Georgiadis_Modelling the Shear Strength of Soils in the General Stress Space

8/12/2019 Georgiadis_Modelling the Shear Strength of Soils in the General Stress Space

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Modelling the shear strength of soils in the general stress space

K. Georgiadis, D.M. Potts, L. Zdravkovic *

Department of Civil and Environmental Engineering, Imperial College London, London SW7 2AZ, UK

Received 15 January 2004; received in revised form 5 May 2004; accepted 8 May 2004

Available online 26 June 2004

Abstract

The shear strength of loose soils and soils which have reached critical state is not significantly affected by the value of the in-termediate principal stress and can therefore be approximated sufficiently by the MohrCoulomb hexagon in the deviatoric plane.

The MohrCoulomb hexagon, however, tends to underestimate the peak shear strength of dense soils for any stress state other than

triaxial compression. Other failure surfaces have been proposed which take account of the influence of the relative magnitude of the

intermediate principal stress. Two of the best-known surfaces, the MatsuokaNakai and Lade surfaces, are considered in this paper.

These two surfaces were evaluated for plane strain conditions. Comparison with experimental results showed that the Matsuoka

Nakai surface approximates well the peak shear strength of dense soils, while the Lade yield surface overestimates it. Based on this

study a new yield surface expression is proposed which can equally be applied to loose or dense soils. More importantly the

proposed expression allows the yield surface to change shape as shearing progresses and reduces to the MohrCoulomb hexagon

once critical state is reached.

2004 Elsevier Ltd. All rights reserved.

Keywords: Deviatoric plane; Yield surface; Constitutive modelling

1. Introduction

The influence of the magnitude of the intermediate

principal stress,r02, relative to that of the major, r01, and

minor, r03, principal stresses, commonly expressed interms of the parameterb ( r02r03=r01r03), on theshear strength of soils has been investigated in the past

by many authors (e.g. Bishop [1], Ochiai and Lade [9]).

It has been found that for values of b other than zero

(triaxial compression) the peak strength is generally

greater than that given by the MohrCoulomb failure

criterion. This corresponds to an increased value of the

peak angle of shearing resistance, /0p, compared to thepeak triaxial compression angle, /0tcp . For plane strainconditions in particular (/0psp ) this increase can be morethan 5%. To take account of this increase other failure

surfaces have been proposed in the past, which apart

from overcoming some numerical problems associated

with the corners of the MohrCoulomb surface also

relate the soil strength to the relative magnitude of the

intermediate principal stress. Matsuoka and Nakais [5]

and Lades [6] are the best known (Fig. 1).

On the other hand tests results reported by many

researchers (e.g. Cornforth [3], Rowe [14], Bolton [2],

Schanz and Vermeer [15]) demonstrate that little or no

difference is usually observed between the plane strain

and triaxial compression angles of shearing resistance

for very loose soils and consequently between the critical

state angles, / 0pscs

and / 0tccs

respectively. This implies that

the MohrCoulomb hexagon is well suited for very

loose soils and for critical state conditions [15].

This paper presents the calculation of the plane

strain angle /0psp using the MatsuokaNakai and Ladesurfaces, assuming an associated flow rule, and pro-

duces charts relating this angle to the triaxial com-

pression angle /0tcp . These charts are compared to theempirical relationship proposed by Schanz and Ver-

meer [15], to the approximate solution proposed by

Wroth [16] and to six sets of experimental results. Fi-

nally, based on these comparisons a new yield surface

expression is proposed.

* Corresponding author. Tel.: +44-207-594-6076; fax: +44-207-225-

2716.

E-mail address: [email protected](L. Zdravkovic).

0266-352X/$ - see front matter 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compgeo.2004.05.002

Computers and Geotechnics 31 (2004) 357364

www.elsevier.com/locate/compgeo
http://mail%20to:%[email protected]/http://mail%20to:%[email protected]/


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2. Calculation of the Lodes angle at failure for plane

strain conditions

The relative magnitude of the intermediate principal

stress can be expressed in terms of the Lodes angle of

principal stresses,

htan1 1ffiffiffi3

p 2 r02r03

r01r03

" 1

!#; 1

which is more appropriate for constitutive modelling and

numerical analysis than using b, and will therefore be

adopted for the remainder of this paper. The other two

stress invariants adopted in this paper are the following.

