding surface one time formed is normally non modified by subsequent distortions of comparatively little magnitude.

The residuary strength of a dirt is chiefly dependent on the mineralogy of the dirt: the clay fraction. Clay fraction uc defined as the weight of the clay particles less than 0.002 millimeter in size over the entire weight of the dirt sample. uc is defined as

I‰ = G0.002 ( 58 )

cG

Some experimental informations on the fluctuation of the critical province strength and residuary strength with clay fraction is shown In Fig. 15 ( From Skempton, 1984 ) . The undermentioned information can be obtained this information.

( 1 ) For a dirt with a clay fraction less than 25 % the concluding strength is the critical province strength. The strength is independent on clay fraction. Otherwise, the concluding strength of the dirt is its residuary strength.

0 20 40 60 80 10 0

Clash angle ( A° )

34

28

22

16

10

4

residuary province critical province

Clay fraction ( % )

Fig. 15 Variation of the critical province and

residuary province strengths with clay fraction

35

With 50 % & gt ; uc & gt ; 25 % , both critical province and residuary province strengths vary with clay fraction.

With uc & gt ; 50 % , critical province and residuary province strengths are different, but remains with any alteration of clay fraction in the scope.

The concluding strength of a dirt, expressed as a clash angle, may change from 32A° to every bit low as 6A° with the fluctuation of the clay fraction.

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