Mean effective stress:

p0 r01 r02r03=3: 2

Deviatoric stress:

J r01h r022 r02 r032 r03 r012.6i1=2:

3The Lodes angle, h, of the stress state at failure for

plane strain deformation is controlled by the shape of

the plastic potential surface in the deviatoric plane and

the angle of dilation, t. It can be shown that h is related

to the slope of plastic potential surface in theJp0 plane,

gh, and t through the following equation [11]: sin tffiffiffi

3p

og h oh

cos hg h sin hog h oh

sin hg h cos h ; 4

or alternatively

og h oh

g h sin hcos h sin tffiffi

3p

cos hsin h sin tffiffi3

pA: 5

The Lodes angle at failure for plane strain conditions

can be calculated from the above equation adopting

either the MatsuokaNakai or the Lade expressions for

the plastic potential surface.

2.1. MatsuokaNakai surface

The value of gh can be obtained from the Mats-uokaNakai criterion, which is given by the relationship

[13]:

2ffiffiffiffiffi27p C sin 3h g

3

h C 3 g2

h C 9 0;6

in which

C 93M2

J

2ffiffi

3p

M3J

9 M2J1

; 7

where MJ is the value ofgh corresponding to triaxialcompression, h 30.

The derivative og h =oh, obtained from Eq. (6) isog h oh

Ccos 3h g2 h ffiffiffi3p C3 Csin 3h g h

:

8

Combining Eqs. (5) and (8) gives

g h ffiffiffi

3p

C3 AC A sin 3h cos 3h : 9

Furthermore, combining Eqs. (6) and (9) gives the

following equation from which the Lodes angle, h, can

be calculated:

A sin 3h cos 3h 3A sin 3h 3 co s 3h A2

C3 3C2 C9 : 10

From Eqs. (5), (7) and (10) it can be seen that the

value of h at failure for plane strain conditions, hps,

depends on the dilation,t, and the parameter MJ which

is related to the triaxial compression angle of shearing

resistance, /0tc, through the following equation:

MJ 2ffiffiffi

3p

sin /0tc

3 sin /0tc : 11

Eq. (10) was solved numerically for four different

values oft and values of/0ps ranging from 25 to 50.The resulting values of hps are plotted in Fig. 2, illus-

trating that the Lodes angle at failure for plane strain

conditions decreases with the increase of the value of the

triaxial compression angle of shearing resistance. Ad-ditionally, increase of the angle of dilation results in a

considerable increase ofh.

2.2. Lade surface

The Lade surface is given by the following equation

[13]:

2ffiffiffiffiffi27

p sin 3h g3 h g2 h CL0; 12

in which

Fig. 1. Failure surfaces in the deviatoric plane.

358 K. Georgiadis et al. / Computers and Geotechnics 31 (2004) 357364


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CLg127

pa3p0

m1 g1

27

pa3p0

m; 13where g1 and m are material properties, and pa is the

atmospheric pressure. Since all comparisons in this

study are performed on the same deviatoric plane (i.e.

constant p0), CL is constant and can be obtained fromEq. (12) for triaxial compression, h 30:

CLM2J 1 2MJ

ffiffiffiffiffi27p : 14The derivative og h =oh, obtained from Eq. (12), is

og h oh

cos 3h g2 h ffiffiffi

3p sin 3h g h : 15

Combining Eqs. (5) and (15) gives

g h ffiffiffi

3p

A

A sin 3h cos 3h : 16

Combining Eqs. (12) and (16) gives the following rela-

tionship from which the Lodes angle, hps, can be cal-

culated:

A2 A sin 3h 3 co s 3h A sin 3h cos 3h 3 CL: 17

Eq. (17) is also solved numerically for four different

values of t and values of/0tc ranging from 25 to 50.The resulting values of hps are plotted in Fig. 3. The

variation of the Lodes angle at failure for plane strain

conditions with /0tc and t is similar to that calculated

-30

-25

-20

-15

-10

-5

0

25 30 35 40 45 50

TRIAXIAL COMPRESSION ANGLE OF SHEARING RESISTANCE

LODEANGLEATFAILURE

v=0

v=10

v=20

v=30

Fig. 2. Variation of the Lode angle at failure for plane strain conditions with the triaxial compression angle of shearing resistance for different angles

of dilation calculated using the MatsuokaNakai equation for the plastic potential surface.

-30

-25

-20

-15

-10

-5

0

25 30 35 40 45 50


LODEANGLEA

TFAILURE

v=0

v=10

v=20

v=30

Fig. 3. Variation of the Lode angle at failure for plane strain conditions with the triaxial compression angle of shearing resistance for different angles

of dilation calculated using the Lade equation for the plastic potential surface.

K. Georgiadis et al. / Computers and Geotechnics 31 (2004) 357364 359


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from the MatsuokaNakai expression, however the

predicted values are significantly higher.

3. Calculation of the angle of shearing resistance for plane

strain conditions

As mentioned in Section 1, while the MohrCoulomb

criterion assumes a constant value of/ 0 independent ofh, defining in this way the well-known hexagon in the

deviatoric plane (Fig. 1), the MatsuokaNakai and

Lades expressions imply varying values of/ 0 with h. Inorder to calculate the value of the angle of shearing re-

sistance for plane strain conditions, / 0ps, when either ofthese two expressions is adopted for the yield function,

the MohrCoulomb hexagon must be defined which

crosses the yield surface under consideration at the point

corresponding to the calculated plane strain value ofh.

This can simply be done by substituting h and gh(which is calculated either from Eq. (9) or (16), de-

pending on the adopted surface once the value of h is

established) into the following equation:

sin /0 g h cos h1g h sin hffiffi

3p

; 18

which defines the MohrCoulomb hexagon in the devi-

atoric plane. This calculation was performed for both

yield surface expressions and for the whole range of

values of the Lodes angle calculated in the previous

section.

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60

25 30 35 40 45 50


PLANESTRAINA

NGLEOFSHEARING

RESISTANCE

v=0

v=10

v=20

v=30

Wrothps= tc

Fig. 4. Variation of the plane strain angle of shearing resistance with the triaxial compression angle of shearing resistance for different angles of

dilation calculated using the MatsuokaNakai equation for the yield surface.

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30

35

40

45

50

55

60

25 30 35 40 45 50


PLANESTRAINA

NGL

EOFSHEARING

RESISTA

NCE

v=0

v=10

v=20

v=30

Wrothps= tc

Fig. 5. Variation of the plane strain angle of shearing resistance with the triaxial compression angle of shearing resistance for different angles of

dilation calculated using the Lade equation for the yield surface.



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Fig. 4 shows the variation of /0ps with /0tc for dif-ferent values of dilation, calculated adopting the Mats-

uokaNakai expression for both the yield function and

the plastic potential. The calculated plane strain values

of the angle of shearing resistance are 514% higher than

the equivalent triaxial compression values over the range

of values plotted in the figure. This increase is higher forhigher values of the angle of dilation.

Also plotted in Fig. 4 is the relationship between / 0ps

with /0tc given from the approximate solution of theMatsuokaNakai failure criterion for plane strain con-

ditions proposed by Wroth [16]:

8/0ps 9/0tc: 19The above relationship is in relatively good agreement

with the exact solution for low values of the angle of

shearing resistance, but overestimates the plane strain

angle of shearing resistance for higher values of/0tc.

The variation of/0ps

with /0tc

for different values ofdilation, calculated adopting the Lade expression is

shown in Fig. 5. A similar type of variation is observed

to that calculated with the MatsuokaNakai expression,

however the values of/0ps are significantly higher. Theobtained plane strain angles of shearing resistance are

1020% higher than the equivalent triaxial compression

angles, for the range of values plotted in Fig. 5.

4. Comparison with experimental results

The analytical expressions on which the calculations

of the plane strain angle of shearing resistance presented

above were based do not take account of the soil den-

sity, nor do they distinguish between peak and critical

state values. As noted in the introduction, such differ-

ences between the plane strain and triaxial compression

values of the angle of shearing resistance, as those pre-

sented in the above figures, are only expected for the

peak values of dense soils. For loose soils and for soils

that have reached critical state it is expected that:

/0ps

/

0tc

20

and therefore the MohrCoulomb hexagon in the devi-

atoric plane is sufficient to model soil strength.

Fig. 6 presents experimental results between the plane

strain and triaxial compression angles of shearing re-

sistance for various soils, as reported by Cornforth [3],

Leussink et al. [8], Rowe [14], Green and Reades [4],

Lam and Tatsuoka [7] and Schanz and Vermeer [15].

Despite the relatively wide scatter, a general trend can be

observed for all sets of data. The difference between /0ps

and / 0tc reduces with the reduction of/0tc and tends tozero at the value corresponding to the critical state angle

of shearing resistance, /0cv. A linear relationship thatagrees with this trend was proposed by Schanz andVermeer [15] based on the findings by Bolton [2]:

/0psp 1

3 5/0tcp

2/0cv: 21

The above relationship is also plotted in Fig. 6 con-

sidering an average value of the estimated critical state

angles of shearing resistance for each set of data,

/0cv31. As seen, Eq. (21) is in good agreement withthe experimental results, although slightly overestimat-

ing the increase of/0ps. It should be noted that part ofthe observed scatter in the experimental results could be

attributed to the different values of/0cv corresponding toeach set of data.

The experimental results are also compared in Fig. 6

with the values calculated in the previous section based

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60

25 30 35 40 45 50


PLANESTRAINA

NGL

EOFSHEARING

RESISTAN

CE

Cornforth (1964)Leussink et al (1966)Rowe (1969)Green & Reades (1975)Lam & Tatsuoka (1988)Schanz & Vermeer (1996)v=0 Matsuoka-Nakaiv=30 Matsuoka-Nakaiv=0 Lade

v=30 LadeEquation 21phi(PS) = phi(TX)

ps= 'tc'

Fig. 6. Comparison of experimental results with Schanz and Vermeers empirical equation and the predictions of the Matsuoka and Nakai and Lade

yield surface equations.



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on the Matsuoka and Nakais and Lades yield surface

expressions. Both expressions seem to overpredict con-

siderably the plane strain angle of shearing resistance at

low values of /0tc (loose soil states close to criticalstate). At high values of/0tc (dense soil state) however,the Matsuoka and Nakai yield surface is in good

agreement with the experimental observations.

5. Proposed expression for the yield surface in the

deviatoric plane

From the above discussion it can be concluded that

the MohrCoulomb hexagon in the deviatoric plane is

well suited for loose soils and for critical state condi-

tions, while the MatsuokaNakai failure criterion ap-

proximates better the peak strength of dense soils. For

intermediate soil states the empirical relationship pro-

posed by Schanz and Vermeer [15] seems to be in good

agreement with experimental results, but is limited to

plane strain conditions without specifying or requiring

the specification of the principal stress combination at

which these occur.

Based on these findings the following more general

yield surface expression is proposed:

F aF1 1 aF2; 22where F1 is the MatsuokaNakai yield surface expres-

sion given by Eq. (6), F2 is the MohrCoulomb yield

surface expression given by

F2g h sin /0cvcos h sin h sin/0cvffiffi

3p

23

and a represents the current soil state.

The value of a can be obtained from the following

expression:

ab /0tc /0cv/0cv

; 24

wherebis a parameter that controls the expansion of the

yield surface in the deviatoric plane. For a loose soil

where /0tc /0cv,a is equal to zero and the yield surfacein the deviatoric plane given by Eq. (22) reduces to the

MohrCoulomb hexagon. For denser soils the value ofa

depends on the parameterb and the value of the current

triaxial compression angle of shearing resistance, /0tc,relative to the critical state angle, /0cv. The maximumvalue ofa is obtained at peak strength, when / 0tc /0tcp ,giving the maximum divergence of the failure surface

from the MohrCoulomb hexagon and consequently themaximum angle of shearing resistance for all values ofh.

As the soil moves past peak, /0tc and a reduce and theyield surface shrinks towards its critical state (Mohr

Coulomb hexagon). In the special case in which a1Eq. (22) reduces to F F1 giving the MatsuokaNakaiyield surface. The influence of a on the shape and po-

sition of the yield surface in the deviatoric plane is

= 0

= +30

= 1

= 2

Mohr-Coulomb hexagon

1


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shown schematically in Fig. 7 and in the Jp0 plane inFig. 8.

The angle of dilation is expected to vary from 0 at

critical state to a maximum value of tp at peak and

the following expression is used in conjunction with

Eq. (22):

tatp: 25It is clear that in order to use the full potential of

Eq. (22) this should be used in conjunction with a con-

stitutive model, which allows the variation of the triaxial

compression angle of shearing resistance with the strain

level. A model of this type is described by Potts and

Zdravkovic [13] and was used to analyse the progressive

failure of Carsington Dam [10] and to investigate the

delayed collapse of cut slopes in stiff clay [12].

It is noted that any yield surface expression giving an

increase of the angle of shearing resistance in the devi-

atoric plane can potentially be used in place of the

MatsuokaNakai criterion in Eq. (22). Based on thecomparisons made in the previous section, however, this

criterion seems to be better suited. Both the Matsuoka

Nakai and Lade criteria are employed in the following

section to justify this selection.

6. Evaluation of the new expression and comparison with

experimental results

In order to evaluate the predictions of Eq. (22) the

plane strain angles of shearing resistance, /0ps, were

calculated for a wide range of triaxial compressionangles of shearing resistance, /0tc. Both the MatsuokaNakai and Lade expressions (Eqs. (6) and (12), respec-

tively) were introduced as F1 in Eq. (22) and the same

expressions were adopted for the plastic potential sur-

face for each respective case, leading to non-associated

flow rules.

The plane strain angle of shearing resistance, /0ps,resulting from the adoption of Eq. (22) for the yield

surface can be calculated from Eq. (18) in which the

Lodes angle, h, of the stress state at failure for planestrain conditions is obtained from Eqs. (10) or (17)

(depending on the adopted plastic potential surface) and

gh is obtained from Eq. (22).Fig. 9 shows the variation of/0ps with /0tc when F1 is

given by the MatsuokaNakai expression for the values

of the parameterb of 1, 2 and 3. The computations were

based on the average value of the estimated critical state

angles of shearing resistance for each of the sets of data

considered in this paper of/ 0cv31, and a peak angleof dilation of 20.

As seen in Fig. 9 the difference between / 0ps with/ 0tc

is zero at the critical state value of 31(a

0) and then

rises gradually as /0tc increases (increasing value ofa).As expected, increase of the value ofb results in higher

values of /0ps. Also shown in the same figure are theexperimental results discussed in previous sections.

Eq. (22) with F1 given by the MatsuokaNakai

expression is in generally good agreement with the re-

sults, with b3 giving the best fit, while b1 appearsto underestimate the observed soil strength particularly

at high values of/ 0tc. This is expected as the value ofbcontrols the ratio of (/0tc /0cv=/0cv at which the yieldsurface expression reduces to the MatsuokaNakai

expression (a1).The same calculations were performed using the Lade

surface expression asF1 in Eq. (22). The resulting curves

are shown in Fig. 10 together with the experimental

results. Although for a value of b2 the predicted

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PLANESTRAINA

NGLEO

FSHEARING

RESISTANC

E

Cornforth (1964)Leussink et al (1966)Rowe (1969)Green & Reades (1975)Lam & Tatsuoka (1988)Sch Vanz & ermeer (1996)

ab=1b=2b=3

'ps

= 'tc

Fig. 9. Comparison of the predictions of Eq. (22) incorporating the MatsuokaNakai criterion for different values ofb with experimental results.



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plane strain angles of shearing resistance are in good

agreement with the experimental results, the proposed

expression proves to be very sensitive to the selected

value of b and in addition gives unrealistic results for

values ofa greater than 1.

7. Conclusions

The analytical calculation of the Lodes angle of

the stress state at failure for plane strain conditions

based on the MatsuokaNakai and Lades surfaces in

the deviatoric plane was presented and charts wereproduced relating this angle to the angle of shearing

resistance in triaxial compression and to the angle of

dilation. Based on these calculations the relation be-

tween the angles of shearing resistance in plane strain

and triaxial conditions was established for the two

yield surfaces. Comparison with experimental results

demonstrated that the MatsuokaNakai expression

approximates well the peak strength of dense soils

while the Lade expression overpredicts soil strength in

plane strain.

A new yield surface expression for the deviatoric

plane was presented which is a combination of theMohrCoulomb hexagon equation and either the

MatsuokaNakai or the Lade failure criteria. The yield

surface described by the expression reduces to the

MohrCoulomb hexagon at critical state conditions and

expands in the deviatoric plane as the angle of shearing

resistance in triaxial compression deviates from the

critical state value. Plane strain calculations showed that

when the MatsuokaNakai criterion is incorporated in

the expression very good agreement is achieved with

experimental results, while use of the Lade criterion

proved to be less effective.

References

[1] Bishop AW. Roscoe Memorial Conference, 1971.

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1986;36(1):6578.

[3] Cornforth DH. Some experiments on the influence of strain

conditions on the shear strength of sand. Geotechnique

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sample shape effects on the stressstrain behaviour of sand in

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[12] Potts DM, Kovacevic N, Vaughan PR. Delayed collapse of cut

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[13] Potts DM, Zdravkovic L. Finite element analysis in geotechnical

engineering: theory. London: Thomas Telford; 1999.

[14] Rowe PW. The relationship between the shear strength of sands in

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[15] Schanz T, Vermeer PA. Angles of friction and dilatancy of sand.

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[16] Wroth CP. The interpretation of in situ soil tests. Geotechnique

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PLANESTRAINA

NG

LEOFSHEARING

RESISTA

NCE

Cornforth (1964)Leussink et al (1966)Rowe (1969)Green & Reades (1975)Lam & Tatsuoka (1988)Scha Vnz & ermeer (1996)

b=1

b=2b=3

'ps= '

tc

Fig. 10. Comparison of the predictions of Eq. (22) incorporating the Lade criterion for different values ofb with experimental results.


Georgiadis_Modelling the Shear Strength of Soils in the General Stress Space

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